From f6a734f3d48907b4d30ceb3d86cf3143a46eb297 Mon Sep 17 00:00:00 2001 From: Takayuki Murooka Date: Tue, 16 Apr 2024 19:22:34 +0900 Subject: [PATCH] fix(path_smoother): fix mathjax visualization error (#6821) Signed-off-by: Takayuki Murooka --- planning/path_smoother/docs/eb.md | 100 +++++++++++++++--------------- 1 file changed, 50 insertions(+), 50 deletions(-) diff --git a/planning/path_smoother/docs/eb.md b/planning/path_smoother/docs/eb.md index 2c232cfa4de81..d62a6de5e3f54 100644 --- a/planning/path_smoother/docs/eb.md +++ b/planning/path_smoother/docs/eb.md @@ -78,65 +78,65 @@ We formulate a quadratic problem minimizing the diagonal length of the rhombus o ![eb](../media/eb.svg){: style="width:600px"} -Assuming that $k$'th point is $\boldsymbol{p}_k = (x_k, y_k)$, the objective function is as follows. +Assuming that $k$'th point is $\mathbf{p}_k = (x_k, y_k)$, the objective function is as follows. $$ \begin{align} -\ J & = \min \sum_{k=1}^{n-2} ||(\boldsymbol{p}_{k+1} - \boldsymbol{p}_{k}) - (\boldsymbol{p}_{k} - \boldsymbol{p}_{k-1})||^2 \\ -\ & = \min \sum_{k=1}^{n-2} ||\boldsymbol{p}_{k+1} - 2 \boldsymbol{p}_{k} + \boldsymbol{p}_{k-1}||^2 \\ -\ & = \min \sum_{k=1}^{n-2} \{(x_{k+1} - x_k + x_{k-1})^2 + (y_{k+1} - y_k + y_{k-1})^2\} \\ +\ J & = \min \sum_{k=1}^{n-2} ||(\mathbf{p}_{k+1} - \mathbf{p}_{k}) - (\mathbf{p}_{k} - \mathbf{p}_{k-1})||^2 \\\ +\ & = \min \sum_{k=1}^{n-2} ||\mathbf{p}_{k+1} - 2 \mathbf{p}_{k} + \mathbf{p}_{k-1}||^2 \\\ +\ & = \min \sum_{k=1}^{n-2} \{(x_{k+1} - x_k + x_{k-1})^2 + (y_{k+1} - y_k + y_{k-1})^2\} \\\ \ & = \min \begin{pmatrix} - \ x_0 \\ - \ x_1 \\ - \ x_2 \\ - \vdots \\ - \ x_{n-3}\\ - \ x_{n-2} \\ - \ x_{n-1} \\ - \ y_0 \\ - \ y_1 \\ - \ y_2 \\ - \vdots \\ - \ y_{n-3}\\ - \ y_{n-2} \\ - \ y_{n-1} \\ + \ x_0 \\\ + \ x_1 \\\ + \ x_2 \\\ + \vdots \\\ + \ x_{n-3}\\\ + \ x_{n-2} \\\ + \ x_{n-1} \\\ + \ y_0 \\\ + \ y_1 \\\ + \ y_2 \\\ + \vdots \\\ + \ y_{n-3}\\\ + \ y_{n-2} \\\ + \ y_{n-1} \\\ \end{pmatrix}^T \begin{pmatrix} - 1 & -2 & 1 & 0 & \dots& \\ - -2 & 5 & -4 & 1 & 0 &\dots \\ - 1 & -4 & 6 & -4 & 1 & \\ - 0 & 1 & -4 & 6 & -4 & \\ - \vdots & 0 & \ddots&\ddots& \ddots \\ - & \vdots & & & \\ - & & & 1 & -4 & 6 & -4 & 1 \\ - & & & & 1 & -4 & 5 & -2 \\ - & & & & & 1 & -2 & 1& \\ - & & & & & & & &1 & -2 & 1 & 0 & \dots& \\ - & & & & & & & &-2 & 5 & -4 & 1 & 0 &\dots \\ - & & & & & & & &1 & -4 & 6 & -4 & 1 & \\ - & & & & & & & &0 & 1 & -4 & 6 & -4 & \\ - & & & & & & & &\vdots & 0 & \ddots&\ddots& \ddots \\ - & & & & & & & & & \vdots & & & \\ - & & & & & & & & & & & 1 & -4 & 6 & -4 & 1 \\ - & & & & & & & & & & & & 1 & -4 & 5 & -2 \\ - & & & & & & & & & & & & & 1 & -2 & 1& \\ + 1 & -2 & 1 & 0 & \dots& \\\ + -2 & 5 & -4 & 1 & 0 &\dots \\\ + 1 & -4 & 6 & -4 & 1 & \\\ + 0 & 1 & -4 & 6 & -4 & \\\ + \vdots & 0 & \ddots&\ddots& \ddots \\\ + & \vdots & & & \\\ + & & & 1 & -4 & 6 & -4 & 1 \\\ + & & & & 1 & -4 & 5 & -2 \\\ + & & & & & 1 & -2 & 1& \\\ + & & & & & & & &1 & -2 & 1 & 0 & \dots& \\\ + & & & & & & & &-2 & 5 & -4 & 1 & 0 &\dots \\\ + & & & & & & & &1 & -4 & 6 & -4 & 1 & \\\ + & & & & & & & &0 & 1 & -4 & 6 & -4 & \\\ + & & & & & & & &\vdots & 0 & \ddots&\ddots& \ddots \\\ + & & & & & & & & & \vdots & & & \\\ + & & & & & & & & & & & 1 & -4 & 6 & -4 & 1 \\\ + & & & & & & & & & & & & 1 & -4 & 5 & -2 \\\ + & & & & & & & & & & & & & 1 & -2 & 1& \\\ \end{pmatrix} \begin{pmatrix} - \ x_0 \\ - \ x_1 \\ - \ x_2 \\ - \vdots \\ - \ x_{n-3}\\ - \ x_{n-2} \\ - \ x_{n-1} \\ - \ y_0 \\ - \ y_1 \\ - \ y_2 \\ - \vdots \\ - \ y_{n-3}\\ - \ y_{n-2} \\ - \ y_{n-1} \\ + \ x_0 \\\ + \ x_1 \\\ + \ x_2 \\\ + \vdots \\\ + \ x_{n-3}\\\ + \ x_{n-2} \\\ + \ x_{n-1} \\\ + \ y_0 \\\ + \ y_1 \\\ + \ y_2 \\\ + \vdots \\\ + \ y_{n-3}\\\ + \ y_{n-2} \\\ + \ y_{n-1} \\\ \end{pmatrix} \end{align} $$