From 3d68decde19f566fafdf29ccca9db2a755949c9d Mon Sep 17 00:00:00 2001
From: pyposebot <109780734+pyposebot@users.noreply.github.com>
Date: Wed, 11 Sep 2024 19:46:55 +0000
Subject: [PATCH] =?UTF-8?q?Deploying=20to=20gh-pages=20from=20@=20pypose/p?=
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+              
+  <h1>Source code for pypose.basics.ops</h1><div class="highlight"><pre>
+<span></span><span class="kn">import</span> <span class="nn">math</span><span class="o">,</span> <span class="nn">torch</span>
+
+
+<div class="viewcode-block" id="pm"><a class="viewcode-back" href="../../../../generated/pypose.pm/#pypose.pm">[docs]</a><span class="k">def</span> <span class="nf">pm</span><span class="p">(</span><span class="nb">input</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Returns plus or minus (:math:`\pm`) states for tensor.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        input (:obj:`Tensor`): the input tensor.</span>
+
+<span class="sd">    Return:</span>
+<span class="sd">        :obj:`Tensor`: the output tensor contains only :math:`-1` or :math:`+1`.</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        The :meth:`pm` function is different from :meth:`torch.sign`, which returns</span>
+<span class="sd">        :math:`0` for zero inputs, it will return :math:`+1` if an input element is zero.</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        &gt;&gt;&gt; pp.pm(torch.tensor([0.1, 0, -0.2]))</span>
+<span class="sd">        tensor([ 1.,  1., -1.])</span>
+<span class="sd">        &gt;&gt;&gt; pp.pm(torch.tensor([0.1, 0, -0.2], dtype=torch.float64))</span>
+<span class="sd">        tensor([ 1.,  1., -1.], dtype=torch.float64)</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">return</span> <span class="n">torch</span><span class="o">.</span><span class="n">sign</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">sign</span><span class="p">(</span><span class="nb">input</span><span class="p">)</span> <span class="o">*</span> <span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="cumops_"><a class="viewcode-back" href="../../../../generated/pypose.cumops_/#pypose.cumops_">[docs]</a><span class="k">def</span> <span class="nf">cumops_</span><span class="p">(</span><span class="nb">input</span><span class="p">,</span> <span class="n">dim</span><span class="p">,</span> <span class="n">ops</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Inplace version of :meth:`pypose.cumops`</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="n">L</span><span class="p">,</span> <span class="n">v</span> <span class="o">=</span> <span class="nb">input</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="n">dim</span><span class="p">],</span> <span class="nb">input</span>
+    <span class="k">assert</span> <span class="n">dim</span> <span class="o">!=</span> <span class="o">-</span><span class="mi">1</span> <span class="ow">or</span> <span class="n">dim</span> <span class="o">!=</span> <span class="n">v</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="s2">&quot;Invalid dim&quot;</span>
+    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">torch</span><span class="o">.</span><span class="n">pow</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">torch</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">math</span><span class="o">.</span><span class="n">log2</span><span class="p">(</span><span class="n">L</span><span class="p">)</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="n">v</span><span class="o">.</span><span class="n">device</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">int64</span><span class="p">)):</span>
+        <span class="n">index</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">L</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="n">v</span><span class="o">.</span><span class="n">device</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">int64</span><span class="p">)</span>
+        <span class="n">v</span><span class="o">.</span><span class="n">index_copy_</span><span class="p">(</span><span class="n">dim</span><span class="p">,</span> <span class="n">index</span><span class="p">,</span> <span class="n">ops</span><span class="p">(</span><span class="n">v</span><span class="o">.</span><span class="n">index_select</span><span class="p">(</span><span class="n">dim</span><span class="p">,</span> <span class="n">index</span><span class="o">-</span><span class="n">i</span><span class="p">),</span> <span class="n">v</span><span class="o">.</span><span class="n">index_select</span><span class="p">(</span><span class="n">dim</span><span class="p">,</span> <span class="n">index</span><span class="p">)))</span>
+    <span class="k">return</span> <span class="n">v</span></div>
+
+
+<div class="viewcode-block" id="cummul_"><a class="viewcode-back" href="../../../../generated/pypose.cummul_/#pypose.cummul_">[docs]</a><span class="k">def</span> <span class="nf">cummul_</span><span class="p">(</span><span class="nb">input</span><span class="p">,</span> <span class="n">dim</span><span class="p">,</span> <span class="n">left</span> <span class="o">=</span> <span class="kc">True</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Inplace version of :meth:`pypose.cummul`</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">if</span> <span class="n">left</span><span class="p">:</span>
+        <span class="k">return</span> <span class="n">cumops_</span><span class="p">(</span><span class="nb">input</span><span class="p">,</span> <span class="n">dim</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="p">:</span> <span class="n">b</span> <span class="o">*</span> <span class="n">a</span><span class="p">)</span>
+    <span class="k">else</span><span class="p">:</span>
+        <span class="k">return</span> <span class="n">cumops_</span><span class="p">(</span><span class="nb">input</span><span class="p">,</span> <span class="n">dim</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="p">:</span> <span class="n">a</span> <span class="o">*</span> <span class="n">b</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="cumprod_"><a class="viewcode-back" href="../../../../generated/pypose.cumprod_/#pypose.cumprod_">[docs]</a><span class="k">def</span> <span class="nf">cumprod_</span><span class="p">(</span><span class="nb">input</span><span class="p">,</span> <span class="n">dim</span><span class="p">,</span> <span class="n">left</span> <span class="o">=</span> <span class="kc">True</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Inplace version of :meth:`pypose.cumprod`</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">if</span> <span class="n">left</span><span class="p">:</span>
+        <span class="k">return</span> <span class="n">cumops_</span><span class="p">(</span><span class="nb">input</span><span class="p">,</span> <span class="n">dim</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="p">:</span> <span class="n">b</span> <span class="o">@</span> <span class="n">a</span><span class="p">)</span>
+    <span class="k">else</span><span class="p">:</span>
+        <span class="k">return</span> <span class="n">cumops_</span><span class="p">(</span><span class="nb">input</span><span class="p">,</span> <span class="n">dim</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="p">:</span> <span class="n">a</span> <span class="o">@</span> <span class="n">b</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="cumops"><a class="viewcode-back" href="../../../../generated/pypose.cumops/#pypose.cumops">[docs]</a><span class="k">def</span> <span class="nf">cumops</span><span class="p">(</span><span class="nb">input</span><span class="p">,</span> <span class="n">dim</span><span class="p">,</span> <span class="n">ops</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&quot;&quot;&quot;Returns the cumulative user-defined operation of LieTensor along a dimension.</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        y_i = x_1~\mathrm{\circ}~x_2 ~\mathrm{\circ}~ \cdots ~\mathrm{\circ}~ x_i,</span>
+
+<span class="sd">    where :math:`\mathrm{\circ}` is the user-defined operation and :math:`x_i,~y_i`</span>
+<span class="sd">    are the :math:`i`-th LieType item along the :obj:`dim` dimension of input and</span>
+<span class="sd">    output, respectively.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        input (LieTensor): the input LieTensor</span>
+<span class="sd">        dim (int): the dimension to do the operation over</span>
+<span class="sd">        ops (func): the user-defined operation or function</span>
+
+<span class="sd">    Returns:</span>
+<span class="sd">        LieTensor: LieTensor</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        - The users are supposed to provide meaningful operation.</span>
+<span class="sd">        - This function doesn&#39;t check whether the results are valid for mathematical</span>
+<span class="sd">          definition of LieTensor, e.g., quaternion.</span>
+<span class="sd">        - The time complexity of the function is :math:`\mathcal{O}(\log N)`, where</span>
+<span class="sd">          :math:`N` is the LieTensor size along the :obj:`dim` dimension.</span>
+
+<span class="sd">    Examples:</span>
+<span class="sd">        &gt;&gt;&gt; # The following operations are equivalent.</span>
+<span class="sd">        &gt;&gt;&gt; input = pp.randn_SE3(2)</span>
+<span class="sd">        &gt;&gt;&gt; input.cumprod(dim = 0, left=True)</span>
+<span class="sd">        SE3Type LieTensor:</span>
+<span class="sd">        tensor([[-0.6466,  0.2956,  2.4055, -0.4428,  0.1893,  0.3933,  0.7833],</span>
+<span class="sd">                [ 1.2711,  1.2020,  0.0651, -0.0685,  0.6732,  0.7331, -0.0685]])</span>
+<span class="sd">        &gt;&gt;&gt; pp.cumops(input, 0, lambda a, b : b @ a) # Left multiplication</span>
+<span class="sd">        SE3Type LieTensor:</span>
+<span class="sd">        tensor([[-0.6466,  0.2956,  2.4055, -0.4428,  0.1893,  0.3933,  0.7833],</span>
+<span class="sd">                [ 1.2711,  1.2020,  0.0651, -0.0685,  0.6732,  0.7331, -0.0685]])</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+    <span class="k">return</span> <span class="n">cumops_</span><span class="p">(</span><span class="nb">input</span><span class="o">.</span><span class="n">clone</span><span class="p">(),</span> <span class="n">dim</span><span class="p">,</span> <span class="n">ops</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="cummul"><a class="viewcode-back" href="../../../../generated/pypose.cummul/#pypose.cummul">[docs]</a><span class="k">def</span> <span class="nf">cummul</span><span class="p">(</span><span class="nb">input</span><span class="p">,</span> <span class="n">dim</span><span class="p">,</span> <span class="n">left</span> <span class="o">=</span> <span class="kc">True</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&quot;&quot;&quot;Returns the cumulative multiplication (*) of LieTensor along a dimension.</span>
+
+<span class="sd">    * Left multiplication:</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        y_i = x_i * x_{i-1} * \cdots * x_1,</span>
+
+<span class="sd">    * Right multiplication:</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        y_i = x_1 * x_2 * \cdots * x_i,</span>
+
+<span class="sd">    where :math:`x_i,~y_i` are the :math:`i`-th LieType item along the :obj:`dim`</span>
+<span class="sd">    dimension of input and output, respectively.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        input (LieTensor): the input LieTensor</span>
+<span class="sd">        dim (int): the dimension to do the multiplication over</span>
+<span class="sd">        left (bool, optional): whether perform left multiplication in :obj:`cummul`.</span>
+<span class="sd">            If set it to :obj:`False`, this function performs right multiplication.</span>
+<span class="sd">            Defaul: ``True``</span>
+
+<span class="sd">    Returns:</span>
+<span class="sd">        LieTensor: The LieTensor</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        - The time complexity of the function is :math:`\mathcal{O}(\log N)`, where</span>
+<span class="sd">          :math:`N` is the LieTensor size along the :obj:`dim` dimension.</span>
+
+<span class="sd">    Example:</span>
+
+<span class="sd">        * Left multiplication with :math:`\text{input} \in` :obj:`SE3`</span>
+
+<span class="sd">        &gt;&gt;&gt; input = pp.randn_SE3(2)</span>
+<span class="sd">        &gt;&gt;&gt; pp.cummul(input, dim=0)</span>
+<span class="sd">        SE3Type LieTensor:</span>
+<span class="sd">        tensor([[-1.9615, -0.1246,  0.3666,  0.0165,  0.2853,  0.3126,  0.9059],</span>
+<span class="sd">                [ 0.7139,  1.3988, -0.1909, -0.1780,  0.4405, -0.6571,  0.5852]])</span>
+
+<span class="sd">        * Left multiplication with :math:`\text{input} \in` :obj:`SO3`</span>
+
+<span class="sd">        &gt;&gt;&gt; input = pp.randn_SO3(1,2)</span>
+<span class="sd">        &gt;&gt;&gt; pp.cummul(input, dim=1, left=False)</span>
+<span class="sd">        SO3Type LieTensor:</span>
+<span class="sd">        tensor([[[-1.8252e-01,  1.6198e-01,  8.3683e-01,  4.9007e-01],</span>
+<span class="sd">                [ 2.0905e-04,  5.2031e-01,  8.4301e-01, -1.3642e-01]]])</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+    <span class="k">if</span> <span class="n">left</span><span class="p">:</span>
+        <span class="k">return</span> <span class="n">cumops</span><span class="p">(</span><span class="nb">input</span><span class="p">,</span> <span class="n">dim</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="p">:</span> <span class="n">b</span> <span class="o">*</span> <span class="n">a</span><span class="p">)</span>
+    <span class="k">else</span><span class="p">:</span>
+        <span class="k">return</span> <span class="n">cumops</span><span class="p">(</span><span class="nb">input</span><span class="p">,</span> <span class="n">dim</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="p">:</span> <span class="n">a</span> <span class="o">*</span> <span class="n">b</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="cumprod"><a class="viewcode-back" href="../../../../generated/pypose.cumprod/#pypose.cumprod">[docs]</a><span class="k">def</span> <span class="nf">cumprod</span><span class="p">(</span><span class="nb">input</span><span class="p">,</span> <span class="n">dim</span><span class="p">,</span> <span class="n">left</span> <span class="o">=</span> <span class="kc">True</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&quot;&quot;&quot;Returns the cumulative product (``@``) of LieTensor along a dimension.</span>
+
+<span class="sd">    * Left product:</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        y_i = x_i ~\times~ x_{i-1} ~\times~ \cdots ~\times~ x_1,</span>
+
+<span class="sd">    * Right product:</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        y_i = x_1 ~\times~ x_2 ~\times~ \cdots ~\times~ x_i,</span>
+
+<span class="sd">    where :math:`\times` denotes the group product (``@``), :math:`x_i,~y_i` are the</span>
+<span class="sd">    :math:`i`-th item along the :obj:`dim` dimension of the input and output LieTensor,</span>
+<span class="sd">    respectively.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        input (LieTensor): the input LieTensor</span>
+<span class="sd">        dim (int): the dimension to do the operation over</span>
+<span class="sd">        left (bool, optional): whether perform left product in :obj:`cumprod`. If set</span>
+<span class="sd">            it to :obj:`False`, this function performs right product. Defaul: ``True``</span>
+
+<span class="sd">    Returns:</span>
+<span class="sd">        LieTensor: The LieTensor</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        - The time complexity of the function is :math:`\mathcal{O}(\log N)`, where</span>
+<span class="sd">          :math:`N` is the LieTensor size along the :obj:`dim` dimension.</span>
+
+<span class="sd">    Example:</span>
+
+<span class="sd">        * Left product with :math:`\text{input} \in` :obj:`SE3`</span>
+
+<span class="sd">        &gt;&gt;&gt; input = pp.randn_SE3(2)</span>
+<span class="sd">        &gt;&gt;&gt; pp.cumprod(input, dim=0)</span>
+<span class="sd">        SE3Type LieTensor:</span>
+<span class="sd">        tensor([[-1.9615, -0.1246,  0.3666,  0.0165,  0.2853,  0.3126,  0.9059],</span>
+<span class="sd">                [ 0.7139,  1.3988, -0.1909, -0.1780,  0.4405, -0.6571,  0.5852]])</span>
+
+<span class="sd">        * Right product with :math:`\text{input} \in` :obj:`SO3`</span>
+
+<span class="sd">        &gt;&gt;&gt; input = pp.randn_SO3(1,2)</span>
+<span class="sd">        &gt;&gt;&gt; pp.cumprod(input, dim=1, left=False)</span>
+<span class="sd">        SO3Type LieTensor:</span>
+<span class="sd">        tensor([[[ 0.5798, -0.1189, -0.2429,  0.7686],</span>
+<span class="sd">                [ 0.7515, -0.1920,  0.5072,  0.3758]]])</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+    <span class="k">if</span> <span class="n">left</span><span class="p">:</span>
+        <span class="k">return</span> <span class="n">cumops</span><span class="p">(</span><span class="nb">input</span><span class="p">,</span> <span class="n">dim</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="p">:</span> <span class="n">b</span> <span class="o">@</span> <span class="n">a</span><span class="p">)</span>
+    <span class="k">else</span><span class="p">:</span>
+        <span class="k">return</span> <span class="n">cumops</span><span class="p">(</span><span class="nb">input</span><span class="p">,</span> <span class="n">dim</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="p">:</span> <span class="n">a</span> <span class="o">@</span> <span class="n">b</span><span class="p">)</span></div>
+</pre></div>
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+            
+            
+              
+            
+            
+              <p class="caption" role="heading"><span class="caption-text">Contents:</span></p>
+<ul>
+<li class="toctree-l1"><a class="reference internal" href="../../../../lietensor/">LieTensor</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../../../func/">Func</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../../../utils/">Utils</a></li>
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+              
+  <h1>Source code for pypose.func.jac</h1><div class="highlight"><pre>
+<span></span><span class="kn">import</span> <span class="nn">torch</span>
+<span class="kn">from</span> <span class="nn">..</span> <span class="kn">import</span> <span class="n">retain_ltype</span>
+<span class="kn">from</span> <span class="nn">typing</span> <span class="kn">import</span> <span class="n">Callable</span><span class="p">,</span> <span class="n">Union</span><span class="p">,</span> <span class="n">Tuple</span><span class="p">,</span> <span class="n">Optional</span>
+
+
+<div class="viewcode-block" id="jacrev"><a class="viewcode-back" href="../../../../generated/pypose.func.jacrev/#pypose.func.jacrev">[docs]</a><span class="k">def</span> <span class="nf">jacrev</span><span class="p">(</span><span class="n">func</span><span class="p">:</span> <span class="n">Callable</span><span class="p">,</span> <span class="n">argnums</span><span class="p">:</span> <span class="n">Union</span><span class="p">[</span><span class="nb">int</span><span class="p">,</span> <span class="n">Tuple</span><span class="p">[</span><span class="nb">int</span><span class="p">]]</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="o">*</span><span class="p">,</span> <span class="n">has_aux</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span>
+           <span class="n">chunk_size</span><span class="p">:</span> <span class="n">Optional</span><span class="p">[</span><span class="nb">int</span><span class="p">]</span> <span class="o">=</span> <span class="kc">None</span><span class="p">,</span>
+           <span class="n">_preallocate_and_copy</span><span class="o">=</span><span class="kc">False</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">    This function provides the exact same functionality as `torch.func.jacrev()</span>
+<span class="sd">    &lt;https://pytorch.org/docs/stable/generated/torch.func.jacrev.html#torch.func.jacrev&gt;`_,</span>
+<span class="sd">    except that it allows LieTensor to be used as input when calculating the jacobian.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        func (function): A Python function that takes one or more arguments,</span>
+<span class="sd">            one of which must be a Tensor, and returns one or more Tensors</span>
+<span class="sd">        argnums (int or Tuple[int]): Optional, integer or tuple of integers,</span>
+<span class="sd">            saying which arguments to get the Jacobian with respect to.</span>
+<span class="sd">            Default: 0.</span>
+<span class="sd">        has_aux (bool): Flag indicating that ``func`` returns a</span>
+<span class="sd">            ``(output, aux)`` tuple where the first element is the output of</span>
+<span class="sd">            the function to be differentiated and the second element is</span>
+<span class="sd">            auxiliary objects that will not be differentiated.</span>
+<span class="sd">            Default: False.</span>
+<span class="sd">        chunk_size (None or int): If None (default), use the maximum chunk size</span>
+<span class="sd">            (equivalent to doing a single vmap over vjp to compute the jacobian).</span>
+<span class="sd">            If 1, then compute the jacobian row-by-row with a for-loop.</span>
+<span class="sd">            If not None, then compute the jacobian :attr:`chunk_size` rows at a time</span>
+<span class="sd">            (equivalent to doing multiple vmap over vjp). If you run into memory issues</span>
+<span class="sd">            computing the jacobian, please try to specify a non-None chunk_size.</span>
+
+<span class="sd">    Returns:</span>
+<span class="sd">        Returns a function that takes in the same inputs as ``func`` and</span>
+<span class="sd">        returns the Jacobian of ``func`` with respect to the arg(s) at</span>
+<span class="sd">        ``argnums``. If ``has_aux is True``, then the returned function</span>
+<span class="sd">        instead returns a ``(jacobian, aux)`` tuple where ``jacobian``</span>
+<span class="sd">        is the Jacobian and ``aux`` is auxiliary objects returned by ``func``.</span>
+
+<span class="sd">    A basic usage with our LieTensor type would be the transformation function.</span>
+
+<span class="sd">        &gt;&gt;&gt; import pypose as pp</span>
+<span class="sd">        &gt;&gt;&gt; import torch</span>
+<span class="sd">        &gt;&gt;&gt; def func(pose, points):</span>
+<span class="sd">        ...     return pose @ points</span>
+<span class="sd">        &gt;&gt;&gt; pose = pp.randn_SE3(1)</span>
+<span class="sd">        &gt;&gt;&gt; points = torch.randn(1, 3)</span>
+<span class="sd">        &gt;&gt;&gt; jacobian = pp.func.jacrev(func)(pose, points)</span>
+<span class="sd">        &gt;&gt;&gt; jacobian</span>
+<span class="sd">        tensor([[[[ 1.0000,  0.0000,  0.0000,  0.0000,  1.5874, -0.2061,  0.0000]],</span>
+<span class="sd">                [[ 0.0000,  1.0000,  0.0000, -1.5874,  0.0000, -1.4273,  0.0000]],</span>
+<span class="sd">                [[ 0.0000,  0.0000,  1.0000,  0.2061,  1.4273,  0.0000,  0.0000]]]])</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+    <span class="n">jac_func</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">func</span><span class="o">.</span><span class="n">jacrev</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="n">argnums</span><span class="p">,</span> <span class="n">has_aux</span><span class="o">=</span><span class="n">has_aux</span><span class="p">,</span> <span class="n">chunk_size</span><span class="o">=</span><span class="n">chunk_size</span><span class="p">,</span>
+        <span class="n">_preallocate_and_copy</span><span class="o">=</span><span class="n">_preallocate_and_copy</span><span class="p">)</span>
+    <span class="nd">@retain_ltype</span><span class="p">()</span>
+    <span class="k">def</span> <span class="nf">wrapper_fn</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+        <span class="k">return</span> <span class="n">jac_func</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+    <span class="k">return</span> <span class="n">wrapper_fn</span></div>
+</pre></div>
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+            
+              <p class="caption" role="heading"><span class="caption-text">Contents:</span></p>
+<ul>
+<li class="toctree-l1"><a class="reference internal" href="../../../../lietensor/">LieTensor</a></li>
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+              
+  <h1>Source code for pypose.function.checking</h1><div class="highlight"><pre>
+<span></span><span class="kn">import</span> <span class="nn">torch</span><span class="o">,</span> <span class="nn">math</span>
+<span class="kn">from</span> <span class="nn">..</span> <span class="kn">import</span> <span class="n">lietensor</span>
+<span class="kn">from</span> <span class="nn">..</span> <span class="kn">import</span> <span class="n">LieTensor</span>
+
+
+<div class="viewcode-block" id="is_lietensor"><a class="viewcode-back" href="../../../../generated/pypose.is_lietensor/#pypose.is_lietensor">[docs]</a><span class="k">def</span> <span class="nf">is_lietensor</span><span class="p">(</span><span class="n">obj</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Check whether an instance or object is a LieTensor or not.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        obj (``obj``): a Python object or instantance.</span>
+
+<span class="sd">    Return:</span>
+<span class="sd">        ``bool``: ``True`` if obj is a LieTensor object otherwise ``False``.</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">return</span> <span class="kc">True</span> <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">obj</span><span class="p">,</span> <span class="n">LieTensor</span><span class="p">)</span> <span class="k">else</span> <span class="kc">False</span></div>
+
+
+<div class="viewcode-block" id="is_SE3"><a class="viewcode-back" href="../../../../generated/pypose.is_SE3/#pypose.is_SE3">[docs]</a><span class="k">def</span> <span class="nf">is_SE3</span><span class="p">(</span><span class="n">obj</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Check whether an instance or object is an SE3 Type LieTensor or not.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        obj (``obj``): a Python object or instantance.</span>
+
+<span class="sd">    Return:</span>
+<span class="sd">        ``bool``: ``True`` if obj is a SE3 Type LieTensor object otherwise ``False``.</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">return</span> <span class="kc">True</span> <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">obj</span><span class="o">.</span><span class="n">ltype</span><span class="p">,</span> <span class="n">lietensor</span><span class="o">.</span><span class="n">lietensor</span><span class="o">.</span><span class="n">SE3Type</span><span class="p">)</span> <span class="k">else</span> <span class="kc">False</span></div>
+
+
+<div class="viewcode-block" id="hasnan"><a class="viewcode-back" href="../../../../generated/pypose.hasnan/#pypose.hasnan">[docs]</a><span class="k">def</span> <span class="nf">hasnan</span><span class="p">(</span><span class="n">obj</span><span class="p">:</span><span class="nb">list</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Checks whether a deep nested list of tensors contains Nan or not.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        obj (``obj``): a Python object that can be a list of nested list.</span>
+
+<span class="sd">    Return:</span>
+<span class="sd">        ``bool``: ``True`` if the list contains a tensor with ``Nan`` otherwise ``False``.</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        &gt;&gt;&gt; L1 = [[1, 3], [4, [5, 6]], 7, [8, torch.tensor([0, -1.0999])]]</span>
+<span class="sd">        &gt;&gt;&gt; hasnan(L1)</span>
+<span class="sd">        False</span>
+<span class="sd">        &gt;&gt;&gt; L2 = [[torch.tensor([float(&#39;nan&#39;), -1.0999]), 3], [4, [5, 6]], 7, [8, 9]]</span>
+<span class="sd">        &gt;&gt;&gt; hasnan(L2)</span>
+<span class="sd">        True</span>
+<span class="sd">        &gt;&gt;&gt; L3 = [[torch.tensor([1, -1.0999]), 3], [4, [float(&#39;nan&#39;), 6]], 7, [8, 9]]</span>
+<span class="sd">        &gt;&gt;&gt; hasnan(L3)</span>
+<span class="sd">        True</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">obj</span><span class="p">,</span> <span class="nb">list</span><span class="p">)</span> <span class="ow">or</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">obj</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">):</span>
+        <span class="k">for</span> <span class="n">l</span> <span class="ow">in</span> <span class="n">obj</span><span class="p">:</span>
+            <span class="k">if</span> <span class="n">hasnan</span><span class="p">(</span><span class="n">l</span><span class="p">):</span>
+                <span class="k">return</span> <span class="kc">True</span>
+        <span class="k">return</span> <span class="kc">False</span>
+    <span class="k">else</span><span class="p">:</span>
+        <span class="k">return</span> <span class="n">torch</span><span class="o">.</span><span class="n">isnan</span><span class="p">(</span><span class="n">obj</span><span class="p">)</span><span class="o">.</span><span class="n">any</span><span class="p">()</span> <span class="k">if</span> <span class="n">torch</span><span class="o">.</span><span class="n">is_tensor</span><span class="p">(</span><span class="n">obj</span><span class="p">)</span> <span class="k">else</span> <span class="n">math</span><span class="o">.</span><span class="n">isnan</span><span class="p">(</span><span class="n">obj</span><span class="p">)</span></div>
+</pre></div>
+
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diff --git a/docs/preview/pr/352/_modules/pypose/function/geometry/index.html b/docs/preview/pr/352/_modules/pypose/function/geometry/index.html
new file mode 100644
index 000000000..4d49460d2
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@@ -0,0 +1,807 @@
+
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+    You are viewing unstable developer preview docs.
+    Click here to view docs for latest stable release.
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+            
+            
+              
+            
+            
+              <p class="caption" role="heading"><span class="caption-text">Contents:</span></p>
+<ul>
+<li class="toctree-l1"><a class="reference internal" href="../../../../lietensor/">LieTensor</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../../../func/">Func</a></li>
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+              
+  <h1>Source code for pypose.function.geometry</h1><div class="highlight"><pre>
+<span></span><span class="kn">import</span> <span class="nn">torch</span>
+<span class="kn">from</span> <span class="nn">..</span> <span class="kn">import</span> <span class="n">mat2SE3</span>
+<span class="kn">from</span> <span class="nn">..basics</span> <span class="kn">import</span> <span class="n">pm</span>
+<span class="kn">from</span> <span class="nn">.checking</span> <span class="kn">import</span> <span class="n">is_lietensor</span>
+
+
+<div class="viewcode-block" id="cart2homo"><a class="viewcode-back" href="../../../../generated/pypose.cart2homo/#pypose.cart2homo">[docs]</a><span class="k">def</span> <span class="nf">cart2homo</span><span class="p">(</span><span class="n">coordinates</span><span class="p">:</span><span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Converts batched Cartesian coordinates to Homogeneous coordinates</span>
+<span class="sd">    by adding ones to last dimension.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        coordinates (``torch.Tensor``): the Cartesian coordinates to be converted.</span>
+
+<span class="sd">    Returns:</span>
+<span class="sd">        ``torch.Tensor``: the coordinates in Homogeneous space.</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        The last dimension of the input coordinates can be any dimension.</span>
+
+<span class="sd">    Example:</span>
+
+<span class="sd">        &gt;&gt;&gt; points = torch.randn(2, 2)</span>
+<span class="sd">        &gt;&gt;&gt; cart2homo(points)</span>
+<span class="sd">        tensor([[ 2.0598,  1.5351,  1.0000],</span>
+<span class="sd">                [-0.8484,  1.2390,  1.0000]])</span>
+<span class="sd">        &gt;&gt;&gt; points = torch.randn(2, 3)</span>
+<span class="sd">        &gt;&gt;&gt; cart2homo(points)</span>
+<span class="sd">        tensor([[ 1.7946,  0.3548, -0.4446,  1.0000],</span>
+<span class="sd">                [ 0.3010, -2.2748, -0.4708,  1.0000]])</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="n">ones</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">ones_like</span><span class="p">(</span><span class="n">coordinates</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="p">:</span><span class="mi">1</span><span class="p">])</span>
+    <span class="k">return</span> <span class="n">torch</span><span class="o">.</span><span class="n">cat</span><span class="p">([</span><span class="n">coordinates</span><span class="p">,</span> <span class="n">ones</span><span class="p">],</span> <span class="n">dim</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="homo2cart"><a class="viewcode-back" href="../../../../generated/pypose.homo2cart/#pypose.homo2cart">[docs]</a><span class="k">def</span> <span class="nf">homo2cart</span><span class="p">(</span><span class="n">coordinates</span><span class="p">:</span><span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Converts batched Homogeneous coordinates to Cartesian coordinates</span>
+<span class="sd">    by dividing the last row. Size of the last dimension will be reduced by 1.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        coordinates (``torch.Tensor``): the Homogeneous coordinates to be converted.</span>
+
+<span class="sd">    Returns:</span>
+<span class="sd">        ``torch.Tensor``: the coordinates in Cartesian space.</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        &gt;&gt;&gt; points = torch.tensor([[4., 3., 2., 1.], [8., 6., 4., 2.]])</span>
+<span class="sd">        &gt;&gt;&gt; homo2cart(points)</span>
+<span class="sd">        tensor([[4., 3., 2.],</span>
+<span class="sd">                [4., 3., 2.]])</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="n">tiny</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">finfo</span><span class="p">(</span><span class="n">coordinates</span><span class="o">.</span><span class="n">dtype</span><span class="p">)</span><span class="o">.</span><span class="n">tiny</span>
+    <span class="n">denum</span> <span class="o">=</span> <span class="n">coordinates</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">:]</span><span class="o">.</span><span class="n">abs</span><span class="p">()</span><span class="o">.</span><span class="n">clamp_</span><span class="p">(</span><span class="nb">min</span><span class="o">=</span><span class="n">tiny</span><span class="p">)</span>
+    <span class="n">denum</span> <span class="o">=</span> <span class="n">pm</span><span class="p">(</span><span class="n">coordinates</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">:])</span> <span class="o">*</span> <span class="n">denum</span>
+    <span class="k">return</span> <span class="n">coordinates</span><span class="p">[</span><span class="o">...</span><span class="p">,:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">/</span> <span class="n">denum</span></div>
+
+
+<div class="viewcode-block" id="point2pixel"><a class="viewcode-back" href="../../../../generated/pypose.point2pixel/#pypose.point2pixel">[docs]</a><span class="k">def</span> <span class="nf">point2pixel</span><span class="p">(</span><span class="n">points</span><span class="p">,</span> <span class="n">intrinsics</span><span class="p">,</span> <span class="n">extrinsics</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Project batched sets of points (either in camera or world frame) to pixels.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        points (``torch.Tensor``): The 3D coordinate of points. Assumed to be in the</span>
+<span class="sd">            camera frame if ``extrinsics`` is ``None``, otherwiwse in the world frame.</span>
+<span class="sd">            The shape has to be (..., N, 3).</span>
+<span class="sd">        intrinsics (``torch.Tensor``): The intrinsic parameters of cameras.</span>
+<span class="sd">            The shape has to be (..., 3, 3).</span>
+<span class="sd">        extrinsics (``pypose.LieTensor``, optional): The extrinsic parameters of cameras.</span>
+<span class="sd">            The shape has to be (..., 7). Default: ``None``.</span>
+
+<span class="sd">    Returns:</span>
+<span class="sd">        ``torch.Tensor``: The associated pixel with shape (..., N, 2).</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        &gt;&gt;&gt; import torch, pypose as pp</span>
+<span class="sd">        &gt;&gt;&gt; f, (H, W) = 2, (9, 9) # focal length and image height, width</span>
+<span class="sd">        &gt;&gt;&gt; intrinsics = torch.tensor([[f, 0, H / 2],</span>
+<span class="sd">        ...                            [0, f, W / 2],</span>
+<span class="sd">        ...                            [0, 0,   1  ]])</span>
+<span class="sd">        &gt;&gt;&gt; object = torch.tensor([[2., 0., 2.],</span>
+<span class="sd">        ...                        [1., 0., 2.],</span>
+<span class="sd">        ...                        [0., 1., 1.],</span>
+<span class="sd">        ...                        [0., 0., 1.],</span>
+<span class="sd">        ...                        [1., 0., 1.],</span>
+<span class="sd">        ...                        [5., 5., 3.]])</span>
+<span class="sd">        &gt;&gt;&gt; pixels = pp.point2pixel(object, intrinsics)</span>
+<span class="sd">        tensor([[6.5000, 4.5000],</span>
+<span class="sd">                [5.5000, 4.5000],</span>
+<span class="sd">                [4.5000, 6.5000],</span>
+<span class="sd">                [4.5000, 4.5000],</span>
+<span class="sd">                [6.5000, 4.5000],</span>
+<span class="sd">                [7.8333, 7.8333]])</span>
+<span class="sd">        &gt;&gt;&gt; pose = pp.SE3([ 0., -8,  0.,  0., -0.3827,  0.,  0.9239])</span>
+<span class="sd">        &gt;&gt;&gt; pixels = pp.point2pixel(object, intrinsics, pose)</span>
+<span class="sd">        tensor([[  4.4999,  -1.1568],</span>
+<span class="sd">                [  3.8332,  -3.0425],</span>
+<span class="sd">                [  2.4998, -15.2997],</span>
+<span class="sd">                [  2.4998, -18.1282],</span>
+<span class="sd">                [  4.4999,  -6.8135],</span>
+<span class="sd">                [  4.9999,   3.4394]])</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">assert</span> <span class="n">points</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">==</span> <span class="mi">3</span><span class="p">,</span> <span class="s2">&quot;Points shape incorrect&quot;</span>
+    <span class="k">assert</span> <span class="n">intrinsics</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">==</span> <span class="n">intrinsics</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span> <span class="o">==</span> <span class="mi">3</span><span class="p">,</span> <span class="s2">&quot;Intrinsics shape incorrect.&quot;</span>
+    <span class="k">if</span> <span class="n">extrinsics</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
+        <span class="n">torch</span><span class="o">.</span><span class="n">broadcast_shapes</span><span class="p">(</span><span class="n">points</span><span class="o">.</span><span class="n">shape</span><span class="p">[:</span><span class="o">-</span><span class="mi">2</span><span class="p">],</span> <span class="n">intrinsics</span><span class="o">.</span><span class="n">shape</span><span class="p">[:</span><span class="o">-</span><span class="mi">2</span><span class="p">])</span>
+    <span class="k">else</span><span class="p">:</span>
+        <span class="k">assert</span> <span class="n">is_lietensor</span><span class="p">(</span><span class="n">extrinsics</span><span class="p">)</span> <span class="ow">and</span> <span class="n">extrinsics</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="mi">7</span><span class="p">,</span> <span class="s2">&quot;Type incorrect.&quot;</span>
+        <span class="n">torch</span><span class="o">.</span><span class="n">broadcast_shapes</span><span class="p">(</span><span class="n">points</span><span class="o">.</span><span class="n">shape</span><span class="p">[:</span><span class="o">-</span><span class="mi">2</span><span class="p">],</span> <span class="n">intrinsics</span><span class="o">.</span><span class="n">shape</span><span class="p">[:</span><span class="o">-</span><span class="mi">2</span><span class="p">],</span> <span class="n">extrinsics</span><span class="o">.</span><span class="n">shape</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
+        <span class="n">points</span> <span class="o">=</span> <span class="n">extrinsics</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span> <span class="o">@</span> <span class="n">points</span>
+    <span class="k">return</span> <span class="n">homo2cart</span><span class="p">(</span><span class="n">points</span> <span class="o">@</span> <span class="n">intrinsics</span><span class="o">.</span><span class="n">mT</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="pixel2point"><a class="viewcode-back" href="../../../../generated/pypose.pixel2point/#pypose.pixel2point">[docs]</a><span class="k">def</span> <span class="nf">pixel2point</span><span class="p">(</span><span class="n">pixels</span><span class="p">,</span> <span class="n">depth</span><span class="p">,</span> <span class="n">intrinsics</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Convert batch of pixels with depth into points (in camera coordinate)</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        pixels: (``torch.Tensor``) The 2d coordinates of pixels in the camera</span>
+<span class="sd">            pixel coordinate.</span>
+<span class="sd">            Shape has to be (..., N, 2)</span>
+
+<span class="sd">        depth: (``torch.Tensor``) The depths of pixels with respect to the</span>
+<span class="sd">            sensor plane.</span>
+<span class="sd">            Shape has to be (..., N)</span>
+
+<span class="sd">        intrinsics: (``torch.Tensor``): The intrinsic parameters of cameras.</span>
+<span class="sd">            The shape has to be (..., 3, 3).</span>
+
+<span class="sd">    Returns:</span>
+<span class="sd">        ``torch.Tensor`` The associated 3D-points with shape (..., N, 3)</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        &gt;&gt;&gt; import torch, pypose as pp</span>
+<span class="sd">        &gt;&gt;&gt; f, (H, W) = 2, (9, 9) # focal length and image height, width</span>
+<span class="sd">        &gt;&gt;&gt; intrinsics = torch.tensor([[f, 0, H / 2],</span>
+<span class="sd">        ...                            [0, f, W / 2],</span>
+<span class="sd">        ...                            [0, 0,   1  ]])</span>
+<span class="sd">        &gt;&gt;&gt; pixels = torch.tensor([[0.5, 0.0],</span>
+<span class="sd">        ...                        [1.0, 0.0],</span>
+<span class="sd">        ...                        [0.0, 1.3],</span>
+<span class="sd">        ...                        [1.0, 0.0],</span>
+<span class="sd">        ...                        [0.5, 1.5],</span>
+<span class="sd">        ...                        [5.0, 1.5]])</span>
+<span class="sd">        &gt;&gt;&gt; depths = torch.tensor([5.0, 3.0, 6.5, 2.0, 0.5, 0.7])</span>
+<span class="sd">        &gt;&gt;&gt; points = pp.pixel2point(pixels, depths, intrinsics)</span>
+<span class="sd">        tensor([[-10.0000, -11.2500,   5.0000],</span>
+<span class="sd">                [ -5.2500,  -6.7500,   3.0000],</span>
+<span class="sd">                [-14.6250, -10.4000,   6.5000],</span>
+<span class="sd">                [ -3.5000,  -4.5000,   2.0000],</span>
+<span class="sd">                [ -1.0000,  -0.7500,   0.5000],</span>
+<span class="sd">                [  0.1750,  -1.0500,   0.7000]])</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">assert</span> <span class="n">pixels</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="p">,</span> <span class="s2">&quot;Pixels shape incorrect&quot;</span>
+    <span class="k">assert</span> <span class="n">depth</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">==</span> <span class="n">pixels</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">),</span> <span class="s2">&quot;Depth shape does not match pixels&quot;</span>
+    <span class="k">assert</span> <span class="n">intrinsics</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">==</span> <span class="n">intrinsics</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span> <span class="o">==</span> <span class="mi">3</span><span class="p">,</span> <span class="s2">&quot;Intrinsics shape incorrect.&quot;</span>
+
+    <span class="n">fx</span><span class="p">,</span> <span class="n">fy</span> <span class="o">=</span> <span class="n">intrinsics</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">intrinsics</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
+    <span class="n">cx</span><span class="p">,</span> <span class="n">cy</span> <span class="o">=</span> <span class="n">intrinsics</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="n">intrinsics</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span>
+
+    <span class="k">assert</span> <span class="ow">not</span> <span class="n">torch</span><span class="o">.</span><span class="n">any</span><span class="p">(</span><span class="n">fx</span> <span class="o">==</span> <span class="mi">0</span><span class="p">),</span> <span class="s2">&quot;fx Cannot contain zero&quot;</span>
+    <span class="k">assert</span> <span class="ow">not</span> <span class="n">torch</span><span class="o">.</span><span class="n">any</span><span class="p">(</span><span class="n">fy</span> <span class="o">==</span> <span class="mi">0</span><span class="p">),</span> <span class="s2">&quot;fy Cannot contain zero&quot;</span>
+
+    <span class="n">pts3d_z</span> <span class="o">=</span> <span class="n">depth</span>
+    <span class="n">pts3d_x</span> <span class="o">=</span> <span class="p">((</span><span class="n">pixels</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="n">cx</span><span class="p">)</span> <span class="o">*</span> <span class="n">pts3d_z</span><span class="p">)</span> <span class="o">/</span> <span class="n">fx</span>
+    <span class="n">pts3d_y</span> <span class="o">=</span> <span class="p">((</span><span class="n">pixels</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">cy</span><span class="p">)</span> <span class="o">*</span> <span class="n">pts3d_z</span><span class="p">)</span> <span class="o">/</span> <span class="n">fy</span>
+    <span class="k">return</span> <span class="n">torch</span><span class="o">.</span><span class="n">stack</span><span class="p">([</span><span class="n">pts3d_x</span><span class="p">,</span> <span class="n">pts3d_y</span><span class="p">,</span> <span class="n">pts3d_z</span><span class="p">],</span> <span class="n">dim</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="reprojerr"><a class="viewcode-back" href="../../../../generated/pypose.reprojerr/#pypose.reprojerr">[docs]</a><span class="k">def</span> <span class="nf">reprojerr</span><span class="p">(</span><span class="n">points</span><span class="p">,</span> <span class="n">pixels</span><span class="p">,</span> <span class="n">intrinsics</span><span class="p">,</span> <span class="n">extrinsics</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">reduction</span><span class="o">=</span><span class="s1">&#39;none&#39;</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Calculates batched per-pixel reprojection error (pixel distance) for points either in</span>
+<span class="sd">    the camera or world frame given camera intrinsics or extrinsics, respectively.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        points (``torch.Tensor``): The 3D coordinate of points. Assumed to be in the</span>
+<span class="sd">            camera frame if ``extrinsics`` is ``None``, otherwiwse in the world frame.</span>
+<span class="sd">            The shape has to be (..., N, 3).</span>
+<span class="sd">        pixels (``torch.Tensor``): The image points. The associated pixel.</span>
+<span class="sd">            The shape has to be (..., N, 2).</span>
+<span class="sd">        intrinsics (``torch.Tensor``): intrinsic matrices.</span>
+<span class="sd">            The shape has to be (..., 3, 3).</span>
+<span class="sd">        extrinsics (``LieTensor``, optional): The camera extrinsics.</span>
+<span class="sd">            The shape has to be (..., 7). Default: ``None``.</span>
+<span class="sd">        reduction (``str``, optional): The reduction to apply on the output: ``&#39;none&#39;``</span>
+<span class="sd">            | ``&#39;sum&#39;`` | ``&#39;norm&#39;``</span>
+
+<span class="sd">            ``&#39;none&#39;``: No reduction is applied</span>
+
+<span class="sd">            ``&#39;sum&#39;``: The reprojection error on each component (u, v) is summed for</span>
+<span class="sd">            each pixel (L1 Norm)</span>
+
+<span class="sd">            ``&#39;norm&#39;``: The reprojection error&#39;s L2 norm for each pixel</span>
+<span class="sd">    Returns:</span>
+<span class="sd">        Per-pixel reprojection error.</span>
+
+<span class="sd">        The shape is (..., N) if reduction is ``&#39;sum&#39;`` or ``&#39;norm&#39;``.</span>
+
+<span class="sd">        The shape is (..., N, 2) if reduction is ``&#39;none&#39;``.</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        &gt;&gt;&gt; import torch, pypose as pp</span>
+<span class="sd">        &gt;&gt;&gt; f, (H, W) = 2, (9, 9) # focal length and image height, width</span>
+<span class="sd">        &gt;&gt;&gt; intrinsics = torch.tensor([[f, 0, H / 2],</span>
+<span class="sd">        ...                            [0, f, W / 2],</span>
+<span class="sd">        ...                            [0, 0,   1  ]])</span>
+<span class="sd">        &gt;&gt;&gt; object = torch.randn(6, 3)</span>
+<span class="sd">        &gt;&gt;&gt; pose = pp.randn_SE3()</span>
+<span class="sd">        &gt;&gt;&gt; pixels = pp.point2pixel(object, intrinsics, pose)</span>
+<span class="sd">        &gt;&gt;&gt; err = pp.reprojerr(object, pixels, intrinsics, pose, reduction=&#39;norm&#39;)</span>
+<span class="sd">        tensor([0., 0., 0., 0., 0., 0.])</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="n">torch</span><span class="o">.</span><span class="n">broadcast_shapes</span><span class="p">(</span><span class="n">points</span><span class="o">.</span><span class="n">shape</span><span class="p">[:</span><span class="o">-</span><span class="mi">2</span><span class="p">],</span> <span class="n">pixels</span><span class="o">.</span><span class="n">shape</span><span class="p">[:</span><span class="o">-</span><span class="mi">2</span><span class="p">],</span> <span class="n">intrinsics</span><span class="o">.</span><span class="n">shape</span><span class="p">[:</span><span class="o">-</span><span class="mi">2</span><span class="p">])</span>
+    <span class="k">assert</span> <span class="n">points</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">==</span> <span class="mi">3</span> <span class="ow">and</span> <span class="n">pixels</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span> <span class="ow">and</span> \
+           <span class="n">intrinsics</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">==</span> <span class="n">intrinsics</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span> <span class="o">==</span> <span class="mi">3</span><span class="p">,</span> <span class="s2">&quot;Shape not compatible.&quot;</span>
+    <span class="k">assert</span> <span class="n">reduction</span> <span class="ow">in</span> <span class="p">{</span><span class="s1">&#39;norm&#39;</span><span class="p">,</span> <span class="s1">&#39;sum&#39;</span><span class="p">,</span> <span class="s1">&#39;none&#39;</span><span class="p">},</span> \
+           <span class="s2">&quot;Reduction method can only be &#39;norm&#39;|&#39;sum&#39;|&#39;none&#39;.&quot;</span>
+    <span class="n">img_repj</span> <span class="o">=</span> <span class="n">point2pixel</span><span class="p">(</span><span class="n">points</span><span class="p">,</span> <span class="n">intrinsics</span><span class="p">,</span> <span class="n">extrinsics</span><span class="p">)</span>
+
+    <span class="k">if</span> <span class="n">reduction</span> <span class="o">==</span> <span class="s1">&#39;norm&#39;</span><span class="p">:</span>
+        <span class="k">return</span> <span class="p">(</span><span class="n">img_repj</span> <span class="o">-</span> <span class="n">pixels</span><span class="p">)</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">dim</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span>
+    <span class="k">elif</span> <span class="n">reduction</span> <span class="o">==</span> <span class="s1">&#39;sum&#39;</span><span class="p">:</span>
+        <span class="k">return</span> <span class="p">(</span><span class="n">img_repj</span> <span class="o">-</span> <span class="n">pixels</span><span class="p">)</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">dim</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span>
+    <span class="k">return</span> <span class="n">img_repj</span> <span class="o">-</span> <span class="n">pixels</span></div>
+
+
+<div class="viewcode-block" id="knn"><a class="viewcode-back" href="../../../../generated/pypose.knn/#pypose.knn">[docs]</a><span class="k">def</span> <span class="nf">knn</span><span class="p">(</span><span class="n">ref</span><span class="p">,</span> <span class="n">nbr</span><span class="p">,</span> <span class="n">k</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="nb">ord</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">dim</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">largest</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="nb">sorted</span><span class="o">=</span><span class="kc">True</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Select the k nearest neighbor points of reference from neighbors in each batch.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        ref (``torch.Tensor``): the coordinates of the reference point sets.</span>
+<span class="sd">            The shape has to be (..., N1, :).</span>
+<span class="sd">        nbr (``torch.Tensor``): the coordinates of the neighbors point sets.</span>
+<span class="sd">            The shape has to be (..., N2, :).</span>
+<span class="sd">        k (``int``, optional): the number of the nearest neighbors to be selected.</span>
+<span class="sd">            k has to be k :math:`\leq` N2. Default: ``1``.</span>
+<span class="sd">        ord (``int``, optional): the order of norm to use for distance calculation.</span>
+<span class="sd">            Default: ``2`` (Euclidean distance).</span>
+<span class="sd">        dim (``int``, optional): the dimension encompassing the point cloud coordinates,</span>
+<span class="sd">            utilized for calculating distance and sorting.</span>
+<span class="sd">            Default: ``-1`` (The last dimension).</span>
+<span class="sd">        largest (``bool``, optional): controls whether to return largest (furthest) or</span>
+<span class="sd">            smallest (nearest) neighbors. Default: ``False``.</span>
+<span class="sd">        sorted (``bool``, optional): controls whether to return the neighbors in sorted</span>
+<span class="sd">            order. Default: ``True``.</span>
+
+<span class="sd">    Returns:</span>
+<span class="sd">        ``torch.return_types.topk(values: torch.Tensor, indices: torch.LongTensor)``:</span>
+<span class="sd">        The named tuple of (values, indices).</span>
+
+<span class="sd">        ``values``: The ord-norm distance between each point in ref and its sorted k</span>
+<span class="sd">        nearest neighbors in nbr. The shape is (..., N1, k).</span>
+
+<span class="sd">        ``indices``: The index of the k nearest neighbor points in neighbors point sets</span>
+<span class="sd">        (nbr). The shape is (..., N1, k).</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        If ``sorted`` is set to ``False``, the output will be unspecified and not</span>
+<span class="sd">        necessarily sorted along the index of the input point cloud.</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        &gt;&gt;&gt; import torch, pypose as pp</span>
+<span class="sd">        &gt;&gt;&gt; ref = torch.tensor([[9., 2., 2.],</span>
+<span class="sd">        ...                     [1., 0., 2.],</span>
+<span class="sd">        ...                     [0., 1., 1.],</span>
+<span class="sd">        ...                     [5., 0., 1.],</span>
+<span class="sd">        ...                     [1., 0., 1.],</span>
+<span class="sd">        ...                     [5., 5., 3.]])</span>
+<span class="sd">        &gt;&gt;&gt; nbr = torch.tensor([[1., 0., 1.],</span>
+<span class="sd">        ...                     [1., 6., 2.],</span>
+<span class="sd">        ...                     [5., 1., 0.],</span>
+<span class="sd">        ...                     [9., 0., 2.]])</span>
+<span class="sd">        &gt;&gt;&gt; pp.knn(ref, nbr)</span>
+<span class="sd">        torch.return_types.topk(</span>
+<span class="sd">        values=tensor([[2.0000],</span>
+<span class="sd">                [1.0000],</span>
+<span class="sd">                [1.4142],</span>
+<span class="sd">                [1.4142],</span>
+<span class="sd">                [0.0000],</span>
+<span class="sd">                [4.2426]]),</span>
+<span class="sd">        indices=tensor([[3],</span>
+<span class="sd">                [0],</span>
+<span class="sd">                [0],</span>
+<span class="sd">                [2],</span>
+<span class="sd">                [0],</span>
+<span class="sd">                [1]]))</span>
+<span class="sd">        &gt;&gt;&gt; pp.knn(ref, nbr, k=2, ord=2)</span>
+<span class="sd">        torch.return_types.topk(</span>
+<span class="sd">        values=tensor([[2.0000, 4.5826],</span>
+<span class="sd">                [1.0000, 4.5826],</span>
+<span class="sd">                [1.4142, 5.0990],</span>
+<span class="sd">                [1.4142, 4.0000],</span>
+<span class="sd">                [0.0000, 4.2426],</span>
+<span class="sd">                [4.2426, 5.0000]]),</span>
+<span class="sd">        indices=tensor([[3, 2],</span>
+<span class="sd">                [0, 2],</span>
+<span class="sd">                [0, 2],</span>
+<span class="sd">                [2, 0],</span>
+<span class="sd">                [0, 2],</span>
+<span class="sd">                [1, 2]]))</span>
+<span class="sd">        &gt;&gt;&gt; pp.knn(ref, nbr, k=2, ord=2).values</span>
+<span class="sd">        tensor([[2.0000, 4.5826],</span>
+<span class="sd">                [1.0000, 4.5826],</span>
+<span class="sd">                [1.4142, 5.0990],</span>
+<span class="sd">                [1.4142, 4.0000],</span>
+<span class="sd">                [0.0000, 4.2426],</span>
+<span class="sd">                [4.2426, 5.0000]])</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="n">diff</span> <span class="o">=</span> <span class="n">ref</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span> <span class="o">-</span> <span class="n">nbr</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">)</span>
+    <span class="n">dist</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">diff</span><span class="p">,</span> <span class="n">dim</span><span class="o">=</span><span class="n">dim</span><span class="p">,</span> <span class="nb">ord</span><span class="o">=</span><span class="nb">ord</span><span class="p">)</span>
+    <span class="k">return</span> <span class="n">dist</span><span class="o">.</span><span class="n">topk</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">dim</span><span class="o">=</span><span class="n">dim</span><span class="p">,</span> <span class="n">largest</span><span class="o">=</span><span class="n">largest</span><span class="p">,</span> <span class="nb">sorted</span><span class="o">=</span><span class="nb">sorted</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="svdtf"><a class="viewcode-back" href="../../../../generated/pypose.svdtf/#pypose.svdtf">[docs]</a><span class="k">def</span> <span class="nf">svdtf</span><span class="p">(</span><span class="n">source</span><span class="p">,</span> <span class="n">target</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Computes the rigid transformation ( :math:`SE(3)` ) between two sets of associated</span>
+<span class="sd">    point clouds (source and target) using Singular Value Decomposition (SVD).</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        source (``torch.Tensor``): the coordinates of the source point cloud.</span>
+<span class="sd">            The shape has to be (..., N, 3).</span>
+<span class="sd">        target (``torch.Tensor``): the coordinates of the target point cloud.</span>
+<span class="sd">            The shape has to be (..., N, 3).</span>
+
+<span class="sd">    Returns:</span>
+<span class="sd">        ``LieTensor``: The rigid transformation matrix in ``SE3Type``  that</span>
+<span class="sd">        minimizes the mean squared error between the input point sets.</span>
+
+<span class="sd">    Warning:</span>
+<span class="sd">        The number of points N has to be the same for both point clouds.</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        &gt;&gt;&gt; import torch, pypose as pp</span>
+<span class="sd">        &gt;&gt;&gt; source = torch.tensor([[0., 0., 0.],</span>
+<span class="sd">        ...                     [1., 0., 0.],</span>
+<span class="sd">        ...                     [0., 1., 0.]])</span>
+<span class="sd">        &gt;&gt;&gt; target = torch.tensor([[1., 1., 1.],</span>
+<span class="sd">        ...                     [2., 1., 1.],</span>
+<span class="sd">        ...                     [1., 2., 1.]])</span>
+<span class="sd">        &gt;&gt;&gt; pp.svdtf(source, target)</span>
+<span class="sd">        SE3Type LieTensor:</span>
+<span class="sd">        LieTensor([1., 1., 1., 0., 0., 0., 1.])</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">assert</span> <span class="n">source</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span> <span class="o">==</span> <span class="n">target</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">),</span> <span class="p">{</span>
+        <span class="s2">&quot;The number of points N has to be the same for both point clouds.&quot;</span><span class="p">}</span>
+    <span class="n">ctnsource</span> <span class="o">=</span> <span class="n">source</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">dim</span><span class="o">=-</span><span class="mi">2</span><span class="p">,</span> <span class="n">keepdim</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+    <span class="n">ctntarget</span> <span class="o">=</span> <span class="n">target</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">dim</span><span class="o">=-</span><span class="mi">2</span><span class="p">,</span> <span class="n">keepdim</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+    <span class="n">source</span> <span class="o">=</span> <span class="n">source</span> <span class="o">-</span> <span class="n">ctnsource</span>
+    <span class="n">target</span> <span class="o">=</span> <span class="n">target</span> <span class="o">-</span> <span class="n">ctntarget</span>
+    <span class="n">M</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">einsum</span><span class="p">(</span><span class="s1">&#39;...Na, ...Nb -&gt; ...ab&#39;</span><span class="p">,</span> <span class="n">target</span><span class="p">,</span> <span class="n">source</span><span class="p">)</span>
+    <span class="n">U</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">Vh</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">svd</span><span class="p">(</span><span class="n">M</span><span class="p">)</span>
+    <span class="n">R</span> <span class="o">=</span> <span class="n">U</span> <span class="o">@</span> <span class="n">Vh</span>
+    <span class="n">mask</span> <span class="o">=</span> <span class="p">(</span><span class="n">R</span><span class="o">.</span><span class="n">det</span><span class="p">()</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">abs</span><span class="p">()</span> <span class="o">&lt;</span> <span class="mf">1e-6</span>
+    <span class="n">R</span><span class="p">[</span><span class="n">mask</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span> <span class="n">R</span><span class="p">[</span><span class="n">mask</span><span class="p">]</span>
+    <span class="n">t</span> <span class="o">=</span> <span class="n">ctntarget</span><span class="o">.</span><span class="n">mT</span> <span class="o">-</span> <span class="n">R</span> <span class="o">@</span> <span class="n">ctnsource</span><span class="o">.</span><span class="n">mT</span>
+    <span class="n">T</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">cat</span><span class="p">((</span><span class="n">R</span><span class="p">,</span> <span class="n">t</span><span class="p">),</span> <span class="n">dim</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span>
+    <span class="k">return</span> <span class="n">mat2SE3</span><span class="p">(</span><span class="n">T</span><span class="p">,</span> <span class="n">check</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span></div>
+</pre></div>
+
+             </article>
+             
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+              <p class="caption" role="heading"><span class="caption-text">Contents:</span></p>
+<ul>
+<li class="toctree-l1"><a class="reference internal" href="../../../../lietensor/">LieTensor</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../../../func/">Func</a></li>
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+              
+  <h1>Source code for pypose.function.linalg</h1><div class="highlight"><pre>
+<span></span><span class="kn">import</span> <span class="nn">torch</span>
+<span class="kn">from</span> <span class="nn">..</span> <span class="kn">import</span> <span class="n">LieTensor</span>
+
+
+<div class="viewcode-block" id="bvv"><a class="viewcode-back" href="../../../../generated/pypose.bvv/#pypose.bvv">[docs]</a><span class="k">def</span> <span class="nf">bvv</span><span class="p">(</span><span class="n">lvec</span><span class="p">,</span> <span class="n">rvec</span><span class="p">,</span> <span class="o">*</span><span class="p">,</span> <span class="n">out</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Performs batched vector-vector product, which results in matrices.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        lvec (:obj:`Tensor`): left vectors to be multiplied.</span>
+<span class="sd">        rvec (:obj:`Tensor`): right vectors to be multiplied.</span>
+
+<span class="sd">    Return:</span>
+<span class="sd">        out (:obj:`Tensor`): the output tensor.</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        This function is broadcastable and supports batched product.</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        &gt;&gt;&gt; lvec = torch.randn(2, 1, 3)</span>
+<span class="sd">        &gt;&gt;&gt; rvec = torch.randn(1, 2, 2)</span>
+<span class="sd">        &gt;&gt;&gt; out = pp.bvv(lvec, rvec)</span>
+<span class="sd">        &gt;&gt;&gt; out.shape</span>
+<span class="sd">        torch.Size([2, 2, 3, 2])</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="n">lvec</span> <span class="o">=</span> <span class="n">lvec</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">lvec</span><span class="p">,</span> <span class="n">LieTensor</span><span class="p">)</span> <span class="k">else</span> <span class="n">lvec</span>
+    <span class="n">rvec</span> <span class="o">=</span> <span class="n">rvec</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">rvec</span><span class="p">,</span> <span class="n">LieTensor</span><span class="p">)</span> <span class="k">else</span> <span class="n">rvec</span>
+    <span class="n">lvec</span><span class="p">,</span> <span class="n">rvec</span> <span class="o">=</span> <span class="n">lvec</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">),</span> <span class="n">rvec</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
+    <span class="k">return</span> <span class="n">torch</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">lvec</span><span class="p">,</span> <span class="n">rvec</span><span class="o">.</span><span class="n">mT</span><span class="p">,</span> <span class="n">out</span><span class="o">=</span><span class="n">out</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="bmv"><a class="viewcode-back" href="../../../../generated/pypose.bmv/#pypose.bmv">[docs]</a><span class="k">def</span> <span class="nf">bmv</span><span class="p">(</span><span class="n">mat</span><span class="p">,</span> <span class="n">vec</span><span class="p">,</span> <span class="o">*</span><span class="p">,</span> <span class="n">out</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Performs batched matrix-vector product.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        mat (:obj:`Tensor`): matrices to be multiplied.</span>
+<span class="sd">        vec (:obj:`Tensor`): vectors to be multiplied.</span>
+
+<span class="sd">    Return:</span>
+<span class="sd">        out (:obj:`Tensor`): the output tensor.</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        The ``mat`` has to be a (:math:`\cdots\times n \times m`) tensor,</span>
+<span class="sd">        the ``vec`` has to be a (:math:`\cdots\times m`) tensor,</span>
+<span class="sd">        and ``out`` will be a (:math:`\cdots\times n`) tensor.</span>
+<span class="sd">        Different from ``torch.mv``, which is not broadcast, this function</span>
+<span class="sd">        is broadcast and supports batched product.</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        &gt;&gt;&gt; matrix = torch.randn(2, 1, 3, 2)</span>
+<span class="sd">        &gt;&gt;&gt; vec = torch.randn(1, 2, 2)</span>
+<span class="sd">        &gt;&gt;&gt; out = pp.bmv(matrix, vec)</span>
+<span class="sd">        &gt;&gt;&gt; out.shape</span>
+<span class="sd">        torch.Size([2, 2, 3])</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">assert</span> <span class="n">mat</span><span class="o">.</span><span class="n">ndim</span> <span class="o">&gt;=</span> <span class="mi">2</span> <span class="ow">and</span> <span class="n">vec</span><span class="o">.</span><span class="n">ndim</span> <span class="o">&gt;=</span> <span class="mi">1</span><span class="p">,</span> <span class="s1">&#39;Input arguments invalid&#39;</span>
+    <span class="k">assert</span> <span class="n">mat</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="n">vec</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;matrix-vector shape invalid&#39;</span>
+    <span class="n">mat</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mat</span><span class="p">,</span> <span class="n">LieTensor</span><span class="p">)</span> <span class="k">else</span> <span class="n">mat</span>
+    <span class="n">vec</span> <span class="o">=</span> <span class="n">vec</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">vec</span><span class="p">,</span> <span class="n">LieTensor</span><span class="p">)</span> <span class="k">else</span> <span class="n">vec</span>
+    <span class="k">return</span> <span class="n">torch</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">mat</span><span class="p">,</span> <span class="n">vec</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">),</span> <span class="n">out</span><span class="o">=</span><span class="n">out</span><span class="p">)</span><span class="o">.</span><span class="n">squeeze_</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="bvmv"><a class="viewcode-back" href="../../../../generated/pypose.bvmv/#pypose.bvmv">[docs]</a><span class="k">def</span> <span class="nf">bvmv</span><span class="p">(</span><span class="n">lvec</span><span class="p">,</span> <span class="n">mat</span><span class="p">,</span> <span class="n">rvec</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Performs batched vector-matrix-vector product.</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \text{out}_i = \mathbf{v}_i^{l\text{T}} \times \mathbf{M}_i \times \mathbf{v}_i^r</span>
+
+<span class="sd">    where :math:`\text{out}_i` is a scalar and :math:`\mathbf{v}_i^l, \mathbf{M}_i,</span>
+<span class="sd">    \mathbf{v}_i^r` are the i-th batched tensors with shape (n), (n, m), and (m),</span>
+<span class="sd">    respectively.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        lvec (:obj:`Tensor`): left vectors to be multiplied.</span>
+<span class="sd">        mat (:obj:`Tensor`): matrices to be multiplied.</span>
+<span class="sd">        rvec (:obj:`Tensor`): right vectors to be multiplied.</span>
+
+<span class="sd">    Return:</span>
+<span class="sd">        :obj:`Tensor`: the output tensor.</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        the ``lvec`` has to be a (:math:`\cdots\times n`) tensor,</span>
+<span class="sd">        The ``mat`` has to be a (:math:`\cdots\times n \times m`) tensor,</span>
+<span class="sd">        the ``rvec`` has to be a (:math:`\cdots\times m`) tensor,</span>
+<span class="sd">        and ``out`` will be a (:math:`\cdots`) or at least a 1D tensor.</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        &gt;&gt;&gt; v1 = torch.randn(4)</span>
+<span class="sd">        &gt;&gt;&gt; mat = torch.randn(4, 5)</span>
+<span class="sd">        &gt;&gt;&gt; v2 = torch.randn(5)</span>
+<span class="sd">        &gt;&gt;&gt; out = pp.bvmv(v1, mat, v2)</span>
+<span class="sd">        &gt;&gt;&gt; out.shape</span>
+<span class="sd">        torch.Size([1])</span>
+
+<span class="sd">        &gt;&gt;&gt; v1 = torch.randn(1, 2, 4)</span>
+<span class="sd">        &gt;&gt;&gt; mat = torch.randn(2, 2, 4, 5)</span>
+<span class="sd">        &gt;&gt;&gt; v2 = torch.randn(2, 1, 5)</span>
+<span class="sd">        &gt;&gt;&gt; out = pp.bvmv(v1, mat, v2)</span>
+<span class="sd">        &gt;&gt;&gt; out.shape</span>
+<span class="sd">        torch.Size([2, 2])</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">assert</span> <span class="n">mat</span><span class="o">.</span><span class="n">ndim</span> <span class="o">&gt;=</span> <span class="mi">2</span> <span class="ow">and</span> <span class="n">lvec</span><span class="o">.</span><span class="n">ndim</span> <span class="o">&gt;=</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">rvec</span><span class="o">.</span><span class="n">ndim</span> <span class="o">&gt;=</span> <span class="mi">1</span><span class="p">,</span> <span class="s1">&#39;Shape invalid&#39;</span>
+    <span class="k">assert</span> <span class="n">lvec</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="n">mat</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span> <span class="ow">and</span> <span class="n">mat</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="n">rvec</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+    <span class="n">lvec</span> <span class="o">=</span> <span class="n">lvec</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">lvec</span><span class="p">,</span> <span class="n">LieTensor</span><span class="p">)</span> <span class="k">else</span> <span class="n">lvec</span>
+    <span class="n">mat</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mat</span><span class="p">,</span> <span class="n">LieTensor</span><span class="p">)</span> <span class="k">else</span> <span class="n">mat</span>
+    <span class="n">rvec</span> <span class="o">=</span> <span class="n">rvec</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">rvec</span><span class="p">,</span> <span class="n">LieTensor</span><span class="p">)</span> <span class="k">else</span> <span class="n">rvec</span>
+    <span class="n">lvec</span><span class="p">,</span> <span class="n">rvec</span> <span class="o">=</span> <span class="n">lvec</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">),</span> <span class="n">rvec</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
+    <span class="k">return</span> <span class="n">torch</span><span class="o">.</span><span class="n">atleast_1d</span><span class="p">((</span><span class="n">lvec</span><span class="o">.</span><span class="n">mT</span> <span class="o">@</span> <span class="n">mat</span> <span class="o">@</span> <span class="n">rvec</span><span class="p">)</span><span class="o">.</span><span class="n">squeeze_</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">squeeze_</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">))</span></div>
+</pre></div>
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+              
+  <h1>Source code for pypose.function.spline</h1><div class="highlight"><pre>
+<span></span><span class="kn">import</span> <span class="nn">torch</span>
+<span class="kn">from</span> <span class="nn">.</span> <span class="kn">import</span> <span class="n">is_SE3</span>
+
+
+<div class="viewcode-block" id="chspline"><a class="viewcode-back" href="../../../../generated/pypose.chspline/#pypose.chspline">[docs]</a><span class="k">def</span> <span class="nf">chspline</span><span class="p">(</span><span class="n">points</span><span class="p">,</span> <span class="n">interval</span><span class="o">=</span><span class="mf">0.1</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">    Cubic Hermite Spline, a piecewise-cubic interpolator matching values and first</span>
+<span class="sd">    derivatives.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        points (:obj:`Tensor`): the sequence of points for interpolation with shape</span>
+<span class="sd">            [..., point_num, dim].</span>
+<span class="sd">        interval (:obj:`float`): the unit interval between interpolated points.</span>
+<span class="sd">            We assume the interval of adjacent input points is 1. Therefore, if we set</span>
+<span class="sd">            ``interval`` as ``0.1``, the interpolated points between the points at</span>
+<span class="sd">            :math:`t` and :math:`t+1` will be :math:`[t, t+0.1, ..., t+0.9, t+1]`.</span>
+<span class="sd">            Default: ``0.1``.</span>
+
+<span class="sd">    Returns:</span>
+<span class="sd">       ``Tensor``: the interpolated points.</span>
+
+<span class="sd">    The interpolated points are evenly distributed between a start point and a end point</span>
+<span class="sd">    according to the number of interpolated points. In this function, the interpolated points</span>
+<span class="sd">    are also evenly distributed between the start point and end point.</span>
+
+<span class="sd">    Denote the starting point :math:`p_0` with the starting tangent :math:`m_0` and the</span>
+<span class="sd">    ending point :math:`p_1` with the ending tagent :math:`m_1`. The interpolated point at</span>
+<span class="sd">    :math:`t` can be defined as:</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \begin{aligned}</span>
+<span class="sd">            p(t) &amp;= (1-3t^2+2t^3)p_0 + (t-2t^2+t^3)m_0+(3t^2-2t^3)p_1+(-t^2+t^3)m_1\\</span>
+<span class="sd">                 &amp;= \begin{bmatrix}</span>
+<span class="sd">                        p_0, m_0, p_1, m_1</span>
+<span class="sd">                    \end{bmatrix}</span>
+<span class="sd">                    \begin{bmatrix}</span>
+<span class="sd">                        1&amp; 0&amp;-3&amp; 2\\</span>
+<span class="sd">                        0&amp; 1&amp;-2&amp; 1\\</span>
+<span class="sd">                        0&amp; 0&amp; 3&amp;-2\\</span>
+<span class="sd">                        0&amp; 0&amp;-1&amp; 1</span>
+<span class="sd">                    \end{bmatrix}</span>
+<span class="sd">                    \begin{bmatrix}1\\ t\\ t^2\\ t^3\end{bmatrix}</span>
+
+<span class="sd">        \end{aligned}</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        The implementation is based on wiki of `Cubic Hermite spline</span>
+<span class="sd">        &lt;https://en.wikipedia.org/wiki/Cubic_Hermite_spline&gt;`_.</span>
+
+<span class="sd">    Examples:</span>
+<span class="sd">        &gt;&gt;&gt; import torch</span>
+<span class="sd">        &gt;&gt;&gt; import pypose as pp</span>
+<span class="sd">        &gt;&gt;&gt; points = torch.tensor([[[0., 0., 0.],</span>
+<span class="sd">        ...                         [1., .5, 0.1],</span>
+<span class="sd">        ...                         [0., 1., 0.2],</span>
+<span class="sd">        ...                         [1., 1.5, 0.4],</span>
+<span class="sd">        ...                         [1.5, 0., 0.],</span>
+<span class="sd">        ...                         [2., 1.5, 0.4],</span>
+<span class="sd">        ...                         [2.5, 0., 0.],</span>
+<span class="sd">        ...                         [1.75, 0.75, 0.2],</span>
+<span class="sd">        ...                         [2.25, 0.75, 0.2],</span>
+<span class="sd">        ...                         [3., 1.5, 0.4],</span>
+<span class="sd">        ...                         [3., 0., 0.],</span>
+<span class="sd">        ...                         [4., 0., 0.],</span>
+<span class="sd">        ...                         [4., 1.5, 0.4],</span>
+<span class="sd">        ...                         [5., 1., 0.2],</span>
+<span class="sd">        ...                         [4., 0.75, 0.2],</span>
+<span class="sd">        ...                         [5., 0., 0.]]])</span>
+<span class="sd">        &gt;&gt;&gt; waypoints = pp.chspline(points, 0.1)</span>
+
+<span class="sd">    .. figure:: /_static/img/module/liespline/CsplineR3.png</span>
+<span class="sd">        :width: 600</span>
+
+<span class="sd">        Fig. 1. Result of Cubic Spline Interpolation in 3D space.</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+    <span class="k">assert</span> <span class="n">points</span><span class="o">.</span><span class="n">dim</span><span class="p">()</span> <span class="o">&gt;=</span> <span class="mi">2</span><span class="p">,</span> <span class="s2">&quot;Dimension of points should be [..., N, C]&quot;</span>
+    <span class="k">assert</span> <span class="n">interval</span> <span class="o">&lt;</span> <span class="mf">1.0</span><span class="p">,</span> <span class="s2">&quot;The interval should be smaller than 1.&quot;</span>
+    <span class="n">batch</span><span class="p">,</span> <span class="n">N</span> <span class="o">=</span> <span class="n">points</span><span class="o">.</span><span class="n">shape</span><span class="p">[:</span><span class="o">-</span><span class="mi">2</span><span class="p">],</span> <span class="n">points</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span>
+    <span class="n">dargs</span> <span class="o">=</span> <span class="p">{</span><span class="s1">&#39;device&#39;</span><span class="p">:</span> <span class="n">points</span><span class="o">.</span><span class="n">device</span><span class="p">,</span> <span class="s1">&#39;dtype&#39;</span><span class="p">:</span> <span class="n">points</span><span class="o">.</span><span class="n">dtype</span><span class="p">}</span>
+    <span class="n">intervals</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">interval</span><span class="p">,</span> <span class="o">**</span><span class="n">dargs</span><span class="p">)</span>
+    <span class="n">timeline</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="o">**</span><span class="n">dargs</span><span class="p">)</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
+    <span class="n">timeline</span> <span class="o">=</span> <span class="p">(</span><span class="n">timeline</span> <span class="o">+</span> <span class="n">intervals</span><span class="p">)</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)[:</span><span class="o">-</span><span class="p">(</span><span class="n">intervals</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">-</span><span class="mi">1</span><span class="p">)]</span>
+    <span class="n">x</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="o">**</span><span class="n">dargs</span><span class="p">)</span>
+    <span class="n">idxs</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">searchsorted</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="o">...</span><span class="p">,</span><span class="mi">1</span><span class="p">:],</span> <span class="n">timeline</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="p">:])</span>
+    <span class="n">x</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">batch</span><span class="o">+</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,))</span>
+    <span class="n">xs</span> <span class="o">=</span> <span class="n">timeline</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">batch</span><span class="o">+</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,))</span>
+    <span class="n">m</span> <span class="o">=</span> <span class="p">(</span><span class="n">points</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">1</span><span class="p">:,</span> <span class="p">:]</span> <span class="o">-</span> <span class="n">points</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="p">:</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="p">:])</span>
+    <span class="n">m</span> <span class="o">/=</span> <span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">1</span><span class="p">:]</span> <span class="o">-</span> <span class="n">x</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="p">:</span><span class="o">-</span><span class="mi">1</span><span class="p">])[</span><span class="o">...</span><span class="p">,</span> <span class="kc">None</span><span class="p">]</span>
+    <span class="n">m</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">cat</span><span class="p">([</span><span class="n">m</span><span class="p">[</span><span class="o">...</span><span class="p">,[</span><span class="mi">0</span><span class="p">],:],</span> <span class="p">(</span><span class="n">m</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">1</span><span class="p">:,:]</span> <span class="o">+</span> <span class="n">m</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="p">:</span><span class="o">-</span><span class="mi">1</span><span class="p">,:])</span> <span class="o">/</span> <span class="mi">2</span><span class="p">,</span> <span class="n">m</span><span class="p">[</span><span class="o">...</span><span class="p">,[</span><span class="o">-</span><span class="mi">1</span><span class="p">],:]],</span> <span class="o">-</span><span class="mi">2</span><span class="p">)</span>
+    <span class="n">dx</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="n">idxs</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">x</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="n">idxs</span><span class="p">]</span>
+    <span class="n">t</span> <span class="o">=</span> <span class="p">(</span><span class="n">xs</span> <span class="o">-</span> <span class="n">x</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="n">idxs</span><span class="p">])</span> <span class="o">/</span> <span class="n">dx</span>
+    <span class="n">alpha</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="o">**</span><span class="n">dargs</span><span class="p">)</span>
+    <span class="n">tt</span> <span class="o">=</span> <span class="n">t</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</span> <span class="p">:]</span><span class="o">**</span><span class="n">alpha</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="kc">None</span><span class="p">]</span>
+    <span class="n">A</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span>
+                      <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span>
+                      <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">],</span>
+                      <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]],</span> <span class="o">**</span><span class="n">dargs</span><span class="p">)</span>
+    <span class="n">hh</span> <span class="o">=</span> <span class="p">(</span><span class="n">A</span> <span class="o">@</span> <span class="n">tt</span><span class="p">)</span><span class="o">.</span><span class="n">mT</span>
+    <span class="n">interpoints</span> <span class="o">=</span> <span class="n">hh</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="p">:</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">points</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="n">idxs</span><span class="p">,</span> <span class="p">:]</span>
+    <span class="n">interpoints</span> <span class="o">+=</span> <span class="n">hh</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">1</span><span class="p">:</span><span class="mi">2</span><span class="p">]</span> <span class="o">*</span> <span class="n">m</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="n">idxs</span><span class="p">,</span> <span class="p">:]</span> <span class="o">*</span> <span class="n">dx</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="kc">None</span><span class="p">]</span>
+    <span class="n">interpoints</span> <span class="o">+=</span> <span class="n">hh</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">2</span><span class="p">:</span><span class="mi">3</span><span class="p">]</span> <span class="o">*</span> <span class="n">points</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="n">idxs</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="p">:]</span>
+    <span class="n">interpoints</span> <span class="o">+=</span> <span class="n">hh</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">3</span><span class="p">:</span><span class="mi">4</span><span class="p">]</span> <span class="o">*</span> <span class="n">m</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="n">idxs</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="p">:]</span> <span class="o">*</span> <span class="n">dx</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="kc">None</span><span class="p">]</span>
+    <span class="k">return</span> <span class="n">interpoints</span></div>
+
+<div class="viewcode-block" id="bspline"><a class="viewcode-back" href="../../../../generated/pypose.bspline/#pypose.bspline">[docs]</a><span class="k">def</span> <span class="nf">bspline</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">interval</span><span class="o">=</span><span class="mf">0.1</span><span class="p">,</span> <span class="n">extrapolate</span><span class="o">=</span><span class="kc">False</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    B-spline interpolation, which currently only supports SE3 LieTensor.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        data (:obj:`LieTensor`): the input sparse poses with</span>
+<span class="sd">            [batch_size, poses_num, dim] shape.</span>
+<span class="sd">        interval (:obj:`float`): the unit interval between interpolated poses.</span>
+<span class="sd">            We assume the interval of adjacent input poses is 1. Therefore, if we set</span>
+<span class="sd">            ``interval`` as ``0.1``, the interpolated poses between the poses at</span>
+<span class="sd">            :math:`t` and :math:`t+1` will be at :math:`[t, t+0.1, ..., t+0.9, t+1]`.</span>
+<span class="sd">            Default: ``0.1``.</span>
+<span class="sd">        extrapolate(``bool``): flag to determine whether the interpolate poses pass the</span>
+<span class="sd">            start and end poses. Default: ``False``.</span>
+
+<span class="sd">    Returns:</span>
+<span class="sd">        :obj:`LieTensor`: the interpolated SE3 LieTensor.</span>
+
+<span class="sd">    A poses query at :math:`t \in [t_i, t_{i+1})` (i.e. a segment of the spline)</span>
+<span class="sd">    only relies on the poses at four steps :math:`\{t_{i-1},t_i,t_{i+1},t_{i+2}\}`.</span>
+<span class="sd">    It means that the interpolation between adjacent poses needs four consecutive poses.</span>
+<span class="sd">    In this function, the interpolated poses are evenly distributed between a</span>
+<span class="sd">    pose query according to the interval parameter.</span>
+
+<span class="sd">    The absolute pose of the spline :math:`T_{s}^w(t)`, where :math:`w` denotes the world</span>
+<span class="sd">    and :math:`s` is the spline coordinate frame, can be calculated:</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \begin{aligned}</span>
+<span class="sd">            &amp;T^w_s(t) = T^w_{i-1} \prod^{i+2}_{j=i}\delta T_j,\\</span>
+<span class="sd">            &amp;\delta T_j= \exp(\lambda_{j-i}(t)\delta \hat{\xi}_j),\\</span>
+<span class="sd">            &amp;\lambda(t) = MU(t),</span>
+<span class="sd">        \end{aligned}</span>
+
+<span class="sd">    :math:`M` is a predefined matrix, shown as follows:</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        M = \frac{1}{6}\begin{bmatrix}</span>
+<span class="sd">            5&amp; 3&amp; -3&amp; 1 \\</span>
+<span class="sd">            1&amp; 3&amp; 3&amp; -2\\</span>
+<span class="sd">            0&amp; 0&amp; 0&amp; 1 \\</span>
+<span class="sd">        \end{bmatrix}</span>
+
+<span class="sd">    :math:`U(t)` is a vector:</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        U(t) = \begin{bmatrix}</span>
+<span class="sd">                1\\</span>
+<span class="sd">                u(t)\\</span>
+<span class="sd">                u(t)^2\\</span>
+<span class="sd">                u(t)^3\\</span>
+<span class="sd">            \end{bmatrix}</span>
+
+<span class="sd">    where :math:`u(t)=(t-t_i)/\Delta t`, :math:`\Delta t = t_{i+1} - t_{i}`. :math:`t_i`</span>
+<span class="sd">    is the time point of the :math:`i_{th}` given pose. :math:`t` is the interpolation</span>
+<span class="sd">    time point :math:`t \in [t_{i},t_{i+1})`.</span>
+
+<span class="sd">    :math:`\delta \hat{\xi}_j` is the transformation between :math:`\hat{\xi}_{j-1}`</span>
+<span class="sd">    and :math:`\hat{\xi}_{j}`</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \begin{aligned}</span>
+<span class="sd">            \delta \hat{\xi}_j :&amp;= \log(\exp(\hat{\xi} _w^{j-1})\exp(\hat{\xi}_j^w))</span>
+<span class="sd">                                &amp;= \log(T_j^{j-1})\in \mathfrak{se3}</span>
+<span class="sd">        \end{aligned}</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        The implementation is based on Eq. (3), (4), (5), and (6) of this paper:</span>
+<span class="sd">        David Hug, et al.,</span>
+<span class="sd">        `HyperSLAM: A Generic and Modular Approach to Sensor Fusion and Simultaneous</span>
+<span class="sd">        Localization And Mapping in Continuous-Time</span>
+<span class="sd">        &lt;https://ieeexplore.ieee.org/abstract/document/9320417&gt;`_,</span>
+<span class="sd">        2020 International Conference on 3D Vision (3DV), Fukuoka, Japan, 2020.</span>
+
+<span class="sd">    Examples:</span>
+<span class="sd">        &gt;&gt;&gt; import torch, pypose as pp</span>
+<span class="sd">        &gt;&gt;&gt; a1 = pp.euler2SO3(torch.Tensor([0., 0., 0.]))</span>
+<span class="sd">        &gt;&gt;&gt; a2 = pp.euler2SO3(torch.Tensor([torch.pi / 4., torch.pi / 3., torch.pi / 2.]))</span>
+<span class="sd">        &gt;&gt;&gt; poses = pp.SE3([[[0., 4., 0., a1[0], a1[1], a1[2], a1[3]],</span>
+<span class="sd">        ...                  [0., 3., 0., a1[0], a1[1], a1[2], a1[3]],</span>
+<span class="sd">        ...                  [0., 2., 0., a1[0], a1[1], a1[2], a1[3]],</span>
+<span class="sd">        ...                  [0., 1., 0., a1[0], a1[1], a1[2], a1[3]],</span>
+<span class="sd">        ...                  [1., 0., 1., a2[0], a2[1], a2[2], a2[3]],</span>
+<span class="sd">        ...                  [2., 0., 1., a2[0], a2[1], a2[2], a2[3]],</span>
+<span class="sd">        ...                  [3., 0., 1., a2[0], a2[1], a2[2], a2[3]],</span>
+<span class="sd">        ...                  [4., 0., 1., a2[0], a2[1], a2[2], a2[3]]]])</span>
+<span class="sd">        &gt;&gt;&gt; wayposes = pp.bspline(poses, 0.1)</span>
+
+<span class="sd">    .. figure:: /_static/img/module/liespline/BsplineSE3.png</span>
+
+<span class="sd">        Fig. 1. Result of B Spline Interpolation in SE3.</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">assert</span> <span class="n">is_SE3</span><span class="p">(</span><span class="n">data</span><span class="p">),</span> <span class="s2">&quot;The input poses are not SE3Type.&quot;</span>
+    <span class="k">assert</span> <span class="n">data</span><span class="o">.</span><span class="n">dim</span><span class="p">()</span> <span class="o">&gt;=</span> <span class="mi">2</span><span class="p">,</span> <span class="s2">&quot;Dimension of data should be [..., N, C].&quot;</span>
+    <span class="k">assert</span> <span class="n">interval</span> <span class="o">&lt;</span> <span class="mf">1.0</span><span class="p">,</span> <span class="s2">&quot;The interval should be smaller than 1.&quot;</span>
+    <span class="n">batch</span> <span class="o">=</span> <span class="n">data</span><span class="o">.</span><span class="n">shape</span><span class="p">[:</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span>
+    <span class="k">if</span> <span class="n">extrapolate</span><span class="p">:</span>
+        <span class="n">data</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">cat</span><span class="p">((</span><span class="n">data</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="p">:</span><span class="mi">1</span><span class="p">,</span> <span class="p">:]</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">batch</span><span class="o">+</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">)),</span>
+                          <span class="n">data</span><span class="p">,</span>
+                          <span class="n">data</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">:,</span> <span class="p">:]</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">batch</span><span class="o">+</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">))),</span> <span class="n">dim</span><span class="o">=-</span><span class="mi">2</span><span class="p">)</span>
+    <span class="k">else</span><span class="p">:</span>
+        <span class="k">assert</span> <span class="n">data</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span><span class="o">&gt;=</span><span class="mi">4</span><span class="p">,</span> <span class="s2">&quot;Number of poses is less than 4.&quot;</span>
+    <span class="n">Bth</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="n">D</span> <span class="o">=</span> <span class="n">data</span><span class="o">.</span><span class="n">shape</span><span class="p">[:</span><span class="o">-</span><span class="mi">2</span><span class="p">],</span> <span class="n">data</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">],</span> <span class="n">data</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+    <span class="n">dargs</span> <span class="o">=</span> <span class="p">{</span><span class="s1">&#39;dtype&#39;</span><span class="p">:</span> <span class="n">data</span><span class="o">.</span><span class="n">dtype</span><span class="p">,</span> <span class="s1">&#39;device&#39;</span><span class="p">:</span> <span class="n">data</span><span class="o">.</span><span class="n">device</span><span class="p">}</span>
+    <span class="n">timeline</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">interval</span><span class="p">,</span> <span class="o">**</span><span class="n">dargs</span><span class="p">)</span>
+    <span class="n">tt</span> <span class="o">=</span> <span class="n">timeline</span> <span class="o">**</span> <span class="n">torch</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="o">**</span><span class="n">dargs</span><span class="p">)</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
+    <span class="n">B</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([[</span><span class="mi">5</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span>
+                      <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span><span class="o">-</span><span class="mi">2</span><span class="p">],</span>
+                      <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]],</span> <span class="o">**</span><span class="n">dargs</span><span class="p">)</span> <span class="o">/</span> <span class="mi">6</span>
+    <span class="n">dP</span> <span class="o">=</span> <span class="n">data</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">0</span><span class="p">:</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="p">:]</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span>
+    <span class="n">w</span> <span class="o">=</span> <span class="p">(</span><span class="n">B</span> <span class="o">@</span> <span class="n">tt</span><span class="p">)</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
+    <span class="n">index</span> <span class="o">=</span> <span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">N</span><span class="o">-</span><span class="mi">3</span><span class="p">)</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="n">torch</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
+    <span class="n">P</span> <span class="o">=</span> <span class="n">data</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="n">index</span><span class="p">,</span> <span class="p">:]</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="n">Bth</span> <span class="o">+</span> <span class="p">(</span><span class="n">N</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="n">D</span><span class="p">))</span>
+    <span class="n">P</span> <span class="o">=</span> <span class="p">(</span><span class="n">P</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="p">:</span><span class="mi">3</span><span class="p">,</span> <span class="p">:,</span> <span class="p">:]</span><span class="o">.</span><span class="n">Inv</span><span class="p">()</span> <span class="o">*</span> <span class="n">P</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">1</span><span class="p">:,</span> <span class="p">:,</span> <span class="p">:])</span><span class="o">.</span><span class="n">Log</span><span class="p">()</span>
+    <span class="n">A</span> <span class="o">=</span> <span class="p">(</span><span class="n">P</span> <span class="o">*</span> <span class="n">w</span><span class="p">)</span><span class="o">.</span><span class="n">Exp</span><span class="p">()</span>
+    <span class="n">Aend</span> <span class="o">=</span> <span class="p">(</span><span class="n">P</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="p">:]</span> <span class="o">*</span> <span class="p">((</span><span class="n">B</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">dim</span><span class="o">=</span><span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)))</span><span class="o">.</span><span class="n">Exp</span><span class="p">()</span>
+    <span class="n">Aend</span> <span class="o">=</span> <span class="n">Aend</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="p">:]</span> <span class="o">*</span> <span class="n">Aend</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="p">:]</span> <span class="o">*</span> <span class="n">Aend</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="p">:]</span>
+    <span class="n">A</span> <span class="o">=</span> <span class="n">A</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="p">:,</span> <span class="p">:]</span> <span class="o">*</span> <span class="n">A</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="p">:,</span> <span class="p">:]</span> <span class="o">*</span> <span class="n">A</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="p">:,</span> <span class="p">:]</span>
+    <span class="n">ps</span><span class="p">,</span> <span class="n">pend</span> <span class="o">=</span> <span class="n">dP</span> <span class="o">*</span> <span class="n">A</span><span class="p">,</span><span class="n">dP</span><span class="p">[</span><span class="o">...</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">,:,:]</span><span class="o">*</span><span class="n">Aend</span><span class="p">[</span><span class="o">...</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">,:,:]</span>
+    <span class="n">poses</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">cat</span><span class="p">((</span><span class="n">ps</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="n">Bth</span> <span class="o">+</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">D</span><span class="p">)),</span> <span class="n">pend</span><span class="p">),</span> <span class="n">dim</span><span class="o">=-</span><span class="mi">2</span><span class="p">)</span>
+    <span class="k">return</span> <span class="n">poses</span></div>
+</pre></div>
+
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diff --git a/docs/preview/pr/352/_modules/pypose/lietensor/basics/index.html b/docs/preview/pr/352/_modules/pypose/lietensor/basics/index.html
new file mode 100644
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--- /dev/null
+++ b/docs/preview/pr/352/_modules/pypose/lietensor/basics/index.html
@@ -0,0 +1,778 @@
+
+
+
+<!DOCTYPE html>
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+              <p class="caption" role="heading"><span class="caption-text">Contents:</span></p>
+<ul>
+<li class="toctree-l1"><a class="reference internal" href="../../../../lietensor/">LieTensor</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../../../func/">Func</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../../../functions/">Functions</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../../../utils/">Utils</a></li>
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+              
+  <h1>Source code for pypose.lietensor.basics</h1><div class="highlight"><pre>
+<span></span><span class="kn">import</span> <span class="nn">math</span>
+<span class="kn">import</span> <span class="nn">torch</span>
+<span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">This basics file includes functions needed to implement LieTensor.</span>
+<span class="sd">&#39;&#39;&#39;</span>
+
+<div class="viewcode-block" id="vec2skew"><a class="viewcode-back" href="../../../../generated/pypose.vec2skew/#pypose.vec2skew">[docs]</a><span class="k">def</span> <span class="nf">vec2skew</span><span class="p">(</span><span class="nb">input</span><span class="p">:</span><span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">:</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">    Convert batched vectors to skew matrices.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        input (Tensor): the tensor :math:`\mathbf{x}` to convert.</span>
+
+<span class="sd">    Return:</span>
+<span class="sd">        Tensor: the skew matrices :math:`\mathbf{y}`.</span>
+
+<span class="sd">    Shape:</span>
+<span class="sd">        Input: :obj:`(*, 3)`</span>
+
+<span class="sd">        Output: :obj:`(*, 3, 3)`</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        {\displaystyle \mathbf{y}_i={\begin{bmatrix}\,\,</span>
+<span class="sd">        0&amp;\!-x_{i,3}&amp;\,\,\,x_{i,2}\\\,\,\,x_{i,3}&amp;0&amp;\!-x_{i,1}</span>
+<span class="sd">        \\\!-x_{i,2}&amp;\,\,x_{i,1}&amp;\,\,0\end{bmatrix}}}</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        The last dimension of the input tensor has to be 3.</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        &gt;&gt;&gt; pp.vec2skew(torch.randn(1,3))</span>
+<span class="sd">        tensor([[[ 0.0000, -2.2059, -1.2761],</span>
+<span class="sd">                [ 2.2059,  0.0000,  0.2929],</span>
+<span class="sd">                [ 1.2761, -0.2929,  0.0000]]])</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+    <span class="n">v</span> <span class="o">=</span> <span class="nb">input</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="nb">input</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="nb">input</span>
+    <span class="k">assert</span> <span class="n">v</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="mi">3</span><span class="p">,</span> <span class="s2">&quot;Last dim should be 3&quot;</span>
+    <span class="n">O</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">v</span><span class="o">.</span><span class="n">shape</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">device</span><span class="o">=</span><span class="n">v</span><span class="o">.</span><span class="n">device</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">v</span><span class="o">.</span><span class="n">dtype</span><span class="p">,</span> <span class="n">requires_grad</span><span class="o">=</span><span class="n">v</span><span class="o">.</span><span class="n">requires_grad</span><span class="p">)</span>
+    <span class="k">return</span> <span class="n">torch</span><span class="o">.</span><span class="n">stack</span><span class="p">([</span><span class="n">torch</span><span class="o">.</span><span class="n">stack</span><span class="p">([</span>        <span class="n">O</span><span class="p">,</span> <span class="o">-</span><span class="n">v</span><span class="p">[</span><span class="o">...</span><span class="p">,</span><span class="mi">2</span><span class="p">],</span>  <span class="n">v</span><span class="p">[</span><span class="o">...</span><span class="p">,</span><span class="mi">1</span><span class="p">]],</span> <span class="n">dim</span><span class="o">=-</span><span class="mi">1</span><span class="p">),</span>
+                        <span class="n">torch</span><span class="o">.</span><span class="n">stack</span><span class="p">([</span> <span class="n">v</span><span class="p">[</span><span class="o">...</span><span class="p">,</span><span class="mi">2</span><span class="p">],</span>         <span class="n">O</span><span class="p">,</span> <span class="o">-</span><span class="n">v</span><span class="p">[</span><span class="o">...</span><span class="p">,</span><span class="mi">0</span><span class="p">]],</span> <span class="n">dim</span><span class="o">=-</span><span class="mi">1</span><span class="p">),</span>
+                        <span class="n">torch</span><span class="o">.</span><span class="n">stack</span><span class="p">([</span><span class="o">-</span><span class="n">v</span><span class="p">[</span><span class="o">...</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span>  <span class="n">v</span><span class="p">[</span><span class="o">...</span><span class="p">,</span><span class="mi">0</span><span class="p">],</span>         <span class="n">O</span><span class="p">],</span> <span class="n">dim</span><span class="o">=-</span><span class="mi">1</span><span class="p">)],</span> <span class="n">dim</span><span class="o">=-</span><span class="mi">2</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="add_"><a class="viewcode-back" href="../../../../generated/pypose.add_/#pypose.add_">[docs]</a><span class="k">def</span> <span class="nf">add_</span><span class="p">(</span><span class="nb">input</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Inplace version of :meth:`pypose.add`.</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">return</span> <span class="nb">input</span><span class="o">.</span><span class="n">add_</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">alpha</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="add"><a class="viewcode-back" href="../../../../generated/pypose.add/#pypose.add">[docs]</a><span class="k">def</span> <span class="nf">add</span><span class="p">(</span><span class="nb">input</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Adds other, scaled by alpha, to input LieTensor.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        input (:obj:`LieTensor`): the input LieTensor (Lie Algebra or Lie Group).</span>
+
+<span class="sd">        other (:obj:`Tensor`): the tensor to add to input. The last dimension has to be no less</span>
+<span class="sd">            than the shape of the corresponding Lie Algebra of the input.</span>
+
+<span class="sd">        alpha (:obj:`Number`): the multiplier for other.</span>
+
+<span class="sd">    Return:</span>
+<span class="sd">        :obj:`LieTensor`: the output LieTensor.</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \bm{y}_i =</span>
+<span class="sd">        \begin{cases}</span>
+<span class="sd">        \alpha * \bm{a}_i + \bm{x}_i &amp; \text{if}~\bm{x}_i~\text{is a Lie Algebra} \\</span>
+<span class="sd">        \mathrm{Exp}(\alpha * \bm{a}_i) \times \bm{x}_i &amp; \text{if}~\bm{x}_i~\text{is a Lie Group}</span>
+<span class="sd">        \end{cases}</span>
+
+<span class="sd">    where :math:`\bm{x}` is the ``input`` LieTensor, :math:`\bm{a}` is the ``other`` Tensor to add,</span>
+<span class="sd">    and :math:`\bm{y}` is the output LieTensor.</span>
+
+<span class="sd">    Warning:</span>
+<span class="sd">        For better readability, we suggest calling ``a + b`` instead of ``a.add(b)`` or</span>
+<span class="sd">        ``pypose.add(a, b)``, which have the same behaviors.</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        A Lie Group normally requires a larger space than its corresponding Lie Algebra, thus</span>
+<span class="sd">        the elements in the last dimension of the ``other`` Tensor (treated as a Lie Algebra</span>
+<span class="sd">        in this function) beyond the expected shape of the Lie Algebra are ignored. This is</span>
+<span class="sd">        because the gradient of a Lie Group is computed as a left perturbation (a Lie Algebra)</span>
+<span class="sd">        in its tangent space and is stored in the LieGroup&#39;s :obj:`LieTensor.grad`, which has</span>
+<span class="sd">        the same storage space with the LieGroup.</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \begin{align*}</span>
+<span class="sd">                \frac{D f(\mathcal{X})}{D \mathcal{X}} &amp; \overset{\underset{\mathrm{def}}{}}{=}</span>
+<span class="sd">                \displaystyle \lim_{\bm{\tau} \to \bm{0}} \frac{f(\bm{\tau} \oplus \mathcal{X})</span>
+<span class="sd">                    \ominus f(\mathcal{X})}{\bm{\tau}} \\</span>
+<span class="sd">                &amp; = \left.\frac{\partial\mathrm{Log}(f(\mathrm{Exp}(\bm{\tau})\times\mathcal{X}))</span>
+<span class="sd">                    \times f(\mathcal{X})^{-1})}{\partial \bm{\tau}}\right|_{\bm{\tau=\bm{0}}}</span>
+<span class="sd">            \end{align*},</span>
+
+<span class="sd">        where :math:`\mathcal{X}` is a Lie Group and :math:`\bm{\tau}` is its left perturbation.</span>
+
+<span class="sd">        See Eq.(44) in `Micro Lie theory &lt;https://arxiv.org/abs/1812.01537&gt;`_ for more details of</span>
+<span class="sd">        the gradient for a Lie Group.</span>
+
+<span class="sd">        This provides convenience to work with PyTorch optimizers like :obj:`torch.optim.SGD`,</span>
+<span class="sd">        which calls function :meth:`.add_` of a Lie Group to adjust parameters by gradients</span>
+<span class="sd">        (:obj:`LieTensor.grad`, where the last element is often zero since tangent vector requires</span>
+<span class="sd">        smaller storage space).</span>
+
+<span class="sd">    See :meth:`LieTensor` for types of Lie Algebra and Lie Group.</span>
+
+<span class="sd">    See :meth:`Exp` for Exponential mapping of Lie Algebra.</span>
+
+<span class="sd">    Examples:</span>
+<span class="sd">        The following operations are equivalent.</span>
+
+<span class="sd">        &gt;&gt;&gt; x = pp.randn_SE3()</span>
+<span class="sd">        &gt;&gt;&gt; a = torch.randn(6)</span>
+<span class="sd">        &gt;&gt;&gt; x + a</span>
+<span class="sd">        SE3Type LieTensor:</span>
+<span class="sd">        tensor([-1.6089,  0.4184,  0.6621, -0.2098,  0.5383,  0.4794,  0.6606])</span>
+<span class="sd">        &gt;&gt;&gt; pp.add(x, a)</span>
+<span class="sd">        SE3Type LieTensor:</span>
+<span class="sd">        tensor([-1.6089,  0.4184,  0.6621, -0.2098,  0.5383,  0.4794,  0.6606])</span>
+<span class="sd">        &gt;&gt;&gt; pp.se3(a).Exp() @ x</span>
+<span class="sd">        SE3Type LieTensor:</span>
+<span class="sd">        tensor([-1.6089,  0.4184,  0.6621, -0.2098,  0.5383,  0.4794,  0.6606])</span>
+<span class="sd">        &gt;&gt;&gt; x + torch.cat([a, torch.randn(1)])</span>
+<span class="sd">        SE3Type LieTensor:</span>
+<span class="sd">        tensor([-1.6089,  0.4184,  0.6621, -0.2098,  0.5383,  0.4794,  0.6606])</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">return</span> <span class="nb">input</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">alpha</span><span class="p">)</span></div>
+
+<div class="viewcode-block" id="mul"><a class="viewcode-back" href="../../../../generated/pypose.mul/#pypose.mul">[docs]</a><span class="k">def</span> <span class="nf">mul</span><span class="p">(</span><span class="nb">input</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Multiply input LieTensor by other.</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \bm{y}_i =</span>
+<span class="sd">        \bm{x}_i \ast \bm{a}_i</span>
+
+<span class="sd">    where :math:`\bm{x}` is the ``input`` LieTensor, :math:`\bm{a}` is the ``other`` Tensor or</span>
+<span class="sd">    LieTensor, and :math:`\bm{y}` is the output.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        input (:obj:`LieTensor`): the input LieTensor (Lie Group or Lie Algebra).</span>
+
+<span class="sd">        other (:obj:`Number`, :obj:`Tensor`, or :obj:`LieTensor`): the value for input to be</span>
+<span class="sd">            multiplied by.</span>
+
+<span class="sd">    Return:</span>
+<span class="sd">        :obj:`Tensor`/:obj:`LieTensor`: the product of ``input`` and ``other``.</span>
+
+<span class="sd">    .. list-table:: List of :obj:`pypose.mul` cases</span>
+<span class="sd">        :widths: 25 30 25</span>
+<span class="sd">        :header-rows: 1</span>
+
+<span class="sd">        * - input :obj:`LieTensor`</span>
+<span class="sd">          - other</span>
+<span class="sd">          - output</span>
+<span class="sd">        * - Lie Algebra</span>
+<span class="sd">          - :obj:`Number`</span>
+<span class="sd">          - :obj:`Lie Algebra`</span>
+<span class="sd">        * - Lie Group</span>
+<span class="sd">          - :obj:`Tensor` :math:`\in \mathbb{R^{*\times3}}`</span>
+<span class="sd">          - :obj:`Tensor`</span>
+<span class="sd">        * - Lie Group</span>
+<span class="sd">          - :obj:`Tensor` :math:`\in \mathbb{R^{*\times4}}`</span>
+<span class="sd">          - :obj:`Tensor`</span>
+<span class="sd">        * - Lie Group</span>
+<span class="sd">          - :obj:`Lie Group`</span>
+<span class="sd">          - :obj:`Lie Group`</span>
+
+<span class="sd">    When multiplying a Lie Group by another Lie Group, they must have the</span>
+<span class="sd">    same Lie type.</span>
+
+<span class="sd">    Warning:</span>
+<span class="sd">        For better readability, we suggest calling ``a * b`` or ``a @ b`` instead of</span>
+<span class="sd">        ``a.mul(b)`` or ``pypose.mul(a, b)``, which have the same behaviors.</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        - When ``other`` is a Tensor, this operator is equivalent to :meth:`Act`.</span>
+<span class="sd">        - When ``other`` is a number and ``input`` is a Lie Algebra, this operator performs</span>
+<span class="sd">          simple element-wise multiplication.</span>
+<span class="sd">        - When ``input`` is a Lie Group, more details are shown below.</span>
+
+<span class="sd">    * Input :math:`\bm{x}`&#39;s :obj:`ltype` is :obj:`SO3_type`</span>
+<span class="sd">      (input :math:`\bm{x}` is an instance of :meth:`SO3`):</span>
+
+<span class="sd">      .. math::</span>
+<span class="sd">        {\displaystyle \bm{y}_i={\begin{bmatrix}</span>
+<span class="sd">              q_i^wq_i^{w&#39;} - q_i^xq_i^{x&#39;} -q_i^yq_i^{y&#39;} - q_i^zq_i^{z&#39;}\\</span>
+<span class="sd">              q_i^wq_i^{x&#39;} + q_i^xq_i^{w&#39;} +q_i^yq_i^{z&#39;} - q_i^zq_i^{y&#39;}\\</span>
+<span class="sd">              q_i^wq_i^{y&#39;} - q_i^xq_i^{z&#39;} +q_i^yq_i^{w&#39;} + q_i^zq_i^{x&#39;}\\</span>
+<span class="sd">              q_i^wq_i^{z&#39;} + q_i^xq_i^{y&#39;} -q_i^yq_i^{x&#39;} + q_i^zq_i^{w&#39;}</span>
+<span class="sd">              \end{bmatrix}</span>
+<span class="sd">        }^T},</span>
+
+<span class="sd">      where :math:`\bm{x}_i = [q_i^x, q_i^y, q_i^z, q_i^w]` and</span>
+<span class="sd">      :math:`\bm{a}_i = [q_i^{x&#39;}, q_i^{y&#39;}, q_i^{z&#39;}, q_i^{w&#39;}]` are the ``input``</span>
+<span class="sd">      and ``other`` LieTensor, respectively.</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        :math:`\mathbf{y}_i` can be simply derived by taking the complex number multiplication.</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \bm{y}_i =</span>
+<span class="sd">            (q_i^x\mathbf{i} + q_i^y\mathbf{j} + q_i^z\mathbf{k} + q_i^w) \ast</span>
+<span class="sd">            (q_i^{x&#39;}\mathbf{i} + q_i^{y&#39;}\mathbf{j} + q_i^{z&#39;}\mathbf{k} + q_i^{w&#39;}),</span>
+
+<span class="sd">        where and :math:`\mathbf{i}` :math:`\mathbf{j}`, and :math:`\mathbf{k}` are the imaginary</span>
+<span class="sd">        units.</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathbf{i}^2 = \mathbf{j}^2 = \mathbf{k}^2 = \mathbf{ijk} = -1</span>
+
+<span class="sd">    * Input :math:`\bm{x}`&#39;s :obj:`ltype` is :obj:`SE3_type`</span>
+<span class="sd">      (input :math:`\bm{x}` is an instance of :meth:`SE3`):</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \bm{y}_i = [\mathbf{q}_i * \mathbf{t}_i&#39; + \mathbf{t}_i,</span>
+<span class="sd">                        \mathbf{q}_i * \mathbf{q}_i&#39;]</span>
+
+<span class="sd">        where :math:`\bm{x}_i = [\mathbf{t}_i, \mathbf{q}_i]` and</span>
+<span class="sd">        :math:`\bm{a}_i = [\mathbf{t}_i&#39;, \mathbf{q}_i&#39;]` are the ``input`` and ``other``</span>
+<span class="sd">        LieTensor, respectively; :math:`\mathbf{t}_i`, :math:`\mathbf{t}_i&#39;` and</span>
+<span class="sd">        :math:`\mathbf{q}_i`, :math:`\mathbf{q}_i&#39;` are their translation and :obj:`SO3`</span>
+<span class="sd">        parts, respectively; the operator :math:`\ast` denotes the obj:`SO3_type`</span>
+<span class="sd">        multiplication introduced above.</span>
+
+<span class="sd">    * Input :math:`\bm{x}`&#39;s :obj:`ltype` is :obj:`RxSO3_type`</span>
+<span class="sd">      (input :math:`\bm{x}` is an instance of :meth:`RxSO3`)</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \bm{y}_i = [\mathbf{q}_i * \mathbf{q}_i&#39;, s_is_i&#39;]</span>
+
+<span class="sd">        where :math:`s_i` and :math:`s_i&#39;` are the scale parts of the ``input`` and</span>
+<span class="sd">        ``other`` LieTensor, respectively.</span>
+
+<span class="sd">    * Input :math:`\bm{x}`&#39;s :obj:`ltype` is :obj:`Sim3_type`</span>
+<span class="sd">      (input :math:`\bm{x}` is an instance of :meth:`Sim3`):</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \bm{y}_i = [\mathbf{q}_i * \mathbf{t}_i&#39; + \mathbf{t}_i,</span>
+<span class="sd">                            \mathbf{q}_i * \mathbf{q}_i&#39;, s_is_i&#39;]</span>
+
+<span class="sd">        where :math:`\bm{x}_i = [\mathbf{t}_i, \mathbf{q}_i, s_i]` and</span>
+<span class="sd">        :math:`\bm{a}_i = [\mathbf{t}_i&#39;, \mathbf{q}_i&#39;, s_i&#39;]` are the ``input`` and</span>
+<span class="sd">        ``other`` LieTensor, respectively.</span>
+
+<span class="sd">    Examples:</span>
+<span class="sd">        * :obj:`Lie Algebra` :math:`*` :obj:`Number` :math:`\mapsto` :obj:`Lie Algebra`</span>
+
+<span class="sd">            &gt;&gt;&gt; x = pp.randn_so3()</span>
+<span class="sd">            &gt;&gt;&gt; x</span>
+<span class="sd">            so3Type LieTensor:</span>
+<span class="sd">            LieTensor([ 0.3018, -1.0246,  0.7784])</span>
+<span class="sd">            &gt;&gt;&gt; a = 5</span>
+<span class="sd">            &gt;&gt;&gt; # The following two operations are equivalent.</span>
+<span class="sd">            &gt;&gt;&gt; x * a</span>
+<span class="sd">            so3Type LieTensor:</span>
+<span class="sd">            LieTensor([ 1.5090, -5.1231,  3.8919])</span>
+<span class="sd">            &gt;&gt;&gt; pp.mul(x, 5)</span>
+<span class="sd">            so3Type LieTensor:</span>
+<span class="sd">            LieTensor([ 1.5090, -5.1231,  3.8919])</span>
+
+<span class="sd">        * :obj:`Lie Group` :math:`*` :obj:`Tensor` :math:`\mapsto` :obj:`Tensor`</span>
+
+<span class="sd">            &gt;&gt;&gt; x = pp.randn_SO3()</span>
+<span class="sd">            &gt;&gt;&gt; a = torch.randn(3)</span>
+<span class="sd">            &gt;&gt;&gt; x, a</span>
+<span class="sd">            (SO3Type LieTensor:</span>
+<span class="sd">            LieTensor([ 0.6047, -0.2129, -0.1781,  0.7465]),</span>
+<span class="sd">            tensor([-0.1811, -0.2278, -1.9956]))</span>
+<span class="sd">            &gt;&gt;&gt; x * a</span>
+<span class="sd">            tensor([ 0.9089,  1.6984, -0.5969])</span>
+<span class="sd">            &gt;&gt;&gt; a = torch.randn(4)</span>
+<span class="sd">            &gt;&gt;&gt; a</span>
+<span class="sd">            tensor([ 1.5236, -1.2757, -0.7140,  0.2467])</span>
+<span class="sd">            &gt;&gt;&gt; x * a</span>
+<span class="sd">            tensor([ 1.6588, -0.4687, -1.2196,  0.2467]</span>
+
+<span class="sd">        * :obj:`SO3_type` :math:`*` :obj:`SO3_type` :math:`\mapsto` :obj:`SO3_type`</span>
+
+<span class="sd">            &gt;&gt;&gt; a = pp.randn_SO3()</span>
+<span class="sd">            &gt;&gt;&gt; a</span>
+<span class="sd">            SO3Type LieTensor:</span>
+<span class="sd">            LieTensor([ 0.0118, -0.7042, -0.4516,  0.5478])</span>
+<span class="sd">            &gt;&gt;&gt; x * a</span>
+<span class="sd">            SO3Type LieTensor:</span>
+<span class="sd">            LieTensor([ 0.3108, -0.3714, -0.8579,  0.1715])</span>
+
+<span class="sd">        * :obj:`SE3_type` :math:`*` :obj:`SE3_type` :math:`\mapsto` :obj:`SE3_type`</span>
+
+<span class="sd">            &gt;&gt;&gt; x = pp.randn_SE3()</span>
+<span class="sd">            &gt;&gt;&gt; a = pp.randn_SE3()</span>
+<span class="sd">            &gt;&gt;&gt; x, a</span>
+<span class="sd">            (SE3Type LieTensor:</span>
+<span class="sd">            LieTensor([ 0.7819,  1.8541, -0.2857, -0.1970,  0.4742,  0.1109,  0.8509]),</span>
+<span class="sd">            SE3Type LieTensor:</span>
+<span class="sd">            LieTensor([ 0.6039, -1.4076,  0.3496,  0.7297,  0.3971,  0.2849,  0.4783]))</span>
+<span class="sd">            &gt;&gt;&gt; x * a</span>
+<span class="sd">            SE3Type LieTensor:</span>
+<span class="sd">            LieTensor([ 1.8949,  0.7456, -0.3104,  0.6177,  0.7017, -0.1287,  0.3308])</span>
+
+<span class="sd">        * :obj:`RxSO3_type` :math:`*` :obj:`RxSO3_type` :math:`\mapsto` :obj:`RxSO3_type`</span>
+
+<span class="sd">            &gt;&gt;&gt; x = pp.randn_RxSO3()</span>
+<span class="sd">            &gt;&gt;&gt; a = pp.randn_RxSO3()</span>
+<span class="sd">            &gt;&gt;&gt; x, a</span>
+<span class="sd">            (RxSO3Type LieTensor:</span>
+<span class="sd">            LieTensor([-0.7518, -0.6481,  0.0933, -0.0775,  1.5791]),</span>
+<span class="sd">            RxSO3Type LieTensor:</span>
+<span class="sd">            LieTensor([ 0.2757,  0.3102, -0.4086,  0.8129,  0.6593]))</span>
+<span class="sd">            &gt;&gt;&gt; x * a</span>
+<span class="sd">            RxSO3Type LieTensor:</span>
+<span class="sd">            LieTensor([-0.3967, -0.8323,  0.0530,  0.3835,  1.0411])</span>
+
+<span class="sd">        * :obj:`Sim3_type` :math:`*` :obj:`Sim3_type` :math:`\mapsto` :obj:`Sim3_type`</span>
+
+<span class="sd">            &gt;&gt;&gt; x = pp.randn_Sim3()</span>
+<span class="sd">            &gt;&gt;&gt; a = pp.randn_Sim3()</span>
+<span class="sd">            &gt;&gt;&gt; x, a</span>
+<span class="sd">            (Sim3Type LieTensor:</span>
+<span class="sd">            LieTensor([-0.3439, -0.2309, -0.6571,  0.3170, -0.6594, -0.1100,  0.6728, 0.6296]),</span>
+<span class="sd">            Sim3Type LieTensor:</span>
+<span class="sd">            LieTensor([-0.7434,  1.8613, -2.1315,  0.7688, -0.0268,  0.0520,  0.6367, 1.7745]))</span>
+<span class="sd">            &gt;&gt;&gt; x * a</span>
+<span class="sd">            Sim3Type LieTensor:</span>
+<span class="sd">            LieTensor([ 0.5740,  1.3197, -0.2752,  0.6819, -0.5389,  0.4634,  0.1727, 1.1172])</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">return</span> <span class="nb">input</span> <span class="o">*</span> <span class="n">other</span></div>
+</pre></div>
+
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@@ -0,0 +1,1275 @@
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+  <h1>Source code for pypose.lietensor.convert</h1><div class="highlight"><pre>
+<span></span><span class="kn">import</span> <span class="nn">torch</span><span class="o">,</span> <span class="nn">warnings</span>
+<span class="kn">from</span> <span class="nn">.utils</span> <span class="kn">import</span> <span class="n">SO3</span><span class="p">,</span> <span class="n">so3</span><span class="p">,</span> <span class="n">SE3</span><span class="p">,</span> <span class="n">RxSO3</span><span class="p">,</span> <span class="n">Sim3</span>
+<span class="kn">from</span> <span class="nn">.lietensor</span> <span class="kn">import</span> <span class="n">LieTensor</span><span class="p">,</span> <span class="n">SE3_type</span><span class="p">,</span> <span class="n">SO3_type</span><span class="p">,</span> <span class="n">Sim3_type</span><span class="p">,</span> <span class="n">RxSO3_type</span>
+
+
+<div class="viewcode-block" id="mat2SO3"><a class="viewcode-back" href="../../../../generated/pypose.mat2SO3/#pypose.mat2SO3">[docs]</a><span class="k">def</span> <span class="nf">mat2SO3</span><span class="p">(</span><span class="n">mat</span><span class="p">,</span> <span class="n">check</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">rtol</span><span class="o">=</span><span class="mf">1e-5</span><span class="p">,</span> <span class="n">atol</span><span class="o">=</span><span class="mf">1e-5</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&quot;&quot;&quot;Convert batched rotation or transformation matrices to SO3Type LieTensor.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        mat (Tensor): the batched matrices to convert. If input is of shape</span>
+<span class="sd">            :obj:`(*, 3, 4)` or :obj:`(*, 4, 4)`, only the top left 3x3 submatrix is used.</span>
+<span class="sd">        check (bool, optional): flag to check if the input is valid rotation matrices</span>
+<span class="sd">            (orthogonal and with a determinant of one). Set to ``False`` if less</span>
+<span class="sd">            computation is needed. Default: ``True``.</span>
+<span class="sd">        rtol (float, optional): relative tolerance when check is enabled. Default: 1e-05</span>
+<span class="sd">        atol (float, optional): absolute tolerance when check is enabled. Default: 1e-05</span>
+
+<span class="sd">    Return:</span>
+<span class="sd">        LieTensor: the converted SO3Type LieTensor.</span>
+
+<span class="sd">    Shape:</span>
+<span class="sd">        Input: :obj:`(*, 3, 3)` or :obj:`(*, 3, 4)` or :obj:`(*, 4, 4)`</span>
+
+<span class="sd">        Output: :obj:`(*, 4)`</span>
+
+<span class="sd">    Let the input be matrix :math:`\mathbf{R}`, :math:`\mathbf{R}_i` represents each individual</span>
+<span class="sd">    matrix in the batch. :math:`\mathbf{R}^{m,n}_i` represents the :math:`m^{\mathrm{th}}` row</span>
+<span class="sd">    and :math:`n^{\mathrm{th}}` column of :math:`\mathbf{R}_i`, :math:`m,n\geq 1`, then the</span>
+<span class="sd">    quaternion can be computed by:</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \left\{\begin{aligned}</span>
+<span class="sd">        q^x_i &amp;= \mathrm{sign}(\mathbf{R}^{2,3}_i - \mathbf{R}^{3,2}_i) \frac{1}{2}</span>
+<span class="sd">            \sqrt{1 + \mathbf{R}^{1,1}_i - \mathbf{R}^{2,2}_i - \mathbf{R}^{3,3}_i}\\</span>
+<span class="sd">        q^y_i &amp;= \mathrm{sign}(\mathbf{R}^{3,1}_i - \mathbf{R}^{1,3}_i) \frac{1}{2}</span>
+<span class="sd">            \sqrt{1 - \mathbf{R}^{1,1}_i + \mathbf{R}^{2,2}_i - \mathbf{R}^{3,3}_i}\\</span>
+<span class="sd">        q^z_i &amp;= \mathrm{sign}(\mathbf{R}^{1,2}_i - \mathbf{R}^{2,1}_i) \frac{1}{2}</span>
+<span class="sd">            \sqrt{1 - \mathbf{R}^{1,1}_i - \mathbf{R}^{2,2}_i + \mathbf{R}^{3,3}_i}\\</span>
+<span class="sd">        q^w_i &amp;= \frac{1}{2} \sqrt{1 + \mathbf{R}^{1,1}_i + \mathbf{R}^{2,2}_i +</span>
+<span class="sd">            \mathbf{R}^{3,3}_i}</span>
+<span class="sd">        \end{aligned}\right.,</span>
+
+<span class="sd">    In summary, the output LieTensor should be of format:</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \textbf{y}_i = [q^x_i, q^y_i, q^z_i, q^w_i]</span>
+
+<span class="sd">    Warning:</span>
+<span class="sd">        Numerically, a transformation matrix is considered legal if:</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            |{\rm det}(\mathbf{R}) - 1| \leq \texttt{atol} + \texttt{rtol}\times 1\\</span>
+<span class="sd">            |\mathbf{RR}^{T}-\mathbf{I}|\leq\texttt{atol} + \texttt{rtol}\times\mathbf{I}</span>
+
+<span class="sd">        where :math:`|\cdot |` is element-wise absolute function. When ``check`` is set to</span>
+<span class="sd">        ``True``, illegal input will raise a ``ValueError``. Otherwise, no data validation</span>
+<span class="sd">        is performed. Illegal input will output an irrelevant result, which likely</span>
+<span class="sd">        contains ``nan``.</span>
+
+<span class="sd">    Examples:</span>
+
+<span class="sd">        &gt;&gt;&gt; input = torch.tensor([[0., -1.,  0.],</span>
+<span class="sd">        ...                       [1.,  0.,  0.],</span>
+<span class="sd">        ...                       [0.,  0.,  1.]])</span>
+<span class="sd">        &gt;&gt;&gt; pp.mat2SO3(input)</span>
+<span class="sd">        SO3Type LieTensor:</span>
+<span class="sd">        tensor([0.0000, 0.0000, 0.7071, 0.7071])</span>
+
+<span class="sd">    See :meth:`pypose.SO3` for more details of the output LieTensor format.</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+
+    <span class="k">if</span> <span class="ow">not</span> <span class="n">torch</span><span class="o">.</span><span class="n">is_tensor</span><span class="p">(</span><span class="n">mat</span><span class="p">):</span>
+        <span class="n">mat</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">(</span><span class="n">mat</span><span class="p">)</span>
+
+    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">mat</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="o">&lt;</span> <span class="mi">2</span><span class="p">:</span>
+        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">&quot;Input size must be at least 2 dimensions. Got </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">mat</span><span class="o">.</span><span class="n">shape</span><span class="p">))</span>
+
+    <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">mat</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">:]</span> <span class="o">==</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span> <span class="ow">or</span> <span class="n">mat</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">:]</span> <span class="o">==</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span> <span class="ow">or</span> <span class="n">mat</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">:]</span> <span class="o">==</span> <span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">)):</span>
+        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">&quot;Input size must be a * x 3 x 3 or * x 3 x 4 or * x 4 x 4 tensor. </span><span class="se">\</span>
+<span class="s2">                Got </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">mat</span><span class="o">.</span><span class="n">shape</span><span class="p">))</span>
+
+    <span class="n">mat</span> <span class="o">=</span> <span class="n">mat</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="p">:</span><span class="mi">3</span><span class="p">,</span> <span class="p">:</span><span class="mi">3</span><span class="p">]</span>
+    <span class="n">shape</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">shape</span>
+
+    <span class="k">with</span> <span class="n">torch</span><span class="o">.</span><span class="n">no_grad</span><span class="p">():</span>
+        <span class="k">if</span> <span class="n">check</span><span class="p">:</span>
+            <span class="n">e0</span> <span class="o">=</span> <span class="n">mat</span> <span class="o">@</span> <span class="n">mat</span><span class="o">.</span><span class="n">mT</span>
+            <span class="n">e1</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">mat</span><span class="o">.</span><span class="n">dtype</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="n">mat</span><span class="o">.</span><span class="n">device</span><span class="p">)</span>
+            <span class="k">if</span> <span class="ow">not</span> <span class="n">torch</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">e0</span><span class="p">,</span> <span class="n">e1</span><span class="o">.</span><span class="n">expand_as</span><span class="p">(</span><span class="n">e0</span><span class="p">),</span> <span class="n">rtol</span><span class="o">=</span><span class="n">rtol</span><span class="p">,</span> <span class="n">atol</span><span class="o">=</span><span class="n">atol</span><span class="p">):</span>
+                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">&quot;Input rotation matrices are not all orthogonal matrix&quot;</span><span class="p">)</span>
+
+            <span class="n">ones</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">shape</span><span class="p">[:</span><span class="o">-</span><span class="mi">2</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">mat</span><span class="o">.</span><span class="n">dtype</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="n">mat</span><span class="o">.</span><span class="n">device</span><span class="p">)</span>
+            <span class="k">if</span> <span class="ow">not</span> <span class="n">torch</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">det</span><span class="p">(</span><span class="n">mat</span><span class="p">),</span> <span class="n">ones</span><span class="p">,</span> <span class="n">rtol</span><span class="o">=</span><span class="n">rtol</span><span class="p">,</span> <span class="n">atol</span><span class="o">=</span><span class="n">atol</span><span class="p">):</span>
+                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">&quot;Input rotation matrices&#39; determinant are not all equal to 1&quot;</span><span class="p">)</span>
+
+    <span class="n">rmat_t</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">mT</span>
+
+    <span class="n">mask_d2</span> <span class="o">=</span> <span class="n">rmat_t</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span> <span class="o">&lt;</span> <span class="n">atol</span>
+
+    <span class="n">mask_d0_d1</span> <span class="o">=</span> <span class="n">rmat_t</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">&gt;</span> <span class="n">rmat_t</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
+    <span class="n">mask_d0_nd1</span> <span class="o">=</span> <span class="n">rmat_t</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">&lt;</span> <span class="o">-</span><span class="n">rmat_t</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
+
+    <span class="n">t0</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">+</span> <span class="n">rmat_t</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="n">rmat_t</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">rmat_t</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span>
+    <span class="n">q0</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">stack</span><span class="p">([</span><span class="n">rmat_t</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span> <span class="o">-</span> <span class="n">rmat_t</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span>
+                      <span class="n">t0</span><span class="p">,</span> <span class="n">rmat_t</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="n">rmat_t</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
+                      <span class="n">rmat_t</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="n">rmat_t</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">]],</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
+    <span class="n">t0_rep</span> <span class="o">=</span> <span class="n">t0</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">repeat</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">shape</span><span class="p">))</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="p">,)</span><span class="o">+</span><span class="p">(</span><span class="mi">4</span><span class="p">,))</span>
+
+    <span class="n">t1</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">rmat_t</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="n">rmat_t</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">rmat_t</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span>
+    <span class="n">q1</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">stack</span><span class="p">([</span><span class="n">rmat_t</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="n">rmat_t</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span>
+                      <span class="n">rmat_t</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="n">rmat_t</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
+                      <span class="n">t1</span><span class="p">,</span> <span class="n">rmat_t</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span> <span class="o">+</span> <span class="n">rmat_t</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">]],</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
+    <span class="n">t1_rep</span> <span class="o">=</span> <span class="n">t1</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">repeat</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">shape</span><span class="p">))</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="p">,)</span><span class="o">+</span><span class="p">(</span><span class="mi">4</span><span class="p">,))</span>
+
+    <span class="n">t2</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">rmat_t</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="n">rmat_t</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="n">rmat_t</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span>
+    <span class="n">q2</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">stack</span><span class="p">([</span><span class="n">rmat_t</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">rmat_t</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
+                      <span class="n">rmat_t</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="n">rmat_t</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span>
+                      <span class="n">rmat_t</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span> <span class="o">+</span> <span class="n">rmat_t</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">t2</span><span class="p">],</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
+    <span class="n">t2_rep</span> <span class="o">=</span> <span class="n">t2</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">repeat</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">shape</span><span class="p">))</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="p">,)</span><span class="o">+</span><span class="p">(</span><span class="mi">4</span><span class="p">,))</span>
+
+    <span class="n">t3</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">+</span> <span class="n">rmat_t</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="n">rmat_t</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="n">rmat_t</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span>
+    <span class="n">q3</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">stack</span><span class="p">([</span><span class="n">t3</span><span class="p">,</span> <span class="n">rmat_t</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span> <span class="o">-</span> <span class="n">rmat_t</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span>
+                      <span class="n">rmat_t</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="n">rmat_t</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span>
+                      <span class="n">rmat_t</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">rmat_t</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]],</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
+    <span class="n">t3_rep</span> <span class="o">=</span> <span class="n">t3</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">repeat</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">shape</span><span class="p">))</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="p">,)</span><span class="o">+</span><span class="p">(</span><span class="mi">4</span><span class="p">,))</span>
+
+    <span class="n">mask_c0</span> <span class="o">=</span> <span class="n">mask_d2</span> <span class="o">*</span> <span class="n">mask_d0_d1</span>
+    <span class="n">mask_c1</span> <span class="o">=</span> <span class="n">mask_d2</span> <span class="o">*</span> <span class="o">~</span><span class="n">mask_d0_d1</span>
+    <span class="n">mask_c2</span> <span class="o">=</span> <span class="o">~</span><span class="n">mask_d2</span> <span class="o">*</span> <span class="n">mask_d0_nd1</span>
+    <span class="n">mask_c3</span> <span class="o">=</span> <span class="o">~</span><span class="n">mask_d2</span> <span class="o">*</span> <span class="o">~</span><span class="n">mask_d0_nd1</span>
+    <span class="n">mask_c0</span> <span class="o">=</span> <span class="n">mask_c0</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">type_as</span><span class="p">(</span><span class="n">q0</span><span class="p">)</span>
+    <span class="n">mask_c1</span> <span class="o">=</span> <span class="n">mask_c1</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">type_as</span><span class="p">(</span><span class="n">q1</span><span class="p">)</span>
+    <span class="n">mask_c2</span> <span class="o">=</span> <span class="n">mask_c2</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">type_as</span><span class="p">(</span><span class="n">q2</span><span class="p">)</span>
+    <span class="n">mask_c3</span> <span class="o">=</span> <span class="n">mask_c3</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">type_as</span><span class="p">(</span><span class="n">q3</span><span class="p">)</span>
+
+    <span class="n">q</span> <span class="o">=</span> <span class="n">q0</span> <span class="o">*</span> <span class="n">mask_c0</span> <span class="o">+</span> <span class="n">q1</span> <span class="o">*</span> <span class="n">mask_c1</span> <span class="o">+</span> <span class="n">q2</span> <span class="o">*</span> <span class="n">mask_c2</span> <span class="o">+</span> <span class="n">q3</span> <span class="o">*</span> <span class="n">mask_c3</span>
+    <span class="n">q</span> <span class="o">/=</span> <span class="mi">2</span><span class="o">*</span><span class="n">torch</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">t0_rep</span> <span class="o">*</span> <span class="n">mask_c0</span> <span class="o">+</span> <span class="n">t1_rep</span> <span class="o">*</span> <span class="n">mask_c1</span> <span class="o">+</span>  <span class="c1"># noqa</span>
+                    <span class="n">t2_rep</span> <span class="o">*</span> <span class="n">mask_c2</span> <span class="o">+</span> <span class="n">t3_rep</span> <span class="o">*</span> <span class="n">mask_c3</span><span class="p">)</span>  <span class="c1"># noqa</span>
+
+    <span class="n">q</span> <span class="o">=</span> <span class="n">q</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="n">shape</span><span class="p">[:</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span><span class="o">+</span><span class="p">(</span><span class="mi">4</span><span class="p">,))</span>
+    <span class="c1"># wxyz -&gt; xyzw</span>
+    <span class="n">q</span> <span class="o">=</span> <span class="n">q</span><span class="o">.</span><span class="n">index_select</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">device</span><span class="o">=</span><span class="n">q</span><span class="o">.</span><span class="n">device</span><span class="p">))</span>
+
+    <span class="k">return</span> <span class="n">SO3</span><span class="p">(</span><span class="n">q</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="mat2SE3"><a class="viewcode-back" href="../../../../generated/pypose.mat2SE3/#pypose.mat2SE3">[docs]</a><span class="k">def</span> <span class="nf">mat2SE3</span><span class="p">(</span><span class="n">mat</span><span class="p">,</span> <span class="n">check</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">rtol</span><span class="o">=</span><span class="mf">1e-5</span><span class="p">,</span> <span class="n">atol</span><span class="o">=</span><span class="mf">1e-5</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&quot;&quot;&quot;Convert batched rotation or transformation matrices to SE3Type LieTensor.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        mat (Tensor): the batched matrices to convert. If input is of shape :obj:`(*, 3, 3)`, then</span>
+<span class="sd">            translation will be filled with zero. For input with shape :obj:`(*, 3, 4)`, the last</span>
+<span class="sd">            row will be treated as ``[0, 0, 0, 1]``.</span>
+<span class="sd">        check (bool, optional): flag to check if the input is valid rotation matrices (orthogonal</span>
+<span class="sd">            and with a determinant of one). Set to ``False`` if less computation is needed.</span>
+<span class="sd">            Default: ``True``.</span>
+<span class="sd">        rtol (float, optional): relative tolerance when check is enabled. Default: 1e-05</span>
+<span class="sd">        atol (float, optional): absolute tolerance when check is enabled. Default: 1e-05</span>
+
+<span class="sd">    Return:</span>
+<span class="sd">        LieTensor: the converted SE3Type LieTensor.</span>
+
+<span class="sd">    Shape:</span>
+<span class="sd">        Input: :obj:`(*, 3, 3)` or :obj:`(*, 3, 4)` or :obj:`(*, 4, 4)`</span>
+
+<span class="sd">        Output: :obj:`(*, 7)`</span>
+
+<span class="sd">    Let the input be matrix :math:`\mathbf{T}`,  :math:`\mathbf{T}_i` represents each individual</span>
+<span class="sd">    matrix in the batch. :math:`\mathbf{R}_i\in\mathbb{R}^{3\times 3}` be the top left 3x3 block</span>
+<span class="sd">    matrix of :math:`\mathbf{T}_i`. :math:`\mathbf{T}^{m,n}_i` represents the :math:`m^{\mathrm{th}}`</span>
+<span class="sd">    row and :math:`n^{\mathrm{th}}` column of :math:`\mathbf{T}_i`, :math:`m,n\geq 1`, then the</span>
+<span class="sd">    translation and quaternion can be computed by:</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \left\{\begin{aligned}</span>
+<span class="sd">        t^x_i &amp;= \mathbf{T}^{1,4}_i\\</span>
+<span class="sd">        t^y_i &amp;= \mathbf{T}^{2,4}_i\\</span>
+<span class="sd">        t^z_i &amp;= \mathbf{T}^{3,4}_i\\</span>
+<span class="sd">        q^x_i &amp;= \mathrm{sign}(\mathbf{R}^{2,3}_i - \mathbf{R}^{3,2}_i) \frac{1}{2}</span>
+<span class="sd">            \sqrt{1 + \mathbf{R}^{1,1}_i - \mathbf{R}^{2,2}_i - \mathbf{R}^{3,3}_i}\\</span>
+<span class="sd">        q^y_i &amp;= \mathrm{sign}(\mathbf{R}^{3,1}_i - \mathbf{R}^{1,3}_i) \frac{1}{2}</span>
+<span class="sd">            \sqrt{1 - \mathbf{R}^{1,1}_i + \mathbf{R}^{2,2}_i - \mathbf{R}^{3,3}_i}\\</span>
+<span class="sd">        q^z_i &amp;= \mathrm{sign}(\mathbf{R}^{1,2}_i - \mathbf{R}^{2,1}_i) \frac{1}{2}</span>
+<span class="sd">            \sqrt{1 - \mathbf{R}^{1,1}_i - \mathbf{R}^{2,2}_i + \mathbf{R}^{3,3}_i}\\</span>
+<span class="sd">        q^w_i &amp;= \frac{1}{2} \sqrt{1 + \mathbf{R}^{1,1}_i + \mathbf{R}^{2,2}_i + \mathbf{R}^{3,3}_i}</span>
+<span class="sd">        \end{aligned}\right.,</span>
+
+<span class="sd">    In summary, the output LieTensor should be of format:</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \textbf{y}_i = [t^x_i, t^y_i, t^z_i, q^x_i, q^y_i, q^z_i, q^w_i]</span>
+
+<span class="sd">    Warning:</span>
+<span class="sd">        Numerically, a transformation matrix is considered legal if:</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            |{\rm det}(\mathbf{R}) - 1| \leq \texttt{atol} + \texttt{rtol}\times 1\\</span>
+<span class="sd">            |\mathbf{RR}^{T} - \mathbf{I}| \leq \texttt{atol} + \texttt{rtol}\times \mathbf{I}</span>
+
+<span class="sd">        where :math:`|\cdot |` is element-wise absolute function. When ``check`` is set to ``True``,</span>
+<span class="sd">        illegal input will raise a ``ValueError``. Otherwise, no data validation is performed.</span>
+<span class="sd">        Illegal input will output an irrelevant result, which likely contains ``nan``.</span>
+
+<span class="sd">        For input with shape :obj:`(*, 4, 4)`, when ``check`` is set to ``True`` and the last row</span>
+<span class="sd">        of the each individual matrix is not ``[0, 0, 0, 1]``, a warning will be triggered.</span>
+<span class="sd">        Even though the last row is not used in the computation, it is worth noting that a matrix not</span>
+<span class="sd">        satisfying this condition is not a valid transformation matrix.</span>
+
+<span class="sd">    Examples:</span>
+
+<span class="sd">        &gt;&gt;&gt; input = torch.tensor([[0., -1., 0., 0.1],</span>
+<span class="sd">        ...                       [1.,  0., 0., 0.2],</span>
+<span class="sd">        ...                       [0.,  0., 1., 0.3],</span>
+<span class="sd">        ...                       [0.,  0., 0.,  1.]])</span>
+<span class="sd">        &gt;&gt;&gt; pp.mat2SE3(input)</span>
+<span class="sd">        SE3Type LieTensor:</span>
+<span class="sd">        tensor([0.1000, 0.2000, 0.3000, 0.0000, 0.0000, 0.7071, 0.7071])</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        The individual matrix in a batch can be written as:</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \begin{bmatrix}</span>
+<span class="sd">                    \mathbf{R}_{3\times3} &amp; \mathbf{t}_{3\times1}\\</span>
+<span class="sd">                    \textbf{0} &amp; 1</span>
+<span class="sd">            \end{bmatrix},</span>
+
+<span class="sd">        where :math:`\mathbf{R}` is the rotation matrix. The translation vector :math:`\mathbf{t}` defines the</span>
+<span class="sd">        displacement between the original position and the transformed position.</span>
+
+
+<span class="sd">    See :meth:`pypose.SE3` for more details of the output LieTensor format.</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+    <span class="k">if</span> <span class="ow">not</span> <span class="n">torch</span><span class="o">.</span><span class="n">is_tensor</span><span class="p">(</span><span class="n">mat</span><span class="p">):</span>
+        <span class="n">mat</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">(</span><span class="n">mat</span><span class="p">)</span>
+
+    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">mat</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="o">&lt;</span> <span class="mi">2</span><span class="p">:</span>
+        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">&quot;Input size must be at least 2 dimensions. Got </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">mat</span><span class="o">.</span><span class="n">shape</span><span class="p">))</span>
+
+    <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">mat</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">:]</span> <span class="o">==</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span> <span class="ow">or</span> <span class="n">mat</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">:]</span> <span class="o">==</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span> <span class="ow">or</span> <span class="n">mat</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">:]</span> <span class="o">==</span> <span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">)):</span>
+        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">&quot;Input size must be a * x 3 x 3 or * x 3 x 4 or * x 4 x 4  tensor. </span><span class="se">\</span>
+<span class="s2">                Got </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">mat</span><span class="o">.</span><span class="n">shape</span><span class="p">))</span>
+
+    <span class="n">shape</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">shape</span>
+    <span class="k">if</span> <span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">:]</span> <span class="o">==</span> <span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span> <span class="ow">and</span> <span class="n">check</span> <span class="o">==</span> <span class="kc">True</span><span class="p">:</span>
+        <span class="n">zerosone</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">mat</span><span class="o">.</span><span class="n">dtype</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="n">mat</span><span class="o">.</span><span class="n">device</span><span class="p">)</span>
+        <span class="k">if</span> <span class="ow">not</span> <span class="n">torch</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">mat</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="p">:],</span> <span class="n">zerosone</span><span class="o">.</span><span class="n">expand_as</span><span class="p">(</span><span class="n">mat</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="p">:]),</span> <span class="n">rtol</span><span class="o">=</span><span class="n">rtol</span><span class="p">,</span> <span class="n">atol</span><span class="o">=</span><span class="n">atol</span><span class="p">):</span>
+            <span class="n">warnings</span><span class="o">.</span><span class="n">warn</span><span class="p">(</span><span class="s2">&quot;input of shape 4x4 last rows are not all equal [0, 0, 0, 1]&quot;</span><span class="p">)</span>
+
+    <span class="n">q</span> <span class="o">=</span> <span class="n">mat2SO3</span><span class="p">(</span><span class="n">mat</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="p">:</span><span class="mi">3</span><span class="p">,</span> <span class="p">:</span><span class="mi">3</span><span class="p">],</span> <span class="n">check</span><span class="o">=</span><span class="n">check</span><span class="p">,</span> <span class="n">rtol</span><span class="o">=</span><span class="n">rtol</span><span class="p">,</span> <span class="n">atol</span><span class="o">=</span><span class="n">atol</span><span class="p">)</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span>
+    <span class="k">if</span> <span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
+        <span class="n">t</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">shape</span><span class="p">[:</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span><span class="o">+</span><span class="p">(</span><span class="mi">3</span><span class="p">,),</span> <span class="n">dtype</span><span class="o">=</span><span class="n">mat</span><span class="o">.</span><span class="n">dtype</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="n">mat</span><span class="o">.</span><span class="n">device</span><span class="p">,</span> <span class="n">requires_grad</span><span class="o">=</span><span class="n">mat</span><span class="o">.</span><span class="n">requires_grad</span><span class="p">)</span>
+    <span class="k">else</span><span class="p">:</span>
+        <span class="n">t</span> <span class="o">=</span> <span class="n">mat</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="p">:</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">]</span>
+    <span class="n">vec</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">cat</span><span class="p">([</span><span class="n">t</span><span class="p">,</span> <span class="n">q</span><span class="p">],</span> <span class="n">dim</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span>
+
+    <span class="k">return</span> <span class="n">SE3</span><span class="p">(</span><span class="n">vec</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="mat2Sim3"><a class="viewcode-back" href="../../../../generated/pypose.mat2Sim3/#pypose.mat2Sim3">[docs]</a><span class="k">def</span> <span class="nf">mat2Sim3</span><span class="p">(</span><span class="n">mat</span><span class="p">,</span> <span class="n">check</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">rtol</span><span class="o">=</span><span class="mf">1e-5</span><span class="p">,</span> <span class="n">atol</span><span class="o">=</span><span class="mf">1e-5</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&quot;&quot;&quot;Convert batched rotation or transformation matrices to Sim3Type LieTensor.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        mat (Tensor): the batched matrices to convert. If input is of shape :obj:`(*, 3, 3)`,</span>
+<span class="sd">            then translation will be filled with zero. For input with shape :obj:`(*, 3, 4)`,</span>
+<span class="sd">            the last row will be treated as ``[0, 0, 0, 1]``.</span>
+<span class="sd">        check (bool, optional): flag to check if the input is valid rotation matrices (orthogonal</span>
+<span class="sd">            and with a determinant of one). Set to ``False`` if less computation is needed.</span>
+<span class="sd">            Default: ``True``.</span>
+<span class="sd">        rtol (float, optional): relative tolerance when check is enabled. Default: 1e-05</span>
+<span class="sd">        atol (float, optional): absolute tolerance when check is enabled. Default: 1e-05</span>
+
+<span class="sd">    Return:</span>
+<span class="sd">        LieTensor: the converted Sim3Type LieTensor.</span>
+
+<span class="sd">    Shape:</span>
+<span class="sd">        Input: :obj:`(*, 3, 3)` or :obj:`(*, 3, 4)` or :obj:`(*, 4, 4)`</span>
+
+<span class="sd">        Output: :obj:`(*, 8)`</span>
+
+<span class="sd">    Let the input be matrix :math:`\mathbf{T}`,  :math:`\mathbf{T}_i` represents each individual</span>
+<span class="sd">    matrix in the batch. :math:`\mathbf{U}_i\in\mathbb{R}^{3\times 3}` be the top left 3x3 block</span>
+<span class="sd">    matrix of :math:`\mathbf{T}_i`. Let :math:`\mathbf{T}^{m,n}_i` represents the</span>
+<span class="sd">    :math:`m^{\mathrm{th}}` row and :math:`n^{\mathrm{th}}` column of :math:`\mathbf{T}_i`,</span>
+<span class="sd">    :math:`m,n\geq 1`, then the scaling factor :math:`s_i\in\mathbb{R}` and the rotation matrix</span>
+<span class="sd">    :math:`\mathbf{R}_i\in\mathbb{R}^{3\times 3}` can be computed as:</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \begin{aligned}</span>
+<span class="sd">            s_i &amp;= \sqrt[3]{\vert \mathbf{U}_i \vert}\\</span>
+<span class="sd">            \mathbf{R}_i &amp;= \mathbf{U}_i/s_i</span>
+<span class="sd">        \end{aligned}</span>
+
+<span class="sd">    the translation and quaternion can be computed by:</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \left\{\begin{aligned}</span>
+<span class="sd">        t^x_i &amp;= \mathbf{T}^{1,4}_i\\</span>
+<span class="sd">        t^y_i &amp;= \mathbf{T}^{2,4}_i\\</span>
+<span class="sd">        t^z_i &amp;= \mathbf{T}^{3,4}_i\\</span>
+<span class="sd">        q^x_i &amp;= \mathrm{sign}(\mathbf{R}^{2,3}_i - \mathbf{R}^{3,2}_i) \frac{1}{2}</span>
+<span class="sd">            \sqrt{1 + \mathbf{R}^{1,1}_i - \mathbf{R}^{2,2}_i - \mathbf{R}^{3,3}_i}\\</span>
+<span class="sd">        q^y_i &amp;= \mathrm{sign}(\mathbf{R}^{3,1}_i - \mathbf{R}^{1,3}_i) \frac{1}{2}</span>
+<span class="sd">            \sqrt{1 - \mathbf{R}^{1,1}_i + \mathbf{R}^{2,2}_i - \mathbf{R}^{3,3}_i}\\</span>
+<span class="sd">        q^z_i &amp;= \mathrm{sign}(\mathbf{R}^{1,2}_i - \mathbf{R}^{2,1}_i) \frac{1}{2}</span>
+<span class="sd">            \sqrt{1 - \mathbf{R}^{1,1}_i - \mathbf{R}^{2,2}_i + \mathbf{R}^{3,3}_i}\\</span>
+<span class="sd">        q^w_i &amp;= \frac{1}{2} \sqrt{1 + \mathbf{R}^{1,1}_i + \mathbf{R}^{2,2}_i +</span>
+<span class="sd">            \mathbf{R}^{3,3}_i}</span>
+<span class="sd">        \end{aligned}\right.,</span>
+
+<span class="sd">    In summary, the output LieTensor should be of format:</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \textbf{y}_i = [t^x_i, t^y_i, t^z_i, q^x_i, q^y_i, q^z_i, q^w_i, s_i]</span>
+
+<span class="sd">    Warning:</span>
+<span class="sd">        Numerically, a transformation matrix is considered legal if:</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \vert s \vert &gt; \texttt{atol} \\</span>
+<span class="sd">            |{\rm det}(\mathbf{R}) - 1| \leq \texttt{atol} + \texttt{rtol}\times 1\\</span>
+<span class="sd">            |\mathbf{RR}^{T} - \mathbf{I}| \leq \texttt{atol} + \texttt{rtol}\times \mathbf{I}</span>
+
+<span class="sd">        where :math:`|\cdot |` is element-wise absolute function. When ``check`` is set to ``True``,</span>
+<span class="sd">        illegal input will raise a ``ValueError``. Otherwise, no data validation is performed.</span>
+<span class="sd">        Illegal input will output an irrelevant result, which likely contains ``nan``.</span>
+
+<span class="sd">        For input with shape :obj:`(*, 4, 4)`, when ``check`` is set to ``True`` and the last row</span>
+<span class="sd">        of the each individual matrix is not ``[0, 0, 0, 1]``, a warning will be triggered.</span>
+<span class="sd">        Even though the last row is not used in the computation, it is worth noting that a matrix not</span>
+<span class="sd">        satisfying this condition is not a valid transformation matrix.</span>
+
+<span class="sd">    Examples:</span>
+<span class="sd">        &gt;&gt;&gt; input = torch.tensor([[ 0.,-0.5,  0., 0.1],</span>
+<span class="sd">        ...                       [0.5,  0.,  0., 0.2],</span>
+<span class="sd">        ...                       [ 0.,  0., 0.5, 0.3],</span>
+<span class="sd">        ...                       [ 0.,  0.,  0.,  1.]])</span>
+<span class="sd">        &gt;&gt;&gt; pp.mat2Sim3(input)</span>
+<span class="sd">        Sim3Type LieTensor:</span>
+<span class="sd">        tensor([0.1000, 0.2000, 0.3000, 0.0000, 0.0000, 0.7071, 0.7071, 0.5000])</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        We follow the convention below to express Sim3:</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \begin{bmatrix}</span>
+<span class="sd">                    s\mathbf{R}_{3\times3} &amp; \mathbf{t}_{3\times1}\\</span>
+<span class="sd">                    \textbf{0} &amp; 1</span>
+<span class="sd">            \end{bmatrix},</span>
+
+<span class="sd">        referred to this paper:</span>
+
+<span class="sd">        * J. Sola et al., `A micro Lie theory for state estimation in</span>
+<span class="sd">          robotics &lt;https://arxiv.org/abs/1812.01537&gt;`_, arXiv preprint arXiv:1812.01537 (2018),</span>
+
+<span class="sd">        where :math:`\mathbf{R}` is the individual matrix in a batch. The scaling factor</span>
+<span class="sd">        :math:`s` defines a linear transformation that enlarges or diminishes the object in the</span>
+<span class="sd">        same ratio across 3 dimensions, the translation vector :math:`\mathbf{t}` defines the</span>
+<span class="sd">        displacement between the original position and the transformed position.</span>
+
+<span class="sd">        We also notice that there is another popular convention:</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \begin{bmatrix}</span>
+<span class="sd">                    \mathbf{R}_{3\times3} &amp; \mathbf{t}_{3\times1}\\</span>
+<span class="sd">                    \textbf{0} &amp; 1/s</span>
+<span class="sd">            \end{bmatrix},</span>
+
+<span class="sd">        referred to this tutorial:</span>
+
+<span class="sd">        * `Lie Groups for 2D and 3D Transformations.</span>
+<span class="sd">          &lt;https://www.ethaneade.org/lie.pdf&gt;`_, by Ethan Eade.</span>
+
+<span class="sd">        Please make sure your own convention before using this function.</span>
+
+<span class="sd">    See :meth:`pypose.Sim3` for more details of the output LieTensor format.</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+    <span class="k">if</span> <span class="ow">not</span> <span class="n">torch</span><span class="o">.</span><span class="n">is_tensor</span><span class="p">(</span><span class="n">mat</span><span class="p">):</span>
+        <span class="n">mat</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">(</span><span class="n">mat</span><span class="p">)</span>
+
+    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">mat</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="o">&lt;</span> <span class="mi">2</span><span class="p">:</span>
+        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">&quot;Input size must be at least 2 dimensions. Got </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">mat</span><span class="o">.</span><span class="n">shape</span><span class="p">))</span>
+
+    <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">mat</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">:]</span> <span class="o">==</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span> <span class="ow">or</span> <span class="n">mat</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">:]</span> <span class="o">==</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span> <span class="ow">or</span> <span class="n">mat</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">:]</span> <span class="o">==</span> <span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">)):</span>
+        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">&quot;Input size must be a * x 3 x 3 or * x 3 x 4 or * x 4 x 4  tensor. </span><span class="se">\</span>
+<span class="s2">                Got </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">mat</span><span class="o">.</span><span class="n">shape</span><span class="p">))</span>
+
+    <span class="n">shape</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">shape</span>
+    <span class="k">if</span> <span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">:]</span> <span class="o">==</span> <span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span> <span class="ow">and</span> <span class="n">check</span> <span class="o">==</span> <span class="kc">True</span><span class="p">:</span>
+        <span class="n">zerosone</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">mat</span><span class="o">.</span><span class="n">dtype</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="n">mat</span><span class="o">.</span><span class="n">device</span><span class="p">)</span>
+        <span class="k">if</span> <span class="ow">not</span> <span class="n">torch</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">mat</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="p">:],</span> <span class="n">zerosone</span><span class="o">.</span><span class="n">expand_as</span><span class="p">(</span><span class="n">mat</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="p">:]),</span> <span class="n">rtol</span><span class="o">=</span><span class="n">rtol</span><span class="p">,</span> <span class="n">atol</span><span class="o">=</span><span class="n">atol</span><span class="p">):</span>
+            <span class="n">warnings</span><span class="o">.</span><span class="n">warn</span><span class="p">(</span><span class="s2">&quot;Input of shape 4x4 last rows are not all equal [0, 0, 0, 1]&quot;</span><span class="p">)</span>
+
+    <span class="n">shape</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">shape</span>
+    <span class="n">rot</span> <span class="o">=</span> <span class="n">mat</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="p">:</span><span class="mi">3</span><span class="p">,</span> <span class="p">:</span><span class="mi">3</span><span class="p">]</span>
+
+    <span class="n">s</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">pow</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">det</span><span class="p">(</span><span class="n">rot</span><span class="p">),</span> <span class="mi">1</span><span class="o">/</span><span class="mi">3</span><span class="p">)</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
+    <span class="n">zeros</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">shape</span><span class="p">[:</span><span class="o">-</span><span class="mi">2</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">mat</span><span class="o">.</span><span class="n">dtype</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="n">mat</span><span class="o">.</span><span class="n">device</span><span class="p">)</span>
+    <span class="k">if</span> <span class="n">torch</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">s</span><span class="p">,</span>  <span class="n">zeros</span><span class="p">,</span> <span class="n">rtol</span><span class="o">=</span><span class="n">rtol</span><span class="p">,</span> <span class="n">atol</span><span class="o">=</span><span class="n">atol</span><span class="p">):</span>
+        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">&quot;Rotation matrix not full rank.&quot;</span><span class="p">)</span>
+
+    <span class="n">q</span> <span class="o">=</span> <span class="n">mat2SO3</span><span class="p">(</span><span class="n">rot</span><span class="o">/</span><span class="n">s</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">),</span> <span class="n">check</span><span class="o">=</span><span class="n">check</span><span class="p">,</span> <span class="n">rtol</span><span class="o">=</span><span class="n">rtol</span><span class="p">,</span> <span class="n">atol</span><span class="o">=</span><span class="n">atol</span><span class="p">)</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span>
+
+    <span class="k">if</span> <span class="n">mat</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
+        <span class="n">t</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">mat</span><span class="o">.</span><span class="n">shape</span><span class="p">[:</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span><span class="o">+</span><span class="p">(</span><span class="mi">3</span><span class="p">,),</span> <span class="n">dtype</span><span class="o">=</span><span class="n">mat</span><span class="o">.</span><span class="n">dtype</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="n">mat</span><span class="o">.</span><span class="n">device</span><span class="p">,</span> <span class="n">requires_grad</span><span class="o">=</span><span class="n">mat</span><span class="o">.</span><span class="n">requires_grad</span><span class="p">)</span>
+    <span class="k">else</span><span class="p">:</span>
+        <span class="n">t</span> <span class="o">=</span> <span class="n">mat</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="p">:</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">]</span>
+
+    <span class="n">vec</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">cat</span><span class="p">([</span><span class="n">t</span><span class="p">,</span> <span class="n">q</span><span class="p">,</span> <span class="n">s</span><span class="p">],</span> <span class="n">dim</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span>
+
+    <span class="k">return</span> <span class="n">Sim3</span><span class="p">(</span><span class="n">vec</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="mat2RxSO3"><a class="viewcode-back" href="../../../../generated/pypose.mat2RxSO3/#pypose.mat2RxSO3">[docs]</a><span class="k">def</span> <span class="nf">mat2RxSO3</span><span class="p">(</span><span class="n">mat</span><span class="p">,</span> <span class="n">check</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">rtol</span><span class="o">=</span><span class="mf">1e-5</span><span class="p">,</span> <span class="n">atol</span><span class="o">=</span><span class="mf">1e-5</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&quot;&quot;&quot;Convert batched rotation or transformation matrices to RxSO3Type LieTensor.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        mat (Tensor): the batched matrices to convert. If input is of shape :obj:`(*, 3, 4)`</span>
+<span class="sd">            or :obj:`(*, 4, 4)`, only the top left 3x3 submatrix is used.</span>
+<span class="sd">        check (bool, optional): flag to check if the input is valid rotation matrices (orthogonal</span>
+<span class="sd">            and with a determinant of one). Set to ``False`` if less computation is needed.</span>
+<span class="sd">            Default: ``True``.</span>
+<span class="sd">        rtol (float, optional): relative tolerance when check is enabled. Default: 1e-05</span>
+<span class="sd">        atol (float, optional): absolute tolerance when check is enabled. Default: 1e-05</span>
+
+<span class="sd">    Return:</span>
+<span class="sd">        LieTensor: the converted RxSO3Type LieTensor.</span>
+
+<span class="sd">    Shape:</span>
+<span class="sd">        Input: :obj:`(*, 3, 3)` or :obj:`(*, 3, 4)` or :obj:`(*, 4, 4)`</span>
+
+<span class="sd">        Output: :obj:`(*, 5)`</span>
+
+<span class="sd">    Let the input be matrix :math:`\mathbf{T}`, :math:`\mathbf{T}_i` represents each individual</span>
+<span class="sd">    matrix in the batch. :math:`\mathbf{T}^{m,n}_i` represents the :math:`m^{\mathrm{th}}` row and</span>
+<span class="sd">    :math:`n^{\mathrm{th}}` column of :math:`\mathbf{T}_i`, :math:`m,n\geq 1`, then the scaling factor</span>
+<span class="sd">    :math:`s_i\in\mathbb{R}` and the rotation matrix :math:`\mathbf{R}_i\in\mathbb{R}^{3\times 3}`</span>
+<span class="sd">    can be computed as:</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \begin{aligned}</span>
+<span class="sd">            s_i &amp;= \sqrt[3]{\vert \mathbf{T_i} \vert}\\</span>
+<span class="sd">            \mathbf{R}_i &amp;= \mathbf{R}_i/s_i</span>
+<span class="sd">        \end{aligned},</span>
+
+<span class="sd">    the translation and quaternion can be computed by:</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \left\{\begin{aligned}</span>
+<span class="sd">        q^x_i &amp;= \mathrm{sign}(\mathbf{R}^{2,3}_i - \mathbf{R}^{3,2}_i) \frac{1}{2}</span>
+<span class="sd">            \sqrt{1 + \mathbf{R}^{1,1}_i - \mathbf{R}^{2,2}_i - \mathbf{R}^{3,3}_i}\\</span>
+<span class="sd">        q^y_i &amp;= \mathrm{sign}(\mathbf{R}^{3,1}_i - \mathbf{R}^{1,3}_i) \frac{1}{2}</span>
+<span class="sd">            \sqrt{1 - \mathbf{R}^{1,1}_i + \mathbf{R}^{2,2}_i - \mathbf{R}^{3,3}_i}\\</span>
+<span class="sd">        q^z_i &amp;= \mathrm{sign}(\mathbf{R}^{1,2}_i - \mathbf{R}^{2,1}_i) \frac{1}{2}</span>
+<span class="sd">            \sqrt{1 - \mathbf{R}^{1,1}_i - \mathbf{R}^{2,2}_i + \mathbf{R}^{3,3}_i}\\</span>
+<span class="sd">        q^w_i &amp;= \frac{1}{2} \sqrt{1 + \mathbf{R}^{1,1}_i + \mathbf{R}^{2,2}_i + \mathbf{R}^{3,3}_i}</span>
+<span class="sd">        \end{aligned}\right.,</span>
+
+<span class="sd">    In summary, the output LieTensor should be of format:</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \textbf{y}_i = [q^x_i, q^y_i, q^z_i, q^w_i, s_i]</span>
+
+<span class="sd">    Warning:</span>
+<span class="sd">        Numerically, a transformation matrix is considered legal if:</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \vert s \vert &gt; \texttt{atol} \\</span>
+<span class="sd">            |{\rm det}(\mathbf{R}) - 1| \leq \texttt{atol} + \texttt{rtol}\times 1\\</span>
+<span class="sd">            |\mathbf{RR}^{T} - \mathbf{I}| \leq \texttt{atol} + \texttt{rtol}\times \mathbf{I}</span>
+
+<span class="sd">        where :math:`|\cdot |` is element-wise absolute function. When ``check`` is set to ``True``,</span>
+<span class="sd">        illegal input will raise a ``ValueError``. Otherwise, no data validation is performed.</span>
+<span class="sd">        Illegal input will output an irrelevant result, which likely contains ``nan``.</span>
+
+<span class="sd">    Examples:</span>
+<span class="sd">        &gt;&gt;&gt; input = torch.tensor([[ 0., -0.5,  0.],</span>
+<span class="sd">        ...                       [0.5,   0.,  0.],</span>
+<span class="sd">        ...                       [ 0.,   0., 0.5]])</span>
+<span class="sd">        &gt;&gt;&gt; pp.mat2RxSO3(input)</span>
+<span class="sd">        RxSO3Type LieTensor:</span>
+<span class="sd">        tensor([0.0000, 0.0000, 0.7071, 0.7071, 0.5000])</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        The individual matrix in a batch can be written as: :math:`s\mathbf{R}_{3\times3}`,</span>
+<span class="sd">        where :math:`\mathbf{R}` is the rotation matrix. where  the scaling factor</span>
+<span class="sd">        :math:`s` defines a linear transformation that enlarges or diminishes the object</span>
+<span class="sd">        in the same ratio across 3 dimensions.</span>
+
+<span class="sd">    See :meth:`pypose.RxSO3` for more details of the output LieTensor format.</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+    <span class="k">if</span> <span class="ow">not</span> <span class="n">torch</span><span class="o">.</span><span class="n">is_tensor</span><span class="p">(</span><span class="n">mat</span><span class="p">):</span>
+        <span class="n">mat</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">(</span><span class="n">mat</span><span class="p">)</span>
+
+    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">mat</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="o">&lt;</span> <span class="mi">2</span><span class="p">:</span>
+        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">&quot;Input size must be at least 2 dimensions. Got </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">mat</span><span class="o">.</span><span class="n">shape</span><span class="p">))</span>
+
+    <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">mat</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">:]</span> <span class="o">==</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span> <span class="ow">or</span> <span class="n">mat</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">:]</span> <span class="o">==</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span> <span class="ow">or</span> <span class="n">mat</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">:]</span> <span class="o">==</span> <span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">)):</span>
+        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">&quot;Input size must be a * x 3 x 3 or * x 3 x 4 or * x 4 x 4  tensor. </span><span class="se">\</span>
+<span class="s2">                Got </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">mat</span><span class="o">.</span><span class="n">shape</span><span class="p">))</span>
+
+    <span class="n">shape</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">shape</span>
+    <span class="n">rot</span> <span class="o">=</span> <span class="n">mat</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="p">:</span><span class="mi">3</span><span class="p">,</span> <span class="p">:</span><span class="mi">3</span><span class="p">]</span>
+
+    <span class="n">s</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">pow</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">det</span><span class="p">(</span><span class="n">rot</span><span class="p">),</span> <span class="mi">1</span><span class="o">/</span><span class="mi">3</span><span class="p">)</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
+    <span class="k">if</span> <span class="n">torch</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">s</span><span class="p">,</span>  <span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">shape</span><span class="p">[:</span><span class="o">-</span><span class="mi">2</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">mat</span><span class="o">.</span><span class="n">dtype</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="n">mat</span><span class="o">.</span><span class="n">device</span><span class="p">),</span> <span class="n">rtol</span><span class="o">=</span><span class="n">rtol</span><span class="p">,</span> <span class="n">atol</span><span class="o">=</span><span class="n">atol</span><span class="p">):</span>
+        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">&quot;Rotation matrix not full rank.&quot;</span><span class="p">)</span>
+
+    <span class="n">q</span> <span class="o">=</span> <span class="n">mat2SO3</span><span class="p">(</span><span class="n">rot</span><span class="o">/</span><span class="n">s</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">),</span> <span class="n">check</span><span class="o">=</span><span class="n">check</span><span class="p">,</span> <span class="n">rtol</span><span class="o">=</span><span class="n">rtol</span><span class="p">,</span> <span class="n">atol</span><span class="o">=</span><span class="n">atol</span><span class="p">)</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span>
+    <span class="n">vec</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">cat</span><span class="p">([</span><span class="n">q</span><span class="p">,</span> <span class="n">s</span><span class="p">],</span> <span class="n">dim</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span>
+
+    <span class="k">return</span> <span class="n">RxSO3</span><span class="p">(</span><span class="n">vec</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="from_matrix"><a class="viewcode-back" href="../../../../generated/pypose.from_matrix/#pypose.from_matrix">[docs]</a><span class="k">def</span> <span class="nf">from_matrix</span><span class="p">(</span><span class="n">mat</span><span class="p">,</span> <span class="n">ltype</span><span class="p">,</span> <span class="n">check</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">rtol</span><span class="o">=</span><span class="mf">1e-5</span><span class="p">,</span> <span class="n">atol</span><span class="o">=</span><span class="mf">1e-5</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&quot;&quot;&quot;Convert batched rotation or transformation matrices to LieTensor.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        mat (Tensor): the matrix to convert.</span>
+<span class="sd">        ltype (ltype): specify the LieTensor type, chosen from :class:`pypose.SO3_type`,</span>
+<span class="sd">            :class:`pypose.SE3_type`, :class:`pypose.Sim3_type`, or :class:`pypose.RxSO3_type`.</span>
+<span class="sd">            See more details in :meth:`LieTensor`</span>
+<span class="sd">        check (bool, optional): flag to check if the input is valid rotation matrices (orthogonal</span>
+<span class="sd">            and with a determinant of one). Set to ``False`` if less computation is needed.</span>
+<span class="sd">            Default: ``True``.</span>
+<span class="sd">        rtol (float, optional): relative tolerance when check is enabled. Default: 1e-05</span>
+<span class="sd">        atol (float, optional): absolute tolerance when check is enabled. Default: 1e-05</span>
+
+<span class="sd">    Warning:</span>
+<span class="sd">        Numerically, a transformation matrix is considered legal if:</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            |{\rm det}(\mathbf{R}) - 1| \leq \texttt{atol} + \texttt{rtol}\times 1\\</span>
+<span class="sd">            |\mathbf{RR}^{T} - \mathbf{I}| \leq \texttt{atol} + \texttt{rtol}\times \mathbf{I}</span>
+
+<span class="sd">        where :math:`|\cdot |` is element-wise absolute function. When ``check`` is set to ``True``,</span>
+<span class="sd">        illegal input will raise a ``ValueError``. Otherwise, no data validation is performed.</span>
+<span class="sd">        Illegal input will output an irrelevant result, which likely contains ``nan``.</span>
+
+<span class="sd">    Return:</span>
+<span class="sd">        LieTensor: the converted LieTensor.</span>
+<span class="sd">    Examples:</span>
+
+<span class="sd">        - :class:`pypose.SO3_type`</span>
+
+<span class="sd">        &gt;&gt;&gt; pp.from_matrix(torch.tensor([[0., -1., 0.],</span>
+<span class="sd">        ...                              [1.,  0., 0.],</span>
+<span class="sd">        ...                              [0.,  0., 1.]]), ltype=pp.SO3_type)</span>
+<span class="sd">        SO3Type LieTensor:</span>
+<span class="sd">        tensor([0.0000, 0.0000, 0.7071, 0.7071])</span>
+
+<span class="sd">        - :class:`pypose.SE3_type`</span>
+
+<span class="sd">        &gt;&gt;&gt; pp.from_matrix(torch.tensor([[0., -1., 0., 0.1],</span>
+<span class="sd">        ...                              [1.,  0., 0., 0.2],</span>
+<span class="sd">        ...                              [0.,  0., 1., 0.3],</span>
+<span class="sd">        ...                              [0.,  0., 0.,  1.]]), ltype=pp.SE3_type)</span>
+<span class="sd">        SE3Type LieTensor:</span>
+<span class="sd">        tensor([0.1000, 0.2000, 0.3000, 0.0000, 0.0000, 0.7071, 0.7071])</span>
+
+<span class="sd">        - :class:`pypose.Sim3_type`</span>
+
+<span class="sd">        &gt;&gt;&gt; pp.from_matrix(torch.tensor([[ 0.,-0.5,  0., 0.1],</span>
+<span class="sd">        ...                              [0.5,  0.,  0., 0.2],</span>
+<span class="sd">        ...                              [ 0.,  0., 0.5, 0.3],</span>
+<span class="sd">        ...                              [ 0.,  0.,  0.,  1.]]), ltype=pp.Sim3_type)</span>
+<span class="sd">        Sim3Type LieTensor:</span>
+<span class="sd">        tensor([0.1000, 0.2000, 0.3000, 0.0000, 0.0000, 0.7071, 0.7071, 0.5000])</span>
+
+<span class="sd">        - :class:`pypose.RxSO3_type`</span>
+
+<span class="sd">        &gt;&gt;&gt; pp.from_matrix(torch.tensor([[0., -0.5, 0.],</span>
+<span class="sd">        ...                              [0.5, 0.,  0.],</span>
+<span class="sd">        ...                              [0.,  0., 0.5]]), ltype=pp.RxSO3_type)</span>
+<span class="sd">        RxSO3Type LieTensor:</span>
+<span class="sd">        tensor([0.0000, 0.0000, 0.7071, 0.7071, 0.5000])</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+    <span class="k">if</span> <span class="ow">not</span> <span class="n">torch</span><span class="o">.</span><span class="n">is_tensor</span><span class="p">(</span><span class="n">mat</span><span class="p">):</span>
+        <span class="n">mat</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">(</span><span class="n">mat</span><span class="p">)</span>
+
+    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">mat</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="o">&lt;</span> <span class="mi">2</span><span class="p">:</span>
+        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">&quot;Input size must be at least 2 dimensions. Got </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">mat</span><span class="o">.</span><span class="n">shape</span><span class="p">))</span>
+
+    <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">mat</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">:]</span> <span class="o">==</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span> <span class="ow">or</span> <span class="n">mat</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">:]</span> <span class="o">==</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span> <span class="ow">or</span> <span class="n">mat</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">:]</span> <span class="o">==</span> <span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">)):</span>
+        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">&quot;Input size must be a * x 3 x 3 or * x 3 x 4 or * x 4 x 4  tensor. </span><span class="se">\</span>
+<span class="s2">                Got </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">mat</span><span class="o">.</span><span class="n">shape</span><span class="p">))</span>
+
+    <span class="k">if</span> <span class="n">ltype</span> <span class="o">==</span> <span class="n">SO3_type</span><span class="p">:</span>
+        <span class="k">return</span> <span class="n">mat2SO3</span><span class="p">(</span><span class="n">mat</span><span class="p">,</span> <span class="n">check</span><span class="o">=</span><span class="n">check</span><span class="p">,</span> <span class="n">rtol</span><span class="o">=</span><span class="n">rtol</span><span class="p">,</span> <span class="n">atol</span><span class="o">=</span><span class="n">atol</span><span class="p">)</span>
+    <span class="k">elif</span> <span class="n">ltype</span> <span class="o">==</span> <span class="n">SE3_type</span><span class="p">:</span>
+        <span class="k">return</span> <span class="n">mat2SE3</span><span class="p">(</span><span class="n">mat</span><span class="p">,</span> <span class="n">check</span><span class="o">=</span><span class="n">check</span><span class="p">,</span> <span class="n">rtol</span><span class="o">=</span><span class="n">rtol</span><span class="p">,</span> <span class="n">atol</span><span class="o">=</span><span class="n">atol</span><span class="p">)</span>
+    <span class="k">elif</span> <span class="n">ltype</span> <span class="o">==</span> <span class="n">Sim3_type</span><span class="p">:</span>
+        <span class="k">return</span> <span class="n">mat2Sim3</span><span class="p">(</span><span class="n">mat</span><span class="p">,</span> <span class="n">check</span><span class="o">=</span><span class="n">check</span><span class="p">,</span> <span class="n">rtol</span><span class="o">=</span><span class="n">rtol</span><span class="p">,</span> <span class="n">atol</span><span class="o">=</span><span class="n">atol</span><span class="p">)</span>
+    <span class="k">elif</span> <span class="n">ltype</span> <span class="o">==</span> <span class="n">RxSO3_type</span><span class="p">:</span>
+        <span class="k">return</span> <span class="n">mat2RxSO3</span><span class="p">(</span><span class="n">mat</span><span class="p">,</span> <span class="n">check</span><span class="o">=</span><span class="n">check</span><span class="p">,</span> <span class="n">rtol</span><span class="o">=</span><span class="n">rtol</span><span class="p">,</span> <span class="n">atol</span><span class="o">=</span><span class="n">atol</span><span class="p">)</span>
+    <span class="k">else</span><span class="p">:</span>
+        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">&quot;Input ltype must be one of SO3_type, SE3_type, Sim3_type or RxSO3_type.</span><span class="se">\</span>
+<span class="s2">                Got </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">ltype</span><span class="p">))</span></div>
+
+
+<span class="k">def</span> <span class="nf">matrix</span><span class="p">(</span><span class="n">lietensor</span><span class="p">):</span>
+    <span class="k">assert</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">lietensor</span><span class="p">,</span> <span class="n">LieTensor</span><span class="p">)</span>
+    <span class="k">return</span> <span class="n">lietensor</span><span class="o">.</span><span class="n">matrix</span><span class="p">()</span>
+
+
+<div class="viewcode-block" id="euler2SO3"><a class="viewcode-back" href="../../../../generated/pypose.euler2SO3/#pypose.euler2SO3">[docs]</a><span class="k">def</span> <span class="nf">euler2SO3</span><span class="p">(</span><span class="n">euler</span><span class="p">:</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&quot;&quot;&quot;Convert batched Euler angles (roll, pitch, and yaw) to SO3Type LieTensor.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        euler (Tensor): the euler angles in radians to convert.</span>
+
+<span class="sd">    Return:</span>
+<span class="sd">        LieTensor: the converted SO3Type LieTensor.</span>
+
+<span class="sd">    Shape:</span>
+<span class="sd">        Input: :obj:`(*, 3)`</span>
+
+<span class="sd">        Output: :obj:`(*, 4)`</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        {\displaystyle \mathbf{y}_i={</span>
+<span class="sd">        \begin{bmatrix}\,</span>
+<span class="sd">        \sin(\alpha_i)\cos(\beta_i)\cos(\gamma_i) - \cos(\alpha_i)\sin(\beta_i)\sin(\gamma_i)\\</span>
+<span class="sd">        \cos(\alpha_i)\sin(\beta_i)\cos(\gamma_i) + \sin(\alpha_i)\cos(\beta_i)\sin(\gamma_i)\\</span>
+<span class="sd">        \cos(\alpha_i)\cos(\beta_i)\sin(\gamma_i) - \sin(\alpha_i)\sin(\beta_i)\cos(\gamma_i)\\</span>
+<span class="sd">        \cos(\alpha_i)\cos(\beta_i)\cos(\gamma_i) + \sin(\alpha_i)\sin(\beta_i)\sin(\gamma_i)</span>
+<span class="sd">        \end{bmatrix}}},</span>
+
+<span class="sd">    where the :math:`i`-th item of input :math:`\mathbf{x}_i = [\alpha_i, \beta_i, \gamma_i]`</span>
+<span class="sd">    are roll, pitch, and yaw, respectively.</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        The last dimension of the input tensor has to be 3. The Euler angle takes the rotation</span>
+<span class="sd">        sequence of x (roll), y (pitch), then z (yaw) axis (counterclockwise).</span>
+
+<span class="sd">    Warning:</span>
+<span class="sd">        Any given rotation has two possible quaternion representations. If one is known, the other</span>
+<span class="sd">        is just the negative of all four terms. This function only returns one of them.</span>
+
+<span class="sd">    Examples:</span>
+<span class="sd">        &gt;&gt;&gt; input = torch.randn(2, 3, requires_grad=True, dtype=torch.float64)</span>
+<span class="sd">        &gt;&gt;&gt; pp.euler2SO3(input)</span>
+<span class="sd">        SO3Type LieTensor:</span>
+<span class="sd">        tensor([[-0.4873,  0.1162,  0.4829,  0.7182],</span>
+<span class="sd">                [ 0.3813,  0.4059, -0.2966,  0.7758]], dtype=torch.float64, grad_fn=&lt;AliasBackward0&gt;)</span>
+
+<span class="sd">    See :obj:`euler` for more information.</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+    <span class="k">if</span> <span class="ow">not</span> <span class="n">torch</span><span class="o">.</span><span class="n">is_tensor</span><span class="p">(</span><span class="n">euler</span><span class="p">):</span>
+        <span class="n">euler</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">(</span><span class="n">euler</span><span class="p">)</span>
+    <span class="k">assert</span> <span class="n">euler</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="mi">3</span>
+    <span class="n">shape</span><span class="p">,</span> <span class="n">euler</span> <span class="o">=</span> <span class="n">euler</span><span class="o">.</span><span class="n">shape</span><span class="p">,</span> <span class="n">euler</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
+    <span class="n">roll</span><span class="p">,</span> <span class="n">pitch</span><span class="p">,</span> <span class="n">yaw</span> <span class="o">=</span> <span class="n">euler</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">euler</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">euler</span><span class="p">[:,</span> <span class="mi">2</span><span class="p">]</span>
+    <span class="n">cy</span><span class="p">,</span> <span class="n">sy</span> <span class="o">=</span> <span class="p">(</span><span class="n">yaw</span> <span class="o">*</span> <span class="mf">0.5</span><span class="p">)</span><span class="o">.</span><span class="n">cos</span><span class="p">(),</span> <span class="p">(</span><span class="n">yaw</span> <span class="o">*</span> <span class="mf">0.5</span><span class="p">)</span><span class="o">.</span><span class="n">sin</span><span class="p">()</span>
+    <span class="n">cp</span><span class="p">,</span> <span class="n">sp</span> <span class="o">=</span> <span class="p">(</span><span class="n">pitch</span> <span class="o">*</span> <span class="mf">0.5</span><span class="p">)</span><span class="o">.</span><span class="n">cos</span><span class="p">(),</span> <span class="p">(</span><span class="n">pitch</span> <span class="o">*</span> <span class="mf">0.5</span><span class="p">)</span><span class="o">.</span><span class="n">sin</span><span class="p">()</span>
+    <span class="n">cr</span><span class="p">,</span> <span class="n">sr</span> <span class="o">=</span> <span class="p">(</span><span class="n">roll</span> <span class="o">*</span> <span class="mf">0.5</span><span class="p">)</span><span class="o">.</span><span class="n">cos</span><span class="p">(),</span> <span class="p">(</span><span class="n">roll</span> <span class="o">*</span> <span class="mf">0.5</span><span class="p">)</span><span class="o">.</span><span class="n">sin</span><span class="p">()</span>
+
+    <span class="n">q</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">stack</span><span class="p">([</span><span class="n">sr</span> <span class="o">*</span> <span class="n">cp</span> <span class="o">*</span> <span class="n">cy</span> <span class="o">-</span> <span class="n">cr</span> <span class="o">*</span> <span class="n">sp</span> <span class="o">*</span> <span class="n">sy</span><span class="p">,</span>
+                     <span class="n">cr</span> <span class="o">*</span> <span class="n">sp</span> <span class="o">*</span> <span class="n">cy</span> <span class="o">+</span> <span class="n">sr</span> <span class="o">*</span> <span class="n">cp</span> <span class="o">*</span> <span class="n">sy</span><span class="p">,</span>
+                     <span class="n">cr</span> <span class="o">*</span> <span class="n">cp</span> <span class="o">*</span> <span class="n">sy</span> <span class="o">-</span> <span class="n">sr</span> <span class="o">*</span> <span class="n">sp</span> <span class="o">*</span> <span class="n">cy</span><span class="p">,</span>
+                     <span class="n">cr</span> <span class="o">*</span> <span class="n">cp</span> <span class="o">*</span> <span class="n">cy</span> <span class="o">+</span> <span class="n">sr</span> <span class="o">*</span> <span class="n">sp</span> <span class="o">*</span> <span class="n">sy</span><span class="p">],</span> <span class="n">dim</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span>
+    <span class="k">return</span> <span class="n">SO3</span><span class="p">(</span><span class="n">q</span><span class="p">)</span><span class="o">.</span><span class="n">lview</span><span class="p">(</span><span class="o">*</span><span class="n">shape</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span></div>
+
+
+<div class="viewcode-block" id="tensor"><a class="viewcode-back" href="../../../../generated/pypose.tensor/#pypose.tensor">[docs]</a><span class="k">def</span> <span class="nf">tensor</span><span class="p">(</span><span class="n">inputs</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Convert a :obj:`LieTensor` into a :obj:`torch.Tensor` without changing data.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        inputs (:obj:`LieTensor`): the input LieTensor.</span>
+
+<span class="sd">    Return:</span>
+<span class="sd">        Tensor: the torch.Tensor form of LieTensor.</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        &gt;&gt;&gt; x = pp.randn_SO3(2)</span>
+<span class="sd">        &gt;&gt;&gt; x.tensor()</span>
+<span class="sd">        tensor([[ 0.1196,  0.2339, -0.6824,  0.6822],</span>
+<span class="sd">                [ 0.9198, -0.2704, -0.2395,  0.1532]])</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">return</span> <span class="n">inputs</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span></div>
+
+
+<div class="viewcode-block" id="translation"><a class="viewcode-back" href="../../../../generated/pypose.translation/#pypose.translation">[docs]</a><span class="k">def</span> <span class="nf">translation</span><span class="p">(</span><span class="n">inputs</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Extract the translation part from a :obj:`LieTensor`.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        inputs (:obj:`LieTensor`): the input LieTensor.</span>
+
+<span class="sd">    Return:</span>
+<span class="sd">        Tensor: the batched translation vectors.</span>
+
+<span class="sd">    Warning:</span>
+<span class="sd">        The :obj:`SO3`, :obj:`so3`, :obj:`RxSO3`, and :obj:`rxso3` types do not contain</span>
+<span class="sd">        translation. Calling :obj:`translation()` on these types will return zero vector(s).</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        &gt;&gt;&gt; x = pp.randn_SE3(2)</span>
+<span class="sd">        &gt;&gt;&gt; x.translation()</span>
+<span class="sd">        tensor([[-0.5358, -1.5421, -0.7224],</span>
+<span class="sd">                [ 0.8331, -1.4412,  0.0863]])</span>
+<span class="sd">        &gt;&gt;&gt; y = pp.randn_SO3(2)</span>
+<span class="sd">        &gt;&gt;&gt; y.translation()</span>
+<span class="sd">        tensor([[0., 0., 0.],</span>
+<span class="sd">                [0., 0., 0.]])</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">return</span> <span class="n">inputs</span><span class="o">.</span><span class="n">translation</span><span class="p">()</span></div>
+
+
+<div class="viewcode-block" id="rotation"><a class="viewcode-back" href="../../../../generated/pypose.rotation/#pypose.rotation">[docs]</a><span class="k">def</span> <span class="nf">rotation</span><span class="p">(</span><span class="n">inputs</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Extract the rotation part from a :obj:`LieTensor`.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        inputs (:obj:`LieTensor`): the input LieTensor.</span>
+
+<span class="sd">    Return:</span>
+<span class="sd">        SO3: the batched quaternions.</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        &gt;&gt;&gt; x = pp.randn_SE3(2)</span>
+<span class="sd">        &gt;&gt;&gt; x.rotation()</span>
+<span class="sd">        SO3Type LieTensor:</span>
+<span class="sd">        tensor([[-0.8302,  0.5200, -0.0056,  0.2006],</span>
+<span class="sd">                [-0.2541, -0.3184,  0.6305,  0.6607]])</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">return</span> <span class="n">inputs</span><span class="o">.</span><span class="n">rotation</span><span class="p">()</span></div>
+
+
+<div class="viewcode-block" id="scale"><a class="viewcode-back" href="../../../../generated/pypose.scale/#pypose.scale">[docs]</a><span class="k">def</span> <span class="nf">scale</span><span class="p">(</span><span class="n">inputs</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Extract the scale part from a :obj:`LieTensor`.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        inputs (:obj:`LieTensor`): the input LieTensor.</span>
+
+<span class="sd">    Return:</span>
+<span class="sd">        Tensor: the batched scale scalars.</span>
+
+<span class="sd">    Warning:</span>
+<span class="sd">        The :obj:`SO3`, :obj:`so3`, :obj:`SE3`, and :obj:`se3` types do not contain scale.</span>
+<span class="sd">        Calling :obj:`scale()` on these types will return one(s).</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        &gt;&gt;&gt; x = pp.randn_Sim3(4)</span>
+<span class="sd">        &gt;&gt;&gt; x.scale()</span>
+<span class="sd">        tensor([[10.9577],</span>
+<span class="sd">                [ 1.0248],</span>
+<span class="sd">                [ 0.0947],</span>
+<span class="sd">                [ 1.1989]])</span>
+<span class="sd">        &gt;&gt;&gt; y = pp.randn_SE3(2)</span>
+<span class="sd">        &gt;&gt;&gt; y.scale()</span>
+<span class="sd">        tensor([[1.],</span>
+<span class="sd">                [1.]])</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+    <span class="k">return</span> <span class="n">inputs</span><span class="o">.</span><span class="n">scale</span><span class="p">()</span></div>
+
+
+<div class="viewcode-block" id="matrix"><a class="viewcode-back" href="../../../../generated/pypose.matrix/#pypose.matrix">[docs]</a><span class="k">def</span> <span class="nf">matrix</span><span class="p">(</span><span class="n">inputs</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Convert a :obj:`LieTensor` into matrix form.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        inputs (:obj:`LieTensor`): the input LieTensor.</span>
+
+<span class="sd">    Return:</span>
+<span class="sd">        Tensor: the batched matrix form (torch.Tensor) of LieTensor.</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        &gt;&gt;&gt; x = pp.randn_SO3(2)</span>
+<span class="sd">        &gt;&gt;&gt; x.matrix()</span>
+<span class="sd">        tensor([[[ 0.9285, -0.0040, -0.3713],</span>
+<span class="sd">                 [ 0.2503,  0.7454,  0.6178],</span>
+<span class="sd">                 [ 0.2743, -0.6666,  0.6931]],</span>
+<span class="sd">                [[ 0.4805,  0.8602, -0.1706],</span>
+<span class="sd">                 [-0.7465,  0.2991, -0.5944],</span>
+<span class="sd">                 [-0.4603,  0.4130,  0.7858]]])</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">return</span> <span class="n">inputs</span><span class="o">.</span><span class="n">matrix</span><span class="p">()</span></div>
+
+
+<div class="viewcode-block" id="euler"><a class="viewcode-back" href="../../../../generated/pypose.euler/#pypose.euler">[docs]</a><span class="k">def</span> <span class="nf">euler</span><span class="p">(</span><span class="n">inputs</span><span class="p">,</span> <span class="n">eps</span><span class="o">=</span><span class="mf">2e-4</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Convert batched LieTensor into Euler angles (roll, pitch, yaw).</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        inputs (:obj:`LieTensor`): the input LieTensor.</span>
+<span class="sd">        eps (:obj:`float`, optional): the threshold to avoid the sigularity caused</span>
+<span class="sd">            by gimbal lock. Default: 2e-4.</span>
+
+<span class="sd">    Return:</span>
+<span class="sd">        :obj:`Tensor`: the batched Euler angles in radians.</span>
+
+<span class="sd">    Supported input type: :obj:`so3`, :obj:`SO3`, :obj:`se3`, :obj:`SE3`,</span>
+<span class="sd">    :obj:`sim3`, :obj:`Sim3`, :obj:`rxso3`, and :obj:`RxSO3`.</span>
+
+<span class="sd">    Warning:</span>
+<span class="sd">        The Euler angle takes the rotation sequence of x (roll), y (pitch),</span>
+<span class="sd">        and z (yaw) axis (counterclockwise). There is always more than one</span>
+<span class="sd">        solution that can result in the same orientation, while this function</span>
+<span class="sd">        only returns one of them.</span>
+
+<span class="sd">        When the pitch angle is around :math:`\pm \frac{\pi}{2}` (north/south pole),</span>
+<span class="sd">        there will be sigularity problem due to the `gimbal lock</span>
+<span class="sd">        &lt;https://en.wikipedia.org/wiki/Gimbal_lock&gt;`_ and some information can be</span>
+<span class="sd">        found in `this pape &lt;https://tinyurl.com/py8frs6v&gt;`_.</span>
+
+<span class="sd">    Example:</span>
+
+<span class="sd">        &gt;&gt;&gt; x = pp.randn_SO3()</span>
+<span class="sd">        &gt;&gt;&gt; x.euler()  # equivalent to pp.euler(x)</span>
+<span class="sd">        tensor([-0.6599,  0.2749, -0.3263])</span>
+
+<span class="sd">        &gt;&gt;&gt; x = pp.randn_Sim3(2)</span>
+<span class="sd">        &gt;&gt;&gt; x.euler()  # equivalent to pp.euler(x)</span>
+<span class="sd">        tensor([[-0.2701, -0.8006, -0.4150],</span>
+<span class="sd">                [ 2.1550,  0.1768,  0.9368]])</span>
+
+<span class="sd">        &gt;&gt;&gt; x = pp.randn_rxso3()</span>
+<span class="sd">        &gt;&gt;&gt; x.euler()  # equivalent to pp.euler(x)</span>
+<span class="sd">        tensor([ 1.2676, -0.4783, -0.3596])</span>
+
+<span class="sd">    See :obj:`euler2SO3` for more information.</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">return</span> <span class="n">inputs</span><span class="o">.</span><span class="n">euler</span><span class="p">(</span><span class="n">eps</span><span class="o">=</span><span class="n">eps</span><span class="p">)</span></div>
+</pre></div>
+
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+            
+            
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+            
+            
+              <p class="caption" role="heading"><span class="caption-text">Contents:</span></p>
+<ul>
+<li class="toctree-l1"><a class="reference internal" href="../../../../lietensor/">LieTensor</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../../../func/">Func</a></li>
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+              
+  <h1>Source code for pypose.lietensor.lietensor</h1><div class="highlight"><pre>
+<span></span><span class="kn">import</span> <span class="nn">torch</span>
+<span class="kn">from</span> <span class="nn">torch</span> <span class="kn">import</span> <span class="n">nn</span>
+<span class="kn">from</span> <span class="nn">.basics</span> <span class="kn">import</span> <span class="n">vec2skew</span>
+<span class="kn">from</span> <span class="nn">contextlib</span> <span class="kn">import</span> <span class="n">contextmanager</span>
+<span class="kn">from</span> <span class="nn">.operation</span> <span class="kn">import</span> <span class="n">broadcast_inputs</span>
+<span class="kn">import</span> <span class="nn">collections</span><span class="o">,</span> <span class="nn">numbers</span><span class="o">,</span> <span class="nn">warnings</span><span class="o">,</span> <span class="nn">importlib</span>
+<span class="kn">from</span> <span class="nn">torch.utils._pytree</span> <span class="kn">import</span> <span class="n">tree_map</span><span class="p">,</span> <span class="n">tree_flatten</span>
+<span class="kn">from</span> <span class="nn">.operation</span> <span class="kn">import</span> <span class="n">SO3_Log</span><span class="p">,</span> <span class="n">SE3_Log</span><span class="p">,</span> <span class="n">RxSO3_Log</span><span class="p">,</span> <span class="n">Sim3_Log</span>
+<span class="kn">from</span> <span class="nn">.operation</span> <span class="kn">import</span> <span class="n">so3_Exp</span><span class="p">,</span> <span class="n">se3_Exp</span><span class="p">,</span> <span class="n">rxso3_Exp</span><span class="p">,</span> <span class="n">sim3_Exp</span>
+<span class="kn">from</span> <span class="nn">.operation</span> <span class="kn">import</span> <span class="n">SO3_Act</span><span class="p">,</span> <span class="n">SE3_Act</span><span class="p">,</span> <span class="n">RxSO3_Act</span><span class="p">,</span> <span class="n">Sim3_Act</span>
+<span class="kn">from</span> <span class="nn">.operation</span> <span class="kn">import</span> <span class="n">SO3_Mul</span><span class="p">,</span> <span class="n">SE3_Mul</span><span class="p">,</span> <span class="n">RxSO3_Mul</span><span class="p">,</span> <span class="n">Sim3_Mul</span>
+<span class="kn">from</span> <span class="nn">.operation</span> <span class="kn">import</span> <span class="n">SO3_Inv</span><span class="p">,</span> <span class="n">SE3_Inv</span><span class="p">,</span> <span class="n">RxSO3_Inv</span><span class="p">,</span> <span class="n">Sim3_Inv</span>
+<span class="kn">from</span> <span class="nn">.operation</span> <span class="kn">import</span> <span class="n">SO3_Act4</span><span class="p">,</span> <span class="n">SE3_Act4</span><span class="p">,</span> <span class="n">RxSO3_Act4</span><span class="p">,</span> <span class="n">Sim3_Act4</span>
+<span class="kn">from</span> <span class="nn">.operation</span> <span class="kn">import</span> <span class="n">SO3_AdjXa</span><span class="p">,</span> <span class="n">SE3_AdjXa</span><span class="p">,</span> <span class="n">RxSO3_AdjXa</span><span class="p">,</span> <span class="n">Sim3_AdjXa</span>
+<span class="kn">from</span> <span class="nn">.operation</span> <span class="kn">import</span> <span class="n">SO3_AdjTXa</span><span class="p">,</span> <span class="n">SE3_AdjTXa</span><span class="p">,</span> <span class="n">RxSO3_AdjTXa</span><span class="p">,</span> <span class="n">Sim3_AdjTXa</span>
+<span class="kn">from</span> <span class="nn">.operation</span> <span class="kn">import</span> <span class="n">so3_Jl_inv</span><span class="p">,</span> <span class="n">se3_Jl_inv</span><span class="p">,</span> <span class="n">rxso3_Jl_inv</span><span class="p">,</span> <span class="n">sim3_Jl_inv</span>
+<span class="kn">from</span> <span class="nn">..basics</span> <span class="kn">import</span> <span class="n">pm</span><span class="p">,</span> <span class="n">cumops_</span><span class="p">,</span> <span class="n">cummul_</span><span class="p">,</span> <span class="n">cumprod_</span><span class="p">,</span> <span class="n">cumops</span><span class="p">,</span> <span class="n">cummul</span><span class="p">,</span> <span class="n">cumprod</span>
+<span class="kn">from</span> <span class="nn">torch.nn.modules.utils</span> <span class="kn">import</span> <span class="n">_single</span><span class="p">,</span> <span class="n">_pair</span><span class="p">,</span> <span class="n">_triple</span><span class="p">,</span> <span class="n">_quadruple</span><span class="p">,</span> <span class="n">_ntuple</span>
+
+
+<span class="n">HANDLED_FUNCTIONS</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;__getitem__&#39;</span><span class="p">,</span> <span class="s1">&#39;__setitem__&#39;</span><span class="p">,</span> <span class="s1">&#39;cpu&#39;</span><span class="p">,</span> <span class="s1">&#39;cuda&#39;</span><span class="p">,</span> <span class="s1">&#39;float&#39;</span><span class="p">,</span> <span class="s1">&#39;double&#39;</span><span class="p">,</span>
+                     <span class="s1">&#39;to&#39;</span><span class="p">,</span> <span class="s1">&#39;detach&#39;</span><span class="p">,</span> <span class="s1">&#39;view&#39;</span><span class="p">,</span> <span class="s1">&#39;view_as&#39;</span><span class="p">,</span> <span class="s1">&#39;squeeze&#39;</span><span class="p">,</span> <span class="s1">&#39;unsqueeze&#39;</span><span class="p">,</span> <span class="s1">&#39;cat&#39;</span><span class="p">,</span>
+                     <span class="s1">&#39;stack&#39;</span><span class="p">,</span> <span class="s1">&#39;split&#39;</span><span class="p">,</span> <span class="s1">&#39;hsplit&#39;</span><span class="p">,</span> <span class="s1">&#39;dsplit&#39;</span><span class="p">,</span> <span class="s1">&#39;vsplit&#39;</span><span class="p">,</span> <span class="s1">&#39;tensor_split&#39;</span><span class="p">,</span>
+                     <span class="s1">&#39;chunk&#39;</span><span class="p">,</span> <span class="s1">&#39;concat&#39;</span><span class="p">,</span> <span class="s1">&#39;column_stack&#39;</span><span class="p">,</span> <span class="s1">&#39;dstack&#39;</span><span class="p">,</span> <span class="s1">&#39;vstack&#39;</span><span class="p">,</span> <span class="s1">&#39;hstack&#39;</span><span class="p">,</span>
+                     <span class="s1">&#39;index_select&#39;</span><span class="p">,</span> <span class="s1">&#39;masked_select&#39;</span><span class="p">,</span> <span class="s1">&#39;movedim&#39;</span><span class="p">,</span> <span class="s1">&#39;moveaxis&#39;</span><span class="p">,</span> <span class="s1">&#39;narrow&#39;</span><span class="p">,</span>
+                     <span class="s1">&#39;permute&#39;</span><span class="p">,</span> <span class="s1">&#39;reshape&#39;</span><span class="p">,</span> <span class="s1">&#39;row_stack&#39;</span><span class="p">,</span> <span class="s1">&#39;scatter&#39;</span><span class="p">,</span> <span class="s1">&#39;scatter_add&#39;</span><span class="p">,</span> <span class="s1">&#39;clone&#39;</span><span class="p">,</span>
+                     <span class="s1">&#39;swapaxes&#39;</span><span class="p">,</span> <span class="s1">&#39;swapdims&#39;</span><span class="p">,</span> <span class="s1">&#39;take&#39;</span><span class="p">,</span> <span class="s1">&#39;take_along_dim&#39;</span><span class="p">,</span> <span class="s1">&#39;tile&#39;</span><span class="p">,</span> <span class="s1">&#39;copy&#39;</span><span class="p">,</span>
+                     <span class="s1">&#39;transpose&#39;</span><span class="p">,</span> <span class="s1">&#39;unbind&#39;</span><span class="p">,</span> <span class="s1">&#39;gather&#39;</span><span class="p">,</span> <span class="s1">&#39;repeat&#39;</span><span class="p">,</span> <span class="s1">&#39;expand&#39;</span><span class="p">,</span> <span class="s1">&#39;expand_as&#39;</span><span class="p">,</span>
+                     <span class="s1">&#39;index_select&#39;</span><span class="p">,</span> <span class="s1">&#39;masked_select&#39;</span><span class="p">,</span> <span class="s1">&#39;index_copy&#39;</span><span class="p">,</span> <span class="s1">&#39;index_copy_&#39;</span><span class="p">,</span>
+                     <span class="s1">&#39;select&#39;</span><span class="p">,</span> <span class="s1">&#39;select_scatter&#39;</span><span class="p">,</span> <span class="s1">&#39;index_put&#39;</span><span class="p">,</span><span class="s1">&#39;index_put_&#39;</span><span class="p">,</span> <span class="s1">&#39;copy_&#39;</span><span class="p">]</span>
+
+<span class="k">class</span> <span class="nc">LieType</span><span class="p">:</span>
+<span class="w">    </span><span class="sd">&#39;&#39;&#39;LieTensor Type Base Class&#39;&#39;&#39;</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">dimension</span><span class="p">,</span> <span class="n">embedding</span><span class="p">,</span> <span class="n">manifold</span><span class="p">):</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">_dimension</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">Size</span><span class="p">([</span><span class="n">dimension</span><span class="p">])</span> <span class="c1"># Data dimension</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">_embedding</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">Size</span><span class="p">([</span><span class="n">embedding</span><span class="p">])</span> <span class="c1"># Embedding dimension</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">_manifold</span>  <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">Size</span><span class="p">([</span><span class="n">manifold</span><span class="p">])</span>  <span class="c1"># Manifold dimension</span>
+
+    <span class="nd">@property</span>
+    <span class="k">def</span> <span class="nf">dimension</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_dimension</span>
+
+    <span class="nd">@property</span>
+    <span class="k">def</span> <span class="nf">embedding</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_embedding</span>
+
+    <span class="nd">@property</span>
+    <span class="k">def</span> <span class="nf">manifold</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_manifold</span>
+
+    <span class="nd">@property</span>
+    <span class="k">def</span> <span class="nf">on_manifold</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">dimension</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">manifold</span>
+
+    <span class="k">def</span> <span class="nf">add_</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">input</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">on_manifold</span><span class="p">:</span>
+            <span class="n">other1</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="o">.</span><span class="n">as_subclass</span><span class="p">(</span><span class="nb">input</span><span class="p">,</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">)</span>
+            <span class="n">other2</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="o">.</span><span class="n">as_subclass</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">)</span>
+            <span class="k">return</span> <span class="nb">input</span><span class="o">.</span><span class="n">copy_</span><span class="p">(</span><span class="n">other1</span> <span class="o">+</span> <span class="n">other2</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="p">:</span><span class="bp">self</span><span class="o">.</span><span class="n">manifold</span><span class="p">[</span><span class="mi">0</span><span class="p">]])</span>
+        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s2">&quot;Instance has no add_ attribute.&quot;</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">Log</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">):</span>
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">on_manifold</span><span class="p">:</span>
+            <span class="k">raise</span> <span class="ne">AttributeError</span><span class="p">(</span><span class="s2">&quot;Lie Algebra has no Log attribute&quot;</span><span class="p">)</span>
+        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s2">&quot;Instance has no Log attribute.&quot;</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">Exp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
+        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">on_manifold</span><span class="p">:</span>
+            <span class="k">raise</span> <span class="ne">AttributeError</span><span class="p">(</span><span class="s2">&quot;Lie Group has no Exp attribute&quot;</span><span class="p">)</span>
+        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s2">&quot;Instance has no Exp attribute.&quot;</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">Inv</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">on_manifold</span><span class="p">:</span>
+            <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">,</span> <span class="n">ltype</span><span class="o">=</span><span class="n">x</span><span class="o">.</span><span class="n">ltype</span><span class="p">)</span>
+        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s2">&quot;Instance has no Inv attribute.&quot;</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">Act</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot; action on a points tensor(*, 3[4]) (homogeneous)&quot;&quot;&quot;</span>
+        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">on_manifold</span><span class="p">:</span>
+            <span class="k">raise</span> <span class="ne">AttributeError</span><span class="p">(</span><span class="s2">&quot;Lie Group has no Act attribute&quot;</span><span class="p">)</span>
+        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s2">&quot;Instance has no Act attribute.&quot;</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">Mul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">):</span>
+        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">on_manifold</span><span class="p">:</span>
+            <span class="k">raise</span> <span class="ne">AttributeError</span><span class="p">(</span><span class="s2">&quot;Lie Group has no Mul attribute&quot;</span><span class="p">)</span>
+        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s2">&quot;Instance has no Mul attribute.&quot;</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">Retr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">a</span><span class="p">):</span>
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">on_manifold</span><span class="p">:</span>
+            <span class="k">raise</span> <span class="ne">AttributeError</span><span class="p">(</span><span class="s2">&quot;Has no Retr attribute&quot;</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">a</span><span class="o">.</span><span class="n">Exp</span><span class="p">()</span> <span class="o">*</span> <span class="n">X</span>
+
+    <span class="k">def</span> <span class="nf">Adj</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">a</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&#39;&#39;&#39; X * Exp(a) = Exp(Adj) * X &#39;&#39;&#39;</span>
+        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">on_manifold</span><span class="p">:</span>
+            <span class="k">raise</span> <span class="ne">AttributeError</span><span class="p">(</span><span class="s2">&quot;Lie Group has no Adj attribute&quot;</span><span class="p">)</span>
+        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s2">&quot;Instance has no Adj attribute.&quot;</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">AdjT</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">a</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&#39;&#39;&#39; Exp(a) * X = X * Exp(AdjT) &#39;&#39;&#39;</span>
+        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">on_manifold</span><span class="p">:</span>
+            <span class="k">raise</span> <span class="ne">AttributeError</span><span class="p">(</span><span class="s2">&quot;Lie Group has no AdjT attribute&quot;</span><span class="p">)</span>
+        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s2">&quot;Instance has no AdjT attribute.&quot;</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">Jinvp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
+        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">on_manifold</span><span class="p">:</span>
+            <span class="k">raise</span> <span class="ne">AttributeError</span><span class="p">(</span><span class="s2">&quot;Lie Group has no Jinvp attribute&quot;</span><span class="p">)</span>
+        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s2">&quot;Instance has no Jinvp attribute.&quot;</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">input</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot; To 4x4 matrix &quot;&quot;&quot;</span>
+        <span class="n">X</span> <span class="o">=</span> <span class="nb">input</span><span class="o">.</span><span class="n">Exp</span><span class="p">()</span> <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">on_manifold</span> <span class="k">else</span> <span class="nb">input</span>
+        <span class="n">I</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">X</span><span class="o">.</span><span class="n">dtype</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="n">X</span><span class="o">.</span><span class="n">device</span><span class="p">)</span>
+        <span class="n">I</span> <span class="o">=</span> <span class="n">I</span><span class="o">.</span><span class="n">view</span><span class="p">([</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">X</span><span class="o">.</span><span class="n">dim</span><span class="p">()</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">])</span>
+        <span class="k">return</span> <span class="n">X</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">Act</span><span class="p">(</span><span class="n">I</span><span class="p">)</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">rotation</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">input</span><span class="p">):</span>
+        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s2">&quot;Rotation is not implemented for the instance.&quot;</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">translation</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">input</span><span class="p">):</span>
+        <span class="n">warnings</span><span class="o">.</span><span class="n">warn</span><span class="p">(</span><span class="s2">&quot;Instance has no translation. Zero vector(s) is returned.&quot;</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="nb">input</span><span class="o">.</span><span class="n">lshape</span> <span class="o">+</span> <span class="p">(</span><span class="mi">3</span><span class="p">,),</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">input</span><span class="o">.</span><span class="n">dtype</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="nb">input</span><span class="o">.</span><span class="n">device</span><span class="p">,</span>
+            <span class="n">requires_grad</span><span class="o">=</span><span class="nb">input</span><span class="o">.</span><span class="n">requires_grad</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">scale</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">input</span><span class="p">):</span>
+        <span class="n">warnings</span><span class="o">.</span><span class="n">warn</span><span class="p">(</span><span class="s2">&quot;Instance has no scale. Scalar one(s) is returned.&quot;</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">torch</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="nb">input</span><span class="o">.</span><span class="n">lshape</span> <span class="o">+</span> <span class="p">(</span><span class="mi">1</span><span class="p">,),</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">input</span><span class="o">.</span><span class="n">dtype</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="nb">input</span><span class="o">.</span><span class="n">device</span><span class="p">,</span>
+            <span class="n">requires_grad</span><span class="o">=</span><span class="nb">input</span><span class="o">.</span><span class="n">requires_grad</span><span class="p">)</span>
+
+    <span class="nd">@classmethod</span>
+    <span class="k">def</span> <span class="nf">to_tuple</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="nb">input</span><span class="p">):</span>
+        <span class="n">out</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">()</span>
+        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">input</span><span class="p">:</span>
+            <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">collections</span><span class="o">.</span><span class="n">abc</span><span class="o">.</span><span class="n">Iterable</span><span class="p">):</span>
+                <span class="n">out</span> <span class="o">+=</span> <span class="p">(</span><span class="n">i</span><span class="p">,)</span>
+            <span class="k">else</span><span class="p">:</span>
+                <span class="n">out</span> <span class="o">+=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">out</span>
+
+    <span class="nd">@classmethod</span>
+    <span class="k">def</span> <span class="nf">identity</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s2">&quot;Instance has no identity.&quot;</span><span class="p">)</span>
+
+    <span class="nd">@classmethod</span>
+    <span class="k">def</span> <span class="nf">identity_like</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+        <span class="k">return</span> <span class="bp">cls</span><span class="o">.</span><span class="n">identity</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">randn_like</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="n">sigma</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">randn</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s2">&quot;randn not implemented yet&quot;</span><span class="p">)</span>
+
+    <span class="nd">@classmethod</span>
+    <span class="k">def</span> <span class="nf">cumops</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">dim</span><span class="p">,</span> <span class="n">ops</span><span class="p">):</span>
+        <span class="k">return</span> <span class="n">cumops</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">dim</span><span class="p">,</span> <span class="n">ops</span><span class="p">)</span>
+
+    <span class="nd">@classmethod</span>
+    <span class="k">def</span> <span class="nf">cummul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">dim</span><span class="p">,</span> <span class="n">left</span> <span class="o">=</span> <span class="kc">True</span><span class="p">):</span>
+        <span class="k">return</span> <span class="n">cummul</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">dim</span><span class="p">,</span> <span class="n">left</span><span class="p">)</span>
+
+    <span class="nd">@classmethod</span>
+    <span class="k">def</span> <span class="nf">cumprod</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">dim</span><span class="p">,</span> <span class="n">left</span> <span class="o">=</span> <span class="kc">True</span><span class="p">):</span>
+        <span class="k">return</span> <span class="n">cumprod</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">dim</span><span class="p">,</span> <span class="n">left</span><span class="p">)</span>
+
+    <span class="nd">@classmethod</span>
+    <span class="k">def</span> <span class="nf">cumops_</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">dim</span><span class="p">,</span> <span class="n">ops</span><span class="p">):</span>
+        <span class="k">return</span> <span class="n">cumops_</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">dim</span><span class="p">,</span> <span class="n">ops</span><span class="p">)</span>
+
+    <span class="nd">@classmethod</span>
+    <span class="k">def</span> <span class="nf">cummul_</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">dim</span><span class="p">,</span> <span class="n">left</span> <span class="o">=</span> <span class="kc">True</span><span class="p">):</span>
+        <span class="k">return</span> <span class="n">cummul_</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">dim</span><span class="p">,</span> <span class="n">left</span><span class="p">)</span>
+
+    <span class="nd">@classmethod</span>
+    <span class="k">def</span> <span class="nf">cumprod_</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">dim</span><span class="p">,</span> <span class="n">left</span> <span class="o">=</span> <span class="kc">True</span><span class="p">):</span>
+        <span class="k">return</span> <span class="n">cumprod_</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">dim</span><span class="p">,</span> <span class="n">left</span><span class="p">)</span>
+
+
+<span class="k">class</span> <span class="nc">SO3Type</span><span class="p">(</span><span class="n">LieType</span><span class="p">):</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">Log</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">):</span>
+        <span class="n">X</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">X</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="n">SO3_Log</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">ltype</span><span class="o">=</span><span class="n">so3_type</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">Act</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
+        <span class="k">assert</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">on_manifold</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">)</span>
+        <span class="k">assert</span> <span class="n">p</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">==</span><span class="mi">3</span> <span class="ow">or</span> <span class="n">p</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">==</span><span class="mi">4</span><span class="p">,</span> <span class="s2">&quot;Invalid Tensor Dimension&quot;</span>
+        <span class="n">X</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">X</span>
+        <span class="nb">input</span><span class="p">,</span> <span class="n">out_shape</span> <span class="o">=</span> <span class="n">broadcast_inputs</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
+        <span class="k">if</span> <span class="n">p</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">==</span><span class="mi">3</span><span class="p">:</span>
+            <span class="n">out</span> <span class="o">=</span> <span class="n">SO3_Act</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="o">*</span><span class="nb">input</span><span class="p">)</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">out</span> <span class="o">=</span> <span class="n">SO3_Act4</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="o">*</span><span class="nb">input</span><span class="p">)</span>
+        <span class="n">dim</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span> <span class="k">if</span> <span class="n">out</span><span class="o">.</span><span class="n">nelement</span><span class="p">()</span> <span class="o">!=</span> <span class="mi">0</span> <span class="k">else</span> <span class="n">X</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+        <span class="k">return</span> <span class="n">out</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="n">out_shape</span> <span class="o">+</span> <span class="p">(</span><span class="n">dim</span><span class="p">,))</span>
+
+    <span class="k">def</span> <span class="nf">Mul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">):</span>
+        <span class="c1"># Transform on transform</span>
+        <span class="n">X</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">X</span>
+        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">on_manifold</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">Y</span><span class="p">,</span> <span class="n">LieTensor</span><span class="p">)</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">Y</span><span class="o">.</span><span class="n">ltype</span><span class="o">.</span><span class="n">on_manifold</span><span class="p">:</span>
+            <span class="n">Y</span> <span class="o">=</span> <span class="n">Y</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">Y</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">Y</span>
+            <span class="nb">input</span><span class="p">,</span> <span class="n">out_shape</span> <span class="o">=</span> <span class="n">broadcast_inputs</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">)</span>
+            <span class="n">out</span> <span class="o">=</span> <span class="n">SO3_Mul</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="o">*</span><span class="nb">input</span><span class="p">)</span>
+            <span class="n">dim</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span> <span class="k">if</span> <span class="n">out</span><span class="o">.</span><span class="n">nelement</span><span class="p">()</span> <span class="o">!=</span> <span class="mi">0</span> <span class="k">else</span> <span class="n">X</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+            <span class="n">out</span> <span class="o">=</span> <span class="n">out</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="n">out_shape</span> <span class="o">+</span> <span class="p">(</span><span class="n">dim</span><span class="p">,))</span>
+            <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">out</span><span class="p">,</span> <span class="n">ltype</span><span class="o">=</span><span class="n">SO3_type</span><span class="p">)</span>
+        <span class="c1"># Transform on points</span>
+        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">on_manifold</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">Y</span><span class="p">,</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">):</span>
+            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">Act</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">)</span>
+        <span class="c1"># (scalar or tensor) * manifold</span>
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">on_manifold</span><span class="p">:</span>
+            <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">mul</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">),</span> <span class="n">ltype</span><span class="o">=</span><span class="n">SO3_type</span><span class="p">)</span>
+        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s1">&#39;Invalid __mul__ operation&#39;</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">Inv</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">):</span>
+        <span class="n">X</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">X</span>
+        <span class="n">out</span> <span class="o">=</span> <span class="n">SO3_Inv</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">out</span><span class="p">,</span> <span class="n">ltype</span><span class="o">=</span><span class="n">SO3_type</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">Adj</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">a</span><span class="p">):</span>
+        <span class="n">X</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">X</span>
+        <span class="n">a</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">a</span>
+        <span class="nb">input</span><span class="p">,</span> <span class="n">out_shape</span> <span class="o">=</span> <span class="n">broadcast_inputs</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
+        <span class="n">out</span> <span class="o">=</span> <span class="n">SO3_AdjXa</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="o">*</span><span class="nb">input</span><span class="p">)</span>
+        <span class="n">dim</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span> <span class="k">if</span> <span class="n">out</span><span class="o">.</span><span class="n">nelement</span><span class="p">()</span> <span class="o">!=</span> <span class="mi">0</span> <span class="k">else</span> <span class="n">X</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+        <span class="n">out</span> <span class="o">=</span> <span class="n">out</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="n">out_shape</span> <span class="o">+</span> <span class="p">(</span><span class="n">dim</span><span class="p">,))</span>
+        <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">out</span><span class="p">,</span> <span class="n">ltype</span><span class="o">=</span><span class="n">so3_type</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">AdjT</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">a</span><span class="p">):</span>
+        <span class="n">X</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">X</span>
+        <span class="n">a</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">a</span>
+        <span class="nb">input</span><span class="p">,</span> <span class="n">out_shape</span> <span class="o">=</span> <span class="n">broadcast_inputs</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
+        <span class="n">out</span> <span class="o">=</span> <span class="n">SO3_AdjTXa</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="o">*</span><span class="nb">input</span><span class="p">)</span>
+        <span class="n">dim</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span> <span class="k">if</span> <span class="n">out</span><span class="o">.</span><span class="n">nelement</span><span class="p">()</span> <span class="o">!=</span> <span class="mi">0</span> <span class="k">else</span> <span class="n">X</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+        <span class="n">out</span> <span class="o">=</span> <span class="n">out</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="n">out_shape</span> <span class="o">+</span> <span class="p">(</span><span class="n">dim</span><span class="p">,))</span>
+        <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">out</span><span class="p">,</span> <span class="n">ltype</span><span class="o">=</span><span class="n">so3_type</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">Jinvp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">a</span><span class="p">):</span>
+        <span class="n">X</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">X</span>
+        <span class="n">a</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">a</span>
+        <span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">a</span><span class="p">),</span> <span class="n">out_shape</span> <span class="o">=</span> <span class="n">broadcast_inputs</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
+        <span class="n">out</span> <span class="o">=</span> <span class="p">(</span><span class="n">so3_Jl_inv</span><span class="p">(</span><span class="n">SO3_Log</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="n">X</span><span class="p">))</span> <span class="o">@</span> <span class="n">a</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
+        <span class="n">dim</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span> <span class="k">if</span> <span class="n">out</span><span class="o">.</span><span class="n">nelement</span><span class="p">()</span> <span class="o">!=</span> <span class="mi">0</span> <span class="k">else</span> <span class="n">X</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+        <span class="n">out</span> <span class="o">=</span> <span class="n">out</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="n">out_shape</span> <span class="o">+</span> <span class="p">(</span><span class="n">dim</span><span class="p">,))</span>
+        <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">out</span><span class="p">,</span> <span class="n">ltype</span><span class="o">=</span><span class="n">so3_type</span><span class="p">)</span>
+
+    <span class="nd">@classmethod</span>
+    <span class="k">def</span> <span class="nf">identity</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">size</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+        <span class="n">data</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">],</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">data</span><span class="o">.</span><span class="n">repeat</span><span class="p">(</span><span class="n">size</span><span class="o">+</span><span class="p">(</span><span class="mi">1</span><span class="p">,)),</span> <span class="n">ltype</span><span class="o">=</span><span class="n">SO3_type</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">randn</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">size</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">requires_grad</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+        <span class="n">data</span> <span class="o">=</span> <span class="n">so3_type</span><span class="o">.</span><span class="n">Exp</span><span class="p">(</span><span class="n">so3_type</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="o">*</span><span class="n">size</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="n">sigma</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span><span class="o">.</span><span class="n">detach</span><span class="p">()</span>
+        <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">ltype</span><span class="o">=</span><span class="n">SO3_type</span><span class="p">)</span><span class="o">.</span><span class="n">requires_grad_</span><span class="p">(</span><span class="n">requires_grad</span><span class="p">)</span>
+
+    <span class="nd">@classmethod</span>
+    <span class="k">def</span> <span class="nf">add_</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="nb">input</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+        <span class="k">return</span> <span class="nb">input</span><span class="o">.</span><span class="n">copy_</span><span class="p">(</span><span class="n">LieTensor</span><span class="p">(</span><span class="n">other</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="p">:</span><span class="mi">3</span><span class="p">],</span> <span class="n">ltype</span><span class="o">=</span><span class="n">so3_type</span><span class="p">)</span><span class="o">.</span><span class="n">Exp</span><span class="p">()</span> <span class="o">*</span> <span class="nb">input</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">input</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot; To 3x3 matrix &quot;&quot;&quot;</span>
+        <span class="n">I</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">input</span><span class="o">.</span><span class="n">dtype</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="nb">input</span><span class="o">.</span><span class="n">device</span><span class="p">)</span>
+        <span class="n">I</span> <span class="o">=</span> <span class="n">I</span><span class="o">.</span><span class="n">view</span><span class="p">([</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="nb">input</span><span class="o">.</span><span class="n">dim</span><span class="p">()</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
+        <span class="k">return</span> <span class="nb">input</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">Act</span><span class="p">(</span><span class="n">I</span><span class="p">)</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">rotation</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">input</span><span class="p">):</span>
+        <span class="k">return</span> <span class="nb">input</span>
+
+    <span class="k">def</span> <span class="nf">identity_</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">):</span>
+        <span class="n">X</span><span class="o">.</span><span class="n">fill_</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
+        <span class="n">X</span><span class="o">.</span><span class="n">index_fill_</span><span class="p">(</span><span class="n">dim</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">index</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">device</span><span class="o">=</span><span class="n">X</span><span class="o">.</span><span class="n">device</span><span class="p">),</span> <span class="n">value</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">X</span>
+
+    <span class="k">def</span> <span class="nf">Jr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Right jacobian of SO(3)</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">return</span> <span class="n">X</span><span class="o">.</span><span class="n">Log</span><span class="p">()</span><span class="o">.</span><span class="n">Jr</span><span class="p">()</span>
+
+
+<span class="k">class</span> <span class="nc">so3Type</span><span class="p">(</span><span class="n">LieType</span><span class="p">):</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">Exp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">x</span>
+        <span class="n">X</span> <span class="o">=</span> <span class="n">so3_Exp</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">ltype</span><span class="o">=</span><span class="n">SO3_type</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">Mul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">):</span>
+        <span class="n">X</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">X</span>
+        <span class="c1"># (scalar or tensor) * manifold</span>
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">on_manifold</span><span class="p">:</span>
+            <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">mul</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">),</span> <span class="n">ltype</span><span class="o">=</span><span class="n">so3_type</span><span class="p">)</span>
+        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s1">&#39;Invalid __mul__ operation&#39;</span><span class="p">)</span>
+
+    <span class="nd">@classmethod</span>
+    <span class="k">def</span> <span class="nf">identity</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">size</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+        <span class="k">return</span> <span class="n">SO3_type</span><span class="o">.</span><span class="n">Log</span><span class="p">(</span><span class="n">SO3_type</span><span class="o">.</span><span class="n">identity</span><span class="p">(</span><span class="o">*</span><span class="n">size</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
+
+    <span class="k">def</span> <span class="nf">randn</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">size</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">requires_grad</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+        <span class="k">assert</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">sigma</span><span class="p">,</span> <span class="n">numbers</span><span class="o">.</span><span class="n">Number</span><span class="p">),</span> <span class="s1">&#39;Only accepts sigma as a single number&#39;</span>
+        <span class="n">size</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">to_tuple</span><span class="p">(</span><span class="n">size</span><span class="p">)</span>
+        <span class="n">data</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="n">size</span> <span class="o">+</span> <span class="n">torch</span><span class="o">.</span><span class="n">Size</span><span class="p">([</span><span class="mi">3</span><span class="p">])),</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+        <span class="n">dist</span> <span class="o">=</span> <span class="n">data</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">dim</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">keepdim</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+        <span class="n">theta</span> <span class="o">=</span> <span class="n">sigma</span> <span class="o">*</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="n">size</span> <span class="o">+</span> <span class="n">torch</span><span class="o">.</span><span class="n">Size</span><span class="p">([</span><span class="mi">1</span><span class="p">])),</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">data</span> <span class="o">/</span> <span class="n">dist</span> <span class="o">*</span> <span class="n">theta</span><span class="p">,</span> <span class="n">ltype</span><span class="o">=</span><span class="n">so3_type</span><span class="p">)</span><span class="o">.</span><span class="n">requires_grad_</span><span class="p">(</span><span class="n">requires_grad</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">input</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot; To 3x3 matrix &quot;&quot;&quot;</span>
+        <span class="n">X</span> <span class="o">=</span> <span class="nb">input</span><span class="o">.</span><span class="n">Exp</span><span class="p">()</span>
+        <span class="n">I</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">X</span><span class="o">.</span><span class="n">dtype</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="n">X</span><span class="o">.</span><span class="n">device</span><span class="p">)</span>
+        <span class="n">I</span> <span class="o">=</span> <span class="n">I</span><span class="o">.</span><span class="n">view</span><span class="p">([</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">X</span><span class="o">.</span><span class="n">dim</span><span class="p">()</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
+        <span class="k">return</span> <span class="n">X</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">Act</span><span class="p">(</span><span class="n">I</span><span class="p">)</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">rotation</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">input</span><span class="p">):</span>
+        <span class="k">return</span> <span class="nb">input</span><span class="o">.</span><span class="n">Exp</span><span class="p">()</span><span class="o">.</span><span class="n">rotation</span><span class="p">()</span>
+
+    <span class="k">def</span> <span class="nf">Jr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Right jacobian of so(3)</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">K</span> <span class="o">=</span> <span class="n">vec2skew</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
+        <span class="n">theta</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">dim</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">keepdim</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
+        <span class="n">I</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="n">x</span><span class="o">.</span><span class="n">device</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">x</span><span class="o">.</span><span class="n">dtype</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">lshape</span><span class="o">+</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
+        <span class="n">Jr</span> <span class="o">=</span> <span class="n">I</span> <span class="o">-</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">theta</span><span class="o">.</span><span class="n">cos</span><span class="p">())</span><span class="o">/</span><span class="n">theta</span><span class="o">**</span><span class="mi">2</span> <span class="o">*</span> <span class="n">K</span> <span class="o">+</span> <span class="p">(</span><span class="n">theta</span> <span class="o">-</span> <span class="n">theta</span><span class="o">.</span><span class="n">sin</span><span class="p">())</span><span class="o">/</span><span class="n">theta</span><span class="o">**</span><span class="mi">3</span> <span class="o">*</span> <span class="n">K</span><span class="nd">@K</span>
+        <span class="k">return</span> <span class="n">torch</span><span class="o">.</span><span class="n">where</span><span class="p">(</span><span class="n">theta</span><span class="o">&gt;</span><span class="n">torch</span><span class="o">.</span><span class="n">finfo</span><span class="p">(</span><span class="n">theta</span><span class="o">.</span><span class="n">dtype</span><span class="p">)</span><span class="o">.</span><span class="n">eps</span><span class="p">,</span> <span class="n">Jr</span><span class="p">,</span> <span class="n">I</span><span class="p">)</span>
+
+
+<span class="k">class</span> <span class="nc">SE3Type</span><span class="p">(</span><span class="n">LieType</span><span class="p">):</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">(</span><span class="mi">7</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">6</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">Log</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">):</span>
+        <span class="n">X</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">X</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="n">SE3_Log</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">ltype</span><span class="o">=</span><span class="n">se3_type</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">Act</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
+        <span class="k">assert</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">on_manifold</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">)</span>
+        <span class="k">assert</span> <span class="n">p</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">==</span><span class="mi">3</span> <span class="ow">or</span> <span class="n">p</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">==</span><span class="mi">4</span><span class="p">,</span> <span class="s2">&quot;Invalid Tensor Dimension&quot;</span>
+        <span class="n">X</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">X</span>
+        <span class="nb">input</span><span class="p">,</span> <span class="n">out_shape</span> <span class="o">=</span> <span class="n">broadcast_inputs</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
+        <span class="k">if</span> <span class="n">p</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">==</span><span class="mi">3</span><span class="p">:</span>
+            <span class="n">out</span> <span class="o">=</span> <span class="n">SE3_Act</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="o">*</span><span class="nb">input</span><span class="p">)</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">out</span> <span class="o">=</span> <span class="n">SE3_Act4</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="o">*</span><span class="nb">input</span><span class="p">)</span>
+        <span class="n">dim</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span> <span class="k">if</span> <span class="n">out</span><span class="o">.</span><span class="n">nelement</span><span class="p">()</span> <span class="o">!=</span> <span class="mi">0</span> <span class="k">else</span> <span class="n">X</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+        <span class="k">return</span> <span class="n">out</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="n">out_shape</span> <span class="o">+</span> <span class="p">(</span><span class="n">dim</span><span class="p">,))</span>
+
+    <span class="k">def</span> <span class="nf">Mul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">):</span>
+        <span class="c1"># Transform on transform</span>
+        <span class="n">X</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">X</span>
+        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">on_manifold</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">Y</span><span class="p">,</span> <span class="n">LieTensor</span><span class="p">)</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">Y</span><span class="o">.</span><span class="n">ltype</span><span class="o">.</span><span class="n">on_manifold</span><span class="p">:</span>
+            <span class="n">Y</span> <span class="o">=</span> <span class="n">Y</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">Y</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">Y</span>
+            <span class="nb">input</span><span class="p">,</span> <span class="n">out_shape</span> <span class="o">=</span> <span class="n">broadcast_inputs</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">)</span>
+            <span class="n">out</span> <span class="o">=</span> <span class="n">SE3_Mul</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="o">*</span><span class="nb">input</span><span class="p">)</span>
+            <span class="n">dim</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span> <span class="k">if</span> <span class="n">out</span><span class="o">.</span><span class="n">nelement</span><span class="p">()</span> <span class="o">!=</span> <span class="mi">0</span> <span class="k">else</span> <span class="n">X</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+            <span class="n">out</span> <span class="o">=</span> <span class="n">out</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="n">out_shape</span> <span class="o">+</span> <span class="p">(</span><span class="n">dim</span><span class="p">,))</span>
+            <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">out</span><span class="p">,</span> <span class="n">ltype</span><span class="o">=</span><span class="n">SE3_type</span><span class="p">)</span>
+        <span class="c1"># Transform on points</span>
+        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">on_manifold</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">Y</span><span class="p">,</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">):</span>
+            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">Act</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">)</span>
+        <span class="c1"># (scalar or tensor) * manifold</span>
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">on_manifold</span><span class="p">:</span>
+            <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">mul</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">),</span> <span class="n">ltype</span><span class="o">=</span><span class="n">SE3_type</span><span class="p">)</span>
+        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s1">&#39;Invalid __mul__ operation&#39;</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">Inv</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">):</span>
+        <span class="n">X</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">X</span>
+        <span class="n">out</span> <span class="o">=</span> <span class="n">SE3_Inv</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">out</span><span class="p">,</span> <span class="n">ltype</span><span class="o">=</span><span class="n">SE3_type</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">rotation</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">input</span><span class="p">):</span>
+        <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="nb">input</span><span class="o">.</span><span class="n">tensor</span><span class="p">()[</span><span class="o">...</span><span class="p">,</span> <span class="mi">3</span><span class="p">:</span><span class="mi">7</span><span class="p">],</span> <span class="n">ltype</span><span class="o">=</span><span class="n">SO3_type</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">translation</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">input</span><span class="p">):</span>
+        <span class="k">return</span> <span class="nb">input</span><span class="o">.</span><span class="n">tensor</span><span class="p">()[</span><span class="o">...</span><span class="p">,</span> <span class="mi">0</span><span class="p">:</span><span class="mi">3</span><span class="p">]</span>
+
+    <span class="k">def</span> <span class="nf">Adj</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">a</span><span class="p">):</span>
+        <span class="n">X</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">X</span>
+        <span class="n">a</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">a</span>
+        <span class="nb">input</span><span class="p">,</span> <span class="n">out_shape</span> <span class="o">=</span> <span class="n">broadcast_inputs</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
+        <span class="n">out</span> <span class="o">=</span> <span class="n">SE3_AdjXa</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="o">*</span><span class="nb">input</span><span class="p">)</span>
+        <span class="n">dim</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span> <span class="k">if</span> <span class="n">out</span><span class="o">.</span><span class="n">nelement</span><span class="p">()</span> <span class="o">!=</span> <span class="mi">0</span> <span class="k">else</span> <span class="n">X</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+        <span class="n">out</span> <span class="o">=</span> <span class="n">out</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="n">out_shape</span> <span class="o">+</span> <span class="p">(</span><span class="n">dim</span><span class="p">,))</span>
+        <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">out</span><span class="p">,</span> <span class="n">ltype</span><span class="o">=</span><span class="n">se3_type</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">AdjT</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">a</span><span class="p">):</span>
+        <span class="n">X</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">X</span>
+        <span class="n">a</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">a</span>
+        <span class="nb">input</span><span class="p">,</span> <span class="n">out_shape</span> <span class="o">=</span> <span class="n">broadcast_inputs</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
+        <span class="n">out</span> <span class="o">=</span> <span class="n">SE3_AdjTXa</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="o">*</span><span class="nb">input</span><span class="p">)</span>
+        <span class="n">dim</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span> <span class="k">if</span> <span class="n">out</span><span class="o">.</span><span class="n">nelement</span><span class="p">()</span> <span class="o">!=</span> <span class="mi">0</span> <span class="k">else</span> <span class="n">X</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+        <span class="n">out</span> <span class="o">=</span> <span class="n">out</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="n">out_shape</span> <span class="o">+</span> <span class="p">(</span><span class="n">dim</span><span class="p">,))</span>
+        <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">out</span><span class="p">,</span> <span class="n">ltype</span><span class="o">=</span><span class="n">se3_type</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">Jinvp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">a</span><span class="p">):</span>
+        <span class="n">X</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">X</span>
+        <span class="n">a</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">a</span>
+        <span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">a</span><span class="p">),</span> <span class="n">out_shape</span> <span class="o">=</span> <span class="n">broadcast_inputs</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
+        <span class="n">out</span> <span class="o">=</span> <span class="p">(</span><span class="n">se3_Jl_inv</span><span class="p">(</span><span class="n">SE3_Log</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="n">X</span><span class="p">))</span> <span class="o">@</span> <span class="n">a</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
+        <span class="n">dim</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span> <span class="k">if</span> <span class="n">out</span><span class="o">.</span><span class="n">nelement</span><span class="p">()</span> <span class="o">!=</span> <span class="mi">0</span> <span class="k">else</span> <span class="n">X</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+        <span class="n">out</span> <span class="o">=</span> <span class="n">out</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="n">out_shape</span> <span class="o">+</span> <span class="p">(</span><span class="n">dim</span><span class="p">,))</span>
+        <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">out</span><span class="p">,</span> <span class="n">ltype</span><span class="o">=</span><span class="n">se3_type</span><span class="p">)</span>
+
+    <span class="nd">@classmethod</span>
+    <span class="k">def</span> <span class="nf">identity</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">size</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+        <span class="n">data</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">],</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">data</span><span class="o">.</span><span class="n">repeat</span><span class="p">(</span><span class="n">size</span><span class="o">+</span><span class="p">(</span><span class="mi">1</span><span class="p">,)),</span> <span class="n">ltype</span><span class="o">=</span><span class="n">SE3_type</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">randn</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">size</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">requires_grad</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+        <span class="n">data</span> <span class="o">=</span> <span class="n">se3_type</span><span class="o">.</span><span class="n">Exp</span><span class="p">(</span><span class="n">se3_type</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="o">*</span><span class="n">size</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="n">sigma</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span><span class="o">.</span><span class="n">detach</span><span class="p">()</span>
+        <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">ltype</span><span class="o">=</span><span class="n">SE3_type</span><span class="p">)</span><span class="o">.</span><span class="n">requires_grad_</span><span class="p">(</span><span class="n">requires_grad</span><span class="p">)</span>
+
+    <span class="nd">@classmethod</span>
+    <span class="k">def</span> <span class="nf">add_</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="nb">input</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+        <span class="k">return</span> <span class="nb">input</span><span class="o">.</span><span class="n">copy_</span><span class="p">(</span><span class="n">LieTensor</span><span class="p">(</span><span class="n">other</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="p">:</span><span class="mi">6</span><span class="p">],</span> <span class="n">ltype</span><span class="o">=</span><span class="n">se3_type</span><span class="p">)</span><span class="o">.</span><span class="n">Exp</span><span class="p">()</span> <span class="o">*</span> <span class="nb">input</span><span class="p">)</span>
+
+
+<span class="k">class</span> <span class="nc">se3Type</span><span class="p">(</span><span class="n">LieType</span><span class="p">):</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">6</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">Exp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">x</span>
+        <span class="n">X</span> <span class="o">=</span> <span class="n">se3_Exp</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">ltype</span><span class="o">=</span><span class="n">SE3_type</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">Mul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">):</span>
+        <span class="n">X</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">X</span>
+        <span class="c1"># (scalar or tensor) * manifold</span>
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">on_manifold</span><span class="p">:</span>
+            <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">mul</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">),</span> <span class="n">ltype</span><span class="o">=</span><span class="n">se3_type</span><span class="p">)</span>
+        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s1">&#39;Invalid __mul__ operation&#39;</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">rotation</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">input</span><span class="p">):</span>
+        <span class="k">return</span> <span class="nb">input</span><span class="o">.</span><span class="n">Exp</span><span class="p">()</span><span class="o">.</span><span class="n">rotation</span><span class="p">()</span>
+
+    <span class="k">def</span> <span class="nf">translation</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">input</span><span class="p">):</span>
+        <span class="k">return</span> <span class="nb">input</span><span class="o">.</span><span class="n">Exp</span><span class="p">()</span><span class="o">.</span><span class="n">translation</span><span class="p">()</span>
+
+    <span class="nd">@classmethod</span>
+    <span class="k">def</span> <span class="nf">identity</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">size</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+        <span class="k">return</span> <span class="n">SE3_type</span><span class="o">.</span><span class="n">Log</span><span class="p">(</span><span class="n">SE3_type</span><span class="o">.</span><span class="n">identity</span><span class="p">(</span><span class="o">*</span><span class="n">size</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
+
+    <span class="k">def</span> <span class="nf">randn</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">size</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">requires_grad</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+        <span class="c1">#  convert different types of inputs to SE3 sigma</span>
+        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">sigma</span><span class="p">,</span> <span class="n">collections</span><span class="o">.</span><span class="n">abc</span><span class="o">.</span><span class="n">Iterable</span><span class="p">):</span>
+            <span class="n">sigma</span> <span class="o">=</span> <span class="n">_quadruple</span><span class="p">(</span><span class="n">sigma</span><span class="p">)</span>
+        <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">sigma</span><span class="p">)</span><span class="o">==</span><span class="mi">2</span><span class="p">:</span>
+            <span class="n">rotation_sigma</span> <span class="o">=</span> <span class="n">_single</span><span class="p">(</span><span class="n">sigma</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
+            <span class="n">translation_sigma</span> <span class="o">=</span> <span class="n">_triple</span><span class="p">(</span><span class="n">sigma</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+            <span class="n">sigma</span> <span class="o">=</span> <span class="n">translation_sigma</span> <span class="o">+</span> <span class="n">rotation_sigma</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="k">assert</span> <span class="nb">len</span><span class="p">(</span><span class="n">sigma</span><span class="p">)</span><span class="o">==</span><span class="mi">4</span><span class="p">,</span> <span class="s1">&#39;Only accepts a tuple of sigma in size 1, 2, or 4.&#39;</span>
+        <span class="n">size</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">to_tuple</span><span class="p">(</span><span class="n">size</span><span class="p">)</span>
+        <span class="n">rotation</span> <span class="o">=</span> <span class="n">so3_type</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="o">*</span><span class="n">size</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="n">sigma</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span><span class="o">.</span><span class="n">detach</span><span class="p">()</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span>
+        <span class="n">sigma</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="n">sigma</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">sigma</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">sigma</span><span class="p">[</span><span class="mi">2</span><span class="p">]],</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+        <span class="n">translation</span> <span class="o">=</span> <span class="n">sigma</span> <span class="o">*</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="n">size</span> <span class="o">+</span> <span class="n">torch</span><span class="o">.</span><span class="n">Size</span><span class="p">([</span><span class="mi">3</span><span class="p">])),</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+        <span class="n">data</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">cat</span><span class="p">([</span><span class="n">translation</span><span class="p">,</span> <span class="n">rotation</span><span class="p">],</span> <span class="n">dim</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">ltype</span><span class="o">=</span><span class="n">se3_type</span><span class="p">)</span><span class="o">.</span><span class="n">requires_grad_</span><span class="p">(</span><span class="n">requires_grad</span><span class="p">)</span>
+
+
+<span class="k">class</span> <span class="nc">Sim3Type</span><span class="p">(</span><span class="n">LieType</span><span class="p">):</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">7</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">Log</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">):</span>
+        <span class="n">X</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">X</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="n">Sim3_Log</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">ltype</span><span class="o">=</span><span class="n">sim3_type</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">Act</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
+        <span class="k">assert</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">on_manifold</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">)</span>
+        <span class="k">assert</span> <span class="n">p</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">==</span><span class="mi">3</span> <span class="ow">or</span> <span class="n">p</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">==</span><span class="mi">4</span><span class="p">,</span> <span class="s2">&quot;Invalid Tensor Dimension&quot;</span>
+        <span class="n">X</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">X</span>
+        <span class="nb">input</span><span class="p">,</span> <span class="n">out_shape</span> <span class="o">=</span> <span class="n">broadcast_inputs</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
+        <span class="k">if</span> <span class="n">p</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">==</span><span class="mi">3</span><span class="p">:</span>
+            <span class="n">out</span> <span class="o">=</span> <span class="n">Sim3_Act</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="o">*</span><span class="nb">input</span><span class="p">)</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">out</span> <span class="o">=</span> <span class="n">Sim3_Act4</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="o">*</span><span class="nb">input</span><span class="p">)</span>
+        <span class="n">dim</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span> <span class="k">if</span> <span class="n">out</span><span class="o">.</span><span class="n">nelement</span><span class="p">()</span> <span class="o">!=</span> <span class="mi">0</span> <span class="k">else</span> <span class="n">X</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+        <span class="k">return</span> <span class="n">out</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="n">out_shape</span> <span class="o">+</span> <span class="p">(</span><span class="n">dim</span><span class="p">,))</span>
+
+    <span class="k">def</span> <span class="nf">Mul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">):</span>
+        <span class="c1"># Transform on transform</span>
+        <span class="n">X</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">X</span>
+        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">on_manifold</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">Y</span><span class="p">,</span> <span class="n">LieTensor</span><span class="p">)</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">Y</span><span class="o">.</span><span class="n">ltype</span><span class="o">.</span><span class="n">on_manifold</span><span class="p">:</span>
+            <span class="n">Y</span> <span class="o">=</span> <span class="n">Y</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">Y</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">Y</span>
+            <span class="nb">input</span><span class="p">,</span> <span class="n">out_shape</span> <span class="o">=</span> <span class="n">broadcast_inputs</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">)</span>
+            <span class="n">out</span> <span class="o">=</span> <span class="n">Sim3_Mul</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="o">*</span><span class="nb">input</span><span class="p">)</span>
+            <span class="n">dim</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span> <span class="k">if</span> <span class="n">out</span><span class="o">.</span><span class="n">nelement</span><span class="p">()</span> <span class="o">!=</span> <span class="mi">0</span> <span class="k">else</span> <span class="n">X</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+            <span class="n">out</span> <span class="o">=</span> <span class="n">out</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="n">out_shape</span> <span class="o">+</span> <span class="p">(</span><span class="n">dim</span><span class="p">,))</span>
+            <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">out</span><span class="p">,</span> <span class="n">ltype</span><span class="o">=</span><span class="n">Sim3_type</span><span class="p">)</span>
+        <span class="c1"># Transform on points</span>
+        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">on_manifold</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">Y</span><span class="p">,</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">):</span>
+            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">Act</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">)</span>
+        <span class="c1"># (scalar or tensor) * manifold</span>
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">on_manifold</span><span class="p">:</span>
+            <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">mul</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">),</span> <span class="n">ltype</span><span class="o">=</span><span class="n">Sim3_type</span><span class="p">)</span>
+        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s1">&#39;Invalid __mul__ operation&#39;</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">Inv</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">):</span>
+        <span class="n">X</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">X</span>
+        <span class="n">out</span> <span class="o">=</span> <span class="n">Sim3_Inv</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">out</span><span class="p">,</span> <span class="n">ltype</span><span class="o">=</span><span class="n">Sim3_type</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">Adj</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">a</span><span class="p">):</span>
+        <span class="n">X</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">X</span>
+        <span class="n">a</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">a</span>
+        <span class="nb">input</span><span class="p">,</span> <span class="n">out_shape</span> <span class="o">=</span> <span class="n">broadcast_inputs</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
+        <span class="n">out</span> <span class="o">=</span> <span class="n">Sim3_AdjXa</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="o">*</span><span class="nb">input</span><span class="p">)</span>
+        <span class="n">dim</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span> <span class="k">if</span> <span class="n">out</span><span class="o">.</span><span class="n">nelement</span><span class="p">()</span> <span class="o">!=</span> <span class="mi">0</span> <span class="k">else</span> <span class="n">X</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+        <span class="n">out</span> <span class="o">=</span> <span class="n">out</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="n">out_shape</span> <span class="o">+</span> <span class="p">(</span><span class="n">dim</span><span class="p">,))</span>
+        <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">out</span><span class="p">,</span> <span class="n">ltype</span><span class="o">=</span><span class="n">sim3_type</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">AdjT</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">a</span><span class="p">):</span>
+        <span class="n">X</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">X</span>
+        <span class="n">a</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">a</span>
+        <span class="nb">input</span><span class="p">,</span> <span class="n">out_shape</span> <span class="o">=</span> <span class="n">broadcast_inputs</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
+        <span class="n">out</span> <span class="o">=</span> <span class="n">Sim3_AdjTXa</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="o">*</span><span class="nb">input</span><span class="p">)</span>
+        <span class="n">dim</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span> <span class="k">if</span> <span class="n">out</span><span class="o">.</span><span class="n">nelement</span><span class="p">()</span> <span class="o">!=</span> <span class="mi">0</span> <span class="k">else</span> <span class="n">X</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+        <span class="n">out</span> <span class="o">=</span> <span class="n">out</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="n">out_shape</span> <span class="o">+</span> <span class="p">(</span><span class="n">dim</span><span class="p">,))</span>
+        <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">out</span><span class="p">,</span> <span class="n">ltype</span><span class="o">=</span><span class="n">sim3_type</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">Jinvp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">a</span><span class="p">):</span>
+        <span class="n">X</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">X</span>
+        <span class="n">a</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">a</span>
+        <span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">a</span><span class="p">),</span> <span class="n">out_shape</span> <span class="o">=</span> <span class="n">broadcast_inputs</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
+        <span class="n">out</span> <span class="o">=</span> <span class="p">(</span><span class="n">sim3_Jl_inv</span><span class="p">(</span><span class="n">Sim3_Log</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="n">X</span><span class="p">))</span> <span class="o">@</span> <span class="n">a</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
+        <span class="n">dim</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span> <span class="k">if</span> <span class="n">out</span><span class="o">.</span><span class="n">nelement</span><span class="p">()</span> <span class="o">!=</span> <span class="mi">0</span> <span class="k">else</span> <span class="n">X</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+        <span class="n">out</span> <span class="o">=</span> <span class="n">out</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="n">out_shape</span> <span class="o">+</span> <span class="p">(</span><span class="n">dim</span><span class="p">,))</span>
+        <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">out</span><span class="p">,</span> <span class="n">ltype</span><span class="o">=</span><span class="n">sim3_type</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">rotation</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">input</span><span class="p">):</span>
+        <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="nb">input</span><span class="o">.</span><span class="n">tensor</span><span class="p">()[</span><span class="o">...</span><span class="p">,</span> <span class="mi">3</span><span class="p">:</span><span class="mi">7</span><span class="p">],</span> <span class="n">ltype</span><span class="o">=</span><span class="n">SO3_type</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">translation</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">input</span><span class="p">):</span>
+        <span class="k">return</span> <span class="nb">input</span><span class="o">.</span><span class="n">tensor</span><span class="p">()[</span><span class="o">...</span><span class="p">,</span> <span class="mi">0</span><span class="p">:</span><span class="mi">3</span><span class="p">]</span>
+
+    <span class="k">def</span> <span class="nf">scale</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">input</span><span class="p">):</span>
+        <span class="k">return</span> <span class="nb">input</span><span class="o">.</span><span class="n">tensor</span><span class="p">()[</span><span class="o">...</span><span class="p">,</span> <span class="mi">7</span><span class="p">:</span><span class="mi">8</span><span class="p">]</span>
+
+    <span class="nd">@classmethod</span>
+    <span class="k">def</span> <span class="nf">identity</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">size</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+        <span class="n">data</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">],</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">data</span><span class="o">.</span><span class="n">repeat</span><span class="p">(</span><span class="n">size</span><span class="o">+</span><span class="p">(</span><span class="mi">1</span><span class="p">,)),</span> <span class="n">ltype</span><span class="o">=</span><span class="n">Sim3_type</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">randn</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">size</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">requires_grad</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+        <span class="n">data</span> <span class="o">=</span> <span class="n">sim3_type</span><span class="o">.</span><span class="n">Exp</span><span class="p">(</span><span class="n">sim3_type</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="o">*</span><span class="n">size</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="n">sigma</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span><span class="o">.</span><span class="n">detach</span><span class="p">()</span>
+        <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">ltype</span><span class="o">=</span><span class="n">Sim3_type</span><span class="p">)</span><span class="o">.</span><span class="n">requires_grad_</span><span class="p">(</span><span class="n">requires_grad</span><span class="p">)</span>
+
+    <span class="nd">@classmethod</span>
+    <span class="k">def</span> <span class="nf">add_</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="nb">input</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+        <span class="k">return</span> <span class="nb">input</span><span class="o">.</span><span class="n">copy_</span><span class="p">(</span><span class="n">LieTensor</span><span class="p">(</span><span class="n">other</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="p">:</span><span class="mi">7</span><span class="p">],</span> <span class="n">ltype</span><span class="o">=</span><span class="n">sim3_type</span><span class="p">)</span><span class="o">.</span><span class="n">Exp</span><span class="p">()</span> <span class="o">*</span> <span class="nb">input</span><span class="p">)</span>
+
+
+<span class="k">class</span> <span class="nc">sim3Type</span><span class="p">(</span><span class="n">LieType</span><span class="p">):</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">(</span><span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">7</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">Exp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">x</span>
+        <span class="n">X</span> <span class="o">=</span> <span class="n">sim3_Exp</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">ltype</span><span class="o">=</span><span class="n">Sim3_type</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">Mul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">):</span>
+        <span class="n">X</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">X</span>
+        <span class="c1"># (scalar or tensor) * manifold</span>
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">on_manifold</span><span class="p">:</span>
+            <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">mul</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">),</span> <span class="n">ltype</span><span class="o">=</span><span class="n">sim3_type</span><span class="p">)</span>
+        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s1">&#39;Invalid __mul__ operation&#39;</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">rotation</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">input</span><span class="p">):</span>
+        <span class="k">return</span> <span class="nb">input</span><span class="o">.</span><span class="n">Exp</span><span class="p">()</span><span class="o">.</span><span class="n">rotation</span><span class="p">()</span>
+
+    <span class="k">def</span> <span class="nf">translation</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">input</span><span class="p">):</span>
+        <span class="k">return</span> <span class="nb">input</span><span class="o">.</span><span class="n">Exp</span><span class="p">()</span><span class="o">.</span><span class="n">translation</span><span class="p">()</span>
+
+    <span class="k">def</span> <span class="nf">scale</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">input</span><span class="p">):</span>
+        <span class="k">return</span> <span class="nb">input</span><span class="o">.</span><span class="n">Exp</span><span class="p">()</span><span class="o">.</span><span class="n">scale</span><span class="p">()</span>
+
+    <span class="nd">@classmethod</span>
+    <span class="k">def</span> <span class="nf">identity</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">size</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+        <span class="k">return</span> <span class="n">Sim3_type</span><span class="o">.</span><span class="n">Log</span><span class="p">(</span><span class="n">Sim3_type</span><span class="o">.</span><span class="n">identity</span><span class="p">(</span><span class="o">*</span><span class="n">size</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
+
+    <span class="k">def</span> <span class="nf">randn</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">size</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">requires_grad</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">sigma</span><span class="p">,</span> <span class="n">collections</span><span class="o">.</span><span class="n">abc</span><span class="o">.</span><span class="n">Iterable</span><span class="p">):</span>
+            <span class="n">sigma</span> <span class="o">=</span> <span class="n">_ntuple</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="s2">&quot;_penta&quot;</span><span class="p">)(</span><span class="n">sigma</span><span class="p">)</span>
+        <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">sigma</span><span class="p">)</span><span class="o">==</span><span class="mi">3</span><span class="p">:</span>
+            <span class="n">rotation_sigma</span> <span class="o">=</span> <span class="n">_single</span><span class="p">(</span><span class="n">sigma</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">])</span>
+            <span class="n">scale_sigma</span> <span class="o">=</span> <span class="n">_single</span><span class="p">(</span><span class="n">sigma</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
+            <span class="n">translation_sigma</span> <span class="o">=</span> <span class="n">_triple</span><span class="p">(</span><span class="n">sigma</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+            <span class="n">sigma</span> <span class="o">=</span> <span class="n">translation_sigma</span><span class="o">+</span><span class="n">rotation_sigma</span><span class="o">+</span><span class="n">scale_sigma</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="k">assert</span> <span class="nb">len</span><span class="p">(</span><span class="n">sigma</span><span class="p">)</span><span class="o">==</span><span class="mi">5</span><span class="p">,</span> <span class="s1">&#39;Only accepts a tuple of sigma in size 1, 3, or 5.&#39;</span>
+        <span class="n">size</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">to_tuple</span><span class="p">(</span><span class="n">size</span><span class="p">)</span>
+        <span class="n">rotation</span> <span class="o">=</span> <span class="n">so3_type</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="o">*</span><span class="n">size</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="n">sigma</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">],</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span><span class="o">.</span><span class="n">detach</span><span class="p">()</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span>
+        <span class="n">scale</span> <span class="o">=</span> <span class="n">sigma</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="n">size</span> <span class="o">+</span> <span class="n">torch</span><span class="o">.</span><span class="n">Size</span><span class="p">([</span><span class="mi">1</span><span class="p">])),</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+        <span class="n">sigma</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="n">sigma</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">sigma</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">sigma</span><span class="p">[</span><span class="mi">2</span><span class="p">]],</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+        <span class="n">translation</span> <span class="o">=</span> <span class="n">sigma</span> <span class="o">*</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="n">size</span> <span class="o">+</span> <span class="n">torch</span><span class="o">.</span><span class="n">Size</span><span class="p">([</span><span class="mi">3</span><span class="p">])),</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+        <span class="n">data</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">cat</span><span class="p">([</span><span class="n">translation</span><span class="p">,</span> <span class="n">rotation</span><span class="p">,</span> <span class="n">scale</span><span class="p">],</span> <span class="n">dim</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">ltype</span><span class="o">=</span><span class="n">sim3_type</span><span class="p">)</span><span class="o">.</span><span class="n">requires_grad_</span><span class="p">(</span><span class="n">requires_grad</span><span class="p">)</span>
+
+
+<span class="k">class</span> <span class="nc">RxSO3Type</span><span class="p">(</span><span class="n">LieType</span><span class="p">):</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">Log</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">):</span>
+        <span class="n">X</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">X</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="n">RxSO3_Log</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">ltype</span><span class="o">=</span><span class="n">rxso3_type</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">Act</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
+        <span class="k">assert</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">on_manifold</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">)</span>
+        <span class="k">assert</span> <span class="n">p</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">==</span><span class="mi">3</span> <span class="ow">or</span> <span class="n">p</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">==</span><span class="mi">4</span><span class="p">,</span> <span class="s2">&quot;Invalid Tensor Dimension&quot;</span>
+        <span class="n">X</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">X</span>
+        <span class="nb">input</span><span class="p">,</span> <span class="n">out_shape</span> <span class="o">=</span> <span class="n">broadcast_inputs</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
+        <span class="k">if</span> <span class="n">p</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">==</span><span class="mi">3</span><span class="p">:</span>
+            <span class="n">out</span> <span class="o">=</span> <span class="n">RxSO3_Act</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="o">*</span><span class="nb">input</span><span class="p">)</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">out</span> <span class="o">=</span> <span class="n">RxSO3_Act4</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="o">*</span><span class="nb">input</span><span class="p">)</span>
+        <span class="n">dim</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span> <span class="k">if</span> <span class="n">out</span><span class="o">.</span><span class="n">nelement</span><span class="p">()</span> <span class="o">!=</span> <span class="mi">0</span> <span class="k">else</span> <span class="n">X</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+        <span class="k">return</span> <span class="n">out</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="n">out_shape</span> <span class="o">+</span> <span class="p">(</span><span class="n">dim</span><span class="p">,))</span>
+
+    <span class="k">def</span> <span class="nf">Mul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">):</span>
+        <span class="c1"># Transform on transform</span>
+        <span class="n">X</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">X</span>
+        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">on_manifold</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">Y</span><span class="p">,</span> <span class="n">LieTensor</span><span class="p">)</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">Y</span><span class="o">.</span><span class="n">ltype</span><span class="o">.</span><span class="n">on_manifold</span><span class="p">:</span>
+            <span class="n">Y</span> <span class="o">=</span> <span class="n">Y</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">Y</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">Y</span>
+            <span class="nb">input</span><span class="p">,</span> <span class="n">out_shape</span> <span class="o">=</span> <span class="n">broadcast_inputs</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">)</span>
+            <span class="n">out</span> <span class="o">=</span> <span class="n">RxSO3_Mul</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="o">*</span><span class="nb">input</span><span class="p">)</span>
+            <span class="n">dim</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span> <span class="k">if</span> <span class="n">out</span><span class="o">.</span><span class="n">nelement</span><span class="p">()</span> <span class="o">!=</span> <span class="mi">0</span> <span class="k">else</span> <span class="n">X</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+            <span class="n">out</span> <span class="o">=</span> <span class="n">out</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="n">out_shape</span> <span class="o">+</span> <span class="p">(</span><span class="n">dim</span><span class="p">,))</span>
+            <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">out</span><span class="p">,</span> <span class="n">ltype</span><span class="o">=</span><span class="n">RxSO3_type</span><span class="p">)</span>
+        <span class="c1"># Transform on points</span>
+        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">on_manifold</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">Y</span><span class="p">,</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">):</span>
+            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">Act</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">)</span>
+        <span class="c1"># (scalar or tensor) * manifold</span>
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">on_manifold</span><span class="p">:</span>
+            <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">mul</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">),</span> <span class="n">ltype</span><span class="o">=</span><span class="n">RxSO3_type</span><span class="p">)</span>
+        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s1">&#39;Invalid __mul__ operation&#39;</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">Inv</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">):</span>
+        <span class="n">X</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">X</span>
+        <span class="n">out</span> <span class="o">=</span> <span class="n">RxSO3_Inv</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">out</span><span class="p">,</span> <span class="n">ltype</span><span class="o">=</span><span class="n">RxSO3_type</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">Adj</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">a</span><span class="p">):</span>
+        <span class="n">X</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">X</span>
+        <span class="n">a</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">a</span>
+        <span class="nb">input</span><span class="p">,</span> <span class="n">out_shape</span> <span class="o">=</span> <span class="n">broadcast_inputs</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
+        <span class="n">out</span> <span class="o">=</span> <span class="n">RxSO3_AdjXa</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="o">*</span><span class="nb">input</span><span class="p">)</span>
+        <span class="n">dim</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span> <span class="k">if</span> <span class="n">out</span><span class="o">.</span><span class="n">nelement</span><span class="p">()</span> <span class="o">!=</span> <span class="mi">0</span> <span class="k">else</span> <span class="n">X</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+        <span class="n">out</span> <span class="o">=</span> <span class="n">out</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="n">out_shape</span> <span class="o">+</span> <span class="p">(</span><span class="n">dim</span><span class="p">,))</span>
+        <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">out</span><span class="p">,</span> <span class="n">ltype</span><span class="o">=</span><span class="n">rxso3_type</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">AdjT</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">a</span><span class="p">):</span>
+        <span class="n">X</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">X</span>
+        <span class="n">a</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">a</span>
+        <span class="nb">input</span><span class="p">,</span> <span class="n">out_shape</span> <span class="o">=</span> <span class="n">broadcast_inputs</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
+        <span class="n">out</span> <span class="o">=</span> <span class="n">RxSO3_AdjTXa</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="o">*</span><span class="nb">input</span><span class="p">)</span>
+        <span class="n">dim</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span> <span class="k">if</span> <span class="n">out</span><span class="o">.</span><span class="n">nelement</span><span class="p">()</span> <span class="o">!=</span> <span class="mi">0</span> <span class="k">else</span> <span class="n">X</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+        <span class="n">out</span> <span class="o">=</span> <span class="n">out</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="n">out_shape</span> <span class="o">+</span> <span class="p">(</span><span class="n">dim</span><span class="p">,))</span>
+        <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">out</span><span class="p">,</span> <span class="n">ltype</span><span class="o">=</span><span class="n">rxso3_type</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">Jinvp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">a</span><span class="p">):</span>
+        <span class="n">X</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">X</span>
+        <span class="n">a</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">a</span>
+        <span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">a</span><span class="p">),</span> <span class="n">out_shape</span> <span class="o">=</span> <span class="n">broadcast_inputs</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
+        <span class="n">out</span> <span class="o">=</span> <span class="p">(</span><span class="n">rxso3_Jl_inv</span><span class="p">(</span><span class="n">RxSO3_Log</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="n">X</span><span class="p">))</span> <span class="o">@</span> <span class="n">a</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
+        <span class="n">dim</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span> <span class="k">if</span> <span class="n">out</span><span class="o">.</span><span class="n">nelement</span><span class="p">()</span> <span class="o">!=</span> <span class="mi">0</span> <span class="k">else</span> <span class="n">X</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+        <span class="n">out</span> <span class="o">=</span> <span class="n">out</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="n">out_shape</span> <span class="o">+</span> <span class="p">(</span><span class="n">dim</span><span class="p">,))</span>
+        <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">out</span><span class="p">,</span> <span class="n">ltype</span><span class="o">=</span><span class="n">rxso3_type</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">rotation</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">input</span><span class="p">):</span>
+        <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="nb">input</span><span class="o">.</span><span class="n">tensor</span><span class="p">()[</span><span class="o">...</span><span class="p">,</span> <span class="mi">0</span><span class="p">:</span><span class="mi">4</span><span class="p">],</span> <span class="n">ltype</span><span class="o">=</span><span class="n">SO3_type</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">scale</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">input</span><span class="p">):</span>
+        <span class="k">return</span> <span class="nb">input</span><span class="o">.</span><span class="n">tensor</span><span class="p">()[</span><span class="o">...</span><span class="p">,</span> <span class="mi">4</span><span class="p">:</span><span class="mi">5</span><span class="p">]</span>
+
+    <span class="nd">@classmethod</span>
+    <span class="k">def</span> <span class="nf">identity</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">size</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+        <span class="n">data</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">],</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">data</span><span class="o">.</span><span class="n">repeat</span><span class="p">(</span><span class="n">size</span><span class="o">+</span><span class="p">(</span><span class="mi">1</span><span class="p">,)),</span> <span class="n">ltype</span><span class="o">=</span><span class="n">RxSO3_type</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">randn</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">size</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">requires_grad</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+        <span class="n">data</span> <span class="o">=</span> <span class="n">rxso3_type</span><span class="o">.</span><span class="n">Exp</span><span class="p">(</span><span class="n">rxso3_type</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="o">*</span><span class="n">size</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="n">sigma</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span><span class="o">.</span><span class="n">detach</span><span class="p">()</span>
+        <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">ltype</span><span class="o">=</span><span class="n">RxSO3_type</span><span class="p">)</span><span class="o">.</span><span class="n">requires_grad_</span><span class="p">(</span><span class="n">requires_grad</span><span class="p">)</span>
+
+    <span class="nd">@classmethod</span>
+    <span class="k">def</span> <span class="nf">add_</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="nb">input</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+        <span class="k">return</span> <span class="nb">input</span><span class="o">.</span><span class="n">copy_</span><span class="p">(</span><span class="n">LieTensor</span><span class="p">(</span><span class="n">other</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="p">:</span><span class="mi">4</span><span class="p">],</span> <span class="n">ltype</span><span class="o">=</span><span class="n">rxso3_type</span><span class="p">)</span><span class="o">.</span><span class="n">Exp</span><span class="p">()</span> <span class="o">*</span> <span class="nb">input</span><span class="p">)</span>
+
+
+<span class="k">class</span> <span class="nc">rxso3Type</span><span class="p">(</span><span class="n">LieType</span><span class="p">):</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">Exp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">x</span>
+        <span class="n">X</span> <span class="o">=</span> <span class="n">rxso3_Exp</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">ltype</span><span class="o">=</span><span class="n">RxSO3_type</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">Mul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">):</span>
+        <span class="n">X</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">X</span>
+        <span class="c1"># (scalar or tensor) * manifold</span>
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">on_manifold</span><span class="p">:</span>
+            <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">mul</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">),</span> <span class="n">ltype</span><span class="o">=</span><span class="n">rxso3_type</span><span class="p">)</span>
+        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s1">&#39;Invalid __mul__ operation&#39;</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">rotation</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">input</span><span class="p">):</span>
+        <span class="k">return</span> <span class="nb">input</span><span class="o">.</span><span class="n">Exp</span><span class="p">()</span><span class="o">.</span><span class="n">rotation</span><span class="p">()</span>
+
+    <span class="k">def</span> <span class="nf">scale</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">input</span><span class="p">):</span>
+        <span class="k">return</span> <span class="nb">input</span><span class="o">.</span><span class="n">Exp</span><span class="p">()</span><span class="o">.</span><span class="n">scale</span><span class="p">()</span>
+
+    <span class="nd">@classmethod</span>
+    <span class="k">def</span> <span class="nf">identity</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">size</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+        <span class="k">return</span> <span class="n">RxSO3_type</span><span class="o">.</span><span class="n">Log</span><span class="p">(</span><span class="n">RxSO3_type</span><span class="o">.</span><span class="n">identity</span><span class="p">(</span><span class="o">*</span><span class="n">size</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
+
+    <span class="k">def</span> <span class="nf">randn</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">size</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">requires_grad</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">sigma</span><span class="p">,</span> <span class="n">collections</span><span class="o">.</span><span class="n">abc</span><span class="o">.</span><span class="n">Iterable</span><span class="p">):</span>
+            <span class="n">sigma</span> <span class="o">=</span> <span class="n">_pair</span><span class="p">(</span><span class="n">sigma</span><span class="p">)</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="k">assert</span> <span class="nb">len</span><span class="p">(</span><span class="n">sigma</span><span class="p">)</span><span class="o">==</span><span class="mi">2</span><span class="p">,</span> <span class="s1">&#39;Only accepts a tuple of sigma in size 1 or 2.&#39;</span>
+        <span class="n">size</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">to_tuple</span><span class="p">(</span><span class="n">size</span><span class="p">)</span>
+        <span class="n">rotation</span> <span class="o">=</span> <span class="n">so3_type</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="o">*</span><span class="n">size</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="n">sigma</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span>
+        <span class="n">scale</span> <span class="o">=</span> <span class="n">sigma</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="n">size</span> <span class="o">+</span> <span class="n">torch</span><span class="o">.</span><span class="n">Size</span><span class="p">([</span><span class="mi">1</span><span class="p">])),</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+        <span class="n">data</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">cat</span><span class="p">([</span><span class="n">rotation</span><span class="p">,</span> <span class="n">scale</span><span class="p">],</span> <span class="n">dim</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">LieTensor</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">ltype</span><span class="o">=</span><span class="n">rxso3_type</span><span class="p">)</span><span class="o">.</span><span class="n">requires_grad_</span><span class="p">(</span><span class="n">requires_grad</span><span class="p">)</span>
+
+
+<span class="n">SO3_type</span><span class="p">,</span> <span class="n">so3_type</span> <span class="o">=</span> <span class="n">SO3Type</span><span class="p">(),</span> <span class="n">so3Type</span><span class="p">()</span>
+<span class="n">SE3_type</span><span class="p">,</span> <span class="n">se3_type</span> <span class="o">=</span> <span class="n">SE3Type</span><span class="p">(),</span> <span class="n">se3Type</span><span class="p">()</span>
+<span class="n">Sim3_type</span><span class="p">,</span> <span class="n">sim3_type</span> <span class="o">=</span> <span class="n">Sim3Type</span><span class="p">(),</span> <span class="n">sim3Type</span><span class="p">()</span>
+<span class="n">RxSO3_type</span><span class="p">,</span> <span class="n">rxso3_type</span> <span class="o">=</span> <span class="n">RxSO3Type</span><span class="p">(),</span> <span class="n">rxso3Type</span><span class="p">()</span>
+
+
+<div class="viewcode-block" id="LieTensor"><a class="viewcode-back" href="../../../../generated/pypose.LieTensor/#pypose.LieTensor">[docs]</a><span class="k">class</span> <span class="nc">LieTensor</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&quot;&quot;&quot; A sub-class of :obj:`torch.Tensor` to represent Lie Algebra and Lie Group.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        data (:obj:`Tensor`, or :obj:`list`, or &#39;:obj:`int`...&#39;): A</span>
+<span class="sd">            :obj:`Tensor` object, or constructing a :obj:`Tensor`</span>
+<span class="sd">            object from :obj:`list`, which defines tensor data, or from</span>
+<span class="sd">            &#39;:obj:`int`...&#39;, which defines tensor shape.</span>
+
+<span class="sd">            The shape of :obj:`Tensor` object should be compatible</span>
+<span class="sd">            with Lie Type :obj:`ltype`, otherwise error will be raised.</span>
+
+<span class="sd">        ltype (:obj:`ltype`): Lie Type, either **Lie Group** or **Lie Algebra** is listed below:</span>
+
+<span class="sd">    Returns:</span>
+<span class="sd">        LieTensor corresponding to Lie Type :obj:`ltype`.</span>
+
+<span class="sd">    .. list-table:: List of :obj:`ltype` for **Lie Group**</span>
+<span class="sd">        :widths: 25 25 30 30</span>
+<span class="sd">        :header-rows: 1</span>
+
+<span class="sd">        * - Representation</span>
+<span class="sd">          - :obj:`ltype`</span>
+<span class="sd">          - :obj:`shape`</span>
+<span class="sd">          - Alias Class</span>
+<span class="sd">        * - Rotation</span>
+<span class="sd">          - :obj:`SO3_type`</span>
+<span class="sd">          - :obj:`(*, 4)`</span>
+<span class="sd">          - :meth:`SO3`</span>
+<span class="sd">        * - Translation + Rotation</span>
+<span class="sd">          - :obj:`SE3_type`</span>
+<span class="sd">          - :obj:`(*, 7)`</span>
+<span class="sd">          - :meth:`SE3`</span>
+<span class="sd">        * - Translation + Rotation + Scale</span>
+<span class="sd">          - :obj:`Sim3_type`</span>
+<span class="sd">          - :obj:`(*, 8)`</span>
+<span class="sd">          - :meth:`Sim3`</span>
+<span class="sd">        * - Rotation + Scale</span>
+<span class="sd">          - :obj:`RxSO3_type`</span>
+<span class="sd">          - :obj:`(*, 5)`</span>
+<span class="sd">          - :meth:`RxSO3`</span>
+
+<span class="sd">    .. list-table:: List of :obj:`ltype` for **Lie Algebra**</span>
+<span class="sd">        :widths: 25 25 30 30</span>
+<span class="sd">        :header-rows: 1</span>
+
+<span class="sd">        * - Representation</span>
+<span class="sd">          - :obj:`ltype`</span>
+<span class="sd">          - :obj:`shape`</span>
+<span class="sd">          - Alias Class</span>
+<span class="sd">        * - Rotation</span>
+<span class="sd">          - :obj:`so3_type`</span>
+<span class="sd">          - :obj:`(*, 3)`</span>
+<span class="sd">          - :meth:`so3`</span>
+<span class="sd">        * - Translation + Rotation</span>
+<span class="sd">          - :obj:`se3_type`</span>
+<span class="sd">          - :obj:`(*, 6)`</span>
+<span class="sd">          - :meth:`se3`</span>
+<span class="sd">        * - Translation + Rotation + Scale</span>
+<span class="sd">          - :obj:`sim3_type`</span>
+<span class="sd">          - :obj:`(*, 7)`</span>
+<span class="sd">          - :meth:`sim3`</span>
+<span class="sd">        * - Rotation + Scale</span>
+<span class="sd">          - :obj:`rxso3_type`</span>
+<span class="sd">          - :obj:`(*, 4)`</span>
+<span class="sd">          - :meth:`rxso3`</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        Two attributes :obj:`shape` and :obj:`lshape` are available for LieTensor.</span>
+<span class="sd">        The only differece is the :obj:`lshape` hides the last dimension of :obj:`shape`,</span>
+<span class="sd">        since :obj:`lshape` takes the data in the last dimension as a single :obj:`ltype` item.</span>
+
+<span class="sd">        See LieTensor method :meth:`lview` for more details.</span>
+
+<span class="sd">    Examples:</span>
+<span class="sd">        &gt;&gt;&gt; import torch</span>
+<span class="sd">        &gt;&gt;&gt; import pypose as pp</span>
+<span class="sd">        &gt;&gt;&gt; data = torch.randn(3, 3, requires_grad=True, device=&#39;cuda:0&#39;)</span>
+<span class="sd">        &gt;&gt;&gt; pp.LieTensor(data, ltype=pp.so3_type)</span>
+<span class="sd">        so3Type LieTensor:</span>
+<span class="sd">        tensor([[ 0.9520,  0.4517,  0.5834],</span>
+<span class="sd">                [-0.8106,  0.8197,  0.7077],</span>
+<span class="sd">                [-0.5743,  0.8182, -1.2104]], device=&#39;cuda:0&#39;, grad_fn=&lt;AliasBackward0&gt;)</span>
+
+<span class="sd">        Alias class for specific LieTensor is recommended:</span>
+
+<span class="sd">        &gt;&gt;&gt; pp.so3(data)</span>
+<span class="sd">        so3Type LieTensor:</span>
+<span class="sd">        tensor([[ 0.9520,  0.4517,  0.5834],</span>
+<span class="sd">                [-0.8106,  0.8197,  0.7077],</span>
+<span class="sd">                [-0.5743,  0.8182, -1.2104]], device=&#39;cuda:0&#39;, grad_fn=&lt;AliasBackward0&gt;)</span>
+
+<span class="sd">        See more alias classes at `Table 1 for Lie Group &lt;#id1&gt;`_  and `Table 2 for Lie Algebra &lt;#id2&gt;`_.</span>
+
+<span class="sd">        Other constructors:</span>
+
+<span class="sd">            - From list.</span>
+
+<span class="sd">            &gt;&gt;&gt; pp.so3([0, 0, 0])</span>
+<span class="sd">            so3Type LieTensor:</span>
+<span class="sd">            tensor([0., 0., 0.])</span>
+
+<span class="sd">            - From ints.</span>
+
+<span class="sd">            &gt;&gt;&gt; pp.so3(2, 3)</span>
+<span class="sd">            so3Type LieTensor:</span>
+<span class="sd">            tensor([[0., 0., 0.],</span>
+<span class="sd">                    [0., 0., 0.]])</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        Alias class for LieTensor is recommended.</span>
+<span class="sd">        For example, the following usage is equivalent:</span>
+
+<span class="sd">        - :obj:`pp.LieTensor(tensor, ltype=pp.so3_type)`</span>
+
+<span class="sd">        - :obj:`pp.so3(tensor)` (This is preferred).</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        All attributes from Tensor are available for LieTensor, e.g., :obj:`dtype`,</span>
+<span class="sd">        :obj:`device`, and :obj:`requires_grad`. See more details at</span>
+<span class="sd">        `tensor attributes &lt;https://pytorch.org/docs/stable/tensor_attributes.html&gt;`_.</span>
+
+<span class="sd">        Example:</span>
+
+<span class="sd">            &gt;&gt;&gt; data = torch.randn(1, 3, dtype=torch.float64, device=&quot;cuda&quot;, requires_grad=True)</span>
+<span class="sd">            &gt;&gt;&gt; pp.so3(data) # All Tensor attributes are available for LieTensor</span>
+<span class="sd">            so3Type LieTensor:</span>
+<span class="sd">            tensor([[-1.5948,  0.3113, -0.9807]], device=&#39;cuda:0&#39;, dtype=torch.float64,</span>
+<span class="sd">                grad_fn=&lt;AliasBackward0&gt;)</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        In most of the cases, Lie Group is expected to be used, therefore we only provide</span>
+<span class="sd">        `converting functions &lt;https://pypose.org/docs/main/convert/&gt;`_ between Lie Groups</span>
+<span class="sd">        and other data structures, e.g., transformation matrix, Euler angle, etc. The users</span>
+<span class="sd">        can convert data between Lie Group and Lie algebra with :obj:`Exp` and :obj:`Log`.</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">data</span><span class="p">,</span> <span class="n">ltype</span><span class="p">:</span><span class="n">LieType</span><span class="p">):</span>
+        <span class="k">assert</span> <span class="bp">self</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">:]</span> <span class="o">==</span> <span class="n">ltype</span><span class="o">.</span><span class="n">dimension</span><span class="p">,</span> <span class="s1">&#39;The last dimension of a LieTensor has to be &#39;</span> \
+            <span class="s1">&#39;corresponding to their LieType. More details go to </span><span class="si">{}</span><span class="s1">. If this error happens in an &#39;</span> \
+            <span class="s1">&#39;optimization process, where LieType is not a necessary structure, we suggest to &#39;</span>    \
+            <span class="s1">&#39;call .tensor() to convert a LieTensor to Tensor before passing it to an optimizer. &#39;</span> \
+            <span class="s1">&#39;If this still happens, create an issue on GitHub please.&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span>
+            <span class="s1">&#39;https://pypose.org/docs/main/generated/pypose.LieTensor&#39;</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">ltype</span> <span class="o">=</span> <span class="n">ltype</span>
+
+    <span class="nd">@staticmethod</span>
+    <span class="k">def</span> <span class="fm">__new__</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">data</span><span class="p">,</span> <span class="n">ltype</span><span class="p">):</span>
+        <span class="n">tensor</span> <span class="o">=</span> <span class="n">data</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">data</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">)</span> <span class="k">else</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">(</span><span class="o">*</span><span class="n">data</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="o">.</span><span class="n">as_subclass</span><span class="p">(</span><span class="n">tensor</span><span class="p">,</span> <span class="n">LieTensor</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="fm">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="s1">&#39;ltype&#39;</span><span class="p">):</span>
+            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">ltype</span><span class="o">.</span><span class="vm">__class__</span><span class="o">.</span><span class="vm">__name__</span> <span class="o">+</span> \
+                   <span class="s1">&#39; </span><span class="si">%s</span><span class="s1">:</span><span class="se">\n</span><span class="s1">&#39;</span><span class="o">%</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="vm">__class__</span><span class="o">.</span><span class="vm">__name__</span><span class="p">)</span> <span class="o">+</span> <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__repr__</span><span class="p">()</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="k">return</span> <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__repr__</span><span class="p">()</span>
+
+<div class="viewcode-block" id="LieTensor.new_empty"><a class="viewcode-back" href="../../../../generated/pypose.LieTensor/#pypose.LieTensor.new_empty">[docs]</a>    <span class="k">def</span> <span class="nf">new_empty</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">shape</span><span class="p">):</span>
+        <span class="k">return</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="o">.</span><span class="n">as_subclass</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">empty</span><span class="p">(</span><span class="n">shape</span><span class="p">),</span> <span class="n">LieTensor</span><span class="p">)</span></div>
+
+    <span class="nd">@classmethod</span>
+    <span class="k">def</span> <span class="nf">__torch_function__</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">types</span><span class="p">,</span> <span class="n">args</span><span class="o">=</span><span class="p">(),</span> <span class="n">kwargs</span><span class="o">=</span><span class="p">{}):</span>
+        <span class="n">ltypes</span> <span class="o">=</span> <span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span> <span class="k">if</span> <span class="n">t</span> <span class="ow">is</span> <span class="n">LieTensor</span> <span class="ow">or</span> <span class="n">Parameter</span> <span class="k">else</span> <span class="n">t</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">types</span><span class="p">)</span>
+        <span class="n">data</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="o">.</span><span class="n">__torch_function__</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="n">ltypes</span><span class="p">,</span> <span class="n">args</span><span class="p">,</span> <span class="n">kwargs</span><span class="p">)</span>
+        <span class="k">if</span> <span class="n">data</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span> <span class="ow">and</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="s1">&#39;__name__&#39;</span><span class="p">)</span> <span class="ow">and</span> <span class="n">func</span><span class="o">.</span><span class="vm">__name__</span> <span class="ow">in</span> <span class="n">HANDLED_FUNCTIONS</span><span class="p">:</span>
+            <span class="n">args</span><span class="p">,</span> <span class="n">spec</span> <span class="o">=</span> <span class="n">tree_flatten</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
+            <span class="n">ltype</span> <span class="o">=</span> <span class="p">[</span><span class="n">arg</span><span class="o">.</span><span class="n">ltype</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">args</span> <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">LieTensor</span><span class="p">)][</span><span class="mi">0</span><span class="p">]</span>
+            <span class="k">def</span> <span class="nf">wrap</span><span class="p">(</span><span class="n">t</span><span class="p">):</span>
+                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">)</span> <span class="ow">and</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="bp">cls</span><span class="p">):</span>
+                    <span class="n">lt</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="o">.</span><span class="n">as_subclass</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">LieTensor</span><span class="p">)</span>
+                    <span class="n">lt</span><span class="o">.</span><span class="n">ltype</span> <span class="o">=</span> <span class="n">ltype</span>
+                    <span class="k">if</span> <span class="n">lt</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">:]</span> <span class="o">!=</span> <span class="n">lt</span><span class="o">.</span><span class="n">ltype</span><span class="o">.</span><span class="n">dimension</span><span class="p">:</span>
+                        <span class="n">link</span> <span class="o">=</span> <span class="s1">&#39;https://pypose.org/docs/main/generated/pypose.LieTensor&#39;</span>
+                        <span class="n">warnings</span><span class="o">.</span><span class="n">warn</span><span class="p">(</span><span class="s1">&#39;Tensor Shape Invalid by calling </span><span class="si">{}</span><span class="s1">, &#39;</span> \
+                            <span class="s1">&#39;go to </span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="n">link</span><span class="p">))</span>
+                    <span class="k">return</span> <span class="n">lt</span>
+                <span class="k">return</span> <span class="n">t</span>
+            <span class="k">return</span> <span class="n">tree_map</span><span class="p">(</span><span class="n">wrap</span><span class="p">,</span> <span class="n">data</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">data</span>
+
+    <span class="nd">@property</span>
+    <span class="k">def</span> <span class="nf">lshape</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="n">torch</span><span class="o">.</span><span class="n">Size</span><span class="p">:</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        LieTensor Shape (shape of torch.Tensor by ignoring the last dimension)</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            torch.Size</span>
+
+<span class="sd">        Note:</span>
+<span class="sd">            - The only difference from :obj:`shape` is the last dimension is hidden,</span>
+<span class="sd">              since :obj:`lshape` takes the last dimension as a single :obj:`ltype` item.</span>
+
+<span class="sd">            - The last dimension can also be accessed via :obj:`LieTensor.ltype.dimension`.</span>
+
+<span class="sd">        Examples:</span>
+<span class="sd">            &gt;&gt;&gt; x = pp.randn_SE3(2)</span>
+<span class="sd">            &gt;&gt;&gt; x.lshape</span>
+<span class="sd">            torch.Size([2])</span>
+<span class="sd">            &gt;&gt;&gt; x.shape</span>
+<span class="sd">            torch.Size([2, 7])</span>
+<span class="sd">            &gt;&gt;&gt; x.ltype.dimension</span>
+<span class="sd">            torch.Size([7])</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">shape</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+
+<div class="viewcode-block" id="LieTensor.lview"><a class="viewcode-back" href="../../../../generated/pypose.LieTensor/#pypose.LieTensor.lview">[docs]</a>    <span class="k">def</span> <span class="nf">lview</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">shape</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Returns a new LieTensor with the same data as the self tensor but of a different</span>
+<span class="sd">        :obj:`lshape`.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            shape (torch.Size or int...): the desired size</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            A new lieGroup tensor sharing with the same data as the self tensor but of a</span>
+<span class="sd">            different shape.</span>
+
+<span class="sd">        Note:</span>
+<span class="sd">            The only difference from :meth:`view` is the last dimension is hidden.</span>
+
+<span class="sd">            See `Tensor.view &lt;https://tinyurl.com/mrds8nmd&gt;`_</span>
+<span class="sd">            for its usage.</span>
+
+<span class="sd">        Examples:</span>
+<span class="sd">            &gt;&gt;&gt; x = pp.randn_so3(2,2)</span>
+<span class="sd">            &gt;&gt;&gt; x.shape</span>
+<span class="sd">            torch.Size([2, 2, 3])</span>
+<span class="sd">            &gt;&gt;&gt; x.lview(-1).lshape</span>
+<span class="sd">            torch.Size([4])</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="o">*</span><span class="n">shape</span><span class="o">+</span><span class="bp">self</span><span class="o">.</span><span class="n">ltype</span><span class="o">.</span><span class="n">dimension</span><span class="p">)</span></div>
+
+<div class="viewcode-block" id="LieTensor.Exp"><a class="viewcode-back" href="../../../../generated/pypose.LieTensor/#pypose.LieTensor.Exp">[docs]</a>    <span class="k">def</span> <span class="nf">Exp</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        See :meth:`pypose.Exp`</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">ltype</span><span class="o">.</span><span class="n">Exp</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span></div>
+
+<div class="viewcode-block" id="LieTensor.Log"><a class="viewcode-back" href="../../../../generated/pypose.LieTensor/#pypose.LieTensor.Log">[docs]</a>    <span class="k">def</span> <span class="nf">Log</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        See :meth:`pypose.Log`</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">ltype</span><span class="o">.</span><span class="n">Log</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span></div>
+
+<div class="viewcode-block" id="LieTensor.Inv"><a class="viewcode-back" href="../../../../generated/pypose.LieTensor/#pypose.LieTensor.Inv">[docs]</a>    <span class="k">def</span> <span class="nf">Inv</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        See :meth:`pypose.Inv`</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">ltype</span><span class="o">.</span><span class="n">Inv</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span></div>
+
+<div class="viewcode-block" id="LieTensor.Act"><a class="viewcode-back" href="../../../../generated/pypose.LieTensor/#pypose.LieTensor.Act">[docs]</a>    <span class="k">def</span> <span class="nf">Act</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        See :meth:`pypose.Act`</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">ltype</span><span class="o">.</span><span class="n">Act</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span></div>
+
+<div class="viewcode-block" id="LieTensor.add"><a class="viewcode-back" href="../../../../generated/pypose.LieTensor/#pypose.LieTensor.add">[docs]</a>    <span class="k">def</span> <span class="nf">add</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        See :meth:`pypose.add`</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">clone</span><span class="p">()</span><span class="o">.</span><span class="n">add_</span><span class="p">(</span><span class="n">other</span> <span class="o">=</span> <span class="n">alpha</span> <span class="o">*</span> <span class="n">other</span><span class="p">)</span></div>
+
+<div class="viewcode-block" id="LieTensor.add_"><a class="viewcode-back" href="../../../../generated/pypose.LieTensor/#pypose.LieTensor.add_">[docs]</a>    <span class="k">def</span> <span class="nf">add_</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        See :meth:`pypose.add_`</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">ltype</span><span class="o">.</span><span class="n">add_</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span> <span class="o">=</span> <span class="n">alpha</span> <span class="o">*</span> <span class="n">other</span><span class="p">)</span></div>
+
+    <span class="k">def</span> <span class="fm">__add__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">other</span><span class="o">=</span><span class="n">other</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="fm">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        See :meth:`pypose.mul`</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">ltype</span><span class="o">.</span><span class="n">Mul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
+
+<div class="viewcode-block" id="LieTensor.mul"><a class="viewcode-back" href="../../../../generated/pypose.LieTensor/#pypose.LieTensor.mul">[docs]</a>    <span class="k">def</span> <span class="nf">mul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        See :meth:`pypose.mul`</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">ltype</span><span class="o">.</span><span class="n">Mul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span></div>
+
+    <span class="k">def</span> <span class="fm">__matmul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        See :meth:`pypose.matmul`</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">LieTensor</span><span class="p">):</span>
+            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">ltype</span><span class="o">.</span><span class="n">Mul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
+        <span class="k">else</span><span class="p">:</span> <span class="c1"># Same with: self.ltype.matrix(self) @ other</span>
+            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">Act</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
+
+<div class="viewcode-block" id="LieTensor.Retr"><a class="viewcode-back" href="../../../../generated/pypose.LieTensor/#pypose.LieTensor.Retr">[docs]</a>    <span class="k">def</span> <span class="nf">Retr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">a</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        See :meth:`pypose.Retr`</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">ltype</span><span class="o">.</span><span class="n">Retr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span></div>
+
+<div class="viewcode-block" id="LieTensor.Adj"><a class="viewcode-back" href="../../../../generated/pypose.LieTensor/#pypose.LieTensor.Adj">[docs]</a>    <span class="k">def</span> <span class="nf">Adj</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">a</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        See :meth:`pypose.Adj`</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">ltype</span><span class="o">.</span><span class="n">Adj</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span></div>
+
+<div class="viewcode-block" id="LieTensor.AdjT"><a class="viewcode-back" href="../../../../generated/pypose.LieTensor/#pypose.LieTensor.AdjT">[docs]</a>    <span class="k">def</span> <span class="nf">AdjT</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">a</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        See :meth:`pypose.AdjT`</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">ltype</span><span class="o">.</span><span class="n">AdjT</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span></div>
+
+<div class="viewcode-block" id="LieTensor.Jinvp"><a class="viewcode-back" href="../../../../generated/pypose.LieTensor/#pypose.LieTensor.Jinvp">[docs]</a>    <span class="k">def</span> <span class="nf">Jinvp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        See :meth:`pypose.Jinvp`</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">ltype</span><span class="o">.</span><span class="n">Jinvp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span></div>
+
+<div class="viewcode-block" id="LieTensor.Jr"><a class="viewcode-back" href="../../../../generated/pypose.LieTensor/#pypose.LieTensor.Jr">[docs]</a>    <span class="k">def</span> <span class="nf">Jr</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        See :meth:`pypose.Jr`</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">ltype</span><span class="o">.</span><span class="n">Jr</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span></div>
+
+<div class="viewcode-block" id="LieTensor.tensor"><a class="viewcode-back" href="../../../../generated/pypose.LieTensor/#pypose.LieTensor.tensor">[docs]</a>    <span class="k">def</span> <span class="nf">tensor</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">:</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        See :meth:`pypose.tensor`</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">return</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="o">.</span><span class="n">as_subclass</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">)</span></div>
+
+<div class="viewcode-block" id="LieTensor.matrix"><a class="viewcode-back" href="../../../../generated/pypose.LieTensor/#pypose.LieTensor.matrix">[docs]</a>    <span class="k">def</span> <span class="nf">matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">:</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        See :meth:`pypose.matrix`</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">ltype</span><span class="o">.</span><span class="n">matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span></div>
+
+<div class="viewcode-block" id="LieTensor.translation"><a class="viewcode-back" href="../../../../generated/pypose.LieTensor/#pypose.LieTensor.translation">[docs]</a>    <span class="k">def</span> <span class="nf">translation</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">:</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        See :meth:`pypose.translation`</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">ltype</span><span class="o">.</span><span class="n">translation</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span></div>
+
+<div class="viewcode-block" id="LieTensor.rotation"><a class="viewcode-back" href="../../../../generated/pypose.LieTensor/#pypose.LieTensor.rotation">[docs]</a>    <span class="k">def</span> <span class="nf">rotation</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        See :meth:`pypose.rotation`</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">ltype</span><span class="o">.</span><span class="n">rotation</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span></div>
+
+<div class="viewcode-block" id="LieTensor.scale"><a class="viewcode-back" href="../../../../generated/pypose.LieTensor/#pypose.LieTensor.scale">[docs]</a>    <span class="k">def</span> <span class="nf">scale</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">:</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        See :meth:`pypose.scale`</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">ltype</span><span class="o">.</span><span class="n">scale</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span></div>
+
+<div class="viewcode-block" id="LieTensor.euler"><a class="viewcode-back" href="../../../../generated/pypose.LieTensor/#pypose.LieTensor.euler">[docs]</a>    <span class="k">def</span> <span class="nf">euler</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">eps</span><span class="o">=</span><span class="mf">2e-4</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">:</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        See :meth:`pypose.euler`</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="n">data</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">rotation</span><span class="p">()</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span>
+        <span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">data</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">data</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
+        <span class="n">z</span><span class="p">,</span> <span class="n">w</span> <span class="o">=</span> <span class="n">data</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="n">data</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">3</span><span class="p">]</span>
+        <span class="n">xx</span><span class="p">,</span> <span class="n">yy</span><span class="p">,</span> <span class="n">zz</span><span class="p">,</span> <span class="n">ww</span> <span class="o">=</span> <span class="n">x</span><span class="o">*</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="o">*</span><span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="o">*</span><span class="n">z</span><span class="p">,</span> <span class="n">w</span><span class="o">*</span><span class="n">w</span>
+
+        <span class="n">t0</span> <span class="o">=</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">(</span><span class="n">w</span> <span class="o">*</span> <span class="n">x</span> <span class="o">+</span> <span class="n">y</span> <span class="o">*</span> <span class="n">z</span><span class="p">)</span>
+        <span class="n">t1</span> <span class="o">=</span> <span class="p">(</span><span class="n">ww</span> <span class="o">+</span> <span class="n">zz</span><span class="p">)</span> <span class="o">-</span> <span class="p">(</span><span class="n">xx</span> <span class="o">+</span> <span class="n">yy</span><span class="p">)</span>
+        <span class="n">t2</span> <span class="o">=</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">(</span><span class="n">w</span> <span class="o">*</span> <span class="n">y</span> <span class="o">-</span> <span class="n">z</span> <span class="o">*</span> <span class="n">x</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">xx</span> <span class="o">+</span> <span class="n">yy</span> <span class="o">+</span> <span class="n">zz</span> <span class="o">+</span> <span class="n">ww</span><span class="p">)</span>
+        <span class="n">t3</span> <span class="o">=</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">(</span><span class="n">w</span> <span class="o">*</span> <span class="n">z</span> <span class="o">+</span> <span class="n">x</span> <span class="o">*</span> <span class="n">y</span><span class="p">)</span>
+        <span class="n">t4</span> <span class="o">=</span> <span class="p">(</span><span class="n">ww</span> <span class="o">+</span> <span class="n">xx</span><span class="p">)</span> <span class="o">-</span> <span class="p">(</span><span class="n">yy</span> <span class="o">+</span> <span class="n">zz</span><span class="p">)</span>
+
+        <span class="c1"># sigularity when pitch angle ~ +/-pi/2</span>
+        <span class="n">roll1</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">atan2</span><span class="p">(</span><span class="n">t0</span><span class="p">,</span> <span class="n">t1</span><span class="p">)</span>
+        <span class="n">roll2</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">zeros_like</span><span class="p">(</span><span class="n">t0</span><span class="p">)</span>
+        <span class="n">flag</span> <span class="o">=</span> <span class="n">t2</span><span class="o">.</span><span class="n">abs</span><span class="p">()</span> <span class="o">&lt;</span> <span class="mf">1.</span> <span class="o">-</span> <span class="n">eps</span>
+        <span class="n">yaw1</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">atan2</span><span class="p">(</span><span class="n">t3</span><span class="p">,</span> <span class="n">t4</span><span class="p">)</span>
+        <span class="n">yaw2</span> <span class="o">=</span> <span class="o">-</span><span class="mi">2</span> <span class="o">*</span> <span class="n">pm</span><span class="p">(</span><span class="n">t2</span><span class="p">)</span> <span class="o">*</span> <span class="n">torch</span><span class="o">.</span><span class="n">atan2</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">w</span><span class="p">)</span>
+
+        <span class="n">roll</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">where</span><span class="p">(</span><span class="n">flag</span><span class="p">,</span> <span class="n">roll1</span><span class="p">,</span> <span class="n">roll2</span><span class="p">)</span>
+        <span class="n">pitch</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">asin</span><span class="p">(</span><span class="n">t2</span><span class="o">.</span><span class="n">clamp</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
+        <span class="n">yaw</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">where</span><span class="p">(</span><span class="n">flag</span><span class="p">,</span> <span class="n">yaw1</span><span class="p">,</span> <span class="n">yaw2</span><span class="p">)</span>
+
+        <span class="k">return</span> <span class="n">torch</span><span class="o">.</span><span class="n">stack</span><span class="p">([</span><span class="n">roll</span><span class="p">,</span> <span class="n">pitch</span><span class="p">,</span> <span class="n">yaw</span><span class="p">],</span> <span class="n">dim</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span></div>
+
+<div class="viewcode-block" id="LieTensor.identity_"><a class="viewcode-back" href="../../../../generated/pypose.LieTensor/#pypose.LieTensor.identity_">[docs]</a>    <span class="k">def</span> <span class="nf">identity_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Inplace set the LieTensor to identity.</span>
+
+<span class="sd">        Return:</span>
+<span class="sd">            LieTensor: the :obj:`self` LieTensor</span>
+
+<span class="sd">        Note:</span>
+<span class="sd">            The translation part, if there is, is set to zeros, while the</span>
+<span class="sd">            rotation part is set to identity quaternion.</span>
+
+<span class="sd">        Example:</span>
+<span class="sd">            &gt;&gt;&gt; x = pp.randn_SO3(2)</span>
+<span class="sd">            &gt;&gt;&gt; x</span>
+<span class="sd">            SO3Type LieTensor:</span>
+<span class="sd">            tensor([[-0.0724,  0.1970,  0.0022,  0.9777],</span>
+<span class="sd">                    [ 0.3492,  0.4998, -0.5310,  0.5885]])</span>
+<span class="sd">            &gt;&gt;&gt; x.identity_()</span>
+<span class="sd">            SO3Type LieTensor:</span>
+<span class="sd">            tensor([[0., 0., 0., 1.],</span>
+<span class="sd">                    [0., 0., 0., 1.]])</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">ltype</span><span class="o">.</span><span class="n">identity_</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span></div>
+
+<div class="viewcode-block" id="LieTensor.cumops"><a class="viewcode-back" href="../../../../generated/pypose.LieTensor/#pypose.LieTensor.cumops">[docs]</a>    <span class="k">def</span> <span class="nf">cumops</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">dim</span><span class="p">,</span> <span class="n">ops</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        See :func:`pypose.cumops`</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">ltype</span><span class="o">.</span><span class="n">cumops</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">dim</span><span class="p">,</span> <span class="n">ops</span><span class="p">)</span></div>
+
+<div class="viewcode-block" id="LieTensor.cummul"><a class="viewcode-back" href="../../../../generated/pypose.LieTensor/#pypose.LieTensor.cummul">[docs]</a>    <span class="k">def</span> <span class="nf">cummul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">dim</span><span class="p">,</span> <span class="n">left</span> <span class="o">=</span> <span class="kc">True</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        See :func:`pypose.cummul`</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">ltype</span><span class="o">.</span><span class="n">cummul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">dim</span><span class="p">,</span> <span class="n">left</span><span class="p">)</span></div>
+
+<div class="viewcode-block" id="LieTensor.cumprod"><a class="viewcode-back" href="../../../../generated/pypose.LieTensor/#pypose.LieTensor.cumprod">[docs]</a>    <span class="k">def</span> <span class="nf">cumprod</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">dim</span><span class="p">,</span> <span class="n">left</span> <span class="o">=</span> <span class="kc">True</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        See :func:`pypose.cumprod`</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">ltype</span><span class="o">.</span><span class="n">cumprod</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">dim</span><span class="p">,</span> <span class="n">left</span><span class="p">)</span></div>
+
+<div class="viewcode-block" id="LieTensor.cumops_"><a class="viewcode-back" href="../../../../generated/pypose.LieTensor/#pypose.LieTensor.cumops_">[docs]</a>    <span class="k">def</span> <span class="nf">cumops_</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">dim</span><span class="p">,</span> <span class="n">ops</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Inplace version of :func:`pypose.cumops`</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">ltype</span><span class="o">.</span><span class="n">cumops_</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">dim</span><span class="p">,</span> <span class="n">ops</span><span class="p">)</span></div>
+
+<div class="viewcode-block" id="LieTensor.cummul_"><a class="viewcode-back" href="../../../../generated/pypose.LieTensor/#pypose.LieTensor.cummul_">[docs]</a>    <span class="k">def</span> <span class="nf">cummul_</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">dim</span><span class="p">,</span> <span class="n">left</span> <span class="o">=</span> <span class="kc">True</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Inplace version of :func:`pypose.cummul`</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">ltype</span><span class="o">.</span><span class="n">cummul_</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">dim</span><span class="p">,</span> <span class="n">left</span><span class="p">)</span></div>
+
+<div class="viewcode-block" id="LieTensor.cumprod_"><a class="viewcode-back" href="../../../../generated/pypose.LieTensor/#pypose.LieTensor.cumprod_">[docs]</a>    <span class="k">def</span> <span class="nf">cumprod_</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">dim</span><span class="p">,</span> <span class="n">left</span> <span class="o">=</span> <span class="kc">True</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Inplace version of :func:`pypose.cumprod`</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">ltype</span><span class="o">.</span><span class="n">cumprod_</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">dim</span><span class="p">,</span> <span class="n">left</span><span class="p">)</span></div></div>
+
+
+<div class="viewcode-block" id="Parameter"><a class="viewcode-back" href="../../../../generated/pypose.Parameter/#pypose.Parameter">[docs]</a><span class="k">class</span> <span class="nc">Parameter</span><span class="p">(</span><span class="n">LieTensor</span><span class="p">,</span> <span class="n">nn</span><span class="o">.</span><span class="n">Parameter</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    A kind of LieTensor that is to be considered a module parameter.</span>
+
+<span class="sd">    Parameters are of :meth:`LieTensor` and :meth:`torch.nn.Parameter`,</span>
+<span class="sd">    that have a very special property when used with Modules: when</span>
+<span class="sd">    they are assigned as Module attributes they are automatically</span>
+<span class="sd">    added to the list of its parameters, and will appear, e.g., in</span>
+<span class="sd">    :meth:`parameters()` iterator.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        data (LieTensor): parameter LieTensor.</span>
+<span class="sd">        requires_grad (bool, optional): if the parameter requires</span>
+<span class="sd">            gradient. Default: ``True``</span>
+
+<span class="sd">    Examples:</span>
+<span class="sd">        &gt;&gt;&gt; import torch, pypose as pp</span>
+<span class="sd">        &gt;&gt;&gt; x = pp.Parameter(pp.randn_SO3(2))</span>
+<span class="sd">        &gt;&gt;&gt; x.Log().sum().backward()</span>
+<span class="sd">        &gt;&gt;&gt; x.grad</span>
+<span class="sd">        tensor([[0.8590, 1.4069, 0.6261, 0.0000],</span>
+<span class="sd">                [1.2869, 1.0748, 0.5385, 0.0000]])</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">data</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">ltype</span> <span class="o">=</span> <span class="n">data</span><span class="o">.</span><span class="n">ltype</span>
+
+    <span class="k">def</span> <span class="fm">__new__</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="n">data</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">requires_grad</span><span class="o">=</span><span class="kc">True</span><span class="p">):</span>
+        <span class="k">if</span> <span class="n">data</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
+            <span class="n">data</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([])</span>
+        <span class="k">return</span> <span class="n">LieTensor</span><span class="o">.</span><span class="n">_make_subclass</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="n">data</span><span class="p">,</span> <span class="n">requires_grad</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">__deepcopy__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">memo</span><span class="p">):</span>
+        <span class="k">if</span> <span class="nb">id</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="ow">in</span> <span class="n">memo</span><span class="p">:</span>
+            <span class="k">return</span> <span class="n">memo</span><span class="p">[</span><span class="nb">id</span><span class="p">(</span><span class="bp">self</span><span class="p">)]</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">result</span> <span class="o">=</span> <span class="nb">type</span><span class="p">(</span><span class="bp">self</span><span class="p">)(</span><span class="bp">self</span><span class="o">.</span><span class="n">clone</span><span class="p">(</span><span class="n">memory_format</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">preserve_format</span><span class="p">))</span>
+            <span class="n">memo</span><span class="p">[</span><span class="nb">id</span><span class="p">(</span><span class="bp">self</span><span class="p">)]</span> <span class="o">=</span> <span class="n">result</span>
+            <span class="k">return</span> <span class="n">result</span></div>
+
+<span class="nd">@contextmanager</span>
+<span class="k">def</span> <span class="nf">retain_ltype</span><span class="p">():</span>
+    <span class="c1"># save the original PyTorch functions</span>
+    <span class="n">TO_BE_WRAPPED</span> <span class="o">=</span> <span class="p">{</span>
+        <span class="n">torch</span><span class="o">.</span><span class="n">autograd</span><span class="o">.</span><span class="n">forward_ad</span><span class="o">.</span><span class="n">make_dual</span><span class="p">,</span>
+        <span class="n">torch</span><span class="o">.</span><span class="n">_functorch</span><span class="o">.</span><span class="n">eager_transforms</span><span class="o">.</span><span class="n">_wrap_tensor_for_grad</span><span class="p">,</span>
+        <span class="n">torch</span><span class="o">.</span><span class="n">_functorch</span><span class="o">.</span><span class="n">vmap</span><span class="o">.</span><span class="n">_add_batch_dim</span><span class="p">,</span>
+    <span class="p">}</span>
+    <span class="n">torch</span><span class="o">.</span><span class="n">_functorch</span><span class="o">.</span><span class="n">vmap</span><span class="o">.</span><span class="n">_add_batch_dim</span><span class="o">.</span><span class="vm">__module__</span> <span class="o">=</span> <span class="s1">&#39;torch._functorch.vmap&#39;</span>
+
+    <span class="k">def</span> <span class="nf">wrap_function</span><span class="p">(</span><span class="n">func</span><span class="p">):</span>
+        <span class="k">def</span> <span class="nf">wrapper</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+            <span class="n">ltype</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">ltype</span> <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">LieTensor</span><span class="p">)</span> <span class="k">else</span> <span class="kc">None</span>
+            <span class="n">res</span> <span class="o">=</span> <span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+            <span class="k">if</span> <span class="n">ltype</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
+                <span class="n">res</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="o">.</span><span class="n">as_subclass</span><span class="p">(</span><span class="n">res</span><span class="p">,</span> <span class="n">LieTensor</span><span class="p">)</span>
+                <span class="n">res</span><span class="o">.</span><span class="n">ltype</span> <span class="o">=</span> <span class="n">ltype</span>
+            <span class="k">return</span> <span class="n">res</span>
+        <span class="k">return</span> <span class="n">wrapper</span>
+
+    <span class="k">try</span><span class="p">:</span>
+        <span class="c1"># swap the original PyTorch functions with the wrapper</span>
+        <span class="k">for</span> <span class="n">func</span> <span class="ow">in</span> <span class="n">TO_BE_WRAPPED</span><span class="p">:</span>
+            <span class="n">module</span><span class="p">,</span> <span class="n">name</span> <span class="o">=</span> <span class="n">func</span><span class="o">.</span><span class="vm">__module__</span><span class="p">,</span> <span class="n">func</span><span class="o">.</span><span class="vm">__name__</span>
+            <span class="n">module</span> <span class="o">=</span> <span class="n">importlib</span><span class="o">.</span><span class="n">import_module</span><span class="p">(</span><span class="n">module</span><span class="p">)</span>
+            <span class="nb">setattr</span><span class="p">(</span><span class="n">module</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">wrap_function</span><span class="p">(</span><span class="n">func</span><span class="p">))</span>
+        <span class="k">yield</span>
+    <span class="k">finally</span><span class="p">:</span>
+        <span class="k">for</span> <span class="n">func</span> <span class="ow">in</span> <span class="n">TO_BE_WRAPPED</span><span class="p">:</span>
+            <span class="n">module</span><span class="p">,</span> <span class="n">name</span> <span class="o">=</span> <span class="n">func</span><span class="o">.</span><span class="vm">__module__</span><span class="p">,</span> <span class="n">func</span><span class="o">.</span><span class="vm">__name__</span>
+            <span class="n">module</span> <span class="o">=</span> <span class="n">importlib</span><span class="o">.</span><span class="n">import_module</span><span class="p">(</span><span class="n">module</span><span class="p">)</span>
+            <span class="nb">setattr</span><span class="p">(</span><span class="n">module</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">func</span><span class="p">)</span>
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+            
+            
+              
+            
+            
+              <p class="caption" role="heading"><span class="caption-text">Contents:</span></p>
+<ul>
+<li class="toctree-l1"><a class="reference internal" href="../../../../lietensor/">LieTensor</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../../../func/">Func</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../../../functions/">Functions</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../../../optim/">Optimization</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../../../utils/">Utils</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../../../testing/">Testing</a></li>
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+              
+  <h1>Source code for pypose.lietensor.utils</h1><div class="highlight"><pre>
+<span></span><span class="kn">import</span> <span class="nn">torch</span><span class="o">,</span> <span class="nn">functools</span>
+<span class="kn">from</span> <span class="nn">.lietensor</span> <span class="kn">import</span> <span class="n">LieTensor</span><span class="p">,</span> <span class="n">LieType</span>
+<span class="kn">from</span> <span class="nn">.lietensor</span> <span class="kn">import</span> <span class="n">SE3_type</span><span class="p">,</span> <span class="n">se3_type</span>
+<span class="kn">from</span> <span class="nn">.lietensor</span> <span class="kn">import</span> <span class="n">SO3_type</span><span class="p">,</span> <span class="n">so3_type</span>
+<span class="kn">from</span> <span class="nn">.lietensor</span> <span class="kn">import</span> <span class="n">Sim3_type</span><span class="p">,</span> <span class="n">sim3_type</span>
+<span class="kn">from</span> <span class="nn">.lietensor</span> <span class="kn">import</span> <span class="n">RxSO3_type</span><span class="p">,</span> <span class="n">rxso3_type</span>
+
+
+<span class="k">def</span> <span class="nf">_LieTensor_wrapper_add_docstr</span><span class="p">(</span><span class="n">wrapper</span><span class="p">:</span> <span class="n">functools</span><span class="o">.</span><span class="n">partial</span><span class="p">,</span> <span class="n">embedding_doc</span><span class="p">):</span>
+    <span class="n">ltype</span><span class="p">:</span> <span class="n">LieType</span> <span class="o">=</span> <span class="n">wrapper</span><span class="o">.</span><span class="n">keywords</span><span class="p">[</span><span class="s1">&#39;ltype&#39;</span><span class="p">]</span>
+    <span class="n">type_name</span> <span class="o">=</span> <span class="nb">type</span><span class="p">(</span><span class="n">ltype</span><span class="p">)</span><span class="o">.</span><span class="vm">__name__</span><span class="p">[:</span><span class="o">-</span><span class="mi">4</span><span class="p">]</span>
+    <span class="n">type_dim</span> <span class="o">=</span> <span class="n">ltype</span><span class="o">.</span><span class="n">dimension</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
+    <span class="n">see_method</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;Exp&#39;</span><span class="p">,</span> <span class="s1">&#39;Inv&#39;</span><span class="p">]</span> <span class="k">if</span> <span class="n">ltype</span><span class="o">.</span><span class="n">on_manifold</span> <span class="k">else</span> \
+        <span class="p">[</span><span class="s1">&#39;Log&#39;</span><span class="p">,</span> <span class="s1">&#39;Inv&#39;</span><span class="p">,</span> <span class="s1">&#39;Act&#39;</span><span class="p">,</span> <span class="s1">&#39;Retr&#39;</span><span class="p">,</span> <span class="s1">&#39;Adj&#39;</span><span class="p">,</span> <span class="s1">&#39;AdjT&#39;</span><span class="p">,</span> <span class="s1">&#39;Jinvp&#39;</span><span class="p">]</span>
+    <span class="n">wrapper</span><span class="o">.</span><span class="vm">__doc__</span> <span class="o">=</span> <span class="sa">fr</span><span class="s1">&#39;&#39;&#39;Alias of </span><span class="si">{</span><span class="n">type_name</span><span class="si">}</span><span class="s1"> type :obj:`LieTensor`.</span>
+
+<span class="s1">    Args:</span>
+<span class="s1">        data (:obj:`Tensor`, or :obj:`list`, or &#39;:obj:`int`...&#39;): A</span>
+<span class="s1">            :obj:`Tensor` object, or constructing a :obj:`Tensor`</span>
+<span class="s1">            object from :obj:`list`, which defines tensor data (see below), or</span>
+<span class="s1">            from &#39;:obj:`int`...&#39;, which defines tensor shape.</span>
+
+<span class="s1">            The shape of :obj:`Tensor` object must be ``(*, </span><span class="si">{</span><span class="n">type_dim</span><span class="si">}</span><span class="s1">)``,</span>
+<span class="s1">            where ``*`` is empty, one, or more batched dimensions (the</span>
+<span class="s1">            :obj:`~LieTensor.lshape` of this LieTensor), otherwise error will</span>
+<span class="s1">            be raised.</span>
+
+<span class="s1">    </span><span class="si">{</span><span class="n">embedding_doc</span><span class="si">}</span>
+
+<span class="s1">    If ``data`` is tensor-like, the last dimension should correspond to the</span>
+<span class="s1">    </span><span class="si">{</span><span class="n">type_dim</span><span class="si">}</span><span class="s1"> elements of the above embedding.</span>
+
+<span class="s1">    Note:</span>
+<span class="s1">        It is not advised to construct </span><span class="si">{</span><span class="n">type_name</span><span class="si">}</span><span class="s1"> Tensors by specifying storage</span>
+<span class="s1">        sizes with &#39;:obj:`int`...&#39;, which does not initialize data.</span>
+
+<span class="s1">        Consider using :obj:`pypose.randn_</span><span class="si">{</span><span class="n">type_name</span><span class="si">}</span><span class="s1">` or</span>
+<span class="s1">        :obj:`pypose.identity_</span><span class="si">{</span><span class="n">type_name</span><span class="si">}</span><span class="s1">` instead.</span>
+
+<span class="s1">    See </span><span class="si">{</span><span class="s1">&#39;, &#39;</span><span class="o">.</span><span class="n">join</span><span class="p">([</span><span class="sa">f</span><span class="s1">&#39;:obj:`pypose.</span><span class="si">{</span><span class="n">m</span><span class="si">}</span><span class="s1">`&#39;</span><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="ow">in</span><span class="w"> </span><span class="n">see_method</span><span class="p">])</span><span class="si">}</span><span class="s1"> for</span>
+<span class="s1">    implementations of relevant operations.</span>
+<span class="s1">    &#39;&#39;&#39;</span>
+    <span class="k">return</span> <span class="n">wrapper</span>
+
+<span class="n">SO3</span> <span class="o">=</span> <span class="n">_LieTensor_wrapper_add_docstr</span><span class="p">(</span><span class="n">functools</span><span class="o">.</span><span class="n">partial</span><span class="p">(</span><span class="n">LieTensor</span><span class="p">,</span> <span class="n">ltype</span><span class="o">=</span><span class="n">SO3_type</span><span class="p">),</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;Internally, SO3 LieTensors are stored as unit quaternions:</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \mathrm{data}[*, :] = [q_x, q_y, q_z, q_w],</span>
+
+<span class="sd">    where :math:`q_x^2 + q_y^2 + q_z^2 + q_w^2 = 1`.</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        Normalization is not required at initialization as it is done internally</span>
+<span class="sd">        by the library right before further computation. However, the normalized</span>
+<span class="sd">        quaternion will not be written back to the tensor storage to prevent</span>
+<span class="sd">        in-place data alteration.</span>
+
+<span class="sd">    Examples:</span>
+<span class="sd">        &gt;&gt;&gt; pp.SO3(torch.randn(2, 4))</span>
+<span class="sd">        SO3Type LieTensor:</span>
+<span class="sd">        tensor([[-1.0722, -0.9440,  0.9437, -0.8485],</span>
+<span class="sd">                [-0.2725,  0.8414, -1.0730,  1.3270]])</span>
+<span class="sd">        &gt;&gt;&gt; pp.SO3([0, 0, 0, 1])</span>
+<span class="sd">        SO3Type LieTensor:</span>
+<span class="sd">        tensor([0., 0., 0., 1.])</span>
+<span class="sd">    &#39;&#39;&#39;</span><span class="p">)</span>
+
+<span class="n">so3</span> <span class="o">=</span> <span class="n">_LieTensor_wrapper_add_docstr</span><span class="p">(</span><span class="n">functools</span><span class="o">.</span><span class="n">partial</span><span class="p">(</span><span class="n">LieTensor</span><span class="p">,</span> <span class="n">ltype</span><span class="o">=</span><span class="n">so3_type</span><span class="p">),</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;Internally, so3 LieTensors are stored in the</span>
+<span class="sd">    `axis-angle &lt;https://en.wikipedia.org/wiki/Axis-angle_representation&gt;`_ format:</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \mathrm{data}[*, :] = [\delta_x, \delta_y, \delta_z],</span>
+
+<span class="sd">    with :math:`\delta = \begin{pmatrix} \delta_x &amp; \delta_y &amp; \delta_z \end{pmatrix}\top`</span>
+<span class="sd">    being the axis of rotation and :math:`\theta = \|{\delta}\|` being the angle.</span>
+
+<span class="sd">    Examples:</span>
+<span class="sd">        &gt;&gt;&gt; pp.so3(torch.randn(2, 3))</span>
+<span class="sd">        so3Type LieTensor:</span>
+<span class="sd">        tensor([[ 0.1571,  0.2203, -0.2457],</span>
+<span class="sd">                [-0.3311,  0.5412, -0.7028]])</span>
+<span class="sd">        &gt;&gt;&gt; pp.so3([0, 0, 1])</span>
+<span class="sd">        so3Type LieTensor:</span>
+<span class="sd">        tensor([0., 0., 1.])</span>
+<span class="sd">    &#39;&#39;&#39;</span><span class="p">)</span>
+
+<span class="n">SE3</span> <span class="o">=</span> <span class="n">_LieTensor_wrapper_add_docstr</span><span class="p">(</span><span class="n">functools</span><span class="o">.</span><span class="n">partial</span><span class="p">(</span><span class="n">LieTensor</span><span class="p">,</span> <span class="n">ltype</span><span class="o">=</span><span class="n">SE3_type</span><span class="p">),</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;Internally, SE3 LieTensors are stored by concatenating the unit quaternion</span>
+<span class="sd">    representing the rotation with a vector representing the translation.</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \mathrm{data}[*, :] = [t_x, t_y, t_z, q_x, q_y, q_z, q_w],</span>
+
+<span class="sd">    where :math:`\begin{pmatrix} t_x &amp; t_y &amp; t_z \end{pmatrix}\top \in \mathbb{R}^3` is</span>
+<span class="sd">    the translation and</span>
+<span class="sd">    :math:`\begin{pmatrix} q_x &amp; q_y &amp; q_z &amp; q_w \end{pmatrix}\top` is the unit</span>
+<span class="sd">    quaternion as in :obj:`pypose.SO3`.</span>
+
+<span class="sd">    Examples:</span>
+<span class="sd">        &gt;&gt;&gt; pp.SE3(torch.randn(2, 7))</span>
+<span class="sd">        SE3Type LieTensor:</span>
+<span class="sd">        tensor([[ 0.1626,  1.6349,  0.3607,  0.2848, -0.0948,  0.1541,  1.0003],</span>
+<span class="sd">                [ 1.4034, -1.3085, -0.8886, -1.6703,  0.7381,  1.5575,  0.6280]])</span>
+<span class="sd">        &gt;&gt;&gt; pp.SE3([0, 0, 0, 0, 0, 0, 1])</span>
+<span class="sd">        SE3Type LieTensor:</span>
+<span class="sd">        tensor([0., 0., 0., 0., 0., 0., 1.])</span>
+<span class="sd">    &#39;&#39;&#39;</span><span class="p">)</span>
+
+<span class="n">se3</span> <span class="o">=</span> <span class="n">_LieTensor_wrapper_add_docstr</span><span class="p">(</span><span class="n">functools</span><span class="o">.</span><span class="n">partial</span><span class="p">(</span><span class="n">LieTensor</span><span class="p">,</span> <span class="n">ltype</span><span class="o">=</span><span class="n">se3_type</span><span class="p">),</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;Internally, se3 LieTensors are stored by concatenating the &quot;velocity&quot; vector</span>
+<span class="sd">    with the axis-angle representation of the rotation:</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \mathrm{data}[*, :] = [\tau_x, \tau_y, \tau_z, \delta_x, \delta_y, \delta_z],</span>
+
+<span class="sd">    where :math:`\begin{pmatrix} \tau_x &amp; \tau_y &amp; \tau_z \end{pmatrix}^T =</span>
+<span class="sd">    \mathbf{J}^{-1} \begin{pmatrix} t_x &amp; t_y &amp; t_z \end{pmatrix}^T` is the product</span>
+<span class="sd">    between the left Jacobian inverse (of SO3&#39;s logarithm map) and the translation</span>
+<span class="sd">    vector, and :math:`\begin{pmatrix} \delta_x &amp; \delta_y &amp; \delta_z \end{pmatrix}^T`</span>
+<span class="sd">    is the axis-angle vector as in :obj:`pypose.so3`.</span>
+<span class="sd">    More details go to :meth:`pypose.Log` with :obj:`SE3_type` input.</span>
+
+<span class="sd">    Examples:</span>
+<span class="sd">        &gt;&gt;&gt; pp.se3(torch.randn(2, 6))</span>
+<span class="sd">        se3Type LieTensor:</span>
+<span class="sd">        tensor([[-0.8710, -1.4994, -0.2843,  1.0185, -0.3932, -0.4839],</span>
+<span class="sd">                [-0.4750, -0.4804, -0.7083, -1.8141, -1.4409, -0.3125]])</span>
+<span class="sd">        &gt;&gt;&gt; pp.se3([0, 0, 0, 0, 0, 1])</span>
+<span class="sd">        se3Type LieTensor:</span>
+<span class="sd">        tensor([0., 0., 0., 0., 0., 1.])</span>
+<span class="sd">    &#39;&#39;&#39;</span><span class="p">)</span>
+
+<span class="n">RxSO3</span> <span class="o">=</span> <span class="n">_LieTensor_wrapper_add_docstr</span><span class="p">(</span><span class="n">functools</span><span class="o">.</span><span class="n">partial</span><span class="p">(</span><span class="n">LieTensor</span><span class="p">,</span> <span class="n">ltype</span><span class="o">=</span><span class="n">RxSO3_type</span><span class="p">),</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;Internally, RxSO3 LieTensors are stored by concatenating the unit quaternion</span>
+<span class="sd">    representing the rotation with a scaling factor:</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \mathrm{data}[*, :] = [q_x, q_y, q_z, q_w, s],</span>
+
+<span class="sd">    where :math:`\begin{pmatrix} q_x &amp; q_y &amp; q_z &amp; q_w \end{pmatrix}^T`</span>
+<span class="sd">    is the unit quaternion as in :obj:`pypose.SO3` and</span>
+<span class="sd">    :math:`s \in \mathbb{R}` is the scaling factor.</span>
+
+<span class="sd">    Examples:</span>
+<span class="sd">        &gt;&gt;&gt; pp.RxSO3(torch.randn(2, 5))</span>
+<span class="sd">        RxSO3Type LieTensor:</span>
+<span class="sd">        tensor([[-0.3693,  2.5155, -0.5384, -0.8119, -0.4798],</span>
+<span class="sd">                [-0.4058, -0.5909, -0.4918, -0.2994,  0.5440]])</span>
+<span class="sd">        &gt;&gt;&gt; pp.RxSO3([0, 0, 0, 1, 1])</span>
+<span class="sd">        RxSO3Type LieTensor:</span>
+<span class="sd">        tensor([0., 0., 0., 1., 1.])</span>
+<span class="sd">    &#39;&#39;&#39;</span><span class="p">)</span>
+
+<span class="n">rxso3</span> <span class="o">=</span> <span class="n">_LieTensor_wrapper_add_docstr</span><span class="p">(</span><span class="n">functools</span><span class="o">.</span><span class="n">partial</span><span class="p">(</span><span class="n">LieTensor</span><span class="p">,</span> <span class="n">ltype</span><span class="o">=</span><span class="n">rxso3_type</span><span class="p">),</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;Internally, rxso3 LieTensors are stored by concatenating the axis-angle</span>
+<span class="sd">    representation of the rotation with the log scale:</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \mathrm{data}[*, :] = [\delta_x, \delta_y, \delta_z, \log s],</span>
+
+<span class="sd">    where :math:`\begin{pmatrix} \delta_x &amp; \delta_y &amp; \delta_z \end{pmatrix}^T`</span>
+<span class="sd">    is the axis-angle vector in :obj:`pypose.so3`, and</span>
+<span class="sd">    :math:`s \in \mathbb{R}` is the scaling factor in :obj:`pypose.RxSO3`.</span>
+
+<span class="sd">    Examples:</span>
+<span class="sd">        &gt;&gt;&gt; pp.rxso3(torch.randn(2, 4))</span>
+<span class="sd">        rxso3Type LieTensor:</span>
+<span class="sd">        tensor([[ 0.3752, -0.1576,  1.2057,  0.6086],</span>
+<span class="sd">                [ 0.8434,  0.2449,  0.0488, -0.1202]])</span>
+<span class="sd">        &gt;&gt;&gt; pp.rxso3([0, 0, 0, 0, 1])</span>
+<span class="sd">        RxSO3Type LieTensor:</span>
+<span class="sd">        tensor([0., 0., 0., 0., 1.])</span>
+<span class="sd">    &#39;&#39;&#39;</span><span class="p">)</span>
+
+<span class="n">Sim3</span> <span class="o">=</span> <span class="n">_LieTensor_wrapper_add_docstr</span><span class="p">(</span><span class="n">functools</span><span class="o">.</span><span class="n">partial</span><span class="p">(</span><span class="n">LieTensor</span><span class="p">,</span> <span class="n">ltype</span><span class="o">=</span><span class="n">Sim3_type</span><span class="p">),</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;Internally, Sim3 LieTensors are stored by concatenating the translation</span>
+<span class="sd">    vector with an RxSO3:</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \mathrm{data}[*, :] = [t_x, t_y, t_z, q_x, q_y, q_z, q_w, s],</span>
+
+<span class="sd">    where :math:`\begin{pmatrix} t_x &amp; t_y &amp; t_z \end{pmatrix}^T \in \mathbb{R}^3`</span>
+<span class="sd">    is the translation vector and</span>
+<span class="sd">    :math:`\begin{pmatrix} q_x &amp; q_y &amp; q_z &amp; q_w \end{pmatrix}^T` and</span>
+<span class="sd">    :math:`s \in \mathbb{R}` are the unit quaternion and the scaling factor</span>
+<span class="sd">    as in :obj:`pp.RxSO3`, respectively.</span>
+
+<span class="sd">    Examples:</span>
+<span class="sd">        &gt;&gt;&gt; pp.Sim3(torch.randn(2, 8))</span>
+<span class="sd">        Sim3Type LieTensor:</span>
+<span class="sd">        tensor([[ 0.0175,  0.8657, -0.2274,  2.2380, -0.0297, -0.3799, -0.0664,  0.9995],</span>
+<span class="sd">                [ 0.8744,  0.4114,  1.2041, -0.5687, -0.5630,  0.6025, -0.6137,  1.1185]])</span>
+<span class="sd">        &gt;&gt;&gt; pp.Sim3([0, 0, 0, 0, 0, 0, 1, 1])</span>
+<span class="sd">        Sim3Type LieTensor:</span>
+<span class="sd">        tensor([0., 0., 0., 0., 0., 0., 1., 1.])</span>
+<span class="sd">    &#39;&#39;&#39;</span><span class="p">)</span>
+
+<span class="n">sim3</span> <span class="o">=</span> <span class="n">_LieTensor_wrapper_add_docstr</span><span class="p">(</span><span class="n">functools</span><span class="o">.</span><span class="n">partial</span><span class="p">(</span><span class="n">LieTensor</span><span class="p">,</span> <span class="n">ltype</span><span class="o">=</span><span class="n">sim3_type</span><span class="p">),</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;Internally, sim3 LieTensors are stored by concatenating the log translation</span>
+<span class="sd">    vector with the corresponding rxso3:</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \mathrm{data}[*, :] = [\tau_x, \tau_y, \tau_z, \delta_x, \delta_y, \delta_z, \log s],</span>
+
+<span class="sd">    where :math:`\begin{pmatrix} \tau_x &amp; \tau_y &amp; \tau_z \end{pmatrix}^T = \mathbf{W}^{-1}</span>
+<span class="sd">    \begin{pmatrix} t_x &amp; t_y &amp; t_z \end{pmatrix}^T` is the product between the</span>
+<span class="sd">    inverse of the :math:`\mathbf{W}`-matrix and the translation vector, and</span>
+<span class="sd">    :math:`\begin{pmatrix} \delta_x &amp; \delta_y &amp; \delta_z &amp; \log s \end{pmatrix}^T`</span>
+<span class="sd">    represents the rotation and scaling, as in :obj:`pypose.rxso3`.</span>
+<span class="sd">    More details about :math:`\mathbf{W}`-matrix go to :meth:`pypose.Log` with</span>
+<span class="sd">    :obj:`Sim3_type` input.</span>
+
+<span class="sd">    Examples:</span>
+<span class="sd">        &gt;&gt;&gt; pp.Sim3(torch.randn(2, 7))</span>
+<span class="sd">        sim3Type LieTensor:</span>
+<span class="sd">        sim3Type LieTensor:</span>
+<span class="sd">        tensor([[ 0.1477, -1.3500, -2.1571,  0.8893, -0.7821, -0.9889, -0.7887],</span>
+<span class="sd">                [ 0.2251,  0.3512,  0.0485,  0.0163, -1.7090, -0.0417, -0.3842]])</span>
+<span class="sd">        &gt;&gt;&gt; pp.sim3([0, 0, 0, 0, 0, 0, 1])</span>
+<span class="sd">        sim3Type LieTensor:</span>
+<span class="sd">        tensor([0., 0., 0., 0., 0., 0., 1.])</span>
+<span class="sd">    &#39;&#39;&#39;</span><span class="p">)</span>
+
+<div class="viewcode-block" id="randn_like"><a class="viewcode-back" href="../../../../generated/pypose.randn_like/#pypose.randn_like">[docs]</a><span class="k">def</span> <span class="nf">randn_like</span><span class="p">(</span><span class="nb">input</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Returns a LieTensor with the same size as input that is filled with random</span>
+<span class="sd">    LieTensor with the corresponding :obj:`input.ltype`.</span>
+
+<span class="sd">    The corresponding random generator can be</span>
+
+<span class="sd">    .. list-table:: List of available random LieTensor generator of input :obj:`ltype`.</span>
+<span class="sd">        :widths: 25 25 30 30 30</span>
+<span class="sd">        :header-rows: 1</span>
+
+<span class="sd">        * - Name</span>
+<span class="sd">          - ltype</span>
+<span class="sd">          - randn function</span>
+<span class="sd">          - Manifold</span>
+<span class="sd">          - randn function</span>
+<span class="sd">        * - Orthogonal Group</span>
+<span class="sd">          - :obj:`SO3_type`</span>
+<span class="sd">          - :meth:`randn_SO3`</span>
+<span class="sd">          - :obj:`so3_type`</span>
+<span class="sd">          - :meth:`randn_so3`</span>
+<span class="sd">        * - Euclidean Group</span>
+<span class="sd">          - :obj:`SE3_type`</span>
+<span class="sd">          - :meth:`randn_SE3`</span>
+<span class="sd">          - :obj:`se3_type`</span>
+<span class="sd">          - :meth:`randn_se3`</span>
+<span class="sd">        * - Similarity Group</span>
+<span class="sd">          - :obj:`Sim3_type`</span>
+<span class="sd">          - :meth:`randn_Sim3`</span>
+<span class="sd">          - :obj:`sim3_type`</span>
+<span class="sd">          - :meth:`randn_sim3`</span>
+<span class="sd">        * - Scaling Orthogonal</span>
+<span class="sd">          - :obj:`RxSO3_type`</span>
+<span class="sd">          - :meth:`randn_RxSO3`</span>
+<span class="sd">          - :obj:`rxso3_type`</span>
+<span class="sd">          - :meth:`randn_rxso3`</span>
+
+<span class="sd">    Args:</span>
+
+<span class="sd">        input (LieTensor): the size of input will determine size of the output tensor.</span>
+
+<span class="sd">        sigma (float or (float...), optional): standard deviation for generating random</span>
+<span class="sd">            LieTensors. Default: ``1.0``.</span>
+
+<span class="sd">        dtype (torch.dtype, optional): the desired data type of returned Tensor.</span>
+<span class="sd">            Default: if ``None``, defaults to the dtype of input.</span>
+
+<span class="sd">        layout (torch.layout, optional): the desired layout of returned tensor.</span>
+<span class="sd">            Default: if ``None``, defaults to the layout of input.</span>
+
+<span class="sd">        device (torch.device, optional): the desired device of returned tensor.</span>
+<span class="sd">            Default: if ``None``, defaults to the device of input.</span>
+
+<span class="sd">        requires_grad (bool, optional): If autograd should record operations on the returned</span>
+<span class="sd">            tensor. Default: ``False``.</span>
+
+<span class="sd">        memory_format (torch.memory_format, optional): the desired memory format of returned</span>
+<span class="sd">            Tensor. Default: ``torch.preserve_format``.</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        The parameter sigma (:math:`\sigma`) can either be:</span>
+
+<span class="sd">        - a single ``float`` -- in which all the elements in the LieType share the same</span>
+<span class="sd">          sigma.</span>
+
+<span class="sd">        - a ``tuple`` of a number of floats -- in which case, the specific sigma for</span>
+<span class="sd">          each element can be assigned independently.</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        If we have:</span>
+
+<span class="sd">        .. code::</span>
+
+<span class="sd">            import pypose as pp</span>
+<span class="sd">            x = pp.SO3([0, 0, 0, 1])</span>
+
+<span class="sd">        Then the following two usages are equivalent:</span>
+
+<span class="sd">        .. code::</span>
+
+<span class="sd">            pp.randn_like(x)</span>
+<span class="sd">            pp.randn_SO3(x.lshape, dtype=x.dtype, layout=x.layout, device=x.device)</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        &gt;&gt;&gt; x = pp.so3(torch.tensor([[0, 0, 0]]))</span>
+<span class="sd">        &gt;&gt;&gt; pp.randn_like(x)</span>
+<span class="sd">        so3Type LieTensor:</span>
+<span class="sd">        tensor([[ 0.5162, -0.4600, -0.9085]])</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">return</span> <span class="nb">input</span><span class="o">.</span><span class="n">ltype</span><span class="o">.</span><span class="n">randn_like</span><span class="p">(</span><span class="o">*</span><span class="nb">input</span><span class="o">.</span><span class="n">lshape</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="n">sigma</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="randn_so3"><a class="viewcode-back" href="../../../../generated/pypose.randn_so3/#pypose.randn_so3">[docs]</a><span class="k">def</span> <span class="nf">randn_so3</span><span class="p">(</span><span class="o">*</span><span class="n">lsize</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Returns :obj:`so3_type` LieTensor filled with random numbers.</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \begin{aligned}</span>
+<span class="sd">        \mathrm{data}[*, :] &amp;= [\delta_x, \delta_y, \delta_z] \\</span>
+<span class="sd">                            &amp;= [\delta_x&#39;, \delta_y&#39;, \delta_z&#39;] \cdot \theta,</span>
+<span class="sd">        \end{aligned}</span>
+
+<span class="sd">    where :math:`[\delta_x&#39;, \delta_y&#39;, \delta_z&#39;]` is randomly generated from uniform</span>
+<span class="sd">    distribution :math:`\mathcal{U}_{\mathrm{s}}` on a standard sphere and :math:`\theta`</span>
+<span class="sd">    is generated from a normal distribution :math:`\mathcal{N}(0, \sigma)`, where sigma</span>
+<span class="sd">    (:math:`\sigma`) is the standard deviation.</span>
+
+<span class="sd">    The generated LieTensor satisfies that the corresponding rotation axis follows uniform</span>
+<span class="sd">    distribution on the standard sphere and the rotation angle follows normal distribution.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        lsize (int...): a sequence of integers defining the lshape of the output tensor.</span>
+<span class="sd">            Can be a variable number of arguments or a collection like a list or tuple.</span>
+
+<span class="sd">        sigma (float, optional): standard deviation for the angle of the generated rotation.</span>
+<span class="sd">            Default: ``1.0``.</span>
+
+<span class="sd">        requires_grad (bool, optional): If autograd should record operations on</span>
+<span class="sd">            the returned tensor. Default: ``False``.</span>
+
+<span class="sd">        generator (torch.Generator, optional): a pseudorandom number generator for sampling</span>
+
+<span class="sd">        dtype (torch.dtype, optional): the desired data type of returned tensor.</span>
+<span class="sd">            Default: ``None``. If ``None``, uses a global default</span>
+<span class="sd">            (see :meth:`torch.set_default_tensor_type()`).</span>
+
+<span class="sd">        layout (torch.layout, optional): the desired layout of returned Tensor.</span>
+<span class="sd">            Default: ``torch.strided``.</span>
+
+<span class="sd">        device (torch.device, optional): the desired device of returned tensor.</span>
+<span class="sd">            Default: ``None``. If ``None``, uses the current device for the default tensor</span>
+<span class="sd">            type (see :meth:`torch.set_default_tensor_type()`). Device will be the CPU</span>
+<span class="sd">            for CPU tensor types and the current CUDA device for CUDA tensor types.</span>
+
+<span class="sd">    Returns:</span>
+<span class="sd">        LieTensor: a :obj:`so3_type` LieTensor</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        The uniform distributed points :math:`[\delta_x&#39;, \delta_y&#39;, \delta_z&#39;]` on a sphere</span>
+<span class="sd">        are sampled using:</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \delta_x&#39; = \frac{x_0}{d}, \delta_y&#39; = \frac{y_0}{d}, \delta_z&#39; = \frac{z_0}{d},</span>
+
+<span class="sd">        where</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            x_0, y_0, z_0 \sim \mathcal{N}(0, \sigma),</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            d = \sqrt{(x^2_0+y^2_0+z^2_0)}.</span>
+
+<span class="sd">        Note that the point :math:`[x_0, y_0, z_0]` follows 3-D Gaussian distribution,</span>
+<span class="sd">        which is centrosymmetry with respect to :math:`(0, 0, 0)`. Therefore, we have</span>
+<span class="sd">        :math:`\delta_x&#39;^2+\delta_y&#39;^2+\delta_z&#39;^2=1` and the normalized points follow</span>
+<span class="sd">        uniform distribution on a standard sphere, i.e.,</span>
+<span class="sd">        :math:`[\delta_x&#39;, \delta_y&#39;, \delta_z&#39;] \sim \mathcal{U}_{\mathrm{s}}`.</span>
+<span class="sd">        The point :math:`[\delta_x&#39;, \delta_y&#39;, \delta_z&#39;]` can also represent an evenly</span>
+<span class="sd">        sampled rotation axis.</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        &gt;&gt;&gt; pp.randn_so3()</span>
+<span class="sd">        so3Type LieTensor:</span>
+<span class="sd">        LieTensor([ 0.4811, -0.1487, -0.5949])  # Shape (3)</span>
+<span class="sd">        &gt;&gt;&gt; pp.randn_so3(1)</span>
+<span class="sd">        so3Type LieTensor:</span>
+<span class="sd">        LieTensor([[0.0235, 0.0334, 1.2601]])   # Shape (1, 3)</span>
+<span class="sd">        &gt;&gt;&gt; pp.randn_so3(2, sigma=0.1, requires_grad=True, dtype=torch.float64)</span>
+<span class="sd">        so3Type LieTensor:</span>
+<span class="sd">        tensor([[-0.0344,  0.0177,  0.0252],</span>
+<span class="sd">                [-0.0040,  0.0032,  0.0149]], dtype=torch.float64, requires_grad=True)</span>
+
+<span class="sd">    .. figure:: /_static/img/lietensor/randn/randn_so3.svg</span>
+<span class="sd">        :height: 1000px</span>
+<span class="sd">        :width: 1000px</span>
+<span class="sd">        :scale: 30 %</span>
+<span class="sd">        :alt: map to buried treasure</span>
+<span class="sd">        :align: center</span>
+
+<span class="sd">        Visualization of :meth:`pypose.randn_so3()`. A total of 5000 random rotations</span>
+<span class="sd">        are sampled using :meth:`pypose.randn_so3()` and applied to the basepoint</span>
+<span class="sd">        [0, 0, 1] (shown in blue).</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">return</span> <span class="n">so3_type</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="o">*</span><span class="n">lsize</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="n">sigma</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="randn_SO3"><a class="viewcode-back" href="../../../../generated/pypose.randn_SO3/#pypose.randn_SO3">[docs]</a><span class="k">def</span> <span class="nf">randn_SO3</span><span class="p">(</span><span class="o">*</span><span class="n">lsize</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Returns :obj:`SO3_type` LieTensor filled with the Exponential map of the random</span>
+<span class="sd">    :obj:`so3_type` LieTensor.</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \begin{aligned}</span>
+<span class="sd">        \mathrm{data}[*, :] &amp;= \mathrm{Exp}([\delta_x, \delta_y, \delta_z]) \\</span>
+<span class="sd">                            &amp;= \mathrm{Exp}([\delta_x&#39;, \delta_y&#39;, \delta_z&#39;] \cdot \theta),</span>
+<span class="sd">        \end{aligned}</span>
+
+<span class="sd">    where :math:`[\delta_x&#39;, \delta_y&#39;, \delta_z&#39;]` is generated from uniform distribution</span>
+<span class="sd">    :math:`\mathcal{U}_{\mathrm{s}}` on a standard sphere, :math:`\mathrm{Exp}()` is the</span>
+<span class="sd">    Exponential map, :math:`\theta` is generated from a normal distribution</span>
+<span class="sd">    :math:`\mathcal{N}(0, \sigma)` where :math:`\sigma` (``sigma``) is the standard deviation.</span>
+
+<span class="sd">    For detailed explanation, please see :meth:`pypose.randn_so3()` and :meth:`pypose.Exp()`.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        lsize (int...): a sequence of integers defining the lshape of the output tensor.</span>
+<span class="sd">            Can be a variable number of arguments or a collection like a list or tuple.</span>
+
+<span class="sd">        sigma (float, optional): standard deviation for the angle of the generated rotation.</span>
+<span class="sd">            Default: ``1.0``.</span>
+
+<span class="sd">        requires_grad (bool, optional): If autograd should record operations on</span>
+<span class="sd">            the returned tensor. Default: ``False``.</span>
+
+<span class="sd">        generator (torch.Generator, optional): a pseudorandom number generator for sampling</span>
+
+<span class="sd">        dtype (torch.dtype, optional): the desired data type of returned tensor.</span>
+<span class="sd">            Default: ``None``.</span>
+<span class="sd">            If ``None``, uses a global default (see :meth:`torch.set_default_tensor_type()`).</span>
+
+<span class="sd">        layout (torch.layout, optional): the desired layout of returned Tensor.</span>
+<span class="sd">            Default: ``torch.strided``.</span>
+
+<span class="sd">        device (torch.device, optional): the desired device of returned tensor.</span>
+<span class="sd">            Default: ``None``. If ``None``, uses the current device for the default tensor</span>
+<span class="sd">            type (see :meth:`torch.set_default_tensor_type()`). Device will be the CPU</span>
+<span class="sd">            for CPU tensor types and the current CUDA device for CUDA tensor types.</span>
+
+<span class="sd">    Returns:</span>
+<span class="sd">        LieTensor: a :obj:`SO3_type` LieTensor.</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        &gt;&gt;&gt; pp.randn_SO3(sigma=0.1, requires_grad=True)</span>
+<span class="sd">        SO3Type LieTensor:</span>
+<span class="sd">        LieTensor([ 0.0082,  0.0077, -0.0018,  0.9999], requires_grad=True)</span>
+<span class="sd">        &gt;&gt;&gt; pp.randn_SO3(2, sigma=0.1, requires_grad=True, dtype=torch.float64)</span>
+<span class="sd">        SO3Type LieTensor:</span>
+<span class="sd">        tensor([[ 2.6857e-03, -8.3274e-03, -6.0457e-04,  9.9996e-01],</span>
+<span class="sd">                [ 3.8711e-02, -6.3148e-02, -1.9388e-02,  9.9706e-01]],</span>
+<span class="sd">                dtype=torch.float64, requires_grad=True)</span>
+
+<span class="sd">    .. figure:: /_static/img/lietensor/randn/randn_so3.svg</span>
+<span class="sd">        :height: 1000px</span>
+<span class="sd">        :width: 1000px</span>
+<span class="sd">        :scale: 30 %</span>
+<span class="sd">        :alt: map to buried treasure</span>
+<span class="sd">        :align: center</span>
+
+<span class="sd">        Visualization of :meth:`pypose.randn_SO3()`. A total of 5000 random rotations</span>
+<span class="sd">        are sampled using :meth:`pypose.randn_SO3()` and applied to the basepoint</span>
+<span class="sd">        [0, 0, 1] (shown in blue).</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">return</span> <span class="n">SO3_type</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="o">*</span><span class="n">lsize</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="n">sigma</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="randn_se3"><a class="viewcode-back" href="../../../../generated/pypose.randn_se3/#pypose.randn_se3">[docs]</a><span class="k">def</span> <span class="nf">randn_se3</span><span class="p">(</span><span class="o">*</span><span class="n">lsize</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Returns :obj:`se3_type` LieTensor filled with random numbers.</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \mathrm{data}[*, :] = [\tau_x, \tau_y, \tau_z, \delta_x, \delta_y, \delta_z],</span>
+
+<span class="sd">    where translation :math:`[\tau_x, \tau_y, \tau_z]` is generated from a normal distribution</span>
+<span class="sd">    :math:`\mathcal{N}(0, \sigma_t)`, rotation :math:`[\delta_x, \delta_y, \delta_z]` is</span>
+<span class="sd">    generated using :meth:`pypose.randn_so3()` with standard deviation :math:`\sigma_r`.</span>
+<span class="sd">    Note that standard deviations :math:`\sigma_t` and :math:`\sigma_r` are</span>
+<span class="sd">    specified by ``sigma`` (:math:`\sigma`), where :math:`\sigma = (\sigma_t, \sigma_r)`.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        lsize (int...): a sequence of integers defining the lshape of the output tensor.</span>
+<span class="sd">            Can be a variable number of arguments or a collection like a list or tuple.</span>
+
+<span class="sd">        sigma (float or (float...), optional): standard deviation</span>
+<span class="sd">            (:math:`\sigma_t` and :math:`\sigma_r`)</span>
+<span class="sd">            for the two normal distribution. Default: ``1.0``.</span>
+
+<span class="sd">        requires_grad (bool, optional): If autograd should record operations on</span>
+<span class="sd">            the returned tensor. Default: ``False``.</span>
+
+<span class="sd">        generator (torch.Generator, optional): a pseudorandom number generator for sampling</span>
+
+<span class="sd">        dtype (torch.dtype, optional): the desired data type of returned tensor.</span>
+<span class="sd">            Default: ``None``. If ``None``, uses a global default</span>
+<span class="sd">            (see :meth:`torch.set_default_tensor_type()`).</span>
+
+<span class="sd">        layout (torch.layout, optional): the desired layout of returned Tensor.</span>
+<span class="sd">            Default: ``torch.strided``.</span>
+
+<span class="sd">        device (torch.device, optional): the desired device of returned tensor.</span>
+<span class="sd">            Default: ``None``. If ``None``, uses the current device for the default tensor</span>
+<span class="sd">            type (see :meth:`torch.set_default_tensor_type()`). Device will be the CPU</span>
+<span class="sd">            for CPU tensor types and the current CUDA device for CUDA tensor types.</span>
+
+<span class="sd">    Returns:</span>
+<span class="sd">        LieTensor: a :obj:`se3_type` LieTensor.</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        The parameter :math:`\sigma` can either be:</span>
+
+<span class="sd">        - a single ``float`` -- in which all the elements in the :obj:`se3_type`</span>
+<span class="sd">          share the same sigma, i.e.,</span>
+<span class="sd">          :math:`\sigma_{\rm{t}}` = :math:`\sigma_{\rm{r}}` = :math:`\sigma`.</span>
+<span class="sd">        - a ``tuple`` of two floats -- in which case, the specific sigmas are</span>
+<span class="sd">          assigned independently, i.e.,</span>
+<span class="sd">          :math:`\sigma` = (:math:`\sigma_{\rm{t}}`, :math:`\sigma_{\rm{r}}`).</span>
+<span class="sd">        - a ``tuple`` of four floats -- in which case, the specific sigmas for</span>
+<span class="sd">          each translation data are assigned independently, i.e.,</span>
+<span class="sd">          :math:`\sigma` = (:math:`\sigma_{\rm{tx}}`, :math:`\sigma_{\rm{ty}}`,</span>
+<span class="sd">          :math:`\sigma_{\rm{tz}}`, :math:`\sigma_{\rm{r}}`).</span>
+
+<span class="sd">    Example:</span>
+
+<span class="sd">        For :math:`\sigma = (\sigma_t, \sigma_r)`</span>
+
+<span class="sd">        &gt;&gt;&gt; pp.randn_se3(2, sigma=(1.0, 0.5))</span>
+<span class="sd">        se3Type LieTensor:</span>
+<span class="sd">        tensor([[-0.4226,  0.4028, -1.3824,  0.4433, -0.2029, -0.1193],</span>
+<span class="sd">                [-0.8423, -1.0435,  0.8311, -0.4733,  0.0175,  0.1400]])</span>
+
+<span class="sd">        For :math:`\sigma = (\sigma_{tx}, \sigma_{ty}, \sigma_{tz}, \sigma_{r})`</span>
+
+<span class="sd">        &gt;&gt;&gt; pp.randn_se3(2, sigma=(1.0, 2.0, 3.0, 0.5))</span>
+<span class="sd">        se3Type LieTensor:</span>
+<span class="sd">        tensor([[ 1.1209,  1.4211, -0.7237, -0.1168,  0.0128,  0.1479],</span>
+<span class="sd">                [ 0.1765,  0.3891,  3.4799, -0.0411, -0.2616, -0.1028]])</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">return</span> <span class="n">se3_type</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="o">*</span><span class="n">lsize</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="n">sigma</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="randn_SE3"><a class="viewcode-back" href="../../../../generated/pypose.randn_SE3/#pypose.randn_SE3">[docs]</a><span class="k">def</span> <span class="nf">randn_SE3</span><span class="p">(</span><span class="o">*</span><span class="n">lsize</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Returns :obj:`SE3_type` LieTensor filled with the Exponential map of the random</span>
+<span class="sd">    :obj:`se3_type` LieTensor generated using :meth:`pypose.randn_se3()`.</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \mathrm{data}[*, :] = \mathrm{Exp}([\tau_x, \tau_y, \tau_z, \delta_x, \delta_y, \delta_z]),</span>
+
+<span class="sd">    where :math:`[\tau_x, \tau_y, \tau_z]` is generated from a normal distribution</span>
+<span class="sd">    :math:`\mathcal{N}(0, \sigma_t)`, :math:`[\delta_x, \delta_y, \delta_z]` is</span>
+<span class="sd">    generated using :meth:`pypose.randn_so3()` with with standard deviation</span>
+<span class="sd">    :math:`\sigma_r`, and :math:`\mathrm{Exp}()` is the Exponential map. Note that standard</span>
+<span class="sd">    deviations :math:`\sigma_t` and :math:`\sigma_r` are specified by ``sigma`` (:math:`\sigma`),</span>
+<span class="sd">    where :math:`\sigma = (\sigma_t, \sigma_r)`.</span>
+
+<span class="sd">    For detailed explanation, please see :meth:`pypose.randn_se3()` and :meth:`pypose.Exp()`.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        lsize (int...): a sequence of integers defining the lshape of the output tensor.</span>
+<span class="sd">            Can be a variable number of arguments or a collection like a list or tuple.</span>
+
+<span class="sd">        sigma (float or (float...), optional): standard deviation</span>
+<span class="sd">            (:math:`\sigma_t` and :math:`\sigma_r`) for the two normal distribution.</span>
+<span class="sd">            Default: ``1.0``.</span>
+
+<span class="sd">        requires_grad (bool, optional): If autograd should record operations on</span>
+<span class="sd">            the returned tensor. Default: ``False``.</span>
+
+<span class="sd">        generator (torch.Generator, optional): a pseudorandom number generator for sampling</span>
+
+<span class="sd">        dtype (torch.dtype, optional): the desired data type of returned tensor.</span>
+<span class="sd">            Default: ``None``. If None, uses a global default</span>
+<span class="sd">            (see :meth:`torch.set_default_tensor_type()`).</span>
+
+<span class="sd">        layout (torch.layout, optional): the desired layout of returned Tensor.</span>
+<span class="sd">            Default: ``torch.strided``.</span>
+
+<span class="sd">        device (torch.device, optional): the desired device of returned tensor.</span>
+<span class="sd">            Default: ``None``. If ``None``, uses the current device for the default tensor</span>
+<span class="sd">            type (see :meth:`torch.set_default_tensor_type()`). Device will be the CPU</span>
+<span class="sd">            for CPU tensor types and the current CUDA device for CUDA tensor types.</span>
+
+<span class="sd">    Returns:</span>
+<span class="sd">        LieTensor: a :obj:`SE3_type` LieTensor</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        The parameter :math:`\sigma` can either be:</span>
+
+<span class="sd">        - a single ``float`` -- in which all the elements in the :obj:`SE3_type`</span>
+<span class="sd">          share the same sigma, i.e.,</span>
+<span class="sd">          :math:`\sigma_{\rm{t}}` = :math:`\sigma_{\rm{r}}` = :math:`\sigma`.</span>
+<span class="sd">        - a ``tuple`` of two floats -- in which case, the specific sigmas are</span>
+<span class="sd">          assigned independently, i.e.,</span>
+<span class="sd">          :math:`\sigma` = (:math:`\sigma_{\rm{t}}`, :math:`\sigma_{\rm{r}}`).</span>
+<span class="sd">        - a ``tuple`` of four floats -- in which case, the specific sigmas for</span>
+<span class="sd">          each translation data are assigned independently, i.e.,</span>
+<span class="sd">          :math:`\sigma` = (:math:`\sigma_{\rm{tx}}`, :math:`\sigma_{\rm{ty}}`,</span>
+<span class="sd">          :math:`\sigma_{\rm{tz}}`, :math:`\sigma_{\rm{r}}`).</span>
+
+<span class="sd">    Example:</span>
+
+<span class="sd">        For :math:`\sigma = (\sigma_t, \sigma_r)`</span>
+
+<span class="sd">        &gt;&gt;&gt; pp.randn_SE3(2, sigma=(1.0, 2.0))</span>
+<span class="sd">        SE3Type LieTensor:</span>
+<span class="sd">        tensor([[ 0.2947, -1.6990, -0.5535,  0.4439,  0.2777,  0.0518,  0.8504],</span>
+<span class="sd">                [ 0.6825,  0.2963,  0.3410,  0.3375, -0.2355,  0.7389, -0.5335]])</span>
+
+<span class="sd">        For :math:`\sigma = (\sigma_{tx}, \sigma_{ty}, \sigma_{tz}, \sigma_{r})`</span>
+
+<span class="sd">        &gt;&gt;&gt; pp.randn_SE3(2, sigma=(1.0, 1.5, 2.0, 2.0))</span>
+<span class="sd">        SE3Type LieTensor:</span>
+<span class="sd">        tensor([[-1.5689, -0.6772,  0.3580, -0.2509,  0.8257, -0.4950,  0.1018],</span>
+<span class="sd">                [ 0.2613, -2.7613,  0.2151, -0.8802,  0.2619,  0.3044,  0.2531]])</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">return</span> <span class="n">SE3_type</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="o">*</span><span class="n">lsize</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="n">sigma</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="randn_sim3"><a class="viewcode-back" href="../../../../generated/pypose.randn_sim3/#pypose.randn_sim3">[docs]</a><span class="k">def</span> <span class="nf">randn_sim3</span><span class="p">(</span><span class="o">*</span><span class="n">lsize</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Returns :obj:`sim3_type` LieTensor filled with random numbers.</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \mathrm{data}[*, :] = [\tau_x, \tau_y, \tau_z, \delta_x, \delta_y, \delta_z, \log s],</span>
+
+<span class="sd">    where translation :math:`[\tau_x, \tau_y, \tau_z]` is generated from a normal distribution</span>
+<span class="sd">    :math:`\mathcal{N}(0, \sigma_t)`, rotation :math:`[\delta_x, \delta_y, \delta_z]` is</span>
+<span class="sd">    generated using :meth:`pypose.randn_so3()` with standard deviation :math:`\sigma_r`,</span>
+<span class="sd">    scale :math:`\log s` is generated from a normal distribution :math:`\mathcal{N}(0, \sigma_s)`.</span>
+<span class="sd">    Note that standard deviations :math:`\sigma_t`, :math:`\sigma_r`, and :math:`\sigma_s` are</span>
+<span class="sd">    specified by ``sigma`` (:math:`\sigma`), where :math:`\sigma=(\sigma_t,\sigma_r,\sigma_s)`.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        lsize (int...): a sequence of integers defining the lshape of the output tensor.</span>
+<span class="sd">            Can be a variable number of arguments or a collection like a list or tuple.</span>
+
+<span class="sd">        sigma (float or (float...), optional): standard deviation (:math:`\sigma_t`,</span>
+<span class="sd">            :math:`\sigma_r`, and :math:`\sigma_s`) for the three normal distribution.</span>
+<span class="sd">            Default: ``1.0``.</span>
+
+<span class="sd">        requires_grad (bool, optional): If autograd should record operations on</span>
+<span class="sd">            the returned tensor. Default: ``False``.</span>
+
+<span class="sd">        generator (torch.Generator, optional): a pseudorandom number generator for sampling.</span>
+
+<span class="sd">        dtype (torch.dtype, optional): the desired data type of returned tensor.</span>
+<span class="sd">            Default: ``None``. If ``None``, uses a global default</span>
+<span class="sd">            (see :meth:`torch.set_default_tensor_type()`).</span>
+
+<span class="sd">        layout (torch.layout, optional): the desired layout of returned Tensor.</span>
+<span class="sd">            Default: ``torch.strided``.</span>
+
+<span class="sd">        device (torch.device, optional): the desired device of returned tensor.</span>
+<span class="sd">            Default: ``None``. If ``None``, uses the current device for the default tensor</span>
+<span class="sd">            type (see :meth:`torch.set_default_tensor_type()`). Device will be the CPU</span>
+<span class="sd">            for CPU tensor types and the current CUDA device for CUDA tensor types.</span>
+
+<span class="sd">    Returns:</span>
+<span class="sd">        LieTensor: a ``sim3_type`` LieTensor.</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        The parameter :math:`\sigma` can either be:</span>
+
+<span class="sd">        - a single ``float`` -- in which all the elements in the :obj:`sim3_type`</span>
+<span class="sd">          share the same sigma, i.e.,</span>
+<span class="sd">          :math:`\sigma_{\rm{t}}` = :math:`\sigma_{\rm{r}}` = :math:`\sigma_{\rm{s}}`</span>
+<span class="sd">          = :math:`\sigma`.</span>
+<span class="sd">        - a ``tuple`` of three floats -- in which case, the specific sigmas for</span>
+<span class="sd">          the three parts are assigned independently, i.e.,</span>
+<span class="sd">          :math:`\sigma` = (:math:`\sigma_{\rm{t}}`, :math:`\sigma_{\rm{r}}`,</span>
+<span class="sd">          :math:`\sigma_{\rm{s}}`).</span>
+<span class="sd">        - a ``tuple`` of five floats -- in which case, the specific sigmas for</span>
+<span class="sd">          each translation data are also assigned independently, i.e.,</span>
+<span class="sd">          :math:`\sigma` = (:math:`\sigma_{\rm{tx}}`, :math:`\sigma_{\rm{ty}}`,</span>
+<span class="sd">          :math:`\sigma_{\rm{tz}}`, :math:`\sigma_{\rm{r}}`, :math:`\sigma_{\rm{s}}`).</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        For :math:`\sigma = (\sigma_{\rm{t}}, \sigma_{\rm{r}}, \sigma_{\rm{s}})`</span>
+
+<span class="sd">        &gt;&gt;&gt; pp.randn_sim3(sigma=(1.0, 1.0, 2.0))</span>
+<span class="sd">        sim3Type LieTensor:</span>
+<span class="sd">        LieTensor([ 1.1994, -1.6163, -0.7566, -0.1805,  0.2199, -0.7044, -3.9935])</span>
+
+<span class="sd">        For :math:`\sigma = (\sigma_{\rm{tx}}, \sigma_{\rm{ty}}, \sigma_{\rm{tz}},</span>
+<span class="sd">        \sigma_{\rm{r}}, \sigma_{\rm{s})}`</span>
+
+<span class="sd">        &gt;&gt;&gt; pp.randn_sim3(2, sigma=(1.0, 1.0, 2.0, 1.0, 2.0))</span>
+<span class="sd">        sim3Type LieTensor:</span>
+<span class="sd">        tensor([[ 0.3995, -1.9705,  2.6748,  0.5061, -1.4121,  1.1144,  0.5393],</span>
+<span class="sd">                [ 0.7968,  0.5076,  1.0034,  0.8263,  1.3350, -0.0851,  2.2611]])</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">return</span> <span class="n">sim3_type</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="o">*</span><span class="n">lsize</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="n">sigma</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="randn_Sim3"><a class="viewcode-back" href="../../../../generated/pypose.randn_Sim3/#pypose.randn_Sim3">[docs]</a><span class="k">def</span> <span class="nf">randn_Sim3</span><span class="p">(</span><span class="o">*</span><span class="n">lsize</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Returns :obj:`Sim3_type` LieTensor filled with the Exponential map of the random</span>
+<span class="sd">    :obj:`sim3_type` LieTensor generated using :meth:`randn_sim3()`.</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \mathrm{data}[*, :] = \mathrm{Exp}([\tau_x, \tau_y, \tau_z, \delta_x,</span>
+<span class="sd">        \delta_y, \delta_z, \log s]),</span>
+
+<span class="sd">    where translation :math:`[\tau_x, \tau_y, \tau_z]` is generated from a normal distribution</span>
+<span class="sd">    :math:`\mathcal{N}(0, \sigma_t)`, rotation :math:`[\delta_x, \delta_y, \delta_z]` is</span>
+<span class="sd">    generated using :meth:`pypose.randn_so3()` with with standard deviation</span>
+<span class="sd">    :math:`\sigma_r`, scale :math:`\log s` is generated from a normal distribution</span>
+<span class="sd">    :math:`\mathcal{N}(0, \sigma_s)`, and :math:`\mathrm{Exp}()` is the Exponential</span>
+<span class="sd">    map. Note that standard deviations :math:`\sigma_t`, :math:`\sigma_r`, and</span>
+<span class="sd">    :math:`\sigma_s` are specified by ``sigma`` (:math:`\sigma`), where</span>
+<span class="sd">    :math:`\sigma = (\sigma_t, \sigma_r, \sigma_s)`.</span>
+
+<span class="sd">    For detailed explanation, please see :meth:`pypose.randn_sim3()` and :meth:`pypose.Exp()`.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        lsize (int...): a sequence of integers defining the lshape of the output tensor.</span>
+<span class="sd">            Can be a variable number of arguments or a collection like a list or tuple.</span>
+
+<span class="sd">        sigma (float or (float...), optional): standard deviation</span>
+<span class="sd">            (:math:`\sigma_t`, :math:`\sigma_r`, and :math:`\sigma_s`)</span>
+<span class="sd">            for the three normal distribution. Default: ``1.0``.</span>
+
+<span class="sd">        requires_grad (bool, optional): If autograd should record operations on</span>
+<span class="sd">            the returned tensor. Default: ``False``.</span>
+
+<span class="sd">        generator (torch.Generator, optional): a pseudorandom number generator for sampling</span>
+
+<span class="sd">        dtype (torch.dtype, optional): the desired data type of returned tensor.</span>
+<span class="sd">            Default: ``None``. If ``None``, uses a global default</span>
+<span class="sd">            (see :meth:`torch.set_default_tensor_type()`).</span>
+
+<span class="sd">        layout (torch.layout, optional): the desired layout of returned Tensor.</span>
+<span class="sd">            Default: ``torch.strided``.</span>
+
+<span class="sd">        device (torch.device, optional): the desired device of returned tensor.</span>
+<span class="sd">            Default: ``None``. If ``None``, uses the current device for the default tensor</span>
+<span class="sd">            type (see :meth:`torch.set_default_tensor_type()`). Device will be the CPU</span>
+<span class="sd">            for CPU tensor types and the current CUDA device for CUDA tensor types.</span>
+
+<span class="sd">    Returns:</span>
+<span class="sd">        LieTensor: a :obj:`Sim_type` LieTensor</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        The parameter :math:`\sigma` can either be:</span>
+
+<span class="sd">        - a single ``float`` -- in which all the elements in the :obj:`Sim3_type`</span>
+<span class="sd">          share the same sigma, i.e.,</span>
+<span class="sd">          :math:`\sigma_{\rm{t}}` = :math:`\sigma_{\rm{r}}`</span>
+<span class="sd">          = :math:`\sigma_{\rm{s}}` = :math:`\sigma`.</span>
+<span class="sd">        - a ``tuple`` of three floats -- in which case, the specific sigmas</span>
+<span class="sd">          for the three parts are assigned independently, i.e.,</span>
+<span class="sd">          :math:`\sigma` = (:math:`\sigma_{\rm{t}}`, :math:`\sigma_{\rm{r}}`,</span>
+<span class="sd">          :math:`\sigma_{\rm{s}}`).</span>
+<span class="sd">        - a ``tuple`` of five floats -- in which case, the specific sigmas</span>
+<span class="sd">          for each translation data are also assigned independently, i.e.,</span>
+<span class="sd">          :math:`\sigma` = (:math:`\sigma_{\rm{tx}}`, :math:`\sigma_{\rm{ty}}`,</span>
+<span class="sd">          :math:`\sigma_{\rm{tz}}`, :math:`\sigma_{\rm{r}}`, :math:`\sigma_{\rm{s}}`).</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        For :math:`\sigma = (\sigma_{\rm{t}}, \sigma_{\rm{r}}, \sigma_{\rm{s}})`</span>
+
+<span class="sd">        &gt;&gt;&gt; pp.randn_Sim3(sigma=(1.0, 1.0, 2.0))</span>
+<span class="sd">        Sim3Type LieTensor:</span>
+<span class="sd">        LieTensor([-0.7667, -0.0981,  0.8168,  0.0931,  0.0917,  0.0939,  0.9870, 0.2391])</span>
+
+<span class="sd">        For :math:`\sigma = (\sigma_{\rm{tx}}, \sigma_{\rm{ty}}, \sigma_{\rm{tz}},</span>
+<span class="sd">        \sigma_{\rm{r}}, \sigma_{\rm{s})}`</span>
+
+<span class="sd">        &gt;&gt;&gt; pp.randn_Sim3(2, sigma=(1.0, 1.0, 2.0, 1.0, 2.0))</span>
+<span class="sd">        Sim3Type LieTensor:</span>
+<span class="sd">        tensor([[-0.0117,  0.0708,  1.6853,  0.1089,  0.4186, -0.0877,  0.8973,  0.3969],</span>
+<span class="sd">                [ 0.2106, -0.0694, -0.0574, -0.2902, -0.4806, -0.0815,  0.8235,  0.0134]])</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">return</span> <span class="n">Sim3_type</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="o">*</span><span class="n">lsize</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="n">sigma</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="randn_rxso3"><a class="viewcode-back" href="../../../../generated/pypose.randn_rxso3/#pypose.randn_rxso3">[docs]</a><span class="k">def</span> <span class="nf">randn_rxso3</span><span class="p">(</span><span class="o">*</span><span class="n">lsize</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Returns :obj:`rxso3_type` LieTensor filled with random numbers.</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \mathrm{data}[*, :] = [\delta_x, \delta_y, \delta_z, \log s],</span>
+
+<span class="sd">    where rotation :math:`[\delta_x, \delta_y, \delta_z]` is</span>
+<span class="sd">    generated using :meth:`pypose.randn_so3()` with standard deviation :math:`\sigma_r`,</span>
+<span class="sd">    scale :math:`\log s` is generated from a normal distribution :math:`\mathcal{N}(0, \sigma_s)`.</span>
+<span class="sd">    Note that standard deviations :math:`\sigma_r` and :math:`\sigma_s` are</span>
+<span class="sd">    specified by ``sigma`` (:math:`\sigma`), where :math:`\sigma = (\sigma_r, \sigma_s)`.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        lsize (int...): a sequence of integers defining the lshape of the output tensor.</span>
+<span class="sd">            Can be a variable number of arguments or a collection like a list or tuple.</span>
+
+<span class="sd">        sigma (float or (float...), optional): standard deviation (:math:`\sigma_r`,</span>
+<span class="sd">            and :math:`\sigma_s`) for the two normal distribution. Default: ``1.0``.</span>
+
+<span class="sd">        requires_grad (bool, optional): If autograd should record operations on</span>
+<span class="sd">            the returned tensor. Default: ``False``.</span>
+
+<span class="sd">        generator (torch.Generator, optional): a pseudorandom number generator for sampling</span>
+
+<span class="sd">        dtype (torch.dtype, optional): the desired data type of returned tensor.</span>
+<span class="sd">            Default: ``None``. If ``None``, uses a global default</span>
+<span class="sd">            (see :meth:`torch.set_default_tensor_type()`).</span>
+
+<span class="sd">        layout (torch.layout, optional): the desired layout of returned Tensor.</span>
+<span class="sd">            Default: ``torch.strided``.</span>
+
+<span class="sd">        device (torch.device, optional): the desired device of returned tensor.</span>
+<span class="sd">            Default: ``None``. If ``None``, uses the current device for the default tensor</span>
+<span class="sd">            type (see :meth:`torch.set_default_tensor_type()`). Device will be the CPU</span>
+<span class="sd">            for CPU tensor types and the current CUDA device for CUDA tensor types.</span>
+
+<span class="sd">    Returns:</span>
+<span class="sd">        LieTensor: a :obj:`rxso3_type` LieTensor</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        The parameter :math:`\sigma` can either be:</span>
+
+<span class="sd">        - a single ``float`` -- in which all the elements in the :obj:`rxso3_type` share</span>
+<span class="sd">          the same sigma, i.e., :math:`\sigma_{\rm{r}}` = :math:`\sigma_{\rm{s}}`</span>
+<span class="sd">          = :math:`\sigma`.</span>
+<span class="sd">        - a ``tuple`` of two floats -- in which case, the specific sigmas for the two parts</span>
+<span class="sd">          are assigned independently, i.e., :math:`\sigma` = (:math:`\sigma_{\rm{r}}`,</span>
+<span class="sd">          :math:`\sigma_{\rm{s}}`).</span>
+
+<span class="sd">    Example:</span>
+
+<span class="sd">        For :math:`\sigma = (\sigma_r, \sigma_s)`</span>
+
+<span class="sd">        &gt;&gt;&gt; pp.randn_rxso3(2, sigma=(1.0, 2.0))</span>
+<span class="sd">        rxso3Type LieTensor:</span>
+<span class="sd">        tensor([[-0.5033, -0.4102, -0.6213, -3.5049],</span>
+<span class="sd">                [-0.3185,  0.1053, -0.0816, -1.1907]])</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">return</span> <span class="n">rxso3_type</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="o">*</span><span class="n">lsize</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="n">sigma</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="randn_RxSO3"><a class="viewcode-back" href="../../../../generated/pypose.randn_RxSO3/#pypose.randn_RxSO3">[docs]</a><span class="k">def</span> <span class="nf">randn_RxSO3</span><span class="p">(</span><span class="o">*</span><span class="n">lsize</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Returns :obj:`RxSO3_type` LieTensor filled with the Exponential map of the random</span>
+<span class="sd">    :obj:`rxso3_type` LieTensor. The :obj:`rxso3_type` LieTensor is generated using</span>
+<span class="sd">    :meth:`randn_rxso3()`.</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \mathrm{data}[*, :] = \mathrm{Exp}([\delta_x, \delta_y, \delta_z, \log s]),</span>
+
+<span class="sd">    where rotation :math:`[\delta_x, \delta_y, \delta_z]` is generated using</span>
+<span class="sd">    :meth:`pypose.randn_so3()` with standard deviation :math:`\sigma_r`, scale :math:`\log s`</span>
+<span class="sd">    is generated from a normal distribution :math:`\mathcal{N}(0, \sigma_s)`, and</span>
+<span class="sd">    :math:`\mathrm{Exp}()` is the Exponential map. Note that standard deviations</span>
+<span class="sd">    :math:`\sigma_r` and :math:`\sigma_s` are specified by ``sigma`` (:math:`\sigma`), where</span>
+<span class="sd">    :math:`\sigma = (\sigma_r, \sigma_s)`.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        lsize (int...): a sequence of integers defining the lshape of the output tensor.</span>
+<span class="sd">            Can be a variable number of arguments or a collection like a list or tuple.</span>
+
+<span class="sd">        sigma (float or (float...), optional): standard deviation (:math:`\sigma_r`,</span>
+<span class="sd">            and :math:`\sigma_s`) for the two normal distribution. Default: ``1.0``.</span>
+
+<span class="sd">        requires_grad (bool, optional): If autograd should record operations on</span>
+<span class="sd">            the returned tensor. Default: ``False``.</span>
+
+<span class="sd">        generator (torch.Generator, optional): a pseudorandom number generator for sampling</span>
+
+<span class="sd">        dtype (torch.dtype, optional): the desired data type of returned tensor.</span>
+<span class="sd">            Default: ``None``. If ``None``, uses a global default</span>
+<span class="sd">            (see :meth:`torch.set_default_tensor_type()`).</span>
+
+<span class="sd">        layout (torch.layout, optional): the desired layout of returned Tensor.</span>
+<span class="sd">            Default: ``torch.strided``.</span>
+
+<span class="sd">        device (torch.device, optional): the desired device of returned tensor.</span>
+<span class="sd">            Default: &quot;None&quot;. If None, uses the current device for the default tensor</span>
+<span class="sd">            type (see :meth:`torch.set_default_tensor_type()`). Device will be the CPU</span>
+<span class="sd">            for CPU tensor types and the current CUDA device for CUDA tensor types.</span>
+
+<span class="sd">    Returns:</span>
+<span class="sd">        LieTensor: a :obj:`RxSO3_type` LieTensor</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        The parameter :math:`\sigma` can either be:</span>
+
+<span class="sd">        - a single ``float`` -- in which all the elements in the :obj:`RxSO3_type` share</span>
+<span class="sd">          the same sigma, i.e., :math:`\sigma_{\rm{r}}` = :math:`\sigma_{\rm{s}}` =</span>
+<span class="sd">          :math:`\sigma`.</span>
+<span class="sd">        - a ``tuple`` of two floats -- in which case, the specific sigmas for the two parts</span>
+<span class="sd">          are assigned independently, i.e., :math:`\sigma` = (:math:`\sigma_{\rm{r}}`,</span>
+<span class="sd">          :math:`\sigma_{\rm{s}}`).</span>
+
+<span class="sd">    Example:</span>
+
+<span class="sd">        For :math:`\sigma = (\sigma_r, \sigma_s)`</span>
+
+<span class="sd">        &gt;&gt;&gt; pp.randn_RxSO3(2, sigma=(1.0, 2.0))</span>
+<span class="sd">        RxSO3Type LieTensor:</span>
+<span class="sd">        tensor([[-0.1929, -0.0141,  0.2859,  0.9385,  4.5562],</span>
+<span class="sd">                [-0.2871,  0.0134, -0.2903,  0.9128,  3.1044]])</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">return</span> <span class="n">RxSO3_type</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="o">*</span><span class="n">lsize</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="n">sigma</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="identity_like"><a class="viewcode-back" href="../../../../generated/pypose.identity_like/#pypose.identity_like">[docs]</a><span class="k">def</span> <span class="nf">identity_like</span><span class="p">(</span><span class="n">liegroup</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">     Returns identity LieTensor with the same :obj:`lsize` and :obj:`ltype` as the given LieTensor.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        liegroup (LieTensor): the size of liegroup will determine the size of the output tensor.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        requires_grad (bool, optional): If autograd should record operations on</span>
+<span class="sd">            the returned tensor. Default: False.</span>
+
+<span class="sd">        generator (torch.Generator, optional): a pseudorandom number generator for sampling</span>
+
+<span class="sd">        dtype (torch.dtype, optional): the desired data type of returned tensor.</span>
+<span class="sd">            Default: if None, uses a global default (see :meth:`torch.set_default_tensor_type()`).</span>
+
+<span class="sd">        layout (torch.layout, optional): the desired layout of returned Tensor.</span>
+<span class="sd">            Default: torch.strided.</span>
+
+<span class="sd">        device (torch.device, optional): the desired device of returned tensor.</span>
+<span class="sd">            Default: if None, uses the current device for the default tensor</span>
+<span class="sd">            type (see :meth:`torch.set_default_tensor_type()`). device will be the CPU</span>
+<span class="sd">            for CPU tensor types and the current CUDA device for CUDA tensor types.</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        &gt;&gt;&gt; x = pp.randn_SO3(3, device=&quot;cuda:0&quot;, dtype=torch.double, requires_grad=True)</span>
+<span class="sd">        &gt;&gt;&gt; pp.identity_like(x, device=&quot;cpu&quot;)</span>
+<span class="sd">        SO3Type LieTensor:</span>
+<span class="sd">        tensor([[0., 0., 0., 1.],</span>
+<span class="sd">                [0., 0., 0., 1.],</span>
+<span class="sd">                [0., 0., 0., 1.]])</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">return</span> <span class="n">liegroup</span><span class="o">.</span><span class="n">ltype</span><span class="o">.</span><span class="n">identity_like</span><span class="p">(</span><span class="o">*</span><span class="n">liegroup</span><span class="o">.</span><span class="n">lshape</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="identity_SO3"><a class="viewcode-back" href="../../../../generated/pypose.identity_SO3/#pypose.identity_SO3">[docs]</a><span class="k">def</span> <span class="nf">identity_SO3</span><span class="p">(</span><span class="o">*</span><span class="n">lsize</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Returns identity :obj:`SO3_type` LieTensor with the given :obj:`lsize`.</span>
+
+<span class="sd">    See :obj:`SO3()` for implementation details.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        lsize (int..., optional): a sequence of integers defining the :obj:`LieTensor.lshape` of</span>
+<span class="sd">            the output LieTensor. Can be a variable number of arguments or a collection like a</span>
+<span class="sd">            list or tuple. If not given, a single :obj:`SO3_type` item will be returned.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        requires_grad (bool, optional): If autograd should record operations on</span>
+<span class="sd">            the returned tensor. Default: False.</span>
+
+<span class="sd">        generator (torch.Generator, optional): a pseudorandom number generator for sampling</span>
+
+<span class="sd">        dtype (torch.dtype, optional): the desired data type of returned tensor.</span>
+<span class="sd">            Default: if None, uses a global default (see :meth:`torch.set_default_tensor_type()`).</span>
+
+<span class="sd">        layout (torch.layout, optional): the desired layout of returned Tensor.</span>
+<span class="sd">            Default: torch.strided.</span>
+
+<span class="sd">        device (torch.device, optional): the desired device of returned tensor.</span>
+<span class="sd">            Default: if None, uses the current device for the default tensor</span>
+<span class="sd">            type (see :meth:`torch.set_default_tensor_type()`). device will be the CPU</span>
+<span class="sd">            for CPU tensor types and the current CUDA device for CUDA tensor types.</span>
+
+<span class="sd">    Returns:</span>
+<span class="sd">        LieTensor: a :obj:`SO3_type` LieTensor</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        &gt;&gt;&gt; pp.identity_SO3()</span>
+<span class="sd">        SO3Type LieTensor:</span>
+<span class="sd">        tensor([0., 0., 0., 1.])</span>
+
+<span class="sd">        &gt;&gt;&gt; pp.identity_SO3(2)</span>
+<span class="sd">        SO3Type LieTensor:</span>
+<span class="sd">        tensor([[0., 0., 0., 1.],</span>
+<span class="sd">                [0., 0., 0., 1.]])</span>
+
+<span class="sd">        &gt;&gt;&gt; pp.identity_SO3(2, 1)</span>
+<span class="sd">        SO3Type LieTensor:</span>
+<span class="sd">        tensor([[[0., 0., 0., 1.]],</span>
+<span class="sd">                [[0., 0., 0., 1.]]])</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">return</span> <span class="n">SO3_type</span><span class="o">.</span><span class="n">identity</span><span class="p">(</span><span class="o">*</span><span class="n">lsize</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="identity_so3"><a class="viewcode-back" href="../../../../generated/pypose.identity_so3/#pypose.identity_so3">[docs]</a><span class="k">def</span> <span class="nf">identity_so3</span><span class="p">(</span><span class="o">*</span><span class="n">lsize</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Returns identity :obj:`so3_type` LieTensor with the given :obj:`lsize`.</span>
+
+<span class="sd">    See :obj:`so3()` for implementation details.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        lsize (int..., optional): a sequence of integers defining the :obj:`LieTensor.lshape` of</span>
+<span class="sd">            the output LieTensor. Can be a variable number of arguments or a collection like a</span>
+<span class="sd">            list or tuple. If not given, a single :obj:`so3_type` item will be returned.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        requires_grad (bool, optional): If autograd should record operations on</span>
+<span class="sd">            the returned tensor. Default: False.</span>
+
+<span class="sd">        generator (torch.Generator, optional): a pseudorandom number generator for sampling</span>
+
+<span class="sd">        dtype (torch.dtype, optional): the desired data type of returned tensor.</span>
+<span class="sd">            Default: if None, uses a global default (see :meth:`torch.set_default_tensor_type()`).</span>
+
+<span class="sd">        layout (torch.layout, optional): the desired layout of returned Tensor.</span>
+<span class="sd">            Default: torch.strided.</span>
+
+<span class="sd">        device (torch.device, optional): the desired device of returned tensor.</span>
+<span class="sd">            Default: if None, uses the current device for the default tensor</span>
+<span class="sd">            type (see :meth:`torch.set_default_tensor_type()`). device will be the CPU</span>
+<span class="sd">            for CPU tensor types and the current CUDA device for CUDA tensor types.</span>
+
+<span class="sd">    Returns:</span>
+<span class="sd">        LieTensor: a :obj:`so3_type` LieTensor</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        &gt;&gt;&gt; pp.identity_so3()</span>
+<span class="sd">        so3Type LieTensor:</span>
+<span class="sd">        tensor([0., 0., 0.])</span>
+
+<span class="sd">        &gt;&gt;&gt; pp.identity_so3(2)</span>
+<span class="sd">        so3Type LieTensor:</span>
+<span class="sd">        tensor([[0., 0., 0.],</span>
+<span class="sd">                [0., 0., 0.]])</span>
+
+<span class="sd">        &gt;&gt;&gt; pp.identity_so3(2,1)</span>
+<span class="sd">        so3Type LieTensor:</span>
+<span class="sd">        tensor([[[0., 0., 0.]],</span>
+<span class="sd">                [[0., 0., 0.]]])</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">return</span> <span class="n">so3_type</span><span class="o">.</span><span class="n">identity</span><span class="p">(</span><span class="o">*</span><span class="n">lsize</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="identity_SE3"><a class="viewcode-back" href="../../../../generated/pypose.identity_SE3/#pypose.identity_SE3">[docs]</a><span class="k">def</span> <span class="nf">identity_SE3</span><span class="p">(</span><span class="o">*</span><span class="n">lsize</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Returns identity :obj:`SE3_type` LieTensor with the given :obj:`lsize`.</span>
+
+<span class="sd">    See :obj:`SE3()` for implementation details.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        lsize (int..., optional): a sequence of integers defining the :obj:`LieTensor.lshape` of</span>
+<span class="sd">            the output LieTensor. Can be a variable number of arguments or a collection like a</span>
+<span class="sd">            list or tuple. If not given, a single :obj:`SE3_type` item will be returned.</span>
+
+<span class="sd">    Args:</span>
+
+<span class="sd">        requires_grad (bool, optional): If autograd should record operations on</span>
+<span class="sd">            the returned tensor. Default: False.</span>
+
+<span class="sd">        generator (torch.Generator, optional): a pseudorandom number generator for sampling</span>
+
+<span class="sd">        dtype (torch.dtype, optional): the desired data type of returned tensor.</span>
+<span class="sd">            Default: if None, uses a global default (see :meth:`torch.set_default_tensor_type()`).</span>
+
+<span class="sd">        layout (torch.layout, optional): the desired layout of returned Tensor.</span>
+<span class="sd">            Default: torch.strided.</span>
+
+<span class="sd">        device (torch.device, optional): the desired device of returned tensor.</span>
+<span class="sd">            Default: if None, uses the current device for the default tensor</span>
+<span class="sd">            type (see :meth:`torch.set_default_tensor_type()`). device will be the CPU</span>
+<span class="sd">            for CPU tensor types and the current CUDA device for CUDA tensor types.</span>
+
+<span class="sd">    Returns:</span>
+<span class="sd">        LieTensor: a :obj:`SE3_type` LieTensor</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        &gt;&gt;&gt; pp.identity_SE3()</span>
+<span class="sd">        SE3Type LieTensor:</span>
+<span class="sd">        tensor([0., 0., 0., 0., 0., 0., 1.])</span>
+
+<span class="sd">        &gt;&gt;&gt; pp.identity_SE3(2)</span>
+<span class="sd">        SE3Type LieTensor:</span>
+<span class="sd">        tensor([[0., 0., 0., 0., 0., 0., 1.],</span>
+<span class="sd">                [0., 0., 0., 0., 0., 0., 1.]])</span>
+
+<span class="sd">        &gt;&gt;&gt; pp.identity_SE3(2,1)</span>
+<span class="sd">        SE3Type LieTensor:</span>
+<span class="sd">        tensor([[[0., 0., 0., 0., 0., 0., 1.]],</span>
+<span class="sd">                [[0., 0., 0., 0., 0., 0., 1.]]])</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">return</span> <span class="n">SE3_type</span><span class="o">.</span><span class="n">identity</span><span class="p">(</span><span class="o">*</span><span class="n">lsize</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="identity_se3"><a class="viewcode-back" href="../../../../generated/pypose.identity_se3/#pypose.identity_se3">[docs]</a><span class="k">def</span> <span class="nf">identity_se3</span><span class="p">(</span><span class="o">*</span><span class="n">lsize</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Returns identity :obj:`se3_type` LieTensor with the given :obj:`lsize`.</span>
+
+<span class="sd">    See :obj:`se3()` for implementation details.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        lsize (int..., optional): a sequence of integers defining the :obj:`LieTensor.lshape` of</span>
+<span class="sd">            the output LieTensor. Can be a variable number of arguments or a collection like a</span>
+<span class="sd">            list or tuple. If not given, a single :obj:`se3_type` item will be returned.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        requires_grad (bool, optional): If autograd should record operations on</span>
+<span class="sd">            the returned tensor. Default: False.</span>
+
+<span class="sd">        generator (torch.Generator, optional): a pseudorandom number generator for sampling</span>
+
+<span class="sd">        dtype (torch.dtype, optional): the desired data type of returned tensor.</span>
+<span class="sd">            Default: if None, uses a global default (see :meth:`torch.set_default_tensor_type()`).</span>
+
+<span class="sd">        layout (torch.layout, optional): the desired layout of returned Tensor.</span>
+<span class="sd">            Default: torch.strided.</span>
+
+<span class="sd">        device (torch.device, optional): the desired device of returned tensor.</span>
+<span class="sd">            Default: if None, uses the current device for the default tensor</span>
+<span class="sd">            type (see :meth:`torch.set_default_tensor_type()`). device will be the CPU</span>
+<span class="sd">            for CPU tensor types and the current CUDA device for CUDA tensor types.</span>
+
+<span class="sd">    Returns:</span>
+<span class="sd">        LieTensor: a :obj:`se3_type` LieTensor</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        &gt;&gt;&gt; pp.identity_se3()</span>
+<span class="sd">        se3Type LieTensor:</span>
+<span class="sd">        tensor([0., 0., 0., 0., 0., 0.])</span>
+
+<span class="sd">        &gt;&gt;&gt; pp.identity_se3(2)</span>
+<span class="sd">        se3Type LieTensor:</span>
+<span class="sd">        tensor([[0., 0., 0., 0., 0., 0.],</span>
+<span class="sd">                [0., 0., 0., 0., 0., 0.]])</span>
+
+<span class="sd">        &gt;&gt;&gt; pp.identity_se3(2,1)</span>
+<span class="sd">        se3Type LieTensor:</span>
+<span class="sd">        tensor([[0., 0., 0., 0., 0., 0.],</span>
+<span class="sd">                [0., 0., 0., 0., 0., 0.]])</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">return</span> <span class="n">se3_type</span><span class="o">.</span><span class="n">identity</span><span class="p">(</span><span class="o">*</span><span class="n">lsize</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="identity_sim3"><a class="viewcode-back" href="../../../../generated/pypose.identity_sim3/#pypose.identity_sim3">[docs]</a><span class="k">def</span> <span class="nf">identity_sim3</span><span class="p">(</span><span class="o">*</span><span class="n">lsize</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Returns identity :obj:`sim3_type` LieTensor with the given :obj:`lsize`.</span>
+
+<span class="sd">    See :obj:`sim3()` for implementation details.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        lsize (int..., optional): a sequence of integers defining the :obj:`LieTensor.lshape` of</span>
+<span class="sd">            the output LieTensor. Can be a variable number of arguments or a collection like a</span>
+<span class="sd">            list or tuple. If not given, a single :obj:`sim3_type` item will be returned.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        requires_grad (bool, optional): If autograd should record operations on</span>
+<span class="sd">            the returned tensor. Default: False.</span>
+
+<span class="sd">        generator (torch.Generator, optional): a pseudorandom number generator for sampling</span>
+
+<span class="sd">        dtype (torch.dtype, optional): the desired data type of returned tensor.</span>
+<span class="sd">            Default: if None, uses a global default (see :meth:`torch.set_default_tensor_type()`).</span>
+
+<span class="sd">        layout (torch.layout, optional): the desired layout of returned Tensor.</span>
+<span class="sd">            Default: torch.strided.</span>
+
+<span class="sd">        device (torch.device, optional): the desired device of returned tensor.</span>
+<span class="sd">            Default: if None, uses the current device for the default tensor</span>
+<span class="sd">            type (see :meth:`torch.set_default_tensor_type()`). device will be the CPU</span>
+<span class="sd">            for CPU tensor types and the current CUDA device for CUDA tensor types.</span>
+
+<span class="sd">    Returns:</span>
+<span class="sd">        LieTensor: a :obj:`sim3_type` LieTensor</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        &gt;&gt;&gt; pp.identity_sim3()</span>
+<span class="sd">        sim3Type LieTensor:</span>
+<span class="sd">        tensor([0., 0., 0., 0., 0., 0., 0.])</span>
+
+<span class="sd">        &gt;&gt;&gt; pp.identity_sim3(2)</span>
+<span class="sd">        sim3Type LieTensor:</span>
+<span class="sd">        tensor([[0., 0., 0., 0., 0., 0., 0.],</span>
+<span class="sd">                [0., 0., 0., 0., 0., 0., 0.]])</span>
+
+<span class="sd">        &gt;&gt;&gt; pp.identity_sim3(2,1)</span>
+<span class="sd">        sim3Type LieTensor:</span>
+<span class="sd">        tensor([[[0., 0., 0., 0., 0., 0., 0.]],</span>
+<span class="sd">                [[0., 0., 0., 0., 0., 0., 0.]]])</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">return</span> <span class="n">sim3_type</span><span class="o">.</span><span class="n">identity</span><span class="p">(</span><span class="o">*</span><span class="n">lsize</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="identity_Sim3"><a class="viewcode-back" href="../../../../generated/pypose.identity_Sim3/#pypose.identity_Sim3">[docs]</a><span class="k">def</span> <span class="nf">identity_Sim3</span><span class="p">(</span><span class="o">*</span><span class="n">lsize</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Returns identity :obj:`Sim3_type` LieTensor with the given :obj:`lsize`.</span>
+
+<span class="sd">    See :obj:`Sim3()` for implementation details.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        lsize (int..., optional): a sequence of integers defining the :obj:`LieTensor.lshape` of</span>
+<span class="sd">            the output LieTensor. Can be a variable number of arguments or a collection like a</span>
+<span class="sd">            list or tuple. If not given, a single :obj:`Sim3_type` item will be returned.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        requires_grad (bool, optional): If autograd should record operations on</span>
+<span class="sd">            the returned tensor. Default: False.</span>
+
+<span class="sd">        generator (torch.Generator, optional): a pseudorandom number generator for sampling</span>
+
+<span class="sd">        dtype (torch.dtype, optional): the desired data type of returned tensor.</span>
+<span class="sd">            Default: if None, uses a global default (see :meth:`torch.set_default_tensor_type()`).</span>
+
+<span class="sd">        layout (torch.layout, optional): the desired layout of returned Tensor.</span>
+<span class="sd">            Default: torch.strided.</span>
+
+<span class="sd">        device (torch.device, optional): the desired device of returned tensor.</span>
+<span class="sd">            Default: if None, uses the current device for the default tensor</span>
+<span class="sd">            type (see :meth:`torch.set_default_tensor_type()`). device will be the CPU</span>
+<span class="sd">            for CPU tensor types and the current CUDA device for CUDA tensor types.</span>
+
+<span class="sd">    Returns:</span>
+<span class="sd">        LieTensor: a :obj:`Sim3_type` LieTensor</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        &gt;&gt;&gt; pp.identity_Sim3()</span>
+<span class="sd">        Sim3Type LieTensor:</span>
+<span class="sd">        tensor([0., 0., 0., 0., 0., 0., 1., 1.])</span>
+
+<span class="sd">        &gt;&gt;&gt; pp.identity_Sim3(2)</span>
+<span class="sd">        Sim3Type LieTensor:</span>
+<span class="sd">        tensor([[0., 0., 0., 0., 0., 0., 1., 1.],</span>
+<span class="sd">                [0., 0., 0., 0., 0., 0., 1., 1.]])</span>
+
+<span class="sd">        &gt;&gt;&gt; pp.identity_Sim3(2, 1)</span>
+<span class="sd">        Sim3Type LieTensor:</span>
+<span class="sd">        tensor([[[0., 0., 0., 0., 0., 0., 1., 1.]],</span>
+<span class="sd">                [[0., 0., 0., 0., 0., 0., 1., 1.]]])</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">return</span> <span class="n">Sim3_type</span><span class="o">.</span><span class="n">identity</span><span class="p">(</span><span class="o">*</span><span class="n">lsize</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="identity_rxso3"><a class="viewcode-back" href="../../../../generated/pypose.identity_rxso3/#pypose.identity_rxso3">[docs]</a><span class="k">def</span> <span class="nf">identity_rxso3</span><span class="p">(</span><span class="o">*</span><span class="n">size</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Returns identity :obj:`rxso3_type` LieTensor with the given :obj:`lsize`.</span>
+
+<span class="sd">    See :obj:`rxSO3()` for implementation details.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        lsize (int..., optional): a sequence of integers defining the :obj:`LieTensor.lshape` of</span>
+<span class="sd">            the output LieTensor. Can be a variable number of arguments or a collection like a</span>
+<span class="sd">            list or tuple. If not given, a single :obj:`rxso3_type` item will be returned.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        requires_grad (bool, optional): If autograd should record operations on</span>
+<span class="sd">            the returned tensor. Default: False.</span>
+
+<span class="sd">        generator (torch.Generator, optional): a pseudorandom number generator for sampling</span>
+
+<span class="sd">        dtype (torch.dtype, optional): the desired data type of returned tensor.</span>
+<span class="sd">            Default: if None, uses a global default (see :meth:`torch.set_default_tensor_type()`).</span>
+
+<span class="sd">        layout (torch.layout, optional): the desired layout of returned Tensor.</span>
+<span class="sd">            Default: torch.strided.</span>
+
+<span class="sd">        device (torch.device, optional): the desired device of returned tensor.</span>
+<span class="sd">            Default: if None, uses the current device for the default tensor</span>
+<span class="sd">            type (see :meth:`torch.set_default_tensor_type()`). device will be the CPU</span>
+<span class="sd">            for CPU tensor types and the current CUDA device for CUDA tensor types.</span>
+
+<span class="sd">    Returns:</span>
+<span class="sd">        LieTensor: a :obj:`rxso3_type` LieTensor</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        &gt;&gt;&gt; pp.identity_rxso3()</span>
+<span class="sd">        rxso3Type LieTensor:</span>
+<span class="sd">        tensor([0., 0., 0., 0.])</span>
+
+<span class="sd">        &gt;&gt;&gt; pp.identity_rxso3(2)</span>
+<span class="sd">        rxso3Type LieTensor:</span>
+<span class="sd">        tensor([[0., 0., 0., 0.],</span>
+<span class="sd">                [0., 0., 0., 0.]])</span>
+
+<span class="sd">        &gt;&gt;&gt; pp.identity_rxso3(2, 1)</span>
+<span class="sd">        rxso3Type LieTensor:</span>
+<span class="sd">        tensor([[[0., 0., 0., 0.]],</span>
+<span class="sd">                [[0., 0., 0., 0.]]])</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">return</span> <span class="n">rxso3_type</span><span class="o">.</span><span class="n">identity</span><span class="p">(</span><span class="o">*</span><span class="n">size</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="identity_RxSO3"><a class="viewcode-back" href="../../../../generated/pypose.identity_RxSO3/#pypose.identity_RxSO3">[docs]</a><span class="k">def</span> <span class="nf">identity_RxSO3</span><span class="p">(</span><span class="o">*</span><span class="n">size</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;Returns identity :obj:`RxSO3_type` LieTensor with the given :obj:`lsize`.</span>
+
+<span class="sd">    See :obj:`RxSO3()` for implementation details.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        lsize (int..., optional): a sequence of integers defining the :obj:`LieTensor.lshape` of</span>
+<span class="sd">            the output LieTensor. Can be a variable number of arguments or a collection like a</span>
+<span class="sd">            list or tuple. If not given, a single :obj:`RxSO3_type` item will be returned.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        requires_grad (bool, optional): If autograd should record operations on</span>
+<span class="sd">            the returned tensor. Default: False.</span>
+
+<span class="sd">        generator (torch.Generator, optional): a pseudorandom number generator for sampling</span>
+
+<span class="sd">        dtype (torch.dtype, optional): the desired data type of returned tensor.</span>
+<span class="sd">            Default: if None, uses a global default (see :meth:`torch.set_default_tensor_type()`).</span>
+
+<span class="sd">        layout (torch.layout, optional): the desired layout of returned Tensor.</span>
+<span class="sd">            Default: torch.strided.</span>
+
+<span class="sd">        device (torch.device, optional): the desired device of returned tensor.</span>
+<span class="sd">            Default: if None, uses the current device for the default tensor</span>
+<span class="sd">            type (see :meth:`torch.set_default_tensor_type()`). device will be the CPU</span>
+<span class="sd">            for CPU tensor types and the current CUDA device for CUDA tensor types.</span>
+
+<span class="sd">    Returns:</span>
+<span class="sd">        LieTensor: a :obj:`RxSO3_type` LieTensor</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        &gt;&gt;&gt; pp.identity_RxSO3()</span>
+<span class="sd">        RxSO3Type LieTensor:</span>
+<span class="sd">        tensor([0., 0., 0., 1., 1.])</span>
+
+<span class="sd">        &gt;&gt;&gt; pp.identity_RxSO3(2)</span>
+<span class="sd">        RxSO3Type LieTensor:</span>
+<span class="sd">        tensor([[0., 0., 0., 1., 1.],</span>
+<span class="sd">                [0., 0., 0., 1., 1.]])</span>
+
+<span class="sd">        &gt;&gt;&gt; pp.identity_RxSO3(2, 1)</span>
+<span class="sd">        RxSO3Type LieTensor:</span>
+<span class="sd">        tensor([[[0., 0., 0., 1., 1.]],</span>
+<span class="sd">                [[0., 0., 0., 1., 1.]]])</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">return</span> <span class="n">RxSO3_type</span><span class="o">.</span><span class="n">identity</span><span class="p">(</span><span class="o">*</span><span class="n">size</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span></div>
+
+
+<span class="k">def</span> <span class="nf">assert_ltype</span><span class="p">(</span><span class="n">func</span><span class="p">):</span>
+    <span class="nd">@functools</span><span class="o">.</span><span class="n">wraps</span><span class="p">(</span><span class="n">func</span><span class="p">)</span>
+    <span class="k">def</span> <span class="nf">checker</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+        <span class="k">assert</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">LieTensor</span><span class="p">),</span> <span class="s2">&quot;Invalid LieTensor Type.&quot;</span>
+        <span class="n">out</span> <span class="o">=</span> <span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">out</span>
+    <span class="k">return</span> <span class="n">checker</span>
+
+
+<div class="viewcode-block" id="Exp"><a class="viewcode-back" href="../../../../generated/pypose.Exp/#pypose.Exp">[docs]</a><span class="nd">@assert_ltype</span>
+<span class="k">def</span> <span class="nf">Exp</span><span class="p">(</span><span class="nb">input</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&quot;&quot;&quot;The Exponential map for :obj:`LieTensor` (Lie Algebra).</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \mathrm{Exp}: \mathcal{g} \mapsto \mathcal{G}</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        input (LieTensor): the input LieTensor (Lie Algebra)</span>
+
+<span class="sd">    Return:</span>
+<span class="sd">        LieTensor: the output LieTensor (Lie Group)</span>
+
+<span class="sd">    .. list-table:: List of supported :math:`\mathrm{Exp}` map</span>
+<span class="sd">        :widths: 20 20 8 20 20</span>
+<span class="sd">        :header-rows: 1</span>
+
+<span class="sd">        * - input :obj:`ltype`</span>
+<span class="sd">          - :math:`\mathcal{g}` (Lie Algebra)</span>
+<span class="sd">          - :math:`\mapsto`</span>
+<span class="sd">          - :math:`\mathcal{G}` (Lie Group)</span>
+<span class="sd">          - output :obj:`ltype`</span>
+<span class="sd">        * - :obj:`so3_type`</span>
+<span class="sd">          - :math:`\mathcal{g}\in\mathbb{R}^{*\times3}`</span>
+<span class="sd">          - :math:`\mapsto`</span>
+<span class="sd">          - :math:`\mathcal{G}\in\mathbb{R}^{*\times4}`</span>
+<span class="sd">          - :obj:`SO3_type`</span>
+<span class="sd">        * - :obj:`se3_type`</span>
+<span class="sd">          - :math:`\mathcal{g}\in\mathbb{R}^{*\times6}`</span>
+<span class="sd">          - :math:`\mapsto`</span>
+<span class="sd">          - :math:`\mathcal{G}\in\mathbb{R}^{*\times7}`</span>
+<span class="sd">          - :obj:`SE3_type`</span>
+<span class="sd">        * - :obj:`sim3_type`</span>
+<span class="sd">          - :math:`\mathcal{g}\in\mathbb{R}^{*\times7}`</span>
+<span class="sd">          - :math:`\mapsto`</span>
+<span class="sd">          - :math:`\mathcal{G}\in\mathbb{R}^{*\times8}`</span>
+<span class="sd">          - :obj:`Sim3_type`</span>
+<span class="sd">        * - :obj:`rxso3_type`</span>
+<span class="sd">          - :math:`\mathcal{g}\in\mathbb{R}^{*\times4}`</span>
+<span class="sd">          - :math:`\mapsto`</span>
+<span class="sd">          - :math:`\mathcal{G}\in\mathbb{R}^{*\times5}`</span>
+<span class="sd">          - :obj:`RxSO3_type`</span>
+
+<span class="sd">    Warning:</span>
+<span class="sd">        This function :func:`Exp()` is different from :func:`exp()`, which returns</span>
+<span class="sd">        a new torch tensor with the exponential of the elements of the input tensor.</span>
+
+<span class="sd">    * Input :math:`\mathbf{x}`&#39;s :obj:`ltype` is :obj:`so3_type`</span>
+<span class="sd">      (input :math:`\mathbf{x}` is an instance of :meth:`so3`):</span>
+
+<span class="sd">        If :math:`\|\mathbf{x}_i\| &gt; \text{eps}`:</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathbf{y}_i = \left[\mathbf{x}_{i,1}\theta_i,</span>
+<span class="sd">            \mathbf{x}_{i,2}\theta_i,</span>
+<span class="sd">            \mathbf{x}_{i,3}\theta_i,</span>
+<span class="sd">            \cos(\frac{\|\mathbf{x}_i\|}{2})\right],</span>
+
+<span class="sd">        where :math:`\theta_i = \frac{1}{\|\mathbf{x}_i\|}\sin(\frac{\|\mathbf{x}_i\|}{2})`,</span>
+
+<span class="sd">        otherwise:</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathbf{y}_i = \left[\mathbf{x}_{i,1}\theta_i,~</span>
+<span class="sd">            \mathbf{x}_{i,2}\theta_i,~</span>
+<span class="sd">            \mathbf{x}_{i,3}\theta_i,~</span>
+<span class="sd">            1 - \frac{\|\mathbf{x}_i\|^2}{8} + \frac{\|\mathbf{x}_i\|^4}{384} \right],</span>
+
+<span class="sd">        where :math:`\theta_i = \frac{1}{2} - \frac{1}{48} \|\mathbf{x}_i\|^2</span>
+<span class="sd">        + \frac{1}{3840} \|\mathbf{x}_i\|^4`.</span>
+
+<span class="sd">    * Input :math:`\mathbf{x}`&#39;s :obj:`ltype` is :obj:`se3_type`</span>
+<span class="sd">      (input :math:`\mathbf{x}` is an instance of :meth:`se3`):</span>
+
+<span class="sd">        Let :math:`\bm{\tau}_i`, :math:`\bm{\phi}_i` be the translation and rotation parts</span>
+<span class="sd">        of :math:`\mathbf{x}_i`, respectively; :math:`\mathbf{y}` be the output.</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathbf{y}_i = \left[\mathbf{J}_i\bm{\tau}_i, \mathrm{Exp}(\bm{\phi}_i)\right],</span>
+
+<span class="sd">        where :math:`\mathrm{Exp}` is the Exponential map for :obj:`so3_type` input and</span>
+<span class="sd">        :math:`\mathbf{J}_i` is the left Jacobian for :obj:`so3_type` input.</span>
+
+<span class="sd">    * Input :math:`\mathbf{x}`&#39;s :obj:`ltype` is :obj:`rxso3_type`</span>
+<span class="sd">      (input :math:`\mathbf{x}` is an instance of :meth:`rxso3`):</span>
+
+<span class="sd">        Let :math:`\bm{\phi}_i`, :math:`\sigma_i` be the rotation and scale parts of</span>
+<span class="sd">        :math:`\mathbf{x}_i`, respectively; :math:`\mathbf{y}` be the output.</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathbf{y}_i = \left[\mathrm{Exp}(\bm{\phi}_i), \mathrm{exp}(\sigma_i)\right],</span>
+
+<span class="sd">        where :math:`\mathrm{exp}` is the exponential function.</span>
+
+<span class="sd">    * Input :math:`\mathbf{x}`&#39;s :obj:`ltype` is :obj:`sim3_type`</span>
+<span class="sd">      (input :math:`\mathbf{x}` is an instance of :meth:`sim3`):</span>
+
+<span class="sd">        Let :math:`\bm{\tau}_i`, :math:`^{s}\bm{\phi}_i` be the translation and</span>
+<span class="sd">        :meth:`rxso3` parts of :math:`\mathbf{x}_i`, respectively.</span>
+<span class="sd">        :math:`\bm{\phi}_i = \theta_i\mathbf{n}_i`, :math:`\sigma_i` be the rotation</span>
+<span class="sd">        and scale parts of :math:`^{s}\bm{\phi}_i`, :math:`\boldsymbol{\Phi}_i` be the skew matrix</span>
+<span class="sd">        of :math:`\bm{\phi}_i`; :math:`s_i = e^\sigma_i`, :math:`\mathbf{y}` be the output.</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathbf{y}_i = \left[^{s}\mathbf{W}_i\bm{\tau}_i, \mathrm{Exp}(^{s}\bm{\phi}_i)\right],</span>
+
+<span class="sd">        where</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            ^s\mathbf{W}_i = A\boldsymbol{\Phi}_i + B\boldsymbol{\Phi}_i^2 + C\mathbf{I}</span>
+
+<span class="sd">        in which if :math:`\|\sigma_i\| \geq \text{eps}`:</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            A = \left\{</span>
+<span class="sd">                    \begin{array}{ll}</span>
+<span class="sd">                        \frac{s_i\sin\theta_i\sigma_i + (1-s_i\cos\theta_i)\theta_i}</span>
+<span class="sd">                        {\theta_i(\sigma_i^2 + \theta_i^2)}, \quad \|\theta_i\| \geq \text{eps}, \\</span>
+<span class="sd">                        \frac{(\sigma_i-1)s_i+1}{\sigma_i^2}, \quad \|\theta_i\| &lt; \text{eps},</span>
+<span class="sd">                    \end{array}</span>
+<span class="sd">                \right.</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            B =</span>
+<span class="sd">            \left\{</span>
+<span class="sd">                \begin{array}{ll}</span>
+<span class="sd">                    \left( C - \frac{(s_i\cos\theta_i-1)\sigma+ s_i\sin\theta_i\sigma_i}</span>
+<span class="sd">                    {\theta_i^2+\sigma_i^2}\right)\frac{1}{\theta_i^2}, \quad \|\theta_i\|</span>
+<span class="sd">                        \geq \text{eps}, \\</span>
+<span class="sd">                    \frac{s_i\sigma_i^2/2 + s_i-1-\sigma_i s_i}{\sigma_i^3}, \quad \|\theta_i\|</span>
+<span class="sd">                        &lt; \text{eps},</span>
+<span class="sd">                \end{array}</span>
+<span class="sd">            \right.</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            C = \frac{e^{\sigma_i} - 1}{\sigma_i}\mathbf{I}</span>
+
+<span class="sd">        otherwise:</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            A = \left\{</span>
+<span class="sd">                    \begin{array}{ll}</span>
+<span class="sd">                        \frac{1-\cos\theta_i}{\theta_i^2}, \quad \|\theta_i\| \geq \text{eps}, \\</span>
+<span class="sd">                        \frac{1}{2}, \quad \|\theta_i\| &lt; \text{eps},</span>
+<span class="sd">                    \end{array}</span>
+<span class="sd">                \right.</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            B = \left\{</span>
+<span class="sd">                    \begin{array}{ll}</span>
+<span class="sd">                        \frac{\theta_i - \sin\theta_i}{\theta_i^3}, \quad \|\theta_i\|</span>
+<span class="sd">                            \geq \text{eps}, \\</span>
+<span class="sd">                        \frac{1}{6}, \quad \|\theta_i\| &lt; \text{eps},</span>
+<span class="sd">                    \end{array}</span>
+<span class="sd">                \right.</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            C = 1</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        The detailed explanation of the above :math:`\mathrm{Exp}`: calculation can be found</span>
+<span class="sd">        in the paper:</span>
+
+<span class="sd">        * Grassia, F. Sebastian., `Practical Parameterization of Rotations using the</span>
+<span class="sd">          Exponential Map &lt;https://www.tandfonline.com/doi/pdf/10.1080/10867651.1998.10487493&gt;`_,</span>
+<span class="sd">          Journal of graphics tools, 1998.</span>
+
+<span class="sd">        Assume we have a unit rotation axis :math:`\mathbf{n}~(\|\mathbf{n}\|=1)` and rotation</span>
+<span class="sd">        angle :math:`\theta~(0\leq\theta&lt;2\pi)`, let :math:`\mathbf{x}=\theta\mathbf{n}`, then</span>
+<span class="sd">        the corresponding quaternion with unit norm :math:`\mathbf{q}` can be represented as:</span>
+
+<span class="sd">            .. math::</span>
+<span class="sd">                \mathbf{q} = \left[\frac{\sin(\theta/2)}{\theta} \mathbf{x}, \cos(\theta/2) \right].</span>
+
+<span class="sd">        Given :math:`\mathbf{x}=\theta\mathbf{n}`, to find its corresponding quaternion</span>
+<span class="sd">        :math:`\mathbf{q}`, we first calculate the rotation angle :math:`\theta` using:</span>
+
+<span class="sd">            .. math::</span>
+<span class="sd">                \theta = \|\mathbf{x}\|.</span>
+
+<span class="sd">        Then, the corresponding quaternion is:</span>
+
+<span class="sd">            .. math::</span>
+<span class="sd">                \mathbf{q} = \left[\frac{\sin(\|\mathbf{x}\|/2)}{\|\mathbf{x}\|} \mathbf{x},</span>
+<span class="sd">                    \cos(\|\mathbf{x}\|/2) \right].</span>
+
+<span class="sd">        If :math:`\|\mathbf{x}\|` is small (:math:`\|\mathbf{x}\|\le \text{eps}`),</span>
+<span class="sd">        we use the Taylor Expansion form of :math:`\sin(\|\mathbf{x}\|/2)` and</span>
+<span class="sd">        :math:`\cos(\|\mathbf{x}\|/2)`.</span>
+
+<span class="sd">        More details about :math:`^s\mathbf{W}_i` in :obj:`sim3_type` can be found in Eq. (5.7):</span>
+
+<span class="sd">        * H. Strasdat, `Local Accuracy and Global Consistency for Efficient Visual SLAM</span>
+<span class="sd">          &lt;http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.640.199&amp;rep=rep1&amp;type=pdf&gt;`_,</span>
+<span class="sd">          Dissertation. Department of Computing, Imperial College London, 2012.</span>
+
+<span class="sd">    Examples:</span>
+
+<span class="sd">        * :math:`\mathrm{Exp}`: :obj:`so3` :math:`\mapsto` :obj:`SO3`</span>
+
+<span class="sd">            &gt;&gt;&gt; x = pp.randn_so3()</span>
+<span class="sd">            &gt;&gt;&gt; x.Exp() # equivalent to: pp.Exp(x)</span>
+<span class="sd">            SO3Type LieTensor:</span>
+<span class="sd">            LieTensor([-0.6627, -0.0447,  0.3492,  0.6610])</span>
+
+<span class="sd">        * :math:`\mathrm{Exp}`: :obj:`se3` :math:`\mapsto` :obj:`SE3`</span>
+
+<span class="sd">            &gt;&gt;&gt; x = pp.randn_se3(2, requires_grad=True)</span>
+<span class="sd">            se3Type LieTensor:</span>
+<span class="sd">            tensor([[ 1.1912,  1.2425, -0.9696,  0.9540, -0.4061, -0.7204],</span>
+<span class="sd">                    [ 0.5964, -1.1894,  0.6451,  1.1373, -2.6733,  0.4142]])</span>
+<span class="sd">            &gt;&gt;&gt; x.Exp() # equivalent to: pp.Exp(x)</span>
+<span class="sd">            SE3Type LieTensor:</span>
+<span class="sd">            tensor([[ 1.6575,  0.8838, -0.1499,  0.4459, -0.1898, -0.3367,  0.8073],</span>
+<span class="sd">                    [ 0.2654, -1.3860,  0.2852,  0.3855, -0.9061,  0.1404,  0.1034]],</span>
+<span class="sd">                    grad_fn=&lt;AliasBackward0&gt;)</span>
+
+<span class="sd">        * :math:`\mathrm{Exp}`: :obj:`rxso3` :math:`\mapsto` :obj:`RxSO3`</span>
+
+<span class="sd">            &gt;&gt;&gt; x = pp.randn_rxso3(2)</span>
+<span class="sd">            &gt;&gt;&gt; x.Exp() # equivalent to: pp.Exp(x)</span>
+<span class="sd">            RxSO3Type LieTensor:</span>
+<span class="sd">            tensor([[-0.5633, -0.4281,  0.1112,  0.6979,  0.7408],</span>
+<span class="sd">                    [ 0.5089,  0.2016, -0.2015,  0.8122,  1.1692]])</span>
+
+<span class="sd">        * :math:`\mathrm{Exp}`: :obj:`sim3` :math:`\mapsto` :obj:`Sim3`</span>
+
+<span class="sd">            &gt;&gt;&gt; x = pp.randn_sim3(2)</span>
+<span class="sd">            &gt;&gt;&gt; x.Exp() # equivalent to: pp.Exp(x)</span>
+<span class="sd">            Sim3Type LieTensor:</span>
+<span class="sd">            tensor([[-1.5811,  1.8128, -0.5835,  0.5849,  0.1142, -0.3438,  0.7257,  2.4443],</span>
+<span class="sd">                    [ 0.9574, -0.9265, -0.2385, -0.7309, -0.3875,  0.1404,  0.5440,  1.1945]])</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+    <span class="k">return</span> <span class="nb">input</span><span class="o">.</span><span class="n">Exp</span><span class="p">()</span></div>
+
+
+<div class="viewcode-block" id="Log"><a class="viewcode-back" href="../../../../generated/pypose.Log/#pypose.Log">[docs]</a><span class="nd">@assert_ltype</span>
+<span class="k">def</span> <span class="nf">Log</span><span class="p">(</span><span class="nb">input</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&quot;&quot;&quot;The Logarithm map for :obj:`LieTensor` (Lie Group).</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \mathrm{Log}: \mathcal{G} \mapsto \mathcal{g}</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        input (LieTensor): the input LieTensor (Lie Group)</span>
+
+<span class="sd">    Return:</span>
+<span class="sd">        LieTensor: the output LieTensor (Lie Algebra)</span>
+
+<span class="sd">    .. list-table:: List of supported :math:`\mathrm{Log}` map</span>
+<span class="sd">        :widths: 20 20 8 20 20</span>
+<span class="sd">        :header-rows: 1</span>
+
+<span class="sd">        * - input :obj:`ltype`</span>
+<span class="sd">          - :math:`\mathcal{G}` (Lie Group)</span>
+<span class="sd">          - :math:`\mapsto`</span>
+<span class="sd">          - :math:`\mathcal{g}` (Lie Algebra)</span>
+<span class="sd">          - output :obj:`ltype`</span>
+<span class="sd">        * - :obj:`SO3_type`</span>
+<span class="sd">          - :math:`\mathcal{G}\in\mathbb{R}^{*\times4}`</span>
+<span class="sd">          - :math:`\mapsto`</span>
+<span class="sd">          - :math:`\mathcal{g}\in\mathbb{R}^{*\times3}`</span>
+<span class="sd">          - :obj:`so3_type`</span>
+<span class="sd">        * - :obj:`SE3_type`</span>
+<span class="sd">          - :math:`\mathcal{G}\in\mathbb{R}^{*\times7}`</span>
+<span class="sd">          - :math:`\mapsto`</span>
+<span class="sd">          - :math:`\mathcal{g}\in\mathbb{R}^{*\times6}`</span>
+<span class="sd">          - :obj:`se3_type`</span>
+<span class="sd">        * - :obj:`Sim3_type`</span>
+<span class="sd">          - :math:`\mathcal{G}\in\mathbb{R}^{*\times8}`</span>
+<span class="sd">          - :math:`\mapsto`</span>
+<span class="sd">          - :math:`\mathcal{g}\in\mathbb{R}^{*\times7}`</span>
+<span class="sd">          - :obj:`sim3_type`</span>
+<span class="sd">        * - :obj:`RxSO3_type`</span>
+<span class="sd">          - :math:`\mathcal{G}\in\mathbb{R}^{*\times5}`</span>
+<span class="sd">          - :math:`\mapsto`</span>
+<span class="sd">          - :math:`\mathcal{g}\in\mathbb{R}^{*\times4}`</span>
+<span class="sd">          - :obj:`rxso3_type`</span>
+
+<span class="sd">    Warning:</span>
+<span class="sd">        This function :func:`Log()` is different from :func:`log()`, which returns</span>
+<span class="sd">        a new torch tensor with the logarithm of the elements of the input tensor.</span>
+
+<span class="sd">    * If input :math:`\mathbf{x}`&#39;s :obj:`ltype` is :obj:`SO3_type`</span>
+<span class="sd">      (input :math:`\mathbf{x}` is an instance of :meth:`SO3`):</span>
+
+<span class="sd">        Let :math:`w_i`, :math:`\boldsymbol{\nu}_i` be the scalar and vector parts of</span>
+<span class="sd">        :math:`\mathbf{x}_i`, respectively; :math:`\mathbf{y}` be the output.</span>
+
+<span class="sd">        If :math:`\|\boldsymbol{\nu}_i\| &gt; \text{eps}`:</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathbf{y}_i =</span>
+<span class="sd">                \left\{</span>
+<span class="sd">                    \begin{array}{ll}</span>
+<span class="sd">                        2\frac{\mathrm{arctan}(\|\boldsymbol{\nu}_i\|/w_i)}{\|</span>
+<span class="sd">                        \boldsymbol{\nu}_i\|}\boldsymbol{\nu}_i,</span>
+<span class="sd">                            \quad \|w_i\| &gt; \text{eps}, \\</span>
+<span class="sd">                        \mathrm{pm}(w_i) \frac{\pi}{\|\boldsymbol{\nu}_i\|}</span>
+<span class="sd">                            \boldsymbol{\nu}_i, \quad \|w_i\| \leq \text{eps},</span>
+<span class="sd">                    \end{array}</span>
+<span class="sd">                \right.</span>
+
+<span class="sd">        otherwise:</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathbf{y}_i = 2\left( \frac{1}{w_i} -</span>
+<span class="sd">                \frac{\|\boldsymbol{\nu}_i\|^2}{3w_i^3}\right)\boldsymbol{\nu}_i.</span>
+
+<span class="sd">        where :math:`\mathrm{pm}` is the plus or minus (:math:`\pm`) operator</span>
+<span class="sd">        (refer to :meth:`pm`).</span>
+
+<span class="sd">    * If input :math:`\mathbf{x}`&#39;s :obj:`ltype` is :obj:`SE3_type`</span>
+<span class="sd">      (input :math:`\mathbf{x}` is an instance of :meth:`SE3`):</span>
+
+<span class="sd">        Let :math:`\mathbf{t}_i`, :math:`\mathbf{q}_i` be the translation and rotation parts of</span>
+<span class="sd">        :math:`\mathbf{x}_i`, respectively; :math:`\mathbf{y}` be the output.</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathbf{y}_i = \left[\mathbf{J}_i^{-1}\mathbf{t}_i, \mathrm{Log}(\mathbf{q}_i) \right],</span>
+
+<span class="sd">        where :math:`\mathrm{Log}` is the Logarithm map for :obj:`SO3_type` input and</span>
+<span class="sd">        :math:`\mathbf{J}_i` is the left Jacobian for :obj:`SO3_type` input.</span>
+
+<span class="sd">    * If input :math:`\mathbf{x}`&#39;s :obj:`ltype` is :obj:`RxSO3_type`</span>
+<span class="sd">      (input :math:`\mathbf{x}` is an instance of :meth:`RxSO3`):</span>
+
+<span class="sd">        Let :math:`\mathbf{q}_i`, :math:`s_i` be the rotation and scale parts of</span>
+<span class="sd">        :math:`\mathbf{x}_i`, respectively; :math:`\mathbf{y}` be the output.</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathbf{y}_i = \left[\mathrm{Log}(\mathbf{q}_i), \log(s_i) \right].</span>
+
+<span class="sd">    * If input :math:`\mathbf{x}`&#39;s :obj:`ltype` is :obj:`Sim3_type` (input :math:`\mathbf{x}`</span>
+<span class="sd">      is an instance of :meth:`Sim3`):</span>
+
+<span class="sd">        Let :math:`\mathbf{t}_i`, :math:`^s\mathbf{q}_i` be the translation and :obj:`RxSO3`</span>
+<span class="sd">        parts of :math:`\mathbf{x}_i`, respectively; :math:`\boldsymbol{\phi}_i`,</span>
+<span class="sd">        :math:`\sigma_i` be the corresponding Lie Algebra of the SO3 and scale part of</span>
+<span class="sd">        :math:`^s\mathbf{q}_i`, :math:`\boldsymbol{\Phi}_i` be the skew matrix of</span>
+<span class="sd">        :math:`\boldsymbol{\phi}_i`, :math:`\boldsymbol{\phi}_i` can be represented as</span>
+<span class="sd">        :math:`\theta_i\mathbf{n}_i`, :math:`s_i = e^\sigma_i`, :math:`\mathbf{y}` be the</span>
+<span class="sd">        output.</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathbf{y}_i = \left[^s\mathbf{W}_i^{-1}\mathbf{t}_i,</span>
+<span class="sd">                \mathrm{Log}(^s\mathbf{q}_i) \right],</span>
+
+<span class="sd">        where</span>
+
+<span class="sd">            .. math::</span>
+<span class="sd">               ^s\mathbf{W}_i = A\boldsymbol{\Phi}_i + B\boldsymbol{\Phi}_i^2 + C\mathbf{I}</span>
+
+<span class="sd">        in which if :math:`\|\sigma_i\| &gt; \text{eps}`:</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            A = \left\{</span>
+<span class="sd">                    \begin{array}{ll}</span>
+<span class="sd">                        \frac{s_i\sin\theta_i\sigma_i + (1-s_i\cos\theta_i)\theta_i}</span>
+<span class="sd">                        {\theta_i(\sigma_i^2 + \theta_i^2)}, \quad \|\theta_i\| &gt; \text{eps}, \\</span>
+<span class="sd">                        \frac{(\sigma_i-1)s_i+1}{\sigma_i^2}, \quad \|\theta_i\| \leq \text{eps},</span>
+<span class="sd">                    \end{array}</span>
+<span class="sd">                \right.</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            B =</span>
+<span class="sd">            \left\{</span>
+<span class="sd">                \begin{array}{ll}</span>
+<span class="sd">                    \left( C - \frac{(s_i\cos\theta_i-1)\sigma+ s_i\sin\theta_i\sigma_i}</span>
+<span class="sd">                    {\theta_i^2+\sigma_i^2}\right)\frac{1}{\theta_i^2},</span>
+<span class="sd">                        \quad \|\theta_i\| &gt; \text{eps}, \\</span>
+<span class="sd">                    \frac{s_i\sigma_i^2/2 + s_i-1-\sigma_i s_i}{\sigma_i^3},</span>
+<span class="sd">                        \quad \|\theta_i\| \leq \text{eps},</span>
+<span class="sd">                \end{array}</span>
+<span class="sd">            \right.</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            C = \frac{e^{\sigma_i} - 1}{\sigma_i}\mathbf{I}</span>
+
+<span class="sd">        otherwise:</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            A = \left\{</span>
+<span class="sd">                    \begin{array}{ll}</span>
+<span class="sd">                        \frac{1-\cos\theta_i}{\theta_i^2}, \quad \|\theta_i\| &gt; \text{eps}, \\</span>
+<span class="sd">                        \frac{1}{2}, \quad \|\theta_i\| \leq \text{eps},</span>
+<span class="sd">                    \end{array}</span>
+<span class="sd">                \right.</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            B = \left\{</span>
+<span class="sd">                    \begin{array}{ll}</span>
+<span class="sd">                        \frac{\theta_i - \sin\theta_i}{\theta_i^3},</span>
+<span class="sd">                            \quad \|\theta_i\| &gt; \text{eps}, \\</span>
+<span class="sd">                        \frac{1}{6}, \quad \|\theta_i\| \leq \text{eps},</span>
+<span class="sd">                    \end{array}</span>
+<span class="sd">                \right.</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            C = 1</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        The :math:`\mathrm{arctan}`-based Logarithm map implementation thanks to the paper:</span>
+
+<span class="sd">        * C. Hertzberg et al., `Integrating Generic Sensor Fusion Algorithms with Sound State</span>
+<span class="sd">          Representation through Encapsulation of Manifolds</span>
+<span class="sd">          &lt;https://doi.org/10.1016/j.inffus.2011.08.003&gt;`_, Information Fusion, 2013.</span>
+
+<span class="sd">        Assume we have a unit rotation axis :math:`\mathbf{n}` and rotation angle</span>
+<span class="sd">        :math:`\theta~(0\leq\theta&lt;2\pi)`, then the corresponding quaternion with</span>
+<span class="sd">        unit norm :math:`\mathbf{q}` can be represented as</span>
+
+<span class="sd">            .. math::</span>
+<span class="sd">                \mathbf{q} = \left[\sin(\theta/2) \mathbf{n}, \cos(\theta/2) \right]</span>
+
+<span class="sd">        Therefore, given a quaternion :math:`\mathbf{q}=[\boldsymbol{\nu}, w]`, where</span>
+<span class="sd">        :math:`\boldsymbol{\nu}` is the vector part, :math:`w` is the scalar part, to find</span>
+<span class="sd">        the corresponding rotation vector, the rotation angle :math:`\theta` can be obtained as</span>
+
+<span class="sd">            .. math::</span>
+<span class="sd">                \theta = 2\mathrm{arctan}(\|\boldsymbol{\nu}\|/w),~\|\boldsymbol{\nu}\|</span>
+<span class="sd">                = \sin(\theta/2),</span>
+
+<span class="sd">        The unit rotation axis :math:`\mathbf{n}` can be obtained as :math:`\mathbf{n} =</span>
+<span class="sd">        \frac{\boldsymbol{\nu}}{{\|\boldsymbol{\nu}\|}}`.</span>
+<span class="sd">        Hence, the corresponding rotation vector is</span>
+
+<span class="sd">            .. math::</span>
+<span class="sd">                \theta \mathbf{n} = 2\frac{\mathrm{arctan}</span>
+<span class="sd">                (\|\boldsymbol{\nu}\|/w)}{\|\boldsymbol{\nu}\|}\boldsymbol{\nu}.</span>
+
+<span class="sd">        More details about :math:`^s\mathbf{W}_i` in :obj:`Sim3_type` can be found in Eq. (5.7):</span>
+
+<span class="sd">        * H. Strasdat, `Local accuracy and global consistency for efficient visual SLAM</span>
+<span class="sd">          &lt;http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.640.199&amp;rep=rep1&amp;type=pdf&gt;`_,</span>
+<span class="sd">          Dissertation. Department of Computing, Imperial College London, 2012.</span>
+
+<span class="sd">    Example:</span>
+
+<span class="sd">        * :math:`\mathrm{Log}`: :obj:`SO3` :math:`\mapsto` :obj:`so3`</span>
+
+<span class="sd">            &gt;&gt;&gt; x = pp.randn_SO3()</span>
+<span class="sd">            &gt;&gt;&gt; x.Log() # equivalent to: pp.Log(x)</span>
+<span class="sd">            so3Type LieTensor:</span>
+<span class="sd">            tensor([-0.3060,  0.2344,  1.2724])</span>
+
+<span class="sd">        * :math:`\mathrm{Log}`: :obj:`SE3` :math:`\mapsto` :obj:`se3`</span>
+
+<span class="sd">            &gt;&gt;&gt; x = pp.randn_SE3(2)</span>
+<span class="sd">            &gt;&gt;&gt; x.Log() # equivalent to: pp.Log(x)</span>
+<span class="sd">            se3Type LieTensor:</span>
+<span class="sd">            tensor([[ 0.2958, -0.0840, -1.4733,  0.7004,  0.4483, -0.9009],</span>
+<span class="sd">                    [ 0.0850, -0.1020, -1.2616, -1.0524, -1.2031,  0.8377]])</span>
+
+<span class="sd">        * :math:`\mathrm{Log}`: :obj:`RxSO3` :math:`\mapsto` :obj:`rxso3`</span>
+
+<span class="sd">            &gt;&gt;&gt; x = pp.randn_RxSO3(2)</span>
+<span class="sd">            &gt;&gt;&gt; x.Log() # equivalent to: pp.Log(x)</span>
+<span class="sd">            rxso3Type LieTensor:</span>
+<span class="sd">            tensor([[-1.3755,  0.3525, -2.2367,  0.5409],</span>
+<span class="sd">                    [ 0.5929, -0.3250, -0.7394,  1.0965]])</span>
+
+<span class="sd">        * :math:`\mathrm{Log}`: :obj:`Sim3` :math:`\mapsto` :obj:`sim3`</span>
+
+<span class="sd">            &gt;&gt;&gt; x = pp.randn_Sim3(2)</span>
+<span class="sd">            &gt;&gt;&gt; x.Log() # equivalent to: pp.Log(x)</span>
+<span class="sd">            sim3Type LieTensor:</span>
+<span class="sd">            tensor([[-0.1747, -0.3698,  0.2000,  0.1735,  0.6220,  1.1852, -0.6402],</span>
+<span class="sd">                    [-0.8685, -0.1717,  1.2139, -0.8385, -2.2957, -1.9545,  0.8474]])</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+    <span class="k">return</span> <span class="nb">input</span><span class="o">.</span><span class="n">Log</span><span class="p">()</span></div>
+
+
+<div class="viewcode-block" id="Inv"><a class="viewcode-back" href="../../../../generated/pypose.Inv/#pypose.Inv">[docs]</a><span class="nd">@assert_ltype</span>
+<span class="k">def</span> <span class="nf">Inv</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">    The inverse of the input lieTensor.</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \mathrm{Inv}: Y_i = X_i^{-1}</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        input (LieTensor): the input LieTensor (Lie Group or Lie Algebra)</span>
+
+<span class="sd">    Return:</span>
+<span class="sd">        LieTensor: the output LieTensor (Lie Group or Lie Algebra)</span>
+
+<span class="sd">    .. list-table:: List of supported :math:`\mathrm{Inv}` map</span>
+<span class="sd">        :widths: 20 20 8 20 20</span>
+<span class="sd">        :header-rows: 1</span>
+
+<span class="sd">        * - input :obj:`ltype`</span>
+<span class="sd">          - LieTensor</span>
+<span class="sd">          - :math:`\mapsto`</span>
+<span class="sd">          - LieTensor</span>
+<span class="sd">          - output :obj:`ltype`</span>
+<span class="sd">        * - :obj:`SO3_type`</span>
+<span class="sd">          - :math:`\mathcal{G}\in\mathbb{R}^{*\times4}`</span>
+<span class="sd">          - :math:`\mapsto`</span>
+<span class="sd">          - :math:`\mathcal{G}\in\mathbb{R}^{*\times4}`</span>
+<span class="sd">          - :obj:`SO3_type`</span>
+<span class="sd">        * - :obj:`SE3_type`</span>
+<span class="sd">          - :math:`\mathcal{G}\in\mathbb{R}^{*\times7}`</span>
+<span class="sd">          - :math:`\mapsto`</span>
+<span class="sd">          - :math:`\mathcal{G}\in\mathbb{R}^{*\times7}`</span>
+<span class="sd">          - :obj:`SE3_type`</span>
+<span class="sd">        * - :obj:`Sim3_type`</span>
+<span class="sd">          - :math:`\mathcal{G}\in\mathbb{R}^{*\times8}`</span>
+<span class="sd">          - :math:`\mapsto`</span>
+<span class="sd">          - :math:`\mathcal{G}\in\mathbb{R}^{*\times8}`</span>
+<span class="sd">          - :obj:`Sim3_type`</span>
+<span class="sd">        * - :obj:`RxSO3_type`</span>
+<span class="sd">          - :math:`\mathcal{G}\in\mathbb{R}^{*\times5}`</span>
+<span class="sd">          - :math:`\mapsto`</span>
+<span class="sd">          - :math:`\mathcal{G}\in\mathbb{R}^{*\times5}`</span>
+<span class="sd">          - :obj:`RxSO3_type`</span>
+<span class="sd">        * - :obj:`so3_type`</span>
+<span class="sd">          - :math:`\mathcal{g}\in\mathbb{R}^{*\times3}`</span>
+<span class="sd">          - :math:`\mapsto`</span>
+<span class="sd">          - :math:`\mathcal{g}\in\mathbb{R}^{*\times3}`</span>
+<span class="sd">          - :obj:`so3_type`</span>
+<span class="sd">        * - :obj:`se3_type`</span>
+<span class="sd">          - :math:`\mathcal{g}\in\mathbb{R}^{*\times6}`</span>
+<span class="sd">          - :math:`\mapsto`</span>
+<span class="sd">          - :math:`\mathcal{g}\in\mathbb{R}^{*\times6}`</span>
+<span class="sd">          - :obj:`se3_type`</span>
+<span class="sd">        * - :obj:`sim3_type`</span>
+<span class="sd">          - :math:`\mathcal{g}\in\mathbb{R}^{*\times7}`</span>
+<span class="sd">          - :math:`\mapsto`</span>
+<span class="sd">          - :math:`\mathcal{g}\in\mathbb{R}^{*\times7}`</span>
+<span class="sd">          - :obj:`sim3_type`</span>
+<span class="sd">        * - :obj:`rxso3_type`</span>
+<span class="sd">          - :math:`\mathcal{g}\in\mathbb{R}^{*\times4}`</span>
+<span class="sd">          - :math:`\mapsto`</span>
+<span class="sd">          - :math:`\mathcal{g}\in\mathbb{R}^{*\times4}`</span>
+<span class="sd">          - :obj:`rxso3_type`</span>
+
+<span class="sd">    * If input :math:`\mathbf{x}`&#39;s :obj:`ltype` is :obj:`SO3_type`</span>
+<span class="sd">      (input :math:`\mathbf{x}` is an instance of :meth:`SO3`):</span>
+
+<span class="sd">        Let :math:`w_i`, :math:`\boldsymbol{\nu}_i` be the scalar and vector parts of</span>
+<span class="sd">        :math:`\mathbf{x}_i`, respectively. :math:`\mathbf{x}_i=\left[\boldsymbol{\nu}_i,</span>
+<span class="sd">        \ w_i \right]`. :math:`\mathbf{y}` be the output.</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathbf{y}_i = \mathrm{conj}(\mathbf{x}_i)=[-\boldsymbol{\nu}_i, \ w_i]</span>
+
+<span class="sd">    * If input :math:`\mathbf{x}`&#39;s :obj:`ltype` is :obj:`SE3_type`</span>
+<span class="sd">      (input :math:`\mathbf{x}` is an instance of :meth:`SE3`):</span>
+
+<span class="sd">        Let :math:`\mathbf{t}_i`, :math:`\mathbf{q}_i` be the translation and rotation parts of</span>
+<span class="sd">        :math:`\mathbf{x}_i`, respectively; :math:`\mathbf{y}` be the output.</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathbf{y}_i = \left[-\mathrm{Inv}(\mathbf{q}_i)*\mathbf{t}_i,</span>
+<span class="sd">            \mathrm{Inv}(\mathbf{q}_i) \right]</span>
+
+<span class="sd">    * If input :math:`\mathbf{x}`&#39;s :obj:`ltype` is :obj:`RxSO3_type`</span>
+<span class="sd">      (input :math:`\mathbf{x}` is an instance of :meth:`RxSO3`):</span>
+
+<span class="sd">        Let :math:`\mathbf{q}_i`, :math:`s_i` be the rotation and scale parts of</span>
+<span class="sd">        :math:`\mathbf{x}_i`, respectively; :math:`\mathbf{y}` be the output.</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathbf{y}_i = \left[\mathrm{conj}(\mathbf{q}_i), \ 1/s_i \right]</span>
+
+<span class="sd">    * If input :math:`\mathbf{x}`&#39;s :obj:`ltype` is :obj:`Sim3_type` (input :math:`\mathbf{x}`</span>
+<span class="sd">      is an instance of :meth:`Sim3`):</span>
+
+<span class="sd">        Let :math:`\mathbf{t}_i`, :math:`^s\mathbf{q}_i` be the translation and :obj:`RxSO3` parts</span>
+<span class="sd">        of :math:`\mathbf{x}_i`, respectively;  :math:`\mathbf{y}` be the output.</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathbf{y}_i = \left[-\mathrm{Inv}(^s\mathbf{q}_i)*\mathbf{t}_i,</span>
+<span class="sd">            \mathrm{Inv}(^s\mathbf{q}_i) \right]</span>
+
+<span class="sd">    * If input :math:`\mathbf{x}`&#39;s :obj:`ltype` is :obj:`so3_type` or :obj:`se3_type` or</span>
+<span class="sd">      :obj:`sim3_type` or :obj:`rxso3_type` (input :math:`\mathbf{x}` is an instance of :meth:`so3`</span>
+<span class="sd">      or :meth:`se3` or :meth:`sim3` or :meth:`rxso3`):</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathbf{y}_i = -\mathbf{x}_i</span>
+
+<span class="sd">    Example:</span>
+
+<span class="sd">        * :math:`\mathrm{Inv}`: :obj:`SO3` :math:`\mapsto` :obj:`SO3`</span>
+
+<span class="sd">            &gt;&gt;&gt; x = pp.randn_SO3()</span>
+<span class="sd">            SO3Type LieTensor:</span>
+<span class="sd">            tensor([-0.1402, -0.2827,  0.2996,  0.9004])</span>
+<span class="sd">            &gt;&gt;&gt; x.Inv() # equivalent to: pp.Inv(x)</span>
+<span class="sd">            SO3Type LieTensor:</span>
+<span class="sd">            tensor([ 0.1402,  0.2827, -0.2996,  0.9004])</span>
+
+<span class="sd">        * :math:`\mathrm{Inv}`: :obj:`SE3` :math:`\mapsto` :obj:`SE3`</span>
+
+<span class="sd">            &gt;&gt;&gt; x = pp.randn_SE3()</span>
+<span class="sd">            SE3Type LieTensor:</span>
+<span class="sd">            tensor([ 0.6074, -0.7596,  0.8703, -0.3092,  0.2932,  0.9027,  0.0598])</span>
+<span class="sd">            &gt;&gt;&gt; x.Inv() # equivalent to: pp.Inv(x)</span>
+<span class="sd">            SE3Type LieTensor:</span>
+<span class="sd">            tensor([ 0.9475, -0.8764,  0.1938,  0.3092, -0.2932, -0.9027,  0.0598])</span>
+
+<span class="sd">        * :math:`\mathrm{Inv}`: :obj:`RxSO3` :math:`\mapsto` :obj:`RxSO3`</span>
+
+<span class="sd">            &gt;&gt;&gt; x = pp.randn_RxSO3()</span>
+<span class="sd">            RxSO3Type LieTensor:</span>
+<span class="sd">            tensor([-0.5103,  0.4707, -0.3494,  0.6292,  0.9199])</span>
+<span class="sd">            &gt;&gt;&gt; x.Inv() # equivalent to: pp.Inv(x)</span>
+<span class="sd">            RxSO3Type LieTensor:</span>
+<span class="sd">            tensor([ 0.5103, -0.4707,  0.3494,  0.6292,  1.0871])</span>
+
+<span class="sd">        * :math:`\mathrm{Inv}`: :obj:`Sim3` :math:`\mapsto` :obj:`Sim3`</span>
+
+<span class="sd">            &gt;&gt;&gt; x = pp.randn_Sim3()</span>
+<span class="sd">            Sim3Type LieTensor:</span>
+<span class="sd">            tensor([ 0.7056,  1.3140, -0.1995, -0.2444, -0.5250,  0.5504,  0.6014,  1.0543])</span>
+<span class="sd">            &gt;&gt;&gt; x.Inv() # equivalent to: pp.Inv(x)</span>
+<span class="sd">            Sim3Type LieTensor:</span>
+<span class="sd">            tensor([-0.9712, -0.2361,  1.0188,  0.2444,  0.5250, -0.5504,  0.6014,  0.9485])</span>
+
+<span class="sd">        * :math:`\mathrm{Inv}`: :obj:`so3` :math:`\mapsto` :obj:`so3`</span>
+
+<span class="sd">            &gt;&gt;&gt; x = pp.randn_so3()</span>
+<span class="sd">            so3Type LieTensor:</span>
+<span class="sd">            tensor([ 0.0612, -0.7190,  2.6897])</span>
+<span class="sd">            &gt;&gt;&gt; x.Inv() # equivalent to: pp.Inv(x)</span>
+<span class="sd">            so3Type LieTensor:</span>
+<span class="sd">            tensor([-0.0612,  0.7190, -2.6897])</span>
+
+<span class="sd">        * :math:`\mathrm{Inv}`: :obj:`se3` :math:`\mapsto` :obj:`se3`</span>
+
+<span class="sd">            &gt;&gt;&gt; x = pp.randn_se3()</span>
+<span class="sd">            se3Type LieTensor:</span>
+<span class="sd">            tensor([ 0.2837, -1.8318,  1.0104,  2.2385, -0.1980, -0.9487])</span>
+<span class="sd">            &gt;&gt;&gt; x.Inv() # equivalent to: pp.Inv(x)</span>
+<span class="sd">            se3Type LieTensor:</span>
+<span class="sd">            tensor([-0.2837,  1.8318, -1.0104, -2.2385,  0.1980,  0.9487])</span>
+
+<span class="sd">        * :math:`\mathrm{Inv}`: :obj:`rxso3` :math:`\mapsto` :obj:`rxso3`</span>
+
+<span class="sd">            &gt;&gt;&gt; x = pp.randn_rxso3()</span>
+<span class="sd">            rxso3Type LieTensor:</span>
+<span class="sd">            tensor([ 1.0414, -0.0087, -0.4427, -1.1343])</span>
+<span class="sd">            &gt;&gt;&gt; x.Inv() # equivalent to: pp.Inv(x)</span>
+<span class="sd">            rxso3Type LieTensor:</span>
+<span class="sd">            tensor([-1.0414,  0.0087,  0.4427,  1.1343])</span>
+
+<span class="sd">        * :math:`\mathrm{Inv}`: :obj:`sim3` :math:`\mapsto` :obj:`sim3`</span>
+
+<span class="sd">            &gt;&gt;&gt; x = pp.randn_sim3()</span>
+<span class="sd">            sim3Type LieTensor:</span>
+<span class="sd">            tensor([-0.0724,  1.8174,  2.1810, -0.9324, -0.0952, -0.5792,  0.4318])</span>
+<span class="sd">            &gt;&gt;&gt; x.Inv() # equivalent to: pp.Inv(x)</span>
+<span class="sd">            sim3Type LieTensor:</span>
+<span class="sd">            tensor([ 0.0724, -1.8174, -2.1810,  0.9324,  0.0952,  0.5792, -0.4318])</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        Mathematically, only Lie Group has an inverse. For the convenience, we also provide the</span>
+<span class="sd">        inverse for the corresponding Lie Algebra. One can validate the results by:</span>
+
+<span class="sd">        if input :math:`\mathbf{x}` is the LieTensor from Lie Group, for example</span>
+<span class="sd">        :math:`\mathbf{x}`&#39;s :obj:`ltype` is :obj:`SO3_type` :</span>
+
+<span class="sd">            &gt;&gt;&gt; x = pp.randn_SO3()</span>
+<span class="sd">            &gt;&gt;&gt; x * x.Inv()</span>
+<span class="sd">            SO3Type LieTensor:</span>
+<span class="sd">            LieTensor([0., 0., 0., 1.])</span>
+
+<span class="sd">        if input :math:`\mathbf{x}` is the LieTensor from Lie Algebra, for example</span>
+<span class="sd">        :math:`\mathbf{x}`&#39;s :obj:`ltype` is :obj:`so3_type` :</span>
+
+<span class="sd">            &gt;&gt;&gt; x = pp.randn_so3()</span>
+<span class="sd">            &gt;&gt;&gt; x + x.Inv()</span>
+<span class="sd">            tensor([0., 0., 0.])</span>
+
+<span class="sd">        One can also verify:</span>
+
+<span class="sd">            &gt;&gt;&gt; x = pp.randn_SO3()</span>
+<span class="sd">            &gt;&gt;&gt; x.Log() + x.Inv().Log()</span>
+<span class="sd">            so3Type LieTensor:</span>
+<span class="sd">            LieTensor([0., 0., 0.])</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+    <span class="k">return</span> <span class="n">x</span><span class="o">.</span><span class="n">Inv</span><span class="p">()</span></div>
+
+
+<span class="nd">@assert_ltype</span>
+<span class="k">def</span> <span class="nf">Mul</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">):</span>
+    <span class="k">return</span> <span class="n">x</span> <span class="o">@</span> <span class="n">y</span>
+
+
+<div class="viewcode-block" id="Retr"><a class="viewcode-back" href="../../../../generated/pypose.Retr/#pypose.Retr">[docs]</a><span class="nd">@assert_ltype</span>
+<span class="k">def</span> <span class="nf">Retr</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">a</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&quot;&quot;&quot;Perform batched retraction with a given direction.</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        Y_i = \mathrm{Exp}(a_i) * X_i,</span>
+
+<span class="sd">    where :math:`\mathrm{Exp}` means the exponetial map. See :obj:`pypose.Exp` for more details.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        X (LieTensor): the input LieTensor to retract (Lie Group)</span>
+
+<span class="sd">        a (LieTensor): the direction of the retraction (Lie Algebra)</span>
+
+<span class="sd">    Return:</span>
+<span class="sd">        LieTensor: The retraction of the inputs (Lie Group)</span>
+
+<span class="sd">    Examples:</span>
+
+<span class="sd">        * :math:`\mathrm{Retr}`: (:obj:`SO3`, :obj:`so3`) :math:`\mapsto` :obj:`SO3`</span>
+
+<span class="sd">            &gt;&gt;&gt; a = pp.randn_so3()</span>
+<span class="sd">            &gt;&gt;&gt; X = pp.randn_SO3()</span>
+<span class="sd">            &gt;&gt;&gt; X.Retr(a) # equivalent to: pp.Retr(X, a)</span>
+<span class="sd">            SO3Type LieTensor:</span>
+<span class="sd">            tensor([0.6399, 0.0898, 0.1656, 0.7451])</span>
+
+<span class="sd">        * :math:`\mathrm{Retr}`: (:obj:`SE3`, :obj:`se3`) :math:`\mapsto` :obj:`SE3`</span>
+
+<span class="sd">            &gt;&gt;&gt; a = pp.randn_se3()</span>
+<span class="sd">            &gt;&gt;&gt; X = pp.randn_SE3()</span>
+<span class="sd">            &gt;&gt;&gt; X.Retr(a)  # equivalent to: pp.Retr(X, a)</span>
+<span class="sd">            SE3Type LieTensor:</span>
+<span class="sd">            tensor([-0.6754,  1.8240,  0.2109, -0.4649, -0.7254, -0.0943,  0.4987])</span>
+
+<span class="sd">        * :math:`\mathrm{Retr}`: (:obj:`Sim3`, :obj:`sim3`) :math:`\mapsto` :obj:`Sim3`</span>
+
+<span class="sd">            &gt;&gt;&gt; a = pp.randn_sim3()</span>
+<span class="sd">            &gt;&gt;&gt; X = pp.randn_Sim3()</span>
+<span class="sd">            &gt;&gt;&gt; X.Retr(a)  # equivalent to: pp.Retr(X, a)</span>
+<span class="sd">            Sim3Type LieTensor:</span>
+<span class="sd">            tensor([-0.6057, -1.6370,  1.1379,  0.7037,  0.6164,  0.3525, -0.0262,  0.3141])</span>
+
+<span class="sd">        * :math:`\mathrm{Retr}`: (:obj:`RxSO3`, :obj:`rxsso3`) :math:`\mapsto` :obj:`RxSO3`</span>
+
+<span class="sd">            &gt;&gt;&gt; a = pp.randn_rxso3()</span>
+<span class="sd">            &gt;&gt;&gt; X = pp.randn_RxSO3()</span>
+<span class="sd">            &gt;&gt;&gt; X.Retr(a)  # equivalent to: pp.Retr(X, a)</span>
+<span class="sd">            RxSO3Type LieTensor:</span>
+<span class="sd">            tensor([-0.0787,  0.4052, -0.7509,  0.5155,  0.1217])</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+    <span class="k">return</span> <span class="n">X</span><span class="o">.</span><span class="n">Retr</span><span class="p">(</span><span class="n">a</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="Act"><a class="viewcode-back" href="../../../../generated/pypose.Act/#pypose.Act">[docs]</a><span class="nd">@assert_ltype</span>
+<span class="k">def</span> <span class="nf">Act</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&quot;&quot;&quot;Apply the batched transform to points in Euclidean or homogeneous coordinates.</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        y_i = X_i * p_i,</span>
+
+<span class="sd">    where :math:`X` is the batched transform and :math:`p_i \in \mathbb{R^{*\times3}}` or</span>
+<span class="sd">    :math:`p_i \in \mathbb{R^{*\times4}}` denotes the points to be transformed.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        X (LieTensor): the input LieTensor (Lie Group).</span>
+
+<span class="sd">        p (Tensor): the points to be transformed.</span>
+
+<span class="sd">    Return:</span>
+<span class="sd">        Tensor: the transformed points in Euclidean or homogeneous coordinates.</span>
+
+<span class="sd">    Examples:</span>
+
+<span class="sd">        * :math:`\mathrm{Act}`: (:obj:`SO3`, :obj:`Tensor`) :math:`\mapsto` :obj:`Tensor`</span>
+
+<span class="sd">            &gt;&gt;&gt; p = torch.randn(3)     # batch size 1, Euclidean coordinates</span>
+<span class="sd">            &gt;&gt;&gt; X = pp.identity_SO3(2) # batch size 2</span>
+<span class="sd">            &gt;&gt;&gt; X.Act(p)               # equivalent to: pp.Act(X, p)</span>
+<span class="sd">            tensor([[ 1.7576,  1.1503, -0.9920],</span>
+<span class="sd">                    [ 1.7576,  1.1503, -0.9920]])</span>
+
+<span class="sd">        * :math:`\mathrm{Act}`: (:obj:`SE3`, :obj:`Tensor`) :math:`\mapsto` :obj:`Tensor`</span>
+
+<span class="sd">            &gt;&gt;&gt; p = torch.tensor([[0, 0, 0, 1.], [0, 0, 0, 1.]]) # batch size 2, homogeneous coordinates</span>
+<span class="sd">            &gt;&gt;&gt; X = pp.randn_SE3()                               # batch size 1</span>
+<span class="sd">            &gt;&gt;&gt; X.Act(p)                                         # apply same transform</span>
+<span class="sd">            tensor([[-0.5676, -0.0452, -0.0929,  1.0000],</span>
+<span class="sd">                    [-0.5676, -0.0452, -0.0929,  1.0000]])</span>
+
+<span class="sd">        * :math:`\mathrm{Act}`: (:obj:`Sim3`, :obj:`Tensor`) :math:`\mapsto` :obj:`Tensor`</span>
+
+<span class="sd">            &gt;&gt;&gt; p = torch.tensor([[0, 0, 0, 1.], [0, 0, 0, 1.]])  # batch size 2</span>
+<span class="sd">            &gt;&gt;&gt; X = pp.randn_Sim3(2)                              # batch size 2</span>
+<span class="sd">            &gt;&gt;&gt; X.Act(p)                                          # apply transform respectively.</span>
+<span class="sd">            tensor([[ 0.1551,  2.2930,  0.4531,  1.0000],</span>
+<span class="sd">                    [-0.6140, -1.1263,  2.7607,  1.0000]])</span>
+
+<span class="sd">        * :math:`\mathrm{Act}`: (:obj:`RxSO3`, :obj:`Tensor`) :math:`\mapsto` :obj:`Tensor`</span>
+
+<span class="sd">            &gt;&gt;&gt; p = torch.tensor([[0, 0, 0.], [0, 0, 0.]])  # batch size 2, Euclidean coordinates</span>
+<span class="sd">            &gt;&gt;&gt; X = pp.randn_RxSO3(2)                       # batch size 2</span>
+<span class="sd">            &gt;&gt;&gt; X.Act(p)                                    # apply transform respectively.</span>
+<span class="sd">            tensor([[0., 0., 0.],</span>
+<span class="sd">                    [0., 0., 0.]])</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+    <span class="k">return</span> <span class="n">X</span><span class="o">.</span><span class="n">Act</span><span class="p">(</span><span class="n">p</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="Adj"><a class="viewcode-back" href="../../../../generated/pypose.Adj/#pypose.Adj">[docs]</a><span class="nd">@assert_ltype</span>
+<span class="k">def</span> <span class="nf">Adj</span><span class="p">(</span><span class="nb">input</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">    The dot product between the Adjoint matrix at the point given by an input (Lie Group) and</span>
+<span class="sd">    the second point (Lie Algebra).</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \mathrm{Adj}: (\mathcal{G}, \mathcal{g}) \mapsto \mathcal{g}</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        input (LieTensor): the input LieTensor (Lie Group)</span>
+<span class="sd">        p (LieTensor): the second LieTensor (Lie Algebra)</span>
+
+<span class="sd">    Return:</span>
+<span class="sd">        LieTensor: the output LieTensor (Lie Algebra)</span>
+
+<span class="sd">    .. list-table:: List of supported :math:`\mathrm{Adj}` map</span>
+<span class="sd">        :widths: 20 20 8 20 20</span>
+<span class="sd">        :header-rows: 1</span>
+
+<span class="sd">        * - input :obj:`ltype`</span>
+<span class="sd">          - :math:`(\mathcal{G}, \mathcal{g})` (Lie Group, Lie Algebra)</span>
+<span class="sd">          - :math:`\mapsto`</span>
+<span class="sd">          - :math:`\mathcal{g}` (Lie Algebra)</span>
+<span class="sd">          - output :obj:`ltype`</span>
+<span class="sd">        * - (:obj:`SO3_type`, :obj:`so3_type`)</span>
+<span class="sd">          - :math:`(\mathcal{G}\in\mathbb{R}^{*\times4}, \mathcal{g}\in\mathbb{R}^{*\times3})`</span>
+<span class="sd">          - :math:`\mapsto`</span>
+<span class="sd">          - :math:`\mathcal{g}\in\mathbb{R}^{*\times3}`</span>
+<span class="sd">          - :obj:`so3_type`</span>
+<span class="sd">        * - (:obj:`SE3_type`, :obj:`se3_type`)</span>
+<span class="sd">          - :math:`(\mathcal{G}\in\mathbb{R}^{*\times7}, \mathcal{g}\in\mathbb{R}^{*\times6})`</span>
+<span class="sd">          - :math:`\mapsto`</span>
+<span class="sd">          - :math:`\mathcal{g}\in\mathbb{R}^{*\times6}`</span>
+<span class="sd">          - :obj:`se3_type`</span>
+<span class="sd">        * - (:obj:`Sim3_type`, :obj:`sim3_type`)</span>
+<span class="sd">          - :math:`(\mathcal{G}\in\mathbb{R}^{*\times8}, \mathcal{g}\in\mathbb{R}^{*\times7})`</span>
+<span class="sd">          - :math:`\mapsto`</span>
+<span class="sd">          - :math:`\mathcal{g}\in\mathbb{R}^{*\times7}`</span>
+<span class="sd">          - :obj:`sim3_type`</span>
+<span class="sd">        * - (:obj:`RxSO3_type`, :obj:`rxso3_type`)</span>
+<span class="sd">          - :math:`(\mathcal{G}\in\mathbb{R}^{*\times5}, \mathcal{g}\in\mathbb{R}^{*\times4})`</span>
+<span class="sd">          - :math:`\mapsto`</span>
+<span class="sd">          - :math:`\mathcal{g}\in\mathbb{R}^{*\times4}`</span>
+<span class="sd">          - :obj:`rxso3_type`</span>
+
+<span class="sd">    Let the input be (:math:`\mathbf{x}`, :math:`\mathbf{p}`), :math:`\mathbf{y}` be the output.</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathbf{y}_i = \mathrm{Adj}(\mathbf{x}_i)\mathbf{p}_i,</span>
+
+<span class="sd">        where, :math:`\mathrm{Adj}(\mathbf{x}_i)` is the adjoint matrix of the Lie group of</span>
+<span class="sd">        :math:`\mathbf{x}_i`.</span>
+
+<span class="sd">    * If input (:math:`\mathbf{x}`, :math:`\mathbf{p}`)&#39;s :obj:`ltype` are :obj:`SO3_type` and</span>
+<span class="sd">      :obj:`so3_type` (input :math:`\mathbf{x}` is an instance of :meth:`SO3`, :math:`\mathbf{p}`</span>
+<span class="sd">      is an instance of :meth:`so3`). Given :math:`\mathbf{x}_i \in` :math:`\textrm{SO(3)}`.</span>
+<span class="sd">      The adjoint transformation is given by:</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathrm{Adj}(\mathbf{x}_i) = \mathbf{x}_i</span>
+
+<span class="sd">        In the case of :math:`\textrm{SO3}`, the adjoint transformation for an element is the same</span>
+<span class="sd">        rotation matrix used to represent the element. Rotating a tangent vector by an element</span>
+<span class="sd">        &quot;moves&quot; it from the tangent space on the right side of the element to the tangent space on</span>
+<span class="sd">        the left.</span>
+
+<span class="sd">    * If input (:math:`\mathbf{x}`, :math:`\mathbf{p}`)&#39;s :obj:`ltype` are :obj:`SE3_type`</span>
+<span class="sd">      and :obj:`se3_type` (input :math:`\mathbf{x}` is an instance of :meth:`SE3`,</span>
+<span class="sd">      :math:`\mathbf{p}` is an instance of :meth:`se3`). Let :math:`\mathbf{R}_i \in`</span>
+<span class="sd">      :math:`\textrm{SO(3)}` and :math:`\mathbf{t}_i \in \mathbb{R}^{3\times3}` represent the</span>
+<span class="sd">      rotation and translation part of the group. Let :math:`\mathbf{t}_{i\times}` be the skew</span>
+<span class="sd">      matrix (:meth:`pypose.vec2skew`) of :math:`\mathbf{t}_i`.</span>
+<span class="sd">      The adjoint transformation is given by:</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathrm{Adj}(\mathbf{x}_i) = \left[</span>
+<span class="sd">                                \begin{array}{cc}</span>
+<span class="sd">                                    \mathbf{R}_i &amp; \mathbf{t}_{i\times}\mathbf{R}_i \\</span>
+<span class="sd">                                    0 &amp; \mathbf{R}_i</span>
+<span class="sd">                                \end{array}</span>
+<span class="sd">                             \right] \in \mathbb{R}^{6\times6}</span>
+
+<span class="sd">        where,</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathbf{x}_i = \left[</span>
+<span class="sd">                                \begin{array}{cc}</span>
+<span class="sd">                                    \mathbf{R}_i&amp; \mathbf{t}_i \\</span>
+<span class="sd">                                    0 &amp; 1</span>
+<span class="sd">                                \end{array}</span>
+<span class="sd">                             \right] \in \mathrm{SE(3)}</span>
+
+<span class="sd">    * If input (:math:`\mathbf{x}`, :math:`\mathbf{p}`)&#39;s :obj:`ltype` are :obj:`Sim3_type` and</span>
+<span class="sd">      :obj:`sim3_type` (input :math:`\mathbf{x}` is an instance of :meth:`Sim3`, :math:`\mathbf{p}`</span>
+<span class="sd">      is an instance of :meth:`sim3`). Let :math:`\mathbf{R}_i\in` :math:`\textrm{SO(3)}`,</span>
+<span class="sd">      :math:`\mathbf{t}_i \in \mathbb{R}^{3\times3}`, and :math:`s_i \in \mathbb{R}^+` represent</span>
+<span class="sd">      the rotation, translation, and scale parts of the group. Let :math:`\mathbf{t}_{i\times}`</span>
+<span class="sd">      be the skew matrix (:meth:`pypose.vec2skew`) of :math:`\mathbf{t}_i`. The adjoint</span>
+<span class="sd">      transformation is given by:</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathrm{Adj}(\mathbf{x}_i) = \left[</span>
+<span class="sd">                        \begin{array}{cc}</span>
+<span class="sd">                            s_i\mathbf{R}_i&amp; \mathbf{t}_{i\times}\mathbf{R}_i&amp; -\mathbf{t}_i \\</span>
+<span class="sd">                            0 &amp; \mathbf{R}_i&amp; 0 \\</span>
+<span class="sd">                            0 &amp; 0 &amp; 1</span>
+<span class="sd">                        \end{array}</span>
+<span class="sd">                        \right] \in \mathbb{R}^{7\times7}</span>
+
+<span class="sd">        where,</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathbf{x}_i = \left[</span>
+<span class="sd">                                \begin{array}{cc}</span>
+<span class="sd">                                    s_i\mathbf{R}_i &amp; \mathbf{t}_i \\</span>
+<span class="sd">                                    0 &amp; 1</span>
+<span class="sd">                                \end{array}</span>
+<span class="sd">                             \right] \in \textrm{Sim(3)}</span>
+
+<span class="sd">    * If input (:math:`\mathbf{x}`, :math:`\mathbf{p}`)&#39;s :obj:`ltype` are :obj:`RxSO3_type`</span>
+<span class="sd">      and :obj:`rxso3_type` (input :math:`\mathbf{x}` is an instance of :meth:`RxSO3`,</span>
+<span class="sd">      :math:`\mathbf{p}` is an instance of :meth:`rxso3`). Let :math:`\mathbf{R}_i \in`</span>
+<span class="sd">      :math:`\textrm{SO(3)}`, and :math:`s_i \in \mathbb{R}^+` represent the rotation and</span>
+<span class="sd">      scale parts of the group. The adjoint transformation is given by:</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathrm{Adj}(\mathbf{x}_i) = \left[</span>
+<span class="sd">                                \begin{array}{cc}</span>
+<span class="sd">                                    \mathbf{R}_i &amp; 0 \\</span>
+<span class="sd">                                    0 &amp; 1</span>
+<span class="sd">                                \end{array}</span>
+<span class="sd">                             \right] \in \mathbb{R}^{4\times4}</span>
+
+<span class="sd">        where,</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathbf{x}_i = \left[</span>
+<span class="sd">                                \begin{array}{cc}</span>
+<span class="sd">                                    s_i\mathbf{R}_i &amp; 0 \\</span>
+<span class="sd">                                    0 &amp; 1</span>
+<span class="sd">                                \end{array}</span>
+<span class="sd">                             \right] \in \mathrm{RxSO(3)}</span>
+
+<span class="sd">        In the case of :math:`\textrm{RxSO3}` group, the adjoint transformation is the same as</span>
+<span class="sd">        the rotation matrix of the group i.e. the :math:`\textrm{SO3}` part of the group.</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        The adjoint operator is a linear map which moves an element</span>
+<span class="sd">        :math:`\mathbf{p} \in \mathcal{g}` in the right tangent space of</span>
+<span class="sd">        :math:`\mathbf{x} \in \mathcal{G}` to the left tangent space.</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathrm{Exp}(\mathrm{Adj}(\mathbf{x}) \mathbf{p}) * \mathbf{x}</span>
+<span class="sd">            = \mathbf{x} * \mathrm{Exp}(\mathbf{p})</span>
+
+<span class="sd">        It can be easily verified:</span>
+
+<span class="sd">            &gt;&gt;&gt; x, p = pp.randn_SO3(), pp.randn_so3()</span>
+<span class="sd">            &gt;&gt;&gt; torch.allclose(x*p.Exp(), x.Adj(p).Exp()*x)</span>
+<span class="sd">            True</span>
+
+<span class="sd">        One can refer to Eq. (8) of the following paper:</span>
+
+<span class="sd">        * Zachary Teed et al., `Tangent Space Backpropagation for 3D Transformation Groups</span>
+<span class="sd">          &lt;https://arxiv.org/pdf/2103.12032.pdf&gt;`_, IEEE/CVF Conference on Computer Vision</span>
+<span class="sd">          and Pattern Recognition (CVPR), 2021.</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        :math:`\mathrm{Adj}` is generally used to transform a tangent vector from the tangent</span>
+<span class="sd">        space around one element to the tangent space of another.</span>
+<span class="sd">        One can refer to this paper for more details:</span>
+
+<span class="sd">        * J. Sola et al., `A micro Lie theory for state estimation in</span>
+<span class="sd">          robotics &lt;https://arxiv.org/abs/1812.01537&gt;`_, arXiv preprint arXiv:1812.01537 (2018).</span>
+
+<span class="sd">        The following thesis and the tutorial serve as a good reading material to learn more</span>
+<span class="sd">        about deriving the adjoint matrices for different transformation groups.</span>
+
+<span class="sd">        * Strasdat, H., 2012. `Local accuracy and global consistency for efficient visual SLAM</span>
+<span class="sd">          &lt;https://www.doc.ic.ac.uk/~ajd/Publications/Strasdat-H-2012-PhD-Thesis.pdf&gt;`_,</span>
+<span class="sd">          (Doctoral dissertation, Department of Computing, Imperial College London).</span>
+
+<span class="sd">        * `Lie Groups for 2D and 3D Transformations.</span>
+<span class="sd">          &lt;https://www.ethaneade.org/lie.pdf&gt;`_, by Ethan Eade.</span>
+
+<span class="sd">    Example:</span>
+
+<span class="sd">        * :math:`\mathrm{Adj}`: (:obj:`SO3`, :obj:`so3`) :math:`\mapsto` :obj:`so3`</span>
+
+<span class="sd">            &gt;&gt;&gt; x = pp.randn_SO3(2)</span>
+<span class="sd">            &gt;&gt;&gt; p = pp.randn_so3(2)</span>
+<span class="sd">            &gt;&gt;&gt; x.Adj(p) # equivalent to: pp.Adj(x, p)</span>
+<span class="sd">            so3Type LieTensor:</span>
+<span class="sd">            tensor([[-0.4171,  2.1218,  0.9951],</span>
+<span class="sd">                    [ 1.8415, -1.2185, -0.4082]])</span>
+
+<span class="sd">        * :math:`\mathrm{Adj}`: (:obj:`SE3`, :obj:`se3`) :math:`\mapsto` :obj:`se3`</span>
+
+<span class="sd">            &gt;&gt;&gt; x = pp.randn_SE3(2)</span>
+<span class="sd">            &gt;&gt;&gt; p = pp.randn_se3(2)</span>
+<span class="sd">            &gt;&gt;&gt; x.Adj(p) # equivalent to: pp.Adj(x, p)</span>
+<span class="sd">            se3Type LieTensor:</span>
+<span class="sd">            tensor([[-0.8536, -0.1984, -0.4554, -0.4868,  0.3231,  0.8535],</span>
+<span class="sd">                    [ 0.1577, -1.7625,  1.7997, -1.5085, -0.2098,  0.3538]])</span>
+
+<span class="sd">        * :math:`\mathrm{Adj}`: (:obj:`Sim3`, :obj:`sim3`) :math:`\mapsto` :obj:`sim3`</span>
+
+<span class="sd">            &gt;&gt;&gt; x = pp.randn_Sim3(2)</span>
+<span class="sd">            &gt;&gt;&gt; p = pp.randn_sim3(2)</span>
+<span class="sd">            &gt;&gt;&gt; x.Adj(p) # equivalent to: pp.Adj(x, p)</span>
+<span class="sd">            sim3Type LieTensor:</span>
+<span class="sd">            tensor([[ 0.1455, -0.5653, -0.1845,  0.0502,  1.3125,  1.5217, -0.8964],</span>
+<span class="sd">                    [-4.8724, -0.5254,  3.9559,  1.5170,  1.7610,  0.4375,  0.4248]])</span>
+
+<span class="sd">        * :math:`\mathrm{Adj}`: (:obj:`RxSO3`, :obj:`rxso3`) :math:`\mapsto` :obj:`rxso3`</span>
+
+<span class="sd">            &gt;&gt;&gt; x = pp.randn_RxSO3(2)</span>
+<span class="sd">            &gt;&gt;&gt; p = pp.randn_rxso3(2)</span>
+<span class="sd">            &gt;&gt;&gt; x.Adj(p) # equivalent to: pp.Adj(x, p)</span>
+<span class="sd">            rxso3Type LieTensor:</span>
+<span class="sd">            tensor([[-1.3590, -0.4314, -0.0297,  1.0166],</span>
+<span class="sd">                    [-0.3378, -0.4942, -2.0083, -0.4321]])</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+    <span class="k">return</span> <span class="nb">input</span><span class="o">.</span><span class="n">Adj</span><span class="p">(</span><span class="n">p</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="AdjT"><a class="viewcode-back" href="../../../../generated/pypose.AdjT/#pypose.AdjT">[docs]</a><span class="nd">@assert_ltype</span>
+<span class="k">def</span> <span class="nf">AdjT</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
+    <span class="k">return</span> <span class="n">X</span><span class="o">.</span><span class="n">AdjT</span><span class="p">(</span><span class="n">p</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="Jinvp"><a class="viewcode-back" href="../../../../generated/pypose.Jinvp/#pypose.Jinvp">[docs]</a><span class="nd">@assert_ltype</span>
+<span class="k">def</span> <span class="nf">Jinvp</span><span class="p">(</span><span class="nb">input</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">    The dot product between left Jacobian inverse at the point given</span>
+<span class="sd">    by input (Lie Group) and second point (Lie Algebra).</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \mathrm{Jinvp}: (\mathcal{G}, \mathcal{g}) \mapsto \mathcal{g}</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        input (LieTensor): the input LieTensor (Lie Group)</span>
+<span class="sd">        p (LieTensor): the second LieTensor (Lie Algebra)</span>
+
+<span class="sd">    Return:</span>
+<span class="sd">        LieTensor: the output LieTensor (Lie Algebra)</span>
+
+<span class="sd">    .. list-table:: List of supported :math:`\mathrm{Jinvp}` map</span>
+<span class="sd">        :widths: 20 20 8 20 20</span>
+<span class="sd">        :header-rows: 1</span>
+
+<span class="sd">        * - input :obj:`ltype`</span>
+<span class="sd">          - :math:`(\mathcal{G}, \mathcal{g})` (Lie Group, Lie Algebra)</span>
+<span class="sd">          - :math:`\mapsto`</span>
+<span class="sd">          - :math:`\mathcal{g}` (Lie Algebra)</span>
+<span class="sd">          - output :obj:`ltype`</span>
+<span class="sd">        * - (:obj:`SO3_type`, :obj:`so3_type`)</span>
+<span class="sd">          - :math:`(\mathcal{G}\in\mathbb{R}^{*\times4}, \mathcal{g}\in\mathbb{R}^{*\times3})`</span>
+<span class="sd">          - :math:`\mapsto`</span>
+<span class="sd">          - :math:`\mathcal{g}\in\mathbb{R}^{*\times3}`</span>
+<span class="sd">          - :obj:`so3_type`</span>
+<span class="sd">        * - (:obj:`SE3_type`, :obj:`se3_type`)</span>
+<span class="sd">          - :math:`(\mathcal{G}\in\mathbb{R}^{*\times7}, \mathcal{g}\in\mathbb{R}^{*\times6})`</span>
+<span class="sd">          - :math:`\mapsto`</span>
+<span class="sd">          - :math:`\mathcal{g}\in\mathbb{R}^{*\times6}`</span>
+<span class="sd">          - :obj:`se3_type`</span>
+<span class="sd">        * - (:obj:`Sim3_type`, :obj:`sim3_type`)</span>
+<span class="sd">          - :math:`(\mathcal{G}\in\mathbb{R}^{*\times8}, \mathcal{g}\in\mathbb{R}^{*\times7})`</span>
+<span class="sd">          - :math:`\mapsto`</span>
+<span class="sd">          - :math:`\mathcal{g}\in\mathbb{R}^{*\times7}`</span>
+<span class="sd">          - :obj:`sim3_type`</span>
+<span class="sd">        * - (:obj:`RxSO3_type`, :obj:`rxso3_type`)</span>
+<span class="sd">          - :math:`(\mathcal{G}\in\mathbb{R}^{*\times5}, \mathcal{g}\in\mathbb{R}^{*\times4})`</span>
+<span class="sd">          - :math:`\mapsto`</span>
+<span class="sd">          - :math:`\mathcal{g}\in\mathbb{R}^{*\times4}`</span>
+<span class="sd">          - :obj:`rxso3_type`</span>
+
+<span class="sd">    Let the input be (:math:`\mathbf{x}`, :math:`\mathbf{p}`), :math:`\mathbf{y}` be the output.</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathbf{y}_i = \mathbf{J}^{-1}_i(\mathbf{x}_i)\mathbf{p}_i,</span>
+
+<span class="sd">        where :math:`\mathbf{J}^{-1}_i(\mathbf{x}_i)` is the inverse of left Jacobian of</span>
+<span class="sd">        :math:`\mathbf{x}_i`.</span>
+
+<span class="sd">    * If input (:math:`\mathbf{x}`, :math:`\mathbf{p}`)&#39;s :obj:`ltype` are :obj:`SO3_type`</span>
+<span class="sd">      and :obj:`so3_type` (input :math:`\mathbf{x}` is an instance of :meth:`SO3`,</span>
+<span class="sd">      :math:`\mathbf{p}` is an instance of :meth:`so3`).</span>
+<span class="sd">      Let :math:`\boldsymbol{\phi}_i = \theta_i\mathbf{n}_i` be the corresponding Lie Algebra</span>
+<span class="sd">      of :math:`\mathbf{x}_i`, :math:`\boldsymbol{\Phi}_i` be the skew matrix</span>
+<span class="sd">      (:meth:`pypose.vec2skew`) of :math:`\boldsymbol{\phi}_i`:</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathbf{J}^{-1}_i(\mathbf{x}_i) = \mathbf{I} - \frac{1}{2}\boldsymbol{\Phi}_i +</span>
+<span class="sd">            \mathrm{coef}\boldsymbol{\Phi}_i^2</span>
+
+<span class="sd">      where :math:`\mathbf{I}` is the identity matrix with the same dimension as</span>
+<span class="sd">      :math:`\boldsymbol{\Phi}_i`, and</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathrm{coef} =</span>
+<span class="sd">                \left\{</span>
+<span class="sd">                \begin{array}{ll}</span>
+<span class="sd">                    \frac{1}{\theta_i^2} -</span>
+<span class="sd">                    \frac{\cos{\frac{\theta_i}{2}}}{2\theta\sin{\frac{\theta_i}{2}}},</span>
+<span class="sd">                    \quad \|\theta_i\| &gt; \text{eps}, \\</span>
+<span class="sd">                    \frac{1}{12},</span>
+<span class="sd">                    \quad \|\theta_i\| \leq \text{eps}</span>
+<span class="sd">                \end{array}</span>
+<span class="sd">                \right.</span>
+
+<span class="sd">    * If input (:math:`\mathbf{x}`, :math:`\mathbf{p}`)&#39;s :obj:`ltype` are :obj:`SE3_type`</span>
+<span class="sd">      and :obj:`se3_type` (input :math:`\mathbf{x}` is an instance of :meth:`SE3`,</span>
+<span class="sd">      :math:`\mathbf{p}` is an instance of :meth:`se3`).</span>
+<span class="sd">      Let :math:`\boldsymbol{\phi}_i = \theta_i\mathbf{n}_i` be the corresponding Lie Algebra</span>
+<span class="sd">      of the SO3 part of :math:`\mathbf{x}_i`, :math:`\boldsymbol{\tau}_i` be the Lie Algebra</span>
+<span class="sd">      of the translation part of :math:`\mathbf{x}_i`; :math:`\boldsymbol{\Phi}_i` and</span>
+<span class="sd">      :math:`\boldsymbol{\Tau}_i` be the skew matrices, respectively:</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathbf{J}^{-1}_i(\mathbf{x}_i) =</span>
+<span class="sd">                \left[</span>
+<span class="sd">                \begin{array}{cc}</span>
+<span class="sd">                    \mathbf{J}_i^{-1}(\boldsymbol{\Phi}_i) &amp;</span>
+<span class="sd">                            - \mathbf{J}_i^{-1}(\boldsymbol{\Phi}_i)</span>
+<span class="sd">                    \mathbf{Q}_i(\boldsymbol{\tau}_i, \boldsymbol{\phi}_i)</span>
+<span class="sd">                        \mathbf{J}_i^{-1}(\boldsymbol{\Phi}_i) \\</span>
+<span class="sd">                    \mathbf{0} &amp; \mathbf{J}_i^{-1}(\boldsymbol{\Phi}_i)</span>
+<span class="sd">                \end{array}</span>
+<span class="sd">                \right]</span>
+
+<span class="sd">        where :math:`\mathbf{J}_i^{-1}(\boldsymbol{\Phi}_i)` is the inverse of left Jacobian</span>
+<span class="sd">        of the SO3 part of :math:`\mathbf{x}_i`.</span>
+<span class="sd">        :math:`\mathbf{Q}_i(\boldsymbol{\tau}_i, \boldsymbol{\phi}_i)` is</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \begin{align*}</span>
+<span class="sd">                \mathbf{Q}_i(\boldsymbol{\tau}_i, \boldsymbol{\phi}_i) =</span>
+<span class="sd">                    \frac{1}{2}\boldsymbol{\Tau}_i &amp;+ c_1</span>
+<span class="sd">                (\boldsymbol{\Phi_i\Tau_i} + \boldsymbol{\Tau_i\Phi_i}</span>
+<span class="sd">                    + \boldsymbol{\Phi_i\Tau_i\Phi_i}) \\</span>
+<span class="sd">                 &amp;+ c_2 (\boldsymbol{\Phi_i^2\Tau_i} + \boldsymbol{\Tau_i\Phi_i^2}</span>
+<span class="sd">                    - 3\boldsymbol{\Phi_i\Tau_i\Phi_i})\\</span>
+<span class="sd">                 &amp;+ c_3 (\boldsymbol{\Phi_i\Tau_i\Phi_i^2} + \boldsymbol{\Phi_i^2\Tau_i\Phi_i})</span>
+<span class="sd">            \end{align*}</span>
+
+<span class="sd">        where,</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            c_1 = \left\{</span>
+<span class="sd">                    \begin{array}{ll}</span>
+<span class="sd">                        \frac{\theta_i - \sin\theta_i}{\theta_i^3},</span>
+<span class="sd">                            \quad \|\theta_i\| &gt; \text{eps}, \\</span>
+<span class="sd">                        \frac{1}{6}-\frac{1}{120}\theta_i^2,</span>
+<span class="sd">                        \quad \|\theta_i\| \leq \text{eps}</span>
+<span class="sd">                    \end{array}</span>
+<span class="sd">                    \right.</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            c_2 = \left\{</span>
+<span class="sd">                    \begin{array}{ll}</span>
+<span class="sd">                        \frac{\theta_i^2 +2\cos\theta_i - 2}{2\theta_i^4},</span>
+<span class="sd">                            \quad \|\theta_i\| &gt; \text{eps}, \\</span>
+<span class="sd">                        \frac{1}{24}-\frac{1}{720}\theta_i^2,</span>
+<span class="sd">                        \quad \|\theta_i\| \leq \text{eps}</span>
+<span class="sd">                    \end{array}</span>
+<span class="sd">                    \right.</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            c_3 = \left\{</span>
+<span class="sd">                    \begin{array}{ll}</span>
+<span class="sd">                        \frac{2\theta_i - 3\sin\theta_i + \theta_i\cos\theta_i}{2\theta_i^5},</span>
+<span class="sd">                        \quad \|\theta_i\| &gt; \text{eps}, \\</span>
+<span class="sd">                        \frac{1}{120}-\frac{1}{2520}\theta_i^2,</span>
+<span class="sd">                        \quad \|\theta_i\| \leq \text{eps}</span>
+<span class="sd">                    \end{array}</span>
+<span class="sd">                    \right.</span>
+
+<span class="sd">    * If input (:math:`\mathbf{x}`, :math:`\mathbf{p}`)&#39;s :obj:`ltype` are :obj:`Sim3_type`</span>
+<span class="sd">      and :obj:`sim3_type` (input :math:`\mathbf{x}` is an instance of :meth:`Sim3`,</span>
+<span class="sd">      :math:`\mathbf{p}` is an instance of :meth:`sim3`). The inverse of left Jacobian can</span>
+<span class="sd">      be approximated as:</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathbf{J}^{-1}_i(\mathbf{x}_i) =</span>
+<span class="sd">                \sum_{n=0}(-1)^n\frac{B_n}{n!}(\boldsymbol{\xi}_i^{\curlywedge})^n</span>
+
+<span class="sd">        where :math:`B_n` is the Bernoulli number: :math:`B_0 = 1`, :math:`B_1 = -\frac{1}{2}`,</span>
+<span class="sd">        :math:`B_2 = \frac{1}{6}`, :math:`B_3 = 0`, :math:`B_4 = -\frac{1}{30}`.</span>
+<span class="sd">        :math:`\boldsymbol{\xi}_i^{\curlywedge} = \mathrm{adj}(\boldsymbol{\xi}_i^{\wedge})`</span>
+<span class="sd">        and :math:`\mathrm{adj}` is the adjoint of the Lie algebra :math:`\mathfrak{sim}(3)`,</span>
+<span class="sd">        e.g., :math:`\boldsymbol{\xi}_i \in \mathfrak{sim}(3)`. Notice that if notate</span>
+<span class="sd">        :math:`\boldsymbol{X}_i = \mathrm{Adj}(\mathbf{x}_i)` and :math:`\mathrm{Adj}` is the</span>
+<span class="sd">        adjoint of the Lie group :math:`\mathrm{Sim}(3)`, there is a nice property:</span>
+<span class="sd">        :math:`\mathrm{Adj}(\mathrm{Exp}(\boldsymbol{\xi}_i^{\curlywedge})) =</span>
+<span class="sd">        \mathrm{Exp}(\mathrm{adj}(\boldsymbol{\xi}_i^{\wedge}))`,</span>
+<span class="sd">        or :math:`\boldsymbol{X}_i = \mathrm{Exp}(\boldsymbol{\xi}_i^{\curlywedge})`.</span>
+
+
+<span class="sd">    * If input (:math:`\mathbf{x}`, :math:`\mathbf{p}`)&#39;s :obj:`ltype` are :obj:`RxSO3_type`</span>
+<span class="sd">      and :obj:`rxso3_type` (input :math:`\mathbf{x}` is an instance of :meth:`RxSO3`,</span>
+<span class="sd">      :math:`\mathbf{p}` is an instance of :meth:`rxso3`). Let :math:`\boldsymbol{\phi}_i`</span>
+<span class="sd">      be the corresponding Lie Algebra of the SO3 part of :math:`\mathbf{x}_i`,</span>
+<span class="sd">      :math:`\boldsymbol{\Phi}_i` be the skew matrix (:meth:`pypose.vec2skew`), The inverse</span>
+<span class="sd">      of left Jacobian of :math:`\mathbf{x}_i` is the same as that for the SO3 part of</span>
+<span class="sd">      :math:`\mathbf{x}_i`.</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathbf{J}^{-1}_i(\mathbf{x}_i) = \left[</span>
+<span class="sd">                                \begin{array}{cc}</span>
+<span class="sd">                                    \mathbf{J}_i^{-1}(\boldsymbol{\Phi}_i) &amp; \mathbf{0} \\</span>
+<span class="sd">                                    \mathbf{0} &amp; 1</span>
+<span class="sd">                                \end{array}</span>
+<span class="sd">                             \right]</span>
+
+<span class="sd">        where :math:`\mathbf{J}_i^{-1}(\boldsymbol{\Phi}_i)` is the</span>
+<span class="sd">        inverse of left Jacobian of the SO3 part of :math:`\mathbf{x}_i`.</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        :math:`\mathrm{Jinvp}` is usually used in the Baker-Campbell-Hausdorff formula</span>
+<span class="sd">        (BCH formula) when performing LieTensor multiplication.</span>
+<span class="sd">        One can refer to this paper for more details:</span>
+
+<span class="sd">        * J. Sola et al., `A micro Lie theory for state estimation in</span>
+<span class="sd">          robotics &lt;https://arxiv.org/abs/1812.01537&gt;`_, arXiv preprint arXiv:1812.01537 (2018).</span>
+
+<span class="sd">        In particular, Eq. (146) is the math used in the :obj:`SO3_type`, :obj:`so3_type` scenario;</span>
+<span class="sd">        Eq. (179b) and Eq. (180) are the math used in the :obj:`SE3_type`, :obj:`se3_type` scenario.</span>
+
+<span class="sd">        For Lie groups such as :obj:`Sim3_type`, :obj:`sim3_type`,</span>
+<span class="sd">        there is no analytic expression for the left Jacobian and its inverse.</span>
+<span class="sd">        Numerical approximation is used based on series expansion.</span>
+<span class="sd">        One can refer to Eq. (26) of this paper for more details about the approximation:</span>
+
+<span class="sd">        * Z. Teed et al., `Tangent Space Backpropagation for 3D Transformation Groups.</span>
+<span class="sd">          &lt;https://arxiv.org/pdf/2103.12032.pdf&gt;`_, in IEEE/CVF Conference on Computer Vision and</span>
+<span class="sd">          Pattern Recognition (CVPR) (2021).</span>
+
+<span class="sd">        In particular, the Bernoulli numbers can be obtained from Eq. (7.72) of this famous book:</span>
+
+<span class="sd">        * T. Barfoot, `State Estimation for Robotics.</span>
+<span class="sd">          &lt;https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.708.1086&amp;rep=rep1&amp;type=pdf&gt;`_,</span>
+<span class="sd">          Cambridge University Press (2017).</span>
+
+<span class="sd">    Example:</span>
+
+<span class="sd">        * :math:`\mathrm{Jinvp}`: (:obj:`SO3`, :obj:`so3`) :math:`\mapsto` :obj:`so3`</span>
+
+<span class="sd">        &gt;&gt;&gt; x = pp.randn_SO3()</span>
+<span class="sd">        &gt;&gt;&gt; p = pp.randn_so3()</span>
+<span class="sd">        &gt;&gt;&gt; x.Jinvp(p) # equivalent to: pp.Jinvp(x, p)</span>
+<span class="sd">        so3Type LieTensor:</span>
+<span class="sd">        tensor([-2.0248,  1.1116, -0.0251])</span>
+
+<span class="sd">        * :math:`\mathrm{Jinvp}`: (:obj:`SE3`, :obj:`se3`) :math:`\mapsto` :obj:`se3`</span>
+
+<span class="sd">        &gt;&gt;&gt; x = pp.randn_SE3(2)</span>
+<span class="sd">        &gt;&gt;&gt; p = pp.randn_se3(2)</span>
+<span class="sd">        &gt;&gt;&gt; x.Jinvp(p) # equivalent to: pp.Jinvp(x, p)</span>
+<span class="sd">        se3Type LieTensor:</span>
+<span class="sd">        tensor([[ 0.4304,  2.0565,  1.0256,  0.0666, -0.2252, -0.7425],</span>
+<span class="sd">                [-0.9317, -1.7806,  0.8660, -2.0028,  0.6098, -0.6517]])</span>
+
+<span class="sd">        * :math:`\mathrm{Jinvp}`: (:obj:`Sim3`, :obj:`sim3`) :math:`\mapsto` :obj:`sim3`</span>
+
+<span class="sd">        &gt;&gt;&gt; x = pp.randn_Sim3(2)</span>
+<span class="sd">        &gt;&gt;&gt; p = pp.randn_sim3(2)</span>
+<span class="sd">        &gt;&gt;&gt; x.Jinvp(p) # equivalent to: pp.Jinvp(x, p)</span>
+<span class="sd">        sim3Type LieTensor:</span>
+<span class="sd">        tensor([[-1.7231, -1.6650, -0.0202, -0.3731,  0.8441, -0.5438,  0.2879],</span>
+<span class="sd">                [ 0.9965,  0.6337, -0.7320, -0.1874,  0.6312,  0.3919,  0.6938]])</span>
+
+<span class="sd">        * :math:`\mathrm{Jinvp}`: (:obj:`RxSO3`, :obj:`rxso3`) :math:`\mapsto` :obj:`rxso3`</span>
+
+<span class="sd">        &gt;&gt;&gt; x = pp.randn_RxSO3(2)</span>
+<span class="sd">        &gt;&gt;&gt; p = pp.randn_rxso3(2)</span>
+<span class="sd">        &gt;&gt;&gt; x.Jinvp(p) # equivalent to: pp.Jinvp(x, p)</span>
+<span class="sd">        rxso3Type LieTensor:</span>
+<span class="sd">        tensor([[ 0.9308, -1.4965, -0.1347,  0.4894],</span>
+<span class="sd">                [ 0.6558,  1.2221, -0.8190,  0.2108]])</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+    <span class="k">return</span> <span class="nb">input</span><span class="o">.</span><span class="n">Jinvp</span><span class="p">(</span><span class="n">p</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="Jr"><a class="viewcode-back" href="../../../../generated/pypose.Jr/#pypose.Jr">[docs]</a><span class="nd">@assert_ltype</span>
+<span class="k">def</span> <span class="nf">Jr</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">    The batched right Jacobian of a LieTensor.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        input (LieTensor): the input LieTensor (either Lie Group or Lie Algebra)</span>
+
+<span class="sd">    Return:</span>
+<span class="sd">        Tensor: the right Jocobian Matrices</span>
+
+<span class="sd">    Examples:</span>
+
+<span class="sd">        * :math:`\mathrm{Jr}`: :meth:`so3` :math:`\mapsto` :math:`\mathcal{R}^{*\times 3\times 3}`</span>
+
+<span class="sd">            &gt;&gt;&gt; x = pp.randn_so3(requires_grad=True)</span>
+<span class="sd">            &gt;&gt;&gt; x.Jr()</span>
+<span class="sd">            tensor([[ 0.9289, -0.3053, -0.0895],</span>
+<span class="sd">                    [ 0.3180,  0.9082,  0.1667],</span>
+<span class="sd">                    [ 0.0104, -0.1889,  0.9757]], grad_fn=&lt;SWhereBackward0&gt;)</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+    <span class="k">return</span> <span class="n">x</span><span class="o">.</span><span class="n">Jr</span><span class="p">()</span></div>
+</pre></div>
+
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+  <h1>Source code for pypose.module.dynamics</h1><div class="highlight"><pre>
+<span></span><span class="kn">import</span> <span class="nn">torch</span>
+<span class="kn">from</span> <span class="nn">..</span> <span class="kn">import</span> <span class="n">bmv</span>
+<span class="kn">from</span> <span class="nn">torch</span> <span class="kn">import</span> <span class="n">nn</span>
+<span class="kn">from</span> <span class="nn">torch.autograd.functional</span> <span class="kn">import</span> <span class="n">jacobian</span>
+
+
+<div class="viewcode-block" id="System"><a class="viewcode-back" href="../../../../generated/pypose.module.System/#pypose.module.System">[docs]</a><span class="k">class</span> <span class="nc">System</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    The base class for a system dynamics model.</span>
+
+<span class="sd">    In most of the cases, users only need to subclass a specific dynamic system,</span>
+<span class="sd">    such as linear time invariant system :meth:`LTI`, Linear Time-Variant :meth:`LTV`,</span>
+<span class="sd">    and a non-linear system :meth:`NLS`.</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">register_buffer</span><span class="p">(</span><span class="s1">&#39;_t&#39;</span><span class="p">,</span><span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">int64</span><span class="p">))</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">register_forward_hook</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">forward_hook</span><span class="p">)</span>
+
+<div class="viewcode-block" id="System.forward_hook"><a class="viewcode-back" href="../../../../generated/pypose.module.System/#pypose.module.System.forward_hook">[docs]</a>    <span class="k">def</span> <span class="nf">forward_hook</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">module</span><span class="p">,</span> <span class="n">inputs</span><span class="p">,</span> <span class="n">outputs</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Automatically advances the time step.</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">_t</span><span class="o">.</span><span class="n">add_</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span></div>
+
+    <span class="k">def</span> <span class="nf">reset</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">t</span><span class="o">=</span><span class="mi">0</span><span class="p">):</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">_t</span><span class="o">.</span><span class="n">fill_</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
+        <span class="k">return</span> <span class="bp">self</span>
+
+<div class="viewcode-block" id="System.forward"><a class="viewcode-back" href="../../../../generated/pypose.module.System/#pypose.module.System.forward">[docs]</a>    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">state</span><span class="p">,</span> <span class="nb">input</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Defines the computation performed at every call that advances the system by one time step.</span>
+
+<span class="sd">        Note:</span>
+<span class="sd">            The :obj:`forward` method implicitly increments the time step via :obj:`forward_hook`.</span>
+<span class="sd">            :obj:`state_transition` and :obj:`observation` still accept time for the flexiblity</span>
+<span class="sd">            such as time-varying system. One can directly access the current system time via the</span>
+<span class="sd">            property :obj:`systime` or :obj:`_t`.</span>
+
+<span class="sd">        Note:</span>
+<span class="sd">            To introduce noise in a model, redefine this method via</span>
+<span class="sd">            subclassing. See example in ``examples/module/ekf/tank_robot.py``.</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">state</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">input</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">atleast_1d</span><span class="p">(</span><span class="n">state</span><span class="p">),</span> <span class="n">torch</span><span class="o">.</span><span class="n">atleast_1d</span><span class="p">(</span><span class="nb">input</span><span class="p">)</span>
+        <span class="n">state</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">state_transition</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">state</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">input</span><span class="p">)</span>
+        <span class="n">obs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">observation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">state</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">input</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">state</span><span class="p">,</span> <span class="n">obs</span></div>
+
+<div class="viewcode-block" id="System.state_transition"><a class="viewcode-back" href="../../../../generated/pypose.module.System/#pypose.module.System.state_transition">[docs]</a>    <span class="k">def</span> <span class="nf">state_transition</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">state</span><span class="p">,</span> <span class="nb">input</span><span class="p">,</span> <span class="n">t</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Args:</span>
+<span class="sd">            state (:obj:`Tensor`): The state of the dynamical system</span>
+<span class="sd">            input (:obj:`Tensor`): The input to the dynamical system</span>
+<span class="sd">            t (:obj:`Tensor`): The time step of the dynamical system.  Default: ``None``.</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            Tensor: The state of the system at next time step</span>
+
+<span class="sd">        Note:</span>
+<span class="sd">            The users need to define this method and can access the current time via the property</span>
+<span class="sd">            :obj:`systime`. Don&#39;t introduce system transision noise in this function, as it will</span>
+<span class="sd">            be used for linearizing the system automaticalluy.</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s2">&quot;The users need to define their own state transition method&quot;</span><span class="p">)</span></div>
+
+<div class="viewcode-block" id="System.observation"><a class="viewcode-back" href="../../../../generated/pypose.module.System/#pypose.module.System.observation">[docs]</a>    <span class="k">def</span> <span class="nf">observation</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">state</span><span class="p">,</span> <span class="nb">input</span><span class="p">,</span> <span class="n">t</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Args:</span>
+<span class="sd">            state (:obj:`Tensor`): The state of the dynamical system</span>
+<span class="sd">            input (:obj:`Tensor`): The input to the dynamical system</span>
+<span class="sd">            t (:obj:`Tensor`): The time step of the dynamical system.  Default: ``None``.</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            Tensor: The observation of the system at the current step</span>
+
+<span class="sd">        Note:</span>
+<span class="sd">            The users need to define this method and can access the current system time via the</span>
+<span class="sd">            property :obj:`systime`. Don&#39;t introduce system transision noise in this function,</span>
+<span class="sd">            as it will be used for linearizing the system automaticalluy.</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s2">&quot;The users need to define their own observation method&quot;</span><span class="p">)</span></div>
+
+<div class="viewcode-block" id="System.set_refpoint"><a class="viewcode-back" href="../../../../generated/pypose.module.System/#pypose.module.System.set_refpoint">[docs]</a>    <span class="k">def</span> <span class="nf">set_refpoint</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">state</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="nb">input</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">t</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Function to set the reference point for linearization.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            state (:obj:`Tensor`): The reference state of the dynamical system. If ``None``,</span>
+<span class="sd">                the the most recent state is taken. Default: ``None``.</span>
+<span class="sd">            input (:obj:`Tensor`): The reference input to the dynamical system. If ``None``,</span>
+<span class="sd">                the the most recent input is taken. Default: ``None``.</span>
+<span class="sd">            t (:obj:`Tensor`): The reference time step of the dynamical system. If ``None``,</span>
+<span class="sd">                the the most recent timestamp is taken. Default: ``None``.</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            The ``self`` module.</span>
+
+<span class="sd">        Warning:</span>
+<span class="sd">            For nonlinear systems, the users have to call this function before getting the</span>
+<span class="sd">            linearized system.</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">return</span> <span class="bp">self</span></div>
+
+    <span class="nd">@property</span>
+    <span class="k">def</span> <span class="nf">systime</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">            System time, automatically advanced by :obj:`forward_hook`.</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_t</span>
+
+    <span class="nd">@systime</span><span class="o">.</span><span class="n">setter</span>
+    <span class="k">def</span> <span class="nf">systime</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">t</span><span class="p">):</span>
+        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">):</span>
+            <span class="n">t</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">_t</span><span class="o">.</span><span class="n">copy_</span><span class="p">(</span><span class="n">t</span><span class="p">)</span></div>
+
+
+<div class="viewcode-block" id="LTI"><a class="viewcode-back" href="../../../../generated/pypose.module.LTI/#pypose.module.LTI">[docs]</a><span class="k">class</span> <span class="nc">LTI</span><span class="p">(</span><span class="n">System</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Discrete-time Linear Time-Invariant (LTI) system.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        A (:obj:`Tensor`): The state matrix of LTI system.</span>
+<span class="sd">        B (:obj:`Tensor`): The input matrix of LTI system.</span>
+<span class="sd">        C (:obj:`Tensor`): The output matrix of LTI system.</span>
+<span class="sd">        D (:obj:`Tensor`): The observation matrix of LTI system,</span>
+<span class="sd">        c1 (:obj:`Tensor`, optional): The constant input of LTI system. Default: ``None``</span>
+<span class="sd">        c2 (:obj:`Tensor`, optional): The constant output of LTI system. Default: ``None``</span>
+
+<span class="sd">    A linear time-invariant lumped system can be described by state-space equation of the form:</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \begin{align*}</span>
+<span class="sd">            \mathbf{x}_{k+1} = \mathbf{A}\mathbf{x}_k + \mathbf{B}\mathbf{u}_k + \mathbf{c}_1 \\</span>
+<span class="sd">            \mathbf{y}_k     = \mathbf{C}\mathbf{x}_k + \mathbf{D}\mathbf{u}_k + \mathbf{c}_2 \\</span>
+<span class="sd">        \end{align*}</span>
+
+<span class="sd">    where :math:`\mathbf{x}` and :math:`\mathbf{u}` are state and input of the current</span>
+<span class="sd">    timestamp of LTI system.</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        The variables including state and input are row vectors, which is the last dimension of</span>
+<span class="sd">        a Tensor. :obj:`A`, :obj:`B`, :obj:`C`, :obj:`D`, :obj:`x`, :obj:`u` could be a single</span>
+<span class="sd">        matrix or batched matrices. In the batch case, their dimensions must be consistent so that</span>
+<span class="sd">        they can be multiplied for each channel.</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        &gt;&gt;&gt; # Batch, State, Input, Observe Dimension</span>
+<span class="sd">        &gt;&gt;&gt; Bd, Sd, Id, Od = 2, 3, 2, 2</span>
+<span class="sd">        &gt;&gt;&gt; # Linear System Matrices</span>
+<span class="sd">        &gt;&gt;&gt; device = torch.device(&quot;cuda&quot; if torch.cuda.is_available() else &quot;cpu&quot;)</span>
+<span class="sd">        &gt;&gt;&gt; A = torch.randn(Bd, Sd, Sd)</span>
+<span class="sd">        &gt;&gt;&gt; B = torch.randn(Bd, Sd, Id)</span>
+<span class="sd">        &gt;&gt;&gt; C = torch.randn(Bd, Od, Sd)</span>
+<span class="sd">        &gt;&gt;&gt; D = torch.randn(Bd, Od, Id)</span>
+<span class="sd">        &gt;&gt;&gt; c1 = torch.randn(Bd, Sd)</span>
+<span class="sd">        &gt;&gt;&gt; c2 = torch.randn(Bd, Od)</span>
+<span class="sd">        ...</span>
+<span class="sd">        &gt;&gt;&gt; lti = pp.module.LTI(A, B, C, D, c1, c2).to(device)</span>
+<span class="sd">        ...</span>
+<span class="sd">        &gt;&gt;&gt; state = torch.randn(Bd, Sd, device=device)</span>
+<span class="sd">        &gt;&gt;&gt; input = torch.randn(Bd, Id, device=device)</span>
+<span class="sd">        &gt;&gt;&gt; state, observation = lti(state, input)</span>
+<span class="sd">        tensor([[[-8.5639,  0.0523, -0.2576]],</span>
+<span class="sd">                [[ 4.1013, -1.5452, -0.0233]]]),</span>
+<span class="sd">        tensor([[[-3.5780, -2.2970, -2.9314]],</span>
+<span class="sd">                [[-0.4358,  1.7306,  2.7514]]]))</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        In this general example, all variables are in a batch. User definable as appropriate.</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        More practical examples can be found at `examples/module/dynamics</span>
+<span class="sd">        &lt;https://github.com/pypose/pypose/tree/main/examples/module/dynamics&gt;`_.</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="n">D</span><span class="p">,</span> <span class="n">c1</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">c2</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">register_buffer</span><span class="p">(</span><span class="s1">&#39;_A&#39;</span><span class="p">,</span> <span class="n">A</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">register_buffer</span><span class="p">(</span><span class="s1">&#39;_B&#39;</span><span class="p">,</span> <span class="n">B</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">register_buffer</span><span class="p">(</span><span class="s1">&#39;_C&#39;</span><span class="p">,</span> <span class="n">C</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">register_buffer</span><span class="p">(</span><span class="s1">&#39;_D&#39;</span><span class="p">,</span> <span class="n">D</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">register_buffer</span><span class="p">(</span><span class="s1">&#39;_c1&#39;</span><span class="p">,</span> <span class="n">c1</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">register_buffer</span><span class="p">(</span><span class="s1">&#39;_c2&#39;</span><span class="p">,</span> <span class="n">c2</span><span class="p">)</span>
+
+<div class="viewcode-block" id="LTI.forward"><a class="viewcode-back" href="../../../../generated/pypose.module.LTI/#pypose.module.LTI.forward">[docs]</a>    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">state</span><span class="p">,</span> <span class="nb">input</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Perform one step advance for the LTI system.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            state (:obj:`Tensor`): The state (:math:`\mathbf{x}`) of the system</span>
+<span class="sd">            input (:obj:`Tensor`): The input (:math:`\mathbf{u}`) of the system.</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            ``tuple`` of Tensor: The state and observation of the system in next time step.</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">return</span> <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="n">forward</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="nb">input</span><span class="p">)</span></div>
+
+<div class="viewcode-block" id="LTI.state_transition"><a class="viewcode-back" href="../../../../generated/pypose.module.LTI/#pypose.module.LTI.state_transition">[docs]</a>    <span class="k">def</span> <span class="nf">state_transition</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">state</span><span class="p">,</span> <span class="nb">input</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Perform one step of LTI state transition.</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathbf{x}_{k+1} = \mathbf{A}\mathbf{x}_k + \mathbf{B}\mathbf{u}_k + \mathbf{c}_1</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            state (:obj:`Tensor`): The state (:math:`\mathbf{x}`) of the system</span>
+<span class="sd">            input (:obj:`Tensor`): The input (:math:`\mathbf{u}`) of the system.</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            ``Tensor``: The state the system in next time step.</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="n">z</span> <span class="o">=</span> <span class="n">bmv</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">A</span><span class="p">,</span> <span class="n">state</span><span class="p">)</span> <span class="o">+</span> <span class="n">bmv</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">B</span><span class="p">,</span> <span class="nb">input</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">z</span> <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">c1</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">z</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">c1</span></div>
+
+<div class="viewcode-block" id="LTI.observation"><a class="viewcode-back" href="../../../../generated/pypose.module.LTI/#pypose.module.LTI.observation">[docs]</a>    <span class="k">def</span> <span class="nf">observation</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">state</span><span class="p">,</span> <span class="nb">input</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Return the observation of LTI system.</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathbf{y}_k = \mathbf{C}\mathbf{x}_k + \mathbf{D}\mathbf{u}_k + \mathbf{c}_2</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            state (:obj:`Tensor`): The state (:math:`\mathbf{x}`) of the system</span>
+<span class="sd">            input (:obj:`Tensor`): The input (:math:`\mathbf{u}`) of the system.</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            ``Tensor``: The observation of the system in next time step.</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="n">y</span> <span class="o">=</span> <span class="n">bmv</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">C</span><span class="p">,</span> <span class="n">state</span><span class="p">)</span> <span class="o">+</span> <span class="n">bmv</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">D</span><span class="p">,</span> <span class="nb">input</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">y</span> <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">c2</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">y</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">c2</span></div>
+
+    <span class="nd">@property</span>
+    <span class="k">def</span> <span class="nf">A</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;System transision matrix.&#39;&#39;&#39;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_A</span>
+
+    <span class="nd">@property</span>
+    <span class="k">def</span> <span class="nf">B</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;System input matrix.&#39;&#39;&#39;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_B</span>
+
+    <span class="nd">@property</span>
+    <span class="k">def</span> <span class="nf">C</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;System output matrix.&#39;&#39;&#39;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_C</span>
+
+    <span class="nd">@property</span>
+    <span class="k">def</span> <span class="nf">D</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;System observation matrix.&#39;&#39;&#39;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_D</span>
+
+    <span class="nd">@property</span>
+    <span class="k">def</span> <span class="nf">c1</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;Constant term generated by state-transition.&#39;&#39;&#39;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_c1</span>
+
+    <span class="nd">@property</span>
+    <span class="k">def</span> <span class="nf">c2</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;Constant term generated by observation.</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_c2</span></div>
+
+
+<div class="viewcode-block" id="LTV"><a class="viewcode-back" href="../../../../generated/pypose.module.LTV/#pypose.module.LTV">[docs]</a><span class="k">class</span> <span class="nc">LTV</span><span class="p">(</span><span class="n">LTI</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Discrete-time Linear Time-Variant (LTV) system.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        A (:obj:`Tensor`, optional): The stacked state matrix of LTI system. Default: ``None``</span>
+<span class="sd">        B (:obj:`Tensor`, optional): The stacked input matrix of LTI system. Default: ``None``</span>
+<span class="sd">        C (:obj:`Tensor`, optional): The stacked output matrix of LTI system. Default: ``None``</span>
+<span class="sd">        D (:obj:`Tensor`, optional): The stacked observation matrix of LTI system. Default: ``None``</span>
+<span class="sd">        c1 (:obj:`Tensor`, optional): The stacked constant input of LTI system. Default: ``None``</span>
+<span class="sd">        c2 (:obj:`Tensor`, optional): The stacked constant output of LTI system. Default: ``None``</span>
+
+<span class="sd">    A linear time-variant lumped system can be described by state-space equation of the form:</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \begin{align*}</span>
+<span class="sd">          \mathbf{x}_{k+1} = \mathbf{A}_k\mathbf{x}_k + \mathbf{B}\mathbf{u}_k + \mathbf{c}^1_k \\</span>
+<span class="sd">          \mathbf{y}_k     = \mathbf{C}_k\mathbf{x}_k + \mathbf{D}\mathbf{u}_k + \mathbf{c}^2_k \\</span>
+<span class="sd">        \end{align*}</span>
+
+<span class="sd">    where :math:`\mathbf{x}_k` and :math:`\mathbf{u}_k` are state and input at the</span>
+<span class="sd">    timestamp :math:`k` of LTI system.</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        The :obj:`forward` method implicitly increments the time step via :obj:`forward_hook`.</span>
+<span class="sd">        :obj:`state_transition` and :obj:`observation` still accept time for the flexiblity</span>
+<span class="sd">        such as time-varying system. One can directly access the current system time via the</span>
+<span class="sd">        property :obj:`systime` or :obj:`_t`.</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        The variables including state and input are row vectors, which is the last dimension of</span>
+<span class="sd">        a Tensor. :obj:`A`, :obj:`B`, :obj:`C`, :obj:`D`, :obj:`x`, :obj:`u` could be a single</span>
+<span class="sd">        matrix or batched matrices. In the batch case, their dimensions must be consistent so that</span>
+<span class="sd">        they can be multiplied for each channel.</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        A periodic linear time variant system.</span>
+
+<span class="sd">        &gt;&gt;&gt; n_batch, n_state, n_ctrl, T = 2, 4, 3, 5</span>
+<span class="sd">        &gt;&gt;&gt; n_sc = n_state + n_ctrl</span>
+<span class="sd">        &gt;&gt;&gt; A = torch.randn(n_batch, T, n_state, n_state)</span>
+<span class="sd">        &gt;&gt;&gt; B = torch.randn(n_batch, T, n_state, n_ctrl)</span>
+<span class="sd">        &gt;&gt;&gt; C = torch.tile(torch.eye(n_state), (n_batch, T, 1, 1))</span>
+<span class="sd">        &gt;&gt;&gt; D = torch.tile(torch.zeros(n_state, n_ctrl), (n_batch, T, 1, 1))</span>
+<span class="sd">        &gt;&gt;&gt; x = torch.randn(n_state)</span>
+<span class="sd">        &gt;&gt;&gt; u = torch.randn(T, n_ctrl)</span>
+<span class="sd">        ...</span>
+<span class="sd">        &gt;&gt;&gt; class MyLTV(pp.module.LTV):</span>
+<span class="sd">        ...     def __init__(self, A, B, C, D, T):</span>
+<span class="sd">        ...         super().__init__(A, B, C, D)</span>
+<span class="sd">        ...         self.T = T</span>
+<span class="sd">        ...</span>
+<span class="sd">        ...     @property</span>
+<span class="sd">        ...     def A(self):</span>
+<span class="sd">        ...         return self._A[...,self._t % self.T,:,:]</span>
+<span class="sd">        ...</span>
+<span class="sd">        ...     @property</span>
+<span class="sd">        ...     def B(self):</span>
+<span class="sd">        ...         return self._B[...,self._t % self.T,:,:]</span>
+<span class="sd">        ...</span>
+<span class="sd">        ...     @property</span>
+<span class="sd">        ...     def C(self):</span>
+<span class="sd">        ...         return self._C[...,self._t % self.T,:,:]</span>
+<span class="sd">        ...</span>
+<span class="sd">        ...     @property</span>
+<span class="sd">        ...     def D(self):</span>
+<span class="sd">        ...         return self._D[...,self._t % self.T,:,:]</span>
+<span class="sd">        ...</span>
+<span class="sd">        &gt;&gt;&gt; ltv = MyLTV(A, B, C, D, T)</span>
+<span class="sd">        &gt;&gt;&gt; for t in range(T):</span>
+<span class="sd">        ...     x, y = ltv(x, u[t])</span>
+
+<span class="sd">        One may also generate the system matrices with the time variable :obj:`_t`.</span>
+
+<span class="sd">        &gt;&gt;&gt; n_batch, n_state, n_ctrl, T = 2, 4, 3, 5</span>
+<span class="sd">        &gt;&gt;&gt; n_sc = n_state + n_ctrl</span>
+<span class="sd">        &gt;&gt;&gt; x = torch.randn(n_state)</span>
+<span class="sd">        &gt;&gt;&gt; u = torch.randn(T, n_ctrl)</span>
+<span class="sd">        ...</span>
+<span class="sd">        &gt;&gt;&gt; class MyLTV(pp.module.LTV):</span>
+<span class="sd">        ...     def __init__(self, A, B, C, D, T):</span>
+<span class="sd">        ...         super().__init__(A, B, C, D)</span>
+<span class="sd">        ...         self.T = T</span>
+<span class="sd">        ...</span>
+<span class="sd">        ...     @property</span>
+<span class="sd">        ...     def A(self):</span>
+<span class="sd">        ...         return torch.eye(4, 4) * self._t.cos()</span>
+<span class="sd">        ...</span>
+<span class="sd">        ...     @property</span>
+<span class="sd">        ...     def B(self):</span>
+<span class="sd">        ...         return torch.eye(4, 3) * self._t.sin()</span>
+<span class="sd">        ...</span>
+<span class="sd">        ...     @property</span>
+<span class="sd">        ...     def C(self):</span>
+<span class="sd">        ...         return torch.eye(4, 4) * self._t.tan()</span>
+<span class="sd">        ...</span>
+<span class="sd">        ...     @property</span>
+<span class="sd">        ...     def D(self):</span>
+<span class="sd">        ...         return torch.eye(4, 3)</span>
+<span class="sd">        ...</span>
+<span class="sd">        &gt;&gt;&gt; ltv = MyLTV()</span>
+<span class="sd">        &gt;&gt;&gt; for t in range(T):</span>
+<span class="sd">        ...     x, y = ltv(x, u[t])</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        More practical examples can be found at `examples/module/dynamics</span>
+<span class="sd">        &lt;https://github.com/pypose/pypose/tree/main/examples/module/dynamics&gt;`_.</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">A</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">B</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">C</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">D</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">c1</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">c2</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="n">D</span><span class="p">,</span> <span class="n">c1</span><span class="p">,</span> <span class="n">c2</span><span class="p">)</span>
+
+<div class="viewcode-block" id="LTV.set_refpoint"><a class="viewcode-back" href="../../../../generated/pypose.module.LTV/#pypose.module.LTV.set_refpoint">[docs]</a>    <span class="k">def</span> <span class="nf">set_refpoint</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">state</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="nb">input</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">t</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Function to set the reference point for linearization.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            state (:obj:`Tensor`): The reference state of the dynamical system. If ``None``,</span>
+<span class="sd">                the the most recent state is taken. Default: ``None``.</span>
+<span class="sd">            input (:obj:`Tensor`): The reference input to the dynamical system. If ``None``,</span>
+<span class="sd">                the the most recent input is taken. Default: ``None``.</span>
+<span class="sd">            t (:obj:`Tensor`): The reference time step of the dynamical system. If ``None``,</span>
+<span class="sd">                the the most recent timestamp is taken. Default: ``None``.</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            The ``self`` module.</span>
+
+<span class="sd">        Warning:</span>
+<span class="sd">            For nonlinear systems, the users have to call this function before getting the</span>
+<span class="sd">            linearized system.</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">systime</span> <span class="o">=</span> <span class="n">t</span>
+        <span class="k">return</span> <span class="bp">self</span></div></div>
+
+
+<div class="viewcode-block" id="NLS"><a class="viewcode-back" href="../../../../generated/pypose.module.NLS/#pypose.module.NLS">[docs]</a><span class="k">class</span> <span class="nc">NLS</span><span class="p">(</span><span class="n">System</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Dynamics model for discrete-time non-linear system (NLS).</span>
+
+<span class="sd">    The state transision function :math:`\mathbf{f}` and observation function</span>
+<span class="sd">    :math:`\mathbf{g}` are given by:</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \begin{aligned}</span>
+<span class="sd">            \mathbf{x}_{k+1} &amp;= \mathbf{f}(\mathbf{x}_k, \mathbf{u}_k, t_k), \\</span>
+<span class="sd">            \mathbf{y}_{k}   &amp;= \mathbf{g}(\mathbf{x}_k, \mathbf{u}_k, t_k),</span>
+<span class="sd">        \end{aligned}</span>
+
+<span class="sd">    where :math:`k`, :math:`\mathbf{x}`, :math:`\mathbf{u}`, :math:`\mathbf{y}` are the time</span>
+<span class="sd">    step, state(s), input(s), and observation(s), respectively.</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        To use the class, users need to inherit this class and define methods</span>
+<span class="sd">        :obj:`state_transition` and :obj:`observation`, which are automatically called by</span>
+<span class="sd">        internal :obj:`forward` method.</span>
+<span class="sd">        The system timestamp (starting from **0**) is also self-added automatically once</span>
+<span class="sd">        the :obj:`forward` method is called.</span>
+
+<span class="sd">    Note:</span>
+
+<span class="sd">        This class provides automatic **linearlization** at a reference point</span>
+<span class="sd">        :math:`\chi^*=(\mathbf{x}^*, \mathbf{u}^*, t^*)` along a trajectory.</span>
+<span class="sd">        One can directly call those linearized system matrices as properties including</span>
+<span class="sd">        :obj:`A`, :obj:`B`, :obj:`C`, :obj:`D`, :obj:`c1`, and :obj:`c2`, after calling</span>
+<span class="sd">        a method :obj:`set_refpoint`.</span>
+
+<span class="sd">        Consider a point</span>
+<span class="sd">        :math:`\chi=(\mathbf{x}^*+\delta\mathbf{x}, \mathbf{u}^*+\delta\mathbf{u}, t^*)` near</span>
+<span class="sd">        :math:`\chi^*`. We have</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \begin{aligned}</span>
+<span class="sd">            \mathbf{f}(\mathbf{x}, \mathbf{u}, t^*) &amp;\approx \mathbf{f}(\mathbf{x}^*,</span>
+<span class="sd">                \mathbf{u}^*, t^*) +  \left. \frac{\partial \mathbf{f}}{\partial \mathbf{x}}</span>
+<span class="sd">                \right|_{\chi^*} \delta \mathbf{x} + \left. \frac{\partial \mathbf{f}}</span>
+<span class="sd">                {\partial \mathbf{u}} \right|_{\chi^*} \delta \mathbf{u} \\</span>
+<span class="sd">            &amp;= \mathbf{f}(\mathbf{x}^*, \mathbf{u}^*, t^*) + \mathbf{A}(\mathbf{x}</span>
+<span class="sd">                - \mathbf{x}^*) + \mathbf{B}(\mathbf{u}-\mathbf{u}^*) \\</span>
+<span class="sd">            &amp;= \mathbf{A}\mathbf{x} + \mathbf{B}\mathbf{u} + \mathbf{c}_1</span>
+<span class="sd">            \end{aligned}</span>
+
+<span class="sd">        and</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathbf{g}(\mathbf{x}, \mathbf{u}, t^*) \approx \mathbf{C}\mathbf{x} \</span>
+<span class="sd">                        + \mathbf{D}\mathbf{u} + \mathbf{c}_2</span>
+
+<span class="sd">        The notion of linearization is slightly different from that in dynamical system</span>
+<span class="sd">        theory. First, the linearization can be done for arbitrary point(s), not limited to</span>
+<span class="sd">        the system&#39;s equilibrium point(s), and therefore the extra constant terms :math:`\mathbf{c}_1`</span>
+<span class="sd">        and :math:`\mathbf{c}_2` are produced. Second, the linearized equations are represented</span>
+<span class="sd">        by the full states and inputs: :math:`\mathbf{x}` and :math:`\mathbf{u}`, rather than</span>
+<span class="sd">        the perturbation format: :math:`\delta \mathbf{x}` and :math:`\delta \mathbf{u}`</span>
+<span class="sd">        so that the model is consistent with, e.g., the LTI model and the iterative LQR</span>
+<span class="sd">        solver. For more details go to :meth:`LTI`.</span>
+
+<span class="sd">    Example:</span>
+
+<span class="sd">        A simple linear time-varying system, but defined via NLS. Here we show an example</span>
+<span class="sd">        for advancing one time step of the system at a given time step and computing the</span>
+<span class="sd">        linearization.</span>
+
+<span class="sd">        &gt;&gt;&gt; import math, torch</span>
+<span class="sd">        &gt;&gt;&gt; import pypose as pp</span>
+<span class="sd">        ...</span>
+<span class="sd">        &gt;&gt;&gt; class Floquet(pp.module.NLS):</span>
+<span class="sd">        ...     def __init__(self):</span>
+<span class="sd">        ...         super().__init__()</span>
+<span class="sd">        ...</span>
+<span class="sd">        ...     def state_transition(self, state, input, t):</span>
+<span class="sd">        ...</span>
+<span class="sd">        ...         cc = (2 * math.pi * t / 100).cos()</span>
+<span class="sd">        ...         ss = (2 * math.pi * t / 100).sin()</span>
+<span class="sd">        ...</span>
+<span class="sd">        ...         A = torch.tensor([[   1., cc/10],</span>
+<span class="sd">        ...                           [cc/10,    1.]], device=t.device)</span>
+<span class="sd">        ...         B = torch.tensor([[ss],</span>
+<span class="sd">        ...                           [1.]], device=t.device)</span>
+<span class="sd">        ...</span>
+<span class="sd">        ...         return pp.bmv(A, state) + pp.bmv(B, input)</span>
+<span class="sd">        ...</span>
+<span class="sd">        ...     def observation(self, state, input, t):</span>
+<span class="sd">        ...         return state + t</span>
+<span class="sd">        ...</span>
+<span class="sd">        &gt;&gt;&gt; # Start from t = 8, and advance one step to t = 9.</span>
+<span class="sd">        &gt;&gt;&gt; step, current = 8, torch.tensor([1., 1.])</span>
+<span class="sd">        &gt;&gt;&gt; input = torch.tensor(2 * math.pi / 50 * step).sin()</span>
+<span class="sd">        ...</span>
+<span class="sd">        &gt;&gt;&gt; system = Floquet().reset(t = step)</span>
+<span class="sd">        &gt;&gt;&gt; next, observation = system(current, input)</span>
+<span class="sd">        &gt;&gt;&gt; system.set_refpoint()</span>
+<span class="sd">        ...</span>
+<span class="sd">        &gt;&gt;&gt; print(next)        # Next state</span>
+<span class="sd">        &gt;&gt;&gt; print(observation) # Observation</span>
+<span class="sd">        &gt;&gt;&gt; print(system.A)    # Linearized state matrix</span>
+<span class="sd">        &gt;&gt;&gt; print(system.B)    # Linearized input matrix</span>
+<span class="sd">        tensor([1.4944, 1.9320])</span>
+<span class="sd">        tensor([9., 9.])</span>
+<span class="sd">        tensor([[1.0000, 0.0844],</span>
+<span class="sd">                [0.0844, 1.0000]])</span>
+<span class="sd">        tensor([[0.5358],</span>
+<span class="sd">                [1.0000]])</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        For generating one trajecotry given a series of inputs, advanced use of</span>
+<span class="sd">        linearization, and more practical examples can be found at `examples/module/dynamics</span>
+<span class="sd">        &lt;https://github.com/pypose/pypose/tree/main/examples/module/dynamics&gt;`_.</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">jacargs</span> <span class="o">=</span> <span class="p">{</span><span class="s1">&#39;vectorize&#39;</span><span class="p">:</span><span class="kc">True</span><span class="p">,</span> <span class="s1">&#39;strategy&#39;</span><span class="p">:</span><span class="s1">&#39;reverse-mode&#39;</span><span class="p">}</span>
+
+<div class="viewcode-block" id="NLS.forward"><a class="viewcode-back" href="../../../../generated/pypose.module.NLS/#pypose.module.NLS.forward">[docs]</a>    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">state</span><span class="p">,</span> <span class="nb">input</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Defines the computation performed at every call that advances the system by one time step.</span>
+
+<span class="sd">        Note:</span>
+<span class="sd">            The :obj:`forward` method implicitly increments the time step via :obj:`forward_hook`.</span>
+<span class="sd">            :obj:`state_transition` and :obj:`observation` still accept time for the flexiblity</span>
+<span class="sd">            such as time-varying system. One can directly access the current system time via the</span>
+<span class="sd">            property :obj:`systime`.</span>
+
+<span class="sd">        Note:</span>
+<span class="sd">            To introduce noise in a model, redefine this method via</span>
+<span class="sd">            subclassing. See example in ``examples/module/ekf/tank_robot.py``.</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">state</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">input</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">atleast_1d</span><span class="p">(</span><span class="n">state</span><span class="p">),</span> <span class="n">torch</span><span class="o">.</span><span class="n">atleast_1d</span><span class="p">(</span><span class="nb">input</span><span class="p">)</span>
+        <span class="n">state</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">state_transition</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">state</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">input</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">systime</span><span class="p">)</span>
+        <span class="n">obs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">observation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">state</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">input</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">systime</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">state</span><span class="p">,</span> <span class="n">obs</span></div>
+
+<div class="viewcode-block" id="NLS.set_refpoint"><a class="viewcode-back" href="../../../../generated/pypose.module.NLS/#pypose.module.NLS.set_refpoint">[docs]</a>    <span class="k">def</span> <span class="nf">set_refpoint</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">state</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="nb">input</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">t</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Function to set the reference point for linearization.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            state (:obj:`Tensor`): The reference state of the dynamical system. If ``None``,</span>
+<span class="sd">                the the most recent state is taken. Default: ``None``.</span>
+<span class="sd">            input (:obj:`Tensor`): The reference input to the dynamical system. If ``None``,</span>
+<span class="sd">                the the most recent input is taken. Default: ``None``.</span>
+<span class="sd">            t (:obj:`Tensor`): The reference time step of the dynamical system. If ``None``,</span>
+<span class="sd">                the the most recent timestamp is taken. Default: ``None``.</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            The ``self`` module.</span>
+
+<span class="sd">        Warning:</span>
+<span class="sd">            For nonlinear systems, the users have to call this function before getting the</span>
+<span class="sd">            linearized system.</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">_ref_state</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">state</span> <span class="k">if</span> <span class="n">state</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">torch</span><span class="o">.</span><span class="n">atleast_1d</span><span class="p">(</span><span class="n">state</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">_ref_input</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">input</span> <span class="k">if</span> <span class="nb">input</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">torch</span><span class="o">.</span><span class="n">atleast_1d</span><span class="p">(</span><span class="nb">input</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">_ref_t</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">systime</span> <span class="k">if</span> <span class="n">t</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">torch</span><span class="o">.</span><span class="n">atleast_1d</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">_ref_f</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">state_transition</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_ref_state</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_ref_input</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_ref_t</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">_ref_g</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">observation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_ref_state</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_ref_input</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_ref_t</span><span class="p">)</span>
+        <span class="k">return</span> <span class="bp">self</span></div>
+
+    <span class="nd">@property</span>
+    <span class="k">def</span> <span class="nf">A</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Linear/linearized system state matrix.</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathbf{A} = \left. \frac{\partial \mathbf{f}}{\partial \mathbf{x}} \right|_{\chi^*}</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="n">func</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">state_transition</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_ref_input</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_ref_t</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">jacobian</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_ref_state</span><span class="p">,</span> <span class="o">**</span><span class="bp">self</span><span class="o">.</span><span class="n">jacargs</span><span class="p">)</span>
+
+    <span class="nd">@property</span>
+    <span class="k">def</span> <span class="nf">B</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Linear/linearized system input matrix.</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathbf{B} = \left. \frac{\partial \mathbf{f}}{\partial \mathbf{u}} \right|_{\chi^*}</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="n">func</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">state_transition</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_ref_state</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_ref_t</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">jacobian</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_ref_input</span><span class="p">,</span> <span class="o">**</span><span class="bp">self</span><span class="o">.</span><span class="n">jacargs</span><span class="p">)</span>
+
+    <span class="nd">@property</span>
+    <span class="k">def</span> <span class="nf">C</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Linear/linearized system output matrix.</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathbf{C} = \left. \frac{\partial \mathbf{g}}{\partial \mathbf{x}} \right|_{\chi^*}</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="n">func</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">observation</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_ref_input</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_ref_t</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">jacobian</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_ref_state</span><span class="p">,</span> <span class="o">**</span><span class="bp">self</span><span class="o">.</span><span class="n">jacargs</span><span class="p">)</span>
+
+    <span class="nd">@property</span>
+    <span class="k">def</span> <span class="nf">D</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Linear/Linearized system observation matrix.</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathbf{D} = \left. \frac{\partial \mathbf{g}}</span>
+<span class="sd">                                {\partial \mathbf{u}} \right|_{\chi^*}</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="n">func</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">observation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_ref_state</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_ref_t</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">jacobian</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_ref_input</span><span class="p">,</span> <span class="o">**</span><span class="bp">self</span><span class="o">.</span><span class="n">jacargs</span><span class="p">)</span>
+
+    <span class="nd">@property</span>
+    <span class="k">def</span> <span class="nf">c1</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Constant term generated by state-transition.</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathbf{c}_1 = \mathbf{f}(\mathbf{x}^*, \mathbf{u}^*, t^*)</span>
+<span class="sd">                           - \mathbf{A}\mathbf{x}^* - \mathbf{B}\mathbf{u}^*</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="c1"># Potential performance loss here - self.A and self.B involves jacobian eval</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_ref_f</span> <span class="o">-</span> <span class="n">bmv</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">A</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_ref_state</span><span class="p">)</span> <span class="o">-</span> <span class="n">bmv</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">B</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_ref_input</span><span class="p">)</span>
+
+    <span class="nd">@property</span>
+    <span class="k">def</span> <span class="nf">c2</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Constant term generated by observation.</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathbf{c}_2 = \mathbf{g}(\mathbf{x}^*, \mathbf{u}^*, t^*)</span>
+<span class="sd">                           - \mathbf{C}\mathbf{x}^* - \mathbf{D}\mathbf{u}^*</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="c1"># Potential performance loss here - self.C and self.D involves jacobian eval</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_ref_g</span> <span class="o">-</span> <span class="n">bmv</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">C</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_ref_state</span><span class="p">)</span> <span class="o">-</span> <span class="n">bmv</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">D</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_ref_input</span><span class="p">)</span></div>
+
+
+<span class="k">def</span> <span class="nf">toBTN</span><span class="p">(</span><span class="n">vec</span><span class="p">,</span> <span class="n">T</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Reshape the input tensor of shape ``[..., N]`` to ``[B, T, N]``, where B, T, N</span>
+<span class="sd">    normally refer to the dimension of batch, time step, and state, respectively.</span>
+
+<span class="sd">    Returns:</span>
+<span class="sd">        The reshaped tensor in shape of ``[B, T, N]``.</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">if</span> <span class="n">vec</span><span class="o">.</span><span class="n">ndim</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
+        <span class="n">vec</span> <span class="o">=</span> <span class="n">vec</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
+
+    <span class="k">if</span> <span class="n">vec</span><span class="o">.</span><span class="n">ndim</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
+        <span class="n">vec</span> <span class="o">=</span> <span class="n">vec</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
+
+    <span class="k">if</span> <span class="n">vec</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
+        <span class="n">vec</span> <span class="o">=</span> <span class="n">vec</span><span class="o">.</span><span class="n">repeat</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
+
+    <span class="k">return</span> <span class="n">vec</span>
+
+
+<span class="k">def</span> <span class="nf">runsys</span><span class="p">(</span><span class="n">system</span><span class="p">:</span> <span class="n">System</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">x_traj</span><span class="p">,</span> <span class="n">u_traj</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Run the system for T steps, given state and input trajectories or vectors.</span>
+
+<span class="sd">    Returns:</span>
+<span class="sd">        The state trajectory of the system based on the state and input trajectories.</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+
+    <span class="c1">#make initial states trajectories if not given</span>
+    <span class="n">x_traj</span> <span class="o">=</span> <span class="n">toBTN</span><span class="p">(</span><span class="n">x_traj</span><span class="p">,</span> <span class="n">T</span><span class="p">)</span>
+    <span class="n">u_traj</span> <span class="o">=</span> <span class="n">toBTN</span><span class="p">(</span><span class="n">u_traj</span><span class="p">,</span> <span class="n">T</span><span class="p">)</span>
+
+    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">T</span><span class="o">-</span><span class="mi">1</span><span class="p">):</span>
+        <span class="n">x_traj</span><span class="p">[</span><span class="o">...</span><span class="p">,</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">,:],</span> <span class="n">_</span> <span class="o">=</span> <span class="n">system</span><span class="p">(</span><span class="n">x_traj</span><span class="p">[</span><span class="o">...</span><span class="p">,</span><span class="n">i</span><span class="p">,:],</span> <span class="n">u_traj</span><span class="p">[</span><span class="o">...</span><span class="p">,</span><span class="n">i</span><span class="p">,:])</span>
+
+    <span class="k">return</span> <span class="n">x_traj</span>
+</pre></div>
+
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new file mode 100644
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+<ul>
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+          
+            <div role="main" class="main-content" itemscope="itemscope" itemtype="http://schema.org/Article">
+             <article itemprop="articleBody" id="pytorch-article" class="pytorch-article">
+              
+  <h1>Source code for pypose.module.ekf</h1><div class="highlight"><pre>
+<span></span><span class="kn">import</span> <span class="nn">torch</span>
+<span class="kn">from</span> <span class="nn">..</span> <span class="kn">import</span> <span class="n">bmv</span>
+<span class="kn">from</span> <span class="nn">torch</span> <span class="kn">import</span> <span class="n">nn</span>
+<span class="kn">from</span> <span class="nn">torch.linalg</span> <span class="kn">import</span> <span class="n">pinv</span>
+
+
+<div class="viewcode-block" id="EKF"><a class="viewcode-back" href="../../../../generated/pypose.module.EKF/#pypose.module.EKF">[docs]</a><span class="k">class</span> <span class="nc">EKF</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Performs Batched Extended Kalman Filter (EKF).</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        model (:obj:`System`): The system model to be estimated, a subclass of</span>
+<span class="sd">            :obj:`pypose.module.NLS`.</span>
+<span class="sd">        Q (:obj:`Tensor`, optional): The covariance matrices of system transition noise.</span>
+<span class="sd">            Ignored if provided during each iteration. Default: ``None``</span>
+<span class="sd">        R (:obj:`Tensor`, optional): The covariance matrices of system observation noise.</span>
+<span class="sd">            Ignored if provided during each iteration. Default: ``None``</span>
+
+<span class="sd">    A non-linear system can be described as</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \begin{aligned}</span>
+<span class="sd">            \mathbf{x}_{k+1} &amp;= \mathbf{f}(\mathbf{x}_k, \mathbf{u}_k, t_k) + \mathbf{w}_k,</span>
+<span class="sd">            \quad \mathbf{w}_k \sim \mathcal{N}(\mathbf{0}, \mathbf{Q})  \\</span>
+<span class="sd">            \mathbf{y}_{k} &amp;= \mathbf{g}(\mathbf{x}_k, \mathbf{u}_k, t_k) + \mathbf{v}_k,</span>
+<span class="sd">            \quad \mathbf{v}_k \sim \mathcal{N}(\mathbf{0}, \mathbf{R})</span>
+<span class="sd">        \end{aligned}</span>
+
+<span class="sd">    It will be linearized automatically:</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \begin{align*}</span>
+<span class="sd">            \mathbf{z}_{k+1} = \mathbf{A}_{k}\mathbf{x}_{k} + \mathbf{B}_{k}\mathbf{u}_{k}</span>
+<span class="sd">                             + \mathbf{c}_{k}^1 + \mathbf{w}_k\\</span>
+<span class="sd">            \mathbf{y}_{k} = \mathbf{C}_{k}\mathbf{x}_{k} + \mathbf{D}_{k}\mathbf{u}_{k}</span>
+<span class="sd">                           + \mathbf{c}_{k}^2 + \mathbf{v}_k\\</span>
+<span class="sd">        \end{align*}</span>
+
+<span class="sd">    EKF can be described as the following five equations, where the subscript :math:`\cdot_{k}`</span>
+<span class="sd">    is omited for simplicity.</span>
+
+<span class="sd">    1. Priori State Estimation.</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathbf{x}^{-} = \mathbf{A}\mathbf{x} + \mathbf{B}\mathbf{u}_k + \mathbf{c}_1</span>
+
+<span class="sd">    2. Priori Covariance Propagation.</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathbf{P}^{-} = \mathbf{A}\mathbf{P}\mathbf{A}^{T} + \mathbf{Q}</span>
+
+<span class="sd">    3. Update Kalman Gain</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathbf{K} = \mathbf{P}\mathbf{C}^{T}</span>
+<span class="sd">                        (\mathbf{C}\mathbf{P} \mathbf{C}^{T} + \mathbf{R})^{-1}</span>
+
+<span class="sd">    4. Posteriori State Estimation</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathbf{x}^{+} = \mathbf{x}^{-} + \mathbf{K} (\mathbf{y} -</span>
+<span class="sd">                        \mathbf{C}\mathbf{x}^{-} - \mathbf{D}\mathbf{u} - \mathbf{c}_2)</span>
+
+<span class="sd">    5. Posteriori Covariance Estimation</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathbf{P} = (\mathbf{I} - \mathbf{K}\mathbf{C}) \mathbf{P}^{-}</span>
+
+<span class="sd">    where superscript :math:`\cdot^{-}` and :math:`\cdot^{+}` denote the priori and</span>
+<span class="sd">    posteriori estimation, respectively.</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        1. Define a discrete-time non-linear system (NLS) model</span>
+
+<span class="sd">        &gt;&gt;&gt; import torch, pypose as pp</span>
+<span class="sd">        &gt;&gt;&gt; class NLS(pp.module.NLS):</span>
+<span class="sd">        ...     def __init__(self):</span>
+<span class="sd">        ...         super().__init__()</span>
+<span class="sd">        ...</span>
+<span class="sd">        ...     def state_transition(self, state, input, t=None):</span>
+<span class="sd">        ...         return state.cos() + input</span>
+<span class="sd">        ...</span>
+<span class="sd">        ...     def observation(self, state, input, t):</span>
+<span class="sd">        ...         return state.sin() + input</span>
+
+<span class="sd">        2. Create a model and filter</span>
+
+<span class="sd">        &gt;&gt;&gt; model = NLS()</span>
+<span class="sd">        &gt;&gt;&gt; ekf = pp.module.EKF(model)</span>
+
+<span class="sd">        3. Prepare data</span>
+
+<span class="sd">        &gt;&gt;&gt; T, N = 5, 2 # steps, state dim</span>
+<span class="sd">        &gt;&gt;&gt; states = torch.zeros(T, N)</span>
+<span class="sd">        &gt;&gt;&gt; inputs = torch.randn(T, N)</span>
+<span class="sd">        &gt;&gt;&gt; observ = torch.zeros(T, N)</span>
+<span class="sd">        &gt;&gt;&gt; # std of transition, observation, and estimation</span>
+<span class="sd">        &gt;&gt;&gt; q, r, p = 0.1, 0.1, 10</span>
+<span class="sd">        &gt;&gt;&gt; Q = torch.eye(N) * q**2</span>
+<span class="sd">        &gt;&gt;&gt; R = torch.eye(N) * r**2</span>
+<span class="sd">        &gt;&gt;&gt; P = torch.eye(N).repeat(T, 1, 1) * p**2</span>
+<span class="sd">        &gt;&gt;&gt; estim = torch.randn(T, N) * p</span>
+
+<span class="sd">        4. Perform EKF prediction. Note that estimation error becomes smaller with more steps.</span>
+
+<span class="sd">        &gt;&gt;&gt; for i in range(T - 1):</span>
+<span class="sd">        ...     w = q * torch.randn(N) # transition noise</span>
+<span class="sd">        ...     v = r * torch.randn(N) # observation noise</span>
+<span class="sd">        ...     states[i+1], observ[i] = model(states[i] + w, inputs[i])</span>
+<span class="sd">        ...     estim[i+1], P[i+1] = ekf(estim[i], observ[i] + v, inputs[i], P[i], Q, R)</span>
+<span class="sd">        ... print(&#39;Est error:&#39;, (states - estim).norm(dim=-1))</span>
+<span class="sd">        Est error: tensor([5.7655, 5.3436, 3.5947, 0.3359, 0.0639])</span>
+
+<span class="sd">    Warning:</span>
+<span class="sd">        Don&#39;t introduce noise in ``System`` methods ``state_transition`` and ``observation``</span>
+<span class="sd">        for filter testing, as those methods are used for automatically linearizing the system</span>
+<span class="sd">        by the parent class ``pypose.module.NLS``, unless your system model explicitly</span>
+<span class="sd">        introduces noise.</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        Implementation is based on Section 5.1 of this book</span>
+
+<span class="sd">        * Dan Simon, `Optimal State Estimation: Kalman, H∞, and Nonlinear Approaches</span>
+<span class="sd">          &lt;https://onlinelibrary.wiley.com/doi/book/10.1002/0470045345&gt;`_,</span>
+<span class="sd">          Cleveland State University, 2006</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">model</span><span class="p">,</span> <span class="n">Q</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">R</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">set_uncertainty</span><span class="p">(</span><span class="n">Q</span><span class="o">=</span><span class="n">Q</span><span class="p">,</span> <span class="n">R</span><span class="o">=</span><span class="n">R</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">model</span> <span class="o">=</span> <span class="n">model</span>
+
+<div class="viewcode-block" id="EKF.forward"><a class="viewcode-back" href="../../../../generated/pypose.module.EKF/#pypose.module.EKF.forward">[docs]</a>    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">Q</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">R</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">t</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Performs one step estimation.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            x (:obj:`Tensor`): estimated system state of previous step.</span>
+<span class="sd">            y (:obj:`Tensor`): system observation at current step (measurement).</span>
+<span class="sd">            u (:obj:`Tensor`): system input at current step.</span>
+<span class="sd">            P (:obj:`Tensor`): state estimation covariance of previous step.</span>
+<span class="sd">            Q (:obj:`Tensor`, optional): covariance of system transition model. Default: ``None``</span>
+<span class="sd">            R (:obj:`Tensor`, optional): covariance of system observation model. Default: ``None``</span>
+<span class="sd">            t (:obj:`Tensor`, optional): timestep of system (only for time variant system).</span>
+<span class="sd">                Default: ``None``</span>
+
+<span class="sd">        Return:</span>
+<span class="sd">            list of :obj:`Tensor`: posteriori state and covariance estimation</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="c1"># Upper cases are matrices, lower cases are vectors</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">model</span><span class="o">.</span><span class="n">set_refpoint</span><span class="p">(</span><span class="n">state</span><span class="o">=</span><span class="n">x</span><span class="p">,</span> <span class="nb">input</span><span class="o">=</span><span class="n">u</span><span class="p">,</span> <span class="n">t</span><span class="o">=</span><span class="n">t</span><span class="p">)</span>
+        <span class="n">I</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">P</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">device</span><span class="o">=</span><span class="n">P</span><span class="o">.</span><span class="n">device</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">P</span><span class="o">.</span><span class="n">dtype</span><span class="p">)</span>
+        <span class="n">A</span><span class="p">,</span> <span class="n">B</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">model</span><span class="o">.</span><span class="n">A</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">model</span><span class="o">.</span><span class="n">B</span>
+        <span class="n">C</span><span class="p">,</span> <span class="n">D</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">model</span><span class="o">.</span><span class="n">C</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">model</span><span class="o">.</span><span class="n">D</span>
+        <span class="n">c1</span><span class="p">,</span> <span class="n">c2</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">model</span><span class="o">.</span><span class="n">c1</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">model</span><span class="o">.</span><span class="n">c2</span>
+        <span class="n">Q</span> <span class="o">=</span> <span class="n">Q</span> <span class="k">if</span> <span class="n">Q</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span> <span class="k">else</span> <span class="bp">self</span><span class="o">.</span><span class="n">Q</span>
+        <span class="n">R</span> <span class="o">=</span> <span class="n">R</span> <span class="k">if</span> <span class="n">R</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span> <span class="k">else</span> <span class="bp">self</span><span class="o">.</span><span class="n">R</span>
+
+        <span class="n">x</span> <span class="o">=</span> <span class="n">bmv</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">bmv</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span> <span class="o">+</span> <span class="n">c1</span>        <span class="c1"># 1. System transition</span>
+        <span class="n">P</span> <span class="o">=</span> <span class="n">A</span> <span class="o">@</span> <span class="n">P</span> <span class="o">@</span> <span class="n">A</span><span class="o">.</span><span class="n">mT</span> <span class="o">+</span> <span class="n">Q</span>                  <span class="c1"># 2. Covariance predict</span>
+        <span class="n">K</span> <span class="o">=</span> <span class="n">P</span> <span class="o">@</span> <span class="n">C</span><span class="o">.</span><span class="n">mT</span> <span class="o">@</span> <span class="n">pinv</span><span class="p">(</span><span class="n">C</span> <span class="o">@</span> <span class="n">P</span> <span class="o">@</span> <span class="n">C</span><span class="o">.</span><span class="n">mT</span> <span class="o">+</span> <span class="n">R</span><span class="p">)</span> <span class="c1"># 3. Kalman gain</span>
+        <span class="n">e</span> <span class="o">=</span> <span class="n">y</span> <span class="o">-</span> <span class="n">bmv</span><span class="p">(</span><span class="n">C</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="o">-</span> <span class="n">bmv</span><span class="p">(</span><span class="n">D</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span> <span class="o">-</span> <span class="n">c2</span>    <span class="c1">#    predicted observation error</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="n">x</span> <span class="o">+</span> <span class="n">bmv</span><span class="p">(</span><span class="n">K</span><span class="p">,</span> <span class="n">e</span><span class="p">)</span>                     <span class="c1"># 4. Posteriori state</span>
+        <span class="n">P</span> <span class="o">=</span> <span class="p">(</span><span class="n">I</span> <span class="o">-</span> <span class="n">K</span> <span class="o">@</span> <span class="n">C</span><span class="p">)</span> <span class="o">@</span> <span class="n">P</span>                   <span class="c1"># 5. Posteriori covariance</span>
+        <span class="k">return</span> <span class="n">x</span><span class="p">,</span> <span class="n">P</span></div>
+
+    <span class="nd">@property</span>
+    <span class="k">def</span> <span class="nf">Q</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        The covariance of system transition noise.</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">if</span> <span class="ow">not</span> <span class="nb">hasattr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="s1">&#39;_Q&#39;</span><span class="p">):</span>
+            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s1">&#39;Call set_uncertainty() to define</span><span class="se">\</span>
+<span class="s1">                                        transition covariance Q.&#39;</span><span class="p">)</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_Q</span>
+
+    <span class="nd">@property</span>
+    <span class="k">def</span> <span class="nf">R</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        The covariance of system observation noise.</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">if</span> <span class="ow">not</span> <span class="nb">hasattr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="s1">&#39;_R&#39;</span><span class="p">):</span>
+            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s1">&#39;Call set_uncertainty() to define</span><span class="se">\</span>
+<span class="s1">                                        transition covariance R.&#39;</span><span class="p">)</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_R</span>
+
+<div class="viewcode-block" id="EKF.set_uncertainty"><a class="viewcode-back" href="../../../../generated/pypose.module.EKF/#pypose.module.EKF.set_uncertainty">[docs]</a>    <span class="k">def</span> <span class="nf">set_uncertainty</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">Q</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">R</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Set the covariance matrices of transition noise and observation noise.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            Q (:obj:`Tensor`): batched square covariance matrices of transition noise.</span>
+<span class="sd">            R (:obj:`Tensor`): batched square covariance matrices of observation noise.</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">if</span> <span class="n">Q</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">register_buffer</span><span class="p">(</span><span class="s2">&quot;_Q&quot;</span><span class="p">,</span> <span class="n">Q</span><span class="p">)</span>
+        <span class="k">if</span> <span class="n">R</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">register_buffer</span><span class="p">(</span><span class="s2">&quot;_R&quot;</span><span class="p">,</span> <span class="n">R</span><span class="p">)</span></div></div>
+</pre></div>
+
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+              
+  <h1>Source code for pypose.module.icp</h1><div class="highlight"><pre>
+<span></span><span class="kn">import</span> <span class="nn">torch</span>
+<span class="kn">from</span> <span class="nn">torch</span> <span class="kn">import</span> <span class="n">nn</span>
+<span class="kn">from</span> <span class="nn">..</span> <span class="kn">import</span> <span class="n">knn</span><span class="p">,</span> <span class="n">svdtf</span><span class="p">,</span> <span class="n">is_SE3</span>
+<span class="kn">from</span> <span class="nn">..utils.stepper</span> <span class="kn">import</span> <span class="n">ReduceToBason</span>
+
+
+<div class="viewcode-block" id="ICP"><a class="viewcode-back" href="../../../../generated/pypose.module.ICP/#pypose.module.ICP">[docs]</a><span class="k">class</span> <span class="nc">ICP</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Batched Iterative Closest Point (ICP) algorithm to find a rigid transformation</span>
+<span class="sd">    between two sets of points using Singular Value Decomposition (SVD).</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        init (``LieTensor``, optional): the initial transformation :math:`T_{\text{init}}`</span>
+<span class="sd">            in ``SE3type LieTensor``. Default: ``None``.</span>
+<span class="sd">        stepper (``Planner``, optional): the stepper to stop a loop. If ``None``,</span>
+<span class="sd">            the ``pypose.utils.ReduceToBason`` with a maximum of 200 steps are used.</span>
+<span class="sd">            Default: ``None``.</span>
+
+<span class="sd">    The algorithm takes two input point clouds (source and target) and finds the optimal</span>
+<span class="sd">    transformation ( :math:`T` ) to minimize the error between the transformed source</span>
+<span class="sd">    point cloud and the target point cloud:</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \begin{align*}</span>
+<span class="sd">            \underset{T}{\operatorname{arg\,min}} \sum_i \| p_{\mathrm{target, j}} -</span>
+<span class="sd">            T \cdot p_{\mathrm{source, i}}\|,</span>
+<span class="sd">        \end{align*}</span>
+
+<span class="sd">    where :math:`p_{\mathrm{source, i}}` is the i-th point in the source point cloud, and</span>
+<span class="sd">    :math:`p_{\mathrm{target, j}}` is the cloest point of :math:`p_{\mathrm{source, i}}`</span>
+<span class="sd">    in the target point clouds with index j. The algorithm consists of the following steps:</span>
+
+<span class="sd">    1. For each point in source, the nearest neighbor algorithm (KNN) is used to select</span>
+<span class="sd">       its closest point in target to form the matched point pairs.</span>
+
+<span class="sd">    2. Singular value decomposition (SVD) algorithm is used to compute the transformation</span>
+<span class="sd">       from the matched point pairs.</span>
+
+<span class="sd">    3. The source point cloud is updated using the obtained transformation.</span>
+<span class="sd">       The distance between the updated source and target is calculated.</span>
+
+<span class="sd">    4. The algorithm continues to iterate through these steps until the ``stepper``</span>
+<span class="sd">       condition is satisfied.</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        &gt;&gt;&gt; import torch, pypose as pp</span>
+<span class="sd">        &gt;&gt;&gt; source = torch.tensor([[[0., 0., 0.],</span>
+<span class="sd">        ...                         [1., 0., 0.],</span>
+<span class="sd">        ...                         [2.,  0, 0.]]])</span>
+<span class="sd">        &gt;&gt;&gt; target = torch.tensor([[[0.2,      0.1,  0.],</span>
+<span class="sd">        ...                         [1.1397, 0.442,  0.],</span>
+<span class="sd">        ...                         [2.0794, 0.7840, 0.]]])</span>
+<span class="sd">        &gt;&gt;&gt; stepper = pp.utils.ReduceToBason(steps=10, verbose=True)</span>
+<span class="sd">        &gt;&gt;&gt; icp = pp.module.ICP(stepper=stepper)</span>
+<span class="sd">        &gt;&gt;&gt; icp(source, target)</span>
+<span class="sd">        ReduceToBason step 0 loss tensor([0.4917])</span>
+<span class="sd">        ReduceToBason step 1 loss tensor([7.4711e-08])</span>
+<span class="sd">        ReduceToBason step 2 loss tensor([1.0450e-07])</span>
+<span class="sd">        ReduceToBason step 3 loss tensor([2.8322e-07])</span>
+<span class="sd">        ReduceToBason: Maximum patience steps reached, Quiting..</span>
+<span class="sd">        SE3Type LieTensor:</span>
+<span class="sd">        LieTensor([[0.2000, 0.1000, 0.0000, 0.0000, 0.0000, 0.1736, 0.9848]])</span>
+
+<span class="sd">    Warning:</span>
+<span class="sd">        It&#39;s important to note that the solution is sensitive to the initialization.</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">init</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">stepper</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">stepper</span> <span class="o">=</span> <span class="n">ReduceToBason</span><span class="p">(</span><span class="n">steps</span><span class="o">=</span><span class="mi">200</span><span class="p">)</span> <span class="k">if</span> <span class="n">stepper</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">stepper</span>
+        <span class="k">assert</span> <span class="n">init</span> <span class="ow">is</span> <span class="kc">None</span> <span class="ow">or</span> <span class="n">is_SE3</span><span class="p">(</span><span class="n">init</span><span class="p">),</span> <span class="s2">&quot;The initial transformation is not SE3Type.&quot;</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">init</span> <span class="o">=</span> <span class="n">init</span>
+
+<div class="viewcode-block" id="ICP.forward"><a class="viewcode-back" href="../../../../generated/pypose.module.ICP/#pypose.module.ICP.forward">[docs]</a>    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">source</span><span class="p">,</span> <span class="n">target</span><span class="p">,</span> <span class="nb">ord</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">dim</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">init</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Args:</span>
+<span class="sd">            source (``torch.Tensor``): the source point clouds with shape</span>
+<span class="sd">                (..., points_num, 3).</span>
+<span class="sd">            target (``torch.Tensor``): the target point clouds with shape</span>
+<span class="sd">                (..., points_num, 3).</span>
+<span class="sd">            ord (``int``, optional): the order of norm to use for distance calculation.</span>
+<span class="sd">                Default: ``2`` (Euclidean distance).</span>
+<span class="sd">            dim (``int``, optional): the dimension encompassing the point cloud</span>
+<span class="sd">                coordinates, utilized for calculating distance.</span>
+<span class="sd">                Default: ``-1`` (The last dimension).</span>
+<span class="sd">            init (``LieTensor``, optional): the initial transformation :math:`T_{init}` in</span>
+<span class="sd">                ``SE3type``. If not ``None``, it will suppress the ``init`` given by the</span>
+<span class="sd">                class constructor. Default: ``None``.</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            ``LieTensor``: The estimated transformation (``SE3type``) from source to</span>
+<span class="sd">            target point cloud.</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="n">temporal</span> <span class="o">=</span> <span class="n">source</span>
+        <span class="n">init</span> <span class="o">=</span> <span class="n">init</span> <span class="k">if</span> <span class="n">init</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span> <span class="k">else</span> <span class="bp">self</span><span class="o">.</span><span class="n">init</span>
+        <span class="k">if</span> <span class="n">init</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
+            <span class="k">assert</span> <span class="n">is_SE3</span><span class="p">(</span><span class="n">init</span><span class="p">),</span> <span class="s2">&quot;The initial transformation is not SE3Type LieTensor.&quot;</span>
+            <span class="n">temporal</span> <span class="o">=</span> <span class="n">init</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span> <span class="o">@</span> <span class="n">temporal</span>
+        <span class="n">batch</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">broadcast_shapes</span><span class="p">(</span><span class="n">source</span><span class="o">.</span><span class="n">shape</span><span class="p">[:</span><span class="o">-</span><span class="mi">2</span><span class="p">],</span> <span class="n">target</span><span class="o">.</span><span class="n">shape</span><span class="p">[:</span><span class="o">-</span><span class="mi">2</span><span class="p">])</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">stepper</span><span class="o">.</span><span class="n">reset</span><span class="p">()</span>
+        <span class="k">while</span> <span class="bp">self</span><span class="o">.</span><span class="n">stepper</span><span class="o">.</span><span class="n">continual</span><span class="p">():</span>
+            <span class="n">knndist</span><span class="p">,</span> <span class="n">knnidx</span> <span class="o">=</span> <span class="n">knn</span><span class="p">(</span><span class="n">temporal</span><span class="p">,</span> <span class="n">target</span><span class="p">,</span> <span class="n">k</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="nb">ord</span><span class="o">=</span><span class="nb">ord</span><span class="p">,</span> <span class="n">dim</span><span class="o">=</span><span class="n">dim</span><span class="p">)</span>
+            <span class="n">error</span> <span class="o">=</span> <span class="n">knndist</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">dim</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span>
+            <span class="n">target</span> <span class="o">=</span> <span class="n">target</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">batch</span> <span class="o">+</span> <span class="n">target</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">:])</span>
+            <span class="n">knnidx</span> <span class="o">=</span> <span class="n">knnidx</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">batch</span> <span class="o">+</span> <span class="n">source</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">:])</span>
+            <span class="n">knntarget</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">gather</span><span class="p">(</span><span class="n">target</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="n">knnidx</span><span class="p">)</span>
+            <span class="n">T</span> <span class="o">=</span> <span class="n">svdtf</span><span class="p">(</span><span class="n">temporal</span><span class="p">,</span> <span class="n">knntarget</span><span class="p">)</span>
+            <span class="n">temporal</span> <span class="o">=</span> <span class="n">T</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span> <span class="o">@</span> <span class="n">temporal</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">stepper</span><span class="o">.</span><span class="n">step</span><span class="p">(</span><span class="n">error</span><span class="p">)</span>
+
+        <span class="k">return</span> <span class="n">svdtf</span><span class="p">(</span><span class="n">source</span><span class="p">,</span> <span class="n">temporal</span><span class="p">)</span></div></div>
+</pre></div>
+
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+            
+            
+              
+            
+            
+              <p class="caption" role="heading"><span class="caption-text">Contents:</span></p>
+<ul>
+<li class="toctree-l1"><a class="reference internal" href="../../../../lietensor/">LieTensor</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../../../func/">Func</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../../../functions/">Functions</a></li>
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+              
+  <h1>Source code for pypose.module.imu_preintegrator</h1><div class="highlight"><pre>
+<span></span><span class="kn">import</span> <span class="nn">torch</span>
+<span class="kn">from</span> <span class="nn">torch</span> <span class="kn">import</span> <span class="n">nn</span>
+<span class="kn">from</span> <span class="nn">..basics</span> <span class="kn">import</span> <span class="n">cumprod</span>
+<span class="kn">from</span> <span class="nn">..</span> <span class="kn">import</span> <span class="n">LieTensor</span><span class="p">,</span> <span class="n">so3</span><span class="p">,</span> <span class="n">SO3</span>
+<span class="kn">from</span> <span class="nn">..</span> <span class="kn">import</span> <span class="n">identity_SO3</span><span class="p">,</span> <span class="n">vec2skew</span>
+
+
+<div class="viewcode-block" id="IMUPreintegrator"><a class="viewcode-back" href="../../../../generated/pypose.module.IMUPreintegrator/#pypose.module.IMUPreintegrator">[docs]</a><span class="k">class</span> <span class="nc">IMUPreintegrator</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Applies preintegration over IMU input signals.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        pos (``torch.Tensor``, optional): initial position. Default: :obj:`torch.zeros(3)`.</span>
+<span class="sd">        rot (``pypose.SO3``, optional): initial rotation. Default: :meth:`pypose.identity_SO3`.</span>
+<span class="sd">        vel (``torch.Tensor``, optional): initial position. Default: ``torch.zeros(3)``.</span>
+<span class="sd">        gravity (``float``, optional): the gravity acceleration. Default: ``9.81007``.</span>
+<span class="sd">        gyro_cov (``torch.Tensor`` or ``float``, optional): covariance of the gyroscope.</span>
+<span class="sd">            Use a three-element tensor, if the covariance on the three axes are different.</span>
+<span class="sd">            Default: ``(3.2e-3)**2``.</span>
+<span class="sd">        acc_cov (``torch.Tensor``  or ``float``, optional): covariance of the accelerator.</span>
+<span class="sd">            Use a three-element tensor, if the covariance on the three axes are different.</span>
+<span class="sd">            Default: ``(8e-2)**2``.</span>
+<span class="sd">        prop_cov (``bool``, optional): flag to propagate the covariance matrix. Default: ``True``.</span>
+<span class="sd">        reset (``bool``, optional): flag to reset the initial states after the :obj:`forward`</span>
+<span class="sd">            function is called. If ``False``, the integration starts from the last integration.</span>
+<span class="sd">            This flag is ignored if :obj:`init_state` is not ``None``. Default: ``False``.</span>
+
+<span class="sd">    Example:</span>
+
+<span class="sd">        1. Preintegrator Initialization</span>
+
+<span class="sd">        &gt;&gt;&gt; import torch</span>
+<span class="sd">        &gt;&gt;&gt; import pypose as pp</span>
+<span class="sd">        &gt;&gt;&gt; p = torch.zeros(3)    # Initial Position</span>
+<span class="sd">        &gt;&gt;&gt; r = pp.identity_SO3() # Initial rotation</span>
+<span class="sd">        &gt;&gt;&gt; v = torch.zeros(3)    # Initial Velocity</span>
+<span class="sd">        &gt;&gt;&gt; integrator = pp.module.IMUPreintegrator(p, r, v)</span>
+
+<span class="sd">        2. Get IMU measurement</span>
+
+<span class="sd">        &gt;&gt;&gt; ang = torch.tensor([0.1,0.1,0.1]) # angular velocity</span>
+<span class="sd">        &gt;&gt;&gt; acc = torch.tensor([0.1,0.1,0.1]) # acceleration</span>
+<span class="sd">        &gt;&gt;&gt; rot = pp.mat2SO3(torch.eye(3))    # Rotation (Optional)</span>
+<span class="sd">        &gt;&gt;&gt; dt = torch.tensor([0.002])        # Time difference between two measurements</span>
+
+<span class="sd">        3. Preintegrating IMU measurements.</span>
+<span class="sd">        Takes as input the IMU values and calculates the preintegrated IMU measurements.</span>
+
+<span class="sd">        &gt;&gt;&gt; states = integrator(dt, ang, acc, rot)</span>
+<span class="sd">        {&#39;rot&#39;: SO3Type LieTensor:</span>
+<span class="sd">        tensor([[[1.0000e-04, 1.0000e-04, 1.0000e-04, 1.0000e+00]]]),</span>
+<span class="sd">        &#39;vel&#39;: tensor([[[ 0.0002,  0.0002, -0.0194]]]),</span>
+<span class="sd">        &#39;pos&#39;: tensor([[[ 2.0000e-07,  2.0000e-07, -1.9420e-05]]]),</span>
+<span class="sd">        &#39;cov&#39;: tensor([[[ 5.7583e-11, -5.6826e-19, -5.6827e-19,  0.0000e+00,  0.0000e+00,</span>
+<span class="sd">                          0.0000e+00,  0.0000e+00,  0.0000e+00,  0.0000e+00],</span>
+<span class="sd">                        [-5.6826e-19,  5.7583e-11, -5.6827e-19,  0.0000e+00,  0.0000e+00,</span>
+<span class="sd">                          0.0000e+00,  0.0000e+00,  0.0000e+00,  0.0000e+00],</span>
+<span class="sd">                        [-5.6827e-19, -5.6827e-19,  5.7583e-11,  0.0000e+00,  0.0000e+00,</span>
+<span class="sd">                          0.0000e+00,  0.0000e+00,  0.0000e+00,  0.0000e+00],</span>
+<span class="sd">                        [ 0.0000e+00,  0.0000e+00,  0.0000e+00,  8.0000e-09, -3.3346e-20,</span>
+<span class="sd">                          -1.0588e-19,  8.0000e-12,  1.5424e-23, -1.0340e-22],</span>
+<span class="sd">                        [ 0.0000e+00,  0.0000e+00,  0.0000e+00, -1.3922e-19,  8.0000e-09,</span>
+<span class="sd">                          0.0000e+00, -8.7974e-23,  8.0000e-12,  0.0000e+00],</span>
+<span class="sd">                        [ 0.0000e+00,  0.0000e+00,  0.0000e+00,  0.0000e+00, -1.0588e-19,</span>
+<span class="sd">                          8.0000e-09,  0.0000e+00, -1.0340e-22,  8.0000e-12],</span>
+<span class="sd">                        [ 0.0000e+00,  0.0000e+00,  0.0000e+00,  8.0000e-12,  1.5424e-23,</span>
+<span class="sd">                          -1.0340e-22,  8.0000e-15, -1.2868e-26,  0.0000e+00],</span>
+<span class="sd">                        [ 0.0000e+00,  0.0000e+00,  0.0000e+00, -8.7974e-23,  8.0000e-12,</span>
+<span class="sd">                          0.0000e+00, -1.2868e-26,  8.0000e-15,  0.0000e+00],</span>
+<span class="sd">                        [ 0.0000e+00,  0.0000e+00,  0.0000e+00,  0.0000e+00, -1.0340e-22,</span>
+<span class="sd">                          8.0000e-12,  0.0000e+00,  0.0000e+00,  8.0000e-15]]])}</span>
+
+<span class="sd">    Preintegrated IMU odometry from the KITTI dataset with and without known rotation.</span>
+
+<span class="sd">    .. list-table::</span>
+
+<span class="sd">        * - .. figure:: /_static/img/module/imu/imu-known-rot.png</span>
+<span class="sd">                :width: 300</span>
+
+<span class="sd">            Fig. 1. Known Rotation.</span>
+
+<span class="sd">          - .. figure:: /_static/img/module/imu/imu-unknown-rot.png</span>
+<span class="sd">                :width: 300</span>
+
+<span class="sd">            Fig. 2. Estimated Rotation.</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        The examples generating the above figures can be found at `examples/module/imu</span>
+<span class="sd">        &lt;https://github.com/pypose/pypose/tree/main/examples/module/imu&gt;`_.</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">pos</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="mi">3</span><span class="p">),</span>
+                       <span class="n">rot</span> <span class="o">=</span> <span class="n">identity_SO3</span><span class="p">(),</span>
+                       <span class="n">vel</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="mi">3</span><span class="p">),</span>
+                       <span class="n">gravity</span> <span class="o">=</span> <span class="mf">9.81007</span><span class="p">,</span>
+                       <span class="n">gyro_cov</span> <span class="o">=</span> <span class="p">(</span><span class="mf">3.2e-3</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span>
+                       <span class="n">acc_cov</span> <span class="o">=</span> <span class="p">(</span><span class="mf">8e-2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span>
+                       <span class="n">prop_cov</span> <span class="o">=</span> <span class="kc">True</span><span class="p">,</span>
+                       <span class="n">reset</span> <span class="o">=</span> <span class="kc">False</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">reset</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">prop_cov</span> <span class="o">=</span> <span class="n">reset</span><span class="p">,</span> <span class="n">prop_cov</span>
+
+        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">acc_cov</span><span class="p">,</span> <span class="nb">float</span><span class="p">):</span>
+            <span class="n">acc_cov</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([[</span><span class="n">acc_cov</span><span class="p">,</span> <span class="n">acc_cov</span><span class="p">,</span> <span class="n">acc_cov</span><span class="p">]])</span>
+        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">gyro_cov</span><span class="p">,</span> <span class="nb">float</span><span class="p">):</span>
+            <span class="n">gyro_cov</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([[</span><span class="n">gyro_cov</span><span class="p">,</span> <span class="n">gyro_cov</span><span class="p">,</span> <span class="n">gyro_cov</span><span class="p">]])</span>
+
+        <span class="c1"># Initial status of IMU: (pos)ition, (rot)ation, (vel)ocity, (cov)ariance</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">register_buffer</span><span class="p">(</span><span class="s1">&#39;gravity&#39;</span><span class="p">,</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">gravity</span><span class="p">]),</span> <span class="n">persistent</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">register_buffer</span><span class="p">(</span><span class="s1">&#39;pos&#39;</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_check</span><span class="p">(</span><span class="n">pos</span><span class="p">)</span><span class="o">.</span><span class="n">clone</span><span class="p">(),</span> <span class="n">persistent</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">register_buffer</span><span class="p">(</span><span class="s1">&#39;rot&#39;</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_check</span><span class="p">(</span><span class="n">rot</span><span class="p">)</span><span class="o">.</span><span class="n">clone</span><span class="p">(),</span> <span class="n">persistent</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">register_buffer</span><span class="p">(</span><span class="s1">&#39;vel&#39;</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_check</span><span class="p">(</span><span class="n">vel</span><span class="p">)</span><span class="o">.</span><span class="n">clone</span><span class="p">(),</span> <span class="n">persistent</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">register_buffer</span><span class="p">(</span><span class="s1">&#39;cov&#39;</span><span class="p">,</span> <span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">9</span><span class="p">,</span> <span class="mi">9</span><span class="p">),</span> <span class="n">persistent</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">register_buffer</span><span class="p">(</span><span class="s1">&#39;gyro_cov&#39;</span><span class="p">,</span> <span class="n">gyro_cov</span><span class="p">,</span> <span class="n">persistent</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">register_buffer</span><span class="p">(</span><span class="s1">&#39;acc_cov&#39;</span><span class="p">,</span> <span class="n">acc_cov</span><span class="p">,</span> <span class="n">persistent</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">Rij</span> <span class="o">=</span> <span class="kc">None</span> <span class="c1"># rotation corresponding to the &quot;zero-state&quot; covariance Sigma_ii.</span>
+
+    <span class="k">def</span> <span class="nf">_check</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">obj</span><span class="p">):</span>
+        <span class="k">if</span> <span class="n">obj</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
+            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">obj</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
+                <span class="n">obj</span> <span class="o">=</span> <span class="n">obj</span><span class="p">[</span><span class="kc">None</span><span class="p">,</span> <span class="o">...</span><span class="p">]</span>
+
+            <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">obj</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
+                <span class="n">obj</span> <span class="o">=</span> <span class="n">obj</span><span class="p">[</span><span class="kc">None</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</span> <span class="o">...</span><span class="p">]</span>
+        <span class="k">return</span> <span class="n">obj</span>
+
+<div class="viewcode-block" id="IMUPreintegrator.forward"><a class="viewcode-back" href="../../../../generated/pypose.module.IMUPreintegrator/#pypose.module.IMUPreintegrator.forward">[docs]</a>    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">gyro</span><span class="p">,</span> <span class="n">acc</span><span class="p">,</span> <span class="n">rot</span><span class="p">:</span><span class="n">SO3</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">gyro_cov</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">acc_cov</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">init_state</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Propagate IMU states from duration (:math:`\delta t`), angular rate</span>
+<span class="sd">        (:math:`\omega`), linear acceleration (:math:`\mathbf{a}`) in body frame, as well as</span>
+<span class="sd">        their measurement covariance for gyroscope :math:`C_{g}` and acceleration</span>
+<span class="sd">        :math:`C_{\mathbf{a}}`. Known IMU rotation estimation :math:`R` can be provided for</span>
+<span class="sd">        better precision.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            dt (``torch.Tensor``): time interval from last update.</span>
+<span class="sd">            gyro (``torch.Tensor``): angular rate (:math:`\omega`) in IMU body frame.</span>
+<span class="sd">            acc (``torch.Tensor``): linear acceleration (:math:`\mathbf{a}`) in IMU body frame</span>
+<span class="sd">                (raw sensor input with gravity).</span>
+<span class="sd">            rot (:obj:`pypose.SO3`, optional): known IMU rotation on the body frame.</span>
+<span class="sd">            gyro_cov (``torch.Tensor``, optional): covariance matrix of angular rate.</span>
+<span class="sd">                If not given, the default state in constructor will be used.</span>
+<span class="sd">            acc_cov (``torch.Tensor``, optional): covariance matrix of linear acceleration.</span>
+<span class="sd">                If not given, the default state in constructor will be used.</span>
+<span class="sd">            init_state (``dict``, optional): the initial state of the integration. The dictionary</span>
+<span class="sd">                should be in form of :obj:`{&#39;pos&#39;: torch.Tensor, &#39;rot&#39;: pypose.SO3, &#39;vel&#39;:</span>
+<span class="sd">                torch.Tensor}`. If not given, the initial state in constructor will be used.</span>
+
+<span class="sd">        Shape:</span>
+<span class="sd">            - input (:obj:`dt`, :obj:`gyro`, :obj:`acc`): This layer supports the input shape with</span>
+<span class="sd">              :math:`(B, F, H_{in})`, :math:`(F, H_{in})` and :math:`(H_{in})`, where :math:`B` is</span>
+<span class="sd">              the batch size (or the number of IMU), :math:`F` is the number of frames</span>
+<span class="sd">              (measurements), and :math:`H_{in}` is the raw sensor signals.</span>
+
+<span class="sd">            - init_state (Optional): The initial state of the integration. It contains</span>
+<span class="sd">              :code:`pos`: initial position, :code:`rot`: initial rotation, :code:`vel`: initial</span>
+<span class="sd">              velocity, with the shape :math:`(B, H_{in})`.</span>
+
+<span class="sd">            - output: a :obj:`dict` of integrated state including ``pos``: position,</span>
+<span class="sd">              ``rot``: rotation, and ``vel``: velocity, each of which has a shape</span>
+<span class="sd">              :math:`(B, F, H_{out})`, where :math:`H_{out}` is the signal dimension.</span>
+<span class="sd">              If :obj:`prop_cov` is ``True``, it will also include ``cov``: covariance</span>
+<span class="sd">              matrix in shape of :math:`(B, 9, 9)`.</span>
+
+<span class="sd">        IMU Measurements Integration:</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \begin{align*}</span>
+<span class="sd">                {\Delta}R_{ik+1} &amp;= {\Delta}R_{ik} \mathrm{Exp} (w_k {\Delta}t) \\</span>
+<span class="sd">                {\Delta}v_{ik+1} &amp;= {\Delta}v_{ik} + {\Delta}R_{ik} a_k {\Delta}t \\</span>
+<span class="sd">                {\Delta}p_{ik+1} &amp;= {\Delta}p_{ik} + {\Delta}v_{ik} {\Delta}t</span>
+<span class="sd">                    + \frac{1}{2} {\Delta}R_{ik} a_k {\Delta}t^2</span>
+<span class="sd">            \end{align*}</span>
+
+<span class="sd">        where:</span>
+
+<span class="sd">            - :math:`{\Delta}R_{ik}` is the preintegrated rotation between the :math:`i`-th</span>
+<span class="sd">              and :math:`k`-th time step.</span>
+
+<span class="sd">            - :math:`{\Delta}v_{ik}` is the preintegrated velocity between the :math:`i`-th</span>
+<span class="sd">              and :math:`k`-th time step.</span>
+
+<span class="sd">            - :math:`{\Delta}p_{ik}` is the preintegrated position between the :math:`i`-th</span>
+<span class="sd">              and :math:`k`-th time step.</span>
+
+<span class="sd">            - :math:`a_k` is linear acceleration at the :math:`k`-th time step.</span>
+
+<span class="sd">            - :math:`w_k` is angular rate at the :math:`k`-th time step.</span>
+
+<span class="sd">            - :math:`{\Delta}t` is the time interval from time step :math:`k`-th to time</span>
+<span class="sd">              step :math:`{k+1}`-th time step.</span>
+
+<span class="sd">        Uncertainty Propagation:</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \begin{align*}</span>
+<span class="sd">                C_{ik+1} &amp;= A C_{ik} A^T + B \mathrm{diag}(C_g, C_a) B^T \\</span>
+<span class="sd">                  &amp;= A C A^T + B_g C_g B_g^T + B_a C_a B_a^T</span>
+<span class="sd">            \end{align*},</span>
+
+<span class="sd">        where</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            A = \begin{bmatrix}</span>
+<span class="sd">                  {\Delta}R_{ik+1}^T &amp; 0_{3*3} \\</span>
+<span class="sd">                  -{\Delta}R_{ik} (a_k^{\wedge}) {\Delta}t &amp; I_{3*3} &amp; 0_{3*3} \\</span>
+<span class="sd">                  -1/2{\Delta}R_{ik} (a_k^{\wedge}) {\Delta}t^2 &amp; I_{3*3} {\Delta}t &amp; I_{3*3}</span>
+<span class="sd">                \end{bmatrix},</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            B = [B_g, B_a] \\</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            B_g = \begin{bmatrix}</span>
+<span class="sd">                    J_r^k \Delta t  \\</span>
+<span class="sd">                    0_{3*3}  \\</span>
+<span class="sd">                    0_{3*3}</span>
+<span class="sd">                \end{bmatrix},</span>
+
+<span class="sd">            B_a = \begin{bmatrix}</span>
+<span class="sd">                    0_{3*3} \\</span>
+<span class="sd">                    {\Delta}R_{ik} {\Delta}t  \\</span>
+<span class="sd">                    1/2 {\Delta}R_{ik} {\Delta}t^2</span>
+<span class="sd">                \end{bmatrix},</span>
+
+<span class="sd">        where :math:`\cdot^\wedge` is the skew matrix (:meth:`pypose.vec2skew`),</span>
+<span class="sd">        :math:`C \in\mathbf{R}^{9\times 9}` is the covariance matrix,</span>
+<span class="sd">        and :math:`J_r^k` is the right jacobian (:meth:`pypose.Jr`) of integrated rotation</span>
+<span class="sd">        :math:`\mathrm{Exp}(w_k{\Delta}t)` at :math:`k`-th time step,</span>
+<span class="sd">        :math:`C_{g}` and :math:`C_{\mathbf{a}}` are measurement covariance of angular rate</span>
+<span class="sd">        and acceleration, respectively.</span>
+
+<span class="sd">        Note:</span>
+<span class="sd">            Output covariance (Shape: :math:`(B, 9, 9)`) is in the order of rotation, velocity,</span>
+<span class="sd">            and position.</span>
+
+<span class="sd">        With IMU preintegration, the propagated IMU status:</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \begin{align*}</span>
+<span class="sd">                R_j &amp;= {\Delta}R_{ij} * R_i                                                     \\</span>
+<span class="sd">                v_j &amp;= R_i * {\Delta}v_{ij}   + v_i + g \Delta t_{ij}                           \\</span>
+<span class="sd">                p_j &amp;= R_i * {\Delta}p_{ij}   + p_i + v_i \Delta t_{ij} + 1/2 g \Delta t_{ij}^2 \\</span>
+<span class="sd">            \end{align*}</span>
+
+<span class="sd">        where:</span>
+
+<span class="sd">            - :math:`{\Delta}R_{ij}`, :math:`{\Delta}v_{ij}`, :math:`{\Delta}p_{ij}`</span>
+<span class="sd">              are the preintegrated measurements.</span>
+<span class="sd">            - :math:`R_i`, :math:`v_i`, and :math:`p_i` are the initial state. Default initial values</span>
+<span class="sd">              are used if :obj:`reset` is True.</span>
+<span class="sd">            - :math:`R_j`, :math:`v_j`, and :math:`p_j` are the propagated state variables.</span>
+<span class="sd">            - :math:`{\Delta}t_{ij}` is the time interval from frame i to j.</span>
+
+<span class="sd">        Note:</span>
+<span class="sd">            The implementation is based on Eq. (A7), (A8), (A9), and (A10) of this report:</span>
+
+<span class="sd">            * Christian Forster, et al., `IMU Preintegration on Manifold for Efficient Visual-Inertial</span>
+<span class="sd">              Maximum-a-posteriori Estimation</span>
+<span class="sd">              &lt;https://rpg.ifi.uzh.ch/docs/RSS15_Forster_Supplementary.pdf&gt;`_, Technical Report</span>
+<span class="sd">              GT-IRIM-CP&amp;R-2015-001, 2015.</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">assert</span><span class="p">(</span><span class="mi">0</span> <span class="o">&lt;</span> <span class="nb">len</span><span class="p">(</span><span class="n">acc</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">dt</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">gyro</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="mi">3</span><span class="p">)</span>
+        <span class="n">acc</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_check</span><span class="p">(</span><span class="n">acc</span><span class="p">);</span> <span class="n">gyro</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_check</span><span class="p">(</span><span class="n">gyro</span><span class="p">)</span>
+        <span class="n">dt</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_check</span><span class="p">(</span><span class="n">dt</span><span class="p">);</span> <span class="n">rot</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_check</span><span class="p">(</span><span class="n">rot</span><span class="p">)</span>
+        <span class="n">B</span> <span class="o">=</span> <span class="n">dt</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
+
+        <span class="k">if</span> <span class="n">init_state</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
+            <span class="n">init_state</span> <span class="o">=</span> <span class="p">{</span><span class="s1">&#39;pos&#39;</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">pos</span><span class="p">,</span> <span class="s1">&#39;rot&#39;</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">rot</span><span class="p">,</span> <span class="s1">&#39;vel&#39;</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">vel</span><span class="p">}</span>
+
+        <span class="n">inte_state</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">integrate</span><span class="p">(</span><span class="n">dt</span><span class="p">,</span> <span class="n">gyro</span><span class="p">,</span> <span class="n">acc</span><span class="p">,</span> <span class="n">rot</span> <span class="o">=</span> <span class="n">rot</span><span class="p">,</span> <span class="n">init_rot</span> <span class="o">=</span> <span class="n">init_state</span><span class="p">[</span><span class="s1">&#39;rot&#39;</span><span class="p">])</span>
+        <span class="n">predict</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">init_state</span><span class="p">,</span> <span class="n">inte_state</span><span class="p">)</span>
+
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">prop_cov</span><span class="p">:</span>
+            <span class="k">if</span> <span class="n">gyro_cov</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
+                <span class="n">gyro_cov</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">gyro_cov</span><span class="o">.</span><span class="n">repeat</span><span class="p">([</span><span class="n">B</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
+            <span class="k">if</span> <span class="n">acc_cov</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
+                <span class="n">acc_cov</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">acc_cov</span><span class="o">.</span><span class="n">repeat</span><span class="p">([</span><span class="n">B</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
+            <span class="k">if</span> <span class="s1">&#39;cov&#39;</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">init_state</span> <span class="ow">or</span> <span class="n">init_state</span><span class="p">[</span><span class="s1">&#39;cov&#39;</span><span class="p">]</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
+                <span class="n">init_cov</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">cov</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="mi">9</span><span class="p">,</span> <span class="mi">9</span><span class="p">)</span>
+            <span class="k">else</span><span class="p">:</span>
+                <span class="n">init_cov</span> <span class="o">=</span> <span class="n">init_state</span><span class="p">[</span><span class="s1">&#39;cov&#39;</span><span class="p">]</span>
+            <span class="k">if</span> <span class="s1">&#39;Rij&#39;</span> <span class="ow">in</span> <span class="n">init_state</span><span class="p">:</span>
+                <span class="n">Rij</span> <span class="o">=</span> <span class="n">init_state</span><span class="p">[</span><span class="s1">&#39;Rij&#39;</span><span class="p">]</span>
+            <span class="k">else</span><span class="p">:</span>
+                <span class="n">Rij</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">Rij</span> <span class="c1"># default is None</span>
+
+            <span class="k">if</span> <span class="n">Rij</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
+                <span class="n">Rij</span> <span class="o">=</span> <span class="n">Rij</span> <span class="o">*</span> <span class="n">inte_state</span><span class="p">[</span><span class="s1">&#39;Dr&#39;</span><span class="p">]</span>
+            <span class="k">else</span><span class="p">:</span>
+                <span class="n">Rij</span> <span class="o">=</span> <span class="n">inte_state</span><span class="p">[</span><span class="s1">&#39;Dr&#39;</span><span class="p">]</span>
+
+            <span class="n">cov_input_state</span> <span class="o">=</span><span class="p">{</span>
+                <span class="s1">&#39;Rij&#39;</span><span class="p">:</span> <span class="n">Rij</span><span class="o">.</span><span class="n">detach</span><span class="p">(),</span>
+                <span class="s1">&#39;Rk&#39;</span><span class="p">:</span> <span class="n">inte_state</span><span class="p">[</span><span class="s1">&#39;w&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">detach</span><span class="p">(),</span>
+                <span class="s1">&#39;Ha&#39;</span><span class="p">:</span> <span class="n">vec2skew</span><span class="p">(</span><span class="n">inte_state</span><span class="p">[</span><span class="s1">&#39;a&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">detach</span><span class="p">()),</span>
+                <span class="s1">&#39;dt&#39;</span><span class="p">:</span> <span class="n">dt</span><span class="o">.</span><span class="n">detach</span><span class="p">()</span>
+            <span class="p">}</span>
+            <span class="n">cov</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">propagate_cov</span><span class="p">(</span><span class="n">cov_input</span> <span class="o">=</span> <span class="n">cov_input_state</span><span class="p">,</span> <span class="n">init_cov</span> <span class="o">=</span> <span class="n">init_cov</span><span class="p">,</span>
+                                    <span class="n">gyro_cov</span> <span class="o">=</span> <span class="n">gyro_cov</span><span class="p">,</span> <span class="n">acc_cov</span> <span class="o">=</span> <span class="n">acc_cov</span><span class="p">)</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">cov</span> <span class="o">=</span> <span class="p">{</span><span class="s1">&#39;cov&#39;</span><span class="p">:</span> <span class="kc">None</span><span class="p">}</span>
+
+        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">reset</span><span class="p">:</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">pos</span> <span class="o">=</span> <span class="n">predict</span><span class="p">[</span><span class="s1">&#39;pos&#39;</span><span class="p">][</span><span class="o">...</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">:,</span> <span class="p">:]</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">rot</span> <span class="o">=</span> <span class="n">predict</span><span class="p">[</span><span class="s1">&#39;rot&#39;</span><span class="p">][</span><span class="o">...</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">:,</span> <span class="p">:]</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">vel</span> <span class="o">=</span> <span class="n">predict</span><span class="p">[</span><span class="s1">&#39;vel&#39;</span><span class="p">][</span><span class="o">...</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">:,</span> <span class="p">:]</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">cov</span> <span class="o">=</span> <span class="n">cov</span><span class="p">[</span><span class="s1">&#39;cov&#39;</span><span class="p">]</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">Rij</span> <span class="o">=</span> <span class="n">Rij</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">:,</span> <span class="p">:]</span>
+
+        <span class="k">return</span> <span class="p">{</span><span class="o">**</span><span class="n">predict</span><span class="p">,</span> <span class="o">**</span><span class="n">cov</span><span class="p">}</span></div>
+
+<div class="viewcode-block" id="IMUPreintegrator.integrate"><a class="viewcode-back" href="../../../../generated/pypose.module.IMUPreintegrator/#pypose.module.IMUPreintegrator.integrate">[docs]</a>    <span class="k">def</span> <span class="nf">integrate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">gyro</span><span class="p">,</span> <span class="n">acc</span><span class="p">,</span> <span class="n">rot</span><span class="p">:</span><span class="n">SO3</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">init_rot</span><span class="p">:</span><span class="n">SO3</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Integrate the IMU sensor signals gyroscope (angular rate</span>
+<span class="sd">        :math:`\omega`), linear acceleration (:math:`\mathbf{a}`) in body frame to</span>
+<span class="sd">        calculate the increments on the rotation (:math:`\Delta r`), velocity (:math:`\Delta v`)</span>
+<span class="sd">        and position (:math:`\Delta p`) of the IMU states.</span>
+<span class="sd">        The IMU rotation of the body frame (:code:`\rot`) is optional,</span>
+<span class="sd">        which can be utilized to compensate the gravity.</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \begin{align*}</span>
+<span class="sd">                {\Delta}r_{ij} &amp;= \int_{t \in [t_i, t_j]} w_k \ dt \\</span>
+<span class="sd">                {\Delta}v_{ij} &amp;= \int_{t \in [t_i, t_j]} R_{k} a_k \ dt \\</span>
+<span class="sd">                {\Delta}p_{ij} &amp;= \int\int_{t \in [t_i, t_j]} R_{k} a_k\ dt^2\  \\</span>
+<span class="sd">            \end{align*}</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            dt (``torch.Tensor``): time interval from last update.</span>
+<span class="sd">            gyro (torch.Tensor): angular rate (:math:`\omega`) in IMU body frame.</span>
+<span class="sd">            acc (``torch.Tensor``): linear acceleration (:math:`\mathbf{a}`) in IMU body frame</span>
+<span class="sd">                (raw sensor input with gravity).</span>
+<span class="sd">            rot (:obj:`pypose.SO3`, optional): known IMU rotation on the body frame.</span>
+<span class="sd">            init_rot (:obj:`pypose.SO3`, optional): the initial orientation of the IMU state.</span>
+<span class="sd">                If not given, the initial state in constructor will be used.</span>
+
+<span class="sd">        Shape:</span>
+<span class="sd">            - input (:obj:`dt`, :obj:`gyro`, :obj:`acc`): This layer supports the input shape</span>
+<span class="sd">              with :math:`(B, F, H_{in})`, :math:`(F, H_{in})` and :math:`(H_{in})`, where</span>
+<span class="sd">              :math:`B` is the batch size (or the number of IMU), :math:`F` is the number of</span>
+<span class="sd">              frames (measurements), and :math:`H_{in}` is the raw sensor signals.</span>
+
+<span class="sd">            - init_rot: The initial orientation of the integration, which helps to</span>
+<span class="sd">              compensate for the gravity. It contains the shape :math:`(B, H_{in})`. d</span>
+
+<span class="sd">            - rot: The ground truth orientation of the integration. If this parameter is</span>
+<span class="sd">              given, the integrator will use the ground truth orientation to compensate the</span>
+<span class="sd">              gravity.</span>
+
+<span class="sd">        Return:</span>
+<span class="sd">            ``dict``: integrated states including ``a``: acceleration in the body frame</span>
+<span class="sd">            without gravity ``Dp``: position increments, ``Dr``: rotation increments,</span>
+<span class="sd">            ``Dv``: velocity increments, ``w``: rotation velocity and ``Dt``: time increments,</span>
+<span class="sd">            each of which has a shape :math:`(B, F, H_{out})`, where :math:`H_{out}` is the</span>
+<span class="sd">            signal dimension.</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">B</span><span class="p">,</span> <span class="n">F</span> <span class="o">=</span> <span class="n">dt</span><span class="o">.</span><span class="n">shape</span><span class="p">[:</span><span class="mi">2</span><span class="p">]</span>
+        <span class="n">dr</span> <span class="o">=</span>  <span class="n">so3</span><span class="p">(</span><span class="n">gyro</span><span class="o">*</span><span class="n">dt</span><span class="p">)</span><span class="o">.</span><span class="n">Exp</span><span class="p">()</span>
+        <span class="n">w</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">cat</span><span class="p">([</span><span class="n">identity_SO3</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">dt</span><span class="o">.</span><span class="n">dtype</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="n">dt</span><span class="o">.</span><span class="n">device</span><span class="p">),</span> <span class="n">dr</span><span class="p">],</span> <span class="n">dim</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+        <span class="n">incre_r</span> <span class="o">=</span> <span class="n">cumprod</span><span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="n">dim</span> <span class="o">=</span> <span class="mi">1</span><span class="p">,</span> <span class="n">left</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+
+        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">rot</span><span class="p">,</span> <span class="n">LieTensor</span><span class="p">):</span>
+            <span class="n">a</span> <span class="o">=</span> <span class="n">acc</span> <span class="o">-</span> <span class="n">rot</span><span class="o">.</span><span class="n">Inv</span><span class="p">()</span> <span class="o">@</span> <span class="bp">self</span><span class="o">.</span><span class="n">gravity</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="k">if</span> <span class="n">init_rot</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
+                <span class="n">init_rot</span> <span class="o">=</span> <span class="n">identity_SO3</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">dt</span><span class="o">.</span><span class="n">dtype</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="n">dt</span><span class="o">.</span><span class="n">device</span><span class="p">)</span>
+            <span class="n">inte_rot</span> <span class="o">=</span> <span class="n">init_rot</span> <span class="o">*</span> <span class="n">incre_r</span>
+            <span class="n">a</span> <span class="o">=</span> <span class="n">acc</span> <span class="o">-</span> <span class="n">inte_rot</span><span class="p">[:,</span><span class="mi">1</span><span class="p">:,:]</span><span class="o">.</span><span class="n">Inv</span><span class="p">()</span> <span class="o">@</span> <span class="bp">self</span><span class="o">.</span><span class="n">gravity</span>
+
+        <span class="n">dv</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">dt</span><span class="o">.</span><span class="n">dtype</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="n">dt</span><span class="o">.</span><span class="n">device</span><span class="p">)</span>
+        <span class="n">dv</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">cat</span><span class="p">([</span><span class="n">dv</span><span class="p">,</span> <span class="n">incre_r</span><span class="p">[:,:</span><span class="n">F</span><span class="p">,:]</span> <span class="o">@</span> <span class="n">a</span> <span class="o">*</span> <span class="n">dt</span><span class="p">],</span> <span class="n">dim</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+        <span class="n">incre_v</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">cumsum</span><span class="p">(</span><span class="n">dv</span><span class="p">,</span> <span class="n">dim</span> <span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+
+        <span class="n">dp</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">dt</span><span class="o">.</span><span class="n">dtype</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="n">dt</span><span class="o">.</span><span class="n">device</span><span class="p">)</span>
+        <span class="n">dp</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">cat</span><span class="p">([</span><span class="n">dp</span><span class="p">,</span> <span class="n">incre_v</span><span class="p">[:,:</span><span class="n">F</span><span class="p">,:]</span> <span class="o">*</span> <span class="n">dt</span> <span class="o">+</span> <span class="n">incre_r</span><span class="p">[:,:</span><span class="n">F</span><span class="p">,:]</span> <span class="o">@</span> <span class="n">a</span> <span class="o">*</span> <span class="mf">0.5</span> <span class="o">*</span> <span class="n">dt</span><span class="o">**</span><span class="mi">2</span><span class="p">],</span> <span class="n">dim</span> <span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+        <span class="n">incre_p</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">cumsum</span><span class="p">(</span><span class="n">dp</span><span class="p">,</span> <span class="n">dim</span> <span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+
+        <span class="n">incre_t</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">cumsum</span><span class="p">(</span><span class="n">dt</span><span class="p">,</span> <span class="n">dim</span> <span class="o">=</span> <span class="mi">1</span><span class="p">)</span>
+        <span class="n">incre_t</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">cat</span><span class="p">([</span><span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">dt</span><span class="o">.</span><span class="n">dtype</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="n">dt</span><span class="o">.</span><span class="n">device</span><span class="p">),</span> <span class="n">incre_t</span><span class="p">],</span> <span class="n">dim</span> <span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+
+        <span class="k">return</span> <span class="p">{</span><span class="s1">&#39;a&#39;</span><span class="p">:</span><span class="n">a</span><span class="p">,</span> <span class="s1">&#39;Dp&#39;</span><span class="p">:</span><span class="n">incre_p</span><span class="p">[:,</span><span class="mi">1</span><span class="p">:,:],</span> <span class="s1">&#39;Dv&#39;</span><span class="p">:</span><span class="n">incre_v</span><span class="p">[</span><span class="o">...</span><span class="p">,</span><span class="mi">1</span><span class="p">:,:],</span> <span class="s1">&#39;Dr&#39;</span><span class="p">:</span><span class="n">incre_r</span><span class="p">[:,</span><span class="mi">1</span><span class="p">:,:],</span>
+                <span class="s1">&#39;Dt&#39;</span><span class="p">:</span><span class="n">incre_t</span><span class="p">[</span><span class="o">...</span><span class="p">,</span><span class="mi">1</span><span class="p">:,:],</span> <span class="s1">&#39;w&#39;</span><span class="p">:</span> <span class="n">w</span><span class="p">[:,</span><span class="mi">1</span><span class="p">:,:]}</span></div>
+
+<div class="viewcode-block" id="IMUPreintegrator.predict"><a class="viewcode-back" href="../../../../generated/pypose.module.IMUPreintegrator/#pypose.module.IMUPreintegrator.predict">[docs]</a>    <span class="nd">@classmethod</span>
+    <span class="k">def</span> <span class="nf">predict</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="n">init_state</span><span class="p">,</span> <span class="n">integrate</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Propogate the next IMU state from the initial IMU state (:obj:`init_state`) with</span>
+<span class="sd">        the preintegrated IMU measurements (:math:`\Delta{p}`, :math:`\Delta{v}` and</span>
+<span class="sd">        :math:`\Delta{r}`).</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \begin{align*}</span>
+<span class="sd">                R_j &amp;= {\Delta}R_{ij} * R_i                                                     \\</span>
+<span class="sd">                v_j &amp;= R_i * {\Delta}v_{ij}   + v_i + g \Delta t_{ij}                           \\</span>
+<span class="sd">                p_j &amp;= R_i * {\Delta}p_{ij}   + p_i + v_i \Delta t_{ij} + 1/2 g \Delta t_{ij}^2 \\</span>
+<span class="sd">            \end{align*}</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            init_state (``dict``): the initial state of the integration. The dictionary</span>
+<span class="sd">                should be in form of :obj:`{&#39;pos&#39;: torch.Tensor, &#39;rot&#39;: pypose.SO3, &#39;vel&#39;:</span>
+<span class="sd">                torch.Tensor}`.</span>
+<span class="sd">            integrate (``dict``): the preintegrated IMU measurements. The dictionary</span>
+<span class="sd">                should be in form of :obj:`{&#39;Dp&#39;: torch.Tensor, &#39;Dr&#39;: pypose.SO3, &#39;Dv&#39;:</span>
+<span class="sd">                torch.Tensor, &#39;Dt&#39;: torch.Tensor}`.</span>
+
+<span class="sd">        Shape:</span>
+<span class="sd">            - init_state: The initial state of the integration. It contains :code:`pos`: initial</span>
+<span class="sd">              position, :code:`rot`: initial rotation, :code:`vel`: initial velocity, with the</span>
+<span class="sd">              shape :math:`(B, H_{in})`.</span>
+
+<span class="sd">            - integrate: The preintegrated IMU measurements. It contains :obj:`Dp`, :obj:`Dv`,</span>
+<span class="sd">              :obj:`Dr`, and :obj:`Dt`, with the shape :math:`(B, F, H_{out})`. It follows the</span>
+<span class="sd">              output of the function :obj:`integrate`.</span>
+
+<span class="sd">        Return:</span>
+<span class="sd">            ``dict``: integrated states including ``pos``: position, ``rot``: rotation, and</span>
+<span class="sd">            ``vel``: velocity, each of which has a shape :math:`(B, F, H_{out})`, where</span>
+<span class="sd">            :math:`H_{out}` is the signal dimension.</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">return</span> <span class="p">{</span>
+            <span class="s1">&#39;rot&#39;</span><span class="p">:</span> <span class="n">init_state</span><span class="p">[</span><span class="s1">&#39;rot&#39;</span><span class="p">]</span> <span class="o">*</span> <span class="n">integrate</span><span class="p">[</span><span class="s1">&#39;Dr&#39;</span><span class="p">],</span>
+            <span class="s1">&#39;vel&#39;</span><span class="p">:</span> <span class="n">init_state</span><span class="p">[</span><span class="s1">&#39;vel&#39;</span><span class="p">]</span> <span class="o">+</span> <span class="n">init_state</span><span class="p">[</span><span class="s1">&#39;rot&#39;</span><span class="p">]</span> <span class="o">*</span> <span class="n">integrate</span><span class="p">[</span><span class="s1">&#39;Dv&#39;</span><span class="p">],</span>
+            <span class="s1">&#39;pos&#39;</span><span class="p">:</span> <span class="n">init_state</span><span class="p">[</span><span class="s1">&#39;pos&#39;</span><span class="p">]</span> <span class="o">+</span> <span class="n">init_state</span><span class="p">[</span><span class="s1">&#39;rot&#39;</span><span class="p">]</span> <span class="o">*</span> <span class="n">integrate</span><span class="p">[</span><span class="s1">&#39;Dp&#39;</span><span class="p">]</span> <span class="o">+</span> <span class="n">init_state</span><span class="p">[</span><span class="s1">&#39;vel&#39;</span><span class="p">]</span> <span class="o">*</span> <span class="n">integrate</span><span class="p">[</span><span class="s1">&#39;Dt&#39;</span><span class="p">],</span>
+        <span class="p">}</span></div>
+
+    <span class="nd">@classmethod</span>
+    <span class="k">def</span> <span class="nf">propagate_cov</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="n">cov_input</span><span class="p">,</span> <span class="n">init_cov</span><span class="p">,</span> <span class="n">gyro_cov</span><span class="p">,</span> <span class="n">acc_cov</span><span class="p">):</span>
+        <span class="c1">## The input acc_cov and gyro cov should be a vector of 3</span>
+        <span class="n">B</span><span class="p">,</span> <span class="n">F</span> <span class="o">=</span> <span class="n">cov_input</span><span class="p">[</span><span class="s1">&#39;dt&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span><span class="p">[:</span><span class="mi">2</span><span class="p">]</span>
+        <span class="n">device</span> <span class="o">=</span> <span class="n">cov_input</span><span class="p">[</span><span class="s1">&#39;dt&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">device</span><span class="p">;</span> <span class="n">dtype</span> <span class="o">=</span> <span class="n">cov_input</span><span class="p">[</span><span class="s1">&#39;dt&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">dtype</span>
+
+        <span class="n">Cg</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">diag_embed</span><span class="p">(</span><span class="n">gyro_cov</span><span class="p">)</span>
+        <span class="n">Ca</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">diag_embed</span><span class="p">(</span><span class="n">acc_cov</span><span class="p">)</span>
+
+        <span class="c1"># constructing the propagate</span>
+        <span class="n">A</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="mi">9</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="n">device</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">dtype</span><span class="p">)</span><span class="o">.</span><span class="n">repeat</span><span class="p">([</span><span class="n">B</span><span class="p">,</span> <span class="n">F</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
+        <span class="n">Bg</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="mi">9</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="n">device</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">dtype</span><span class="p">)</span>
+        <span class="n">Ba</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="mi">9</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="n">device</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">dtype</span><span class="p">)</span>
+
+        <span class="n">A</span><span class="p">[:,</span> <span class="p">:</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">:</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">:</span><span class="mi">3</span><span class="p">]</span> <span class="o">=</span> <span class="n">cov_input</span><span class="p">[</span><span class="s1">&#39;Rk&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">matrix</span><span class="p">()</span><span class="o">.</span><span class="n">mT</span> <span class="c1"># R_{k,k+1}^T</span>
+        <span class="n">A</span><span class="p">[:,</span> <span class="p">:</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">:</span><span class="mi">6</span><span class="p">,</span> <span class="mi">0</span><span class="p">:</span><span class="mi">3</span><span class="p">]</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">einsum</span><span class="p">(</span><span class="s1">&#39;...xy,...t -&gt; ...xy&#39;</span><span class="p">,</span> \
+            <span class="o">-</span> <span class="n">cov_input</span><span class="p">[</span><span class="s1">&#39;Rij&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">matrix</span><span class="p">()</span> <span class="o">@</span> <span class="n">cov_input</span><span class="p">[</span><span class="s1">&#39;Ha&#39;</span><span class="p">],</span> <span class="n">cov_input</span><span class="p">[</span><span class="s1">&#39;dt&#39;</span><span class="p">])</span>
+        <span class="n">A</span><span class="p">[:,</span> <span class="p">:</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">6</span><span class="p">:</span><span class="mi">9</span><span class="p">,</span> <span class="mi">0</span><span class="p">:</span><span class="mi">3</span><span class="p">]</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">einsum</span><span class="p">(</span><span class="s1">&#39;...xy,...t -&gt; ...xy&#39;</span><span class="p">,</span> \
+            <span class="o">-</span> <span class="mf">0.5</span> <span class="o">*</span> <span class="n">cov_input</span><span class="p">[</span><span class="s1">&#39;Rij&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">matrix</span><span class="p">()</span> <span class="o">@</span> <span class="n">cov_input</span><span class="p">[</span><span class="s1">&#39;Ha&#39;</span><span class="p">],</span> <span class="n">cov_input</span><span class="p">[</span><span class="s1">&#39;dt&#39;</span><span class="p">]</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
+        <span class="n">A</span><span class="p">[:,</span> <span class="p">:</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">6</span><span class="p">:</span><span class="mi">9</span><span class="p">,</span> <span class="mi">3</span><span class="p">:</span><span class="mi">6</span><span class="p">]</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">einsum</span><span class="p">(</span><span class="s1">&#39;...xy,...t -&gt; ...xy&#39;</span><span class="p">,</span> \
+            <span class="n">torch</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="n">device</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">dtype</span><span class="p">)</span><span class="o">.</span><span class="n">repeat</span><span class="p">([</span><span class="n">B</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]),</span> <span class="n">cov_input</span><span class="p">[</span><span class="s1">&#39;dt&#39;</span><span class="p">])</span>
+
+        <span class="c1"># [J dt, 0, 0]^T</span>
+        <span class="n">Bg</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">0</span><span class="p">:</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">:</span><span class="mi">3</span><span class="p">]</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">einsum</span><span class="p">(</span><span class="s1">&#39;...xy,...t -&gt; ...xy&#39;</span><span class="p">,</span> <span class="n">cov_input</span><span class="p">[</span><span class="s1">&#39;Rk&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">Jr</span><span class="p">(),</span> <span class="n">cov_input</span><span class="p">[</span><span class="s1">&#39;dt&#39;</span><span class="p">])</span>
+
+        <span class="c1"># [0, R_{ik}dt, 1/2R_{ik}dt^2]^T</span>
+        <span class="n">Ba</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">3</span><span class="p">:</span><span class="mi">6</span><span class="p">,</span> <span class="mi">0</span><span class="p">:</span><span class="mi">3</span><span class="p">]</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">einsum</span><span class="p">(</span><span class="s1">&#39;...xy,...t -&gt; ...xy&#39;</span><span class="p">,</span> <span class="n">cov_input</span><span class="p">[</span><span class="s1">&#39;Rij&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">matrix</span><span class="p">(),</span> <span class="n">cov_input</span><span class="p">[</span><span class="s1">&#39;dt&#39;</span><span class="p">])</span>
+        <span class="n">Ba</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">6</span><span class="p">:</span><span class="mi">9</span><span class="p">,</span> <span class="mi">0</span><span class="p">:</span><span class="mi">3</span><span class="p">]</span> <span class="o">=</span> <span class="mf">0.5</span> <span class="o">*</span> <span class="n">torch</span><span class="o">.</span><span class="n">einsum</span><span class="p">(</span><span class="s1">&#39;...xy,...t -&gt; ...xy&#39;</span><span class="p">,</span>
+                                                <span class="n">cov_input</span><span class="p">[</span><span class="s1">&#39;Rij&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">matrix</span><span class="p">(),</span> <span class="n">cov_input</span><span class="p">[</span><span class="s1">&#39;dt&#39;</span><span class="p">]</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
+
+        <span class="c1"># the term of B</span>
+        <span class="n">B_cov</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">einsum</span><span class="p">(</span><span class="s1">&#39;...xy,...t -&gt; ...xy&#39;</span><span class="p">,</span> <span class="n">Bg</span> <span class="o">@</span> <span class="n">Cg</span> <span class="o">@</span> <span class="n">Bg</span><span class="o">.</span><span class="n">mT</span> <span class="o">+</span> <span class="n">Ba</span> <span class="o">@</span> <span class="n">Ca</span> <span class="o">@</span> <span class="n">Ba</span><span class="o">.</span><span class="n">mT</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="n">cov_input</span><span class="p">[</span><span class="s1">&#39;dt&#39;</span><span class="p">])</span>
+        <span class="n">B_cov</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">cat</span><span class="p">([</span><span class="n">init_cov</span><span class="p">[:,</span><span class="kc">None</span><span class="p">,</span><span class="o">...</span><span class="p">],</span> <span class="n">B_cov</span><span class="p">],</span> <span class="n">dim</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+
+        <span class="n">A_left_cum</span> <span class="o">=</span> <span class="n">cumprod</span><span class="p">(</span><span class="n">A</span><span class="o">.</span><span class="n">flip</span><span class="p">([</span><span class="mi">1</span><span class="p">]),</span> <span class="n">dim</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">flip</span><span class="p">([</span><span class="mi">1</span><span class="p">])</span> <span class="c1"># cum from An to I, then flip</span>
+        <span class="n">A_right_cum</span> <span class="o">=</span> <span class="n">A_left_cum</span><span class="o">.</span><span class="n">mT</span>
+        <span class="n">cov</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">A_left_cum</span> <span class="o">@</span> <span class="n">B_cov</span> <span class="o">@</span> <span class="n">A_right_cum</span><span class="p">,</span> <span class="n">dim</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+        <span class="k">return</span> <span class="p">{</span><span class="s1">&#39;cov&#39;</span><span class="p">:</span> <span class="n">cov</span><span class="p">,</span> <span class="s1">&#39;Rij&#39;</span><span class="p">:</span> <span class="n">cov_input</span><span class="p">[</span><span class="s1">&#39;Rij&#39;</span><span class="p">][</span><span class="o">...</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">:,</span> <span class="p">:]}</span></div>
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+  <h1>Source code for pypose.module.lqr</h1><div class="highlight"><pre>
+<span></span><span class="kn">import</span> <span class="nn">torch</span>
+<span class="kn">from</span> <span class="nn">torch</span> <span class="kn">import</span> <span class="n">nn</span>
+<span class="kn">from</span> <span class="nn">..</span> <span class="kn">import</span> <span class="n">bmv</span><span class="p">,</span> <span class="n">bvmv</span>
+<span class="kn">from</span> <span class="nn">.dynamics</span> <span class="kn">import</span> <span class="n">runsys</span>
+<span class="kn">from</span> <span class="nn">torch.linalg</span> <span class="kn">import</span> <span class="n">cholesky</span><span class="p">,</span> <span class="n">vecdot</span>
+
+
+<div class="viewcode-block" id="LQR"><a class="viewcode-back" href="../../../../generated/pypose.module.LQR/#pypose.module.LQR">[docs]</a><span class="k">class</span> <span class="nc">LQR</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Linear Quadratic Regulator (LQR) with Dynamic Programming.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        system (:obj:`instance`): The system to be soved by LQR.</span>
+<span class="sd">        Q (:obj:`Tensor`): The weight matrix of the quadratic term.</span>
+<span class="sd">        p (:obj:`Tensor`): The weight vector of the first-order term.</span>
+<span class="sd">        T (:obj:`int`): Time steps of system.</span>
+
+<span class="sd">    A discrete-time linear system can be described as:</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \begin{align*}</span>
+<span class="sd">            \mathbf{x}_{t+1} &amp;= \mathbf{A}_t\mathbf{x}_t + \mathbf{B}_t\mathbf{u}_t</span>
+<span class="sd">                                                         + \mathbf{c}_{1t}          \\</span>
+<span class="sd">            \mathbf{y}_t &amp;= \mathbf{C}_t\mathbf{x}_t + \mathbf{D}_t\mathbf{u}_t</span>
+<span class="sd">                                                     + \mathbf{c}_{2t}              \\</span>
+<span class="sd">        \end{align*}</span>
+
+<span class="sd">    where :math:`\mathbf{x}`, :math:`\mathbf{u}` are the state and input of the linear</span>
+<span class="sd">    system; :math:`\mathbf{y}` is the observation of the linear system; :math:`\mathbf{A}`</span>
+<span class="sd">    and :math:`\mathbf{B}` are the state matrix and input matrix of the linear system;</span>
+<span class="sd">    :math:`\mathbf{C}`, :math:`\mathbf{D}` are the output matrix and observation matrix</span>
+<span class="sd">    of the linear system; :math:`\mathbf{c}_{1}`, :math:`\mathbf{c}_{2}` are the constant</span>
+<span class="sd">    input and constant output of the linear system. The subscript :math:`\cdot_{t}`</span>
+<span class="sd">    denotes the time step.</span>
+
+<span class="sd">    LQR finds the optimal nominal trajectory :math:`\mathbf{\tau}_{1:T}^*` =</span>
+<span class="sd">    :math:`\begin{Bmatrix} \mathbf{x}_t, \mathbf{u}_t \end{Bmatrix}_{1:T}`</span>
+<span class="sd">    for the linear system of the optimization problem:</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \begin{align*}</span>
+<span class="sd">          \mathbf{\tau}_{1:T}^* = \mathop{\arg\min}\limits_{\tau_{1:T}}</span>
+<span class="sd">            &amp;\sum\limits_t\frac{1}{2}</span>
+<span class="sd">          \mathbf{\tau}_t^\top\mathbf{Q}_t\mathbf{\tau}_t</span>
+<span class="sd">            + \mathbf{p}_t^\top\mathbf{\tau}_t \\</span>
+<span class="sd">          \mathrm{s.t.} \quad \mathbf{x}_1 &amp;= \mathbf{x}_{\text{init}}, \\</span>
+<span class="sd">          \mathbf{x}_{t+1} &amp;= \mathbf{F}_t\mathbf{\tau}_t + \mathbf{c}_{1t} \\</span>
+<span class="sd">        \end{align*}</span>
+
+<span class="sd">    where :math:`\mathbf{\tau}_t` = :math:`\begin{bmatrix} \mathbf{x}_t \\ \mathbf{u}_t</span>
+<span class="sd">    \end{bmatrix}`, :math:`\mathbf{F}_t` = :math:`\begin{bmatrix} \mathbf{A}_t &amp;</span>
+<span class="sd">    \mathbf{B}_t \end{bmatrix}`.</span>
+
+<span class="sd">    One way to solve the LQR problem is to use the dynamic programming, where the process</span>
+<span class="sd">    can be summarised as a backward and a forward recursion.</span>
+
+<span class="sd">    - The backward recursion.</span>
+
+<span class="sd">      For :math:`t` = :math:`T` to 1:</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \begin{align*}</span>
+<span class="sd">            \mathbf{Q}_t &amp;= \mathbf{Q}_t +</span>
+<span class="sd">                \mathbf{F}_t^\top\mathbf{V}_{t+1} \mathbf{F}_t \\</span>
+<span class="sd">            \mathbf{q}_t &amp;= \mathbf{p}_t + \mathbf{F}_t^\top\mathbf{V}_{t+1}</span>
+<span class="sd">                            \mathbf{c}_{1t} + \mathbf{F}_t^\top\mathbf{v}_{t+1}  \\</span>
+<span class="sd">            \mathbf{K}_t &amp;= -\mathbf{Q}_{\mathbf{u}_t, \mathbf{u}_t}^{-1}</span>
+<span class="sd">                                        \mathbf{Q}_{\mathbf{u}_t, \mathbf{x}_t} \\</span>
+<span class="sd">            \mathbf{k}_t &amp;= -\mathbf{Q}_{\mathbf{u}_t, \mathbf{u}_t}^{-1}</span>
+<span class="sd">                                                        \mathbf{q}_{\mathbf{u}_t}  \\</span>
+<span class="sd">            \mathbf{V}_t &amp;= \mathbf{Q}_{\mathbf{x}_t, \mathbf{x}_t}</span>
+<span class="sd">                + \mathbf{Q}_{\mathbf{x}_t, \mathbf{u}_t}\mathbf{K}_t</span>
+<span class="sd">                + \mathbf{K}_t^\top\mathbf{Q}_{\mathbf{u}_t, \mathbf{x}_t}</span>
+<span class="sd">                + \mathbf{K}_t^\top\mathbf{Q}_{\mathbf{u}_t, \mathbf{u}_t}</span>
+<span class="sd">                    \mathbf{K}_t \\</span>
+<span class="sd">            \mathbf{v}_t &amp;= \mathbf{q}_{\mathbf{x}_t}</span>
+<span class="sd">                + \mathbf{Q}_{\mathbf{x}_t, \mathbf{u}_t}\mathbf{k}_t</span>
+<span class="sd">                + \mathbf{K}_t^\top\mathbf{q}_{\mathbf{u}_t}</span>
+<span class="sd">                + \mathbf{K}_t^\top\mathbf{Q}_{\mathbf{u}_t,</span>
+<span class="sd">                    \mathbf{u}_t}\mathbf{k}_t  \\</span>
+<span class="sd">        \end{align*}</span>
+
+<span class="sd">    - The forward recursion.</span>
+
+<span class="sd">      For :math:`t` = 1 to :math:`T`:</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \begin{align*}</span>
+<span class="sd">                \mathbf{u}_t &amp;= \mathbf{K}_t\mathbf{x}_t + \mathbf{k}_t \\</span>
+<span class="sd">                \mathbf{x}_{t+1} &amp;= \mathbf{A}_t\mathbf{x}_t + \mathbf{B}_t\mathbf{u}_t</span>
+<span class="sd">                                                                + \mathbf{c}_{1t} \\</span>
+<span class="sd">            \end{align*}</span>
+
+<span class="sd">    Then quadratic costs of the system over the time horizon:</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \mathbf{c} \left( \mathbf{\tau}_t \right) = \frac{1}{2}</span>
+<span class="sd">        \mathbf{\tau}_t^\top\mathbf{Q}_t\mathbf{\tau}_t</span>
+<span class="sd">            + \mathbf{p}_t^\top\mathbf{\tau}_t</span>
+
+<span class="sd">    For the **non-linear system**, sometimes people want to solve MPC problem with</span>
+<span class="sd">    **iterative LQR**. A discrete-time non-linear system can be described as:</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \begin{aligned}</span>
+<span class="sd">            \mathbf{x}_{t+1} &amp;= \mathbf{f}(\mathbf{x}_t, \mathbf{u}_t, t_t) \\</span>
+<span class="sd">            \mathbf{y}_{t} &amp;= \mathbf{g}(\mathbf{x}_t, \mathbf{u}_t, t_t) \\</span>
+<span class="sd">        \end{aligned}</span>
+
+<span class="sd">    We can do a linear approximation at current point :math:`\chi^*=(\mathbf{x}^*,</span>
+<span class="sd">    \mathbf{u}^*, t^*)` along a trajectory with small perturbation</span>
+<span class="sd">    :math:`\chi=(\mathbf{x}^*+\delta\mathbf{x}, \mathbf{u}^* +\delta\mathbf{u}, t^*)`</span>
+<span class="sd">    near :math:`\chi^*` for both dynamics and cost:</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">            \begin{aligned}</span>
+<span class="sd">            \mathbf{f}(\mathbf{x}, \mathbf{u}, t^*) &amp;\approx \mathbf{f}(\mathbf{x}^*,</span>
+<span class="sd">                \mathbf{u}^*, t^*) +  \left.\frac{\partial \mathbf{f}}{\partial\mathbf{x}}</span>
+<span class="sd">                \right|_{\chi^*} \delta \mathbf{x} + \left. \frac{\partial \mathbf{f}}</span>
+<span class="sd">                {\partial \mathbf{u}} \right|_{\chi^*} \delta \mathbf{u} \\</span>
+<span class="sd">            &amp;= \mathbf{f}(\mathbf{x}^*, \mathbf{u}^*, t^*) + \mathbf{A} \delta \mathbf{x}</span>
+<span class="sd">                + \mathbf{B} \delta \mathbf{u} \\</span>
+<span class="sd">            \delta \mathbf{x}_{t+1} &amp;= \mathbf{A}_t \delta \mathbf{x}_t + \mathbf{B}_t</span>
+<span class="sd">                \delta \mathbf{u}_t \\</span>
+<span class="sd">            &amp;= \mathbf{F}_t \delta \mathbf{\tau}_t \\</span>
+<span class="sd">            \mathbf{c} \left( \mathbf{\tau}, t^* \right) &amp;\approx</span>
+<span class="sd">                \mathbf{c} \left( \mathbf{\tau}^*, t^* \right) + \frac{1}{2} \delta</span>
+<span class="sd">                \mathbf{\tau}^\top\nabla^2_{\mathbf{\tau}}\mathbf{c}\left(\mathbf{\tau}^*,</span>
+<span class="sd">                t^* \right) \delta \mathbf{\tau} + \nabla_{\mathbf{\tau}}</span>
+<span class="sd">                \mathbf{c} \left( \mathbf{\tau}^*, t^* \right)^\top \delta \mathbf{\tau}\\</span>
+<span class="sd">            \bar{\mathbf{c}} \left( \delta \mathbf{\tau} \right) &amp;= \frac{1}{2} \delta</span>
+<span class="sd">                \mathbf{\tau}_t^\top \bar{\mathbf{Q}}_t \delta \mathbf{\tau}_t +</span>
+<span class="sd">                \bar{\mathbf{p}}_t^\top \delta \mathbf{\tau}_t \\</span>
+<span class="sd">            \end{aligned}</span>
+
+<span class="sd">    where :math:`\delta \mathbf{\tau}_t` = :math:`\begin{bmatrix} \delta \mathbf{x}_t \\</span>
+<span class="sd">    \delta \mathbf{u}_t \end{bmatrix}`, :math:`\mathbf{F}_t` = :math:`\begin{bmatrix}</span>
+<span class="sd">    \mathbf{A}_t &amp; \mathbf{B}_t \end{bmatrix}`, :math:`\bar{\mathbf{Q}}_t = \mathbf{Q}_t`,</span>
+<span class="sd">    :math:`\bar{\mathbf{p}}_t` = :math:`\mathbf{Q}_t \mathbf{\tau}^*_t + \mathbf{p}_t`.</span>
+
+<span class="sd">    Then, LQR can be performed on a linear quadractic problem with</span>
+<span class="sd">    :math:`\delta \mathbf{\tau}_t`, :math:`\mathbf{F}_t`,</span>
+<span class="sd">    :math:`\bar{\mathbf{Q}}_t` and :math:`\bar{\mathbf{p}}_t`.</span>
+
+<span class="sd">    - The backward recursion.</span>
+
+<span class="sd">      For :math:`t` = :math:`T` to 1:</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \begin{align*}</span>
+<span class="sd">            \mathbf{Q}_t &amp;= \bar{\mathbf{Q}}_t + \mathbf{F}_t^\top\mathbf{V}_{t+1}</span>
+<span class="sd">                                \mathbf{F}_t \\</span>
+<span class="sd">            \mathbf{q}_t &amp;= \bar{\mathbf{p}}_t + \mathbf{F}_t^\top\mathbf{v}_{t+1}  \\</span>
+<span class="sd">            \mathbf{K}_t &amp;= -\mathbf{Q}_{\delta \mathbf{u}_t, \delta \mathbf{u}_t}^{-1}</span>
+<span class="sd">                                \mathbf{Q}_{\delta \mathbf{u}_t, \delta \mathbf{x}_t} \\</span>
+<span class="sd">            \mathbf{k}_t &amp;= -\mathbf{Q}_{\delta \mathbf{u}_t, \delta \mathbf{u}_t}^{-1}</span>
+<span class="sd">                                \mathbf{q}_{\delta \mathbf{u}_t}         \\</span>
+<span class="sd">            \mathbf{V}_t &amp;= \mathbf{Q}_{\delta \mathbf{x}_t, \delta \mathbf{x}_t}</span>
+<span class="sd">                + \mathbf{Q}_{\delta \mathbf{x}_t, \delta \mathbf{u}_t}\mathbf{K}_t</span>
+<span class="sd">                + \mathbf{K}_t^\top\mathbf{Q}_{\delta \mathbf{u}_t, \delta \mathbf{x}_t}</span>
+<span class="sd">                + \mathbf{K}_t^\top\mathbf{Q}_{\delta \mathbf{u}_t, \delta \mathbf{u}_t}</span>
+<span class="sd">                    \mathbf{K}_t \\</span>
+<span class="sd">            \mathbf{v}_t &amp;= \mathbf{q}_{\delta \mathbf{x}_t}</span>
+<span class="sd">                + \mathbf{Q}_{\delta \mathbf{x}_t, \delta \mathbf{u}_t}\mathbf{k}_t</span>
+<span class="sd">                + \mathbf{K}_t^\top\mathbf{q}_{\delta \mathbf{u}_t}</span>
+<span class="sd">                + \mathbf{K}_t^\top\mathbf{Q}_{\delta \mathbf{u}_t,</span>
+<span class="sd">                    \delta \mathbf{u}_t}\mathbf{k}_t   \\</span>
+<span class="sd">        \end{align*}</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        Because we made a linear approximation, here :math:`\bar{\mathbf{p}}_t` leads to a</span>
+<span class="sd">        difference with :math:`{\mathbf{q}}_t` and :math:`{\mathbf{k}}_t` relative to the</span>
+<span class="sd">        linear backward recursion, and this change will be compensated back by</span>
+<span class="sd">        :math:`\mathbf{u}_t^*` in the forward recursion.</span>
+
+<span class="sd">    - The forward recursion.</span>
+
+<span class="sd">      For :math:`t` = 1 to :math:`T`:</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \begin{align*}</span>
+<span class="sd">                \delta \mathbf{u}_t &amp;= \mathbf{K}_t \delta \mathbf{x}_t + \mathbf{k}_t \\</span>
+<span class="sd">                \mathbf{u}_t &amp;= \delta \mathbf{u}_t + \mathbf{u}_t^* \\</span>
+<span class="sd">                \mathbf{x}_{t+1} &amp;= \mathbf{f}(\mathbf{x}_t, \mathbf{u}_t) \\</span>
+<span class="sd">            \end{align*}</span>
+
+<span class="sd">    Then quadratic costs of the system over the time horizon:</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathbf{c} \left( \mathbf{\tau}_t \right) = \frac{1}{2}</span>
+<span class="sd">            \mathbf{\tau}_t^\top\mathbf{Q}_t\mathbf{\tau}_t</span>
+<span class="sd">                + \mathbf{p}_t^\top\mathbf{\tau}_t</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        The discrete-time system to be solved by LQR can be either linear time-invariant</span>
+<span class="sd">        (:meth:`LTI`) system, or linear time-varying (:meth:`LTV`) system. For non-linear</span>
+<span class="sd">        system, one can approximate it as a linear system via Taylor expansion, using</span>
+<span class="sd">        iterative LQR algorithm for MPC. Here we provide a unified general format for the</span>
+<span class="sd">        implementation.</span>
+
+<span class="sd">    From the learning perspective, this can be interpreted as a module with unknown</span>
+<span class="sd">    parameters :math:`\begin{Bmatrix} \mathbf{Q}, \mathbf{p}, \mathbf{F}, \mathbf{f}</span>
+<span class="sd">    \end{Bmatrix}`, which can be integrated into an end-to-end learning system.</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        The implementation of LQR is based on page 24-32 of the slides:</span>
+
+<span class="sd">        * `Optimal Control and Planning &lt;https://tinyurl.com/y5ck36vw&gt;`_.</span>
+
+<span class="sd">        The implementation of iterative LQR is based on Eq. (1)~(19) of this paper:</span>
+
+<span class="sd">        * Li Weiwei, and Emanuel Todorov, `Iterative linear quadratic regulator design for</span>
+<span class="sd">          nonlinear biological movement systems &lt;https://tinyurl.com/bdma36s6&gt;`_, ICINCO</span>
+<span class="sd">          (1), 2004.</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        &gt;&gt;&gt; torch.manual_seed(0)</span>
+<span class="sd">        &gt;&gt;&gt; n_batch, T = 2, 5</span>
+<span class="sd">        &gt;&gt;&gt; n_state, n_ctrl = 4, 3</span>
+<span class="sd">        &gt;&gt;&gt; n_sc = n_state + n_ctrl</span>
+<span class="sd">        &gt;&gt;&gt; Q = torch.randn(n_batch, T, n_sc, n_sc)</span>
+<span class="sd">        &gt;&gt;&gt; Q = torch.matmul(Q.mT, Q)</span>
+<span class="sd">        &gt;&gt;&gt; p = torch.randn(n_batch, T, n_sc)</span>
+<span class="sd">        &gt;&gt;&gt; r = 0.2 * torch.randn(n_state, n_state)</span>
+<span class="sd">        &gt;&gt;&gt; A = torch.tile(torch.eye(n_state) + r, (n_batch, 1, 1))</span>
+<span class="sd">        &gt;&gt;&gt; B = torch.randn(n_batch, n_state, n_ctrl)</span>
+<span class="sd">        &gt;&gt;&gt; C = torch.tile(torch.eye(n_state), (n_batch, 1, 1))</span>
+<span class="sd">        &gt;&gt;&gt; D = torch.tile(torch.zeros(n_state, n_ctrl), (n_batch, 1, 1))</span>
+<span class="sd">        &gt;&gt;&gt; c1 = torch.tile(torch.randn(n_state), (n_batch, 1))</span>
+<span class="sd">        &gt;&gt;&gt; c2 = torch.tile(torch.zeros(n_state), (n_batch, 1))</span>
+<span class="sd">        &gt;&gt;&gt; x_init = torch.randn(n_batch, n_state)</span>
+<span class="sd">        &gt;&gt;&gt; u_traj = torch.zeros(n_batch, T, n_ctrl, device=device)</span>
+<span class="sd">        &gt;&gt;&gt; lti = pp.module.LTI(A, B, C, D, c1, c2)</span>
+<span class="sd">        &gt;&gt;&gt; dt = 1</span>
+<span class="sd">        &gt;&gt;&gt; LQR = pp.module.LQR(lti, Q, p, T)</span>
+<span class="sd">        &gt;&gt;&gt; x, u, cost = LQR(x_init, dt)</span>
+<span class="sd">        &gt;&gt;&gt; print(&quot;x = &quot;, x)</span>
+<span class="sd">        &gt;&gt;&gt; print(&quot;u = &quot;, u)</span>
+<span class="sd">        x = tensor([[[-0.2633, -0.3466,  2.3803, -0.0423],</span>
+<span class="sd">                     [ 0.1849, -1.3884,  1.0898, -1.6229],</span>
+<span class="sd">                     [ 1.2138, -0.7161,  0.2954, -0.6819],</span>
+<span class="sd">                     [ 1.4840, -1.1249, -1.0302,  0.9805],</span>
+<span class="sd">                     [-0.3477, -1.7063,  4.6494,  2.6780],</span>
+<span class="sd">                     [ 7.2346,  4.9958, 17.9926, -7.7881]],</span>
+<span class="sd">                    [[-0.9744,  0.4976,  0.0603, -0.5258],</span>
+<span class="sd">                     [-0.6356,  0.0539,  0.7264, -0.5048],</span>
+<span class="sd">                     [-0.2275, -0.1649,  0.3872, -0.4614],</span>
+<span class="sd">                     [ 0.2697, -0.3577,  0.0999, -0.4594],</span>
+<span class="sd">                     [ 0.3916, -2.0832,  0.0701, -0.5407],</span>
+<span class="sd">                     [ 1.0404, -1.3799, -2.0913, -0.1459]]])</span>
+<span class="sd">        u = tensor([[[ 1.0405,  0.1586, -0.1282],</span>
+<span class="sd">                     [-1.4845, -0.5745,  0.2523],</span>
+<span class="sd">                     [-0.6322, -0.3281, -0.3620],</span>
+<span class="sd">                     [-1.6768,  2.4054, -0.1047],</span>
+<span class="sd">                     [-1.7948,  3.5269,  9.0703]],</span>
+<span class="sd">                    [[-0.1795,  0.9153,  1.7066],</span>
+<span class="sd">                     [ 0.0814,  0.4004,  0.7114],</span>
+<span class="sd">                     [ 0.0436,  0.5782,  1.0127],</span>
+<span class="sd">                     [-0.3017, -0.2897,  0.7251],</span>
+<span class="sd">                     [-0.0728,  0.7290, -0.3117]]])</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">system</span><span class="p">,</span> <span class="n">Q</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">T</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">system</span> <span class="o">=</span> <span class="n">system</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">Q</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">T</span> <span class="o">=</span> <span class="n">Q</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">T</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">x_traj</span> <span class="o">=</span> <span class="kc">None</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">u_traj</span> <span class="o">=</span> <span class="kc">None</span>
+
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">Q</span><span class="o">.</span><span class="n">ndim</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">Q</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tile</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">Q</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
+
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="o">.</span><span class="n">ndim</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">p</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tile</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">n_batch</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="o">.</span><span class="n">shape</span><span class="p">[:</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span>
+
+        <span class="k">assert</span> <span class="bp">self</span><span class="o">.</span><span class="n">Q</span><span class="o">.</span><span class="n">shape</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="o">.</span><span class="n">shape</span><span class="p">,</span> <span class="s2">&quot;Shape not compatible.&quot;</span>
+        <span class="k">assert</span> <span class="bp">self</span><span class="o">.</span><span class="n">Q</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">Q</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">),</span> <span class="s2">&quot;Shape not compatible.&quot;</span>
+        <span class="k">assert</span> <span class="bp">self</span><span class="o">.</span><span class="n">Q</span><span class="o">.</span><span class="n">ndim</span> <span class="o">==</span> <span class="mi">4</span> <span class="ow">or</span> <span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="o">.</span><span class="n">ndim</span> <span class="o">==</span> <span class="mi">3</span><span class="p">,</span> <span class="s2">&quot;Shape not compatible.&quot;</span>
+        <span class="k">assert</span> <span class="bp">self</span><span class="o">.</span><span class="n">Q</span><span class="o">.</span><span class="n">device</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="o">.</span><span class="n">device</span><span class="p">,</span> <span class="s2">&quot;Device not compatible.&quot;</span>
+        <span class="k">assert</span> <span class="bp">self</span><span class="o">.</span><span class="n">Q</span><span class="o">.</span><span class="n">dtype</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="o">.</span><span class="n">dtype</span><span class="p">,</span> <span class="s2">&quot;Tensor data type not compatible.&quot;</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">dargs</span> <span class="o">=</span> <span class="p">{</span><span class="s1">&#39;dtype&#39;</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="o">.</span><span class="n">dtype</span><span class="p">,</span> <span class="s1">&#39;device&#39;</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="o">.</span><span class="n">device</span><span class="p">}</span>
+
+<div class="viewcode-block" id="LQR.forward"><a class="viewcode-back" href="../../../../generated/pypose.module.LQR/#pypose.module.LQR.forward">[docs]</a>    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x_init</span><span class="p">,</span> <span class="n">dt</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">u_traj</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">u_lower</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">u_upper</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">du</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Performs LQR for the discrete system.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            x_init (:obj:`Tensor`): The initial state of the system.</span>
+<span class="sd">            dt (:obj:`int`): The interval (:math:`\delta t`) between two time steps.</span>
+<span class="sd">                Default: `1`.</span>
+<span class="sd">            u_traj (:obj:`Tensor`, optinal): The current inputs of the system along a</span>
+<span class="sd">                trajectory. Default: ``None``.</span>
+<span class="sd">            u_lower (:obj:`Tensor`, optinal): The lower bounds on the controls.</span>
+<span class="sd">                Default: ``None``.</span>
+<span class="sd">            u_upper (:obj:`Tensor`, optinal): The upper bounds on the controls.</span>
+<span class="sd">                Default: ``None``.</span>
+<span class="sd">            du (:obj:`int`, optinal): The amount each component of the controls</span>
+<span class="sd">                is allowed to change in each LQR iteration. Default: ``None``.</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            List of :obj:`Tensor`: A list of tensors including the solved state sequence</span>
+<span class="sd">            :math:`\mathbf{x}`, the solved input sequence :math:`\mathbf{u}`, and the</span>
+<span class="sd">            associated quadratic costs :math:`\mathbf{c}` over the time horizon.</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="n">K</span><span class="p">,</span> <span class="n">k</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">lqr_backward</span><span class="p">(</span><span class="n">x_init</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">u_traj</span><span class="p">,</span> <span class="n">u_lower</span><span class="p">,</span> <span class="n">u_upper</span><span class="p">,</span> <span class="n">du</span><span class="p">)</span>
+        <span class="n">x</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">cost</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">lqr_forward</span><span class="p">(</span><span class="n">x_init</span><span class="p">,</span> <span class="n">K</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">u_lower</span><span class="p">,</span> <span class="n">u_upper</span><span class="p">,</span> <span class="n">du</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">x</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">cost</span></div>
+
+    <span class="k">def</span> <span class="nf">lqr_backward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x_init</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">u_traj</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">u_lower</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">u_upper</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">du</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
+
+        <span class="n">ns</span><span class="p">,</span> <span class="n">nsc</span> <span class="o">=</span> <span class="n">x_init</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
+        <span class="n">nc</span> <span class="o">=</span> <span class="n">nsc</span> <span class="o">-</span> <span class="n">ns</span>
+
+        <span class="k">if</span> <span class="n">u_traj</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">u_traj</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">n_batch</span> <span class="o">+</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">nc</span><span class="p">),</span> <span class="o">**</span><span class="bp">self</span><span class="o">.</span><span class="n">dargs</span><span class="p">)</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">u_traj</span> <span class="o">=</span> <span class="n">u_traj</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">x_traj</span> <span class="o">=</span> <span class="n">x_init</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">repeat</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">x_traj</span> <span class="o">=</span> <span class="n">runsys</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">system</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">x_traj</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">u_traj</span><span class="p">)</span>
+
+        <span class="n">K</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">n_batch</span> <span class="o">+</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">nc</span><span class="p">,</span> <span class="n">ns</span><span class="p">),</span> <span class="o">**</span><span class="bp">self</span><span class="o">.</span><span class="n">dargs</span><span class="p">)</span>
+        <span class="n">k</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">n_batch</span> <span class="o">+</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">nc</span><span class="p">),</span> <span class="o">**</span><span class="bp">self</span><span class="o">.</span><span class="n">dargs</span><span class="p">)</span>
+
+        <span class="n">xut</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">cat</span><span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">x_traj</span><span class="p">[</span><span class="o">...</span><span class="p">,:</span><span class="bp">self</span><span class="o">.</span><span class="n">T</span><span class="p">,:],</span> <span class="bp">self</span><span class="o">.</span><span class="n">u_traj</span><span class="p">),</span> <span class="n">dim</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span>
+        <span class="n">p</span> <span class="o">=</span> <span class="n">bmv</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">Q</span><span class="p">,</span> <span class="n">xut</span><span class="p">)</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">p</span>
+
+        <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">T</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
+            <span class="k">if</span> <span class="n">t</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">T</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
+                <span class="n">Qt</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">Q</span><span class="p">[</span><span class="o">...</span><span class="p">,</span><span class="n">t</span><span class="p">,:,:]</span>
+                <span class="n">qt</span> <span class="o">=</span> <span class="n">p</span><span class="p">[</span><span class="o">...</span><span class="p">,</span><span class="n">t</span><span class="p">,:]</span>
+            <span class="k">else</span><span class="p">:</span>
+                <span class="bp">self</span><span class="o">.</span><span class="n">system</span><span class="o">.</span><span class="n">set_refpoint</span><span class="p">(</span><span class="n">state</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">x_traj</span><span class="p">[</span><span class="o">...</span><span class="p">,</span><span class="n">t</span><span class="p">,:],</span>
+                                         <span class="nb">input</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">u_traj</span><span class="p">[</span><span class="o">...</span><span class="p">,</span><span class="n">t</span><span class="p">,:],</span>
+                                         <span class="n">t</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">(</span><span class="n">t</span><span class="o">*</span><span class="n">dt</span><span class="p">))</span>
+                <span class="n">A</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">system</span><span class="o">.</span><span class="n">A</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span>
+                <span class="n">B</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">system</span><span class="o">.</span><span class="n">B</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span>
+                <span class="n">F</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">cat</span><span class="p">((</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">),</span> <span class="n">dim</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span>
+                <span class="n">Qt</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">Q</span><span class="p">[</span><span class="o">...</span><span class="p">,</span><span class="n">t</span><span class="p">,:,:]</span> <span class="o">+</span> <span class="n">F</span><span class="o">.</span><span class="n">mT</span> <span class="o">@</span> <span class="n">V</span> <span class="o">@</span> <span class="n">F</span>
+                <span class="n">qt</span> <span class="o">=</span> <span class="n">p</span><span class="p">[</span><span class="o">...</span><span class="p">,</span><span class="n">t</span><span class="p">,:]</span> <span class="o">+</span> <span class="n">bmv</span><span class="p">(</span><span class="n">F</span><span class="o">.</span><span class="n">mT</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
+
+            <span class="n">Qxx</span><span class="p">,</span> <span class="n">Qxu</span> <span class="o">=</span> <span class="n">Qt</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="p">:</span><span class="n">ns</span><span class="p">,</span> <span class="p">:</span><span class="n">ns</span><span class="p">],</span> <span class="n">Qt</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="p">:</span><span class="n">ns</span><span class="p">,</span> <span class="n">ns</span><span class="p">:]</span>
+            <span class="n">Qux</span><span class="p">,</span> <span class="n">Quu</span> <span class="o">=</span> <span class="n">Qt</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="n">ns</span><span class="p">:,</span> <span class="p">:</span><span class="n">ns</span><span class="p">],</span> <span class="n">Qt</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="n">ns</span><span class="p">:,</span> <span class="n">ns</span><span class="p">:]</span>
+            <span class="n">qx</span><span class="p">,</span> <span class="n">qu</span> <span class="o">=</span> <span class="n">qt</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="p">:</span><span class="n">ns</span><span class="p">],</span> <span class="n">qt</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="n">ns</span><span class="p">:]</span>
+
+            <span class="n">L</span> <span class="o">=</span> <span class="n">cholesky</span><span class="p">(</span><span class="n">Quu</span><span class="p">)</span>
+            <span class="n">K</span><span class="p">[</span><span class="o">...</span><span class="p">,</span><span class="n">t</span><span class="p">,:,:]</span> <span class="o">=</span> <span class="n">Kt</span> <span class="o">=</span> <span class="o">-</span><span class="n">torch</span><span class="o">.</span><span class="n">cholesky_solve</span><span class="p">(</span><span class="n">Qux</span><span class="p">,</span> <span class="n">L</span><span class="p">)</span>
+            <span class="n">k</span><span class="p">[</span><span class="o">...</span><span class="p">,</span><span class="n">t</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">kt</span> <span class="o">=</span> <span class="o">-</span><span class="n">torch</span><span class="o">.</span><span class="n">cholesky_solve</span><span class="p">(</span><span class="n">qu</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">),</span> <span class="n">L</span><span class="p">)</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
+
+            <span class="n">V</span> <span class="o">=</span> <span class="n">Qxx</span> <span class="o">+</span> <span class="n">Qxu</span> <span class="o">@</span> <span class="n">Kt</span> <span class="o">+</span> <span class="n">Kt</span><span class="o">.</span><span class="n">mT</span> <span class="o">@</span> <span class="n">Qux</span> <span class="o">+</span> <span class="n">Kt</span><span class="o">.</span><span class="n">mT</span> <span class="o">@</span> <span class="n">Quu</span> <span class="o">@</span> <span class="n">Kt</span>
+            <span class="n">v</span> <span class="o">=</span> <span class="n">qx</span>  <span class="o">+</span> <span class="n">bmv</span><span class="p">(</span><span class="n">Qxu</span><span class="p">,</span> <span class="n">kt</span><span class="p">)</span> <span class="o">+</span> <span class="n">bmv</span><span class="p">(</span><span class="n">Kt</span><span class="o">.</span><span class="n">mT</span><span class="p">,</span> <span class="n">qu</span><span class="p">)</span> <span class="o">+</span> <span class="n">bmv</span><span class="p">(</span><span class="n">Kt</span><span class="o">.</span><span class="n">mT</span> <span class="o">@</span> <span class="n">Quu</span><span class="p">,</span> <span class="n">kt</span><span class="p">)</span>
+
+        <span class="k">return</span> <span class="n">K</span><span class="p">,</span> <span class="n">k</span>
+
+    <span class="k">def</span> <span class="nf">lqr_forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x_init</span><span class="p">,</span> <span class="n">K</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">u_lower</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">u_upper</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">du</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
+
+        <span class="k">assert</span> <span class="n">x_init</span><span class="o">.</span><span class="n">device</span> <span class="o">==</span> <span class="n">K</span><span class="o">.</span><span class="n">device</span> <span class="o">==</span> <span class="n">k</span><span class="o">.</span><span class="n">device</span>
+        <span class="k">assert</span> <span class="n">x_init</span><span class="o">.</span><span class="n">dtype</span> <span class="o">==</span> <span class="n">K</span><span class="o">.</span><span class="n">dtype</span> <span class="o">==</span> <span class="n">k</span><span class="o">.</span><span class="n">dtype</span>
+        <span class="k">assert</span> <span class="n">x_init</span><span class="o">.</span><span class="n">ndim</span> <span class="o">==</span> <span class="mi">2</span><span class="p">,</span> <span class="s2">&quot;Shape not compatible.&quot;</span>
+
+        <span class="n">ns</span><span class="p">,</span> <span class="n">nc</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">x_traj</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">u_traj</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
+
+        <span class="n">u</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">n_batch</span> <span class="o">+</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">nc</span><span class="p">),</span> <span class="o">**</span><span class="bp">self</span><span class="o">.</span><span class="n">dargs</span><span class="p">)</span>
+        <span class="n">delta_u</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">n_batch</span> <span class="o">+</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">nc</span><span class="p">),</span> <span class="o">**</span><span class="bp">self</span><span class="o">.</span><span class="n">dargs</span><span class="p">)</span>
+        <span class="n">cost</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">n_batch</span><span class="p">,</span> <span class="o">**</span><span class="bp">self</span><span class="o">.</span><span class="n">dargs</span><span class="p">)</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">n_batch</span> <span class="o">+</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">T</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span> <span class="n">ns</span><span class="p">),</span> <span class="o">**</span><span class="bp">self</span><span class="o">.</span><span class="n">dargs</span><span class="p">)</span>
+        <span class="n">xt</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">x_init</span>
+
+        <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">T</span><span class="p">):</span>
+            <span class="n">Kt</span><span class="p">,</span> <span class="n">kt</span> <span class="o">=</span> <span class="n">K</span><span class="p">[</span><span class="o">...</span><span class="p">,</span><span class="n">t</span><span class="p">,:,:],</span> <span class="n">k</span><span class="p">[</span><span class="o">...</span><span class="p">,</span><span class="n">t</span><span class="p">,:]</span>
+            <span class="n">delta_xt</span> <span class="o">=</span> <span class="n">xt</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">x_traj</span><span class="p">[</span><span class="o">...</span><span class="p">,</span><span class="n">t</span><span class="p">,:]</span>
+            <span class="n">delta_u</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">bmv</span><span class="p">(</span><span class="n">Kt</span><span class="p">,</span> <span class="n">delta_xt</span><span class="p">)</span> <span class="o">+</span> <span class="n">kt</span>
+            <span class="n">u</span><span class="p">[</span><span class="o">...</span><span class="p">,</span><span class="n">t</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">ut</span> <span class="o">=</span> <span class="n">delta_u</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="p">:]</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">u_traj</span><span class="p">[</span><span class="o">...</span><span class="p">,</span><span class="n">t</span><span class="p">,:]</span>
+            <span class="n">xut</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">cat</span><span class="p">((</span><span class="n">xt</span><span class="p">,</span> <span class="n">ut</span><span class="p">),</span> <span class="n">dim</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span>
+            <span class="n">x</span><span class="p">[</span><span class="o">...</span><span class="p">,</span><span class="n">t</span><span class="o">+</span><span class="mi">1</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">xt</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">system</span><span class="p">(</span><span class="n">xt</span><span class="p">,</span> <span class="n">ut</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
+            <span class="n">cost</span> <span class="o">+=</span> <span class="mf">0.5</span> <span class="o">*</span> <span class="n">bvmv</span><span class="p">(</span><span class="n">xut</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">Q</span><span class="p">[</span><span class="o">...</span><span class="p">,</span><span class="n">t</span><span class="p">,:,:],</span> <span class="n">xut</span><span class="p">)</span> <span class="o">+</span> <span class="n">vecdot</span><span class="p">(</span><span class="n">xut</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="p">[</span><span class="o">...</span><span class="p">,</span><span class="n">t</span><span class="p">,:])</span>
+
+        <span class="k">return</span> <span class="n">x</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">cost</span></div>
+</pre></div>
+
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+  <h1>Source code for pypose.module.mpc</h1><div class="highlight"><pre>
+<span></span><span class="kn">import</span> <span class="nn">torch</span>
+<span class="kn">from</span> <span class="nn">.lqr</span> <span class="kn">import</span> <span class="n">LQR</span>
+<span class="kn">from</span> <span class="nn">torch</span> <span class="kn">import</span> <span class="n">nn</span>
+<span class="kn">from</span> <span class="nn">..utils.stepper</span> <span class="kn">import</span> <span class="n">ReduceToBason</span>
+
+
+<div class="viewcode-block" id="MPC"><a class="viewcode-back" href="../../../../generated/pypose.module.MPC/#pypose.module.MPC">[docs]</a><span class="k">class</span> <span class="nc">MPC</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Model Predictive Control (MPC) based on iterative LQR.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        system (:obj:`instance`): The system to be soved by MPC.</span>
+<span class="sd">        Q (:obj:`Tensor`): The weight matrix of the quadratic term.</span>
+<span class="sd">        p (:obj:`Tensor`): The weight vector of the first-order term.</span>
+<span class="sd">        T (:obj:`int`): Time steps of system.</span>
+<span class="sd">        stepper (``Planner``, optional): the stepper to stop iterations. If ``None``,</span>
+<span class="sd">            the ``pypose.utils.ReduceToBason`` with a maximum of 10 steps are used.</span>
+<span class="sd">            Default: ``None``.</span>
+
+<span class="sd">    **Model Predictive Control**, also known as Receding Horizon Control (RHC), uses the</span>
+<span class="sd">    mathematical model of the system to solve a finite, moving horizon, and</span>
+<span class="sd">    closed loop optimal control problem. Thus, the MPC scheme is able to utilize the</span>
+<span class="sd">    information about the current state of the system to predict future states</span>
+<span class="sd">    and control inputs for the system. At each time stamp, MPC solves the</span>
+<span class="sd">    optimization problem on a time horizon :math:`T`:</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \begin{align*}</span>
+<span class="sd">          \mathop{\arg\min}\limits_{\mathbf{x}_{1:T}, \mathbf{u}_{1:T}} \sum\limits_t</span>
+<span class="sd">          &amp;\mathbf{c}_t (\mathbf{x}_t, \mathbf{u}_t) \\</span>
+<span class="sd">          \mathrm{s.t.}\quad \mathbf{x}_{t+1} &amp;= \mathbf{f}(\mathbf{x}_t, \mathbf{u}_t),\\</span>
+<span class="sd">          \mathbf{x}_1 &amp;= \mathbf{x}_{\text{init}} \\</span>
+<span class="sd">        \end{align*}</span>
+
+<span class="sd">    where :math:`\mathbf{c}` is the cost function; :math:`\mathbf{x}`, :math:`\mathbf{u}`</span>
+<span class="sd">    are the state and input of the linear system; :math:`\mathbf{f}` is the dynamics</span>
+<span class="sd">    model; :math:`\mathbf{x}_{\text{init}}` is the initial state of the system.</span>
+
+<span class="sd">    For the linear system with quadratic cost, MPC is equivalent to solving an LQR problem</span>
+<span class="sd">    on each horizon as well as considering the dynamics as constraints. For nonlinear</span>
+<span class="sd">    system, one way to solve the MPC problem is to use iterative LQR, which uses linear</span>
+<span class="sd">    approximations of the dynamics and quadratic approximations of the cost function to</span>
+<span class="sd">    iteratively compute a local optimal solution based on the current states and control</span>
+<span class="sd">    sequences. Then, analytical derivative for the backward can be computed using one</span>
+<span class="sd">    additional forward pass of iLQR for the learning problem, such as learning the</span>
+<span class="sd">    parameters of the dynamic system.</span>
+
+<span class="sd">    Specifically, the discrete-time non-linear system can be described as:</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \begin{aligned}</span>
+<span class="sd">            \mathbf{x}_{t+1} &amp;= \mathbf{f}(\mathbf{x}_t, \mathbf{u}_t, t_t) \\</span>
+<span class="sd">            \mathbf{y}_{t} &amp;= \mathbf{g}(\mathbf{x}_t, \mathbf{u}_t, t_t) \\</span>
+<span class="sd">        \end{aligned}</span>
+
+<span class="sd">    We can do a linear approximation at current point :math:`\chi^*=(\mathbf{x}^*,</span>
+<span class="sd">    \mathbf{u}^*, t^*)` along a trajectory with small perturbation</span>
+<span class="sd">    :math:`\chi=(\mathbf{x}^*+\delta\mathbf{x}, \mathbf{u}^* +\delta\mathbf{u}, t^*)`</span>
+<span class="sd">    near :math:`\chi^*` for both dynamics and cost:</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">            \begin{aligned}</span>
+<span class="sd">            \mathbf{f}(\mathbf{x}, \mathbf{u}, t^*) &amp;\approx \mathbf{f}(\mathbf{x}^*,</span>
+<span class="sd">                \mathbf{u}^*, t^*) + \left.\frac{\partial \mathbf{f}}{\partial \mathbf{x}}</span>
+<span class="sd">                \right|_{\chi^*} \delta \mathbf{x} + \left. \frac{\partial \mathbf{f}}</span>
+<span class="sd">                {\partial \mathbf{u}} \right|_{\chi^*} \delta \mathbf{u} \\</span>
+<span class="sd">            &amp;= \mathbf{f}(\mathbf{x}^*, \mathbf{u}^*, t^*) + \mathbf{A} \delta \mathbf{x}</span>
+<span class="sd">                + \mathbf{B} \delta \mathbf{u} \\</span>
+<span class="sd">            \delta \mathbf{x}_{t+1} &amp;= \mathbf{A}_t \delta \mathbf{x}_t + \mathbf{B}_t</span>
+<span class="sd">                \delta \mathbf{u}_t \\</span>
+<span class="sd">            &amp;= \mathbf{F}_t \delta \mathbf{\tau}_t \\</span>
+<span class="sd">            \mathbf{c} \left( \mathbf{\tau}, t^* \right) &amp;\approx</span>
+<span class="sd">                \mathbf{c} \left(\mathbf{\tau}^*, t^* \right) + \frac{1}{2} \delta</span>
+<span class="sd">                \mathbf{\tau}^\top\nabla^2_{\mathbf{\tau}}\mathbf{c}\left(\mathbf{\tau}^*,</span>
+<span class="sd">                t^* \right) \delta \mathbf{\tau} + \nabla_{\mathbf{\tau}}</span>
+<span class="sd">                \mathbf{c} \left( \mathbf{\tau}^*, t^* \right)^\top \delta \mathbf{\tau}\\</span>
+<span class="sd">            \bar{\mathbf{c}} \left( \delta \mathbf{\tau} \right) &amp;= \frac{1}{2} \delta</span>
+<span class="sd">                \mathbf{\tau}_t^\top \bar{\mathbf{Q}}_t \delta \mathbf{\tau}_t +</span>
+<span class="sd">                \bar{\mathbf{p}}_t^\top \delta \mathbf{\tau}_t \\</span>
+<span class="sd">            \end{aligned}</span>
+
+<span class="sd">    where :math:`\mathbf{F}_t` = :math:`\begin{bmatrix}</span>
+<span class="sd">    \mathbf{A}_t &amp; \mathbf{B}_t \end{bmatrix}`.</span>
+
+<span class="sd">    Then, LQR can be performed on a linear quadractic problem with</span>
+<span class="sd">    :math:`\delta \mathbf{\tau}_t`, :math:`\mathbf{F}_t`, :math:`\bar{\mathbf{Q}}_t` and</span>
+<span class="sd">    :math:`\bar{\mathbf{p}}_t`.</span>
+
+<span class="sd">    - The backward recursion.</span>
+
+<span class="sd">      For :math:`t` = :math:`T` to 1:</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \begin{align*}</span>
+<span class="sd">                \mathbf{Q}_t &amp;= \bar{\mathbf{Q}}_t + \mathbf{F}_t^\top\mathbf{V}_{t+1}</span>
+<span class="sd">                                    \mathbf{F}_t \\</span>
+<span class="sd">                \mathbf{q}_t &amp;= \bar{\mathbf{p}}_t + \mathbf{F}_t^\top\mathbf{v}_{t+1}  \\</span>
+<span class="sd">                \mathbf{K}_t &amp;= -\mathbf{Q}_{\delta \mathbf{u}_t,\delta \mathbf{u}_t}^{-1}</span>
+<span class="sd">                                 \mathbf{Q}_{\delta \mathbf{u}_t,\delta \mathbf{x}_t} \\</span>
+<span class="sd">                \mathbf{k}_t &amp;= -\mathbf{Q}_{\delta \mathbf{u}_t,\delta \mathbf{u}_t}^{-1}</span>
+<span class="sd">                                    \mathbf{q}_{\delta \mathbf{u}_t}         \\</span>
+<span class="sd">                \mathbf{V}_t &amp;= \mathbf{Q}_{\delta \mathbf{x}_t, \delta \mathbf{x}_t}</span>
+<span class="sd">                    + \mathbf{Q}_{\delta \mathbf{x}_t, \delta \mathbf{u}_t}\mathbf{K}_t</span>
+<span class="sd">                    + \mathbf{K}_t^\top\mathbf{Q}_{\delta\mathbf{u}_t,\delta \mathbf{x}_t}</span>
+<span class="sd">                    + \mathbf{K}_t^\top\mathbf{Q}_{\delta\mathbf{u}_t,\delta \mathbf{u}_t}</span>
+<span class="sd">                        \mathbf{K}_t   \\</span>
+<span class="sd">                \mathbf{v}_t &amp;= \mathbf{q}_{\delta \mathbf{x}_t}</span>
+<span class="sd">                    + \mathbf{Q}_{\delta \mathbf{x}_t, \delta \mathbf{u}_t}\mathbf{k}_t</span>
+<span class="sd">                    + \mathbf{K}_t^\top\mathbf{q}_{\delta \mathbf{u}_t}</span>
+<span class="sd">                    + \mathbf{K}_t^\top\mathbf{Q}_{\delta \mathbf{u}_t,</span>
+<span class="sd">                        \delta \mathbf{u}_t}\mathbf{k}_t   \\</span>
+<span class="sd">            \end{align*}</span>
+
+<span class="sd">    - The forward recursion.</span>
+
+<span class="sd">      For :math:`t` = 1 to :math:`T`:</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \begin{align*}</span>
+<span class="sd">                \delta \mathbf{u}_t &amp;= \mathbf{K}_t \delta \mathbf{x}_t + \mathbf{k}_t \\</span>
+<span class="sd">                \mathbf{u}_t &amp;= \delta \mathbf{u}_t + \mathbf{u}_t^* \\</span>
+<span class="sd">                \mathbf{x}_{t+1} &amp;= \mathbf{f}(\mathbf{x}_t, \mathbf{u}_t) \\</span>
+<span class="sd">            \end{align*}</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        For the details of lqr solver and the iterative LQR, please refer to :meth:`LQR`.</span>
+<span class="sd">        Please note that the linear system with quadratic cost only requires one single</span>
+<span class="sd">        iteration.</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        The definition of MPC is cited from this book:</span>
+
+<span class="sd">        * George Nikolakopoulos, Sina Sharif Mansouri, Christoforos Kanellakis, `Aerial</span>
+<span class="sd">          Robotic Workers: Design, Modeling, Control, Vision and Their Applications</span>
+<span class="sd">          &lt;https://doi.org/10.1016/C2017-0-02260-7&gt;`_, Butterworth-Heinemann, 2023.</span>
+
+<span class="sd">        The implementation of MPC is based on Eq. (10)~(13) of this paper:</span>
+
+<span class="sd">        * Amos, Brandon, et al, `Differentiable mpc for end-to-end planning and control</span>
+<span class="sd">          &lt;https://tinyurl.com/3h9tsuyb&gt;`_, Advances in neural information processing</span>
+<span class="sd">          systems, 31, 2018.</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        &gt;&gt;&gt; import torch, pypose as pp</span>
+<span class="sd">        &gt;&gt;&gt; class CartPole(pp.module.NLS):</span>
+<span class="sd">        ...     def __init__(self, dt, length, cartmass, polemass, gravity):</span>
+<span class="sd">        ...         super().__init__()</span>
+<span class="sd">        ...         self.tau = dt</span>
+<span class="sd">        ...         self.length = length</span>
+<span class="sd">        ...         self.cartmass = cartmass</span>
+<span class="sd">        ...         self.polemass = polemass</span>
+<span class="sd">        ...         self.gravity = gravity</span>
+<span class="sd">        ...         self.poleml = self.polemass * self.length</span>
+<span class="sd">        ...         self.totalMass = self.cartmass + self.polemass</span>
+<span class="sd">        ...</span>
+<span class="sd">        ...     def state_transition(self, state, input, t=None):</span>
+<span class="sd">        ...         x, xDot, theta, thetaDot = state.squeeze()</span>
+<span class="sd">        ...         force = input.squeeze()</span>
+<span class="sd">        ...         costheta = torch.cos(theta)</span>
+<span class="sd">        ...         sintheta = torch.sin(theta)</span>
+<span class="sd">        ...         temp = (force + self.poleml * thetaDot**2 * sintheta) / self.totalMass</span>
+<span class="sd">        ...         thetaAcc = (self.gravity * sintheta - costheta * temp) /</span>
+<span class="sd">        ...                     (self.length * (4.0 / 3.0</span>
+<span class="sd">        ...                            - self.polemass * costheta**2 / self.totalMass))</span>
+<span class="sd">        ...         xAcc = temp - self.poleml * thetaAcc * costheta / self.totalMass</span>
+<span class="sd">        ...         _dstate = torch.stack((xDot, xAcc, thetaDot, thetaAcc))</span>
+<span class="sd">        ...         return (state.squeeze() + torch.mul(_dstate, self.tau)).unsqueeze(0)</span>
+<span class="sd">        ...</span>
+<span class="sd">        ...     def observation(self, state, input, t=None):</span>
+<span class="sd">        ...         return state</span>
+<span class="sd">        ...</span>
+<span class="sd">        &gt;&gt;&gt; torch.manual_seed(0)</span>
+<span class="sd">        &gt;&gt;&gt; dt, len, m_cart, m_pole, g = 0.01, 1.5, 20, 10, 9.81</span>
+<span class="sd">        &gt;&gt;&gt; n_batch, T = 1, 5</span>
+<span class="sd">        &gt;&gt;&gt; n_state, n_ctrl = 4, 1</span>
+<span class="sd">        &gt;&gt;&gt; n_sc = n_state + n_ctrl</span>
+<span class="sd">        &gt;&gt;&gt; Q = torch.tile(torch.eye(n_state + n_ctrl, device=device), (n_batch, T, 1, 1))</span>
+<span class="sd">        &gt;&gt;&gt; p = torch.randn(n_batch, T, n_sc)</span>
+<span class="sd">        &gt;&gt;&gt; time  = torch.arange(0, T, device=device) * dt</span>
+<span class="sd">        &gt;&gt;&gt; current_u = torch.sin(time).unsqueeze(1).unsqueeze(0)</span>
+<span class="sd">        &gt;&gt;&gt; x_init = torch.tensor([[0, 0, torch.pi, 0]])</span>
+<span class="sd">        &gt;&gt;&gt; stepper = pp.utils.ReduceToBason(steps=15, verbose=True)</span>
+<span class="sd">        &gt;&gt;&gt; cartPoleSolver = CartPole(dt, len, m_cart, m_pole, g).to(device)</span>
+<span class="sd">        &gt;&gt;&gt; MPC = pp.module.MPC(cartPoleSolver, Q, p, T, stepper=stepper).to(device)</span>
+<span class="sd">        &gt;&gt;&gt; x, u, cost = MPC(dt, x_init, u_init=current_u)</span>
+<span class="sd">        &gt;&gt;&gt; print(&quot;x = &quot;, x)</span>
+<span class="sd">        &gt;&gt;&gt; print(&quot;u = &quot;, u)</span>
+<span class="sd">        x =  tensor([[[ 0.0000e+00,  0.0000e+00,  3.1416e+00,  0.0000e+00],</span>
+<span class="sd">                      [ 0.0000e+00, -3.7711e-04,  3.1416e+00, -1.8856e-04],</span>
+<span class="sd">                      [-3.7711e-06, -3.0693e-04,  3.1416e+00, -1.5347e-04],</span>
+<span class="sd">                      [-6.8404e-06, -2.4288e-04,  3.1416e+00, -1.2136e-04],</span>
+<span class="sd">                      [-9.2692e-06, -2.6634e-04,  3.1416e+00, -1.3293e-04],</span>
+<span class="sd">                      [-1.1933e-05, -2.2073e-04,  3.1416e+00, -1.0991e-04]]])</span>
+<span class="sd">        u =  tensor([[[-0.8485],</span>
+<span class="sd">                      [ 0.1579],</span>
+<span class="sd">                      [ 0.1440],</span>
+<span class="sd">                      [-0.0530],</span>
+<span class="sd">                      [ 0.1023]]])</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">system</span><span class="p">,</span> <span class="n">Q</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">stepper</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">stepper</span> <span class="o">=</span> <span class="n">ReduceToBason</span><span class="p">(</span><span class="n">steps</span><span class="o">=</span><span class="mi">10</span><span class="p">)</span> <span class="k">if</span> <span class="n">stepper</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">stepper</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">stepper</span><span class="o">.</span><span class="n">max_steps</span> <span class="o">-=</span> <span class="mi">1</span> <span class="c1"># n-1 loops, 1 loop with gradient</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">lqr</span> <span class="o">=</span> <span class="n">LQR</span><span class="p">(</span><span class="n">system</span><span class="p">,</span> <span class="n">Q</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">T</span><span class="p">)</span>
+
+<div class="viewcode-block" id="MPC.forward"><a class="viewcode-back" href="../../../../generated/pypose.module.MPC/#pypose.module.MPC.forward">[docs]</a>    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">x_init</span><span class="p">,</span> <span class="n">u_init</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">u_lower</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">u_upper</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">du</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Performs MPC for the discrete system.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            dt (:obj:`int`): The interval (:math:`\delta t`) between two time steps.</span>
+<span class="sd">            x_init (:obj:`Tensor`): The initial state of the system.</span>
+<span class="sd">            u_init (:obj:`Tensor`, optinal): The current inputs of the system along a</span>
+<span class="sd">                trajectory. Default: ``None``.</span>
+<span class="sd">            u_lower (:obj:`Tensor`, optinal): The lower bounds on the controls.</span>
+<span class="sd">                Default: ``None``.</span>
+<span class="sd">            u_upper (:obj:`Tensor`, optinal): The upper bounds on the controls.</span>
+<span class="sd">                Default: ``None``.</span>
+<span class="sd">            du (:obj:`int`, optinal): The amount each component of the controls</span>
+<span class="sd">                is allowed to change in each LQR iteration. Default: ``None``.</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            List of :obj:`Tensor`: A list of tensors including the solved state sequence</span>
+<span class="sd">            :math:`\mathbf{x}`, the solved input sequence :math:`\mathbf{u}`, and the</span>
+<span class="sd">            associated quadratic costs :math:`\mathbf{c}` over the time horizon.</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="n">x</span><span class="p">,</span> <span class="n">u</span> <span class="o">=</span> <span class="kc">None</span><span class="p">,</span> <span class="n">u_init</span>
+        <span class="n">best</span> <span class="o">=</span> <span class="p">{</span><span class="s1">&#39;x&#39;</span><span class="p">:</span> <span class="n">x</span><span class="p">,</span> <span class="s1">&#39;u&#39;</span><span class="p">:</span> <span class="n">u</span><span class="p">,</span> <span class="s1">&#39;cost&#39;</span><span class="p">:</span> <span class="kc">None</span><span class="p">}</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">stepper</span><span class="o">.</span><span class="n">reset</span><span class="p">()</span>
+        <span class="k">with</span> <span class="n">torch</span><span class="o">.</span><span class="n">no_grad</span><span class="p">():</span>
+            <span class="k">while</span> <span class="bp">self</span><span class="o">.</span><span class="n">stepper</span><span class="o">.</span><span class="n">continual</span><span class="p">():</span>
+                <span class="n">x</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">cost</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">lqr</span><span class="p">(</span><span class="n">x_init</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
+                <span class="bp">self</span><span class="o">.</span><span class="n">stepper</span><span class="o">.</span><span class="n">step</span><span class="p">(</span><span class="n">cost</span><span class="p">)</span>
+
+                <span class="k">if</span> <span class="n">best</span><span class="p">[</span><span class="s1">&#39;cost&#39;</span><span class="p">]</span> <span class="o">==</span> <span class="kc">None</span> <span class="ow">or</span> <span class="n">cost</span> <span class="o">&lt;</span> <span class="n">best</span><span class="p">[</span><span class="s1">&#39;cost&#39;</span><span class="p">]:</span>
+                    <span class="n">best</span> <span class="o">=</span> <span class="p">{</span><span class="s1">&#39;x&#39;</span><span class="p">:</span> <span class="n">x</span><span class="p">,</span> <span class="s1">&#39;u&#39;</span><span class="p">:</span> <span class="n">u</span><span class="p">,</span> <span class="s1">&#39;cost&#39;</span><span class="p">:</span> <span class="n">cost</span><span class="p">}</span>
+
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">lqr</span><span class="p">(</span><span class="n">x_init</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">u_traj</span><span class="o">=</span><span class="n">best</span><span class="p">[</span><span class="s1">&#39;u&#39;</span><span class="p">])</span></div></div>
+</pre></div>
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new file mode 100644
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+              <p class="caption" role="heading"><span class="caption-text">Contents:</span></p>
+<ul>
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+          
+            <div role="main" class="main-content" itemscope="itemscope" itemtype="http://schema.org/Article">
+             <article itemprop="articleBody" id="pytorch-article" class="pytorch-article">
+              
+  <h1>Source code for pypose.module.pf</h1><div class="highlight"><pre>
+<span></span><span class="kn">import</span> <span class="nn">torch</span>
+<span class="kn">from</span> <span class="nn">.</span> <span class="kn">import</span> <span class="n">EKF</span>
+<span class="kn">from</span> <span class="nn">..</span> <span class="kn">import</span> <span class="n">bmv</span><span class="p">,</span> <span class="n">bvv</span>
+<span class="kn">import</span> <span class="nn">torch.nn.functional</span> <span class="k">as</span> <span class="nn">F</span>
+<span class="kn">from</span> <span class="nn">torch.distributions</span> <span class="kn">import</span> <span class="n">MultivariateNormal</span>
+
+
+<div class="viewcode-block" id="PF"><a class="viewcode-back" href="../../../../generated/pypose.module.PF/#pypose.module.PF">[docs]</a><span class="k">class</span> <span class="nc">PF</span><span class="p">(</span><span class="n">EKF</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Performs Batched Particle Filter (PF).</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        model (:obj:`System`): The system model to be estimated, a subclass of</span>
+<span class="sd">            :obj:`pypose.module.NLS`.</span>
+<span class="sd">        Q (:obj:`Tensor`, optional): The covariance matrices of system transition noise.</span>
+<span class="sd">            Ignored if provided during each iteration. Default: ``None``</span>
+<span class="sd">        R (:obj:`Tensor`, optional): The covariance matrices of system observation noise.</span>
+<span class="sd">            Ignored if provided during each iteration. Default: ``None``</span>
+<span class="sd">        particles (:obj:`Int`, optional): The number of particle. Default: ``1000``</span>
+
+<span class="sd">    A non-linear system can be described as</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \begin{aligned}</span>
+<span class="sd">            \mathbf{x}_{k+1} &amp;= \mathbf{f}(\mathbf{x}_k, \mathbf{u}_k, t_k) + \mathbf{w}_k,</span>
+<span class="sd">            \quad \mathbf{w}_k \sim \mathcal{N}(\mathbf{0}, \mathbf{Q})  \\</span>
+<span class="sd">            \mathbf{y}_{k} &amp;= \mathbf{g}(\mathbf{x}_k, \mathbf{u}_k, t_k) + \mathbf{v}_k,</span>
+<span class="sd">            \quad \mathbf{v}_k \sim \mathcal{N}(\mathbf{0}, \mathbf{R})</span>
+<span class="sd">        \end{aligned}</span>
+
+<span class="sd">    Particle filter can be described as the following equations, where the subscript</span>
+<span class="sd">    :math:`\cdot_{k}` is omited for simplicity.</span>
+
+<span class="sd">    1. Generate Particles.</span>
+
+<span class="sd">       .. math::</span>
+<span class="sd">           \begin{aligned}</span>
+<span class="sd">             &amp;\mathbf{x}_{k}=\mathbf{p}(\mathbf{x},n\cdot\mathbf{P}_{k},N)&amp;\quad k=1,...,N \\</span>
+<span class="sd">             &amp;\mathbf{P}_{k}=\mathbf{P} &amp;\quad k=1,...,N</span>
+<span class="sd">           \end{aligned}</span>
+
+<span class="sd">       where :math:`N` is the number of Particles and :math:`\mathbf{p}` is the probability</span>
+<span class="sd">       density function (PDF), :math:`n` is the dimension of state.</span>
+
+<span class="sd">    2. Priori State Estimation.</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \mathbf{x}^{-}_{k} = f(\mathbf{x}_{k}, \mathbf{u}_{k}, t)</span>
+
+<span class="sd">    3. Relative Likelihood.</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \begin{aligned}</span>
+<span class="sd">                &amp; \mathbf{q} = \mathbf{p} (\mathbf{y} |\mathbf{x}^{-}_{k}) \\</span>
+<span class="sd">                &amp; \mathbf{q}_{i} = \frac{\mathbf{q}_{i}}{\sum_{j=1}^{N}\mathbf{q}_{j}}</span>
+<span class="sd">            \end{aligned}</span>
+
+<span class="sd">    4. Resample Particles.</span>
+
+<span class="sd">       a. Generate random numbers :math:`r_i \sim U(0,1), i=1,\cdots,N`, where :math:`U`</span>
+<span class="sd">          is uniform distribution.</span>
+
+<span class="sd">       b. Find :math:`j` satisfying :math:`\sum_{m=1}^{j-1}q_m\leq r_i&lt;\sum_{m=1}^{j}q_m`,</span>
+<span class="sd">          then new particle :math:`\mathbf{x}^{r-}_{i}` is set equal to old particle</span>
+<span class="sd">          :math:`\mathbf{x}^{-}_{j}`.</span>
+
+<span class="sd">    5. Refine Posteriori And Covariances.</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \begin{aligned}</span>
+<span class="sd">                &amp; \mathbf{x}^{+} =\frac{1}{N}  \sum_{i=1}^{n}\mathbf{x}^{r-}_{i}   \\</span>
+<span class="sd">                &amp; P^{+} = \sum_{i=1}^{N} (\mathbf{x}^{+} - \mathbf{x}^{r-}_{i})</span>
+<span class="sd">                (\mathbf{x}^{+} - \mathbf{x}^{r-}_{i})^{T} + \mathbf{Q}</span>
+<span class="sd">            \end{aligned}</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        1. Define a discrete-time non-linear system (NLS) model</span>
+
+<span class="sd">        &gt;&gt;&gt; import torch, pypose as pp</span>
+<span class="sd">        &gt;&gt;&gt; class NLS(pp.module.NLS):</span>
+<span class="sd">        ...     def __init__(self):</span>
+<span class="sd">        ...         super().__init__()</span>
+<span class="sd">        ...</span>
+<span class="sd">        ...     def state_transition(self, state, input, t=None):</span>
+<span class="sd">        ...         return state.cos() + input</span>
+<span class="sd">        ...</span>
+<span class="sd">        ...     def observation(self, state, input, t):</span>
+<span class="sd">        ...         return state.sin() + input</span>
+
+<span class="sd">        2. Create a model and filter</span>
+
+<span class="sd">        &gt;&gt;&gt; model = NLS()</span>
+<span class="sd">        &gt;&gt;&gt; pf = pp.module.PF(model)</span>
+
+<span class="sd">        3. Prepare data</span>
+
+<span class="sd">        &gt;&gt;&gt; B, T, N = 3, 5, 2 # batchsize, steps, state dim</span>
+<span class="sd">        &gt;&gt;&gt; states = torch.zeros(B, T, N)</span>
+<span class="sd">        &gt;&gt;&gt; inputs = torch.randn(B, T, N)</span>
+<span class="sd">        &gt;&gt;&gt; observ = torch.zeros(B, T, N)</span>
+<span class="sd">        &gt;&gt;&gt; # std of transition, observation, and estimation</span>
+<span class="sd">        &gt;&gt;&gt; q, r, p = 0.1, 0.1, 10</span>
+<span class="sd">        &gt;&gt;&gt; Q = torch.eye(N).repeat(B, 1, 1) * q**2</span>
+<span class="sd">        &gt;&gt;&gt; R = torch.eye(N).repeat(B, 1, 1) * r**2</span>
+<span class="sd">        &gt;&gt;&gt; P = torch.eye(N).repeat(B, T, 1, 1) * p**2</span>
+<span class="sd">        &gt;&gt;&gt; estim = torch.randn(B, T, N) * p</span>
+
+<span class="sd">        4. Perform PF prediction. Note that estimation error becomes smaller with more steps.</span>
+
+<span class="sd">        &gt;&gt;&gt; for i in range(T - 1):</span>
+<span class="sd">        ...     w = q * torch.randn(B, N) # transition noise</span>
+<span class="sd">        ...     v = r * torch.randn(B, N) # observation noise</span>
+<span class="sd">        ...     states[:, i+1], observ[:, i] = model(states[:, i] + w, inputs[:, i])</span>
+<span class="sd">        ...     estim[:, i+1], P[:, i+1] = pf(estim[:, i], observ[:, i] + v, inputs[:, i], P[:, i], Q, R)</span>
+<span class="sd">        ... print(&#39;Est error:&#39;, (states - estim).norm(dim=-1))</span>
+<span class="sd">        Est error: tensor([[13.5084,  1.3604,  1.1626,  0.4157,  0.7117],</span>
+<span class="sd">                           [14.4350,  2.0775,  1.0280,  0.2258,  0.4144],</span>
+<span class="sd">                           [16.8538,  2.4473,  1.1300,  0.5104,  0.1647]])</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        Implementation is based on Section 15.2 of this book</span>
+
+<span class="sd">        * Dan Simon, `Optimal State Estimation: Kalman, H∞, and Nonlinear Approaches</span>
+<span class="sd">          &lt;https://onlinelibrary.wiley.com/doi/book/10.1002/0470045345&gt;`_,</span>
+<span class="sd">          Cleveland State University, 2006</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">model</span><span class="p">,</span> <span class="n">Q</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">R</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">particles</span><span class="o">=</span><span class="mi">1000</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">(</span><span class="n">model</span><span class="p">,</span> <span class="n">Q</span><span class="p">,</span> <span class="n">R</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">particles</span> <span class="o">=</span> <span class="n">particles</span>
+
+<div class="viewcode-block" id="PF.forward"><a class="viewcode-back" href="../../../../generated/pypose.module.PF/#pypose.module.PF.forward">[docs]</a>    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">Q</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">R</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">t</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Performs one step estimation.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            x (:obj:`Tensor`): estimated system state of previous step.</span>
+<span class="sd">            y (:obj:`Tensor`): system observation at current step (measurement).</span>
+<span class="sd">            u (:obj:`Tensor`): system input at current step.</span>
+<span class="sd">            P (:obj:`Tensor`): state estimation covariance of previous step.</span>
+<span class="sd">            Q (:obj:`Tensor`, optional): covariance of system transition model.</span>
+<span class="sd">                Default: ``None``.</span>
+<span class="sd">            R (:obj:`Tensor`, optional): covariance of system observation model.</span>
+<span class="sd">                Default: ``None``.</span>
+<span class="sd">            t (:obj:`int`, optional): set system timestamp for estimation.</span>
+<span class="sd">                If ``None``, current system time is used. Default: ``None``.</span>
+
+<span class="sd">        Return:</span>
+<span class="sd">            list of :obj:`Tensor`: posteriori state and covariance estimation</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="c1"># Upper cases are matrices, lower cases are vectors</span>
+        <span class="n">Q</span> <span class="o">=</span> <span class="n">Q</span> <span class="k">if</span> <span class="n">Q</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span> <span class="k">else</span> <span class="bp">self</span><span class="o">.</span><span class="n">Q</span>
+        <span class="n">R</span> <span class="o">=</span> <span class="n">R</span> <span class="k">if</span> <span class="n">R</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span> <span class="k">else</span> <span class="bp">self</span><span class="o">.</span><span class="n">R</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">model</span><span class="o">.</span><span class="n">set_refpoint</span><span class="p">(</span><span class="n">state</span><span class="o">=</span><span class="n">x</span><span class="p">,</span> <span class="nb">input</span><span class="o">=</span><span class="n">u</span><span class="p">,</span> <span class="n">t</span><span class="o">=</span><span class="n">t</span><span class="p">)</span>
+
+        <span class="n">n</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
+        <span class="n">xp</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">generate_particles</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span> <span class="o">*</span> <span class="n">P</span><span class="p">)</span>
+        <span class="n">xs</span><span class="p">,</span> <span class="n">ye</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">model</span><span class="p">(</span><span class="n">xp</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
+        <span class="n">q</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">relative_likelihood</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">ye</span><span class="p">,</span> <span class="n">R</span><span class="p">)</span>
+        <span class="n">xr</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">resample_particles</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="n">xs</span><span class="p">)</span>
+
+        <span class="n">x</span> <span class="o">=</span> <span class="n">xr</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">dim</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
+        <span class="n">ex</span> <span class="o">=</span> <span class="n">xr</span> <span class="o">-</span> <span class="n">x</span>
+        <span class="n">P</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">compute_cov</span><span class="p">(</span><span class="n">ex</span><span class="p">,</span> <span class="n">ex</span><span class="p">,</span> <span class="n">Q</span><span class="p">)</span>
+
+        <span class="k">return</span> <span class="n">x</span><span class="p">,</span> <span class="n">P</span></div>
+
+<div class="viewcode-block" id="PF.generate_particles"><a class="viewcode-back" href="../../../../generated/pypose.module.PF/#pypose.module.PF.generate_particles">[docs]</a>    <span class="k">def</span> <span class="nf">generate_particles</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">P</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Randomly generate particles</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            x (:obj:`Tensor`): estimated system state of previous step</span>
+<span class="sd">            P (:obj:`Tensor`): state estimation covariance of previous step</span>
+
+<span class="sd">        Return:</span>
+<span class="sd">            list of :obj:`Tensor`: particles</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="n">m</span> <span class="o">=</span> <span class="n">MultivariateNormal</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">P</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">m</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">Size</span><span class="p">([</span><span class="bp">self</span><span class="o">.</span><span class="n">particles</span><span class="p">]))</span></div>
+
+<div class="viewcode-block" id="PF.relative_likelihood"><a class="viewcode-back" href="../../../../generated/pypose.module.PF/#pypose.module.PF.relative_likelihood">[docs]</a>    <span class="k">def</span> <span class="nf">relative_likelihood</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">ye</span><span class="p">,</span> <span class="n">R</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Compute the relative likelihood</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">return</span> <span class="n">F</span><span class="o">.</span><span class="n">softmax</span><span class="p">(</span><span class="n">MultivariateNormal</span><span class="p">(</span><span class="n">ye</span><span class="p">,</span> <span class="n">R</span><span class="p">)</span><span class="o">.</span><span class="n">log_prob</span><span class="p">(</span><span class="n">y</span><span class="p">),</span> <span class="n">dim</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span></div>
+
+<div class="viewcode-block" id="PF.resample_particles"><a class="viewcode-back" href="../../../../generated/pypose.module.PF/#pypose.module.PF.resample_particles">[docs]</a>    <span class="k">def</span> <span class="nf">resample_particles</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">q</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Resample the set of a posteriori particles</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="n">r</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">shape</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">x</span><span class="o">.</span><span class="n">dtype</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="n">x</span><span class="o">.</span><span class="n">device</span><span class="p">)</span>
+        <span class="n">index</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">searchsorted</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">cumsum</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="n">dim</span><span class="o">=</span><span class="mi">0</span><span class="p">),</span> <span class="n">r</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">x</span><span class="p">[</span><span class="n">index</span><span class="p">,</span> <span class="n">torch</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="mi">0</span><span class="p">)]</span> <span class="k">if</span> <span class="n">x</span><span class="o">.</span><span class="n">dim</span><span class="p">()</span><span class="o">==</span><span class="mi">3</span> <span class="k">else</span> <span class="n">x</span><span class="p">[</span><span class="n">index</span><span class="p">]</span></div>
+
+<div class="viewcode-block" id="PF.compute_cov"><a class="viewcode-back" href="../../../../generated/pypose.module.PF/#pypose.module.PF.compute_cov">[docs]</a>    <span class="k">def</span> <span class="nf">compute_cov</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">Q</span><span class="o">=</span><span class="mi">0</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&#39;&#39;&#39;Compute covariance of two set of variables.&#39;&#39;&#39;</span>
+        <span class="k">return</span> <span class="n">Q</span> <span class="o">+</span> <span class="n">bvv</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">dim</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span></div></div>
+</pre></div>
+
+             </article>
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+              
+  <h1>Source code for pypose.module.pnp</h1><div class="highlight"><pre>
+<span></span><span class="kn">import</span> <span class="nn">torch</span>
+<span class="kn">from</span> <span class="nn">torch</span> <span class="kn">import</span> <span class="n">nn</span>
+<span class="kn">from</span> <span class="nn">torch.linalg</span> <span class="kn">import</span> <span class="n">vecdot</span>
+<span class="kn">from</span> <span class="nn">torch</span> <span class="kn">import</span> <span class="n">broadcast_shapes</span>
+
+<span class="kn">from</span> <span class="nn">..</span> <span class="kn">import</span> <span class="n">bmv</span>
+<span class="kn">from</span> <span class="nn">..optim</span> <span class="kn">import</span> <span class="n">GaussNewton</span>
+<span class="kn">from</span> <span class="nn">..optim.solver</span> <span class="kn">import</span> <span class="n">LSTSQ</span>
+<span class="kn">from</span> <span class="nn">..optim.scheduler</span> <span class="kn">import</span> <span class="n">StopOnPlateau</span>
+<span class="kn">from</span> <span class="nn">..function</span> <span class="kn">import</span> <span class="n">reprojerr</span><span class="p">,</span> <span class="n">cart2homo</span><span class="p">,</span> <span class="n">svdtf</span>
+
+
+<span class="k">class</span> <span class="nc">BetaObjective</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+    <span class="c1"># Optimize the beta according to the objective in the ePnP paper.</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">beta</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">beta</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">nn</span><span class="o">.</span><span class="n">Parameter</span><span class="p">(</span><span class="n">beta</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">i</span> <span class="o">=</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">j</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">base_w</span><span class="p">,</span> <span class="n">nullv</span><span class="p">):</span>
+        <span class="c1"># See Eq. 15 in the paper</span>
+        <span class="n">base_c</span> <span class="o">=</span> <span class="n">bmv</span><span class="p">(</span><span class="n">nullv</span><span class="o">.</span><span class="n">mT</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">beta</span><span class="p">)</span><span class="o">.</span><span class="n">unflatten</span><span class="p">(</span><span class="n">dim</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">sizes</span><span class="o">=</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
+        <span class="n">dist_c</span> <span class="o">=</span> <span class="p">(</span><span class="n">base_c</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span> <span class="o">-</span> <span class="n">base_c</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">j</span><span class="p">,</span> <span class="p">:])</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">dim</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span>
+        <span class="n">dist_w</span> <span class="o">=</span> <span class="p">(</span><span class="n">base_w</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span> <span class="o">-</span> <span class="n">base_w</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">j</span><span class="p">,</span> <span class="p">:])</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">dim</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">dist_w</span> <span class="o">-</span> <span class="n">dist_c</span>
+
+
+<div class="viewcode-block" id="EPnP"><a class="viewcode-back" href="../../../../generated/pypose.module.EPnP/#pypose.module.EPnP">[docs]</a><span class="k">class</span> <span class="nc">EPnP</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Batched EPnP Solver - a non-iterative :math:`\mathcal{O}(n)` solution to the</span>
+<span class="sd">    Perspective-:math:`n`-Point (PnP) problem for :math:`n \geq 4`.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        intrinsics (``torch.Tensor``, optional): The camera intrinsics.</span>
+<span class="sd">            The shape is (..., 3, 3). Default: None</span>
+<span class="sd">        refine (``bool``, optional): refine the solution with Gaussian-Newton optimizer.</span>
+<span class="sd">            Default: ``True``.</span>
+
+<span class="sd">    Assume each of the :math:`n` points in the world coordinate is :math:`p^w_i` and in</span>
+<span class="sd">    camera coordinate is :math:`p^c_i`.</span>
+<span class="sd">    They are represented by weighted sums of the four virtual control points,</span>
+<span class="sd">    :math:`c^w_j` and :math:`c^c_j` in the world and camera coordinate, respectively.</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \begin{aligned}</span>
+<span class="sd">            &amp; p^w_i = \sum^4_{j=1}{\alpha_{ij}c^w_j} \\</span>
+<span class="sd">            &amp; p^c_i = \sum^4_{j=1}{\alpha_{ij}c^c_j} \\</span>
+<span class="sd">            &amp; \sum^4_{j=1}{\alpha_{ij}} = 1</span>
+<span class="sd">        \end{aligned}</span>
+
+<span class="sd">    Let the projection matrix be :math:`P = K[R|T]`, where :math:`K` is the camera</span>
+<span class="sd">    intrinsics, :math:`R` is the rotation matrix and :math:`T` is the translation</span>
+<span class="sd">    vector. Then we have</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \begin{aligned}</span>
+<span class="sd">            s_i p^{\text{img}}_i &amp;= K\sum^4_{j=1}{\alpha_{ij}c^c_j},</span>
+<span class="sd">        \end{aligned}</span>
+
+<span class="sd">    where :math:`p^{\text{img}}_i` is pixels in homogeneous form :math:`(u_i, v_i, 1)`,</span>
+<span class="sd">    :math:`s_i` is the scale factor. Let the control point in camera coordinate</span>
+<span class="sd">    represented by :math:`c^c_j = (x^c_j, y^c_j, z^c_j)`. Rearranging the projection</span>
+<span class="sd">    equation yields two linear equations for each of the :math:`n` points:</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \begin{aligned}</span>
+<span class="sd">            \sum^4_{j=1}{\alpha_{ij}f_xx^c_j + \alpha_{ij}(u_0 - u_i)z^c_j} &amp;= 0 \\</span>
+<span class="sd">            \sum^4_{j=1}{\alpha_{ij}f_yy^c_j + \alpha_{ij}(v_0 - v_i)z^c_j} &amp;= 0</span>
+<span class="sd">        \end{aligned}</span>
+
+<span class="sd">    Assume :math:`\mathbf{x} = \begin{bmatrix} c^{c^T}_1 &amp; c^{c^T}_2 &amp; c^{c^T}_3 &amp;</span>
+<span class="sd">    c^{c^T}_4 \end{bmatrix}^T`, then the two equations form a system :math:`Mx = 0`</span>
+<span class="sd">    considering all of the :math:`n` points. Its solution can be expressed as</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \begin{aligned}</span>
+<span class="sd">            x &amp;= \sum^4_{i=1}{\beta_iv_i},</span>
+<span class="sd">        \end{aligned}</span>
+
+<span class="sd">    where :math:`v_i` is the null vectors of matrix :math:`M^T M` corresponding to its</span>
+<span class="sd">    least 4 eigenvalues.</span>
+
+<span class="sd">    The final step involves calculating the coefficients :math:`\beta_i`. Optionally, the</span>
+<span class="sd">    Gauss-Newton algorithm can be used to refine the solution of :math:`\beta_i`.</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        &gt;&gt;&gt; import torch, pypose as pp</span>
+<span class="sd">        &gt;&gt;&gt; f, (H, W) = 2, (9, 9) # focal length and image height, width</span>
+<span class="sd">        &gt;&gt;&gt; intrinsics = torch.tensor([[f, 0, H / 2],</span>
+<span class="sd">        ...                            [0, f, W / 2],</span>
+<span class="sd">        ...                            [0, 0,   1  ]])</span>
+<span class="sd">        &gt;&gt;&gt; object = torch.tensor([[2., 0., 2.],</span>
+<span class="sd">        ...                        [1., 0., 2.],</span>
+<span class="sd">        ...                        [0., 1., 1.],</span>
+<span class="sd">        ...                        [0., 0., 1.],</span>
+<span class="sd">        ...                        [1., 0., 1.],</span>
+<span class="sd">        ...                        [5., 5., 3.]])</span>
+<span class="sd">        &gt;&gt;&gt; pixels = pp.point2pixel(object, intrinsics)</span>
+<span class="sd">        &gt;&gt;&gt; pose = pp.SE3([ 0., -8,  0.,  0., -0.3827,  0.,  0.9239])</span>
+<span class="sd">        &gt;&gt;&gt; points = pose.Inv() @ object</span>
+<span class="sd">        ...</span>
+<span class="sd">        &gt;&gt;&gt; epnp = pp.module.EPnP(intrinsics)</span>
+<span class="sd">        &gt;&gt;&gt; pose = epnp(points, pixels)</span>
+<span class="sd">        SE3Type LieTensor:</span>
+<span class="sd">        LieTensor([ 3.9816e-05, -8.0000e+00,  5.8174e-05, -3.3186e-06, -3.8271e-01,</span>
+<span class="sd">                    3.6321e-06,  9.2387e-01])</span>
+
+<span class="sd">    Warning:</span>
+<span class="sd">        Currently this module only supports batched rectified camera intrinsics, which can</span>
+<span class="sd">        be defined in the form:</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            K = \begin{pmatrix}</span>
+<span class="sd">                    f_x &amp;   0 &amp; c_x \\</span>
+<span class="sd">                    0   &amp; f_y &amp; c_y \\</span>
+<span class="sd">                    0   &amp;   0 &amp;   1</span>
+<span class="sd">                \end{pmatrix}</span>
+
+<span class="sd">        The full form of camera intrinsics will be supported in a future release.</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        The implementation is based on the paper</span>
+
+<span class="sd">        * Francesc Moreno-Noguer, Vincent Lepetit, and Pascal Fua, `EPnP: An Accurate O(n)</span>
+<span class="sd">          Solution to the PnP Problem &lt;https://doi.org/10.1007/s11263-008-0152-6&gt;`_,</span>
+<span class="sd">          International Journal of Computer Vision (IJCV), 2009.</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">intrinsics</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">refine</span><span class="o">=</span><span class="kc">True</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">refine</span> <span class="o">=</span> <span class="n">refine</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">solver</span> <span class="o">=</span> <span class="n">LSTSQ</span><span class="p">()</span>
+        <span class="k">if</span> <span class="n">intrinsics</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">register_buffer</span><span class="p">(</span><span class="s1">&#39;intrinsics&#39;</span><span class="p">,</span> <span class="n">intrinsics</span><span class="p">)</span>
+
+<div class="viewcode-block" id="EPnP.forward"><a class="viewcode-back" href="../../../../generated/pypose.module.EPnP/#pypose.module.EPnP.forward">[docs]</a>    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">points</span><span class="p">,</span> <span class="n">pixels</span><span class="p">,</span> <span class="n">intrinsics</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Args:</span>
+<span class="sd">            points (``torch.Tensor``): 3D object points in the world coordinates.</span>
+<span class="sd">                Shape (..., N, 3)</span>
+<span class="sd">            pixels (``torch.Tensor``): 2D image points, which are the projection of</span>
+<span class="sd">                object points. Shape (..., N, 2)</span>
+<span class="sd">            intrinsics (torch.Tensor, optional): camera intrinsics. Shape (..., 3, 3).</span>
+<span class="sd">                Setting it to any non-``None`` value will override the default intrinsics</span>
+<span class="sd">                kept in the module.</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            ``LieTensor``: estimated pose (``SE3type``) for the camera.</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">assert</span> <span class="n">pixels</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span> <span class="o">==</span> <span class="n">points</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span> <span class="o">&gt;=</span> <span class="mi">4</span><span class="p">,</span> \
+            <span class="s2">&quot;Number of points/pixels cannot be smaller than 4.&quot;</span>
+        <span class="n">intrinsics</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">intrinsics</span> <span class="k">if</span> <span class="n">intrinsics</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">intrinsics</span>
+        <span class="n">broadcast_shapes</span><span class="p">(</span><span class="n">points</span><span class="o">.</span><span class="n">shape</span><span class="p">[:</span><span class="o">-</span><span class="mi">2</span><span class="p">],</span> <span class="n">pixels</span><span class="o">.</span><span class="n">shape</span><span class="p">[:</span><span class="o">-</span><span class="mi">2</span><span class="p">],</span> <span class="n">intrinsics</span><span class="o">.</span><span class="n">shape</span><span class="p">[:</span><span class="o">-</span><span class="mi">2</span><span class="p">])</span>
+
+        <span class="c1"># Select naive and calculate alpha in the world coordinate</span>
+        <span class="n">bases</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_svd_basis</span><span class="p">(</span><span class="n">points</span><span class="p">)</span>
+        <span class="n">alpha</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_compute_alpha</span><span class="p">(</span><span class="n">points</span><span class="p">,</span> <span class="n">bases</span><span class="p">)</span>
+        <span class="n">nullv</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_compute_nullv</span><span class="p">(</span><span class="n">pixels</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">intrinsics</span><span class="p">)</span>
+        <span class="n">l_mat</span><span class="p">,</span> <span class="n">rho</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_compute_lrho</span><span class="p">(</span><span class="n">nullv</span><span class="p">,</span> <span class="n">bases</span><span class="p">)</span>
+        <span class="n">betas</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_compute_betas</span><span class="p">(</span><span class="n">l_mat</span><span class="p">,</span> <span class="n">rho</span><span class="p">)</span>
+        <span class="n">poses</span><span class="p">,</span> <span class="n">scales</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_compute_solution</span><span class="p">(</span><span class="n">betas</span><span class="p">,</span> <span class="n">nullv</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">points</span><span class="p">)</span>
+        <span class="n">errors</span> <span class="o">=</span> <span class="n">reprojerr</span><span class="p">(</span><span class="n">points</span><span class="p">,</span> <span class="n">pixels</span><span class="p">,</span> <span class="n">intrinsics</span><span class="p">,</span> <span class="n">poses</span><span class="p">,</span> <span class="n">reduction</span><span class="o">=</span><span class="s1">&#39;norm&#39;</span><span class="p">)</span>
+        <span class="n">pose</span><span class="p">,</span> <span class="n">beta</span><span class="p">,</span> <span class="n">scale</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_best_solution</span><span class="p">(</span><span class="n">errors</span><span class="p">,</span> <span class="n">poses</span><span class="p">,</span> <span class="n">betas</span><span class="p">,</span> <span class="n">scales</span><span class="p">)</span>
+
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">refine</span><span class="p">:</span>
+            <span class="n">beta</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_refine</span><span class="p">(</span><span class="n">beta</span> <span class="o">*</span> <span class="n">scale</span><span class="p">,</span> <span class="n">nullv</span><span class="p">,</span> <span class="n">bases</span><span class="p">)</span>
+            <span class="n">pose</span><span class="p">,</span> <span class="n">scale</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_compute_solution</span><span class="p">(</span><span class="n">beta</span><span class="p">,</span> <span class="n">nullv</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">points</span><span class="p">)</span>
+
+        <span class="k">return</span> <span class="n">pose</span></div>
+
+    <span class="k">def</span> <span class="nf">_compute_solution</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">beta</span><span class="p">,</span> <span class="n">nullv</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">points</span><span class="p">):</span>
+        <span class="n">bases</span> <span class="o">=</span> <span class="n">bmv</span><span class="p">(</span><span class="n">nullv</span><span class="o">.</span><span class="n">mT</span><span class="p">,</span> <span class="n">beta</span><span class="p">)</span>
+        <span class="n">bases</span><span class="p">,</span> <span class="n">transp</span><span class="p">,</span> <span class="n">scale</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_compute_scale</span><span class="p">(</span><span class="n">bases</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">points</span><span class="p">)</span>
+        <span class="n">pose</span> <span class="o">=</span> <span class="n">svdtf</span><span class="p">(</span><span class="n">points</span><span class="p">,</span> <span class="n">transp</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">pose</span><span class="p">,</span> <span class="n">scale</span>
+
+    <span class="nd">@staticmethod</span>
+    <span class="k">def</span> <span class="nf">_best_solution</span><span class="p">(</span><span class="n">errors</span><span class="p">,</span> <span class="n">poses</span><span class="p">,</span> <span class="n">betas</span><span class="p">,</span> <span class="n">scales</span><span class="p">):</span>
+        <span class="n">_</span><span class="p">,</span> <span class="n">idx</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">min</span><span class="p">(</span><span class="n">errors</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">dim</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">keepdim</span><span class="o">=</span><span class="kc">True</span><span class="p">),</span> <span class="n">dim</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">keepdim</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+        <span class="n">pose</span> <span class="o">=</span> <span class="n">poses</span><span class="o">.</span><span class="n">gather</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">index</span><span class="o">=</span><span class="n">idx</span><span class="o">.</span><span class="n">tile</span><span class="p">(</span><span class="n">poses</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)))</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
+        <span class="n">beta</span> <span class="o">=</span> <span class="n">betas</span><span class="o">.</span><span class="n">gather</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">index</span><span class="o">=</span><span class="n">idx</span><span class="o">.</span><span class="n">tile</span><span class="p">(</span><span class="n">betas</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)))</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
+        <span class="n">scale</span> <span class="o">=</span> <span class="n">scales</span><span class="o">.</span><span class="n">gather</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">index</span><span class="o">=</span><span class="n">idx</span><span class="p">)</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">pose</span><span class="p">,</span> <span class="n">beta</span><span class="p">,</span> <span class="n">scale</span>
+
+    <span class="nd">@staticmethod</span>
+    <span class="k">def</span> <span class="nf">_refine</span><span class="p">(</span><span class="n">beta</span><span class="p">,</span> <span class="n">nullv</span><span class="p">,</span> <span class="n">bases</span><span class="p">):</span>
+        <span class="c1"># Refine beta according to Eq 15 in the paper</span>
+        <span class="n">model</span> <span class="o">=</span> <span class="n">BetaObjective</span><span class="p">(</span><span class="n">beta</span><span class="p">)</span>
+        <span class="n">optim</span> <span class="o">=</span> <span class="n">GaussNewton</span><span class="p">(</span><span class="n">model</span><span class="p">,</span> <span class="n">solver</span><span class="o">=</span><span class="n">LSTSQ</span><span class="p">())</span>
+        <span class="n">scheduler</span> <span class="o">=</span> <span class="n">StopOnPlateau</span><span class="p">(</span><span class="n">optim</span><span class="p">,</span> <span class="n">steps</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="n">patience</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
+        <span class="n">scheduler</span><span class="o">.</span><span class="n">optimize</span><span class="p">(</span><span class="nb">input</span><span class="o">=</span><span class="p">(</span><span class="n">bases</span><span class="p">,</span> <span class="n">nullv</span><span class="p">))</span>
+        <span class="c1"># Retain the grad of initial beta after optimization.</span>
+        <span class="k">return</span> <span class="n">beta</span> <span class="o">+</span> <span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">beta</span> <span class="o">-</span> <span class="n">beta</span><span class="p">)</span><span class="o">.</span><span class="n">detach</span><span class="p">()</span>
+
+    <span class="nd">@staticmethod</span>
+    <span class="k">def</span> <span class="nf">_svd_basis</span><span class="p">(</span><span class="n">points</span><span class="p">):</span>
+        <span class="c1"># Select 4 virtual control points with SVD</span>
+        <span class="n">center</span> <span class="o">=</span> <span class="n">points</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">dim</span><span class="o">=-</span><span class="mi">2</span><span class="p">,</span> <span class="n">keepdim</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+        <span class="n">translated</span> <span class="o">=</span> <span class="n">points</span> <span class="o">-</span> <span class="n">center</span>
+        <span class="n">u</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">vh</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">svd</span><span class="p">(</span><span class="n">translated</span><span class="o">.</span><span class="n">mT</span> <span class="o">@</span> <span class="n">translated</span><span class="p">)</span>
+        <span class="n">controls</span> <span class="o">=</span> <span class="n">center</span> <span class="o">+</span> <span class="n">s</span><span class="o">.</span><span class="n">sqrt</span><span class="p">()</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">vh</span><span class="o">.</span><span class="n">mT</span>
+        <span class="k">return</span> <span class="n">torch</span><span class="o">.</span><span class="n">cat</span><span class="p">([</span><span class="n">center</span><span class="p">,</span> <span class="n">controls</span><span class="p">],</span> <span class="n">dim</span><span class="o">=-</span><span class="mi">2</span><span class="p">)</span>
+
+    <span class="nd">@staticmethod</span>
+    <span class="k">def</span> <span class="nf">_compute_alpha</span><span class="p">(</span><span class="n">points</span><span class="p">,</span> <span class="n">bases</span><span class="p">):</span>
+        <span class="c1"># Compute weights (..., N, 4) corresponded to bases (..., N, 4)</span>
+        <span class="c1"># for points (..., N, 3). Check equation 1 in paper for details.</span>
+        <span class="n">points</span><span class="p">,</span> <span class="n">bases</span> <span class="o">=</span> <span class="n">cart2homo</span><span class="p">(</span><span class="n">points</span><span class="p">),</span> <span class="n">cart2homo</span><span class="p">(</span><span class="n">bases</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">solve</span><span class="p">(</span><span class="n">A</span><span class="o">=</span><span class="n">bases</span><span class="p">,</span> <span class="n">B</span><span class="o">=</span><span class="n">points</span><span class="p">,</span> <span class="n">left</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+
+    <span class="nd">@staticmethod</span>
+    <span class="k">def</span> <span class="nf">_compute_nullv</span><span class="p">(</span><span class="n">pixels</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">intrinsics</span><span class="p">,</span> <span class="n">least</span><span class="o">=</span><span class="mi">4</span><span class="p">):</span>
+        <span class="c1"># Construct M matrix and find its null eigenvectors with the least eigenvalues</span>
+        <span class="c1"># Check equation 7 in paper for more details.</span>
+        <span class="c1"># pixels (..., N, 2); alpha (..., N, 4); intrinsics (..., 3, 3)</span>
+        <span class="n">batch</span><span class="p">,</span> <span class="n">point</span> <span class="o">=</span> <span class="n">pixels</span><span class="o">.</span><span class="n">shape</span><span class="p">[:</span><span class="o">-</span><span class="mi">2</span><span class="p">],</span> <span class="n">pixels</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span>
+        <span class="n">u</span><span class="p">,</span> <span class="n">v</span> <span class="o">=</span> <span class="n">pixels</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">pixels</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
+        <span class="n">fu</span><span class="p">,</span> <span class="n">u0</span> <span class="o">=</span> <span class="n">intrinsics</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="kc">None</span><span class="p">],</span> <span class="n">intrinsics</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="kc">None</span><span class="p">]</span>
+        <span class="n">fv</span><span class="p">,</span> <span class="n">v0</span> <span class="o">=</span> <span class="n">intrinsics</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="kc">None</span><span class="p">],</span> <span class="n">intrinsics</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="kc">None</span><span class="p">]</span>
+        <span class="n">a0</span><span class="p">,</span> <span class="n">a1</span><span class="p">,</span> <span class="n">a2</span><span class="p">,</span> <span class="n">a3</span> <span class="o">=</span> <span class="n">alpha</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">alpha</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">alpha</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="n">alpha</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">3</span><span class="p">]</span>
+        <span class="n">O</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">zeros_like</span><span class="p">(</span><span class="n">a1</span><span class="p">)</span>
+        <span class="n">M</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">stack</span><span class="p">([</span><span class="n">a0</span> <span class="o">*</span> <span class="n">fu</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">a0</span> <span class="o">*</span> <span class="p">(</span><span class="n">u0</span> <span class="o">-</span> <span class="n">u</span><span class="p">),</span>
+                         <span class="n">a1</span> <span class="o">*</span> <span class="n">fu</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">a1</span> <span class="o">*</span> <span class="p">(</span><span class="n">u0</span> <span class="o">-</span> <span class="n">u</span><span class="p">),</span>
+                         <span class="n">a2</span> <span class="o">*</span> <span class="n">fu</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">a2</span> <span class="o">*</span> <span class="p">(</span><span class="n">u0</span> <span class="o">-</span> <span class="n">u</span><span class="p">),</span>
+                         <span class="n">a3</span> <span class="o">*</span> <span class="n">fu</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">a3</span> <span class="o">*</span> <span class="p">(</span><span class="n">u0</span> <span class="o">-</span> <span class="n">u</span><span class="p">),</span>
+                         <span class="n">O</span><span class="p">,</span> <span class="n">a0</span> <span class="o">*</span> <span class="n">fv</span><span class="p">,</span> <span class="n">a0</span> <span class="o">*</span> <span class="p">(</span><span class="n">v0</span> <span class="o">-</span> <span class="n">v</span><span class="p">),</span>
+                         <span class="n">O</span><span class="p">,</span> <span class="n">a1</span> <span class="o">*</span> <span class="n">fv</span><span class="p">,</span> <span class="n">a1</span> <span class="o">*</span> <span class="p">(</span><span class="n">v0</span> <span class="o">-</span> <span class="n">v</span><span class="p">),</span>
+                         <span class="n">O</span><span class="p">,</span> <span class="n">a2</span> <span class="o">*</span> <span class="n">fv</span><span class="p">,</span> <span class="n">a2</span> <span class="o">*</span> <span class="p">(</span><span class="n">v0</span> <span class="o">-</span> <span class="n">v</span><span class="p">),</span>
+                         <span class="n">O</span><span class="p">,</span> <span class="n">a3</span> <span class="o">*</span> <span class="n">fv</span><span class="p">,</span> <span class="n">a3</span> <span class="o">*</span> <span class="p">(</span><span class="n">v0</span> <span class="o">-</span> <span class="n">v</span><span class="p">)],</span> <span class="n">dim</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="o">*</span><span class="n">batch</span><span class="p">,</span> <span class="n">point</span> <span class="o">*</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">12</span><span class="p">)</span>
+        <span class="n">eigenvalues</span><span class="p">,</span> <span class="n">eigenvectors</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">eig</span><span class="p">(</span><span class="n">M</span><span class="o">.</span><span class="n">mT</span> <span class="o">@</span> <span class="n">M</span><span class="p">)</span>
+        <span class="n">eigenvalues</span><span class="p">,</span> <span class="n">eigenvectors</span> <span class="o">=</span> <span class="n">eigenvalues</span><span class="o">.</span><span class="n">real</span><span class="p">,</span> <span class="n">eigenvectors</span><span class="o">.</span><span class="n">real</span>
+        <span class="n">_</span><span class="p">,</span> <span class="n">index</span> <span class="o">=</span> <span class="n">eigenvalues</span><span class="o">.</span><span class="n">topk</span><span class="p">(</span><span class="n">k</span><span class="o">=</span><span class="n">least</span><span class="p">,</span> <span class="n">largest</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="nb">sorted</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>  <span class="c1"># (batch, 4)</span>
+        <span class="n">index</span> <span class="o">=</span> <span class="n">index</span><span class="o">.</span><span class="n">flip</span><span class="p">(</span><span class="n">dims</span><span class="o">=</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">tile</span><span class="p">((</span><span class="mi">1</span><span class="p">,)</span> <span class="o">*</span> <span class="nb">len</span><span class="p">(</span><span class="n">batch</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
+        <span class="k">return</span> <span class="n">torch</span><span class="o">.</span><span class="n">gather</span><span class="p">(</span><span class="n">eigenvectors</span><span class="p">,</span> <span class="n">dim</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">index</span><span class="o">=</span><span class="n">index</span><span class="p">)</span><span class="o">.</span><span class="n">mT</span>  <span class="c1"># (batch, 4, 12)</span>
+
+    <span class="nd">@staticmethod</span>
+    <span class="k">def</span> <span class="nf">_compute_lrho</span><span class="p">(</span><span class="n">nullv</span><span class="p">,</span> <span class="n">bases</span><span class="p">):</span>
+        <span class="c1"># prepare l_mat and rho to compute beta</span>
+        <span class="n">nullv</span> <span class="o">=</span> <span class="n">nullv</span><span class="o">.</span><span class="n">unflatten</span><span class="p">(</span><span class="n">dim</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">sizes</span><span class="o">=</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
+        <span class="n">i</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
+        <span class="n">j</span> <span class="o">=</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
+        <span class="n">dv</span> <span class="o">=</span> <span class="n">nullv</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="p">:]</span> <span class="o">-</span> <span class="n">nullv</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="p">:]</span>
+        <span class="n">a</span> <span class="o">=</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
+        <span class="n">b</span> <span class="o">=</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
+        <span class="n">dp</span> <span class="o">=</span> <span class="p">(</span><span class="n">dv</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="p">:,</span> <span class="p">:]</span> <span class="o">*</span> <span class="n">dv</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="p">:,</span> <span class="p">:])</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">dim</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span>
+        <span class="n">m</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">device</span><span class="o">=</span><span class="n">dp</span><span class="o">.</span><span class="n">device</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">dp</span><span class="o">.</span><span class="n">dtype</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">dp</span><span class="o">.</span><span class="n">mT</span> <span class="o">*</span> <span class="n">m</span><span class="p">,</span> <span class="p">(</span><span class="n">bases</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="p">:]</span> <span class="o">-</span> <span class="n">bases</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="p">:])</span><span class="o">.</span><span class="n">pow</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">_compute_betas</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">l_mat</span><span class="p">,</span> <span class="n">rho</span><span class="p">):</span>
+        <span class="c1"># Given the L matrix and rho vector, compute the betas vector.</span>
+        <span class="c1"># Check Eq 10 - 14 in paper.</span>
+        <span class="c1"># l_mat (..., 6, 10); rho (..., 6); betas (..., 4); return betas (4, ..., 4)</span>
+        <span class="n">betas</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">4</span><span class="p">,)</span><span class="o">+</span><span class="n">rho</span><span class="o">.</span><span class="n">shape</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">+</span><span class="p">(</span><span class="mi">4</span><span class="p">,),</span> <span class="n">device</span><span class="o">=</span><span class="n">rho</span><span class="o">.</span><span class="n">device</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">rho</span><span class="o">.</span><span class="n">dtype</span><span class="p">)</span>
+        <span class="c1"># dim == 1:</span>
+        <span class="n">betas</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="o">...</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
+        <span class="c1"># dim == 2:</span>
+        <span class="n">L</span> <span class="o">=</span> <span class="n">l_mat</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">9</span><span class="p">)]</span>
+        <span class="n">S</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">solver</span><span class="p">(</span><span class="n">L</span><span class="p">,</span> <span class="n">rho</span><span class="p">)</span>  <span class="c1"># (b, 3)</span>
+        <span class="n">betas</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="o">...</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="n">S</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">abs</span><span class="p">()</span><span class="o">.</span><span class="n">sqrt</span><span class="p">()</span>
+        <span class="n">betas</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="o">...</span><span class="p">,</span> <span class="mi">3</span><span class="p">]</span> <span class="o">=</span> <span class="n">S</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">abs</span><span class="p">()</span><span class="o">.</span><span class="n">sqrt</span><span class="p">()</span> <span class="o">*</span> <span class="n">S</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">sign</span><span class="p">()</span> <span class="o">*</span> <span class="n">S</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">sign</span><span class="p">()</span>
+        <span class="c1"># dim == 3:</span>
+        <span class="n">L</span> <span class="o">=</span> <span class="n">l_mat</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">9</span><span class="p">)]</span>
+        <span class="n">S</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">solver</span><span class="p">(</span><span class="n">L</span><span class="p">,</span> <span class="n">rho</span><span class="p">)</span>  <span class="c1"># (b, 6)</span>
+        <span class="n">betas</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="o">...</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">S</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">abs</span><span class="p">()</span><span class="o">.</span><span class="n">sqrt</span><span class="p">()</span>
+        <span class="n">betas</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="o">...</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="n">S</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">3</span><span class="p">]</span><span class="o">.</span><span class="n">abs</span><span class="p">()</span><span class="o">.</span><span class="n">sqrt</span><span class="p">()</span> <span class="o">*</span> <span class="n">S</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">sign</span><span class="p">()</span> <span class="o">*</span> <span class="n">S</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">sign</span><span class="p">()</span>
+        <span class="n">betas</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="o">...</span><span class="p">,</span> <span class="mi">3</span><span class="p">]</span> <span class="o">=</span> <span class="n">S</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">5</span><span class="p">]</span><span class="o">.</span><span class="n">abs</span><span class="p">()</span><span class="o">.</span><span class="n">sqrt</span><span class="p">()</span> <span class="o">*</span> <span class="n">S</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">sign</span><span class="p">()</span> <span class="o">*</span> <span class="n">S</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">sign</span><span class="p">()</span>
+        <span class="c1"># dim == 4:</span>
+        <span class="n">S</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">solver</span><span class="p">(</span><span class="n">l_mat</span><span class="p">,</span> <span class="n">rho</span><span class="p">)</span>  <span class="c1"># (b, 10)</span>
+        <span class="n">betas</span><span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="o">...</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">S</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">9</span><span class="p">]</span><span class="o">.</span><span class="n">abs</span><span class="p">()</span><span class="o">.</span><span class="n">sqrt</span><span class="p">()</span> <span class="o">*</span> <span class="n">S</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">6</span><span class="p">]</span><span class="o">.</span><span class="n">sign</span><span class="p">()</span> <span class="o">*</span> <span class="n">S</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">sign</span><span class="p">()</span>
+        <span class="n">betas</span><span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="o">...</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">S</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">5</span><span class="p">]</span><span class="o">.</span><span class="n">abs</span><span class="p">()</span><span class="o">.</span><span class="n">sqrt</span><span class="p">()</span> <span class="o">*</span> <span class="n">S</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">3</span><span class="p">]</span><span class="o">.</span><span class="n">sign</span><span class="p">()</span> <span class="o">*</span> <span class="n">S</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">sign</span><span class="p">()</span>
+        <span class="n">betas</span><span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="o">...</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="n">S</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">abs</span><span class="p">()</span><span class="o">.</span><span class="n">sqrt</span><span class="p">()</span> <span class="o">*</span> <span class="n">S</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">sign</span><span class="p">()</span> <span class="o">*</span> <span class="n">S</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">sign</span><span class="p">()</span>
+        <span class="n">betas</span><span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="o">...</span><span class="p">,</span> <span class="mi">3</span><span class="p">]</span> <span class="o">=</span> <span class="n">S</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">abs</span><span class="p">()</span><span class="o">.</span><span class="n">sqrt</span><span class="p">()</span>
+        <span class="k">return</span> <span class="n">betas</span>
+
+    <span class="nd">@staticmethod</span>
+    <span class="k">def</span> <span class="nf">_compute_scale</span><span class="p">(</span><span class="n">bases</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">points</span><span class="p">):</span>
+        <span class="c1"># Compute the scaling factor and the sign of the scaling factor</span>
+        <span class="c1"># input:  bases (4, ..., 12); alpha (..., N, 4); points (..., N, 3);</span>
+        <span class="c1"># return: bases (4, ..., 4, 3); scalep (4, ..., N, 3); scale (4, ..., 1)</span>
+        <span class="n">bases</span> <span class="o">=</span> <span class="n">bases</span><span class="o">.</span><span class="n">unflatten</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
+        <span class="n">transp</span> <span class="o">=</span> <span class="n">alpha</span> <span class="o">@</span> <span class="n">bases</span> <span class="c1"># transformed points</span>
+        <span class="n">dw</span> <span class="o">=</span> <span class="p">(</span><span class="n">points</span> <span class="o">-</span> <span class="n">points</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">dim</span><span class="o">=-</span><span class="mi">2</span><span class="p">,</span> <span class="n">keepdim</span><span class="o">=</span><span class="kc">True</span><span class="p">))</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">dim</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span>
+        <span class="n">dc</span> <span class="o">=</span> <span class="p">(</span><span class="n">transp</span> <span class="o">-</span> <span class="n">transp</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">dim</span><span class="o">=-</span><span class="mi">2</span><span class="p">,</span> <span class="n">keepdim</span><span class="o">=</span><span class="kc">True</span><span class="p">))</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">dim</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span>
+        <span class="n">scale</span> <span class="o">=</span> <span class="n">vecdot</span><span class="p">(</span><span class="n">dc</span><span class="p">,</span> <span class="n">dw</span><span class="p">)</span> <span class="o">/</span> <span class="n">vecdot</span><span class="p">(</span><span class="n">dc</span><span class="p">,</span> <span class="n">dc</span><span class="p">)</span>
+        <span class="n">bases</span> <span class="o">=</span> <span class="n">bases</span> <span class="o">*</span> <span class="n">scale</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</span> <span class="kc">None</span><span class="p">]</span> <span class="c1"># the real position</span>
+        <span class="n">scalep</span> <span class="o">=</span> <span class="n">alpha</span> <span class="o">@</span> <span class="n">bases</span> <span class="c1"># scaled transformed points</span>
+        <span class="n">mask</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">any</span><span class="p">(</span><span class="n">scalep</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">,</span> <span class="n">dim</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span> <span class="c1"># negate when z &lt; 0</span>
+        <span class="n">sign</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">ones_like</span><span class="p">(</span><span class="n">scale</span><span class="p">)</span> <span class="o">-</span> <span class="n">mask</span> <span class="o">*</span> <span class="mi">2</span>     <span class="c1"># 1 or -1</span>
+        <span class="n">scalep</span> <span class="o">=</span> <span class="n">sign</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</span> <span class="kc">None</span><span class="p">]</span> <span class="o">*</span> <span class="n">scalep</span>
+        <span class="n">scale</span> <span class="o">=</span> <span class="p">(</span><span class="n">sign</span> <span class="o">*</span> <span class="n">scale</span><span class="p">)</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">bases</span><span class="p">,</span> <span class="n">scalep</span><span class="p">,</span> <span class="n">scale</span></div>
+</pre></div>
+
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new file mode 100644
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+  <h1>Source code for pypose.module.ukf</h1><div class="highlight"><pre>
+<span></span><span class="kn">import</span> <span class="nn">torch</span>
+<span class="kn">from</span> <span class="nn">..</span> <span class="kn">import</span> <span class="n">bmv</span>
+<span class="kn">from</span> <span class="nn">.ekf</span> <span class="kn">import</span> <span class="n">EKF</span>
+<span class="kn">from</span> <span class="nn">torch.linalg</span> <span class="kn">import</span> <span class="n">pinv</span>
+
+
+<div class="viewcode-block" id="UKF"><a class="viewcode-back" href="../../../../generated/pypose.module.UKF/#pypose.module.UKF">[docs]</a><span class="k">class</span> <span class="nc">UKF</span><span class="p">(</span><span class="n">EKF</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Performs Batched Unscented Kalman Filter (UKF).</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        model (:obj:`System`): The system model to be estimated, a subclass of</span>
+<span class="sd">            :obj:`pypose.module.NLS`.</span>
+<span class="sd">        Q (:obj:`Tensor`, optional): The covariance matrices of system transition noise.</span>
+<span class="sd">            Ignored if provided during each iteration. Default: ``None``</span>
+<span class="sd">        R (:obj:`Tensor`, optional): The covariance matrices of system observation noise.</span>
+<span class="sd">            Ignored if provided during each iteration. Default: ``None``</span>
+<span class="sd">        msqrt (:obj:`Function`, optional): The method for computing the square root of matrices.</span>
+<span class="sd">            Use ``torch.linalg.cholesky`` if not provided. Default: ``None``</span>
+<span class="sd">        correct_PD_eps (:obj:`Float`, optional): The eps used in positive definite correction of the state covariance matrices P.</span>
+<span class="sd">            If provided a small positive number, P will be projected to be positive definite by clamping its smaller eigenvalues to eps.</span>
+<span class="sd">            If provided a negative number, no correction will be made. Default: ``-1``</span>
+
+<span class="sd">    A non-linear system can be described as</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \begin{aligned}</span>
+<span class="sd">            \mathbf{x}_{k+1} &amp;= \mathbf{f}(\mathbf{x}_k, \mathbf{u}_k, t_k) + \mathbf{w}_k,</span>
+<span class="sd">            \quad \mathbf{w}_k \sim \mathcal{N}(\mathbf{0}, \mathbf{Q})  \\</span>
+<span class="sd">            \mathbf{y}_{k} &amp;= \mathbf{g}(\mathbf{x}_k, \mathbf{u}_k, t_k) + \mathbf{v}_k,</span>
+<span class="sd">            \quad \mathbf{v}_k \sim \mathcal{N}(\mathbf{0}, \mathbf{R})</span>
+<span class="sd">        \end{aligned}</span>
+
+<span class="sd">    UKF can be described as the following equations, where the subscript :math:`\cdot_{k}`</span>
+<span class="sd">    is omited for simplicity.</span>
+
+<span class="sd">    1. Sigma Points.</span>
+
+<span class="sd">      .. math::</span>
+<span class="sd">          \begin{aligned}</span>
+<span class="sd">              &amp; \mathbf{x}^{\left ( i \right ) } = \mathbf{x}^{+} +</span>
+<span class="sd">                  \mathbf{\check{x}}^{\left(i \right)}, &amp; \quad i=0,...,2n\\</span>
+<span class="sd">              &amp; \mathbf{\check{x}}^{\left (i \right)} = \mathbf{0}, &amp; \quad i=0 \\</span>
+<span class="sd">              &amp; \mathbf{\check{x}}^{\left (i \right)} = \left(\sqrt{(n+\kappa)P} \right)_{i}^{T},</span>
+<span class="sd">               &amp; \quad i= 1, ..., n \\</span>
+<span class="sd">              &amp; \mathbf{\check{x}}^{\left(i \right)} = -\left(\sqrt{(n+\kappa)P} \right)_{i}^{T},</span>
+<span class="sd">                   &amp; \quad i=n+1, ..., 2n \\</span>
+<span class="sd">         \end{aligned}</span>
+
+<span class="sd">    where :math:`\left(\sqrt{(n+\kappa)P}\right) _{i}` is the :math:`i`-th row of</span>
+<span class="sd">    :math:`\left(\sqrt{(n+\kappa)P}\right)` and :math:`n` is the dimension of state :math:`\mathbf{x}`.</span>
+
+<span class="sd">    Their weighting cofficients are given as</span>
+
+<span class="sd">      .. math::</span>
+<span class="sd">          \begin{aligned}</span>
+<span class="sd">              &amp;W^{(0)} = \frac{\kappa}{n+\kappa}, &amp; \quad i = 0\\</span>
+<span class="sd">              &amp;W^{(i)} = \frac{1}{2(n+\kappa)}, &amp; \quad i = 1,...,2n\\</span>
+<span class="sd">          \end{aligned}</span>
+
+<span class="sd">    2. Priori State Estimation.</span>
+
+<span class="sd">      .. math::</span>
+<span class="sd">           \mathbf{x}^{-} = \sum_{i=0}^{2n} W^{(i)} f(\mathbf{x}^{(i)}, \mathbf{u}, t)</span>
+
+<span class="sd">    3. Priori Covariance.</span>
+
+<span class="sd">      .. math::</span>
+<span class="sd">          \mathbf{P}^{-} = \sum_{i=0}^{2n} W^{(i)}</span>
+<span class="sd">              \left(\mathbf{x}^{(i)} - \mathbf{x}^{-} \right)</span>
+<span class="sd">              \left(\mathbf{x}^{(i)} - \mathbf{x}^- \right)^T + \mathbf{Q}</span>
+
+<span class="sd">    4. Observational Estimation.</span>
+
+<span class="sd">      .. math::</span>
+<span class="sd">          \begin{aligned}</span>
+<span class="sd">              &amp; \mathbf{y}^{(i)} = g \left( \mathbf{x}^{(i)}, \mathbf{u}, t \right),</span>
+<span class="sd">                  &amp; \quad i = 0, \cdots, 2n \\</span>
+<span class="sd">              &amp; \bar{\mathbf{y}} = \sum_{i=0}^{2n} W^{(i)} \mathbf{y}^{(i)}</span>
+<span class="sd">          \end{aligned}</span>
+
+<span class="sd">    5. Observational Covariance.</span>
+
+<span class="sd">      .. math::</span>
+<span class="sd">          \mathbf{P}_{y} = \sum_{i=0}^{2n} W^{(i)}</span>
+<span class="sd">              \left(\mathbf{y}^{(i)} - \bar{\mathbf{y}} \right)</span>
+<span class="sd">              \left( \mathbf{y}^{(i)} - \bar{\mathbf{y}} \right)^T + \mathbf{R}</span>
+
+<span class="sd">    6. Priori and Observation Covariance:</span>
+
+<span class="sd">      .. math::</span>
+<span class="sd">          \mathbf{P}_{xy} = \sum_{i=0}^{2n} W^{(i)}</span>
+<span class="sd">              \left( \mathbf{x}^{(i)} - \mathbf{x}^- \right)</span>
+<span class="sd">              \left( \mathbf{y}^{(i)} - \bar{\mathbf{y}} \right)^T</span>
+
+<span class="sd">    7. Kalman Gain</span>
+
+<span class="sd">      .. math::</span>
+<span class="sd">          \mathbf{K} = \mathbf{P}_{xy}\mathbf{P}_{y}^{-1}</span>
+
+<span class="sd">    8. Posteriori State Estimation.</span>
+
+<span class="sd">      .. math::</span>
+<span class="sd">          \mathbf{x}^{+} = \mathbf{x}^{-} + \mathbf{K} (\mathbf{y}- \bar{\mathbf{y}})</span>
+
+<span class="sd">    9. Posteriori Covariance Estimation.</span>
+
+<span class="sd">      .. math::</span>
+<span class="sd">          \mathbf{P} = \mathbf{P}^{-} - \mathbf{K}\mathbf{P}_{y}\mathbf{K}^T</span>
+
+<span class="sd">    where superscript :math:`\cdot^{-}` and :math:`\cdot^{+}` denote the priori and</span>
+<span class="sd">    posteriori estimation, respectively.</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        1. Define a discrete-time non-linear system (NLS) model</span>
+
+<span class="sd">        &gt;&gt;&gt; import torch, pypose as pp</span>
+<span class="sd">        &gt;&gt;&gt; class NLS(pp.module.NLS):</span>
+<span class="sd">        ...     def __init__(self):</span>
+<span class="sd">        ...         super().__init__()</span>
+<span class="sd">        ...</span>
+<span class="sd">        ...     def state_transition(self, state, input, t=None):</span>
+<span class="sd">        ...         return state.cos() + input</span>
+<span class="sd">        ...</span>
+<span class="sd">        ...     def observation(self, state, input, t):</span>
+<span class="sd">        ...         return state.sin() + input</span>
+
+<span class="sd">        2. Create a model and filter</span>
+
+<span class="sd">        &gt;&gt;&gt; model = NLS()</span>
+<span class="sd">        &gt;&gt;&gt; ukf = pp.module.UKF(model)</span>
+
+<span class="sd">        3. Prepare data</span>
+
+<span class="sd">        &gt;&gt;&gt; B, T, N = 3, 5, 2 # batchsize, steps, state dim</span>
+<span class="sd">        &gt;&gt;&gt; states = torch.zeros(B, T, N)</span>
+<span class="sd">        &gt;&gt;&gt; inputs = torch.randn(B, T, N)</span>
+<span class="sd">        &gt;&gt;&gt; observ = torch.zeros(B, T, N)</span>
+<span class="sd">        &gt;&gt;&gt; # std of transition, observation, and estimation</span>
+<span class="sd">        &gt;&gt;&gt; q, r, p = 0.1, 0.1, 10</span>
+<span class="sd">        &gt;&gt;&gt; Q = torch.eye(N).repeat(B, 1, 1) * q**2</span>
+<span class="sd">        &gt;&gt;&gt; R = torch.eye(N).repeat(B, 1, 1) * r**2</span>
+<span class="sd">        &gt;&gt;&gt; P = torch.eye(N).repeat(B, T, 1, 1) * p**2</span>
+<span class="sd">        &gt;&gt;&gt; estim = torch.randn(B, T, N) * p</span>
+
+<span class="sd">        4. Perform UKF prediction. Note that estimation error becomes smaller with more steps.</span>
+
+<span class="sd">        &gt;&gt;&gt; for i in range(T - 1):</span>
+<span class="sd">        ...     w = q * torch.randn(B, N) # transition noise</span>
+<span class="sd">        ...     v = r * torch.randn(B, N) # observation noise</span>
+<span class="sd">        ...     states[:, i+1], observ[:, i] = model(states[:, i] + w, inputs[:, i])</span>
+<span class="sd">        ...     estim[:, i+1], P[:, i+1] = ukf(estim[:, i], observ[:, i] + v, inputs[:, i], P[:, i], Q, R)</span>
+<span class="sd">        ... print(&#39;Est error:&#39;, (states - estim).norm(dim=-1))</span>
+<span class="sd">        Est error: tensor([[19.2216,  0.9972,  0.7375,  0.8644,  0.9604],</span>
+<span class="sd">                           [16.6286,  2.1883,  1.7114,  0.8491,  1.2250],</span>
+<span class="sd">                           [10.5133,  2.1080,  1.9851,  0.5087,  1.3921]])</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        Implementation is based on Section 14.3 of this book</span>
+
+<span class="sd">        * Dan Simon, `Optimal State Estimation: Kalman, H∞, and Nonlinear Approaches</span>
+<span class="sd">          &lt;https://onlinelibrary.wiley.com/doi/book/10.1002/0470045345&gt;`_,</span>
+<span class="sd">          Cleveland State University, 2006</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">model</span><span class="p">,</span> <span class="n">Q</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">R</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">msqrt</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">correct_PD_eps</span><span class="o">=-</span><span class="mi">1</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">(</span><span class="n">model</span><span class="p">,</span> <span class="n">Q</span><span class="p">,</span> <span class="n">R</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">msqrt</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">cholesky</span> <span class="k">if</span> <span class="n">msqrt</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">msqrt</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">correct_PD_eps</span> <span class="o">=</span> <span class="n">correct_PD_eps</span>
+
+<div class="viewcode-block" id="UKF.forward"><a class="viewcode-back" href="../../../../generated/pypose.module.UKF/#pypose.module.UKF.forward">[docs]</a>    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">Q</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">R</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">t</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">k</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Performs one step estimation.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            x (:obj:`Tensor`): estimated system state of previous step.</span>
+<span class="sd">            y (:obj:`Tensor`): system observation at current step (measurement).</span>
+<span class="sd">            u (:obj:`Tensor`): system input at current step.</span>
+<span class="sd">            P (:obj:`Tensor`): state estimation covariance of previous step.</span>
+<span class="sd">            Q (:obj:`Tensor`, optional): covariance of system transition model.</span>
+<span class="sd">            R (:obj:`Tensor`, optional): covariance of system observation model.</span>
+<span class="sd">            t (:obj:`Tensor`, optional): timestamp for ths system to estimate. Default: ``None``.</span>
+<span class="sd">            k (``int``, optional): a parameter for weighting the sigma points.</span>
+<span class="sd">                If ``None`` is given, then ``3 - n`` is adopted (``n`` is the state dimension),</span>
+<span class="sd">                which is best for Gaussian noise model. Default: ``None``.</span>
+
+<span class="sd">        Return:</span>
+<span class="sd">            list of :obj:`Tensor`: posteriori state and covariance estimation</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="c1"># Upper cases are matrices, lower cases are vectors</span>
+        <span class="n">k</span> <span class="o">=</span> <span class="mi">3</span> <span class="o">-</span> <span class="n">x</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="k">if</span> <span class="n">k</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">k</span>
+        <span class="n">Q</span> <span class="o">=</span> <span class="n">Q</span> <span class="k">if</span> <span class="n">Q</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span> <span class="k">else</span> <span class="bp">self</span><span class="o">.</span><span class="n">Q</span>
+        <span class="n">R</span> <span class="o">=</span> <span class="n">R</span> <span class="k">if</span> <span class="n">R</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span> <span class="k">else</span> <span class="bp">self</span><span class="o">.</span><span class="n">R</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">model</span><span class="o">.</span><span class="n">set_refpoint</span><span class="p">(</span><span class="n">state</span><span class="o">=</span><span class="n">x</span><span class="p">,</span> <span class="nb">input</span><span class="o">=</span><span class="n">u</span><span class="p">,</span> <span class="n">t</span><span class="o">=</span><span class="n">t</span><span class="p">)</span>
+
+        <span class="n">xs</span><span class="p">,</span> <span class="n">w</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">sigma_weight_points</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span>
+        <span class="n">xs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">model</span><span class="o">.</span><span class="n">state_transition</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span>
+        <span class="n">xe</span> <span class="o">=</span> <span class="p">(</span><span class="n">w</span> <span class="o">*</span> <span class="n">xs</span><span class="p">)</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">dim</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
+        <span class="n">ex</span> <span class="o">=</span> <span class="n">xe</span> <span class="o">-</span> <span class="n">xs</span>
+        <span class="n">P</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">compute_cov</span><span class="p">(</span><span class="n">ex</span><span class="p">,</span> <span class="n">ex</span><span class="p">,</span> <span class="n">w</span><span class="p">,</span> <span class="n">Q</span><span class="p">)</span>
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">correct_PD_eps</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">:</span>
+            <span class="n">P</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">project_to_positive_definite</span><span class="p">(</span><span class="n">P</span><span class="p">)</span>
+
+        <span class="n">xs</span><span class="p">,</span> <span class="n">w</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">sigma_weight_points</span><span class="p">(</span><span class="n">xe</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span>
+        <span class="n">ys</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">model</span><span class="o">.</span><span class="n">observation</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span>
+        <span class="n">ye</span> <span class="o">=</span> <span class="p">(</span><span class="n">w</span> <span class="o">*</span> <span class="n">ys</span><span class="p">)</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">dim</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
+        <span class="n">ey</span> <span class="o">=</span> <span class="n">ye</span> <span class="o">-</span> <span class="n">ys</span>
+        <span class="n">Py</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">compute_cov</span><span class="p">(</span><span class="n">ey</span><span class="p">,</span> <span class="n">ey</span><span class="p">,</span> <span class="n">w</span><span class="p">,</span> <span class="n">R</span><span class="p">)</span>
+
+        <span class="n">Pxy</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">compute_cov</span><span class="p">(</span><span class="n">ex</span><span class="p">,</span> <span class="n">ey</span><span class="p">,</span> <span class="n">w</span><span class="p">)</span>
+        <span class="n">K</span> <span class="o">=</span> <span class="n">Pxy</span> <span class="o">@</span> <span class="n">pinv</span><span class="p">(</span><span class="n">Py</span><span class="p">)</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="n">xe</span> <span class="o">+</span> <span class="n">bmv</span><span class="p">(</span><span class="n">K</span><span class="p">,</span> <span class="n">y</span> <span class="o">-</span> <span class="n">ye</span><span class="p">)</span>
+        <span class="n">P</span> <span class="o">=</span> <span class="n">P</span> <span class="o">-</span> <span class="n">K</span> <span class="o">@</span> <span class="n">Py</span> <span class="o">@</span> <span class="n">K</span><span class="o">.</span><span class="n">mT</span>
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">correct_PD_eps</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">:</span>
+            <span class="n">P</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">project_to_positive_definite</span><span class="p">(</span><span class="n">P</span><span class="p">)</span>
+
+        <span class="k">return</span> <span class="n">x</span><span class="p">,</span> <span class="n">P</span></div>
+
+<div class="viewcode-block" id="UKF.sigma_weight_points"><a class="viewcode-back" href="../../../../generated/pypose.module.UKF/#pypose.module.UKF.sigma_weight_points">[docs]</a>    <span class="k">def</span> <span class="nf">sigma_weight_points</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Compute sigma point and its weights</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            x (:obj:`Tensor`): estimated system state of previous step</span>
+<span class="sd">            P (:obj:`Tensor`): state estimation covariance of previous step</span>
+<span class="sd">            k (:obj:`int`): parameter for weighting sigma points.</span>
+
+<span class="sd">        Return:</span>
+<span class="sd">            tuple of :obj:`Tensor`: sigma points and their weights</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">assert</span> <span class="n">x</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">==</span> <span class="n">P</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">==</span> <span class="n">P</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">),</span> <span class="s1">&#39;Invalid shape&#39;</span>
+        <span class="n">n</span><span class="p">,</span> <span class="n">xe</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">),</span> <span class="n">x</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
+        <span class="n">xr</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">msqrt</span><span class="p">((</span><span class="n">n</span> <span class="o">+</span> <span class="n">k</span><span class="p">)</span> <span class="o">*</span> <span class="n">P</span><span class="p">)</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">)</span>
+        <span class="n">we</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">full</span><span class="p">(</span><span class="n">xe</span><span class="o">.</span><span class="n">shape</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">k</span> <span class="o">/</span> <span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="n">k</span><span class="p">),</span> <span class="n">dtype</span><span class="o">=</span><span class="n">x</span><span class="o">.</span><span class="n">dtype</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="n">x</span><span class="o">.</span><span class="n">device</span><span class="p">)</span>
+        <span class="n">wr</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">full</span><span class="p">(</span><span class="n">xr</span><span class="o">.</span><span class="n">shape</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="mi">1</span> <span class="o">/</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="n">k</span><span class="p">)),</span> <span class="n">dtype</span><span class="o">=</span><span class="n">x</span><span class="o">.</span><span class="n">dtype</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="n">x</span><span class="o">.</span><span class="n">device</span><span class="p">)</span>
+        <span class="n">p</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">cat</span><span class="p">((</span><span class="n">xe</span><span class="p">,</span> <span class="n">xe</span> <span class="o">+</span> <span class="n">xr</span><span class="p">,</span> <span class="n">xe</span> <span class="o">-</span> <span class="n">xr</span><span class="p">),</span> <span class="n">dim</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
+        <span class="n">w</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">cat</span><span class="p">((</span><span class="n">we</span><span class="p">,</span> <span class="n">wr</span><span class="p">,</span> <span class="n">wr</span><span class="p">),</span> <span class="n">dim</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">p</span><span class="p">,</span> <span class="n">w</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span></div>
+
+<div class="viewcode-block" id="UKF.compute_cov"><a class="viewcode-back" href="../../../../generated/pypose.module.UKF/#pypose.module.UKF.compute_cov">[docs]</a>    <span class="k">def</span> <span class="nf">compute_cov</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">w</span><span class="p">,</span> <span class="n">Q</span><span class="o">=</span><span class="mi">0</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&#39;&#39;&#39;Compute covariance of two set of variables.&#39;&#39;&#39;</span>
+        <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">),</span> <span class="n">b</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">Q</span> <span class="o">+</span> <span class="p">(</span><span class="n">w</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">a</span> <span class="o">@</span> <span class="n">b</span><span class="o">.</span><span class="n">mT</span><span class="p">)</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">dim</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span></div>
+
+<div class="viewcode-block" id="UKF.project_to_positive_definite"><a class="viewcode-back" href="../../../../generated/pypose.module.UKF/#pypose.module.UKF.project_to_positive_definite">[docs]</a>    <span class="k">def</span> <span class="nf">project_to_positive_definite</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">A</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&#39;&#39;&#39;Project matrix to positive definite by clamping its eigenvalues to positive.&#39;&#39;&#39;</span>
+        <span class="n">eigenvalues</span><span class="p">,</span> <span class="n">eigenvectors</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">eigh</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
+        <span class="n">eigenvalues</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">clamp</span><span class="p">(</span><span class="n">eigenvalues</span><span class="p">,</span> <span class="nb">min</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">correct_PD_eps</span><span class="p">)</span>
+        <span class="k">return</span> <span class="p">(</span><span class="n">eigenvectors</span> <span class="o">@</span> <span class="n">torch</span><span class="o">.</span><span class="n">diag_embed</span><span class="p">(</span><span class="n">eigenvalues</span><span class="p">))</span> <span class="o">@</span> <span class="n">eigenvectors</span><span class="o">.</span><span class="n">mT</span></div></div>
+</pre></div>
+
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+              
+  <h1>Source code for pypose.optim.corrector</h1><div class="highlight"><pre>
+<span></span><span class="kn">import</span> <span class="nn">torch</span><span class="o">,</span> <span class="nn">warnings</span>
+<span class="kn">from</span> <span class="nn">torch</span> <span class="kn">import</span> <span class="n">Tensor</span><span class="p">,</span> <span class="n">nn</span>
+<span class="kn">from</span> <span class="nn">torch.autograd</span> <span class="kn">import</span> <span class="n">grad</span>
+<span class="kn">from</span> <span class="nn">torch.autograd.functional</span> <span class="kn">import</span> <span class="n">jacobian</span>
+
+
+<div class="viewcode-block" id="FastTriggs"><a class="viewcode-back" href="../../../../generated/pypose.optim.corrector.FastTriggs/#pypose.optim.corrector.FastTriggs">[docs]</a><span class="k">class</span> <span class="nc">FastTriggs</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Faster yet stable version of Triggs correction of model residual and Jacobian.</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \begin{align*}</span>
+<span class="sd">            \mathbf{R}_i^\rho &amp;= \sqrt{\rho&#39;(c_i)} \mathbf{R}_i\\</span>
+<span class="sd">            \mathbf{J}_i^\rho &amp;= \sqrt{\rho&#39;(c_i)} \mathbf{J}_i</span>
+<span class="sd">        \end{align*},</span>
+<span class="sd">    </span>
+<span class="sd">    where :math:`\mathbf{R}_i` and :math:`\mathbf{J}_i` are the :math:`i`-th item of the</span>
+<span class="sd">    model residual and Jacobian, respectively. :math:`\rho()` is the kernel function and</span>
+<span class="sd">    :math:`c_i = \mathbf{R}_i^T\mathbf{R}_i` is the point to compute the gradient.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        kernel (nn.Module): the robust kernel (cost) function.</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        This implementation has a faster and numerically stable solution than :meth:`Triggs`.</span>
+<span class="sd">        It removes the kernel&#39;s 2nd order derivatives (often negative), which can lead a 2nd</span>
+<span class="sd">        order optimizer unstable. It basically aims to solve</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \bm{\theta}^* = \arg\min_{\bm{\theta}} \mathbf{g}(\bm{x})</span>
+<span class="sd">                        = \arg\min_{\bm{\theta}} \sum_i \rho(\mathbf{R}_i^T \mathbf{R}_i),</span>
+
+<span class="sd">        where :math:`\mathbf{R}_i = \bm{f}(\bm{\theta},\bm{x}_i) - \bm{y}_i` and</span>
+<span class="sd">        :math:`\bm{f}(\bm{\theta}, \bm{x})` is the model, :math:`\bm{\theta}` is the parameters</span>
+<span class="sd">        to be optimized, :math:`\bm{x}` is the model inputs, :math:`\bm{y}` is the model targets.</span>
+<span class="sd">        Considering the 1st order Taylor expansion of the model</span>
+<span class="sd">        :math:`\bm{f}(\bm{\theta}+\bm{\delta})\approx\bm{f}(\bm{\theta})+\mathbf{J}_i\bm{\delta}`.</span>
+<span class="sd">        If we take :math:`c_i = \mathbf{R}_i^T \mathbf{R}_i` and set the first derivative of</span>
+<span class="sd">        :math:`\mathbf{g}(\bm{\delta})` to zero, we have</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \frac{\partial \bm{g}}{\partial \bm{\delta}} </span>
+<span class="sd">            = \sum_i \frac{\partial \rho}{\partial c_i} \frac{\partial c_i}{\partial \bm{\delta}}</span>
+<span class="sd">            = \bm{0}</span>
+
+<span class="sd">        This leads to</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \sum_i \frac{\partial \rho}{\partial c_i} \mathbf{J}_i^T \mathbf{J}_i \bm{\delta}</span>
+<span class="sd">            = - \sum_i \frac{\partial \rho}{\partial c_i} \mathbf{J}_i^T \mathbf{R}_i</span>
+
+<span class="sd">        Rearrange the gradient of :math:`\rho`, we have</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \sum_i \left(\sqrt{\frac{\partial \rho}{\partial c_i}} \mathbf{J}_i\right)^T </span>
+<span class="sd">                \left(\sqrt{\frac{\partial \rho}{\partial c_i}} \mathbf{J}_i\right) \bm{\delta}</span>
+<span class="sd">            = - \sum_i \left(\sqrt{\frac{\partial \rho}{\partial c_i}} \mathbf{J}_i\right)^T </span>
+<span class="sd">                \left(\sqrt{\frac{\partial \rho}{\partial c_i}} \mathbf{R}_i\right)</span>
+
+<span class="sd">        This gives us the corrected model residual :math:`\mathbf{R}_i^\rho` and Jacobian</span>
+<span class="sd">        :math:`\mathbf{J}_i^\rho`, which ends with the same problem formulation as the</span>
+<span class="sd">        standard 2nd order optimizers such as :meth:`pypose.optim.GN` and</span>
+<span class="sd">        :meth:`pypose.optim.LM`.</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \sum_i {\mathbf{J}_i^\rho}^T \mathbf{J}_i^\rho \bm{\delta}</span>
+<span class="sd">            = - \sum_i {\mathbf{J}_i^\rho}^T \mathbf{R}_i^\rho</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">kernel</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">func</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">kernel</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span>
+
+<div class="viewcode-block" id="FastTriggs.forward"><a class="viewcode-back" href="../../../../generated/pypose.optim.corrector.FastTriggs/#pypose.optim.corrector.FastTriggs.forward">[docs]</a>    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">R</span><span class="p">:</span> <span class="n">Tensor</span><span class="p">,</span> <span class="n">J</span><span class="p">:</span> <span class="n">Tensor</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Args:</span>
+<span class="sd">            R (Tensor): the model residual.</span>
+<span class="sd">            J (Tensor): the model Jacobian.</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            tuple of Tensors: the corrected model residual and model Jacobian.</span>
+
+<span class="sd">        Note:</span>
+<span class="sd">            The users basically only need to call the constructor, while the :obj:`.forward()`</span>
+<span class="sd">            function is not supposed to be directly called by PyPose users. It will be called</span>
+<span class="sd">            internally by optimizers such as :meth:`pypose.optim.GN` and :meth:`pypose.optim.LM`.</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="n">R</span><span class="o">.</span><span class="n">square</span><span class="p">()</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">keepdim</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+        <span class="n">s</span> <span class="o">=</span> <span class="n">jacobian</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">sqrt</span><span class="p">()</span>
+        <span class="n">sj</span> <span class="o">=</span> <span class="n">s</span><span class="o">.</span><span class="n">expand_as</span><span class="p">(</span><span class="n">R</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">s</span> <span class="o">*</span> <span class="n">R</span><span class="p">,</span> <span class="n">sj</span> <span class="o">*</span> <span class="n">J</span></div></div>
+
+
+<div class="viewcode-block" id="Triggs"><a class="viewcode-back" href="../../../../generated/pypose.optim.corrector.Triggs/#pypose.optim.corrector.Triggs">[docs]</a><span class="k">class</span> <span class="nc">Triggs</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;The Triggs correction of model residual and Jacobian.</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \begin{align*}</span>
+<span class="sd">            \mathbf{R}_i^\rho &amp;= \frac{\sqrt{\rho&#39;(c_i)}}{1 - \alpha} \mathbf{R}_i,\\</span>
+<span class="sd">            \mathbf{J}_i^\rho &amp;= \sqrt{\rho&#39;(c_i)} \left(\mathbf{I} - \alpha</span>
+<span class="sd">                \frac{\mathbf{R}_i^T\mathbf{R}_i}{\|\mathbf{R}_i\|^2} \right) \mathbf{J}_i,</span>
+<span class="sd">        \end{align*}</span>
+<span class="sd">    </span>
+<span class="sd">    where :math:`\alpha` is a root of</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \frac{1}{2} \alpha^2 - \alpha - \frac{\rho&#39;&#39;}{\rho&#39;} \|\mathbf{R}_i\|^2 = 0.</span>
+
+<span class="sd">    :math:`\mathbf{R}_i` and :math:`\mathbf{J}_i` are the :math:`i`-th item of the model</span>
+<span class="sd">    residual and Jacobian, respectively. :math:`\rho()` is the kernel function and</span>
+<span class="sd">    :math:`c_i = \mathbf{R}_i^T\mathbf{R}_i` is the evaluation point.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        kernel (nn.Module): the robust kernel (cost) function.</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        This implementation thanks to Eq. (11) of the following paper.</span>
+
+<span class="sd">        * Bill Triggs, etc., `Bundle Adjustment -- A Modern Synthesis</span>
+<span class="sd">          &lt;https://link.springer.com/chapter/10.1007/3-540-44480-7_21&gt;`_, International</span>
+<span class="sd">          Workshop on Vision Algorithms, 1999.</span>
+<span class="sd">    </span>
+<span class="sd">    Warning:</span>
+
+<span class="sd">        The :meth:`FastTriggs` corrector is preferred when the kernel function has a</span>
+<span class="sd">        negative 2nd order derivative.</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">kernel</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">kernel</span> <span class="o">=</span> <span class="n">kernel</span>
+
+    <span class="nd">@torch</span><span class="o">.</span><span class="n">enable_grad</span><span class="p">()</span>
+    <span class="k">def</span> <span class="nf">compute_grads</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">R</span><span class="p">):</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="n">R</span><span class="o">.</span><span class="n">square</span><span class="p">()</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">keepdim</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span><span class="o">.</span><span class="n">requires_grad_</span><span class="p">(</span><span class="kc">True</span><span class="p">)</span>
+        <span class="n">y</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">kernel</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span>
+        <span class="n">g1</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">create_graph</span><span class="o">=</span><span class="kc">True</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
+        <span class="n">g2</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">g1</span><span class="o">.</span><span class="n">sum</span><span class="p">(),</span> <span class="n">x</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
+        <span class="k">return</span> <span class="n">x</span><span class="o">.</span><span class="n">detach_</span><span class="p">(),</span> <span class="n">g1</span><span class="o">.</span><span class="n">detach_</span><span class="p">(),</span> <span class="n">g2</span><span class="o">.</span><span class="n">detach_</span><span class="p">()</span>
+
+<div class="viewcode-block" id="Triggs.forward"><a class="viewcode-back" href="../../../../generated/pypose.optim.corrector.Triggs/#pypose.optim.corrector.Triggs.forward">[docs]</a>    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">R</span><span class="p">:</span> <span class="n">Tensor</span><span class="p">,</span> <span class="n">J</span><span class="p">:</span> <span class="n">Tensor</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Args:</span>
+<span class="sd">            R (Tensor): the model residual.</span>
+<span class="sd">            J (Tensor): the model Jacobian.</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            tuple of Tensors: the corrected model residual and model Jacobian.</span>
+
+<span class="sd">        Note:</span>
+<span class="sd">            The users basically only need to call the constructor, while the :obj:`.forward()`</span>
+<span class="sd">            function is not supposed to be directly called by PyPose users. It will be called</span>
+<span class="sd">            internally by optimizers such as :meth:`pypose.optim.GN` and :meth:`pypose.optim.LM`.</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="n">x</span><span class="p">,</span> <span class="n">g1</span><span class="p">,</span> <span class="n">g2</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">compute_grads</span><span class="p">(</span><span class="n">R</span><span class="p">)</span>
+        <span class="n">se</span> <span class="o">=</span> <span class="n">g1</span><span class="o">.</span><span class="n">sqrt</span><span class="p">()</span>
+        <span class="n">sj</span> <span class="o">=</span> <span class="n">se</span><span class="o">.</span><span class="n">expand_as</span><span class="p">(</span><span class="n">R</span><span class="p">)</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
+        <span class="n">sR</span><span class="p">,</span> <span class="n">sJ</span> <span class="o">=</span> <span class="n">se</span> <span class="o">*</span> <span class="n">R</span><span class="p">,</span> <span class="n">sj</span> <span class="o">*</span> <span class="n">J</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="n">R</span><span class="o">.</span><span class="n">shape</span> <span class="o">+</span> <span class="p">(</span><span class="n">J</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],))</span>
+        <span class="n">M</span> <span class="o">=</span> <span class="o">~</span><span class="p">((</span><span class="n">x</span><span class="o">==</span><span class="mi">0</span><span class="p">)</span><span class="o">|</span><span class="p">(</span><span class="n">g2</span> <span class="o">&lt;=</span><span class="mi">0</span><span class="p">))</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
+        <span class="n">alpha</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">-</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="n">M</span><span class="p">]</span><span class="o">*</span><span class="n">g2</span><span class="p">[</span><span class="n">M</span><span class="p">]</span><span class="o">/</span><span class="n">g1</span><span class="p">[</span><span class="n">M</span><span class="p">])</span><span class="o">.</span><span class="n">clamp</span><span class="p">(</span><span class="nb">min</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span><span class="o">.</span><span class="n">sqrt</span><span class="p">()</span>
+        <span class="n">sR</span><span class="p">[</span><span class="n">M</span><span class="p">]</span> <span class="o">=</span> <span class="n">se</span><span class="p">[</span><span class="n">M</span><span class="p">]</span> <span class="o">/</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">alpha</span><span class="p">)</span>
+        <span class="n">Q</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">einsum</span><span class="p">(</span><span class="s1">&#39;...d,...k,...kl-&gt;...dl&#39;</span><span class="p">,</span> <span class="n">R</span><span class="p">[</span><span class="n">M</span><span class="p">],</span> <span class="n">R</span><span class="p">[</span><span class="n">M</span><span class="p">],</span> <span class="n">sJ</span><span class="p">[</span><span class="n">M</span><span class="p">])</span>
+        <span class="n">sJ</span><span class="p">[</span><span class="n">M</span><span class="p">]</span> <span class="o">=</span> <span class="n">sJ</span><span class="p">[</span><span class="n">M</span><span class="p">]</span> <span class="o">-</span> <span class="p">(</span><span class="n">alpha</span> <span class="o">/</span> <span class="n">x</span><span class="p">[</span><span class="n">M</span><span class="p">])</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">Q</span>
+        <span class="k">return</span> <span class="n">sR</span><span class="p">,</span> <span class="n">sJ</span><span class="o">.</span><span class="n">view_as</span><span class="p">(</span><span class="n">J</span><span class="p">)</span></div></div>
+</pre></div>
+
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+              
+  <h1>Source code for pypose.optim.functional</h1><div class="highlight"><pre>
+<span></span><span class="kn">import</span> <span class="nn">torch</span>
+<span class="kn">from</span> <span class="nn">..</span> <span class="kn">import</span> <span class="n">hasnan</span>
+<span class="kn">from</span> <span class="nn">functools</span> <span class="kn">import</span> <span class="n">partial</span>
+<span class="kn">from</span> <span class="nn">torch.autograd.functional</span> <span class="kn">import</span> <span class="n">jacobian</span>
+<span class="kn">from</span> <span class="nn">torch.func</span> <span class="kn">import</span> <span class="n">jacrev</span><span class="p">,</span> <span class="n">jacfwd</span><span class="p">,</span> <span class="n">functional_call</span>
+
+
+<div class="viewcode-block" id="modjac"><a class="viewcode-back" href="../../../../generated/pypose.optim.functional.modjac/#pypose.optim.functional.modjac">[docs]</a><span class="k">def</span> <span class="nf">modjac</span><span class="p">(</span><span class="n">model</span><span class="p">,</span> <span class="nb">input</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">create_graph</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">strict</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">vectorize</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> \
+                    <span class="n">strategy</span><span class="o">=</span><span class="s1">&#39;reverse-mode&#39;</span><span class="p">,</span> <span class="n">flatten</span><span class="o">=</span><span class="kc">False</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Compute the model Jacobian with respect to the model parameters.</span>
+
+<span class="sd">    For a parametric model :math:`\bm{f}(\bm{\theta}, \bm{x})`, where :math:`\bm{\theta}`</span>
+<span class="sd">    is the learnable parameter and :math:`\bm{x}` is the input, it computes the</span>
+<span class="sd">    Jacobian of the :math:`i`-th output and :math:`j`-th parameter as</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        {\displaystyle \mathbf{J}_{i,j} =</span>
+<span class="sd">        {\begin{bmatrix}</span>
+<span class="sd">            {\dfrac {\partial \bm{f}_{i,1}}{\partial \bm{\theta}_{j,1}}} &amp;amp;</span>
+<span class="sd">            \cdots&amp;amp;{\dfrac {\partial \bm{f}_{i,1}}{\partial \bm{\theta}_{j,n}}}\\</span>
+<span class="sd">            \vdots &amp;amp;\ddots &amp;amp;\vdots \\</span>
+<span class="sd">            {\dfrac {\partial \bm{f}_{i,m}}{\partial \bm{\theta}_{j,1}}}&amp;amp;</span>
+<span class="sd">            \cdots &amp;amp;{\dfrac {\partial \bm{f}_{i,m}}{\partial \bm{\theta}_{j,n}}}</span>
+<span class="sd">        \end{bmatrix}}}.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        model (torch.nn.Module): a PyTorch model that takes Tensor or LieTensor</span>
+<span class="sd">            input and returns a tuple of Tensors/LieTensors or a Tensor/LieTensor.</span>
+<span class="sd">        input (tuple of Tensors/LieTensors or Tensor/LieTensor): input to the</span>
+<span class="sd">            model. Defaults to ``None``.</span>
+<span class="sd">        create_graph (bool, optional): If ``True``, the Jacobian will be</span>
+<span class="sd">            computed in a differentiable manner. Note that when ``strict`` is</span>
+<span class="sd">            ``False``, the result can not require gradients or be disconnected</span>
+<span class="sd">            from the input.  Defaults to ``False``.</span>
+<span class="sd">        strict (bool, optional): If ``True``, an error will be raised when we</span>
+<span class="sd">            detect that there exists an input such that all the outputs are</span>
+<span class="sd">            independent of it. If ``False``, we return a Tensor of zeros as the</span>
+<span class="sd">            jacobian for said input, which is the expected mathematical value.</span>
+<span class="sd">            Defaults to ``False``.</span>
+<span class="sd">        vectorize (bool, optional): When computing the jacobian, usually we invoke</span>
+<span class="sd">            ``autograd.grad`` once per row of the jacobian. If this flag is</span>
+<span class="sd">            ``True``, we perform only a single ``autograd.grad`` call with</span>
+<span class="sd">            ``batched_grad=True`` which uses the vmap prototype feature.</span>
+<span class="sd">            Though this should lead to performance improvements in many cases,</span>
+<span class="sd">            because this feature is still experimental, there may be performance</span>
+<span class="sd">            cliffs. See :func:`torch.autograd.grad`&#39;s ``batched_grad`` parameter for</span>
+<span class="sd">            more information.</span>
+<span class="sd">        strategy (str, optional): Set to ``&quot;forward-mode&quot;`` or ``&quot;reverse-mode&quot;`` to</span>
+<span class="sd">            determine whether the Jacobian will be computed with forward or reverse</span>
+<span class="sd">            mode AD. Currently, ``&quot;forward-mode&quot;`` requires ``vectorized=True``.</span>
+<span class="sd">            Defaults to ``&quot;reverse-mode&quot;``. If ``func`` has more outputs than</span>
+<span class="sd">            input, ``&quot;forward-mode&quot;`` tends to be more performant. Otherwise,</span>
+<span class="sd">            prefer to use ``&quot;reverse-mode&quot;``.</span>
+<span class="sd">        flatten (bool, optional): If ``True``, all module parameters and outputs</span>
+<span class="sd">            are flattened and concatenated to form a single vector. The Jacobian</span>
+<span class="sd">            will be computed with respect to this single flattened vectors, thus</span>
+<span class="sd">            a single Tensor will be returned.</span>
+
+<span class="sd">    Returns:</span>
+<span class="sd">        Jacobian (Tensor or nested tuple of Tensors): if there is a single</span>
+<span class="sd">        parameter and output, this will be a single Tensor containing the</span>
+<span class="sd">        Jacobian for the linearized parameter and output. If there are more</span>
+<span class="sd">        than one parameters, then the Jacobian will be a tuple of Tensors.</span>
+<span class="sd">        If there are more than one outputs (even if there is only one parameter),</span>
+<span class="sd">        then the Jacobian will be a tuple of tuple of Tensors where ``Jacobian[i][j]``</span>
+<span class="sd">        will contain the Jacobian of the ``i``\th output and ``j``\th parameter</span>
+<span class="sd">        and will have as size the concatenation of the sizes of the corresponding</span>
+<span class="sd">        output and the corresponding parameter and will have same dtype and device as the</span>
+<span class="sd">        corresponding parameter. If strategy is ``forward-mode``, the dtype will be</span>
+<span class="sd">        that of the output; otherwise, the parameters.</span>
+
+<span class="sd">    Warning:</span>
+<span class="sd">        The function :obj:`modjac` calculate Jacobian of model parameters.</span>
+<span class="sd">        This is in contrast to PyTorch&#39;s function `jacobian</span>
+<span class="sd">        &lt;https://pytorch.org/docs/stable/generated/torch.autograd.functional.jacobian.html&gt;`_,</span>
+<span class="sd">        which computes the Jacobian of a given Python function.</span>
+
+<span class="sd">    Example:</span>
+
+<span class="sd">        Calculates Jacobian with respect to all model parameters.</span>
+
+<span class="sd">        &gt;&gt;&gt; model = nn.Conv2d(in_channels=1, out_channels=1, kernel_size=1)</span>
+<span class="sd">        &gt;&gt;&gt; input = torch.randn(1, 1, 1)</span>
+<span class="sd">        &gt;&gt;&gt; J = pp.optim.functional.modjac(model, input)</span>
+<span class="sd">        (tensor([[[[[[[0.3365]]]]]]]), tensor([[[[1.]]]]))</span>
+<span class="sd">        &gt;&gt;&gt; [j.shape for j in J]</span>
+<span class="sd">        [torch.Size([1, 1, 1, 1, 1, 1, 1]), torch.Size([1, 1, 1, 1])]</span>
+
+<span class="sd">        Function with flattened parameters returns a combined Jacobian.</span>
+
+<span class="sd">        &gt;&gt;&gt; input = torch.randn(2, 2, 2)</span>
+<span class="sd">        &gt;&gt;&gt; model = nn.Conv2d(in_channels=2, out_channels=2, kernel_size=1)</span>
+<span class="sd">        &gt;&gt;&gt; J = pp.optim.functional.modjac(model, input, flatten=True)</span>
+<span class="sd">        tensor([[-0.4162,  0.0968,  0.0000,  0.0000,  1.0000,  0.0000],</span>
+<span class="sd">                [-0.6042,  1.1886,  0.0000,  0.0000,  1.0000,  0.0000],</span>
+<span class="sd">                [ 1.4623,  0.7389,  0.0000,  0.0000,  1.0000,  0.0000],</span>
+<span class="sd">                [ 1.0716,  2.4293,  0.0000,  0.0000,  1.0000,  0.0000],</span>
+<span class="sd">                [ 0.0000,  0.0000, -0.4162,  0.0968,  0.0000,  1.0000],</span>
+<span class="sd">                [ 0.0000,  0.0000, -0.6042,  1.1886,  0.0000,  1.0000],</span>
+<span class="sd">                [ 0.0000,  0.0000,  1.4623,  0.7389,  0.0000,  1.0000],</span>
+<span class="sd">                [ 0.0000,  0.0000,  1.0716,  2.4293,  0.0000,  1.0000]])</span>
+<span class="sd">        &gt;&gt;&gt; J.shape</span>
+<span class="sd">        torch.Size([8, 6])</span>
+
+<span class="sd">        Calculate Jacobian with respect to parameter of :obj:`pypose.LieTensor`.</span>
+
+<span class="sd">        &gt;&gt;&gt; class PoseTransform(torch.nn.Module):</span>
+<span class="sd">        ...     def __init__(self):</span>
+<span class="sd">        ...         super().__init__()</span>
+<span class="sd">        ...         self.p = pp.Parameter(pp.randn_so3(2))</span>
+<span class="sd">        ...</span>
+<span class="sd">        ...     def forward(self, x):</span>
+<span class="sd">        ...         return self.p.Exp() * x</span>
+<span class="sd">        ...</span>
+<span class="sd">        &gt;&gt;&gt; model, input = PoseTransform(), pp.randn_SO3()</span>
+<span class="sd">        &gt;&gt;&gt; J = pp.optim.functional.modjac(model, input, flatten=True)</span>
+<span class="sd">        tensor([[ 0.4670,  0.7041,  0.0029,  0.0000,  0.0000,  0.0000],</span>
+<span class="sd">                [-0.6591,  0.4554, -0.2566,  0.0000,  0.0000,  0.0000],</span>
+<span class="sd">                [-0.2477,  0.0670,  0.9535,  0.0000,  0.0000,  0.0000],</span>
+<span class="sd">                [ 0.0000,  0.0000,  0.0000,  0.0000,  0.0000,  0.0000],</span>
+<span class="sd">                [ 0.0000,  0.0000,  0.0000,  0.8593,  0.2672,  0.3446],</span>
+<span class="sd">                [ 0.0000,  0.0000,  0.0000, -0.2417,  0.9503, -0.1154],</span>
+<span class="sd">                [ 0.0000,  0.0000,  0.0000, -0.3630, -0.0179,  0.9055],</span>
+<span class="sd">                [ 0.0000,  0.0000,  0.0000,  0.0000,  0.0000,  0.0000]])</span>
+<span class="sd">        &gt;&gt;&gt; J.shape</span>
+<span class="sd">        torch.Size([8, 6])</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="n">params</span><span class="p">,</span> <span class="n">buffers</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">named_parameters</span><span class="p">()),</span> <span class="nb">dict</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">named_buffers</span><span class="p">())</span>
+    <span class="n">params_names</span><span class="p">,</span> <span class="n">params_values</span> <span class="o">=</span> <span class="n">params</span><span class="o">.</span><span class="n">keys</span><span class="p">(),</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">params</span><span class="o">.</span><span class="n">values</span><span class="p">())</span>
+
+    <span class="k">if</span> <span class="nb">input</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
+        <span class="nb">input</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">()</span>
+
+    <span class="k">def</span> <span class="nf">func_param</span><span class="p">(</span><span class="o">*</span><span class="n">new_params_values</span><span class="p">):</span>
+        <span class="n">new_params_dict</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">params_names</span><span class="p">,</span> <span class="n">new_params_values</span><span class="p">))</span>
+        <span class="k">return</span> <span class="n">functional_call</span><span class="p">(</span><span class="n">model</span><span class="p">,</span> <span class="p">(</span><span class="n">new_params_dict</span><span class="p">,</span> <span class="n">buffers</span><span class="p">),</span> <span class="nb">input</span><span class="p">)</span>
+
+    <span class="n">J</span> <span class="o">=</span> <span class="n">jacobian</span><span class="p">(</span><span class="n">func_param</span><span class="p">,</span> <span class="n">params_values</span><span class="p">,</span> <span class="n">create_graph</span><span class="o">=</span><span class="n">create_graph</span><span class="p">,</span> <span class="n">strict</span><span class="o">=</span><span class="n">strict</span><span class="p">,</span> \
+                    <span class="n">vectorize</span><span class="o">=</span><span class="n">vectorize</span><span class="p">,</span> <span class="n">strategy</span><span class="o">=</span><span class="n">strategy</span><span class="p">)</span>
+
+    <span class="k">assert</span> <span class="ow">not</span> <span class="n">hasnan</span><span class="p">(</span><span class="n">J</span><span class="p">),</span> <span class="s1">&#39;Jacobian contains Nan! Check your model and input!&#39;</span>
+
+    <span class="k">if</span> <span class="n">flatten</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">J</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">):</span>
+        <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">)</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">J</span><span class="p">):</span>
+            <span class="n">J</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">cat</span><span class="p">([</span><span class="n">torch</span><span class="o">.</span><span class="n">cat</span><span class="p">([</span><span class="n">j</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">p</span><span class="o">.</span><span class="n">numel</span><span class="p">())</span> \
+                    <span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="n">p</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">Jr</span><span class="p">,</span> <span class="n">params_values</span><span class="p">)],</span> <span class="n">dim</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span> <span class="k">for</span> <span class="n">Jr</span> <span class="ow">in</span> <span class="n">J</span><span class="p">])</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">J</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">cat</span><span class="p">([</span><span class="n">j</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">p</span><span class="o">.</span><span class="n">numel</span><span class="p">())</span> \
+                           <span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="n">p</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">J</span><span class="p">,</span> <span class="n">params_values</span><span class="p">)],</span> <span class="n">dim</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+
+    <span class="k">return</span> <span class="n">J</span></div>
+
+
+<span class="k">def</span> <span class="nf">modjacrev</span><span class="p">(</span><span class="n">model</span><span class="p">,</span> <span class="nb">input</span><span class="p">,</span> <span class="n">argnums</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="o">*</span><span class="p">,</span> <span class="n">has_aux</span><span class="o">=</span><span class="kc">False</span><span class="p">):</span>
+    <span class="n">params</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">named_parameters</span><span class="p">())</span>
+    <span class="n">func</span> <span class="o">=</span> <span class="n">partial</span><span class="p">(</span><span class="n">functional_call</span><span class="p">,</span> <span class="n">model</span><span class="p">)</span>
+    <span class="k">return</span> <span class="n">jacrev</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="n">argnums</span><span class="o">=</span><span class="n">argnums</span><span class="p">,</span> <span class="n">has_aux</span><span class="o">=</span><span class="n">has_aux</span><span class="p">)(</span><span class="n">params</span><span class="p">,</span> <span class="nb">input</span><span class="p">)</span>
+
+
+<span class="k">def</span> <span class="nf">modjacfwd</span><span class="p">(</span><span class="n">model</span><span class="p">,</span> <span class="nb">input</span><span class="p">,</span> <span class="n">argnums</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="o">*</span><span class="p">,</span> <span class="n">has_aux</span><span class="o">=</span><span class="kc">False</span><span class="p">):</span>
+    <span class="n">params</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">named_parameters</span><span class="p">())</span>
+    <span class="n">func</span> <span class="o">=</span> <span class="n">partial</span><span class="p">(</span><span class="n">functional_call</span><span class="p">,</span> <span class="n">model</span><span class="p">)</span>
+    <span class="k">return</span> <span class="n">jacfwd</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="n">argnums</span><span class="o">=</span><span class="n">argnums</span><span class="p">,</span> <span class="n">has_aux</span><span class="o">=</span><span class="n">has_aux</span><span class="p">)(</span><span class="n">params</span><span class="p">,</span> <span class="nb">input</span><span class="p">)</span>
+</pre></div>
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+              <p class="caption" role="heading"><span class="caption-text">Contents:</span></p>
+<ul>
+<li class="toctree-l1"><a class="reference internal" href="../../../../lietensor/">LieTensor</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../../../utils/">Utils</a></li>
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+              
+  <h1>Source code for pypose.optim.kernel</h1><div class="highlight"><pre>
+<span></span><span class="kn">import</span> <span class="nn">torch</span><span class="o">,</span> <span class="nn">math</span>
+<span class="kn">from</span> <span class="nn">torch</span> <span class="kn">import</span> <span class="n">nn</span><span class="p">,</span> <span class="n">Tensor</span>
+
+
+<div class="viewcode-block" id="Huber"><a class="viewcode-back" href="../../../../generated/pypose.optim.kernel.Huber/#pypose.optim.kernel.Huber">[docs]</a><span class="k">class</span> <span class="nc">Huber</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&quot;&quot;&quot;The robust Huber kernel cost function.</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \bm{y}_i = \begin{cases}</span>
+<span class="sd">            \bm{x}_i                            &amp; \text{if } \sqrt{\bm{x}_i} &lt; \delta \\</span>
+<span class="sd">            2 \delta \sqrt{\bm{x}_i} - \delta^2 &amp; \text{otherwise }</span>
+<span class="sd">        \end{cases},</span>
+
+<span class="sd">    where :math:`\delta` (delta) is a threshold, :math:`\bm{x}` and :math:`\bm{y}` are the input</span>
+<span class="sd">    and output tensors, respectively.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        delta (float, optional): Specify the threshold at which to scale the input. The value must</span>
+<span class="sd">            be positive. Default: 1.0</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        The input has to be a non-negative tensor and the output tensor has the same shape with the</span>
+<span class="sd">        input. Use `torch.nn.HuberLoss</span>
+<span class="sd">        &lt;https://pytorch.org/docs/stable/generated/torch.nn.HuberLoss.html&gt;`_ instead, if a scalar</span>
+<span class="sd">        Huber loss function is needed.</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        &gt;&gt;&gt; import pypose.optim.kernel as ppok</span>
+<span class="sd">        &gt;&gt;&gt; kernel = ppok.Huber()</span>
+<span class="sd">        &gt;&gt;&gt; input = torch.tensor([0, 0.5, 1, 2, 3])</span>
+<span class="sd">        &gt;&gt;&gt; kernel(input)</span>
+<span class="sd">        tensor([0.0000, 0.5000, 1.0000, 1.8284, 2.4641])</span>
+
+<span class="sd">    .. figure:: /_static/img/optim/kernel/huber.png</span>
+<span class="sd">                        :width: 600</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">delta</span><span class="p">:</span> <span class="nb">float</span> <span class="o">=</span> <span class="mf">1.0</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kc">None</span><span class="p">:</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+        <span class="k">assert</span> <span class="n">delta</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">,</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">&quot;delta has to be positive: </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">delta</span><span class="p">))</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">delta</span> <span class="o">=</span> <span class="n">delta</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">delta2</span> <span class="o">=</span> <span class="n">delta</span><span class="o">**</span><span class="mi">2</span>
+
+<div class="viewcode-block" id="Huber.forward"><a class="viewcode-back" href="../../../../generated/pypose.optim.kernel.Huber/#pypose.optim.kernel.Huber.forward">[docs]</a>    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">input</span><span class="p">:</span> <span class="n">Tensor</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="n">Tensor</span><span class="p">:</span>
+<span class="w">        </span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Args:</span>
+<span class="sd">            input (torch.Tensor): the input tensor (non-negative).</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">assert</span> <span class="n">torch</span><span class="o">.</span><span class="n">all</span><span class="p">(</span><span class="nb">input</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">),</span> <span class="s1">&#39;input has to be non-negative.&#39;</span>
+        <span class="n">mask</span> <span class="o">=</span> <span class="nb">input</span><span class="o">.</span><span class="n">sqrt</span><span class="p">()</span> <span class="o">&lt;</span> <span class="bp">self</span><span class="o">.</span><span class="n">delta</span>
+        <span class="n">output</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">zeros_like</span><span class="p">(</span><span class="nb">input</span><span class="p">)</span>
+        <span class="n">output</span><span class="p">[</span><span class="n">mask</span><span class="p">]</span> <span class="o">=</span> <span class="nb">input</span><span class="p">[</span><span class="n">mask</span><span class="p">]</span>
+        <span class="n">output</span><span class="p">[</span><span class="o">~</span><span class="n">mask</span><span class="p">]</span> <span class="o">=</span> <span class="mi">2</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">delta</span> <span class="o">*</span> <span class="nb">input</span><span class="p">[</span><span class="o">~</span><span class="n">mask</span><span class="p">]</span><span class="o">.</span><span class="n">sqrt</span><span class="p">()</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">delta2</span>
+        <span class="k">return</span> <span class="n">output</span></div></div>
+
+
+<div class="viewcode-block" id="PseudoHuber"><a class="viewcode-back" href="../../../../generated/pypose.optim.kernel.PseudoHuber/#pypose.optim.kernel.PseudoHuber">[docs]</a><span class="k">class</span> <span class="nc">PseudoHuber</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&quot;&quot;&quot;The robust pseudo Huber kernel cost function.</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \bm{y}_i = 2\delta^2 \left(\sqrt{1 + \bm{x}_i/\delta^2} - 1\right),</span>
+
+<span class="sd">    where :math:`\delta` (delta) defines the steepness of the slope, :math:`\bm{x}` and</span>
+<span class="sd">    :math:`\bm{y}` are the input and output tensors, respectively.  It can be used as a smooth</span>
+<span class="sd">    version of :obj:`Huber`.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        delta (float, optional): Specify the slope. The value must be positive. Default: 1.0</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        The input has to be a non-negative tensor and the output tensor has the same shape with the</span>
+<span class="sd">        input.</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        &gt;&gt;&gt; import pypose.optim.kernel as ppok</span>
+<span class="sd">        &gt;&gt;&gt; kernel = ppok.PseudoHuber()</span>
+<span class="sd">        &gt;&gt;&gt; input = torch.tensor([0, 0.5, 1, 2, 3])</span>
+<span class="sd">        &gt;&gt;&gt; kernel(input)</span>
+<span class="sd">        tensor([0.0000, 0.4495, 0.8284, 1.4641, 2.0000])</span>
+
+<span class="sd">    .. figure:: /_static/img/optim/kernel/pseudohuber.png</span>
+<span class="sd">                        :width: 600</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">delta</span><span class="p">:</span> <span class="nb">float</span> <span class="o">=</span> <span class="mf">1.0</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kc">None</span><span class="p">:</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+        <span class="k">assert</span> <span class="n">delta</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">,</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">&quot;delta has to be positive: </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">delta</span><span class="p">))</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">delta2</span> <span class="o">=</span> <span class="n">delta</span><span class="o">**</span><span class="mi">2</span>
+
+<div class="viewcode-block" id="PseudoHuber.forward"><a class="viewcode-back" href="../../../../generated/pypose.optim.kernel.PseudoHuber/#pypose.optim.kernel.PseudoHuber.forward">[docs]</a>    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">input</span><span class="p">:</span> <span class="n">Tensor</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="n">Tensor</span><span class="p">:</span>
+<span class="w">        </span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Args:</span>
+<span class="sd">            input (torch.Tensor): the input tensor (non-negative).</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">assert</span> <span class="n">torch</span><span class="o">.</span><span class="n">all</span><span class="p">(</span><span class="nb">input</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">),</span> <span class="s1">&#39;input has to be non-negative&#39;</span>
+        <span class="k">return</span> <span class="mi">2</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">delta2</span> <span class="o">*</span> <span class="p">((</span><span class="nb">input</span><span class="o">/</span><span class="bp">self</span><span class="o">.</span><span class="n">delta2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">sqrt</span><span class="p">()</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span></div></div>
+
+
+<div class="viewcode-block" id="Cauchy"><a class="viewcode-back" href="../../../../generated/pypose.optim.kernel.Cauchy/#pypose.optim.kernel.Cauchy">[docs]</a><span class="k">class</span> <span class="nc">Cauchy</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&quot;&quot;&quot;The robust Cauchy kernel cost function.</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \bm{y}_i = \delta^2 \log\left(1 + \frac{\bm{x}_i}{\delta^2}\right),</span>
+
+<span class="sd">    where :math:`\delta` (delta) is a hyperparameter, :math:`\bm{x}`</span>
+<span class="sd">    and :math:`\bm{y}` are the input and output tensors, respectively.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        delta (float, optional): Specify the Cauchy cost. The value must be positive. Default: 1.0</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        The input has to be a non-negative tensor and the output tensor</span>
+<span class="sd">        has the same shape with the input.</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        &gt;&gt;&gt; import pypose.optim.kernel as ppok</span>
+<span class="sd">        &gt;&gt;&gt; kernel = ppok.Cauchy()</span>
+<span class="sd">        &gt;&gt;&gt; input = torch.tensor([0, 0.5, 1, 2, 3])</span>
+<span class="sd">        &gt;&gt;&gt; kernel(input)</span>
+<span class="sd">        tensor([0.0000, 0.4055, 0.6931, 1.0986, 1.3863])</span>
+
+<span class="sd">    .. figure:: /_static/img/optim/kernel/cauchy.png</span>
+<span class="sd">                        :width: 600</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">delta</span><span class="p">:</span> <span class="nb">float</span> <span class="o">=</span> <span class="mf">1.0</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kc">None</span><span class="p">:</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+        <span class="k">assert</span> <span class="n">delta</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">,</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">&quot;delta has to be positive: </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">delta</span><span class="p">))</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">delta2</span> <span class="o">=</span> <span class="n">delta</span><span class="o">**</span><span class="mi">2</span>
+
+<div class="viewcode-block" id="Cauchy.forward"><a class="viewcode-back" href="../../../../generated/pypose.optim.kernel.Cauchy/#pypose.optim.kernel.Cauchy.forward">[docs]</a>    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">input</span><span class="p">:</span> <span class="n">Tensor</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="n">Tensor</span><span class="p">:</span>
+<span class="w">        </span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Args:</span>
+<span class="sd">            input (torch.Tensor): the input tensor (non-negative).</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">assert</span> <span class="n">torch</span><span class="o">.</span><span class="n">all</span><span class="p">(</span><span class="nb">input</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">),</span> <span class="s1">&#39;input has to be non-negative&#39;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">delta2</span> <span class="o">*</span> <span class="p">(</span><span class="nb">input</span><span class="o">/</span><span class="bp">self</span><span class="o">.</span><span class="n">delta2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">log</span><span class="p">()</span></div></div>
+
+
+<div class="viewcode-block" id="SoftLOne"><a class="viewcode-back" href="../../../../generated/pypose.optim.kernel.SoftLOne/#pypose.optim.kernel.SoftLOne">[docs]</a><span class="k">class</span> <span class="nc">SoftLOne</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&quot;&quot;&quot;The robust SoftLOne kernel cost function.</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \bm{y}_i=2\left ( \delta \sqrt{\frac{1}{{\delta{}}^{2}}+\bm{x}_i}- 1\right )</span>
+
+<span class="sd">    where :math:`\delta` (delta) is a hyperparameter, :math:`\bm{x}` and :math:`\bm{y}`</span>
+<span class="sd">    are the input and output tensors, respectively.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        delta (float, optional): Specify the SoftLOne cost. The value must be positive. Default: 1.0</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        The input has to be a non-negative tensor and the output tensor has the same shape with the</span>
+<span class="sd">        input.</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        &gt;&gt;&gt; import pypose.optim.kernel as ppok</span>
+<span class="sd">        &gt;&gt;&gt; kernel = ppok.SoftLOne()</span>
+<span class="sd">        &gt;&gt;&gt; input = torch.tensor([0, 0.5, 1, 2, 3])</span>
+<span class="sd">        &gt;&gt;&gt; kernel(input)</span>
+<span class="sd">        tensor([0.0000, 0.4495, 0.8284, 1.4641, 2.0000])</span>
+
+<span class="sd">     .. figure:: /_static/img/optim/kernel/softlone.png</span>
+<span class="sd">                        :width: 600</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">delta</span><span class="p">:</span> <span class="nb">float</span> <span class="o">=</span> <span class="mf">1.0</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kc">None</span><span class="p">:</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+        <span class="k">assert</span> <span class="n">delta</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">,</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">&quot;delta has to be positive: </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">delta</span><span class="p">))</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">delta1</span> <span class="o">=</span> <span class="n">delta</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">delta2</span> <span class="o">=</span> <span class="n">delta</span><span class="o">**</span><span class="mi">2</span>
+
+<div class="viewcode-block" id="SoftLOne.forward"><a class="viewcode-back" href="../../../../generated/pypose.optim.kernel.SoftLOne/#pypose.optim.kernel.SoftLOne.forward">[docs]</a>    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">input</span><span class="p">:</span> <span class="n">Tensor</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="n">Tensor</span><span class="p">:</span>
+<span class="w">        </span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Args:</span>
+<span class="sd">            input (torch.Tensor): the input tensor (non-negative).</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">assert</span> <span class="n">torch</span><span class="o">.</span><span class="n">all</span><span class="p">(</span><span class="nb">input</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">),</span> <span class="s1">&#39;input has to be non-negative&#39;</span>
+        <span class="k">return</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">delta1</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">/</span> <span class="bp">self</span><span class="o">.</span><span class="n">delta2</span> <span class="o">+</span> <span class="nb">input</span><span class="p">)</span><span class="o">.</span><span class="n">sqrt</span><span class="p">()</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span></div></div>
+
+
+<div class="viewcode-block" id="Arctan"><a class="viewcode-back" href="../../../../generated/pypose.optim.kernel.Arctan/#pypose.optim.kernel.Arctan">[docs]</a><span class="k">class</span> <span class="nc">Arctan</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&quot;&quot;&quot;The robust Arctan kernel cost function.</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \bm{y}_i=\delta ^{2}\arctan \left ( \frac{\bm{x}_i}{\delta ^{2}}\right )</span>
+
+<span class="sd">    where :math:`\delta` (delta) is a hyperparameter, :math:`\bm{x}` and :math:`\bm{y}` are the</span>
+<span class="sd">    input and output tensors, respectively.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        delta (float, optional): Specify the Arctan cost. Default: 1.0</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        The input has to be a non-negative tensor and the output tensor has the same shape with the</span>
+<span class="sd">        input.</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        &gt;&gt;&gt; import pypose.optim.kernel as ppok</span>
+<span class="sd">        &gt;&gt;&gt; kernel = ppok.Arctan()</span>
+<span class="sd">        &gt;&gt;&gt; input = torch.tensor([0, 0.5, 1, 2, 3])</span>
+<span class="sd">        &gt;&gt;&gt; kernel(input)</span>
+<span class="sd">        tensor([0.0000, 0.4636, 0.7854, 1.1071, 1.2490])</span>
+
+<span class="sd">    .. figure:: /_static/img/optim/kernel/arctan.png</span>
+<span class="sd">                        :width: 600</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">delta</span><span class="p">:</span> <span class="nb">float</span> <span class="o">=</span> <span class="mf">1.0</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kc">None</span><span class="p">:</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">delta2</span> <span class="o">=</span> <span class="n">delta</span><span class="o">**</span><span class="mi">2</span>
+
+<div class="viewcode-block" id="Arctan.forward"><a class="viewcode-back" href="../../../../generated/pypose.optim.kernel.Arctan/#pypose.optim.kernel.Arctan.forward">[docs]</a>    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">input</span><span class="p">:</span> <span class="n">Tensor</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="n">Tensor</span><span class="p">:</span>
+<span class="w">        </span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Args:</span>
+<span class="sd">            input (torch.Tensor): the input tensor (non-negative).</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">assert</span> <span class="n">torch</span><span class="o">.</span><span class="n">all</span><span class="p">(</span><span class="nb">input</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">),</span> <span class="s1">&#39;input has to be non-negative&#39;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">delta2</span> <span class="o">*</span> <span class="p">(</span><span class="nb">input</span> <span class="o">/</span> <span class="bp">self</span><span class="o">.</span><span class="n">delta2</span><span class="p">)</span><span class="o">.</span><span class="n">arctan</span><span class="p">()</span></div></div>
+
+
+<div class="viewcode-block" id="Tolerant"><a class="viewcode-back" href="../../../../generated/pypose.optim.kernel.Tolerant/#pypose.optim.kernel.Tolerant">[docs]</a><span class="k">class</span> <span class="nc">Tolerant</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&quot;&quot;&quot;The robust Tolerant kernel cost function.</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \bm{y}_i = b\log (1+e^{\frac{\bm{x}_i-a}{b}})-b\log (1+e^{\frac{-a}{b}})</span>
+
+<span class="sd">    where :math:`\bm{a}`  and :math:`\bm{b}` are hyperparameters, :math:`\bm{x}`</span>
+<span class="sd">    and :math:`\bm{y}` are the input and output tensors, respectively.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        a (float): Specify the Tolerant cost. The value must be positive. Default: 1.0</span>
+<span class="sd">        b (float): Specify the Tolerant cost. The value must be negative. Default: -1.0</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        The input has to be a non-negative tensor and the output tensor has the same shape with</span>
+<span class="sd">        the input.</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        &gt;&gt;&gt; import pypose.optim.kernel as ppok</span>
+<span class="sd">        &gt;&gt;&gt; kernel = ppok.Tolerant()</span>
+<span class="sd">        &gt;&gt;&gt; input = torch.tensor([0, 0.5, 1, 2, 3])</span>
+<span class="sd">        &gt;&gt;&gt; kernel(input)</span>
+<span class="sd">        tensor([0.0000, 0.4636, 0.7854, 1.1071, 1.2490])</span>
+
+<span class="sd">    .. figure:: /_static/img/optim/kernel/tolerant.png</span>
+<span class="sd">                        :width: 600</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">a</span><span class="p">:</span> <span class="nb">float</span> <span class="o">=</span> <span class="mf">1.0</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="nb">float</span> <span class="o">=</span> <span class="o">-</span><span class="mf">1.0</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kc">None</span><span class="p">:</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+        <span class="k">assert</span> <span class="n">a</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">,</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">&quot;a has to be positive: </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">a</span><span class="p">))</span>
+        <span class="k">assert</span> <span class="n">b</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">,</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">&quot;b has to be negative: </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">b</span><span class="p">))</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">a</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">b</span> <span class="o">=</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span>
+
+<div class="viewcode-block" id="Tolerant.forward"><a class="viewcode-back" href="../../../../generated/pypose.optim.kernel.Tolerant/#pypose.optim.kernel.Tolerant.forward">[docs]</a>    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">input</span><span class="p">:</span> <span class="n">Tensor</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="n">Tensor</span><span class="p">:</span>
+<span class="w">        </span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Args:</span>
+<span class="sd">            input (torch.Tensor): the input tensor (non-negative).</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">assert</span> <span class="n">torch</span><span class="o">.</span><span class="n">all</span><span class="p">(</span><span class="nb">input</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">),</span> <span class="s1">&#39;input has to be non-negative&#39;</span>
+        <span class="n">result</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">b</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="p">((</span><span class="nb">input</span><span class="o">-</span><span class="bp">self</span><span class="o">.</span><span class="n">a</span><span class="p">)</span> <span class="o">/</span> <span class="bp">self</span><span class="o">.</span><span class="n">b</span><span class="p">)</span><span class="o">.</span><span class="n">exp</span><span class="p">())</span><span class="o">.</span><span class="n">log</span><span class="p">()</span>
+        <span class="n">offset</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">b</span> <span class="o">*</span> <span class="n">math</span><span class="o">.</span><span class="n">log</span><span class="p">((</span><span class="mi">1</span> <span class="o">+</span> <span class="n">math</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="bp">self</span><span class="o">.</span><span class="n">a</span> <span class="o">/</span> <span class="bp">self</span><span class="o">.</span><span class="n">b</span><span class="p">)))</span> <span class="c1"># constant, no grad.</span>
+        <span class="k">return</span> <span class="n">result</span> <span class="o">-</span> <span class="n">offset</span></div></div>
+
+
+<div class="viewcode-block" id="Scale"><a class="viewcode-back" href="../../../../generated/pypose.optim.kernel.Scale/#pypose.optim.kernel.Scale">[docs]</a><span class="k">class</span> <span class="nc">Scale</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&quot;&quot;&quot;The robust Scale kernel cost function.</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \bm{y}_i=\delta*\bm{x}_i</span>
+
+<span class="sd">    where :math:`\delta` (delta) is a scalar, :math:`\bm{x}` and :math:`\bm{y}` are the input</span>
+<span class="sd">    and output tensors, respectively.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        delta (float): Specify the scale factor. Should be between 0 and 1.  Default: 1.0</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        The input has to be a non-negative tensor and the output tensor has the same shape with</span>
+<span class="sd">        the input.</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        &gt;&gt;&gt; import pypose.optim.kernel as ppok</span>
+<span class="sd">        &gt;&gt;&gt; kernel = ppok.Scale()</span>
+<span class="sd">        &gt;&gt;&gt; input = torch.tensor([0, 0.5, 1, 2, 3])</span>
+<span class="sd">        &gt;&gt;&gt; kernel(input)</span>
+<span class="sd">        tensor([0.0000, 0.5000, 1.0000, 2.0000, 3.0000])</span>
+
+<span class="sd">     .. figure:: /_static/img/optim/kernel/scale.png</span>
+<span class="sd">                        :width: 600</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">delta</span><span class="p">:</span> <span class="nb">float</span> <span class="o">=</span> <span class="mf">1.0</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kc">None</span><span class="p">:</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+        <span class="k">assert</span> <span class="mi">0</span> <span class="o">&lt;</span> <span class="n">delta</span> <span class="o">&lt;=</span> <span class="mi">1</span> <span class="p">,</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">&quot;delta has to be between 0 and 1: </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">delta</span><span class="p">))</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">delta</span> <span class="o">=</span> <span class="n">delta</span>
+
+<div class="viewcode-block" id="Scale.forward"><a class="viewcode-back" href="../../../../generated/pypose.optim.kernel.Scale/#pypose.optim.kernel.Scale.forward">[docs]</a>    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">input</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Args:</span>
+<span class="sd">            input (torch.Tensor): the input tensor (non-negative).</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">delta</span> <span class="o">*</span> <span class="nb">input</span></div></div>
+</pre></div>
+
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+  <h1>Source code for pypose.optim.optimizer</h1><div class="highlight"><pre>
+<span></span><span class="kn">import</span> <span class="nn">torch</span>
+<span class="kn">from</span> <span class="nn">..</span> <span class="kn">import</span> <span class="n">bmv</span>
+<span class="kn">from</span> <span class="nn">torch</span> <span class="kn">import</span> <span class="n">nn</span><span class="p">,</span> <span class="n">finfo</span>
+<span class="kn">from</span> <span class="nn">.functional</span> <span class="kn">import</span> <span class="n">modjac</span>
+<span class="kn">from</span> <span class="nn">.strategy</span> <span class="kn">import</span> <span class="n">TrustRegion</span>
+<span class="kn">from</span> <span class="nn">torch.optim</span> <span class="kn">import</span> <span class="n">Optimizer</span>
+<span class="kn">from</span> <span class="nn">.solver</span> <span class="kn">import</span> <span class="n">PINV</span><span class="p">,</span> <span class="n">Cholesky</span>
+<span class="kn">from</span> <span class="nn">torch.linalg</span> <span class="kn">import</span> <span class="n">cholesky_ex</span>
+<span class="kn">from</span> <span class="nn">.corrector</span> <span class="kn">import</span> <span class="n">FastTriggs</span>
+
+
+<span class="k">class</span> <span class="nc">Trivial</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">    A trivial module. Get anything, return anything.</span>
+<span class="sd">    Not supposed to be called by PyPose users.</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+
+    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+        <span class="n">out</span> <span class="o">=</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">*</span><span class="n">kwargs</span><span class="o">.</span><span class="n">values</span><span class="p">()</span>
+        <span class="k">return</span> <span class="n">out</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">out</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="k">else</span> <span class="n">out</span>
+
+
+<span class="k">class</span> <span class="nc">RobustModel</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Standardize a model for least square problems with an option of square-rooting kernel.</span>
+<span class="sd">    Then model regression becomes minimizing the output of the standardized model.</span>
+<span class="sd">    This class is used during optimization but is not designed to expose to PyPose users.</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">model</span><span class="p">,</span> <span class="n">kernel</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">auto</span><span class="o">=</span><span class="kc">False</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">model</span> <span class="o">=</span> <span class="n">model</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">kernel</span> <span class="o">=</span> <span class="p">[</span><span class="n">Trivial</span><span class="p">()]</span> <span class="k">if</span> <span class="n">kernel</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">kernel</span>
+
+    <span class="k">def</span> <span class="nf">flatten_row_jacobian</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">J</span><span class="p">,</span> <span class="n">params_values</span><span class="p">):</span>
+        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">J</span><span class="p">,</span> <span class="p">(</span><span class="nb">tuple</span><span class="p">,</span> <span class="nb">list</span><span class="p">)):</span>
+            <span class="n">J</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">cat</span><span class="p">([</span><span class="n">j</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">p</span><span class="o">.</span><span class="n">numel</span><span class="p">())</span> <span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="n">p</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">J</span><span class="p">,</span> <span class="n">params_values</span><span class="p">)],</span> <span class="mi">1</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">J</span>
+
+    <span class="k">def</span> <span class="nf">normalize_RWJ</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">R</span><span class="p">,</span> <span class="n">weight</span><span class="p">,</span> <span class="n">J</span><span class="p">):</span>
+        <span class="n">weight_diag</span> <span class="o">=</span> <span class="kc">None</span>
+        <span class="k">if</span> <span class="n">weight</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
+            <span class="n">weight</span> <span class="o">=</span> <span class="n">weight</span> <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">weight</span><span class="p">,</span> <span class="p">(</span><span class="nb">tuple</span><span class="p">,</span> <span class="nb">list</span><span class="p">))</span> <span class="k">else</span> <span class="p">[</span><span class="n">weight</span><span class="p">]</span>
+            <span class="k">assert</span> <span class="nb">len</span><span class="p">(</span><span class="n">R</span><span class="p">)</span><span class="o">==</span><span class="nb">len</span><span class="p">(</span><span class="n">weight</span><span class="p">)</span>
+            <span class="n">weight_diag</span> <span class="o">=</span> <span class="p">[]</span>
+            <span class="k">for</span> <span class="n">w</span><span class="p">,</span> <span class="n">r</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">weight</span><span class="p">,</span> <span class="n">R</span><span class="p">):</span>
+                <span class="n">ni</span> <span class="o">=</span> <span class="n">r</span><span class="o">.</span><span class="n">numel</span><span class="p">()</span> <span class="o">*</span> <span class="n">w</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">/</span> <span class="n">w</span><span class="o">.</span><span class="n">numel</span><span class="p">()</span>
+                <span class="n">w</span> <span class="o">=</span> <span class="n">w</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="o">*</span><span class="n">w</span><span class="o">.</span><span class="n">shape</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="k">if</span> <span class="n">r</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span> <span class="k">else</span> <span class="n">w</span>
+                <span class="n">ws</span> <span class="o">=</span> <span class="n">w</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">w</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">],</span> <span class="n">w</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
+                <span class="n">ws</span> <span class="o">=</span> <span class="p">[</span><span class="n">wsi</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span> <span class="k">for</span> <span class="n">wsi</span> <span class="ow">in</span> <span class="n">ws</span><span class="p">]</span>
+                <span class="n">weight_diag</span> <span class="o">+=</span> <span class="n">ws</span> <span class="o">*</span> <span class="nb">int</span><span class="p">(</span><span class="n">ni</span><span class="p">)</span>
+            <span class="n">weight_diag</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">block_diag</span><span class="p">(</span><span class="o">*</span><span class="n">weight_diag</span><span class="p">)</span>
+        <span class="n">R</span> <span class="o">=</span> <span class="p">[</span><span class="n">r</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">R</span><span class="p">]</span>
+        <span class="n">J</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">cat</span><span class="p">(</span><span class="n">J</span><span class="p">)</span> <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">J</span><span class="p">,</span> <span class="p">(</span><span class="nb">tuple</span><span class="p">,</span> <span class="nb">list</span><span class="p">))</span> <span class="k">else</span> <span class="n">J</span>
+        <span class="k">return</span> <span class="n">torch</span><span class="o">.</span><span class="n">cat</span><span class="p">(</span><span class="n">R</span><span class="p">),</span> <span class="n">weight_diag</span><span class="p">,</span> <span class="n">J</span>
+
+    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">input</span><span class="p">,</span> <span class="n">target</span><span class="p">):</span>
+        <span class="n">output</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">model_forward</span><span class="p">(</span><span class="nb">input</span><span class="p">)</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">residuals</span><span class="p">(</span><span class="n">output</span><span class="p">,</span> <span class="n">target</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">model_forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">input</span><span class="p">):</span>
+        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="nb">input</span><span class="p">,</span> <span class="nb">dict</span><span class="p">):</span>
+            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">model</span><span class="p">(</span><span class="o">**</span><span class="nb">input</span><span class="p">)</span>
+        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="nb">input</span><span class="p">,</span> <span class="p">(</span><span class="nb">tuple</span><span class="p">,</span> <span class="nb">list</span><span class="p">)):</span>
+            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">model</span><span class="p">(</span><span class="o">*</span><span class="nb">input</span><span class="p">)</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">model</span><span class="p">(</span><span class="nb">input</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">residual</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">output</span><span class="p">,</span> <span class="n">target</span><span class="p">):</span>
+        <span class="k">return</span> <span class="n">output</span> <span class="k">if</span> <span class="n">target</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">output</span> <span class="o">-</span> <span class="n">target</span>
+
+    <span class="k">def</span> <span class="nf">residuals</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">outputs</span><span class="p">,</span> <span class="n">targets</span><span class="p">):</span>
+        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">outputs</span><span class="p">,</span> <span class="p">(</span><span class="nb">tuple</span><span class="p">,</span> <span class="nb">list</span><span class="p">)):</span>
+            <span class="n">targets</span> <span class="o">=</span> <span class="p">[</span><span class="kc">None</span><span class="p">]</span> <span class="o">*</span> <span class="nb">len</span><span class="p">(</span><span class="n">outputs</span><span class="p">)</span> <span class="k">if</span> <span class="n">targets</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">targets</span>
+            <span class="k">return</span> <span class="nb">tuple</span><span class="p">([</span><span class="bp">self</span><span class="o">.</span><span class="n">residual</span><span class="p">(</span><span class="n">out</span><span class="p">,</span> <span class="n">targets</span><span class="p">[</span><span class="n">i</span><span class="p">])</span> <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">out</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">outputs</span><span class="p">)])</span>
+        <span class="k">return</span> <span class="nb">tuple</span><span class="p">([</span><span class="bp">self</span><span class="o">.</span><span class="n">residual</span><span class="p">(</span><span class="n">outputs</span><span class="p">,</span> <span class="n">targets</span><span class="p">)])</span>
+
+    <span class="k">def</span> <span class="nf">loss</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">input</span><span class="p">,</span> <span class="n">target</span><span class="p">):</span>
+        <span class="n">output</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">model_forward</span><span class="p">(</span><span class="nb">input</span><span class="p">)</span>
+        <span class="n">residuals</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">residuals</span><span class="p">(</span><span class="n">output</span><span class="p">,</span> <span class="n">target</span><span class="p">)</span>
+        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">kernel</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
+            <span class="n">residuals</span> <span class="o">=</span> <span class="p">[</span><span class="n">k</span><span class="p">(</span><span class="n">r</span><span class="o">.</span><span class="n">square</span><span class="p">()</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span> <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">r</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">kernel</span><span class="p">,</span> <span class="n">residuals</span><span class="p">)]</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">residuals</span> <span class="o">=</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">kernel</span><span class="p">[</span><span class="mi">0</span><span class="p">](</span><span class="n">r</span><span class="o">.</span><span class="n">square</span><span class="p">()</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span> <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">residuals</span><span class="p">]</span>
+        <span class="k">return</span> <span class="nb">sum</span><span class="p">(</span><span class="n">residuals</span><span class="p">)</span>
+
+
+<span class="k">class</span> <span class="nc">_Optimizer</span><span class="p">(</span><span class="n">Optimizer</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Base class for all second order optimizers.</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">update_parameter</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">params</span><span class="p">,</span> <span class="n">step</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        params will be updated by calling this function</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="n">steps</span> <span class="o">=</span> <span class="n">step</span><span class="o">.</span><span class="n">split</span><span class="p">([</span><span class="n">p</span><span class="o">.</span><span class="n">numel</span><span class="p">()</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">params</span> <span class="k">if</span> <span class="n">p</span><span class="o">.</span><span class="n">requires_grad</span><span class="p">])</span>
+        <span class="p">[</span><span class="n">p</span><span class="o">.</span><span class="n">add_</span><span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">shape</span><span class="p">))</span> <span class="k">for</span> <span class="n">p</span><span class="p">,</span> <span class="n">d</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">params</span><span class="p">,</span> <span class="n">steps</span><span class="p">)</span> <span class="k">if</span> <span class="n">p</span><span class="o">.</span><span class="n">requires_grad</span><span class="p">]</span>
+
+
+<div class="viewcode-block" id="GaussNewton"><a class="viewcode-back" href="../../../../generated/pypose.optim.GaussNewton/#pypose.optim.GaussNewton">[docs]</a><span class="k">class</span> <span class="nc">GaussNewton</span><span class="p">(</span><span class="n">_Optimizer</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    The Gauss-Newton (GN) algorithm solving non-linear least squares problems. This implementation</span>
+<span class="sd">    is for optimizing the model parameters to approximate the target, which can be a</span>
+<span class="sd">    Tensor/LieTensor or a tuple of Tensors/LieTensors.</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \bm{\theta}^* = \arg\min_{\bm{\theta}} \sum_i</span>
+<span class="sd">            \rho\left((\bm{f}(\bm{\theta},\bm{x}_i)-\bm{y}_i)^T \mathbf{W}_i</span>
+<span class="sd">            (\bm{f}(\bm{\theta},\bm{x}_i)-\bm{y}_i)\right),</span>
+
+<span class="sd">    where :math:`\bm{f}()` is the model, :math:`\bm{\theta}` is the parameters to be optimized,</span>
+<span class="sd">    :math:`\bm{x}` is the model input, :math:`\mathbf{W}_i` is a weighted square matrix (positive</span>
+<span class="sd">    definite), and :math:`\rho` is a robust kernel function to reduce the effect of outliers.</span>
+<span class="sd">    :math:`\rho(x) = x` is used by default.</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">       \begin{aligned}</span>
+<span class="sd">            &amp;\rule{113mm}{0.4pt}                                                                 \\</span>
+<span class="sd">            &amp;\textbf{input}: \bm{\theta}_0~\text{(params)}, \bm{f}~\text{(model)},</span>
+<span class="sd">                \bm{x}~(\text{input}), \bm{y}~(\text{target}), \rho~(\text{kernel})              \\</span>
+<span class="sd">            &amp;\rule{113mm}{0.4pt}                                                                 \\</span>
+<span class="sd">            &amp;\textbf{for} \: t=1 \: \textbf{to} \: \ldots \: \textbf{do}                         \\</span>
+<span class="sd">            &amp;\hspace{5mm} \mathbf{J} \leftarrow {\dfrac {\partial \bm{f} } {\partial \bm{\theta}_{t-1}}}             \\</span>
+<span class="sd">            &amp;\hspace{5mm} \mathbf{R} = \bm{f(\bm{\theta}_{t-1}, \bm{x})}-\bm{y}                  \\</span>
+<span class="sd">            &amp;\hspace{5mm} \mathbf{R}, \mathbf{J}=\mathrm{corrector}(\rho, \mathbf{R}, \mathbf{J})\\</span>
+<span class="sd">            &amp;\hspace{5mm} \bm{\delta} = \mathrm{solver}(\mathbf{W}\mathbf{J}, -\mathbf{W}\mathbf{R})                 \\</span>
+<span class="sd">            &amp;\hspace{5mm} \bm{\theta}_t \leftarrow \bm{\theta}_{t-1} + \bm{\delta}               \\</span>
+<span class="sd">            &amp;\rule{113mm}{0.4pt}                                                          \\[-1.ex]</span>
+<span class="sd">            &amp;\bf{return} \:  \theta_t                                                     \\[-1.ex]</span>
+<span class="sd">            &amp;\rule{113mm}{0.4pt}                                                          \\[-1.ex]</span>
+<span class="sd">       \end{aligned}</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        model (nn.Module): a module containing learnable parameters.</span>
+<span class="sd">        solver (nn.Module, optional): a linear solver. Available linear solvers include</span>
+<span class="sd">            :meth:`solver.PINV` and :meth:`solver.LSTSQ`. If ``None``, :meth:`solver.PINV` is used.</span>
+<span class="sd">            Default: ``None``.</span>
+<span class="sd">        kernel (nn.Module, or :obj:`list`, optional): the robust kernel function. If a :obj:`list`,</span>
+<span class="sd">            the element must be nn.Module or ``None`` and the length must be 1 or the number of residuals.</span>
+<span class="sd">            Default: ``None``.</span>
+<span class="sd">        corrector: (nn.Module, or :obj:`list`, optional): the Jacobian and model residual corrector to</span>
+<span class="sd">            fit the kernel function. If a :obj:`list`, the element must be nn.Module or ``None`` and</span>
+<span class="sd">            the length must be 1 or the number of residuals.</span>
+<span class="sd">            If a kernel is given but a corrector is not specified, auto correction is</span>
+<span class="sd">            used. Auto correction can be unstable when the robust model has indefinite Hessian.</span>
+<span class="sd">            Default: ``None``.</span>
+<span class="sd">        weight (:obj:`Tensor`, or :obj:`list`, optional): the square positive definite matrix defining</span>
+<span class="sd">            the weight of model residual. If a :obj:`list`, the element must be :obj:`Tensor` and</span>
+<span class="sd">            the length must be equal to the number of residuals.</span>
+<span class="sd">            The corresponding residual and weight should be</span>
+<span class="sd">            `broadcastable &lt;https://pytorch.org/docs/stable/notes/broadcasting.html#broadcasting-semantics&gt;`_.</span>
+<span class="sd">            For example, if the shape of a residual is ``B*M*N*R``, the shape of its weight can</span>
+<span class="sd">            be ``R*R``, ``N*R*R``, ``M*N*R*R`` or ``B*M*N*R*R``.</span>
+<span class="sd">            Use this only when all inputs shared the same weight matrices. This is</span>
+<span class="sd">            ignored when weight is given when calling :meth:`.step` or :meth:`.optimize` method.</span>
+<span class="sd">            Default: ``None``.</span>
+<span class="sd">        vectorize (bool, optional): the method of computing Jacobian. If ``True``, the</span>
+<span class="sd">            gradient of each scalar in output with respect to the model parameters will be</span>
+<span class="sd">            computed in parallel with ``&quot;reverse-mode&quot;``. More details go to</span>
+<span class="sd">            :meth:`pypose.optim.functional.modjac`. Default: ``True``.</span>
+
+<span class="sd">    Available solvers: :meth:`solver.PINV`; :meth:`solver.LSTSQ`.</span>
+
+<span class="sd">    Available kernels: :meth:`kernel.Huber`; :meth:`kernel.PseudoHuber`; :meth:`kernel.Cauchy`.</span>
+
+<span class="sd">    Available correctors: :meth:`corrector.FastTriggs`; :meth:`corrector.Triggs`.</span>
+
+<span class="sd">    Warning:</span>
+<span class="sd">        The output of model :math:`\bm{f}(\bm{\theta},\bm{x}_i)` and target :math:`\bm{y}_i`</span>
+<span class="sd">        can be any shape, while their **last dimension** :math:`d` is always taken as the</span>
+<span class="sd">        dimension of model residual, whose inner product is the input to the kernel function.</span>
+<span class="sd">        This is useful for residuals like re-projection error, whose last dimension is 2.</span>
+
+<span class="sd">        Note that **auto correction** is equivalent to the method of &#39;square-rooting the kernel&#39;</span>
+<span class="sd">        mentioned in Section 3.3 of the following paper. It replaces the</span>
+<span class="sd">        :math:`d`-dimensional residual with a one-dimensional one, which loses</span>
+<span class="sd">        residual-level structural information.</span>
+
+<span class="sd">        * Christopher Zach, `Robust Bundle Adjustment Revisited</span>
+<span class="sd">          &lt;https://link.springer.com/chapter/10.1007/978-3-319-10602-1_50&gt;`_, European</span>
+<span class="sd">          Conference on Computer Vision (ECCV), 2014.</span>
+
+<span class="sd">        **Therefore, the users need to keep the last dimension of model output and target to</span>
+<span class="sd">        1, even if the model residual is a scalar. If the users flatten all sample residuals</span>
+<span class="sd">        into a vector (residual inner product will be a scalar), the model Jacobian will be a</span>
+<span class="sd">        row vector, instead of a matrix, which loses sample-level structural information,</span>
+<span class="sd">        although computing Jacobian vector is faster.**</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        Instead of solving :math:`\mathbf{J}^T\mathbf{J}\delta = -\mathbf{J}^T\mathbf{R}`, we solve</span>
+<span class="sd">        :math:`\mathbf{J}\delta = -\mathbf{R}` via pseudo inversion, which is more numerically</span>
+<span class="sd">        advisable. Therefore, only solvers with pseudo inversion (inverting non-square matrices)</span>
+<span class="sd">        such as :meth:`solver.PINV` and :meth:`solver.LSTSQ` are available.</span>
+<span class="sd">        More details are in Eq. (5) of the paper &quot;`Robust Bundle Adjustment Revisited`_&quot;.</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">model</span><span class="p">,</span> <span class="n">solver</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">kernel</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">corrector</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">vectorize</span><span class="o">=</span><span class="kc">True</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">parameters</span><span class="p">(),</span> <span class="n">defaults</span><span class="o">=</span><span class="p">{})</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">jackwargs</span> <span class="o">=</span> <span class="p">{</span><span class="s1">&#39;vectorize&#39;</span><span class="p">:</span> <span class="n">vectorize</span><span class="p">}</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">solver</span> <span class="o">=</span> <span class="n">PINV</span><span class="p">()</span> <span class="k">if</span> <span class="n">solver</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">solver</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">weight</span> <span class="o">=</span> <span class="n">weight</span>
+        <span class="k">if</span> <span class="n">kernel</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
+            <span class="n">kernel</span> <span class="o">=</span> <span class="p">[</span><span class="n">kernel</span><span class="p">]</span> <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">kernel</span><span class="p">,</span> <span class="p">(</span><span class="nb">tuple</span><span class="p">,</span> <span class="nb">list</span><span class="p">))</span> <span class="k">else</span> <span class="n">kernel</span>
+            <span class="n">kernel</span> <span class="o">=</span> <span class="p">[</span><span class="n">k</span> <span class="k">if</span> <span class="n">k</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">Trivial</span><span class="p">()</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">kernel</span><span class="p">]</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">corrector</span> <span class="o">=</span> <span class="p">[</span><span class="n">FastTriggs</span><span class="p">(</span><span class="n">k</span><span class="p">)</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">kernel</span><span class="p">]</span> <span class="k">if</span> <span class="n">corrector</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">corrector</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">corrector</span> <span class="o">=</span> <span class="p">[</span><span class="n">Trivial</span><span class="p">()]</span> <span class="k">if</span> <span class="n">corrector</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">corrector</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">corrector</span> <span class="o">=</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">corrector</span><span class="p">]</span> <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">corrector</span><span class="p">,</span> <span class="p">(</span><span class="nb">tuple</span><span class="p">,</span> <span class="nb">list</span><span class="p">))</span> <span class="k">else</span> <span class="bp">self</span><span class="o">.</span><span class="n">corrector</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">corrector</span> <span class="o">=</span> <span class="p">[</span><span class="n">c</span> <span class="k">if</span> <span class="n">c</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">Trivial</span><span class="p">()</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">corrector</span><span class="p">]</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">model</span> <span class="o">=</span> <span class="n">RobustModel</span><span class="p">(</span><span class="n">model</span><span class="p">,</span> <span class="n">kernel</span><span class="p">)</span>
+
+
+<div class="viewcode-block" id="GaussNewton.step"><a class="viewcode-back" href="../../../../generated/pypose.optim.GaussNewton/#pypose.optim.GaussNewton.step">[docs]</a>    <span class="nd">@torch</span><span class="o">.</span><span class="n">no_grad</span><span class="p">()</span>
+    <span class="k">def</span> <span class="nf">step</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">input</span><span class="p">,</span> <span class="n">target</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Performs a single optimization step.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            input (Tensor/LieTensor, tuple or a dict of Tensors/LieTensors): the input to the model.</span>
+<span class="sd">            target (Tensor/LieTensor): the model target to approximate.</span>
+<span class="sd">                If not given, the model output is minimized. Default: ``None``.</span>
+<span class="sd">            weight (:obj:`Tensor`, or :obj:`list`, optional): the square positive definite matrix defining</span>
+<span class="sd">                the weight of model residual. If a :obj:`list`, the element must be :obj:`Tensor` and</span>
+<span class="sd">                the length must be equal to the number of residuals. Default: ``None``.</span>
+
+<span class="sd">        Return:</span>
+<span class="sd">            Tensor: the minimized model loss.</span>
+
+<span class="sd">        Note:</span>
+<span class="sd">            Different from PyTorch optimizers like</span>
+<span class="sd">            `SGD &lt;https://pytorch.org/docs/stable/generated/torch.optim.SGD.html&gt;`_, where the model</span>
+<span class="sd">            error has to be a scalar, the output of model :math:`\bm{f}` can be a Tensor/LieTensor or a</span>
+<span class="sd">            tuple of Tensors/LieTensors.</span>
+
+<span class="sd">            See more details of</span>
+<span class="sd">            `Gauss-Newton (GN) algorithm &lt;https://en.wikipedia.org/wiki/Gauss-Newton_algorithm&gt;`_ on</span>
+<span class="sd">            Wikipedia.</span>
+
+<span class="sd">        Example:</span>
+<span class="sd">            Optimizing a simple module to **approximate pose inversion**.</span>
+
+<span class="sd">            &gt;&gt;&gt; class PoseInv(nn.Module):</span>
+<span class="sd">            ...     def __init__(self, *dim):</span>
+<span class="sd">            ...         super().__init__()</span>
+<span class="sd">            ...         self.pose = pp.Parameter(pp.randn_se3(*dim))</span>
+<span class="sd">            ...</span>
+<span class="sd">            ...     def forward(self, input):</span>
+<span class="sd">            ...         # the last dimension of the output is 6,</span>
+<span class="sd">            ...         # which will be the residual dimension.</span>
+<span class="sd">            ...         return (self.pose.Exp() @ input).Log()</span>
+<span class="sd">            ...</span>
+<span class="sd">            &gt;&gt;&gt; posinv = PoseInv(2, 2)</span>
+<span class="sd">            &gt;&gt;&gt; input = pp.randn_SE3(2, 2)</span>
+<span class="sd">            &gt;&gt;&gt; optimizer = pp.optim.GN(posinv)</span>
+<span class="sd">            ...</span>
+<span class="sd">            &gt;&gt;&gt; for idx in range(10):</span>
+<span class="sd">            ...     error = optimizer.step(input)</span>
+<span class="sd">            ...     print(&#39;Pose Inversion error %.7f @ %d it&#39;%(error, idx))</span>
+<span class="sd">            ...     if error &lt; 1e-5:</span>
+<span class="sd">            ...         print(&#39;Early Stopping with error:&#39;, error.item())</span>
+<span class="sd">            ...         break</span>
+<span class="sd">            ...</span>
+<span class="sd">            Pose Inversion error: 1.6865690 @ 0 it</span>
+<span class="sd">            Pose Inversion error: 0.1065131 @ 1 it</span>
+<span class="sd">            Pose Inversion error: 0.0002673 @ 2 it</span>
+<span class="sd">            Pose Inversion error: 0.0000005 @ 3 it</span>
+<span class="sd">            Early Stopping with error: 5.21540641784668e-07</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">for</span> <span class="n">pg</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">param_groups</span><span class="p">:</span>
+            <span class="n">weight</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">weight</span> <span class="k">if</span> <span class="n">weight</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">weight</span>
+            <span class="n">R</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">model</span><span class="p">(</span><span class="nb">input</span><span class="p">,</span> <span class="n">target</span><span class="p">))</span>
+            <span class="n">J</span> <span class="o">=</span> <span class="n">modjac</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">model</span><span class="p">,</span> <span class="nb">input</span><span class="o">=</span><span class="p">(</span><span class="nb">input</span><span class="p">,</span> <span class="n">target</span><span class="p">),</span> <span class="n">flatten</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="o">**</span><span class="bp">self</span><span class="o">.</span><span class="n">jackwargs</span><span class="p">)</span>
+            <span class="n">params</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">model</span><span class="o">.</span><span class="n">named_parameters</span><span class="p">())</span>
+            <span class="n">params_values</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">params</span><span class="o">.</span><span class="n">values</span><span class="p">())</span>
+            <span class="n">J</span> <span class="o">=</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">model</span><span class="o">.</span><span class="n">flatten_row_jacobian</span><span class="p">(</span><span class="n">Jr</span><span class="p">,</span> <span class="n">params_values</span><span class="p">)</span> <span class="k">for</span> <span class="n">Jr</span> <span class="ow">in</span> <span class="n">J</span><span class="p">]</span>
+            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">R</span><span class="p">)):</span>
+                <span class="n">R</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">J</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">corrector</span><span class="p">[</span><span class="mi">0</span><span class="p">](</span><span class="n">R</span> <span class="o">=</span> <span class="n">R</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">J</span> <span class="o">=</span> <span class="n">J</span><span class="p">[</span><span class="n">i</span><span class="p">])</span> <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">corrector</span><span class="p">)</span> <span class="o">==</span><span class="mi">1</span> \
+                    <span class="k">else</span> <span class="bp">self</span><span class="o">.</span><span class="n">corrector</span><span class="p">[</span><span class="n">i</span><span class="p">](</span><span class="n">R</span> <span class="o">=</span> <span class="n">R</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">J</span> <span class="o">=</span> <span class="n">J</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
+            <span class="n">R</span><span class="p">,</span> <span class="n">weight</span><span class="p">,</span> <span class="n">J</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">model</span><span class="o">.</span><span class="n">normalize_RWJ</span><span class="p">(</span><span class="n">R</span><span class="p">,</span> <span class="n">weight</span><span class="p">,</span> <span class="n">J</span><span class="p">)</span>
+            <span class="n">A</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="p">(</span><span class="n">J</span><span class="p">,</span> <span class="o">-</span><span class="n">R</span><span class="p">)</span> <span class="k">if</span> <span class="n">weight</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="p">(</span><span class="n">weight</span> <span class="o">@</span> <span class="n">J</span><span class="p">,</span> <span class="o">-</span><span class="n">weight</span> <span class="o">@</span> <span class="n">R</span><span class="p">)</span>
+            <span class="n">D</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">solver</span><span class="p">(</span><span class="n">A</span> <span class="o">=</span> <span class="n">A</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">last</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">loss</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="s1">&#39;loss&#39;</span><span class="p">)</span> \
+                        <span class="k">else</span> <span class="bp">self</span><span class="o">.</span><span class="n">model</span><span class="o">.</span><span class="n">loss</span><span class="p">(</span><span class="nb">input</span><span class="p">,</span> <span class="n">target</span><span class="p">)</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">update_parameter</span><span class="p">(</span><span class="n">params</span> <span class="o">=</span> <span class="n">pg</span><span class="p">[</span><span class="s1">&#39;params&#39;</span><span class="p">],</span> <span class="n">step</span> <span class="o">=</span> <span class="n">D</span><span class="p">)</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">loss</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">model</span><span class="o">.</span><span class="n">loss</span><span class="p">(</span><span class="nb">input</span><span class="p">,</span> <span class="n">target</span><span class="p">)</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">loss</span></div></div>
+
+
+<div class="viewcode-block" id="LevenbergMarquardt"><a class="viewcode-back" href="../../../../generated/pypose.optim.LevenbergMarquardt/#pypose.optim.LevenbergMarquardt">[docs]</a><span class="k">class</span> <span class="nc">LevenbergMarquardt</span><span class="p">(</span><span class="n">_Optimizer</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    The Levenberg-Marquardt (LM) algorithm solving non-linear least squares problems. It</span>
+<span class="sd">    is also known as the damped least squares (DLS) method. This implementation is for</span>
+<span class="sd">    optimizing the model parameters to approximate the target, which can be a</span>
+<span class="sd">    Tensor/LieTensor or a tuple of Tensors/LieTensors.</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \bm{\theta}^* = \arg\min_{\bm{\theta}} \sum_i</span>
+<span class="sd">            \rho\left((\bm{f}(\bm{\theta},\bm{x}_i)-\bm{y}_i)^T \mathbf{W}_i</span>
+<span class="sd">            (\bm{f}(\bm{\theta},\bm{x}_i)-\bm{y}_i)\right),</span>
+
+<span class="sd">    where :math:`\bm{f}()` is the model, :math:`\bm{\theta}` is the parameters to be optimized,</span>
+<span class="sd">    :math:`\bm{x}` is the model input, :math:`\mathbf{W}_i` is a weighted square matrix (positive</span>
+<span class="sd">    definite), and :math:`\rho` is a robust kernel function to reduce the effect of outliers.</span>
+<span class="sd">    :math:`\rho(x) = x` is used by default.</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">       \begin{aligned}</span>
+<span class="sd">            &amp;\rule{113mm}{0.4pt}                                                                 \\</span>
+<span class="sd">            &amp;\textbf{input}: \lambda~\text{(damping)}, \bm{\theta}_0~\text{(params)},</span>
+<span class="sd">                \bm{f}~\text{(model)}, \bm{x}~(\text{input}), \bm{y}~(\text{target})             \\</span>
+<span class="sd">            &amp;\hspace{12mm} \rho~(\text{kernel}), \epsilon_{s}~(\text{min}),</span>
+<span class="sd">                           \epsilon_{l}~(\text{max})                                             \\</span>
+<span class="sd">            &amp;\rule{113mm}{0.4pt}                                                                 \\</span>
+<span class="sd">            &amp;\textbf{for} \: t=1 \: \textbf{to} \: \ldots \: \textbf{do}                         \\</span>
+<span class="sd">            &amp;\hspace{5mm} \mathbf{J} \leftarrow {\dfrac {\partial \bm{f}} {\partial \bm{\theta}_{t-1}}}       \\</span>
+<span class="sd">            &amp;\hspace{5mm} \mathbf{A} \leftarrow (\mathbf{J}^T \mathbf{W} \mathbf{J})</span>
+<span class="sd">                                     .\mathrm{diagnal\_clamp(\epsilon_{s}, \epsilon_{l})}        \\</span>
+<span class="sd">            &amp;\hspace{5mm} \mathbf{R} = \bm{f(\bm{\theta}_{t-1}, \bm{x})}-\bm{y}                               \\</span>
+<span class="sd">            &amp;\hspace{5mm} \mathbf{R}, \mathbf{J}=\mathrm{corrector}(\rho, \mathbf{R}, \mathbf{J})\\</span>
+<span class="sd">            &amp;\hspace{5mm} \textbf{while}~\text{first iteration}~\textbf{or}~</span>
+<span class="sd">                                         \text{loss not decreasing}                              \\</span>
+<span class="sd">            &amp;\hspace{10mm} \mathbf{A} \leftarrow \mathbf{A} + \lambda \mathrm{diag}(\mathbf{A})  \\</span>
+<span class="sd">            &amp;\hspace{10mm} \bm{\delta} = \mathrm{solver}(\mathbf{A}, -\mathbf{J}^T \mathbf{W} \mathbf{R})     \\</span>
+<span class="sd">            &amp;\hspace{10mm} \lambda \leftarrow \mathrm{strategy}(\lambda,\text{model information})\\</span>
+<span class="sd">            &amp;\hspace{10mm} \bm{\theta}_t \leftarrow \bm{\theta}_{t-1} + \bm{\delta}              \\</span>
+<span class="sd">            &amp;\hspace{10mm} \textbf{if}~\text{loss not decreasing}~\textbf{and}~</span>
+<span class="sd">                                       \text{maximum reject step not reached}                    \\</span>
+<span class="sd">            &amp;\hspace{15mm} \bm{\theta}_t \leftarrow \bm{\theta}_{t-1} - \bm{\delta}</span>
+<span class="sd">                                                               ~(\text{reject step})             \\</span>
+<span class="sd">            &amp;\rule{113mm}{0.4pt}                                                          \\[-1.ex]</span>
+<span class="sd">            &amp;\bf{return} \:  \theta_t                                                     \\[-1.ex]</span>
+<span class="sd">            &amp;\rule{113mm}{0.4pt}                                                          \\[-1.ex]</span>
+<span class="sd">       \end{aligned}</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        model (nn.Module): a module containing learnable parameters.</span>
+<span class="sd">        solver (nn.Module, optional): a linear solver. If ``None``, :meth:`solver.Cholesky` is used.</span>
+<span class="sd">            Default: ``None``.</span>
+<span class="sd">        strategy (object, optional): strategy for adjusting the damping factor. If ``None``, the</span>
+<span class="sd">            :meth:`strategy.TrustRegion` will be used. Defult: ``None``.</span>
+<span class="sd">        kernel (nn.Module, or :obj:`list`, optional): the robust kernel function. If a :obj:`list`,</span>
+<span class="sd">            the element must be nn.Module or ``None`` and the length must be 1 or the number of residuals.</span>
+<span class="sd">            Default: ``None``.</span>
+<span class="sd">        corrector: (nn.Module, or :obj:`list`, optional): the Jacobian and model residual corrector to</span>
+<span class="sd">            fit the kernel function. If a :obj:`list`, the element must be nn.Module or ``None`` and</span>
+<span class="sd">            the length must be 1 or the number of residuals.</span>
+<span class="sd">            If a kernel is given but a corrector is not specified, auto correction is</span>
+<span class="sd">            used. Auto correction can be unstable when the robust model has indefinite Hessian.</span>
+<span class="sd">            Default: ``None``.</span>
+<span class="sd">        weight (:obj:`Tensor`, or :obj:`list`, optional): the square positive definite matrix defining</span>
+<span class="sd">            the weight of model residual. If a :obj:`list`, the element must be :obj:`Tensor` and</span>
+<span class="sd">            the length must be equal to the number of residuals.</span>
+<span class="sd">            The corresponding residual and weight should be</span>
+<span class="sd">            `broadcastable &lt;https://pytorch.org/docs/stable/notes/broadcasting.html#broadcasting-semantics&gt;`_.</span>
+<span class="sd">            For example, if the shape of a residual is ``B*M*N*R``, the shape of its weight can</span>
+<span class="sd">            be ``R*R``, ``N*R*R``, ``M*N*R*R`` or ``B*M*N*R*R``.</span>
+<span class="sd">            Use this only when all inputs shared the same weight matrices. This is</span>
+<span class="sd">            ignored when weight is given when calling :meth:`.step` or :meth:`.optimize` method.</span>
+<span class="sd">            Default: ``None``.</span>
+<span class="sd">        reject (integer, optional): the maximum number of rejecting unsuccessfull steps.</span>
+<span class="sd">            Default: 16.</span>
+<span class="sd">        min (float, optional): the lower-bound of the Hessian diagonal. Default: 1e-6.</span>
+<span class="sd">        max (float, optional): the upper-bound of the Hessian diagonal. Default: 1e32.</span>
+<span class="sd">        vectorize (bool, optional): the method of computing Jacobian. If ``True``, the</span>
+<span class="sd">            gradient of each scalar in output with respect to the model parameters will be</span>
+<span class="sd">            computed in parallel with ``&quot;reverse-mode&quot;``. More details go to</span>
+<span class="sd">            :meth:`pypose.optim.functional.modjac`. Default: ``True``.</span>
+
+<span class="sd">    Available solvers: :meth:`solver.PINV`; :meth:`solver.LSTSQ`, :meth:`solver.Cholesky`.</span>
+
+<span class="sd">    Available kernels: :meth:`kernel.Huber`; :meth:`kernel.PseudoHuber`; :meth:`kernel.Cauchy`.</span>
+
+<span class="sd">    Available correctors: :meth:`corrector.FastTriggs`, :meth:`corrector.Triggs`.</span>
+
+<span class="sd">    Available strategies: :meth:`strategy.Constant`; :meth:`strategy.Adaptive`;</span>
+<span class="sd">    :meth:`strategy.TrustRegion`;</span>
+
+<span class="sd">    Warning:</span>
+<span class="sd">        The output of model :math:`\bm{f}(\bm{\theta},\bm{x}_i)` and target :math:`\bm{y}_i`</span>
+<span class="sd">        can be any shape, while their **last dimension** :math:`d` is always taken as the</span>
+<span class="sd">        dimension of model residual, whose inner product will be input to the kernel</span>
+<span class="sd">        function. This is useful for residuals like re-projection error, whose last</span>
+<span class="sd">        dimension is 2.</span>
+
+<span class="sd">        Note that **auto correction** is equivalent to the method of &#39;square-rooting the kernel&#39;</span>
+<span class="sd">        mentioned in Section 3.3 of the following paper. It replaces the</span>
+<span class="sd">        :math:`d`-dimensional residual with a one-dimensional one, which loses</span>
+<span class="sd">        residual-level structural information.</span>
+
+<span class="sd">        * Christopher Zach, `Robust Bundle Adjustment Revisited</span>
+<span class="sd">          &lt;https://link.springer.com/chapter/10.1007/978-3-319-10602-1_50&gt;`_, European</span>
+<span class="sd">          Conference on Computer Vision (ECCV), 2014.</span>
+
+<span class="sd">        **Therefore, the users need to keep the last dimension of model output and target to</span>
+<span class="sd">        1, even if the model residual is a scalar. If the model output only has one dimension,</span>
+<span class="sd">        the model Jacobian will be a row vector, instead of a matrix, which loses sample-level</span>
+<span class="sd">        structural information, although computing Jacobian vector is faster.**</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">model</span><span class="p">,</span> <span class="n">solver</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">strategy</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">kernel</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">corrector</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> \
+                       <span class="n">weight</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">reject</span><span class="o">=</span><span class="mi">16</span><span class="p">,</span> <span class="nb">min</span><span class="o">=</span><span class="mf">1e-6</span><span class="p">,</span> <span class="nb">max</span><span class="o">=</span><span class="mf">1e32</span><span class="p">,</span> <span class="n">vectorize</span><span class="o">=</span><span class="kc">True</span><span class="p">):</span>
+        <span class="k">assert</span> <span class="nb">min</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">,</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">&quot;min value has to be positive: </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="nb">min</span><span class="p">))</span>
+        <span class="k">assert</span> <span class="nb">max</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">,</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">&quot;max value has to be positive: </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="nb">max</span><span class="p">))</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">strategy</span> <span class="o">=</span> <span class="n">TrustRegion</span><span class="p">()</span> <span class="k">if</span> <span class="n">strategy</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">strategy</span>
+        <span class="n">defaults</span> <span class="o">=</span> <span class="p">{</span><span class="o">**</span><span class="p">{</span><span class="s1">&#39;min&#39;</span><span class="p">:</span><span class="nb">min</span><span class="p">,</span> <span class="s1">&#39;max&#39;</span><span class="p">:</span><span class="nb">max</span><span class="p">},</span> <span class="o">**</span><span class="bp">self</span><span class="o">.</span><span class="n">strategy</span><span class="o">.</span><span class="n">defaults</span><span class="p">}</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">parameters</span><span class="p">(),</span> <span class="n">defaults</span><span class="o">=</span><span class="n">defaults</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">jackwargs</span> <span class="o">=</span> <span class="p">{</span><span class="s1">&#39;vectorize&#39;</span><span class="p">:</span> <span class="n">vectorize</span><span class="p">}</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">solver</span> <span class="o">=</span> <span class="n">Cholesky</span><span class="p">()</span> <span class="k">if</span> <span class="n">solver</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">solver</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">reject</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">reject_count</span> <span class="o">=</span> <span class="n">reject</span><span class="p">,</span> <span class="mi">0</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">weight</span> <span class="o">=</span> <span class="n">weight</span>
+        <span class="k">if</span> <span class="n">kernel</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
+            <span class="n">kernel</span> <span class="o">=</span> <span class="p">[</span><span class="n">kernel</span><span class="p">]</span> <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">kernel</span><span class="p">,</span> <span class="p">(</span><span class="nb">tuple</span><span class="p">,</span> <span class="nb">list</span><span class="p">))</span> <span class="k">else</span> <span class="n">kernel</span>
+            <span class="n">kernel</span> <span class="o">=</span> <span class="p">[</span><span class="n">k</span> <span class="k">if</span> <span class="n">k</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">Trivial</span><span class="p">()</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">kernel</span><span class="p">]</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">corrector</span> <span class="o">=</span> <span class="p">[</span><span class="n">FastTriggs</span><span class="p">(</span><span class="n">k</span><span class="p">)</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">kernel</span><span class="p">]</span> <span class="k">if</span> <span class="n">corrector</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">corrector</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">corrector</span> <span class="o">=</span> <span class="p">[</span><span class="n">Trivial</span><span class="p">()]</span> <span class="k">if</span> <span class="n">corrector</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">corrector</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">corrector</span> <span class="o">=</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">corrector</span><span class="p">]</span> <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">corrector</span><span class="p">,</span> <span class="p">(</span><span class="nb">tuple</span><span class="p">,</span> <span class="nb">list</span><span class="p">))</span> <span class="k">else</span> <span class="bp">self</span><span class="o">.</span><span class="n">corrector</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">corrector</span> <span class="o">=</span> <span class="p">[</span><span class="n">c</span> <span class="k">if</span> <span class="n">c</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">Trivial</span><span class="p">()</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">corrector</span><span class="p">]</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">model</span> <span class="o">=</span> <span class="n">RobustModel</span><span class="p">(</span><span class="n">model</span><span class="p">,</span> <span class="n">kernel</span><span class="p">)</span>
+
+
+<div class="viewcode-block" id="LevenbergMarquardt.step"><a class="viewcode-back" href="../../../../generated/pypose.optim.LevenbergMarquardt/#pypose.optim.LevenbergMarquardt.step">[docs]</a>    <span class="nd">@torch</span><span class="o">.</span><span class="n">no_grad</span><span class="p">()</span>
+    <span class="k">def</span> <span class="nf">step</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">input</span><span class="p">,</span> <span class="n">target</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Performs a single optimization step.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            input (Tensor/LieTensor, tuple or a dict of Tensors/LieTensors): the input to the model.</span>
+<span class="sd">            target (Tensor/LieTensor): the model target to optimize.</span>
+<span class="sd">                If not given, the squared model output is minimized. Defaults: ``None``.</span>
+<span class="sd">            weight (:obj:`Tensor`, or :obj:`list`, optional): the square positive definite matrix defining</span>
+<span class="sd">                the weight of model residual. If a :obj:`list`, the element must be :obj:`Tensor` and</span>
+<span class="sd">                the length must be equal to the number of residuals. Default: ``None``.</span>
+
+<span class="sd">        Return:</span>
+<span class="sd">            Tensor: the minimized model loss.</span>
+
+<span class="sd">        Note:</span>
+<span class="sd">            The (non-negative) damping factor :math:`\lambda` can be adjusted at each iteration.</span>
+<span class="sd">            If the residual reduces rapidly, a smaller value can be used, bringing the algorithm</span>
+<span class="sd">            closer to the Gauss-Newton algorithm, whereas if an iteration gives insufficient</span>
+<span class="sd">            residual reduction, :math:`\lambda` can be increased, giving a step closer to the</span>
+<span class="sd">            gradient descent direction.</span>
+
+<span class="sd">            See more details of `Levenberg-Marquardt (LM) algorithm</span>
+<span class="sd">            &lt;https://en.wikipedia.org/wiki/Levenberg-Marquardt_algorithm&gt;`_ on Wikipedia.</span>
+
+<span class="sd">        Note:</span>
+<span class="sd">            Different from PyTorch optimizers like</span>
+<span class="sd">            `SGD &lt;https://pytorch.org/docs/stable/generated/torch.optim.SGD.html&gt;`_, where the</span>
+<span class="sd">            model error has to be a scalar, the output of model :math:`\bm{f}` can be a</span>
+<span class="sd">            Tensor/LieTensor or a tuple of Tensors/LieTensors.</span>
+
+<span class="sd">        Example:</span>
+<span class="sd">            Optimizing a simple module to **approximate pose inversion**.</span>
+
+<span class="sd">            &gt;&gt;&gt; class PoseInv(nn.Module):</span>
+<span class="sd">            ...     def __init__(self, *dim):</span>
+<span class="sd">            ...         super().__init__()</span>
+<span class="sd">            ...         self.pose = pp.Parameter(pp.randn_se3(*dim))</span>
+<span class="sd">            ...</span>
+<span class="sd">            ...     def forward(self, input):</span>
+<span class="sd">            ...         # the last dimension of the output is 6,</span>
+<span class="sd">            ...         # which will be the residual dimension.</span>
+<span class="sd">            ...         return (self.pose.Exp() @ input).Log()</span>
+<span class="sd">            ...</span>
+<span class="sd">            &gt;&gt;&gt; posinv = PoseInv(2, 2)</span>
+<span class="sd">            &gt;&gt;&gt; input = pp.randn_SE3(2, 2)</span>
+<span class="sd">            &gt;&gt;&gt; strategy = pp.optim.strategy.Adaptive(damping=1e-6)</span>
+<span class="sd">            &gt;&gt;&gt; optimizer = pp.optim.LM(posinv, strategy=strategy)</span>
+<span class="sd">            ...</span>
+<span class="sd">            &gt;&gt;&gt; for idx in range(10):</span>
+<span class="sd">            ...     loss = optimizer.step(input)</span>
+<span class="sd">            ...     print(&#39;Pose Inversion loss %.7f @ %d it&#39;%(loss, idx))</span>
+<span class="sd">            ...     if loss &lt; 1e-5:</span>
+<span class="sd">            ...         print(&#39;Early Stopping with loss:&#39;, loss.item())</span>
+<span class="sd">            ...         break</span>
+<span class="sd">            ...</span>
+<span class="sd">            Pose Inversion error: 1.6600330 @ 0 it</span>
+<span class="sd">            Pose Inversion error: 0.1296970 @ 1 it</span>
+<span class="sd">            Pose Inversion error: 0.0008593 @ 2 it</span>
+<span class="sd">            Pose Inversion error: 0.0000004 @ 3 it</span>
+<span class="sd">            Early Stopping with error: 4.443569991963159e-07</span>
+
+<span class="sd">        Note:</span>
+<span class="sd">            More practical examples, e.g., pose graph optimization (PGO), can be found at</span>
+<span class="sd">            `examples/module/pgo</span>
+<span class="sd">            &lt;https://github.com/pypose/pypose/tree/main/examples/module/pgo&gt;`_.</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">for</span> <span class="n">pg</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">param_groups</span><span class="p">:</span>
+            <span class="n">weight</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">weight</span> <span class="k">if</span> <span class="n">weight</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">weight</span>
+            <span class="n">R</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">model</span><span class="p">(</span><span class="nb">input</span><span class="p">,</span> <span class="n">target</span><span class="p">))</span>
+            <span class="n">J</span> <span class="o">=</span> <span class="n">modjac</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">model</span><span class="p">,</span> <span class="nb">input</span><span class="o">=</span><span class="p">(</span><span class="nb">input</span><span class="p">,</span> <span class="n">target</span><span class="p">),</span> <span class="n">flatten</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="o">**</span><span class="bp">self</span><span class="o">.</span><span class="n">jackwargs</span><span class="p">)</span>
+            <span class="n">params</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">model</span><span class="o">.</span><span class="n">named_parameters</span><span class="p">())</span>
+            <span class="n">params_values</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">params</span><span class="o">.</span><span class="n">values</span><span class="p">())</span>
+            <span class="n">J</span> <span class="o">=</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">model</span><span class="o">.</span><span class="n">flatten_row_jacobian</span><span class="p">(</span><span class="n">Jr</span><span class="p">,</span> <span class="n">params_values</span><span class="p">)</span> <span class="k">for</span> <span class="n">Jr</span> <span class="ow">in</span> <span class="n">J</span><span class="p">]</span>
+            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">R</span><span class="p">)):</span>
+                <span class="n">R</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">J</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">corrector</span><span class="p">[</span><span class="mi">0</span><span class="p">](</span><span class="n">R</span> <span class="o">=</span> <span class="n">R</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">J</span> <span class="o">=</span> <span class="n">J</span><span class="p">[</span><span class="n">i</span><span class="p">])</span> <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">corrector</span><span class="p">)</span> <span class="o">==</span><span class="mi">1</span> \
+                    <span class="k">else</span> <span class="bp">self</span><span class="o">.</span><span class="n">corrector</span><span class="p">[</span><span class="n">i</span><span class="p">](</span><span class="n">R</span> <span class="o">=</span> <span class="n">R</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">J</span> <span class="o">=</span> <span class="n">J</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
+            <span class="n">R</span><span class="p">,</span> <span class="n">weight</span><span class="p">,</span> <span class="n">J</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">model</span><span class="o">.</span><span class="n">normalize_RWJ</span><span class="p">(</span><span class="n">R</span><span class="p">,</span> <span class="n">weight</span><span class="p">,</span> <span class="n">J</span><span class="p">)</span>
+
+            <span class="bp">self</span><span class="o">.</span><span class="n">last</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">loss</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">loss</span> <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="s1">&#39;loss&#39;</span><span class="p">)</span> \
+                                    <span class="k">else</span> <span class="bp">self</span><span class="o">.</span><span class="n">model</span><span class="o">.</span><span class="n">loss</span><span class="p">(</span><span class="nb">input</span><span class="p">,</span> <span class="n">target</span><span class="p">)</span>
+            <span class="n">J_T</span> <span class="o">=</span> <span class="n">J</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">weight</span> <span class="k">if</span> <span class="n">weight</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">J</span><span class="o">.</span><span class="n">T</span>
+            <span class="n">A</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">reject_count</span> <span class="o">=</span> <span class="n">J_T</span> <span class="o">@</span> <span class="n">J</span><span class="p">,</span> <span class="mi">0</span>
+            <span class="n">A</span><span class="o">.</span><span class="n">diagonal</span><span class="p">()</span><span class="o">.</span><span class="n">clamp_</span><span class="p">(</span><span class="n">pg</span><span class="p">[</span><span class="s1">&#39;min&#39;</span><span class="p">],</span> <span class="n">pg</span><span class="p">[</span><span class="s1">&#39;max&#39;</span><span class="p">])</span>
+            <span class="k">while</span> <span class="bp">self</span><span class="o">.</span><span class="n">last</span> <span class="o">&lt;=</span> <span class="bp">self</span><span class="o">.</span><span class="n">loss</span><span class="p">:</span>
+                <span class="n">A</span><span class="o">.</span><span class="n">diagonal</span><span class="p">()</span><span class="o">.</span><span class="n">add_</span><span class="p">(</span><span class="n">A</span><span class="o">.</span><span class="n">diagonal</span><span class="p">()</span> <span class="o">*</span> <span class="n">pg</span><span class="p">[</span><span class="s1">&#39;damping&#39;</span><span class="p">])</span>
+                <span class="k">try</span><span class="p">:</span>
+                    <span class="n">D</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">solver</span><span class="p">(</span><span class="n">A</span> <span class="o">=</span> <span class="n">A</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="o">-</span><span class="n">J_T</span> <span class="o">@</span> <span class="n">R</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
+                <span class="k">except</span> <span class="ne">Exception</span> <span class="k">as</span> <span class="n">e</span><span class="p">:</span>
+                    <span class="nb">print</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="s2">&quot;</span><span class="se">\n</span><span class="s2">Linear solver failed. Breaking optimization step...&quot;</span><span class="p">)</span>
+                    <span class="k">break</span>
+                <span class="bp">self</span><span class="o">.</span><span class="n">update_parameter</span><span class="p">(</span><span class="n">pg</span><span class="p">[</span><span class="s1">&#39;params&#39;</span><span class="p">],</span> <span class="n">D</span><span class="p">)</span>
+                <span class="bp">self</span><span class="o">.</span><span class="n">loss</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">model</span><span class="o">.</span><span class="n">loss</span><span class="p">(</span><span class="nb">input</span><span class="p">,</span> <span class="n">target</span><span class="p">)</span>
+                <span class="bp">self</span><span class="o">.</span><span class="n">strategy</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="n">pg</span><span class="p">,</span> <span class="n">last</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">last</span><span class="p">,</span> <span class="n">loss</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">loss</span><span class="p">,</span> <span class="n">J</span><span class="o">=</span><span class="n">J</span><span class="p">,</span> <span class="n">D</span><span class="o">=</span><span class="n">D</span><span class="p">,</span> <span class="n">R</span><span class="o">=</span><span class="n">R</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
+                <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">last</span> <span class="o">&lt;</span> <span class="bp">self</span><span class="o">.</span><span class="n">loss</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">reject_count</span> <span class="o">&lt;</span> <span class="bp">self</span><span class="o">.</span><span class="n">reject</span><span class="p">:</span> <span class="c1"># reject step</span>
+                    <span class="bp">self</span><span class="o">.</span><span class="n">update_parameter</span><span class="p">(</span><span class="n">params</span> <span class="o">=</span> <span class="n">pg</span><span class="p">[</span><span class="s1">&#39;params&#39;</span><span class="p">],</span> <span class="n">step</span> <span class="o">=</span> <span class="o">-</span><span class="n">D</span><span class="p">)</span>
+                    <span class="bp">self</span><span class="o">.</span><span class="n">loss</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">reject_count</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">last</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">reject_count</span> <span class="o">+</span> <span class="mi">1</span>
+                <span class="k">else</span><span class="p">:</span>
+                    <span class="k">break</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">loss</span></div></div>
+</pre></div>
+
+             </article>
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+  <h1>Source code for pypose.optim.scheduler</h1><div class="highlight"><pre>
+<span></span><span class="kn">import</span> <span class="nn">torch</span>
+<span class="kn">from</span> <span class="nn">.optimizer</span> <span class="kn">import</span> <span class="n">_Optimizer</span>
+
+
+<span class="k">class</span> <span class="nc">_Scheduler</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
+    <span class="k">class</span> <span class="nc">Continual</span><span class="p">:</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        From PyPose v0.3.6, we change scheduler.continual to scheduler.continual().</span>
+<span class="sd">        This is a temporary workaround for triggering an error when users call continual</span>
+<span class="sd">        attribute of the scheduler. This wrapper will be removed in a future release.</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">optimizer</span><span class="p">):</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">optimizer</span> <span class="o">=</span> <span class="n">optimizer</span>
+
+        <span class="k">def</span> <span class="fm">__call__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">            </span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">            Determining whether to stop an optimizer should be provided here.</span>
+<span class="sd">            This function is only for temporarailiy replacing Scheduler.continual().</span>
+<span class="sd">            &#39;&#39;&#39;</span>
+            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">optimizer</span><span class="o">.</span><span class="n">iscontinual</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+
+        <span class="k">def</span> <span class="fm">__bool__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+            <span class="k">raise</span> <span class="ne">RuntimeError</span><span class="p">(</span><span class="s1">&#39;Calling scheduler.continual is deprecated, &#39;</span>
+                               <span class="s1">&#39;please call scheduler.continual() instead. &#39;</span>
+                               <span class="s1">&#39;This error msg will be removed in a future release.&#39;</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">optimizer</span><span class="p">,</span> <span class="n">max_steps</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="kc">False</span><span class="p">):</span>
+
+        <span class="c1"># Attach optimizer</span>
+        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">optimizer</span><span class="p">,</span> <span class="n">_Optimizer</span><span class="p">):</span>
+            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s1">&#39;</span><span class="si">{}</span><span class="s1"> is not an Optimizer&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span>
+                <span class="nb">type</span><span class="p">(</span><span class="n">optimizer</span><span class="p">)</span><span class="o">.</span><span class="vm">__name__</span><span class="p">))</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">optimizer</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">verbose</span> <span class="o">=</span> <span class="n">optimizer</span><span class="p">,</span> <span class="n">verbose</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">max_steps</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">steps</span> <span class="o">=</span> <span class="n">max_steps</span><span class="p">,</span> <span class="mi">0</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">continual</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">Continual</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">_continual</span> <span class="o">=</span> <span class="kc">True</span>
+
+    <span class="k">def</span> <span class="nf">iscontinual</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        This is a temporary function.</span>
+<span class="sd">        We will change to continual() in a future release.</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_continual</span>
+
+    <span class="k">def</span> <span class="nf">state_dict</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;Returns the state of the scheduler as a :class:`dict`.</span>
+
+<span class="sd">        It contains an entry for every variable in self.__dict__ which</span>
+<span class="sd">        is not the optimizer.</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">return</span> <span class="p">{</span><span class="n">key</span><span class="p">:</span> <span class="n">value</span> <span class="k">for</span> <span class="n">key</span><span class="p">,</span> <span class="n">value</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="vm">__dict__</span><span class="o">.</span><span class="n">items</span><span class="p">()</span> <span class="k">if</span> <span class="n">key</span> <span class="o">!=</span> <span class="s1">&#39;optimizer&#39;</span><span class="p">}</span>
+
+    <span class="k">def</span> <span class="nf">load_state_dict</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">state_dict</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;Loads the schedulers state.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            state_dict (dict): scheduler state. Should be an object returned</span>
+<span class="sd">                from a call to :meth:`state_dict`.</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="bp">self</span><span class="o">.</span><span class="vm">__dict__</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="n">state_dict</span><span class="p">)</span>
+
+
+<div class="viewcode-block" id="StopOnPlateau"><a class="viewcode-back" href="../../../../generated/pypose.optim.scheduler.StopOnPlateau/#pypose.optim.scheduler.StopOnPlateau">[docs]</a><span class="k">class</span> <span class="nc">StopOnPlateau</span><span class="p">(</span><span class="n">_Scheduler</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    A scheduler to stop an optimizer when no relative loss &#39;decreasing&#39; is seen for a &#39;patience&#39;</span>
+<span class="sd">    number of steps.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        optimizer (Optimizer): Wrapped optimizer.</span>
+<span class="sd">        steps (int): Maximum number of interations optimizer will step.</span>
+<span class="sd">        patience (int): Number of steps with no loss &#39;decreasing&#39; is seen. For example, if</span>
+<span class="sd">            ``patience = 2``, then it ignores the first 2 steps with no improvement, and stop</span>
+<span class="sd">            the optimizer after the 3rd step if the loss has no decerasing. Default: 5.</span>
+<span class="sd">        decreasing (float): relative loss decreasing used to count the number of patience steps.</span>
+<span class="sd">            Default: 1e-3.</span>
+<span class="sd">        verbose (bool): if ``True``, prints a message to stdout for each step. Default: ``False``.</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        The users have two options to use a scheduler.</span>
+<span class="sd">        The first one is to call the :meth:`step` method for multiple times, which is easy to be</span>
+<span class="sd">        extended for customized operation in each iteration.</span>
+<span class="sd">        The second one is to call the :meth:`optimize` method, which interally calls :meth:`step`</span>
+<span class="sd">        multiple times to perform full optimization.</span>
+<span class="sd">        See examples below.</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">optimizer</span><span class="p">,</span> <span class="n">steps</span><span class="p">,</span> <span class="n">patience</span><span class="o">=</span><span class="mi">5</span><span class="p">,</span> <span class="n">decreasing</span><span class="o">=</span><span class="mf">1e-3</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="kc">False</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">(</span><span class="n">optimizer</span><span class="p">,</span> <span class="n">steps</span><span class="p">,</span> <span class="n">verbose</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">decreasing</span> <span class="o">=</span> <span class="n">decreasing</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">patience</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">patience_count</span> <span class="o">=</span> <span class="n">patience</span><span class="p">,</span> <span class="mi">0</span>
+
+<div class="viewcode-block" id="StopOnPlateau.step"><a class="viewcode-back" href="../../../../generated/pypose.optim.scheduler.StopOnPlateau/#pypose.optim.scheduler.StopOnPlateau.step">[docs]</a>    <span class="k">def</span> <span class="nf">step</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">loss</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Performs a scheduler step.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            loss (float): the model loss after one optimizer step.</span>
+
+<span class="sd">        Example:</span>
+
+<span class="sd">            &gt;&gt;&gt; class PoseInv(nn.Module):</span>
+<span class="sd">            ...</span>
+<span class="sd">            ...     def __init__(self, *dim):</span>
+<span class="sd">            ...         super().__init__()</span>
+<span class="sd">            ...         self.pose = pp.Parameter(pp.randn_SE3(*dim))</span>
+<span class="sd">            ...</span>
+<span class="sd">            ...     def forward(self, input):</span>
+<span class="sd">            ...         return (self.pose @ input).Log().tensor()</span>
+<span class="sd">            ...</span>
+<span class="sd">            &gt;&gt;&gt; device = torch.device(&quot;cuda&quot; if torch.cuda.is_available() else &quot;cpu&quot;)</span>
+<span class="sd">            &gt;&gt;&gt; input = pp.randn_SE3(2, 2).to(device)</span>
+<span class="sd">            &gt;&gt;&gt; invnet = PoseInv(2, 2).to(device)</span>
+<span class="sd">            &gt;&gt;&gt; strategy = pp.optim.strategy.Constant(damping=1e-4)</span>
+<span class="sd">            &gt;&gt;&gt; optimizer = pp.optim.LM(invnet, strategy=strategy)</span>
+<span class="sd">            &gt;&gt;&gt; scheduler = pp.optim.scheduler.StopOnPlateau(optimizer, steps=10, \</span>
+<span class="sd">            &gt;&gt;&gt;                     patience=3, decreasing=1e-3, verbose=True)</span>
+<span class="sd">            ...</span>
+<span class="sd">            &gt;&gt;&gt; while scheduler.continual():</span>
+<span class="sd">            ...     loss = optimizer.step(input)</span>
+<span class="sd">            ...     scheduler.step(loss)</span>
+<span class="sd">            StopOnPlateau on step 0 Loss 9.337769e+01 --&gt; Loss 3.502787e-05 (reduction/loss: 1.0000e+00).</span>
+<span class="sd">            StopOnPlateau on step 1 Loss 3.502787e-05 --&gt; Loss 4.527339e-13 (reduction/loss: 1.0000e+00).</span>
+<span class="sd">            StopOnPlateau on step 2 Loss 4.527339e-13 --&gt; Loss 7.112640e-14 (reduction/loss: 8.4290e-01).</span>
+<span class="sd">            StopOnPlateau on step 3 Loss 7.112640e-14 --&gt; Loss 3.693307e-14 (reduction/loss: 4.8074e-01).</span>
+<span class="sd">            StopOnPlateau: Maximum patience steps reached, Quiting..</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">assert</span> <span class="bp">self</span><span class="o">.</span><span class="n">optimizer</span><span class="o">.</span><span class="n">loss</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">,</span> \
+            <span class="s1">&#39;scheduler.step() should be called after optimizer.step()&#39;</span>
+
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">verbose</span><span class="p">:</span>
+            <span class="nb">print</span><span class="p">(</span><span class="s1">&#39;StopOnPlateau on step </span><span class="si">{}</span><span class="s1"> Loss </span><span class="si">{:.6e}</span><span class="s1"> --&gt; Loss </span><span class="si">{:.6e}</span><span class="s1"> &#39;</span>\
+                  <span class="s1">&#39;(reduction/loss: </span><span class="si">{:.4e}</span><span class="s1">).&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">steps</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">optimizer</span><span class="o">.</span><span class="n">last</span><span class="p">,</span>
+                  <span class="bp">self</span><span class="o">.</span><span class="n">optimizer</span><span class="o">.</span><span class="n">loss</span><span class="p">,</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">optimizer</span><span class="o">.</span><span class="n">last</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">optimizer</span><span class="o">.</span><span class="n">loss</span><span class="p">)</span> <span class="o">/</span>\
+                  <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">optimizer</span><span class="o">.</span><span class="n">last</span> <span class="o">+</span> <span class="mf">1e-31</span><span class="p">)))</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">steps</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">steps</span> <span class="o">+</span> <span class="mi">1</span>
+
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">steps</span> <span class="o">&gt;=</span> <span class="bp">self</span><span class="o">.</span><span class="n">max_steps</span><span class="p">:</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">_continual</span> <span class="o">=</span> <span class="kc">False</span>
+            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">verbose</span><span class="p">:</span>
+                <span class="nb">print</span><span class="p">(</span><span class="s2">&quot;StopOnPlateau: Maximum steps reached, Quiting..&quot;</span><span class="p">)</span>
+
+        <span class="k">if</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">optimizer</span><span class="o">.</span><span class="n">last</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">optimizer</span><span class="o">.</span><span class="n">loss</span><span class="p">)</span> <span class="o">&lt;</span> <span class="bp">self</span><span class="o">.</span><span class="n">decreasing</span><span class="p">:</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">patience_count</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">patience_count</span> <span class="o">+</span> <span class="mi">1</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">patience_count</span> <span class="o">=</span> <span class="mi">0</span>
+
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">patience_count</span> <span class="o">&gt;=</span> <span class="bp">self</span><span class="o">.</span><span class="n">patience</span><span class="p">:</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">_continual</span> <span class="o">=</span> <span class="kc">False</span>
+            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">verbose</span><span class="p">:</span>
+                <span class="nb">print</span><span class="p">(</span><span class="s2">&quot;StopOnPlateau: Maximum patience steps reached, Quiting..&quot;</span><span class="p">)</span>
+
+        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">optimizer</span><span class="p">,</span> <span class="s1">&#39;reject_count&#39;</span><span class="p">):</span>
+            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">optimizer</span><span class="o">.</span><span class="n">reject_count</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
+                <span class="bp">self</span><span class="o">.</span><span class="n">_continual</span> <span class="o">=</span> <span class="kc">False</span>
+                <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">verbose</span><span class="p">:</span>
+                    <span class="nb">print</span><span class="p">(</span><span class="s2">&quot;StopOnPlateau: Maximum rejected steps reached, Quiting..&quot;</span><span class="p">)</span></div>
+
+<div class="viewcode-block" id="StopOnPlateau.optimize"><a class="viewcode-back" href="../../../../generated/pypose.optim.scheduler.StopOnPlateau/#pypose.optim.scheduler.StopOnPlateau.optimize">[docs]</a>    <span class="nd">@torch</span><span class="o">.</span><span class="n">no_grad</span><span class="p">()</span>
+    <span class="k">def</span> <span class="nf">optimize</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">input</span><span class="p">,</span> <span class="n">target</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Perform full optimization steps.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            input (Tensor/LieTensor or tuple of Tensors/LieTensors): the input to the model.</span>
+<span class="sd">            target (Tensor/LieTensor): the model target to optimize.</span>
+<span class="sd">                If not given, the squared model output is minimized. Defaults: ``None``.</span>
+<span class="sd">            weight (Tensor, optional): a square positive definite matrix defining the weight of</span>
+<span class="sd">                model residual. Default: ``None``.</span>
+
+<span class="sd">        The above arguments are sent to optimizers. More details go to</span>
+<span class="sd">        :obj:`pypose.optim.LevenbergMarquardt` or :obj:`pypose.optim.GaussNewton`.</span>
+
+<span class="sd">        Example:</span>
+<span class="sd">            &gt;&gt;&gt; class PoseInv(nn.Module):</span>
+<span class="sd">            ...</span>
+<span class="sd">            ...     def __init__(self, *dim):</span>
+<span class="sd">            ...         super().__init__()</span>
+<span class="sd">            ...         self.pose = pp.Parameter(pp.randn_SE3(*dim))</span>
+<span class="sd">            ...</span>
+<span class="sd">            ...     def forward(self, input):</span>
+<span class="sd">            ...         return (self.pose @ input).Log().tensor()</span>
+<span class="sd">            ...</span>
+<span class="sd">            &gt;&gt;&gt; device = torch.device(&quot;cuda&quot; if torch.cuda.is_available() else &quot;cpu&quot;)</span>
+<span class="sd">            &gt;&gt;&gt; input = pp.randn_SE3(2, 2).to(device)</span>
+<span class="sd">            &gt;&gt;&gt; invnet = PoseInv(2, 2).to(device)</span>
+<span class="sd">            &gt;&gt;&gt; strategy = pp.optim.strategy.Constant(damping=1e-4)</span>
+<span class="sd">            &gt;&gt;&gt; optimizer = pp.optim.LM(invnet, strategy=strategy)</span>
+<span class="sd">            &gt;&gt;&gt; scheduler = pp.optim.scheduler.StopOnPlateau(optimizer, steps=10, \</span>
+<span class="sd">            &gt;&gt;&gt;                     patience=3, decreasing=1e-3, verbose=True)</span>
+<span class="sd">            ...</span>
+<span class="sd">            &gt;&gt;&gt; scheduler.optimize(input=input)</span>
+<span class="sd">            StopOnPlateau on step 0 Loss 5.199298e+01 --&gt; Loss 8.425808e-06 (reduction/loss: 1.0000e+00).</span>
+<span class="sd">            StopOnPlateau on step 1 Loss 8.425808e-06 --&gt; Loss 3.456247e-13 (reduction/loss: 1.0000e+00).</span>
+<span class="sd">            StopOnPlateau on step 2 Loss 3.456247e-13 --&gt; Loss 1.525355e-13 (reduction/loss: 5.5867e-01).</span>
+<span class="sd">            StopOnPlateau on step 3 Loss 1.525355e-13 --&gt; Loss 6.769275e-14 (reduction/loss: 5.5622e-01).</span>
+<span class="sd">            StopOnPlateau: Maximum patience steps reached, Quiting..</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">while</span> <span class="bp">self</span><span class="o">.</span><span class="n">continual</span><span class="p">():</span>
+            <span class="n">loss</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">optimizer</span><span class="o">.</span><span class="n">step</span><span class="p">(</span><span class="nb">input</span><span class="p">,</span> <span class="n">target</span><span class="p">,</span> <span class="n">weight</span><span class="p">)</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">step</span><span class="p">(</span><span class="n">loss</span><span class="p">)</span></div></div>
+</pre></div>
+
+             </article>
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+              
+  <h1>Source code for pypose.optim.solver</h1><div class="highlight"><pre>
+<span></span><span class="kn">import</span> <span class="nn">torch</span><span class="o">,</span> <span class="nn">warnings</span>
+<span class="kn">from</span> <span class="nn">typing</span> <span class="kn">import</span> <span class="n">Optional</span>
+<span class="kn">from</span> <span class="nn">torch</span> <span class="kn">import</span> <span class="n">Tensor</span><span class="p">,</span> <span class="n">nn</span>
+<span class="kn">from</span> <span class="nn">functools</span> <span class="kn">import</span> <span class="n">partial</span>
+<span class="kn">from</span> <span class="nn">..function.linalg</span> <span class="kn">import</span> <span class="n">bmv</span>
+<span class="kn">from</span> <span class="nn">torch.linalg</span> <span class="kn">import</span> <span class="n">pinv</span><span class="p">,</span> <span class="n">lstsq</span><span class="p">,</span> <span class="n">cholesky_ex</span><span class="p">,</span> <span class="n">vecdot</span>
+
+
+<div class="viewcode-block" id="PINV"><a class="viewcode-back" href="../../../../generated/pypose.optim.solver.PINV/#pypose.optim.solver.PINV">[docs]</a><span class="k">class</span> <span class="nc">PINV</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;The batched linear solver with pseudo inversion.</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \mathbf{A}_i \bm{x}_i = \mathbf{b}_i,</span>
+
+<span class="sd">    where :math:`\mathbf{A}_i \in \mathbb{C}^{M \times N}` and :math:`\bm{b}_i \in</span>
+<span class="sd">    \mathbb{C}^{M \times 1}` are the :math:`i`-th item of batched linear equations.</span>
+
+<span class="sd">    The solution is given by</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \bm{x}_i = \mathrm{pinv}(\mathbf{A}_i) \mathbf{b}_i,</span>
+
+<span class="sd">    where :math:`\mathrm{pinv}()` is the `pseudo inversion</span>
+<span class="sd">    &lt;https://en.wikipedia.org/wiki/Moore-Penrose_inverse&gt;`_ function.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        atol (float, Tensor, optional): the absolute tolerance value. When None it&#39;s considered to</span>
+<span class="sd">            be zero. Default: ``None``.</span>
+
+<span class="sd">        rtol (float, Tensor, optional): the relative tolerance value. Default: ``None``.</span>
+
+<span class="sd">        hermitian (bool, optional): indicates whether :math:`\mathbf{A}` is Hermitian if complex or</span>
+<span class="sd">            symmetric if real. Default: ``False``.</span>
+
+<span class="sd">    More details go to `torch.linalg.pinv</span>
+<span class="sd">    &lt;https://pytorch.org/docs/stable/generated/torch.linalg.pinv.html&gt;`_.</span>
+
+<span class="sd">    Warning:</span>
+<span class="sd">        It is always preferred to use :meth:`LSTSQ`, which is faster and more numerically stable.</span>
+
+<span class="sd">    Examples:</span>
+<span class="sd">        &gt;&gt;&gt; import pypose.optim.solver as ppos</span>
+<span class="sd">        &gt;&gt;&gt; A, b = torch.randn(2, 3, 3), torch.randn(2, 3, 1)</span>
+<span class="sd">        &gt;&gt;&gt; solver = ppos.PINV()</span>
+<span class="sd">        &gt;&gt;&gt; x = solver(A, b)</span>
+<span class="sd">        tensor([[[-0.2675],</span>
+<span class="sd">                 [-0.1897],</span>
+<span class="sd">                 [ 0.2708]],</span>
+<span class="sd">                [[-0.3472],</span>
+<span class="sd">                 [ 1.1191],</span>
+<span class="sd">                 [ 0.3301]]])</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">atol</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">rtol</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">hermitian</span><span class="o">=</span><span class="kc">False</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">atol</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">rtol</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">hermitian</span> <span class="o">=</span> <span class="n">atol</span><span class="p">,</span> <span class="n">rtol</span><span class="p">,</span> <span class="n">hermitian</span>
+
+<div class="viewcode-block" id="PINV.forward"><a class="viewcode-back" href="../../../../generated/pypose.optim.solver.PINV/#pypose.optim.solver.PINV.forward">[docs]</a>    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">A</span><span class="p">:</span> <span class="n">Tensor</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">Tensor</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="n">Tensor</span><span class="p">:</span>
+<span class="w">        </span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Args:</span>
+<span class="sd">            A (Tensor): the input batched tensor.</span>
+<span class="sd">            b (Tensor): the batched tensor on the right hand side.</span>
+
+<span class="sd">        Return:</span>
+<span class="sd">            Tensor: the solved batched tensor.</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">return</span> <span class="n">pinv</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">atol</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">atol</span><span class="p">,</span> <span class="n">rtol</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">rtol</span><span class="p">,</span> <span class="n">hermitian</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">hermitian</span><span class="p">)</span> <span class="o">@</span> <span class="n">b</span></div></div>
+
+
+
+<div class="viewcode-block" id="LSTSQ"><a class="viewcode-back" href="../../../../generated/pypose.optim.solver.LSTSQ/#pypose.optim.solver.LSTSQ">[docs]</a><span class="k">class</span> <span class="nc">LSTSQ</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;The batched linear solver with fast pseudo inversion.</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \mathbf{A}_i \bm{x}_i = \mathbf{b}_i,</span>
+
+<span class="sd">    where :math:`\mathbf{A}_i \in \mathbb{C}^{M \times N}` and :math:`\bm{b}_i \in</span>
+<span class="sd">    \mathbb{C}^{M \times 1}` are the :math:`i`-th item of batched linear equations.</span>
+
+<span class="sd">    The solution is given by</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \bm{x}_i = \mathrm{lstsq}(\mathbf{A}_i, \mathbf{b}_i),</span>
+
+<span class="sd">    where :math:`\mathrm{lstsq}()` computes a solution to the least squares problem</span>
+<span class="sd">    of a system of linear equations. More details go to `torch.linalg.lstsq</span>
+<span class="sd">    &lt;https://pytorch.org/docs/stable/generated/torch.linalg.lstsq.html&gt;`_.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        rcond (float, optional): Cut-off ratio for small singular values. For the purposes of</span>
+<span class="sd">            rank determination, singular values are treated as zero if they are smaller than</span>
+<span class="sd">            rcond times the largest singular value. It is used only when the fast model is</span>
+<span class="sd">            enabled. If ``None``, rcond is set to the machine precision of the dtype of</span>
+<span class="sd">            :math:`\mathbf{A}`. Default: ``None``.</span>
+<span class="sd">        driver (string, optional): chooses the LAPACK/MAGMA function that will be used. It is</span>
+<span class="sd">            used only when the fast model is enabled. For CPU users, the valid values are ``gels``,</span>
+<span class="sd">            ``gelsy``, ``gelsd``, ``gelss``. For CUDA users, the only valid driver is ``gels``,</span>
+<span class="sd">            which assumes that input matrices (:math:`\mathbf{A}`) are full-rank. If ``None``,</span>
+<span class="sd">            ``gelsy`` is used for CPU inputs and ``gels`` for CUDA inputs. Default: ``None``.</span>
+<span class="sd">            To choose the best driver on CPU consider:</span>
+
+<span class="sd">            - If input matrices (:math:`\mathbf{A}`) are well-conditioned (`condition number</span>
+<span class="sd">              &lt;https://en.wikipedia.org/wiki/Condition_number&gt;`_ is not too large), or you do not</span>
+<span class="sd">              mind some precision loss.</span>
+
+<span class="sd">                - For a general matrix: ``gelsy`` (QR with pivoting) (default)</span>
+
+<span class="sd">                - If A is full-rank: ``gels`` (QR)</span>
+
+<span class="sd">            - If input matrices (:math:`\mathbf{A}`) are not well-conditioned.</span>
+
+<span class="sd">                - ``gelsd`` (tridiagonal reduction and SVD)</span>
+
+<span class="sd">                - But if you run into memory issues: ``gelss`` (full SVD).</span>
+
+<span class="sd">            See full description of `drivers &lt;https://www.netlib.org/lapack/lug/node27.html&gt;`_.</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        This solver is faster and more numerically stable than :meth:`PINV`.</span>
+
+<span class="sd">        It is also preferred to use :meth:`Cholesky` if input matrices (:math:`\mathbf{A}`)</span>
+<span class="sd">        are guaranteed to complex Hermitian or real symmetric positive-definite.</span>
+
+<span class="sd">    Examples:</span>
+<span class="sd">        &gt;&gt;&gt; import pypose.optim.solver as ppos</span>
+<span class="sd">        &gt;&gt;&gt; A, b = torch.randn(2, 3, 3), torch.randn(2, 3, 1)</span>
+<span class="sd">        &gt;&gt;&gt; solver = ppos.LSTSQ(driver=&#39;gels&#39;)</span>
+<span class="sd">        &gt;&gt;&gt; x = solver(A, b)</span>
+<span class="sd">        tensor([[[ 0.9997],</span>
+<span class="sd">                 [-1.3288],</span>
+<span class="sd">                 [-1.6327]],</span>
+<span class="sd">                [[ 3.1639],</span>
+<span class="sd">                 [-0.5379],</span>
+<span class="sd">                 [-1.2872]]])</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">rcond</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">driver</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">rcond</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">driver</span> <span class="o">=</span> <span class="n">rcond</span><span class="p">,</span> <span class="n">driver</span>
+
+<div class="viewcode-block" id="LSTSQ.forward"><a class="viewcode-back" href="../../../../generated/pypose.optim.solver.LSTSQ/#pypose.optim.solver.LSTSQ.forward">[docs]</a>    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">A</span><span class="p">:</span> <span class="n">Tensor</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">Tensor</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="n">Tensor</span><span class="p">:</span>
+<span class="w">        </span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Args:</span>
+<span class="sd">            A (Tensor): the input batched tensor.</span>
+<span class="sd">            b (Tensor): the batched tensor on the right hand side.</span>
+
+<span class="sd">        Return:</span>
+<span class="sd">            Tensor: the solved batched tensor.</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">out</span> <span class="o">=</span> <span class="n">lstsq</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">rcond</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">rcond</span><span class="p">,</span> <span class="n">driver</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">driver</span><span class="p">)</span>
+        <span class="k">assert</span> <span class="ow">not</span> <span class="n">torch</span><span class="o">.</span><span class="n">any</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">isnan</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">out</span><span class="o">.</span><span class="n">solution</span><span class="p">)),</span> \
+            <span class="s1">&#39;Linear Solver Failed Using LSTSQ. Using PINV() instead&#39;</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">out</span><span class="o">.</span><span class="n">solution</span></div></div>
+
+
+<div class="viewcode-block" id="Cholesky"><a class="viewcode-back" href="../../../../generated/pypose.optim.solver.Cholesky/#pypose.optim.solver.Cholesky">[docs]</a><span class="k">class</span> <span class="nc">Cholesky</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;The batched linear solver with Cholesky decomposition.</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \mathbf{A}_i \bm{x}_i = \mathbf{b}_i,</span>
+
+<span class="sd">    where :math:`\mathbf{A}_i \in \mathbb{C}^{M \times N}` and :math:`\bm{b}_i \in</span>
+<span class="sd">    \mathbb{C}^{M \times 1}` are the :math:`i`-th item of batched linear equations.</span>
+<span class="sd">    Note that :math:`\mathbf{A}_i` has to be a complex Hermitian or a real symmetric</span>
+<span class="sd">    positive-definite matrix.</span>
+
+<span class="sd">    The solution is given by</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \begin{align*}</span>
+<span class="sd">            \bm{L}_i &amp;= \mathrm{cholesky}(\mathbf{A}_i), \\</span>
+<span class="sd">            \bm{x}_i &amp;= \mathrm{cholesky\_solve}(\mathbf{b}_i, \bm{L}_i), \\</span>
+<span class="sd">        \end{align*}</span>
+
+<span class="sd">    where :math:`\mathrm{cholesky}()` is the `Cholesky decomposition</span>
+<span class="sd">    &lt;https://en.wikipedia.org/wiki/Cholesky_decomposition&gt;`_ function.</span>
+
+<span class="sd">    More details go to</span>
+<span class="sd">    `torch.linalg.cholesky &lt;https://pytorch.org/docs/stable/generated/torch.linalg.cholesky.html&gt;`_</span>
+<span class="sd">    and</span>
+<span class="sd">    `torch.cholesky_solve &lt;https://pytorch.org/docs/stable/generated/torch.cholesky_solve.html&gt;`_.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        upper (bool, optional): whether use an upper triangular matrix in Cholesky decomposition.</span>
+<span class="sd">            Default: ``False``.</span>
+
+<span class="sd">    Examples:</span>
+<span class="sd">        &gt;&gt;&gt; import pypose.optim.solver as ppos</span>
+<span class="sd">        &gt;&gt;&gt; A = torch.tensor([[[1.00, 0.10, 0.00], [0.10, 1.00, 0.20], [0.00, 0.20, 1.00]],</span>
+<span class="sd">                              [[1.00, 0.20, 0.10], [0.20, 1.00, 0.20], [0.10, 0.20, 1.00]]])</span>
+<span class="sd">        &gt;&gt;&gt; b = torch.tensor([[[1.], [2.], [3.]], [[1.], [2.], [3.]]])</span>
+<span class="sd">        &gt;&gt;&gt; solver = ppos.Cholesky()</span>
+<span class="sd">        &gt;&gt;&gt; x = solver(A, b)</span>
+<span class="sd">        tensor([[[0.8632],</span>
+<span class="sd">                 [1.3684],</span>
+<span class="sd">                 [2.7263]],</span>
+<span class="sd">                [[0.4575],</span>
+<span class="sd">                 [1.3725],</span>
+<span class="sd">                 [2.6797]]])</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">upper</span><span class="o">=</span><span class="kc">False</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">upper</span> <span class="o">=</span> <span class="n">upper</span>
+
+<div class="viewcode-block" id="Cholesky.forward"><a class="viewcode-back" href="../../../../generated/pypose.optim.solver.Cholesky/#pypose.optim.solver.Cholesky.forward">[docs]</a>    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">A</span><span class="p">:</span> <span class="n">Tensor</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">Tensor</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="n">Tensor</span><span class="p">:</span>
+<span class="w">        </span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Args:</span>
+<span class="sd">            A (Tensor): the input batched tensor.</span>
+<span class="sd">            b (Tensor): the batched tensor on the right hand side.</span>
+
+<span class="sd">        Return:</span>
+<span class="sd">            Tensor: the solved batched tensor.</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="n">L</span><span class="p">,</span> <span class="n">info</span> <span class="o">=</span> <span class="n">cholesky_ex</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">upper</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">upper</span><span class="p">)</span>
+        <span class="k">assert</span> <span class="ow">not</span> <span class="n">torch</span><span class="o">.</span><span class="n">any</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">isnan</span><span class="p">(</span><span class="n">L</span><span class="p">)),</span> \
+            <span class="s1">&#39;Cholesky decomposition failed. Check your matrix (may not be positive-definite)&#39;</span>
+        <span class="k">return</span> <span class="n">b</span><span class="o">.</span><span class="n">cholesky_solve</span><span class="p">(</span><span class="n">L</span><span class="p">,</span> <span class="n">upper</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">upper</span><span class="p">)</span></div></div>
+
+
+<div class="viewcode-block" id="CG"><a class="viewcode-back" href="../../../../generated/pypose.optim.solver.CG/#pypose.optim.solver.CG">[docs]</a><span class="k">class</span> <span class="nc">CG</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;The batched linear solver with conjugate gradient method.</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">        \mathbf{A}_i \bm{x}_i = \mathbf{b}_i,</span>
+
+<span class="sd">    where :math:`\mathbf{A}_i \in \mathbb{C}^{M \times N}` and :math:`\bm{b}_i \in</span>
+<span class="sd">    \mathbb{C}^{M \times 1}` are the :math:`i`-th item of batched linear equations.</span>
+
+<span class="sd">    This function is a 1:1 replica of `scipy.sparse.linalg.cg &lt;https://docs.scipy.org/doc</span>
+<span class="sd">    /scipy/reference/generated/scipy.sparse.linalg.cg.html&gt;`_.</span>
+<span class="sd">    The solution is consistent with the scipy version up to numerical precision.</span>
+<span class="sd">    Variable names are kept the same as the scipy version for easy reference.</span>
+<span class="sd">    We recommend using only non-batched or batch size 1 input for this solver, as</span>
+<span class="sd">    the batched version was not appeared in the original scipy version. When handling</span>
+<span class="sd">    sparse matrices, the batched computation may introduce additional overhead.</span>
+
+<span class="sd">    Examples:</span>
+<span class="sd">        &gt;&gt;&gt; # dense example</span>
+<span class="sd">        &gt;&gt;&gt; import pypose.optim.solver as ppos</span>
+<span class="sd">        &gt;&gt;&gt; A = torch.tensor([[0.1802967, 0.3151198, 0.4548111, 0.3860016, 0.2870615],</span>
+<span class="sd">                              [0.3151198, 1.4575327, 1.5533425, 1.0540756, 1.0795838],</span>
+<span class="sd">                              [0.4548111, 1.5533425, 2.3674474, 1.1222278, 1.2365348],</span>
+<span class="sd">                              [0.3860016, 1.0540756, 1.1222278, 1.3748058, 1.2223261],</span>
+<span class="sd">                              [0.2870615, 1.0795838, 1.2365348, 1.2223261, 1.2577004]])</span>
+<span class="sd">        &gt;&gt;&gt; b = torch.tensor([[ 2.64306851],</span>
+<span class="sd">                              [-0.03593633],</span>
+<span class="sd">                              [ 0.73612658],</span>
+<span class="sd">                              [ 0.51501254],</span>
+<span class="sd">                              [-0.26689271]])</span>
+<span class="sd">        &gt;&gt;&gt; solver = ppos.CG()</span>
+<span class="sd">        &gt;&gt;&gt; x = solver(A, b)</span>
+<span class="sd">        tensor([[246.4098],</span>
+<span class="sd">                [ 22.6997],</span>
+<span class="sd">                [-56.9239],</span>
+<span class="sd">                [-161.7914],</span>
+<span class="sd">                [137.2683]])</span>
+
+<span class="sd">        &gt;&gt;&gt; # sparse csr example</span>
+<span class="sd">        &gt;&gt;&gt; import pypose.optim.solver as ppos</span>
+<span class="sd">        &gt;&gt;&gt; crow_indices = torch.tensor([0, 2, 4])</span>
+<span class="sd">        &gt;&gt;&gt; col_indices = torch.tensor([0, 1, 0, 1])</span>
+<span class="sd">        &gt;&gt;&gt; values = torch.tensor([1, 2, 3, 4], dtype=torch.float)</span>
+<span class="sd">        &gt;&gt;&gt; A = torch.sparse_csr_tensor(crow_indices, col_indices, values)</span>
+<span class="sd">        &gt;&gt;&gt; A.to_dense()  # visualize</span>
+<span class="sd">        tensor([[1., 2.],</span>
+<span class="sd">                [3., 4.]])</span>
+<span class="sd">        &gt;&gt;&gt; b = torch.tensor([[1.], [2.]])</span>
+<span class="sd">        &gt;&gt;&gt; solver = ppos.CG()</span>
+<span class="sd">        &gt;&gt;&gt; x = solver(A, b)</span>
+<span class="sd">        tensor([-4.4052e-05,  5.0003e-01])</span>
+
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">maxiter</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">tol</span><span class="o">=</span><span class="mf">1e-5</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">maxiter</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">tol</span> <span class="o">=</span> <span class="n">maxiter</span><span class="p">,</span> <span class="n">tol</span>
+
+<div class="viewcode-block" id="CG.forward"><a class="viewcode-back" href="../../../../generated/pypose.optim.solver.CG/#pypose.optim.solver.CG.forward">[docs]</a>    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">A</span><span class="p">:</span> <span class="n">Tensor</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">Tensor</span><span class="p">,</span> <span class="n">x</span><span class="p">:</span> <span class="n">Optional</span><span class="p">[</span><span class="n">Tensor</span><span class="p">]</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span>
+                <span class="n">M</span><span class="p">:</span> <span class="n">Optional</span><span class="p">[</span><span class="n">Tensor</span><span class="p">]</span><span class="o">=</span><span class="kc">None</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="n">Tensor</span><span class="p">:</span>
+<span class="w">        </span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Args:</span>
+<span class="sd">            A (Tensor): the input tensor. It is assumed to be a symmetric</span>
+<span class="sd">                positive-definite matrix. Layout is allowed to be COO, CSR, BSR, or dense.</span>
+<span class="sd">            b (Tensor): the tensor on the right hand side. Layout could be sparse or dense</span>
+<span class="sd">                but is only allowed to be a type that is compatible with the layout of A.</span>
+<span class="sd">                In other words, `A @ b` operation must be supported by the layout of A.</span>
+<span class="sd">            x (Tensor, optional): the initial guess for the solution. Default: ``None``.</span>
+<span class="sd">            M (Tensor, optional): the preconditioner for A. Layout is allowed to be COO,</span>
+<span class="sd">                CSR, BSR, or dense. Default: ``None``.</span>
+
+<span class="sd">        Return:</span>
+<span class="sd">            Tensor: the solved tensor. Layout is the same as the layout of b.</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">if</span> <span class="n">A</span><span class="o">.</span><span class="n">ndim</span> <span class="o">==</span> <span class="n">b</span><span class="o">.</span><span class="n">ndim</span><span class="p">:</span>
+            <span class="n">b</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="k">assert</span> <span class="n">A</span><span class="o">.</span><span class="n">ndim</span> <span class="o">==</span> <span class="n">b</span><span class="o">.</span><span class="n">ndim</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> \
+                <span class="s1">&#39;The number of dimensions of A and b must differ by 1&#39;</span>
+        <span class="k">if</span> <span class="n">x</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
+            <span class="n">x</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">zeros_like</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
+        <span class="n">bnrm2</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">dim</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span>
+        <span class="k">if</span> <span class="p">(</span><span class="n">bnrm2</span> <span class="o">==</span> <span class="mi">0</span><span class="p">)</span><span class="o">.</span><span class="n">all</span><span class="p">():</span>
+            <span class="k">return</span> <span class="n">b</span>
+        <span class="n">atol</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">tol</span> <span class="o">*</span> <span class="n">bnrm2</span>
+        <span class="n">n</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">maxiter</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
+            <span class="n">maxiter</span> <span class="o">=</span> <span class="n">n</span> <span class="o">*</span> <span class="mi">10</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">maxiter</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">maxiter</span>
+        <span class="n">r</span> <span class="o">=</span> <span class="n">b</span> <span class="o">-</span> <span class="n">bmv</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="k">if</span> <span class="n">x</span><span class="o">.</span><span class="n">any</span><span class="p">()</span> <span class="k">else</span> <span class="n">b</span><span class="o">.</span><span class="n">clone</span><span class="p">()</span>
+        <span class="n">rho_prev</span><span class="p">,</span> <span class="n">p</span> <span class="o">=</span> <span class="kc">None</span><span class="p">,</span> <span class="kc">None</span>
+
+        <span class="k">for</span> <span class="n">iteration</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">maxiter</span><span class="p">):</span>
+            <span class="k">if</span> <span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">dim</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span> <span class="o">&lt;</span> <span class="n">atol</span><span class="p">)</span><span class="o">.</span><span class="n">all</span><span class="p">():</span>
+                <span class="k">return</span> <span class="n">x</span>
+
+            <span class="n">z</span> <span class="o">=</span> <span class="n">bmv</span><span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="n">r</span><span class="p">)</span> <span class="k">if</span> <span class="n">M</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">r</span>
+            <span class="n">rho_cur</span> <span class="o">=</span> <span class="n">vecdot</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
+            <span class="k">if</span> <span class="n">iteration</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
+                <span class="n">beta</span> <span class="o">=</span> <span class="n">rho_cur</span> <span class="o">/</span> <span class="n">rho_prev</span>
+                <span class="n">p</span> <span class="o">=</span> <span class="n">p</span> <span class="o">*</span> <span class="n">beta</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="n">z</span>
+            <span class="k">else</span><span class="p">:</span>  <span class="c1"># First spin</span>
+                <span class="n">p</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">empty_like</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
+                <span class="n">p</span><span class="p">[:]</span> <span class="o">=</span> <span class="n">z</span><span class="p">[:]</span>
+
+            <span class="n">q</span> <span class="o">=</span> <span class="n">bmv</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
+            <span class="n">alpha</span> <span class="o">=</span> <span class="n">rho_cur</span> <span class="o">/</span> <span class="n">vecdot</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">)</span>
+            <span class="n">x</span> <span class="o">+=</span> <span class="n">alpha</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">p</span>
+            <span class="n">r</span> <span class="o">-=</span> <span class="n">alpha</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">q</span>
+            <span class="n">rho_prev</span> <span class="o">=</span> <span class="n">rho_cur</span>
+
+        <span class="k">return</span> <span class="n">x</span></div></div>
+</pre></div>
+
+             </article>
+             
+            </div>
+            <footer>
+  
+
+  
+
+    <hr>
+
+  
+
+  <div role="contentinfo">
+    <p>
+        &copy; Copyright 2022, PyPose Contributors.
+
+    </p>
+  </div>
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+  <h1>Source code for pypose.optim.strategy</h1><div class="highlight"><pre>
+<span></span><span class="kn">import</span> <span class="nn">torch</span>
+<span class="kn">from</span> <span class="nn">torch</span> <span class="kn">import</span> <span class="n">nn</span><span class="p">,</span> <span class="n">finfo</span>
+
+
+<div class="viewcode-block" id="Constant"><a class="viewcode-back" href="../../../../generated/pypose.optim.strategy.Constant/#pypose.optim.strategy.Constant">[docs]</a><span class="k">class</span> <span class="nc">Constant</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Constant damping strategy used in the Levenberg-Marquardt (LM) algorithm.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        damping (float, optional): damping factor of LM optimizer. Default: 1e-6.</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        &gt;&gt;&gt; class PoseInv(nn.Module):</span>
+<span class="sd">        ...     def __init__(self, *dim):</span>
+<span class="sd">        ...         super().__init__()</span>
+<span class="sd">        ...         self.pose = pp.Parameter(pp.randn_SE3(*dim))</span>
+<span class="sd">        ... </span>
+<span class="sd">        ...     def forward(self, inputs):</span>
+<span class="sd">        ...         return (self.pose @ inputs).Log().tensor()</span>
+<span class="sd">        ... </span>
+<span class="sd">        ... device = torch.device(&quot;cuda&quot; if torch.cuda.is_available() else &quot;cpu&quot;)</span>
+<span class="sd">        ... inputs = pp.randn_SE3(2, 2).to(device)</span>
+<span class="sd">        ... invnet = PoseInv(2, 2).to(device)</span>
+<span class="sd">        ... strategy = pp.optim.strategy.Constant(damping=1e-6)</span>
+<span class="sd">        ... optimizer = pp.optim.LM(invnet, strategy=strategy)</span>
+<span class="sd">        ... </span>
+<span class="sd">        ... for idx in range(10):</span>
+<span class="sd">        ...     loss = optimizer.step(inputs)</span>
+<span class="sd">        ...     print(&#39;Pose loss %.7f @ %dit&#39;%(loss, idx))</span>
+<span class="sd">        ...     if loss &lt; 1e-5:</span>
+<span class="sd">        ...         print(&#39;Early Stoping!&#39;)</span>
+<span class="sd">        ...         print(&#39;Optimization Early Done with loss:&#39;, loss.item())</span>
+<span class="sd">        ...         break</span>
+<span class="sd">        Pose loss 0.0000000 @ 0it</span>
+<span class="sd">        Early Stoping!</span>
+<span class="sd">        Optimization Early Done with loss: 9.236661990819073e-10</span>
+<span class="sd">    </span>
+<span class="sd">    Note:</span>
+<span class="sd">        More details about optimization go to :meth:`pypose.optim.LevenbergMarquardt`.</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">damping</span><span class="o">=</span><span class="mf">1e-6</span><span class="p">):</span>
+        <span class="k">assert</span> <span class="n">damping</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">,</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">&quot;damping has to be positive: </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">damping</span><span class="p">))</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">defaults</span> <span class="o">=</span> <span class="p">{</span><span class="s1">&#39;damping&#39;</span><span class="p">:</span> <span class="n">damping</span><span class="p">}</span>
+
+    <span class="k">def</span> <span class="nf">update</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">pg</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+        <span class="n">pg</span><span class="p">[</span><span class="s1">&#39;damping&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">pg</span><span class="p">[</span><span class="s1">&#39;damping&#39;</span><span class="p">]</span></div>
+
+
+<div class="viewcode-block" id="Adaptive"><a class="viewcode-back" href="../../../../generated/pypose.optim.strategy.Adaptive/#pypose.optim.strategy.Adaptive">[docs]</a><span class="k">class</span> <span class="nc">Adaptive</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Adaptive damping strategy used in the Levenberg-Marquardt (LM) algorithm.</span>
+
+<span class="sd">    It scales down the damping factor if the optimization step is very successful, unchanges the</span>
+<span class="sd">    factor if the step is successful, and scales the factor up if the step is unsuccessful.</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">       \begin{aligned}</span>
+<span class="sd">            &amp;\rule{113mm}{0.4pt}                                                                 \\</span>
+<span class="sd">            &amp;\textbf{input}: \lambda ~\text{(damping)}, \bm{f}(\bm{\theta})~(\text{model}),</span>
+<span class="sd">                             \theta_h~(\text{high}), \theta_l~(\text{low}), \delta_u~(\text{up}),</span>
+<span class="sd">                             \delta_d~(\text{down}),                                             \\</span>
+<span class="sd">            &amp;\hspace{13mm}   \epsilon_{s}~(\text{min}), \epsilon_{l}~(\text{max})                \\</span>
+<span class="sd">            &amp;\rule{113mm}{0.4pt}                                                                 \\</span>
+<span class="sd">            &amp;\rho = \frac{ \|\bm{f}(\bm{\theta})\|^2 - \|\bm{f}(\bm{\theta} + \delta)\|^2}</span>
+<span class="sd">                    {\|\bm{f}(\bm{\theta})\|^2 - \|\bm{f}(\bm{\theta}) + \mathbf{J}\delta\|^2}</span>
+<span class="sd">                    ~\text{(step quality)}                                                       \\</span>
+<span class="sd">            &amp;\textbf{if} ~~ \rho &gt; \theta_h ~ \text{(``very successful&#39;&#39; step)}                  \\</span>
+<span class="sd">            &amp;\hspace{5mm} \lambda \leftarrow \delta_d \cdot \lambda                              \\</span>
+<span class="sd">            &amp;\textbf{elif} ~~ \rho &gt; \theta_l ~ \text{(``successful&#39;&#39; step)}                     \\</span>
+<span class="sd">            &amp;\hspace{5mm} \lambda \leftarrow \lambda                                             \\</span>
+<span class="sd">            &amp;\textbf{else} ~ \text{(``unsuccessful&#39;&#39; step)}                                      \\</span>
+<span class="sd">            &amp;\hspace{5mm} \lambda \leftarrow \delta_u  \cdot \lambda                             \\</span>
+<span class="sd">            &amp;\lambda \leftarrow \mathrm{min}(\mathrm{max}(\lambda, \epsilon_{s}), \epsilon_{l})  \\</span>
+<span class="sd">            &amp;\rule{113mm}{0.4pt}                                                          \\[-1.ex]</span>
+<span class="sd">            &amp;\textbf{return} \:  \lambda                                                  \\[-1.ex]</span>
+<span class="sd">            &amp;\rule{113mm}{0.4pt}                                                          \\[-1.ex]</span>
+<span class="sd">       \end{aligned}</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        damping (float, optional): initial damping factor of LM optimizer. Default: 1e-6.</span>
+<span class="sd">        high (float, optional): high threshold for scaling down the damping factor.</span>
+<span class="sd">            Default: 0.5.</span>
+<span class="sd">        low (float, optional): low threshold for scaling up the damping factor.</span>
+<span class="sd">            Default: 1e-3.</span>
+<span class="sd">        down (float, optional): the down scaling factor in the range of :math:`(0,1)`.</span>
+<span class="sd">            Default: 0.5.</span>
+<span class="sd">        up (float, optional): the up scaling factor in the range of :math:`(1,\infty)`.</span>
+<span class="sd">            Default: 2.0.</span>
+<span class="sd">        min (float, optional): the lower-bound of damping factor. Default: 1e-6.</span>
+<span class="sd">        max (float, optional): the upper-bound of damping factor. Default: 1e16.</span>
+
+<span class="sd">    More details about the optimization process go to :meth:`pypose.optim.LevenbergMarquardt`.</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        For efficiency, we calculate the denominator of the step quality :math:`\rho` as</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \begin{aligned}</span>
+<span class="sd">            &amp; \|\bm{f}(\bm{\theta})\|^2 - \|\bm{f}(\bm{\theta}) + \mathbf{J}\delta\|^2 \\</span>
+<span class="sd">            &amp; = \bm{f}^T\bm{f} - \left( \bm{f}^T\bm{f} + 2 \bm{f}^T \delta +</span>
+<span class="sd">                \delta^T \mathbf{J}^T\mathbf{J}\delta \right) \\</span>
+<span class="sd">            &amp; = -(\mathbf{J} \delta)^T (2\bm{f} + \mathbf{J}\delta)</span>
+<span class="sd">            \end{aligned}</span>
+
+<span class="sd">        where :math:`\mathbf{J}` is the Jacobian of the model :math:`\bm{f}` at evaluation</span>
+<span class="sd">        point :math:`\bm{\theta}`.</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        &gt;&gt;&gt; class PoseInv(nn.Module):</span>
+<span class="sd">        ...     def __init__(self, *dim):</span>
+<span class="sd">        ...         super().__init__()</span>
+<span class="sd">        ...         self.pose = pp.Parameter(pp.randn_SE3(*dim))</span>
+<span class="sd">        ... </span>
+<span class="sd">        ...     def forward(self, inputs):</span>
+<span class="sd">        ...         return (self.pose @ inputs).Log().tensor()</span>
+<span class="sd">        ... </span>
+<span class="sd">        ... device = torch.device(&quot;cuda&quot; if torch.cuda.is_available() else &quot;cpu&quot;)</span>
+<span class="sd">        ... inputs = pp.randn_SE3(2, 2).to(device)</span>
+<span class="sd">        ... invnet = PoseInv(2, 2).to(device)</span>
+<span class="sd">        ... strategy = pp.optim.strategy.Adaptive(damping=1e-6)</span>
+<span class="sd">        ... optimizer = pp.optim.LM(invnet, strategy=strategy)</span>
+<span class="sd">        ... </span>
+<span class="sd">        ... for idx in range(10):</span>
+<span class="sd">        ...     loss = optimizer.step(inputs)</span>
+<span class="sd">        ...     print(&#39;Pose loss %.7f @ %dit&#39;%(loss, idx))</span>
+<span class="sd">        ...     if loss &lt; 1e-5:</span>
+<span class="sd">        ...         print(&#39;Early Stoping!&#39;)</span>
+<span class="sd">        ...         print(&#39;Optimization Early Done with loss:&#39;, loss.item())</span>
+<span class="sd">        ...         break</span>
+<span class="sd">        Pose loss 0.0000000 @ 0it</span>
+<span class="sd">        Early Stoping!</span>
+<span class="sd">        Optimization Early Done with loss: 9.236661990819073e-10</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">damping</span><span class="o">=</span><span class="mf">1e-6</span><span class="p">,</span> <span class="n">high</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">low</span><span class="o">=</span><span class="mf">1e-3</span><span class="p">,</span> <span class="n">up</span><span class="o">=</span><span class="mf">2.</span><span class="p">,</span> <span class="n">down</span><span class="o">=</span><span class="mf">.5</span><span class="p">,</span> <span class="nb">min</span><span class="o">=</span><span class="mf">1e-6</span><span class="p">,</span> <span class="nb">max</span><span class="o">=</span><span class="mf">1e16</span><span class="p">):</span>
+        <span class="k">assert</span> <span class="n">damping</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">,</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">&quot;damping has to be positive: </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">damping</span><span class="p">))</span>
+        <span class="k">assert</span> <span class="n">high</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">,</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">&quot;high has to be positive: </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">high</span><span class="p">))</span>
+        <span class="k">assert</span> <span class="n">low</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">,</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">&quot;low for decrease has to be positive: </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">low</span><span class="p">))</span>
+        <span class="k">assert</span> <span class="mi">0</span> <span class="o">&lt;</span> <span class="n">down</span> <span class="o">&lt;</span> <span class="mi">1</span><span class="p">,</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">&quot;down factor has to be smaller than 1: </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">down</span><span class="p">))</span>
+        <span class="k">assert</span> <span class="mi">1</span> <span class="o">&lt;</span> <span class="n">up</span><span class="p">,</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">&quot;up factor has to be larger than 1: </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">up</span><span class="p">))</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">defaults</span> <span class="o">=</span> <span class="p">{</span><span class="s1">&#39;damping&#39;</span><span class="p">:</span> <span class="n">damping</span><span class="p">,</span> <span class="s1">&#39;high&#39;</span><span class="p">:</span> <span class="n">high</span><span class="p">,</span> <span class="s1">&#39;low&#39;</span><span class="p">:</span> <span class="n">low</span><span class="p">,</span> <span class="s1">&#39;up&#39;</span><span class="p">:</span> <span class="n">up</span><span class="p">,</span> <span class="s1">&#39;down&#39;</span><span class="p">:</span> <span class="n">down</span><span class="p">}</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">min</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">max</span> <span class="o">=</span> <span class="nb">min</span><span class="p">,</span> <span class="nb">max</span>
+
+    <span class="k">def</span> <span class="nf">update</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">pg</span><span class="p">,</span> <span class="n">last</span><span class="p">,</span> <span class="n">loss</span><span class="p">,</span> <span class="n">J</span><span class="p">,</span> <span class="n">D</span><span class="p">,</span> <span class="n">R</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+        <span class="n">quality</span> <span class="o">=</span> <span class="p">(</span><span class="n">last</span> <span class="o">-</span> <span class="n">loss</span><span class="p">)</span> <span class="o">/</span> <span class="o">-</span><span class="p">((</span><span class="n">J</span> <span class="o">@</span> <span class="n">D</span><span class="p">)</span><span class="o">.</span><span class="n">mT</span> <span class="o">@</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">R</span> <span class="o">+</span> <span class="n">J</span> <span class="o">@</span> <span class="n">D</span><span class="p">))</span><span class="o">.</span><span class="n">squeeze</span><span class="p">()</span>
+        <span class="k">if</span> <span class="n">quality</span> <span class="o">&gt;</span> <span class="n">pg</span><span class="p">[</span><span class="s1">&#39;high&#39;</span><span class="p">]:</span>
+            <span class="n">pg</span><span class="p">[</span><span class="s1">&#39;damping&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">pg</span><span class="p">[</span><span class="s1">&#39;damping&#39;</span><span class="p">]</span> <span class="o">*</span> <span class="n">pg</span><span class="p">[</span><span class="s1">&#39;down&#39;</span><span class="p">]</span>
+        <span class="k">elif</span> <span class="n">quality</span> <span class="o">&gt;</span> <span class="n">pg</span><span class="p">[</span><span class="s1">&#39;low&#39;</span><span class="p">]:</span>
+            <span class="n">pg</span><span class="p">[</span><span class="s1">&#39;damping&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">pg</span><span class="p">[</span><span class="s1">&#39;damping&#39;</span><span class="p">]</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">pg</span><span class="p">[</span><span class="s1">&#39;damping&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">pg</span><span class="p">[</span><span class="s1">&#39;damping&#39;</span><span class="p">]</span> <span class="o">*</span> <span class="n">pg</span><span class="p">[</span><span class="s1">&#39;up&#39;</span><span class="p">]</span>
+        <span class="n">pg</span><span class="p">[</span><span class="s1">&#39;damping&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">min</span><span class="p">,</span> <span class="nb">min</span><span class="p">(</span><span class="n">pg</span><span class="p">[</span><span class="s1">&#39;damping&#39;</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">max</span><span class="p">))</span></div>
+
+
+<div class="viewcode-block" id="TrustRegion"><a class="viewcode-back" href="../../../../generated/pypose.optim.strategy.TrustRegion/#pypose.optim.strategy.TrustRegion">[docs]</a><span class="k">class</span> <span class="nc">TrustRegion</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    The trust region (TR) algorithm used in the Levenberg-Marquardt (LM) algorithm.</span>
+
+<span class="sd">    .. math::</span>
+<span class="sd">       \begin{aligned}</span>
+<span class="sd">            &amp;\rule{113mm}{0.4pt}                                                                 \\</span>
+<span class="sd">            &amp;\textbf{input}: \Delta ~\text{(radius)}, \bm{f}(\bm{\theta})~(\text{model}),</span>
+<span class="sd">                             \theta_h~(\text{high}), \theta_l~(\text{low}), \delta_u~(\text{up}),</span>
+<span class="sd">                             \delta_d~(\text{down}),                                             \\</span>
+<span class="sd">            &amp;\hspace{13mm}   \sigma~(\text{factor}),  \epsilon_{s}~(\text{min}),</span>
+<span class="sd">                             \epsilon_{l}~(\text{max})                                           \\</span>
+<span class="sd">            &amp;\rule{113mm}{0.4pt}                                                                 \\</span>
+<span class="sd">            &amp; \rho = \frac{ \|\bm{f}(\bm{\theta})\|^2 - \|\bm{f}(\bm{\theta} + \delta)\|^2}</span>
+<span class="sd">                      {\|\bm{f}(\bm{\theta})\|^2 - \|\bm{f}(\bm{\theta}) + \mathbf{J}\delta\|^2}</span>
+<span class="sd">                    ~\text{(step quality)}                                                       \\</span>
+<span class="sd">            &amp;\textbf{if} ~~ \rho &gt; \theta_h ~ \text{(``very successful&#39;&#39; step)}                  \\</span>
+<span class="sd">            &amp;\hspace{5mm} \Delta \leftarrow \delta_u \cdot \Delta                                \\</span>
+<span class="sd">            &amp;\hspace{5mm} \delta_d \leftarrow \delta_d^{\text{init}}                             \\</span>
+<span class="sd">            &amp;\textbf{elif} ~~ \rho &gt; \theta_l ~ \text{(``successful&#39;&#39; step)}                     \\</span>
+<span class="sd">            &amp;\hspace{5mm} \Delta \leftarrow \Delta                                               \\</span>
+<span class="sd">            &amp;\hspace{5mm} \delta_d \leftarrow \delta_d^{\text{init}}                             \\</span>
+<span class="sd">            &amp;\textbf{else} ~ \text{(``unsuccessful&#39;&#39; step)}                                      \\</span>
+<span class="sd">            &amp;\hspace{5mm} \Delta \leftarrow \delta_d \cdot \Delta                                \\</span>
+<span class="sd">            &amp;\hspace{5mm} \delta_d \leftarrow \sigma \cdot \delta_d                              \\</span>
+<span class="sd">            &amp;\Delta \leftarrow \mathrm{min}(\mathrm{max}(\Delta, \epsilon_{s}), \epsilon_{l})    \\</span>
+<span class="sd">            &amp;\delta_d \leftarrow \mathrm{min}(\mathrm{max}(\delta_d, \epsilon_{s}), \epsilon_{l})\\</span>
+<span class="sd">            &amp;\rule{113mm}{0.4pt}                                                          \\[-1.ex]</span>
+<span class="sd">            &amp;\textbf{return} \:  \Delta, \delta_d                                         \\[-1.ex]</span>
+<span class="sd">            &amp;\rule{113mm}{0.4pt}                                                          \\[-1.ex]</span>
+<span class="sd">       \end{aligned}</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        radius (float, optional): the initial radius of the trust region (positive number).</span>
+<span class="sd">            Default: 1e6.</span>
+<span class="sd">        high (float, optional): high threshold for scaling down the damping factor.</span>
+<span class="sd">            Default: 0.5.</span>
+<span class="sd">        low (float, optional): low threshold for scaling up the damping factor.</span>
+<span class="sd">            Default: 1e-3.</span>
+<span class="sd">        up (float, optional): the up scaling factor in the range of :math:`(1,\infty)`.</span>
+<span class="sd">            Default: 2.0.</span>
+<span class="sd">        down (float, optional): the initial down scaling factor in range of :math:`(0,1)`.</span>
+<span class="sd">            Default: 0.5.</span>
+<span class="sd">        factor (float, optional): exponential factor for shrinking of the trust region.</span>
+<span class="sd">            Default: 0.5.</span>
+<span class="sd">        min (float, optional): the lower-bound of trust region radius and down scaling factor.</span>
+<span class="sd">            Default: 1e-6.</span>
+<span class="sd">        max (float, optional): the upper-bound of trust region radius and down scaling factor.</span>
+<span class="sd">            Default: 1e16.</span>
+
+<span class="sd">    More details about the optimization process go to :meth:`pypose.optim.LevenbergMarquardt`.</span>
+
+<span class="sd">    Note:</span>
+<span class="sd">        This implementation is an improved version of `TR in Ceres &lt;https://tinyurl.com/38r8e77x&gt;`_.</span>
+
+<span class="sd">        For efficiency, we calculate the denominator of the step quality :math:`\rho` as</span>
+
+<span class="sd">        .. math::</span>
+<span class="sd">            \begin{aligned}</span>
+<span class="sd">            &amp; \|\bm{f}(\bm{\theta})\|^2 - \|\bm{f}(\bm{\theta}) + \mathbf{J}\delta\|^2 \\</span>
+<span class="sd">            &amp; = \bm{f}^T\bm{f} - \left( \bm{f}^T\bm{f} + 2 \bm{f}^T \delta +</span>
+<span class="sd">                \delta^T \mathbf{J}^T\mathbf{J}\delta \right) \\</span>
+<span class="sd">            &amp; = -(\mathbf{J} \delta)^T (2\bm{f} + \mathbf{J}\delta)</span>
+<span class="sd">            \end{aligned}</span>
+
+<span class="sd">        where :math:`\mathbf{J}` is the Jacobian of the model :math:`\bm{f}` at evaluation</span>
+<span class="sd">        point :math:`\bm{\theta}`.</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        &gt;&gt;&gt; class PoseInv(nn.Module):</span>
+<span class="sd">        ...     def __init__(self, *dim):</span>
+<span class="sd">        ...         super().__init__()</span>
+<span class="sd">        ...         self.pose = pp.Parameter(pp.randn_SE3(*dim))</span>
+<span class="sd">        ... </span>
+<span class="sd">        ...     def forward(self, inputs):</span>
+<span class="sd">        ...         return (self.pose @ inputs).Log().tensor()</span>
+<span class="sd">        ... </span>
+<span class="sd">        ... device = torch.device(&quot;cuda&quot; if torch.cuda.is_available() else &quot;cpu&quot;)</span>
+<span class="sd">        ... inputs = pp.randn_SE3(2, 2).to(device)</span>
+<span class="sd">        ... invnet = PoseInv(2, 2).to(device)</span>
+<span class="sd">        ... strategy = pp.optim.strategy.TrustRegion(radius=1e6)</span>
+<span class="sd">        ... optimizer = pp.optim.LM(invnet, strategy=strategy)</span>
+<span class="sd">        ... </span>
+<span class="sd">        ... for idx in range(10):</span>
+<span class="sd">        ...     loss = optimizer.step(inputs)</span>
+<span class="sd">        ...     print(&#39;Pose loss %.7f @ %dit&#39;%(loss, idx))</span>
+<span class="sd">        ...     if loss &lt; 1e-5:</span>
+<span class="sd">        ...         print(&#39;Early Stoping!&#39;)</span>
+<span class="sd">        ...         print(&#39;Optimization Early Done with loss:&#39;, loss.item())</span>
+<span class="sd">        ...         break</span>
+<span class="sd">        Pose loss 0.0000000 @ 0it</span>
+<span class="sd">        Early Stoping!</span>
+<span class="sd">        Optimization Early Done with loss: 7.462681583803032e-10</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">radius</span><span class="o">=</span><span class="mf">1e6</span><span class="p">,</span> <span class="n">high</span><span class="o">=</span><span class="mf">.5</span><span class="p">,</span> <span class="n">low</span><span class="o">=</span><span class="mf">1e-3</span><span class="p">,</span> <span class="n">up</span><span class="o">=</span><span class="mf">2.</span><span class="p">,</span> <span class="n">down</span><span class="o">=</span><span class="mf">.5</span><span class="p">,</span> <span class="n">factor</span><span class="o">=</span><span class="mf">.5</span><span class="p">,</span> <span class="nb">min</span><span class="o">=</span><span class="mf">1e-6</span><span class="p">,</span> <span class="nb">max</span><span class="o">=</span><span class="mf">1e16</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+        <span class="k">assert</span> <span class="n">radius</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">,</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">&quot;trust region radius has to be positive: </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">radius</span><span class="p">))</span>
+        <span class="k">assert</span> <span class="n">high</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">,</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">&quot;high has to be positive: </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">high</span><span class="p">))</span>
+        <span class="k">assert</span> <span class="n">low</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">,</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">&quot;low for decrease has to be positive: </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">low</span><span class="p">))</span>
+        <span class="k">assert</span> <span class="mi">0</span> <span class="o">&lt;</span> <span class="n">down</span> <span class="o">&lt;</span> <span class="mi">1</span><span class="p">,</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">&quot;down factor has to be smaller than 1: </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">down</span><span class="p">))</span>
+        <span class="k">assert</span> <span class="mi">1</span> <span class="o">&lt;</span> <span class="n">up</span><span class="p">,</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">&quot;up factor has to be larger than 1: </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">up</span><span class="p">))</span>
+        <span class="k">assert</span> <span class="mi">0</span> <span class="o">&lt;</span> <span class="n">factor</span> <span class="o">&lt;</span> <span class="mi">1</span><span class="p">,</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">&quot;factor has to be smaller than 1: </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">factor</span><span class="p">))</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">min</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">max</span><span class="p">,</span> <span class="n">damping</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">down</span> <span class="o">=</span> <span class="nb">min</span><span class="p">,</span> <span class="nb">max</span><span class="p">,</span> <span class="mi">1</span> <span class="o">/</span> <span class="n">radius</span><span class="p">,</span> <span class="n">down</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">defaults</span> <span class="o">=</span> <span class="p">{</span><span class="s1">&#39;radius&#39;</span><span class="p">:</span><span class="n">radius</span><span class="p">,</span> <span class="s1">&#39;damping&#39;</span><span class="p">:</span><span class="n">damping</span><span class="p">,</span> <span class="s1">&#39;high&#39;</span><span class="p">:</span><span class="n">high</span><span class="p">,</span> <span class="s1">&#39;low&#39;</span><span class="p">:</span><span class="n">low</span><span class="p">,</span>
+                         <span class="s1">&#39;up&#39;</span><span class="p">:</span><span class="n">up</span><span class="p">,</span> <span class="s1">&#39;down&#39;</span><span class="p">:</span> <span class="n">down</span><span class="p">,</span> <span class="s1">&#39;factor&#39;</span><span class="p">:</span><span class="n">factor</span><span class="p">}</span>
+
+    <span class="k">def</span> <span class="nf">update</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">pg</span><span class="p">,</span> <span class="n">last</span><span class="p">,</span> <span class="n">loss</span><span class="p">,</span> <span class="n">J</span><span class="p">,</span> <span class="n">D</span><span class="p">,</span> <span class="n">R</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+        <span class="n">quality</span> <span class="o">=</span> <span class="p">(</span><span class="n">last</span> <span class="o">-</span> <span class="n">loss</span><span class="p">)</span> <span class="o">/</span> <span class="o">-</span><span class="p">((</span><span class="n">J</span> <span class="o">@</span> <span class="n">D</span><span class="p">)</span><span class="o">.</span><span class="n">mT</span> <span class="o">@</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">R</span> <span class="o">+</span> <span class="n">J</span> <span class="o">@</span> <span class="n">D</span><span class="p">))</span><span class="o">.</span><span class="n">squeeze</span><span class="p">()</span>
+        <span class="n">pg</span><span class="p">[</span><span class="s1">&#39;radius&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="mf">1.</span> <span class="o">/</span> <span class="n">pg</span><span class="p">[</span><span class="s1">&#39;damping&#39;</span><span class="p">]</span>
+        <span class="k">if</span> <span class="n">quality</span> <span class="o">&gt;</span> <span class="n">pg</span><span class="p">[</span><span class="s1">&#39;high&#39;</span><span class="p">]:</span>
+            <span class="n">pg</span><span class="p">[</span><span class="s1">&#39;radius&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">pg</span><span class="p">[</span><span class="s1">&#39;up&#39;</span><span class="p">]</span> <span class="o">*</span> <span class="n">pg</span><span class="p">[</span><span class="s1">&#39;radius&#39;</span><span class="p">]</span>
+            <span class="n">pg</span><span class="p">[</span><span class="s1">&#39;down&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">down</span>
+        <span class="k">elif</span> <span class="n">quality</span> <span class="o">&gt;</span> <span class="n">pg</span><span class="p">[</span><span class="s1">&#39;low&#39;</span><span class="p">]:</span>
+            <span class="n">pg</span><span class="p">[</span><span class="s1">&#39;radius&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">pg</span><span class="p">[</span><span class="s1">&#39;radius&#39;</span><span class="p">]</span>
+            <span class="n">pg</span><span class="p">[</span><span class="s1">&#39;down&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">down</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">pg</span><span class="p">[</span><span class="s1">&#39;radius&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">pg</span><span class="p">[</span><span class="s1">&#39;radius&#39;</span><span class="p">]</span> <span class="o">*</span> <span class="n">pg</span><span class="p">[</span><span class="s1">&#39;down&#39;</span><span class="p">]</span>
+            <span class="n">pg</span><span class="p">[</span><span class="s1">&#39;down&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">pg</span><span class="p">[</span><span class="s1">&#39;down&#39;</span><span class="p">]</span> <span class="o">*</span> <span class="n">pg</span><span class="p">[</span><span class="s1">&#39;factor&#39;</span><span class="p">]</span>
+        <span class="n">pg</span><span class="p">[</span><span class="s1">&#39;down&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">min</span><span class="p">,</span> <span class="nb">min</span><span class="p">(</span><span class="n">pg</span><span class="p">[</span><span class="s1">&#39;down&#39;</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">max</span><span class="p">))</span>
+        <span class="n">pg</span><span class="p">[</span><span class="s1">&#39;radius&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">min</span><span class="p">,</span> <span class="nb">min</span><span class="p">(</span><span class="n">pg</span><span class="p">[</span><span class="s1">&#39;radius&#39;</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">max</span><span class="p">))</span>
+        <span class="n">pg</span><span class="p">[</span><span class="s1">&#39;damping&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="mf">1.</span> <span class="o">/</span> <span class="n">pg</span><span class="p">[</span><span class="s1">&#39;radius&#39;</span><span class="p">]</span></div>
+</pre></div>
+
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+              
+  <h1>Source code for pypose.testing.comparison</h1><div class="highlight"><pre>
+<span></span><span class="kn">import</span> <span class="nn">torch</span>
+<span class="kn">from</span> <span class="nn">..function.checking</span> <span class="kn">import</span> <span class="n">is_lietensor</span>
+
+
+<div class="viewcode-block" id="assert_close"><a class="viewcode-back" href="../../../../generated/pypose.testing.assert_close/#pypose.testing.assert_close">[docs]</a><span class="k">def</span> <span class="nf">assert_close</span><span class="p">(</span><span class="n">actual</span><span class="p">,</span> <span class="n">expected</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    Asserts that ``actual`` and ``expected`` are close. This function is exactly the same</span>
+<span class="sd">    with `torch.testing.assert_close &lt;https://tinyurl.com/3bm33ps7&gt;`_ except for that it</span>
+<span class="sd">    also accepts ``pypose.LieTensor``.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        actual(:obj:`Tensor` or :obj:`LieTensor`): Actual input.</span>
+<span class="sd">        expected(:obj:`Tensor` or :obj:`LieTensor`): Expected input.</span>
+<span class="sd">        rtol (Optional[float]): Relative tolerance. If specified ``atol`` must also be</span>
+<span class="sd">            specified. If omitted, default values based on the :attr:`~torch.Tensor.dtype`</span>
+<span class="sd">            are selected with the below table.</span>
+<span class="sd">        atol (Optional[float]): Absolute tolerance. If specified ``rtol`` must also be</span>
+<span class="sd">            specified. If omitted, default values based on the :attr:`~torch.Tensor.dtype`</span>
+<span class="sd">            are selected with the below table.</span>
+
+<span class="sd">    If :math:`T_e` and :math:`T_a` are Lietensor, they are considered close if</span>
+<span class="sd">    :math:`T_e*T_a^{-1} = \mathbf{O}`, where :math:`\mathbf{O}` is close to zero tensor in</span>
+<span class="sd">    the sense of ``torch.testing.assert_close`` is ``True.``</span>
+
+<span class="sd">    Warning:</span>
+<span class="sd">        The prerequisites for the other arguments align precisely with</span>
+<span class="sd">        `torch.testing.assert_close &lt;https://tinyurl.com/3bm33ps7&gt;`_. Kindly consult it</span>
+<span class="sd">        for further details.</span>
+
+<span class="sd">    Examples:</span>
+<span class="sd">        &gt;&gt;&gt; import pypose as pp</span>
+<span class="sd">        &gt;&gt;&gt; actual = pp.randn_SE3(3)</span>
+<span class="sd">        &gt;&gt;&gt; expected = actual.Log().Exp()</span>
+<span class="sd">        &gt;&gt;&gt; pp.testing.assert_close(actual, expected, rtol=1e-5, atol=1e-5)</span>
+
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">if</span> <span class="n">is_lietensor</span><span class="p">(</span><span class="n">actual</span><span class="p">)</span> <span class="ow">and</span> <span class="n">is_lietensor</span><span class="p">(</span><span class="n">expected</span><span class="p">):</span>
+        <span class="n">source</span> <span class="o">=</span> <span class="p">(</span><span class="n">actual</span><span class="o">.</span><span class="n">Inv</span><span class="p">()</span> <span class="o">@</span> <span class="n">expected</span><span class="p">)</span><span class="o">.</span><span class="n">Log</span><span class="p">()</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span>
+        <span class="n">target</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">zeros_like</span><span class="p">(</span><span class="n">source</span><span class="p">)</span>
+    <span class="k">else</span><span class="p">:</span>
+        <span class="n">source</span><span class="p">,</span> <span class="n">target</span> <span class="o">=</span> <span class="n">actual</span><span class="p">,</span> <span class="n">expected</span>
+    <span class="n">torch</span><span class="o">.</span><span class="n">testing</span><span class="o">.</span><span class="n">assert_close</span><span class="p">(</span><span class="n">source</span><span class="p">,</span> <span class="n">target</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span></div>
+</pre></div>
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+  <h1>Source code for pypose.utils.stepper</h1><div class="highlight"><pre>
+<span></span><span class="kn">import</span> <span class="nn">torch</span>
+
+
+<span class="k">class</span> <span class="nc">_Stepper</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
+
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">max_steps</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="kc">False</span><span class="p">):</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">max_steps</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">verbose</span> <span class="o">=</span> <span class="n">max_steps</span><span class="p">,</span> <span class="n">verbose</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">reset</span><span class="p">()</span>
+
+    <span class="k">def</span> <span class="nf">continual</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_continual</span>
+
+    <span class="k">def</span> <span class="nf">reset</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">last</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">(</span><span class="nb">float</span><span class="p">(</span><span class="s1">&#39;inf&#39;</span><span class="p">))</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">steps</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_continual</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="kc">True</span>
+
+
+<div class="viewcode-block" id="ReduceToBason"><a class="viewcode-back" href="../../../../generated/pypose.utils.ReduceToBason/#pypose.utils.ReduceToBason">[docs]</a><span class="k">class</span> <span class="nc">ReduceToBason</span><span class="p">(</span><span class="n">_Stepper</span><span class="p">):</span>
+<span class="w">    </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">    A stepper to stop a loop when no relative loss &#39;decreasing&#39; is seen for a &#39;patience&#39;</span>
+<span class="sd">    number of steps.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        steps (``int``): Maximum number of iterations a loop will step.</span>
+<span class="sd">        patience (``int``, optional): Number of steps with no loss &#39;decreasing&#39; is seen.</span>
+<span class="sd">            For example, if ``patience = 2``, then it ignores the first 2 steps with no</span>
+<span class="sd">            improvement, and stop the loop after the 3rd step if the loss has no</span>
+<span class="sd">            decerasing. Default: ``5``.</span>
+<span class="sd">        decreasing (``float``, optional): relative loss decreasing used to count the</span>
+<span class="sd">            number of patience steps. Default: ``1e-3``.</span>
+<span class="sd">        tol (``float``, optional): the minimum loss tolerance to stop a loop.</span>
+<span class="sd">            Default: ``1e-5``.</span>
+<span class="sd">        verbose (``bool``, optional): if ``True``, prints a message to stdout for each</span>
+<span class="sd">            step. Default: ``False``.</span>
+
+<span class="sd">    Warning:</span>
+<span class="sd">        Remember to call `stepper.reset()` if you want to re-use a stepper.</span>
+
+<span class="sd">    Example:</span>
+<span class="sd">        &gt;&gt;&gt; from pypose.utils import ReduceToBason</span>
+<span class="sd">        &gt;&gt;&gt; x = 0.9</span>
+<span class="sd">        &gt;&gt;&gt; stepper = ReduceToBason(steps=5, patience=2, decreasing=0.1, verbose=True)</span>
+<span class="sd">        &gt;&gt;&gt; while stepper.continual():</span>
+<span class="sd">        ...     x = x ** 2</span>
+<span class="sd">        ...     stepper.step(x)</span>
+<span class="sd">        ReduceToBason step 0 loss 8.099999e-01.</span>
+<span class="sd">        ReduceToBason step 1 loss 6.560999e-01.</span>
+<span class="sd">        ReduceToBason step 2 loss 4.304671e-01.</span>
+<span class="sd">        ReduceToBason step 3 loss 1.853019e-01.</span>
+<span class="sd">        ReduceToBason step 4 loss 3.433681e-02.</span>
+<span class="sd">        ReduceToBason: Maximum steps reached, Quiting..</span>
+<span class="sd">    &#39;&#39;&#39;</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">steps</span><span class="p">,</span> <span class="n">patience</span><span class="o">=</span><span class="mi">5</span><span class="p">,</span> <span class="n">decreasing</span><span class="o">=</span><span class="mf">1e-3</span><span class="p">,</span> <span class="n">tol</span><span class="o">=</span><span class="mf">1e-5</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="kc">False</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">(</span><span class="n">steps</span><span class="p">,</span> <span class="n">verbose</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">decreasing</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">tol</span> <span class="o">=</span> <span class="n">decreasing</span><span class="p">,</span> <span class="n">tol</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">patience</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">patience_count</span> <span class="o">=</span> <span class="n">patience</span><span class="p">,</span> <span class="mi">0</span>
+
+<div class="viewcode-block" id="ReduceToBason.step"><a class="viewcode-back" href="../../../../generated/pypose.utils.ReduceToBason/#pypose.utils.ReduceToBason.step">[docs]</a>    <span class="k">def</span> <span class="nf">step</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">loss</span><span class="p">):</span>
+<span class="w">        </span><span class="sa">r</span><span class="sd">&#39;&#39;&#39;</span>
+<span class="sd">        Performs a stepper step.</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            loss (``float`` or ``torch.Tensor``): the model loss after one loop.</span>
+<span class="sd">                Can be a batched tensor. If batched tensor, all losses in the batch has to</span>
+<span class="sd">                satisfy the condition to stop a loop.</span>
+<span class="sd">        &#39;&#39;&#39;</span>
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">verbose</span><span class="p">:</span>
+            <span class="nb">print</span><span class="p">(</span><span class="s1">&#39;ReduceToBason step&#39;</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">steps</span><span class="p">,</span> <span class="s1">&#39;loss&#39;</span><span class="p">,</span> <span class="n">loss</span><span class="p">)</span>
+
+        <span class="k">if</span> <span class="ow">not</span> <span class="n">torch</span><span class="o">.</span><span class="n">is_tensor</span><span class="p">(</span><span class="n">loss</span><span class="p">):</span>
+            <span class="n">loss</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">(</span><span class="n">loss</span><span class="p">)</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">steps</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">steps</span> <span class="o">+</span> <span class="mi">1</span>
+
+        <span class="k">if</span> <span class="n">torch</span><span class="o">.</span><span class="n">all</span><span class="p">(</span><span class="n">loss</span> <span class="o">&lt;</span> <span class="bp">self</span><span class="o">.</span><span class="n">tol</span><span class="p">):</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">_continual</span> <span class="o">=</span> <span class="kc">False</span>
+            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">verbose</span><span class="p">:</span>
+                <span class="nb">print</span><span class="p">(</span><span class="s2">&quot;ReduceToBason: Loss tol reached, Quiting..&quot;</span><span class="p">)</span>
+
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">steps</span> <span class="o">&gt;=</span> <span class="bp">self</span><span class="o">.</span><span class="n">max_steps</span><span class="p">:</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">_continual</span> <span class="o">=</span> <span class="kc">False</span>
+            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">verbose</span><span class="p">:</span>
+                <span class="nb">print</span><span class="p">(</span><span class="s2">&quot;ReduceToBason: Maximum steps reached, Quiting..&quot;</span><span class="p">)</span>
+
+        <span class="k">if</span> <span class="n">torch</span><span class="o">.</span><span class="n">all</span><span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">last</span> <span class="o">-</span> <span class="n">loss</span><span class="p">)</span><span class="o">/</span><span class="n">loss</span> <span class="o">&lt;</span> <span class="bp">self</span><span class="o">.</span><span class="n">decreasing</span><span class="p">):</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">patience_count</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">patience_count</span> <span class="o">+</span> <span class="mi">1</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">patience_count</span> <span class="o">=</span> <span class="mi">0</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">last</span> <span class="o">=</span> <span class="n">loss</span>
+
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">patience_count</span> <span class="o">&gt;=</span> <span class="bp">self</span><span class="o">.</span><span class="n">patience</span><span class="p">:</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">_continual</span> <span class="o">=</span> <span class="kc">False</span>
+            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">verbose</span><span class="p">:</span>
+                <span class="nb">print</span><span class="p">(</span><span class="s2">&quot;ReduceToBason: Maximum patience steps reached, Quiting..&quot;</span><span class="p">)</span></div></div>
+</pre></div>
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+      filterTags.bind();
+
+      // Add class to links that have code blocks, since we cannot create links in code blocks
+      $("article.pytorch-article a span.pre").each(function(e) {
+        $(this).closest("a").addClass("has-code");
+      });
+    })
+  </script>
+</body>
+</html>
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/convert.rst.txt b/docs/preview/pr/352/_sources/convert.rst.txt
new file mode 100644
index 000000000..21e429bf8
--- /dev/null
+++ b/docs/preview/pr/352/_sources/convert.rst.txt
@@ -0,0 +1,26 @@
+Convert
+=======
+
+.. currentmodule:: pypose
+
+.. autosummary::
+    :toctree: generated
+    :nosignatures:
+
+    mat2SO3
+    mat2SE3
+    mat2Sim3
+    mat2RxSO3
+    from_matrix
+    euler2SO3
+    vec2skew
+    tensor
+    translation
+    rotation
+    euler
+    scale
+    matrix
+    cart2homo
+    homo2cart
+    point2pixel
+    pixel2point
diff --git a/docs/preview/pr/352/_sources/func.rst.txt b/docs/preview/pr/352/_sources/func.rst.txt
new file mode 100644
index 000000000..ce38a0f03
--- /dev/null
+++ b/docs/preview/pr/352/_sources/func.rst.txt
@@ -0,0 +1,10 @@
+Func
+=======
+
+.. currentmodule:: pypose
+
+.. autosummary::
+    :toctree: generated
+    :nosignatures:
+
+    func.jacrev
diff --git a/docs/preview/pr/352/_sources/functions.rst.txt b/docs/preview/pr/352/_sources/functions.rst.txt
new file mode 100644
index 000000000..44db85a05
--- /dev/null
+++ b/docs/preview/pr/352/_sources/functions.rst.txt
@@ -0,0 +1,58 @@
+Functions
+=========
+
+.. currentmodule:: pypose
+
+.. autosummary::
+    :toctree: generated
+    :nosignatures:
+
+    :template: autosummary/class.rst
+    is_lietensor
+    is_SE3
+    hasnan
+    add
+    add_
+    mul
+    bvv
+    bmv
+    bvmv
+    pm
+    Exp
+    Log
+    Inv
+    Retr
+    Act
+    Adj
+    AdjT
+    Jinvp
+    Jr
+    randn_so3
+    randn_SO3
+    randn_se3
+    randn_SE3
+    randn_sim3
+    randn_Sim3
+    randn_rxso3
+    randn_RxSO3
+    randn_like
+    identity_so3
+    identity_SO3
+    identity_se3
+    identity_SE3
+    identity_sim3
+    identity_Sim3
+    identity_rxso3
+    identity_RxSO3
+    identity_like
+    cumprod
+    cumprod_
+    cummul
+    cummul_
+    cumops
+    cumops_
+    reprojerr
+    knn
+    svdtf
+    chspline
+    bspline
diff --git a/docs/preview/pr/352/_sources/generated/pypose.Act.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.Act.rst.txt
new file mode 100644
index 000000000..2d30b2482
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.Act.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.Act
+==========
+
+.. autoclass:: Act
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.Adj.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.Adj.rst.txt
new file mode 100644
index 000000000..420fa14db
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.Adj.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.Adj
+==========
+
+.. autoclass:: Adj
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.AdjT.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.AdjT.rst.txt
new file mode 100644
index 000000000..bb4c4d0c6
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.AdjT.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.AdjT
+===========
+
+.. autoclass:: AdjT
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.Exp.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.Exp.rst.txt
new file mode 100644
index 000000000..c912e85c9
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.Exp.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.Exp
+==========
+
+.. autoclass:: Exp
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.Inv.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.Inv.rst.txt
new file mode 100644
index 000000000..70dcbd663
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.Inv.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.Inv
+==========
+
+.. autoclass:: Inv
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.Jinvp.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.Jinvp.rst.txt
new file mode 100644
index 000000000..0a9a8fa95
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.Jinvp.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.Jinvp
+============
+
+.. autoclass:: Jinvp
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.Jr.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.Jr.rst.txt
new file mode 100644
index 000000000..402917468
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.Jr.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.Jr
+=========
+
+.. autoclass:: Jr
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.LieTensor.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.LieTensor.rst.txt
new file mode 100644
index 000000000..2265bcd4e
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.LieTensor.rst.txt
@@ -0,0 +1,8 @@
+.. currentmodule:: pypose
+
+
+pypose.LieTensor
+================
+
+.. autoclass:: LieTensor
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.Log.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.Log.rst.txt
new file mode 100644
index 000000000..460f5509c
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.Log.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.Log
+==========
+
+.. autoclass:: Log
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.Parameter.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.Parameter.rst.txt
new file mode 100644
index 000000000..6d12f1114
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.Parameter.rst.txt
@@ -0,0 +1,8 @@
+.. currentmodule:: pypose
+
+
+pypose.Parameter
+================
+
+.. autoclass:: Parameter
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.Retr.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.Retr.rst.txt
new file mode 100644
index 000000000..ff1e38470
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.Retr.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.Retr
+===========
+
+.. autoclass:: Retr
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.RxSO3.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.RxSO3.rst.txt
new file mode 100644
index 000000000..362a731d0
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.RxSO3.rst.txt
@@ -0,0 +1,6 @@
+pypose.RxSO3
+============
+
+.. currentmodule:: pypose
+
+.. autodata:: RxSO3
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.SE3.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.SE3.rst.txt
new file mode 100644
index 000000000..b4e3fd7c7
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.SE3.rst.txt
@@ -0,0 +1,6 @@
+pypose.SE3
+==========
+
+.. currentmodule:: pypose
+
+.. autodata:: SE3
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.SO3.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.SO3.rst.txt
new file mode 100644
index 000000000..525c24aee
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.SO3.rst.txt
@@ -0,0 +1,6 @@
+pypose.SO3
+==========
+
+.. currentmodule:: pypose
+
+.. autodata:: SO3
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.Sim3.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.Sim3.rst.txt
new file mode 100644
index 000000000..19ab0f274
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.Sim3.rst.txt
@@ -0,0 +1,6 @@
+pypose.Sim3
+===========
+
+.. currentmodule:: pypose
+
+.. autodata:: Sim3
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.add.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.add.rst.txt
new file mode 100644
index 000000000..84bc2a657
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.add.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.add
+==========
+
+.. autoclass:: add
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.add_.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.add_.rst.txt
new file mode 100644
index 000000000..e341ced8e
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.add_.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.add\_
+============
+
+.. autoclass:: add_
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.bmv.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.bmv.rst.txt
new file mode 100644
index 000000000..0a7e95772
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.bmv.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.bmv
+==========
+
+.. autoclass:: bmv
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.bspline.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.bspline.rst.txt
new file mode 100644
index 000000000..071725827
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.bspline.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.bspline
+==============
+
+.. autoclass:: bspline
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.bvmv.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.bvmv.rst.txt
new file mode 100644
index 000000000..42bb194db
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.bvmv.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.bvmv
+===========
+
+.. autoclass:: bvmv
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.bvv.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.bvv.rst.txt
new file mode 100644
index 000000000..ce90322e6
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.bvv.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.bvv
+==========
+
+.. autoclass:: bvv
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.cart2homo.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.cart2homo.rst.txt
new file mode 100644
index 000000000..582f9c431
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.cart2homo.rst.txt
@@ -0,0 +1,6 @@
+pypose.cart2homo
+================
+
+.. currentmodule:: pypose
+
+.. autofunction:: cart2homo
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.chspline.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.chspline.rst.txt
new file mode 100644
index 000000000..92e8117e4
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.chspline.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.chspline
+===============
+
+.. autoclass:: chspline
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.cummul.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.cummul.rst.txt
new file mode 100644
index 000000000..32772812b
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.cummul.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.cummul
+=============
+
+.. autoclass:: cummul
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.cummul_.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.cummul_.rst.txt
new file mode 100644
index 000000000..95acb6f60
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.cummul_.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.cummul\_
+===============
+
+.. autoclass:: cummul_
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.cumops.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.cumops.rst.txt
new file mode 100644
index 000000000..faae1d366
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.cumops.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.cumops
+=============
+
+.. autoclass:: cumops
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.cumops_.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.cumops_.rst.txt
new file mode 100644
index 000000000..8a9c96d93
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.cumops_.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.cumops\_
+===============
+
+.. autoclass:: cumops_
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.cumprod.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.cumprod.rst.txt
new file mode 100644
index 000000000..cfc08cc3e
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.cumprod.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.cumprod
+==============
+
+.. autoclass:: cumprod
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.cumprod_.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.cumprod_.rst.txt
new file mode 100644
index 000000000..3c6ee0a52
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.cumprod_.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.cumprod\_
+================
+
+.. autoclass:: cumprod_
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.euler.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.euler.rst.txt
new file mode 100644
index 000000000..2a2593c19
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.euler.rst.txt
@@ -0,0 +1,6 @@
+pypose.euler
+============
+
+.. currentmodule:: pypose
+
+.. autofunction:: euler
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.euler2SO3.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.euler2SO3.rst.txt
new file mode 100644
index 000000000..48f247e69
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.euler2SO3.rst.txt
@@ -0,0 +1,6 @@
+pypose.euler2SO3
+================
+
+.. currentmodule:: pypose
+
+.. autofunction:: euler2SO3
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.from_matrix.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.from_matrix.rst.txt
new file mode 100644
index 000000000..cb6b86e06
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.from_matrix.rst.txt
@@ -0,0 +1,6 @@
+pypose.from\_matrix
+===================
+
+.. currentmodule:: pypose
+
+.. autofunction:: from_matrix
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.func.jacrev.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.func.jacrev.rst.txt
new file mode 100644
index 000000000..8fd8911f7
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.func.jacrev.rst.txt
@@ -0,0 +1,6 @@
+pypose.func.jacrev
+==================
+
+.. currentmodule:: pypose.func
+
+.. autofunction:: jacrev
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.hasnan.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.hasnan.rst.txt
new file mode 100644
index 000000000..a9827d950
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.hasnan.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.hasnan
+=============
+
+.. autoclass:: hasnan
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.homo2cart.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.homo2cart.rst.txt
new file mode 100644
index 000000000..86ec58194
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.homo2cart.rst.txt
@@ -0,0 +1,6 @@
+pypose.homo2cart
+================
+
+.. currentmodule:: pypose
+
+.. autofunction:: homo2cart
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.identity_RxSO3.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.identity_RxSO3.rst.txt
new file mode 100644
index 000000000..d87b66f28
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.identity_RxSO3.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.identity\_RxSO3
+======================
+
+.. autoclass:: identity_RxSO3
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.identity_SE3.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.identity_SE3.rst.txt
new file mode 100644
index 000000000..e989605af
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.identity_SE3.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.identity\_SE3
+====================
+
+.. autoclass:: identity_SE3
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.identity_SO3.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.identity_SO3.rst.txt
new file mode 100644
index 000000000..ab0fd7735
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.identity_SO3.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.identity\_SO3
+====================
+
+.. autoclass:: identity_SO3
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.identity_Sim3.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.identity_Sim3.rst.txt
new file mode 100644
index 000000000..80b9e35a9
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.identity_Sim3.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.identity\_Sim3
+=====================
+
+.. autoclass:: identity_Sim3
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.identity_like.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.identity_like.rst.txt
new file mode 100644
index 000000000..ad73063dc
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.identity_like.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.identity\_like
+=====================
+
+.. autoclass:: identity_like
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.identity_rxso3.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.identity_rxso3.rst.txt
new file mode 100644
index 000000000..72441c7c6
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.identity_rxso3.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.identity\_rxso3
+======================
+
+.. autoclass:: identity_rxso3
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.identity_se3.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.identity_se3.rst.txt
new file mode 100644
index 000000000..135dbc9d2
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.identity_se3.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.identity\_se3
+====================
+
+.. autoclass:: identity_se3
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.identity_sim3.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.identity_sim3.rst.txt
new file mode 100644
index 000000000..e8f83510a
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.identity_sim3.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.identity\_sim3
+=====================
+
+.. autoclass:: identity_sim3
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.identity_so3.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.identity_so3.rst.txt
new file mode 100644
index 000000000..d0487c57a
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.identity_so3.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.identity\_so3
+====================
+
+.. autoclass:: identity_so3
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.is_SE3.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.is_SE3.rst.txt
new file mode 100644
index 000000000..d44d9c5c3
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.is_SE3.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.is\_SE3
+==============
+
+.. autoclass:: is_SE3
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.is_lietensor.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.is_lietensor.rst.txt
new file mode 100644
index 000000000..f1070d5ac
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.is_lietensor.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.is\_lietensor
+====================
+
+.. autoclass:: is_lietensor
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.knn.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.knn.rst.txt
new file mode 100644
index 000000000..49d00ef10
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.knn.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.knn
+==========
+
+.. autoclass:: knn
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.mat2RxSO3.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.mat2RxSO3.rst.txt
new file mode 100644
index 000000000..7afaf9752
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.mat2RxSO3.rst.txt
@@ -0,0 +1,6 @@
+pypose.mat2RxSO3
+================
+
+.. currentmodule:: pypose
+
+.. autofunction:: mat2RxSO3
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.mat2SE3.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.mat2SE3.rst.txt
new file mode 100644
index 000000000..70988e514
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.mat2SE3.rst.txt
@@ -0,0 +1,6 @@
+pypose.mat2SE3
+==============
+
+.. currentmodule:: pypose
+
+.. autofunction:: mat2SE3
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.mat2SO3.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.mat2SO3.rst.txt
new file mode 100644
index 000000000..ec8acd9b3
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.mat2SO3.rst.txt
@@ -0,0 +1,6 @@
+pypose.mat2SO3
+==============
+
+.. currentmodule:: pypose
+
+.. autofunction:: mat2SO3
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.mat2Sim3.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.mat2Sim3.rst.txt
new file mode 100644
index 000000000..8816fc0a4
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.mat2Sim3.rst.txt
@@ -0,0 +1,6 @@
+pypose.mat2Sim3
+===============
+
+.. currentmodule:: pypose
+
+.. autofunction:: mat2Sim3
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.matrix.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.matrix.rst.txt
new file mode 100644
index 000000000..635b242dd
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.matrix.rst.txt
@@ -0,0 +1,6 @@
+pypose.matrix
+=============
+
+.. currentmodule:: pypose
+
+.. autofunction:: matrix
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.module.EKF.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.module.EKF.rst.txt
new file mode 100644
index 000000000..f93638ad2
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.module.EKF.rst.txt
@@ -0,0 +1,8 @@
+.. currentmodule:: pypose.module
+
+
+pypose.module.EKF
+=================
+
+.. autoclass:: EKF
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.module.EPnP.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.module.EPnP.rst.txt
new file mode 100644
index 000000000..8200755b3
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.module.EPnP.rst.txt
@@ -0,0 +1,8 @@
+.. currentmodule:: pypose.module
+
+
+pypose.module.EPnP
+==================
+
+.. autoclass:: EPnP
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.module.ICP.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.module.ICP.rst.txt
new file mode 100644
index 000000000..c86f4743e
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.module.ICP.rst.txt
@@ -0,0 +1,8 @@
+.. currentmodule:: pypose.module
+
+
+pypose.module.ICP
+=================
+
+.. autoclass:: ICP
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.module.IMUPreintegrator.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.module.IMUPreintegrator.rst.txt
new file mode 100644
index 000000000..4e67b2fdc
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.module.IMUPreintegrator.rst.txt
@@ -0,0 +1,8 @@
+.. currentmodule:: pypose.module
+
+
+pypose.module.IMUPreintegrator
+==============================
+
+.. autoclass:: IMUPreintegrator
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.module.LQR.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.module.LQR.rst.txt
new file mode 100644
index 000000000..ec502c333
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.module.LQR.rst.txt
@@ -0,0 +1,8 @@
+.. currentmodule:: pypose.module
+
+
+pypose.module.LQR
+=================
+
+.. autoclass:: LQR
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.module.LTI.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.module.LTI.rst.txt
new file mode 100644
index 000000000..4a095f66d
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.module.LTI.rst.txt
@@ -0,0 +1,8 @@
+.. currentmodule:: pypose.module
+
+
+pypose.module.LTI
+=================
+
+.. autoclass:: LTI
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.module.LTV.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.module.LTV.rst.txt
new file mode 100644
index 000000000..c56f22087
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.module.LTV.rst.txt
@@ -0,0 +1,8 @@
+.. currentmodule:: pypose.module
+
+
+pypose.module.LTV
+=================
+
+.. autoclass:: LTV
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.module.MPC.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.module.MPC.rst.txt
new file mode 100644
index 000000000..d5ee8b18a
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.module.MPC.rst.txt
@@ -0,0 +1,8 @@
+.. currentmodule:: pypose.module
+
+
+pypose.module.MPC
+=================
+
+.. autoclass:: MPC
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.module.NLS.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.module.NLS.rst.txt
new file mode 100644
index 000000000..6d32546ae
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.module.NLS.rst.txt
@@ -0,0 +1,8 @@
+.. currentmodule:: pypose.module
+
+
+pypose.module.NLS
+=================
+
+.. autoclass:: NLS
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.module.PF.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.module.PF.rst.txt
new file mode 100644
index 000000000..f01ba988b
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.module.PF.rst.txt
@@ -0,0 +1,8 @@
+.. currentmodule:: pypose.module
+
+
+pypose.module.PF
+================
+
+.. autoclass:: PF
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.module.System.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.module.System.rst.txt
new file mode 100644
index 000000000..c3f98da8c
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.module.System.rst.txt
@@ -0,0 +1,8 @@
+.. currentmodule:: pypose.module
+
+
+pypose.module.System
+====================
+
+.. autoclass:: System
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.module.UKF.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.module.UKF.rst.txt
new file mode 100644
index 000000000..24079751b
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.module.UKF.rst.txt
@@ -0,0 +1,8 @@
+.. currentmodule:: pypose.module
+
+
+pypose.module.UKF
+=================
+
+.. autoclass:: UKF
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.mul.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.mul.rst.txt
new file mode 100644
index 000000000..bbbbab3fb
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.mul.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.mul
+==========
+
+.. autoclass:: mul
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.optim.GN.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.optim.GN.rst.txt
new file mode 100644
index 000000000..a16ca50d5
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.optim.GN.rst.txt
@@ -0,0 +1,8 @@
+.. currentmodule:: pypose.optim
+
+
+pypose.optim.GN
+===============
+
+.. autoclass:: GN
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.optim.GaussNewton.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.optim.GaussNewton.rst.txt
new file mode 100644
index 000000000..575f303c8
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.optim.GaussNewton.rst.txt
@@ -0,0 +1,8 @@
+.. currentmodule:: pypose.optim
+
+
+pypose.optim.GaussNewton
+========================
+
+.. autoclass:: GaussNewton
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.optim.LM.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.optim.LM.rst.txt
new file mode 100644
index 000000000..178197211
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.optim.LM.rst.txt
@@ -0,0 +1,8 @@
+.. currentmodule:: pypose.optim
+
+
+pypose.optim.LM
+===============
+
+.. autoclass:: LM
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.optim.LevenbergMarquardt.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.optim.LevenbergMarquardt.rst.txt
new file mode 100644
index 000000000..1b2fa764f
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.optim.LevenbergMarquardt.rst.txt
@@ -0,0 +1,8 @@
+.. currentmodule:: pypose.optim
+
+
+pypose.optim.LevenbergMarquardt
+===============================
+
+.. autoclass:: LevenbergMarquardt
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.optim.corrector.FastTriggs.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.optim.corrector.FastTriggs.rst.txt
new file mode 100644
index 000000000..513613b2a
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.optim.corrector.FastTriggs.rst.txt
@@ -0,0 +1,8 @@
+.. currentmodule:: pypose.optim.corrector
+
+
+pypose.optim.corrector.FastTriggs
+=================================
+
+.. autoclass:: FastTriggs
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.optim.corrector.Triggs.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.optim.corrector.Triggs.rst.txt
new file mode 100644
index 000000000..c5cd21846
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.optim.corrector.Triggs.rst.txt
@@ -0,0 +1,8 @@
+.. currentmodule:: pypose.optim.corrector
+
+
+pypose.optim.corrector.Triggs
+=============================
+
+.. autoclass:: Triggs
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.optim.functional.modjac.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.optim.functional.modjac.rst.txt
new file mode 100644
index 000000000..c1561d62c
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.optim.functional.modjac.rst.txt
@@ -0,0 +1,8 @@
+.. currentmodule:: pypose.optim.functional
+
+
+pypose.optim.functional.modjac
+==============================
+
+.. autoclass:: modjac
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.optim.kernel.Arctan.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.optim.kernel.Arctan.rst.txt
new file mode 100644
index 000000000..d364f808a
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.optim.kernel.Arctan.rst.txt
@@ -0,0 +1,8 @@
+.. currentmodule:: pypose.optim.kernel
+
+
+pypose.optim.kernel.Arctan
+==========================
+
+.. autoclass:: Arctan
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.optim.kernel.Cauchy.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.optim.kernel.Cauchy.rst.txt
new file mode 100644
index 000000000..81866ec7f
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.optim.kernel.Cauchy.rst.txt
@@ -0,0 +1,8 @@
+.. currentmodule:: pypose.optim.kernel
+
+
+pypose.optim.kernel.Cauchy
+==========================
+
+.. autoclass:: Cauchy
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.optim.kernel.Huber.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.optim.kernel.Huber.rst.txt
new file mode 100644
index 000000000..8ed0e15d8
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.optim.kernel.Huber.rst.txt
@@ -0,0 +1,8 @@
+.. currentmodule:: pypose.optim.kernel
+
+
+pypose.optim.kernel.Huber
+=========================
+
+.. autoclass:: Huber
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.optim.kernel.PseudoHuber.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.optim.kernel.PseudoHuber.rst.txt
new file mode 100644
index 000000000..823aa8271
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.optim.kernel.PseudoHuber.rst.txt
@@ -0,0 +1,8 @@
+.. currentmodule:: pypose.optim.kernel
+
+
+pypose.optim.kernel.PseudoHuber
+===============================
+
+.. autoclass:: PseudoHuber
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.optim.kernel.Scale.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.optim.kernel.Scale.rst.txt
new file mode 100644
index 000000000..56af323b4
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.optim.kernel.Scale.rst.txt
@@ -0,0 +1,8 @@
+.. currentmodule:: pypose.optim.kernel
+
+
+pypose.optim.kernel.Scale
+=========================
+
+.. autoclass:: Scale
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.optim.kernel.SoftLOne.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.optim.kernel.SoftLOne.rst.txt
new file mode 100644
index 000000000..ca64f1163
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.optim.kernel.SoftLOne.rst.txt
@@ -0,0 +1,8 @@
+.. currentmodule:: pypose.optim.kernel
+
+
+pypose.optim.kernel.SoftLOne
+============================
+
+.. autoclass:: SoftLOne
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.optim.kernel.Tolerant.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.optim.kernel.Tolerant.rst.txt
new file mode 100644
index 000000000..2e2f98a51
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.optim.kernel.Tolerant.rst.txt
@@ -0,0 +1,8 @@
+.. currentmodule:: pypose.optim.kernel
+
+
+pypose.optim.kernel.Tolerant
+============================
+
+.. autoclass:: Tolerant
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.optim.scheduler.StopOnPlateau.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.optim.scheduler.StopOnPlateau.rst.txt
new file mode 100644
index 000000000..3e49054b6
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.optim.scheduler.StopOnPlateau.rst.txt
@@ -0,0 +1,8 @@
+.. currentmodule:: pypose.optim.scheduler
+
+
+pypose.optim.scheduler.StopOnPlateau
+====================================
+
+.. autoclass:: StopOnPlateau
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.optim.solver.CG.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.optim.solver.CG.rst.txt
new file mode 100644
index 000000000..ab720229a
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.optim.solver.CG.rst.txt
@@ -0,0 +1,8 @@
+.. currentmodule:: pypose.optim.solver
+
+
+pypose.optim.solver.CG
+======================
+
+.. autoclass:: CG
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.optim.solver.Cholesky.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.optim.solver.Cholesky.rst.txt
new file mode 100644
index 000000000..33fdc93b8
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.optim.solver.Cholesky.rst.txt
@@ -0,0 +1,8 @@
+.. currentmodule:: pypose.optim.solver
+
+
+pypose.optim.solver.Cholesky
+============================
+
+.. autoclass:: Cholesky
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.optim.solver.LSTSQ.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.optim.solver.LSTSQ.rst.txt
new file mode 100644
index 000000000..7daf79b74
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.optim.solver.LSTSQ.rst.txt
@@ -0,0 +1,8 @@
+.. currentmodule:: pypose.optim.solver
+
+
+pypose.optim.solver.LSTSQ
+=========================
+
+.. autoclass:: LSTSQ
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.optim.solver.PINV.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.optim.solver.PINV.rst.txt
new file mode 100644
index 000000000..1cab60708
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.optim.solver.PINV.rst.txt
@@ -0,0 +1,8 @@
+.. currentmodule:: pypose.optim.solver
+
+
+pypose.optim.solver.PINV
+========================
+
+.. autoclass:: PINV
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.optim.strategy.Adaptive.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.optim.strategy.Adaptive.rst.txt
new file mode 100644
index 000000000..5dc985b77
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.optim.strategy.Adaptive.rst.txt
@@ -0,0 +1,8 @@
+.. currentmodule:: pypose.optim.strategy
+
+
+pypose.optim.strategy.Adaptive
+==============================
+
+.. autoclass:: Adaptive
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.optim.strategy.Constant.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.optim.strategy.Constant.rst.txt
new file mode 100644
index 000000000..b13d268d6
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.optim.strategy.Constant.rst.txt
@@ -0,0 +1,8 @@
+.. currentmodule:: pypose.optim.strategy
+
+
+pypose.optim.strategy.Constant
+==============================
+
+.. autoclass:: Constant
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.optim.strategy.TrustRegion.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.optim.strategy.TrustRegion.rst.txt
new file mode 100644
index 000000000..84b987d87
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.optim.strategy.TrustRegion.rst.txt
@@ -0,0 +1,8 @@
+.. currentmodule:: pypose.optim.strategy
+
+
+pypose.optim.strategy.TrustRegion
+=================================
+
+.. autoclass:: TrustRegion
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.pixel2point.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.pixel2point.rst.txt
new file mode 100644
index 000000000..9d7b3f6dd
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.pixel2point.rst.txt
@@ -0,0 +1,6 @@
+pypose.pixel2point
+==================
+
+.. currentmodule:: pypose
+
+.. autofunction:: pixel2point
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.pm.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.pm.rst.txt
new file mode 100644
index 000000000..01d999b59
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.pm.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.pm
+=========
+
+.. autoclass:: pm
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.point2pixel.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.point2pixel.rst.txt
new file mode 100644
index 000000000..d09e95e21
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.point2pixel.rst.txt
@@ -0,0 +1,6 @@
+pypose.point2pixel
+==================
+
+.. currentmodule:: pypose
+
+.. autofunction:: point2pixel
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.randn_RxSO3.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.randn_RxSO3.rst.txt
new file mode 100644
index 000000000..8b160fa81
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.randn_RxSO3.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.randn\_RxSO3
+===================
+
+.. autoclass:: randn_RxSO3
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.randn_SE3.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.randn_SE3.rst.txt
new file mode 100644
index 000000000..37b9d7729
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.randn_SE3.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.randn\_SE3
+=================
+
+.. autoclass:: randn_SE3
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.randn_SO3.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.randn_SO3.rst.txt
new file mode 100644
index 000000000..63d559a34
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.randn_SO3.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.randn\_SO3
+=================
+
+.. autoclass:: randn_SO3
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.randn_Sim3.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.randn_Sim3.rst.txt
new file mode 100644
index 000000000..e460b64f5
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.randn_Sim3.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.randn\_Sim3
+==================
+
+.. autoclass:: randn_Sim3
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.randn_like.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.randn_like.rst.txt
new file mode 100644
index 000000000..c65c0326f
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.randn_like.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.randn\_like
+==================
+
+.. autoclass:: randn_like
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.randn_rxso3.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.randn_rxso3.rst.txt
new file mode 100644
index 000000000..5c240a30e
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.randn_rxso3.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.randn\_rxso3
+===================
+
+.. autoclass:: randn_rxso3
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.randn_se3.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.randn_se3.rst.txt
new file mode 100644
index 000000000..5f48b2d26
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.randn_se3.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.randn\_se3
+=================
+
+.. autoclass:: randn_se3
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.randn_sim3.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.randn_sim3.rst.txt
new file mode 100644
index 000000000..6951cfb16
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.randn_sim3.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.randn\_sim3
+==================
+
+.. autoclass:: randn_sim3
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.randn_so3.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.randn_so3.rst.txt
new file mode 100644
index 000000000..a3fa61f9c
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.randn_so3.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.randn\_so3
+=================
+
+.. autoclass:: randn_so3
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.reprojerr.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.reprojerr.rst.txt
new file mode 100644
index 000000000..cc8dcef0d
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.reprojerr.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.reprojerr
+================
+
+.. autoclass:: reprojerr
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.rotation.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.rotation.rst.txt
new file mode 100644
index 000000000..d6575f892
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.rotation.rst.txt
@@ -0,0 +1,6 @@
+pypose.rotation
+===============
+
+.. currentmodule:: pypose
+
+.. autofunction:: rotation
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.rxso3.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.rxso3.rst.txt
new file mode 100644
index 000000000..f9ef4eca6
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.rxso3.rst.txt
@@ -0,0 +1,6 @@
+pypose.rxso3
+============
+
+.. currentmodule:: pypose
+
+.. autodata:: rxso3
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.scale.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.scale.rst.txt
new file mode 100644
index 000000000..825ce78f0
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.scale.rst.txt
@@ -0,0 +1,6 @@
+pypose.scale
+============
+
+.. currentmodule:: pypose
+
+.. autofunction:: scale
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.se3.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.se3.rst.txt
new file mode 100644
index 000000000..56682dc15
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.se3.rst.txt
@@ -0,0 +1,6 @@
+pypose.se3
+==========
+
+.. currentmodule:: pypose
+
+.. autodata:: se3
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.sim3.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.sim3.rst.txt
new file mode 100644
index 000000000..92a60488f
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.sim3.rst.txt
@@ -0,0 +1,6 @@
+pypose.sim3
+===========
+
+.. currentmodule:: pypose
+
+.. autodata:: sim3
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.so3.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.so3.rst.txt
new file mode 100644
index 000000000..9ae6d9ba9
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.so3.rst.txt
@@ -0,0 +1,6 @@
+pypose.so3
+==========
+
+.. currentmodule:: pypose
+
+.. autodata:: so3
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.svdtf.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.svdtf.rst.txt
new file mode 100644
index 000000000..d1b947a32
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.svdtf.rst.txt
@@ -0,0 +1,9 @@
+.. currentmodule:: pypose
+
+
+pypose.svdtf
+============
+
+.. autoclass:: svdtf
+    :inherited-members:
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.tensor.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.tensor.rst.txt
new file mode 100644
index 000000000..64ef41ba7
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.tensor.rst.txt
@@ -0,0 +1,6 @@
+pypose.tensor
+=============
+
+.. currentmodule:: pypose
+
+.. autofunction:: tensor
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.testing.assert_close.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.testing.assert_close.rst.txt
new file mode 100644
index 000000000..d77dcb8d3
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.testing.assert_close.rst.txt
@@ -0,0 +1,8 @@
+.. currentmodule:: pypose.testing
+
+
+pypose.testing.assert\_close
+============================
+
+.. autoclass:: assert_close
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.translation.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.translation.rst.txt
new file mode 100644
index 000000000..55c133366
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.translation.rst.txt
@@ -0,0 +1,6 @@
+pypose.translation
+==================
+
+.. currentmodule:: pypose
+
+.. autofunction:: translation
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.utils.ReduceToBason.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.utils.ReduceToBason.rst.txt
new file mode 100644
index 000000000..3eb480183
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.utils.ReduceToBason.rst.txt
@@ -0,0 +1,8 @@
+.. currentmodule:: pypose.utils
+
+
+pypose.utils.ReduceToBason
+==========================
+
+.. autoclass:: ReduceToBason
+    :members:
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/generated/pypose.vec2skew.rst.txt b/docs/preview/pr/352/_sources/generated/pypose.vec2skew.rst.txt
new file mode 100644
index 000000000..bac9af920
--- /dev/null
+++ b/docs/preview/pr/352/_sources/generated/pypose.vec2skew.rst.txt
@@ -0,0 +1,6 @@
+pypose.vec2skew
+===============
+
+.. currentmodule:: pypose
+
+.. autofunction:: vec2skew
\ No newline at end of file
diff --git a/docs/preview/pr/352/_sources/index.rst.txt b/docs/preview/pr/352/_sources/index.rst.txt
new file mode 100644
index 000000000..320b09669
--- /dev/null
+++ b/docs/preview/pr/352/_sources/index.rst.txt
@@ -0,0 +1,25 @@
+.. PyPose documentation master file
+
+PyPose Documentation
+==================================
+
+.. toctree::
+   :maxdepth: 2
+   :caption: Contents:
+
+   lietensor
+   func
+   functions
+   convert
+   modules
+   optim
+   utils
+   testing
+
+
+Indices and tables
+==================
+
+* :ref:`genindex`
+* :ref:`modindex`
+* :ref:`search`
diff --git a/docs/preview/pr/352/_sources/lietensor.rst.txt b/docs/preview/pr/352/_sources/lietensor.rst.txt
new file mode 100644
index 000000000..240b21ae8
--- /dev/null
+++ b/docs/preview/pr/352/_sources/lietensor.rst.txt
@@ -0,0 +1,22 @@
+LieTensor
+=========
+
+.. currentmodule:: pypose
+
+.. autosummary::
+   :toctree: generated
+   :nosignatures:
+
+   :template: autosummary/class-no-inherit.rst
+   LieTensor
+   Parameter
+
+   :template: autosummary/base.rst
+   SO3
+   so3
+   SE3
+   se3
+   Sim3
+   sim3
+   RxSO3
+   rxso3
diff --git a/docs/preview/pr/352/_sources/modules.rst.txt b/docs/preview/pr/352/_sources/modules.rst.txt
new file mode 100644
index 000000000..217f8492b
--- /dev/null
+++ b/docs/preview/pr/352/_sources/modules.rst.txt
@@ -0,0 +1,21 @@
+Modules
+========
+
+.. currentmodule:: pypose
+.. autosummary::
+   :toctree: generated
+   :nosignatures:
+
+   :template: autosummary/class-no-inherit.rst
+   module.System
+   module.LTI
+   module.LTV
+   module.NLS
+   module.EKF
+   module.UKF
+   module.PF
+   module.LQR
+   module.IMUPreintegrator
+   module.EPnP
+   module.MPC
+   module.ICP
diff --git a/docs/preview/pr/352/_sources/optim.rst.txt b/docs/preview/pr/352/_sources/optim.rst.txt
new file mode 100644
index 000000000..84db24984
--- /dev/null
+++ b/docs/preview/pr/352/_sources/optim.rst.txt
@@ -0,0 +1,31 @@
+Optimization
+============
+
+.. currentmodule:: pypose
+.. autosummary::
+   :toctree: generated
+   :nosignatures:
+
+   :template: autosummary/class-no-inherit.rst
+   optim.GaussNewton
+   optim.GN
+   optim.LevenbergMarquardt
+   optim.LM
+   optim.solver.PINV
+   optim.solver.LSTSQ
+   optim.solver.Cholesky
+   optim.solver.CG
+   optim.strategy.Constant
+   optim.strategy.Adaptive
+   optim.strategy.TrustRegion
+   optim.kernel.Scale
+   optim.kernel.Huber
+   optim.kernel.PseudoHuber
+   optim.kernel.SoftLOne
+   optim.kernel.Cauchy
+   optim.kernel.Arctan
+   optim.kernel.Tolerant
+   optim.corrector.FastTriggs
+   optim.corrector.Triggs
+   optim.scheduler.StopOnPlateau
+   optim.functional.modjac
diff --git a/docs/preview/pr/352/_sources/testing.rst.txt b/docs/preview/pr/352/_sources/testing.rst.txt
new file mode 100644
index 000000000..eae182bc9
--- /dev/null
+++ b/docs/preview/pr/352/_sources/testing.rst.txt
@@ -0,0 +1,11 @@
+Testing
+=========
+
+
+.. currentmodule:: pypose
+.. autosummary::
+   :toctree: generated
+   :nosignatures:
+
+   :template: autosummary/class-no-inherit.rst
+   testing.assert_close
diff --git a/docs/preview/pr/352/_sources/utils.rst.txt b/docs/preview/pr/352/_sources/utils.rst.txt
new file mode 100644
index 000000000..91d210b79
--- /dev/null
+++ b/docs/preview/pr/352/_sources/utils.rst.txt
@@ -0,0 +1,10 @@
+Utils
+========
+
+.. currentmodule:: pypose
+.. autosummary::
+   :toctree: generated
+   :nosignatures:
+
+   :template: autosummary/class-no-inherit.rst
+   utils.ReduceToBason
diff --git a/docs/preview/pr/352/_static/auto-render.min.js b/docs/preview/pr/352/_static/auto-render.min.js
new file mode 100644
index 000000000..46d62af28
--- /dev/null
+++ b/docs/preview/pr/352/_static/auto-render.min.js
@@ -0,0 +1 @@
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new file mode 100644
index 000000000..f348fdca8
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@@ -0,0 +1,927 @@
+/*
+ * basic.css
+ * ~~~~~~~~~
+ *
+ * Sphinx stylesheet -- basic theme.
+ *
+ * :copyright: Copyright 2007-2022 by the Sphinx team, see AUTHORS.
+ * :license: BSD, see LICENSE for details.
+ *
+ */
+
+/* -- main layout ----------------------------------------------------------- */
+
+div.clearer {
+    clear: both;
+}
+
+div.section::after {
+    display: block;
+    content: '';
+    clear: left;
+}
+
+/* -- relbar ---------------------------------------------------------------- */
+
+div.related {
+    width: 100%;
+    font-size: 90%;
+}
+
+div.related h3 {
+    display: none;
+}
+
+div.related ul {
+    margin: 0;
+    padding: 0 0 0 10px;
+    list-style: none;
+}
+
+div.related li {
+    display: inline;
+}
+
+div.related li.right {
+    float: right;
+    margin-right: 5px;
+}
+
+/* -- sidebar --------------------------------------------------------------- */
+
+div.sphinxsidebarwrapper {
+    padding: 10px 5px 0 10px;
+}
+
+div.sphinxsidebar {
+    float: left;
+    width: 230px;
+    margin-left: -100%;
+    font-size: 90%;
+    word-wrap: break-word;
+    overflow-wrap : break-word;
+}
+
+div.sphinxsidebar ul {
+    list-style: none;
+}
+
+div.sphinxsidebar ul ul,
+div.sphinxsidebar ul.want-points {
+    margin-left: 20px;
+    list-style: square;
+}
+
+div.sphinxsidebar ul ul {
+    margin-top: 0;
+    margin-bottom: 0;
+}
+
+div.sphinxsidebar form {
+    margin-top: 10px;
+}
+
+div.sphinxsidebar input {
+    border: 1px solid #98dbcc;
+    font-family: sans-serif;
+    font-size: 1em;
+}
+
+div.sphinxsidebar #searchbox form.search {
+    overflow: hidden;
+}
+
+div.sphinxsidebar #searchbox input[type="text"] {
+    float: left;
+    width: 80%;
+    padding: 0.25em;
+    box-sizing: border-box;
+}
+
+div.sphinxsidebar #searchbox input[type="submit"] {
+    float: left;
+    width: 20%;
+    border-left: none;
+    padding: 0.25em;
+    box-sizing: border-box;
+}
+
+
+img {
+    border: 0;
+    max-width: 100%;
+}
+
+/* -- search page ----------------------------------------------------------- */
+
+ul.search {
+    margin: 10px 0 0 20px;
+    padding: 0;
+}
+
+ul.search li {
+    padding: 5px 0 5px 20px;
+    background-image: url(file.png);
+    background-repeat: no-repeat;
+    background-position: 0 7px;
+}
+
+ul.search li a {
+    font-weight: bold;
+}
+
+ul.search li p.context {
+    color: #888;
+    margin: 2px 0 0 30px;
+    text-align: left;
+}
+
+ul.keywordmatches li.goodmatch a {
+    font-weight: bold;
+}
+
+/* -- index page ------------------------------------------------------------ */
+
+table.contentstable {
+    width: 90%;
+    margin-left: auto;
+    margin-right: auto;
+}
+
+table.contentstable p.biglink {
+    line-height: 150%;
+}
+
+a.biglink {
+    font-size: 1.3em;
+}
+
+span.linkdescr {
+    font-style: italic;
+    padding-top: 5px;
+    font-size: 90%;
+}
+
+/* -- general index --------------------------------------------------------- */
+
+table.indextable {
+    width: 100%;
+}
+
+table.indextable td {
+    text-align: left;
+    vertical-align: top;
+}
+
+table.indextable ul {
+    margin-top: 0;
+    margin-bottom: 0;
+    list-style-type: none;
+}
+
+table.indextable > tbody > tr > td > ul {
+    padding-left: 0em;
+}
+
+table.indextable tr.pcap {
+    height: 10px;
+}
+
+table.indextable tr.cap {
+    margin-top: 10px;
+    background-color: #f2f2f2;
+}
+
+img.toggler {
+    margin-right: 3px;
+    margin-top: 3px;
+    cursor: pointer;
+}
+
+div.modindex-jumpbox {
+    border-top: 1px solid #ddd;
+    border-bottom: 1px solid #ddd;
+    margin: 1em 0 1em 0;
+    padding: 0.4em;
+}
+
+div.genindex-jumpbox {
+    border-top: 1px solid #ddd;
+    border-bottom: 1px solid #ddd;
+    margin: 1em 0 1em 0;
+    padding: 0.4em;
+}
+
+/* -- domain module index --------------------------------------------------- */
+
+table.modindextable td {
+    padding: 2px;
+    border-collapse: collapse;
+}
+
+/* -- general body styles --------------------------------------------------- */
+
+div.body {
+    min-width: 360px;
+    max-width: 800px;
+}
+
+div.body p, div.body dd, div.body li, div.body blockquote {
+    -ms-hyphens: auto;
+    -webkit-hyphens: auto;
+    hyphens: auto;
+}
+
+a.headerlink {
+    visibility: hidden;
+}
+
+a.brackets:before,
+span.brackets > a:before{
+    content: "[";
+}
+
+a.brackets:after,
+span.brackets > a:after {
+    content: "]";
+}
+
+h1:hover > a.headerlink,
+h2:hover > a.headerlink,
+h3:hover > a.headerlink,
+h4:hover > a.headerlink,
+h5:hover > a.headerlink,
+h6:hover > a.headerlink,
+dt:hover > a.headerlink,
+caption:hover > a.headerlink,
+p.caption:hover > a.headerlink,
+div.code-block-caption:hover > a.headerlink {
+    visibility: visible;
+}
+
+div.body p.caption {
+    text-align: inherit;
+}
+
+div.body td {
+    text-align: left;
+}
+
+.first {
+    margin-top: 0 !important;
+}
+
+p.rubric {
+    margin-top: 30px;
+    font-weight: bold;
+}
+
+img.align-left, figure.align-left, .figure.align-left, object.align-left {
+    clear: left;
+    float: left;
+    margin-right: 1em;
+}
+
+img.align-right, figure.align-right, .figure.align-right, object.align-right {
+    clear: right;
+    float: right;
+    margin-left: 1em;
+}
+
+img.align-center, figure.align-center, .figure.align-center, object.align-center {
+  display: block;
+  margin-left: auto;
+  margin-right: auto;
+}
+
+img.align-default, figure.align-default, .figure.align-default {
+  display: block;
+  margin-left: auto;
+  margin-right: auto;
+}
+
+.align-left {
+    text-align: left;
+}
+
+.align-center {
+    text-align: center;
+}
+
+.align-default {
+    text-align: center;
+}
+
+.align-right {
+    text-align: right;
+}
+
+/* -- sidebars -------------------------------------------------------------- */
+
+div.sidebar,
+aside.sidebar {
+    margin: 0 0 0.5em 1em;
+    border: 1px solid #ddb;
+    padding: 7px;
+    background-color: #ffe;
+    width: 40%;
+    float: right;
+    clear: right;
+    overflow-x: auto;
+}
+
+p.sidebar-title {
+    font-weight: bold;
+}
+
+div.admonition, div.topic, blockquote {
+    clear: left;
+}
+
+/* -- topics ---------------------------------------------------------------- */
+
+div.topic {
+    border: 1px solid #ccc;
+    padding: 7px;
+    margin: 10px 0 10px 0;
+}
+
+p.topic-title {
+    font-size: 1.1em;
+    font-weight: bold;
+    margin-top: 10px;
+}
+
+/* -- admonitions ----------------------------------------------------------- */
+
+div.admonition {
+    margin-top: 10px;
+    margin-bottom: 10px;
+    padding: 7px;
+}
+
+div.admonition dt {
+    font-weight: bold;
+}
+
+p.admonition-title {
+    margin: 0px 10px 5px 0px;
+    font-weight: bold;
+}
+
+div.body p.centered {
+    text-align: center;
+    margin-top: 25px;
+}
+
+/* -- content of sidebars/topics/admonitions -------------------------------- */
+
+div.sidebar > :last-child,
+aside.sidebar > :last-child,
+div.topic > :last-child,
+div.admonition > :last-child {
+    margin-bottom: 0;
+}
+
+div.sidebar::after,
+aside.sidebar::after,
+div.topic::after,
+div.admonition::after,
+blockquote::after {
+    display: block;
+    content: '';
+    clear: both;
+}
+
+/* -- tables ---------------------------------------------------------------- */
+
+table.docutils {
+    margin-top: 10px;
+    margin-bottom: 10px;
+    border: 0;
+    border-collapse: collapse;
+}
+
+table.align-center {
+    margin-left: auto;
+    margin-right: auto;
+}
+
+table.align-default {
+    margin-left: auto;
+    margin-right: auto;
+}
+
+table caption span.caption-number {
+    font-style: italic;
+}
+
+table caption span.caption-text {
+}
+
+table.docutils td, table.docutils th {
+    padding: 1px 8px 1px 5px;
+    border-top: 0;
+    border-left: 0;
+    border-right: 0;
+    border-bottom: 1px solid #aaa;
+}
+
+th {
+    text-align: left;
+    padding-right: 5px;
+}
+
+table.citation {
+    border-left: solid 1px gray;
+    margin-left: 1px;
+}
+
+table.citation td {
+    border-bottom: none;
+}
+
+th > :first-child,
+td > :first-child {
+    margin-top: 0px;
+}
+
+th > :last-child,
+td > :last-child {
+    margin-bottom: 0px;
+}
+
+/* -- figures --------------------------------------------------------------- */
+
+div.figure, figure {
+    margin: 0.5em;
+    padding: 0.5em;
+}
+
+div.figure p.caption, figcaption {
+    padding: 0.3em;
+}
+
+div.figure p.caption span.caption-number,
+figcaption span.caption-number {
+    font-style: italic;
+}
+
+div.figure p.caption span.caption-text,
+figcaption span.caption-text {
+}
+
+/* -- field list styles ----------------------------------------------------- */
+
+table.field-list td, table.field-list th {
+    border: 0 !important;
+}
+
+.field-list ul {
+    margin: 0;
+    padding-left: 1em;
+}
+
+.field-list p {
+    margin: 0;
+}
+
+.field-name {
+    -ms-hyphens: manual;
+    -webkit-hyphens: manual;
+    hyphens: manual;
+}
+
+/* -- hlist styles ---------------------------------------------------------- */
+
+table.hlist {
+    margin: 1em 0;
+}
+
+table.hlist td {
+    vertical-align: top;
+}
+
+/* -- object description styles --------------------------------------------- */
+
+.sig {
+	font-family: 'Consolas', 'Menlo', 'DejaVu Sans Mono', 'Bitstream Vera Sans Mono', monospace;
+}
+
+.sig-name, code.descname {
+    background-color: transparent;
+    font-weight: bold;
+}
+
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+	font-size: 1.1em;
+}
+
+code.descname {
+    font-size: 1.2em;
+}
+
+.sig-prename, code.descclassname {
+    background-color: transparent;
+}
+
+.optional {
+    font-size: 1.3em;
+}
+
+.sig-paren {
+    font-size: larger;
+}
+
+.sig-param.n {
+	font-style: italic;
+}
+
+/* C++ specific styling */
+
+.sig-inline.c-texpr,
+.sig-inline.cpp-texpr {
+	font-family: unset;
+}
+
+.sig.c   .k, .sig.c   .kt,
+.sig.cpp .k, .sig.cpp .kt {
+	color: #0033B3;
+}
+
+.sig.c   .m,
+.sig.cpp .m {
+	color: #1750EB;
+}
+
+.sig.c   .s, .sig.c   .sc,
+.sig.cpp .s, .sig.cpp .sc {
+	color: #067D17;
+}
+
+
+/* -- other body styles ----------------------------------------------------- */
+
+ol.arabic {
+    list-style: decimal;
+}
+
+ol.loweralpha {
+    list-style: lower-alpha;
+}
+
+ol.upperalpha {
+    list-style: upper-alpha;
+}
+
+ol.lowerroman {
+    list-style: lower-roman;
+}
+
+ol.upperroman {
+    list-style: upper-roman;
+}
+
+:not(li) > ol > li:first-child > :first-child,
+:not(li) > ul > li:first-child > :first-child {
+    margin-top: 0px;
+}
+
+:not(li) > ol > li:last-child > :last-child,
+:not(li) > ul > li:last-child > :last-child {
+    margin-bottom: 0px;
+}
+
+ol.simple ol p,
+ol.simple ul p,
+ul.simple ol p,
+ul.simple ul p {
+    margin-top: 0;
+}
+
+ol.simple > li:not(:first-child) > p,
+ul.simple > li:not(:first-child) > p {
+    margin-top: 0;
+}
+
+ol.simple p,
+ul.simple p {
+    margin-bottom: 0;
+}
+
+/* Docutils 0.17 and older (footnotes & citations) */
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+dl.citation > dt {
+    float: left;
+    margin-right: 0.5em;
+}
+
+dl.footnote > dd,
+dl.citation > dd {
+    margin-bottom: 0em;
+}
+
+dl.footnote > dd:after,
+dl.citation > dd:after {
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+    clear: both;
+}
+
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+div.citation > span {
+    float: left;
+}
+aside.footnote > span:last-of-type,
+div.citation > span:last-of-type {
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+}
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+}
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+}
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+div.citation > p:last-of-type {
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+}
+aside.footnote > p:last-of-type:after,
+div.citation > p:last-of-type:after {
+    content: "";
+    clear: both;
+}
+
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+
+dl.field-list {
+    display: grid;
+    grid-template-columns: fit-content(30%) auto;
+}
+
+dl.field-list > dt {
+    font-weight: bold;
+    word-break: break-word;
+    padding-left: 0.5em;
+    padding-right: 5px;
+}
+
+dl.field-list > dt:after {
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+}
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+    margin-top: 0em;
+    margin-left: 0em;
+    margin-bottom: 0em;
+}
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+}
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+dd > :first-child {
+    margin-top: 0px;
+}
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+dd ul, dd table {
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+}
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+    margin-bottom: 10px;
+    margin-left: 30px;
+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+    margin-bottom: 1em;
+}
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+}
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+}
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+}
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+}
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+}
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+    cursor: help;
+}
+
+/* -- code displays --------------------------------------------------------- */
+
+pre {
+    overflow: auto;
+    overflow-y: hidden;  /* fixes display issues on Chrome browsers */
+}
+
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+    clear: both;
+}
+
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+    -webkit-hyphens: none;
+    hyphens: none;
+    white-space: nowrap;
+}
+
+div[class*="highlight-"] {
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+}
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+    color: #aaa;
+}
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+}
+
+table.highlighttable tbody {
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+}
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+}
+
+table.highlighttable td {
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+    padding: 0;
+}
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+}
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+    overflow: hidden;
+}
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+}
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+div.highlight pre,
+table.highlighttable pre {
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+}
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+div.code-block-caption + div {
+    margin-top: 0;
+}
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+    padding: 2px 5px;
+    font-size: small;
+}
+
+div.code-block-caption code {
+    background-color: transparent;
+}
+
+table.highlighttable td.linenos,
+span.linenos,
+div.highlight span.gp {  /* gp: Generic.Prompt */
+  user-select: none;
+  -webkit-user-select: text; /* Safari fallback only */
+  -webkit-user-select: none; /* Chrome/Safari */
+  -moz-user-select: none; /* Firefox */
+  -ms-user-select: none; /* IE10+ */
+}
+
+div.code-block-caption span.caption-number {
+    padding: 0.1em 0.3em;
+    font-style: italic;
+}
+
+div.code-block-caption span.caption-text {
+}
+
+div.literal-block-wrapper {
+    margin: 1em 0;
+}
+
+code.xref, a code {
+    background-color: transparent;
+    font-weight: bold;
+}
+
+h1 code, h2 code, h3 code, h4 code, h5 code, h6 code {
+    background-color: transparent;
+}
+
+.viewcode-link {
+    float: right;
+}
+
+.viewcode-back {
+    float: right;
+    font-family: sans-serif;
+}
+
+div.viewcode-block:target {
+    margin: -1px -10px;
+    padding: 0 10px;
+}
+
+/* -- math display ---------------------------------------------------------- */
+
+img.math {
+    vertical-align: middle;
+}
+
+div.body div.math p {
+    text-align: center;
+}
+
+span.eqno {
+    float: right;
+}
+
+span.eqno a.headerlink {
+    position: absolute;
+    z-index: 1;
+}
+
+div.math:hover a.headerlink {
+    visibility: visible;
+}
+
+/* -- printout stylesheet --------------------------------------------------- */
+
+@media print {
+    div.document,
+    div.documentwrapper,
+    div.bodywrapper {
+        margin: 0 !important;
+        width: 100%;
+    }
+
+    div.sphinxsidebar,
+    div.related,
+    div.footer,
+    #top-link {
+        display: none;
+    }
+}
\ No newline at end of file
diff --git a/docs/preview/pr/352/_static/css/theme.css b/docs/preview/pr/352/_static/css/theme.css
new file mode 100644
index 000000000..6f3330389
--- /dev/null
+++ b/docs/preview/pr/352/_static/css/theme.css
@@ -0,0 +1,11482 @@
+@charset "UTF-8";
+/*!
+ * Bootstrap v4.0.0 (https://getbootstrap.com)
+ * Copyright 2011-2018 The Bootstrap Authors
+ * Copyright 2011-2018 Twitter, Inc.
+ * Licensed under MIT (https://github.com/twbs/bootstrap/blob/master/LICENSE)
+ */
+:root {
+  --blue: #007bff;
+  --indigo: #6610f2;
+  --purple: #6f42c1;
+  --pink: #e83e8c;
+  --red: #dc3545;
+  --orange: #fd7e14;
+  --yellow: #ffc107;
+  --green: #28a745;
+  --teal: #20c997;
+  --cyan: #17a2b8;
+  --white: #fff;
+  --gray: #6c757d;
+  --gray-dark: #343a40;
+  --primary: #007bff;
+  --secondary: #6c757d;
+  --success: #28a745;
+  --info: #17a2b8;
+  --warning: #ffc107;
+  --danger: #dc3545;
+  --light: #f8f9fa;
+  --dark: #343a40;
+  --breakpoint-xs: 0;
+  --breakpoint-sm: 576px;
+  --breakpoint-md: 768px;
+  --breakpoint-lg: 992px;
+  --breakpoint-xl: 1200px;
+  --font-family-sans-serif: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, "Helvetica Neue", Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol";
+  --font-family-monospace: SFMono-Regular, Menlo, Monaco, Consolas, "Liberation Mono", "Courier New", monospace;
+}
+
+*,
+*::before,
+*::after {
+  box-sizing: border-box;
+}
+
+html {
+  font-family: sans-serif;
+  line-height: 1.15;
+  -webkit-text-size-adjust: 100%;
+  -ms-text-size-adjust: 100%;
+  -ms-overflow-style: scrollbar;
+  -webkit-tap-highlight-color: rgba(0, 0, 0, 0);
+}
+
+@-ms-viewport {
+  width: device-width;
+}
+article, aside, dialog, figcaption, figure, footer, header, hgroup, main, nav, section {
+  display: block;
+}
+
+body {
+  margin: 0;
+  font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, "Helvetica Neue", Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol";
+  font-size: 1rem;
+  font-weight: 400;
+  line-height: 1.5;
+  color: #212529;
+  text-align: left;
+  background-color: #fff;
+}
+
+[tabindex="-1"]:focus {
+  outline: 0 !important;
+}
+
+hr {
+  box-sizing: content-box;
+  height: 0;
+  overflow: visible;
+}
+
+h1, h2, h3, h4, h5, h6 {
+  margin-top: 0;
+  margin-bottom: 0.5rem;
+}
+
+p {
+  margin-top: 0;
+  margin-bottom: 1rem;
+}
+
+abbr[title],
+abbr[data-original-title] {
+  text-decoration: underline;
+  -webkit-text-decoration: underline dotted;
+          text-decoration: underline dotted;
+  cursor: help;
+  border-bottom: 0;
+}
+
+address {
+  margin-bottom: 1rem;
+  font-style: normal;
+  line-height: inherit;
+}
+
+ol,
+ul,
+dl {
+  margin-top: 0;
+  margin-bottom: 1rem;
+}
+
+ol ol,
+ul ul,
+ol ul,
+ul ol {
+  margin-bottom: 0;
+}
+
+dt {
+  font-weight: 700;
+}
+
+dd {
+  margin-bottom: .5rem;
+  margin-left: 0;
+}
+
+blockquote {
+  margin: 0 0 1rem;
+}
+
+dfn {
+  font-style: italic;
+}
+
+b,
+strong {
+  font-weight: bolder;
+}
+
+small {
+  font-size: 80%;
+}
+
+sub,
+sup {
+  position: relative;
+  font-size: 75%;
+  line-height: 0;
+  vertical-align: baseline;
+}
+
+sub {
+  bottom: -.25em;
+}
+
+sup {
+  top: -.5em;
+}
+
+a {
+  color: #007bff;
+  text-decoration: none;
+  background-color: transparent;
+  -webkit-text-decoration-skip: objects;
+}
+a:hover {
+  color: #0056b3;
+  text-decoration: underline;
+}
+
+a:not([href]):not([tabindex]) {
+  color: inherit;
+  text-decoration: none;
+}
+a:not([href]):not([tabindex]):hover, a:not([href]):not([tabindex]):focus {
+  color: inherit;
+  text-decoration: none;
+}
+a:not([href]):not([tabindex]):focus {
+  outline: 0;
+}
+
+pre,
+code,
+kbd,
+samp {
+  font-family: monospace, monospace;
+  font-size: 1em;
+}
+
+pre {
+  margin-top: 0;
+  margin-bottom: 1rem;
+  overflow: auto;
+  -ms-overflow-style: scrollbar;
+}
+
+figure {
+  margin: 0 0 1rem;
+}
+
+img {
+  vertical-align: middle;
+  border-style: none;
+}
+
+svg:not(:root) {
+  overflow: hidden;
+}
+
+table {
+  border-collapse: collapse;
+}
+
+caption {
+  padding-top: 0.75rem;
+  padding-bottom: 0.75rem;
+  color: #6c757d;
+  text-align: left;
+  caption-side: bottom;
+}
+
+th {
+  text-align: inherit;
+}
+
+label {
+  display: inline-block;
+  margin-bottom: .5rem;
+}
+
+button {
+  border-radius: 0;
+}
+
+button:focus {
+  outline: 1px dotted;
+  outline: 5px auto -webkit-focus-ring-color;
+}
+
+input,
+button,
+select,
+optgroup,
+textarea {
+  margin: 0;
+  font-family: inherit;
+  font-size: inherit;
+  line-height: inherit;
+}
+
+button,
+input {
+  overflow: visible;
+}
+
+button,
+select {
+  text-transform: none;
+}
+
+button,
+html [type="button"],
+[type="reset"],
+[type="submit"] {
+  -webkit-appearance: button;
+}
+
+button::-moz-focus-inner,
+[type="button"]::-moz-focus-inner,
+[type="reset"]::-moz-focus-inner,
+[type="submit"]::-moz-focus-inner {
+  padding: 0;
+  border-style: none;
+}
+
+input[type="radio"],
+input[type="checkbox"] {
+  box-sizing: border-box;
+  padding: 0;
+}
+
+input[type="date"],
+input[type="time"],
+input[type="datetime-local"],
+input[type="month"] {
+  -webkit-appearance: listbox;
+}
+
+textarea {
+  overflow: auto;
+  resize: vertical;
+}
+
+fieldset {
+  min-width: 0;
+  padding: 0;
+  margin: 0;
+  border: 0;
+}
+
+legend {
+  display: block;
+  width: 100%;
+  max-width: 100%;
+  padding: 0;
+  margin-bottom: .5rem;
+  font-size: 1.5rem;
+  line-height: inherit;
+  color: inherit;
+  white-space: normal;
+}
+
+progress {
+  vertical-align: baseline;
+}
+
+[type="number"]::-webkit-inner-spin-button,
+[type="number"]::-webkit-outer-spin-button {
+  height: auto;
+}
+
+[type="search"] {
+  outline-offset: -2px;
+  -webkit-appearance: none;
+}
+
+[type="search"]::-webkit-search-cancel-button,
+[type="search"]::-webkit-search-decoration {
+  -webkit-appearance: none;
+}
+
+::-webkit-file-upload-button {
+  font: inherit;
+  -webkit-appearance: button;
+}
+
+output {
+  display: inline-block;
+}
+
+summary {
+  display: list-item;
+  cursor: pointer;
+}
+
+template {
+  display: none;
+}
+
+[hidden] {
+  display: none !important;
+}
+
+h1, h2, h3, h4, h5, h6,
+.h1, .h2, .h3, .h4, .h5, .h6 {
+  margin-bottom: 0.5rem;
+  font-family: inherit;
+  font-weight: 500;
+  line-height: 1.2;
+  color: inherit;
+}
+
+h1, .h1 {
+  font-size: 2.5rem;
+}
+
+h2, .h2 {
+  font-size: 2rem;
+}
+
+h3, .h3 {
+  font-size: 1.75rem;
+}
+
+h4, .h4 {
+  font-size: 1.5rem;
+}
+
+h5, .h5 {
+  font-size: 1.25rem;
+}
+
+h6, .h6 {
+  font-size: 1rem;
+}
+
+.lead {
+  font-size: 1.25rem;
+  font-weight: 300;
+}
+
+.display-1 {
+  font-size: 6rem;
+  font-weight: 300;
+  line-height: 1.2;
+}
+
+.display-2 {
+  font-size: 5.5rem;
+  font-weight: 300;
+  line-height: 1.2;
+}
+
+.display-3 {
+  font-size: 4.5rem;
+  font-weight: 300;
+  line-height: 1.2;
+}
+
+.display-4 {
+  font-size: 3.5rem;
+  font-weight: 300;
+  line-height: 1.2;
+}
+
+hr {
+  margin-top: 1rem;
+  margin-bottom: 1rem;
+  border: 0;
+  border-top: 1px solid rgba(0, 0, 0, 0.1);
+}
+
+small,
+.small {
+  font-size: 80%;
+  font-weight: 400;
+}
+
+mark,
+.mark {
+  padding: 0.2em;
+  background-color: #fcf8e3;
+}
+
+.list-unstyled {
+  padding-left: 0;
+  list-style: none;
+}
+
+.list-inline {
+  padding-left: 0;
+  list-style: none;
+}
+
+.list-inline-item {
+  display: inline-block;
+}
+.list-inline-item:not(:last-child) {
+  margin-right: 0.5rem;
+}
+
+.initialism {
+  font-size: 90%;
+  text-transform: uppercase;
+}
+
+.blockquote {
+  margin-bottom: 1rem;
+  font-size: 1.25rem;
+}
+
+.blockquote-footer {
+  display: block;
+  font-size: 80%;
+  color: #6c757d;
+}
+.blockquote-footer::before {
+  content: "\2014 \00A0";
+}
+
+.img-fluid {
+  max-width: 100%;
+  height: auto;
+}
+
+.img-thumbnail {
+  padding: 0.25rem;
+  background-color: #fff;
+  border: 1px solid #dee2e6;
+  border-radius: 0.25rem;
+  max-width: 100%;
+  height: auto;
+}
+
+.figure {
+  display: inline-block;
+}
+
+.figure-img {
+  margin-bottom: 0.5rem;
+  line-height: 1;
+}
+
+.figure-caption {
+  font-size: 90%;
+  color: #6c757d;
+}
+
+code,
+kbd,
+pre,
+samp {
+  font-family: SFMono-Regular, Menlo, Monaco, Consolas, "Liberation Mono", "Courier New", monospace;
+}
+
+code {
+  font-size: 87.5%;
+  color: #e83e8c;
+  word-break: break-word;
+}
+a > code {
+  color: inherit;
+}
+
+kbd {
+  padding: 0.2rem 0.4rem;
+  font-size: 87.5%;
+  color: #fff;
+  background-color: #212529;
+  border-radius: 0.2rem;
+}
+kbd kbd {
+  padding: 0;
+  font-size: 100%;
+  font-weight: 700;
+}
+
+pre {
+  display: block;
+  font-size: 87.5%;
+  color: #212529;
+}
+pre code {
+  font-size: inherit;
+  color: inherit;
+  word-break: normal;
+}
+
+.pre-scrollable {
+  max-height: 340px;
+  overflow-y: scroll;
+}
+
+.container {
+  width: 100%;
+  padding-right: 15px;
+  padding-left: 15px;
+  margin-right: auto;
+  margin-left: auto;
+}
+@media (min-width: 576px) {
+  .container {
+    max-width: 540px;
+  }
+}
+@media (min-width: 768px) {
+  .container {
+    max-width: 720px;
+  }
+}
+@media (min-width: 992px) {
+  .container {
+    max-width: 960px;
+  }
+}
+@media (min-width: 1200px) {
+  .container {
+    max-width: 1140px;
+  }
+}
+
+.container-fluid {
+  width: 100%;
+  padding-right: 15px;
+  padding-left: 15px;
+  margin-right: auto;
+  margin-left: auto;
+}
+
+.row {
+  display: flex;
+  flex-wrap: wrap;
+  margin-right: -15px;
+  margin-left: -15px;
+}
+
+.no-gutters {
+  margin-right: 0;
+  margin-left: 0;
+}
+.no-gutters > .col,
+.no-gutters > [class*="col-"] {
+  padding-right: 0;
+  padding-left: 0;
+}
+
+.col-1, .col-2, .col-3, .col-4, .col-5, .col-6, .col-7, .col-8, .col-9, .col-10, .col-11, .col-12, .col,
+.col-auto, .col-sm-1, .col-sm-2, .col-sm-3, .col-sm-4, .col-sm-5, .col-sm-6, .col-sm-7, .col-sm-8, .col-sm-9, .col-sm-10, .col-sm-11, .col-sm-12, .col-sm,
+.col-sm-auto, .col-md-1, .col-md-2, .col-md-3, .col-md-4, .col-md-5, .col-md-6, .col-md-7, .col-md-8, .col-md-9, .col-md-10, .col-md-11, .col-md-12, .col-md,
+.col-md-auto, .col-lg-1, .col-lg-2, .col-lg-3, .col-lg-4, .col-lg-5, .col-lg-6, .col-lg-7, .col-lg-8, .col-lg-9, .col-lg-10, .col-lg-11, .col-lg-12, .col-lg,
+.col-lg-auto, .col-xl-1, .col-xl-2, .col-xl-3, .col-xl-4, .col-xl-5, .col-xl-6, .col-xl-7, .col-xl-8, .col-xl-9, .col-xl-10, .col-xl-11, .col-xl-12, .col-xl,
+.col-xl-auto {
+  position: relative;
+  width: 100%;
+  min-height: 1px;
+  padding-right: 15px;
+  padding-left: 15px;
+}
+
+.col {
+  flex-basis: 0;
+  flex-grow: 1;
+  max-width: 100%;
+}
+
+.col-auto {
+  flex: 0 0 auto;
+  width: auto;
+  max-width: none;
+}
+
+.col-1 {
+  flex: 0 0 8.3333333333%;
+  max-width: 8.3333333333%;
+}
+
+.col-2 {
+  flex: 0 0 16.6666666667%;
+  max-width: 16.6666666667%;
+}
+
+.col-3 {
+  flex: 0 0 25%;
+  max-width: 25%;
+}
+
+.col-4 {
+  flex: 0 0 33.3333333333%;
+  max-width: 33.3333333333%;
+}
+
+.col-5 {
+  flex: 0 0 41.6666666667%;
+  max-width: 41.6666666667%;
+}
+
+.col-6 {
+  flex: 0 0 50%;
+  max-width: 50%;
+}
+
+.col-7 {
+  flex: 0 0 58.3333333333%;
+  max-width: 58.3333333333%;
+}
+
+.col-8 {
+  flex: 0 0 66.6666666667%;
+  max-width: 66.6666666667%;
+}
+
+.col-9 {
+  flex: 0 0 75%;
+  max-width: 75%;
+}
+
+.col-10 {
+  flex: 0 0 83.3333333333%;
+  max-width: 83.3333333333%;
+}
+
+.col-11 {
+  flex: 0 0 91.6666666667%;
+  max-width: 91.6666666667%;
+}
+
+.col-12 {
+  flex: 0 0 100%;
+  max-width: 100%;
+}
+
+.order-first {
+  order: -1;
+}
+
+.order-last {
+  order: 13;
+}
+
+.order-0 {
+  order: 0;
+}
+
+.order-1 {
+  order: 1;
+}
+
+.order-2 {
+  order: 2;
+}
+
+.order-3 {
+  order: 3;
+}
+
+.order-4 {
+  order: 4;
+}
+
+.order-5 {
+  order: 5;
+}
+
+.order-6 {
+  order: 6;
+}
+
+.order-7 {
+  order: 7;
+}
+
+.order-8 {
+  order: 8;
+}
+
+.order-9 {
+  order: 9;
+}
+
+.order-10 {
+  order: 10;
+}
+
+.order-11 {
+  order: 11;
+}
+
+.order-12 {
+  order: 12;
+}
+
+.offset-1 {
+  margin-left: 8.3333333333%;
+}
+
+.offset-2 {
+  margin-left: 16.6666666667%;
+}
+
+.offset-3 {
+  margin-left: 25%;
+}
+
+.offset-4 {
+  margin-left: 33.3333333333%;
+}
+
+.offset-5 {
+  margin-left: 41.6666666667%;
+}
+
+.offset-6 {
+  margin-left: 50%;
+}
+
+.offset-7 {
+  margin-left: 58.3333333333%;
+}
+
+.offset-8 {
+  margin-left: 66.6666666667%;
+}
+
+.offset-9 {
+  margin-left: 75%;
+}
+
+.offset-10 {
+  margin-left: 83.3333333333%;
+}
+
+.offset-11 {
+  margin-left: 91.6666666667%;
+}
+
+@media (min-width: 576px) {
+  .col-sm {
+    flex-basis: 0;
+    flex-grow: 1;
+    max-width: 100%;
+  }
+
+  .col-sm-auto {
+    flex: 0 0 auto;
+    width: auto;
+    max-width: none;
+  }
+
+  .col-sm-1 {
+    flex: 0 0 8.3333333333%;
+    max-width: 8.3333333333%;
+  }
+
+  .col-sm-2 {
+    flex: 0 0 16.6666666667%;
+    max-width: 16.6666666667%;
+  }
+
+  .col-sm-3 {
+    flex: 0 0 25%;
+    max-width: 25%;
+  }
+
+  .col-sm-4 {
+    flex: 0 0 33.3333333333%;
+    max-width: 33.3333333333%;
+  }
+
+  .col-sm-5 {
+    flex: 0 0 41.6666666667%;
+    max-width: 41.6666666667%;
+  }
+
+  .col-sm-6 {
+    flex: 0 0 50%;
+    max-width: 50%;
+  }
+
+  .col-sm-7 {
+    flex: 0 0 58.3333333333%;
+    max-width: 58.3333333333%;
+  }
+
+  .col-sm-8 {
+    flex: 0 0 66.6666666667%;
+    max-width: 66.6666666667%;
+  }
+
+  .col-sm-9 {
+    flex: 0 0 75%;
+    max-width: 75%;
+  }
+
+  .col-sm-10 {
+    flex: 0 0 83.3333333333%;
+    max-width: 83.3333333333%;
+  }
+
+  .col-sm-11 {
+    flex: 0 0 91.6666666667%;
+    max-width: 91.6666666667%;
+  }
+
+  .col-sm-12 {
+    flex: 0 0 100%;
+    max-width: 100%;
+  }
+
+  .order-sm-first {
+    order: -1;
+  }
+
+  .order-sm-last {
+    order: 13;
+  }
+
+  .order-sm-0 {
+    order: 0;
+  }
+
+  .order-sm-1 {
+    order: 1;
+  }
+
+  .order-sm-2 {
+    order: 2;
+  }
+
+  .order-sm-3 {
+    order: 3;
+  }
+
+  .order-sm-4 {
+    order: 4;
+  }
+
+  .order-sm-5 {
+    order: 5;
+  }
+
+  .order-sm-6 {
+    order: 6;
+  }
+
+  .order-sm-7 {
+    order: 7;
+  }
+
+  .order-sm-8 {
+    order: 8;
+  }
+
+  .order-sm-9 {
+    order: 9;
+  }
+
+  .order-sm-10 {
+    order: 10;
+  }
+
+  .order-sm-11 {
+    order: 11;
+  }
+
+  .order-sm-12 {
+    order: 12;
+  }
+
+  .offset-sm-0 {
+    margin-left: 0;
+  }
+
+  .offset-sm-1 {
+    margin-left: 8.3333333333%;
+  }
+
+  .offset-sm-2 {
+    margin-left: 16.6666666667%;
+  }
+
+  .offset-sm-3 {
+    margin-left: 25%;
+  }
+
+  .offset-sm-4 {
+    margin-left: 33.3333333333%;
+  }
+
+  .offset-sm-5 {
+    margin-left: 41.6666666667%;
+  }
+
+  .offset-sm-6 {
+    margin-left: 50%;
+  }
+
+  .offset-sm-7 {
+    margin-left: 58.3333333333%;
+  }
+
+  .offset-sm-8 {
+    margin-left: 66.6666666667%;
+  }
+
+  .offset-sm-9 {
+    margin-left: 75%;
+  }
+
+  .offset-sm-10 {
+    margin-left: 83.3333333333%;
+  }
+
+  .offset-sm-11 {
+    margin-left: 91.6666666667%;
+  }
+}
+@media (min-width: 768px) {
+  .col-md {
+    flex-basis: 0;
+    flex-grow: 1;
+    max-width: 100%;
+  }
+
+  .col-md-auto {
+    flex: 0 0 auto;
+    width: auto;
+    max-width: none;
+  }
+
+  .col-md-1 {
+    flex: 0 0 8.3333333333%;
+    max-width: 8.3333333333%;
+  }
+
+  .col-md-2 {
+    flex: 0 0 16.6666666667%;
+    max-width: 16.6666666667%;
+  }
+
+  .col-md-3 {
+    flex: 0 0 25%;
+    max-width: 25%;
+  }
+
+  .col-md-4 {
+    flex: 0 0 33.3333333333%;
+    max-width: 33.3333333333%;
+  }
+
+  .col-md-5 {
+    flex: 0 0 41.6666666667%;
+    max-width: 41.6666666667%;
+  }
+
+  .col-md-6 {
+    flex: 0 0 50%;
+    max-width: 50%;
+  }
+
+  .col-md-7 {
+    flex: 0 0 58.3333333333%;
+    max-width: 58.3333333333%;
+  }
+
+  .col-md-8 {
+    flex: 0 0 66.6666666667%;
+    max-width: 66.6666666667%;
+  }
+
+  .col-md-9 {
+    flex: 0 0 75%;
+    max-width: 75%;
+  }
+
+  .col-md-10 {
+    flex: 0 0 83.3333333333%;
+    max-width: 83.3333333333%;
+  }
+
+  .col-md-11 {
+    flex: 0 0 91.6666666667%;
+    max-width: 91.6666666667%;
+  }
+
+  .col-md-12 {
+    flex: 0 0 100%;
+    max-width: 100%;
+  }
+
+  .order-md-first {
+    order: -1;
+  }
+
+  .order-md-last {
+    order: 13;
+  }
+
+  .order-md-0 {
+    order: 0;
+  }
+
+  .order-md-1 {
+    order: 1;
+  }
+
+  .order-md-2 {
+    order: 2;
+  }
+
+  .order-md-3 {
+    order: 3;
+  }
+
+  .order-md-4 {
+    order: 4;
+  }
+
+  .order-md-5 {
+    order: 5;
+  }
+
+  .order-md-6 {
+    order: 6;
+  }
+
+  .order-md-7 {
+    order: 7;
+  }
+
+  .order-md-8 {
+    order: 8;
+  }
+
+  .order-md-9 {
+    order: 9;
+  }
+
+  .order-md-10 {
+    order: 10;
+  }
+
+  .order-md-11 {
+    order: 11;
+  }
+
+  .order-md-12 {
+    order: 12;
+  }
+
+  .offset-md-0 {
+    margin-left: 0;
+  }
+
+  .offset-md-1 {
+    margin-left: 8.3333333333%;
+  }
+
+  .offset-md-2 {
+    margin-left: 16.6666666667%;
+  }
+
+  .offset-md-3 {
+    margin-left: 25%;
+  }
+
+  .offset-md-4 {
+    margin-left: 33.3333333333%;
+  }
+
+  .offset-md-5 {
+    margin-left: 41.6666666667%;
+  }
+
+  .offset-md-6 {
+    margin-left: 50%;
+  }
+
+  .offset-md-7 {
+    margin-left: 58.3333333333%;
+  }
+
+  .offset-md-8 {
+    margin-left: 66.6666666667%;
+  }
+
+  .offset-md-9 {
+    margin-left: 75%;
+  }
+
+  .offset-md-10 {
+    margin-left: 83.3333333333%;
+  }
+
+  .offset-md-11 {
+    margin-left: 91.6666666667%;
+  }
+}
+@media (min-width: 992px) {
+  .col-lg {
+    flex-basis: 0;
+    flex-grow: 1;
+    max-width: 100%;
+  }
+
+  .col-lg-auto {
+    flex: 0 0 auto;
+    width: auto;
+    max-width: none;
+  }
+
+  .col-lg-1 {
+    flex: 0 0 8.3333333333%;
+    max-width: 8.3333333333%;
+  }
+
+  .col-lg-2 {
+    flex: 0 0 16.6666666667%;
+    max-width: 16.6666666667%;
+  }
+
+  .col-lg-3 {
+    flex: 0 0 25%;
+    max-width: 25%;
+  }
+
+  .col-lg-4 {
+    flex: 0 0 33.3333333333%;
+    max-width: 33.3333333333%;
+  }
+
+  .col-lg-5 {
+    flex: 0 0 41.6666666667%;
+    max-width: 41.6666666667%;
+  }
+
+  .col-lg-6 {
+    flex: 0 0 50%;
+    max-width: 50%;
+  }
+
+  .col-lg-7 {
+    flex: 0 0 58.3333333333%;
+    max-width: 58.3333333333%;
+  }
+
+  .col-lg-8 {
+    flex: 0 0 66.6666666667%;
+    max-width: 66.6666666667%;
+  }
+
+  .col-lg-9 {
+    flex: 0 0 75%;
+    max-width: 75%;
+  }
+
+  .col-lg-10 {
+    flex: 0 0 83.3333333333%;
+    max-width: 83.3333333333%;
+  }
+
+  .col-lg-11 {
+    flex: 0 0 91.6666666667%;
+    max-width: 91.6666666667%;
+  }
+
+  .col-lg-12 {
+    flex: 0 0 100%;
+    max-width: 100%;
+  }
+
+  .order-lg-first {
+    order: -1;
+  }
+
+  .order-lg-last {
+    order: 13;
+  }
+
+  .order-lg-0 {
+    order: 0;
+  }
+
+  .order-lg-1 {
+    order: 1;
+  }
+
+  .order-lg-2 {
+    order: 2;
+  }
+
+  .order-lg-3 {
+    order: 3;
+  }
+
+  .order-lg-4 {
+    order: 4;
+  }
+
+  .order-lg-5 {
+    order: 5;
+  }
+
+  .order-lg-6 {
+    order: 6;
+  }
+
+  .order-lg-7 {
+    order: 7;
+  }
+
+  .order-lg-8 {
+    order: 8;
+  }
+
+  .order-lg-9 {
+    order: 9;
+  }
+
+  .order-lg-10 {
+    order: 10;
+  }
+
+  .order-lg-11 {
+    order: 11;
+  }
+
+  .order-lg-12 {
+    order: 12;
+  }
+
+  .offset-lg-0 {
+    margin-left: 0;
+  }
+
+  .offset-lg-1 {
+    margin-left: 8.3333333333%;
+  }
+
+  .offset-lg-2 {
+    margin-left: 16.6666666667%;
+  }
+
+  .offset-lg-3 {
+    margin-left: 25%;
+  }
+
+  .offset-lg-4 {
+    margin-left: 33.3333333333%;
+  }
+
+  .offset-lg-5 {
+    margin-left: 41.6666666667%;
+  }
+
+  .offset-lg-6 {
+    margin-left: 50%;
+  }
+
+  .offset-lg-7 {
+    margin-left: 58.3333333333%;
+  }
+
+  .offset-lg-8 {
+    margin-left: 66.6666666667%;
+  }
+
+  .offset-lg-9 {
+    margin-left: 75%;
+  }
+
+  .offset-lg-10 {
+    margin-left: 83.3333333333%;
+  }
+
+  .offset-lg-11 {
+    margin-left: 91.6666666667%;
+  }
+}
+@media (min-width: 1200px) {
+  .col-xl {
+    flex-basis: 0;
+    flex-grow: 1;
+    max-width: 100%;
+  }
+
+  .col-xl-auto {
+    flex: 0 0 auto;
+    width: auto;
+    max-width: none;
+  }
+
+  .col-xl-1 {
+    flex: 0 0 8.3333333333%;
+    max-width: 8.3333333333%;
+  }
+
+  .col-xl-2 {
+    flex: 0 0 16.6666666667%;
+    max-width: 16.6666666667%;
+  }
+
+  .col-xl-3 {
+    flex: 0 0 25%;
+    max-width: 25%;
+  }
+
+  .col-xl-4 {
+    flex: 0 0 33.3333333333%;
+    max-width: 33.3333333333%;
+  }
+
+  .col-xl-5 {
+    flex: 0 0 41.6666666667%;
+    max-width: 41.6666666667%;
+  }
+
+  .col-xl-6 {
+    flex: 0 0 50%;
+    max-width: 50%;
+  }
+
+  .col-xl-7 {
+    flex: 0 0 58.3333333333%;
+    max-width: 58.3333333333%;
+  }
+
+  .col-xl-8 {
+    flex: 0 0 66.6666666667%;
+    max-width: 66.6666666667%;
+  }
+
+  .col-xl-9 {
+    flex: 0 0 75%;
+    max-width: 75%;
+  }
+
+  .col-xl-10 {
+    flex: 0 0 83.3333333333%;
+    max-width: 83.3333333333%;
+  }
+
+  .col-xl-11 {
+    flex: 0 0 91.6666666667%;
+    max-width: 91.6666666667%;
+  }
+
+  .col-xl-12 {
+    flex: 0 0 100%;
+    max-width: 100%;
+  }
+
+  .order-xl-first {
+    order: -1;
+  }
+
+  .order-xl-last {
+    order: 13;
+  }
+
+  .order-xl-0 {
+    order: 0;
+  }
+
+  .order-xl-1 {
+    order: 1;
+  }
+
+  .order-xl-2 {
+    order: 2;
+  }
+
+  .order-xl-3 {
+    order: 3;
+  }
+
+  .order-xl-4 {
+    order: 4;
+  }
+
+  .order-xl-5 {
+    order: 5;
+  }
+
+  .order-xl-6 {
+    order: 6;
+  }
+
+  .order-xl-7 {
+    order: 7;
+  }
+
+  .order-xl-8 {
+    order: 8;
+  }
+
+  .order-xl-9 {
+    order: 9;
+  }
+
+  .order-xl-10 {
+    order: 10;
+  }
+
+  .order-xl-11 {
+    order: 11;
+  }
+
+  .order-xl-12 {
+    order: 12;
+  }
+
+  .offset-xl-0 {
+    margin-left: 0;
+  }
+
+  .offset-xl-1 {
+    margin-left: 8.3333333333%;
+  }
+
+  .offset-xl-2 {
+    margin-left: 16.6666666667%;
+  }
+
+  .offset-xl-3 {
+    margin-left: 25%;
+  }
+
+  .offset-xl-4 {
+    margin-left: 33.3333333333%;
+  }
+
+  .offset-xl-5 {
+    margin-left: 41.6666666667%;
+  }
+
+  .offset-xl-6 {
+    margin-left: 50%;
+  }
+
+  .offset-xl-7 {
+    margin-left: 58.3333333333%;
+  }
+
+  .offset-xl-8 {
+    margin-left: 66.6666666667%;
+  }
+
+  .offset-xl-9 {
+    margin-left: 75%;
+  }
+
+  .offset-xl-10 {
+    margin-left: 83.3333333333%;
+  }
+
+  .offset-xl-11 {
+    margin-left: 91.6666666667%;
+  }
+}
+.table {
+  width: 100%;
+  max-width: 100%;
+  margin-bottom: 1rem;
+  background-color: transparent;
+}
+.table th,
+.table td {
+  padding: 0.75rem;
+  vertical-align: top;
+  border-top: 1px solid #dee2e6;
+}
+.table thead th {
+  vertical-align: bottom;
+  border-bottom: 2px solid #dee2e6;
+}
+.table tbody + tbody {
+  border-top: 2px solid #dee2e6;
+}
+.table .table {
+  background-color: #fff;
+}
+
+.table-sm th,
+.table-sm td {
+  padding: 0.3rem;
+}
+
+.table-bordered {
+  border: 1px solid #dee2e6;
+}
+.table-bordered th,
+.table-bordered td {
+  border: 1px solid #dee2e6;
+}
+.table-bordered thead th,
+.table-bordered thead td {
+  border-bottom-width: 2px;
+}
+
+.table-striped tbody tr:nth-of-type(odd) {
+  background-color: rgba(0, 0, 0, 0.05);
+}
+
+.table-hover tbody tr:hover {
+  background-color: rgba(0, 0, 0, 0.075);
+}
+
+.table-primary,
+.table-primary > th,
+.table-primary > td {
+  background-color: #b8daff;
+}
+
+.table-hover .table-primary:hover {
+  background-color: #9fcdff;
+}
+.table-hover .table-primary:hover > td,
+.table-hover .table-primary:hover > th {
+  background-color: #9fcdff;
+}
+
+.table-secondary,
+.table-secondary > th,
+.table-secondary > td {
+  background-color: #d6d8db;
+}
+
+.table-hover .table-secondary:hover {
+  background-color: #c8cbcf;
+}
+.table-hover .table-secondary:hover > td,
+.table-hover .table-secondary:hover > th {
+  background-color: #c8cbcf;
+}
+
+.table-success,
+.table-success > th,
+.table-success > td {
+  background-color: #c3e6cb;
+}
+
+.table-hover .table-success:hover {
+  background-color: #b1dfbb;
+}
+.table-hover .table-success:hover > td,
+.table-hover .table-success:hover > th {
+  background-color: #b1dfbb;
+}
+
+.table-info,
+.table-info > th,
+.table-info > td {
+  background-color: #bee5eb;
+}
+
+.table-hover .table-info:hover {
+  background-color: #abdde5;
+}
+.table-hover .table-info:hover > td,
+.table-hover .table-info:hover > th {
+  background-color: #abdde5;
+}
+
+.table-warning,
+.table-warning > th,
+.table-warning > td {
+  background-color: #ffeeba;
+}
+
+.table-hover .table-warning:hover {
+  background-color: #ffe8a1;
+}
+.table-hover .table-warning:hover > td,
+.table-hover .table-warning:hover > th {
+  background-color: #ffe8a1;
+}
+
+.table-danger,
+.table-danger > th,
+.table-danger > td {
+  background-color: #f5c6cb;
+}
+
+.table-hover .table-danger:hover {
+  background-color: #f1b0b7;
+}
+.table-hover .table-danger:hover > td,
+.table-hover .table-danger:hover > th {
+  background-color: #f1b0b7;
+}
+
+.table-light,
+.table-light > th,
+.table-light > td {
+  background-color: #fdfdfe;
+}
+
+.table-hover .table-light:hover {
+  background-color: #ececf6;
+}
+.table-hover .table-light:hover > td,
+.table-hover .table-light:hover > th {
+  background-color: #ececf6;
+}
+
+.table-dark,
+.table-dark > th,
+.table-dark > td {
+  background-color: #c6c8ca;
+}
+
+.table-hover .table-dark:hover {
+  background-color: #b9bbbe;
+}
+.table-hover .table-dark:hover > td,
+.table-hover .table-dark:hover > th {
+  background-color: #b9bbbe;
+}
+
+.table-active,
+.table-active > th,
+.table-active > td {
+  background-color: rgba(0, 0, 0, 0.075);
+}
+
+.table-hover .table-active:hover {
+  background-color: rgba(0, 0, 0, 0.075);
+}
+.table-hover .table-active:hover > td,
+.table-hover .table-active:hover > th {
+  background-color: rgba(0, 0, 0, 0.075);
+}
+
+.table .thead-dark th {
+  color: #fff;
+  background-color: #212529;
+  border-color: #32383e;
+}
+.table .thead-light th {
+  color: #495057;
+  background-color: #e9ecef;
+  border-color: #dee2e6;
+}
+
+.table-dark {
+  color: #fff;
+  background-color: #212529;
+}
+.table-dark th,
+.table-dark td,
+.table-dark thead th {
+  border-color: #32383e;
+}
+.table-dark.table-bordered {
+  border: 0;
+}
+.table-dark.table-striped tbody tr:nth-of-type(odd) {
+  background-color: rgba(255, 255, 255, 0.05);
+}
+.table-dark.table-hover tbody tr:hover {
+  background-color: rgba(255, 255, 255, 0.075);
+}
+
+@media (max-width: 575.98px) {
+  .table-responsive-sm {
+    display: block;
+    width: 100%;
+    overflow-x: auto;
+    -webkit-overflow-scrolling: touch;
+    -ms-overflow-style: -ms-autohiding-scrollbar;
+  }
+  .table-responsive-sm > .table-bordered {
+    border: 0;
+  }
+}
+@media (max-width: 767.98px) {
+  .table-responsive-md {
+    display: block;
+    width: 100%;
+    overflow-x: auto;
+    -webkit-overflow-scrolling: touch;
+    -ms-overflow-style: -ms-autohiding-scrollbar;
+  }
+  .table-responsive-md > .table-bordered {
+    border: 0;
+  }
+}
+@media (max-width: 991.98px) {
+  .table-responsive-lg {
+    display: block;
+    width: 100%;
+    overflow-x: auto;
+    -webkit-overflow-scrolling: touch;
+    -ms-overflow-style: -ms-autohiding-scrollbar;
+  }
+  .table-responsive-lg > .table-bordered {
+    border: 0;
+  }
+}
+@media (max-width: 1199.98px) {
+  .table-responsive-xl {
+    display: block;
+    width: 100%;
+    overflow-x: auto;
+    -webkit-overflow-scrolling: touch;
+    -ms-overflow-style: -ms-autohiding-scrollbar;
+  }
+  .table-responsive-xl > .table-bordered {
+    border: 0;
+  }
+}
+.table-responsive {
+  display: block;
+  width: 100%;
+  overflow-x: auto;
+  -webkit-overflow-scrolling: touch;
+  -ms-overflow-style: -ms-autohiding-scrollbar;
+}
+.table-responsive > .table-bordered {
+  border: 0;
+}
+
+.form-control {
+  display: block;
+  width: 100%;
+  padding: 0.375rem 0.75rem;
+  font-size: 1rem;
+  line-height: 1.5;
+  color: #495057;
+  background-color: #fff;
+  background-clip: padding-box;
+  border: 1px solid #ced4da;
+  border-radius: 0.25rem;
+  transition: border-color 0.15s ease-in-out, box-shadow 0.15s ease-in-out;
+}
+.form-control::-ms-expand {
+  background-color: transparent;
+  border: 0;
+}
+.form-control:focus {
+  color: #495057;
+  background-color: #fff;
+  border-color: #80bdff;
+  outline: 0;
+  box-shadow: 0 0 0 0.2rem rgba(0, 123, 255, 0.25);
+}
+.form-control::-moz-placeholder {
+  color: #6c757d;
+  opacity: 1;
+}
+.form-control:-ms-input-placeholder {
+  color: #6c757d;
+  opacity: 1;
+}
+.form-control::-ms-input-placeholder {
+  color: #6c757d;
+  opacity: 1;
+}
+.form-control::placeholder {
+  color: #6c757d;
+  opacity: 1;
+}
+.form-control:disabled, .form-control[readonly] {
+  background-color: #e9ecef;
+  opacity: 1;
+}
+
+select.form-control:not([size]):not([multiple]) {
+  height: calc(2.25rem + 2px);
+}
+select.form-control:focus::-ms-value {
+  color: #495057;
+  background-color: #fff;
+}
+
+.form-control-file,
+.form-control-range {
+  display: block;
+  width: 100%;
+}
+
+.col-form-label {
+  padding-top: calc(0.375rem + 1px);
+  padding-bottom: calc(0.375rem + 1px);
+  margin-bottom: 0;
+  font-size: inherit;
+  line-height: 1.5;
+}
+
+.col-form-label-lg {
+  padding-top: calc(0.5rem + 1px);
+  padding-bottom: calc(0.5rem + 1px);
+  font-size: 1.25rem;
+  line-height: 1.5;
+}
+
+.col-form-label-sm {
+  padding-top: calc(0.25rem + 1px);
+  padding-bottom: calc(0.25rem + 1px);
+  font-size: 0.875rem;
+  line-height: 1.5;
+}
+
+.form-control-plaintext {
+  display: block;
+  width: 100%;
+  padding-top: 0.375rem;
+  padding-bottom: 0.375rem;
+  margin-bottom: 0;
+  line-height: 1.5;
+  background-color: transparent;
+  border: solid transparent;
+  border-width: 1px 0;
+}
+.form-control-plaintext.form-control-sm, .input-group-sm > .form-control-plaintext.form-control,
+.input-group-sm > .input-group-prepend > .form-control-plaintext.input-group-text,
+.input-group-sm > .input-group-append > .form-control-plaintext.input-group-text,
+.input-group-sm > .input-group-prepend > .form-control-plaintext.btn,
+.input-group-sm > .input-group-append > .form-control-plaintext.btn, .form-control-plaintext.form-control-lg, .input-group-lg > .form-control-plaintext.form-control,
+.input-group-lg > .input-group-prepend > .form-control-plaintext.input-group-text,
+.input-group-lg > .input-group-append > .form-control-plaintext.input-group-text,
+.input-group-lg > .input-group-prepend > .form-control-plaintext.btn,
+.input-group-lg > .input-group-append > .form-control-plaintext.btn {
+  padding-right: 0;
+  padding-left: 0;
+}
+
+.form-control-sm, .input-group-sm > .form-control,
+.input-group-sm > .input-group-prepend > .input-group-text,
+.input-group-sm > .input-group-append > .input-group-text,
+.input-group-sm > .input-group-prepend > .btn,
+.input-group-sm > .input-group-append > .btn {
+  padding: 0.25rem 0.5rem;
+  font-size: 0.875rem;
+  line-height: 1.5;
+  border-radius: 0.2rem;
+}
+
+select.form-control-sm:not([size]):not([multiple]), .input-group-sm > select.form-control:not([size]):not([multiple]),
+.input-group-sm > .input-group-prepend > select.input-group-text:not([size]):not([multiple]),
+.input-group-sm > .input-group-append > select.input-group-text:not([size]):not([multiple]),
+.input-group-sm > .input-group-prepend > select.btn:not([size]):not([multiple]),
+.input-group-sm > .input-group-append > select.btn:not([size]):not([multiple]) {
+  height: calc(1.8125rem + 2px);
+}
+
+.form-control-lg, .input-group-lg > .form-control,
+.input-group-lg > .input-group-prepend > .input-group-text,
+.input-group-lg > .input-group-append > .input-group-text,
+.input-group-lg > .input-group-prepend > .btn,
+.input-group-lg > .input-group-append > .btn {
+  padding: 0.5rem 1rem;
+  font-size: 1.25rem;
+  line-height: 1.5;
+  border-radius: 0.3rem;
+}
+
+select.form-control-lg:not([size]):not([multiple]), .input-group-lg > select.form-control:not([size]):not([multiple]),
+.input-group-lg > .input-group-prepend > select.input-group-text:not([size]):not([multiple]),
+.input-group-lg > .input-group-append > select.input-group-text:not([size]):not([multiple]),
+.input-group-lg > .input-group-prepend > select.btn:not([size]):not([multiple]),
+.input-group-lg > .input-group-append > select.btn:not([size]):not([multiple]) {
+  height: calc(2.875rem + 2px);
+}
+
+.form-group {
+  margin-bottom: 1rem;
+}
+
+.form-text {
+  display: block;
+  margin-top: 0.25rem;
+}
+
+.form-row {
+  display: flex;
+  flex-wrap: wrap;
+  margin-right: -5px;
+  margin-left: -5px;
+}
+.form-row > .col,
+.form-row > [class*="col-"] {
+  padding-right: 5px;
+  padding-left: 5px;
+}
+
+.form-check {
+  position: relative;
+  display: block;
+  padding-left: 1.25rem;
+}
+
+.form-check-input {
+  position: absolute;
+  margin-top: 0.3rem;
+  margin-left: -1.25rem;
+}
+.form-check-input:disabled ~ .form-check-label {
+  color: #6c757d;
+}
+
+.form-check-label {
+  margin-bottom: 0;
+}
+
+.form-check-inline {
+  display: inline-flex;
+  align-items: center;
+  padding-left: 0;
+  margin-right: 0.75rem;
+}
+.form-check-inline .form-check-input {
+  position: static;
+  margin-top: 0;
+  margin-right: 0.3125rem;
+  margin-left: 0;
+}
+
+.valid-feedback {
+  display: none;
+  width: 100%;
+  margin-top: 0.25rem;
+  font-size: 80%;
+  color: #28a745;
+}
+
+.valid-tooltip {
+  position: absolute;
+  top: 100%;
+  z-index: 5;
+  display: none;
+  max-width: 100%;
+  padding: .5rem;
+  margin-top: .1rem;
+  font-size: .875rem;
+  line-height: 1;
+  color: #fff;
+  background-color: rgba(40, 167, 69, 0.8);
+  border-radius: .2rem;
+}
+
+.was-validated .form-control:valid, .form-control.is-valid,
+.was-validated .custom-select:valid,
+.custom-select.is-valid {
+  border-color: #28a745;
+}
+.was-validated .form-control:valid:focus, .form-control.is-valid:focus,
+.was-validated .custom-select:valid:focus,
+.custom-select.is-valid:focus {
+  border-color: #28a745;
+  box-shadow: 0 0 0 0.2rem rgba(40, 167, 69, 0.25);
+}
+.was-validated .form-control:valid ~ .valid-feedback,
+.was-validated .form-control:valid ~ .valid-tooltip, .form-control.is-valid ~ .valid-feedback,
+.form-control.is-valid ~ .valid-tooltip,
+.was-validated .custom-select:valid ~ .valid-feedback,
+.was-validated .custom-select:valid ~ .valid-tooltip,
+.custom-select.is-valid ~ .valid-feedback,
+.custom-select.is-valid ~ .valid-tooltip {
+  display: block;
+}
+
+.was-validated .form-check-input:valid ~ .form-check-label, .form-check-input.is-valid ~ .form-check-label {
+  color: #28a745;
+}
+.was-validated .form-check-input:valid ~ .valid-feedback,
+.was-validated .form-check-input:valid ~ .valid-tooltip, .form-check-input.is-valid ~ .valid-feedback,
+.form-check-input.is-valid ~ .valid-tooltip {
+  display: block;
+}
+
+.was-validated .custom-control-input:valid ~ .custom-control-label, .custom-control-input.is-valid ~ .custom-control-label {
+  color: #28a745;
+}
+.was-validated .custom-control-input:valid ~ .custom-control-label::before, .custom-control-input.is-valid ~ .custom-control-label::before {
+  background-color: #71dd8a;
+}
+.was-validated .custom-control-input:valid ~ .valid-feedback,
+.was-validated .custom-control-input:valid ~ .valid-tooltip, .custom-control-input.is-valid ~ .valid-feedback,
+.custom-control-input.is-valid ~ .valid-tooltip {
+  display: block;
+}
+.was-validated .custom-control-input:valid:checked ~ .custom-control-label::before, .custom-control-input.is-valid:checked ~ .custom-control-label::before {
+  background-color: #34ce57;
+}
+.was-validated .custom-control-input:valid:focus ~ .custom-control-label::before, .custom-control-input.is-valid:focus ~ .custom-control-label::before {
+  box-shadow: 0 0 0 1px #fff, 0 0 0 0.2rem rgba(40, 167, 69, 0.25);
+}
+
+.was-validated .custom-file-input:valid ~ .custom-file-label, .custom-file-input.is-valid ~ .custom-file-label {
+  border-color: #28a745;
+}
+.was-validated .custom-file-input:valid ~ .custom-file-label::before, .custom-file-input.is-valid ~ .custom-file-label::before {
+  border-color: inherit;
+}
+.was-validated .custom-file-input:valid ~ .valid-feedback,
+.was-validated .custom-file-input:valid ~ .valid-tooltip, .custom-file-input.is-valid ~ .valid-feedback,
+.custom-file-input.is-valid ~ .valid-tooltip {
+  display: block;
+}
+.was-validated .custom-file-input:valid:focus ~ .custom-file-label, .custom-file-input.is-valid:focus ~ .custom-file-label {
+  box-shadow: 0 0 0 0.2rem rgba(40, 167, 69, 0.25);
+}
+
+.invalid-feedback {
+  display: none;
+  width: 100%;
+  margin-top: 0.25rem;
+  font-size: 80%;
+  color: #dc3545;
+}
+
+.invalid-tooltip {
+  position: absolute;
+  top: 100%;
+  z-index: 5;
+  display: none;
+  max-width: 100%;
+  padding: .5rem;
+  margin-top: .1rem;
+  font-size: .875rem;
+  line-height: 1;
+  color: #fff;
+  background-color: rgba(220, 53, 69, 0.8);
+  border-radius: .2rem;
+}
+
+.was-validated .form-control:invalid, .form-control.is-invalid,
+.was-validated .custom-select:invalid,
+.custom-select.is-invalid {
+  border-color: #dc3545;
+}
+.was-validated .form-control:invalid:focus, .form-control.is-invalid:focus,
+.was-validated .custom-select:invalid:focus,
+.custom-select.is-invalid:focus {
+  border-color: #dc3545;
+  box-shadow: 0 0 0 0.2rem rgba(220, 53, 69, 0.25);
+}
+.was-validated .form-control:invalid ~ .invalid-feedback,
+.was-validated .form-control:invalid ~ .invalid-tooltip, .form-control.is-invalid ~ .invalid-feedback,
+.form-control.is-invalid ~ .invalid-tooltip,
+.was-validated .custom-select:invalid ~ .invalid-feedback,
+.was-validated .custom-select:invalid ~ .invalid-tooltip,
+.custom-select.is-invalid ~ .invalid-feedback,
+.custom-select.is-invalid ~ .invalid-tooltip {
+  display: block;
+}
+
+.was-validated .form-check-input:invalid ~ .form-check-label, .form-check-input.is-invalid ~ .form-check-label {
+  color: #dc3545;
+}
+.was-validated .form-check-input:invalid ~ .invalid-feedback,
+.was-validated .form-check-input:invalid ~ .invalid-tooltip, .form-check-input.is-invalid ~ .invalid-feedback,
+.form-check-input.is-invalid ~ .invalid-tooltip {
+  display: block;
+}
+
+.was-validated .custom-control-input:invalid ~ .custom-control-label, .custom-control-input.is-invalid ~ .custom-control-label {
+  color: #dc3545;
+}
+.was-validated .custom-control-input:invalid ~ .custom-control-label::before, .custom-control-input.is-invalid ~ .custom-control-label::before {
+  background-color: #efa2a9;
+}
+.was-validated .custom-control-input:invalid ~ .invalid-feedback,
+.was-validated .custom-control-input:invalid ~ .invalid-tooltip, .custom-control-input.is-invalid ~ .invalid-feedback,
+.custom-control-input.is-invalid ~ .invalid-tooltip {
+  display: block;
+}
+.was-validated .custom-control-input:invalid:checked ~ .custom-control-label::before, .custom-control-input.is-invalid:checked ~ .custom-control-label::before {
+  background-color: #e4606d;
+}
+.was-validated .custom-control-input:invalid:focus ~ .custom-control-label::before, .custom-control-input.is-invalid:focus ~ .custom-control-label::before {
+  box-shadow: 0 0 0 1px #fff, 0 0 0 0.2rem rgba(220, 53, 69, 0.25);
+}
+
+.was-validated .custom-file-input:invalid ~ .custom-file-label, .custom-file-input.is-invalid ~ .custom-file-label {
+  border-color: #dc3545;
+}
+.was-validated .custom-file-input:invalid ~ .custom-file-label::before, .custom-file-input.is-invalid ~ .custom-file-label::before {
+  border-color: inherit;
+}
+.was-validated .custom-file-input:invalid ~ .invalid-feedback,
+.was-validated .custom-file-input:invalid ~ .invalid-tooltip, .custom-file-input.is-invalid ~ .invalid-feedback,
+.custom-file-input.is-invalid ~ .invalid-tooltip {
+  display: block;
+}
+.was-validated .custom-file-input:invalid:focus ~ .custom-file-label, .custom-file-input.is-invalid:focus ~ .custom-file-label {
+  box-shadow: 0 0 0 0.2rem rgba(220, 53, 69, 0.25);
+}
+
+.form-inline {
+  display: flex;
+  flex-flow: row wrap;
+  align-items: center;
+}
+.form-inline .form-check {
+  width: 100%;
+}
+@media (min-width: 576px) {
+  .form-inline label {
+    display: flex;
+    align-items: center;
+    justify-content: center;
+    margin-bottom: 0;
+  }
+  .form-inline .form-group {
+    display: flex;
+    flex: 0 0 auto;
+    flex-flow: row wrap;
+    align-items: center;
+    margin-bottom: 0;
+  }
+  .form-inline .form-control {
+    display: inline-block;
+    width: auto;
+    vertical-align: middle;
+  }
+  .form-inline .form-control-plaintext {
+    display: inline-block;
+  }
+  .form-inline .input-group {
+    width: auto;
+  }
+  .form-inline .form-check {
+    display: flex;
+    align-items: center;
+    justify-content: center;
+    width: auto;
+    padding-left: 0;
+  }
+  .form-inline .form-check-input {
+    position: relative;
+    margin-top: 0;
+    margin-right: 0.25rem;
+    margin-left: 0;
+  }
+  .form-inline .custom-control {
+    align-items: center;
+    justify-content: center;
+  }
+  .form-inline .custom-control-label {
+    margin-bottom: 0;
+  }
+}
+
+.btn {
+  display: inline-block;
+  font-weight: 400;
+  text-align: center;
+  white-space: nowrap;
+  vertical-align: middle;
+  -webkit-user-select: none;
+     -moz-user-select: none;
+      -ms-user-select: none;
+          user-select: none;
+  border: 1px solid transparent;
+  padding: 0.375rem 0.75rem;
+  font-size: 1rem;
+  line-height: 1.5;
+  border-radius: 0.25rem;
+  transition: color 0.15s ease-in-out, background-color 0.15s ease-in-out, border-color 0.15s ease-in-out, box-shadow 0.15s ease-in-out;
+}
+.btn:hover, .btn:focus {
+  text-decoration: none;
+}
+.btn:focus, .btn.focus {
+  outline: 0;
+  box-shadow: 0 0 0 0.2rem rgba(0, 123, 255, 0.25);
+}
+.btn.disabled, .btn:disabled {
+  opacity: 0.65;
+}
+.btn:not(:disabled):not(.disabled) {
+  cursor: pointer;
+}
+.btn:not(:disabled):not(.disabled):active, .btn:not(:disabled):not(.disabled).active {
+  background-image: none;
+}
+
+a.btn.disabled,
+fieldset:disabled a.btn {
+  pointer-events: none;
+}
+
+.btn-primary {
+  color: #fff;
+  background-color: #007bff;
+  border-color: #007bff;
+}
+.btn-primary:hover {
+  color: #fff;
+  background-color: #0069d9;
+  border-color: #0062cc;
+}
+.btn-primary:focus, .btn-primary.focus {
+  box-shadow: 0 0 0 0.2rem rgba(0, 123, 255, 0.5);
+}
+.btn-primary.disabled, .btn-primary:disabled {
+  color: #fff;
+  background-color: #007bff;
+  border-color: #007bff;
+}
+.btn-primary:not(:disabled):not(.disabled):active, .btn-primary:not(:disabled):not(.disabled).active, .show > .btn-primary.dropdown-toggle {
+  color: #fff;
+  background-color: #0062cc;
+  border-color: #005cbf;
+}
+.btn-primary:not(:disabled):not(.disabled):active:focus, .btn-primary:not(:disabled):not(.disabled).active:focus, .show > .btn-primary.dropdown-toggle:focus {
+  box-shadow: 0 0 0 0.2rem rgba(0, 123, 255, 0.5);
+}
+
+.btn-secondary {
+  color: #fff;
+  background-color: #6c757d;
+  border-color: #6c757d;
+}
+.btn-secondary:hover {
+  color: #fff;
+  background-color: #5a6268;
+  border-color: #545b62;
+}
+.btn-secondary:focus, .btn-secondary.focus {
+  box-shadow: 0 0 0 0.2rem rgba(108, 117, 125, 0.5);
+}
+.btn-secondary.disabled, .btn-secondary:disabled {
+  color: #fff;
+  background-color: #6c757d;
+  border-color: #6c757d;
+}
+.btn-secondary:not(:disabled):not(.disabled):active, .btn-secondary:not(:disabled):not(.disabled).active, .show > .btn-secondary.dropdown-toggle {
+  color: #fff;
+  background-color: #545b62;
+  border-color: #4e555b;
+}
+.btn-secondary:not(:disabled):not(.disabled):active:focus, .btn-secondary:not(:disabled):not(.disabled).active:focus, .show > .btn-secondary.dropdown-toggle:focus {
+  box-shadow: 0 0 0 0.2rem rgba(108, 117, 125, 0.5);
+}
+
+.btn-success {
+  color: #fff;
+  background-color: #28a745;
+  border-color: #28a745;
+}
+.btn-success:hover {
+  color: #fff;
+  background-color: #218838;
+  border-color: #1e7e34;
+}
+.btn-success:focus, .btn-success.focus {
+  box-shadow: 0 0 0 0.2rem rgba(40, 167, 69, 0.5);
+}
+.btn-success.disabled, .btn-success:disabled {
+  color: #fff;
+  background-color: #28a745;
+  border-color: #28a745;
+}
+.btn-success:not(:disabled):not(.disabled):active, .btn-success:not(:disabled):not(.disabled).active, .show > .btn-success.dropdown-toggle {
+  color: #fff;
+  background-color: #1e7e34;
+  border-color: #1c7430;
+}
+.btn-success:not(:disabled):not(.disabled):active:focus, .btn-success:not(:disabled):not(.disabled).active:focus, .show > .btn-success.dropdown-toggle:focus {
+  box-shadow: 0 0 0 0.2rem rgba(40, 167, 69, 0.5);
+}
+
+.btn-info {
+  color: #fff;
+  background-color: #17a2b8;
+  border-color: #17a2b8;
+}
+.btn-info:hover {
+  color: #fff;
+  background-color: #138496;
+  border-color: #117a8b;
+}
+.btn-info:focus, .btn-info.focus {
+  box-shadow: 0 0 0 0.2rem rgba(23, 162, 184, 0.5);
+}
+.btn-info.disabled, .btn-info:disabled {
+  color: #fff;
+  background-color: #17a2b8;
+  border-color: #17a2b8;
+}
+.btn-info:not(:disabled):not(.disabled):active, .btn-info:not(:disabled):not(.disabled).active, .show > .btn-info.dropdown-toggle {
+  color: #fff;
+  background-color: #117a8b;
+  border-color: #10707f;
+}
+.btn-info:not(:disabled):not(.disabled):active:focus, .btn-info:not(:disabled):not(.disabled).active:focus, .show > .btn-info.dropdown-toggle:focus {
+  box-shadow: 0 0 0 0.2rem rgba(23, 162, 184, 0.5);
+}
+
+.btn-warning {
+  color: #212529;
+  background-color: #ffc107;
+  border-color: #ffc107;
+}
+.btn-warning:hover {
+  color: #212529;
+  background-color: #e0a800;
+  border-color: #d39e00;
+}
+.btn-warning:focus, .btn-warning.focus {
+  box-shadow: 0 0 0 0.2rem rgba(255, 193, 7, 0.5);
+}
+.btn-warning.disabled, .btn-warning:disabled {
+  color: #212529;
+  background-color: #ffc107;
+  border-color: #ffc107;
+}
+.btn-warning:not(:disabled):not(.disabled):active, .btn-warning:not(:disabled):not(.disabled).active, .show > .btn-warning.dropdown-toggle {
+  color: #212529;
+  background-color: #d39e00;
+  border-color: #c69500;
+}
+.btn-warning:not(:disabled):not(.disabled):active:focus, .btn-warning:not(:disabled):not(.disabled).active:focus, .show > .btn-warning.dropdown-toggle:focus {
+  box-shadow: 0 0 0 0.2rem rgba(255, 193, 7, 0.5);
+}
+
+.btn-danger {
+  color: #fff;
+  background-color: #dc3545;
+  border-color: #dc3545;
+}
+.btn-danger:hover {
+  color: #fff;
+  background-color: #c82333;
+  border-color: #bd2130;
+}
+.btn-danger:focus, .btn-danger.focus {
+  box-shadow: 0 0 0 0.2rem rgba(220, 53, 69, 0.5);
+}
+.btn-danger.disabled, .btn-danger:disabled {
+  color: #fff;
+  background-color: #dc3545;
+  border-color: #dc3545;
+}
+.btn-danger:not(:disabled):not(.disabled):active, .btn-danger:not(:disabled):not(.disabled).active, .show > .btn-danger.dropdown-toggle {
+  color: #fff;
+  background-color: #bd2130;
+  border-color: #b21f2d;
+}
+.btn-danger:not(:disabled):not(.disabled):active:focus, .btn-danger:not(:disabled):not(.disabled).active:focus, .show > .btn-danger.dropdown-toggle:focus {
+  box-shadow: 0 0 0 0.2rem rgba(220, 53, 69, 0.5);
+}
+
+.btn-light {
+  color: #212529;
+  background-color: #f8f9fa;
+  border-color: #f8f9fa;
+}
+.btn-light:hover {
+  color: #212529;
+  background-color: #e2e6ea;
+  border-color: #dae0e5;
+}
+.btn-light:focus, .btn-light.focus {
+  box-shadow: 0 0 0 0.2rem rgba(248, 249, 250, 0.5);
+}
+.btn-light.disabled, .btn-light:disabled {
+  color: #212529;
+  background-color: #f8f9fa;
+  border-color: #f8f9fa;
+}
+.btn-light:not(:disabled):not(.disabled):active, .btn-light:not(:disabled):not(.disabled).active, .show > .btn-light.dropdown-toggle {
+  color: #212529;
+  background-color: #dae0e5;
+  border-color: #d3d9df;
+}
+.btn-light:not(:disabled):not(.disabled):active:focus, .btn-light:not(:disabled):not(.disabled).active:focus, .show > .btn-light.dropdown-toggle:focus {
+  box-shadow: 0 0 0 0.2rem rgba(248, 249, 250, 0.5);
+}
+
+.btn-dark {
+  color: #fff;
+  background-color: #343a40;
+  border-color: #343a40;
+}
+.btn-dark:hover {
+  color: #fff;
+  background-color: #23272b;
+  border-color: #1d2124;
+}
+.btn-dark:focus, .btn-dark.focus {
+  box-shadow: 0 0 0 0.2rem rgba(52, 58, 64, 0.5);
+}
+.btn-dark.disabled, .btn-dark:disabled {
+  color: #fff;
+  background-color: #343a40;
+  border-color: #343a40;
+}
+.btn-dark:not(:disabled):not(.disabled):active, .btn-dark:not(:disabled):not(.disabled).active, .show > .btn-dark.dropdown-toggle {
+  color: #fff;
+  background-color: #1d2124;
+  border-color: #171a1d;
+}
+.btn-dark:not(:disabled):not(.disabled):active:focus, .btn-dark:not(:disabled):not(.disabled).active:focus, .show > .btn-dark.dropdown-toggle:focus {
+  box-shadow: 0 0 0 0.2rem rgba(52, 58, 64, 0.5);
+}
+
+.btn-outline-primary {
+  color: #007bff;
+  background-color: transparent;
+  background-image: none;
+  border-color: #007bff;
+}
+.btn-outline-primary:hover {
+  color: #fff;
+  background-color: #007bff;
+  border-color: #007bff;
+}
+.btn-outline-primary:focus, .btn-outline-primary.focus {
+  box-shadow: 0 0 0 0.2rem rgba(0, 123, 255, 0.5);
+}
+.btn-outline-primary.disabled, .btn-outline-primary:disabled {
+  color: #007bff;
+  background-color: transparent;
+}
+.btn-outline-primary:not(:disabled):not(.disabled):active, .btn-outline-primary:not(:disabled):not(.disabled).active, .show > .btn-outline-primary.dropdown-toggle {
+  color: #fff;
+  background-color: #007bff;
+  border-color: #007bff;
+}
+.btn-outline-primary:not(:disabled):not(.disabled):active:focus, .btn-outline-primary:not(:disabled):not(.disabled).active:focus, .show > .btn-outline-primary.dropdown-toggle:focus {
+  box-shadow: 0 0 0 0.2rem rgba(0, 123, 255, 0.5);
+}
+
+.btn-outline-secondary {
+  color: #6c757d;
+  background-color: transparent;
+  background-image: none;
+  border-color: #6c757d;
+}
+.btn-outline-secondary:hover {
+  color: #fff;
+  background-color: #6c757d;
+  border-color: #6c757d;
+}
+.btn-outline-secondary:focus, .btn-outline-secondary.focus {
+  box-shadow: 0 0 0 0.2rem rgba(108, 117, 125, 0.5);
+}
+.btn-outline-secondary.disabled, .btn-outline-secondary:disabled {
+  color: #6c757d;
+  background-color: transparent;
+}
+.btn-outline-secondary:not(:disabled):not(.disabled):active, .btn-outline-secondary:not(:disabled):not(.disabled).active, .show > .btn-outline-secondary.dropdown-toggle {
+  color: #fff;
+  background-color: #6c757d;
+  border-color: #6c757d;
+}
+.btn-outline-secondary:not(:disabled):not(.disabled):active:focus, .btn-outline-secondary:not(:disabled):not(.disabled).active:focus, .show > .btn-outline-secondary.dropdown-toggle:focus {
+  box-shadow: 0 0 0 0.2rem rgba(108, 117, 125, 0.5);
+}
+
+.btn-outline-success {
+  color: #28a745;
+  background-color: transparent;
+  background-image: none;
+  border-color: #28a745;
+}
+.btn-outline-success:hover {
+  color: #fff;
+  background-color: #28a745;
+  border-color: #28a745;
+}
+.btn-outline-success:focus, .btn-outline-success.focus {
+  box-shadow: 0 0 0 0.2rem rgba(40, 167, 69, 0.5);
+}
+.btn-outline-success.disabled, .btn-outline-success:disabled {
+  color: #28a745;
+  background-color: transparent;
+}
+.btn-outline-success:not(:disabled):not(.disabled):active, .btn-outline-success:not(:disabled):not(.disabled).active, .show > .btn-outline-success.dropdown-toggle {
+  color: #fff;
+  background-color: #28a745;
+  border-color: #28a745;
+}
+.btn-outline-success:not(:disabled):not(.disabled):active:focus, .btn-outline-success:not(:disabled):not(.disabled).active:focus, .show > .btn-outline-success.dropdown-toggle:focus {
+  box-shadow: 0 0 0 0.2rem rgba(40, 167, 69, 0.5);
+}
+
+.btn-outline-info {
+  color: #17a2b8;
+  background-color: transparent;
+  background-image: none;
+  border-color: #17a2b8;
+}
+.btn-outline-info:hover {
+  color: #fff;
+  background-color: #17a2b8;
+  border-color: #17a2b8;
+}
+.btn-outline-info:focus, .btn-outline-info.focus {
+  box-shadow: 0 0 0 0.2rem rgba(23, 162, 184, 0.5);
+}
+.btn-outline-info.disabled, .btn-outline-info:disabled {
+  color: #17a2b8;
+  background-color: transparent;
+}
+.btn-outline-info:not(:disabled):not(.disabled):active, .btn-outline-info:not(:disabled):not(.disabled).active, .show > .btn-outline-info.dropdown-toggle {
+  color: #fff;
+  background-color: #17a2b8;
+  border-color: #17a2b8;
+}
+.btn-outline-info:not(:disabled):not(.disabled):active:focus, .btn-outline-info:not(:disabled):not(.disabled).active:focus, .show > .btn-outline-info.dropdown-toggle:focus {
+  box-shadow: 0 0 0 0.2rem rgba(23, 162, 184, 0.5);
+}
+
+.btn-outline-warning {
+  color: #ffc107;
+  background-color: transparent;
+  background-image: none;
+  border-color: #ffc107;
+}
+.btn-outline-warning:hover {
+  color: #212529;
+  background-color: #ffc107;
+  border-color: #ffc107;
+}
+.btn-outline-warning:focus, .btn-outline-warning.focus {
+  box-shadow: 0 0 0 0.2rem rgba(255, 193, 7, 0.5);
+}
+.btn-outline-warning.disabled, .btn-outline-warning:disabled {
+  color: #ffc107;
+  background-color: transparent;
+}
+.btn-outline-warning:not(:disabled):not(.disabled):active, .btn-outline-warning:not(:disabled):not(.disabled).active, .show > .btn-outline-warning.dropdown-toggle {
+  color: #212529;
+  background-color: #ffc107;
+  border-color: #ffc107;
+}
+.btn-outline-warning:not(:disabled):not(.disabled):active:focus, .btn-outline-warning:not(:disabled):not(.disabled).active:focus, .show > .btn-outline-warning.dropdown-toggle:focus {
+  box-shadow: 0 0 0 0.2rem rgba(255, 193, 7, 0.5);
+}
+
+.btn-outline-danger {
+  color: #dc3545;
+  background-color: transparent;
+  background-image: none;
+  border-color: #dc3545;
+}
+.btn-outline-danger:hover {
+  color: #fff;
+  background-color: #dc3545;
+  border-color: #dc3545;
+}
+.btn-outline-danger:focus, .btn-outline-danger.focus {
+  box-shadow: 0 0 0 0.2rem rgba(220, 53, 69, 0.5);
+}
+.btn-outline-danger.disabled, .btn-outline-danger:disabled {
+  color: #dc3545;
+  background-color: transparent;
+}
+.btn-outline-danger:not(:disabled):not(.disabled):active, .btn-outline-danger:not(:disabled):not(.disabled).active, .show > .btn-outline-danger.dropdown-toggle {
+  color: #fff;
+  background-color: #dc3545;
+  border-color: #dc3545;
+}
+.btn-outline-danger:not(:disabled):not(.disabled):active:focus, .btn-outline-danger:not(:disabled):not(.disabled).active:focus, .show > .btn-outline-danger.dropdown-toggle:focus {
+  box-shadow: 0 0 0 0.2rem rgba(220, 53, 69, 0.5);
+}
+
+.btn-outline-light {
+  color: #f8f9fa;
+  background-color: transparent;
+  background-image: none;
+  border-color: #f8f9fa;
+}
+.btn-outline-light:hover {
+  color: #212529;
+  background-color: #f8f9fa;
+  border-color: #f8f9fa;
+}
+.btn-outline-light:focus, .btn-outline-light.focus {
+  box-shadow: 0 0 0 0.2rem rgba(248, 249, 250, 0.5);
+}
+.btn-outline-light.disabled, .btn-outline-light:disabled {
+  color: #f8f9fa;
+  background-color: transparent;
+}
+.btn-outline-light:not(:disabled):not(.disabled):active, .btn-outline-light:not(:disabled):not(.disabled).active, .show > .btn-outline-light.dropdown-toggle {
+  color: #212529;
+  background-color: #f8f9fa;
+  border-color: #f8f9fa;
+}
+.btn-outline-light:not(:disabled):not(.disabled):active:focus, .btn-outline-light:not(:disabled):not(.disabled).active:focus, .show > .btn-outline-light.dropdown-toggle:focus {
+  box-shadow: 0 0 0 0.2rem rgba(248, 249, 250, 0.5);
+}
+
+.btn-outline-dark {
+  color: #343a40;
+  background-color: transparent;
+  background-image: none;
+  border-color: #343a40;
+}
+.btn-outline-dark:hover {
+  color: #fff;
+  background-color: #343a40;
+  border-color: #343a40;
+}
+.btn-outline-dark:focus, .btn-outline-dark.focus {
+  box-shadow: 0 0 0 0.2rem rgba(52, 58, 64, 0.5);
+}
+.btn-outline-dark.disabled, .btn-outline-dark:disabled {
+  color: #343a40;
+  background-color: transparent;
+}
+.btn-outline-dark:not(:disabled):not(.disabled):active, .btn-outline-dark:not(:disabled):not(.disabled).active, .show > .btn-outline-dark.dropdown-toggle {
+  color: #fff;
+  background-color: #343a40;
+  border-color: #343a40;
+}
+.btn-outline-dark:not(:disabled):not(.disabled):active:focus, .btn-outline-dark:not(:disabled):not(.disabled).active:focus, .show > .btn-outline-dark.dropdown-toggle:focus {
+  box-shadow: 0 0 0 0.2rem rgba(52, 58, 64, 0.5);
+}
+
+.btn-link {
+  font-weight: 400;
+  color: #007bff;
+  background-color: transparent;
+}
+.btn-link:hover {
+  color: #0056b3;
+  text-decoration: underline;
+  background-color: transparent;
+  border-color: transparent;
+}
+.btn-link:focus, .btn-link.focus {
+  text-decoration: underline;
+  border-color: transparent;
+  box-shadow: none;
+}
+.btn-link:disabled, .btn-link.disabled {
+  color: #6c757d;
+}
+
+.btn-lg, .btn-group-lg > .btn {
+  padding: 0.5rem 1rem;
+  font-size: 1.25rem;
+  line-height: 1.5;
+  border-radius: 0.3rem;
+}
+
+.btn-sm, .btn-group-sm > .btn {
+  padding: 0.25rem 0.5rem;
+  font-size: 0.875rem;
+  line-height: 1.5;
+  border-radius: 0.2rem;
+}
+
+.btn-block {
+  display: block;
+  width: 100%;
+}
+.btn-block + .btn-block {
+  margin-top: 0.5rem;
+}
+
+input[type="submit"].btn-block,
+input[type="reset"].btn-block,
+input[type="button"].btn-block {
+  width: 100%;
+}
+
+.fade {
+  opacity: 0;
+  transition: opacity 0.15s linear;
+}
+.fade.show {
+  opacity: 1;
+}
+
+.collapse {
+  display: none;
+}
+.collapse.show {
+  display: block;
+}
+
+tr.collapse.show {
+  display: table-row;
+}
+
+tbody.collapse.show {
+  display: table-row-group;
+}
+
+.collapsing {
+  position: relative;
+  height: 0;
+  overflow: hidden;
+  transition: height 0.35s ease;
+}
+
+.dropup,
+.dropdown {
+  position: relative;
+}
+
+.dropdown-toggle::after {
+  display: inline-block;
+  width: 0;
+  height: 0;
+  margin-left: 0.255em;
+  vertical-align: 0.255em;
+  content: "";
+  border-top: 0.3em solid;
+  border-right: 0.3em solid transparent;
+  border-bottom: 0;
+  border-left: 0.3em solid transparent;
+}
+.dropdown-toggle:empty::after {
+  margin-left: 0;
+}
+
+.dropdown-menu {
+  position: absolute;
+  top: 100%;
+  left: 0;
+  z-index: 1000;
+  display: none;
+  float: left;
+  min-width: 10rem;
+  padding: 0.5rem 0;
+  margin: 0.125rem 0 0;
+  font-size: 1rem;
+  color: #212529;
+  text-align: left;
+  list-style: none;
+  background-color: #fff;
+  background-clip: padding-box;
+  border: 1px solid rgba(0, 0, 0, 0.15);
+  border-radius: 0.25rem;
+}
+
+.dropup .dropdown-menu {
+  margin-top: 0;
+  margin-bottom: 0.125rem;
+}
+.dropup .dropdown-toggle::after {
+  display: inline-block;
+  width: 0;
+  height: 0;
+  margin-left: 0.255em;
+  vertical-align: 0.255em;
+  content: "";
+  border-top: 0;
+  border-right: 0.3em solid transparent;
+  border-bottom: 0.3em solid;
+  border-left: 0.3em solid transparent;
+}
+.dropup .dropdown-toggle:empty::after {
+  margin-left: 0;
+}
+
+.dropright .dropdown-menu {
+  margin-top: 0;
+  margin-left: 0.125rem;
+}
+.dropright .dropdown-toggle::after {
+  display: inline-block;
+  width: 0;
+  height: 0;
+  margin-left: 0.255em;
+  vertical-align: 0.255em;
+  content: "";
+  border-top: 0.3em solid transparent;
+  border-bottom: 0.3em solid transparent;
+  border-left: 0.3em solid;
+}
+.dropright .dropdown-toggle:empty::after {
+  margin-left: 0;
+}
+.dropright .dropdown-toggle::after {
+  vertical-align: 0;
+}
+
+.dropleft .dropdown-menu {
+  margin-top: 0;
+  margin-right: 0.125rem;
+}
+.dropleft .dropdown-toggle::after {
+  display: inline-block;
+  width: 0;
+  height: 0;
+  margin-left: 0.255em;
+  vertical-align: 0.255em;
+  content: "";
+}
+.dropleft .dropdown-toggle::after {
+  display: none;
+}
+.dropleft .dropdown-toggle::before {
+  display: inline-block;
+  width: 0;
+  height: 0;
+  margin-right: 0.255em;
+  vertical-align: 0.255em;
+  content: "";
+  border-top: 0.3em solid transparent;
+  border-right: 0.3em solid;
+  border-bottom: 0.3em solid transparent;
+}
+.dropleft .dropdown-toggle:empty::after {
+  margin-left: 0;
+}
+.dropleft .dropdown-toggle::before {
+  vertical-align: 0;
+}
+
+.dropdown-divider {
+  height: 0;
+  margin: 0.5rem 0;
+  overflow: hidden;
+  border-top: 1px solid #e9ecef;
+}
+
+.dropdown-item {
+  display: block;
+  width: 100%;
+  padding: 0.25rem 1.5rem;
+  clear: both;
+  font-weight: 400;
+  color: #212529;
+  text-align: inherit;
+  white-space: nowrap;
+  background-color: transparent;
+  border: 0;
+}
+.dropdown-item:hover, .dropdown-item:focus {
+  color: #16181b;
+  text-decoration: none;
+  background-color: #f8f9fa;
+}
+.dropdown-item.active, .dropdown-item:active {
+  color: #fff;
+  text-decoration: none;
+  background-color: #007bff;
+}
+.dropdown-item.disabled, .dropdown-item:disabled {
+  color: #6c757d;
+  background-color: transparent;
+}
+
+.dropdown-menu.show {
+  display: block;
+}
+
+.dropdown-header {
+  display: block;
+  padding: 0.5rem 1.5rem;
+  margin-bottom: 0;
+  font-size: 0.875rem;
+  color: #6c757d;
+  white-space: nowrap;
+}
+
+.btn-group,
+.btn-group-vertical {
+  position: relative;
+  display: inline-flex;
+  vertical-align: middle;
+}
+.btn-group > .btn,
+.btn-group-vertical > .btn {
+  position: relative;
+  flex: 0 1 auto;
+}
+.btn-group > .btn:hover,
+.btn-group-vertical > .btn:hover {
+  z-index: 1;
+}
+.btn-group > .btn:focus, .btn-group > .btn:active, .btn-group > .btn.active,
+.btn-group-vertical > .btn:focus,
+.btn-group-vertical > .btn:active,
+.btn-group-vertical > .btn.active {
+  z-index: 1;
+}
+.btn-group .btn + .btn,
+.btn-group .btn + .btn-group,
+.btn-group .btn-group + .btn,
+.btn-group .btn-group + .btn-group,
+.btn-group-vertical .btn + .btn,
+.btn-group-vertical .btn + .btn-group,
+.btn-group-vertical .btn-group + .btn,
+.btn-group-vertical .btn-group + .btn-group {
+  margin-left: -1px;
+}
+
+.btn-toolbar {
+  display: flex;
+  flex-wrap: wrap;
+  justify-content: flex-start;
+}
+.btn-toolbar .input-group {
+  width: auto;
+}
+
+.btn-group > .btn:first-child {
+  margin-left: 0;
+}
+.btn-group > .btn:not(:last-child):not(.dropdown-toggle),
+.btn-group > .btn-group:not(:last-child) > .btn {
+  border-top-right-radius: 0;
+  border-bottom-right-radius: 0;
+}
+.btn-group > .btn:not(:first-child),
+.btn-group > .btn-group:not(:first-child) > .btn {
+  border-top-left-radius: 0;
+  border-bottom-left-radius: 0;
+}
+
+.dropdown-toggle-split {
+  padding-right: 0.5625rem;
+  padding-left: 0.5625rem;
+}
+.dropdown-toggle-split::after {
+  margin-left: 0;
+}
+
+.btn-sm + .dropdown-toggle-split, .btn-group-sm > .btn + .dropdown-toggle-split {
+  padding-right: 0.375rem;
+  padding-left: 0.375rem;
+}
+
+.btn-lg + .dropdown-toggle-split, .btn-group-lg > .btn + .dropdown-toggle-split {
+  padding-right: 0.75rem;
+  padding-left: 0.75rem;
+}
+
+.btn-group-vertical {
+  flex-direction: column;
+  align-items: flex-start;
+  justify-content: center;
+}
+.btn-group-vertical .btn,
+.btn-group-vertical .btn-group {
+  width: 100%;
+}
+.btn-group-vertical > .btn + .btn,
+.btn-group-vertical > .btn + .btn-group,
+.btn-group-vertical > .btn-group + .btn,
+.btn-group-vertical > .btn-group + .btn-group {
+  margin-top: -1px;
+  margin-left: 0;
+}
+.btn-group-vertical > .btn:not(:last-child):not(.dropdown-toggle),
+.btn-group-vertical > .btn-group:not(:last-child) > .btn {
+  border-bottom-right-radius: 0;
+  border-bottom-left-radius: 0;
+}
+.btn-group-vertical > .btn:not(:first-child),
+.btn-group-vertical > .btn-group:not(:first-child) > .btn {
+  border-top-left-radius: 0;
+  border-top-right-radius: 0;
+}
+
+.btn-group-toggle > .btn,
+.btn-group-toggle > .btn-group > .btn {
+  margin-bottom: 0;
+}
+.btn-group-toggle > .btn input[type="radio"],
+.btn-group-toggle > .btn input[type="checkbox"],
+.btn-group-toggle > .btn-group > .btn input[type="radio"],
+.btn-group-toggle > .btn-group > .btn input[type="checkbox"] {
+  position: absolute;
+  clip: rect(0, 0, 0, 0);
+  pointer-events: none;
+}
+
+.input-group {
+  position: relative;
+  display: flex;
+  flex-wrap: wrap;
+  align-items: stretch;
+  width: 100%;
+}
+.input-group > .form-control,
+.input-group > .custom-select,
+.input-group > .custom-file {
+  position: relative;
+  flex: 1 1 auto;
+  width: 1%;
+  margin-bottom: 0;
+}
+.input-group > .form-control:focus,
+.input-group > .custom-select:focus,
+.input-group > .custom-file:focus {
+  z-index: 3;
+}
+.input-group > .form-control + .form-control,
+.input-group > .form-control + .custom-select,
+.input-group > .form-control + .custom-file,
+.input-group > .custom-select + .form-control,
+.input-group > .custom-select + .custom-select,
+.input-group > .custom-select + .custom-file,
+.input-group > .custom-file + .form-control,
+.input-group > .custom-file + .custom-select,
+.input-group > .custom-file + .custom-file {
+  margin-left: -1px;
+}
+.input-group > .form-control:not(:last-child),
+.input-group > .custom-select:not(:last-child) {
+  border-top-right-radius: 0;
+  border-bottom-right-radius: 0;
+}
+.input-group > .form-control:not(:first-child),
+.input-group > .custom-select:not(:first-child) {
+  border-top-left-radius: 0;
+  border-bottom-left-radius: 0;
+}
+.input-group > .custom-file {
+  display: flex;
+  align-items: center;
+}
+.input-group > .custom-file:not(:last-child) .custom-file-label, .input-group > .custom-file:not(:last-child) .custom-file-label::before {
+  border-top-right-radius: 0;
+  border-bottom-right-radius: 0;
+}
+.input-group > .custom-file:not(:first-child) .custom-file-label, .input-group > .custom-file:not(:first-child) .custom-file-label::before {
+  border-top-left-radius: 0;
+  border-bottom-left-radius: 0;
+}
+
+.input-group-prepend,
+.input-group-append {
+  display: flex;
+}
+.input-group-prepend .btn,
+.input-group-append .btn {
+  position: relative;
+  z-index: 2;
+}
+.input-group-prepend .btn + .btn,
+.input-group-prepend .btn + .input-group-text,
+.input-group-prepend .input-group-text + .input-group-text,
+.input-group-prepend .input-group-text + .btn,
+.input-group-append .btn + .btn,
+.input-group-append .btn + .input-group-text,
+.input-group-append .input-group-text + .input-group-text,
+.input-group-append .input-group-text + .btn {
+  margin-left: -1px;
+}
+
+.input-group-prepend {
+  margin-right: -1px;
+}
+
+.input-group-append {
+  margin-left: -1px;
+}
+
+.input-group-text {
+  display: flex;
+  align-items: center;
+  padding: 0.375rem 0.75rem;
+  margin-bottom: 0;
+  font-size: 1rem;
+  font-weight: 400;
+  line-height: 1.5;
+  color: #495057;
+  text-align: center;
+  white-space: nowrap;
+  background-color: #e9ecef;
+  border: 1px solid #ced4da;
+  border-radius: 0.25rem;
+}
+.input-group-text input[type="radio"],
+.input-group-text input[type="checkbox"] {
+  margin-top: 0;
+}
+
+.input-group > .input-group-prepend > .btn,
+.input-group > .input-group-prepend > .input-group-text,
+.input-group > .input-group-append:not(:last-child) > .btn,
+.input-group > .input-group-append:not(:last-child) > .input-group-text,
+.input-group > .input-group-append:last-child > .btn:not(:last-child):not(.dropdown-toggle),
+.input-group > .input-group-append:last-child > .input-group-text:not(:last-child) {
+  border-top-right-radius: 0;
+  border-bottom-right-radius: 0;
+}
+
+.input-group > .input-group-append > .btn,
+.input-group > .input-group-append > .input-group-text,
+.input-group > .input-group-prepend:not(:first-child) > .btn,
+.input-group > .input-group-prepend:not(:first-child) > .input-group-text,
+.input-group > .input-group-prepend:first-child > .btn:not(:first-child),
+.input-group > .input-group-prepend:first-child > .input-group-text:not(:first-child) {
+  border-top-left-radius: 0;
+  border-bottom-left-radius: 0;
+}
+
+.custom-control {
+  position: relative;
+  display: block;
+  min-height: 1.5rem;
+  padding-left: 1.5rem;
+}
+
+.custom-control-inline {
+  display: inline-flex;
+  margin-right: 1rem;
+}
+
+.custom-control-input {
+  position: absolute;
+  z-index: -1;
+  opacity: 0;
+}
+.custom-control-input:checked ~ .custom-control-label::before {
+  color: #fff;
+  background-color: #007bff;
+}
+.custom-control-input:focus ~ .custom-control-label::before {
+  box-shadow: 0 0 0 1px #fff, 0 0 0 0.2rem rgba(0, 123, 255, 0.25);
+}
+.custom-control-input:active ~ .custom-control-label::before {
+  color: #fff;
+  background-color: #b3d7ff;
+}
+.custom-control-input:disabled ~ .custom-control-label {
+  color: #6c757d;
+}
+.custom-control-input:disabled ~ .custom-control-label::before {
+  background-color: #e9ecef;
+}
+
+.custom-control-label {
+  margin-bottom: 0;
+}
+.custom-control-label::before {
+  position: absolute;
+  top: 0.25rem;
+  left: 0;
+  display: block;
+  width: 1rem;
+  height: 1rem;
+  pointer-events: none;
+  content: "";
+  -webkit-user-select: none;
+     -moz-user-select: none;
+      -ms-user-select: none;
+          user-select: none;
+  background-color: #dee2e6;
+}
+.custom-control-label::after {
+  position: absolute;
+  top: 0.25rem;
+  left: 0;
+  display: block;
+  width: 1rem;
+  height: 1rem;
+  content: "";
+  background-repeat: no-repeat;
+  background-position: center center;
+  background-size: 50% 50%;
+}
+
+.custom-checkbox .custom-control-label::before {
+  border-radius: 0.25rem;
+}
+.custom-checkbox .custom-control-input:checked ~ .custom-control-label::before {
+  background-color: #007bff;
+}
+.custom-checkbox .custom-control-input:checked ~ .custom-control-label::after {
+  background-image: url("data:image/svg+xml;charset=utf8,%3Csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 8 8'%3E%3Cpath fill='%23fff' d='M6.564.75l-3.59 3.612-1.538-1.55L0 4.26 2.974 7.25 8 2.193z'/%3E%3C/svg%3E");
+}
+.custom-checkbox .custom-control-input:indeterminate ~ .custom-control-label::before {
+  background-color: #007bff;
+}
+.custom-checkbox .custom-control-input:indeterminate ~ .custom-control-label::after {
+  background-image: url("data:image/svg+xml;charset=utf8,%3Csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 4 4'%3E%3Cpath stroke='%23fff' d='M0 2h4'/%3E%3C/svg%3E");
+}
+.custom-checkbox .custom-control-input:disabled:checked ~ .custom-control-label::before {
+  background-color: rgba(0, 123, 255, 0.5);
+}
+.custom-checkbox .custom-control-input:disabled:indeterminate ~ .custom-control-label::before {
+  background-color: rgba(0, 123, 255, 0.5);
+}
+
+.custom-radio .custom-control-label::before {
+  border-radius: 50%;
+}
+.custom-radio .custom-control-input:checked ~ .custom-control-label::before {
+  background-color: #007bff;
+}
+.custom-radio .custom-control-input:checked ~ .custom-control-label::after {
+  background-image: url("data:image/svg+xml;charset=utf8,%3Csvg xmlns='http://www.w3.org/2000/svg' viewBox='-4 -4 8 8'%3E%3Ccircle r='3' fill='%23fff'/%3E%3C/svg%3E");
+}
+.custom-radio .custom-control-input:disabled:checked ~ .custom-control-label::before {
+  background-color: rgba(0, 123, 255, 0.5);
+}
+
+.custom-select {
+  display: inline-block;
+  width: 100%;
+  height: calc(2.25rem + 2px);
+  padding: 0.375rem 1.75rem 0.375rem 0.75rem;
+  line-height: 1.5;
+  color: #495057;
+  vertical-align: middle;
+  background: #fff url("data:image/svg+xml;charset=utf8,%3Csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 4 5'%3E%3Cpath fill='%23343a40' d='M2 0L0 2h4zm0 5L0 3h4z'/%3E%3C/svg%3E") no-repeat right 0.75rem center;
+  background-size: 8px 10px;
+  border: 1px solid #ced4da;
+  border-radius: 0.25rem;
+  -webkit-appearance: none;
+     -moz-appearance: none;
+          appearance: none;
+}
+.custom-select:focus {
+  border-color: #80bdff;
+  outline: 0;
+  box-shadow: inset 0 1px 2px rgba(0, 0, 0, 0.075), 0 0 5px rgba(128, 189, 255, 0.5);
+}
+.custom-select:focus::-ms-value {
+  color: #495057;
+  background-color: #fff;
+}
+.custom-select[multiple], .custom-select[size]:not([size="1"]) {
+  height: auto;
+  padding-right: 0.75rem;
+  background-image: none;
+}
+.custom-select:disabled {
+  color: #6c757d;
+  background-color: #e9ecef;
+}
+.custom-select::-ms-expand {
+  opacity: 0;
+}
+
+.custom-select-sm {
+  height: calc(1.8125rem + 2px);
+  padding-top: 0.375rem;
+  padding-bottom: 0.375rem;
+  font-size: 75%;
+}
+
+.custom-select-lg {
+  height: calc(2.875rem + 2px);
+  padding-top: 0.375rem;
+  padding-bottom: 0.375rem;
+  font-size: 125%;
+}
+
+.custom-file {
+  position: relative;
+  display: inline-block;
+  width: 100%;
+  height: calc(2.25rem + 2px);
+  margin-bottom: 0;
+}
+
+.custom-file-input {
+  position: relative;
+  z-index: 2;
+  width: 100%;
+  height: calc(2.25rem + 2px);
+  margin: 0;
+  opacity: 0;
+}
+.custom-file-input:focus ~ .custom-file-control {
+  border-color: #80bdff;
+  box-shadow: 0 0 0 0.2rem rgba(0, 123, 255, 0.25);
+}
+.custom-file-input:focus ~ .custom-file-control::before {
+  border-color: #80bdff;
+}
+.custom-file-input:lang(en) ~ .custom-file-label::after {
+  content: "Browse";
+}
+
+.custom-file-label {
+  position: absolute;
+  top: 0;
+  right: 0;
+  left: 0;
+  z-index: 1;
+  height: calc(2.25rem + 2px);
+  padding: 0.375rem 0.75rem;
+  line-height: 1.5;
+  color: #495057;
+  background-color: #fff;
+  border: 1px solid #ced4da;
+  border-radius: 0.25rem;
+}
+.custom-file-label::after {
+  position: absolute;
+  top: 0;
+  right: 0;
+  bottom: 0;
+  z-index: 3;
+  display: block;
+  height: calc(calc(2.25rem + 2px) - 1px * 2);
+  padding: 0.375rem 0.75rem;
+  line-height: 1.5;
+  color: #495057;
+  content: "Browse";
+  background-color: #e9ecef;
+  border-left: 1px solid #ced4da;
+  border-radius: 0 0.25rem 0.25rem 0;
+}
+
+.nav {
+  display: flex;
+  flex-wrap: wrap;
+  padding-left: 0;
+  margin-bottom: 0;
+  list-style: none;
+}
+
+.nav-link {
+  display: block;
+  padding: 0.5rem 1rem;
+}
+.nav-link:hover, .nav-link:focus {
+  text-decoration: none;
+}
+.nav-link.disabled {
+  color: #6c757d;
+}
+
+.nav-tabs {
+  border-bottom: 1px solid #dee2e6;
+}
+.nav-tabs .nav-item {
+  margin-bottom: -1px;
+}
+.nav-tabs .nav-link {
+  border: 1px solid transparent;
+  border-top-left-radius: 0.25rem;
+  border-top-right-radius: 0.25rem;
+}
+.nav-tabs .nav-link:hover, .nav-tabs .nav-link:focus {
+  border-color: #e9ecef #e9ecef #dee2e6;
+}
+.nav-tabs .nav-link.disabled {
+  color: #6c757d;
+  background-color: transparent;
+  border-color: transparent;
+}
+.nav-tabs .nav-link.active,
+.nav-tabs .nav-item.show .nav-link {
+  color: #495057;
+  background-color: #fff;
+  border-color: #dee2e6 #dee2e6 #fff;
+}
+.nav-tabs .dropdown-menu {
+  margin-top: -1px;
+  border-top-left-radius: 0;
+  border-top-right-radius: 0;
+}
+
+.nav-pills .nav-link {
+  border-radius: 0.25rem;
+}
+.nav-pills .nav-link.active,
+.nav-pills .show > .nav-link {
+  color: #fff;
+  background-color: #007bff;
+}
+
+.nav-fill .nav-item {
+  flex: 1 1 auto;
+  text-align: center;
+}
+
+.nav-justified .nav-item {
+  flex-basis: 0;
+  flex-grow: 1;
+  text-align: center;
+}
+
+.tab-content > .tab-pane {
+  display: none;
+}
+.tab-content > .active {
+  display: block;
+}
+
+.navbar {
+  position: relative;
+  display: flex;
+  flex-wrap: wrap;
+  align-items: center;
+  justify-content: space-between;
+  padding: 0.5rem 1rem;
+}
+.navbar > .container,
+.navbar > .container-fluid {
+  display: flex;
+  flex-wrap: wrap;
+  align-items: center;
+  justify-content: space-between;
+}
+
+.navbar-brand {
+  display: inline-block;
+  padding-top: 0.3125rem;
+  padding-bottom: 0.3125rem;
+  margin-right: 1rem;
+  font-size: 1.25rem;
+  line-height: inherit;
+  white-space: nowrap;
+}
+.navbar-brand:hover, .navbar-brand:focus {
+  text-decoration: none;
+}
+
+.navbar-nav {
+  display: flex;
+  flex-direction: column;
+  padding-left: 0;
+  margin-bottom: 0;
+  list-style: none;
+}
+.navbar-nav .nav-link {
+  padding-right: 0;
+  padding-left: 0;
+}
+.navbar-nav .dropdown-menu {
+  position: static;
+  float: none;
+}
+
+.navbar-text {
+  display: inline-block;
+  padding-top: 0.5rem;
+  padding-bottom: 0.5rem;
+}
+
+.navbar-collapse {
+  flex-basis: 100%;
+  flex-grow: 1;
+  align-items: center;
+}
+
+.navbar-toggler {
+  padding: 0.25rem 0.75rem;
+  font-size: 1.25rem;
+  line-height: 1;
+  background-color: transparent;
+  border: 1px solid transparent;
+  border-radius: 0.25rem;
+}
+.navbar-toggler:hover, .navbar-toggler:focus {
+  text-decoration: none;
+}
+.navbar-toggler:not(:disabled):not(.disabled) {
+  cursor: pointer;
+}
+
+.navbar-toggler-icon {
+  display: inline-block;
+  width: 1.5em;
+  height: 1.5em;
+  vertical-align: middle;
+  content: "";
+  background: no-repeat center center;
+  background-size: 100% 100%;
+}
+
+@media (max-width: 575.98px) {
+  .navbar-expand-sm > .container,
+  .navbar-expand-sm > .container-fluid {
+    padding-right: 0;
+    padding-left: 0;
+  }
+}
+@media (min-width: 576px) {
+  .navbar-expand-sm {
+    flex-flow: row nowrap;
+    justify-content: flex-start;
+  }
+  .navbar-expand-sm .navbar-nav {
+    flex-direction: row;
+  }
+  .navbar-expand-sm .navbar-nav .dropdown-menu {
+    position: absolute;
+  }
+  .navbar-expand-sm .navbar-nav .dropdown-menu-right {
+    right: 0;
+    left: auto;
+  }
+  .navbar-expand-sm .navbar-nav .nav-link {
+    padding-right: 0.5rem;
+    padding-left: 0.5rem;
+  }
+  .navbar-expand-sm > .container,
+  .navbar-expand-sm > .container-fluid {
+    flex-wrap: nowrap;
+  }
+  .navbar-expand-sm .navbar-collapse {
+    display: flex !important;
+    flex-basis: auto;
+  }
+  .navbar-expand-sm .navbar-toggler {
+    display: none;
+  }
+  .navbar-expand-sm .dropup .dropdown-menu {
+    top: auto;
+    bottom: 100%;
+  }
+}
+@media (max-width: 767.98px) {
+  .navbar-expand-md > .container,
+  .navbar-expand-md > .container-fluid {
+    padding-right: 0;
+    padding-left: 0;
+  }
+}
+@media (min-width: 768px) {
+  .navbar-expand-md {
+    flex-flow: row nowrap;
+    justify-content: flex-start;
+  }
+  .navbar-expand-md .navbar-nav {
+    flex-direction: row;
+  }
+  .navbar-expand-md .navbar-nav .dropdown-menu {
+    position: absolute;
+  }
+  .navbar-expand-md .navbar-nav .dropdown-menu-right {
+    right: 0;
+    left: auto;
+  }
+  .navbar-expand-md .navbar-nav .nav-link {
+    padding-right: 0.5rem;
+    padding-left: 0.5rem;
+  }
+  .navbar-expand-md > .container,
+  .navbar-expand-md > .container-fluid {
+    flex-wrap: nowrap;
+  }
+  .navbar-expand-md .navbar-collapse {
+    display: flex !important;
+    flex-basis: auto;
+  }
+  .navbar-expand-md .navbar-toggler {
+    display: none;
+  }
+  .navbar-expand-md .dropup .dropdown-menu {
+    top: auto;
+    bottom: 100%;
+  }
+}
+@media (max-width: 991.98px) {
+  .navbar-expand-lg > .container,
+  .navbar-expand-lg > .container-fluid {
+    padding-right: 0;
+    padding-left: 0;
+  }
+}
+@media (min-width: 992px) {
+  .navbar-expand-lg {
+    flex-flow: row nowrap;
+    justify-content: flex-start;
+  }
+  .navbar-expand-lg .navbar-nav {
+    flex-direction: row;
+  }
+  .navbar-expand-lg .navbar-nav .dropdown-menu {
+    position: absolute;
+  }
+  .navbar-expand-lg .navbar-nav .dropdown-menu-right {
+    right: 0;
+    left: auto;
+  }
+  .navbar-expand-lg .navbar-nav .nav-link {
+    padding-right: 0.5rem;
+    padding-left: 0.5rem;
+  }
+  .navbar-expand-lg > .container,
+  .navbar-expand-lg > .container-fluid {
+    flex-wrap: nowrap;
+  }
+  .navbar-expand-lg .navbar-collapse {
+    display: flex !important;
+    flex-basis: auto;
+  }
+  .navbar-expand-lg .navbar-toggler {
+    display: none;
+  }
+  .navbar-expand-lg .dropup .dropdown-menu {
+    top: auto;
+    bottom: 100%;
+  }
+}
+@media (max-width: 1199.98px) {
+  .navbar-expand-xl > .container,
+  .navbar-expand-xl > .container-fluid {
+    padding-right: 0;
+    padding-left: 0;
+  }
+}
+@media (min-width: 1200px) {
+  .navbar-expand-xl {
+    flex-flow: row nowrap;
+    justify-content: flex-start;
+  }
+  .navbar-expand-xl .navbar-nav {
+    flex-direction: row;
+  }
+  .navbar-expand-xl .navbar-nav .dropdown-menu {
+    position: absolute;
+  }
+  .navbar-expand-xl .navbar-nav .dropdown-menu-right {
+    right: 0;
+    left: auto;
+  }
+  .navbar-expand-xl .navbar-nav .nav-link {
+    padding-right: 0.5rem;
+    padding-left: 0.5rem;
+  }
+  .navbar-expand-xl > .container,
+  .navbar-expand-xl > .container-fluid {
+    flex-wrap: nowrap;
+  }
+  .navbar-expand-xl .navbar-collapse {
+    display: flex !important;
+    flex-basis: auto;
+  }
+  .navbar-expand-xl .navbar-toggler {
+    display: none;
+  }
+  .navbar-expand-xl .dropup .dropdown-menu {
+    top: auto;
+    bottom: 100%;
+  }
+}
+.navbar-expand {
+  flex-flow: row nowrap;
+  justify-content: flex-start;
+}
+.navbar-expand > .container,
+.navbar-expand > .container-fluid {
+  padding-right: 0;
+  padding-left: 0;
+}
+.navbar-expand .navbar-nav {
+  flex-direction: row;
+}
+.navbar-expand .navbar-nav .dropdown-menu {
+  position: absolute;
+}
+.navbar-expand .navbar-nav .dropdown-menu-right {
+  right: 0;
+  left: auto;
+}
+.navbar-expand .navbar-nav .nav-link {
+  padding-right: 0.5rem;
+  padding-left: 0.5rem;
+}
+.navbar-expand > .container,
+.navbar-expand > .container-fluid {
+  flex-wrap: nowrap;
+}
+.navbar-expand .navbar-collapse {
+  display: flex !important;
+  flex-basis: auto;
+}
+.navbar-expand .navbar-toggler {
+  display: none;
+}
+.navbar-expand .dropup .dropdown-menu {
+  top: auto;
+  bottom: 100%;
+}
+
+.navbar-light .navbar-brand {
+  color: rgba(0, 0, 0, 0.9);
+}
+.navbar-light .navbar-brand:hover, .navbar-light .navbar-brand:focus {
+  color: rgba(0, 0, 0, 0.9);
+}
+.navbar-light .navbar-nav .nav-link {
+  color: rgba(0, 0, 0, 0.5);
+}
+.navbar-light .navbar-nav .nav-link:hover, .navbar-light .navbar-nav .nav-link:focus {
+  color: rgba(0, 0, 0, 0.7);
+}
+.navbar-light .navbar-nav .nav-link.disabled {
+  color: rgba(0, 0, 0, 0.3);
+}
+.navbar-light .navbar-nav .show > .nav-link,
+.navbar-light .navbar-nav .active > .nav-link,
+.navbar-light .navbar-nav .nav-link.show,
+.navbar-light .navbar-nav .nav-link.active {
+  color: rgba(0, 0, 0, 0.9);
+}
+.navbar-light .navbar-toggler {
+  color: rgba(0, 0, 0, 0.5);
+  border-color: rgba(0, 0, 0, 0.1);
+}
+.navbar-light .navbar-toggler-icon {
+  background-image: url("data:image/svg+xml;charset=utf8,%3Csvg viewBox='0 0 30 30' xmlns='http://www.w3.org/2000/svg'%3E%3Cpath stroke='rgba(0, 0, 0, 0.5)' stroke-width='2' stroke-linecap='round' stroke-miterlimit='10' d='M4 7h22M4 15h22M4 23h22'/%3E%3C/svg%3E");
+}
+.navbar-light .navbar-text {
+  color: rgba(0, 0, 0, 0.5);
+}
+.navbar-light .navbar-text a {
+  color: rgba(0, 0, 0, 0.9);
+}
+.navbar-light .navbar-text a:hover, .navbar-light .navbar-text a:focus {
+  color: rgba(0, 0, 0, 0.9);
+}
+
+.navbar-dark .navbar-brand {
+  color: #fff;
+}
+.navbar-dark .navbar-brand:hover, .navbar-dark .navbar-brand:focus {
+  color: #fff;
+}
+.navbar-dark .navbar-nav .nav-link {
+  color: rgba(255, 255, 255, 0.5);
+}
+.navbar-dark .navbar-nav .nav-link:hover, .navbar-dark .navbar-nav .nav-link:focus {
+  color: rgba(255, 255, 255, 0.75);
+}
+.navbar-dark .navbar-nav .nav-link.disabled {
+  color: rgba(255, 255, 255, 0.25);
+}
+.navbar-dark .navbar-nav .show > .nav-link,
+.navbar-dark .navbar-nav .active > .nav-link,
+.navbar-dark .navbar-nav .nav-link.show,
+.navbar-dark .navbar-nav .nav-link.active {
+  color: #fff;
+}
+.navbar-dark .navbar-toggler {
+  color: rgba(255, 255, 255, 0.5);
+  border-color: rgba(255, 255, 255, 0.1);
+}
+.navbar-dark .navbar-toggler-icon {
+  background-image: url("data:image/svg+xml;charset=utf8,%3Csvg viewBox='0 0 30 30' xmlns='http://www.w3.org/2000/svg'%3E%3Cpath stroke='rgba(255, 255, 255, 0.5)' stroke-width='2' stroke-linecap='round' stroke-miterlimit='10' d='M4 7h22M4 15h22M4 23h22'/%3E%3C/svg%3E");
+}
+.navbar-dark .navbar-text {
+  color: rgba(255, 255, 255, 0.5);
+}
+.navbar-dark .navbar-text a {
+  color: #fff;
+}
+.navbar-dark .navbar-text a:hover, .navbar-dark .navbar-text a:focus {
+  color: #fff;
+}
+
+.card {
+  position: relative;
+  display: flex;
+  flex-direction: column;
+  min-width: 0;
+  word-wrap: break-word;
+  background-color: #fff;
+  background-clip: border-box;
+  border: 1px solid rgba(0, 0, 0, 0.125);
+  border-radius: 0.25rem;
+}
+.card > hr {
+  margin-right: 0;
+  margin-left: 0;
+}
+.card > .list-group:first-child .list-group-item:first-child {
+  border-top-left-radius: 0.25rem;
+  border-top-right-radius: 0.25rem;
+}
+.card > .list-group:last-child .list-group-item:last-child {
+  border-bottom-right-radius: 0.25rem;
+  border-bottom-left-radius: 0.25rem;
+}
+
+.card-body {
+  flex: 1 1 auto;
+  padding: 1.25rem;
+}
+
+.card-title {
+  margin-bottom: 0.75rem;
+}
+
+.card-subtitle {
+  margin-top: -0.375rem;
+  margin-bottom: 0;
+}
+
+.card-text:last-child {
+  margin-bottom: 0;
+}
+
+.card-link:hover {
+  text-decoration: none;
+}
+.card-link + .card-link {
+  margin-left: 1.25rem;
+}
+
+.card-header {
+  padding: 0.75rem 1.25rem;
+  margin-bottom: 0;
+  background-color: rgba(0, 0, 0, 0.03);
+  border-bottom: 1px solid rgba(0, 0, 0, 0.125);
+}
+.card-header:first-child {
+  border-radius: calc(0.25rem - 1px) calc(0.25rem - 1px) 0 0;
+}
+.card-header + .list-group .list-group-item:first-child {
+  border-top: 0;
+}
+
+.card-footer {
+  padding: 0.75rem 1.25rem;
+  background-color: rgba(0, 0, 0, 0.03);
+  border-top: 1px solid rgba(0, 0, 0, 0.125);
+}
+.card-footer:last-child {
+  border-radius: 0 0 calc(0.25rem - 1px) calc(0.25rem - 1px);
+}
+
+.card-header-tabs {
+  margin-right: -0.625rem;
+  margin-bottom: -0.75rem;
+  margin-left: -0.625rem;
+  border-bottom: 0;
+}
+
+.card-header-pills {
+  margin-right: -0.625rem;
+  margin-left: -0.625rem;
+}
+
+.card-img-overlay {
+  position: absolute;
+  top: 0;
+  right: 0;
+  bottom: 0;
+  left: 0;
+  padding: 1.25rem;
+}
+
+.card-img {
+  width: 100%;
+  border-radius: calc(0.25rem - 1px);
+}
+
+.card-img-top {
+  width: 100%;
+  border-top-left-radius: calc(0.25rem - 1px);
+  border-top-right-radius: calc(0.25rem - 1px);
+}
+
+.card-img-bottom {
+  width: 100%;
+  border-bottom-right-radius: calc(0.25rem - 1px);
+  border-bottom-left-radius: calc(0.25rem - 1px);
+}
+
+.card-deck {
+  display: flex;
+  flex-direction: column;
+}
+.card-deck .card {
+  margin-bottom: 15px;
+}
+@media (min-width: 576px) {
+  .card-deck {
+    flex-flow: row wrap;
+    margin-right: -15px;
+    margin-left: -15px;
+  }
+  .card-deck .card {
+    display: flex;
+    flex: 1 0 0%;
+    flex-direction: column;
+    margin-right: 15px;
+    margin-bottom: 0;
+    margin-left: 15px;
+  }
+}
+
+.card-group {
+  display: flex;
+  flex-direction: column;
+}
+.card-group > .card {
+  margin-bottom: 15px;
+}
+@media (min-width: 576px) {
+  .card-group {
+    flex-flow: row wrap;
+  }
+  .card-group > .card {
+    flex: 1 0 0%;
+    margin-bottom: 0;
+  }
+  .card-group > .card + .card {
+    margin-left: 0;
+    border-left: 0;
+  }
+  .card-group > .card:first-child {
+    border-top-right-radius: 0;
+    border-bottom-right-radius: 0;
+  }
+  .card-group > .card:first-child .card-img-top,
+  .card-group > .card:first-child .card-header {
+    border-top-right-radius: 0;
+  }
+  .card-group > .card:first-child .card-img-bottom,
+  .card-group > .card:first-child .card-footer {
+    border-bottom-right-radius: 0;
+  }
+  .card-group > .card:last-child {
+    border-top-left-radius: 0;
+    border-bottom-left-radius: 0;
+  }
+  .card-group > .card:last-child .card-img-top,
+  .card-group > .card:last-child .card-header {
+    border-top-left-radius: 0;
+  }
+  .card-group > .card:last-child .card-img-bottom,
+  .card-group > .card:last-child .card-footer {
+    border-bottom-left-radius: 0;
+  }
+  .card-group > .card:only-child {
+    border-radius: 0.25rem;
+  }
+  .card-group > .card:only-child .card-img-top,
+  .card-group > .card:only-child .card-header {
+    border-top-left-radius: 0.25rem;
+    border-top-right-radius: 0.25rem;
+  }
+  .card-group > .card:only-child .card-img-bottom,
+  .card-group > .card:only-child .card-footer {
+    border-bottom-right-radius: 0.25rem;
+    border-bottom-left-radius: 0.25rem;
+  }
+  .card-group > .card:not(:first-child):not(:last-child):not(:only-child) {
+    border-radius: 0;
+  }
+  .card-group > .card:not(:first-child):not(:last-child):not(:only-child) .card-img-top,
+  .card-group > .card:not(:first-child):not(:last-child):not(:only-child) .card-img-bottom,
+  .card-group > .card:not(:first-child):not(:last-child):not(:only-child) .card-header,
+  .card-group > .card:not(:first-child):not(:last-child):not(:only-child) .card-footer {
+    border-radius: 0;
+  }
+}
+
+.card-columns .card {
+  margin-bottom: 0.75rem;
+}
+@media (min-width: 576px) {
+  .card-columns {
+    -moz-column-count: 3;
+         column-count: 3;
+    -moz-column-gap: 1.25rem;
+         column-gap: 1.25rem;
+  }
+  .card-columns .card {
+    display: inline-block;
+    width: 100%;
+  }
+}
+
+.breadcrumb {
+  display: flex;
+  flex-wrap: wrap;
+  padding: 0.75rem 1rem;
+  margin-bottom: 1rem;
+  list-style: none;
+  background-color: #e9ecef;
+  border-radius: 0.25rem;
+}
+
+.breadcrumb-item + .breadcrumb-item::before {
+  display: inline-block;
+  padding-right: 0.5rem;
+  padding-left: 0.5rem;
+  color: #6c757d;
+  content: "/";
+}
+.breadcrumb-item + .breadcrumb-item:hover::before {
+  text-decoration: underline;
+}
+.breadcrumb-item + .breadcrumb-item:hover::before {
+  text-decoration: none;
+}
+.breadcrumb-item.active {
+  color: #6c757d;
+}
+
+.pagination {
+  display: flex;
+  padding-left: 0;
+  list-style: none;
+  border-radius: 0.25rem;
+}
+
+.page-link {
+  position: relative;
+  display: block;
+  padding: 0.5rem 0.75rem;
+  margin-left: -1px;
+  line-height: 1.25;
+  color: #007bff;
+  background-color: #fff;
+  border: 1px solid #dee2e6;
+}
+.page-link:hover {
+  color: #0056b3;
+  text-decoration: none;
+  background-color: #e9ecef;
+  border-color: #dee2e6;
+}
+.page-link:focus {
+  z-index: 2;
+  outline: 0;
+  box-shadow: 0 0 0 0.2rem rgba(0, 123, 255, 0.25);
+}
+.page-link:not(:disabled):not(.disabled) {
+  cursor: pointer;
+}
+
+.page-item:first-child .page-link {
+  margin-left: 0;
+  border-top-left-radius: 0.25rem;
+  border-bottom-left-radius: 0.25rem;
+}
+.page-item:last-child .page-link {
+  border-top-right-radius: 0.25rem;
+  border-bottom-right-radius: 0.25rem;
+}
+.page-item.active .page-link {
+  z-index: 1;
+  color: #fff;
+  background-color: #007bff;
+  border-color: #007bff;
+}
+.page-item.disabled .page-link {
+  color: #6c757d;
+  pointer-events: none;
+  cursor: auto;
+  background-color: #fff;
+  border-color: #dee2e6;
+}
+
+.pagination-lg .page-link {
+  padding: 0.75rem 1.5rem;
+  font-size: 1.25rem;
+  line-height: 1.5;
+}
+.pagination-lg .page-item:first-child .page-link {
+  border-top-left-radius: 0.3rem;
+  border-bottom-left-radius: 0.3rem;
+}
+.pagination-lg .page-item:last-child .page-link {
+  border-top-right-radius: 0.3rem;
+  border-bottom-right-radius: 0.3rem;
+}
+
+.pagination-sm .page-link {
+  padding: 0.25rem 0.5rem;
+  font-size: 0.875rem;
+  line-height: 1.5;
+}
+.pagination-sm .page-item:first-child .page-link {
+  border-top-left-radius: 0.2rem;
+  border-bottom-left-radius: 0.2rem;
+}
+.pagination-sm .page-item:last-child .page-link {
+  border-top-right-radius: 0.2rem;
+  border-bottom-right-radius: 0.2rem;
+}
+
+.badge {
+  display: inline-block;
+  padding: 0.25em 0.4em;
+  font-size: 75%;
+  font-weight: 700;
+  line-height: 1;
+  text-align: center;
+  white-space: nowrap;
+  vertical-align: baseline;
+  border-radius: 0.25rem;
+}
+.badge:empty {
+  display: none;
+}
+
+.btn .badge {
+  position: relative;
+  top: -1px;
+}
+
+.badge-pill {
+  padding-right: 0.6em;
+  padding-left: 0.6em;
+  border-radius: 10rem;
+}
+
+.badge-primary {
+  color: #fff;
+  background-color: #007bff;
+}
+.badge-primary[href]:hover, .badge-primary[href]:focus {
+  color: #fff;
+  text-decoration: none;
+  background-color: #0062cc;
+}
+
+.badge-secondary {
+  color: #fff;
+  background-color: #6c757d;
+}
+.badge-secondary[href]:hover, .badge-secondary[href]:focus {
+  color: #fff;
+  text-decoration: none;
+  background-color: #545b62;
+}
+
+.badge-success {
+  color: #fff;
+  background-color: #28a745;
+}
+.badge-success[href]:hover, .badge-success[href]:focus {
+  color: #fff;
+  text-decoration: none;
+  background-color: #1e7e34;
+}
+
+.badge-info {
+  color: #fff;
+  background-color: #17a2b8;
+}
+.badge-info[href]:hover, .badge-info[href]:focus {
+  color: #fff;
+  text-decoration: none;
+  background-color: #117a8b;
+}
+
+.badge-warning {
+  color: #212529;
+  background-color: #ffc107;
+}
+.badge-warning[href]:hover, .badge-warning[href]:focus {
+  color: #212529;
+  text-decoration: none;
+  background-color: #d39e00;
+}
+
+.badge-danger {
+  color: #fff;
+  background-color: #dc3545;
+}
+.badge-danger[href]:hover, .badge-danger[href]:focus {
+  color: #fff;
+  text-decoration: none;
+  background-color: #bd2130;
+}
+
+.badge-light {
+  color: #212529;
+  background-color: #f8f9fa;
+}
+.badge-light[href]:hover, .badge-light[href]:focus {
+  color: #212529;
+  text-decoration: none;
+  background-color: #dae0e5;
+}
+
+.badge-dark {
+  color: #fff;
+  background-color: #343a40;
+}
+.badge-dark[href]:hover, .badge-dark[href]:focus {
+  color: #fff;
+  text-decoration: none;
+  background-color: #1d2124;
+}
+
+.jumbotron {
+  padding: 2rem 1rem;
+  margin-bottom: 2rem;
+  background-color: #e9ecef;
+  border-radius: 0.3rem;
+}
+@media (min-width: 576px) {
+  .jumbotron {
+    padding: 4rem 2rem;
+  }
+}
+
+.jumbotron-fluid {
+  padding-right: 0;
+  padding-left: 0;
+  border-radius: 0;
+}
+
+.alert {
+  position: relative;
+  padding: 0.75rem 1.25rem;
+  margin-bottom: 1rem;
+  border: 1px solid transparent;
+  border-radius: 0.25rem;
+}
+
+.alert-heading {
+  color: inherit;
+}
+
+.alert-link {
+  font-weight: 700;
+}
+
+.alert-dismissible {
+  padding-right: 4rem;
+}
+.alert-dismissible .close {
+  position: absolute;
+  top: 0;
+  right: 0;
+  padding: 0.75rem 1.25rem;
+  color: inherit;
+}
+
+.alert-primary {
+  color: #004085;
+  background-color: #cce5ff;
+  border-color: #b8daff;
+}
+.alert-primary hr {
+  border-top-color: #9fcdff;
+}
+.alert-primary .alert-link {
+  color: #002752;
+}
+
+.alert-secondary {
+  color: #383d41;
+  background-color: #e2e3e5;
+  border-color: #d6d8db;
+}
+.alert-secondary hr {
+  border-top-color: #c8cbcf;
+}
+.alert-secondary .alert-link {
+  color: #202326;
+}
+
+.alert-success {
+  color: #155724;
+  background-color: #d4edda;
+  border-color: #c3e6cb;
+}
+.alert-success hr {
+  border-top-color: #b1dfbb;
+}
+.alert-success .alert-link {
+  color: #0b2e13;
+}
+
+.alert-info {
+  color: #0c5460;
+  background-color: #d1ecf1;
+  border-color: #bee5eb;
+}
+.alert-info hr {
+  border-top-color: #abdde5;
+}
+.alert-info .alert-link {
+  color: #062c33;
+}
+
+.alert-warning {
+  color: #856404;
+  background-color: #fff3cd;
+  border-color: #ffeeba;
+}
+.alert-warning hr {
+  border-top-color: #ffe8a1;
+}
+.alert-warning .alert-link {
+  color: #533f03;
+}
+
+.alert-danger {
+  color: #721c24;
+  background-color: #f8d7da;
+  border-color: #f5c6cb;
+}
+.alert-danger hr {
+  border-top-color: #f1b0b7;
+}
+.alert-danger .alert-link {
+  color: #491217;
+}
+
+.alert-light {
+  color: #818182;
+  background-color: #fefefe;
+  border-color: #fdfdfe;
+}
+.alert-light hr {
+  border-top-color: #ececf6;
+}
+.alert-light .alert-link {
+  color: #686868;
+}
+
+.alert-dark {
+  color: #1b1e21;
+  background-color: #d6d8d9;
+  border-color: #c6c8ca;
+}
+.alert-dark hr {
+  border-top-color: #b9bbbe;
+}
+.alert-dark .alert-link {
+  color: #040505;
+}
+
+@-webkit-keyframes progress-bar-stripes {
+  from {
+    background-position: 1rem 0;
+  }
+  to {
+    background-position: 0 0;
+  }
+}
+
+@keyframes progress-bar-stripes {
+  from {
+    background-position: 1rem 0;
+  }
+  to {
+    background-position: 0 0;
+  }
+}
+.progress {
+  display: flex;
+  height: 1rem;
+  overflow: hidden;
+  font-size: 0.75rem;
+  background-color: #e9ecef;
+  border-radius: 0.25rem;
+}
+
+.progress-bar {
+  display: flex;
+  flex-direction: column;
+  justify-content: center;
+  color: #fff;
+  text-align: center;
+  background-color: #007bff;
+  transition: width 0.6s ease;
+}
+
+.progress-bar-striped {
+  background-image: linear-gradient(45deg, rgba(255, 255, 255, 0.15) 25%, transparent 25%, transparent 50%, rgba(255, 255, 255, 0.15) 50%, rgba(255, 255, 255, 0.15) 75%, transparent 75%, transparent);
+  background-size: 1rem 1rem;
+}
+
+.progress-bar-animated {
+  -webkit-animation: progress-bar-stripes 1s linear infinite;
+          animation: progress-bar-stripes 1s linear infinite;
+}
+
+.media {
+  display: flex;
+  align-items: flex-start;
+}
+
+.media-body {
+  flex: 1;
+}
+
+.list-group {
+  display: flex;
+  flex-direction: column;
+  padding-left: 0;
+  margin-bottom: 0;
+}
+
+.list-group-item-action {
+  width: 100%;
+  color: #495057;
+  text-align: inherit;
+}
+.list-group-item-action:hover, .list-group-item-action:focus {
+  color: #495057;
+  text-decoration: none;
+  background-color: #f8f9fa;
+}
+.list-group-item-action:active {
+  color: #212529;
+  background-color: #e9ecef;
+}
+
+.list-group-item {
+  position: relative;
+  display: block;
+  padding: 0.75rem 1.25rem;
+  margin-bottom: -1px;
+  background-color: #fff;
+  border: 1px solid rgba(0, 0, 0, 0.125);
+}
+.list-group-item:first-child {
+  border-top-left-radius: 0.25rem;
+  border-top-right-radius: 0.25rem;
+}
+.list-group-item:last-child {
+  margin-bottom: 0;
+  border-bottom-right-radius: 0.25rem;
+  border-bottom-left-radius: 0.25rem;
+}
+.list-group-item:hover, .list-group-item:focus {
+  z-index: 1;
+  text-decoration: none;
+}
+.list-group-item.disabled, .list-group-item:disabled {
+  color: #6c757d;
+  background-color: #fff;
+}
+.list-group-item.active {
+  z-index: 2;
+  color: #fff;
+  background-color: #007bff;
+  border-color: #007bff;
+}
+
+.list-group-flush .list-group-item {
+  border-right: 0;
+  border-left: 0;
+  border-radius: 0;
+}
+.list-group-flush:first-child .list-group-item:first-child {
+  border-top: 0;
+}
+.list-group-flush:last-child .list-group-item:last-child {
+  border-bottom: 0;
+}
+
+.list-group-item-primary {
+  color: #004085;
+  background-color: #b8daff;
+}
+.list-group-item-primary.list-group-item-action:hover, .list-group-item-primary.list-group-item-action:focus {
+  color: #004085;
+  background-color: #9fcdff;
+}
+.list-group-item-primary.list-group-item-action.active {
+  color: #fff;
+  background-color: #004085;
+  border-color: #004085;
+}
+
+.list-group-item-secondary {
+  color: #383d41;
+  background-color: #d6d8db;
+}
+.list-group-item-secondary.list-group-item-action:hover, .list-group-item-secondary.list-group-item-action:focus {
+  color: #383d41;
+  background-color: #c8cbcf;
+}
+.list-group-item-secondary.list-group-item-action.active {
+  color: #fff;
+  background-color: #383d41;
+  border-color: #383d41;
+}
+
+.list-group-item-success {
+  color: #155724;
+  background-color: #c3e6cb;
+}
+.list-group-item-success.list-group-item-action:hover, .list-group-item-success.list-group-item-action:focus {
+  color: #155724;
+  background-color: #b1dfbb;
+}
+.list-group-item-success.list-group-item-action.active {
+  color: #fff;
+  background-color: #155724;
+  border-color: #155724;
+}
+
+.list-group-item-info {
+  color: #0c5460;
+  background-color: #bee5eb;
+}
+.list-group-item-info.list-group-item-action:hover, .list-group-item-info.list-group-item-action:focus {
+  color: #0c5460;
+  background-color: #abdde5;
+}
+.list-group-item-info.list-group-item-action.active {
+  color: #fff;
+  background-color: #0c5460;
+  border-color: #0c5460;
+}
+
+.list-group-item-warning {
+  color: #856404;
+  background-color: #ffeeba;
+}
+.list-group-item-warning.list-group-item-action:hover, .list-group-item-warning.list-group-item-action:focus {
+  color: #856404;
+  background-color: #ffe8a1;
+}
+.list-group-item-warning.list-group-item-action.active {
+  color: #fff;
+  background-color: #856404;
+  border-color: #856404;
+}
+
+.list-group-item-danger {
+  color: #721c24;
+  background-color: #f5c6cb;
+}
+.list-group-item-danger.list-group-item-action:hover, .list-group-item-danger.list-group-item-action:focus {
+  color: #721c24;
+  background-color: #f1b0b7;
+}
+.list-group-item-danger.list-group-item-action.active {
+  color: #fff;
+  background-color: #721c24;
+  border-color: #721c24;
+}
+
+.list-group-item-light {
+  color: #818182;
+  background-color: #fdfdfe;
+}
+.list-group-item-light.list-group-item-action:hover, .list-group-item-light.list-group-item-action:focus {
+  color: #818182;
+  background-color: #ececf6;
+}
+.list-group-item-light.list-group-item-action.active {
+  color: #fff;
+  background-color: #818182;
+  border-color: #818182;
+}
+
+.list-group-item-dark {
+  color: #1b1e21;
+  background-color: #c6c8ca;
+}
+.list-group-item-dark.list-group-item-action:hover, .list-group-item-dark.list-group-item-action:focus {
+  color: #1b1e21;
+  background-color: #b9bbbe;
+}
+.list-group-item-dark.list-group-item-action.active {
+  color: #fff;
+  background-color: #1b1e21;
+  border-color: #1b1e21;
+}
+
+.close {
+  float: right;
+  font-size: 1.5rem;
+  font-weight: 700;
+  line-height: 1;
+  color: #000;
+  text-shadow: 0 1px 0 #fff;
+  opacity: .5;
+}
+.close:hover, .close:focus {
+  color: #000;
+  text-decoration: none;
+  opacity: .75;
+}
+.close:not(:disabled):not(.disabled) {
+  cursor: pointer;
+}
+
+button.close {
+  padding: 0;
+  background-color: transparent;
+  border: 0;
+  -webkit-appearance: none;
+}
+
+.modal-open {
+  overflow: hidden;
+}
+
+.modal {
+  position: fixed;
+  top: 0;
+  right: 0;
+  bottom: 0;
+  left: 0;
+  z-index: 1050;
+  display: none;
+  overflow: hidden;
+  outline: 0;
+}
+.modal-open .modal {
+  overflow-x: hidden;
+  overflow-y: auto;
+}
+
+.modal-dialog {
+  position: relative;
+  width: auto;
+  margin: 0.5rem;
+  pointer-events: none;
+}
+.modal.fade .modal-dialog {
+  transition: transform 0.3s ease-out;
+  transform: translate(0, -25%);
+}
+.modal.show .modal-dialog {
+  transform: translate(0, 0);
+}
+
+.modal-dialog-centered {
+  display: flex;
+  align-items: center;
+  min-height: calc(100% - (0.5rem * 2));
+}
+
+.modal-content {
+  position: relative;
+  display: flex;
+  flex-direction: column;
+  width: 100%;
+  pointer-events: auto;
+  background-color: #fff;
+  background-clip: padding-box;
+  border: 1px solid rgba(0, 0, 0, 0.2);
+  border-radius: 0.3rem;
+  outline: 0;
+}
+
+.modal-backdrop {
+  position: fixed;
+  top: 0;
+  right: 0;
+  bottom: 0;
+  left: 0;
+  z-index: 1040;
+  background-color: #000;
+}
+.modal-backdrop.fade {
+  opacity: 0;
+}
+.modal-backdrop.show {
+  opacity: 0.5;
+}
+
+.modal-header {
+  display: flex;
+  align-items: flex-start;
+  justify-content: space-between;
+  padding: 1rem;
+  border-bottom: 1px solid #e9ecef;
+  border-top-left-radius: 0.3rem;
+  border-top-right-radius: 0.3rem;
+}
+.modal-header .close {
+  padding: 1rem;
+  margin: -1rem -1rem -1rem auto;
+}
+
+.modal-title {
+  margin-bottom: 0;
+  line-height: 1.5;
+}
+
+.modal-body {
+  position: relative;
+  flex: 1 1 auto;
+  padding: 1rem;
+}
+
+.modal-footer {
+  display: flex;
+  align-items: center;
+  justify-content: flex-end;
+  padding: 1rem;
+  border-top: 1px solid #e9ecef;
+}
+.modal-footer > :not(:first-child) {
+  margin-left: .25rem;
+}
+.modal-footer > :not(:last-child) {
+  margin-right: .25rem;
+}
+
+.modal-scrollbar-measure {
+  position: absolute;
+  top: -9999px;
+  width: 50px;
+  height: 50px;
+  overflow: scroll;
+}
+
+@media (min-width: 576px) {
+  .modal-dialog {
+    max-width: 500px;
+    margin: 1.75rem auto;
+  }
+
+  .modal-dialog-centered {
+    min-height: calc(100% - (1.75rem * 2));
+  }
+
+  .modal-sm {
+    max-width: 300px;
+  }
+}
+@media (min-width: 992px) {
+  .modal-lg {
+    max-width: 800px;
+  }
+}
+.tooltip {
+  position: absolute;
+  z-index: 1070;
+  display: block;
+  margin: 0;
+  font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, "Helvetica Neue", Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol";
+  font-style: normal;
+  font-weight: 400;
+  line-height: 1.5;
+  text-align: left;
+  text-align: start;
+  text-decoration: none;
+  text-shadow: none;
+  text-transform: none;
+  letter-spacing: normal;
+  word-break: normal;
+  word-spacing: normal;
+  white-space: normal;
+  line-break: auto;
+  font-size: 0.875rem;
+  word-wrap: break-word;
+  opacity: 0;
+}
+.tooltip.show {
+  opacity: 0.9;
+}
+.tooltip .arrow {
+  position: absolute;
+  display: block;
+  width: 0.8rem;
+  height: 0.4rem;
+}
+.tooltip .arrow::before {
+  position: absolute;
+  content: "";
+  border-color: transparent;
+  border-style: solid;
+}
+
+.bs-tooltip-top, .bs-tooltip-auto[x-placement^="top"] {
+  padding: 0.4rem 0;
+}
+.bs-tooltip-top .arrow, .bs-tooltip-auto[x-placement^="top"] .arrow {
+  bottom: 0;
+}
+.bs-tooltip-top .arrow::before, .bs-tooltip-auto[x-placement^="top"] .arrow::before {
+  top: 0;
+  border-width: 0.4rem 0.4rem 0;
+  border-top-color: #000;
+}
+
+.bs-tooltip-right, .bs-tooltip-auto[x-placement^="right"] {
+  padding: 0 0.4rem;
+}
+.bs-tooltip-right .arrow, .bs-tooltip-auto[x-placement^="right"] .arrow {
+  left: 0;
+  width: 0.4rem;
+  height: 0.8rem;
+}
+.bs-tooltip-right .arrow::before, .bs-tooltip-auto[x-placement^="right"] .arrow::before {
+  right: 0;
+  border-width: 0.4rem 0.4rem 0.4rem 0;
+  border-right-color: #000;
+}
+
+.bs-tooltip-bottom, .bs-tooltip-auto[x-placement^="bottom"] {
+  padding: 0.4rem 0;
+}
+.bs-tooltip-bottom .arrow, .bs-tooltip-auto[x-placement^="bottom"] .arrow {
+  top: 0;
+}
+.bs-tooltip-bottom .arrow::before, .bs-tooltip-auto[x-placement^="bottom"] .arrow::before {
+  bottom: 0;
+  border-width: 0 0.4rem 0.4rem;
+  border-bottom-color: #000;
+}
+
+.bs-tooltip-left, .bs-tooltip-auto[x-placement^="left"] {
+  padding: 0 0.4rem;
+}
+.bs-tooltip-left .arrow, .bs-tooltip-auto[x-placement^="left"] .arrow {
+  right: 0;
+  width: 0.4rem;
+  height: 0.8rem;
+}
+.bs-tooltip-left .arrow::before, .bs-tooltip-auto[x-placement^="left"] .arrow::before {
+  left: 0;
+  border-width: 0.4rem 0 0.4rem 0.4rem;
+  border-left-color: #000;
+}
+
+.tooltip-inner {
+  max-width: 200px;
+  padding: 0.25rem 0.5rem;
+  color: #fff;
+  text-align: center;
+  background-color: #000;
+  border-radius: 0.25rem;
+}
+
+.popover {
+  position: absolute;
+  top: 0;
+  left: 0;
+  z-index: 1060;
+  display: block;
+  max-width: 276px;
+  font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, "Helvetica Neue", Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol";
+  font-style: normal;
+  font-weight: 400;
+  line-height: 1.5;
+  text-align: left;
+  text-align: start;
+  text-decoration: none;
+  text-shadow: none;
+  text-transform: none;
+  letter-spacing: normal;
+  word-break: normal;
+  word-spacing: normal;
+  white-space: normal;
+  line-break: auto;
+  font-size: 0.875rem;
+  word-wrap: break-word;
+  background-color: #fff;
+  background-clip: padding-box;
+  border: 1px solid rgba(0, 0, 0, 0.2);
+  border-radius: 0.3rem;
+}
+.popover .arrow {
+  position: absolute;
+  display: block;
+  width: 1rem;
+  height: 0.5rem;
+  margin: 0 0.3rem;
+}
+.popover .arrow::before, .popover .arrow::after {
+  position: absolute;
+  display: block;
+  content: "";
+  border-color: transparent;
+  border-style: solid;
+}
+
+.bs-popover-top, .bs-popover-auto[x-placement^="top"] {
+  margin-bottom: 0.5rem;
+}
+.bs-popover-top .arrow, .bs-popover-auto[x-placement^="top"] .arrow {
+  bottom: calc((0.5rem + 1px) * -1);
+}
+.bs-popover-top .arrow::before, .bs-popover-auto[x-placement^="top"] .arrow::before,
+.bs-popover-top .arrow::after,
+.bs-popover-auto[x-placement^="top"] .arrow::after {
+  border-width: 0.5rem 0.5rem 0;
+}
+.bs-popover-top .arrow::before, .bs-popover-auto[x-placement^="top"] .arrow::before {
+  bottom: 0;
+  border-top-color: rgba(0, 0, 0, 0.25);
+}
+.bs-popover-top .arrow::after, .bs-popover-auto[x-placement^="top"] .arrow::after {
+  bottom: 1px;
+  border-top-color: #fff;
+}
+
+.bs-popover-right, .bs-popover-auto[x-placement^="right"] {
+  margin-left: 0.5rem;
+}
+.bs-popover-right .arrow, .bs-popover-auto[x-placement^="right"] .arrow {
+  left: calc((0.5rem + 1px) * -1);
+  width: 0.5rem;
+  height: 1rem;
+  margin: 0.3rem 0;
+}
+.bs-popover-right .arrow::before, .bs-popover-auto[x-placement^="right"] .arrow::before,
+.bs-popover-right .arrow::after,
+.bs-popover-auto[x-placement^="right"] .arrow::after {
+  border-width: 0.5rem 0.5rem 0.5rem 0;
+}
+.bs-popover-right .arrow::before, .bs-popover-auto[x-placement^="right"] .arrow::before {
+  left: 0;
+  border-right-color: rgba(0, 0, 0, 0.25);
+}
+.bs-popover-right .arrow::after, .bs-popover-auto[x-placement^="right"] .arrow::after {
+  left: 1px;
+  border-right-color: #fff;
+}
+
+.bs-popover-bottom, .bs-popover-auto[x-placement^="bottom"] {
+  margin-top: 0.5rem;
+}
+.bs-popover-bottom .arrow, .bs-popover-auto[x-placement^="bottom"] .arrow {
+  top: calc((0.5rem + 1px) * -1);
+}
+.bs-popover-bottom .arrow::before, .bs-popover-auto[x-placement^="bottom"] .arrow::before,
+.bs-popover-bottom .arrow::after,
+.bs-popover-auto[x-placement^="bottom"] .arrow::after {
+  border-width: 0 0.5rem 0.5rem 0.5rem;
+}
+.bs-popover-bottom .arrow::before, .bs-popover-auto[x-placement^="bottom"] .arrow::before {
+  top: 0;
+  border-bottom-color: rgba(0, 0, 0, 0.25);
+}
+.bs-popover-bottom .arrow::after, .bs-popover-auto[x-placement^="bottom"] .arrow::after {
+  top: 1px;
+  border-bottom-color: #fff;
+}
+.bs-popover-bottom .popover-header::before, .bs-popover-auto[x-placement^="bottom"] .popover-header::before {
+  position: absolute;
+  top: 0;
+  left: 50%;
+  display: block;
+  width: 1rem;
+  margin-left: -0.5rem;
+  content: "";
+  border-bottom: 1px solid #f7f7f7;
+}
+
+.bs-popover-left, .bs-popover-auto[x-placement^="left"] {
+  margin-right: 0.5rem;
+}
+.bs-popover-left .arrow, .bs-popover-auto[x-placement^="left"] .arrow {
+  right: calc((0.5rem + 1px) * -1);
+  width: 0.5rem;
+  height: 1rem;
+  margin: 0.3rem 0;
+}
+.bs-popover-left .arrow::before, .bs-popover-auto[x-placement^="left"] .arrow::before,
+.bs-popover-left .arrow::after,
+.bs-popover-auto[x-placement^="left"] .arrow::after {
+  border-width: 0.5rem 0 0.5rem 0.5rem;
+}
+.bs-popover-left .arrow::before, .bs-popover-auto[x-placement^="left"] .arrow::before {
+  right: 0;
+  border-left-color: rgba(0, 0, 0, 0.25);
+}
+.bs-popover-left .arrow::after, .bs-popover-auto[x-placement^="left"] .arrow::after {
+  right: 1px;
+  border-left-color: #fff;
+}
+
+.popover-header {
+  padding: 0.5rem 0.75rem;
+  margin-bottom: 0;
+  font-size: 1rem;
+  color: inherit;
+  background-color: #f7f7f7;
+  border-bottom: 1px solid #ebebeb;
+  border-top-left-radius: calc(0.3rem - 1px);
+  border-top-right-radius: calc(0.3rem - 1px);
+}
+.popover-header:empty {
+  display: none;
+}
+
+.popover-body {
+  padding: 0.5rem 0.75rem;
+  color: #212529;
+}
+
+.carousel {
+  position: relative;
+}
+
+.carousel-inner {
+  position: relative;
+  width: 100%;
+  overflow: hidden;
+}
+
+.carousel-item {
+  position: relative;
+  display: none;
+  align-items: center;
+  width: 100%;
+  transition: transform 0.6s ease;
+  -webkit-backface-visibility: hidden;
+          backface-visibility: hidden;
+  perspective: 1000px;
+}
+
+.carousel-item.active,
+.carousel-item-next,
+.carousel-item-prev {
+  display: block;
+}
+
+.carousel-item-next,
+.carousel-item-prev {
+  position: absolute;
+  top: 0;
+}
+
+.carousel-item-next.carousel-item-left,
+.carousel-item-prev.carousel-item-right {
+  transform: translateX(0);
+}
+@supports (transform-style: preserve-3d) {
+  .carousel-item-next.carousel-item-left,
+  .carousel-item-prev.carousel-item-right {
+    transform: translate3d(0, 0, 0);
+  }
+}
+
+.carousel-item-next,
+.active.carousel-item-right {
+  transform: translateX(100%);
+}
+@supports (transform-style: preserve-3d) {
+  .carousel-item-next,
+  .active.carousel-item-right {
+    transform: translate3d(100%, 0, 0);
+  }
+}
+
+.carousel-item-prev,
+.active.carousel-item-left {
+  transform: translateX(-100%);
+}
+@supports (transform-style: preserve-3d) {
+  .carousel-item-prev,
+  .active.carousel-item-left {
+    transform: translate3d(-100%, 0, 0);
+  }
+}
+
+.carousel-control-prev,
+.carousel-control-next {
+  position: absolute;
+  top: 0;
+  bottom: 0;
+  display: flex;
+  align-items: center;
+  justify-content: center;
+  width: 15%;
+  color: #fff;
+  text-align: center;
+  opacity: 0.5;
+}
+.carousel-control-prev:hover, .carousel-control-prev:focus,
+.carousel-control-next:hover,
+.carousel-control-next:focus {
+  color: #fff;
+  text-decoration: none;
+  outline: 0;
+  opacity: .9;
+}
+
+.carousel-control-prev {
+  left: 0;
+}
+
+.carousel-control-next {
+  right: 0;
+}
+
+.carousel-control-prev-icon,
+.carousel-control-next-icon {
+  display: inline-block;
+  width: 20px;
+  height: 20px;
+  background: transparent no-repeat center center;
+  background-size: 100% 100%;
+}
+
+.carousel-control-prev-icon {
+  background-image: url("data:image/svg+xml;charset=utf8,%3Csvg xmlns='http://www.w3.org/2000/svg' fill='%23fff' viewBox='0 0 8 8'%3E%3Cpath d='M5.25 0l-4 4 4 4 1.5-1.5-2.5-2.5 2.5-2.5-1.5-1.5z'/%3E%3C/svg%3E");
+}
+
+.carousel-control-next-icon {
+  background-image: url("data:image/svg+xml;charset=utf8,%3Csvg xmlns='http://www.w3.org/2000/svg' fill='%23fff' viewBox='0 0 8 8'%3E%3Cpath d='M2.75 0l-1.5 1.5 2.5 2.5-2.5 2.5 1.5 1.5 4-4-4-4z'/%3E%3C/svg%3E");
+}
+
+.carousel-indicators {
+  position: absolute;
+  right: 0;
+  bottom: 10px;
+  left: 0;
+  z-index: 15;
+  display: flex;
+  justify-content: center;
+  padding-left: 0;
+  margin-right: 15%;
+  margin-left: 15%;
+  list-style: none;
+}
+.carousel-indicators li {
+  position: relative;
+  flex: 0 1 auto;
+  width: 30px;
+  height: 3px;
+  margin-right: 3px;
+  margin-left: 3px;
+  text-indent: -999px;
+  background-color: rgba(255, 255, 255, 0.5);
+}
+.carousel-indicators li::before {
+  position: absolute;
+  top: -10px;
+  left: 0;
+  display: inline-block;
+  width: 100%;
+  height: 10px;
+  content: "";
+}
+.carousel-indicators li::after {
+  position: absolute;
+  bottom: -10px;
+  left: 0;
+  display: inline-block;
+  width: 100%;
+  height: 10px;
+  content: "";
+}
+.carousel-indicators .active {
+  background-color: #fff;
+}
+
+.carousel-caption {
+  position: absolute;
+  right: 15%;
+  bottom: 20px;
+  left: 15%;
+  z-index: 10;
+  padding-top: 20px;
+  padding-bottom: 20px;
+  color: #fff;
+  text-align: center;
+}
+
+.align-baseline {
+  vertical-align: baseline !important;
+}
+
+.align-top {
+  vertical-align: top !important;
+}
+
+.align-middle {
+  vertical-align: middle !important;
+}
+
+.align-bottom {
+  vertical-align: bottom !important;
+}
+
+.align-text-bottom {
+  vertical-align: text-bottom !important;
+}
+
+.align-text-top {
+  vertical-align: text-top !important;
+}
+
+.bg-primary {
+  background-color: #007bff !important;
+}
+
+a.bg-primary:hover, a.bg-primary:focus,
+button.bg-primary:hover,
+button.bg-primary:focus {
+  background-color: #0062cc !important;
+}
+
+.bg-secondary {
+  background-color: #6c757d !important;
+}
+
+a.bg-secondary:hover, a.bg-secondary:focus,
+button.bg-secondary:hover,
+button.bg-secondary:focus {
+  background-color: #545b62 !important;
+}
+
+.bg-success {
+  background-color: #28a745 !important;
+}
+
+a.bg-success:hover, a.bg-success:focus,
+button.bg-success:hover,
+button.bg-success:focus {
+  background-color: #1e7e34 !important;
+}
+
+.bg-info {
+  background-color: #17a2b8 !important;
+}
+
+a.bg-info:hover, a.bg-info:focus,
+button.bg-info:hover,
+button.bg-info:focus {
+  background-color: #117a8b !important;
+}
+
+.bg-warning {
+  background-color: #ffc107 !important;
+}
+
+a.bg-warning:hover, a.bg-warning:focus,
+button.bg-warning:hover,
+button.bg-warning:focus {
+  background-color: #d39e00 !important;
+}
+
+.bg-danger {
+  background-color: #dc3545 !important;
+}
+
+a.bg-danger:hover, a.bg-danger:focus,
+button.bg-danger:hover,
+button.bg-danger:focus {
+  background-color: #bd2130 !important;
+}
+
+.bg-light {
+  background-color: #f8f9fa !important;
+}
+
+a.bg-light:hover, a.bg-light:focus,
+button.bg-light:hover,
+button.bg-light:focus {
+  background-color: #dae0e5 !important;
+}
+
+.bg-dark {
+  background-color: #343a40 !important;
+}
+
+a.bg-dark:hover, a.bg-dark:focus,
+button.bg-dark:hover,
+button.bg-dark:focus {
+  background-color: #1d2124 !important;
+}
+
+.bg-white {
+  background-color: #fff !important;
+}
+
+.bg-transparent {
+  background-color: transparent !important;
+}
+
+.border {
+  border: 1px solid #dee2e6 !important;
+}
+
+.border-top {
+  border-top: 1px solid #dee2e6 !important;
+}
+
+.border-right {
+  border-right: 1px solid #dee2e6 !important;
+}
+
+.border-bottom {
+  border-bottom: 1px solid #dee2e6 !important;
+}
+
+.border-left {
+  border-left: 1px solid #dee2e6 !important;
+}
+
+.border-0 {
+  border: 0 !important;
+}
+
+.border-top-0 {
+  border-top: 0 !important;
+}
+
+.border-right-0 {
+  border-right: 0 !important;
+}
+
+.border-bottom-0 {
+  border-bottom: 0 !important;
+}
+
+.border-left-0 {
+  border-left: 0 !important;
+}
+
+.border-primary {
+  border-color: #007bff !important;
+}
+
+.border-secondary {
+  border-color: #6c757d !important;
+}
+
+.border-success {
+  border-color: #28a745 !important;
+}
+
+.border-info {
+  border-color: #17a2b8 !important;
+}
+
+.border-warning {
+  border-color: #ffc107 !important;
+}
+
+.border-danger {
+  border-color: #dc3545 !important;
+}
+
+.border-light {
+  border-color: #f8f9fa !important;
+}
+
+.border-dark {
+  border-color: #343a40 !important;
+}
+
+.border-white {
+  border-color: #fff !important;
+}
+
+.rounded {
+  border-radius: 0.25rem !important;
+}
+
+.rounded-top {
+  border-top-left-radius: 0.25rem !important;
+  border-top-right-radius: 0.25rem !important;
+}
+
+.rounded-right {
+  border-top-right-radius: 0.25rem !important;
+  border-bottom-right-radius: 0.25rem !important;
+}
+
+.rounded-bottom {
+  border-bottom-right-radius: 0.25rem !important;
+  border-bottom-left-radius: 0.25rem !important;
+}
+
+.rounded-left {
+  border-top-left-radius: 0.25rem !important;
+  border-bottom-left-radius: 0.25rem !important;
+}
+
+.rounded-circle {
+  border-radius: 50% !important;
+}
+
+.rounded-0 {
+  border-radius: 0 !important;
+}
+
+.clearfix::after {
+  display: block;
+  clear: both;
+  content: "";
+}
+
+.d-none {
+  display: none !important;
+}
+
+.d-inline {
+  display: inline !important;
+}
+
+.d-inline-block {
+  display: inline-block !important;
+}
+
+.d-block {
+  display: block !important;
+}
+
+.d-table {
+  display: table !important;
+}
+
+.d-table-row {
+  display: table-row !important;
+}
+
+.d-table-cell {
+  display: table-cell !important;
+}
+
+.d-flex {
+  display: flex !important;
+}
+
+.d-inline-flex {
+  display: inline-flex !important;
+}
+
+@media (min-width: 576px) {
+  .d-sm-none {
+    display: none !important;
+  }
+
+  .d-sm-inline {
+    display: inline !important;
+  }
+
+  .d-sm-inline-block {
+    display: inline-block !important;
+  }
+
+  .d-sm-block {
+    display: block !important;
+  }
+
+  .d-sm-table {
+    display: table !important;
+  }
+
+  .d-sm-table-row {
+    display: table-row !important;
+  }
+
+  .d-sm-table-cell {
+    display: table-cell !important;
+  }
+
+  .d-sm-flex {
+    display: flex !important;
+  }
+
+  .d-sm-inline-flex {
+    display: inline-flex !important;
+  }
+}
+@media (min-width: 768px) {
+  .d-md-none {
+    display: none !important;
+  }
+
+  .d-md-inline {
+    display: inline !important;
+  }
+
+  .d-md-inline-block {
+    display: inline-block !important;
+  }
+
+  .d-md-block {
+    display: block !important;
+  }
+
+  .d-md-table {
+    display: table !important;
+  }
+
+  .d-md-table-row {
+    display: table-row !important;
+  }
+
+  .d-md-table-cell {
+    display: table-cell !important;
+  }
+
+  .d-md-flex {
+    display: flex !important;
+  }
+
+  .d-md-inline-flex {
+    display: inline-flex !important;
+  }
+}
+@media (min-width: 992px) {
+  .d-lg-none {
+    display: none !important;
+  }
+
+  .d-lg-inline {
+    display: inline !important;
+  }
+
+  .d-lg-inline-block {
+    display: inline-block !important;
+  }
+
+  .d-lg-block {
+    display: block !important;
+  }
+
+  .d-lg-table {
+    display: table !important;
+  }
+
+  .d-lg-table-row {
+    display: table-row !important;
+  }
+
+  .d-lg-table-cell {
+    display: table-cell !important;
+  }
+
+  .d-lg-flex {
+    display: flex !important;
+  }
+
+  .d-lg-inline-flex {
+    display: inline-flex !important;
+  }
+}
+@media (min-width: 1200px) {
+  .d-xl-none {
+    display: none !important;
+  }
+
+  .d-xl-inline {
+    display: inline !important;
+  }
+
+  .d-xl-inline-block {
+    display: inline-block !important;
+  }
+
+  .d-xl-block {
+    display: block !important;
+  }
+
+  .d-xl-table {
+    display: table !important;
+  }
+
+  .d-xl-table-row {
+    display: table-row !important;
+  }
+
+  .d-xl-table-cell {
+    display: table-cell !important;
+  }
+
+  .d-xl-flex {
+    display: flex !important;
+  }
+
+  .d-xl-inline-flex {
+    display: inline-flex !important;
+  }
+}
+@media print {
+  .d-print-none {
+    display: none !important;
+  }
+
+  .d-print-inline {
+    display: inline !important;
+  }
+
+  .d-print-inline-block {
+    display: inline-block !important;
+  }
+
+  .d-print-block {
+    display: block !important;
+  }
+
+  .d-print-table {
+    display: table !important;
+  }
+
+  .d-print-table-row {
+    display: table-row !important;
+  }
+
+  .d-print-table-cell {
+    display: table-cell !important;
+  }
+
+  .d-print-flex {
+    display: flex !important;
+  }
+
+  .d-print-inline-flex {
+    display: inline-flex !important;
+  }
+}
+.embed-responsive {
+  position: relative;
+  display: block;
+  width: 100%;
+  padding: 0;
+  overflow: hidden;
+}
+.embed-responsive::before {
+  display: block;
+  content: "";
+}
+.embed-responsive .embed-responsive-item,
+.embed-responsive iframe,
+.embed-responsive embed,
+.embed-responsive object,
+.embed-responsive video {
+  position: absolute;
+  top: 0;
+  bottom: 0;
+  left: 0;
+  width: 100%;
+  height: 100%;
+  border: 0;
+}
+
+.embed-responsive-21by9::before {
+  padding-top: 42.8571428571%;
+}
+
+.embed-responsive-16by9::before {
+  padding-top: 56.25%;
+}
+
+.embed-responsive-4by3::before {
+  padding-top: 75%;
+}
+
+.embed-responsive-1by1::before {
+  padding-top: 100%;
+}
+
+.flex-row {
+  flex-direction: row !important;
+}
+
+.flex-column {
+  flex-direction: column !important;
+}
+
+.flex-row-reverse {
+  flex-direction: row-reverse !important;
+}
+
+.flex-column-reverse {
+  flex-direction: column-reverse !important;
+}
+
+.flex-wrap {
+  flex-wrap: wrap !important;
+}
+
+.flex-nowrap {
+  flex-wrap: nowrap !important;
+}
+
+.flex-wrap-reverse {
+  flex-wrap: wrap-reverse !important;
+}
+
+.justify-content-start {
+  justify-content: flex-start !important;
+}
+
+.justify-content-end {
+  justify-content: flex-end !important;
+}
+
+.justify-content-center {
+  justify-content: center !important;
+}
+
+.justify-content-between {
+  justify-content: space-between !important;
+}
+
+.justify-content-around {
+  justify-content: space-around !important;
+}
+
+.align-items-start {
+  align-items: flex-start !important;
+}
+
+.align-items-end {
+  align-items: flex-end !important;
+}
+
+.align-items-center {
+  align-items: center !important;
+}
+
+.align-items-baseline {
+  align-items: baseline !important;
+}
+
+.align-items-stretch {
+  align-items: stretch !important;
+}
+
+.align-content-start {
+  align-content: flex-start !important;
+}
+
+.align-content-end {
+  align-content: flex-end !important;
+}
+
+.align-content-center {
+  align-content: center !important;
+}
+
+.align-content-between {
+  align-content: space-between !important;
+}
+
+.align-content-around {
+  align-content: space-around !important;
+}
+
+.align-content-stretch {
+  align-content: stretch !important;
+}
+
+.align-self-auto {
+  align-self: auto !important;
+}
+
+.align-self-start {
+  align-self: flex-start !important;
+}
+
+.align-self-end {
+  align-self: flex-end !important;
+}
+
+.align-self-center {
+  align-self: center !important;
+}
+
+.align-self-baseline {
+  align-self: baseline !important;
+}
+
+.align-self-stretch {
+  align-self: stretch !important;
+}
+
+@media (min-width: 576px) {
+  .flex-sm-row {
+    flex-direction: row !important;
+  }
+
+  .flex-sm-column {
+    flex-direction: column !important;
+  }
+
+  .flex-sm-row-reverse {
+    flex-direction: row-reverse !important;
+  }
+
+  .flex-sm-column-reverse {
+    flex-direction: column-reverse !important;
+  }
+
+  .flex-sm-wrap {
+    flex-wrap: wrap !important;
+  }
+
+  .flex-sm-nowrap {
+    flex-wrap: nowrap !important;
+  }
+
+  .flex-sm-wrap-reverse {
+    flex-wrap: wrap-reverse !important;
+  }
+
+  .justify-content-sm-start {
+    justify-content: flex-start !important;
+  }
+
+  .justify-content-sm-end {
+    justify-content: flex-end !important;
+  }
+
+  .justify-content-sm-center {
+    justify-content: center !important;
+  }
+
+  .justify-content-sm-between {
+    justify-content: space-between !important;
+  }
+
+  .justify-content-sm-around {
+    justify-content: space-around !important;
+  }
+
+  .align-items-sm-start {
+    align-items: flex-start !important;
+  }
+
+  .align-items-sm-end {
+    align-items: flex-end !important;
+  }
+
+  .align-items-sm-center {
+    align-items: center !important;
+  }
+
+  .align-items-sm-baseline {
+    align-items: baseline !important;
+  }
+
+  .align-items-sm-stretch {
+    align-items: stretch !important;
+  }
+
+  .align-content-sm-start {
+    align-content: flex-start !important;
+  }
+
+  .align-content-sm-end {
+    align-content: flex-end !important;
+  }
+
+  .align-content-sm-center {
+    align-content: center !important;
+  }
+
+  .align-content-sm-between {
+    align-content: space-between !important;
+  }
+
+  .align-content-sm-around {
+    align-content: space-around !important;
+  }
+
+  .align-content-sm-stretch {
+    align-content: stretch !important;
+  }
+
+  .align-self-sm-auto {
+    align-self: auto !important;
+  }
+
+  .align-self-sm-start {
+    align-self: flex-start !important;
+  }
+
+  .align-self-sm-end {
+    align-self: flex-end !important;
+  }
+
+  .align-self-sm-center {
+    align-self: center !important;
+  }
+
+  .align-self-sm-baseline {
+    align-self: baseline !important;
+  }
+
+  .align-self-sm-stretch {
+    align-self: stretch !important;
+  }
+}
+@media (min-width: 768px) {
+  .flex-md-row {
+    flex-direction: row !important;
+  }
+
+  .flex-md-column {
+    flex-direction: column !important;
+  }
+
+  .flex-md-row-reverse {
+    flex-direction: row-reverse !important;
+  }
+
+  .flex-md-column-reverse {
+    flex-direction: column-reverse !important;
+  }
+
+  .flex-md-wrap {
+    flex-wrap: wrap !important;
+  }
+
+  .flex-md-nowrap {
+    flex-wrap: nowrap !important;
+  }
+
+  .flex-md-wrap-reverse {
+    flex-wrap: wrap-reverse !important;
+  }
+
+  .justify-content-md-start {
+    justify-content: flex-start !important;
+  }
+
+  .justify-content-md-end {
+    justify-content: flex-end !important;
+  }
+
+  .justify-content-md-center {
+    justify-content: center !important;
+  }
+
+  .justify-content-md-between {
+    justify-content: space-between !important;
+  }
+
+  .justify-content-md-around {
+    justify-content: space-around !important;
+  }
+
+  .align-items-md-start {
+    align-items: flex-start !important;
+  }
+
+  .align-items-md-end {
+    align-items: flex-end !important;
+  }
+
+  .align-items-md-center {
+    align-items: center !important;
+  }
+
+  .align-items-md-baseline {
+    align-items: baseline !important;
+  }
+
+  .align-items-md-stretch {
+    align-items: stretch !important;
+  }
+
+  .align-content-md-start {
+    align-content: flex-start !important;
+  }
+
+  .align-content-md-end {
+    align-content: flex-end !important;
+  }
+
+  .align-content-md-center {
+    align-content: center !important;
+  }
+
+  .align-content-md-between {
+    align-content: space-between !important;
+  }
+
+  .align-content-md-around {
+    align-content: space-around !important;
+  }
+
+  .align-content-md-stretch {
+    align-content: stretch !important;
+  }
+
+  .align-self-md-auto {
+    align-self: auto !important;
+  }
+
+  .align-self-md-start {
+    align-self: flex-start !important;
+  }
+
+  .align-self-md-end {
+    align-self: flex-end !important;
+  }
+
+  .align-self-md-center {
+    align-self: center !important;
+  }
+
+  .align-self-md-baseline {
+    align-self: baseline !important;
+  }
+
+  .align-self-md-stretch {
+    align-self: stretch !important;
+  }
+}
+@media (min-width: 992px) {
+  .flex-lg-row {
+    flex-direction: row !important;
+  }
+
+  .flex-lg-column {
+    flex-direction: column !important;
+  }
+
+  .flex-lg-row-reverse {
+    flex-direction: row-reverse !important;
+  }
+
+  .flex-lg-column-reverse {
+    flex-direction: column-reverse !important;
+  }
+
+  .flex-lg-wrap {
+    flex-wrap: wrap !important;
+  }
+
+  .flex-lg-nowrap {
+    flex-wrap: nowrap !important;
+  }
+
+  .flex-lg-wrap-reverse {
+    flex-wrap: wrap-reverse !important;
+  }
+
+  .justify-content-lg-start {
+    justify-content: flex-start !important;
+  }
+
+  .justify-content-lg-end {
+    justify-content: flex-end !important;
+  }
+
+  .justify-content-lg-center {
+    justify-content: center !important;
+  }
+
+  .justify-content-lg-between {
+    justify-content: space-between !important;
+  }
+
+  .justify-content-lg-around {
+    justify-content: space-around !important;
+  }
+
+  .align-items-lg-start {
+    align-items: flex-start !important;
+  }
+
+  .align-items-lg-end {
+    align-items: flex-end !important;
+  }
+
+  .align-items-lg-center {
+    align-items: center !important;
+  }
+
+  .align-items-lg-baseline {
+    align-items: baseline !important;
+  }
+
+  .align-items-lg-stretch {
+    align-items: stretch !important;
+  }
+
+  .align-content-lg-start {
+    align-content: flex-start !important;
+  }
+
+  .align-content-lg-end {
+    align-content: flex-end !important;
+  }
+
+  .align-content-lg-center {
+    align-content: center !important;
+  }
+
+  .align-content-lg-between {
+    align-content: space-between !important;
+  }
+
+  .align-content-lg-around {
+    align-content: space-around !important;
+  }
+
+  .align-content-lg-stretch {
+    align-content: stretch !important;
+  }
+
+  .align-self-lg-auto {
+    align-self: auto !important;
+  }
+
+  .align-self-lg-start {
+    align-self: flex-start !important;
+  }
+
+  .align-self-lg-end {
+    align-self: flex-end !important;
+  }
+
+  .align-self-lg-center {
+    align-self: center !important;
+  }
+
+  .align-self-lg-baseline {
+    align-self: baseline !important;
+  }
+
+  .align-self-lg-stretch {
+    align-self: stretch !important;
+  }
+}
+@media (min-width: 1200px) {
+  .flex-xl-row {
+    flex-direction: row !important;
+  }
+
+  .flex-xl-column {
+    flex-direction: column !important;
+  }
+
+  .flex-xl-row-reverse {
+    flex-direction: row-reverse !important;
+  }
+
+  .flex-xl-column-reverse {
+    flex-direction: column-reverse !important;
+  }
+
+  .flex-xl-wrap {
+    flex-wrap: wrap !important;
+  }
+
+  .flex-xl-nowrap {
+    flex-wrap: nowrap !important;
+  }
+
+  .flex-xl-wrap-reverse {
+    flex-wrap: wrap-reverse !important;
+  }
+
+  .justify-content-xl-start {
+    justify-content: flex-start !important;
+  }
+
+  .justify-content-xl-end {
+    justify-content: flex-end !important;
+  }
+
+  .justify-content-xl-center {
+    justify-content: center !important;
+  }
+
+  .justify-content-xl-between {
+    justify-content: space-between !important;
+  }
+
+  .justify-content-xl-around {
+    justify-content: space-around !important;
+  }
+
+  .align-items-xl-start {
+    align-items: flex-start !important;
+  }
+
+  .align-items-xl-end {
+    align-items: flex-end !important;
+  }
+
+  .align-items-xl-center {
+    align-items: center !important;
+  }
+
+  .align-items-xl-baseline {
+    align-items: baseline !important;
+  }
+
+  .align-items-xl-stretch {
+    align-items: stretch !important;
+  }
+
+  .align-content-xl-start {
+    align-content: flex-start !important;
+  }
+
+  .align-content-xl-end {
+    align-content: flex-end !important;
+  }
+
+  .align-content-xl-center {
+    align-content: center !important;
+  }
+
+  .align-content-xl-between {
+    align-content: space-between !important;
+  }
+
+  .align-content-xl-around {
+    align-content: space-around !important;
+  }
+
+  .align-content-xl-stretch {
+    align-content: stretch !important;
+  }
+
+  .align-self-xl-auto {
+    align-self: auto !important;
+  }
+
+  .align-self-xl-start {
+    align-self: flex-start !important;
+  }
+
+  .align-self-xl-end {
+    align-self: flex-end !important;
+  }
+
+  .align-self-xl-center {
+    align-self: center !important;
+  }
+
+  .align-self-xl-baseline {
+    align-self: baseline !important;
+  }
+
+  .align-self-xl-stretch {
+    align-self: stretch !important;
+  }
+}
+.float-left {
+  float: left !important;
+}
+
+.float-right {
+  float: right !important;
+}
+
+.float-none {
+  float: none !important;
+}
+
+@media (min-width: 576px) {
+  .float-sm-left {
+    float: left !important;
+  }
+
+  .float-sm-right {
+    float: right !important;
+  }
+
+  .float-sm-none {
+    float: none !important;
+  }
+}
+@media (min-width: 768px) {
+  .float-md-left {
+    float: left !important;
+  }
+
+  .float-md-right {
+    float: right !important;
+  }
+
+  .float-md-none {
+    float: none !important;
+  }
+}
+@media (min-width: 992px) {
+  .float-lg-left {
+    float: left !important;
+  }
+
+  .float-lg-right {
+    float: right !important;
+  }
+
+  .float-lg-none {
+    float: none !important;
+  }
+}
+@media (min-width: 1200px) {
+  .float-xl-left {
+    float: left !important;
+  }
+
+  .float-xl-right {
+    float: right !important;
+  }
+
+  .float-xl-none {
+    float: none !important;
+  }
+}
+.position-static {
+  position: static !important;
+}
+
+.position-relative {
+  position: relative !important;
+}
+
+.position-absolute {
+  position: absolute !important;
+}
+
+.position-fixed {
+  position: fixed !important;
+}
+
+.position-sticky {
+  position: -webkit-sticky !important;
+  position: sticky !important;
+}
+
+.fixed-top {
+  position: fixed;
+  top: 0;
+  right: 0;
+  left: 0;
+  z-index: 1030;
+}
+
+.fixed-bottom {
+  position: fixed;
+  right: 0;
+  bottom: 0;
+  left: 0;
+  z-index: 1030;
+}
+
+@supports ((position: -webkit-sticky) or (position: sticky)) {
+  .sticky-top {
+    position: -webkit-sticky;
+    position: sticky;
+    top: 0;
+    z-index: 1020;
+  }
+}
+
+.sr-only {
+  position: absolute;
+  width: 1px;
+  height: 1px;
+  padding: 0;
+  overflow: hidden;
+  clip: rect(0, 0, 0, 0);
+  white-space: nowrap;
+  -webkit-clip-path: inset(50%);
+          clip-path: inset(50%);
+  border: 0;
+}
+
+.sr-only-focusable:active, .sr-only-focusable:focus {
+  position: static;
+  width: auto;
+  height: auto;
+  overflow: visible;
+  clip: auto;
+  white-space: normal;
+  -webkit-clip-path: none;
+          clip-path: none;
+}
+
+.w-25 {
+  width: 25% !important;
+}
+
+.w-50 {
+  width: 50% !important;
+}
+
+.w-75 {
+  width: 75% !important;
+}
+
+.w-100 {
+  width: 100% !important;
+}
+
+.h-25 {
+  height: 25% !important;
+}
+
+.h-50 {
+  height: 50% !important;
+}
+
+.h-75 {
+  height: 75% !important;
+}
+
+.h-100 {
+  height: 100% !important;
+}
+
+.mw-100 {
+  max-width: 100% !important;
+}
+
+.mh-100 {
+  max-height: 100% !important;
+}
+
+.m-0 {
+  margin: 0 !important;
+}
+
+.mt-0,
+.my-0 {
+  margin-top: 0 !important;
+}
+
+.mr-0,
+.mx-0 {
+  margin-right: 0 !important;
+}
+
+.mb-0,
+.my-0 {
+  margin-bottom: 0 !important;
+}
+
+.ml-0,
+.mx-0 {
+  margin-left: 0 !important;
+}
+
+.m-1 {
+  margin: 0.25rem !important;
+}
+
+.mt-1,
+.my-1 {
+  margin-top: 0.25rem !important;
+}
+
+.mr-1,
+.mx-1 {
+  margin-right: 0.25rem !important;
+}
+
+.mb-1,
+.my-1 {
+  margin-bottom: 0.25rem !important;
+}
+
+.ml-1,
+.mx-1 {
+  margin-left: 0.25rem !important;
+}
+
+.m-2 {
+  margin: 0.5rem !important;
+}
+
+.mt-2,
+.my-2 {
+  margin-top: 0.5rem !important;
+}
+
+.mr-2,
+.mx-2 {
+  margin-right: 0.5rem !important;
+}
+
+.mb-2,
+.my-2 {
+  margin-bottom: 0.5rem !important;
+}
+
+.ml-2,
+.mx-2 {
+  margin-left: 0.5rem !important;
+}
+
+.m-3 {
+  margin: 1rem !important;
+}
+
+.mt-3,
+.my-3 {
+  margin-top: 1rem !important;
+}
+
+.mr-3,
+.mx-3 {
+  margin-right: 1rem !important;
+}
+
+.mb-3,
+.my-3 {
+  margin-bottom: 1rem !important;
+}
+
+.ml-3,
+.mx-3 {
+  margin-left: 1rem !important;
+}
+
+.m-4 {
+  margin: 1.5rem !important;
+}
+
+.mt-4,
+.my-4 {
+  margin-top: 1.5rem !important;
+}
+
+.mr-4,
+.mx-4 {
+  margin-right: 1.5rem !important;
+}
+
+.mb-4,
+.my-4 {
+  margin-bottom: 1.5rem !important;
+}
+
+.ml-4,
+.mx-4 {
+  margin-left: 1.5rem !important;
+}
+
+.m-5 {
+  margin: 3rem !important;
+}
+
+.mt-5,
+.my-5 {
+  margin-top: 3rem !important;
+}
+
+.mr-5,
+.mx-5 {
+  margin-right: 3rem !important;
+}
+
+.mb-5,
+.my-5 {
+  margin-bottom: 3rem !important;
+}
+
+.ml-5,
+.mx-5 {
+  margin-left: 3rem !important;
+}
+
+.p-0 {
+  padding: 0 !important;
+}
+
+.pt-0,
+.py-0 {
+  padding-top: 0 !important;
+}
+
+.pr-0,
+.px-0 {
+  padding-right: 0 !important;
+}
+
+.pb-0,
+.py-0 {
+  padding-bottom: 0 !important;
+}
+
+.pl-0,
+.px-0 {
+  padding-left: 0 !important;
+}
+
+.p-1 {
+  padding: 0.25rem !important;
+}
+
+.pt-1,
+.py-1 {
+  padding-top: 0.25rem !important;
+}
+
+.pr-1,
+.px-1 {
+  padding-right: 0.25rem !important;
+}
+
+.pb-1,
+.py-1 {
+  padding-bottom: 0.25rem !important;
+}
+
+.pl-1,
+.px-1 {
+  padding-left: 0.25rem !important;
+}
+
+.p-2 {
+  padding: 0.5rem !important;
+}
+
+.pt-2,
+.py-2 {
+  padding-top: 0.5rem !important;
+}
+
+.pr-2,
+.px-2 {
+  padding-right: 0.5rem !important;
+}
+
+.pb-2,
+.py-2 {
+  padding-bottom: 0.5rem !important;
+}
+
+.pl-2,
+.px-2 {
+  padding-left: 0.5rem !important;
+}
+
+.p-3 {
+  padding: 1rem !important;
+}
+
+.pt-3,
+.py-3 {
+  padding-top: 1rem !important;
+}
+
+.pr-3,
+.px-3 {
+  padding-right: 1rem !important;
+}
+
+.pb-3,
+.py-3 {
+  padding-bottom: 1rem !important;
+}
+
+.pl-3,
+.px-3 {
+  padding-left: 1rem !important;
+}
+
+.p-4 {
+  padding: 1.5rem !important;
+}
+
+.pt-4,
+.py-4 {
+  padding-top: 1.5rem !important;
+}
+
+.pr-4,
+.px-4 {
+  padding-right: 1.5rem !important;
+}
+
+.pb-4,
+.py-4 {
+  padding-bottom: 1.5rem !important;
+}
+
+.pl-4,
+.px-4 {
+  padding-left: 1.5rem !important;
+}
+
+.p-5 {
+  padding: 3rem !important;
+}
+
+.pt-5,
+.py-5 {
+  padding-top: 3rem !important;
+}
+
+.pr-5,
+.px-5 {
+  padding-right: 3rem !important;
+}
+
+.pb-5,
+.py-5 {
+  padding-bottom: 3rem !important;
+}
+
+.pl-5,
+.px-5 {
+  padding-left: 3rem !important;
+}
+
+.m-auto {
+  margin: auto !important;
+}
+
+.mt-auto,
+.my-auto {
+  margin-top: auto !important;
+}
+
+.mr-auto,
+.mx-auto {
+  margin-right: auto !important;
+}
+
+.mb-auto,
+.my-auto {
+  margin-bottom: auto !important;
+}
+
+.ml-auto,
+.mx-auto {
+  margin-left: auto !important;
+}
+
+@media (min-width: 576px) {
+  .m-sm-0 {
+    margin: 0 !important;
+  }
+
+  .mt-sm-0,
+  .my-sm-0 {
+    margin-top: 0 !important;
+  }
+
+  .mr-sm-0,
+  .mx-sm-0 {
+    margin-right: 0 !important;
+  }
+
+  .mb-sm-0,
+  .my-sm-0 {
+    margin-bottom: 0 !important;
+  }
+
+  .ml-sm-0,
+  .mx-sm-0 {
+    margin-left: 0 !important;
+  }
+
+  .m-sm-1 {
+    margin: 0.25rem !important;
+  }
+
+  .mt-sm-1,
+  .my-sm-1 {
+    margin-top: 0.25rem !important;
+  }
+
+  .mr-sm-1,
+  .mx-sm-1 {
+    margin-right: 0.25rem !important;
+  }
+
+  .mb-sm-1,
+  .my-sm-1 {
+    margin-bottom: 0.25rem !important;
+  }
+
+  .ml-sm-1,
+  .mx-sm-1 {
+    margin-left: 0.25rem !important;
+  }
+
+  .m-sm-2 {
+    margin: 0.5rem !important;
+  }
+
+  .mt-sm-2,
+  .my-sm-2 {
+    margin-top: 0.5rem !important;
+  }
+
+  .mr-sm-2,
+  .mx-sm-2 {
+    margin-right: 0.5rem !important;
+  }
+
+  .mb-sm-2,
+  .my-sm-2 {
+    margin-bottom: 0.5rem !important;
+  }
+
+  .ml-sm-2,
+  .mx-sm-2 {
+    margin-left: 0.5rem !important;
+  }
+
+  .m-sm-3 {
+    margin: 1rem !important;
+  }
+
+  .mt-sm-3,
+  .my-sm-3 {
+    margin-top: 1rem !important;
+  }
+
+  .mr-sm-3,
+  .mx-sm-3 {
+    margin-right: 1rem !important;
+  }
+
+  .mb-sm-3,
+  .my-sm-3 {
+    margin-bottom: 1rem !important;
+  }
+
+  .ml-sm-3,
+  .mx-sm-3 {
+    margin-left: 1rem !important;
+  }
+
+  .m-sm-4 {
+    margin: 1.5rem !important;
+  }
+
+  .mt-sm-4,
+  .my-sm-4 {
+    margin-top: 1.5rem !important;
+  }
+
+  .mr-sm-4,
+  .mx-sm-4 {
+    margin-right: 1.5rem !important;
+  }
+
+  .mb-sm-4,
+  .my-sm-4 {
+    margin-bottom: 1.5rem !important;
+  }
+
+  .ml-sm-4,
+  .mx-sm-4 {
+    margin-left: 1.5rem !important;
+  }
+
+  .m-sm-5 {
+    margin: 3rem !important;
+  }
+
+  .mt-sm-5,
+  .my-sm-5 {
+    margin-top: 3rem !important;
+  }
+
+  .mr-sm-5,
+  .mx-sm-5 {
+    margin-right: 3rem !important;
+  }
+
+  .mb-sm-5,
+  .my-sm-5 {
+    margin-bottom: 3rem !important;
+  }
+
+  .ml-sm-5,
+  .mx-sm-5 {
+    margin-left: 3rem !important;
+  }
+
+  .p-sm-0 {
+    padding: 0 !important;
+  }
+
+  .pt-sm-0,
+  .py-sm-0 {
+    padding-top: 0 !important;
+  }
+
+  .pr-sm-0,
+  .px-sm-0 {
+    padding-right: 0 !important;
+  }
+
+  .pb-sm-0,
+  .py-sm-0 {
+    padding-bottom: 0 !important;
+  }
+
+  .pl-sm-0,
+  .px-sm-0 {
+    padding-left: 0 !important;
+  }
+
+  .p-sm-1 {
+    padding: 0.25rem !important;
+  }
+
+  .pt-sm-1,
+  .py-sm-1 {
+    padding-top: 0.25rem !important;
+  }
+
+  .pr-sm-1,
+  .px-sm-1 {
+    padding-right: 0.25rem !important;
+  }
+
+  .pb-sm-1,
+  .py-sm-1 {
+    padding-bottom: 0.25rem !important;
+  }
+
+  .pl-sm-1,
+  .px-sm-1 {
+    padding-left: 0.25rem !important;
+  }
+
+  .p-sm-2 {
+    padding: 0.5rem !important;
+  }
+
+  .pt-sm-2,
+  .py-sm-2 {
+    padding-top: 0.5rem !important;
+  }
+
+  .pr-sm-2,
+  .px-sm-2 {
+    padding-right: 0.5rem !important;
+  }
+
+  .pb-sm-2,
+  .py-sm-2 {
+    padding-bottom: 0.5rem !important;
+  }
+
+  .pl-sm-2,
+  .px-sm-2 {
+    padding-left: 0.5rem !important;
+  }
+
+  .p-sm-3 {
+    padding: 1rem !important;
+  }
+
+  .pt-sm-3,
+  .py-sm-3 {
+    padding-top: 1rem !important;
+  }
+
+  .pr-sm-3,
+  .px-sm-3 {
+    padding-right: 1rem !important;
+  }
+
+  .pb-sm-3,
+  .py-sm-3 {
+    padding-bottom: 1rem !important;
+  }
+
+  .pl-sm-3,
+  .px-sm-3 {
+    padding-left: 1rem !important;
+  }
+
+  .p-sm-4 {
+    padding: 1.5rem !important;
+  }
+
+  .pt-sm-4,
+  .py-sm-4 {
+    padding-top: 1.5rem !important;
+  }
+
+  .pr-sm-4,
+  .px-sm-4 {
+    padding-right: 1.5rem !important;
+  }
+
+  .pb-sm-4,
+  .py-sm-4 {
+    padding-bottom: 1.5rem !important;
+  }
+
+  .pl-sm-4,
+  .px-sm-4 {
+    padding-left: 1.5rem !important;
+  }
+
+  .p-sm-5 {
+    padding: 3rem !important;
+  }
+
+  .pt-sm-5,
+  .py-sm-5 {
+    padding-top: 3rem !important;
+  }
+
+  .pr-sm-5,
+  .px-sm-5 {
+    padding-right: 3rem !important;
+  }
+
+  .pb-sm-5,
+  .py-sm-5 {
+    padding-bottom: 3rem !important;
+  }
+
+  .pl-sm-5,
+  .px-sm-5 {
+    padding-left: 3rem !important;
+  }
+
+  .m-sm-auto {
+    margin: auto !important;
+  }
+
+  .mt-sm-auto,
+  .my-sm-auto {
+    margin-top: auto !important;
+  }
+
+  .mr-sm-auto,
+  .mx-sm-auto {
+    margin-right: auto !important;
+  }
+
+  .mb-sm-auto,
+  .my-sm-auto {
+    margin-bottom: auto !important;
+  }
+
+  .ml-sm-auto,
+  .mx-sm-auto {
+    margin-left: auto !important;
+  }
+}
+@media (min-width: 768px) {
+  .m-md-0 {
+    margin: 0 !important;
+  }
+
+  .mt-md-0,
+  .my-md-0 {
+    margin-top: 0 !important;
+  }
+
+  .mr-md-0,
+  .mx-md-0 {
+    margin-right: 0 !important;
+  }
+
+  .mb-md-0,
+  .my-md-0 {
+    margin-bottom: 0 !important;
+  }
+
+  .ml-md-0,
+  .mx-md-0 {
+    margin-left: 0 !important;
+  }
+
+  .m-md-1 {
+    margin: 0.25rem !important;
+  }
+
+  .mt-md-1,
+  .my-md-1 {
+    margin-top: 0.25rem !important;
+  }
+
+  .mr-md-1,
+  .mx-md-1 {
+    margin-right: 0.25rem !important;
+  }
+
+  .mb-md-1,
+  .my-md-1 {
+    margin-bottom: 0.25rem !important;
+  }
+
+  .ml-md-1,
+  .mx-md-1 {
+    margin-left: 0.25rem !important;
+  }
+
+  .m-md-2 {
+    margin: 0.5rem !important;
+  }
+
+  .mt-md-2,
+  .my-md-2 {
+    margin-top: 0.5rem !important;
+  }
+
+  .mr-md-2,
+  .mx-md-2 {
+    margin-right: 0.5rem !important;
+  }
+
+  .mb-md-2,
+  .my-md-2 {
+    margin-bottom: 0.5rem !important;
+  }
+
+  .ml-md-2,
+  .mx-md-2 {
+    margin-left: 0.5rem !important;
+  }
+
+  .m-md-3 {
+    margin: 1rem !important;
+  }
+
+  .mt-md-3,
+  .my-md-3 {
+    margin-top: 1rem !important;
+  }
+
+  .mr-md-3,
+  .mx-md-3 {
+    margin-right: 1rem !important;
+  }
+
+  .mb-md-3,
+  .my-md-3 {
+    margin-bottom: 1rem !important;
+  }
+
+  .ml-md-3,
+  .mx-md-3 {
+    margin-left: 1rem !important;
+  }
+
+  .m-md-4 {
+    margin: 1.5rem !important;
+  }
+
+  .mt-md-4,
+  .my-md-4 {
+    margin-top: 1.5rem !important;
+  }
+
+  .mr-md-4,
+  .mx-md-4 {
+    margin-right: 1.5rem !important;
+  }
+
+  .mb-md-4,
+  .my-md-4 {
+    margin-bottom: 1.5rem !important;
+  }
+
+  .ml-md-4,
+  .mx-md-4 {
+    margin-left: 1.5rem !important;
+  }
+
+  .m-md-5 {
+    margin: 3rem !important;
+  }
+
+  .mt-md-5,
+  .my-md-5 {
+    margin-top: 3rem !important;
+  }
+
+  .mr-md-5,
+  .mx-md-5 {
+    margin-right: 3rem !important;
+  }
+
+  .mb-md-5,
+  .my-md-5 {
+    margin-bottom: 3rem !important;
+  }
+
+  .ml-md-5,
+  .mx-md-5 {
+    margin-left: 3rem !important;
+  }
+
+  .p-md-0 {
+    padding: 0 !important;
+  }
+
+  .pt-md-0,
+  .py-md-0 {
+    padding-top: 0 !important;
+  }
+
+  .pr-md-0,
+  .px-md-0 {
+    padding-right: 0 !important;
+  }
+
+  .pb-md-0,
+  .py-md-0 {
+    padding-bottom: 0 !important;
+  }
+
+  .pl-md-0,
+  .px-md-0 {
+    padding-left: 0 !important;
+  }
+
+  .p-md-1 {
+    padding: 0.25rem !important;
+  }
+
+  .pt-md-1,
+  .py-md-1 {
+    padding-top: 0.25rem !important;
+  }
+
+  .pr-md-1,
+  .px-md-1 {
+    padding-right: 0.25rem !important;
+  }
+
+  .pb-md-1,
+  .py-md-1 {
+    padding-bottom: 0.25rem !important;
+  }
+
+  .pl-md-1,
+  .px-md-1 {
+    padding-left: 0.25rem !important;
+  }
+
+  .p-md-2 {
+    padding: 0.5rem !important;
+  }
+
+  .pt-md-2,
+  .py-md-2 {
+    padding-top: 0.5rem !important;
+  }
+
+  .pr-md-2,
+  .px-md-2 {
+    padding-right: 0.5rem !important;
+  }
+
+  .pb-md-2,
+  .py-md-2 {
+    padding-bottom: 0.5rem !important;
+  }
+
+  .pl-md-2,
+  .px-md-2 {
+    padding-left: 0.5rem !important;
+  }
+
+  .p-md-3 {
+    padding: 1rem !important;
+  }
+
+  .pt-md-3,
+  .py-md-3 {
+    padding-top: 1rem !important;
+  }
+
+  .pr-md-3,
+  .px-md-3 {
+    padding-right: 1rem !important;
+  }
+
+  .pb-md-3,
+  .py-md-3 {
+    padding-bottom: 1rem !important;
+  }
+
+  .pl-md-3,
+  .px-md-3 {
+    padding-left: 1rem !important;
+  }
+
+  .p-md-4 {
+    padding: 1.5rem !important;
+  }
+
+  .pt-md-4,
+  .py-md-4 {
+    padding-top: 1.5rem !important;
+  }
+
+  .pr-md-4,
+  .px-md-4 {
+    padding-right: 1.5rem !important;
+  }
+
+  .pb-md-4,
+  .py-md-4 {
+    padding-bottom: 1.5rem !important;
+  }
+
+  .pl-md-4,
+  .px-md-4 {
+    padding-left: 1.5rem !important;
+  }
+
+  .p-md-5 {
+    padding: 3rem !important;
+  }
+
+  .pt-md-5,
+  .py-md-5 {
+    padding-top: 3rem !important;
+  }
+
+  .pr-md-5,
+  .px-md-5 {
+    padding-right: 3rem !important;
+  }
+
+  .pb-md-5,
+  .py-md-5 {
+    padding-bottom: 3rem !important;
+  }
+
+  .pl-md-5,
+  .px-md-5 {
+    padding-left: 3rem !important;
+  }
+
+  .m-md-auto {
+    margin: auto !important;
+  }
+
+  .mt-md-auto,
+  .my-md-auto {
+    margin-top: auto !important;
+  }
+
+  .mr-md-auto,
+  .mx-md-auto {
+    margin-right: auto !important;
+  }
+
+  .mb-md-auto,
+  .my-md-auto {
+    margin-bottom: auto !important;
+  }
+
+  .ml-md-auto,
+  .mx-md-auto {
+    margin-left: auto !important;
+  }
+}
+@media (min-width: 992px) {
+  .m-lg-0 {
+    margin: 0 !important;
+  }
+
+  .mt-lg-0,
+  .my-lg-0 {
+    margin-top: 0 !important;
+  }
+
+  .mr-lg-0,
+  .mx-lg-0 {
+    margin-right: 0 !important;
+  }
+
+  .mb-lg-0,
+  .my-lg-0 {
+    margin-bottom: 0 !important;
+  }
+
+  .ml-lg-0,
+  .mx-lg-0 {
+    margin-left: 0 !important;
+  }
+
+  .m-lg-1 {
+    margin: 0.25rem !important;
+  }
+
+  .mt-lg-1,
+  .my-lg-1 {
+    margin-top: 0.25rem !important;
+  }
+
+  .mr-lg-1,
+  .mx-lg-1 {
+    margin-right: 0.25rem !important;
+  }
+
+  .mb-lg-1,
+  .my-lg-1 {
+    margin-bottom: 0.25rem !important;
+  }
+
+  .ml-lg-1,
+  .mx-lg-1 {
+    margin-left: 0.25rem !important;
+  }
+
+  .m-lg-2 {
+    margin: 0.5rem !important;
+  }
+
+  .mt-lg-2,
+  .my-lg-2 {
+    margin-top: 0.5rem !important;
+  }
+
+  .mr-lg-2,
+  .mx-lg-2 {
+    margin-right: 0.5rem !important;
+  }
+
+  .mb-lg-2,
+  .my-lg-2 {
+    margin-bottom: 0.5rem !important;
+  }
+
+  .ml-lg-2,
+  .mx-lg-2 {
+    margin-left: 0.5rem !important;
+  }
+
+  .m-lg-3 {
+    margin: 1rem !important;
+  }
+
+  .mt-lg-3,
+  .my-lg-3 {
+    margin-top: 1rem !important;
+  }
+
+  .mr-lg-3,
+  .mx-lg-3 {
+    margin-right: 1rem !important;
+  }
+
+  .mb-lg-3,
+  .my-lg-3 {
+    margin-bottom: 1rem !important;
+  }
+
+  .ml-lg-3,
+  .mx-lg-3 {
+    margin-left: 1rem !important;
+  }
+
+  .m-lg-4 {
+    margin: 1.5rem !important;
+  }
+
+  .mt-lg-4,
+  .my-lg-4 {
+    margin-top: 1.5rem !important;
+  }
+
+  .mr-lg-4,
+  .mx-lg-4 {
+    margin-right: 1.5rem !important;
+  }
+
+  .mb-lg-4,
+  .my-lg-4 {
+    margin-bottom: 1.5rem !important;
+  }
+
+  .ml-lg-4,
+  .mx-lg-4 {
+    margin-left: 1.5rem !important;
+  }
+
+  .m-lg-5 {
+    margin: 3rem !important;
+  }
+
+  .mt-lg-5,
+  .my-lg-5 {
+    margin-top: 3rem !important;
+  }
+
+  .mr-lg-5,
+  .mx-lg-5 {
+    margin-right: 3rem !important;
+  }
+
+  .mb-lg-5,
+  .my-lg-5 {
+    margin-bottom: 3rem !important;
+  }
+
+  .ml-lg-5,
+  .mx-lg-5 {
+    margin-left: 3rem !important;
+  }
+
+  .p-lg-0 {
+    padding: 0 !important;
+  }
+
+  .pt-lg-0,
+  .py-lg-0 {
+    padding-top: 0 !important;
+  }
+
+  .pr-lg-0,
+  .px-lg-0 {
+    padding-right: 0 !important;
+  }
+
+  .pb-lg-0,
+  .py-lg-0 {
+    padding-bottom: 0 !important;
+  }
+
+  .pl-lg-0,
+  .px-lg-0 {
+    padding-left: 0 !important;
+  }
+
+  .p-lg-1 {
+    padding: 0.25rem !important;
+  }
+
+  .pt-lg-1,
+  .py-lg-1 {
+    padding-top: 0.25rem !important;
+  }
+
+  .pr-lg-1,
+  .px-lg-1 {
+    padding-right: 0.25rem !important;
+  }
+
+  .pb-lg-1,
+  .py-lg-1 {
+    padding-bottom: 0.25rem !important;
+  }
+
+  .pl-lg-1,
+  .px-lg-1 {
+    padding-left: 0.25rem !important;
+  }
+
+  .p-lg-2 {
+    padding: 0.5rem !important;
+  }
+
+  .pt-lg-2,
+  .py-lg-2 {
+    padding-top: 0.5rem !important;
+  }
+
+  .pr-lg-2,
+  .px-lg-2 {
+    padding-right: 0.5rem !important;
+  }
+
+  .pb-lg-2,
+  .py-lg-2 {
+    padding-bottom: 0.5rem !important;
+  }
+
+  .pl-lg-2,
+  .px-lg-2 {
+    padding-left: 0.5rem !important;
+  }
+
+  .p-lg-3 {
+    padding: 1rem !important;
+  }
+
+  .pt-lg-3,
+  .py-lg-3 {
+    padding-top: 1rem !important;
+  }
+
+  .pr-lg-3,
+  .px-lg-3 {
+    padding-right: 1rem !important;
+  }
+
+  .pb-lg-3,
+  .py-lg-3 {
+    padding-bottom: 1rem !important;
+  }
+
+  .pl-lg-3,
+  .px-lg-3 {
+    padding-left: 1rem !important;
+  }
+
+  .p-lg-4 {
+    padding: 1.5rem !important;
+  }
+
+  .pt-lg-4,
+  .py-lg-4 {
+    padding-top: 1.5rem !important;
+  }
+
+  .pr-lg-4,
+  .px-lg-4 {
+    padding-right: 1.5rem !important;
+  }
+
+  .pb-lg-4,
+  .py-lg-4 {
+    padding-bottom: 1.5rem !important;
+  }
+
+  .pl-lg-4,
+  .px-lg-4 {
+    padding-left: 1.5rem !important;
+  }
+
+  .p-lg-5 {
+    padding: 3rem !important;
+  }
+
+  .pt-lg-5,
+  .py-lg-5 {
+    padding-top: 3rem !important;
+  }
+
+  .pr-lg-5,
+  .px-lg-5 {
+    padding-right: 3rem !important;
+  }
+
+  .pb-lg-5,
+  .py-lg-5 {
+    padding-bottom: 3rem !important;
+  }
+
+  .pl-lg-5,
+  .px-lg-5 {
+    padding-left: 3rem !important;
+  }
+
+  .m-lg-auto {
+    margin: auto !important;
+  }
+
+  .mt-lg-auto,
+  .my-lg-auto {
+    margin-top: auto !important;
+  }
+
+  .mr-lg-auto,
+  .mx-lg-auto {
+    margin-right: auto !important;
+  }
+
+  .mb-lg-auto,
+  .my-lg-auto {
+    margin-bottom: auto !important;
+  }
+
+  .ml-lg-auto,
+  .mx-lg-auto {
+    margin-left: auto !important;
+  }
+}
+@media (min-width: 1200px) {
+  .m-xl-0 {
+    margin: 0 !important;
+  }
+
+  .mt-xl-0,
+  .my-xl-0 {
+    margin-top: 0 !important;
+  }
+
+  .mr-xl-0,
+  .mx-xl-0 {
+    margin-right: 0 !important;
+  }
+
+  .mb-xl-0,
+  .my-xl-0 {
+    margin-bottom: 0 !important;
+  }
+
+  .ml-xl-0,
+  .mx-xl-0 {
+    margin-left: 0 !important;
+  }
+
+  .m-xl-1 {
+    margin: 0.25rem !important;
+  }
+
+  .mt-xl-1,
+  .my-xl-1 {
+    margin-top: 0.25rem !important;
+  }
+
+  .mr-xl-1,
+  .mx-xl-1 {
+    margin-right: 0.25rem !important;
+  }
+
+  .mb-xl-1,
+  .my-xl-1 {
+    margin-bottom: 0.25rem !important;
+  }
+
+  .ml-xl-1,
+  .mx-xl-1 {
+    margin-left: 0.25rem !important;
+  }
+
+  .m-xl-2 {
+    margin: 0.5rem !important;
+  }
+
+  .mt-xl-2,
+  .my-xl-2 {
+    margin-top: 0.5rem !important;
+  }
+
+  .mr-xl-2,
+  .mx-xl-2 {
+    margin-right: 0.5rem !important;
+  }
+
+  .mb-xl-2,
+  .my-xl-2 {
+    margin-bottom: 0.5rem !important;
+  }
+
+  .ml-xl-2,
+  .mx-xl-2 {
+    margin-left: 0.5rem !important;
+  }
+
+  .m-xl-3 {
+    margin: 1rem !important;
+  }
+
+  .mt-xl-3,
+  .my-xl-3 {
+    margin-top: 1rem !important;
+  }
+
+  .mr-xl-3,
+  .mx-xl-3 {
+    margin-right: 1rem !important;
+  }
+
+  .mb-xl-3,
+  .my-xl-3 {
+    margin-bottom: 1rem !important;
+  }
+
+  .ml-xl-3,
+  .mx-xl-3 {
+    margin-left: 1rem !important;
+  }
+
+  .m-xl-4 {
+    margin: 1.5rem !important;
+  }
+
+  .mt-xl-4,
+  .my-xl-4 {
+    margin-top: 1.5rem !important;
+  }
+
+  .mr-xl-4,
+  .mx-xl-4 {
+    margin-right: 1.5rem !important;
+  }
+
+  .mb-xl-4,
+  .my-xl-4 {
+    margin-bottom: 1.5rem !important;
+  }
+
+  .ml-xl-4,
+  .mx-xl-4 {
+    margin-left: 1.5rem !important;
+  }
+
+  .m-xl-5 {
+    margin: 3rem !important;
+  }
+
+  .mt-xl-5,
+  .my-xl-5 {
+    margin-top: 3rem !important;
+  }
+
+  .mr-xl-5,
+  .mx-xl-5 {
+    margin-right: 3rem !important;
+  }
+
+  .mb-xl-5,
+  .my-xl-5 {
+    margin-bottom: 3rem !important;
+  }
+
+  .ml-xl-5,
+  .mx-xl-5 {
+    margin-left: 3rem !important;
+  }
+
+  .p-xl-0 {
+    padding: 0 !important;
+  }
+
+  .pt-xl-0,
+  .py-xl-0 {
+    padding-top: 0 !important;
+  }
+
+  .pr-xl-0,
+  .px-xl-0 {
+    padding-right: 0 !important;
+  }
+
+  .pb-xl-0,
+  .py-xl-0 {
+    padding-bottom: 0 !important;
+  }
+
+  .pl-xl-0,
+  .px-xl-0 {
+    padding-left: 0 !important;
+  }
+
+  .p-xl-1 {
+    padding: 0.25rem !important;
+  }
+
+  .pt-xl-1,
+  .py-xl-1 {
+    padding-top: 0.25rem !important;
+  }
+
+  .pr-xl-1,
+  .px-xl-1 {
+    padding-right: 0.25rem !important;
+  }
+
+  .pb-xl-1,
+  .py-xl-1 {
+    padding-bottom: 0.25rem !important;
+  }
+
+  .pl-xl-1,
+  .px-xl-1 {
+    padding-left: 0.25rem !important;
+  }
+
+  .p-xl-2 {
+    padding: 0.5rem !important;
+  }
+
+  .pt-xl-2,
+  .py-xl-2 {
+    padding-top: 0.5rem !important;
+  }
+
+  .pr-xl-2,
+  .px-xl-2 {
+    padding-right: 0.5rem !important;
+  }
+
+  .pb-xl-2,
+  .py-xl-2 {
+    padding-bottom: 0.5rem !important;
+  }
+
+  .pl-xl-2,
+  .px-xl-2 {
+    padding-left: 0.5rem !important;
+  }
+
+  .p-xl-3 {
+    padding: 1rem !important;
+  }
+
+  .pt-xl-3,
+  .py-xl-3 {
+    padding-top: 1rem !important;
+  }
+
+  .pr-xl-3,
+  .px-xl-3 {
+    padding-right: 1rem !important;
+  }
+
+  .pb-xl-3,
+  .py-xl-3 {
+    padding-bottom: 1rem !important;
+  }
+
+  .pl-xl-3,
+  .px-xl-3 {
+    padding-left: 1rem !important;
+  }
+
+  .p-xl-4 {
+    padding: 1.5rem !important;
+  }
+
+  .pt-xl-4,
+  .py-xl-4 {
+    padding-top: 1.5rem !important;
+  }
+
+  .pr-xl-4,
+  .px-xl-4 {
+    padding-right: 1.5rem !important;
+  }
+
+  .pb-xl-4,
+  .py-xl-4 {
+    padding-bottom: 1.5rem !important;
+  }
+
+  .pl-xl-4,
+  .px-xl-4 {
+    padding-left: 1.5rem !important;
+  }
+
+  .p-xl-5 {
+    padding: 3rem !important;
+  }
+
+  .pt-xl-5,
+  .py-xl-5 {
+    padding-top: 3rem !important;
+  }
+
+  .pr-xl-5,
+  .px-xl-5 {
+    padding-right: 3rem !important;
+  }
+
+  .pb-xl-5,
+  .py-xl-5 {
+    padding-bottom: 3rem !important;
+  }
+
+  .pl-xl-5,
+  .px-xl-5 {
+    padding-left: 3rem !important;
+  }
+
+  .m-xl-auto {
+    margin: auto !important;
+  }
+
+  .mt-xl-auto,
+  .my-xl-auto {
+    margin-top: auto !important;
+  }
+
+  .mr-xl-auto,
+  .mx-xl-auto {
+    margin-right: auto !important;
+  }
+
+  .mb-xl-auto,
+  .my-xl-auto {
+    margin-bottom: auto !important;
+  }
+
+  .ml-xl-auto,
+  .mx-xl-auto {
+    margin-left: auto !important;
+  }
+}
+.text-justify {
+  text-align: justify !important;
+}
+
+.text-nowrap {
+  white-space: nowrap !important;
+}
+
+.text-truncate {
+  overflow: hidden;
+  text-overflow: ellipsis;
+  white-space: nowrap;
+}
+
+.text-left {
+  text-align: left !important;
+}
+
+.text-right {
+  text-align: right !important;
+}
+
+.text-center {
+  text-align: center !important;
+}
+
+@media (min-width: 576px) {
+  .text-sm-left {
+    text-align: left !important;
+  }
+
+  .text-sm-right {
+    text-align: right !important;
+  }
+
+  .text-sm-center {
+    text-align: center !important;
+  }
+}
+@media (min-width: 768px) {
+  .text-md-left {
+    text-align: left !important;
+  }
+
+  .text-md-right {
+    text-align: right !important;
+  }
+
+  .text-md-center {
+    text-align: center !important;
+  }
+}
+@media (min-width: 992px) {
+  .text-lg-left {
+    text-align: left !important;
+  }
+
+  .text-lg-right {
+    text-align: right !important;
+  }
+
+  .text-lg-center {
+    text-align: center !important;
+  }
+}
+@media (min-width: 1200px) {
+  .text-xl-left {
+    text-align: left !important;
+  }
+
+  .text-xl-right {
+    text-align: right !important;
+  }
+
+  .text-xl-center {
+    text-align: center !important;
+  }
+}
+.text-lowercase {
+  text-transform: lowercase !important;
+}
+
+.text-uppercase {
+  text-transform: uppercase !important;
+}
+
+.text-capitalize {
+  text-transform: capitalize !important;
+}
+
+.font-weight-light {
+  font-weight: 300 !important;
+}
+
+.font-weight-normal {
+  font-weight: 400 !important;
+}
+
+.font-weight-bold {
+  font-weight: 700 !important;
+}
+
+.font-italic {
+  font-style: italic !important;
+}
+
+.text-white {
+  color: #fff !important;
+}
+
+.text-primary {
+  color: #007bff !important;
+}
+
+a.text-primary:hover, a.text-primary:focus {
+  color: #0062cc !important;
+}
+
+.text-secondary {
+  color: #6c757d !important;
+}
+
+a.text-secondary:hover, a.text-secondary:focus {
+  color: #545b62 !important;
+}
+
+.text-success {
+  color: #28a745 !important;
+}
+
+a.text-success:hover, a.text-success:focus {
+  color: #1e7e34 !important;
+}
+
+.text-info {
+  color: #17a2b8 !important;
+}
+
+a.text-info:hover, a.text-info:focus {
+  color: #117a8b !important;
+}
+
+.text-warning {
+  color: #ffc107 !important;
+}
+
+a.text-warning:hover, a.text-warning:focus {
+  color: #d39e00 !important;
+}
+
+.text-danger {
+  color: #dc3545 !important;
+}
+
+a.text-danger:hover, a.text-danger:focus {
+  color: #bd2130 !important;
+}
+
+.text-light {
+  color: #f8f9fa !important;
+}
+
+a.text-light:hover, a.text-light:focus {
+  color: #dae0e5 !important;
+}
+
+.text-dark {
+  color: #343a40 !important;
+}
+
+a.text-dark:hover, a.text-dark:focus {
+  color: #1d2124 !important;
+}
+
+.text-muted {
+  color: #6c757d !important;
+}
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+  font: 0/0 a;
+  color: transparent;
+  text-shadow: none;
+  background-color: transparent;
+  border: 0;
+}
+
+.visible {
+  visibility: visible !important;
+}
+
+.invisible {
+  visibility: hidden !important;
+}
+
+@media print {
+  *,
+  *::before,
+  *::after {
+    text-shadow: none !important;
+    box-shadow: none !important;
+  }
+
+  a:not(.btn) {
+    text-decoration: underline;
+  }
+
+  abbr[title]::after {
+    content: " (" attr(title) ")";
+  }
+
+  pre {
+    white-space: pre-wrap !important;
+  }
+
+  pre,
+  blockquote {
+    border: 1px solid #999;
+    page-break-inside: avoid;
+  }
+
+  thead {
+    display: table-header-group;
+  }
+
+  tr,
+  img {
+    page-break-inside: avoid;
+  }
+
+  p,
+  h2,
+  h3 {
+    orphans: 3;
+    widows: 3;
+  }
+
+  h2,
+  h3 {
+    page-break-after: avoid;
+  }
+
+  @page {
+    size: a3;
+  }
+  body {
+    min-width: 992px !important;
+  }
+
+  .container {
+    min-width: 992px !important;
+  }
+
+  .navbar {
+    display: none;
+  }
+
+  .badge {
+    border: 1px solid #000;
+  }
+
+  .table {
+    border-collapse: collapse !important;
+  }
+  .table td,
+  .table th {
+    background-color: #fff !important;
+  }
+
+  .table-bordered th,
+  .table-bordered td {
+    border: 1px solid #ddd !important;
+  }
+}
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+  padding: 5px;
+}
+
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+  margin: 0;
+}
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+  font-style: italic;
+}
+
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+  font-weight: bold;
+}
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+.highlight .c1 {
+  color: #999988;
+  font-style: italic;
+}
+
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+  font-weight: bold;
+  font-style: italic;
+}
+
+.highlight .c, .highlight .cd {
+  color: #999988;
+  font-style: italic;
+}
+
+.highlight .err {
+  color: #a61717;
+  background-color: #e3d2d2;
+}
+
+.highlight .gd {
+  color: #000000;
+  background-color: #ffdddd;
+}
+
+.highlight .ge {
+  color: #000000;
+  font-style: italic;
+}
+
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+  color: #aa0000;
+}
+
+.highlight .gh {
+  color: #999999;
+}
+
+.highlight .gi {
+  color: #000000;
+  background-color: #ddffdd;
+}
+
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+  color: #888888;
+}
+
+.highlight .gp {
+  color: #555555;
+}
+
+.highlight .gs {
+  font-weight: bold;
+}
+
+.highlight .gu {
+  color: #aaaaaa;
+}
+
+.highlight .gt {
+  color: #aa0000;
+}
+
+.highlight .kc {
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+  font-weight: bold;
+}
+
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+  font-weight: bold;
+}
+
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+  font-weight: bold;
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+  font-weight: bold;
+}
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+  font-weight: bold;
+}
+
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+  font-weight: bold;
+}
+
+.highlight .k, .highlight .kv {
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+  font-weight: bold;
+}
+
+.highlight .mf {
+  color: #009999;
+}
+
+.highlight .mh {
+  color: #009999;
+}
+
+.highlight .il {
+  color: #009999;
+}
+
+.highlight .mi {
+  color: #009999;
+}
+
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+}
+
+.highlight .m, .highlight .mb, .highlight .mx {
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+}
+
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+  color: #d14;
+}
+
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+  color: #d14;
+}
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+}
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+}
+
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+}
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+}
+
+.highlight .si {
+  color: #d14;
+}
+
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+  color: #d14;
+}
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+  color: #009926;
+}
+
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+}
+
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+}
+
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+}
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+}
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+}
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+
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+  font-weight: bold;
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+  font-weight: bold;
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+}
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+}
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+}
+
+@font-face {
+  font-family: FreightSans;
+  font-weight: 700;
+  font-style: normal;
+  src: url("../fonts/FreightSans/freight-sans-bold.woff2") format("woff2"), url("../fonts/FreightSans/freight-sans-bold.woff") format("woff");
+}
+@font-face {
+  font-family: FreightSans;
+  font-weight: 700;
+  font-style: italic;
+  src: url("../fonts/FreightSans/freight-sans-bold-italic.woff2") format("woff2"), url("../fonts/FreightSans/freight-sans-bold-italic.woff") format("woff");
+}
+@font-face {
+  font-family: FreightSans;
+  font-weight: 500;
+  font-style: normal;
+  src: url("../fonts/FreightSans/freight-sans-medium.woff2") format("woff2"), url("../fonts/FreightSans/freight-sans-medium.woff") format("woff");
+}
+@font-face {
+  font-family: FreightSans;
+  font-weight: 500;
+  font-style: italic;
+  src: url("../fonts/FreightSans/freight-sans-medium-italic.woff2") format("woff2"), url("../fonts/FreightSans/freight-sans-medium-italic.woff") format("woff");
+}
+@font-face {
+  font-family: FreightSans;
+  font-weight: 100;
+  font-style: normal;
+  src: url("../fonts/FreightSans/freight-sans-light.woff2") format("woff2"), url("../fonts/FreightSans/freight-sans-light.woff") format("woff");
+}
+@font-face {
+  font-family: FreightSans;
+  font-weight: 100;
+  font-style: italic;
+  src: url("../fonts/FreightSans/freight-sans-light-italic.woff2") format("woff2"), url("../fonts/FreightSans/freight-sans-light-italic.woff") format("woff");
+}
+@font-face {
+  font-family: FreightSans;
+  font-weight: 400;
+  font-style: italic;
+  src: url("../fonts/FreightSans/freight-sans-book-italic.woff2") format("woff2"), url("../fonts/FreightSans/freight-sans-book-italic.woff") format("woff");
+}
+@font-face {
+  font-family: FreightSans;
+  font-weight: 400;
+  font-style: normal;
+  src: url("../fonts/FreightSans/freight-sans-book.woff2") format("woff2"), url("../fonts/FreightSans/freight-sans-book.woff") format("woff");
+}
+@font-face {
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+  font-weight: 600;
+  font-style: normal;
+  unicode-range: u+0020-007f;
+  src: local("IBMPlexMono-SemiBold"), url("../fonts/IBMPlexMono/IBMPlexMono-SemiBold.woff2") format("woff2"), url("../fonts/IBMPlexMono/IBMPlexMono-SemiBold.woff") format("woff");
+}
+@font-face {
+  font-family: IBMPlexMono;
+  font-weight: 500;
+  font-style: normal;
+  unicode-range: u+0020-007f;
+  src: local("IBMPlexMono-Medium"), url("../fonts/IBMPlexMono/IBMPlexMono-Medium.woff2") format("woff2"), url("../fonts/IBMPlexMono/IBMPlexMono-Medium.woff") format("woff");
+}
+@font-face {
+  font-family: IBMPlexMono;
+  font-weight: 400;
+  font-style: normal;
+  unicode-range: u+0020-007f;
+  src: local("IBMPlexMono-Regular"), url("../fonts/IBMPlexMono/IBMPlexMono-Regular.woff2") format("woff2"), url("../fonts/IBMPlexMono/IBMPlexMono-Regular.woff") format("woff");
+}
+@font-face {
+  font-family: IBMPlexMono;
+  font-weight: 300;
+  font-style: normal;
+  unicode-range: u+0020-007f;
+  src: local("IBMPlexMono-Light"), url("../fonts/IBMPlexMono/IBMPlexMono-Light.woff2") format("woff2"), url("../fonts/IBMPlexMono/IBMPlexMono-Light.woff") format("woff");
+}
+html {
+  position: relative;
+  min-height: 100%;
+  font-size: 12px;
+}
+@media screen and (min-width: 768px) {
+  html {
+    font-size: 16px;
+  }
+}
+
+* {
+  box-sizing: border-box;
+}
+
+body {
+  font-family: FreightSans, Helvetica Neue, Helvetica, Arial, sans-serif;
+}
+
+a:link,
+a:visited,
+a:hover {
+  text-decoration: none;
+  color: #e44c2c;
+}
+
+a.with-right-arrow, .btn.with-right-arrow {
+  padding-right: 1.375rem;
+  position: relative;
+  background-image: url("../images/chevron-right-orange.svg");
+  background-size: 6px 13px;
+  background-position: center right 5px;
+  background-repeat: no-repeat;
+}
+@media screen and (min-width: 768px) {
+  a.with-right-arrow, .btn.with-right-arrow {
+    background-size: 8px 14px;
+    background-position: center right 12px;
+    padding-right: 2rem;
+  }
+}
+
+::-webkit-input-placeholder {
+  color: #e44c2c;
+}
+
+::-moz-placeholder {
+  color: #e44c2c;
+}
+
+:-ms-input-placeholder {
+  color: #e44c2c;
+}
+
+:-moz-placeholder {
+  color: #e44c2c;
+}
+
+.email-subscribe-form input.email {
+  color: #e44c2c;
+  border: none;
+  border-bottom: 1px solid #939393;
+  width: 100%;
+  background-color: transparent;
+  outline: none;
+  font-size: 1.125rem;
+  letter-spacing: 0.25px;
+  line-height: 2.25rem;
+}
+.email-subscribe-form input[type="submit"] {
+  position: absolute;
+  right: 0;
+  top: 10px;
+  height: 15px;
+  width: 15px;
+  background-image: url("../images/arrow-right-with-tail.svg");
+  background-color: transparent;
+  background-repeat: no-repeat;
+  background-size: 15px 15px;
+  background-position: center center;
+  -webkit-appearance: none;
+  -moz-appearance: none;
+  appearance: none;
+  border: 0;
+}
+
+.email-subscribe-form-fields-wrapper {
+  position: relative;
+}
+
+.anchorjs-link {
+  color: #6c6c6d !important;
+}
+@media screen and (min-width: 768px) {
+  .anchorjs-link:hover {
+    color: inherit;
+    text-decoration: none !important;
+  }
+}
+
+.pytorch-article #table-of-contents {
+  display: none;
+}
+
+code, kbd, pre, samp {
+  font-family: IBMPlexMono,SFMono-Regular,Menlo,Monaco,Consolas,"Liberation Mono","Courier New",monospace;
+}
+code span, kbd span, pre span, samp span {
+  font-family: IBMPlexMono,SFMono-Regular,Menlo,Monaco,Consolas,"Liberation Mono","Courier New",monospace;
+}
+
+pre {
+  padding: 1.125rem;
+}
+pre code {
+  font-size: 0.875rem;
+}
+pre.highlight {
+  background-color: #f3f4f7;
+  line-height: 1.3125rem;
+}
+
+code.highlighter-rouge {
+  color: #6c6c6d;
+  background-color: #f3f4f7;
+  padding: 2px 6px;
+}
+
+a:link code.highlighter-rouge,
+a:visited code.highlighter-rouge,
+a:hover code.highlighter-rouge {
+  color: #4974D1;
+}
+a:link.has-code,
+a:visited.has-code,
+a:hover.has-code {
+  color: #4974D1;
+}
+
+p code,
+h1 code,
+h2 code,
+h3 code,
+h4 code,
+h5 code,
+h6 code {
+  font-size: 78.5%;
+}
+
+pre {
+  white-space: pre-wrap;
+  white-space: -moz-pre-wrap;
+  white-space: -pre-wrap;
+  white-space: -o-pre-wrap;
+  word-wrap: break-word;
+}
+
+.header-holder {
+  height: 68px;
+  align-items: center;
+  display: flex;
+  left: 0;
+  margin-left: auto;
+  margin-right: auto;
+  position: fixed;
+  right: 0;
+  top: 0;
+  width: 100%;
+  z-index: 9999;
+  background-color: #ffffff;
+  border-bottom: 1px solid #e2e2e2;
+}
+@media screen and (min-width: 1100px) {
+  .header-holder {
+    height: 90px;
+  }
+}
+
+.header-container {
+  position: relative;
+  display: flex;
+  align-items: center;
+}
+.header-container:before, .header-container:after {
+  content: "";
+  display: table;
+}
+.header-container:after {
+  clear: both;
+}
+.header-container {
+  *zoom: 1;
+}
+@media screen and (min-width: 1100px) {
+  .header-container {
+    display: block;
+  }
+}
+
+.header-logo {
+  height: 63px;
+  width: 93px;
+  background-image: url("../images/pypose.svg");
+  background-repeat: no-repeat;
+  background-size: 93px 63px;
+  display: block;
+  float: left;
+  z-index: 10;
+}
+@media screen and (min-width: 1100px) {
+  .header-logo {
+    background-size: 108px 87px;
+    position: absolute;
+    height: 87px;
+    width: 108px;
+    top: 50%;
+    transform: translateY(-50%);
+    float: none;
+  }
+}
+
+.main-menu-open-button {
+  background-image: url("../images/icon-menu-dots.svg");
+  background-position: center center;
+  background-size: 25px 7px;
+  background-repeat: no-repeat;
+  width: 25px;
+  height: 17px;
+  position: absolute;
+  right: 0;
+  top: 4px;
+}
+@media screen and (min-width: 1100px) {
+  .main-menu-open-button {
+    display: none;
+  }
+}
+
+.header-holder .main-menu {
+  display: none;
+}
+@media screen and (min-width: 1100px) {
+  .header-holder .main-menu {
+    display: flex;
+    align-items: center;
+    justify-content: flex-end;
+  }
+}
+.header-holder .main-menu ul {
+  display: flex;
+  align-items: center;
+  margin: 0;
+}
+.header-holder .main-menu ul li {
+  display: inline-block;
+  margin-right: 40px;
+  position: relative;
+}
+.header-holder .main-menu ul li.active:after {
+  content: "•";
+  bottom: -24px;
+  color: #e44c2c;
+  font-size: 1.375rem;
+  left: 0;
+  position: absolute;
+  right: 0;
+  text-align: center;
+}
+.header-holder .main-menu ul li.active a {
+  color: #e44c2c;
+}
+.header-holder .main-menu ul li.docs-active:after {
+  content: "•";
+  bottom: -24px;
+  color: #e44c2c;
+  font-size: 1.375rem;
+  left: -24px;
+  position: absolute;
+  right: 0;
+  text-align: center;
+}
+.header-holder .main-menu ul li:last-of-type {
+  margin-right: 0;
+}
+.header-holder .main-menu ul li a {
+  color: #ffffff;
+  font-size: 1.3rem;
+  letter-spacing: 0;
+  line-height: 2.125rem;
+  text-align: center;
+  text-decoration: none;
+}
+@media screen and (min-width: 1100px) {
+  .header-holder .main-menu ul li a:hover {
+    color: #e44c2c;
+  }
+}
+
+.mobile-main-menu {
+  display: none;
+}
+.mobile-main-menu.open {
+  background-color: #262626;
+  display: block;
+  height: 100%;
+  left: 0;
+  margin-left: auto;
+  margin-right: auto;
+  min-height: 100%;
+  position: fixed;
+  right: 0;
+  top: 0;
+  width: 100%;
+  z-index: 99999;
+}
+
+.mobile-main-menu .container-fluid {
+  align-items: center;
+  display: flex;
+  height: 68px;
+  position: relative;
+}
+.mobile-main-menu .container-fluid:before, .mobile-main-menu .container-fluid:after {
+  content: "";
+  display: table;
+}
+.mobile-main-menu .container-fluid:after {
+  clear: both;
+}
+.mobile-main-menu .container-fluid {
+  *zoom: 1;
+}
+
+.mobile-main-menu.open ul {
+  list-style-type: none;
+  padding: 0;
+}
+.mobile-main-menu.open ul li a, .mobile-main-menu.open .resources-mobile-menu-title {
+  font-size: 2rem;
+  color: #ffffff;
+  letter-spacing: 0;
+  line-height: 4rem;
+  text-decoration: none;
+}
+.mobile-main-menu.open ul li.active a {
+  color: #e44c2c;
+}
+
+.main-menu-close-button {
+  background-image: url("../images/icon-close.svg");
+  background-position: center center;
+  background-repeat: no-repeat;
+  background-size: 24px 24px;
+  height: 24px;
+  position: absolute;
+  right: 0;
+  width: 24px;
+  top: -4px;
+}
+
+.mobile-main-menu-header-container {
+  position: relative;
+}
+
+.mobile-main-menu-links-container {
+  display: flex;
+  align-items: center;
+  padding-left: 2.8125rem;
+  height: 90vh;
+  margin-top: -25px;
+  padding-top: 50%;
+  overflow-y: scroll;
+}
+.mobile-main-menu-links-container .main-menu {
+  height: 100vh;
+}
+
+.mobile-main-menu-links-container ul.resources-mobile-menu-items li {
+  padding-left: 15px;
+}
+
+.site-footer {
+  padding: 2.5rem 0;
+  width: 100%;
+  background: #000000;
+  background-size: 100%;
+  margin-left: 0;
+  margin-right: 0;
+  position: relative;
+  z-index: 201;
+}
+@media screen and (min-width: 768px) {
+  .site-footer {
+    padding: 5rem 0;
+  }
+}
+
+.site-footer p {
+  color: #ffffff;
+}
+.site-footer ul {
+  list-style-type: none;
+  padding-left: 0;
+  margin-bottom: 0;
+}
+
+.site-footer ul li {
+  font-size: 1.125rem;
+  line-height: 2rem;
+  color: #A0A0A1;
+  padding-bottom: 0.375rem;
+}
+.site-footer ul li.list-title {
+  padding-bottom: 0.75rem;
+  color: #ffffff;
+}
+.site-footer a:link,
+.site-footer a:visited {
+  color: inherit;
+}
+@media screen and (min-width: 768px) {
+  .site-footer a:hover {
+    color: #e44c2c;
+  }
+}
+
+.site-footer .privacy-policy {
+    background: #000000;
+    display: flex;
+    border-bottom: 1px solid white;
+    padding-bottom: 10px;
+}
+
+.site-footer .privacy-policy-links {
+    background: #000000;
+    display: flex; 
+    padding-top: 1rem;
+    padding-right: 1rem;
+    display: inline-flex;
+    color: white;
+}
+
+.site-footer .footer-links-wrapper {
+    display: flex;
+    flex-wrap: wrap;
+    border-bottom: 1px solid white;
+    padding-bottom: 1rem;
+}
+
+.site-footer .copyright {
+    padding-top: 1rem;
+    padding-right: 1rem;
+    display: inline-flex;
+    color: white;
+}
+
+.site-footer .copyright p {
+    color: white;
+}
+
+.site-footer .copyright a {
+    color: red;
+}
+
+.docs-tutorials-resources {
+  background-color: #262626;
+  color: #ffffff;
+  padding-top: 2.5rem;
+  padding-bottom: 2.5rem;
+  position: relative;
+  z-index: 201;
+}
+@media screen and (min-width: 768px) {
+  .docs-tutorials-resources {
+    padding-top: 5rem;
+    padding-bottom: 5rem;
+  }
+}
+.docs-tutorials-resources p {
+  color: #929292;
+  font-size: 1.125rem;
+}
+.docs-tutorials-resources h2 {
+  font-size: 1.5rem;
+  letter-spacing: -0.25px;
+  text-transform: none;
+  margin-bottom: 0.25rem;
+}
+@media screen and (min-width: 768px) {
+  .docs-tutorials-resources h2 {
+    margin-bottom: 1.25rem;
+  }
+}
+.docs-tutorials-resources .col-md-4 {
+  margin-bottom: 2rem;
+  text-align: center;
+}
+@media screen and (min-width: 768px) {
+  .docs-tutorials-resources .col-md-4 {
+    margin-bottom: 0;
+  }
+}
+.docs-tutorials-resources .with-right-arrow {
+  margin-left: 12px;
+}
+.docs-tutorials-resources .with-right-arrow:hover {
+  background-image: url("../images/chevron-right-white.svg");
+}
+.docs-tutorials-resources p {
+  font-size: 1rem;
+  line-height: 1.5rem;
+  letter-spacing: 0.22px;
+  color: #939393;
+  margin-bottom: 0;
+}
+@media screen and (min-width: 768px) {
+  .docs-tutorials-resources p {
+    margin-bottom: 1.25rem;
+  }
+}
+.docs-tutorials-resources a {
+  font-size: 1.125rem;
+  color: #e44c2c;
+}
+.docs-tutorials-resources a:hover {
+  color: #ffffff;
+}
+
+.footer-container {
+  position: relative;
+}
+
+@media screen and (min-width: 768px) {
+  .footer-logo-wrapper {
+    position: absolute;
+    top: 0;
+    left: 30px;
+  }
+}
+
+.footer-logo {
+  background-image: url("../images/pypose.svg");
+  background-position: center;
+  background-repeat: no-repeat;
+  background-size: 30px 34px;
+  display: block;
+  height: 34px;
+  margin-bottom: rem(45px);
+  width: 30px;
+}
+@media screen and (min-width: 768px) {
+  .footer-logo {
+    background-size: 39px 46px;
+    height: 46px;
+    margin-bottom: 0;
+    margin-bottom: 0;
+    width: 39px;
+  }
+}
+
+.footer-links-wrapper {
+  display: flex;
+  flex-wrap: wrap;
+}
+@media screen and (min-width: 768px) {
+  .footer-links-wrapper {
+    flex-wrap: initial;
+    justify-content: flex-end;
+  }
+}
+
+.footer-links-col {
+  margin-bottom: 3.75rem;
+  width: 50%;
+}
+@media screen and (min-width: 768px) {
+  .footer-links-col {
+    margin-bottom: 0;
+    width: 14%;
+    margin-right: 23px;
+  }
+  .footer-links-col.follow-us-col {
+    width: 18%;
+    margin-right: 0;
+  }
+}
+@media (min-width: 768px) and (max-width: 1239px) {
+  .footer-links-col {
+    width: 18%;
+    margin-right: 30px;
+  }
+}
+
+.footer-social-icons {
+  margin: 8.5625rem 0 2.5rem 0;
+}
+.footer-social-icons a {
+  height: 32px;
+  width: 32px;
+  display: inline-block;
+  background-color: #CCCDD1;
+  border-radius: 50%;
+  margin-right: 5px;
+}
+.footer-social-icons a.facebook {
+  background-image: url("../images/logo-facebook-dark.svg");
+  background-position: center center;
+  background-size: 9px 18px;
+  background-repeat: no-repeat;
+}
+.footer-social-icons a.twitter {
+  background-image: url("../images/logo-twitter-dark.svg");
+  background-position: center center;
+  background-size: 17px 17px;
+  background-repeat: no-repeat;
+}
+.footer-social-icons a.youtube {
+  background-image: url("../images/logo-youtube-dark.svg");
+  background-position: center center;
+  background-repeat: no-repeat;
+}
+
+.site-footer .mc-field-group {
+  margin-top: -2px;
+}
+
+article.pytorch-article {
+  max-width: 920px;
+  margin: 0 auto;
+}
+article.pytorch-article h2,
+article.pytorch-article h3,
+article.pytorch-article h4,
+article.pytorch-article h5,
+article.pytorch-article h6 {
+  margin: 1.375rem 0;
+  color: #262626;
+}
+article.pytorch-article h2 {
+  font-size: 1.625rem;
+  letter-spacing: 1.33px;
+  line-height: 2rem;
+  text-transform: none;
+}
+article.pytorch-article h3 {
+  font-size: 1.5rem;
+  letter-spacing: -0.25px;
+  line-height: 1.875rem;
+  text-transform: none;
+}
+article.pytorch-article h4,
+article.pytorch-article h5,
+article.pytorch-article h6 {
+  font-size: 1.125rem;
+  letter-spacing: -0.19px;
+  line-height: 1.875rem;
+}
+article.pytorch-article p {
+  margin-bottom: 1.125rem;
+}
+article.pytorch-article p,
+article.pytorch-article ul li,
+article.pytorch-article ol li,
+article.pytorch-article dl dt,
+article.pytorch-article dl dd,
+article.pytorch-article blockquote {
+  font-size: 1rem;
+  line-height: 1.375rem;
+  color: #262626;
+  letter-spacing: 0.01px;
+  font-weight: 500;
+}
+article.pytorch-article table {
+  margin-bottom: 2.5rem;
+  width: 100%;
+}
+article.pytorch-article table thead {
+  border-bottom: 1px solid #cacaca;
+}
+article.pytorch-article table th {
+  padding: 0.625rem;
+  color: #262626;
+}
+article.pytorch-article table td {
+  padding: 0.3125rem;
+}
+article.pytorch-article table tr th:first-of-type,
+article.pytorch-article table tr td:first-of-type {
+  padding-left: 0;
+}
+article.pytorch-article table.docutils.field-list th.field-name {
+  padding: 0.3125rem;
+  padding-left: 0;
+}
+article.pytorch-article table.docutils.field-list td.field-body {
+  padding: 0.3125rem;
+}
+article.pytorch-article table.docutils.field-list td.field-body p:last-of-type {
+  margin-bottom: 0;
+}
+article.pytorch-article ul,
+article.pytorch-article ol {
+  margin: 1.5rem 0 3.125rem 0;
+}
+@media screen and (min-width: 768px) {
+  article.pytorch-article ul,
+  article.pytorch-article ol {
+    padding-left: 6.25rem;
+  }
+}
+article.pytorch-article ul li,
+article.pytorch-article ol li {
+  margin-bottom: 0.625rem;
+}
+article.pytorch-article dl {
+  margin-bottom: 1.5rem;
+}
+article.pytorch-article dl dt {
+  margin-bottom: 0.75rem;
+}
+article.pytorch-article pre {
+  margin-bottom: 2.5rem;
+}
+article.pytorch-article hr {
+  margin-top: 4.6875rem;
+  margin-bottom: 4.6875rem;
+}
+article.pytorch-article blockquote {
+  margin: 0 auto;
+  margin-bottom: 2.5rem;
+  width: 65%;
+}
+article.pytorch-article img {
+  width: 100%;
+}
+
+html {
+  height: 100%;
+}
+@media screen and (min-width: 768px) {
+  html {
+    font-size: 16px;
+  }
+}
+
+body {
+  background: #ffffff;
+  height: 100%;
+  margin: 0;
+}
+body.no-scroll {
+  height: 100%;
+  overflow: hidden;
+}
+
+p {
+  margin-top: 0;
+  margin-bottom: 1.125rem;
+}
+p a:link,
+p a:visited,
+p a:hover {
+  color: #e44c2c;
+  text-decoration: none;
+}
+@media screen and (min-width: 768px) {
+  p a:hover {
+    text-decoration: underline;
+  }
+}
+p a:link,
+p a:visited,
+p a:hover {
+  color: #ee4c2c;
+}
+
+.wy-breadcrumbs li a {
+  color: #ee4c2c;
+}
+
+ul.pytorch-breadcrumbs {
+  padding-left: 0;
+  list-style-type: none;
+}
+ul.pytorch-breadcrumbs li {
+  display: inline-block;
+  font-size: 0.875rem;
+}
+ul.pytorch-breadcrumbs a {
+  color: #ee4c2c;
+  text-decoration: none;
+}
+
+.table-of-contents-link-wrapper {
+  display: block;
+  margin-top: 0;
+  padding: 1.25rem 1.875rem;
+  background-color: #f3f4f7;
+  position: relative;
+  color: #262626;
+  font-size: 1.25rem;
+}
+.table-of-contents-link-wrapper.is-open .toggle-table-of-contents {
+  transform: rotate(180deg);
+}
+@media screen and (min-width: 1100px) {
+  .table-of-contents-link-wrapper {
+    display: none;
+  }
+}
+
+.toggle-table-of-contents {
+  background-image: url("../images/chevron-down-grey.svg");
+  background-position: center center;
+  background-repeat: no-repeat;
+  background-size: 18px 18px;
+  height: 100%;
+  position: absolute;
+  right: 21px;
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+  margin-top: 1.125rem;
+}
+article.pytorch-article .note .admonition-title {
+  background: #54c7ec;
+}
+article.pytorch-article .warning .admonition-title {
+  background: #e94f3b;
+}
+article.pytorch-article .tip .admonition-title {
+  background: #6bcebb;
+}
+article.pytorch-article .seealso .admonition-title {
+  background: #6bcebb;
+}
+article.pytorch-article .hint .admonition-title {
+  background: #a2cdde;
+}
+article.pytorch-article .important .admonition-title {
+  background: #5890ff;
+}
+article.pytorch-article .caution .admonition-title {
+  background: #f7923a;
+}
+article.pytorch-article .danger .admonition-title {
+  background: #db2c49;
+}
+article.pytorch-article .attention .admonition-title {
+  background: #f5a623;
+}
+article.pytorch-article .error .admonition-title {
+  background: #cc2f90;
+}
+article.pytorch-article .sphx-glr-download-link-note.admonition.note,
+article.pytorch-article .reference.download.internal, article.pytorch-article .sphx-glr-signature {
+  display: none;
+}
+article.pytorch-article .admonition > p:last-of-type {
+  margin-bottom: 0;
+  padding-bottom: 1.125rem !important;
+}
+
+.pytorch-article div.sphx-glr-download a {
+  background-color: #f3f4f7;
+  background-image: url("../images/arrow-down-orange.svg");
+  background-repeat: no-repeat;
+  background-position: left 10px center;
+  background-size: 15px 15px;
+  border-radius: 0;
+  border: none;
+  display: block;
+  text-align: left;
+  padding: 0.9375rem 3.125rem;
+  position: relative;
+  margin: 1.25rem auto;
+}
+@media screen and (min-width: 768px) {
+  .pytorch-article div.sphx-glr-download a:after {
+    content: "";
+    display: block;
+    width: 0;
+    height: 1px;
+    position: absolute;
+    bottom: 0;
+    left: 0;
+    background-color: #e44c2c;
+    transition: width .250s ease-in-out;
+  }
+  .pytorch-article div.sphx-glr-download a:hover:after {
+    width: 100%;
+  }
+}
+@media screen and (min-width: 768px) {
+  .pytorch-article div.sphx-glr-download a:after {
+    background-color: #ee4c2c;
+  }
+}
+@media screen and (min-width: 768px) {
+  .pytorch-article div.sphx-glr-download a {
+    background-position: left 20px center;
+  }
+}
+.pytorch-article div.sphx-glr-download a:hover {
+  box-shadow: none;
+  text-decoration: none;
+  background-image: url("../images/arrow-down-orange.svg");
+  background-color: #f3f4f7;
+}
+.pytorch-article div.sphx-glr-download a span.pre {
+  background-color: transparent;
+  font-size: 1.125rem;
+  padding: 0;
+  color: #262626;
+}
+.pytorch-article div.sphx-glr-download a code, .pytorch-article div.sphx-glr-download a kbd, .pytorch-article div.sphx-glr-download a pre, .pytorch-article div.sphx-glr-download a samp, .pytorch-article div.sphx-glr-download a span.pre {
+  font-family: FreightSans, Helvetica Neue, Helvetica, Arial, sans-serif;
+}
+
+.pytorch-article p.sphx-glr-script-out {
+  margin-bottom: 1.125rem;
+}
+
+.pytorch-article div.sphx-glr-script-out {
+  margin-bottom: 2.5rem;
+}
+.pytorch-article div.sphx-glr-script-out .highlight {
+  margin-left: 0;
+  margin-top: 0;
+}
+.pytorch-article div.sphx-glr-script-out .highlight pre {
+  background-color: #fdede9;
+  padding: 1.5625rem;
+  color: #837b79;
+}
+.pytorch-article div.sphx-glr-script-out + p {
+  margin-top: unset;
+}
+
+article.pytorch-article .wy-table-responsive table {
+  border: none;
+  border-color: #ffffff !important;
+  table-layout: fixed;
+}
+article.pytorch-article .wy-table-responsive table thead tr {
+  border-bottom: 2px solid #6c6c6d;
+}
+article.pytorch-article .wy-table-responsive table thead th {
+  line-height: 1.75rem;
+  padding-left: 0.9375rem;
+  padding-right: 0.9375rem;
+}
+article.pytorch-article .wy-table-responsive table tbody .row-odd {
+  background-color: #f3f4f7;
+}
+article.pytorch-article .wy-table-responsive table tbody td {
+  color: #6c6c6d;
+  white-space: normal;
+  padding: 0.9375rem;
+  font-size: 1rem;
+  line-height: 1.375rem;
+}
+article.pytorch-article .wy-table-responsive table tbody td .pre {
+  background: #ffffff;
+  color: #ee4c2c;
+  font-size: 87.5%;
+}
+article.pytorch-article .wy-table-responsive table tbody td code {
+  font-size: 87.5%;
+}
+
+a[rel~="prev"], a[rel~="next"] {
+  padding: 0.375rem 0 0 0;
+}
+
+img.next-page,
+img.previous-page {
+  width: 8px;
+  height: 10px;
+  position: relative;
+  top: -1px;
+}
+
+img.previous-page {
+  transform: scaleX(-1);
+}
+
+.rst-footer-buttons {
+  margin-top: 1.875rem;
+  margin-bottom: 1.875rem;
+}
+.rst-footer-buttons .btn:focus,
+.rst-footer-buttons .btn.focus {
+  box-shadow: none;
+}
+
+article.pytorch-article blockquote {
+  margin-left: 3.75rem;
+  color: #6c6c6d;
+}
+
+article.pytorch-article .caption {
+  color: #6c6c6d;
+  letter-spacing: 0.25px;
+  line-height: 2.125rem;
+}
+
+article.pytorch-article .math {
+  color: #262626;
+  width: auto;
+  text-align: center;
+}
+article.pytorch-article .math img {
+  width: auto;
+}
+
+.pytorch-breadcrumbs-wrapper {
+  width: 100%;
+}
+@media screen and (min-width: 1101px) {
+  .pytorch-breadcrumbs-wrapper {
+    float: left;
+    margin-left: 3%;
+    width: 75%;
+  }
+}
+@media screen and (min-width: 1600px) {
+  .pytorch-breadcrumbs-wrapper {
+    width: 850px;
+    margin-left: 1.875rem;
+  }
+}
+.pytorch-breadcrumbs-wrapper .pytorch-breadcrumbs-aside {
+  float: right;
+}
+.pytorch-breadcrumbs-wrapper .pytorch-breadcrumbs-aside .fa.fa-github {
+  margin-top: 5px;
+  display: block;
+}
+
+.pytorch-article .container {
+  padding-left: 0;
+  padding-right: 0;
+  max-width: none;
+}
+
+a:link,
+a:visited,
+a:hover {
+  color: #ee4c2c;
+}
+
+::-webkit-input-placeholder {
+  color: #ee4c2c;
+}
+
+::-moz-placeholder {
+  color: #ee4c2c;
+}
+
+:-ms-input-placeholder {
+  color: #ee4c2c;
+}
+
+:-moz-placeholder {
+  color: #ee4c2c;
+}
+
+@media screen and (min-width: 768px) {
+  .site-footer a:hover {
+    color: #ee4c2c;
+  }
+}
+
+.docs-tutorials-resources a {
+  color: #ee4c2c;
+}
+
+.header-holder {
+  position: relative;
+  z-index: 201;
+}
+
+.header-holder .main-menu ul li.active:after {
+  color: #ee4c2c;
+}
+.header-holder .main-menu ul li.active a {
+  color: #ee4c2c;
+}
+@media screen and (min-width: 1100px) {
+  .header-holder .main-menu ul li a:hover {
+    color: #ee4c2c;
+  }
+}
+
+.mobile-main-menu.open ul li.active a {
+  color: #ee4c2c;
+}
+
+.version {
+  padding-bottom: 1rem;
+}
+
+.pytorch-call-to-action-links {
+  padding-top: 0;
+  display: flex;
+}
+@media screen and (min-width: 768px) {
+  .pytorch-call-to-action-links {
+    padding-top: 2.5rem;
+  }
+}
+@media (min-width: 768px) and (max-width: 1239px) {
+  .pytorch-call-to-action-links {
+    padding-top: 0;
+  }
+}
+@media (min-width: 1100px) and (max-width: 1239px) {
+  .pytorch-call-to-action-links {
+    padding-top: 2.5rem;
+  }
+}
+.pytorch-call-to-action-links #tutorial-type {
+  display: none;
+}
+.pytorch-call-to-action-links .call-to-action-img, .pytorch-call-to-action-links .call-to-action-notebook-img {
+  height: 1.375rem;
+  width: 1.375rem;
+  margin-right: 10px;
+}
+.pytorch-call-to-action-links .call-to-action-notebook-img {
+  height: 1rem;
+}
+.pytorch-call-to-action-links a {
+  padding-right: 1.25rem;
+  color: #000000;
+  cursor: pointer;
+}
+.pytorch-call-to-action-links a:hover {
+  color: #e44c2c;
+}
+.pytorch-call-to-action-links a .call-to-action-desktop-view {
+  display: none;
+}
+@media screen and (min-width: 768px) {
+  .pytorch-call-to-action-links a .call-to-action-desktop-view {
+    display: block;
+  }
+}
+.pytorch-call-to-action-links a .call-to-action-mobile-view {
+  display: block;
+}
+@media screen and (min-width: 768px) {
+  .pytorch-call-to-action-links a .call-to-action-mobile-view {
+    display: none;
+  }
+}
+.pytorch-call-to-action-links a #google-colab-link, .pytorch-call-to-action-links a #download-notebook-link,
+.pytorch-call-to-action-links a #github-view-link {
+  padding-bottom: 0.625rem;
+  border-bottom: 1px solid #f3f4f7;
+  padding-right: 2.5rem;
+  display: flex;
+  align-items: center;
+}
+.pytorch-call-to-action-links a #google-colab-link:hover, .pytorch-call-to-action-links a #download-notebook-link:hover,
+.pytorch-call-to-action-links a #github-view-link:hover {
+  border-bottom-color: #e44c2c;
+  color: #e44c2c;
+}
+
+#tutorial-cards-container #tutorial-cards {
+  width: 100%;
+}
+#tutorial-cards-container .tutorials-nav {
+  padding-left: 0;
+  padding-right: 0;
+  padding-bottom: 0;
+}
+#tutorial-cards-container .tutorials-hr {
+  margin-top: 1rem;
+  margin-bottom: 1rem;
+}
+#tutorial-cards-container .card.tutorials-card {
+  border-radius: 0;
+  border-color: #f3f4f7;
+  height: 98px;
+  margin-bottom: 1.25rem;
+  margin-bottom: 1.875rem;
+  overflow: scroll;
+  background-color: #f3f4f7;
+  cursor: pointer;
+}
+@media screen and (min-width: 1240px) {
+  #tutorial-cards-container .card.tutorials-card {
+    height: 200px;
+    overflow: inherit;
+  }
+}
+@media (min-width: 768px) and (max-width: 1239px) {
+  #tutorial-cards-container .card.tutorials-card {
+    height: 200px;
+    overflow: scroll;
+  }
+}
+#tutorial-cards-container .card.tutorials-card .tutorials-image {
+  position: absolute;
+  top: 0px;
+  right: 0px;
+  height: 96px;
+  width: 96px;
+  opacity: 0.5;
+}
+#tutorial-cards-container .card.tutorials-card .tutorials-image img {
+  height: 100%;
+  width: 100%;
+}
+@media screen and (min-width: 768px) {
+  #tutorial-cards-container .card.tutorials-card .tutorials-image {
+    height: 100%;
+    width: 25%;
+  }
+}
+@media (min-width: 768px) and (max-width: 1239px) {
+  #tutorial-cards-container .card.tutorials-card .tutorials-image {
+    height: 100%;
+    width: 198px;
+  }
+}
+#tutorial-cards-container .card.tutorials-card .tutorials-image:before {
+  content: '';
+  position: absolute;
+  top: 0;
+  left: 0;
+  bottom: 0;
+  right: 0;
+  z-index: 1;
+  background: #000000;
+  opacity: .075;
+}
+#tutorial-cards-container .card.tutorials-card .card-title-container {
+  width: 70%;
+  display: inline-flex;
+}
+@media screen and (min-width: 768px) {
+  #tutorial-cards-container .card.tutorials-card .card-title-container {
+    width: 75%;
+  }
+}
+@media (min-width: 768px) and (max-width: 1239px) {
+  #tutorial-cards-container .card.tutorials-card .card-title-container {
+    width: 70%;
+  }
+}
+#tutorial-cards-container .card.tutorials-card .card-title-container h4 {
+  margin-bottom: 1.125rem;
+  margin-top: 0;
+  font-size: 1.5rem;
+}
+#tutorial-cards-container .card.tutorials-card p.card-summary, #tutorial-cards-container .card.tutorials-card p.tags {
+  font-size: 0.9375rem;
+  line-height: 1.5rem;
+  margin-bottom: 0;
+  color: #6c6c6d;
+  font-weight: 400;
+  width: 70%;
+}
+@media screen and (min-width: 768px) {
+  #tutorial-cards-container .card.tutorials-card p.card-summary, #tutorial-cards-container .card.tutorials-card p.tags {
+    width: 75%;
+  }
+}
+@media (min-width: 768px) and (max-width: 1239px) {
+  #tutorial-cards-container .card.tutorials-card p.card-summary, #tutorial-cards-container .card.tutorials-card p.tags {
+    width: 70%;
+  }
+}
+#tutorial-cards-container .card.tutorials-card p.tags {
+  margin-top: 30px;
+  text-overflow: ellipsis;
+  white-space: nowrap;
+  overflow: hidden;
+}
+#tutorial-cards-container .card.tutorials-card h4 {
+  color: #262626;
+  margin-bottom: 1.125rem;
+}
+#tutorial-cards-container .card.tutorials-card a {
+  height: 100%;
+}
+@media screen and (min-width: 768px) {
+  #tutorial-cards-container .card.tutorials-card a {
+    min-height: 190px;
+  }
+}
+@media (min-width: 768px) and (max-width: 1239px) {
+  #tutorial-cards-container .card.tutorials-card a {
+    min-height: 234px;
+  }
+}
+@media screen and (min-width: 768px) {
+  #tutorial-cards-container .card.tutorials-card:after {
+    content: "";
+    display: block;
+    width: 0;
+    height: 1px;
+    position: absolute;
+    bottom: 0;
+    left: 0;
+    background-color: #e44c2c;
+    transition: width .250s ease-in-out;
+  }
+  #tutorial-cards-container .card.tutorials-card:hover:after {
+    width: 100%;
+  }
+}
+#tutorial-cards-container .card.tutorials-card:hover {
+  background-color: #ffffff;
+  border: 1px solid #e2e2e2;
+  border-bottom: none;
+}
+#tutorial-cards-container .card.tutorials-card:hover p.card-summary {
+  color: #262626;
+}
+#tutorial-cards-container .card.tutorials-card:hover .tutorials-image {
+  opacity: unset;
+}
+#tutorial-cards-container .tutorial-tags-container {
+  width: 75%;
+}
+#tutorial-cards-container .tutorial-tags-container.active {
+  width: 0;
+}
+#tutorial-cards-container .tutorial-filter-menu ul {
+  list-style-type: none;
+  padding-left: 1.25rem;
+}
+#tutorial-cards-container .tutorial-filter-menu ul li {
+  padding-right: 1.25rem;
+  word-break: break-all;
+}
+#tutorial-cards-container .tutorial-filter-menu ul li a {
+  color: #979797;
+}
+#tutorial-cards-container .tutorial-filter-menu ul li a:hover {
+  color: #e44c2c;
+}
+#tutorial-cards-container .tutorial-filter {
+  cursor: pointer;
+}
+#tutorial-cards-container .filter-btn {
+  color: #979797;
+  border: 1px solid #979797;
+  display: inline-block;
+  text-align: center;
+  white-space: nowrap;
+  vertical-align: middle;
+  padding: 0.375rem 0.75rem;
+  font-size: 1rem;
+  line-height: 1.5;
+  margin-bottom: 5px;
+}
+#tutorial-cards-container .filter-btn:hover {
+  border: 1px solid #e44c2c;
+  color: #e44c2c;
+}
+#tutorial-cards-container .filter-btn.selected {
+  background-color: #e44c2c;
+  border: 1px solid #e44c2c;
+  color: #ffffff;
+}
+#tutorial-cards-container .all-tag-selected {
+  background-color: #979797;
+  color: #ffffff;
+}
+#tutorial-cards-container .all-tag-selected:hover {
+  border-color: #979797;
+  color: #ffffff;
+}
+#tutorial-cards-container .pagination .page {
+  border: 1px solid #dee2e6;
+  padding: 0.5rem 0.75rem;
+}
+#tutorial-cards-container .pagination .active .page {
+  background-color: #dee2e6;
+}
+
+article.pytorch-article .tutorials-callout-container {
+  padding-bottom: 50px;
+}
+article.pytorch-article .tutorials-callout-container .col-md-6 {
+  padding-bottom: 10px;
+}
+article.pytorch-article .tutorials-callout-container .text-container {
+  padding: 10px 0px 30px 0px;
+  padding-bottom: 10px;
+}
+article.pytorch-article .tutorials-callout-container .text-container .body-paragraph {
+  color: #666666;
+  font-weight: 300;
+  font-size: 1.125rem;
+  line-height: 1.875rem;
+}
+article.pytorch-article .tutorials-callout-container .btn.callout-button {
+  font-size: 1.125rem;
+  border-radius: 0;
+  border: none;
+  background-color: #f3f4f7;
+  color: #6c6c6d;
+  font-weight: 400;
+  position: relative;
+  letter-spacing: 0.25px;
+}
+@media screen and (min-width: 768px) {
+  article.pytorch-article .tutorials-callout-container .btn.callout-button:after {
+    content: "";
+    display: block;
+    width: 0;
+    height: 1px;
+    position: absolute;
+    bottom: 0;
+    left: 0;
+    background-color: #e44c2c;
+    transition: width .250s ease-in-out;
+  }
+  article.pytorch-article .tutorials-callout-container .btn.callout-button:hover:after {
+    width: 100%;
+  }
+}
+article.pytorch-article .tutorials-callout-container .btn.callout-button a {
+  color: inherit;
+}
+
+.pytorch-container {
+  margin: 0 auto;
+  padding: 0 1.875rem;
+  width: auto;
+  position: relative;
+}
+@media screen and (min-width: 1100px) {
+  .pytorch-container {
+    padding: 0;
+  }
+}
+@media screen and (min-width: 1101px) {
+  .pytorch-container {
+    margin-left: 20%;
+  }
+}
+@media screen and (min-width: 1600px) {
+  .pytorch-container {
+    margin-left: 350px;
+  }
+}
+.pytorch-container:before, .pytorch-container:after {
+  content: "";
+  display: table;
+}
+.pytorch-container:after {
+  clear: both;
+}
+.pytorch-container {
+  *zoom: 1;
+}
+
+.pytorch-content-wrap {
+  background-color: #ffffff;
+  display: flex;
+  position: relative;
+  padding-top: 0;
+}
+.pytorch-content-wrap:before, .pytorch-content-wrap:after {
+  content: "";
+  display: table;
+}
+.pytorch-content-wrap:after {
+  clear: both;
+}
+.pytorch-content-wrap {
+  *zoom: 1;
+}
+@media screen and (min-width: 1101px) {
+  .pytorch-content-wrap {
+    padding-top: 45px;
+    float: left;
+    width: 100%;
+    display: block;
+  }
+}
+@media screen and (min-width: 1600px) {
+  .pytorch-content-wrap {
+    width: 100%;
+  }
+}
+
+.pytorch-content {
+  background: #ffffff;
+  width: 100%;
+  max-width: 700px;
+  position: relative;
+}
+
+.pytorch-content-left {
+  min-height: 100vh;
+  margin-top: 2.5rem;
+  width: 100%;
+}
+@media screen and (min-width: 1101px) {
+  .pytorch-content-left {
+    margin-top: 0;
+    margin-left: 3%;
+    width: 70%;
+    float: left;
+  }
+}
+@media screen and (min-width: 1600px) {
+  .pytorch-content-left {
+    width: 850px;
+    margin-left: 30px;
+  }
+}
+.pytorch-content-left .main-content {
+  padding-top: 0.9375rem;
+}
+.pytorch-content-left .main-content ul.simple {
+  padding-bottom: 1.25rem;
+}
+.pytorch-content-left .main-content .note:nth-child(1), .pytorch-content-left .main-content .warning:nth-child(1) {
+  margin-top: 0;
+}
+
+.pytorch-content-right {
+  display: none;
+  position: relative;
+  overflow-x: hidden;
+  overflow-y: hidden;
+}
+@media screen and (min-width: 1101px) {
+  .pytorch-content-right {
+    display: block;
+    margin-left: 0;
+    width: 27%;
+    float: left;
+    height: 100%;
+  }
+}
+@media screen and (min-width: 1600px) {
+  .pytorch-content-right {
+    width: 380px;
+  }
+}
+
+@media screen and (min-width: 1101px) {
+  .pytorch-side-scroll {
+    position: relative;
+    overflow-x: hidden;
+    overflow-y: scroll;
+    height: 100%;
+  }
+}
+
+.pytorch-menu-vertical {
+  padding: 1.25rem 1.875rem 2.5rem 1.875rem;
+}
+@media screen and (min-width: 1101px) {
+  .pytorch-menu-vertical {
+    display: block;
+    padding-top: 0;
+    padding-right: 13.5%;
+    padding-bottom: 5.625rem;
+  }
+}
+@media screen and (min-width: 1600px) {
+  .pytorch-menu-vertical {
+    padding-left: 0;
+    padding-right: 1.5625rem;
+  }
+}
+
+.pytorch-left-menu {
+  display: none;
+  background-color: #f3f4f7;
+  color: #262626;
+  overflow: scroll;
+}
+@media screen and (min-width: 1101px) {
+  .pytorch-left-menu {
+    display: block;
+    overflow-x: hidden;
+    overflow-y: hidden;
+    padding-bottom: 110px;
+    padding: 0 1.875rem 0 0;
+    width: 20%;
+    z-index: 200;
+    float: left;
+  }
+  .pytorch-left-menu.make-fixed {
+    position: fixed;
+    top: 0;
+    bottom: 0;
+    left: 0;
+    float: none;
+  }
+}
+@media screen and (min-width: 1600px) {
+  .pytorch-left-menu {
+    padding: 0 0 0 1.875rem;
+    width: 350px;
+  }
+}
+
+.expand-menu, .hide-menu {
+  color: #6c6c6d;
+  padding-left: 10px;
+  cursor: pointer;
+}
+
+.collapse {
+  display: none;
+}
+
+.left-nav-top-caption {
+  padding-top: 1rem;
+}
+
+.pytorch-left-menu p.caption {
+  color: #262626;
+  display: block;
+  font-size: 1rem;
+  line-height: 1.375rem;
+  margin-bottom: 1rem;
+  text-transform: none;
+  white-space: normal;
+}
+
+.pytorch-left-menu-search {
+  margin-bottom: 2.5rem;
+}
+@media screen and (min-width: 1101px) {
+  .pytorch-left-menu-search {
+    margin: 1.25rem 0.625rem 1.875rem 0;
+  }
+}
+.pytorch-left-menu-search ::-moz-placeholder {
+  color: #262626;
+}
+.pytorch-left-menu-search :-ms-input-placeholder {
+  color: #262626;
+}
+.pytorch-left-menu-search ::-ms-input-placeholder {
+  color: #262626;
+}
+.pytorch-left-menu-search ::placeholder {
+  color: #262626;
+}
+.pytorch-left-menu-search :focus::-moz-placeholder {
+  color: transparent;
+}
+.pytorch-left-menu-search :focus:-ms-input-placeholder {
+  color: transparent;
+}
+.pytorch-left-menu-search :focus::-ms-input-placeholder {
+  color: transparent;
+}
+.pytorch-left-menu-search :focus::placeholder {
+  color: transparent;
+}
+
+.pytorch-left-menu-search input[type=text] {
+  border-radius: 0;
+  padding: 0.5rem 0.75rem;
+  border-color: #ffffff;
+  color: #262626;
+  border-style: solid;
+  font-size: 1rem;
+  width: 100%;
+  background-color: #f3f4f7;
+  background-image: url("../images/search-icon.svg");
+  background-repeat: no-repeat;
+  background-size: 18px 18px;
+  background-position: 12px 10px;
+  padding-left: 40px;
+  background-color: #ffffff;
+}
+.pytorch-left-menu-search input[type=text]:focus {
+  outline: 0;
+}
+
+@media screen and (min-width: 1101px) {
+  .pytorch-left-menu .pytorch-side-scroll {
+    width: 120%;
+  }
+}
+@media screen and (min-width: 1600px) {
+  .pytorch-left-menu .pytorch-side-scroll {
+    width: 340px;
+  }
+}
+
+.pytorch-right-menu {
+  min-height: 100px;
+  overflow-x: hidden;
+  overflow-y: hidden;
+  left: 0;
+  z-index: 200;
+  padding-top: 0;
+  position: relative;
+}
+@media screen and (min-width: 1101px) {
+  .pytorch-right-menu {
+    width: 100%;
+  }
+  .pytorch-right-menu.scrolling-fixed {
+    position: fixed;
+    top: 45px;
+    left: 83.5%;
+    width: 14%;
+  }
+  .pytorch-right-menu.scrolling-absolute {
+    position: absolute;
+    left: 0;
+  }
+}
+@media screen and (min-width: 1600px) {
+  .pytorch-right-menu {
+    left: 0;
+    width: 380px;
+  }
+  .pytorch-right-menu.scrolling-fixed {
+    position: fixed;
+    top: 45px;
+    left: 1230px;
+  }
+  .pytorch-right-menu.scrolling-absolute {
+    position: absolute;
+    left: 0;
+  }
+}
+
+.pytorch-left-menu ul,
+.pytorch-right-menu ul {
+  list-style-type: none;
+  padding-left: 0;
+  margin-bottom: 2.5rem;
+}
+.pytorch-left-menu > ul,
+.pytorch-right-menu > ul {
+  margin-bottom: 2.5rem;
+}
+.pytorch-left-menu a:link,
+.pytorch-left-menu a:visited,
+.pytorch-left-menu a:hover,
+.pytorch-right-menu a:link,
+.pytorch-right-menu a:visited,
+.pytorch-right-menu a:hover {
+  color: #6c6c6d;
+  font-size: 0.875rem;
+  line-height: 1rem;
+  padding: 0;
+  text-decoration: none;
+}
+.pytorch-left-menu a:link.reference.internal,
+.pytorch-left-menu a:visited.reference.internal,
+.pytorch-left-menu a:hover.reference.internal,
+.pytorch-right-menu a:link.reference.internal,
+.pytorch-right-menu a:visited.reference.internal,
+.pytorch-right-menu a:hover.reference.internal {
+  margin-bottom: 0.3125rem;
+  position: relative;
+}
+.pytorch-left-menu li code,
+.pytorch-right-menu li code {
+  border: none;
+  background: inherit;
+  color: inherit;
+  padding-left: 0;
+  padding-right: 0;
+}
+.pytorch-left-menu li span.toctree-expand,
+.pytorch-right-menu li span.toctree-expand {
+  display: block;
+  float: left;
+  margin-left: -1.2em;
+  font-size: 0.8em;
+  line-height: 1.6em;
+}
+.pytorch-left-menu li.on a, .pytorch-left-menu li.current > a,
+.pytorch-right-menu li.on a,
+.pytorch-right-menu li.current > a {
+  position: relative;
+  border: none;
+}
+.pytorch-left-menu li.on a span.toctree-expand, .pytorch-left-menu li.current > a span.toctree-expand,
+.pytorch-right-menu li.on a span.toctree-expand,
+.pytorch-right-menu li.current > a span.toctree-expand {
+  display: block;
+  font-size: 0.8em;
+  line-height: 1.6em;
+}
+.pytorch-left-menu li.toctree-l1.current > a,
+.pytorch-right-menu li.toctree-l1.current > a {
+  color: #ee4c2c;
+}
+.pytorch-left-menu li.toctree-l1.current > a:before,
+.pytorch-right-menu li.toctree-l1.current > a:before {
+  content: "\2022";
+  display: inline-block;
+  position: absolute;
+  left: -15px;
+  top: -10%;
+  font-size: 1.375rem;
+  color: #ee4c2c;
+}
+@media screen and (min-width: 1101px) {
+  .pytorch-left-menu li.toctree-l1.current > a:before,
+  .pytorch-right-menu li.toctree-l1.current > a:before {
+    left: -20px;
+  }
+}
+.pytorch-left-menu li.toctree-l1.current li.toctree-l2 > ul, .pytorch-left-menu li.toctree-l2.current li.toctree-l3 > ul,
+.pytorch-right-menu li.toctree-l1.current li.toctree-l2 > ul,
+.pytorch-right-menu li.toctree-l2.current li.toctree-l3 > ul {
+  display: none;
+}
+.pytorch-left-menu li.toctree-l1.current li.toctree-l2.current > ul, .pytorch-left-menu li.toctree-l2.current li.toctree-l3.current > ul,
+.pytorch-right-menu li.toctree-l1.current li.toctree-l2.current > ul,
+.pytorch-right-menu li.toctree-l2.current li.toctree-l3.current > ul {
+  display: block;
+}
+.pytorch-left-menu li.toctree-l2.current li.toctree-l3 > a,
+.pytorch-right-menu li.toctree-l2.current li.toctree-l3 > a {
+  display: block;
+}
+.pytorch-left-menu li.toctree-l3,
+.pytorch-right-menu li.toctree-l3 {
+  font-size: 0.9em;
+}
+.pytorch-left-menu li.toctree-l3.current li.toctree-l4 > a,
+.pytorch-right-menu li.toctree-l3.current li.toctree-l4 > a {
+  display: block;
+}
+.pytorch-left-menu li.toctree-l4,
+.pytorch-right-menu li.toctree-l4 {
+  font-size: 0.9em;
+}
+.pytorch-left-menu li.current ul,
+.pytorch-right-menu li.current ul {
+  display: block;
+}
+.pytorch-left-menu li ul,
+.pytorch-right-menu li ul {
+  margin-bottom: 0;
+  display: none;
+}
+.pytorch-left-menu li ul li a,
+.pytorch-right-menu li ul li a {
+  margin-bottom: 0;
+}
+.pytorch-left-menu a,
+.pytorch-right-menu a {
+  display: inline-block;
+  position: relative;
+}
+.pytorch-left-menu a:hover,
+.pytorch-right-menu a:hover {
+  cursor: pointer;
+}
+.pytorch-left-menu a:active,
+.pytorch-right-menu a:active {
+  cursor: pointer;
+}
+
+.pytorch-left-menu ul {
+  padding-left: 0;
+}
+
+.pytorch-right-menu a:link,
+.pytorch-right-menu a:visited,
+.pytorch-right-menu a:hover {
+  color: #6c6c6d;
+}
+.pytorch-right-menu a:link span.pre,
+.pytorch-right-menu a:visited span.pre,
+.pytorch-right-menu a:hover span.pre {
+  color: #6c6c6d;
+}
+.pytorch-right-menu a.reference.internal.expanded:before {
+  content: "-";
+  font-family: monospace;
+  position: absolute;
+  left: -12px;
+}
+.pytorch-right-menu a.reference.internal.not-expanded:before {
+  content: "+";
+  font-family: monospace;
+  position: absolute;
+  left: -12px;
+}
+.pytorch-right-menu li.active > a {
+  color: #ee4c2c;
+}
+.pytorch-right-menu li.active > a span.pre, .pytorch-right-menu li.active > a:before {
+  color: #ee4c2c;
+}
+.pytorch-right-menu li.active > a:after {
+  content: "\2022";
+  color: #e44c2c;
+  display: inline-block;
+  font-size: 1.375rem;
+  left: -17px;
+  position: absolute;
+  top: 1px;
+}
+.pytorch-right-menu .pytorch-side-scroll > ul > li > ul > li {
+  margin-bottom: 0;
+}
+.pytorch-right-menu ul ul {
+  padding-left: 0;
+}
+.pytorch-right-menu ul ul li {
+  padding-left: 0px;
+}
+.pytorch-right-menu ul ul li a.reference.internal {
+  padding-left: 0;
+}
+.pytorch-right-menu ul ul li ul {
+  display: none;
+  padding-left: 10px;
+}
+.pytorch-right-menu ul ul li li a.reference.internal {
+  padding-left: 0;
+}
+.pytorch-right-menu li ul {
+  display: block;
+}
+
+.pytorch-right-menu .pytorch-side-scroll {
+  padding-top: 20px;
+}
+@media screen and (min-width: 1101px) {
+  .pytorch-right-menu .pytorch-side-scroll {
+    width: 120%;
+  }
+}
+@media screen and (min-width: 1600px) {
+  .pytorch-right-menu .pytorch-side-scroll {
+    width: 400px;
+  }
+}
+.pytorch-right-menu .pytorch-side-scroll > ul {
+  padding-left: 10%;
+  padding-right: 10%;
+  margin-bottom: 0;
+}
+@media screen and (min-width: 1600px) {
+  .pytorch-right-menu .pytorch-side-scroll > ul {
+    padding-left: 25px;
+  }
+}
+.pytorch-right-menu .pytorch-side-scroll > ul > li > a.reference.internal {
+  color: #262626;
+  font-weight: 500;
+}
+.pytorch-right-menu .pytorch-side-scroll ul li {
+  position: relative;
+}
+
+#pytorch-right-menu .side-scroll-highlight {
+  color: #ee4c2c;
+}
+
+.header-container {
+  max-width: none;
+  margin-top: 4px;
+}
+@media screen and (min-width: 1101px) {
+  .header-container {
+    margin-top: 0;
+  }
+}
+@media screen and (min-width: 1600px) {
+  .header-container {
+    margin-top: 0;
+  }
+}
+
+.container-fluid.header-holder {
+  padding-right: 0;
+  padding-left: 0;
+}
+
+.header-holder .container {
+  max-width: none;
+  padding-right: 1.875rem;
+  padding-left: 1.875rem;
+}
+@media screen and (min-width: 1101px) {
+  .header-holder .container {
+    padding-right: 1.875rem;
+    padding-left: 1.875rem;
+  }
+}
+
+.header-holder .main-menu {
+  justify-content: unset;
+  position: relative;
+}
+@media screen and (min-width: 1101px) {
+  .header-holder .main-menu ul {
+    padding-left: 0;
+    margin-left: 26%;
+  }
+}
+@media screen and (min-width: 1600px) {
+  .header-holder .main-menu ul {
+    padding-left: 38px;
+    margin-left: 310px;
+  }
+}
+
+.pytorch-page-level-bar {
+  display: none;
+  align-items: center;
+  background-color: #ffffff;
+  border-bottom: 1px solid #e2e2e2;
+  width: 100%;
+  z-index: 201;
+}
+@media screen and (min-width: 1101px) {
+  .pytorch-page-level-bar {
+    left: 0;
+    display: flex;
+    height: 45px;
+    padding-left: 0;
+    width: 100%;
+    position: absolute;
+    z-index: 1;
+  }
+  .pytorch-page-level-bar.left-menu-is-fixed {
+    position: fixed;
+    top: 0;
+    left: 20%;
+    padding-left: 0;
+    right: 0;
+    width: 70%;
+  }
+}
+@media screen and (min-width: 1600px) {
+  .pytorch-page-level-bar {
+    left: 0;
+    right: 0;
+    width: auto;
+    z-index: 1;
+  }
+  .pytorch-page-level-bar.left-menu-is-fixed {
+    left: 350px;
+    right: 0;
+    width: auto;
+  }
+}
+.pytorch-page-level-bar ul, .pytorch-page-level-bar li {
+  margin: 0;
+}
+
+.pytorch-shortcuts-wrapper {
+  display: none;
+}
+@media screen and (min-width: 1101px) {
+  .pytorch-shortcuts-wrapper {
+    font-size: 0.875rem;
+    float: left;
+    margin-left: 2%;
+  }
+}
+@media screen and (min-width: 1600px) {
+  .pytorch-shortcuts-wrapper {
+    margin-left: 1.875rem;
+  }
+}
+
+.cookie-banner-wrapper {
+  display: none;
+}
+.cookie-banner-wrapper .container {
+  padding-left: 1.875rem;
+  padding-right: 1.875rem;
+  max-width: 1240px;
+}
+.cookie-banner-wrapper.is-visible {
+  display: block;
+  position: fixed;
+  bottom: 0;
+  background-color: #f3f4f7;
+  min-height: 100px;
+  width: 100%;
+  z-index: 401;
+  border-top: 3px solid #ededee;
+}
+.cookie-banner-wrapper .gdpr-notice {
+  color: #6c6c6d;
+  margin-top: 1.5625rem;
+  text-align: left;
+  max-width: 1440px;
+}
+@media screen and (min-width: 768px) {
+  .cookie-banner-wrapper .gdpr-notice {
+    width: 77%;
+  }
+}
+@media (min-width: 768px) and (max-width: 1239px) {
+  .cookie-banner-wrapper .gdpr-notice {
+    width: inherit;
+  }
+}
+.cookie-banner-wrapper .gdpr-notice .cookie-policy-link {
+  color: #343434;
+}
+.cookie-banner-wrapper .close-button {
+  -webkit-appearance: none;
+     -moz-appearance: none;
+          appearance: none;
+  background: transparent;
+  border: 1px solid #f3f4f7;
+  height: 1.3125rem;
+  position: absolute;
+  bottom: 42px;
+  right: 0;
+  top: 0;
+  cursor: pointer;
+  outline: none;
+}
+@media screen and (min-width: 768px) {
+  .cookie-banner-wrapper .close-button {
+    right: 20%;
+    top: inherit;
+  }
+}
+@media (min-width: 768px) and (max-width: 1239px) {
+  .cookie-banner-wrapper .close-button {
+    right: 0;
+    top: 0;
+  }
+}
+
+.main-menu ul li .resources-dropdown a {
+  cursor: pointer;
+}
+.main-menu ul li .dropdown-menu {
+  border-radius: 0;
+  padding: 0;
+}
+.main-menu ul li .dropdown-menu .dropdown-item {
+  color: #6c6c6d;
+  border-bottom: 1px solid #e2e2e2;
+}
+.main-menu ul li .dropdown-menu .dropdown-item:last-of-type {
+  border-bottom-color: transparent;
+}
+.main-menu ul li .dropdown-menu .dropdown-item:hover {
+  background-color: #e44c2c;
+}
+.main-menu ul li .dropdown-menu .dropdown-item p {
+  font-size: 1rem;
+  color: #979797;
+}
+.main-menu ul li .dropdown-menu a.dropdown-item:hover {
+  color: #ffffff;
+}
+.main-menu ul li .dropdown-menu a.dropdown-item:hover p {
+  color: #ffffff;
+}
+
+.resources-dropdown-menu {
+  left: -75px;
+  width: 226px;
+  display: none;
+  position: absolute;
+  z-index: 1000;
+  display: none;
+  float: left;
+  min-width: 10rem;
+  padding: 0.5rem 0;
+  font-size: 1rem;
+  color: #212529;
+  text-align: left;
+  list-style: none;
+  background-color: #ffffff;
+  background-clip: padding-box;
+  border: 1px solid rgba(0, 0, 0, 0.15);
+  border-radius: 0.25rem;
+}
+
+.resources-dropdown:hover .resources-dropdown-menu {
+  display: block;
+}
+
+.main-menu ul li .resources-dropdown-menu {
+  border-radius: 0;
+  padding: 0;
+}
+.main-menu ul li.active:hover .resources-dropdown-menu {
+  display: block;
+}
+
+.main-menu ul li .resources-dropdown-menu .dropdown-item {
+  color: #6c6c6d;
+  border-bottom: 1px solid #e2e2e2;
+}
+
+.resources-dropdown .with-down-orange-arrow {
+  padding-right: 2rem;
+  position: relative;
+  background: url("../images/chevron-down-orange.svg");
+  background-size: 14px 18px;
+  background-position: top 7px right 10px;
+  background-repeat: no-repeat;
+}
+
+.with-down-arrow {
+  padding-right: 2rem;
+  position: relative;
+  background-image: url("../images/chevron-down-black.svg");
+  background-size: 14px 18px;
+  background-position: top 7px right 10px;
+  background-repeat: no-repeat;
+}
+.with-down-arrow:hover {
+  background-image: url("../images/chevron-down-orange.svg");
+  background-repeat: no-repeat;
+}
+
+.header-holder .main-menu ul li .resources-dropdown .doc-dropdown-option {
+  padding-top: 1rem;
+}
+
+.header-holder .main-menu ul li a.nav-dropdown-item {
+  display: block;
+  font-size: 1rem;
+  line-height: 1.3125rem;
+  width: 100%;
+  padding: 0.25rem 1.5rem;
+  clear: both;
+  font-weight: 400;
+  color: #979797;
+  text-align: center;
+  background-color: transparent;
+  border-bottom: 1px solid #e2e2e2;
+}
+.header-holder .main-menu ul li a.nav-dropdown-item:last-of-type {
+  border-bottom-color: transparent;
+}
+.header-holder .main-menu ul li a.nav-dropdown-item:hover {
+  background-color: #e44c2c;
+  color: white;
+}
+.header-holder .main-menu ul li a.nav-dropdown-item .dropdown-title {
+  font-size: 1.125rem;
+  color: #6c6c6d;
+  letter-spacing: 0;
+  line-height: 34px;
+}
+
+.header-holder .main-menu ul li a.nav-dropdown-item:hover .dropdown-title {
+  background-color: #e44c2c;
+  color: white;
+}
diff --git a/docs/preview/pr/352/_static/doctools.js b/docs/preview/pr/352/_static/doctools.js
new file mode 100644
index 000000000..c3db08d1c
--- /dev/null
+++ b/docs/preview/pr/352/_static/doctools.js
@@ -0,0 +1,264 @@
+/*
+ * doctools.js
+ * ~~~~~~~~~~~
+ *
+ * Base JavaScript utilities for all Sphinx HTML documentation.
+ *
+ * :copyright: Copyright 2007-2022 by the Sphinx team, see AUTHORS.
+ * :license: BSD, see LICENSE for details.
+ *
+ */
+"use strict";
+
+const _ready = (callback) => {
+  if (document.readyState !== "loading") {
+    callback();
+  } else {
+    document.addEventListener("DOMContentLoaded", callback);
+  }
+};
+
+/**
+ * highlight a given string on a node by wrapping it in
+ * span elements with the given class name.
+ */
+const _highlight = (node, addItems, text, className) => {
+  if (node.nodeType === Node.TEXT_NODE) {
+    const val = node.nodeValue;
+    const parent = node.parentNode;
+    const pos = val.toLowerCase().indexOf(text);
+    if (
+      pos >= 0 &&
+      !parent.classList.contains(className) &&
+      !parent.classList.contains("nohighlight")
+    ) {
+      let span;
+
+      const closestNode = parent.closest("body, svg, foreignObject");
+      const isInSVG = closestNode && closestNode.matches("svg");
+      if (isInSVG) {
+        span = document.createElementNS("http://www.w3.org/2000/svg", "tspan");
+      } else {
+        span = document.createElement("span");
+        span.classList.add(className);
+      }
+
+      span.appendChild(document.createTextNode(val.substr(pos, text.length)));
+      parent.insertBefore(
+        span,
+        parent.insertBefore(
+          document.createTextNode(val.substr(pos + text.length)),
+          node.nextSibling
+        )
+      );
+      node.nodeValue = val.substr(0, pos);
+
+      if (isInSVG) {
+        const rect = document.createElementNS(
+          "http://www.w3.org/2000/svg",
+          "rect"
+        );
+        const bbox = parent.getBBox();
+        rect.x.baseVal.value = bbox.x;
+        rect.y.baseVal.value = bbox.y;
+        rect.width.baseVal.value = bbox.width;
+        rect.height.baseVal.value = bbox.height;
+        rect.setAttribute("class", className);
+        addItems.push({ parent: parent, target: rect });
+      }
+    }
+  } else if (node.matches && !node.matches("button, select, textarea")) {
+    node.childNodes.forEach((el) => _highlight(el, addItems, text, className));
+  }
+};
+const _highlightText = (thisNode, text, className) => {
+  let addItems = [];
+  _highlight(thisNode, addItems, text, className);
+  addItems.forEach((obj) =>
+    obj.parent.insertAdjacentElement("beforebegin", obj.target)
+  );
+};
+
+/**
+ * Small JavaScript module for the documentation.
+ */
+const Documentation = {
+  init: () => {
+    Documentation.highlightSearchWords();
+    Documentation.initDomainIndexTable();
+    Documentation.initOnKeyListeners();
+  },
+
+  /**
+   * i18n support
+   */
+  TRANSLATIONS: {},
+  PLURAL_EXPR: (n) => (n === 1 ? 0 : 1),
+  LOCALE: "unknown",
+
+  // gettext and ngettext don't access this so that the functions
+  // can safely bound to a different name (_ = Documentation.gettext)
+  gettext: (string) => {
+    const translated = Documentation.TRANSLATIONS[string];
+    switch (typeof translated) {
+      case "undefined":
+        return string; // no translation
+      case "string":
+        return translated; // translation exists
+      default:
+        return translated[0]; // (singular, plural) translation tuple exists
+    }
+  },
+
+  ngettext: (singular, plural, n) => {
+    const translated = Documentation.TRANSLATIONS[singular];
+    if (typeof translated !== "undefined")
+      return translated[Documentation.PLURAL_EXPR(n)];
+    return n === 1 ? singular : plural;
+  },
+
+  addTranslations: (catalog) => {
+    Object.assign(Documentation.TRANSLATIONS, catalog.messages);
+    Documentation.PLURAL_EXPR = new Function(
+      "n",
+      `return (${catalog.plural_expr})`
+    );
+    Documentation.LOCALE = catalog.locale;
+  },
+
+  /**
+   * highlight the search words provided in the url in the text
+   */
+  highlightSearchWords: () => {
+    const highlight =
+      new URLSearchParams(window.location.search).get("highlight") || "";
+    const terms = highlight.toLowerCase().split(/\s+/).filter(x => x);
+    if (terms.length === 0) return; // nothing to do
+
+    // There should never be more than one element matching "div.body"
+    const divBody = document.querySelectorAll("div.body");
+    const body = divBody.length ? divBody[0] : document.querySelector("body");
+    window.setTimeout(() => {
+      terms.forEach((term) => _highlightText(body, term, "highlighted"));
+    }, 10);
+
+    const searchBox = document.getElementById("searchbox");
+    if (searchBox === null) return;
+    searchBox.appendChild(
+      document
+        .createRange()
+        .createContextualFragment(
+          '<p class="highlight-link">' +
+            '<a href="javascript:Documentation.hideSearchWords()">' +
+            Documentation.gettext("Hide Search Matches") +
+            "</a></p>"
+        )
+    );
+  },
+
+  /**
+   * helper function to hide the search marks again
+   */
+  hideSearchWords: () => {
+    document
+      .querySelectorAll("#searchbox .highlight-link")
+      .forEach((el) => el.remove());
+    document
+      .querySelectorAll("span.highlighted")
+      .forEach((el) => el.classList.remove("highlighted"));
+    const url = new URL(window.location);
+    url.searchParams.delete("highlight");
+    window.history.replaceState({}, "", url);
+  },
+
+  /**
+   * helper function to focus on search bar
+   */
+  focusSearchBar: () => {
+    document.querySelectorAll("input[name=q]")[0]?.focus();
+  },
+
+  /**
+   * Initialise the domain index toggle buttons
+   */
+  initDomainIndexTable: () => {
+    const toggler = (el) => {
+      const idNumber = el.id.substr(7);
+      const toggledRows = document.querySelectorAll(`tr.cg-${idNumber}`);
+      if (el.src.substr(-9) === "minus.png") {
+        el.src = `${el.src.substr(0, el.src.length - 9)}plus.png`;
+        toggledRows.forEach((el) => (el.style.display = "none"));
+      } else {
+        el.src = `${el.src.substr(0, el.src.length - 8)}minus.png`;
+        toggledRows.forEach((el) => (el.style.display = ""));
+      }
+    };
+
+    const togglerElements = document.querySelectorAll("img.toggler");
+    togglerElements.forEach((el) =>
+      el.addEventListener("click", (event) => toggler(event.currentTarget))
+    );
+    togglerElements.forEach((el) => (el.style.display = ""));
+    if (DOCUMENTATION_OPTIONS.COLLAPSE_INDEX) togglerElements.forEach(toggler);
+  },
+
+  initOnKeyListeners: () => {
+    // only install a listener if it is really needed
+    if (
+      !DOCUMENTATION_OPTIONS.NAVIGATION_WITH_KEYS &&
+      !DOCUMENTATION_OPTIONS.ENABLE_SEARCH_SHORTCUTS
+    )
+      return;
+
+    const blacklistedElements = new Set([
+      "TEXTAREA",
+      "INPUT",
+      "SELECT",
+      "BUTTON",
+    ]);
+    document.addEventListener("keydown", (event) => {
+      if (blacklistedElements.has(document.activeElement.tagName)) return; // bail for input elements
+      if (event.altKey || event.ctrlKey || event.metaKey) return; // bail with special keys
+
+      if (!event.shiftKey) {
+        switch (event.key) {
+          case "ArrowLeft":
+            if (!DOCUMENTATION_OPTIONS.NAVIGATION_WITH_KEYS) break;
+
+            const prevLink = document.querySelector('link[rel="prev"]');
+            if (prevLink && prevLink.href) {
+              window.location.href = prevLink.href;
+              event.preventDefault();
+            }
+            break;
+          case "ArrowRight":
+            if (!DOCUMENTATION_OPTIONS.NAVIGATION_WITH_KEYS) break;
+
+            const nextLink = document.querySelector('link[rel="next"]');
+            if (nextLink && nextLink.href) {
+              window.location.href = nextLink.href;
+              event.preventDefault();
+            }
+            break;
+          case "Escape":
+            if (!DOCUMENTATION_OPTIONS.ENABLE_SEARCH_SHORTCUTS) break;
+            Documentation.hideSearchWords();
+            event.preventDefault();
+        }
+      }
+
+      // some keyboard layouts may need Shift to get /
+      switch (event.key) {
+        case "/":
+          if (!DOCUMENTATION_OPTIONS.ENABLE_SEARCH_SHORTCUTS) break;
+          Documentation.focusSearchBar();
+          event.preventDefault();
+      }
+    });
+  },
+};
+
+// quick alias for translations
+const _ = Documentation.gettext;
+
+_ready(Documentation.init);
diff --git a/docs/preview/pr/352/_static/documentation_options.js b/docs/preview/pr/352/_static/documentation_options.js
new file mode 100644
index 000000000..88cb93e55
--- /dev/null
+++ b/docs/preview/pr/352/_static/documentation_options.js
@@ -0,0 +1,14 @@
+var DOCUMENTATION_OPTIONS = {
+    URL_ROOT: document.getElementById("documentation_options").getAttribute('data-url_root'),
+    VERSION: 'main',
+    LANGUAGE: 'en',
+    COLLAPSE_INDEX: false,
+    BUILDER: 'dirhtml',
+    FILE_SUFFIX: '.html',
+    LINK_SUFFIX: '.html',
+    HAS_SOURCE: true,
+    SOURCELINK_SUFFIX: '.txt',
+    NAVIGATION_WITH_KEYS: false,
+    SHOW_SEARCH_SUMMARY: true,
+    ENABLE_SEARCH_SHORTCUTS: false,
+};
\ No newline at end of file
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diff --git a/docs/preview/pr/352/_static/fonts/FreightSans/freight-sans-bold.woff b/docs/preview/pr/352/_static/fonts/FreightSans/freight-sans-bold.woff
new file mode 100644
index 0000000000000000000000000000000000000000..de46625edfc8c12d6d2594f630350a55e92d49b8
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diff --git a/docs/preview/pr/352/_static/fonts/FreightSans/freight-sans-book.woff b/docs/preview/pr/352/_static/fonts/FreightSans/freight-sans-book.woff
new file mode 100644
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new file mode 100644
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new file mode 100644
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diff --git a/docs/preview/pr/352/_static/fonts/FreightSans/freight-sans-medium.woff2 b/docs/preview/pr/352/_static/fonts/FreightSans/freight-sans-medium.woff2
new file mode 100644
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new file mode 100644
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+    <desc>Created with Sketch.</desc>
+    <defs></defs>
+    <g id="desktop" stroke="none" stroke-width="1" fill="none" fill-rule="evenodd">
+        <g id="Cloud-Button-Sequence" transform="translate(-1311.000000, -3818.000000)" stroke="#262626" stroke-width="2">
+            <g id="GetStarted-Copy-3" transform="translate(98.000000, 3249.000000)">
+                <g id="Via-CLoud" transform="translate(855.000000, 212.000000)">
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+                        <polyline id="Page-1" transform="translate(364.566667, 32.500000) rotate(90.000000) translate(-364.566667, -32.500000) " points="362 27 367.133333 32.4998402 362 38"></polyline>
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+                </g>
+            </g>
+        </g>
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+</svg>
diff --git a/docs/preview/pr/352/_static/images/chevron-down-grey.svg b/docs/preview/pr/352/_static/images/chevron-down-grey.svg
new file mode 100644
index 000000000..82d6514f2
--- /dev/null
+++ b/docs/preview/pr/352/_static/images/chevron-down-grey.svg
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+<?xml version="1.0" encoding="utf-8"?>
+<!-- Generator: Adobe Illustrator 17.1.0, SVG Export Plug-In . SVG Version: 6.00 Build 0)  -->
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+<svg version="1.1" id="Layer_1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0px" y="0px"
+	 viewBox="0.4 0.2 12.4 7.2" enable-background="new 0.4 0.2 12.4 7.2" xml:space="preserve">
+<desc>Created with Sketch.</desc>
+<g id="desktop">
+	<g id="Cloud-Button-Sequence" transform="translate(-1311.000000, -3818.000000)">
+		<g id="GetStarted-Copy-3" transform="translate(98.000000, 3249.000000)">
+			<g id="Via-CLoud" transform="translate(855.000000, 212.000000)">
+				<g id="Button" transform="translate(0.000000, 328.000000)">
+					<polyline id="Page-1" fill="none" stroke="#C7C7C7" stroke-width="2" points="370.1,29.9 364.6,35.1 359.1,29.9 					"/>
+				</g>
+			</g>
+		</g>
+	</g>
+</g>
+</svg>
diff --git a/docs/preview/pr/352/_static/images/chevron-down-orange.svg b/docs/preview/pr/352/_static/images/chevron-down-orange.svg
new file mode 100644
index 000000000..fd79a5785
--- /dev/null
+++ b/docs/preview/pr/352/_static/images/chevron-down-orange.svg
@@ -0,0 +1,16 @@
+<svg width="13px" height="8px" viewBox="0 0 13 8" version="1.1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
+    <!-- Generator: Sketch 51.2 (57519) - http://www.bohemiancoding.com/sketch -->
+    <desc>Created with Sketch.</desc>
+    <defs></defs>
+    <g id="desktop" stroke="none" stroke-width="1" fill="none" fill-rule="evenodd">
+        <g id="Cloud-Button-Sequence" transform="translate(-1311.000000, -3818.000000)" stroke="#EE4C2C" stroke-width="2">
+            <g id="GetStarted-Copy-3" transform="translate(98.000000, 3249.000000)">
+                <g id="Via-CLoud" transform="translate(855.000000, 212.000000)">
+                    <g id="Button" transform="translate(0.000000, 328.000000)">
+                        <polyline id="Page-1" transform="translate(364.566667, 32.500000) rotate(90.000000) translate(-364.566667, -32.500000) " points="362 27 367.133333 32.4998402 362 38"></polyline>
+                    </g>
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+            </g>
+        </g>
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+</svg>
diff --git a/docs/preview/pr/352/_static/images/chevron-down-white.svg b/docs/preview/pr/352/_static/images/chevron-down-white.svg
new file mode 100644
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--- /dev/null
+++ b/docs/preview/pr/352/_static/images/chevron-down-white.svg
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+<svg width="13px" height="8px" viewBox="0 0 13 8" version="1.1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
+    <!-- Generator: Sketch 51.2 (57519) - http://www.bohemiancoding.com/sketch -->
+    <desc>Created with Sketch.</desc>
+    <defs></defs>
+    <g id="desktop" stroke="none" stroke-width="1" fill="none" fill-rule="evenodd">
+        <g id="Cloud-Button-Sequence" transform="translate(-1311.000000, -3818.000000)" stroke="#FFFFFF" stroke-width="2">
+            <g id="GetStarted-Copy-3" transform="translate(98.000000, 3249.000000)">
+                <g id="Via-CLoud" transform="translate(855.000000, 212.000000)">
+                    <g id="Button" transform="translate(0.000000, 328.000000)">
+                        <polyline id="Page-1" transform="translate(364.566667, 32.500000) rotate(90.000000) translate(-364.566667, -32.500000) " points="362 27 367.133333 32.4998402 362 38"></polyline>
+                    </g>
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+</svg>
diff --git a/docs/preview/pr/352/_static/images/chevron-right-orange.svg b/docs/preview/pr/352/_static/images/chevron-right-orange.svg
new file mode 100644
index 000000000..7033fc93b
--- /dev/null
+++ b/docs/preview/pr/352/_static/images/chevron-right-orange.svg
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+	 viewBox="0.3 0.3 8.2 14.4" enable-background="new 0.3 0.3 8.2 14.4" xml:space="preserve">
+<title>Page 1</title>
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+<g id="desktop">
+	<g id="_x30_1_x5F_Home" transform="translate(-864.000000, -1683.000000)">
+		<g id="community" transform="translate(-18.000000, 1391.000000)">
+			<g id="cta" transform="translate(741.000000, 277.000000)">
+				<polyline id="Page-1" fill="none" stroke="#EE4C2C" stroke-width="2" points="142,16 148.1,22.5 142,29 				"/>
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+		</g>
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diff --git a/docs/preview/pr/352/_static/images/chevron-right-white.svg b/docs/preview/pr/352/_static/images/chevron-right-white.svg
new file mode 100644
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HcmV?d00001

diff --git a/docs/preview/pr/352/_static/jquery-3.6.0.js b/docs/preview/pr/352/_static/jquery-3.6.0.js
new file mode 100644
index 000000000..fc6c299b7
--- /dev/null
+++ b/docs/preview/pr/352/_static/jquery-3.6.0.js
@@ -0,0 +1,10881 @@
+/*!
+ * jQuery JavaScript Library v3.6.0
+ * https://jquery.com/
+ *
+ * Includes Sizzle.js
+ * https://sizzlejs.com/
+ *
+ * Copyright OpenJS Foundation and other contributors
+ * Released under the MIT license
+ * https://jquery.org/license
+ *
+ * Date: 2021-03-02T17:08Z
+ */
+( function( global, factory ) {
+
+	"use strict";
+
+	if ( typeof module === "object" && typeof module.exports === "object" ) {
+
+		// For CommonJS and CommonJS-like environments where a proper `window`
+		// is present, execute the factory and get jQuery.
+		// For environments that do not have a `window` with a `document`
+		// (such as Node.js), expose a factory as module.exports.
+		// This accentuates the need for the creation of a real `window`.
+		// e.g. var jQuery = require("jquery")(window);
+		// See ticket #14549 for more info.
+		module.exports = global.document ?
+			factory( global, true ) :
+			function( w ) {
+				if ( !w.document ) {
+					throw new Error( "jQuery requires a window with a document" );
+				}
+				return factory( w );
+			};
+	} else {
+		factory( global );
+	}
+
+// Pass this if window is not defined yet
+} )( typeof window !== "undefined" ? window : this, function( window, noGlobal ) {
+
+// Edge <= 12 - 13+, Firefox <=18 - 45+, IE 10 - 11, Safari 5.1 - 9+, iOS 6 - 9.1
+// throw exceptions when non-strict code (e.g., ASP.NET 4.5) accesses strict mode
+// arguments.callee.caller (trac-13335). But as of jQuery 3.0 (2016), strict mode should be common
+// enough that all such attempts are guarded in a try block.
+"use strict";
+
+var arr = [];
+
+var getProto = Object.getPrototypeOf;
+
+var slice = arr.slice;
+
+var flat = arr.flat ? function( array ) {
+	return arr.flat.call( array );
+} : function( array ) {
+	return arr.concat.apply( [], array );
+};
+
+
+var push = arr.push;
+
+var indexOf = arr.indexOf;
+
+var class2type = {};
+
+var toString = class2type.toString;
+
+var hasOwn = class2type.hasOwnProperty;
+
+var fnToString = hasOwn.toString;
+
+var ObjectFunctionString = fnToString.call( Object );
+
+var support = {};
+
+var isFunction = function isFunction( obj ) {
+
+		// Support: Chrome <=57, Firefox <=52
+		// In some browsers, typeof returns "function" for HTML <object> elements
+		// (i.e., `typeof document.createElement( "object" ) === "function"`).
+		// We don't want to classify *any* DOM node as a function.
+		// Support: QtWeb <=3.8.5, WebKit <=534.34, wkhtmltopdf tool <=0.12.5
+		// Plus for old WebKit, typeof returns "function" for HTML collections
+		// (e.g., `typeof document.getElementsByTagName("div") === "function"`). (gh-4756)
+		return typeof obj === "function" && typeof obj.nodeType !== "number" &&
+			typeof obj.item !== "function";
+	};
+
+
+var isWindow = function isWindow( obj ) {
+		return obj != null && obj === obj.window;
+	};
+
+
+var document = window.document;
+
+
+
+	var preservedScriptAttributes = {
+		type: true,
+		src: true,
+		nonce: true,
+		noModule: true
+	};
+
+	function DOMEval( code, node, doc ) {
+		doc = doc || document;
+
+		var i, val,
+			script = doc.createElement( "script" );
+
+		script.text = code;
+		if ( node ) {
+			for ( i in preservedScriptAttributes ) {
+
+				// Support: Firefox 64+, Edge 18+
+				// Some browsers don't support the "nonce" property on scripts.
+				// On the other hand, just using `getAttribute` is not enough as
+				// the `nonce` attribute is reset to an empty string whenever it
+				// becomes browsing-context connected.
+				// See https://github.com/whatwg/html/issues/2369
+				// See https://html.spec.whatwg.org/#nonce-attributes
+				// The `node.getAttribute` check was added for the sake of
+				// `jQuery.globalEval` so that it can fake a nonce-containing node
+				// via an object.
+				val = node[ i ] || node.getAttribute && node.getAttribute( i );
+				if ( val ) {
+					script.setAttribute( i, val );
+				}
+			}
+		}
+		doc.head.appendChild( script ).parentNode.removeChild( script );
+	}
+
+
+function toType( obj ) {
+	if ( obj == null ) {
+		return obj + "";
+	}
+
+	// Support: Android <=2.3 only (functionish RegExp)
+	return typeof obj === "object" || typeof obj === "function" ?
+		class2type[ toString.call( obj ) ] || "object" :
+		typeof obj;
+}
+/* global Symbol */
+// Defining this global in .eslintrc.json would create a danger of using the global
+// unguarded in another place, it seems safer to define global only for this module
+
+
+
+var
+	version = "3.6.0",
+
+	// Define a local copy of jQuery
+	jQuery = function( selector, context ) {
+
+		// The jQuery object is actually just the init constructor 'enhanced'
+		// Need init if jQuery is called (just allow error to be thrown if not included)
+		return new jQuery.fn.init( selector, context );
+	};
+
+jQuery.fn = jQuery.prototype = {
+
+	// The current version of jQuery being used
+	jquery: version,
+
+	constructor: jQuery,
+
+	// The default length of a jQuery object is 0
+	length: 0,
+
+	toArray: function() {
+		return slice.call( this );
+	},
+
+	// Get the Nth element in the matched element set OR
+	// Get the whole matched element set as a clean array
+	get: function( num ) {
+
+		// Return all the elements in a clean array
+		if ( num == null ) {
+			return slice.call( this );
+		}
+
+		// Return just the one element from the set
+		return num < 0 ? this[ num + this.length ] : this[ num ];
+	},
+
+	// Take an array of elements and push it onto the stack
+	// (returning the new matched element set)
+	pushStack: function( elems ) {
+
+		// Build a new jQuery matched element set
+		var ret = jQuery.merge( this.constructor(), elems );
+
+		// Add the old object onto the stack (as a reference)
+		ret.prevObject = this;
+
+		// Return the newly-formed element set
+		return ret;
+	},
+
+	// Execute a callback for every element in the matched set.
+	each: function( callback ) {
+		return jQuery.each( this, callback );
+	},
+
+	map: function( callback ) {
+		return this.pushStack( jQuery.map( this, function( elem, i ) {
+			return callback.call( elem, i, elem );
+		} ) );
+	},
+
+	slice: function() {
+		return this.pushStack( slice.apply( this, arguments ) );
+	},
+
+	first: function() {
+		return this.eq( 0 );
+	},
+
+	last: function() {
+		return this.eq( -1 );
+	},
+
+	even: function() {
+		return this.pushStack( jQuery.grep( this, function( _elem, i ) {
+			return ( i + 1 ) % 2;
+		} ) );
+	},
+
+	odd: function() {
+		return this.pushStack( jQuery.grep( this, function( _elem, i ) {
+			return i % 2;
+		} ) );
+	},
+
+	eq: function( i ) {
+		var len = this.length,
+			j = +i + ( i < 0 ? len : 0 );
+		return this.pushStack( j >= 0 && j < len ? [ this[ j ] ] : [] );
+	},
+
+	end: function() {
+		return this.prevObject || this.constructor();
+	},
+
+	// For internal use only.
+	// Behaves like an Array's method, not like a jQuery method.
+	push: push,
+	sort: arr.sort,
+	splice: arr.splice
+};
+
+jQuery.extend = jQuery.fn.extend = function() {
+	var options, name, src, copy, copyIsArray, clone,
+		target = arguments[ 0 ] || {},
+		i = 1,
+		length = arguments.length,
+		deep = false;
+
+	// Handle a deep copy situation
+	if ( typeof target === "boolean" ) {
+		deep = target;
+
+		// Skip the boolean and the target
+		target = arguments[ i ] || {};
+		i++;
+	}
+
+	// Handle case when target is a string or something (possible in deep copy)
+	if ( typeof target !== "object" && !isFunction( target ) ) {
+		target = {};
+	}
+
+	// Extend jQuery itself if only one argument is passed
+	if ( i === length ) {
+		target = this;
+		i--;
+	}
+
+	for ( ; i < length; i++ ) {
+
+		// Only deal with non-null/undefined values
+		if ( ( options = arguments[ i ] ) != null ) {
+
+			// Extend the base object
+			for ( name in options ) {
+				copy = options[ name ];
+
+				// Prevent Object.prototype pollution
+				// Prevent never-ending loop
+				if ( name === "__proto__" || target === copy ) {
+					continue;
+				}
+
+				// Recurse if we're merging plain objects or arrays
+				if ( deep && copy && ( jQuery.isPlainObject( copy ) ||
+					( copyIsArray = Array.isArray( copy ) ) ) ) {
+					src = target[ name ];
+
+					// Ensure proper type for the source value
+					if ( copyIsArray && !Array.isArray( src ) ) {
+						clone = [];
+					} else if ( !copyIsArray && !jQuery.isPlainObject( src ) ) {
+						clone = {};
+					} else {
+						clone = src;
+					}
+					copyIsArray = false;
+
+					// Never move original objects, clone them
+					target[ name ] = jQuery.extend( deep, clone, copy );
+
+				// Don't bring in undefined values
+				} else if ( copy !== undefined ) {
+					target[ name ] = copy;
+				}
+			}
+		}
+	}
+
+	// Return the modified object
+	return target;
+};
+
+jQuery.extend( {
+
+	// Unique for each copy of jQuery on the page
+	expando: "jQuery" + ( version + Math.random() ).replace( /\D/g, "" ),
+
+	// Assume jQuery is ready without the ready module
+	isReady: true,
+
+	error: function( msg ) {
+		throw new Error( msg );
+	},
+
+	noop: function() {},
+
+	isPlainObject: function( obj ) {
+		var proto, Ctor;
+
+		// Detect obvious negatives
+		// Use toString instead of jQuery.type to catch host objects
+		if ( !obj || toString.call( obj ) !== "[object Object]" ) {
+			return false;
+		}
+
+		proto = getProto( obj );
+
+		// Objects with no prototype (e.g., `Object.create( null )`) are plain
+		if ( !proto ) {
+			return true;
+		}
+
+		// Objects with prototype are plain iff they were constructed by a global Object function
+		Ctor = hasOwn.call( proto, "constructor" ) && proto.constructor;
+		return typeof Ctor === "function" && fnToString.call( Ctor ) === ObjectFunctionString;
+	},
+
+	isEmptyObject: function( obj ) {
+		var name;
+
+		for ( name in obj ) {
+			return false;
+		}
+		return true;
+	},
+
+	// Evaluates a script in a provided context; falls back to the global one
+	// if not specified.
+	globalEval: function( code, options, doc ) {
+		DOMEval( code, { nonce: options && options.nonce }, doc );
+	},
+
+	each: function( obj, callback ) {
+		var length, i = 0;
+
+		if ( isArrayLike( obj ) ) {
+			length = obj.length;
+			for ( ; i < length; i++ ) {
+				if ( callback.call( obj[ i ], i, obj[ i ] ) === false ) {
+					break;
+				}
+			}
+		} else {
+			for ( i in obj ) {
+				if ( callback.call( obj[ i ], i, obj[ i ] ) === false ) {
+					break;
+				}
+			}
+		}
+
+		return obj;
+	},
+
+	// results is for internal usage only
+	makeArray: function( arr, results ) {
+		var ret = results || [];
+
+		if ( arr != null ) {
+			if ( isArrayLike( Object( arr ) ) ) {
+				jQuery.merge( ret,
+					typeof arr === "string" ?
+						[ arr ] : arr
+				);
+			} else {
+				push.call( ret, arr );
+			}
+		}
+
+		return ret;
+	},
+
+	inArray: function( elem, arr, i ) {
+		return arr == null ? -1 : indexOf.call( arr, elem, i );
+	},
+
+	// Support: Android <=4.0 only, PhantomJS 1 only
+	// push.apply(_, arraylike) throws on ancient WebKit
+	merge: function( first, second ) {
+		var len = +second.length,
+			j = 0,
+			i = first.length;
+
+		for ( ; j < len; j++ ) {
+			first[ i++ ] = second[ j ];
+		}
+
+		first.length = i;
+
+		return first;
+	},
+
+	grep: function( elems, callback, invert ) {
+		var callbackInverse,
+			matches = [],
+			i = 0,
+			length = elems.length,
+			callbackExpect = !invert;
+
+		// Go through the array, only saving the items
+		// that pass the validator function
+		for ( ; i < length; i++ ) {
+			callbackInverse = !callback( elems[ i ], i );
+			if ( callbackInverse !== callbackExpect ) {
+				matches.push( elems[ i ] );
+			}
+		}
+
+		return matches;
+	},
+
+	// arg is for internal usage only
+	map: function( elems, callback, arg ) {
+		var length, value,
+			i = 0,
+			ret = [];
+
+		// Go through the array, translating each of the items to their new values
+		if ( isArrayLike( elems ) ) {
+			length = elems.length;
+			for ( ; i < length; i++ ) {
+				value = callback( elems[ i ], i, arg );
+
+				if ( value != null ) {
+					ret.push( value );
+				}
+			}
+
+		// Go through every key on the object,
+		} else {
+			for ( i in elems ) {
+				value = callback( elems[ i ], i, arg );
+
+				if ( value != null ) {
+					ret.push( value );
+				}
+			}
+		}
+
+		// Flatten any nested arrays
+		return flat( ret );
+	},
+
+	// A global GUID counter for objects
+	guid: 1,
+
+	// jQuery.support is not used in Core but other projects attach their
+	// properties to it so it needs to exist.
+	support: support
+} );
+
+if ( typeof Symbol === "function" ) {
+	jQuery.fn[ Symbol.iterator ] = arr[ Symbol.iterator ];
+}
+
+// Populate the class2type map
+jQuery.each( "Boolean Number String Function Array Date RegExp Object Error Symbol".split( " " ),
+	function( _i, name ) {
+		class2type[ "[object " + name + "]" ] = name.toLowerCase();
+	} );
+
+function isArrayLike( obj ) {
+
+	// Support: real iOS 8.2 only (not reproducible in simulator)
+	// `in` check used to prevent JIT error (gh-2145)
+	// hasOwn isn't used here due to false negatives
+	// regarding Nodelist length in IE
+	var length = !!obj && "length" in obj && obj.length,
+		type = toType( obj );
+
+	if ( isFunction( obj ) || isWindow( obj ) ) {
+		return false;
+	}
+
+	return type === "array" || length === 0 ||
+		typeof length === "number" && length > 0 && ( length - 1 ) in obj;
+}
+var Sizzle =
+/*!
+ * Sizzle CSS Selector Engine v2.3.6
+ * https://sizzlejs.com/
+ *
+ * Copyright JS Foundation and other contributors
+ * Released under the MIT license
+ * https://js.foundation/
+ *
+ * Date: 2021-02-16
+ */
+( function( window ) {
+var i,
+	support,
+	Expr,
+	getText,
+	isXML,
+	tokenize,
+	compile,
+	select,
+	outermostContext,
+	sortInput,
+	hasDuplicate,
+
+	// Local document vars
+	setDocument,
+	document,
+	docElem,
+	documentIsHTML,
+	rbuggyQSA,
+	rbuggyMatches,
+	matches,
+	contains,
+
+	// Instance-specific data
+	expando = "sizzle" + 1 * new Date(),
+	preferredDoc = window.document,
+	dirruns = 0,
+	done = 0,
+	classCache = createCache(),
+	tokenCache = createCache(),
+	compilerCache = createCache(),
+	nonnativeSelectorCache = createCache(),
+	sortOrder = function( a, b ) {
+		if ( a === b ) {
+			hasDuplicate = true;
+		}
+		return 0;
+	},
+
+	// Instance methods
+	hasOwn = ( {} ).hasOwnProperty,
+	arr = [],
+	pop = arr.pop,
+	pushNative = arr.push,
+	push = arr.push,
+	slice = arr.slice,
+
+	// Use a stripped-down indexOf as it's faster than native
+	// https://jsperf.com/thor-indexof-vs-for/5
+	indexOf = function( list, elem ) {
+		var i = 0,
+			len = list.length;
+		for ( ; i < len; i++ ) {
+			if ( list[ i ] === elem ) {
+				return i;
+			}
+		}
+		return -1;
+	},
+
+	booleans = "checked|selected|async|autofocus|autoplay|controls|defer|disabled|hidden|" +
+		"ismap|loop|multiple|open|readonly|required|scoped",
+
+	// Regular expressions
+
+	// http://www.w3.org/TR/css3-selectors/#whitespace
+	whitespace = "[\\x20\\t\\r\\n\\f]",
+
+	// https://www.w3.org/TR/css-syntax-3/#ident-token-diagram
+	identifier = "(?:\\\\[\\da-fA-F]{1,6}" + whitespace +
+		"?|\\\\[^\\r\\n\\f]|[\\w-]|[^\0-\\x7f])+",
+
+	// Attribute selectors: http://www.w3.org/TR/selectors/#attribute-selectors
+	attributes = "\\[" + whitespace + "*(" + identifier + ")(?:" + whitespace +
+
+		// Operator (capture 2)
+		"*([*^$|!~]?=)" + whitespace +
+
+		// "Attribute values must be CSS identifiers [capture 5]
+		// or strings [capture 3 or capture 4]"
+		"*(?:'((?:\\\\.|[^\\\\'])*)'|\"((?:\\\\.|[^\\\\\"])*)\"|(" + identifier + "))|)" +
+		whitespace + "*\\]",
+
+	pseudos = ":(" + identifier + ")(?:\\((" +
+
+		// To reduce the number of selectors needing tokenize in the preFilter, prefer arguments:
+		// 1. quoted (capture 3; capture 4 or capture 5)
+		"('((?:\\\\.|[^\\\\'])*)'|\"((?:\\\\.|[^\\\\\"])*)\")|" +
+
+		// 2. simple (capture 6)
+		"((?:\\\\.|[^\\\\()[\\]]|" + attributes + ")*)|" +
+
+		// 3. anything else (capture 2)
+		".*" +
+		")\\)|)",
+
+	// Leading and non-escaped trailing whitespace, capturing some non-whitespace characters preceding the latter
+	rwhitespace = new RegExp( whitespace + "+", "g" ),
+	rtrim = new RegExp( "^" + whitespace + "+|((?:^|[^\\\\])(?:\\\\.)*)" +
+		whitespace + "+$", "g" ),
+
+	rcomma = new RegExp( "^" + whitespace + "*," + whitespace + "*" ),
+	rcombinators = new RegExp( "^" + whitespace + "*([>+~]|" + whitespace + ")" + whitespace +
+		"*" ),
+	rdescend = new RegExp( whitespace + "|>" ),
+
+	rpseudo = new RegExp( pseudos ),
+	ridentifier = new RegExp( "^" + identifier + "$" ),
+
+	matchExpr = {
+		"ID": new RegExp( "^#(" + identifier + ")" ),
+		"CLASS": new RegExp( "^\\.(" + identifier + ")" ),
+		"TAG": new RegExp( "^(" + identifier + "|[*])" ),
+		"ATTR": new RegExp( "^" + attributes ),
+		"PSEUDO": new RegExp( "^" + pseudos ),
+		"CHILD": new RegExp( "^:(only|first|last|nth|nth-last)-(child|of-type)(?:\\(" +
+			whitespace + "*(even|odd|(([+-]|)(\\d*)n|)" + whitespace + "*(?:([+-]|)" +
+			whitespace + "*(\\d+)|))" + whitespace + "*\\)|)", "i" ),
+		"bool": new RegExp( "^(?:" + booleans + ")$", "i" ),
+
+		// For use in libraries implementing .is()
+		// We use this for POS matching in `select`
+		"needsContext": new RegExp( "^" + whitespace +
+			"*[>+~]|:(even|odd|eq|gt|lt|nth|first|last)(?:\\(" + whitespace +
+			"*((?:-\\d)?\\d*)" + whitespace + "*\\)|)(?=[^-]|$)", "i" )
+	},
+
+	rhtml = /HTML$/i,
+	rinputs = /^(?:input|select|textarea|button)$/i,
+	rheader = /^h\d$/i,
+
+	rnative = /^[^{]+\{\s*\[native \w/,
+
+	// Easily-parseable/retrievable ID or TAG or CLASS selectors
+	rquickExpr = /^(?:#([\w-]+)|(\w+)|\.([\w-]+))$/,
+
+	rsibling = /[+~]/,
+
+	// CSS escapes
+	// http://www.w3.org/TR/CSS21/syndata.html#escaped-characters
+	runescape = new RegExp( "\\\\[\\da-fA-F]{1,6}" + whitespace + "?|\\\\([^\\r\\n\\f])", "g" ),
+	funescape = function( escape, nonHex ) {
+		var high = "0x" + escape.slice( 1 ) - 0x10000;
+
+		return nonHex ?
+
+			// Strip the backslash prefix from a non-hex escape sequence
+			nonHex :
+
+			// Replace a hexadecimal escape sequence with the encoded Unicode code point
+			// Support: IE <=11+
+			// For values outside the Basic Multilingual Plane (BMP), manually construct a
+			// surrogate pair
+			high < 0 ?
+				String.fromCharCode( high + 0x10000 ) :
+				String.fromCharCode( high >> 10 | 0xD800, high & 0x3FF | 0xDC00 );
+	},
+
+	// CSS string/identifier serialization
+	// https://drafts.csswg.org/cssom/#common-serializing-idioms
+	rcssescape = /([\0-\x1f\x7f]|^-?\d)|^-$|[^\0-\x1f\x7f-\uFFFF\w-]/g,
+	fcssescape = function( ch, asCodePoint ) {
+		if ( asCodePoint ) {
+
+			// U+0000 NULL becomes U+FFFD REPLACEMENT CHARACTER
+			if ( ch === "\0" ) {
+				return "\uFFFD";
+			}
+
+			// Control characters and (dependent upon position) numbers get escaped as code points
+			return ch.slice( 0, -1 ) + "\\" +
+				ch.charCodeAt( ch.length - 1 ).toString( 16 ) + " ";
+		}
+
+		// Other potentially-special ASCII characters get backslash-escaped
+		return "\\" + ch;
+	},
+
+	// Used for iframes
+	// See setDocument()
+	// Removing the function wrapper causes a "Permission Denied"
+	// error in IE
+	unloadHandler = function() {
+		setDocument();
+	},
+
+	inDisabledFieldset = addCombinator(
+		function( elem ) {
+			return elem.disabled === true && elem.nodeName.toLowerCase() === "fieldset";
+		},
+		{ dir: "parentNode", next: "legend" }
+	);
+
+// Optimize for push.apply( _, NodeList )
+try {
+	push.apply(
+		( arr = slice.call( preferredDoc.childNodes ) ),
+		preferredDoc.childNodes
+	);
+
+	// Support: Android<4.0
+	// Detect silently failing push.apply
+	// eslint-disable-next-line no-unused-expressions
+	arr[ preferredDoc.childNodes.length ].nodeType;
+} catch ( e ) {
+	push = { apply: arr.length ?
+
+		// Leverage slice if possible
+		function( target, els ) {
+			pushNative.apply( target, slice.call( els ) );
+		} :
+
+		// Support: IE<9
+		// Otherwise append directly
+		function( target, els ) {
+			var j = target.length,
+				i = 0;
+
+			// Can't trust NodeList.length
+			while ( ( target[ j++ ] = els[ i++ ] ) ) {}
+			target.length = j - 1;
+		}
+	};
+}
+
+function Sizzle( selector, context, results, seed ) {
+	var m, i, elem, nid, match, groups, newSelector,
+		newContext = context && context.ownerDocument,
+
+		// nodeType defaults to 9, since context defaults to document
+		nodeType = context ? context.nodeType : 9;
+
+	results = results || [];
+
+	// Return early from calls with invalid selector or context
+	if ( typeof selector !== "string" || !selector ||
+		nodeType !== 1 && nodeType !== 9 && nodeType !== 11 ) {
+
+		return results;
+	}
+
+	// Try to shortcut find operations (as opposed to filters) in HTML documents
+	if ( !seed ) {
+		setDocument( context );
+		context = context || document;
+
+		if ( documentIsHTML ) {
+
+			// If the selector is sufficiently simple, try using a "get*By*" DOM method
+			// (excepting DocumentFragment context, where the methods don't exist)
+			if ( nodeType !== 11 && ( match = rquickExpr.exec( selector ) ) ) {
+
+				// ID selector
+				if ( ( m = match[ 1 ] ) ) {
+
+					// Document context
+					if ( nodeType === 9 ) {
+						if ( ( elem = context.getElementById( m ) ) ) {
+
+							// Support: IE, Opera, Webkit
+							// TODO: identify versions
+							// getElementById can match elements by name instead of ID
+							if ( elem.id === m ) {
+								results.push( elem );
+								return results;
+							}
+						} else {
+							return results;
+						}
+
+					// Element context
+					} else {
+
+						// Support: IE, Opera, Webkit
+						// TODO: identify versions
+						// getElementById can match elements by name instead of ID
+						if ( newContext && ( elem = newContext.getElementById( m ) ) &&
+							contains( context, elem ) &&
+							elem.id === m ) {
+
+							results.push( elem );
+							return results;
+						}
+					}
+
+				// Type selector
+				} else if ( match[ 2 ] ) {
+					push.apply( results, context.getElementsByTagName( selector ) );
+					return results;
+
+				// Class selector
+				} else if ( ( m = match[ 3 ] ) && support.getElementsByClassName &&
+					context.getElementsByClassName ) {
+
+					push.apply( results, context.getElementsByClassName( m ) );
+					return results;
+				}
+			}
+
+			// Take advantage of querySelectorAll
+			if ( support.qsa &&
+				!nonnativeSelectorCache[ selector + " " ] &&
+				( !rbuggyQSA || !rbuggyQSA.test( selector ) ) &&
+
+				// Support: IE 8 only
+				// Exclude object elements
+				( nodeType !== 1 || context.nodeName.toLowerCase() !== "object" ) ) {
+
+				newSelector = selector;
+				newContext = context;
+
+				// qSA considers elements outside a scoping root when evaluating child or
+				// descendant combinators, which is not what we want.
+				// In such cases, we work around the behavior by prefixing every selector in the
+				// list with an ID selector referencing the scope context.
+				// The technique has to be used as well when a leading combinator is used
+				// as such selectors are not recognized by querySelectorAll.
+				// Thanks to Andrew Dupont for this technique.
+				if ( nodeType === 1 &&
+					( rdescend.test( selector ) || rcombinators.test( selector ) ) ) {
+
+					// Expand context for sibling selectors
+					newContext = rsibling.test( selector ) && testContext( context.parentNode ) ||
+						context;
+
+					// We can use :scope instead of the ID hack if the browser
+					// supports it & if we're not changing the context.
+					if ( newContext !== context || !support.scope ) {
+
+						// Capture the context ID, setting it first if necessary
+						if ( ( nid = context.getAttribute( "id" ) ) ) {
+							nid = nid.replace( rcssescape, fcssescape );
+						} else {
+							context.setAttribute( "id", ( nid = expando ) );
+						}
+					}
+
+					// Prefix every selector in the list
+					groups = tokenize( selector );
+					i = groups.length;
+					while ( i-- ) {
+						groups[ i ] = ( nid ? "#" + nid : ":scope" ) + " " +
+							toSelector( groups[ i ] );
+					}
+					newSelector = groups.join( "," );
+				}
+
+				try {
+					push.apply( results,
+						newContext.querySelectorAll( newSelector )
+					);
+					return results;
+				} catch ( qsaError ) {
+					nonnativeSelectorCache( selector, true );
+				} finally {
+					if ( nid === expando ) {
+						context.removeAttribute( "id" );
+					}
+				}
+			}
+		}
+	}
+
+	// All others
+	return select( selector.replace( rtrim, "$1" ), context, results, seed );
+}
+
+/**
+ * Create key-value caches of limited size
+ * @returns {function(string, object)} Returns the Object data after storing it on itself with
+ *	property name the (space-suffixed) string and (if the cache is larger than Expr.cacheLength)
+ *	deleting the oldest entry
+ */
+function createCache() {
+	var keys = [];
+
+	function cache( key, value ) {
+
+		// Use (key + " ") to avoid collision with native prototype properties (see Issue #157)
+		if ( keys.push( key + " " ) > Expr.cacheLength ) {
+
+			// Only keep the most recent entries
+			delete cache[ keys.shift() ];
+		}
+		return ( cache[ key + " " ] = value );
+	}
+	return cache;
+}
+
+/**
+ * Mark a function for special use by Sizzle
+ * @param {Function} fn The function to mark
+ */
+function markFunction( fn ) {
+	fn[ expando ] = true;
+	return fn;
+}
+
+/**
+ * Support testing using an element
+ * @param {Function} fn Passed the created element and returns a boolean result
+ */
+function assert( fn ) {
+	var el = document.createElement( "fieldset" );
+
+	try {
+		return !!fn( el );
+	} catch ( e ) {
+		return false;
+	} finally {
+
+		// Remove from its parent by default
+		if ( el.parentNode ) {
+			el.parentNode.removeChild( el );
+		}
+
+		// release memory in IE
+		el = null;
+	}
+}
+
+/**
+ * Adds the same handler for all of the specified attrs
+ * @param {String} attrs Pipe-separated list of attributes
+ * @param {Function} handler The method that will be applied
+ */
+function addHandle( attrs, handler ) {
+	var arr = attrs.split( "|" ),
+		i = arr.length;
+
+	while ( i-- ) {
+		Expr.attrHandle[ arr[ i ] ] = handler;
+	}
+}
+
+/**
+ * Checks document order of two siblings
+ * @param {Element} a
+ * @param {Element} b
+ * @returns {Number} Returns less than 0 if a precedes b, greater than 0 if a follows b
+ */
+function siblingCheck( a, b ) {
+	var cur = b && a,
+		diff = cur && a.nodeType === 1 && b.nodeType === 1 &&
+			a.sourceIndex - b.sourceIndex;
+
+	// Use IE sourceIndex if available on both nodes
+	if ( diff ) {
+		return diff;
+	}
+
+	// Check if b follows a
+	if ( cur ) {
+		while ( ( cur = cur.nextSibling ) ) {
+			if ( cur === b ) {
+				return -1;
+			}
+		}
+	}
+
+	return a ? 1 : -1;
+}
+
+/**
+ * Returns a function to use in pseudos for input types
+ * @param {String} type
+ */
+function createInputPseudo( type ) {
+	return function( elem ) {
+		var name = elem.nodeName.toLowerCase();
+		return name === "input" && elem.type === type;
+	};
+}
+
+/**
+ * Returns a function to use in pseudos for buttons
+ * @param {String} type
+ */
+function createButtonPseudo( type ) {
+	return function( elem ) {
+		var name = elem.nodeName.toLowerCase();
+		return ( name === "input" || name === "button" ) && elem.type === type;
+	};
+}
+
+/**
+ * Returns a function to use in pseudos for :enabled/:disabled
+ * @param {Boolean} disabled true for :disabled; false for :enabled
+ */
+function createDisabledPseudo( disabled ) {
+
+	// Known :disabled false positives: fieldset[disabled] > legend:nth-of-type(n+2) :can-disable
+	return function( elem ) {
+
+		// Only certain elements can match :enabled or :disabled
+		// https://html.spec.whatwg.org/multipage/scripting.html#selector-enabled
+		// https://html.spec.whatwg.org/multipage/scripting.html#selector-disabled
+		if ( "form" in elem ) {
+
+			// Check for inherited disabledness on relevant non-disabled elements:
+			// * listed form-associated elements in a disabled fieldset
+			//   https://html.spec.whatwg.org/multipage/forms.html#category-listed
+			//   https://html.spec.whatwg.org/multipage/forms.html#concept-fe-disabled
+			// * option elements in a disabled optgroup
+			//   https://html.spec.whatwg.org/multipage/forms.html#concept-option-disabled
+			// All such elements have a "form" property.
+			if ( elem.parentNode && elem.disabled === false ) {
+
+				// Option elements defer to a parent optgroup if present
+				if ( "label" in elem ) {
+					if ( "label" in elem.parentNode ) {
+						return elem.parentNode.disabled === disabled;
+					} else {
+						return elem.disabled === disabled;
+					}
+				}
+
+				// Support: IE 6 - 11
+				// Use the isDisabled shortcut property to check for disabled fieldset ancestors
+				return elem.isDisabled === disabled ||
+
+					// Where there is no isDisabled, check manually
+					/* jshint -W018 */
+					elem.isDisabled !== !disabled &&
+					inDisabledFieldset( elem ) === disabled;
+			}
+
+			return elem.disabled === disabled;
+
+		// Try to winnow out elements that can't be disabled before trusting the disabled property.
+		// Some victims get caught in our net (label, legend, menu, track), but it shouldn't
+		// even exist on them, let alone have a boolean value.
+		} else if ( "label" in elem ) {
+			return elem.disabled === disabled;
+		}
+
+		// Remaining elements are neither :enabled nor :disabled
+		return false;
+	};
+}
+
+/**
+ * Returns a function to use in pseudos for positionals
+ * @param {Function} fn
+ */
+function createPositionalPseudo( fn ) {
+	return markFunction( function( argument ) {
+		argument = +argument;
+		return markFunction( function( seed, matches ) {
+			var j,
+				matchIndexes = fn( [], seed.length, argument ),
+				i = matchIndexes.length;
+
+			// Match elements found at the specified indexes
+			while ( i-- ) {
+				if ( seed[ ( j = matchIndexes[ i ] ) ] ) {
+					seed[ j ] = !( matches[ j ] = seed[ j ] );
+				}
+			}
+		} );
+	} );
+}
+
+/**
+ * Checks a node for validity as a Sizzle context
+ * @param {Element|Object=} context
+ * @returns {Element|Object|Boolean} The input node if acceptable, otherwise a falsy value
+ */
+function testContext( context ) {
+	return context && typeof context.getElementsByTagName !== "undefined" && context;
+}
+
+// Expose support vars for convenience
+support = Sizzle.support = {};
+
+/**
+ * Detects XML nodes
+ * @param {Element|Object} elem An element or a document
+ * @returns {Boolean} True iff elem is a non-HTML XML node
+ */
+isXML = Sizzle.isXML = function( elem ) {
+	var namespace = elem && elem.namespaceURI,
+		docElem = elem && ( elem.ownerDocument || elem ).documentElement;
+
+	// Support: IE <=8
+	// Assume HTML when documentElement doesn't yet exist, such as inside loading iframes
+	// https://bugs.jquery.com/ticket/4833
+	return !rhtml.test( namespace || docElem && docElem.nodeName || "HTML" );
+};
+
+/**
+ * Sets document-related variables once based on the current document
+ * @param {Element|Object} [doc] An element or document object to use to set the document
+ * @returns {Object} Returns the current document
+ */
+setDocument = Sizzle.setDocument = function( node ) {
+	var hasCompare, subWindow,
+		doc = node ? node.ownerDocument || node : preferredDoc;
+
+	// Return early if doc is invalid or already selected
+	// Support: IE 11+, Edge 17 - 18+
+	// IE/Edge sometimes throw a "Permission denied" error when strict-comparing
+	// two documents; shallow comparisons work.
+	// eslint-disable-next-line eqeqeq
+	if ( doc == document || doc.nodeType !== 9 || !doc.documentElement ) {
+		return document;
+	}
+
+	// Update global variables
+	document = doc;
+	docElem = document.documentElement;
+	documentIsHTML = !isXML( document );
+
+	// Support: IE 9 - 11+, Edge 12 - 18+
+	// Accessing iframe documents after unload throws "permission denied" errors (jQuery #13936)
+	// Support: IE 11+, Edge 17 - 18+
+	// IE/Edge sometimes throw a "Permission denied" error when strict-comparing
+	// two documents; shallow comparisons work.
+	// eslint-disable-next-line eqeqeq
+	if ( preferredDoc != document &&
+		( subWindow = document.defaultView ) && subWindow.top !== subWindow ) {
+
+		// Support: IE 11, Edge
+		if ( subWindow.addEventListener ) {
+			subWindow.addEventListener( "unload", unloadHandler, false );
+
+		// Support: IE 9 - 10 only
+		} else if ( subWindow.attachEvent ) {
+			subWindow.attachEvent( "onunload", unloadHandler );
+		}
+	}
+
+	// Support: IE 8 - 11+, Edge 12 - 18+, Chrome <=16 - 25 only, Firefox <=3.6 - 31 only,
+	// Safari 4 - 5 only, Opera <=11.6 - 12.x only
+	// IE/Edge & older browsers don't support the :scope pseudo-class.
+	// Support: Safari 6.0 only
+	// Safari 6.0 supports :scope but it's an alias of :root there.
+	support.scope = assert( function( el ) {
+		docElem.appendChild( el ).appendChild( document.createElement( "div" ) );
+		return typeof el.querySelectorAll !== "undefined" &&
+			!el.querySelectorAll( ":scope fieldset div" ).length;
+	} );
+
+	/* Attributes
+	---------------------------------------------------------------------- */
+
+	// Support: IE<8
+	// Verify that getAttribute really returns attributes and not properties
+	// (excepting IE8 booleans)
+	support.attributes = assert( function( el ) {
+		el.className = "i";
+		return !el.getAttribute( "className" );
+	} );
+
+	/* getElement(s)By*
+	---------------------------------------------------------------------- */
+
+	// Check if getElementsByTagName("*") returns only elements
+	support.getElementsByTagName = assert( function( el ) {
+		el.appendChild( document.createComment( "" ) );
+		return !el.getElementsByTagName( "*" ).length;
+	} );
+
+	// Support: IE<9
+	support.getElementsByClassName = rnative.test( document.getElementsByClassName );
+
+	// Support: IE<10
+	// Check if getElementById returns elements by name
+	// The broken getElementById methods don't pick up programmatically-set names,
+	// so use a roundabout getElementsByName test
+	support.getById = assert( function( el ) {
+		docElem.appendChild( el ).id = expando;
+		return !document.getElementsByName || !document.getElementsByName( expando ).length;
+	} );
+
+	// ID filter and find
+	if ( support.getById ) {
+		Expr.filter[ "ID" ] = function( id ) {
+			var attrId = id.replace( runescape, funescape );
+			return function( elem ) {
+				return elem.getAttribute( "id" ) === attrId;
+			};
+		};
+		Expr.find[ "ID" ] = function( id, context ) {
+			if ( typeof context.getElementById !== "undefined" && documentIsHTML ) {
+				var elem = context.getElementById( id );
+				return elem ? [ elem ] : [];
+			}
+		};
+	} else {
+		Expr.filter[ "ID" ] =  function( id ) {
+			var attrId = id.replace( runescape, funescape );
+			return function( elem ) {
+				var node = typeof elem.getAttributeNode !== "undefined" &&
+					elem.getAttributeNode( "id" );
+				return node && node.value === attrId;
+			};
+		};
+
+		// Support: IE 6 - 7 only
+		// getElementById is not reliable as a find shortcut
+		Expr.find[ "ID" ] = function( id, context ) {
+			if ( typeof context.getElementById !== "undefined" && documentIsHTML ) {
+				var node, i, elems,
+					elem = context.getElementById( id );
+
+				if ( elem ) {
+
+					// Verify the id attribute
+					node = elem.getAttributeNode( "id" );
+					if ( node && node.value === id ) {
+						return [ elem ];
+					}
+
+					// Fall back on getElementsByName
+					elems = context.getElementsByName( id );
+					i = 0;
+					while ( ( elem = elems[ i++ ] ) ) {
+						node = elem.getAttributeNode( "id" );
+						if ( node && node.value === id ) {
+							return [ elem ];
+						}
+					}
+				}
+
+				return [];
+			}
+		};
+	}
+
+	// Tag
+	Expr.find[ "TAG" ] = support.getElementsByTagName ?
+		function( tag, context ) {
+			if ( typeof context.getElementsByTagName !== "undefined" ) {
+				return context.getElementsByTagName( tag );
+
+			// DocumentFragment nodes don't have gEBTN
+			} else if ( support.qsa ) {
+				return context.querySelectorAll( tag );
+			}
+		} :
+
+		function( tag, context ) {
+			var elem,
+				tmp = [],
+				i = 0,
+
+				// By happy coincidence, a (broken) gEBTN appears on DocumentFragment nodes too
+				results = context.getElementsByTagName( tag );
+
+			// Filter out possible comments
+			if ( tag === "*" ) {
+				while ( ( elem = results[ i++ ] ) ) {
+					if ( elem.nodeType === 1 ) {
+						tmp.push( elem );
+					}
+				}
+
+				return tmp;
+			}
+			return results;
+		};
+
+	// Class
+	Expr.find[ "CLASS" ] = support.getElementsByClassName && function( className, context ) {
+		if ( typeof context.getElementsByClassName !== "undefined" && documentIsHTML ) {
+			return context.getElementsByClassName( className );
+		}
+	};
+
+	/* QSA/matchesSelector
+	---------------------------------------------------------------------- */
+
+	// QSA and matchesSelector support
+
+	// matchesSelector(:active) reports false when true (IE9/Opera 11.5)
+	rbuggyMatches = [];
+
+	// qSa(:focus) reports false when true (Chrome 21)
+	// We allow this because of a bug in IE8/9 that throws an error
+	// whenever `document.activeElement` is accessed on an iframe
+	// So, we allow :focus to pass through QSA all the time to avoid the IE error
+	// See https://bugs.jquery.com/ticket/13378
+	rbuggyQSA = [];
+
+	if ( ( support.qsa = rnative.test( document.querySelectorAll ) ) ) {
+
+		// Build QSA regex
+		// Regex strategy adopted from Diego Perini
+		assert( function( el ) {
+
+			var input;
+
+			// Select is set to empty string on purpose
+			// This is to test IE's treatment of not explicitly
+			// setting a boolean content attribute,
+			// since its presence should be enough
+			// https://bugs.jquery.com/ticket/12359
+			docElem.appendChild( el ).innerHTML = "<a id='" + expando + "'></a>" +
+				"<select id='" + expando + "-\r\\' msallowcapture=''>" +
+				"<option selected=''></option></select>";
+
+			// Support: IE8, Opera 11-12.16
+			// Nothing should be selected when empty strings follow ^= or $= or *=
+			// The test attribute must be unknown in Opera but "safe" for WinRT
+			// https://msdn.microsoft.com/en-us/library/ie/hh465388.aspx#attribute_section
+			if ( el.querySelectorAll( "[msallowcapture^='']" ).length ) {
+				rbuggyQSA.push( "[*^$]=" + whitespace + "*(?:''|\"\")" );
+			}
+
+			// Support: IE8
+			// Boolean attributes and "value" are not treated correctly
+			if ( !el.querySelectorAll( "[selected]" ).length ) {
+				rbuggyQSA.push( "\\[" + whitespace + "*(?:value|" + booleans + ")" );
+			}
+
+			// Support: Chrome<29, Android<4.4, Safari<7.0+, iOS<7.0+, PhantomJS<1.9.8+
+			if ( !el.querySelectorAll( "[id~=" + expando + "-]" ).length ) {
+				rbuggyQSA.push( "~=" );
+			}
+
+			// Support: IE 11+, Edge 15 - 18+
+			// IE 11/Edge don't find elements on a `[name='']` query in some cases.
+			// Adding a temporary attribute to the document before the selection works
+			// around the issue.
+			// Interestingly, IE 10 & older don't seem to have the issue.
+			input = document.createElement( "input" );
+			input.setAttribute( "name", "" );
+			el.appendChild( input );
+			if ( !el.querySelectorAll( "[name='']" ).length ) {
+				rbuggyQSA.push( "\\[" + whitespace + "*name" + whitespace + "*=" +
+					whitespace + "*(?:''|\"\")" );
+			}
+
+			// Webkit/Opera - :checked should return selected option elements
+			// http://www.w3.org/TR/2011/REC-css3-selectors-20110929/#checked
+			// IE8 throws error here and will not see later tests
+			if ( !el.querySelectorAll( ":checked" ).length ) {
+				rbuggyQSA.push( ":checked" );
+			}
+
+			// Support: Safari 8+, iOS 8+
+			// https://bugs.webkit.org/show_bug.cgi?id=136851
+			// In-page `selector#id sibling-combinator selector` fails
+			if ( !el.querySelectorAll( "a#" + expando + "+*" ).length ) {
+				rbuggyQSA.push( ".#.+[+~]" );
+			}
+
+			// Support: Firefox <=3.6 - 5 only
+			// Old Firefox doesn't throw on a badly-escaped identifier.
+			el.querySelectorAll( "\\\f" );
+			rbuggyQSA.push( "[\\r\\n\\f]" );
+		} );
+
+		assert( function( el ) {
+			el.innerHTML = "<a href='' disabled='disabled'></a>" +
+				"<select disabled='disabled'><option/></select>";
+
+			// Support: Windows 8 Native Apps
+			// The type and name attributes are restricted during .innerHTML assignment
+			var input = document.createElement( "input" );
+			input.setAttribute( "type", "hidden" );
+			el.appendChild( input ).setAttribute( "name", "D" );
+
+			// Support: IE8
+			// Enforce case-sensitivity of name attribute
+			if ( el.querySelectorAll( "[name=d]" ).length ) {
+				rbuggyQSA.push( "name" + whitespace + "*[*^$|!~]?=" );
+			}
+
+			// FF 3.5 - :enabled/:disabled and hidden elements (hidden elements are still enabled)
+			// IE8 throws error here and will not see later tests
+			if ( el.querySelectorAll( ":enabled" ).length !== 2 ) {
+				rbuggyQSA.push( ":enabled", ":disabled" );
+			}
+
+			// Support: IE9-11+
+			// IE's :disabled selector does not pick up the children of disabled fieldsets
+			docElem.appendChild( el ).disabled = true;
+			if ( el.querySelectorAll( ":disabled" ).length !== 2 ) {
+				rbuggyQSA.push( ":enabled", ":disabled" );
+			}
+
+			// Support: Opera 10 - 11 only
+			// Opera 10-11 does not throw on post-comma invalid pseudos
+			el.querySelectorAll( "*,:x" );
+			rbuggyQSA.push( ",.*:" );
+		} );
+	}
+
+	if ( ( support.matchesSelector = rnative.test( ( matches = docElem.matches ||
+		docElem.webkitMatchesSelector ||
+		docElem.mozMatchesSelector ||
+		docElem.oMatchesSelector ||
+		docElem.msMatchesSelector ) ) ) ) {
+
+		assert( function( el ) {
+
+			// Check to see if it's possible to do matchesSelector
+			// on a disconnected node (IE 9)
+			support.disconnectedMatch = matches.call( el, "*" );
+
+			// This should fail with an exception
+			// Gecko does not error, returns false instead
+			matches.call( el, "[s!='']:x" );
+			rbuggyMatches.push( "!=", pseudos );
+		} );
+	}
+
+	rbuggyQSA = rbuggyQSA.length && new RegExp( rbuggyQSA.join( "|" ) );
+	rbuggyMatches = rbuggyMatches.length && new RegExp( rbuggyMatches.join( "|" ) );
+
+	/* Contains
+	---------------------------------------------------------------------- */
+	hasCompare = rnative.test( docElem.compareDocumentPosition );
+
+	// Element contains another
+	// Purposefully self-exclusive
+	// As in, an element does not contain itself
+	contains = hasCompare || rnative.test( docElem.contains ) ?
+		function( a, b ) {
+			var adown = a.nodeType === 9 ? a.documentElement : a,
+				bup = b && b.parentNode;
+			return a === bup || !!( bup && bup.nodeType === 1 && (
+				adown.contains ?
+					adown.contains( bup ) :
+					a.compareDocumentPosition && a.compareDocumentPosition( bup ) & 16
+			) );
+		} :
+		function( a, b ) {
+			if ( b ) {
+				while ( ( b = b.parentNode ) ) {
+					if ( b === a ) {
+						return true;
+					}
+				}
+			}
+			return false;
+		};
+
+	/* Sorting
+	---------------------------------------------------------------------- */
+
+	// Document order sorting
+	sortOrder = hasCompare ?
+	function( a, b ) {
+
+		// Flag for duplicate removal
+		if ( a === b ) {
+			hasDuplicate = true;
+			return 0;
+		}
+
+		// Sort on method existence if only one input has compareDocumentPosition
+		var compare = !a.compareDocumentPosition - !b.compareDocumentPosition;
+		if ( compare ) {
+			return compare;
+		}
+
+		// Calculate position if both inputs belong to the same document
+		// Support: IE 11+, Edge 17 - 18+
+		// IE/Edge sometimes throw a "Permission denied" error when strict-comparing
+		// two documents; shallow comparisons work.
+		// eslint-disable-next-line eqeqeq
+		compare = ( a.ownerDocument || a ) == ( b.ownerDocument || b ) ?
+			a.compareDocumentPosition( b ) :
+
+			// Otherwise we know they are disconnected
+			1;
+
+		// Disconnected nodes
+		if ( compare & 1 ||
+			( !support.sortDetached && b.compareDocumentPosition( a ) === compare ) ) {
+
+			// Choose the first element that is related to our preferred document
+			// Support: IE 11+, Edge 17 - 18+
+			// IE/Edge sometimes throw a "Permission denied" error when strict-comparing
+			// two documents; shallow comparisons work.
+			// eslint-disable-next-line eqeqeq
+			if ( a == document || a.ownerDocument == preferredDoc &&
+				contains( preferredDoc, a ) ) {
+				return -1;
+			}
+
+			// Support: IE 11+, Edge 17 - 18+
+			// IE/Edge sometimes throw a "Permission denied" error when strict-comparing
+			// two documents; shallow comparisons work.
+			// eslint-disable-next-line eqeqeq
+			if ( b == document || b.ownerDocument == preferredDoc &&
+				contains( preferredDoc, b ) ) {
+				return 1;
+			}
+
+			// Maintain original order
+			return sortInput ?
+				( indexOf( sortInput, a ) - indexOf( sortInput, b ) ) :
+				0;
+		}
+
+		return compare & 4 ? -1 : 1;
+	} :
+	function( a, b ) {
+
+		// Exit early if the nodes are identical
+		if ( a === b ) {
+			hasDuplicate = true;
+			return 0;
+		}
+
+		var cur,
+			i = 0,
+			aup = a.parentNode,
+			bup = b.parentNode,
+			ap = [ a ],
+			bp = [ b ];
+
+		// Parentless nodes are either documents or disconnected
+		if ( !aup || !bup ) {
+
+			// Support: IE 11+, Edge 17 - 18+
+			// IE/Edge sometimes throw a "Permission denied" error when strict-comparing
+			// two documents; shallow comparisons work.
+			/* eslint-disable eqeqeq */
+			return a == document ? -1 :
+				b == document ? 1 :
+				/* eslint-enable eqeqeq */
+				aup ? -1 :
+				bup ? 1 :
+				sortInput ?
+				( indexOf( sortInput, a ) - indexOf( sortInput, b ) ) :
+				0;
+
+		// If the nodes are siblings, we can do a quick check
+		} else if ( aup === bup ) {
+			return siblingCheck( a, b );
+		}
+
+		// Otherwise we need full lists of their ancestors for comparison
+		cur = a;
+		while ( ( cur = cur.parentNode ) ) {
+			ap.unshift( cur );
+		}
+		cur = b;
+		while ( ( cur = cur.parentNode ) ) {
+			bp.unshift( cur );
+		}
+
+		// Walk down the tree looking for a discrepancy
+		while ( ap[ i ] === bp[ i ] ) {
+			i++;
+		}
+
+		return i ?
+
+			// Do a sibling check if the nodes have a common ancestor
+			siblingCheck( ap[ i ], bp[ i ] ) :
+
+			// Otherwise nodes in our document sort first
+			// Support: IE 11+, Edge 17 - 18+
+			// IE/Edge sometimes throw a "Permission denied" error when strict-comparing
+			// two documents; shallow comparisons work.
+			/* eslint-disable eqeqeq */
+			ap[ i ] == preferredDoc ? -1 :
+			bp[ i ] == preferredDoc ? 1 :
+			/* eslint-enable eqeqeq */
+			0;
+	};
+
+	return document;
+};
+
+Sizzle.matches = function( expr, elements ) {
+	return Sizzle( expr, null, null, elements );
+};
+
+Sizzle.matchesSelector = function( elem, expr ) {
+	setDocument( elem );
+
+	if ( support.matchesSelector && documentIsHTML &&
+		!nonnativeSelectorCache[ expr + " " ] &&
+		( !rbuggyMatches || !rbuggyMatches.test( expr ) ) &&
+		( !rbuggyQSA     || !rbuggyQSA.test( expr ) ) ) {
+
+		try {
+			var ret = matches.call( elem, expr );
+
+			// IE 9's matchesSelector returns false on disconnected nodes
+			if ( ret || support.disconnectedMatch ||
+
+				// As well, disconnected nodes are said to be in a document
+				// fragment in IE 9
+				elem.document && elem.document.nodeType !== 11 ) {
+				return ret;
+			}
+		} catch ( e ) {
+			nonnativeSelectorCache( expr, true );
+		}
+	}
+
+	return Sizzle( expr, document, null, [ elem ] ).length > 0;
+};
+
+Sizzle.contains = function( context, elem ) {
+
+	// Set document vars if needed
+	// Support: IE 11+, Edge 17 - 18+
+	// IE/Edge sometimes throw a "Permission denied" error when strict-comparing
+	// two documents; shallow comparisons work.
+	// eslint-disable-next-line eqeqeq
+	if ( ( context.ownerDocument || context ) != document ) {
+		setDocument( context );
+	}
+	return contains( context, elem );
+};
+
+Sizzle.attr = function( elem, name ) {
+
+	// Set document vars if needed
+	// Support: IE 11+, Edge 17 - 18+
+	// IE/Edge sometimes throw a "Permission denied" error when strict-comparing
+	// two documents; shallow comparisons work.
+	// eslint-disable-next-line eqeqeq
+	if ( ( elem.ownerDocument || elem ) != document ) {
+		setDocument( elem );
+	}
+
+	var fn = Expr.attrHandle[ name.toLowerCase() ],
+
+		// Don't get fooled by Object.prototype properties (jQuery #13807)
+		val = fn && hasOwn.call( Expr.attrHandle, name.toLowerCase() ) ?
+			fn( elem, name, !documentIsHTML ) :
+			undefined;
+
+	return val !== undefined ?
+		val :
+		support.attributes || !documentIsHTML ?
+			elem.getAttribute( name ) :
+			( val = elem.getAttributeNode( name ) ) && val.specified ?
+				val.value :
+				null;
+};
+
+Sizzle.escape = function( sel ) {
+	return ( sel + "" ).replace( rcssescape, fcssescape );
+};
+
+Sizzle.error = function( msg ) {
+	throw new Error( "Syntax error, unrecognized expression: " + msg );
+};
+
+/**
+ * Document sorting and removing duplicates
+ * @param {ArrayLike} results
+ */
+Sizzle.uniqueSort = function( results ) {
+	var elem,
+		duplicates = [],
+		j = 0,
+		i = 0;
+
+	// Unless we *know* we can detect duplicates, assume their presence
+	hasDuplicate = !support.detectDuplicates;
+	sortInput = !support.sortStable && results.slice( 0 );
+	results.sort( sortOrder );
+
+	if ( hasDuplicate ) {
+		while ( ( elem = results[ i++ ] ) ) {
+			if ( elem === results[ i ] ) {
+				j = duplicates.push( i );
+			}
+		}
+		while ( j-- ) {
+			results.splice( duplicates[ j ], 1 );
+		}
+	}
+
+	// Clear input after sorting to release objects
+	// See https://github.com/jquery/sizzle/pull/225
+	sortInput = null;
+
+	return results;
+};
+
+/**
+ * Utility function for retrieving the text value of an array of DOM nodes
+ * @param {Array|Element} elem
+ */
+getText = Sizzle.getText = function( elem ) {
+	var node,
+		ret = "",
+		i = 0,
+		nodeType = elem.nodeType;
+
+	if ( !nodeType ) {
+
+		// If no nodeType, this is expected to be an array
+		while ( ( node = elem[ i++ ] ) ) {
+
+			// Do not traverse comment nodes
+			ret += getText( node );
+		}
+	} else if ( nodeType === 1 || nodeType === 9 || nodeType === 11 ) {
+
+		// Use textContent for elements
+		// innerText usage removed for consistency of new lines (jQuery #11153)
+		if ( typeof elem.textContent === "string" ) {
+			return elem.textContent;
+		} else {
+
+			// Traverse its children
+			for ( elem = elem.firstChild; elem; elem = elem.nextSibling ) {
+				ret += getText( elem );
+			}
+		}
+	} else if ( nodeType === 3 || nodeType === 4 ) {
+		return elem.nodeValue;
+	}
+
+	// Do not include comment or processing instruction nodes
+
+	return ret;
+};
+
+Expr = Sizzle.selectors = {
+
+	// Can be adjusted by the user
+	cacheLength: 50,
+
+	createPseudo: markFunction,
+
+	match: matchExpr,
+
+	attrHandle: {},
+
+	find: {},
+
+	relative: {
+		">": { dir: "parentNode", first: true },
+		" ": { dir: "parentNode" },
+		"+": { dir: "previousSibling", first: true },
+		"~": { dir: "previousSibling" }
+	},
+
+	preFilter: {
+		"ATTR": function( match ) {
+			match[ 1 ] = match[ 1 ].replace( runescape, funescape );
+
+			// Move the given value to match[3] whether quoted or unquoted
+			match[ 3 ] = ( match[ 3 ] || match[ 4 ] ||
+				match[ 5 ] || "" ).replace( runescape, funescape );
+
+			if ( match[ 2 ] === "~=" ) {
+				match[ 3 ] = " " + match[ 3 ] + " ";
+			}
+
+			return match.slice( 0, 4 );
+		},
+
+		"CHILD": function( match ) {
+
+			/* matches from matchExpr["CHILD"]
+				1 type (only|nth|...)
+				2 what (child|of-type)
+				3 argument (even|odd|\d*|\d*n([+-]\d+)?|...)
+				4 xn-component of xn+y argument ([+-]?\d*n|)
+				5 sign of xn-component
+				6 x of xn-component
+				7 sign of y-component
+				8 y of y-component
+			*/
+			match[ 1 ] = match[ 1 ].toLowerCase();
+
+			if ( match[ 1 ].slice( 0, 3 ) === "nth" ) {
+
+				// nth-* requires argument
+				if ( !match[ 3 ] ) {
+					Sizzle.error( match[ 0 ] );
+				}
+
+				// numeric x and y parameters for Expr.filter.CHILD
+				// remember that false/true cast respectively to 0/1
+				match[ 4 ] = +( match[ 4 ] ?
+					match[ 5 ] + ( match[ 6 ] || 1 ) :
+					2 * ( match[ 3 ] === "even" || match[ 3 ] === "odd" ) );
+				match[ 5 ] = +( ( match[ 7 ] + match[ 8 ] ) || match[ 3 ] === "odd" );
+
+				// other types prohibit arguments
+			} else if ( match[ 3 ] ) {
+				Sizzle.error( match[ 0 ] );
+			}
+
+			return match;
+		},
+
+		"PSEUDO": function( match ) {
+			var excess,
+				unquoted = !match[ 6 ] && match[ 2 ];
+
+			if ( matchExpr[ "CHILD" ].test( match[ 0 ] ) ) {
+				return null;
+			}
+
+			// Accept quoted arguments as-is
+			if ( match[ 3 ] ) {
+				match[ 2 ] = match[ 4 ] || match[ 5 ] || "";
+
+			// Strip excess characters from unquoted arguments
+			} else if ( unquoted && rpseudo.test( unquoted ) &&
+
+				// Get excess from tokenize (recursively)
+				( excess = tokenize( unquoted, true ) ) &&
+
+				// advance to the next closing parenthesis
+				( excess = unquoted.indexOf( ")", unquoted.length - excess ) - unquoted.length ) ) {
+
+				// excess is a negative index
+				match[ 0 ] = match[ 0 ].slice( 0, excess );
+				match[ 2 ] = unquoted.slice( 0, excess );
+			}
+
+			// Return only captures needed by the pseudo filter method (type and argument)
+			return match.slice( 0, 3 );
+		}
+	},
+
+	filter: {
+
+		"TAG": function( nodeNameSelector ) {
+			var nodeName = nodeNameSelector.replace( runescape, funescape ).toLowerCase();
+			return nodeNameSelector === "*" ?
+				function() {
+					return true;
+				} :
+				function( elem ) {
+					return elem.nodeName && elem.nodeName.toLowerCase() === nodeName;
+				};
+		},
+
+		"CLASS": function( className ) {
+			var pattern = classCache[ className + " " ];
+
+			return pattern ||
+				( pattern = new RegExp( "(^|" + whitespace +
+					")" + className + "(" + whitespace + "|$)" ) ) && classCache(
+						className, function( elem ) {
+							return pattern.test(
+								typeof elem.className === "string" && elem.className ||
+								typeof elem.getAttribute !== "undefined" &&
+									elem.getAttribute( "class" ) ||
+								""
+							);
+				} );
+		},
+
+		"ATTR": function( name, operator, check ) {
+			return function( elem ) {
+				var result = Sizzle.attr( elem, name );
+
+				if ( result == null ) {
+					return operator === "!=";
+				}
+				if ( !operator ) {
+					return true;
+				}
+
+				result += "";
+
+				/* eslint-disable max-len */
+
+				return operator === "=" ? result === check :
+					operator === "!=" ? result !== check :
+					operator === "^=" ? check && result.indexOf( check ) === 0 :
+					operator === "*=" ? check && result.indexOf( check ) > -1 :
+					operator === "$=" ? check && result.slice( -check.length ) === check :
+					operator === "~=" ? ( " " + result.replace( rwhitespace, " " ) + " " ).indexOf( check ) > -1 :
+					operator === "|=" ? result === check || result.slice( 0, check.length + 1 ) === check + "-" :
+					false;
+				/* eslint-enable max-len */
+
+			};
+		},
+
+		"CHILD": function( type, what, _argument, first, last ) {
+			var simple = type.slice( 0, 3 ) !== "nth",
+				forward = type.slice( -4 ) !== "last",
+				ofType = what === "of-type";
+
+			return first === 1 && last === 0 ?
+
+				// Shortcut for :nth-*(n)
+				function( elem ) {
+					return !!elem.parentNode;
+				} :
+
+				function( elem, _context, xml ) {
+					var cache, uniqueCache, outerCache, node, nodeIndex, start,
+						dir = simple !== forward ? "nextSibling" : "previousSibling",
+						parent = elem.parentNode,
+						name = ofType && elem.nodeName.toLowerCase(),
+						useCache = !xml && !ofType,
+						diff = false;
+
+					if ( parent ) {
+
+						// :(first|last|only)-(child|of-type)
+						if ( simple ) {
+							while ( dir ) {
+								node = elem;
+								while ( ( node = node[ dir ] ) ) {
+									if ( ofType ?
+										node.nodeName.toLowerCase() === name :
+										node.nodeType === 1 ) {
+
+										return false;
+									}
+								}
+
+								// Reverse direction for :only-* (if we haven't yet done so)
+								start = dir = type === "only" && !start && "nextSibling";
+							}
+							return true;
+						}
+
+						start = [ forward ? parent.firstChild : parent.lastChild ];
+
+						// non-xml :nth-child(...) stores cache data on `parent`
+						if ( forward && useCache ) {
+
+							// Seek `elem` from a previously-cached index
+
+							// ...in a gzip-friendly way
+							node = parent;
+							outerCache = node[ expando ] || ( node[ expando ] = {} );
+
+							// Support: IE <9 only
+							// Defend against cloned attroperties (jQuery gh-1709)
+							uniqueCache = outerCache[ node.uniqueID ] ||
+								( outerCache[ node.uniqueID ] = {} );
+
+							cache = uniqueCache[ type ] || [];
+							nodeIndex = cache[ 0 ] === dirruns && cache[ 1 ];
+							diff = nodeIndex && cache[ 2 ];
+							node = nodeIndex && parent.childNodes[ nodeIndex ];
+
+							while ( ( node = ++nodeIndex && node && node[ dir ] ||
+
+								// Fallback to seeking `elem` from the start
+								( diff = nodeIndex = 0 ) || start.pop() ) ) {
+
+								// When found, cache indexes on `parent` and break
+								if ( node.nodeType === 1 && ++diff && node === elem ) {
+									uniqueCache[ type ] = [ dirruns, nodeIndex, diff ];
+									break;
+								}
+							}
+
+						} else {
+
+							// Use previously-cached element index if available
+							if ( useCache ) {
+
+								// ...in a gzip-friendly way
+								node = elem;
+								outerCache = node[ expando ] || ( node[ expando ] = {} );
+
+								// Support: IE <9 only
+								// Defend against cloned attroperties (jQuery gh-1709)
+								uniqueCache = outerCache[ node.uniqueID ] ||
+									( outerCache[ node.uniqueID ] = {} );
+
+								cache = uniqueCache[ type ] || [];
+								nodeIndex = cache[ 0 ] === dirruns && cache[ 1 ];
+								diff = nodeIndex;
+							}
+
+							// xml :nth-child(...)
+							// or :nth-last-child(...) or :nth(-last)?-of-type(...)
+							if ( diff === false ) {
+
+								// Use the same loop as above to seek `elem` from the start
+								while ( ( node = ++nodeIndex && node && node[ dir ] ||
+									( diff = nodeIndex = 0 ) || start.pop() ) ) {
+
+									if ( ( ofType ?
+										node.nodeName.toLowerCase() === name :
+										node.nodeType === 1 ) &&
+										++diff ) {
+
+										// Cache the index of each encountered element
+										if ( useCache ) {
+											outerCache = node[ expando ] ||
+												( node[ expando ] = {} );
+
+											// Support: IE <9 only
+											// Defend against cloned attroperties (jQuery gh-1709)
+											uniqueCache = outerCache[ node.uniqueID ] ||
+												( outerCache[ node.uniqueID ] = {} );
+
+											uniqueCache[ type ] = [ dirruns, diff ];
+										}
+
+										if ( node === elem ) {
+											break;
+										}
+									}
+								}
+							}
+						}
+
+						// Incorporate the offset, then check against cycle size
+						diff -= last;
+						return diff === first || ( diff % first === 0 && diff / first >= 0 );
+					}
+				};
+		},
+
+		"PSEUDO": function( pseudo, argument ) {
+
+			// pseudo-class names are case-insensitive
+			// http://www.w3.org/TR/selectors/#pseudo-classes
+			// Prioritize by case sensitivity in case custom pseudos are added with uppercase letters
+			// Remember that setFilters inherits from pseudos
+			var args,
+				fn = Expr.pseudos[ pseudo ] || Expr.setFilters[ pseudo.toLowerCase() ] ||
+					Sizzle.error( "unsupported pseudo: " + pseudo );
+
+			// The user may use createPseudo to indicate that
+			// arguments are needed to create the filter function
+			// just as Sizzle does
+			if ( fn[ expando ] ) {
+				return fn( argument );
+			}
+
+			// But maintain support for old signatures
+			if ( fn.length > 1 ) {
+				args = [ pseudo, pseudo, "", argument ];
+				return Expr.setFilters.hasOwnProperty( pseudo.toLowerCase() ) ?
+					markFunction( function( seed, matches ) {
+						var idx,
+							matched = fn( seed, argument ),
+							i = matched.length;
+						while ( i-- ) {
+							idx = indexOf( seed, matched[ i ] );
+							seed[ idx ] = !( matches[ idx ] = matched[ i ] );
+						}
+					} ) :
+					function( elem ) {
+						return fn( elem, 0, args );
+					};
+			}
+
+			return fn;
+		}
+	},
+
+	pseudos: {
+
+		// Potentially complex pseudos
+		"not": markFunction( function( selector ) {
+
+			// Trim the selector passed to compile
+			// to avoid treating leading and trailing
+			// spaces as combinators
+			var input = [],
+				results = [],
+				matcher = compile( selector.replace( rtrim, "$1" ) );
+
+			return matcher[ expando ] ?
+				markFunction( function( seed, matches, _context, xml ) {
+					var elem,
+						unmatched = matcher( seed, null, xml, [] ),
+						i = seed.length;
+
+					// Match elements unmatched by `matcher`
+					while ( i-- ) {
+						if ( ( elem = unmatched[ i ] ) ) {
+							seed[ i ] = !( matches[ i ] = elem );
+						}
+					}
+				} ) :
+				function( elem, _context, xml ) {
+					input[ 0 ] = elem;
+					matcher( input, null, xml, results );
+
+					// Don't keep the element (issue #299)
+					input[ 0 ] = null;
+					return !results.pop();
+				};
+		} ),
+
+		"has": markFunction( function( selector ) {
+			return function( elem ) {
+				return Sizzle( selector, elem ).length > 0;
+			};
+		} ),
+
+		"contains": markFunction( function( text ) {
+			text = text.replace( runescape, funescape );
+			return function( elem ) {
+				return ( elem.textContent || getText( elem ) ).indexOf( text ) > -1;
+			};
+		} ),
+
+		// "Whether an element is represented by a :lang() selector
+		// is based solely on the element's language value
+		// being equal to the identifier C,
+		// or beginning with the identifier C immediately followed by "-".
+		// The matching of C against the element's language value is performed case-insensitively.
+		// The identifier C does not have to be a valid language name."
+		// http://www.w3.org/TR/selectors/#lang-pseudo
+		"lang": markFunction( function( lang ) {
+
+			// lang value must be a valid identifier
+			if ( !ridentifier.test( lang || "" ) ) {
+				Sizzle.error( "unsupported lang: " + lang );
+			}
+			lang = lang.replace( runescape, funescape ).toLowerCase();
+			return function( elem ) {
+				var elemLang;
+				do {
+					if ( ( elemLang = documentIsHTML ?
+						elem.lang :
+						elem.getAttribute( "xml:lang" ) || elem.getAttribute( "lang" ) ) ) {
+
+						elemLang = elemLang.toLowerCase();
+						return elemLang === lang || elemLang.indexOf( lang + "-" ) === 0;
+					}
+				} while ( ( elem = elem.parentNode ) && elem.nodeType === 1 );
+				return false;
+			};
+		} ),
+
+		// Miscellaneous
+		"target": function( elem ) {
+			var hash = window.location && window.location.hash;
+			return hash && hash.slice( 1 ) === elem.id;
+		},
+
+		"root": function( elem ) {
+			return elem === docElem;
+		},
+
+		"focus": function( elem ) {
+			return elem === document.activeElement &&
+				( !document.hasFocus || document.hasFocus() ) &&
+				!!( elem.type || elem.href || ~elem.tabIndex );
+		},
+
+		// Boolean properties
+		"enabled": createDisabledPseudo( false ),
+		"disabled": createDisabledPseudo( true ),
+
+		"checked": function( elem ) {
+
+			// In CSS3, :checked should return both checked and selected elements
+			// http://www.w3.org/TR/2011/REC-css3-selectors-20110929/#checked
+			var nodeName = elem.nodeName.toLowerCase();
+			return ( nodeName === "input" && !!elem.checked ) ||
+				( nodeName === "option" && !!elem.selected );
+		},
+
+		"selected": function( elem ) {
+
+			// Accessing this property makes selected-by-default
+			// options in Safari work properly
+			if ( elem.parentNode ) {
+				// eslint-disable-next-line no-unused-expressions
+				elem.parentNode.selectedIndex;
+			}
+
+			return elem.selected === true;
+		},
+
+		// Contents
+		"empty": function( elem ) {
+
+			// http://www.w3.org/TR/selectors/#empty-pseudo
+			// :empty is negated by element (1) or content nodes (text: 3; cdata: 4; entity ref: 5),
+			//   but not by others (comment: 8; processing instruction: 7; etc.)
+			// nodeType < 6 works because attributes (2) do not appear as children
+			for ( elem = elem.firstChild; elem; elem = elem.nextSibling ) {
+				if ( elem.nodeType < 6 ) {
+					return false;
+				}
+			}
+			return true;
+		},
+
+		"parent": function( elem ) {
+			return !Expr.pseudos[ "empty" ]( elem );
+		},
+
+		// Element/input types
+		"header": function( elem ) {
+			return rheader.test( elem.nodeName );
+		},
+
+		"input": function( elem ) {
+			return rinputs.test( elem.nodeName );
+		},
+
+		"button": function( elem ) {
+			var name = elem.nodeName.toLowerCase();
+			return name === "input" && elem.type === "button" || name === "button";
+		},
+
+		"text": function( elem ) {
+			var attr;
+			return elem.nodeName.toLowerCase() === "input" &&
+				elem.type === "text" &&
+
+				// Support: IE<8
+				// New HTML5 attribute values (e.g., "search") appear with elem.type === "text"
+				( ( attr = elem.getAttribute( "type" ) ) == null ||
+					attr.toLowerCase() === "text" );
+		},
+
+		// Position-in-collection
+		"first": createPositionalPseudo( function() {
+			return [ 0 ];
+		} ),
+
+		"last": createPositionalPseudo( function( _matchIndexes, length ) {
+			return [ length - 1 ];
+		} ),
+
+		"eq": createPositionalPseudo( function( _matchIndexes, length, argument ) {
+			return [ argument < 0 ? argument + length : argument ];
+		} ),
+
+		"even": createPositionalPseudo( function( matchIndexes, length ) {
+			var i = 0;
+			for ( ; i < length; i += 2 ) {
+				matchIndexes.push( i );
+			}
+			return matchIndexes;
+		} ),
+
+		"odd": createPositionalPseudo( function( matchIndexes, length ) {
+			var i = 1;
+			for ( ; i < length; i += 2 ) {
+				matchIndexes.push( i );
+			}
+			return matchIndexes;
+		} ),
+
+		"lt": createPositionalPseudo( function( matchIndexes, length, argument ) {
+			var i = argument < 0 ?
+				argument + length :
+				argument > length ?
+					length :
+					argument;
+			for ( ; --i >= 0; ) {
+				matchIndexes.push( i );
+			}
+			return matchIndexes;
+		} ),
+
+		"gt": createPositionalPseudo( function( matchIndexes, length, argument ) {
+			var i = argument < 0 ? argument + length : argument;
+			for ( ; ++i < length; ) {
+				matchIndexes.push( i );
+			}
+			return matchIndexes;
+		} )
+	}
+};
+
+Expr.pseudos[ "nth" ] = Expr.pseudos[ "eq" ];
+
+// Add button/input type pseudos
+for ( i in { radio: true, checkbox: true, file: true, password: true, image: true } ) {
+	Expr.pseudos[ i ] = createInputPseudo( i );
+}
+for ( i in { submit: true, reset: true } ) {
+	Expr.pseudos[ i ] = createButtonPseudo( i );
+}
+
+// Easy API for creating new setFilters
+function setFilters() {}
+setFilters.prototype = Expr.filters = Expr.pseudos;
+Expr.setFilters = new setFilters();
+
+tokenize = Sizzle.tokenize = function( selector, parseOnly ) {
+	var matched, match, tokens, type,
+		soFar, groups, preFilters,
+		cached = tokenCache[ selector + " " ];
+
+	if ( cached ) {
+		return parseOnly ? 0 : cached.slice( 0 );
+	}
+
+	soFar = selector;
+	groups = [];
+	preFilters = Expr.preFilter;
+
+	while ( soFar ) {
+
+		// Comma and first run
+		if ( !matched || ( match = rcomma.exec( soFar ) ) ) {
+			if ( match ) {
+
+				// Don't consume trailing commas as valid
+				soFar = soFar.slice( match[ 0 ].length ) || soFar;
+			}
+			groups.push( ( tokens = [] ) );
+		}
+
+		matched = false;
+
+		// Combinators
+		if ( ( match = rcombinators.exec( soFar ) ) ) {
+			matched = match.shift();
+			tokens.push( {
+				value: matched,
+
+				// Cast descendant combinators to space
+				type: match[ 0 ].replace( rtrim, " " )
+			} );
+			soFar = soFar.slice( matched.length );
+		}
+
+		// Filters
+		for ( type in Expr.filter ) {
+			if ( ( match = matchExpr[ type ].exec( soFar ) ) && ( !preFilters[ type ] ||
+				( match = preFilters[ type ]( match ) ) ) ) {
+				matched = match.shift();
+				tokens.push( {
+					value: matched,
+					type: type,
+					matches: match
+				} );
+				soFar = soFar.slice( matched.length );
+			}
+		}
+
+		if ( !matched ) {
+			break;
+		}
+	}
+
+	// Return the length of the invalid excess
+	// if we're just parsing
+	// Otherwise, throw an error or return tokens
+	return parseOnly ?
+		soFar.length :
+		soFar ?
+			Sizzle.error( selector ) :
+
+			// Cache the tokens
+			tokenCache( selector, groups ).slice( 0 );
+};
+
+function toSelector( tokens ) {
+	var i = 0,
+		len = tokens.length,
+		selector = "";
+	for ( ; i < len; i++ ) {
+		selector += tokens[ i ].value;
+	}
+	return selector;
+}
+
+function addCombinator( matcher, combinator, base ) {
+	var dir = combinator.dir,
+		skip = combinator.next,
+		key = skip || dir,
+		checkNonElements = base && key === "parentNode",
+		doneName = done++;
+
+	return combinator.first ?
+
+		// Check against closest ancestor/preceding element
+		function( elem, context, xml ) {
+			while ( ( elem = elem[ dir ] ) ) {
+				if ( elem.nodeType === 1 || checkNonElements ) {
+					return matcher( elem, context, xml );
+				}
+			}
+			return false;
+		} :
+
+		// Check against all ancestor/preceding elements
+		function( elem, context, xml ) {
+			var oldCache, uniqueCache, outerCache,
+				newCache = [ dirruns, doneName ];
+
+			// We can't set arbitrary data on XML nodes, so they don't benefit from combinator caching
+			if ( xml ) {
+				while ( ( elem = elem[ dir ] ) ) {
+					if ( elem.nodeType === 1 || checkNonElements ) {
+						if ( matcher( elem, context, xml ) ) {
+							return true;
+						}
+					}
+				}
+			} else {
+				while ( ( elem = elem[ dir ] ) ) {
+					if ( elem.nodeType === 1 || checkNonElements ) {
+						outerCache = elem[ expando ] || ( elem[ expando ] = {} );
+
+						// Support: IE <9 only
+						// Defend against cloned attroperties (jQuery gh-1709)
+						uniqueCache = outerCache[ elem.uniqueID ] ||
+							( outerCache[ elem.uniqueID ] = {} );
+
+						if ( skip && skip === elem.nodeName.toLowerCase() ) {
+							elem = elem[ dir ] || elem;
+						} else if ( ( oldCache = uniqueCache[ key ] ) &&
+							oldCache[ 0 ] === dirruns && oldCache[ 1 ] === doneName ) {
+
+							// Assign to newCache so results back-propagate to previous elements
+							return ( newCache[ 2 ] = oldCache[ 2 ] );
+						} else {
+
+							// Reuse newcache so results back-propagate to previous elements
+							uniqueCache[ key ] = newCache;
+
+							// A match means we're done; a fail means we have to keep checking
+							if ( ( newCache[ 2 ] = matcher( elem, context, xml ) ) ) {
+								return true;
+							}
+						}
+					}
+				}
+			}
+			return false;
+		};
+}
+
+function elementMatcher( matchers ) {
+	return matchers.length > 1 ?
+		function( elem, context, xml ) {
+			var i = matchers.length;
+			while ( i-- ) {
+				if ( !matchers[ i ]( elem, context, xml ) ) {
+					return false;
+				}
+			}
+			return true;
+		} :
+		matchers[ 0 ];
+}
+
+function multipleContexts( selector, contexts, results ) {
+	var i = 0,
+		len = contexts.length;
+	for ( ; i < len; i++ ) {
+		Sizzle( selector, contexts[ i ], results );
+	}
+	return results;
+}
+
+function condense( unmatched, map, filter, context, xml ) {
+	var elem,
+		newUnmatched = [],
+		i = 0,
+		len = unmatched.length,
+		mapped = map != null;
+
+	for ( ; i < len; i++ ) {
+		if ( ( elem = unmatched[ i ] ) ) {
+			if ( !filter || filter( elem, context, xml ) ) {
+				newUnmatched.push( elem );
+				if ( mapped ) {
+					map.push( i );
+				}
+			}
+		}
+	}
+
+	return newUnmatched;
+}
+
+function setMatcher( preFilter, selector, matcher, postFilter, postFinder, postSelector ) {
+	if ( postFilter && !postFilter[ expando ] ) {
+		postFilter = setMatcher( postFilter );
+	}
+	if ( postFinder && !postFinder[ expando ] ) {
+		postFinder = setMatcher( postFinder, postSelector );
+	}
+	return markFunction( function( seed, results, context, xml ) {
+		var temp, i, elem,
+			preMap = [],
+			postMap = [],
+			preexisting = results.length,
+
+			// Get initial elements from seed or context
+			elems = seed || multipleContexts(
+				selector || "*",
+				context.nodeType ? [ context ] : context,
+				[]
+			),
+
+			// Prefilter to get matcher input, preserving a map for seed-results synchronization
+			matcherIn = preFilter && ( seed || !selector ) ?
+				condense( elems, preMap, preFilter, context, xml ) :
+				elems,
+
+			matcherOut = matcher ?
+
+				// If we have a postFinder, or filtered seed, or non-seed postFilter or preexisting results,
+				postFinder || ( seed ? preFilter : preexisting || postFilter ) ?
+
+					// ...intermediate processing is necessary
+					[] :
+
+					// ...otherwise use results directly
+					results :
+				matcherIn;
+
+		// Find primary matches
+		if ( matcher ) {
+			matcher( matcherIn, matcherOut, context, xml );
+		}
+
+		// Apply postFilter
+		if ( postFilter ) {
+			temp = condense( matcherOut, postMap );
+			postFilter( temp, [], context, xml );
+
+			// Un-match failing elements by moving them back to matcherIn
+			i = temp.length;
+			while ( i-- ) {
+				if ( ( elem = temp[ i ] ) ) {
+					matcherOut[ postMap[ i ] ] = !( matcherIn[ postMap[ i ] ] = elem );
+				}
+			}
+		}
+
+		if ( seed ) {
+			if ( postFinder || preFilter ) {
+				if ( postFinder ) {
+
+					// Get the final matcherOut by condensing this intermediate into postFinder contexts
+					temp = [];
+					i = matcherOut.length;
+					while ( i-- ) {
+						if ( ( elem = matcherOut[ i ] ) ) {
+
+							// Restore matcherIn since elem is not yet a final match
+							temp.push( ( matcherIn[ i ] = elem ) );
+						}
+					}
+					postFinder( null, ( matcherOut = [] ), temp, xml );
+				}
+
+				// Move matched elements from seed to results to keep them synchronized
+				i = matcherOut.length;
+				while ( i-- ) {
+					if ( ( elem = matcherOut[ i ] ) &&
+						( temp = postFinder ? indexOf( seed, elem ) : preMap[ i ] ) > -1 ) {
+
+						seed[ temp ] = !( results[ temp ] = elem );
+					}
+				}
+			}
+
+		// Add elements to results, through postFinder if defined
+		} else {
+			matcherOut = condense(
+				matcherOut === results ?
+					matcherOut.splice( preexisting, matcherOut.length ) :
+					matcherOut
+			);
+			if ( postFinder ) {
+				postFinder( null, results, matcherOut, xml );
+			} else {
+				push.apply( results, matcherOut );
+			}
+		}
+	} );
+}
+
+function matcherFromTokens( tokens ) {
+	var checkContext, matcher, j,
+		len = tokens.length,
+		leadingRelative = Expr.relative[ tokens[ 0 ].type ],
+		implicitRelative = leadingRelative || Expr.relative[ " " ],
+		i = leadingRelative ? 1 : 0,
+
+		// The foundational matcher ensures that elements are reachable from top-level context(s)
+		matchContext = addCombinator( function( elem ) {
+			return elem === checkContext;
+		}, implicitRelative, true ),
+		matchAnyContext = addCombinator( function( elem ) {
+			return indexOf( checkContext, elem ) > -1;
+		}, implicitRelative, true ),
+		matchers = [ function( elem, context, xml ) {
+			var ret = ( !leadingRelative && ( xml || context !== outermostContext ) ) || (
+				( checkContext = context ).nodeType ?
+					matchContext( elem, context, xml ) :
+					matchAnyContext( elem, context, xml ) );
+
+			// Avoid hanging onto element (issue #299)
+			checkContext = null;
+			return ret;
+		} ];
+
+	for ( ; i < len; i++ ) {
+		if ( ( matcher = Expr.relative[ tokens[ i ].type ] ) ) {
+			matchers = [ addCombinator( elementMatcher( matchers ), matcher ) ];
+		} else {
+			matcher = Expr.filter[ tokens[ i ].type ].apply( null, tokens[ i ].matches );
+
+			// Return special upon seeing a positional matcher
+			if ( matcher[ expando ] ) {
+
+				// Find the next relative operator (if any) for proper handling
+				j = ++i;
+				for ( ; j < len; j++ ) {
+					if ( Expr.relative[ tokens[ j ].type ] ) {
+						break;
+					}
+				}
+				return setMatcher(
+					i > 1 && elementMatcher( matchers ),
+					i > 1 && toSelector(
+
+					// If the preceding token was a descendant combinator, insert an implicit any-element `*`
+					tokens
+						.slice( 0, i - 1 )
+						.concat( { value: tokens[ i - 2 ].type === " " ? "*" : "" } )
+					).replace( rtrim, "$1" ),
+					matcher,
+					i < j && matcherFromTokens( tokens.slice( i, j ) ),
+					j < len && matcherFromTokens( ( tokens = tokens.slice( j ) ) ),
+					j < len && toSelector( tokens )
+				);
+			}
+			matchers.push( matcher );
+		}
+	}
+
+	return elementMatcher( matchers );
+}
+
+function matcherFromGroupMatchers( elementMatchers, setMatchers ) {
+	var bySet = setMatchers.length > 0,
+		byElement = elementMatchers.length > 0,
+		superMatcher = function( seed, context, xml, results, outermost ) {
+			var elem, j, matcher,
+				matchedCount = 0,
+				i = "0",
+				unmatched = seed && [],
+				setMatched = [],
+				contextBackup = outermostContext,
+
+				// We must always have either seed elements or outermost context
+				elems = seed || byElement && Expr.find[ "TAG" ]( "*", outermost ),
+
+				// Use integer dirruns iff this is the outermost matcher
+				dirrunsUnique = ( dirruns += contextBackup == null ? 1 : Math.random() || 0.1 ),
+				len = elems.length;
+
+			if ( outermost ) {
+
+				// Support: IE 11+, Edge 17 - 18+
+				// IE/Edge sometimes throw a "Permission denied" error when strict-comparing
+				// two documents; shallow comparisons work.
+				// eslint-disable-next-line eqeqeq
+				outermostContext = context == document || context || outermost;
+			}
+
+			// Add elements passing elementMatchers directly to results
+			// Support: IE<9, Safari
+			// Tolerate NodeList properties (IE: "length"; Safari: <number>) matching elements by id
+			for ( ; i !== len && ( elem = elems[ i ] ) != null; i++ ) {
+				if ( byElement && elem ) {
+					j = 0;
+
+					// Support: IE 11+, Edge 17 - 18+
+					// IE/Edge sometimes throw a "Permission denied" error when strict-comparing
+					// two documents; shallow comparisons work.
+					// eslint-disable-next-line eqeqeq
+					if ( !context && elem.ownerDocument != document ) {
+						setDocument( elem );
+						xml = !documentIsHTML;
+					}
+					while ( ( matcher = elementMatchers[ j++ ] ) ) {
+						if ( matcher( elem, context || document, xml ) ) {
+							results.push( elem );
+							break;
+						}
+					}
+					if ( outermost ) {
+						dirruns = dirrunsUnique;
+					}
+				}
+
+				// Track unmatched elements for set filters
+				if ( bySet ) {
+
+					// They will have gone through all possible matchers
+					if ( ( elem = !matcher && elem ) ) {
+						matchedCount--;
+					}
+
+					// Lengthen the array for every element, matched or not
+					if ( seed ) {
+						unmatched.push( elem );
+					}
+				}
+			}
+
+			// `i` is now the count of elements visited above, and adding it to `matchedCount`
+			// makes the latter nonnegative.
+			matchedCount += i;
+
+			// Apply set filters to unmatched elements
+			// NOTE: This can be skipped if there are no unmatched elements (i.e., `matchedCount`
+			// equals `i`), unless we didn't visit _any_ elements in the above loop because we have
+			// no element matchers and no seed.
+			// Incrementing an initially-string "0" `i` allows `i` to remain a string only in that
+			// case, which will result in a "00" `matchedCount` that differs from `i` but is also
+			// numerically zero.
+			if ( bySet && i !== matchedCount ) {
+				j = 0;
+				while ( ( matcher = setMatchers[ j++ ] ) ) {
+					matcher( unmatched, setMatched, context, xml );
+				}
+
+				if ( seed ) {
+
+					// Reintegrate element matches to eliminate the need for sorting
+					if ( matchedCount > 0 ) {
+						while ( i-- ) {
+							if ( !( unmatched[ i ] || setMatched[ i ] ) ) {
+								setMatched[ i ] = pop.call( results );
+							}
+						}
+					}
+
+					// Discard index placeholder values to get only actual matches
+					setMatched = condense( setMatched );
+				}
+
+				// Add matches to results
+				push.apply( results, setMatched );
+
+				// Seedless set matches succeeding multiple successful matchers stipulate sorting
+				if ( outermost && !seed && setMatched.length > 0 &&
+					( matchedCount + setMatchers.length ) > 1 ) {
+
+					Sizzle.uniqueSort( results );
+				}
+			}
+
+			// Override manipulation of globals by nested matchers
+			if ( outermost ) {
+				dirruns = dirrunsUnique;
+				outermostContext = contextBackup;
+			}
+
+			return unmatched;
+		};
+
+	return bySet ?
+		markFunction( superMatcher ) :
+		superMatcher;
+}
+
+compile = Sizzle.compile = function( selector, match /* Internal Use Only */ ) {
+	var i,
+		setMatchers = [],
+		elementMatchers = [],
+		cached = compilerCache[ selector + " " ];
+
+	if ( !cached ) {
+
+		// Generate a function of recursive functions that can be used to check each element
+		if ( !match ) {
+			match = tokenize( selector );
+		}
+		i = match.length;
+		while ( i-- ) {
+			cached = matcherFromTokens( match[ i ] );
+			if ( cached[ expando ] ) {
+				setMatchers.push( cached );
+			} else {
+				elementMatchers.push( cached );
+			}
+		}
+
+		// Cache the compiled function
+		cached = compilerCache(
+			selector,
+			matcherFromGroupMatchers( elementMatchers, setMatchers )
+		);
+
+		// Save selector and tokenization
+		cached.selector = selector;
+	}
+	return cached;
+};
+
+/**
+ * A low-level selection function that works with Sizzle's compiled
+ *  selector functions
+ * @param {String|Function} selector A selector or a pre-compiled
+ *  selector function built with Sizzle.compile
+ * @param {Element} context
+ * @param {Array} [results]
+ * @param {Array} [seed] A set of elements to match against
+ */
+select = Sizzle.select = function( selector, context, results, seed ) {
+	var i, tokens, token, type, find,
+		compiled = typeof selector === "function" && selector,
+		match = !seed && tokenize( ( selector = compiled.selector || selector ) );
+
+	results = results || [];
+
+	// Try to minimize operations if there is only one selector in the list and no seed
+	// (the latter of which guarantees us context)
+	if ( match.length === 1 ) {
+
+		// Reduce context if the leading compound selector is an ID
+		tokens = match[ 0 ] = match[ 0 ].slice( 0 );
+		if ( tokens.length > 2 && ( token = tokens[ 0 ] ).type === "ID" &&
+			context.nodeType === 9 && documentIsHTML && Expr.relative[ tokens[ 1 ].type ] ) {
+
+			context = ( Expr.find[ "ID" ]( token.matches[ 0 ]
+				.replace( runescape, funescape ), context ) || [] )[ 0 ];
+			if ( !context ) {
+				return results;
+
+			// Precompiled matchers will still verify ancestry, so step up a level
+			} else if ( compiled ) {
+				context = context.parentNode;
+			}
+
+			selector = selector.slice( tokens.shift().value.length );
+		}
+
+		// Fetch a seed set for right-to-left matching
+		i = matchExpr[ "needsContext" ].test( selector ) ? 0 : tokens.length;
+		while ( i-- ) {
+			token = tokens[ i ];
+
+			// Abort if we hit a combinator
+			if ( Expr.relative[ ( type = token.type ) ] ) {
+				break;
+			}
+			if ( ( find = Expr.find[ type ] ) ) {
+
+				// Search, expanding context for leading sibling combinators
+				if ( ( seed = find(
+					token.matches[ 0 ].replace( runescape, funescape ),
+					rsibling.test( tokens[ 0 ].type ) && testContext( context.parentNode ) ||
+						context
+				) ) ) {
+
+					// If seed is empty or no tokens remain, we can return early
+					tokens.splice( i, 1 );
+					selector = seed.length && toSelector( tokens );
+					if ( !selector ) {
+						push.apply( results, seed );
+						return results;
+					}
+
+					break;
+				}
+			}
+		}
+	}
+
+	// Compile and execute a filtering function if one is not provided
+	// Provide `match` to avoid retokenization if we modified the selector above
+	( compiled || compile( selector, match ) )(
+		seed,
+		context,
+		!documentIsHTML,
+		results,
+		!context || rsibling.test( selector ) && testContext( context.parentNode ) || context
+	);
+	return results;
+};
+
+// One-time assignments
+
+// Sort stability
+support.sortStable = expando.split( "" ).sort( sortOrder ).join( "" ) === expando;
+
+// Support: Chrome 14-35+
+// Always assume duplicates if they aren't passed to the comparison function
+support.detectDuplicates = !!hasDuplicate;
+
+// Initialize against the default document
+setDocument();
+
+// Support: Webkit<537.32 - Safari 6.0.3/Chrome 25 (fixed in Chrome 27)
+// Detached nodes confoundingly follow *each other*
+support.sortDetached = assert( function( el ) {
+
+	// Should return 1, but returns 4 (following)
+	return el.compareDocumentPosition( document.createElement( "fieldset" ) ) & 1;
+} );
+
+// Support: IE<8
+// Prevent attribute/property "interpolation"
+// https://msdn.microsoft.com/en-us/library/ms536429%28VS.85%29.aspx
+if ( !assert( function( el ) {
+	el.innerHTML = "<a href='#'></a>";
+	return el.firstChild.getAttribute( "href" ) === "#";
+} ) ) {
+	addHandle( "type|href|height|width", function( elem, name, isXML ) {
+		if ( !isXML ) {
+			return elem.getAttribute( name, name.toLowerCase() === "type" ? 1 : 2 );
+		}
+	} );
+}
+
+// Support: IE<9
+// Use defaultValue in place of getAttribute("value")
+if ( !support.attributes || !assert( function( el ) {
+	el.innerHTML = "<input/>";
+	el.firstChild.setAttribute( "value", "" );
+	return el.firstChild.getAttribute( "value" ) === "";
+} ) ) {
+	addHandle( "value", function( elem, _name, isXML ) {
+		if ( !isXML && elem.nodeName.toLowerCase() === "input" ) {
+			return elem.defaultValue;
+		}
+	} );
+}
+
+// Support: IE<9
+// Use getAttributeNode to fetch booleans when getAttribute lies
+if ( !assert( function( el ) {
+	return el.getAttribute( "disabled" ) == null;
+} ) ) {
+	addHandle( booleans, function( elem, name, isXML ) {
+		var val;
+		if ( !isXML ) {
+			return elem[ name ] === true ? name.toLowerCase() :
+				( val = elem.getAttributeNode( name ) ) && val.specified ?
+					val.value :
+					null;
+		}
+	} );
+}
+
+return Sizzle;
+
+} )( window );
+
+
+
+jQuery.find = Sizzle;
+jQuery.expr = Sizzle.selectors;
+
+// Deprecated
+jQuery.expr[ ":" ] = jQuery.expr.pseudos;
+jQuery.uniqueSort = jQuery.unique = Sizzle.uniqueSort;
+jQuery.text = Sizzle.getText;
+jQuery.isXMLDoc = Sizzle.isXML;
+jQuery.contains = Sizzle.contains;
+jQuery.escapeSelector = Sizzle.escape;
+
+
+
+
+var dir = function( elem, dir, until ) {
+	var matched = [],
+		truncate = until !== undefined;
+
+	while ( ( elem = elem[ dir ] ) && elem.nodeType !== 9 ) {
+		if ( elem.nodeType === 1 ) {
+			if ( truncate && jQuery( elem ).is( until ) ) {
+				break;
+			}
+			matched.push( elem );
+		}
+	}
+	return matched;
+};
+
+
+var siblings = function( n, elem ) {
+	var matched = [];
+
+	for ( ; n; n = n.nextSibling ) {
+		if ( n.nodeType === 1 && n !== elem ) {
+			matched.push( n );
+		}
+	}
+
+	return matched;
+};
+
+
+var rneedsContext = jQuery.expr.match.needsContext;
+
+
+
+function nodeName( elem, name ) {
+
+	return elem.nodeName && elem.nodeName.toLowerCase() === name.toLowerCase();
+
+}
+var rsingleTag = ( /^<([a-z][^\/\0>:\x20\t\r\n\f]*)[\x20\t\r\n\f]*\/?>(?:<\/\1>|)$/i );
+
+
+
+// Implement the identical functionality for filter and not
+function winnow( elements, qualifier, not ) {
+	if ( isFunction( qualifier ) ) {
+		return jQuery.grep( elements, function( elem, i ) {
+			return !!qualifier.call( elem, i, elem ) !== not;
+		} );
+	}
+
+	// Single element
+	if ( qualifier.nodeType ) {
+		return jQuery.grep( elements, function( elem ) {
+			return ( elem === qualifier ) !== not;
+		} );
+	}
+
+	// Arraylike of elements (jQuery, arguments, Array)
+	if ( typeof qualifier !== "string" ) {
+		return jQuery.grep( elements, function( elem ) {
+			return ( indexOf.call( qualifier, elem ) > -1 ) !== not;
+		} );
+	}
+
+	// Filtered directly for both simple and complex selectors
+	return jQuery.filter( qualifier, elements, not );
+}
+
+jQuery.filter = function( expr, elems, not ) {
+	var elem = elems[ 0 ];
+
+	if ( not ) {
+		expr = ":not(" + expr + ")";
+	}
+
+	if ( elems.length === 1 && elem.nodeType === 1 ) {
+		return jQuery.find.matchesSelector( elem, expr ) ? [ elem ] : [];
+	}
+
+	return jQuery.find.matches( expr, jQuery.grep( elems, function( elem ) {
+		return elem.nodeType === 1;
+	} ) );
+};
+
+jQuery.fn.extend( {
+	find: function( selector ) {
+		var i, ret,
+			len = this.length,
+			self = this;
+
+		if ( typeof selector !== "string" ) {
+			return this.pushStack( jQuery( selector ).filter( function() {
+				for ( i = 0; i < len; i++ ) {
+					if ( jQuery.contains( self[ i ], this ) ) {
+						return true;
+					}
+				}
+			} ) );
+		}
+
+		ret = this.pushStack( [] );
+
+		for ( i = 0; i < len; i++ ) {
+			jQuery.find( selector, self[ i ], ret );
+		}
+
+		return len > 1 ? jQuery.uniqueSort( ret ) : ret;
+	},
+	filter: function( selector ) {
+		return this.pushStack( winnow( this, selector || [], false ) );
+	},
+	not: function( selector ) {
+		return this.pushStack( winnow( this, selector || [], true ) );
+	},
+	is: function( selector ) {
+		return !!winnow(
+			this,
+
+			// If this is a positional/relative selector, check membership in the returned set
+			// so $("p:first").is("p:last") won't return true for a doc with two "p".
+			typeof selector === "string" && rneedsContext.test( selector ) ?
+				jQuery( selector ) :
+				selector || [],
+			false
+		).length;
+	}
+} );
+
+
+// Initialize a jQuery object
+
+
+// A central reference to the root jQuery(document)
+var rootjQuery,
+
+	// A simple way to check for HTML strings
+	// Prioritize #id over <tag> to avoid XSS via location.hash (#9521)
+	// Strict HTML recognition (#11290: must start with <)
+	// Shortcut simple #id case for speed
+	rquickExpr = /^(?:\s*(<[\w\W]+>)[^>]*|#([\w-]+))$/,
+
+	init = jQuery.fn.init = function( selector, context, root ) {
+		var match, elem;
+
+		// HANDLE: $(""), $(null), $(undefined), $(false)
+		if ( !selector ) {
+			return this;
+		}
+
+		// Method init() accepts an alternate rootjQuery
+		// so migrate can support jQuery.sub (gh-2101)
+		root = root || rootjQuery;
+
+		// Handle HTML strings
+		if ( typeof selector === "string" ) {
+			if ( selector[ 0 ] === "<" &&
+				selector[ selector.length - 1 ] === ">" &&
+				selector.length >= 3 ) {
+
+				// Assume that strings that start and end with <> are HTML and skip the regex check
+				match = [ null, selector, null ];
+
+			} else {
+				match = rquickExpr.exec( selector );
+			}
+
+			// Match html or make sure no context is specified for #id
+			if ( match && ( match[ 1 ] || !context ) ) {
+
+				// HANDLE: $(html) -> $(array)
+				if ( match[ 1 ] ) {
+					context = context instanceof jQuery ? context[ 0 ] : context;
+
+					// Option to run scripts is true for back-compat
+					// Intentionally let the error be thrown if parseHTML is not present
+					jQuery.merge( this, jQuery.parseHTML(
+						match[ 1 ],
+						context && context.nodeType ? context.ownerDocument || context : document,
+						true
+					) );
+
+					// HANDLE: $(html, props)
+					if ( rsingleTag.test( match[ 1 ] ) && jQuery.isPlainObject( context ) ) {
+						for ( match in context ) {
+
+							// Properties of context are called as methods if possible
+							if ( isFunction( this[ match ] ) ) {
+								this[ match ]( context[ match ] );
+
+							// ...and otherwise set as attributes
+							} else {
+								this.attr( match, context[ match ] );
+							}
+						}
+					}
+
+					return this;
+
+				// HANDLE: $(#id)
+				} else {
+					elem = document.getElementById( match[ 2 ] );
+
+					if ( elem ) {
+
+						// Inject the element directly into the jQuery object
+						this[ 0 ] = elem;
+						this.length = 1;
+					}
+					return this;
+				}
+
+			// HANDLE: $(expr, $(...))
+			} else if ( !context || context.jquery ) {
+				return ( context || root ).find( selector );
+
+			// HANDLE: $(expr, context)
+			// (which is just equivalent to: $(context).find(expr)
+			} else {
+				return this.constructor( context ).find( selector );
+			}
+
+		// HANDLE: $(DOMElement)
+		} else if ( selector.nodeType ) {
+			this[ 0 ] = selector;
+			this.length = 1;
+			return this;
+
+		// HANDLE: $(function)
+		// Shortcut for document ready
+		} else if ( isFunction( selector ) ) {
+			return root.ready !== undefined ?
+				root.ready( selector ) :
+
+				// Execute immediately if ready is not present
+				selector( jQuery );
+		}
+
+		return jQuery.makeArray( selector, this );
+	};
+
+// Give the init function the jQuery prototype for later instantiation
+init.prototype = jQuery.fn;
+
+// Initialize central reference
+rootjQuery = jQuery( document );
+
+
+var rparentsprev = /^(?:parents|prev(?:Until|All))/,
+
+	// Methods guaranteed to produce a unique set when starting from a unique set
+	guaranteedUnique = {
+		children: true,
+		contents: true,
+		next: true,
+		prev: true
+	};
+
+jQuery.fn.extend( {
+	has: function( target ) {
+		var targets = jQuery( target, this ),
+			l = targets.length;
+
+		return this.filter( function() {
+			var i = 0;
+			for ( ; i < l; i++ ) {
+				if ( jQuery.contains( this, targets[ i ] ) ) {
+					return true;
+				}
+			}
+		} );
+	},
+
+	closest: function( selectors, context ) {
+		var cur,
+			i = 0,
+			l = this.length,
+			matched = [],
+			targets = typeof selectors !== "string" && jQuery( selectors );
+
+		// Positional selectors never match, since there's no _selection_ context
+		if ( !rneedsContext.test( selectors ) ) {
+			for ( ; i < l; i++ ) {
+				for ( cur = this[ i ]; cur && cur !== context; cur = cur.parentNode ) {
+
+					// Always skip document fragments
+					if ( cur.nodeType < 11 && ( targets ?
+						targets.index( cur ) > -1 :
+
+						// Don't pass non-elements to Sizzle
+						cur.nodeType === 1 &&
+							jQuery.find.matchesSelector( cur, selectors ) ) ) {
+
+						matched.push( cur );
+						break;
+					}
+				}
+			}
+		}
+
+		return this.pushStack( matched.length > 1 ? jQuery.uniqueSort( matched ) : matched );
+	},
+
+	// Determine the position of an element within the set
+	index: function( elem ) {
+
+		// No argument, return index in parent
+		if ( !elem ) {
+			return ( this[ 0 ] && this[ 0 ].parentNode ) ? this.first().prevAll().length : -1;
+		}
+
+		// Index in selector
+		if ( typeof elem === "string" ) {
+			return indexOf.call( jQuery( elem ), this[ 0 ] );
+		}
+
+		// Locate the position of the desired element
+		return indexOf.call( this,
+
+			// If it receives a jQuery object, the first element is used
+			elem.jquery ? elem[ 0 ] : elem
+		);
+	},
+
+	add: function( selector, context ) {
+		return this.pushStack(
+			jQuery.uniqueSort(
+				jQuery.merge( this.get(), jQuery( selector, context ) )
+			)
+		);
+	},
+
+	addBack: function( selector ) {
+		return this.add( selector == null ?
+			this.prevObject : this.prevObject.filter( selector )
+		);
+	}
+} );
+
+function sibling( cur, dir ) {
+	while ( ( cur = cur[ dir ] ) && cur.nodeType !== 1 ) {}
+	return cur;
+}
+
+jQuery.each( {
+	parent: function( elem ) {
+		var parent = elem.parentNode;
+		return parent && parent.nodeType !== 11 ? parent : null;
+	},
+	parents: function( elem ) {
+		return dir( elem, "parentNode" );
+	},
+	parentsUntil: function( elem, _i, until ) {
+		return dir( elem, "parentNode", until );
+	},
+	next: function( elem ) {
+		return sibling( elem, "nextSibling" );
+	},
+	prev: function( elem ) {
+		return sibling( elem, "previousSibling" );
+	},
+	nextAll: function( elem ) {
+		return dir( elem, "nextSibling" );
+	},
+	prevAll: function( elem ) {
+		return dir( elem, "previousSibling" );
+	},
+	nextUntil: function( elem, _i, until ) {
+		return dir( elem, "nextSibling", until );
+	},
+	prevUntil: function( elem, _i, until ) {
+		return dir( elem, "previousSibling", until );
+	},
+	siblings: function( elem ) {
+		return siblings( ( elem.parentNode || {} ).firstChild, elem );
+	},
+	children: function( elem ) {
+		return siblings( elem.firstChild );
+	},
+	contents: function( elem ) {
+		if ( elem.contentDocument != null &&
+
+			// Support: IE 11+
+			// <object> elements with no `data` attribute has an object
+			// `contentDocument` with a `null` prototype.
+			getProto( elem.contentDocument ) ) {
+
+			return elem.contentDocument;
+		}
+
+		// Support: IE 9 - 11 only, iOS 7 only, Android Browser <=4.3 only
+		// Treat the template element as a regular one in browsers that
+		// don't support it.
+		if ( nodeName( elem, "template" ) ) {
+			elem = elem.content || elem;
+		}
+
+		return jQuery.merge( [], elem.childNodes );
+	}
+}, function( name, fn ) {
+	jQuery.fn[ name ] = function( until, selector ) {
+		var matched = jQuery.map( this, fn, until );
+
+		if ( name.slice( -5 ) !== "Until" ) {
+			selector = until;
+		}
+
+		if ( selector && typeof selector === "string" ) {
+			matched = jQuery.filter( selector, matched );
+		}
+
+		if ( this.length > 1 ) {
+
+			// Remove duplicates
+			if ( !guaranteedUnique[ name ] ) {
+				jQuery.uniqueSort( matched );
+			}
+
+			// Reverse order for parents* and prev-derivatives
+			if ( rparentsprev.test( name ) ) {
+				matched.reverse();
+			}
+		}
+
+		return this.pushStack( matched );
+	};
+} );
+var rnothtmlwhite = ( /[^\x20\t\r\n\f]+/g );
+
+
+
+// Convert String-formatted options into Object-formatted ones
+function createOptions( options ) {
+	var object = {};
+	jQuery.each( options.match( rnothtmlwhite ) || [], function( _, flag ) {
+		object[ flag ] = true;
+	} );
+	return object;
+}
+
+/*
+ * Create a callback list using the following parameters:
+ *
+ *	options: an optional list of space-separated options that will change how
+ *			the callback list behaves or a more traditional option object
+ *
+ * By default a callback list will act like an event callback list and can be
+ * "fired" multiple times.
+ *
+ * Possible options:
+ *
+ *	once:			will ensure the callback list can only be fired once (like a Deferred)
+ *
+ *	memory:			will keep track of previous values and will call any callback added
+ *					after the list has been fired right away with the latest "memorized"
+ *					values (like a Deferred)
+ *
+ *	unique:			will ensure a callback can only be added once (no duplicate in the list)
+ *
+ *	stopOnFalse:	interrupt callings when a callback returns false
+ *
+ */
+jQuery.Callbacks = function( options ) {
+
+	// Convert options from String-formatted to Object-formatted if needed
+	// (we check in cache first)
+	options = typeof options === "string" ?
+		createOptions( options ) :
+		jQuery.extend( {}, options );
+
+	var // Flag to know if list is currently firing
+		firing,
+
+		// Last fire value for non-forgettable lists
+		memory,
+
+		// Flag to know if list was already fired
+		fired,
+
+		// Flag to prevent firing
+		locked,
+
+		// Actual callback list
+		list = [],
+
+		// Queue of execution data for repeatable lists
+		queue = [],
+
+		// Index of currently firing callback (modified by add/remove as needed)
+		firingIndex = -1,
+
+		// Fire callbacks
+		fire = function() {
+
+			// Enforce single-firing
+			locked = locked || options.once;
+
+			// Execute callbacks for all pending executions,
+			// respecting firingIndex overrides and runtime changes
+			fired = firing = true;
+			for ( ; queue.length; firingIndex = -1 ) {
+				memory = queue.shift();
+				while ( ++firingIndex < list.length ) {
+
+					// Run callback and check for early termination
+					if ( list[ firingIndex ].apply( memory[ 0 ], memory[ 1 ] ) === false &&
+						options.stopOnFalse ) {
+
+						// Jump to end and forget the data so .add doesn't re-fire
+						firingIndex = list.length;
+						memory = false;
+					}
+				}
+			}
+
+			// Forget the data if we're done with it
+			if ( !options.memory ) {
+				memory = false;
+			}
+
+			firing = false;
+
+			// Clean up if we're done firing for good
+			if ( locked ) {
+
+				// Keep an empty list if we have data for future add calls
+				if ( memory ) {
+					list = [];
+
+				// Otherwise, this object is spent
+				} else {
+					list = "";
+				}
+			}
+		},
+
+		// Actual Callbacks object
+		self = {
+
+			// Add a callback or a collection of callbacks to the list
+			add: function() {
+				if ( list ) {
+
+					// If we have memory from a past run, we should fire after adding
+					if ( memory && !firing ) {
+						firingIndex = list.length - 1;
+						queue.push( memory );
+					}
+
+					( function add( args ) {
+						jQuery.each( args, function( _, arg ) {
+							if ( isFunction( arg ) ) {
+								if ( !options.unique || !self.has( arg ) ) {
+									list.push( arg );
+								}
+							} else if ( arg && arg.length && toType( arg ) !== "string" ) {
+
+								// Inspect recursively
+								add( arg );
+							}
+						} );
+					} )( arguments );
+
+					if ( memory && !firing ) {
+						fire();
+					}
+				}
+				return this;
+			},
+
+			// Remove a callback from the list
+			remove: function() {
+				jQuery.each( arguments, function( _, arg ) {
+					var index;
+					while ( ( index = jQuery.inArray( arg, list, index ) ) > -1 ) {
+						list.splice( index, 1 );
+
+						// Handle firing indexes
+						if ( index <= firingIndex ) {
+							firingIndex--;
+						}
+					}
+				} );
+				return this;
+			},
+
+			// Check if a given callback is in the list.
+			// If no argument is given, return whether or not list has callbacks attached.
+			has: function( fn ) {
+				return fn ?
+					jQuery.inArray( fn, list ) > -1 :
+					list.length > 0;
+			},
+
+			// Remove all callbacks from the list
+			empty: function() {
+				if ( list ) {
+					list = [];
+				}
+				return this;
+			},
+
+			// Disable .fire and .add
+			// Abort any current/pending executions
+			// Clear all callbacks and values
+			disable: function() {
+				locked = queue = [];
+				list = memory = "";
+				return this;
+			},
+			disabled: function() {
+				return !list;
+			},
+
+			// Disable .fire
+			// Also disable .add unless we have memory (since it would have no effect)
+			// Abort any pending executions
+			lock: function() {
+				locked = queue = [];
+				if ( !memory && !firing ) {
+					list = memory = "";
+				}
+				return this;
+			},
+			locked: function() {
+				return !!locked;
+			},
+
+			// Call all callbacks with the given context and arguments
+			fireWith: function( context, args ) {
+				if ( !locked ) {
+					args = args || [];
+					args = [ context, args.slice ? args.slice() : args ];
+					queue.push( args );
+					if ( !firing ) {
+						fire();
+					}
+				}
+				return this;
+			},
+
+			// Call all the callbacks with the given arguments
+			fire: function() {
+				self.fireWith( this, arguments );
+				return this;
+			},
+
+			// To know if the callbacks have already been called at least once
+			fired: function() {
+				return !!fired;
+			}
+		};
+
+	return self;
+};
+
+
+function Identity( v ) {
+	return v;
+}
+function Thrower( ex ) {
+	throw ex;
+}
+
+function adoptValue( value, resolve, reject, noValue ) {
+	var method;
+
+	try {
+
+		// Check for promise aspect first to privilege synchronous behavior
+		if ( value && isFunction( ( method = value.promise ) ) ) {
+			method.call( value ).done( resolve ).fail( reject );
+
+		// Other thenables
+		} else if ( value && isFunction( ( method = value.then ) ) ) {
+			method.call( value, resolve, reject );
+
+		// Other non-thenables
+		} else {
+
+			// Control `resolve` arguments by letting Array#slice cast boolean `noValue` to integer:
+			// * false: [ value ].slice( 0 ) => resolve( value )
+			// * true: [ value ].slice( 1 ) => resolve()
+			resolve.apply( undefined, [ value ].slice( noValue ) );
+		}
+
+	// For Promises/A+, convert exceptions into rejections
+	// Since jQuery.when doesn't unwrap thenables, we can skip the extra checks appearing in
+	// Deferred#then to conditionally suppress rejection.
+	} catch ( value ) {
+
+		// Support: Android 4.0 only
+		// Strict mode functions invoked without .call/.apply get global-object context
+		reject.apply( undefined, [ value ] );
+	}
+}
+
+jQuery.extend( {
+
+	Deferred: function( func ) {
+		var tuples = [
+
+				// action, add listener, callbacks,
+				// ... .then handlers, argument index, [final state]
+				[ "notify", "progress", jQuery.Callbacks( "memory" ),
+					jQuery.Callbacks( "memory" ), 2 ],
+				[ "resolve", "done", jQuery.Callbacks( "once memory" ),
+					jQuery.Callbacks( "once memory" ), 0, "resolved" ],
+				[ "reject", "fail", jQuery.Callbacks( "once memory" ),
+					jQuery.Callbacks( "once memory" ), 1, "rejected" ]
+			],
+			state = "pending",
+			promise = {
+				state: function() {
+					return state;
+				},
+				always: function() {
+					deferred.done( arguments ).fail( arguments );
+					return this;
+				},
+				"catch": function( fn ) {
+					return promise.then( null, fn );
+				},
+
+				// Keep pipe for back-compat
+				pipe: function( /* fnDone, fnFail, fnProgress */ ) {
+					var fns = arguments;
+
+					return jQuery.Deferred( function( newDefer ) {
+						jQuery.each( tuples, function( _i, tuple ) {
+
+							// Map tuples (progress, done, fail) to arguments (done, fail, progress)
+							var fn = isFunction( fns[ tuple[ 4 ] ] ) && fns[ tuple[ 4 ] ];
+
+							// deferred.progress(function() { bind to newDefer or newDefer.notify })
+							// deferred.done(function() { bind to newDefer or newDefer.resolve })
+							// deferred.fail(function() { bind to newDefer or newDefer.reject })
+							deferred[ tuple[ 1 ] ]( function() {
+								var returned = fn && fn.apply( this, arguments );
+								if ( returned && isFunction( returned.promise ) ) {
+									returned.promise()
+										.progress( newDefer.notify )
+										.done( newDefer.resolve )
+										.fail( newDefer.reject );
+								} else {
+									newDefer[ tuple[ 0 ] + "With" ](
+										this,
+										fn ? [ returned ] : arguments
+									);
+								}
+							} );
+						} );
+						fns = null;
+					} ).promise();
+				},
+				then: function( onFulfilled, onRejected, onProgress ) {
+					var maxDepth = 0;
+					function resolve( depth, deferred, handler, special ) {
+						return function() {
+							var that = this,
+								args = arguments,
+								mightThrow = function() {
+									var returned, then;
+
+									// Support: Promises/A+ section 2.3.3.3.3
+									// https://promisesaplus.com/#point-59
+									// Ignore double-resolution attempts
+									if ( depth < maxDepth ) {
+										return;
+									}
+
+									returned = handler.apply( that, args );
+
+									// Support: Promises/A+ section 2.3.1
+									// https://promisesaplus.com/#point-48
+									if ( returned === deferred.promise() ) {
+										throw new TypeError( "Thenable self-resolution" );
+									}
+
+									// Support: Promises/A+ sections 2.3.3.1, 3.5
+									// https://promisesaplus.com/#point-54
+									// https://promisesaplus.com/#point-75
+									// Retrieve `then` only once
+									then = returned &&
+
+										// Support: Promises/A+ section 2.3.4
+										// https://promisesaplus.com/#point-64
+										// Only check objects and functions for thenability
+										( typeof returned === "object" ||
+											typeof returned === "function" ) &&
+										returned.then;
+
+									// Handle a returned thenable
+									if ( isFunction( then ) ) {
+
+										// Special processors (notify) just wait for resolution
+										if ( special ) {
+											then.call(
+												returned,
+												resolve( maxDepth, deferred, Identity, special ),
+												resolve( maxDepth, deferred, Thrower, special )
+											);
+
+										// Normal processors (resolve) also hook into progress
+										} else {
+
+											// ...and disregard older resolution values
+											maxDepth++;
+
+											then.call(
+												returned,
+												resolve( maxDepth, deferred, Identity, special ),
+												resolve( maxDepth, deferred, Thrower, special ),
+												resolve( maxDepth, deferred, Identity,
+													deferred.notifyWith )
+											);
+										}
+
+									// Handle all other returned values
+									} else {
+
+										// Only substitute handlers pass on context
+										// and multiple values (non-spec behavior)
+										if ( handler !== Identity ) {
+											that = undefined;
+											args = [ returned ];
+										}
+
+										// Process the value(s)
+										// Default process is resolve
+										( special || deferred.resolveWith )( that, args );
+									}
+								},
+
+								// Only normal processors (resolve) catch and reject exceptions
+								process = special ?
+									mightThrow :
+									function() {
+										try {
+											mightThrow();
+										} catch ( e ) {
+
+											if ( jQuery.Deferred.exceptionHook ) {
+												jQuery.Deferred.exceptionHook( e,
+													process.stackTrace );
+											}
+
+											// Support: Promises/A+ section 2.3.3.3.4.1
+											// https://promisesaplus.com/#point-61
+											// Ignore post-resolution exceptions
+											if ( depth + 1 >= maxDepth ) {
+
+												// Only substitute handlers pass on context
+												// and multiple values (non-spec behavior)
+												if ( handler !== Thrower ) {
+													that = undefined;
+													args = [ e ];
+												}
+
+												deferred.rejectWith( that, args );
+											}
+										}
+									};
+
+							// Support: Promises/A+ section 2.3.3.3.1
+							// https://promisesaplus.com/#point-57
+							// Re-resolve promises immediately to dodge false rejection from
+							// subsequent errors
+							if ( depth ) {
+								process();
+							} else {
+
+								// Call an optional hook to record the stack, in case of exception
+								// since it's otherwise lost when execution goes async
+								if ( jQuery.Deferred.getStackHook ) {
+									process.stackTrace = jQuery.Deferred.getStackHook();
+								}
+								window.setTimeout( process );
+							}
+						};
+					}
+
+					return jQuery.Deferred( function( newDefer ) {
+
+						// progress_handlers.add( ... )
+						tuples[ 0 ][ 3 ].add(
+							resolve(
+								0,
+								newDefer,
+								isFunction( onProgress ) ?
+									onProgress :
+									Identity,
+								newDefer.notifyWith
+							)
+						);
+
+						// fulfilled_handlers.add( ... )
+						tuples[ 1 ][ 3 ].add(
+							resolve(
+								0,
+								newDefer,
+								isFunction( onFulfilled ) ?
+									onFulfilled :
+									Identity
+							)
+						);
+
+						// rejected_handlers.add( ... )
+						tuples[ 2 ][ 3 ].add(
+							resolve(
+								0,
+								newDefer,
+								isFunction( onRejected ) ?
+									onRejected :
+									Thrower
+							)
+						);
+					} ).promise();
+				},
+
+				// Get a promise for this deferred
+				// If obj is provided, the promise aspect is added to the object
+				promise: function( obj ) {
+					return obj != null ? jQuery.extend( obj, promise ) : promise;
+				}
+			},
+			deferred = {};
+
+		// Add list-specific methods
+		jQuery.each( tuples, function( i, tuple ) {
+			var list = tuple[ 2 ],
+				stateString = tuple[ 5 ];
+
+			// promise.progress = list.add
+			// promise.done = list.add
+			// promise.fail = list.add
+			promise[ tuple[ 1 ] ] = list.add;
+
+			// Handle state
+			if ( stateString ) {
+				list.add(
+					function() {
+
+						// state = "resolved" (i.e., fulfilled)
+						// state = "rejected"
+						state = stateString;
+					},
+
+					// rejected_callbacks.disable
+					// fulfilled_callbacks.disable
+					tuples[ 3 - i ][ 2 ].disable,
+
+					// rejected_handlers.disable
+					// fulfilled_handlers.disable
+					tuples[ 3 - i ][ 3 ].disable,
+
+					// progress_callbacks.lock
+					tuples[ 0 ][ 2 ].lock,
+
+					// progress_handlers.lock
+					tuples[ 0 ][ 3 ].lock
+				);
+			}
+
+			// progress_handlers.fire
+			// fulfilled_handlers.fire
+			// rejected_handlers.fire
+			list.add( tuple[ 3 ].fire );
+
+			// deferred.notify = function() { deferred.notifyWith(...) }
+			// deferred.resolve = function() { deferred.resolveWith(...) }
+			// deferred.reject = function() { deferred.rejectWith(...) }
+			deferred[ tuple[ 0 ] ] = function() {
+				deferred[ tuple[ 0 ] + "With" ]( this === deferred ? undefined : this, arguments );
+				return this;
+			};
+
+			// deferred.notifyWith = list.fireWith
+			// deferred.resolveWith = list.fireWith
+			// deferred.rejectWith = list.fireWith
+			deferred[ tuple[ 0 ] + "With" ] = list.fireWith;
+		} );
+
+		// Make the deferred a promise
+		promise.promise( deferred );
+
+		// Call given func if any
+		if ( func ) {
+			func.call( deferred, deferred );
+		}
+
+		// All done!
+		return deferred;
+	},
+
+	// Deferred helper
+	when: function( singleValue ) {
+		var
+
+			// count of uncompleted subordinates
+			remaining = arguments.length,
+
+			// count of unprocessed arguments
+			i = remaining,
+
+			// subordinate fulfillment data
+			resolveContexts = Array( i ),
+			resolveValues = slice.call( arguments ),
+
+			// the primary Deferred
+			primary = jQuery.Deferred(),
+
+			// subordinate callback factory
+			updateFunc = function( i ) {
+				return function( value ) {
+					resolveContexts[ i ] = this;
+					resolveValues[ i ] = arguments.length > 1 ? slice.call( arguments ) : value;
+					if ( !( --remaining ) ) {
+						primary.resolveWith( resolveContexts, resolveValues );
+					}
+				};
+			};
+
+		// Single- and empty arguments are adopted like Promise.resolve
+		if ( remaining <= 1 ) {
+			adoptValue( singleValue, primary.done( updateFunc( i ) ).resolve, primary.reject,
+				!remaining );
+
+			// Use .then() to unwrap secondary thenables (cf. gh-3000)
+			if ( primary.state() === "pending" ||
+				isFunction( resolveValues[ i ] && resolveValues[ i ].then ) ) {
+
+				return primary.then();
+			}
+		}
+
+		// Multiple arguments are aggregated like Promise.all array elements
+		while ( i-- ) {
+			adoptValue( resolveValues[ i ], updateFunc( i ), primary.reject );
+		}
+
+		return primary.promise();
+	}
+} );
+
+
+// These usually indicate a programmer mistake during development,
+// warn about them ASAP rather than swallowing them by default.
+var rerrorNames = /^(Eval|Internal|Range|Reference|Syntax|Type|URI)Error$/;
+
+jQuery.Deferred.exceptionHook = function( error, stack ) {
+
+	// Support: IE 8 - 9 only
+	// Console exists when dev tools are open, which can happen at any time
+	if ( window.console && window.console.warn && error && rerrorNames.test( error.name ) ) {
+		window.console.warn( "jQuery.Deferred exception: " + error.message, error.stack, stack );
+	}
+};
+
+
+
+
+jQuery.readyException = function( error ) {
+	window.setTimeout( function() {
+		throw error;
+	} );
+};
+
+
+
+
+// The deferred used on DOM ready
+var readyList = jQuery.Deferred();
+
+jQuery.fn.ready = function( fn ) {
+
+	readyList
+		.then( fn )
+
+		// Wrap jQuery.readyException in a function so that the lookup
+		// happens at the time of error handling instead of callback
+		// registration.
+		.catch( function( error ) {
+			jQuery.readyException( error );
+		} );
+
+	return this;
+};
+
+jQuery.extend( {
+
+	// Is the DOM ready to be used? Set to true once it occurs.
+	isReady: false,
+
+	// A counter to track how many items to wait for before
+	// the ready event fires. See #6781
+	readyWait: 1,
+
+	// Handle when the DOM is ready
+	ready: function( wait ) {
+
+		// Abort if there are pending holds or we're already ready
+		if ( wait === true ? --jQuery.readyWait : jQuery.isReady ) {
+			return;
+		}
+
+		// Remember that the DOM is ready
+		jQuery.isReady = true;
+
+		// If a normal DOM Ready event fired, decrement, and wait if need be
+		if ( wait !== true && --jQuery.readyWait > 0 ) {
+			return;
+		}
+
+		// If there are functions bound, to execute
+		readyList.resolveWith( document, [ jQuery ] );
+	}
+} );
+
+jQuery.ready.then = readyList.then;
+
+// The ready event handler and self cleanup method
+function completed() {
+	document.removeEventListener( "DOMContentLoaded", completed );
+	window.removeEventListener( "load", completed );
+	jQuery.ready();
+}
+
+// Catch cases where $(document).ready() is called
+// after the browser event has already occurred.
+// Support: IE <=9 - 10 only
+// Older IE sometimes signals "interactive" too soon
+if ( document.readyState === "complete" ||
+	( document.readyState !== "loading" && !document.documentElement.doScroll ) ) {
+
+	// Handle it asynchronously to allow scripts the opportunity to delay ready
+	window.setTimeout( jQuery.ready );
+
+} else {
+
+	// Use the handy event callback
+	document.addEventListener( "DOMContentLoaded", completed );
+
+	// A fallback to window.onload, that will always work
+	window.addEventListener( "load", completed );
+}
+
+
+
+
+// Multifunctional method to get and set values of a collection
+// The value/s can optionally be executed if it's a function
+var access = function( elems, fn, key, value, chainable, emptyGet, raw ) {
+	var i = 0,
+		len = elems.length,
+		bulk = key == null;
+
+	// Sets many values
+	if ( toType( key ) === "object" ) {
+		chainable = true;
+		for ( i in key ) {
+			access( elems, fn, i, key[ i ], true, emptyGet, raw );
+		}
+
+	// Sets one value
+	} else if ( value !== undefined ) {
+		chainable = true;
+
+		if ( !isFunction( value ) ) {
+			raw = true;
+		}
+
+		if ( bulk ) {
+
+			// Bulk operations run against the entire set
+			if ( raw ) {
+				fn.call( elems, value );
+				fn = null;
+
+			// ...except when executing function values
+			} else {
+				bulk = fn;
+				fn = function( elem, _key, value ) {
+					return bulk.call( jQuery( elem ), value );
+				};
+			}
+		}
+
+		if ( fn ) {
+			for ( ; i < len; i++ ) {
+				fn(
+					elems[ i ], key, raw ?
+						value :
+						value.call( elems[ i ], i, fn( elems[ i ], key ) )
+				);
+			}
+		}
+	}
+
+	if ( chainable ) {
+		return elems;
+	}
+
+	// Gets
+	if ( bulk ) {
+		return fn.call( elems );
+	}
+
+	return len ? fn( elems[ 0 ], key ) : emptyGet;
+};
+
+
+// Matches dashed string for camelizing
+var rmsPrefix = /^-ms-/,
+	rdashAlpha = /-([a-z])/g;
+
+// Used by camelCase as callback to replace()
+function fcamelCase( _all, letter ) {
+	return letter.toUpperCase();
+}
+
+// Convert dashed to camelCase; used by the css and data modules
+// Support: IE <=9 - 11, Edge 12 - 15
+// Microsoft forgot to hump their vendor prefix (#9572)
+function camelCase( string ) {
+	return string.replace( rmsPrefix, "ms-" ).replace( rdashAlpha, fcamelCase );
+}
+var acceptData = function( owner ) {
+
+	// Accepts only:
+	//  - Node
+	//    - Node.ELEMENT_NODE
+	//    - Node.DOCUMENT_NODE
+	//  - Object
+	//    - Any
+	return owner.nodeType === 1 || owner.nodeType === 9 || !( +owner.nodeType );
+};
+
+
+
+
+function Data() {
+	this.expando = jQuery.expando + Data.uid++;
+}
+
+Data.uid = 1;
+
+Data.prototype = {
+
+	cache: function( owner ) {
+
+		// Check if the owner object already has a cache
+		var value = owner[ this.expando ];
+
+		// If not, create one
+		if ( !value ) {
+			value = {};
+
+			// We can accept data for non-element nodes in modern browsers,
+			// but we should not, see #8335.
+			// Always return an empty object.
+			if ( acceptData( owner ) ) {
+
+				// If it is a node unlikely to be stringify-ed or looped over
+				// use plain assignment
+				if ( owner.nodeType ) {
+					owner[ this.expando ] = value;
+
+				// Otherwise secure it in a non-enumerable property
+				// configurable must be true to allow the property to be
+				// deleted when data is removed
+				} else {
+					Object.defineProperty( owner, this.expando, {
+						value: value,
+						configurable: true
+					} );
+				}
+			}
+		}
+
+		return value;
+	},
+	set: function( owner, data, value ) {
+		var prop,
+			cache = this.cache( owner );
+
+		// Handle: [ owner, key, value ] args
+		// Always use camelCase key (gh-2257)
+		if ( typeof data === "string" ) {
+			cache[ camelCase( data ) ] = value;
+
+		// Handle: [ owner, { properties } ] args
+		} else {
+
+			// Copy the properties one-by-one to the cache object
+			for ( prop in data ) {
+				cache[ camelCase( prop ) ] = data[ prop ];
+			}
+		}
+		return cache;
+	},
+	get: function( owner, key ) {
+		return key === undefined ?
+			this.cache( owner ) :
+
+			// Always use camelCase key (gh-2257)
+			owner[ this.expando ] && owner[ this.expando ][ camelCase( key ) ];
+	},
+	access: function( owner, key, value ) {
+
+		// In cases where either:
+		//
+		//   1. No key was specified
+		//   2. A string key was specified, but no value provided
+		//
+		// Take the "read" path and allow the get method to determine
+		// which value to return, respectively either:
+		//
+		//   1. The entire cache object
+		//   2. The data stored at the key
+		//
+		if ( key === undefined ||
+				( ( key && typeof key === "string" ) && value === undefined ) ) {
+
+			return this.get( owner, key );
+		}
+
+		// When the key is not a string, or both a key and value
+		// are specified, set or extend (existing objects) with either:
+		//
+		//   1. An object of properties
+		//   2. A key and value
+		//
+		this.set( owner, key, value );
+
+		// Since the "set" path can have two possible entry points
+		// return the expected data based on which path was taken[*]
+		return value !== undefined ? value : key;
+	},
+	remove: function( owner, key ) {
+		var i,
+			cache = owner[ this.expando ];
+
+		if ( cache === undefined ) {
+			return;
+		}
+
+		if ( key !== undefined ) {
+
+			// Support array or space separated string of keys
+			if ( Array.isArray( key ) ) {
+
+				// If key is an array of keys...
+				// We always set camelCase keys, so remove that.
+				key = key.map( camelCase );
+			} else {
+				key = camelCase( key );
+
+				// If a key with the spaces exists, use it.
+				// Otherwise, create an array by matching non-whitespace
+				key = key in cache ?
+					[ key ] :
+					( key.match( rnothtmlwhite ) || [] );
+			}
+
+			i = key.length;
+
+			while ( i-- ) {
+				delete cache[ key[ i ] ];
+			}
+		}
+
+		// Remove the expando if there's no more data
+		if ( key === undefined || jQuery.isEmptyObject( cache ) ) {
+
+			// Support: Chrome <=35 - 45
+			// Webkit & Blink performance suffers when deleting properties
+			// from DOM nodes, so set to undefined instead
+			// https://bugs.chromium.org/p/chromium/issues/detail?id=378607 (bug restricted)
+			if ( owner.nodeType ) {
+				owner[ this.expando ] = undefined;
+			} else {
+				delete owner[ this.expando ];
+			}
+		}
+	},
+	hasData: function( owner ) {
+		var cache = owner[ this.expando ];
+		return cache !== undefined && !jQuery.isEmptyObject( cache );
+	}
+};
+var dataPriv = new Data();
+
+var dataUser = new Data();
+
+
+
+//	Implementation Summary
+//
+//	1. Enforce API surface and semantic compatibility with 1.9.x branch
+//	2. Improve the module's maintainability by reducing the storage
+//		paths to a single mechanism.
+//	3. Use the same single mechanism to support "private" and "user" data.
+//	4. _Never_ expose "private" data to user code (TODO: Drop _data, _removeData)
+//	5. Avoid exposing implementation details on user objects (eg. expando properties)
+//	6. Provide a clear path for implementation upgrade to WeakMap in 2014
+
+var rbrace = /^(?:\{[\w\W]*\}|\[[\w\W]*\])$/,
+	rmultiDash = /[A-Z]/g;
+
+function getData( data ) {
+	if ( data === "true" ) {
+		return true;
+	}
+
+	if ( data === "false" ) {
+		return false;
+	}
+
+	if ( data === "null" ) {
+		return null;
+	}
+
+	// Only convert to a number if it doesn't change the string
+	if ( data === +data + "" ) {
+		return +data;
+	}
+
+	if ( rbrace.test( data ) ) {
+		return JSON.parse( data );
+	}
+
+	return data;
+}
+
+function dataAttr( elem, key, data ) {
+	var name;
+
+	// If nothing was found internally, try to fetch any
+	// data from the HTML5 data-* attribute
+	if ( data === undefined && elem.nodeType === 1 ) {
+		name = "data-" + key.replace( rmultiDash, "-$&" ).toLowerCase();
+		data = elem.getAttribute( name );
+
+		if ( typeof data === "string" ) {
+			try {
+				data = getData( data );
+			} catch ( e ) {}
+
+			// Make sure we set the data so it isn't changed later
+			dataUser.set( elem, key, data );
+		} else {
+			data = undefined;
+		}
+	}
+	return data;
+}
+
+jQuery.extend( {
+	hasData: function( elem ) {
+		return dataUser.hasData( elem ) || dataPriv.hasData( elem );
+	},
+
+	data: function( elem, name, data ) {
+		return dataUser.access( elem, name, data );
+	},
+
+	removeData: function( elem, name ) {
+		dataUser.remove( elem, name );
+	},
+
+	// TODO: Now that all calls to _data and _removeData have been replaced
+	// with direct calls to dataPriv methods, these can be deprecated.
+	_data: function( elem, name, data ) {
+		return dataPriv.access( elem, name, data );
+	},
+
+	_removeData: function( elem, name ) {
+		dataPriv.remove( elem, name );
+	}
+} );
+
+jQuery.fn.extend( {
+	data: function( key, value ) {
+		var i, name, data,
+			elem = this[ 0 ],
+			attrs = elem && elem.attributes;
+
+		// Gets all values
+		if ( key === undefined ) {
+			if ( this.length ) {
+				data = dataUser.get( elem );
+
+				if ( elem.nodeType === 1 && !dataPriv.get( elem, "hasDataAttrs" ) ) {
+					i = attrs.length;
+					while ( i-- ) {
+
+						// Support: IE 11 only
+						// The attrs elements can be null (#14894)
+						if ( attrs[ i ] ) {
+							name = attrs[ i ].name;
+							if ( name.indexOf( "data-" ) === 0 ) {
+								name = camelCase( name.slice( 5 ) );
+								dataAttr( elem, name, data[ name ] );
+							}
+						}
+					}
+					dataPriv.set( elem, "hasDataAttrs", true );
+				}
+			}
+
+			return data;
+		}
+
+		// Sets multiple values
+		if ( typeof key === "object" ) {
+			return this.each( function() {
+				dataUser.set( this, key );
+			} );
+		}
+
+		return access( this, function( value ) {
+			var data;
+
+			// The calling jQuery object (element matches) is not empty
+			// (and therefore has an element appears at this[ 0 ]) and the
+			// `value` parameter was not undefined. An empty jQuery object
+			// will result in `undefined` for elem = this[ 0 ] which will
+			// throw an exception if an attempt to read a data cache is made.
+			if ( elem && value === undefined ) {
+
+				// Attempt to get data from the cache
+				// The key will always be camelCased in Data
+				data = dataUser.get( elem, key );
+				if ( data !== undefined ) {
+					return data;
+				}
+
+				// Attempt to "discover" the data in
+				// HTML5 custom data-* attrs
+				data = dataAttr( elem, key );
+				if ( data !== undefined ) {
+					return data;
+				}
+
+				// We tried really hard, but the data doesn't exist.
+				return;
+			}
+
+			// Set the data...
+			this.each( function() {
+
+				// We always store the camelCased key
+				dataUser.set( this, key, value );
+			} );
+		}, null, value, arguments.length > 1, null, true );
+	},
+
+	removeData: function( key ) {
+		return this.each( function() {
+			dataUser.remove( this, key );
+		} );
+	}
+} );
+
+
+jQuery.extend( {
+	queue: function( elem, type, data ) {
+		var queue;
+
+		if ( elem ) {
+			type = ( type || "fx" ) + "queue";
+			queue = dataPriv.get( elem, type );
+
+			// Speed up dequeue by getting out quickly if this is just a lookup
+			if ( data ) {
+				if ( !queue || Array.isArray( data ) ) {
+					queue = dataPriv.access( elem, type, jQuery.makeArray( data ) );
+				} else {
+					queue.push( data );
+				}
+			}
+			return queue || [];
+		}
+	},
+
+	dequeue: function( elem, type ) {
+		type = type || "fx";
+
+		var queue = jQuery.queue( elem, type ),
+			startLength = queue.length,
+			fn = queue.shift(),
+			hooks = jQuery._queueHooks( elem, type ),
+			next = function() {
+				jQuery.dequeue( elem, type );
+			};
+
+		// If the fx queue is dequeued, always remove the progress sentinel
+		if ( fn === "inprogress" ) {
+			fn = queue.shift();
+			startLength--;
+		}
+
+		if ( fn ) {
+
+			// Add a progress sentinel to prevent the fx queue from being
+			// automatically dequeued
+			if ( type === "fx" ) {
+				queue.unshift( "inprogress" );
+			}
+
+			// Clear up the last queue stop function
+			delete hooks.stop;
+			fn.call( elem, next, hooks );
+		}
+
+		if ( !startLength && hooks ) {
+			hooks.empty.fire();
+		}
+	},
+
+	// Not public - generate a queueHooks object, or return the current one
+	_queueHooks: function( elem, type ) {
+		var key = type + "queueHooks";
+		return dataPriv.get( elem, key ) || dataPriv.access( elem, key, {
+			empty: jQuery.Callbacks( "once memory" ).add( function() {
+				dataPriv.remove( elem, [ type + "queue", key ] );
+			} )
+		} );
+	}
+} );
+
+jQuery.fn.extend( {
+	queue: function( type, data ) {
+		var setter = 2;
+
+		if ( typeof type !== "string" ) {
+			data = type;
+			type = "fx";
+			setter--;
+		}
+
+		if ( arguments.length < setter ) {
+			return jQuery.queue( this[ 0 ], type );
+		}
+
+		return data === undefined ?
+			this :
+			this.each( function() {
+				var queue = jQuery.queue( this, type, data );
+
+				// Ensure a hooks for this queue
+				jQuery._queueHooks( this, type );
+
+				if ( type === "fx" && queue[ 0 ] !== "inprogress" ) {
+					jQuery.dequeue( this, type );
+				}
+			} );
+	},
+	dequeue: function( type ) {
+		return this.each( function() {
+			jQuery.dequeue( this, type );
+		} );
+	},
+	clearQueue: function( type ) {
+		return this.queue( type || "fx", [] );
+	},
+
+	// Get a promise resolved when queues of a certain type
+	// are emptied (fx is the type by default)
+	promise: function( type, obj ) {
+		var tmp,
+			count = 1,
+			defer = jQuery.Deferred(),
+			elements = this,
+			i = this.length,
+			resolve = function() {
+				if ( !( --count ) ) {
+					defer.resolveWith( elements, [ elements ] );
+				}
+			};
+
+		if ( typeof type !== "string" ) {
+			obj = type;
+			type = undefined;
+		}
+		type = type || "fx";
+
+		while ( i-- ) {
+			tmp = dataPriv.get( elements[ i ], type + "queueHooks" );
+			if ( tmp && tmp.empty ) {
+				count++;
+				tmp.empty.add( resolve );
+			}
+		}
+		resolve();
+		return defer.promise( obj );
+	}
+} );
+var pnum = ( /[+-]?(?:\d*\.|)\d+(?:[eE][+-]?\d+|)/ ).source;
+
+var rcssNum = new RegExp( "^(?:([+-])=|)(" + pnum + ")([a-z%]*)$", "i" );
+
+
+var cssExpand = [ "Top", "Right", "Bottom", "Left" ];
+
+var documentElement = document.documentElement;
+
+
+
+	var isAttached = function( elem ) {
+			return jQuery.contains( elem.ownerDocument, elem );
+		},
+		composed = { composed: true };
+
+	// Support: IE 9 - 11+, Edge 12 - 18+, iOS 10.0 - 10.2 only
+	// Check attachment across shadow DOM boundaries when possible (gh-3504)
+	// Support: iOS 10.0-10.2 only
+	// Early iOS 10 versions support `attachShadow` but not `getRootNode`,
+	// leading to errors. We need to check for `getRootNode`.
+	if ( documentElement.getRootNode ) {
+		isAttached = function( elem ) {
+			return jQuery.contains( elem.ownerDocument, elem ) ||
+				elem.getRootNode( composed ) === elem.ownerDocument;
+		};
+	}
+var isHiddenWithinTree = function( elem, el ) {
+
+		// isHiddenWithinTree might be called from jQuery#filter function;
+		// in that case, element will be second argument
+		elem = el || elem;
+
+		// Inline style trumps all
+		return elem.style.display === "none" ||
+			elem.style.display === "" &&
+
+			// Otherwise, check computed style
+			// Support: Firefox <=43 - 45
+			// Disconnected elements can have computed display: none, so first confirm that elem is
+			// in the document.
+			isAttached( elem ) &&
+
+			jQuery.css( elem, "display" ) === "none";
+	};
+
+
+
+function adjustCSS( elem, prop, valueParts, tween ) {
+	var adjusted, scale,
+		maxIterations = 20,
+		currentValue = tween ?
+			function() {
+				return tween.cur();
+			} :
+			function() {
+				return jQuery.css( elem, prop, "" );
+			},
+		initial = currentValue(),
+		unit = valueParts && valueParts[ 3 ] || ( jQuery.cssNumber[ prop ] ? "" : "px" ),
+
+		// Starting value computation is required for potential unit mismatches
+		initialInUnit = elem.nodeType &&
+			( jQuery.cssNumber[ prop ] || unit !== "px" && +initial ) &&
+			rcssNum.exec( jQuery.css( elem, prop ) );
+
+	if ( initialInUnit && initialInUnit[ 3 ] !== unit ) {
+
+		// Support: Firefox <=54
+		// Halve the iteration target value to prevent interference from CSS upper bounds (gh-2144)
+		initial = initial / 2;
+
+		// Trust units reported by jQuery.css
+		unit = unit || initialInUnit[ 3 ];
+
+		// Iteratively approximate from a nonzero starting point
+		initialInUnit = +initial || 1;
+
+		while ( maxIterations-- ) {
+
+			// Evaluate and update our best guess (doubling guesses that zero out).
+			// Finish if the scale equals or crosses 1 (making the old*new product non-positive).
+			jQuery.style( elem, prop, initialInUnit + unit );
+			if ( ( 1 - scale ) * ( 1 - ( scale = currentValue() / initial || 0.5 ) ) <= 0 ) {
+				maxIterations = 0;
+			}
+			initialInUnit = initialInUnit / scale;
+
+		}
+
+		initialInUnit = initialInUnit * 2;
+		jQuery.style( elem, prop, initialInUnit + unit );
+
+		// Make sure we update the tween properties later on
+		valueParts = valueParts || [];
+	}
+
+	if ( valueParts ) {
+		initialInUnit = +initialInUnit || +initial || 0;
+
+		// Apply relative offset (+=/-=) if specified
+		adjusted = valueParts[ 1 ] ?
+			initialInUnit + ( valueParts[ 1 ] + 1 ) * valueParts[ 2 ] :
+			+valueParts[ 2 ];
+		if ( tween ) {
+			tween.unit = unit;
+			tween.start = initialInUnit;
+			tween.end = adjusted;
+		}
+	}
+	return adjusted;
+}
+
+
+var defaultDisplayMap = {};
+
+function getDefaultDisplay( elem ) {
+	var temp,
+		doc = elem.ownerDocument,
+		nodeName = elem.nodeName,
+		display = defaultDisplayMap[ nodeName ];
+
+	if ( display ) {
+		return display;
+	}
+
+	temp = doc.body.appendChild( doc.createElement( nodeName ) );
+	display = jQuery.css( temp, "display" );
+
+	temp.parentNode.removeChild( temp );
+
+	if ( display === "none" ) {
+		display = "block";
+	}
+	defaultDisplayMap[ nodeName ] = display;
+
+	return display;
+}
+
+function showHide( elements, show ) {
+	var display, elem,
+		values = [],
+		index = 0,
+		length = elements.length;
+
+	// Determine new display value for elements that need to change
+	for ( ; index < length; index++ ) {
+		elem = elements[ index ];
+		if ( !elem.style ) {
+			continue;
+		}
+
+		display = elem.style.display;
+		if ( show ) {
+
+			// Since we force visibility upon cascade-hidden elements, an immediate (and slow)
+			// check is required in this first loop unless we have a nonempty display value (either
+			// inline or about-to-be-restored)
+			if ( display === "none" ) {
+				values[ index ] = dataPriv.get( elem, "display" ) || null;
+				if ( !values[ index ] ) {
+					elem.style.display = "";
+				}
+			}
+			if ( elem.style.display === "" && isHiddenWithinTree( elem ) ) {
+				values[ index ] = getDefaultDisplay( elem );
+			}
+		} else {
+			if ( display !== "none" ) {
+				values[ index ] = "none";
+
+				// Remember what we're overwriting
+				dataPriv.set( elem, "display", display );
+			}
+		}
+	}
+
+	// Set the display of the elements in a second loop to avoid constant reflow
+	for ( index = 0; index < length; index++ ) {
+		if ( values[ index ] != null ) {
+			elements[ index ].style.display = values[ index ];
+		}
+	}
+
+	return elements;
+}
+
+jQuery.fn.extend( {
+	show: function() {
+		return showHide( this, true );
+	},
+	hide: function() {
+		return showHide( this );
+	},
+	toggle: function( state ) {
+		if ( typeof state === "boolean" ) {
+			return state ? this.show() : this.hide();
+		}
+
+		return this.each( function() {
+			if ( isHiddenWithinTree( this ) ) {
+				jQuery( this ).show();
+			} else {
+				jQuery( this ).hide();
+			}
+		} );
+	}
+} );
+var rcheckableType = ( /^(?:checkbox|radio)$/i );
+
+var rtagName = ( /<([a-z][^\/\0>\x20\t\r\n\f]*)/i );
+
+var rscriptType = ( /^$|^module$|\/(?:java|ecma)script/i );
+
+
+
+( function() {
+	var fragment = document.createDocumentFragment(),
+		div = fragment.appendChild( document.createElement( "div" ) ),
+		input = document.createElement( "input" );
+
+	// Support: Android 4.0 - 4.3 only
+	// Check state lost if the name is set (#11217)
+	// Support: Windows Web Apps (WWA)
+	// `name` and `type` must use .setAttribute for WWA (#14901)
+	input.setAttribute( "type", "radio" );
+	input.setAttribute( "checked", "checked" );
+	input.setAttribute( "name", "t" );
+
+	div.appendChild( input );
+
+	// Support: Android <=4.1 only
+	// Older WebKit doesn't clone checked state correctly in fragments
+	support.checkClone = div.cloneNode( true ).cloneNode( true ).lastChild.checked;
+
+	// Support: IE <=11 only
+	// Make sure textarea (and checkbox) defaultValue is properly cloned
+	div.innerHTML = "<textarea>x</textarea>";
+	support.noCloneChecked = !!div.cloneNode( true ).lastChild.defaultValue;
+
+	// Support: IE <=9 only
+	// IE <=9 replaces <option> tags with their contents when inserted outside of
+	// the select element.
+	div.innerHTML = "<option></option>";
+	support.option = !!div.lastChild;
+} )();
+
+
+// We have to close these tags to support XHTML (#13200)
+var wrapMap = {
+
+	// XHTML parsers do not magically insert elements in the
+	// same way that tag soup parsers do. So we cannot shorten
+	// this by omitting <tbody> or other required elements.
+	thead: [ 1, "<table>", "</table>" ],
+	col: [ 2, "<table><colgroup>", "</colgroup></table>" ],
+	tr: [ 2, "<table><tbody>", "</tbody></table>" ],
+	td: [ 3, "<table><tbody><tr>", "</tr></tbody></table>" ],
+
+	_default: [ 0, "", "" ]
+};
+
+wrapMap.tbody = wrapMap.tfoot = wrapMap.colgroup = wrapMap.caption = wrapMap.thead;
+wrapMap.th = wrapMap.td;
+
+// Support: IE <=9 only
+if ( !support.option ) {
+	wrapMap.optgroup = wrapMap.option = [ 1, "<select multiple='multiple'>", "</select>" ];
+}
+
+
+function getAll( context, tag ) {
+
+	// Support: IE <=9 - 11 only
+	// Use typeof to avoid zero-argument method invocation on host objects (#15151)
+	var ret;
+
+	if ( typeof context.getElementsByTagName !== "undefined" ) {
+		ret = context.getElementsByTagName( tag || "*" );
+
+	} else if ( typeof context.querySelectorAll !== "undefined" ) {
+		ret = context.querySelectorAll( tag || "*" );
+
+	} else {
+		ret = [];
+	}
+
+	if ( tag === undefined || tag && nodeName( context, tag ) ) {
+		return jQuery.merge( [ context ], ret );
+	}
+
+	return ret;
+}
+
+
+// Mark scripts as having already been evaluated
+function setGlobalEval( elems, refElements ) {
+	var i = 0,
+		l = elems.length;
+
+	for ( ; i < l; i++ ) {
+		dataPriv.set(
+			elems[ i ],
+			"globalEval",
+			!refElements || dataPriv.get( refElements[ i ], "globalEval" )
+		);
+	}
+}
+
+
+var rhtml = /<|&#?\w+;/;
+
+function buildFragment( elems, context, scripts, selection, ignored ) {
+	var elem, tmp, tag, wrap, attached, j,
+		fragment = context.createDocumentFragment(),
+		nodes = [],
+		i = 0,
+		l = elems.length;
+
+	for ( ; i < l; i++ ) {
+		elem = elems[ i ];
+
+		if ( elem || elem === 0 ) {
+
+			// Add nodes directly
+			if ( toType( elem ) === "object" ) {
+
+				// Support: Android <=4.0 only, PhantomJS 1 only
+				// push.apply(_, arraylike) throws on ancient WebKit
+				jQuery.merge( nodes, elem.nodeType ? [ elem ] : elem );
+
+			// Convert non-html into a text node
+			} else if ( !rhtml.test( elem ) ) {
+				nodes.push( context.createTextNode( elem ) );
+
+			// Convert html into DOM nodes
+			} else {
+				tmp = tmp || fragment.appendChild( context.createElement( "div" ) );
+
+				// Deserialize a standard representation
+				tag = ( rtagName.exec( elem ) || [ "", "" ] )[ 1 ].toLowerCase();
+				wrap = wrapMap[ tag ] || wrapMap._default;
+				tmp.innerHTML = wrap[ 1 ] + jQuery.htmlPrefilter( elem ) + wrap[ 2 ];
+
+				// Descend through wrappers to the right content
+				j = wrap[ 0 ];
+				while ( j-- ) {
+					tmp = tmp.lastChild;
+				}
+
+				// Support: Android <=4.0 only, PhantomJS 1 only
+				// push.apply(_, arraylike) throws on ancient WebKit
+				jQuery.merge( nodes, tmp.childNodes );
+
+				// Remember the top-level container
+				tmp = fragment.firstChild;
+
+				// Ensure the created nodes are orphaned (#12392)
+				tmp.textContent = "";
+			}
+		}
+	}
+
+	// Remove wrapper from fragment
+	fragment.textContent = "";
+
+	i = 0;
+	while ( ( elem = nodes[ i++ ] ) ) {
+
+		// Skip elements already in the context collection (trac-4087)
+		if ( selection && jQuery.inArray( elem, selection ) > -1 ) {
+			if ( ignored ) {
+				ignored.push( elem );
+			}
+			continue;
+		}
+
+		attached = isAttached( elem );
+
+		// Append to fragment
+		tmp = getAll( fragment.appendChild( elem ), "script" );
+
+		// Preserve script evaluation history
+		if ( attached ) {
+			setGlobalEval( tmp );
+		}
+
+		// Capture executables
+		if ( scripts ) {
+			j = 0;
+			while ( ( elem = tmp[ j++ ] ) ) {
+				if ( rscriptType.test( elem.type || "" ) ) {
+					scripts.push( elem );
+				}
+			}
+		}
+	}
+
+	return fragment;
+}
+
+
+var rtypenamespace = /^([^.]*)(?:\.(.+)|)/;
+
+function returnTrue() {
+	return true;
+}
+
+function returnFalse() {
+	return false;
+}
+
+// Support: IE <=9 - 11+
+// focus() and blur() are asynchronous, except when they are no-op.
+// So expect focus to be synchronous when the element is already active,
+// and blur to be synchronous when the element is not already active.
+// (focus and blur are always synchronous in other supported browsers,
+// this just defines when we can count on it).
+function expectSync( elem, type ) {
+	return ( elem === safeActiveElement() ) === ( type === "focus" );
+}
+
+// Support: IE <=9 only
+// Accessing document.activeElement can throw unexpectedly
+// https://bugs.jquery.com/ticket/13393
+function safeActiveElement() {
+	try {
+		return document.activeElement;
+	} catch ( err ) { }
+}
+
+function on( elem, types, selector, data, fn, one ) {
+	var origFn, type;
+
+	// Types can be a map of types/handlers
+	if ( typeof types === "object" ) {
+
+		// ( types-Object, selector, data )
+		if ( typeof selector !== "string" ) {
+
+			// ( types-Object, data )
+			data = data || selector;
+			selector = undefined;
+		}
+		for ( type in types ) {
+			on( elem, type, selector, data, types[ type ], one );
+		}
+		return elem;
+	}
+
+	if ( data == null && fn == null ) {
+
+		// ( types, fn )
+		fn = selector;
+		data = selector = undefined;
+	} else if ( fn == null ) {
+		if ( typeof selector === "string" ) {
+
+			// ( types, selector, fn )
+			fn = data;
+			data = undefined;
+		} else {
+
+			// ( types, data, fn )
+			fn = data;
+			data = selector;
+			selector = undefined;
+		}
+	}
+	if ( fn === false ) {
+		fn = returnFalse;
+	} else if ( !fn ) {
+		return elem;
+	}
+
+	if ( one === 1 ) {
+		origFn = fn;
+		fn = function( event ) {
+
+			// Can use an empty set, since event contains the info
+			jQuery().off( event );
+			return origFn.apply( this, arguments );
+		};
+
+		// Use same guid so caller can remove using origFn
+		fn.guid = origFn.guid || ( origFn.guid = jQuery.guid++ );
+	}
+	return elem.each( function() {
+		jQuery.event.add( this, types, fn, data, selector );
+	} );
+}
+
+/*
+ * Helper functions for managing events -- not part of the public interface.
+ * Props to Dean Edwards' addEvent library for many of the ideas.
+ */
+jQuery.event = {
+
+	global: {},
+
+	add: function( elem, types, handler, data, selector ) {
+
+		var handleObjIn, eventHandle, tmp,
+			events, t, handleObj,
+			special, handlers, type, namespaces, origType,
+			elemData = dataPriv.get( elem );
+
+		// Only attach events to objects that accept data
+		if ( !acceptData( elem ) ) {
+			return;
+		}
+
+		// Caller can pass in an object of custom data in lieu of the handler
+		if ( handler.handler ) {
+			handleObjIn = handler;
+			handler = handleObjIn.handler;
+			selector = handleObjIn.selector;
+		}
+
+		// Ensure that invalid selectors throw exceptions at attach time
+		// Evaluate against documentElement in case elem is a non-element node (e.g., document)
+		if ( selector ) {
+			jQuery.find.matchesSelector( documentElement, selector );
+		}
+
+		// Make sure that the handler has a unique ID, used to find/remove it later
+		if ( !handler.guid ) {
+			handler.guid = jQuery.guid++;
+		}
+
+		// Init the element's event structure and main handler, if this is the first
+		if ( !( events = elemData.events ) ) {
+			events = elemData.events = Object.create( null );
+		}
+		if ( !( eventHandle = elemData.handle ) ) {
+			eventHandle = elemData.handle = function( e ) {
+
+				// Discard the second event of a jQuery.event.trigger() and
+				// when an event is called after a page has unloaded
+				return typeof jQuery !== "undefined" && jQuery.event.triggered !== e.type ?
+					jQuery.event.dispatch.apply( elem, arguments ) : undefined;
+			};
+		}
+
+		// Handle multiple events separated by a space
+		types = ( types || "" ).match( rnothtmlwhite ) || [ "" ];
+		t = types.length;
+		while ( t-- ) {
+			tmp = rtypenamespace.exec( types[ t ] ) || [];
+			type = origType = tmp[ 1 ];
+			namespaces = ( tmp[ 2 ] || "" ).split( "." ).sort();
+
+			// There *must* be a type, no attaching namespace-only handlers
+			if ( !type ) {
+				continue;
+			}
+
+			// If event changes its type, use the special event handlers for the changed type
+			special = jQuery.event.special[ type ] || {};
+
+			// If selector defined, determine special event api type, otherwise given type
+			type = ( selector ? special.delegateType : special.bindType ) || type;
+
+			// Update special based on newly reset type
+			special = jQuery.event.special[ type ] || {};
+
+			// handleObj is passed to all event handlers
+			handleObj = jQuery.extend( {
+				type: type,
+				origType: origType,
+				data: data,
+				handler: handler,
+				guid: handler.guid,
+				selector: selector,
+				needsContext: selector && jQuery.expr.match.needsContext.test( selector ),
+				namespace: namespaces.join( "." )
+			}, handleObjIn );
+
+			// Init the event handler queue if we're the first
+			if ( !( handlers = events[ type ] ) ) {
+				handlers = events[ type ] = [];
+				handlers.delegateCount = 0;
+
+				// Only use addEventListener if the special events handler returns false
+				if ( !special.setup ||
+					special.setup.call( elem, data, namespaces, eventHandle ) === false ) {
+
+					if ( elem.addEventListener ) {
+						elem.addEventListener( type, eventHandle );
+					}
+				}
+			}
+
+			if ( special.add ) {
+				special.add.call( elem, handleObj );
+
+				if ( !handleObj.handler.guid ) {
+					handleObj.handler.guid = handler.guid;
+				}
+			}
+
+			// Add to the element's handler list, delegates in front
+			if ( selector ) {
+				handlers.splice( handlers.delegateCount++, 0, handleObj );
+			} else {
+				handlers.push( handleObj );
+			}
+
+			// Keep track of which events have ever been used, for event optimization
+			jQuery.event.global[ type ] = true;
+		}
+
+	},
+
+	// Detach an event or set of events from an element
+	remove: function( elem, types, handler, selector, mappedTypes ) {
+
+		var j, origCount, tmp,
+			events, t, handleObj,
+			special, handlers, type, namespaces, origType,
+			elemData = dataPriv.hasData( elem ) && dataPriv.get( elem );
+
+		if ( !elemData || !( events = elemData.events ) ) {
+			return;
+		}
+
+		// Once for each type.namespace in types; type may be omitted
+		types = ( types || "" ).match( rnothtmlwhite ) || [ "" ];
+		t = types.length;
+		while ( t-- ) {
+			tmp = rtypenamespace.exec( types[ t ] ) || [];
+			type = origType = tmp[ 1 ];
+			namespaces = ( tmp[ 2 ] || "" ).split( "." ).sort();
+
+			// Unbind all events (on this namespace, if provided) for the element
+			if ( !type ) {
+				for ( type in events ) {
+					jQuery.event.remove( elem, type + types[ t ], handler, selector, true );
+				}
+				continue;
+			}
+
+			special = jQuery.event.special[ type ] || {};
+			type = ( selector ? special.delegateType : special.bindType ) || type;
+			handlers = events[ type ] || [];
+			tmp = tmp[ 2 ] &&
+				new RegExp( "(^|\\.)" + namespaces.join( "\\.(?:.*\\.|)" ) + "(\\.|$)" );
+
+			// Remove matching events
+			origCount = j = handlers.length;
+			while ( j-- ) {
+				handleObj = handlers[ j ];
+
+				if ( ( mappedTypes || origType === handleObj.origType ) &&
+					( !handler || handler.guid === handleObj.guid ) &&
+					( !tmp || tmp.test( handleObj.namespace ) ) &&
+					( !selector || selector === handleObj.selector ||
+						selector === "**" && handleObj.selector ) ) {
+					handlers.splice( j, 1 );
+
+					if ( handleObj.selector ) {
+						handlers.delegateCount--;
+					}
+					if ( special.remove ) {
+						special.remove.call( elem, handleObj );
+					}
+				}
+			}
+
+			// Remove generic event handler if we removed something and no more handlers exist
+			// (avoids potential for endless recursion during removal of special event handlers)
+			if ( origCount && !handlers.length ) {
+				if ( !special.teardown ||
+					special.teardown.call( elem, namespaces, elemData.handle ) === false ) {
+
+					jQuery.removeEvent( elem, type, elemData.handle );
+				}
+
+				delete events[ type ];
+			}
+		}
+
+		// Remove data and the expando if it's no longer used
+		if ( jQuery.isEmptyObject( events ) ) {
+			dataPriv.remove( elem, "handle events" );
+		}
+	},
+
+	dispatch: function( nativeEvent ) {
+
+		var i, j, ret, matched, handleObj, handlerQueue,
+			args = new Array( arguments.length ),
+
+			// Make a writable jQuery.Event from the native event object
+			event = jQuery.event.fix( nativeEvent ),
+
+			handlers = (
+				dataPriv.get( this, "events" ) || Object.create( null )
+			)[ event.type ] || [],
+			special = jQuery.event.special[ event.type ] || {};
+
+		// Use the fix-ed jQuery.Event rather than the (read-only) native event
+		args[ 0 ] = event;
+
+		for ( i = 1; i < arguments.length; i++ ) {
+			args[ i ] = arguments[ i ];
+		}
+
+		event.delegateTarget = this;
+
+		// Call the preDispatch hook for the mapped type, and let it bail if desired
+		if ( special.preDispatch && special.preDispatch.call( this, event ) === false ) {
+			return;
+		}
+
+		// Determine handlers
+		handlerQueue = jQuery.event.handlers.call( this, event, handlers );
+
+		// Run delegates first; they may want to stop propagation beneath us
+		i = 0;
+		while ( ( matched = handlerQueue[ i++ ] ) && !event.isPropagationStopped() ) {
+			event.currentTarget = matched.elem;
+
+			j = 0;
+			while ( ( handleObj = matched.handlers[ j++ ] ) &&
+				!event.isImmediatePropagationStopped() ) {
+
+				// If the event is namespaced, then each handler is only invoked if it is
+				// specially universal or its namespaces are a superset of the event's.
+				if ( !event.rnamespace || handleObj.namespace === false ||
+					event.rnamespace.test( handleObj.namespace ) ) {
+
+					event.handleObj = handleObj;
+					event.data = handleObj.data;
+
+					ret = ( ( jQuery.event.special[ handleObj.origType ] || {} ).handle ||
+						handleObj.handler ).apply( matched.elem, args );
+
+					if ( ret !== undefined ) {
+						if ( ( event.result = ret ) === false ) {
+							event.preventDefault();
+							event.stopPropagation();
+						}
+					}
+				}
+			}
+		}
+
+		// Call the postDispatch hook for the mapped type
+		if ( special.postDispatch ) {
+			special.postDispatch.call( this, event );
+		}
+
+		return event.result;
+	},
+
+	handlers: function( event, handlers ) {
+		var i, handleObj, sel, matchedHandlers, matchedSelectors,
+			handlerQueue = [],
+			delegateCount = handlers.delegateCount,
+			cur = event.target;
+
+		// Find delegate handlers
+		if ( delegateCount &&
+
+			// Support: IE <=9
+			// Black-hole SVG <use> instance trees (trac-13180)
+			cur.nodeType &&
+
+			// Support: Firefox <=42
+			// Suppress spec-violating clicks indicating a non-primary pointer button (trac-3861)
+			// https://www.w3.org/TR/DOM-Level-3-Events/#event-type-click
+			// Support: IE 11 only
+			// ...but not arrow key "clicks" of radio inputs, which can have `button` -1 (gh-2343)
+			!( event.type === "click" && event.button >= 1 ) ) {
+
+			for ( ; cur !== this; cur = cur.parentNode || this ) {
+
+				// Don't check non-elements (#13208)
+				// Don't process clicks on disabled elements (#6911, #8165, #11382, #11764)
+				if ( cur.nodeType === 1 && !( event.type === "click" && cur.disabled === true ) ) {
+					matchedHandlers = [];
+					matchedSelectors = {};
+					for ( i = 0; i < delegateCount; i++ ) {
+						handleObj = handlers[ i ];
+
+						// Don't conflict with Object.prototype properties (#13203)
+						sel = handleObj.selector + " ";
+
+						if ( matchedSelectors[ sel ] === undefined ) {
+							matchedSelectors[ sel ] = handleObj.needsContext ?
+								jQuery( sel, this ).index( cur ) > -1 :
+								jQuery.find( sel, this, null, [ cur ] ).length;
+						}
+						if ( matchedSelectors[ sel ] ) {
+							matchedHandlers.push( handleObj );
+						}
+					}
+					if ( matchedHandlers.length ) {
+						handlerQueue.push( { elem: cur, handlers: matchedHandlers } );
+					}
+				}
+			}
+		}
+
+		// Add the remaining (directly-bound) handlers
+		cur = this;
+		if ( delegateCount < handlers.length ) {
+			handlerQueue.push( { elem: cur, handlers: handlers.slice( delegateCount ) } );
+		}
+
+		return handlerQueue;
+	},
+
+	addProp: function( name, hook ) {
+		Object.defineProperty( jQuery.Event.prototype, name, {
+			enumerable: true,
+			configurable: true,
+
+			get: isFunction( hook ) ?
+				function() {
+					if ( this.originalEvent ) {
+						return hook( this.originalEvent );
+					}
+				} :
+				function() {
+					if ( this.originalEvent ) {
+						return this.originalEvent[ name ];
+					}
+				},
+
+			set: function( value ) {
+				Object.defineProperty( this, name, {
+					enumerable: true,
+					configurable: true,
+					writable: true,
+					value: value
+				} );
+			}
+		} );
+	},
+
+	fix: function( originalEvent ) {
+		return originalEvent[ jQuery.expando ] ?
+			originalEvent :
+			new jQuery.Event( originalEvent );
+	},
+
+	special: {
+		load: {
+
+			// Prevent triggered image.load events from bubbling to window.load
+			noBubble: true
+		},
+		click: {
+
+			// Utilize native event to ensure correct state for checkable inputs
+			setup: function( data ) {
+
+				// For mutual compressibility with _default, replace `this` access with a local var.
+				// `|| data` is dead code meant only to preserve the variable through minification.
+				var el = this || data;
+
+				// Claim the first handler
+				if ( rcheckableType.test( el.type ) &&
+					el.click && nodeName( el, "input" ) ) {
+
+					// dataPriv.set( el, "click", ... )
+					leverageNative( el, "click", returnTrue );
+				}
+
+				// Return false to allow normal processing in the caller
+				return false;
+			},
+			trigger: function( data ) {
+
+				// For mutual compressibility with _default, replace `this` access with a local var.
+				// `|| data` is dead code meant only to preserve the variable through minification.
+				var el = this || data;
+
+				// Force setup before triggering a click
+				if ( rcheckableType.test( el.type ) &&
+					el.click && nodeName( el, "input" ) ) {
+
+					leverageNative( el, "click" );
+				}
+
+				// Return non-false to allow normal event-path propagation
+				return true;
+			},
+
+			// For cross-browser consistency, suppress native .click() on links
+			// Also prevent it if we're currently inside a leveraged native-event stack
+			_default: function( event ) {
+				var target = event.target;
+				return rcheckableType.test( target.type ) &&
+					target.click && nodeName( target, "input" ) &&
+					dataPriv.get( target, "click" ) ||
+					nodeName( target, "a" );
+			}
+		},
+
+		beforeunload: {
+			postDispatch: function( event ) {
+
+				// Support: Firefox 20+
+				// Firefox doesn't alert if the returnValue field is not set.
+				if ( event.result !== undefined && event.originalEvent ) {
+					event.originalEvent.returnValue = event.result;
+				}
+			}
+		}
+	}
+};
+
+// Ensure the presence of an event listener that handles manually-triggered
+// synthetic events by interrupting progress until reinvoked in response to
+// *native* events that it fires directly, ensuring that state changes have
+// already occurred before other listeners are invoked.
+function leverageNative( el, type, expectSync ) {
+
+	// Missing expectSync indicates a trigger call, which must force setup through jQuery.event.add
+	if ( !expectSync ) {
+		if ( dataPriv.get( el, type ) === undefined ) {
+			jQuery.event.add( el, type, returnTrue );
+		}
+		return;
+	}
+
+	// Register the controller as a special universal handler for all event namespaces
+	dataPriv.set( el, type, false );
+	jQuery.event.add( el, type, {
+		namespace: false,
+		handler: function( event ) {
+			var notAsync, result,
+				saved = dataPriv.get( this, type );
+
+			if ( ( event.isTrigger & 1 ) && this[ type ] ) {
+
+				// Interrupt processing of the outer synthetic .trigger()ed event
+				// Saved data should be false in such cases, but might be a leftover capture object
+				// from an async native handler (gh-4350)
+				if ( !saved.length ) {
+
+					// Store arguments for use when handling the inner native event
+					// There will always be at least one argument (an event object), so this array
+					// will not be confused with a leftover capture object.
+					saved = slice.call( arguments );
+					dataPriv.set( this, type, saved );
+
+					// Trigger the native event and capture its result
+					// Support: IE <=9 - 11+
+					// focus() and blur() are asynchronous
+					notAsync = expectSync( this, type );
+					this[ type ]();
+					result = dataPriv.get( this, type );
+					if ( saved !== result || notAsync ) {
+						dataPriv.set( this, type, false );
+					} else {
+						result = {};
+					}
+					if ( saved !== result ) {
+
+						// Cancel the outer synthetic event
+						event.stopImmediatePropagation();
+						event.preventDefault();
+
+						// Support: Chrome 86+
+						// In Chrome, if an element having a focusout handler is blurred by
+						// clicking outside of it, it invokes the handler synchronously. If
+						// that handler calls `.remove()` on the element, the data is cleared,
+						// leaving `result` undefined. We need to guard against this.
+						return result && result.value;
+					}
+
+				// If this is an inner synthetic event for an event with a bubbling surrogate
+				// (focus or blur), assume that the surrogate already propagated from triggering the
+				// native event and prevent that from happening again here.
+				// This technically gets the ordering wrong w.r.t. to `.trigger()` (in which the
+				// bubbling surrogate propagates *after* the non-bubbling base), but that seems
+				// less bad than duplication.
+				} else if ( ( jQuery.event.special[ type ] || {} ).delegateType ) {
+					event.stopPropagation();
+				}
+
+			// If this is a native event triggered above, everything is now in order
+			// Fire an inner synthetic event with the original arguments
+			} else if ( saved.length ) {
+
+				// ...and capture the result
+				dataPriv.set( this, type, {
+					value: jQuery.event.trigger(
+
+						// Support: IE <=9 - 11+
+						// Extend with the prototype to reset the above stopImmediatePropagation()
+						jQuery.extend( saved[ 0 ], jQuery.Event.prototype ),
+						saved.slice( 1 ),
+						this
+					)
+				} );
+
+				// Abort handling of the native event
+				event.stopImmediatePropagation();
+			}
+		}
+	} );
+}
+
+jQuery.removeEvent = function( elem, type, handle ) {
+
+	// This "if" is needed for plain objects
+	if ( elem.removeEventListener ) {
+		elem.removeEventListener( type, handle );
+	}
+};
+
+jQuery.Event = function( src, props ) {
+
+	// Allow instantiation without the 'new' keyword
+	if ( !( this instanceof jQuery.Event ) ) {
+		return new jQuery.Event( src, props );
+	}
+
+	// Event object
+	if ( src && src.type ) {
+		this.originalEvent = src;
+		this.type = src.type;
+
+		// Events bubbling up the document may have been marked as prevented
+		// by a handler lower down the tree; reflect the correct value.
+		this.isDefaultPrevented = src.defaultPrevented ||
+				src.defaultPrevented === undefined &&
+
+				// Support: Android <=2.3 only
+				src.returnValue === false ?
+			returnTrue :
+			returnFalse;
+
+		// Create target properties
+		// Support: Safari <=6 - 7 only
+		// Target should not be a text node (#504, #13143)
+		this.target = ( src.target && src.target.nodeType === 3 ) ?
+			src.target.parentNode :
+			src.target;
+
+		this.currentTarget = src.currentTarget;
+		this.relatedTarget = src.relatedTarget;
+
+	// Event type
+	} else {
+		this.type = src;
+	}
+
+	// Put explicitly provided properties onto the event object
+	if ( props ) {
+		jQuery.extend( this, props );
+	}
+
+	// Create a timestamp if incoming event doesn't have one
+	this.timeStamp = src && src.timeStamp || Date.now();
+
+	// Mark it as fixed
+	this[ jQuery.expando ] = true;
+};
+
+// jQuery.Event is based on DOM3 Events as specified by the ECMAScript Language Binding
+// https://www.w3.org/TR/2003/WD-DOM-Level-3-Events-20030331/ecma-script-binding.html
+jQuery.Event.prototype = {
+	constructor: jQuery.Event,
+	isDefaultPrevented: returnFalse,
+	isPropagationStopped: returnFalse,
+	isImmediatePropagationStopped: returnFalse,
+	isSimulated: false,
+
+	preventDefault: function() {
+		var e = this.originalEvent;
+
+		this.isDefaultPrevented = returnTrue;
+
+		if ( e && !this.isSimulated ) {
+			e.preventDefault();
+		}
+	},
+	stopPropagation: function() {
+		var e = this.originalEvent;
+
+		this.isPropagationStopped = returnTrue;
+
+		if ( e && !this.isSimulated ) {
+			e.stopPropagation();
+		}
+	},
+	stopImmediatePropagation: function() {
+		var e = this.originalEvent;
+
+		this.isImmediatePropagationStopped = returnTrue;
+
+		if ( e && !this.isSimulated ) {
+			e.stopImmediatePropagation();
+		}
+
+		this.stopPropagation();
+	}
+};
+
+// Includes all common event props including KeyEvent and MouseEvent specific props
+jQuery.each( {
+	altKey: true,
+	bubbles: true,
+	cancelable: true,
+	changedTouches: true,
+	ctrlKey: true,
+	detail: true,
+	eventPhase: true,
+	metaKey: true,
+	pageX: true,
+	pageY: true,
+	shiftKey: true,
+	view: true,
+	"char": true,
+	code: true,
+	charCode: true,
+	key: true,
+	keyCode: true,
+	button: true,
+	buttons: true,
+	clientX: true,
+	clientY: true,
+	offsetX: true,
+	offsetY: true,
+	pointerId: true,
+	pointerType: true,
+	screenX: true,
+	screenY: true,
+	targetTouches: true,
+	toElement: true,
+	touches: true,
+	which: true
+}, jQuery.event.addProp );
+
+jQuery.each( { focus: "focusin", blur: "focusout" }, function( type, delegateType ) {
+	jQuery.event.special[ type ] = {
+
+		// Utilize native event if possible so blur/focus sequence is correct
+		setup: function() {
+
+			// Claim the first handler
+			// dataPriv.set( this, "focus", ... )
+			// dataPriv.set( this, "blur", ... )
+			leverageNative( this, type, expectSync );
+
+			// Return false to allow normal processing in the caller
+			return false;
+		},
+		trigger: function() {
+
+			// Force setup before trigger
+			leverageNative( this, type );
+
+			// Return non-false to allow normal event-path propagation
+			return true;
+		},
+
+		// Suppress native focus or blur as it's already being fired
+		// in leverageNative.
+		_default: function() {
+			return true;
+		},
+
+		delegateType: delegateType
+	};
+} );
+
+// Create mouseenter/leave events using mouseover/out and event-time checks
+// so that event delegation works in jQuery.
+// Do the same for pointerenter/pointerleave and pointerover/pointerout
+//
+// Support: Safari 7 only
+// Safari sends mouseenter too often; see:
+// https://bugs.chromium.org/p/chromium/issues/detail?id=470258
+// for the description of the bug (it existed in older Chrome versions as well).
+jQuery.each( {
+	mouseenter: "mouseover",
+	mouseleave: "mouseout",
+	pointerenter: "pointerover",
+	pointerleave: "pointerout"
+}, function( orig, fix ) {
+	jQuery.event.special[ orig ] = {
+		delegateType: fix,
+		bindType: fix,
+
+		handle: function( event ) {
+			var ret,
+				target = this,
+				related = event.relatedTarget,
+				handleObj = event.handleObj;
+
+			// For mouseenter/leave call the handler if related is outside the target.
+			// NB: No relatedTarget if the mouse left/entered the browser window
+			if ( !related || ( related !== target && !jQuery.contains( target, related ) ) ) {
+				event.type = handleObj.origType;
+				ret = handleObj.handler.apply( this, arguments );
+				event.type = fix;
+			}
+			return ret;
+		}
+	};
+} );
+
+jQuery.fn.extend( {
+
+	on: function( types, selector, data, fn ) {
+		return on( this, types, selector, data, fn );
+	},
+	one: function( types, selector, data, fn ) {
+		return on( this, types, selector, data, fn, 1 );
+	},
+	off: function( types, selector, fn ) {
+		var handleObj, type;
+		if ( types && types.preventDefault && types.handleObj ) {
+
+			// ( event )  dispatched jQuery.Event
+			handleObj = types.handleObj;
+			jQuery( types.delegateTarget ).off(
+				handleObj.namespace ?
+					handleObj.origType + "." + handleObj.namespace :
+					handleObj.origType,
+				handleObj.selector,
+				handleObj.handler
+			);
+			return this;
+		}
+		if ( typeof types === "object" ) {
+
+			// ( types-object [, selector] )
+			for ( type in types ) {
+				this.off( type, selector, types[ type ] );
+			}
+			return this;
+		}
+		if ( selector === false || typeof selector === "function" ) {
+
+			// ( types [, fn] )
+			fn = selector;
+			selector = undefined;
+		}
+		if ( fn === false ) {
+			fn = returnFalse;
+		}
+		return this.each( function() {
+			jQuery.event.remove( this, types, fn, selector );
+		} );
+	}
+} );
+
+
+var
+
+	// Support: IE <=10 - 11, Edge 12 - 13 only
+	// In IE/Edge using regex groups here causes severe slowdowns.
+	// See https://connect.microsoft.com/IE/feedback/details/1736512/
+	rnoInnerhtml = /<script|<style|<link/i,
+
+	// checked="checked" or checked
+	rchecked = /checked\s*(?:[^=]|=\s*.checked.)/i,
+	rcleanScript = /^\s*<!(?:\[CDATA\[|--)|(?:\]\]|--)>\s*$/g;
+
+// Prefer a tbody over its parent table for containing new rows
+function manipulationTarget( elem, content ) {
+	if ( nodeName( elem, "table" ) &&
+		nodeName( content.nodeType !== 11 ? content : content.firstChild, "tr" ) ) {
+
+		return jQuery( elem ).children( "tbody" )[ 0 ] || elem;
+	}
+
+	return elem;
+}
+
+// Replace/restore the type attribute of script elements for safe DOM manipulation
+function disableScript( elem ) {
+	elem.type = ( elem.getAttribute( "type" ) !== null ) + "/" + elem.type;
+	return elem;
+}
+function restoreScript( elem ) {
+	if ( ( elem.type || "" ).slice( 0, 5 ) === "true/" ) {
+		elem.type = elem.type.slice( 5 );
+	} else {
+		elem.removeAttribute( "type" );
+	}
+
+	return elem;
+}
+
+function cloneCopyEvent( src, dest ) {
+	var i, l, type, pdataOld, udataOld, udataCur, events;
+
+	if ( dest.nodeType !== 1 ) {
+		return;
+	}
+
+	// 1. Copy private data: events, handlers, etc.
+	if ( dataPriv.hasData( src ) ) {
+		pdataOld = dataPriv.get( src );
+		events = pdataOld.events;
+
+		if ( events ) {
+			dataPriv.remove( dest, "handle events" );
+
+			for ( type in events ) {
+				for ( i = 0, l = events[ type ].length; i < l; i++ ) {
+					jQuery.event.add( dest, type, events[ type ][ i ] );
+				}
+			}
+		}
+	}
+
+	// 2. Copy user data
+	if ( dataUser.hasData( src ) ) {
+		udataOld = dataUser.access( src );
+		udataCur = jQuery.extend( {}, udataOld );
+
+		dataUser.set( dest, udataCur );
+	}
+}
+
+// Fix IE bugs, see support tests
+function fixInput( src, dest ) {
+	var nodeName = dest.nodeName.toLowerCase();
+
+	// Fails to persist the checked state of a cloned checkbox or radio button.
+	if ( nodeName === "input" && rcheckableType.test( src.type ) ) {
+		dest.checked = src.checked;
+
+	// Fails to return the selected option to the default selected state when cloning options
+	} else if ( nodeName === "input" || nodeName === "textarea" ) {
+		dest.defaultValue = src.defaultValue;
+	}
+}
+
+function domManip( collection, args, callback, ignored ) {
+
+	// Flatten any nested arrays
+	args = flat( args );
+
+	var fragment, first, scripts, hasScripts, node, doc,
+		i = 0,
+		l = collection.length,
+		iNoClone = l - 1,
+		value = args[ 0 ],
+		valueIsFunction = isFunction( value );
+
+	// We can't cloneNode fragments that contain checked, in WebKit
+	if ( valueIsFunction ||
+			( l > 1 && typeof value === "string" &&
+				!support.checkClone && rchecked.test( value ) ) ) {
+		return collection.each( function( index ) {
+			var self = collection.eq( index );
+			if ( valueIsFunction ) {
+				args[ 0 ] = value.call( this, index, self.html() );
+			}
+			domManip( self, args, callback, ignored );
+		} );
+	}
+
+	if ( l ) {
+		fragment = buildFragment( args, collection[ 0 ].ownerDocument, false, collection, ignored );
+		first = fragment.firstChild;
+
+		if ( fragment.childNodes.length === 1 ) {
+			fragment = first;
+		}
+
+		// Require either new content or an interest in ignored elements to invoke the callback
+		if ( first || ignored ) {
+			scripts = jQuery.map( getAll( fragment, "script" ), disableScript );
+			hasScripts = scripts.length;
+
+			// Use the original fragment for the last item
+			// instead of the first because it can end up
+			// being emptied incorrectly in certain situations (#8070).
+			for ( ; i < l; i++ ) {
+				node = fragment;
+
+				if ( i !== iNoClone ) {
+					node = jQuery.clone( node, true, true );
+
+					// Keep references to cloned scripts for later restoration
+					if ( hasScripts ) {
+
+						// Support: Android <=4.0 only, PhantomJS 1 only
+						// push.apply(_, arraylike) throws on ancient WebKit
+						jQuery.merge( scripts, getAll( node, "script" ) );
+					}
+				}
+
+				callback.call( collection[ i ], node, i );
+			}
+
+			if ( hasScripts ) {
+				doc = scripts[ scripts.length - 1 ].ownerDocument;
+
+				// Reenable scripts
+				jQuery.map( scripts, restoreScript );
+
+				// Evaluate executable scripts on first document insertion
+				for ( i = 0; i < hasScripts; i++ ) {
+					node = scripts[ i ];
+					if ( rscriptType.test( node.type || "" ) &&
+						!dataPriv.access( node, "globalEval" ) &&
+						jQuery.contains( doc, node ) ) {
+
+						if ( node.src && ( node.type || "" ).toLowerCase()  !== "module" ) {
+
+							// Optional AJAX dependency, but won't run scripts if not present
+							if ( jQuery._evalUrl && !node.noModule ) {
+								jQuery._evalUrl( node.src, {
+									nonce: node.nonce || node.getAttribute( "nonce" )
+								}, doc );
+							}
+						} else {
+							DOMEval( node.textContent.replace( rcleanScript, "" ), node, doc );
+						}
+					}
+				}
+			}
+		}
+	}
+
+	return collection;
+}
+
+function remove( elem, selector, keepData ) {
+	var node,
+		nodes = selector ? jQuery.filter( selector, elem ) : elem,
+		i = 0;
+
+	for ( ; ( node = nodes[ i ] ) != null; i++ ) {
+		if ( !keepData && node.nodeType === 1 ) {
+			jQuery.cleanData( getAll( node ) );
+		}
+
+		if ( node.parentNode ) {
+			if ( keepData && isAttached( node ) ) {
+				setGlobalEval( getAll( node, "script" ) );
+			}
+			node.parentNode.removeChild( node );
+		}
+	}
+
+	return elem;
+}
+
+jQuery.extend( {
+	htmlPrefilter: function( html ) {
+		return html;
+	},
+
+	clone: function( elem, dataAndEvents, deepDataAndEvents ) {
+		var i, l, srcElements, destElements,
+			clone = elem.cloneNode( true ),
+			inPage = isAttached( elem );
+
+		// Fix IE cloning issues
+		if ( !support.noCloneChecked && ( elem.nodeType === 1 || elem.nodeType === 11 ) &&
+				!jQuery.isXMLDoc( elem ) ) {
+
+			// We eschew Sizzle here for performance reasons: https://jsperf.com/getall-vs-sizzle/2
+			destElements = getAll( clone );
+			srcElements = getAll( elem );
+
+			for ( i = 0, l = srcElements.length; i < l; i++ ) {
+				fixInput( srcElements[ i ], destElements[ i ] );
+			}
+		}
+
+		// Copy the events from the original to the clone
+		if ( dataAndEvents ) {
+			if ( deepDataAndEvents ) {
+				srcElements = srcElements || getAll( elem );
+				destElements = destElements || getAll( clone );
+
+				for ( i = 0, l = srcElements.length; i < l; i++ ) {
+					cloneCopyEvent( srcElements[ i ], destElements[ i ] );
+				}
+			} else {
+				cloneCopyEvent( elem, clone );
+			}
+		}
+
+		// Preserve script evaluation history
+		destElements = getAll( clone, "script" );
+		if ( destElements.length > 0 ) {
+			setGlobalEval( destElements, !inPage && getAll( elem, "script" ) );
+		}
+
+		// Return the cloned set
+		return clone;
+	},
+
+	cleanData: function( elems ) {
+		var data, elem, type,
+			special = jQuery.event.special,
+			i = 0;
+
+		for ( ; ( elem = elems[ i ] ) !== undefined; i++ ) {
+			if ( acceptData( elem ) ) {
+				if ( ( data = elem[ dataPriv.expando ] ) ) {
+					if ( data.events ) {
+						for ( type in data.events ) {
+							if ( special[ type ] ) {
+								jQuery.event.remove( elem, type );
+
+							// This is a shortcut to avoid jQuery.event.remove's overhead
+							} else {
+								jQuery.removeEvent( elem, type, data.handle );
+							}
+						}
+					}
+
+					// Support: Chrome <=35 - 45+
+					// Assign undefined instead of using delete, see Data#remove
+					elem[ dataPriv.expando ] = undefined;
+				}
+				if ( elem[ dataUser.expando ] ) {
+
+					// Support: Chrome <=35 - 45+
+					// Assign undefined instead of using delete, see Data#remove
+					elem[ dataUser.expando ] = undefined;
+				}
+			}
+		}
+	}
+} );
+
+jQuery.fn.extend( {
+	detach: function( selector ) {
+		return remove( this, selector, true );
+	},
+
+	remove: function( selector ) {
+		return remove( this, selector );
+	},
+
+	text: function( value ) {
+		return access( this, function( value ) {
+			return value === undefined ?
+				jQuery.text( this ) :
+				this.empty().each( function() {
+					if ( this.nodeType === 1 || this.nodeType === 11 || this.nodeType === 9 ) {
+						this.textContent = value;
+					}
+				} );
+		}, null, value, arguments.length );
+	},
+
+	append: function() {
+		return domManip( this, arguments, function( elem ) {
+			if ( this.nodeType === 1 || this.nodeType === 11 || this.nodeType === 9 ) {
+				var target = manipulationTarget( this, elem );
+				target.appendChild( elem );
+			}
+		} );
+	},
+
+	prepend: function() {
+		return domManip( this, arguments, function( elem ) {
+			if ( this.nodeType === 1 || this.nodeType === 11 || this.nodeType === 9 ) {
+				var target = manipulationTarget( this, elem );
+				target.insertBefore( elem, target.firstChild );
+			}
+		} );
+	},
+
+	before: function() {
+		return domManip( this, arguments, function( elem ) {
+			if ( this.parentNode ) {
+				this.parentNode.insertBefore( elem, this );
+			}
+		} );
+	},
+
+	after: function() {
+		return domManip( this, arguments, function( elem ) {
+			if ( this.parentNode ) {
+				this.parentNode.insertBefore( elem, this.nextSibling );
+			}
+		} );
+	},
+
+	empty: function() {
+		var elem,
+			i = 0;
+
+		for ( ; ( elem = this[ i ] ) != null; i++ ) {
+			if ( elem.nodeType === 1 ) {
+
+				// Prevent memory leaks
+				jQuery.cleanData( getAll( elem, false ) );
+
+				// Remove any remaining nodes
+				elem.textContent = "";
+			}
+		}
+
+		return this;
+	},
+
+	clone: function( dataAndEvents, deepDataAndEvents ) {
+		dataAndEvents = dataAndEvents == null ? false : dataAndEvents;
+		deepDataAndEvents = deepDataAndEvents == null ? dataAndEvents : deepDataAndEvents;
+
+		return this.map( function() {
+			return jQuery.clone( this, dataAndEvents, deepDataAndEvents );
+		} );
+	},
+
+	html: function( value ) {
+		return access( this, function( value ) {
+			var elem = this[ 0 ] || {},
+				i = 0,
+				l = this.length;
+
+			if ( value === undefined && elem.nodeType === 1 ) {
+				return elem.innerHTML;
+			}
+
+			// See if we can take a shortcut and just use innerHTML
+			if ( typeof value === "string" && !rnoInnerhtml.test( value ) &&
+				!wrapMap[ ( rtagName.exec( value ) || [ "", "" ] )[ 1 ].toLowerCase() ] ) {
+
+				value = jQuery.htmlPrefilter( value );
+
+				try {
+					for ( ; i < l; i++ ) {
+						elem = this[ i ] || {};
+
+						// Remove element nodes and prevent memory leaks
+						if ( elem.nodeType === 1 ) {
+							jQuery.cleanData( getAll( elem, false ) );
+							elem.innerHTML = value;
+						}
+					}
+
+					elem = 0;
+
+				// If using innerHTML throws an exception, use the fallback method
+				} catch ( e ) {}
+			}
+
+			if ( elem ) {
+				this.empty().append( value );
+			}
+		}, null, value, arguments.length );
+	},
+
+	replaceWith: function() {
+		var ignored = [];
+
+		// Make the changes, replacing each non-ignored context element with the new content
+		return domManip( this, arguments, function( elem ) {
+			var parent = this.parentNode;
+
+			if ( jQuery.inArray( this, ignored ) < 0 ) {
+				jQuery.cleanData( getAll( this ) );
+				if ( parent ) {
+					parent.replaceChild( elem, this );
+				}
+			}
+
+		// Force callback invocation
+		}, ignored );
+	}
+} );
+
+jQuery.each( {
+	appendTo: "append",
+	prependTo: "prepend",
+	insertBefore: "before",
+	insertAfter: "after",
+	replaceAll: "replaceWith"
+}, function( name, original ) {
+	jQuery.fn[ name ] = function( selector ) {
+		var elems,
+			ret = [],
+			insert = jQuery( selector ),
+			last = insert.length - 1,
+			i = 0;
+
+		for ( ; i <= last; i++ ) {
+			elems = i === last ? this : this.clone( true );
+			jQuery( insert[ i ] )[ original ]( elems );
+
+			// Support: Android <=4.0 only, PhantomJS 1 only
+			// .get() because push.apply(_, arraylike) throws on ancient WebKit
+			push.apply( ret, elems.get() );
+		}
+
+		return this.pushStack( ret );
+	};
+} );
+var rnumnonpx = new RegExp( "^(" + pnum + ")(?!px)[a-z%]+$", "i" );
+
+var getStyles = function( elem ) {
+
+		// Support: IE <=11 only, Firefox <=30 (#15098, #14150)
+		// IE throws on elements created in popups
+		// FF meanwhile throws on frame elements through "defaultView.getComputedStyle"
+		var view = elem.ownerDocument.defaultView;
+
+		if ( !view || !view.opener ) {
+			view = window;
+		}
+
+		return view.getComputedStyle( elem );
+	};
+
+var swap = function( elem, options, callback ) {
+	var ret, name,
+		old = {};
+
+	// Remember the old values, and insert the new ones
+	for ( name in options ) {
+		old[ name ] = elem.style[ name ];
+		elem.style[ name ] = options[ name ];
+	}
+
+	ret = callback.call( elem );
+
+	// Revert the old values
+	for ( name in options ) {
+		elem.style[ name ] = old[ name ];
+	}
+
+	return ret;
+};
+
+
+var rboxStyle = new RegExp( cssExpand.join( "|" ), "i" );
+
+
+
+( function() {
+
+	// Executing both pixelPosition & boxSizingReliable tests require only one layout
+	// so they're executed at the same time to save the second computation.
+	function computeStyleTests() {
+
+		// This is a singleton, we need to execute it only once
+		if ( !div ) {
+			return;
+		}
+
+		container.style.cssText = "position:absolute;left:-11111px;width:60px;" +
+			"margin-top:1px;padding:0;border:0";
+		div.style.cssText =
+			"position:relative;display:block;box-sizing:border-box;overflow:scroll;" +
+			"margin:auto;border:1px;padding:1px;" +
+			"width:60%;top:1%";
+		documentElement.appendChild( container ).appendChild( div );
+
+		var divStyle = window.getComputedStyle( div );
+		pixelPositionVal = divStyle.top !== "1%";
+
+		// Support: Android 4.0 - 4.3 only, Firefox <=3 - 44
+		reliableMarginLeftVal = roundPixelMeasures( divStyle.marginLeft ) === 12;
+
+		// Support: Android 4.0 - 4.3 only, Safari <=9.1 - 10.1, iOS <=7.0 - 9.3
+		// Some styles come back with percentage values, even though they shouldn't
+		div.style.right = "60%";
+		pixelBoxStylesVal = roundPixelMeasures( divStyle.right ) === 36;
+
+		// Support: IE 9 - 11 only
+		// Detect misreporting of content dimensions for box-sizing:border-box elements
+		boxSizingReliableVal = roundPixelMeasures( divStyle.width ) === 36;
+
+		// Support: IE 9 only
+		// Detect overflow:scroll screwiness (gh-3699)
+		// Support: Chrome <=64
+		// Don't get tricked when zoom affects offsetWidth (gh-4029)
+		div.style.position = "absolute";
+		scrollboxSizeVal = roundPixelMeasures( div.offsetWidth / 3 ) === 12;
+
+		documentElement.removeChild( container );
+
+		// Nullify the div so it wouldn't be stored in the memory and
+		// it will also be a sign that checks already performed
+		div = null;
+	}
+
+	function roundPixelMeasures( measure ) {
+		return Math.round( parseFloat( measure ) );
+	}
+
+	var pixelPositionVal, boxSizingReliableVal, scrollboxSizeVal, pixelBoxStylesVal,
+		reliableTrDimensionsVal, reliableMarginLeftVal,
+		container = document.createElement( "div" ),
+		div = document.createElement( "div" );
+
+	// Finish early in limited (non-browser) environments
+	if ( !div.style ) {
+		return;
+	}
+
+	// Support: IE <=9 - 11 only
+	// Style of cloned element affects source element cloned (#8908)
+	div.style.backgroundClip = "content-box";
+	div.cloneNode( true ).style.backgroundClip = "";
+	support.clearCloneStyle = div.style.backgroundClip === "content-box";
+
+	jQuery.extend( support, {
+		boxSizingReliable: function() {
+			computeStyleTests();
+			return boxSizingReliableVal;
+		},
+		pixelBoxStyles: function() {
+			computeStyleTests();
+			return pixelBoxStylesVal;
+		},
+		pixelPosition: function() {
+			computeStyleTests();
+			return pixelPositionVal;
+		},
+		reliableMarginLeft: function() {
+			computeStyleTests();
+			return reliableMarginLeftVal;
+		},
+		scrollboxSize: function() {
+			computeStyleTests();
+			return scrollboxSizeVal;
+		},
+
+		// Support: IE 9 - 11+, Edge 15 - 18+
+		// IE/Edge misreport `getComputedStyle` of table rows with width/height
+		// set in CSS while `offset*` properties report correct values.
+		// Behavior in IE 9 is more subtle than in newer versions & it passes
+		// some versions of this test; make sure not to make it pass there!
+		//
+		// Support: Firefox 70+
+		// Only Firefox includes border widths
+		// in computed dimensions. (gh-4529)
+		reliableTrDimensions: function() {
+			var table, tr, trChild, trStyle;
+			if ( reliableTrDimensionsVal == null ) {
+				table = document.createElement( "table" );
+				tr = document.createElement( "tr" );
+				trChild = document.createElement( "div" );
+
+				table.style.cssText = "position:absolute;left:-11111px;border-collapse:separate";
+				tr.style.cssText = "border:1px solid";
+
+				// Support: Chrome 86+
+				// Height set through cssText does not get applied.
+				// Computed height then comes back as 0.
+				tr.style.height = "1px";
+				trChild.style.height = "9px";
+
+				// Support: Android 8 Chrome 86+
+				// In our bodyBackground.html iframe,
+				// display for all div elements is set to "inline",
+				// which causes a problem only in Android 8 Chrome 86.
+				// Ensuring the div is display: block
+				// gets around this issue.
+				trChild.style.display = "block";
+
+				documentElement
+					.appendChild( table )
+					.appendChild( tr )
+					.appendChild( trChild );
+
+				trStyle = window.getComputedStyle( tr );
+				reliableTrDimensionsVal = ( parseInt( trStyle.height, 10 ) +
+					parseInt( trStyle.borderTopWidth, 10 ) +
+					parseInt( trStyle.borderBottomWidth, 10 ) ) === tr.offsetHeight;
+
+				documentElement.removeChild( table );
+			}
+			return reliableTrDimensionsVal;
+		}
+	} );
+} )();
+
+
+function curCSS( elem, name, computed ) {
+	var width, minWidth, maxWidth, ret,
+
+		// Support: Firefox 51+
+		// Retrieving style before computed somehow
+		// fixes an issue with getting wrong values
+		// on detached elements
+		style = elem.style;
+
+	computed = computed || getStyles( elem );
+
+	// getPropertyValue is needed for:
+	//   .css('filter') (IE 9 only, #12537)
+	//   .css('--customProperty) (#3144)
+	if ( computed ) {
+		ret = computed.getPropertyValue( name ) || computed[ name ];
+
+		if ( ret === "" && !isAttached( elem ) ) {
+			ret = jQuery.style( elem, name );
+		}
+
+		// A tribute to the "awesome hack by Dean Edwards"
+		// Android Browser returns percentage for some values,
+		// but width seems to be reliably pixels.
+		// This is against the CSSOM draft spec:
+		// https://drafts.csswg.org/cssom/#resolved-values
+		if ( !support.pixelBoxStyles() && rnumnonpx.test( ret ) && rboxStyle.test( name ) ) {
+
+			// Remember the original values
+			width = style.width;
+			minWidth = style.minWidth;
+			maxWidth = style.maxWidth;
+
+			// Put in the new values to get a computed value out
+			style.minWidth = style.maxWidth = style.width = ret;
+			ret = computed.width;
+
+			// Revert the changed values
+			style.width = width;
+			style.minWidth = minWidth;
+			style.maxWidth = maxWidth;
+		}
+	}
+
+	return ret !== undefined ?
+
+		// Support: IE <=9 - 11 only
+		// IE returns zIndex value as an integer.
+		ret + "" :
+		ret;
+}
+
+
+function addGetHookIf( conditionFn, hookFn ) {
+
+	// Define the hook, we'll check on the first run if it's really needed.
+	return {
+		get: function() {
+			if ( conditionFn() ) {
+
+				// Hook not needed (or it's not possible to use it due
+				// to missing dependency), remove it.
+				delete this.get;
+				return;
+			}
+
+			// Hook needed; redefine it so that the support test is not executed again.
+			return ( this.get = hookFn ).apply( this, arguments );
+		}
+	};
+}
+
+
+var cssPrefixes = [ "Webkit", "Moz", "ms" ],
+	emptyStyle = document.createElement( "div" ).style,
+	vendorProps = {};
+
+// Return a vendor-prefixed property or undefined
+function vendorPropName( name ) {
+
+	// Check for vendor prefixed names
+	var capName = name[ 0 ].toUpperCase() + name.slice( 1 ),
+		i = cssPrefixes.length;
+
+	while ( i-- ) {
+		name = cssPrefixes[ i ] + capName;
+		if ( name in emptyStyle ) {
+			return name;
+		}
+	}
+}
+
+// Return a potentially-mapped jQuery.cssProps or vendor prefixed property
+function finalPropName( name ) {
+	var final = jQuery.cssProps[ name ] || vendorProps[ name ];
+
+	if ( final ) {
+		return final;
+	}
+	if ( name in emptyStyle ) {
+		return name;
+	}
+	return vendorProps[ name ] = vendorPropName( name ) || name;
+}
+
+
+var
+
+	// Swappable if display is none or starts with table
+	// except "table", "table-cell", or "table-caption"
+	// See here for display values: https://developer.mozilla.org/en-US/docs/CSS/display
+	rdisplayswap = /^(none|table(?!-c[ea]).+)/,
+	rcustomProp = /^--/,
+	cssShow = { position: "absolute", visibility: "hidden", display: "block" },
+	cssNormalTransform = {
+		letterSpacing: "0",
+		fontWeight: "400"
+	};
+
+function setPositiveNumber( _elem, value, subtract ) {
+
+	// Any relative (+/-) values have already been
+	// normalized at this point
+	var matches = rcssNum.exec( value );
+	return matches ?
+
+		// Guard against undefined "subtract", e.g., when used as in cssHooks
+		Math.max( 0, matches[ 2 ] - ( subtract || 0 ) ) + ( matches[ 3 ] || "px" ) :
+		value;
+}
+
+function boxModelAdjustment( elem, dimension, box, isBorderBox, styles, computedVal ) {
+	var i = dimension === "width" ? 1 : 0,
+		extra = 0,
+		delta = 0;
+
+	// Adjustment may not be necessary
+	if ( box === ( isBorderBox ? "border" : "content" ) ) {
+		return 0;
+	}
+
+	for ( ; i < 4; i += 2 ) {
+
+		// Both box models exclude margin
+		if ( box === "margin" ) {
+			delta += jQuery.css( elem, box + cssExpand[ i ], true, styles );
+		}
+
+		// If we get here with a content-box, we're seeking "padding" or "border" or "margin"
+		if ( !isBorderBox ) {
+
+			// Add padding
+			delta += jQuery.css( elem, "padding" + cssExpand[ i ], true, styles );
+
+			// For "border" or "margin", add border
+			if ( box !== "padding" ) {
+				delta += jQuery.css( elem, "border" + cssExpand[ i ] + "Width", true, styles );
+
+			// But still keep track of it otherwise
+			} else {
+				extra += jQuery.css( elem, "border" + cssExpand[ i ] + "Width", true, styles );
+			}
+
+		// If we get here with a border-box (content + padding + border), we're seeking "content" or
+		// "padding" or "margin"
+		} else {
+
+			// For "content", subtract padding
+			if ( box === "content" ) {
+				delta -= jQuery.css( elem, "padding" + cssExpand[ i ], true, styles );
+			}
+
+			// For "content" or "padding", subtract border
+			if ( box !== "margin" ) {
+				delta -= jQuery.css( elem, "border" + cssExpand[ i ] + "Width", true, styles );
+			}
+		}
+	}
+
+	// Account for positive content-box scroll gutter when requested by providing computedVal
+	if ( !isBorderBox && computedVal >= 0 ) {
+
+		// offsetWidth/offsetHeight is a rounded sum of content, padding, scroll gutter, and border
+		// Assuming integer scroll gutter, subtract the rest and round down
+		delta += Math.max( 0, Math.ceil(
+			elem[ "offset" + dimension[ 0 ].toUpperCase() + dimension.slice( 1 ) ] -
+			computedVal -
+			delta -
+			extra -
+			0.5
+
+		// If offsetWidth/offsetHeight is unknown, then we can't determine content-box scroll gutter
+		// Use an explicit zero to avoid NaN (gh-3964)
+		) ) || 0;
+	}
+
+	return delta;
+}
+
+function getWidthOrHeight( elem, dimension, extra ) {
+
+	// Start with computed style
+	var styles = getStyles( elem ),
+
+		// To avoid forcing a reflow, only fetch boxSizing if we need it (gh-4322).
+		// Fake content-box until we know it's needed to know the true value.
+		boxSizingNeeded = !support.boxSizingReliable() || extra,
+		isBorderBox = boxSizingNeeded &&
+			jQuery.css( elem, "boxSizing", false, styles ) === "border-box",
+		valueIsBorderBox = isBorderBox,
+
+		val = curCSS( elem, dimension, styles ),
+		offsetProp = "offset" + dimension[ 0 ].toUpperCase() + dimension.slice( 1 );
+
+	// Support: Firefox <=54
+	// Return a confounding non-pixel value or feign ignorance, as appropriate.
+	if ( rnumnonpx.test( val ) ) {
+		if ( !extra ) {
+			return val;
+		}
+		val = "auto";
+	}
+
+
+	// Support: IE 9 - 11 only
+	// Use offsetWidth/offsetHeight for when box sizing is unreliable.
+	// In those cases, the computed value can be trusted to be border-box.
+	if ( ( !support.boxSizingReliable() && isBorderBox ||
+
+		// Support: IE 10 - 11+, Edge 15 - 18+
+		// IE/Edge misreport `getComputedStyle` of table rows with width/height
+		// set in CSS while `offset*` properties report correct values.
+		// Interestingly, in some cases IE 9 doesn't suffer from this issue.
+		!support.reliableTrDimensions() && nodeName( elem, "tr" ) ||
+
+		// Fall back to offsetWidth/offsetHeight when value is "auto"
+		// This happens for inline elements with no explicit setting (gh-3571)
+		val === "auto" ||
+
+		// Support: Android <=4.1 - 4.3 only
+		// Also use offsetWidth/offsetHeight for misreported inline dimensions (gh-3602)
+		!parseFloat( val ) && jQuery.css( elem, "display", false, styles ) === "inline" ) &&
+
+		// Make sure the element is visible & connected
+		elem.getClientRects().length ) {
+
+		isBorderBox = jQuery.css( elem, "boxSizing", false, styles ) === "border-box";
+
+		// Where available, offsetWidth/offsetHeight approximate border box dimensions.
+		// Where not available (e.g., SVG), assume unreliable box-sizing and interpret the
+		// retrieved value as a content box dimension.
+		valueIsBorderBox = offsetProp in elem;
+		if ( valueIsBorderBox ) {
+			val = elem[ offsetProp ];
+		}
+	}
+
+	// Normalize "" and auto
+	val = parseFloat( val ) || 0;
+
+	// Adjust for the element's box model
+	return ( val +
+		boxModelAdjustment(
+			elem,
+			dimension,
+			extra || ( isBorderBox ? "border" : "content" ),
+			valueIsBorderBox,
+			styles,
+
+			// Provide the current computed size to request scroll gutter calculation (gh-3589)
+			val
+		)
+	) + "px";
+}
+
+jQuery.extend( {
+
+	// Add in style property hooks for overriding the default
+	// behavior of getting and setting a style property
+	cssHooks: {
+		opacity: {
+			get: function( elem, computed ) {
+				if ( computed ) {
+
+					// We should always get a number back from opacity
+					var ret = curCSS( elem, "opacity" );
+					return ret === "" ? "1" : ret;
+				}
+			}
+		}
+	},
+
+	// Don't automatically add "px" to these possibly-unitless properties
+	cssNumber: {
+		"animationIterationCount": true,
+		"columnCount": true,
+		"fillOpacity": true,
+		"flexGrow": true,
+		"flexShrink": true,
+		"fontWeight": true,
+		"gridArea": true,
+		"gridColumn": true,
+		"gridColumnEnd": true,
+		"gridColumnStart": true,
+		"gridRow": true,
+		"gridRowEnd": true,
+		"gridRowStart": true,
+		"lineHeight": true,
+		"opacity": true,
+		"order": true,
+		"orphans": true,
+		"widows": true,
+		"zIndex": true,
+		"zoom": true
+	},
+
+	// Add in properties whose names you wish to fix before
+	// setting or getting the value
+	cssProps: {},
+
+	// Get and set the style property on a DOM Node
+	style: function( elem, name, value, extra ) {
+
+		// Don't set styles on text and comment nodes
+		if ( !elem || elem.nodeType === 3 || elem.nodeType === 8 || !elem.style ) {
+			return;
+		}
+
+		// Make sure that we're working with the right name
+		var ret, type, hooks,
+			origName = camelCase( name ),
+			isCustomProp = rcustomProp.test( name ),
+			style = elem.style;
+
+		// Make sure that we're working with the right name. We don't
+		// want to query the value if it is a CSS custom property
+		// since they are user-defined.
+		if ( !isCustomProp ) {
+			name = finalPropName( origName );
+		}
+
+		// Gets hook for the prefixed version, then unprefixed version
+		hooks = jQuery.cssHooks[ name ] || jQuery.cssHooks[ origName ];
+
+		// Check if we're setting a value
+		if ( value !== undefined ) {
+			type = typeof value;
+
+			// Convert "+=" or "-=" to relative numbers (#7345)
+			if ( type === "string" && ( ret = rcssNum.exec( value ) ) && ret[ 1 ] ) {
+				value = adjustCSS( elem, name, ret );
+
+				// Fixes bug #9237
+				type = "number";
+			}
+
+			// Make sure that null and NaN values aren't set (#7116)
+			if ( value == null || value !== value ) {
+				return;
+			}
+
+			// If a number was passed in, add the unit (except for certain CSS properties)
+			// The isCustomProp check can be removed in jQuery 4.0 when we only auto-append
+			// "px" to a few hardcoded values.
+			if ( type === "number" && !isCustomProp ) {
+				value += ret && ret[ 3 ] || ( jQuery.cssNumber[ origName ] ? "" : "px" );
+			}
+
+			// background-* props affect original clone's values
+			if ( !support.clearCloneStyle && value === "" && name.indexOf( "background" ) === 0 ) {
+				style[ name ] = "inherit";
+			}
+
+			// If a hook was provided, use that value, otherwise just set the specified value
+			if ( !hooks || !( "set" in hooks ) ||
+				( value = hooks.set( elem, value, extra ) ) !== undefined ) {
+
+				if ( isCustomProp ) {
+					style.setProperty( name, value );
+				} else {
+					style[ name ] = value;
+				}
+			}
+
+		} else {
+
+			// If a hook was provided get the non-computed value from there
+			if ( hooks && "get" in hooks &&
+				( ret = hooks.get( elem, false, extra ) ) !== undefined ) {
+
+				return ret;
+			}
+
+			// Otherwise just get the value from the style object
+			return style[ name ];
+		}
+	},
+
+	css: function( elem, name, extra, styles ) {
+		var val, num, hooks,
+			origName = camelCase( name ),
+			isCustomProp = rcustomProp.test( name );
+
+		// Make sure that we're working with the right name. We don't
+		// want to modify the value if it is a CSS custom property
+		// since they are user-defined.
+		if ( !isCustomProp ) {
+			name = finalPropName( origName );
+		}
+
+		// Try prefixed name followed by the unprefixed name
+		hooks = jQuery.cssHooks[ name ] || jQuery.cssHooks[ origName ];
+
+		// If a hook was provided get the computed value from there
+		if ( hooks && "get" in hooks ) {
+			val = hooks.get( elem, true, extra );
+		}
+
+		// Otherwise, if a way to get the computed value exists, use that
+		if ( val === undefined ) {
+			val = curCSS( elem, name, styles );
+		}
+
+		// Convert "normal" to computed value
+		if ( val === "normal" && name in cssNormalTransform ) {
+			val = cssNormalTransform[ name ];
+		}
+
+		// Make numeric if forced or a qualifier was provided and val looks numeric
+		if ( extra === "" || extra ) {
+			num = parseFloat( val );
+			return extra === true || isFinite( num ) ? num || 0 : val;
+		}
+
+		return val;
+	}
+} );
+
+jQuery.each( [ "height", "width" ], function( _i, dimension ) {
+	jQuery.cssHooks[ dimension ] = {
+		get: function( elem, computed, extra ) {
+			if ( computed ) {
+
+				// Certain elements can have dimension info if we invisibly show them
+				// but it must have a current display style that would benefit
+				return rdisplayswap.test( jQuery.css( elem, "display" ) ) &&
+
+					// Support: Safari 8+
+					// Table columns in Safari have non-zero offsetWidth & zero
+					// getBoundingClientRect().width unless display is changed.
+					// Support: IE <=11 only
+					// Running getBoundingClientRect on a disconnected node
+					// in IE throws an error.
+					( !elem.getClientRects().length || !elem.getBoundingClientRect().width ) ?
+					swap( elem, cssShow, function() {
+						return getWidthOrHeight( elem, dimension, extra );
+					} ) :
+					getWidthOrHeight( elem, dimension, extra );
+			}
+		},
+
+		set: function( elem, value, extra ) {
+			var matches,
+				styles = getStyles( elem ),
+
+				// Only read styles.position if the test has a chance to fail
+				// to avoid forcing a reflow.
+				scrollboxSizeBuggy = !support.scrollboxSize() &&
+					styles.position === "absolute",
+
+				// To avoid forcing a reflow, only fetch boxSizing if we need it (gh-3991)
+				boxSizingNeeded = scrollboxSizeBuggy || extra,
+				isBorderBox = boxSizingNeeded &&
+					jQuery.css( elem, "boxSizing", false, styles ) === "border-box",
+				subtract = extra ?
+					boxModelAdjustment(
+						elem,
+						dimension,
+						extra,
+						isBorderBox,
+						styles
+					) :
+					0;
+
+			// Account for unreliable border-box dimensions by comparing offset* to computed and
+			// faking a content-box to get border and padding (gh-3699)
+			if ( isBorderBox && scrollboxSizeBuggy ) {
+				subtract -= Math.ceil(
+					elem[ "offset" + dimension[ 0 ].toUpperCase() + dimension.slice( 1 ) ] -
+					parseFloat( styles[ dimension ] ) -
+					boxModelAdjustment( elem, dimension, "border", false, styles ) -
+					0.5
+				);
+			}
+
+			// Convert to pixels if value adjustment is needed
+			if ( subtract && ( matches = rcssNum.exec( value ) ) &&
+				( matches[ 3 ] || "px" ) !== "px" ) {
+
+				elem.style[ dimension ] = value;
+				value = jQuery.css( elem, dimension );
+			}
+
+			return setPositiveNumber( elem, value, subtract );
+		}
+	};
+} );
+
+jQuery.cssHooks.marginLeft = addGetHookIf( support.reliableMarginLeft,
+	function( elem, computed ) {
+		if ( computed ) {
+			return ( parseFloat( curCSS( elem, "marginLeft" ) ) ||
+				elem.getBoundingClientRect().left -
+					swap( elem, { marginLeft: 0 }, function() {
+						return elem.getBoundingClientRect().left;
+					} )
+			) + "px";
+		}
+	}
+);
+
+// These hooks are used by animate to expand properties
+jQuery.each( {
+	margin: "",
+	padding: "",
+	border: "Width"
+}, function( prefix, suffix ) {
+	jQuery.cssHooks[ prefix + suffix ] = {
+		expand: function( value ) {
+			var i = 0,
+				expanded = {},
+
+				// Assumes a single number if not a string
+				parts = typeof value === "string" ? value.split( " " ) : [ value ];
+
+			for ( ; i < 4; i++ ) {
+				expanded[ prefix + cssExpand[ i ] + suffix ] =
+					parts[ i ] || parts[ i - 2 ] || parts[ 0 ];
+			}
+
+			return expanded;
+		}
+	};
+
+	if ( prefix !== "margin" ) {
+		jQuery.cssHooks[ prefix + suffix ].set = setPositiveNumber;
+	}
+} );
+
+jQuery.fn.extend( {
+	css: function( name, value ) {
+		return access( this, function( elem, name, value ) {
+			var styles, len,
+				map = {},
+				i = 0;
+
+			if ( Array.isArray( name ) ) {
+				styles = getStyles( elem );
+				len = name.length;
+
+				for ( ; i < len; i++ ) {
+					map[ name[ i ] ] = jQuery.css( elem, name[ i ], false, styles );
+				}
+
+				return map;
+			}
+
+			return value !== undefined ?
+				jQuery.style( elem, name, value ) :
+				jQuery.css( elem, name );
+		}, name, value, arguments.length > 1 );
+	}
+} );
+
+
+function Tween( elem, options, prop, end, easing ) {
+	return new Tween.prototype.init( elem, options, prop, end, easing );
+}
+jQuery.Tween = Tween;
+
+Tween.prototype = {
+	constructor: Tween,
+	init: function( elem, options, prop, end, easing, unit ) {
+		this.elem = elem;
+		this.prop = prop;
+		this.easing = easing || jQuery.easing._default;
+		this.options = options;
+		this.start = this.now = this.cur();
+		this.end = end;
+		this.unit = unit || ( jQuery.cssNumber[ prop ] ? "" : "px" );
+	},
+	cur: function() {
+		var hooks = Tween.propHooks[ this.prop ];
+
+		return hooks && hooks.get ?
+			hooks.get( this ) :
+			Tween.propHooks._default.get( this );
+	},
+	run: function( percent ) {
+		var eased,
+			hooks = Tween.propHooks[ this.prop ];
+
+		if ( this.options.duration ) {
+			this.pos = eased = jQuery.easing[ this.easing ](
+				percent, this.options.duration * percent, 0, 1, this.options.duration
+			);
+		} else {
+			this.pos = eased = percent;
+		}
+		this.now = ( this.end - this.start ) * eased + this.start;
+
+		if ( this.options.step ) {
+			this.options.step.call( this.elem, this.now, this );
+		}
+
+		if ( hooks && hooks.set ) {
+			hooks.set( this );
+		} else {
+			Tween.propHooks._default.set( this );
+		}
+		return this;
+	}
+};
+
+Tween.prototype.init.prototype = Tween.prototype;
+
+Tween.propHooks = {
+	_default: {
+		get: function( tween ) {
+			var result;
+
+			// Use a property on the element directly when it is not a DOM element,
+			// or when there is no matching style property that exists.
+			if ( tween.elem.nodeType !== 1 ||
+				tween.elem[ tween.prop ] != null && tween.elem.style[ tween.prop ] == null ) {
+				return tween.elem[ tween.prop ];
+			}
+
+			// Passing an empty string as a 3rd parameter to .css will automatically
+			// attempt a parseFloat and fallback to a string if the parse fails.
+			// Simple values such as "10px" are parsed to Float;
+			// complex values such as "rotate(1rad)" are returned as-is.
+			result = jQuery.css( tween.elem, tween.prop, "" );
+
+			// Empty strings, null, undefined and "auto" are converted to 0.
+			return !result || result === "auto" ? 0 : result;
+		},
+		set: function( tween ) {
+
+			// Use step hook for back compat.
+			// Use cssHook if its there.
+			// Use .style if available and use plain properties where available.
+			if ( jQuery.fx.step[ tween.prop ] ) {
+				jQuery.fx.step[ tween.prop ]( tween );
+			} else if ( tween.elem.nodeType === 1 && (
+				jQuery.cssHooks[ tween.prop ] ||
+					tween.elem.style[ finalPropName( tween.prop ) ] != null ) ) {
+				jQuery.style( tween.elem, tween.prop, tween.now + tween.unit );
+			} else {
+				tween.elem[ tween.prop ] = tween.now;
+			}
+		}
+	}
+};
+
+// Support: IE <=9 only
+// Panic based approach to setting things on disconnected nodes
+Tween.propHooks.scrollTop = Tween.propHooks.scrollLeft = {
+	set: function( tween ) {
+		if ( tween.elem.nodeType && tween.elem.parentNode ) {
+			tween.elem[ tween.prop ] = tween.now;
+		}
+	}
+};
+
+jQuery.easing = {
+	linear: function( p ) {
+		return p;
+	},
+	swing: function( p ) {
+		return 0.5 - Math.cos( p * Math.PI ) / 2;
+	},
+	_default: "swing"
+};
+
+jQuery.fx = Tween.prototype.init;
+
+// Back compat <1.8 extension point
+jQuery.fx.step = {};
+
+
+
+
+var
+	fxNow, inProgress,
+	rfxtypes = /^(?:toggle|show|hide)$/,
+	rrun = /queueHooks$/;
+
+function schedule() {
+	if ( inProgress ) {
+		if ( document.hidden === false && window.requestAnimationFrame ) {
+			window.requestAnimationFrame( schedule );
+		} else {
+			window.setTimeout( schedule, jQuery.fx.interval );
+		}
+
+		jQuery.fx.tick();
+	}
+}
+
+// Animations created synchronously will run synchronously
+function createFxNow() {
+	window.setTimeout( function() {
+		fxNow = undefined;
+	} );
+	return ( fxNow = Date.now() );
+}
+
+// Generate parameters to create a standard animation
+function genFx( type, includeWidth ) {
+	var which,
+		i = 0,
+		attrs = { height: type };
+
+	// If we include width, step value is 1 to do all cssExpand values,
+	// otherwise step value is 2 to skip over Left and Right
+	includeWidth = includeWidth ? 1 : 0;
+	for ( ; i < 4; i += 2 - includeWidth ) {
+		which = cssExpand[ i ];
+		attrs[ "margin" + which ] = attrs[ "padding" + which ] = type;
+	}
+
+	if ( includeWidth ) {
+		attrs.opacity = attrs.width = type;
+	}
+
+	return attrs;
+}
+
+function createTween( value, prop, animation ) {
+	var tween,
+		collection = ( Animation.tweeners[ prop ] || [] ).concat( Animation.tweeners[ "*" ] ),
+		index = 0,
+		length = collection.length;
+	for ( ; index < length; index++ ) {
+		if ( ( tween = collection[ index ].call( animation, prop, value ) ) ) {
+
+			// We're done with this property
+			return tween;
+		}
+	}
+}
+
+function defaultPrefilter( elem, props, opts ) {
+	var prop, value, toggle, hooks, oldfire, propTween, restoreDisplay, display,
+		isBox = "width" in props || "height" in props,
+		anim = this,
+		orig = {},
+		style = elem.style,
+		hidden = elem.nodeType && isHiddenWithinTree( elem ),
+		dataShow = dataPriv.get( elem, "fxshow" );
+
+	// Queue-skipping animations hijack the fx hooks
+	if ( !opts.queue ) {
+		hooks = jQuery._queueHooks( elem, "fx" );
+		if ( hooks.unqueued == null ) {
+			hooks.unqueued = 0;
+			oldfire = hooks.empty.fire;
+			hooks.empty.fire = function() {
+				if ( !hooks.unqueued ) {
+					oldfire();
+				}
+			};
+		}
+		hooks.unqueued++;
+
+		anim.always( function() {
+
+			// Ensure the complete handler is called before this completes
+			anim.always( function() {
+				hooks.unqueued--;
+				if ( !jQuery.queue( elem, "fx" ).length ) {
+					hooks.empty.fire();
+				}
+			} );
+		} );
+	}
+
+	// Detect show/hide animations
+	for ( prop in props ) {
+		value = props[ prop ];
+		if ( rfxtypes.test( value ) ) {
+			delete props[ prop ];
+			toggle = toggle || value === "toggle";
+			if ( value === ( hidden ? "hide" : "show" ) ) {
+
+				// Pretend to be hidden if this is a "show" and
+				// there is still data from a stopped show/hide
+				if ( value === "show" && dataShow && dataShow[ prop ] !== undefined ) {
+					hidden = true;
+
+				// Ignore all other no-op show/hide data
+				} else {
+					continue;
+				}
+			}
+			orig[ prop ] = dataShow && dataShow[ prop ] || jQuery.style( elem, prop );
+		}
+	}
+
+	// Bail out if this is a no-op like .hide().hide()
+	propTween = !jQuery.isEmptyObject( props );
+	if ( !propTween && jQuery.isEmptyObject( orig ) ) {
+		return;
+	}
+
+	// Restrict "overflow" and "display" styles during box animations
+	if ( isBox && elem.nodeType === 1 ) {
+
+		// Support: IE <=9 - 11, Edge 12 - 15
+		// Record all 3 overflow attributes because IE does not infer the shorthand
+		// from identically-valued overflowX and overflowY and Edge just mirrors
+		// the overflowX value there.
+		opts.overflow = [ style.overflow, style.overflowX, style.overflowY ];
+
+		// Identify a display type, preferring old show/hide data over the CSS cascade
+		restoreDisplay = dataShow && dataShow.display;
+		if ( restoreDisplay == null ) {
+			restoreDisplay = dataPriv.get( elem, "display" );
+		}
+		display = jQuery.css( elem, "display" );
+		if ( display === "none" ) {
+			if ( restoreDisplay ) {
+				display = restoreDisplay;
+			} else {
+
+				// Get nonempty value(s) by temporarily forcing visibility
+				showHide( [ elem ], true );
+				restoreDisplay = elem.style.display || restoreDisplay;
+				display = jQuery.css( elem, "display" );
+				showHide( [ elem ] );
+			}
+		}
+
+		// Animate inline elements as inline-block
+		if ( display === "inline" || display === "inline-block" && restoreDisplay != null ) {
+			if ( jQuery.css( elem, "float" ) === "none" ) {
+
+				// Restore the original display value at the end of pure show/hide animations
+				if ( !propTween ) {
+					anim.done( function() {
+						style.display = restoreDisplay;
+					} );
+					if ( restoreDisplay == null ) {
+						display = style.display;
+						restoreDisplay = display === "none" ? "" : display;
+					}
+				}
+				style.display = "inline-block";
+			}
+		}
+	}
+
+	if ( opts.overflow ) {
+		style.overflow = "hidden";
+		anim.always( function() {
+			style.overflow = opts.overflow[ 0 ];
+			style.overflowX = opts.overflow[ 1 ];
+			style.overflowY = opts.overflow[ 2 ];
+		} );
+	}
+
+	// Implement show/hide animations
+	propTween = false;
+	for ( prop in orig ) {
+
+		// General show/hide setup for this element animation
+		if ( !propTween ) {
+			if ( dataShow ) {
+				if ( "hidden" in dataShow ) {
+					hidden = dataShow.hidden;
+				}
+			} else {
+				dataShow = dataPriv.access( elem, "fxshow", { display: restoreDisplay } );
+			}
+
+			// Store hidden/visible for toggle so `.stop().toggle()` "reverses"
+			if ( toggle ) {
+				dataShow.hidden = !hidden;
+			}
+
+			// Show elements before animating them
+			if ( hidden ) {
+				showHide( [ elem ], true );
+			}
+
+			/* eslint-disable no-loop-func */
+
+			anim.done( function() {
+
+				/* eslint-enable no-loop-func */
+
+				// The final step of a "hide" animation is actually hiding the element
+				if ( !hidden ) {
+					showHide( [ elem ] );
+				}
+				dataPriv.remove( elem, "fxshow" );
+				for ( prop in orig ) {
+					jQuery.style( elem, prop, orig[ prop ] );
+				}
+			} );
+		}
+
+		// Per-property setup
+		propTween = createTween( hidden ? dataShow[ prop ] : 0, prop, anim );
+		if ( !( prop in dataShow ) ) {
+			dataShow[ prop ] = propTween.start;
+			if ( hidden ) {
+				propTween.end = propTween.start;
+				propTween.start = 0;
+			}
+		}
+	}
+}
+
+function propFilter( props, specialEasing ) {
+	var index, name, easing, value, hooks;
+
+	// camelCase, specialEasing and expand cssHook pass
+	for ( index in props ) {
+		name = camelCase( index );
+		easing = specialEasing[ name ];
+		value = props[ index ];
+		if ( Array.isArray( value ) ) {
+			easing = value[ 1 ];
+			value = props[ index ] = value[ 0 ];
+		}
+
+		if ( index !== name ) {
+			props[ name ] = value;
+			delete props[ index ];
+		}
+
+		hooks = jQuery.cssHooks[ name ];
+		if ( hooks && "expand" in hooks ) {
+			value = hooks.expand( value );
+			delete props[ name ];
+
+			// Not quite $.extend, this won't overwrite existing keys.
+			// Reusing 'index' because we have the correct "name"
+			for ( index in value ) {
+				if ( !( index in props ) ) {
+					props[ index ] = value[ index ];
+					specialEasing[ index ] = easing;
+				}
+			}
+		} else {
+			specialEasing[ name ] = easing;
+		}
+	}
+}
+
+function Animation( elem, properties, options ) {
+	var result,
+		stopped,
+		index = 0,
+		length = Animation.prefilters.length,
+		deferred = jQuery.Deferred().always( function() {
+
+			// Don't match elem in the :animated selector
+			delete tick.elem;
+		} ),
+		tick = function() {
+			if ( stopped ) {
+				return false;
+			}
+			var currentTime = fxNow || createFxNow(),
+				remaining = Math.max( 0, animation.startTime + animation.duration - currentTime ),
+
+				// Support: Android 2.3 only
+				// Archaic crash bug won't allow us to use `1 - ( 0.5 || 0 )` (#12497)
+				temp = remaining / animation.duration || 0,
+				percent = 1 - temp,
+				index = 0,
+				length = animation.tweens.length;
+
+			for ( ; index < length; index++ ) {
+				animation.tweens[ index ].run( percent );
+			}
+
+			deferred.notifyWith( elem, [ animation, percent, remaining ] );
+
+			// If there's more to do, yield
+			if ( percent < 1 && length ) {
+				return remaining;
+			}
+
+			// If this was an empty animation, synthesize a final progress notification
+			if ( !length ) {
+				deferred.notifyWith( elem, [ animation, 1, 0 ] );
+			}
+
+			// Resolve the animation and report its conclusion
+			deferred.resolveWith( elem, [ animation ] );
+			return false;
+		},
+		animation = deferred.promise( {
+			elem: elem,
+			props: jQuery.extend( {}, properties ),
+			opts: jQuery.extend( true, {
+				specialEasing: {},
+				easing: jQuery.easing._default
+			}, options ),
+			originalProperties: properties,
+			originalOptions: options,
+			startTime: fxNow || createFxNow(),
+			duration: options.duration,
+			tweens: [],
+			createTween: function( prop, end ) {
+				var tween = jQuery.Tween( elem, animation.opts, prop, end,
+					animation.opts.specialEasing[ prop ] || animation.opts.easing );
+				animation.tweens.push( tween );
+				return tween;
+			},
+			stop: function( gotoEnd ) {
+				var index = 0,
+
+					// If we are going to the end, we want to run all the tweens
+					// otherwise we skip this part
+					length = gotoEnd ? animation.tweens.length : 0;
+				if ( stopped ) {
+					return this;
+				}
+				stopped = true;
+				for ( ; index < length; index++ ) {
+					animation.tweens[ index ].run( 1 );
+				}
+
+				// Resolve when we played the last frame; otherwise, reject
+				if ( gotoEnd ) {
+					deferred.notifyWith( elem, [ animation, 1, 0 ] );
+					deferred.resolveWith( elem, [ animation, gotoEnd ] );
+				} else {
+					deferred.rejectWith( elem, [ animation, gotoEnd ] );
+				}
+				return this;
+			}
+		} ),
+		props = animation.props;
+
+	propFilter( props, animation.opts.specialEasing );
+
+	for ( ; index < length; index++ ) {
+		result = Animation.prefilters[ index ].call( animation, elem, props, animation.opts );
+		if ( result ) {
+			if ( isFunction( result.stop ) ) {
+				jQuery._queueHooks( animation.elem, animation.opts.queue ).stop =
+					result.stop.bind( result );
+			}
+			return result;
+		}
+	}
+
+	jQuery.map( props, createTween, animation );
+
+	if ( isFunction( animation.opts.start ) ) {
+		animation.opts.start.call( elem, animation );
+	}
+
+	// Attach callbacks from options
+	animation
+		.progress( animation.opts.progress )
+		.done( animation.opts.done, animation.opts.complete )
+		.fail( animation.opts.fail )
+		.always( animation.opts.always );
+
+	jQuery.fx.timer(
+		jQuery.extend( tick, {
+			elem: elem,
+			anim: animation,
+			queue: animation.opts.queue
+		} )
+	);
+
+	return animation;
+}
+
+jQuery.Animation = jQuery.extend( Animation, {
+
+	tweeners: {
+		"*": [ function( prop, value ) {
+			var tween = this.createTween( prop, value );
+			adjustCSS( tween.elem, prop, rcssNum.exec( value ), tween );
+			return tween;
+		} ]
+	},
+
+	tweener: function( props, callback ) {
+		if ( isFunction( props ) ) {
+			callback = props;
+			props = [ "*" ];
+		} else {
+			props = props.match( rnothtmlwhite );
+		}
+
+		var prop,
+			index = 0,
+			length = props.length;
+
+		for ( ; index < length; index++ ) {
+			prop = props[ index ];
+			Animation.tweeners[ prop ] = Animation.tweeners[ prop ] || [];
+			Animation.tweeners[ prop ].unshift( callback );
+		}
+	},
+
+	prefilters: [ defaultPrefilter ],
+
+	prefilter: function( callback, prepend ) {
+		if ( prepend ) {
+			Animation.prefilters.unshift( callback );
+		} else {
+			Animation.prefilters.push( callback );
+		}
+	}
+} );
+
+jQuery.speed = function( speed, easing, fn ) {
+	var opt = speed && typeof speed === "object" ? jQuery.extend( {}, speed ) : {
+		complete: fn || !fn && easing ||
+			isFunction( speed ) && speed,
+		duration: speed,
+		easing: fn && easing || easing && !isFunction( easing ) && easing
+	};
+
+	// Go to the end state if fx are off
+	if ( jQuery.fx.off ) {
+		opt.duration = 0;
+
+	} else {
+		if ( typeof opt.duration !== "number" ) {
+			if ( opt.duration in jQuery.fx.speeds ) {
+				opt.duration = jQuery.fx.speeds[ opt.duration ];
+
+			} else {
+				opt.duration = jQuery.fx.speeds._default;
+			}
+		}
+	}
+
+	// Normalize opt.queue - true/undefined/null -> "fx"
+	if ( opt.queue == null || opt.queue === true ) {
+		opt.queue = "fx";
+	}
+
+	// Queueing
+	opt.old = opt.complete;
+
+	opt.complete = function() {
+		if ( isFunction( opt.old ) ) {
+			opt.old.call( this );
+		}
+
+		if ( opt.queue ) {
+			jQuery.dequeue( this, opt.queue );
+		}
+	};
+
+	return opt;
+};
+
+jQuery.fn.extend( {
+	fadeTo: function( speed, to, easing, callback ) {
+
+		// Show any hidden elements after setting opacity to 0
+		return this.filter( isHiddenWithinTree ).css( "opacity", 0 ).show()
+
+			// Animate to the value specified
+			.end().animate( { opacity: to }, speed, easing, callback );
+	},
+	animate: function( prop, speed, easing, callback ) {
+		var empty = jQuery.isEmptyObject( prop ),
+			optall = jQuery.speed( speed, easing, callback ),
+			doAnimation = function() {
+
+				// Operate on a copy of prop so per-property easing won't be lost
+				var anim = Animation( this, jQuery.extend( {}, prop ), optall );
+
+				// Empty animations, or finishing resolves immediately
+				if ( empty || dataPriv.get( this, "finish" ) ) {
+					anim.stop( true );
+				}
+			};
+
+		doAnimation.finish = doAnimation;
+
+		return empty || optall.queue === false ?
+			this.each( doAnimation ) :
+			this.queue( optall.queue, doAnimation );
+	},
+	stop: function( type, clearQueue, gotoEnd ) {
+		var stopQueue = function( hooks ) {
+			var stop = hooks.stop;
+			delete hooks.stop;
+			stop( gotoEnd );
+		};
+
+		if ( typeof type !== "string" ) {
+			gotoEnd = clearQueue;
+			clearQueue = type;
+			type = undefined;
+		}
+		if ( clearQueue ) {
+			this.queue( type || "fx", [] );
+		}
+
+		return this.each( function() {
+			var dequeue = true,
+				index = type != null && type + "queueHooks",
+				timers = jQuery.timers,
+				data = dataPriv.get( this );
+
+			if ( index ) {
+				if ( data[ index ] && data[ index ].stop ) {
+					stopQueue( data[ index ] );
+				}
+			} else {
+				for ( index in data ) {
+					if ( data[ index ] && data[ index ].stop && rrun.test( index ) ) {
+						stopQueue( data[ index ] );
+					}
+				}
+			}
+
+			for ( index = timers.length; index--; ) {
+				if ( timers[ index ].elem === this &&
+					( type == null || timers[ index ].queue === type ) ) {
+
+					timers[ index ].anim.stop( gotoEnd );
+					dequeue = false;
+					timers.splice( index, 1 );
+				}
+			}
+
+			// Start the next in the queue if the last step wasn't forced.
+			// Timers currently will call their complete callbacks, which
+			// will dequeue but only if they were gotoEnd.
+			if ( dequeue || !gotoEnd ) {
+				jQuery.dequeue( this, type );
+			}
+		} );
+	},
+	finish: function( type ) {
+		if ( type !== false ) {
+			type = type || "fx";
+		}
+		return this.each( function() {
+			var index,
+				data = dataPriv.get( this ),
+				queue = data[ type + "queue" ],
+				hooks = data[ type + "queueHooks" ],
+				timers = jQuery.timers,
+				length = queue ? queue.length : 0;
+
+			// Enable finishing flag on private data
+			data.finish = true;
+
+			// Empty the queue first
+			jQuery.queue( this, type, [] );
+
+			if ( hooks && hooks.stop ) {
+				hooks.stop.call( this, true );
+			}
+
+			// Look for any active animations, and finish them
+			for ( index = timers.length; index--; ) {
+				if ( timers[ index ].elem === this && timers[ index ].queue === type ) {
+					timers[ index ].anim.stop( true );
+					timers.splice( index, 1 );
+				}
+			}
+
+			// Look for any animations in the old queue and finish them
+			for ( index = 0; index < length; index++ ) {
+				if ( queue[ index ] && queue[ index ].finish ) {
+					queue[ index ].finish.call( this );
+				}
+			}
+
+			// Turn off finishing flag
+			delete data.finish;
+		} );
+	}
+} );
+
+jQuery.each( [ "toggle", "show", "hide" ], function( _i, name ) {
+	var cssFn = jQuery.fn[ name ];
+	jQuery.fn[ name ] = function( speed, easing, callback ) {
+		return speed == null || typeof speed === "boolean" ?
+			cssFn.apply( this, arguments ) :
+			this.animate( genFx( name, true ), speed, easing, callback );
+	};
+} );
+
+// Generate shortcuts for custom animations
+jQuery.each( {
+	slideDown: genFx( "show" ),
+	slideUp: genFx( "hide" ),
+	slideToggle: genFx( "toggle" ),
+	fadeIn: { opacity: "show" },
+	fadeOut: { opacity: "hide" },
+	fadeToggle: { opacity: "toggle" }
+}, function( name, props ) {
+	jQuery.fn[ name ] = function( speed, easing, callback ) {
+		return this.animate( props, speed, easing, callback );
+	};
+} );
+
+jQuery.timers = [];
+jQuery.fx.tick = function() {
+	var timer,
+		i = 0,
+		timers = jQuery.timers;
+
+	fxNow = Date.now();
+
+	for ( ; i < timers.length; i++ ) {
+		timer = timers[ i ];
+
+		// Run the timer and safely remove it when done (allowing for external removal)
+		if ( !timer() && timers[ i ] === timer ) {
+			timers.splice( i--, 1 );
+		}
+	}
+
+	if ( !timers.length ) {
+		jQuery.fx.stop();
+	}
+	fxNow = undefined;
+};
+
+jQuery.fx.timer = function( timer ) {
+	jQuery.timers.push( timer );
+	jQuery.fx.start();
+};
+
+jQuery.fx.interval = 13;
+jQuery.fx.start = function() {
+	if ( inProgress ) {
+		return;
+	}
+
+	inProgress = true;
+	schedule();
+};
+
+jQuery.fx.stop = function() {
+	inProgress = null;
+};
+
+jQuery.fx.speeds = {
+	slow: 600,
+	fast: 200,
+
+	// Default speed
+	_default: 400
+};
+
+
+// Based off of the plugin by Clint Helfers, with permission.
+// https://web.archive.org/web/20100324014747/http://blindsignals.com/index.php/2009/07/jquery-delay/
+jQuery.fn.delay = function( time, type ) {
+	time = jQuery.fx ? jQuery.fx.speeds[ time ] || time : time;
+	type = type || "fx";
+
+	return this.queue( type, function( next, hooks ) {
+		var timeout = window.setTimeout( next, time );
+		hooks.stop = function() {
+			window.clearTimeout( timeout );
+		};
+	} );
+};
+
+
+( function() {
+	var input = document.createElement( "input" ),
+		select = document.createElement( "select" ),
+		opt = select.appendChild( document.createElement( "option" ) );
+
+	input.type = "checkbox";
+
+	// Support: Android <=4.3 only
+	// Default value for a checkbox should be "on"
+	support.checkOn = input.value !== "";
+
+	// Support: IE <=11 only
+	// Must access selectedIndex to make default options select
+	support.optSelected = opt.selected;
+
+	// Support: IE <=11 only
+	// An input loses its value after becoming a radio
+	input = document.createElement( "input" );
+	input.value = "t";
+	input.type = "radio";
+	support.radioValue = input.value === "t";
+} )();
+
+
+var boolHook,
+	attrHandle = jQuery.expr.attrHandle;
+
+jQuery.fn.extend( {
+	attr: function( name, value ) {
+		return access( this, jQuery.attr, name, value, arguments.length > 1 );
+	},
+
+	removeAttr: function( name ) {
+		return this.each( function() {
+			jQuery.removeAttr( this, name );
+		} );
+	}
+} );
+
+jQuery.extend( {
+	attr: function( elem, name, value ) {
+		var ret, hooks,
+			nType = elem.nodeType;
+
+		// Don't get/set attributes on text, comment and attribute nodes
+		if ( nType === 3 || nType === 8 || nType === 2 ) {
+			return;
+		}
+
+		// Fallback to prop when attributes are not supported
+		if ( typeof elem.getAttribute === "undefined" ) {
+			return jQuery.prop( elem, name, value );
+		}
+
+		// Attribute hooks are determined by the lowercase version
+		// Grab necessary hook if one is defined
+		if ( nType !== 1 || !jQuery.isXMLDoc( elem ) ) {
+			hooks = jQuery.attrHooks[ name.toLowerCase() ] ||
+				( jQuery.expr.match.bool.test( name ) ? boolHook : undefined );
+		}
+
+		if ( value !== undefined ) {
+			if ( value === null ) {
+				jQuery.removeAttr( elem, name );
+				return;
+			}
+
+			if ( hooks && "set" in hooks &&
+				( ret = hooks.set( elem, value, name ) ) !== undefined ) {
+				return ret;
+			}
+
+			elem.setAttribute( name, value + "" );
+			return value;
+		}
+
+		if ( hooks && "get" in hooks && ( ret = hooks.get( elem, name ) ) !== null ) {
+			return ret;
+		}
+
+		ret = jQuery.find.attr( elem, name );
+
+		// Non-existent attributes return null, we normalize to undefined
+		return ret == null ? undefined : ret;
+	},
+
+	attrHooks: {
+		type: {
+			set: function( elem, value ) {
+				if ( !support.radioValue && value === "radio" &&
+					nodeName( elem, "input" ) ) {
+					var val = elem.value;
+					elem.setAttribute( "type", value );
+					if ( val ) {
+						elem.value = val;
+					}
+					return value;
+				}
+			}
+		}
+	},
+
+	removeAttr: function( elem, value ) {
+		var name,
+			i = 0,
+
+			// Attribute names can contain non-HTML whitespace characters
+			// https://html.spec.whatwg.org/multipage/syntax.html#attributes-2
+			attrNames = value && value.match( rnothtmlwhite );
+
+		if ( attrNames && elem.nodeType === 1 ) {
+			while ( ( name = attrNames[ i++ ] ) ) {
+				elem.removeAttribute( name );
+			}
+		}
+	}
+} );
+
+// Hooks for boolean attributes
+boolHook = {
+	set: function( elem, value, name ) {
+		if ( value === false ) {
+
+			// Remove boolean attributes when set to false
+			jQuery.removeAttr( elem, name );
+		} else {
+			elem.setAttribute( name, name );
+		}
+		return name;
+	}
+};
+
+jQuery.each( jQuery.expr.match.bool.source.match( /\w+/g ), function( _i, name ) {
+	var getter = attrHandle[ name ] || jQuery.find.attr;
+
+	attrHandle[ name ] = function( elem, name, isXML ) {
+		var ret, handle,
+			lowercaseName = name.toLowerCase();
+
+		if ( !isXML ) {
+
+			// Avoid an infinite loop by temporarily removing this function from the getter
+			handle = attrHandle[ lowercaseName ];
+			attrHandle[ lowercaseName ] = ret;
+			ret = getter( elem, name, isXML ) != null ?
+				lowercaseName :
+				null;
+			attrHandle[ lowercaseName ] = handle;
+		}
+		return ret;
+	};
+} );
+
+
+
+
+var rfocusable = /^(?:input|select|textarea|button)$/i,
+	rclickable = /^(?:a|area)$/i;
+
+jQuery.fn.extend( {
+	prop: function( name, value ) {
+		return access( this, jQuery.prop, name, value, arguments.length > 1 );
+	},
+
+	removeProp: function( name ) {
+		return this.each( function() {
+			delete this[ jQuery.propFix[ name ] || name ];
+		} );
+	}
+} );
+
+jQuery.extend( {
+	prop: function( elem, name, value ) {
+		var ret, hooks,
+			nType = elem.nodeType;
+
+		// Don't get/set properties on text, comment and attribute nodes
+		if ( nType === 3 || nType === 8 || nType === 2 ) {
+			return;
+		}
+
+		if ( nType !== 1 || !jQuery.isXMLDoc( elem ) ) {
+
+			// Fix name and attach hooks
+			name = jQuery.propFix[ name ] || name;
+			hooks = jQuery.propHooks[ name ];
+		}
+
+		if ( value !== undefined ) {
+			if ( hooks && "set" in hooks &&
+				( ret = hooks.set( elem, value, name ) ) !== undefined ) {
+				return ret;
+			}
+
+			return ( elem[ name ] = value );
+		}
+
+		if ( hooks && "get" in hooks && ( ret = hooks.get( elem, name ) ) !== null ) {
+			return ret;
+		}
+
+		return elem[ name ];
+	},
+
+	propHooks: {
+		tabIndex: {
+			get: function( elem ) {
+
+				// Support: IE <=9 - 11 only
+				// elem.tabIndex doesn't always return the
+				// correct value when it hasn't been explicitly set
+				// https://web.archive.org/web/20141116233347/http://fluidproject.org/blog/2008/01/09/getting-setting-and-removing-tabindex-values-with-javascript/
+				// Use proper attribute retrieval(#12072)
+				var tabindex = jQuery.find.attr( elem, "tabindex" );
+
+				if ( tabindex ) {
+					return parseInt( tabindex, 10 );
+				}
+
+				if (
+					rfocusable.test( elem.nodeName ) ||
+					rclickable.test( elem.nodeName ) &&
+					elem.href
+				) {
+					return 0;
+				}
+
+				return -1;
+			}
+		}
+	},
+
+	propFix: {
+		"for": "htmlFor",
+		"class": "className"
+	}
+} );
+
+// Support: IE <=11 only
+// Accessing the selectedIndex property
+// forces the browser to respect setting selected
+// on the option
+// The getter ensures a default option is selected
+// when in an optgroup
+// eslint rule "no-unused-expressions" is disabled for this code
+// since it considers such accessions noop
+if ( !support.optSelected ) {
+	jQuery.propHooks.selected = {
+		get: function( elem ) {
+
+			/* eslint no-unused-expressions: "off" */
+
+			var parent = elem.parentNode;
+			if ( parent && parent.parentNode ) {
+				parent.parentNode.selectedIndex;
+			}
+			return null;
+		},
+		set: function( elem ) {
+
+			/* eslint no-unused-expressions: "off" */
+
+			var parent = elem.parentNode;
+			if ( parent ) {
+				parent.selectedIndex;
+
+				if ( parent.parentNode ) {
+					parent.parentNode.selectedIndex;
+				}
+			}
+		}
+	};
+}
+
+jQuery.each( [
+	"tabIndex",
+	"readOnly",
+	"maxLength",
+	"cellSpacing",
+	"cellPadding",
+	"rowSpan",
+	"colSpan",
+	"useMap",
+	"frameBorder",
+	"contentEditable"
+], function() {
+	jQuery.propFix[ this.toLowerCase() ] = this;
+} );
+
+
+
+
+	// Strip and collapse whitespace according to HTML spec
+	// https://infra.spec.whatwg.org/#strip-and-collapse-ascii-whitespace
+	function stripAndCollapse( value ) {
+		var tokens = value.match( rnothtmlwhite ) || [];
+		return tokens.join( " " );
+	}
+
+
+function getClass( elem ) {
+	return elem.getAttribute && elem.getAttribute( "class" ) || "";
+}
+
+function classesToArray( value ) {
+	if ( Array.isArray( value ) ) {
+		return value;
+	}
+	if ( typeof value === "string" ) {
+		return value.match( rnothtmlwhite ) || [];
+	}
+	return [];
+}
+
+jQuery.fn.extend( {
+	addClass: function( value ) {
+		var classes, elem, cur, curValue, clazz, j, finalValue,
+			i = 0;
+
+		if ( isFunction( value ) ) {
+			return this.each( function( j ) {
+				jQuery( this ).addClass( value.call( this, j, getClass( this ) ) );
+			} );
+		}
+
+		classes = classesToArray( value );
+
+		if ( classes.length ) {
+			while ( ( elem = this[ i++ ] ) ) {
+				curValue = getClass( elem );
+				cur = elem.nodeType === 1 && ( " " + stripAndCollapse( curValue ) + " " );
+
+				if ( cur ) {
+					j = 0;
+					while ( ( clazz = classes[ j++ ] ) ) {
+						if ( cur.indexOf( " " + clazz + " " ) < 0 ) {
+							cur += clazz + " ";
+						}
+					}
+
+					// Only assign if different to avoid unneeded rendering.
+					finalValue = stripAndCollapse( cur );
+					if ( curValue !== finalValue ) {
+						elem.setAttribute( "class", finalValue );
+					}
+				}
+			}
+		}
+
+		return this;
+	},
+
+	removeClass: function( value ) {
+		var classes, elem, cur, curValue, clazz, j, finalValue,
+			i = 0;
+
+		if ( isFunction( value ) ) {
+			return this.each( function( j ) {
+				jQuery( this ).removeClass( value.call( this, j, getClass( this ) ) );
+			} );
+		}
+
+		if ( !arguments.length ) {
+			return this.attr( "class", "" );
+		}
+
+		classes = classesToArray( value );
+
+		if ( classes.length ) {
+			while ( ( elem = this[ i++ ] ) ) {
+				curValue = getClass( elem );
+
+				// This expression is here for better compressibility (see addClass)
+				cur = elem.nodeType === 1 && ( " " + stripAndCollapse( curValue ) + " " );
+
+				if ( cur ) {
+					j = 0;
+					while ( ( clazz = classes[ j++ ] ) ) {
+
+						// Remove *all* instances
+						while ( cur.indexOf( " " + clazz + " " ) > -1 ) {
+							cur = cur.replace( " " + clazz + " ", " " );
+						}
+					}
+
+					// Only assign if different to avoid unneeded rendering.
+					finalValue = stripAndCollapse( cur );
+					if ( curValue !== finalValue ) {
+						elem.setAttribute( "class", finalValue );
+					}
+				}
+			}
+		}
+
+		return this;
+	},
+
+	toggleClass: function( value, stateVal ) {
+		var type = typeof value,
+			isValidValue = type === "string" || Array.isArray( value );
+
+		if ( typeof stateVal === "boolean" && isValidValue ) {
+			return stateVal ? this.addClass( value ) : this.removeClass( value );
+		}
+
+		if ( isFunction( value ) ) {
+			return this.each( function( i ) {
+				jQuery( this ).toggleClass(
+					value.call( this, i, getClass( this ), stateVal ),
+					stateVal
+				);
+			} );
+		}
+
+		return this.each( function() {
+			var className, i, self, classNames;
+
+			if ( isValidValue ) {
+
+				// Toggle individual class names
+				i = 0;
+				self = jQuery( this );
+				classNames = classesToArray( value );
+
+				while ( ( className = classNames[ i++ ] ) ) {
+
+					// Check each className given, space separated list
+					if ( self.hasClass( className ) ) {
+						self.removeClass( className );
+					} else {
+						self.addClass( className );
+					}
+				}
+
+			// Toggle whole class name
+			} else if ( value === undefined || type === "boolean" ) {
+				className = getClass( this );
+				if ( className ) {
+
+					// Store className if set
+					dataPriv.set( this, "__className__", className );
+				}
+
+				// If the element has a class name or if we're passed `false`,
+				// then remove the whole classname (if there was one, the above saved it).
+				// Otherwise bring back whatever was previously saved (if anything),
+				// falling back to the empty string if nothing was stored.
+				if ( this.setAttribute ) {
+					this.setAttribute( "class",
+						className || value === false ?
+							"" :
+							dataPriv.get( this, "__className__" ) || ""
+					);
+				}
+			}
+		} );
+	},
+
+	hasClass: function( selector ) {
+		var className, elem,
+			i = 0;
+
+		className = " " + selector + " ";
+		while ( ( elem = this[ i++ ] ) ) {
+			if ( elem.nodeType === 1 &&
+				( " " + stripAndCollapse( getClass( elem ) ) + " " ).indexOf( className ) > -1 ) {
+				return true;
+			}
+		}
+
+		return false;
+	}
+} );
+
+
+
+
+var rreturn = /\r/g;
+
+jQuery.fn.extend( {
+	val: function( value ) {
+		var hooks, ret, valueIsFunction,
+			elem = this[ 0 ];
+
+		if ( !arguments.length ) {
+			if ( elem ) {
+				hooks = jQuery.valHooks[ elem.type ] ||
+					jQuery.valHooks[ elem.nodeName.toLowerCase() ];
+
+				if ( hooks &&
+					"get" in hooks &&
+					( ret = hooks.get( elem, "value" ) ) !== undefined
+				) {
+					return ret;
+				}
+
+				ret = elem.value;
+
+				// Handle most common string cases
+				if ( typeof ret === "string" ) {
+					return ret.replace( rreturn, "" );
+				}
+
+				// Handle cases where value is null/undef or number
+				return ret == null ? "" : ret;
+			}
+
+			return;
+		}
+
+		valueIsFunction = isFunction( value );
+
+		return this.each( function( i ) {
+			var val;
+
+			if ( this.nodeType !== 1 ) {
+				return;
+			}
+
+			if ( valueIsFunction ) {
+				val = value.call( this, i, jQuery( this ).val() );
+			} else {
+				val = value;
+			}
+
+			// Treat null/undefined as ""; convert numbers to string
+			if ( val == null ) {
+				val = "";
+
+			} else if ( typeof val === "number" ) {
+				val += "";
+
+			} else if ( Array.isArray( val ) ) {
+				val = jQuery.map( val, function( value ) {
+					return value == null ? "" : value + "";
+				} );
+			}
+
+			hooks = jQuery.valHooks[ this.type ] || jQuery.valHooks[ this.nodeName.toLowerCase() ];
+
+			// If set returns undefined, fall back to normal setting
+			if ( !hooks || !( "set" in hooks ) || hooks.set( this, val, "value" ) === undefined ) {
+				this.value = val;
+			}
+		} );
+	}
+} );
+
+jQuery.extend( {
+	valHooks: {
+		option: {
+			get: function( elem ) {
+
+				var val = jQuery.find.attr( elem, "value" );
+				return val != null ?
+					val :
+
+					// Support: IE <=10 - 11 only
+					// option.text throws exceptions (#14686, #14858)
+					// Strip and collapse whitespace
+					// https://html.spec.whatwg.org/#strip-and-collapse-whitespace
+					stripAndCollapse( jQuery.text( elem ) );
+			}
+		},
+		select: {
+			get: function( elem ) {
+				var value, option, i,
+					options = elem.options,
+					index = elem.selectedIndex,
+					one = elem.type === "select-one",
+					values = one ? null : [],
+					max = one ? index + 1 : options.length;
+
+				if ( index < 0 ) {
+					i = max;
+
+				} else {
+					i = one ? index : 0;
+				}
+
+				// Loop through all the selected options
+				for ( ; i < max; i++ ) {
+					option = options[ i ];
+
+					// Support: IE <=9 only
+					// IE8-9 doesn't update selected after form reset (#2551)
+					if ( ( option.selected || i === index ) &&
+
+							// Don't return options that are disabled or in a disabled optgroup
+							!option.disabled &&
+							( !option.parentNode.disabled ||
+								!nodeName( option.parentNode, "optgroup" ) ) ) {
+
+						// Get the specific value for the option
+						value = jQuery( option ).val();
+
+						// We don't need an array for one selects
+						if ( one ) {
+							return value;
+						}
+
+						// Multi-Selects return an array
+						values.push( value );
+					}
+				}
+
+				return values;
+			},
+
+			set: function( elem, value ) {
+				var optionSet, option,
+					options = elem.options,
+					values = jQuery.makeArray( value ),
+					i = options.length;
+
+				while ( i-- ) {
+					option = options[ i ];
+
+					/* eslint-disable no-cond-assign */
+
+					if ( option.selected =
+						jQuery.inArray( jQuery.valHooks.option.get( option ), values ) > -1
+					) {
+						optionSet = true;
+					}
+
+					/* eslint-enable no-cond-assign */
+				}
+
+				// Force browsers to behave consistently when non-matching value is set
+				if ( !optionSet ) {
+					elem.selectedIndex = -1;
+				}
+				return values;
+			}
+		}
+	}
+} );
+
+// Radios and checkboxes getter/setter
+jQuery.each( [ "radio", "checkbox" ], function() {
+	jQuery.valHooks[ this ] = {
+		set: function( elem, value ) {
+			if ( Array.isArray( value ) ) {
+				return ( elem.checked = jQuery.inArray( jQuery( elem ).val(), value ) > -1 );
+			}
+		}
+	};
+	if ( !support.checkOn ) {
+		jQuery.valHooks[ this ].get = function( elem ) {
+			return elem.getAttribute( "value" ) === null ? "on" : elem.value;
+		};
+	}
+} );
+
+
+
+
+// Return jQuery for attributes-only inclusion
+
+
+support.focusin = "onfocusin" in window;
+
+
+var rfocusMorph = /^(?:focusinfocus|focusoutblur)$/,
+	stopPropagationCallback = function( e ) {
+		e.stopPropagation();
+	};
+
+jQuery.extend( jQuery.event, {
+
+	trigger: function( event, data, elem, onlyHandlers ) {
+
+		var i, cur, tmp, bubbleType, ontype, handle, special, lastElement,
+			eventPath = [ elem || document ],
+			type = hasOwn.call( event, "type" ) ? event.type : event,
+			namespaces = hasOwn.call( event, "namespace" ) ? event.namespace.split( "." ) : [];
+
+		cur = lastElement = tmp = elem = elem || document;
+
+		// Don't do events on text and comment nodes
+		if ( elem.nodeType === 3 || elem.nodeType === 8 ) {
+			return;
+		}
+
+		// focus/blur morphs to focusin/out; ensure we're not firing them right now
+		if ( rfocusMorph.test( type + jQuery.event.triggered ) ) {
+			return;
+		}
+
+		if ( type.indexOf( "." ) > -1 ) {
+
+			// Namespaced trigger; create a regexp to match event type in handle()
+			namespaces = type.split( "." );
+			type = namespaces.shift();
+			namespaces.sort();
+		}
+		ontype = type.indexOf( ":" ) < 0 && "on" + type;
+
+		// Caller can pass in a jQuery.Event object, Object, or just an event type string
+		event = event[ jQuery.expando ] ?
+			event :
+			new jQuery.Event( type, typeof event === "object" && event );
+
+		// Trigger bitmask: & 1 for native handlers; & 2 for jQuery (always true)
+		event.isTrigger = onlyHandlers ? 2 : 3;
+		event.namespace = namespaces.join( "." );
+		event.rnamespace = event.namespace ?
+			new RegExp( "(^|\\.)" + namespaces.join( "\\.(?:.*\\.|)" ) + "(\\.|$)" ) :
+			null;
+
+		// Clean up the event in case it is being reused
+		event.result = undefined;
+		if ( !event.target ) {
+			event.target = elem;
+		}
+
+		// Clone any incoming data and prepend the event, creating the handler arg list
+		data = data == null ?
+			[ event ] :
+			jQuery.makeArray( data, [ event ] );
+
+		// Allow special events to draw outside the lines
+		special = jQuery.event.special[ type ] || {};
+		if ( !onlyHandlers && special.trigger && special.trigger.apply( elem, data ) === false ) {
+			return;
+		}
+
+		// Determine event propagation path in advance, per W3C events spec (#9951)
+		// Bubble up to document, then to window; watch for a global ownerDocument var (#9724)
+		if ( !onlyHandlers && !special.noBubble && !isWindow( elem ) ) {
+
+			bubbleType = special.delegateType || type;
+			if ( !rfocusMorph.test( bubbleType + type ) ) {
+				cur = cur.parentNode;
+			}
+			for ( ; cur; cur = cur.parentNode ) {
+				eventPath.push( cur );
+				tmp = cur;
+			}
+
+			// Only add window if we got to document (e.g., not plain obj or detached DOM)
+			if ( tmp === ( elem.ownerDocument || document ) ) {
+				eventPath.push( tmp.defaultView || tmp.parentWindow || window );
+			}
+		}
+
+		// Fire handlers on the event path
+		i = 0;
+		while ( ( cur = eventPath[ i++ ] ) && !event.isPropagationStopped() ) {
+			lastElement = cur;
+			event.type = i > 1 ?
+				bubbleType :
+				special.bindType || type;
+
+			// jQuery handler
+			handle = ( dataPriv.get( cur, "events" ) || Object.create( null ) )[ event.type ] &&
+				dataPriv.get( cur, "handle" );
+			if ( handle ) {
+				handle.apply( cur, data );
+			}
+
+			// Native handler
+			handle = ontype && cur[ ontype ];
+			if ( handle && handle.apply && acceptData( cur ) ) {
+				event.result = handle.apply( cur, data );
+				if ( event.result === false ) {
+					event.preventDefault();
+				}
+			}
+		}
+		event.type = type;
+
+		// If nobody prevented the default action, do it now
+		if ( !onlyHandlers && !event.isDefaultPrevented() ) {
+
+			if ( ( !special._default ||
+				special._default.apply( eventPath.pop(), data ) === false ) &&
+				acceptData( elem ) ) {
+
+				// Call a native DOM method on the target with the same name as the event.
+				// Don't do default actions on window, that's where global variables be (#6170)
+				if ( ontype && isFunction( elem[ type ] ) && !isWindow( elem ) ) {
+
+					// Don't re-trigger an onFOO event when we call its FOO() method
+					tmp = elem[ ontype ];
+
+					if ( tmp ) {
+						elem[ ontype ] = null;
+					}
+
+					// Prevent re-triggering of the same event, since we already bubbled it above
+					jQuery.event.triggered = type;
+
+					if ( event.isPropagationStopped() ) {
+						lastElement.addEventListener( type, stopPropagationCallback );
+					}
+
+					elem[ type ]();
+
+					if ( event.isPropagationStopped() ) {
+						lastElement.removeEventListener( type, stopPropagationCallback );
+					}
+
+					jQuery.event.triggered = undefined;
+
+					if ( tmp ) {
+						elem[ ontype ] = tmp;
+					}
+				}
+			}
+		}
+
+		return event.result;
+	},
+
+	// Piggyback on a donor event to simulate a different one
+	// Used only for `focus(in | out)` events
+	simulate: function( type, elem, event ) {
+		var e = jQuery.extend(
+			new jQuery.Event(),
+			event,
+			{
+				type: type,
+				isSimulated: true
+			}
+		);
+
+		jQuery.event.trigger( e, null, elem );
+	}
+
+} );
+
+jQuery.fn.extend( {
+
+	trigger: function( type, data ) {
+		return this.each( function() {
+			jQuery.event.trigger( type, data, this );
+		} );
+	},
+	triggerHandler: function( type, data ) {
+		var elem = this[ 0 ];
+		if ( elem ) {
+			return jQuery.event.trigger( type, data, elem, true );
+		}
+	}
+} );
+
+
+// Support: Firefox <=44
+// Firefox doesn't have focus(in | out) events
+// Related ticket - https://bugzilla.mozilla.org/show_bug.cgi?id=687787
+//
+// Support: Chrome <=48 - 49, Safari <=9.0 - 9.1
+// focus(in | out) events fire after focus & blur events,
+// which is spec violation - http://www.w3.org/TR/DOM-Level-3-Events/#events-focusevent-event-order
+// Related ticket - https://bugs.chromium.org/p/chromium/issues/detail?id=449857
+if ( !support.focusin ) {
+	jQuery.each( { focus: "focusin", blur: "focusout" }, function( orig, fix ) {
+
+		// Attach a single capturing handler on the document while someone wants focusin/focusout
+		var handler = function( event ) {
+			jQuery.event.simulate( fix, event.target, jQuery.event.fix( event ) );
+		};
+
+		jQuery.event.special[ fix ] = {
+			setup: function() {
+
+				// Handle: regular nodes (via `this.ownerDocument`), window
+				// (via `this.document`) & document (via `this`).
+				var doc = this.ownerDocument || this.document || this,
+					attaches = dataPriv.access( doc, fix );
+
+				if ( !attaches ) {
+					doc.addEventListener( orig, handler, true );
+				}
+				dataPriv.access( doc, fix, ( attaches || 0 ) + 1 );
+			},
+			teardown: function() {
+				var doc = this.ownerDocument || this.document || this,
+					attaches = dataPriv.access( doc, fix ) - 1;
+
+				if ( !attaches ) {
+					doc.removeEventListener( orig, handler, true );
+					dataPriv.remove( doc, fix );
+
+				} else {
+					dataPriv.access( doc, fix, attaches );
+				}
+			}
+		};
+	} );
+}
+var location = window.location;
+
+var nonce = { guid: Date.now() };
+
+var rquery = ( /\?/ );
+
+
+
+// Cross-browser xml parsing
+jQuery.parseXML = function( data ) {
+	var xml, parserErrorElem;
+	if ( !data || typeof data !== "string" ) {
+		return null;
+	}
+
+	// Support: IE 9 - 11 only
+	// IE throws on parseFromString with invalid input.
+	try {
+		xml = ( new window.DOMParser() ).parseFromString( data, "text/xml" );
+	} catch ( e ) {}
+
+	parserErrorElem = xml && xml.getElementsByTagName( "parsererror" )[ 0 ];
+	if ( !xml || parserErrorElem ) {
+		jQuery.error( "Invalid XML: " + (
+			parserErrorElem ?
+				jQuery.map( parserErrorElem.childNodes, function( el ) {
+					return el.textContent;
+				} ).join( "\n" ) :
+				data
+		) );
+	}
+	return xml;
+};
+
+
+var
+	rbracket = /\[\]$/,
+	rCRLF = /\r?\n/g,
+	rsubmitterTypes = /^(?:submit|button|image|reset|file)$/i,
+	rsubmittable = /^(?:input|select|textarea|keygen)/i;
+
+function buildParams( prefix, obj, traditional, add ) {
+	var name;
+
+	if ( Array.isArray( obj ) ) {
+
+		// Serialize array item.
+		jQuery.each( obj, function( i, v ) {
+			if ( traditional || rbracket.test( prefix ) ) {
+
+				// Treat each array item as a scalar.
+				add( prefix, v );
+
+			} else {
+
+				// Item is non-scalar (array or object), encode its numeric index.
+				buildParams(
+					prefix + "[" + ( typeof v === "object" && v != null ? i : "" ) + "]",
+					v,
+					traditional,
+					add
+				);
+			}
+		} );
+
+	} else if ( !traditional && toType( obj ) === "object" ) {
+
+		// Serialize object item.
+		for ( name in obj ) {
+			buildParams( prefix + "[" + name + "]", obj[ name ], traditional, add );
+		}
+
+	} else {
+
+		// Serialize scalar item.
+		add( prefix, obj );
+	}
+}
+
+// Serialize an array of form elements or a set of
+// key/values into a query string
+jQuery.param = function( a, traditional ) {
+	var prefix,
+		s = [],
+		add = function( key, valueOrFunction ) {
+
+			// If value is a function, invoke it and use its return value
+			var value = isFunction( valueOrFunction ) ?
+				valueOrFunction() :
+				valueOrFunction;
+
+			s[ s.length ] = encodeURIComponent( key ) + "=" +
+				encodeURIComponent( value == null ? "" : value );
+		};
+
+	if ( a == null ) {
+		return "";
+	}
+
+	// If an array was passed in, assume that it is an array of form elements.
+	if ( Array.isArray( a ) || ( a.jquery && !jQuery.isPlainObject( a ) ) ) {
+
+		// Serialize the form elements
+		jQuery.each( a, function() {
+			add( this.name, this.value );
+		} );
+
+	} else {
+
+		// If traditional, encode the "old" way (the way 1.3.2 or older
+		// did it), otherwise encode params recursively.
+		for ( prefix in a ) {
+			buildParams( prefix, a[ prefix ], traditional, add );
+		}
+	}
+
+	// Return the resulting serialization
+	return s.join( "&" );
+};
+
+jQuery.fn.extend( {
+	serialize: function() {
+		return jQuery.param( this.serializeArray() );
+	},
+	serializeArray: function() {
+		return this.map( function() {
+
+			// Can add propHook for "elements" to filter or add form elements
+			var elements = jQuery.prop( this, "elements" );
+			return elements ? jQuery.makeArray( elements ) : this;
+		} ).filter( function() {
+			var type = this.type;
+
+			// Use .is( ":disabled" ) so that fieldset[disabled] works
+			return this.name && !jQuery( this ).is( ":disabled" ) &&
+				rsubmittable.test( this.nodeName ) && !rsubmitterTypes.test( type ) &&
+				( this.checked || !rcheckableType.test( type ) );
+		} ).map( function( _i, elem ) {
+			var val = jQuery( this ).val();
+
+			if ( val == null ) {
+				return null;
+			}
+
+			if ( Array.isArray( val ) ) {
+				return jQuery.map( val, function( val ) {
+					return { name: elem.name, value: val.replace( rCRLF, "\r\n" ) };
+				} );
+			}
+
+			return { name: elem.name, value: val.replace( rCRLF, "\r\n" ) };
+		} ).get();
+	}
+} );
+
+
+var
+	r20 = /%20/g,
+	rhash = /#.*$/,
+	rantiCache = /([?&])_=[^&]*/,
+	rheaders = /^(.*?):[ \t]*([^\r\n]*)$/mg,
+
+	// #7653, #8125, #8152: local protocol detection
+	rlocalProtocol = /^(?:about|app|app-storage|.+-extension|file|res|widget):$/,
+	rnoContent = /^(?:GET|HEAD)$/,
+	rprotocol = /^\/\//,
+
+	/* Prefilters
+	 * 1) They are useful to introduce custom dataTypes (see ajax/jsonp.js for an example)
+	 * 2) These are called:
+	 *    - BEFORE asking for a transport
+	 *    - AFTER param serialization (s.data is a string if s.processData is true)
+	 * 3) key is the dataType
+	 * 4) the catchall symbol "*" can be used
+	 * 5) execution will start with transport dataType and THEN continue down to "*" if needed
+	 */
+	prefilters = {},
+
+	/* Transports bindings
+	 * 1) key is the dataType
+	 * 2) the catchall symbol "*" can be used
+	 * 3) selection will start with transport dataType and THEN go to "*" if needed
+	 */
+	transports = {},
+
+	// Avoid comment-prolog char sequence (#10098); must appease lint and evade compression
+	allTypes = "*/".concat( "*" ),
+
+	// Anchor tag for parsing the document origin
+	originAnchor = document.createElement( "a" );
+
+originAnchor.href = location.href;
+
+// Base "constructor" for jQuery.ajaxPrefilter and jQuery.ajaxTransport
+function addToPrefiltersOrTransports( structure ) {
+
+	// dataTypeExpression is optional and defaults to "*"
+	return function( dataTypeExpression, func ) {
+
+		if ( typeof dataTypeExpression !== "string" ) {
+			func = dataTypeExpression;
+			dataTypeExpression = "*";
+		}
+
+		var dataType,
+			i = 0,
+			dataTypes = dataTypeExpression.toLowerCase().match( rnothtmlwhite ) || [];
+
+		if ( isFunction( func ) ) {
+
+			// For each dataType in the dataTypeExpression
+			while ( ( dataType = dataTypes[ i++ ] ) ) {
+
+				// Prepend if requested
+				if ( dataType[ 0 ] === "+" ) {
+					dataType = dataType.slice( 1 ) || "*";
+					( structure[ dataType ] = structure[ dataType ] || [] ).unshift( func );
+
+				// Otherwise append
+				} else {
+					( structure[ dataType ] = structure[ dataType ] || [] ).push( func );
+				}
+			}
+		}
+	};
+}
+
+// Base inspection function for prefilters and transports
+function inspectPrefiltersOrTransports( structure, options, originalOptions, jqXHR ) {
+
+	var inspected = {},
+		seekingTransport = ( structure === transports );
+
+	function inspect( dataType ) {
+		var selected;
+		inspected[ dataType ] = true;
+		jQuery.each( structure[ dataType ] || [], function( _, prefilterOrFactory ) {
+			var dataTypeOrTransport = prefilterOrFactory( options, originalOptions, jqXHR );
+			if ( typeof dataTypeOrTransport === "string" &&
+				!seekingTransport && !inspected[ dataTypeOrTransport ] ) {
+
+				options.dataTypes.unshift( dataTypeOrTransport );
+				inspect( dataTypeOrTransport );
+				return false;
+			} else if ( seekingTransport ) {
+				return !( selected = dataTypeOrTransport );
+			}
+		} );
+		return selected;
+	}
+
+	return inspect( options.dataTypes[ 0 ] ) || !inspected[ "*" ] && inspect( "*" );
+}
+
+// A special extend for ajax options
+// that takes "flat" options (not to be deep extended)
+// Fixes #9887
+function ajaxExtend( target, src ) {
+	var key, deep,
+		flatOptions = jQuery.ajaxSettings.flatOptions || {};
+
+	for ( key in src ) {
+		if ( src[ key ] !== undefined ) {
+			( flatOptions[ key ] ? target : ( deep || ( deep = {} ) ) )[ key ] = src[ key ];
+		}
+	}
+	if ( deep ) {
+		jQuery.extend( true, target, deep );
+	}
+
+	return target;
+}
+
+/* Handles responses to an ajax request:
+ * - finds the right dataType (mediates between content-type and expected dataType)
+ * - returns the corresponding response
+ */
+function ajaxHandleResponses( s, jqXHR, responses ) {
+
+	var ct, type, finalDataType, firstDataType,
+		contents = s.contents,
+		dataTypes = s.dataTypes;
+
+	// Remove auto dataType and get content-type in the process
+	while ( dataTypes[ 0 ] === "*" ) {
+		dataTypes.shift();
+		if ( ct === undefined ) {
+			ct = s.mimeType || jqXHR.getResponseHeader( "Content-Type" );
+		}
+	}
+
+	// Check if we're dealing with a known content-type
+	if ( ct ) {
+		for ( type in contents ) {
+			if ( contents[ type ] && contents[ type ].test( ct ) ) {
+				dataTypes.unshift( type );
+				break;
+			}
+		}
+	}
+
+	// Check to see if we have a response for the expected dataType
+	if ( dataTypes[ 0 ] in responses ) {
+		finalDataType = dataTypes[ 0 ];
+	} else {
+
+		// Try convertible dataTypes
+		for ( type in responses ) {
+			if ( !dataTypes[ 0 ] || s.converters[ type + " " + dataTypes[ 0 ] ] ) {
+				finalDataType = type;
+				break;
+			}
+			if ( !firstDataType ) {
+				firstDataType = type;
+			}
+		}
+
+		// Or just use first one
+		finalDataType = finalDataType || firstDataType;
+	}
+
+	// If we found a dataType
+	// We add the dataType to the list if needed
+	// and return the corresponding response
+	if ( finalDataType ) {
+		if ( finalDataType !== dataTypes[ 0 ] ) {
+			dataTypes.unshift( finalDataType );
+		}
+		return responses[ finalDataType ];
+	}
+}
+
+/* Chain conversions given the request and the original response
+ * Also sets the responseXXX fields on the jqXHR instance
+ */
+function ajaxConvert( s, response, jqXHR, isSuccess ) {
+	var conv2, current, conv, tmp, prev,
+		converters = {},
+
+		// Work with a copy of dataTypes in case we need to modify it for conversion
+		dataTypes = s.dataTypes.slice();
+
+	// Create converters map with lowercased keys
+	if ( dataTypes[ 1 ] ) {
+		for ( conv in s.converters ) {
+			converters[ conv.toLowerCase() ] = s.converters[ conv ];
+		}
+	}
+
+	current = dataTypes.shift();
+
+	// Convert to each sequential dataType
+	while ( current ) {
+
+		if ( s.responseFields[ current ] ) {
+			jqXHR[ s.responseFields[ current ] ] = response;
+		}
+
+		// Apply the dataFilter if provided
+		if ( !prev && isSuccess && s.dataFilter ) {
+			response = s.dataFilter( response, s.dataType );
+		}
+
+		prev = current;
+		current = dataTypes.shift();
+
+		if ( current ) {
+
+			// There's only work to do if current dataType is non-auto
+			if ( current === "*" ) {
+
+				current = prev;
+
+			// Convert response if prev dataType is non-auto and differs from current
+			} else if ( prev !== "*" && prev !== current ) {
+
+				// Seek a direct converter
+				conv = converters[ prev + " " + current ] || converters[ "* " + current ];
+
+				// If none found, seek a pair
+				if ( !conv ) {
+					for ( conv2 in converters ) {
+
+						// If conv2 outputs current
+						tmp = conv2.split( " " );
+						if ( tmp[ 1 ] === current ) {
+
+							// If prev can be converted to accepted input
+							conv = converters[ prev + " " + tmp[ 0 ] ] ||
+								converters[ "* " + tmp[ 0 ] ];
+							if ( conv ) {
+
+								// Condense equivalence converters
+								if ( conv === true ) {
+									conv = converters[ conv2 ];
+
+								// Otherwise, insert the intermediate dataType
+								} else if ( converters[ conv2 ] !== true ) {
+									current = tmp[ 0 ];
+									dataTypes.unshift( tmp[ 1 ] );
+								}
+								break;
+							}
+						}
+					}
+				}
+
+				// Apply converter (if not an equivalence)
+				if ( conv !== true ) {
+
+					// Unless errors are allowed to bubble, catch and return them
+					if ( conv && s.throws ) {
+						response = conv( response );
+					} else {
+						try {
+							response = conv( response );
+						} catch ( e ) {
+							return {
+								state: "parsererror",
+								error: conv ? e : "No conversion from " + prev + " to " + current
+							};
+						}
+					}
+				}
+			}
+		}
+	}
+
+	return { state: "success", data: response };
+}
+
+jQuery.extend( {
+
+	// Counter for holding the number of active queries
+	active: 0,
+
+	// Last-Modified header cache for next request
+	lastModified: {},
+	etag: {},
+
+	ajaxSettings: {
+		url: location.href,
+		type: "GET",
+		isLocal: rlocalProtocol.test( location.protocol ),
+		global: true,
+		processData: true,
+		async: true,
+		contentType: "application/x-www-form-urlencoded; charset=UTF-8",
+
+		/*
+		timeout: 0,
+		data: null,
+		dataType: null,
+		username: null,
+		password: null,
+		cache: null,
+		throws: false,
+		traditional: false,
+		headers: {},
+		*/
+
+		accepts: {
+			"*": allTypes,
+			text: "text/plain",
+			html: "text/html",
+			xml: "application/xml, text/xml",
+			json: "application/json, text/javascript"
+		},
+
+		contents: {
+			xml: /\bxml\b/,
+			html: /\bhtml/,
+			json: /\bjson\b/
+		},
+
+		responseFields: {
+			xml: "responseXML",
+			text: "responseText",
+			json: "responseJSON"
+		},
+
+		// Data converters
+		// Keys separate source (or catchall "*") and destination types with a single space
+		converters: {
+
+			// Convert anything to text
+			"* text": String,
+
+			// Text to html (true = no transformation)
+			"text html": true,
+
+			// Evaluate text as a json expression
+			"text json": JSON.parse,
+
+			// Parse text as xml
+			"text xml": jQuery.parseXML
+		},
+
+		// For options that shouldn't be deep extended:
+		// you can add your own custom options here if
+		// and when you create one that shouldn't be
+		// deep extended (see ajaxExtend)
+		flatOptions: {
+			url: true,
+			context: true
+		}
+	},
+
+	// Creates a full fledged settings object into target
+	// with both ajaxSettings and settings fields.
+	// If target is omitted, writes into ajaxSettings.
+	ajaxSetup: function( target, settings ) {
+		return settings ?
+
+			// Building a settings object
+			ajaxExtend( ajaxExtend( target, jQuery.ajaxSettings ), settings ) :
+
+			// Extending ajaxSettings
+			ajaxExtend( jQuery.ajaxSettings, target );
+	},
+
+	ajaxPrefilter: addToPrefiltersOrTransports( prefilters ),
+	ajaxTransport: addToPrefiltersOrTransports( transports ),
+
+	// Main method
+	ajax: function( url, options ) {
+
+		// If url is an object, simulate pre-1.5 signature
+		if ( typeof url === "object" ) {
+			options = url;
+			url = undefined;
+		}
+
+		// Force options to be an object
+		options = options || {};
+
+		var transport,
+
+			// URL without anti-cache param
+			cacheURL,
+
+			// Response headers
+			responseHeadersString,
+			responseHeaders,
+
+			// timeout handle
+			timeoutTimer,
+
+			// Url cleanup var
+			urlAnchor,
+
+			// Request state (becomes false upon send and true upon completion)
+			completed,
+
+			// To know if global events are to be dispatched
+			fireGlobals,
+
+			// Loop variable
+			i,
+
+			// uncached part of the url
+			uncached,
+
+			// Create the final options object
+			s = jQuery.ajaxSetup( {}, options ),
+
+			// Callbacks context
+			callbackContext = s.context || s,
+
+			// Context for global events is callbackContext if it is a DOM node or jQuery collection
+			globalEventContext = s.context &&
+				( callbackContext.nodeType || callbackContext.jquery ) ?
+				jQuery( callbackContext ) :
+				jQuery.event,
+
+			// Deferreds
+			deferred = jQuery.Deferred(),
+			completeDeferred = jQuery.Callbacks( "once memory" ),
+
+			// Status-dependent callbacks
+			statusCode = s.statusCode || {},
+
+			// Headers (they are sent all at once)
+			requestHeaders = {},
+			requestHeadersNames = {},
+
+			// Default abort message
+			strAbort = "canceled",
+
+			// Fake xhr
+			jqXHR = {
+				readyState: 0,
+
+				// Builds headers hashtable if needed
+				getResponseHeader: function( key ) {
+					var match;
+					if ( completed ) {
+						if ( !responseHeaders ) {
+							responseHeaders = {};
+							while ( ( match = rheaders.exec( responseHeadersString ) ) ) {
+								responseHeaders[ match[ 1 ].toLowerCase() + " " ] =
+									( responseHeaders[ match[ 1 ].toLowerCase() + " " ] || [] )
+										.concat( match[ 2 ] );
+							}
+						}
+						match = responseHeaders[ key.toLowerCase() + " " ];
+					}
+					return match == null ? null : match.join( ", " );
+				},
+
+				// Raw string
+				getAllResponseHeaders: function() {
+					return completed ? responseHeadersString : null;
+				},
+
+				// Caches the header
+				setRequestHeader: function( name, value ) {
+					if ( completed == null ) {
+						name = requestHeadersNames[ name.toLowerCase() ] =
+							requestHeadersNames[ name.toLowerCase() ] || name;
+						requestHeaders[ name ] = value;
+					}
+					return this;
+				},
+
+				// Overrides response content-type header
+				overrideMimeType: function( type ) {
+					if ( completed == null ) {
+						s.mimeType = type;
+					}
+					return this;
+				},
+
+				// Status-dependent callbacks
+				statusCode: function( map ) {
+					var code;
+					if ( map ) {
+						if ( completed ) {
+
+							// Execute the appropriate callbacks
+							jqXHR.always( map[ jqXHR.status ] );
+						} else {
+
+							// Lazy-add the new callbacks in a way that preserves old ones
+							for ( code in map ) {
+								statusCode[ code ] = [ statusCode[ code ], map[ code ] ];
+							}
+						}
+					}
+					return this;
+				},
+
+				// Cancel the request
+				abort: function( statusText ) {
+					var finalText = statusText || strAbort;
+					if ( transport ) {
+						transport.abort( finalText );
+					}
+					done( 0, finalText );
+					return this;
+				}
+			};
+
+		// Attach deferreds
+		deferred.promise( jqXHR );
+
+		// Add protocol if not provided (prefilters might expect it)
+		// Handle falsy url in the settings object (#10093: consistency with old signature)
+		// We also use the url parameter if available
+		s.url = ( ( url || s.url || location.href ) + "" )
+			.replace( rprotocol, location.protocol + "//" );
+
+		// Alias method option to type as per ticket #12004
+		s.type = options.method || options.type || s.method || s.type;
+
+		// Extract dataTypes list
+		s.dataTypes = ( s.dataType || "*" ).toLowerCase().match( rnothtmlwhite ) || [ "" ];
+
+		// A cross-domain request is in order when the origin doesn't match the current origin.
+		if ( s.crossDomain == null ) {
+			urlAnchor = document.createElement( "a" );
+
+			// Support: IE <=8 - 11, Edge 12 - 15
+			// IE throws exception on accessing the href property if url is malformed,
+			// e.g. http://example.com:80x/
+			try {
+				urlAnchor.href = s.url;
+
+				// Support: IE <=8 - 11 only
+				// Anchor's host property isn't correctly set when s.url is relative
+				urlAnchor.href = urlAnchor.href;
+				s.crossDomain = originAnchor.protocol + "//" + originAnchor.host !==
+					urlAnchor.protocol + "//" + urlAnchor.host;
+			} catch ( e ) {
+
+				// If there is an error parsing the URL, assume it is crossDomain,
+				// it can be rejected by the transport if it is invalid
+				s.crossDomain = true;
+			}
+		}
+
+		// Convert data if not already a string
+		if ( s.data && s.processData && typeof s.data !== "string" ) {
+			s.data = jQuery.param( s.data, s.traditional );
+		}
+
+		// Apply prefilters
+		inspectPrefiltersOrTransports( prefilters, s, options, jqXHR );
+
+		// If request was aborted inside a prefilter, stop there
+		if ( completed ) {
+			return jqXHR;
+		}
+
+		// We can fire global events as of now if asked to
+		// Don't fire events if jQuery.event is undefined in an AMD-usage scenario (#15118)
+		fireGlobals = jQuery.event && s.global;
+
+		// Watch for a new set of requests
+		if ( fireGlobals && jQuery.active++ === 0 ) {
+			jQuery.event.trigger( "ajaxStart" );
+		}
+
+		// Uppercase the type
+		s.type = s.type.toUpperCase();
+
+		// Determine if request has content
+		s.hasContent = !rnoContent.test( s.type );
+
+		// Save the URL in case we're toying with the If-Modified-Since
+		// and/or If-None-Match header later on
+		// Remove hash to simplify url manipulation
+		cacheURL = s.url.replace( rhash, "" );
+
+		// More options handling for requests with no content
+		if ( !s.hasContent ) {
+
+			// Remember the hash so we can put it back
+			uncached = s.url.slice( cacheURL.length );
+
+			// If data is available and should be processed, append data to url
+			if ( s.data && ( s.processData || typeof s.data === "string" ) ) {
+				cacheURL += ( rquery.test( cacheURL ) ? "&" : "?" ) + s.data;
+
+				// #9682: remove data so that it's not used in an eventual retry
+				delete s.data;
+			}
+
+			// Add or update anti-cache param if needed
+			if ( s.cache === false ) {
+				cacheURL = cacheURL.replace( rantiCache, "$1" );
+				uncached = ( rquery.test( cacheURL ) ? "&" : "?" ) + "_=" + ( nonce.guid++ ) +
+					uncached;
+			}
+
+			// Put hash and anti-cache on the URL that will be requested (gh-1732)
+			s.url = cacheURL + uncached;
+
+		// Change '%20' to '+' if this is encoded form body content (gh-2658)
+		} else if ( s.data && s.processData &&
+			( s.contentType || "" ).indexOf( "application/x-www-form-urlencoded" ) === 0 ) {
+			s.data = s.data.replace( r20, "+" );
+		}
+
+		// Set the If-Modified-Since and/or If-None-Match header, if in ifModified mode.
+		if ( s.ifModified ) {
+			if ( jQuery.lastModified[ cacheURL ] ) {
+				jqXHR.setRequestHeader( "If-Modified-Since", jQuery.lastModified[ cacheURL ] );
+			}
+			if ( jQuery.etag[ cacheURL ] ) {
+				jqXHR.setRequestHeader( "If-None-Match", jQuery.etag[ cacheURL ] );
+			}
+		}
+
+		// Set the correct header, if data is being sent
+		if ( s.data && s.hasContent && s.contentType !== false || options.contentType ) {
+			jqXHR.setRequestHeader( "Content-Type", s.contentType );
+		}
+
+		// Set the Accepts header for the server, depending on the dataType
+		jqXHR.setRequestHeader(
+			"Accept",
+			s.dataTypes[ 0 ] && s.accepts[ s.dataTypes[ 0 ] ] ?
+				s.accepts[ s.dataTypes[ 0 ] ] +
+					( s.dataTypes[ 0 ] !== "*" ? ", " + allTypes + "; q=0.01" : "" ) :
+				s.accepts[ "*" ]
+		);
+
+		// Check for headers option
+		for ( i in s.headers ) {
+			jqXHR.setRequestHeader( i, s.headers[ i ] );
+		}
+
+		// Allow custom headers/mimetypes and early abort
+		if ( s.beforeSend &&
+			( s.beforeSend.call( callbackContext, jqXHR, s ) === false || completed ) ) {
+
+			// Abort if not done already and return
+			return jqXHR.abort();
+		}
+
+		// Aborting is no longer a cancellation
+		strAbort = "abort";
+
+		// Install callbacks on deferreds
+		completeDeferred.add( s.complete );
+		jqXHR.done( s.success );
+		jqXHR.fail( s.error );
+
+		// Get transport
+		transport = inspectPrefiltersOrTransports( transports, s, options, jqXHR );
+
+		// If no transport, we auto-abort
+		if ( !transport ) {
+			done( -1, "No Transport" );
+		} else {
+			jqXHR.readyState = 1;
+
+			// Send global event
+			if ( fireGlobals ) {
+				globalEventContext.trigger( "ajaxSend", [ jqXHR, s ] );
+			}
+
+			// If request was aborted inside ajaxSend, stop there
+			if ( completed ) {
+				return jqXHR;
+			}
+
+			// Timeout
+			if ( s.async && s.timeout > 0 ) {
+				timeoutTimer = window.setTimeout( function() {
+					jqXHR.abort( "timeout" );
+				}, s.timeout );
+			}
+
+			try {
+				completed = false;
+				transport.send( requestHeaders, done );
+			} catch ( e ) {
+
+				// Rethrow post-completion exceptions
+				if ( completed ) {
+					throw e;
+				}
+
+				// Propagate others as results
+				done( -1, e );
+			}
+		}
+
+		// Callback for when everything is done
+		function done( status, nativeStatusText, responses, headers ) {
+			var isSuccess, success, error, response, modified,
+				statusText = nativeStatusText;
+
+			// Ignore repeat invocations
+			if ( completed ) {
+				return;
+			}
+
+			completed = true;
+
+			// Clear timeout if it exists
+			if ( timeoutTimer ) {
+				window.clearTimeout( timeoutTimer );
+			}
+
+			// Dereference transport for early garbage collection
+			// (no matter how long the jqXHR object will be used)
+			transport = undefined;
+
+			// Cache response headers
+			responseHeadersString = headers || "";
+
+			// Set readyState
+			jqXHR.readyState = status > 0 ? 4 : 0;
+
+			// Determine if successful
+			isSuccess = status >= 200 && status < 300 || status === 304;
+
+			// Get response data
+			if ( responses ) {
+				response = ajaxHandleResponses( s, jqXHR, responses );
+			}
+
+			// Use a noop converter for missing script but not if jsonp
+			if ( !isSuccess &&
+				jQuery.inArray( "script", s.dataTypes ) > -1 &&
+				jQuery.inArray( "json", s.dataTypes ) < 0 ) {
+				s.converters[ "text script" ] = function() {};
+			}
+
+			// Convert no matter what (that way responseXXX fields are always set)
+			response = ajaxConvert( s, response, jqXHR, isSuccess );
+
+			// If successful, handle type chaining
+			if ( isSuccess ) {
+
+				// Set the If-Modified-Since and/or If-None-Match header, if in ifModified mode.
+				if ( s.ifModified ) {
+					modified = jqXHR.getResponseHeader( "Last-Modified" );
+					if ( modified ) {
+						jQuery.lastModified[ cacheURL ] = modified;
+					}
+					modified = jqXHR.getResponseHeader( "etag" );
+					if ( modified ) {
+						jQuery.etag[ cacheURL ] = modified;
+					}
+				}
+
+				// if no content
+				if ( status === 204 || s.type === "HEAD" ) {
+					statusText = "nocontent";
+
+				// if not modified
+				} else if ( status === 304 ) {
+					statusText = "notmodified";
+
+				// If we have data, let's convert it
+				} else {
+					statusText = response.state;
+					success = response.data;
+					error = response.error;
+					isSuccess = !error;
+				}
+			} else {
+
+				// Extract error from statusText and normalize for non-aborts
+				error = statusText;
+				if ( status || !statusText ) {
+					statusText = "error";
+					if ( status < 0 ) {
+						status = 0;
+					}
+				}
+			}
+
+			// Set data for the fake xhr object
+			jqXHR.status = status;
+			jqXHR.statusText = ( nativeStatusText || statusText ) + "";
+
+			// Success/Error
+			if ( isSuccess ) {
+				deferred.resolveWith( callbackContext, [ success, statusText, jqXHR ] );
+			} else {
+				deferred.rejectWith( callbackContext, [ jqXHR, statusText, error ] );
+			}
+
+			// Status-dependent callbacks
+			jqXHR.statusCode( statusCode );
+			statusCode = undefined;
+
+			if ( fireGlobals ) {
+				globalEventContext.trigger( isSuccess ? "ajaxSuccess" : "ajaxError",
+					[ jqXHR, s, isSuccess ? success : error ] );
+			}
+
+			// Complete
+			completeDeferred.fireWith( callbackContext, [ jqXHR, statusText ] );
+
+			if ( fireGlobals ) {
+				globalEventContext.trigger( "ajaxComplete", [ jqXHR, s ] );
+
+				// Handle the global AJAX counter
+				if ( !( --jQuery.active ) ) {
+					jQuery.event.trigger( "ajaxStop" );
+				}
+			}
+		}
+
+		return jqXHR;
+	},
+
+	getJSON: function( url, data, callback ) {
+		return jQuery.get( url, data, callback, "json" );
+	},
+
+	getScript: function( url, callback ) {
+		return jQuery.get( url, undefined, callback, "script" );
+	}
+} );
+
+jQuery.each( [ "get", "post" ], function( _i, method ) {
+	jQuery[ method ] = function( url, data, callback, type ) {
+
+		// Shift arguments if data argument was omitted
+		if ( isFunction( data ) ) {
+			type = type || callback;
+			callback = data;
+			data = undefined;
+		}
+
+		// The url can be an options object (which then must have .url)
+		return jQuery.ajax( jQuery.extend( {
+			url: url,
+			type: method,
+			dataType: type,
+			data: data,
+			success: callback
+		}, jQuery.isPlainObject( url ) && url ) );
+	};
+} );
+
+jQuery.ajaxPrefilter( function( s ) {
+	var i;
+	for ( i in s.headers ) {
+		if ( i.toLowerCase() === "content-type" ) {
+			s.contentType = s.headers[ i ] || "";
+		}
+	}
+} );
+
+
+jQuery._evalUrl = function( url, options, doc ) {
+	return jQuery.ajax( {
+		url: url,
+
+		// Make this explicit, since user can override this through ajaxSetup (#11264)
+		type: "GET",
+		dataType: "script",
+		cache: true,
+		async: false,
+		global: false,
+
+		// Only evaluate the response if it is successful (gh-4126)
+		// dataFilter is not invoked for failure responses, so using it instead
+		// of the default converter is kludgy but it works.
+		converters: {
+			"text script": function() {}
+		},
+		dataFilter: function( response ) {
+			jQuery.globalEval( response, options, doc );
+		}
+	} );
+};
+
+
+jQuery.fn.extend( {
+	wrapAll: function( html ) {
+		var wrap;
+
+		if ( this[ 0 ] ) {
+			if ( isFunction( html ) ) {
+				html = html.call( this[ 0 ] );
+			}
+
+			// The elements to wrap the target around
+			wrap = jQuery( html, this[ 0 ].ownerDocument ).eq( 0 ).clone( true );
+
+			if ( this[ 0 ].parentNode ) {
+				wrap.insertBefore( this[ 0 ] );
+			}
+
+			wrap.map( function() {
+				var elem = this;
+
+				while ( elem.firstElementChild ) {
+					elem = elem.firstElementChild;
+				}
+
+				return elem;
+			} ).append( this );
+		}
+
+		return this;
+	},
+
+	wrapInner: function( html ) {
+		if ( isFunction( html ) ) {
+			return this.each( function( i ) {
+				jQuery( this ).wrapInner( html.call( this, i ) );
+			} );
+		}
+
+		return this.each( function() {
+			var self = jQuery( this ),
+				contents = self.contents();
+
+			if ( contents.length ) {
+				contents.wrapAll( html );
+
+			} else {
+				self.append( html );
+			}
+		} );
+	},
+
+	wrap: function( html ) {
+		var htmlIsFunction = isFunction( html );
+
+		return this.each( function( i ) {
+			jQuery( this ).wrapAll( htmlIsFunction ? html.call( this, i ) : html );
+		} );
+	},
+
+	unwrap: function( selector ) {
+		this.parent( selector ).not( "body" ).each( function() {
+			jQuery( this ).replaceWith( this.childNodes );
+		} );
+		return this;
+	}
+} );
+
+
+jQuery.expr.pseudos.hidden = function( elem ) {
+	return !jQuery.expr.pseudos.visible( elem );
+};
+jQuery.expr.pseudos.visible = function( elem ) {
+	return !!( elem.offsetWidth || elem.offsetHeight || elem.getClientRects().length );
+};
+
+
+
+
+jQuery.ajaxSettings.xhr = function() {
+	try {
+		return new window.XMLHttpRequest();
+	} catch ( e ) {}
+};
+
+var xhrSuccessStatus = {
+
+		// File protocol always yields status code 0, assume 200
+		0: 200,
+
+		// Support: IE <=9 only
+		// #1450: sometimes IE returns 1223 when it should be 204
+		1223: 204
+	},
+	xhrSupported = jQuery.ajaxSettings.xhr();
+
+support.cors = !!xhrSupported && ( "withCredentials" in xhrSupported );
+support.ajax = xhrSupported = !!xhrSupported;
+
+jQuery.ajaxTransport( function( options ) {
+	var callback, errorCallback;
+
+	// Cross domain only allowed if supported through XMLHttpRequest
+	if ( support.cors || xhrSupported && !options.crossDomain ) {
+		return {
+			send: function( headers, complete ) {
+				var i,
+					xhr = options.xhr();
+
+				xhr.open(
+					options.type,
+					options.url,
+					options.async,
+					options.username,
+					options.password
+				);
+
+				// Apply custom fields if provided
+				if ( options.xhrFields ) {
+					for ( i in options.xhrFields ) {
+						xhr[ i ] = options.xhrFields[ i ];
+					}
+				}
+
+				// Override mime type if needed
+				if ( options.mimeType && xhr.overrideMimeType ) {
+					xhr.overrideMimeType( options.mimeType );
+				}
+
+				// X-Requested-With header
+				// For cross-domain requests, seeing as conditions for a preflight are
+				// akin to a jigsaw puzzle, we simply never set it to be sure.
+				// (it can always be set on a per-request basis or even using ajaxSetup)
+				// For same-domain requests, won't change header if already provided.
+				if ( !options.crossDomain && !headers[ "X-Requested-With" ] ) {
+					headers[ "X-Requested-With" ] = "XMLHttpRequest";
+				}
+
+				// Set headers
+				for ( i in headers ) {
+					xhr.setRequestHeader( i, headers[ i ] );
+				}
+
+				// Callback
+				callback = function( type ) {
+					return function() {
+						if ( callback ) {
+							callback = errorCallback = xhr.onload =
+								xhr.onerror = xhr.onabort = xhr.ontimeout =
+									xhr.onreadystatechange = null;
+
+							if ( type === "abort" ) {
+								xhr.abort();
+							} else if ( type === "error" ) {
+
+								// Support: IE <=9 only
+								// On a manual native abort, IE9 throws
+								// errors on any property access that is not readyState
+								if ( typeof xhr.status !== "number" ) {
+									complete( 0, "error" );
+								} else {
+									complete(
+
+										// File: protocol always yields status 0; see #8605, #14207
+										xhr.status,
+										xhr.statusText
+									);
+								}
+							} else {
+								complete(
+									xhrSuccessStatus[ xhr.status ] || xhr.status,
+									xhr.statusText,
+
+									// Support: IE <=9 only
+									// IE9 has no XHR2 but throws on binary (trac-11426)
+									// For XHR2 non-text, let the caller handle it (gh-2498)
+									( xhr.responseType || "text" ) !== "text"  ||
+									typeof xhr.responseText !== "string" ?
+										{ binary: xhr.response } :
+										{ text: xhr.responseText },
+									xhr.getAllResponseHeaders()
+								);
+							}
+						}
+					};
+				};
+
+				// Listen to events
+				xhr.onload = callback();
+				errorCallback = xhr.onerror = xhr.ontimeout = callback( "error" );
+
+				// Support: IE 9 only
+				// Use onreadystatechange to replace onabort
+				// to handle uncaught aborts
+				if ( xhr.onabort !== undefined ) {
+					xhr.onabort = errorCallback;
+				} else {
+					xhr.onreadystatechange = function() {
+
+						// Check readyState before timeout as it changes
+						if ( xhr.readyState === 4 ) {
+
+							// Allow onerror to be called first,
+							// but that will not handle a native abort
+							// Also, save errorCallback to a variable
+							// as xhr.onerror cannot be accessed
+							window.setTimeout( function() {
+								if ( callback ) {
+									errorCallback();
+								}
+							} );
+						}
+					};
+				}
+
+				// Create the abort callback
+				callback = callback( "abort" );
+
+				try {
+
+					// Do send the request (this may raise an exception)
+					xhr.send( options.hasContent && options.data || null );
+				} catch ( e ) {
+
+					// #14683: Only rethrow if this hasn't been notified as an error yet
+					if ( callback ) {
+						throw e;
+					}
+				}
+			},
+
+			abort: function() {
+				if ( callback ) {
+					callback();
+				}
+			}
+		};
+	}
+} );
+
+
+
+
+// Prevent auto-execution of scripts when no explicit dataType was provided (See gh-2432)
+jQuery.ajaxPrefilter( function( s ) {
+	if ( s.crossDomain ) {
+		s.contents.script = false;
+	}
+} );
+
+// Install script dataType
+jQuery.ajaxSetup( {
+	accepts: {
+		script: "text/javascript, application/javascript, " +
+			"application/ecmascript, application/x-ecmascript"
+	},
+	contents: {
+		script: /\b(?:java|ecma)script\b/
+	},
+	converters: {
+		"text script": function( text ) {
+			jQuery.globalEval( text );
+			return text;
+		}
+	}
+} );
+
+// Handle cache's special case and crossDomain
+jQuery.ajaxPrefilter( "script", function( s ) {
+	if ( s.cache === undefined ) {
+		s.cache = false;
+	}
+	if ( s.crossDomain ) {
+		s.type = "GET";
+	}
+} );
+
+// Bind script tag hack transport
+jQuery.ajaxTransport( "script", function( s ) {
+
+	// This transport only deals with cross domain or forced-by-attrs requests
+	if ( s.crossDomain || s.scriptAttrs ) {
+		var script, callback;
+		return {
+			send: function( _, complete ) {
+				script = jQuery( "<script>" )
+					.attr( s.scriptAttrs || {} )
+					.prop( { charset: s.scriptCharset, src: s.url } )
+					.on( "load error", callback = function( evt ) {
+						script.remove();
+						callback = null;
+						if ( evt ) {
+							complete( evt.type === "error" ? 404 : 200, evt.type );
+						}
+					} );
+
+				// Use native DOM manipulation to avoid our domManip AJAX trickery
+				document.head.appendChild( script[ 0 ] );
+			},
+			abort: function() {
+				if ( callback ) {
+					callback();
+				}
+			}
+		};
+	}
+} );
+
+
+
+
+var oldCallbacks = [],
+	rjsonp = /(=)\?(?=&|$)|\?\?/;
+
+// Default jsonp settings
+jQuery.ajaxSetup( {
+	jsonp: "callback",
+	jsonpCallback: function() {
+		var callback = oldCallbacks.pop() || ( jQuery.expando + "_" + ( nonce.guid++ ) );
+		this[ callback ] = true;
+		return callback;
+	}
+} );
+
+// Detect, normalize options and install callbacks for jsonp requests
+jQuery.ajaxPrefilter( "json jsonp", function( s, originalSettings, jqXHR ) {
+
+	var callbackName, overwritten, responseContainer,
+		jsonProp = s.jsonp !== false && ( rjsonp.test( s.url ) ?
+			"url" :
+			typeof s.data === "string" &&
+				( s.contentType || "" )
+					.indexOf( "application/x-www-form-urlencoded" ) === 0 &&
+				rjsonp.test( s.data ) && "data"
+		);
+
+	// Handle iff the expected data type is "jsonp" or we have a parameter to set
+	if ( jsonProp || s.dataTypes[ 0 ] === "jsonp" ) {
+
+		// Get callback name, remembering preexisting value associated with it
+		callbackName = s.jsonpCallback = isFunction( s.jsonpCallback ) ?
+			s.jsonpCallback() :
+			s.jsonpCallback;
+
+		// Insert callback into url or form data
+		if ( jsonProp ) {
+			s[ jsonProp ] = s[ jsonProp ].replace( rjsonp, "$1" + callbackName );
+		} else if ( s.jsonp !== false ) {
+			s.url += ( rquery.test( s.url ) ? "&" : "?" ) + s.jsonp + "=" + callbackName;
+		}
+
+		// Use data converter to retrieve json after script execution
+		s.converters[ "script json" ] = function() {
+			if ( !responseContainer ) {
+				jQuery.error( callbackName + " was not called" );
+			}
+			return responseContainer[ 0 ];
+		};
+
+		// Force json dataType
+		s.dataTypes[ 0 ] = "json";
+
+		// Install callback
+		overwritten = window[ callbackName ];
+		window[ callbackName ] = function() {
+			responseContainer = arguments;
+		};
+
+		// Clean-up function (fires after converters)
+		jqXHR.always( function() {
+
+			// If previous value didn't exist - remove it
+			if ( overwritten === undefined ) {
+				jQuery( window ).removeProp( callbackName );
+
+			// Otherwise restore preexisting value
+			} else {
+				window[ callbackName ] = overwritten;
+			}
+
+			// Save back as free
+			if ( s[ callbackName ] ) {
+
+				// Make sure that re-using the options doesn't screw things around
+				s.jsonpCallback = originalSettings.jsonpCallback;
+
+				// Save the callback name for future use
+				oldCallbacks.push( callbackName );
+			}
+
+			// Call if it was a function and we have a response
+			if ( responseContainer && isFunction( overwritten ) ) {
+				overwritten( responseContainer[ 0 ] );
+			}
+
+			responseContainer = overwritten = undefined;
+		} );
+
+		// Delegate to script
+		return "script";
+	}
+} );
+
+
+
+
+// Support: Safari 8 only
+// In Safari 8 documents created via document.implementation.createHTMLDocument
+// collapse sibling forms: the second one becomes a child of the first one.
+// Because of that, this security measure has to be disabled in Safari 8.
+// https://bugs.webkit.org/show_bug.cgi?id=137337
+support.createHTMLDocument = ( function() {
+	var body = document.implementation.createHTMLDocument( "" ).body;
+	body.innerHTML = "<form></form><form></form>";
+	return body.childNodes.length === 2;
+} )();
+
+
+// Argument "data" should be string of html
+// context (optional): If specified, the fragment will be created in this context,
+// defaults to document
+// keepScripts (optional): If true, will include scripts passed in the html string
+jQuery.parseHTML = function( data, context, keepScripts ) {
+	if ( typeof data !== "string" ) {
+		return [];
+	}
+	if ( typeof context === "boolean" ) {
+		keepScripts = context;
+		context = false;
+	}
+
+	var base, parsed, scripts;
+
+	if ( !context ) {
+
+		// Stop scripts or inline event handlers from being executed immediately
+		// by using document.implementation
+		if ( support.createHTMLDocument ) {
+			context = document.implementation.createHTMLDocument( "" );
+
+			// Set the base href for the created document
+			// so any parsed elements with URLs
+			// are based on the document's URL (gh-2965)
+			base = context.createElement( "base" );
+			base.href = document.location.href;
+			context.head.appendChild( base );
+		} else {
+			context = document;
+		}
+	}
+
+	parsed = rsingleTag.exec( data );
+	scripts = !keepScripts && [];
+
+	// Single tag
+	if ( parsed ) {
+		return [ context.createElement( parsed[ 1 ] ) ];
+	}
+
+	parsed = buildFragment( [ data ], context, scripts );
+
+	if ( scripts && scripts.length ) {
+		jQuery( scripts ).remove();
+	}
+
+	return jQuery.merge( [], parsed.childNodes );
+};
+
+
+/**
+ * Load a url into a page
+ */
+jQuery.fn.load = function( url, params, callback ) {
+	var selector, type, response,
+		self = this,
+		off = url.indexOf( " " );
+
+	if ( off > -1 ) {
+		selector = stripAndCollapse( url.slice( off ) );
+		url = url.slice( 0, off );
+	}
+
+	// If it's a function
+	if ( isFunction( params ) ) {
+
+		// We assume that it's the callback
+		callback = params;
+		params = undefined;
+
+	// Otherwise, build a param string
+	} else if ( params && typeof params === "object" ) {
+		type = "POST";
+	}
+
+	// If we have elements to modify, make the request
+	if ( self.length > 0 ) {
+		jQuery.ajax( {
+			url: url,
+
+			// If "type" variable is undefined, then "GET" method will be used.
+			// Make value of this field explicit since
+			// user can override it through ajaxSetup method
+			type: type || "GET",
+			dataType: "html",
+			data: params
+		} ).done( function( responseText ) {
+
+			// Save response for use in complete callback
+			response = arguments;
+
+			self.html( selector ?
+
+				// If a selector was specified, locate the right elements in a dummy div
+				// Exclude scripts to avoid IE 'Permission Denied' errors
+				jQuery( "<div>" ).append( jQuery.parseHTML( responseText ) ).find( selector ) :
+
+				// Otherwise use the full result
+				responseText );
+
+		// If the request succeeds, this function gets "data", "status", "jqXHR"
+		// but they are ignored because response was set above.
+		// If it fails, this function gets "jqXHR", "status", "error"
+		} ).always( callback && function( jqXHR, status ) {
+			self.each( function() {
+				callback.apply( this, response || [ jqXHR.responseText, status, jqXHR ] );
+			} );
+		} );
+	}
+
+	return this;
+};
+
+
+
+
+jQuery.expr.pseudos.animated = function( elem ) {
+	return jQuery.grep( jQuery.timers, function( fn ) {
+		return elem === fn.elem;
+	} ).length;
+};
+
+
+
+
+jQuery.offset = {
+	setOffset: function( elem, options, i ) {
+		var curPosition, curLeft, curCSSTop, curTop, curOffset, curCSSLeft, calculatePosition,
+			position = jQuery.css( elem, "position" ),
+			curElem = jQuery( elem ),
+			props = {};
+
+		// Set position first, in-case top/left are set even on static elem
+		if ( position === "static" ) {
+			elem.style.position = "relative";
+		}
+
+		curOffset = curElem.offset();
+		curCSSTop = jQuery.css( elem, "top" );
+		curCSSLeft = jQuery.css( elem, "left" );
+		calculatePosition = ( position === "absolute" || position === "fixed" ) &&
+			( curCSSTop + curCSSLeft ).indexOf( "auto" ) > -1;
+
+		// Need to be able to calculate position if either
+		// top or left is auto and position is either absolute or fixed
+		if ( calculatePosition ) {
+			curPosition = curElem.position();
+			curTop = curPosition.top;
+			curLeft = curPosition.left;
+
+		} else {
+			curTop = parseFloat( curCSSTop ) || 0;
+			curLeft = parseFloat( curCSSLeft ) || 0;
+		}
+
+		if ( isFunction( options ) ) {
+
+			// Use jQuery.extend here to allow modification of coordinates argument (gh-1848)
+			options = options.call( elem, i, jQuery.extend( {}, curOffset ) );
+		}
+
+		if ( options.top != null ) {
+			props.top = ( options.top - curOffset.top ) + curTop;
+		}
+		if ( options.left != null ) {
+			props.left = ( options.left - curOffset.left ) + curLeft;
+		}
+
+		if ( "using" in options ) {
+			options.using.call( elem, props );
+
+		} else {
+			curElem.css( props );
+		}
+	}
+};
+
+jQuery.fn.extend( {
+
+	// offset() relates an element's border box to the document origin
+	offset: function( options ) {
+
+		// Preserve chaining for setter
+		if ( arguments.length ) {
+			return options === undefined ?
+				this :
+				this.each( function( i ) {
+					jQuery.offset.setOffset( this, options, i );
+				} );
+		}
+
+		var rect, win,
+			elem = this[ 0 ];
+
+		if ( !elem ) {
+			return;
+		}
+
+		// Return zeros for disconnected and hidden (display: none) elements (gh-2310)
+		// Support: IE <=11 only
+		// Running getBoundingClientRect on a
+		// disconnected node in IE throws an error
+		if ( !elem.getClientRects().length ) {
+			return { top: 0, left: 0 };
+		}
+
+		// Get document-relative position by adding viewport scroll to viewport-relative gBCR
+		rect = elem.getBoundingClientRect();
+		win = elem.ownerDocument.defaultView;
+		return {
+			top: rect.top + win.pageYOffset,
+			left: rect.left + win.pageXOffset
+		};
+	},
+
+	// position() relates an element's margin box to its offset parent's padding box
+	// This corresponds to the behavior of CSS absolute positioning
+	position: function() {
+		if ( !this[ 0 ] ) {
+			return;
+		}
+
+		var offsetParent, offset, doc,
+			elem = this[ 0 ],
+			parentOffset = { top: 0, left: 0 };
+
+		// position:fixed elements are offset from the viewport, which itself always has zero offset
+		if ( jQuery.css( elem, "position" ) === "fixed" ) {
+
+			// Assume position:fixed implies availability of getBoundingClientRect
+			offset = elem.getBoundingClientRect();
+
+		} else {
+			offset = this.offset();
+
+			// Account for the *real* offset parent, which can be the document or its root element
+			// when a statically positioned element is identified
+			doc = elem.ownerDocument;
+			offsetParent = elem.offsetParent || doc.documentElement;
+			while ( offsetParent &&
+				( offsetParent === doc.body || offsetParent === doc.documentElement ) &&
+				jQuery.css( offsetParent, "position" ) === "static" ) {
+
+				offsetParent = offsetParent.parentNode;
+			}
+			if ( offsetParent && offsetParent !== elem && offsetParent.nodeType === 1 ) {
+
+				// Incorporate borders into its offset, since they are outside its content origin
+				parentOffset = jQuery( offsetParent ).offset();
+				parentOffset.top += jQuery.css( offsetParent, "borderTopWidth", true );
+				parentOffset.left += jQuery.css( offsetParent, "borderLeftWidth", true );
+			}
+		}
+
+		// Subtract parent offsets and element margins
+		return {
+			top: offset.top - parentOffset.top - jQuery.css( elem, "marginTop", true ),
+			left: offset.left - parentOffset.left - jQuery.css( elem, "marginLeft", true )
+		};
+	},
+
+	// This method will return documentElement in the following cases:
+	// 1) For the element inside the iframe without offsetParent, this method will return
+	//    documentElement of the parent window
+	// 2) For the hidden or detached element
+	// 3) For body or html element, i.e. in case of the html node - it will return itself
+	//
+	// but those exceptions were never presented as a real life use-cases
+	// and might be considered as more preferable results.
+	//
+	// This logic, however, is not guaranteed and can change at any point in the future
+	offsetParent: function() {
+		return this.map( function() {
+			var offsetParent = this.offsetParent;
+
+			while ( offsetParent && jQuery.css( offsetParent, "position" ) === "static" ) {
+				offsetParent = offsetParent.offsetParent;
+			}
+
+			return offsetParent || documentElement;
+		} );
+	}
+} );
+
+// Create scrollLeft and scrollTop methods
+jQuery.each( { scrollLeft: "pageXOffset", scrollTop: "pageYOffset" }, function( method, prop ) {
+	var top = "pageYOffset" === prop;
+
+	jQuery.fn[ method ] = function( val ) {
+		return access( this, function( elem, method, val ) {
+
+			// Coalesce documents and windows
+			var win;
+			if ( isWindow( elem ) ) {
+				win = elem;
+			} else if ( elem.nodeType === 9 ) {
+				win = elem.defaultView;
+			}
+
+			if ( val === undefined ) {
+				return win ? win[ prop ] : elem[ method ];
+			}
+
+			if ( win ) {
+				win.scrollTo(
+					!top ? val : win.pageXOffset,
+					top ? val : win.pageYOffset
+				);
+
+			} else {
+				elem[ method ] = val;
+			}
+		}, method, val, arguments.length );
+	};
+} );
+
+// Support: Safari <=7 - 9.1, Chrome <=37 - 49
+// Add the top/left cssHooks using jQuery.fn.position
+// Webkit bug: https://bugs.webkit.org/show_bug.cgi?id=29084
+// Blink bug: https://bugs.chromium.org/p/chromium/issues/detail?id=589347
+// getComputedStyle returns percent when specified for top/left/bottom/right;
+// rather than make the css module depend on the offset module, just check for it here
+jQuery.each( [ "top", "left" ], function( _i, prop ) {
+	jQuery.cssHooks[ prop ] = addGetHookIf( support.pixelPosition,
+		function( elem, computed ) {
+			if ( computed ) {
+				computed = curCSS( elem, prop );
+
+				// If curCSS returns percentage, fallback to offset
+				return rnumnonpx.test( computed ) ?
+					jQuery( elem ).position()[ prop ] + "px" :
+					computed;
+			}
+		}
+	);
+} );
+
+
+// Create innerHeight, innerWidth, height, width, outerHeight and outerWidth methods
+jQuery.each( { Height: "height", Width: "width" }, function( name, type ) {
+	jQuery.each( {
+		padding: "inner" + name,
+		content: type,
+		"": "outer" + name
+	}, function( defaultExtra, funcName ) {
+
+		// Margin is only for outerHeight, outerWidth
+		jQuery.fn[ funcName ] = function( margin, value ) {
+			var chainable = arguments.length && ( defaultExtra || typeof margin !== "boolean" ),
+				extra = defaultExtra || ( margin === true || value === true ? "margin" : "border" );
+
+			return access( this, function( elem, type, value ) {
+				var doc;
+
+				if ( isWindow( elem ) ) {
+
+					// $( window ).outerWidth/Height return w/h including scrollbars (gh-1729)
+					return funcName.indexOf( "outer" ) === 0 ?
+						elem[ "inner" + name ] :
+						elem.document.documentElement[ "client" + name ];
+				}
+
+				// Get document width or height
+				if ( elem.nodeType === 9 ) {
+					doc = elem.documentElement;
+
+					// Either scroll[Width/Height] or offset[Width/Height] or client[Width/Height],
+					// whichever is greatest
+					return Math.max(
+						elem.body[ "scroll" + name ], doc[ "scroll" + name ],
+						elem.body[ "offset" + name ], doc[ "offset" + name ],
+						doc[ "client" + name ]
+					);
+				}
+
+				return value === undefined ?
+
+					// Get width or height on the element, requesting but not forcing parseFloat
+					jQuery.css( elem, type, extra ) :
+
+					// Set width or height on the element
+					jQuery.style( elem, type, value, extra );
+			}, type, chainable ? margin : undefined, chainable );
+		};
+	} );
+} );
+
+
+jQuery.each( [
+	"ajaxStart",
+	"ajaxStop",
+	"ajaxComplete",
+	"ajaxError",
+	"ajaxSuccess",
+	"ajaxSend"
+], function( _i, type ) {
+	jQuery.fn[ type ] = function( fn ) {
+		return this.on( type, fn );
+	};
+} );
+
+
+
+
+jQuery.fn.extend( {
+
+	bind: function( types, data, fn ) {
+		return this.on( types, null, data, fn );
+	},
+	unbind: function( types, fn ) {
+		return this.off( types, null, fn );
+	},
+
+	delegate: function( selector, types, data, fn ) {
+		return this.on( types, selector, data, fn );
+	},
+	undelegate: function( selector, types, fn ) {
+
+		// ( namespace ) or ( selector, types [, fn] )
+		return arguments.length === 1 ?
+			this.off( selector, "**" ) :
+			this.off( types, selector || "**", fn );
+	},
+
+	hover: function( fnOver, fnOut ) {
+		return this.mouseenter( fnOver ).mouseleave( fnOut || fnOver );
+	}
+} );
+
+jQuery.each(
+	( "blur focus focusin focusout resize scroll click dblclick " +
+	"mousedown mouseup mousemove mouseover mouseout mouseenter mouseleave " +
+	"change select submit keydown keypress keyup contextmenu" ).split( " " ),
+	function( _i, name ) {
+
+		// Handle event binding
+		jQuery.fn[ name ] = function( data, fn ) {
+			return arguments.length > 0 ?
+				this.on( name, null, data, fn ) :
+				this.trigger( name );
+		};
+	}
+);
+
+
+
+
+// Support: Android <=4.0 only
+// Make sure we trim BOM and NBSP
+var rtrim = /^[\s\uFEFF\xA0]+|[\s\uFEFF\xA0]+$/g;
+
+// Bind a function to a context, optionally partially applying any
+// arguments.
+// jQuery.proxy is deprecated to promote standards (specifically Function#bind)
+// However, it is not slated for removal any time soon
+jQuery.proxy = function( fn, context ) {
+	var tmp, args, proxy;
+
+	if ( typeof context === "string" ) {
+		tmp = fn[ context ];
+		context = fn;
+		fn = tmp;
+	}
+
+	// Quick check to determine if target is callable, in the spec
+	// this throws a TypeError, but we will just return undefined.
+	if ( !isFunction( fn ) ) {
+		return undefined;
+	}
+
+	// Simulated bind
+	args = slice.call( arguments, 2 );
+	proxy = function() {
+		return fn.apply( context || this, args.concat( slice.call( arguments ) ) );
+	};
+
+	// Set the guid of unique handler to the same of original handler, so it can be removed
+	proxy.guid = fn.guid = fn.guid || jQuery.guid++;
+
+	return proxy;
+};
+
+jQuery.holdReady = function( hold ) {
+	if ( hold ) {
+		jQuery.readyWait++;
+	} else {
+		jQuery.ready( true );
+	}
+};
+jQuery.isArray = Array.isArray;
+jQuery.parseJSON = JSON.parse;
+jQuery.nodeName = nodeName;
+jQuery.isFunction = isFunction;
+jQuery.isWindow = isWindow;
+jQuery.camelCase = camelCase;
+jQuery.type = toType;
+
+jQuery.now = Date.now;
+
+jQuery.isNumeric = function( obj ) {
+
+	// As of jQuery 3.0, isNumeric is limited to
+	// strings and numbers (primitives or objects)
+	// that can be coerced to finite numbers (gh-2662)
+	var type = jQuery.type( obj );
+	return ( type === "number" || type === "string" ) &&
+
+		// parseFloat NaNs numeric-cast false positives ("")
+		// ...but misinterprets leading-number strings, particularly hex literals ("0x...")
+		// subtraction forces infinities to NaN
+		!isNaN( obj - parseFloat( obj ) );
+};
+
+jQuery.trim = function( text ) {
+	return text == null ?
+		"" :
+		( text + "" ).replace( rtrim, "" );
+};
+
+
+
+// Register as a named AMD module, since jQuery can be concatenated with other
+// files that may use define, but not via a proper concatenation script that
+// understands anonymous AMD modules. A named AMD is safest and most robust
+// way to register. Lowercase jquery is used because AMD module names are
+// derived from file names, and jQuery is normally delivered in a lowercase
+// file name. Do this after creating the global so that if an AMD module wants
+// to call noConflict to hide this version of jQuery, it will work.
+
+// Note that for maximum portability, libraries that are not jQuery should
+// declare themselves as anonymous modules, and avoid setting a global if an
+// AMD loader is present. jQuery is a special case. For more information, see
+// https://github.com/jrburke/requirejs/wiki/Updating-existing-libraries#wiki-anon
+
+if ( typeof define === "function" && define.amd ) {
+	define( "jquery", [], function() {
+		return jQuery;
+	} );
+}
+
+
+
+
+var
+
+	// Map over jQuery in case of overwrite
+	_jQuery = window.jQuery,
+
+	// Map over the $ in case of overwrite
+	_$ = window.$;
+
+jQuery.noConflict = function( deep ) {
+	if ( window.$ === jQuery ) {
+		window.$ = _$;
+	}
+
+	if ( deep && window.jQuery === jQuery ) {
+		window.jQuery = _jQuery;
+	}
+
+	return jQuery;
+};
+
+// Expose jQuery and $ identifiers, even in AMD
+// (#7102#comment:10, https://github.com/jquery/jquery/pull/557)
+// and CommonJS for browser emulators (#13566)
+if ( typeof noGlobal === "undefined" ) {
+	window.jQuery = window.$ = jQuery;
+}
+
+
+
+
+return jQuery;
+} );
diff --git a/docs/preview/pr/352/_static/jquery.js b/docs/preview/pr/352/_static/jquery.js
new file mode 100644
index 000000000..c4c6022f2
--- /dev/null
+++ b/docs/preview/pr/352/_static/jquery.js
@@ -0,0 +1,2 @@
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new file mode 100644
index 000000000..6a9ce27b3
--- /dev/null
+++ b/docs/preview/pr/352/_static/js/theme.js
@@ -0,0 +1,1128 @@
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+window.utilities = {
+  scrollTop: function() {
+    var supportPageOffset = window.pageXOffset !== undefined;
+    var isCSS1Compat = ((document.compatMode || "") === "CSS1Compat");
+    var scrollLeft = supportPageOffset ? window.pageXOffset : isCSS1Compat ? document.documentElement.scrollLeft : document.body.scrollLeft;
+    return supportPageOffset ? window.pageYOffset : isCSS1Compat ? document.documentElement.scrollTop : document.body.scrollTop;
+  },
+
+  // Modified from https://stackoverflow.com/a/27078401
+  throttle: function(func, wait, options) {
+    var context, args, result;
+    var timeout = null;
+    var previous = 0;
+    if (!options) options = {};
+    var later = function() {
+      previous = options.leading === false ? 0 : Date.now();
+      timeout = null;
+      result = func.apply(context, args);
+      if (!timeout) context = args = null;
+    };
+    return function() {
+      var now = Date.now();
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+      var remaining = wait - (now - previous);
+      context = this;
+      args = arguments;
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+        }
+        previous = now;
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+        timeout = setTimeout(later, remaining);
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+      return result;
+    };
+  },
+
+  closest: function (el, selector) {
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+
+    // find vendor prefix
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+
+    var parent;
+
+    // traverse parents
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+      if (parent && parent[matchesFn](selector)) {
+        return parent;
+      }
+      el = parent;
+    }
+
+    return null;
+  },
+
+  // Modified from https://stackoverflow.com/a/18953277
+  offset: function(elem) {
+    if (!elem) {
+      return;
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+
+    rect = elem.getBoundingClientRect();
+
+    // Make sure element is not hidden (display: none) or disconnected
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+      var doc = elem.ownerDocument;
+      var docElem = doc.documentElement;
+
+      return {
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+        left: rect.left + window.pageXOffset - docElem.clientLeft
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+    }
+  },
+
+  headersHeight: function() {
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+    } else {
+      return document.getElementById("header-holder").offsetHeight +
+             document.getElementById("pytorch-page-level-bar").offsetHeight;
+    }
+  },
+
+  windowHeight: function() {
+    return window.innerHeight ||
+           document.documentElement.clientHeight ||
+           document.body.clientHeight;
+  }
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+
+},{}],2:[function(require,module,exports){
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+    cookieBanner.bind();
+
+    var cookieExists = cookieBanner.cookieExists();
+
+    if (!cookieExists) {
+      cookieBanner.setCookie();
+      cookieBanner.showCookieNotice();
+    }
+  },
+
+  bind: function() {
+    $(".close-button").on("click", cookieBanner.hideCookieNotice);
+  },
+
+  cookieExists: function() {
+    var cookie = localStorage.getItem("returningPytorchUser");
+
+    if (cookie) {
+      return true;
+    } else {
+      return false;
+    }
+  },
+
+  setCookie: function() {
+    localStorage.setItem("returningPytorchUser", true);
+  },
+
+  showCookieNotice: function() {
+    $(".cookie-banner-wrapper").addClass("is-visible");
+  },
+
+  hideCookieNotice: function() {
+    $(".cookie-banner-wrapper").removeClass("is-visible");
+  }
+};
+
+$(function() {
+  cookieBanner.init();
+});
+
+},{}],3:[function(require,module,exports){
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+  bind: function() {
+    var options = {
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+      page: "6",
+      pagination: true
+    };
+
+    var tutorialList = new List("tutorial-cards", options);
+
+    function filterSelectedTags(cardTags, selectedTags) {
+      return cardTags.some(function(tag) {
+        return selectedTags.some(function(selectedTag) {
+          return selectedTag == tag;
+        });
+      });
+    }
+
+    function updateList() {
+      var selectedTags = [];
+
+      $(".selected").each(function() {
+        selectedTags.push($(this).data("tag"));
+      });
+
+      tutorialList.filter(function(item) {
+        var cardTags;
+
+        if (item.values().tags == null) {
+          cardTags = [""];
+        } else {
+          cardTags = item.values().tags.split(",");
+        }
+
+        if (selectedTags.length == 0) {
+          return true;
+        } else {
+          return filterSelectedTags(cardTags, selectedTags);
+        }
+      });
+    }
+
+    $(".filter-btn").on("click", function() {
+      if ($(this).data("tag") == "all") {
+        $(this).addClass("all-tag-selected");
+        $(".filter").removeClass("selected");
+      } else {
+        $(this).toggleClass("selected");
+        $("[data-tag='all']").removeClass("all-tag-selected");
+      }
+
+      // If no tags are selected then highlight the 'All' tag
+
+      if (!$(".selected")[0]) {
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+      }
+
+      updateList();
+    });
+  }
+};
+
+},{}],4:[function(require,module,exports){
+// Modified from https://stackoverflow.com/a/32396543
+window.highlightNavigation = {
+  navigationListItems: document.querySelectorAll("#pytorch-right-menu li"),
+  sections: document.querySelectorAll(".pytorch-article .section"),
+  sectionIdTonavigationLink: {},
+
+  bind: function() {
+    if (!sideMenus.displayRightMenu) {
+      return;
+    };
+
+    for (var i = 0; i < highlightNavigation.sections.length; i++) {
+      var id = highlightNavigation.sections[i].id;
+      highlightNavigation.sectionIdTonavigationLink[id] =
+        document.querySelectorAll('#pytorch-right-menu li a[href="#' + id + '"]')[0];
+    }
+
+    $(window).scroll(utilities.throttle(highlightNavigation.highlight, 100));
+  },
+
+  highlight: function() {
+    var rightMenu = document.getElementById("pytorch-right-menu");
+
+    // If right menu is not on the screen don't bother
+    if (rightMenu.offsetWidth === 0 && rightMenu.offsetHeight === 0) {
+      return;
+    }
+
+    var scrollPosition = utilities.scrollTop();
+    var OFFSET_TOP_PADDING = 25;
+    var offset = document.getElementById("header-holder").offsetHeight +
+                 document.getElementById("pytorch-page-level-bar").offsetHeight +
+                 OFFSET_TOP_PADDING;
+
+    var sections = highlightNavigation.sections;
+
+    for (var i = (sections.length - 1); i >= 0; i--) {
+      var currentSection = sections[i];
+      var sectionTop = utilities.offset(currentSection).top;
+
+      if (scrollPosition >= sectionTop - offset) {
+        var navigationLink = highlightNavigation.sectionIdTonavigationLink[currentSection.id];
+        var navigationListItem = utilities.closest(navigationLink, "li");
+
+        if (navigationListItem && !navigationListItem.classList.contains("active")) {
+          for (var i = 0; i < highlightNavigation.navigationListItems.length; i++) {
+            var el = highlightNavigation.navigationListItems[i];
+            if (el.classList.contains("active")) {
+              el.classList.remove("active");
+            }
+          }
+
+          navigationListItem.classList.add("active");
+
+          // Scroll to active item. Not a requested feature but we could revive it. Needs work.
+
+          // var menuTop = $("#pytorch-right-menu").position().top;
+          // var itemTop = navigationListItem.getBoundingClientRect().top;
+          // var TOP_PADDING = 20
+          // var newActiveTop = $("#pytorch-side-scroll-right").scrollTop() + itemTop - menuTop - TOP_PADDING;
+
+          // $("#pytorch-side-scroll-right").animate({
+          //   scrollTop: newActiveTop
+          // }, 100);
+        }
+
+        break;
+      }
+    }
+  }
+};
+
+},{}],5:[function(require,module,exports){
+window.mainMenuDropdown = {
+  bind: function() {
+    $("[data-toggle='ecosystem-dropdown']").on("click", function() {
+      toggleDropdown($(this).attr("data-toggle"));
+    });
+
+    $("[data-toggle='resources-dropdown']").on("click", function() {
+      toggleDropdown($(this).attr("data-toggle"));
+    });
+
+    function toggleDropdown(menuToggle) {
+      var showMenuClass = "show-menu";
+      var menuClass = "." + menuToggle + "-menu";
+
+      if ($(menuClass).hasClass(showMenuClass)) {
+        $(menuClass).removeClass(showMenuClass);
+      } else {
+        $("[data-toggle=" + menuToggle + "].show-menu").removeClass(
+          showMenuClass
+        );
+        $(menuClass).addClass(showMenuClass);
+      }
+    }
+  }
+};
+
+},{}],6:[function(require,module,exports){
+window.mobileMenu = {
+  bind: function() {
+    $("[data-behavior='open-mobile-menu']").on('click', function(e) {
+      e.preventDefault();
+      $(".mobile-main-menu").addClass("open");
+      $("body").addClass('no-scroll');
+
+      mobileMenu.listenForResize();
+    });
+
+    $("[data-behavior='close-mobile-menu']").on('click', function(e) {
+      e.preventDefault();
+      mobileMenu.close();
+    });
+  },
+
+  listenForResize: function() {
+    $(window).on('resize.ForMobileMenu', function() {
+      if ($(this).width() > 768) {
+        mobileMenu.close();
+      }
+    });
+  },
+
+  close: function() {
+    $(".mobile-main-menu").removeClass("open");
+    $("body").removeClass('no-scroll');
+    $(window).off('resize.ForMobileMenu');
+  }
+};
+
+},{}],7:[function(require,module,exports){
+window.mobileTOC = {
+  bind: function() {
+    $("[data-behavior='toggle-table-of-contents']").on("click", function(e) {
+      e.preventDefault();
+
+      var $parent = $(this).parent();
+
+      if ($parent.hasClass("is-open")) {
+        $parent.removeClass("is-open");
+        $(".pytorch-left-menu").slideUp(200, function() {
+          $(this).css({display: ""});
+        });
+      } else {
+        $parent.addClass("is-open");
+        $(".pytorch-left-menu").slideDown(200);
+      }
+    });
+  }
+}
+
+},{}],8:[function(require,module,exports){
+window.pytorchAnchors = {
+  bind: function() {
+    // Replace Sphinx-generated anchors with anchorjs ones
+    $(".headerlink").text("");
+
+    window.anchors.add(".pytorch-article .headerlink");
+
+    $(".anchorjs-link").each(function() {
+      var $headerLink = $(this).closest(".headerlink");
+      var href = $headerLink.attr("href");
+      var clone = this.outerHTML;
+
+      $clone = $(clone).attr("href", href);
+      $headerLink.before($clone);
+      $headerLink.remove();
+    });
+  }
+};
+
+},{}],9:[function(require,module,exports){
+// Modified from https://stackoverflow.com/a/13067009
+// Going for a JS solution to scrolling to an anchor so we can benefit from
+// less hacky css and smooth scrolling.
+
+window.scrollToAnchor = {
+  bind: function() {
+    var document = window.document;
+    var history = window.history;
+    var location = window.location
+    var HISTORY_SUPPORT = !!(history && history.pushState);
+
+    var anchorScrolls = {
+      ANCHOR_REGEX: /^#[^ ]+$/,
+      offsetHeightPx: function() {
+        var OFFSET_HEIGHT_PADDING = 20;
+        // TODO: this is a little janky. We should try to not rely on JS for this
+        return utilities.headersHeight() + OFFSET_HEIGHT_PADDING;
+      },
+
+      /**
+       * Establish events, and fix initial scroll position if a hash is provided.
+       */
+      init: function() {
+        this.scrollToCurrent();
+        // This interferes with clicks below it, causing a double fire
+        // $(window).on('hashchange', $.proxy(this, 'scrollToCurrent'));
+        $('body').on('click', 'a', $.proxy(this, 'delegateAnchors'));
+        $('body').on('click', '#pytorch-right-menu li span', $.proxy(this, 'delegateSpans'));
+      },
+
+      /**
+       * Return the offset amount to deduct from the normal scroll position.
+       * Modify as appropriate to allow for dynamic calculations
+       */
+      getFixedOffset: function() {
+        return this.offsetHeightPx();
+      },
+
+      /**
+       * If the provided href is an anchor which resolves to an element on the
+       * page, scroll to it.
+       * @param  {String} href
+       * @return {Boolean} - Was the href an anchor.
+       */
+      scrollIfAnchor: function(href, pushToHistory) {
+        var match, anchorOffset;
+
+        if(!this.ANCHOR_REGEX.test(href)) {
+          return false;
+        }
+
+        match = document.getElementById(href.slice(1));
+
+        if(match) {
+          var anchorOffset = $(match).offset().top - this.getFixedOffset();
+
+          $('html, body').scrollTop(anchorOffset);
+
+          // Add the state to history as-per normal anchor links
+          if(HISTORY_SUPPORT && pushToHistory) {
+            history.pushState({}, document.title, location.pathname + href);
+          }
+        }
+
+        return !!match;
+      },
+
+      /**
+       * Attempt to scroll to the current location's hash.
+       */
+      scrollToCurrent: function(e) {
+        if(this.scrollIfAnchor(window.location.hash) && e) {
+          e.preventDefault();
+        }
+      },
+
+      delegateSpans: function(e) {
+        var elem = utilities.closest(e.target, "a");
+
+        if(this.scrollIfAnchor(elem.getAttribute('href'), true)) {
+          e.preventDefault();
+        }
+      },
+
+      /**
+       * If the click event's target was an anchor, fix the scroll position.
+       */
+      delegateAnchors: function(e) {
+        var elem = e.target;
+
+        if(this.scrollIfAnchor(elem.getAttribute('href'), true)) {
+          e.preventDefault();
+        }
+      }
+    };
+
+    $(document).ready($.proxy(anchorScrolls, 'init'));
+  }
+};
+
+},{}],10:[function(require,module,exports){
+window.sideMenus = {
+  rightMenuIsOnScreen: function() {
+    return document.getElementById("pytorch-content-right").offsetParent !== null;
+  },
+
+  isFixedToBottom: false,
+
+  bind: function() {
+    sideMenus.handleLeftMenu();
+
+    var rightMenuLinks = document.querySelectorAll("#pytorch-right-menu li");
+    var rightMenuHasLinks = rightMenuLinks.length > 1;
+
+    if (!rightMenuHasLinks) {
+      for (var i = 0; i < rightMenuLinks.length; i++) {
+        rightMenuLinks[i].style.display = "none";
+      }
+    }
+
+    if (rightMenuHasLinks) {
+      // Don't show the Shortcuts menu title text unless there are menu items
+      document.getElementById("pytorch-shortcuts-wrapper").style.display = "block";
+
+      // We are hiding the titles of the pages in the right side menu but there are a few
+      // pages that include other pages in the right side menu (see 'torch.nn' in the docs)
+      // so if we exclude those it looks confusing. Here we add a 'title-link' class to these
+      // links so we can exclude them from normal right side menu link operations
+      var titleLinks = document.querySelectorAll(
+        "#pytorch-right-menu #pytorch-side-scroll-right \
+         > ul > li > a.reference.internal"
+      );
+
+      for (var i = 0; i < titleLinks.length; i++) {
+        var link = titleLinks[i];
+
+        link.classList.add("title-link");
+
+        if (
+          link.nextElementSibling &&
+          link.nextElementSibling.tagName === "UL" &&
+          link.nextElementSibling.children.length > 0
+        ) {
+          link.classList.add("has-children");
+        }
+      }
+
+      // Add + expansion signifiers to normal right menu links that have sub menus
+      var menuLinks = document.querySelectorAll(
+        "#pytorch-right-menu ul li ul li a.reference.internal"
+      );
+
+      for (var i = 0; i < menuLinks.length; i++) {
+        if (
+          menuLinks[i].nextElementSibling &&
+          menuLinks[i].nextElementSibling.tagName === "UL"
+        ) {
+          menuLinks[i].classList.add("not-expanded");
+        }
+      }
+
+      // If a hash is present on page load recursively expand menu items leading to selected item
+      var linkWithHash =
+        document.querySelector(
+          "#pytorch-right-menu a[href=\"" + window.location.hash + "\"]"
+        );
+
+      if (linkWithHash) {
+        // Expand immediate sibling list if present
+        if (
+          linkWithHash.nextElementSibling &&
+          linkWithHash.nextElementSibling.tagName === "UL" &&
+          linkWithHash.nextElementSibling.children.length > 0
+        ) {
+          linkWithHash.nextElementSibling.style.display = "block";
+          linkWithHash.classList.add("expanded");
+        }
+
+        // Expand ancestor lists if any
+        sideMenus.expandClosestUnexpandedParentList(linkWithHash);
+      }
+
+      // Bind click events on right menu links
+      $("#pytorch-right-menu a.reference.internal").on("click", function() {
+        if (this.classList.contains("expanded")) {
+          this.nextElementSibling.style.display = "none";
+          this.classList.remove("expanded");
+          this.classList.add("not-expanded");
+        } else if (this.classList.contains("not-expanded")) {
+          this.nextElementSibling.style.display = "block";
+          this.classList.remove("not-expanded");
+          this.classList.add("expanded");
+        }
+      });
+
+      sideMenus.handleRightMenu();
+    }
+
+    $(window).on('resize scroll', function(e) {
+      sideMenus.handleNavBar();
+
+      sideMenus.handleLeftMenu();
+
+      if (sideMenus.rightMenuIsOnScreen()) {
+        sideMenus.handleRightMenu();
+      }
+    });
+  },
+
+  leftMenuIsFixed: function() {
+    return document.getElementById("pytorch-left-menu").classList.contains("make-fixed");
+  },
+
+  handleNavBar: function() {
+    var mainHeaderHeight = document.getElementById('header-holder').offsetHeight;
+
+    // If we are scrolled past the main navigation header fix the sub menu bar to top of page
+    if (utilities.scrollTop() >= mainHeaderHeight) {
+      document.getElementById("pytorch-left-menu").classList.add("make-fixed");
+      document.getElementById("pytorch-page-level-bar").classList.add("left-menu-is-fixed");
+    } else {
+      document.getElementById("pytorch-left-menu").classList.remove("make-fixed");
+      document.getElementById("pytorch-page-level-bar").classList.remove("left-menu-is-fixed");
+    }
+  },
+
+  expandClosestUnexpandedParentList: function (el) {
+    var closestParentList = utilities.closest(el, "ul");
+
+    if (closestParentList) {
+      var closestParentLink = closestParentList.previousElementSibling;
+      var closestParentLinkExists = closestParentLink &&
+                                    closestParentLink.tagName === "A" &&
+                                    closestParentLink.classList.contains("reference");
+
+      if (closestParentLinkExists) {
+        // Don't add expansion class to any title links
+         if (closestParentLink.classList.contains("title-link")) {
+           return;
+         }
+
+        closestParentList.style.display = "block";
+        closestParentLink.classList.remove("not-expanded");
+        closestParentLink.classList.add("expanded");
+        sideMenus.expandClosestUnexpandedParentList(closestParentLink);
+      }
+    }
+  },
+
+  handleLeftMenu: function () {
+    var windowHeight = utilities.windowHeight();
+    var topOfFooterRelativeToWindow = document.getElementById("docs-tutorials-resources").getBoundingClientRect().top;
+
+    if (topOfFooterRelativeToWindow >= windowHeight) {
+      document.getElementById("pytorch-left-menu").style.height = "100%";
+    } else {
+      var howManyPixelsOfTheFooterAreInTheWindow = windowHeight - topOfFooterRelativeToWindow;
+      var leftMenuDifference = howManyPixelsOfTheFooterAreInTheWindow;
+      document.getElementById("pytorch-left-menu").style.height = (windowHeight - leftMenuDifference) + "px";
+    }
+  },
+
+  handleRightMenu: function() {
+    var rightMenuWrapper = document.getElementById("pytorch-content-right");
+    var rightMenu = document.getElementById("pytorch-right-menu");
+    var rightMenuList = rightMenu.getElementsByTagName("ul")[0];
+    var article = document.getElementById("pytorch-article");
+    var articleHeight = article.offsetHeight;
+    var articleBottom = utilities.offset(article).top + articleHeight;
+    var mainHeaderHeight = document.getElementById('header-holder').offsetHeight;
+
+    if (utilities.scrollTop() < mainHeaderHeight) {
+      rightMenuWrapper.style.height = "100%";
+      rightMenu.style.top = 0;
+      rightMenu.classList.remove("scrolling-fixed");
+      rightMenu.classList.remove("scrolling-absolute");
+    } else {
+      if (rightMenu.classList.contains("scrolling-fixed")) {
+        var rightMenuBottom =
+          utilities.offset(rightMenuList).top + rightMenuList.offsetHeight;
+
+        if (rightMenuBottom >= articleBottom) {
+          rightMenuWrapper.style.height = articleHeight + mainHeaderHeight + "px";
+          rightMenu.style.top = utilities.scrollTop() - mainHeaderHeight + "px";
+          rightMenu.classList.add("scrolling-absolute");
+          rightMenu.classList.remove("scrolling-fixed");
+        }
+      } else {
+        rightMenuWrapper.style.height = articleHeight + mainHeaderHeight + "px";
+        rightMenu.style.top =
+          articleBottom - mainHeaderHeight - rightMenuList.offsetHeight + "px";
+        rightMenu.classList.add("scrolling-absolute");
+      }
+
+      if (utilities.scrollTop() < articleBottom - rightMenuList.offsetHeight) {
+        rightMenuWrapper.style.height = "100%";
+        rightMenu.style.top = "";
+        rightMenu.classList.remove("scrolling-absolute");
+        rightMenu.classList.add("scrolling-fixed");
+      }
+    }
+
+    var rightMenuSideScroll = document.getElementById("pytorch-side-scroll-right");
+    var sideScrollFromWindowTop = rightMenuSideScroll.getBoundingClientRect().top;
+
+    rightMenuSideScroll.style.height = utilities.windowHeight() - sideScrollFromWindowTop + "px";
+  }
+};
+
+},{}],"pypose-sphinx-theme":[function(require,module,exports){
+var jQuery = (typeof(window) != 'undefined') ? window.jQuery : require('jquery');
+
+// Sphinx theme nav state
+function ThemeNav () {
+
+    var nav = {
+        navBar: null,
+        win: null,
+        winScroll: false,
+        winResize: false,
+        linkScroll: false,
+        winPosition: 0,
+        winHeight: null,
+        docHeight: null,
+        isRunning: false
+    };
+
+    nav.enable = function (withStickyNav) {
+        var self = this;
+
+        // TODO this can likely be removed once the theme javascript is broken
+        // out from the RTD assets. This just ensures old projects that are
+        // calling `enable()` get the sticky menu on by default. All other cals
+        // to `enable` should include an argument for enabling the sticky menu.
+        if (typeof(withStickyNav) == 'undefined') {
+            withStickyNav = true;
+        }
+
+        if (self.isRunning) {
+            // Only allow enabling nav logic once
+            return;
+        }
+
+        self.isRunning = true;
+        jQuery(function ($) {
+            self.init($);
+
+            self.reset();
+            self.win.on('hashchange', self.reset);
+
+            if (withStickyNav) {
+                // Set scroll monitor
+                self.win.on('scroll', function () {
+                    if (!self.linkScroll) {
+                        if (!self.winScroll) {
+                            self.winScroll = true;
+                            requestAnimationFrame(function() { self.onScroll(); });
+                        }
+                    }
+                });
+            }
+
+            // Set resize monitor
+            self.win.on('resize', function () {
+                if (!self.winResize) {
+                    self.winResize = true;
+                    requestAnimationFrame(function() { self.onResize(); });
+                }
+            });
+
+            self.onResize();
+        });
+
+    };
+
+    // TODO remove this with a split in theme and Read the Docs JS logic as
+    // well, it's only here to support 0.3.0 installs of our theme.
+    nav.enableSticky = function() {
+        this.enable(true);
+    };
+
+    nav.init = function ($) {
+        var doc = $(document),
+            self = this;
+
+        this.navBar = $('div.pytorch-side-scroll:first');
+        this.win = $(window);
+
+        // Set up javascript UX bits
+        $(document)
+            // Shift nav in mobile when clicking the menu.
+            .on('click', "[data-toggle='pytorch-left-menu-nav-top']", function() {
+                $("[data-toggle='wy-nav-shift']").toggleClass("shift");
+                $("[data-toggle='rst-versions']").toggleClass("shift");
+            })
+
+            // Nav menu link click operations
+            .on('click', ".pytorch-menu-vertical .current ul li a", function() {
+                var target = $(this);
+                // Close menu when you click a link.
+                $("[data-toggle='wy-nav-shift']").removeClass("shift");
+                $("[data-toggle='rst-versions']").toggleClass("shift");
+                // Handle dynamic display of l3 and l4 nav lists
+                self.toggleCurrent(target);
+                self.hashChange();
+            })
+            .on('click', "[data-toggle='rst-current-version']", function() {
+                $("[data-toggle='rst-versions']").toggleClass("shift-up");
+            })
+
+        // Make tables responsive
+        $("table.docutils:not(.field-list,.footnote,.citation)")
+            .wrap("<div class='wy-table-responsive'></div>");
+
+        // Add extra class to responsive tables that contain
+        // footnotes or citations so that we can target them for styling
+        $("table.docutils.footnote")
+            .wrap("<div class='wy-table-responsive footnote'></div>");
+        $("table.docutils.citation")
+            .wrap("<div class='wy-table-responsive citation'></div>");
+
+        // Add expand links to all parents of nested ul
+        $('.pytorch-menu-vertical ul').not('.simple').siblings('a').each(function () {
+            var link = $(this);
+                expand = $('<span class="toctree-expand"></span>');
+            expand.on('click', function (ev) {
+                self.toggleCurrent(link);
+                ev.stopPropagation();
+                return false;
+            });
+            link.prepend(expand);
+        });
+    };
+
+    nav.reset = function () {
+        // Get anchor from URL and open up nested nav
+        var anchor = encodeURI(window.location.hash) || '#';
+
+        try {
+            var vmenu = $('.pytorch-menu-vertical');
+            var link = vmenu.find('[href="' + anchor + '"]');
+            if (link.length === 0) {
+                // this link was not found in the sidebar.
+                // Find associated id element, then its closest section
+                // in the document and try with that one.
+                var id_elt = $('.document [id="' + anchor.substring(1) + '"]');
+                var closest_section = id_elt.closest('div.section');
+                link = vmenu.find('[href="#' + closest_section.attr("id") + '"]');
+                if (link.length === 0) {
+                    // still not found in the sidebar. fall back to main section
+                    link = vmenu.find('[href="#"]');
+                }
+            }
+            // If we found a matching link then reset current and re-apply
+            // otherwise retain the existing match
+            if (link.length > 0) {
+                $('.pytorch-menu-vertical .current').removeClass('current');
+                link.addClass('current');
+                link.closest('li.toctree-l1').addClass('current');
+                link.closest('li.toctree-l1').parent().addClass('current');
+                link.closest('li.toctree-l1').addClass('current');
+                link.closest('li.toctree-l2').addClass('current');
+                link.closest('li.toctree-l3').addClass('current');
+                link.closest('li.toctree-l4').addClass('current');
+            }
+        }
+        catch (err) {
+            console.log("Error expanding nav for anchor", err);
+        }
+
+    };
+
+    nav.onScroll = function () {
+        this.winScroll = false;
+        var newWinPosition = this.win.scrollTop(),
+            winBottom = newWinPosition + this.winHeight,
+            navPosition = this.navBar.scrollTop(),
+            newNavPosition = navPosition + (newWinPosition - this.winPosition);
+        if (newWinPosition < 0 || winBottom > this.docHeight) {
+            return;
+        }
+        this.navBar.scrollTop(newNavPosition);
+        this.winPosition = newWinPosition;
+    };
+
+    nav.onResize = function () {
+        this.winResize = false;
+        this.winHeight = this.win.height();
+        this.docHeight = $(document).height();
+    };
+
+    nav.hashChange = function () {
+        this.linkScroll = true;
+        this.win.one('hashchange', function () {
+            this.linkScroll = false;
+        });
+    };
+
+    nav.toggleCurrent = function (elem) {
+        var parent_li = elem.closest('li');
+        parent_li.siblings('li.current').removeClass('current');
+        parent_li.siblings().find('li.current').removeClass('current');
+        parent_li.find('> ul li.current').removeClass('current');
+        parent_li.toggleClass('current');
+    }
+
+    return nav;
+};
+
+module.exports.ThemeNav = ThemeNav();
+
+if (typeof(window) != 'undefined') {
+    window.SphinxRtdTheme = {
+        Navigation: module.exports.ThemeNav,
+        // TODO remove this once static assets are split up between the theme
+        // and Read the Docs. For now, this patches 0.3.0 to be backwards
+        // compatible with a pre-0.3.0 layout.html
+        StickyNav: module.exports.ThemeNav,
+    };
+}
+
+
+// requestAnimationFrame polyfill by Erik Möller. fixes from Paul Irish and Tino Zijdel
+// https://gist.github.com/paulirish/1579671
+// MIT license
+
+(function() {
+    var lastTime = 0;
+    var vendors = ['ms', 'moz', 'webkit', 'o'];
+    for(var x = 0; x < vendors.length && !window.requestAnimationFrame; ++x) {
+        window.requestAnimationFrame = window[vendors[x]+'RequestAnimationFrame'];
+        window.cancelAnimationFrame = window[vendors[x]+'CancelAnimationFrame']
+                                   || window[vendors[x]+'CancelRequestAnimationFrame'];
+    }
+
+    if (!window.requestAnimationFrame)
+        window.requestAnimationFrame = function(callback, element) {
+            var currTime = new Date().getTime();
+            var timeToCall = Math.max(0, 16 - (currTime - lastTime));
+            var id = window.setTimeout(function() { callback(currTime + timeToCall); },
+              timeToCall);
+            lastTime = currTime + timeToCall;
+            return id;
+        };
+
+    if (!window.cancelAnimationFrame)
+        window.cancelAnimationFrame = function(id) {
+            clearTimeout(id);
+        };
+}());
+
+$(".sphx-glr-thumbcontainer").removeAttr("tooltip");
+$("table").removeAttr("border");
+
+// This code replaces the default sphinx gallery download buttons
+// with the 3 download buttons at the top of the page
+
+var downloadNote = $(".sphx-glr-download-link-note.admonition.note");
+if (downloadNote.length >= 1) {
+    var tutorialUrlArray = $("#tutorial-type").text().split('/');
+        tutorialUrlArray[0] = tutorialUrlArray[0] + "_source"
+
+    var githubLink = "https://github.com/pypose/tutorials/blob/main/" + tutorialUrlArray.join("/") + ".py",
+        notebookLink = $(".reference.download")[1].href,
+        notebookDownloadPath = notebookLink.split('_downloads')[1],
+        colabLink = "https://colab.research.google.com/github/pypose/tutorials/blob/gh-pages/_downloads" + notebookDownloadPath;
+
+    $("#google-colab-link").wrap("<a href=" + colabLink + " data-behavior='call-to-action-event' data-response='Run in Google Colab' target='_blank'/>");
+    $("#download-notebook-link").wrap("<a href=" + notebookLink + " data-behavior='call-to-action-event' data-response='Download Notebook'/>");
+    $("#github-view-link").wrap("<a href=" + githubLink + " data-behavior='call-to-action-event' data-response='View on Github' target='_blank'/>");
+} else {
+    $(".pytorch-call-to-action-links").hide();
+}
+
+//This code handles the Expand/Hide toggle for the Docs/Tutorials left nav items
+
+$(document).ready(function() {
+  var caption = "#pytorch-left-menu p.caption";
+  var collapseAdded = $(this).not("checked");
+  $(caption).each(function () {
+    var menuName = this.innerText.replace(/[^\w\s]/gi, "").trim();
+    $(this).find("span").addClass("checked");
+    if (collapsedSections.includes(menuName) == true && collapseAdded && sessionStorage.getItem(menuName) !== "expand" || sessionStorage.getItem(menuName) == "collapse") {
+      $(this.firstChild).after("<span class='expand-menu'>[ + ]</span>");
+      $(this.firstChild).after("<span class='hide-menu collapse'>[ - ]</span>");
+      $(this).next("ul").hide();
+    } else if (collapsedSections.includes(menuName) == false && collapseAdded || sessionStorage.getItem(menuName) == "expand") {
+      $(this.firstChild).after("<span class='expand-menu collapse'>[ + ]</span>");
+      $(this.firstChild).after("<span class='hide-menu'>[ - ]</span>");
+    }
+  });
+
+  $(".expand-menu").on("click", function () {
+    $(this).prev(".hide-menu").toggle();
+    $(this).parent().next("ul").toggle();
+    var menuName = $(this).parent().text().replace(/[^\w\s]/gi, "").trim();
+    if (sessionStorage.getItem(menuName) == "collapse") {
+      sessionStorage.removeItem(menuName);
+    }
+    sessionStorage.setItem(menuName, "expand");
+    toggleList(this);
+  });
+
+  $(".hide-menu").on("click", function () {
+    $(this).next(".expand-menu").toggle();
+    $(this).parent().next("ul").toggle();
+    var menuName = $(this).parent().text().replace(/[^\w\s]/gi, "").trim();
+    if (sessionStorage.getItem(menuName) == "expand") {
+      sessionStorage.removeItem(menuName);
+    }
+    sessionStorage.setItem(menuName, "collapse");
+    toggleList(this);
+  });
+
+  function toggleList(menuCommand) {
+    $(menuCommand).toggle();
+  }
+});
+
+// Build an array from each tag that's present
+
+var tagList = $(".tutorials-card-container").map(function() {
+    return $(this).data("tags").split(",").map(function(item) {
+        return item.trim();
+      });
+}).get();
+
+function unique(value, index, self) {
+      return self.indexOf(value) == index && value != ""
+    }
+
+// Only return unique tags
+
+var tags = tagList.sort().filter(unique);
+
+// Add filter buttons to the top of the page for each tag
+
+function createTagMenu() {
+    tags.forEach(function(item){
+    $(".tutorial-filter-menu").append(" <div class='tutorial-filter filter-btn filter' data-tag='" + item + "'>" + item + "</div>")
+  })
+};
+
+createTagMenu();
+
+// Remove hyphens if they are present in the filter buttons
+
+$(".tags").each(function(){
+    var tags = $(this).text().split(",");
+    tags.forEach(function(tag, i ) {
+       tags[i] = tags[i].replace(/-/, ' ')
+    })
+    $(this).html(tags.join(", "));
+});
+
+// Remove hyphens if they are present in the card body
+
+$(".tutorial-filter").each(function(){
+    var tag = $(this).text();
+    $(this).html(tag.replace(/-/, ' '))
+})
+
+// Remove any empty p tags that Sphinx adds
+
+$("#tutorial-cards p").each(function(index, item) {
+    if(!$(item).text().trim()) {
+        $(item).remove();
+    }
+});
+
+// Jump back to top on pagination click
+
+$(document).on("click", ".page", function() {
+    $('html, body').animate(
+      {scrollTop: $("#dropdown-filter-tags").position().top},
+      'slow'
+    );
+});
+
+var link = $("a[href='intermediate/speech_command_recognition_with_torchaudio.html']");
+
+if (link.text() == "SyntaxError") {
+    console.log("There is an issue with the intermediate/speech_command_recognition_with_torchaudio.html menu item.");
+    link.text("Speech Command Recognition with torchaudio");
+}
+
+$(".stars-outer > i").hover(function() {
+    $(this).prevAll().addBack().toggleClass("fas star-fill");
+});
+
+$(".stars-outer > i").on("click", function() {
+    $(this).prevAll().each(function() {
+        $(this).addBack().addClass("fas star-fill");
+    });
+
+    $(".stars-outer > i").each(function() {
+        $(this).unbind("mouseenter mouseleave").css({
+            "pointer-events": "none"
+        });
+    });
+})
+
+$("#pytorch-side-scroll-right li a").on("click", function (e) {
+  var href = $(this).attr("href");
+  $('html, body').stop().animate({
+    scrollTop: $(href).offset().top - 100
+  }, 850);
+  e.preventDefault;
+});
+
+var lastId,
+  topMenu = $("#pytorch-side-scroll-right"),
+  topMenuHeight = topMenu.outerHeight() + 1,
+  // All sidenav items
+  menuItems = topMenu.find("a"),
+  // Anchors for menu items
+  scrollItems = menuItems.map(function () {
+    var item = $(this).attr("href");
+    if (item.length) {
+      return item;
+    }
+  });
+
+$(window).scroll(function () {
+  var fromTop = $(this).scrollTop() + topMenuHeight;
+  var article = ".section";
+
+  $(article).each(function (i) {
+    var offsetScroll = $(this).offset().top - $(window).scrollTop();
+    if (
+      offsetScroll <= topMenuHeight + 200 &&
+      offsetScroll >= topMenuHeight - 200 &&
+      scrollItems[i] == "#" + $(this).attr("id") &&
+      $(".hidden:visible")
+    ) {
+      $(menuItems).removeClass("side-scroll-highlight");
+      $(menuItems[i]).addClass("side-scroll-highlight");
+    }
+  });
+});
+
+
+},{"jquery":"jquery"}]},{},[1,2,3,4,5,6,7,8,9,10,"pypose-sphinx-theme"]);
diff --git a/docs/preview/pr/352/_static/js/vendor/anchor.min.js b/docs/preview/pr/352/_static/js/vendor/anchor.min.js
new file mode 100644
index 000000000..1216eeac2
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+++ b/docs/preview/pr/352/_static/js/vendor/anchor.min.js
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+// @license magnet:?xt=urn:btih:d3d9a9a6595521f9666a5e94cc830dab83b65699&dn=expat.txt Expat
+//
+// AnchorJS - v4.3.0 - 2020-10-21
+// https://www.bryanbraun.com/anchorjs/
+// Copyright (c) 2020 Bryan Braun; Licensed MIT
+//
+// @license magnet:?xt=urn:btih:d3d9a9a6595521f9666a5e94cc830dab83b65699&dn=expat.txt Expat
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37:t.preventDefault(),this.prev();break;case 39:t.preventDefault(),this.next()}},C._getItemIndex=function(e){return this._items=t.makeArray(t(e).parent().find(y.ITEM)),this._items.indexOf(e)},C._getItemByDirection=function(t,e){var n=t===h,i=t===c,s=this._getItemIndex(e),r=this._items.length-1;if((i&&0===s||n&&s===r)&&!this._config.wrap)return e;var o=(s+(t===c?-1:1))%this._items.length;return-1===o?this._items[this._items.length-1]:this._items[o]},C._triggerSlideEvent=function(e,n){var i=this._getItemIndex(e),s=this._getItemIndex(t(this._element).find(y.ACTIVE_ITEM)[0]),r=t.Event(d.SLIDE,{relatedTarget:e,direction:n,from:s,to:i});return t(this._element).trigger(r),r},C._setActiveIndicatorElement=function(e){if(this._indicatorsElement){t(this._indicatorsElement).find(y.ACTIVE).removeClass(g);var n=this._indicatorsElement.children[this._getItemIndex(e)];n&&t(n).addClass(g)}},C._slide=function(e,n){var i,s,r,o=this,a=t(this._element).find(y.ACTIVE_ITEM)[0],l=this._getItemIndex(a),c=n||a&&this._getItemByDirection(e,a),_=this._getItemIndex(c),C=Boolean(this._interval);if(e===h?(i=v,s=E,r=u):(i=m,s=T,r=f),c&&t(c).hasClass(g))this._isSliding=!1;else if(!this._triggerSlideEvent(c,r).isDefaultPrevented()&&a&&c){this._isSliding=!0,C&&this.pause(),this._setActiveIndicatorElement(c);var I=t.Event(d.SLID,{relatedTarget:c,direction:r,from:l,to:_});P.supportsTransitionEnd()&&t(this._element).hasClass(p)?(t(c).addClass(s),P.reflow(c),t(a).addClass(i),t(c).addClass(i),t(a).one(P.TRANSITION_END,function(){t(c).removeClass(i+" "+s).addClass(g),t(a).removeClass(g+" "+s+" "+i),o._isSliding=!1,setTimeout(function(){return t(o._element).trigger(I)},0)}).emulateTransitionEnd(600)):(t(a).removeClass(g),t(c).addClass(g),this._isSliding=!1,t(this._element).trigger(I)),C&&this.cycle()}},o._jQueryInterface=function(e){return this.each(function(){var i=t(this).data(n),s=r({},a,t(this).data());"object"==typeof 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t.fn[e]=o,C._jQueryInterface},C}(e),H=function(t){var e="collapse",n="bs.collapse",i="."+n,o=t.fn[e],a={toggle:!0,parent:""},l={toggle:"boolean",parent:"(string|element)"},h={SHOW:"show"+i,SHOWN:"shown"+i,HIDE:"hide"+i,HIDDEN:"hidden"+i,CLICK_DATA_API:"click"+i+".data-api"},c="show",u="collapse",f="collapsing",d="collapsed",_="width",g="height",p={ACTIVES:".show, .collapsing",DATA_TOGGLE:'[data-toggle="collapse"]'},m=function(){function i(e,n){this._isTransitioning=!1,this._element=e,this._config=this._getConfig(n),this._triggerArray=t.makeArray(t('[data-toggle="collapse"][href="#'+e.id+'"],[data-toggle="collapse"][data-target="#'+e.id+'"]'));for(var i=t(p.DATA_TOGGLE),s=0;s<i.length;s++){var r=i[s],o=P.getSelectorFromElement(r);null!==o&&t(o).filter(e).length>0&&(this._selector=o,this._triggerArray.push(r))}this._parent=this._config.parent?this._getParent():null,this._config.parent||this._addAriaAndCollapsedClass(this._element,this._triggerArray),this._config.toggle&&this.toggle()}var o=i.prototype;return o.toggle=function(){t(this._element).hasClass(c)?this.hide():this.show()},o.show=function(){var e,s,r=this;if(!this._isTransitioning&&!t(this._element).hasClass(c)&&(this._parent&&0===(e=t.makeArray(t(this._parent).find(p.ACTIVES).filter('[data-parent="'+this._config.parent+'"]'))).length&&(e=null),!(e&&(s=t(e).not(this._selector).data(n))&&s._isTransitioning))){var o=t.Event(h.SHOW);if(t(this._element).trigger(o),!o.isDefaultPrevented()){e&&(i._jQueryInterface.call(t(e).not(this._selector),"hide"),s||t(e).data(n,null));var a=this._getDimension();t(this._element).removeClass(u).addClass(f),this._element.style[a]=0,this._triggerArray.length>0&&t(this._triggerArray).removeClass(d).attr("aria-expanded",!0),this.setTransitioning(!0);var l=function(){t(r._element).removeClass(f).addClass(u).addClass(c),r._element.style[a]="",r.setTransitioning(!1),t(r._element).trigger(h.SHOWN)};if(P.supportsTransitionEnd()){var _="scroll"+(a[0].toUpperCase()+a.slice(1));t(this._element).one(P.TRANSITION_END,l).emulateTransitionEnd(600),this._element.style[a]=this._element[_]+"px"}else l()}}},o.hide=function(){var e=this;if(!this._isTransitioning&&t(this._element).hasClass(c)){var n=t.Event(h.HIDE);if(t(this._element).trigger(n),!n.isDefaultPrevented()){var i=this._getDimension();if(this._element.style[i]=this._element.getBoundingClientRect()[i]+"px",P.reflow(this._element),t(this._element).addClass(f).removeClass(u).removeClass(c),this._triggerArray.length>0)for(var s=0;s<this._triggerArray.length;s++){var r=this._triggerArray[s],o=P.getSelectorFromElement(r);if(null!==o)t(o).hasClass(c)||t(r).addClass(d).attr("aria-expanded",!1)}this.setTransitioning(!0);var a=function(){e.setTransitioning(!1),t(e._element).removeClass(f).addClass(u).trigger(h.HIDDEN)};this._element.style[i]="",P.supportsTransitionEnd()?t(this._element).one(P.TRANSITION_END,a).emulateTransitionEnd(600):a()}}},o.setTransitioning=function(t){this._isTransitioning=t},o.dispose=function(){t.removeData(this._element,n),this._config=null,this._parent=null,this._element=null,this._triggerArray=null,this._isTransitioning=null},o._getConfig=function(t){return(t=r({},a,t)).toggle=Boolean(t.toggle),P.typeCheckConfig(e,t,l),t},o._getDimension=function(){return t(this._element).hasClass(_)?_:g},o._getParent=function(){var e=this,n=null;P.isElement(this._config.parent)?(n=this._config.parent,"undefined"!=typeof this._config.parent.jquery&&(n=this._config.parent[0])):n=t(this._config.parent)[0];var s='[data-toggle="collapse"][data-parent="'+this._config.parent+'"]';return t(n).find(s).each(function(t,n){e._addAriaAndCollapsedClass(i._getTargetFromElement(n),[n])}),n},o._addAriaAndCollapsedClass=function(e,n){if(e){var i=t(e).hasClass(c);n.length>0&&t(n).toggleClass(d,!i).attr("aria-expanded",i)}},i._getTargetFromElement=function(e){var n=P.getSelectorFromElement(e);return n?t(n)[0]:null},i._jQueryInterface=function(e){return this.each(function(){var s=t(this),o=s.data(n),l=r({},a,s.data(),"object"==typeof e&&e);if(!o&&l.toggle&&/show|hide/.test(e)&&(l.toggle=!1),o||(o=new i(this,l),s.data(n,o)),"string"==typeof e){if("undefined"==typeof o[e])throw new TypeError('No method named "'+e+'"');o[e]()}})},s(i,null,[{key:"VERSION",get:function(){return"4.0.0"}},{key:"Default",get:function(){return a}}]),i}();return t(document).on(h.CLICK_DATA_API,p.DATA_TOGGLE,function(e){"A"===e.currentTarget.tagName&&e.preventDefault();var i=t(this),s=P.getSelectorFromElement(this);t(s).each(function(){var e=t(this),s=e.data(n)?"toggle":i.data();m._jQueryInterface.call(e,s)})}),t.fn[e]=m._jQueryInterface,t.fn[e].Constructor=m,t.fn[e].noConflict=function(){return t.fn[e]=o,m._jQueryInterface},m}(e),W=function(t){var e="dropdown",i="bs.dropdown",o="."+i,a=".data-api",l=t.fn[e],h=new RegExp("38|40|27"),c={HIDE:"hide"+o,HIDDEN:"hidden"+o,SHOW:"show"+o,SHOWN:"shown"+o,CLICK:"click"+o,CLICK_DATA_API:"click"+o+a,KEYDOWN_DATA_API:"keydown"+o+a,KEYUP_DATA_API:"keyup"+o+a},u="disabled",f="show",d="dropup",_="dropright",g="dropleft",p="dropdown-menu-right",m="dropdown-menu-left",v="position-static",E='[data-toggle="dropdown"]',T=".dropdown form",y=".dropdown-menu",C=".navbar-nav",I=".dropdown-menu .dropdown-item:not(.disabled)",A="top-start",b="top-end",D="bottom-start",S="bottom-end",w="right-start",N="left-start",O={offset:0,flip:!0,boundary:"scrollParent"},k={offset:"(number|string|function)",flip:"boolean",boundary:"(string|element)"},L=function(){function a(t,e){this._element=t,this._popper=null,this._config=this._getConfig(e),this._menu=this._getMenuElement(),this._inNavbar=this._detectNavbar(),this._addEventListeners()}var l=a.prototype;return l.toggle=function(){if(!this._element.disabled&&!t(this._element).hasClass(u)){var e=a._getParentFromElement(this._element),i=t(this._menu).hasClass(f);if(a._clearMenus(),!i){var s={relatedTarget:this._element},r=t.Event(c.SHOW,s);if(t(e).trigger(r),!r.isDefaultPrevented()){if(!this._inNavbar){if("undefined"==typeof n)throw new TypeError("Bootstrap dropdown require Popper.js (https://popper.js.org)");var o=this._element;t(e).hasClass(d)&&(t(this._menu).hasClass(m)||t(this._menu).hasClass(p))&&(o=e),"scrollParent"!==this._config.boundary&&t(e).addClass(v),this._popper=new n(o,this._menu,this._getPopperConfig())}"ontouchstart"in document.documentElement&&0===t(e).closest(C).length&&t("body").children().on("mouseover",null,t.noop),this._element.focus(),this._element.setAttribute("aria-expanded",!0),t(this._menu).toggleClass(f),t(e).toggleClass(f).trigger(t.Event(c.SHOWN,s))}}}},l.dispose=function(){t.removeData(this._element,i),t(this._element).off(o),this._element=null,this._menu=null,null!==this._popper&&(this._popper.destroy(),this._popper=null)},l.update=function(){this._inNavbar=this._detectNavbar(),null!==this._popper&&this._popper.scheduleUpdate()},l._addEventListeners=function(){var e=this;t(this._element).on(c.CLICK,function(t){t.preventDefault(),t.stopPropagation(),e.toggle()})},l._getConfig=function(n){return n=r({},this.constructor.Default,t(this._element).data(),n),P.typeCheckConfig(e,n,this.constructor.DefaultType),n},l._getMenuElement=function(){if(!this._menu){var e=a._getParentFromElement(this._element);this._menu=t(e).find(y)[0]}return this._menu},l._getPlacement=function(){var e=t(this._element).parent(),n=D;return e.hasClass(d)?(n=A,t(this._menu).hasClass(p)&&(n=b)):e.hasClass(_)?n=w:e.hasClass(g)?n=N:t(this._menu).hasClass(p)&&(n=S),n},l._detectNavbar=function(){return t(this._element).closest(".navbar").length>0},l._getPopperConfig=function(){var t=this,e={};return"function"==typeof this._config.offset?e.fn=function(e){return e.offsets=r({},e.offsets,t._config.offset(e.offsets)||{}),e}:e.offset=this._config.offset,{placement:this._getPlacement(),modifiers:{offset:e,flip:{enabled:this._config.flip},preventOverflow:{boundariesElement:this._config.boundary}}}},a._jQueryInterface=function(e){return this.each(function(){var n=t(this).data(i);if(n||(n=new a(this,"object"==typeof e?e:null),t(this).data(i,n)),"string"==typeof e){if("undefined"==typeof n[e])throw new TypeError('No method named "'+e+'"');n[e]()}})},a._clearMenus=function(e){if(!e||3!==e.which&&("keyup"!==e.type||9===e.which))for(var n=t.makeArray(t(E)),s=0;s<n.length;s++){var r=a._getParentFromElement(n[s]),o=t(n[s]).data(i),l={relatedTarget:n[s]};if(o){var h=o._menu;if(t(r).hasClass(f)&&!(e&&("click"===e.type&&/input|textarea/i.test(e.target.tagName)||"keyup"===e.type&&9===e.which)&&t.contains(r,e.target))){var u=t.Event(c.HIDE,l);t(r).trigger(u),u.isDefaultPrevented()||("ontouchstart"in document.documentElement&&t("body").children().off("mouseover",null,t.noop),n[s].setAttribute("aria-expanded","false"),t(h).removeClass(f),t(r).removeClass(f).trigger(t.Event(c.HIDDEN,l)))}}}},a._getParentFromElement=function(e){var n,i=P.getSelectorFromElement(e);return i&&(n=t(i)[0]),n||e.parentNode},a._dataApiKeydownHandler=function(e){if((/input|textarea/i.test(e.target.tagName)?!(32===e.which||27!==e.which&&(40!==e.which&&38!==e.which||t(e.target).closest(y).length)):h.test(e.which))&&(e.preventDefault(),e.stopPropagation(),!this.disabled&&!t(this).hasClass(u))){var n=a._getParentFromElement(this),i=t(n).hasClass(f);if((i||27===e.which&&32===e.which)&&(!i||27!==e.which&&32!==e.which)){var s=t(n).find(I).get();if(0!==s.length){var r=s.indexOf(e.target);38===e.which&&r>0&&r--,40===e.which&&r<s.length-1&&r++,r<0&&(r=0),s[r].focus()}}else{if(27===e.which){var o=t(n).find(E)[0];t(o).trigger("focus")}t(this).trigger("click")}}},s(a,null,[{key:"VERSION",get:function(){return"4.0.0"}},{key:"Default",get:function(){return O}},{key:"DefaultType",get:function(){return k}}]),a}();return t(document).on(c.KEYDOWN_DATA_API,E,L._dataApiKeydownHandler).on(c.KEYDOWN_DATA_API,y,L._dataApiKeydownHandler).on(c.CLICK_DATA_API+" "+c.KEYUP_DATA_API,L._clearMenus).on(c.CLICK_DATA_API,E,function(e){e.preventDefault(),e.stopPropagation(),L._jQueryInterface.call(t(this),"toggle")}).on(c.CLICK_DATA_API,T,function(t){t.stopPropagation()}),t.fn[e]=L._jQueryInterface,t.fn[e].Constructor=L,t.fn[e].noConflict=function(){return t.fn[e]=l,L._jQueryInterface},L}(e),M=function(t){var e="modal",n="bs.modal",i="."+n,o=t.fn.modal,a={backdrop:!0,keyboard:!0,focus:!0,show:!0},l={backdrop:"(boolean|string)",keyboard:"boolean",focus:"boolean",show:"boolean"},h={HIDE:"hide"+i,HIDDEN:"hidden"+i,SHOW:"show"+i,SHOWN:"shown"+i,FOCUSIN:"focusin"+i,RESIZE:"resize"+i,CLICK_DISMISS:"click.dismiss"+i,KEYDOWN_DISMISS:"keydown.dismiss"+i,MOUSEUP_DISMISS:"mouseup.dismiss"+i,MOUSEDOWN_DISMISS:"mousedown.dismiss"+i,CLICK_DATA_API:"click"+i+".data-api"},c="modal-scrollbar-measure",u="modal-backdrop",f="modal-open",d="fade",_="show",g={DIALOG:".modal-dialog",DATA_TOGGLE:'[data-toggle="modal"]',DATA_DISMISS:'[data-dismiss="modal"]',FIXED_CONTENT:".fixed-top, .fixed-bottom, .is-fixed, .sticky-top",STICKY_CONTENT:".sticky-top",NAVBAR_TOGGLER:".navbar-toggler"},p=function(){function o(e,n){this._config=this._getConfig(n),this._element=e,this._dialog=t(e).find(g.DIALOG)[0],this._backdrop=null,this._isShown=!1,this._isBodyOverflowing=!1,this._ignoreBackdropClick=!1,this._originalBodyPadding=0,this._scrollbarWidth=0}var p=o.prototype;return p.toggle=function(t){return this._isShown?this.hide():this.show(t)},p.show=function(e){var n=this;if(!this._isTransitioning&&!this._isShown){P.supportsTransitionEnd()&&t(this._element).hasClass(d)&&(this._isTransitioning=!0);var i=t.Event(h.SHOW,{relatedTarget:e});t(this._element).trigger(i),this._isShown||i.isDefaultPrevented()||(this._isShown=!0,this._checkScrollbar(),this._setScrollbar(),this._adjustDialog(),t(document.body).addClass(f),this._setEscapeEvent(),this._setResizeEvent(),t(this._element).on(h.CLICK_DISMISS,g.DATA_DISMISS,function(t){return n.hide(t)}),t(this._dialog).on(h.MOUSEDOWN_DISMISS,function(){t(n._element).one(h.MOUSEUP_DISMISS,function(e){t(e.target).is(n._element)&&(n._ignoreBackdropClick=!0)})}),this._showBackdrop(function(){return n._showElement(e)}))}},p.hide=function(e){var n=this;if(e&&e.preventDefault(),!this._isTransitioning&&this._isShown){var i=t.Event(h.HIDE);if(t(this._element).trigger(i),this._isShown&&!i.isDefaultPrevented()){this._isShown=!1;var s=P.supportsTransitionEnd()&&t(this._element).hasClass(d);s&&(this._isTransitioning=!0),this._setEscapeEvent(),this._setResizeEvent(),t(document).off(h.FOCUSIN),t(this._element).removeClass(_),t(this._element).off(h.CLICK_DISMISS),t(this._dialog).off(h.MOUSEDOWN_DISMISS),s?t(this._element).one(P.TRANSITION_END,function(t){return 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s=t.Event(h.SHOWN,{relatedTarget:e}),r=function(){n._config.focus&&n._element.focus(),n._isTransitioning=!1,t(n._element).trigger(s)};i?t(this._dialog).one(P.TRANSITION_END,r).emulateTransitionEnd(300):r()},p._enforceFocus=function(){var e=this;t(document).off(h.FOCUSIN).on(h.FOCUSIN,function(n){document!==n.target&&e._element!==n.target&&0===t(e._element).has(n.target).length&&e._element.focus()})},p._setEscapeEvent=function(){var e=this;this._isShown&&this._config.keyboard?t(this._element).on(h.KEYDOWN_DISMISS,function(t){27===t.which&&(t.preventDefault(),e.hide())}):this._isShown||t(this._element).off(h.KEYDOWN_DISMISS)},p._setResizeEvent=function(){var e=this;this._isShown?t(window).on(h.RESIZE,function(t){return e.handleUpdate(t)}):t(window).off(h.RESIZE)},p._hideModal=function(){var 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diff --git a/docs/preview/pr/352/_static/js/vendor/popper.min.js b/docs/preview/pr/352/_static/js/vendor/popper.min.js
new file mode 100644
index 000000000..bb1aaae3e
--- /dev/null
+++ b/docs/preview/pr/352/_static/js/vendor/popper.min.js
@@ -0,0 +1,5 @@
+/*
+ Copyright (C) Federico Zivolo 2020
+ Distributed under the MIT License (license terms are at http://opensource.org/licenses/MIT).
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+//# sourceMappingURL=popper.min.js.map
diff --git a/docs/preview/pr/352/_static/katex-math.css b/docs/preview/pr/352/_static/katex-math.css
new file mode 100644
index 000000000..bdd1634d8
--- /dev/null
+++ b/docs/preview/pr/352/_static/katex-math.css
@@ -0,0 +1,50 @@
+/* Responsives: make equations scrollable on small screens.
+ * See: https://github.com/Khan/KaTeX/issues/327 */
+.katex-display > .katex {
+    max-width: 100%;
+}
+.katex-display > .katex > .katex-html {
+    max-width: 100%;
+    overflow-x: auto;
+    overflow-y: hidden;
+    padding-left: 2px;
+    padding-right: 2px;
+    padding-bottom: 1px;
+    padding-top: 3px;
+}
+/* Increase margin around equations */
+.katex-display {
+    margin: 1.2em 0;
+}
+/* Equation number floats to the right and shows permalink for mouse hover
+   on the right side of equation number. */
+div.math {
+    position: relative;
+    padding-right: 2.5em;
+}
+.eqno {
+    height: 100%;
+    position: absolute;
+    right: 0;
+    padding-left: 5px;
+    padding-bottom: 5px;
+    padding-right: 1px;
+}
+.eqno:before {
+    /* Force vertical alignment of number */
+    display: inline-block;
+    height: 100%;
+    vertical-align: middle;
+    content: "";
+}
+.eqno .headerlink {
+    display: none;
+    visibility: hidden;
+    font-size: 14px;
+    padding-left: .3em;
+}
+.eqno:hover .headerlink {
+    display: inline-block;
+    visibility: visible;
+    margin-right: -1.05em;
+}
diff --git a/docs/preview/pr/352/_static/katex.min.js b/docs/preview/pr/352/_static/katex.min.js
new file mode 100644
index 000000000..e701cfde7
--- /dev/null
+++ b/docs/preview/pr/352/_static/katex.min.js
@@ -0,0 +1 @@
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Vn{constructor(e,t){this.mode=void 0,this.gullet=void 0,this.settings=void 0,this.leftrightDepth=void 0,this.nextToken=void 0,this.mode="math",this.gullet=new Hn(e,t,this.mode),this.settings=t,this.leftrightDepth=0}expect(e,t){if(void 0===t&&(t=!0),this.fetch().text!==e)throw new n("Expected '"+e+"', got '"+this.fetch().text+"'",this.fetch());t&&this.consume()}consume(){this.nextToken=null}fetch(){return null==this.nextToken&&(this.nextToken=this.gullet.expandNextToken()),this.nextToken}switchMode(e){this.mode=e,this.gullet.switchMode(e)}parse(){this.settings.globalGroup||this.gullet.beginGroup(),this.settings.colorIsTextColor&&this.gullet.macros.set("\\color","\\textcolor");try{const e=this.parseExpression(!1);return this.expect("EOF"),this.settings.globalGroup||this.gullet.endGroup(),e}finally{this.gullet.endGroups()}}subparse(e){const t=this.nextToken;this.consume(),this.gullet.pushToken(new Nr("}")),this.gullet.pushTokens(e);const r=this.parseExpression(!1);return this.expect("}"),this.nextToken=t,r}parseExpression(e,t){const r=[];for(;;){"math"===this.mode&&this.consumeSpaces();const n=this.fetch();if(-1!==Vn.endOfExpression.indexOf(n.text))break;if(t&&n.text===t)break;if(e&&yn[n.text]&&yn[n.text].infix)break;const o=this.parseAtom(t);if(!o)break;"internal"!==o.type&&r.push(o)}return"text"===this.mode&&this.formLigatures(r),this.handleInfixNodes(r)}handleInfixNodes(e){let t,r=-1;for(let o=0;o<e.length;o++)if("infix"===e[o].type){if(-1!==r)throw new n("only one infix operator per group",e[o].token);r=o,t=e[o].replaceWith}if(-1!==r&&t){let n,o;const s=e.slice(0,r),i=e.slice(r+1);let a;return n=1===s.length&&"ordgroup"===s[0].type?s[0]:{type:"ordgroup",mode:this.mode,body:s},o=1===i.length&&"ordgroup"===i[0].type?i[0]:{type:"ordgroup",mode:this.mode,body:i},a="\\\\abovefrac"===t?this.callFunction(t,[n,e[r],o],[]):this.callFunction(t,[n,o],[]),[a]}return e}handleSupSubscript(e){const t=this.fetch(),r=t.text;this.consume(),this.consumeSpaces();const o=this.parseGroup(e);if(!o)throw new n("Expected group after '"+r+"'",t);return o}formatUnsupportedCmd(e){const t=[];for(let r=0;r<e.length;r++)t.push({type:"textord",mode:"text",text:e[r]});const r={type:"text",mode:this.mode,body:t};return{type:"color",mode:this.mode,color:this.settings.errorColor,body:[r]}}parseAtom(e){const t=this.parseGroup("atom",e);if("text"===this.mode)return t;let r,o;for(;;){this.consumeSpaces();const e=this.fetch();if("\\limits"===e.text||"\\nolimits"===e.text){if(t&&"op"===t.type){const r="\\limits"===e.text;t.limits=r,t.alwaysHandleSupSub=!0}else{if(!t||"operatorname"!==t.type)throw new n("Limit controls must follow a math operator",e);t.alwaysHandleSupSub&&(t.limits="\\limits"===e.text)}this.consume()}else if("^"===e.text){if(r)throw new n("Double superscript",e);r=this.handleSupSubscript("superscript")}else if("_"===e.text){if(o)throw new n("Double subscript",e);o=this.handleSupSubscript("subscript")}else if("'"===e.text){if(r)throw new n("Double 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t=this.parseStringGroup("url",e);if(this.gullet.lexer.setCatcode("%",14),this.gullet.lexer.setCatcode("~",13),null==t)return null;const r=t.text.replace(/\\([#$%&~_^{}])/g,"$1");return{type:"url",mode:this.mode,url:r}}parseArgumentGroup(e,t){const r=this.gullet.scanArgument(e);if(null==r)return null;const n=this.mode;t&&this.switchMode(t),this.gullet.beginGroup();const o=this.parseExpression(!1,"EOF");this.expect("EOF"),this.gullet.endGroup();const s={type:"ordgroup",mode:this.mode,loc:r.loc,body:o};return t&&this.switchMode(n),s}parseGroup(e,t){const r=this.fetch(),o=r.text;let s;if("{"===o||"\\begingroup"===o){this.consume();const e="{"===o?"}":"\\endgroup";this.gullet.beginGroup();const t=this.parseExpression(!1,e),n=this.fetch();this.expect(e),this.gullet.endGroup(),s={type:"ordgroup",mode:this.mode,loc:Cr.range(r,n),body:t,semisimple:"\\begingroup"===o||void 0}}else if(s=this.parseFunction(t,e)||this.parseSymbol(),null==s&&"\\"===o[0]&&!Rn.hasOwnProperty(o)){if(this.settings.throwOnError)throw new n("Undefined control sequence: "+o,r);s=this.formatUnsupportedCmd(o),this.consume()}return s}formLigatures(e){let t=e.length-1;for(let r=0;r<t;++r){const n=e[r],o=n.text;"-"===o&&"-"===e[r+1].text&&(r+1<t&&"-"===e[r+2].text?(e.splice(r,3,{type:"textord",mode:"text",loc:Cr.range(n,e[r+2]),text:"---"}),t-=2):(e.splice(r,2,{type:"textord",mode:"text",loc:Cr.range(n,e[r+1]),text:"--"}),t-=1)),"'"!==o&&"`"!==o||e[r+1].text!==o||(e.splice(r,2,{type:"textord",mode:"text",loc:Cr.range(n,e[r+1]),text:o+o}),t-=1)}}parseSymbol(){const e=this.fetch();let t=e.text;if(/^\\verb[^a-zA-Z]/.test(t)){this.consume();let e=t.slice(5);const r="*"===e.charAt(0);if(r&&(e=e.slice(1)),e.length<2||e.charAt(0)!==e.slice(-1))throw new n("\\verb assertion failed --\n                    please report what input caused this bug");return 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mode',e):this.settings.reportNonstrict("unknownSymbol",'Unrecognized Unicode character "'+t[0]+'" ('+t.charCodeAt(0)+")",e)),o={type:"textord",mode:"text",loc:Cr.range(e),text:t}}if(this.consume(),r)for(let t=0;t<r[0].length;t++){const s=r[0][t];if(!Ln[s])throw new n("Unknown accent ' "+s+"'",e);const i=Ln[s][this.mode]||Ln[s].text;if(!i)throw new n("Accent "+s+" unsupported in "+this.mode+" mode",e);o={type:"accent",mode:this.mode,loc:Cr.range(e),label:i,isStretchy:!1,isShifty:!0,base:o}}return o}}Vn.endOfExpression=["}","\\endgroup","\\end","\\right","&"];var Pn=function(e,t){if(!("string"==typeof e||e instanceof String))throw new TypeError("KaTeX can only parse string typed expression");const r=new Vn(e,t);delete r.gullet.macros.current["\\df@tag"];let o=r.parse();if(delete r.gullet.macros.current["\\current@color"],delete r.gullet.macros.current["\\color"],r.gullet.macros.get("\\df@tag")){if(!t.displayMode)throw new n("\\tag works only in display equations");o=[{type:"tag",mode:"text",body:o,tag:r.subparse([new Nr("\\df@tag")])}]}return o};let Fn=function(e,t,r){t.textContent="";const n=Un(e,r).toNode();t.appendChild(n)};"undefined"!=typeof document&&"CSS1Compat"!==document.compatMode&&("undefined"!=typeof console&&console.warn("Warning: KaTeX doesn't work in quirks mode. Make sure your website has a suitable doctype."),Fn=function(){throw new n("KaTeX doesn't work in quirks mode.")});const Gn=function(e,t,r){if(r.throwOnError||!(e instanceof n))throw e;const o=Ve.makeSpan(["katex-error"],[new Z(t)]);return o.setAttribute("title",e.toString()),o.setAttribute("style","color:"+r.errorColor),o},Un=function(e,t){const r=new m(t);try{const t=Pn(e,r);return zt(t,e,r)}catch(t){return Gn(t,e,r)}};var Yn={version:"0.16.10",render:Fn,renderToString:function(e,t){return Un(e,t).toMarkup()},ParseError:n,SETTINGS_SCHEMA:h,__parse:function(e,t){const r=new m(t);return Pn(e,r)},__renderToDomTree:Un,__renderToHTMLTree:function(e,t){const r=new m(t);try{return function(e,t,r){const n=mt(e,St(r)),o=Ve.makeSpan(["katex"],[n]);return Mt(o,r)}(Pn(e,r),0,r)}catch(t){return Gn(t,e,r)}},__setFontMetrics:function(e,t){T[e]=t},__defineSymbol:se,__defineFunction:je,__defineMacro:Br,__domTree:{Span:W,Anchor:_,SymbolNode:Z,SvgNode:K,PathNode:J,LineNode:Q}};return t=t.default}()}));
\ No newline at end of file
diff --git a/docs/preview/pr/352/_static/katex_autorenderer.js b/docs/preview/pr/352/_static/katex_autorenderer.js
new file mode 100644
index 000000000..406069e22
--- /dev/null
+++ b/docs/preview/pr/352/_static/katex_autorenderer.js
@@ -0,0 +1,10 @@
+katex_options = {
+
+delimiters: [
+        { left: "\\(", right: "\\)", display: false },
+        { left: "\\[", right: "\\]", display: true }
+        ]
+}
+document.addEventListener("DOMContentLoaded", function() {
+  renderMathInElement(document.body, katex_options);
+});
diff --git a/docs/preview/pr/352/_static/language_data.js b/docs/preview/pr/352/_static/language_data.js
new file mode 100644
index 000000000..2e22b06ab
--- /dev/null
+++ b/docs/preview/pr/352/_static/language_data.js
@@ -0,0 +1,199 @@
+/*
+ * language_data.js
+ * ~~~~~~~~~~~~~~~~
+ *
+ * This script contains the language-specific data used by searchtools.js,
+ * namely the list of stopwords, stemmer, scorer and splitter.
+ *
+ * :copyright: Copyright 2007-2022 by the Sphinx team, see AUTHORS.
+ * :license: BSD, see LICENSE for details.
+ *
+ */
+
+var stopwords = ["a", "and", "are", "as", "at", "be", "but", "by", "for", "if", "in", "into", "is", "it", "near", "no", "not", "of", "on", "or", "such", "that", "the", "their", "then", "there", "these", "they", "this", "to", "was", "will", "with"];
+
+
+/* Non-minified version is copied as a separate JS file, is available */
+
+/**
+ * Porter Stemmer
+ */
+var Stemmer = function() {
+
+  var step2list = {
+    ational: 'ate',
+    tional: 'tion',
+    enci: 'ence',
+    anci: 'ance',
+    izer: 'ize',
+    bli: 'ble',
+    alli: 'al',
+    entli: 'ent',
+    eli: 'e',
+    ousli: 'ous',
+    ization: 'ize',
+    ation: 'ate',
+    ator: 'ate',
+    alism: 'al',
+    iveness: 'ive',
+    fulness: 'ful',
+    ousness: 'ous',
+    aliti: 'al',
+    iviti: 'ive',
+    biliti: 'ble',
+    logi: 'log'
+  };
+
+  var step3list = {
+    icate: 'ic',
+    ative: '',
+    alize: 'al',
+    iciti: 'ic',
+    ical: 'ic',
+    ful: '',
+    ness: ''
+  };
+
+  var c = "[^aeiou]";          // consonant
+  var v = "[aeiouy]";          // vowel
+  var C = c + "[^aeiouy]*";    // consonant sequence
+  var V = v + "[aeiou]*";      // vowel sequence
+
+  var mgr0 = "^(" + C + ")?" + V + C;                      // [C]VC... is m>0
+  var meq1 = "^(" + C + ")?" + V + C + "(" + V + ")?$";    // [C]VC[V] is m=1
+  var mgr1 = "^(" + C + ")?" + V + C + V + C;              // [C]VCVC... is m>1
+  var s_v   = "^(" + C + ")?" + v;                         // vowel in stem
+
+  this.stemWord = function (w) {
+    var stem;
+    var suffix;
+    var firstch;
+    var origword = w;
+
+    if (w.length < 3)
+      return w;
+
+    var re;
+    var re2;
+    var re3;
+    var re4;
+
+    firstch = w.substr(0,1);
+    if (firstch == "y")
+      w = firstch.toUpperCase() + w.substr(1);
+
+    // Step 1a
+    re = /^(.+?)(ss|i)es$/;
+    re2 = /^(.+?)([^s])s$/;
+
+    if (re.test(w))
+      w = w.replace(re,"$1$2");
+    else if (re2.test(w))
+      w = w.replace(re2,"$1$2");
+
+    // Step 1b
+    re = /^(.+?)eed$/;
+    re2 = /^(.+?)(ed|ing)$/;
+    if (re.test(w)) {
+      var fp = re.exec(w);
+      re = new RegExp(mgr0);
+      if (re.test(fp[1])) {
+        re = /.$/;
+        w = w.replace(re,"");
+      }
+    }
+    else if (re2.test(w)) {
+      var fp = re2.exec(w);
+      stem = fp[1];
+      re2 = new RegExp(s_v);
+      if (re2.test(stem)) {
+        w = stem;
+        re2 = /(at|bl|iz)$/;
+        re3 = new RegExp("([^aeiouylsz])\\1$");
+        re4 = new RegExp("^" + C + v + "[^aeiouwxy]$");
+        if (re2.test(w))
+          w = w + "e";
+        else if (re3.test(w)) {
+          re = /.$/;
+          w = w.replace(re,"");
+        }
+        else if (re4.test(w))
+          w = w + "e";
+      }
+    }
+
+    // Step 1c
+    re = /^(.+?)y$/;
+    if (re.test(w)) {
+      var fp = re.exec(w);
+      stem = fp[1];
+      re = new RegExp(s_v);
+      if (re.test(stem))
+        w = stem + "i";
+    }
+
+    // Step 2
+    re = /^(.+?)(ational|tional|enci|anci|izer|bli|alli|entli|eli|ousli|ization|ation|ator|alism|iveness|fulness|ousness|aliti|iviti|biliti|logi)$/;
+    if (re.test(w)) {
+      var fp = re.exec(w);
+      stem = fp[1];
+      suffix = fp[2];
+      re = new RegExp(mgr0);
+      if (re.test(stem))
+        w = stem + step2list[suffix];
+    }
+
+    // Step 3
+    re = /^(.+?)(icate|ative|alize|iciti|ical|ful|ness)$/;
+    if (re.test(w)) {
+      var fp = re.exec(w);
+      stem = fp[1];
+      suffix = fp[2];
+      re = new RegExp(mgr0);
+      if (re.test(stem))
+        w = stem + step3list[suffix];
+    }
+
+    // Step 4
+    re = /^(.+?)(al|ance|ence|er|ic|able|ible|ant|ement|ment|ent|ou|ism|ate|iti|ous|ive|ize)$/;
+    re2 = /^(.+?)(s|t)(ion)$/;
+    if (re.test(w)) {
+      var fp = re.exec(w);
+      stem = fp[1];
+      re = new RegExp(mgr1);
+      if (re.test(stem))
+        w = stem;
+    }
+    else if (re2.test(w)) {
+      var fp = re2.exec(w);
+      stem = fp[1] + fp[2];
+      re2 = new RegExp(mgr1);
+      if (re2.test(stem))
+        w = stem;
+    }
+
+    // Step 5
+    re = /^(.+?)e$/;
+    if (re.test(w)) {
+      var fp = re.exec(w);
+      stem = fp[1];
+      re = new RegExp(mgr1);
+      re2 = new RegExp(meq1);
+      re3 = new RegExp("^" + C + v + "[^aeiouwxy]$");
+      if (re.test(stem) || (re2.test(stem) && !(re3.test(stem))))
+        w = stem;
+    }
+    re = /ll$/;
+    re2 = new RegExp(mgr1);
+    if (re.test(w) && re2.test(w)) {
+      re = /.$/;
+      w = w.replace(re,"");
+    }
+
+    // and turn initial Y back to y
+    if (firstch == "y")
+      w = firstch.toLowerCase() + w.substr(1);
+    return w;
+  }
+}
+
diff --git a/docs/preview/pr/352/_static/minus.png b/docs/preview/pr/352/_static/minus.png
new file mode 100644
index 0000000000000000000000000000000000000000..d96755fdaf8bb2214971e0db9c1fd3077d7c419d
GIT binary patch
literal 90
zcmeAS@N?(olHy`uVBq!ia0vp^+#t*WBp7;*Yy1LIik>cxAr*|t7R?Mi>2?kWtu=nj
kDsEF_5m^0CR;1wuP-*O&G^0G}KYk!hp00i_>zopr08q^qX#fBK

literal 0
HcmV?d00001

diff --git a/docs/preview/pr/352/_static/plus.png b/docs/preview/pr/352/_static/plus.png
new file mode 100644
index 0000000000000000000000000000000000000000..7107cec93a979b9a5f64843235a16651d563ce2d
GIT binary patch
literal 90
zcmeAS@N?(olHy`uVBq!ia0vp^+#t*WBp7;*Yy1LIik>cxAr*|t7R?Mi>2?kWtu>-2
m3q%Vub%g%s<8sJhVPMczOq}xhg9DJoz~JfX=d#Wzp$Pyb1r*Kz

literal 0
HcmV?d00001

diff --git a/docs/preview/pr/352/_static/pygments.css b/docs/preview/pr/352/_static/pygments.css
new file mode 100644
index 000000000..84ab3030a
--- /dev/null
+++ b/docs/preview/pr/352/_static/pygments.css
@@ -0,0 +1,75 @@
+pre { line-height: 125%; }
+td.linenos .normal { color: inherit; background-color: transparent; padding-left: 5px; padding-right: 5px; }
+span.linenos { color: inherit; background-color: transparent; padding-left: 5px; padding-right: 5px; }
+td.linenos .special { color: #000000; background-color: #ffffc0; padding-left: 5px; padding-right: 5px; }
+span.linenos.special { color: #000000; background-color: #ffffc0; padding-left: 5px; padding-right: 5px; }
+.highlight .hll { background-color: #ffffcc }
+.highlight { background: #f8f8f8; }
+.highlight .c { color: #3D7B7B; font-style: italic } /* Comment */
+.highlight .err { border: 1px solid #FF0000 } /* Error */
+.highlight .k { color: #008000; font-weight: bold } /* Keyword */
+.highlight .o { color: #666666 } /* Operator */
+.highlight .ch { color: #3D7B7B; font-style: italic } /* Comment.Hashbang */
+.highlight .cm { color: #3D7B7B; font-style: italic } /* Comment.Multiline */
+.highlight .cp { color: #9C6500 } /* Comment.Preproc */
+.highlight .cpf { color: #3D7B7B; font-style: italic } /* Comment.PreprocFile */
+.highlight .c1 { color: #3D7B7B; font-style: italic } /* Comment.Single */
+.highlight .cs { color: #3D7B7B; font-style: italic } /* Comment.Special */
+.highlight .gd { color: #A00000 } /* Generic.Deleted */
+.highlight .ge { font-style: italic } /* Generic.Emph */
+.highlight .ges { font-weight: bold; font-style: italic } /* Generic.EmphStrong */
+.highlight .gr { color: #E40000 } /* Generic.Error */
+.highlight .gh { color: #000080; font-weight: bold } /* Generic.Heading */
+.highlight .gi { color: #008400 } /* Generic.Inserted */
+.highlight .go { color: #717171 } /* Generic.Output */
+.highlight .gp { color: #000080; font-weight: bold } /* Generic.Prompt */
+.highlight .gs { font-weight: bold } /* Generic.Strong */
+.highlight .gu { color: #800080; font-weight: bold } /* Generic.Subheading */
+.highlight .gt { color: #0044DD } /* Generic.Traceback */
+.highlight .kc { color: #008000; font-weight: bold } /* Keyword.Constant */
+.highlight .kd { color: #008000; font-weight: bold } /* Keyword.Declaration */
+.highlight .kn { color: #008000; font-weight: bold } /* Keyword.Namespace */
+.highlight .kp { color: #008000 } /* Keyword.Pseudo */
+.highlight .kr { color: #008000; font-weight: bold } /* Keyword.Reserved */
+.highlight .kt { color: #B00040 } /* Keyword.Type */
+.highlight .m { color: #666666 } /* Literal.Number */
+.highlight .s { color: #BA2121 } /* Literal.String */
+.highlight .na { color: #687822 } /* Name.Attribute */
+.highlight .nb { color: #008000 } /* Name.Builtin */
+.highlight .nc { color: #0000FF; font-weight: bold } /* Name.Class */
+.highlight .no { color: #880000 } /* Name.Constant */
+.highlight .nd { color: #AA22FF } /* Name.Decorator */
+.highlight .ni { color: #717171; font-weight: bold } /* Name.Entity */
+.highlight .ne { color: #CB3F38; font-weight: bold } /* Name.Exception */
+.highlight .nf { color: #0000FF } /* Name.Function */
+.highlight .nl { color: #767600 } /* Name.Label */
+.highlight .nn { color: #0000FF; font-weight: bold } /* Name.Namespace */
+.highlight .nt { color: #008000; font-weight: bold } /* Name.Tag */
+.highlight .nv { color: #19177C } /* Name.Variable */
+.highlight .ow { color: #AA22FF; font-weight: bold } /* Operator.Word */
+.highlight .w { color: #bbbbbb } /* Text.Whitespace */
+.highlight .mb { color: #666666 } /* Literal.Number.Bin */
+.highlight .mf { color: #666666 } /* Literal.Number.Float */
+.highlight .mh { color: #666666 } /* Literal.Number.Hex */
+.highlight .mi { color: #666666 } /* Literal.Number.Integer */
+.highlight .mo { color: #666666 } /* Literal.Number.Oct */
+.highlight .sa { color: #BA2121 } /* Literal.String.Affix */
+.highlight .sb { color: #BA2121 } /* Literal.String.Backtick */
+.highlight .sc { color: #BA2121 } /* Literal.String.Char */
+.highlight .dl { color: #BA2121 } /* Literal.String.Delimiter */
+.highlight .sd { color: #BA2121; font-style: italic } /* Literal.String.Doc */
+.highlight .s2 { color: #BA2121 } /* Literal.String.Double */
+.highlight .se { color: #AA5D1F; font-weight: bold } /* Literal.String.Escape */
+.highlight .sh { color: #BA2121 } /* Literal.String.Heredoc */
+.highlight .si { color: #A45A77; font-weight: bold } /* Literal.String.Interpol */
+.highlight .sx { color: #008000 } /* Literal.String.Other */
+.highlight .sr { color: #A45A77 } /* Literal.String.Regex */
+.highlight .s1 { color: #BA2121 } /* Literal.String.Single */
+.highlight .ss { color: #19177C } /* Literal.String.Symbol */
+.highlight .bp { color: #008000 } /* Name.Builtin.Pseudo */
+.highlight .fm { color: #0000FF } /* Name.Function.Magic */
+.highlight .vc { color: #19177C } /* Name.Variable.Class */
+.highlight .vg { color: #19177C } /* Name.Variable.Global */
+.highlight .vi { color: #19177C } /* Name.Variable.Instance */
+.highlight .vm { color: #19177C } /* Name.Variable.Magic */
+.highlight .il { color: #666666 } /* Literal.Number.Integer.Long */
\ No newline at end of file
diff --git a/docs/preview/pr/352/_static/searchtools.js b/docs/preview/pr/352/_static/searchtools.js
new file mode 100644
index 000000000..ac4d5861f
--- /dev/null
+++ b/docs/preview/pr/352/_static/searchtools.js
@@ -0,0 +1,531 @@
+/*
+ * searchtools.js
+ * ~~~~~~~~~~~~~~~~
+ *
+ * Sphinx JavaScript utilities for the full-text search.
+ *
+ * :copyright: Copyright 2007-2022 by the Sphinx team, see AUTHORS.
+ * :license: BSD, see LICENSE for details.
+ *
+ */
+"use strict";
+
+/**
+ * Simple result scoring code.
+ */
+if (typeof Scorer === "undefined") {
+  var Scorer = {
+    // Implement the following function to further tweak the score for each result
+    // The function takes a result array [docname, title, anchor, descr, score, filename]
+    // and returns the new score.
+    /*
+    score: result => {
+      const [docname, title, anchor, descr, score, filename] = result
+      return score
+    },
+    */
+
+    // query matches the full name of an object
+    objNameMatch: 11,
+    // or matches in the last dotted part of the object name
+    objPartialMatch: 6,
+    // Additive scores depending on the priority of the object
+    objPrio: {
+      0: 15, // used to be importantResults
+      1: 5, // used to be objectResults
+      2: -5, // used to be unimportantResults
+    },
+    //  Used when the priority is not in the mapping.
+    objPrioDefault: 0,
+
+    // query found in title
+    title: 15,
+    partialTitle: 7,
+    // query found in terms
+    term: 5,
+    partialTerm: 2,
+  };
+}
+
+const _removeChildren = (element) => {
+  while (element && element.lastChild) element.removeChild(element.lastChild);
+};
+
+/**
+ * See https://developer.mozilla.org/en-US/docs/Web/JavaScript/Guide/Regular_Expressions#escaping
+ */
+const _escapeRegExp = (string) =>
+  string.replace(/[.*+\-?^${}()|[\]\\]/g, "\\$&"); // $& means the whole matched string
+
+const _displayItem = (item, highlightTerms, searchTerms) => {
+  const docBuilder = DOCUMENTATION_OPTIONS.BUILDER;
+  const docUrlRoot = DOCUMENTATION_OPTIONS.URL_ROOT;
+  const docFileSuffix = DOCUMENTATION_OPTIONS.FILE_SUFFIX;
+  const docLinkSuffix = DOCUMENTATION_OPTIONS.LINK_SUFFIX;
+  const showSearchSummary = DOCUMENTATION_OPTIONS.SHOW_SEARCH_SUMMARY;
+
+  const [docName, title, anchor, descr] = item;
+
+  let listItem = document.createElement("li");
+  let requestUrl;
+  let linkUrl;
+  if (docBuilder === "dirhtml") {
+    // dirhtml builder
+    let dirname = docName + "/";
+    if (dirname.match(/\/index\/$/))
+      dirname = dirname.substring(0, dirname.length - 6);
+    else if (dirname === "index/") dirname = "";
+    requestUrl = docUrlRoot + dirname;
+    linkUrl = requestUrl;
+  } else {
+    // normal html builders
+    requestUrl = docUrlRoot + docName + docFileSuffix;
+    linkUrl = docName + docLinkSuffix;
+  }
+  const params = new URLSearchParams();
+  params.set("highlight", [...highlightTerms].join(" "));
+  let linkEl = listItem.appendChild(document.createElement("a"));
+  linkEl.href = linkUrl + "?" + params.toString() + anchor;
+  linkEl.innerHTML = title;
+  if (descr)
+    listItem.appendChild(document.createElement("span")).innerText =
+      " (" + descr + ")";
+  else if (showSearchSummary)
+    fetch(requestUrl)
+      .then((responseData) => responseData.text())
+      .then((data) => {
+        if (data)
+          listItem.appendChild(
+            Search.makeSearchSummary(data, searchTerms, highlightTerms)
+          );
+      });
+  Search.output.appendChild(listItem);
+};
+const _finishSearch = (resultCount) => {
+  Search.stopPulse();
+  Search.title.innerText = _("Search Results");
+  if (!resultCount)
+    Search.status.innerText = Documentation.gettext(
+      "Your search did not match any documents. Please make sure that all words are spelled correctly and that you've selected enough categories."
+    );
+  else
+    Search.status.innerText = _(
+      `Search finished, found ${resultCount} page(s) matching the search query.`
+    );
+};
+const _displayNextItem = (
+  results,
+  resultCount,
+  highlightTerms,
+  searchTerms
+) => {
+  // results left, load the summary and display it
+  // this is intended to be dynamic (don't sub resultsCount)
+  if (results.length) {
+    _displayItem(results.pop(), highlightTerms, searchTerms);
+    setTimeout(
+      () => _displayNextItem(results, resultCount, highlightTerms, searchTerms),
+      5
+    );
+  }
+  // search finished, update title and status message
+  else _finishSearch(resultCount);
+};
+
+/**
+ * Default splitQuery function. Can be overridden in ``sphinx.search`` with a
+ * custom function per language.
+ *
+ * The regular expression works by splitting the string on consecutive characters
+ * that are not Unicode letters, numbers, underscores, or emoji characters.
+ * This is the same as ``\W+`` in Python, preserving the surrogate pair area.
+ */
+if (typeof splitQuery === "undefined") {
+  var splitQuery = (query) => query
+      .split(/[^\p{Letter}\p{Number}_\p{Emoji_Presentation}]+/gu)
+      .filter(term => term)  // remove remaining empty strings
+}
+
+/**
+ * Search Module
+ */
+const Search = {
+  _index: null,
+  _queued_query: null,
+  _pulse_status: -1,
+
+  htmlToText: (htmlString) => {
+    const htmlElement = document
+      .createRange()
+      .createContextualFragment(htmlString);
+    _removeChildren(htmlElement.querySelectorAll(".headerlink"));
+    const docContent = htmlElement.querySelector('[role="main"]');
+    if (docContent !== undefined) return docContent.textContent;
+    console.warn(
+      "Content block not found. Sphinx search tries to obtain it via '[role=main]'. Could you check your theme or template."
+    );
+    return "";
+  },
+
+  init: () => {
+    const query = new URLSearchParams(window.location.search).get("q");
+    document
+      .querySelectorAll('input[name="q"]')
+      .forEach((el) => (el.value = query));
+    if (query) Search.performSearch(query);
+  },
+
+  loadIndex: (url) =>
+    (document.body.appendChild(document.createElement("script")).src = url),
+
+  setIndex: (index) => {
+    Search._index = index;
+    if (Search._queued_query !== null) {
+      const query = Search._queued_query;
+      Search._queued_query = null;
+      Search.query(query);
+    }
+  },
+
+  hasIndex: () => Search._index !== null,
+
+  deferQuery: (query) => (Search._queued_query = query),
+
+  stopPulse: () => (Search._pulse_status = -1),
+
+  startPulse: () => {
+    if (Search._pulse_status >= 0) return;
+
+    const pulse = () => {
+      Search._pulse_status = (Search._pulse_status + 1) % 4;
+      Search.dots.innerText = ".".repeat(Search._pulse_status);
+      if (Search._pulse_status >= 0) window.setTimeout(pulse, 500);
+    };
+    pulse();
+  },
+
+  /**
+   * perform a search for something (or wait until index is loaded)
+   */
+  performSearch: (query) => {
+    // create the required interface elements
+    const searchText = document.createElement("h2");
+    searchText.textContent = _("Searching");
+    const searchSummary = document.createElement("p");
+    searchSummary.classList.add("search-summary");
+    searchSummary.innerText = "";
+    const searchList = document.createElement("ul");
+    searchList.classList.add("search");
+
+    const out = document.getElementById("search-results");
+    Search.title = out.appendChild(searchText);
+    Search.dots = Search.title.appendChild(document.createElement("span"));
+    Search.status = out.appendChild(searchSummary);
+    Search.output = out.appendChild(searchList);
+
+    const searchProgress = document.getElementById("search-progress");
+    // Some themes don't use the search progress node
+    if (searchProgress) {
+      searchProgress.innerText = _("Preparing search...");
+    }
+    Search.startPulse();
+
+    // index already loaded, the browser was quick!
+    if (Search.hasIndex()) Search.query(query);
+    else Search.deferQuery(query);
+  },
+
+  /**
+   * execute search (requires search index to be loaded)
+   */
+  query: (query) => {
+    // stem the search terms and add them to the correct list
+    const stemmer = new Stemmer();
+    const searchTerms = new Set();
+    const excludedTerms = new Set();
+    const highlightTerms = new Set();
+    const objectTerms = new Set(splitQuery(query.toLowerCase().trim()));
+    splitQuery(query.trim()).forEach((queryTerm) => {
+      const queryTermLower = queryTerm.toLowerCase();
+
+      // maybe skip this "word"
+      // stopwords array is from language_data.js
+      if (
+        stopwords.indexOf(queryTermLower) !== -1 ||
+        queryTerm.match(/^\d+$/)
+      )
+        return;
+
+      // stem the word
+      let word = stemmer.stemWord(queryTermLower);
+      // select the correct list
+      if (word[0] === "-") excludedTerms.add(word.substr(1));
+      else {
+        searchTerms.add(word);
+        highlightTerms.add(queryTermLower);
+      }
+    });
+
+    // console.debug("SEARCH: searching for:");
+    // console.info("required: ", [...searchTerms]);
+    // console.info("excluded: ", [...excludedTerms]);
+
+    // array of [docname, title, anchor, descr, score, filename]
+    let results = [];
+    _removeChildren(document.getElementById("search-progress"));
+
+    // lookup as object
+    objectTerms.forEach((term) =>
+      results.push(...Search.performObjectSearch(term, objectTerms))
+    );
+
+    // lookup as search terms in fulltext
+    results.push(...Search.performTermsSearch(searchTerms, excludedTerms));
+
+    // let the scorer override scores with a custom scoring function
+    if (Scorer.score) results.forEach((item) => (item[4] = Scorer.score(item)));
+
+    // now sort the results by score (in opposite order of appearance, since the
+    // display function below uses pop() to retrieve items) and then
+    // alphabetically
+    results.sort((a, b) => {
+      const leftScore = a[4];
+      const rightScore = b[4];
+      if (leftScore === rightScore) {
+        // same score: sort alphabetically
+        const leftTitle = a[1].toLowerCase();
+        const rightTitle = b[1].toLowerCase();
+        if (leftTitle === rightTitle) return 0;
+        return leftTitle > rightTitle ? -1 : 1; // inverted is intentional
+      }
+      return leftScore > rightScore ? 1 : -1;
+    });
+
+    // remove duplicate search results
+    // note the reversing of results, so that in the case of duplicates, the highest-scoring entry is kept
+    let seen = new Set();
+    results = results.reverse().reduce((acc, result) => {
+      let resultStr = result.slice(0, 4).concat([result[5]]).map(v => String(v)).join(',');
+      if (!seen.has(resultStr)) {
+        acc.push(result);
+        seen.add(resultStr);
+      }
+      return acc;
+    }, []);
+
+    results = results.reverse();
+
+    // for debugging
+    //Search.lastresults = results.slice();  // a copy
+    // console.info("search results:", Search.lastresults);
+
+    // print the results
+    _displayNextItem(results, results.length, highlightTerms, searchTerms);
+  },
+
+  /**
+   * search for object names
+   */
+  performObjectSearch: (object, objectTerms) => {
+    const filenames = Search._index.filenames;
+    const docNames = Search._index.docnames;
+    const objects = Search._index.objects;
+    const objNames = Search._index.objnames;
+    const titles = Search._index.titles;
+
+    const results = [];
+
+    const objectSearchCallback = (prefix, match) => {
+      const name = match[4]
+      const fullname = (prefix ? prefix + "." : "") + name;
+      const fullnameLower = fullname.toLowerCase();
+      if (fullnameLower.indexOf(object) < 0) return;
+
+      let score = 0;
+      const parts = fullnameLower.split(".");
+
+      // check for different match types: exact matches of full name or
+      // "last name" (i.e. last dotted part)
+      if (fullnameLower === object || parts.slice(-1)[0] === object)
+        score += Scorer.objNameMatch;
+      else if (parts.slice(-1)[0].indexOf(object) > -1)
+        score += Scorer.objPartialMatch; // matches in last name
+
+      const objName = objNames[match[1]][2];
+      const title = titles[match[0]];
+
+      // If more than one term searched for, we require other words to be
+      // found in the name/title/description
+      const otherTerms = new Set(objectTerms);
+      otherTerms.delete(object);
+      if (otherTerms.size > 0) {
+        const haystack = `${prefix} ${name} ${objName} ${title}`.toLowerCase();
+        if (
+          [...otherTerms].some((otherTerm) => haystack.indexOf(otherTerm) < 0)
+        )
+          return;
+      }
+
+      let anchor = match[3];
+      if (anchor === "") anchor = fullname;
+      else if (anchor === "-") anchor = objNames[match[1]][1] + "-" + fullname;
+
+      const descr = objName + _(", in ") + title;
+
+      // add custom score for some objects according to scorer
+      if (Scorer.objPrio.hasOwnProperty(match[2]))
+        score += Scorer.objPrio[match[2]];
+      else score += Scorer.objPrioDefault;
+
+      results.push([
+        docNames[match[0]],
+        fullname,
+        "#" + anchor,
+        descr,
+        score,
+        filenames[match[0]],
+      ]);
+    };
+    Object.keys(objects).forEach((prefix) =>
+      objects[prefix].forEach((array) =>
+        objectSearchCallback(prefix, array)
+      )
+    );
+    return results;
+  },
+
+  /**
+   * search for full-text terms in the index
+   */
+  performTermsSearch: (searchTerms, excludedTerms) => {
+    // prepare search
+    const terms = Search._index.terms;
+    const titleTerms = Search._index.titleterms;
+    const docNames = Search._index.docnames;
+    const filenames = Search._index.filenames;
+    const titles = Search._index.titles;
+
+    const scoreMap = new Map();
+    const fileMap = new Map();
+
+    // perform the search on the required terms
+    searchTerms.forEach((word) => {
+      const files = [];
+      const arr = [
+        { files: terms[word], score: Scorer.term },
+        { files: titleTerms[word], score: Scorer.title },
+      ];
+      // add support for partial matches
+      if (word.length > 2) {
+        const escapedWord = _escapeRegExp(word);
+        Object.keys(terms).forEach((term) => {
+          if (term.match(escapedWord) && !terms[word])
+            arr.push({ files: terms[term], score: Scorer.partialTerm });
+        });
+        Object.keys(titleTerms).forEach((term) => {
+          if (term.match(escapedWord) && !titleTerms[word])
+            arr.push({ files: titleTerms[word], score: Scorer.partialTitle });
+        });
+      }
+
+      // no match but word was a required one
+      if (arr.every((record) => record.files === undefined)) return;
+
+      // found search word in contents
+      arr.forEach((record) => {
+        if (record.files === undefined) return;
+
+        let recordFiles = record.files;
+        if (recordFiles.length === undefined) recordFiles = [recordFiles];
+        files.push(...recordFiles);
+
+        // set score for the word in each file
+        recordFiles.forEach((file) => {
+          if (!scoreMap.has(file)) scoreMap.set(file, {});
+          scoreMap.get(file)[word] = record.score;
+        });
+      });
+
+      // create the mapping
+      files.forEach((file) => {
+        if (fileMap.has(file) && fileMap.get(file).indexOf(word) === -1)
+          fileMap.get(file).push(word);
+        else fileMap.set(file, [word]);
+      });
+    });
+
+    // now check if the files don't contain excluded terms
+    const results = [];
+    for (const [file, wordList] of fileMap) {
+      // check if all requirements are matched
+
+      // as search terms with length < 3 are discarded
+      const filteredTermCount = [...searchTerms].filter(
+        (term) => term.length > 2
+      ).length;
+      if (
+        wordList.length !== searchTerms.size &&
+        wordList.length !== filteredTermCount
+      )
+        continue;
+
+      // ensure that none of the excluded terms is in the search result
+      if (
+        [...excludedTerms].some(
+          (term) =>
+            terms[term] === file ||
+            titleTerms[term] === file ||
+            (terms[term] || []).includes(file) ||
+            (titleTerms[term] || []).includes(file)
+        )
+      )
+        break;
+
+      // select one (max) score for the file.
+      const score = Math.max(...wordList.map((w) => scoreMap.get(file)[w]));
+      // add result to the result list
+      results.push([
+        docNames[file],
+        titles[file],
+        "",
+        null,
+        score,
+        filenames[file],
+      ]);
+    }
+    return results;
+  },
+
+  /**
+   * helper function to return a node containing the
+   * search summary for a given text. keywords is a list
+   * of stemmed words, highlightWords is the list of normal, unstemmed
+   * words. the first one is used to find the occurrence, the
+   * latter for highlighting it.
+   */
+  makeSearchSummary: (htmlText, keywords, highlightWords) => {
+    const text = Search.htmlToText(htmlText).toLowerCase();
+    if (text === "") return null;
+
+    const actualStartPosition = [...keywords]
+      .map((k) => text.indexOf(k.toLowerCase()))
+      .filter((i) => i > -1)
+      .slice(-1)[0];
+    const startWithContext = Math.max(actualStartPosition - 120, 0);
+
+    const top = startWithContext === 0 ? "" : "...";
+    const tail = startWithContext + 240 < text.length ? "..." : "";
+
+    let summary = document.createElement("div");
+    summary.classList.add("context");
+    summary.innerText = top + text.substr(startWithContext, 240).trim() + tail;
+
+    highlightWords.forEach((highlightWord) =>
+      _highlightText(summary, highlightWord, "highlighted")
+    );
+
+    return summary;
+  },
+};
+
+_ready(Search.init);
diff --git a/docs/preview/pr/352/_static/underscore-1.13.1.js b/docs/preview/pr/352/_static/underscore-1.13.1.js
new file mode 100644
index 000000000..ffd77af96
--- /dev/null
+++ b/docs/preview/pr/352/_static/underscore-1.13.1.js
@@ -0,0 +1,2042 @@
+(function (global, factory) {
+  typeof exports === 'object' && typeof module !== 'undefined' ? module.exports = factory() :
+  typeof define === 'function' && define.amd ? define('underscore', factory) :
+  (global = typeof globalThis !== 'undefined' ? globalThis : global || self, (function () {
+    var current = global._;
+    var exports = global._ = factory();
+    exports.noConflict = function () { global._ = current; return exports; };
+  }()));
+}(this, (function () {
+  //     Underscore.js 1.13.1
+  //     https://underscorejs.org
+  //     (c) 2009-2021 Jeremy Ashkenas, Julian Gonggrijp, and DocumentCloud and Investigative Reporters & Editors
+  //     Underscore may be freely distributed under the MIT license.
+
+  // Current version.
+  var VERSION = '1.13.1';
+
+  // Establish the root object, `window` (`self`) in the browser, `global`
+  // on the server, or `this` in some virtual machines. We use `self`
+  // instead of `window` for `WebWorker` support.
+  var root = typeof self == 'object' && self.self === self && self ||
+            typeof global == 'object' && global.global === global && global ||
+            Function('return this')() ||
+            {};
+
+  // Save bytes in the minified (but not gzipped) version:
+  var ArrayProto = Array.prototype, ObjProto = Object.prototype;
+  var SymbolProto = typeof Symbol !== 'undefined' ? Symbol.prototype : null;
+
+  // Create quick reference variables for speed access to core prototypes.
+  var push = ArrayProto.push,
+      slice = ArrayProto.slice,
+      toString = ObjProto.toString,
+      hasOwnProperty = ObjProto.hasOwnProperty;
+
+  // Modern feature detection.
+  var supportsArrayBuffer = typeof ArrayBuffer !== 'undefined',
+      supportsDataView = typeof DataView !== 'undefined';
+
+  // All **ECMAScript 5+** native function implementations that we hope to use
+  // are declared here.
+  var nativeIsArray = Array.isArray,
+      nativeKeys = Object.keys,
+      nativeCreate = Object.create,
+      nativeIsView = supportsArrayBuffer && ArrayBuffer.isView;
+
+  // Create references to these builtin functions because we override them.
+  var _isNaN = isNaN,
+      _isFinite = isFinite;
+
+  // Keys in IE < 9 that won't be iterated by `for key in ...` and thus missed.
+  var hasEnumBug = !{toString: null}.propertyIsEnumerable('toString');
+  var nonEnumerableProps = ['valueOf', 'isPrototypeOf', 'toString',
+    'propertyIsEnumerable', 'hasOwnProperty', 'toLocaleString'];
+
+  // The largest integer that can be represented exactly.
+  var MAX_ARRAY_INDEX = Math.pow(2, 53) - 1;
+
+  // Some functions take a variable number of arguments, or a few expected
+  // arguments at the beginning and then a variable number of values to operate
+  // on. This helper accumulates all remaining arguments past the function’s
+  // argument length (or an explicit `startIndex`), into an array that becomes
+  // the last argument. Similar to ES6’s "rest parameter".
+  function restArguments(func, startIndex) {
+    startIndex = startIndex == null ? func.length - 1 : +startIndex;
+    return function() {
+      var length = Math.max(arguments.length - startIndex, 0),
+          rest = Array(length),
+          index = 0;
+      for (; index < length; index++) {
+        rest[index] = arguments[index + startIndex];
+      }
+      switch (startIndex) {
+        case 0: return func.call(this, rest);
+        case 1: return func.call(this, arguments[0], rest);
+        case 2: return func.call(this, arguments[0], arguments[1], rest);
+      }
+      var args = Array(startIndex + 1);
+      for (index = 0; index < startIndex; index++) {
+        args[index] = arguments[index];
+      }
+      args[startIndex] = rest;
+      return func.apply(this, args);
+    };
+  }
+
+  // Is a given variable an object?
+  function isObject(obj) {
+    var type = typeof obj;
+    return type === 'function' || type === 'object' && !!obj;
+  }
+
+  // Is a given value equal to null?
+  function isNull(obj) {
+    return obj === null;
+  }
+
+  // Is a given variable undefined?
+  function isUndefined(obj) {
+    return obj === void 0;
+  }
+
+  // Is a given value a boolean?
+  function isBoolean(obj) {
+    return obj === true || obj === false || toString.call(obj) === '[object Boolean]';
+  }
+
+  // Is a given value a DOM element?
+  function isElement(obj) {
+    return !!(obj && obj.nodeType === 1);
+  }
+
+  // Internal function for creating a `toString`-based type tester.
+  function tagTester(name) {
+    var tag = '[object ' + name + ']';
+    return function(obj) {
+      return toString.call(obj) === tag;
+    };
+  }
+
+  var isString = tagTester('String');
+
+  var isNumber = tagTester('Number');
+
+  var isDate = tagTester('Date');
+
+  var isRegExp = tagTester('RegExp');
+
+  var isError = tagTester('Error');
+
+  var isSymbol = tagTester('Symbol');
+
+  var isArrayBuffer = tagTester('ArrayBuffer');
+
+  var isFunction = tagTester('Function');
+
+  // Optimize `isFunction` if appropriate. Work around some `typeof` bugs in old
+  // v8, IE 11 (#1621), Safari 8 (#1929), and PhantomJS (#2236).
+  var nodelist = root.document && root.document.childNodes;
+  if (typeof /./ != 'function' && typeof Int8Array != 'object' && typeof nodelist != 'function') {
+    isFunction = function(obj) {
+      return typeof obj == 'function' || false;
+    };
+  }
+
+  var isFunction$1 = isFunction;
+
+  var hasObjectTag = tagTester('Object');
+
+  // In IE 10 - Edge 13, `DataView` has string tag `'[object Object]'`.
+  // In IE 11, the most common among them, this problem also applies to
+  // `Map`, `WeakMap` and `Set`.
+  var hasStringTagBug = (
+        supportsDataView && hasObjectTag(new DataView(new ArrayBuffer(8)))
+      ),
+      isIE11 = (typeof Map !== 'undefined' && hasObjectTag(new Map));
+
+  var isDataView = tagTester('DataView');
+
+  // In IE 10 - Edge 13, we need a different heuristic
+  // to determine whether an object is a `DataView`.
+  function ie10IsDataView(obj) {
+    return obj != null && isFunction$1(obj.getInt8) && isArrayBuffer(obj.buffer);
+  }
+
+  var isDataView$1 = (hasStringTagBug ? ie10IsDataView : isDataView);
+
+  // Is a given value an array?
+  // Delegates to ECMA5's native `Array.isArray`.
+  var isArray = nativeIsArray || tagTester('Array');
+
+  // Internal function to check whether `key` is an own property name of `obj`.
+  function has$1(obj, key) {
+    return obj != null && hasOwnProperty.call(obj, key);
+  }
+
+  var isArguments = tagTester('Arguments');
+
+  // Define a fallback version of the method in browsers (ahem, IE < 9), where
+  // there isn't any inspectable "Arguments" type.
+  (function() {
+    if (!isArguments(arguments)) {
+      isArguments = function(obj) {
+        return has$1(obj, 'callee');
+      };
+    }
+  }());
+
+  var isArguments$1 = isArguments;
+
+  // Is a given object a finite number?
+  function isFinite$1(obj) {
+    return !isSymbol(obj) && _isFinite(obj) && !isNaN(parseFloat(obj));
+  }
+
+  // Is the given value `NaN`?
+  function isNaN$1(obj) {
+    return isNumber(obj) && _isNaN(obj);
+  }
+
+  // Predicate-generating function. Often useful outside of Underscore.
+  function constant(value) {
+    return function() {
+      return value;
+    };
+  }
+
+  // Common internal logic for `isArrayLike` and `isBufferLike`.
+  function createSizePropertyCheck(getSizeProperty) {
+    return function(collection) {
+      var sizeProperty = getSizeProperty(collection);
+      return typeof sizeProperty == 'number' && sizeProperty >= 0 && sizeProperty <= MAX_ARRAY_INDEX;
+    }
+  }
+
+  // Internal helper to generate a function to obtain property `key` from `obj`.
+  function shallowProperty(key) {
+    return function(obj) {
+      return obj == null ? void 0 : obj[key];
+    };
+  }
+
+  // Internal helper to obtain the `byteLength` property of an object.
+  var getByteLength = shallowProperty('byteLength');
+
+  // Internal helper to determine whether we should spend extensive checks against
+  // `ArrayBuffer` et al.
+  var isBufferLike = createSizePropertyCheck(getByteLength);
+
+  // Is a given value a typed array?
+  var typedArrayPattern = /\[object ((I|Ui)nt(8|16|32)|Float(32|64)|Uint8Clamped|Big(I|Ui)nt64)Array\]/;
+  function isTypedArray(obj) {
+    // `ArrayBuffer.isView` is the most future-proof, so use it when available.
+    // Otherwise, fall back on the above regular expression.
+    return nativeIsView ? (nativeIsView(obj) && !isDataView$1(obj)) :
+                  isBufferLike(obj) && typedArrayPattern.test(toString.call(obj));
+  }
+
+  var isTypedArray$1 = supportsArrayBuffer ? isTypedArray : constant(false);
+
+  // Internal helper to obtain the `length` property of an object.
+  var getLength = shallowProperty('length');
+
+  // Internal helper to create a simple lookup structure.
+  // `collectNonEnumProps` used to depend on `_.contains`, but this led to
+  // circular imports. `emulatedSet` is a one-off solution that only works for
+  // arrays of strings.
+  function emulatedSet(keys) {
+    var hash = {};
+    for (var l = keys.length, i = 0; i < l; ++i) hash[keys[i]] = true;
+    return {
+      contains: function(key) { return hash[key]; },
+      push: function(key) {
+        hash[key] = true;
+        return keys.push(key);
+      }
+    };
+  }
+
+  // Internal helper. Checks `keys` for the presence of keys in IE < 9 that won't
+  // be iterated by `for key in ...` and thus missed. Extends `keys` in place if
+  // needed.
+  function collectNonEnumProps(obj, keys) {
+    keys = emulatedSet(keys);
+    var nonEnumIdx = nonEnumerableProps.length;
+    var constructor = obj.constructor;
+    var proto = isFunction$1(constructor) && constructor.prototype || ObjProto;
+
+    // Constructor is a special case.
+    var prop = 'constructor';
+    if (has$1(obj, prop) && !keys.contains(prop)) keys.push(prop);
+
+    while (nonEnumIdx--) {
+      prop = nonEnumerableProps[nonEnumIdx];
+      if (prop in obj && obj[prop] !== proto[prop] && !keys.contains(prop)) {
+        keys.push(prop);
+      }
+    }
+  }
+
+  // Retrieve the names of an object's own properties.
+  // Delegates to **ECMAScript 5**'s native `Object.keys`.
+  function keys(obj) {
+    if (!isObject(obj)) return [];
+    if (nativeKeys) return nativeKeys(obj);
+    var keys = [];
+    for (var key in obj) if (has$1(obj, key)) keys.push(key);
+    // Ahem, IE < 9.
+    if (hasEnumBug) collectNonEnumProps(obj, keys);
+    return keys;
+  }
+
+  // Is a given array, string, or object empty?
+  // An "empty" object has no enumerable own-properties.
+  function isEmpty(obj) {
+    if (obj == null) return true;
+    // Skip the more expensive `toString`-based type checks if `obj` has no
+    // `.length`.
+    var length = getLength(obj);
+    if (typeof length == 'number' && (
+      isArray(obj) || isString(obj) || isArguments$1(obj)
+    )) return length === 0;
+    return getLength(keys(obj)) === 0;
+  }
+
+  // Returns whether an object has a given set of `key:value` pairs.
+  function isMatch(object, attrs) {
+    var _keys = keys(attrs), length = _keys.length;
+    if (object == null) return !length;
+    var obj = Object(object);
+    for (var i = 0; i < length; i++) {
+      var key = _keys[i];
+      if (attrs[key] !== obj[key] || !(key in obj)) return false;
+    }
+    return true;
+  }
+
+  // If Underscore is called as a function, it returns a wrapped object that can
+  // be used OO-style. This wrapper holds altered versions of all functions added
+  // through `_.mixin`. Wrapped objects may be chained.
+  function _$1(obj) {
+    if (obj instanceof _$1) return obj;
+    if (!(this instanceof _$1)) return new _$1(obj);
+    this._wrapped = obj;
+  }
+
+  _$1.VERSION = VERSION;
+
+  // Extracts the result from a wrapped and chained object.
+  _$1.prototype.value = function() {
+    return this._wrapped;
+  };
+
+  // Provide unwrapping proxies for some methods used in engine operations
+  // such as arithmetic and JSON stringification.
+  _$1.prototype.valueOf = _$1.prototype.toJSON = _$1.prototype.value;
+
+  _$1.prototype.toString = function() {
+    return String(this._wrapped);
+  };
+
+  // Internal function to wrap or shallow-copy an ArrayBuffer,
+  // typed array or DataView to a new view, reusing the buffer.
+  function toBufferView(bufferSource) {
+    return new Uint8Array(
+      bufferSource.buffer || bufferSource,
+      bufferSource.byteOffset || 0,
+      getByteLength(bufferSource)
+    );
+  }
+
+  // We use this string twice, so give it a name for minification.
+  var tagDataView = '[object DataView]';
+
+  // Internal recursive comparison function for `_.isEqual`.
+  function eq(a, b, aStack, bStack) {
+    // Identical objects are equal. `0 === -0`, but they aren't identical.
+    // See the [Harmony `egal` proposal](https://wiki.ecmascript.org/doku.php?id=harmony:egal).
+    if (a === b) return a !== 0 || 1 / a === 1 / b;
+    // `null` or `undefined` only equal to itself (strict comparison).
+    if (a == null || b == null) return false;
+    // `NaN`s are equivalent, but non-reflexive.
+    if (a !== a) return b !== b;
+    // Exhaust primitive checks
+    var type = typeof a;
+    if (type !== 'function' && type !== 'object' && typeof b != 'object') return false;
+    return deepEq(a, b, aStack, bStack);
+  }
+
+  // Internal recursive comparison function for `_.isEqual`.
+  function deepEq(a, b, aStack, bStack) {
+    // Unwrap any wrapped objects.
+    if (a instanceof _$1) a = a._wrapped;
+    if (b instanceof _$1) b = b._wrapped;
+    // Compare `[[Class]]` names.
+    var className = toString.call(a);
+    if (className !== toString.call(b)) return false;
+    // Work around a bug in IE 10 - Edge 13.
+    if (hasStringTagBug && className == '[object Object]' && isDataView$1(a)) {
+      if (!isDataView$1(b)) return false;
+      className = tagDataView;
+    }
+    switch (className) {
+      // These types are compared by value.
+      case '[object RegExp]':
+        // RegExps are coerced to strings for comparison (Note: '' + /a/i === '/a/i')
+      case '[object String]':
+        // Primitives and their corresponding object wrappers are equivalent; thus, `"5"` is
+        // equivalent to `new String("5")`.
+        return '' + a === '' + b;
+      case '[object Number]':
+        // `NaN`s are equivalent, but non-reflexive.
+        // Object(NaN) is equivalent to NaN.
+        if (+a !== +a) return +b !== +b;
+        // An `egal` comparison is performed for other numeric values.
+        return +a === 0 ? 1 / +a === 1 / b : +a === +b;
+      case '[object Date]':
+      case '[object Boolean]':
+        // Coerce dates and booleans to numeric primitive values. Dates are compared by their
+        // millisecond representations. Note that invalid dates with millisecond representations
+        // of `NaN` are not equivalent.
+        return +a === +b;
+      case '[object Symbol]':
+        return SymbolProto.valueOf.call(a) === SymbolProto.valueOf.call(b);
+      case '[object ArrayBuffer]':
+      case tagDataView:
+        // Coerce to typed array so we can fall through.
+        return deepEq(toBufferView(a), toBufferView(b), aStack, bStack);
+    }
+
+    var areArrays = className === '[object Array]';
+    if (!areArrays && isTypedArray$1(a)) {
+        var byteLength = getByteLength(a);
+        if (byteLength !== getByteLength(b)) return false;
+        if (a.buffer === b.buffer && a.byteOffset === b.byteOffset) return true;
+        areArrays = true;
+    }
+    if (!areArrays) {
+      if (typeof a != 'object' || typeof b != 'object') return false;
+
+      // Objects with different constructors are not equivalent, but `Object`s or `Array`s
+      // from different frames are.
+      var aCtor = a.constructor, bCtor = b.constructor;
+      if (aCtor !== bCtor && !(isFunction$1(aCtor) && aCtor instanceof aCtor &&
+                               isFunction$1(bCtor) && bCtor instanceof bCtor)
+                          && ('constructor' in a && 'constructor' in b)) {
+        return false;
+      }
+    }
+    // Assume equality for cyclic structures. The algorithm for detecting cyclic
+    // structures is adapted from ES 5.1 section 15.12.3, abstract operation `JO`.
+
+    // Initializing stack of traversed objects.
+    // It's done here since we only need them for objects and arrays comparison.
+    aStack = aStack || [];
+    bStack = bStack || [];
+    var length = aStack.length;
+    while (length--) {
+      // Linear search. Performance is inversely proportional to the number of
+      // unique nested structures.
+      if (aStack[length] === a) return bStack[length] === b;
+    }
+
+    // Add the first object to the stack of traversed objects.
+    aStack.push(a);
+    bStack.push(b);
+
+    // Recursively compare objects and arrays.
+    if (areArrays) {
+      // Compare array lengths to determine if a deep comparison is necessary.
+      length = a.length;
+      if (length !== b.length) return false;
+      // Deep compare the contents, ignoring non-numeric properties.
+      while (length--) {
+        if (!eq(a[length], b[length], aStack, bStack)) return false;
+      }
+    } else {
+      // Deep compare objects.
+      var _keys = keys(a), key;
+      length = _keys.length;
+      // Ensure that both objects contain the same number of properties before comparing deep equality.
+      if (keys(b).length !== length) return false;
+      while (length--) {
+        // Deep compare each member
+        key = _keys[length];
+        if (!(has$1(b, key) && eq(a[key], b[key], aStack, bStack))) return false;
+      }
+    }
+    // Remove the first object from the stack of traversed objects.
+    aStack.pop();
+    bStack.pop();
+    return true;
+  }
+
+  // Perform a deep comparison to check if two objects are equal.
+  function isEqual(a, b) {
+    return eq(a, b);
+  }
+
+  // Retrieve all the enumerable property names of an object.
+  function allKeys(obj) {
+    if (!isObject(obj)) return [];
+    var keys = [];
+    for (var key in obj) keys.push(key);
+    // Ahem, IE < 9.
+    if (hasEnumBug) collectNonEnumProps(obj, keys);
+    return keys;
+  }
+
+  // Since the regular `Object.prototype.toString` type tests don't work for
+  // some types in IE 11, we use a fingerprinting heuristic instead, based
+  // on the methods. It's not great, but it's the best we got.
+  // The fingerprint method lists are defined below.
+  function ie11fingerprint(methods) {
+    var length = getLength(methods);
+    return function(obj) {
+      if (obj == null) return false;
+      // `Map`, `WeakMap` and `Set` have no enumerable keys.
+      var keys = allKeys(obj);
+      if (getLength(keys)) return false;
+      for (var i = 0; i < length; i++) {
+        if (!isFunction$1(obj[methods[i]])) return false;
+      }
+      // If we are testing against `WeakMap`, we need to ensure that
+      // `obj` doesn't have a `forEach` method in order to distinguish
+      // it from a regular `Map`.
+      return methods !== weakMapMethods || !isFunction$1(obj[forEachName]);
+    };
+  }
+
+  // In the interest of compact minification, we write
+  // each string in the fingerprints only once.
+  var forEachName = 'forEach',
+      hasName = 'has',
+      commonInit = ['clear', 'delete'],
+      mapTail = ['get', hasName, 'set'];
+
+  // `Map`, `WeakMap` and `Set` each have slightly different
+  // combinations of the above sublists.
+  var mapMethods = commonInit.concat(forEachName, mapTail),
+      weakMapMethods = commonInit.concat(mapTail),
+      setMethods = ['add'].concat(commonInit, forEachName, hasName);
+
+  var isMap = isIE11 ? ie11fingerprint(mapMethods) : tagTester('Map');
+
+  var isWeakMap = isIE11 ? ie11fingerprint(weakMapMethods) : tagTester('WeakMap');
+
+  var isSet = isIE11 ? ie11fingerprint(setMethods) : tagTester('Set');
+
+  var isWeakSet = tagTester('WeakSet');
+
+  // Retrieve the values of an object's properties.
+  function values(obj) {
+    var _keys = keys(obj);
+    var length = _keys.length;
+    var values = Array(length);
+    for (var i = 0; i < length; i++) {
+      values[i] = obj[_keys[i]];
+    }
+    return values;
+  }
+
+  // Convert an object into a list of `[key, value]` pairs.
+  // The opposite of `_.object` with one argument.
+  function pairs(obj) {
+    var _keys = keys(obj);
+    var length = _keys.length;
+    var pairs = Array(length);
+    for (var i = 0; i < length; i++) {
+      pairs[i] = [_keys[i], obj[_keys[i]]];
+    }
+    return pairs;
+  }
+
+  // Invert the keys and values of an object. The values must be serializable.
+  function invert(obj) {
+    var result = {};
+    var _keys = keys(obj);
+    for (var i = 0, length = _keys.length; i < length; i++) {
+      result[obj[_keys[i]]] = _keys[i];
+    }
+    return result;
+  }
+
+  // Return a sorted list of the function names available on the object.
+  function functions(obj) {
+    var names = [];
+    for (var key in obj) {
+      if (isFunction$1(obj[key])) names.push(key);
+    }
+    return names.sort();
+  }
+
+  // An internal function for creating assigner functions.
+  function createAssigner(keysFunc, defaults) {
+    return function(obj) {
+      var length = arguments.length;
+      if (defaults) obj = Object(obj);
+      if (length < 2 || obj == null) return obj;
+      for (var index = 1; index < length; index++) {
+        var source = arguments[index],
+            keys = keysFunc(source),
+            l = keys.length;
+        for (var i = 0; i < l; i++) {
+          var key = keys[i];
+          if (!defaults || obj[key] === void 0) obj[key] = source[key];
+        }
+      }
+      return obj;
+    };
+  }
+
+  // Extend a given object with all the properties in passed-in object(s).
+  var extend = createAssigner(allKeys);
+
+  // Assigns a given object with all the own properties in the passed-in
+  // object(s).
+  // (https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Object/assign)
+  var extendOwn = createAssigner(keys);
+
+  // Fill in a given object with default properties.
+  var defaults = createAssigner(allKeys, true);
+
+  // Create a naked function reference for surrogate-prototype-swapping.
+  function ctor() {
+    return function(){};
+  }
+
+  // An internal function for creating a new object that inherits from another.
+  function baseCreate(prototype) {
+    if (!isObject(prototype)) return {};
+    if (nativeCreate) return nativeCreate(prototype);
+    var Ctor = ctor();
+    Ctor.prototype = prototype;
+    var result = new Ctor;
+    Ctor.prototype = null;
+    return result;
+  }
+
+  // Creates an object that inherits from the given prototype object.
+  // If additional properties are provided then they will be added to the
+  // created object.
+  function create(prototype, props) {
+    var result = baseCreate(prototype);
+    if (props) extendOwn(result, props);
+    return result;
+  }
+
+  // Create a (shallow-cloned) duplicate of an object.
+  function clone(obj) {
+    if (!isObject(obj)) return obj;
+    return isArray(obj) ? obj.slice() : extend({}, obj);
+  }
+
+  // Invokes `interceptor` with the `obj` and then returns `obj`.
+  // The primary purpose of this method is to "tap into" a method chain, in
+  // order to perform operations on intermediate results within the chain.
+  function tap(obj, interceptor) {
+    interceptor(obj);
+    return obj;
+  }
+
+  // Normalize a (deep) property `path` to array.
+  // Like `_.iteratee`, this function can be customized.
+  function toPath$1(path) {
+    return isArray(path) ? path : [path];
+  }
+  _$1.toPath = toPath$1;
+
+  // Internal wrapper for `_.toPath` to enable minification.
+  // Similar to `cb` for `_.iteratee`.
+  function toPath(path) {
+    return _$1.toPath(path);
+  }
+
+  // Internal function to obtain a nested property in `obj` along `path`.
+  function deepGet(obj, path) {
+    var length = path.length;
+    for (var i = 0; i < length; i++) {
+      if (obj == null) return void 0;
+      obj = obj[path[i]];
+    }
+    return length ? obj : void 0;
+  }
+
+  // Get the value of the (deep) property on `path` from `object`.
+  // If any property in `path` does not exist or if the value is
+  // `undefined`, return `defaultValue` instead.
+  // The `path` is normalized through `_.toPath`.
+  function get(object, path, defaultValue) {
+    var value = deepGet(object, toPath(path));
+    return isUndefined(value) ? defaultValue : value;
+  }
+
+  // Shortcut function for checking if an object has a given property directly on
+  // itself (in other words, not on a prototype). Unlike the internal `has`
+  // function, this public version can also traverse nested properties.
+  function has(obj, path) {
+    path = toPath(path);
+    var length = path.length;
+    for (var i = 0; i < length; i++) {
+      var key = path[i];
+      if (!has$1(obj, key)) return false;
+      obj = obj[key];
+    }
+    return !!length;
+  }
+
+  // Keep the identity function around for default iteratees.
+  function identity(value) {
+    return value;
+  }
+
+  // Returns a predicate for checking whether an object has a given set of
+  // `key:value` pairs.
+  function matcher(attrs) {
+    attrs = extendOwn({}, attrs);
+    return function(obj) {
+      return isMatch(obj, attrs);
+    };
+  }
+
+  // Creates a function that, when passed an object, will traverse that object’s
+  // properties down the given `path`, specified as an array of keys or indices.
+  function property(path) {
+    path = toPath(path);
+    return function(obj) {
+      return deepGet(obj, path);
+    };
+  }
+
+  // Internal function that returns an efficient (for current engines) version
+  // of the passed-in callback, to be repeatedly applied in other Underscore
+  // functions.
+  function optimizeCb(func, context, argCount) {
+    if (context === void 0) return func;
+    switch (argCount == null ? 3 : argCount) {
+      case 1: return function(value) {
+        return func.call(context, value);
+      };
+      // The 2-argument case is omitted because we’re not using it.
+      case 3: return function(value, index, collection) {
+        return func.call(context, value, index, collection);
+      };
+      case 4: return function(accumulator, value, index, collection) {
+        return func.call(context, accumulator, value, index, collection);
+      };
+    }
+    return function() {
+      return func.apply(context, arguments);
+    };
+  }
+
+  // An internal function to generate callbacks that can be applied to each
+  // element in a collection, returning the desired result — either `_.identity`,
+  // an arbitrary callback, a property matcher, or a property accessor.
+  function baseIteratee(value, context, argCount) {
+    if (value == null) return identity;
+    if (isFunction$1(value)) return optimizeCb(value, context, argCount);
+    if (isObject(value) && !isArray(value)) return matcher(value);
+    return property(value);
+  }
+
+  // External wrapper for our callback generator. Users may customize
+  // `_.iteratee` if they want additional predicate/iteratee shorthand styles.
+  // This abstraction hides the internal-only `argCount` argument.
+  function iteratee(value, context) {
+    return baseIteratee(value, context, Infinity);
+  }
+  _$1.iteratee = iteratee;
+
+  // The function we call internally to generate a callback. It invokes
+  // `_.iteratee` if overridden, otherwise `baseIteratee`.
+  function cb(value, context, argCount) {
+    if (_$1.iteratee !== iteratee) return _$1.iteratee(value, context);
+    return baseIteratee(value, context, argCount);
+  }
+
+  // Returns the results of applying the `iteratee` to each element of `obj`.
+  // In contrast to `_.map` it returns an object.
+  function mapObject(obj, iteratee, context) {
+    iteratee = cb(iteratee, context);
+    var _keys = keys(obj),
+        length = _keys.length,
+        results = {};
+    for (var index = 0; index < length; index++) {
+      var currentKey = _keys[index];
+      results[currentKey] = iteratee(obj[currentKey], currentKey, obj);
+    }
+    return results;
+  }
+
+  // Predicate-generating function. Often useful outside of Underscore.
+  function noop(){}
+
+  // Generates a function for a given object that returns a given property.
+  function propertyOf(obj) {
+    if (obj == null) return noop;
+    return function(path) {
+      return get(obj, path);
+    };
+  }
+
+  // Run a function **n** times.
+  function times(n, iteratee, context) {
+    var accum = Array(Math.max(0, n));
+    iteratee = optimizeCb(iteratee, context, 1);
+    for (var i = 0; i < n; i++) accum[i] = iteratee(i);
+    return accum;
+  }
+
+  // Return a random integer between `min` and `max` (inclusive).
+  function random(min, max) {
+    if (max == null) {
+      max = min;
+      min = 0;
+    }
+    return min + Math.floor(Math.random() * (max - min + 1));
+  }
+
+  // A (possibly faster) way to get the current timestamp as an integer.
+  var now = Date.now || function() {
+    return new Date().getTime();
+  };
+
+  // Internal helper to generate functions for escaping and unescaping strings
+  // to/from HTML interpolation.
+  function createEscaper(map) {
+    var escaper = function(match) {
+      return map[match];
+    };
+    // Regexes for identifying a key that needs to be escaped.
+    var source = '(?:' + keys(map).join('|') + ')';
+    var testRegexp = RegExp(source);
+    var replaceRegexp = RegExp(source, 'g');
+    return function(string) {
+      string = string == null ? '' : '' + string;
+      return testRegexp.test(string) ? string.replace(replaceRegexp, escaper) : string;
+    };
+  }
+
+  // Internal list of HTML entities for escaping.
+  var escapeMap = {
+    '&': '&amp;',
+    '<': '&lt;',
+    '>': '&gt;',
+    '"': '&quot;',
+    "'": '&#x27;',
+    '`': '&#x60;'
+  };
+
+  // Function for escaping strings to HTML interpolation.
+  var _escape = createEscaper(escapeMap);
+
+  // Internal list of HTML entities for unescaping.
+  var unescapeMap = invert(escapeMap);
+
+  // Function for unescaping strings from HTML interpolation.
+  var _unescape = createEscaper(unescapeMap);
+
+  // By default, Underscore uses ERB-style template delimiters. Change the
+  // following template settings to use alternative delimiters.
+  var templateSettings = _$1.templateSettings = {
+    evaluate: /<%([\s\S]+?)%>/g,
+    interpolate: /<%=([\s\S]+?)%>/g,
+    escape: /<%-([\s\S]+?)%>/g
+  };
+
+  // When customizing `_.templateSettings`, if you don't want to define an
+  // interpolation, evaluation or escaping regex, we need one that is
+  // guaranteed not to match.
+  var noMatch = /(.)^/;
+
+  // Certain characters need to be escaped so that they can be put into a
+  // string literal.
+  var escapes = {
+    "'": "'",
+    '\\': '\\',
+    '\r': 'r',
+    '\n': 'n',
+    '\u2028': 'u2028',
+    '\u2029': 'u2029'
+  };
+
+  var escapeRegExp = /\\|'|\r|\n|\u2028|\u2029/g;
+
+  function escapeChar(match) {
+    return '\\' + escapes[match];
+  }
+
+  // In order to prevent third-party code injection through
+  // `_.templateSettings.variable`, we test it against the following regular
+  // expression. It is intentionally a bit more liberal than just matching valid
+  // identifiers, but still prevents possible loopholes through defaults or
+  // destructuring assignment.
+  var bareIdentifier = /^\s*(\w|\$)+\s*$/;
+
+  // JavaScript micro-templating, similar to John Resig's implementation.
+  // Underscore templating handles arbitrary delimiters, preserves whitespace,
+  // and correctly escapes quotes within interpolated code.
+  // NB: `oldSettings` only exists for backwards compatibility.
+  function template(text, settings, oldSettings) {
+    if (!settings && oldSettings) settings = oldSettings;
+    settings = defaults({}, settings, _$1.templateSettings);
+
+    // Combine delimiters into one regular expression via alternation.
+    var matcher = RegExp([
+      (settings.escape || noMatch).source,
+      (settings.interpolate || noMatch).source,
+      (settings.evaluate || noMatch).source
+    ].join('|') + '|$', 'g');
+
+    // Compile the template source, escaping string literals appropriately.
+    var index = 0;
+    var source = "__p+='";
+    text.replace(matcher, function(match, escape, interpolate, evaluate, offset) {
+      source += text.slice(index, offset).replace(escapeRegExp, escapeChar);
+      index = offset + match.length;
+
+      if (escape) {
+        source += "'+\n((__t=(" + escape + "))==null?'':_.escape(__t))+\n'";
+      } else if (interpolate) {
+        source += "'+\n((__t=(" + interpolate + "))==null?'':__t)+\n'";
+      } else if (evaluate) {
+        source += "';\n" + evaluate + "\n__p+='";
+      }
+
+      // Adobe VMs need the match returned to produce the correct offset.
+      return match;
+    });
+    source += "';\n";
+
+    var argument = settings.variable;
+    if (argument) {
+      // Insure against third-party code injection. (CVE-2021-23358)
+      if (!bareIdentifier.test(argument)) throw new Error(
+        'variable is not a bare identifier: ' + argument
+      );
+    } else {
+      // If a variable is not specified, place data values in local scope.
+      source = 'with(obj||{}){\n' + source + '}\n';
+      argument = 'obj';
+    }
+
+    source = "var __t,__p='',__j=Array.prototype.join," +
+      "print=function(){__p+=__j.call(arguments,'');};\n" +
+      source + 'return __p;\n';
+
+    var render;
+    try {
+      render = new Function(argument, '_', source);
+    } catch (e) {
+      e.source = source;
+      throw e;
+    }
+
+    var template = function(data) {
+      return render.call(this, data, _$1);
+    };
+
+    // Provide the compiled source as a convenience for precompilation.
+    template.source = 'function(' + argument + '){\n' + source + '}';
+
+    return template;
+  }
+
+  // Traverses the children of `obj` along `path`. If a child is a function, it
+  // is invoked with its parent as context. Returns the value of the final
+  // child, or `fallback` if any child is undefined.
+  function result(obj, path, fallback) {
+    path = toPath(path);
+    var length = path.length;
+    if (!length) {
+      return isFunction$1(fallback) ? fallback.call(obj) : fallback;
+    }
+    for (var i = 0; i < length; i++) {
+      var prop = obj == null ? void 0 : obj[path[i]];
+      if (prop === void 0) {
+        prop = fallback;
+        i = length; // Ensure we don't continue iterating.
+      }
+      obj = isFunction$1(prop) ? prop.call(obj) : prop;
+    }
+    return obj;
+  }
+
+  // Generate a unique integer id (unique within the entire client session).
+  // Useful for temporary DOM ids.
+  var idCounter = 0;
+  function uniqueId(prefix) {
+    var id = ++idCounter + '';
+    return prefix ? prefix + id : id;
+  }
+
+  // Start chaining a wrapped Underscore object.
+  function chain(obj) {
+    var instance = _$1(obj);
+    instance._chain = true;
+    return instance;
+  }
+
+  // Internal function to execute `sourceFunc` bound to `context` with optional
+  // `args`. Determines whether to execute a function as a constructor or as a
+  // normal function.
+  function executeBound(sourceFunc, boundFunc, context, callingContext, args) {
+    if (!(callingContext instanceof boundFunc)) return sourceFunc.apply(context, args);
+    var self = baseCreate(sourceFunc.prototype);
+    var result = sourceFunc.apply(self, args);
+    if (isObject(result)) return result;
+    return self;
+  }
+
+  // Partially apply a function by creating a version that has had some of its
+  // arguments pre-filled, without changing its dynamic `this` context. `_` acts
+  // as a placeholder by default, allowing any combination of arguments to be
+  // pre-filled. Set `_.partial.placeholder` for a custom placeholder argument.
+  var partial = restArguments(function(func, boundArgs) {
+    var placeholder = partial.placeholder;
+    var bound = function() {
+      var position = 0, length = boundArgs.length;
+      var args = Array(length);
+      for (var i = 0; i < length; i++) {
+        args[i] = boundArgs[i] === placeholder ? arguments[position++] : boundArgs[i];
+      }
+      while (position < arguments.length) args.push(arguments[position++]);
+      return executeBound(func, bound, this, this, args);
+    };
+    return bound;
+  });
+
+  partial.placeholder = _$1;
+
+  // Create a function bound to a given object (assigning `this`, and arguments,
+  // optionally).
+  var bind = restArguments(function(func, context, args) {
+    if (!isFunction$1(func)) throw new TypeError('Bind must be called on a function');
+    var bound = restArguments(function(callArgs) {
+      return executeBound(func, bound, context, this, args.concat(callArgs));
+    });
+    return bound;
+  });
+
+  // Internal helper for collection methods to determine whether a collection
+  // should be iterated as an array or as an object.
+  // Related: https://people.mozilla.org/~jorendorff/es6-draft.html#sec-tolength
+  // Avoids a very nasty iOS 8 JIT bug on ARM-64. #2094
+  var isArrayLike = createSizePropertyCheck(getLength);
+
+  // Internal implementation of a recursive `flatten` function.
+  function flatten$1(input, depth, strict, output) {
+    output = output || [];
+    if (!depth && depth !== 0) {
+      depth = Infinity;
+    } else if (depth <= 0) {
+      return output.concat(input);
+    }
+    var idx = output.length;
+    for (var i = 0, length = getLength(input); i < length; i++) {
+      var value = input[i];
+      if (isArrayLike(value) && (isArray(value) || isArguments$1(value))) {
+        // Flatten current level of array or arguments object.
+        if (depth > 1) {
+          flatten$1(value, depth - 1, strict, output);
+          idx = output.length;
+        } else {
+          var j = 0, len = value.length;
+          while (j < len) output[idx++] = value[j++];
+        }
+      } else if (!strict) {
+        output[idx++] = value;
+      }
+    }
+    return output;
+  }
+
+  // Bind a number of an object's methods to that object. Remaining arguments
+  // are the method names to be bound. Useful for ensuring that all callbacks
+  // defined on an object belong to it.
+  var bindAll = restArguments(function(obj, keys) {
+    keys = flatten$1(keys, false, false);
+    var index = keys.length;
+    if (index < 1) throw new Error('bindAll must be passed function names');
+    while (index--) {
+      var key = keys[index];
+      obj[key] = bind(obj[key], obj);
+    }
+    return obj;
+  });
+
+  // Memoize an expensive function by storing its results.
+  function memoize(func, hasher) {
+    var memoize = function(key) {
+      var cache = memoize.cache;
+      var address = '' + (hasher ? hasher.apply(this, arguments) : key);
+      if (!has$1(cache, address)) cache[address] = func.apply(this, arguments);
+      return cache[address];
+    };
+    memoize.cache = {};
+    return memoize;
+  }
+
+  // Delays a function for the given number of milliseconds, and then calls
+  // it with the arguments supplied.
+  var delay = restArguments(function(func, wait, args) {
+    return setTimeout(function() {
+      return func.apply(null, args);
+    }, wait);
+  });
+
+  // Defers a function, scheduling it to run after the current call stack has
+  // cleared.
+  var defer = partial(delay, _$1, 1);
+
+  // Returns a function, that, when invoked, will only be triggered at most once
+  // during a given window of time. Normally, the throttled function will run
+  // as much as it can, without ever going more than once per `wait` duration;
+  // but if you'd like to disable the execution on the leading edge, pass
+  // `{leading: false}`. To disable execution on the trailing edge, ditto.
+  function throttle(func, wait, options) {
+    var timeout, context, args, result;
+    var previous = 0;
+    if (!options) options = {};
+
+    var later = function() {
+      previous = options.leading === false ? 0 : now();
+      timeout = null;
+      result = func.apply(context, args);
+      if (!timeout) context = args = null;
+    };
+
+    var throttled = function() {
+      var _now = now();
+      if (!previous && options.leading === false) previous = _now;
+      var remaining = wait - (_now - previous);
+      context = this;
+      args = arguments;
+      if (remaining <= 0 || remaining > wait) {
+        if (timeout) {
+          clearTimeout(timeout);
+          timeout = null;
+        }
+        previous = _now;
+        result = func.apply(context, args);
+        if (!timeout) context = args = null;
+      } else if (!timeout && options.trailing !== false) {
+        timeout = setTimeout(later, remaining);
+      }
+      return result;
+    };
+
+    throttled.cancel = function() {
+      clearTimeout(timeout);
+      previous = 0;
+      timeout = context = args = null;
+    };
+
+    return throttled;
+  }
+
+  // When a sequence of calls of the returned function ends, the argument
+  // function is triggered. The end of a sequence is defined by the `wait`
+  // parameter. If `immediate` is passed, the argument function will be
+  // triggered at the beginning of the sequence instead of at the end.
+  function debounce(func, wait, immediate) {
+    var timeout, previous, args, result, context;
+
+    var later = function() {
+      var passed = now() - previous;
+      if (wait > passed) {
+        timeout = setTimeout(later, wait - passed);
+      } else {
+        timeout = null;
+        if (!immediate) result = func.apply(context, args);
+        // This check is needed because `func` can recursively invoke `debounced`.
+        if (!timeout) args = context = null;
+      }
+    };
+
+    var debounced = restArguments(function(_args) {
+      context = this;
+      args = _args;
+      previous = now();
+      if (!timeout) {
+        timeout = setTimeout(later, wait);
+        if (immediate) result = func.apply(context, args);
+      }
+      return result;
+    });
+
+    debounced.cancel = function() {
+      clearTimeout(timeout);
+      timeout = args = context = null;
+    };
+
+    return debounced;
+  }
+
+  // Returns the first function passed as an argument to the second,
+  // allowing you to adjust arguments, run code before and after, and
+  // conditionally execute the original function.
+  function wrap(func, wrapper) {
+    return partial(wrapper, func);
+  }
+
+  // Returns a negated version of the passed-in predicate.
+  function negate(predicate) {
+    return function() {
+      return !predicate.apply(this, arguments);
+    };
+  }
+
+  // Returns a function that is the composition of a list of functions, each
+  // consuming the return value of the function that follows.
+  function compose() {
+    var args = arguments;
+    var start = args.length - 1;
+    return function() {
+      var i = start;
+      var result = args[start].apply(this, arguments);
+      while (i--) result = args[i].call(this, result);
+      return result;
+    };
+  }
+
+  // Returns a function that will only be executed on and after the Nth call.
+  function after(times, func) {
+    return function() {
+      if (--times < 1) {
+        return func.apply(this, arguments);
+      }
+    };
+  }
+
+  // Returns a function that will only be executed up to (but not including) the
+  // Nth call.
+  function before(times, func) {
+    var memo;
+    return function() {
+      if (--times > 0) {
+        memo = func.apply(this, arguments);
+      }
+      if (times <= 1) func = null;
+      return memo;
+    };
+  }
+
+  // Returns a function that will be executed at most one time, no matter how
+  // often you call it. Useful for lazy initialization.
+  var once = partial(before, 2);
+
+  // Returns the first key on an object that passes a truth test.
+  function findKey(obj, predicate, context) {
+    predicate = cb(predicate, context);
+    var _keys = keys(obj), key;
+    for (var i = 0, length = _keys.length; i < length; i++) {
+      key = _keys[i];
+      if (predicate(obj[key], key, obj)) return key;
+    }
+  }
+
+  // Internal function to generate `_.findIndex` and `_.findLastIndex`.
+  function createPredicateIndexFinder(dir) {
+    return function(array, predicate, context) {
+      predicate = cb(predicate, context);
+      var length = getLength(array);
+      var index = dir > 0 ? 0 : length - 1;
+      for (; index >= 0 && index < length; index += dir) {
+        if (predicate(array[index], index, array)) return index;
+      }
+      return -1;
+    };
+  }
+
+  // Returns the first index on an array-like that passes a truth test.
+  var findIndex = createPredicateIndexFinder(1);
+
+  // Returns the last index on an array-like that passes a truth test.
+  var findLastIndex = createPredicateIndexFinder(-1);
+
+  // Use a comparator function to figure out the smallest index at which
+  // an object should be inserted so as to maintain order. Uses binary search.
+  function sortedIndex(array, obj, iteratee, context) {
+    iteratee = cb(iteratee, context, 1);
+    var value = iteratee(obj);
+    var low = 0, high = getLength(array);
+    while (low < high) {
+      var mid = Math.floor((low + high) / 2);
+      if (iteratee(array[mid]) < value) low = mid + 1; else high = mid;
+    }
+    return low;
+  }
+
+  // Internal function to generate the `_.indexOf` and `_.lastIndexOf` functions.
+  function createIndexFinder(dir, predicateFind, sortedIndex) {
+    return function(array, item, idx) {
+      var i = 0, length = getLength(array);
+      if (typeof idx == 'number') {
+        if (dir > 0) {
+          i = idx >= 0 ? idx : Math.max(idx + length, i);
+        } else {
+          length = idx >= 0 ? Math.min(idx + 1, length) : idx + length + 1;
+        }
+      } else if (sortedIndex && idx && length) {
+        idx = sortedIndex(array, item);
+        return array[idx] === item ? idx : -1;
+      }
+      if (item !== item) {
+        idx = predicateFind(slice.call(array, i, length), isNaN$1);
+        return idx >= 0 ? idx + i : -1;
+      }
+      for (idx = dir > 0 ? i : length - 1; idx >= 0 && idx < length; idx += dir) {
+        if (array[idx] === item) return idx;
+      }
+      return -1;
+    };
+  }
+
+  // Return the position of the first occurrence of an item in an array,
+  // or -1 if the item is not included in the array.
+  // If the array is large and already in sort order, pass `true`
+  // for **isSorted** to use binary search.
+  var indexOf = createIndexFinder(1, findIndex, sortedIndex);
+
+  // Return the position of the last occurrence of an item in an array,
+  // or -1 if the item is not included in the array.
+  var lastIndexOf = createIndexFinder(-1, findLastIndex);
+
+  // Return the first value which passes a truth test.
+  function find(obj, predicate, context) {
+    var keyFinder = isArrayLike(obj) ? findIndex : findKey;
+    var key = keyFinder(obj, predicate, context);
+    if (key !== void 0 && key !== -1) return obj[key];
+  }
+
+  // Convenience version of a common use case of `_.find`: getting the first
+  // object containing specific `key:value` pairs.
+  function findWhere(obj, attrs) {
+    return find(obj, matcher(attrs));
+  }
+
+  // The cornerstone for collection functions, an `each`
+  // implementation, aka `forEach`.
+  // Handles raw objects in addition to array-likes. Treats all
+  // sparse array-likes as if they were dense.
+  function each(obj, iteratee, context) {
+    iteratee = optimizeCb(iteratee, context);
+    var i, length;
+    if (isArrayLike(obj)) {
+      for (i = 0, length = obj.length; i < length; i++) {
+        iteratee(obj[i], i, obj);
+      }
+    } else {
+      var _keys = keys(obj);
+      for (i = 0, length = _keys.length; i < length; i++) {
+        iteratee(obj[_keys[i]], _keys[i], obj);
+      }
+    }
+    return obj;
+  }
+
+  // Return the results of applying the iteratee to each element.
+  function map(obj, iteratee, context) {
+    iteratee = cb(iteratee, context);
+    var _keys = !isArrayLike(obj) && keys(obj),
+        length = (_keys || obj).length,
+        results = Array(length);
+    for (var index = 0; index < length; index++) {
+      var currentKey = _keys ? _keys[index] : index;
+      results[index] = iteratee(obj[currentKey], currentKey, obj);
+    }
+    return results;
+  }
+
+  // Internal helper to create a reducing function, iterating left or right.
+  function createReduce(dir) {
+    // Wrap code that reassigns argument variables in a separate function than
+    // the one that accesses `arguments.length` to avoid a perf hit. (#1991)
+    var reducer = function(obj, iteratee, memo, initial) {
+      var _keys = !isArrayLike(obj) && keys(obj),
+          length = (_keys || obj).length,
+          index = dir > 0 ? 0 : length - 1;
+      if (!initial) {
+        memo = obj[_keys ? _keys[index] : index];
+        index += dir;
+      }
+      for (; index >= 0 && index < length; index += dir) {
+        var currentKey = _keys ? _keys[index] : index;
+        memo = iteratee(memo, obj[currentKey], currentKey, obj);
+      }
+      return memo;
+    };
+
+    return function(obj, iteratee, memo, context) {
+      var initial = arguments.length >= 3;
+      return reducer(obj, optimizeCb(iteratee, context, 4), memo, initial);
+    };
+  }
+
+  // **Reduce** builds up a single result from a list of values, aka `inject`,
+  // or `foldl`.
+  var reduce = createReduce(1);
+
+  // The right-associative version of reduce, also known as `foldr`.
+  var reduceRight = createReduce(-1);
+
+  // Return all the elements that pass a truth test.
+  function filter(obj, predicate, context) {
+    var results = [];
+    predicate = cb(predicate, context);
+    each(obj, function(value, index, list) {
+      if (predicate(value, index, list)) results.push(value);
+    });
+    return results;
+  }
+
+  // Return all the elements for which a truth test fails.
+  function reject(obj, predicate, context) {
+    return filter(obj, negate(cb(predicate)), context);
+  }
+
+  // Determine whether all of the elements pass a truth test.
+  function every(obj, predicate, context) {
+    predicate = cb(predicate, context);
+    var _keys = !isArrayLike(obj) && keys(obj),
+        length = (_keys || obj).length;
+    for (var index = 0; index < length; index++) {
+      var currentKey = _keys ? _keys[index] : index;
+      if (!predicate(obj[currentKey], currentKey, obj)) return false;
+    }
+    return true;
+  }
+
+  // Determine if at least one element in the object passes a truth test.
+  function some(obj, predicate, context) {
+    predicate = cb(predicate, context);
+    var _keys = !isArrayLike(obj) && keys(obj),
+        length = (_keys || obj).length;
+    for (var index = 0; index < length; index++) {
+      var currentKey = _keys ? _keys[index] : index;
+      if (predicate(obj[currentKey], currentKey, obj)) return true;
+    }
+    return false;
+  }
+
+  // Determine if the array or object contains a given item (using `===`).
+  function contains(obj, item, fromIndex, guard) {
+    if (!isArrayLike(obj)) obj = values(obj);
+    if (typeof fromIndex != 'number' || guard) fromIndex = 0;
+    return indexOf(obj, item, fromIndex) >= 0;
+  }
+
+  // Invoke a method (with arguments) on every item in a collection.
+  var invoke = restArguments(function(obj, path, args) {
+    var contextPath, func;
+    if (isFunction$1(path)) {
+      func = path;
+    } else {
+      path = toPath(path);
+      contextPath = path.slice(0, -1);
+      path = path[path.length - 1];
+    }
+    return map(obj, function(context) {
+      var method = func;
+      if (!method) {
+        if (contextPath && contextPath.length) {
+          context = deepGet(context, contextPath);
+        }
+        if (context == null) return void 0;
+        method = context[path];
+      }
+      return method == null ? method : method.apply(context, args);
+    });
+  });
+
+  // Convenience version of a common use case of `_.map`: fetching a property.
+  function pluck(obj, key) {
+    return map(obj, property(key));
+  }
+
+  // Convenience version of a common use case of `_.filter`: selecting only
+  // objects containing specific `key:value` pairs.
+  function where(obj, attrs) {
+    return filter(obj, matcher(attrs));
+  }
+
+  // Return the maximum element (or element-based computation).
+  function max(obj, iteratee, context) {
+    var result = -Infinity, lastComputed = -Infinity,
+        value, computed;
+    if (iteratee == null || typeof iteratee == 'number' && typeof obj[0] != 'object' && obj != null) {
+      obj = isArrayLike(obj) ? obj : values(obj);
+      for (var i = 0, length = obj.length; i < length; i++) {
+        value = obj[i];
+        if (value != null && value > result) {
+          result = value;
+        }
+      }
+    } else {
+      iteratee = cb(iteratee, context);
+      each(obj, function(v, index, list) {
+        computed = iteratee(v, index, list);
+        if (computed > lastComputed || computed === -Infinity && result === -Infinity) {
+          result = v;
+          lastComputed = computed;
+        }
+      });
+    }
+    return result;
+  }
+
+  // Return the minimum element (or element-based computation).
+  function min(obj, iteratee, context) {
+    var result = Infinity, lastComputed = Infinity,
+        value, computed;
+    if (iteratee == null || typeof iteratee == 'number' && typeof obj[0] != 'object' && obj != null) {
+      obj = isArrayLike(obj) ? obj : values(obj);
+      for (var i = 0, length = obj.length; i < length; i++) {
+        value = obj[i];
+        if (value != null && value < result) {
+          result = value;
+        }
+      }
+    } else {
+      iteratee = cb(iteratee, context);
+      each(obj, function(v, index, list) {
+        computed = iteratee(v, index, list);
+        if (computed < lastComputed || computed === Infinity && result === Infinity) {
+          result = v;
+          lastComputed = computed;
+        }
+      });
+    }
+    return result;
+  }
+
+  // Sample **n** random values from a collection using the modern version of the
+  // [Fisher-Yates shuffle](https://en.wikipedia.org/wiki/Fisher–Yates_shuffle).
+  // If **n** is not specified, returns a single random element.
+  // The internal `guard` argument allows it to work with `_.map`.
+  function sample(obj, n, guard) {
+    if (n == null || guard) {
+      if (!isArrayLike(obj)) obj = values(obj);
+      return obj[random(obj.length - 1)];
+    }
+    var sample = isArrayLike(obj) ? clone(obj) : values(obj);
+    var length = getLength(sample);
+    n = Math.max(Math.min(n, length), 0);
+    var last = length - 1;
+    for (var index = 0; index < n; index++) {
+      var rand = random(index, last);
+      var temp = sample[index];
+      sample[index] = sample[rand];
+      sample[rand] = temp;
+    }
+    return sample.slice(0, n);
+  }
+
+  // Shuffle a collection.
+  function shuffle(obj) {
+    return sample(obj, Infinity);
+  }
+
+  // Sort the object's values by a criterion produced by an iteratee.
+  function sortBy(obj, iteratee, context) {
+    var index = 0;
+    iteratee = cb(iteratee, context);
+    return pluck(map(obj, function(value, key, list) {
+      return {
+        value: value,
+        index: index++,
+        criteria: iteratee(value, key, list)
+      };
+    }).sort(function(left, right) {
+      var a = left.criteria;
+      var b = right.criteria;
+      if (a !== b) {
+        if (a > b || a === void 0) return 1;
+        if (a < b || b === void 0) return -1;
+      }
+      return left.index - right.index;
+    }), 'value');
+  }
+
+  // An internal function used for aggregate "group by" operations.
+  function group(behavior, partition) {
+    return function(obj, iteratee, context) {
+      var result = partition ? [[], []] : {};
+      iteratee = cb(iteratee, context);
+      each(obj, function(value, index) {
+        var key = iteratee(value, index, obj);
+        behavior(result, value, key);
+      });
+      return result;
+    };
+  }
+
+  // Groups the object's values by a criterion. Pass either a string attribute
+  // to group by, or a function that returns the criterion.
+  var groupBy = group(function(result, value, key) {
+    if (has$1(result, key)) result[key].push(value); else result[key] = [value];
+  });
+
+  // Indexes the object's values by a criterion, similar to `_.groupBy`, but for
+  // when you know that your index values will be unique.
+  var indexBy = group(function(result, value, key) {
+    result[key] = value;
+  });
+
+  // Counts instances of an object that group by a certain criterion. Pass
+  // either a string attribute to count by, or a function that returns the
+  // criterion.
+  var countBy = group(function(result, value, key) {
+    if (has$1(result, key)) result[key]++; else result[key] = 1;
+  });
+
+  // Split a collection into two arrays: one whose elements all pass the given
+  // truth test, and one whose elements all do not pass the truth test.
+  var partition = group(function(result, value, pass) {
+    result[pass ? 0 : 1].push(value);
+  }, true);
+
+  // Safely create a real, live array from anything iterable.
+  var reStrSymbol = /[^\ud800-\udfff]|[\ud800-\udbff][\udc00-\udfff]|[\ud800-\udfff]/g;
+  function toArray(obj) {
+    if (!obj) return [];
+    if (isArray(obj)) return slice.call(obj);
+    if (isString(obj)) {
+      // Keep surrogate pair characters together.
+      return obj.match(reStrSymbol);
+    }
+    if (isArrayLike(obj)) return map(obj, identity);
+    return values(obj);
+  }
+
+  // Return the number of elements in a collection.
+  function size(obj) {
+    if (obj == null) return 0;
+    return isArrayLike(obj) ? obj.length : keys(obj).length;
+  }
+
+  // Internal `_.pick` helper function to determine whether `key` is an enumerable
+  // property name of `obj`.
+  function keyInObj(value, key, obj) {
+    return key in obj;
+  }
+
+  // Return a copy of the object only containing the allowed properties.
+  var pick = restArguments(function(obj, keys) {
+    var result = {}, iteratee = keys[0];
+    if (obj == null) return result;
+    if (isFunction$1(iteratee)) {
+      if (keys.length > 1) iteratee = optimizeCb(iteratee, keys[1]);
+      keys = allKeys(obj);
+    } else {
+      iteratee = keyInObj;
+      keys = flatten$1(keys, false, false);
+      obj = Object(obj);
+    }
+    for (var i = 0, length = keys.length; i < length; i++) {
+      var key = keys[i];
+      var value = obj[key];
+      if (iteratee(value, key, obj)) result[key] = value;
+    }
+    return result;
+  });
+
+  // Return a copy of the object without the disallowed properties.
+  var omit = restArguments(function(obj, keys) {
+    var iteratee = keys[0], context;
+    if (isFunction$1(iteratee)) {
+      iteratee = negate(iteratee);
+      if (keys.length > 1) context = keys[1];
+    } else {
+      keys = map(flatten$1(keys, false, false), String);
+      iteratee = function(value, key) {
+        return !contains(keys, key);
+      };
+    }
+    return pick(obj, iteratee, context);
+  });
+
+  // Returns everything but the last entry of the array. Especially useful on
+  // the arguments object. Passing **n** will return all the values in
+  // the array, excluding the last N.
+  function initial(array, n, guard) {
+    return slice.call(array, 0, Math.max(0, array.length - (n == null || guard ? 1 : n)));
+  }
+
+  // Get the first element of an array. Passing **n** will return the first N
+  // values in the array. The **guard** check allows it to work with `_.map`.
+  function first(array, n, guard) {
+    if (array == null || array.length < 1) return n == null || guard ? void 0 : [];
+    if (n == null || guard) return array[0];
+    return initial(array, array.length - n);
+  }
+
+  // Returns everything but the first entry of the `array`. Especially useful on
+  // the `arguments` object. Passing an **n** will return the rest N values in the
+  // `array`.
+  function rest(array, n, guard) {
+    return slice.call(array, n == null || guard ? 1 : n);
+  }
+
+  // Get the last element of an array. Passing **n** will return the last N
+  // values in the array.
+  function last(array, n, guard) {
+    if (array == null || array.length < 1) return n == null || guard ? void 0 : [];
+    if (n == null || guard) return array[array.length - 1];
+    return rest(array, Math.max(0, array.length - n));
+  }
+
+  // Trim out all falsy values from an array.
+  function compact(array) {
+    return filter(array, Boolean);
+  }
+
+  // Flatten out an array, either recursively (by default), or up to `depth`.
+  // Passing `true` or `false` as `depth` means `1` or `Infinity`, respectively.
+  function flatten(array, depth) {
+    return flatten$1(array, depth, false);
+  }
+
+  // Take the difference between one array and a number of other arrays.
+  // Only the elements present in just the first array will remain.
+  var difference = restArguments(function(array, rest) {
+    rest = flatten$1(rest, true, true);
+    return filter(array, function(value){
+      return !contains(rest, value);
+    });
+  });
+
+  // Return a version of the array that does not contain the specified value(s).
+  var without = restArguments(function(array, otherArrays) {
+    return difference(array, otherArrays);
+  });
+
+  // Produce a duplicate-free version of the array. If the array has already
+  // been sorted, you have the option of using a faster algorithm.
+  // The faster algorithm will not work with an iteratee if the iteratee
+  // is not a one-to-one function, so providing an iteratee will disable
+  // the faster algorithm.
+  function uniq(array, isSorted, iteratee, context) {
+    if (!isBoolean(isSorted)) {
+      context = iteratee;
+      iteratee = isSorted;
+      isSorted = false;
+    }
+    if (iteratee != null) iteratee = cb(iteratee, context);
+    var result = [];
+    var seen = [];
+    for (var i = 0, length = getLength(array); i < length; i++) {
+      var value = array[i],
+          computed = iteratee ? iteratee(value, i, array) : value;
+      if (isSorted && !iteratee) {
+        if (!i || seen !== computed) result.push(value);
+        seen = computed;
+      } else if (iteratee) {
+        if (!contains(seen, computed)) {
+          seen.push(computed);
+          result.push(value);
+        }
+      } else if (!contains(result, value)) {
+        result.push(value);
+      }
+    }
+    return result;
+  }
+
+  // Produce an array that contains the union: each distinct element from all of
+  // the passed-in arrays.
+  var union = restArguments(function(arrays) {
+    return uniq(flatten$1(arrays, true, true));
+  });
+
+  // Produce an array that contains every item shared between all the
+  // passed-in arrays.
+  function intersection(array) {
+    var result = [];
+    var argsLength = arguments.length;
+    for (var i = 0, length = getLength(array); i < length; i++) {
+      var item = array[i];
+      if (contains(result, item)) continue;
+      var j;
+      for (j = 1; j < argsLength; j++) {
+        if (!contains(arguments[j], item)) break;
+      }
+      if (j === argsLength) result.push(item);
+    }
+    return result;
+  }
+
+  // Complement of zip. Unzip accepts an array of arrays and groups
+  // each array's elements on shared indices.
+  function unzip(array) {
+    var length = array && max(array, getLength).length || 0;
+    var result = Array(length);
+
+    for (var index = 0; index < length; index++) {
+      result[index] = pluck(array, index);
+    }
+    return result;
+  }
+
+  // Zip together multiple lists into a single array -- elements that share
+  // an index go together.
+  var zip = restArguments(unzip);
+
+  // Converts lists into objects. Pass either a single array of `[key, value]`
+  // pairs, or two parallel arrays of the same length -- one of keys, and one of
+  // the corresponding values. Passing by pairs is the reverse of `_.pairs`.
+  function object(list, values) {
+    var result = {};
+    for (var i = 0, length = getLength(list); i < length; i++) {
+      if (values) {
+        result[list[i]] = values[i];
+      } else {
+        result[list[i][0]] = list[i][1];
+      }
+    }
+    return result;
+  }
+
+  // Generate an integer Array containing an arithmetic progression. A port of
+  // the native Python `range()` function. See
+  // [the Python documentation](https://docs.python.org/library/functions.html#range).
+  function range(start, stop, step) {
+    if (stop == null) {
+      stop = start || 0;
+      start = 0;
+    }
+    if (!step) {
+      step = stop < start ? -1 : 1;
+    }
+
+    var length = Math.max(Math.ceil((stop - start) / step), 0);
+    var range = Array(length);
+
+    for (var idx = 0; idx < length; idx++, start += step) {
+      range[idx] = start;
+    }
+
+    return range;
+  }
+
+  // Chunk a single array into multiple arrays, each containing `count` or fewer
+  // items.
+  function chunk(array, count) {
+    if (count == null || count < 1) return [];
+    var result = [];
+    var i = 0, length = array.length;
+    while (i < length) {
+      result.push(slice.call(array, i, i += count));
+    }
+    return result;
+  }
+
+  // Helper function to continue chaining intermediate results.
+  function chainResult(instance, obj) {
+    return instance._chain ? _$1(obj).chain() : obj;
+  }
+
+  // Add your own custom functions to the Underscore object.
+  function mixin(obj) {
+    each(functions(obj), function(name) {
+      var func = _$1[name] = obj[name];
+      _$1.prototype[name] = function() {
+        var args = [this._wrapped];
+        push.apply(args, arguments);
+        return chainResult(this, func.apply(_$1, args));
+      };
+    });
+    return _$1;
+  }
+
+  // Add all mutator `Array` functions to the wrapper.
+  each(['pop', 'push', 'reverse', 'shift', 'sort', 'splice', 'unshift'], function(name) {
+    var method = ArrayProto[name];
+    _$1.prototype[name] = function() {
+      var obj = this._wrapped;
+      if (obj != null) {
+        method.apply(obj, arguments);
+        if ((name === 'shift' || name === 'splice') && obj.length === 0) {
+          delete obj[0];
+        }
+      }
+      return chainResult(this, obj);
+    };
+  });
+
+  // Add all accessor `Array` functions to the wrapper.
+  each(['concat', 'join', 'slice'], function(name) {
+    var method = ArrayProto[name];
+    _$1.prototype[name] = function() {
+      var obj = this._wrapped;
+      if (obj != null) obj = method.apply(obj, arguments);
+      return chainResult(this, obj);
+    };
+  });
+
+  // Named Exports
+
+  var allExports = {
+    __proto__: null,
+    VERSION: VERSION,
+    restArguments: restArguments,
+    isObject: isObject,
+    isNull: isNull,
+    isUndefined: isUndefined,
+    isBoolean: isBoolean,
+    isElement: isElement,
+    isString: isString,
+    isNumber: isNumber,
+    isDate: isDate,
+    isRegExp: isRegExp,
+    isError: isError,
+    isSymbol: isSymbol,
+    isArrayBuffer: isArrayBuffer,
+    isDataView: isDataView$1,
+    isArray: isArray,
+    isFunction: isFunction$1,
+    isArguments: isArguments$1,
+    isFinite: isFinite$1,
+    isNaN: isNaN$1,
+    isTypedArray: isTypedArray$1,
+    isEmpty: isEmpty,
+    isMatch: isMatch,
+    isEqual: isEqual,
+    isMap: isMap,
+    isWeakMap: isWeakMap,
+    isSet: isSet,
+    isWeakSet: isWeakSet,
+    keys: keys,
+    allKeys: allKeys,
+    values: values,
+    pairs: pairs,
+    invert: invert,
+    functions: functions,
+    methods: functions,
+    extend: extend,
+    extendOwn: extendOwn,
+    assign: extendOwn,
+    defaults: defaults,
+    create: create,
+    clone: clone,
+    tap: tap,
+    get: get,
+    has: has,
+    mapObject: mapObject,
+    identity: identity,
+    constant: constant,
+    noop: noop,
+    toPath: toPath$1,
+    property: property,
+    propertyOf: propertyOf,
+    matcher: matcher,
+    matches: matcher,
+    times: times,
+    random: random,
+    now: now,
+    escape: _escape,
+    unescape: _unescape,
+    templateSettings: templateSettings,
+    template: template,
+    result: result,
+    uniqueId: uniqueId,
+    chain: chain,
+    iteratee: iteratee,
+    partial: partial,
+    bind: bind,
+    bindAll: bindAll,
+    memoize: memoize,
+    delay: delay,
+    defer: defer,
+    throttle: throttle,
+    debounce: debounce,
+    wrap: wrap,
+    negate: negate,
+    compose: compose,
+    after: after,
+    before: before,
+    once: once,
+    findKey: findKey,
+    findIndex: findIndex,
+    findLastIndex: findLastIndex,
+    sortedIndex: sortedIndex,
+    indexOf: indexOf,
+    lastIndexOf: lastIndexOf,
+    find: find,
+    detect: find,
+    findWhere: findWhere,
+    each: each,
+    forEach: each,
+    map: map,
+    collect: map,
+    reduce: reduce,
+    foldl: reduce,
+    inject: reduce,
+    reduceRight: reduceRight,
+    foldr: reduceRight,
+    filter: filter,
+    select: filter,
+    reject: reject,
+    every: every,
+    all: every,
+    some: some,
+    any: some,
+    contains: contains,
+    includes: contains,
+    include: contains,
+    invoke: invoke,
+    pluck: pluck,
+    where: where,
+    max: max,
+    min: min,
+    shuffle: shuffle,
+    sample: sample,
+    sortBy: sortBy,
+    groupBy: groupBy,
+    indexBy: indexBy,
+    countBy: countBy,
+    partition: partition,
+    toArray: toArray,
+    size: size,
+    pick: pick,
+    omit: omit,
+    first: first,
+    head: first,
+    take: first,
+    initial: initial,
+    last: last,
+    rest: rest,
+    tail: rest,
+    drop: rest,
+    compact: compact,
+    flatten: flatten,
+    without: without,
+    uniq: uniq,
+    unique: uniq,
+    union: union,
+    intersection: intersection,
+    difference: difference,
+    unzip: unzip,
+    transpose: unzip,
+    zip: zip,
+    object: object,
+    range: range,
+    chunk: chunk,
+    mixin: mixin,
+    'default': _$1
+  };
+
+  // Default Export
+
+  // Add all of the Underscore functions to the wrapper object.
+  var _ = mixin(allExports);
+  // Legacy Node.js API.
+  _._ = _;
+
+  return _;
+
+})));
+//# sourceMappingURL=underscore-umd.js.map
diff --git a/docs/preview/pr/352/_static/underscore.js b/docs/preview/pr/352/_static/underscore.js
new file mode 100644
index 000000000..cf177d428
--- /dev/null
+++ b/docs/preview/pr/352/_static/underscore.js
@@ -0,0 +1,6 @@
+!function(n,r){"object"==typeof exports&&"undefined"!=typeof module?module.exports=r():"function"==typeof define&&define.amd?define("underscore",r):(n="undefined"!=typeof globalThis?globalThis:n||self,function(){var t=n._,e=n._=r();e.noConflict=function(){return n._=t,e}}())}(this,(function(){
+//     Underscore.js 1.13.1
+//     https://underscorejs.org
+//     (c) 2009-2021 Jeremy Ashkenas, Julian Gonggrijp, and DocumentCloud and Investigative Reporters & Editors
+//     Underscore may be freely distributed under the MIT license.
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+             <article itemprop="articleBody" id="pytorch-article" class="pytorch-article">
+              
+  <section id="convert">
+<h1>Convert<a class="headerlink" href="#convert" title="Permalink to this heading">¶</a></h1>
+<table class="autosummary longtable docutils align-default">
+<tbody>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.mat2SO3/#pypose.mat2SO3" title="pypose.mat2SO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">mat2SO3</span></code></a></p></td>
+<td><p>Convert batched rotation or transformation matrices to SO3Type LieTensor.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.mat2SE3/#pypose.mat2SE3" title="pypose.mat2SE3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">mat2SE3</span></code></a></p></td>
+<td><p>Convert batched rotation or transformation matrices to SE3Type LieTensor.</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.mat2Sim3/#pypose.mat2Sim3" title="pypose.mat2Sim3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">mat2Sim3</span></code></a></p></td>
+<td><p>Convert batched rotation or transformation matrices to Sim3Type LieTensor.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.mat2RxSO3/#pypose.mat2RxSO3" title="pypose.mat2RxSO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">mat2RxSO3</span></code></a></p></td>
+<td><p>Convert batched rotation or transformation matrices to RxSO3Type LieTensor.</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.from_matrix/#pypose.from_matrix" title="pypose.from_matrix"><code class="xref py py-obj docutils literal notranslate"><span class="pre">from_matrix</span></code></a></p></td>
+<td><p>Convert batched rotation or transformation matrices to LieTensor.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.euler2SO3/#pypose.euler2SO3" title="pypose.euler2SO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">euler2SO3</span></code></a></p></td>
+<td><p>Convert batched Euler angles (roll, pitch, and yaw) to SO3Type LieTensor.</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.vec2skew/#pypose.vec2skew" title="pypose.vec2skew"><code class="xref py py-obj docutils literal notranslate"><span class="pre">vec2skew</span></code></a></p></td>
+<td><p>Convert batched vectors to skew matrices.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.tensor/#pypose.tensor" title="pypose.tensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">tensor</span></code></a></p></td>
+<td><p>Convert a <a class="reference internal" href="../generated/pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code></a> into a <code class="xref py py-obj docutils literal notranslate"><span class="pre">torch.Tensor</span></code> without changing data.</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.translation/#pypose.translation" title="pypose.translation"><code class="xref py py-obj docutils literal notranslate"><span class="pre">translation</span></code></a></p></td>
+<td><p>Extract the translation part from a <a class="reference internal" href="../generated/pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code></a>.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.rotation/#pypose.rotation" title="pypose.rotation"><code class="xref py py-obj docutils literal notranslate"><span class="pre">rotation</span></code></a></p></td>
+<td><p>Extract the rotation part from a <a class="reference internal" href="../generated/pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code></a>.</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.euler/#pypose.euler" title="pypose.euler"><code class="xref py py-obj docutils literal notranslate"><span class="pre">euler</span></code></a></p></td>
+<td><p>Convert batched LieTensor into Euler angles (roll, pitch, yaw).</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.scale/#pypose.scale" title="pypose.scale"><code class="xref py py-obj docutils literal notranslate"><span class="pre">scale</span></code></a></p></td>
+<td><p>Extract the scale part from a <a class="reference internal" href="../generated/pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code></a>.</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.matrix/#pypose.matrix" title="pypose.matrix"><code class="xref py py-obj docutils literal notranslate"><span class="pre">matrix</span></code></a></p></td>
+<td><p>Convert a <a class="reference internal" href="../generated/pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code></a> into matrix form.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.cart2homo/#pypose.cart2homo" title="pypose.cart2homo"><code class="xref py py-obj docutils literal notranslate"><span class="pre">cart2homo</span></code></a></p></td>
+<td><p>Converts batched Cartesian coordinates to Homogeneous coordinates by adding ones to last dimension.</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.homo2cart/#pypose.homo2cart" title="pypose.homo2cart"><code class="xref py py-obj docutils literal notranslate"><span class="pre">homo2cart</span></code></a></p></td>
+<td><p>Converts batched Homogeneous coordinates to Cartesian coordinates by dividing the last row.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.point2pixel/#pypose.point2pixel" title="pypose.point2pixel"><code class="xref py py-obj docutils literal notranslate"><span class="pre">point2pixel</span></code></a></p></td>
+<td><p>Project batched sets of points (either in camera or world frame) to pixels.</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.pixel2point/#pypose.pixel2point" title="pypose.pixel2point"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pixel2point</span></code></a></p></td>
+<td><p>Convert batch of pixels with depth into points (in camera coordinate)</p></td>
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+              <p class="caption" role="heading"><span class="caption-text">Contents:</span></p>
+<ul class="current">
+<li class="toctree-l1"><a class="reference internal" href="../lietensor/">LieTensor</a></li>
+<li class="toctree-l1 current"><a class="current reference internal" href="#">Func</a></li>
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+  <section id="func">
+<h1>Func<a class="headerlink" href="#func" title="Permalink to this heading">¶</a></h1>
+<table class="autosummary longtable docutils align-default">
+<tbody>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.func.jacrev/#pypose.func.jacrev" title="pypose.func.jacrev"><code class="xref py py-obj docutils literal notranslate"><span class="pre">func.jacrev</span></code></a></p></td>
+<td><p>This function provides the exact same functionality as <a class="reference external" href="https://pytorch.org/docs/stable/generated/torch.func.jacrev.html#torch.func.jacrev">torch.func.jacrev()</a>, except that it allows LieTensor to be used as input when calculating the jacobian.</p></td>
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+<ul class="current">
+<li class="toctree-l1"><a class="reference internal" href="../lietensor/">LieTensor</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../utils/">Utils</a></li>
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+              
+  <section id="functions">
+<h1>Functions<a class="headerlink" href="#functions" title="Permalink to this heading">¶</a></h1>
+<table class="autosummary longtable docutils align-default">
+<tbody>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.is_lietensor/#pypose.is_lietensor" title="pypose.is_lietensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">is_lietensor</span></code></a></p></td>
+<td><p>Check whether an instance or object is a LieTensor or not.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.is_SE3/#pypose.is_SE3" title="pypose.is_SE3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">is_SE3</span></code></a></p></td>
+<td><p>Check whether an instance or object is an SE3 Type LieTensor or not.</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.hasnan/#pypose.hasnan" title="pypose.hasnan"><code class="xref py py-obj docutils literal notranslate"><span class="pre">hasnan</span></code></a></p></td>
+<td><p>Checks whether a deep nested list of tensors contains Nan or not.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.add/#pypose.add" title="pypose.add"><code class="xref py py-obj docutils literal notranslate"><span class="pre">add</span></code></a></p></td>
+<td><p>Adds other, scaled by alpha, to input LieTensor.</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.add_/#pypose.add_" title="pypose.add_"><code class="xref py py-obj docutils literal notranslate"><span class="pre">add_</span></code></a></p></td>
+<td><p>Inplace version of <a class="reference internal" href="../generated/pypose.add/#pypose.add" title="pypose.add"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.add()</span></code></a>.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.mul/#pypose.mul" title="pypose.mul"><code class="xref py py-obj docutils literal notranslate"><span class="pre">mul</span></code></a></p></td>
+<td><p>Multiply input LieTensor by other.</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.bvv/#pypose.bvv" title="pypose.bvv"><code class="xref py py-obj docutils literal notranslate"><span class="pre">bvv</span></code></a></p></td>
+<td><p>Performs batched vector-vector product, which results in matrices.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.bmv/#pypose.bmv" title="pypose.bmv"><code class="xref py py-obj docutils literal notranslate"><span class="pre">bmv</span></code></a></p></td>
+<td><p>Performs batched matrix-vector product.</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.bvmv/#pypose.bvmv" title="pypose.bvmv"><code class="xref py py-obj docutils literal notranslate"><span class="pre">bvmv</span></code></a></p></td>
+<td><p>Performs batched vector-matrix-vector product.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.pm/#pypose.pm" title="pypose.pm"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pm</span></code></a></p></td>
+<td><p>Returns plus or minus (<span class="math">\(\pm\)</span>) states for tensor.</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.Exp/#pypose.Exp" title="pypose.Exp"><code class="xref py py-obj docutils literal notranslate"><span class="pre">Exp</span></code></a></p></td>
+<td><p>The Exponential map for <a class="reference internal" href="../generated/pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code></a> (Lie Algebra).</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.Log/#pypose.Log" title="pypose.Log"><code class="xref py py-obj docutils literal notranslate"><span class="pre">Log</span></code></a></p></td>
+<td><p>The Logarithm map for <a class="reference internal" href="../generated/pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code></a> (Lie Group).</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.Inv/#pypose.Inv" title="pypose.Inv"><code class="xref py py-obj docutils literal notranslate"><span class="pre">Inv</span></code></a></p></td>
+<td><p>The inverse of the input lieTensor.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.Retr/#pypose.Retr" title="pypose.Retr"><code class="xref py py-obj docutils literal notranslate"><span class="pre">Retr</span></code></a></p></td>
+<td><p>Perform batched retraction with a given direction.</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.Act/#pypose.Act" title="pypose.Act"><code class="xref py py-obj docutils literal notranslate"><span class="pre">Act</span></code></a></p></td>
+<td><p>Apply the batched transform to points in Euclidean or homogeneous coordinates.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.Adj/#pypose.Adj" title="pypose.Adj"><code class="xref py py-obj docutils literal notranslate"><span class="pre">Adj</span></code></a></p></td>
+<td><p>The dot product between the Adjoint matrix at the point given by an input (Lie Group) and the second point (Lie Algebra).</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.AdjT/#pypose.AdjT" title="pypose.AdjT"><code class="xref py py-obj docutils literal notranslate"><span class="pre">AdjT</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.Jinvp/#pypose.Jinvp" title="pypose.Jinvp"><code class="xref py py-obj docutils literal notranslate"><span class="pre">Jinvp</span></code></a></p></td>
+<td><p>The dot product between left Jacobian inverse at the point given by input (Lie Group) and second point (Lie Algebra).</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.Jr/#pypose.Jr" title="pypose.Jr"><code class="xref py py-obj docutils literal notranslate"><span class="pre">Jr</span></code></a></p></td>
+<td><p>The batched right Jacobian of a LieTensor.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.randn_so3/#pypose.randn_so3" title="pypose.randn_so3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">randn_so3</span></code></a></p></td>
+<td><p>Returns <code class="xref py py-obj docutils literal notranslate"><span class="pre">so3_type</span></code> LieTensor filled with random numbers.</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.randn_SO3/#pypose.randn_SO3" title="pypose.randn_SO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">randn_SO3</span></code></a></p></td>
+<td><p>Returns <code class="xref py py-obj docutils literal notranslate"><span class="pre">SO3_type</span></code> LieTensor filled with the Exponential map of the random <code class="xref py py-obj docutils literal notranslate"><span class="pre">so3_type</span></code> LieTensor.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.randn_se3/#pypose.randn_se3" title="pypose.randn_se3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">randn_se3</span></code></a></p></td>
+<td><p>Returns <code class="xref py py-obj docutils literal notranslate"><span class="pre">se3_type</span></code> LieTensor filled with random numbers.</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.randn_SE3/#pypose.randn_SE3" title="pypose.randn_SE3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">randn_SE3</span></code></a></p></td>
+<td><p>Returns <code class="xref py py-obj docutils literal notranslate"><span class="pre">SE3_type</span></code> LieTensor filled with the Exponential map of the random <code class="xref py py-obj docutils literal notranslate"><span class="pre">se3_type</span></code> LieTensor generated using <a class="reference internal" href="../generated/pypose.randn_se3/#pypose.randn_se3" title="pypose.randn_se3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.randn_se3()</span></code></a>.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.randn_sim3/#pypose.randn_sim3" title="pypose.randn_sim3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">randn_sim3</span></code></a></p></td>
+<td><p>Returns <code class="xref py py-obj docutils literal notranslate"><span class="pre">sim3_type</span></code> LieTensor filled with random numbers.</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.randn_Sim3/#pypose.randn_Sim3" title="pypose.randn_Sim3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">randn_Sim3</span></code></a></p></td>
+<td><p>Returns <code class="xref py py-obj docutils literal notranslate"><span class="pre">Sim3_type</span></code> LieTensor filled with the Exponential map of the random <code class="xref py py-obj docutils literal notranslate"><span class="pre">sim3_type</span></code> LieTensor generated using <a class="reference internal" href="../generated/pypose.randn_sim3/#pypose.randn_sim3" title="pypose.randn_sim3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">randn_sim3()</span></code></a>.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.randn_rxso3/#pypose.randn_rxso3" title="pypose.randn_rxso3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">randn_rxso3</span></code></a></p></td>
+<td><p>Returns <code class="xref py py-obj docutils literal notranslate"><span class="pre">rxso3_type</span></code> LieTensor filled with random numbers.</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.randn_RxSO3/#pypose.randn_RxSO3" title="pypose.randn_RxSO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">randn_RxSO3</span></code></a></p></td>
+<td><p>Returns <code class="xref py py-obj docutils literal notranslate"><span class="pre">RxSO3_type</span></code> LieTensor filled with the Exponential map of the random <code class="xref py py-obj docutils literal notranslate"><span class="pre">rxso3_type</span></code> LieTensor.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.randn_like/#pypose.randn_like" title="pypose.randn_like"><code class="xref py py-obj docutils literal notranslate"><span class="pre">randn_like</span></code></a></p></td>
+<td><p>Returns a LieTensor with the same size as input that is filled with random LieTensor with the corresponding <code class="xref py py-obj docutils literal notranslate"><span class="pre">input.ltype</span></code>.</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.identity_so3/#pypose.identity_so3" title="pypose.identity_so3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">identity_so3</span></code></a></p></td>
+<td><p>Returns identity <code class="xref py py-obj docutils literal notranslate"><span class="pre">so3_type</span></code> LieTensor with the given <code class="xref py py-obj docutils literal notranslate"><span class="pre">lsize</span></code>.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.identity_SO3/#pypose.identity_SO3" title="pypose.identity_SO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">identity_SO3</span></code></a></p></td>
+<td><p>Returns identity <code class="xref py py-obj docutils literal notranslate"><span class="pre">SO3_type</span></code> LieTensor with the given <code class="xref py py-obj docutils literal notranslate"><span class="pre">lsize</span></code>.</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.identity_se3/#pypose.identity_se3" title="pypose.identity_se3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">identity_se3</span></code></a></p></td>
+<td><p>Returns identity <code class="xref py py-obj docutils literal notranslate"><span class="pre">se3_type</span></code> LieTensor with the given <code class="xref py py-obj docutils literal notranslate"><span class="pre">lsize</span></code>.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.identity_SE3/#pypose.identity_SE3" title="pypose.identity_SE3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">identity_SE3</span></code></a></p></td>
+<td><p>Returns identity <code class="xref py py-obj docutils literal notranslate"><span class="pre">SE3_type</span></code> LieTensor with the given <code class="xref py py-obj docutils literal notranslate"><span class="pre">lsize</span></code>.</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.identity_sim3/#pypose.identity_sim3" title="pypose.identity_sim3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">identity_sim3</span></code></a></p></td>
+<td><p>Returns identity <code class="xref py py-obj docutils literal notranslate"><span class="pre">sim3_type</span></code> LieTensor with the given <code class="xref py py-obj docutils literal notranslate"><span class="pre">lsize</span></code>.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.identity_Sim3/#pypose.identity_Sim3" title="pypose.identity_Sim3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">identity_Sim3</span></code></a></p></td>
+<td><p>Returns identity <code class="xref py py-obj docutils literal notranslate"><span class="pre">Sim3_type</span></code> LieTensor with the given <code class="xref py py-obj docutils literal notranslate"><span class="pre">lsize</span></code>.</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.identity_rxso3/#pypose.identity_rxso3" title="pypose.identity_rxso3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">identity_rxso3</span></code></a></p></td>
+<td><p>Returns identity <code class="xref py py-obj docutils literal notranslate"><span class="pre">rxso3_type</span></code> LieTensor with the given <code class="xref py py-obj docutils literal notranslate"><span class="pre">lsize</span></code>.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.identity_RxSO3/#pypose.identity_RxSO3" title="pypose.identity_RxSO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">identity_RxSO3</span></code></a></p></td>
+<td><p>Returns identity <code class="xref py py-obj docutils literal notranslate"><span class="pre">RxSO3_type</span></code> LieTensor with the given <code class="xref py py-obj docutils literal notranslate"><span class="pre">lsize</span></code>.</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.identity_like/#pypose.identity_like" title="pypose.identity_like"><code class="xref py py-obj docutils literal notranslate"><span class="pre">identity_like</span></code></a></p></td>
+<td><p>Returns identity LieTensor with the same <code class="xref py py-obj docutils literal notranslate"><span class="pre">lsize</span></code> and <code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code> as the given LieTensor.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.cumprod/#pypose.cumprod" title="pypose.cumprod"><code class="xref py py-obj docutils literal notranslate"><span class="pre">cumprod</span></code></a></p></td>
+<td><p>Returns the cumulative product (<code class="docutils literal notranslate"><span class="pre">&#64;</span></code>) of LieTensor along a dimension.</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.cumprod_/#pypose.cumprod_" title="pypose.cumprod_"><code class="xref py py-obj docutils literal notranslate"><span class="pre">cumprod_</span></code></a></p></td>
+<td><p>Inplace version of <a class="reference internal" href="../generated/pypose.cumprod/#pypose.cumprod" title="pypose.cumprod"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.cumprod()</span></code></a></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.cummul/#pypose.cummul" title="pypose.cummul"><code class="xref py py-obj docutils literal notranslate"><span class="pre">cummul</span></code></a></p></td>
+<td><p>Returns the cumulative multiplication (*) of LieTensor along a dimension.</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.cummul_/#pypose.cummul_" title="pypose.cummul_"><code class="xref py py-obj docutils literal notranslate"><span class="pre">cummul_</span></code></a></p></td>
+<td><p>Inplace version of <a class="reference internal" href="../generated/pypose.cummul/#pypose.cummul" title="pypose.cummul"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.cummul()</span></code></a></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.cumops/#pypose.cumops" title="pypose.cumops"><code class="xref py py-obj docutils literal notranslate"><span class="pre">cumops</span></code></a></p></td>
+<td><p>Returns the cumulative user-defined operation of LieTensor along a dimension.</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.cumops_/#pypose.cumops_" title="pypose.cumops_"><code class="xref py py-obj docutils literal notranslate"><span class="pre">cumops_</span></code></a></p></td>
+<td><p>Inplace version of <a class="reference internal" href="../generated/pypose.cumops/#pypose.cumops" title="pypose.cumops"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.cumops()</span></code></a></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.reprojerr/#pypose.reprojerr" title="pypose.reprojerr"><code class="xref py py-obj docutils literal notranslate"><span class="pre">reprojerr</span></code></a></p></td>
+<td><p>Calculates batched per-pixel reprojection error (pixel distance) for points either in the camera or world frame given camera intrinsics or extrinsics, respectively.</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.knn/#pypose.knn" title="pypose.knn"><code class="xref py py-obj docutils literal notranslate"><span class="pre">knn</span></code></a></p></td>
+<td><p>Select the k nearest neighbor points of reference from neighbors in each batch.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.svdtf/#pypose.svdtf" title="pypose.svdtf"><code class="xref py py-obj docutils literal notranslate"><span class="pre">svdtf</span></code></a></p></td>
+<td><p>Computes the rigid transformation ( <span class="math">\(SE(3)\)</span> ) between two sets of associated point clouds (source and target) using Singular Value Decomposition (SVD).</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.chspline/#pypose.chspline" title="pypose.chspline"><code class="xref py py-obj docutils literal notranslate"><span class="pre">chspline</span></code></a></p></td>
+<td><p>Cubic Hermite Spline, a piecewise-cubic interpolator matching values and first derivatives.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.bspline/#pypose.bspline" title="pypose.bspline"><code class="xref py py-obj docutils literal notranslate"><span class="pre">bspline</span></code></a></p></td>
+<td><p>B-spline interpolation, which currently only supports SE3 LieTensor.</p></td>
+</tr>
+</tbody>
+</table>
+</section>
+
+
+             </article>
+             
+            </div>
+            <footer>
+  
+    <div class="rst-footer-buttons" role="navigation" aria-label="footer navigation">
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+  <section id="pypose-act">
+<h1>pypose.Act<a class="headerlink" href="#pypose-act" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.Act">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">Act</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">X</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">p</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/utils/#Act"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.Act" title="Permalink to this definition">¶</a></dt>
+<dd><p>Apply the batched transform to points in Euclidean or homogeneous coordinates.</p>
+<div class="math">
+\[y_i = X_i * p_i,
+
+\]</div>
+<p>where <span class="math">\(X\)</span> is the batched transform and <span class="math">\(p_i \in \mathbb{R^{*\times3}}\)</span> or
+<span class="math">\(p_i \in \mathbb{R^{*\times4}}\)</span> denotes the points to be transformed.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>X</strong> (<a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><em>LieTensor</em></a>) – the input LieTensor (Lie Group).</p></li>
+<li><p><strong>p</strong> (<em>Tensor</em>) – the points to be transformed.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>the transformed points in Euclidean or homogeneous coordinates.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>Tensor</p>
+</dd>
+</dl>
+<p class="rubric">Examples</p>
+<ul>
+<li><p><span class="math">\(\mathrm{Act}\)</span>: (<a class="reference internal" href="../pypose.SO3/#pypose.SO3" title="pypose.SO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">SO3</span></code></a>, <code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) <span class="math">\(\mapsto\)</span> <code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code></p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>     <span class="c1"># batch size 1, Euclidean coordinates</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">identity_SO3</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="c1"># batch size 2</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span><span class="o">.</span><span class="n">Act</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>               <span class="c1"># equivalent to: pp.Act(X, p)</span>
+<span class="go">tensor([[ 1.7576,  1.1503, -0.9920],</span>
+<span class="go">        [ 1.7576,  1.1503, -0.9920]])</span>
+</pre></div>
+</div>
+</li>
+<li><p><span class="math">\(\mathrm{Act}\)</span>: (<a class="reference internal" href="../pypose.SE3/#pypose.SE3" title="pypose.SE3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">SE3</span></code></a>, <code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) <span class="math">\(\mapsto\)</span> <code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code></p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mf">1.</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]])</span> <span class="c1"># batch size 2, homogeneous coordinates</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_SE3</span><span class="p">()</span>                               <span class="c1"># batch size 1</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span><span class="o">.</span><span class="n">Act</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>                                         <span class="c1"># apply same transform</span>
+<span class="go">tensor([[-0.5676, -0.0452, -0.0929,  1.0000],</span>
+<span class="go">        [-0.5676, -0.0452, -0.0929,  1.0000]])</span>
+</pre></div>
+</div>
+</li>
+<li><p><span class="math">\(\mathrm{Act}\)</span>: (<a class="reference internal" href="../pypose.Sim3/#pypose.Sim3" title="pypose.Sim3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">Sim3</span></code></a>, <code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) <span class="math">\(\mapsto\)</span> <code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code></p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mf">1.</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]])</span>  <span class="c1"># batch size 2</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_Sim3</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>                              <span class="c1"># batch size 2</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span><span class="o">.</span><span class="n">Act</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>                                          <span class="c1"># apply transform respectively.</span>
+<span class="go">tensor([[ 0.1551,  2.2930,  0.4531,  1.0000],</span>
+<span class="go">        [-0.6140, -1.1263,  2.7607,  1.0000]])</span>
+</pre></div>
+</div>
+</li>
+<li><p><span class="math">\(\mathrm{Act}\)</span>: (<a class="reference internal" href="../pypose.RxSO3/#pypose.RxSO3" title="pypose.RxSO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">RxSO3</span></code></a>, <code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) <span class="math">\(\mapsto\)</span> <code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code></p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mf">0.</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mf">0.</span><span class="p">]])</span>  <span class="c1"># batch size 2, Euclidean coordinates</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_RxSO3</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>                       <span class="c1"># batch size 2</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span><span class="o">.</span><span class="n">Act</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>                                    <span class="c1"># apply transform respectively.</span>
+<span class="go">tensor([[0., 0., 0.],</span>
+<span class="go">        [0., 0., 0.]])</span>
+</pre></div>
+</div>
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+              
+  <section id="pypose-adj">
+<h1>pypose.Adj<a class="headerlink" href="#pypose-adj" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.Adj">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">Adj</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">p</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/utils/#Adj"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.Adj" title="Permalink to this definition">¶</a></dt>
+<dd><p>The dot product between the Adjoint matrix at the point given by an input (Lie Group) and
+the second point (Lie Algebra).</p>
+<div class="math">
+\[\mathrm{Adj}: (\mathcal{G}, \mathcal{g}) \mapsto \mathcal{g}
+
+\]</div>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>input</strong> (<a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><em>LieTensor</em></a>) – the input LieTensor (Lie Group)</p></li>
+<li><p><strong>p</strong> (<a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><em>LieTensor</em></a>) – the second LieTensor (Lie Algebra)</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>the output LieTensor (Lie Algebra)</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor">LieTensor</a></p>
+</dd>
+</dl>
+<table class="docutils align-default" id="id1">
+<caption><span class="caption-text">List of supported <span class="math">\(\mathrm{Adj}\)</span> map</span><a class="headerlink" href="#id1" title="Permalink to this table">¶</a></caption>
+<colgroup>
+<col style="width: 23%" />
+<col style="width: 23%" />
+<col style="width: 9%" />
+<col style="width: 23%" />
+<col style="width: 23%" />
+</colgroup>
+<thead>
+<tr class="row-odd"><th class="head"><p>input <code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code></p></th>
+<th class="head"><p><span class="math">\((\mathcal{G}, \mathcal{g})\)</span> (Lie Group, Lie Algebra)</p></th>
+<th class="head"><p><span class="math">\(\mapsto\)</span></p></th>
+<th class="head"><p><span class="math">\(\mathcal{g}\)</span> (Lie Algebra)</p></th>
+<th class="head"><p>output <code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code></p></th>
+</tr>
+</thead>
+<tbody>
+<tr class="row-even"><td><p>(<code class="xref py py-obj docutils literal notranslate"><span class="pre">SO3_type</span></code>, <code class="xref py py-obj docutils literal notranslate"><span class="pre">so3_type</span></code>)</p></td>
+<td><p><span class="math">\((\mathcal{G}\in\mathbb{R}^{*\times4}, \mathcal{g}\in\mathbb{R}^{*\times3})\)</span></p></td>
+<td><p><span class="math">\(\mapsto\)</span></p></td>
+<td><p><span class="math">\(\mathcal{g}\in\mathbb{R}^{*\times3}\)</span></p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">so3_type</span></code></p></td>
+</tr>
+<tr class="row-odd"><td><p>(<code class="xref py py-obj docutils literal notranslate"><span class="pre">SE3_type</span></code>, <code class="xref py py-obj docutils literal notranslate"><span class="pre">se3_type</span></code>)</p></td>
+<td><p><span class="math">\((\mathcal{G}\in\mathbb{R}^{*\times7}, \mathcal{g}\in\mathbb{R}^{*\times6})\)</span></p></td>
+<td><p><span class="math">\(\mapsto\)</span></p></td>
+<td><p><span class="math">\(\mathcal{g}\in\mathbb{R}^{*\times6}\)</span></p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">se3_type</span></code></p></td>
+</tr>
+<tr class="row-even"><td><p>(<code class="xref py py-obj docutils literal notranslate"><span class="pre">Sim3_type</span></code>, <code class="xref py py-obj docutils literal notranslate"><span class="pre">sim3_type</span></code>)</p></td>
+<td><p><span class="math">\((\mathcal{G}\in\mathbb{R}^{*\times8}, \mathcal{g}\in\mathbb{R}^{*\times7})\)</span></p></td>
+<td><p><span class="math">\(\mapsto\)</span></p></td>
+<td><p><span class="math">\(\mathcal{g}\in\mathbb{R}^{*\times7}\)</span></p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">sim3_type</span></code></p></td>
+</tr>
+<tr class="row-odd"><td><p>(<code class="xref py py-obj docutils literal notranslate"><span class="pre">RxSO3_type</span></code>, <code class="xref py py-obj docutils literal notranslate"><span class="pre">rxso3_type</span></code>)</p></td>
+<td><p><span class="math">\((\mathcal{G}\in\mathbb{R}^{*\times5}, \mathcal{g}\in\mathbb{R}^{*\times4})\)</span></p></td>
+<td><p><span class="math">\(\mapsto\)</span></p></td>
+<td><p><span class="math">\(\mathcal{g}\in\mathbb{R}^{*\times4}\)</span></p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">rxso3_type</span></code></p></td>
+</tr>
+</tbody>
+</table>
+<p>Let the input be (<span class="math">\(\mathbf{x}\)</span>, <span class="math">\(\mathbf{p}\)</span>), <span class="math">\(\mathbf{y}\)</span> be the output.</p>
+<blockquote>
+<div><div class="math">
+\[\mathbf{y}_i = \mathrm{Adj}(\mathbf{x}_i)\mathbf{p}_i,
+
+\]</div>
+<p>where, <span class="math">\(\mathrm{Adj}(\mathbf{x}_i)\)</span> is the adjoint matrix of the Lie group of
+<span class="math">\(\mathbf{x}_i\)</span>.</p>
+</div></blockquote>
+<ul>
+<li><p>If input (<span class="math">\(\mathbf{x}\)</span>, <span class="math">\(\mathbf{p}\)</span>)’s <code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code> are <code class="xref py py-obj docutils literal notranslate"><span class="pre">SO3_type</span></code> and
+<code class="xref py py-obj docutils literal notranslate"><span class="pre">so3_type</span></code> (input <span class="math">\(\mathbf{x}\)</span> is an instance of <a class="reference internal" href="../pypose.SO3/#pypose.SO3" title="pypose.SO3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">SO3()</span></code></a>, <span class="math">\(\mathbf{p}\)</span>
+is an instance of <a class="reference internal" href="../pypose.so3/#pypose.so3" title="pypose.so3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">so3()</span></code></a>). Given <span class="math">\(\mathbf{x}_i \in\)</span> <span class="math">\(\textrm{SO(3)}\)</span>.
+The adjoint transformation is given by:</p>
+<blockquote>
+<div><div class="math">
+\[\mathrm{Adj}(\mathbf{x}_i) = \mathbf{x}_i
+
+\]</div>
+<p>In the case of <span class="math">\(\textrm{SO3}\)</span>, the adjoint transformation for an element is the same
+rotation matrix used to represent the element. Rotating a tangent vector by an element
+“moves” it from the tangent space on the right side of the element to the tangent space on
+the left.</p>
+</div></blockquote>
+</li>
+<li><p>If input (<span class="math">\(\mathbf{x}\)</span>, <span class="math">\(\mathbf{p}\)</span>)’s <code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code> are <code class="xref py py-obj docutils literal notranslate"><span class="pre">SE3_type</span></code>
+and <code class="xref py py-obj docutils literal notranslate"><span class="pre">se3_type</span></code> (input <span class="math">\(\mathbf{x}\)</span> is an instance of <a class="reference internal" href="../pypose.SE3/#pypose.SE3" title="pypose.SE3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">SE3()</span></code></a>,
+<span class="math">\(\mathbf{p}\)</span> is an instance of <a class="reference internal" href="../pypose.se3/#pypose.se3" title="pypose.se3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">se3()</span></code></a>). Let <span class="math">\(\mathbf{R}_i \in\)</span>
+<span class="math">\(\textrm{SO(3)}\)</span> and <span class="math">\(\mathbf{t}_i \in \mathbb{R}^{3\times3}\)</span> represent the
+rotation and translation part of the group. Let <span class="math">\(\mathbf{t}_{i\times}\)</span> be the skew
+matrix (<a class="reference internal" href="../pypose.vec2skew/#pypose.vec2skew" title="pypose.vec2skew"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.vec2skew()</span></code></a>) of <span class="math">\(\mathbf{t}_i\)</span>.
+The adjoint transformation is given by:</p>
+<blockquote>
+<div><div class="math">
+\[\mathrm{Adj}(\mathbf{x}_i) = \left[
+                    \begin{array}{cc}
+                        \mathbf{R}_i & \mathbf{t}_{i\times}\mathbf{R}_i \\
+                        0 & \mathbf{R}_i
+                    \end{array}
+                 \right] \in \mathbb{R}^{6\times6}
+
+\]</div>
+<p>where,</p>
+<div class="math">
+\[\mathbf{x}_i = \left[
+                    \begin{array}{cc}
+                        \mathbf{R}_i& \mathbf{t}_i \\
+                        0 & 1
+                    \end{array}
+                 \right] \in \mathrm{SE(3)}
+
+\]</div>
+</div></blockquote>
+</li>
+<li><p>If input (<span class="math">\(\mathbf{x}\)</span>, <span class="math">\(\mathbf{p}\)</span>)’s <code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code> are <code class="xref py py-obj docutils literal notranslate"><span class="pre">Sim3_type</span></code> and
+<code class="xref py py-obj docutils literal notranslate"><span class="pre">sim3_type</span></code> (input <span class="math">\(\mathbf{x}\)</span> is an instance of <a class="reference internal" href="../pypose.Sim3/#pypose.Sim3" title="pypose.Sim3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">Sim3()</span></code></a>, <span class="math">\(\mathbf{p}\)</span>
+is an instance of <a class="reference internal" href="../pypose.sim3/#pypose.sim3" title="pypose.sim3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">sim3()</span></code></a>). Let <span class="math">\(\mathbf{R}_i\in\)</span> <span class="math">\(\textrm{SO(3)}\)</span>,
+<span class="math">\(\mathbf{t}_i \in \mathbb{R}^{3\times3}\)</span>, and <span class="math">\(s_i \in \mathbb{R}^+\)</span> represent
+the rotation, translation, and scale parts of the group. Let <span class="math">\(\mathbf{t}_{i\times}\)</span>
+be the skew matrix (<a class="reference internal" href="../pypose.vec2skew/#pypose.vec2skew" title="pypose.vec2skew"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.vec2skew()</span></code></a>) of <span class="math">\(\mathbf{t}_i\)</span>. The adjoint
+transformation is given by:</p>
+<blockquote>
+<div><div class="math">
+\[\mathrm{Adj}(\mathbf{x}_i) = \left[
+            \begin{array}{cc}
+                s_i\mathbf{R}_i& \mathbf{t}_{i\times}\mathbf{R}_i& -\mathbf{t}_i \\
+                0 & \mathbf{R}_i& 0 \\
+                0 & 0 & 1
+            \end{array}
+            \right] \in \mathbb{R}^{7\times7}
+
+\]</div>
+<p>where,</p>
+<div class="math">
+\[\mathbf{x}_i = \left[
+                    \begin{array}{cc}
+                        s_i\mathbf{R}_i & \mathbf{t}_i \\
+                        0 & 1
+                    \end{array}
+                 \right] \in \textrm{Sim(3)}
+
+\]</div>
+</div></blockquote>
+</li>
+<li><p>If input (<span class="math">\(\mathbf{x}\)</span>, <span class="math">\(\mathbf{p}\)</span>)’s <code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code> are <code class="xref py py-obj docutils literal notranslate"><span class="pre">RxSO3_type</span></code>
+and <code class="xref py py-obj docutils literal notranslate"><span class="pre">rxso3_type</span></code> (input <span class="math">\(\mathbf{x}\)</span> is an instance of <a class="reference internal" href="../pypose.RxSO3/#pypose.RxSO3" title="pypose.RxSO3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">RxSO3()</span></code></a>,
+<span class="math">\(\mathbf{p}\)</span> is an instance of <a class="reference internal" href="../pypose.rxso3/#pypose.rxso3" title="pypose.rxso3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">rxso3()</span></code></a>). Let <span class="math">\(\mathbf{R}_i \in\)</span>
+<span class="math">\(\textrm{SO(3)}\)</span>, and <span class="math">\(s_i \in \mathbb{R}^+\)</span> represent the rotation and
+scale parts of the group. The adjoint transformation is given by:</p>
+<blockquote>
+<div><div class="math">
+\[\mathrm{Adj}(\mathbf{x}_i) = \left[
+                    \begin{array}{cc}
+                        \mathbf{R}_i & 0 \\
+                        0 & 1
+                    \end{array}
+                 \right] \in \mathbb{R}^{4\times4}
+
+\]</div>
+<p>where,</p>
+<div class="math">
+\[\mathbf{x}_i = \left[
+                    \begin{array}{cc}
+                        s_i\mathbf{R}_i & 0 \\
+                        0 & 1
+                    \end{array}
+                 \right] \in \mathrm{RxSO(3)}
+
+\]</div>
+<p>In the case of <span class="math">\(\textrm{RxSO3}\)</span> group, the adjoint transformation is the same as
+the rotation matrix of the group i.e. the <span class="math">\(\textrm{SO3}\)</span> part of the group.</p>
+</div></blockquote>
+</li>
+</ul>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The adjoint operator is a linear map which moves an element
+<span class="math">\(\mathbf{p} \in \mathcal{g}\)</span> in the right tangent space of
+<span class="math">\(\mathbf{x} \in \mathcal{G}\)</span> to the left tangent space.</p>
+<div class="math">
+\[\mathrm{Exp}(\mathrm{Adj}(\mathbf{x}) \mathbf{p}) * \mathbf{x}
+= \mathbf{x} * \mathrm{Exp}(\mathbf{p})
+
+\]</div>
+<p>It can be easily verified:</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">p</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_SO3</span><span class="p">(),</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_so3</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">torch</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">p</span><span class="o">.</span><span class="n">Exp</span><span class="p">(),</span> <span class="n">x</span><span class="o">.</span><span class="n">Adj</span><span class="p">(</span><span class="n">p</span><span class="p">)</span><span class="o">.</span><span class="n">Exp</span><span class="p">()</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
+<span class="go">True</span>
+</pre></div>
+</div>
+<p>One can refer to Eq. (8) of the following paper:</p>
+<ul class="simple">
+<li><p>Zachary Teed et al., <a class="reference external" href="https://arxiv.org/pdf/2103.12032.pdf">Tangent Space Backpropagation for 3D Transformation Groups</a>, IEEE/CVF Conference on Computer Vision
+and Pattern Recognition (CVPR), 2021.</p></li>
+</ul>
+</div>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p><span class="math">\(\mathrm{Adj}\)</span> is generally used to transform a tangent vector from the tangent
+space around one element to the tangent space of another.
+One can refer to this paper for more details:</p>
+<ul class="simple">
+<li><p>J. Sola et al., <a class="reference external" href="https://arxiv.org/abs/1812.01537">A micro Lie theory for state estimation in
+robotics</a>, arXiv preprint arXiv:1812.01537 (2018).</p></li>
+</ul>
+<p>The following thesis and the tutorial serve as a good reading material to learn more
+about deriving the adjoint matrices for different transformation groups.</p>
+<ul class="simple">
+<li><p>Strasdat, H., 2012. <a class="reference external" href="https://www.doc.ic.ac.uk/~ajd/Publications/Strasdat-H-2012-PhD-Thesis.pdf">Local accuracy and global consistency for efficient visual SLAM</a>,
+(Doctoral dissertation, Department of Computing, Imperial College London).</p></li>
+<li><p><a class="reference external" href="https://www.ethaneade.org/lie.pdf">Lie Groups for 2D and 3D Transformations.</a>, by Ethan Eade.</p></li>
+</ul>
+</div>
+<p class="rubric">Example</p>
+<ul>
+<li><p><span class="math">\(\mathrm{Adj}\)</span>: (<a class="reference internal" href="../pypose.SO3/#pypose.SO3" title="pypose.SO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">SO3</span></code></a>, <a class="reference internal" href="../pypose.so3/#pypose.so3" title="pypose.so3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">so3</span></code></a>) <span class="math">\(\mapsto\)</span> <a class="reference internal" href="../pypose.so3/#pypose.so3" title="pypose.so3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">so3</span></code></a></p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_SO3</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_so3</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">Adj</span><span class="p">(</span><span class="n">p</span><span class="p">)</span> <span class="c1"># equivalent to: pp.Adj(x, p)</span>
+<span class="go">so3Type LieTensor:</span>
+<span class="go">tensor([[-0.4171,  2.1218,  0.9951],</span>
+<span class="go">        [ 1.8415, -1.2185, -0.4082]])</span>
+</pre></div>
+</div>
+</li>
+<li><p><span class="math">\(\mathrm{Adj}\)</span>: (<a class="reference internal" href="../pypose.SE3/#pypose.SE3" title="pypose.SE3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">SE3</span></code></a>, <a class="reference internal" href="../pypose.se3/#pypose.se3" title="pypose.se3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">se3</span></code></a>) <span class="math">\(\mapsto\)</span> <a class="reference internal" href="../pypose.se3/#pypose.se3" title="pypose.se3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">se3</span></code></a></p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_SE3</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_se3</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">Adj</span><span class="p">(</span><span class="n">p</span><span class="p">)</span> <span class="c1"># equivalent to: pp.Adj(x, p)</span>
+<span class="go">se3Type LieTensor:</span>
+<span class="go">tensor([[-0.8536, -0.1984, -0.4554, -0.4868,  0.3231,  0.8535],</span>
+<span class="go">        [ 0.1577, -1.7625,  1.7997, -1.5085, -0.2098,  0.3538]])</span>
+</pre></div>
+</div>
+</li>
+<li><p><span class="math">\(\mathrm{Adj}\)</span>: (<a class="reference internal" href="../pypose.Sim3/#pypose.Sim3" title="pypose.Sim3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">Sim3</span></code></a>, <a class="reference internal" href="../pypose.sim3/#pypose.sim3" title="pypose.sim3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">sim3</span></code></a>) <span class="math">\(\mapsto\)</span> <a class="reference internal" href="../pypose.sim3/#pypose.sim3" title="pypose.sim3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">sim3</span></code></a></p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_Sim3</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_sim3</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">Adj</span><span class="p">(</span><span class="n">p</span><span class="p">)</span> <span class="c1"># equivalent to: pp.Adj(x, p)</span>
+<span class="go">sim3Type LieTensor:</span>
+<span class="go">tensor([[ 0.1455, -0.5653, -0.1845,  0.0502,  1.3125,  1.5217, -0.8964],</span>
+<span class="go">        [-4.8724, -0.5254,  3.9559,  1.5170,  1.7610,  0.4375,  0.4248]])</span>
+</pre></div>
+</div>
+</li>
+<li><p><span class="math">\(\mathrm{Adj}\)</span>: (<a class="reference internal" href="../pypose.RxSO3/#pypose.RxSO3" title="pypose.RxSO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">RxSO3</span></code></a>, <a class="reference internal" href="../pypose.rxso3/#pypose.rxso3" title="pypose.rxso3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">rxso3</span></code></a>) <span class="math">\(\mapsto\)</span> <a class="reference internal" href="../pypose.rxso3/#pypose.rxso3" title="pypose.rxso3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">rxso3</span></code></a></p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_RxSO3</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_rxso3</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">Adj</span><span class="p">(</span><span class="n">p</span><span class="p">)</span> <span class="c1"># equivalent to: pp.Adj(x, p)</span>
+<span class="go">rxso3Type LieTensor:</span>
+<span class="go">tensor([[-1.3590, -0.4314, -0.0297,  1.0166],</span>
+<span class="go">        [-0.3378, -0.4942, -2.0083, -0.4321]])</span>
+</pre></div>
+</div>
+</li>
+</ul>
+</dd></dl>
+
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+  <section id="pypose-adjt">
+<h1>pypose.AdjT<a class="headerlink" href="#pypose-adjt" title="Permalink to this heading">¶</a></h1>
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+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">AdjT</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">X</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">p</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/utils/#AdjT"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.AdjT" title="Permalink to this definition">¶</a></dt>
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+              
+  <section id="pypose-exp">
+<h1>pypose.Exp<a class="headerlink" href="#pypose-exp" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.Exp">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">Exp</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/utils/#Exp"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.Exp" title="Permalink to this definition">¶</a></dt>
+<dd><p>The Exponential map for <a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code></a> (Lie Algebra).</p>
+<div class="math">
+\[\mathrm{Exp}: \mathcal{g} \mapsto \mathcal{G}
+
+\]</div>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>input</strong> (<a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><em>LieTensor</em></a>) – the input LieTensor (Lie Algebra)</p>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>the output LieTensor (Lie Group)</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor">LieTensor</a></p>
+</dd>
+</dl>
+<table class="docutils align-default" id="id1">
+<caption><span class="caption-text">List of supported <span class="math">\(\mathrm{Exp}\)</span> map</span><a class="headerlink" href="#id1" title="Permalink to this table">¶</a></caption>
+<colgroup>
+<col style="width: 23%" />
+<col style="width: 23%" />
+<col style="width: 9%" />
+<col style="width: 23%" />
+<col style="width: 23%" />
+</colgroup>
+<thead>
+<tr class="row-odd"><th class="head"><p>input <code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code></p></th>
+<th class="head"><p><span class="math">\(\mathcal{g}\)</span> (Lie Algebra)</p></th>
+<th class="head"><p><span class="math">\(\mapsto\)</span></p></th>
+<th class="head"><p><span class="math">\(\mathcal{G}\)</span> (Lie Group)</p></th>
+<th class="head"><p>output <code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code></p></th>
+</tr>
+</thead>
+<tbody>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">so3_type</span></code></p></td>
+<td><p><span class="math">\(\mathcal{g}\in\mathbb{R}^{*\times3}\)</span></p></td>
+<td><p><span class="math">\(\mapsto\)</span></p></td>
+<td><p><span class="math">\(\mathcal{G}\in\mathbb{R}^{*\times4}\)</span></p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">SO3_type</span></code></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">se3_type</span></code></p></td>
+<td><p><span class="math">\(\mathcal{g}\in\mathbb{R}^{*\times6}\)</span></p></td>
+<td><p><span class="math">\(\mapsto\)</span></p></td>
+<td><p><span class="math">\(\mathcal{G}\in\mathbb{R}^{*\times7}\)</span></p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">SE3_type</span></code></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">sim3_type</span></code></p></td>
+<td><p><span class="math">\(\mathcal{g}\in\mathbb{R}^{*\times7}\)</span></p></td>
+<td><p><span class="math">\(\mapsto\)</span></p></td>
+<td><p><span class="math">\(\mathcal{G}\in\mathbb{R}^{*\times8}\)</span></p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Sim3_type</span></code></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">rxso3_type</span></code></p></td>
+<td><p><span class="math">\(\mathcal{g}\in\mathbb{R}^{*\times4}\)</span></p></td>
+<td><p><span class="math">\(\mapsto\)</span></p></td>
+<td><p><span class="math">\(\mathcal{G}\in\mathbb{R}^{*\times5}\)</span></p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">RxSO3_type</span></code></p></td>
+</tr>
+</tbody>
+</table>
+<div class="admonition warning">
+<p class="admonition-title">Warning</p>
+<p>This function <a class="reference internal" href="#pypose.Exp" title="pypose.Exp"><code class="xref py py-func docutils literal notranslate"><span class="pre">Exp()</span></code></a> is different from <code class="xref py py-func docutils literal notranslate"><span class="pre">exp()</span></code>, which returns
+a new torch tensor with the exponential of the elements of the input tensor.</p>
+</div>
+<ul>
+<li><p>Input <span class="math">\(\mathbf{x}\)</span>’s <code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code> is <code class="xref py py-obj docutils literal notranslate"><span class="pre">so3_type</span></code>
+(input <span class="math">\(\mathbf{x}\)</span> is an instance of <a class="reference internal" href="../pypose.so3/#pypose.so3" title="pypose.so3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">so3()</span></code></a>):</p>
+<blockquote>
+<div><p>If <span class="math">\(\|\mathbf{x}_i\| &gt; \text{eps}\)</span>:</p>
+<div class="math">
+\[\mathbf{y}_i = \left[\mathbf{x}_{i,1}\theta_i,
+\mathbf{x}_{i,2}\theta_i,
+\mathbf{x}_{i,3}\theta_i,
+\cos(\frac{\|\mathbf{x}_i\|}{2})\right],
+
+\]</div>
+<p>where <span class="math">\(\theta_i = \frac{1}{\|\mathbf{x}_i\|}\sin(\frac{\|\mathbf{x}_i\|}{2})\)</span>,</p>
+<p>otherwise:</p>
+<div class="math">
+\[\mathbf{y}_i = \left[\mathbf{x}_{i,1}\theta_i,~
+\mathbf{x}_{i,2}\theta_i,~
+\mathbf{x}_{i,3}\theta_i,~
+1 - \frac{\|\mathbf{x}_i\|^2}{8} + \frac{\|\mathbf{x}_i\|^4}{384} \right],
+
+\]</div>
+<p>where <span class="math">\(\theta_i = \frac{1}{2} - \frac{1}{48} \|\mathbf{x}_i\|^2
++ \frac{1}{3840} \|\mathbf{x}_i\|^4\)</span>.</p>
+</div></blockquote>
+</li>
+<li><p>Input <span class="math">\(\mathbf{x}\)</span>’s <code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code> is <code class="xref py py-obj docutils literal notranslate"><span class="pre">se3_type</span></code>
+(input <span class="math">\(\mathbf{x}\)</span> is an instance of <a class="reference internal" href="../pypose.se3/#pypose.se3" title="pypose.se3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">se3()</span></code></a>):</p>
+<blockquote>
+<div><p>Let <span class="math">\(\bm{\tau}_i\)</span>, <span class="math">\(\bm{\phi}_i\)</span> be the translation and rotation parts
+of <span class="math">\(\mathbf{x}_i\)</span>, respectively; <span class="math">\(\mathbf{y}\)</span> be the output.</p>
+<div class="math">
+\[\mathbf{y}_i = \left[\mathbf{J}_i\bm{\tau}_i, \mathrm{Exp}(\bm{\phi}_i)\right],
+
+\]</div>
+<p>where <span class="math">\(\mathrm{Exp}\)</span> is the Exponential map for <code class="xref py py-obj docutils literal notranslate"><span class="pre">so3_type</span></code> input and
+<span class="math">\(\mathbf{J}_i\)</span> is the left Jacobian for <code class="xref py py-obj docutils literal notranslate"><span class="pre">so3_type</span></code> input.</p>
+</div></blockquote>
+</li>
+<li><p>Input <span class="math">\(\mathbf{x}\)</span>’s <code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code> is <code class="xref py py-obj docutils literal notranslate"><span class="pre">rxso3_type</span></code>
+(input <span class="math">\(\mathbf{x}\)</span> is an instance of <a class="reference internal" href="../pypose.rxso3/#pypose.rxso3" title="pypose.rxso3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">rxso3()</span></code></a>):</p>
+<blockquote>
+<div><p>Let <span class="math">\(\bm{\phi}_i\)</span>, <span class="math">\(\sigma_i\)</span> be the rotation and scale parts of
+<span class="math">\(\mathbf{x}_i\)</span>, respectively; <span class="math">\(\mathbf{y}\)</span> be the output.</p>
+<div class="math">
+\[\mathbf{y}_i = \left[\mathrm{Exp}(\bm{\phi}_i), \mathrm{exp}(\sigma_i)\right],
+
+\]</div>
+<p>where <span class="math">\(\mathrm{exp}\)</span> is the exponential function.</p>
+</div></blockquote>
+</li>
+<li><p>Input <span class="math">\(\mathbf{x}\)</span>’s <code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code> is <code class="xref py py-obj docutils literal notranslate"><span class="pre">sim3_type</span></code>
+(input <span class="math">\(\mathbf{x}\)</span> is an instance of <a class="reference internal" href="../pypose.sim3/#pypose.sim3" title="pypose.sim3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">sim3()</span></code></a>):</p>
+<blockquote>
+<div><p>Let <span class="math">\(\bm{\tau}_i\)</span>, <span class="math">\(^{s}\bm{\phi}_i\)</span> be the translation and
+<a class="reference internal" href="../pypose.rxso3/#pypose.rxso3" title="pypose.rxso3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">rxso3()</span></code></a> parts of <span class="math">\(\mathbf{x}_i\)</span>, respectively.
+<span class="math">\(\bm{\phi}_i = \theta_i\mathbf{n}_i\)</span>, <span class="math">\(\sigma_i\)</span> be the rotation
+and scale parts of <span class="math">\(^{s}\bm{\phi}_i\)</span>, <span class="math">\(\boldsymbol{\Phi}_i\)</span> be the skew matrix
+of <span class="math">\(\bm{\phi}_i\)</span>; <span class="math">\(s_i = e^\sigma_i\)</span>, <span class="math">\(\mathbf{y}\)</span> be the output.</p>
+<div class="math">
+\[\mathbf{y}_i = \left[^{s}\mathbf{W}_i\bm{\tau}_i, \mathrm{Exp}(^{s}\bm{\phi}_i)\right],
+
+\]</div>
+<p>where</p>
+<div class="math">
+\[^s\mathbf{W}_i = A\boldsymbol{\Phi}_i + B\boldsymbol{\Phi}_i^2 + C\mathbf{I}
+
+\]</div>
+<p>in which if <span class="math">\(\|\sigma_i\| \geq \text{eps}\)</span>:</p>
+<div class="math">
+\[A = \left\{
+        \begin{array}{ll}
+            \frac{s_i\sin\theta_i\sigma_i + (1-s_i\cos\theta_i)\theta_i}
+            {\theta_i(\sigma_i^2 + \theta_i^2)}, \quad \|\theta_i\| \geq \text{eps}, \\
+            \frac{(\sigma_i-1)s_i+1}{\sigma_i^2}, \quad \|\theta_i\| < \text{eps},
+        \end{array}
+    \right.
+
+\]</div>
+<div class="math">
+\[B =
+\left\{
+    \begin{array}{ll}
+        \left( C - \frac{(s_i\cos\theta_i-1)\sigma+ s_i\sin\theta_i\sigma_i}
+        {\theta_i^2+\sigma_i^2}\right)\frac{1}{\theta_i^2}, \quad \|\theta_i\|
+            \geq \text{eps}, \\
+        \frac{s_i\sigma_i^2/2 + s_i-1-\sigma_i s_i}{\sigma_i^3}, \quad \|\theta_i\|
+            < \text{eps},
+    \end{array}
+\right.
+
+\]</div>
+<div class="math">
+\[C = \frac{e^{\sigma_i} - 1}{\sigma_i}\mathbf{I}
+
+\]</div>
+<p>otherwise:</p>
+<div class="math">
+\[A = \left\{
+        \begin{array}{ll}
+            \frac{1-\cos\theta_i}{\theta_i^2}, \quad \|\theta_i\| \geq \text{eps}, \\
+            \frac{1}{2}, \quad \|\theta_i\| < \text{eps},
+        \end{array}
+    \right.
+
+\]</div>
+<div class="math">
+\[B = \left\{
+        \begin{array}{ll}
+            \frac{\theta_i - \sin\theta_i}{\theta_i^3}, \quad \|\theta_i\|
+                \geq \text{eps}, \\
+            \frac{1}{6}, \quad \|\theta_i\| < \text{eps},
+        \end{array}
+    \right.
+
+\]</div>
+<div class="math">
+\[C = 1
+
+\]</div>
+</div></blockquote>
+</li>
+</ul>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The detailed explanation of the above <span class="math">\(\mathrm{Exp}\)</span>: calculation can be found
+in the paper:</p>
+<ul class="simple">
+<li><p>Grassia, F. Sebastian., <a class="reference external" href="https://www.tandfonline.com/doi/pdf/10.1080/10867651.1998.10487493">Practical Parameterization of Rotations using the
+Exponential Map</a>,
+Journal of graphics tools, 1998.</p></li>
+</ul>
+<p>Assume we have a unit rotation axis <span class="math">\(\mathbf{n}~(\|\mathbf{n}\|=1)\)</span> and rotation
+angle <span class="math">\(\theta~(0\leq\theta&lt;2\pi)\)</span>, let <span class="math">\(\mathbf{x}=\theta\mathbf{n}\)</span>, then
+the corresponding quaternion with unit norm <span class="math">\(\mathbf{q}\)</span> can be represented as:</p>
+<blockquote>
+<div><div class="math">
+\[\mathbf{q} = \left[\frac{\sin(\theta/2)}{\theta} \mathbf{x}, \cos(\theta/2) \right].
+
+\]</div>
+</div></blockquote>
+<p>Given <span class="math">\(\mathbf{x}=\theta\mathbf{n}\)</span>, to find its corresponding quaternion
+<span class="math">\(\mathbf{q}\)</span>, we first calculate the rotation angle <span class="math">\(\theta\)</span> using:</p>
+<blockquote>
+<div><div class="math">
+\[\theta = \|\mathbf{x}\|.
+
+\]</div>
+</div></blockquote>
+<p>Then, the corresponding quaternion is:</p>
+<blockquote>
+<div><div class="math">
+\[\mathbf{q} = \left[\frac{\sin(\|\mathbf{x}\|/2)}{\|\mathbf{x}\|} \mathbf{x},
+    \cos(\|\mathbf{x}\|/2) \right].
+
+\]</div>
+</div></blockquote>
+<p>If <span class="math">\(\|\mathbf{x}\|\)</span> is small (<span class="math">\(\|\mathbf{x}\|\le \text{eps}\)</span>),
+we use the Taylor Expansion form of <span class="math">\(\sin(\|\mathbf{x}\|/2)\)</span> and
+<span class="math">\(\cos(\|\mathbf{x}\|/2)\)</span>.</p>
+<p>More details about <span class="math">\(^s\mathbf{W}_i\)</span> in <code class="xref py py-obj docutils literal notranslate"><span class="pre">sim3_type</span></code> can be found in Eq. (5.7):</p>
+<ul class="simple">
+<li><p>H. Strasdat, <a class="reference external" href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.640.199&amp;rep=rep1&amp;type=pdf">Local Accuracy and Global Consistency for Efficient Visual SLAM</a>,
+Dissertation. Department of Computing, Imperial College London, 2012.</p></li>
+</ul>
+</div>
+<p class="rubric">Examples</p>
+<ul>
+<li><p><span class="math">\(\mathrm{Exp}\)</span>: <a class="reference internal" href="../pypose.so3/#pypose.so3" title="pypose.so3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">so3</span></code></a> <span class="math">\(\mapsto\)</span> <a class="reference internal" href="../pypose.SO3/#pypose.SO3" title="pypose.SO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">SO3</span></code></a></p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_so3</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">Exp</span><span class="p">()</span> <span class="c1"># equivalent to: pp.Exp(x)</span>
+<span class="go">SO3Type LieTensor:</span>
+<span class="go">LieTensor([-0.6627, -0.0447,  0.3492,  0.6610])</span>
+</pre></div>
+</div>
+</li>
+<li><p><span class="math">\(\mathrm{Exp}\)</span>: <a class="reference internal" href="../pypose.se3/#pypose.se3" title="pypose.se3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">se3</span></code></a> <span class="math">\(\mapsto\)</span> <a class="reference internal" href="../pypose.SE3/#pypose.SE3" title="pypose.SE3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">SE3</span></code></a></p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_se3</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">requires_grad</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+<span class="go">se3Type LieTensor:</span>
+<span class="go">tensor([[ 1.1912,  1.2425, -0.9696,  0.9540, -0.4061, -0.7204],</span>
+<span class="go">        [ 0.5964, -1.1894,  0.6451,  1.1373, -2.6733,  0.4142]])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">Exp</span><span class="p">()</span> <span class="c1"># equivalent to: pp.Exp(x)</span>
+<span class="go">SE3Type LieTensor:</span>
+<span class="go">tensor([[ 1.6575,  0.8838, -0.1499,  0.4459, -0.1898, -0.3367,  0.8073],</span>
+<span class="go">        [ 0.2654, -1.3860,  0.2852,  0.3855, -0.9061,  0.1404,  0.1034]],</span>
+<span class="go">        grad_fn=&lt;AliasBackward0&gt;)</span>
+</pre></div>
+</div>
+</li>
+<li><p><span class="math">\(\mathrm{Exp}\)</span>: <a class="reference internal" href="../pypose.rxso3/#pypose.rxso3" title="pypose.rxso3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">rxso3</span></code></a> <span class="math">\(\mapsto\)</span> <a class="reference internal" href="../pypose.RxSO3/#pypose.RxSO3" title="pypose.RxSO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">RxSO3</span></code></a></p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_rxso3</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">Exp</span><span class="p">()</span> <span class="c1"># equivalent to: pp.Exp(x)</span>
+<span class="go">RxSO3Type LieTensor:</span>
+<span class="go">tensor([[-0.5633, -0.4281,  0.1112,  0.6979,  0.7408],</span>
+<span class="go">        [ 0.5089,  0.2016, -0.2015,  0.8122,  1.1692]])</span>
+</pre></div>
+</div>
+</li>
+<li><p><span class="math">\(\mathrm{Exp}\)</span>: <a class="reference internal" href="../pypose.sim3/#pypose.sim3" title="pypose.sim3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">sim3</span></code></a> <span class="math">\(\mapsto\)</span> <a class="reference internal" href="../pypose.Sim3/#pypose.Sim3" title="pypose.Sim3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">Sim3</span></code></a></p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_sim3</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">Exp</span><span class="p">()</span> <span class="c1"># equivalent to: pp.Exp(x)</span>
+<span class="go">Sim3Type LieTensor:</span>
+<span class="go">tensor([[-1.5811,  1.8128, -0.5835,  0.5849,  0.1142, -0.3438,  0.7257,  2.4443],</span>
+<span class="go">        [ 0.9574, -0.9265, -0.2385, -0.7309, -0.3875,  0.1404,  0.5440,  1.1945]])</span>
+</pre></div>
+</div>
+</li>
+</ul>
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+             <article itemprop="articleBody" id="pytorch-article" class="pytorch-article">
+              
+  <section id="pypose-inv">
+<h1>pypose.Inv<a class="headerlink" href="#pypose-inv" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.Inv">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">Inv</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/utils/#Inv"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.Inv" title="Permalink to this definition">¶</a></dt>
+<dd><p>The inverse of the input lieTensor.</p>
+<div class="math">
+\[\mathrm{Inv}: Y_i = X_i^{-1}
+
+\]</div>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>input</strong> (<a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><em>LieTensor</em></a>) – the input LieTensor (Lie Group or Lie Algebra)</p>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>the output LieTensor (Lie Group or Lie Algebra)</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor">LieTensor</a></p>
+</dd>
+</dl>
+<table class="docutils align-default" id="id1">
+<caption><span class="caption-text">List of supported <span class="math">\(\mathrm{Inv}\)</span> map</span><a class="headerlink" href="#id1" title="Permalink to this table">¶</a></caption>
+<colgroup>
+<col style="width: 23%" />
+<col style="width: 23%" />
+<col style="width: 9%" />
+<col style="width: 23%" />
+<col style="width: 23%" />
+</colgroup>
+<thead>
+<tr class="row-odd"><th class="head"><p>input <code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code></p></th>
+<th class="head"><p>LieTensor</p></th>
+<th class="head"><p><span class="math">\(\mapsto\)</span></p></th>
+<th class="head"><p>LieTensor</p></th>
+<th class="head"><p>output <code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code></p></th>
+</tr>
+</thead>
+<tbody>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">SO3_type</span></code></p></td>
+<td><p><span class="math">\(\mathcal{G}\in\mathbb{R}^{*\times4}\)</span></p></td>
+<td><p><span class="math">\(\mapsto\)</span></p></td>
+<td><p><span class="math">\(\mathcal{G}\in\mathbb{R}^{*\times4}\)</span></p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">SO3_type</span></code></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">SE3_type</span></code></p></td>
+<td><p><span class="math">\(\mathcal{G}\in\mathbb{R}^{*\times7}\)</span></p></td>
+<td><p><span class="math">\(\mapsto\)</span></p></td>
+<td><p><span class="math">\(\mathcal{G}\in\mathbb{R}^{*\times7}\)</span></p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">SE3_type</span></code></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Sim3_type</span></code></p></td>
+<td><p><span class="math">\(\mathcal{G}\in\mathbb{R}^{*\times8}\)</span></p></td>
+<td><p><span class="math">\(\mapsto\)</span></p></td>
+<td><p><span class="math">\(\mathcal{G}\in\mathbb{R}^{*\times8}\)</span></p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Sim3_type</span></code></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">RxSO3_type</span></code></p></td>
+<td><p><span class="math">\(\mathcal{G}\in\mathbb{R}^{*\times5}\)</span></p></td>
+<td><p><span class="math">\(\mapsto\)</span></p></td>
+<td><p><span class="math">\(\mathcal{G}\in\mathbb{R}^{*\times5}\)</span></p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">RxSO3_type</span></code></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">so3_type</span></code></p></td>
+<td><p><span class="math">\(\mathcal{g}\in\mathbb{R}^{*\times3}\)</span></p></td>
+<td><p><span class="math">\(\mapsto\)</span></p></td>
+<td><p><span class="math">\(\mathcal{g}\in\mathbb{R}^{*\times3}\)</span></p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">so3_type</span></code></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">se3_type</span></code></p></td>
+<td><p><span class="math">\(\mathcal{g}\in\mathbb{R}^{*\times6}\)</span></p></td>
+<td><p><span class="math">\(\mapsto\)</span></p></td>
+<td><p><span class="math">\(\mathcal{g}\in\mathbb{R}^{*\times6}\)</span></p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">se3_type</span></code></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">sim3_type</span></code></p></td>
+<td><p><span class="math">\(\mathcal{g}\in\mathbb{R}^{*\times7}\)</span></p></td>
+<td><p><span class="math">\(\mapsto\)</span></p></td>
+<td><p><span class="math">\(\mathcal{g}\in\mathbb{R}^{*\times7}\)</span></p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">sim3_type</span></code></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">rxso3_type</span></code></p></td>
+<td><p><span class="math">\(\mathcal{g}\in\mathbb{R}^{*\times4}\)</span></p></td>
+<td><p><span class="math">\(\mapsto\)</span></p></td>
+<td><p><span class="math">\(\mathcal{g}\in\mathbb{R}^{*\times4}\)</span></p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">rxso3_type</span></code></p></td>
+</tr>
+</tbody>
+</table>
+<ul>
+<li><p>If input <span class="math">\(\mathbf{x}\)</span>’s <code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code> is <code class="xref py py-obj docutils literal notranslate"><span class="pre">SO3_type</span></code>
+(input <span class="math">\(\mathbf{x}\)</span> is an instance of <a class="reference internal" href="../pypose.SO3/#pypose.SO3" title="pypose.SO3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">SO3()</span></code></a>):</p>
+<blockquote>
+<div><p>Let <span class="math">\(w_i\)</span>, <span class="math">\(\boldsymbol{\nu}_i\)</span> be the scalar and vector parts of
+<span class="math">\(\mathbf{x}_i\)</span>, respectively. <span class="math">\(\mathbf{x}_i=\left[\boldsymbol{\nu}_i,
+\ w_i \right]\)</span>. <span class="math">\(\mathbf{y}\)</span> be the output.</p>
+<div class="math">
+\[\mathbf{y}_i = \mathrm{conj}(\mathbf{x}_i)=[-\boldsymbol{\nu}_i, \ w_i]
+
+\]</div>
+</div></blockquote>
+</li>
+<li><p>If input <span class="math">\(\mathbf{x}\)</span>’s <code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code> is <code class="xref py py-obj docutils literal notranslate"><span class="pre">SE3_type</span></code>
+(input <span class="math">\(\mathbf{x}\)</span> is an instance of <a class="reference internal" href="../pypose.SE3/#pypose.SE3" title="pypose.SE3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">SE3()</span></code></a>):</p>
+<blockquote>
+<div><p>Let <span class="math">\(\mathbf{t}_i\)</span>, <span class="math">\(\mathbf{q}_i\)</span> be the translation and rotation parts of
+<span class="math">\(\mathbf{x}_i\)</span>, respectively; <span class="math">\(\mathbf{y}\)</span> be the output.</p>
+<div class="math">
+\[\mathbf{y}_i = \left[-\mathrm{Inv}(\mathbf{q}_i)*\mathbf{t}_i,
+\mathrm{Inv}(\mathbf{q}_i) \right]
+
+\]</div>
+</div></blockquote>
+</li>
+<li><p>If input <span class="math">\(\mathbf{x}\)</span>’s <code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code> is <code class="xref py py-obj docutils literal notranslate"><span class="pre">RxSO3_type</span></code>
+(input <span class="math">\(\mathbf{x}\)</span> is an instance of <a class="reference internal" href="../pypose.RxSO3/#pypose.RxSO3" title="pypose.RxSO3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">RxSO3()</span></code></a>):</p>
+<blockquote>
+<div><p>Let <span class="math">\(\mathbf{q}_i\)</span>, <span class="math">\(s_i\)</span> be the rotation and scale parts of
+<span class="math">\(\mathbf{x}_i\)</span>, respectively; <span class="math">\(\mathbf{y}\)</span> be the output.</p>
+<div class="math">
+\[\mathbf{y}_i = \left[\mathrm{conj}(\mathbf{q}_i), \ 1/s_i \right]
+
+\]</div>
+</div></blockquote>
+</li>
+<li><p>If input <span class="math">\(\mathbf{x}\)</span>’s <code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code> is <code class="xref py py-obj docutils literal notranslate"><span class="pre">Sim3_type</span></code> (input <span class="math">\(\mathbf{x}\)</span>
+is an instance of <a class="reference internal" href="../pypose.Sim3/#pypose.Sim3" title="pypose.Sim3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">Sim3()</span></code></a>):</p>
+<blockquote>
+<div><p>Let <span class="math">\(\mathbf{t}_i\)</span>, <span class="math">\(^s\mathbf{q}_i\)</span> be the translation and <a class="reference internal" href="../pypose.RxSO3/#pypose.RxSO3" title="pypose.RxSO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">RxSO3</span></code></a> parts
+of <span class="math">\(\mathbf{x}_i\)</span>, respectively;  <span class="math">\(\mathbf{y}\)</span> be the output.</p>
+<div class="math">
+\[\mathbf{y}_i = \left[-\mathrm{Inv}(^s\mathbf{q}_i)*\mathbf{t}_i,
+\mathrm{Inv}(^s\mathbf{q}_i) \right]
+
+\]</div>
+</div></blockquote>
+</li>
+<li><p>If input <span class="math">\(\mathbf{x}\)</span>’s <code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code> is <code class="xref py py-obj docutils literal notranslate"><span class="pre">so3_type</span></code> or <code class="xref py py-obj docutils literal notranslate"><span class="pre">se3_type</span></code> or
+<code class="xref py py-obj docutils literal notranslate"><span class="pre">sim3_type</span></code> or <code class="xref py py-obj docutils literal notranslate"><span class="pre">rxso3_type</span></code> (input <span class="math">\(\mathbf{x}\)</span> is an instance of <a class="reference internal" href="../pypose.so3/#pypose.so3" title="pypose.so3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">so3()</span></code></a>
+or <a class="reference internal" href="../pypose.se3/#pypose.se3" title="pypose.se3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">se3()</span></code></a> or <a class="reference internal" href="../pypose.sim3/#pypose.sim3" title="pypose.sim3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">sim3()</span></code></a> or <a class="reference internal" href="../pypose.rxso3/#pypose.rxso3" title="pypose.rxso3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">rxso3()</span></code></a>):</p>
+<blockquote>
+<div><div class="math">
+\[\mathbf{y}_i = -\mathbf{x}_i
+
+\]</div>
+</div></blockquote>
+</li>
+</ul>
+<p class="rubric">Example</p>
+<ul>
+<li><p><span class="math">\(\mathrm{Inv}\)</span>: <a class="reference internal" href="../pypose.SO3/#pypose.SO3" title="pypose.SO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">SO3</span></code></a> <span class="math">\(\mapsto\)</span> <a class="reference internal" href="../pypose.SO3/#pypose.SO3" title="pypose.SO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">SO3</span></code></a></p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_SO3</span><span class="p">()</span>
+<span class="go">SO3Type LieTensor:</span>
+<span class="go">tensor([-0.1402, -0.2827,  0.2996,  0.9004])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">Inv</span><span class="p">()</span> <span class="c1"># equivalent to: pp.Inv(x)</span>
+<span class="go">SO3Type LieTensor:</span>
+<span class="go">tensor([ 0.1402,  0.2827, -0.2996,  0.9004])</span>
+</pre></div>
+</div>
+</li>
+<li><p><span class="math">\(\mathrm{Inv}\)</span>: <a class="reference internal" href="../pypose.SE3/#pypose.SE3" title="pypose.SE3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">SE3</span></code></a> <span class="math">\(\mapsto\)</span> <a class="reference internal" href="../pypose.SE3/#pypose.SE3" title="pypose.SE3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">SE3</span></code></a></p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_SE3</span><span class="p">()</span>
+<span class="go">SE3Type LieTensor:</span>
+<span class="go">tensor([ 0.6074, -0.7596,  0.8703, -0.3092,  0.2932,  0.9027,  0.0598])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">Inv</span><span class="p">()</span> <span class="c1"># equivalent to: pp.Inv(x)</span>
+<span class="go">SE3Type LieTensor:</span>
+<span class="go">tensor([ 0.9475, -0.8764,  0.1938,  0.3092, -0.2932, -0.9027,  0.0598])</span>
+</pre></div>
+</div>
+</li>
+<li><p><span class="math">\(\mathrm{Inv}\)</span>: <a class="reference internal" href="../pypose.RxSO3/#pypose.RxSO3" title="pypose.RxSO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">RxSO3</span></code></a> <span class="math">\(\mapsto\)</span> <a class="reference internal" href="../pypose.RxSO3/#pypose.RxSO3" title="pypose.RxSO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">RxSO3</span></code></a></p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_RxSO3</span><span class="p">()</span>
+<span class="go">RxSO3Type LieTensor:</span>
+<span class="go">tensor([-0.5103,  0.4707, -0.3494,  0.6292,  0.9199])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">Inv</span><span class="p">()</span> <span class="c1"># equivalent to: pp.Inv(x)</span>
+<span class="go">RxSO3Type LieTensor:</span>
+<span class="go">tensor([ 0.5103, -0.4707,  0.3494,  0.6292,  1.0871])</span>
+</pre></div>
+</div>
+</li>
+<li><p><span class="math">\(\mathrm{Inv}\)</span>: <a class="reference internal" href="../pypose.Sim3/#pypose.Sim3" title="pypose.Sim3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">Sim3</span></code></a> <span class="math">\(\mapsto\)</span> <a class="reference internal" href="../pypose.Sim3/#pypose.Sim3" title="pypose.Sim3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">Sim3</span></code></a></p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_Sim3</span><span class="p">()</span>
+<span class="go">Sim3Type LieTensor:</span>
+<span class="go">tensor([ 0.7056,  1.3140, -0.1995, -0.2444, -0.5250,  0.5504,  0.6014,  1.0543])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">Inv</span><span class="p">()</span> <span class="c1"># equivalent to: pp.Inv(x)</span>
+<span class="go">Sim3Type LieTensor:</span>
+<span class="go">tensor([-0.9712, -0.2361,  1.0188,  0.2444,  0.5250, -0.5504,  0.6014,  0.9485])</span>
+</pre></div>
+</div>
+</li>
+<li><p><span class="math">\(\mathrm{Inv}\)</span>: <a class="reference internal" href="../pypose.so3/#pypose.so3" title="pypose.so3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">so3</span></code></a> <span class="math">\(\mapsto\)</span> <a class="reference internal" href="../pypose.so3/#pypose.so3" title="pypose.so3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">so3</span></code></a></p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_so3</span><span class="p">()</span>
+<span class="go">so3Type LieTensor:</span>
+<span class="go">tensor([ 0.0612, -0.7190,  2.6897])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">Inv</span><span class="p">()</span> <span class="c1"># equivalent to: pp.Inv(x)</span>
+<span class="go">so3Type LieTensor:</span>
+<span class="go">tensor([-0.0612,  0.7190, -2.6897])</span>
+</pre></div>
+</div>
+</li>
+<li><p><span class="math">\(\mathrm{Inv}\)</span>: <a class="reference internal" href="../pypose.se3/#pypose.se3" title="pypose.se3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">se3</span></code></a> <span class="math">\(\mapsto\)</span> <a class="reference internal" href="../pypose.se3/#pypose.se3" title="pypose.se3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">se3</span></code></a></p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_se3</span><span class="p">()</span>
+<span class="go">se3Type LieTensor:</span>
+<span class="go">tensor([ 0.2837, -1.8318,  1.0104,  2.2385, -0.1980, -0.9487])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">Inv</span><span class="p">()</span> <span class="c1"># equivalent to: pp.Inv(x)</span>
+<span class="go">se3Type LieTensor:</span>
+<span class="go">tensor([-0.2837,  1.8318, -1.0104, -2.2385,  0.1980,  0.9487])</span>
+</pre></div>
+</div>
+</li>
+<li><p><span class="math">\(\mathrm{Inv}\)</span>: <a class="reference internal" href="../pypose.rxso3/#pypose.rxso3" title="pypose.rxso3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">rxso3</span></code></a> <span class="math">\(\mapsto\)</span> <a class="reference internal" href="../pypose.rxso3/#pypose.rxso3" title="pypose.rxso3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">rxso3</span></code></a></p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_rxso3</span><span class="p">()</span>
+<span class="go">rxso3Type LieTensor:</span>
+<span class="go">tensor([ 1.0414, -0.0087, -0.4427, -1.1343])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">Inv</span><span class="p">()</span> <span class="c1"># equivalent to: pp.Inv(x)</span>
+<span class="go">rxso3Type LieTensor:</span>
+<span class="go">tensor([-1.0414,  0.0087,  0.4427,  1.1343])</span>
+</pre></div>
+</div>
+</li>
+<li><p><span class="math">\(\mathrm{Inv}\)</span>: <a class="reference internal" href="../pypose.sim3/#pypose.sim3" title="pypose.sim3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">sim3</span></code></a> <span class="math">\(\mapsto\)</span> <a class="reference internal" href="../pypose.sim3/#pypose.sim3" title="pypose.sim3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">sim3</span></code></a></p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_sim3</span><span class="p">()</span>
+<span class="go">sim3Type LieTensor:</span>
+<span class="go">tensor([-0.0724,  1.8174,  2.1810, -0.9324, -0.0952, -0.5792,  0.4318])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">Inv</span><span class="p">()</span> <span class="c1"># equivalent to: pp.Inv(x)</span>
+<span class="go">sim3Type LieTensor:</span>
+<span class="go">tensor([ 0.0724, -1.8174, -2.1810,  0.9324,  0.0952,  0.5792, -0.4318])</span>
+</pre></div>
+</div>
+</li>
+</ul>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Mathematically, only Lie Group has an inverse. For the convenience, we also provide the
+inverse for the corresponding Lie Algebra. One can validate the results by:</p>
+<p>if input <span class="math">\(\mathbf{x}\)</span> is the LieTensor from Lie Group, for example
+<span class="math">\(\mathbf{x}\)</span>’s <code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code> is <code class="xref py py-obj docutils literal notranslate"><span class="pre">SO3_type</span></code> :</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_SO3</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">*</span> <span class="n">x</span><span class="o">.</span><span class="n">Inv</span><span class="p">()</span>
+<span class="go">SO3Type LieTensor:</span>
+<span class="go">LieTensor([0., 0., 0., 1.])</span>
+</pre></div>
+</div>
+<p>if input <span class="math">\(\mathbf{x}\)</span> is the LieTensor from Lie Algebra, for example
+<span class="math">\(\mathbf{x}\)</span>’s <code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code> is <code class="xref py py-obj docutils literal notranslate"><span class="pre">so3_type</span></code> :</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_so3</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">+</span> <span class="n">x</span><span class="o">.</span><span class="n">Inv</span><span class="p">()</span>
+<span class="go">tensor([0., 0., 0.])</span>
+</pre></div>
+</div>
+<p>One can also verify:</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_SO3</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">Log</span><span class="p">()</span> <span class="o">+</span> <span class="n">x</span><span class="o">.</span><span class="n">Inv</span><span class="p">()</span><span class="o">.</span><span class="n">Log</span><span class="p">()</span>
+<span class="go">so3Type LieTensor:</span>
+<span class="go">LieTensor([0., 0., 0.])</span>
+</pre></div>
+</div>
+</div>
+</dd></dl>
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+              
+  <section id="pypose-jinvp">
+<h1>pypose.Jinvp<a class="headerlink" href="#pypose-jinvp" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.Jinvp">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">Jinvp</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">p</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/utils/#Jinvp"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.Jinvp" title="Permalink to this definition">¶</a></dt>
+<dd><p>The dot product between left Jacobian inverse at the point given
+by input (Lie Group) and second point (Lie Algebra).</p>
+<div class="math">
+\[\mathrm{Jinvp}: (\mathcal{G}, \mathcal{g}) \mapsto \mathcal{g}
+
+\]</div>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>input</strong> (<a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><em>LieTensor</em></a>) – the input LieTensor (Lie Group)</p></li>
+<li><p><strong>p</strong> (<a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><em>LieTensor</em></a>) – the second LieTensor (Lie Algebra)</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>the output LieTensor (Lie Algebra)</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor">LieTensor</a></p>
+</dd>
+</dl>
+<table class="docutils align-default" id="id1">
+<caption><span class="caption-text">List of supported <span class="math">\(\mathrm{Jinvp}\)</span> map</span><a class="headerlink" href="#id1" title="Permalink to this table">¶</a></caption>
+<colgroup>
+<col style="width: 23%" />
+<col style="width: 23%" />
+<col style="width: 9%" />
+<col style="width: 23%" />
+<col style="width: 23%" />
+</colgroup>
+<thead>
+<tr class="row-odd"><th class="head"><p>input <code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code></p></th>
+<th class="head"><p><span class="math">\((\mathcal{G}, \mathcal{g})\)</span> (Lie Group, Lie Algebra)</p></th>
+<th class="head"><p><span class="math">\(\mapsto\)</span></p></th>
+<th class="head"><p><span class="math">\(\mathcal{g}\)</span> (Lie Algebra)</p></th>
+<th class="head"><p>output <code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code></p></th>
+</tr>
+</thead>
+<tbody>
+<tr class="row-even"><td><p>(<code class="xref py py-obj docutils literal notranslate"><span class="pre">SO3_type</span></code>, <code class="xref py py-obj docutils literal notranslate"><span class="pre">so3_type</span></code>)</p></td>
+<td><p><span class="math">\((\mathcal{G}\in\mathbb{R}^{*\times4}, \mathcal{g}\in\mathbb{R}^{*\times3})\)</span></p></td>
+<td><p><span class="math">\(\mapsto\)</span></p></td>
+<td><p><span class="math">\(\mathcal{g}\in\mathbb{R}^{*\times3}\)</span></p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">so3_type</span></code></p></td>
+</tr>
+<tr class="row-odd"><td><p>(<code class="xref py py-obj docutils literal notranslate"><span class="pre">SE3_type</span></code>, <code class="xref py py-obj docutils literal notranslate"><span class="pre">se3_type</span></code>)</p></td>
+<td><p><span class="math">\((\mathcal{G}\in\mathbb{R}^{*\times7}, \mathcal{g}\in\mathbb{R}^{*\times6})\)</span></p></td>
+<td><p><span class="math">\(\mapsto\)</span></p></td>
+<td><p><span class="math">\(\mathcal{g}\in\mathbb{R}^{*\times6}\)</span></p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">se3_type</span></code></p></td>
+</tr>
+<tr class="row-even"><td><p>(<code class="xref py py-obj docutils literal notranslate"><span class="pre">Sim3_type</span></code>, <code class="xref py py-obj docutils literal notranslate"><span class="pre">sim3_type</span></code>)</p></td>
+<td><p><span class="math">\((\mathcal{G}\in\mathbb{R}^{*\times8}, \mathcal{g}\in\mathbb{R}^{*\times7})\)</span></p></td>
+<td><p><span class="math">\(\mapsto\)</span></p></td>
+<td><p><span class="math">\(\mathcal{g}\in\mathbb{R}^{*\times7}\)</span></p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">sim3_type</span></code></p></td>
+</tr>
+<tr class="row-odd"><td><p>(<code class="xref py py-obj docutils literal notranslate"><span class="pre">RxSO3_type</span></code>, <code class="xref py py-obj docutils literal notranslate"><span class="pre">rxso3_type</span></code>)</p></td>
+<td><p><span class="math">\((\mathcal{G}\in\mathbb{R}^{*\times5}, \mathcal{g}\in\mathbb{R}^{*\times4})\)</span></p></td>
+<td><p><span class="math">\(\mapsto\)</span></p></td>
+<td><p><span class="math">\(\mathcal{g}\in\mathbb{R}^{*\times4}\)</span></p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">rxso3_type</span></code></p></td>
+</tr>
+</tbody>
+</table>
+<p>Let the input be (<span class="math">\(\mathbf{x}\)</span>, <span class="math">\(\mathbf{p}\)</span>), <span class="math">\(\mathbf{y}\)</span> be the output.</p>
+<blockquote>
+<div><div class="math">
+\[\mathbf{y}_i = \mathbf{J}^{-1}_i(\mathbf{x}_i)\mathbf{p}_i,
+
+\]</div>
+<p>where <span class="math">\(\mathbf{J}^{-1}_i(\mathbf{x}_i)\)</span> is the inverse of left Jacobian of
+<span class="math">\(\mathbf{x}_i\)</span>.</p>
+</div></blockquote>
+<ul>
+<li><p>If input (<span class="math">\(\mathbf{x}\)</span>, <span class="math">\(\mathbf{p}\)</span>)’s <code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code> are <code class="xref py py-obj docutils literal notranslate"><span class="pre">SO3_type</span></code>
+and <code class="xref py py-obj docutils literal notranslate"><span class="pre">so3_type</span></code> (input <span class="math">\(\mathbf{x}\)</span> is an instance of <a class="reference internal" href="../pypose.SO3/#pypose.SO3" title="pypose.SO3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">SO3()</span></code></a>,
+<span class="math">\(\mathbf{p}\)</span> is an instance of <a class="reference internal" href="../pypose.so3/#pypose.so3" title="pypose.so3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">so3()</span></code></a>).
+Let <span class="math">\(\boldsymbol{\phi}_i = \theta_i\mathbf{n}_i\)</span> be the corresponding Lie Algebra
+of <span class="math">\(\mathbf{x}_i\)</span>, <span class="math">\(\boldsymbol{\Phi}_i\)</span> be the skew matrix
+(<a class="reference internal" href="../pypose.vec2skew/#pypose.vec2skew" title="pypose.vec2skew"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.vec2skew()</span></code></a>) of <span class="math">\(\boldsymbol{\phi}_i\)</span>:</p>
+<blockquote>
+<div><div class="math">
+\[\mathbf{J}^{-1}_i(\mathbf{x}_i) = \mathbf{I} - \frac{1}{2}\boldsymbol{\Phi}_i +
+\mathrm{coef}\boldsymbol{\Phi}_i^2
+
+\]</div>
+</div></blockquote>
+<p>where <span class="math">\(\mathbf{I}\)</span> is the identity matrix with the same dimension as
+<span class="math">\(\boldsymbol{\Phi}_i\)</span>, and</p>
+<blockquote>
+<div><div class="math">
+\[\mathrm{coef} =
+    \left\{
+    \begin{array}{ll}
+        \frac{1}{\theta_i^2} -
+        \frac{\cos{\frac{\theta_i}{2}}}{2\theta\sin{\frac{\theta_i}{2}}},
+        \quad \|\theta_i\| > \text{eps}, \\
+        \frac{1}{12},
+        \quad \|\theta_i\| \leq \text{eps}
+    \end{array}
+    \right.
+
+\]</div>
+</div></blockquote>
+</li>
+<li><p>If input (<span class="math">\(\mathbf{x}\)</span>, <span class="math">\(\mathbf{p}\)</span>)’s <code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code> are <code class="xref py py-obj docutils literal notranslate"><span class="pre">SE3_type</span></code>
+and <code class="xref py py-obj docutils literal notranslate"><span class="pre">se3_type</span></code> (input <span class="math">\(\mathbf{x}\)</span> is an instance of <a class="reference internal" href="../pypose.SE3/#pypose.SE3" title="pypose.SE3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">SE3()</span></code></a>,
+<span class="math">\(\mathbf{p}\)</span> is an instance of <a class="reference internal" href="../pypose.se3/#pypose.se3" title="pypose.se3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">se3()</span></code></a>).
+Let <span class="math">\(\boldsymbol{\phi}_i = \theta_i\mathbf{n}_i\)</span> be the corresponding Lie Algebra
+of the SO3 part of <span class="math">\(\mathbf{x}_i\)</span>, <span class="math">\(\boldsymbol{\tau}_i\)</span> be the Lie Algebra
+of the translation part of <span class="math">\(\mathbf{x}_i\)</span>; <span class="math">\(\boldsymbol{\Phi}_i\)</span> and
+<span class="math">\(\boldsymbol{\Tau}_i\)</span> be the skew matrices, respectively:</p>
+<blockquote>
+<div><div class="math">
+\[\mathbf{J}^{-1}_i(\mathbf{x}_i) =
+    \left[
+    \begin{array}{cc}
+        \mathbf{J}_i^{-1}(\boldsymbol{\Phi}_i) &
+                - \mathbf{J}_i^{-1}(\boldsymbol{\Phi}_i)
+        \mathbf{Q}_i(\boldsymbol{\tau}_i, \boldsymbol{\phi}_i)
+            \mathbf{J}_i^{-1}(\boldsymbol{\Phi}_i) \\
+        \mathbf{0} & \mathbf{J}_i^{-1}(\boldsymbol{\Phi}_i)
+    \end{array}
+    \right]
+
+\]</div>
+<p>where <span class="math">\(\mathbf{J}_i^{-1}(\boldsymbol{\Phi}_i)\)</span> is the inverse of left Jacobian
+of the SO3 part of <span class="math">\(\mathbf{x}_i\)</span>.
+<span class="math">\(\mathbf{Q}_i(\boldsymbol{\tau}_i, \boldsymbol{\phi}_i)\)</span> is</p>
+<div class="math">
+\[\begin{align*}
+    \mathbf{Q}_i(\boldsymbol{\tau}_i, \boldsymbol{\phi}_i) =
+        \frac{1}{2}\boldsymbol{\Tau}_i &+ c_1
+    (\boldsymbol{\Phi_i\Tau_i} + \boldsymbol{\Tau_i\Phi_i}
+        + \boldsymbol{\Phi_i\Tau_i\Phi_i}) \\
+     &+ c_2 (\boldsymbol{\Phi_i^2\Tau_i} + \boldsymbol{\Tau_i\Phi_i^2}
+        - 3\boldsymbol{\Phi_i\Tau_i\Phi_i})\\
+     &+ c_3 (\boldsymbol{\Phi_i\Tau_i\Phi_i^2} + \boldsymbol{\Phi_i^2\Tau_i\Phi_i})
+\end{align*}
+
+\]</div>
+<p>where,</p>
+<div class="math">
+\[c_1 = \left\{
+        \begin{array}{ll}
+            \frac{\theta_i - \sin\theta_i}{\theta_i^3},
+                \quad \|\theta_i\| > \text{eps}, \\
+            \frac{1}{6}-\frac{1}{120}\theta_i^2,
+            \quad \|\theta_i\| \leq \text{eps}
+        \end{array}
+        \right.
+
+\]</div>
+<div class="math">
+\[c_2 = \left\{
+        \begin{array}{ll}
+            \frac{\theta_i^2 +2\cos\theta_i - 2}{2\theta_i^4},
+                \quad \|\theta_i\| > \text{eps}, \\
+            \frac{1}{24}-\frac{1}{720}\theta_i^2,
+            \quad \|\theta_i\| \leq \text{eps}
+        \end{array}
+        \right.
+
+\]</div>
+<div class="math">
+\[c_3 = \left\{
+        \begin{array}{ll}
+            \frac{2\theta_i - 3\sin\theta_i + \theta_i\cos\theta_i}{2\theta_i^5},
+            \quad \|\theta_i\| > \text{eps}, \\
+            \frac{1}{120}-\frac{1}{2520}\theta_i^2,
+            \quad \|\theta_i\| \leq \text{eps}
+        \end{array}
+        \right.
+
+\]</div>
+</div></blockquote>
+</li>
+<li><p>If input (<span class="math">\(\mathbf{x}\)</span>, <span class="math">\(\mathbf{p}\)</span>)’s <code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code> are <code class="xref py py-obj docutils literal notranslate"><span class="pre">Sim3_type</span></code>
+and <code class="xref py py-obj docutils literal notranslate"><span class="pre">sim3_type</span></code> (input <span class="math">\(\mathbf{x}\)</span> is an instance of <a class="reference internal" href="../pypose.Sim3/#pypose.Sim3" title="pypose.Sim3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">Sim3()</span></code></a>,
+<span class="math">\(\mathbf{p}\)</span> is an instance of <a class="reference internal" href="../pypose.sim3/#pypose.sim3" title="pypose.sim3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">sim3()</span></code></a>). The inverse of left Jacobian can
+be approximated as:</p>
+<blockquote>
+<div><div class="math">
+\[\mathbf{J}^{-1}_i(\mathbf{x}_i) =
+    \sum_{n=0}(-1)^n\frac{B_n}{n!}(\boldsymbol{\xi}_i^{\curlywedge})^n
+
+\]</div>
+<p>where <span class="math">\(B_n\)</span> is the Bernoulli number: <span class="math">\(B_0 = 1\)</span>, <span class="math">\(B_1 = -\frac{1}{2}\)</span>,
+<span class="math">\(B_2 = \frac{1}{6}\)</span>, <span class="math">\(B_3 = 0\)</span>, <span class="math">\(B_4 = -\frac{1}{30}\)</span>.
+<span class="math">\(\boldsymbol{\xi}_i^{\curlywedge} = \mathrm{adj}(\boldsymbol{\xi}_i^{\wedge})\)</span>
+and <span class="math">\(\mathrm{adj}\)</span> is the adjoint of the Lie algebra <span class="math">\(\mathfrak{sim}(3)\)</span>,
+e.g., <span class="math">\(\boldsymbol{\xi}_i \in \mathfrak{sim}(3)\)</span>. Notice that if notate
+<span class="math">\(\boldsymbol{X}_i = \mathrm{Adj}(\mathbf{x}_i)\)</span> and <span class="math">\(\mathrm{Adj}\)</span> is the
+adjoint of the Lie group <span class="math">\(\mathrm{Sim}(3)\)</span>, there is a nice property:
+<span class="math">\(\mathrm{Adj}(\mathrm{Exp}(\boldsymbol{\xi}_i^{\curlywedge})) =
+\mathrm{Exp}(\mathrm{adj}(\boldsymbol{\xi}_i^{\wedge}))\)</span>,
+or <span class="math">\(\boldsymbol{X}_i = \mathrm{Exp}(\boldsymbol{\xi}_i^{\curlywedge})\)</span>.</p>
+</div></blockquote>
+</li>
+<li><p>If input (<span class="math">\(\mathbf{x}\)</span>, <span class="math">\(\mathbf{p}\)</span>)’s <code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code> are <code class="xref py py-obj docutils literal notranslate"><span class="pre">RxSO3_type</span></code>
+and <code class="xref py py-obj docutils literal notranslate"><span class="pre">rxso3_type</span></code> (input <span class="math">\(\mathbf{x}\)</span> is an instance of <a class="reference internal" href="../pypose.RxSO3/#pypose.RxSO3" title="pypose.RxSO3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">RxSO3()</span></code></a>,
+<span class="math">\(\mathbf{p}\)</span> is an instance of <a class="reference internal" href="../pypose.rxso3/#pypose.rxso3" title="pypose.rxso3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">rxso3()</span></code></a>). Let <span class="math">\(\boldsymbol{\phi}_i\)</span>
+be the corresponding Lie Algebra of the SO3 part of <span class="math">\(\mathbf{x}_i\)</span>,
+<span class="math">\(\boldsymbol{\Phi}_i\)</span> be the skew matrix (<a class="reference internal" href="../pypose.vec2skew/#pypose.vec2skew" title="pypose.vec2skew"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.vec2skew()</span></code></a>), The inverse
+of left Jacobian of <span class="math">\(\mathbf{x}_i\)</span> is the same as that for the SO3 part of
+<span class="math">\(\mathbf{x}_i\)</span>.</p>
+<blockquote>
+<div><div class="math">
+\[\mathbf{J}^{-1}_i(\mathbf{x}_i) = \left[
+                    \begin{array}{cc}
+                        \mathbf{J}_i^{-1}(\boldsymbol{\Phi}_i) & \mathbf{0} \\
+                        \mathbf{0} & 1
+                    \end{array}
+                 \right]
+
+\]</div>
+<p>where <span class="math">\(\mathbf{J}_i^{-1}(\boldsymbol{\Phi}_i)\)</span> is the
+inverse of left Jacobian of the SO3 part of <span class="math">\(\mathbf{x}_i\)</span>.</p>
+</div></blockquote>
+</li>
+</ul>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p><span class="math">\(\mathrm{Jinvp}\)</span> is usually used in the Baker-Campbell-Hausdorff formula
+(BCH formula) when performing LieTensor multiplication.
+One can refer to this paper for more details:</p>
+<ul class="simple">
+<li><p>J. Sola et al., <a class="reference external" href="https://arxiv.org/abs/1812.01537">A micro Lie theory for state estimation in
+robotics</a>, arXiv preprint arXiv:1812.01537 (2018).</p></li>
+</ul>
+<p>In particular, Eq. (146) is the math used in the <code class="xref py py-obj docutils literal notranslate"><span class="pre">SO3_type</span></code>, <code class="xref py py-obj docutils literal notranslate"><span class="pre">so3_type</span></code> scenario;
+Eq. (179b) and Eq. (180) are the math used in the <code class="xref py py-obj docutils literal notranslate"><span class="pre">SE3_type</span></code>, <code class="xref py py-obj docutils literal notranslate"><span class="pre">se3_type</span></code> scenario.</p>
+<p>For Lie groups such as <code class="xref py py-obj docutils literal notranslate"><span class="pre">Sim3_type</span></code>, <code class="xref py py-obj docutils literal notranslate"><span class="pre">sim3_type</span></code>,
+there is no analytic expression for the left Jacobian and its inverse.
+Numerical approximation is used based on series expansion.
+One can refer to Eq. (26) of this paper for more details about the approximation:</p>
+<ul class="simple">
+<li><p>Z. Teed et al., <a class="reference external" href="https://arxiv.org/pdf/2103.12032.pdf">Tangent Space Backpropagation for 3D Transformation Groups.</a>, in IEEE/CVF Conference on Computer Vision and
+Pattern Recognition (CVPR) (2021).</p></li>
+</ul>
+<p>In particular, the Bernoulli numbers can be obtained from Eq. (7.72) of this famous book:</p>
+<ul class="simple">
+<li><p>T. Barfoot, <a class="reference external" href="https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.708.1086&amp;rep=rep1&amp;type=pdf">State Estimation for Robotics.</a>,
+Cambridge University Press (2017).</p></li>
+</ul>
+</div>
+<p class="rubric">Example</p>
+<ul class="simple">
+<li><p><span class="math">\(\mathrm{Jinvp}\)</span>: (<a class="reference internal" href="../pypose.SO3/#pypose.SO3" title="pypose.SO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">SO3</span></code></a>, <a class="reference internal" href="../pypose.so3/#pypose.so3" title="pypose.so3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">so3</span></code></a>) <span class="math">\(\mapsto\)</span> <a class="reference internal" href="../pypose.so3/#pypose.so3" title="pypose.so3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">so3</span></code></a></p></li>
+</ul>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_SO3</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_so3</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">Jinvp</span><span class="p">(</span><span class="n">p</span><span class="p">)</span> <span class="c1"># equivalent to: pp.Jinvp(x, p)</span>
+<span class="go">so3Type LieTensor:</span>
+<span class="go">tensor([-2.0248,  1.1116, -0.0251])</span>
+</pre></div>
+</div>
+<ul class="simple">
+<li><p><span class="math">\(\mathrm{Jinvp}\)</span>: (<a class="reference internal" href="../pypose.SE3/#pypose.SE3" title="pypose.SE3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">SE3</span></code></a>, <a class="reference internal" href="../pypose.se3/#pypose.se3" title="pypose.se3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">se3</span></code></a>) <span class="math">\(\mapsto\)</span> <a class="reference internal" href="../pypose.se3/#pypose.se3" title="pypose.se3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">se3</span></code></a></p></li>
+</ul>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_SE3</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_se3</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">Jinvp</span><span class="p">(</span><span class="n">p</span><span class="p">)</span> <span class="c1"># equivalent to: pp.Jinvp(x, p)</span>
+<span class="go">se3Type LieTensor:</span>
+<span class="go">tensor([[ 0.4304,  2.0565,  1.0256,  0.0666, -0.2252, -0.7425],</span>
+<span class="go">        [-0.9317, -1.7806,  0.8660, -2.0028,  0.6098, -0.6517]])</span>
+</pre></div>
+</div>
+<ul class="simple">
+<li><p><span class="math">\(\mathrm{Jinvp}\)</span>: (<a class="reference internal" href="../pypose.Sim3/#pypose.Sim3" title="pypose.Sim3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">Sim3</span></code></a>, <a class="reference internal" href="../pypose.sim3/#pypose.sim3" title="pypose.sim3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">sim3</span></code></a>) <span class="math">\(\mapsto\)</span> <a class="reference internal" href="../pypose.sim3/#pypose.sim3" title="pypose.sim3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">sim3</span></code></a></p></li>
+</ul>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_Sim3</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_sim3</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">Jinvp</span><span class="p">(</span><span class="n">p</span><span class="p">)</span> <span class="c1"># equivalent to: pp.Jinvp(x, p)</span>
+<span class="go">sim3Type LieTensor:</span>
+<span class="go">tensor([[-1.7231, -1.6650, -0.0202, -0.3731,  0.8441, -0.5438,  0.2879],</span>
+<span class="go">        [ 0.9965,  0.6337, -0.7320, -0.1874,  0.6312,  0.3919,  0.6938]])</span>
+</pre></div>
+</div>
+<ul class="simple">
+<li><p><span class="math">\(\mathrm{Jinvp}\)</span>: (<a class="reference internal" href="../pypose.RxSO3/#pypose.RxSO3" title="pypose.RxSO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">RxSO3</span></code></a>, <a class="reference internal" href="../pypose.rxso3/#pypose.rxso3" title="pypose.rxso3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">rxso3</span></code></a>) <span class="math">\(\mapsto\)</span> <a class="reference internal" href="../pypose.rxso3/#pypose.rxso3" title="pypose.rxso3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">rxso3</span></code></a></p></li>
+</ul>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_RxSO3</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_rxso3</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">Jinvp</span><span class="p">(</span><span class="n">p</span><span class="p">)</span> <span class="c1"># equivalent to: pp.Jinvp(x, p)</span>
+<span class="go">rxso3Type LieTensor:</span>
+<span class="go">tensor([[ 0.9308, -1.4965, -0.1347,  0.4894],</span>
+<span class="go">        [ 0.6558,  1.2221, -0.8190,  0.2108]])</span>
+</pre></div>
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+             <article itemprop="articleBody" id="pytorch-article" class="pytorch-article">
+              
+  <section id="pypose-jr">
+<h1>pypose.Jr<a class="headerlink" href="#pypose-jr" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.Jr">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">Jr</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/utils/#Jr"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.Jr" title="Permalink to this definition">¶</a></dt>
+<dd><p>The batched right Jacobian of a LieTensor.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>input</strong> (<a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><em>LieTensor</em></a>) – the input LieTensor (either Lie Group or Lie Algebra)</p>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>the right Jocobian Matrices</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>Tensor</p>
+</dd>
+</dl>
+<p class="rubric">Examples</p>
+<ul>
+<li><p><span class="math">\(\mathrm{Jr}\)</span>: <a class="reference internal" href="../pypose.so3/#pypose.so3" title="pypose.so3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">so3()</span></code></a> <span class="math">\(\mapsto\)</span> <span class="math">\(\mathcal{R}^{*\times 3\times 3}\)</span></p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_so3</span><span class="p">(</span><span class="n">requires_grad</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">Jr</span><span class="p">()</span>
+<span class="go">tensor([[ 0.9289, -0.3053, -0.0895],</span>
+<span class="go">        [ 0.3180,  0.9082,  0.1667],</span>
+<span class="go">        [ 0.0104, -0.1889,  0.9757]], grad_fn=&lt;SWhereBackward0&gt;)</span>
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+  <section id="pypose-lietensor">
+<h1>pypose.LieTensor<a class="headerlink" href="#pypose-lietensor" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.LieTensor">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">LieTensor</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">data</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">ltype</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/lietensor/#LieTensor"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.LieTensor" title="Permalink to this definition">¶</a></dt>
+<dd><p>A sub-class of <code class="xref py py-obj docutils literal notranslate"><span class="pre">torch.Tensor</span></code> to represent Lie Algebra and Lie Group.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>data</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>, or <code class="xref py py-obj docutils literal notranslate"><span class="pre">list</span></code>, or ‘<code class="xref py py-obj docutils literal notranslate"><span class="pre">int</span></code>…’) – <p>A
+<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code> object, or constructing a <code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>
+object from <code class="xref py py-obj docutils literal notranslate"><span class="pre">list</span></code>, which defines tensor data, or from
+‘<code class="xref py py-obj docutils literal notranslate"><span class="pre">int</span></code>…’, which defines tensor shape.</p>
+<p>The shape of <code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code> object should be compatible
+with Lie Type <code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code>, otherwise error will be raised.</p>
+</p></li>
+<li><p><strong>ltype</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code>) – Lie Type, either <strong>Lie Group</strong> or <strong>Lie Algebra</strong> is listed below:</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>LieTensor corresponding to Lie Type <code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code>.</p>
+</dd>
+</dl>
+<table class="docutils align-default" id="id1">
+<caption><span class="caption-text">List of <code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code> for <strong>Lie Group</strong></span><a class="headerlink" href="#id1" title="Permalink to this table">¶</a></caption>
+<colgroup>
+<col style="width: 23%" />
+<col style="width: 23%" />
+<col style="width: 27%" />
+<col style="width: 27%" />
+</colgroup>
+<thead>
+<tr class="row-odd"><th class="head"><p>Representation</p></th>
+<th class="head"><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code></p></th>
+<th class="head"><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">shape</span></code></p></th>
+<th class="head"><p>Alias Class</p></th>
+</tr>
+</thead>
+<tbody>
+<tr class="row-even"><td><p>Rotation</p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">SO3_type</span></code></p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">(*,</span> <span class="pre">4)</span></code></p></td>
+<td><p><a class="reference internal" href="../pypose.SO3/#pypose.SO3" title="pypose.SO3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">SO3()</span></code></a></p></td>
+</tr>
+<tr class="row-odd"><td><p>Translation + Rotation</p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">SE3_type</span></code></p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">(*,</span> <span class="pre">7)</span></code></p></td>
+<td><p><a class="reference internal" href="../pypose.SE3/#pypose.SE3" title="pypose.SE3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">SE3()</span></code></a></p></td>
+</tr>
+<tr class="row-even"><td><p>Translation + Rotation + Scale</p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Sim3_type</span></code></p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">(*,</span> <span class="pre">8)</span></code></p></td>
+<td><p><a class="reference internal" href="../pypose.Sim3/#pypose.Sim3" title="pypose.Sim3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">Sim3()</span></code></a></p></td>
+</tr>
+<tr class="row-odd"><td><p>Rotation + Scale</p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">RxSO3_type</span></code></p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">(*,</span> <span class="pre">5)</span></code></p></td>
+<td><p><a class="reference internal" href="../pypose.RxSO3/#pypose.RxSO3" title="pypose.RxSO3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">RxSO3()</span></code></a></p></td>
+</tr>
+</tbody>
+</table>
+<table class="docutils align-default" id="id2">
+<caption><span class="caption-text">List of <code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code> for <strong>Lie Algebra</strong></span><a class="headerlink" href="#id2" title="Permalink to this table">¶</a></caption>
+<colgroup>
+<col style="width: 23%" />
+<col style="width: 23%" />
+<col style="width: 27%" />
+<col style="width: 27%" />
+</colgroup>
+<thead>
+<tr class="row-odd"><th class="head"><p>Representation</p></th>
+<th class="head"><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code></p></th>
+<th class="head"><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">shape</span></code></p></th>
+<th class="head"><p>Alias Class</p></th>
+</tr>
+</thead>
+<tbody>
+<tr class="row-even"><td><p>Rotation</p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">so3_type</span></code></p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">(*,</span> <span class="pre">3)</span></code></p></td>
+<td><p><a class="reference internal" href="../pypose.so3/#pypose.so3" title="pypose.so3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">so3()</span></code></a></p></td>
+</tr>
+<tr class="row-odd"><td><p>Translation + Rotation</p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">se3_type</span></code></p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">(*,</span> <span class="pre">6)</span></code></p></td>
+<td><p><a class="reference internal" href="../pypose.se3/#pypose.se3" title="pypose.se3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">se3()</span></code></a></p></td>
+</tr>
+<tr class="row-even"><td><p>Translation + Rotation + Scale</p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">sim3_type</span></code></p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">(*,</span> <span class="pre">7)</span></code></p></td>
+<td><p><a class="reference internal" href="../pypose.sim3/#pypose.sim3" title="pypose.sim3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">sim3()</span></code></a></p></td>
+</tr>
+<tr class="row-odd"><td><p>Rotation + Scale</p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">rxso3_type</span></code></p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">(*,</span> <span class="pre">4)</span></code></p></td>
+<td><p><a class="reference internal" href="../pypose.rxso3/#pypose.rxso3" title="pypose.rxso3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">rxso3()</span></code></a></p></td>
+</tr>
+</tbody>
+</table>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Two attributes <code class="xref py py-obj docutils literal notranslate"><span class="pre">shape</span></code> and <a class="reference internal" href="#pypose.LieTensor.lshape" title="pypose.LieTensor.lshape"><code class="xref py py-obj docutils literal notranslate"><span class="pre">lshape</span></code></a> are available for LieTensor.
+The only differece is the <a class="reference internal" href="#pypose.LieTensor.lshape" title="pypose.LieTensor.lshape"><code class="xref py py-obj docutils literal notranslate"><span class="pre">lshape</span></code></a> hides the last dimension of <code class="xref py py-obj docutils literal notranslate"><span class="pre">shape</span></code>,
+since <a class="reference internal" href="#pypose.LieTensor.lshape" title="pypose.LieTensor.lshape"><code class="xref py py-obj docutils literal notranslate"><span class="pre">lshape</span></code></a> takes the data in the last dimension as a single <code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code> item.</p>
+<p>See LieTensor method <a class="reference internal" href="#pypose.LieTensor.lview" title="pypose.LieTensor.lview"><code class="xref py py-meth docutils literal notranslate"><span class="pre">lview()</span></code></a> for more details.</p>
+</div>
+<p class="rubric">Examples</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">torch</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">pypose</span> <span class="k">as</span> <span class="nn">pp</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">data</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">requires_grad</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="s1">&#39;cuda:0&#39;</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">LieTensor</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">ltype</span><span class="o">=</span><span class="n">pp</span><span class="o">.</span><span class="n">so3_type</span><span class="p">)</span>
+<span class="go">so3Type LieTensor:</span>
+<span class="go">tensor([[ 0.9520,  0.4517,  0.5834],</span>
+<span class="go">        [-0.8106,  0.8197,  0.7077],</span>
+<span class="go">        [-0.5743,  0.8182, -1.2104]], device=&#39;cuda:0&#39;, grad_fn=&lt;AliasBackward0&gt;)</span>
+</pre></div>
+</div>
+<p>Alias class for specific LieTensor is recommended:</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">so3</span><span class="p">(</span><span class="n">data</span><span class="p">)</span>
+<span class="go">so3Type LieTensor:</span>
+<span class="go">tensor([[ 0.9520,  0.4517,  0.5834],</span>
+<span class="go">        [-0.8106,  0.8197,  0.7077],</span>
+<span class="go">        [-0.5743,  0.8182, -1.2104]], device=&#39;cuda:0&#39;, grad_fn=&lt;AliasBackward0&gt;)</span>
+</pre></div>
+</div>
+<p>See more alias classes at <a class="reference external" href="#id1">Table 1 for Lie Group</a>  and <a class="reference external" href="#id2">Table 2 for Lie Algebra</a>.</p>
+<p>Other constructors:</p>
+<blockquote>
+<div><ul class="simple">
+<li><p>From list.</p></li>
+</ul>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">so3</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
+<span class="go">so3Type LieTensor:</span>
+<span class="go">tensor([0., 0., 0.])</span>
+</pre></div>
+</div>
+<ul class="simple">
+<li><p>From ints.</p></li>
+</ul>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">so3</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
+<span class="go">so3Type LieTensor:</span>
+<span class="go">tensor([[0., 0., 0.],</span>
+<span class="go">        [0., 0., 0.]])</span>
+</pre></div>
+</div>
+</div></blockquote>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Alias class for LieTensor is recommended.
+For example, the following usage is equivalent:</p>
+<ul class="simple">
+<li><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">pp.LieTensor(tensor,</span> <span class="pre">ltype=pp.so3_type)</span></code></p></li>
+<li><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">pp.so3(tensor)</span></code> (This is preferred).</p></li>
+</ul>
+</div>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>All attributes from Tensor are available for LieTensor, e.g., <code class="xref py py-obj docutils literal notranslate"><span class="pre">dtype</span></code>,
+<code class="xref py py-obj docutils literal notranslate"><span class="pre">device</span></code>, and <code class="xref py py-obj docutils literal notranslate"><span class="pre">requires_grad</span></code>. See more details at
+<a class="reference external" href="https://pytorch.org/docs/stable/tensor_attributes.html">tensor attributes</a>.</p>
+<p>Example:</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">data</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">float64</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="s2">&quot;cuda&quot;</span><span class="p">,</span> <span class="n">requires_grad</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">so3</span><span class="p">(</span><span class="n">data</span><span class="p">)</span> <span class="c1"># All Tensor attributes are available for LieTensor</span>
+<span class="go">so3Type LieTensor:</span>
+<span class="go">tensor([[-1.5948,  0.3113, -0.9807]], device=&#39;cuda:0&#39;, dtype=torch.float64,</span>
+<span class="go">    grad_fn=&lt;AliasBackward0&gt;)</span>
+</pre></div>
+</div>
+</div>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>In most of the cases, Lie Group is expected to be used, therefore we only provide
+<a class="reference external" href="https://pypose.org/docs/main/convert/">converting functions</a> between Lie Groups
+and other data structures, e.g., transformation matrix, Euler angle, etc. The users
+can convert data between Lie Group and Lie algebra with <a class="reference internal" href="../pypose.Exp/#pypose.Exp" title="pypose.Exp"><code class="xref py py-obj docutils literal notranslate"><span class="pre">Exp</span></code></a> and <a class="reference internal" href="../pypose.Log/#pypose.Log" title="pypose.Log"><code class="xref py py-obj docutils literal notranslate"><span class="pre">Log</span></code></a>.</p>
+</div>
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.LieTensor.Act">
+<span class="sig-name descname"><span class="pre">Act</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">p</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/lietensor/#LieTensor.Act"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.LieTensor.Act" title="Permalink to this definition">¶</a></dt>
+<dd><p>See <a class="reference internal" href="../pypose.Act/#pypose.Act" title="pypose.Act"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.Act()</span></code></a></p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.LieTensor.Adj">
+<span class="sig-name descname"><span class="pre">Adj</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">a</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/lietensor/#LieTensor.Adj"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.LieTensor.Adj" title="Permalink to this definition">¶</a></dt>
+<dd><p>See <a class="reference internal" href="../pypose.Adj/#pypose.Adj" title="pypose.Adj"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.Adj()</span></code></a></p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.LieTensor.AdjT">
+<span class="sig-name descname"><span class="pre">AdjT</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">a</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/lietensor/#LieTensor.AdjT"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.LieTensor.AdjT" title="Permalink to this definition">¶</a></dt>
+<dd><p>See <a class="reference internal" href="../pypose.AdjT/#pypose.AdjT" title="pypose.AdjT"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.AdjT()</span></code></a></p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.LieTensor.Exp">
+<span class="sig-name descname"><span class="pre">Exp</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/lietensor/#LieTensor.Exp"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.LieTensor.Exp" title="Permalink to this definition">¶</a></dt>
+<dd><p>See <a class="reference internal" href="../pypose.Exp/#pypose.Exp" title="pypose.Exp"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.Exp()</span></code></a></p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.LieTensor.Inv">
+<span class="sig-name descname"><span class="pre">Inv</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/lietensor/#LieTensor.Inv"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.LieTensor.Inv" title="Permalink to this definition">¶</a></dt>
+<dd><p>See <a class="reference internal" href="../pypose.Inv/#pypose.Inv" title="pypose.Inv"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.Inv()</span></code></a></p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.LieTensor.Jinvp">
+<span class="sig-name descname"><span class="pre">Jinvp</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">p</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/lietensor/#LieTensor.Jinvp"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.LieTensor.Jinvp" title="Permalink to this definition">¶</a></dt>
+<dd><p>See <a class="reference internal" href="../pypose.Jinvp/#pypose.Jinvp" title="pypose.Jinvp"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.Jinvp()</span></code></a></p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.LieTensor.Jr">
+<span class="sig-name descname"><span class="pre">Jr</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/lietensor/#LieTensor.Jr"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.LieTensor.Jr" title="Permalink to this definition">¶</a></dt>
+<dd><p>See <a class="reference internal" href="../pypose.Jr/#pypose.Jr" title="pypose.Jr"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.Jr()</span></code></a></p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.LieTensor.Log">
+<span class="sig-name descname"><span class="pre">Log</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/lietensor/#LieTensor.Log"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.LieTensor.Log" title="Permalink to this definition">¶</a></dt>
+<dd><p>See <a class="reference internal" href="../pypose.Log/#pypose.Log" title="pypose.Log"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.Log()</span></code></a></p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.LieTensor.Retr">
+<span class="sig-name descname"><span class="pre">Retr</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">a</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/lietensor/#LieTensor.Retr"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.LieTensor.Retr" title="Permalink to this definition">¶</a></dt>
+<dd><p>See <a class="reference internal" href="../pypose.Retr/#pypose.Retr" title="pypose.Retr"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.Retr()</span></code></a></p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.LieTensor.add">
+<span class="sig-name descname"><span class="pre">add</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">other</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">alpha</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/lietensor/#LieTensor.add"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.LieTensor.add" title="Permalink to this definition">¶</a></dt>
+<dd><p>See <a class="reference internal" href="../pypose.add/#pypose.add" title="pypose.add"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.add()</span></code></a></p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.LieTensor.add_">
+<span class="sig-name descname"><span class="pre">add_</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">other</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">alpha</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/lietensor/#LieTensor.add_"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.LieTensor.add_" title="Permalink to this definition">¶</a></dt>
+<dd><p>See <a class="reference internal" href="../pypose.add_/#pypose.add_" title="pypose.add_"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.add_()</span></code></a></p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.LieTensor.cummul">
+<span class="sig-name descname"><span class="pre">cummul</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">dim</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">left</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/lietensor/#LieTensor.cummul"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.LieTensor.cummul" title="Permalink to this definition">¶</a></dt>
+<dd><p>See <a class="reference internal" href="../pypose.cummul/#pypose.cummul" title="pypose.cummul"><code class="xref py py-func docutils literal notranslate"><span class="pre">pypose.cummul()</span></code></a></p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.LieTensor.cummul_">
+<span class="sig-name descname"><span class="pre">cummul_</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">dim</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">left</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/lietensor/#LieTensor.cummul_"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.LieTensor.cummul_" title="Permalink to this definition">¶</a></dt>
+<dd><p>Inplace version of <a class="reference internal" href="../pypose.cummul/#pypose.cummul" title="pypose.cummul"><code class="xref py py-func docutils literal notranslate"><span class="pre">pypose.cummul()</span></code></a></p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.LieTensor.cumops">
+<span class="sig-name descname"><span class="pre">cumops</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">dim</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">ops</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/lietensor/#LieTensor.cumops"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.LieTensor.cumops" title="Permalink to this definition">¶</a></dt>
+<dd><p>See <a class="reference internal" href="../pypose.cumops/#pypose.cumops" title="pypose.cumops"><code class="xref py py-func docutils literal notranslate"><span class="pre">pypose.cumops()</span></code></a></p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.LieTensor.cumops_">
+<span class="sig-name descname"><span class="pre">cumops_</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">dim</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">ops</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/lietensor/#LieTensor.cumops_"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.LieTensor.cumops_" title="Permalink to this definition">¶</a></dt>
+<dd><p>Inplace version of <a class="reference internal" href="../pypose.cumops/#pypose.cumops" title="pypose.cumops"><code class="xref py py-func docutils literal notranslate"><span class="pre">pypose.cumops()</span></code></a></p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.LieTensor.cumprod">
+<span class="sig-name descname"><span class="pre">cumprod</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">dim</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">left</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/lietensor/#LieTensor.cumprod"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.LieTensor.cumprod" title="Permalink to this definition">¶</a></dt>
+<dd><p>See <a class="reference internal" href="../pypose.cumprod/#pypose.cumprod" title="pypose.cumprod"><code class="xref py py-func docutils literal notranslate"><span class="pre">pypose.cumprod()</span></code></a></p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.LieTensor.cumprod_">
+<span class="sig-name descname"><span class="pre">cumprod_</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">dim</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">left</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/lietensor/#LieTensor.cumprod_"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.LieTensor.cumprod_" title="Permalink to this definition">¶</a></dt>
+<dd><p>Inplace version of <a class="reference internal" href="../pypose.cumprod/#pypose.cumprod" title="pypose.cumprod"><code class="xref py py-func docutils literal notranslate"><span class="pre">pypose.cumprod()</span></code></a></p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.LieTensor.euler">
+<span class="sig-name descname"><span class="pre">euler</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">eps</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0.0002</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/lietensor/#LieTensor.euler"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.LieTensor.euler" title="Permalink to this definition">¶</a></dt>
+<dd><p>See <a class="reference internal" href="../pypose.euler/#pypose.euler" title="pypose.euler"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.euler()</span></code></a></p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.LieTensor.identity_">
+<span class="sig-name descname"><span class="pre">identity_</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/lietensor/#LieTensor.identity_"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.LieTensor.identity_" title="Permalink to this definition">¶</a></dt>
+<dd><p>Inplace set the LieTensor to identity.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Returns<span class="colon">:</span></dt>
+<dd class="field-odd"><p>the <code class="xref py py-obj docutils literal notranslate"><span class="pre">self</span></code> LieTensor</p>
+</dd>
+<dt class="field-even">Return type<span class="colon">:</span></dt>
+<dd class="field-even"><p><a class="reference internal" href="#pypose.LieTensor" title="pypose.LieTensor">LieTensor</a></p>
+</dd>
+</dl>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The translation part, if there is, is set to zeros, while the
+rotation part is set to identity quaternion.</p>
+</div>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_SO3</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span>
+<span class="go">SO3Type LieTensor:</span>
+<span class="go">tensor([[-0.0724,  0.1970,  0.0022,  0.9777],</span>
+<span class="go">        [ 0.3492,  0.4998, -0.5310,  0.5885]])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">identity_</span><span class="p">()</span>
+<span class="go">SO3Type LieTensor:</span>
+<span class="go">tensor([[0., 0., 0., 1.],</span>
+<span class="go">        [0., 0., 0., 1.]])</span>
+</pre></div>
+</div>
+</dd></dl>
+
+<dl class="py property">
+<dt class="sig sig-object py" id="pypose.LieTensor.lshape">
+<em class="property"><span class="pre">property</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">lshape</span></span><a class="headerlink" href="#pypose.LieTensor.lshape" title="Permalink to this definition">¶</a></dt>
+<dd><p>LieTensor Shape (shape of torch.Tensor by ignoring the last dimension)</p>
+<dl class="field-list simple">
+<dt class="field-odd">Returns<span class="colon">:</span></dt>
+<dd class="field-odd"><p>torch.Size</p>
+</dd>
+</dl>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<ul class="simple">
+<li><p>The only difference from <code class="xref py py-obj docutils literal notranslate"><span class="pre">shape</span></code> is the last dimension is hidden,
+since <a class="reference internal" href="#pypose.LieTensor.lshape" title="pypose.LieTensor.lshape"><code class="xref py py-obj docutils literal notranslate"><span class="pre">lshape</span></code></a> takes the last dimension as a single <code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code> item.</p></li>
+<li><p>The last dimension can also be accessed via <code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor.ltype.dimension</span></code>.</p></li>
+</ul>
+</div>
+<p class="rubric">Examples</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_SE3</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">lshape</span>
+<span class="go">torch.Size([2])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">shape</span>
+<span class="go">torch.Size([2, 7])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">ltype</span><span class="o">.</span><span class="n">dimension</span>
+<span class="go">torch.Size([7])</span>
+</pre></div>
+</div>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.LieTensor.lview">
+<span class="sig-name descname"><span class="pre">lview</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">shape</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/lietensor/#LieTensor.lview"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.LieTensor.lview" title="Permalink to this definition">¶</a></dt>
+<dd><p>Returns a new LieTensor with the same data as the self tensor but of a different
+<a class="reference internal" href="#pypose.LieTensor.lshape" title="pypose.LieTensor.lshape"><code class="xref py py-obj docutils literal notranslate"><span class="pre">lshape</span></code></a>.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>shape</strong> (<em>torch.Size</em><em> or </em><em>int</em><em>...</em>) – the desired size</p>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>A new lieGroup tensor sharing with the same data as the self tensor but of a
+different shape.</p>
+</dd>
+</dl>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The only difference from <code class="xref py py-meth docutils literal notranslate"><span class="pre">view()</span></code> is the last dimension is hidden.</p>
+<p>See <a class="reference external" href="https://tinyurl.com/mrds8nmd">Tensor.view</a>
+for its usage.</p>
+</div>
+<p class="rubric">Examples</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_so3</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">shape</span>
+<span class="go">torch.Size([2, 2, 3])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">lview</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">lshape</span>
+<span class="go">torch.Size([4])</span>
+</pre></div>
+</div>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.LieTensor.matrix">
+<span class="sig-name descname"><span class="pre">matrix</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/lietensor/#LieTensor.matrix"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.LieTensor.matrix" title="Permalink to this definition">¶</a></dt>
+<dd><p>See <a class="reference internal" href="../pypose.matrix/#pypose.matrix" title="pypose.matrix"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.matrix()</span></code></a></p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.LieTensor.mul">
+<span class="sig-name descname"><span class="pre">mul</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">other</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/lietensor/#LieTensor.mul"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.LieTensor.mul" title="Permalink to this definition">¶</a></dt>
+<dd><p>See <a class="reference internal" href="../pypose.mul/#pypose.mul" title="pypose.mul"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.mul()</span></code></a></p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.LieTensor.new_empty">
+<span class="sig-name descname"><span class="pre">new_empty</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">size</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">dtype</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">device</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">requires_grad</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">layout</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">torch.strided</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">pin_memory</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">&#x2192;</span> <span class="sig-return-typehint"><span class="pre">Tensor</span></span></span><a class="reference internal" href="../../_modules/pypose/lietensor/lietensor/#LieTensor.new_empty"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.LieTensor.new_empty" title="Permalink to this definition">¶</a></dt>
+<dd><p>Returns a Tensor of size <code class="xref py py-attr docutils literal notranslate"><span class="pre">size</span></code> filled with uninitialized data.
+By default, the returned Tensor has the same <code class="xref py py-class docutils literal notranslate"><span class="pre">torch.dtype</span></code> and
+<code class="xref py py-class docutils literal notranslate"><span class="pre">torch.device</span></code> as this tensor.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>size</strong> (<em>int</em><em>...</em>) – a list, tuple, or <code class="xref py py-class docutils literal notranslate"><span class="pre">torch.Size</span></code> of integers defining the
+shape of the output tensor.</p>
+</dd>
+<dt class="field-even">Keyword Arguments<span class="colon">:</span></dt>
+<dd class="field-even"><ul class="simple">
+<li><p><strong>dtype</strong> (<code class="xref py py-class docutils literal notranslate"><span class="pre">torch.dtype</span></code>, optional) – the desired type of returned tensor.
+Default: if None, same <code class="xref py py-class docutils literal notranslate"><span class="pre">torch.dtype</span></code> as this tensor.</p></li>
+<li><p><strong>device</strong> (<code class="xref py py-class docutils literal notranslate"><span class="pre">torch.device</span></code>, optional) – the desired device of returned tensor.
+Default: if None, same <code class="xref py py-class docutils literal notranslate"><span class="pre">torch.device</span></code> as this tensor.</p></li>
+<li><p><strong>requires_grad</strong> (<em>bool</em><em>, </em><em>optional</em>) – If autograd should record operations on the
+returned tensor. Default: <code class="docutils literal notranslate"><span class="pre">False</span></code>.</p></li>
+<li><p><strong>layout</strong> (<code class="xref py py-class docutils literal notranslate"><span class="pre">torch.layout</span></code>, optional) – the desired layout of returned Tensor.
+Default: <code class="docutils literal notranslate"><span class="pre">torch.strided</span></code>.</p></li>
+<li><p><strong>pin_memory</strong> (<em>bool</em><em>, </em><em>optional</em>) – If set, returned tensor would be allocated in
+the pinned memory. Works only for CPU tensors. Default: <code class="docutils literal notranslate"><span class="pre">False</span></code>.</p></li>
+</ul>
+</dd>
+</dl>
+<p>Example:</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">tensor</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">ones</span><span class="p">(())</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">tensor</span><span class="o">.</span><span class="n">new_empty</span><span class="p">((</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
+<span class="go">tensor([[ 5.8182e-18,  4.5765e-41, -1.0545e+30],</span>
+<span class="go">        [ 3.0949e-41,  4.4842e-44,  0.0000e+00]])</span>
+</pre></div>
+</div>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.LieTensor.rotation">
+<span class="sig-name descname"><span class="pre">rotation</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/lietensor/#LieTensor.rotation"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.LieTensor.rotation" title="Permalink to this definition">¶</a></dt>
+<dd><p>See <a class="reference internal" href="../pypose.rotation/#pypose.rotation" title="pypose.rotation"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.rotation()</span></code></a></p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.LieTensor.scale">
+<span class="sig-name descname"><span class="pre">scale</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/lietensor/#LieTensor.scale"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.LieTensor.scale" title="Permalink to this definition">¶</a></dt>
+<dd><p>See <a class="reference internal" href="../pypose.scale/#pypose.scale" title="pypose.scale"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.scale()</span></code></a></p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.LieTensor.tensor">
+<span class="sig-name descname"><span class="pre">tensor</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/lietensor/#LieTensor.tensor"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.LieTensor.tensor" title="Permalink to this definition">¶</a></dt>
+<dd><p>See <a class="reference internal" href="../pypose.tensor/#pypose.tensor" title="pypose.tensor"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.tensor()</span></code></a></p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.LieTensor.translation">
+<span class="sig-name descname"><span class="pre">translation</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/lietensor/#LieTensor.translation"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.LieTensor.translation" title="Permalink to this definition">¶</a></dt>
+<dd><p>See <a class="reference internal" href="../pypose.translation/#pypose.translation" title="pypose.translation"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.translation()</span></code></a></p>
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+              
+  <section id="pypose-log">
+<h1>pypose.Log<a class="headerlink" href="#pypose-log" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.Log">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">Log</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/utils/#Log"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.Log" title="Permalink to this definition">¶</a></dt>
+<dd><p>The Logarithm map for <a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code></a> (Lie Group).</p>
+<div class="math">
+\[\mathrm{Log}: \mathcal{G} \mapsto \mathcal{g}
+
+\]</div>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>input</strong> (<a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><em>LieTensor</em></a>) – the input LieTensor (Lie Group)</p>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>the output LieTensor (Lie Algebra)</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor">LieTensor</a></p>
+</dd>
+</dl>
+<table class="docutils align-default" id="id1">
+<caption><span class="caption-text">List of supported <span class="math">\(\mathrm{Log}\)</span> map</span><a class="headerlink" href="#id1" title="Permalink to this table">¶</a></caption>
+<colgroup>
+<col style="width: 23%" />
+<col style="width: 23%" />
+<col style="width: 9%" />
+<col style="width: 23%" />
+<col style="width: 23%" />
+</colgroup>
+<thead>
+<tr class="row-odd"><th class="head"><p>input <code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code></p></th>
+<th class="head"><p><span class="math">\(\mathcal{G}\)</span> (Lie Group)</p></th>
+<th class="head"><p><span class="math">\(\mapsto\)</span></p></th>
+<th class="head"><p><span class="math">\(\mathcal{g}\)</span> (Lie Algebra)</p></th>
+<th class="head"><p>output <code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code></p></th>
+</tr>
+</thead>
+<tbody>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">SO3_type</span></code></p></td>
+<td><p><span class="math">\(\mathcal{G}\in\mathbb{R}^{*\times4}\)</span></p></td>
+<td><p><span class="math">\(\mapsto\)</span></p></td>
+<td><p><span class="math">\(\mathcal{g}\in\mathbb{R}^{*\times3}\)</span></p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">so3_type</span></code></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">SE3_type</span></code></p></td>
+<td><p><span class="math">\(\mathcal{G}\in\mathbb{R}^{*\times7}\)</span></p></td>
+<td><p><span class="math">\(\mapsto\)</span></p></td>
+<td><p><span class="math">\(\mathcal{g}\in\mathbb{R}^{*\times6}\)</span></p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">se3_type</span></code></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Sim3_type</span></code></p></td>
+<td><p><span class="math">\(\mathcal{G}\in\mathbb{R}^{*\times8}\)</span></p></td>
+<td><p><span class="math">\(\mapsto\)</span></p></td>
+<td><p><span class="math">\(\mathcal{g}\in\mathbb{R}^{*\times7}\)</span></p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">sim3_type</span></code></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">RxSO3_type</span></code></p></td>
+<td><p><span class="math">\(\mathcal{G}\in\mathbb{R}^{*\times5}\)</span></p></td>
+<td><p><span class="math">\(\mapsto\)</span></p></td>
+<td><p><span class="math">\(\mathcal{g}\in\mathbb{R}^{*\times4}\)</span></p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">rxso3_type</span></code></p></td>
+</tr>
+</tbody>
+</table>
+<div class="admonition warning">
+<p class="admonition-title">Warning</p>
+<p>This function <a class="reference internal" href="#pypose.Log" title="pypose.Log"><code class="xref py py-func docutils literal notranslate"><span class="pre">Log()</span></code></a> is different from <code class="xref py py-func docutils literal notranslate"><span class="pre">log()</span></code>, which returns
+a new torch tensor with the logarithm of the elements of the input tensor.</p>
+</div>
+<ul>
+<li><p>If input <span class="math">\(\mathbf{x}\)</span>’s <code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code> is <code class="xref py py-obj docutils literal notranslate"><span class="pre">SO3_type</span></code>
+(input <span class="math">\(\mathbf{x}\)</span> is an instance of <a class="reference internal" href="../pypose.SO3/#pypose.SO3" title="pypose.SO3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">SO3()</span></code></a>):</p>
+<blockquote>
+<div><p>Let <span class="math">\(w_i\)</span>, <span class="math">\(\boldsymbol{\nu}_i\)</span> be the scalar and vector parts of
+<span class="math">\(\mathbf{x}_i\)</span>, respectively; <span class="math">\(\mathbf{y}\)</span> be the output.</p>
+<p>If <span class="math">\(\|\boldsymbol{\nu}_i\| &gt; \text{eps}\)</span>:</p>
+<div class="math">
+\[\mathbf{y}_i =
+    \left\{
+        \begin{array}{ll}
+            2\frac{\mathrm{arctan}(\|\boldsymbol{\nu}_i\|/w_i)}{\|
+            \boldsymbol{\nu}_i\|}\boldsymbol{\nu}_i,
+                \quad \|w_i\| > \text{eps}, \\
+            \mathrm{pm}(w_i) \frac{\pi}{\|\boldsymbol{\nu}_i\|}
+                \boldsymbol{\nu}_i, \quad \|w_i\| \leq \text{eps},
+        \end{array}
+    \right.
+
+\]</div>
+<p>otherwise:</p>
+<div class="math">
+\[\mathbf{y}_i = 2\left( \frac{1}{w_i} -
+    \frac{\|\boldsymbol{\nu}_i\|^2}{3w_i^3}\right)\boldsymbol{\nu}_i.
+
+\]</div>
+<p>where <span class="math">\(\mathrm{pm}\)</span> is the plus or minus (<span class="math">\(\pm\)</span>) operator
+(refer to <a class="reference internal" href="../pypose.pm/#pypose.pm" title="pypose.pm"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pm()</span></code></a>).</p>
+</div></blockquote>
+</li>
+<li><p>If input <span class="math">\(\mathbf{x}\)</span>’s <code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code> is <code class="xref py py-obj docutils literal notranslate"><span class="pre">SE3_type</span></code>
+(input <span class="math">\(\mathbf{x}\)</span> is an instance of <a class="reference internal" href="../pypose.SE3/#pypose.SE3" title="pypose.SE3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">SE3()</span></code></a>):</p>
+<blockquote>
+<div><p>Let <span class="math">\(\mathbf{t}_i\)</span>, <span class="math">\(\mathbf{q}_i\)</span> be the translation and rotation parts of
+<span class="math">\(\mathbf{x}_i\)</span>, respectively; <span class="math">\(\mathbf{y}\)</span> be the output.</p>
+<div class="math">
+\[\mathbf{y}_i = \left[\mathbf{J}_i^{-1}\mathbf{t}_i, \mathrm{Log}(\mathbf{q}_i) \right],
+
+\]</div>
+<p>where <span class="math">\(\mathrm{Log}\)</span> is the Logarithm map for <code class="xref py py-obj docutils literal notranslate"><span class="pre">SO3_type</span></code> input and
+<span class="math">\(\mathbf{J}_i\)</span> is the left Jacobian for <code class="xref py py-obj docutils literal notranslate"><span class="pre">SO3_type</span></code> input.</p>
+</div></blockquote>
+</li>
+<li><p>If input <span class="math">\(\mathbf{x}\)</span>’s <code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code> is <code class="xref py py-obj docutils literal notranslate"><span class="pre">RxSO3_type</span></code>
+(input <span class="math">\(\mathbf{x}\)</span> is an instance of <a class="reference internal" href="../pypose.RxSO3/#pypose.RxSO3" title="pypose.RxSO3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">RxSO3()</span></code></a>):</p>
+<blockquote>
+<div><p>Let <span class="math">\(\mathbf{q}_i\)</span>, <span class="math">\(s_i\)</span> be the rotation and scale parts of
+<span class="math">\(\mathbf{x}_i\)</span>, respectively; <span class="math">\(\mathbf{y}\)</span> be the output.</p>
+<div class="math">
+\[\mathbf{y}_i = \left[\mathrm{Log}(\mathbf{q}_i), \log(s_i) \right].
+
+\]</div>
+</div></blockquote>
+</li>
+<li><p>If input <span class="math">\(\mathbf{x}\)</span>’s <code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code> is <code class="xref py py-obj docutils literal notranslate"><span class="pre">Sim3_type</span></code> (input <span class="math">\(\mathbf{x}\)</span>
+is an instance of <a class="reference internal" href="../pypose.Sim3/#pypose.Sim3" title="pypose.Sim3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">Sim3()</span></code></a>):</p>
+<blockquote>
+<div><p>Let <span class="math">\(\mathbf{t}_i\)</span>, <span class="math">\(^s\mathbf{q}_i\)</span> be the translation and <a class="reference internal" href="../pypose.RxSO3/#pypose.RxSO3" title="pypose.RxSO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">RxSO3</span></code></a>
+parts of <span class="math">\(\mathbf{x}_i\)</span>, respectively; <span class="math">\(\boldsymbol{\phi}_i\)</span>,
+<span class="math">\(\sigma_i\)</span> be the corresponding Lie Algebra of the SO3 and scale part of
+<span class="math">\(^s\mathbf{q}_i\)</span>, <span class="math">\(\boldsymbol{\Phi}_i\)</span> be the skew matrix of
+<span class="math">\(\boldsymbol{\phi}_i\)</span>, <span class="math">\(\boldsymbol{\phi}_i\)</span> can be represented as
+<span class="math">\(\theta_i\mathbf{n}_i\)</span>, <span class="math">\(s_i = e^\sigma_i\)</span>, <span class="math">\(\mathbf{y}\)</span> be the
+output.</p>
+<div class="math">
+\[\mathbf{y}_i = \left[^s\mathbf{W}_i^{-1}\mathbf{t}_i,
+    \mathrm{Log}(^s\mathbf{q}_i) \right],
+
+\]</div>
+<p>where</p>
+<blockquote>
+<div><div class="math">
+\[^s\mathbf{W}_i = A\boldsymbol{\Phi}_i + B\boldsymbol{\Phi}_i^2 + C\mathbf{I}
+
+\]</div>
+</div></blockquote>
+<p>in which if <span class="math">\(\|\sigma_i\| &gt; \text{eps}\)</span>:</p>
+<div class="math">
+\[A = \left\{
+        \begin{array}{ll}
+            \frac{s_i\sin\theta_i\sigma_i + (1-s_i\cos\theta_i)\theta_i}
+            {\theta_i(\sigma_i^2 + \theta_i^2)}, \quad \|\theta_i\| > \text{eps}, \\
+            \frac{(\sigma_i-1)s_i+1}{\sigma_i^2}, \quad \|\theta_i\| \leq \text{eps},
+        \end{array}
+    \right.
+
+\]</div>
+<div class="math">
+\[B =
+\left\{
+    \begin{array}{ll}
+        \left( C - \frac{(s_i\cos\theta_i-1)\sigma+ s_i\sin\theta_i\sigma_i}
+        {\theta_i^2+\sigma_i^2}\right)\frac{1}{\theta_i^2},
+            \quad \|\theta_i\| > \text{eps}, \\
+        \frac{s_i\sigma_i^2/2 + s_i-1-\sigma_i s_i}{\sigma_i^3},
+            \quad \|\theta_i\| \leq \text{eps},
+    \end{array}
+\right.
+
+\]</div>
+<div class="math">
+\[C = \frac{e^{\sigma_i} - 1}{\sigma_i}\mathbf{I}
+
+\]</div>
+<p>otherwise:</p>
+<div class="math">
+\[A = \left\{
+        \begin{array}{ll}
+            \frac{1-\cos\theta_i}{\theta_i^2}, \quad \|\theta_i\| > \text{eps}, \\
+            \frac{1}{2}, \quad \|\theta_i\| \leq \text{eps},
+        \end{array}
+    \right.
+
+\]</div>
+<div class="math">
+\[B = \left\{
+        \begin{array}{ll}
+            \frac{\theta_i - \sin\theta_i}{\theta_i^3},
+                \quad \|\theta_i\| > \text{eps}, \\
+            \frac{1}{6}, \quad \|\theta_i\| \leq \text{eps},
+        \end{array}
+    \right.
+
+\]</div>
+<div class="math">
+\[C = 1
+
+\]</div>
+</div></blockquote>
+</li>
+</ul>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The <span class="math">\(\mathrm{arctan}\)</span>-based Logarithm map implementation thanks to the paper:</p>
+<ul class="simple">
+<li><p>C. Hertzberg et al., <a class="reference external" href="https://doi.org/10.1016/j.inffus.2011.08.003">Integrating Generic Sensor Fusion Algorithms with Sound State
+Representation through Encapsulation of Manifolds</a>, Information Fusion, 2013.</p></li>
+</ul>
+<p>Assume we have a unit rotation axis <span class="math">\(\mathbf{n}\)</span> and rotation angle
+<span class="math">\(\theta~(0\leq\theta&lt;2\pi)\)</span>, then the corresponding quaternion with
+unit norm <span class="math">\(\mathbf{q}\)</span> can be represented as</p>
+<blockquote>
+<div><div class="math">
+\[\mathbf{q} = \left[\sin(\theta/2) \mathbf{n}, \cos(\theta/2) \right]
+
+\]</div>
+</div></blockquote>
+<p>Therefore, given a quaternion <span class="math">\(\mathbf{q}=[\boldsymbol{\nu}, w]\)</span>, where
+<span class="math">\(\boldsymbol{\nu}\)</span> is the vector part, <span class="math">\(w\)</span> is the scalar part, to find
+the corresponding rotation vector, the rotation angle <span class="math">\(\theta\)</span> can be obtained as</p>
+<blockquote>
+<div><div class="math">
+\[\theta = 2\mathrm{arctan}(\|\boldsymbol{\nu}\|/w),~\|\boldsymbol{\nu}\|
+= \sin(\theta/2),
+
+\]</div>
+</div></blockquote>
+<p>The unit rotation axis <span class="math">\(\mathbf{n}\)</span> can be obtained as <span class="math">\(\mathbf{n} =
+\frac{\boldsymbol{\nu}}{{\|\boldsymbol{\nu}\|}}\)</span>.
+Hence, the corresponding rotation vector is</p>
+<blockquote>
+<div><div class="math">
+\[\theta \mathbf{n} = 2\frac{\mathrm{arctan}
+(\|\boldsymbol{\nu}\|/w)}{\|\boldsymbol{\nu}\|}\boldsymbol{\nu}.
+
+\]</div>
+</div></blockquote>
+<p>More details about <span class="math">\(^s\mathbf{W}_i\)</span> in <code class="xref py py-obj docutils literal notranslate"><span class="pre">Sim3_type</span></code> can be found in Eq. (5.7):</p>
+<ul class="simple">
+<li><p>H. Strasdat, <a class="reference external" href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.640.199&amp;rep=rep1&amp;type=pdf">Local accuracy and global consistency for efficient visual SLAM</a>,
+Dissertation. Department of Computing, Imperial College London, 2012.</p></li>
+</ul>
+</div>
+<p class="rubric">Example</p>
+<ul>
+<li><p><span class="math">\(\mathrm{Log}\)</span>: <a class="reference internal" href="../pypose.SO3/#pypose.SO3" title="pypose.SO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">SO3</span></code></a> <span class="math">\(\mapsto\)</span> <a class="reference internal" href="../pypose.so3/#pypose.so3" title="pypose.so3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">so3</span></code></a></p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_SO3</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">Log</span><span class="p">()</span> <span class="c1"># equivalent to: pp.Log(x)</span>
+<span class="go">so3Type LieTensor:</span>
+<span class="go">tensor([-0.3060,  0.2344,  1.2724])</span>
+</pre></div>
+</div>
+</li>
+<li><p><span class="math">\(\mathrm{Log}\)</span>: <a class="reference internal" href="../pypose.SE3/#pypose.SE3" title="pypose.SE3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">SE3</span></code></a> <span class="math">\(\mapsto\)</span> <a class="reference internal" href="../pypose.se3/#pypose.se3" title="pypose.se3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">se3</span></code></a></p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_SE3</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">Log</span><span class="p">()</span> <span class="c1"># equivalent to: pp.Log(x)</span>
+<span class="go">se3Type LieTensor:</span>
+<span class="go">tensor([[ 0.2958, -0.0840, -1.4733,  0.7004,  0.4483, -0.9009],</span>
+<span class="go">        [ 0.0850, -0.1020, -1.2616, -1.0524, -1.2031,  0.8377]])</span>
+</pre></div>
+</div>
+</li>
+<li><p><span class="math">\(\mathrm{Log}\)</span>: <a class="reference internal" href="../pypose.RxSO3/#pypose.RxSO3" title="pypose.RxSO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">RxSO3</span></code></a> <span class="math">\(\mapsto\)</span> <a class="reference internal" href="../pypose.rxso3/#pypose.rxso3" title="pypose.rxso3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">rxso3</span></code></a></p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_RxSO3</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">Log</span><span class="p">()</span> <span class="c1"># equivalent to: pp.Log(x)</span>
+<span class="go">rxso3Type LieTensor:</span>
+<span class="go">tensor([[-1.3755,  0.3525, -2.2367,  0.5409],</span>
+<span class="go">        [ 0.5929, -0.3250, -0.7394,  1.0965]])</span>
+</pre></div>
+</div>
+</li>
+<li><p><span class="math">\(\mathrm{Log}\)</span>: <a class="reference internal" href="../pypose.Sim3/#pypose.Sim3" title="pypose.Sim3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">Sim3</span></code></a> <span class="math">\(\mapsto\)</span> <a class="reference internal" href="../pypose.sim3/#pypose.sim3" title="pypose.sim3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">sim3</span></code></a></p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_Sim3</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">Log</span><span class="p">()</span> <span class="c1"># equivalent to: pp.Log(x)</span>
+<span class="go">sim3Type LieTensor:</span>
+<span class="go">tensor([[-0.1747, -0.3698,  0.2000,  0.1735,  0.6220,  1.1852, -0.6402],</span>
+<span class="go">        [-0.8685, -0.1717,  1.2139, -0.8385, -2.2957, -1.9545,  0.8474]])</span>
+</pre></div>
+</div>
+</li>
+</ul>
+</dd></dl>
+
+</section>
+
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+              <p class="caption" role="heading"><span class="caption-text">Contents:</span></p>
+<ul class="current">
+<li class="toctree-l1 current"><a class="reference internal" href="../../lietensor/">LieTensor</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../func/">Func</a></li>
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+              
+  <section id="pypose-parameter">
+<h1>pypose.Parameter<a class="headerlink" href="#pypose-parameter" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.Parameter">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">Parameter</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">data</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">requires_grad</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/lietensor/#Parameter"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.Parameter" title="Permalink to this definition">¶</a></dt>
+<dd><p>A kind of LieTensor that is to be considered a module parameter.</p>
+<p>Parameters are of <a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-meth docutils literal notranslate"><span class="pre">LieTensor()</span></code></a> and <code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.nn.Parameter()</span></code>,
+that have a very special property when used with Modules: when
+they are assigned as Module attributes they are automatically
+added to the list of its parameters, and will appear, e.g., in
+<code class="xref py py-meth docutils literal notranslate"><span class="pre">parameters()</span></code> iterator.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>data</strong> (<a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><em>LieTensor</em></a>) – parameter LieTensor.</p></li>
+<li><p><strong>requires_grad</strong> (<em>bool</em><em>, </em><em>optional</em>) – if the parameter requires
+gradient. Default: <code class="docutils literal notranslate"><span class="pre">True</span></code></p></li>
+</ul>
+</dd>
+</dl>
+<p class="rubric">Examples</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">torch</span><span class="o">,</span> <span class="nn">pypose</span> <span class="k">as</span> <span class="nn">pp</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">Parameter</span><span class="p">(</span><span class="n">pp</span><span class="o">.</span><span class="n">randn_SO3</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">Log</span><span class="p">()</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span><span class="o">.</span><span class="n">backward</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">grad</span>
+<span class="go">tensor([[0.8590, 1.4069, 0.6261, 0.0000],</span>
+<span class="go">        [1.2869, 1.0748, 0.5385, 0.0000]])</span>
+</pre></div>
+</div>
+</dd></dl>
+
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+  <section id="pypose-retr">
+<h1>pypose.Retr<a class="headerlink" href="#pypose-retr" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.Retr">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">Retr</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">X</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">a</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/utils/#Retr"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.Retr" title="Permalink to this definition">¶</a></dt>
+<dd><p>Perform batched retraction with a given direction.</p>
+<div class="math">
+\[Y_i = \mathrm{Exp}(a_i) * X_i,
+
+\]</div>
+<p>where <span class="math">\(\mathrm{Exp}\)</span> means the exponetial map. See <a class="reference internal" href="../pypose.Exp/#pypose.Exp" title="pypose.Exp"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.Exp</span></code></a> for more details.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>X</strong> (<a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><em>LieTensor</em></a>) – the input LieTensor to retract (Lie Group)</p></li>
+<li><p><strong>a</strong> (<a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><em>LieTensor</em></a>) – the direction of the retraction (Lie Algebra)</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>The retraction of the inputs (Lie Group)</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor">LieTensor</a></p>
+</dd>
+</dl>
+<p class="rubric">Examples</p>
+<ul>
+<li><p><span class="math">\(\mathrm{Retr}\)</span>: (<a class="reference internal" href="../pypose.SO3/#pypose.SO3" title="pypose.SO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">SO3</span></code></a>, <a class="reference internal" href="../pypose.so3/#pypose.so3" title="pypose.so3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">so3</span></code></a>) <span class="math">\(\mapsto\)</span> <a class="reference internal" href="../pypose.SO3/#pypose.SO3" title="pypose.SO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">SO3</span></code></a></p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_so3</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_SO3</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span><span class="o">.</span><span class="n">Retr</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="c1"># equivalent to: pp.Retr(X, a)</span>
+<span class="go">SO3Type LieTensor:</span>
+<span class="go">tensor([0.6399, 0.0898, 0.1656, 0.7451])</span>
+</pre></div>
+</div>
+</li>
+<li><p><span class="math">\(\mathrm{Retr}\)</span>: (<a class="reference internal" href="../pypose.SE3/#pypose.SE3" title="pypose.SE3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">SE3</span></code></a>, <a class="reference internal" href="../pypose.se3/#pypose.se3" title="pypose.se3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">se3</span></code></a>) <span class="math">\(\mapsto\)</span> <a class="reference internal" href="../pypose.SE3/#pypose.SE3" title="pypose.SE3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">SE3</span></code></a></p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_se3</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_SE3</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span><span class="o">.</span><span class="n">Retr</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>  <span class="c1"># equivalent to: pp.Retr(X, a)</span>
+<span class="go">SE3Type LieTensor:</span>
+<span class="go">tensor([-0.6754,  1.8240,  0.2109, -0.4649, -0.7254, -0.0943,  0.4987])</span>
+</pre></div>
+</div>
+</li>
+<li><p><span class="math">\(\mathrm{Retr}\)</span>: (<a class="reference internal" href="../pypose.Sim3/#pypose.Sim3" title="pypose.Sim3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">Sim3</span></code></a>, <a class="reference internal" href="../pypose.sim3/#pypose.sim3" title="pypose.sim3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">sim3</span></code></a>) <span class="math">\(\mapsto\)</span> <a class="reference internal" href="../pypose.Sim3/#pypose.Sim3" title="pypose.Sim3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">Sim3</span></code></a></p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_sim3</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_Sim3</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span><span class="o">.</span><span class="n">Retr</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>  <span class="c1"># equivalent to: pp.Retr(X, a)</span>
+<span class="go">Sim3Type LieTensor:</span>
+<span class="go">tensor([-0.6057, -1.6370,  1.1379,  0.7037,  0.6164,  0.3525, -0.0262,  0.3141])</span>
+</pre></div>
+</div>
+</li>
+<li><p><span class="math">\(\mathrm{Retr}\)</span>: (<a class="reference internal" href="../pypose.RxSO3/#pypose.RxSO3" title="pypose.RxSO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">RxSO3</span></code></a>, <code class="xref py py-obj docutils literal notranslate"><span class="pre">rxsso3</span></code>) <span class="math">\(\mapsto\)</span> <a class="reference internal" href="../pypose.RxSO3/#pypose.RxSO3" title="pypose.RxSO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">RxSO3</span></code></a></p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_rxso3</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_RxSO3</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span><span class="o">.</span><span class="n">Retr</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>  <span class="c1"># equivalent to: pp.Retr(X, a)</span>
+<span class="go">RxSO3Type LieTensor:</span>
+<span class="go">tensor([-0.0787,  0.4052, -0.7509,  0.5155,  0.1217])</span>
+</pre></div>
+</div>
+</li>
+</ul>
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+              
+  <section id="pypose-rxso3">
+<h1>pypose.RxSO3<a class="headerlink" href="#pypose-rxso3" title="Permalink to this heading">¶</a></h1>
+<dl class="py data">
+<dt class="sig sig-object py" id="pypose.RxSO3">
+<span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">RxSO3</span></span><em class="property"><span class="w"> </span><span class="p"><span class="pre">=</span></span><span class="w"> </span><span class="pre">functools.partial(&lt;class</span> <span class="pre">'pypose.lietensor.lietensor.LieTensor'&gt;,</span> <span class="pre">ltype=&lt;pypose.lietensor.lietensor.RxSO3Type</span> <span class="pre">object&gt;)</span></em><a class="headerlink" href="#pypose.RxSO3" title="Permalink to this definition">¶</a></dt>
+<dd><p>Alias of RxSO3 type <a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code></a>.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>data</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>, or <code class="xref py py-obj docutils literal notranslate"><span class="pre">list</span></code>, or ‘<code class="xref py py-obj docutils literal notranslate"><span class="pre">int</span></code>…’) – <p>A
+<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code> object, or constructing a <code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>
+object from <code class="xref py py-obj docutils literal notranslate"><span class="pre">list</span></code>, which defines tensor data (see below), or
+from ‘<code class="xref py py-obj docutils literal notranslate"><span class="pre">int</span></code>…’, which defines tensor shape.</p>
+<p>The shape of <code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code> object must be <code class="docutils literal notranslate"><span class="pre">(*,</span> <span class="pre">5)</span></code>,
+where <code class="docutils literal notranslate"><span class="pre">*</span></code> is empty, one, or more batched dimensions (the
+<a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor.lshape" title="pypose.LieTensor.lshape"><code class="xref py py-obj docutils literal notranslate"><span class="pre">lshape</span></code></a> of this LieTensor), otherwise error will
+be raised.</p>
+</p>
+</dd>
+</dl>
+<p>Internally, RxSO3 LieTensors are stored by concatenating the unit quaternion
+representing the rotation with a scaling factor:</p>
+<div class="math">
+\[\mathrm{data}[*, :] = [q_x, q_y, q_z, q_w, s],
+
+\]</div>
+<p>where <span class="math">\(\begin{pmatrix} q_x &amp; q_y &amp; q_z &amp; q_w \end{pmatrix}^T\)</span>
+is the unit quaternion as in <a class="reference internal" href="../pypose.SO3/#pypose.SO3" title="pypose.SO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.SO3</span></code></a> and
+<span class="math">\(s \in \mathbb{R}\)</span> is the scaling factor.</p>
+<p class="rubric">Examples</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">RxSO3</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>
+<span class="go">RxSO3Type LieTensor:</span>
+<span class="go">tensor([[-0.3693,  2.5155, -0.5384, -0.8119, -0.4798],</span>
+<span class="go">        [-0.4058, -0.5909, -0.4918, -0.2994,  0.5440]])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">RxSO3</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
+<span class="go">RxSO3Type LieTensor:</span>
+<span class="go">tensor([0., 0., 0., 1., 1.])</span>
+</pre></div>
+</div>
+<p>If <code class="docutils literal notranslate"><span class="pre">data</span></code> is tensor-like, the last dimension should correspond to the
+5 elements of the above embedding.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>It is not advised to construct RxSO3 Tensors by specifying storage
+sizes with ‘<code class="xref py py-obj docutils literal notranslate"><span class="pre">int</span></code>…’, which does not initialize data.</p>
+<p>Consider using <a class="reference internal" href="../pypose.randn_RxSO3/#pypose.randn_RxSO3" title="pypose.randn_RxSO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.randn_RxSO3</span></code></a> or
+<a class="reference internal" href="../pypose.identity_RxSO3/#pypose.identity_RxSO3" title="pypose.identity_RxSO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.identity_RxSO3</span></code></a> instead.</p>
+</div>
+<p>See <a class="reference internal" href="../pypose.Log/#pypose.Log" title="pypose.Log"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.Log</span></code></a>, <a class="reference internal" href="../pypose.Inv/#pypose.Inv" title="pypose.Inv"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.Inv</span></code></a>, <a class="reference internal" href="../pypose.Act/#pypose.Act" title="pypose.Act"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.Act</span></code></a>, <a class="reference internal" href="../pypose.Retr/#pypose.Retr" title="pypose.Retr"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.Retr</span></code></a>, <a class="reference internal" href="../pypose.Adj/#pypose.Adj" title="pypose.Adj"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.Adj</span></code></a>, <a class="reference internal" href="../pypose.AdjT/#pypose.AdjT" title="pypose.AdjT"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.AdjT</span></code></a>, <a class="reference internal" href="../pypose.Jinvp/#pypose.Jinvp" title="pypose.Jinvp"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.Jinvp</span></code></a> for
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+              
+  <section id="pypose-se3">
+<h1>pypose.SE3<a class="headerlink" href="#pypose-se3" title="Permalink to this heading">¶</a></h1>
+<dl class="py data">
+<dt class="sig sig-object py" id="pypose.SE3">
+<span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">SE3</span></span><em class="property"><span class="w"> </span><span class="p"><span class="pre">=</span></span><span class="w"> </span><span class="pre">functools.partial(&lt;class</span> <span class="pre">'pypose.lietensor.lietensor.LieTensor'&gt;,</span> <span class="pre">ltype=&lt;pypose.lietensor.lietensor.SE3Type</span> <span class="pre">object&gt;)</span></em><a class="headerlink" href="#pypose.SE3" title="Permalink to this definition">¶</a></dt>
+<dd><p>Alias of SE3 type <a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code></a>.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>data</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>, or <code class="xref py py-obj docutils literal notranslate"><span class="pre">list</span></code>, or ‘<code class="xref py py-obj docutils literal notranslate"><span class="pre">int</span></code>…’) – <p>A
+<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code> object, or constructing a <code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>
+object from <code class="xref py py-obj docutils literal notranslate"><span class="pre">list</span></code>, which defines tensor data (see below), or
+from ‘<code class="xref py py-obj docutils literal notranslate"><span class="pre">int</span></code>…’, which defines tensor shape.</p>
+<p>The shape of <code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code> object must be <code class="docutils literal notranslate"><span class="pre">(*,</span> <span class="pre">7)</span></code>,
+where <code class="docutils literal notranslate"><span class="pre">*</span></code> is empty, one, or more batched dimensions (the
+<a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor.lshape" title="pypose.LieTensor.lshape"><code class="xref py py-obj docutils literal notranslate"><span class="pre">lshape</span></code></a> of this LieTensor), otherwise error will
+be raised.</p>
+</p>
+</dd>
+</dl>
+<p>Internally, SE3 LieTensors are stored by concatenating the unit quaternion
+representing the rotation with a vector representing the translation.</p>
+<div class="math">
+\[\mathrm{data}[*, :] = [t_x, t_y, t_z, q_x, q_y, q_z, q_w],
+
+\]</div>
+<p>where <span class="math">\(\begin{pmatrix} t_x &amp; t_y &amp; t_z \end{pmatrix}\top \in \mathbb{R}^3\)</span> is
+the translation and
+<span class="math">\(\begin{pmatrix} q_x &amp; q_y &amp; q_z &amp; q_w \end{pmatrix}\top\)</span> is the unit
+quaternion as in <a class="reference internal" href="../pypose.SO3/#pypose.SO3" title="pypose.SO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.SO3</span></code></a>.</p>
+<p class="rubric">Examples</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">SE3</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">7</span><span class="p">))</span>
+<span class="go">SE3Type LieTensor:</span>
+<span class="go">tensor([[ 0.1626,  1.6349,  0.3607,  0.2848, -0.0948,  0.1541,  1.0003],</span>
+<span class="go">        [ 1.4034, -1.3085, -0.8886, -1.6703,  0.7381,  1.5575,  0.6280]])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">SE3</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
+<span class="go">SE3Type LieTensor:</span>
+<span class="go">tensor([0., 0., 0., 0., 0., 0., 1.])</span>
+</pre></div>
+</div>
+<p>If <code class="docutils literal notranslate"><span class="pre">data</span></code> is tensor-like, the last dimension should correspond to the
+7 elements of the above embedding.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>It is not advised to construct SE3 Tensors by specifying storage
+sizes with ‘<code class="xref py py-obj docutils literal notranslate"><span class="pre">int</span></code>…’, which does not initialize data.</p>
+<p>Consider using <a class="reference internal" href="../pypose.randn_SE3/#pypose.randn_SE3" title="pypose.randn_SE3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.randn_SE3</span></code></a> or
+<a class="reference internal" href="../pypose.identity_SE3/#pypose.identity_SE3" title="pypose.identity_SE3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.identity_SE3</span></code></a> instead.</p>
+</div>
+<p>See <a class="reference internal" href="../pypose.Log/#pypose.Log" title="pypose.Log"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.Log</span></code></a>, <a class="reference internal" href="../pypose.Inv/#pypose.Inv" title="pypose.Inv"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.Inv</span></code></a>, <a class="reference internal" href="../pypose.Act/#pypose.Act" title="pypose.Act"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.Act</span></code></a>, <a class="reference internal" href="../pypose.Retr/#pypose.Retr" title="pypose.Retr"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.Retr</span></code></a>, <a class="reference internal" href="../pypose.Adj/#pypose.Adj" title="pypose.Adj"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.Adj</span></code></a>, <a class="reference internal" href="../pypose.AdjT/#pypose.AdjT" title="pypose.AdjT"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.AdjT</span></code></a>, <a class="reference internal" href="../pypose.Jinvp/#pypose.Jinvp" title="pypose.Jinvp"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.Jinvp</span></code></a> for
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+              
+  <section id="pypose-so3">
+<h1>pypose.SO3<a class="headerlink" href="#pypose-so3" title="Permalink to this heading">¶</a></h1>
+<dl class="py data">
+<dt class="sig sig-object py" id="pypose.SO3">
+<span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">SO3</span></span><em class="property"><span class="w"> </span><span class="p"><span class="pre">=</span></span><span class="w"> </span><span class="pre">functools.partial(&lt;class</span> <span class="pre">'pypose.lietensor.lietensor.LieTensor'&gt;,</span> <span class="pre">ltype=&lt;pypose.lietensor.lietensor.SO3Type</span> <span class="pre">object&gt;)</span></em><a class="headerlink" href="#pypose.SO3" title="Permalink to this definition">¶</a></dt>
+<dd><p>Alias of SO3 type <a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code></a>.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>data</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>, or <code class="xref py py-obj docutils literal notranslate"><span class="pre">list</span></code>, or ‘<code class="xref py py-obj docutils literal notranslate"><span class="pre">int</span></code>…’) – <p>A
+<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code> object, or constructing a <code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>
+object from <code class="xref py py-obj docutils literal notranslate"><span class="pre">list</span></code>, which defines tensor data (see below), or
+from ‘<code class="xref py py-obj docutils literal notranslate"><span class="pre">int</span></code>…’, which defines tensor shape.</p>
+<p>The shape of <code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code> object must be <code class="docutils literal notranslate"><span class="pre">(*,</span> <span class="pre">4)</span></code>,
+where <code class="docutils literal notranslate"><span class="pre">*</span></code> is empty, one, or more batched dimensions (the
+<a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor.lshape" title="pypose.LieTensor.lshape"><code class="xref py py-obj docutils literal notranslate"><span class="pre">lshape</span></code></a> of this LieTensor), otherwise error will
+be raised.</p>
+</p>
+</dd>
+</dl>
+<p>Internally, SO3 LieTensors are stored as unit quaternions:</p>
+<div class="math">
+\[\mathrm{data}[*, :] = [q_x, q_y, q_z, q_w],
+
+\]</div>
+<p>where <span class="math">\(q_x^2 + q_y^2 + q_z^2 + q_w^2 = 1\)</span>.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Normalization is not required at initialization as it is done internally
+by the library right before further computation. However, the normalized
+quaternion will not be written back to the tensor storage to prevent
+in-place data alteration.</p>
+</div>
+<p class="rubric">Examples</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">SO3</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
+<span class="go">SO3Type LieTensor:</span>
+<span class="go">tensor([[-1.0722, -0.9440,  0.9437, -0.8485],</span>
+<span class="go">        [-0.2725,  0.8414, -1.0730,  1.3270]])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">SO3</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
+<span class="go">SO3Type LieTensor:</span>
+<span class="go">tensor([0., 0., 0., 1.])</span>
+</pre></div>
+</div>
+<p>If <code class="docutils literal notranslate"><span class="pre">data</span></code> is tensor-like, the last dimension should correspond to the
+4 elements of the above embedding.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>It is not advised to construct SO3 Tensors by specifying storage
+sizes with ‘<code class="xref py py-obj docutils literal notranslate"><span class="pre">int</span></code>…’, which does not initialize data.</p>
+<p>Consider using <a class="reference internal" href="../pypose.randn_SO3/#pypose.randn_SO3" title="pypose.randn_SO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.randn_SO3</span></code></a> or
+<a class="reference internal" href="../pypose.identity_SO3/#pypose.identity_SO3" title="pypose.identity_SO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.identity_SO3</span></code></a> instead.</p>
+</div>
+<p>See <a class="reference internal" href="../pypose.Log/#pypose.Log" title="pypose.Log"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.Log</span></code></a>, <a class="reference internal" href="../pypose.Inv/#pypose.Inv" title="pypose.Inv"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.Inv</span></code></a>, <a class="reference internal" href="../pypose.Act/#pypose.Act" title="pypose.Act"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.Act</span></code></a>, <a class="reference internal" href="../pypose.Retr/#pypose.Retr" title="pypose.Retr"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.Retr</span></code></a>, <a class="reference internal" href="../pypose.Adj/#pypose.Adj" title="pypose.Adj"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.Adj</span></code></a>, <a class="reference internal" href="../pypose.AdjT/#pypose.AdjT" title="pypose.AdjT"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.AdjT</span></code></a>, <a class="reference internal" href="../pypose.Jinvp/#pypose.Jinvp" title="pypose.Jinvp"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.Jinvp</span></code></a> for
+implementations of relevant operations.</p>
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+<li class="toctree-l1 current"><a class="reference internal" href="../../lietensor/">LieTensor</a></li>
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+              
+  <section id="pypose-sim3">
+<h1>pypose.Sim3<a class="headerlink" href="#pypose-sim3" title="Permalink to this heading">¶</a></h1>
+<dl class="py data">
+<dt class="sig sig-object py" id="pypose.Sim3">
+<span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">Sim3</span></span><em class="property"><span class="w"> </span><span class="p"><span class="pre">=</span></span><span class="w"> </span><span class="pre">functools.partial(&lt;class</span> <span class="pre">'pypose.lietensor.lietensor.LieTensor'&gt;,</span> <span class="pre">ltype=&lt;pypose.lietensor.lietensor.Sim3Type</span> <span class="pre">object&gt;)</span></em><a class="headerlink" href="#pypose.Sim3" title="Permalink to this definition">¶</a></dt>
+<dd><p>Alias of Sim3 type <a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code></a>.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>data</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>, or <code class="xref py py-obj docutils literal notranslate"><span class="pre">list</span></code>, or ‘<code class="xref py py-obj docutils literal notranslate"><span class="pre">int</span></code>…’) – <p>A
+<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code> object, or constructing a <code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>
+object from <code class="xref py py-obj docutils literal notranslate"><span class="pre">list</span></code>, which defines tensor data (see below), or
+from ‘<code class="xref py py-obj docutils literal notranslate"><span class="pre">int</span></code>…’, which defines tensor shape.</p>
+<p>The shape of <code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code> object must be <code class="docutils literal notranslate"><span class="pre">(*,</span> <span class="pre">8)</span></code>,
+where <code class="docutils literal notranslate"><span class="pre">*</span></code> is empty, one, or more batched dimensions (the
+<a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor.lshape" title="pypose.LieTensor.lshape"><code class="xref py py-obj docutils literal notranslate"><span class="pre">lshape</span></code></a> of this LieTensor), otherwise error will
+be raised.</p>
+</p>
+</dd>
+</dl>
+<p>Internally, Sim3 LieTensors are stored by concatenating the translation
+vector with an RxSO3:</p>
+<div class="math">
+\[\mathrm{data}[*, :] = [t_x, t_y, t_z, q_x, q_y, q_z, q_w, s],
+
+\]</div>
+<p>where <span class="math">\(\begin{pmatrix} t_x &amp; t_y &amp; t_z \end{pmatrix}^T \in \mathbb{R}^3\)</span>
+is the translation vector and
+<span class="math">\(\begin{pmatrix} q_x &amp; q_y &amp; q_z &amp; q_w \end{pmatrix}^T\)</span> and
+<span class="math">\(s \in \mathbb{R}\)</span> are the unit quaternion and the scaling factor
+as in <code class="xref py py-obj docutils literal notranslate"><span class="pre">pp.RxSO3</span></code>, respectively.</p>
+<p class="rubric">Examples</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">Sim3</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>
+<span class="go">Sim3Type LieTensor:</span>
+<span class="go">tensor([[ 0.0175,  0.8657, -0.2274,  2.2380, -0.0297, -0.3799, -0.0664,  0.9995],</span>
+<span class="go">        [ 0.8744,  0.4114,  1.2041, -0.5687, -0.5630,  0.6025, -0.6137,  1.1185]])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">Sim3</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
+<span class="go">Sim3Type LieTensor:</span>
+<span class="go">tensor([0., 0., 0., 0., 0., 0., 1., 1.])</span>
+</pre></div>
+</div>
+<p>If <code class="docutils literal notranslate"><span class="pre">data</span></code> is tensor-like, the last dimension should correspond to the
+8 elements of the above embedding.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>It is not advised to construct Sim3 Tensors by specifying storage
+sizes with ‘<code class="xref py py-obj docutils literal notranslate"><span class="pre">int</span></code>…’, which does not initialize data.</p>
+<p>Consider using <a class="reference internal" href="../pypose.randn_Sim3/#pypose.randn_Sim3" title="pypose.randn_Sim3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.randn_Sim3</span></code></a> or
+<a class="reference internal" href="../pypose.identity_Sim3/#pypose.identity_Sim3" title="pypose.identity_Sim3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.identity_Sim3</span></code></a> instead.</p>
+</div>
+<p>See <a class="reference internal" href="../pypose.Log/#pypose.Log" title="pypose.Log"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.Log</span></code></a>, <a class="reference internal" href="../pypose.Inv/#pypose.Inv" title="pypose.Inv"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.Inv</span></code></a>, <a class="reference internal" href="../pypose.Act/#pypose.Act" title="pypose.Act"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.Act</span></code></a>, <a class="reference internal" href="../pypose.Retr/#pypose.Retr" title="pypose.Retr"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.Retr</span></code></a>, <a class="reference internal" href="../pypose.Adj/#pypose.Adj" title="pypose.Adj"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.Adj</span></code></a>, <a class="reference internal" href="../pypose.AdjT/#pypose.AdjT" title="pypose.AdjT"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.AdjT</span></code></a>, <a class="reference internal" href="../pypose.Jinvp/#pypose.Jinvp" title="pypose.Jinvp"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.Jinvp</span></code></a> for
+implementations of relevant operations.</p>
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+              
+  <section id="pypose-add">
+<h1>pypose.add<a class="headerlink" href="#pypose-add" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.add">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">add</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">other</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">alpha</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/basics/#add"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.add" title="Permalink to this definition">¶</a></dt>
+<dd><p>Adds other, scaled by alpha, to input LieTensor.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>input</strong> (<a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code></a>) – the input LieTensor (Lie Algebra or Lie Group).</p></li>
+<li><p><strong>other</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – the tensor to add to input. The last dimension has to be no less
+than the shape of the corresponding Lie Algebra of the input.</p></li>
+<li><p><strong>alpha</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Number</span></code>) – the multiplier for other.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>the output LieTensor.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code></a></p>
+</dd>
+</dl>
+<div class="math">
+\[\bm{y}_i =
+\begin{cases}
+\alpha * \bm{a}_i + \bm{x}_i & \text{if}~\bm{x}_i~\text{is a Lie Algebra} \\
+\mathrm{Exp}(\alpha * \bm{a}_i) \times \bm{x}_i & \text{if}~\bm{x}_i~\text{is a Lie Group}
+\end{cases}
+
+\]</div>
+<p>where <span class="math">\(\bm{x}\)</span> is the <code class="docutils literal notranslate"><span class="pre">input</span></code> LieTensor, <span class="math">\(\bm{a}\)</span> is the <code class="docutils literal notranslate"><span class="pre">other</span></code> Tensor to add,
+and <span class="math">\(\bm{y}\)</span> is the output LieTensor.</p>
+<div class="admonition warning">
+<p class="admonition-title">Warning</p>
+<p>For better readability, we suggest calling <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">+</span> <span class="pre">b</span></code> instead of <code class="docutils literal notranslate"><span class="pre">a.add(b)</span></code> or
+<code class="docutils literal notranslate"><span class="pre">pypose.add(a,</span> <span class="pre">b)</span></code>, which have the same behaviors.</p>
+</div>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>A Lie Group normally requires a larger space than its corresponding Lie Algebra, thus
+the elements in the last dimension of the <code class="docutils literal notranslate"><span class="pre">other</span></code> Tensor (treated as a Lie Algebra
+in this function) beyond the expected shape of the Lie Algebra are ignored. This is
+because the gradient of a Lie Group is computed as a left perturbation (a Lie Algebra)
+in its tangent space and is stored in the LieGroup’s <code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor.grad</span></code>, which has
+the same storage space with the LieGroup.</p>
+<div class="math">
+\[\begin{align*}
+    \frac{D f(\mathcal{X})}{D \mathcal{X}} & \overset{\underset{\mathrm{def}}{}}{=}
+    \displaystyle \lim_{\bm{\tau} \to \bm{0}} \frac{f(\bm{\tau} \oplus \mathcal{X})
+        \ominus f(\mathcal{X})}{\bm{\tau}} \\
+    & = \left.\frac{\partial\mathrm{Log}(f(\mathrm{Exp}(\bm{\tau})\times\mathcal{X}))
+        \times f(\mathcal{X})^{-1})}{\partial \bm{\tau}}\right|_{\bm{\tau=\bm{0}}}
+\end{align*},
+
+\]</div>
+<p>where <span class="math">\(\mathcal{X}\)</span> is a Lie Group and <span class="math">\(\bm{\tau}\)</span> is its left perturbation.</p>
+<p>See Eq.(44) in <a class="reference external" href="https://arxiv.org/abs/1812.01537">Micro Lie theory</a> for more details of
+the gradient for a Lie Group.</p>
+<p>This provides convenience to work with PyTorch optimizers like <code class="xref py py-obj docutils literal notranslate"><span class="pre">torch.optim.SGD</span></code>,
+which calls function <a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor.add_" title="pypose.LieTensor.add_"><code class="xref py py-meth docutils literal notranslate"><span class="pre">add_()</span></code></a> of a Lie Group to adjust parameters by gradients
+(<code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor.grad</span></code>, where the last element is often zero since tangent vector requires
+smaller storage space).</p>
+</div>
+<p>See <a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-meth docutils literal notranslate"><span class="pre">LieTensor()</span></code></a> for types of Lie Algebra and Lie Group.</p>
+<p>See <a class="reference internal" href="../pypose.Exp/#pypose.Exp" title="pypose.Exp"><code class="xref py py-meth docutils literal notranslate"><span class="pre">Exp()</span></code></a> for Exponential mapping of Lie Algebra.</p>
+<p class="rubric">Examples</p>
+<p>The following operations are equivalent.</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_SE3</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">6</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">+</span> <span class="n">a</span>
+<span class="go">SE3Type LieTensor:</span>
+<span class="go">tensor([-1.6089,  0.4184,  0.6621, -0.2098,  0.5383,  0.4794,  0.6606])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
+<span class="go">SE3Type LieTensor:</span>
+<span class="go">tensor([-1.6089,  0.4184,  0.6621, -0.2098,  0.5383,  0.4794,  0.6606])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">se3</span><span class="p">(</span><span class="n">a</span><span class="p">)</span><span class="o">.</span><span class="n">Exp</span><span class="p">()</span> <span class="o">@</span> <span class="n">x</span>
+<span class="go">SE3Type LieTensor:</span>
+<span class="go">tensor([-1.6089,  0.4184,  0.6621, -0.2098,  0.5383,  0.4794,  0.6606])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">+</span> <span class="n">torch</span><span class="o">.</span><span class="n">cat</span><span class="p">([</span><span class="n">a</span><span class="p">,</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">1</span><span class="p">)])</span>
+<span class="go">SE3Type LieTensor:</span>
+<span class="go">tensor([-1.6089,  0.4184,  0.6621, -0.2098,  0.5383,  0.4794,  0.6606])</span>
+</pre></div>
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+<dd><p>Inplace version of <a class="reference internal" href="../pypose.add/#pypose.add" title="pypose.add"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.add()</span></code></a>.</p>
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+  <section id="pypose-bmv">
+<h1>pypose.bmv<a class="headerlink" href="#pypose-bmv" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.bmv">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">bmv</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">mat</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">vec</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">out</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/function/linalg/#bmv"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.bmv" title="Permalink to this definition">¶</a></dt>
+<dd><p>Performs batched matrix-vector product.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>mat</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – matrices to be multiplied.</p></li>
+<li><p><strong>vec</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – vectors to be multiplied.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>the output tensor.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>out (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>)</p>
+</dd>
+</dl>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The <code class="docutils literal notranslate"><span class="pre">mat</span></code> has to be a (<span class="math">\(\cdots\times n \times m\)</span>) tensor,
+the <code class="docutils literal notranslate"><span class="pre">vec</span></code> has to be a (<span class="math">\(\cdots\times m\)</span>) tensor,
+and <code class="docutils literal notranslate"><span class="pre">out</span></code> will be a (<span class="math">\(\cdots\times n\)</span>) tensor.
+Different from <code class="docutils literal notranslate"><span class="pre">torch.mv</span></code>, which is not broadcast, this function
+is broadcast and supports batched product.</p>
+</div>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">matrix</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">vec</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">out</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">bmv</span><span class="p">(</span><span class="n">matrix</span><span class="p">,</span> <span class="n">vec</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">out</span><span class="o">.</span><span class="n">shape</span>
+<span class="go">torch.Size([2, 2, 3])</span>
+</pre></div>
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+  <section id="pypose-bspline">
+<h1>pypose.bspline<a class="headerlink" href="#pypose-bspline" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.bspline">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">bspline</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">data</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">interval</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0.1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">extrapolate</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/function/spline/#bspline"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.bspline" title="Permalink to this definition">¶</a></dt>
+<dd><p>B-spline interpolation, which currently only supports SE3 LieTensor.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>data</strong> (<a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code></a>) – the input sparse poses with
+[batch_size, poses_num, dim] shape.</p></li>
+<li><p><strong>interval</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">float</span></code>) – the unit interval between interpolated poses.
+We assume the interval of adjacent input poses is 1. Therefore, if we set
+<code class="docutils literal notranslate"><span class="pre">interval</span></code> as <code class="docutils literal notranslate"><span class="pre">0.1</span></code>, the interpolated poses between the poses at
+<span class="math">\(t\)</span> and <span class="math">\(t+1\)</span> will be at <span class="math">\([t, t+0.1, ..., t+0.9, t+1]\)</span>.
+Default: <code class="docutils literal notranslate"><span class="pre">0.1</span></code>.</p></li>
+<li><p><strong>extrapolate</strong> (<code class="docutils literal notranslate"><span class="pre">bool</span></code>) – flag to determine whether the interpolate poses pass the
+start and end poses. Default: <code class="docutils literal notranslate"><span class="pre">False</span></code>.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>the interpolated SE3 LieTensor.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code></a></p>
+</dd>
+</dl>
+<p>A poses query at <span class="math">\(t \in [t_i, t_{i+1})\)</span> (i.e. a segment of the spline)
+only relies on the poses at four steps <span class="math">\(\{t_{i-1},t_i,t_{i+1},t_{i+2}\}\)</span>.
+It means that the interpolation between adjacent poses needs four consecutive poses.
+In this function, the interpolated poses are evenly distributed between a
+pose query according to the interval parameter.</p>
+<p>The absolute pose of the spline <span class="math">\(T_{s}^w(t)\)</span>, where <span class="math">\(w\)</span> denotes the world
+and <span class="math">\(s\)</span> is the spline coordinate frame, can be calculated:</p>
+<div class="math">
+\[\begin{aligned}
+    &T^w_s(t) = T^w_{i-1} \prod^{i+2}_{j=i}\delta T_j,\\
+    &\delta T_j= \exp(\lambda_{j-i}(t)\delta \hat{\xi}_j),\\
+    &\lambda(t) = MU(t),
+\end{aligned}
+
+\]</div>
+<p><span class="math">\(M\)</span> is a predefined matrix, shown as follows:</p>
+<div class="math">
+\[M = \frac{1}{6}\begin{bmatrix}
+    5& 3& -3& 1 \\
+    1& 3& 3& -2\\
+    0& 0& 0& 1 \\
+\end{bmatrix}
+
+\]</div>
+<p><span class="math">\(U(t)\)</span> is a vector:</p>
+<div class="math">
+\[U(t) = \begin{bmatrix}
+        1\\
+        u(t)\\
+        u(t)^2\\
+        u(t)^3\\
+    \end{bmatrix}
+
+\]</div>
+<p>where <span class="math">\(u(t)=(t-t_i)/\Delta t\)</span>, <span class="math">\(\Delta t = t_{i+1} - t_{i}\)</span>. <span class="math">\(t_i\)</span>
+is the time point of the <span class="math">\(i_{th}\)</span> given pose. <span class="math">\(t\)</span> is the interpolation
+time point <span class="math">\(t \in [t_{i},t_{i+1})\)</span>.</p>
+<p><span class="math">\(\delta \hat{\xi}_j\)</span> is the transformation between <span class="math">\(\hat{\xi}_{j-1}\)</span>
+and <span class="math">\(\hat{\xi}_{j}\)</span></p>
+<div class="math">
+\[\begin{aligned}
+    \delta \hat{\xi}_j :&= \log(\exp(\hat{\xi} _w^{j-1})\exp(\hat{\xi}_j^w))
+                        &= \log(T_j^{j-1})\in \mathfrak{se3}
+\end{aligned}
+
+\]</div>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The implementation is based on Eq. (3), (4), (5), and (6) of this paper:
+David Hug, et al.,
+<a class="reference external" href="https://ieeexplore.ieee.org/abstract/document/9320417">HyperSLAM: A Generic and Modular Approach to Sensor Fusion and Simultaneous
+Localization And Mapping in Continuous-Time</a>,
+2020 International Conference on 3D Vision (3DV), Fukuoka, Japan, 2020.</p>
+</div>
+<p class="rubric">Examples</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">torch</span><span class="o">,</span> <span class="nn">pypose</span> <span class="k">as</span> <span class="nn">pp</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">a1</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">euler2SO3</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">([</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">]))</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">a2</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">euler2SO3</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">([</span><span class="n">torch</span><span class="o">.</span><span class="n">pi</span> <span class="o">/</span> <span class="mf">4.</span><span class="p">,</span> <span class="n">torch</span><span class="o">.</span><span class="n">pi</span> <span class="o">/</span> <span class="mf">3.</span><span class="p">,</span> <span class="n">torch</span><span class="o">.</span><span class="n">pi</span> <span class="o">/</span> <span class="mf">2.</span><span class="p">]))</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">poses</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">SE3</span><span class="p">([[[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">4.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="n">a1</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">a1</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">a1</span><span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="n">a1</span><span class="p">[</span><span class="mi">3</span><span class="p">]],</span>
+<span class="gp">... </span>                 <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">3.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="n">a1</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">a1</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">a1</span><span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="n">a1</span><span class="p">[</span><span class="mi">3</span><span class="p">]],</span>
+<span class="gp">... </span>                 <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">2.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="n">a1</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">a1</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">a1</span><span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="n">a1</span><span class="p">[</span><span class="mi">3</span><span class="p">]],</span>
+<span class="gp">... </span>                 <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="n">a1</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">a1</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">a1</span><span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="n">a1</span><span class="p">[</span><span class="mi">3</span><span class="p">]],</span>
+<span class="gp">... </span>                 <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="n">a2</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">a2</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">a2</span><span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="n">a2</span><span class="p">[</span><span class="mi">3</span><span class="p">]],</span>
+<span class="gp">... </span>                 <span class="p">[</span><span class="mf">2.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="n">a2</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">a2</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">a2</span><span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="n">a2</span><span class="p">[</span><span class="mi">3</span><span class="p">]],</span>
+<span class="gp">... </span>                 <span class="p">[</span><span class="mf">3.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="n">a2</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">a2</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">a2</span><span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="n">a2</span><span class="p">[</span><span class="mi">3</span><span class="p">]],</span>
+<span class="gp">... </span>                 <span class="p">[</span><span class="mf">4.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="n">a2</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">a2</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">a2</span><span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="n">a2</span><span class="p">[</span><span class="mi">3</span><span class="p">]]]])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">wayposes</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">bspline</span><span class="p">(</span><span class="n">poses</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">)</span>
+</pre></div>
+</div>
+<figure class="align-default" id="id1">
+<img alt="../../_images/BsplineSE3.png" src="../../_images/BsplineSE3.png" />
+<figcaption>
+<p><span class="caption-text">Fig. 1. Result of B Spline Interpolation in SE3.</span><a class="headerlink" href="#id1" title="Permalink to this image">¶</a></p>
+</figcaption>
+</figure>
+</dd></dl>
+
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+  <section id="pypose-bvmv">
+<h1>pypose.bvmv<a class="headerlink" href="#pypose-bvmv" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.bvmv">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">bvmv</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">lvec</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">mat</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">rvec</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/function/linalg/#bvmv"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.bvmv" title="Permalink to this definition">¶</a></dt>
+<dd><p>Performs batched vector-matrix-vector product.</p>
+<div class="math">
+\[\text{out}_i = \mathbf{v}_i^{l\text{T}} \times \mathbf{M}_i \times \mathbf{v}_i^r
+
+\]</div>
+<p>where <span class="math">\(\text{out}_i\)</span> is a scalar and <span class="math">\(\mathbf{v}_i^l, \mathbf{M}_i,
+\mathbf{v}_i^r\)</span> are the i-th batched tensors with shape (n), (n, m), and (m),
+respectively.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>lvec</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – left vectors to be multiplied.</p></li>
+<li><p><strong>mat</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – matrices to be multiplied.</p></li>
+<li><p><strong>rvec</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – right vectors to be multiplied.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>the output tensor.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code></p>
+</dd>
+</dl>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>the <code class="docutils literal notranslate"><span class="pre">lvec</span></code> has to be a (<span class="math">\(\cdots\times n\)</span>) tensor,
+The <code class="docutils literal notranslate"><span class="pre">mat</span></code> has to be a (<span class="math">\(\cdots\times n \times m\)</span>) tensor,
+the <code class="docutils literal notranslate"><span class="pre">rvec</span></code> has to be a (<span class="math">\(\cdots\times m\)</span>) tensor,
+and <code class="docutils literal notranslate"><span class="pre">out</span></code> will be a (<span class="math">\(\cdots\)</span>) or at least a 1D tensor.</p>
+</div>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">v1</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">mat</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">v2</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">out</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">bvmv</span><span class="p">(</span><span class="n">v1</span><span class="p">,</span> <span class="n">mat</span><span class="p">,</span> <span class="n">v2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">out</span><span class="o">.</span><span class="n">shape</span>
+<span class="go">torch.Size([1])</span>
+</pre></div>
+</div>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">v1</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">mat</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">v2</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">out</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">bvmv</span><span class="p">(</span><span class="n">v1</span><span class="p">,</span> <span class="n">mat</span><span class="p">,</span> <span class="n">v2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">out</span><span class="o">.</span><span class="n">shape</span>
+<span class="go">torch.Size([2, 2])</span>
+</pre></div>
+</div>
+</dd></dl>
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+  <section id="pypose-bvv">
+<h1>pypose.bvv<a class="headerlink" href="#pypose-bvv" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.bvv">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">bvv</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">lvec</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">rvec</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">out</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/function/linalg/#bvv"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.bvv" title="Permalink to this definition">¶</a></dt>
+<dd><p>Performs batched vector-vector product, which results in matrices.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>lvec</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – left vectors to be multiplied.</p></li>
+<li><p><strong>rvec</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – right vectors to be multiplied.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>the output tensor.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>out (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>)</p>
+</dd>
+</dl>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>This function is broadcastable and supports batched product.</p>
+</div>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">lvec</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">rvec</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">out</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">bvv</span><span class="p">(</span><span class="n">lvec</span><span class="p">,</span> <span class="n">rvec</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">out</span><span class="o">.</span><span class="n">shape</span>
+<span class="go">torch.Size([2, 2, 3, 2])</span>
+</pre></div>
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+  <section id="pypose-cart2homo">
+<h1>pypose.cart2homo<a class="headerlink" href="#pypose-cart2homo" title="Permalink to this heading">¶</a></h1>
+<dl class="py function">
+<dt class="sig sig-object py" id="pypose.cart2homo">
+<span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">cart2homo</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">coordinates</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/function/geometry/#cart2homo"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.cart2homo" title="Permalink to this definition">¶</a></dt>
+<dd><p>Converts batched Cartesian coordinates to Homogeneous coordinates
+by adding ones to last dimension.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>coordinates</strong> (<code class="docutils literal notranslate"><span class="pre">torch.Tensor</span></code>) – the Cartesian coordinates to be converted.</p>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>the coordinates in Homogeneous space.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><code class="docutils literal notranslate"><span class="pre">torch.Tensor</span></code></p>
+</dd>
+</dl>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The last dimension of the input coordinates can be any dimension.</p>
+</div>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">points</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">cart2homo</span><span class="p">(</span><span class="n">points</span><span class="p">)</span>
+<span class="go">tensor([[ 2.0598,  1.5351,  1.0000],</span>
+<span class="go">        [-0.8484,  1.2390,  1.0000]])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">points</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">cart2homo</span><span class="p">(</span><span class="n">points</span><span class="p">)</span>
+<span class="go">tensor([[ 1.7946,  0.3548, -0.4446,  1.0000],</span>
+<span class="go">        [ 0.3010, -2.2748, -0.4708,  1.0000]])</span>
+</pre></div>
+</div>
+</dd></dl>
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+  <section id="pypose-chspline">
+<h1>pypose.chspline<a class="headerlink" href="#pypose-chspline" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.chspline">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">chspline</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">points</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">interval</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0.1</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/function/spline/#chspline"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.chspline" title="Permalink to this definition">¶</a></dt>
+<dd><p>Cubic Hermite Spline, a piecewise-cubic interpolator matching values and first
+derivatives.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>points</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – the sequence of points for interpolation with shape
+[…, point_num, dim].</p></li>
+<li><p><strong>interval</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">float</span></code>) – the unit interval between interpolated points.
+We assume the interval of adjacent input points is 1. Therefore, if we set
+<code class="docutils literal notranslate"><span class="pre">interval</span></code> as <code class="docutils literal notranslate"><span class="pre">0.1</span></code>, the interpolated points between the points at
+<span class="math">\(t\)</span> and <span class="math">\(t+1\)</span> will be <span class="math">\([t, t+0.1, ..., t+0.9, t+1]\)</span>.
+Default: <code class="docutils literal notranslate"><span class="pre">0.1</span></code>.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>the interpolated points.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><code class="docutils literal notranslate"><span class="pre">Tensor</span></code></p>
+</dd>
+</dl>
+<p>The interpolated points are evenly distributed between a start point and a end point
+according to the number of interpolated points. In this function, the interpolated points
+are also evenly distributed between the start point and end point.</p>
+<p>Denote the starting point <span class="math">\(p_0\)</span> with the starting tangent <span class="math">\(m_0\)</span> and the
+ending point <span class="math">\(p_1\)</span> with the ending tagent <span class="math">\(m_1\)</span>. The interpolated point at
+<span class="math">\(t\)</span> can be defined as:</p>
+<div class="math">
+\[\begin{aligned}
+    p(t) &= (1-3t^2+2t^3)p_0 + (t-2t^2+t^3)m_0+(3t^2-2t^3)p_1+(-t^2+t^3)m_1\\
+         &= \begin{bmatrix}
+                p_0, m_0, p_1, m_1
+            \end{bmatrix}
+            \begin{bmatrix}
+                1& 0&-3& 2\\
+                0& 1&-2& 1\\
+                0& 0& 3&-2\\
+                0& 0&-1& 1
+            \end{bmatrix}
+            \begin{bmatrix}1\\ t\\ t^2\\ t^3\end{bmatrix}
+
+\end{aligned}\]</div>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The implementation is based on wiki of <a class="reference external" href="https://en.wikipedia.org/wiki/Cubic_Hermite_spline">Cubic Hermite spline</a>.</p>
+</div>
+<p class="rubric">Examples</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">torch</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">pypose</span> <span class="k">as</span> <span class="nn">pp</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">points</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([[[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">],</span>
+<span class="gp">... </span>                        <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">.5</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">],</span>
+<span class="gp">... </span>                        <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">],</span>
+<span class="gp">... </span>                        <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">,</span> <span class="mf">0.4</span><span class="p">],</span>
+<span class="gp">... </span>                        <span class="p">[</span><span class="mf">1.5</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">],</span>
+<span class="gp">... </span>                        <span class="p">[</span><span class="mf">2.</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">,</span> <span class="mf">0.4</span><span class="p">],</span>
+<span class="gp">... </span>                        <span class="p">[</span><span class="mf">2.5</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">],</span>
+<span class="gp">... </span>                        <span class="p">[</span><span class="mf">1.75</span><span class="p">,</span> <span class="mf">0.75</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">],</span>
+<span class="gp">... </span>                        <span class="p">[</span><span class="mf">2.25</span><span class="p">,</span> <span class="mf">0.75</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">],</span>
+<span class="gp">... </span>                        <span class="p">[</span><span class="mf">3.</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">,</span> <span class="mf">0.4</span><span class="p">],</span>
+<span class="gp">... </span>                        <span class="p">[</span><span class="mf">3.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">],</span>
+<span class="gp">... </span>                        <span class="p">[</span><span class="mf">4.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">],</span>
+<span class="gp">... </span>                        <span class="p">[</span><span class="mf">4.</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">,</span> <span class="mf">0.4</span><span class="p">],</span>
+<span class="gp">... </span>                        <span class="p">[</span><span class="mf">5.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">],</span>
+<span class="gp">... </span>                        <span class="p">[</span><span class="mf">4.</span><span class="p">,</span> <span class="mf">0.75</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">],</span>
+<span class="gp">... </span>                        <span class="p">[</span><span class="mf">5.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">]]])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">waypoints</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">chspline</span><span class="p">(</span><span class="n">points</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">)</span>
+</pre></div>
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+              <p class="caption" role="heading"><span class="caption-text">Contents:</span></p>
+<ul class="current">
+<li class="toctree-l1"><a class="reference internal" href="../../lietensor/">LieTensor</a></li>
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+              
+  <section id="pypose-cummul">
+<h1>pypose.cummul<a class="headerlink" href="#pypose-cummul" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.cummul">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">cummul</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">dim</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">left</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/basics/ops/#cummul"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.cummul" title="Permalink to this definition">¶</a></dt>
+<dd><p>Returns the cumulative multiplication (*) of LieTensor along a dimension.</p>
+<ul class="simple">
+<li><p>Left multiplication:</p></li>
+</ul>
+<div class="math">
+\[y_i = x_i * x_{i-1} * \cdots * x_1,
+
+\]</div>
+<ul class="simple">
+<li><p>Right multiplication:</p></li>
+</ul>
+<div class="math">
+\[y_i = x_1 * x_2 * \cdots * x_i,
+
+\]</div>
+<p>where <span class="math">\(x_i,~y_i\)</span> are the <span class="math">\(i\)</span>-th LieType item along the <code class="xref py py-obj docutils literal notranslate"><span class="pre">dim</span></code>
+dimension of input and output, respectively.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>input</strong> (<a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><em>LieTensor</em></a>) – the input LieTensor</p></li>
+<li><p><strong>dim</strong> (<em>int</em>) – the dimension to do the multiplication over</p></li>
+<li><p><strong>left</strong> (<em>bool</em><em>, </em><em>optional</em>) – whether perform left multiplication in <a class="reference internal" href="#pypose.cummul" title="pypose.cummul"><code class="xref py py-obj docutils literal notranslate"><span class="pre">cummul</span></code></a>.
+If set it to <code class="xref py py-obj docutils literal notranslate"><span class="pre">False</span></code>, this function performs right multiplication.
+Defaul: <code class="docutils literal notranslate"><span class="pre">True</span></code></p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>The LieTensor</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor">LieTensor</a></p>
+</dd>
+</dl>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<ul class="simple">
+<li><p>The time complexity of the function is <span class="math">\(\mathcal{O}(\log N)\)</span>, where
+<span class="math">\(N\)</span> is the LieTensor size along the <code class="xref py py-obj docutils literal notranslate"><span class="pre">dim</span></code> dimension.</p></li>
+</ul>
+</div>
+<p class="rubric">Example</p>
+<ul class="simple">
+<li><p>Left multiplication with <span class="math">\(\text{input} \in\)</span> <a class="reference internal" href="../pypose.SE3/#pypose.SE3" title="pypose.SE3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">SE3</span></code></a></p></li>
+</ul>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="nb">input</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_SE3</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">cummul</span><span class="p">(</span><span class="nb">input</span><span class="p">,</span> <span class="n">dim</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
+<span class="go">SE3Type LieTensor:</span>
+<span class="go">tensor([[-1.9615, -0.1246,  0.3666,  0.0165,  0.2853,  0.3126,  0.9059],</span>
+<span class="go">        [ 0.7139,  1.3988, -0.1909, -0.1780,  0.4405, -0.6571,  0.5852]])</span>
+</pre></div>
+</div>
+<ul class="simple">
+<li><p>Left multiplication with <span class="math">\(\text{input} \in\)</span> <a class="reference internal" href="../pypose.SO3/#pypose.SO3" title="pypose.SO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">SO3</span></code></a></p></li>
+</ul>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="nb">input</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_SO3</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">cummul</span><span class="p">(</span><span class="nb">input</span><span class="p">,</span> <span class="n">dim</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">left</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+<span class="go">SO3Type LieTensor:</span>
+<span class="go">tensor([[[-1.8252e-01,  1.6198e-01,  8.3683e-01,  4.9007e-01],</span>
+<span class="go">        [ 2.0905e-04,  5.2031e-01,  8.4301e-01, -1.3642e-01]]])</span>
+</pre></div>
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+  <section id="pypose-cummul">
+<h1>pypose.cummul_<a class="headerlink" href="#pypose-cummul" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.cummul_">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">cummul_</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">dim</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">left</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/basics/ops/#cummul_"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.cummul_" title="Permalink to this definition">¶</a></dt>
+<dd><p>Inplace version of <a class="reference internal" href="../pypose.cummul/#pypose.cummul" title="pypose.cummul"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.cummul()</span></code></a></p>
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+<li class="toctree-l1"><a class="reference internal" href="../../lietensor/">LieTensor</a></li>
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+              
+  <section id="pypose-cumops">
+<h1>pypose.cumops<a class="headerlink" href="#pypose-cumops" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.cumops">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">cumops</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">dim</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">ops</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/basics/ops/#cumops"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.cumops" title="Permalink to this definition">¶</a></dt>
+<dd><p>Returns the cumulative user-defined operation of LieTensor along a dimension.</p>
+<div class="math">
+\[y_i = x_1~\mathrm{\circ}~x_2 ~\mathrm{\circ}~ \cdots ~\mathrm{\circ}~ x_i,
+
+\]</div>
+<p>where <span class="math">\(\mathrm{\circ}\)</span> is the user-defined operation and <span class="math">\(x_i,~y_i\)</span>
+are the <span class="math">\(i\)</span>-th LieType item along the <code class="xref py py-obj docutils literal notranslate"><span class="pre">dim</span></code> dimension of input and
+output, respectively.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>input</strong> (<a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><em>LieTensor</em></a>) – the input LieTensor</p></li>
+<li><p><strong>dim</strong> (<em>int</em>) – the dimension to do the operation over</p></li>
+<li><p><strong>ops</strong> (<em>func</em>) – the user-defined operation or function</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>LieTensor</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor">LieTensor</a></p>
+</dd>
+</dl>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<ul class="simple">
+<li><p>The users are supposed to provide meaningful operation.</p></li>
+<li><p>This function doesn’t check whether the results are valid for mathematical
+definition of LieTensor, e.g., quaternion.</p></li>
+<li><p>The time complexity of the function is <span class="math">\(\mathcal{O}(\log N)\)</span>, where
+<span class="math">\(N\)</span> is the LieTensor size along the <code class="xref py py-obj docutils literal notranslate"><span class="pre">dim</span></code> dimension.</p></li>
+</ul>
+</div>
+<p class="rubric">Examples</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="c1"># The following operations are equivalent.</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="nb">input</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_SE3</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="nb">input</span><span class="o">.</span><span class="n">cumprod</span><span class="p">(</span><span class="n">dim</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="n">left</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+<span class="go">SE3Type LieTensor:</span>
+<span class="go">tensor([[-0.6466,  0.2956,  2.4055, -0.4428,  0.1893,  0.3933,  0.7833],</span>
+<span class="go">        [ 1.2711,  1.2020,  0.0651, -0.0685,  0.6732,  0.7331, -0.0685]])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">cumops</span><span class="p">(</span><span class="nb">input</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="p">:</span> <span class="n">b</span> <span class="o">@</span> <span class="n">a</span><span class="p">)</span> <span class="c1"># Left multiplication</span>
+<span class="go">SE3Type LieTensor:</span>
+<span class="go">tensor([[-0.6466,  0.2956,  2.4055, -0.4428,  0.1893,  0.3933,  0.7833],</span>
+<span class="go">        [ 1.2711,  1.2020,  0.0651, -0.0685,  0.6732,  0.7331, -0.0685]])</span>
+</pre></div>
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+  <section id="pypose-cumops">
+<h1>pypose.cumops_<a class="headerlink" href="#pypose-cumops" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.cumops_">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">cumops_</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">dim</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">ops</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/basics/ops/#cumops_"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.cumops_" title="Permalink to this definition">¶</a></dt>
+<dd><p>Inplace version of <a class="reference internal" href="../pypose.cumops/#pypose.cumops" title="pypose.cumops"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.cumops()</span></code></a></p>
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+              
+  <section id="pypose-cumprod">
+<h1>pypose.cumprod<a class="headerlink" href="#pypose-cumprod" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.cumprod">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">cumprod</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">dim</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">left</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/basics/ops/#cumprod"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.cumprod" title="Permalink to this definition">¶</a></dt>
+<dd><p>Returns the cumulative product (<code class="docutils literal notranslate"><span class="pre">&#64;</span></code>) of LieTensor along a dimension.</p>
+<ul class="simple">
+<li><p>Left product:</p></li>
+</ul>
+<div class="math">
+\[y_i = x_i ~\times~ x_{i-1} ~\times~ \cdots ~\times~ x_1,
+
+\]</div>
+<ul class="simple">
+<li><p>Right product:</p></li>
+</ul>
+<div class="math">
+\[y_i = x_1 ~\times~ x_2 ~\times~ \cdots ~\times~ x_i,
+
+\]</div>
+<p>where <span class="math">\(\times\)</span> denotes the group product (<code class="docutils literal notranslate"><span class="pre">&#64;</span></code>), <span class="math">\(x_i,~y_i\)</span> are the
+<span class="math">\(i\)</span>-th item along the <code class="xref py py-obj docutils literal notranslate"><span class="pre">dim</span></code> dimension of the input and output LieTensor,
+respectively.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>input</strong> (<a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><em>LieTensor</em></a>) – the input LieTensor</p></li>
+<li><p><strong>dim</strong> (<em>int</em>) – the dimension to do the operation over</p></li>
+<li><p><strong>left</strong> (<em>bool</em><em>, </em><em>optional</em>) – whether perform left product in <a class="reference internal" href="#pypose.cumprod" title="pypose.cumprod"><code class="xref py py-obj docutils literal notranslate"><span class="pre">cumprod</span></code></a>. If set
+it to <code class="xref py py-obj docutils literal notranslate"><span class="pre">False</span></code>, this function performs right product. Defaul: <code class="docutils literal notranslate"><span class="pre">True</span></code></p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>The LieTensor</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor">LieTensor</a></p>
+</dd>
+</dl>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<ul class="simple">
+<li><p>The time complexity of the function is <span class="math">\(\mathcal{O}(\log N)\)</span>, where
+<span class="math">\(N\)</span> is the LieTensor size along the <code class="xref py py-obj docutils literal notranslate"><span class="pre">dim</span></code> dimension.</p></li>
+</ul>
+</div>
+<p class="rubric">Example</p>
+<ul class="simple">
+<li><p>Left product with <span class="math">\(\text{input} \in\)</span> <a class="reference internal" href="../pypose.SE3/#pypose.SE3" title="pypose.SE3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">SE3</span></code></a></p></li>
+</ul>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="nb">input</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_SE3</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">cumprod</span><span class="p">(</span><span class="nb">input</span><span class="p">,</span> <span class="n">dim</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
+<span class="go">SE3Type LieTensor:</span>
+<span class="go">tensor([[-1.9615, -0.1246,  0.3666,  0.0165,  0.2853,  0.3126,  0.9059],</span>
+<span class="go">        [ 0.7139,  1.3988, -0.1909, -0.1780,  0.4405, -0.6571,  0.5852]])</span>
+</pre></div>
+</div>
+<ul class="simple">
+<li><p>Right product with <span class="math">\(\text{input} \in\)</span> <a class="reference internal" href="../pypose.SO3/#pypose.SO3" title="pypose.SO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">SO3</span></code></a></p></li>
+</ul>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="nb">input</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_SO3</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">cumprod</span><span class="p">(</span><span class="nb">input</span><span class="p">,</span> <span class="n">dim</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">left</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+<span class="go">SO3Type LieTensor:</span>
+<span class="go">tensor([[[ 0.5798, -0.1189, -0.2429,  0.7686],</span>
+<span class="go">        [ 0.7515, -0.1920,  0.5072,  0.3758]]])</span>
+</pre></div>
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+              
+  <section id="pypose-cumprod">
+<h1>pypose.cumprod_<a class="headerlink" href="#pypose-cumprod" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.cumprod_">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">cumprod_</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">dim</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">left</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/basics/ops/#cumprod_"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.cumprod_" title="Permalink to this definition">¶</a></dt>
+<dd><p>Inplace version of <a class="reference internal" href="../pypose.cumprod/#pypose.cumprod" title="pypose.cumprod"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.cumprod()</span></code></a></p>
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+<li class="toctree-l1"><a class="reference internal" href="../../lietensor/">LieTensor</a></li>
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+              
+  <section id="pypose-euler">
+<h1>pypose.euler<a class="headerlink" href="#pypose-euler" title="Permalink to this heading">¶</a></h1>
+<dl class="py function">
+<dt class="sig sig-object py" id="pypose.euler">
+<span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">euler</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">inputs</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">eps</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0.0002</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/convert/#euler"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.euler" title="Permalink to this definition">¶</a></dt>
+<dd><p>Convert batched LieTensor into Euler angles (roll, pitch, yaw).</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>inputs</strong> (<a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code></a>) – the input LieTensor.</p></li>
+<li><p><strong>eps</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">float</span></code>, optional) – the threshold to avoid the sigularity caused
+by gimbal lock. Default: 2e-4.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>the batched Euler angles in radians.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code></p>
+</dd>
+</dl>
+<p>Supported input type: <a class="reference internal" href="../pypose.so3/#pypose.so3" title="pypose.so3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">so3</span></code></a>, <a class="reference internal" href="../pypose.SO3/#pypose.SO3" title="pypose.SO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">SO3</span></code></a>, <a class="reference internal" href="../pypose.se3/#pypose.se3" title="pypose.se3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">se3</span></code></a>, <a class="reference internal" href="../pypose.SE3/#pypose.SE3" title="pypose.SE3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">SE3</span></code></a>,
+<a class="reference internal" href="../pypose.sim3/#pypose.sim3" title="pypose.sim3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">sim3</span></code></a>, <a class="reference internal" href="../pypose.Sim3/#pypose.Sim3" title="pypose.Sim3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">Sim3</span></code></a>, <a class="reference internal" href="../pypose.rxso3/#pypose.rxso3" title="pypose.rxso3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">rxso3</span></code></a>, and <a class="reference internal" href="../pypose.RxSO3/#pypose.RxSO3" title="pypose.RxSO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">RxSO3</span></code></a>.</p>
+<div class="admonition warning">
+<p class="admonition-title">Warning</p>
+<p>The Euler angle takes the rotation sequence of x (roll), y (pitch),
+and z (yaw) axis (counterclockwise). There is always more than one
+solution that can result in the same orientation, while this function
+only returns one of them.</p>
+<p>When the pitch angle is around <span class="math">\(\pm \frac{\pi}{2}\)</span> (north/south pole),
+there will be sigularity problem due to the <a class="reference external" href="https://en.wikipedia.org/wiki/Gimbal_lock">gimbal lock</a> and some information can be
+found in <a class="reference external" href="https://tinyurl.com/py8frs6v">this pape</a>.</p>
+</div>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_SO3</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">euler</span><span class="p">()</span>  <span class="c1"># equivalent to pp.euler(x)</span>
+<span class="go">tensor([-0.6599,  0.2749, -0.3263])</span>
+</pre></div>
+</div>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_Sim3</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">euler</span><span class="p">()</span>  <span class="c1"># equivalent to pp.euler(x)</span>
+<span class="go">tensor([[-0.2701, -0.8006, -0.4150],</span>
+<span class="go">        [ 2.1550,  0.1768,  0.9368]])</span>
+</pre></div>
+</div>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_rxso3</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">euler</span><span class="p">()</span>  <span class="c1"># equivalent to pp.euler(x)</span>
+<span class="go">tensor([ 1.2676, -0.4783, -0.3596])</span>
+</pre></div>
+</div>
+<p>See <a class="reference internal" href="../pypose.euler2SO3/#pypose.euler2SO3" title="pypose.euler2SO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">euler2SO3</span></code></a> for more information.</p>
+</dd></dl>
+
+</section>
+
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+              <p class="caption" role="heading"><span class="caption-text">Contents:</span></p>
+<ul class="current">
+<li class="toctree-l1"><a class="reference internal" href="../../lietensor/">LieTensor</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../func/">Func</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../functions/">Functions</a></li>
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+              
+  <section id="pypose-euler2so3">
+<h1>pypose.euler2SO3<a class="headerlink" href="#pypose-euler2so3" title="Permalink to this heading">¶</a></h1>
+<dl class="py function">
+<dt class="sig sig-object py" id="pypose.euler2SO3">
+<span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">euler2SO3</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">euler</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/convert/#euler2SO3"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.euler2SO3" title="Permalink to this definition">¶</a></dt>
+<dd><p>Convert batched Euler angles (roll, pitch, and yaw) to SO3Type LieTensor.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>euler</strong> (<em>Tensor</em>) – the euler angles in radians to convert.</p>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>the converted SO3Type LieTensor.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor">LieTensor</a></p>
+</dd>
+</dl>
+<dl>
+<dt>Shape:</dt><dd><p>Input: <code class="xref py py-obj docutils literal notranslate"><span class="pre">(*,</span> <span class="pre">3)</span></code></p>
+<p>Output: <code class="xref py py-obj docutils literal notranslate"><span class="pre">(*,</span> <span class="pre">4)</span></code></p>
+</dd>
+</dl>
+<div class="math">
+\[{\displaystyle \mathbf{y}_i={
+\begin{bmatrix}\,
+\sin(\alpha_i)\cos(\beta_i)\cos(\gamma_i) - \cos(\alpha_i)\sin(\beta_i)\sin(\gamma_i)\\
+\cos(\alpha_i)\sin(\beta_i)\cos(\gamma_i) + \sin(\alpha_i)\cos(\beta_i)\sin(\gamma_i)\\
+\cos(\alpha_i)\cos(\beta_i)\sin(\gamma_i) - \sin(\alpha_i)\sin(\beta_i)\cos(\gamma_i)\\
+\cos(\alpha_i)\cos(\beta_i)\cos(\gamma_i) + \sin(\alpha_i)\sin(\beta_i)\sin(\gamma_i)
+\end{bmatrix}}},
+
+\]</div>
+<p>where the <span class="math">\(i\)</span>-th item of input <span class="math">\(\mathbf{x}_i = [\alpha_i, \beta_i, \gamma_i]\)</span>
+are roll, pitch, and yaw, respectively.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The last dimension of the input tensor has to be 3. The Euler angle takes the rotation
+sequence of x (roll), y (pitch), then z (yaw) axis (counterclockwise).</p>
+</div>
+<div class="admonition warning">
+<p class="admonition-title">Warning</p>
+<p>Any given rotation has two possible quaternion representations. If one is known, the other
+is just the negative of all four terms. This function only returns one of them.</p>
+</div>
+<p class="rubric">Examples</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="nb">input</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">requires_grad</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">float64</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">euler2SO3</span><span class="p">(</span><span class="nb">input</span><span class="p">)</span>
+<span class="go">SO3Type LieTensor:</span>
+<span class="go">tensor([[-0.4873,  0.1162,  0.4829,  0.7182],</span>
+<span class="go">        [ 0.3813,  0.4059, -0.2966,  0.7758]], dtype=torch.float64, grad_fn=&lt;AliasBackward0&gt;)</span>
+</pre></div>
+</div>
+<p>See <a class="reference internal" href="../pypose.euler/#pypose.euler" title="pypose.euler"><code class="xref py py-obj docutils literal notranslate"><span class="pre">euler</span></code></a> for more information.</p>
+</dd></dl>
+
+</section>
+
+
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+              <p class="caption" role="heading"><span class="caption-text">Contents:</span></p>
+<ul class="current">
+<li class="toctree-l1"><a class="reference internal" href="../../lietensor/">LieTensor</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../func/">Func</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../functions/">Functions</a></li>
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+  <section id="pypose-from-matrix">
+<h1>pypose.from_matrix<a class="headerlink" href="#pypose-from-matrix" title="Permalink to this heading">¶</a></h1>
+<dl class="py function">
+<dt class="sig sig-object py" id="pypose.from_matrix">
+<span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">from_matrix</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">mat</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">ltype</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">check</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">rtol</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1e-05</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">atol</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1e-05</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/convert/#from_matrix"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.from_matrix" title="Permalink to this definition">¶</a></dt>
+<dd><p>Convert batched rotation or transformation matrices to LieTensor.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>mat</strong> (<em>Tensor</em>) – the matrix to convert.</p></li>
+<li><p><strong>ltype</strong> (<em>ltype</em>) – specify the LieTensor type, chosen from <code class="xref py py-class docutils literal notranslate"><span class="pre">pypose.SO3_type</span></code>,
+<code class="xref py py-class docutils literal notranslate"><span class="pre">pypose.SE3_type</span></code>, <code class="xref py py-class docutils literal notranslate"><span class="pre">pypose.Sim3_type</span></code>, or <code class="xref py py-class docutils literal notranslate"><span class="pre">pypose.RxSO3_type</span></code>.
+See more details in <a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-meth docutils literal notranslate"><span class="pre">LieTensor()</span></code></a></p></li>
+<li><p><strong>check</strong> (<em>bool</em><em>, </em><em>optional</em>) – flag to check if the input is valid rotation matrices (orthogonal
+and with a determinant of one). Set to <code class="docutils literal notranslate"><span class="pre">False</span></code> if less computation is needed.
+Default: <code class="docutils literal notranslate"><span class="pre">True</span></code>.</p></li>
+<li><p><strong>rtol</strong> (<em>float</em><em>, </em><em>optional</em>) – relative tolerance when check is enabled. Default: 1e-05</p></li>
+<li><p><strong>atol</strong> (<em>float</em><em>, </em><em>optional</em>) – absolute tolerance when check is enabled. Default: 1e-05</p></li>
+</ul>
+</dd>
+</dl>
+<div class="admonition warning">
+<p class="admonition-title">Warning</p>
+<p>Numerically, a transformation matrix is considered legal if:</p>
+<div class="math">
+\[|{\rm det}(\mathbf{R}) - 1| \leq \texttt{atol} + \texttt{rtol}\times 1\\
+|\mathbf{RR}^{T} - \mathbf{I}| \leq \texttt{atol} + \texttt{rtol}\times \mathbf{I}
+
+\]</div>
+<p>where <span class="math">\(|\cdot |\)</span> is element-wise absolute function. When <code class="docutils literal notranslate"><span class="pre">check</span></code> is set to <code class="docutils literal notranslate"><span class="pre">True</span></code>,
+illegal input will raise a <code class="docutils literal notranslate"><span class="pre">ValueError</span></code>. Otherwise, no data validation is performed.
+Illegal input will output an irrelevant result, which likely contains <code class="docutils literal notranslate"><span class="pre">nan</span></code>.</p>
+</div>
+<dl class="field-list simple">
+<dt class="field-odd">Returns<span class="colon">:</span></dt>
+<dd class="field-odd"><p>the converted LieTensor.</p>
+</dd>
+<dt class="field-even">Return type<span class="colon">:</span></dt>
+<dd class="field-even"><p><a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor">LieTensor</a></p>
+</dd>
+</dl>
+<p class="rubric">Examples</p>
+<ul class="simple">
+<li><p><code class="xref py py-class docutils literal notranslate"><span class="pre">pypose.SO3_type</span></code></p></li>
+</ul>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">from_matrix</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([[</span><span class="mf">0.</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">],</span>
+<span class="gp">... </span>                             <span class="p">[</span><span class="mf">1.</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">],</span>
+<span class="gp">... </span>                             <span class="p">[</span><span class="mf">0.</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]]),</span> <span class="n">ltype</span><span class="o">=</span><span class="n">pp</span><span class="o">.</span><span class="n">SO3_type</span><span class="p">)</span>
+<span class="go">SO3Type LieTensor:</span>
+<span class="go">tensor([0.0000, 0.0000, 0.7071, 0.7071])</span>
+</pre></div>
+</div>
+<ul class="simple">
+<li><p><code class="xref py py-class docutils literal notranslate"><span class="pre">pypose.SE3_type</span></code></p></li>
+</ul>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">from_matrix</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([[</span><span class="mf">0.</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">],</span>
+<span class="gp">... </span>                             <span class="p">[</span><span class="mf">1.</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">],</span>
+<span class="gp">... </span>                             <span class="p">[</span><span class="mf">0.</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">],</span>
+<span class="gp">... </span>                             <span class="p">[</span><span class="mf">0.</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span>  <span class="mf">1.</span><span class="p">]]),</span> <span class="n">ltype</span><span class="o">=</span><span class="n">pp</span><span class="o">.</span><span class="n">SE3_type</span><span class="p">)</span>
+<span class="go">SE3Type LieTensor:</span>
+<span class="go">tensor([0.1000, 0.2000, 0.3000, 0.0000, 0.0000, 0.7071, 0.7071])</span>
+</pre></div>
+</div>
+<ul class="simple">
+<li><p><code class="xref py py-class docutils literal notranslate"><span class="pre">pypose.Sim3_type</span></code></p></li>
+</ul>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">from_matrix</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([[</span> <span class="mf">0.</span><span class="p">,</span><span class="o">-</span><span class="mf">0.5</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">],</span>
+<span class="gp">... </span>                             <span class="p">[</span><span class="mf">0.5</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">],</span>
+<span class="gp">... </span>                             <span class="p">[</span> <span class="mf">0.</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">],</span>
+<span class="gp">... </span>                             <span class="p">[</span> <span class="mf">0.</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">,</span>  <span class="mf">1.</span><span class="p">]]),</span> <span class="n">ltype</span><span class="o">=</span><span class="n">pp</span><span class="o">.</span><span class="n">Sim3_type</span><span class="p">)</span>
+<span class="go">Sim3Type LieTensor:</span>
+<span class="go">tensor([0.1000, 0.2000, 0.3000, 0.0000, 0.0000, 0.7071, 0.7071, 0.5000])</span>
+</pre></div>
+</div>
+<ul class="simple">
+<li><p><code class="xref py py-class docutils literal notranslate"><span class="pre">pypose.RxSO3_type</span></code></p></li>
+</ul>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">from_matrix</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([[</span><span class="mf">0.</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.</span><span class="p">],</span>
+<span class="gp">... </span>                             <span class="p">[</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">],</span>
+<span class="gp">... </span>                             <span class="p">[</span><span class="mf">0.</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">]]),</span> <span class="n">ltype</span><span class="o">=</span><span class="n">pp</span><span class="o">.</span><span class="n">RxSO3_type</span><span class="p">)</span>
+<span class="go">RxSO3Type LieTensor:</span>
+<span class="go">tensor([0.0000, 0.0000, 0.7071, 0.7071, 0.5000])</span>
+</pre></div>
+</div>
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+              
+  <section id="pypose-func-jacrev">
+<h1>pypose.func.jacrev<a class="headerlink" href="#pypose-func-jacrev" title="Permalink to this heading">¶</a></h1>
+<dl class="py function">
+<dt class="sig sig-object py" id="pypose.func.jacrev">
+<span class="sig-prename descclassname"><span class="pre">pypose.func.</span></span><span class="sig-name descname"><span class="pre">jacrev</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">func</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">argnums</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">has_aux</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">chunk_size</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">_preallocate_and_copy</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/func/jac/#jacrev"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.func.jacrev" title="Permalink to this definition">¶</a></dt>
+<dd><p>This function provides the exact same functionality as <a class="reference external" href="https://pytorch.org/docs/stable/generated/torch.func.jacrev.html#torch.func.jacrev">torch.func.jacrev()</a>,
+except that it allows LieTensor to be used as input when calculating the jacobian.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>func</strong> (<em>function</em>) – A Python function that takes one or more arguments,
+one of which must be a Tensor, and returns one or more Tensors</p></li>
+<li><p><strong>argnums</strong> (<em>int</em><em> or </em><em>Tuple</em><em>[</em><em>int</em><em>]</em>) – Optional, integer or tuple of integers,
+saying which arguments to get the Jacobian with respect to.
+Default: 0.</p></li>
+<li><p><strong>has_aux</strong> (<em>bool</em>) – Flag indicating that <code class="docutils literal notranslate"><span class="pre">func</span></code> returns a
+<code class="docutils literal notranslate"><span class="pre">(output,</span> <span class="pre">aux)</span></code> tuple where the first element is the output of
+the function to be differentiated and the second element is
+auxiliary objects that will not be differentiated.
+Default: False.</p></li>
+<li><p><strong>chunk_size</strong> (<em>None</em><em> or </em><em>int</em>) – If None (default), use the maximum chunk size
+(equivalent to doing a single vmap over vjp to compute the jacobian).
+If 1, then compute the jacobian row-by-row with a for-loop.
+If not None, then compute the jacobian <code class="xref py py-attr docutils literal notranslate"><span class="pre">chunk_size</span></code> rows at a time
+(equivalent to doing multiple vmap over vjp). If you run into memory issues
+computing the jacobian, please try to specify a non-None chunk_size.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>Returns a function that takes in the same inputs as <code class="docutils literal notranslate"><span class="pre">func</span></code> and
+returns the Jacobian of <code class="docutils literal notranslate"><span class="pre">func</span></code> with respect to the arg(s) at
+<code class="docutils literal notranslate"><span class="pre">argnums</span></code>. If <code class="docutils literal notranslate"><span class="pre">has_aux</span> <span class="pre">is</span> <span class="pre">True</span></code>, then the returned function
+instead returns a <code class="docutils literal notranslate"><span class="pre">(jacobian,</span> <span class="pre">aux)</span></code> tuple where <code class="docutils literal notranslate"><span class="pre">jacobian</span></code>
+is the Jacobian and <code class="docutils literal notranslate"><span class="pre">aux</span></code> is auxiliary objects returned by <code class="docutils literal notranslate"><span class="pre">func</span></code>.</p>
+</dd>
+</dl>
+<p>A basic usage with our LieTensor type would be the transformation function.</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">pypose</span> <span class="k">as</span> <span class="nn">pp</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">torch</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="k">def</span> <span class="nf">func</span><span class="p">(</span><span class="n">pose</span><span class="p">,</span> <span class="n">points</span><span class="p">):</span>
+<span class="gp">... </span>    <span class="k">return</span> <span class="n">pose</span> <span class="o">@</span> <span class="n">points</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">pose</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_SE3</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">points</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">jacobian</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">func</span><span class="o">.</span><span class="n">jacrev</span><span class="p">(</span><span class="n">func</span><span class="p">)(</span><span class="n">pose</span><span class="p">,</span> <span class="n">points</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">jacobian</span>
+<span class="go">tensor([[[[ 1.0000,  0.0000,  0.0000,  0.0000,  1.5874, -0.2061,  0.0000]],</span>
+<span class="go">        [[ 0.0000,  1.0000,  0.0000, -1.5874,  0.0000, -1.4273,  0.0000]],</span>
+<span class="go">        [[ 0.0000,  0.0000,  1.0000,  0.2061,  1.4273,  0.0000,  0.0000]]]])</span>
+</pre></div>
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+              
+  <section id="pypose-hasnan">
+<h1>pypose.hasnan<a class="headerlink" href="#pypose-hasnan" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.hasnan">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">hasnan</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">obj</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/function/checking/#hasnan"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.hasnan" title="Permalink to this definition">¶</a></dt>
+<dd><p>Checks whether a deep nested list of tensors contains Nan or not.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>obj</strong> (<code class="docutils literal notranslate"><span class="pre">obj</span></code>) – a Python object that can be a list of nested list.</p>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p><code class="docutils literal notranslate"><span class="pre">True</span></code> if the list contains a tensor with <code class="docutils literal notranslate"><span class="pre">Nan</span></code> otherwise <code class="docutils literal notranslate"><span class="pre">False</span></code>.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><code class="docutils literal notranslate"><span class="pre">bool</span></code></p>
+</dd>
+</dl>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">L1</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="p">[</span><span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">]],</span> <span class="mi">7</span><span class="p">,</span> <span class="p">[</span><span class="mi">8</span><span class="p">,</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.0999</span><span class="p">])]]</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">hasnan</span><span class="p">(</span><span class="n">L1</span><span class="p">)</span>
+<span class="go">False</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">L2</span> <span class="o">=</span> <span class="p">[[</span><span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="nb">float</span><span class="p">(</span><span class="s1">&#39;nan&#39;</span><span class="p">),</span> <span class="o">-</span><span class="mf">1.0999</span><span class="p">]),</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="p">[</span><span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">]],</span> <span class="mi">7</span><span class="p">,</span> <span class="p">[</span><span class="mi">8</span><span class="p">,</span> <span class="mi">9</span><span class="p">]]</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">hasnan</span><span class="p">(</span><span class="n">L2</span><span class="p">)</span>
+<span class="go">True</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">L3</span> <span class="o">=</span> <span class="p">[[</span><span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.0999</span><span class="p">]),</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="p">[</span><span class="nb">float</span><span class="p">(</span><span class="s1">&#39;nan&#39;</span><span class="p">),</span> <span class="mi">6</span><span class="p">]],</span> <span class="mi">7</span><span class="p">,</span> <span class="p">[</span><span class="mi">8</span><span class="p">,</span> <span class="mi">9</span><span class="p">]]</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">hasnan</span><span class="p">(</span><span class="n">L3</span><span class="p">)</span>
+<span class="go">True</span>
+</pre></div>
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+              
+  <section id="pypose-homo2cart">
+<h1>pypose.homo2cart<a class="headerlink" href="#pypose-homo2cart" title="Permalink to this heading">¶</a></h1>
+<dl class="py function">
+<dt class="sig sig-object py" id="pypose.homo2cart">
+<span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">homo2cart</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">coordinates</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/function/geometry/#homo2cart"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.homo2cart" title="Permalink to this definition">¶</a></dt>
+<dd><p>Converts batched Homogeneous coordinates to Cartesian coordinates
+by dividing the last row. Size of the last dimension will be reduced by 1.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>coordinates</strong> (<code class="docutils literal notranslate"><span class="pre">torch.Tensor</span></code>) – the Homogeneous coordinates to be converted.</p>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>the coordinates in Cartesian space.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><code class="docutils literal notranslate"><span class="pre">torch.Tensor</span></code></p>
+</dd>
+</dl>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">points</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([[</span><span class="mf">4.</span><span class="p">,</span> <span class="mf">3.</span><span class="p">,</span> <span class="mf">2.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">],</span> <span class="p">[</span><span class="mf">8.</span><span class="p">,</span> <span class="mf">6.</span><span class="p">,</span> <span class="mf">4.</span><span class="p">,</span> <span class="mf">2.</span><span class="p">]])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">homo2cart</span><span class="p">(</span><span class="n">points</span><span class="p">)</span>
+<span class="go">tensor([[4., 3., 2.],</span>
+<span class="go">        [4., 3., 2.]])</span>
+</pre></div>
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+  <section id="pypose-identity-rxso3">
+<h1>pypose.identity_RxSO3<a class="headerlink" href="#pypose-identity-rxso3" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.identity_RxSO3">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">identity_RxSO3</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">size</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/utils/#identity_RxSO3"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.identity_RxSO3" title="Permalink to this definition">¶</a></dt>
+<dd><p>Returns identity <code class="xref py py-obj docutils literal notranslate"><span class="pre">RxSO3_type</span></code> LieTensor with the given <code class="xref py py-obj docutils literal notranslate"><span class="pre">lsize</span></code>.</p>
+<p>See <a class="reference internal" href="../pypose.RxSO3/#pypose.RxSO3" title="pypose.RxSO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">RxSO3()</span></code></a> for implementation details.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>lsize</strong> (<em>int</em><em>...</em><em>, </em><em>optional</em>) – a sequence of integers defining the <a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor.lshape" title="pypose.LieTensor.lshape"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor.lshape</span></code></a> of
+the output LieTensor. Can be a variable number of arguments or a collection like a
+list or tuple. If not given, a single <code class="xref py py-obj docutils literal notranslate"><span class="pre">RxSO3_type</span></code> item will be returned.</p></li>
+<li><p><strong>requires_grad</strong> (<em>bool</em><em>, </em><em>optional</em>) – If autograd should record operations on
+the returned tensor. Default: False.</p></li>
+<li><p><strong>generator</strong> (<em>torch.Generator</em><em>, </em><em>optional</em>) – a pseudorandom number generator for sampling</p></li>
+<li><p><strong>dtype</strong> (<em>torch.dtype</em><em>, </em><em>optional</em>) – the desired data type of returned tensor.
+Default: if None, uses a global default (see <code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.set_default_tensor_type()</span></code>).</p></li>
+<li><p><strong>layout</strong> (<em>torch.layout</em><em>, </em><em>optional</em>) – the desired layout of returned Tensor.
+Default: torch.strided.</p></li>
+<li><p><strong>device</strong> (<em>torch.device</em><em>, </em><em>optional</em>) – the desired device of returned tensor.
+Default: if None, uses the current device for the default tensor
+type (see <code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.set_default_tensor_type()</span></code>). device will be the CPU
+for CPU tensor types and the current CUDA device for CUDA tensor types.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>a <code class="xref py py-obj docutils literal notranslate"><span class="pre">RxSO3_type</span></code> LieTensor</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor">LieTensor</a></p>
+</dd>
+</dl>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">identity_RxSO3</span><span class="p">()</span>
+<span class="go">RxSO3Type LieTensor:</span>
+<span class="go">tensor([0., 0., 0., 1., 1.])</span>
+</pre></div>
+</div>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">identity_RxSO3</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
+<span class="go">RxSO3Type LieTensor:</span>
+<span class="go">tensor([[0., 0., 0., 1., 1.],</span>
+<span class="go">        [0., 0., 0., 1., 1.]])</span>
+</pre></div>
+</div>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">identity_RxSO3</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
+<span class="go">RxSO3Type LieTensor:</span>
+<span class="go">tensor([[[0., 0., 0., 1., 1.]],</span>
+<span class="go">        [[0., 0., 0., 1., 1.]]])</span>
+</pre></div>
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+  <section id="pypose-identity-se3">
+<h1>pypose.identity_SE3<a class="headerlink" href="#pypose-identity-se3" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.identity_SE3">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">identity_SE3</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">lsize</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/utils/#identity_SE3"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.identity_SE3" title="Permalink to this definition">¶</a></dt>
+<dd><p>Returns identity <code class="xref py py-obj docutils literal notranslate"><span class="pre">SE3_type</span></code> LieTensor with the given <code class="xref py py-obj docutils literal notranslate"><span class="pre">lsize</span></code>.</p>
+<p>See <a class="reference internal" href="../pypose.SE3/#pypose.SE3" title="pypose.SE3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">SE3()</span></code></a> for implementation details.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>lsize</strong> (<em>int</em><em>...</em><em>, </em><em>optional</em>) – a sequence of integers defining the <a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor.lshape" title="pypose.LieTensor.lshape"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor.lshape</span></code></a> of
+the output LieTensor. Can be a variable number of arguments or a collection like a
+list or tuple. If not given, a single <code class="xref py py-obj docutils literal notranslate"><span class="pre">SE3_type</span></code> item will be returned.</p></li>
+<li><p><strong>requires_grad</strong> (<em>bool</em><em>, </em><em>optional</em>) – If autograd should record operations on
+the returned tensor. Default: False.</p></li>
+<li><p><strong>generator</strong> (<em>torch.Generator</em><em>, </em><em>optional</em>) – a pseudorandom number generator for sampling</p></li>
+<li><p><strong>dtype</strong> (<em>torch.dtype</em><em>, </em><em>optional</em>) – the desired data type of returned tensor.
+Default: if None, uses a global default (see <code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.set_default_tensor_type()</span></code>).</p></li>
+<li><p><strong>layout</strong> (<em>torch.layout</em><em>, </em><em>optional</em>) – the desired layout of returned Tensor.
+Default: torch.strided.</p></li>
+<li><p><strong>device</strong> (<em>torch.device</em><em>, </em><em>optional</em>) – the desired device of returned tensor.
+Default: if None, uses the current device for the default tensor
+type (see <code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.set_default_tensor_type()</span></code>). device will be the CPU
+for CPU tensor types and the current CUDA device for CUDA tensor types.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>a <code class="xref py py-obj docutils literal notranslate"><span class="pre">SE3_type</span></code> LieTensor</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor">LieTensor</a></p>
+</dd>
+</dl>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">identity_SE3</span><span class="p">()</span>
+<span class="go">SE3Type LieTensor:</span>
+<span class="go">tensor([0., 0., 0., 0., 0., 0., 1.])</span>
+</pre></div>
+</div>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">identity_SE3</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
+<span class="go">SE3Type LieTensor:</span>
+<span class="go">tensor([[0., 0., 0., 0., 0., 0., 1.],</span>
+<span class="go">        [0., 0., 0., 0., 0., 0., 1.]])</span>
+</pre></div>
+</div>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">identity_SE3</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
+<span class="go">SE3Type LieTensor:</span>
+<span class="go">tensor([[[0., 0., 0., 0., 0., 0., 1.]],</span>
+<span class="go">        [[0., 0., 0., 0., 0., 0., 1.]]])</span>
+</pre></div>
+</div>
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+  <section id="pypose-identity-so3">
+<h1>pypose.identity_SO3<a class="headerlink" href="#pypose-identity-so3" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.identity_SO3">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">identity_SO3</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">lsize</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/utils/#identity_SO3"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.identity_SO3" title="Permalink to this definition">¶</a></dt>
+<dd><p>Returns identity <code class="xref py py-obj docutils literal notranslate"><span class="pre">SO3_type</span></code> LieTensor with the given <code class="xref py py-obj docutils literal notranslate"><span class="pre">lsize</span></code>.</p>
+<p>See <a class="reference internal" href="../pypose.SO3/#pypose.SO3" title="pypose.SO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">SO3()</span></code></a> for implementation details.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>lsize</strong> (<em>int</em><em>...</em><em>, </em><em>optional</em>) – a sequence of integers defining the <a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor.lshape" title="pypose.LieTensor.lshape"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor.lshape</span></code></a> of
+the output LieTensor. Can be a variable number of arguments or a collection like a
+list or tuple. If not given, a single <code class="xref py py-obj docutils literal notranslate"><span class="pre">SO3_type</span></code> item will be returned.</p></li>
+<li><p><strong>requires_grad</strong> (<em>bool</em><em>, </em><em>optional</em>) – If autograd should record operations on
+the returned tensor. Default: False.</p></li>
+<li><p><strong>generator</strong> (<em>torch.Generator</em><em>, </em><em>optional</em>) – a pseudorandom number generator for sampling</p></li>
+<li><p><strong>dtype</strong> (<em>torch.dtype</em><em>, </em><em>optional</em>) – the desired data type of returned tensor.
+Default: if None, uses a global default (see <code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.set_default_tensor_type()</span></code>).</p></li>
+<li><p><strong>layout</strong> (<em>torch.layout</em><em>, </em><em>optional</em>) – the desired layout of returned Tensor.
+Default: torch.strided.</p></li>
+<li><p><strong>device</strong> (<em>torch.device</em><em>, </em><em>optional</em>) – the desired device of returned tensor.
+Default: if None, uses the current device for the default tensor
+type (see <code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.set_default_tensor_type()</span></code>). device will be the CPU
+for CPU tensor types and the current CUDA device for CUDA tensor types.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>a <code class="xref py py-obj docutils literal notranslate"><span class="pre">SO3_type</span></code> LieTensor</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor">LieTensor</a></p>
+</dd>
+</dl>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">identity_SO3</span><span class="p">()</span>
+<span class="go">SO3Type LieTensor:</span>
+<span class="go">tensor([0., 0., 0., 1.])</span>
+</pre></div>
+</div>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">identity_SO3</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
+<span class="go">SO3Type LieTensor:</span>
+<span class="go">tensor([[0., 0., 0., 1.],</span>
+<span class="go">        [0., 0., 0., 1.]])</span>
+</pre></div>
+</div>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">identity_SO3</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
+<span class="go">SO3Type LieTensor:</span>
+<span class="go">tensor([[[0., 0., 0., 1.]],</span>
+<span class="go">        [[0., 0., 0., 1.]]])</span>
+</pre></div>
+</div>
+</dd></dl>
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+  <section id="pypose-identity-sim3">
+<h1>pypose.identity_Sim3<a class="headerlink" href="#pypose-identity-sim3" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.identity_Sim3">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">identity_Sim3</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">lsize</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/utils/#identity_Sim3"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.identity_Sim3" title="Permalink to this definition">¶</a></dt>
+<dd><p>Returns identity <code class="xref py py-obj docutils literal notranslate"><span class="pre">Sim3_type</span></code> LieTensor with the given <code class="xref py py-obj docutils literal notranslate"><span class="pre">lsize</span></code>.</p>
+<p>See <a class="reference internal" href="../pypose.Sim3/#pypose.Sim3" title="pypose.Sim3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">Sim3()</span></code></a> for implementation details.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>lsize</strong> (<em>int</em><em>...</em><em>, </em><em>optional</em>) – a sequence of integers defining the <a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor.lshape" title="pypose.LieTensor.lshape"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor.lshape</span></code></a> of
+the output LieTensor. Can be a variable number of arguments or a collection like a
+list or tuple. If not given, a single <code class="xref py py-obj docutils literal notranslate"><span class="pre">Sim3_type</span></code> item will be returned.</p></li>
+<li><p><strong>requires_grad</strong> (<em>bool</em><em>, </em><em>optional</em>) – If autograd should record operations on
+the returned tensor. Default: False.</p></li>
+<li><p><strong>generator</strong> (<em>torch.Generator</em><em>, </em><em>optional</em>) – a pseudorandom number generator for sampling</p></li>
+<li><p><strong>dtype</strong> (<em>torch.dtype</em><em>, </em><em>optional</em>) – the desired data type of returned tensor.
+Default: if None, uses a global default (see <code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.set_default_tensor_type()</span></code>).</p></li>
+<li><p><strong>layout</strong> (<em>torch.layout</em><em>, </em><em>optional</em>) – the desired layout of returned Tensor.
+Default: torch.strided.</p></li>
+<li><p><strong>device</strong> (<em>torch.device</em><em>, </em><em>optional</em>) – the desired device of returned tensor.
+Default: if None, uses the current device for the default tensor
+type (see <code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.set_default_tensor_type()</span></code>). device will be the CPU
+for CPU tensor types and the current CUDA device for CUDA tensor types.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>a <code class="xref py py-obj docutils literal notranslate"><span class="pre">Sim3_type</span></code> LieTensor</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor">LieTensor</a></p>
+</dd>
+</dl>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">identity_Sim3</span><span class="p">()</span>
+<span class="go">Sim3Type LieTensor:</span>
+<span class="go">tensor([0., 0., 0., 0., 0., 0., 1., 1.])</span>
+</pre></div>
+</div>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">identity_Sim3</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
+<span class="go">Sim3Type LieTensor:</span>
+<span class="go">tensor([[0., 0., 0., 0., 0., 0., 1., 1.],</span>
+<span class="go">        [0., 0., 0., 0., 0., 0., 1., 1.]])</span>
+</pre></div>
+</div>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">identity_Sim3</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
+<span class="go">Sim3Type LieTensor:</span>
+<span class="go">tensor([[[0., 0., 0., 0., 0., 0., 1., 1.]],</span>
+<span class="go">        [[0., 0., 0., 0., 0., 0., 1., 1.]]])</span>
+</pre></div>
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+              
+  <section id="pypose-identity-like">
+<h1>pypose.identity_like<a class="headerlink" href="#pypose-identity-like" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.identity_like">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">identity_like</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">liegroup</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/utils/#identity_like"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.identity_like" title="Permalink to this definition">¶</a></dt>
+<dd><blockquote>
+<div><p>Returns identity LieTensor with the same <code class="xref py py-obj docutils literal notranslate"><span class="pre">lsize</span></code> and <code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code> as the given LieTensor.</p>
+</div></blockquote>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>liegroup</strong> (<a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><em>LieTensor</em></a>) – the size of liegroup will determine the size of the output tensor.</p></li>
+<li><p><strong>requires_grad</strong> (<em>bool</em><em>, </em><em>optional</em>) – If autograd should record operations on
+the returned tensor. Default: False.</p></li>
+<li><p><strong>generator</strong> (<em>torch.Generator</em><em>, </em><em>optional</em>) – a pseudorandom number generator for sampling</p></li>
+<li><p><strong>dtype</strong> (<em>torch.dtype</em><em>, </em><em>optional</em>) – the desired data type of returned tensor.
+Default: if None, uses a global default (see <code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.set_default_tensor_type()</span></code>).</p></li>
+<li><p><strong>layout</strong> (<em>torch.layout</em><em>, </em><em>optional</em>) – the desired layout of returned Tensor.
+Default: torch.strided.</p></li>
+<li><p><strong>device</strong> (<em>torch.device</em><em>, </em><em>optional</em>) – the desired device of returned tensor.
+Default: if None, uses the current device for the default tensor
+type (see <code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.set_default_tensor_type()</span></code>). device will be the CPU
+for CPU tensor types and the current CUDA device for CUDA tensor types.</p></li>
+</ul>
+</dd>
+</dl>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_SO3</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="s2">&quot;cuda:0&quot;</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">double</span><span class="p">,</span> <span class="n">requires_grad</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">identity_like</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="s2">&quot;cpu&quot;</span><span class="p">)</span>
+<span class="go">SO3Type LieTensor:</span>
+<span class="go">tensor([[0., 0., 0., 1.],</span>
+<span class="go">        [0., 0., 0., 1.],</span>
+<span class="go">        [0., 0., 0., 1.]])</span>
+</pre></div>
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+  <section id="pypose-identity-rxso3">
+<h1>pypose.identity_rxso3<a class="headerlink" href="#pypose-identity-rxso3" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.identity_rxso3">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">identity_rxso3</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">size</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/utils/#identity_rxso3"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.identity_rxso3" title="Permalink to this definition">¶</a></dt>
+<dd><p>Returns identity <code class="xref py py-obj docutils literal notranslate"><span class="pre">rxso3_type</span></code> LieTensor with the given <code class="xref py py-obj docutils literal notranslate"><span class="pre">lsize</span></code>.</p>
+<p>See <code class="xref py py-obj docutils literal notranslate"><span class="pre">rxSO3()</span></code> for implementation details.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>lsize</strong> (<em>int</em><em>...</em><em>, </em><em>optional</em>) – a sequence of integers defining the <a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor.lshape" title="pypose.LieTensor.lshape"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor.lshape</span></code></a> of
+the output LieTensor. Can be a variable number of arguments or a collection like a
+list or tuple. If not given, a single <code class="xref py py-obj docutils literal notranslate"><span class="pre">rxso3_type</span></code> item will be returned.</p></li>
+<li><p><strong>requires_grad</strong> (<em>bool</em><em>, </em><em>optional</em>) – If autograd should record operations on
+the returned tensor. Default: False.</p></li>
+<li><p><strong>generator</strong> (<em>torch.Generator</em><em>, </em><em>optional</em>) – a pseudorandom number generator for sampling</p></li>
+<li><p><strong>dtype</strong> (<em>torch.dtype</em><em>, </em><em>optional</em>) – the desired data type of returned tensor.
+Default: if None, uses a global default (see <code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.set_default_tensor_type()</span></code>).</p></li>
+<li><p><strong>layout</strong> (<em>torch.layout</em><em>, </em><em>optional</em>) – the desired layout of returned Tensor.
+Default: torch.strided.</p></li>
+<li><p><strong>device</strong> (<em>torch.device</em><em>, </em><em>optional</em>) – the desired device of returned tensor.
+Default: if None, uses the current device for the default tensor
+type (see <code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.set_default_tensor_type()</span></code>). device will be the CPU
+for CPU tensor types and the current CUDA device for CUDA tensor types.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>a <code class="xref py py-obj docutils literal notranslate"><span class="pre">rxso3_type</span></code> LieTensor</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor">LieTensor</a></p>
+</dd>
+</dl>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">identity_rxso3</span><span class="p">()</span>
+<span class="go">rxso3Type LieTensor:</span>
+<span class="go">tensor([0., 0., 0., 0.])</span>
+</pre></div>
+</div>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">identity_rxso3</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
+<span class="go">rxso3Type LieTensor:</span>
+<span class="go">tensor([[0., 0., 0., 0.],</span>
+<span class="go">        [0., 0., 0., 0.]])</span>
+</pre></div>
+</div>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">identity_rxso3</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
+<span class="go">rxso3Type LieTensor:</span>
+<span class="go">tensor([[[0., 0., 0., 0.]],</span>
+<span class="go">        [[0., 0., 0., 0.]]])</span>
+</pre></div>
+</div>
+</dd></dl>
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+  <section id="pypose-identity-se3">
+<h1>pypose.identity_se3<a class="headerlink" href="#pypose-identity-se3" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.identity_se3">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">identity_se3</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">lsize</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/utils/#identity_se3"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.identity_se3" title="Permalink to this definition">¶</a></dt>
+<dd><p>Returns identity <code class="xref py py-obj docutils literal notranslate"><span class="pre">se3_type</span></code> LieTensor with the given <code class="xref py py-obj docutils literal notranslate"><span class="pre">lsize</span></code>.</p>
+<p>See <a class="reference internal" href="../pypose.se3/#pypose.se3" title="pypose.se3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">se3()</span></code></a> for implementation details.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>lsize</strong> (<em>int</em><em>...</em><em>, </em><em>optional</em>) – a sequence of integers defining the <a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor.lshape" title="pypose.LieTensor.lshape"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor.lshape</span></code></a> of
+the output LieTensor. Can be a variable number of arguments or a collection like a
+list or tuple. If not given, a single <code class="xref py py-obj docutils literal notranslate"><span class="pre">se3_type</span></code> item will be returned.</p></li>
+<li><p><strong>requires_grad</strong> (<em>bool</em><em>, </em><em>optional</em>) – If autograd should record operations on
+the returned tensor. Default: False.</p></li>
+<li><p><strong>generator</strong> (<em>torch.Generator</em><em>, </em><em>optional</em>) – a pseudorandom number generator for sampling</p></li>
+<li><p><strong>dtype</strong> (<em>torch.dtype</em><em>, </em><em>optional</em>) – the desired data type of returned tensor.
+Default: if None, uses a global default (see <code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.set_default_tensor_type()</span></code>).</p></li>
+<li><p><strong>layout</strong> (<em>torch.layout</em><em>, </em><em>optional</em>) – the desired layout of returned Tensor.
+Default: torch.strided.</p></li>
+<li><p><strong>device</strong> (<em>torch.device</em><em>, </em><em>optional</em>) – the desired device of returned tensor.
+Default: if None, uses the current device for the default tensor
+type (see <code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.set_default_tensor_type()</span></code>). device will be the CPU
+for CPU tensor types and the current CUDA device for CUDA tensor types.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>a <code class="xref py py-obj docutils literal notranslate"><span class="pre">se3_type</span></code> LieTensor</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor">LieTensor</a></p>
+</dd>
+</dl>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">identity_se3</span><span class="p">()</span>
+<span class="go">se3Type LieTensor:</span>
+<span class="go">tensor([0., 0., 0., 0., 0., 0.])</span>
+</pre></div>
+</div>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">identity_se3</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
+<span class="go">se3Type LieTensor:</span>
+<span class="go">tensor([[0., 0., 0., 0., 0., 0.],</span>
+<span class="go">        [0., 0., 0., 0., 0., 0.]])</span>
+</pre></div>
+</div>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">identity_se3</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
+<span class="go">se3Type LieTensor:</span>
+<span class="go">tensor([[0., 0., 0., 0., 0., 0.],</span>
+<span class="go">        [0., 0., 0., 0., 0., 0.]])</span>
+</pre></div>
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+  <section id="pypose-identity-sim3">
+<h1>pypose.identity_sim3<a class="headerlink" href="#pypose-identity-sim3" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.identity_sim3">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">identity_sim3</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">lsize</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/utils/#identity_sim3"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.identity_sim3" title="Permalink to this definition">¶</a></dt>
+<dd><p>Returns identity <code class="xref py py-obj docutils literal notranslate"><span class="pre">sim3_type</span></code> LieTensor with the given <code class="xref py py-obj docutils literal notranslate"><span class="pre">lsize</span></code>.</p>
+<p>See <a class="reference internal" href="../pypose.sim3/#pypose.sim3" title="pypose.sim3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">sim3()</span></code></a> for implementation details.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>lsize</strong> (<em>int</em><em>...</em><em>, </em><em>optional</em>) – a sequence of integers defining the <a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor.lshape" title="pypose.LieTensor.lshape"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor.lshape</span></code></a> of
+the output LieTensor. Can be a variable number of arguments or a collection like a
+list or tuple. If not given, a single <code class="xref py py-obj docutils literal notranslate"><span class="pre">sim3_type</span></code> item will be returned.</p></li>
+<li><p><strong>requires_grad</strong> (<em>bool</em><em>, </em><em>optional</em>) – If autograd should record operations on
+the returned tensor. Default: False.</p></li>
+<li><p><strong>generator</strong> (<em>torch.Generator</em><em>, </em><em>optional</em>) – a pseudorandom number generator for sampling</p></li>
+<li><p><strong>dtype</strong> (<em>torch.dtype</em><em>, </em><em>optional</em>) – the desired data type of returned tensor.
+Default: if None, uses a global default (see <code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.set_default_tensor_type()</span></code>).</p></li>
+<li><p><strong>layout</strong> (<em>torch.layout</em><em>, </em><em>optional</em>) – the desired layout of returned Tensor.
+Default: torch.strided.</p></li>
+<li><p><strong>device</strong> (<em>torch.device</em><em>, </em><em>optional</em>) – the desired device of returned tensor.
+Default: if None, uses the current device for the default tensor
+type (see <code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.set_default_tensor_type()</span></code>). device will be the CPU
+for CPU tensor types and the current CUDA device for CUDA tensor types.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>a <code class="xref py py-obj docutils literal notranslate"><span class="pre">sim3_type</span></code> LieTensor</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor">LieTensor</a></p>
+</dd>
+</dl>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">identity_sim3</span><span class="p">()</span>
+<span class="go">sim3Type LieTensor:</span>
+<span class="go">tensor([0., 0., 0., 0., 0., 0., 0.])</span>
+</pre></div>
+</div>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">identity_sim3</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
+<span class="go">sim3Type LieTensor:</span>
+<span class="go">tensor([[0., 0., 0., 0., 0., 0., 0.],</span>
+<span class="go">        [0., 0., 0., 0., 0., 0., 0.]])</span>
+</pre></div>
+</div>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">identity_sim3</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
+<span class="go">sim3Type LieTensor:</span>
+<span class="go">tensor([[[0., 0., 0., 0., 0., 0., 0.]],</span>
+<span class="go">        [[0., 0., 0., 0., 0., 0., 0.]]])</span>
+</pre></div>
+</div>
+</dd></dl>
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+<li class="toctree-l1"><a class="reference internal" href="../../lietensor/">LieTensor</a></li>
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+  <section id="pypose-identity-so3">
+<h1>pypose.identity_so3<a class="headerlink" href="#pypose-identity-so3" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.identity_so3">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">identity_so3</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">lsize</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/utils/#identity_so3"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.identity_so3" title="Permalink to this definition">¶</a></dt>
+<dd><p>Returns identity <code class="xref py py-obj docutils literal notranslate"><span class="pre">so3_type</span></code> LieTensor with the given <code class="xref py py-obj docutils literal notranslate"><span class="pre">lsize</span></code>.</p>
+<p>See <a class="reference internal" href="../pypose.so3/#pypose.so3" title="pypose.so3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">so3()</span></code></a> for implementation details.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>lsize</strong> (<em>int</em><em>...</em><em>, </em><em>optional</em>) – a sequence of integers defining the <a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor.lshape" title="pypose.LieTensor.lshape"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor.lshape</span></code></a> of
+the output LieTensor. Can be a variable number of arguments or a collection like a
+list or tuple. If not given, a single <code class="xref py py-obj docutils literal notranslate"><span class="pre">so3_type</span></code> item will be returned.</p></li>
+<li><p><strong>requires_grad</strong> (<em>bool</em><em>, </em><em>optional</em>) – If autograd should record operations on
+the returned tensor. Default: False.</p></li>
+<li><p><strong>generator</strong> (<em>torch.Generator</em><em>, </em><em>optional</em>) – a pseudorandom number generator for sampling</p></li>
+<li><p><strong>dtype</strong> (<em>torch.dtype</em><em>, </em><em>optional</em>) – the desired data type of returned tensor.
+Default: if None, uses a global default (see <code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.set_default_tensor_type()</span></code>).</p></li>
+<li><p><strong>layout</strong> (<em>torch.layout</em><em>, </em><em>optional</em>) – the desired layout of returned Tensor.
+Default: torch.strided.</p></li>
+<li><p><strong>device</strong> (<em>torch.device</em><em>, </em><em>optional</em>) – the desired device of returned tensor.
+Default: if None, uses the current device for the default tensor
+type (see <code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.set_default_tensor_type()</span></code>). device will be the CPU
+for CPU tensor types and the current CUDA device for CUDA tensor types.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>a <code class="xref py py-obj docutils literal notranslate"><span class="pre">so3_type</span></code> LieTensor</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor">LieTensor</a></p>
+</dd>
+</dl>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">identity_so3</span><span class="p">()</span>
+<span class="go">so3Type LieTensor:</span>
+<span class="go">tensor([0., 0., 0.])</span>
+</pre></div>
+</div>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">identity_so3</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
+<span class="go">so3Type LieTensor:</span>
+<span class="go">tensor([[0., 0., 0.],</span>
+<span class="go">        [0., 0., 0.]])</span>
+</pre></div>
+</div>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">identity_so3</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
+<span class="go">so3Type LieTensor:</span>
+<span class="go">tensor([[[0., 0., 0.]],</span>
+<span class="go">        [[0., 0., 0.]]])</span>
+</pre></div>
+</div>
+</dd></dl>
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+<ul class="current">
+<li class="toctree-l1"><a class="reference internal" href="../../lietensor/">LieTensor</a></li>
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+  <section id="pypose-is-se3">
+<h1>pypose.is_SE3<a class="headerlink" href="#pypose-is-se3" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.is_SE3">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">is_SE3</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">obj</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/function/checking/#is_SE3"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.is_SE3" title="Permalink to this definition">¶</a></dt>
+<dd><p>Check whether an instance or object is an SE3 Type LieTensor or not.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>obj</strong> (<code class="docutils literal notranslate"><span class="pre">obj</span></code>) – a Python object or instantance.</p>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p><code class="docutils literal notranslate"><span class="pre">True</span></code> if obj is a SE3 Type LieTensor object otherwise <code class="docutils literal notranslate"><span class="pre">False</span></code>.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><code class="docutils literal notranslate"><span class="pre">bool</span></code></p>
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+              <p class="caption" role="heading"><span class="caption-text">Contents:</span></p>
+<ul class="current">
+<li class="toctree-l1"><a class="reference internal" href="../../lietensor/">LieTensor</a></li>
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+  <section id="pypose-is-lietensor">
+<h1>pypose.is_lietensor<a class="headerlink" href="#pypose-is-lietensor" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.is_lietensor">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">is_lietensor</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">obj</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/function/checking/#is_lietensor"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.is_lietensor" title="Permalink to this definition">¶</a></dt>
+<dd><p>Check whether an instance or object is a LieTensor or not.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>obj</strong> (<code class="docutils literal notranslate"><span class="pre">obj</span></code>) – a Python object or instantance.</p>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p><code class="docutils literal notranslate"><span class="pre">True</span></code> if obj is a LieTensor object otherwise <code class="docutils literal notranslate"><span class="pre">False</span></code>.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><code class="docutils literal notranslate"><span class="pre">bool</span></code></p>
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+  <section id="pypose-knn">
+<h1>pypose.knn<a class="headerlink" href="#pypose-knn" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.knn">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">knn</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">ref</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">nbr</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">k</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">ord</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">2</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">dim</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">-</span> <span class="pre">1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">largest</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">sorted</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/function/geometry/#knn"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.knn" title="Permalink to this definition">¶</a></dt>
+<dd><p>Select the k nearest neighbor points of reference from neighbors in each batch.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>ref</strong> (<code class="docutils literal notranslate"><span class="pre">torch.Tensor</span></code>) – the coordinates of the reference point sets.
+The shape has to be (…, N1, :).</p></li>
+<li><p><strong>nbr</strong> (<code class="docutils literal notranslate"><span class="pre">torch.Tensor</span></code>) – the coordinates of the neighbors point sets.
+The shape has to be (…, N2, :).</p></li>
+<li><p><strong>k</strong> (<code class="docutils literal notranslate"><span class="pre">int</span></code>, optional) – the number of the nearest neighbors to be selected.
+k has to be k <span class="math">\(\leq\)</span> N2. Default: <code class="docutils literal notranslate"><span class="pre">1</span></code>.</p></li>
+<li><p><strong>ord</strong> (<code class="docutils literal notranslate"><span class="pre">int</span></code>, optional) – the order of norm to use for distance calculation.
+Default: <code class="docutils literal notranslate"><span class="pre">2</span></code> (Euclidean distance).</p></li>
+<li><p><strong>dim</strong> (<code class="docutils literal notranslate"><span class="pre">int</span></code>, optional) – the dimension encompassing the point cloud coordinates,
+utilized for calculating distance and sorting.
+Default: <code class="docutils literal notranslate"><span class="pre">-1</span></code> (The last dimension).</p></li>
+<li><p><strong>largest</strong> (<code class="docutils literal notranslate"><span class="pre">bool</span></code>, optional) – controls whether to return largest (furthest) or
+smallest (nearest) neighbors. Default: <code class="docutils literal notranslate"><span class="pre">False</span></code>.</p></li>
+<li><p><strong>sorted</strong> (<code class="docutils literal notranslate"><span class="pre">bool</span></code>, optional) – controls whether to return the neighbors in sorted
+order. Default: <code class="docutils literal notranslate"><span class="pre">True</span></code>.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p><p>The named tuple of (values, indices).</p>
+<p><code class="docutils literal notranslate"><span class="pre">values</span></code>: The ord-norm distance between each point in ref and its sorted k
+nearest neighbors in nbr. The shape is (…, N1, k).</p>
+<p><code class="docutils literal notranslate"><span class="pre">indices</span></code>: The index of the k nearest neighbor points in neighbors point sets
+(nbr). The shape is (…, N1, k).</p>
+</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><code class="docutils literal notranslate"><span class="pre">torch.return_types.topk(values:</span> <span class="pre">torch.Tensor,</span> <span class="pre">indices:</span> <span class="pre">torch.LongTensor)</span></code></p>
+</dd>
+</dl>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>If <code class="docutils literal notranslate"><span class="pre">sorted</span></code> is set to <code class="docutils literal notranslate"><span class="pre">False</span></code>, the output will be unspecified and not
+necessarily sorted along the index of the input point cloud.</p>
+</div>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">torch</span><span class="o">,</span> <span class="nn">pypose</span> <span class="k">as</span> <span class="nn">pp</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">ref</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([[</span><span class="mf">9.</span><span class="p">,</span> <span class="mf">2.</span><span class="p">,</span> <span class="mf">2.</span><span class="p">],</span>
+<span class="gp">... </span>                    <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">2.</span><span class="p">],</span>
+<span class="gp">... </span>                    <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">],</span>
+<span class="gp">... </span>                    <span class="p">[</span><span class="mf">5.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">],</span>
+<span class="gp">... </span>                    <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">],</span>
+<span class="gp">... </span>                    <span class="p">[</span><span class="mf">5.</span><span class="p">,</span> <span class="mf">5.</span><span class="p">,</span> <span class="mf">3.</span><span class="p">]])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">nbr</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">],</span>
+<span class="gp">... </span>                    <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">6.</span><span class="p">,</span> <span class="mf">2.</span><span class="p">],</span>
+<span class="gp">... </span>                    <span class="p">[</span><span class="mf">5.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">],</span>
+<span class="gp">... </span>                    <span class="p">[</span><span class="mf">9.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">2.</span><span class="p">]])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">knn</span><span class="p">(</span><span class="n">ref</span><span class="p">,</span> <span class="n">nbr</span><span class="p">)</span>
+<span class="go">torch.return_types.topk(</span>
+<span class="go">values=tensor([[2.0000],</span>
+<span class="go">        [1.0000],</span>
+<span class="go">        [1.4142],</span>
+<span class="go">        [1.4142],</span>
+<span class="go">        [0.0000],</span>
+<span class="go">        [4.2426]]),</span>
+<span class="go">indices=tensor([[3],</span>
+<span class="go">        [0],</span>
+<span class="go">        [0],</span>
+<span class="go">        [2],</span>
+<span class="go">        [0],</span>
+<span class="go">        [1]]))</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">knn</span><span class="p">(</span><span class="n">ref</span><span class="p">,</span> <span class="n">nbr</span><span class="p">,</span> <span class="n">k</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="nb">ord</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
+<span class="go">torch.return_types.topk(</span>
+<span class="go">values=tensor([[2.0000, 4.5826],</span>
+<span class="go">        [1.0000, 4.5826],</span>
+<span class="go">        [1.4142, 5.0990],</span>
+<span class="go">        [1.4142, 4.0000],</span>
+<span class="go">        [0.0000, 4.2426],</span>
+<span class="go">        [4.2426, 5.0000]]),</span>
+<span class="go">indices=tensor([[3, 2],</span>
+<span class="go">        [0, 2],</span>
+<span class="go">        [0, 2],</span>
+<span class="go">        [2, 0],</span>
+<span class="go">        [0, 2],</span>
+<span class="go">        [1, 2]]))</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">knn</span><span class="p">(</span><span class="n">ref</span><span class="p">,</span> <span class="n">nbr</span><span class="p">,</span> <span class="n">k</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="nb">ord</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">values</span>
+<span class="go">tensor([[2.0000, 4.5826],</span>
+<span class="go">        [1.0000, 4.5826],</span>
+<span class="go">        [1.4142, 5.0990],</span>
+<span class="go">        [1.4142, 4.0000],</span>
+<span class="go">        [0.0000, 4.2426],</span>
+<span class="go">        [4.2426, 5.0000]])</span>
+</pre></div>
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+  <section id="pypose-mat2rxso3">
+<h1>pypose.mat2RxSO3<a class="headerlink" href="#pypose-mat2rxso3" title="Permalink to this heading">¶</a></h1>
+<dl class="py function">
+<dt class="sig sig-object py" id="pypose.mat2RxSO3">
+<span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">mat2RxSO3</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">mat</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">check</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">rtol</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1e-05</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">atol</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1e-05</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/convert/#mat2RxSO3"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.mat2RxSO3" title="Permalink to this definition">¶</a></dt>
+<dd><p>Convert batched rotation or transformation matrices to RxSO3Type LieTensor.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>mat</strong> (<em>Tensor</em>) – the batched matrices to convert. If input is of shape <code class="xref py py-obj docutils literal notranslate"><span class="pre">(*,</span> <span class="pre">3,</span> <span class="pre">4)</span></code>
+or <code class="xref py py-obj docutils literal notranslate"><span class="pre">(*,</span> <span class="pre">4,</span> <span class="pre">4)</span></code>, only the top left 3x3 submatrix is used.</p></li>
+<li><p><strong>check</strong> (<em>bool</em><em>, </em><em>optional</em>) – flag to check if the input is valid rotation matrices (orthogonal
+and with a determinant of one). Set to <code class="docutils literal notranslate"><span class="pre">False</span></code> if less computation is needed.
+Default: <code class="docutils literal notranslate"><span class="pre">True</span></code>.</p></li>
+<li><p><strong>rtol</strong> (<em>float</em><em>, </em><em>optional</em>) – relative tolerance when check is enabled. Default: 1e-05</p></li>
+<li><p><strong>atol</strong> (<em>float</em><em>, </em><em>optional</em>) – absolute tolerance when check is enabled. Default: 1e-05</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>the converted RxSO3Type LieTensor.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor">LieTensor</a></p>
+</dd>
+</dl>
+<dl>
+<dt>Shape:</dt><dd><p>Input: <code class="xref py py-obj docutils literal notranslate"><span class="pre">(*,</span> <span class="pre">3,</span> <span class="pre">3)</span></code> or <code class="xref py py-obj docutils literal notranslate"><span class="pre">(*,</span> <span class="pre">3,</span> <span class="pre">4)</span></code> or <code class="xref py py-obj docutils literal notranslate"><span class="pre">(*,</span> <span class="pre">4,</span> <span class="pre">4)</span></code></p>
+<p>Output: <code class="xref py py-obj docutils literal notranslate"><span class="pre">(*,</span> <span class="pre">5)</span></code></p>
+</dd>
+</dl>
+<p>Let the input be matrix <span class="math">\(\mathbf{T}\)</span>, <span class="math">\(\mathbf{T}_i\)</span> represents each individual
+matrix in the batch. <span class="math">\(\mathbf{T}^{m,n}_i\)</span> represents the <span class="math">\(m^{\mathrm{th}}\)</span> row and
+<span class="math">\(n^{\mathrm{th}}\)</span> column of <span class="math">\(\mathbf{T}_i\)</span>, <span class="math">\(m,n\geq 1\)</span>, then the scaling factor
+<span class="math">\(s_i\in\mathbb{R}\)</span> and the rotation matrix <span class="math">\(\mathbf{R}_i\in\mathbb{R}^{3\times 3}\)</span>
+can be computed as:</p>
+<div class="math">
+\[\begin{aligned}
+    s_i &= \sqrt[3]{\vert \mathbf{T_i} \vert}\\
+    \mathbf{R}_i &= \mathbf{R}_i/s_i
+\end{aligned},
+
+\]</div>
+<p>the translation and quaternion can be computed by:</p>
+<div class="math">
+\[\left\{\begin{aligned}
+q^x_i &= \mathrm{sign}(\mathbf{R}^{2,3}_i - \mathbf{R}^{3,2}_i) \frac{1}{2}
+    \sqrt{1 + \mathbf{R}^{1,1}_i - \mathbf{R}^{2,2}_i - \mathbf{R}^{3,3}_i}\\
+q^y_i &= \mathrm{sign}(\mathbf{R}^{3,1}_i - \mathbf{R}^{1,3}_i) \frac{1}{2}
+    \sqrt{1 - \mathbf{R}^{1,1}_i + \mathbf{R}^{2,2}_i - \mathbf{R}^{3,3}_i}\\
+q^z_i &= \mathrm{sign}(\mathbf{R}^{1,2}_i - \mathbf{R}^{2,1}_i) \frac{1}{2}
+    \sqrt{1 - \mathbf{R}^{1,1}_i - \mathbf{R}^{2,2}_i + \mathbf{R}^{3,3}_i}\\
+q^w_i &= \frac{1}{2} \sqrt{1 + \mathbf{R}^{1,1}_i + \mathbf{R}^{2,2}_i + \mathbf{R}^{3,3}_i}
+\end{aligned}\right.,
+
+\]</div>
+<p>In summary, the output LieTensor should be of format:</p>
+<div class="math">
+\[\textbf{y}_i = [q^x_i, q^y_i, q^z_i, q^w_i, s_i]
+
+\]</div>
+<div class="admonition warning">
+<p class="admonition-title">Warning</p>
+<p>Numerically, a transformation matrix is considered legal if:</p>
+<div class="math">
+\[\vert s \vert > \texttt{atol} \\
+|{\rm det}(\mathbf{R}) - 1| \leq \texttt{atol} + \texttt{rtol}\times 1\\
+|\mathbf{RR}^{T} - \mathbf{I}| \leq \texttt{atol} + \texttt{rtol}\times \mathbf{I}
+
+\]</div>
+<p>where <span class="math">\(|\cdot |\)</span> is element-wise absolute function. When <code class="docutils literal notranslate"><span class="pre">check</span></code> is set to <code class="docutils literal notranslate"><span class="pre">True</span></code>,
+illegal input will raise a <code class="docutils literal notranslate"><span class="pre">ValueError</span></code>. Otherwise, no data validation is performed.
+Illegal input will output an irrelevant result, which likely contains <code class="docutils literal notranslate"><span class="pre">nan</span></code>.</p>
+</div>
+<p class="rubric">Examples</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="nb">input</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([[</span> <span class="mf">0.</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.5</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">],</span>
+<span class="gp">... </span>                      <span class="p">[</span><span class="mf">0.5</span><span class="p">,</span>   <span class="mf">0.</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">],</span>
+<span class="gp">... </span>                      <span class="p">[</span> <span class="mf">0.</span><span class="p">,</span>   <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">]])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">mat2RxSO3</span><span class="p">(</span><span class="nb">input</span><span class="p">)</span>
+<span class="go">RxSO3Type LieTensor:</span>
+<span class="go">tensor([0.0000, 0.0000, 0.7071, 0.7071, 0.5000])</span>
+</pre></div>
+</div>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The individual matrix in a batch can be written as: <span class="math">\(s\mathbf{R}_{3\times3}\)</span>,
+where <span class="math">\(\mathbf{R}\)</span> is the rotation matrix. where  the scaling factor
+<span class="math">\(s\)</span> defines a linear transformation that enlarges or diminishes the object
+in the same ratio across 3 dimensions.</p>
+</div>
+<p>See <a class="reference internal" href="../pypose.RxSO3/#pypose.RxSO3" title="pypose.RxSO3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.RxSO3()</span></code></a> for more details of the output LieTensor format.</p>
+</dd></dl>
+
+</section>
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+              <p class="caption" role="heading"><span class="caption-text">Contents:</span></p>
+<ul class="current">
+<li class="toctree-l1"><a class="reference internal" href="../../lietensor/">LieTensor</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../func/">Func</a></li>
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+  <section id="pypose-mat2se3">
+<h1>pypose.mat2SE3<a class="headerlink" href="#pypose-mat2se3" title="Permalink to this heading">¶</a></h1>
+<dl class="py function">
+<dt class="sig sig-object py" id="pypose.mat2SE3">
+<span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">mat2SE3</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">mat</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">check</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">rtol</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1e-05</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">atol</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1e-05</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/convert/#mat2SE3"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.mat2SE3" title="Permalink to this definition">¶</a></dt>
+<dd><p>Convert batched rotation or transformation matrices to SE3Type LieTensor.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>mat</strong> (<em>Tensor</em>) – the batched matrices to convert. If input is of shape <code class="xref py py-obj docutils literal notranslate"><span class="pre">(*,</span> <span class="pre">3,</span> <span class="pre">3)</span></code>, then
+translation will be filled with zero. For input with shape <code class="xref py py-obj docutils literal notranslate"><span class="pre">(*,</span> <span class="pre">3,</span> <span class="pre">4)</span></code>, the last
+row will be treated as <code class="docutils literal notranslate"><span class="pre">[0,</span> <span class="pre">0,</span> <span class="pre">0,</span> <span class="pre">1]</span></code>.</p></li>
+<li><p><strong>check</strong> (<em>bool</em><em>, </em><em>optional</em>) – flag to check if the input is valid rotation matrices (orthogonal
+and with a determinant of one). Set to <code class="docutils literal notranslate"><span class="pre">False</span></code> if less computation is needed.
+Default: <code class="docutils literal notranslate"><span class="pre">True</span></code>.</p></li>
+<li><p><strong>rtol</strong> (<em>float</em><em>, </em><em>optional</em>) – relative tolerance when check is enabled. Default: 1e-05</p></li>
+<li><p><strong>atol</strong> (<em>float</em><em>, </em><em>optional</em>) – absolute tolerance when check is enabled. Default: 1e-05</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>the converted SE3Type LieTensor.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor">LieTensor</a></p>
+</dd>
+</dl>
+<dl>
+<dt>Shape:</dt><dd><p>Input: <code class="xref py py-obj docutils literal notranslate"><span class="pre">(*,</span> <span class="pre">3,</span> <span class="pre">3)</span></code> or <code class="xref py py-obj docutils literal notranslate"><span class="pre">(*,</span> <span class="pre">3,</span> <span class="pre">4)</span></code> or <code class="xref py py-obj docutils literal notranslate"><span class="pre">(*,</span> <span class="pre">4,</span> <span class="pre">4)</span></code></p>
+<p>Output: <code class="xref py py-obj docutils literal notranslate"><span class="pre">(*,</span> <span class="pre">7)</span></code></p>
+</dd>
+</dl>
+<p>Let the input be matrix <span class="math">\(\mathbf{T}\)</span>,  <span class="math">\(\mathbf{T}_i\)</span> represents each individual
+matrix in the batch. <span class="math">\(\mathbf{R}_i\in\mathbb{R}^{3\times 3}\)</span> be the top left 3x3 block
+matrix of <span class="math">\(\mathbf{T}_i\)</span>. <span class="math">\(\mathbf{T}^{m,n}_i\)</span> represents the <span class="math">\(m^{\mathrm{th}}\)</span>
+row and <span class="math">\(n^{\mathrm{th}}\)</span> column of <span class="math">\(\mathbf{T}_i\)</span>, <span class="math">\(m,n\geq 1\)</span>, then the
+translation and quaternion can be computed by:</p>
+<div class="math">
+\[\left\{\begin{aligned}
+t^x_i &= \mathbf{T}^{1,4}_i\\
+t^y_i &= \mathbf{T}^{2,4}_i\\
+t^z_i &= \mathbf{T}^{3,4}_i\\
+q^x_i &= \mathrm{sign}(\mathbf{R}^{2,3}_i - \mathbf{R}^{3,2}_i) \frac{1}{2}
+    \sqrt{1 + \mathbf{R}^{1,1}_i - \mathbf{R}^{2,2}_i - \mathbf{R}^{3,3}_i}\\
+q^y_i &= \mathrm{sign}(\mathbf{R}^{3,1}_i - \mathbf{R}^{1,3}_i) \frac{1}{2}
+    \sqrt{1 - \mathbf{R}^{1,1}_i + \mathbf{R}^{2,2}_i - \mathbf{R}^{3,3}_i}\\
+q^z_i &= \mathrm{sign}(\mathbf{R}^{1,2}_i - \mathbf{R}^{2,1}_i) \frac{1}{2}
+    \sqrt{1 - \mathbf{R}^{1,1}_i - \mathbf{R}^{2,2}_i + \mathbf{R}^{3,3}_i}\\
+q^w_i &= \frac{1}{2} \sqrt{1 + \mathbf{R}^{1,1}_i + \mathbf{R}^{2,2}_i + \mathbf{R}^{3,3}_i}
+\end{aligned}\right.,
+
+\]</div>
+<p>In summary, the output LieTensor should be of format:</p>
+<div class="math">
+\[\textbf{y}_i = [t^x_i, t^y_i, t^z_i, q^x_i, q^y_i, q^z_i, q^w_i]
+
+\]</div>
+<div class="admonition warning">
+<p class="admonition-title">Warning</p>
+<p>Numerically, a transformation matrix is considered legal if:</p>
+<div class="math">
+\[|{\rm det}(\mathbf{R}) - 1| \leq \texttt{atol} + \texttt{rtol}\times 1\\
+|\mathbf{RR}^{T} - \mathbf{I}| \leq \texttt{atol} + \texttt{rtol}\times \mathbf{I}
+
+\]</div>
+<p>where <span class="math">\(|\cdot |\)</span> is element-wise absolute function. When <code class="docutils literal notranslate"><span class="pre">check</span></code> is set to <code class="docutils literal notranslate"><span class="pre">True</span></code>,
+illegal input will raise a <code class="docutils literal notranslate"><span class="pre">ValueError</span></code>. Otherwise, no data validation is performed.
+Illegal input will output an irrelevant result, which likely contains <code class="docutils literal notranslate"><span class="pre">nan</span></code>.</p>
+<p>For input with shape <code class="xref py py-obj docutils literal notranslate"><span class="pre">(*,</span> <span class="pre">4,</span> <span class="pre">4)</span></code>, when <code class="docutils literal notranslate"><span class="pre">check</span></code> is set to <code class="docutils literal notranslate"><span class="pre">True</span></code> and the last row
+of the each individual matrix is not <code class="docutils literal notranslate"><span class="pre">[0,</span> <span class="pre">0,</span> <span class="pre">0,</span> <span class="pre">1]</span></code>, a warning will be triggered.
+Even though the last row is not used in the computation, it is worth noting that a matrix not
+satisfying this condition is not a valid transformation matrix.</p>
+</div>
+<p class="rubric">Examples</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="nb">input</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([[</span><span class="mf">0.</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">],</span>
+<span class="gp">... </span>                      <span class="p">[</span><span class="mf">1.</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">],</span>
+<span class="gp">... </span>                      <span class="p">[</span><span class="mf">0.</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">],</span>
+<span class="gp">... </span>                      <span class="p">[</span><span class="mf">0.</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span>  <span class="mf">1.</span><span class="p">]])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">mat2SE3</span><span class="p">(</span><span class="nb">input</span><span class="p">)</span>
+<span class="go">SE3Type LieTensor:</span>
+<span class="go">tensor([0.1000, 0.2000, 0.3000, 0.0000, 0.0000, 0.7071, 0.7071])</span>
+</pre></div>
+</div>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The individual matrix in a batch can be written as:</p>
+<div class="math">
+\[\begin{bmatrix}
+        \mathbf{R}_{3\times3} & \mathbf{t}_{3\times1}\\
+        \textbf{0} & 1
+\end{bmatrix},
+
+\]</div>
+<p>where <span class="math">\(\mathbf{R}\)</span> is the rotation matrix. The translation vector <span class="math">\(\mathbf{t}\)</span> defines the
+displacement between the original position and the transformed position.</p>
+</div>
+<p>See <a class="reference internal" href="../pypose.SE3/#pypose.SE3" title="pypose.SE3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.SE3()</span></code></a> for more details of the output LieTensor format.</p>
+</dd></dl>
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+<ul class="current">
+<li class="toctree-l1"><a class="reference internal" href="../../lietensor/">LieTensor</a></li>
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+  <section id="pypose-mat2so3">
+<h1>pypose.mat2SO3<a class="headerlink" href="#pypose-mat2so3" title="Permalink to this heading">¶</a></h1>
+<dl class="py function">
+<dt class="sig sig-object py" id="pypose.mat2SO3">
+<span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">mat2SO3</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">mat</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">check</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">rtol</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1e-05</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">atol</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1e-05</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/convert/#mat2SO3"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.mat2SO3" title="Permalink to this definition">¶</a></dt>
+<dd><p>Convert batched rotation or transformation matrices to SO3Type LieTensor.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>mat</strong> (<em>Tensor</em>) – the batched matrices to convert. If input is of shape
+<code class="xref py py-obj docutils literal notranslate"><span class="pre">(*,</span> <span class="pre">3,</span> <span class="pre">4)</span></code> or <code class="xref py py-obj docutils literal notranslate"><span class="pre">(*,</span> <span class="pre">4,</span> <span class="pre">4)</span></code>, only the top left 3x3 submatrix is used.</p></li>
+<li><p><strong>check</strong> (<em>bool</em><em>, </em><em>optional</em>) – flag to check if the input is valid rotation matrices
+(orthogonal and with a determinant of one). Set to <code class="docutils literal notranslate"><span class="pre">False</span></code> if less
+computation is needed. Default: <code class="docutils literal notranslate"><span class="pre">True</span></code>.</p></li>
+<li><p><strong>rtol</strong> (<em>float</em><em>, </em><em>optional</em>) – relative tolerance when check is enabled. Default: 1e-05</p></li>
+<li><p><strong>atol</strong> (<em>float</em><em>, </em><em>optional</em>) – absolute tolerance when check is enabled. Default: 1e-05</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>the converted SO3Type LieTensor.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor">LieTensor</a></p>
+</dd>
+</dl>
+<dl>
+<dt>Shape:</dt><dd><p>Input: <code class="xref py py-obj docutils literal notranslate"><span class="pre">(*,</span> <span class="pre">3,</span> <span class="pre">3)</span></code> or <code class="xref py py-obj docutils literal notranslate"><span class="pre">(*,</span> <span class="pre">3,</span> <span class="pre">4)</span></code> or <code class="xref py py-obj docutils literal notranslate"><span class="pre">(*,</span> <span class="pre">4,</span> <span class="pre">4)</span></code></p>
+<p>Output: <code class="xref py py-obj docutils literal notranslate"><span class="pre">(*,</span> <span class="pre">4)</span></code></p>
+</dd>
+</dl>
+<p>Let the input be matrix <span class="math">\(\mathbf{R}\)</span>, <span class="math">\(\mathbf{R}_i\)</span> represents each individual
+matrix in the batch. <span class="math">\(\mathbf{R}^{m,n}_i\)</span> represents the <span class="math">\(m^{\mathrm{th}}\)</span> row
+and <span class="math">\(n^{\mathrm{th}}\)</span> column of <span class="math">\(\mathbf{R}_i\)</span>, <span class="math">\(m,n\geq 1\)</span>, then the
+quaternion can be computed by:</p>
+<div class="math">
+\[\left\{\begin{aligned}
+q^x_i &= \mathrm{sign}(\mathbf{R}^{2,3}_i - \mathbf{R}^{3,2}_i) \frac{1}{2}
+    \sqrt{1 + \mathbf{R}^{1,1}_i - \mathbf{R}^{2,2}_i - \mathbf{R}^{3,3}_i}\\
+q^y_i &= \mathrm{sign}(\mathbf{R}^{3,1}_i - \mathbf{R}^{1,3}_i) \frac{1}{2}
+    \sqrt{1 - \mathbf{R}^{1,1}_i + \mathbf{R}^{2,2}_i - \mathbf{R}^{3,3}_i}\\
+q^z_i &= \mathrm{sign}(\mathbf{R}^{1,2}_i - \mathbf{R}^{2,1}_i) \frac{1}{2}
+    \sqrt{1 - \mathbf{R}^{1,1}_i - \mathbf{R}^{2,2}_i + \mathbf{R}^{3,3}_i}\\
+q^w_i &= \frac{1}{2} \sqrt{1 + \mathbf{R}^{1,1}_i + \mathbf{R}^{2,2}_i +
+    \mathbf{R}^{3,3}_i}
+\end{aligned}\right.,
+
+\]</div>
+<p>In summary, the output LieTensor should be of format:</p>
+<div class="math">
+\[\textbf{y}_i = [q^x_i, q^y_i, q^z_i, q^w_i]
+
+\]</div>
+<div class="admonition warning">
+<p class="admonition-title">Warning</p>
+<p>Numerically, a transformation matrix is considered legal if:</p>
+<div class="math">
+\[|{\rm det}(\mathbf{R}) - 1| \leq \texttt{atol} + \texttt{rtol}\times 1\\
+|\mathbf{RR}^{T}-\mathbf{I}|\leq\texttt{atol} + \texttt{rtol}\times\mathbf{I}
+
+\]</div>
+<p>where <span class="math">\(|\cdot |\)</span> is element-wise absolute function. When <code class="docutils literal notranslate"><span class="pre">check</span></code> is set to
+<code class="docutils literal notranslate"><span class="pre">True</span></code>, illegal input will raise a <code class="docutils literal notranslate"><span class="pre">ValueError</span></code>. Otherwise, no data validation
+is performed. Illegal input will output an irrelevant result, which likely
+contains <code class="docutils literal notranslate"><span class="pre">nan</span></code>.</p>
+</div>
+<p class="rubric">Examples</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="nb">input</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([[</span><span class="mf">0.</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">],</span>
+<span class="gp">... </span>                      <span class="p">[</span><span class="mf">1.</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">],</span>
+<span class="gp">... </span>                      <span class="p">[</span><span class="mf">0.</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">,</span>  <span class="mf">1.</span><span class="p">]])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">mat2SO3</span><span class="p">(</span><span class="nb">input</span><span class="p">)</span>
+<span class="go">SO3Type LieTensor:</span>
+<span class="go">tensor([0.0000, 0.0000, 0.7071, 0.7071])</span>
+</pre></div>
+</div>
+<p>See <a class="reference internal" href="../pypose.SO3/#pypose.SO3" title="pypose.SO3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.SO3()</span></code></a> for more details of the output LieTensor format.</p>
+</dd></dl>
+
+</section>
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+  <section id="pypose-mat2sim3">
+<h1>pypose.mat2Sim3<a class="headerlink" href="#pypose-mat2sim3" title="Permalink to this heading">¶</a></h1>
+<dl class="py function">
+<dt class="sig sig-object py" id="pypose.mat2Sim3">
+<span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">mat2Sim3</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">mat</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">check</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">rtol</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1e-05</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">atol</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1e-05</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/convert/#mat2Sim3"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.mat2Sim3" title="Permalink to this definition">¶</a></dt>
+<dd><p>Convert batched rotation or transformation matrices to Sim3Type LieTensor.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>mat</strong> (<em>Tensor</em>) – the batched matrices to convert. If input is of shape <code class="xref py py-obj docutils literal notranslate"><span class="pre">(*,</span> <span class="pre">3,</span> <span class="pre">3)</span></code>,
+then translation will be filled with zero. For input with shape <code class="xref py py-obj docutils literal notranslate"><span class="pre">(*,</span> <span class="pre">3,</span> <span class="pre">4)</span></code>,
+the last row will be treated as <code class="docutils literal notranslate"><span class="pre">[0,</span> <span class="pre">0,</span> <span class="pre">0,</span> <span class="pre">1]</span></code>.</p></li>
+<li><p><strong>check</strong> (<em>bool</em><em>, </em><em>optional</em>) – flag to check if the input is valid rotation matrices (orthogonal
+and with a determinant of one). Set to <code class="docutils literal notranslate"><span class="pre">False</span></code> if less computation is needed.
+Default: <code class="docutils literal notranslate"><span class="pre">True</span></code>.</p></li>
+<li><p><strong>rtol</strong> (<em>float</em><em>, </em><em>optional</em>) – relative tolerance when check is enabled. Default: 1e-05</p></li>
+<li><p><strong>atol</strong> (<em>float</em><em>, </em><em>optional</em>) – absolute tolerance when check is enabled. Default: 1e-05</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>the converted Sim3Type LieTensor.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor">LieTensor</a></p>
+</dd>
+</dl>
+<dl>
+<dt>Shape:</dt><dd><p>Input: <code class="xref py py-obj docutils literal notranslate"><span class="pre">(*,</span> <span class="pre">3,</span> <span class="pre">3)</span></code> or <code class="xref py py-obj docutils literal notranslate"><span class="pre">(*,</span> <span class="pre">3,</span> <span class="pre">4)</span></code> or <code class="xref py py-obj docutils literal notranslate"><span class="pre">(*,</span> <span class="pre">4,</span> <span class="pre">4)</span></code></p>
+<p>Output: <code class="xref py py-obj docutils literal notranslate"><span class="pre">(*,</span> <span class="pre">8)</span></code></p>
+</dd>
+</dl>
+<p>Let the input be matrix <span class="math">\(\mathbf{T}\)</span>,  <span class="math">\(\mathbf{T}_i\)</span> represents each individual
+matrix in the batch. <span class="math">\(\mathbf{U}_i\in\mathbb{R}^{3\times 3}\)</span> be the top left 3x3 block
+matrix of <span class="math">\(\mathbf{T}_i\)</span>. Let <span class="math">\(\mathbf{T}^{m,n}_i\)</span> represents the
+<span class="math">\(m^{\mathrm{th}}\)</span> row and <span class="math">\(n^{\mathrm{th}}\)</span> column of <span class="math">\(\mathbf{T}_i\)</span>,
+<span class="math">\(m,n\geq 1\)</span>, then the scaling factor <span class="math">\(s_i\in\mathbb{R}\)</span> and the rotation matrix
+<span class="math">\(\mathbf{R}_i\in\mathbb{R}^{3\times 3}\)</span> can be computed as:</p>
+<div class="math">
+\[\begin{aligned}
+    s_i &= \sqrt[3]{\vert \mathbf{U}_i \vert}\\
+    \mathbf{R}_i &= \mathbf{U}_i/s_i
+\end{aligned}
+
+\]</div>
+<p>the translation and quaternion can be computed by:</p>
+<div class="math">
+\[\left\{\begin{aligned}
+t^x_i &= \mathbf{T}^{1,4}_i\\
+t^y_i &= \mathbf{T}^{2,4}_i\\
+t^z_i &= \mathbf{T}^{3,4}_i\\
+q^x_i &= \mathrm{sign}(\mathbf{R}^{2,3}_i - \mathbf{R}^{3,2}_i) \frac{1}{2}
+    \sqrt{1 + \mathbf{R}^{1,1}_i - \mathbf{R}^{2,2}_i - \mathbf{R}^{3,3}_i}\\
+q^y_i &= \mathrm{sign}(\mathbf{R}^{3,1}_i - \mathbf{R}^{1,3}_i) \frac{1}{2}
+    \sqrt{1 - \mathbf{R}^{1,1}_i + \mathbf{R}^{2,2}_i - \mathbf{R}^{3,3}_i}\\
+q^z_i &= \mathrm{sign}(\mathbf{R}^{1,2}_i - \mathbf{R}^{2,1}_i) \frac{1}{2}
+    \sqrt{1 - \mathbf{R}^{1,1}_i - \mathbf{R}^{2,2}_i + \mathbf{R}^{3,3}_i}\\
+q^w_i &= \frac{1}{2} \sqrt{1 + \mathbf{R}^{1,1}_i + \mathbf{R}^{2,2}_i +
+    \mathbf{R}^{3,3}_i}
+\end{aligned}\right.,
+
+\]</div>
+<p>In summary, the output LieTensor should be of format:</p>
+<div class="math">
+\[\textbf{y}_i = [t^x_i, t^y_i, t^z_i, q^x_i, q^y_i, q^z_i, q^w_i, s_i]
+
+\]</div>
+<div class="admonition warning">
+<p class="admonition-title">Warning</p>
+<p>Numerically, a transformation matrix is considered legal if:</p>
+<div class="math">
+\[\vert s \vert > \texttt{atol} \\
+|{\rm det}(\mathbf{R}) - 1| \leq \texttt{atol} + \texttt{rtol}\times 1\\
+|\mathbf{RR}^{T} - \mathbf{I}| \leq \texttt{atol} + \texttt{rtol}\times \mathbf{I}
+
+\]</div>
+<p>where <span class="math">\(|\cdot |\)</span> is element-wise absolute function. When <code class="docutils literal notranslate"><span class="pre">check</span></code> is set to <code class="docutils literal notranslate"><span class="pre">True</span></code>,
+illegal input will raise a <code class="docutils literal notranslate"><span class="pre">ValueError</span></code>. Otherwise, no data validation is performed.
+Illegal input will output an irrelevant result, which likely contains <code class="docutils literal notranslate"><span class="pre">nan</span></code>.</p>
+<p>For input with shape <code class="xref py py-obj docutils literal notranslate"><span class="pre">(*,</span> <span class="pre">4,</span> <span class="pre">4)</span></code>, when <code class="docutils literal notranslate"><span class="pre">check</span></code> is set to <code class="docutils literal notranslate"><span class="pre">True</span></code> and the last row
+of the each individual matrix is not <code class="docutils literal notranslate"><span class="pre">[0,</span> <span class="pre">0,</span> <span class="pre">0,</span> <span class="pre">1]</span></code>, a warning will be triggered.
+Even though the last row is not used in the computation, it is worth noting that a matrix not
+satisfying this condition is not a valid transformation matrix.</p>
+</div>
+<p class="rubric">Examples</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="nb">input</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([[</span> <span class="mf">0.</span><span class="p">,</span><span class="o">-</span><span class="mf">0.5</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">],</span>
+<span class="gp">... </span>                      <span class="p">[</span><span class="mf">0.5</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">],</span>
+<span class="gp">... </span>                      <span class="p">[</span> <span class="mf">0.</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">],</span>
+<span class="gp">... </span>                      <span class="p">[</span> <span class="mf">0.</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">,</span>  <span class="mf">1.</span><span class="p">]])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">mat2Sim3</span><span class="p">(</span><span class="nb">input</span><span class="p">)</span>
+<span class="go">Sim3Type LieTensor:</span>
+<span class="go">tensor([0.1000, 0.2000, 0.3000, 0.0000, 0.0000, 0.7071, 0.7071, 0.5000])</span>
+</pre></div>
+</div>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>We follow the convention below to express Sim3:</p>
+<div class="math">
+\[\begin{bmatrix}
+        s\mathbf{R}_{3\times3} & \mathbf{t}_{3\times1}\\
+        \textbf{0} & 1
+\end{bmatrix},
+
+\]</div>
+<p>referred to this paper:</p>
+<ul class="simple">
+<li><p>J. Sola et al., <a class="reference external" href="https://arxiv.org/abs/1812.01537">A micro Lie theory for state estimation in
+robotics</a>, arXiv preprint arXiv:1812.01537 (2018),</p></li>
+</ul>
+<p>where <span class="math">\(\mathbf{R}\)</span> is the individual matrix in a batch. The scaling factor
+<span class="math">\(s\)</span> defines a linear transformation that enlarges or diminishes the object in the
+same ratio across 3 dimensions, the translation vector <span class="math">\(\mathbf{t}\)</span> defines the
+displacement between the original position and the transformed position.</p>
+<p>We also notice that there is another popular convention:</p>
+<div class="math">
+\[\begin{bmatrix}
+        \mathbf{R}_{3\times3} & \mathbf{t}_{3\times1}\\
+        \textbf{0} & 1/s
+\end{bmatrix},
+
+\]</div>
+<p>referred to this tutorial:</p>
+<ul class="simple">
+<li><p><a class="reference external" href="https://www.ethaneade.org/lie.pdf">Lie Groups for 2D and 3D Transformations.</a>, by Ethan Eade.</p></li>
+</ul>
+<p>Please make sure your own convention before using this function.</p>
+</div>
+<p>See <a class="reference internal" href="../pypose.Sim3/#pypose.Sim3" title="pypose.Sim3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.Sim3()</span></code></a> for more details of the output LieTensor format.</p>
+</dd></dl>
+
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+              <p class="caption" role="heading"><span class="caption-text">Contents:</span></p>
+<ul class="current">
+<li class="toctree-l1"><a class="reference internal" href="../../lietensor/">LieTensor</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../func/">Func</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../functions/">Functions</a></li>
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+              
+  <section id="pypose-matrix">
+<h1>pypose.matrix<a class="headerlink" href="#pypose-matrix" title="Permalink to this heading">¶</a></h1>
+<dl class="py function">
+<dt class="sig sig-object py" id="pypose.matrix">
+<span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">matrix</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">inputs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/convert/#matrix"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.matrix" title="Permalink to this definition">¶</a></dt>
+<dd><p>Convert a <a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code></a> into matrix form.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>inputs</strong> (<a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code></a>) – the input LieTensor.</p>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>the batched matrix form (torch.Tensor) of LieTensor.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>Tensor</p>
+</dd>
+</dl>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_SO3</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">matrix</span><span class="p">()</span>
+<span class="go">tensor([[[ 0.9285, -0.0040, -0.3713],</span>
+<span class="go">         [ 0.2503,  0.7454,  0.6178],</span>
+<span class="go">         [ 0.2743, -0.6666,  0.6931]],</span>
+<span class="go">        [[ 0.4805,  0.8602, -0.1706],</span>
+<span class="go">         [-0.7465,  0.2991, -0.5944],</span>
+<span class="go">         [-0.4603,  0.4130,  0.7858]]])</span>
+</pre></div>
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+  <section id="pypose-module-ekf">
+<h1>pypose.module.EKF<a class="headerlink" href="#pypose-module-ekf" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.module.EKF">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.module.</span></span><span class="sig-name descname"><span class="pre">EKF</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">model</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Q</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">R</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/module/ekf/#EKF"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.module.EKF" title="Permalink to this definition">¶</a></dt>
+<dd><p>Performs Batched Extended Kalman Filter (EKF).</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>model</strong> (<a class="reference internal" href="../pypose.module.System/#pypose.module.System" title="pypose.module.System"><code class="xref py py-obj docutils literal notranslate"><span class="pre">System</span></code></a>) – The system model to be estimated, a subclass of
+<a class="reference internal" href="../pypose.module.NLS/#pypose.module.NLS" title="pypose.module.NLS"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.module.NLS</span></code></a>.</p></li>
+<li><p><strong>Q</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>, optional) – The covariance matrices of system transition noise.
+Ignored if provided during each iteration. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code></p></li>
+<li><p><strong>R</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>, optional) – The covariance matrices of system observation noise.
+Ignored if provided during each iteration. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code></p></li>
+</ul>
+</dd>
+</dl>
+<p>A non-linear system can be described as</p>
+<div class="math">
+\[\begin{aligned}
+    \mathbf{x}_{k+1} &= \mathbf{f}(\mathbf{x}_k, \mathbf{u}_k, t_k) + \mathbf{w}_k,
+    \quad \mathbf{w}_k \sim \mathcal{N}(\mathbf{0}, \mathbf{Q})  \\
+    \mathbf{y}_{k} &= \mathbf{g}(\mathbf{x}_k, \mathbf{u}_k, t_k) + \mathbf{v}_k,
+    \quad \mathbf{v}_k \sim \mathcal{N}(\mathbf{0}, \mathbf{R})
+\end{aligned}
+
+\]</div>
+<p>It will be linearized automatically:</p>
+<div class="math">
+\[\begin{align*}
+    \mathbf{z}_{k+1} = \mathbf{A}_{k}\mathbf{x}_{k} + \mathbf{B}_{k}\mathbf{u}_{k}
+                     + \mathbf{c}_{k}^1 + \mathbf{w}_k\\
+    \mathbf{y}_{k} = \mathbf{C}_{k}\mathbf{x}_{k} + \mathbf{D}_{k}\mathbf{u}_{k}
+                   + \mathbf{c}_{k}^2 + \mathbf{v}_k\\
+\end{align*}
+
+\]</div>
+<p>EKF can be described as the following five equations, where the subscript <span class="math">\(\cdot_{k}\)</span>
+is omited for simplicity.</p>
+<ol class="arabic">
+<li><p>Priori State Estimation.</p>
+<blockquote>
+<div><div class="math">
+\[\mathbf{x}^{-} = \mathbf{A}\mathbf{x} + \mathbf{B}\mathbf{u}_k + \mathbf{c}_1
+
+\]</div>
+</div></blockquote>
+</li>
+<li><p>Priori Covariance Propagation.</p>
+<blockquote>
+<div><div class="math">
+\[\mathbf{P}^{-} = \mathbf{A}\mathbf{P}\mathbf{A}^{T} + \mathbf{Q}
+
+\]</div>
+</div></blockquote>
+</li>
+<li><p>Update Kalman Gain</p>
+<blockquote>
+<div><div class="math">
+\[\mathbf{K} = \mathbf{P}\mathbf{C}^{T}
+            (\mathbf{C}\mathbf{P} \mathbf{C}^{T} + \mathbf{R})^{-1}
+
+\]</div>
+</div></blockquote>
+</li>
+<li><p>Posteriori State Estimation</p>
+<blockquote>
+<div><div class="math">
+\[\mathbf{x}^{+} = \mathbf{x}^{-} + \mathbf{K} (\mathbf{y} -
+            \mathbf{C}\mathbf{x}^{-} - \mathbf{D}\mathbf{u} - \mathbf{c}_2)
+
+\]</div>
+</div></blockquote>
+</li>
+<li><p>Posteriori Covariance Estimation</p>
+<blockquote>
+<div><div class="math">
+\[\mathbf{P} = (\mathbf{I} - \mathbf{K}\mathbf{C}) \mathbf{P}^{-}
+
+\]</div>
+</div></blockquote>
+</li>
+</ol>
+<p>where superscript <span class="math">\(\cdot^{-}\)</span> and <span class="math">\(\cdot^{+}\)</span> denote the priori and
+posteriori estimation, respectively.</p>
+<p class="rubric">Example</p>
+<ol class="arabic simple">
+<li><p>Define a discrete-time non-linear system (NLS) model</p></li>
+</ol>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">torch</span><span class="o">,</span> <span class="nn">pypose</span> <span class="k">as</span> <span class="nn">pp</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="k">class</span> <span class="nc">NLS</span><span class="p">(</span><span class="n">pp</span><span class="o">.</span><span class="n">module</span><span class="o">.</span><span class="n">NLS</span><span class="p">):</span>
+<span class="gp">... </span>    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="gp">... </span>        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+<span class="gp">...</span>
+<span class="gp">... </span>    <span class="k">def</span> <span class="nf">state_transition</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">state</span><span class="p">,</span> <span class="nb">input</span><span class="p">,</span> <span class="n">t</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
+<span class="gp">... </span>        <span class="k">return</span> <span class="n">state</span><span class="o">.</span><span class="n">cos</span><span class="p">()</span> <span class="o">+</span> <span class="nb">input</span>
+<span class="gp">...</span>
+<span class="gp">... </span>    <span class="k">def</span> <span class="nf">observation</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">state</span><span class="p">,</span> <span class="nb">input</span><span class="p">,</span> <span class="n">t</span><span class="p">):</span>
+<span class="gp">... </span>        <span class="k">return</span> <span class="n">state</span><span class="o">.</span><span class="n">sin</span><span class="p">()</span> <span class="o">+</span> <span class="nb">input</span>
+</pre></div>
+</div>
+<ol class="arabic simple" start="2">
+<li><p>Create a model and filter</p></li>
+</ol>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">model</span> <span class="o">=</span> <span class="n">NLS</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">ekf</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">module</span><span class="o">.</span><span class="n">EKF</span><span class="p">(</span><span class="n">model</span><span class="p">)</span>
+</pre></div>
+</div>
+<ol class="arabic simple" start="3">
+<li><p>Prepare data</p></li>
+</ol>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">T</span><span class="p">,</span> <span class="n">N</span> <span class="o">=</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">2</span> <span class="c1"># steps, state dim</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">states</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">T</span><span class="p">,</span> <span class="n">N</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">inputs</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">T</span><span class="p">,</span> <span class="n">N</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">observ</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">T</span><span class="p">,</span> <span class="n">N</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="c1"># std of transition, observation, and estimation</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">p</span> <span class="o">=</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mi">10</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">Q</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">N</span><span class="p">)</span> <span class="o">*</span> <span class="n">q</span><span class="o">**</span><span class="mi">2</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">R</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">N</span><span class="p">)</span> <span class="o">*</span> <span class="n">r</span><span class="o">**</span><span class="mi">2</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">N</span><span class="p">)</span><span class="o">.</span><span class="n">repeat</span><span class="p">(</span><span class="n">T</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">p</span><span class="o">**</span><span class="mi">2</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">estim</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">T</span><span class="p">,</span> <span class="n">N</span><span class="p">)</span> <span class="o">*</span> <span class="n">p</span>
+</pre></div>
+</div>
+<ol class="arabic simple" start="4">
+<li><p>Perform EKF prediction. Note that estimation error becomes smaller with more steps.</p></li>
+</ol>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">T</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
+<span class="gp">... </span>    <span class="n">w</span> <span class="o">=</span> <span class="n">q</span> <span class="o">*</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">N</span><span class="p">)</span> <span class="c1"># transition noise</span>
+<span class="gp">... </span>    <span class="n">v</span> <span class="o">=</span> <span class="n">r</span> <span class="o">*</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">N</span><span class="p">)</span> <span class="c1"># observation noise</span>
+<span class="gp">... </span>    <span class="n">states</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">],</span> <span class="n">observ</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">model</span><span class="p">(</span><span class="n">states</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="n">w</span><span class="p">,</span> <span class="n">inputs</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
+<span class="gp">... </span>    <span class="n">estim</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">],</span> <span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">ekf</span><span class="p">(</span><span class="n">estim</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">observ</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="n">v</span><span class="p">,</span> <span class="n">inputs</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">Q</span><span class="p">,</span> <span class="n">R</span><span class="p">)</span>
+<span class="gp">... </span><span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Est error:&#39;</span><span class="p">,</span> <span class="p">(</span><span class="n">states</span> <span class="o">-</span> <span class="n">estim</span><span class="p">)</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">dim</span><span class="o">=-</span><span class="mi">1</span><span class="p">))</span>
+<span class="go">Est error: tensor([5.7655, 5.3436, 3.5947, 0.3359, 0.0639])</span>
+</pre></div>
+</div>
+<div class="admonition warning">
+<p class="admonition-title">Warning</p>
+<p>Don’t introduce noise in <code class="docutils literal notranslate"><span class="pre">System</span></code> methods <code class="docutils literal notranslate"><span class="pre">state_transition</span></code> and <code class="docutils literal notranslate"><span class="pre">observation</span></code>
+for filter testing, as those methods are used for automatically linearizing the system
+by the parent class <code class="docutils literal notranslate"><span class="pre">pypose.module.NLS</span></code>, unless your system model explicitly
+introduces noise.</p>
+</div>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Implementation is based on Section 5.1 of this book</p>
+<ul class="simple">
+<li><p>Dan Simon, <a class="reference external" href="https://onlinelibrary.wiley.com/doi/book/10.1002/0470045345">Optimal State Estimation: Kalman, H∞, and Nonlinear Approaches</a>,
+Cleveland State University, 2006</p></li>
+</ul>
+</div>
+<dl class="py property">
+<dt class="sig sig-object py" id="pypose.module.EKF.Q">
+<em class="property"><span class="pre">property</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">Q</span></span><a class="headerlink" href="#pypose.module.EKF.Q" title="Permalink to this definition">¶</a></dt>
+<dd><p>The covariance of system transition noise.</p>
+</dd></dl>
+
+<dl class="py property">
+<dt class="sig sig-object py" id="pypose.module.EKF.R">
+<em class="property"><span class="pre">property</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">R</span></span><a class="headerlink" href="#pypose.module.EKF.R" title="Permalink to this definition">¶</a></dt>
+<dd><p>The covariance of system observation noise.</p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.module.EKF.forward">
+<span class="sig-name descname"><span class="pre">forward</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">y</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">u</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">P</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Q</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">R</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">t</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/module/ekf/#EKF.forward"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.module.EKF.forward" title="Permalink to this definition">¶</a></dt>
+<dd><p>Performs one step estimation.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>x</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – estimated system state of previous step.</p></li>
+<li><p><strong>y</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – system observation at current step (measurement).</p></li>
+<li><p><strong>u</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – system input at current step.</p></li>
+<li><p><strong>P</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – state estimation covariance of previous step.</p></li>
+<li><p><strong>Q</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>, optional) – covariance of system transition model. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code></p></li>
+<li><p><strong>R</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>, optional) – covariance of system observation model. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code></p></li>
+<li><p><strong>t</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>, optional) – timestep of system (only for time variant system).
+Default: <code class="docutils literal notranslate"><span class="pre">None</span></code></p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>posteriori state and covariance estimation</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>list of <code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code></p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.module.EKF.set_uncertainty">
+<span class="sig-name descname"><span class="pre">set_uncertainty</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">Q</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">R</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/module/ekf/#EKF.set_uncertainty"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.module.EKF.set_uncertainty" title="Permalink to this definition">¶</a></dt>
+<dd><p>Set the covariance matrices of transition noise and observation noise.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>Q</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – batched square covariance matrices of transition noise.</p></li>
+<li><p><strong>R</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – batched square covariance matrices of observation noise.</p></li>
+</ul>
+</dd>
+</dl>
+</dd></dl>
+
+</dd></dl>
+
+</section>
+
+
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+  <section id="pypose-module-epnp">
+<h1>pypose.module.EPnP<a class="headerlink" href="#pypose-module-epnp" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.module.EPnP">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.module.</span></span><span class="sig-name descname"><span class="pre">EPnP</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">intrinsics</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">refine</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/module/pnp/#EPnP"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.module.EPnP" title="Permalink to this definition">¶</a></dt>
+<dd><p>Batched EPnP Solver - a non-iterative <span class="math">\(\mathcal{O}(n)\)</span> solution to the
+Perspective-<span class="math">\(n\)</span>-Point (PnP) problem for <span class="math">\(n \geq 4\)</span>.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>intrinsics</strong> (<code class="docutils literal notranslate"><span class="pre">torch.Tensor</span></code>, optional) – The camera intrinsics.
+The shape is (…, 3, 3). Default: None</p></li>
+<li><p><strong>refine</strong> (<code class="docutils literal notranslate"><span class="pre">bool</span></code>, optional) – refine the solution with Gaussian-Newton optimizer.
+Default: <code class="docutils literal notranslate"><span class="pre">True</span></code>.</p></li>
+</ul>
+</dd>
+</dl>
+<p>Assume each of the <span class="math">\(n\)</span> points in the world coordinate is <span class="math">\(p^w_i\)</span> and in
+camera coordinate is <span class="math">\(p^c_i\)</span>.
+They are represented by weighted sums of the four virtual control points,
+<span class="math">\(c^w_j\)</span> and <span class="math">\(c^c_j\)</span> in the world and camera coordinate, respectively.</p>
+<div class="math">
+\[\begin{aligned}
+    & p^w_i = \sum^4_{j=1}{\alpha_{ij}c^w_j} \\
+    & p^c_i = \sum^4_{j=1}{\alpha_{ij}c^c_j} \\
+    & \sum^4_{j=1}{\alpha_{ij}} = 1
+\end{aligned}
+
+\]</div>
+<p>Let the projection matrix be <span class="math">\(P = K[R|T]\)</span>, where <span class="math">\(K\)</span> is the camera
+intrinsics, <span class="math">\(R\)</span> is the rotation matrix and <span class="math">\(T\)</span> is the translation
+vector. Then we have</p>
+<div class="math">
+\[\begin{aligned}
+    s_i p^{\text{img}}_i &= K\sum^4_{j=1}{\alpha_{ij}c^c_j},
+\end{aligned}
+
+\]</div>
+<p>where <span class="math">\(p^{\text{img}}_i\)</span> is pixels in homogeneous form <span class="math">\((u_i, v_i, 1)\)</span>,
+<span class="math">\(s_i\)</span> is the scale factor. Let the control point in camera coordinate
+represented by <span class="math">\(c^c_j = (x^c_j, y^c_j, z^c_j)\)</span>. Rearranging the projection
+equation yields two linear equations for each of the <span class="math">\(n\)</span> points:</p>
+<div class="math">
+\[\begin{aligned}
+    \sum^4_{j=1}{\alpha_{ij}f_xx^c_j + \alpha_{ij}(u_0 - u_i)z^c_j} &= 0 \\
+    \sum^4_{j=1}{\alpha_{ij}f_yy^c_j + \alpha_{ij}(v_0 - v_i)z^c_j} &= 0
+\end{aligned}
+
+\]</div>
+<p>Assume <span class="math">\(\mathbf{x} = \begin{bmatrix} c^{c^T}_1 &amp; c^{c^T}_2 &amp; c^{c^T}_3 &amp;
+c^{c^T}_4 \end{bmatrix}^T\)</span>, then the two equations form a system <span class="math">\(Mx = 0\)</span>
+considering all of the <span class="math">\(n\)</span> points. Its solution can be expressed as</p>
+<div class="math">
+\[\begin{aligned}
+    x &= \sum^4_{i=1}{\beta_iv_i},
+\end{aligned}
+
+\]</div>
+<p>where <span class="math">\(v_i\)</span> is the null vectors of matrix <span class="math">\(M^T M\)</span> corresponding to its
+least 4 eigenvalues.</p>
+<p>The final step involves calculating the coefficients <span class="math">\(\beta_i\)</span>. Optionally, the
+Gauss-Newton algorithm can be used to refine the solution of <span class="math">\(\beta_i\)</span>.</p>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">torch</span><span class="o">,</span> <span class="nn">pypose</span> <span class="k">as</span> <span class="nn">pp</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">,</span> <span class="p">(</span><span class="n">H</span><span class="p">,</span> <span class="n">W</span><span class="p">)</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="mi">9</span><span class="p">,</span> <span class="mi">9</span><span class="p">)</span> <span class="c1"># focal length and image height, width</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">intrinsics</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([[</span><span class="n">f</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">H</span> <span class="o">/</span> <span class="mi">2</span><span class="p">],</span>
+<span class="gp">... </span>                           <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">W</span> <span class="o">/</span> <span class="mi">2</span><span class="p">],</span>
+<span class="gp">... </span>                           <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span>   <span class="mi">1</span>  <span class="p">]])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="nb">object</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([[</span><span class="mf">2.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">2.</span><span class="p">],</span>
+<span class="gp">... </span>                       <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">2.</span><span class="p">],</span>
+<span class="gp">... </span>                       <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">],</span>
+<span class="gp">... </span>                       <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">],</span>
+<span class="gp">... </span>                       <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">],</span>
+<span class="gp">... </span>                       <span class="p">[</span><span class="mf">5.</span><span class="p">,</span> <span class="mf">5.</span><span class="p">,</span> <span class="mf">3.</span><span class="p">]])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">pixels</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">point2pixel</span><span class="p">(</span><span class="nb">object</span><span class="p">,</span> <span class="n">intrinsics</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">pose</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">SE3</span><span class="p">([</span> <span class="mf">0.</span><span class="p">,</span> <span class="o">-</span><span class="mi">8</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.3827</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">,</span>  <span class="mf">0.9239</span><span class="p">])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">points</span> <span class="o">=</span> <span class="n">pose</span><span class="o">.</span><span class="n">Inv</span><span class="p">()</span> <span class="o">@</span> <span class="nb">object</span>
+<span class="gp">...</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">epnp</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">module</span><span class="o">.</span><span class="n">EPnP</span><span class="p">(</span><span class="n">intrinsics</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">pose</span> <span class="o">=</span> <span class="n">epnp</span><span class="p">(</span><span class="n">points</span><span class="p">,</span> <span class="n">pixels</span><span class="p">)</span>
+<span class="go">SE3Type LieTensor:</span>
+<span class="go">LieTensor([ 3.9816e-05, -8.0000e+00,  5.8174e-05, -3.3186e-06, -3.8271e-01,</span>
+<span class="go">            3.6321e-06,  9.2387e-01])</span>
+</pre></div>
+</div>
+<div class="admonition warning">
+<p class="admonition-title">Warning</p>
+<p>Currently this module only supports batched rectified camera intrinsics, which can
+be defined in the form:</p>
+<div class="math">
+\[K = \begin{pmatrix}
+        f_x &   0 & c_x \\
+        0   & f_y & c_y \\
+        0   &   0 &   1
+    \end{pmatrix}
+
+\]</div>
+<p>The full form of camera intrinsics will be supported in a future release.</p>
+</div>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The implementation is based on the paper</p>
+<ul class="simple">
+<li><p>Francesc Moreno-Noguer, Vincent Lepetit, and Pascal Fua, <a class="reference external" href="https://doi.org/10.1007/s11263-008-0152-6">EPnP: An Accurate O(n)
+Solution to the PnP Problem</a>,
+International Journal of Computer Vision (IJCV), 2009.</p></li>
+</ul>
+</div>
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.module.EPnP.forward">
+<span class="sig-name descname"><span class="pre">forward</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">points</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">pixels</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">intrinsics</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/module/pnp/#EPnP.forward"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.module.EPnP.forward" title="Permalink to this definition">¶</a></dt>
+<dd><dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>points</strong> (<code class="docutils literal notranslate"><span class="pre">torch.Tensor</span></code>) – 3D object points in the world coordinates.
+Shape (…, N, 3)</p></li>
+<li><p><strong>pixels</strong> (<code class="docutils literal notranslate"><span class="pre">torch.Tensor</span></code>) – 2D image points, which are the projection of
+object points. Shape (…, N, 2)</p></li>
+<li><p><strong>intrinsics</strong> (<em>torch.Tensor</em><em>, </em><em>optional</em>) – camera intrinsics. Shape (…, 3, 3).
+Setting it to any non-<code class="docutils literal notranslate"><span class="pre">None</span></code> value will override the default intrinsics
+kept in the module.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>estimated pose (<code class="docutils literal notranslate"><span class="pre">SE3type</span></code>) for the camera.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><code class="docutils literal notranslate"><span class="pre">LieTensor</span></code></p>
+</dd>
+</dl>
+</dd></dl>
+
+</dd></dl>
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+              
+  <section id="pypose-module-icp">
+<h1>pypose.module.ICP<a class="headerlink" href="#pypose-module-icp" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.module.ICP">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.module.</span></span><span class="sig-name descname"><span class="pre">ICP</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">init</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">stepper</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/module/icp/#ICP"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.module.ICP" title="Permalink to this definition">¶</a></dt>
+<dd><p>Batched Iterative Closest Point (ICP) algorithm to find a rigid transformation
+between two sets of points using Singular Value Decomposition (SVD).</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>init</strong> (<code class="docutils literal notranslate"><span class="pre">LieTensor</span></code>, optional) – the initial transformation <span class="math">\(T_{\text{init}}\)</span>
+in <code class="docutils literal notranslate"><span class="pre">SE3type</span> <span class="pre">LieTensor</span></code>. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+<li><p><strong>stepper</strong> (<code class="docutils literal notranslate"><span class="pre">Planner</span></code>, optional) – the stepper to stop a loop. If <code class="docutils literal notranslate"><span class="pre">None</span></code>,
+the <code class="docutils literal notranslate"><span class="pre">pypose.utils.ReduceToBason</span></code> with a maximum of 200 steps are used.
+Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+</ul>
+</dd>
+</dl>
+<p>The algorithm takes two input point clouds (source and target) and finds the optimal
+transformation ( <span class="math">\(T\)</span> ) to minimize the error between the transformed source
+point cloud and the target point cloud:</p>
+<div class="math">
+\[\begin{align*}
+    \underset{T}{\operatorname{arg\,min}} \sum_i \| p_{\mathrm{target, j}} -
+    T \cdot p_{\mathrm{source, i}}\|,
+\end{align*}
+
+\]</div>
+<p>where <span class="math">\(p_{\mathrm{source, i}}\)</span> is the i-th point in the source point cloud, and
+<span class="math">\(p_{\mathrm{target, j}}\)</span> is the cloest point of <span class="math">\(p_{\mathrm{source, i}}\)</span>
+in the target point clouds with index j. The algorithm consists of the following steps:</p>
+<ol class="arabic simple">
+<li><p>For each point in source, the nearest neighbor algorithm (KNN) is used to select
+its closest point in target to form the matched point pairs.</p></li>
+<li><p>Singular value decomposition (SVD) algorithm is used to compute the transformation
+from the matched point pairs.</p></li>
+<li><p>The source point cloud is updated using the obtained transformation.
+The distance between the updated source and target is calculated.</p></li>
+<li><p>The algorithm continues to iterate through these steps until the <code class="docutils literal notranslate"><span class="pre">stepper</span></code>
+condition is satisfied.</p></li>
+</ol>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">torch</span><span class="o">,</span> <span class="nn">pypose</span> <span class="k">as</span> <span class="nn">pp</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">source</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([[[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">],</span>
+<span class="gp">... </span>                        <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">],</span>
+<span class="gp">... </span>                        <span class="p">[</span><span class="mf">2.</span><span class="p">,</span>  <span class="mi">0</span><span class="p">,</span> <span class="mf">0.</span><span class="p">]]])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">target</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([[[</span><span class="mf">0.2</span><span class="p">,</span>      <span class="mf">0.1</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">],</span>
+<span class="gp">... </span>                        <span class="p">[</span><span class="mf">1.1397</span><span class="p">,</span> <span class="mf">0.442</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">],</span>
+<span class="gp">... </span>                        <span class="p">[</span><span class="mf">2.0794</span><span class="p">,</span> <span class="mf">0.7840</span><span class="p">,</span> <span class="mf">0.</span><span class="p">]]])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">stepper</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">utils</span><span class="o">.</span><span class="n">ReduceToBason</span><span class="p">(</span><span class="n">steps</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">icp</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">module</span><span class="o">.</span><span class="n">ICP</span><span class="p">(</span><span class="n">stepper</span><span class="o">=</span><span class="n">stepper</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">icp</span><span class="p">(</span><span class="n">source</span><span class="p">,</span> <span class="n">target</span><span class="p">)</span>
+<span class="go">ReduceToBason step 0 loss tensor([0.4917])</span>
+<span class="go">ReduceToBason step 1 loss tensor([7.4711e-08])</span>
+<span class="go">ReduceToBason step 2 loss tensor([1.0450e-07])</span>
+<span class="go">ReduceToBason step 3 loss tensor([2.8322e-07])</span>
+<span class="go">ReduceToBason: Maximum patience steps reached, Quiting..</span>
+<span class="go">SE3Type LieTensor:</span>
+<span class="go">LieTensor([[0.2000, 0.1000, 0.0000, 0.0000, 0.0000, 0.1736, 0.9848]])</span>
+</pre></div>
+</div>
+<div class="admonition warning">
+<p class="admonition-title">Warning</p>
+<p>It’s important to note that the solution is sensitive to the initialization.</p>
+</div>
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.module.ICP.forward">
+<span class="sig-name descname"><span class="pre">forward</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">source</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">target</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">ord</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">2</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">dim</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">-</span> <span class="pre">1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">init</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/module/icp/#ICP.forward"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.module.ICP.forward" title="Permalink to this definition">¶</a></dt>
+<dd><dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>source</strong> (<code class="docutils literal notranslate"><span class="pre">torch.Tensor</span></code>) – the source point clouds with shape
+(…, points_num, 3).</p></li>
+<li><p><strong>target</strong> (<code class="docutils literal notranslate"><span class="pre">torch.Tensor</span></code>) – the target point clouds with shape
+(…, points_num, 3).</p></li>
+<li><p><strong>ord</strong> (<code class="docutils literal notranslate"><span class="pre">int</span></code>, optional) – the order of norm to use for distance calculation.
+Default: <code class="docutils literal notranslate"><span class="pre">2</span></code> (Euclidean distance).</p></li>
+<li><p><strong>dim</strong> (<code class="docutils literal notranslate"><span class="pre">int</span></code>, optional) – the dimension encompassing the point cloud
+coordinates, utilized for calculating distance.
+Default: <code class="docutils literal notranslate"><span class="pre">-1</span></code> (The last dimension).</p></li>
+<li><p><strong>init</strong> (<code class="docutils literal notranslate"><span class="pre">LieTensor</span></code>, optional) – the initial transformation <span class="math">\(T_{init}\)</span> in
+<code class="docutils literal notranslate"><span class="pre">SE3type</span></code>. If not <code class="docutils literal notranslate"><span class="pre">None</span></code>, it will suppress the <code class="docutils literal notranslate"><span class="pre">init</span></code> given by the
+class constructor. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
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+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
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+              
+  <section id="pypose-module-imupreintegrator">
+<h1>pypose.module.IMUPreintegrator<a class="headerlink" href="#pypose-module-imupreintegrator" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.module.IMUPreintegrator">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.module.</span></span><span class="sig-name descname"><span class="pre">IMUPreintegrator</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">pos=tensor([0.</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">0.</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">0.])</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">rot=SO3Type</span> <span class="pre">LieTensor:</span> <span class="pre">LieTensor([0.</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">0.</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">0.</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">1.])</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">vel=tensor([0.</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">0.</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">0.])</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">gravity=9.81007</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">gyro_cov=1.024e-05</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">acc_cov=0.0064</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">prop_cov=True</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">reset=False</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/module/imu_preintegrator/#IMUPreintegrator"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.module.IMUPreintegrator" title="Permalink to this definition">¶</a></dt>
+<dd><p>Applies preintegration over IMU input signals.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>pos</strong> (<code class="docutils literal notranslate"><span class="pre">torch.Tensor</span></code>, optional) – initial position. Default: <code class="xref py py-obj docutils literal notranslate"><span class="pre">torch.zeros(3)</span></code>.</p></li>
+<li><p><strong>rot</strong> (<code class="docutils literal notranslate"><span class="pre">pypose.SO3</span></code>, optional) – initial rotation. Default: <a class="reference internal" href="../pypose.identity_SO3/#pypose.identity_SO3" title="pypose.identity_SO3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.identity_SO3()</span></code></a>.</p></li>
+<li><p><strong>vel</strong> (<code class="docutils literal notranslate"><span class="pre">torch.Tensor</span></code>, optional) – initial position. Default: <code class="docutils literal notranslate"><span class="pre">torch.zeros(3)</span></code>.</p></li>
+<li><p><strong>gravity</strong> (<code class="docutils literal notranslate"><span class="pre">float</span></code>, optional) – the gravity acceleration. Default: <code class="docutils literal notranslate"><span class="pre">9.81007</span></code>.</p></li>
+<li><p><strong>gyro_cov</strong> (<code class="docutils literal notranslate"><span class="pre">torch.Tensor</span></code> or <code class="docutils literal notranslate"><span class="pre">float</span></code>, optional) – covariance of the gyroscope.
+Use a three-element tensor, if the covariance on the three axes are different.
+Default: <code class="docutils literal notranslate"><span class="pre">(3.2e-3)**2</span></code>.</p></li>
+<li><p><strong>acc_cov</strong> (<code class="docutils literal notranslate"><span class="pre">torch.Tensor</span></code>  or <code class="docutils literal notranslate"><span class="pre">float</span></code>, optional) – covariance of the accelerator.
+Use a three-element tensor, if the covariance on the three axes are different.
+Default: <code class="docutils literal notranslate"><span class="pre">(8e-2)**2</span></code>.</p></li>
+<li><p><strong>prop_cov</strong> (<code class="docutils literal notranslate"><span class="pre">bool</span></code>, optional) – flag to propagate the covariance matrix. Default: <code class="docutils literal notranslate"><span class="pre">True</span></code>.</p></li>
+<li><p><strong>reset</strong> (<code class="docutils literal notranslate"><span class="pre">bool</span></code>, optional) – flag to reset the initial states after the <a class="reference internal" href="#pypose.module.IMUPreintegrator.forward" title="pypose.module.IMUPreintegrator.forward"><code class="xref py py-obj docutils literal notranslate"><span class="pre">forward</span></code></a>
+function is called. If <code class="docutils literal notranslate"><span class="pre">False</span></code>, the integration starts from the last integration.
+This flag is ignored if <code class="xref py py-obj docutils literal notranslate"><span class="pre">init_state</span></code> is not <code class="docutils literal notranslate"><span class="pre">None</span></code>. Default: <code class="docutils literal notranslate"><span class="pre">False</span></code>.</p></li>
+</ul>
+</dd>
+</dl>
+<p class="rubric">Example</p>
+<ol class="arabic simple">
+<li><p>Preintegrator Initialization</p></li>
+</ol>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">torch</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">pypose</span> <span class="k">as</span> <span class="nn">pp</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>    <span class="c1"># Initial Position</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">identity_SO3</span><span class="p">()</span> <span class="c1"># Initial rotation</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">v</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>    <span class="c1"># Initial Velocity</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">integrator</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">module</span><span class="o">.</span><span class="n">IMUPreintegrator</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
+</pre></div>
+</div>
+<ol class="arabic simple" start="2">
+<li><p>Get IMU measurement</p></li>
+</ol>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">ang</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mf">0.1</span><span class="p">,</span><span class="mf">0.1</span><span class="p">,</span><span class="mf">0.1</span><span class="p">])</span> <span class="c1"># angular velocity</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">acc</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mf">0.1</span><span class="p">,</span><span class="mf">0.1</span><span class="p">,</span><span class="mf">0.1</span><span class="p">])</span> <span class="c1"># acceleration</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">rot</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">mat2SO3</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>    <span class="c1"># Rotation (Optional)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">dt</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mf">0.002</span><span class="p">])</span>        <span class="c1"># Time difference between two measurements</span>
+</pre></div>
+</div>
+<p>3. Preintegrating IMU measurements.
+Takes as input the IMU values and calculates the preintegrated IMU measurements.</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">states</span> <span class="o">=</span> <span class="n">integrator</span><span class="p">(</span><span class="n">dt</span><span class="p">,</span> <span class="n">ang</span><span class="p">,</span> <span class="n">acc</span><span class="p">,</span> <span class="n">rot</span><span class="p">)</span>
+<span class="go">{&#39;rot&#39;: SO3Type LieTensor:</span>
+<span class="go">tensor([[[1.0000e-04, 1.0000e-04, 1.0000e-04, 1.0000e+00]]]),</span>
+<span class="go">&#39;vel&#39;: tensor([[[ 0.0002,  0.0002, -0.0194]]]),</span>
+<span class="go">&#39;pos&#39;: tensor([[[ 2.0000e-07,  2.0000e-07, -1.9420e-05]]]),</span>
+<span class="go">&#39;cov&#39;: tensor([[[ 5.7583e-11, -5.6826e-19, -5.6827e-19,  0.0000e+00,  0.0000e+00,</span>
+<span class="go">                  0.0000e+00,  0.0000e+00,  0.0000e+00,  0.0000e+00],</span>
+<span class="go">                [-5.6826e-19,  5.7583e-11, -5.6827e-19,  0.0000e+00,  0.0000e+00,</span>
+<span class="go">                  0.0000e+00,  0.0000e+00,  0.0000e+00,  0.0000e+00],</span>
+<span class="go">                [-5.6827e-19, -5.6827e-19,  5.7583e-11,  0.0000e+00,  0.0000e+00,</span>
+<span class="go">                  0.0000e+00,  0.0000e+00,  0.0000e+00,  0.0000e+00],</span>
+<span class="go">                [ 0.0000e+00,  0.0000e+00,  0.0000e+00,  8.0000e-09, -3.3346e-20,</span>
+<span class="go">                  -1.0588e-19,  8.0000e-12,  1.5424e-23, -1.0340e-22],</span>
+<span class="go">                [ 0.0000e+00,  0.0000e+00,  0.0000e+00, -1.3922e-19,  8.0000e-09,</span>
+<span class="go">                  0.0000e+00, -8.7974e-23,  8.0000e-12,  0.0000e+00],</span>
+<span class="go">                [ 0.0000e+00,  0.0000e+00,  0.0000e+00,  0.0000e+00, -1.0588e-19,</span>
+<span class="go">                  8.0000e-09,  0.0000e+00, -1.0340e-22,  8.0000e-12],</span>
+<span class="go">                [ 0.0000e+00,  0.0000e+00,  0.0000e+00,  8.0000e-12,  1.5424e-23,</span>
+<span class="go">                  -1.0340e-22,  8.0000e-15, -1.2868e-26,  0.0000e+00],</span>
+<span class="go">                [ 0.0000e+00,  0.0000e+00,  0.0000e+00, -8.7974e-23,  8.0000e-12,</span>
+<span class="go">                  0.0000e+00, -1.2868e-26,  8.0000e-15,  0.0000e+00],</span>
+<span class="go">                [ 0.0000e+00,  0.0000e+00,  0.0000e+00,  0.0000e+00, -1.0340e-22,</span>
+<span class="go">                  8.0000e-12,  0.0000e+00,  0.0000e+00,  8.0000e-15]]])}</span>
+</pre></div>
+</div>
+<p>Preintegrated IMU odometry from the KITTI dataset with and without known rotation.</p>
+<table class="docutils align-default">
+<tbody>
+<tr class="row-odd"><td><figure class="align-default">
+<a class="reference internal image-reference" href="../../_images/imu-known-rot.png"><img alt="../../_images/imu-known-rot.png" src="../../_images/imu-known-rot.png" style="width: 300px;" /></a>
+</figure>
+<p>Fig. 1. Known Rotation.</p>
+</td>
+<td><figure class="align-default">
+<a class="reference internal image-reference" href="../../_images/imu-unknown-rot.png"><img alt="../../_images/imu-unknown-rot.png" src="../../_images/imu-unknown-rot.png" style="width: 300px;" /></a>
+</figure>
+<p>Fig. 2. Estimated Rotation.</p>
+</td>
+</tr>
+</tbody>
+</table>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The examples generating the above figures can be found at <a class="reference external" href="https://github.com/pypose/pypose/tree/main/examples/module/imu">examples/module/imu</a>.</p>
+</div>
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.module.IMUPreintegrator.forward">
+<span class="sig-name descname"><span class="pre">forward</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">dt</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">gyro</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">acc</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">rot</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">gyro_cov</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">acc_cov</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">init_state</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/module/imu_preintegrator/#IMUPreintegrator.forward"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.module.IMUPreintegrator.forward" title="Permalink to this definition">¶</a></dt>
+<dd><p>Propagate IMU states from duration (<span class="math">\(\delta t\)</span>), angular rate
+(<span class="math">\(\omega\)</span>), linear acceleration (<span class="math">\(\mathbf{a}\)</span>) in body frame, as well as
+their measurement covariance for gyroscope <span class="math">\(C_{g}\)</span> and acceleration
+<span class="math">\(C_{\mathbf{a}}\)</span>. Known IMU rotation estimation <span class="math">\(R\)</span> can be provided for
+better precision.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>dt</strong> (<code class="docutils literal notranslate"><span class="pre">torch.Tensor</span></code>) – time interval from last update.</p></li>
+<li><p><strong>gyro</strong> (<code class="docutils literal notranslate"><span class="pre">torch.Tensor</span></code>) – angular rate (<span class="math">\(\omega\)</span>) in IMU body frame.</p></li>
+<li><p><strong>acc</strong> (<code class="docutils literal notranslate"><span class="pre">torch.Tensor</span></code>) – linear acceleration (<span class="math">\(\mathbf{a}\)</span>) in IMU body frame
+(raw sensor input with gravity).</p></li>
+<li><p><strong>rot</strong> (<a class="reference internal" href="../pypose.SO3/#pypose.SO3" title="pypose.SO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.SO3</span></code></a>, optional) – known IMU rotation on the body frame.</p></li>
+<li><p><strong>gyro_cov</strong> (<code class="docutils literal notranslate"><span class="pre">torch.Tensor</span></code>, optional) – covariance matrix of angular rate.
+If not given, the default state in constructor will be used.</p></li>
+<li><p><strong>acc_cov</strong> (<code class="docutils literal notranslate"><span class="pre">torch.Tensor</span></code>, optional) – covariance matrix of linear acceleration.
+If not given, the default state in constructor will be used.</p></li>
+<li><p><strong>init_state</strong> (<code class="docutils literal notranslate"><span class="pre">dict</span></code>, optional) – the initial state of the integration. The dictionary
+should be in form of <code class="xref py py-obj docutils literal notranslate"><span class="pre">{'pos':</span> <span class="pre">torch.Tensor,</span> <span class="pre">'rot':</span> <span class="pre">pypose.SO3,</span> <span class="pre">'vel':</span>
+<span class="pre">torch.Tensor}</span></code>. If not given, the initial state in constructor will be used.</p></li>
+</ul>
+</dd>
+</dl>
+<dl class="simple">
+<dt>Shape:</dt><dd><ul class="simple">
+<li><p>input (<code class="xref py py-obj docutils literal notranslate"><span class="pre">dt</span></code>, <code class="xref py py-obj docutils literal notranslate"><span class="pre">gyro</span></code>, <code class="xref py py-obj docutils literal notranslate"><span class="pre">acc</span></code>): This layer supports the input shape with
+<span class="math">\((B, F, H_{in})\)</span>, <span class="math">\((F, H_{in})\)</span> and <span class="math">\((H_{in})\)</span>, where <span class="math">\(B\)</span> is
+the batch size (or the number of IMU), <span class="math">\(F\)</span> is the number of frames
+(measurements), and <span class="math">\(H_{in}\)</span> is the raw sensor signals.</p></li>
+<li><p>init_state (Optional): The initial state of the integration. It contains
+<code class="code docutils literal notranslate"><span class="pre">pos</span></code>: initial position, <code class="code docutils literal notranslate"><span class="pre">rot</span></code>: initial rotation, <code class="code docutils literal notranslate"><span class="pre">vel</span></code>: initial
+velocity, with the shape <span class="math">\((B, H_{in})\)</span>.</p></li>
+<li><p>output: a <code class="xref py py-obj docutils literal notranslate"><span class="pre">dict</span></code> of integrated state including <code class="docutils literal notranslate"><span class="pre">pos</span></code>: position,
+<code class="docutils literal notranslate"><span class="pre">rot</span></code>: rotation, and <code class="docutils literal notranslate"><span class="pre">vel</span></code>: velocity, each of which has a shape
+<span class="math">\((B, F, H_{out})\)</span>, where <span class="math">\(H_{out}\)</span> is the signal dimension.
+If <code class="xref py py-obj docutils literal notranslate"><span class="pre">prop_cov</span></code> is <code class="docutils literal notranslate"><span class="pre">True</span></code>, it will also include <code class="docutils literal notranslate"><span class="pre">cov</span></code>: covariance
+matrix in shape of <span class="math">\((B, 9, 9)\)</span>.</p></li>
+</ul>
+</dd>
+</dl>
+<p>IMU Measurements Integration:</p>
+<div class="math">
+\[\begin{align*}
+    {\Delta}R_{ik+1} &= {\Delta}R_{ik} \mathrm{Exp} (w_k {\Delta}t) \\
+    {\Delta}v_{ik+1} &= {\Delta}v_{ik} + {\Delta}R_{ik} a_k {\Delta}t \\
+    {\Delta}p_{ik+1} &= {\Delta}p_{ik} + {\Delta}v_{ik} {\Delta}t
+        + \frac{1}{2} {\Delta}R_{ik} a_k {\Delta}t^2
+\end{align*}
+
+\]</div>
+<p>where:</p>
+<blockquote>
+<div><ul class="simple">
+<li><p><span class="math">\({\Delta}R_{ik}\)</span> is the preintegrated rotation between the <span class="math">\(i\)</span>-th
+and <span class="math">\(k\)</span>-th time step.</p></li>
+<li><p><span class="math">\({\Delta}v_{ik}\)</span> is the preintegrated velocity between the <span class="math">\(i\)</span>-th
+and <span class="math">\(k\)</span>-th time step.</p></li>
+<li><p><span class="math">\({\Delta}p_{ik}\)</span> is the preintegrated position between the <span class="math">\(i\)</span>-th
+and <span class="math">\(k\)</span>-th time step.</p></li>
+<li><p><span class="math">\(a_k\)</span> is linear acceleration at the <span class="math">\(k\)</span>-th time step.</p></li>
+<li><p><span class="math">\(w_k\)</span> is angular rate at the <span class="math">\(k\)</span>-th time step.</p></li>
+<li><p><span class="math">\({\Delta}t\)</span> is the time interval from time step <span class="math">\(k\)</span>-th to time
+step <span class="math">\({k+1}\)</span>-th time step.</p></li>
+</ul>
+</div></blockquote>
+<p>Uncertainty Propagation:</p>
+<div class="math">
+\[\begin{align*}
+    C_{ik+1} &= A C_{ik} A^T + B \mathrm{diag}(C_g, C_a) B^T \\
+      &= A C A^T + B_g C_g B_g^T + B_a C_a B_a^T
+\end{align*},
+
+\]</div>
+<p>where</p>
+<div class="math">
+\[A = \begin{bmatrix}
+      {\Delta}R_{ik+1}^T & 0_{3*3} \\
+      -{\Delta}R_{ik} (a_k^{\wedge}) {\Delta}t & I_{3*3} & 0_{3*3} \\
+      -1/2{\Delta}R_{ik} (a_k^{\wedge}) {\Delta}t^2 & I_{3*3} {\Delta}t & I_{3*3}
+    \end{bmatrix},
+
+\]</div>
+<div class="math">
+\[B = [B_g, B_a] \\
+
+\]</div>
+<div class="math">
+\[B_g = \begin{bmatrix}
+        J_r^k \Delta t  \\
+        0_{3*3}  \\
+        0_{3*3}
+    \end{bmatrix},
+
+B_a = \begin{bmatrix}
+        0_{3*3} \\
+        {\Delta}R_{ik} {\Delta}t  \\
+        1/2 {\Delta}R_{ik} {\Delta}t^2
+    \end{bmatrix},\]</div>
+<p>where <span class="math">\(\cdot^\wedge\)</span> is the skew matrix (<a class="reference internal" href="../pypose.vec2skew/#pypose.vec2skew" title="pypose.vec2skew"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.vec2skew()</span></code></a>),
+<span class="math">\(C \in\mathbf{R}^{9\times 9}\)</span> is the covariance matrix,
+and <span class="math">\(J_r^k\)</span> is the right jacobian (<a class="reference internal" href="../pypose.Jr/#pypose.Jr" title="pypose.Jr"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.Jr()</span></code></a>) of integrated rotation
+<span class="math">\(\mathrm{Exp}(w_k{\Delta}t)\)</span> at <span class="math">\(k\)</span>-th time step,
+<span class="math">\(C_{g}\)</span> and <span class="math">\(C_{\mathbf{a}}\)</span> are measurement covariance of angular rate
+and acceleration, respectively.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Output covariance (Shape: <span class="math">\((B, 9, 9)\)</span>) is in the order of rotation, velocity,
+and position.</p>
+</div>
+<p>With IMU preintegration, the propagated IMU status:</p>
+<div class="math">
+\[\begin{align*}
+    R_j &= {\Delta}R_{ij} * R_i                                                     \\
+    v_j &= R_i * {\Delta}v_{ij}   + v_i + g \Delta t_{ij}                           \\
+    p_j &= R_i * {\Delta}p_{ij}   + p_i + v_i \Delta t_{ij} + 1/2 g \Delta t_{ij}^2 \\
+\end{align*}
+
+\]</div>
+<p>where:</p>
+<blockquote>
+<div><ul class="simple">
+<li><p><span class="math">\({\Delta}R_{ij}\)</span>, <span class="math">\({\Delta}v_{ij}\)</span>, <span class="math">\({\Delta}p_{ij}\)</span>
+are the preintegrated measurements.</p></li>
+<li><p><span class="math">\(R_i\)</span>, <span class="math">\(v_i\)</span>, and <span class="math">\(p_i\)</span> are the initial state. Default initial values
+are used if <code class="xref py py-obj docutils literal notranslate"><span class="pre">reset</span></code> is True.</p></li>
+<li><p><span class="math">\(R_j\)</span>, <span class="math">\(v_j\)</span>, and <span class="math">\(p_j\)</span> are the propagated state variables.</p></li>
+<li><p><span class="math">\({\Delta}t_{ij}\)</span> is the time interval from frame i to j.</p></li>
+</ul>
+</div></blockquote>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The implementation is based on Eq. (A7), (A8), (A9), and (A10) of this report:</p>
+<ul class="simple">
+<li><p>Christian Forster, et al., <a class="reference external" href="https://rpg.ifi.uzh.ch/docs/RSS15_Forster_Supplementary.pdf">IMU Preintegration on Manifold for Efficient Visual-Inertial
+Maximum-a-posteriori Estimation</a>, Technical Report
+GT-IRIM-CP&amp;R-2015-001, 2015.</p></li>
+</ul>
+</div>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.module.IMUPreintegrator.integrate">
+<span class="sig-name descname"><span class="pre">integrate</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">dt</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">gyro</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">acc</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">rot</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">init_rot</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/module/imu_preintegrator/#IMUPreintegrator.integrate"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.module.IMUPreintegrator.integrate" title="Permalink to this definition">¶</a></dt>
+<dd><p>Integrate the IMU sensor signals gyroscope (angular rate
+<span class="math">\(\omega\)</span>), linear acceleration (<span class="math">\(\mathbf{a}\)</span>) in body frame to
+calculate the increments on the rotation (<span class="math">\(\Delta r\)</span>), velocity (<span class="math">\(\Delta v\)</span>)
+and position (<span class="math">\(\Delta p\)</span>) of the IMU states.
+The IMU rotation of the body frame (<code class="code docutils literal notranslate"><span class="pre">rot</span></code>) is optional,
+which can be utilized to compensate the gravity.</p>
+<div class="math">
+\[\begin{align*}
+    {\Delta}r_{ij} &= \int_{t \in [t_i, t_j]} w_k \ dt \\
+    {\Delta}v_{ij} &= \int_{t \in [t_i, t_j]} R_{k} a_k \ dt \\
+    {\Delta}p_{ij} &= \int\int_{t \in [t_i, t_j]} R_{k} a_k\ dt^2\  \\
+\end{align*}
+
+\]</div>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>dt</strong> (<code class="docutils literal notranslate"><span class="pre">torch.Tensor</span></code>) – time interval from last update.</p></li>
+<li><p><strong>gyro</strong> (<em>torch.Tensor</em>) – angular rate (<span class="math">\(\omega\)</span>) in IMU body frame.</p></li>
+<li><p><strong>acc</strong> (<code class="docutils literal notranslate"><span class="pre">torch.Tensor</span></code>) – linear acceleration (<span class="math">\(\mathbf{a}\)</span>) in IMU body frame
+(raw sensor input with gravity).</p></li>
+<li><p><strong>rot</strong> (<a class="reference internal" href="../pypose.SO3/#pypose.SO3" title="pypose.SO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.SO3</span></code></a>, optional) – known IMU rotation on the body frame.</p></li>
+<li><p><strong>init_rot</strong> (<a class="reference internal" href="../pypose.SO3/#pypose.SO3" title="pypose.SO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.SO3</span></code></a>, optional) – the initial orientation of the IMU state.
+If not given, the initial state in constructor will be used.</p></li>
+</ul>
+</dd>
+</dl>
+<dl class="simple">
+<dt>Shape:</dt><dd><ul class="simple">
+<li><p>input (<code class="xref py py-obj docutils literal notranslate"><span class="pre">dt</span></code>, <code class="xref py py-obj docutils literal notranslate"><span class="pre">gyro</span></code>, <code class="xref py py-obj docutils literal notranslate"><span class="pre">acc</span></code>): This layer supports the input shape
+with <span class="math">\((B, F, H_{in})\)</span>, <span class="math">\((F, H_{in})\)</span> and <span class="math">\((H_{in})\)</span>, where
+<span class="math">\(B\)</span> is the batch size (or the number of IMU), <span class="math">\(F\)</span> is the number of
+frames (measurements), and <span class="math">\(H_{in}\)</span> is the raw sensor signals.</p></li>
+<li><p>init_rot: The initial orientation of the integration, which helps to
+compensate for the gravity. It contains the shape <span class="math">\((B, H_{in})\)</span>. d</p></li>
+<li><p>rot: The ground truth orientation of the integration. If this parameter is
+given, the integrator will use the ground truth orientation to compensate the
+gravity.</p></li>
+</ul>
+</dd>
+</dl>
+<dl class="field-list simple">
+<dt class="field-odd">Returns<span class="colon">:</span></dt>
+<dd class="field-odd"><p>integrated states including <code class="docutils literal notranslate"><span class="pre">a</span></code>: acceleration in the body frame
+without gravity <code class="docutils literal notranslate"><span class="pre">Dp</span></code>: position increments, <code class="docutils literal notranslate"><span class="pre">Dr</span></code>: rotation increments,
+<code class="docutils literal notranslate"><span class="pre">Dv</span></code>: velocity increments, <code class="docutils literal notranslate"><span class="pre">w</span></code>: rotation velocity and <code class="docutils literal notranslate"><span class="pre">Dt</span></code>: time increments,
+each of which has a shape <span class="math">\((B, F, H_{out})\)</span>, where <span class="math">\(H_{out}\)</span> is the
+signal dimension.</p>
+</dd>
+<dt class="field-even">Return type<span class="colon">:</span></dt>
+<dd class="field-even"><p><code class="docutils literal notranslate"><span class="pre">dict</span></code></p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.module.IMUPreintegrator.predict">
+<em class="property"><span class="pre">classmethod</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">predict</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">init_state</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">integrate</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/module/imu_preintegrator/#IMUPreintegrator.predict"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.module.IMUPreintegrator.predict" title="Permalink to this definition">¶</a></dt>
+<dd><p>Propogate the next IMU state from the initial IMU state (<code class="xref py py-obj docutils literal notranslate"><span class="pre">init_state</span></code>) with
+the preintegrated IMU measurements (<span class="math">\(\Delta{p}\)</span>, <span class="math">\(\Delta{v}\)</span> and
+<span class="math">\(\Delta{r}\)</span>).</p>
+<div class="math">
+\[\begin{align*}
+    R_j &= {\Delta}R_{ij} * R_i                                                     \\
+    v_j &= R_i * {\Delta}v_{ij}   + v_i + g \Delta t_{ij}                           \\
+    p_j &= R_i * {\Delta}p_{ij}   + p_i + v_i \Delta t_{ij} + 1/2 g \Delta t_{ij}^2 \\
+\end{align*}
+
+\]</div>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>init_state</strong> (<code class="docutils literal notranslate"><span class="pre">dict</span></code>) – the initial state of the integration. The dictionary
+should be in form of <code class="xref py py-obj docutils literal notranslate"><span class="pre">{'pos':</span> <span class="pre">torch.Tensor,</span> <span class="pre">'rot':</span> <span class="pre">pypose.SO3,</span> <span class="pre">'vel':</span>
+<span class="pre">torch.Tensor}</span></code>.</p></li>
+<li><p><strong>integrate</strong> (<code class="docutils literal notranslate"><span class="pre">dict</span></code>) – the preintegrated IMU measurements. The dictionary
+should be in form of <code class="xref py py-obj docutils literal notranslate"><span class="pre">{'Dp':</span> <span class="pre">torch.Tensor,</span> <span class="pre">'Dr':</span> <span class="pre">pypose.SO3,</span> <span class="pre">'Dv':</span>
+<span class="pre">torch.Tensor,</span> <span class="pre">'Dt':</span> <span class="pre">torch.Tensor}</span></code>.</p></li>
+</ul>
+</dd>
+</dl>
+<dl class="simple">
+<dt>Shape:</dt><dd><ul class="simple">
+<li><p>init_state: The initial state of the integration. It contains <code class="code docutils literal notranslate"><span class="pre">pos</span></code>: initial
+position, <code class="code docutils literal notranslate"><span class="pre">rot</span></code>: initial rotation, <code class="code docutils literal notranslate"><span class="pre">vel</span></code>: initial velocity, with the
+shape <span class="math">\((B, H_{in})\)</span>.</p></li>
+<li><p>integrate: The preintegrated IMU measurements. It contains <code class="xref py py-obj docutils literal notranslate"><span class="pre">Dp</span></code>, <code class="xref py py-obj docutils literal notranslate"><span class="pre">Dv</span></code>,
+<code class="xref py py-obj docutils literal notranslate"><span class="pre">Dr</span></code>, and <code class="xref py py-obj docutils literal notranslate"><span class="pre">Dt</span></code>, with the shape <span class="math">\((B, F, H_{out})\)</span>. It follows the
+output of the function <a class="reference internal" href="#pypose.module.IMUPreintegrator.integrate" title="pypose.module.IMUPreintegrator.integrate"><code class="xref py py-obj docutils literal notranslate"><span class="pre">integrate</span></code></a>.</p></li>
+</ul>
+</dd>
+</dl>
+<dl class="field-list simple">
+<dt class="field-odd">Returns<span class="colon">:</span></dt>
+<dd class="field-odd"><p>integrated states including <code class="docutils literal notranslate"><span class="pre">pos</span></code>: position, <code class="docutils literal notranslate"><span class="pre">rot</span></code>: rotation, and
+<code class="docutils literal notranslate"><span class="pre">vel</span></code>: velocity, each of which has a shape <span class="math">\((B, F, H_{out})\)</span>, where
+<span class="math">\(H_{out}\)</span> is the signal dimension.</p>
+</dd>
+<dt class="field-even">Return type<span class="colon">:</span></dt>
+<dd class="field-even"><p><code class="docutils literal notranslate"><span class="pre">dict</span></code></p>
+</dd>
+</dl>
+</dd></dl>
+
+</dd></dl>
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+  <section id="pypose-module-lqr">
+<h1>pypose.module.LQR<a class="headerlink" href="#pypose-module-lqr" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.module.LQR">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.module.</span></span><span class="sig-name descname"><span class="pre">LQR</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">system</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Q</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">p</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">T</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/module/lqr/#LQR"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.module.LQR" title="Permalink to this definition">¶</a></dt>
+<dd><p>Linear Quadratic Regulator (LQR) with Dynamic Programming.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>system</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">instance</span></code>) – The system to be soved by LQR.</p></li>
+<li><p><strong>Q</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – The weight matrix of the quadratic term.</p></li>
+<li><p><strong>p</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – The weight vector of the first-order term.</p></li>
+<li><p><strong>T</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">int</span></code>) – Time steps of system.</p></li>
+</ul>
+</dd>
+</dl>
+<p>A discrete-time linear system can be described as:</p>
+<div class="math">
+\[\begin{align*}
+    \mathbf{x}_{t+1} &= \mathbf{A}_t\mathbf{x}_t + \mathbf{B}_t\mathbf{u}_t
+                                                 + \mathbf{c}_{1t}          \\
+    \mathbf{y}_t &= \mathbf{C}_t\mathbf{x}_t + \mathbf{D}_t\mathbf{u}_t
+                                             + \mathbf{c}_{2t}              \\
+\end{align*}
+
+\]</div>
+<p>where <span class="math">\(\mathbf{x}\)</span>, <span class="math">\(\mathbf{u}\)</span> are the state and input of the linear
+system; <span class="math">\(\mathbf{y}\)</span> is the observation of the linear system; <span class="math">\(\mathbf{A}\)</span>
+and <span class="math">\(\mathbf{B}\)</span> are the state matrix and input matrix of the linear system;
+<span class="math">\(\mathbf{C}\)</span>, <span class="math">\(\mathbf{D}\)</span> are the output matrix and observation matrix
+of the linear system; <span class="math">\(\mathbf{c}_{1}\)</span>, <span class="math">\(\mathbf{c}_{2}\)</span> are the constant
+input and constant output of the linear system. The subscript <span class="math">\(\cdot_{t}\)</span>
+denotes the time step.</p>
+<p>LQR finds the optimal nominal trajectory <span class="math">\(\mathbf{\tau}_{1:T}^*\)</span> =
+<span class="math">\(\begin{Bmatrix} \mathbf{x}_t, \mathbf{u}_t \end{Bmatrix}_{1:T}\)</span>
+for the linear system of the optimization problem:</p>
+<div class="math">
+\[\begin{align*}
+  \mathbf{\tau}_{1:T}^* = \mathop{\arg\min}\limits_{\tau_{1:T}}
+    &\sum\limits_t\frac{1}{2}
+  \mathbf{\tau}_t^\top\mathbf{Q}_t\mathbf{\tau}_t
+    + \mathbf{p}_t^\top\mathbf{\tau}_t \\
+  \mathrm{s.t.} \quad \mathbf{x}_1 &= \mathbf{x}_{\text{init}}, \\
+  \mathbf{x}_{t+1} &= \mathbf{F}_t\mathbf{\tau}_t + \mathbf{c}_{1t} \\
+\end{align*}
+
+\]</div>
+<p>where <span class="math">\(\mathbf{\tau}_t\)</span> = <span class="math">\(\begin{bmatrix} \mathbf{x}_t \\ \mathbf{u}_t
+\end{bmatrix}\)</span>, <span class="math">\(\mathbf{F}_t\)</span> = <span class="math">\(\begin{bmatrix} \mathbf{A}_t &amp;
+\mathbf{B}_t \end{bmatrix}\)</span>.</p>
+<p>One way to solve the LQR problem is to use the dynamic programming, where the process
+can be summarised as a backward and a forward recursion.</p>
+<ul>
+<li><p>The backward recursion.</p>
+<p>For <span class="math">\(t\)</span> = <span class="math">\(T\)</span> to 1:</p>
+</li>
+</ul>
+<div class="math">
+\[\begin{align*}
+    \mathbf{Q}_t &= \mathbf{Q}_t +
+        \mathbf{F}_t^\top\mathbf{V}_{t+1} \mathbf{F}_t \\
+    \mathbf{q}_t &= \mathbf{p}_t + \mathbf{F}_t^\top\mathbf{V}_{t+1}
+                    \mathbf{c}_{1t} + \mathbf{F}_t^\top\mathbf{v}_{t+1}  \\
+    \mathbf{K}_t &= -\mathbf{Q}_{\mathbf{u}_t, \mathbf{u}_t}^{-1}
+                                \mathbf{Q}_{\mathbf{u}_t, \mathbf{x}_t} \\
+    \mathbf{k}_t &= -\mathbf{Q}_{\mathbf{u}_t, \mathbf{u}_t}^{-1}
+                                                \mathbf{q}_{\mathbf{u}_t}  \\
+    \mathbf{V}_t &= \mathbf{Q}_{\mathbf{x}_t, \mathbf{x}_t}
+        + \mathbf{Q}_{\mathbf{x}_t, \mathbf{u}_t}\mathbf{K}_t
+        + \mathbf{K}_t^\top\mathbf{Q}_{\mathbf{u}_t, \mathbf{x}_t}
+        + \mathbf{K}_t^\top\mathbf{Q}_{\mathbf{u}_t, \mathbf{u}_t}
+            \mathbf{K}_t \\
+    \mathbf{v}_t &= \mathbf{q}_{\mathbf{x}_t}
+        + \mathbf{Q}_{\mathbf{x}_t, \mathbf{u}_t}\mathbf{k}_t
+        + \mathbf{K}_t^\top\mathbf{q}_{\mathbf{u}_t}
+        + \mathbf{K}_t^\top\mathbf{Q}_{\mathbf{u}_t,
+            \mathbf{u}_t}\mathbf{k}_t  \\
+\end{align*}
+
+\]</div>
+<ul>
+<li><p>The forward recursion.</p>
+<p>For <span class="math">\(t\)</span> = 1 to <span class="math">\(T\)</span>:</p>
+<blockquote>
+<div><div class="math">
+\[\begin{align*}
+    \mathbf{u}_t &= \mathbf{K}_t\mathbf{x}_t + \mathbf{k}_t \\
+    \mathbf{x}_{t+1} &= \mathbf{A}_t\mathbf{x}_t + \mathbf{B}_t\mathbf{u}_t
+                                                    + \mathbf{c}_{1t} \\
+\end{align*}
+
+\]</div>
+</div></blockquote>
+</li>
+</ul>
+<p>Then quadratic costs of the system over the time horizon:</p>
+<div class="math">
+\[\mathbf{c} \left( \mathbf{\tau}_t \right) = \frac{1}{2}
+\mathbf{\tau}_t^\top\mathbf{Q}_t\mathbf{\tau}_t
+    + \mathbf{p}_t^\top\mathbf{\tau}_t
+
+\]</div>
+<p>For the <strong>non-linear system</strong>, sometimes people want to solve MPC problem with
+<strong>iterative LQR</strong>. A discrete-time non-linear system can be described as:</p>
+<div class="math">
+\[\begin{aligned}
+    \mathbf{x}_{t+1} &= \mathbf{f}(\mathbf{x}_t, \mathbf{u}_t, t_t) \\
+    \mathbf{y}_{t} &= \mathbf{g}(\mathbf{x}_t, \mathbf{u}_t, t_t) \\
+\end{aligned}
+
+\]</div>
+<p>We can do a linear approximation at current point <span class="math">\(\chi^*=(\mathbf{x}^*,
+\mathbf{u}^*, t^*)\)</span> along a trajectory with small perturbation
+<span class="math">\(\chi=(\mathbf{x}^*+\delta\mathbf{x}, \mathbf{u}^* +\delta\mathbf{u}, t^*)\)</span>
+near <span class="math">\(\chi^*\)</span> for both dynamics and cost:</p>
+<div class="math">
+\[\begin{aligned}
+\mathbf{f}(\mathbf{x}, \mathbf{u}, t^*) &\approx \mathbf{f}(\mathbf{x}^*,
+    \mathbf{u}^*, t^*) +  \left.\frac{\partial \mathbf{f}}{\partial\mathbf{x}}
+    \right|_{\chi^*} \delta \mathbf{x} + \left. \frac{\partial \mathbf{f}}
+    {\partial \mathbf{u}} \right|_{\chi^*} \delta \mathbf{u} \\
+&= \mathbf{f}(\mathbf{x}^*, \mathbf{u}^*, t^*) + \mathbf{A} \delta \mathbf{x}
+    + \mathbf{B} \delta \mathbf{u} \\
+\delta \mathbf{x}_{t+1} &= \mathbf{A}_t \delta \mathbf{x}_t + \mathbf{B}_t
+    \delta \mathbf{u}_t \\
+&= \mathbf{F}_t \delta \mathbf{\tau}_t \\
+\mathbf{c} \left( \mathbf{\tau}, t^* \right) &\approx
+    \mathbf{c} \left( \mathbf{\tau}^*, t^* \right) + \frac{1}{2} \delta
+    \mathbf{\tau}^\top\nabla^2_{\mathbf{\tau}}\mathbf{c}\left(\mathbf{\tau}^*,
+    t^* \right) \delta \mathbf{\tau} + \nabla_{\mathbf{\tau}}
+    \mathbf{c} \left( \mathbf{\tau}^*, t^* \right)^\top \delta \mathbf{\tau}\\
+\bar{\mathbf{c}} \left( \delta \mathbf{\tau} \right) &= \frac{1}{2} \delta
+    \mathbf{\tau}_t^\top \bar{\mathbf{Q}}_t \delta \mathbf{\tau}_t +
+    \bar{\mathbf{p}}_t^\top \delta \mathbf{\tau}_t \\
+\end{aligned}
+
+\]</div>
+<p>where <span class="math">\(\delta \mathbf{\tau}_t\)</span> = <span class="math">\(\begin{bmatrix} \delta \mathbf{x}_t \\
+\delta \mathbf{u}_t \end{bmatrix}\)</span>, <span class="math">\(\mathbf{F}_t\)</span> = <span class="math">\(\begin{bmatrix}
+\mathbf{A}_t &amp; \mathbf{B}_t \end{bmatrix}\)</span>, <span class="math">\(\bar{\mathbf{Q}}_t = \mathbf{Q}_t\)</span>,
+<span class="math">\(\bar{\mathbf{p}}_t\)</span> = <span class="math">\(\mathbf{Q}_t \mathbf{\tau}^*_t + \mathbf{p}_t\)</span>.</p>
+<p>Then, LQR can be performed on a linear quadractic problem with
+<span class="math">\(\delta \mathbf{\tau}_t\)</span>, <span class="math">\(\mathbf{F}_t\)</span>,
+<span class="math">\(\bar{\mathbf{Q}}_t\)</span> and <span class="math">\(\bar{\mathbf{p}}_t\)</span>.</p>
+<ul>
+<li><p>The backward recursion.</p>
+<p>For <span class="math">\(t\)</span> = <span class="math">\(T\)</span> to 1:</p>
+</li>
+</ul>
+<div class="math">
+\[\begin{align*}
+    \mathbf{Q}_t &= \bar{\mathbf{Q}}_t + \mathbf{F}_t^\top\mathbf{V}_{t+1}
+                        \mathbf{F}_t \\
+    \mathbf{q}_t &= \bar{\mathbf{p}}_t + \mathbf{F}_t^\top\mathbf{v}_{t+1}  \\
+    \mathbf{K}_t &= -\mathbf{Q}_{\delta \mathbf{u}_t, \delta \mathbf{u}_t}^{-1}
+                        \mathbf{Q}_{\delta \mathbf{u}_t, \delta \mathbf{x}_t} \\
+    \mathbf{k}_t &= -\mathbf{Q}_{\delta \mathbf{u}_t, \delta \mathbf{u}_t}^{-1}
+                        \mathbf{q}_{\delta \mathbf{u}_t}         \\
+    \mathbf{V}_t &= \mathbf{Q}_{\delta \mathbf{x}_t, \delta \mathbf{x}_t}
+        + \mathbf{Q}_{\delta \mathbf{x}_t, \delta \mathbf{u}_t}\mathbf{K}_t
+        + \mathbf{K}_t^\top\mathbf{Q}_{\delta \mathbf{u}_t, \delta \mathbf{x}_t}
+        + \mathbf{K}_t^\top\mathbf{Q}_{\delta \mathbf{u}_t, \delta \mathbf{u}_t}
+            \mathbf{K}_t \\
+    \mathbf{v}_t &= \mathbf{q}_{\delta \mathbf{x}_t}
+        + \mathbf{Q}_{\delta \mathbf{x}_t, \delta \mathbf{u}_t}\mathbf{k}_t
+        + \mathbf{K}_t^\top\mathbf{q}_{\delta \mathbf{u}_t}
+        + \mathbf{K}_t^\top\mathbf{Q}_{\delta \mathbf{u}_t,
+            \delta \mathbf{u}_t}\mathbf{k}_t   \\
+\end{align*}
+
+\]</div>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Because we made a linear approximation, here <span class="math">\(\bar{\mathbf{p}}_t\)</span> leads to a
+difference with <span class="math">\({\mathbf{q}}_t\)</span> and <span class="math">\({\mathbf{k}}_t\)</span> relative to the
+linear backward recursion, and this change will be compensated back by
+<span class="math">\(\mathbf{u}_t^*\)</span> in the forward recursion.</p>
+</div>
+<ul>
+<li><p>The forward recursion.</p>
+<p>For <span class="math">\(t\)</span> = 1 to <span class="math">\(T\)</span>:</p>
+<blockquote>
+<div><div class="math">
+\[\begin{align*}
+    \delta \mathbf{u}_t &= \mathbf{K}_t \delta \mathbf{x}_t + \mathbf{k}_t \\
+    \mathbf{u}_t &= \delta \mathbf{u}_t + \mathbf{u}_t^* \\
+    \mathbf{x}_{t+1} &= \mathbf{f}(\mathbf{x}_t, \mathbf{u}_t) \\
+\end{align*}
+
+\]</div>
+</div></blockquote>
+</li>
+</ul>
+<p>Then quadratic costs of the system over the time horizon:</p>
+<blockquote>
+<div><div class="math">
+\[\mathbf{c} \left( \mathbf{\tau}_t \right) = \frac{1}{2}
+\mathbf{\tau}_t^\top\mathbf{Q}_t\mathbf{\tau}_t
+    + \mathbf{p}_t^\top\mathbf{\tau}_t
+
+\]</div>
+</div></blockquote>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The discrete-time system to be solved by LQR can be either linear time-invariant
+(<a class="reference internal" href="../pypose.module.LTI/#pypose.module.LTI" title="pypose.module.LTI"><code class="xref py py-meth docutils literal notranslate"><span class="pre">LTI()</span></code></a>) system, or linear time-varying (<a class="reference internal" href="../pypose.module.LTV/#pypose.module.LTV" title="pypose.module.LTV"><code class="xref py py-meth docutils literal notranslate"><span class="pre">LTV()</span></code></a>) system. For non-linear
+system, one can approximate it as a linear system via Taylor expansion, using
+iterative LQR algorithm for MPC. Here we provide a unified general format for the
+implementation.</p>
+</div>
+<p>From the learning perspective, this can be interpreted as a module with unknown
+parameters <span class="math">\(\begin{Bmatrix} \mathbf{Q}, \mathbf{p}, \mathbf{F}, \mathbf{f}
+\end{Bmatrix}\)</span>, which can be integrated into an end-to-end learning system.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The implementation of LQR is based on page 24-32 of the slides:</p>
+<ul class="simple">
+<li><p><a class="reference external" href="https://tinyurl.com/y5ck36vw">Optimal Control and Planning</a>.</p></li>
+</ul>
+<p>The implementation of iterative LQR is based on Eq. (1)~(19) of this paper:</p>
+<ul class="simple">
+<li><p>Li Weiwei, and Emanuel Todorov, <a class="reference external" href="https://tinyurl.com/bdma36s6">Iterative linear quadratic regulator design for
+nonlinear biological movement systems</a>, ICINCO
+(1), 2004.</p></li>
+</ul>
+</div>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">torch</span><span class="o">.</span><span class="n">manual_seed</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">n_batch</span><span class="p">,</span> <span class="n">T</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">5</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">n_state</span><span class="p">,</span> <span class="n">n_ctrl</span> <span class="o">=</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">3</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">n_sc</span> <span class="o">=</span> <span class="n">n_state</span> <span class="o">+</span> <span class="n">n_ctrl</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">Q</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n_batch</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">n_sc</span><span class="p">,</span> <span class="n">n_sc</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">Q</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">mT</span><span class="p">,</span> <span class="n">Q</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n_batch</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">n_sc</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span> <span class="o">=</span> <span class="mf">0.2</span> <span class="o">*</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n_state</span><span class="p">,</span> <span class="n">n_state</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tile</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">n_state</span><span class="p">)</span> <span class="o">+</span> <span class="n">r</span><span class="p">,</span> <span class="p">(</span><span class="n">n_batch</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n_batch</span><span class="p">,</span> <span class="n">n_state</span><span class="p">,</span> <span class="n">n_ctrl</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tile</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">n_state</span><span class="p">),</span> <span class="p">(</span><span class="n">n_batch</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">D</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tile</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">n_state</span><span class="p">,</span> <span class="n">n_ctrl</span><span class="p">),</span> <span class="p">(</span><span class="n">n_batch</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">c1</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tile</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n_state</span><span class="p">),</span> <span class="p">(</span><span class="n">n_batch</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">c2</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tile</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">n_state</span><span class="p">),</span> <span class="p">(</span><span class="n">n_batch</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x_init</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n_batch</span><span class="p">,</span> <span class="n">n_state</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">u_traj</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">n_batch</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">n_ctrl</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="n">device</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">lti</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">module</span><span class="o">.</span><span class="n">LTI</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="n">D</span><span class="p">,</span> <span class="n">c1</span><span class="p">,</span> <span class="n">c2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">dt</span> <span class="o">=</span> <span class="mi">1</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">LQR</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">module</span><span class="o">.</span><span class="n">LQR</span><span class="p">(</span><span class="n">lti</span><span class="p">,</span> <span class="n">Q</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">T</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">cost</span> <span class="o">=</span> <span class="n">LQR</span><span class="p">(</span><span class="n">x_init</span><span class="p">,</span> <span class="n">dt</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="nb">print</span><span class="p">(</span><span class="s2">&quot;x = &quot;</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="nb">print</span><span class="p">(</span><span class="s2">&quot;u = &quot;</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
+<span class="go">x = tensor([[[-0.2633, -0.3466,  2.3803, -0.0423],</span>
+<span class="go">             [ 0.1849, -1.3884,  1.0898, -1.6229],</span>
+<span class="go">             [ 1.2138, -0.7161,  0.2954, -0.6819],</span>
+<span class="go">             [ 1.4840, -1.1249, -1.0302,  0.9805],</span>
+<span class="go">             [-0.3477, -1.7063,  4.6494,  2.6780],</span>
+<span class="go">             [ 7.2346,  4.9958, 17.9926, -7.7881]],</span>
+<span class="go">            [[-0.9744,  0.4976,  0.0603, -0.5258],</span>
+<span class="go">             [-0.6356,  0.0539,  0.7264, -0.5048],</span>
+<span class="go">             [-0.2275, -0.1649,  0.3872, -0.4614],</span>
+<span class="go">             [ 0.2697, -0.3577,  0.0999, -0.4594],</span>
+<span class="go">             [ 0.3916, -2.0832,  0.0701, -0.5407],</span>
+<span class="go">             [ 1.0404, -1.3799, -2.0913, -0.1459]]])</span>
+<span class="go">u = tensor([[[ 1.0405,  0.1586, -0.1282],</span>
+<span class="go">             [-1.4845, -0.5745,  0.2523],</span>
+<span class="go">             [-0.6322, -0.3281, -0.3620],</span>
+<span class="go">             [-1.6768,  2.4054, -0.1047],</span>
+<span class="go">             [-1.7948,  3.5269,  9.0703]],</span>
+<span class="go">            [[-0.1795,  0.9153,  1.7066],</span>
+<span class="go">             [ 0.0814,  0.4004,  0.7114],</span>
+<span class="go">             [ 0.0436,  0.5782,  1.0127],</span>
+<span class="go">             [-0.3017, -0.2897,  0.7251],</span>
+<span class="go">             [-0.0728,  0.7290, -0.3117]]])</span>
+</pre></div>
+</div>
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.module.LQR.forward">
+<span class="sig-name descname"><span class="pre">forward</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x_init</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">dt</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">u_traj</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">u_lower</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">u_upper</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">du</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/module/lqr/#LQR.forward"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.module.LQR.forward" title="Permalink to this definition">¶</a></dt>
+<dd><p>Performs LQR for the discrete system.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>x_init</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – The initial state of the system.</p></li>
+<li><p><strong>dt</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">int</span></code>) – The interval (<span class="math">\(\delta t\)</span>) between two time steps.
+Default: <cite>1</cite>.</p></li>
+<li><p><strong>u_traj</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>, optinal) – The current inputs of the system along a
+trajectory. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+<li><p><strong>u_lower</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>, optinal) – The lower bounds on the controls.
+Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+<li><p><strong>u_upper</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>, optinal) – The upper bounds on the controls.
+Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+<li><p><strong>du</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">int</span></code>, optinal) – The amount each component of the controls
+is allowed to change in each LQR iteration. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>A list of tensors including the solved state sequence
+<span class="math">\(\mathbf{x}\)</span>, the solved input sequence <span class="math">\(\mathbf{u}\)</span>, and the
+associated quadratic costs <span class="math">\(\mathbf{c}\)</span> over the time horizon.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>List of <code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code></p>
+</dd>
+</dl>
+</dd></dl>
+
+</dd></dl>
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diff --git a/docs/preview/pr/352/generated/pypose.module.LTI/index.html b/docs/preview/pr/352/generated/pypose.module.LTI/index.html
new file mode 100644
index 000000000..d2d8a1e23
--- /dev/null
+++ b/docs/preview/pr/352/generated/pypose.module.LTI/index.html
@@ -0,0 +1,637 @@
+
+
+
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+  <section id="pypose-module-lti">
+<h1>pypose.module.LTI<a class="headerlink" href="#pypose-module-lti" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.module.LTI">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.module.</span></span><span class="sig-name descname"><span class="pre">LTI</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">A</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">B</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">C</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">D</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">c1</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">c2</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/module/dynamics/#LTI"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.module.LTI" title="Permalink to this definition">¶</a></dt>
+<dd><p>Discrete-time Linear Time-Invariant (LTI) system.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>A</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – The state matrix of LTI system.</p></li>
+<li><p><strong>B</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – The input matrix of LTI system.</p></li>
+<li><p><strong>C</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – The output matrix of LTI system.</p></li>
+<li><p><strong>D</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – The observation matrix of LTI system,</p></li>
+<li><p><strong>c1</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>, optional) – The constant input of LTI system. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code></p></li>
+<li><p><strong>c2</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>, optional) – The constant output of LTI system. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code></p></li>
+</ul>
+</dd>
+</dl>
+<p>A linear time-invariant lumped system can be described by state-space equation of the form:</p>
+<div class="math">
+\[\begin{align*}
+    \mathbf{x}_{k+1} = \mathbf{A}\mathbf{x}_k + \mathbf{B}\mathbf{u}_k + \mathbf{c}_1 \\
+    \mathbf{y}_k     = \mathbf{C}\mathbf{x}_k + \mathbf{D}\mathbf{u}_k + \mathbf{c}_2 \\
+\end{align*}
+
+\]</div>
+<p>where <span class="math">\(\mathbf{x}\)</span> and <span class="math">\(\mathbf{u}\)</span> are state and input of the current
+timestamp of LTI system.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The variables including state and input are row vectors, which is the last dimension of
+a Tensor. <a class="reference internal" href="#pypose.module.LTI.A" title="pypose.module.LTI.A"><code class="xref py py-obj docutils literal notranslate"><span class="pre">A</span></code></a>, <a class="reference internal" href="#pypose.module.LTI.B" title="pypose.module.LTI.B"><code class="xref py py-obj docutils literal notranslate"><span class="pre">B</span></code></a>, <a class="reference internal" href="#pypose.module.LTI.C" title="pypose.module.LTI.C"><code class="xref py py-obj docutils literal notranslate"><span class="pre">C</span></code></a>, <a class="reference internal" href="#pypose.module.LTI.D" title="pypose.module.LTI.D"><code class="xref py py-obj docutils literal notranslate"><span class="pre">D</span></code></a>, <code class="xref py py-obj docutils literal notranslate"><span class="pre">x</span></code>, <code class="xref py py-obj docutils literal notranslate"><span class="pre">u</span></code> could be a single
+matrix or batched matrices. In the batch case, their dimensions must be consistent so that
+they can be multiplied for each channel.</p>
+</div>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="c1"># Batch, State, Input, Observe Dimension</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">Bd</span><span class="p">,</span> <span class="n">Sd</span><span class="p">,</span> <span class="n">Id</span><span class="p">,</span> <span class="n">Od</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="c1"># Linear System Matrices</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">device</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">device</span><span class="p">(</span><span class="s2">&quot;cuda&quot;</span> <span class="k">if</span> <span class="n">torch</span><span class="o">.</span><span class="n">cuda</span><span class="o">.</span><span class="n">is_available</span><span class="p">()</span> <span class="k">else</span> <span class="s2">&quot;cpu&quot;</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">Bd</span><span class="p">,</span> <span class="n">Sd</span><span class="p">,</span> <span class="n">Sd</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">Bd</span><span class="p">,</span> <span class="n">Sd</span><span class="p">,</span> <span class="n">Id</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">Bd</span><span class="p">,</span> <span class="n">Od</span><span class="p">,</span> <span class="n">Sd</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">D</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">Bd</span><span class="p">,</span> <span class="n">Od</span><span class="p">,</span> <span class="n">Id</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">c1</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">Bd</span><span class="p">,</span> <span class="n">Sd</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">c2</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">Bd</span><span class="p">,</span> <span class="n">Od</span><span class="p">)</span>
+<span class="gp">...</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">lti</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">module</span><span class="o">.</span><span class="n">LTI</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="n">D</span><span class="p">,</span> <span class="n">c1</span><span class="p">,</span> <span class="n">c2</span><span class="p">)</span><span class="o">.</span><span class="n">to</span><span class="p">(</span><span class="n">device</span><span class="p">)</span>
+<span class="gp">...</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">state</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">Bd</span><span class="p">,</span> <span class="n">Sd</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="n">device</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="nb">input</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">Bd</span><span class="p">,</span> <span class="n">Id</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="n">device</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">state</span><span class="p">,</span> <span class="n">observation</span> <span class="o">=</span> <span class="n">lti</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="nb">input</span><span class="p">)</span>
+<span class="go">tensor([[[-8.5639,  0.0523, -0.2576]],</span>
+<span class="go">        [[ 4.1013, -1.5452, -0.0233]]]),</span>
+<span class="go">tensor([[[-3.5780, -2.2970, -2.9314]],</span>
+<span class="go">        [[-0.4358,  1.7306,  2.7514]]]))</span>
+</pre></div>
+</div>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>In this general example, all variables are in a batch. User definable as appropriate.</p>
+</div>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>More practical examples can be found at <a class="reference external" href="https://github.com/pypose/pypose/tree/main/examples/module/dynamics">examples/module/dynamics</a>.</p>
+</div>
+<dl class="py property">
+<dt class="sig sig-object py" id="pypose.module.LTI.A">
+<em class="property"><span class="pre">property</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">A</span></span><a class="headerlink" href="#pypose.module.LTI.A" title="Permalink to this definition">¶</a></dt>
+<dd><p>System transision matrix.</p>
+</dd></dl>
+
+<dl class="py property">
+<dt class="sig sig-object py" id="pypose.module.LTI.B">
+<em class="property"><span class="pre">property</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">B</span></span><a class="headerlink" href="#pypose.module.LTI.B" title="Permalink to this definition">¶</a></dt>
+<dd><p>System input matrix.</p>
+</dd></dl>
+
+<dl class="py property">
+<dt class="sig sig-object py" id="pypose.module.LTI.C">
+<em class="property"><span class="pre">property</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">C</span></span><a class="headerlink" href="#pypose.module.LTI.C" title="Permalink to this definition">¶</a></dt>
+<dd><p>System output matrix.</p>
+</dd></dl>
+
+<dl class="py property">
+<dt class="sig sig-object py" id="pypose.module.LTI.D">
+<em class="property"><span class="pre">property</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">D</span></span><a class="headerlink" href="#pypose.module.LTI.D" title="Permalink to this definition">¶</a></dt>
+<dd><p>System observation matrix.</p>
+</dd></dl>
+
+<dl class="py property">
+<dt class="sig sig-object py" id="pypose.module.LTI.c1">
+<em class="property"><span class="pre">property</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">c1</span></span><a class="headerlink" href="#pypose.module.LTI.c1" title="Permalink to this definition">¶</a></dt>
+<dd><p>Constant term generated by state-transition.</p>
+</dd></dl>
+
+<dl class="py property">
+<dt class="sig sig-object py" id="pypose.module.LTI.c2">
+<em class="property"><span class="pre">property</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">c2</span></span><a class="headerlink" href="#pypose.module.LTI.c2" title="Permalink to this definition">¶</a></dt>
+<dd><p>Constant term generated by observation.</p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.module.LTI.forward">
+<span class="sig-name descname"><span class="pre">forward</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">state</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">input</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/module/dynamics/#LTI.forward"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.module.LTI.forward" title="Permalink to this definition">¶</a></dt>
+<dd><p>Perform one step advance for the LTI system.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>state</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – The state (<span class="math">\(\mathbf{x}\)</span>) of the system</p></li>
+<li><p><strong>input</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – The input (<span class="math">\(\mathbf{u}\)</span>) of the system.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>The state and observation of the system in next time step.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><code class="docutils literal notranslate"><span class="pre">tuple</span></code> of Tensor</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.module.LTI.observation">
+<span class="sig-name descname"><span class="pre">observation</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">state</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">input</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/module/dynamics/#LTI.observation"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.module.LTI.observation" title="Permalink to this definition">¶</a></dt>
+<dd><p>Return the observation of LTI system.</p>
+<div class="math">
+\[\mathbf{y}_k = \mathbf{C}\mathbf{x}_k + \mathbf{D}\mathbf{u}_k + \mathbf{c}_2
+
+\]</div>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>state</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – The state (<span class="math">\(\mathbf{x}\)</span>) of the system</p></li>
+<li><p><strong>input</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – The input (<span class="math">\(\mathbf{u}\)</span>) of the system.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>The observation of the system in next time step.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><code class="docutils literal notranslate"><span class="pre">Tensor</span></code></p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.module.LTI.state_transition">
+<span class="sig-name descname"><span class="pre">state_transition</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">state</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">input</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/module/dynamics/#LTI.state_transition"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.module.LTI.state_transition" title="Permalink to this definition">¶</a></dt>
+<dd><p>Perform one step of LTI state transition.</p>
+<div class="math">
+\[\mathbf{x}_{k+1} = \mathbf{A}\mathbf{x}_k + \mathbf{B}\mathbf{u}_k + \mathbf{c}_1
+
+\]</div>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>state</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – The state (<span class="math">\(\mathbf{x}\)</span>) of the system</p></li>
+<li><p><strong>input</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – The input (<span class="math">\(\mathbf{u}\)</span>) of the system.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>The state the system in next time step.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><code class="docutils literal notranslate"><span class="pre">Tensor</span></code></p>
+</dd>
+</dl>
+</dd></dl>
+
+</dd></dl>
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+  <section id="pypose-module-ltv">
+<h1>pypose.module.LTV<a class="headerlink" href="#pypose-module-ltv" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.module.LTV">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.module.</span></span><span class="sig-name descname"><span class="pre">LTV</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">A</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">B</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">C</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">D</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">c1</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">c2</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/module/dynamics/#LTV"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.module.LTV" title="Permalink to this definition">¶</a></dt>
+<dd><p>Discrete-time Linear Time-Variant (LTV) system.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>A</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>, optional) – The stacked state matrix of LTI system. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code></p></li>
+<li><p><strong>B</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>, optional) – The stacked input matrix of LTI system. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code></p></li>
+<li><p><strong>C</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>, optional) – The stacked output matrix of LTI system. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code></p></li>
+<li><p><strong>D</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>, optional) – The stacked observation matrix of LTI system. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code></p></li>
+<li><p><strong>c1</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>, optional) – The stacked constant input of LTI system. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code></p></li>
+<li><p><strong>c2</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>, optional) – The stacked constant output of LTI system. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code></p></li>
+</ul>
+</dd>
+</dl>
+<p>A linear time-variant lumped system can be described by state-space equation of the form:</p>
+<div class="math">
+\[\begin{align*}
+  \mathbf{x}_{k+1} = \mathbf{A}_k\mathbf{x}_k + \mathbf{B}\mathbf{u}_k + \mathbf{c}^1_k \\
+  \mathbf{y}_k     = \mathbf{C}_k\mathbf{x}_k + \mathbf{D}\mathbf{u}_k + \mathbf{c}^2_k \\
+\end{align*}
+
+\]</div>
+<p>where <span class="math">\(\mathbf{x}_k\)</span> and <span class="math">\(\mathbf{u}_k\)</span> are state and input at the
+timestamp <span class="math">\(k\)</span> of LTI system.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The <code class="xref py py-obj docutils literal notranslate"><span class="pre">forward</span></code> method implicitly increments the time step via <code class="xref py py-obj docutils literal notranslate"><span class="pre">forward_hook</span></code>.
+<code class="xref py py-obj docutils literal notranslate"><span class="pre">state_transition</span></code> and <code class="xref py py-obj docutils literal notranslate"><span class="pre">observation</span></code> still accept time for the flexiblity
+such as time-varying system. One can directly access the current system time via the
+property <code class="xref py py-obj docutils literal notranslate"><span class="pre">systime</span></code> or <code class="xref py py-obj docutils literal notranslate"><span class="pre">_t</span></code>.</p>
+</div>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The variables including state and input are row vectors, which is the last dimension of
+a Tensor. <code class="xref py py-obj docutils literal notranslate"><span class="pre">A</span></code>, <code class="xref py py-obj docutils literal notranslate"><span class="pre">B</span></code>, <code class="xref py py-obj docutils literal notranslate"><span class="pre">C</span></code>, <code class="xref py py-obj docutils literal notranslate"><span class="pre">D</span></code>, <code class="xref py py-obj docutils literal notranslate"><span class="pre">x</span></code>, <code class="xref py py-obj docutils literal notranslate"><span class="pre">u</span></code> could be a single
+matrix or batched matrices. In the batch case, their dimensions must be consistent so that
+they can be multiplied for each channel.</p>
+</div>
+<p class="rubric">Example</p>
+<p>A periodic linear time variant system.</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">n_batch</span><span class="p">,</span> <span class="n">n_state</span><span class="p">,</span> <span class="n">n_ctrl</span><span class="p">,</span> <span class="n">T</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">n_sc</span> <span class="o">=</span> <span class="n">n_state</span> <span class="o">+</span> <span class="n">n_ctrl</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n_batch</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">n_state</span><span class="p">,</span> <span class="n">n_state</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n_batch</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">n_state</span><span class="p">,</span> <span class="n">n_ctrl</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tile</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">n_state</span><span class="p">),</span> <span class="p">(</span><span class="n">n_batch</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">D</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tile</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">n_state</span><span class="p">,</span> <span class="n">n_ctrl</span><span class="p">),</span> <span class="p">(</span><span class="n">n_batch</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n_state</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">u</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">T</span><span class="p">,</span> <span class="n">n_ctrl</span><span class="p">)</span>
+<span class="gp">...</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="k">class</span> <span class="nc">MyLTV</span><span class="p">(</span><span class="n">pp</span><span class="o">.</span><span class="n">module</span><span class="o">.</span><span class="n">LTV</span><span class="p">):</span>
+<span class="gp">... </span>    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="n">D</span><span class="p">,</span> <span class="n">T</span><span class="p">):</span>
+<span class="gp">... </span>        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="n">D</span><span class="p">)</span>
+<span class="gp">... </span>        <span class="bp">self</span><span class="o">.</span><span class="n">T</span> <span class="o">=</span> <span class="n">T</span>
+<span class="gp">...</span>
+<span class="gp">... </span>    <span class="nd">@property</span>
+<span class="gp">... </span>    <span class="k">def</span> <span class="nf">A</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="gp">... </span>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_A</span><span class="p">[</span><span class="o">...</span><span class="p">,</span><span class="bp">self</span><span class="o">.</span><span class="n">_t</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">T</span><span class="p">,:,:]</span>
+<span class="gp">...</span>
+<span class="gp">... </span>    <span class="nd">@property</span>
+<span class="gp">... </span>    <span class="k">def</span> <span class="nf">B</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="gp">... </span>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_B</span><span class="p">[</span><span class="o">...</span><span class="p">,</span><span class="bp">self</span><span class="o">.</span><span class="n">_t</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">T</span><span class="p">,:,:]</span>
+<span class="gp">...</span>
+<span class="gp">... </span>    <span class="nd">@property</span>
+<span class="gp">... </span>    <span class="k">def</span> <span class="nf">C</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="gp">... </span>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_C</span><span class="p">[</span><span class="o">...</span><span class="p">,</span><span class="bp">self</span><span class="o">.</span><span class="n">_t</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">T</span><span class="p">,:,:]</span>
+<span class="gp">...</span>
+<span class="gp">... </span>    <span class="nd">@property</span>
+<span class="gp">... </span>    <span class="k">def</span> <span class="nf">D</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="gp">... </span>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_D</span><span class="p">[</span><span class="o">...</span><span class="p">,</span><span class="bp">self</span><span class="o">.</span><span class="n">_t</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">T</span><span class="p">,:,:]</span>
+<span class="gp">...</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">ltv</span> <span class="o">=</span> <span class="n">MyLTV</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="n">D</span><span class="p">,</span> <span class="n">T</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">T</span><span class="p">):</span>
+<span class="gp">... </span>    <span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">ltv</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">u</span><span class="p">[</span><span class="n">t</span><span class="p">])</span>
+</pre></div>
+</div>
+<p>One may also generate the system matrices with the time variable <code class="xref py py-obj docutils literal notranslate"><span class="pre">_t</span></code>.</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">n_batch</span><span class="p">,</span> <span class="n">n_state</span><span class="p">,</span> <span class="n">n_ctrl</span><span class="p">,</span> <span class="n">T</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">n_sc</span> <span class="o">=</span> <span class="n">n_state</span> <span class="o">+</span> <span class="n">n_ctrl</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n_state</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">u</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">T</span><span class="p">,</span> <span class="n">n_ctrl</span><span class="p">)</span>
+<span class="gp">...</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="k">class</span> <span class="nc">MyLTV</span><span class="p">(</span><span class="n">pp</span><span class="o">.</span><span class="n">module</span><span class="o">.</span><span class="n">LTV</span><span class="p">):</span>
+<span class="gp">... </span>    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="n">D</span><span class="p">,</span> <span class="n">T</span><span class="p">):</span>
+<span class="gp">... </span>        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="n">D</span><span class="p">)</span>
+<span class="gp">... </span>        <span class="bp">self</span><span class="o">.</span><span class="n">T</span> <span class="o">=</span> <span class="n">T</span>
+<span class="gp">...</span>
+<span class="gp">... </span>    <span class="nd">@property</span>
+<span class="gp">... </span>    <span class="k">def</span> <span class="nf">A</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="gp">... </span>        <span class="k">return</span> <span class="n">torch</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">_t</span><span class="o">.</span><span class="n">cos</span><span class="p">()</span>
+<span class="gp">...</span>
+<span class="gp">... </span>    <span class="nd">@property</span>
+<span class="gp">... </span>    <span class="k">def</span> <span class="nf">B</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="gp">... </span>        <span class="k">return</span> <span class="n">torch</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">_t</span><span class="o">.</span><span class="n">sin</span><span class="p">()</span>
+<span class="gp">...</span>
+<span class="gp">... </span>    <span class="nd">@property</span>
+<span class="gp">... </span>    <span class="k">def</span> <span class="nf">C</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="gp">... </span>        <span class="k">return</span> <span class="n">torch</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">_t</span><span class="o">.</span><span class="n">tan</span><span class="p">()</span>
+<span class="gp">...</span>
+<span class="gp">... </span>    <span class="nd">@property</span>
+<span class="gp">... </span>    <span class="k">def</span> <span class="nf">D</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="gp">... </span>        <span class="k">return</span> <span class="n">torch</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
+<span class="gp">...</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">ltv</span> <span class="o">=</span> <span class="n">MyLTV</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">T</span><span class="p">):</span>
+<span class="gp">... </span>    <span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">ltv</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">u</span><span class="p">[</span><span class="n">t</span><span class="p">])</span>
+</pre></div>
+</div>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>More practical examples can be found at <a class="reference external" href="https://github.com/pypose/pypose/tree/main/examples/module/dynamics">examples/module/dynamics</a>.</p>
+</div>
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.module.LTV.set_refpoint">
+<span class="sig-name descname"><span class="pre">set_refpoint</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">state</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">input</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">t</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/module/dynamics/#LTV.set_refpoint"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.module.LTV.set_refpoint" title="Permalink to this definition">¶</a></dt>
+<dd><p>Function to set the reference point for linearization.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>state</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – The reference state of the dynamical system. If <code class="docutils literal notranslate"><span class="pre">None</span></code>,
+the the most recent state is taken. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+<li><p><strong>input</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – The reference input to the dynamical system. If <code class="docutils literal notranslate"><span class="pre">None</span></code>,
+the the most recent input is taken. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+<li><p><strong>t</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – The reference time step of the dynamical system. If <code class="docutils literal notranslate"><span class="pre">None</span></code>,
+the the most recent timestamp is taken. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+</ul>
+</dd>
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+  <section id="pypose-module-mpc">
+<h1>pypose.module.MPC<a class="headerlink" href="#pypose-module-mpc" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.module.MPC">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.module.</span></span><span class="sig-name descname"><span class="pre">MPC</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">system</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Q</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">p</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">T</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">stepper</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/module/mpc/#MPC"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.module.MPC" title="Permalink to this definition">¶</a></dt>
+<dd><p>Model Predictive Control (MPC) based on iterative LQR.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>system</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">instance</span></code>) – The system to be soved by MPC.</p></li>
+<li><p><strong>Q</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – The weight matrix of the quadratic term.</p></li>
+<li><p><strong>p</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – The weight vector of the first-order term.</p></li>
+<li><p><strong>T</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">int</span></code>) – Time steps of system.</p></li>
+<li><p><strong>stepper</strong> (<code class="docutils literal notranslate"><span class="pre">Planner</span></code>, optional) – the stepper to stop iterations. If <code class="docutils literal notranslate"><span class="pre">None</span></code>,
+the <code class="docutils literal notranslate"><span class="pre">pypose.utils.ReduceToBason</span></code> with a maximum of 10 steps are used.
+Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+</ul>
+</dd>
+</dl>
+<p><strong>Model Predictive Control</strong>, also known as Receding Horizon Control (RHC), uses the
+mathematical model of the system to solve a finite, moving horizon, and
+closed loop optimal control problem. Thus, the MPC scheme is able to utilize the
+information about the current state of the system to predict future states
+and control inputs for the system. At each time stamp, MPC solves the
+optimization problem on a time horizon <span class="math">\(T\)</span>:</p>
+<div class="math">
+\[\begin{align*}
+  \mathop{\arg\min}\limits_{\mathbf{x}_{1:T}, \mathbf{u}_{1:T}} \sum\limits_t
+  &\mathbf{c}_t (\mathbf{x}_t, \mathbf{u}_t) \\
+  \mathrm{s.t.}\quad \mathbf{x}_{t+1} &= \mathbf{f}(\mathbf{x}_t, \mathbf{u}_t),\\
+  \mathbf{x}_1 &= \mathbf{x}_{\text{init}} \\
+\end{align*}
+
+\]</div>
+<p>where <span class="math">\(\mathbf{c}\)</span> is the cost function; <span class="math">\(\mathbf{x}\)</span>, <span class="math">\(\mathbf{u}\)</span>
+are the state and input of the linear system; <span class="math">\(\mathbf{f}\)</span> is the dynamics
+model; <span class="math">\(\mathbf{x}_{\text{init}}\)</span> is the initial state of the system.</p>
+<p>For the linear system with quadratic cost, MPC is equivalent to solving an LQR problem
+on each horizon as well as considering the dynamics as constraints. For nonlinear
+system, one way to solve the MPC problem is to use iterative LQR, which uses linear
+approximations of the dynamics and quadratic approximations of the cost function to
+iteratively compute a local optimal solution based on the current states and control
+sequences. Then, analytical derivative for the backward can be computed using one
+additional forward pass of iLQR for the learning problem, such as learning the
+parameters of the dynamic system.</p>
+<p>Specifically, the discrete-time non-linear system can be described as:</p>
+<div class="math">
+\[\begin{aligned}
+    \mathbf{x}_{t+1} &= \mathbf{f}(\mathbf{x}_t, \mathbf{u}_t, t_t) \\
+    \mathbf{y}_{t} &= \mathbf{g}(\mathbf{x}_t, \mathbf{u}_t, t_t) \\
+\end{aligned}
+
+\]</div>
+<p>We can do a linear approximation at current point <span class="math">\(\chi^*=(\mathbf{x}^*,
+\mathbf{u}^*, t^*)\)</span> along a trajectory with small perturbation
+<span class="math">\(\chi=(\mathbf{x}^*+\delta\mathbf{x}, \mathbf{u}^* +\delta\mathbf{u}, t^*)\)</span>
+near <span class="math">\(\chi^*\)</span> for both dynamics and cost:</p>
+<div class="math">
+\[\begin{aligned}
+\mathbf{f}(\mathbf{x}, \mathbf{u}, t^*) &\approx \mathbf{f}(\mathbf{x}^*,
+    \mathbf{u}^*, t^*) + \left.\frac{\partial \mathbf{f}}{\partial \mathbf{x}}
+    \right|_{\chi^*} \delta \mathbf{x} + \left. \frac{\partial \mathbf{f}}
+    {\partial \mathbf{u}} \right|_{\chi^*} \delta \mathbf{u} \\
+&= \mathbf{f}(\mathbf{x}^*, \mathbf{u}^*, t^*) + \mathbf{A} \delta \mathbf{x}
+    + \mathbf{B} \delta \mathbf{u} \\
+\delta \mathbf{x}_{t+1} &= \mathbf{A}_t \delta \mathbf{x}_t + \mathbf{B}_t
+    \delta \mathbf{u}_t \\
+&= \mathbf{F}_t \delta \mathbf{\tau}_t \\
+\mathbf{c} \left( \mathbf{\tau}, t^* \right) &\approx
+    \mathbf{c} \left(\mathbf{\tau}^*, t^* \right) + \frac{1}{2} \delta
+    \mathbf{\tau}^\top\nabla^2_{\mathbf{\tau}}\mathbf{c}\left(\mathbf{\tau}^*,
+    t^* \right) \delta \mathbf{\tau} + \nabla_{\mathbf{\tau}}
+    \mathbf{c} \left( \mathbf{\tau}^*, t^* \right)^\top \delta \mathbf{\tau}\\
+\bar{\mathbf{c}} \left( \delta \mathbf{\tau} \right) &= \frac{1}{2} \delta
+    \mathbf{\tau}_t^\top \bar{\mathbf{Q}}_t \delta \mathbf{\tau}_t +
+    \bar{\mathbf{p}}_t^\top \delta \mathbf{\tau}_t \\
+\end{aligned}
+
+\]</div>
+<p>where <span class="math">\(\mathbf{F}_t\)</span> = <span class="math">\(\begin{bmatrix}
+\mathbf{A}_t &amp; \mathbf{B}_t \end{bmatrix}\)</span>.</p>
+<p>Then, LQR can be performed on a linear quadractic problem with
+<span class="math">\(\delta \mathbf{\tau}_t\)</span>, <span class="math">\(\mathbf{F}_t\)</span>, <span class="math">\(\bar{\mathbf{Q}}_t\)</span> and
+<span class="math">\(\bar{\mathbf{p}}_t\)</span>.</p>
+<ul>
+<li><p>The backward recursion.</p>
+<p>For <span class="math">\(t\)</span> = <span class="math">\(T\)</span> to 1:</p>
+<blockquote>
+<div><div class="math">
+\[\begin{align*}
+    \mathbf{Q}_t &= \bar{\mathbf{Q}}_t + \mathbf{F}_t^\top\mathbf{V}_{t+1}
+                        \mathbf{F}_t \\
+    \mathbf{q}_t &= \bar{\mathbf{p}}_t + \mathbf{F}_t^\top\mathbf{v}_{t+1}  \\
+    \mathbf{K}_t &= -\mathbf{Q}_{\delta \mathbf{u}_t,\delta \mathbf{u}_t}^{-1}
+                     \mathbf{Q}_{\delta \mathbf{u}_t,\delta \mathbf{x}_t} \\
+    \mathbf{k}_t &= -\mathbf{Q}_{\delta \mathbf{u}_t,\delta \mathbf{u}_t}^{-1}
+                        \mathbf{q}_{\delta \mathbf{u}_t}         \\
+    \mathbf{V}_t &= \mathbf{Q}_{\delta \mathbf{x}_t, \delta \mathbf{x}_t}
+        + \mathbf{Q}_{\delta \mathbf{x}_t, \delta \mathbf{u}_t}\mathbf{K}_t
+        + \mathbf{K}_t^\top\mathbf{Q}_{\delta\mathbf{u}_t,\delta \mathbf{x}_t}
+        + \mathbf{K}_t^\top\mathbf{Q}_{\delta\mathbf{u}_t,\delta \mathbf{u}_t}
+            \mathbf{K}_t   \\
+    \mathbf{v}_t &= \mathbf{q}_{\delta \mathbf{x}_t}
+        + \mathbf{Q}_{\delta \mathbf{x}_t, \delta \mathbf{u}_t}\mathbf{k}_t
+        + \mathbf{K}_t^\top\mathbf{q}_{\delta \mathbf{u}_t}
+        + \mathbf{K}_t^\top\mathbf{Q}_{\delta \mathbf{u}_t,
+            \delta \mathbf{u}_t}\mathbf{k}_t   \\
+\end{align*}
+
+\]</div>
+</div></blockquote>
+</li>
+<li><p>The forward recursion.</p>
+<p>For <span class="math">\(t\)</span> = 1 to <span class="math">\(T\)</span>:</p>
+<blockquote>
+<div><div class="math">
+\[\begin{align*}
+    \delta \mathbf{u}_t &= \mathbf{K}_t \delta \mathbf{x}_t + \mathbf{k}_t \\
+    \mathbf{u}_t &= \delta \mathbf{u}_t + \mathbf{u}_t^* \\
+    \mathbf{x}_{t+1} &= \mathbf{f}(\mathbf{x}_t, \mathbf{u}_t) \\
+\end{align*}
+
+\]</div>
+</div></blockquote>
+</li>
+</ul>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>For the details of lqr solver and the iterative LQR, please refer to <a class="reference internal" href="../pypose.module.LQR/#pypose.module.LQR" title="pypose.module.LQR"><code class="xref py py-meth docutils literal notranslate"><span class="pre">LQR()</span></code></a>.
+Please note that the linear system with quadratic cost only requires one single
+iteration.</p>
+</div>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The definition of MPC is cited from this book:</p>
+<ul class="simple">
+<li><p>George Nikolakopoulos, Sina Sharif Mansouri, Christoforos Kanellakis, <a class="reference external" href="https://doi.org/10.1016/C2017-0-02260-7">Aerial
+Robotic Workers: Design, Modeling, Control, Vision and Their Applications</a>, Butterworth-Heinemann, 2023.</p></li>
+</ul>
+<p>The implementation of MPC is based on Eq. (10)~(13) of this paper:</p>
+<ul class="simple">
+<li><p>Amos, Brandon, et al, <a class="reference external" href="https://tinyurl.com/3h9tsuyb">Differentiable mpc for end-to-end planning and control</a>, Advances in neural information processing
+systems, 31, 2018.</p></li>
+</ul>
+</div>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">torch</span><span class="o">,</span> <span class="nn">pypose</span> <span class="k">as</span> <span class="nn">pp</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="k">class</span> <span class="nc">CartPole</span><span class="p">(</span><span class="n">pp</span><span class="o">.</span><span class="n">module</span><span class="o">.</span><span class="n">NLS</span><span class="p">):</span>
+<span class="gp">... </span>    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">length</span><span class="p">,</span> <span class="n">cartmass</span><span class="p">,</span> <span class="n">polemass</span><span class="p">,</span> <span class="n">gravity</span><span class="p">):</span>
+<span class="gp">... </span>        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+<span class="gp">... </span>        <span class="bp">self</span><span class="o">.</span><span class="n">tau</span> <span class="o">=</span> <span class="n">dt</span>
+<span class="gp">... </span>        <span class="bp">self</span><span class="o">.</span><span class="n">length</span> <span class="o">=</span> <span class="n">length</span>
+<span class="gp">... </span>        <span class="bp">self</span><span class="o">.</span><span class="n">cartmass</span> <span class="o">=</span> <span class="n">cartmass</span>
+<span class="gp">... </span>        <span class="bp">self</span><span class="o">.</span><span class="n">polemass</span> <span class="o">=</span> <span class="n">polemass</span>
+<span class="gp">... </span>        <span class="bp">self</span><span class="o">.</span><span class="n">gravity</span> <span class="o">=</span> <span class="n">gravity</span>
+<span class="gp">... </span>        <span class="bp">self</span><span class="o">.</span><span class="n">poleml</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">polemass</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">length</span>
+<span class="gp">... </span>        <span class="bp">self</span><span class="o">.</span><span class="n">totalMass</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">cartmass</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">polemass</span>
+<span class="gp">...</span>
+<span class="gp">... </span>    <span class="k">def</span> <span class="nf">state_transition</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">state</span><span class="p">,</span> <span class="nb">input</span><span class="p">,</span> <span class="n">t</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
+<span class="gp">... </span>        <span class="n">x</span><span class="p">,</span> <span class="n">xDot</span><span class="p">,</span> <span class="n">theta</span><span class="p">,</span> <span class="n">thetaDot</span> <span class="o">=</span> <span class="n">state</span><span class="o">.</span><span class="n">squeeze</span><span class="p">()</span>
+<span class="gp">... </span>        <span class="n">force</span> <span class="o">=</span> <span class="nb">input</span><span class="o">.</span><span class="n">squeeze</span><span class="p">()</span>
+<span class="gp">... </span>        <span class="n">costheta</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
+<span class="gp">... </span>        <span class="n">sintheta</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
+<span class="gp">... </span>        <span class="n">temp</span> <span class="o">=</span> <span class="p">(</span><span class="n">force</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">poleml</span> <span class="o">*</span> <span class="n">thetaDot</span><span class="o">**</span><span class="mi">2</span> <span class="o">*</span> <span class="n">sintheta</span><span class="p">)</span> <span class="o">/</span> <span class="bp">self</span><span class="o">.</span><span class="n">totalMass</span>
+<span class="gp">... </span>        <span class="n">thetaAcc</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">gravity</span> <span class="o">*</span> <span class="n">sintheta</span> <span class="o">-</span> <span class="n">costheta</span> <span class="o">*</span> <span class="n">temp</span><span class="p">)</span> <span class="o">/</span>
+<span class="gp">... </span>                    <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">length</span> <span class="o">*</span> <span class="p">(</span><span class="mf">4.0</span> <span class="o">/</span> <span class="mf">3.0</span>
+<span class="gp">... </span>                           <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">polemass</span> <span class="o">*</span> <span class="n">costheta</span><span class="o">**</span><span class="mi">2</span> <span class="o">/</span> <span class="bp">self</span><span class="o">.</span><span class="n">totalMass</span><span class="p">))</span>
+<span class="gp">... </span>        <span class="n">xAcc</span> <span class="o">=</span> <span class="n">temp</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">poleml</span> <span class="o">*</span> <span class="n">thetaAcc</span> <span class="o">*</span> <span class="n">costheta</span> <span class="o">/</span> <span class="bp">self</span><span class="o">.</span><span class="n">totalMass</span>
+<span class="gp">... </span>        <span class="n">_dstate</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">stack</span><span class="p">((</span><span class="n">xDot</span><span class="p">,</span> <span class="n">xAcc</span><span class="p">,</span> <span class="n">thetaDot</span><span class="p">,</span> <span class="n">thetaAcc</span><span class="p">))</span>
+<span class="gp">... </span>        <span class="k">return</span> <span class="p">(</span><span class="n">state</span><span class="o">.</span><span class="n">squeeze</span><span class="p">()</span> <span class="o">+</span> <span class="n">torch</span><span class="o">.</span><span class="n">mul</span><span class="p">(</span><span class="n">_dstate</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">tau</span><span class="p">))</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
+<span class="gp">...</span>
+<span class="gp">... </span>    <span class="k">def</span> <span class="nf">observation</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">state</span><span class="p">,</span> <span class="nb">input</span><span class="p">,</span> <span class="n">t</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
+<span class="gp">... </span>        <span class="k">return</span> <span class="n">state</span>
+<span class="gp">...</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">torch</span><span class="o">.</span><span class="n">manual_seed</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">dt</span><span class="p">,</span> <span class="nb">len</span><span class="p">,</span> <span class="n">m_cart</span><span class="p">,</span> <span class="n">m_pole</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="mf">0.01</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">,</span> <span class="mi">20</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mf">9.81</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">n_batch</span><span class="p">,</span> <span class="n">T</span> <span class="o">=</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">5</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">n_state</span><span class="p">,</span> <span class="n">n_ctrl</span> <span class="o">=</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">1</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">n_sc</span> <span class="o">=</span> <span class="n">n_state</span> <span class="o">+</span> <span class="n">n_ctrl</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">Q</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tile</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">n_state</span> <span class="o">+</span> <span class="n">n_ctrl</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="n">device</span><span class="p">),</span> <span class="p">(</span><span class="n">n_batch</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n_batch</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">n_sc</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">time</span>  <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="n">device</span><span class="p">)</span> <span class="o">*</span> <span class="n">dt</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">current_u</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">time</span><span class="p">)</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x_init</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">torch</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">stepper</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">utils</span><span class="o">.</span><span class="n">ReduceToBason</span><span class="p">(</span><span class="n">steps</span><span class="o">=</span><span class="mi">15</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">cartPoleSolver</span> <span class="o">=</span> <span class="n">CartPole</span><span class="p">(</span><span class="n">dt</span><span class="p">,</span> <span class="nb">len</span><span class="p">,</span> <span class="n">m_cart</span><span class="p">,</span> <span class="n">m_pole</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span><span class="o">.</span><span class="n">to</span><span class="p">(</span><span class="n">device</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">MPC</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">module</span><span class="o">.</span><span class="n">MPC</span><span class="p">(</span><span class="n">cartPoleSolver</span><span class="p">,</span> <span class="n">Q</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">stepper</span><span class="o">=</span><span class="n">stepper</span><span class="p">)</span><span class="o">.</span><span class="n">to</span><span class="p">(</span><span class="n">device</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">cost</span> <span class="o">=</span> <span class="n">MPC</span><span class="p">(</span><span class="n">dt</span><span class="p">,</span> <span class="n">x_init</span><span class="p">,</span> <span class="n">u_init</span><span class="o">=</span><span class="n">current_u</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="nb">print</span><span class="p">(</span><span class="s2">&quot;x = &quot;</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="nb">print</span><span class="p">(</span><span class="s2">&quot;u = &quot;</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
+<span class="go">x =  tensor([[[ 0.0000e+00,  0.0000e+00,  3.1416e+00,  0.0000e+00],</span>
+<span class="go">              [ 0.0000e+00, -3.7711e-04,  3.1416e+00, -1.8856e-04],</span>
+<span class="go">              [-3.7711e-06, -3.0693e-04,  3.1416e+00, -1.5347e-04],</span>
+<span class="go">              [-6.8404e-06, -2.4288e-04,  3.1416e+00, -1.2136e-04],</span>
+<span class="go">              [-9.2692e-06, -2.6634e-04,  3.1416e+00, -1.3293e-04],</span>
+<span class="go">              [-1.1933e-05, -2.2073e-04,  3.1416e+00, -1.0991e-04]]])</span>
+<span class="go">u =  tensor([[[-0.8485],</span>
+<span class="go">              [ 0.1579],</span>
+<span class="go">              [ 0.1440],</span>
+<span class="go">              [-0.0530],</span>
+<span class="go">              [ 0.1023]]])</span>
+</pre></div>
+</div>
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.module.MPC.forward">
+<span class="sig-name descname"><span class="pre">forward</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">dt</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">x_init</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">u_init</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">u_lower</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">u_upper</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">du</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/module/mpc/#MPC.forward"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.module.MPC.forward" title="Permalink to this definition">¶</a></dt>
+<dd><p>Performs MPC for the discrete system.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>dt</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">int</span></code>) – The interval (<span class="math">\(\delta t\)</span>) between two time steps.</p></li>
+<li><p><strong>x_init</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – The initial state of the system.</p></li>
+<li><p><strong>u_init</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>, optinal) – The current inputs of the system along a
+trajectory. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+<li><p><strong>u_lower</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>, optinal) – The lower bounds on the controls.
+Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+<li><p><strong>u_upper</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>, optinal) – The upper bounds on the controls.
+Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+<li><p><strong>du</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">int</span></code>, optinal) – The amount each component of the controls
+is allowed to change in each LQR iteration. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>A list of tensors including the solved state sequence
+<span class="math">\(\mathbf{x}\)</span>, the solved input sequence <span class="math">\(\mathbf{u}\)</span>, and the
+associated quadratic costs <span class="math">\(\mathbf{c}\)</span> over the time horizon.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>List of <code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code></p>
+</dd>
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+  <section id="pypose-module-nls">
+<h1>pypose.module.NLS<a class="headerlink" href="#pypose-module-nls" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.module.NLS">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.module.</span></span><span class="sig-name descname"><span class="pre">NLS</span></span><a class="reference internal" href="../../_modules/pypose/module/dynamics/#NLS"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.module.NLS" title="Permalink to this definition">¶</a></dt>
+<dd><p>Dynamics model for discrete-time non-linear system (NLS).</p>
+<p>The state transision function <span class="math">\(\mathbf{f}\)</span> and observation function
+<span class="math">\(\mathbf{g}\)</span> are given by:</p>
+<div class="math">
+\[\begin{aligned}
+    \mathbf{x}_{k+1} &= \mathbf{f}(\mathbf{x}_k, \mathbf{u}_k, t_k), \\
+    \mathbf{y}_{k}   &= \mathbf{g}(\mathbf{x}_k, \mathbf{u}_k, t_k),
+\end{aligned}
+
+\]</div>
+<p>where <span class="math">\(k\)</span>, <span class="math">\(\mathbf{x}\)</span>, <span class="math">\(\mathbf{u}\)</span>, <span class="math">\(\mathbf{y}\)</span> are the time
+step, state(s), input(s), and observation(s), respectively.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>To use the class, users need to inherit this class and define methods
+<code class="xref py py-obj docutils literal notranslate"><span class="pre">state_transition</span></code> and <code class="xref py py-obj docutils literal notranslate"><span class="pre">observation</span></code>, which are automatically called by
+internal <a class="reference internal" href="#pypose.module.NLS.forward" title="pypose.module.NLS.forward"><code class="xref py py-obj docutils literal notranslate"><span class="pre">forward</span></code></a> method.
+The system timestamp (starting from <strong>0</strong>) is also self-added automatically once
+the <a class="reference internal" href="#pypose.module.NLS.forward" title="pypose.module.NLS.forward"><code class="xref py py-obj docutils literal notranslate"><span class="pre">forward</span></code></a> method is called.</p>
+</div>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>This class provides automatic <strong>linearlization</strong> at a reference point
+<span class="math">\(\chi^*=(\mathbf{x}^*, \mathbf{u}^*, t^*)\)</span> along a trajectory.
+One can directly call those linearized system matrices as properties including
+<a class="reference internal" href="#pypose.module.NLS.A" title="pypose.module.NLS.A"><code class="xref py py-obj docutils literal notranslate"><span class="pre">A</span></code></a>, <a class="reference internal" href="#pypose.module.NLS.B" title="pypose.module.NLS.B"><code class="xref py py-obj docutils literal notranslate"><span class="pre">B</span></code></a>, <a class="reference internal" href="#pypose.module.NLS.C" title="pypose.module.NLS.C"><code class="xref py py-obj docutils literal notranslate"><span class="pre">C</span></code></a>, <a class="reference internal" href="#pypose.module.NLS.D" title="pypose.module.NLS.D"><code class="xref py py-obj docutils literal notranslate"><span class="pre">D</span></code></a>, <a class="reference internal" href="#pypose.module.NLS.c1" title="pypose.module.NLS.c1"><code class="xref py py-obj docutils literal notranslate"><span class="pre">c1</span></code></a>, and <a class="reference internal" href="#pypose.module.NLS.c2" title="pypose.module.NLS.c2"><code class="xref py py-obj docutils literal notranslate"><span class="pre">c2</span></code></a>, after calling
+a method <a class="reference internal" href="#pypose.module.NLS.set_refpoint" title="pypose.module.NLS.set_refpoint"><code class="xref py py-obj docutils literal notranslate"><span class="pre">set_refpoint</span></code></a>.</p>
+<p>Consider a point
+<span class="math">\(\chi=(\mathbf{x}^*+\delta\mathbf{x}, \mathbf{u}^*+\delta\mathbf{u}, t^*)\)</span> near
+<span class="math">\(\chi^*\)</span>. We have</p>
+<div class="math">
+\[\begin{aligned}
+\mathbf{f}(\mathbf{x}, \mathbf{u}, t^*) &\approx \mathbf{f}(\mathbf{x}^*,
+    \mathbf{u}^*, t^*) +  \left. \frac{\partial \mathbf{f}}{\partial \mathbf{x}}
+    \right|_{\chi^*} \delta \mathbf{x} + \left. \frac{\partial \mathbf{f}}
+    {\partial \mathbf{u}} \right|_{\chi^*} \delta \mathbf{u} \\
+&= \mathbf{f}(\mathbf{x}^*, \mathbf{u}^*, t^*) + \mathbf{A}(\mathbf{x}
+    - \mathbf{x}^*) + \mathbf{B}(\mathbf{u}-\mathbf{u}^*) \\
+&= \mathbf{A}\mathbf{x} + \mathbf{B}\mathbf{u} + \mathbf{c}_1
+\end{aligned}
+
+\]</div>
+<p>and</p>
+<div class="math">
+\[\mathbf{g}(\mathbf{x}, \mathbf{u}, t^*) \approx \mathbf{C}\mathbf{x} \
+            + \mathbf{D}\mathbf{u} + \mathbf{c}_2
+
+\]</div>
+<p>The notion of linearization is slightly different from that in dynamical system
+theory. First, the linearization can be done for arbitrary point(s), not limited to
+the system’s equilibrium point(s), and therefore the extra constant terms <span class="math">\(\mathbf{c}_1\)</span>
+and <span class="math">\(\mathbf{c}_2\)</span> are produced. Second, the linearized equations are represented
+by the full states and inputs: <span class="math">\(\mathbf{x}\)</span> and <span class="math">\(\mathbf{u}\)</span>, rather than
+the perturbation format: <span class="math">\(\delta \mathbf{x}\)</span> and <span class="math">\(\delta \mathbf{u}\)</span>
+so that the model is consistent with, e.g., the LTI model and the iterative LQR
+solver. For more details go to <a class="reference internal" href="../pypose.module.LTI/#pypose.module.LTI" title="pypose.module.LTI"><code class="xref py py-meth docutils literal notranslate"><span class="pre">LTI()</span></code></a>.</p>
+</div>
+<p class="rubric">Example</p>
+<p>A simple linear time-varying system, but defined via NLS. Here we show an example
+for advancing one time step of the system at a given time step and computing the
+linearization.</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">math</span><span class="o">,</span> <span class="nn">torch</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">pypose</span> <span class="k">as</span> <span class="nn">pp</span>
+<span class="gp">...</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="k">class</span> <span class="nc">Floquet</span><span class="p">(</span><span class="n">pp</span><span class="o">.</span><span class="n">module</span><span class="o">.</span><span class="n">NLS</span><span class="p">):</span>
+<span class="gp">... </span>    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="gp">... </span>        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+<span class="gp">...</span>
+<span class="gp">... </span>    <span class="k">def</span> <span class="nf">state_transition</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">state</span><span class="p">,</span> <span class="nb">input</span><span class="p">,</span> <span class="n">t</span><span class="p">):</span>
+<span class="gp">...</span>
+<span class="gp">... </span>        <span class="n">cc</span> <span class="o">=</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">math</span><span class="o">.</span><span class="n">pi</span> <span class="o">*</span> <span class="n">t</span> <span class="o">/</span> <span class="mi">100</span><span class="p">)</span><span class="o">.</span><span class="n">cos</span><span class="p">()</span>
+<span class="gp">... </span>        <span class="n">ss</span> <span class="o">=</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">math</span><span class="o">.</span><span class="n">pi</span> <span class="o">*</span> <span class="n">t</span> <span class="o">/</span> <span class="mi">100</span><span class="p">)</span><span class="o">.</span><span class="n">sin</span><span class="p">()</span>
+<span class="gp">...</span>
+<span class="gp">... </span>        <span class="n">A</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([[</span>   <span class="mf">1.</span><span class="p">,</span> <span class="n">cc</span><span class="o">/</span><span class="mi">10</span><span class="p">],</span>
+<span class="gp">... </span>                          <span class="p">[</span><span class="n">cc</span><span class="o">/</span><span class="mi">10</span><span class="p">,</span>    <span class="mf">1.</span><span class="p">]],</span> <span class="n">device</span><span class="o">=</span><span class="n">t</span><span class="o">.</span><span class="n">device</span><span class="p">)</span>
+<span class="gp">... </span>        <span class="n">B</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([[</span><span class="n">ss</span><span class="p">],</span>
+<span class="gp">... </span>                          <span class="p">[</span><span class="mf">1.</span><span class="p">]],</span> <span class="n">device</span><span class="o">=</span><span class="n">t</span><span class="o">.</span><span class="n">device</span><span class="p">)</span>
+<span class="gp">...</span>
+<span class="gp">... </span>        <span class="k">return</span> <span class="n">pp</span><span class="o">.</span><span class="n">bmv</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">state</span><span class="p">)</span> <span class="o">+</span> <span class="n">pp</span><span class="o">.</span><span class="n">bmv</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="nb">input</span><span class="p">)</span>
+<span class="gp">...</span>
+<span class="gp">... </span>    <span class="k">def</span> <span class="nf">observation</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">state</span><span class="p">,</span> <span class="nb">input</span><span class="p">,</span> <span class="n">t</span><span class="p">):</span>
+<span class="gp">... </span>        <span class="k">return</span> <span class="n">state</span> <span class="o">+</span> <span class="n">t</span>
+<span class="gp">...</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="c1"># Start from t = 8, and advance one step to t = 9.</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">step</span><span class="p">,</span> <span class="n">current</span> <span class="o">=</span> <span class="mi">8</span><span class="p">,</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="nb">input</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">math</span><span class="o">.</span><span class="n">pi</span> <span class="o">/</span> <span class="mi">50</span> <span class="o">*</span> <span class="n">step</span><span class="p">)</span><span class="o">.</span><span class="n">sin</span><span class="p">()</span>
+<span class="gp">...</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">system</span> <span class="o">=</span> <span class="n">Floquet</span><span class="p">()</span><span class="o">.</span><span class="n">reset</span><span class="p">(</span><span class="n">t</span> <span class="o">=</span> <span class="n">step</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="nb">next</span><span class="p">,</span> <span class="n">observation</span> <span class="o">=</span> <span class="n">system</span><span class="p">(</span><span class="n">current</span><span class="p">,</span> <span class="nb">input</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">system</span><span class="o">.</span><span class="n">set_refpoint</span><span class="p">()</span>
+<span class="gp">...</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="nb">print</span><span class="p">(</span><span class="nb">next</span><span class="p">)</span>        <span class="c1"># Next state</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="nb">print</span><span class="p">(</span><span class="n">observation</span><span class="p">)</span> <span class="c1"># Observation</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="nb">print</span><span class="p">(</span><span class="n">system</span><span class="o">.</span><span class="n">A</span><span class="p">)</span>    <span class="c1"># Linearized state matrix</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="nb">print</span><span class="p">(</span><span class="n">system</span><span class="o">.</span><span class="n">B</span><span class="p">)</span>    <span class="c1"># Linearized input matrix</span>
+<span class="go">tensor([1.4944, 1.9320])</span>
+<span class="go">tensor([9., 9.])</span>
+<span class="go">tensor([[1.0000, 0.0844],</span>
+<span class="go">        [0.0844, 1.0000]])</span>
+<span class="go">tensor([[0.5358],</span>
+<span class="go">        [1.0000]])</span>
+</pre></div>
+</div>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>For generating one trajecotry given a series of inputs, advanced use of
+linearization, and more practical examples can be found at <a class="reference external" href="https://github.com/pypose/pypose/tree/main/examples/module/dynamics">examples/module/dynamics</a>.</p>
+</div>
+<dl class="py property">
+<dt class="sig sig-object py" id="pypose.module.NLS.A">
+<em class="property"><span class="pre">property</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">A</span></span><a class="headerlink" href="#pypose.module.NLS.A" title="Permalink to this definition">¶</a></dt>
+<dd><p>Linear/linearized system state matrix.</p>
+<div class="math">
+\[\mathbf{A} = \left. \frac{\partial \mathbf{f}}{\partial \mathbf{x}} \right|_{\chi^*}
+
+\]</div>
+</dd></dl>
+
+<dl class="py property">
+<dt class="sig sig-object py" id="pypose.module.NLS.B">
+<em class="property"><span class="pre">property</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">B</span></span><a class="headerlink" href="#pypose.module.NLS.B" title="Permalink to this definition">¶</a></dt>
+<dd><p>Linear/linearized system input matrix.</p>
+<div class="math">
+\[\mathbf{B} = \left. \frac{\partial \mathbf{f}}{\partial \mathbf{u}} \right|_{\chi^*}
+
+\]</div>
+</dd></dl>
+
+<dl class="py property">
+<dt class="sig sig-object py" id="pypose.module.NLS.C">
+<em class="property"><span class="pre">property</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">C</span></span><a class="headerlink" href="#pypose.module.NLS.C" title="Permalink to this definition">¶</a></dt>
+<dd><p>Linear/linearized system output matrix.</p>
+<div class="math">
+\[\mathbf{C} = \left. \frac{\partial \mathbf{g}}{\partial \mathbf{x}} \right|_{\chi^*}
+
+\]</div>
+</dd></dl>
+
+<dl class="py property">
+<dt class="sig sig-object py" id="pypose.module.NLS.D">
+<em class="property"><span class="pre">property</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">D</span></span><a class="headerlink" href="#pypose.module.NLS.D" title="Permalink to this definition">¶</a></dt>
+<dd><p>Linear/Linearized system observation matrix.</p>
+<div class="math">
+\[\mathbf{D} = \left. \frac{\partial \mathbf{g}}
+                    {\partial \mathbf{u}} \right|_{\chi^*}
+
+\]</div>
+</dd></dl>
+
+<dl class="py property">
+<dt class="sig sig-object py" id="pypose.module.NLS.c1">
+<em class="property"><span class="pre">property</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">c1</span></span><a class="headerlink" href="#pypose.module.NLS.c1" title="Permalink to this definition">¶</a></dt>
+<dd><p>Constant term generated by state-transition.</p>
+<div class="math">
+\[\mathbf{c}_1 = \mathbf{f}(\mathbf{x}^*, \mathbf{u}^*, t^*)
+               - \mathbf{A}\mathbf{x}^* - \mathbf{B}\mathbf{u}^*
+
+\]</div>
+</dd></dl>
+
+<dl class="py property">
+<dt class="sig sig-object py" id="pypose.module.NLS.c2">
+<em class="property"><span class="pre">property</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">c2</span></span><a class="headerlink" href="#pypose.module.NLS.c2" title="Permalink to this definition">¶</a></dt>
+<dd><p>Constant term generated by observation.</p>
+<div class="math">
+\[\mathbf{c}_2 = \mathbf{g}(\mathbf{x}^*, \mathbf{u}^*, t^*)
+               - \mathbf{C}\mathbf{x}^* - \mathbf{D}\mathbf{u}^*
+
+\]</div>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.module.NLS.forward">
+<span class="sig-name descname"><span class="pre">forward</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">state</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">input</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/module/dynamics/#NLS.forward"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.module.NLS.forward" title="Permalink to this definition">¶</a></dt>
+<dd><p>Defines the computation performed at every call that advances the system by one time step.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The <a class="reference internal" href="#pypose.module.NLS.forward" title="pypose.module.NLS.forward"><code class="xref py py-obj docutils literal notranslate"><span class="pre">forward</span></code></a> method implicitly increments the time step via <code class="xref py py-obj docutils literal notranslate"><span class="pre">forward_hook</span></code>.
+<code class="xref py py-obj docutils literal notranslate"><span class="pre">state_transition</span></code> and <code class="xref py py-obj docutils literal notranslate"><span class="pre">observation</span></code> still accept time for the flexiblity
+such as time-varying system. One can directly access the current system time via the
+property <code class="xref py py-obj docutils literal notranslate"><span class="pre">systime</span></code>.</p>
+</div>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>To introduce noise in a model, redefine this method via
+subclassing. See example in <code class="docutils literal notranslate"><span class="pre">examples/module/ekf/tank_robot.py</span></code>.</p>
+</div>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.module.NLS.set_refpoint">
+<span class="sig-name descname"><span class="pre">set_refpoint</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">state</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">input</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">t</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/module/dynamics/#NLS.set_refpoint"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.module.NLS.set_refpoint" title="Permalink to this definition">¶</a></dt>
+<dd><p>Function to set the reference point for linearization.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>state</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – The reference state of the dynamical system. If <code class="docutils literal notranslate"><span class="pre">None</span></code>,
+the the most recent state is taken. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+<li><p><strong>input</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – The reference input to the dynamical system. If <code class="docutils literal notranslate"><span class="pre">None</span></code>,
+the the most recent input is taken. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+<li><p><strong>t</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – The reference time step of the dynamical system. If <code class="docutils literal notranslate"><span class="pre">None</span></code>,
+the the most recent timestamp is taken. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>The <code class="docutils literal notranslate"><span class="pre">self</span></code> module.</p>
+</dd>
+</dl>
+<div class="admonition warning">
+<p class="admonition-title">Warning</p>
+<p>For nonlinear systems, the users have to call this function before getting the
+linearized system.</p>
+</div>
+</dd></dl>
+
+</dd></dl>
+
+</section>
+
+
+             </article>
+             
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+  <section id="pypose-module-pf">
+<h1>pypose.module.PF<a class="headerlink" href="#pypose-module-pf" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.module.PF">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.module.</span></span><span class="sig-name descname"><span class="pre">PF</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">model</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Q</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">R</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">particles</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1000</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/module/pf/#PF"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.module.PF" title="Permalink to this definition">¶</a></dt>
+<dd><p>Performs Batched Particle Filter (PF).</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>model</strong> (<a class="reference internal" href="../pypose.module.System/#pypose.module.System" title="pypose.module.System"><code class="xref py py-obj docutils literal notranslate"><span class="pre">System</span></code></a>) – The system model to be estimated, a subclass of
+<a class="reference internal" href="../pypose.module.NLS/#pypose.module.NLS" title="pypose.module.NLS"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.module.NLS</span></code></a>.</p></li>
+<li><p><strong>Q</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>, optional) – The covariance matrices of system transition noise.
+Ignored if provided during each iteration. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code></p></li>
+<li><p><strong>R</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>, optional) – The covariance matrices of system observation noise.
+Ignored if provided during each iteration. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code></p></li>
+<li><p><strong>particles</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Int</span></code>, optional) – The number of particle. Default: <code class="docutils literal notranslate"><span class="pre">1000</span></code></p></li>
+</ul>
+</dd>
+</dl>
+<p>A non-linear system can be described as</p>
+<div class="math">
+\[\begin{aligned}
+    \mathbf{x}_{k+1} &= \mathbf{f}(\mathbf{x}_k, \mathbf{u}_k, t_k) + \mathbf{w}_k,
+    \quad \mathbf{w}_k \sim \mathcal{N}(\mathbf{0}, \mathbf{Q})  \\
+    \mathbf{y}_{k} &= \mathbf{g}(\mathbf{x}_k, \mathbf{u}_k, t_k) + \mathbf{v}_k,
+    \quad \mathbf{v}_k \sim \mathcal{N}(\mathbf{0}, \mathbf{R})
+\end{aligned}
+
+\]</div>
+<p>Particle filter can be described as the following equations, where the subscript
+<span class="math">\(\cdot_{k}\)</span> is omited for simplicity.</p>
+<ol class="arabic">
+<li><p>Generate Particles.</p>
+<div class="math">
+\[\begin{aligned}
+  &\mathbf{x}_{k}=\mathbf{p}(\mathbf{x},n\cdot\mathbf{P}_{k},N)&\quad k=1,...,N \\
+  &\mathbf{P}_{k}=\mathbf{P} &\quad k=1,...,N
+\end{aligned}
+
+\]</div>
+<p>where <span class="math">\(N\)</span> is the number of Particles and <span class="math">\(\mathbf{p}\)</span> is the probability
+density function (PDF), <span class="math">\(n\)</span> is the dimension of state.</p>
+</li>
+<li><p>Priori State Estimation.</p>
+<blockquote>
+<div><div class="math">
+\[\mathbf{x}^{-}_{k} = f(\mathbf{x}_{k}, \mathbf{u}_{k}, t)
+
+\]</div>
+</div></blockquote>
+</li>
+<li><p>Relative Likelihood.</p>
+<blockquote>
+<div><div class="math">
+\[\begin{aligned}
+    & \mathbf{q} = \mathbf{p} (\mathbf{y} |\mathbf{x}^{-}_{k}) \\
+    & \mathbf{q}_{i} = \frac{\mathbf{q}_{i}}{\sum_{j=1}^{N}\mathbf{q}_{j}}
+\end{aligned}
+
+\]</div>
+</div></blockquote>
+</li>
+<li><p>Resample Particles.</p>
+<ol class="loweralpha simple">
+<li><p>Generate random numbers <span class="math">\(r_i \sim U(0,1), i=1,\cdots,N\)</span>, where <span class="math">\(U\)</span>
+is uniform distribution.</p></li>
+<li><p>Find <span class="math">\(j\)</span> satisfying <span class="math">\(\sum_{m=1}^{j-1}q_m\leq r_i&lt;\sum_{m=1}^{j}q_m\)</span>,
+then new particle <span class="math">\(\mathbf{x}^{r-}_{i}\)</span> is set equal to old particle
+<span class="math">\(\mathbf{x}^{-}_{j}\)</span>.</p></li>
+</ol>
+</li>
+<li><p>Refine Posteriori And Covariances.</p>
+<blockquote>
+<div><div class="math">
+\[\begin{aligned}
+    & \mathbf{x}^{+} =\frac{1}{N}  \sum_{i=1}^{n}\mathbf{x}^{r-}_{i}   \\
+    & P^{+} = \sum_{i=1}^{N} (\mathbf{x}^{+} - \mathbf{x}^{r-}_{i})
+    (\mathbf{x}^{+} - \mathbf{x}^{r-}_{i})^{T} + \mathbf{Q}
+\end{aligned}
+
+\]</div>
+</div></blockquote>
+</li>
+</ol>
+<p class="rubric">Example</p>
+<ol class="arabic simple">
+<li><p>Define a discrete-time non-linear system (NLS) model</p></li>
+</ol>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">torch</span><span class="o">,</span> <span class="nn">pypose</span> <span class="k">as</span> <span class="nn">pp</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="k">class</span> <span class="nc">NLS</span><span class="p">(</span><span class="n">pp</span><span class="o">.</span><span class="n">module</span><span class="o">.</span><span class="n">NLS</span><span class="p">):</span>
+<span class="gp">... </span>    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="gp">... </span>        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+<span class="gp">...</span>
+<span class="gp">... </span>    <span class="k">def</span> <span class="nf">state_transition</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">state</span><span class="p">,</span> <span class="nb">input</span><span class="p">,</span> <span class="n">t</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
+<span class="gp">... </span>        <span class="k">return</span> <span class="n">state</span><span class="o">.</span><span class="n">cos</span><span class="p">()</span> <span class="o">+</span> <span class="nb">input</span>
+<span class="gp">...</span>
+<span class="gp">... </span>    <span class="k">def</span> <span class="nf">observation</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">state</span><span class="p">,</span> <span class="nb">input</span><span class="p">,</span> <span class="n">t</span><span class="p">):</span>
+<span class="gp">... </span>        <span class="k">return</span> <span class="n">state</span><span class="o">.</span><span class="n">sin</span><span class="p">()</span> <span class="o">+</span> <span class="nb">input</span>
+</pre></div>
+</div>
+<ol class="arabic simple" start="2">
+<li><p>Create a model and filter</p></li>
+</ol>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">model</span> <span class="o">=</span> <span class="n">NLS</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">pf</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">module</span><span class="o">.</span><span class="n">PF</span><span class="p">(</span><span class="n">model</span><span class="p">)</span>
+</pre></div>
+</div>
+<ol class="arabic simple" start="3">
+<li><p>Prepare data</p></li>
+</ol>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">N</span> <span class="o">=</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">2</span> <span class="c1"># batchsize, steps, state dim</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">states</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">N</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">inputs</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">N</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">observ</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">N</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="c1"># std of transition, observation, and estimation</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">p</span> <span class="o">=</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mi">10</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">Q</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">N</span><span class="p">)</span><span class="o">.</span><span class="n">repeat</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">q</span><span class="o">**</span><span class="mi">2</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">R</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">N</span><span class="p">)</span><span class="o">.</span><span class="n">repeat</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">r</span><span class="o">**</span><span class="mi">2</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">N</span><span class="p">)</span><span class="o">.</span><span class="n">repeat</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">p</span><span class="o">**</span><span class="mi">2</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">estim</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">N</span><span class="p">)</span> <span class="o">*</span> <span class="n">p</span>
+</pre></div>
+</div>
+<ol class="arabic simple" start="4">
+<li><p>Perform PF prediction. Note that estimation error becomes smaller with more steps.</p></li>
+</ol>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">T</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
+<span class="gp">... </span>    <span class="n">w</span> <span class="o">=</span> <span class="n">q</span> <span class="o">*</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">N</span><span class="p">)</span> <span class="c1"># transition noise</span>
+<span class="gp">... </span>    <span class="n">v</span> <span class="o">=</span> <span class="n">r</span> <span class="o">*</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">N</span><span class="p">)</span> <span class="c1"># observation noise</span>
+<span class="gp">... </span>    <span class="n">states</span><span class="p">[:,</span> <span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">],</span> <span class="n">observ</span><span class="p">[:,</span> <span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">model</span><span class="p">(</span><span class="n">states</span><span class="p">[:,</span> <span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="n">w</span><span class="p">,</span> <span class="n">inputs</span><span class="p">[:,</span> <span class="n">i</span><span class="p">])</span>
+<span class="gp">... </span>    <span class="n">estim</span><span class="p">[:,</span> <span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">],</span> <span class="n">P</span><span class="p">[:,</span> <span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">pf</span><span class="p">(</span><span class="n">estim</span><span class="p">[:,</span> <span class="n">i</span><span class="p">],</span> <span class="n">observ</span><span class="p">[:,</span> <span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="n">v</span><span class="p">,</span> <span class="n">inputs</span><span class="p">[:,</span> <span class="n">i</span><span class="p">],</span> <span class="n">P</span><span class="p">[:,</span> <span class="n">i</span><span class="p">],</span> <span class="n">Q</span><span class="p">,</span> <span class="n">R</span><span class="p">)</span>
+<span class="gp">... </span><span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Est error:&#39;</span><span class="p">,</span> <span class="p">(</span><span class="n">states</span> <span class="o">-</span> <span class="n">estim</span><span class="p">)</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">dim</span><span class="o">=-</span><span class="mi">1</span><span class="p">))</span>
+<span class="go">Est error: tensor([[13.5084,  1.3604,  1.1626,  0.4157,  0.7117],</span>
+<span class="go">                   [14.4350,  2.0775,  1.0280,  0.2258,  0.4144],</span>
+<span class="go">                   [16.8538,  2.4473,  1.1300,  0.5104,  0.1647]])</span>
+</pre></div>
+</div>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Implementation is based on Section 15.2 of this book</p>
+<ul class="simple">
+<li><p>Dan Simon, <a class="reference external" href="https://onlinelibrary.wiley.com/doi/book/10.1002/0470045345">Optimal State Estimation: Kalman, H∞, and Nonlinear Approaches</a>,
+Cleveland State University, 2006</p></li>
+</ul>
+</div>
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.module.PF.compute_cov">
+<span class="sig-name descname"><span class="pre">compute_cov</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">a</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">b</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Q</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/module/pf/#PF.compute_cov"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.module.PF.compute_cov" title="Permalink to this definition">¶</a></dt>
+<dd><p>Compute covariance of two set of variables.</p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.module.PF.forward">
+<span class="sig-name descname"><span class="pre">forward</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">y</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">u</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">P</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Q</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">R</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">t</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/module/pf/#PF.forward"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.module.PF.forward" title="Permalink to this definition">¶</a></dt>
+<dd><p>Performs one step estimation.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>x</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – estimated system state of previous step.</p></li>
+<li><p><strong>y</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – system observation at current step (measurement).</p></li>
+<li><p><strong>u</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – system input at current step.</p></li>
+<li><p><strong>P</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – state estimation covariance of previous step.</p></li>
+<li><p><strong>Q</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>, optional) – covariance of system transition model.
+Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+<li><p><strong>R</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>, optional) – covariance of system observation model.
+Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+<li><p><strong>t</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">int</span></code>, optional) – set system timestamp for estimation.
+If <code class="docutils literal notranslate"><span class="pre">None</span></code>, current system time is used. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>posteriori state and covariance estimation</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>list of <code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code></p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.module.PF.generate_particles">
+<span class="sig-name descname"><span class="pre">generate_particles</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">P</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/module/pf/#PF.generate_particles"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.module.PF.generate_particles" title="Permalink to this definition">¶</a></dt>
+<dd><p>Randomly generate particles</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>x</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – estimated system state of previous step</p></li>
+<li><p><strong>P</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – state estimation covariance of previous step</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>particles</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>list of <code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code></p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.module.PF.relative_likelihood">
+<span class="sig-name descname"><span class="pre">relative_likelihood</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">y</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">ye</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">R</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/module/pf/#PF.relative_likelihood"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.module.PF.relative_likelihood" title="Permalink to this definition">¶</a></dt>
+<dd><p>Compute the relative likelihood</p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.module.PF.resample_particles">
+<span class="sig-name descname"><span class="pre">resample_particles</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">q</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/module/pf/#PF.resample_particles"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.module.PF.resample_particles" title="Permalink to this definition">¶</a></dt>
+<dd><p>Resample the set of a posteriori particles</p>
+</dd></dl>
+
+</dd></dl>
+
+</section>
+
+
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+  <section id="pypose-module-system">
+<h1>pypose.module.System<a class="headerlink" href="#pypose-module-system" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.module.System">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.module.</span></span><span class="sig-name descname"><span class="pre">System</span></span><a class="reference internal" href="../../_modules/pypose/module/dynamics/#System"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.module.System" title="Permalink to this definition">¶</a></dt>
+<dd><p>The base class for a system dynamics model.</p>
+<p>In most of the cases, users only need to subclass a specific dynamic system,
+such as linear time invariant system <a class="reference internal" href="../pypose.module.LTI/#pypose.module.LTI" title="pypose.module.LTI"><code class="xref py py-meth docutils literal notranslate"><span class="pre">LTI()</span></code></a>, Linear Time-Variant <a class="reference internal" href="../pypose.module.LTV/#pypose.module.LTV" title="pypose.module.LTV"><code class="xref py py-meth docutils literal notranslate"><span class="pre">LTV()</span></code></a>,
+and a non-linear system <a class="reference internal" href="../pypose.module.NLS/#pypose.module.NLS" title="pypose.module.NLS"><code class="xref py py-meth docutils literal notranslate"><span class="pre">NLS()</span></code></a>.</p>
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.module.System.forward">
+<span class="sig-name descname"><span class="pre">forward</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">state</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">input</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/module/dynamics/#System.forward"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.module.System.forward" title="Permalink to this definition">¶</a></dt>
+<dd><p>Defines the computation performed at every call that advances the system by one time step.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The <a class="reference internal" href="#pypose.module.System.forward" title="pypose.module.System.forward"><code class="xref py py-obj docutils literal notranslate"><span class="pre">forward</span></code></a> method implicitly increments the time step via <a class="reference internal" href="#pypose.module.System.forward_hook" title="pypose.module.System.forward_hook"><code class="xref py py-obj docutils literal notranslate"><span class="pre">forward_hook</span></code></a>.
+<a class="reference internal" href="#pypose.module.System.state_transition" title="pypose.module.System.state_transition"><code class="xref py py-obj docutils literal notranslate"><span class="pre">state_transition</span></code></a> and <a class="reference internal" href="#pypose.module.System.observation" title="pypose.module.System.observation"><code class="xref py py-obj docutils literal notranslate"><span class="pre">observation</span></code></a> still accept time for the flexiblity
+such as time-varying system. One can directly access the current system time via the
+property <a class="reference internal" href="#pypose.module.System.systime" title="pypose.module.System.systime"><code class="xref py py-obj docutils literal notranslate"><span class="pre">systime</span></code></a> or <code class="xref py py-obj docutils literal notranslate"><span class="pre">_t</span></code>.</p>
+</div>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>To introduce noise in a model, redefine this method via
+subclassing. See example in <code class="docutils literal notranslate"><span class="pre">examples/module/ekf/tank_robot.py</span></code>.</p>
+</div>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.module.System.forward_hook">
+<span class="sig-name descname"><span class="pre">forward_hook</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">module</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">inputs</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">outputs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/module/dynamics/#System.forward_hook"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.module.System.forward_hook" title="Permalink to this definition">¶</a></dt>
+<dd><p>Automatically advances the time step.</p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.module.System.observation">
+<span class="sig-name descname"><span class="pre">observation</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">state</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">input</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">t</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/module/dynamics/#System.observation"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.module.System.observation" title="Permalink to this definition">¶</a></dt>
+<dd><dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>state</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – The state of the dynamical system</p></li>
+<li><p><strong>input</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – The input to the dynamical system</p></li>
+<li><p><strong>t</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – The time step of the dynamical system.  Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>The observation of the system at the current step</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>Tensor</p>
+</dd>
+</dl>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The users need to define this method and can access the current system time via the
+property <a class="reference internal" href="#pypose.module.System.systime" title="pypose.module.System.systime"><code class="xref py py-obj docutils literal notranslate"><span class="pre">systime</span></code></a>. Don’t introduce system transision noise in this function,
+as it will be used for linearizing the system automaticalluy.</p>
+</div>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.module.System.set_refpoint">
+<span class="sig-name descname"><span class="pre">set_refpoint</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">state</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">input</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">t</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/module/dynamics/#System.set_refpoint"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.module.System.set_refpoint" title="Permalink to this definition">¶</a></dt>
+<dd><p>Function to set the reference point for linearization.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>state</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – The reference state of the dynamical system. If <code class="docutils literal notranslate"><span class="pre">None</span></code>,
+the the most recent state is taken. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+<li><p><strong>input</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – The reference input to the dynamical system. If <code class="docutils literal notranslate"><span class="pre">None</span></code>,
+the the most recent input is taken. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+<li><p><strong>t</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – The reference time step of the dynamical system. If <code class="docutils literal notranslate"><span class="pre">None</span></code>,
+the the most recent timestamp is taken. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>The <code class="docutils literal notranslate"><span class="pre">self</span></code> module.</p>
+</dd>
+</dl>
+<div class="admonition warning">
+<p class="admonition-title">Warning</p>
+<p>For nonlinear systems, the users have to call this function before getting the
+linearized system.</p>
+</div>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.module.System.state_transition">
+<span class="sig-name descname"><span class="pre">state_transition</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">state</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">input</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">t</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/module/dynamics/#System.state_transition"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.module.System.state_transition" title="Permalink to this definition">¶</a></dt>
+<dd><dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>state</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – The state of the dynamical system</p></li>
+<li><p><strong>input</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – The input to the dynamical system</p></li>
+<li><p><strong>t</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – The time step of the dynamical system.  Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>The state of the system at next time step</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>Tensor</p>
+</dd>
+</dl>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The users need to define this method and can access the current time via the property
+<a class="reference internal" href="#pypose.module.System.systime" title="pypose.module.System.systime"><code class="xref py py-obj docutils literal notranslate"><span class="pre">systime</span></code></a>. Don’t introduce system transision noise in this function, as it will
+be used for linearizing the system automaticalluy.</p>
+</div>
+</dd></dl>
+
+<dl class="py property">
+<dt class="sig sig-object py" id="pypose.module.System.systime">
+<em class="property"><span class="pre">property</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">systime</span></span><a class="headerlink" href="#pypose.module.System.systime" title="Permalink to this definition">¶</a></dt>
+<dd><p>System time, automatically advanced by <a class="reference internal" href="#pypose.module.System.forward_hook" title="pypose.module.System.forward_hook"><code class="xref py py-obj docutils literal notranslate"><span class="pre">forward_hook</span></code></a>.</p>
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new file mode 100644
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+  <section id="pypose-module-ukf">
+<h1>pypose.module.UKF<a class="headerlink" href="#pypose-module-ukf" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.module.UKF">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.module.</span></span><span class="sig-name descname"><span class="pre">UKF</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">model</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Q</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">R</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">msqrt</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">correct_PD_eps</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">-</span> <span class="pre">1</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/module/ukf/#UKF"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.module.UKF" title="Permalink to this definition">¶</a></dt>
+<dd><p>Performs Batched Unscented Kalman Filter (UKF).</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>model</strong> (<a class="reference internal" href="../pypose.module.System/#pypose.module.System" title="pypose.module.System"><code class="xref py py-obj docutils literal notranslate"><span class="pre">System</span></code></a>) – The system model to be estimated, a subclass of
+<a class="reference internal" href="../pypose.module.NLS/#pypose.module.NLS" title="pypose.module.NLS"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.module.NLS</span></code></a>.</p></li>
+<li><p><strong>Q</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>, optional) – The covariance matrices of system transition noise.
+Ignored if provided during each iteration. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code></p></li>
+<li><p><strong>R</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>, optional) – The covariance matrices of system observation noise.
+Ignored if provided during each iteration. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code></p></li>
+<li><p><strong>msqrt</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Function</span></code>, optional) – The method for computing the square root of matrices.
+Use <code class="docutils literal notranslate"><span class="pre">torch.linalg.cholesky</span></code> if not provided. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code></p></li>
+<li><p><strong>correct_PD_eps</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Float</span></code>, optional) – The eps used in positive definite correction of the state covariance matrices P.
+If provided a small positive number, P will be projected to be positive definite by clamping its smaller eigenvalues to eps.
+If provided a negative number, no correction will be made. Default: <code class="docutils literal notranslate"><span class="pre">-1</span></code></p></li>
+</ul>
+</dd>
+</dl>
+<p>A non-linear system can be described as</p>
+<div class="math">
+\[\begin{aligned}
+    \mathbf{x}_{k+1} &= \mathbf{f}(\mathbf{x}_k, \mathbf{u}_k, t_k) + \mathbf{w}_k,
+    \quad \mathbf{w}_k \sim \mathcal{N}(\mathbf{0}, \mathbf{Q})  \\
+    \mathbf{y}_{k} &= \mathbf{g}(\mathbf{x}_k, \mathbf{u}_k, t_k) + \mathbf{v}_k,
+    \quad \mathbf{v}_k \sim \mathcal{N}(\mathbf{0}, \mathbf{R})
+\end{aligned}
+
+\]</div>
+<p>UKF can be described as the following equations, where the subscript <span class="math">\(\cdot_{k}\)</span>
+is omited for simplicity.</p>
+<ol class="arabic simple">
+<li><p>Sigma Points.</p></li>
+</ol>
+<blockquote>
+<div><div class="math">
+\[\begin{aligned}
+     & \mathbf{x}^{\left ( i \right ) } = \mathbf{x}^{+} +
+         \mathbf{\check{x}}^{\left(i \right)}, & \quad i=0,...,2n\\
+     & \mathbf{\check{x}}^{\left (i \right)} = \mathbf{0}, & \quad i=0 \\
+     & \mathbf{\check{x}}^{\left (i \right)} = \left(\sqrt{(n+\kappa)P} \right)_{i}^{T},
+      & \quad i= 1, ..., n \\
+     & \mathbf{\check{x}}^{\left(i \right)} = -\left(\sqrt{(n+\kappa)P} \right)_{i}^{T},
+          & \quad i=n+1, ..., 2n \\
+\end{aligned}
+
+\]</div>
+</div></blockquote>
+<p>where <span class="math">\(\left(\sqrt{(n+\kappa)P}\right) _{i}\)</span> is the <span class="math">\(i\)</span>-th row of
+<span class="math">\(\left(\sqrt{(n+\kappa)P}\right)\)</span> and <span class="math">\(n\)</span> is the dimension of state <span class="math">\(\mathbf{x}\)</span>.</p>
+<p>Their weighting cofficients are given as</p>
+<blockquote>
+<div><div class="math">
+\[\begin{aligned}
+    &W^{(0)} = \frac{\kappa}{n+\kappa}, & \quad i = 0\\
+    &W^{(i)} = \frac{1}{2(n+\kappa)}, & \quad i = 1,...,2n\\
+\end{aligned}
+
+\]</div>
+</div></blockquote>
+<ol class="arabic simple" start="2">
+<li><p>Priori State Estimation.</p></li>
+</ol>
+<blockquote>
+<div><div class="math">
+\[\mathbf{x}^{-} = \sum_{i=0}^{2n} W^{(i)} f(\mathbf{x}^{(i)}, \mathbf{u}, t)
+
+\]</div>
+</div></blockquote>
+<ol class="arabic simple" start="3">
+<li><p>Priori Covariance.</p></li>
+</ol>
+<blockquote>
+<div><div class="math">
+\[\mathbf{P}^{-} = \sum_{i=0}^{2n} W^{(i)}
+    \left(\mathbf{x}^{(i)} - \mathbf{x}^{-} \right)
+    \left(\mathbf{x}^{(i)} - \mathbf{x}^- \right)^T + \mathbf{Q}
+
+\]</div>
+</div></blockquote>
+<ol class="arabic simple" start="4">
+<li><p>Observational Estimation.</p></li>
+</ol>
+<blockquote>
+<div><div class="math">
+\[\begin{aligned}
+    & \mathbf{y}^{(i)} = g \left( \mathbf{x}^{(i)}, \mathbf{u}, t \right),
+        & \quad i = 0, \cdots, 2n \\
+    & \bar{\mathbf{y}} = \sum_{i=0}^{2n} W^{(i)} \mathbf{y}^{(i)}
+\end{aligned}
+
+\]</div>
+</div></blockquote>
+<ol class="arabic simple" start="5">
+<li><p>Observational Covariance.</p></li>
+</ol>
+<blockquote>
+<div><div class="math">
+\[\mathbf{P}_{y} = \sum_{i=0}^{2n} W^{(i)}
+    \left(\mathbf{y}^{(i)} - \bar{\mathbf{y}} \right)
+    \left( \mathbf{y}^{(i)} - \bar{\mathbf{y}} \right)^T + \mathbf{R}
+
+\]</div>
+</div></blockquote>
+<ol class="arabic simple" start="6">
+<li><p>Priori and Observation Covariance:</p></li>
+</ol>
+<blockquote>
+<div><div class="math">
+\[\mathbf{P}_{xy} = \sum_{i=0}^{2n} W^{(i)}
+    \left( \mathbf{x}^{(i)} - \mathbf{x}^- \right)
+    \left( \mathbf{y}^{(i)} - \bar{\mathbf{y}} \right)^T
+
+\]</div>
+</div></blockquote>
+<ol class="arabic simple" start="7">
+<li><p>Kalman Gain</p></li>
+</ol>
+<blockquote>
+<div><div class="math">
+\[\mathbf{K} = \mathbf{P}_{xy}\mathbf{P}_{y}^{-1}
+
+\]</div>
+</div></blockquote>
+<ol class="arabic simple" start="8">
+<li><p>Posteriori State Estimation.</p></li>
+</ol>
+<blockquote>
+<div><div class="math">
+\[\mathbf{x}^{+} = \mathbf{x}^{-} + \mathbf{K} (\mathbf{y}- \bar{\mathbf{y}})
+
+\]</div>
+</div></blockquote>
+<ol class="arabic simple" start="9">
+<li><p>Posteriori Covariance Estimation.</p></li>
+</ol>
+<blockquote>
+<div><div class="math">
+\[\mathbf{P} = \mathbf{P}^{-} - \mathbf{K}\mathbf{P}_{y}\mathbf{K}^T
+
+\]</div>
+</div></blockquote>
+<p>where superscript <span class="math">\(\cdot^{-}\)</span> and <span class="math">\(\cdot^{+}\)</span> denote the priori and
+posteriori estimation, respectively.</p>
+<p class="rubric">Example</p>
+<ol class="arabic simple">
+<li><p>Define a discrete-time non-linear system (NLS) model</p></li>
+</ol>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">torch</span><span class="o">,</span> <span class="nn">pypose</span> <span class="k">as</span> <span class="nn">pp</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="k">class</span> <span class="nc">NLS</span><span class="p">(</span><span class="n">pp</span><span class="o">.</span><span class="n">module</span><span class="o">.</span><span class="n">NLS</span><span class="p">):</span>
+<span class="gp">... </span>    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="gp">... </span>        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+<span class="gp">...</span>
+<span class="gp">... </span>    <span class="k">def</span> <span class="nf">state_transition</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">state</span><span class="p">,</span> <span class="nb">input</span><span class="p">,</span> <span class="n">t</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
+<span class="gp">... </span>        <span class="k">return</span> <span class="n">state</span><span class="o">.</span><span class="n">cos</span><span class="p">()</span> <span class="o">+</span> <span class="nb">input</span>
+<span class="gp">...</span>
+<span class="gp">... </span>    <span class="k">def</span> <span class="nf">observation</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">state</span><span class="p">,</span> <span class="nb">input</span><span class="p">,</span> <span class="n">t</span><span class="p">):</span>
+<span class="gp">... </span>        <span class="k">return</span> <span class="n">state</span><span class="o">.</span><span class="n">sin</span><span class="p">()</span> <span class="o">+</span> <span class="nb">input</span>
+</pre></div>
+</div>
+<ol class="arabic simple" start="2">
+<li><p>Create a model and filter</p></li>
+</ol>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">model</span> <span class="o">=</span> <span class="n">NLS</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">ukf</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">module</span><span class="o">.</span><span class="n">UKF</span><span class="p">(</span><span class="n">model</span><span class="p">)</span>
+</pre></div>
+</div>
+<ol class="arabic simple" start="3">
+<li><p>Prepare data</p></li>
+</ol>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">N</span> <span class="o">=</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">2</span> <span class="c1"># batchsize, steps, state dim</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">states</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">N</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">inputs</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">N</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">observ</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">N</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="c1"># std of transition, observation, and estimation</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">p</span> <span class="o">=</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mi">10</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">Q</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">N</span><span class="p">)</span><span class="o">.</span><span class="n">repeat</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">q</span><span class="o">**</span><span class="mi">2</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">R</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">N</span><span class="p">)</span><span class="o">.</span><span class="n">repeat</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">r</span><span class="o">**</span><span class="mi">2</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">N</span><span class="p">)</span><span class="o">.</span><span class="n">repeat</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">p</span><span class="o">**</span><span class="mi">2</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">estim</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">N</span><span class="p">)</span> <span class="o">*</span> <span class="n">p</span>
+</pre></div>
+</div>
+<ol class="arabic simple" start="4">
+<li><p>Perform UKF prediction. Note that estimation error becomes smaller with more steps.</p></li>
+</ol>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">T</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
+<span class="gp">... </span>    <span class="n">w</span> <span class="o">=</span> <span class="n">q</span> <span class="o">*</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">N</span><span class="p">)</span> <span class="c1"># transition noise</span>
+<span class="gp">... </span>    <span class="n">v</span> <span class="o">=</span> <span class="n">r</span> <span class="o">*</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">N</span><span class="p">)</span> <span class="c1"># observation noise</span>
+<span class="gp">... </span>    <span class="n">states</span><span class="p">[:,</span> <span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">],</span> <span class="n">observ</span><span class="p">[:,</span> <span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">model</span><span class="p">(</span><span class="n">states</span><span class="p">[:,</span> <span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="n">w</span><span class="p">,</span> <span class="n">inputs</span><span class="p">[:,</span> <span class="n">i</span><span class="p">])</span>
+<span class="gp">... </span>    <span class="n">estim</span><span class="p">[:,</span> <span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">],</span> <span class="n">P</span><span class="p">[:,</span> <span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">ukf</span><span class="p">(</span><span class="n">estim</span><span class="p">[:,</span> <span class="n">i</span><span class="p">],</span> <span class="n">observ</span><span class="p">[:,</span> <span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="n">v</span><span class="p">,</span> <span class="n">inputs</span><span class="p">[:,</span> <span class="n">i</span><span class="p">],</span> <span class="n">P</span><span class="p">[:,</span> <span class="n">i</span><span class="p">],</span> <span class="n">Q</span><span class="p">,</span> <span class="n">R</span><span class="p">)</span>
+<span class="gp">... </span><span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Est error:&#39;</span><span class="p">,</span> <span class="p">(</span><span class="n">states</span> <span class="o">-</span> <span class="n">estim</span><span class="p">)</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">dim</span><span class="o">=-</span><span class="mi">1</span><span class="p">))</span>
+<span class="go">Est error: tensor([[19.2216,  0.9972,  0.7375,  0.8644,  0.9604],</span>
+<span class="go">                   [16.6286,  2.1883,  1.7114,  0.8491,  1.2250],</span>
+<span class="go">                   [10.5133,  2.1080,  1.9851,  0.5087,  1.3921]])</span>
+</pre></div>
+</div>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Implementation is based on Section 14.3 of this book</p>
+<ul class="simple">
+<li><p>Dan Simon, <a class="reference external" href="https://onlinelibrary.wiley.com/doi/book/10.1002/0470045345">Optimal State Estimation: Kalman, H∞, and Nonlinear Approaches</a>,
+Cleveland State University, 2006</p></li>
+</ul>
+</div>
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.module.UKF.compute_cov">
+<span class="sig-name descname"><span class="pre">compute_cov</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">a</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">b</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">w</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Q</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/module/ukf/#UKF.compute_cov"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.module.UKF.compute_cov" title="Permalink to this definition">¶</a></dt>
+<dd><p>Compute covariance of two set of variables.</p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.module.UKF.forward">
+<span class="sig-name descname"><span class="pre">forward</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">y</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">u</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">P</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Q</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">R</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">t</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">k</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/module/ukf/#UKF.forward"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.module.UKF.forward" title="Permalink to this definition">¶</a></dt>
+<dd><p>Performs one step estimation.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>x</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – estimated system state of previous step.</p></li>
+<li><p><strong>y</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – system observation at current step (measurement).</p></li>
+<li><p><strong>u</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – system input at current step.</p></li>
+<li><p><strong>P</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – state estimation covariance of previous step.</p></li>
+<li><p><strong>Q</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>, optional) – covariance of system transition model.</p></li>
+<li><p><strong>R</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>, optional) – covariance of system observation model.</p></li>
+<li><p><strong>t</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>, optional) – timestamp for ths system to estimate. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+<li><p><strong>k</strong> (<code class="docutils literal notranslate"><span class="pre">int</span></code>, optional) – a parameter for weighting the sigma points.
+If <code class="docutils literal notranslate"><span class="pre">None</span></code> is given, then <code class="docutils literal notranslate"><span class="pre">3</span> <span class="pre">-</span> <span class="pre">n</span></code> is adopted (<code class="docutils literal notranslate"><span class="pre">n</span></code> is the state dimension),
+which is best for Gaussian noise model. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>posteriori state and covariance estimation</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>list of <code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code></p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.module.UKF.project_to_positive_definite">
+<span class="sig-name descname"><span class="pre">project_to_positive_definite</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">A</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/module/ukf/#UKF.project_to_positive_definite"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.module.UKF.project_to_positive_definite" title="Permalink to this definition">¶</a></dt>
+<dd><p>Project matrix to positive definite by clamping its eigenvalues to positive.</p>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.module.UKF.sigma_weight_points">
+<span class="sig-name descname"><span class="pre">sigma_weight_points</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">P</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">k</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/module/ukf/#UKF.sigma_weight_points"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.module.UKF.sigma_weight_points" title="Permalink to this definition">¶</a></dt>
+<dd><p>Compute sigma point and its weights</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>x</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – estimated system state of previous step</p></li>
+<li><p><strong>P</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – state estimation covariance of previous step</p></li>
+<li><p><strong>k</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">int</span></code>) – parameter for weighting sigma points.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>sigma points and their weights</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>tuple of <code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code></p>
+</dd>
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+          <div class="rst-content">
+          
+            <div role="main" class="main-content" itemscope="itemscope" itemtype="http://schema.org/Article">
+             <article itemprop="articleBody" id="pytorch-article" class="pytorch-article">
+              
+  <section id="pypose-mul">
+<h1>pypose.mul<a class="headerlink" href="#pypose-mul" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.mul">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">mul</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">other</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/basics/#mul"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.mul" title="Permalink to this definition">¶</a></dt>
+<dd><p>Multiply input LieTensor by other.</p>
+<div class="math">
+\[\bm{y}_i =
+\bm{x}_i \ast \bm{a}_i
+
+\]</div>
+<p>where <span class="math">\(\bm{x}\)</span> is the <code class="docutils literal notranslate"><span class="pre">input</span></code> LieTensor, <span class="math">\(\bm{a}\)</span> is the <code class="docutils literal notranslate"><span class="pre">other</span></code> Tensor or
+LieTensor, and <span class="math">\(\bm{y}\)</span> is the output.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>input</strong> (<a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code></a>) – the input LieTensor (Lie Group or Lie Algebra).</p></li>
+<li><p><strong>other</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Number</span></code>, <code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>, or <a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code></a>) – the value for input to be
+multiplied by.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>the product of <code class="docutils literal notranslate"><span class="pre">input</span></code> and <code class="docutils literal notranslate"><span class="pre">other</span></code>.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>/<a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code></a></p>
+</dd>
+</dl>
+<table class="docutils align-default" id="id1">
+<caption><span class="caption-text">List of <a class="reference internal" href="#pypose.mul" title="pypose.mul"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.mul</span></code></a> cases</span><a class="headerlink" href="#id1" title="Permalink to this table">¶</a></caption>
+<colgroup>
+<col style="width: 31%" />
+<col style="width: 38%" />
+<col style="width: 31%" />
+</colgroup>
+<thead>
+<tr class="row-odd"><th class="head"><p>input <a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code></a></p></th>
+<th class="head"><p>other</p></th>
+<th class="head"><p>output</p></th>
+</tr>
+</thead>
+<tbody>
+<tr class="row-even"><td><p>Lie Algebra</p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Number</span></code></p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Lie</span> <span class="pre">Algebra</span></code></p></td>
+</tr>
+<tr class="row-odd"><td><p>Lie Group</p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code> <span class="math">\(\in \mathbb{R^{*\times3}}\)</span></p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code></p></td>
+</tr>
+<tr class="row-even"><td><p>Lie Group</p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code> <span class="math">\(\in \mathbb{R^{*\times4}}\)</span></p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code></p></td>
+</tr>
+<tr class="row-odd"><td><p>Lie Group</p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Lie</span> <span class="pre">Group</span></code></p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Lie</span> <span class="pre">Group</span></code></p></td>
+</tr>
+</tbody>
+</table>
+<p>When multiplying a Lie Group by another Lie Group, they must have the
+same Lie type.</p>
+<div class="admonition warning">
+<p class="admonition-title">Warning</p>
+<p>For better readability, we suggest calling <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">*</span> <span class="pre">b</span></code> or <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">&#64;</span> <span class="pre">b</span></code> instead of
+<code class="docutils literal notranslate"><span class="pre">a.mul(b)</span></code> or <code class="docutils literal notranslate"><span class="pre">pypose.mul(a,</span> <span class="pre">b)</span></code>, which have the same behaviors.</p>
+</div>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<ul class="simple">
+<li><p>When <code class="docutils literal notranslate"><span class="pre">other</span></code> is a Tensor, this operator is equivalent to <a class="reference internal" href="../pypose.Act/#pypose.Act" title="pypose.Act"><code class="xref py py-meth docutils literal notranslate"><span class="pre">Act()</span></code></a>.</p></li>
+<li><p>When <code class="docutils literal notranslate"><span class="pre">other</span></code> is a number and <code class="docutils literal notranslate"><span class="pre">input</span></code> is a Lie Algebra, this operator performs
+simple element-wise multiplication.</p></li>
+<li><p>When <code class="docutils literal notranslate"><span class="pre">input</span></code> is a Lie Group, more details are shown below.</p></li>
+</ul>
+</div>
+<ul>
+<li><p>Input <span class="math">\(\bm{x}\)</span>’s <code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code> is <code class="xref py py-obj docutils literal notranslate"><span class="pre">SO3_type</span></code>
+(input <span class="math">\(\bm{x}\)</span> is an instance of <a class="reference internal" href="../pypose.SO3/#pypose.SO3" title="pypose.SO3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">SO3()</span></code></a>):</p>
+<div class="math">
+\[{\displaystyle \bm{y}_i={\begin{bmatrix}
+      q_i^wq_i^{w'} - q_i^xq_i^{x'} -q_i^yq_i^{y'} - q_i^zq_i^{z'}\\
+      q_i^wq_i^{x'} + q_i^xq_i^{w'} +q_i^yq_i^{z'} - q_i^zq_i^{y'}\\
+      q_i^wq_i^{y'} - q_i^xq_i^{z'} +q_i^yq_i^{w'} + q_i^zq_i^{x'}\\
+      q_i^wq_i^{z'} + q_i^xq_i^{y'} -q_i^yq_i^{x'} + q_i^zq_i^{w'}
+      \end{bmatrix}
+}^T},
+
+\]</div>
+<p>where <span class="math">\(\bm{x}_i = [q_i^x, q_i^y, q_i^z, q_i^w]\)</span> and
+<span class="math">\(\bm{a}_i = [q_i^{x'}, q_i^{y'}, q_i^{z'}, q_i^{w'}]\)</span> are the <code class="docutils literal notranslate"><span class="pre">input</span></code>
+and <code class="docutils literal notranslate"><span class="pre">other</span></code> LieTensor, respectively.</p>
+</li>
+</ul>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p><span class="math">\(\mathbf{y}_i\)</span> can be simply derived by taking the complex number multiplication.</p>
+<div class="math">
+\[\bm{y}_i =
+(q_i^x\mathbf{i} + q_i^y\mathbf{j} + q_i^z\mathbf{k} + q_i^w) \ast
+(q_i^{x'}\mathbf{i} + q_i^{y'}\mathbf{j} + q_i^{z'}\mathbf{k} + q_i^{w'}),
+
+\]</div>
+<p>where and <span class="math">\(\mathbf{i}\)</span> <span class="math">\(\mathbf{j}\)</span>, and <span class="math">\(\mathbf{k}\)</span> are the imaginary
+units.</p>
+<div class="math">
+\[\mathbf{i}^2 = \mathbf{j}^2 = \mathbf{k}^2 = \mathbf{ijk} = -1
+
+\]</div>
+</div>
+<ul>
+<li><p>Input <span class="math">\(\bm{x}\)</span>’s <code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code> is <code class="xref py py-obj docutils literal notranslate"><span class="pre">SE3_type</span></code>
+(input <span class="math">\(\bm{x}\)</span> is an instance of <a class="reference internal" href="../pypose.SE3/#pypose.SE3" title="pypose.SE3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">SE3()</span></code></a>):</p>
+<blockquote>
+<div><div class="math">
+\[\bm{y}_i = [\mathbf{q}_i * \mathbf{t}_i' + \mathbf{t}_i,
+            \mathbf{q}_i * \mathbf{q}_i']
+
+\]</div>
+<p>where <span class="math">\(\bm{x}_i = [\mathbf{t}_i, \mathbf{q}_i]\)</span> and
+<span class="math">\(\bm{a}_i = [\mathbf{t}_i', \mathbf{q}_i']\)</span> are the <code class="docutils literal notranslate"><span class="pre">input</span></code> and <code class="docutils literal notranslate"><span class="pre">other</span></code>
+LieTensor, respectively; <span class="math">\(\mathbf{t}_i\)</span>, <span class="math">\(\mathbf{t}_i'\)</span> and
+<span class="math">\(\mathbf{q}_i\)</span>, <span class="math">\(\mathbf{q}_i'\)</span> are their translation and <a class="reference internal" href="../pypose.SO3/#pypose.SO3" title="pypose.SO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">SO3</span></code></a>
+parts, respectively; the operator <span class="math">\(\ast\)</span> denotes the obj:<cite>SO3_type</cite>
+multiplication introduced above.</p>
+</div></blockquote>
+</li>
+<li><p>Input <span class="math">\(\bm{x}\)</span>’s <code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code> is <code class="xref py py-obj docutils literal notranslate"><span class="pre">RxSO3_type</span></code>
+(input <span class="math">\(\bm{x}\)</span> is an instance of <a class="reference internal" href="../pypose.RxSO3/#pypose.RxSO3" title="pypose.RxSO3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">RxSO3()</span></code></a>)</p>
+<blockquote>
+<div><div class="math">
+\[\bm{y}_i = [\mathbf{q}_i * \mathbf{q}_i', s_is_i']
+
+\]</div>
+<p>where <span class="math">\(s_i\)</span> and <span class="math">\(s_i'\)</span> are the scale parts of the <code class="docutils literal notranslate"><span class="pre">input</span></code> and
+<code class="docutils literal notranslate"><span class="pre">other</span></code> LieTensor, respectively.</p>
+</div></blockquote>
+</li>
+<li><p>Input <span class="math">\(\bm{x}\)</span>’s <code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code> is <code class="xref py py-obj docutils literal notranslate"><span class="pre">Sim3_type</span></code>
+(input <span class="math">\(\bm{x}\)</span> is an instance of <a class="reference internal" href="../pypose.Sim3/#pypose.Sim3" title="pypose.Sim3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">Sim3()</span></code></a>):</p>
+<blockquote>
+<div><div class="math">
+\[\bm{y}_i = [\mathbf{q}_i * \mathbf{t}_i' + \mathbf{t}_i,
+                \mathbf{q}_i * \mathbf{q}_i', s_is_i']
+
+\]</div>
+<p>where <span class="math">\(\bm{x}_i = [\mathbf{t}_i, \mathbf{q}_i, s_i]\)</span> and
+<span class="math">\(\bm{a}_i = [\mathbf{t}_i', \mathbf{q}_i', s_i']\)</span> are the <code class="docutils literal notranslate"><span class="pre">input</span></code> and
+<code class="docutils literal notranslate"><span class="pre">other</span></code> LieTensor, respectively.</p>
+</div></blockquote>
+</li>
+</ul>
+<p class="rubric">Examples</p>
+<ul>
+<li><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Lie</span> <span class="pre">Algebra</span></code> <span class="math">\(*\)</span> <code class="xref py py-obj docutils literal notranslate"><span class="pre">Number</span></code> <span class="math">\(\mapsto\)</span> <code class="xref py py-obj docutils literal notranslate"><span class="pre">Lie</span> <span class="pre">Algebra</span></code></p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_so3</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span>
+<span class="go">so3Type LieTensor:</span>
+<span class="go">LieTensor([ 0.3018, -1.0246,  0.7784])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="mi">5</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="c1"># The following two operations are equivalent.</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">*</span> <span class="n">a</span>
+<span class="go">so3Type LieTensor:</span>
+<span class="go">LieTensor([ 1.5090, -5.1231,  3.8919])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">mul</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
+<span class="go">so3Type LieTensor:</span>
+<span class="go">LieTensor([ 1.5090, -5.1231,  3.8919])</span>
+</pre></div>
+</div>
+</li>
+<li><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Lie</span> <span class="pre">Group</span></code> <span class="math">\(*\)</span> <code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code> <span class="math">\(\mapsto\)</span> <code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code></p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_SO3</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">a</span>
+<span class="go">(SO3Type LieTensor:</span>
+<span class="go">LieTensor([ 0.6047, -0.2129, -0.1781,  0.7465]),</span>
+<span class="go">tensor([-0.1811, -0.2278, -1.9956]))</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">*</span> <span class="n">a</span>
+<span class="go">tensor([ 0.9089,  1.6984, -0.5969])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span>
+<span class="go">tensor([ 1.5236, -1.2757, -0.7140,  0.2467])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">*</span> <span class="n">a</span>
+<span class="go">tensor([ 1.6588, -0.4687, -1.2196,  0.2467]</span>
+</pre></div>
+</div>
+</li>
+<li><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">SO3_type</span></code> <span class="math">\(*\)</span> <code class="xref py py-obj docutils literal notranslate"><span class="pre">SO3_type</span></code> <span class="math">\(\mapsto\)</span> <code class="xref py py-obj docutils literal notranslate"><span class="pre">SO3_type</span></code></p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_SO3</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span>
+<span class="go">SO3Type LieTensor:</span>
+<span class="go">LieTensor([ 0.0118, -0.7042, -0.4516,  0.5478])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">*</span> <span class="n">a</span>
+<span class="go">SO3Type LieTensor:</span>
+<span class="go">LieTensor([ 0.3108, -0.3714, -0.8579,  0.1715])</span>
+</pre></div>
+</div>
+</li>
+<li><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">SE3_type</span></code> <span class="math">\(*\)</span> <code class="xref py py-obj docutils literal notranslate"><span class="pre">SE3_type</span></code> <span class="math">\(\mapsto\)</span> <code class="xref py py-obj docutils literal notranslate"><span class="pre">SE3_type</span></code></p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_SE3</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_SE3</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">a</span>
+<span class="go">(SE3Type LieTensor:</span>
+<span class="go">LieTensor([ 0.7819,  1.8541, -0.2857, -0.1970,  0.4742,  0.1109,  0.8509]),</span>
+<span class="go">SE3Type LieTensor:</span>
+<span class="go">LieTensor([ 0.6039, -1.4076,  0.3496,  0.7297,  0.3971,  0.2849,  0.4783]))</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">*</span> <span class="n">a</span>
+<span class="go">SE3Type LieTensor:</span>
+<span class="go">LieTensor([ 1.8949,  0.7456, -0.3104,  0.6177,  0.7017, -0.1287,  0.3308])</span>
+</pre></div>
+</div>
+</li>
+<li><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">RxSO3_type</span></code> <span class="math">\(*\)</span> <code class="xref py py-obj docutils literal notranslate"><span class="pre">RxSO3_type</span></code> <span class="math">\(\mapsto\)</span> <code class="xref py py-obj docutils literal notranslate"><span class="pre">RxSO3_type</span></code></p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_RxSO3</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_RxSO3</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">a</span>
+<span class="go">(RxSO3Type LieTensor:</span>
+<span class="go">LieTensor([-0.7518, -0.6481,  0.0933, -0.0775,  1.5791]),</span>
+<span class="go">RxSO3Type LieTensor:</span>
+<span class="go">LieTensor([ 0.2757,  0.3102, -0.4086,  0.8129,  0.6593]))</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">*</span> <span class="n">a</span>
+<span class="go">RxSO3Type LieTensor:</span>
+<span class="go">LieTensor([-0.3967, -0.8323,  0.0530,  0.3835,  1.0411])</span>
+</pre></div>
+</div>
+</li>
+<li><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Sim3_type</span></code> <span class="math">\(*\)</span> <code class="xref py py-obj docutils literal notranslate"><span class="pre">Sim3_type</span></code> <span class="math">\(\mapsto\)</span> <code class="xref py py-obj docutils literal notranslate"><span class="pre">Sim3_type</span></code></p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_Sim3</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_Sim3</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">a</span>
+<span class="go">(Sim3Type LieTensor:</span>
+<span class="go">LieTensor([-0.3439, -0.2309, -0.6571,  0.3170, -0.6594, -0.1100,  0.6728, 0.6296]),</span>
+<span class="go">Sim3Type LieTensor:</span>
+<span class="go">LieTensor([-0.7434,  1.8613, -2.1315,  0.7688, -0.0268,  0.0520,  0.6367, 1.7745]))</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">*</span> <span class="n">a</span>
+<span class="go">Sim3Type LieTensor:</span>
+<span class="go">LieTensor([ 0.5740,  1.3197, -0.2752,  0.6819, -0.5389,  0.4634,  0.1727, 1.1172])</span>
+</pre></div>
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+  <section id="pypose-optim-gn">
+<h1>pypose.optim.GN<a class="headerlink" href="#pypose-optim-gn" title="Permalink to this heading">¶</a></h1>
+<dl class="py attribute">
+<dt class="sig sig-object py" id="pypose.optim.GN">
+<span class="sig-prename descclassname"><span class="pre">pypose.optim.</span></span><span class="sig-name descname"><span class="pre">GN</span></span><a class="headerlink" href="#pypose.optim.GN" title="Permalink to this definition">¶</a></dt>
+<dd><p>alias of <a class="reference internal" href="../pypose.optim.GaussNewton/#pypose.optim.GaussNewton" title="pypose.optim.optimizer.GaussNewton"><code class="xref py py-class docutils literal notranslate"><span class="pre">GaussNewton</span></code></a></p>
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+  <section id="pypose-optim-gaussnewton">
+<h1>pypose.optim.GaussNewton<a class="headerlink" href="#pypose-optim-gaussnewton" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.optim.GaussNewton">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.optim.</span></span><span class="sig-name descname"><span class="pre">GaussNewton</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">model</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">solver</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">kernel</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">corrector</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">weight</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">vectorize</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/optim/optimizer/#GaussNewton"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.optim.GaussNewton" title="Permalink to this definition">¶</a></dt>
+<dd><p>The Gauss-Newton (GN) algorithm solving non-linear least squares problems. This implementation
+is for optimizing the model parameters to approximate the target, which can be a
+Tensor/LieTensor or a tuple of Tensors/LieTensors.</p>
+<div class="math">
+\[\bm{\theta}^* = \arg\min_{\bm{\theta}} \sum_i
+    \rho\left((\bm{f}(\bm{\theta},\bm{x}_i)-\bm{y}_i)^T \mathbf{W}_i
+    (\bm{f}(\bm{\theta},\bm{x}_i)-\bm{y}_i)\right),
+
+\]</div>
+<p>where <span class="math">\(\bm{f}()\)</span> is the model, <span class="math">\(\bm{\theta}\)</span> is the parameters to be optimized,
+<span class="math">\(\bm{x}\)</span> is the model input, <span class="math">\(\mathbf{W}_i\)</span> is a weighted square matrix (positive
+definite), and <span class="math">\(\rho\)</span> is a robust kernel function to reduce the effect of outliers.
+<span class="math">\(\rho(x) = x\)</span> is used by default.</p>
+<div class="math">
+\[\begin{aligned}
+     &\rule{113mm}{0.4pt}                                                                 \\
+     &\textbf{input}: \bm{\theta}_0~\text{(params)}, \bm{f}~\text{(model)},
+         \bm{x}~(\text{input}), \bm{y}~(\text{target}), \rho~(\text{kernel})              \\
+     &\rule{113mm}{0.4pt}                                                                 \\
+     &\textbf{for} \: t=1 \: \textbf{to} \: \ldots \: \textbf{do}                         \\
+     &\hspace{5mm} \mathbf{J} \leftarrow {\dfrac {\partial \bm{f} } {\partial \bm{\theta}_{t-1}}}             \\
+     &\hspace{5mm} \mathbf{R} = \bm{f(\bm{\theta}_{t-1}, \bm{x})}-\bm{y}                  \\
+     &\hspace{5mm} \mathbf{R}, \mathbf{J}=\mathrm{corrector}(\rho, \mathbf{R}, \mathbf{J})\\
+     &\hspace{5mm} \bm{\delta} = \mathrm{solver}(\mathbf{W}\mathbf{J}, -\mathbf{W}\mathbf{R})                 \\
+     &\hspace{5mm} \bm{\theta}_t \leftarrow \bm{\theta}_{t-1} + \bm{\delta}               \\
+     &\rule{113mm}{0.4pt}                                                          \\[-1.ex]
+     &\bf{return} \:  \theta_t                                                     \\[-1.ex]
+     &\rule{113mm}{0.4pt}                                                          \\[-1.ex]
+\end{aligned}
+
+\]</div>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>model</strong> (<em>nn.Module</em>) – a module containing learnable parameters.</p></li>
+<li><p><strong>solver</strong> (<em>nn.Module</em><em>, </em><em>optional</em>) – a linear solver. Available linear solvers include
+<a class="reference internal" href="../pypose.optim.solver.PINV/#pypose.optim.solver.PINV" title="pypose.optim.solver.PINV"><code class="xref py py-meth docutils literal notranslate"><span class="pre">solver.PINV()</span></code></a> and <a class="reference internal" href="../pypose.optim.solver.LSTSQ/#pypose.optim.solver.LSTSQ" title="pypose.optim.solver.LSTSQ"><code class="xref py py-meth docutils literal notranslate"><span class="pre">solver.LSTSQ()</span></code></a>. If <code class="docutils literal notranslate"><span class="pre">None</span></code>, <a class="reference internal" href="../pypose.optim.solver.PINV/#pypose.optim.solver.PINV" title="pypose.optim.solver.PINV"><code class="xref py py-meth docutils literal notranslate"><span class="pre">solver.PINV()</span></code></a> is used.
+Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+<li><p><strong>kernel</strong> (nn.Module, or <code class="xref py py-obj docutils literal notranslate"><span class="pre">list</span></code>, optional) – the robust kernel function. If a <code class="xref py py-obj docutils literal notranslate"><span class="pre">list</span></code>,
+the element must be nn.Module or <code class="docutils literal notranslate"><span class="pre">None</span></code> and the length must be 1 or the number of residuals.
+Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+<li><p><strong>corrector</strong> – (nn.Module, or <code class="xref py py-obj docutils literal notranslate"><span class="pre">list</span></code>, optional): the Jacobian and model residual corrector to
+fit the kernel function. If a <code class="xref py py-obj docutils literal notranslate"><span class="pre">list</span></code>, the element must be nn.Module or <code class="docutils literal notranslate"><span class="pre">None</span></code> and
+the length must be 1 or the number of residuals.
+If a kernel is given but a corrector is not specified, auto correction is
+used. Auto correction can be unstable when the robust model has indefinite Hessian.
+Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+<li><p><strong>weight</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>, or <code class="xref py py-obj docutils literal notranslate"><span class="pre">list</span></code>, optional) – the square positive definite matrix defining
+the weight of model residual. If a <code class="xref py py-obj docutils literal notranslate"><span class="pre">list</span></code>, the element must be <code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code> and
+the length must be equal to the number of residuals.
+The corresponding residual and weight should be
+<a class="reference external" href="https://pytorch.org/docs/stable/notes/broadcasting.html#broadcasting-semantics">broadcastable</a>.
+For example, if the shape of a residual is <code class="docutils literal notranslate"><span class="pre">B*M*N*R</span></code>, the shape of its weight can
+be <code class="docutils literal notranslate"><span class="pre">R*R</span></code>, <code class="docutils literal notranslate"><span class="pre">N*R*R</span></code>, <code class="docutils literal notranslate"><span class="pre">M*N*R*R</span></code> or <code class="docutils literal notranslate"><span class="pre">B*M*N*R*R</span></code>.
+Use this only when all inputs shared the same weight matrices. This is
+ignored when weight is given when calling <a class="reference internal" href="#pypose.optim.GaussNewton.step" title="pypose.optim.GaussNewton.step"><code class="xref py py-meth docutils literal notranslate"><span class="pre">step()</span></code></a> or <a class="reference internal" href="../pypose.optim.scheduler.StopOnPlateau/#pypose.optim.scheduler.StopOnPlateau.optimize" title="pypose.optim.scheduler.StopOnPlateau.optimize"><code class="xref py py-meth docutils literal notranslate"><span class="pre">optimize()</span></code></a> method.
+Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+<li><p><strong>vectorize</strong> (<em>bool</em><em>, </em><em>optional</em>) – the method of computing Jacobian. If <code class="docutils literal notranslate"><span class="pre">True</span></code>, the
+gradient of each scalar in output with respect to the model parameters will be
+computed in parallel with <code class="docutils literal notranslate"><span class="pre">&quot;reverse-mode&quot;</span></code>. More details go to
+<a class="reference internal" href="../pypose.optim.functional.modjac/#pypose.optim.functional.modjac" title="pypose.optim.functional.modjac"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.optim.functional.modjac()</span></code></a>. Default: <code class="docutils literal notranslate"><span class="pre">True</span></code>.</p></li>
+</ul>
+</dd>
+</dl>
+<p>Available solvers: <a class="reference internal" href="../pypose.optim.solver.PINV/#pypose.optim.solver.PINV" title="pypose.optim.solver.PINV"><code class="xref py py-meth docutils literal notranslate"><span class="pre">solver.PINV()</span></code></a>; <a class="reference internal" href="../pypose.optim.solver.LSTSQ/#pypose.optim.solver.LSTSQ" title="pypose.optim.solver.LSTSQ"><code class="xref py py-meth docutils literal notranslate"><span class="pre">solver.LSTSQ()</span></code></a>.</p>
+<p>Available kernels: <a class="reference internal" href="../pypose.optim.kernel.Huber/#pypose.optim.kernel.Huber" title="pypose.optim.kernel.Huber"><code class="xref py py-meth docutils literal notranslate"><span class="pre">kernel.Huber()</span></code></a>; <a class="reference internal" href="../pypose.optim.kernel.PseudoHuber/#pypose.optim.kernel.PseudoHuber" title="pypose.optim.kernel.PseudoHuber"><code class="xref py py-meth docutils literal notranslate"><span class="pre">kernel.PseudoHuber()</span></code></a>; <a class="reference internal" href="../pypose.optim.kernel.Cauchy/#pypose.optim.kernel.Cauchy" title="pypose.optim.kernel.Cauchy"><code class="xref py py-meth docutils literal notranslate"><span class="pre">kernel.Cauchy()</span></code></a>.</p>
+<p>Available correctors: <a class="reference internal" href="../pypose.optim.corrector.FastTriggs/#pypose.optim.corrector.FastTriggs" title="pypose.optim.corrector.FastTriggs"><code class="xref py py-meth docutils literal notranslate"><span class="pre">corrector.FastTriggs()</span></code></a>; <a class="reference internal" href="../pypose.optim.corrector.Triggs/#pypose.optim.corrector.Triggs" title="pypose.optim.corrector.Triggs"><code class="xref py py-meth docutils literal notranslate"><span class="pre">corrector.Triggs()</span></code></a>.</p>
+<div class="admonition warning">
+<p class="admonition-title">Warning</p>
+<p>The output of model <span class="math">\(\bm{f}(\bm{\theta},\bm{x}_i)\)</span> and target <span class="math">\(\bm{y}_i\)</span>
+can be any shape, while their <strong>last dimension</strong> <span class="math">\(d\)</span> is always taken as the
+dimension of model residual, whose inner product is the input to the kernel function.
+This is useful for residuals like re-projection error, whose last dimension is 2.</p>
+<p>Note that <strong>auto correction</strong> is equivalent to the method of ‘square-rooting the kernel’
+mentioned in Section 3.3 of the following paper. It replaces the
+<span class="math">\(d\)</span>-dimensional residual with a one-dimensional one, which loses
+residual-level structural information.</p>
+<ul class="simple">
+<li><p>Christopher Zach, <a class="reference external" href="https://link.springer.com/chapter/10.1007/978-3-319-10602-1_50">Robust Bundle Adjustment Revisited</a>, European
+Conference on Computer Vision (ECCV), 2014.</p></li>
+</ul>
+<p><strong>Therefore, the users need to keep the last dimension of model output and target to
+1, even if the model residual is a scalar. If the users flatten all sample residuals
+into a vector (residual inner product will be a scalar), the model Jacobian will be a
+row vector, instead of a matrix, which loses sample-level structural information,
+although computing Jacobian vector is faster.</strong></p>
+</div>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Instead of solving <span class="math">\(\mathbf{J}^T\mathbf{J}\delta = -\mathbf{J}^T\mathbf{R}\)</span>, we solve
+<span class="math">\(\mathbf{J}\delta = -\mathbf{R}\)</span> via pseudo inversion, which is more numerically
+advisable. Therefore, only solvers with pseudo inversion (inverting non-square matrices)
+such as <a class="reference internal" href="../pypose.optim.solver.PINV/#pypose.optim.solver.PINV" title="pypose.optim.solver.PINV"><code class="xref py py-meth docutils literal notranslate"><span class="pre">solver.PINV()</span></code></a> and <a class="reference internal" href="../pypose.optim.solver.LSTSQ/#pypose.optim.solver.LSTSQ" title="pypose.optim.solver.LSTSQ"><code class="xref py py-meth docutils literal notranslate"><span class="pre">solver.LSTSQ()</span></code></a> are available.
+More details are in Eq. (5) of the paper “<a class="reference external" href="https://link.springer.com/chapter/10.1007/978-3-319-10602-1_50">Robust Bundle Adjustment Revisited</a>”.</p>
+</div>
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.optim.GaussNewton.step">
+<span class="sig-name descname"><span class="pre">step</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">target</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">weight</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/optim/optimizer/#GaussNewton.step"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.optim.GaussNewton.step" title="Permalink to this definition">¶</a></dt>
+<dd><p>Performs a single optimization step.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>input</strong> (<em>Tensor/LieTensor</em><em>, </em><em>tuple</em><em> or </em><em>a dict of Tensors/LieTensors</em>) – the input to the model.</p></li>
+<li><p><strong>target</strong> (<em>Tensor/LieTensor</em>) – the model target to approximate.
+If not given, the model output is minimized. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+<li><p><strong>weight</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>, or <code class="xref py py-obj docutils literal notranslate"><span class="pre">list</span></code>, optional) – the square positive definite matrix defining
+the weight of model residual. If a <code class="xref py py-obj docutils literal notranslate"><span class="pre">list</span></code>, the element must be <code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code> and
+the length must be equal to the number of residuals. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>the minimized model loss.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>Tensor</p>
+</dd>
+</dl>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Different from PyTorch optimizers like
+<a class="reference external" href="https://pytorch.org/docs/stable/generated/torch.optim.SGD.html">SGD</a>, where the model
+error has to be a scalar, the output of model <span class="math">\(\bm{f}\)</span> can be a Tensor/LieTensor or a
+tuple of Tensors/LieTensors.</p>
+<p>See more details of
+<a class="reference external" href="https://en.wikipedia.org/wiki/Gauss-Newton_algorithm">Gauss-Newton (GN) algorithm</a> on
+Wikipedia.</p>
+</div>
+<p class="rubric">Example</p>
+<p>Optimizing a simple module to <strong>approximate pose inversion</strong>.</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="k">class</span> <span class="nc">PoseInv</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+<span class="gp">... </span>    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">dim</span><span class="p">):</span>
+<span class="gp">... </span>        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+<span class="gp">... </span>        <span class="bp">self</span><span class="o">.</span><span class="n">pose</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">Parameter</span><span class="p">(</span><span class="n">pp</span><span class="o">.</span><span class="n">randn_se3</span><span class="p">(</span><span class="o">*</span><span class="n">dim</span><span class="p">))</span>
+<span class="gp">...</span>
+<span class="gp">... </span>    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">input</span><span class="p">):</span>
+<span class="gp">... </span>        <span class="c1"># the last dimension of the output is 6,</span>
+<span class="gp">... </span>        <span class="c1"># which will be the residual dimension.</span>
+<span class="gp">... </span>        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">pose</span><span class="o">.</span><span class="n">Exp</span><span class="p">()</span> <span class="o">@</span> <span class="nb">input</span><span class="p">)</span><span class="o">.</span><span class="n">Log</span><span class="p">()</span>
+<span class="gp">...</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">posinv</span> <span class="o">=</span> <span class="n">PoseInv</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="nb">input</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_SE3</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">optimizer</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">optim</span><span class="o">.</span><span class="n">GN</span><span class="p">(</span><span class="n">posinv</span><span class="p">)</span>
+<span class="gp">...</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">idx</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10</span><span class="p">):</span>
+<span class="gp">... </span>    <span class="n">error</span> <span class="o">=</span> <span class="n">optimizer</span><span class="o">.</span><span class="n">step</span><span class="p">(</span><span class="nb">input</span><span class="p">)</span>
+<span class="gp">... </span>    <span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Pose Inversion error </span><span class="si">%.7f</span><span class="s1"> @ </span><span class="si">%d</span><span class="s1"> it&#39;</span><span class="o">%</span><span class="p">(</span><span class="n">error</span><span class="p">,</span> <span class="n">idx</span><span class="p">))</span>
+<span class="gp">... </span>    <span class="k">if</span> <span class="n">error</span> <span class="o">&lt;</span> <span class="mf">1e-5</span><span class="p">:</span>
+<span class="gp">... </span>        <span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Early Stopping with error:&#39;</span><span class="p">,</span> <span class="n">error</span><span class="o">.</span><span class="n">item</span><span class="p">())</span>
+<span class="gp">... </span>        <span class="k">break</span>
+<span class="gp">...</span>
+<span class="go">Pose Inversion error: 1.6865690 @ 0 it</span>
+<span class="go">Pose Inversion error: 0.1065131 @ 1 it</span>
+<span class="go">Pose Inversion error: 0.0002673 @ 2 it</span>
+<span class="go">Pose Inversion error: 0.0000005 @ 3 it</span>
+<span class="go">Early Stopping with error: 5.21540641784668e-07</span>
+</pre></div>
+</div>
+</dd></dl>
+
+</dd></dl>
+
+</section>
+
+
+             </article>
+             
+            </div>
+            <footer>
+  
+    <div class="rst-footer-buttons" role="navigation" aria-label="footer navigation">
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+  <section id="pypose-optim-lm">
+<h1>pypose.optim.LM<a class="headerlink" href="#pypose-optim-lm" title="Permalink to this heading">¶</a></h1>
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+<dt class="sig sig-object py" id="pypose.optim.LM">
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+<dd><p>alias of <a class="reference internal" href="../pypose.optim.LevenbergMarquardt/#pypose.optim.LevenbergMarquardt" title="pypose.optim.optimizer.LevenbergMarquardt"><code class="xref py py-class docutils literal notranslate"><span class="pre">LevenbergMarquardt</span></code></a></p>
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+  <section id="pypose-optim-levenbergmarquardt">
+<h1>pypose.optim.LevenbergMarquardt<a class="headerlink" href="#pypose-optim-levenbergmarquardt" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.optim.LevenbergMarquardt">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.optim.</span></span><span class="sig-name descname"><span class="pre">LevenbergMarquardt</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">model</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">solver</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">strategy</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">kernel</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">corrector</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">weight</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">reject</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">16</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">min</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1e-06</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">max</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1e+32</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">vectorize</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/optim/optimizer/#LevenbergMarquardt"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.optim.LevenbergMarquardt" title="Permalink to this definition">¶</a></dt>
+<dd><p>The Levenberg-Marquardt (LM) algorithm solving non-linear least squares problems. It
+is also known as the damped least squares (DLS) method. This implementation is for
+optimizing the model parameters to approximate the target, which can be a
+Tensor/LieTensor or a tuple of Tensors/LieTensors.</p>
+<div class="math">
+\[\bm{\theta}^* = \arg\min_{\bm{\theta}} \sum_i
+    \rho\left((\bm{f}(\bm{\theta},\bm{x}_i)-\bm{y}_i)^T \mathbf{W}_i
+    (\bm{f}(\bm{\theta},\bm{x}_i)-\bm{y}_i)\right),
+
+\]</div>
+<p>where <span class="math">\(\bm{f}()\)</span> is the model, <span class="math">\(\bm{\theta}\)</span> is the parameters to be optimized,
+<span class="math">\(\bm{x}\)</span> is the model input, <span class="math">\(\mathbf{W}_i\)</span> is a weighted square matrix (positive
+definite), and <span class="math">\(\rho\)</span> is a robust kernel function to reduce the effect of outliers.
+<span class="math">\(\rho(x) = x\)</span> is used by default.</p>
+<div class="math">
+\[\begin{aligned}
+     &\rule{113mm}{0.4pt}                                                                 \\
+     &\textbf{input}: \lambda~\text{(damping)}, \bm{\theta}_0~\text{(params)},
+         \bm{f}~\text{(model)}, \bm{x}~(\text{input}), \bm{y}~(\text{target})             \\
+     &\hspace{12mm} \rho~(\text{kernel}), \epsilon_{s}~(\text{min}),
+                    \epsilon_{l}~(\text{max})                                             \\
+     &\rule{113mm}{0.4pt}                                                                 \\
+     &\textbf{for} \: t=1 \: \textbf{to} \: \ldots \: \textbf{do}                         \\
+     &\hspace{5mm} \mathbf{J} \leftarrow {\dfrac {\partial \bm{f}} {\partial \bm{\theta}_{t-1}}}       \\
+     &\hspace{5mm} \mathbf{A} \leftarrow (\mathbf{J}^T \mathbf{W} \mathbf{J})
+                              .\mathrm{diagnal\_clamp(\epsilon_{s}, \epsilon_{l})}        \\
+     &\hspace{5mm} \mathbf{R} = \bm{f(\bm{\theta}_{t-1}, \bm{x})}-\bm{y}                               \\
+     &\hspace{5mm} \mathbf{R}, \mathbf{J}=\mathrm{corrector}(\rho, \mathbf{R}, \mathbf{J})\\
+     &\hspace{5mm} \textbf{while}~\text{first iteration}~\textbf{or}~
+                                  \text{loss not decreasing}                              \\
+     &\hspace{10mm} \mathbf{A} \leftarrow \mathbf{A} + \lambda \mathrm{diag}(\mathbf{A})  \\
+     &\hspace{10mm} \bm{\delta} = \mathrm{solver}(\mathbf{A}, -\mathbf{J}^T \mathbf{W} \mathbf{R})     \\
+     &\hspace{10mm} \lambda \leftarrow \mathrm{strategy}(\lambda,\text{model information})\\
+     &\hspace{10mm} \bm{\theta}_t \leftarrow \bm{\theta}_{t-1} + \bm{\delta}              \\
+     &\hspace{10mm} \textbf{if}~\text{loss not decreasing}~\textbf{and}~
+                                \text{maximum reject step not reached}                    \\
+     &\hspace{15mm} \bm{\theta}_t \leftarrow \bm{\theta}_{t-1} - \bm{\delta}
+                                                        ~(\text{reject step})             \\
+     &\rule{113mm}{0.4pt}                                                          \\[-1.ex]
+     &\bf{return} \:  \theta_t                                                     \\[-1.ex]
+     &\rule{113mm}{0.4pt}                                                          \\[-1.ex]
+\end{aligned}
+
+\]</div>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>model</strong> (<em>nn.Module</em>) – a module containing learnable parameters.</p></li>
+<li><p><strong>solver</strong> (<em>nn.Module</em><em>, </em><em>optional</em>) – a linear solver. If <code class="docutils literal notranslate"><span class="pre">None</span></code>, <a class="reference internal" href="../pypose.optim.solver.Cholesky/#pypose.optim.solver.Cholesky" title="pypose.optim.solver.Cholesky"><code class="xref py py-meth docutils literal notranslate"><span class="pre">solver.Cholesky()</span></code></a> is used.
+Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+<li><p><strong>strategy</strong> (<em>object</em><em>, </em><em>optional</em>) – strategy for adjusting the damping factor. If <code class="docutils literal notranslate"><span class="pre">None</span></code>, the
+<a class="reference internal" href="../pypose.optim.strategy.TrustRegion/#pypose.optim.strategy.TrustRegion" title="pypose.optim.strategy.TrustRegion"><code class="xref py py-meth docutils literal notranslate"><span class="pre">strategy.TrustRegion()</span></code></a> will be used. Defult: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+<li><p><strong>kernel</strong> (nn.Module, or <code class="xref py py-obj docutils literal notranslate"><span class="pre">list</span></code>, optional) – the robust kernel function. If a <code class="xref py py-obj docutils literal notranslate"><span class="pre">list</span></code>,
+the element must be nn.Module or <code class="docutils literal notranslate"><span class="pre">None</span></code> and the length must be 1 or the number of residuals.
+Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+<li><p><strong>corrector</strong> – (nn.Module, or <code class="xref py py-obj docutils literal notranslate"><span class="pre">list</span></code>, optional): the Jacobian and model residual corrector to
+fit the kernel function. If a <code class="xref py py-obj docutils literal notranslate"><span class="pre">list</span></code>, the element must be nn.Module or <code class="docutils literal notranslate"><span class="pre">None</span></code> and
+the length must be 1 or the number of residuals.
+If a kernel is given but a corrector is not specified, auto correction is
+used. Auto correction can be unstable when the robust model has indefinite Hessian.
+Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+<li><p><strong>weight</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>, or <code class="xref py py-obj docutils literal notranslate"><span class="pre">list</span></code>, optional) – the square positive definite matrix defining
+the weight of model residual. If a <code class="xref py py-obj docutils literal notranslate"><span class="pre">list</span></code>, the element must be <code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code> and
+the length must be equal to the number of residuals.
+The corresponding residual and weight should be
+<a class="reference external" href="https://pytorch.org/docs/stable/notes/broadcasting.html#broadcasting-semantics">broadcastable</a>.
+For example, if the shape of a residual is <code class="docutils literal notranslate"><span class="pre">B*M*N*R</span></code>, the shape of its weight can
+be <code class="docutils literal notranslate"><span class="pre">R*R</span></code>, <code class="docutils literal notranslate"><span class="pre">N*R*R</span></code>, <code class="docutils literal notranslate"><span class="pre">M*N*R*R</span></code> or <code class="docutils literal notranslate"><span class="pre">B*M*N*R*R</span></code>.
+Use this only when all inputs shared the same weight matrices. This is
+ignored when weight is given when calling <a class="reference internal" href="#pypose.optim.LevenbergMarquardt.step" title="pypose.optim.LevenbergMarquardt.step"><code class="xref py py-meth docutils literal notranslate"><span class="pre">step()</span></code></a> or <a class="reference internal" href="../pypose.optim.scheduler.StopOnPlateau/#pypose.optim.scheduler.StopOnPlateau.optimize" title="pypose.optim.scheduler.StopOnPlateau.optimize"><code class="xref py py-meth docutils literal notranslate"><span class="pre">optimize()</span></code></a> method.
+Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+<li><p><strong>reject</strong> (<em>integer</em><em>, </em><em>optional</em>) – the maximum number of rejecting unsuccessfull steps.
+Default: 16.</p></li>
+<li><p><strong>min</strong> (<em>float</em><em>, </em><em>optional</em>) – the lower-bound of the Hessian diagonal. Default: 1e-6.</p></li>
+<li><p><strong>max</strong> (<em>float</em><em>, </em><em>optional</em>) – the upper-bound of the Hessian diagonal. Default: 1e32.</p></li>
+<li><p><strong>vectorize</strong> (<em>bool</em><em>, </em><em>optional</em>) – the method of computing Jacobian. If <code class="docutils literal notranslate"><span class="pre">True</span></code>, the
+gradient of each scalar in output with respect to the model parameters will be
+computed in parallel with <code class="docutils literal notranslate"><span class="pre">&quot;reverse-mode&quot;</span></code>. More details go to
+<a class="reference internal" href="../pypose.optim.functional.modjac/#pypose.optim.functional.modjac" title="pypose.optim.functional.modjac"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.optim.functional.modjac()</span></code></a>. Default: <code class="docutils literal notranslate"><span class="pre">True</span></code>.</p></li>
+</ul>
+</dd>
+</dl>
+<p>Available solvers: <a class="reference internal" href="../pypose.optim.solver.PINV/#pypose.optim.solver.PINV" title="pypose.optim.solver.PINV"><code class="xref py py-meth docutils literal notranslate"><span class="pre">solver.PINV()</span></code></a>; <a class="reference internal" href="../pypose.optim.solver.LSTSQ/#pypose.optim.solver.LSTSQ" title="pypose.optim.solver.LSTSQ"><code class="xref py py-meth docutils literal notranslate"><span class="pre">solver.LSTSQ()</span></code></a>, <a class="reference internal" href="../pypose.optim.solver.Cholesky/#pypose.optim.solver.Cholesky" title="pypose.optim.solver.Cholesky"><code class="xref py py-meth docutils literal notranslate"><span class="pre">solver.Cholesky()</span></code></a>.</p>
+<p>Available kernels: <a class="reference internal" href="../pypose.optim.kernel.Huber/#pypose.optim.kernel.Huber" title="pypose.optim.kernel.Huber"><code class="xref py py-meth docutils literal notranslate"><span class="pre">kernel.Huber()</span></code></a>; <a class="reference internal" href="../pypose.optim.kernel.PseudoHuber/#pypose.optim.kernel.PseudoHuber" title="pypose.optim.kernel.PseudoHuber"><code class="xref py py-meth docutils literal notranslate"><span class="pre">kernel.PseudoHuber()</span></code></a>; <a class="reference internal" href="../pypose.optim.kernel.Cauchy/#pypose.optim.kernel.Cauchy" title="pypose.optim.kernel.Cauchy"><code class="xref py py-meth docutils literal notranslate"><span class="pre">kernel.Cauchy()</span></code></a>.</p>
+<p>Available correctors: <a class="reference internal" href="../pypose.optim.corrector.FastTriggs/#pypose.optim.corrector.FastTriggs" title="pypose.optim.corrector.FastTriggs"><code class="xref py py-meth docutils literal notranslate"><span class="pre">corrector.FastTriggs()</span></code></a>, <a class="reference internal" href="../pypose.optim.corrector.Triggs/#pypose.optim.corrector.Triggs" title="pypose.optim.corrector.Triggs"><code class="xref py py-meth docutils literal notranslate"><span class="pre">corrector.Triggs()</span></code></a>.</p>
+<p>Available strategies: <a class="reference internal" href="../pypose.optim.strategy.Constant/#pypose.optim.strategy.Constant" title="pypose.optim.strategy.Constant"><code class="xref py py-meth docutils literal notranslate"><span class="pre">strategy.Constant()</span></code></a>; <a class="reference internal" href="../pypose.optim.strategy.Adaptive/#pypose.optim.strategy.Adaptive" title="pypose.optim.strategy.Adaptive"><code class="xref py py-meth docutils literal notranslate"><span class="pre">strategy.Adaptive()</span></code></a>;
+<a class="reference internal" href="../pypose.optim.strategy.TrustRegion/#pypose.optim.strategy.TrustRegion" title="pypose.optim.strategy.TrustRegion"><code class="xref py py-meth docutils literal notranslate"><span class="pre">strategy.TrustRegion()</span></code></a>;</p>
+<div class="admonition warning">
+<p class="admonition-title">Warning</p>
+<p>The output of model <span class="math">\(\bm{f}(\bm{\theta},\bm{x}_i)\)</span> and target <span class="math">\(\bm{y}_i\)</span>
+can be any shape, while their <strong>last dimension</strong> <span class="math">\(d\)</span> is always taken as the
+dimension of model residual, whose inner product will be input to the kernel
+function. This is useful for residuals like re-projection error, whose last
+dimension is 2.</p>
+<p>Note that <strong>auto correction</strong> is equivalent to the method of ‘square-rooting the kernel’
+mentioned in Section 3.3 of the following paper. It replaces the
+<span class="math">\(d\)</span>-dimensional residual with a one-dimensional one, which loses
+residual-level structural information.</p>
+<ul class="simple">
+<li><p>Christopher Zach, <a class="reference external" href="https://link.springer.com/chapter/10.1007/978-3-319-10602-1_50">Robust Bundle Adjustment Revisited</a>, European
+Conference on Computer Vision (ECCV), 2014.</p></li>
+</ul>
+<p><strong>Therefore, the users need to keep the last dimension of model output and target to
+1, even if the model residual is a scalar. If the model output only has one dimension,
+the model Jacobian will be a row vector, instead of a matrix, which loses sample-level
+structural information, although computing Jacobian vector is faster.</strong></p>
+</div>
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.optim.LevenbergMarquardt.step">
+<span class="sig-name descname"><span class="pre">step</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">target</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">weight</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/optim/optimizer/#LevenbergMarquardt.step"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.optim.LevenbergMarquardt.step" title="Permalink to this definition">¶</a></dt>
+<dd><p>Performs a single optimization step.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>input</strong> (<em>Tensor/LieTensor</em><em>, </em><em>tuple</em><em> or </em><em>a dict of Tensors/LieTensors</em>) – the input to the model.</p></li>
+<li><p><strong>target</strong> (<em>Tensor/LieTensor</em>) – the model target to optimize.
+If not given, the squared model output is minimized. Defaults: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+<li><p><strong>weight</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>, or <code class="xref py py-obj docutils literal notranslate"><span class="pre">list</span></code>, optional) – the square positive definite matrix defining
+the weight of model residual. If a <code class="xref py py-obj docutils literal notranslate"><span class="pre">list</span></code>, the element must be <code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code> and
+the length must be equal to the number of residuals. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>the minimized model loss.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>Tensor</p>
+</dd>
+</dl>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The (non-negative) damping factor <span class="math">\(\lambda\)</span> can be adjusted at each iteration.
+If the residual reduces rapidly, a smaller value can be used, bringing the algorithm
+closer to the Gauss-Newton algorithm, whereas if an iteration gives insufficient
+residual reduction, <span class="math">\(\lambda\)</span> can be increased, giving a step closer to the
+gradient descent direction.</p>
+<p>See more details of <a class="reference external" href="https://en.wikipedia.org/wiki/Levenberg-Marquardt_algorithm">Levenberg-Marquardt (LM) algorithm</a> on Wikipedia.</p>
+</div>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Different from PyTorch optimizers like
+<a class="reference external" href="https://pytorch.org/docs/stable/generated/torch.optim.SGD.html">SGD</a>, where the
+model error has to be a scalar, the output of model <span class="math">\(\bm{f}\)</span> can be a
+Tensor/LieTensor or a tuple of Tensors/LieTensors.</p>
+</div>
+<p class="rubric">Example</p>
+<p>Optimizing a simple module to <strong>approximate pose inversion</strong>.</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="k">class</span> <span class="nc">PoseInv</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+<span class="gp">... </span>    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">dim</span><span class="p">):</span>
+<span class="gp">... </span>        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+<span class="gp">... </span>        <span class="bp">self</span><span class="o">.</span><span class="n">pose</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">Parameter</span><span class="p">(</span><span class="n">pp</span><span class="o">.</span><span class="n">randn_se3</span><span class="p">(</span><span class="o">*</span><span class="n">dim</span><span class="p">))</span>
+<span class="gp">...</span>
+<span class="gp">... </span>    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">input</span><span class="p">):</span>
+<span class="gp">... </span>        <span class="c1"># the last dimension of the output is 6,</span>
+<span class="gp">... </span>        <span class="c1"># which will be the residual dimension.</span>
+<span class="gp">... </span>        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">pose</span><span class="o">.</span><span class="n">Exp</span><span class="p">()</span> <span class="o">@</span> <span class="nb">input</span><span class="p">)</span><span class="o">.</span><span class="n">Log</span><span class="p">()</span>
+<span class="gp">...</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">posinv</span> <span class="o">=</span> <span class="n">PoseInv</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="nb">input</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_SE3</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">strategy</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">optim</span><span class="o">.</span><span class="n">strategy</span><span class="o">.</span><span class="n">Adaptive</span><span class="p">(</span><span class="n">damping</span><span class="o">=</span><span class="mf">1e-6</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">optimizer</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">optim</span><span class="o">.</span><span class="n">LM</span><span class="p">(</span><span class="n">posinv</span><span class="p">,</span> <span class="n">strategy</span><span class="o">=</span><span class="n">strategy</span><span class="p">)</span>
+<span class="gp">...</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">idx</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10</span><span class="p">):</span>
+<span class="gp">... </span>    <span class="n">loss</span> <span class="o">=</span> <span class="n">optimizer</span><span class="o">.</span><span class="n">step</span><span class="p">(</span><span class="nb">input</span><span class="p">)</span>
+<span class="gp">... </span>    <span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Pose Inversion loss </span><span class="si">%.7f</span><span class="s1"> @ </span><span class="si">%d</span><span class="s1"> it&#39;</span><span class="o">%</span><span class="p">(</span><span class="n">loss</span><span class="p">,</span> <span class="n">idx</span><span class="p">))</span>
+<span class="gp">... </span>    <span class="k">if</span> <span class="n">loss</span> <span class="o">&lt;</span> <span class="mf">1e-5</span><span class="p">:</span>
+<span class="gp">... </span>        <span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Early Stopping with loss:&#39;</span><span class="p">,</span> <span class="n">loss</span><span class="o">.</span><span class="n">item</span><span class="p">())</span>
+<span class="gp">... </span>        <span class="k">break</span>
+<span class="gp">...</span>
+<span class="go">Pose Inversion error: 1.6600330 @ 0 it</span>
+<span class="go">Pose Inversion error: 0.1296970 @ 1 it</span>
+<span class="go">Pose Inversion error: 0.0008593 @ 2 it</span>
+<span class="go">Pose Inversion error: 0.0000004 @ 3 it</span>
+<span class="go">Early Stopping with error: 4.443569991963159e-07</span>
+</pre></div>
+</div>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>More practical examples, e.g., pose graph optimization (PGO), can be found at
+<a class="reference external" href="https://github.com/pypose/pypose/tree/main/examples/module/pgo">examples/module/pgo</a>.</p>
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+  <section id="pypose-optim-corrector-fasttriggs">
+<h1>pypose.optim.corrector.FastTriggs<a class="headerlink" href="#pypose-optim-corrector-fasttriggs" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.optim.corrector.FastTriggs">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.optim.corrector.</span></span><span class="sig-name descname"><span class="pre">FastTriggs</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">kernel</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/optim/corrector/#FastTriggs"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.optim.corrector.FastTriggs" title="Permalink to this definition">¶</a></dt>
+<dd><p>Faster yet stable version of Triggs correction of model residual and Jacobian.</p>
+<div class="math">
+\[\begin{align*}
+    \mathbf{R}_i^\rho &= \sqrt{\rho'(c_i)} \mathbf{R}_i\\
+    \mathbf{J}_i^\rho &= \sqrt{\rho'(c_i)} \mathbf{J}_i
+\end{align*},
+
+\]</div>
+<p>where <span class="math">\(\mathbf{R}_i\)</span> and <span class="math">\(\mathbf{J}_i\)</span> are the <span class="math">\(i\)</span>-th item of the
+model residual and Jacobian, respectively. <span class="math">\(\rho()\)</span> is the kernel function and
+<span class="math">\(c_i = \mathbf{R}_i^T\mathbf{R}_i\)</span> is the point to compute the gradient.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>kernel</strong> (<em>nn.Module</em>) – the robust kernel (cost) function.</p>
+</dd>
+</dl>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>This implementation has a faster and numerically stable solution than <a class="reference internal" href="../pypose.optim.corrector.Triggs/#pypose.optim.corrector.Triggs" title="pypose.optim.corrector.Triggs"><code class="xref py py-meth docutils literal notranslate"><span class="pre">Triggs()</span></code></a>.
+It removes the kernel’s 2nd order derivatives (often negative), which can lead a 2nd
+order optimizer unstable. It basically aims to solve</p>
+<div class="math">
+\[\bm{\theta}^* = \arg\min_{\bm{\theta}} \mathbf{g}(\bm{x})
+            = \arg\min_{\bm{\theta}} \sum_i \rho(\mathbf{R}_i^T \mathbf{R}_i),
+
+\]</div>
+<p>where <span class="math">\(\mathbf{R}_i = \bm{f}(\bm{\theta},\bm{x}_i) - \bm{y}_i\)</span> and
+<span class="math">\(\bm{f}(\bm{\theta}, \bm{x})\)</span> is the model, <span class="math">\(\bm{\theta}\)</span> is the parameters
+to be optimized, <span class="math">\(\bm{x}\)</span> is the model inputs, <span class="math">\(\bm{y}\)</span> is the model targets.
+Considering the 1st order Taylor expansion of the model
+<span class="math">\(\bm{f}(\bm{\theta}+\bm{\delta})\approx\bm{f}(\bm{\theta})+\mathbf{J}_i\bm{\delta}\)</span>.
+If we take <span class="math">\(c_i = \mathbf{R}_i^T \mathbf{R}_i\)</span> and set the first derivative of
+<span class="math">\(\mathbf{g}(\bm{\delta})\)</span> to zero, we have</p>
+<div class="math">
+\[\frac{\partial \bm{g}}{\partial \bm{\delta}}
+= \sum_i \frac{\partial \rho}{\partial c_i} \frac{\partial c_i}{\partial \bm{\delta}}
+= \bm{0}
+
+\]</div>
+<p>This leads to</p>
+<div class="math">
+\[\sum_i \frac{\partial \rho}{\partial c_i} \mathbf{J}_i^T \mathbf{J}_i \bm{\delta}
+= - \sum_i \frac{\partial \rho}{\partial c_i} \mathbf{J}_i^T \mathbf{R}_i
+
+\]</div>
+<p>Rearrange the gradient of <span class="math">\(\rho\)</span>, we have</p>
+<div class="math">
+\[\sum_i \left(\sqrt{\frac{\partial \rho}{\partial c_i}} \mathbf{J}_i\right)^T
+    \left(\sqrt{\frac{\partial \rho}{\partial c_i}} \mathbf{J}_i\right) \bm{\delta}
+= - \sum_i \left(\sqrt{\frac{\partial \rho}{\partial c_i}} \mathbf{J}_i\right)^T
+    \left(\sqrt{\frac{\partial \rho}{\partial c_i}} \mathbf{R}_i\right)
+
+\]</div>
+<p>This gives us the corrected model residual <span class="math">\(\mathbf{R}_i^\rho\)</span> and Jacobian
+<span class="math">\(\mathbf{J}_i^\rho\)</span>, which ends with the same problem formulation as the
+standard 2nd order optimizers such as <a class="reference internal" href="../pypose.optim.GN/#pypose.optim.GN" title="pypose.optim.GN"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.optim.GN()</span></code></a> and
+<a class="reference internal" href="../pypose.optim.LM/#pypose.optim.LM" title="pypose.optim.LM"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.optim.LM()</span></code></a>.</p>
+<div class="math">
+\[\sum_i {\mathbf{J}_i^\rho}^T \mathbf{J}_i^\rho \bm{\delta}
+= - \sum_i {\mathbf{J}_i^\rho}^T \mathbf{R}_i^\rho
+
+\]</div>
+</div>
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.optim.corrector.FastTriggs.forward">
+<span class="sig-name descname"><span class="pre">forward</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">R</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">J</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/optim/corrector/#FastTriggs.forward"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.optim.corrector.FastTriggs.forward" title="Permalink to this definition">¶</a></dt>
+<dd><dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>R</strong> (<em>Tensor</em>) – the model residual.</p></li>
+<li><p><strong>J</strong> (<em>Tensor</em>) – the model Jacobian.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>the corrected model residual and model Jacobian.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>tuple of Tensors</p>
+</dd>
+</dl>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The users basically only need to call the constructor, while the <a class="reference internal" href="#pypose.optim.corrector.FastTriggs.forward" title="pypose.optim.corrector.FastTriggs.forward"><code class="xref py py-obj docutils literal notranslate"><span class="pre">forward()</span></code></a>
+function is not supposed to be directly called by PyPose users. It will be called
+internally by optimizers such as <a class="reference internal" href="../pypose.optim.GN/#pypose.optim.GN" title="pypose.optim.GN"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.optim.GN()</span></code></a> and <a class="reference internal" href="../pypose.optim.LM/#pypose.optim.LM" title="pypose.optim.LM"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.optim.LM()</span></code></a>.</p>
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+  <section id="pypose-optim-corrector-triggs">
+<h1>pypose.optim.corrector.Triggs<a class="headerlink" href="#pypose-optim-corrector-triggs" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.optim.corrector.Triggs">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.optim.corrector.</span></span><span class="sig-name descname"><span class="pre">Triggs</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">kernel</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/optim/corrector/#Triggs"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.optim.corrector.Triggs" title="Permalink to this definition">¶</a></dt>
+<dd><p>The Triggs correction of model residual and Jacobian.</p>
+<div class="math">
+\[\begin{align*}
+    \mathbf{R}_i^\rho &= \frac{\sqrt{\rho'(c_i)}}{1 - \alpha} \mathbf{R}_i,\\
+    \mathbf{J}_i^\rho &= \sqrt{\rho'(c_i)} \left(\mathbf{I} - \alpha
+        \frac{\mathbf{R}_i^T\mathbf{R}_i}{\|\mathbf{R}_i\|^2} \right) \mathbf{J}_i,
+\end{align*}
+
+\]</div>
+<p>where <span class="math">\(\alpha\)</span> is a root of</p>
+<div class="math">
+\[\frac{1}{2} \alpha^2 - \alpha - \frac{\rho''}{\rho'} \|\mathbf{R}_i\|^2 = 0.
+
+\]</div>
+<p><span class="math">\(\mathbf{R}_i\)</span> and <span class="math">\(\mathbf{J}_i\)</span> are the <span class="math">\(i\)</span>-th item of the model
+residual and Jacobian, respectively. <span class="math">\(\rho()\)</span> is the kernel function and
+<span class="math">\(c_i = \mathbf{R}_i^T\mathbf{R}_i\)</span> is the evaluation point.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>kernel</strong> (<em>nn.Module</em>) – the robust kernel (cost) function.</p>
+</dd>
+</dl>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>This implementation thanks to Eq. (11) of the following paper.</p>
+<ul class="simple">
+<li><p>Bill Triggs, etc., <a class="reference external" href="https://link.springer.com/chapter/10.1007/3-540-44480-7_21">Bundle Adjustment – A Modern Synthesis</a>, International
+Workshop on Vision Algorithms, 1999.</p></li>
+</ul>
+</div>
+<div class="admonition warning">
+<p class="admonition-title">Warning</p>
+<p>The <a class="reference internal" href="../pypose.optim.corrector.FastTriggs/#pypose.optim.corrector.FastTriggs" title="pypose.optim.corrector.FastTriggs"><code class="xref py py-meth docutils literal notranslate"><span class="pre">FastTriggs()</span></code></a> corrector is preferred when the kernel function has a
+negative 2nd order derivative.</p>
+</div>
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.optim.corrector.Triggs.forward">
+<span class="sig-name descname"><span class="pre">forward</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">R</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">J</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/optim/corrector/#Triggs.forward"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.optim.corrector.Triggs.forward" title="Permalink to this definition">¶</a></dt>
+<dd><dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>R</strong> (<em>Tensor</em>) – the model residual.</p></li>
+<li><p><strong>J</strong> (<em>Tensor</em>) – the model Jacobian.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>the corrected model residual and model Jacobian.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>tuple of Tensors</p>
+</dd>
+</dl>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The users basically only need to call the constructor, while the <a class="reference internal" href="#pypose.optim.corrector.Triggs.forward" title="pypose.optim.corrector.Triggs.forward"><code class="xref py py-obj docutils literal notranslate"><span class="pre">forward()</span></code></a>
+function is not supposed to be directly called by PyPose users. It will be called
+internally by optimizers such as <a class="reference internal" href="../pypose.optim.GN/#pypose.optim.GN" title="pypose.optim.GN"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.optim.GN()</span></code></a> and <a class="reference internal" href="../pypose.optim.LM/#pypose.optim.LM" title="pypose.optim.LM"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.optim.LM()</span></code></a>.</p>
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+  <section id="pypose-optim-functional-modjac">
+<h1>pypose.optim.functional.modjac<a class="headerlink" href="#pypose-optim-functional-modjac" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.optim.functional.modjac">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.optim.functional.</span></span><span class="sig-name descname"><span class="pre">modjac</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">model</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">input</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">create_graph</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">strict</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">vectorize</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">strategy</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'reverse-mode'</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">flatten</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/optim/functional/#modjac"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.optim.functional.modjac" title="Permalink to this definition">¶</a></dt>
+<dd><p>Compute the model Jacobian with respect to the model parameters.</p>
+<p>For a parametric model <span class="math">\(\bm{f}(\bm{\theta}, \bm{x})\)</span>, where <span class="math">\(\bm{\theta}\)</span>
+is the learnable parameter and <span class="math">\(\bm{x}\)</span> is the input, it computes the
+Jacobian of the <span class="math">\(i\)</span>-th output and <span class="math">\(j\)</span>-th parameter as</p>
+<div class="math">
+\[{\displaystyle \mathbf{J}_{i,j} =
+{\begin{bmatrix}
+    {\dfrac {\partial \bm{f}_{i,1}}{\partial \bm{\theta}_{j,1}}} &amp;
+    \cdots&amp;{\dfrac {\partial \bm{f}_{i,1}}{\partial \bm{\theta}_{j,n}}}\\
+    \vdots &amp;\ddots &amp;\vdots \\
+    {\dfrac {\partial \bm{f}_{i,m}}{\partial \bm{\theta}_{j,1}}}&amp;
+    \cdots &amp;{\dfrac {\partial \bm{f}_{i,m}}{\partial \bm{\theta}_{j,n}}}
+\end{bmatrix}}}.
+
+\]</div>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>model</strong> (<em>torch.nn.Module</em>) – a PyTorch model that takes Tensor or LieTensor
+input and returns a tuple of Tensors/LieTensors or a Tensor/LieTensor.</p></li>
+<li><p><strong>input</strong> (<em>tuple of Tensors/LieTensors</em><em> or </em><em>Tensor/LieTensor</em>) – input to the
+model. Defaults to <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+<li><p><strong>create_graph</strong> (<em>bool</em><em>, </em><em>optional</em>) – If <code class="docutils literal notranslate"><span class="pre">True</span></code>, the Jacobian will be
+computed in a differentiable manner. Note that when <code class="docutils literal notranslate"><span class="pre">strict</span></code> is
+<code class="docutils literal notranslate"><span class="pre">False</span></code>, the result can not require gradients or be disconnected
+from the input.  Defaults to <code class="docutils literal notranslate"><span class="pre">False</span></code>.</p></li>
+<li><p><strong>strict</strong> (<em>bool</em><em>, </em><em>optional</em>) – If <code class="docutils literal notranslate"><span class="pre">True</span></code>, an error will be raised when we
+detect that there exists an input such that all the outputs are
+independent of it. If <code class="docutils literal notranslate"><span class="pre">False</span></code>, we return a Tensor of zeros as the
+jacobian for said input, which is the expected mathematical value.
+Defaults to <code class="docutils literal notranslate"><span class="pre">False</span></code>.</p></li>
+<li><p><strong>vectorize</strong> (<em>bool</em><em>, </em><em>optional</em>) – When computing the jacobian, usually we invoke
+<code class="docutils literal notranslate"><span class="pre">autograd.grad</span></code> once per row of the jacobian. If this flag is
+<code class="docutils literal notranslate"><span class="pre">True</span></code>, we perform only a single <code class="docutils literal notranslate"><span class="pre">autograd.grad</span></code> call with
+<code class="docutils literal notranslate"><span class="pre">batched_grad=True</span></code> which uses the vmap prototype feature.
+Though this should lead to performance improvements in many cases,
+because this feature is still experimental, there may be performance
+cliffs. See <code class="xref py py-func docutils literal notranslate"><span class="pre">torch.autograd.grad()</span></code>’s <code class="docutils literal notranslate"><span class="pre">batched_grad</span></code> parameter for
+more information.</p></li>
+<li><p><strong>strategy</strong> (<em>str</em><em>, </em><em>optional</em>) – Set to <code class="docutils literal notranslate"><span class="pre">&quot;forward-mode&quot;</span></code> or <code class="docutils literal notranslate"><span class="pre">&quot;reverse-mode&quot;</span></code> to
+determine whether the Jacobian will be computed with forward or reverse
+mode AD. Currently, <code class="docutils literal notranslate"><span class="pre">&quot;forward-mode&quot;</span></code> requires <code class="docutils literal notranslate"><span class="pre">vectorized=True</span></code>.
+Defaults to <code class="docutils literal notranslate"><span class="pre">&quot;reverse-mode&quot;</span></code>. If <code class="docutils literal notranslate"><span class="pre">func</span></code> has more outputs than
+input, <code class="docutils literal notranslate"><span class="pre">&quot;forward-mode&quot;</span></code> tends to be more performant. Otherwise,
+prefer to use <code class="docutils literal notranslate"><span class="pre">&quot;reverse-mode&quot;</span></code>.</p></li>
+<li><p><strong>flatten</strong> (<em>bool</em><em>, </em><em>optional</em>) – If <code class="docutils literal notranslate"><span class="pre">True</span></code>, all module parameters and outputs
+are flattened and concatenated to form a single vector. The Jacobian
+will be computed with respect to this single flattened vectors, thus
+a single Tensor will be returned.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>if there is a single
+parameter and output, this will be a single Tensor containing the
+Jacobian for the linearized parameter and output. If there are more
+than one parameters, then the Jacobian will be a tuple of Tensors.
+If there are more than one outputs (even if there is only one parameter),
+then the Jacobian will be a tuple of tuple of Tensors where <code class="docutils literal notranslate"><span class="pre">Jacobian[i][j]</span></code>
+will contain the Jacobian of the <code class="docutils literal notranslate"><span class="pre">i</span></code>th output and <code class="docutils literal notranslate"><span class="pre">j</span></code>th parameter
+and will have as size the concatenation of the sizes of the corresponding
+output and the corresponding parameter and will have same dtype and device as the
+corresponding parameter. If strategy is <code class="docutils literal notranslate"><span class="pre">forward-mode</span></code>, the dtype will be
+that of the output; otherwise, the parameters.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>Jacobian (Tensor or nested tuple of Tensors)</p>
+</dd>
+</dl>
+<div class="admonition warning">
+<p class="admonition-title">Warning</p>
+<p>The function <a class="reference internal" href="#pypose.optim.functional.modjac" title="pypose.optim.functional.modjac"><code class="xref py py-obj docutils literal notranslate"><span class="pre">modjac</span></code></a> calculate Jacobian of model parameters.
+This is in contrast to PyTorch’s function <a class="reference external" href="https://pytorch.org/docs/stable/generated/torch.autograd.functional.jacobian.html">jacobian</a>,
+which computes the Jacobian of a given Python function.</p>
+</div>
+<p class="rubric">Example</p>
+<p>Calculates Jacobian with respect to all model parameters.</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">model</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Conv2d</span><span class="p">(</span><span class="n">in_channels</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">out_channels</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">kernel_size</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="nb">input</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">J</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">optim</span><span class="o">.</span><span class="n">functional</span><span class="o">.</span><span class="n">modjac</span><span class="p">(</span><span class="n">model</span><span class="p">,</span> <span class="nb">input</span><span class="p">)</span>
+<span class="go">(tensor([[[[[[[0.3365]]]]]]]), tensor([[[[1.]]]]))</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="n">j</span><span class="o">.</span><span class="n">shape</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">J</span><span class="p">]</span>
+<span class="go">[torch.Size([1, 1, 1, 1, 1, 1, 1]), torch.Size([1, 1, 1, 1])]</span>
+</pre></div>
+</div>
+<p>Function with flattened parameters returns a combined Jacobian.</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="nb">input</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">model</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Conv2d</span><span class="p">(</span><span class="n">in_channels</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">out_channels</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">kernel_size</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">J</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">optim</span><span class="o">.</span><span class="n">functional</span><span class="o">.</span><span class="n">modjac</span><span class="p">(</span><span class="n">model</span><span class="p">,</span> <span class="nb">input</span><span class="p">,</span> <span class="n">flatten</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+<span class="go">tensor([[-0.4162,  0.0968,  0.0000,  0.0000,  1.0000,  0.0000],</span>
+<span class="go">        [-0.6042,  1.1886,  0.0000,  0.0000,  1.0000,  0.0000],</span>
+<span class="go">        [ 1.4623,  0.7389,  0.0000,  0.0000,  1.0000,  0.0000],</span>
+<span class="go">        [ 1.0716,  2.4293,  0.0000,  0.0000,  1.0000,  0.0000],</span>
+<span class="go">        [ 0.0000,  0.0000, -0.4162,  0.0968,  0.0000,  1.0000],</span>
+<span class="go">        [ 0.0000,  0.0000, -0.6042,  1.1886,  0.0000,  1.0000],</span>
+<span class="go">        [ 0.0000,  0.0000,  1.4623,  0.7389,  0.0000,  1.0000],</span>
+<span class="go">        [ 0.0000,  0.0000,  1.0716,  2.4293,  0.0000,  1.0000]])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">J</span><span class="o">.</span><span class="n">shape</span>
+<span class="go">torch.Size([8, 6])</span>
+</pre></div>
+</div>
+<p>Calculate Jacobian with respect to parameter of <a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.LieTensor</span></code></a>.</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="k">class</span> <span class="nc">PoseTransform</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+<span class="gp">... </span>    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+<span class="gp">... </span>        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+<span class="gp">... </span>        <span class="bp">self</span><span class="o">.</span><span class="n">p</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">Parameter</span><span class="p">(</span><span class="n">pp</span><span class="o">.</span><span class="n">randn_so3</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
+<span class="gp">...</span>
+<span class="gp">... </span>    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
+<span class="gp">... </span>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="o">.</span><span class="n">Exp</span><span class="p">()</span> <span class="o">*</span> <span class="n">x</span>
+<span class="gp">...</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">model</span><span class="p">,</span> <span class="nb">input</span> <span class="o">=</span> <span class="n">PoseTransform</span><span class="p">(),</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_SO3</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">J</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">optim</span><span class="o">.</span><span class="n">functional</span><span class="o">.</span><span class="n">modjac</span><span class="p">(</span><span class="n">model</span><span class="p">,</span> <span class="nb">input</span><span class="p">,</span> <span class="n">flatten</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+<span class="go">tensor([[ 0.4670,  0.7041,  0.0029,  0.0000,  0.0000,  0.0000],</span>
+<span class="go">        [-0.6591,  0.4554, -0.2566,  0.0000,  0.0000,  0.0000],</span>
+<span class="go">        [-0.2477,  0.0670,  0.9535,  0.0000,  0.0000,  0.0000],</span>
+<span class="go">        [ 0.0000,  0.0000,  0.0000,  0.0000,  0.0000,  0.0000],</span>
+<span class="go">        [ 0.0000,  0.0000,  0.0000,  0.8593,  0.2672,  0.3446],</span>
+<span class="go">        [ 0.0000,  0.0000,  0.0000, -0.2417,  0.9503, -0.1154],</span>
+<span class="go">        [ 0.0000,  0.0000,  0.0000, -0.3630, -0.0179,  0.9055],</span>
+<span class="go">        [ 0.0000,  0.0000,  0.0000,  0.0000,  0.0000,  0.0000]])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">J</span><span class="o">.</span><span class="n">shape</span>
+<span class="go">torch.Size([8, 6])</span>
+</pre></div>
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+  <section id="pypose-optim-kernel-arctan">
+<h1>pypose.optim.kernel.Arctan<a class="headerlink" href="#pypose-optim-kernel-arctan" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.optim.kernel.Arctan">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.optim.kernel.</span></span><span class="sig-name descname"><span class="pre">Arctan</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">delta</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1.0</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/optim/kernel/#Arctan"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.optim.kernel.Arctan" title="Permalink to this definition">¶</a></dt>
+<dd><p>The robust Arctan kernel cost function.</p>
+<div class="math">
+\[\bm{y}_i=\delta ^{2}\arctan \left ( \frac{\bm{x}_i}{\delta ^{2}}\right )
+
+\]</div>
+<p>where <span class="math">\(\delta\)</span> (delta) is a hyperparameter, <span class="math">\(\bm{x}\)</span> and <span class="math">\(\bm{y}\)</span> are the
+input and output tensors, respectively.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>delta</strong> (<em>float</em><em>, </em><em>optional</em>) – Specify the Arctan cost. Default: 1.0</p>
+</dd>
+</dl>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The input has to be a non-negative tensor and the output tensor has the same shape with the
+input.</p>
+</div>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">pypose.optim.kernel</span> <span class="k">as</span> <span class="nn">ppok</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">kernel</span> <span class="o">=</span> <span class="n">ppok</span><span class="o">.</span><span class="n">Arctan</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="nb">input</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">kernel</span><span class="p">(</span><span class="nb">input</span><span class="p">)</span>
+<span class="go">tensor([0.0000, 0.4636, 0.7854, 1.1071, 1.2490])</span>
+</pre></div>
+</div>
+<figure class="align-default">
+<a class="reference internal image-reference" href="../../_images/arctan.png"><img alt="../../_images/arctan.png" src="../../_images/arctan.png" style="width: 600px;" /></a>
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+<span class="sig-name descname"><span class="pre">forward</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/optim/kernel/#Arctan.forward"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.optim.kernel.Arctan.forward" title="Permalink to this definition">¶</a></dt>
+<dd><dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>input</strong> (<em>torch.Tensor</em>) – the input tensor (non-negative).</p>
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+  <section id="pypose-optim-kernel-cauchy">
+<h1>pypose.optim.kernel.Cauchy<a class="headerlink" href="#pypose-optim-kernel-cauchy" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.optim.kernel.Cauchy">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.optim.kernel.</span></span><span class="sig-name descname"><span class="pre">Cauchy</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">delta</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1.0</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/optim/kernel/#Cauchy"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.optim.kernel.Cauchy" title="Permalink to this definition">¶</a></dt>
+<dd><p>The robust Cauchy kernel cost function.</p>
+<div class="math">
+\[\bm{y}_i = \delta^2 \log\left(1 + \frac{\bm{x}_i}{\delta^2}\right),
+
+\]</div>
+<p>where <span class="math">\(\delta\)</span> (delta) is a hyperparameter, <span class="math">\(\bm{x}\)</span>
+and <span class="math">\(\bm{y}\)</span> are the input and output tensors, respectively.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>delta</strong> (<em>float</em><em>, </em><em>optional</em>) – Specify the Cauchy cost. The value must be positive. Default: 1.0</p>
+</dd>
+</dl>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The input has to be a non-negative tensor and the output tensor
+has the same shape with the input.</p>
+</div>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">pypose.optim.kernel</span> <span class="k">as</span> <span class="nn">ppok</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">kernel</span> <span class="o">=</span> <span class="n">ppok</span><span class="o">.</span><span class="n">Cauchy</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="nb">input</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">kernel</span><span class="p">(</span><span class="nb">input</span><span class="p">)</span>
+<span class="go">tensor([0.0000, 0.4055, 0.6931, 1.0986, 1.3863])</span>
+</pre></div>
+</div>
+<figure class="align-default">
+<a class="reference internal image-reference" href="../../_images/cauchy.png"><img alt="../../_images/cauchy.png" src="../../_images/cauchy.png" style="width: 600px;" /></a>
+</figure>
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+<span class="sig-name descname"><span class="pre">forward</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/optim/kernel/#Cauchy.forward"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.optim.kernel.Cauchy.forward" title="Permalink to this definition">¶</a></dt>
+<dd><dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>input</strong> (<em>torch.Tensor</em>) – the input tensor (non-negative).</p>
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+  <section id="pypose-optim-kernel-huber">
+<h1>pypose.optim.kernel.Huber<a class="headerlink" href="#pypose-optim-kernel-huber" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.optim.kernel.Huber">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.optim.kernel.</span></span><span class="sig-name descname"><span class="pre">Huber</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">delta</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1.0</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/optim/kernel/#Huber"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.optim.kernel.Huber" title="Permalink to this definition">¶</a></dt>
+<dd><p>The robust Huber kernel cost function.</p>
+<div class="math">
+\[\bm{y}_i = \begin{cases}
+    \bm{x}_i                            & \text{if } \sqrt{\bm{x}_i} < \delta \\
+    2 \delta \sqrt{\bm{x}_i} - \delta^2 & \text{otherwise }
+\end{cases},
+
+\]</div>
+<p>where <span class="math">\(\delta\)</span> (delta) is a threshold, <span class="math">\(\bm{x}\)</span> and <span class="math">\(\bm{y}\)</span> are the input
+and output tensors, respectively.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>delta</strong> (<em>float</em><em>, </em><em>optional</em>) – Specify the threshold at which to scale the input. The value must
+be positive. Default: 1.0</p>
+</dd>
+</dl>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The input has to be a non-negative tensor and the output tensor has the same shape with the
+input. Use <a class="reference external" href="https://pytorch.org/docs/stable/generated/torch.nn.HuberLoss.html">torch.nn.HuberLoss</a> instead, if a scalar
+Huber loss function is needed.</p>
+</div>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">pypose.optim.kernel</span> <span class="k">as</span> <span class="nn">ppok</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">kernel</span> <span class="o">=</span> <span class="n">ppok</span><span class="o">.</span><span class="n">Huber</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="nb">input</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">kernel</span><span class="p">(</span><span class="nb">input</span><span class="p">)</span>
+<span class="go">tensor([0.0000, 0.5000, 1.0000, 1.8284, 2.4641])</span>
+</pre></div>
+</div>
+<figure class="align-default">
+<a class="reference internal image-reference" href="../../_images/huber.png"><img alt="../../_images/huber.png" src="../../_images/huber.png" style="width: 600px;" /></a>
+</figure>
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+<dt class="sig sig-object py" id="pypose.optim.kernel.Huber.forward">
+<span class="sig-name descname"><span class="pre">forward</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/optim/kernel/#Huber.forward"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.optim.kernel.Huber.forward" title="Permalink to this definition">¶</a></dt>
+<dd><dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>input</strong> (<em>torch.Tensor</em>) – the input tensor (non-negative).</p>
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+  <section id="pypose-optim-kernel-pseudohuber">
+<h1>pypose.optim.kernel.PseudoHuber<a class="headerlink" href="#pypose-optim-kernel-pseudohuber" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.optim.kernel.PseudoHuber">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.optim.kernel.</span></span><span class="sig-name descname"><span class="pre">PseudoHuber</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">delta</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1.0</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/optim/kernel/#PseudoHuber"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.optim.kernel.PseudoHuber" title="Permalink to this definition">¶</a></dt>
+<dd><p>The robust pseudo Huber kernel cost function.</p>
+<div class="math">
+\[\bm{y}_i = 2\delta^2 \left(\sqrt{1 + \bm{x}_i/\delta^2} - 1\right),
+
+\]</div>
+<p>where <span class="math">\(\delta\)</span> (delta) defines the steepness of the slope, <span class="math">\(\bm{x}\)</span> and
+<span class="math">\(\bm{y}\)</span> are the input and output tensors, respectively.  It can be used as a smooth
+version of <a class="reference internal" href="../pypose.optim.kernel.Huber/#pypose.optim.kernel.Huber" title="pypose.optim.kernel.Huber"><code class="xref py py-obj docutils literal notranslate"><span class="pre">Huber</span></code></a>.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>delta</strong> (<em>float</em><em>, </em><em>optional</em>) – Specify the slope. The value must be positive. Default: 1.0</p>
+</dd>
+</dl>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The input has to be a non-negative tensor and the output tensor has the same shape with the
+input.</p>
+</div>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">pypose.optim.kernel</span> <span class="k">as</span> <span class="nn">ppok</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">kernel</span> <span class="o">=</span> <span class="n">ppok</span><span class="o">.</span><span class="n">PseudoHuber</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="nb">input</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">kernel</span><span class="p">(</span><span class="nb">input</span><span class="p">)</span>
+<span class="go">tensor([0.0000, 0.4495, 0.8284, 1.4641, 2.0000])</span>
+</pre></div>
+</div>
+<figure class="align-default">
+<a class="reference internal image-reference" href="../../_images/pseudohuber.png"><img alt="../../_images/pseudohuber.png" src="../../_images/pseudohuber.png" style="width: 600px;" /></a>
+</figure>
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.optim.kernel.PseudoHuber.forward">
+<span class="sig-name descname"><span class="pre">forward</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/optim/kernel/#PseudoHuber.forward"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.optim.kernel.PseudoHuber.forward" title="Permalink to this definition">¶</a></dt>
+<dd><dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>input</strong> (<em>torch.Tensor</em>) – the input tensor (non-negative).</p>
+</dd>
+</dl>
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+  <section id="pypose-optim-kernel-scale">
+<h1>pypose.optim.kernel.Scale<a class="headerlink" href="#pypose-optim-kernel-scale" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.optim.kernel.Scale">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.optim.kernel.</span></span><span class="sig-name descname"><span class="pre">Scale</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">delta</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1.0</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/optim/kernel/#Scale"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.optim.kernel.Scale" title="Permalink to this definition">¶</a></dt>
+<dd><p>The robust Scale kernel cost function.</p>
+<div class="math">
+\[\bm{y}_i=\delta*\bm{x}_i
+
+\]</div>
+<p>where <span class="math">\(\delta\)</span> (delta) is a scalar, <span class="math">\(\bm{x}\)</span> and <span class="math">\(\bm{y}\)</span> are the input
+and output tensors, respectively.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>delta</strong> (<em>float</em>) – Specify the scale factor. Should be between 0 and 1.  Default: 1.0</p>
+</dd>
+</dl>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The input has to be a non-negative tensor and the output tensor has the same shape with
+the input.</p>
+</div>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">pypose.optim.kernel</span> <span class="k">as</span> <span class="nn">ppok</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">kernel</span> <span class="o">=</span> <span class="n">ppok</span><span class="o">.</span><span class="n">Scale</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="nb">input</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">kernel</span><span class="p">(</span><span class="nb">input</span><span class="p">)</span>
+<span class="go">tensor([0.0000, 0.5000, 1.0000, 2.0000, 3.0000])</span>
+</pre></div>
+</div>
+<blockquote>
+<div><figure class="align-default">
+<a class="reference internal image-reference" href="../../_images/scale.png"><img alt="../../_images/scale.png" src="../../_images/scale.png" style="width: 600px;" /></a>
+</figure>
+</div></blockquote>
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.optim.kernel.Scale.forward">
+<span class="sig-name descname"><span class="pre">forward</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/optim/kernel/#Scale.forward"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.optim.kernel.Scale.forward" title="Permalink to this definition">¶</a></dt>
+<dd><dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>input</strong> (<em>torch.Tensor</em>) – the input tensor (non-negative).</p>
+</dd>
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+  <section id="pypose-optim-kernel-softlone">
+<h1>pypose.optim.kernel.SoftLOne<a class="headerlink" href="#pypose-optim-kernel-softlone" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.optim.kernel.SoftLOne">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.optim.kernel.</span></span><span class="sig-name descname"><span class="pre">SoftLOne</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">delta</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1.0</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/optim/kernel/#SoftLOne"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.optim.kernel.SoftLOne" title="Permalink to this definition">¶</a></dt>
+<dd><p>The robust SoftLOne kernel cost function.</p>
+<div class="math">
+\[\bm{y}_i=2\left ( \delta \sqrt{\frac{1}{{\delta{}}^{2}}+\bm{x}_i}- 1\right )
+
+\]</div>
+<p>where <span class="math">\(\delta\)</span> (delta) is a hyperparameter, <span class="math">\(\bm{x}\)</span> and <span class="math">\(\bm{y}\)</span>
+are the input and output tensors, respectively.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>delta</strong> (<em>float</em><em>, </em><em>optional</em>) – Specify the SoftLOne cost. The value must be positive. Default: 1.0</p>
+</dd>
+</dl>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The input has to be a non-negative tensor and the output tensor has the same shape with the
+input.</p>
+</div>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">pypose.optim.kernel</span> <span class="k">as</span> <span class="nn">ppok</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">kernel</span> <span class="o">=</span> <span class="n">ppok</span><span class="o">.</span><span class="n">SoftLOne</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="nb">input</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">kernel</span><span class="p">(</span><span class="nb">input</span><span class="p">)</span>
+<span class="go">tensor([0.0000, 0.4495, 0.8284, 1.4641, 2.0000])</span>
+</pre></div>
+</div>
+<blockquote>
+<div><figure class="align-default">
+<a class="reference internal image-reference" href="../../_images/softlone.png"><img alt="../../_images/softlone.png" src="../../_images/softlone.png" style="width: 600px;" /></a>
+</figure>
+</div></blockquote>
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.optim.kernel.SoftLOne.forward">
+<span class="sig-name descname"><span class="pre">forward</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/optim/kernel/#SoftLOne.forward"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.optim.kernel.SoftLOne.forward" title="Permalink to this definition">¶</a></dt>
+<dd><dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>input</strong> (<em>torch.Tensor</em>) – the input tensor (non-negative).</p>
+</dd>
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+  <section id="pypose-optim-kernel-tolerant">
+<h1>pypose.optim.kernel.Tolerant<a class="headerlink" href="#pypose-optim-kernel-tolerant" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.optim.kernel.Tolerant">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.optim.kernel.</span></span><span class="sig-name descname"><span class="pre">Tolerant</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">a</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1.0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">b</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">-</span> <span class="pre">1.0</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/optim/kernel/#Tolerant"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.optim.kernel.Tolerant" title="Permalink to this definition">¶</a></dt>
+<dd><p>The robust Tolerant kernel cost function.</p>
+<div class="math">
+\[\bm{y}_i = b\log (1+e^{\frac{\bm{x}_i-a}{b}})-b\log (1+e^{\frac{-a}{b}})
+
+\]</div>
+<p>where <span class="math">\(\bm{a}\)</span>  and <span class="math">\(\bm{b}\)</span> are hyperparameters, <span class="math">\(\bm{x}\)</span>
+and <span class="math">\(\bm{y}\)</span> are the input and output tensors, respectively.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>a</strong> (<em>float</em>) – Specify the Tolerant cost. The value must be positive. Default: 1.0</p></li>
+<li><p><strong>b</strong> (<em>float</em>) – Specify the Tolerant cost. The value must be negative. Default: -1.0</p></li>
+</ul>
+</dd>
+</dl>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The input has to be a non-negative tensor and the output tensor has the same shape with
+the input.</p>
+</div>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">pypose.optim.kernel</span> <span class="k">as</span> <span class="nn">ppok</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">kernel</span> <span class="o">=</span> <span class="n">ppok</span><span class="o">.</span><span class="n">Tolerant</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="nb">input</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">kernel</span><span class="p">(</span><span class="nb">input</span><span class="p">)</span>
+<span class="go">tensor([0.0000, 0.4636, 0.7854, 1.1071, 1.2490])</span>
+</pre></div>
+</div>
+<figure class="align-default">
+<a class="reference internal image-reference" href="../../_images/tolerant.png"><img alt="../../_images/tolerant.png" src="../../_images/tolerant.png" style="width: 600px;" /></a>
+</figure>
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.optim.kernel.Tolerant.forward">
+<span class="sig-name descname"><span class="pre">forward</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/optim/kernel/#Tolerant.forward"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.optim.kernel.Tolerant.forward" title="Permalink to this definition">¶</a></dt>
+<dd><dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>input</strong> (<em>torch.Tensor</em>) – the input tensor (non-negative).</p>
+</dd>
+</dl>
+</dd></dl>
+
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+
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+  <section id="pypose-optim-scheduler-stoponplateau">
+<h1>pypose.optim.scheduler.StopOnPlateau<a class="headerlink" href="#pypose-optim-scheduler-stoponplateau" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.optim.scheduler.StopOnPlateau">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.optim.scheduler.</span></span><span class="sig-name descname"><span class="pre">StopOnPlateau</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">optimizer</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">steps</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">patience</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">5</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">decreasing</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0.001</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">verbose</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/optim/scheduler/#StopOnPlateau"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.optim.scheduler.StopOnPlateau" title="Permalink to this definition">¶</a></dt>
+<dd><p>A scheduler to stop an optimizer when no relative loss ‘decreasing’ is seen for a ‘patience’
+number of steps.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>optimizer</strong> (<em>Optimizer</em>) – Wrapped optimizer.</p></li>
+<li><p><strong>steps</strong> (<em>int</em>) – Maximum number of interations optimizer will step.</p></li>
+<li><p><strong>patience</strong> (<em>int</em>) – Number of steps with no loss ‘decreasing’ is seen. For example, if
+<code class="docutils literal notranslate"><span class="pre">patience</span> <span class="pre">=</span> <span class="pre">2</span></code>, then it ignores the first 2 steps with no improvement, and stop
+the optimizer after the 3rd step if the loss has no decerasing. Default: 5.</p></li>
+<li><p><strong>decreasing</strong> (<em>float</em>) – relative loss decreasing used to count the number of patience steps.
+Default: 1e-3.</p></li>
+<li><p><strong>verbose</strong> (<em>bool</em>) – if <code class="docutils literal notranslate"><span class="pre">True</span></code>, prints a message to stdout for each step. Default: <code class="docutils literal notranslate"><span class="pre">False</span></code>.</p></li>
+</ul>
+</dd>
+</dl>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The users have two options to use a scheduler.
+The first one is to call the <a class="reference internal" href="#pypose.optim.scheduler.StopOnPlateau.step" title="pypose.optim.scheduler.StopOnPlateau.step"><code class="xref py py-meth docutils literal notranslate"><span class="pre">step()</span></code></a> method for multiple times, which is easy to be
+extended for customized operation in each iteration.
+The second one is to call the <a class="reference internal" href="#pypose.optim.scheduler.StopOnPlateau.optimize" title="pypose.optim.scheduler.StopOnPlateau.optimize"><code class="xref py py-meth docutils literal notranslate"><span class="pre">optimize()</span></code></a> method, which interally calls <a class="reference internal" href="#pypose.optim.scheduler.StopOnPlateau.step" title="pypose.optim.scheduler.StopOnPlateau.step"><code class="xref py py-meth docutils literal notranslate"><span class="pre">step()</span></code></a>
+multiple times to perform full optimization.
+See examples below.</p>
+</div>
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.optim.scheduler.StopOnPlateau.optimize">
+<span class="sig-name descname"><span class="pre">optimize</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">target</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">weight</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/optim/scheduler/#StopOnPlateau.optimize"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.optim.scheduler.StopOnPlateau.optimize" title="Permalink to this definition">¶</a></dt>
+<dd><p>Perform full optimization steps.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>input</strong> (<em>Tensor/LieTensor</em><em> or </em><em>tuple of Tensors/LieTensors</em>) – the input to the model.</p></li>
+<li><p><strong>target</strong> (<em>Tensor/LieTensor</em>) – the model target to optimize.
+If not given, the squared model output is minimized. Defaults: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+<li><p><strong>weight</strong> (<em>Tensor</em><em>, </em><em>optional</em>) – a square positive definite matrix defining the weight of
+model residual. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+</ul>
+</dd>
+</dl>
+<p>The above arguments are sent to optimizers. More details go to
+<a class="reference internal" href="../pypose.optim.LevenbergMarquardt/#pypose.optim.LevenbergMarquardt" title="pypose.optim.LevenbergMarquardt"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.optim.LevenbergMarquardt</span></code></a> or <a class="reference internal" href="../pypose.optim.GaussNewton/#pypose.optim.GaussNewton" title="pypose.optim.GaussNewton"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.optim.GaussNewton</span></code></a>.</p>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="k">class</span> <span class="nc">PoseInv</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+<span class="gp">...</span>
+<span class="gp">... </span>    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">dim</span><span class="p">):</span>
+<span class="gp">... </span>        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+<span class="gp">... </span>        <span class="bp">self</span><span class="o">.</span><span class="n">pose</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">Parameter</span><span class="p">(</span><span class="n">pp</span><span class="o">.</span><span class="n">randn_SE3</span><span class="p">(</span><span class="o">*</span><span class="n">dim</span><span class="p">))</span>
+<span class="gp">...</span>
+<span class="gp">... </span>    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">input</span><span class="p">):</span>
+<span class="gp">... </span>        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">pose</span> <span class="o">@</span> <span class="nb">input</span><span class="p">)</span><span class="o">.</span><span class="n">Log</span><span class="p">()</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span>
+<span class="gp">...</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">device</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">device</span><span class="p">(</span><span class="s2">&quot;cuda&quot;</span> <span class="k">if</span> <span class="n">torch</span><span class="o">.</span><span class="n">cuda</span><span class="o">.</span><span class="n">is_available</span><span class="p">()</span> <span class="k">else</span> <span class="s2">&quot;cpu&quot;</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="nb">input</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_SE3</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">to</span><span class="p">(</span><span class="n">device</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">invnet</span> <span class="o">=</span> <span class="n">PoseInv</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">to</span><span class="p">(</span><span class="n">device</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">strategy</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">optim</span><span class="o">.</span><span class="n">strategy</span><span class="o">.</span><span class="n">Constant</span><span class="p">(</span><span class="n">damping</span><span class="o">=</span><span class="mf">1e-4</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">optimizer</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">optim</span><span class="o">.</span><span class="n">LM</span><span class="p">(</span><span class="n">invnet</span><span class="p">,</span> <span class="n">strategy</span><span class="o">=</span><span class="n">strategy</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">scheduler</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">optim</span><span class="o">.</span><span class="n">scheduler</span><span class="o">.</span><span class="n">StopOnPlateau</span><span class="p">(</span><span class="n">optimizer</span><span class="p">,</span> <span class="n">steps</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> \
+<span class="gp">&gt;&gt;&gt; </span>                    <span class="n">patience</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">decreasing</span><span class="o">=</span><span class="mf">1e-3</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+<span class="gp">...</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">scheduler</span><span class="o">.</span><span class="n">optimize</span><span class="p">(</span><span class="nb">input</span><span class="o">=</span><span class="nb">input</span><span class="p">)</span>
+<span class="go">StopOnPlateau on step 0 Loss 5.199298e+01 --&gt; Loss 8.425808e-06 (reduction/loss: 1.0000e+00).</span>
+<span class="go">StopOnPlateau on step 1 Loss 8.425808e-06 --&gt; Loss 3.456247e-13 (reduction/loss: 1.0000e+00).</span>
+<span class="go">StopOnPlateau on step 2 Loss 3.456247e-13 --&gt; Loss 1.525355e-13 (reduction/loss: 5.5867e-01).</span>
+<span class="go">StopOnPlateau on step 3 Loss 1.525355e-13 --&gt; Loss 6.769275e-14 (reduction/loss: 5.5622e-01).</span>
+<span class="go">StopOnPlateau: Maximum patience steps reached, Quiting..</span>
+</pre></div>
+</div>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.optim.scheduler.StopOnPlateau.step">
+<span class="sig-name descname"><span class="pre">step</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">loss</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/optim/scheduler/#StopOnPlateau.step"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.optim.scheduler.StopOnPlateau.step" title="Permalink to this definition">¶</a></dt>
+<dd><p>Performs a scheduler step.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>loss</strong> (<em>float</em>) – the model loss after one optimizer step.</p>
+</dd>
+</dl>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="k">class</span> <span class="nc">PoseInv</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+<span class="gp">...</span>
+<span class="gp">... </span>    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">dim</span><span class="p">):</span>
+<span class="gp">... </span>        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+<span class="gp">... </span>        <span class="bp">self</span><span class="o">.</span><span class="n">pose</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">Parameter</span><span class="p">(</span><span class="n">pp</span><span class="o">.</span><span class="n">randn_SE3</span><span class="p">(</span><span class="o">*</span><span class="n">dim</span><span class="p">))</span>
+<span class="gp">...</span>
+<span class="gp">... </span>    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">input</span><span class="p">):</span>
+<span class="gp">... </span>        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">pose</span> <span class="o">@</span> <span class="nb">input</span><span class="p">)</span><span class="o">.</span><span class="n">Log</span><span class="p">()</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span>
+<span class="gp">...</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">device</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">device</span><span class="p">(</span><span class="s2">&quot;cuda&quot;</span> <span class="k">if</span> <span class="n">torch</span><span class="o">.</span><span class="n">cuda</span><span class="o">.</span><span class="n">is_available</span><span class="p">()</span> <span class="k">else</span> <span class="s2">&quot;cpu&quot;</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="nb">input</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_SE3</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">to</span><span class="p">(</span><span class="n">device</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">invnet</span> <span class="o">=</span> <span class="n">PoseInv</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">to</span><span class="p">(</span><span class="n">device</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">strategy</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">optim</span><span class="o">.</span><span class="n">strategy</span><span class="o">.</span><span class="n">Constant</span><span class="p">(</span><span class="n">damping</span><span class="o">=</span><span class="mf">1e-4</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">optimizer</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">optim</span><span class="o">.</span><span class="n">LM</span><span class="p">(</span><span class="n">invnet</span><span class="p">,</span> <span class="n">strategy</span><span class="o">=</span><span class="n">strategy</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">scheduler</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">optim</span><span class="o">.</span><span class="n">scheduler</span><span class="o">.</span><span class="n">StopOnPlateau</span><span class="p">(</span><span class="n">optimizer</span><span class="p">,</span> <span class="n">steps</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> \
+<span class="gp">&gt;&gt;&gt; </span>                    <span class="n">patience</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">decreasing</span><span class="o">=</span><span class="mf">1e-3</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+<span class="gp">...</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="k">while</span> <span class="n">scheduler</span><span class="o">.</span><span class="n">continual</span><span class="p">():</span>
+<span class="gp">... </span>    <span class="n">loss</span> <span class="o">=</span> <span class="n">optimizer</span><span class="o">.</span><span class="n">step</span><span class="p">(</span><span class="nb">input</span><span class="p">)</span>
+<span class="gp">... </span>    <span class="n">scheduler</span><span class="o">.</span><span class="n">step</span><span class="p">(</span><span class="n">loss</span><span class="p">)</span>
+<span class="go">StopOnPlateau on step 0 Loss 9.337769e+01 --&gt; Loss 3.502787e-05 (reduction/loss: 1.0000e+00).</span>
+<span class="go">StopOnPlateau on step 1 Loss 3.502787e-05 --&gt; Loss 4.527339e-13 (reduction/loss: 1.0000e+00).</span>
+<span class="go">StopOnPlateau on step 2 Loss 4.527339e-13 --&gt; Loss 7.112640e-14 (reduction/loss: 8.4290e-01).</span>
+<span class="go">StopOnPlateau on step 3 Loss 7.112640e-14 --&gt; Loss 3.693307e-14 (reduction/loss: 4.8074e-01).</span>
+<span class="go">StopOnPlateau: Maximum patience steps reached, Quiting..</span>
+</pre></div>
+</div>
+</dd></dl>
+
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+  <section id="pypose-optim-solver-cg">
+<h1>pypose.optim.solver.CG<a class="headerlink" href="#pypose-optim-solver-cg" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.optim.solver.CG">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.optim.solver.</span></span><span class="sig-name descname"><span class="pre">CG</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">maxiter</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">tol</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1e-05</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/optim/solver/#CG"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.optim.solver.CG" title="Permalink to this definition">¶</a></dt>
+<dd><p>The batched linear solver with conjugate gradient method.</p>
+<div class="math">
+\[\mathbf{A}_i \bm{x}_i = \mathbf{b}_i,
+
+\]</div>
+<p>where <span class="math">\(\mathbf{A}_i \in \mathbb{C}^{M \times N}\)</span> and <span class="math">\(\bm{b}_i \in
+\mathbb{C}^{M \times 1}\)</span> are the <span class="math">\(i\)</span>-th item of batched linear equations.</p>
+<p>This function is a 1:1 replica of <a class="reference external" href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.sparse.linalg.cg.html">scipy.sparse.linalg.cg</a>.
+The solution is consistent with the scipy version up to numerical precision.
+Variable names are kept the same as the scipy version for easy reference.
+We recommend using only non-batched or batch size 1 input for this solver, as
+the batched version was not appeared in the original scipy version. When handling
+sparse matrices, the batched computation may introduce additional overhead.</p>
+<p class="rubric">Examples</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="c1"># dense example</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">pypose.optim.solver</span> <span class="k">as</span> <span class="nn">ppos</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([[</span><span class="mf">0.1802967</span><span class="p">,</span> <span class="mf">0.3151198</span><span class="p">,</span> <span class="mf">0.4548111</span><span class="p">,</span> <span class="mf">0.3860016</span><span class="p">,</span> <span class="mf">0.2870615</span><span class="p">],</span>
+<span class="go">                      [0.3151198, 1.4575327, 1.5533425, 1.0540756, 1.0795838],</span>
+<span class="go">                      [0.4548111, 1.5533425, 2.3674474, 1.1222278, 1.2365348],</span>
+<span class="go">                      [0.3860016, 1.0540756, 1.1222278, 1.3748058, 1.2223261],</span>
+<span class="go">                      [0.2870615, 1.0795838, 1.2365348, 1.2223261, 1.2577004]])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([[</span> <span class="mf">2.64306851</span><span class="p">],</span>
+<span class="go">                      [-0.03593633],</span>
+<span class="go">                      [ 0.73612658],</span>
+<span class="go">                      [ 0.51501254],</span>
+<span class="go">                      [-0.26689271]])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">solver</span> <span class="o">=</span> <span class="n">ppos</span><span class="o">.</span><span class="n">CG</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">solver</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
+<span class="go">tensor([[246.4098],</span>
+<span class="go">        [ 22.6997],</span>
+<span class="go">        [-56.9239],</span>
+<span class="go">        [-161.7914],</span>
+<span class="go">        [137.2683]])</span>
+</pre></div>
+</div>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="c1"># sparse csr example</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">pypose.optim.solver</span> <span class="k">as</span> <span class="nn">ppos</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">crow_indices</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">col_indices</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">values</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">float</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">sparse_csr_tensor</span><span class="p">(</span><span class="n">crow_indices</span><span class="p">,</span> <span class="n">col_indices</span><span class="p">,</span> <span class="n">values</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">to_dense</span><span class="p">()</span>  <span class="c1"># visualize</span>
+<span class="go">tensor([[1., 2.],</span>
+<span class="go">        [3., 4.]])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([[</span><span class="mf">1.</span><span class="p">],</span> <span class="p">[</span><span class="mf">2.</span><span class="p">]])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">solver</span> <span class="o">=</span> <span class="n">ppos</span><span class="o">.</span><span class="n">CG</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">solver</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
+<span class="go">tensor([-4.4052e-05,  5.0003e-01])</span>
+</pre></div>
+</div>
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.optim.solver.CG.forward">
+<span class="sig-name descname"><span class="pre">forward</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">A</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">b</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">x</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">M</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/optim/solver/#CG.forward"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.optim.solver.CG.forward" title="Permalink to this definition">¶</a></dt>
+<dd><dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>A</strong> (<em>Tensor</em>) – the input tensor. It is assumed to be a symmetric
+positive-definite matrix. Layout is allowed to be COO, CSR, BSR, or dense.</p></li>
+<li><p><strong>b</strong> (<em>Tensor</em>) – the tensor on the right hand side. Layout could be sparse or dense
+but is only allowed to be a type that is compatible with the layout of A.
+In other words, <cite>A &#64; b</cite> operation must be supported by the layout of A.</p></li>
+<li><p><strong>x</strong> (<em>Tensor</em><em>, </em><em>optional</em>) – the initial guess for the solution. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+<li><p><strong>M</strong> (<em>Tensor</em><em>, </em><em>optional</em>) – the preconditioner for A. Layout is allowed to be COO,
+CSR, BSR, or dense. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>the solved tensor. Layout is the same as the layout of b.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>Tensor</p>
+</dd>
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+  <section id="pypose-optim-solver-cholesky">
+<h1>pypose.optim.solver.Cholesky<a class="headerlink" href="#pypose-optim-solver-cholesky" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.optim.solver.Cholesky">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.optim.solver.</span></span><span class="sig-name descname"><span class="pre">Cholesky</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">upper</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/optim/solver/#Cholesky"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.optim.solver.Cholesky" title="Permalink to this definition">¶</a></dt>
+<dd><p>The batched linear solver with Cholesky decomposition.</p>
+<div class="math">
+\[\mathbf{A}_i \bm{x}_i = \mathbf{b}_i,
+
+\]</div>
+<p>where <span class="math">\(\mathbf{A}_i \in \mathbb{C}^{M \times N}\)</span> and <span class="math">\(\bm{b}_i \in
+\mathbb{C}^{M \times 1}\)</span> are the <span class="math">\(i\)</span>-th item of batched linear equations.
+Note that <span class="math">\(\mathbf{A}_i\)</span> has to be a complex Hermitian or a real symmetric
+positive-definite matrix.</p>
+<p>The solution is given by</p>
+<div class="math">
+\[\begin{align*}
+    \bm{L}_i &= \mathrm{cholesky}(\mathbf{A}_i), \\
+    \bm{x}_i &= \mathrm{cholesky\_solve}(\mathbf{b}_i, \bm{L}_i), \\
+\end{align*}
+
+\]</div>
+<p>where <span class="math">\(\mathrm{cholesky}()\)</span> is the <a class="reference external" href="https://en.wikipedia.org/wiki/Cholesky_decomposition">Cholesky decomposition</a> function.</p>
+<p>More details go to
+<a class="reference external" href="https://pytorch.org/docs/stable/generated/torch.linalg.cholesky.html">torch.linalg.cholesky</a>
+and
+<a class="reference external" href="https://pytorch.org/docs/stable/generated/torch.cholesky_solve.html">torch.cholesky_solve</a>.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>upper</strong> (<em>bool</em><em>, </em><em>optional</em>) – whether use an upper triangular matrix in Cholesky decomposition.
+Default: <code class="docutils literal notranslate"><span class="pre">False</span></code>.</p>
+</dd>
+</dl>
+<p class="rubric">Examples</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">pypose.optim.solver</span> <span class="k">as</span> <span class="nn">ppos</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([[[</span><span class="mf">1.00</span><span class="p">,</span> <span class="mf">0.10</span><span class="p">,</span> <span class="mf">0.00</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.10</span><span class="p">,</span> <span class="mf">1.00</span><span class="p">,</span> <span class="mf">0.20</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.00</span><span class="p">,</span> <span class="mf">0.20</span><span class="p">,</span> <span class="mf">1.00</span><span class="p">]],</span>
+<span class="go">                      [[1.00, 0.20, 0.10], [0.20, 1.00, 0.20], [0.10, 0.20, 1.00]]])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([[[</span><span class="mf">1.</span><span class="p">],</span> <span class="p">[</span><span class="mf">2.</span><span class="p">],</span> <span class="p">[</span><span class="mf">3.</span><span class="p">]],</span> <span class="p">[[</span><span class="mf">1.</span><span class="p">],</span> <span class="p">[</span><span class="mf">2.</span><span class="p">],</span> <span class="p">[</span><span class="mf">3.</span><span class="p">]]])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">solver</span> <span class="o">=</span> <span class="n">ppos</span><span class="o">.</span><span class="n">Cholesky</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">solver</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
+<span class="go">tensor([[[0.8632],</span>
+<span class="go">         [1.3684],</span>
+<span class="go">         [2.7263]],</span>
+<span class="go">        [[0.4575],</span>
+<span class="go">         [1.3725],</span>
+<span class="go">         [2.6797]]])</span>
+</pre></div>
+</div>
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.optim.solver.Cholesky.forward">
+<span class="sig-name descname"><span class="pre">forward</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">A</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">b</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/optim/solver/#Cholesky.forward"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.optim.solver.Cholesky.forward" title="Permalink to this definition">¶</a></dt>
+<dd><dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>A</strong> (<em>Tensor</em>) – the input batched tensor.</p></li>
+<li><p><strong>b</strong> (<em>Tensor</em>) – the batched tensor on the right hand side.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>the solved batched tensor.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>Tensor</p>
+</dd>
+</dl>
+</dd></dl>
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+              
+  <section id="pypose-optim-solver-lstsq">
+<h1>pypose.optim.solver.LSTSQ<a class="headerlink" href="#pypose-optim-solver-lstsq" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.optim.solver.LSTSQ">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.optim.solver.</span></span><span class="sig-name descname"><span class="pre">LSTSQ</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">rcond</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">driver</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/optim/solver/#LSTSQ"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.optim.solver.LSTSQ" title="Permalink to this definition">¶</a></dt>
+<dd><p>The batched linear solver with fast pseudo inversion.</p>
+<div class="math">
+\[\mathbf{A}_i \bm{x}_i = \mathbf{b}_i,
+
+\]</div>
+<p>where <span class="math">\(\mathbf{A}_i \in \mathbb{C}^{M \times N}\)</span> and <span class="math">\(\bm{b}_i \in
+\mathbb{C}^{M \times 1}\)</span> are the <span class="math">\(i\)</span>-th item of batched linear equations.</p>
+<p>The solution is given by</p>
+<div class="math">
+\[\bm{x}_i = \mathrm{lstsq}(\mathbf{A}_i, \mathbf{b}_i),
+
+\]</div>
+<p>where <span class="math">\(\mathrm{lstsq}()\)</span> computes a solution to the least squares problem
+of a system of linear equations. More details go to <a class="reference external" href="https://pytorch.org/docs/stable/generated/torch.linalg.lstsq.html">torch.linalg.lstsq</a>.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>rcond</strong> (<em>float</em><em>, </em><em>optional</em>) – Cut-off ratio for small singular values. For the purposes of
+rank determination, singular values are treated as zero if they are smaller than
+rcond times the largest singular value. It is used only when the fast model is
+enabled. If <code class="docutils literal notranslate"><span class="pre">None</span></code>, rcond is set to the machine precision of the dtype of
+<span class="math">\(\mathbf{A}\)</span>. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+<li><p><strong>driver</strong> (<em>string</em><em>, </em><em>optional</em>) – <p>chooses the LAPACK/MAGMA function that will be used. It is
+used only when the fast model is enabled. For CPU users, the valid values are <code class="docutils literal notranslate"><span class="pre">gels</span></code>,
+<code class="docutils literal notranslate"><span class="pre">gelsy</span></code>, <code class="docutils literal notranslate"><span class="pre">gelsd</span></code>, <code class="docutils literal notranslate"><span class="pre">gelss</span></code>. For CUDA users, the only valid driver is <code class="docutils literal notranslate"><span class="pre">gels</span></code>,
+which assumes that input matrices (<span class="math">\(\mathbf{A}\)</span>) are full-rank. If <code class="docutils literal notranslate"><span class="pre">None</span></code>,
+<code class="docutils literal notranslate"><span class="pre">gelsy</span></code> is used for CPU inputs and <code class="docutils literal notranslate"><span class="pre">gels</span></code> for CUDA inputs. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.
+To choose the best driver on CPU consider:</p>
+<ul>
+<li><p>If input matrices (<span class="math">\(\mathbf{A}\)</span>) are well-conditioned (<a class="reference external" href="https://en.wikipedia.org/wiki/Condition_number">condition number</a> is not too large), or you do not
+mind some precision loss.</p>
+<blockquote>
+<div><ul class="simple">
+<li><p>For a general matrix: <code class="docutils literal notranslate"><span class="pre">gelsy</span></code> (QR with pivoting) (default)</p></li>
+<li><p>If A is full-rank: <code class="docutils literal notranslate"><span class="pre">gels</span></code> (QR)</p></li>
+</ul>
+</div></blockquote>
+</li>
+<li><p>If input matrices (<span class="math">\(\mathbf{A}\)</span>) are not well-conditioned.</p>
+<blockquote>
+<div><ul class="simple">
+<li><p><code class="docutils literal notranslate"><span class="pre">gelsd</span></code> (tridiagonal reduction and SVD)</p></li>
+<li><p>But if you run into memory issues: <code class="docutils literal notranslate"><span class="pre">gelss</span></code> (full SVD).</p></li>
+</ul>
+</div></blockquote>
+</li>
+</ul>
+<p>See full description of <a class="reference external" href="https://www.netlib.org/lapack/lug/node27.html">drivers</a>.</p>
+</p></li>
+</ul>
+</dd>
+</dl>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>This solver is faster and more numerically stable than <a class="reference internal" href="../pypose.optim.solver.PINV/#pypose.optim.solver.PINV" title="pypose.optim.solver.PINV"><code class="xref py py-meth docutils literal notranslate"><span class="pre">PINV()</span></code></a>.</p>
+<p>It is also preferred to use <a class="reference internal" href="../pypose.optim.solver.Cholesky/#pypose.optim.solver.Cholesky" title="pypose.optim.solver.Cholesky"><code class="xref py py-meth docutils literal notranslate"><span class="pre">Cholesky()</span></code></a> if input matrices (<span class="math">\(\mathbf{A}\)</span>)
+are guaranteed to complex Hermitian or real symmetric positive-definite.</p>
+</div>
+<p class="rubric">Examples</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">pypose.optim.solver</span> <span class="k">as</span> <span class="nn">ppos</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">solver</span> <span class="o">=</span> <span class="n">ppos</span><span class="o">.</span><span class="n">LSTSQ</span><span class="p">(</span><span class="n">driver</span><span class="o">=</span><span class="s1">&#39;gels&#39;</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">solver</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
+<span class="go">tensor([[[ 0.9997],</span>
+<span class="go">         [-1.3288],</span>
+<span class="go">         [-1.6327]],</span>
+<span class="go">        [[ 3.1639],</span>
+<span class="go">         [-0.5379],</span>
+<span class="go">         [-1.2872]]])</span>
+</pre></div>
+</div>
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.optim.solver.LSTSQ.forward">
+<span class="sig-name descname"><span class="pre">forward</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">A</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">b</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/optim/solver/#LSTSQ.forward"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.optim.solver.LSTSQ.forward" title="Permalink to this definition">¶</a></dt>
+<dd><dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>A</strong> (<em>Tensor</em>) – the input batched tensor.</p></li>
+<li><p><strong>b</strong> (<em>Tensor</em>) – the batched tensor on the right hand side.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>the solved batched tensor.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>Tensor</p>
+</dd>
+</dl>
+</dd></dl>
+
+</dd></dl>
+
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+  <section id="pypose-optim-solver-pinv">
+<h1>pypose.optim.solver.PINV<a class="headerlink" href="#pypose-optim-solver-pinv" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.optim.solver.PINV">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.optim.solver.</span></span><span class="sig-name descname"><span class="pre">PINV</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">atol</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">rtol</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">hermitian</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/optim/solver/#PINV"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.optim.solver.PINV" title="Permalink to this definition">¶</a></dt>
+<dd><p>The batched linear solver with pseudo inversion.</p>
+<div class="math">
+\[\mathbf{A}_i \bm{x}_i = \mathbf{b}_i,
+
+\]</div>
+<p>where <span class="math">\(\mathbf{A}_i \in \mathbb{C}^{M \times N}\)</span> and <span class="math">\(\bm{b}_i \in
+\mathbb{C}^{M \times 1}\)</span> are the <span class="math">\(i\)</span>-th item of batched linear equations.</p>
+<p>The solution is given by</p>
+<div class="math">
+\[\bm{x}_i = \mathrm{pinv}(\mathbf{A}_i) \mathbf{b}_i,
+
+\]</div>
+<p>where <span class="math">\(\mathrm{pinv}()\)</span> is the <a class="reference external" href="https://en.wikipedia.org/wiki/Moore-Penrose_inverse">pseudo inversion</a> function.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>atol</strong> (<em>float</em><em>, </em><em>Tensor</em><em>, </em><em>optional</em>) – the absolute tolerance value. When None it’s considered to
+be zero. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+<li><p><strong>rtol</strong> (<em>float</em><em>, </em><em>Tensor</em><em>, </em><em>optional</em>) – the relative tolerance value. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+<li><p><strong>hermitian</strong> (<em>bool</em><em>, </em><em>optional</em>) – indicates whether <span class="math">\(\mathbf{A}\)</span> is Hermitian if complex or
+symmetric if real. Default: <code class="docutils literal notranslate"><span class="pre">False</span></code>.</p></li>
+</ul>
+</dd>
+</dl>
+<p>More details go to <a class="reference external" href="https://pytorch.org/docs/stable/generated/torch.linalg.pinv.html">torch.linalg.pinv</a>.</p>
+<div class="admonition warning">
+<p class="admonition-title">Warning</p>
+<p>It is always preferred to use <a class="reference internal" href="../pypose.optim.solver.LSTSQ/#pypose.optim.solver.LSTSQ" title="pypose.optim.solver.LSTSQ"><code class="xref py py-meth docutils literal notranslate"><span class="pre">LSTSQ()</span></code></a>, which is faster and more numerically stable.</p>
+</div>
+<p class="rubric">Examples</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">pypose.optim.solver</span> <span class="k">as</span> <span class="nn">ppos</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">solver</span> <span class="o">=</span> <span class="n">ppos</span><span class="o">.</span><span class="n">PINV</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">solver</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
+<span class="go">tensor([[[-0.2675],</span>
+<span class="go">         [-0.1897],</span>
+<span class="go">         [ 0.2708]],</span>
+<span class="go">        [[-0.3472],</span>
+<span class="go">         [ 1.1191],</span>
+<span class="go">         [ 0.3301]]])</span>
+</pre></div>
+</div>
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.optim.solver.PINV.forward">
+<span class="sig-name descname"><span class="pre">forward</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">A</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">b</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/optim/solver/#PINV.forward"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.optim.solver.PINV.forward" title="Permalink to this definition">¶</a></dt>
+<dd><dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>A</strong> (<em>Tensor</em>) – the input batched tensor.</p></li>
+<li><p><strong>b</strong> (<em>Tensor</em>) – the batched tensor on the right hand side.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>the solved batched tensor.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>Tensor</p>
+</dd>
+</dl>
+</dd></dl>
+
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+  <section id="pypose-optim-strategy-adaptive">
+<h1>pypose.optim.strategy.Adaptive<a class="headerlink" href="#pypose-optim-strategy-adaptive" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.optim.strategy.Adaptive">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.optim.strategy.</span></span><span class="sig-name descname"><span class="pre">Adaptive</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">damping</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1e-06</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">high</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0.5</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">low</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0.001</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">up</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">2.0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">down</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0.5</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">min</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1e-06</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">max</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1e+16</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/optim/strategy/#Adaptive"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.optim.strategy.Adaptive" title="Permalink to this definition">¶</a></dt>
+<dd><p>Adaptive damping strategy used in the Levenberg-Marquardt (LM) algorithm.</p>
+<p>It scales down the damping factor if the optimization step is very successful, unchanges the
+factor if the step is successful, and scales the factor up if the step is unsuccessful.</p>
+<div class="math">
+\[\begin{aligned}
+     &\rule{113mm}{0.4pt}                                                                 \\
+     &\textbf{input}: \lambda ~\text{(damping)}, \bm{f}(\bm{\theta})~(\text{model}),
+                      \theta_h~(\text{high}), \theta_l~(\text{low}), \delta_u~(\text{up}),
+                      \delta_d~(\text{down}),                                             \\
+     &\hspace{13mm}   \epsilon_{s}~(\text{min}), \epsilon_{l}~(\text{max})                \\
+     &\rule{113mm}{0.4pt}                                                                 \\
+     &\rho = \frac{ \|\bm{f}(\bm{\theta})\|^2 - \|\bm{f}(\bm{\theta} + \delta)\|^2}
+             {\|\bm{f}(\bm{\theta})\|^2 - \|\bm{f}(\bm{\theta}) + \mathbf{J}\delta\|^2}
+             ~\text{(step quality)}                                                       \\
+     &\textbf{if} ~~ \rho > \theta_h ~ \text{(``very successful'' step)}                  \\
+     &\hspace{5mm} \lambda \leftarrow \delta_d \cdot \lambda                              \\
+     &\textbf{elif} ~~ \rho > \theta_l ~ \text{(``successful'' step)}                     \\
+     &\hspace{5mm} \lambda \leftarrow \lambda                                             \\
+     &\textbf{else} ~ \text{(``unsuccessful'' step)}                                      \\
+     &\hspace{5mm} \lambda \leftarrow \delta_u  \cdot \lambda                             \\
+     &\lambda \leftarrow \mathrm{min}(\mathrm{max}(\lambda, \epsilon_{s}), \epsilon_{l})  \\
+     &\rule{113mm}{0.4pt}                                                          \\[-1.ex]
+     &\textbf{return} \:  \lambda                                                  \\[-1.ex]
+     &\rule{113mm}{0.4pt}                                                          \\[-1.ex]
+\end{aligned}
+
+\]</div>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>damping</strong> (<em>float</em><em>, </em><em>optional</em>) – initial damping factor of LM optimizer. Default: 1e-6.</p></li>
+<li><p><strong>high</strong> (<em>float</em><em>, </em><em>optional</em>) – high threshold for scaling down the damping factor.
+Default: 0.5.</p></li>
+<li><p><strong>low</strong> (<em>float</em><em>, </em><em>optional</em>) – low threshold for scaling up the damping factor.
+Default: 1e-3.</p></li>
+<li><p><strong>down</strong> (<em>float</em><em>, </em><em>optional</em>) – the down scaling factor in the range of <span class="math">\((0,1)\)</span>.
+Default: 0.5.</p></li>
+<li><p><strong>up</strong> (<em>float</em><em>, </em><em>optional</em>) – the up scaling factor in the range of <span class="math">\((1,\infty)\)</span>.
+Default: 2.0.</p></li>
+<li><p><strong>min</strong> (<em>float</em><em>, </em><em>optional</em>) – the lower-bound of damping factor. Default: 1e-6.</p></li>
+<li><p><strong>max</strong> (<em>float</em><em>, </em><em>optional</em>) – the upper-bound of damping factor. Default: 1e16.</p></li>
+</ul>
+</dd>
+</dl>
+<p>More details about the optimization process go to <a class="reference internal" href="../pypose.optim.LevenbergMarquardt/#pypose.optim.LevenbergMarquardt" title="pypose.optim.LevenbergMarquardt"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.optim.LevenbergMarquardt()</span></code></a>.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>For efficiency, we calculate the denominator of the step quality <span class="math">\(\rho\)</span> as</p>
+<div class="math">
+\[\begin{aligned}
+& \|\bm{f}(\bm{\theta})\|^2 - \|\bm{f}(\bm{\theta}) + \mathbf{J}\delta\|^2 \\
+& = \bm{f}^T\bm{f} - \left( \bm{f}^T\bm{f} + 2 \bm{f}^T \delta +
+    \delta^T \mathbf{J}^T\mathbf{J}\delta \right) \\
+& = -(\mathbf{J} \delta)^T (2\bm{f} + \mathbf{J}\delta)
+\end{aligned}
+
+\]</div>
+<p>where <span class="math">\(\mathbf{J}\)</span> is the Jacobian of the model <span class="math">\(\bm{f}\)</span> at evaluation
+point <span class="math">\(\bm{\theta}\)</span>.</p>
+</div>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="k">class</span> <span class="nc">PoseInv</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+<span class="gp">... </span>    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">dim</span><span class="p">):</span>
+<span class="gp">... </span>        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+<span class="gp">... </span>        <span class="bp">self</span><span class="o">.</span><span class="n">pose</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">Parameter</span><span class="p">(</span><span class="n">pp</span><span class="o">.</span><span class="n">randn_SE3</span><span class="p">(</span><span class="o">*</span><span class="n">dim</span><span class="p">))</span>
+<span class="gp">...</span>
+<span class="gp">... </span>    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">inputs</span><span class="p">):</span>
+<span class="gp">... </span>        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">pose</span> <span class="o">@</span> <span class="n">inputs</span><span class="p">)</span><span class="o">.</span><span class="n">Log</span><span class="p">()</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span>
+<span class="gp">...</span>
+<span class="gp">... </span><span class="n">device</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">device</span><span class="p">(</span><span class="s2">&quot;cuda&quot;</span> <span class="k">if</span> <span class="n">torch</span><span class="o">.</span><span class="n">cuda</span><span class="o">.</span><span class="n">is_available</span><span class="p">()</span> <span class="k">else</span> <span class="s2">&quot;cpu&quot;</span><span class="p">)</span>
+<span class="gp">... </span><span class="n">inputs</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_SE3</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">to</span><span class="p">(</span><span class="n">device</span><span class="p">)</span>
+<span class="gp">... </span><span class="n">invnet</span> <span class="o">=</span> <span class="n">PoseInv</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">to</span><span class="p">(</span><span class="n">device</span><span class="p">)</span>
+<span class="gp">... </span><span class="n">strategy</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">optim</span><span class="o">.</span><span class="n">strategy</span><span class="o">.</span><span class="n">Adaptive</span><span class="p">(</span><span class="n">damping</span><span class="o">=</span><span class="mf">1e-6</span><span class="p">)</span>
+<span class="gp">... </span><span class="n">optimizer</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">optim</span><span class="o">.</span><span class="n">LM</span><span class="p">(</span><span class="n">invnet</span><span class="p">,</span> <span class="n">strategy</span><span class="o">=</span><span class="n">strategy</span><span class="p">)</span>
+<span class="gp">...</span>
+<span class="gp">... </span><span class="k">for</span> <span class="n">idx</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10</span><span class="p">):</span>
+<span class="gp">... </span>    <span class="n">loss</span> <span class="o">=</span> <span class="n">optimizer</span><span class="o">.</span><span class="n">step</span><span class="p">(</span><span class="n">inputs</span><span class="p">)</span>
+<span class="gp">... </span>    <span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Pose loss </span><span class="si">%.7f</span><span class="s1"> @ </span><span class="si">%d</span><span class="s1">it&#39;</span><span class="o">%</span><span class="p">(</span><span class="n">loss</span><span class="p">,</span> <span class="n">idx</span><span class="p">))</span>
+<span class="gp">... </span>    <span class="k">if</span> <span class="n">loss</span> <span class="o">&lt;</span> <span class="mf">1e-5</span><span class="p">:</span>
+<span class="gp">... </span>        <span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Early Stoping!&#39;</span><span class="p">)</span>
+<span class="gp">... </span>        <span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Optimization Early Done with loss:&#39;</span><span class="p">,</span> <span class="n">loss</span><span class="o">.</span><span class="n">item</span><span class="p">())</span>
+<span class="gp">... </span>        <span class="k">break</span>
+<span class="go">Pose loss 0.0000000 @ 0it</span>
+<span class="go">Early Stoping!</span>
+<span class="go">Optimization Early Done with loss: 9.236661990819073e-10</span>
+</pre></div>
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+  <section id="pypose-optim-strategy-constant">
+<h1>pypose.optim.strategy.Constant<a class="headerlink" href="#pypose-optim-strategy-constant" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.optim.strategy.Constant">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.optim.strategy.</span></span><span class="sig-name descname"><span class="pre">Constant</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">damping</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1e-06</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/optim/strategy/#Constant"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.optim.strategy.Constant" title="Permalink to this definition">¶</a></dt>
+<dd><p>Constant damping strategy used in the Levenberg-Marquardt (LM) algorithm.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>damping</strong> (<em>float</em><em>, </em><em>optional</em>) – damping factor of LM optimizer. Default: 1e-6.</p>
+</dd>
+</dl>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="k">class</span> <span class="nc">PoseInv</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+<span class="gp">... </span>    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">dim</span><span class="p">):</span>
+<span class="gp">... </span>        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+<span class="gp">... </span>        <span class="bp">self</span><span class="o">.</span><span class="n">pose</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">Parameter</span><span class="p">(</span><span class="n">pp</span><span class="o">.</span><span class="n">randn_SE3</span><span class="p">(</span><span class="o">*</span><span class="n">dim</span><span class="p">))</span>
+<span class="gp">...</span>
+<span class="gp">... </span>    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">inputs</span><span class="p">):</span>
+<span class="gp">... </span>        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">pose</span> <span class="o">@</span> <span class="n">inputs</span><span class="p">)</span><span class="o">.</span><span class="n">Log</span><span class="p">()</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span>
+<span class="gp">...</span>
+<span class="gp">... </span><span class="n">device</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">device</span><span class="p">(</span><span class="s2">&quot;cuda&quot;</span> <span class="k">if</span> <span class="n">torch</span><span class="o">.</span><span class="n">cuda</span><span class="o">.</span><span class="n">is_available</span><span class="p">()</span> <span class="k">else</span> <span class="s2">&quot;cpu&quot;</span><span class="p">)</span>
+<span class="gp">... </span><span class="n">inputs</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_SE3</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">to</span><span class="p">(</span><span class="n">device</span><span class="p">)</span>
+<span class="gp">... </span><span class="n">invnet</span> <span class="o">=</span> <span class="n">PoseInv</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">to</span><span class="p">(</span><span class="n">device</span><span class="p">)</span>
+<span class="gp">... </span><span class="n">strategy</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">optim</span><span class="o">.</span><span class="n">strategy</span><span class="o">.</span><span class="n">Constant</span><span class="p">(</span><span class="n">damping</span><span class="o">=</span><span class="mf">1e-6</span><span class="p">)</span>
+<span class="gp">... </span><span class="n">optimizer</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">optim</span><span class="o">.</span><span class="n">LM</span><span class="p">(</span><span class="n">invnet</span><span class="p">,</span> <span class="n">strategy</span><span class="o">=</span><span class="n">strategy</span><span class="p">)</span>
+<span class="gp">...</span>
+<span class="gp">... </span><span class="k">for</span> <span class="n">idx</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10</span><span class="p">):</span>
+<span class="gp">... </span>    <span class="n">loss</span> <span class="o">=</span> <span class="n">optimizer</span><span class="o">.</span><span class="n">step</span><span class="p">(</span><span class="n">inputs</span><span class="p">)</span>
+<span class="gp">... </span>    <span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Pose loss </span><span class="si">%.7f</span><span class="s1"> @ </span><span class="si">%d</span><span class="s1">it&#39;</span><span class="o">%</span><span class="p">(</span><span class="n">loss</span><span class="p">,</span> <span class="n">idx</span><span class="p">))</span>
+<span class="gp">... </span>    <span class="k">if</span> <span class="n">loss</span> <span class="o">&lt;</span> <span class="mf">1e-5</span><span class="p">:</span>
+<span class="gp">... </span>        <span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Early Stoping!&#39;</span><span class="p">)</span>
+<span class="gp">... </span>        <span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Optimization Early Done with loss:&#39;</span><span class="p">,</span> <span class="n">loss</span><span class="o">.</span><span class="n">item</span><span class="p">())</span>
+<span class="gp">... </span>        <span class="k">break</span>
+<span class="go">Pose loss 0.0000000 @ 0it</span>
+<span class="go">Early Stoping!</span>
+<span class="go">Optimization Early Done with loss: 9.236661990819073e-10</span>
+</pre></div>
+</div>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
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+  <section id="pypose-optim-strategy-trustregion">
+<h1>pypose.optim.strategy.TrustRegion<a class="headerlink" href="#pypose-optim-strategy-trustregion" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.optim.strategy.TrustRegion">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.optim.strategy.</span></span><span class="sig-name descname"><span class="pre">TrustRegion</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">radius</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1000000.0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">high</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0.5</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">low</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0.001</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">up</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">2.0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">down</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0.5</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">factor</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0.5</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">min</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1e-06</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">max</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1e+16</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/optim/strategy/#TrustRegion"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.optim.strategy.TrustRegion" title="Permalink to this definition">¶</a></dt>
+<dd><p>The trust region (TR) algorithm used in the Levenberg-Marquardt (LM) algorithm.</p>
+<div class="math">
+\[\begin{aligned}
+     &\rule{113mm}{0.4pt}                                                                 \\
+     &\textbf{input}: \Delta ~\text{(radius)}, \bm{f}(\bm{\theta})~(\text{model}),
+                      \theta_h~(\text{high}), \theta_l~(\text{low}), \delta_u~(\text{up}),
+                      \delta_d~(\text{down}),                                             \\
+     &\hspace{13mm}   \sigma~(\text{factor}),  \epsilon_{s}~(\text{min}),
+                      \epsilon_{l}~(\text{max})                                           \\
+     &\rule{113mm}{0.4pt}                                                                 \\
+     & \rho = \frac{ \|\bm{f}(\bm{\theta})\|^2 - \|\bm{f}(\bm{\theta} + \delta)\|^2}
+               {\|\bm{f}(\bm{\theta})\|^2 - \|\bm{f}(\bm{\theta}) + \mathbf{J}\delta\|^2}
+             ~\text{(step quality)}                                                       \\
+     &\textbf{if} ~~ \rho > \theta_h ~ \text{(``very successful'' step)}                  \\
+     &\hspace{5mm} \Delta \leftarrow \delta_u \cdot \Delta                                \\
+     &\hspace{5mm} \delta_d \leftarrow \delta_d^{\text{init}}                             \\
+     &\textbf{elif} ~~ \rho > \theta_l ~ \text{(``successful'' step)}                     \\
+     &\hspace{5mm} \Delta \leftarrow \Delta                                               \\
+     &\hspace{5mm} \delta_d \leftarrow \delta_d^{\text{init}}                             \\
+     &\textbf{else} ~ \text{(``unsuccessful'' step)}                                      \\
+     &\hspace{5mm} \Delta \leftarrow \delta_d \cdot \Delta                                \\
+     &\hspace{5mm} \delta_d \leftarrow \sigma \cdot \delta_d                              \\
+     &\Delta \leftarrow \mathrm{min}(\mathrm{max}(\Delta, \epsilon_{s}), \epsilon_{l})    \\
+     &\delta_d \leftarrow \mathrm{min}(\mathrm{max}(\delta_d, \epsilon_{s}), \epsilon_{l})\\
+     &\rule{113mm}{0.4pt}                                                          \\[-1.ex]
+     &\textbf{return} \:  \Delta, \delta_d                                         \\[-1.ex]
+     &\rule{113mm}{0.4pt}                                                          \\[-1.ex]
+\end{aligned}
+
+\]</div>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>radius</strong> (<em>float</em><em>, </em><em>optional</em>) – the initial radius of the trust region (positive number).
+Default: 1e6.</p></li>
+<li><p><strong>high</strong> (<em>float</em><em>, </em><em>optional</em>) – high threshold for scaling down the damping factor.
+Default: 0.5.</p></li>
+<li><p><strong>low</strong> (<em>float</em><em>, </em><em>optional</em>) – low threshold for scaling up the damping factor.
+Default: 1e-3.</p></li>
+<li><p><strong>up</strong> (<em>float</em><em>, </em><em>optional</em>) – the up scaling factor in the range of <span class="math">\((1,\infty)\)</span>.
+Default: 2.0.</p></li>
+<li><p><strong>down</strong> (<em>float</em><em>, </em><em>optional</em>) – the initial down scaling factor in range of <span class="math">\((0,1)\)</span>.
+Default: 0.5.</p></li>
+<li><p><strong>factor</strong> (<em>float</em><em>, </em><em>optional</em>) – exponential factor for shrinking of the trust region.
+Default: 0.5.</p></li>
+<li><p><strong>min</strong> (<em>float</em><em>, </em><em>optional</em>) – the lower-bound of trust region radius and down scaling factor.
+Default: 1e-6.</p></li>
+<li><p><strong>max</strong> (<em>float</em><em>, </em><em>optional</em>) – the upper-bound of trust region radius and down scaling factor.
+Default: 1e16.</p></li>
+</ul>
+</dd>
+</dl>
+<p>More details about the optimization process go to <a class="reference internal" href="../pypose.optim.LevenbergMarquardt/#pypose.optim.LevenbergMarquardt" title="pypose.optim.LevenbergMarquardt"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.optim.LevenbergMarquardt()</span></code></a>.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>This implementation is an improved version of <a class="reference external" href="https://tinyurl.com/38r8e77x">TR in Ceres</a>.</p>
+<p>For efficiency, we calculate the denominator of the step quality <span class="math">\(\rho\)</span> as</p>
+<div class="math">
+\[\begin{aligned}
+& \|\bm{f}(\bm{\theta})\|^2 - \|\bm{f}(\bm{\theta}) + \mathbf{J}\delta\|^2 \\
+& = \bm{f}^T\bm{f} - \left( \bm{f}^T\bm{f} + 2 \bm{f}^T \delta +
+    \delta^T \mathbf{J}^T\mathbf{J}\delta \right) \\
+& = -(\mathbf{J} \delta)^T (2\bm{f} + \mathbf{J}\delta)
+\end{aligned}
+
+\]</div>
+<p>where <span class="math">\(\mathbf{J}\)</span> is the Jacobian of the model <span class="math">\(\bm{f}\)</span> at evaluation
+point <span class="math">\(\bm{\theta}\)</span>.</p>
+</div>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="k">class</span> <span class="nc">PoseInv</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+<span class="gp">... </span>    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">dim</span><span class="p">):</span>
+<span class="gp">... </span>        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+<span class="gp">... </span>        <span class="bp">self</span><span class="o">.</span><span class="n">pose</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">Parameter</span><span class="p">(</span><span class="n">pp</span><span class="o">.</span><span class="n">randn_SE3</span><span class="p">(</span><span class="o">*</span><span class="n">dim</span><span class="p">))</span>
+<span class="gp">...</span>
+<span class="gp">... </span>    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">inputs</span><span class="p">):</span>
+<span class="gp">... </span>        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">pose</span> <span class="o">@</span> <span class="n">inputs</span><span class="p">)</span><span class="o">.</span><span class="n">Log</span><span class="p">()</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span>
+<span class="gp">...</span>
+<span class="gp">... </span><span class="n">device</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">device</span><span class="p">(</span><span class="s2">&quot;cuda&quot;</span> <span class="k">if</span> <span class="n">torch</span><span class="o">.</span><span class="n">cuda</span><span class="o">.</span><span class="n">is_available</span><span class="p">()</span> <span class="k">else</span> <span class="s2">&quot;cpu&quot;</span><span class="p">)</span>
+<span class="gp">... </span><span class="n">inputs</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_SE3</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">to</span><span class="p">(</span><span class="n">device</span><span class="p">)</span>
+<span class="gp">... </span><span class="n">invnet</span> <span class="o">=</span> <span class="n">PoseInv</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">to</span><span class="p">(</span><span class="n">device</span><span class="p">)</span>
+<span class="gp">... </span><span class="n">strategy</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">optim</span><span class="o">.</span><span class="n">strategy</span><span class="o">.</span><span class="n">TrustRegion</span><span class="p">(</span><span class="n">radius</span><span class="o">=</span><span class="mf">1e6</span><span class="p">)</span>
+<span class="gp">... </span><span class="n">optimizer</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">optim</span><span class="o">.</span><span class="n">LM</span><span class="p">(</span><span class="n">invnet</span><span class="p">,</span> <span class="n">strategy</span><span class="o">=</span><span class="n">strategy</span><span class="p">)</span>
+<span class="gp">...</span>
+<span class="gp">... </span><span class="k">for</span> <span class="n">idx</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10</span><span class="p">):</span>
+<span class="gp">... </span>    <span class="n">loss</span> <span class="o">=</span> <span class="n">optimizer</span><span class="o">.</span><span class="n">step</span><span class="p">(</span><span class="n">inputs</span><span class="p">)</span>
+<span class="gp">... </span>    <span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Pose loss </span><span class="si">%.7f</span><span class="s1"> @ </span><span class="si">%d</span><span class="s1">it&#39;</span><span class="o">%</span><span class="p">(</span><span class="n">loss</span><span class="p">,</span> <span class="n">idx</span><span class="p">))</span>
+<span class="gp">... </span>    <span class="k">if</span> <span class="n">loss</span> <span class="o">&lt;</span> <span class="mf">1e-5</span><span class="p">:</span>
+<span class="gp">... </span>        <span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Early Stoping!&#39;</span><span class="p">)</span>
+<span class="gp">... </span>        <span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Optimization Early Done with loss:&#39;</span><span class="p">,</span> <span class="n">loss</span><span class="o">.</span><span class="n">item</span><span class="p">())</span>
+<span class="gp">... </span>        <span class="k">break</span>
+<span class="go">Pose loss 0.0000000 @ 0it</span>
+<span class="go">Early Stoping!</span>
+<span class="go">Optimization Early Done with loss: 7.462681583803032e-10</span>
+</pre></div>
+</div>
+</dd></dl>
+
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+              
+  <section id="pypose-pixel2point">
+<h1>pypose.pixel2point<a class="headerlink" href="#pypose-pixel2point" title="Permalink to this heading">¶</a></h1>
+<dl class="py function">
+<dt class="sig sig-object py" id="pypose.pixel2point">
+<span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">pixel2point</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">pixels</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">depth</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">intrinsics</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/function/geometry/#pixel2point"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.pixel2point" title="Permalink to this definition">¶</a></dt>
+<dd><p>Convert batch of pixels with depth into points (in camera coordinate)</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>pixels</strong> – (<code class="docutils literal notranslate"><span class="pre">torch.Tensor</span></code>) The 2d coordinates of pixels in the camera
+pixel coordinate.
+Shape has to be (…, N, 2)</p></li>
+<li><p><strong>depth</strong> – (<code class="docutils literal notranslate"><span class="pre">torch.Tensor</span></code>) The depths of pixels with respect to the
+sensor plane.
+Shape has to be (…, N)</p></li>
+<li><p><strong>intrinsics</strong> – (<code class="docutils literal notranslate"><span class="pre">torch.Tensor</span></code>): The intrinsic parameters of cameras.
+The shape has to be (…, 3, 3).</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p><code class="docutils literal notranslate"><span class="pre">torch.Tensor</span></code> The associated 3D-points with shape (…, N, 3)</p>
+</dd>
+</dl>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">torch</span><span class="o">,</span> <span class="nn">pypose</span> <span class="k">as</span> <span class="nn">pp</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">,</span> <span class="p">(</span><span class="n">H</span><span class="p">,</span> <span class="n">W</span><span class="p">)</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="mi">9</span><span class="p">,</span> <span class="mi">9</span><span class="p">)</span> <span class="c1"># focal length and image height, width</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">intrinsics</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([[</span><span class="n">f</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">H</span> <span class="o">/</span> <span class="mi">2</span><span class="p">],</span>
+<span class="gp">... </span>                           <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">W</span> <span class="o">/</span> <span class="mi">2</span><span class="p">],</span>
+<span class="gp">... </span>                           <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span>   <span class="mi">1</span>  <span class="p">]])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">pixels</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([[</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">],</span>
+<span class="gp">... </span>                       <span class="p">[</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">],</span>
+<span class="gp">... </span>                       <span class="p">[</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">1.3</span><span class="p">],</span>
+<span class="gp">... </span>                       <span class="p">[</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">],</span>
+<span class="gp">... </span>                       <span class="p">[</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">],</span>
+<span class="gp">... </span>                       <span class="p">[</span><span class="mf">5.0</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">]])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">depths</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mf">5.0</span><span class="p">,</span> <span class="mf">3.0</span><span class="p">,</span> <span class="mf">6.5</span><span class="p">,</span> <span class="mf">2.0</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">points</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">pixel2point</span><span class="p">(</span><span class="n">pixels</span><span class="p">,</span> <span class="n">depths</span><span class="p">,</span> <span class="n">intrinsics</span><span class="p">)</span>
+<span class="go">tensor([[-10.0000, -11.2500,   5.0000],</span>
+<span class="go">        [ -5.2500,  -6.7500,   3.0000],</span>
+<span class="go">        [-14.6250, -10.4000,   6.5000],</span>
+<span class="go">        [ -3.5000,  -4.5000,   2.0000],</span>
+<span class="go">        [ -1.0000,  -0.7500,   0.5000],</span>
+<span class="go">        [  0.1750,  -1.0500,   0.7000]])</span>
+</pre></div>
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+              
+  <section id="pypose-pm">
+<h1>pypose.pm<a class="headerlink" href="#pypose-pm" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.pm">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">pm</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/basics/ops/#pm"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.pm" title="Permalink to this definition">¶</a></dt>
+<dd><p>Returns plus or minus (<span class="math">\(\pm\)</span>) states for tensor.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>input</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>) – the input tensor.</p>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>the output tensor contains only <span class="math">\(-1\)</span> or <span class="math">\(+1\)</span>.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code></p>
+</dd>
+</dl>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The <a class="reference internal" href="#pypose.pm" title="pypose.pm"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pm()</span></code></a> function is different from <code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.sign()</span></code>, which returns
+<span class="math">\(0\)</span> for zero inputs, it will return <span class="math">\(+1\)</span> if an input element is zero.</p>
+</div>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">pm</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mf">0.1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.2</span><span class="p">]))</span>
+<span class="go">tensor([ 1.,  1., -1.])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">pm</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mf">0.1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.2</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">float64</span><span class="p">))</span>
+<span class="go">tensor([ 1.,  1., -1.], dtype=torch.float64)</span>
+</pre></div>
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+  <section id="pypose-point2pixel">
+<h1>pypose.point2pixel<a class="headerlink" href="#pypose-point2pixel" title="Permalink to this heading">¶</a></h1>
+<dl class="py function">
+<dt class="sig sig-object py" id="pypose.point2pixel">
+<span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">point2pixel</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">points</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">intrinsics</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">extrinsics</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/function/geometry/#point2pixel"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.point2pixel" title="Permalink to this definition">¶</a></dt>
+<dd><p>Project batched sets of points (either in camera or world frame) to pixels.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>points</strong> (<code class="docutils literal notranslate"><span class="pre">torch.Tensor</span></code>) – The 3D coordinate of points. Assumed to be in the
+camera frame if <code class="docutils literal notranslate"><span class="pre">extrinsics</span></code> is <code class="docutils literal notranslate"><span class="pre">None</span></code>, otherwiwse in the world frame.
+The shape has to be (…, N, 3).</p></li>
+<li><p><strong>intrinsics</strong> (<code class="docutils literal notranslate"><span class="pre">torch.Tensor</span></code>) – The intrinsic parameters of cameras.
+The shape has to be (…, 3, 3).</p></li>
+<li><p><strong>extrinsics</strong> (<code class="docutils literal notranslate"><span class="pre">pypose.LieTensor</span></code>, optional) – The extrinsic parameters of cameras.
+The shape has to be (…, 7). Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>The associated pixel with shape (…, N, 2).</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><code class="docutils literal notranslate"><span class="pre">torch.Tensor</span></code></p>
+</dd>
+</dl>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">torch</span><span class="o">,</span> <span class="nn">pypose</span> <span class="k">as</span> <span class="nn">pp</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">,</span> <span class="p">(</span><span class="n">H</span><span class="p">,</span> <span class="n">W</span><span class="p">)</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="mi">9</span><span class="p">,</span> <span class="mi">9</span><span class="p">)</span> <span class="c1"># focal length and image height, width</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">intrinsics</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([[</span><span class="n">f</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">H</span> <span class="o">/</span> <span class="mi">2</span><span class="p">],</span>
+<span class="gp">... </span>                           <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">W</span> <span class="o">/</span> <span class="mi">2</span><span class="p">],</span>
+<span class="gp">... </span>                           <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span>   <span class="mi">1</span>  <span class="p">]])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="nb">object</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([[</span><span class="mf">2.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">2.</span><span class="p">],</span>
+<span class="gp">... </span>                       <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">2.</span><span class="p">],</span>
+<span class="gp">... </span>                       <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">],</span>
+<span class="gp">... </span>                       <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">],</span>
+<span class="gp">... </span>                       <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">],</span>
+<span class="gp">... </span>                       <span class="p">[</span><span class="mf">5.</span><span class="p">,</span> <span class="mf">5.</span><span class="p">,</span> <span class="mf">3.</span><span class="p">]])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">pixels</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">point2pixel</span><span class="p">(</span><span class="nb">object</span><span class="p">,</span> <span class="n">intrinsics</span><span class="p">)</span>
+<span class="go">tensor([[6.5000, 4.5000],</span>
+<span class="go">        [5.5000, 4.5000],</span>
+<span class="go">        [4.5000, 6.5000],</span>
+<span class="go">        [4.5000, 4.5000],</span>
+<span class="go">        [6.5000, 4.5000],</span>
+<span class="go">        [7.8333, 7.8333]])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">pose</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">SE3</span><span class="p">([</span> <span class="mf">0.</span><span class="p">,</span> <span class="o">-</span><span class="mi">8</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.3827</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">,</span>  <span class="mf">0.9239</span><span class="p">])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">pixels</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">point2pixel</span><span class="p">(</span><span class="nb">object</span><span class="p">,</span> <span class="n">intrinsics</span><span class="p">,</span> <span class="n">pose</span><span class="p">)</span>
+<span class="go">tensor([[  4.4999,  -1.1568],</span>
+<span class="go">        [  3.8332,  -3.0425],</span>
+<span class="go">        [  2.4998, -15.2997],</span>
+<span class="go">        [  2.4998, -18.1282],</span>
+<span class="go">        [  4.4999,  -6.8135],</span>
+<span class="go">        [  4.9999,   3.4394]])</span>
+</pre></div>
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+  <section id="pypose-randn-rxso3">
+<h1>pypose.randn_RxSO3<a class="headerlink" href="#pypose-randn-rxso3" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.randn_RxSO3">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">randn_RxSO3</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">lsize</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">sigma</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1.0</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/utils/#randn_RxSO3"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.randn_RxSO3" title="Permalink to this definition">¶</a></dt>
+<dd><p>Returns <code class="xref py py-obj docutils literal notranslate"><span class="pre">RxSO3_type</span></code> LieTensor filled with the Exponential map of the random
+<code class="xref py py-obj docutils literal notranslate"><span class="pre">rxso3_type</span></code> LieTensor. The <code class="xref py py-obj docutils literal notranslate"><span class="pre">rxso3_type</span></code> LieTensor is generated using
+<a class="reference internal" href="../pypose.randn_rxso3/#pypose.randn_rxso3" title="pypose.randn_rxso3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">randn_rxso3()</span></code></a>.</p>
+<div class="math">
+\[\mathrm{data}[*, :] = \mathrm{Exp}([\delta_x, \delta_y, \delta_z, \log s]),
+
+\]</div>
+<p>where rotation <span class="math">\([\delta_x, \delta_y, \delta_z]\)</span> is generated using
+<a class="reference internal" href="../pypose.randn_so3/#pypose.randn_so3" title="pypose.randn_so3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.randn_so3()</span></code></a> with standard deviation <span class="math">\(\sigma_r\)</span>, scale <span class="math">\(\log s\)</span>
+is generated from a normal distribution <span class="math">\(\mathcal{N}(0, \sigma_s)\)</span>, and
+<span class="math">\(\mathrm{Exp}()\)</span> is the Exponential map. Note that standard deviations
+<span class="math">\(\sigma_r\)</span> and <span class="math">\(\sigma_s\)</span> are specified by <code class="docutils literal notranslate"><span class="pre">sigma</span></code> (<span class="math">\(\sigma\)</span>), where
+<span class="math">\(\sigma = (\sigma_r, \sigma_s)\)</span>.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>lsize</strong> (<em>int</em><em>...</em>) – a sequence of integers defining the lshape of the output tensor.
+Can be a variable number of arguments or a collection like a list or tuple.</p></li>
+<li><p><strong>sigma</strong> (<em>float</em><em> or </em><em>(</em><em>float</em><em>...</em><em>)</em><em>, </em><em>optional</em>) – standard deviation (<span class="math">\(\sigma_r\)</span>,
+and <span class="math">\(\sigma_s\)</span>) for the two normal distribution. Default: <code class="docutils literal notranslate"><span class="pre">1.0</span></code>.</p></li>
+<li><p><strong>requires_grad</strong> (<em>bool</em><em>, </em><em>optional</em>) – If autograd should record operations on
+the returned tensor. Default: <code class="docutils literal notranslate"><span class="pre">False</span></code>.</p></li>
+<li><p><strong>generator</strong> (<em>torch.Generator</em><em>, </em><em>optional</em>) – a pseudorandom number generator for sampling</p></li>
+<li><p><strong>dtype</strong> (<em>torch.dtype</em><em>, </em><em>optional</em>) – the desired data type of returned tensor.
+Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>. If <code class="docutils literal notranslate"><span class="pre">None</span></code>, uses a global default
+(see <code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.set_default_tensor_type()</span></code>).</p></li>
+<li><p><strong>layout</strong> (<em>torch.layout</em><em>, </em><em>optional</em>) – the desired layout of returned Tensor.
+Default: <code class="docutils literal notranslate"><span class="pre">torch.strided</span></code>.</p></li>
+<li><p><strong>device</strong> (<em>torch.device</em><em>, </em><em>optional</em>) – the desired device of returned tensor.
+Default: “None”. If None, uses the current device for the default tensor
+type (see <code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.set_default_tensor_type()</span></code>). Device will be the CPU
+for CPU tensor types and the current CUDA device for CUDA tensor types.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>a <code class="xref py py-obj docutils literal notranslate"><span class="pre">RxSO3_type</span></code> LieTensor</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor">LieTensor</a></p>
+</dd>
+</dl>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The parameter <span class="math">\(\sigma\)</span> can either be:</p>
+<ul class="simple">
+<li><p>a single <code class="docutils literal notranslate"><span class="pre">float</span></code> – in which all the elements in the <code class="xref py py-obj docutils literal notranslate"><span class="pre">RxSO3_type</span></code> share
+the same sigma, i.e., <span class="math">\(\sigma_{\rm{r}}\)</span> = <span class="math">\(\sigma_{\rm{s}}\)</span> =
+<span class="math">\(\sigma\)</span>.</p></li>
+<li><p>a <code class="docutils literal notranslate"><span class="pre">tuple</span></code> of two floats – in which case, the specific sigmas for the two parts
+are assigned independently, i.e., <span class="math">\(\sigma\)</span> = (<span class="math">\(\sigma_{\rm{r}}\)</span>,
+<span class="math">\(\sigma_{\rm{s}}\)</span>).</p></li>
+</ul>
+</div>
+<p class="rubric">Example</p>
+<p>For <span class="math">\(\sigma = (\sigma_r, \sigma_s)\)</span></p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">randn_RxSO3</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="p">(</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">2.0</span><span class="p">))</span>
+<span class="go">RxSO3Type LieTensor:</span>
+<span class="go">tensor([[-0.1929, -0.0141,  0.2859,  0.9385,  4.5562],</span>
+<span class="go">        [-0.2871,  0.0134, -0.2903,  0.9128,  3.1044]])</span>
+</pre></div>
+</div>
+</dd></dl>
+
+</section>
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+              
+  <section id="pypose-randn-se3">
+<h1>pypose.randn_SE3<a class="headerlink" href="#pypose-randn-se3" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.randn_SE3">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">randn_SE3</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">lsize</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">sigma</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1.0</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/utils/#randn_SE3"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.randn_SE3" title="Permalink to this definition">¶</a></dt>
+<dd><p>Returns <code class="xref py py-obj docutils literal notranslate"><span class="pre">SE3_type</span></code> LieTensor filled with the Exponential map of the random
+<code class="xref py py-obj docutils literal notranslate"><span class="pre">se3_type</span></code> LieTensor generated using <a class="reference internal" href="../pypose.randn_se3/#pypose.randn_se3" title="pypose.randn_se3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.randn_se3()</span></code></a>.</p>
+<div class="math">
+\[\mathrm{data}[*, :] = \mathrm{Exp}([\tau_x, \tau_y, \tau_z, \delta_x, \delta_y, \delta_z]),
+
+\]</div>
+<p>where <span class="math">\([\tau_x, \tau_y, \tau_z]\)</span> is generated from a normal distribution
+<span class="math">\(\mathcal{N}(0, \sigma_t)\)</span>, <span class="math">\([\delta_x, \delta_y, \delta_z]\)</span> is
+generated using <a class="reference internal" href="../pypose.randn_so3/#pypose.randn_so3" title="pypose.randn_so3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.randn_so3()</span></code></a> with with standard deviation
+<span class="math">\(\sigma_r\)</span>, and <span class="math">\(\mathrm{Exp}()\)</span> is the Exponential map. Note that standard
+deviations <span class="math">\(\sigma_t\)</span> and <span class="math">\(\sigma_r\)</span> are specified by <code class="docutils literal notranslate"><span class="pre">sigma</span></code> (<span class="math">\(\sigma\)</span>),
+where <span class="math">\(\sigma = (\sigma_t, \sigma_r)\)</span>.</p>
+<p>For detailed explanation, please see <a class="reference internal" href="../pypose.randn_se3/#pypose.randn_se3" title="pypose.randn_se3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.randn_se3()</span></code></a> and <a class="reference internal" href="../pypose.Exp/#pypose.Exp" title="pypose.Exp"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.Exp()</span></code></a>.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>lsize</strong> (<em>int</em><em>...</em>) – a sequence of integers defining the lshape of the output tensor.
+Can be a variable number of arguments or a collection like a list or tuple.</p></li>
+<li><p><strong>sigma</strong> (<em>float</em><em> or </em><em>(</em><em>float</em><em>...</em><em>)</em><em>, </em><em>optional</em>) – standard deviation
+(<span class="math">\(\sigma_t\)</span> and <span class="math">\(\sigma_r\)</span>) for the two normal distribution.
+Default: <code class="docutils literal notranslate"><span class="pre">1.0</span></code>.</p></li>
+<li><p><strong>requires_grad</strong> (<em>bool</em><em>, </em><em>optional</em>) – If autograd should record operations on
+the returned tensor. Default: <code class="docutils literal notranslate"><span class="pre">False</span></code>.</p></li>
+<li><p><strong>generator</strong> (<em>torch.Generator</em><em>, </em><em>optional</em>) – a pseudorandom number generator for sampling</p></li>
+<li><p><strong>dtype</strong> (<em>torch.dtype</em><em>, </em><em>optional</em>) – the desired data type of returned tensor.
+Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>. If None, uses a global default
+(see <code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.set_default_tensor_type()</span></code>).</p></li>
+<li><p><strong>layout</strong> (<em>torch.layout</em><em>, </em><em>optional</em>) – the desired layout of returned Tensor.
+Default: <code class="docutils literal notranslate"><span class="pre">torch.strided</span></code>.</p></li>
+<li><p><strong>device</strong> (<em>torch.device</em><em>, </em><em>optional</em>) – the desired device of returned tensor.
+Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>. If <code class="docutils literal notranslate"><span class="pre">None</span></code>, uses the current device for the default tensor
+type (see <code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.set_default_tensor_type()</span></code>). Device will be the CPU
+for CPU tensor types and the current CUDA device for CUDA tensor types.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>a <code class="xref py py-obj docutils literal notranslate"><span class="pre">SE3_type</span></code> LieTensor</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor">LieTensor</a></p>
+</dd>
+</dl>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The parameter <span class="math">\(\sigma\)</span> can either be:</p>
+<ul class="simple">
+<li><p>a single <code class="docutils literal notranslate"><span class="pre">float</span></code> – in which all the elements in the <code class="xref py py-obj docutils literal notranslate"><span class="pre">SE3_type</span></code>
+share the same sigma, i.e.,
+<span class="math">\(\sigma_{\rm{t}}\)</span> = <span class="math">\(\sigma_{\rm{r}}\)</span> = <span class="math">\(\sigma\)</span>.</p></li>
+<li><p>a <code class="docutils literal notranslate"><span class="pre">tuple</span></code> of two floats – in which case, the specific sigmas are
+assigned independently, i.e.,
+<span class="math">\(\sigma\)</span> = (<span class="math">\(\sigma_{\rm{t}}\)</span>, <span class="math">\(\sigma_{\rm{r}}\)</span>).</p></li>
+<li><p>a <code class="docutils literal notranslate"><span class="pre">tuple</span></code> of four floats – in which case, the specific sigmas for
+each translation data are assigned independently, i.e.,
+<span class="math">\(\sigma\)</span> = (<span class="math">\(\sigma_{\rm{tx}}\)</span>, <span class="math">\(\sigma_{\rm{ty}}\)</span>,
+<span class="math">\(\sigma_{\rm{tz}}\)</span>, <span class="math">\(\sigma_{\rm{r}}\)</span>).</p></li>
+</ul>
+</div>
+<p class="rubric">Example</p>
+<p>For <span class="math">\(\sigma = (\sigma_t, \sigma_r)\)</span></p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">randn_SE3</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="p">(</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">2.0</span><span class="p">))</span>
+<span class="go">SE3Type LieTensor:</span>
+<span class="go">tensor([[ 0.2947, -1.6990, -0.5535,  0.4439,  0.2777,  0.0518,  0.8504],</span>
+<span class="go">        [ 0.6825,  0.2963,  0.3410,  0.3375, -0.2355,  0.7389, -0.5335]])</span>
+</pre></div>
+</div>
+<p>For <span class="math">\(\sigma = (\sigma_{tx}, \sigma_{ty}, \sigma_{tz}, \sigma_{r})\)</span></p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">randn_SE3</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="p">(</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">,</span> <span class="mf">2.0</span><span class="p">,</span> <span class="mf">2.0</span><span class="p">))</span>
+<span class="go">SE3Type LieTensor:</span>
+<span class="go">tensor([[-1.5689, -0.6772,  0.3580, -0.2509,  0.8257, -0.4950,  0.1018],</span>
+<span class="go">        [ 0.2613, -2.7613,  0.2151, -0.8802,  0.2619,  0.3044,  0.2531]])</span>
+</pre></div>
+</div>
+</dd></dl>
+
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+  <section id="pypose-randn-so3">
+<h1>pypose.randn_SO3<a class="headerlink" href="#pypose-randn-so3" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.randn_SO3">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">randn_SO3</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">lsize</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">sigma</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1.0</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/utils/#randn_SO3"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.randn_SO3" title="Permalink to this definition">¶</a></dt>
+<dd><p>Returns <code class="xref py py-obj docutils literal notranslate"><span class="pre">SO3_type</span></code> LieTensor filled with the Exponential map of the random
+<code class="xref py py-obj docutils literal notranslate"><span class="pre">so3_type</span></code> LieTensor.</p>
+<div class="math">
+\[\begin{aligned}
+\mathrm{data}[*, :] &= \mathrm{Exp}([\delta_x, \delta_y, \delta_z]) \\
+                    &= \mathrm{Exp}([\delta_x', \delta_y', \delta_z'] \cdot \theta),
+\end{aligned}
+
+\]</div>
+<p>where <span class="math">\([\delta_x', \delta_y', \delta_z']\)</span> is generated from uniform distribution
+<span class="math">\(\mathcal{U}_{\mathrm{s}}\)</span> on a standard sphere, <span class="math">\(\mathrm{Exp}()\)</span> is the
+Exponential map, <span class="math">\(\theta\)</span> is generated from a normal distribution
+<span class="math">\(\mathcal{N}(0, \sigma)\)</span> where <span class="math">\(\sigma\)</span> (<code class="docutils literal notranslate"><span class="pre">sigma</span></code>) is the standard deviation.</p>
+<p>For detailed explanation, please see <a class="reference internal" href="../pypose.randn_so3/#pypose.randn_so3" title="pypose.randn_so3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.randn_so3()</span></code></a> and <a class="reference internal" href="../pypose.Exp/#pypose.Exp" title="pypose.Exp"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.Exp()</span></code></a>.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>lsize</strong> (<em>int</em><em>...</em>) – a sequence of integers defining the lshape of the output tensor.
+Can be a variable number of arguments or a collection like a list or tuple.</p></li>
+<li><p><strong>sigma</strong> (<em>float</em><em>, </em><em>optional</em>) – standard deviation for the angle of the generated rotation.
+Default: <code class="docutils literal notranslate"><span class="pre">1.0</span></code>.</p></li>
+<li><p><strong>requires_grad</strong> (<em>bool</em><em>, </em><em>optional</em>) – If autograd should record operations on
+the returned tensor. Default: <code class="docutils literal notranslate"><span class="pre">False</span></code>.</p></li>
+<li><p><strong>generator</strong> (<em>torch.Generator</em><em>, </em><em>optional</em>) – a pseudorandom number generator for sampling</p></li>
+<li><p><strong>dtype</strong> (<em>torch.dtype</em><em>, </em><em>optional</em>) – the desired data type of returned tensor.
+Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.
+If <code class="docutils literal notranslate"><span class="pre">None</span></code>, uses a global default (see <code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.set_default_tensor_type()</span></code>).</p></li>
+<li><p><strong>layout</strong> (<em>torch.layout</em><em>, </em><em>optional</em>) – the desired layout of returned Tensor.
+Default: <code class="docutils literal notranslate"><span class="pre">torch.strided</span></code>.</p></li>
+<li><p><strong>device</strong> (<em>torch.device</em><em>, </em><em>optional</em>) – the desired device of returned tensor.
+Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>. If <code class="docutils literal notranslate"><span class="pre">None</span></code>, uses the current device for the default tensor
+type (see <code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.set_default_tensor_type()</span></code>). Device will be the CPU
+for CPU tensor types and the current CUDA device for CUDA tensor types.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>a <code class="xref py py-obj docutils literal notranslate"><span class="pre">SO3_type</span></code> LieTensor.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor">LieTensor</a></p>
+</dd>
+</dl>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">randn_SO3</span><span class="p">(</span><span class="n">sigma</span><span class="o">=</span><span class="mf">0.1</span><span class="p">,</span> <span class="n">requires_grad</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+<span class="go">SO3Type LieTensor:</span>
+<span class="go">LieTensor([ 0.0082,  0.0077, -0.0018,  0.9999], requires_grad=True)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">randn_SO3</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">0.1</span><span class="p">,</span> <span class="n">requires_grad</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">float64</span><span class="p">)</span>
+<span class="go">SO3Type LieTensor:</span>
+<span class="go">tensor([[ 2.6857e-03, -8.3274e-03, -6.0457e-04,  9.9996e-01],</span>
+<span class="go">        [ 3.8711e-02, -6.3148e-02, -1.9388e-02,  9.9706e-01]],</span>
+<span class="go">        dtype=torch.float64, requires_grad=True)</span>
+</pre></div>
+</div>
+<figure class="align-center" id="id1">
+<a class="reference internal image-reference" href="../../_images/randn_so3.svg"><img alt="map to buried treasure" height="300px" src="../../_images/randn_so3.svg" width="300px" /></a>
+<figcaption>
+<p><span class="caption-text">Visualization of <a class="reference internal" href="#pypose.randn_SO3" title="pypose.randn_SO3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.randn_SO3()</span></code></a>. A total of 5000 random rotations
+are sampled using <a class="reference internal" href="#pypose.randn_SO3" title="pypose.randn_SO3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.randn_SO3()</span></code></a> and applied to the basepoint
+[0, 0, 1] (shown in blue).</span><a class="headerlink" href="#id1" title="Permalink to this image">¶</a></p>
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+  <section id="pypose-randn-sim3">
+<h1>pypose.randn_Sim3<a class="headerlink" href="#pypose-randn-sim3" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.randn_Sim3">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">randn_Sim3</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">lsize</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">sigma</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1.0</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/utils/#randn_Sim3"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.randn_Sim3" title="Permalink to this definition">¶</a></dt>
+<dd><p>Returns <code class="xref py py-obj docutils literal notranslate"><span class="pre">Sim3_type</span></code> LieTensor filled with the Exponential map of the random
+<code class="xref py py-obj docutils literal notranslate"><span class="pre">sim3_type</span></code> LieTensor generated using <a class="reference internal" href="../pypose.randn_sim3/#pypose.randn_sim3" title="pypose.randn_sim3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">randn_sim3()</span></code></a>.</p>
+<div class="math">
+\[\mathrm{data}[*, :] = \mathrm{Exp}([\tau_x, \tau_y, \tau_z, \delta_x,
+\delta_y, \delta_z, \log s]),
+
+\]</div>
+<p>where translation <span class="math">\([\tau_x, \tau_y, \tau_z]\)</span> is generated from a normal distribution
+<span class="math">\(\mathcal{N}(0, \sigma_t)\)</span>, rotation <span class="math">\([\delta_x, \delta_y, \delta_z]\)</span> is
+generated using <a class="reference internal" href="../pypose.randn_so3/#pypose.randn_so3" title="pypose.randn_so3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.randn_so3()</span></code></a> with with standard deviation
+<span class="math">\(\sigma_r\)</span>, scale <span class="math">\(\log s\)</span> is generated from a normal distribution
+<span class="math">\(\mathcal{N}(0, \sigma_s)\)</span>, and <span class="math">\(\mathrm{Exp}()\)</span> is the Exponential
+map. Note that standard deviations <span class="math">\(\sigma_t\)</span>, <span class="math">\(\sigma_r\)</span>, and
+<span class="math">\(\sigma_s\)</span> are specified by <code class="docutils literal notranslate"><span class="pre">sigma</span></code> (<span class="math">\(\sigma\)</span>), where
+<span class="math">\(\sigma = (\sigma_t, \sigma_r, \sigma_s)\)</span>.</p>
+<p>For detailed explanation, please see <a class="reference internal" href="../pypose.randn_sim3/#pypose.randn_sim3" title="pypose.randn_sim3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.randn_sim3()</span></code></a> and <a class="reference internal" href="../pypose.Exp/#pypose.Exp" title="pypose.Exp"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.Exp()</span></code></a>.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>lsize</strong> (<em>int</em><em>...</em>) – a sequence of integers defining the lshape of the output tensor.
+Can be a variable number of arguments or a collection like a list or tuple.</p></li>
+<li><p><strong>sigma</strong> (<em>float</em><em> or </em><em>(</em><em>float</em><em>...</em><em>)</em><em>, </em><em>optional</em>) – standard deviation
+(<span class="math">\(\sigma_t\)</span>, <span class="math">\(\sigma_r\)</span>, and <span class="math">\(\sigma_s\)</span>)
+for the three normal distribution. Default: <code class="docutils literal notranslate"><span class="pre">1.0</span></code>.</p></li>
+<li><p><strong>requires_grad</strong> (<em>bool</em><em>, </em><em>optional</em>) – If autograd should record operations on
+the returned tensor. Default: <code class="docutils literal notranslate"><span class="pre">False</span></code>.</p></li>
+<li><p><strong>generator</strong> (<em>torch.Generator</em><em>, </em><em>optional</em>) – a pseudorandom number generator for sampling</p></li>
+<li><p><strong>dtype</strong> (<em>torch.dtype</em><em>, </em><em>optional</em>) – the desired data type of returned tensor.
+Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>. If <code class="docutils literal notranslate"><span class="pre">None</span></code>, uses a global default
+(see <code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.set_default_tensor_type()</span></code>).</p></li>
+<li><p><strong>layout</strong> (<em>torch.layout</em><em>, </em><em>optional</em>) – the desired layout of returned Tensor.
+Default: <code class="docutils literal notranslate"><span class="pre">torch.strided</span></code>.</p></li>
+<li><p><strong>device</strong> (<em>torch.device</em><em>, </em><em>optional</em>) – the desired device of returned tensor.
+Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>. If <code class="docutils literal notranslate"><span class="pre">None</span></code>, uses the current device for the default tensor
+type (see <code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.set_default_tensor_type()</span></code>). Device will be the CPU
+for CPU tensor types and the current CUDA device for CUDA tensor types.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>a <code class="xref py py-obj docutils literal notranslate"><span class="pre">Sim_type</span></code> LieTensor</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor">LieTensor</a></p>
+</dd>
+</dl>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The parameter <span class="math">\(\sigma\)</span> can either be:</p>
+<ul class="simple">
+<li><p>a single <code class="docutils literal notranslate"><span class="pre">float</span></code> – in which all the elements in the <code class="xref py py-obj docutils literal notranslate"><span class="pre">Sim3_type</span></code>
+share the same sigma, i.e.,
+<span class="math">\(\sigma_{\rm{t}}\)</span> = <span class="math">\(\sigma_{\rm{r}}\)</span>
+= <span class="math">\(\sigma_{\rm{s}}\)</span> = <span class="math">\(\sigma\)</span>.</p></li>
+<li><p>a <code class="docutils literal notranslate"><span class="pre">tuple</span></code> of three floats – in which case, the specific sigmas
+for the three parts are assigned independently, i.e.,
+<span class="math">\(\sigma\)</span> = (<span class="math">\(\sigma_{\rm{t}}\)</span>, <span class="math">\(\sigma_{\rm{r}}\)</span>,
+<span class="math">\(\sigma_{\rm{s}}\)</span>).</p></li>
+<li><p>a <code class="docutils literal notranslate"><span class="pre">tuple</span></code> of five floats – in which case, the specific sigmas
+for each translation data are also assigned independently, i.e.,
+<span class="math">\(\sigma\)</span> = (<span class="math">\(\sigma_{\rm{tx}}\)</span>, <span class="math">\(\sigma_{\rm{ty}}\)</span>,
+<span class="math">\(\sigma_{\rm{tz}}\)</span>, <span class="math">\(\sigma_{\rm{r}}\)</span>, <span class="math">\(\sigma_{\rm{s}}\)</span>).</p></li>
+</ul>
+</div>
+<p class="rubric">Example</p>
+<p>For <span class="math">\(\sigma = (\sigma_{\rm{t}}, \sigma_{\rm{r}}, \sigma_{\rm{s}})\)</span></p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">randn_Sim3</span><span class="p">(</span><span class="n">sigma</span><span class="o">=</span><span class="p">(</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">,</span> <span class="mf">2.0</span><span class="p">))</span>
+<span class="go">Sim3Type LieTensor:</span>
+<span class="go">LieTensor([-0.7667, -0.0981,  0.8168,  0.0931,  0.0917,  0.0939,  0.9870, 0.2391])</span>
+</pre></div>
+</div>
+<p>For <span class="math">\(\sigma = (\sigma_{\rm{tx}}, \sigma_{\rm{ty}}, \sigma_{\rm{tz}},
+\sigma_{\rm{r}}, \sigma_{\rm{s})}\)</span></p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">randn_Sim3</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="p">(</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">,</span> <span class="mf">2.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">,</span> <span class="mf">2.0</span><span class="p">))</span>
+<span class="go">Sim3Type LieTensor:</span>
+<span class="go">tensor([[-0.0117,  0.0708,  1.6853,  0.1089,  0.4186, -0.0877,  0.8973,  0.3969],</span>
+<span class="go">        [ 0.2106, -0.0694, -0.0574, -0.2902, -0.4806, -0.0815,  0.8235,  0.0134]])</span>
+</pre></div>
+</div>
+</dd></dl>
+
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+              
+  <section id="pypose-randn-like">
+<h1>pypose.randn_like<a class="headerlink" href="#pypose-randn-like" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.randn_like">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">randn_like</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">sigma</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1.0</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/utils/#randn_like"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.randn_like" title="Permalink to this definition">¶</a></dt>
+<dd><p>Returns a LieTensor with the same size as input that is filled with random
+LieTensor with the corresponding <code class="xref py py-obj docutils literal notranslate"><span class="pre">input.ltype</span></code>.</p>
+<p>The corresponding random generator can be</p>
+<table class="docutils align-default" id="id1">
+<caption><span class="caption-text">List of available random LieTensor generator of input <code class="xref py py-obj docutils literal notranslate"><span class="pre">ltype</span></code>.</span><a class="headerlink" href="#id1" title="Permalink to this table">¶</a></caption>
+<colgroup>
+<col style="width: 18%" />
+<col style="width: 18%" />
+<col style="width: 21%" />
+<col style="width: 21%" />
+<col style="width: 21%" />
+</colgroup>
+<thead>
+<tr class="row-odd"><th class="head"><p>Name</p></th>
+<th class="head"><p>ltype</p></th>
+<th class="head"><p>randn function</p></th>
+<th class="head"><p>Manifold</p></th>
+<th class="head"><p>randn function</p></th>
+</tr>
+</thead>
+<tbody>
+<tr class="row-even"><td><p>Orthogonal Group</p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">SO3_type</span></code></p></td>
+<td><p><a class="reference internal" href="../pypose.randn_SO3/#pypose.randn_SO3" title="pypose.randn_SO3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">randn_SO3()</span></code></a></p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">so3_type</span></code></p></td>
+<td><p><a class="reference internal" href="../pypose.randn_so3/#pypose.randn_so3" title="pypose.randn_so3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">randn_so3()</span></code></a></p></td>
+</tr>
+<tr class="row-odd"><td><p>Euclidean Group</p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">SE3_type</span></code></p></td>
+<td><p><a class="reference internal" href="../pypose.randn_SE3/#pypose.randn_SE3" title="pypose.randn_SE3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">randn_SE3()</span></code></a></p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">se3_type</span></code></p></td>
+<td><p><a class="reference internal" href="../pypose.randn_se3/#pypose.randn_se3" title="pypose.randn_se3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">randn_se3()</span></code></a></p></td>
+</tr>
+<tr class="row-even"><td><p>Similarity Group</p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Sim3_type</span></code></p></td>
+<td><p><a class="reference internal" href="../pypose.randn_Sim3/#pypose.randn_Sim3" title="pypose.randn_Sim3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">randn_Sim3()</span></code></a></p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">sim3_type</span></code></p></td>
+<td><p><a class="reference internal" href="../pypose.randn_sim3/#pypose.randn_sim3" title="pypose.randn_sim3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">randn_sim3()</span></code></a></p></td>
+</tr>
+<tr class="row-odd"><td><p>Scaling Orthogonal</p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">RxSO3_type</span></code></p></td>
+<td><p><a class="reference internal" href="../pypose.randn_RxSO3/#pypose.randn_RxSO3" title="pypose.randn_RxSO3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">randn_RxSO3()</span></code></a></p></td>
+<td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">rxso3_type</span></code></p></td>
+<td><p><a class="reference internal" href="../pypose.randn_rxso3/#pypose.randn_rxso3" title="pypose.randn_rxso3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">randn_rxso3()</span></code></a></p></td>
+</tr>
+</tbody>
+</table>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>input</strong> (<a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><em>LieTensor</em></a>) – the size of input will determine size of the output tensor.</p></li>
+<li><p><strong>sigma</strong> (<em>float</em><em> or </em><em>(</em><em>float</em><em>...</em><em>)</em><em>, </em><em>optional</em>) – standard deviation for generating random
+LieTensors. Default: <code class="docutils literal notranslate"><span class="pre">1.0</span></code>.</p></li>
+<li><p><strong>dtype</strong> (<em>torch.dtype</em><em>, </em><em>optional</em>) – the desired data type of returned Tensor.
+Default: if <code class="docutils literal notranslate"><span class="pre">None</span></code>, defaults to the dtype of input.</p></li>
+<li><p><strong>layout</strong> (<em>torch.layout</em><em>, </em><em>optional</em>) – the desired layout of returned tensor.
+Default: if <code class="docutils literal notranslate"><span class="pre">None</span></code>, defaults to the layout of input.</p></li>
+<li><p><strong>device</strong> (<em>torch.device</em><em>, </em><em>optional</em>) – the desired device of returned tensor.
+Default: if <code class="docutils literal notranslate"><span class="pre">None</span></code>, defaults to the device of input.</p></li>
+<li><p><strong>requires_grad</strong> (<em>bool</em><em>, </em><em>optional</em>) – If autograd should record operations on the returned
+tensor. Default: <code class="docutils literal notranslate"><span class="pre">False</span></code>.</p></li>
+<li><p><strong>memory_format</strong> (<em>torch.memory_format</em><em>, </em><em>optional</em>) – the desired memory format of returned
+Tensor. Default: <code class="docutils literal notranslate"><span class="pre">torch.preserve_format</span></code>.</p></li>
+</ul>
+</dd>
+</dl>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The parameter sigma (<span class="math">\(\sigma\)</span>) can either be:</p>
+<ul class="simple">
+<li><p>a single <code class="docutils literal notranslate"><span class="pre">float</span></code> – in which all the elements in the LieType share the same
+sigma.</p></li>
+<li><p>a <code class="docutils literal notranslate"><span class="pre">tuple</span></code> of a number of floats – in which case, the specific sigma for
+each element can be assigned independently.</p></li>
+</ul>
+</div>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>If we have:</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">pypose</span> <span class="k">as</span> <span class="nn">pp</span>
+<span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">SO3</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
+</pre></div>
+</div>
+<p>Then the following two usages are equivalent:</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">pp</span><span class="o">.</span><span class="n">randn_like</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
+<span class="n">pp</span><span class="o">.</span><span class="n">randn_SO3</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">lshape</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">x</span><span class="o">.</span><span class="n">dtype</span><span class="p">,</span> <span class="n">layout</span><span class="o">=</span><span class="n">x</span><span class="o">.</span><span class="n">layout</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="n">x</span><span class="o">.</span><span class="n">device</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">so3</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]))</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">randn_like</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
+<span class="go">so3Type LieTensor:</span>
+<span class="go">tensor([[ 0.5162, -0.4600, -0.9085]])</span>
+</pre></div>
+</div>
+</dd></dl>
+
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+  <section id="pypose-randn-rxso3">
+<h1>pypose.randn_rxso3<a class="headerlink" href="#pypose-randn-rxso3" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.randn_rxso3">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">randn_rxso3</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">lsize</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">sigma</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1.0</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/utils/#randn_rxso3"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.randn_rxso3" title="Permalink to this definition">¶</a></dt>
+<dd><p>Returns <code class="xref py py-obj docutils literal notranslate"><span class="pre">rxso3_type</span></code> LieTensor filled with random numbers.</p>
+<div class="math">
+\[\mathrm{data}[*, :] = [\delta_x, \delta_y, \delta_z, \log s],
+
+\]</div>
+<p>where rotation <span class="math">\([\delta_x, \delta_y, \delta_z]\)</span> is
+generated using <a class="reference internal" href="../pypose.randn_so3/#pypose.randn_so3" title="pypose.randn_so3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.randn_so3()</span></code></a> with standard deviation <span class="math">\(\sigma_r\)</span>,
+scale <span class="math">\(\log s\)</span> is generated from a normal distribution <span class="math">\(\mathcal{N}(0, \sigma_s)\)</span>.
+Note that standard deviations <span class="math">\(\sigma_r\)</span> and <span class="math">\(\sigma_s\)</span> are
+specified by <code class="docutils literal notranslate"><span class="pre">sigma</span></code> (<span class="math">\(\sigma\)</span>), where <span class="math">\(\sigma = (\sigma_r, \sigma_s)\)</span>.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>lsize</strong> (<em>int</em><em>...</em>) – a sequence of integers defining the lshape of the output tensor.
+Can be a variable number of arguments or a collection like a list or tuple.</p></li>
+<li><p><strong>sigma</strong> (<em>float</em><em> or </em><em>(</em><em>float</em><em>...</em><em>)</em><em>, </em><em>optional</em>) – standard deviation (<span class="math">\(\sigma_r\)</span>,
+and <span class="math">\(\sigma_s\)</span>) for the two normal distribution. Default: <code class="docutils literal notranslate"><span class="pre">1.0</span></code>.</p></li>
+<li><p><strong>requires_grad</strong> (<em>bool</em><em>, </em><em>optional</em>) – If autograd should record operations on
+the returned tensor. Default: <code class="docutils literal notranslate"><span class="pre">False</span></code>.</p></li>
+<li><p><strong>generator</strong> (<em>torch.Generator</em><em>, </em><em>optional</em>) – a pseudorandom number generator for sampling</p></li>
+<li><p><strong>dtype</strong> (<em>torch.dtype</em><em>, </em><em>optional</em>) – the desired data type of returned tensor.
+Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>. If <code class="docutils literal notranslate"><span class="pre">None</span></code>, uses a global default
+(see <code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.set_default_tensor_type()</span></code>).</p></li>
+<li><p><strong>layout</strong> (<em>torch.layout</em><em>, </em><em>optional</em>) – the desired layout of returned Tensor.
+Default: <code class="docutils literal notranslate"><span class="pre">torch.strided</span></code>.</p></li>
+<li><p><strong>device</strong> (<em>torch.device</em><em>, </em><em>optional</em>) – the desired device of returned tensor.
+Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>. If <code class="docutils literal notranslate"><span class="pre">None</span></code>, uses the current device for the default tensor
+type (see <code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.set_default_tensor_type()</span></code>). Device will be the CPU
+for CPU tensor types and the current CUDA device for CUDA tensor types.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>a <code class="xref py py-obj docutils literal notranslate"><span class="pre">rxso3_type</span></code> LieTensor</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor">LieTensor</a></p>
+</dd>
+</dl>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The parameter <span class="math">\(\sigma\)</span> can either be:</p>
+<ul class="simple">
+<li><p>a single <code class="docutils literal notranslate"><span class="pre">float</span></code> – in which all the elements in the <code class="xref py py-obj docutils literal notranslate"><span class="pre">rxso3_type</span></code> share
+the same sigma, i.e., <span class="math">\(\sigma_{\rm{r}}\)</span> = <span class="math">\(\sigma_{\rm{s}}\)</span>
+= <span class="math">\(\sigma\)</span>.</p></li>
+<li><p>a <code class="docutils literal notranslate"><span class="pre">tuple</span></code> of two floats – in which case, the specific sigmas for the two parts
+are assigned independently, i.e., <span class="math">\(\sigma\)</span> = (<span class="math">\(\sigma_{\rm{r}}\)</span>,
+<span class="math">\(\sigma_{\rm{s}}\)</span>).</p></li>
+</ul>
+</div>
+<p class="rubric">Example</p>
+<p>For <span class="math">\(\sigma = (\sigma_r, \sigma_s)\)</span></p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">randn_rxso3</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="p">(</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">2.0</span><span class="p">))</span>
+<span class="go">rxso3Type LieTensor:</span>
+<span class="go">tensor([[-0.5033, -0.4102, -0.6213, -3.5049],</span>
+<span class="go">        [-0.3185,  0.1053, -0.0816, -1.1907]])</span>
+</pre></div>
+</div>
+</dd></dl>
+
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+  <section id="pypose-randn-se3">
+<h1>pypose.randn_se3<a class="headerlink" href="#pypose-randn-se3" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.randn_se3">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">randn_se3</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">lsize</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">sigma</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1.0</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/utils/#randn_se3"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.randn_se3" title="Permalink to this definition">¶</a></dt>
+<dd><p>Returns <code class="xref py py-obj docutils literal notranslate"><span class="pre">se3_type</span></code> LieTensor filled with random numbers.</p>
+<div class="math">
+\[\mathrm{data}[*, :] = [\tau_x, \tau_y, \tau_z, \delta_x, \delta_y, \delta_z],
+
+\]</div>
+<p>where translation <span class="math">\([\tau_x, \tau_y, \tau_z]\)</span> is generated from a normal distribution
+<span class="math">\(\mathcal{N}(0, \sigma_t)\)</span>, rotation <span class="math">\([\delta_x, \delta_y, \delta_z]\)</span> is
+generated using <a class="reference internal" href="../pypose.randn_so3/#pypose.randn_so3" title="pypose.randn_so3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.randn_so3()</span></code></a> with standard deviation <span class="math">\(\sigma_r\)</span>.
+Note that standard deviations <span class="math">\(\sigma_t\)</span> and <span class="math">\(\sigma_r\)</span> are
+specified by <code class="docutils literal notranslate"><span class="pre">sigma</span></code> (<span class="math">\(\sigma\)</span>), where <span class="math">\(\sigma = (\sigma_t, \sigma_r)\)</span>.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>lsize</strong> (<em>int</em><em>...</em>) – a sequence of integers defining the lshape of the output tensor.
+Can be a variable number of arguments or a collection like a list or tuple.</p></li>
+<li><p><strong>sigma</strong> (<em>float</em><em> or </em><em>(</em><em>float</em><em>...</em><em>)</em><em>, </em><em>optional</em>) – standard deviation
+(<span class="math">\(\sigma_t\)</span> and <span class="math">\(\sigma_r\)</span>)
+for the two normal distribution. Default: <code class="docutils literal notranslate"><span class="pre">1.0</span></code>.</p></li>
+<li><p><strong>requires_grad</strong> (<em>bool</em><em>, </em><em>optional</em>) – If autograd should record operations on
+the returned tensor. Default: <code class="docutils literal notranslate"><span class="pre">False</span></code>.</p></li>
+<li><p><strong>generator</strong> (<em>torch.Generator</em><em>, </em><em>optional</em>) – a pseudorandom number generator for sampling</p></li>
+<li><p><strong>dtype</strong> (<em>torch.dtype</em><em>, </em><em>optional</em>) – the desired data type of returned tensor.
+Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>. If <code class="docutils literal notranslate"><span class="pre">None</span></code>, uses a global default
+(see <code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.set_default_tensor_type()</span></code>).</p></li>
+<li><p><strong>layout</strong> (<em>torch.layout</em><em>, </em><em>optional</em>) – the desired layout of returned Tensor.
+Default: <code class="docutils literal notranslate"><span class="pre">torch.strided</span></code>.</p></li>
+<li><p><strong>device</strong> (<em>torch.device</em><em>, </em><em>optional</em>) – the desired device of returned tensor.
+Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>. If <code class="docutils literal notranslate"><span class="pre">None</span></code>, uses the current device for the default tensor
+type (see <code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.set_default_tensor_type()</span></code>). Device will be the CPU
+for CPU tensor types and the current CUDA device for CUDA tensor types.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>a <code class="xref py py-obj docutils literal notranslate"><span class="pre">se3_type</span></code> LieTensor.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor">LieTensor</a></p>
+</dd>
+</dl>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The parameter <span class="math">\(\sigma\)</span> can either be:</p>
+<ul class="simple">
+<li><p>a single <code class="docutils literal notranslate"><span class="pre">float</span></code> – in which all the elements in the <code class="xref py py-obj docutils literal notranslate"><span class="pre">se3_type</span></code>
+share the same sigma, i.e.,
+<span class="math">\(\sigma_{\rm{t}}\)</span> = <span class="math">\(\sigma_{\rm{r}}\)</span> = <span class="math">\(\sigma\)</span>.</p></li>
+<li><p>a <code class="docutils literal notranslate"><span class="pre">tuple</span></code> of two floats – in which case, the specific sigmas are
+assigned independently, i.e.,
+<span class="math">\(\sigma\)</span> = (<span class="math">\(\sigma_{\rm{t}}\)</span>, <span class="math">\(\sigma_{\rm{r}}\)</span>).</p></li>
+<li><p>a <code class="docutils literal notranslate"><span class="pre">tuple</span></code> of four floats – in which case, the specific sigmas for
+each translation data are assigned independently, i.e.,
+<span class="math">\(\sigma\)</span> = (<span class="math">\(\sigma_{\rm{tx}}\)</span>, <span class="math">\(\sigma_{\rm{ty}}\)</span>,
+<span class="math">\(\sigma_{\rm{tz}}\)</span>, <span class="math">\(\sigma_{\rm{r}}\)</span>).</p></li>
+</ul>
+</div>
+<p class="rubric">Example</p>
+<p>For <span class="math">\(\sigma = (\sigma_t, \sigma_r)\)</span></p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">randn_se3</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="p">(</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">))</span>
+<span class="go">se3Type LieTensor:</span>
+<span class="go">tensor([[-0.4226,  0.4028, -1.3824,  0.4433, -0.2029, -0.1193],</span>
+<span class="go">        [-0.8423, -1.0435,  0.8311, -0.4733,  0.0175,  0.1400]])</span>
+</pre></div>
+</div>
+<p>For <span class="math">\(\sigma = (\sigma_{tx}, \sigma_{ty}, \sigma_{tz}, \sigma_{r})\)</span></p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">randn_se3</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="p">(</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">2.0</span><span class="p">,</span> <span class="mf">3.0</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">))</span>
+<span class="go">se3Type LieTensor:</span>
+<span class="go">tensor([[ 1.1209,  1.4211, -0.7237, -0.1168,  0.0128,  0.1479],</span>
+<span class="go">        [ 0.1765,  0.3891,  3.4799, -0.0411, -0.2616, -0.1028]])</span>
+</pre></div>
+</div>
+</dd></dl>
+
+</section>
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+<li class="toctree-l1"><a class="reference internal" href="../../lietensor/">LieTensor</a></li>
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+              
+  <section id="pypose-randn-sim3">
+<h1>pypose.randn_sim3<a class="headerlink" href="#pypose-randn-sim3" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.randn_sim3">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">randn_sim3</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">lsize</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">sigma</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1.0</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/utils/#randn_sim3"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.randn_sim3" title="Permalink to this definition">¶</a></dt>
+<dd><p>Returns <code class="xref py py-obj docutils literal notranslate"><span class="pre">sim3_type</span></code> LieTensor filled with random numbers.</p>
+<div class="math">
+\[\mathrm{data}[*, :] = [\tau_x, \tau_y, \tau_z, \delta_x, \delta_y, \delta_z, \log s],
+
+\]</div>
+<p>where translation <span class="math">\([\tau_x, \tau_y, \tau_z]\)</span> is generated from a normal distribution
+<span class="math">\(\mathcal{N}(0, \sigma_t)\)</span>, rotation <span class="math">\([\delta_x, \delta_y, \delta_z]\)</span> is
+generated using <a class="reference internal" href="../pypose.randn_so3/#pypose.randn_so3" title="pypose.randn_so3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.randn_so3()</span></code></a> with standard deviation <span class="math">\(\sigma_r\)</span>,
+scale <span class="math">\(\log s\)</span> is generated from a normal distribution <span class="math">\(\mathcal{N}(0, \sigma_s)\)</span>.
+Note that standard deviations <span class="math">\(\sigma_t\)</span>, <span class="math">\(\sigma_r\)</span>, and <span class="math">\(\sigma_s\)</span> are
+specified by <code class="docutils literal notranslate"><span class="pre">sigma</span></code> (<span class="math">\(\sigma\)</span>), where <span class="math">\(\sigma=(\sigma_t,\sigma_r,\sigma_s)\)</span>.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>lsize</strong> (<em>int</em><em>...</em>) – a sequence of integers defining the lshape of the output tensor.
+Can be a variable number of arguments or a collection like a list or tuple.</p></li>
+<li><p><strong>sigma</strong> (<em>float</em><em> or </em><em>(</em><em>float</em><em>...</em><em>)</em><em>, </em><em>optional</em>) – standard deviation (<span class="math">\(\sigma_t\)</span>,
+<span class="math">\(\sigma_r\)</span>, and <span class="math">\(\sigma_s\)</span>) for the three normal distribution.
+Default: <code class="docutils literal notranslate"><span class="pre">1.0</span></code>.</p></li>
+<li><p><strong>requires_grad</strong> (<em>bool</em><em>, </em><em>optional</em>) – If autograd should record operations on
+the returned tensor. Default: <code class="docutils literal notranslate"><span class="pre">False</span></code>.</p></li>
+<li><p><strong>generator</strong> (<em>torch.Generator</em><em>, </em><em>optional</em>) – a pseudorandom number generator for sampling.</p></li>
+<li><p><strong>dtype</strong> (<em>torch.dtype</em><em>, </em><em>optional</em>) – the desired data type of returned tensor.
+Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>. If <code class="docutils literal notranslate"><span class="pre">None</span></code>, uses a global default
+(see <code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.set_default_tensor_type()</span></code>).</p></li>
+<li><p><strong>layout</strong> (<em>torch.layout</em><em>, </em><em>optional</em>) – the desired layout of returned Tensor.
+Default: <code class="docutils literal notranslate"><span class="pre">torch.strided</span></code>.</p></li>
+<li><p><strong>device</strong> (<em>torch.device</em><em>, </em><em>optional</em>) – the desired device of returned tensor.
+Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>. If <code class="docutils literal notranslate"><span class="pre">None</span></code>, uses the current device for the default tensor
+type (see <code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.set_default_tensor_type()</span></code>). Device will be the CPU
+for CPU tensor types and the current CUDA device for CUDA tensor types.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>a <code class="docutils literal notranslate"><span class="pre">sim3_type</span></code> LieTensor.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor">LieTensor</a></p>
+</dd>
+</dl>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The parameter <span class="math">\(\sigma\)</span> can either be:</p>
+<ul class="simple">
+<li><p>a single <code class="docutils literal notranslate"><span class="pre">float</span></code> – in which all the elements in the <code class="xref py py-obj docutils literal notranslate"><span class="pre">sim3_type</span></code>
+share the same sigma, i.e.,
+<span class="math">\(\sigma_{\rm{t}}\)</span> = <span class="math">\(\sigma_{\rm{r}}\)</span> = <span class="math">\(\sigma_{\rm{s}}\)</span>
+= <span class="math">\(\sigma\)</span>.</p></li>
+<li><p>a <code class="docutils literal notranslate"><span class="pre">tuple</span></code> of three floats – in which case, the specific sigmas for
+the three parts are assigned independently, i.e.,
+<span class="math">\(\sigma\)</span> = (<span class="math">\(\sigma_{\rm{t}}\)</span>, <span class="math">\(\sigma_{\rm{r}}\)</span>,
+<span class="math">\(\sigma_{\rm{s}}\)</span>).</p></li>
+<li><p>a <code class="docutils literal notranslate"><span class="pre">tuple</span></code> of five floats – in which case, the specific sigmas for
+each translation data are also assigned independently, i.e.,
+<span class="math">\(\sigma\)</span> = (<span class="math">\(\sigma_{\rm{tx}}\)</span>, <span class="math">\(\sigma_{\rm{ty}}\)</span>,
+<span class="math">\(\sigma_{\rm{tz}}\)</span>, <span class="math">\(\sigma_{\rm{r}}\)</span>, <span class="math">\(\sigma_{\rm{s}}\)</span>).</p></li>
+</ul>
+</div>
+<p class="rubric">Example</p>
+<p>For <span class="math">\(\sigma = (\sigma_{\rm{t}}, \sigma_{\rm{r}}, \sigma_{\rm{s}})\)</span></p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">randn_sim3</span><span class="p">(</span><span class="n">sigma</span><span class="o">=</span><span class="p">(</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">,</span> <span class="mf">2.0</span><span class="p">))</span>
+<span class="go">sim3Type LieTensor:</span>
+<span class="go">LieTensor([ 1.1994, -1.6163, -0.7566, -0.1805,  0.2199, -0.7044, -3.9935])</span>
+</pre></div>
+</div>
+<p>For <span class="math">\(\sigma = (\sigma_{\rm{tx}}, \sigma_{\rm{ty}}, \sigma_{\rm{tz}},
+\sigma_{\rm{r}}, \sigma_{\rm{s})}\)</span></p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">randn_sim3</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="p">(</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">,</span> <span class="mf">2.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">,</span> <span class="mf">2.0</span><span class="p">))</span>
+<span class="go">sim3Type LieTensor:</span>
+<span class="go">tensor([[ 0.3995, -1.9705,  2.6748,  0.5061, -1.4121,  1.1144,  0.5393],</span>
+<span class="go">        [ 0.7968,  0.5076,  1.0034,  0.8263,  1.3350, -0.0851,  2.2611]])</span>
+</pre></div>
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+<li class="toctree-l1"><a class="reference internal" href="../../lietensor/">LieTensor</a></li>
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+  <section id="pypose-randn-so3">
+<h1>pypose.randn_so3<a class="headerlink" href="#pypose-randn-so3" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.randn_so3">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">randn_so3</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">lsize</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">sigma</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1.0</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/utils/#randn_so3"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.randn_so3" title="Permalink to this definition">¶</a></dt>
+<dd><p>Returns <code class="xref py py-obj docutils literal notranslate"><span class="pre">so3_type</span></code> LieTensor filled with random numbers.</p>
+<div class="math">
+\[\begin{aligned}
+\mathrm{data}[*, :] &= [\delta_x, \delta_y, \delta_z] \\
+                    &= [\delta_x', \delta_y', \delta_z'] \cdot \theta,
+\end{aligned}
+
+\]</div>
+<p>where <span class="math">\([\delta_x', \delta_y', \delta_z']\)</span> is randomly generated from uniform
+distribution <span class="math">\(\mathcal{U}_{\mathrm{s}}\)</span> on a standard sphere and <span class="math">\(\theta\)</span>
+is generated from a normal distribution <span class="math">\(\mathcal{N}(0, \sigma)\)</span>, where sigma
+(<span class="math">\(\sigma\)</span>) is the standard deviation.</p>
+<p>The generated LieTensor satisfies that the corresponding rotation axis follows uniform
+distribution on the standard sphere and the rotation angle follows normal distribution.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>lsize</strong> (<em>int</em><em>...</em>) – a sequence of integers defining the lshape of the output tensor.
+Can be a variable number of arguments or a collection like a list or tuple.</p></li>
+<li><p><strong>sigma</strong> (<em>float</em><em>, </em><em>optional</em>) – standard deviation for the angle of the generated rotation.
+Default: <code class="docutils literal notranslate"><span class="pre">1.0</span></code>.</p></li>
+<li><p><strong>requires_grad</strong> (<em>bool</em><em>, </em><em>optional</em>) – If autograd should record operations on
+the returned tensor. Default: <code class="docutils literal notranslate"><span class="pre">False</span></code>.</p></li>
+<li><p><strong>generator</strong> (<em>torch.Generator</em><em>, </em><em>optional</em>) – a pseudorandom number generator for sampling</p></li>
+<li><p><strong>dtype</strong> (<em>torch.dtype</em><em>, </em><em>optional</em>) – the desired data type of returned tensor.
+Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>. If <code class="docutils literal notranslate"><span class="pre">None</span></code>, uses a global default
+(see <code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.set_default_tensor_type()</span></code>).</p></li>
+<li><p><strong>layout</strong> (<em>torch.layout</em><em>, </em><em>optional</em>) – the desired layout of returned Tensor.
+Default: <code class="docutils literal notranslate"><span class="pre">torch.strided</span></code>.</p></li>
+<li><p><strong>device</strong> (<em>torch.device</em><em>, </em><em>optional</em>) – the desired device of returned tensor.
+Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>. If <code class="docutils literal notranslate"><span class="pre">None</span></code>, uses the current device for the default tensor
+type (see <code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.set_default_tensor_type()</span></code>). Device will be the CPU
+for CPU tensor types and the current CUDA device for CUDA tensor types.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>a <code class="xref py py-obj docutils literal notranslate"><span class="pre">so3_type</span></code> LieTensor</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor">LieTensor</a></p>
+</dd>
+</dl>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The uniform distributed points <span class="math">\([\delta_x', \delta_y', \delta_z']\)</span> on a sphere
+are sampled using:</p>
+<div class="math">
+\[\delta_x' = \frac{x_0}{d}, \delta_y' = \frac{y_0}{d}, \delta_z' = \frac{z_0}{d},
+
+\]</div>
+<p>where</p>
+<div class="math">
+\[x_0, y_0, z_0 \sim \mathcal{N}(0, \sigma),
+
+\]</div>
+<div class="math">
+\[d = \sqrt{(x^2_0+y^2_0+z^2_0)}.
+
+\]</div>
+<p>Note that the point <span class="math">\([x_0, y_0, z_0]\)</span> follows 3-D Gaussian distribution,
+which is centrosymmetry with respect to <span class="math">\((0, 0, 0)\)</span>. Therefore, we have
+<span class="math">\(\delta_x'^2+\delta_y'^2+\delta_z'^2=1\)</span> and the normalized points follow
+uniform distribution on a standard sphere, i.e.,
+<span class="math">\([\delta_x', \delta_y', \delta_z'] \sim \mathcal{U}_{\mathrm{s}}\)</span>.
+The point <span class="math">\([\delta_x', \delta_y', \delta_z']\)</span> can also represent an evenly
+sampled rotation axis.</p>
+</div>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">randn_so3</span><span class="p">()</span>
+<span class="go">so3Type LieTensor:</span>
+<span class="go">LieTensor([ 0.4811, -0.1487, -0.5949])  # Shape (3)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">randn_so3</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
+<span class="go">so3Type LieTensor:</span>
+<span class="go">LieTensor([[0.0235, 0.0334, 1.2601]])   # Shape (1, 3)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">randn_so3</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">0.1</span><span class="p">,</span> <span class="n">requires_grad</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">float64</span><span class="p">)</span>
+<span class="go">so3Type LieTensor:</span>
+<span class="go">tensor([[-0.0344,  0.0177,  0.0252],</span>
+<span class="go">        [-0.0040,  0.0032,  0.0149]], dtype=torch.float64, requires_grad=True)</span>
+</pre></div>
+</div>
+<figure class="align-center" id="id1">
+<a class="reference internal image-reference" href="../../_images/randn_so3.svg"><img alt="map to buried treasure" height="300px" src="../../_images/randn_so3.svg" width="300px" /></a>
+<figcaption>
+<p><span class="caption-text">Visualization of <a class="reference internal" href="#pypose.randn_so3" title="pypose.randn_so3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.randn_so3()</span></code></a>. A total of 5000 random rotations
+are sampled using <a class="reference internal" href="#pypose.randn_so3" title="pypose.randn_so3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.randn_so3()</span></code></a> and applied to the basepoint
+[0, 0, 1] (shown in blue).</span><a class="headerlink" href="#id1" title="Permalink to this image">¶</a></p>
+</figcaption>
+</figure>
+</dd></dl>
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+              <p class="caption" role="heading"><span class="caption-text">Contents:</span></p>
+<ul class="current">
+<li class="toctree-l1"><a class="reference internal" href="../../lietensor/">LieTensor</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../func/">Func</a></li>
+<li class="toctree-l1 current"><a class="reference internal" href="../../functions/">Functions</a></li>
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+  <section id="pypose-reprojerr">
+<h1>pypose.reprojerr<a class="headerlink" href="#pypose-reprojerr" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.reprojerr">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">reprojerr</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">points</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">pixels</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">intrinsics</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">extrinsics</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">reduction</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'none'</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/function/geometry/#reprojerr"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.reprojerr" title="Permalink to this definition">¶</a></dt>
+<dd><p>Calculates batched per-pixel reprojection error (pixel distance) for points either in
+the camera or world frame given camera intrinsics or extrinsics, respectively.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>points</strong> (<code class="docutils literal notranslate"><span class="pre">torch.Tensor</span></code>) – The 3D coordinate of points. Assumed to be in the
+camera frame if <code class="docutils literal notranslate"><span class="pre">extrinsics</span></code> is <code class="docutils literal notranslate"><span class="pre">None</span></code>, otherwiwse in the world frame.
+The shape has to be (…, N, 3).</p></li>
+<li><p><strong>pixels</strong> (<code class="docutils literal notranslate"><span class="pre">torch.Tensor</span></code>) – The image points. The associated pixel.
+The shape has to be (…, N, 2).</p></li>
+<li><p><strong>intrinsics</strong> (<code class="docutils literal notranslate"><span class="pre">torch.Tensor</span></code>) – intrinsic matrices.
+The shape has to be (…, 3, 3).</p></li>
+<li><p><strong>extrinsics</strong> (<code class="docutils literal notranslate"><span class="pre">LieTensor</span></code>, optional) – The camera extrinsics.
+The shape has to be (…, 7). Default: <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+<li><p><strong>reduction</strong> (<code class="docutils literal notranslate"><span class="pre">str</span></code>, optional) – <p>The reduction to apply on the output: <code class="docutils literal notranslate"><span class="pre">'none'</span></code>
+| <code class="docutils literal notranslate"><span class="pre">'sum'</span></code> | <code class="docutils literal notranslate"><span class="pre">'norm'</span></code></p>
+<p><code class="docutils literal notranslate"><span class="pre">'none'</span></code>: No reduction is applied</p>
+<p><code class="docutils literal notranslate"><span class="pre">'sum'</span></code>: The reprojection error on each component (u, v) is summed for
+each pixel (L1 Norm)</p>
+<p><code class="docutils literal notranslate"><span class="pre">'norm'</span></code>: The reprojection error’s L2 norm for each pixel</p>
+</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p><p>Per-pixel reprojection error.</p>
+<p>The shape is (…, N) if reduction is <code class="docutils literal notranslate"><span class="pre">'sum'</span></code> or <code class="docutils literal notranslate"><span class="pre">'norm'</span></code>.</p>
+<p>The shape is (…, N, 2) if reduction is <code class="docutils literal notranslate"><span class="pre">'none'</span></code>.</p>
+</p>
+</dd>
+</dl>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">torch</span><span class="o">,</span> <span class="nn">pypose</span> <span class="k">as</span> <span class="nn">pp</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">,</span> <span class="p">(</span><span class="n">H</span><span class="p">,</span> <span class="n">W</span><span class="p">)</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="mi">9</span><span class="p">,</span> <span class="mi">9</span><span class="p">)</span> <span class="c1"># focal length and image height, width</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">intrinsics</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([[</span><span class="n">f</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">H</span> <span class="o">/</span> <span class="mi">2</span><span class="p">],</span>
+<span class="gp">... </span>                           <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">W</span> <span class="o">/</span> <span class="mi">2</span><span class="p">],</span>
+<span class="gp">... </span>                           <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span>   <span class="mi">1</span>  <span class="p">]])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="nb">object</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">pose</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_SE3</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">pixels</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">point2pixel</span><span class="p">(</span><span class="nb">object</span><span class="p">,</span> <span class="n">intrinsics</span><span class="p">,</span> <span class="n">pose</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">err</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">reprojerr</span><span class="p">(</span><span class="nb">object</span><span class="p">,</span> <span class="n">pixels</span><span class="p">,</span> <span class="n">intrinsics</span><span class="p">,</span> <span class="n">pose</span><span class="p">,</span> <span class="n">reduction</span><span class="o">=</span><span class="s1">&#39;norm&#39;</span><span class="p">)</span>
+<span class="go">tensor([0., 0., 0., 0., 0., 0.])</span>
+</pre></div>
+</div>
+</dd></dl>
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+<ul class="current">
+<li class="toctree-l1"><a class="reference internal" href="../../lietensor/">LieTensor</a></li>
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+              
+  <section id="pypose-rotation">
+<h1>pypose.rotation<a class="headerlink" href="#pypose-rotation" title="Permalink to this heading">¶</a></h1>
+<dl class="py function">
+<dt class="sig sig-object py" id="pypose.rotation">
+<span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">rotation</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">inputs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/convert/#rotation"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.rotation" title="Permalink to this definition">¶</a></dt>
+<dd><p>Extract the rotation part from a <a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code></a>.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>inputs</strong> (<a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code></a>) – the input LieTensor.</p>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>the batched quaternions.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>SO3</p>
+</dd>
+</dl>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_SE3</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">rotation</span><span class="p">()</span>
+<span class="go">SO3Type LieTensor:</span>
+<span class="go">tensor([[-0.8302,  0.5200, -0.0056,  0.2006],</span>
+<span class="go">        [-0.2541, -0.3184,  0.6305,  0.6607]])</span>
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+              
+  <section id="pypose-rxso3">
+<h1>pypose.rxso3<a class="headerlink" href="#pypose-rxso3" title="Permalink to this heading">¶</a></h1>
+<dl class="py data">
+<dt class="sig sig-object py" id="pypose.rxso3">
+<span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">rxso3</span></span><em class="property"><span class="w"> </span><span class="p"><span class="pre">=</span></span><span class="w"> </span><span class="pre">functools.partial(&lt;class</span> <span class="pre">'pypose.lietensor.lietensor.LieTensor'&gt;,</span> <span class="pre">ltype=&lt;pypose.lietensor.lietensor.rxso3Type</span> <span class="pre">object&gt;)</span></em><a class="headerlink" href="#pypose.rxso3" title="Permalink to this definition">¶</a></dt>
+<dd><p>Alias of rxso3 type <a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code></a>.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>data</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>, or <code class="xref py py-obj docutils literal notranslate"><span class="pre">list</span></code>, or ‘<code class="xref py py-obj docutils literal notranslate"><span class="pre">int</span></code>…’) – <p>A
+<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code> object, or constructing a <code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>
+object from <code class="xref py py-obj docutils literal notranslate"><span class="pre">list</span></code>, which defines tensor data (see below), or
+from ‘<code class="xref py py-obj docutils literal notranslate"><span class="pre">int</span></code>…’, which defines tensor shape.</p>
+<p>The shape of <code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code> object must be <code class="docutils literal notranslate"><span class="pre">(*,</span> <span class="pre">4)</span></code>,
+where <code class="docutils literal notranslate"><span class="pre">*</span></code> is empty, one, or more batched dimensions (the
+<a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor.lshape" title="pypose.LieTensor.lshape"><code class="xref py py-obj docutils literal notranslate"><span class="pre">lshape</span></code></a> of this LieTensor), otherwise error will
+be raised.</p>
+</p>
+</dd>
+</dl>
+<p>Internally, rxso3 LieTensors are stored by concatenating the axis-angle
+representation of the rotation with the log scale:</p>
+<div class="math">
+\[\mathrm{data}[*, :] = [\delta_x, \delta_y, \delta_z, \log s],
+
+\]</div>
+<p>where <span class="math">\(\begin{pmatrix} \delta_x &amp; \delta_y &amp; \delta_z \end{pmatrix}^T\)</span>
+is the axis-angle vector in <a class="reference internal" href="../pypose.so3/#pypose.so3" title="pypose.so3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.so3</span></code></a>, and
+<span class="math">\(s \in \mathbb{R}\)</span> is the scaling factor in <a class="reference internal" href="../pypose.RxSO3/#pypose.RxSO3" title="pypose.RxSO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.RxSO3</span></code></a>.</p>
+<p class="rubric">Examples</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">rxso3</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
+<span class="go">rxso3Type LieTensor:</span>
+<span class="go">tensor([[ 0.3752, -0.1576,  1.2057,  0.6086],</span>
+<span class="go">        [ 0.8434,  0.2449,  0.0488, -0.1202]])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">rxso3</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
+<span class="go">RxSO3Type LieTensor:</span>
+<span class="go">tensor([0., 0., 0., 0., 1.])</span>
+</pre></div>
+</div>
+<p>If <code class="docutils literal notranslate"><span class="pre">data</span></code> is tensor-like, the last dimension should correspond to the
+4 elements of the above embedding.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>It is not advised to construct rxso3 Tensors by specifying storage
+sizes with ‘<code class="xref py py-obj docutils literal notranslate"><span class="pre">int</span></code>…’, which does not initialize data.</p>
+<p>Consider using <a class="reference internal" href="../pypose.randn_rxso3/#pypose.randn_rxso3" title="pypose.randn_rxso3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.randn_rxso3</span></code></a> or
+<a class="reference internal" href="../pypose.identity_rxso3/#pypose.identity_rxso3" title="pypose.identity_rxso3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.identity_rxso3</span></code></a> instead.</p>
+</div>
+<p>See <a class="reference internal" href="../pypose.Exp/#pypose.Exp" title="pypose.Exp"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.Exp</span></code></a>, <a class="reference internal" href="../pypose.Inv/#pypose.Inv" title="pypose.Inv"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.Inv</span></code></a> for
+implementations of relevant operations.</p>
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+              <p class="caption" role="heading"><span class="caption-text">Contents:</span></p>
+<ul class="current">
+<li class="toctree-l1"><a class="reference internal" href="../../lietensor/">LieTensor</a></li>
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+             <article itemprop="articleBody" id="pytorch-article" class="pytorch-article">
+              
+  <section id="pypose-scale">
+<h1>pypose.scale<a class="headerlink" href="#pypose-scale" title="Permalink to this heading">¶</a></h1>
+<dl class="py function">
+<dt class="sig sig-object py" id="pypose.scale">
+<span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">scale</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">inputs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/convert/#scale"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.scale" title="Permalink to this definition">¶</a></dt>
+<dd><p>Extract the scale part from a <a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code></a>.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>inputs</strong> (<a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code></a>) – the input LieTensor.</p>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>the batched scale scalars.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>Tensor</p>
+</dd>
+</dl>
+<div class="admonition warning">
+<p class="admonition-title">Warning</p>
+<p>The <a class="reference internal" href="../pypose.SO3/#pypose.SO3" title="pypose.SO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">SO3</span></code></a>, <a class="reference internal" href="../pypose.so3/#pypose.so3" title="pypose.so3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">so3</span></code></a>, <a class="reference internal" href="../pypose.SE3/#pypose.SE3" title="pypose.SE3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">SE3</span></code></a>, and <a class="reference internal" href="../pypose.se3/#pypose.se3" title="pypose.se3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">se3</span></code></a> types do not contain scale.
+Calling <a class="reference internal" href="#pypose.scale" title="pypose.scale"><code class="xref py py-obj docutils literal notranslate"><span class="pre">scale()</span></code></a> on these types will return one(s).</p>
+</div>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_Sim3</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">scale</span><span class="p">()</span>
+<span class="go">tensor([[10.9577],</span>
+<span class="go">        [ 1.0248],</span>
+<span class="go">        [ 0.0947],</span>
+<span class="go">        [ 1.1989]])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_SE3</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span><span class="o">.</span><span class="n">scale</span><span class="p">()</span>
+<span class="go">tensor([[1.],</span>
+<span class="go">        [1.]])</span>
+</pre></div>
+</div>
+</dd></dl>
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+  <section id="pypose-se3">
+<h1>pypose.se3<a class="headerlink" href="#pypose-se3" title="Permalink to this heading">¶</a></h1>
+<dl class="py data">
+<dt class="sig sig-object py" id="pypose.se3">
+<span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">se3</span></span><em class="property"><span class="w"> </span><span class="p"><span class="pre">=</span></span><span class="w"> </span><span class="pre">functools.partial(&lt;class</span> <span class="pre">'pypose.lietensor.lietensor.LieTensor'&gt;,</span> <span class="pre">ltype=&lt;pypose.lietensor.lietensor.se3Type</span> <span class="pre">object&gt;)</span></em><a class="headerlink" href="#pypose.se3" title="Permalink to this definition">¶</a></dt>
+<dd><p>Alias of se3 type <a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code></a>.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>data</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>, or <code class="xref py py-obj docutils literal notranslate"><span class="pre">list</span></code>, or ‘<code class="xref py py-obj docutils literal notranslate"><span class="pre">int</span></code>…’) – <p>A
+<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code> object, or constructing a <code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>
+object from <code class="xref py py-obj docutils literal notranslate"><span class="pre">list</span></code>, which defines tensor data (see below), or
+from ‘<code class="xref py py-obj docutils literal notranslate"><span class="pre">int</span></code>…’, which defines tensor shape.</p>
+<p>The shape of <code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code> object must be <code class="docutils literal notranslate"><span class="pre">(*,</span> <span class="pre">6)</span></code>,
+where <code class="docutils literal notranslate"><span class="pre">*</span></code> is empty, one, or more batched dimensions (the
+<a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor.lshape" title="pypose.LieTensor.lshape"><code class="xref py py-obj docutils literal notranslate"><span class="pre">lshape</span></code></a> of this LieTensor), otherwise error will
+be raised.</p>
+</p>
+</dd>
+</dl>
+<p>Internally, se3 LieTensors are stored by concatenating the “velocity” vector
+with the axis-angle representation of the rotation:</p>
+<div class="math">
+\[\mathrm{data}[*, :] = [\tau_x, \tau_y, \tau_z, \delta_x, \delta_y, \delta_z],
+
+\]</div>
+<p>where <span class="math">\(\begin{pmatrix} \tau_x &amp; \tau_y &amp; \tau_z \end{pmatrix}^T =
+\mathbf{J}^{-1} \begin{pmatrix} t_x &amp; t_y &amp; t_z \end{pmatrix}^T\)</span> is the product
+between the left Jacobian inverse (of SO3’s logarithm map) and the translation
+vector, and <span class="math">\(\begin{pmatrix} \delta_x &amp; \delta_y &amp; \delta_z \end{pmatrix}^T\)</span>
+is the axis-angle vector as in <a class="reference internal" href="../pypose.so3/#pypose.so3" title="pypose.so3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.so3</span></code></a>.
+More details go to <a class="reference internal" href="../pypose.Log/#pypose.Log" title="pypose.Log"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.Log()</span></code></a> with <code class="xref py py-obj docutils literal notranslate"><span class="pre">SE3_type</span></code> input.</p>
+<p class="rubric">Examples</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">se3</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">6</span><span class="p">))</span>
+<span class="go">se3Type LieTensor:</span>
+<span class="go">tensor([[-0.8710, -1.4994, -0.2843,  1.0185, -0.3932, -0.4839],</span>
+<span class="go">        [-0.4750, -0.4804, -0.7083, -1.8141, -1.4409, -0.3125]])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">se3</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
+<span class="go">se3Type LieTensor:</span>
+<span class="go">tensor([0., 0., 0., 0., 0., 1.])</span>
+</pre></div>
+</div>
+<p>If <code class="docutils literal notranslate"><span class="pre">data</span></code> is tensor-like, the last dimension should correspond to the
+6 elements of the above embedding.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>It is not advised to construct se3 Tensors by specifying storage
+sizes with ‘<code class="xref py py-obj docutils literal notranslate"><span class="pre">int</span></code>…’, which does not initialize data.</p>
+<p>Consider using <a class="reference internal" href="../pypose.randn_se3/#pypose.randn_se3" title="pypose.randn_se3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.randn_se3</span></code></a> or
+<a class="reference internal" href="../pypose.identity_se3/#pypose.identity_se3" title="pypose.identity_se3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.identity_se3</span></code></a> instead.</p>
+</div>
+<p>See <a class="reference internal" href="../pypose.Exp/#pypose.Exp" title="pypose.Exp"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.Exp</span></code></a>, <a class="reference internal" href="../pypose.Inv/#pypose.Inv" title="pypose.Inv"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.Inv</span></code></a> for
+implementations of relevant operations.</p>
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+  <section id="pypose-sim3">
+<h1>pypose.sim3<a class="headerlink" href="#pypose-sim3" title="Permalink to this heading">¶</a></h1>
+<dl class="py data">
+<dt class="sig sig-object py" id="pypose.sim3">
+<span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">sim3</span></span><em class="property"><span class="w"> </span><span class="p"><span class="pre">=</span></span><span class="w"> </span><span class="pre">functools.partial(&lt;class</span> <span class="pre">'pypose.lietensor.lietensor.LieTensor'&gt;,</span> <span class="pre">ltype=&lt;pypose.lietensor.lietensor.sim3Type</span> <span class="pre">object&gt;)</span></em><a class="headerlink" href="#pypose.sim3" title="Permalink to this definition">¶</a></dt>
+<dd><p>Alias of sim3 type <a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code></a>.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>data</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>, or <code class="xref py py-obj docutils literal notranslate"><span class="pre">list</span></code>, or ‘<code class="xref py py-obj docutils literal notranslate"><span class="pre">int</span></code>…’) – <p>A
+<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code> object, or constructing a <code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>
+object from <code class="xref py py-obj docutils literal notranslate"><span class="pre">list</span></code>, which defines tensor data (see below), or
+from ‘<code class="xref py py-obj docutils literal notranslate"><span class="pre">int</span></code>…’, which defines tensor shape.</p>
+<p>The shape of <code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code> object must be <code class="docutils literal notranslate"><span class="pre">(*,</span> <span class="pre">7)</span></code>,
+where <code class="docutils literal notranslate"><span class="pre">*</span></code> is empty, one, or more batched dimensions (the
+<a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor.lshape" title="pypose.LieTensor.lshape"><code class="xref py py-obj docutils literal notranslate"><span class="pre">lshape</span></code></a> of this LieTensor), otherwise error will
+be raised.</p>
+</p>
+</dd>
+</dl>
+<p>Internally, sim3 LieTensors are stored by concatenating the log translation
+vector with the corresponding rxso3:</p>
+<div class="math">
+\[\mathrm{data}[*, :] = [\tau_x, \tau_y, \tau_z, \delta_x, \delta_y, \delta_z, \log s],
+
+\]</div>
+<p>where <span class="math">\(\begin{pmatrix} \tau_x &amp; \tau_y &amp; \tau_z \end{pmatrix}^T = \mathbf{W}^{-1}
+\begin{pmatrix} t_x &amp; t_y &amp; t_z \end{pmatrix}^T\)</span> is the product between the
+inverse of the <span class="math">\(\mathbf{W}\)</span>-matrix and the translation vector, and
+<span class="math">\(\begin{pmatrix} \delta_x &amp; \delta_y &amp; \delta_z &amp; \log s \end{pmatrix}^T\)</span>
+represents the rotation and scaling, as in <a class="reference internal" href="../pypose.rxso3/#pypose.rxso3" title="pypose.rxso3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.rxso3</span></code></a>.
+More details about <span class="math">\(\mathbf{W}\)</span>-matrix go to <a class="reference internal" href="../pypose.Log/#pypose.Log" title="pypose.Log"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pypose.Log()</span></code></a> with
+<code class="xref py py-obj docutils literal notranslate"><span class="pre">Sim3_type</span></code> input.</p>
+<p class="rubric">Examples</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">Sim3</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">7</span><span class="p">))</span>
+<span class="go">sim3Type LieTensor:</span>
+<span class="go">sim3Type LieTensor:</span>
+<span class="go">tensor([[ 0.1477, -1.3500, -2.1571,  0.8893, -0.7821, -0.9889, -0.7887],</span>
+<span class="go">        [ 0.2251,  0.3512,  0.0485,  0.0163, -1.7090, -0.0417, -0.3842]])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">sim3</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
+<span class="go">sim3Type LieTensor:</span>
+<span class="go">tensor([0., 0., 0., 0., 0., 0., 1.])</span>
+</pre></div>
+</div>
+<p>If <code class="docutils literal notranslate"><span class="pre">data</span></code> is tensor-like, the last dimension should correspond to the
+7 elements of the above embedding.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>It is not advised to construct sim3 Tensors by specifying storage
+sizes with ‘<code class="xref py py-obj docutils literal notranslate"><span class="pre">int</span></code>…’, which does not initialize data.</p>
+<p>Consider using <a class="reference internal" href="../pypose.randn_sim3/#pypose.randn_sim3" title="pypose.randn_sim3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.randn_sim3</span></code></a> or
+<a class="reference internal" href="../pypose.identity_sim3/#pypose.identity_sim3" title="pypose.identity_sim3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.identity_sim3</span></code></a> instead.</p>
+</div>
+<p>See <a class="reference internal" href="../pypose.Exp/#pypose.Exp" title="pypose.Exp"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.Exp</span></code></a>, <a class="reference internal" href="../pypose.Inv/#pypose.Inv" title="pypose.Inv"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.Inv</span></code></a> for
+implementations of relevant operations.</p>
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+              
+  <section id="pypose-so3">
+<h1>pypose.so3<a class="headerlink" href="#pypose-so3" title="Permalink to this heading">¶</a></h1>
+<dl class="py data">
+<dt class="sig sig-object py" id="pypose.so3">
+<span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">so3</span></span><em class="property"><span class="w"> </span><span class="p"><span class="pre">=</span></span><span class="w"> </span><span class="pre">functools.partial(&lt;class</span> <span class="pre">'pypose.lietensor.lietensor.LieTensor'&gt;,</span> <span class="pre">ltype=&lt;pypose.lietensor.lietensor.so3Type</span> <span class="pre">object&gt;)</span></em><a class="headerlink" href="#pypose.so3" title="Permalink to this definition">¶</a></dt>
+<dd><p>Alias of so3 type <a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code></a>.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>data</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>, or <code class="xref py py-obj docutils literal notranslate"><span class="pre">list</span></code>, or ‘<code class="xref py py-obj docutils literal notranslate"><span class="pre">int</span></code>…’) – <p>A
+<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code> object, or constructing a <code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>
+object from <code class="xref py py-obj docutils literal notranslate"><span class="pre">list</span></code>, which defines tensor data (see below), or
+from ‘<code class="xref py py-obj docutils literal notranslate"><span class="pre">int</span></code>…’, which defines tensor shape.</p>
+<p>The shape of <code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code> object must be <code class="docutils literal notranslate"><span class="pre">(*,</span> <span class="pre">3)</span></code>,
+where <code class="docutils literal notranslate"><span class="pre">*</span></code> is empty, one, or more batched dimensions (the
+<a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor.lshape" title="pypose.LieTensor.lshape"><code class="xref py py-obj docutils literal notranslate"><span class="pre">lshape</span></code></a> of this LieTensor), otherwise error will
+be raised.</p>
+</p>
+</dd>
+</dl>
+<p>Internally, so3 LieTensors are stored in the
+<a class="reference external" href="https://en.wikipedia.org/wiki/Axis-angle_representation">axis-angle</a> format:</p>
+<div class="math">
+\[\mathrm{data}[*, :] = [\delta_x, \delta_y, \delta_z],
+
+\]</div>
+<p>with <span class="math">\(\delta = \begin{pmatrix} \delta_x &amp; \delta_y &amp; \delta_z \end{pmatrix}\top\)</span>
+being the axis of rotation and <span class="math">\(\theta = \|{\delta}\|\)</span> being the angle.</p>
+<p class="rubric">Examples</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">so3</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
+<span class="go">so3Type LieTensor:</span>
+<span class="go">tensor([[ 0.1571,  0.2203, -0.2457],</span>
+<span class="go">        [-0.3311,  0.5412, -0.7028]])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">so3</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
+<span class="go">so3Type LieTensor:</span>
+<span class="go">tensor([0., 0., 1.])</span>
+</pre></div>
+</div>
+<p>If <code class="docutils literal notranslate"><span class="pre">data</span></code> is tensor-like, the last dimension should correspond to the
+3 elements of the above embedding.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>It is not advised to construct so3 Tensors by specifying storage
+sizes with ‘<code class="xref py py-obj docutils literal notranslate"><span class="pre">int</span></code>…’, which does not initialize data.</p>
+<p>Consider using <a class="reference internal" href="../pypose.randn_so3/#pypose.randn_so3" title="pypose.randn_so3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.randn_so3</span></code></a> or
+<a class="reference internal" href="../pypose.identity_so3/#pypose.identity_so3" title="pypose.identity_so3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.identity_so3</span></code></a> instead.</p>
+</div>
+<p>See <a class="reference internal" href="../pypose.Exp/#pypose.Exp" title="pypose.Exp"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.Exp</span></code></a>, <a class="reference internal" href="../pypose.Inv/#pypose.Inv" title="pypose.Inv"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pypose.Inv</span></code></a> for
+implementations of relevant operations.</p>
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+              
+  <section id="pypose-svdtf">
+<h1>pypose.svdtf<a class="headerlink" href="#pypose-svdtf" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.svdtf">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">svdtf</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">source</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">target</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/function/geometry/#svdtf"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.svdtf" title="Permalink to this definition">¶</a></dt>
+<dd><p>Computes the rigid transformation ( <span class="math">\(SE(3)\)</span> ) between two sets of associated
+point clouds (source and target) using Singular Value Decomposition (SVD).</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>source</strong> (<code class="docutils literal notranslate"><span class="pre">torch.Tensor</span></code>) – the coordinates of the source point cloud.
+The shape has to be (…, N, 3).</p></li>
+<li><p><strong>target</strong> (<code class="docutils literal notranslate"><span class="pre">torch.Tensor</span></code>) – the coordinates of the target point cloud.
+The shape has to be (…, N, 3).</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>The rigid transformation matrix in <code class="docutils literal notranslate"><span class="pre">SE3Type</span></code>  that
+minimizes the mean squared error between the input point sets.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><code class="docutils literal notranslate"><span class="pre">LieTensor</span></code></p>
+</dd>
+</dl>
+<div class="admonition warning">
+<p class="admonition-title">Warning</p>
+<p>The number of points N has to be the same for both point clouds.</p>
+</div>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">torch</span><span class="o">,</span> <span class="nn">pypose</span> <span class="k">as</span> <span class="nn">pp</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">source</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">],</span>
+<span class="gp">... </span>                    <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">],</span>
+<span class="gp">... </span>                    <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">]])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">target</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">],</span>
+<span class="gp">... </span>                    <span class="p">[</span><span class="mf">2.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">],</span>
+<span class="gp">... </span>                    <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">2.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">svdtf</span><span class="p">(</span><span class="n">source</span><span class="p">,</span> <span class="n">target</span><span class="p">)</span>
+<span class="go">SE3Type LieTensor:</span>
+<span class="go">LieTensor([1., 1., 1., 0., 0., 0., 1.])</span>
+</pre></div>
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+              <p class="caption" role="heading"><span class="caption-text">Contents:</span></p>
+<ul class="current">
+<li class="toctree-l1"><a class="reference internal" href="../../lietensor/">LieTensor</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../func/">Func</a></li>
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+              
+  <section id="pypose-tensor">
+<h1>pypose.tensor<a class="headerlink" href="#pypose-tensor" title="Permalink to this heading">¶</a></h1>
+<dl class="py function">
+<dt class="sig sig-object py" id="pypose.tensor">
+<span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">tensor</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">inputs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/convert/#tensor"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.tensor" title="Permalink to this definition">¶</a></dt>
+<dd><p>Convert a <a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code></a> into a <code class="xref py py-obj docutils literal notranslate"><span class="pre">torch.Tensor</span></code> without changing data.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>inputs</strong> (<a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code></a>) – the input LieTensor.</p>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>the torch.Tensor form of LieTensor.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>Tensor</p>
+</dd>
+</dl>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_SO3</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">tensor</span><span class="p">()</span>
+<span class="go">tensor([[ 0.1196,  0.2339, -0.6824,  0.6822],</span>
+<span class="go">        [ 0.9198, -0.2704, -0.2395,  0.1532]])</span>
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+  <section id="pypose-testing-assert-close">
+<h1>pypose.testing.assert_close<a class="headerlink" href="#pypose-testing-assert-close" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.testing.assert_close">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.testing.</span></span><span class="sig-name descname"><span class="pre">assert_close</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">actual</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">expected</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">args</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/testing/comparison/#assert_close"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.testing.assert_close" title="Permalink to this definition">¶</a></dt>
+<dd><p>Asserts that <code class="docutils literal notranslate"><span class="pre">actual</span></code> and <code class="docutils literal notranslate"><span class="pre">expected</span></code> are close. This function is exactly the same
+with <a class="reference external" href="https://tinyurl.com/3bm33ps7">torch.testing.assert_close</a> except for that it
+also accepts <code class="docutils literal notranslate"><span class="pre">pypose.LieTensor</span></code>.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>actual</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code> or <code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code>) – Actual input.</p></li>
+<li><p><strong>expected</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code> or <code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code>) – Expected input.</p></li>
+<li><p><strong>rtol</strong> (<em>Optional</em><em>[</em><em>float</em><em>]</em>) – Relative tolerance. If specified <code class="docutils literal notranslate"><span class="pre">atol</span></code> must also be
+specified. If omitted, default values based on the <code class="xref py py-attr docutils literal notranslate"><span class="pre">dtype</span></code>
+are selected with the below table.</p></li>
+<li><p><strong>atol</strong> (<em>Optional</em><em>[</em><em>float</em><em>]</em>) – Absolute tolerance. If specified <code class="docutils literal notranslate"><span class="pre">rtol</span></code> must also be
+specified. If omitted, default values based on the <code class="xref py py-attr docutils literal notranslate"><span class="pre">dtype</span></code>
+are selected with the below table.</p></li>
+</ul>
+</dd>
+</dl>
+<p>If <span class="math">\(T_e\)</span> and <span class="math">\(T_a\)</span> are Lietensor, they are considered close if
+<span class="math">\(T_e*T_a^{-1} = \mathbf{O}\)</span>, where <span class="math">\(\mathbf{O}\)</span> is close to zero tensor in
+the sense of <code class="docutils literal notranslate"><span class="pre">torch.testing.assert_close</span></code> is <code class="docutils literal notranslate"><span class="pre">True.</span></code></p>
+<div class="admonition warning">
+<p class="admonition-title">Warning</p>
+<p>The prerequisites for the other arguments align precisely with
+<a class="reference external" href="https://tinyurl.com/3bm33ps7">torch.testing.assert_close</a>. Kindly consult it
+for further details.</p>
+</div>
+<p class="rubric">Examples</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">pypose</span> <span class="k">as</span> <span class="nn">pp</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">actual</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_SE3</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">expected</span> <span class="o">=</span> <span class="n">actual</span><span class="o">.</span><span class="n">Log</span><span class="p">()</span><span class="o">.</span><span class="n">Exp</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">testing</span><span class="o">.</span><span class="n">assert_close</span><span class="p">(</span><span class="n">actual</span><span class="p">,</span> <span class="n">expected</span><span class="p">,</span> <span class="n">rtol</span><span class="o">=</span><span class="mf">1e-5</span><span class="p">,</span> <span class="n">atol</span><span class="o">=</span><span class="mf">1e-5</span><span class="p">)</span>
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+  </a>
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+            
+            
+              
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+              <p class="caption" role="heading"><span class="caption-text">Contents:</span></p>
+<ul class="current">
+<li class="toctree-l1"><a class="reference internal" href="../../lietensor/">LieTensor</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../func/">Func</a></li>
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+             <article itemprop="articleBody" id="pytorch-article" class="pytorch-article">
+              
+  <section id="pypose-translation">
+<h1>pypose.translation<a class="headerlink" href="#pypose-translation" title="Permalink to this heading">¶</a></h1>
+<dl class="py function">
+<dt class="sig sig-object py" id="pypose.translation">
+<span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">translation</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">inputs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/convert/#translation"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.translation" title="Permalink to this definition">¶</a></dt>
+<dd><p>Extract the translation part from a <a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code></a>.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>inputs</strong> (<a class="reference internal" href="../pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code></a>) – the input LieTensor.</p>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>the batched translation vectors.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>Tensor</p>
+</dd>
+</dl>
+<div class="admonition warning">
+<p class="admonition-title">Warning</p>
+<p>The <a class="reference internal" href="../pypose.SO3/#pypose.SO3" title="pypose.SO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">SO3</span></code></a>, <a class="reference internal" href="../pypose.so3/#pypose.so3" title="pypose.so3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">so3</span></code></a>, <a class="reference internal" href="../pypose.RxSO3/#pypose.RxSO3" title="pypose.RxSO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">RxSO3</span></code></a>, and <a class="reference internal" href="../pypose.rxso3/#pypose.rxso3" title="pypose.rxso3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">rxso3</span></code></a> types do not contain
+translation. Calling <a class="reference internal" href="#pypose.translation" title="pypose.translation"><code class="xref py py-obj docutils literal notranslate"><span class="pre">translation()</span></code></a> on these types will return zero vector(s).</p>
+</div>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_SE3</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">translation</span><span class="p">()</span>
+<span class="go">tensor([[-0.5358, -1.5421, -0.7224],</span>
+<span class="go">        [ 0.8331, -1.4412,  0.0863]])</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">randn_SO3</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span><span class="o">.</span><span class="n">translation</span><span class="p">()</span>
+<span class="go">tensor([[0., 0., 0.],</span>
+<span class="go">        [0., 0., 0.]])</span>
+</pre></div>
+</div>
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+  <section id="pypose-utils-reducetobason">
+<h1>pypose.utils.ReduceToBason<a class="headerlink" href="#pypose-utils-reducetobason" title="Permalink to this heading">¶</a></h1>
+<dl class="py class">
+<dt class="sig sig-object py" id="pypose.utils.ReduceToBason">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">pypose.utils.</span></span><span class="sig-name descname"><span class="pre">ReduceToBason</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">steps</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">patience</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">5</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">decreasing</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0.001</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">tol</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1e-05</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">verbose</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/utils/stepper/#ReduceToBason"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.utils.ReduceToBason" title="Permalink to this definition">¶</a></dt>
+<dd><p>A stepper to stop a loop when no relative loss ‘decreasing’ is seen for a ‘patience’
+number of steps.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>steps</strong> (<code class="docutils literal notranslate"><span class="pre">int</span></code>) – Maximum number of iterations a loop will step.</p></li>
+<li><p><strong>patience</strong> (<code class="docutils literal notranslate"><span class="pre">int</span></code>, optional) – Number of steps with no loss ‘decreasing’ is seen.
+For example, if <code class="docutils literal notranslate"><span class="pre">patience</span> <span class="pre">=</span> <span class="pre">2</span></code>, then it ignores the first 2 steps with no
+improvement, and stop the loop after the 3rd step if the loss has no
+decerasing. Default: <code class="docutils literal notranslate"><span class="pre">5</span></code>.</p></li>
+<li><p><strong>decreasing</strong> (<code class="docutils literal notranslate"><span class="pre">float</span></code>, optional) – relative loss decreasing used to count the
+number of patience steps. Default: <code class="docutils literal notranslate"><span class="pre">1e-3</span></code>.</p></li>
+<li><p><strong>tol</strong> (<code class="docutils literal notranslate"><span class="pre">float</span></code>, optional) – the minimum loss tolerance to stop a loop.
+Default: <code class="docutils literal notranslate"><span class="pre">1e-5</span></code>.</p></li>
+<li><p><strong>verbose</strong> (<code class="docutils literal notranslate"><span class="pre">bool</span></code>, optional) – if <code class="docutils literal notranslate"><span class="pre">True</span></code>, prints a message to stdout for each
+step. Default: <code class="docutils literal notranslate"><span class="pre">False</span></code>.</p></li>
+</ul>
+</dd>
+</dl>
+<div class="admonition warning">
+<p class="admonition-title">Warning</p>
+<p>Remember to call <cite>stepper.reset()</cite> if you want to re-use a stepper.</p>
+</div>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">pypose.utils</span> <span class="kn">import</span> <span class="n">ReduceToBason</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="mf">0.9</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">stepper</span> <span class="o">=</span> <span class="n">ReduceToBason</span><span class="p">(</span><span class="n">steps</span><span class="o">=</span><span class="mi">5</span><span class="p">,</span> <span class="n">patience</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">decreasing</span><span class="o">=</span><span class="mf">0.1</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="k">while</span> <span class="n">stepper</span><span class="o">.</span><span class="n">continual</span><span class="p">():</span>
+<span class="gp">... </span>    <span class="n">x</span> <span class="o">=</span> <span class="n">x</span> <span class="o">**</span> <span class="mi">2</span>
+<span class="gp">... </span>    <span class="n">stepper</span><span class="o">.</span><span class="n">step</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
+<span class="go">ReduceToBason step 0 loss 8.099999e-01.</span>
+<span class="go">ReduceToBason step 1 loss 6.560999e-01.</span>
+<span class="go">ReduceToBason step 2 loss 4.304671e-01.</span>
+<span class="go">ReduceToBason step 3 loss 1.853019e-01.</span>
+<span class="go">ReduceToBason step 4 loss 3.433681e-02.</span>
+<span class="go">ReduceToBason: Maximum steps reached, Quiting..</span>
+</pre></div>
+</div>
+<dl class="py method">
+<dt class="sig sig-object py" id="pypose.utils.ReduceToBason.step">
+<span class="sig-name descname"><span class="pre">step</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">loss</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/utils/stepper/#ReduceToBason.step"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.utils.ReduceToBason.step" title="Permalink to this definition">¶</a></dt>
+<dd><p>Performs a stepper step.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>loss</strong> (<code class="docutils literal notranslate"><span class="pre">float</span></code> or <code class="docutils literal notranslate"><span class="pre">torch.Tensor</span></code>) – the model loss after one loop.
+Can be a batched tensor. If batched tensor, all losses in the batch has to
+satisfy the condition to stop a loop.</p>
+</dd>
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+  <section id="pypose-vec2skew">
+<h1>pypose.vec2skew<a class="headerlink" href="#pypose-vec2skew" title="Permalink to this heading">¶</a></h1>
+<dl class="py function">
+<dt class="sig sig-object py" id="pypose.vec2skew">
+<span class="sig-prename descclassname"><span class="pre">pypose.</span></span><span class="sig-name descname"><span class="pre">vec2skew</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../_modules/pypose/lietensor/basics/#vec2skew"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#pypose.vec2skew" title="Permalink to this definition">¶</a></dt>
+<dd><p>Convert batched vectors to skew matrices.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>input</strong> (<em>Tensor</em>) – the tensor <span class="math">\(\mathbf{x}\)</span> to convert.</p>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>the skew matrices <span class="math">\(\mathbf{y}\)</span>.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>Tensor</p>
+</dd>
+</dl>
+<dl>
+<dt>Shape:</dt><dd><p>Input: <code class="xref py py-obj docutils literal notranslate"><span class="pre">(*,</span> <span class="pre">3)</span></code></p>
+<p>Output: <code class="xref py py-obj docutils literal notranslate"><span class="pre">(*,</span> <span class="pre">3,</span> <span class="pre">3)</span></code></p>
+</dd>
+</dl>
+<div class="math">
+\[{\displaystyle \mathbf{y}_i={\begin{bmatrix}\,\,
+0&\!-x_{i,3}&\,\,\,x_{i,2}\\\,\,\,x_{i,3}&0&\!-x_{i,1}
+\\\!-x_{i,2}&\,\,x_{i,1}&\,\,0\end{bmatrix}}}
+
+\]</div>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The last dimension of the input tensor has to be 3.</p>
+</div>
+<p class="rubric">Example</p>
+<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pp</span><span class="o">.</span><span class="n">vec2skew</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">))</span>
+<span class="go">tensor([[[ 0.0000, -2.2059, -1.2761],</span>
+<span class="go">        [ 2.2059,  0.0000,  0.2929],</span>
+<span class="go">        [ 1.2761, -0.2929,  0.0000]]])</span>
+</pre></div>
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+<link rel="preload" href="../_static/fonts/FreightSans/freight-sans-medium-italic.woff2" as="font" type="font/woff2" crossorigin="anonymous">
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+
+<!-- Preload the katex fonts -->
+
+<link rel="preload" href="https://cdn.jsdelivr.net/npm/katex@0.10.0/dist/fonts/KaTeX_Math-Italic.woff2" as="font" type="font/woff2" crossorigin="anonymous">
+<link rel="preload" href="https://cdn.jsdelivr.net/npm/katex@0.10.0/dist/fonts/KaTeX_Main-Regular.woff2" as="font" type="font/woff2" crossorigin="anonymous">
+<link rel="preload" href="https://cdn.jsdelivr.net/npm/katex@0.10.0/dist/fonts/KaTeX_Main-Bold.woff2" as="font" type="font/woff2" crossorigin="anonymous">
+<link rel="preload" href="https://cdn.jsdelivr.net/npm/katex@0.10.0/dist/fonts/KaTeX_Size1-Regular.woff2" as="font" type="font/woff2" crossorigin="anonymous">
+<link rel="preload" href="https://cdn.jsdelivr.net/npm/katex@0.10.0/dist/fonts/KaTeX_Size4-Regular.woff2" as="font" type="font/woff2" crossorigin="anonymous">
+<link rel="preload" href="https://cdn.jsdelivr.net/npm/katex@0.10.0/dist/fonts/KaTeX_Size2-Regular.woff2" as="font" type="font/woff2" crossorigin="anonymous">
+<link rel="preload" href="https://cdn.jsdelivr.net/npm/katex@0.10.0/dist/fonts/KaTeX_Size3-Regular.woff2" as="font" type="font/woff2" crossorigin="anonymous">
+<link rel="preload" href="https://cdn.jsdelivr.net/npm/katex@0.10.0/dist/fonts/KaTeX_Caligraphic-Regular.woff2" as="font" type="font/woff2" crossorigin="anonymous">
+  <link rel="stylesheet" href="https://use.fontawesome.com/releases/v5.15.2/css/all.css" integrity="sha384-vSIIfh2YWi9wW0r9iZe7RJPrKwp6bG+s9QZMoITbCckVJqGCCRhc+ccxNcdpHuYu" crossorigin="anonymous">
+</head>
+
+<div class="container-fluid header-holder tutorials-header" id="header-holder">
+  <div class="container">
+    <div class="header-container">
+      <a class="header-logo" href="https://pypose.org/" aria-label="PyTorch"></a>
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+      <span>Table of Contents</span>
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+    </div>
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+    <nav data-toggle="wy-nav-shift" class="pytorch-left-menu" id="pytorch-left-menu">
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+          <div class="pytorch-left-menu-search">
+            
+    <div class="version">
+      <a href='https://pypose.org/docs/versions.html'>0.6.8 &#x25BC</a>
+    </div>
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+  <form id="rtd-search-form" class="wy-form" action="../search/" method="get">
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+<div>
+  <a style="color:#F05732" href="genindex.html">
+    You are viewing unstable developer preview docs.
+    Click here to view docs for latest stable release.
+  </a>
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+            
+            
+              
+            
+            
+              <p class="caption" role="heading"><span class="caption-text">Contents:</span></p>
+<ul>
+<li class="toctree-l1"><a class="reference internal" href="../lietensor/">LieTensor</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../func/">Func</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../functions/">Functions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../convert/">Convert</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../modules/">Modules</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../optim/">Optimization</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../utils/">Utils</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../testing/">Testing</a></li>
+</ul>
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+        
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+          Shortcuts
+        </div>
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+      <section data-toggle="wy-nav-shift" id="pytorch-content-wrap" class="pytorch-content-wrap">
+        <div class="pytorch-content-left">
+
+        
+          
+          <div class="rst-content">
+          
+            <div role="main" class="main-content" itemscope="itemscope" itemtype="http://schema.org/Article">
+             <article itemprop="articleBody" id="pytorch-article" class="pytorch-article">
+              
+
+<h1 id="index">Index</h1>
+
+<div class="genindex-jumpbox">
+ <a href="#A"><strong>A</strong></a>
+ | <a href="#B"><strong>B</strong></a>
+ | <a href="#C"><strong>C</strong></a>
+ | <a href="#D"><strong>D</strong></a>
+ | <a href="#E"><strong>E</strong></a>
+ | <a href="#F"><strong>F</strong></a>
+ | <a href="#G"><strong>G</strong></a>
+ | <a href="#H"><strong>H</strong></a>
+ | <a href="#I"><strong>I</strong></a>
+ | <a href="#J"><strong>J</strong></a>
+ | <a href="#K"><strong>K</strong></a>
+ | <a href="#L"><strong>L</strong></a>
+ | <a href="#M"><strong>M</strong></a>
+ | <a href="#N"><strong>N</strong></a>
+ | <a href="#O"><strong>O</strong></a>
+ | <a href="#P"><strong>P</strong></a>
+ | <a href="#Q"><strong>Q</strong></a>
+ | <a href="#R"><strong>R</strong></a>
+ | <a href="#S"><strong>S</strong></a>
+ | <a href="#T"><strong>T</strong></a>
+ | <a href="#U"><strong>U</strong></a>
+ | <a href="#V"><strong>V</strong></a>
+ 
+</div>
+<h2 id="A">A</h2>
+<table style="width: 100%" class="indextable genindextable"><tr>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="../generated/pypose.module.LTI/#pypose.module.LTI.A">A (pypose.module.LTI property)</a>
+
+      <ul>
+        <li><a href="../generated/pypose.module.NLS/#pypose.module.NLS.A">(pypose.module.NLS property)</a>
+</li>
+      </ul></li>
+      <li><a href="../generated/pypose.Act/#pypose.Act">Act (class in pypose)</a>
+</li>
+      <li><a href="../generated/pypose.LieTensor/#pypose.LieTensor.Act">Act() (pypose.LieTensor method)</a>
+</li>
+      <li><a href="../generated/pypose.optim.strategy.Adaptive/#pypose.optim.strategy.Adaptive">Adaptive (class in pypose.optim.strategy)</a>
+</li>
+      <li><a href="../generated/pypose.add/#pypose.add">add (class in pypose)</a>
+</li>
+      <li><a href="../generated/pypose.LieTensor/#pypose.LieTensor.add">add() (pypose.LieTensor method)</a>
+</li>
+  </ul></td>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="../generated/pypose.add_/#pypose.add_">add_ (class in pypose)</a>
+</li>
+      <li><a href="../generated/pypose.LieTensor/#pypose.LieTensor.add_">add_() (pypose.LieTensor method)</a>
+</li>
+      <li><a href="../generated/pypose.Adj/#pypose.Adj">Adj (class in pypose)</a>
+</li>
+      <li><a href="../generated/pypose.LieTensor/#pypose.LieTensor.Adj">Adj() (pypose.LieTensor method)</a>
+</li>
+      <li><a href="../generated/pypose.AdjT/#pypose.AdjT">AdjT (class in pypose)</a>
+</li>
+      <li><a href="../generated/pypose.LieTensor/#pypose.LieTensor.AdjT">AdjT() (pypose.LieTensor method)</a>
+</li>
+      <li><a href="../generated/pypose.optim.kernel.Arctan/#pypose.optim.kernel.Arctan">Arctan (class in pypose.optim.kernel)</a>
+</li>
+      <li><a href="../generated/pypose.testing.assert_close/#pypose.testing.assert_close">assert_close (class in pypose.testing)</a>
+</li>
+  </ul></td>
+</tr></table>
+
+<h2 id="B">B</h2>
+<table style="width: 100%" class="indextable genindextable"><tr>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="../generated/pypose.module.LTI/#pypose.module.LTI.B">B (pypose.module.LTI property)</a>
+
+      <ul>
+        <li><a href="../generated/pypose.module.NLS/#pypose.module.NLS.B">(pypose.module.NLS property)</a>
+</li>
+      </ul></li>
+      <li><a href="../generated/pypose.bmv/#pypose.bmv">bmv (class in pypose)</a>
+</li>
+  </ul></td>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="../generated/pypose.bspline/#pypose.bspline">bspline (class in pypose)</a>
+</li>
+      <li><a href="../generated/pypose.bvmv/#pypose.bvmv">bvmv (class in pypose)</a>
+</li>
+      <li><a href="../generated/pypose.bvv/#pypose.bvv">bvv (class in pypose)</a>
+</li>
+  </ul></td>
+</tr></table>
+
+<h2 id="C">C</h2>
+<table style="width: 100%" class="indextable genindextable"><tr>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="../generated/pypose.module.LTI/#pypose.module.LTI.C">C (pypose.module.LTI property)</a>
+
+      <ul>
+        <li><a href="../generated/pypose.module.NLS/#pypose.module.NLS.C">(pypose.module.NLS property)</a>
+</li>
+      </ul></li>
+      <li><a href="../generated/pypose.module.LTI/#pypose.module.LTI.c1">c1 (pypose.module.LTI property)</a>
+
+      <ul>
+        <li><a href="../generated/pypose.module.NLS/#pypose.module.NLS.c1">(pypose.module.NLS property)</a>
+</li>
+      </ul></li>
+      <li><a href="../generated/pypose.module.LTI/#pypose.module.LTI.c2">c2 (pypose.module.LTI property)</a>
+
+      <ul>
+        <li><a href="../generated/pypose.module.NLS/#pypose.module.NLS.c2">(pypose.module.NLS property)</a>
+</li>
+      </ul></li>
+      <li><a href="../generated/pypose.cart2homo/#pypose.cart2homo">cart2homo() (in module pypose)</a>
+</li>
+      <li><a href="../generated/pypose.optim.kernel.Cauchy/#pypose.optim.kernel.Cauchy">Cauchy (class in pypose.optim.kernel)</a>
+</li>
+      <li><a href="../generated/pypose.optim.solver.CG/#pypose.optim.solver.CG">CG (class in pypose.optim.solver)</a>
+</li>
+      <li><a href="../generated/pypose.optim.solver.Cholesky/#pypose.optim.solver.Cholesky">Cholesky (class in pypose.optim.solver)</a>
+</li>
+      <li><a href="../generated/pypose.chspline/#pypose.chspline">chspline (class in pypose)</a>
+</li>
+      <li><a href="../generated/pypose.module.PF/#pypose.module.PF.compute_cov">compute_cov() (pypose.module.PF method)</a>
+
+      <ul>
+        <li><a href="../generated/pypose.module.UKF/#pypose.module.UKF.compute_cov">(pypose.module.UKF method)</a>
+</li>
+      </ul></li>
+  </ul></td>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="../generated/pypose.optim.strategy.Constant/#pypose.optim.strategy.Constant">Constant (class in pypose.optim.strategy)</a>
+</li>
+      <li><a href="../generated/pypose.cummul/#pypose.cummul">cummul (class in pypose)</a>
+</li>
+      <li><a href="../generated/pypose.LieTensor/#pypose.LieTensor.cummul">cummul() (pypose.LieTensor method)</a>
+</li>
+      <li><a href="../generated/pypose.cummul_/#pypose.cummul_">cummul_ (class in pypose)</a>
+</li>
+      <li><a href="../generated/pypose.LieTensor/#pypose.LieTensor.cummul_">cummul_() (pypose.LieTensor method)</a>
+</li>
+      <li><a href="../generated/pypose.cumops/#pypose.cumops">cumops (class in pypose)</a>
+</li>
+      <li><a href="../generated/pypose.LieTensor/#pypose.LieTensor.cumops">cumops() (pypose.LieTensor method)</a>
+</li>
+      <li><a href="../generated/pypose.cumops_/#pypose.cumops_">cumops_ (class in pypose)</a>
+</li>
+      <li><a href="../generated/pypose.LieTensor/#pypose.LieTensor.cumops_">cumops_() (pypose.LieTensor method)</a>
+</li>
+      <li><a href="../generated/pypose.cumprod/#pypose.cumprod">cumprod (class in pypose)</a>
+</li>
+      <li><a href="../generated/pypose.LieTensor/#pypose.LieTensor.cumprod">cumprod() (pypose.LieTensor method)</a>
+</li>
+      <li><a href="../generated/pypose.cumprod_/#pypose.cumprod_">cumprod_ (class in pypose)</a>
+</li>
+      <li><a href="../generated/pypose.LieTensor/#pypose.LieTensor.cumprod_">cumprod_() (pypose.LieTensor method)</a>
+</li>
+  </ul></td>
+</tr></table>
+
+<h2 id="D">D</h2>
+<table style="width: 100%" class="indextable genindextable"><tr>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="../generated/pypose.module.LTI/#pypose.module.LTI.D">D (pypose.module.LTI property)</a>
+
+      <ul>
+        <li><a href="../generated/pypose.module.NLS/#pypose.module.NLS.D">(pypose.module.NLS property)</a>
+</li>
+      </ul></li>
+  </ul></td>
+</tr></table>
+
+<h2 id="E">E</h2>
+<table style="width: 100%" class="indextable genindextable"><tr>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="../generated/pypose.module.EKF/#pypose.module.EKF">EKF (class in pypose.module)</a>
+</li>
+      <li><a href="../generated/pypose.module.EPnP/#pypose.module.EPnP">EPnP (class in pypose.module)</a>
+</li>
+      <li><a href="../generated/pypose.euler/#pypose.euler">euler() (in module pypose)</a>
+
+      <ul>
+        <li><a href="../generated/pypose.LieTensor/#pypose.LieTensor.euler">(pypose.LieTensor method)</a>
+</li>
+      </ul></li>
+  </ul></td>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="../generated/pypose.euler2SO3/#pypose.euler2SO3">euler2SO3() (in module pypose)</a>
+</li>
+      <li><a href="../generated/pypose.Exp/#pypose.Exp">Exp (class in pypose)</a>
+</li>
+      <li><a href="../generated/pypose.LieTensor/#pypose.LieTensor.Exp">Exp() (pypose.LieTensor method)</a>
+</li>
+  </ul></td>
+</tr></table>
+
+<h2 id="F">F</h2>
+<table style="width: 100%" class="indextable genindextable"><tr>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="../generated/pypose.optim.corrector.FastTriggs/#pypose.optim.corrector.FastTriggs">FastTriggs (class in pypose.optim.corrector)</a>
+</li>
+      <li><a href="../generated/pypose.module.EKF/#pypose.module.EKF.forward">forward() (pypose.module.EKF method)</a>
+
+      <ul>
+        <li><a href="../generated/pypose.module.EPnP/#pypose.module.EPnP.forward">(pypose.module.EPnP method)</a>
+</li>
+        <li><a href="../generated/pypose.module.ICP/#pypose.module.ICP.forward">(pypose.module.ICP method)</a>
+</li>
+        <li><a href="../generated/pypose.module.IMUPreintegrator/#pypose.module.IMUPreintegrator.forward">(pypose.module.IMUPreintegrator method)</a>
+</li>
+        <li><a href="../generated/pypose.module.LQR/#pypose.module.LQR.forward">(pypose.module.LQR method)</a>
+</li>
+        <li><a href="../generated/pypose.module.LTI/#pypose.module.LTI.forward">(pypose.module.LTI method)</a>
+</li>
+        <li><a href="../generated/pypose.module.MPC/#pypose.module.MPC.forward">(pypose.module.MPC method)</a>
+</li>
+        <li><a href="../generated/pypose.module.NLS/#pypose.module.NLS.forward">(pypose.module.NLS method)</a>
+</li>
+        <li><a href="../generated/pypose.module.PF/#pypose.module.PF.forward">(pypose.module.PF method)</a>
+</li>
+        <li><a href="../generated/pypose.module.System/#pypose.module.System.forward">(pypose.module.System method)</a>
+</li>
+        <li><a href="../generated/pypose.module.UKF/#pypose.module.UKF.forward">(pypose.module.UKF method)</a>
+</li>
+        <li><a href="../generated/pypose.optim.corrector.FastTriggs/#pypose.optim.corrector.FastTriggs.forward">(pypose.optim.corrector.FastTriggs method)</a>
+</li>
+        <li><a href="../generated/pypose.optim.corrector.Triggs/#pypose.optim.corrector.Triggs.forward">(pypose.optim.corrector.Triggs method)</a>
+</li>
+        <li><a href="../generated/pypose.optim.kernel.Arctan/#pypose.optim.kernel.Arctan.forward">(pypose.optim.kernel.Arctan method)</a>
+</li>
+        <li><a href="../generated/pypose.optim.kernel.Cauchy/#pypose.optim.kernel.Cauchy.forward">(pypose.optim.kernel.Cauchy method)</a>
+</li>
+        <li><a href="../generated/pypose.optim.kernel.Huber/#pypose.optim.kernel.Huber.forward">(pypose.optim.kernel.Huber method)</a>
+</li>
+        <li><a href="../generated/pypose.optim.kernel.PseudoHuber/#pypose.optim.kernel.PseudoHuber.forward">(pypose.optim.kernel.PseudoHuber method)</a>
+</li>
+        <li><a href="../generated/pypose.optim.kernel.Scale/#pypose.optim.kernel.Scale.forward">(pypose.optim.kernel.Scale method)</a>
+</li>
+        <li><a href="../generated/pypose.optim.kernel.SoftLOne/#pypose.optim.kernel.SoftLOne.forward">(pypose.optim.kernel.SoftLOne method)</a>
+</li>
+        <li><a href="../generated/pypose.optim.kernel.Tolerant/#pypose.optim.kernel.Tolerant.forward">(pypose.optim.kernel.Tolerant method)</a>
+</li>
+        <li><a href="../generated/pypose.optim.solver.CG/#pypose.optim.solver.CG.forward">(pypose.optim.solver.CG method)</a>
+</li>
+        <li><a href="../generated/pypose.optim.solver.Cholesky/#pypose.optim.solver.Cholesky.forward">(pypose.optim.solver.Cholesky method)</a>
+</li>
+        <li><a href="../generated/pypose.optim.solver.LSTSQ/#pypose.optim.solver.LSTSQ.forward">(pypose.optim.solver.LSTSQ method)</a>
+</li>
+        <li><a href="../generated/pypose.optim.solver.PINV/#pypose.optim.solver.PINV.forward">(pypose.optim.solver.PINV method)</a>
+</li>
+      </ul></li>
+  </ul></td>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="../generated/pypose.module.System/#pypose.module.System.forward_hook">forward_hook() (pypose.module.System method)</a>
+</li>
+      <li><a href="../generated/pypose.from_matrix/#pypose.from_matrix">from_matrix() (in module pypose)</a>
+</li>
+  </ul></td>
+</tr></table>
+
+<h2 id="G">G</h2>
+<table style="width: 100%" class="indextable genindextable"><tr>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="../generated/pypose.optim.GaussNewton/#pypose.optim.GaussNewton">GaussNewton (class in pypose.optim)</a>
+</li>
+  </ul></td>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="../generated/pypose.module.PF/#pypose.module.PF.generate_particles">generate_particles() (pypose.module.PF method)</a>
+</li>
+      <li><a href="../generated/pypose.optim.GN/#pypose.optim.GN">GN (in module pypose.optim)</a>
+</li>
+  </ul></td>
+</tr></table>
+
+<h2 id="H">H</h2>
+<table style="width: 100%" class="indextable genindextable"><tr>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="../generated/pypose.hasnan/#pypose.hasnan">hasnan (class in pypose)</a>
+</li>
+  </ul></td>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="../generated/pypose.homo2cart/#pypose.homo2cart">homo2cart() (in module pypose)</a>
+</li>
+      <li><a href="../generated/pypose.optim.kernel.Huber/#pypose.optim.kernel.Huber">Huber (class in pypose.optim.kernel)</a>
+</li>
+  </ul></td>
+</tr></table>
+
+<h2 id="I">I</h2>
+<table style="width: 100%" class="indextable genindextable"><tr>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="../generated/pypose.module.ICP/#pypose.module.ICP">ICP (class in pypose.module)</a>
+</li>
+      <li><a href="../generated/pypose.LieTensor/#pypose.LieTensor.identity_">identity_() (pypose.LieTensor method)</a>
+</li>
+      <li><a href="../generated/pypose.identity_like/#pypose.identity_like">identity_like (class in pypose)</a>
+</li>
+      <li><a href="../generated/pypose.identity_RxSO3/#pypose.identity_RxSO3">identity_RxSO3 (class in pypose)</a>
+</li>
+      <li><a href="../generated/pypose.identity_rxso3/#pypose.identity_rxso3">identity_rxso3 (class in pypose)</a>
+</li>
+      <li><a href="../generated/pypose.identity_SE3/#pypose.identity_SE3">identity_SE3 (class in pypose)</a>
+</li>
+      <li><a href="../generated/pypose.identity_se3/#pypose.identity_se3">identity_se3 (class in pypose)</a>
+</li>
+      <li><a href="../generated/pypose.identity_Sim3/#pypose.identity_Sim3">identity_Sim3 (class in pypose)</a>
+</li>
+  </ul></td>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="../generated/pypose.identity_sim3/#pypose.identity_sim3">identity_sim3 (class in pypose)</a>
+</li>
+      <li><a href="../generated/pypose.identity_SO3/#pypose.identity_SO3">identity_SO3 (class in pypose)</a>
+</li>
+      <li><a href="../generated/pypose.identity_so3/#pypose.identity_so3">identity_so3 (class in pypose)</a>
+</li>
+      <li><a href="../generated/pypose.module.IMUPreintegrator/#pypose.module.IMUPreintegrator">IMUPreintegrator (class in pypose.module)</a>
+</li>
+      <li><a href="../generated/pypose.module.IMUPreintegrator/#pypose.module.IMUPreintegrator.integrate">integrate() (pypose.module.IMUPreintegrator method)</a>
+</li>
+      <li><a href="../generated/pypose.Inv/#pypose.Inv">Inv (class in pypose)</a>
+</li>
+      <li><a href="../generated/pypose.LieTensor/#pypose.LieTensor.Inv">Inv() (pypose.LieTensor method)</a>
+</li>
+      <li><a href="../generated/pypose.is_lietensor/#pypose.is_lietensor">is_lietensor (class in pypose)</a>
+</li>
+      <li><a href="../generated/pypose.is_SE3/#pypose.is_SE3">is_SE3 (class in pypose)</a>
+</li>
+  </ul></td>
+</tr></table>
+
+<h2 id="J">J</h2>
+<table style="width: 100%" class="indextable genindextable"><tr>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="../generated/pypose.func.jacrev/#pypose.func.jacrev">jacrev() (in module pypose.func)</a>
+</li>
+      <li><a href="../generated/pypose.Jinvp/#pypose.Jinvp">Jinvp (class in pypose)</a>
+</li>
+  </ul></td>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="../generated/pypose.LieTensor/#pypose.LieTensor.Jinvp">Jinvp() (pypose.LieTensor method)</a>
+</li>
+      <li><a href="../generated/pypose.Jr/#pypose.Jr">Jr (class in pypose)</a>
+</li>
+      <li><a href="../generated/pypose.LieTensor/#pypose.LieTensor.Jr">Jr() (pypose.LieTensor method)</a>
+</li>
+  </ul></td>
+</tr></table>
+
+<h2 id="K">K</h2>
+<table style="width: 100%" class="indextable genindextable"><tr>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="../generated/pypose.knn/#pypose.knn">knn (class in pypose)</a>
+</li>
+  </ul></td>
+</tr></table>
+
+<h2 id="L">L</h2>
+<table style="width: 100%" class="indextable genindextable"><tr>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="../generated/pypose.optim.LevenbergMarquardt/#pypose.optim.LevenbergMarquardt">LevenbergMarquardt (class in pypose.optim)</a>
+</li>
+      <li><a href="../generated/pypose.LieTensor/#pypose.LieTensor">LieTensor (class in pypose)</a>
+</li>
+      <li><a href="../generated/pypose.optim.LM/#pypose.optim.LM">LM (in module pypose.optim)</a>
+</li>
+      <li><a href="../generated/pypose.Log/#pypose.Log">Log (class in pypose)</a>
+</li>
+      <li><a href="../generated/pypose.LieTensor/#pypose.LieTensor.Log">Log() (pypose.LieTensor method)</a>
+</li>
+  </ul></td>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="../generated/pypose.module.LQR/#pypose.module.LQR">LQR (class in pypose.module)</a>
+</li>
+      <li><a href="../generated/pypose.LieTensor/#pypose.LieTensor.lshape">lshape (pypose.LieTensor property)</a>
+</li>
+      <li><a href="../generated/pypose.optim.solver.LSTSQ/#pypose.optim.solver.LSTSQ">LSTSQ (class in pypose.optim.solver)</a>
+</li>
+      <li><a href="../generated/pypose.module.LTI/#pypose.module.LTI">LTI (class in pypose.module)</a>
+</li>
+      <li><a href="../generated/pypose.module.LTV/#pypose.module.LTV">LTV (class in pypose.module)</a>
+</li>
+      <li><a href="../generated/pypose.LieTensor/#pypose.LieTensor.lview">lview() (pypose.LieTensor method)</a>
+</li>
+  </ul></td>
+</tr></table>
+
+<h2 id="M">M</h2>
+<table style="width: 100%" class="indextable genindextable"><tr>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="../generated/pypose.mat2RxSO3/#pypose.mat2RxSO3">mat2RxSO3() (in module pypose)</a>
+</li>
+      <li><a href="../generated/pypose.mat2SE3/#pypose.mat2SE3">mat2SE3() (in module pypose)</a>
+</li>
+      <li><a href="../generated/pypose.mat2Sim3/#pypose.mat2Sim3">mat2Sim3() (in module pypose)</a>
+</li>
+      <li><a href="../generated/pypose.mat2SO3/#pypose.mat2SO3">mat2SO3() (in module pypose)</a>
+</li>
+      <li><a href="../generated/pypose.matrix/#pypose.matrix">matrix() (in module pypose)</a>
+
+      <ul>
+        <li><a href="../generated/pypose.LieTensor/#pypose.LieTensor.matrix">(pypose.LieTensor method)</a>
+</li>
+      </ul></li>
+  </ul></td>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="../generated/pypose.optim.functional.modjac/#pypose.optim.functional.modjac">modjac (class in pypose.optim.functional)</a>
+</li>
+      <li><a href="../generated/pypose.module.MPC/#pypose.module.MPC">MPC (class in pypose.module)</a>
+</li>
+      <li><a href="../generated/pypose.mul/#pypose.mul">mul (class in pypose)</a>
+</li>
+      <li><a href="../generated/pypose.LieTensor/#pypose.LieTensor.mul">mul() (pypose.LieTensor method)</a>
+</li>
+  </ul></td>
+</tr></table>
+
+<h2 id="N">N</h2>
+<table style="width: 100%" class="indextable genindextable"><tr>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="../generated/pypose.LieTensor/#pypose.LieTensor.new_empty">new_empty() (pypose.LieTensor method)</a>
+</li>
+  </ul></td>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="../generated/pypose.module.NLS/#pypose.module.NLS">NLS (class in pypose.module)</a>
+</li>
+  </ul></td>
+</tr></table>
+
+<h2 id="O">O</h2>
+<table style="width: 100%" class="indextable genindextable"><tr>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="../generated/pypose.module.LTI/#pypose.module.LTI.observation">observation() (pypose.module.LTI method)</a>
+
+      <ul>
+        <li><a href="../generated/pypose.module.System/#pypose.module.System.observation">(pypose.module.System method)</a>
+</li>
+      </ul></li>
+  </ul></td>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="../generated/pypose.optim.scheduler.StopOnPlateau/#pypose.optim.scheduler.StopOnPlateau.optimize">optimize() (pypose.optim.scheduler.StopOnPlateau method)</a>
+</li>
+  </ul></td>
+</tr></table>
+
+<h2 id="P">P</h2>
+<table style="width: 100%" class="indextable genindextable"><tr>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="../generated/pypose.Parameter/#pypose.Parameter">Parameter (class in pypose)</a>
+</li>
+      <li><a href="../generated/pypose.module.PF/#pypose.module.PF">PF (class in pypose.module)</a>
+</li>
+      <li><a href="../generated/pypose.optim.solver.PINV/#pypose.optim.solver.PINV">PINV (class in pypose.optim.solver)</a>
+</li>
+      <li><a href="../generated/pypose.pixel2point/#pypose.pixel2point">pixel2point() (in module pypose)</a>
+</li>
+  </ul></td>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="../generated/pypose.pm/#pypose.pm">pm (class in pypose)</a>
+</li>
+      <li><a href="../generated/pypose.point2pixel/#pypose.point2pixel">point2pixel() (in module pypose)</a>
+</li>
+      <li><a href="../generated/pypose.module.IMUPreintegrator/#pypose.module.IMUPreintegrator.predict">predict() (pypose.module.IMUPreintegrator class method)</a>
+</li>
+      <li><a href="../generated/pypose.module.UKF/#pypose.module.UKF.project_to_positive_definite">project_to_positive_definite() (pypose.module.UKF method)</a>
+</li>
+      <li><a href="../generated/pypose.optim.kernel.PseudoHuber/#pypose.optim.kernel.PseudoHuber">PseudoHuber (class in pypose.optim.kernel)</a>
+</li>
+  </ul></td>
+</tr></table>
+
+<h2 id="Q">Q</h2>
+<table style="width: 100%" class="indextable genindextable"><tr>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="../generated/pypose.module.EKF/#pypose.module.EKF.Q">Q (pypose.module.EKF property)</a>
+</li>
+  </ul></td>
+</tr></table>
+
+<h2 id="R">R</h2>
+<table style="width: 100%" class="indextable genindextable"><tr>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="../generated/pypose.module.EKF/#pypose.module.EKF.R">R (pypose.module.EKF property)</a>
+</li>
+      <li><a href="../generated/pypose.randn_like/#pypose.randn_like">randn_like (class in pypose)</a>
+</li>
+      <li><a href="../generated/pypose.randn_RxSO3/#pypose.randn_RxSO3">randn_RxSO3 (class in pypose)</a>
+</li>
+      <li><a href="../generated/pypose.randn_rxso3/#pypose.randn_rxso3">randn_rxso3 (class in pypose)</a>
+</li>
+      <li><a href="../generated/pypose.randn_SE3/#pypose.randn_SE3">randn_SE3 (class in pypose)</a>
+</li>
+      <li><a href="../generated/pypose.randn_se3/#pypose.randn_se3">randn_se3 (class in pypose)</a>
+</li>
+      <li><a href="../generated/pypose.randn_Sim3/#pypose.randn_Sim3">randn_Sim3 (class in pypose)</a>
+</li>
+      <li><a href="../generated/pypose.randn_sim3/#pypose.randn_sim3">randn_sim3 (class in pypose)</a>
+</li>
+      <li><a href="../generated/pypose.randn_SO3/#pypose.randn_SO3">randn_SO3 (class in pypose)</a>
+</li>
+      <li><a href="../generated/pypose.randn_so3/#pypose.randn_so3">randn_so3 (class in pypose)</a>
+</li>
+  </ul></td>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="../generated/pypose.utils.ReduceToBason/#pypose.utils.ReduceToBason">ReduceToBason (class in pypose.utils)</a>
+</li>
+      <li><a href="../generated/pypose.module.PF/#pypose.module.PF.relative_likelihood">relative_likelihood() (pypose.module.PF method)</a>
+</li>
+      <li><a href="../generated/pypose.reprojerr/#pypose.reprojerr">reprojerr (class in pypose)</a>
+</li>
+      <li><a href="../generated/pypose.module.PF/#pypose.module.PF.resample_particles">resample_particles() (pypose.module.PF method)</a>
+</li>
+      <li><a href="../generated/pypose.Retr/#pypose.Retr">Retr (class in pypose)</a>
+</li>
+      <li><a href="../generated/pypose.LieTensor/#pypose.LieTensor.Retr">Retr() (pypose.LieTensor method)</a>
+</li>
+      <li><a href="../generated/pypose.rotation/#pypose.rotation">rotation() (in module pypose)</a>
+
+      <ul>
+        <li><a href="../generated/pypose.LieTensor/#pypose.LieTensor.rotation">(pypose.LieTensor method)</a>
+</li>
+      </ul></li>
+      <li><a href="../generated/pypose.RxSO3/#pypose.RxSO3">RxSO3 (in module pypose)</a>
+</li>
+      <li><a href="../generated/pypose.rxso3/#pypose.rxso3">rxso3 (in module pypose)</a>
+</li>
+  </ul></td>
+</tr></table>
+
+<h2 id="S">S</h2>
+<table style="width: 100%" class="indextable genindextable"><tr>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="../generated/pypose.optim.kernel.Scale/#pypose.optim.kernel.Scale">Scale (class in pypose.optim.kernel)</a>
+</li>
+      <li><a href="../generated/pypose.scale/#pypose.scale">scale() (in module pypose)</a>
+
+      <ul>
+        <li><a href="../generated/pypose.LieTensor/#pypose.LieTensor.scale">(pypose.LieTensor method)</a>
+</li>
+      </ul></li>
+      <li><a href="../generated/pypose.SE3/#pypose.SE3">SE3 (in module pypose)</a>
+</li>
+      <li><a href="../generated/pypose.se3/#pypose.se3">se3 (in module pypose)</a>
+</li>
+      <li><a href="../generated/pypose.module.LTV/#pypose.module.LTV.set_refpoint">set_refpoint() (pypose.module.LTV method)</a>
+
+      <ul>
+        <li><a href="../generated/pypose.module.NLS/#pypose.module.NLS.set_refpoint">(pypose.module.NLS method)</a>
+</li>
+        <li><a href="../generated/pypose.module.System/#pypose.module.System.set_refpoint">(pypose.module.System method)</a>
+</li>
+      </ul></li>
+      <li><a href="../generated/pypose.module.EKF/#pypose.module.EKF.set_uncertainty">set_uncertainty() (pypose.module.EKF method)</a>
+</li>
+      <li><a href="../generated/pypose.module.UKF/#pypose.module.UKF.sigma_weight_points">sigma_weight_points() (pypose.module.UKF method)</a>
+</li>
+      <li><a href="../generated/pypose.Sim3/#pypose.Sim3">Sim3 (in module pypose)</a>
+</li>
+      <li><a href="../generated/pypose.sim3/#pypose.sim3">sim3 (in module pypose)</a>
+</li>
+  </ul></td>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="../generated/pypose.SO3/#pypose.SO3">SO3 (in module pypose)</a>
+</li>
+      <li><a href="../generated/pypose.so3/#pypose.so3">so3 (in module pypose)</a>
+</li>
+      <li><a href="../generated/pypose.optim.kernel.SoftLOne/#pypose.optim.kernel.SoftLOne">SoftLOne (class in pypose.optim.kernel)</a>
+</li>
+      <li><a href="../generated/pypose.module.LTI/#pypose.module.LTI.state_transition">state_transition() (pypose.module.LTI method)</a>
+
+      <ul>
+        <li><a href="../generated/pypose.module.System/#pypose.module.System.state_transition">(pypose.module.System method)</a>
+</li>
+      </ul></li>
+      <li><a href="../generated/pypose.optim.GaussNewton/#pypose.optim.GaussNewton.step">step() (pypose.optim.GaussNewton method)</a>
+
+      <ul>
+        <li><a href="../generated/pypose.optim.LevenbergMarquardt/#pypose.optim.LevenbergMarquardt.step">(pypose.optim.LevenbergMarquardt method)</a>
+</li>
+        <li><a href="../generated/pypose.optim.scheduler.StopOnPlateau/#pypose.optim.scheduler.StopOnPlateau.step">(pypose.optim.scheduler.StopOnPlateau method)</a>
+</li>
+        <li><a href="../generated/pypose.utils.ReduceToBason/#pypose.utils.ReduceToBason.step">(pypose.utils.ReduceToBason method)</a>
+</li>
+      </ul></li>
+      <li><a href="../generated/pypose.optim.scheduler.StopOnPlateau/#pypose.optim.scheduler.StopOnPlateau">StopOnPlateau (class in pypose.optim.scheduler)</a>
+</li>
+      <li><a href="../generated/pypose.svdtf/#pypose.svdtf">svdtf (class in pypose)</a>
+</li>
+      <li><a href="../generated/pypose.module.System/#pypose.module.System">System (class in pypose.module)</a>
+</li>
+      <li><a href="../generated/pypose.module.System/#pypose.module.System.systime">systime (pypose.module.System property)</a>
+</li>
+  </ul></td>
+</tr></table>
+
+<h2 id="T">T</h2>
+<table style="width: 100%" class="indextable genindextable"><tr>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="../generated/pypose.tensor/#pypose.tensor">tensor() (in module pypose)</a>
+
+      <ul>
+        <li><a href="../generated/pypose.LieTensor/#pypose.LieTensor.tensor">(pypose.LieTensor method)</a>
+</li>
+      </ul></li>
+      <li><a href="../generated/pypose.optim.kernel.Tolerant/#pypose.optim.kernel.Tolerant">Tolerant (class in pypose.optim.kernel)</a>
+</li>
+  </ul></td>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="../generated/pypose.translation/#pypose.translation">translation() (in module pypose)</a>
+
+      <ul>
+        <li><a href="../generated/pypose.LieTensor/#pypose.LieTensor.translation">(pypose.LieTensor method)</a>
+</li>
+      </ul></li>
+      <li><a href="../generated/pypose.optim.corrector.Triggs/#pypose.optim.corrector.Triggs">Triggs (class in pypose.optim.corrector)</a>
+</li>
+      <li><a href="../generated/pypose.optim.strategy.TrustRegion/#pypose.optim.strategy.TrustRegion">TrustRegion (class in pypose.optim.strategy)</a>
+</li>
+  </ul></td>
+</tr></table>
+
+<h2 id="U">U</h2>
+<table style="width: 100%" class="indextable genindextable"><tr>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="../generated/pypose.module.UKF/#pypose.module.UKF">UKF (class in pypose.module)</a>
+</li>
+  </ul></td>
+</tr></table>
+
+<h2 id="V">V</h2>
+<table style="width: 100%" class="indextable genindextable"><tr>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="../generated/pypose.vec2skew/#pypose.vec2skew">vec2skew() (in module pypose)</a>
+</li>
+  </ul></td>
+</tr></table>
+
+
+
+             </article>
+             
+            </div>
+            <footer>
+  
+
+  
+
+    <hr>
+
+  
+
+  <div role="contentinfo">
+    <p>
+        &copy; Copyright 2022, PyPose Contributors.
+
+    </p>
+  </div>
+    
+      <div>
+        Built with <a href="http://sphinx-doc.org/">Sphinx</a> using a <a href="https://github.com/rtfd/sphinx_rtd_theme">theme</a> provided by <a href="https://readthedocs.org">Read the Docs</a>.
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+     
+
+</footer>
+
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+        </div>
+
+        <div class="pytorch-content-right" id="pytorch-content-right">
+          <div class="pytorch-right-menu" id="pytorch-right-menu">
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+<li class="toctree-l1"><a class="reference internal" href="lietensor/">LieTensor</a><ul>
+<li class="toctree-l2"><a class="reference internal" href="generated/pypose.LieTensor/">pypose.LieTensor</a></li>
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+<li class="toctree-l1 current"><a class="current reference internal" href="#">LieTensor</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../func/">Func</a></li>
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+             <article itemprop="articleBody" id="pytorch-article" class="pytorch-article">
+              
+  <section id="lietensor">
+<h1>LieTensor<a class="headerlink" href="#lietensor" title="Permalink to this heading">¶</a></h1>
+<table class="autosummary longtable docutils align-default">
+<tbody>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code></a></p></td>
+<td><p>A sub-class of <code class="xref py py-obj docutils literal notranslate"><span class="pre">torch.Tensor</span></code> to represent Lie Algebra and Lie Group.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.Parameter/#pypose.Parameter" title="pypose.Parameter"><code class="xref py py-obj docutils literal notranslate"><span class="pre">Parameter</span></code></a></p></td>
+<td><p>A kind of LieTensor that is to be considered a module parameter.</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.SO3/#pypose.SO3" title="pypose.SO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">SO3</span></code></a></p></td>
+<td><p>Alias of SO3 type <a class="reference internal" href="../generated/pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code></a>.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.so3/#pypose.so3" title="pypose.so3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">so3</span></code></a></p></td>
+<td><p>Alias of so3 type <a class="reference internal" href="../generated/pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code></a>.</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.SE3/#pypose.SE3" title="pypose.SE3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">SE3</span></code></a></p></td>
+<td><p>Alias of SE3 type <a class="reference internal" href="../generated/pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code></a>.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.se3/#pypose.se3" title="pypose.se3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">se3</span></code></a></p></td>
+<td><p>Alias of se3 type <a class="reference internal" href="../generated/pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code></a>.</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.Sim3/#pypose.Sim3" title="pypose.Sim3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">Sim3</span></code></a></p></td>
+<td><p>Alias of Sim3 type <a class="reference internal" href="../generated/pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code></a>.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.sim3/#pypose.sim3" title="pypose.sim3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">sim3</span></code></a></p></td>
+<td><p>Alias of sim3 type <a class="reference internal" href="../generated/pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code></a>.</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.RxSO3/#pypose.RxSO3" title="pypose.RxSO3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">RxSO3</span></code></a></p></td>
+<td><p>Alias of RxSO3 type <a class="reference internal" href="../generated/pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code></a>.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.rxso3/#pypose.rxso3" title="pypose.rxso3"><code class="xref py py-obj docutils literal notranslate"><span class="pre">rxso3</span></code></a></p></td>
+<td><p>Alias of rxso3 type <a class="reference internal" href="../generated/pypose.LieTensor/#pypose.LieTensor" title="pypose.LieTensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieTensor</span></code></a>.</p></td>
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+             <article itemprop="articleBody" id="pytorch-article" class="pytorch-article">
+              
+  <section id="modules">
+<h1>Modules<a class="headerlink" href="#modules" title="Permalink to this heading">¶</a></h1>
+<table class="autosummary longtable docutils align-default">
+<tbody>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.module.System/#pypose.module.System" title="pypose.module.System"><code class="xref py py-obj docutils literal notranslate"><span class="pre">module.System</span></code></a></p></td>
+<td><p>The base class for a system dynamics model.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.module.LTI/#pypose.module.LTI" title="pypose.module.LTI"><code class="xref py py-obj docutils literal notranslate"><span class="pre">module.LTI</span></code></a></p></td>
+<td><p>Discrete-time Linear Time-Invariant (LTI) system.</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.module.LTV/#pypose.module.LTV" title="pypose.module.LTV"><code class="xref py py-obj docutils literal notranslate"><span class="pre">module.LTV</span></code></a></p></td>
+<td><p>Discrete-time Linear Time-Variant (LTV) system.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.module.NLS/#pypose.module.NLS" title="pypose.module.NLS"><code class="xref py py-obj docutils literal notranslate"><span class="pre">module.NLS</span></code></a></p></td>
+<td><p>Dynamics model for discrete-time non-linear system (NLS).</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.module.EKF/#pypose.module.EKF" title="pypose.module.EKF"><code class="xref py py-obj docutils literal notranslate"><span class="pre">module.EKF</span></code></a></p></td>
+<td><p>Performs Batched Extended Kalman Filter (EKF).</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.module.UKF/#pypose.module.UKF" title="pypose.module.UKF"><code class="xref py py-obj docutils literal notranslate"><span class="pre">module.UKF</span></code></a></p></td>
+<td><p>Performs Batched Unscented Kalman Filter (UKF).</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.module.PF/#pypose.module.PF" title="pypose.module.PF"><code class="xref py py-obj docutils literal notranslate"><span class="pre">module.PF</span></code></a></p></td>
+<td><p>Performs Batched Particle Filter (PF).</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.module.LQR/#pypose.module.LQR" title="pypose.module.LQR"><code class="xref py py-obj docutils literal notranslate"><span class="pre">module.LQR</span></code></a></p></td>
+<td><p>Linear Quadratic Regulator (LQR) with Dynamic Programming.</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.module.IMUPreintegrator/#pypose.module.IMUPreintegrator" title="pypose.module.IMUPreintegrator"><code class="xref py py-obj docutils literal notranslate"><span class="pre">module.IMUPreintegrator</span></code></a></p></td>
+<td><p>Applies preintegration over IMU input signals.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.module.EPnP/#pypose.module.EPnP" title="pypose.module.EPnP"><code class="xref py py-obj docutils literal notranslate"><span class="pre">module.EPnP</span></code></a></p></td>
+<td><p>Batched EPnP Solver - a non-iterative <span class="math">\(\mathcal{O}(n)\)</span> solution to the Perspective-<span class="math">\(n\)</span>-Point (PnP) problem for <span class="math">\(n \geq 4\)</span>.</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.module.MPC/#pypose.module.MPC" title="pypose.module.MPC"><code class="xref py py-obj docutils literal notranslate"><span class="pre">module.MPC</span></code></a></p></td>
+<td><p>Model Predictive Control (MPC) based on iterative LQR.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.module.ICP/#pypose.module.ICP" title="pypose.module.ICP"><code class="xref py py-obj docutils literal notranslate"><span class="pre">module.ICP</span></code></a></p></td>
+<td><p>Batched Iterative Closest Point (ICP) algorithm to find a rigid transformation between two sets of points using Singular Value Decomposition (SVD).</p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+              
+  <section id="optimization">
+<h1>Optimization<a class="headerlink" href="#optimization" title="Permalink to this heading">¶</a></h1>
+<table class="autosummary longtable docutils align-default">
+<tbody>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.optim.GaussNewton/#pypose.optim.GaussNewton" title="pypose.optim.GaussNewton"><code class="xref py py-obj docutils literal notranslate"><span class="pre">optim.GaussNewton</span></code></a></p></td>
+<td><p>The Gauss-Newton (GN) algorithm solving non-linear least squares problems.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.optim.GN/#pypose.optim.GN" title="pypose.optim.GN"><code class="xref py py-obj docutils literal notranslate"><span class="pre">optim.GN</span></code></a></p></td>
+<td><p>alias of <a class="reference internal" href="../generated/pypose.optim.GaussNewton/#pypose.optim.GaussNewton" title="pypose.optim.optimizer.GaussNewton"><code class="xref py py-class docutils literal notranslate"><span class="pre">GaussNewton</span></code></a></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.optim.LevenbergMarquardt/#pypose.optim.LevenbergMarquardt" title="pypose.optim.LevenbergMarquardt"><code class="xref py py-obj docutils literal notranslate"><span class="pre">optim.LevenbergMarquardt</span></code></a></p></td>
+<td><p>The Levenberg-Marquardt (LM) algorithm solving non-linear least squares problems.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.optim.LM/#pypose.optim.LM" title="pypose.optim.LM"><code class="xref py py-obj docutils literal notranslate"><span class="pre">optim.LM</span></code></a></p></td>
+<td><p>alias of <a class="reference internal" href="../generated/pypose.optim.LevenbergMarquardt/#pypose.optim.LevenbergMarquardt" title="pypose.optim.optimizer.LevenbergMarquardt"><code class="xref py py-class docutils literal notranslate"><span class="pre">LevenbergMarquardt</span></code></a></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.optim.solver.PINV/#pypose.optim.solver.PINV" title="pypose.optim.solver.PINV"><code class="xref py py-obj docutils literal notranslate"><span class="pre">optim.solver.PINV</span></code></a></p></td>
+<td><p>The batched linear solver with pseudo inversion.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.optim.solver.LSTSQ/#pypose.optim.solver.LSTSQ" title="pypose.optim.solver.LSTSQ"><code class="xref py py-obj docutils literal notranslate"><span class="pre">optim.solver.LSTSQ</span></code></a></p></td>
+<td><p>The batched linear solver with fast pseudo inversion.</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.optim.solver.Cholesky/#pypose.optim.solver.Cholesky" title="pypose.optim.solver.Cholesky"><code class="xref py py-obj docutils literal notranslate"><span class="pre">optim.solver.Cholesky</span></code></a></p></td>
+<td><p>The batched linear solver with Cholesky decomposition.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.optim.solver.CG/#pypose.optim.solver.CG" title="pypose.optim.solver.CG"><code class="xref py py-obj docutils literal notranslate"><span class="pre">optim.solver.CG</span></code></a></p></td>
+<td><p>The batched linear solver with conjugate gradient method.</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.optim.strategy.Constant/#pypose.optim.strategy.Constant" title="pypose.optim.strategy.Constant"><code class="xref py py-obj docutils literal notranslate"><span class="pre">optim.strategy.Constant</span></code></a></p></td>
+<td><p>Constant damping strategy used in the Levenberg-Marquardt (LM) algorithm.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.optim.strategy.Adaptive/#pypose.optim.strategy.Adaptive" title="pypose.optim.strategy.Adaptive"><code class="xref py py-obj docutils literal notranslate"><span class="pre">optim.strategy.Adaptive</span></code></a></p></td>
+<td><p>Adaptive damping strategy used in the Levenberg-Marquardt (LM) algorithm.</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.optim.strategy.TrustRegion/#pypose.optim.strategy.TrustRegion" title="pypose.optim.strategy.TrustRegion"><code class="xref py py-obj docutils literal notranslate"><span class="pre">optim.strategy.TrustRegion</span></code></a></p></td>
+<td><p>The trust region (TR) algorithm used in the Levenberg-Marquardt (LM) algorithm.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.optim.kernel.Scale/#pypose.optim.kernel.Scale" title="pypose.optim.kernel.Scale"><code class="xref py py-obj docutils literal notranslate"><span class="pre">optim.kernel.Scale</span></code></a></p></td>
+<td><p>The robust Scale kernel cost function.</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.optim.kernel.Huber/#pypose.optim.kernel.Huber" title="pypose.optim.kernel.Huber"><code class="xref py py-obj docutils literal notranslate"><span class="pre">optim.kernel.Huber</span></code></a></p></td>
+<td><p>The robust Huber kernel cost function.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.optim.kernel.PseudoHuber/#pypose.optim.kernel.PseudoHuber" title="pypose.optim.kernel.PseudoHuber"><code class="xref py py-obj docutils literal notranslate"><span class="pre">optim.kernel.PseudoHuber</span></code></a></p></td>
+<td><p>The robust pseudo Huber kernel cost function.</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.optim.kernel.SoftLOne/#pypose.optim.kernel.SoftLOne" title="pypose.optim.kernel.SoftLOne"><code class="xref py py-obj docutils literal notranslate"><span class="pre">optim.kernel.SoftLOne</span></code></a></p></td>
+<td><p>The robust SoftLOne kernel cost function.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.optim.kernel.Cauchy/#pypose.optim.kernel.Cauchy" title="pypose.optim.kernel.Cauchy"><code class="xref py py-obj docutils literal notranslate"><span class="pre">optim.kernel.Cauchy</span></code></a></p></td>
+<td><p>The robust Cauchy kernel cost function.</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.optim.kernel.Arctan/#pypose.optim.kernel.Arctan" title="pypose.optim.kernel.Arctan"><code class="xref py py-obj docutils literal notranslate"><span class="pre">optim.kernel.Arctan</span></code></a></p></td>
+<td><p>The robust Arctan kernel cost function.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.optim.kernel.Tolerant/#pypose.optim.kernel.Tolerant" title="pypose.optim.kernel.Tolerant"><code class="xref py py-obj docutils literal notranslate"><span class="pre">optim.kernel.Tolerant</span></code></a></p></td>
+<td><p>The robust Tolerant kernel cost function.</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.optim.corrector.FastTriggs/#pypose.optim.corrector.FastTriggs" title="pypose.optim.corrector.FastTriggs"><code class="xref py py-obj docutils literal notranslate"><span class="pre">optim.corrector.FastTriggs</span></code></a></p></td>
+<td><p>Faster yet stable version of Triggs correction of model residual and Jacobian.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.optim.corrector.Triggs/#pypose.optim.corrector.Triggs" title="pypose.optim.corrector.Triggs"><code class="xref py py-obj docutils literal notranslate"><span class="pre">optim.corrector.Triggs</span></code></a></p></td>
+<td><p>The Triggs correction of model residual and Jacobian.</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.optim.scheduler.StopOnPlateau/#pypose.optim.scheduler.StopOnPlateau" title="pypose.optim.scheduler.StopOnPlateau"><code class="xref py py-obj docutils literal notranslate"><span class="pre">optim.scheduler.StopOnPlateau</span></code></a></p></td>
+<td><p>A scheduler to stop an optimizer when no relative loss 'decreasing' is seen for a 'patience' number of steps.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../generated/pypose.optim.functional.modjac/#pypose.optim.functional.modjac" title="pypose.optim.functional.modjac"><code class="xref py py-obj docutils literal notranslate"><span class="pre">optim.functional.modjac</span></code></a></p></td>
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diff --git a/docs/preview/pr/352/searchindex.js b/docs/preview/pr/352/searchindex.js
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78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 104, 106, 107, 108, 109, 110, 111, 113, 114], "randn": [3, 10, 14, 15, 16, 17, 18, 20, 22, 23, 24, 33, 35, 55, 59, 60, 61, 62, 64, 66, 67, 74, 85, 86, 97, 102, 104, 106, 107, 108, 114], "3": [3, 4, 6, 8, 9, 10, 11, 15, 17, 20, 21, 23, 24, 25, 33, 34, 35, 36, 37, 42, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 64, 66, 67, 69, 71, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 89, 90, 92, 93, 95, 98, 99, 100, 101, 102, 108, 109, 111, 113, 114], "size": [3, 10, 14, 15, 16, 17, 20, 22, 23, 26, 28, 30, 35, 37, 38, 42, 43, 58, 74, 83, 97, 104, 106, 107, 108], "1": [3, 4, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 30, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 69, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 104, 105, 106, 107, 108, 109, 111, 112, 113, 114], "pp": [3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 25, 26, 28, 30, 32, 33, 34, 35, 38, 39, 40, 41, 42, 43, 44, 45, 46, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 69, 71, 74, 82, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 114], "identity_so3": [3, 16, 58, 108, 115], "2": [3, 4, 6, 7, 8, 10, 11, 12, 14, 15, 16, 17, 20, 21, 22, 23, 24, 25, 26, 28, 30, 32, 33, 34, 37, 38, 39, 40, 41, 43, 44, 45, 46, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 69, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 112, 113, 114], "equival": [3, 4, 6, 7, 8, 10, 11, 13, 18, 28, 32, 35, 62, 67, 69, 71, 97], "7576": 3, "1503": 3, "0": [3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 21, 24, 25, 26, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 66, 67, 69, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 112, 113, 114], "9920": 3, "se3": [3, 4, 6, 7, 8, 10, 11, 13, 18, 21, 26, 30, 32, 39, 44, 47, 51, 56, 67, 92, 105, 115], "randn_se3": [3, 4, 6, 7, 8, 10, 11, 13, 15, 18, 26, 28, 30, 35, 67, 69, 71, 82, 87, 88, 89, 97, 102, 103, 105, 106, 111, 112, 115], "same": [3, 4, 8, 10, 18, 32, 35, 42, 50, 53, 67, 69, 71, 72, 74, 75, 76, 77, 78, 79, 80, 81, 83, 93, 94, 96, 97, 98, 99, 100, 109, 111], "5676": 3, "0452": 3, "0929": 3, "0000": [3, 12, 24, 34, 35, 49, 50, 51, 52, 53, 57, 63, 74, 75, 76, 77, 78, 79, 80, 81, 90, 114], "sim3": [3, 4, 6, 7, 8, 10, 11, 13, 32, 41, 45, 53, 67, 115], "randn_sim3": [3, 4, 6, 7, 8, 11, 13, 17, 32, 67, 97, 105, 107, 115], "respect": [3, 6, 7, 8, 11, 17, 22, 26, 28, 30, 33, 35, 55, 56, 58, 63, 66, 67, 69, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 90, 101, 102], "1551": 3, "2930": 3, "4531": 3, "6140": 3, "1263": 3, "7607": 3, "rxso3": [3, 4, 6, 7, 8, 10, 11, 13, 17, 32, 38, 43, 50, 67, 107, 112, 115], "randn_rxso3": [3, 4, 6, 7, 8, 11, 13, 14, 32, 67, 97, 104, 115], "The": [4, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 25, 26, 28, 30, 32, 33, 49, 50, 51, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 89, 90, 91, 92, 93, 94, 96, 97, 98, 99, 100, 101, 102, 104, 105, 106, 107, 108, 109, 111, 112, 114], "dot": [4, 8], "product": [4, 8, 20, 22, 23, 30, 67, 69, 71, 106, 107], "between": [4, 8, 10, 21, 25, 49, 51, 53, 57, 58, 59, 62, 79, 106, 107, 109], "adjoint": [4, 8], "matrix": [4, 6, 8, 10, 11, 20, 21, 22, 34, 50, 51, 52, 53, 56, 58, 59, 60, 61, 62, 63, 66, 69, 71, 82, 83, 84, 85, 107, 109, 115], "given": [4, 6, 8, 11, 13, 21, 33, 38, 39, 40, 41, 42, 43, 44, 45, 46, 57, 58, 63, 66, 69, 71, 74, 82, 84, 85, 86, 102], "an": [4, 6, 7, 8, 11, 17, 34, 47, 48, 50, 51, 52, 53, 56, 59, 62, 63, 67, 71, 74, 82, 84, 89, 91, 101], "second": [4, 8, 35, 63, 82], "algebra": [4, 6, 7, 8, 9, 11, 13, 18, 67], "mathcal": [4, 6, 7, 8, 9, 11, 18, 26, 28, 30, 55, 56, 64, 66, 93, 94, 95, 96, 98, 99, 100, 101], "g": [4, 6, 7, 8, 10, 11, 12, 28, 55, 58, 59, 62, 63, 64, 66, 71, 72], "output": [4, 6, 7, 8, 10, 11, 18, 20, 22, 23, 26, 28, 30, 33, 34, 35, 38, 39, 40, 41, 42, 43, 44, 45, 46, 49, 50, 51, 52, 53, 58, 59, 60, 61, 63, 65, 67, 69, 71, 74, 75, 76, 77, 78, 79, 80, 81, 82, 91, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 114], "ltype": [4, 6, 7, 8, 11, 14, 15, 16, 17, 34, 42, 67, 104, 106, 107, 108], "so3_typ": [4, 6, 7, 8, 10, 11, 34, 40, 46, 67, 95, 97, 101], "se3_typ": [4, 6, 7, 8, 10, 11, 34, 39, 44, 67, 94, 97, 99, 106], "times7": [4, 6, 7, 8, 11], "times6": [4, 6, 7, 8, 11], "sim3_typ": [4, 6, 7, 8, 10, 11, 34, 41, 45, 67, 96, 97, 100, 107], "times8": [4, 6, 7, 8, 11], "rxso3_typ": [4, 6, 7, 8, 10, 11, 34, 38, 43, 67, 93, 97, 98], "times5": [4, 6, 7, 8, 11], "let": [4, 6, 7, 8, 11, 50, 51, 52, 53, 56], "mathbf": [4, 6, 7, 8, 11, 22, 33, 34, 50, 51, 52, 53, 55, 56, 58, 59, 60, 61, 62, 63, 64, 66, 67, 69, 71, 72, 73, 74, 83, 84, 85, 86, 87, 89, 106, 107, 111, 114], "y": [4, 6, 7, 8, 11, 18, 32, 33, 50, 51, 52, 53, 55, 56, 59, 60, 61, 62, 63, 64, 66, 67, 69, 71, 72, 75, 76, 77, 78, 79, 80, 81, 101, 105, 112, 114], "_i": [4, 6, 7, 8, 11, 18, 22, 33, 50, 51, 52, 53, 56, 67, 69, 71, 72, 73, 75, 76, 77, 78, 79, 80, 81, 83, 84, 85, 86, 114], "If": [4, 6, 7, 8, 10, 11, 14, 15, 16, 17, 26, 30, 33, 35, 38, 39, 40, 41, 42, 43, 44, 45, 46, 49, 50, 51, 52, 53, 57, 58, 61, 62, 63, 64, 65, 66, 69, 71, 72, 74, 82, 85, 93, 94, 95, 96, 97, 98, 99, 100, 101, 104, 106, 107, 108, 111, 113], "s": [4, 6, 7, 8, 11, 14, 17, 18, 21, 35, 50, 53, 57, 59, 62, 63, 67, 71, 72, 74, 86, 87, 89, 93, 95, 96, 98, 100, 101, 102, 104, 105, 106, 107, 112], "ar": [4, 8, 10, 12, 14, 15, 16, 17, 18, 21, 22, 25, 26, 28, 30, 33, 55, 56, 57, 58, 59, 60, 61, 62, 63, 66, 67, 69, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 93, 94, 95, 96, 97, 98, 99, 100, 101, 104, 106, 107, 108, 111], "instanc": [4, 6, 7, 8, 11, 47, 48, 59, 62, 67], "textrm": 4, "so": [4, 60, 61, 63], "In": [4, 8, 10, 21, 25, 50, 51, 52, 53, 60, 61, 65, 83], "case": [4, 10, 18, 60, 61, 65, 74, 77, 93, 94, 96, 97, 98, 99, 100], "element": [4, 6, 11, 14, 15, 16, 17, 18, 34, 35, 50, 51, 52, 53, 58, 67, 69, 71, 91, 93, 94, 96, 97, 98, 99, 100, 104, 106, 107, 108], "rotat": [4, 6, 7, 10, 11, 14, 15, 32, 33, 34, 50, 51, 52, 53, 56, 58, 93, 95, 96, 98, 99, 100, 101, 104, 106, 107, 108, 115], "us": [4, 6, 8, 10, 12, 14, 15, 16, 17, 35, 38, 39, 40, 41, 42, 43, 44, 45, 46, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 62, 63, 64, 65, 66, 69, 71, 72, 74, 77, 78, 82, 83, 84, 85, 86, 87, 88, 89, 93, 94, 95, 96, 98, 99, 100, 101, 104, 106, 107, 108, 109, 113], "repres": [4, 6, 10, 11, 14, 15, 50, 51, 52, 53, 56, 63, 101, 107], "tangent": [4, 8, 18, 25], "vector": [4, 7, 11, 15, 17, 18, 20, 21, 22, 23, 51, 53, 56, 59, 60, 61, 62, 69, 71, 74, 104, 106, 107, 112, 114], "move": [4, 62], "from": [4, 6, 7, 8, 10, 11, 14, 15, 16, 17, 20, 34, 49, 57, 58, 59, 62, 63, 69, 71, 74, 91, 93, 94, 95, 96, 98, 99, 100, 101, 103, 104, 105, 106, 107, 108, 112, 113], "space": [4, 8, 18, 24, 25, 37, 60, 61], "right": [4, 6, 7, 8, 9, 11, 16, 18, 22, 23, 26, 30, 50, 51, 52, 53, 58, 59, 62, 63, 66, 69, 71, 72, 73, 75, 76, 78, 80, 83, 84, 85, 86, 87, 89], "side": [4, 83, 84, 85, 86], "left": [4, 6, 7, 8, 10, 11, 18, 22, 23, 26, 27, 28, 30, 31, 50, 51, 52, 53, 59, 62, 63, 66, 69, 71, 72, 73, 75, 76, 78, 80, 87, 89, 106], "t": [4, 7, 8, 11, 14, 17, 21, 22, 25, 28, 34, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 69, 71, 72, 73, 87, 89, 94, 96, 99, 100, 104, 106, 107], "translat": [4, 6, 7, 8, 10, 11, 15, 17, 50, 51, 53, 56, 67, 94, 96, 99, 100, 106, 107, 115], "part": [4, 6, 7, 8, 10, 11, 67, 93, 96, 98, 100, 103, 105, 112], "_": [4, 6, 18, 21, 50, 51, 53, 55, 59, 60, 61, 62, 63, 64, 66, 69, 71, 74, 95, 101], "i": [4, 6, 8, 11, 21, 22, 26, 28, 30, 33, 34, 50, 51, 52, 53, 55, 56, 57, 58, 64, 66, 67, 72, 73, 74, 83, 84, 85, 86, 93, 94, 96, 98, 99, 100, 101, 114], "time": [4, 9, 18, 20, 21, 22, 26, 28, 30, 34, 35, 50, 51, 52, 53, 55, 58, 59, 60, 61, 62, 63, 64, 65, 66, 82, 83, 84, 85, 86], "skew": [4, 6, 8, 11, 58, 114], "vec2skew": [4, 8, 58, 115], "begin": [4, 6, 8, 11, 14, 15, 17, 18, 21, 25, 33, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 69, 71, 72, 73, 74, 77, 84, 87, 89, 95, 101, 104, 106, 107, 108, 114], "arrai": [4, 6, 8, 11], "cc": [4, 8, 63], "end": [4, 6, 8, 11, 14, 15, 17, 18, 21, 25, 33, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 69, 71, 72, 73, 74, 77, 84, 87, 89, 95, 101, 104, 106, 107, 108, 114], "6": [4, 6, 8, 10, 11, 18, 21, 36, 37, 49, 62, 69, 71, 74, 82, 87, 88, 89, 90, 92, 95, 102, 106, 113], "se": [4, 109], "s_i": [4, 6, 7, 11, 50, 53, 56, 67], "scale": [4, 6, 7, 10, 11, 14, 17, 18, 50, 53, 56, 67, 77, 87, 89, 93, 96, 97, 98, 100, 104, 107, 115], "7": [4, 6, 8, 10, 11, 15, 36, 51, 57, 59, 82, 89, 90, 92, 102, 107], "sim": [4, 8, 55, 64, 66, 101], "4": [4, 6, 8, 10, 16, 21, 22, 25, 26, 32, 33, 36, 37, 49, 50, 51, 52, 53, 56, 59, 60, 61, 62, 67, 71, 82, 83, 90, 92, 93, 104, 105, 113], "rxso": 4, "e": [4, 6, 8, 10, 11, 12, 21, 28, 63, 71, 81, 93, 94, 96, 98, 99, 100, 101], "oper": [4, 10, 11, 14, 15, 16, 17, 18, 28, 30, 38, 39, 40, 41, 42, 43, 44, 45, 46, 67, 82, 83, 93, 94, 95, 96, 97, 98, 99, 100, 101, 104, 106, 107, 108], "linear": [4, 50, 53, 55, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 69, 71, 74, 83, 84, 85, 86], "which": [4, 6, 10, 11, 14, 15, 16, 17, 18, 20, 21, 23, 34, 35, 50, 51, 52, 53, 56, 58, 59, 60, 61, 62, 63, 66, 67, 69, 71, 72, 74, 77, 82, 85, 86, 91, 93, 94, 96, 97, 98, 99, 100, 101, 104, 106, 107, 108], "exp": [4, 8, 10, 13, 18, 21, 58, 69, 71, 74, 93, 94, 95, 96, 104, 106, 107, 108, 111, 115], "It": [4, 14, 15, 16, 17, 21, 55, 57, 58, 69, 71, 72, 73, 78, 83, 85, 86, 87, 104, 106, 107, 108], "can": [4, 6, 7, 8, 10, 11, 21, 24, 25, 32, 36, 38, 39, 40, 41, 43, 44, 45, 46, 50, 51, 52, 53, 55, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 71, 72, 74, 78, 93, 94, 95, 96, 97, 98, 99, 100, 101, 113], "easili": 4, "verifi": [4, 7], "randn_so3": [4, 6, 7, 8, 9, 10, 11, 12, 13, 16, 26, 30, 32, 42, 54, 67, 74, 93, 94, 96, 97, 98, 99, 100, 108, 110, 112, 115], "allclos": 4, "true": [4, 6, 9, 10, 12, 26, 27, 28, 30, 31, 33, 34, 35, 36, 42, 47, 48, 49, 50, 51, 52, 53, 56, 57, 58, 62, 69, 71, 74, 82, 95, 101, 111, 113], "One": [4, 7, 8, 59, 61, 63, 65], "refer": [4, 8, 11, 49, 53, 61, 62, 63, 65, 83], "eq": [4, 6, 8, 11, 18, 21, 58, 59, 62, 69, 73], "8": [4, 6, 10, 17, 26, 36, 37, 53, 56, 58, 60, 63, 74, 82, 92, 95, 113], "follow": [4, 10, 18, 21, 28, 53, 55, 57, 58, 64, 66, 67, 69, 71, 73, 97, 101], "paper": [4, 6, 8, 11, 21, 53, 56, 59, 62, 69, 71, 73], "zachari": 4, "teed": [4, 8], "et": [4, 8, 11, 21, 53, 58, 62], "al": [4, 8, 11, 21, 53, 58, 62], "backpropag": [4, 8], "3d": [4, 8, 21, 25, 53, 56, 90, 92, 102], "ieee": [4, 8], "cvf": [4, 8], "confer": [4, 8, 21, 69, 71], "comput": [4, 6, 8, 11, 16, 18, 34, 35, 50, 51, 52, 53, 56, 57, 62, 63, 64, 65, 66, 69, 71, 72, 74, 83, 85, 109], "vision": [4, 8, 21, 56, 62, 69, 71, 73], "pattern": [4, 8], "recognit": [4, 8], "cvpr": [4, 8], "2021": [4, 8], "gener": [4, 11, 21, 38, 39, 40, 41, 42, 43, 44, 45, 46, 58, 59, 60, 61, 63, 64, 85, 93, 94, 95, 96, 98, 99, 100, 101], "around": [4, 32], "one": [4, 14, 15, 16, 17, 32, 33, 34, 35, 50, 51, 52, 53, 55, 59, 60, 62, 63, 64, 65, 66, 69, 71, 74, 82, 104, 105, 106, 107, 108, 113], "anoth": [4, 53, 67], "thi": [4, 6, 8, 10, 11, 14, 15, 16, 17, 18, 20, 21, 23, 25, 26, 28, 30, 32, 33, 35, 51, 53, 55, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 71, 72, 73, 74, 83, 85, 89, 104, 106, 107, 108, 111], "more": [4, 6, 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+<table class="autosummary longtable docutils align-default">
+<tbody>
+<tr class="row-odd"><td><p><a class="reference internal" href="../generated/pypose.utils.ReduceToBason/#pypose.utils.ReduceToBason" title="pypose.utils.ReduceToBason"><code class="xref py py-obj docutils literal notranslate"><span class="pre">utils.ReduceToBason</span></code></a></p></td>
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+            <a href="https://pypose.org/get-started">Get Started</a>
+          </li>
+          <li>
+            <a href="https://pypose.org/tutorials">Tutorials</a>
+          </li>
+          <li>
+            <a href="https://pypose.org/docs/main/index.html">Docs</a>
+          </li>
+          <li>
+            <a href="https://pypose.org/about-us">About Us</a>
+          </li>
+          <li>
+            <a href="https://github.com/pypose/pypose">Github</a>
+          </li>
+        </ul>
+      </div>
+    </div>
+  </div>
+
+  <!-- End Mobile Menu -->
+
+  <script type="text/javascript" src="../_static/js/vendor/anchor.min.js"></script>
+
+  <script type="text/javascript">
+    $(document).ready(function() {
+      mobileMenu.bind();
+      mobileTOC.bind();
+      pytorchAnchors.bind();
+      sideMenus.bind();
+      scrollToAnchor.bind();
+      highlightNavigation.bind();
+      mainMenuDropdown.bind();
+      filterTags.bind();
+
+      // Add class to links that have code blocks, since we cannot create links in code blocks
+      $("article.pytorch-article a span.pre").each(function(e) {
+        $(this).closest("a").addClass("has-code");
+      });
+    })
+  </script>
+</body>
+</html>
\ No newline at end of file
diff --git a/feed.xml b/feed.xml
index 987211b46..2bcb9d83e 100644
--- a/feed.xml
+++ b/feed.xml
@@ -4,7 +4,7 @@
   <generator uri="https://jekyllrb.com/" version="3.9.2">Jekyll</generator>
   <link href="https://pypose.org/feed.xml" rel="self" type="application/atom+xml" />
   <link href="https://pypose.org/" rel="alternate" type="text/html" />
-  <updated>2024-07-10T09:16:14-04:00</updated>
+  <updated>2024-09-11T15:38:49-04:00</updated>
   <id>https://pypose.org/feed.xml</id>