{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0 & 0 & - v \\sin{\\left(\\theta \\right)} & 0 & 0\\\\0 & 0 & v \\cos{\\left(\\theta \\right)} & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & - v \\cos{\\left(\\psi \\right)} & 0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[0, 0, -v*sin(theta),           0, 0],\n",
       "[0, 0,  v*cos(theta),           0, 0],\n",
       "[0, 0,             0,           0, 0],\n",
       "[0, 0,             0,           0, 0],\n",
       "[0, 0,             0, -v*cos(psi), 0]])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import sympy as sp\n",
    "\n",
    "x,y,theta,psi,cte,v,w = sp.symbols(\"x y theta psi cte v w\")\n",
    "\n",
    "gs = sp.Matrix([[ sp.cos(theta)*v],\n",
    "             [ sp.sin(theta)*v],\n",
    "             [w],\n",
    "             [w],\n",
    "             [ v*sp.sin(-psi)]])\n",
    "\n",
    "state = sp.Matrix([x,y,theta,psi,cte])\n",
    "\n",
    "gs.jacobian(state)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\cos{\\left(\\theta \\right)} & 0\\\\\\sin{\\left(\\theta \\right)} & 0\\\\0 & 1\\\\0 & 1\\\\- \\sin{\\left(\\psi \\right)} & 0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[cos(theta), 0],\n",
       "[sin(theta), 0],\n",
       "[         0, 1],\n",
       "[         0, 1],\n",
       "[ -sin(psi), 0]])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "state = sp.Matrix([v,w])\n",
    "\n",
    "gs.jacobian(state)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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  "language_info": {
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   "file_extension": ".py",
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