Fix markdown equations syntax (AtsushiSakai#535)

Author: ksvbka

PathTracking/model_predictive_speed_and_steer_control/Model_predictive_speed_and_steering_control.ipynb

@@ -2,16 +2,15 @@
  "cells": [
   {
    "cell_type": "markdown",
-   "metadata": {},
    "source": [
     "## Model predictive speed and steering control\n",
     "\n",
     "![Model predictive speed and steering control](https://github.com/AtsushiSakai/PythonRoboticsGifs/raw/master/PathTracking/model_predictive_speed_and_steer_control/animation.gif?raw=true)\n"
-   ]
+   ],
+   "metadata": {}
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
    "source": [
     "\n",
     "\n",
@@ -26,11 +25,11 @@
     "This code uses cvxpy as an optimization modeling tool.\n",
     "\n",
     "- [Welcome to CVXPY 1\\.0 — CVXPY 1\\.0\\.6 documentation](http://www.cvxpy.org/)"
-   ]
+   ],
+   "metadata": {}
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
    "source": [
     "### MPC modeling\n",
     "\n",
@@ -46,22 +45,22 @@
     "\n",
     "a: accellation, δ: steering angle\n",
     "\n"
-   ]
+   ],
+   "metadata": {}
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
    "source": [
     "The MPC cotroller minimize this cost function for path tracking:\n",
     "\n",
     "$$min\\ Q_f(z_{T,ref}-z_{T})^2+Q\\Sigma({z_{t,ref}-z_{t}})^2+R\\Sigma{u_t}^2+R_d\\Sigma({u_{t+1}-u_{t}})^2$$\n",
     "\n",
     "z_ref come from target path and speed."
-   ]
+   ],
+   "metadata": {}
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
    "source": [
     "subject to:\n",
     "\n",
@@ -89,20 +88,20 @@
     "\n",
     "$$u_{min} < u_t < u_{max}$$\n",
     "\n"
-   ]
+   ],
+   "metadata": {}
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
    "source": [
     "This is implemented at \n",
     "\n",
     "[PythonRobotics/model\\_predictive\\_speed\\_and\\_steer\\_control\\.py at f51a73f47cb922a12659f8ce2d544c347a2a8156 · AtsushiSakai/PythonRobotics](https://github.com/AtsushiSakai/PythonRobotics/blob/f51a73f47cb922a12659f8ce2d544c347a2a8156/PathTracking/model_predictive_speed_and_steer_control/model_predictive_speed_and_steer_control.py#L247-L301)"
-   ]
+   ],
+   "metadata": {}
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
    "source": [
     "### Vehicle model linearization\n",
     "\n",
@@ -120,21 +119,21 @@
     "\n",
     "\n",
     "\n"
-   ]
+   ],
+   "metadata": {}
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
    "source": [
     "ODE is\n",
     "\n",
     "$$ \\dot{z} =\\frac{\\partial }{\\partial z} z = f(z, u) = A'z+B'u$$\n",
     "\n"
-   ]
+   ],
+   "metadata": {}
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
    "source": [
     "where\n",
     "\n",
@@ -168,11 +167,11 @@
     "\\end{bmatrix}\n",
     "\\end{equation*}$\n",
     "\n"
-   ]
+   ],
+   "metadata": {}
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
    "source": [
     "$\\begin{equation*}\n",
     "B' =\n",
@@ -196,11 +195,11 @@
     "\\end{bmatrix}\n",
     "\\end{equation*}$\n",
     "\n"
-   ]
+   ],
+   "metadata": {}
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
    "source": [
     "You can get a discrete-time mode with Forward Euler Discretization with sampling time dt.\n",
     "\n",
@@ -210,19 +209,19 @@
     "$$z_{k+1}=z_k+(f(\\bar{z},\\bar{u})+A'z_k+B'u_k-A'\\bar{z}-B'\\bar{u})dt$$\n",
     "\n",
     "$$z_{k+1}=(I + dtA')z_k+(dtB')u_k + (f(\\bar{z},\\bar{u})-A'\\bar{z}-B'\\bar{u})dt$$\n"
-   ]
+   ],
+   "metadata": {}
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
    "source": [
     "So, \n",
     "\n",
     "$$z_{k+1}=Az_k+Bu_k +C$$\n",
     "\n",
     "where,\n",
     "\n",
-    "$\\begin{equation*}\n",
+    "$$\\begin{equation*}\n",
     "A = (I + dtA')\\\\\n",
     "=\n",
     "\\begin{bmatrix} \n",
@@ -231,14 +230,14 @@
     "0 & 0 & 1 & 0 \\\\\n",
     "0 & 0 &\\frac{tan(\\bar{\\delta})}{L}dt & 1 \\\\\n",
     "\\end{bmatrix}\n",
-    "\\end{equation*}$"
-   ]
+    "\\end{equation*}$$"
+   ],
+   "metadata": {}
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
    "source": [
-    "$\\begin{equation*}\n",
+    "$$\\begin{equation*}\n",
     "B = dtB'\\\\\n",
     "=\n",
     "\\begin{bmatrix} \n",
@@ -247,14 +246,14 @@
     "dt & 0 \\\\\n",
     "0 & \\frac{\\bar{v}}{Lcos^2(\\bar{\\delta})}dt \\\\\n",
     "\\end{bmatrix}\n",
-    "\\end{equation*}$"
-   ]
+    "\\end{equation*}$$"
+   ],
+   "metadata": {}
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
    "source": [
-    "$\\begin{equation*}\n",
+    "$$\\begin{equation*}\n",
     "C = (f(\\bar{z},\\bar{u})-A'\\bar{z}-B'\\bar{u})dt\\\\\n",
     "= dt(\n",
     "\\begin{bmatrix} \n",
@@ -285,28 +284,29 @@
     "0\\\\\n",
     "-\\frac{\\bar{v}\\bar{\\delta}}{Lcos^2(\\bar{\\delta})}dt\\\\\n",
     "\\end{bmatrix}\n",
-    "\\end{equation*}$"
-   ]
+    "\\end{equation*}$$"
+   ],
+   "metadata": {}
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
    "source": [
     "This equation is implemented at \n",
     "\n",
     "[PythonRobotics/model\\_predictive\\_speed\\_and\\_steer\\_control\\.py at eb6d1cbe6fc90c7be9210bf153b3a04f177cc138 · AtsushiSakai/PythonRobotics](https://github.com/AtsushiSakai/PythonRobotics/blob/eb6d1cbe6fc90c7be9210bf153b3a04f177cc138/PathTracking/model_predictive_speed_and_steer_control/model_predictive_speed_and_steer_control.py#L80-L102)\n"
-   ]
+   ],
+   "metadata": {}
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
    "source": [
     "### Reference\n",
     "\n",
     "- [Vehicle Dynamics and Control \\| Rajesh Rajamani \\| Springer](http://www.springer.com/us/book/9781461414322)\n",
     "\n",
     "- [MPC Course Material \\- MPC Lab @ UC\\-Berkeley](http://www.mpc.berkeley.edu/mpc-course-material)\n"
-   ]
+   ],
+   "metadata": {}
   }
  ],
  "metadata": {
@@ -330,4 +330,4 @@
  },
  "nbformat": 4,
  "nbformat_minor": 2
-}
+}