{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Ackermann Kinematics model\n",
    "\n",
    "### Jacobians"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0 & 0 & \\cos{\\left(\\theta \\right)} & - v \\sin{\\left(\\theta \\right)}\\\\0 & 0 & \\sin{\\left(\\theta \\right)} & v \\cos{\\left(\\theta \\right)}\\\\0 & 0 & 0 & 0\\\\0 & 0 & \\frac{\\tan{\\left(\\delta \\right)}}{L} & 0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[0, 0,   cos(theta), -v*sin(theta)],\n",
       "[0, 0,   sin(theta),  v*cos(theta)],\n",
       "[0, 0,            0,             0],\n",
       "[0, 0, tan(delta)/L,             0]])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0 & 0\\\\0 & 0\\\\1 & 0\\\\0 & \\frac{v \\left(\\tan^{2}{\\left(\\delta \\right)} + 1\\right)}{L}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[0,                       0],\n",
       "[0,                       0],\n",
       "[1,                       0],\n",
       "[0, v*(tan(delta)**2 + 1)/L]])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x, y, theta, v, delta, L, a = sp.symbols(\"x y theta v delta L a\")\n",
    "\n",
    "gs = sp.Matrix([[sp.cos(theta) * v], [sp.sin(theta) * v], [a], [v * sp.tan(delta) / L]])\n",
    "\n",
    "X = sp.Matrix([x, y, v, theta])\n",
    "\n",
    "# A\n",
    "A = gs.jacobian(X)\n",
    "\n",
    "U = sp.Matrix([a, delta])\n",
    "\n",
    "# B\n",
    "B = gs.jacobian(U)\n",
    "display(A)\n",
    "display(B)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Discretized and Linearized model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 0 & dt \\cos{\\left(\\theta \\right)} & - dt v \\sin{\\left(\\theta \\right)}\\\\0 & 1 & dt \\sin{\\left(\\theta \\right)} & dt v \\cos{\\left(\\theta \\right)}\\\\0 & 0 & 1 & 0\\\\0 & 0 & \\frac{dt \\tan{\\left(\\delta \\right)}}{L} & 1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[1, 0,   dt*cos(theta), -dt*v*sin(theta)],\n",
       "[0, 1,   dt*sin(theta),  dt*v*cos(theta)],\n",
       "[0, 0,               1,                0],\n",
       "[0, 0, dt*tan(delta)/L,                1]])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0 & 0\\\\0 & 0\\\\dt & 0\\\\0 & \\frac{dt v \\left(\\tan^{2}{\\left(\\delta \\right)} + 1\\right)}{L}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[ 0,                          0],\n",
       "[ 0,                          0],\n",
       "[dt,                          0],\n",
       "[ 0, dt*v*(tan(delta)**2 + 1)/L]])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}dt \\theta v \\sin{\\left(\\theta \\right)}\\\\- dt \\theta v \\cos{\\left(\\theta \\right)}\\\\0\\\\- \\frac{\\delta dt v \\left(\\tan^{2}{\\left(\\delta \\right)} + 1\\right)}{L}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[            dt*theta*v*sin(theta)],\n",
       "[           -dt*theta*v*cos(theta)],\n",
       "[                                0],\n",
       "[-delta*dt*v*(tan(delta)**2 + 1)/L]])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "DT = sp.symbols(\"dt\")\n",
    "\n",
    "display(sp.eye(4) + A * DT)\n",
    "display(B * DT)\n",
    "display(DT * (gs - A * X - B * U))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# ADD DELAY (for real time implementation)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It is necessary to take *actuation latency* into account: so instead of using the actual state as estimated, the delay factored in using the kinematic model\n",
    "\n",
    "Starting State is :\n",
    "\n",
    "* $x_{delay} = 0.0 + v * dt$\n",
    "* $y_{delay} = 0.0$\n",
    "* $psi_{delay} = 0.0 + w * dt$\n",
    "* $cte_{delay} = cte + v * sin(epsi) * dt$\n",
    "* $epsi_{delay} = epsi - w * dt$\n",
    "\n",
    "Note that the starting position and heading is always 0; this is becouse the path is parametrized to **vehicle reference frame**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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