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+{
+ "cells": [
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "from scipy.integrate import odeint\n",
+    "from scipy.interpolate import interp1d\n",
+    "import cvxpy as cp\n",
+    "\n",
+    "import matplotlib.pyplot as plt\n",
+    "plt.style.use(\"ggplot\")\n",
+    "\n",
+    "import time"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### kinematics model equations\n",
+    "\n",
+    "The variables of the model are:\n",
+    "\n",
+    "* $x$ coordinate of the robot\n",
+    "* $y$ coordinate of the robot\n",
+    "* $\\theta$ heading of the robot\n",
+    "* $\\psi$ heading error = $\\psi = \\theta_{ref} - \\theta$\n",
+    "* $cte$ crosstrack error = lateral distance of the robot from the path \n",
+    "\n",
+    "The inputs of the model are:\n",
+    "\n",
+    "* $v$ linear velocity of the robot\n",
+    "* $w$ angular velocity of the robot\n",
+    "\n",
+    "These are the differential equations f(x,u) of the model:\n",
+    "\n",
+    "* $\\dot{x} = v\\cos{\\theta}$ \n",
+    "* $\\dot{y} = v\\sin{\\theta}$\n",
+    "* $\\dot{\\theta} = w$\n",
+    "* $\\dot{\\psi} = -w$\n",
+    "* $\\dot{cte} = v\\sin{-\\psi}$\n",
+    "\n",
+    "The **Continuous** State Space Model is\n",
+    "\n",
+    "$ {\\dot{x}} = Ax + Bu $\n",
+    "\n",
+    "with:\n",
+    "\n",
+    "$ A =\n",
+    "\\quad\n",
+    "\\begin{bmatrix}\n",
+    "\\frac{\\partial f(x,u)}{\\partial x} & \\frac{\\partial f(x,u)}{\\partial y} & \\frac{\\partial f(x,u)}{\\partial \\theta} & \\frac{\\partial f(x,u)}{\\partial \\psi} & \\frac{\\partial f(x,u)}{\\partial cte} \\\\\n",
+    "\\end{bmatrix}\n",
+    "\\quad\n",
+    "=\n",
+    "\\quad\n",
+    "\\begin{bmatrix}\n",
+    "0 & 0 & -vsin(\\theta) & 0 & 0 \\\\\n",
+    "0 & 0 & vcos(\\theta) & 0 & 0 \\\\\n",
+    "0 & 0 & 0 & 0 & 0 \\\\\n",
+    "0 & 0 & 0 & 0 & 0 \\\\\n",
+    "0 & 0 & 0 & vcos(-\\psi) & 0 \n",
+    "\\end{bmatrix}\n",
+    "\\quad $\n",
+    "\n",
+    "\n",
+    "$ B = \\quad\n",
+    "\\begin{bmatrix}\n",
+    "\\frac{\\partial f(x,u)}{\\partial v} & \\frac{\\partial f(x,u)}{\\partial w} \\\\\n",
+    "\\end{bmatrix}\n",
+    "\\quad\n",
+    "=\n",
+    "\\quad\n",
+    "\\begin{bmatrix}\n",
+    "\\cos{\\theta_t} & 0 \\\\\n",
+    "\\sin{\\theta_t} & 0 \\\\\n",
+    "0 & 1 \\\\\n",
+    "0 & -1 \\\\\n",
+    "\\sin{(-\\psi_t)} & 0  \n",
+    "\\end{bmatrix}\n",
+    "\\quad $\n",
+    "\n",
+    "discretize with forward Euler Integration for time step dt:\n",
+    "\n",
+    "* ${x_{t+1}} = x_{t} + v_t\\cos{\\theta}*dt$\n",
+    "* ${y_{t+1}} = y_{t} + v_t\\sin{\\theta}*dt$\n",
+    "* ${\\theta_{t+1}} = \\theta_{t} + w_t*dt$\n",
+    "* ${\\psi_{t+1}} = \\psi_{t} - w_t*dt$\n",
+    "* ${cte_{t+1}} = cte_{t} + v_t\\sin{-\\psi}*dt$\n",
+    "\n",
+    "The **Discrete** State Space Model is then:\n",
+    "\n",
+    "${x_{t+1}} = Ax_t + Bu_t $\n",
+    "\n",
+    "with:\n",
+    "\n",
+    "$\n",
+    "A = \\quad\n",
+    "\\begin{bmatrix}\n",
+    "1 & 0 & -v\\sin{\\theta}dt & 0 & 0 \\\\\n",
+    "0 & 1 & v\\cos{\\theta}dt & 0 & 0 \\\\\n",
+    "0 & 0 & 1 & 0 & 0 \\\\\n",
+    "0 & 0 & 0 & 1 & 0 \\\\\n",
+    "0 & 0 & 0 & vcos{(-\\psi)}dt & 1 \n",
+    "\\end{bmatrix}\n",
+    "\\quad\n",
+    "$\n",
+    "\n",
+    "$\n",
+    "B = \\quad\n",
+    "\\begin{bmatrix}\n",
+    "\\cos{\\theta_t}dt & 0 \\\\\n",
+    "\\sin{\\theta_t}dt & 0 \\\\\n",
+    "0 & dt \\\\\n",
+    "0 & -dt \\\\\n",
+    "\\sin{-\\psi_t}dt & 0  \n",
+    "\\end{bmatrix}\n",
+    "\\quad\n",
+    "$\n",
+    "\n",
+    "This State Space Model is not linear (A,B are time changing), to linearize it the **Taylor's series expansion** is used around $\\bar{x}$ and $\\bar{u}$:\n",
+    "\n",
+    "$ \\dot{x} = f(x,u) \\approx f(\\bar{x},\\bar{u}) +  A(x-\\bar{x}) + B(u-\\bar{u})$\n",
+    "\n",
+    "So:\n",
+    "\n",
+    "$ x{t+1} = x_t + (f(\\bar{x},\\bar{u}) +  A(x-\\bar{x}) + B(u-\\bar{u}) )dt $\n",
+    "\n",
+    "$ x_{t+1} = (I+dtA)x_t + dtBu_t +dt(f(\\bar{x},\\bar{u}) - A\\bar{x} - B\\bar{u}))$\n",
+    "\n",
+    "The Discrete linearized kinematics model is\n",
+    "\n",
+    "$ x_{t+1} = A'x_t + B' u_t + C' $\n",
+    "\n",
+    "with:\n",
+    "\n",
+    "A' = I+dtA\n",
+    "\n",
+    "B' = dtB\n",
+    "\n",
+    "C' = dt(f(\\bar{x},\\bar{u}) - A\\bar{x} - B\\bar{u})"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "------------------\n",
+    "NB: psi and cte are expressed w.r.t. the reference frame.\n",
+    "in the reference frame of the veicle the equtions would be:\n",
+    "* psi_dot = w\n",
+    "* cte_dot = sin(psi)\n",
+    "-----------------"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Control problem statement.\n",
+    "\n",
+    "N = 5 #number of state variables\n",
+    "M = 2 #number of control variables\n",
+    "T = 20 #Prediction Horizon\n",
+    "dt = 0.25 #discretization step\n",
+    "\n",
+    "x = cp.Variable((N, T+1))\n",
+    "u = cp.Variable((M, T))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def get_linear_model(x_bar,u_bar):\n",
+    "    \n",
+    "    x = x_bar[0]\n",
+    "    y = x_bar[1]\n",
+    "    theta = x_bar[2]\n",
+    "    psi = x_bar[3]\n",
+    "    cte = x_bar[4]\n",
+    "    \n",
+    "    v = u_bar[0]\n",
+    "    w = u_bar[1]\n",
+    "    \n",
+    "    A = np.zeros((N,N))\n",
+    "    A[0,2]=-v*np.sin(theta)\n",
+    "    A[1,2]=v*np.cos(theta)\n",
+    "    A[4,3]=v*np.cos(-psi)\n",
+    "    A_lin=np.eye(N)+dt*A\n",
+    "    \n",
+    "    B = np.zeros((N,M))\n",
+    "    B[0,0]=np.cos(theta)\n",
+    "    B[1,0]=np.sin(theta)\n",
+    "    B[2,1]=1\n",
+    "    B[3,1]=-1\n",
+    "    B[4,0]=np.sin(-psi)\n",
+    "    B_lin=dt*B\n",
+    "    \n",
+    "    f_xu=np.array([v*np.cos(theta),v*np.sin(theta),w,-w,v*np.sin(-psi)]).reshape(N,1)\n",
+    "    C_lin = dt*(f_xu - np.dot(A,x_bar.reshape(N,1)) - np.dot(B,u_bar.reshape(M,1)))\n",
+    "    \n",
+    "    return A_lin,B_lin,C_lin"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Motion Prediction: using scipy intergration"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Define process model\n",
+    "def kinematics_model(x,t,u):\n",
+    "\n",
+    "    dxdt = u[0]*np.cos(x[2])\n",
+    "    dydt = u[0]*np.sin(x[2])\n",
+    "    dthetadt = u[1]\n",
+    "    dpsidt = -u[1]\n",
+    "    dctedt = u[0]*np.sin(-x[3])\n",
+    "\n",
+    "    dqdt = [dxdt,\n",
+    "            dydt,\n",
+    "            dthetadt,\n",
+    "            dpsidt,\n",
+    "            dctedt]\n",
+    "\n",
+    "    return dqdt\n",
+    "\n",
+    "def predict(x0,u):\n",
+    "    \"\"\"\n",
+    "    x0(5)\n",
+    "    u(2,T)\n",
+    "    \n",
+    "    x_bar(5,T+1)\n",
+    "    \"\"\"\n",
+    "    \n",
+    "    x_bar = np.zeros((N,T+1))\n",
+    "    \n",
+    "    x_bar[:,0] = x0\n",
+    "    \n",
+    "    # solve ODE\n",
+    "    for t in range(1,T+1):\n",
+    "\n",
+    "        tspan = [0,dt]\n",
+    "        x_next = odeint(kinematics_model,\n",
+    "                         x0,\n",
+    "                         tspan,\n",
+    "                         args=(u[:,t-1],))\n",
+    "\n",
+    "        x0 = x_next[1]\n",
+    "        x_bar[:,t]=x_next[1]\n",
+    "        \n",
+    "    return x_bar"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Validate the model, here the status w.r.t a straight line with constant heading 0"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "CPU times: user 4.05 ms, sys: 38 µs, total: 4.09 ms\n",
+      "Wall time: 3.75 ms\n"
+     ]
+    }
+   ],
+   "source": [
+    "%%time\n",
+    "\n",
+    "u_bar = np.zeros((M,T))\n",
+    "u_bar[0,:] = 1 #m/s\n",
+    "u_bar[1,:] = np.radians(-10) #rad/s\n",
+    "\n",
+    "x0 = np.zeros(N)\n",
+    "x0[0] = 0\n",
+    "x0[1] = 1\n",
+    "x0[2] = np.radians(0)\n",
+    "x0[3] = 0\n",
+    "x0[4] = 1\n",
+    "\n",
+    "x_bar=predict(x0,u_bar)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Check the model prediction"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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klCuf+CwJlkxZPM/Eo3PjMuCwyEgOe/uYoIVdcSCqIZFU8jDFS2A6XIFr+h/8SaP4656bME1aYC7qjjkgPsduYRV1gRKRa6211wPtCD4Z3matXQRMBmaJyKY4ZcyeIc8Ohbfckjo3CrNzaw3+Yw/AD98E01Vf1B1TrHgoWRIhVbJE26FwbyIyevdja21lEVmVw8vKZX+dSi2m5uF4tw2nxDsvs1mewH39STDNfIs2OvNBjOTrpoeI7BKRF0TkIqA5UIlgRtBV1toJ1toch1KJQqHvVGgqHoh34z2YMyzuvfn499yI+yOnv2kqBX2Xy/K0u8da2JjMTEpfeCne4BFQtQZu0ij8hwbiVq8MO1payFcjCWttWYI5bToDRwMzgZ7Ar0A/ghu/R+9HjsVAHWttTYLC1BG4eD+2k9JMRgbmvM64uvXxHx2Gf2dfvO43Yuo3DjuaKph/fZyOHEt+CFlUHJjK1fBuuBv3zqu4mU8GjZ/aX4xp0x6ToR1891d+Gkk8C5wOLADGAc9lv75ure1L0EIp30Rkp7W2N/AKwb2uxyMdDQslU68x3m0P4Y+5C3/EUEzHbngnnxV2LJVP1trfCMbaK2Gt/XWv1RWB6YlPpeLFeB6mZVvc0cfhTxuHe/ZJ3AcL8LpeozMd7Kf8nEEtBHrnci0dEfGttQftbxARmUfQ4kkBplJlvFuG4U94EDd9PP7aP4Lm6NoUPZV0Jjh7mgdckm25A34XkW9DSaXiypSviNdzACx5D3/ao/h39cOcei7mnIuCGRFU1PLTSOKBKF6zJa/XqOiZYsXxrr4ZN+Mx3Kuzg6nqL79Op6lOESLyNgT9ofTYKFyMMdCkBd4RDXHPPoF7ZRZuyXtBB98j9ucuSOGkH8eTnPEyMBf1wPz3Utzid/DH3oPbsT3sWCoP1tprrbXFIPcPbtbaYtbaaxObTCWSKVUar+s1eH3vAMB/8LZgtPTNcW/0nBaSYiQJtW/GGMzp5+OXKIl76hH8kbfj9boVU7xE2NFU7ioDS62184C3gW/5p6Pu4UArgi4bk8MKqBLHHNkw6OD7/Azcq7Nxny3Gu7gHHHOCNknfBz2DSiHeSW0xl18H336BP/pOHRU9iYnIAKAx8D1wBUEL1y8I7kddDnwDNBaR20ILqRLKFC2Gd0FXvFsfhHIV8cfdF1wRWb827GhJS8+gUozXvDU+Bvf4cPxH7sXrOWCPkZVV8hCRNcADkS+lADDVa+MNeAD3+hzcnGn4g3thLrgU85/TtBHUXvTdSEFe81aYzj3hi4+CVn6+TvOjVCoxGRl4p5+PN2Qk1DgM99RY/AcG4FbFavS49KBnUCnKO+l0/L+3BVMAyOPQoZtey05SkU65QwgGiM0iW8ddEakeUiyVBMyBVfD63oF7bz5OJuIPvRZzVkfM6edpa130DCqleae2x7RpH4yq/PrcsOOo3I0FjgFuByoA1xCMvjI8zFAqORhj8Fq0wbt9LKZRc9xzT+Hf2Rf3U24jZBUeWqBSnLnwMjjmBNwzj+O++yLsOCpnpwEXiMgcYFfkewf27LyrCjlzQHm8Hv3xet0Kmzfh33Mj/tMTcNu2hh0tNHqJL8UZz8O74nrcW0dAbZ2TJkl5/DMM2CZr7QHASuCw8CKpZGUaNcOr2wA3azLu9bm4jxfide6JOeqYsKMlnJ5BpQFTtBjeaefqoJTJ61OC+08A7xBc8nuE3Ec5V4WcKVESr9NVeP3vhSJF8UcMwZ84HLdxQ9jREkoLlFLxdyXwc+RxH2ArUI5gOnilcmXq1MMb9DDmrA64xQvwB/XEX/Q2zrmwoyWEXuJTKv4qicgiABFZTWQWXWtt01BTqZRgihTFtO+Ea9ICf/Jo3IQHcYvexut0NaZipbDjxZWeQSkVf6/lsvzlhKZQKc1UOxTv5vswHa+E777AH9wLf/4Lad0PUs+glIoTa61H0OfJWGsNe05cWBvYGUowlbKMl4E55Wxco2b4T43FzRiP++BtvC7XYKqmX5c6PYNSKn52AtuBkpHHO7J9fUXQWEKpfDMVD8S7djDmir6wegX+Hdfhz5mWdjMd6BmUUvFTk+Cs6W3gpMhjF/n6Q0QKbwcXVWDGGEzzVrj6jYMRZV6YwdpPFuI6XY05LD26nGiBUipOROSXyMMa8P+X/A4SkZXhpVLpxpQ5AHNFX1yzlrhp4/DvvxnTqh3mvC6YEiXDjlcgWqCUijNrbTmCy3n/Jbi8V8paew7QNFbTbVhr+xGMml5JRNZE7nmNAM4AtgCXisiSWOxLJSdzVBMqjJjKmokjcG+8gPvkg6ClX8Pjwo6230IvUNbaYcDZBNfqfwAuE5E/w02lVEyNA9YTnEl9FVn2PvAgUOACZa09hGA4pV+zLW4H1Il8NSPoGNysoPtSyc0rURKv45W4pifhTx6NP/oOzHH/wXTshilbPux4+ZYMjSReA44SkaMJetbfEnIepWLtFODayKU9ByAifwAHxmj7w4H+u7cd0R6YLCJORBYC5ay1B8dofyrJmVp18W57KOg/9fH7+AN74f9vfsp18A39DEpEXs32dCHBZRCl0slfBNNs/P+9J2tt9ezP95e1tj2wXEQ+tdZmX1UV+C3b82WRZf/ap7W2O9AdQETIysrKcV+ZmZm5rks0zZKzf2W5tBc725zJhrH3sePJERRZ8j/KXH0TmZWrJj7L/mwjRlli5XLg6dxW6oFUMJolZwnIMgGYaa29FfCstccDdxNc+suTtfZ1oHIOq24FBhBc3ttvIjIeGB956tasWZPj67KysshtXaJplpzlmKV4adx1QzELXmH7zCdZ26cTpn0nzCnnxHX8zn29L1WqVIlqGwkpUPs6wCJTDxA5eHcCU3Pbjh5IBaNZchaLAykP9xGMvzcGKAI8DjxK0IghTyLSJqfl1toGBE3Zd589VQOWRIZQWg4cku3l1SLLVCFkPA/Tqh2uYVP8qY/gnnkC98E7eF16Y6rXCjterkwyXJO01l4K9ABOEZEtUf6YW7FiRY4rUuWPX6JplpxFUaBSYqpia+3PwLGRVnxnAr0JWvE1A0aKSDRj/+lxlU+plsU5B0vew5/2KGzaEMzee1ZHTNFiCcsS7XEV+iU+a21bghu8LfNRnJRKKdbaukBDoHT25SLyeJx2OY+gOC0laGZ+WZz2o1KMMQaatMA74ujgTOqlmbiP3gvOpuo2CDveHkIvUMBooBjwWuQyxUIRuSrcSErFjrV2ADCIYF6o7B/CHMHlvpgQkUOzPXZAr1htW6UfU6oM5tJrcc1a4k8Zg//ArZj/nIb576WYkqXz3kAChF6gRERnFVXp7jqCTrmfhR1Eqb2ZIxviDR6Fe34a7tU5uM8W413cA3PMCWFHS4p+UEqlu63AN2GHUCo3plgxvP9ehnfrg1C2HP4j97Jr7N24P9eGmiv0Myil0lFk3L3dBgKjrLVDgN+zv05E/ETmUmpfTI3aeAMexL0+Bzd3Ov6gXsElvxNPw3iJP5/RAqVUfOzkn5EddrdW6pZt/e6RzePXEUWp/WAyMzFtL8Adczz+5DG4KWODGXwv6YWpXC2hWbRAKRUfNcMOoFRBmAOr4PW7E/e/13HPPI4/tA/mrA6Y08/HZCamdGiBUioOsk21gbX2BhF5YO/XWGv7Ag8lNJhS+WCMwZx4Kq7Bsbjp43HPPYX78N1gBt+adeK+f20koVT8DcpleUym2lAq3swB5fGuugmv1wDYtAH/nhvxn56I+3tbXPerZ1BKxYm19uTIwwxrbWv27DlfC9iY+FRK7T/TqDne4Q1wsycHDSk+fj+4N1W/cVz2pwVKqfiZGPlenD075DpgFXBNwhMpVUCmZClMp6txTVviTx6F//BgzPGtMfYKTOmyMd2XFiil4kREagJYayeLSJew8ygVS6ZOPbxBI3AvCu7lmbgvlmA6dMM0PSkYTikG9B6UUnGmxUmlK1OkKN65nfFuGw5ZB+EmPIg/6g7c2j9isn0tUEoppQrEVDsU7+b7MB2ugG8/xx/cmy3znsX5BeuHrgVKKaVUgRkvA69Ne7yho6H2EWx98yX+6au+f/QelFJKqZgxWQfhXTeE8iWKsW7b9gJtS8+glFJKxZQxBi8GLfq0QCmllEpKWqCUUkolJeNcwW5ihShlg6uUE5tOHalBjyuVKHkeV6l8BmVy+7LWfrSv9Yn80ixpkaUwSZf/M82S/FnylMoFSimlVBrTAqWUUioppWuBGh92gGw0S840S+pJpvdJs+QsrbKkciMJpZRSaSxdz6CUUkqluJQe6sha2xYYAWQAE0Tk3r3WFwMmA02AtUAHEfk5DjkOieznIIJmuuNFZMRer2kFzAF+iiyaJSK3xzpLZF8/E0yGtwvYKSLH7rXeELxvZwBbgEtFZEkcctQFns62qBYwSEQezvaaVsTpfbHWPg6cBawWkaMiyypEMh0K/AxYEVmfw8925Z8Zb+8UkUmxyJSq8jrWEpzlZ/bx+x3nfe/371SCsgwBrgR2Dyc+QETmJSBLjn8DC/repOwZlLU2AxgDtAPqARdZa+vt9bIrgPUichgwHLgvTnF2Av1EpB7QHOiVQxaAd0SkUeQrLsUpm9aR/eR08LYD6kS+ugOPxCOAiHy7+99L8CFhCzA7h5fG6315Emi717KbgfkiUgeYH3m+h8hBNRhoBjQFBltry8cwV0qJ8lhLtH39fsfTk+zH71QCswAMz3Y8xb04ReT2N7BA703KFiiCPxxLReRHEdkOzADa7/Wa9sDuT77PAqdEzh5iSkRW7j4DEZGNwNdA1VjvJ4baA5NFxInIQqCctfbgOO/zFOAHEfklzvv5fyKyAFi31+LsvxOTgHNz+NHTgddEZF3k095r5PyHoLCI5lgrFArwO5WoLKHYx9/AAr03qXyJryrwW7bnywg+8eb4GhHZaa39C6gIrIlXKGvtoUBjYFEOq4+31n4KrABuEJEv4xTDAa9aax3wqIjs3Zomp/euKrAyTnkAOgLTc1mXqPcF4CAR2f3vXEVwSWJvub0/hVU0x1oi5fX7nWjR/E4lUm9rbRfgQ4Kzmrhfbsxur7+BBXpvUvkMKulYa0sDM4HrRGTDXquXADVEpCEwCngujlFOFJFjCC7J9LLWnhTHfeXJWlsUOAd4JofViXxf9iAiDh3aJxUl1e93dknwO/UIUBtoRPCB88FE7nxffwP3571J5QK1HDgk2/NqkWU5vsZamwkcQNBYIuastUUI/mOmisisvdeLyAYR2RR5PA8oYq3NikcWEVke+b6a4J5P071eEs17F0vtgCUi8vveKxL5vkT8vvtyZuT76hxek+j3J9kl1fsRxe93okXzO5UQIvK7iOwSER94jAS+N7n8DSzQe5PKBWoxUMdaWzPyCb0jMHev18wFukYe/xd4I1LFYypyX2si8LWIPJTLayrvvv9lrW1K8N7HvFhaa0tZa8vsfgycBnyx18vmAl2stcZa2xz4K9tpeDxcRC6X9xL1vmST/XeiK0ELwr29ApxmrS0faRxxWmRZYRXNsZYQUf5+J1o0v1MJsde95PNI0Huzj7+BBXpvUrqjrrX2DOBhgqavj4vIXdba24EPRWSutbY4MIXgeug6oKOI/BiHHCcC7wCfA35k8QCgOoCIjLPW9gauJmjtshXoKyLvxSFLLf5pKZcJTIu8L1dly2KA0QQ3/rcAl4nIh7HOEslTCvgVqCUif0WWZc8St/fFWjsdaAVkAb8TtMx7DhCC/5tfCJq9rrPWHgtcJSLdIj97OcH/IcBdIvJELDKlqpyOtZBy5Pj7ncD9R/07FVKWVgSX9xxBs+4ecf7wuTtLbn8DF1GA9yalC5RSSqn0lcqX+JRSSqUxLVBKKaWSkhYopZRSSUkLlFJKqaSkBUoppVRS0gKllFIqKWmBUkoplZS0QCmllEpKqTyauYqCtbY2wVA1bURkibW2CvApcKGIvBVqOKWU2gcdSaIQsNZeCVwPHEswTMznInJDuKmUUmrf9BJfISAijwFLCcbFOhi4NdxESimVNy1QhcdjwFHAKBH5O+wwSimVF73EVwhEJhH7FHiTYG6mBokYbVkppQpCz6AKhxEEU5B0A14ExoWcRyml8qQFKs1Za9sTzPt0dWRRX+AYa22n8FIppVSb74FpAAAgAElEQVTe9BKfUkqppKRnUEoppZKSFiillFJJSQuUUkqppKQFSimlVFLSAqWUUiopaYFSSimVlLRAKaWUSkpaoJRSSiUlLVBKKaWSkhYopZRSSUkLlFJKqaSkBUoppVRS0gKllFIqKWmBUkoplZS0QCmllEpKWqCUUkolpcywAxSAzrSoEsWEHSCB9LhSiZLncZXKBYoVK1bkuDwrK4s1a9YkOE3ONEvOUiVLlSpVEpwmtqy1jwNnAatF5KhofkaPq/zRLDmLxXGll/iUSm9PAm3DDqHU/tACpQo1t24N27/5POwYcSMiC4B1Bd2O+/l7tn/xMe6n73DLfsL9vgK3bg1u0wbc39twTq8MqthL6Ut8Su0v5/u4t1/GzZrEhgqVcINHYrzC+XnNWtsd6A4gImRlZf3rNeuGD2L9V5/kvpHMInjlK5JRIQuvQiW8ClmRx1lkHFiFzOo18UqXjVnmzMzMHHOGQbPkLBZZtECpQset/A1/8mhY+jUc2ZBy197Gn4W0OAGIyHhgfOSpy+m+gbvwcsp78NeaP2DHdtyO7bB9O+z+vnkj7q917PhzHfz0PXy8ELZt3XMj5SpCleqYqtWhag1MlepQpQamWLF8Z06Vey2JlipZor0HpQVKFRpu5w7cyzNxLwoULY65rA/m+JPJrFQJkuSgTlam2qEUzcrCRN6naJo1um1b4M918Mcq3PJfYPmvuBW/4N56KShyABkZUL02pk59TJ16UKceplSZeP5TVArRAqUKBffDN8FZ04pfMcf9B9OxG6Zs+bBjpTVTvCRULgmVq2EaHPv/y52/C/74HZb/jPvpe9z3X+LmP497dXbwgirVMXXqYY5sCPUbB9tRhZIWKJXW3LatuOeewr3xApSriNf7NkzDpmHHShhr7XSgFZBlrV0GDBaRiWFmMl4GHFQFDqqCOeYEANz2v+Gn73FLvwoK1qK3cW+/DJmZULcBpmEzTMOmmArJcX9FJYYWKJW23Ocf4T81FtavwbRshzm/C6ZE4fo0LiIXhZ0hGqZoMah7FKZu0FXL7doFP3yD+3QR7pNFuGnjcNPGBZcDGzZl5ylnQKkDQk6t4k0LlEo7buNfuKcn4Ba9DQcfgtf/Hsxh9cKOpfLBZGTA4fUxh9fH/fcyWLU8KFaffoB7YQZrn58eFKsTTsY0bYkpE7sWgip5aIFSacM5h1v0Fu7pCbB1K+bsjph2F2KKFAk7mioAYwwcXA1zcDVoewFuw5+U+vpjNr72PG7GY7hnHocGx+GdcDI0aILJ1P/vdKEFSqUFt+b34HLelx9Drbp4Xa4JmjOrtGPKlqPkmReypVlr3PJfcO+9gVv4Jv4nC6F0WUzz1piTz8RUqhx2VFVAWqBUSnP+LtwbL+BmPwXGYDp2x7RuF9yIV2nPVK2BufAy3Pld4KuPcf+bj3vzBdz856FRU7w250Cd+sFZmEo5WqBUynLLfsKfNBp+/h4aHIvX6SpMxQPDjqVCYDIyoMGxmAbH4tavxb01D/f2y/gfL4TqtTCnnBN0L9DLvSlFC5RKOW7HdtwLgntlJpQohenWD9P0JP2UrAAw5StizrsEd4bFLXoT9/rzuCcexs2ahGl9Jubkswpda85UpQVKpRT33Zf4U0bDquWY41tjLrxCW3CpHJlixTAntcX953T46hP8154L+sS9Ngdz2rnBfSrtBJzUtECplOC2bMbNmhR03qx4IN51QzH1G4cdS6UAYwzUb0xG/ca4n77Hf346bvYU3GvPYU47LzirKl4i7JgqB3kWKGvt5VFua6eITC5gHqX+xX2yCH/qOPhrPaZNe8y5nTDFiocdS6UgU7MOGdcOwv34bVCoZk3Gvfoc5vRIodLfq6QSzRnUeOCdKF53HBBVgbLWZgAfAstF5CxrbU1gBlAR+Ai4RES2R7Mtlb7cX+tx08fjPvofVDsUr+cATM06YcdSacDUqktGnyHBGI3PT8fNnIR7fS7m3M5B519tBZoUoilQW0WkdV4vstauz8d++wBfA7tvHtwHDBeRGdbaccAVwCP52J5KI8453P9eDzpgbt8e/NE4/XxMpl6RVrFlah9BxnVDcUu/wn/mCdykUbg3X8Sz3f5/2CUVnmgmwTkmym0dF82LrLXVgDOBCZHnBjgZeDbykknAuVHuU6UZt3ol/kMDcZNGBWdNg0fgnWm1OKm4MofVw7v5fky3frBpA/4DA9j1yD241SvDjlao5XnUi8j30WxIRJZGuc+Hgf7A7klfKgJ/isjOyPNlQNWcfjCamT8h/WaVjJVkzuJ27WTL3KfZNOMxTGYRSl/dnxJtzknILLfJ9L6o8BhjMM1a4ho3x702B/fSs/ifLcacfDbmTIspWSrsiIVOvj6WWmtvz2XV3wSF5WUR+X0fP38WsFpEPrLWtsrPviG6mT8hdWaVTLRkzeJ++QF/8ij49Udo1Axz8VVsKV+RLevWJTzL3qKd+VOlD1O0GOZMi2vRBvdc0NrPLXwT0/FKzLEnan+7BMrvx9PDgZuA1sBhke83AY2Bq4EfrbVt9/HzLYBzrLU/EzSKOBkYAZSz1u4ultWA5fnMpVKQ+/tv/GefxL+7H/y1Hu+qm4OGEOUrhh1NKUy5CniX9sG79UEon4UbPwx/1B24tavDjlZo5LdAeUBHEfmPiFwsIv8BLLBLRJoDPYF7c/thEblFRKqJyKFAR+ANEekEvAn8N/KyrsCcfOZSKWb7Zx/iD70G98osTIs2eEPHYJqcoJ9OVdIxNQ7DGzAM06EbfPcF/qBe+K8+F8xZpeIqvwXqdGDuXsteANpFHj8F1NqPHDcBfa21SwnuSYU646eKH7d5E/6TI1k/+FowBq/fnXhdemNKlQ47mlK5Ml4GXptz8IaOgSOOxj3zOP7dN+B+ifbWu9of+W0a9QPBpbzR2ZZdFVkOkAVsiWZDIvIW8Fbk8Y9A4ZmHuxByzsFH/8OfPh42baDkeZ3Y1ubcYCZVpVKEqVgJr/dtsOQ9/Onj8e+6gY3ndMCdfgGmSNGw46Wd/BaobsAsa+1NBPeJqgK7gPMj6+sCA2MXT6UDt24N/rRx8OkHUL02Xp/BlDmmGX8nSYMNpfLDGANNWuAd2RA3czJb5kyHxf/Du+J6TPXaYcdLK/kqUCKyxFpbB2gOVAFWAu+LyI7I+gXAgpinVCnJ+T5uwcu4mZPA34X572WYNucEUyMoleJMydKYS3pSpuVp/DnyTvy7b8SccxGm7fk6EkWMFKiTSaQgFbXWagcBtQe3chn+sAG4qeOg5uF4g0fhnX6eFieVdood0xxvyEhM4+a42VPw779FO/jGSH77QTUgaCTxN0Fz8KeBlgQt7zrEPJ1KOW7nDtzLs3AvPg1Fi2Mu7ROMbaat81QaM6XLQvcboVEz3NRx+Lf3wdgrMP85TX/3CyC/Z1CPAINE5AhgR2TZ28CJMU2lUpL78Vv8O/vi5kzFND4e744xeC1O0QNUFQrGGLxmLfGGjIRadXFTxuCPuQu3eWPY0VJWfgtUfYKm5AAOQEQ2AzqZSiHmtm3Fn/EY/r39YfMmvF634nW/EVO2fNjRlEo4U6FSMF9Zh27wxRL826/D/fBN2LFSUn5b8f0MNCGYKgMAa21TQDsDFFLui4/wn3oE1q7GtDoDc34XnU5bFXrG8zBtzsEddiT+o/fjD7sFc35XzKnt9YpCPuS3QA0EXoxMiVHUWnsLQT+oK2OeTCU1t3EDTibgFr4Flavh9b8XU6de2LGUSirm0Dp4A4fjPzES98zjuO+/xLu0j3ZMj1K+LvGJyAtAW6ASwb2nGsD5IvJqHLKpJOScw1/4Fv6gnrjF72LO6oA3aIQWJ6VyYUqWxut5C6bDFfD5h/h3XIf7KapJIgq9fE+yIyIfE4y5pwoZt3Y1/lNj4YslQdPxrtdgqtYIO1bKstYeSDB8WEOgHPAn8CnwmoisCjObii1jDKZNe1ytI4JLfvfdhLGXB9PM6yW/XOVZoPYxxcYeRGRQweOoZOT8Xbg3XsQ9F7SPMR2vxLQ+Qzsj7idr7ZHAHQSzAXxEMLv0KoI50i4BHrbWvknQYvar0IKqmDO16uINehj/iRG46ePh1x+gU09MkSJhR0tK0ZxBHZLtcXHgAmAx8AtQnWAMvZmxj6aSgVv+C/6kUfDTd3BUE7zOV2MqHhh2rFT3JDAM6CQif++90lpbDDiHYNDk4xMbTcWbKVUGr+cA3PMzcC/MwK1chnf1LZhyFcKOlnSimVH3st2PrbUzgItEZGa2ZecDF8YnngqL27EDN09wLz0LJUphuvXDND1JL0fEgIg0y2P938AzkS+VhoznYdpfjKt2KP4TD+Pf1TcoUrXqhh0tqeS3H1Q74Lm9ls0FzohNHJUM3Pdf4d/eB/fC05jjTsK7fSxes5ZanOLAWpvj3GfW2lmJzqISzzQ5Ae/m+yAjE3/YLfjvzQ87UlLJb4FaCvTaa9nV/DPdhkphbusW/KmP4N9/M+zYjtdnSDBCc5myYUdLZ61zWd4qkSFUeEy1mni3PgS1j8Q9MQL/6Qk6GWLE/ky3Mdta259/ptvYyT/TbagU5T5ZhD91HPy1HtOmPab9xZjiOkBIvGRrfFQ0h4ZItQju8apCwpQpi3fdUNyzT+Ben4tb8Stej5swJQv3ONz5nW7j431Nt6FSj9uwHjf9MdyH70LVGkF/jZqHhx2rMNjd+Mhjz4ZIDvgNGJLoQCpcJjMT0/FK/Ko1cJErGd61gzAVKoUdLTT70w9qB/BOHLKoBHLO4d6bj5PHYfs2TPtOwTw2mdrcNRF2Nz6y1r4nIo/Faz/W2rbACCADmCAi98ZrXyo2vP+chqt4IP64e/HvvhHv2oGFdiLEPO9BWWvfimZD1lq9u5ci3OqV+MMH4Z4cCVWr4w0aiXdWBy1OCWKtPWj3430Vp+yv28/9ZABjCBo31QMustbqkB8pwNRrhNf/XvA8/PsH4D7/KOxIoYjmDKqZtfYyIK8mXMfGII+KI7drJ/4rs3Fzp0JGJqZzz2C+Gq9A81aq/HvDWvs2MAVYJCL+7hXWWo+gb2EX4CTgqALspymwVER+jGx7BtAe0M6/KcBUOxRvwDD8kbfjj74D0+kqvJPahh0roaIpUIsIDpa8LCxgFhVH7tcfWXfPI7gfv4VGzfAuvgpTvmLYsQqrxkB34DGgprX2R2AjwUgSNQlayz4KXFfA/VQluJ+12zLgX32wrLXdI3kQEbKysnLcWGZmZq7rEq3QZMnKwr9vPH8NG8j2KWMptnkjpTv1yPVDZbq9L9F01G1VoD2oULntfwc91l+dDWXL4V11ExxzgvZpCpGIbAdGA6OttYcADQjG4lsPfCYiyxOcZzwwPvLUrVmzJsfXZWVlkdu6RCtsWVyPmzDTxrFl1hS2LvsFc1mfHC/Jp8r7UqVKlai2ke9GEip1uG8+w58yBlavxJx4KhV79GPdtu1hx1LZiMhv7HmWE0vL2bOFYLXIMpViTEYGdO4JFQ/EzZ6C27IJ76pbMMWKhR0trrRApSG3eVPQn+Ld16BSZby+d2CObIhXuixsS45PVyoQGXdvEHARUFFEDrDWngYcLiKjC7j5xUAda21NgsLUEbi4gNtUITHGYM64EL90WdxTj+APH4h3zaC0nltK746nEecc7qP/BXM1vTcfc/r5eINHYY5sGHY0lbvhBA0hOhH0gQL4kmCElgIRkZ1Ab+AVghHTRUS+LOh2Vbi8k07H63Ej/LwUf9gtuD/XhR0pbvQMKk249Wvxp42DTxZB9Vp41w7G1CicfSdSzHnAYSKy2VrrA4jIcmtt1VhsXETmAfNisS2VPEyTFnglSuKPvSfo0Hv97ZhKlcOOFXP5KlCRG7p7TK4WuYauQuJ8H7fgFdysSbBzJ+aCrphTzw2uWatUsJ29jkNrbSVgbThxVKow9Rrj9b0jaIZ+38141w2BJGnBFyvRTFhYBOgR+apF0AR2d5PYw6y1PwHjgPGR1kn72tYhwGTgIILLGeNFZIS1tgLwNHAo8DNgRWT9fv6bCg23ahn+5NHw/VdwxNF4l/TEHBhd6xiVNJ4BJllrrwew1h4MPAzMCDWVSgmmVl28G+/Bf3gQ/rBb2D7wIcg6OOxYMRPNPahPgSMIClRZEWkoIieKSEOgLHBlZP3HUWxrJ9BPROoRjOfXK9Kz/WZgvojUAeZHnqtcuJ078F8U/KHXwvJfMF2vCRpCaHFKRQOAn4DPCa5MfA+sAKKayVopU7U63k33Qemy/Dn0Oty3X4QdKWaiucTXSkRW57RCRHYB7wPvRy5L7JOIrCQYYBYR2Wit/ZqgM2F7/pleYBLwFnBTFNkKHffTd8EMt8t/wTRpgbmoO+aA8mHHUvspctXheuD6yDG0RkRcHj+m1B5M1kF4N96DGTGEXSOH4PUemBaNo4xz0R8L1tobROSBHJb3FZGH8rNja+2hwAKCFky/iki5yHIDrN/9fK+fyd7jvcn27TlfUczMzGTnzp35iRM3scritm1l07TxbHnxGbzyFSlzZT+KNzsplCyxkCpZihYtCnkP87XfrLXrRORfc31ba1eLyIHx2u8+uBUrVuS4IlU6gSZaMmUpn+mx5rZesHolXq9bMfUbh5Ylio66eR5X+W3FNwj4V4ECbgOiLlDW2tLATOA6Edlgrf3/dSLirLU5Vs3C2uPdfbEE/6mxsHY1plU7OK8Lm0qWYlM+t5tu70usxKLHewH8aziAyH1fbeWi8i2jXAW8fnfiPzQQf/SdwfQ5DVJ3mNSoCpS19uTIwwxrbWv2rHy1CBpNRCVy8M0EporI7mmtf7fWHiwiKyM3iXO8pFjYuI0bcDIRt/BNqFwNr/+9mDo6GHU6sNa+Q9BQqLi1dsFeq6sRXDpXKt9MmQOCIjV8EP7Yu/GuuhnTsGnYsfZLtGdQEyPfiwOPZ1vugFXANdFsJHL5biLw9V6XBOcCXYF7I9/nRJkrLTnncB8swM14DLZuxpzVAXPGhZgiRcOOpmJnAsEHveP45/iC4Jj6naCxkFL7xZQui9f3TvyHB+M/cg9e9/6YY44PO1a+RVWgRKQmgLV2sohEM7J5bloAlwCfW2s/iSwbQFCYxFp7BcFU1zaXn097bu0f+FMfgc8/hJqH43Xpjal2aNixVIyJyCQAa21n4HsReW/3OmvtCQSX0gs6mrkqxEyp0njX344/Ygj+o/dhut2Ad9yJYcfKl/xO+V6Q4oSIvEvuN8ZOKci2U53zd+HefAk3ezIApkM3zMlnYjy9FZHmGgIf7rXsI+A5tECpAjIlS+FdNxR/5FDchAdwGRkpdSYVTUfdxcD9wJycOuJaa4sC5xL0b/rXXDMqb275r/iTR8GP38JRx+B17ompGEYDLhUCx78bRGSg42SqGDElSuL1GYw/fDD++GF4vQakTMOJaM6guhJ0GnzEWrsE+JZ/RpI4HDgGeAO4NE4Z05bbsQM37xncS89CiRKYK/pimrXUuZoKl3eAO6y1/UXEj8yoOySyXKmYMMUjRerBgfhj78G7ZiCmXqOwY+UpmgkLvwL+a62tDJxKMLlaFsHkapOBS3LryKty55Z+hT9pNKxahmneCmOvwJQ5IOxYKvH6AC8AK621vwDVCTqznx1qKpV2TMnSeNcPxX/gVvwxd+L1GYI5/KiwY+1T1PegRGQVMCWOWQoFt3ULbtZk3FvzoEKlYNTxBk3CjqVCIiLLrLXHAE0JJhf8DfhARPxwk6l0FLTuuwN/2AD8kXfgXT8UU/uIsGPlKpp7UCeJyILI45Nze52IvBHLYOnIfbo4aKH351rMKWdjzu2MKV4i7FgqZJFitDDypVRcmbLl8PpFitSIIcE4nofWCTtWjqI5gxpLMBwR7NlfIztH0GFX5cBtWI+bMQG3+B2oWgPvqpswteqGHUspVUiZchWDzrz334I/fDDeDXdhDqkZdqx/ieYe1FHZHiffvyCJOefY+saL+BNHwPZtmPadMG3Px2T+a3QbpZRKKFOhUlCkhg3AHz4I76b7MAcl14wIBWrKaq1tba3N34ilhYT7YxX+8EFsGHUXVKmON2gk3lkdtDgppZKGqVQZr+/t4Bz+8EG49ck1T2a+CpS19m1rbYvI45sIJlWbbq0dEI9wqcjt2oX/6mz8Ib3hp+8o0+MGvBvvxhxcLexoSin1L6ZytWA23s0bgyK1aUPYkf5ffs+gjuKfG7lXAq0JJh68KpahUpX79Uf8e27EPfMEHNkIb+gYSrY9H+Npn0ulVPIyNQ7D630b/LEKf+TtuG1bw44E5L9AeYCz1tYGjIh8JSK/AYV6xjy3/W/8WZPw7+oL69fg9egfzMVSISvsaEopFRVTtwFe9xvh56X4j9yD27Ej7Ej5ng/qXWA0cDAwGyBSrJJjYp8QuG+/wJ88GlavwLRog7nwMkypMmHHUkqpfDONm2O69sY9ORI38SHofkOo44Hmt0BdCvQD/gCGRZYdAYyIYaaU4LZswj37JO6dV6FS5aAvQRpMsayUKty8Fm3wN28MblVMLQ2de4Y2/Fp+RzNfSzA9RvZlL8Y0UQpwS97Dn/YobPgLc/p5mLMvxhQrFnYspZSKCe+08/A3bQzGCS1dFnPeJaHkyFeBioxcfhtwEVAFWEHQku8uEdkW+3jJxa1fGxSmTxbCITXxrhmEqVE77FhKKRVz5rxLYNMG3Lxn8MtVxGt9RsIz5PcS3yNAXeBagokFaxCcUVUFLo9ttOThfB/3zqu4mU/Czp2YC7pi2rTHZOb37VNKqdRgjIFOV+P+Wo+bPh5XvgKmUfOEZsjvX9hzgdoi8mfk+VfW2kXAUtK0QLlVy/CnjIHvvoS6DfAu6ZV0va2VUioeTEYGXvcbgxHQH3sAr++dCR1cNr/NzFcBJfdaVoJgeoC04nbuxH9R8If2gWU/Y7r0xut3pxYnpVShYooVx7tmIBxQAX/0nbjfVyRs3/k9g5oCvGytHQUsI5geoBcwOftI56k+srn76Xv8SSNh+S+YJi0wF3XHHFCou3oppQoxU7Yc3nVD8O/pH4yAfvP9mLLl4r7f/BaoHpHvew9tdBX/jCaRsiObu7+34Z6bipv/PBxQLpgaOcHXXJVSKhmZA6vgXTMQ/8Fb8UfdEYyAXqx4XPeZ32bmaTuaufvy4+Be09rVmJZtMed3xZQsFXYspZRKGqZWXbwrb8Qfew/++GF4PQdgMuLXkbfQDxLnNm3Af3w4/sODoUgRvBvvwevcU4uTUkrlwDRqhrm4B3y2GDftUZxzcdtXoW0n7ZzDfbAA9/QE2LIJc6YNvooUDTuaUkolNa9VO/x1q3EvzYSDqmBOOzcu+ymUBcqt/SOYev3zD6Hm4Xhd7sBUOzTsWEoplTLMuZfgVq/EPfsE7sCDMY2axXwfhapAOX8X7q2XcLOmgPMxHbphTj4z1MEQlVIqFRnPw7vsevy1fwR9pG66F1M9tiPrFJoC5Zb/ij9lNPzwDdRvHNxnyjoo7FhKxYW19kJgCHAk0FREPgw3kUpHplgxvN634d/dD3/UnXgDHsCUrxiz7ad9Iwm3Ywf+3Gn4d1wHvy/HXH49Xp8hWpxUuvsCOB9YEHYQld7MAeWDjrxbt+CPvgP3d+yGZU2aMyhrbVuCaTsygAkicm9Bt+mWfh3M1bTyN0yzlsElvTIHFDirUslORL4GsNaGHUUVAqZaTbzuN+CPvgt/woN4V98ck+0mRYGy1mYAY4BTCUaoWGytnSsiX+3P9vytm/GnjcO99RKUz8K7djCmQZNYRlYqbVhruwPdAUSErKycZ4LOzMzMdV2iaZachZrl5HZs2bKRjRMfpvi8Z8jsdl2BsyRFgQKaAktF5EcAa+0MoD2Q7wLlPl3M2umP4tb9gTn5LMy5nTHFS8Q4rlLhs9a+DlTOYdWtIjIn2u2IyHhgfOSpW7Mm5wmys7KyyG1dommWnIWdxTVrjfnhO7bMmUZGlUPYckyLHF9XpUp0Y5omS4GqCvyW7fky4F9tFvP6pOd8n3XzBEqXoVz/uyhyeP04Ro6OfrrKmWYpOBFpE3YGpbIzxkDHK3FrVrFlznRcg6aYIkX2e3vJUqCiEs0nPdfjJrJq1mbtX39BEnyqCfsTTXaaJWf7yhLtJz2lVCCYoqM/FcqWYd2OXQXaVrK04ltOMDL6btUiy/LNVMgqUMVWKh1Ya8+z1i4DjgdetNa+EnYmVXiYEiXxYjADRLKcQS0G6lhraxIUpo7AxeFGUip1ichsYHbYOZQqiKQ4gxKRnUBv4BXg62CRfBluKqWUUmEy8RyJNs5SNrhKOSbsAAmkx5VKlDyPq6Q4g9pPJrcva+1H+1qfyC/NkhZZCpN0+T/TLMmfJU+pXKCUUkqlMS1QSimlklK6Fqjxeb8kYTRLzjRL6kmm90mz5CytsqRyIwmllFJpLF3PoJRSSqU4LVBKKaWSUrKMJBET8ZhTqoB5fgY2AruAnSJybAL3/ThwFrBaRI6KLKsAPA0cCvwMWBFZH1KWIcCVwB+Rlw0QkXlxznEIMBk4iKC/z3gRGRHW+5JKkunY0uNqn1mGkODjKrLfuBxbaXMGlW1OqXZAPeAia229cFMB0FpEGiXyIIp4Emi717KbgfkiUgeYH3keVhaA4ZH3plEiDiJgJ9BPROoBzYFekd+RsN6XlJCkx5YeV8lzXEGcjq20KVBkm1NKRLYDu+eUKpREZAGwbq/F7YFJkceTgHNDzJJwIrJSRJZEHm8kGFarKiG9LylEj60IPa5yFq9jK50u8UU1p1SCOeBVa60DHo1MFxKmg0RkZeTxKoLT8TD1ttZ2AT4k+PSVsMtq1tpDgcbA/7V3tzwNBEEYx/8kgAEcBoOAL0AQqIZUIXCYsZUIDBqDQkJwCAIOSCaBQL8CCg0JGkFIkXhexG6hpmB6t8Pl+Zlr7sRNJn0y6fZye0e8vkQTLVvK1e+K5QpGm60m/YKKqOXuy6SlkS0zWy1dUJ+7f1L2vWtHwJ7v1WAAAAFDSURBVCKwBLwA+3Xd2MymgUtg293fBq8F6Iv8TbkarliuYPTZatKAGtmeUqPi7s/5+Era+mClZD1Az8zmAPLxtVQh7t5z93d3/wCOqak3ZjZBCtCZu1/l02H6ElSobClXw5XKFVSTrSYNqO89pcxskrSnVLdUMWY2ZWYz/c/AGvBQqp6sC3Ty5w5wU6qQ/pc226CG3pjZGHACPLr7wcClMH0JKky2lKvflchVvm8l2WrUmyTMbB04JD0Ke+ruewVrWeBnw7hx4LzOeszsAmgDs0AP2AWuAQfmgSfSI5+V/8k6pJY2aRnik/T46ebAWnVVdbSAW+Ae+Mind0hr5bX35T+Jki3l6s9a2tScq1xLJdlq1IASEZHmaNISn4iINIgGlIiIhKQBJSIiIWlAiYhISBpQIiISkgaUiIiEpAElIiIhfQE1wHSB/VkiEAAAAABJRU5ErkJggg==\n",
+      "text/plain": [
+       "<Figure size 432x288 with 4 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "#plot trajectory\n",
+    "plt.subplot(2, 2, 1)\n",
+    "plt.plot(x_bar[0,:],x_bar[1,:])\n",
+    "plt.plot(np.linspace(0,10,T+1),np.zeros(T+1),\"b-\")\n",
+    "plt.axis('equal')\n",
+    "plt.ylabel('y')\n",
+    "plt.xlabel('x')\n",
+    "\n",
+    "plt.subplot(2, 2, 2)\n",
+    "plt.plot(np.degrees(x_bar[2,:]))\n",
+    "plt.ylabel('theta(t) [deg]')\n",
+    "#plt.xlabel('time')\n",
+    "\n",
+    "plt.subplot(2, 2, 3)\n",
+    "plt.plot(np.degrees(x_bar[3,:]))\n",
+    "plt.ylabel('psi(t) [deg]')\n",
+    "#plt.xlabel('time')\n",
+    "\n",
+    "plt.subplot(2, 2, 4)\n",
+    "plt.plot(x_bar[4,:])\n",
+    "plt.ylabel('cte(t)')\n",
+    "\n",
+    "plt.tight_layout()\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "the results seems valid:\n",
+    "* the cte is correct\n",
+    "* theta error is correct"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Motion Prediction: using the state space model"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "CPU times: user 2.53 ms, sys: 2.36 ms, total: 4.9 ms\n",
+      "Wall time: 1.44 ms\n"
+     ]
+    }
+   ],
+   "source": [
+    "%%time\n",
+    "\n",
+    "u_bar = np.zeros((M,T))\n",
+    "u_bar[0,:] = 1 #m/s\n",
+    "u_bar[1,:] = np.radians(-10) #rad/s\n",
+    "\n",
+    "x0 = np.zeros(N)\n",
+    "x0[0] = 0\n",
+    "x0[1] = 1\n",
+    "x0[2] = np.radians(0)\n",
+    "x0[3] = 0\n",
+    "x0[4] = 1\n",
+    "\n",
+    "x_bar=np.zeros((N,T+1))\n",
+    "x_bar[:,0]=x0\n",
+    "\n",
+    "for t in range (1,T+1):\n",
+    "    xt=x_bar[:,t-1].reshape(5,1)\n",
+    "    ut=u_bar[:,t-1].reshape(2,1)\n",
+    "    \n",
+    "    A,B,C=get_linear_model(xt,ut)\n",
+    "    \n",
+    "    xt_plus_one = np.dot(A,xt)+np.dot(B,ut)+C\n",
+    "    \n",
+    "    x_bar[:,t]= np.squeeze(xt_plus_one)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 8,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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\n",
+      "text/plain": [
+       "<Figure size 432x288 with 4 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "#plot trajectory\n",
+    "plt.subplot(2, 2, 1)\n",
+    "plt.plot(x_bar[0,:],x_bar[1,:])\n",
+    "plt.plot(np.linspace(0,10,T+1),np.zeros(T+1),\"b-\")\n",
+    "plt.axis('equal')\n",
+    "plt.ylabel('y')\n",
+    "plt.xlabel('x')\n",
+    "\n",
+    "plt.subplot(2, 2, 2)\n",
+    "plt.plot(x_bar[2,:])\n",
+    "plt.ylabel('theta(t)')\n",
+    "#plt.xlabel('time')\n",
+    "\n",
+    "plt.subplot(2, 2, 3)\n",
+    "plt.plot(x_bar[3,:])\n",
+    "plt.ylabel('psi(t)')\n",
+    "#plt.xlabel('time')\n",
+    "\n",
+    "plt.subplot(2, 2, 4)\n",
+    "plt.plot(x_bar[4,:])\n",
+    "plt.ylabel('cte(t)')\n",
+    "\n",
+    "plt.tight_layout()\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The results are the same as expected"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "------------------\n",
+    "\n",
+    "the kinematics model predictits psi and cte for constant heading references. So, for non-constant paths appropriate functions have to be developed.\n",
+    "\n",
+    "-----------------"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 9,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def calc_err(state,path):\n",
+    "    \"\"\"\n",
+    "    Finds psi and cte w.r.t. the closest waypoint.\n",
+    "\n",
+    "    :param state: array_like, state of the vehicle [x_pos, y_pos, theta]\n",
+    "    :param path: array_like, reference path ((x1, x2, ...), (y1, y2, ...), (th1 ,th2, ...)]\n",
+    "    :returns: (float,float)\n",
+    "    \"\"\"\n",
+    "\n",
+    "    dx = state[0]-path[0,:]\n",
+    "    dy = state[1]-path[1,:]\n",
+    "    dist = np.sqrt(dx**2 + dy**2)\n",
+    "    nn_idx = np.argmin(dist)\n",
+    "\n",
+    "    try:\n",
+    "        v = [path[0,nn_idx+1] - path[0,nn_idx],\n",
+    "             path[1,nn_idx+1] - path[1,nn_idx]]   \n",
+    "        v /= np.linalg.norm(v)\n",
+    "\n",
+    "        d = [path[0,nn_idx] - state[0],\n",
+    "             path[1,nn_idx] - state[1]]\n",
+    "\n",
+    "        if np.dot(d,v) > 0:\n",
+    "            target_idx = nn_idx\n",
+    "        else:\n",
+    "            target_idx = nn_idx+1\n",
+    "\n",
+    "    except IndexError as e:\n",
+    "        target_idx = nn_idx\n",
+    "\n",
+    "#     front_axle_vect = [np.cos(state[2] - np.pi / 2),\n",
+    "#                        np.sin(state[2] - np.pi / 2)]\n",
+    "    path_ref_vect = [np.cos(path[2,target_idx] + np.pi / 2),\n",
+    "                     np.sin(path[2,target_idx] + np.pi / 2)]\n",
+    "    \n",
+    "    #heading error w.r.t path frame\n",
+    "    psi = path[2,target_idx] - state[2]\n",
+    "        \n",
+    "    # the cross-track error is given by the scalar projection of the car->wp vector onto the faxle versor\n",
+    "    #cte = np.dot([dx[target_idx], dy[target_idx]],front_axle_vect)\n",
+    "    cte = np.dot([dx[target_idx], dy[target_idx]],path_ref_vect)\n",
+    "\n",
+    "    return target_idx,psi,cte\n",
+    "\n",
+    "def compute_path_from_wp(start_xp, start_yp, step = 0.1):\n",
+    "    '''\n",
+    "    Interpolation range is computed to assure one point every fixed distance step [m].\n",
+    "    \n",
+    "    :param start_xp: array_like, list of starting x coordinates\n",
+    "    :param start_yp: array_like, list of starting y coordinates\n",
+    "    :param step: float, interpolation distance [m] between consecutive waypoints\n",
+    "    :returns: array_like, of shape (3,N)\n",
+    "    '''\n",
+    "\n",
+    "    final_xp=[]\n",
+    "    final_yp=[]\n",
+    "    delta = step #[m]\n",
+    "\n",
+    "    for idx in range(len(start_xp)-1):\n",
+    "        section_len = np.sum(np.sqrt(np.power(np.diff(start_xp[idx:idx+2]),2)+np.power(np.diff(start_yp[idx:idx+2]),2)))\n",
+    "\n",
+    "        interp_range = np.linspace(0,1,section_len/delta)\n",
+    "        \n",
+    "        fx=interp1d(np.linspace(0,1,2),start_xp[idx:idx+2],kind=1)\n",
+    "        fy=interp1d(np.linspace(0,1,2),start_yp[idx:idx+2],kind=1)\n",
+    "        \n",
+    "        final_xp=np.append(final_xp,fx(interp_range))\n",
+    "        final_yp=np.append(final_yp,fy(interp_range))\n",
+    "        \n",
+    "    dx = np.append(0, np.diff(final_xp))\n",
+    "    dy = np.append(0, np.diff(final_yp))\n",
+    "    theta = np.arctan2(dy, dx)\n",
+    "\n",
+    "    return np.vstack((final_xp,final_yp,theta))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "test it"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 10,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "track = compute_path_from_wp([0,5],[0,0])\n",
+    "\n",
+    "u_bar = np.zeros((M,T))\n",
+    "u_bar[0,:] = 1 #m/s\n",
+    "u_bar[1,:] = np.radians(-10) #rad/s\n",
+    "\n",
+    "x0 = np.zeros(N)\n",
+    "x0[0] = 0\n",
+    "x0[1] = 1\n",
+    "x0[2] = np.radians(0)\n",
+    "_,psi,cte = calc_err(x0,track)\n",
+    "x0[3]=psi\n",
+    "x0[4]=cte\n",
+    "\n",
+    "x_bar=np.zeros((N,T+1))\n",
+    "x_bar[:,0]=x0\n",
+    "\n",
+    "for t in range (1,T+1):\n",
+    "    xt=x_bar[:,t-1].reshape(5,1)\n",
+    "    ut=u_bar[:,t-1].reshape(2,1)\n",
+    "    \n",
+    "    A,B,C=get_linear_model(xt,ut)\n",
+    "    \n",
+    "    xt_plus_one = np.squeeze(np.dot(A,xt)+np.dot(B,ut)+C)\n",
+    "    \n",
+    "    _,psi,cte = calc_err(xt_plus_one,track)\n",
+    "    xt_plus_one[3]=psi\n",
+    "    xt_plus_one[4]=cte\n",
+    "    \n",
+    "    x_bar[:,t]= xt_plus_one"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 11,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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X9/G72i09e2cCna21LwP7Iv924cDBuyqBmHIV8G64E3PFAFizCn94f/wFrwXTiqmoSpgqPhHpAIwDUoAp1tpRIYdUpJn04phLrsS1OjXojvvYKGjRGq/7tZgKlcIOL554BAsVAmwXkXLAz0D8T4SocmSMwZz4D1yjo/FnTMDNegy35D28nv1BFyuMmqjcQYnIGBFpEY1z5XD+FGAiwdiRJkA3EdG5SOKAqdMAb/A/MZ17wrIl+EOux1/0pt5N/WUpQfsTwCKCKr9HCWY5VwnOVMzAGzAMc9n1sPJb/GF9+f3tV/X3P0qidQeVAswTkY3ATIKVQqM5cKYVsNxauxJARGYDnQiWLDioIUPK8t13qezZk7zf6tPS4uHz9cHt7gXr18DCHVB6B1SpgUkt2K9YrD9bkyZ7GDFiazRPeTV/dYy4EbgPKE+wHLxKAsYYzKkdcI1b4E8fz9aJ90Gz4/Au74spX9AVVYq2qCQoa21/ERlIcIdzKXCXiLxPMFL+RWttQVvRawA/ZXq+GmideQcR6QP0icRDRuQ2u0SJFIwxpKUl7yjwuPl8aWm4ekfg/7KBfevXwqrlpNSojVem3CGfMtafrUSJVDIy8j//2kFUtta+D2Ct3UBkFV0RaRXNN1HhM5Wr4t18L6U+WMi2mZPwh/bFdL8G0+pUXS3gEEWtDcpauw94FXhVRI4CZgHTgUmRO56h1to10Xq/bN5/MjA58tTtHyA2aFDyD5yLx8/nVv+BP+WfsGYVpm2HYFG4Q1jGIIzPtv/tIoMJC+otsl+w8A0KvmChijPG8yh5nrCj7pH408bipvwT98l7eD2uwxTgi1pRFbUEFRnvcQnBMgNHAy8A1xN0qb2ZYLqXow/x9GuAzKPiaka2qThlatbFu3N0MLDxrTm4rz/Hu+Y2TK16YYcWEyLiEVTtmciS6pm/QtcH9oYSmIoJU7UG3u2jcG/Owb38NP7QZUGSanli2KEllKgkKBF5HjgL+D/gMWCOtXZXptdv4q+eTIfiQ6ChiNQjSExdge4FOJ+KAZOWhrnkClyzY/GfGI1//62YS6/DO+n0sEOLhb0Eaz/tf5yZD4ws6BuISEXgWaAu8AMg1tot2ey3D/g88vRHa23HyPZ6wGyCBRQ/Bi6z1u4uaFwqYLwUTIfOuGbH4U8di//oKEzrtphu12BKxW66sEQWrXFQi4GG1tpzrbXPZk5OANZaH6hyqCe31u4F+gLzgK+CTXZZQQJWsWMaHY139xio3wg3fVzQLXf3rtwPTGz1CO6UVgOHRx4fHtle1lo7LArvcQcw31rbEJgfeZ6d3621LSI/HTNtfwAYY61tAGwhmNhWRZmpUQdv0EOY87vhPvov/tC+uM8/CjushBCtThIP52GfAi3aZq19jWA2aJWATNkKeAOH415+Bveaxf2wHO+62zGHRaWdJ+5Ya1dFHtaBP6v8qlhrozntQCeCpTsAZgALgdvzcmCk2vEf/FUTMYNgSqZHoxifijCpqZiO3XDNW+FPG4s/fgTm5DMw0htTomTY4cWthBmoqxKf8VIwF/bA1T8S/4kx+PfehHflQEyL1rkfnKBEpDzB2KeLgT1AKRHpCLSKwnIbmRPeOnKupSguIh8RVDWOstbOIajW+zVSOwHBnV6NHD5Dtj1ks0pNTc3xtaLkoOWQkYE7egbbZz/BzjlPY775jDJ97yT96ONiG2QMROP3QROUijlz9PF4d48JVu+ddB/mossxZ12UrF1xHyOoPqvDX+P23gP+CeSaoETkbYIFELO6M/MTa60TkZxGh9ax1q4RkcOBd0Tkc/LRJpxTD9ms4rE3aRjyVA5nX4J3RDP8aeP4dWh/TLtzMJ17HVJP13iVh+U2cqUJSoXCZFTBu/V+3PRxuBdmwLrV0OP6ZFy19HSgurV2z/4EYq3dKCKH5eVga237nF4TkfUiUs1a+7OIVAM25HCONZF/V4rIQoLFFF8AyotIauQuSnvGxpip3wjv7rG4OTNxb8/FffFJsIxHA50kZz+dLFaFxqSnY/rcijm/K+5/8/FH343bFtVZHOLBbwTLbPxJRGoTzMdXUHOBnpHHPYGXs+4gIhVEJD3yOAM4CfjSWuuABQRVjzkerwqXSU/H63IV3i33gXP4Dw7Cf24qbo92pgRNUCpkxhi8jt0xV98C33+Hf/8tuLU/hh1WNE0BXhCRdoAnIm0IOiQ8FoVzjwLOEJHvgPaR54jIcSIyJbJPY+AjEVlKkJBGWWv3VzXeDtwkIssJ2qSeiEJM6hCYI5viDR2HOeUs3Jtz8O8ZiPv+u7DDCp0u+Z4EkuXzuZXf4E8cCXv24PW9C3PEUQm/5Hukt1x/4BqCdqgfgceBcZG7mERT5JZ8z6+CloNbtgR/xgT4bTOmw8WY87skZNV3NJZ81wSVBJLp87lfNuKPHQK/bMS75jYqn35OQieoJKQJKhfRKAe3cztu9hTce+9AzXpB21SCzcISjQSlnSRUXDGVKuPd9gD++OH4k+7jdwMcndjzqorIkUBz4IDpA6y1U8OJSMU7U7I05soBuJZt8GdOxB95M+b8rpgOnTEpRWeJeW2DUnHHlCmLd/M9cGQztk64F3/eS2GHdMhEZDDBmlA3E6yiu/+nR5hxqcRgWrTGG/4IpmUb3Jyn8Efdhvv5p9wPTBJ6B6XikileEq/fENKemsiu56fhb/sN07lnIo6VGkAwKPezsANRicmULovpcyv+MW1wsx7FHzEAc2EPTPuOGC+576Y0Qam4ZdLSKHfTcDYWS8fNexH27QXpnWhJ6nfg67CDUInPO/5k3JFH4c+chHtuGm7J+3hX3Ig5rFrYoRUaTVAqrpmUFEz3ayElFff23GBjnCepyLx7+90NTBCRYcD6zPtFJlFWKs9M2Qp41w/GLV6Ie2Yy/vD+mIuvCNZc85KvxUYTlIp7xhjochVAoiSpzEttQNBb6aoszx2Q3PUzqlAYYzBt2uEaHY3/5ATcrMdwS97D69kfU6ly2OFFlSYolRASLEnt7w9sCBbxtFleN0DnmEakko6pUAmv/1Dcojdxdir+sL6YLldhTmofr9dFvmmCUgnj70nKgFwZdxdjpqU2EJG7rbUPZd1HRO4kmDBWqUNmjMGcehauSQv86eNxMyYES8xffgOmfKWwwyswTVAqoRyYpF6GUqUw53UNOaq/E5F/RB6mRqY5ypxFDwe2xT4qlaxMRhW8m+7BLfg37sUZ+EP7Ybpfg2l1atx9gcsPTVAq4RhjQHrDzh24l2fhly6Hd9rZYYeV1f557dKBzANyHcHaTf1iHpFKasbzMKefjzuqJf70cbgp/8R98i7epddhypYPO7xDoglKJSTjeXB5X9yObUEjcekymONODjusP1lr6wGIyJPW2svDjkcVHaZqDbzb7se9OQf38tP4332J1+M6TMsTww4t3+I+QYnIJQRLUTcmGPD4UbgRqXhhUlPx+tyGP3Yo/pTReCVLY5q0CDusA2hyUmEwXgqmQ2dcs+Pwp47Ff3QUplVbTPc+mFJlwg4vzxKh4/wXwEXA/4UdiIo/Jj0dr+9dULUG/qT7cT/oEgVK7Wdq1MEb9BCmY3fcx//FH9oP99mHYYeVZ3GfoKy1X1lrvwk7DhW/TKnSeAOGQekyuP+8EXY4SsUVk5qKd35XvEEPQ+ky+BPuCXr8/b4z7NByFfdVfErlhSlfCe+OB6BMYjYGK1XYTJ36eHeOxr0yC/fGS7ivluL16o9p3Dzs0HIUFwlKRN4Gqmbz0p3W2jwtQy0ifYA+ANZaMjL+WmU7NTX1gOfJJpk/X74+W5KWQU5EpCLwLFAX+AEQa+2WLPu0A8Zk2tQI6GqtnSMi04G2BMvSA/Sy1n5ayGGrEJm0NMxFPXHNW+NPG4c/+m5Mu3MwnXth0ouHHd7fxEWCsta2j8I5JgOTI09d5oWykn0htWT+fHGwom48uwOYb60dJSJ3RJ7fnnkHa+0CoAX8mdCWA29m2uVWa+3zMYpXxQlTvxHe3WNxc57CzZ+L++KTYFHEBk3CDu0Acd8GpZTKUSdgRuTxDOCCXPa/GHjdWhv/jQ+q0Jn0dLwuvfFuGQnO4T84CP+5qbg9u8MO7U9xn6BE5EIRWQ20Af4tIvPCjkmpOFHFWvtz5PE6oEou+3cFnsmybaSIfCYiY0QkPeoRqrhnjmiKN3Q8pm0H3Jtz8O8ZiPs+PnrDxkUV38FYa18CEndJVaUK4GDts5mfWGudiLhs9tt/nmpAMyDzF7xBBImtGEH1+O3AiGyOzbF9N7NkbgvNj4QthxvvZlfbs9j6yH34o26j1EU9KCVXYtLSDul00SgH41yOv9OJzK1du/bPJ8ncRgPJ/fnioA0qbicyE5FvgNOstT9HEtBCa+2ROex7I3CUtbZPDq+fBtxirT0vl7c94NrKLJl/D/Mj0cvB7dyBs1Nw/5sPNesFbVO16uV+YBYHK4e8XltxX8WnlMrRXKBn5HFP4GA9XruRpXovktQQEUPQfvVFIcSoEowpWQqv1414fe+Gbb/ij7wZ/9Vncfv2xTwWTVBKJa5RwBki8h3QPvIcETlORKbs30lE6gK1gP9kOf5pEfkc+BzIAO6NRdAqMZjmx+MNm4Bp2SaY0+/+W3Frf4xtDFrFl/iS+fNpFV/c0Sq+XCRjObiP/ov/9KPwxx+YC3pgzuiI8Q6+ILRW8SmllCp05riT8YY/Ak2PxT0/Df+hwbgN2X9RiSZNUEoppXJlylbAu34Q5sqBsOZH/OE34r/zKs73C+09NUEppZTKE2MMXpt2eMMmQMMmuGcm448ZgvtlQ6G8nyYopZRS+WIqZuDdOAxz2fXw/Xf4w/rhL3qTaPdp0ASllFIq34wxeKd2wBs2Huo0wD35CP74Ebhff4nae2iCUkopdchMRhW8m+7BdO0D336OP7Qv/uIFUbmb0gSllFKqQIzn4Z1+Ht6Q8VCtFu6JMfz2wGDc1l8LdF5NUEoppaLCVKmOd9v9mIt7sXvpB7ClYOPB4n6yWKWUUonDeCmYsy6iYseubN5VsKU79A5KKaVU1Hllyhb8HFGIQymllIo6TVBKKaXiUtJOFht2ACpp6GSxB9JrS0VLkZ0s1mT+EZGPs25Lpp9k/nxx8NnUgeL5/youfrQc8lwOuUrWBKWUUirBaYJSSikVl4pKgpocdgCFLJk/XzJ/tmSj/1cBLYdAgcshWTtJKKWUSnBF5Q5KKaVUgkn6qY5EpAMwDkgBplhrR4UcUlSISC3gSaAKQdffydbaceFGFV0ikgJ8BKyx1p4Xdjwqe8l6jeWFiEwFzgM2WGubRrZVBJ4F6gI/AGKt3RJWjLGQ09+jgpZFUt9BRf7ATQTOBpoA3USkSbhRRc1e4GZrbRPgBOCGJPps+90IfBV2ECpnSX6N5cV0oEOWbXcA8621DYH5kefJLqe/RwUqi6ROUEArYLm1dqW1djcwG+gUckxRYa392Vr7SeTxNoI/5DXCjSp6RKQmcC4wJexY1EEl7TWWF9ba/wM2Z9ncCZgReTwDuCCmQYXgIH+PClQWyZ6gagA/ZXq+miT6I76fiNQFjgHeDzmUaBoL3Ab4YQeiDqpIXGP5VMVa+3Pk8TqCaq8iI8vfowKVRbInqKQnIqWBF4AB1tqtYccTDSKyv07/47BjUaogrLWOIjQ91MH+Hh1KWSR7gloD1Mr0vGZkW1IQkTSCX4anrbUvhh1PFJ0EdBSRHwiqjP4hIk+FG5LKQVJfY4dovYhUA4j8uyHkeGIih79HBSqLZO/F9yHQUETqEVw0XYHu4YYUHSJigCeAr6y1o8OOJ5qstYOAQQAichpwi7W2R6hBqZwk7TVWAHOBnsCoyL8vhxtO4TvI36MClUXSD9QVkXMI2jNSgKnW2pEhhxQVInIysAj4nL/aaQZba18LL6roy5SgtJt5nErWaywvROQZ4DQgA1gPDAXmABaoDawi6FqdtSNFUsnp7xFBO9Qhl0XSJyillFKJKdnboJRSSiUoTVBKKaXikiYopZRScUkTlFJKqbikCUoppVRc0gSllFIqLmmCUkopFZc0QSmllIpLyT7VkcpEROoTTE3T3lr7iYhUB5YCl1hrF4YanFJKZaEzSRQxInI1MBA4DngJ+Nxae0u4USml1N9pFV+NSnIjAAAgAElEQVQRY639F7CcYI6sasCd4UaklFLZ0wRVNP0LaApMsNbuCjsYpZTKjlbxFTGRBcWWAguAs4FmyT7TslIqMekdVNEzDvjIWnsV8G/gsZDjUUqpbGmCKkJEpBPQAbgusukmoKWIXBpeVEoplT2t4lNKKRWX9A5KKaVUXNIEpZRSKi5pglJKKRWXNEEppZSKS5qglFJKxSVNUEoppeKSJiillFJxSROUUkqpuKQJSimlVFzSBKWUUiouaYJSSikVlzRBKaWUikuaoJRSSsUlTVBKKaXikiYopZRScUkTlFJKqbiUGnYAhURXYVTRYsIOIM7otaWiJddrK1kTFGvXrs12e0ZGBps2bYpxNPFHy+EvOZVF9erVQ4gmtkRkKnAesMFa2zQvx+i1dXBaDoGDlUNery2t4lOqaJsOdAg7CKWyk7R3UErlhfviY/zWJ4cdRmistf8nInULeh63/Et2ry2J++23v7/oeVCiFJQuA6XLYoqlF/TtVBGhCUoVSW7nDtyzU3Dvzmdn16vg9I5hhxS3RKQP0AfAWktGRsbf9tk4aAxbNq3P2wmLpeOVKRf8VMwgtXotUmrUJrV6bVKq18arVBljErfpLzU1NdsyKmqiUQ6aoFSR475cgj9jAmzZjDnnEkpd1IM/ftsadlhxy1o7GZgceeqybVe4+hYqlCrJb9ndQfk+7NyO274NdmyD7Vtx27exd8c22LiO3V98Art3/bV/enGoUh1Tuz40aIyp3wiq1EiYpKVtUIFotEFpglJFhvvjd9wL03ELX4eqNfEGPYipdwQmrVjYoSU8c/iRFMvIwBzkD3NO6cU5B1t+gfVrcOvWBP/+vBr3yXvw37eCboOlykD9Rpj6jTANGsPhjTCp+ucr2en/sCoS3Ldf4E8fD5vWY87ohLmgh7aFxAljDFTMgIoZmMbN/9zufD9IVsu/ghVf41Z+g/vswyBhlSyFOaolNG+FaXosplTp0OJXhUcTlEpqbvcu3EtP4ebPhYwqeLfchzniqLDDihsi8gxwGpAhIquBodbaJ8KNKmA8D6rVwlSrBaecCYDbsQ2++QL32Qe4zz6CDxfhPA8aHoU5+njMMSdgKlcNOXIVLZqgVNJyK7/BnzYW1q3BnHYOpnNPTPESYYcVV6y13cKOIT9MqTLQsg2mZZvgDuv7b4O7qqUf4J6bintuKhzRFHPi6ZhjT9T/7wSnCUolHbdnD+6VZ3BvvAgVKuINHIFp0iLssFSUGc/7s12KCy/DbVyH++D/cO/Ox00fh3vmccyxJ2FOPB0aNgn2VwlFE5RKKu7HFfhTx8KaVZiT2mOkN6ZkqbDDUjFgKlfFnCu4cy6BFV/h3n0H9+Ei3LvzoXJVTNuzMaecqb8PCUQTlEoKbu9e3BvP4159FkqXxet3N+bo48MOS4XAGAMNmmAaNMF1uRq35F3cordwz0/DvTobc/KZmPbnYyodFnaoKheaoFTCc2t+DNqaVi3HtDoV060PpnTZsMNSccCkp2NOaAcntMOtWoF7cw7unVdw77wSVP+dcQGmXsOww1Q5iFmCEpEOwDggBZhirR2V5fUxQLvI05LAYdba8pHX9gGfR1770Vqrw/4Vzt+He+tl3JynoHhJvGtvxxx7UthhqThl6tTHXH0zrvPluPmv4hbNw324CI5oitepO+aIPM2Vq2IoJglKRFKAicAZwGrgQxGZa639cv8+1tqBmfbvBxyT6RS/W2u1lVv9ya1fG9w1rfgajjkBr8f1mLLlww5LJQBTsTLmkitw53XBLXoT9+Yc/IcGw1HH4F3QA1NX76jiRazuoFoBy621KwFEZDbQCfgyh/27AUNjFJtKIM73cQtew704HVLTML1vwrRumzDT4Kj4YUqUxJx5Aa7t2biFr+Fefx5/5M3Qsg1ep0sx1WuHHWKRF6sEVQP4KdPz1UDr7HYUkTpAPeCdTJuLi8hHwF5glLV2TjbH5TqhJehEjvslYjns2/Azv00YyZ4vPqFYyzaUvf4OUipVLvB5E7EsVPSY9HTMWRfiTj0L99Yc3Fsv4y9ZjGl9GqZTd0xGlbBDLLLisZNEV+B5a+2+TNvqWGvXiMjhwDsi8rm1dkXmg/I0oSU6keN+iVQOzrmgKsZOBQPm8r7sPfkMtjgDUfgMRXnBQvUXU6IkpmN3XLvzcG+8gFvwb9zH/8N0uAjTobNOjRWCWI1cWwPUyvS8ZmRbdroCz2TeYK1dE/l3JbCQA9unVBJzW37BHz8cN3Mi1GuIN2wC3ilnapWeKjSmTFm8S67Au/cxTIvWuFdm4w+5Affp4mBiWxUzsbqD+hBoKCL1CBJTV6B71p1EpBFQAXgv07YKwE5r7S4RyQBOAh6MSdQqNM453OKFuNmTYe9eTPdrgoGWOhuAihFTMQPT51Zc2w74sx7Hn3gfNG2J1+VqTNUaYYdXJMTkarfW7gX6AvOAr4JNdpmIjBCRzF3GuwKzrbWZv6Y0Bj4SkaXAAoI2qJw6V6gk4LZuwZ90P27qGKheG2/oOLx252pyUqEwRzbDu3sspstVsOJr/GH98F+cgdu1K/eDVYGYJL1ldWvXrs32hURqeylM8VoO7qP/4j/9KPzxR7AkxhkdMV5Kob5nLm1QWpd4oCJ9bbmtW3AvPBlMn3RYNbye/f82O35RKIe8yMOChbleW/HYSUIVQW77Vtysx4OBk3Ua4F05QLv5qrhjylbAXHEj7sR/4E8fj//wYMw/zsNceBkmvXjY4SUdTVAqdG7pB/gzJ8L2bZhOl2LOvhiTUrh3TUoVhDmyGd7Q8biXZuLmv4L77MPgbupInY0imrRSX4XG7dyBP20c/iP3QplyeIMfxjuviyYnlRBM8RJ43frg3XIfAP7Dg4POFL/vDDmy5KF3UCoUbtkS/BkT4LfNmHMEc34XTGpa2GEplW/myKYH3E1t/nIJrvfNOgltFOgdlIop98fv+E9Nwh87FNLT8e54EO/CHpqcVEIz6cXxul6Nd+t9uH378B+4Df/Nl4JVf9Uh0zsoFTPu2y/wp42DXzZgzugU9NLT0fkqiZgjmlJx9Aw2jhmGe24a7qvPgg4/ZcqFHVpC0jsoVejc7l34zz6B//CdYAzeLffhSW9NTiopeWXK4l03CNP9Wvj6M/zhN+K+Whp2WAlJ76BUoXIrvwmWxVi3BnPaOZjOPTHFS4QdllKFyhiDaXcOrkFj/MkP4o8ZgjnnEsz53bQTUD5oglKFwu3Zg3vlGdwbL0KFingDR2Ca6JJeqmgxterh3TUG98xk3L8tbvlXeNfcplV+eaQJSkWd+3EF/tSxsGYV5qT2GOmNKVkq7LCUCoVJL47p1R//yGa4mRPx770J7/rBmDr1ww4t7mkblIoat3cv/iuz8e+7BbZvxet3N16v/pqclAK8Nu3wbh8FOPwHbsdfvDDskOKe3kGpqHBrfgzamlYtx7Rqi+neB1OqTNhhKRVXTJ0GeHeOxn/8QdwTo/FXrcBc3EvbpXKgCUoViPP34d6cg3v5aShRCu/aOzDHnhh2WErFLVO2PN7AEbjnpuLefhm3+nu8PrdhypQNO7S4owlKHTK3bg3+9HGw4mto2Qbv0uswZcuHHZZScc+kpmK69cGvUx83cxL+yJvw+t2NqVEn7NDiiiYolW/O94PlsF+cAalpmKtuxrQ6VVe5jTIROQw4C2gOlAd+BZYCb1lr14UZm4oO78TTcdVq408cif/A7UENhPZ2/VOeE5SI1CLLhWKt/amwAlPxyW1ajz99PHzzOTQ7Du/yGzDlK4UdVlIRkcbAPUA74GOCRT7XAWWAy4CxIrIAGKKLdyY+U68h3uCH8MePwB8/HHPZDXgntQ87rLhw0AQlImnANZGfw4HlwDaCC6WBiHwPPAZMttbuLuRYVYicc7hFb+LsVDBgLu+LOfkMvWsqHNOBh4BLrbV/W7ZVRNKBjsATQJvYhqYKg6lYGe+2UfiPP4CbPh5/03pMx+5F/vrK7Q5qKfAOQYJ631q7b/8LIpICtAIuBZYAR2V7hr/27wCMA1KAKdbaUVle70VwUa6JbHrEWjsl8lpP4K7I9nuttTNy/WQqatyWX/CfnABffAKNjg66jlc6LOywkpa1tnUur+8Cnov8qCRhSpbC6zcE99Qk3KvPwqb1cHk/TFrRnUg5twR1mrV2Q3YvRJLVe8B7IlL5YCeJJLOJwBnAauBDEZmbTfXEs9bavlmOrQgMBY4DHPBx5NgtucSuCsg5h1u8EDd7Muzdi+l+Dabt2RhPh8/Fioi8bK3tlM32F621F4URkyo8JjUVevaDylVxc57CbfklmNevVOmwQwvFQf/SZE5OInJLdvuIyE3W2o25vE8rYLm1dmWkKnA28LeLLgdnETQKb44kpbeADnk8Vh0it3UL/qT7cFPHQPXaeEPH4bU7V5NT7LXLYftpsQxCxY4xBu9cwfS+CVZ8hf/A7bgtv4QdVijy04tvCPBwNtvvAkbncmwNIHOHitVAdtUYnUXkVOBbYGCkE0Z2x9bIeqCI9AH6AFhrycjIyDaQ1NTUHF8rSg5WDn/87x22Pv4Q/PE7pXv2peT5yb3KbTz+TojIiMjDYpke73c4sCrGIakY8044DVehEv4j9wY9/G66B3NYtbDDiqlcE5SI/CPyMEVE2gGZW+0OJ+g0EQ2vAM9Ya3eJyDXADOAfuRzzJ2vtZGBy5KnbtGlTtvtlZGSQ02tFSXbl4LZvxc16HPfhIqjTAO/KAfxevTa/b0nu2tScfieqV68eQjR/qhX518v0GIJq7p+AYbEOSMWeObIZ3s334o8bhv/gHXgDhmFq1gs7rJjJyx3UE5F/iwNTM213BF1f++XhHGs48CKryV+dIQCw1ma+h50CPJjp2NOyHLswD++p8sEt/QB/5kTYvg3TqTvm7EuS+q4p3llrrwAQkXettf8KOx4VHlO3YdDDb/QQ/IcG4/UfiqnfKOywYiLXBGWtrQcgIk9aay8/xPf5EGgoIvUIEk5XoHvmHUSkmrX258jTjgRjPwDmAfeJSIXI8zOBQYcYh8rC7dyBe3YK7t35ULNu8Mtf+/CwwyrSRKSKtXY9wMGSU+b9VHIz1Wrh3T4Kf8wQ/NF3490wGNPkmLDDKnR5boMqQHLCWrtXRPoSJJsUYKq1dlmkbv0ja+1coL+IdAT2ApuBXpFjN4vIPQRJDmCEtXbzocai/uKWLcGfMQF+24w5RzDnd8GkFt0urXHkHRH5DzCTYHiHv/8FEfEIOh1dDpwKNC3om+U2BETFB5NRJbiTGjsUf/w9eFffkvTzXhrnXI4visiHBFVtL2c3EFdEigEXADfnNnYjxtzatWuzfUHboMD98Tvpr87m93kvQbVaeFcMwNRrGHZYocmlDSrmIyUj11Uf4FqgHrCSvwbI1yMYMP848ERBB8hHhoB8S6YhIEC3g8xQoddWLgq7HNyO7fgTRsDKbzG9+uOdmOem+pg6WDnk9drK7Q6qJzACeFREPgG+4a8L5QigJcFA3l55DVqFy33zBf70cfz+ywbMmRdiLrgUk1Ys7LBUJpGk8wjwSGSKsWYEU4xtAT6z1q452PH59OcQEAAR2T8ERKdQilOmVGm8gSPwJ47ETR+Hj8M78fSwwyoUB01QkW9RF4tIVYJvWM2ADIIL5UngspwG8qr44nbvwr00Ezf/FcioQoV7J7H1sL/11ldxJjLUojDnvMzrEBAVR0x6cby+d+E/cm8wNZID76TkS1J5aoOKzJw8s5BjUYXErfgaf9o4WL8G0+5cTOeeFKtRE7Q6Ju5F5t0bAnQDKllry4nImcAR1tpHYhSDjjHMh1iWgxs6ll/vv43dM8ZTqnQpSpx+XkzeNy+iUQ65TRZ7qrX2/yKPc6zotNa+U6AoVKFwe/bgXpmFe+MlqFApGOjXuHnYYan8GUNwl3Mp8Hpk27LI9mgkqLwMAdExhvkQ63JwfW6DifexdeL9bNu2De/kM2L23geThzaoXOV2BzWJv3oJPZHDPo5gwK6KI+7HFfhTx8KaVcGs49IbU6Jk2GGp/LsQaGCt3SEiPoC1do2IRKt+NtchICq+mWLpeDcMDqYme/IRfIibJFVQubVBNc30uOgMX05gbu9e3GvP4V6zULpcsErn0ceHHZY6dLvJcp1GJmeOyuRsOQ0Bica5VewESerOoOPEjAn4zuGdcmbYYRXYIa+oG5n2aN/+KkAVPrfmR/ypY+DHFZjWbTHd+mBKlQk7LFUwzwEzRGQgBAPagbEEEy5HhbX2NeC1aJ1PhcOkFQuS1P47qdQ0vDY5zTWcGPI8NbWI/EdEToo8vp3gAnlGRAYXVnAqb5y/D//1F/DvHQCbN+JdewfeVTdrckoOg4Hvgc8Jupp/B6wlGP6h1AFMWjG86wdDo6Nx08fhPnkv7JAKJD93UE2BxZHHVxMsA7AN+B9wX5TjUnnk1q3Bnz4OVnwNLdvgXXodpmz5sMNSURIZEzUQGBip2ttkrc15dL0q8v68kxozBH/yQ3h978I0bRl2WIckP4v7eIATkfqAsdZ+GRmjUSGX41QhcL6P//Zc/HtuhJ9/wvS+Ce/aOzQ5JRkR+XNaL2vtxv3JSUR0/KHKkSleAq//UKheC//R+3DfJea46/wkqP8SdGt9GHgJIJKstF9pjLmN6/BH3417dgoc0Qxv2CN4J5yGMTGflUcVvr9NjigiaQQdGpTKkSlVGm/AcKhYGX/CCNyq5WGHlG/5qeLrBdwMbAAeimxrRDDJpIoB5xxu0TycnQYGzOV9gy7kmpiSjogsIhjCUVxEsnZEqgkkduOCiglTtjzewHvwH7wDf+xQvFvvx1SvHXZYeZafBLUN2Af0AG4VkbXAs8C9hRGYOpDbvAn/yQmwbAk0bo7Xsx+m0mFhh6UKzxSCyTSP58AxiA5YD8wPIyiVeEzFDLyb7sF/cBD+6CF4t92fMCvz5idBPUowQWw/guWm6xD0MKoOXBn90BRE7preW4Cb/S/YtxfT/VpM2w4YLz+1syrRWGtnAIhID+A7a+27+18TkRMJqtoHhBSeSjDmsGrBBLMPDwrupO54AFM2/rsP5CdBXQDUt9b+Gnn+pYi8TzD1vyaoQuB+2xKscrv0A2jQBO+K/pjDQl2GXMVec+CjLNs+BuagCUrlg6lRG6//UPyH78QfNwLv1pGY4vE9u0x+voavA7J+mhLAz9nsqwrIffRf/GF9YdkSzCVXBr9MmpyKIsffO0SkkL9rVykATL0j8K69HVZ/j//oA7i9e8IO6aDycwc1E3hDRCYQTMlfC7gBeDLzRLI5TRyb26qdInITcBXBirobgSuttasir+0jGKgI8KO1tmM+4k4obvtW3KzHcR8ugroN8a4cgKlWK/cDVbJaBNwjIrdZa/3IirrDItuVyjfT7DjM5X1x08fjZkyAKwbEbZNBfhLUNZF/s84ccW3kB3KYODayaudEMq3aKSJzs6zauQQ4zlq7U0SuI1jJt0vktd+ttS3yEWtCcks/wH/yEdixHXNBD0yHzpgU7U1cxN0IvAr8LCKrgNoEtRbnhxqVSmjeSe3xf92Mm/MUlKuIubhX2CFlK88JqoCTxea6aqe1dkGm/RcT9BYsEtzO7bjZU3DvvQM16+INGI6ppXPzKrDWrhaRlgTXUC2CxQU/sNb64UamEp055xL4dTNu3ov45SvitY+/iqlDniw2n/K7amdv/lr7BoKxIB8RVP+NstbOiX6I4XDLluDPmAC/bcacK5jzumBS/zY2UxVhkWS0mL+mGlOqwIwx0O1q3NYtOPsEfrmKeMefHHZYB4hVgsqzSLfa44C2mTbXiayBczjwjoh8bq1dkeW4hFr10/99B9unT+T3N+eQUrMu5QaNIq1hk5i9f7yUQzzQslBFlfFS8HrfhD9mKG7qaFzZ8pgjm+Z+YIzEKkHlumongIi0B+4E2lprd+3fbq1dE/l3pYgsBI4BDkhQibTqp/vmi2CC1182YM68EHfBpfyWViymS7DHQznEi5zKIq+rfiqVyEyxdLy+d+GPug1/0n14gx7CVI3WepgFE6uuG3+u2ikixQhW7ZybeQcROQZ4HOhord2QaXsFEUmPPM4ATiJT21Uicbt24c/+F/7Dg8EYvFvvx7vkCkxasbBDU0oVYaZUabz+Q8Dz8Cfcg9u+NeyQgBglKGvtXmD/qp1fBZvsMhEZISL7W+YeAkoDz4nIpyKyP4E1Bj4SkaXAAoI2qIRLUG7F1/j3DMDNfwXT7hy8oeMxMazSU0qpgzGVq+LdcCds3oj/6P24PeGPkTLOJeXSMm7t2rXZvhDrqi23Zw9u7izcvJegQiW8Xv0xjZvH7P1zolV8f8mlik9n4j1Q3Fxb8SrRy8F//z+4Kf/EtGmHuWLAIU9GfbByyOu1FXedJJKJW7UCf9pYWLMqmHVcemNKxPfUIkqpos1r3RZ/48+4l2dBlRqYcyW0WDRBFQK3dy/utedwr1koXQ6v/xBMs+PCDksppfLEnNsF1q/FzXkK/7BqeMefEkocmqCizK1ZhT91LPy4AtO6LaZbH0ypMmGHpZRSeWaMgcv74TZtwE0di6tYGVO/UczjiM8JmBKQ8/fhv/4C/r0DYcsmvOsG4V11syYnpVRCMmlpeNcPhgqV8CeOxP2yMeYxaIKKArduDf4Dd+BenAFHt8Ib/gimZZuww1JKqQIxZcoG3c/37sGfNBK3a1fuB0WRJqgCcL6P//Zc/HtuhHVrMFfdjHft7Zgy5cIOTSmlosJUrYl39S3w0/e4GeOJZc9vbYM6RG7jumAOvW8+h2bH4V1+A6Z8pbDDUkqpqDPNjsNceHlQS1SrHubsi2Pyvpqg8sk5h1s0D2engQHTsx/mpPaHPFZAKaUSgelwEaz+HvfSTFz1Opjmxxf6e2qCyge3eRP+kxNg2RJo3ByvZ39Mpcphh6WUUoXuz55969bgT3kYb/DDhb6YqrZB5YFzDv/dd/CH9YPvvsR0vzZYs0mTk1KqCDHp6Xg3DIa0YviPjMTt2F6o76cJKhfuty1BF8tpY6FGHbyh4/DanRO3SyQrpVRhMhUr410/CH7ZgP+vh3D+vkJ7L/0rexD+h//FH9YXli3BXHIl3q0jMYfpEgxKqaLNNGiC6X4NLFuCe3Fmob2PtkFlw23bipv1GO6j/0LdhnhXDij0ulallEok3qln4f+0EjfvRVy9IzDHnhj199AElYX7dDH+zEmwYzvmgh6YDp0xKSlhh6WUUnHHdLkK9+NK/Gnj8KrXxlSrGdXzaxVfhNu5HX/qGPyJ90HZCnh3/hPvXNHkpJRSOTCpaXjX3A7FigVrSP2xM6rn1wQFuC8+wR/aD/f+fzDndcG782FMrXphh6WUUnHPVMwIZppYtwY3fUJUZ5oo0gnK/bETf+ZE/HHDoERJvEEP4XW6FJOaFnZoSimVMEzj5piLLsN9/D/cWy9H7bwxa4MSkQ7AOCAFmGKtHZXl9XTgSeBY4Begi7X2h8hrg4DewD6gv7V2XkHjcd98jj9tHGzeiDnrQkynSzFpxQp6WqWUKpLMWRfhVn6De2E6rk4DyDitwOeMyR2UiKQAE4GzgSZANxFpkmW33sAWa20DYAzwQOTYJkBX4CigAzApcr5D4nb9gT/7X/gP3wkpKXi33Y938RWanJRSqgCMMXhXDIDK1fAff4B9UVieI1ZVfK2A5dbaldba3cBsoFOWfToBMyKPnwdOFxET2T7bWrvLWvs9sDxyvnxzK77ml5t64ea/gml3Lt6QcZgGWfOkUkWDiFwiIstExBcRXfJZFZgpURLvukGwexe/PXwXbu+eAp0vVlV8NYCfMj1fDbTOaR9r7V4R+Q2oFNm+OMuxNbK+gYj0AfpEjicjI+NvQfz6r1fYu2c3FYaPp9jRRft6TE1NzbaMiqIiXBZfABcBj4cdiEoepkZtTM9+7JkyGu+7L6Fx80M+V9KMg7LWTgYmR566TZs2/W0f17UPlapUZfPO3yGb14uSjIwMsiujoiinsqhePblnDbHWfgUgImGHopKMd/wplG9xPL+mFS/YeaIUT27WAJmnYqgZ2ZbtPiKSCpQj6CyRl2PzxJQph1ey1KEcqpRSKh9SozBoN1Z3UB8CDUWkHkFy6Qp0z7LPXKAn8B5wMfCOtdaJyFxgloiMBqoDDYEPYhS3UglNRN4Gqmbz0p3W2jz1B85L9TkU6arSA2g5BKJRDjFJUJE2pb7APIJu5lOttctEZATwkbV2LvAEMFNElgObCZIYkf0s8CWwF7jBWlt40+cqlUSste2jcI5cq89Bq43303IIHKwc8lp9bmK5vnwMJeWHUqFI+qWSRWQhcIu19qM87K7XloqWXK+tZJ1JwuT0IyIfH+z1ovKj5ZDnskhaInKhiKwG2gD/FpG8DIDX36mC/T4VmZ88lEOukqYXn1Iqf6y1LwEvhR2HUjlJ1jsopZRSCa4oJqjJue9SJGg5/EXLIjq0HANaDoECl0OydpJQSimV4IriHZRSSqkEoAlKKaVUXCpSvfhyW5MqWYnIVOA8YIO1tmlkW0XgWaAu8AMg1totYcUYCyJSi2DNsSoE43kmW2vHFcWyiKaiel2BXlv7Fda1VWTuoPK4JlWymk6wllZmdwDzrbUNgfmR58luL3CztbYJcAJwQ+R3oCiWRVQU8esK9Nrar1CurSKToMjbmlRJyVr7fwTTR2X2/+3doU7EUBCF4R8BCklCMIRXwCPQKNR5DjQGhSW8AoKQTAIhvAIKg0GsRi2LBA+I24UNhYSEdtvunM+0ac3N5J5M2t10ZudvnQH7c11UByJiHBH31fkrMKKMbklXiwalzRU4W1NtZStTg/ppJlVtrlQi6xExrs6fKI/maUjaAraBO5LX4p+cq7rU+6nJbGVqUPaLiHgn0TfWJK0Cl8BBRLzM3stWC2tXtv3UdLYyNajG5kotiImkDYDq+NzxeuZC0jIlQOcRcVVdTlmLhjhXdSn3U/2iTQwAAACrSURBVBvZytSgPmdSSVqhjPO46XhNXZrO36I6/mk20JBJWqKMdRlFxMnMrXS1aJBzVZduP7WVrVRfkpC0B5zyNZPquOMlzYWkC2AXWAMmwBFwDQSwCTxS/v75/cfehSJpB7gFHoC36vIh5V15qlo0KWuuwNmaaitbqRqUmZkNR6ZXfGZmNiBuUGZm1ktuUGZm1ktuUGZm1ktuUGZm1ktuUGZm1ktuUGZm1ksf6yHm+I8x09oAAAAASUVORK5CYII=\n",
+      "text/plain": [
+       "<Figure size 432x288 with 4 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "#plot trajectory\n",
+    "plt.subplot(2, 2, 1)\n",
+    "plt.plot(x_bar[0,:],x_bar[1,:])\n",
+    "plt.plot(track[0,:],track[1,:],\"b-\")\n",
+    "plt.axis('equal')\n",
+    "plt.ylabel('y')\n",
+    "plt.xlabel('x')\n",
+    "\n",
+    "plt.subplot(2, 2, 2)\n",
+    "plt.plot(x_bar[2,:])\n",
+    "plt.ylabel('theta(t)')\n",
+    "#plt.xlabel('time')\n",
+    "\n",
+    "plt.subplot(2, 2, 3)\n",
+    "plt.plot(x_bar[3,:])\n",
+    "plt.ylabel('psi(t)')\n",
+    "#plt.xlabel('time')\n",
+    "\n",
+    "plt.subplot(2, 2, 4)\n",
+    "plt.plot(x_bar[4,:])\n",
+    "plt.ylabel('cte(t)')\n",
+    "\n",
+    "plt.tight_layout()\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 12,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "track = compute_path_from_wp([0,2,4,5,10],[0,0,3,1,1])\n",
+    "\n",
+    "u_bar = np.zeros((M,T))\n",
+    "u_bar[0,:] = 1 #m/s\n",
+    "u_bar[1,:] = np.radians(10) #rad/s\n",
+    "\n",
+    "x0 = np.zeros(N)\n",
+    "x0[0] = 2\n",
+    "x0[1] = 2\n",
+    "x0[2] = np.radians(0)\n",
+    "_,psi,cte = calc_err(x0,track)\n",
+    "x0[3]=psi\n",
+    "x0[4]=cte\n",
+    "\n",
+    "x_bar=np.zeros((N,T+1))\n",
+    "x_bar[:,0]=x0\n",
+    "\n",
+    "for t in range (1,T+1):\n",
+    "    xt=x_bar[:,t-1].reshape(5,1)\n",
+    "    ut=u_bar[:,t-1].reshape(2,1)\n",
+    "    \n",
+    "    A,B,C=get_linear_model(xt,ut)\n",
+    "    \n",
+    "    xt_plus_one = np.squeeze(np.dot(A,xt)+np.dot(B,ut)+C)\n",
+    "    \n",
+    "    _,psi,cte = calc_err(xt_plus_one,track)\n",
+    "    xt_plus_one[3]=psi\n",
+    "    xt_plus_one[4]=cte\n",
+    "    \n",
+    "    x_bar[:,t]= xt_plus_one"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 13,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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\n",
+      "text/plain": [
+       "<Figure size 432x288 with 4 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "#plot trajectory\n",
+    "plt.subplot(2, 2, 1)\n",
+    "plt.plot(x_bar[0,:],x_bar[1,:])\n",
+    "plt.plot(track[0,:],track[1,:],\"b-\")\n",
+    "plt.axis('equal')\n",
+    "plt.ylabel('y')\n",
+    "plt.xlabel('x')\n",
+    "\n",
+    "plt.subplot(2, 2, 2)\n",
+    "plt.plot(x_bar[2,:])\n",
+    "plt.ylabel('theta(t)')\n",
+    "#plt.xlabel('time')\n",
+    "\n",
+    "plt.subplot(2, 2, 3)\n",
+    "plt.plot(x_bar[3,:])\n",
+    "plt.ylabel('psi(t)')\n",
+    "#plt.xlabel('time')\n",
+    "\n",
+    "plt.subplot(2, 2, 4)\n",
+    "plt.plot(x_bar[4,:])\n",
+    "plt.ylabel('cte(t)')\n",
+    "\n",
+    "plt.tight_layout()\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "MPC Problem formulation\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 18,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "CPU times: user 289 ms, sys: 617 µs, total: 290 ms\n",
+      "Wall time: 287 ms\n"
+     ]
+    }
+   ],
+   "source": [
+    "%%time\n",
+    "\n",
+    "MAX_SPEED = 1.2\n",
+    "MIN_SPEED = 0.8\n",
+    "MAX_STEER_SPEED = 0.57\n",
+    "\n",
+    "#starting guess\n",
+    "u_bar = np.zeros((M,T))\n",
+    "u_bar[0,:]=1\n",
+    "u_bar[1,:]=0.01\n",
+    "\n",
+    "track = compute_path_from_wp([0,10],[2,2])\n",
+    "\n",
+    "# Starting Condition\n",
+    "x0 = np.zeros(N)\n",
+    "x0[0] = 2\n",
+    "x0[1] = -0.5\n",
+    "x0[2] = np.radians(-30)\n",
+    "_,psi,cte = calc_err(x0,track)\n",
+    "x0[3]=psi\n",
+    "x0[4]=cte\n",
+    "\n",
+    "# Prediction\n",
+    "x_bar=np.zeros((N,T+1))\n",
+    "x_bar[:,0]=x0\n",
+    "\n",
+    "for t in range (1,T+1):\n",
+    "    xt=x_bar[:,t-1].reshape(5,1)\n",
+    "    ut=u_bar[:,t-1].reshape(2,1)\n",
+    "    \n",
+    "    A,B,C=get_linear_model(xt,ut)\n",
+    "    \n",
+    "    xt_plus_one = np.squeeze(np.dot(A,xt)+np.dot(B,ut)+C)\n",
+    "    \n",
+    "    _,psi,cte = calc_err(xt_plus_one,track)\n",
+    "    xt_plus_one[3]=psi\n",
+    "    xt_plus_one[4]=cte\n",
+    "    \n",
+    "    x_bar[:,t]= xt_plus_one\n",
+    "\n",
+    "\n",
+    "#CVXPY MPC problem statement\n",
+    "cost = 0\n",
+    "constr = []\n",
+    "\n",
+    "for t in range(T):\n",
+    "    \n",
+    "    # Cost function\n",
+    "    #cost += 10*cp.sum_squares( x[3, t]) # psi\n",
+    "    #cost += 50*cp.sum_squares( x[4, t]) # cte\n",
+    "    \n",
+    "    # Tracking\n",
+    "    idx,_,_ = calc_err(x_bar[:,t],track)\n",
+    "    delta_x =track[:,idx]-x[0:3,t]\n",
+    "    cost+= cp.quad_form(delta_x,10*np.eye(3))\n",
+    "    #cost += cp.quad_form( x[:, t],10*np.eye(N))\n",
+    "    \n",
+    "    # Actuation effort\n",
+    "    cost += cp.quad_form( u[:, t],1*np.eye(M))\n",
+    "    \n",
+    "    # Actuation rate of change\n",
+    "    if t < (T - 1):\n",
+    "        cost += cp.quad_form(u[:, t + 1] - u[:, t], 1*np.eye(M))\n",
+    "    \n",
+    "    #constrains\n",
+    "    A,B,C=get_linear_model(x_bar[:,t],u_bar[:,t])\n",
+    "    constr += [x[:,t+1] == A*x[:,t] + B*u[:,t] + C.flatten()]\n",
+    "    \n",
+    "# sums problem objectives and concatenates constraints.\n",
+    "constr += [x[:,0] == x0]\n",
+    "constr += [u[0, :] <= MAX_SPEED]\n",
+    "constr += [u[0, :] >= MIN_SPEED]\n",
+    "constr += [cp.abs(u[1, :]) <= MAX_STEER_SPEED]\n",
+    "\n",
+    "\n",
+    "prob = cp.Problem(cp.Minimize(cost), constr)\n",
+    "solution = prob.solve(solver=cp.ECOS, verbose=False)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Print Results:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 19,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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\n",
+      "text/plain": [
+       "<Figure size 432x288 with 3 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "x_mpc=np.array(x.value[0, :]).flatten()\n",
+    "y_mpc=np.array(x.value[1, :]).flatten()\n",
+    "theta_mpc=np.array(x.value[1, :]).flatten()\n",
+    "psi_mpc=np.array(x.value[1, :]).flatten()\n",
+    "cte_mpc=np.array(x.value[1, :]).flatten()\n",
+    "v_mpc=np.array(u.value[0, :]).flatten()\n",
+    "w_mpc=np.array(u.value[1, :]).flatten()\n",
+    "\n",
+    "#simulate robot state trajectory for optimized U\n",
+    "x_traj=predict(x0, np.vstack((v_mpc,w_mpc)))\n",
+    "\n",
+    "#plot trajectory\n",
+    "plt.subplot(2, 2, 1)\n",
+    "plt.plot(track[0,:],track[1,:],\"b*\")\n",
+    "plt.plot(x_traj[0,:],x_traj[1,:])\n",
+    "plt.axis(\"equal\")\n",
+    "plt.ylabel('y')\n",
+    "plt.xlabel('x')\n",
+    "\n",
+    "#plot v(t)\n",
+    "plt.subplot(2, 2, 2)\n",
+    "plt.plot(v_mpc)\n",
+    "plt.ylabel('v(t)')\n",
+    "#plt.xlabel('time')\n",
+    "\n",
+    "#plot w(t)\n",
+    "plt.subplot(2, 2, 3)\n",
+    "plt.plot(w_mpc)\n",
+    "plt.ylabel('w(t)')\n",
+    "#plt.xlabel('time')\n",
+    "\n",
+    "#plot theta(t)\n",
+    "#plt.subplot(2, 2, 4)\n",
+    "#plt.plot(cte_mpc)\n",
+    "#plt.ylabel('cte(t)')\n",
+    "#plt.xlabel('time')\n",
+    "#plt.legend(loc='best')\n",
+    "\n",
+    "plt.tight_layout()\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "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": "3.5.2"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
diff --git a/MPC_cvxpy.ipynb b/MPC_cvxpy.ipynb
index e929ca3..3de729d 100644
--- a/MPC_cvxpy.ipynb
+++ b/MPC_cvxpy.ipynb
@@ -2,7 +2,7 @@
  "cells": [
   {
    "cell_type": "code",
-   "execution_count": 2,
+   "execution_count": 1,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -29,7 +29,7 @@
     "* $y$ coordinate of the robot\n",
     "* $\\theta$ heading of the robot\n",
     "* $\\psi$ heading error = $\\psi = \\theta_{ref} - \\theta$\n",
-    "* $cte$ lateral error = distance of the robot from the path \n",
+    "* $cte$ crosstrack error = lateral distance of the robot from the path \n",
     "\n",
     "The inputs of the model are:\n",
     "\n",
@@ -42,16 +42,16 @@
     "* $\\dot{y} = v\\sin{\\theta}$\n",
     "* $\\dot{\\theta} = w$\n",
     "* $\\dot{\\psi} = -w$\n",
-    "* $\\dot{cte} = v\\sin{\\psi}$\n",
+    "* $\\dot{cte} = v\\sin{-\\psi}$\n",
     "\n",
     "The **Continuous** State Space Model is\n",
     "\n",
-    "${\\dot{x}} = Ax + Bu $\n",
+    "$ {\\dot{x}} = Ax + Bu $\n",
     "\n",
     "with:\n",
     "\n",
-    "$\n",
-    "A = \\quad\n",
+    "$ A =\n",
+    "\\quad\n",
     "\\begin{bmatrix}\n",
     "\\frac{\\partial f(x,u)}{\\partial x} & \\frac{\\partial f(x,u)}{\\partial y} & \\frac{\\partial f(x,u)}{\\partial \\theta} & \\frac{\\partial f(x,u)}{\\partial \\psi} & \\frac{\\partial f(x,u)}{\\partial cte} \\\\\n",
     "\\end{bmatrix}\n",
@@ -63,13 +63,12 @@
     "0 & 0 & vcos(\\theta) & 0 & 0 \\\\\n",
     "0 & 0 & 0 & 0 & 0 \\\\\n",
     "0 & 0 & 0 & 0 & 0 \\\\\n",
-    "0 & 0 & 0 & 0 & vcos(\\psi) \n",
+    "0 & 0 & 0 & vcos(-\\psi) & 0 \n",
     "\\end{bmatrix}\n",
-    "\\quad\n",
-    "$\n",
+    "\\quad $\n",
     "\n",
-    "$\n",
-    "B = \\quad\n",
+    "\n",
+    "$ B = \\quad\n",
     "\\begin{bmatrix}\n",
     "\\frac{\\partial f(x,u)}{\\partial v} & \\frac{\\partial f(x,u)}{\\partial w} \\\\\n",
     "\\end{bmatrix}\n",
@@ -83,10 +82,9 @@
     "0 & -1 \\\\\n",
     "\\sin{(-\\psi_t)} & 0  \n",
     "\\end{bmatrix}\n",
-    "\\quad\n",
-    "$\n",
+    "\\quad $\n",
     "\n",
-    "discretize with forward Euler Integration\n",
+    "discretize with forward Euler Integration for time step dt:\n",
     "\n",
     "* ${x_{t+1}} = x_{t} + v_t\\cos{\\theta}*dt$\n",
     "* ${y_{t+1}} = y_{t} + v_t\\sin{\\theta}*dt$\n",
@@ -94,7 +92,7 @@
     "* ${\\psi_{t+1}} = \\psi_{t} - w_t*dt$\n",
     "* ${cte_{t+1}} = cte_{t} + v_t\\sin{-\\psi}*dt$\n",
     "\n",
-    "The **Discrete** State Space Model is\n",
+    "The **Discrete** State Space Model is then:\n",
     "\n",
     "${x_{t+1}} = Ax_t + Bu_t $\n",
     "\n",
@@ -107,7 +105,7 @@
     "0 & 1 & v\\cos{\\theta}dt & 0 & 0 \\\\\n",
     "0 & 0 & 1 & 0 & 0 \\\\\n",
     "0 & 0 & 0 & 1 & 0 \\\\\n",
-    "0 & 0 & 0 & 0 & 1 +vcos{-\\psi}dt\n",
+    "0 & 0 & 0 & vcos{(-\\psi)}dt & 1 \n",
     "\\end{bmatrix}\n",
     "\\quad\n",
     "$\n",
@@ -130,24 +128,38 @@
     "\n",
     "So:\n",
     "\n",
-    "$ x_{k+1} = x_k + (f(\\bar{x},\\bar{u}) +  A(x-\\bar{x}) + B(u-\\bar{u}) )dt $\n",
+    "$ x{t+1} = x_t + (f(\\bar{x},\\bar{u}) +  A(x-\\bar{x}) + B(u-\\bar{u}) )dt $\n",
     "\n",
-    "$ x_{k+1} = (I+dtA)x_k + dtBu_k +dt(f(\\bar{x},\\bar{u}) - A\\bar{x} - B\\bar{u}))$"
+    "$ x_{t+1} = (I+dtA)x_t + dtBu_t +dt(f(\\bar{x},\\bar{u}) - A\\bar{x} - B\\bar{u}))$\n",
+    "\n",
+    "The Discrete linearized kinematics model is\n",
+    "\n",
+    "$ x_{t+1} = A'x_t + B' u_t + C' $\n",
+    "\n",
+    "with:\n",
+    "\n",
+    "A' = I+dtA\n",
+    "\n",
+    "B' = dtB\n",
+    "\n",
+    "C' = dt(f(\\bar{x},\\bar{u}) - A\\bar{x} - B\\bar{u})"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "NB: psi and cte are expressed w.r.t. the reference point frame.\n",
+    "------------------\n",
+    "NB: psi and cte are expressed w.r.t. the reference frame.\n",
     "in the reference frame of the veicle the equtions would be:\n",
     "* psi_dot = w\n",
-    "* cte_dot = sin(psi)"
+    "* cte_dot = sin(psi)\n",
+    "-----------------"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 48,
+   "execution_count": 2,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -159,37 +171,12 @@
     "dt = 0.25 #discretization step\n",
     "\n",
     "x = cp.Variable((N, T+1))\n",
-    "u = cp.Variable((M, T))\n",
-    "\n",
-    "def get_model(x,u):\n",
-    "    \n",
-    "    x_t = x[0]\n",
-    "    y_t = x[1]\n",
-    "    theta_t = x[2]\n",
-    "    psi_t = x[3]\n",
-    "    cte_t = x[4]\n",
-    "    \n",
-    "    v_t = u[0]\n",
-    "    u_t = u[1]\n",
-    "    \n",
-    "    A = np.eye(N)\n",
-    "    A[0,2]=-v_t*np.sin(theta_t)*dt\n",
-    "    A[1,2]=v_t*np.cos(theta_t)*dt\n",
-    "    A[4,4]=1+v_t*np.cos(psi_t)*dt\n",
-    "    \n",
-    "    B = np.zeros((N,M))\n",
-    "    B[0,0]=np.cos(theta_t)*dt\n",
-    "    B[1,0]=np.sin(theta_t)*dt\n",
-    "    B[2,1]=dt\n",
-    "    B[3,1]=-dt\n",
-    "    B[4,0]=np.sin(-psi_t)*dt\n",
-    "    \n",
-    "    return A,B"
+    "u = cp.Variable((M, T))"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 71,
+   "execution_count": 3,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -207,7 +194,7 @@
     "    A = np.zeros((N,N))\n",
     "    A[0,2]=-v*np.sin(theta)\n",
     "    A[1,2]=v*np.cos(theta)\n",
-    "    A[4,4]=v*np.cos(-psi)\n",
+    "    A[4,3]=v*np.cos(-psi)\n",
     "    A_lin=np.eye(N)+dt*A\n",
     "    \n",
     "    B = np.zeros((N,M))\n",
@@ -224,34 +211,6 @@
     "    return A_lin,B_lin,C_lin"
    ]
   },
-  {
-   "cell_type": "code",
-   "execution_count": 4,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "(array([[1., 0., 0., 0., 0.],\n",
-       "        [0., 1., 0., 0., 0.],\n",
-       "        [0., 0., 1., 0., 0.],\n",
-       "        [0., 0., 0., 1., 0.],\n",
-       "        [0., 0., 0., 0., 1.]]), array([[ 0.25,  0.  ],\n",
-       "        [ 0.  ,  0.  ],\n",
-       "        [ 0.  ,  0.25],\n",
-       "        [ 0.  , -0.25],\n",
-       "        [-0.  ,  0.  ]]))"
-      ]
-     },
-     "execution_count": 4,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "get_model(np.zeros(N))"
-   ]
-  },
   {
    "cell_type": "markdown",
    "metadata": {},
@@ -261,14 +220,13 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 50,
+   "execution_count": 4,
    "metadata": {},
    "outputs": [],
    "source": [
     "# Define process model\n",
     "def kinematics_model(x,t,u):\n",
     "\n",
-    "    # calculate derivative\n",
     "    dxdt = u[0]*np.cos(x[2])\n",
     "    dydt = u[0]*np.sin(x[2])\n",
     "    dthetadt = u[1]\n",
@@ -319,10 +277,21 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 51,
+   "execution_count": 5,
    "metadata": {},
-   "outputs": [],
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "CPU times: user 4.05 ms, sys: 38 µs, total: 4.09 ms\n",
+      "Wall time: 3.75 ms\n"
+     ]
+    }
+   ],
    "source": [
+    "%%time\n",
+    "\n",
     "u_bar = np.zeros((M,T))\n",
     "u_bar[0,:] = 1 #m/s\n",
     "u_bar[1,:] = np.radians(-10) #rad/s\n",
@@ -346,7 +315,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 57,
+   "execution_count": 6,
    "metadata": {},
    "outputs": [
     {
@@ -407,10 +376,21 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 72,
+   "execution_count": 7,
    "metadata": {},
-   "outputs": [],
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "CPU times: user 2.53 ms, sys: 2.36 ms, total: 4.9 ms\n",
+      "Wall time: 1.44 ms\n"
+     ]
+    }
+   ],
    "source": [
+    "%%time\n",
+    "\n",
     "u_bar = np.zeros((M,T))\n",
     "u_bar[0,:] = 1 #m/s\n",
     "u_bar[1,:] = np.radians(-10) #rad/s\n",
@@ -438,7 +418,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 73,
+   "execution_count": 8,
    "metadata": {},
    "outputs": [
     {
@@ -492,12 +472,16 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "this way of predictiting psi and cte does not work for non-constant heading trajectories, appropriate functions have to be developed."
+    "------------------\n",
+    "\n",
+    "the kinematics model predictits psi and cte for constant heading references. So, for non-constant paths appropriate functions have to be developed.\n",
+    "\n",
+    "-----------------"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 36,
+   "execution_count": 9,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -586,7 +570,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 75,
+   "execution_count": 10,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -624,7 +608,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 76,
+   "execution_count": 11,
    "metadata": {},
    "outputs": [
     {
@@ -669,7 +653,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 79,
+   "execution_count": 12,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -707,7 +691,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 81,
+   "execution_count": 13,
    "metadata": {},
    "outputs": [
     {
@@ -759,19 +743,21 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 109,
+   "execution_count": 18,
    "metadata": {},
    "outputs": [
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "optimal\n",
-      "time elapsed: 0.2521398067474365\n"
+      "CPU times: user 289 ms, sys: 617 µs, total: 290 ms\n",
+      "Wall time: 287 ms\n"
      ]
     }
    ],
    "source": [
+    "%%time\n",
+    "\n",
     "MAX_SPEED = 1.2\n",
     "MIN_SPEED = 0.8\n",
     "MAX_STEER_SPEED = 0.57\n",
@@ -785,15 +771,13 @@
     "\n",
     "# Starting Condition\n",
     "x0 = np.zeros(N)\n",
-    "x0[0] = 0\n",
-    "x0[1] = 1\n",
+    "x0[0] = 2\n",
+    "x0[1] = -0.5\n",
     "x0[2] = np.radians(-30)\n",
     "_,psi,cte = calc_err(x0,track)\n",
     "x0[3]=psi\n",
     "x0[4]=cte\n",
     "\n",
-    "start = time.time()\n",
-    "\n",
     "# Prediction\n",
     "x_bar=np.zeros((N,T+1))\n",
     "x_bar[:,0]=x0\n",
@@ -848,12 +832,7 @@
     "\n",
     "\n",
     "prob = cp.Problem(cp.Minimize(cost), constr)\n",
-    "solution = prob.solve(solver=cp.ECOS, verbose=False)\n",
-    "\n",
-    "end = time.time()\n",
-    "\n",
-    "print(prob.status)\n",
-    "print(\"time elapsed: {}\".format(end-start))"
+    "solution = prob.solve(solver=cp.ECOS, verbose=False)"
    ]
   },
   {
@@ -865,12 +844,12 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 110,
+   "execution_count": 19,
    "metadata": {},
    "outputs": [
     {
      "data": {
-      "image/png": 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\n",
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\n",
       "text/plain": [
        "<Figure size 432x288 with 3 Axes>"
       ]
@@ -890,7 +869,7 @@
     "v_mpc=np.array(u.value[0, :]).flatten()\n",
     "w_mpc=np.array(u.value[1, :]).flatten()\n",
     "\n",
-    "#simulate q for optimized u_hat\n",
+    "#simulate robot state trajectory for optimized U\n",
     "x_traj=predict(x0, np.vstack((v_mpc,w_mpc)))\n",
     "\n",
     "#plot trajectory\n",