CarFollowing-KalmanFilter-noAnimation.ipynb

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     },
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   "source": [
    "import numpy as np\n",
    "import time\n",
    "# from filterpy.common import Q_discrete_white_noise\n",
    "from bokeh.io import show, output_notebook\n",
    "from bokeh.plotting import figure \n",
    "from bokeh.layouts import row, column\n",
    "output_notebook()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### The predict and update steps of the Kalman Filter are defined in the class below"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "class kalman_filter:\n",
    "    def __init__(self, dim_x, dim_z):\n",
    "        self.x = np.zeros([dim_x, 1])\n",
    "        self.F = np.zeros([dim_x, dim_x])\n",
    "        self.H = np.zeros([1, dim_x])\n",
    "        self.P = np.zeros([dim_x, dim_x])\n",
    "        self.R = np.zeros([dim_z])\n",
    "        self.Q = np.zeros([dim_x, dim_x])\n",
    "        self.B = np.zeros([dim_x, 1])\n",
    "        self.u = np.zeros([1])\n",
    "        \n",
    "    def predict(self):\n",
    "        # x_k = A_k * x_k-1 + B_k * u_k-1 (predicted next state)\n",
    "        # P_k = A_k * P_k-1 * (A_k)^T + Q_k (predicted next state covariance)\n",
    "        self.x = np.matmul(self.A, self.x) + (self.B * self.u)\n",
    "        self.P = np.matmul(self.A, np.matmul(self.P, self.A.transpose())) + self.Q\n",
    "        \n",
    "    def update(self, z):\n",
    "        # K_k = P_k-1 * (H_k)^T * (H_k * P_k-1 * (H_k)^T + R_k)^-1 (Kalman Gain)\n",
    "        # x_k = x_k-1 + K_k * (z_k - H_k * x_k-1) (updated next state)\n",
    "        # P_k = P_k-1 * (I - K_k * H_k) (updated next state covariance)\n",
    "        self.K = (np.matmul(self.P, self.H.transpose()) / \n",
    "             (np.matmul(self.H, np.matmul(self.P, self.H.transpose())) + self.R))\n",
    "#         self.K = np.array([[0], [0]])\n",
    "#         self.K = np.array([[1], [1]])\n",
    "        self.x = self.x + self.K * (z - np.matmul(self.H, self.x))\n",
    "        self.P = np.matmul((np.identity(2) - np.matmul(self.K, self.H)), self.P)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Change the initial values in the cell below to simulate different conditions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Initial values\n",
    "car1_initpos = 4  # starting position of the car in the front\n",
    "car2_initpos = 0  # starting position of the following cars (keep at 0)\n",
    "car1_vel = 8  # velocity of the car in the front\n",
    "car2_initvel = 8  # starting velocity of the following cars\n",
    "car_size = 2  # size of the car \n",
    "dist_sep = 2 * car_size  # desired ideal separation between the car in the front and the cars following\n",
    "dist_size = 50  # length of the track in arb. units\n",
    "car1_acc = 0  # accelaration of the car in the front (keep at 0)\n",
    "car2_initacc = 0 # starting accelaration of the following cars\n",
    "car2_maxacc = 1.5  # maximum possible accelaration for the following cars\n",
    "car2_maxbrake = 10  # maximum possible decelaration for the following cars\n",
    "noise_type = 'gauss'  # type of measurement noise (can use 'gauss', 'power', and 'uniform', if anything else is entered measurement noise will be 0\n",
    "measurement_noise = 2  # measurement noise std. dev.\n",
    "process_noise = 2  # process noise std. dev.\n",
    "dt = 0.01  # time step \n",
    "timesteps = int((dist_size - car1_initpos) / car1_vel / dt)  # total number of timesteps "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Initializing the arrays that will be used for plotting"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Initializing arrays for plotting\n",
    "timepoints = np.zeros(timesteps)\n",
    "poscar1 = np.zeros(timesteps)\n",
    "poscar1_est = np.zeros(timesteps)\n",
    "poscar1_meas = np.zeros(timesteps)\n",
    "poscar2 = np.zeros(timesteps)\n",
    "car_sep = np.zeros(timesteps)\n",
    "car2_acc = np.zeros(timesteps)\n",
    "poscar2_kf = np.zeros(timesteps)\n",
    "car_sep_kf = np.zeros(timesteps)\n",
    "car2_acc_kf = np.zeros(timesteps)\n",
    "noiserange = np.zeros(timesteps)\n",
    "pdf_proc = np.zeros(timesteps)\n",
    "pdf_meas = np.zeros(timesteps)\n",
    "kf_gain = np.zeros([timesteps, 2])\n",
    "def InitPlotArrays():    \n",
    "    timepoints.fill(np.nan)\n",
    "    poscar1.fill(np.nan)\n",
    "    poscar1_est.fill(np.nan)\n",
    "    poscar1_meas.fill(np.nan)\n",
    "    poscar2.fill(np.nan)\n",
    "    car_sep.fill(np.nan)\n",
    "    car2_acc.fill(np.nan)\n",
    "    poscar2_kf.fill(np.nan)\n",
    "    car_sep_kf.fill(np.nan)\n",
    "    car2_acc_kf.fill(np.nan)\n",
    "    noiserange.fill(np.nan)\n",
    "    pdf_proc.fill(np.nan)\n",
    "    pdf_meas.fill(np.nan)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### The car following function is defined below. The car in the front is moving with constant velocity. The car following has a proportional controller."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "def RunCarFollowing(PN, MN):\n",
    "    \n",
    "    # Setup initial values\n",
    "    InitPlotArrays()\n",
    "    pos1real = car1_initpos\n",
    "    pos2 = car2_initpos\n",
    "    pos2_kf = car2_initpos\n",
    "    vel2 = car2_initvel\n",
    "    vel2_kf = car2_initvel\n",
    "    acc2 = car2_initacc\n",
    "    acc2_kf = car2_initacc\n",
    "    tt = 0\n",
    "    realt = 0\n",
    "    resp_flag = False\n",
    "    resp_kf_flag = False\n",
    "    \n",
    "    process_noise_level = PN\n",
    "    measurement_noise_level = MN\n",
    "    \n",
    "    # Update noise distributions\n",
    "#     if noise_type == 'gauss':\n",
    "#         nlim = max(process_noise_level, measurement_noise_level)\n",
    "#         global noiserange, pdf_proc, pdf_meas\n",
    "#         noiserange = np.linspace(-4 * nlim, 4 * nlim, timesteps)\n",
    "#         # PDFs of process and measurement noise\n",
    "#         if process_noise_level == 0:\n",
    "#             pdf_proc = np.zeros(np.shape(noiserange))\n",
    "#         else:\n",
    "#             pdf_proc = (1 / (process_noise_level * np.sqrt(2 * np.pi)) \n",
    "#                         * np.exp(-noiserange**2 / (2 * process_noise_level**2)))\n",
    "#         if measurement_noise_level == 0:\n",
    "#             pdf_meas = np.zeros(np.shape(noiserange))\n",
    "#         else:\n",
    "#             pdf_meas = (1 / (measurement_noise_level * np.sqrt(2 * np.pi)) \n",
    "#                         * np.exp(-noiserange**2 / (2 * measurement_noise_level**2)))\n",
    "    \n",
    "    # Initialize the Kalman Filter\n",
    "    # initialize Kalman filter object\n",
    "    f = kalman_filter(dim_x=2, dim_z=1)\n",
    "    # Initial state\n",
    "    f.x = np.array([[car1_initpos], [car1_vel]])\n",
    "    # State transition matrix\n",
    "    f.A = np.array([[1.0, dt],  \n",
    "                    [0.0, 1.0]])\n",
    "    # Control vector\n",
    "    f.u = np.array([[acc2_kf]])\n",
    "    # Control transition matrix\n",
    "    f.B = np.array([[0.5 * dt**2], [dt]])\n",
    "    # Measurement function\n",
    "    f.H = np.array([[1.0, 0.0]]) \n",
    "    # Apriori state covariance\n",
    "    f.P = np.array([[1000.0, 0.0],  \n",
    "                    [0.0, 1000.0]])\n",
    "    # Measurement noise covariance\n",
    "    f.R = np.array([[measurement_noise_level**2]])  \n",
    "    # Process noise\n",
    "    f.Q = np.matmul(f.B, f.B.transpose()) * process_noise_level**2\n",
    "#     if process_noise_level == 0:\n",
    "#         f.Q = np.zeros([2, 2])\n",
    "#     else:\n",
    "#         f.Q = Q_discrete_white_noise(dim=2, dt=dt, var=process_noise_level**2)\n",
    "\n",
    "    \n",
    "    # Loop for updating positions\n",
    "    for tt in range(timesteps):\n",
    "\n",
    "        # Real position of leading car\n",
    "        pos1real = pos1real + car1_vel * dt\n",
    "\n",
    "        # Measurement of position of leading car with noise\n",
    "        if noise_type == 'gauss':\n",
    "            z = np.random.normal(pos1real, measurement_noise_level)\n",
    "        elif noise_type == 'power':\n",
    "            z = (pos1real + 2.0 * measurement_noise_level \n",
    "                 * np.sign((np.random.uniform() - 0.5)) * np.random.power(5))\n",
    "        elif noise_type == 'uniform':\n",
    "            z = pos1real + 4.0 * measurement_noise_level * (np.random.uniform() - 0.5)\n",
    "        else:\n",
    "            z = pos1real\n",
    "\n",
    "        # Kalman Filter predict and update steps\n",
    "        \n",
    "        # predict next state using Kalman filter state propagation equation\n",
    "        # x_k = F_k * x_k-1 + B_k * u_k-1 + w_k (predicted next state)\n",
    "        # P_k = F_k * P_k-1 * (F_k)^T + Q_k (predicted next state covariance)\n",
    "        f.predict()  \n",
    "        \n",
    "        # find updated state using Kalman gain and new measurement (z)\n",
    "        # K_k = P_k-1 * (H_k)^T * (H_k * P_k-1 * (H_k)^T + R_k)^-1 (Kalman Gain)\n",
    "        # x_k = x_k-1 + K_k * (z_k - H_k * x_k-1) (updated next state)\n",
    "        # P_k = P_k-1 * (I - K_k * H_k) (updated next state covariance)\n",
    "        f.update(z)  \n",
    "\n",
    "        # Following car update without Kalman Filter\n",
    "        dx = z - pos2\n",
    "        err = (dx - dist_sep) / dist_sep\n",
    "        if err > 1.0: err = 1.0\n",
    "        # Acceleration model\n",
    "        if err > 0:\n",
    "            acc2 = err * car2_maxacc\n",
    "        else:\n",
    "            acc2 = err * car2_maxbrake\n",
    "        vel2 = vel2 + acc2 * dt\n",
    "        # crash protection \n",
    "        if vel2 < 0:\n",
    "            vel2 = 0\n",
    "            acc2 = 0\n",
    "        # update position\n",
    "        pos2 = pos2 + vel2 * dt + 0.5 * acc2 * dt**2\n",
    "        \n",
    "        # Calculate how much time to get to ideal separation\n",
    "        if tt > int(timesteps / 5) and resp_flag == False:\n",
    "            if err < 0.03 and err > -0.03:\n",
    "                global resp\n",
    "                resp = tt * dt\n",
    "                resp_flag = True\n",
    "\n",
    "        # Following car update with Kalman Filter\n",
    "        dx_kf = f.x[0, 0] - pos2_kf\n",
    "        err_kf = (dx_kf - dist_sep) / dist_sep\n",
    "        if err_kf > 1.0: err_kf = 1.0\n",
    "        # Acceleration model\n",
    "        if err_kf > 0:\n",
    "            acc2_kf = err_kf * car2_maxacc\n",
    "        else:\n",
    "            acc2_kf = err_kf * car2_maxbrake\n",
    "        vel2_kf = vel2_kf + acc2_kf * dt\n",
    "        # crash protection\n",
    "        if vel2_kf < 0:\n",
    "            vel2_kf = 0\n",
    "            acc2_kf = 0\n",
    "        # update position\n",
    "        pos2_kf = pos2_kf + vel2_kf * dt + 0.5 * acc2_kf * dt**2\n",
    "        \n",
    "        # Calculate how much time to get to ideal separation\n",
    "        if tt > int(timesteps / 5) and resp_kf_flag == False:\n",
    "            if err_kf < 0.03 and err > -0.03:\n",
    "                global resp_kf\n",
    "                resp_kf = tt * dt\n",
    "                resp_kf_flag = True\n",
    "\n",
    "        # Update arrays for plotting\n",
    "        global timepoints, poscar1, poscar2, poscar1_est, car_sep\n",
    "        timepoints[tt] = tt * dt\n",
    "        poscar1[tt] = pos1real\n",
    "        poscar1_est[tt] = f.x[0, 0]\n",
    "        poscar1_meas[tt] = z\n",
    "        poscar2[tt] = pos2\n",
    "        poscar2_kf[tt] = pos2_kf\n",
    "        car_sep[tt] = pos1real - pos2\n",
    "        car_sep_kf[tt] = pos1real - pos2_kf  \n",
    "        car2_acc[tt] = acc2\n",
    "        car2_acc_kf[tt] = acc2_kf\n",
    "        kf_gain[tt, :] = f.K.T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Run the car following function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "RunCarFollowing(process_noise, measurement_noise)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Plot the results"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
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Application\",\"version\":\"0.13.0\"}};\n",
       "  var render_items = [{\"docid\":\"6cbe7b6a-3cc8-4180-ab27-fa277fe52f47\",\"roots\":{\"196185c1-dddb-4a34-bc07-d06da8161700\":\"f304cdc3-5815-4dcf-941c-3a2eb9007b8a\"}}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        embed_document(root);\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "      attempts++;\n",
       "      if (attempts > 100) {\n",
       "        console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\")\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "196185c1-dddb-4a34-bc07-d06da8161700"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Setup plots\n",
    "\n",
    "# Figure 1: Noise distributions\n",
    "# p5 = figure(plot_width = 400, plot_height = 200, title='Noise Distributions')\n",
    "# r11 = p5.line(noiserange, pdf_proc, line_color='seagreen', \n",
    "#               line_width=4, legend='Process')\n",
    "# r12 = p5.line(noiserange, pdf_meas, line_color='greenyellow', \n",
    "#               line_width=4, legend='Measured')\n",
    "\n",
    "p5 = figure(plot_width = 400, plot_height = 300, y_range=[0, 1], \n",
    "            title='Kalman Gain')\n",
    "r11 = p5.line(timepoints, kf_gain[:, 0], line_color='seagreen', \n",
    "              line_width=4, legend='Position', alpha=0.6)\n",
    "r12 = p5.line(timepoints, kf_gain[:, 1], line_color='greenyellow', \n",
    "              line_width=4, legend='Velocity', alpha=0.6)\n",
    "p5.toolbar.logo = None\n",
    "p5.toolbar_location = None\n",
    "p5.legend.location = 'top_right'\n",
    "\n",
    "# Figure 2: Car positions\n",
    "p1 = figure(plot_width=400, plot_height=300, x_axis_label='time', y_axis_label='position',\n",
    "            title='Car Positions', x_range=[0, timesteps * dt], y_range=[0, dist_size])\n",
    "r1 = p1.line(timepoints, poscar1, color='darkslategray', line_width=4)\n",
    "r2 = p1.line(timepoints, poscar1_meas, color='royalblue', \n",
    "             legend='Measured', line_width=2, alpha=0.6)\n",
    "r3 = p1.line(timepoints, poscar1_est, color='skyblue', \n",
    "             legend='KF estimated', line_width=2, alpha=0.6)\n",
    "r4 = p1.line(timepoints, poscar2, color='gold', line_width=4, alpha=0.6)\n",
    "r5 = p1.line(timepoints, poscar2_kf, color='crimson', line_width=4, alpha=0.6)\n",
    "p1.legend.location = 'top_left'\n",
    "p1.toolbar.logo = None\n",
    "p1.toolbar_location = None\n",
    "\n",
    "# Figure 3: Car separations\n",
    "p2 = figure(plot_width=400, plot_height=300, x_axis_label='time', \n",
    "            y_axis_label='separation', title='Car Separations', \n",
    "            x_range=[0, timesteps * dt], y_range=[0, 4 * dist_sep])\n",
    "r6 = p2.line(timepoints, car_sep, color='gold', line_width=4, alpha=0.6)\n",
    "r7 = p2.line(timepoints, car_sep_kf, color='crimson', line_width=4, alpha=0.6)\n",
    "p2.toolbar.logo = None\n",
    "p2.toolbar_location = None\n",
    "\n",
    "# Figure 4: Car accelarations\n",
    "p3 = figure(plot_width=400, plot_height=300, x_axis_label='time', \n",
    "            y_axis_label='acceleration', title='Following Car Acceleration', \n",
    "            x_range=[0, timesteps * dt], y_range=[-car2_maxbrake, car2_maxacc])\n",
    "r8 = p3.line(timepoints, car2_acc, color='gold', line_width=4, alpha=0.6)\n",
    "r9 = p3.line(timepoints, car2_acc_kf, color='crimson', line_width=4, alpha=0.6)\n",
    "p3.toolbar.logo = None\n",
    "p3.toolbar_location = None\n",
    "\n",
    "show(row(column(p5, p1), column(p2, p3)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Energy and response time trade-off"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Average energy (without KF) 226.1263395030959 [arb units]\n",
      "Average energy (with KF) 14.307319599056516 [arb units]\n",
      "Average response time (without KF) 1.2182 [s]\n",
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       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
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       "  if (root.Bokeh !== undefined) {\n",
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       "        console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\")\n",
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      "application/vnd.bokehjs_exec.v0+json": ""
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       "id": "845117bc-d4b2-48b3-ae5b-926c90626c73"
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   "source": [
    "# Define arrays for measurement and process noise\n",
    "arr_len = 100\n",
    "measurement_noise = np.linspace(0.1, 1, arr_len)\n",
    "process_noise = np.linspace(0.1, 0.1, arr_len) \n",
    "\n",
    "# Initialize energy and time response arrays\n",
    "car2_acc_int = np.zeros(arr_len)\n",
    "car2_acc_kf_int = np.zeros(arr_len)\n",
    "car2_resp = np.zeros(arr_len)\n",
    "car2_kf_resp = np.zeros(arr_len)\n",
    "\n",
    "# Run the \"RunCarFollowing\" in the loop to simulate all noise values\n",
    "for n in range(arr_len):\n",
    "    RunCarFollowing(process_noise[n], measurement_noise[n])\n",
    "    global car2_acc_int, car2_acc_kf_int\n",
    "    car2_acc_int[n] = np.sum(car2_acc**2)\n",
    "    car2_acc_kf_int[n] = np.sum(car2_acc_kf**2)\n",
    "    car2_resp[n] = resp\n",
    "    car2_kf_resp[n] = resp_kf\n",
    "\n",
    "# Calculate average across all noise values\n",
    "print('Average energy (without KF)', np.mean(car2_acc_int), '[arb units]')\n",
    "print('Average energy (with KF)', np.mean(car2_acc_kf_int), '[arb units]')\n",
    "print('Average response time (without KF)', np.mean(car2_resp), '[s]')\n",
    "print('Average response time (with KF)', np.mean(car2_kf_resp), '[s]')\n",
    "\n",
    "# Plot results\n",
    "# Energy trade-off \n",
    "p6 = figure(plot_width=400, plot_height=400, x_axis_label='Noise Index', \n",
    "            y_axis_label='energy (arb. units)', title='Energy trade off')\n",
    "p6.line(range(arr_len), car2_acc_int, line_color='gold', \n",
    "        line_width=2, legend='without KF')\n",
    "p6.line(range(arr_len), car2_acc_kf_int, line_color='crimson', \n",
    "        line_width=2, legend='with KF')\n",
    "p6.legend.location = 'top_left'\n",
    "\n",
    "# Response time trade-off\n",
    "p7 = figure(plot_width=400, plot_height=400, x_axis_label='Noise Index', \n",
    "            y_axis_label='response (s)', title='Response time trade off')\n",
    "p7.line(range(arr_len), car2_resp, line_color='gold', \n",
    "        line_width=2, legend='without KF')\n",
    "p7.line(range(arr_len), car2_kf_resp, line_color='crimson', \n",
    "        line_width=2, legend='with KF')\n",
    "p7.legend.location = 'top_left'\n",
    "show(row(p6, p7))"
   ]
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