Rocket_Car.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## <center>$\\mathbf{\\text{Modeling of a Variable-Mass Nonholonomic Gyrostatic Rocket Car Using Extended Kane's Equations}}$</center>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### <center>$\\text{Ge, Z.M. and Cheng, Y.H., 1982. Extended Kane’s equations for nonholonomic variable mass system.}$</center>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"Images/fig_1.png\" width=\"500\"/>\n",
    "\n",
    "##### <center>$\\text{Figure 1: Idealized model of a jet racing car that is propelled by a rocket engine at point P, the rocket engine being treated as a variable mass particle at P}$</center>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### <center>$\\text{Here}, \\{a_x: g_3, a_y: g_1, a_z: g_2\\}$</center>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"Images/Gyrostat_def.png\" width=\"750\"/>\n",
    "\n",
    "##### <center>$\\text{Leimanis, E., 2013. The general problem of the motion of coupled rigid bodies about a fixed point (Vol. 7). Springer Science & Business Media. (Page 207)}$</center>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy as sm\n",
    "import sympy.physics.mechanics as me\n",
    "from pydy.system import System\n",
    "import numpy as np\n",
    "from sympy.simplify.fu import TR2\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.integrate import odeint\n",
    "me.init_vprinting()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "N = me.ReferenceFrame('N')\n",
    "\n",
    "q1, q2, q3, q4, q5, q6, q7, q8 = me.dynamicsymbols('q1:9')\n",
    "\n",
    "A2 = N.orientnew('A_2', 'Axis', [q3, N.z])\n",
    "A1 = A2.orientnew('A_1', 'Axis', [q4, A2.z])\n",
    "\n",
    "B1 = A1.orientnew('B_1', 'Axis', [q5, A1.y])\n",
    "B2 = A1.orientnew('B_2', 'Axis', [q6, A1.y])\n",
    "B3 = A2.orientnew('B_3', 'Axis', [q7, A2.y])\n",
    "B4 = A2.orientnew('B_4', 'Axis', [q8, A2.y])\n",
    "\n",
    "t = me.dynamicsymbols._t"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "O = me.Point('O') # fixed point in the inertial reference frame\n",
    "O.set_vel(N, 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "L, l , a, b, r1, r2 = sm.symbols('L, l , a, b, r_1, r_2')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "Q = O.locatenew('Q', q1 * N.x + q2 * N.y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "P = Q.locatenew('P', L * -A2.x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "C = P.locatenew('C', l * A2.x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\dot{q}_{1}\\mathbf{\\hat{n}_x} + \\dot{q}_{2}\\mathbf{\\hat{n}_y}$"
      ],
      "text/plain": [
       "q₁̇ n_x + q₂̇ n_y"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Q.set_vel(N, Q.pos_from(O).dt(N))\n",
    "Q.vel(N)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\dot{q}_{1}\\mathbf{\\hat{n}_x} + \\dot{q}_{2}\\mathbf{\\hat{n}_y} -  L \\dot{q}_{3}\\mathbf{\\hat{a_2}_y}$"
      ],
      "text/plain": [
       "q₁̇ n_x + q₂̇ n_y + -L⋅q₃̇ a_2_y"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P.v2pt_theory(Q, N, A2)\n",
    "P.vel(N)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\dot{q}_{1}\\mathbf{\\hat{n}_x} + \\dot{q}_{2}\\mathbf{\\hat{n}_y} + (- L \\dot{q}_{3} + l \\dot{q}_{3})\\mathbf{\\hat{a_2}_y}$"
      ],
      "text/plain": [
       "q₁̇ n_x + q₂̇ n_y + (-L⋅q₃̇ + l⋅q₃̇) a_2_y"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C.v2pt_theory(P, N, A2)\n",
    "# C.vel(N)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\dot{q}_{4}\\mathbf{\\hat{a_1}_z}$"
      ],
      "text/plain": [
       "q₄̇ a_1_z"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A1.ang_vel_in(A2).express(A1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "u1, u2 = me.dynamicsymbols('u_1:3')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "z1 = sm.Eq(u1, A1.ang_vel_in(A2).dot(A1.z))\n",
    "z2 = sm.Eq(u2, Q.vel(N).dot(A1.x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left\\{ \\operatorname{sin}\\left(q_{3} + q_{4}\\right) \\dot{q}_{2} + \\operatorname{cos}\\left(q_{3} + q_{4}\\right) \\dot{q}_{1} : u_{2}, \\  \\dot{q}_{4} : u_{1}\\right\\}$"
      ],
      "text/plain": [
       "{sin(q₃ + q₄)⋅q₂̇ + cos(q₃ + q₄)⋅q₁̇: u₂, q₄̇: u₁}"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u = sm.trigsimp(sm.solve([z1, z2], A1.ang_vel_in(A2).dot(A1.z), Q.vel(N).dot(A1.x)))\n",
    "u"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### $\\text{Nonholonomic Constraints:}\\ B_1$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACsAAAAXCAYAAACS5bYWAAAACXBIWXMAAA7EAAAOxAGVKw4bAAACjUlEQVRIDc2W4VFUMRDHHwwFIFQAdHCMFSAdgHYgHcj46e6bAx2gFah0IFbgSAdiBeh1cP5+mWwm76Enc5Nj3s7sy2az2f1ns0nexmKx6FrTbDb7jM872vOWvjdaggXcNuC+wvuw8g26Y9omtNnEC04y0J+Id/Ae/AxW/8O2CZnZVjydTidDX3/TDW0e229aBk2yt8TJ1pKxfw6xtW8YfFUZpPqkf87YvNI3FXuZJZAHwxNswHv4Gj5Bf0mbCPkFwhd4jhx1WXTo99CvBXA5YBnEd4JdIJshAV7lvpkLckE9wvYGxS2s3dveYMNOAkswg3g3CtLTHKTsfVkyhfwendk/CqPcxrzJQN+sGzV7gUcBfxp4dnsthR4B+BLehh0X3EFutdvxU5O29N0R43T0V7p7A+xLfHiB1xmMANZnIWwi6AlKt99SMdsCeQ33CHsPo+D07eIsl5VoC2eCkodOXEDHuIASZVvrWnuzW55T5GQz/KC39tNO0LrAlSkyq4NvAy+n9NMCCGgQZbMqUMmM1uTYWsnMegUFkBSMvttp8Mjqc3TX8K8KjbWaDhV65QBrLcvzynapiK3x4tw4L11/2a//GtJpumd1TucDbHZ34Y+wk3SgznqOLAtKvfXnYpwraEvCG0W9dMYcazlRjvGbzi3yYVaXBp0LtsR64+idk8D3HoUycw3C/8AaEhsPs4s9RjZBZvyQ9oy22/QzIooD685J9kMeF1gyaKlZWhNkS8qSiMemq28DxtoTwbxJ6id4H5212dE+qF3UZtNx5/nYFHqymi0RHyGwiPhR8vosNIqaBZzZLrUJOm+cdwVlFkaR2QzUZ9lt9+QLvpdV8Y4is+DwNfRw2fqgPACKvvsDkx54wjTjWjAAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle a\\mathbf{\\hat{a_1}_y}$"
      ],
      "text/plain": [
       "a a_1_y"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B1_center = Q.locatenew('B_1_center', a * A1.y)\n",
    "B1_center.pos_from(Q)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle (- a \\left(\\dot{q}_{3} + \\dot{q}_{4}\\right) + \\operatorname{sin}\\left(q_{3} + q_{4}\\right) \\dot{q}_{2} + \\operatorname{cos}\\left(q_{3} + q_{4}\\right) \\dot{q}_{1})\\mathbf{\\hat{a_1}_x} + (- \\operatorname{sin}\\left(q_{3} + q_{4}\\right) \\dot{q}_{1} + \\operatorname{cos}\\left(q_{3} + q_{4}\\right) \\dot{q}_{2})\\mathbf{\\hat{a_1}_y}$"
      ],
      "text/plain": [
       "(-a⋅(q₃̇ + q₄̇) + sin(q₃ + q₄)⋅q₂̇ + cos(q₃ + q₄)⋅q₁̇) a_1_x + (-sin(q₃ + q₄)⋅\n",
       "q₁̇ + cos(q₃ + q₄)⋅q₂̇) a_1_y"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B1_center.v2pt_theory(Q, N, A1)\n",
    "B1_center.vel(N).express(A1).simplify()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle -  r_{1}\\mathbf{\\hat{a_1}_z}$"
      ],
      "text/plain": [
       "-r₁ a_1_z"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B1_ground = B1_center.locatenew('B_1_ground', r1 * -A1.z)\n",
    "B1_ground.pos_from(B1_center)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\dot{q}_{1}\\mathbf{\\hat{n}_x} + \\dot{q}_{2}\\mathbf{\\hat{n}_y} + (- a \\left(\\dot{q}_{3} + \\dot{q}_{4}\\right) - r_{1} \\dot{q}_{5})\\mathbf{\\hat{a_1}_x}$"
      ],
      "text/plain": [
       "q₁̇ n_x + q₂̇ n_y + (-a⋅(q₃̇ + q₄̇) - r₁⋅q₅̇) a_1_x"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B1_ground.v2pt_theory(B1_center, N, B1)\n",
    "B1_ground.vel(N).simplify()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ - a \\left(\\dot{q}_{3} + \\dot{q}_{4}\\right) - r_{1} \\dot{q}_{5} + \\left(- \\operatorname{sin}\\left(q_{3}\\right) \\operatorname{sin}\\left(q_{4}\\right) + \\operatorname{cos}\\left(q_{3}\\right) \\operatorname{cos}\\left(q_{4}\\right)\\right) \\dot{q}_{1} + \\left(\\operatorname{sin}\\left(q_{3}\\right) \\operatorname{cos}\\left(q_{4}\\right) + \\operatorname{sin}\\left(q_{4}\\right) \\operatorname{cos}\\left(q_{3}\\right)\\right) \\dot{q}_{2}, \\  \\left(- \\operatorname{sin}\\left(q_{3}\\right) \\operatorname{sin}\\left(q_{4}\\right) + \\operatorname{cos}\\left(q_{3}\\right) \\operatorname{cos}\\left(q_{4}\\right)\\right) \\dot{q}_{2} + \\left(- \\operatorname{sin}\\left(q_{3}\\right) \\operatorname{cos}\\left(q_{4}\\right) - \\operatorname{sin}\\left(q_{4}\\right) \\operatorname{cos}\\left(q_{3}\\right)\\right) \\dot{q}_{1}, \\  0\\right]$"
      ],
      "text/plain": [
       "[-a⋅(q₃̇ + q₄̇) - r₁⋅q₅̇ + (-sin(q₃)⋅sin(q₄) + cos(q₃)⋅cos(q₄))⋅q₁̇ + (sin(q₃)\n",
       "⋅cos(q₄) + sin(q₄)⋅cos(q₃))⋅q₂̇, (-sin(q₃)⋅sin(q₄) + cos(q₃)⋅cos(q₄))⋅q₂̇ + (-\n",
       "sin(q₃)⋅cos(q₄) - sin(q₄)⋅cos(q₃))⋅q₁̇, 0]"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B1_cons = [me.dot(B1_ground.vel(N).simplify(), uv) for uv in A1]\n",
    "sm.trigsimp(B1_cons)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - a \\left(u_{1} + \\dot{q}_{3}\\right) - r_{1} \\dot{q}_{5} + u_{2} = 0$"
      ],
      "text/plain": [
       "-a⋅(u₁ + q₃̇) - r₁⋅q₅̇ + u₂ = 0"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq1 = sm.Eq(B1_cons[0].simplify().subs(u), 0)\n",
    "eq1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - \\operatorname{sin}\\left(q_{3} + q_{4}\\right) \\dot{q}_{1} + \\operatorname{cos}\\left(q_{3} + q_{4}\\right) \\dot{q}_{2} = 0$"
      ],
      "text/plain": [
       "-sin(q₃ + q₄)⋅q₁̇ + cos(q₃ + q₄)⋅q₂̇ = 0"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq2 = sm.Eq(B1_cons[1].simplify().subs(u), 0)\n",
    "eq2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### $\\text{Nonholonomic Constraints:}\\ B_2$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADoAAAAXCAYAAABaiVzAAAAACXBIWXMAAA7EAAAOxAGVKw4bAAACtklEQVRYCd2X7U0bQRCGD+QC8lEBpAOjVBDoACcdQAdB+WX/i6CDhApI6CCkgih0AKkgCR04z7PaXe2dIzC+PclmpPHOzn7Nu/Ox5635fN6sE81ms6/Yc0t7UtOurXUBCrBnAPsO78LKV+gOaKvQdpVdem4SQf5im1t4B37uluhvbKuQHl0Hnk6n464d/9N15yzbX5vQreK1ezYZ3TM2yBDh+J6N3xWbh3ykf8LYXaGvKlbzKEZaRKyUGvsbvoQP0Z/RBkLeR/gG3yGnPMw69DvoBwFbpRhFAD8x9BRZzwjuU+zrsUReRouYe4XiGnbeh9ZgxU5voBiqgb59ArRqJlL2PcweQv6MTq+/SZNim9aNO/pq3VE01PervPmHDpiwTi9Ip7Brv9gpyJA0fFvEujPPhB0X2KvYOu+FPyU5l76R4DkN/ZXeVoF643tusiK9ZZ2Pu/sEQk7GmY+Z0CeDD1Easoa3XhbEEdwi5lu4BObeXky6XMTH0ehx09uzIyBBdQ0QfMO4YALFueax8/Vq/sRDDnO6P+jN9RABtF7OytQLaHHqj0JWnMABPMZqoLLeFKSkJ0tybFDqBRQQPhMJRDCUviGo4cmbr9Fdwn8KJOZmKEDolRNQc1c2VJci5npeqhOuC09U3NfaI016v6MaxkbnsF59CV/AHujh6szf5F0Bqd9XD7tWwIaxlVu9dMwaczdQPOMvnWvkhXqCzssyLVrj6F0TgPcGykaD00NANYA5Fj4v6gDZy9XTe7THtM22P0+EUnEzYiT7SX46QPGc6WE6jJFNA8M4fYg0vYoRmw1KGGrFLj8Ld9GZiw3tQq6i1ouOu84PkUwbkaPZ2iUELiD9afCJy7TxOQowvZxzEWRW9o8ZYRQ23qMRpJ+KhqoVVuAtb4p14z0KBr+yLES2fmwsgETf/AP+2K04RWDhxgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle -  a\\mathbf{\\hat{a_1}_y}$"
      ],
      "text/plain": [
       "-a a_1_y"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B2_center = Q.locatenew('B_1_center', a * -A1.y)\n",
    "B2_center.pos_from(Q)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle (a \\left(\\dot{q}_{3} + \\dot{q}_{4}\\right) + \\operatorname{sin}\\left(q_{3} + q_{4}\\right) \\dot{q}_{2} + \\operatorname{cos}\\left(q_{3} + q_{4}\\right) \\dot{q}_{1})\\mathbf{\\hat{a_1}_x} + (- \\operatorname{sin}\\left(q_{3} + q_{4}\\right) \\dot{q}_{1} + \\operatorname{cos}\\left(q_{3} + q_{4}\\right) \\dot{q}_{2})\\mathbf{\\hat{a_1}_y}$"
      ],
      "text/plain": [
       "(a⋅(q₃̇ + q₄̇) + sin(q₃ + q₄)⋅q₂̇ + cos(q₃ + q₄)⋅q₁̇) a_1_x + (-sin(q₃ + q₄)⋅q\n",
       "₁̇ + cos(q₃ + q₄)⋅q₂̇) a_1_y"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B2_center.v2pt_theory(Q, N, A1)\n",
    "B2_center.vel(N).express(A1).simplify()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle -  r_{1}\\mathbf{\\hat{a_1}_z}$"
      ],
      "text/plain": [
       "-r₁ a_1_z"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B2_ground = B2_center.locatenew('B_2_ground', r1 * -A1.z)\n",
    "B2_ground.pos_from(B2_center)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\dot{q}_{1}\\mathbf{\\hat{n}_x} + \\dot{q}_{2}\\mathbf{\\hat{n}_y} + (a \\left(\\dot{q}_{3} + \\dot{q}_{4}\\right) - r_{1} \\dot{q}_{6})\\mathbf{\\hat{a_1}_x}$"
      ],
      "text/plain": [
       "q₁̇ n_x + q₂̇ n_y + (a⋅(q₃̇ + q₄̇) - r₁⋅q₆̇) a_1_x"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B2_ground.v2pt_theory(B2_center, N, B2)\n",
    "B2_ground.vel(N).simplify()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ a \\left(\\dot{q}_{3} + \\dot{q}_{4}\\right) - r_{1} \\dot{q}_{6} + \\left(- \\operatorname{sin}\\left(q_{3}\\right) \\operatorname{sin}\\left(q_{4}\\right) + \\operatorname{cos}\\left(q_{3}\\right) \\operatorname{cos}\\left(q_{4}\\right)\\right) \\dot{q}_{1} + \\left(\\operatorname{sin}\\left(q_{3}\\right) \\operatorname{cos}\\left(q_{4}\\right) + \\operatorname{sin}\\left(q_{4}\\right) \\operatorname{cos}\\left(q_{3}\\right)\\right) \\dot{q}_{2}, \\  \\left(- \\operatorname{sin}\\left(q_{3}\\right) \\operatorname{sin}\\left(q_{4}\\right) + \\operatorname{cos}\\left(q_{3}\\right) \\operatorname{cos}\\left(q_{4}\\right)\\right) \\dot{q}_{2} + \\left(- \\operatorname{sin}\\left(q_{3}\\right) \\operatorname{cos}\\left(q_{4}\\right) - \\operatorname{sin}\\left(q_{4}\\right) \\operatorname{cos}\\left(q_{3}\\right)\\right) \\dot{q}_{1}, \\  0\\right]$"
      ],
      "text/plain": [
       "[a⋅(q₃̇ + q₄̇) - r₁⋅q₆̇ + (-sin(q₃)⋅sin(q₄) + cos(q₃)⋅cos(q₄))⋅q₁̇ + (sin(q₃)⋅\n",
       "cos(q₄) + sin(q₄)⋅cos(q₃))⋅q₂̇, (-sin(q₃)⋅sin(q₄) + cos(q₃)⋅cos(q₄))⋅q₂̇ + (-s\n",
       "in(q₃)⋅cos(q₄) - sin(q₄)⋅cos(q₃))⋅q₁̇, 0]"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B2_cons = [me.dot(B2_ground.vel(N).simplify(), uv) for uv in A1]\n",
    "sm.trigsimp(B2_cons)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle a \\left(u_{1} + \\dot{q}_{3}\\right) - r_{1} \\dot{q}_{6} + u_{2} = 0$"
      ],
      "text/plain": [
       "a⋅(u₁ + q₃̇) - r₁⋅q₆̇ + u₂ = 0"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq3 = sm.Eq(B2_cons[0].simplify().subs(u), 0)\n",
    "eq3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - \\operatorname{sin}\\left(q_{3} + q_{4}\\right) \\dot{q}_{1} + \\operatorname{cos}\\left(q_{3} + q_{4}\\right) \\dot{q}_{2} = 0$"
      ],
      "text/plain": [
       "-sin(q₃ + q₄)⋅q₁̇ + cos(q₃ + q₄)⋅q₂̇ = 0"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq4 = sm.Eq(B2_cons[1].simplify().subs(u), 0)\n",
    "eq4"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### $\\text{Nonholonomic Constraints:}\\ B_3$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle b\\mathbf{\\hat{a_2}_y}$"
      ],
      "text/plain": [
       "b a_2_y"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B3_center = P.locatenew('B_3_center', b * A2.y)\n",
    "B3_center.pos_from(P)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle (- b \\dot{q}_{3} + \\operatorname{sin}\\left(q_{3}\\right) \\dot{q}_{2} + \\operatorname{cos}\\left(q_{3}\\right) \\dot{q}_{1})\\mathbf{\\hat{a_2}_x} + (- L \\dot{q}_{3} - \\operatorname{sin}\\left(q_{3}\\right) \\dot{q}_{1} + \\operatorname{cos}\\left(q_{3}\\right) \\dot{q}_{2})\\mathbf{\\hat{a_2}_y}$"
      ],
      "text/plain": [
       "(-b⋅q₃̇ + sin(q₃)⋅q₂̇ + cos(q₃)⋅q₁̇) a_2_x + (-L⋅q₃̇ - sin(q₃)⋅q₁̇ + cos(q₃)⋅q\n",
       "₂̇) a_2_y"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B3_center.v2pt_theory(P, N, A2)\n",
    "B3_center.vel(N).express(A2).simplify()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle -  r_{2}\\mathbf{\\hat{a_2}_z}$"
      ],
      "text/plain": [
       "-r₂ a_2_z"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B3_ground = B3_center.locatenew('B_3_ground', r2 * -A2.z)\n",
    "B3_ground.pos_from(B3_center)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\dot{q}_{1}\\mathbf{\\hat{n}_x} + \\dot{q}_{2}\\mathbf{\\hat{n}_y} + (- b \\dot{q}_{3} - r_{2} \\dot{q}_{7})\\mathbf{\\hat{a_2}_x} -  L \\dot{q}_{3}\\mathbf{\\hat{a_2}_y}$"
      ],
      "text/plain": [
       "q₁̇ n_x + q₂̇ n_y + (-b⋅q₃̇ - r₂⋅q₇̇) a_2_x + -L⋅q₃̇ a_2_y"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B3_ground.v2pt_theory(B3_center, N, B3)\n",
    "B3_ground.vel(N).simplify()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ - b \\dot{q}_{3} - r_{2} \\dot{q}_{7} + \\operatorname{sin}\\left(q_{3}\\right) \\dot{q}_{2} + \\operatorname{cos}\\left(q_{3}\\right) \\dot{q}_{1}, \\  - L \\dot{q}_{3} - \\operatorname{sin}\\left(q_{3}\\right) \\dot{q}_{1} + \\operatorname{cos}\\left(q_{3}\\right) \\dot{q}_{2}, \\  0\\right]$"
      ],
      "text/plain": [
       "[-b⋅q₃̇ - r₂⋅q₇̇ + sin(q₃)⋅q₂̇ + cos(q₃)⋅q₁̇, -L⋅q₃̇ - sin(q₃)⋅q₁̇ + cos(q₃)⋅q₂̇, 0]"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B3_cons = [me.dot(B3_ground.vel(N).simplify(), uv) for uv in A2]\n",
    "sm.trigsimp(B3_cons)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - b \\dot{q}_{3} - r_{2} \\dot{q}_{7} + \\operatorname{sin}\\left(q_{3}\\right) \\dot{q}_{2} + \\operatorname{cos}\\left(q_{3}\\right) \\dot{q}_{1} = 0$"
      ],
      "text/plain": [
       "-b⋅q₃̇ - r₂⋅q₇̇ + sin(q₃)⋅q₂̇ + cos(q₃)⋅q₁̇ = 0"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq5 = sm.Eq(B3_cons[0].simplify().subs(u), 0)\n",
    "eq5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - L \\dot{q}_{3} - \\operatorname{sin}\\left(q_{3}\\right) \\dot{q}_{1} + \\operatorname{cos}\\left(q_{3}\\right) \\dot{q}_{2} = 0$"
      ],
      "text/plain": [
       "-L⋅q₃̇ - sin(q₃)⋅q₁̇ + cos(q₃)⋅q₂̇ = 0"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq6 = sm.Eq(B3_cons[1].simplify().subs(u), 0)\n",
    "eq6"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### $\\text{Nonholonomic Constraints:}\\ B_4$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle -  b\\mathbf{\\hat{a_2}_y}$"
      ],
      "text/plain": [
       "-b a_2_y"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B4_center = P.locatenew('B_4_center', b * -A2.y)\n",
    "B4_center.pos_from(P)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\dot{q}_{1}\\mathbf{\\hat{n}_x} + \\dot{q}_{2}\\mathbf{\\hat{n}_y} + b \\dot{q}_{3}\\mathbf{\\hat{a_2}_x} -  L \\dot{q}_{3}\\mathbf{\\hat{a_2}_y}$"
      ],
      "text/plain": [
       "q₁̇ n_x + q₂̇ n_y + b⋅q₃̇ a_2_x + -L⋅q₃̇ a_2_y"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B4_center.v2pt_theory(P, N, A2)\n",
    "# B4_center.vel(N).express(A2).simplify() # Invalid Json"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle -  r_{2}\\mathbf{\\hat{a_2}_z}$"
      ],
      "text/plain": [
       "-r₂ a_2_z"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B4_ground = B4_center.locatenew('B_4_ground', r2 * -A2.z)\n",
    "B4_ground.pos_from(B4_center)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\dot{q}_{1}\\mathbf{\\hat{n}_x} + \\dot{q}_{2}\\mathbf{\\hat{n}_y} + (b \\dot{q}_{3} - r_{2} \\dot{q}_{8})\\mathbf{\\hat{a_2}_x} -  L \\dot{q}_{3}\\mathbf{\\hat{a_2}_y}$"
      ],
      "text/plain": [
       "q₁̇ n_x + q₂̇ n_y + (b⋅q₃̇ - r₂⋅q₈̇) a_2_x + -L⋅q₃̇ a_2_y"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B4_ground.v2pt_theory(B4_center, N, B4)\n",
    "B4_ground.vel(N).simplify()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ b \\dot{q}_{3} - r_{2} \\dot{q}_{8} + \\operatorname{sin}\\left(q_{3}\\right) \\dot{q}_{2} + \\operatorname{cos}\\left(q_{3}\\right) \\dot{q}_{1}, \\  - L \\dot{q}_{3} - \\operatorname{sin}\\left(q_{3}\\right) \\dot{q}_{1} + \\operatorname{cos}\\left(q_{3}\\right) \\dot{q}_{2}, \\  0\\right]$"
      ],
      "text/plain": [
       "[b⋅q₃̇ - r₂⋅q₈̇ + sin(q₃)⋅q₂̇ + cos(q₃)⋅q₁̇, -L⋅q₃̇ - sin(q₃)⋅q₁̇ + cos(q₃)⋅q₂̇, 0]"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B4_cons = [me.dot(B4_ground.vel(N).simplify(), uv) for uv in A2]\n",
    "sm.trigsimp(B4_cons)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAWAAAAAVCAYAAAB4+TSKAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAKqElEQVR4Ae2c63EUORDHF5cDMBDBmQwMjgCTAY8IgAyg+ATfXJABXAaGDOAi4CADuAgwmwH3/8lqWaOdh2ZXmvWut6vGerW61Q+1erQDsz9//sx2z04HpX3g9evXh6VpQq8W3Rpr3QaaNfRdg+am6npvtoOdBgpr4M2bNy9E8qgwWSN36Olbe1dW0kBFO+5s6G12g5NDij5Q+6Oee3rO1b7jx3sL4f0QwheVz3sRt2xwk+XW2g9ljm96Xqr+obRpRPOhaB6rfFmattETbQL8T5WfrC+31Jxr6bO5+jE86amqHVexIWvcJDtqrW+9Xn+pJLa+Vd9P+lwGrMZczwO16fzCwBAIn6DNZiZoXxvYArmxGba7W9poXjevVFYLvqxZ9N+pgA9yZIPHv3Y+m60gj+j1VNWOy9qQJW6SHbVWkp2vKkl48Fv2xmfV8cOLAEzFA6+Nn63RV4rAXOM3VRbfyH181z02ldzic6LnWWl5RZMDFrvVeGvhpH9fes0d9OBjmUUHSrNbMk/is+JTxXZNaaq2prLjaBsi9VR2XFXDWif790BleFPza6ft9smeMdHAia9nZcDgemJG4tqUE8lNdjcqw8s1QMX1Pxbt4tcabXJ5PvAbpaOKssfLrGa7mEnF+iR2XNaGyD2RHVdV8SMR+N5C5Kv6OKQPQgBWh7uC2BDBWmTada1TA/IbDnB3rzXhOuD3eEJ+W89qDXbcZhuyJ85bnGbu+072o0GQv8sA/MABcFnMXUVIn+lU+1AF9xhG5FR1Um1+jGuL9hqaHrQWrlOe6DlS/UGy7h9qj87UEhoIdeVk1xqRG1tiHzIx4In67/r182MrNsRenNAzlcyh/5aeMz28gvIjDHCshzss7q/6AFq9b0+iUdp34EfiMGjLCrz7dLHSmNaK3bCB7THoYYOwFyMcflQE2K8f1R9soHqnL7gZ7X+mtuM227Bdw5e9t1wA9sbEWMBTtflRDif4rfKOHpfZqGRjs1HZzNZH+6HaN1ReJSDF5+KbQ4T7FmR6rgfH5hnctLEwmnflZdca3cZVSVAKoLa771WJzbCdbVqHozYHJ3b+rZIgjD1dwFWJX3xT+UmPs7nabcCPse5eq21Qc2voDzmg2wuVePfyXHZQa+WQ4ocb9qELuLZ+lXz5QZKETdh3JBbBJqrj6yQc7/T0+oLmdsHUdly7DaUr/HbQjxKFYQeXwCT91mQfAfEhetFzmRUf7PseY35fRN0ESj0MkwmZQTH6qfqD0UEQuMxX/TgP+NDAAciewJ80MxY/eDs5VLImvju0T+sYc8HX471SGwAPcAfQRfXir8frlR1M4blLd1XhDb/36kt1pe5qwOa5x3r1mPwwC5mT59y1pn81zsEVHEt1HI1p+EjfoYVez0FMQfMZ69WfcJbxHfhBuxMyeUNj0A86mZQdQE/pJ3Ym49yzAqftQOTNlMOSrJIAkOMLQmsAvKa0YxEbIoG3NVdSJFrZHwcIt8YP0ixpCG7vewwyJjaaGXimuhn9tscha2Rjp6+i8cYEB+ex7InrjH/03NTTCZ4XeMazEzcaeKR536N2XMX5znwHGzsoWHNCXf18jxfaqnMSkn1YsFbVwaDsmousYVOojSxslEY26qhFfzxPdJiCO0E1HtYXIbSevsLlWgGH5s2FTcgXLayJjZkLBOE2GLIN6523TVTfoP48zljf4SCxg7ODdR5v6SjoWfUuP2jw8HhFbAdh0UPHR3oae0z9HKDuEFWdcWT+qqcBGrPDkisn3v6W8YWp7VjEhl4vZoshX23orVID3XcBOgZ+7V+U7lvedOORTQFmaE6WcL/EgITGERDWPl07pT8CgndXthXQRIeNm31ihYkdFdFzPFWaQSwYpzOeCSe+NyNQ0MdrXBzcc2R3h5jmG2/eIEzRKd/QFk7Y+KFTFfXzJkHm3tiMMU5HHT2SzSE78nDIfNDTykfjU0CO/pbxnb5gYXLl8M71A6Ppyi6dqn9Z29lhwgf7XWA47JkuOPIDpX0hR5dj7VjEhtI5+5UDCN2vHbQOu0FoOwys76cFYAzGqR+DvYaSVTGBxwKt4bkAp3EXmFXGQQsclEEQWBcgA1nVvGMBBKX04Gmg5squSeiGOziua8g+uI5IddqgXbohnmzOc/hDWyU2Y9NwFcKT2ge0UsDBA78G+DXU8h3onjcYRo1c3poy6AcR2ZpVS1bSN7CYp+Es6DpCwueX9YWp7VjKhpH446rSFfvUkrXcya1voclk4qIdmPEQhw7wxQIwDTPszDsuG5e7FCK5GTvgMEFAgFvY1MJ/oX6+QGDT990bCqUqoFT36tbGpWVtbEScd0Em+hMaDdk1h8ALChkn8rtATMeEwEGKwV3WrHXMVSf7JTu/p6dNLnUXAQJhm7MZ8V79GZLXXa7v4MgpXSMVlylOarvUR/v8IKZbtI699GAjbLUAGuPHUa6UsCs2bfi2+i2IEFCW9YWp7VjEhgvKGtEhvWHvGsAVZFsCypsJAXy+57liyHjzMPFMCPZjFQZvRHONkeFh8Ma1hNozjREA7uvhV1qC0eQgvsjDk2btrWsRPg5Lxt64ClF/lux+/rGfjz5fqG/SDFg8gVfie3BRDX9pp3ZKcUBmMywLBA7kb0Cu/myS8Mf4DrbqtO9Y3qxBc1r9wNY3QcnhwNUT+yuA2mxkdAywt/jHEqw1BnA4+A0v1xdiGlPbsbgNY2HWWZcdiJ+8kRJXHKjOviO5fUrHPn8ENCxz4/WHzLVxuqoPx/hb/RZQLato3QDC4zTnVdj9Kqu6OYW6JgGCL9lsGngWmAsHXJyXT7TmCwh5sqMbC978QMjhxP0yupxKdtaO0dmcFmBvq41t0QUbFjldhqU2tnMb3vczjt3ox3bgMQ7wNsR/smPti97Lv8zpOnBq+Q46Rp4+yOYt2Yb8oI9PkTGtATv9JWLmT+aP+JHbcyrJnvA1rrpsnLXTNn/v9YWexU5tx6I27JFrXUPYiYMQ+wDHevjazMUE97+hue6Rf0TA3XGqdN//qmTD/xcTVx9M+c6PzZu+5ql7/eDXSIBygUWlBSE7YBYWKZxUduRkg/BaGEBtgkPjA/owOFDRXE7NZX6EG6Bcb1hrxtZ9X6c45sJL9Tfad0QDnXPn3ndfuiBsytsQPL1RfmBz01K0Ns52sQxa/yR29DovZkNk8LrHjqP8IpZ/yvreCswINiGzk8CcuOd64sDF6Ub/mZ4rB1qzC5xa2KnqfPlA8OU+CDn6IJUdmW9pPoEkBuhbRhL359TRG88mAQdOzn1aqj/kHOs7ZOhD2W+b7hq8QZDdlvWDNvr0baLtYlmmsmMxG8aL36T6/gqLJbimWS1ZJOm2fUZDut31Wr8C62JT+eb3QA9lAK1/KIh0yc7JS/YA8OofvyK6ztw/orNs4M5lURxPa+YHPzIaMvf4IE55dekvy3egL4LwGLJTypd2G+9l/aCN/kzr2jjbxYJo/dXtKB5Fbejp4Q/YF9/gOoz/csD9IB3Ld5Xqo68gJBCnI0KSLbLJxn7orymbCddZ9lyLSUccaNx9k2k2oJT+RMfdXask08yCUryzmG0BkvRV1Y47G144yegAvAW+tROhsga0uchuTlSmb0grcxZN7o/5Nr0vw16Zz45AuJopbsedDS+963+/YwagtGRwTgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle b \\dot{q}_{3} - r_{2} \\dot{q}_{8} + \\operatorname{sin}\\left(q_{3}\\right) \\dot{q}_{2} + \\operatorname{cos}\\left(q_{3}\\right) \\dot{q}_{1} = 0$"
      ],
      "text/plain": [
       "b⋅q₃̇ - r₂⋅q₈̇ + sin(q₃)⋅q₂̇ + cos(q₃)⋅q₁̇ = 0"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq7 = sm.Eq(B4_cons[0].simplify().subs(u), 0)\n",
    "eq7"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - L \\dot{q}_{3} - \\operatorname{sin}\\left(q_{3}\\right) \\dot{q}_{1} + \\operatorname{cos}\\left(q_{3}\\right) \\dot{q}_{2} = 0$"
      ],
      "text/plain": [
       "-L⋅q₃̇ - sin(q₃)⋅q₁̇ + cos(q₃)⋅q₂̇ = 0"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq8 = sm.Eq(B4_cons[1].simplify().subs(u), 0)\n",
    "eq8"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### $\\text{LHS} \\Longleftrightarrow \\text{RHS}\\ \\text{in}\\ z_1, z_2 \\rightarrow \\text{Equations}\\ 9, 10$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\dot{q}_{4} = u_{1}$"
      ],
      "text/plain": [
       "q₄̇ = u₁"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq9 = sm.Eq(A1.ang_vel_in(A2).dot(A1.z), u1)\n",
    "eq9"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left(- \\operatorname{sin}\\left(q_{3}\\right) \\operatorname{sin}\\left(q_{4}\\right) + \\operatorname{cos}\\left(q_{3}\\right) \\operatorname{cos}\\left(q_{4}\\right)\\right) \\dot{q}_{1} + \\left(\\operatorname{sin}\\left(q_{3}\\right) \\operatorname{cos}\\left(q_{4}\\right) + \\operatorname{sin}\\left(q_{4}\\right) \\operatorname{cos}\\left(q_{3}\\right)\\right) \\dot{q}_{2} = u_{2}$"
      ],
      "text/plain": [
       "(-sin(q₃)⋅sin(q₄) + cos(q₃)⋅cos(q₄))⋅q₁̇ + (sin(q₃)⋅cos(q₄) + sin(q₄)⋅cos(q₃))\n",
       "⋅q₂̇ = u₂"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq10 = sm.Eq(Q.vel(N).dot(A1.x), u2)\n",
    "eq10"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### $\\text{Solution of the System of Linear (in}\\ \\dot{q}_1,...\\text{) Equations}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [],
   "source": [
    "solution = sm.solve([eq1, eq2, eq3, eq4, eq5, eq6, eq7, eq8, eq9, eq10], q1.diff(), q2.diff(), q3.diff(),  q4.diff(), q5.diff(), q6.diff(), q7.diff(), q8.diff())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left\\{ \\dot{q}_{1} : u_{2} \\operatorname{cos}\\left(q_{3} + q_{4}\\right), \\  \\dot{q}_{2} : u_{2} \\operatorname{sin}\\left(q_{3} + q_{4}\\right), \\  \\dot{q}_{3} : \\frac{\\left(L \\operatorname{cos}\\left(q_{4}\\right) + b \\operatorname{sin}\\left(q_{4}\\right)\\right) u_{2} \\operatorname{tan}\\left(q_{4}\\right)}{L \\left(L + b \\operatorname{tan}\\left(q_{4}\\right)\\right)}, \\  \\dot{q}_{4} : u_{1}, \\  \\dot{q}_{5} : \\frac{- L \\left(L + b \\operatorname{tan}\\left(q_{4}\\right)\\right) \\left(a u_{1} + u_{2}\\right) + 2 L \\left(L + b \\operatorname{tan}\\left(q_{4}\\right)\\right) u_{2} - a \\left(L \\operatorname{cos}\\left(q_{4}\\right) + b \\operatorname{sin}\\left(q_{4}\\right)\\right) u_{2} \\operatorname{tan}\\left(q_{4}\\right)}{L r_{1} \\left(L + b \\operatorname{tan}\\left(q_{4}\\right)\\right)}, \\  \\dot{q}_{6} : \\frac{L \\left(L + b \\operatorname{tan}\\left(q_{4}\\right)\\right) \\left(a u_{1} + u_{2}\\right) + a \\left(L \\operatorname{cos}\\left(q_{4}\\right) + b \\operatorname{sin}\\left(q_{4}\\right)\\right) u_{2} \\operatorname{tan}\\left(q_{4}\\right)}{L r_{1} \\left(L + b \\operatorname{tan}\\left(q_{4}\\right)\\right)}, \\  \\dot{q}_{7} : \\frac{\\left(L - b \\operatorname{tan}\\left(q_{4}\\right)\\right) \\left(L \\operatorname{cos}\\left(q_{4}\\right) + b \\operatorname{sin}\\left(q_{4}\\right)\\right) u_{2}}{L r_{2} \\left(L + b \\operatorname{tan}\\left(q_{4}\\right)\\right)}, \\  \\dot{q}_{8} : \\frac{\\left(L \\operatorname{cos}\\left(q_{4}\\right) + b \\operatorname{sin}\\left(q_{4}\\right)\\right) u_{2}}{L r_{2}}\\right\\}$"
      ],
      "text/plain": [
       "⎧                                              (L⋅cos(q₄) + b⋅sin(q₄))⋅u₂⋅tan(\n",
       "⎨q₁̇: u₂⋅cos(q₃ + q₄), q₂̇: u₂⋅sin(q₃ + q₄), q₃̇: ────────────────────────────\n",
       "⎩                                                      L⋅(L + b⋅tan(q₄))      \n",
       "\n",
       "q₄)              -L⋅(L + b⋅tan(q₄))⋅(a⋅u₁ + u₂) + 2⋅L⋅(L + b⋅tan(q₄))⋅u₂ - a⋅(\n",
       "──────, q₄̇: u₁, q₅̇: ────────────────────────────────────────────────────────\n",
       "                                                      L⋅r₁⋅(L + b⋅tan(q₄))    \n",
       "\n",
       "L⋅cos(q₄) + b⋅sin(q₄))⋅u₂⋅tan(q₄)      L⋅(L + b⋅tan(q₄))⋅(a⋅u₁ + u₂) + a⋅(L⋅co\n",
       "──────────────────────────────────────, q₆̇: ─────────────────────────────────\n",
       "                                                               L⋅r₁⋅(L + b⋅tan\n",
       "\n",
       "s(q₄) + b⋅sin(q₄))⋅u₂⋅tan(q₄)      (L - b⋅tan(q₄))⋅(L⋅cos(q₄) + b⋅sin(q₄))⋅u₂ \n",
       "───────────────────────────────────, q₇̇: ────────────────────────────────────\n",
       "(q₄))                                         L⋅r₂⋅(L + b⋅tan(q₄))            \n",
       "\n",
       "     (L⋅cos(q₄) + b⋅sin(q₄))⋅u₂⎫\n",
       "──────, q₈̇: ──────────────────────────⎬\n",
       "                L⋅r₂           ⎭"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\dot{q}_{8}, \\  \\dot{q}_{4}, \\  \\dot{q}_{7}, \\  \\dot{q}_{6}, \\  \\dot{q}_{1}, \\  \\dot{q}_{5}, \\  \\dot{q}_{2}, \\  \\dot{q}_{3}\\right]$"
      ],
      "text/plain": [
       "[q₈̇, q₄̇, q₇̇, q₆̇, q₁̇, q₅̇, q₂̇, q₃̇]"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sollist_keys = list(solution.keys())\n",
    "sollist_keys"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\frac{\\left(L \\operatorname{cos}\\left(q_{4}\\right) + b \\operatorname{sin}\\left(q_{4}\\right)\\right) u_{2}}{L r_{2}}, \\  u_{1}, \\  \\frac{\\left(L - b \\operatorname{tan}\\left(q_{4}\\right)\\right) \\left(L \\operatorname{cos}\\left(q_{4}\\right) + b \\operatorname{sin}\\left(q_{4}\\right)\\right) u_{2}}{L r_{2} \\left(L + b \\operatorname{tan}\\left(q_{4}\\right)\\right)}, \\  \\frac{L \\left(L + b \\operatorname{tan}\\left(q_{4}\\right)\\right) \\left(a u_{1} + u_{2}\\right) + a \\left(L \\operatorname{cos}\\left(q_{4}\\right) + b \\operatorname{sin}\\left(q_{4}\\right)\\right) u_{2} \\operatorname{tan}\\left(q_{4}\\right)}{L r_{1} \\left(L + b \\operatorname{tan}\\left(q_{4}\\right)\\right)}, \\  u_{2} \\operatorname{cos}\\left(q_{3} + q_{4}\\right), \\  \\frac{- L \\left(L + b \\operatorname{tan}\\left(q_{4}\\right)\\right) \\left(a u_{1} + u_{2}\\right) + 2 L \\left(L + b \\operatorname{tan}\\left(q_{4}\\right)\\right) u_{2} - a \\left(L \\operatorname{cos}\\left(q_{4}\\right) + b \\operatorname{sin}\\left(q_{4}\\right)\\right) u_{2} \\operatorname{tan}\\left(q_{4}\\right)}{L r_{1} \\left(L + b \\operatorname{tan}\\left(q_{4}\\right)\\right)}, \\  u_{2} \\operatorname{sin}\\left(q_{3} + q_{4}\\right), \\  \\frac{\\left(L \\operatorname{cos}\\left(q_{4}\\right) + b \\operatorname{sin}\\left(q_{4}\\right)\\right) u_{2} \\operatorname{tan}\\left(q_{4}\\right)}{L \\left(L + b \\operatorname{tan}\\left(q_{4}\\right)\\right)}\\right]$"
      ],
      "text/plain": [
       "⎡(L⋅cos(q₄) + b⋅sin(q₄))⋅u₂      (L - b⋅tan(q₄))⋅(L⋅cos(q₄) + b⋅sin(q₄))⋅u₂  L\n",
       "⎢──────────────────────────, u₁, ──────────────────────────────────────────, ─\n",
       "⎣           L⋅r₂                            L⋅r₂⋅(L + b⋅tan(q₄))              \n",
       "\n",
       "⋅(L + b⋅tan(q₄))⋅(a⋅u₁ + u₂) + a⋅(L⋅cos(q₄) + b⋅sin(q₄))⋅u₂⋅tan(q₄)           \n",
       "───────────────────────────────────────────────────────────────────, u₂⋅cos(q₃\n",
       "                       L⋅r₁⋅(L + b⋅tan(q₄))                                   \n",
       "\n",
       "        -L⋅(L + b⋅tan(q₄))⋅(a⋅u₁ + u₂) + 2⋅L⋅(L + b⋅tan(q₄))⋅u₂ - a⋅(L⋅cos(q₄)\n",
       " + q₄), ──────────────────────────────────────────────────────────────────────\n",
       "                                             L⋅r₁⋅(L + b⋅tan(q₄))             \n",
       "\n",
       " + b⋅sin(q₄))⋅u₂⋅tan(q₄)                   (L⋅cos(q₄) + b⋅sin(q₄))⋅u₂⋅tan(q₄)⎤\n",
       "────────────────────────, u₂⋅sin(q₃ + q₄), ──────────────────────────────────⎥\n",
       "                                                   L⋅(L + b⋅tan(q₄))         ⎦"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sollist_values = list(solution.values())\n",
    "sollist_values"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [],
   "source": [
    "sollist_values_simple = []\n",
    "for i in range(8):\n",
    "   sollist_values_simple.append(sm.factor(TR2(sollist_values[i]).simplify()))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\frac{\\left(L \\operatorname{cos}\\left(q_{4}\\right) + b \\operatorname{sin}\\left(q_{4}\\right)\\right) u_{2}}{L r_{2}}, \\  u_{1}, \\  - \\frac{\\left(- L \\operatorname{cos}\\left(q_{4}\\right) + b \\operatorname{sin}\\left(q_{4}\\right)\\right) u_{2}}{L r_{2}}, \\  \\frac{L a u_{1} + L u_{2} + a u_{2} \\operatorname{sin}\\left(q_{4}\\right)}{L r_{1}}, \\  u_{2} \\operatorname{cos}\\left(q_{3} + q_{4}\\right), \\  - \\frac{L a u_{1} - L u_{2} + a u_{2} \\operatorname{sin}\\left(q_{4}\\right)}{L r_{1}}, \\  u_{2} \\operatorname{sin}\\left(q_{3} + q_{4}\\right), \\  \\frac{u_{2} \\operatorname{sin}\\left(q_{4}\\right)}{L}\\right]$"
      ],
      "text/plain": [
       "⎡(L⋅cos(q₄) + b⋅sin(q₄))⋅u₂      -(-L⋅cos(q₄) + b⋅sin(q₄))⋅u₂   L⋅a⋅u₁ + L⋅u₂ \n",
       "⎢──────────────────────────, u₁, ─────────────────────────────, ──────────────\n",
       "⎣           L⋅r₂                              L⋅r₂                          L⋅\n",
       "\n",
       "+ a⋅u₂⋅sin(q₄)                   -(L⋅a⋅u₁ - L⋅u₂ + a⋅u₂⋅sin(q₄))              \n",
       "──────────────, u₂⋅cos(q₃ + q₄), ────────────────────────────────, u₂⋅sin(q₃ +\n",
       "r₁                                             L⋅r₁                           \n",
       "\n",
       "      u₂⋅sin(q₄)⎤\n",
       " q₄), ──────────⎥\n",
       "          L     ⎦"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sollist_values_simple"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left\\{ \\dot{q}_{1} : u_{2} \\operatorname{cos}\\left(q_{3} + q_{4}\\right), \\  \\dot{q}_{2} : u_{2} \\operatorname{sin}\\left(q_{3} + q_{4}\\right), \\  \\dot{q}_{3} : \\frac{u_{2} \\operatorname{sin}\\left(q_{4}\\right)}{L}, \\  \\dot{q}_{4} : u_{1}, \\  \\dot{q}_{5} : - \\frac{L a u_{1} - L u_{2} + a u_{2} \\operatorname{sin}\\left(q_{4}\\right)}{L r_{1}}, \\  \\dot{q}_{6} : \\frac{L a u_{1} + L u_{2} + a u_{2} \\operatorname{sin}\\left(q_{4}\\right)}{L r_{1}}, \\  \\dot{q}_{7} : - \\frac{\\left(- L \\operatorname{cos}\\left(q_{4}\\right) + b \\operatorname{sin}\\left(q_{4}\\right)\\right) u_{2}}{L r_{2}}, \\  \\dot{q}_{8} : \\frac{\\left(L \\operatorname{cos}\\left(q_{4}\\right) + b \\operatorname{sin}\\left(q_{4}\\right)\\right) u_{2}}{L r_{2}}\\right\\}$"
      ],
      "text/plain": [
       "⎧                                              u₂⋅sin(q₄)              -(L⋅a⋅u\n",
       "⎨q₁̇: u₂⋅cos(q₃ + q₄), q₂̇: u₂⋅sin(q₃ + q₄), q₃̇: ──────────, q₄̇: u₁, q₅̇: ──\n",
       "⎩                                                  L                          \n",
       "\n",
       "₁ - L⋅u₂ + a⋅u₂⋅sin(q₄))       L⋅a⋅u₁ + L⋅u₂ + a⋅u₂⋅sin(q₄)      -(-L⋅cos(q₄) \n",
       "──────────────────────────────, q₆̇: ────────────────────────────, q₇̇: ──────\n",
       "       L⋅r₁                                L⋅r₁                               \n",
       "\n",
       "+ b⋅sin(q₄))⋅u₂       (L⋅cos(q₄) + b⋅sin(q₄))⋅u₂⎫\n",
       "───────────────────────, q₈̇: ──────────────────────────⎬\n",
       "L⋅r₂                             L⋅r₂           ⎭"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "soldict = dict(zip(sollist_keys, sollist_values_simple)) \n",
    "soldict"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### $\\text{Reformulated Velocity and Angular Velocity Expressions}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle u_{2}\\mathbf{\\hat{a_1}_x}$"
      ],
      "text/plain": [
       "u₂ a_1_x"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N_v_Q = Q.vel(N).subs(soldict).express(A1).simplify()\n",
    "N_v_Q"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle u_{2} \\operatorname{cos}\\left(q_{4}\\right)\\mathbf{\\hat{a_2}_x}$"
      ],
      "text/plain": [
       "u₂⋅cos(q₄) a_2_x"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N_v_P = P.vel(N).subs(soldict).express(A2).simplify()\n",
    "N_v_P"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle u_{2} \\operatorname{cos}\\left(q_{4}\\right)\\mathbf{\\hat{a_2}_x} + \\frac{l u_{2} \\operatorname{sin}\\left(q_{4}\\right)}{L}\\mathbf{\\hat{a_2}_y}$"
      ],
      "text/plain": [
       "                   l⋅u₂⋅sin(q₄)\n",
       "u₂⋅cos(q₄) a_2_x + ──────────── a_2_y\n",
       "                        L"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N_v_C = C.vel(N).subs(soldict).express(A2).simplify()\n",
    "N_v_C"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle (u_{1} + \\frac{u_{2} \\operatorname{sin}\\left(q_{4}\\right)}{L})\\mathbf{\\hat{a_1}_z}$"
      ],
      "text/plain": [
       "⎛     u₂⋅sin(q₄)⎞\n",
       "⎜u₁ + ──────────⎟ a_1_z\n",
       "⎝         L     ⎠"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N_w_A1 = A1.ang_vel_in(N).subs(soldict).express(A1).simplify()\n",
    "N_w_A1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{u_{2} \\operatorname{sin}\\left(q_{4}\\right)}{L}\\mathbf{\\hat{a_2}_z}$"
      ],
      "text/plain": [
       "u₂⋅sin(q₄)\n",
       "────────── a_2_z\n",
       "    L"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N_w_A2 = A2.ang_vel_in(N).subs(soldict).express(A2).simplify()\n",
    "N_w_A2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### $\\text{Partial Velocities and Partial Angular Velocities}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOCAYAAAAWo42rAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA0UlEQVQoFXWR0Q2CQBBEhVgAwQ60BLUCsQO0A+3BP/4MJWgLWgIdGCmBDiB2gG+OW0PIuckyO3Ozu5cj6vt+ZlEURenrFlyRJVojLTIjwht+BZ86ABNA2p66ib14BhMzee0Dqukm7ozggawlTOIFzzTdjBlCNzGJaqoii/1dBvr/m2pi6s+te2y3Lb/V48NQvdBE6woZbFurO9rKJOA0bXhHDBW5DBhtYmXP88C0CRjXaLW2OiPFHaEDczNTa+2RPEmb6+ND3RcMdoUtfAd3f+wLrOY/XeyA6s8AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle 0$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "V_1_Q = N_v_Q.diff(u1, N)\n",
    "V_1_Q"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\mathbf{\\hat{a_1}_x}$"
      ],
      "text/plain": [
       "a_1_x"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "V_2_Q = N_v_Q.diff(u2, N)\n",
    "V_2_Q"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOCAYAAAAWo42rAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA0UlEQVQoFXWR0Q2CQBBEhVgAwQ60BLUCsQO0A+3BP/4MJWgLWgIdGCmBDiB2gG+OW0PIuckyO3Ozu5cj6vt+ZlEURenrFlyRJVojLTIjwht+BZ86ABNA2p66ib14BhMzee0Dqukm7ozggawlTOIFzzTdjBlCNzGJaqoii/1dBvr/m2pi6s+te2y3Lb/V48NQvdBE6woZbFurO9rKJOA0bXhHDBW5DBhtYmXP88C0CRjXaLW2OiPFHaEDczNTa+2RPEmb6+ND3RcMdoUtfAd3f+wLrOY/XeyA6s8AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle 0$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "V_1_C = N_v_C.diff(u1, N)\n",
    "V_1_C"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAO8AAAAsCAYAAACNDO+jAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAMf0lEQVR4Ae2c65UctRKAhz0bgFkiwGQAOAJMBtiOAJMBHH7Z/3xMBuAIuJAB3AgAZwA3gms2A/N9cpfoh7rntT3du6M6R6uWVCqVqqtUJalnN2/fvt3UVGVwG3Xg2bNn9+fgey66N83rxaZClcAtlMDz58+/hu2PZ2L9fkN/JvI3Q/Y9V4MKVQIhAZT2Jc9fkO6TPqH8Otp2zelj3z9I3/D8w679dsWDpvw9IP9m1z774kHbxeEv8p/37Xsq/Op5TyXpWzJOYxAa8IbnvQ23mabGe4/0SVO+sQyepPttw+eN0e0Tgv53zTiOt0qoxrvK17I4UxrdoYa7QfF/pf/75F/NMBMXlu9noFsi6ThpISs1Ll1XjXfpN7DO8R/C1o/HsIbhXh/Tf6LvY2jfeCheGq8Zx/FW6X2r8Zbe2hnXoaiGvCa956oA3lxU/joxU473+MRj7jTc5U5YFemcJKCBbDCU0bCZNk95xdO7hld6Qr0HXBr+T6S0AFB+xPOG3D7WX5H+QzIc9eBJeED6DRz3mVMgrclFBRqO60FWeP4XPD+1H22jc6J9DBzvc9JJvP0YE6X6arwlqZx3nYo6aiAYgMb6kly8DJTT/pZcT6UR/5kbeaCs4XxE/je5BvwFz8lYyTXsP8h/Jk151k/BG93v0tcFxQXC8RMdcsuO9R75IeA80oJ2SOc5+1zMSbzSvpUSUFF/meBcA/oUYwiPG6j9K5UxI/ydDg/pn70sz+ERtxmJY76JAdt5w4+G+oLn/thBv91lA57XWdvA8fpz3dbnJO2XJxmlDnIrJIAy6wFV1L4hZv7BMfxUof/2mVxD12Puc+eqAZdgm5Hosa9LHakzDL8HH3lRaPBcEAYhL3i7fuThQmAovjqonnd1r2RRhlT0axS777n6THmVpJFoTBrNn/QZDWdpPwV4qNQJ9+FJo3NB6EQS1DvPbXMEJcHUghE4i+TVeBcR+2oHzftdFLzoBRuD2JD79ZRG/D7J/e5TynruOUGDG/DFuNYNjJQ6jXRDezbqBlc+c504EyDdYqg+0eckTdV4TyLmWzOIyh5eyhPaEqj4uQ1j0FMblhpqux+eEzQivekY9L2pp9Ove8guMv3QuofSKep5+3Q7CEsVVmm8CHfqBR0sq7noHszQOjv6Pe/Au/VY9fPEPo7lvjfr40hGYzgUNESvlToAL9dUOHbWG+pcYFyMMk/UeTU1up+nrQRGF7GgldoXq7tcbOSRgRFw+iCc5jlWO38tkq8oRljYuRpanm6q7Psc1uxMfwFEw1+91ecTc9JQ9LTtL48+oOz1kbLQM7sPTl6YsoovTQ3Letv11tYrN/FsF76i3h8cRPld7b9/7TO2t7bPK/qqP0LoTzI86h3/ijzqE9IOf1wA5PtgYMxZ9GRVvypikq6Mq/+1CHzqUf5LUiF89gS2c+9JXYUZJICcvXd9RN4Phzuj0a7n/Z483e+Su2g8aSH53sQxhPYDkYFHps73+wv5R+R7A/1m1ZOLvTmaqUMz0dX/WqTh83+IwRX8Q5IHNhvqOx8lWFdhFgnoBdMHIVuou5hmA+f9vCZ5yJYSbeFNvRceGG5D28gg8Jqq3TJoariz6slqjLcR0lhItJvEdsdynENfyjV9P+PluPob/plUlLFQb3eujsCEBz98yAdJR5BadVfmaMju9kevOAWd/W4bsZFT6JqhthFfBxr6jjO4I+4gjhToN7uerCZsZrJe+icvNiKPG612PAh+2Aj5RmkvQYx5qIAq2z4nqUuwevSYzFGv9hP5YKtCnYuyhmuYbHS07wckdEmRVNqrQ08jXCWs4sAKASnsfQ8SjhWo43mxP7qywpeHH/19kqeXhl6rfanwd6dB2ZM83PLap/P+KBvqHgXShYD0t75jcBbTkWy8MOFq5qrVZrizkW/hxP7OjbwrYPs43hVPY5SONIUn4HjkPgaGnJlGCYn+hkm+mODvBc8KeZZfizCec0jy4Dn2tda5InvSeme8NvO5dYD8XXw7hntTk4D2TnTBW1RHLp0wTGgYfqT9Jc9p894wpmF6/O9mX6P0yNtrhOwlefY07mPSdySNdfQXJ7SNgdcFsQcZ4EBXITn2KX8tMthTwUcsFMriW9LRqzw0KtxeCSyqI8l4kV3cQ7VP3cJrhqcTp/STLRXYn3PpOa9I6RcnlKMfVVsvxh3rjYh9gI5tju2pYF40Grx8mtjuB578THl60R1P2kWg/w8k2/sRgTxovKYMDe4rKnyhJunbt4bYCOEuwtI6ctkonYrYOeigXkMOL2y7CvkbqQPg6ZWtMzRWUVXafX9xckWfa4kUwND1HnQ7/FGnNx6EN+C5B+kYFuUSaISDlbON6Jgkx3YsabpNCNry3IZ07wtuhNjyEYcnB90TtonHM/SNUOSnD4kf2kvXKL6jydNw2uu/Ee1LtFBGTp3fBVNeTEcu4S8U+P8FXqMqcMYMTLxQaj2eIaUKpvIaRuvFSkpF81Z4DEbH+0FLfvSK6euZoEC9Y/a9czT3c5V9dD7NGPLvKa7jazQuFtaVrmQST/TT2D1Q8aUqB0+AvcbpzIH6gwA6RTlSf9RpM/07SnkQc2fWCZn5zhfTEY03lH3KOwSOBjMG7o2dzBvytBckF1/j80sXUzHMpV36A9pNf+s7RkpZI93Qng2iwXUBGXhjcQsgXaOEATS0PAMQRyPMe1ueB/hNxWfkfn53PYZQ6++OBNagI4bNegmNykOjAdDmt8DudVVK79Xa++IN9cmQqNczaTwacApxmz56XftJf8x4NSL7jUEsHtFuCNin9ZRxUggTSFtyPW+fbnSRVw1XcF5tKPLJ2B1+KCsXabiopUWGXO/oSi0N5R4htqf38qMcPJSrCwCCWDnMoiPOmfdvZKeeqD/qQrrZoF77cnsmPLp4l6evgwzvOuEgZQmEUupZvCKRQBvE0WgCz08cHbQNlrOXbDc0z/Z90K+HjozbLxsMdfKoYWR61GkUnUWF8jYwvO979OjzezyQ5/kyjs/Bi+Fxf56pW1PvIZs85oMz6l0EjXCst7/htPSuSH6x5f94cs4V1i+BWXTEaaMDRo/am+Din3SCPGxMY/41f2FFQUX0tPQNKRTIUDd7J55VNEPIaLcsTjIkco1KL6snkZ7wAcnrpGxsqbb1p+knHRW7A9QFX3FYJj8ahldWXt3Ig0aQwuUG3wOzyT0c7fLYufainIF26bowxUIhH47t/B3fesHL/Byq8yyeIXf6soeyBq+XzXKkvKHs+I6RaFLed/GRTAb6u4CdxRdWedILPyDzuXVE55JsirHUdR2XkVk697iM+VNxzfO2E0kVrXhgIh1oaKCjRipOCexHsr/3xbG6JFTKHb4op+iAPMbRWPRYGppgedOUOx+ZWC/QptDNOwZlXUDTNiaPwWd59qOPdDVsDTr4c0H8ktSHWOT0uoHbxzm7MnLzlN4ozMVIUB90KH5zkBdJG5YG+FF/5tQRHYWOQN02grOcdS8bL5VLgwy6MIwuDg2DMp8NHAH63C5rQBq498IafgkUQhh7qX3vOsZy3DjkMtJo0ygtEu6l9bYqqfuYHF7zfAg417H5HkJvkT7ILc5LvLoyZDxWLovMozTovjoCvld8LuwPyXUKlrMuXZQGWaIOplxVDfs0gil4SGPRU9FXo40Dpll+LTLBWBjuAAW+OkZF2RdhKO2q7WJkxDG2/x7QK1XQ3+hlVZ6pxOcudczDrYZw1FbiHYlV/d1ZR1pc62gEF/l4ThVr8rwypDKr2Dk0sFLgheopNdx7JE/AretMhrLKu02BNe6xUIemw4Cxt/4iChwXF3l2gQre4+DD1dV9sB+63DWlZVp7ge9ZOGpBe0diPX930ZE+t/QJ7zs4N8kHVv1OS5VhVsVWkbcZ4d4sQlPj0UPl0GNvIrXD7BLg/eihjEYmDx1nZ2ShAZi3NuC5SVrgyV3UB/89ZHXGu5C86rArkgDK6n5Xj3Nn9rv7iJd5G2V6cOfti2dAbicH0eIFDRWqBFYjAZQ0QubiucZqGJ2XEbd2HsKa+z3AwHAd/tI/FaoEViSBOO+4U/vdfeSLsbqt2xp1VM+7j1Qr7ikkkDwvCnzOnncnOVfj3UlMFemEEvCaKN/b98fFqJ+SPNA5e6jGe/YqsB4BYJSx3/1xgqvRT1on+tzJpmq8d/K13tpJxX63GDJj3J7Anu1euP9W64FVXyK1vKQE0vfMGOkgbKYurk+2fgyz5AROOXa95z2ltOtYAwlglH4x94rkPrb0WWSnHvzitcmA8BlU/APr1Mfxq9dINwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle \\operatorname{cos}\\left(q_{4}\\right)\\mathbf{\\hat{a_2}_x} + \\frac{l \\operatorname{sin}\\left(q_{4}\\right)}{L}\\mathbf{\\hat{a_2}_y}$"
      ],
      "text/plain": [
       "                l⋅sin(q₄)\n",
       "cos(q₄) a_2_x + ───────── a_2_y\n",
       "                    L"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "V_2_C = N_v_C.diff(u2, N)\n",
    "V_2_C"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOCAYAAAAWo42rAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA0UlEQVQoFXWR0Q2CQBBEhVgAwQ60BLUCsQO0A+3BP/4MJWgLWgIdGCmBDiB2gG+OW0PIuckyO3Ozu5cj6vt+ZlEURenrFlyRJVojLTIjwht+BZ86ABNA2p66ib14BhMzee0Dqukm7ozggawlTOIFzzTdjBlCNzGJaqoii/1dBvr/m2pi6s+te2y3Lb/V48NQvdBE6woZbFurO9rKJOA0bXhHDBW5DBhtYmXP88C0CRjXaLW2OiPFHaEDczNTa+2RPEmb6+ND3RcMdoUtfAd3f+wLrOY/XeyA6s8AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle 0$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "V_1_P = N_v_P.diff(u1, N)\n",
    "V_1_P"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\operatorname{cos}\\left(q_{4}\\right)\\mathbf{\\hat{a_2}_x}$"
      ],
      "text/plain": [
       "cos(q₄) a_2_x"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "V_2_P = N_v_P.diff(u2, N)\n",
    "V_2_P"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAB4AAAAUCAYAAACaq43EAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAB2UlEQVRIDbWV7VHCQBCGgaEALAHtAO1A6UDGDqADGX/BX+lALAE6EDtQOhA7EDrA54m5TBIYQiS8M5vb29vbvf24S3273daOwXg8nqG3Yhweo1+kUy9yjKMWRt6hS0h+gazLeBIah3bHTr/RWUFt6EJ95F+OJ8GID9FoNOrk1/fJ8jpF88JUnxTVgc3NsEb6HuEfwpwxqifjkLVNSl4JG0WM4TusvUEb+FDHRIa8XbXz0Fx2bAY4WiBYQkb+lFmsYBI5xskUW97P25xNu1l0/obqvpnm4gBGdxM7umI03WZjydo1Y4JY17VnhcxL3e2oudgUDNxjwxS/QGZBo30oA/RtRB3ZdB7OkpRCEyNG+Qk5TpgnTyI8ol0gnyBV1z0etjSssanVgDDSNMzEWWCqP1KWbaKooYhGPjhuGR20SekWsuj7B3JP8GFZRK8RG7OR5tAr8xnknR5ANpQ19wBr5Dv1Rr4X6IYs+gbYD+GG+JOZG3ENRmFPfg9KdWtu/xTbPkr2gYdO/GSuU25T4RSDRrWGdq5b2IyO2Qp/MzP741rDz5lh2USPQxixr2D/XxHHqQvPqE1oA0U1ZC15aOB1akOZ8gFkhry6w385ZmMhcKLDEG14YEy7zru/5NL+UFdDrLkAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle \\mathbf{\\hat{a_1}_z}$"
      ],
      "text/plain": [
       "a_1_z"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w_1_A1 = N_w_A1.diff(u1, N)\n",
    "w_1_A1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{\\operatorname{sin}\\left(q_{4}\\right)}{L}\\mathbf{\\hat{a_1}_z}$"
      ],
      "text/plain": [
       "sin(q₄)\n",
       "─────── a_1_z\n",
       "   L"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w_2_A1 = N_w_A1.diff(u2, N)\n",
    "w_2_A1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOCAYAAAAWo42rAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA0UlEQVQoFXWR0Q2CQBBEhVgAwQ60BLUCsQO0A+3BP/4MJWgLWgIdGCmBDiB2gG+OW0PIuckyO3Ozu5cj6vt+ZlEURenrFlyRJVojLTIjwht+BZ86ABNA2p66ib14BhMzee0Dqukm7ozggawlTOIFzzTdjBlCNzGJaqoii/1dBvr/m2pi6s+te2y3Lb/V48NQvdBE6woZbFurO9rKJOA0bXhHDBW5DBhtYmXP88C0CRjXaLW2OiPFHaEDczNTa+2RPEmb6+ND3RcMdoUtfAd3f+wLrOY/XeyA6s8AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle 0$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w_1_A2 = N_w_A2.diff(u1, N)\n",
    "w_1_A2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{\\operatorname{sin}\\left(q_{4}\\right)}{L}\\mathbf{\\hat{a_2}_z}$"
      ],
      "text/plain": [
       "sin(q₄)\n",
       "─────── a_2_z\n",
       "   L"
      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w_2_A2 = N_w_A2.diff(u2, N)\n",
    "w_2_A2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### $\\text{Accelerations and Angular Accelerations}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [],
   "source": [
    "a_1__P, a_2__P, a_3__P, a_1__C, a_2__C, a_3__C, a__Q, alpha__A1, alpha__A2 = sm.symbols('a_1__P, a_2__P, a_3__P, a_1__C, a_2__C, a_3__C, a__Q, alpha__A1, alpha__A2')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle (- u_{1} u_{2} \\operatorname{sin}\\left(q_{4}\\right) + \\operatorname{cos}\\left(q_{4}\\right) \\dot{u}_{2})\\mathbf{\\hat{a_2}_x} + \\frac{u^{2}_{2} \\operatorname{sin}\\left(q_{4}\\right) \\operatorname{cos}\\left(q_{4}\\right)}{L}\\mathbf{\\hat{a_2}_y}$"
      ],
      "text/plain": [
       "                                        2\n",
       "                                      u₂ ⋅sin(q₄)⋅cos(q₄)\n",
       "(-u₁⋅u₂⋅sin(q₄) + cos(q₄)⋅u₂̇) a_2_x + ─────────────────── a_2_y\n",
       "                                               L"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N_a_P = N_v_P.dt(N).subs(soldict)\n",
    "N_a_P"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle (- u_{1} u_{2} \\operatorname{sin}\\left(q_{4}\\right) + \\operatorname{cos}\\left(q_{4}\\right) \\dot{u}_{2} - \\frac{l u^{2}_{2} \\operatorname{sin}^{2}\\left(q_{4}\\right)}{L^{2}})\\mathbf{\\hat{a_2}_x} + (\\frac{l u_{1} u_{2} \\operatorname{cos}\\left(q_{4}\\right)}{L} + \\frac{l \\operatorname{sin}\\left(q_{4}\\right) \\dot{u}_{2}}{L} + \\frac{u^{2}_{2} \\operatorname{sin}\\left(q_{4}\\right) \\operatorname{cos}\\left(q_{4}\\right)}{L})\\mathbf{\\hat{a_2}_y}$"
      ],
      "text/plain": [
       "⎛                                  2    2    ⎞         ⎛\n",
       "⎜                              l⋅u₂ ⋅sin (q₄)⎟         ⎜l⋅u₁⋅u₂⋅cos(q₄)   l⋅si\n",
       "⎜-u₁⋅u₂⋅sin(q₄) + cos(q₄)⋅u₂̇ - ──────────────⎟ a_2_x + ⎜─────────────── + ───\n",
       "⎜                                     2      ⎟         ⎝       L\n",
       "⎝                                    L       ⎠\n",
       "\n",
       "             2                ⎞\n",
       "n(q₄)⋅u₂̇   u₂ ⋅sin(q₄)⋅cos(q₄)⎟\n",
       "───────── + ───────────────────⎟ a_2_y\n",
       " L                  L         ⎠\n"
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N_a_C = N_v_C.dt(N).subs(soldict)\n",
    "N_a_C"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\dot{u}_{2}\\mathbf{\\hat{a_1}_x} + \\left(u_{1} + \\frac{u_{2} \\operatorname{sin}\\left(q_{4}\\right)}{L}\\right) u_{2}\\mathbf{\\hat{a_1}_y}$"
      ],
      "text/plain": [
       "           ⎛     u₂⋅sin(q₄)⎞\n",
       "u₂̇ a_1_x + ⎜u₁ + ──────────⎟⋅u₂ a_1_y\n",
       "           ⎝         L     ⎠"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N_a_Q = N_v_Q.dt(N).subs(soldict)\n",
    "N_a_Q"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle (\\dot{u}_{1} + \\frac{u_{1} u_{2} \\operatorname{cos}\\left(q_{4}\\right)}{L} + \\frac{\\operatorname{sin}\\left(q_{4}\\right) \\dot{u}_{2}}{L})\\mathbf{\\hat{a_1}_z}$"
      ],
      "text/plain": [
       "⎛     u₁⋅u₂⋅cos(q₄)   sin(q₄)⋅u₂̇⎞\n",
       "⎜u₁̇ + ───────────── + ──────────⎟ a_1_z\n",
       "⎝           L             L     ⎠"
      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N_aa_A1 = N_w_A1.dt(N).subs(soldict)\n",
    "N_aa_A1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle (\\frac{u_{1} u_{2} \\operatorname{cos}\\left(q_{4}\\right)}{L} + \\frac{\\operatorname{sin}\\left(q_{4}\\right) \\dot{u}_{2}}{L})\\mathbf{\\hat{a_2}_z}$"
      ],
      "text/plain": [
       "⎛u₁⋅u₂⋅cos(q₄)   sin(q₄)⋅u₂̇⎞\n",
       "⎜───────────── + ──────────⎟ a_2_z\n",
       "⎝      L             L     ⎠"
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N_aa_A2 = N_w_A2.dt(N).subs(soldict)\n",
    "N_aa_A2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### $\\text{Forces}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$(F_r^*)_G = (F_r^*)_{GR} + (F_r^*)_{GI}$\n",
    "    \n",
    "where,<br>\n",
    "\n",
    "$(F_r^*)_{GR} = {V_r}^G \\cdot {F_G}^* + \\omega_r^A \\cdot {T_G}^*$\n",
    "\n",
    "$F_G^* = -m_G {a^G}^*$\n",
    "      \n",
    "$T_G^* \\overset{\\Delta}{=} -[\\alpha_A \\cdot I_G + \\omega_r^A \\times (I_G \\cdot \\omega_r^A)]$\n",
    "\n",
    "$({F_r}^*)_{GI} = -J\\{\\omega_r^A [\\ddot{q_k} g_1 + \\dot{q_k} (\\omega_3^A g_2 - \\omega_2^A g_3)] + C_{kr} (\\dot{\\omega_1^A} + \\ddot{q_k}) \\}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### <center>$\\text{Kane, T.R., 1978. Nonholonomic multibody systems containing gyrostats. In Dynamics of Multibody Systems (pp. 97-107). Springer, Berlin, Heidelberg.}$</center>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\text{Naming Convention:}$\n",
    "\n",
    "$({F_r}^*)_{G_n R}\\ \\text{(rigid)}$\n",
    "\n",
    "$({F_r}^*)_{G_n I}\\ \\text{(internal)}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### $\\text{Masses and Moments of Inertia}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [],
   "source": [
    "M1, M2 = sm.symbols('M_1, M_2')\n",
    "m = me.dynamicsymbols('m')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [],
   "source": [
    "I1x, I1y, I1z = sm.symbols('I_{1_x}, I_{1_y}, I_{1_z}')\n",
    "I2x, I2y, I2z = sm.symbols('I_{2_x}, I_{2_y}, I_{2_z}')\n",
    "J1, J2 = sm.symbols('J_1, J_2')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle I_{1_x}\\mathbf{\\hat{a_1}_x}\\otimes \\mathbf{\\hat{a_1}_x} + I_{1_y}\\mathbf{\\hat{a_1}_y}\\otimes \\mathbf{\\hat{a_1}_y} + I_{1_z}\\mathbf{\\hat{a_1}_z}\\otimes \\mathbf{\\hat{a_1}_z}$"
      ],
      "text/plain": [
       "I_{1_x} a_1_x⊗a_1_x + I_{1_y} a_1_y⊗a_1_y + I_{1_z} a_1_z⊗a_1_z"
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I1 = me.inertia(A1, I1x, I1y, I1z)\n",
    "I1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAZMAAAAXCAYAAADOSqIbAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAJz0lEQVR4Ae2d75ETNxiHfTdXwHFUENIBSSoAOoBJBUAHMPkE3xjoAKggAx1AOiB0EKiA5DogzyMkz3q9tndtab22/M4IrbSS3j/67ftK2vVx9v3799kU6fnz5++Q6wv50ynKl0umWvTMZa++49Rk15p07Tv/u7arwaa5dTxrBxMYvGQibpPuxgn5TP6F9JR75kUJHpcw+It0i+T1R+rukR8VHaue6HXCz0hIPWEov6GP1aZNS5XScSmYJKYwdMvizuDnVFc6j0p+hc9H0sPIzx3KrTHliHyLZTXoiY4n/BRD0Gx2wlB+41ZiUxfoRXzsRdeUYFR3JtL7H9k4/8L3mnSH5G4o0b2GPKmuaA4/Df6IdLPF6BvlN9y/btUPKtqftHc920Ijk7tRA/eb9r0hZfpXjR9thQ2qwxA6Z8FPtN/oGEL+vT+Xh4ybzp0JCj1hQj2u0JG7S6iG0NcH4gHJY72FoEE5OYjPx2gXdLqP3gaTV7tMOP2rxY92Q/8qMYTeWfATbVgdhg4dNxcrnEZ4R4FyxQIJYwuW3xv8ddTyW3LijTZFL+Nk+sL/MdcLgUTG1pFctb8jt7zRPrSZnJ7qUpiqxI82Zb4NJNkwVCl+NGVRDE3NrrlxowHH1vFcph3kA9E8auposn1VNJw7H1fBv5i4fkzyaOkrZQPLPsggIojN/QBggaJcL2Mbd21r5eS+dpyingt6FSjUih9NmQ1DFeNHOxbD0ETtmg03Gm8fOi4FkyiE8vzpP4Woy1G7yjeA6aD/KMR35bDo7Q4ifIbMdVpZzuWkTrkMJAY96QVpk5zz/qEH/9B/r3omOUrl6KcTkKrCjwqje24MVYefaMfSGJqUXQvgRjOOruO5XFsUtpfUbTzCafXrXcR4HhXpsO+0OqVPj9PLt9btosXfkCvxn3GdAsptrp2YZiDx/jV1SxPWlJA2U9SzKWKJ61rxoy2zYqhS/GjHohiaoF2z4kYD7kPHCxm3KKwKEGbhmItyc8eQnOhD6nWqg4l+rxyTJD+Dh58gpyBy1RzQdpTfkuRr+pdksMv5fmUeSBg3EHwd/zWFK3JfyrdpqU+7Af2mpucs6pRWf02Rg925n3ZfzXt+dNBlg2Ybr2vFj7ov4QGb7YQh+teGn1EwNDG7ZseNRhxbxwuZtkiH3rUraa/MdbJ/kwb/DgUlDQi+S/DrD3k5lqt463xv0qbwI0b63fAGuccJttVxDeZPny4yYC1QlNNg6Yt3dygLAZb6pT7NAWL/qek5Q66uYGF9jq9xasWPU7+EB2wq1rfCUOxbG360Y1EMTdCuWXGjAfeh47mMEyFAWq1+SHWN/FHjvtWC3BfoTnxvor2GMwjpuFx1+SL7PckHbhX5QLqLCUYn99NV28s/ybyq71b1jCtPX4q5svTI6+4QXrQ9CD23Ms6KTg37VI8fTYQ9tsZQjfiJNkvPcxEMHYJdd8FNtOFefM+5zBu07qzS1eynRtttL3+lYwgK5O5ImuTD10W+WzHoGEBKkZ/7GuBm5AbIEEgSM+oMYO5OAthj3gX41GWqeib5SuQ140d75sRQjfjRhqUxNEW75sSNNtyLjgs/WsRB/oMgrvbPlGgd0SZsv8kHHTPR3kDyXxz7AeXwK3tyHXj40ynkniH6yXBn8KBeh64jn/+5F+ruUw67JXKPpdKRmDpdkXzPsnJM7s3oY3BTBgOXu5Elot4jNuXzpVlnGztxb7J6Kl8XIbM2dP63+tEi/Q4ZPx6vih/nTdz9hD7heJNrj1kl8dp1BPzjLv9yPwuGGKcIfhSUsbPoOlc6XjDuTviJshXFUGG7+ieExE5adKdd1tzPRVMtZciVBTcOvC8dz549e+ZuIDnhdGSlg1/5F3sRNjn+tc5ZxbqI/omnxvbh9MHRQeucdeZpEtwd+C5lTtFQ/m0ZJ8xJcvLmRNkgY39XOI7pkdpGJ0CbQLG/fO27RNQrqzIu8W43pu1k9WzLahl5BzuDho7qetD4QRflFy9+bOBvnwJx7eInBJdYtTajvRjcGUOMUxI/WXRtGgJ5B+PH/g09R8FQg182/8OY+gVxcoNrFyEGB4N27z9US58suIHnjLGyY4cx1+q4sDNRiE0UhdRQG53pprGG3o/K+LD7jsV3GT4QTtyC46ccVjfcCwGKctj99OVHe0Gg4Qx0BtW0QhV8Nymv3JFwf2difHkX17MtKHy3cgbtcdaV4SHIJ4sf5JsvRrjWEYgFF02dHy2s0jX22wuG4N0XP1l0TTaAb3H8yAs+e8HQOrvGe+EjJa61gwtO/c+gBTd9J+t7Nul4gbK9icFCtCMP55rkc2ce7+lsPfb6RromeVzU53NSmq6nOL4T5IovHTW8pfywo6fyGVCuSKltR7PuKsb3jzn6QKrPXa5tKDB2/iOPDrSO4KWNR9GzQw7nzFSEom4+cF34CY5IxtwPx2zkniWPjR8XCgZyd+vuTiwHecl7E3LvBUPwHYKfLLo2jFIUP/KJ+nViKN533sSSC0j9UJZndpNdua/u+qZkf4o/cEPdZbxv3Vqi3V5wo1DwTrJ3+tiow0odeweTyMgVZdoRyN/VWlql+3JaQ+iAbWMgGfQ+xQG7iHFU0gf8kvSBMtmcFnYlsVY5BZOg8rx7fmTBdS+Ch+AYtKPpNfCaRvvQsykO/AcH3mb/dddRt3X4sbufXofgQXsXKlloiF1pm/6Ip4sIg7rlLoxtlI1+o2IIfoOeE9pn01VjMF4x/MTx1W8Thl4jh/7HFf4ncudgJ2KMIXZ1tyd5cuOphsHNoNb7PWSUecq+p1PHc7XuSTpzA4V5So+i4jPypLxOIDCjLgUaqnYi+RlIlijxTzco6wA8ltIpuaI0yCXlU7Op5sesp7ptwo9zlubKtul61/nqbdfIKOHWxUi63lWGMfoP1VOZkn6HoOtaDKkMz7oO3EDiEWX7d2E22YZ62RV+4tXA46Lao3h9lnbdajFCvzFpZx0HvzNZpR2Ge0LydyMa1JWBq577pBRkVnXNUg8fAeSD4WTKP/xymOv05ZgTav0o8sCrCB2znugmoP1/XlwMzHFUxJAbBo3806JkQ+vDvn1MuqKLO4EX5M7dmP6nufhJQUxfZEDxlKborm0MBKLDWh1zBhMffncFGu8myXcWbjOTYSme6GSB1RYAK352LV2TPK7Y+Il6aJ3hH3j54HseHFbq5OLXo4qjw++x6ope4kfsuMN1Pn23svA1KHUnKmSBbMGkkHynYSuxQHRwHkl6PODRgKu5QV9Q7WIqeLmi1Rn5nk++/t4mvL/h+qioJl2PauImrszFxOU7iVePBdzR/oaj8wu8S/LRAkk0sS923cabe+5+lIGkQl2jyqestAX+B+rzX6XCEt56AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle I_{2_x}\\mathbf{\\hat{a_2}_x}\\otimes \\mathbf{\\hat{a_2}_x} + I_{2_y}\\mathbf{\\hat{a_2}_y}\\otimes \\mathbf{\\hat{a_2}_y} + I_{2_z}\\mathbf{\\hat{a_2}_z}\\otimes \\mathbf{\\hat{a_2}_z}$"
      ],
      "text/plain": [
       "I_{2_x} a_2_x⊗a_2_x + I_{2_y} a_2_y⊗a_2_y + I_{2_z} a_2_z⊗a_2_z"
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I2 = me.inertia(A2, I2x, I2y, I2z)\n",
    "I2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### $\\text{Gyrostat}\\ G_1$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### $\\rightarrow {F_G}^* = -m_G {a^G}^*$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle -  M_{1} \\dot{u}_{2}\\mathbf{\\hat{a_1}_x} -  M_{1} \\left(u_{1} + \\frac{u_{2} \\operatorname{sin}\\left(q_{4}\\right)}{L}\\right) u_{2}\\mathbf{\\hat{a_1}_y}$"
      ],
      "text/plain": [
       "                   ⎛     u₂⋅sin(q₄)⎞\n",
       "-M₁⋅u₂̇ a_1_x + -M₁⋅⎜u₁ + ──────────⎟⋅u₂ a_1_y\n",
       "                   ⎝         L     ⎠"
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Fstar_G1 = -M1 * N_a_Q\n",
    "Fstar_G1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### $\\rightarrow {T_G}^* \\overset{\\Delta}{=} -[\\alpha_A \\cdot I_G + {\\omega_r}^A \\times (I_G \\cdot {\\omega_r}^A)]$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle -  I_{1_z} \\left(\\dot{u}_{1} + \\frac{u_{1} u_{2} \\operatorname{cos}\\left(q_{4}\\right)}{L} + \\frac{\\operatorname{sin}\\left(q_{4}\\right) \\dot{u}_{2}}{L}\\right)\\mathbf{\\hat{a_1}_z}$"
      ],
      "text/plain": [
       "         ⎛     u₁⋅u₂⋅cos(q₄)   sin(q₄)⋅u₂̇⎞\n",
       "-I_{1_z}⋅⎜u₁̇ + ───────────── + ──────────⎟ a_1_z\n",
       "         ⎝           L             L     ⎠"
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Tstar_G1 = -(N_aa_A1.dot(I1) + me.cross(N_w_A1, I1.dot(N_w_A1)))\n",
    "Tstar_G1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### $\\rightarrow ({F_r}^*)_{GR} = {V_r}^G \\cdot {F_G}^* + {\\omega_r}^A \\cdot {T_G}^*$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - I_{1_z} \\alpha^{A1}$"
      ],
      "text/plain": [
       "-I_{1_z}⋅α__A1"
      ]
     },
     "execution_count": 79,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Fstar_1_G1_R = V_1_Q.dot(Fstar_G1) + w_1_A1.dot(Tstar_G1).subs(soldict)\n",
    "Fstar_1_G1_R.subs({N_w_A1.dt(N).subs(soldict).dot(A1.z): alpha__A1})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - \\frac{I_{1_z} \\alpha^{A1} \\operatorname{sin}\\left(q_{4}\\right)}{L} - M_{1} \\dot{u}_{2}$"
      ],
      "text/plain": [
       "  I_{1_z}⋅α__A1⋅sin(q₄)        \n",
       "- ───────────────────── - M₁⋅u₂̇\n",
       "            L                  "
      ]
     },
     "execution_count": 80,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Fstar_2_G1_R = V_2_Q.dot(Fstar_G1) + w_2_A1.dot(Tstar_G1).subs(soldict)\n",
    "Fstar_2_G1_R.subs({N_w_A1.dt(N).subs(soldict).dot(A1.z): alpha__A1})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### $\\rightarrow (F_r^*)_{GI} = -J\\{\\omega_r^A \\cdot [\\ddot{q_k} g_1 + \\dot{q_k} (\\omega_3^A g_2 - \\omega_2^A g_3)] + C_{kr} (\\dot{\\omega}_1^A + \\ddot{q_k}) \\} \\qquad (r=1,...,n-m)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\text{Here}, \\{\\omega_1^A: \\omega_2^A,\\ \\omega_2^A: \\omega_3^A,\\ \\omega_3^A: \\omega_1^A\\}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### $\\rightarrow \\dot{q_k} = \\sum_{s = 1}^{n - m} C_{ks} u_s + D_k \\quad \\text{(Generalized Speeds)}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### $\\omega_i^A \\overset{\\Delta}{=} \\omega^A \\cdot \\hat{g}_i \\quad (i = 1, 2, 3)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( \\frac{J_{1} a \\ddot{q}_{5}}{r_{1}} - \\frac{J_{1} a \\ddot{q}_{6}}{r_{1}}, \\  - \\frac{a}{r_{1}}, \\  \\frac{a}{r_{1}}, \\  - J_{1} \\left(C_{51} \\ddot{q}_{5} + C_{61} \\ddot{q}_{6}\\right)\\right)$"
      ],
      "text/plain": [
       "⎛J₁⋅a⋅q₅̈   J₁⋅a⋅q₆̈  -a   a                        ⎞\n",
       "⎜─────── - ───────, ───, ──, -J₁⋅(C₅₁⋅q₅̈ + C₆₁⋅q₆̈)⎟\n",
       "⎝   r₁        r₁     r₁  r₁                       ⎠"
      ]
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# C_kr\n",
    "C51, C61 = sm.symbols('C_51, C_61')\n",
    "C_51 = soldict[q5.diff()].diff(u1)\n",
    "C_61 = soldict[q6.diff()].diff(u1)\n",
    "Fstar_1_G1_I = -J1 * (N_w_A1.dot(q5.diff().diff() * A1.y + q5.diff()*(N_w_A1.dot(A1.x)*A1.z - N_w_A1.dot(A1.z)*A1.x)) + C_51 * (N_w_A1.dot(A1.y).diff() + q5.diff().diff())) \\\n",
    "               -J1 * (N_w_A1.dot(q6.diff().diff() * A1.y + q6.diff()*(N_w_A1.dot(A1.x)*A1.z - N_w_A1.dot(A1.z)*A1.x)) + C_61 * (N_w_A1.dot(A1.y).diff() + q6.diff().diff()))   # B1 \\ B2\n",
    "\n",
    "Fstar_1_G1_I, C_51, C_61, Fstar_1_G1_I.subs({-C_51: -C51, -C_61: -C61}).simplify()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left( \\frac{J_{1} \\left(- L + a \\operatorname{sin}\\left(q_{4}\\right)\\right) \\ddot{q}_{5}}{L r_{1}} - \\frac{J_{1} \\left(L + a \\operatorname{sin}\\left(q_{4}\\right)\\right) \\ddot{q}_{6}}{L r_{1}}, \\  - \\frac{- L + a \\operatorname{sin}\\left(q_{4}\\right)}{L r_{1}}, \\  \\frac{L + a \\operatorname{sin}\\left(q_{4}\\right)}{L r_{1}}, \\  - J_{1} \\left(C_{52} \\ddot{q}_{5} + C_{62} \\ddot{q}_{6}\\right)\\right)$"
      ],
      "text/plain": [
       "⎛J₁⋅(-L + a⋅sin(q₄))⋅q₅̈   J₁⋅(L + a⋅sin(q₄))⋅q₆̈  -(-L + a⋅sin(q₄))   L + a⋅s\n",
       "⎜────────────────────── - ─────────────────────, ──────────────────, ─────────\n",
       "⎝         L⋅r₁                     L⋅r₁                 L⋅r₁              L⋅r₁\n",
       "\n",
       "in(q₄)                       ⎞\n",
       "────, -J₁⋅(C₅₂⋅q₅̈ + C₆₂⋅q₆̈)⎟\n",
       "                           ⎠"
      ]
     },
     "execution_count": 82,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# C_kr \n",
    "C52, C62 = sm.symbols('C_52, C_62')\n",
    "C_52 = soldict[q5.diff()].diff(u2)\n",
    "C_62 = soldict[q6.diff()].diff(u2)\n",
    "Fstar_2_G1_I = -J1 * (N_w_A1.dot(q5.diff().diff() * A1.y + q5.diff()*(N_w_A1.dot(A1.x)*A1.z - N_w_A1.dot(A1.z)*A1.x)) + C_52 * (N_w_A1.dot(A1.y).diff() + q5.diff().diff())) \\\n",
    "               -J1 * (N_w_A1.dot(q6.diff().diff() * A1.y + q6.diff()*(N_w_A1.dot(A1.x)*A1.z - N_w_A1.dot(A1.z)*A1.x)) + C_62 * (N_w_A1.dot(A1.y).diff() + q6.diff().diff()))   # B1 \\ B2\n",
    "\n",
    "Fstar_2_G1_I, C_52, C_62, Fstar_2_G1_I.subs({-C_52: -C52, -C_62: -C62}).simplify()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### $\\rightarrow (F_r^*)_G = (F_r^*)_{GR} + (F_r^*)_{GI}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - C_{51} J_{1} \\ddot{q}_{5} - C_{61} J_{1} \\ddot{q}_{6} - I_{1_z} \\alpha^{A1}$"
      ],
      "text/plain": [
       "-C₅₁⋅J₁⋅q₅̈ - C₆₁⋅J₁⋅q₆̈ - I_{1_z}⋅α__A1"
      ]
     },
     "execution_count": 83,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Fstar_1_G1 = Fstar_1_G1_R + Fstar_1_G1_I\n",
    "Fstar_1_G1.subs({N_w_A1.dt(N).subs(soldict).dot(A1.z): alpha__A1}).subs({-C_51: -C51, -C_61: -C61}).simplify()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - C_{52} J_{1} \\ddot{q}_{5} - C_{62} J_{1} \\ddot{q}_{6} - \\frac{I_{1_z} \\alpha^{A1} \\operatorname{sin}\\left(q_{4}\\right)}{L} - M_{1} \\dot{u}_{2}$"
      ],
      "text/plain": [
       "                         I_{1_z}⋅α__A1⋅sin(q₄)        \n",
       "-C₅₂⋅J₁⋅q₅̈ - C₆₂⋅J₁⋅q₆̈ - ───────────────────── - M₁⋅u₂̇\n",
       "                                   L                  "
      ]
     },
     "execution_count": 84,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Fstar_2_G1 = Fstar_2_G1_R + Fstar_2_G1_I\n",
    "Fstar_2_G1.subs({N_w_A1.dt(N).subs(soldict).dot(A1.z): alpha__A1}).subs({-C_52: -C52, -C_62: -C62}).simplify()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### $\\text{Gyrostat}\\ G_2$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### $\\rightarrow {F_G}^* = -m_G {a^G}^*$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle -  M_{2} \\left(- u_{1} u_{2} \\operatorname{sin}\\left(q_{4}\\right) + \\operatorname{cos}\\left(q_{4}\\right) \\dot{u}_{2} - \\frac{l u^{2}_{2} \\operatorname{sin}^{2}\\left(q_{4}\\right)}{L^{2}}\\right)\\mathbf{\\hat{a_2}_x} -  M_{2} \\left(\\frac{l u_{1} u_{2} \\operatorname{cos}\\left(q_{4}\\right)}{L} + \\frac{l \\operatorname{sin}\\left(q_{4}\\right) \\dot{u}_{2}}{L} + \\frac{u^{2}_{2} \\operatorname{sin}\\left(q_{4}\\right) \\operatorname{cos}\\left(q_{4}\\right)}{L}\\right)\\mathbf{\\hat{a_2}_y}$"
      ],
      "text/plain": [
       "    ⎛                                  2    2    ⎞             ⎛\n",
       "    ⎜                              l⋅u₂ ⋅sin (q₄)⎟             ⎜l⋅u₁⋅u₂⋅cos(q₄\n",
       "-M₂⋅⎜-u₁⋅u₂⋅sin(q₄) + cos(q₄)⋅u₂̇ - ──────────────⎟ a_2_x + -M₂⋅⎜─────────────\n",
       "    ⎜                                     2      ⎟             ⎝       L\n",
       "    ⎝                                    L       ⎠\n",
       "\n",
       "                     2                ⎞\n",
       ")   l⋅sin(q₄)⋅u₂̇   u₂ ⋅sin(q₄)⋅cos(q₄)⎟\n",
       "── + ──────────── + ───────────────────⎟ a_2_y\n",
       "         L                  L         ⎠\n"
      ]
     },
     "execution_count": 85,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Fstar_G2 = -M2 * N_a_C\n",
    "Fstar_G2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### $\\rightarrow {T_G}^* \\overset{\\Delta}{=} -[\\alpha_A \\cdot I_G + {\\omega_r}^A \\times (I_G \\cdot {\\omega_r}^A)]$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle -  I_{2_z} \\left(\\frac{u_{1} u_{2} \\operatorname{cos}\\left(q_{4}\\right)}{L} + \\frac{\\operatorname{sin}\\left(q_{4}\\right) \\dot{u}_{2}}{L}\\right)\\mathbf{\\hat{a_2}_z}$"
      ],
      "text/plain": [
       "         ⎛u₁⋅u₂⋅cos(q₄)   sin(q₄)⋅u₂̇⎞\n",
       "-I_{2_z}⋅⎜───────────── + ──────────⎟ a_2_z\n",
       "         ⎝      L             L     ⎠"
      ]
     },
     "execution_count": 86,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Tstar_G2 = -(N_aa_A2.dot(I2) + me.cross(N_w_A2, I2.dot(N_w_A2)))\n",
    "Tstar_G2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### $\\rightarrow ({F_r}^*)_{GR} = {V_r}^G \\cdot {F_G}^* + {\\omega_r}^A \\cdot {T_G}^*$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOCAYAAAAWo42rAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA0UlEQVQoFXWR0Q2CQBBEhVgAwQ60BLUCsQO0A+3BP/4MJWgLWgIdGCmBDiB2gG+OW0PIuckyO3Ozu5cj6vt+ZlEURenrFlyRJVojLTIjwht+BZ86ABNA2p66ib14BhMzee0Dqukm7ozggawlTOIFzzTdjBlCNzGJaqoii/1dBvr/m2pi6s+te2y3Lb/V48NQvdBE6woZbFurO9rKJOA0bXhHDBW5DBhtYmXP88C0CRjXaLW2OiPFHaEDczNTa+2RPEmb6+ND3RcMdoUtfAd3f+wLrOY/XeyA6s8AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle 0$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 87,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Fstar_1_G2_R = V_1_C.dot(Fstar_G2) + w_1_A2.dot(Tstar_G2).subs(soldict)\n",
    "Fstar_1_G2_R.subs({N_w_A2.dt(N).subs(soldict).dot(A2.z): alpha__A2})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle - \\frac{I_{2_z} \\alpha^{A2} \\operatorname{sin}\\left(q_{4}\\right)}{L} - M_{2} \\left(- u_{1} u_{2} \\operatorname{sin}\\left(q_{4}\\right) + \\operatorname{cos}\\left(q_{4}\\right) \\dot{u}_{2} - \\frac{l u^{2}_{2} \\operatorname{sin}^{2}\\left(q_{4}\\right)}{L^{2}}\\right) \\operatorname{cos}\\left(q_{4}\\right) - \\frac{M_{2} l \\left(\\frac{l u_{1} u_{2} \\operatorname{cos}\\left(q_{4}\\right)}{L} + \\frac{l \\operatorname{sin}\\left(q_{4}\\right) \\dot{u}_{2}}{L} + \\frac{u^{2}_{2} \\operatorname{sin}\\left(q_{4}\\right) \\operatorname{cos}\\left(q_{4}\\right)}{L}\\right) \\operatorname{sin}\\left(q_{4}\\right)}{L}$"
      ],
      "text/plain": [
       "                                                                              \n",
       "                                                                              \n",
       "                             ⎛                                  2    2    ⎞   \n",
       "  I_{2_z}⋅α__A2⋅sin(q₄)      ⎜                              l⋅u₂ ⋅sin (q₄)⎟   \n",
       "- ───────────────────── - M₂⋅⎜-u₁⋅u₂⋅sin(q₄) + cos(q₄)⋅u₂̇ - ──────────────⎟⋅c\n",
       "            L                ⎜                                     2      ⎟   \n",
       "                             ⎝                                    L       ⎠   \n",
       "\n",
       "             ⎛                                   2                ⎞        \n",
       "             ⎜l⋅u₁⋅u₂⋅cos(q₄)   l⋅sin(q₄)⋅u₂̇   u₂ ⋅sin(q₄)⋅cos(q₄)⎟        \n",
       "        M₂⋅l⋅⎜─────────────── + ──────────── + ───────────────────⎟⋅sin(q₄)\n",
       "             ⎝       L               L                  L         ⎠        \n",
       "os(q₄) - ───────────────────────────────────────────────────────────────────\n",
       "                                         L                                 \n",
       "                                                                           "
      ]
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Fstar_2_G2_R = V_2_C.dot(Fstar_G2) + w_2_A1.dot(Tstar_G2).subs(soldict)\n",
    "Fstar_2_G2_R.subs({N_w_A2.dt(N).subs(soldict).dot(A2.z): alpha__A2})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### $\\rightarrow (F_r^*)_{GI} = -J\\{\\omega_r^A \\cdot [\\ddot{q_k} g_1 + \\dot{q_k} (\\omega_3^A g_2 - \\omega_2^A g_3)] + C_{kr} (\\dot{\\omega}_1^A + \\ddot{q_k}) \\} \\qquad (r=1,...,n-m)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\text{Here}, \\{\\omega_1^A: \\omega_2^A,\\ \\omega_2^A: \\omega_3^A,\\ \\omega_3^A: \\omega_1^A\\}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### $\\rightarrow \\dot{q_k} = \\sum_{s = 1}^{n - m} C_{ks} u_s + D_k \\quad \\text{(Generalized Speeds)}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### $\\omega_i^A \\overset{\\Delta}{=} \\omega^A \\cdot \\hat{g}_i \\quad (i = 1, 2, 3)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( 0, \\  0, \\  0\\right)$"
      ],
      "text/plain": [
       "(0, 0, 0)"
      ]
     },
     "execution_count": 89,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# C_kr\n",
    "C71, C81 = sm.symbols('C_71, C_81')\n",
    "C_71 = soldict[q7.diff()].diff(u1)\n",
    "C_81 = soldict[q8.diff()].diff(u1)\n",
    "Fstar_1_G2_I = -J2 * (N_w_A2.dot(q7.diff().diff() * A2.y + q7.diff()*(N_w_A2.dot(A2.x)*A2.z - N_w_A2.dot(A2.z)*A2.x)) + C_71 * (N_w_A2.dot(A2.y).diff() + q7.diff().diff())) \\\n",
    "               -J2 * (N_w_A2.dot(q8.diff().diff() * A2.y + q8.diff()*(N_w_A2.dot(A2.x)*A2.z - N_w_A2.dot(A2.z)*A2.x)) + C_81 * (N_w_A2.dot(A2.y).diff() + q8.diff().diff()))   # B1 \\ B2\n",
    "\n",
    "Fstar_1_G2_I, C_71, C_81, # Fstar_1_G2_I.subs({-C_71: -C71, -C_81: -C81}).simplify()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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AojusbTM67UE3i5Nzpowdz98nw59NYOh7FsJnedduMW6yBbdVQc0Dqt97W7HhdjdGYKweFHNQvrnDPxUfFf1gwsmqouZ/Y3u7qfIc6Iuzh5VQBCatHBTf+xFMHZXD2fOh7mGmLzChmX3yq3b5YMdBBy9hUCkNB0oMpEUHVe8vWqoeq9X+pvI/KMZwxT59Rx4EFXP4Onl8jMd82uA54kU6zhucY/BHeE8X53r1yaYPI17o8BVlFOmEiaLS4eFpgr6KXAUcZ8hw7fCFGgQP+sOSdDDmRdoVejFVXWRMP55DQM/gEiZBikmPY4N0ViJQjnwCDp+PdMXn+9T7v30yocQScknJJJWf5Ou+a+VfVthFXS4Tt76RPDazS2qbcfuDLsYV10VptisAkQ45+pmmMq7EELuylU2hZ312ZQmbQpspuaTye/miAYX47uSeOQQ0Of8T+85H+00G9c32QpIVDp6HLDMPYdysbTNS9mCWsTzX2BEdz99nkUgWkSHvWQie6V27yrjJktJOCmls8p3HN2v8bt0JZ2ZjAwTwNbHydshK8Mjm77oJ/pwHd3d3FxH5Swm/Kw6esFhqiVhtMLnBaREcBYqjU6FTqVXmD9Xp+tXFJdg8HM1cjFQunJWiOBxErhj840o/+s0knDIo1r+Vf+VQUhoy/EVx3zkIKtIeVI//yqHAY5S3negOU9W/bL1V2UlyUf1emaTyu+Br8lUtpzwm0zhzcQYeOhT42C41pChcsnRY5RbV3wZbrY/iwXalgsxUmUBKNGa3Ky180QbOrdr8Q8/806H1HQRvDvtDQDJbxV7Qc7XVq5s56IiGbUYFKOGxuR2vsHOaW+HucbNDaa8lF7Uz2ZZtBZ945z0N/3xLEvMtz3dB+MbX/aig+thmFhX0fl8oP36zMn/ASRoD36gcqQRPLFDopRMr5caix2KcF4rDd7BicMApM+q7OLfdqeXE32HxEu/BCar4yi/Rh4vKIxt2Dzx4pD8AwNXpdOojNiRPbaF8KAqrm4LXTfec65VSRhimHJdDOwK5GGEYyv+cSw7cV5+REROf18qrGhAllQF50d7YAN0u2mNp7rFerkzgfapcUjJJ5XfhV+OrUei5nqurmBrZx3i0XeqVU64O1/RkhF0Zq59V5m1XqmhMtylQS8kllV/n6P6pqSustGC18WNd1fcC76LXbQSctlsE1rIXADBG95rA2WbUEWmOzaHzwzlkUufoHE8eN/uU81pyOey40Tsb3i+KwQqnxOQFKqL1WHReKWZ3VGdQPu290sU8gXbL+YPucUDBE84z4rnDUxHkioH5SvU5pu8mPjpe4p+joP6ji908pawzAOafChfVefFQMYIihMT728X+8qt7LOEkjtfLFPPKZ8sfK4Mir3p0qCIwACPwj9spqyQuohH+y1ckst0EY1ELSkMGyKLvDK1aneaD6qKwo+s36e31uehjrt6Olova6ZVJKj+BXydfqvdWF8tGjx5slzokOECHO/VENHrtykT9LDkXHduVEo1wM1om1E7JJZVfZ6X21MYX/xUO29ZFl//IMUmtHgVQI+CHfSIgueXO1dp0IHRKNHrtBYVUpvedl4uO6Nhm1MEaLZe5ZFJn5xxPws7jZoeiXkMuNzRusB0ck1QuopggUt7/nccaqQ3+QcZOExbC/E33OENqThA9sxoI5xnvitZvXqWPDqKPk+1JJNB8julTY9F9oYt34uig+reEF3ox1IkZv1OfPVJlFIKwxkqvUkHumxz0l0kxq0r4T5RDOwKdGEnpURKMEkYAD/hFV/DQR1J65sWbckahcJOWrcb2ThJ3yoT+C/M55JKSSSr/ShSZfO16Ge9VpzoS1FfbpQ5siuROHc7Uk5RdGayf/eyeIndpmQBiSi6p/Jog+nRFecw/vJK7hthhH/agm4cFb0HGl5bLIHuwYD+PSnpp+YCLZTRcO5aWy63IhN1bg7ad9YiCc6375gO/qS6+CxxeNWdXgyY+A2gt7t9otDvnI9/sXFPCzeAlWbJl9S9dQ/4pGuX/wYNvvvkm/leNQ+Xn8NBOEUxvXfGHkuP1TDlmeunccuaSGIk2usJ/RqMC3TKUs/VtS5lYZrOJsZPQkvLtbHTljKX6aP0cL8ilZAJHKbmk8sf3yjVvAYEtdfMW8FuqD0vJxfZgHoktJR+4s4zGy2gpudyKTNQPFlSElVe6n/R9XtBi1Xfr1kals1AgbqPsdbKpLI44zvPqc6CNV4wVaop3dlyNPgP7FvFSn9idwzmw2bqmsneq8zNOL5QHJfpAiXZmCAgHI2AEjIARMAJGwAgYASNgBIyAETACRqAdAfkOWK3Goognuu9bedVOoJJa0GKbZG0nEkWUxsIXjmLiHNDkThOVYYUUDqNZF/QUfMBf7Otr3cdFIXO3NdrpVfB5c3ipX/iskCurMLOCyuL0+pXtje8VNd5m1XQhI2AEjIARMAJGwAgYASNgBIyAETACRuDMCLDSC0dUdAJNweK5KuNEawtxxVZXfq1OwU9wQukeJ8nkhT2iQV/ZNsnuuEBPMc8cG/RA1yzt1Doy/mE0XuObrNcUHr141UtnP+HIg+6QgG4+faQ/eEIdjIARMAJGwAgYASNgBIyAETACRsAIGAEj0IuAnBr4EFiBVdtqVqS/KiqTT/hC6SnHGPS6FuHEH1cbczA9Z0DBBw6zmvNL6W9y+C3K4OB6rfsaDaXFFV697ajcmmEKXhf1kdVryAOZsbKOLYXR0Ud6r3xVljIpvHrbUf22gH5Ae0gIdXB6hZ/YFHN0KjuoPEvFHIyAETACVwjIPjy4SiwSlIex4mDFIUYr62eQbZe6UHe6ETg+An12ZWjv5rBDtjdDUXf5MyIw57iN+HnsRSQc3yoCc44b0Vpq3v1ZgT/OjWrgXK640uiie5xNnMWU2paIT6LLHxH8FcpvOpyUdB3os65IC4cXq4OaK4Rw1sFnDr9sq4Pmt4qrAZrR6Zdqp1qvvC/wafJGfvTRlFiWleRoU72+LX6j8RJdjr76WXHVyYWM448JzoJXRjuV7pa38BQdqWVixk1Y6ZVR7rqIGO38qL0u7RQjYASMwD0Csh28hFoPqZyKke3SVARd3wicA4E57JDtzTl0xb3cHwIee/uTiTnaLwJzvO86ehecIKLfXH31Umk/VdJxGJH2TFdcFdVBsjM5rNZR/ejI6iyoMqxywjkS22L7Zc1voWdWLkVnUg6/OPhq/VR9nC+PdXGQPyHVzn2pxt8KH7UcpdMPtkw2HW21ch0PU/BCrmAXnV6csxadaDQ3F16pdmirGeAjqQPNSjw/1AUoF3UGoTkYASNgBIyAETACRsAIGAEjYASMgBEwAidHQD4CHC9/6cJ5VQ2sToqOpWo6zqTfqwmZ9zhZuvwRweEELxm0PlK5Kl81vot+VA/L7+VX5eGJKzq3IgthdZbyozMs1U6st0Y8BS/6+UuB00UxWx1ZrRfDXHil2ontVWPkEHxX1cSM+7c4vd4VBasevIy6LmIEjIARMAJGwAgYgW0Q0ESMyY+DETACB0LA4/ZAwjKru0Fg43HDiiPet/+NgIifmFZ1HoVs5X2vK/oXSMNJwiqoqiMqlG38wZnR5dSKDiXa7Qxqg3KvqwWUFlYskaZ76uPQKfnTfS6/JZ2CPtsLyz6JTpnf1k5RZ61oCl6sLEOuX6kfHGf1ARhFxrnXVeKn9C75lngUdZt49bYT22vE+KuadBtFrh7D6jCcXlGBUWYHI2AEjIARMAJGwAjsGgFNuJiz8J9nzglxMAJG4AAIeNweQEhmcXcI7GDc4NjhkPawza7g5welfav7uMqpFTflP1MGjqacY01o56M2QqKDowOnCedJsfKoFpTGLygGh5fiqkOmLKd05g1fKu7kWXlX/Bb0qFM65JQGD6z0uqKlvGQ7qrdoEA+j8VJdMEAOyOxnXTi/qiu9lHQfirI1+SoN/JN4DWkntqcYnpor7irZnbd/PlJWq2J0VnGGETACRsAIGAEjYAQ2RIBJlS4mdRwsy+W5zIbycNNGIAcBj9sclFzGCNQR2HrcqP1fdbHFkQPOCThE+IGpK4dPyC3+UEe3OKI+1H3OOxpnRqtzBZKiweHqT3SL44uykSYrxDhD7GrVmdKrgXO5Yp1qerhX/T5+cbj9oDIRg7jaqM0B09vOVcMLJYjXsXjRz+ikRM4498CXc9CqK9um4pXVTgMeeImr2BpZnY+PlfMWpxeKQmDpl4MRuEihGdAYtLiEFAWPBqVc3miojIARMAJzIWC7MxeS56EjnWHJfXxPnafjJ+mpbcJtCtrjdnu5emxtL4OhHGw9btT+oO8/lQ8OEcXxsHtWD8V/VrV2X2Vxrl10dR54rzycVmynGxNwXJVOmyoB0e3lt2iX+iHoOaw2U9zm+OtsJ9ZfKy74zsZL5cEh+oZiX5EL8icv4FeUw/nYKl+lI6dOvIr6yXYCA8Wfos5FcXQ4VrNb71UWhxfhHU6vWJGOOBgBlCkuX2UfL3uwo7fX6BgBI2AEFkHAdmcRWM9AlANrWX7vcGMI2CbcmEDr3fG4reOx6pPH1qpwz9nYIcaN9AufAiu22BKJs4uA4yW1EotyrOKhbLajhkqZ4bnK/dQsO5Lf+MuDTXI8t7bTVjCRhuOIa7UgLPjuf6qruYIemb6GEeWNkW8NL9FIttPSafRn6Cqv6N/67yM1+kYXdO3YAAWHgIB0Ihopf0xYJ4yAEVgFAdudVWC+mUakL/wHL55LejP9ckf+j4Btwv+xuJU7j9t9SNJjax9yyOXiYOOGszZ5P9fO3FQfko4sleGQdA6aZztlXJiTC1OqHCuL2n5Vcgy/L0Sra/VbVzsp/mr56n/bKrJamYUeWKHFKq4/CvrvKcaBGR1wc+GVaqfsntrGeYVOJHWorHR/E51e4UwvklAqvJIORiAiwGAmtO1Vvs/xXyNgBIzAvAjY7syL561T+0wToLAy+dY7euL+2SbcnvA9bvchU4+tfcghl4vDjBu9lzl3a0rAGcKKrLB1bgqhal3x9UH1Od4P4VdlWWnE2MGpx+H5F121FWx6bm0ntrf3WPzjF+p0Lik/W74q24lXqp0GTqwcLLdLNvL6HqPT682Du7u7ixpFsV4ozu5EH3XnHR8B6QJeXPZUPzh+b9wDI2AEjoCA7c4RpGQejcB6CNgmrIe1WzoXAh5b55L30Xor/cRZgW+iazXV0bpkfkciIB3g/DTOFRu88k918HHhnHzwqGif1TwkNPdvFtmOTogA2xvDYXV9fZfOUO5zXTjIPtGFkcLjzRLIP/RsYyUgHIyAEchCwHYnCyYXMgKnQcA24TSidkdXRsBja2XA3Vw+Avp+xMHhb8h8yG625ERfAqvywjbRhwVCcX8rGQ4nR0DKFfUgZy8xXviwrFMxSw+/VByXRA49bO7kyLv7RuC8CNjunFf27rkRaEPANqENFacZgekIeGxNx9AUjIAR2DcCsnMsxGEbajiqKaz0UiKH2bMyhxU7PrhcIJw8xD3Uved5SWdQJPSGgGJxwFzcx0xe8NAX5V5RSIFyhC8Knbt/8l8jYATOjoDtztk1wP03AnUEbBPqePjJCMyFgMfWXEiajhEwAntFIC7iCb6t4PQqOP1R8Wd75dp8rYpAUBI5pVIrvZ6KK/SGgDMrrvC6qG55r3R+AaJ81j0rwjgzLDrIdOtgBIzAyRGw3Tm5Arj7RqCBgG1CAxA/GoGZEPDYmglIkzECRmC3CODc/1N+h3AW2MMKmxz0xZlewRBW0n17PgR69/lLR17qCj8lq/hdRWeiA6yJGOWresW2R+rTjoMRMAJGAARsd6wHRsAIVBGwTaii4XsjMB8CHlvzYWlKRsAI7BOBT8UWC21CKJ1eckCwqgdP2Jifg7yn5r+HR6DinPpXT2c4sL76CwroDJ7UuNWxWZVVXvHcuGaen42AETg5ArY7J1cAd98INBCwTWgA4kcjMBMCHlszAWkyRsAI7BYB2Tl+8ZFQ/hjCo/vn8i8rcL5Twa91dTkwysK+uUkE4kZSWrEAAALMSURBVD7/1q2N0ouv1OvmWV+s4uo8C051SoUrEMMJhpMs+euQRXlHRsAI3DYCtju3LV/3zggMRcA2YShiLm8E8hDw2MrDyaWMgBE4LgL8yN7PVX9WudKLPhXOCZxd8dBxkh3OhQBLAdGFK4eU0nCKcpXbGJXGWV5cTUeYkq6DyrOkmjY+vM51ihEwAidFwHbnpIJ3t41ABwK2CR3AONkITETAY2sigK5uBIzAfhGQr4HFOPgmcHyVobnSiwwKsNrrtS6v9iqhut0byfmxeveDrujAuiiNM95iIB1nFaHmNdUzeazaal0ZRoUYVIayOM0+1L11KwLj2AicEAHZANudE8rdXTYCXQjYJnQh43QjMA0Bj61p+Lm2ETACh0IgrvKqHsV0eXB3d3fVCxlHflnvV8U1D9lVQScYgUwEpEvB4aU4nBmnGCcah+DXFDKTnIsZASNgBJII2O4kIXIBI3AqBGwTTiVud3ZFBDy2VgTbTRkBI9CKgOwQq7zYffZE97UFNm0rvSDyha7fVJgVX3ZKgIjDaASkQzi8+PUEzoqLK8Y418tO1dGouqIRMAJ9CNju9KHjPCNwPgRsE84nc/d4HQQ8ttbB2a0YASOQRICdan+XTao5vKjV6vRSwTe6Xiufij57CaQcpiDAykG2MhGXQTqG48vBCBgBI7AEArY7S6BqmkbguAjYJhxXduZ83wh4bO1bPubOCNw8AvIr4Ldip2Lrj+u1bm+MqBSVOa/JK3IiKI6NgBEwAkbACBgBI2AEjIARMAJGwAgYASNgBDZFQL4qfqDjleLOxVoP+zhURc5feqb4ZV855xkBI2AEjIARMAJGwAgYASNgBIyAETACRsAIGIE1EJCfiqOTXun6uK+9XqcXFUXoE0XsjeRgMAcjYASMgBEwAkbACBgBI2AEjIARMAJGwAgYASOwCQLyT8Vfgv9Y91fneFWZ+h+D++i9HuoaZQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle \\left( \\frac{J_{2} \\left(- L \\operatorname{cos}\\left(q_{4}\\right) + b \\operatorname{sin}\\left(q_{4}\\right)\\right) \\ddot{q}_{7}}{L r_{2}} - \\frac{J_{2} \\left(L \\operatorname{cos}\\left(q_{4}\\right) + b \\operatorname{sin}\\left(q_{4}\\right)\\right) \\ddot{q}_{8}}{L r_{2}}, \\  - \\frac{- L \\operatorname{cos}\\left(q_{4}\\right) + b \\operatorname{sin}\\left(q_{4}\\right)}{L r_{2}}, \\  \\frac{L \\operatorname{cos}\\left(q_{4}\\right) + b \\operatorname{sin}\\left(q_{4}\\right)}{L r_{2}}, \\  - J_{2} \\left(C_{72} \\ddot{q}_{7} + C_{82} \\ddot{q}_{8}\\right)\\right)$"
      ],
      "text/plain": [
       "⎛J₂⋅(-L⋅cos(q₄) + b⋅sin(q₄))⋅q₇̈   J₂⋅(L⋅cos(q₄) + b⋅sin(q₄))⋅q₈̈  -(-L⋅cos(q₄\n",
       "⎜────────────────────────────── - ─────────────────────────────, ─────────────\n",
       "⎝             L⋅r₂                             L⋅r₂                         L⋅\n",
       "\n",
       ") + b⋅sin(q₄))   L⋅cos(q₄) + b⋅sin(q₄)                       ⎞\n",
       "─────────────, ─────────────────────, -J₂⋅(C₇₂⋅q₇̈ + C₈₂⋅q₈̈)⎟\n",
       "r₂                      L⋅r₂                               ⎠"
      ]
     },
     "execution_count": 90,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# C_kr \n",
    "C72, C82 = sm.symbols('C_72, C_82')\n",
    "C_72 = soldict[q7.diff()].diff(u2)\n",
    "C_82 = soldict[q8.diff()].diff(u2)\n",
    "Fstar_2_G2_I = -J2 * (N_w_A2.dot(q7.diff().diff() * A2.y + q7.diff()*(N_w_A2.dot(A2.x)*A2.z - N_w_A2.dot(A2.z)*A2.x)) + C_72 * (N_w_A2.dot(A2.y).diff() + q7.diff().diff())) \\\n",
    "               -J2 * (N_w_A2.dot(q8.diff().diff() * A2.y + q8.diff()*(N_w_A2.dot(A2.x)*A2.z - N_w_A2.dot(A2.z)*A2.x)) + C_82 * (N_w_A2.dot(A2.y).diff() + q8.diff().diff()))   # B1 \\ B2\n",
    "\n",
    "Fstar_2_G2_I, C_72, C_82, Fstar_2_G2_I.subs({-C_72: -C72, -C_82: -C82}).simplify()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### $\\rightarrow (F_r^*)_G = (F_r^*)_{GR} + (F_r^*)_{GI}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {},
   "outputs": [],
   "source": [
    "Fstar_1_G2 = Fstar_1_G2_R + Fstar_1_G2_I\n",
    "# Fstar_1_G2.subs({N_w_A2.dt(N).subs(soldict).dot(A2.z): alpha__A2}) # .subs({-C_71: -C71, -C_81: -C81}).simplify()\n",
    "Fstar_1_G2 = 0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\text{Here}, \\{a_1^C: a_2^C,\\ a_2^C: a_3^C,\\ a_3^C: a_1^C\\}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle - C_{72} J_{2} \\ddot{q}_{7} - C_{82} J_{2} \\ddot{q}_{8} - \\frac{I_{2_z} \\alpha^{A2} \\operatorname{sin}\\left(q_{4}\\right)}{L} - M_{2} a^{C}_{3} \\operatorname{cos}\\left(q_{4}\\right) - \\frac{M_{2} a^{C}_{1} l \\operatorname{sin}\\left(q_{4}\\right)}{L}$"
      ],
      "text/plain": [
       "                         I_{2_z}⋅α__A2⋅sin(q₄)                       M₂⋅a_1__C\n",
       "-C₇₂⋅J₂⋅q₇̈ - C₈₂⋅J₂⋅q₈̈ - ───────────────────── - M₂⋅a_3__C⋅cos(q₄) - ───────\n",
       "                                   L                                          \n",
       "\n",
       "⋅l⋅sin(q₄)\n",
       "────────────\n",
       "L         "
      ]
     },
     "execution_count": 92,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Fstar_2_G2 = Fstar_2_G2_R + Fstar_2_G2_I\n",
    "Fstar_2_G2.subs({N_w_A2.dt(N).subs(soldict).dot(A2.z): alpha__A2}).subs({N_v_C.dt(N).subs(soldict).dot(A2.x): a_3__C}).subs({N_v_C.dt(N).subs(soldict).dot(A2.y): a_1__C}).subs({-C_72: -C72, -C_82: -C82}).simplify()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### $\\text{Variable-Mass Particle}\\ P$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### $\\rightarrow {F_G}^* = -m_G {a^G}^*$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle -  \\left(- u_{1} u_{2} \\operatorname{sin}\\left(q_{4}\\right) + \\operatorname{cos}\\left(q_{4}\\right) \\dot{u}_{2}\\right) m\\mathbf{\\hat{a_2}_x} -  \\frac{m u^{2}_{2} \\operatorname{sin}\\left(q_{4}\\right) \\operatorname{cos}\\left(q_{4}\\right)}{L}\\mathbf{\\hat{a_2}_y}$"
      ],
      "text/plain": [
       "                                              2\n",
       "                                         -m⋅u₂ ⋅sin(q₄)⋅cos(q₄)\n",
       "-(-u₁⋅u₂⋅sin(q₄) + cos(q₄)⋅u₂̇)⋅m a_2_x + ─────────────────────── a_2_y\n",
       "                                                    L"
      ]
     },
     "execution_count": 93,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Fstar_P = -m * N_a_P\n",
    "Fstar_P"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### $\\rightarrow ({F_r}^*)_{GR} = {V_r}^G \\cdot {F_G}^*$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOCAYAAAAWo42rAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA0UlEQVQoFXWR0Q2CQBBEhVgAwQ60BLUCsQO0A+3BP/4MJWgLWgIdGCmBDiB2gG+OW0PIuckyO3Ozu5cj6vt+ZlEURenrFlyRJVojLTIjwht+BZ86ABNA2p66ib14BhMzee0Dqukm7ozggawlTOIFzzTdjBlCNzGJaqoii/1dBvr/m2pi6s+te2y3Lb/V48NQvdBE6woZbFurO9rKJOA0bXhHDBW5DBhtYmXP88C0CRjXaLW2OiPFHaEDczNTa+2RPEmb6+ND3RcMdoUtfAd3f+wLrOY/XeyA6s8AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle 0$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 94,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Fstar_1_P_R = V_1_P.dot(Fstar_P)\n",
    "Fstar_1_P_R"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - \\left(- u_{1} u_{2} \\operatorname{sin}\\left(q_{4}\\right) + \\operatorname{cos}\\left(q_{4}\\right) \\dot{u}_{2}\\right) m \\operatorname{cos}\\left(q_{4}\\right)$"
      ],
      "text/plain": [
       "-(-u₁⋅u₂⋅sin(q₄) + cos(q₄)⋅u₂̇)⋅m⋅cos(q₄)"
      ]
     },
     "execution_count": 95,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Fstar_2_P_R = V_2_P.dot(Fstar_P) \n",
    "Fstar_2_P_R"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### $\\rightarrow (F_r^*)_G = (F_r^*)_{GR}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOCAYAAAAWo42rAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA0UlEQVQoFXWR0Q2CQBBEhVgAwQ60BLUCsQO0A+3BP/4MJWgLWgIdGCmBDiB2gG+OW0PIuckyO3Ozu5cj6vt+ZlEURenrFlyRJVojLTIjwht+BZ86ABNA2p66ib14BhMzee0Dqukm7ozggawlTOIFzzTdjBlCNzGJaqoii/1dBvr/m2pi6s+te2y3Lb/V48NQvdBE6woZbFurO9rKJOA0bXhHDBW5DBhtYmXP88C0CRjXaLW2OiPFHaEDczNTa+2RPEmb6+ND3RcMdoUtfAd3f+wLrOY/XeyA6s8AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle 0$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 96,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Fstar_1_P = Fstar_1_P_R\n",
    "Fstar_1_P"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\text{Here}, \\{a_1^P: a_2^P,\\ a_2^P: a_3^P,\\ a_3^P: a_1^P\\}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - a^{P}_{3} m \\operatorname{cos}\\left(q_{4}\\right)$"
      ],
      "text/plain": [
       "-a_3__P⋅m⋅cos(q₄)"
      ]
     },
     "execution_count": 97,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Fstar_2_P = Fstar_2_P_R\n",
    "Fstar_2_P.subs({N_v_P.dt(N).subs(soldict).dot(A2.x): a_3__P}).subs({N_v_P.dt(N).subs(soldict).dot(A2.y): a_1__P}).simplify()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### $\\text{Generalized Inertia Forces}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\rightarrow F_r^* = (F_r^*)_{G_1} + (F_r^*)_{G_2} + (F_r^*)_{P} \\quad (r = 1, 2)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\frac{- I_{1_z} r_{1} \\left(L \\dot{u}_{1} + u_{1} u_{2} \\operatorname{cos}\\left(q_{4}\\right) + \\operatorname{sin}\\left(q_{4}\\right) \\dot{u}_{2}\\right) + J_{1} L a \\left(\\frac{- a \\dot{u}_{1} + \\dot{u}_{2} - \\frac{a u_{2} \\operatorname{cos}\\left(q_{4}\\right) \\dot{q}_{4}}{L} - \\frac{a \\operatorname{sin}\\left(q_{4}\\right) \\dot{u}_{2}}{L}}{r_{1}} - \\frac{a \\dot{u}_{1} + \\dot{u}_{2} + \\frac{a u_{2} \\operatorname{cos}\\left(q_{4}\\right) \\dot{q}_{4}}{L} + \\frac{a \\operatorname{sin}\\left(q_{4}\\right) \\dot{u}_{2}}{L}}{r_{1}}\\right)}{L r_{1}}$"
      ],
      "text/plain": [
       "                                                         ⎛             a⋅u₂⋅co\n",
       "                                                         ⎜-a⋅u₁̇ + u₂̇ - ─────\n",
       "                                                         ⎜                    \n",
       "-I_{1_z}⋅r₁⋅(L⋅u₁̇ + u₁⋅u₂⋅cos(q₄) + sin(q₄)⋅u₂̇) + J₁⋅L⋅a⋅⎜──────────────────\n",
       "                                                         ⎝                    \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                        L⋅r₁  \n",
       "\n",
       "s(q₄)⋅q₄̇   a⋅sin(q₄)⋅u₂̇               a⋅u₂⋅cos(q₄)⋅q₄̇   a⋅sin(q₄)⋅u₂̇⎞\n",
       "────────── - ────────────   a⋅u₁̇ + u₂̇ + ─────────────── + ────────────⎟\n",
       "L               L                            L               L      ⎟\n",
       "───────────────────────── - ──────────────────────────────────────────⎟\n",
       " r₁                                           r₁                    ⎠\n",
       "─────────────────────────────────────────────────────────────────────\n",
       "                                                                     "
      ]
     },
     "execution_count": 98,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Fstar_1 = Fstar_1_G1 + Fstar_1_G2 + Fstar_1_P\n",
    "Fstar_1.subs(soldict).simplify()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\frac{- J_{1} L r_{2} \\left(\\frac{\\left(L - a \\operatorname{sin}\\left(q_{4}\\right)\\right) \\left(- a \\dot{u}_{1} + \\dot{u}_{2} - \\frac{a u_{2} \\operatorname{cos}\\left(q_{4}\\right) \\dot{q}_{4}}{L} - \\frac{a \\operatorname{sin}\\left(q_{4}\\right) \\dot{u}_{2}}{L}\\right)}{r_{1}} + \\frac{\\left(L + a \\operatorname{sin}\\left(q_{4}\\right)\\right) \\left(a \\dot{u}_{1} + \\dot{u}_{2} + \\frac{a u_{2} \\operatorname{cos}\\left(q_{4}\\right) \\dot{q}_{4}}{L} + \\frac{a \\operatorname{sin}\\left(q_{4}\\right) \\dot{u}_{2}}{L}\\right)}{r_{1}}\\right) - J_{2} L r_{1} \\left(\\left(L \\operatorname{cos}\\left(q_{4}\\right) - b \\operatorname{sin}\\left(q_{4}\\right)\\right) \\left(\\frac{\\left(- \\operatorname{sin}\\left(q_{4}\\right) \\dot{q}_{4} - \\frac{b \\operatorname{cos}\\left(q_{4}\\right) \\dot{q}_{4}}{L}\\right) u_{2}}{r_{2}} + \\frac{\\left(\\operatorname{cos}\\left(q_{4}\\right) - \\frac{b \\operatorname{sin}\\left(q_{4}\\right)}{L}\\right) \\dot{u}_{2}}{r_{2}}\\right) + \\left(L \\operatorname{cos}\\left(q_{4}\\right) + b \\operatorname{sin}\\left(q_{4}\\right)\\right) \\left(\\frac{\\left(- \\operatorname{sin}\\left(q_{4}\\right) \\dot{q}_{4} + \\frac{b \\operatorname{cos}\\left(q_{4}\\right) \\dot{q}_{4}}{L}\\right) u_{2}}{r_{2}} + \\frac{\\left(\\operatorname{cos}\\left(q_{4}\\right) + \\frac{b \\operatorname{sin}\\left(q_{4}\\right)}{L}\\right) \\dot{u}_{2}}{r_{2}}\\right)\\right) + \\frac{L^{2} r_{1} r_{2} \\left(- 2 M_{1} \\dot{u}_{2} + \\left(u_{1} u_{2} \\operatorname{sin}\\left(2 q_{4}\\right) - \\operatorname{cos}\\left(2 q_{4}\\right) \\dot{u}_{2} - \\dot{u}_{2}\\right) m\\right)}{2} + \\frac{r_{1} r_{2} \\left(- 2 I_{1_z} L \\operatorname{sin}\\left(q_{4}\\right) \\dot{u}_{1} - I_{1_z} u_{1} u_{2} \\operatorname{sin}\\left(2 q_{4}\\right) + I_{1_z} \\operatorname{cos}\\left(2 q_{4}\\right) \\dot{u}_{2} - I_{1_z} \\dot{u}_{2} - I_{2_z} u_{1} u_{2} \\operatorname{sin}\\left(2 q_{4}\\right) + I_{2_z} \\operatorname{cos}\\left(2 q_{4}\\right) \\dot{u}_{2} - I_{2_z} \\dot{u}_{2} + L^{2} M_{2} u_{1} u_{2} \\operatorname{sin}\\left(2 q_{4}\\right) - L^{2} M_{2} \\operatorname{cos}\\left(2 q_{4}\\right) \\dot{u}_{2} - L^{2} M_{2} \\dot{u}_{2} - M_{2} l^{2} u_{1} u_{2} \\operatorname{sin}\\left(2 q_{4}\\right) + M_{2} l^{2} \\operatorname{cos}\\left(2 q_{4}\\right) \\dot{u}_{2} - M_{2} l^{2} \\dot{u}_{2}\\right)}{2}}{L^{2} r_{1} r_{2}}$"
      ],
      "text/plain": [
       "          ⎛                ⎛             a⋅u₂⋅cos(q₄)⋅q₄̇   a⋅sin(q₄)⋅u₂̇⎞    \n",
       "          ⎜(L - a⋅sin(q₄))⋅⎜-a⋅u₁̇ + u₂̇ - ─────────────── - ────────────⎟   (\n",
       "          ⎜                ⎝                    L               L      ⎠      \n",
       "- J₁⋅L⋅r₂⋅⎜───────────────────────────────────────────────────────────── + ───\n",
       "          ⎝                              r₁                                   \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "               ⎛            a⋅u₂⋅cos(q₄)⋅q₄̇   a⋅sin(q₄)⋅u₂̇⎞⎞           ⎛    \n",
       "L + a⋅sin(q₄))⋅⎜a⋅u₁̇ + u₂̇ + ─────────────── + ────────────⎟⎟           ⎜    \n",
       "             ⎝                   L               L      ⎠⎟           ⎜        \n",
       "─────────────────────────────────────────────────────────⎟ - J₂⋅L⋅r₁⋅⎜(L⋅cos(q\n",
       "                          r₁                             ⎠           ⎝        \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "                    ⎛⎛              b⋅cos(q₄)⋅q₄̇⎞      ⎛          b⋅sin(q₄)⎞ \n",
       "                    ⎜⎜-sin(q₄)⋅q₄̇ - ────────────⎟⋅u₂   ⎜cos(q₄) - ─────────⎟⋅\n",
       "                ⎜⎝                   L      ⎠      ⎝              L    ⎠   ⎟  \n",
       "₄) - b⋅sin(q₄))⋅⎜─────────────────────────────── + ────────────────────────⎟ +\n",
       "                ⎝               r₂                            r₂           ⎠  \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "  ⎞                           ⎛⎛              b⋅cos(q₄)⋅q₄̇⎞      ⎛          b\n",
       "u₂̇⎟                           ⎜⎜-sin(q₄)⋅q₄̇ + ────────────⎟⋅u₂   ⎜cos(q₄) + \n",
       "                         ⎜⎝                   L      ⎠      ⎝              L  \n",
       " (L⋅cos(q₄) + b⋅sin(q₄))⋅⎜─────────────────────────────── + ──────────────────\n",
       "                         ⎝               r₂                            r₂     \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "⋅sin(q₄)⎞   ⎞⎞                                                                \n",
       "─────────⎟⋅u₂̇⎟⎟    2                                                         \n",
       "  ⎠   ⎟⎟   L ⋅r₁⋅r₂⋅(-2⋅M₁⋅u₂̇ + (u₁⋅u₂⋅sin(2⋅q₄) - cos(2⋅q₄)⋅u₂̇ - u₂̇)⋅m)   \n",
       "──────⎟⎟ + ───────────────────────────────────────────────────────────── + ───\n",
       "      ⎠⎠                                 2                                    \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "            2                                                                 \n",
       "           L ⋅r₁⋅r₂                                                           \n",
       "\n",
       "                                                                              \n",
       "           ⎛                                                                  \n",
       "r₁⋅r₂⋅⎝-2⋅I_{1_z}⋅L⋅sin(q₄)⋅u₁̇ - I_{1_z}⋅u₁⋅u₂⋅sin(2⋅q₄) + I_{1_z}⋅cos(2⋅q₄)⋅\n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "                                                                              \n",
       "u₂̇ - I_{1_z}⋅u₂̇ - I_{2_z}⋅u₁⋅u₂⋅sin(2⋅q₄) + I_{2_z}⋅cos(2⋅q₄)⋅u₂̇ - I_{2_z}⋅\n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                    2                         \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "       2                       2                    2             2           \n",
       "u₂̇ + L ⋅M₂⋅u₁⋅u₂⋅sin(2⋅q₄) - L ⋅M₂⋅cos(2⋅q₄)⋅u₂̇ - L ⋅M₂⋅u₂̇ - M₂⋅l ⋅u₁⋅u₂⋅si\n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "                                    \n",
       "            2                    2   ⎞\n",
       "n(2⋅q₄) + M₂⋅l ⋅cos(2⋅q₄)⋅u₂̇ - M₂⋅l ⋅u₂̇⎠\n",
       "──────────────────────────────\n",
       "                              \n",
       "──────────────────────────────\n",
       "                              \n",
       "                              "
      ]
     },
     "execution_count": 99,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Fstar_2 = Fstar_2_G1 + Fstar_2_G2 + Fstar_2_P\n",
    "Fstar_2.subs(soldict).simplify()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\mathbf{\\text{Velocity of material ejected at}\\ P\\ \\text{relative to}\\ A_2 \\rightarrow -C(t)g_3^{'};\\ C(t) \\rightarrow \\text{positive}}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEAAAAAUCAYAAAA9djs/AAAACXBIWXMAAA7EAAAOxAGVKw4bAAADGklEQVRYCdWY0VEbMRCGDeMCknQAdIBDBYEOwqSD0EE8efMbEzoI6SDQAaSDJB0EOiDuwPk+Id3oDtncgZzEO7PWabXS7b/aXem8tVgsRptAs9nsAjtvaKc17d1ahwMw8gVGvoeP4Hlm8BVj5/AuspM+YOJa39B3juteI3PdKrRdZZVsEYz7QPc2io7pN4zsjv4n2iv4V9RZ2qArYNe6gXfgl/AI+aNz1etD1SIgGmuYulMT+vnON7Yg/0zH6NjjWWArCZ19+GeuVJLl40OeazrgBy8W/A4GFsFrGGOHNBe0YTeV/Usa13g5YAzrfdhwXwo+vuuO9mt8Dg1zTJt3mczQv4anPdbLpg1/fHYNwEB3XQBW6MseJhj2pkEg5hgROnCXZ1NnwvMJbJrc0tcZa6NnOwDLNFZqQN13y78AmsN5TuvAFjHu7qsj+I+twcqdGg54G23S6MEE2HMmeba/6UxOBdLUWhuNMUAve84OCTVzPe3iq2hdMnilsb4PnudK9M+i3HQQ8F5sVUvr+zxSj+YLbOTI1pQn1wsdoDHm3VNJAx6AKi3Gu4wWHZWcJyBBWAMcE4ipZFQosw50KVyKmJfuBNYfdXWejhtENVIghH4E8tjLD9DLwbubHp+CNwqO4Eu4FSGdRXWYDnfuiPaMRn2LqE4YRDUcoPelVAvue51fjFPvtCN+TT8Aoe0WUYGWyFqho1Y5qTSvKKtyEcIYwXsL9H5v+DYUxw4QnHaNpi/431HZuhKOUVrrQLpVmjKrbpbuuldrj+EmBXjWJp2uIz15Usp4jbaumLqTKg5goREvEEzzQmWQL/HWt/SEYEwDnScQ9VxH0J4MOiGFdcm56vqt8B0uXsJYX+e4hh9QrmvKqRtsquYAFv2rBADBC8aa4Y3RqHGnBdkiZO66jg6Opd9c2LZbmhvSAYBgPA3yT2qPRp1SovT5bOi3onFc0v6fZRG8Oy9Y/1/IzX2w+3HQAuuuWxd0XHPsb2IEJPDgaBPOmLcloTZZR0yNY1ojwc9r60KgTYyAXp/RgPQSZSE1XWwli6V0yLh1YfoHfMs6F0GxMxAAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle -  C\\mathbf{\\hat{a_2}_x}$"
      ],
      "text/plain": [
       "-C a_2_x"
      ]
     },
     "execution_count": 100,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C = me.dynamicsymbols('C')\n",
    "C_t = -C*A2.x\n",
    "C_t"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### $\\text{Generalized Thrust}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\rightarrow F_r^{\\prime} \\triangleq \\sum_{i=1}^{N} \\mathbf{V}_{r}^{P i} \\cdot \\mathbf{C}^{P i} \\dot{m}_{i} \\quad (r=1, \\ldots, k)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOCAYAAAAWo42rAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA0UlEQVQoFXWR0Q2CQBBEhVgAwQ60BLUCsQO0A+3BP/4MJWgLWgIdGCmBDiB2gG+OW0PIuckyO3Ozu5cj6vt+ZlEURenrFlyRJVojLTIjwht+BZ86ABNA2p66ib14BhMzee0Dqukm7ozggawlTOIFzzTdjBlCNzGJaqoii/1dBvr/m2pi6s+te2y3Lb/V48NQvdBE6woZbFurO9rKJOA0bXhHDBW5DBhtYmXP88C0CRjXaLW2OiPFHaEDczNTa+2RPEmb6+ND3RcMdoUtfAd3f+wLrOY/XeyA6s8AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle 0$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 101,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Fprime_1 = V_1_P.dot(C_t)*m.diff()\n",
    "Fprime_1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - C \\operatorname{cos}\\left(q_{4}\\right) \\dot{m}$"
      ],
      "text/plain": [
       "-C⋅cos(q₄)⋅ṁ"
      ]
     },
     "execution_count": 102,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Fprime_2 = V_2_P.dot(C_t)*m.diff()\n",
    "Fprime_2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### $\\text{Extended Kane's Equations for Variable-Mass Systems}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\rightarrow F_r + F_r^* + F_r^{\\prime} = 0 \\quad (r = 1,..., k)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here, $F_r = 0$ $\\rightarrow$ no forces contributing to generalized active forces"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\frac{- I_{1_z} r_{1} \\left(L \\dot{u}_{1} + u_{1} u_{2} \\operatorname{cos}\\left(q_{4}\\right) + \\operatorname{sin}\\left(q_{4}\\right) \\dot{u}_{2}\\right) + J_{1} L a \\left(\\frac{- a \\dot{u}_{1} + \\dot{u}_{2} - \\frac{a u_{2} \\operatorname{cos}\\left(q_{4}\\right) \\dot{q}_{4}}{L} - \\frac{a \\operatorname{sin}\\left(q_{4}\\right) \\dot{u}_{2}}{L}}{r_{1}} - \\frac{a \\dot{u}_{1} + \\dot{u}_{2} + \\frac{a u_{2} \\operatorname{cos}\\left(q_{4}\\right) \\dot{q}_{4}}{L} + \\frac{a \\operatorname{sin}\\left(q_{4}\\right) \\dot{u}_{2}}{L}}{r_{1}}\\right)}{L r_{1}}$"
      ],
      "text/plain": [
       "                                                         ⎛             a⋅u₂⋅co\n",
       "                                                         ⎜-a⋅u₁̇ + u₂̇ - ─────\n",
       "                                                         ⎜                    \n",
       "-I_{1_z}⋅r₁⋅(L⋅u₁̇ + u₁⋅u₂⋅cos(q₄) + sin(q₄)⋅u₂̇) + J₁⋅L⋅a⋅⎜──────────────────\n",
       "                                                         ⎝                    \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                        L⋅r₁  \n",
       "\n",
       "s(q₄)⋅q₄̇   a⋅sin(q₄)⋅u₂̇               a⋅u₂⋅cos(q₄)⋅q₄̇   a⋅sin(q₄)⋅u₂̇⎞\n",
       "────────── - ────────────   a⋅u₁̇ + u₂̇ + ─────────────── + ────────────⎟\n",
       "L               L                            L               L      ⎟\n",
       "───────────────────────── - ──────────────────────────────────────────⎟\n",
       " r₁                                           r₁                    ⎠\n",
       "─────────────────────────────────────────────────────────────────────\n",
       "                                                                     "
      ]
     },
     "execution_count": 103,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "kane_1 = Fstar_1.simplify() + Fprime_1.simplify()\n",
    "kane_1.subs(soldict).simplify()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\frac{- J_{1} L r_{2} \\left(\\frac{\\left(L - a \\operatorname{sin}\\left(q_{4}\\right)\\right) \\left(- a \\dot{u}_{1} + \\dot{u}_{2} - \\frac{a u_{2} \\operatorname{cos}\\left(q_{4}\\right) \\dot{q}_{4}}{L} - \\frac{a \\operatorname{sin}\\left(q_{4}\\right) \\dot{u}_{2}}{L}\\right)}{r_{1}} + \\frac{\\left(L + a \\operatorname{sin}\\left(q_{4}\\right)\\right) \\left(a \\dot{u}_{1} + \\dot{u}_{2} + \\frac{a u_{2} \\operatorname{cos}\\left(q_{4}\\right) \\dot{q}_{4}}{L} + \\frac{a \\operatorname{sin}\\left(q_{4}\\right) \\dot{u}_{2}}{L}\\right)}{r_{1}}\\right) - J_{2} L r_{1} \\left(\\left(L \\operatorname{cos}\\left(q_{4}\\right) - b \\operatorname{sin}\\left(q_{4}\\right)\\right) \\left(\\frac{\\left(- \\operatorname{sin}\\left(q_{4}\\right) \\dot{q}_{4} - \\frac{b \\operatorname{cos}\\left(q_{4}\\right) \\dot{q}_{4}}{L}\\right) u_{2}}{r_{2}} + \\frac{\\left(\\operatorname{cos}\\left(q_{4}\\right) - \\frac{b \\operatorname{sin}\\left(q_{4}\\right)}{L}\\right) \\dot{u}_{2}}{r_{2}}\\right) + \\left(L \\operatorname{cos}\\left(q_{4}\\right) + b \\operatorname{sin}\\left(q_{4}\\right)\\right) \\left(\\frac{\\left(- \\operatorname{sin}\\left(q_{4}\\right) \\dot{q}_{4} + \\frac{b \\operatorname{cos}\\left(q_{4}\\right) \\dot{q}_{4}}{L}\\right) u_{2}}{r_{2}} + \\frac{\\left(\\operatorname{cos}\\left(q_{4}\\right) + \\frac{b \\operatorname{sin}\\left(q_{4}\\right)}{L}\\right) \\dot{u}_{2}}{r_{2}}\\right)\\right) + \\frac{L^{2} r_{1} r_{2} \\left(- 2 M_{1} \\dot{u}_{2} + \\left(u_{1} u_{2} \\operatorname{sin}\\left(2 q_{4}\\right) - \\operatorname{cos}\\left(2 q_{4}\\right) \\dot{u}_{2} - \\dot{u}_{2}\\right) m - 2 C \\operatorname{cos}\\left(q_{4}\\right) \\dot{m}\\right)}{2} + \\frac{r_{1} r_{2} \\left(- 2 I_{1_z} L \\operatorname{sin}\\left(q_{4}\\right) \\dot{u}_{1} - I_{1_z} u_{1} u_{2} \\operatorname{sin}\\left(2 q_{4}\\right) + I_{1_z} \\operatorname{cos}\\left(2 q_{4}\\right) \\dot{u}_{2} - I_{1_z} \\dot{u}_{2} - I_{2_z} u_{1} u_{2} \\operatorname{sin}\\left(2 q_{4}\\right) + I_{2_z} \\operatorname{cos}\\left(2 q_{4}\\right) \\dot{u}_{2} - I_{2_z} \\dot{u}_{2} + L^{2} M_{2} u_{1} u_{2} \\operatorname{sin}\\left(2 q_{4}\\right) - L^{2} M_{2} \\operatorname{cos}\\left(2 q_{4}\\right) \\dot{u}_{2} - L^{2} M_{2} \\dot{u}_{2} - M_{2} l^{2} u_{1} u_{2} \\operatorname{sin}\\left(2 q_{4}\\right) + M_{2} l^{2} \\operatorname{cos}\\left(2 q_{4}\\right) \\dot{u}_{2} - M_{2} l^{2} \\dot{u}_{2}\\right)}{2}}{L^{2} r_{1} r_{2}}$"
      ],
      "text/plain": [
       "          ⎛                ⎛             a⋅u₂⋅cos(q₄)⋅q₄̇   a⋅sin(q₄)⋅u₂̇⎞    \n",
       "          ⎜(L - a⋅sin(q₄))⋅⎜-a⋅u₁̇ + u₂̇ - ─────────────── - ────────────⎟   (\n",
       "          ⎜                ⎝                    L               L      ⎠      \n",
       "- J₁⋅L⋅r₂⋅⎜───────────────────────────────────────────────────────────── + ───\n",
       "          ⎝                              r₁                                   \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "               ⎛            a⋅u₂⋅cos(q₄)⋅q₄̇   a⋅sin(q₄)⋅u₂̇⎞⎞           ⎛    \n",
       "L + a⋅sin(q₄))⋅⎜a⋅u₁̇ + u₂̇ + ─────────────── + ────────────⎟⎟           ⎜    \n",
       "             ⎝                   L               L      ⎠⎟           ⎜        \n",
       "─────────────────────────────────────────────────────────⎟ - J₂⋅L⋅r₁⋅⎜(L⋅cos(q\n",
       "                          r₁                             ⎠           ⎝        \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "                    ⎛⎛              b⋅cos(q₄)⋅q₄̇⎞      ⎛          b⋅sin(q₄)⎞ \n",
       "                    ⎜⎜-sin(q₄)⋅q₄̇ - ────────────⎟⋅u₂   ⎜cos(q₄) - ─────────⎟⋅\n",
       "                ⎜⎝                   L      ⎠      ⎝              L    ⎠   ⎟  \n",
       "₄) - b⋅sin(q₄))⋅⎜─────────────────────────────── + ────────────────────────⎟ +\n",
       "                ⎝               r₂                            r₂           ⎠  \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "  ⎞                           ⎛⎛              b⋅cos(q₄)⋅q₄̇⎞      ⎛          b\n",
       "u₂̇⎟                           ⎜⎜-sin(q₄)⋅q₄̇ + ────────────⎟⋅u₂   ⎜cos(q₄) + \n",
       "                         ⎜⎝                   L      ⎠      ⎝              L  \n",
       " (L⋅cos(q₄) + b⋅sin(q₄))⋅⎜─────────────────────────────── + ──────────────────\n",
       "                         ⎝               r₂                            r₂     \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "⋅sin(q₄)⎞   ⎞⎞                                                                \n",
       "─────────⎟⋅u₂̇⎟⎟    2                                                         \n",
       "  ⎠   ⎟⎟   L ⋅r₁⋅r₂⋅(-2⋅M₁⋅u₂̇ + (u₁⋅u₂⋅sin(2⋅q₄) - cos(2⋅q₄)⋅u₂̇ - u₂̇)⋅m - 2\n",
       "──────⎟⎟ + ───────────────────────────────────────────────────────────────────\n",
       "      ⎠⎠                                         2                            \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                    2                                                         \n",
       "                   L ⋅r₁⋅r₂                                                   \n",
       "\n",
       "                                                                              \n",
       "                           ⎛                                                  \n",
       "⋅C⋅cos(q₄)⋅ṁ)   r₁⋅r₂⋅⎝-2⋅I_{1_z}⋅L⋅sin(q₄)⋅u₁̇ - I_{1_z}⋅u₁⋅u₂⋅sin(2⋅q₄) + I\n",
       "────────── + ─────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "                                                                              \n",
       "_{1_z}⋅cos(2⋅q₄)⋅u₂̇ - I_{1_z}⋅u₂̇ - I_{2_z}⋅u₁⋅u₂⋅sin(2⋅q₄) + I_{2_z}⋅cos(2⋅q\n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                    2         \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "                       2                       2                    2         \n",
       "₄)⋅u₂̇ - I_{2_z}⋅u₂̇ + L ⋅M₂⋅u₁⋅u₂⋅sin(2⋅q₄) - L ⋅M₂⋅cos(2⋅q₄)⋅u₂̇ - L ⋅M₂⋅u₂̇\n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "                                                    \n",
       "    2                       2                    2   ⎞\n",
       " - M₂⋅l ⋅u₁⋅u₂⋅sin(2⋅q₄) + M₂⋅l ⋅cos(2⋅q₄)⋅u₂̇ - M₂⋅l ⋅u₂̇⎠\n",
       "──────────────────────────────────────────────\n",
       "                                              \n",
       "──────────────────────────────────────────────\n",
       "                                              \n",
       "                                              "
      ]
     },
     "execution_count": 104,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "kane_2 = Fstar_2 + Fprime_2\n",
    "kane_2.subs(soldict).simplify()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle - \\frac{\\left(I_{1_z} r_{1}^{2} + 2 J_{1} a^{2}\\right) \\left(L \\dot{u}_{1} + u_{1} u_{2} \\operatorname{cos}\\left(q_{4}\\right) + \\operatorname{sin}\\left(q_{4}\\right) \\dot{u}_{2}\\right)}{L r_{1}^{2}} = 0$"
      ],
      "text/plain": [
       " ⎛          2         2⎞                                         \n",
       "-⎝I_{1_z}⋅r₁  + 2⋅J₁⋅a ⎠⋅(L⋅u₁̇ + u₁⋅u₂⋅cos(q₄) + sin(q₄)⋅u₂̇)     \n",
       "───────────────────────────────────────────────────────────── = 0\n",
       "                                2                                \n",
       "                            L⋅r₁                                 "
      ]
     },
     "execution_count": 105,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "kane_1_eq = sm.Eq(kane_1.simplify().subs(soldict).simplify().subs(u).simplify(), 0)\n",
    "kane_1_eq"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 107,
   "metadata": {},
   "outputs": [],
   "source": [
    "kane_2_eq = sm.Eq(kane_2.simplify().subs(soldict).simplify().subs(u).simplify(), 0)\n",
    "# kane_2_eq"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.8"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}