Updated for UKF bug fixes. Added summary.

09_Unscented_Kalman_Filter.ipynb

@@ -289,9 +289,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAtUAAAFfCAYAAACBcDBIAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x4502dd8>"
+       "<matplotlib.figure.Figure at 0x541ce80>"
       ]
      },
      "metadata": {},
@@ -351,7 +351,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x451e2b0>"
+       "<matplotlib.figure.Figure at 0x5427358>"
       ]
      },
      "metadata": {},
@@ -397,14 +397,14 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "Difference in mean x=0.039, y=42.884\n"
+      "Difference in mean x=0.042, y=44.008\n"
      ]
     },
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x8554438>"
+       "<matplotlib.figure.Figure at 0x541cf98>"
       ]
      },
      "metadata": {},
@@ -489,7 +489,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x78eab38>"
+       "<matplotlib.figure.Figure at 0x541ccf8>"
       ]
      },
      "metadata": {},
@@ -527,7 +527,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7909be0>"
+       "<matplotlib.figure.Figure at 0x541cf60>"
       ]
      },
      "metadata": {},
@@ -572,7 +572,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x8528da0>"
+       "<matplotlib.figure.Figure at 0x54272b0>"
       ]
      },
      "metadata": {},
@@ -613,7 +613,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x8606320>"
+       "<matplotlib.figure.Figure at 0x8841e80>"
       ]
      },
      "metadata": {},
@@ -726,7 +726,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x78e0358>"
+       "<matplotlib.figure.Figure at 0x54270f0>"
       ]
      },
      "metadata": {},
@@ -765,14 +765,14 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "Difference in mean x=-0.050, y=-0.252\n"
+      "Difference in mean x=-0.019, y=0.088\n"
      ]
     },
     {
      "data": {
-      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x8b71198>"
+       "<matplotlib.figure.Figure at 0x83bf048>"
       ]
      },
      "metadata": {},
@@ -991,7 +991,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa6966a0>"
+       "<matplotlib.figure.Figure at 0xa404748>"
       ]
      },
      "metadata": {},
@@ -1103,7 +1103,7 @@
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x8bda048>"
+       "<matplotlib.figure.Figure at 0xa2e9eb8>"
       ]
      },
      "metadata": {},
@@ -1188,7 +1188,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa66fe48>"
+       "<matplotlib.figure.Figure at 0xa40e438>"
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      "metadata": {},
@@ -1366,7 +1366,7 @@
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x8bda588>"
+       "<matplotlib.figure.Figure at 0x8ec2400>"
       ]
      },
      "metadata": {},
@@ -1376,7 +1376,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x854af28>"
+       "<matplotlib.figure.Figure at 0xa476e10>"
       ]
      },
      "metadata": {},
@@ -1386,7 +1386,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x856f898>"
+       "<matplotlib.figure.Figure at 0x7e2dba8>"
       ]
      },
      "metadata": {},
@@ -1492,7 +1492,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7a405c0>"
+       "<matplotlib.figure.Figure at 0x83bf6d8>"
       ]
      },
      "metadata": {},
@@ -1565,7 +1565,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x8ae4c88>"
+       "<matplotlib.figure.Figure at 0x76a0898>"
       ]
      },
      "metadata": {},
@@ -1664,7 +1664,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x84edcc0>"
+       "<matplotlib.figure.Figure at 0xa6ec080>"
       ]
      },
      "metadata": {},
@@ -1724,7 +1724,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x78b9828>"
+       "<matplotlib.figure.Figure at 0xa3730b8>"
       ]
      },
      "metadata": {},
@@ -1820,7 +1820,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x790b198>"
+       "<matplotlib.figure.Figure at 0xa37c940>"
       ]
      },
      "metadata": {},
@@ -1872,7 +1872,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa6aaeb8>"
+       "<matplotlib.figure.Figure at 0xa321908>"
       ]
      },
      "metadata": {},
@@ -1974,7 +1974,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x8a9ac88>"
+       "<matplotlib.figure.Figure at 0xa6c2080>"
       ]
      },
      "metadata": {},
@@ -2086,7 +2086,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x734a0f0>"
+       "<matplotlib.figure.Figure at 0xa6c2d30>"
       ]
      },
      "metadata": {},
@@ -2127,7 +2127,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x84e40b8>"
+       "<matplotlib.figure.Figure at 0xa6c2908>"
       ]
      },
      "metadata": {},
@@ -2201,7 +2201,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x8b1c390>"
+       "<matplotlib.figure.Figure at 0x826b0b8>"
       ]
      },
      "metadata": {},
@@ -2320,7 +2320,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x8b2c518>"
+       "<matplotlib.figure.Figure at 0xa37cac8>"
       ]
      },
      "metadata": {},
@@ -2383,7 +2383,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x864f2e8>"
+       "<matplotlib.figure.Figure at 0xa399ac8>"
       ]
      },
      "metadata": {},
@@ -2439,7 +2439,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x737da58>"
+       "<matplotlib.figure.Figure at 0xa4f69b0>"
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      "metadata": {},
@@ -2535,7 +2535,7 @@
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x863e048>"
+       "<matplotlib.figure.Figure at 0x7e33b70>"
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      },
      "metadata": {},
@@ -2886,7 +2886,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x78c30f0>"
+       "<matplotlib.figure.Figure at 0xa2e9320>"
       ]
      },
      "metadata": {},
@@ -2896,7 +2896,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x8c343c8>"
+       "<matplotlib.figure.Figure at 0x8ec4400>"
       ]
      },
      "metadata": {},
@@ -2906,7 +2906,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x8bfcb00>"
+       "<matplotlib.figure.Figure at 0xa347f60>"
       ]
      },
      "metadata": {},
@@ -2976,7 +2976,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x78a40b8>"
+       "<matplotlib.figure.Figure at 0xa7700b8>"
       ]
      },
      "metadata": {},
@@ -3024,7 +3024,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x863eac8>"
+       "<matplotlib.figure.Figure at 0xa770320>"
       ]
      },
      "metadata": {},
@@ -3034,7 +3034,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x8aacd68>"
+       "<matplotlib.figure.Figure at 0xa6a9278>"
       ]
      },
      "metadata": {},
@@ -3044,7 +3044,7 @@
      "data": {
       "image/png": 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       "text/plain": [
-       "<matplotlib.figure.Figure at 0x8ae27f0>"
+       "<matplotlib.figure.Figure at 0x826b668>"
       ]
      },
      "metadata": {},
@@ -3343,7 +3343,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x84eebe0>"
+       "<matplotlib.figure.Figure at 0xa2e4d30>"
       ]
      },
      "metadata": {},
@@ -3427,7 +3427,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 76,
+   "execution_count": 50,
    "metadata": {
     "collapsed": true
    },
@@ -3543,7 +3543,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "The state vector has the bearing at position 2, but the measurement vector has it at position 1, so we need to write functions to handle each. These will be passed into the `UnscentedKalmanFilter.__init__()` function."
+    "The state vector has the bearing at position 2, but the measurement vector has it at position 1, so we need to write functions to handle each. Furthermore, the function for the residual in the measurement will be passed an array of several measurements, one per landmark. These will be passed into the `UnscentedKalmanFilter.__init__()` function."
    ]
   },
   {
@@ -3556,7 +3556,9 @@
    "source": [
     "def residual_h(a, b):\n",
     "    y = a - b\n",
-    "    y[1] = normalize_angle(y[1])\n",
+    "    # data in format [dist_1, bearing_1, dist_2, bearing_2,...]\n",
+    "    for i in range(0, len(y), 2):\n",
+    "        y[i + 1] = normalize_angle(y[i + 1])\n",
     "    return y\n",
     "\n",
     "def residual_x(a, b):\n",
@@ -3569,35 +3571,37 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "It is this problem that kept me from implementing the measurement model. The equation is\n",
+    "The modularity of angles kept me from implementing the measurement model. The equation is\n",
     "$$h(x,p)\n",
     "= \\begin{bmatrix}\n",
     "\\sqrt{(p_x - x)^2 + (p_y - y)^2} \\\\\n",
     "\\tan^{-1}(\\frac{p_y - y}{p_x - x}) - \\theta \n",
     "\\end{bmatrix}$$\n",
     "\n",
-    "The expression $\\tan^{-1}(\\frac{p_y - y}{p_x - x}) - \\theta$ can produce a result outside the range $[\\pi, \\pi)$, so we should normalize the angle to that range."
+    "The expression $\\tan^{-1}(\\frac{p_y - y}{p_x - x}) - \\theta$ can produce a result outside the range $[\\pi, \\pi)$, so we should normalize the angle to that range.\n",
+    "\n",
+    "The function will be passed an array of landmarks and needs to produce an array of measurements in the form `[dist_to_1, bearing_to_1, dist_to_2, bearing_to_2, ...]`."
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 74,
+   "execution_count": 54,
    "metadata": {
     "collapsed": true
    },
    "outputs": [],
    "source": [
-    "\n",
-    "def Hx(x, landmark):\n",
+    "def Hx(x, landmarks):\n",
     "    \"\"\" takes a state variable and returns the measurement that would\n",
     "    correspond to that state.\n",
     "    \"\"\"\n",
-    "    px, py = landmark\n",
-    "    dist = math.sqrt((px - x[0])**2 + (py - x[1])**2)\n",
-    "    angle = math.atan2(py - x[1], px - x[0])\n",
-    "\n",
-    "    Hx = np.array([dist, normalize_angle(angle - x[2])])\n",
-    "    return Hx"
+    "    hx = []\n",
+    "    for lmark in landmarks:\n",
+    "        px, py = lmark\n",
+    "        dist = sqrt((px - x[0])**2 + (py - x[1])**2)\n",
+    "        angle = atan2(py - x[1], px - x[0])\n",
+    "        hx.extend([dist, normalize_angle(angle - x[2])])\n",
+    "    return np.array(hx)"
    ]
   },
   {
@@ -3615,7 +3619,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 59,
+   "execution_count": 55,
    "metadata": {
     "collapsed": true
    },
@@ -3623,30 +3627,25 @@
    "source": [
     "def state_mean(sigmas, Wm):\n",
     "    x = np.zeros(3)\n",
-    "    sum_sin, sum_cos = 0., 0.\n",
-    "\n",
-    "    for i in range(len(sigmas)):\n",
-    "        s = sigmas[i]\n",
-    "        x[0] += s[0] * Wm[i]\n",
-    "        x[1] += s[1] * Wm[i]\n",
-    "        sum_sin += sin(s[2])*Wm[i]\n",
-    "        sum_cos += cos(s[2])*Wm[i]\n",
     "\n",
+    "    sum_sin = np.sum(np.dot(np.sin(sigmas[:, 2]), Wm))\n",
+    "    sum_cos = np.sum(np.dot(np.cos(sigmas[:, 2]), Wm))\n",
+    "    x[0] = np.sum(np.dot(sigmas[:, 0], Wm))\n",
+    "    x[1] = np.sum(np.dot(sigmas[:, 1], Wm))\n",
     "    x[2] = atan2(sum_sin, sum_cos)\n",
     "    return x\n",
     "\n",
     "\n",
     "def z_mean(sigmas, Wm):\n",
-    "    x = np.zeros(2)\n",
-    "    sum_sin, sum_cos = 0., 0.\n",
+    "    z_count = sigmas.shape[1]\n",
+    "    x = np.zeros(z_count)\n",
     "\n",
-    "    for i in range(len(sigmas)):\n",
-    "        s = sigmas[i]\n",
-    "        x[0] += s[0] * Wm[i]\n",
-    "        sum_sin += sin(s[1])*Wm[i]\n",
-    "        sum_cos += cos(s[1])*Wm[i]\n",
+    "    for z in range(0, z_count, 2):\n",
+    "        sum_sin = np.sum(np.dot(np.sin(sigmas[:, z+1]), Wm))\n",
+    "        sum_cos = np.sum(np.dot(np.cos(sigmas[:, z+1]), Wm))\n",
     "\n",
-    "    x[1] = atan2(sum_sin, sum_cos)\n",
+    "        x[z] = np.sum(np.dot(sigmas[:,z], Wm))\n",
+    "        x[z+1] = atan2(sum_sin, sum_cos)\n",
     "    return x"
    ]
   },
@@ -3654,22 +3653,23 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "With that done we are now ready to implement the UKF. You've seen this sort of code many times, so I will not describe it in detail. There are two new things here, one trivial, one important. \n",
+    "These functions take advantage of the fact that NumPy's trigometric functions operate on arrays, and `dot` performs element-wise multiplication. NumPy is implemented in C and Fortran, so `sum(dot(sin(x), w))` is much faster than writing the equivalent loop in Python. Once you are used to seeing it, I think it is as readable as the loop form.\n",
     "\n",
-    "First the trivial point. When we construct the filter we have to provide it the functions that we have written to compute the residuals and means.\n",
+    "With that done we are now ready to implement the UKF. You've seen this sort of code many times, so I will not describe it in detail. There is one new thing here that is important to discuss. When we construct the sigma points and filter we have to provide it the functions that we have written to compute the residuals and means.\n",
+    "\n",
+    "    points = MerweScaledSigmaPoints(n=3, alpha=.00001, beta=2, kappa=0, \n",
+    "                                    subtract=residual_x)\n",
     "\n",
     "    ukf = UKF(dim_x=3, dim_z=2, fx=fx, hx=Hx, dt=dt, points=points,\n",
     "             x_mean_fn=state_mean, z_mean_fn=z_mean,\n",
     "             residual_x=residual_x, residual_z=residual_h)\n",
     "\n",
-    "The far more important point is how we update the filter with our measurements. As the robot moves around in its environment various landmarks will come into and go out of range. How should we incorpate a varying number of measurements? It turns out to be very very simple. We will just call `update()` once for each landmark that we sense. In other words, if the robot sees 3 landmarks then one call to `predict()` will be followed by 3 calls to `update()`. If this seems unlikely to work, just think of it as receiving 3 separate measurements with no time between them. If no time passed the robot could not have moved, and thus no change in position or uncertainty happened, and thus there is no need to call `predict()`. This is a very common method in Kalman filtering.\n",
-    "\n",
     "The rest of the code runs the simulation and plots the results, and shouldn't need too much comment by now. I create a variable `landmarks` that contains the coordinates of the landmarks. I update the simulated robot position 10 times a second, but run the EKF only once. This is for two reasons. First, we are not using Runge Kutta to integrate the differental equations of motion, so a narrow time step allows our simulation to be more accurate. Second, it is fairly normal in embedded systems to have limited processing speed. This forces you to run your Kalman filter only as frequently as absolutely needed."
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 94,
+   "execution_count": 56,
    "metadata": {
     "collapsed": false
    },
@@ -3682,23 +3682,21 @@
     "\n",
     "def run_localization(cmds, landmarks, sigma_vel, sigma_steer, sigma_range, \n",
     "                     sigma_bearing, ellipse_step=1, step=10):\n",
-    "    \n",
-    "    \n",
-    "    points = MerweScaledSigmaPoints(n=3, alpha=.00001, beta=2, kappa=0)\n",
-    "    ukf = UKF(dim_x=3, dim_z=2, fx=fx, hx=Hx, dt=dt, points=points,\n",
-    "             x_mean_fn=state_mean, z_mean_fn=z_mean,\n",
+    "\n",
+    "    points = MerweScaledSigmaPoints(n=3, alpha=.00001, beta=2, kappa=0, subtract=residual_x)\n",
+    "    ukf = UKF(dim_x=3, dim_z=2*len(landmarks), fx=fx, hx=Hx, dt=dt,\n",
+    "              points=points, x_mean_fn=state_mean, z_mean_fn=z_mean,\n",
     "             residual_x=residual_x, residual_z=residual_h)\n",
     "    ukf.x = np.array([2, 6, .3])\n",
-    "    ukf.P = np.diag([.1, .1, .1])\n",
-    "    ukf.R = np.diag([sigma_range**2, sigma_bearing**2])\n",
-    "    ukf.Q = np.zeros((3,3))\n",
+    "    ukf.P = np.diag([.1, .1, .05])\n",
+    "    ukf.R = np.diag([sigma_range**2, sigma_bearing**2]*len(landmarks))\n",
     "    ukf.Q = np.eye(3)*0.0001\n",
     "    \n",
-    "    u = np.array([1.1, .01])\n",
     "    sim_pos = ukf.x.copy()\n",
     "    \n",
     "    # plot landmarks\n",
-    "    plt.scatter(landmarks[:, 0], landmarks[:, 1], marker='s', s=60)\n",
+    "    if len(landmarks) > 0:\n",
+    "        plt.scatter(landmarks[:, 0], landmarks[:, 1], marker='s', s=60)\n",
     "    \n",
     "    track = []\n",
     "    for i, u in enumerate(cmds):     \n",
@@ -3713,12 +3711,13 @@
     "                                        facecolor='b', alpha=0.08)\n",
     "\n",
     "            x, y = sim_pos[0], sim_pos[1]\n",
+    "            z = []\n",
     "            for lmark in landmarks:\n",
-    "                d = sqrt((lmark[0] - x)**2 + (lmark[1] - y)**2)\n",
-    "                a = atan2(lmark[1] - y, lmark[0] - x) - sim_pos[2]\n",
-    "                z = np.array([d + randn()*sigma_range, a + randn()*sigma_bearing])\n",
-    "\n",
-    "                ukf.update(z, hx_args=(lmark,))\n",
+    "                d = sqrt((lmark[0] - x)**2 + (lmark[1] - y)**2) + randn()*sigma_range\n",
+    "                bearing = atan2(lmark[1] - y, lmark[0] - x)\n",
+    "                a = normalize_angle(bearing - sim_pos[2] + randn()*sigma_bearing)\n",
+    "                z.extend([d, a])            \n",
+    "            ukf.update(z, hx_args=(landmarks,))\n",
     "\n",
     "            if i % ellipse_step == 0:\n",
     "                plot_covariance_ellipse((ukf.x[0], ukf.x[1]), ukf.P[0:2, 0:2], std=6,\n",
@@ -3733,7 +3732,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 95,
+   "execution_count": 57,
    "metadata": {
     "collapsed": false,
     "scrolled": true
@@ -3741,9 +3740,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAtIAAAEjCAYAAAAfTOaLAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\nAAALEgAACxIB0t1+/AAAIABJREFUeJzs3Xd4XMXZ8OHfOdubtOpdlizLvYBsMGADBmPcqKGEFgwk\nhJCEhAQn5P0ScCAkJBDevAQSSgzBEJpD6GCqGzEGbNxky7Llpt67Vto+3x9CgvVKshEuMjz3de1l\nOGdmzpzV2d1nZ+c8oymlFEIIIYQQQogvRT/aHRBCCCGEEOJYJIG0EEIIIYQQgyCBtBBCCCGEEIMg\ngbQQQgghhBCDIIG0EEIIIYQQgyCBtBBCCCGEEIMggbQQQnwJ11xzDbqus3r16qPdlSj79u1D13Wu\nvfbao92VAeXk5JCbmxux7YknnkDXdZYsWXKUehVN13XOOOOMo90NIcQQJoG0ECLCypUrDyqA0HUd\nXY98CzlQ3XfeeQeXy4XFYuHZZ5+Naqu/x/33339Qfd+/nsFgIC4ujmnTpvHggw8SDAYPqp2j7be/\n/e1XCio1TTvEPTr09u+jpmm9jyNF1/WogH5/x8JzKYQ4eoxHuwNCiKHpYAKI/sr0tf1f//oX1113\nHXa7nWXLlnHmmWdG1Vm0aFGf7Z188skH0ePodoLBIPv27ePFF19k7dq1vPvuu7zyyisH3dbR9k0K\n4i688EJOPvlkUlNTj+hxB3qOi4uLsdvtR7A3QohjjQTSQojD7s9//jO//OUvSU1N5c033+S4447r\ns9ztt99+SI63fzu33XYbBQUFvPbaa6xatYrTTz/9kBzncPsmLTwbExNDTEzM0e5GhJEjRx7tLggh\nhjiZ2iGEOGyUUvz85z/nl7/8JSNHjmTt2rX9BtGHU35+PqeddhoA69evj9q/cuVK5s+fT0JCAlar\nlby8PH72s5/R0NDQb5tKKf75z39y3HHHYbfbSU1N5frrr6eurq7P8nv27OHaa68lMzMTi8VCamoq\n3/72tyksLIwoN2PGDO68804Arr322oipKmVlZYN9CqitreUnP/kJw4cPx2q1kpiYyLnnnssHH3zQ\nb513332X8847j5SUFKxWK1lZWZxzzjm8/vrrvWUCgQAPPvgg8+bNY9iwYVitVuLj4znrrLN44403\nDrp/fc2R7pmPPtBjMP3omYIEn88r73l8cX55f9OU2tvb+c1vfsPo0aOx2WzExcUxc+ZMXn311aiy\nPe2fccYZNDY28v3vf5+0tDSsVivjx4/niSeeOOjnSAgx9MiItBDisAgGgyxYsIBnn32WE088kTfe\neIOEhISj1p+e0V2LxRKxffHixXz/+9/H4XBwySWXkJaWxpo1a7j//vt56aWXWLNmDRkZGVHt3Xff\nfbz//vtcdtllzJ8/n1WrVvHYY4+xYsUKPvnkE+Lj43vLbtiwgZkzZ9LW1sY555zDhAkT2LVrFy++\n+CKvvfYar7zyCrNmzQK6g2dN01i1ahUXXHBBxBeP2NjYQZ17aWkp06dPp7KykhkzZnD55ZdTVVXF\n0qVLWbZsGY899hgLFiyIqLNo0SJ+97vf4XQ6ueCCC8jOzqa6upqPPvqIxx9/nHPOOQeAxsZGbr75\nZqZNm8bs2bNJSkqiqqqK1157jXPPPZeHH36Y73//+wfd1y9OtbjwwgsZPnx4VJndu3fz1FNPRUy7\n+DL9yM3NZdGiRdxxxx3Exsbys5/9rLed/b/o7T/1o7W1lenTp7Nt2zYKCgq4+eabaW5u5t///jcX\nXHABd9xxB7fddltUn1taWpg2bRoWi4VLL70Un8/H0qVLue6669B1nauvvvqgnyMhxBCihBDiC1as\nWKE0TVNnnHHGgOU0TVO6rvdZd8qUKWrWrFlK0zQ1f/581dnZecC2NE1Tv/3tb9WiRYsiHg8//PBB\n972vPimlVFFRkbLb7UrXdbVly5be7WVlZcpsNiuXy6WKiooi6tx2221K0zR1zjnnRGxfsGCB0jRN\nWSwWtWnTpoh9N910k9I0Td1www2928LhsBo7dqzSNE09+eSTEeXfe+89peu6Sk5OjniOFi1apDRN\nU0uWLDnoc1dKqb179ypN09S1114bsX3OnDlK0zR15513RmwvLCxUdrtdWa1WVVFR0bv97bffVpqm\nqdzc3IjtPb64zefzqcrKyqgyra2tavz48So+Pl51dXVF7Bs2bJjKzc2N2PbPf/7zoM65oaFBjRw5\nUhmNRvXqq69+pX70nGN/+nod/OAHP1Capqnvfve7EdsrKipUWlqa0nVdrVu3rnd7z99E0zR1/fXX\nq3A43LuvqKhIGY1GNXbs2AHPWQgxdEkgLYSIcCgC6Z7HyJEjVTAYPOAxv1hn/8fxxx9/0H3fPyD/\n9a9/ra688kpls9mUwWBQv/vd7yLK33XXXUrTNHXrrbdGteX1elV6errSNE1VVVX1bu8JpL/3ve9F\n1WlqalIOh0M5nc7e8/7vf/+rNE1TU6dO7bPPF110kdI0TT377LO92w5lIF1RUaE0TVPZ2dkqEAhE\n1bnllluUpmnq7rvv7t12zjnnKE3T1AsvvPCljr+/++67T2maplavXh2xfbCBdFdXl5o2bZrSNE09\n+OCDX7kfXzaQ9vv9ym63K6fTqRobG6PKP/DAA1FfpHr+Jk6nU7W3t0fVOe2005Su68rj8Rz0+Qgh\nhg6ZIy2EOORGjRpFTk4OJSUl3HjjjQd105ymaYTD4ajHhg0bvvTx77jjDu68807+8Ic/8Mwzz+D1\nernnnnv4zW9+E1Gup+39M4hA9xSQ6dOnA7Bx48ao/X3dsBgXF8eECRPweDzs2LHjgMcAOOuss/o9\nxqHQc/xp06ZhNEbP5uvr+B999BGapjF37tyDOsa2bdu45pprGD58OHa7vXe+8cKFCwGoqqr6qqeB\nUorvfOc7fPjhhyxcuJAf/ehHR7wfxcXFdHV1MWHChIipOz0G+lvm5+fjdDqjtmdlZaGUorm5+Sv1\nTQhxdMgcaSFEhJ6bsMLhcL9levb1lzosLS2Np556ipkzZ7J48WI6OztZsmQJBoPh0Hd4P5qmEQqF\nAPB6vXzyySfccMMN/OIXvyAlJYUrr7yyt2xraytAvynX0tLSIsp9UUpKSp91erb31DnQMXq2t7S0\nDHxigzSY47e0tBATE3NQqd8++ugjzjzzTMLhMDNnzuSCCy4gJiYGXdfZuHEjr7zyCj6f7yufx8KF\nC/nPf/7DpZdeyj333HNU+vFV/pZut7vPOj1fbnquWSHEsUUCaSFEhJ4b2hobG/st05PNor/gACAj\nI4PVq1cza9YsnnnmGbq6unjuuecwmUyHtsMDsFqtnHbaabz11luMGzeOG2+8kZkzZ/YGPD3nWl1d\nzcSJE6PqV1dXR5T7otra2j6P2bO9p07PvzU1NX2WH+gYh8Jgju92u2lqasLj8eBwOAZs/6677sLr\n9bJy5crezCg97r777kOSt/uBBx7gL3/5C9OnT+fJJ588av042n9LIcTQI1M7hBARRo8ejdlsZufO\nnf0G0x9++CEAkyZNGrCt5ORkVq5cyZQpU3jppZc4//zz8Xq9h7zPBzJs2DBuvfVWOjo6InJMT548\nGYAVK1ZE1fH5fKxZswZN0ygoKIjav3Llyqhtzc3NFBYW4nA4GDVqVMQxli9f3mff3n///YhyQO/I\n/aEYpezp+5o1awgEAgd1/JNPPhmlFMuWLTtg+7t27SIhISEqeAVYtWrVYLvd6+WXX+bmm29m1KhR\nvPLKK5jN5kPWjy/+enEwxowZg81mo7CwsM/XRl/PpRDi600CaSFEBIvFwhVXXEEgEOCWW26Jmt/c\n3NzcG4xed911B2wvLi6O999/n+nTp/PWW28xf/58PB7PYen7QH72s5+RmJjIE088QUlJCQBXXXUV\nZrOZv//9771zmnvcfffdVFVVMW/evD5/yn/qqafYtGlTxLbbb7+dzs5Orrzyyt5g+JRTTmHMmDF8\n8sknPP300xHlly9fzosvvkhSUhLnn39+7/aeNIGlpaVf+bwzMjKYPXs25eXlUVMitm3bxkMPPYTV\nauWqq67q3X7TTTcB8Itf/IKKioqoNisrK3v/Ozc3l8bGxqh82I899hjvvPPOV+r7xx9/zBVXXEFy\ncjLLli0jLi6u37KD6UdCQgL19fUH/eXOaDRy9dVX4/F4+J//+Z+IfVVVVdx9993oun5QrwshxNeD\nTO0QQkS57777WL9+PU8++SRr167l7LPPJjY2lqqqKl599VWampr4zne+EzHfeCAul4u3336b888/\nn/fee4+zzz6bZcuWHdGV7JxOJ7/61a9YuHAht912G8899xzZ2dn89a9/5cYbb2TKlClceumlpKSk\n8OGHH7J69WqysrJ46KGH+mxvzpw5TJs2jW9/+9ukpKSwevVq1q5dS15eHn/4wx8iyi5ZsoSzzjqL\nq6++mqVLlzJ+/Hh2797Nf/7zH6xWK08++SRWq7W3/MyZM9F1nf/7v/+jsbGxd971T37yk0E9Zw8/\n/DDTpk3jtttuY/ny5UydOpXq6mqWLl2K3+/n0UcfjciVPWvWLG677TZ+97vfMXbsWM4//3yys7Op\nq6vjo48+YsSIEbz00ksA3Hzzzbz99ttMnz6dSy+9lJiYGNavX8+aNWu4+OKLeeGFF750f3tce+21\neL1eZs2a1efCJV9cDn4w/Tj77LN55plnmDNnDqeeeioWi4XjjjuuN0d2X/74xz/ywQcfsHjxYjZu\n3MjMmTNpaWnh3//+Ny0tLdx+++2ccMIJgz5nIcQxZqCUHqtWrVLnnnuuysjIUJqmqSeeeCJif2tr\nq7rxxhtVZmamstlsatSoUeovf/nL4cowIoQ4gjo7O9U999yjpk6dqmJjY5XJZFLJyLine truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa65a0f0>"
+       "<matplotlib.figure.Figure at 0xa3c9f98>"
       ]
      },
      "metadata": {},
@@ -3753,7 +3752,7 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "[ 0.0133  0.0378  0.0013]\n"
+      "[ 0.0092  0.0187  0.0007]\n"
      ]
     }
    ],
@@ -3783,23 +3782,11 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 96,
+   "execution_count": 58,
    "metadata": {
-    "collapsed": false,
-    "scrolled": true
+    "collapsed": true
    },
-   "outputs": [
-    {
-     "data": {
-      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAtsAAAEjCAYAAADjWc1BAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
-      "text/plain": [
-       "<matplotlib.figure.Figure at 0x8b11a90>"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    }
-   ],
+   "outputs": [],
    "source": [
     "landmarks = np.array([[5, 10], [10, 5], [15, 15], [20, 5],\n",
     "                      [0, 30], [50, 30], [40, 10]])\n",
@@ -3825,11 +3812,94 @@
     "cmds.extend([cmds[-1]]*150)\n",
     "\n",
     "cmds.extend([[v, a] for a in np.linspace(0, np.radians(1), 25)])\n",
-    "cmds.extend([cmds[-1]]*100)\n",
-    "\n",
+    "cmds.extend([cmds[-1]]*100)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 59,
+   "metadata": {
+    "collapsed": false,
+    "scrolled": true
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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truncated
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0xa3c9fd0>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "final covariance [ 0.0013  0.0043  0.0004]\n"
+     ]
+    }
+   ],
+   "source": [
     "ukf = run_localization(cmds, landmarks, sigma_vel=0.1, sigma_steer=np.radians(1),\n",
     "                      sigma_range=0.3, sigma_bearing=0.1, step=1,\n",
-    "                      ellipse_step=20)\n"
+    "                      ellipse_step=20)\n",
+    "print('final covariance', ukf.P.diagonal())"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "You can see that the uncertainty becomes very small very quickly. The covariance ellipses are displaying the $6\\sigma$ covariance, yet the ellipses are so small they are hard to see. We can force some error into the answer by only supplying two landmarks near the start point. When we run this filter the errors increase as the robot gets further from the landmarks."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 60,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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truncated
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x83b35f8>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "final covariance [ 0.0026  0.0657  0.0008]\n"
+     ]
+    }
+   ],
+   "source": [
+    "ukf = run_localization(cmds, landmarks[0:2], sigma_vel=0.1, sigma_steer=np.radians(1),\n",
+    "                      sigma_range=0.3, sigma_bearing=0.1, step=1,\n",
+    "                      ellipse_step=20)\n",
+    "print('final covariance', ukf.P.diagonal())"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Discussion\n",
+    "\n",
+    "Your impression of this chapter probably depends on how many nonlinear Kalman filters you have implemented in the past. If this is your first exposure perhaps the computation of $2n+1$ sigma points and the subsequent writing of the $f(x)$ and $h(x)$ function struck you as a bit finicky. Indeed, I spent more time than I'd care to admit getting everything working. On the other hand, if you have implemented an EKF or linearized Kalman filter you are perhaps bouncing gleefully in your seat. There is a small amount of tedium in writing the functions for the UKF, but the concepts are very basic. As you will see in the next chapter the EKF for the same problems requires some fairly difficult mathematics. In fact, for many problems we cannot find a closed form solution for the equations of the EKF, and we must retreat to some sort of iterated solution.\n",
+    "\n",
+    "However, the main advantage of the UKF over the EKF is not just the relative ease of implementation. It is somewhat premature to discuss this because you haven't learned the EKF yet, but the EKF linearizes the problem at one point and passes that point through a linear Kalman filter. In contrast, the UKF takes $2n+1$ samples. Therefore the UKF is almost always more accurate than the EKF, especially when the problem is highly nonlinear. While it is not true that the UKF is guaranteed to always outperform the EKF, in practice it has been shown to perform at least as well, and usually much better than the EKF. \n",
+    "\n",
+    "Hence my recommendation is to always start by implementing the UKF. If your filter has real world consequences if it diverges (people die, lots of money lost, power plant blows up) of course you will have to engage in a lot of sophisticated analysis and experimentation to chose the best filter. That is beyond the scope of this book, and you should be going to graduate school to learn this theory in much greater detail than this book provides. \n",
+    "\n",
+    "I have spoken of the UKF I presented in this chapter as *the* way to perform sigma point filters. This is not true. The specific version I chose is Julier's scaled unscented filter as parameterized by Van der Merwe in his 2004 dissertation. If you search for Julier, Van der Merwe, Uhlmann, and Wan you will find a family of similar sigma point filters that they developed. Each technique uses a different way of choosing and weighting the sigma points. But the choices don't stop there. For example, there is the SVD Kalman filter that uses singular value decomposition (SVD) to find the approximate mean and covariance of the probability distribution. There are many more. Think of this chapter as an introduction to the sigma point filters, rather than a definitive treatment of how they work. If you have been reading carefully and writing your own code you should be able to read the literature and implement your own filters without needing FilterPy, which does not implement every possible sigma point filter. "
    ]
   },
   {