added notebook on numerical jacobian

notebooks/MPC_racecar_tracking.ipynb

@@ -1273,7 +1273,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.8.5"
+   "version": "3.7.6"
   }
  },
  "nbformat": 4,

notebooks/equations.ipynb

@@ -671,7 +671,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.8.5"
+   "version": "3.7.6"
   }
  },
  "nbformat": 4,

notebooks/numericalJacobian.ipynb

@@ -0,0 +1,127 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Compute the jacobian numerically\n",
+    "\n",
+    "link: --> http://www.maths.lth.se/na/courses/FMN081/FMN081-06/lecture7.pdf\n",
+    "\n",
+    "Often the Jacobian is not **analytically** available and it has to be computed numerically.\n",
+    "It can be computed column wise by finite differences:\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 12,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "\n",
+    "def f(x,u):\n",
+    "    \"\"\"\n",
+    "    :param x:\n",
+    "    :param u:\n",
+    "    \"\"\"\n",
+    "    xx = x[0]\n",
+    "    xy = x[1]\n",
+    "    v =  x[2]\n",
+    "    theta =x[3]\n",
+    "    \n",
+    "    a = u[0]\n",
+    "    delta = u[1]\n",
+    "    \n",
+    "    L=0.3\n",
+    "    \n",
+    "    #vector of ackerman equations\n",
+    "    return np.array([\n",
+    "        [np.cos(theta)*v],\n",
+    "        [np.sin(theta)*v],\n",
+    "        [a],\n",
+    "        [v*np.arctan(delta)/L]\n",
+    "        ])\n",
+    "\n",
+    "def J(f,x,u,epsilon=1e-6):\n",
+    "    \"\"\"\n",
+    "    :param f:\n",
+    "    :param x:\n",
+    "    :param u:\n",
+    "    \"\"\"\n",
+    "    \n",
+    "    \n",
+    "    x_minus = x-epsilon\n",
+    "    u_minus = u-epsilon\n",
+    "    \n",
+    "    x_plus = x+epsilon\n",
+    "    u_plus = u+epsilon\n",
+    "    \n",
+    "    # compute finite differences\n",
+    "    f_plus = f(x_plus, u_plus)\n",
+    "    f_minus = f(x_minus, u_minus)\n",
+    "    \n",
+    "    jac = (f_plus - f_minus)/2*epsilon\n",
+    "    \n",
+    "    return jac\n",
+    "    "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 13,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "array([[1.e-12],\n",
+       "       [0.e+00],\n",
+       "       [1.e-12],\n",
+       "       [0.e+00]])"
+      ]
+     },
+     "execution_count": 13,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "#starting condition\n",
+    "x=np.array([0,0,0,0])\n",
+    "u=np.array([1,0])\n",
+    "\n",
+    "J(f,x,u)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "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": "3.7.6"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}