Expansion of UKF chapter.

Wrote several examples. Still more to do.

10_Unscented_Kalman_Filters/Unscented_Kalman_Filter.ipynb

@@ -1,7 +1,7 @@
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  "metadata": {
   "name": "",
-  "signature": "sha256:71b2584db8f7c39febb9622c86c64ba03269b1ec3d1ac8ede348a51a7357cf44"
+  "signature": "sha256:f640d994c2b3f8b0eec29fa6a65a1916071a1487707aefd49319c2a0b63486fc"
  },
  "nbformat": 3,
  "nbformat_minor": 0,
@@ -261,7 +261,7 @@
        "output_type": "pyout",
        "prompt_number": 1,
        "text": [
-        "<IPython.core.display.HTML at 0x7f47fa1db4e0>"
+        "<IPython.core.display.HTML at 0x7fa311bd04e0>"
        ]
       }
      ],
@@ -307,9 +307,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f47ddd36160>"
+        "<matplotlib.figure.Figure at 0x7fa2f575f2e8>"
        ]
       }
      ],
@@ -345,9 +345,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f47ddd198d0>"
+        "<matplotlib.figure.Figure at 0x7fa2f57440b8>"
        ]
       }
      ],
@@ -392,9 +392,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x7f47fa1e0d68>"
+        "<matplotlib.figure.Figure at 0x7fa311bd54a8>"
        ]
       }
      ],
@@ -492,16 +492,16 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f47dd7c8fd0>"
+        "<matplotlib.figure.Figure at 0x7fa2f52ff080>"
        ]
       },
       {
        "output_type": "stream",
        "stream": "stdout",
        "text": [
-        "Difference in mean x=-0.037, y=42.097\n"
+        "Difference in mean x=-0.091, y=44.968\n"
        ]
       }
      ],
@@ -536,9 +536,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x7f47dda213c8>"
+        "<matplotlib.figure.Figure at 0x7fa2f5306a58>"
        ]
       }
      ],
@@ -576,9 +576,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f47d4278cc0>"
+        "<matplotlib.figure.Figure at 0x7fa2f51cfb38>"
        ]
       }
      ],
@@ -625,9 +625,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f47d4245208>"
+        "<matplotlib.figure.Figure at 0x7fa2f51c0550>"
        ]
       }
      ],
@@ -695,9 +695,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x7f47d424b710>"
+        "<matplotlib.figure.Figure at 0x7fa2f51e9a90>"
        ]
       }
      ],
@@ -1146,13 +1146,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f47d408a320>"
+        "<matplotlib.figure.Figure at 0x7fa2f541ff60>"
        ]
       }
      ],
-     "prompt_number": 37
+     "prompt_number": 15
     },
     {
      "cell_type": "markdown",
@@ -1165,7 +1165,9 @@
      "cell_type": "code",
      "collapsed": false,
      "input": [
-      "plt.ylabel('target X position')\n",
+      "book_format.equal_axis()\n",
+      "\n",
+      "plt.xlabel('target X position')\n",
       "plt.ylabel('target Y position')\n",
       "plt.plot(xs[:,0], xs[:,2])\n",
       "plt.show()"
@@ -1176,13 +1178,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f47d4154da0>"
+        "<matplotlib.figure.Figure at 0x7fa2f51e9080>"
        ]
       }
      ],
-     "prompt_number": 38
+     "prompt_number": 16
     },
     {
      "cell_type": "markdown",
@@ -1225,54 +1227,36 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f47dd779f28>"
+        "<matplotlib.figure.Figure at 0x7fa2f5322fd0>"
        ]
       }
      ],
-     "prompt_number": 21
+     "prompt_number": 17
     },
     {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "We can compute the bearing between a sensor and the target as follows:"
+      "We can compute the bearing between a sensor and the target as follows:\n",
+      "\n",
+      "    def bearing(sensor, target):\n",
+      "        return math.atan2(target[1] - sensor[1], target[0] - sensor[0])"
      ]
     },
-    {
-     "cell_type": "code",
-     "collapsed": false,
-     "input": [
-      "def bearing(sensor, target_pos):\n",
-      "    return math.atan2(target_pos[1] - sensor[1], target_pos[0] - sensor[0])"
-     ],
-     "language": "python",
-     "metadata": {},
-     "outputs": [],
-     "prompt_number": 79
-    },
     {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "So our filter will need to receive a vector of 2 measurements during each update, one for each sensor. We can implement that as:"
+      "So our filter will need to receive a vector of 2 measurements during each update, one for each sensor. We can implement that as:\n",
+      "\n",
+      "    def measurement(A_pos, B_pos, pos):\n",
+      "        angle_a = bearing(A_pos, pos)\n",
+      "        angle_b = bearing(B_pos, pos)\n",
+      "        return [angle_a, angle_b]"
      ]
     },
-    {
-     "cell_type": "code",
-     "collapsed": false,
-     "input": [
-      "def measurement(A_pos, B_pos, pos):\n",
-      "    angle_a = bearing(A_pos, pos)\n",
-      "    angle_b = bearing(B_pos, pos)\n",
-      "    return [angle_a, angle_b]"
-     ],
-     "language": "python",
-     "metadata": {},
-     "outputs": [],
-     "prompt_number": 80
-    },
     {
      "cell_type": "markdown",
      "metadata": {},
@@ -1306,53 +1290,435 @@
       "    pos = (x[0], x[2])\n",
       "    return measurement(sa_pos, sb_pos, pos)\n",
       "\n",
-      "def normalize(a):\n",
-      "    if a > math.pi:\n",
-      "        a -= math.pi*2\n",
       "\n",
-      "    if a < -math.pi:\n",
-      "        a += math.pi*2\n",
-      "    return a\n",
+      "def moving_target_filter(std_noise):\n",
+      "    f = UKF(dim_x=4, dim_z=2, dt=0.1, hx=hx, fx=fx, kappa=2.)\n",
+      "    f.x = np.array([target_pos[0], 1., target_pos[1], 1.])\n",
+      "\n",
+      "    Q = Q_discrete_white_noise(2, dt, 1.1)\n",
+      "    f.Q[0:2,0:2] = Q\n",
+      "    f.Q[2:4, 2:4] = Q\n",
+      "\n",
+      "    f.R *= std_noise**2\n",
+      "    f.P *= 1000\n",
+      "    \n",
+      "    return f\n",
+      "\n",
+      "\n",
+      "def plot_straight_line_target(f, std_noise):\n",
+      "    xs = []\n",
+      "    txs = []\n",
+      "\n",
+      "    for i in range(300):\n",
+      "        target_pos[0] += 1 + randn()*0.0001\n",
+      "        target_pos[1] += 1 + randn()*0.0001\n",
+      "        txs.append(target_pos.copy())\n",
+      "\n",
+      "        z = measurement(sa_pos, sb_pos, target_pos)\n",
+      "        z[0] += randn() * std_noise\n",
+      "        z[1] += randn() * std_noise\n",
+      "\n",
+      "        f.predict()\n",
+      "        f.update(z)\n",
+      "        xs.append(f.x)\n",
+      "\n",
+      "    xs = np.asarray(xs)\n",
+      "    txs = np.asarray(txs)\n",
+      "\n",
+      "    plt.plot(xs[:,0], xs[:,2])\n",
+      "    plt.plot(txs[:,0], txs[:,1])\n",
+      "    plt.show()\n",
+      "   \n",
+      "    \n",
+      "np.random.seed(123)\n",
       "\n",
       "sa_pos = [-40, 0]\n",
       "sb_pos = [40, 0]\n",
-      "\n",
-      "dt = .1\n",
       "target_pos = [40, 20]\n",
       "\n",
       "std_noise = math.radians(1.)\n",
+      "f = moving_target_filter(std_noise)\n",
+      "plot_straight_line_target(f, std_noise)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": "iVBORw0KGgoAAAANSUhEUgAAAl4AAAJVCAYAAAAP5vtFAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x7fa2f53a0a20>"
+       ]
+      }
+     ],
+     "prompt_number": 18
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "This looks quite good to me. There is a very large error at the beginning of the track, but the filter is able to settle down and start producing good data.\n",
       "\n",
-      "f = UKF(dim_x=4, dim_z=2, dt=dt, hx=hx, fx=fx, kappa=2.)\n",
-      "f.x = np.array([target_pos[0], 1., target_pos[1], 1.])\n",
+      "Let's revisit the nonlinearity of the angles. I will position the target between the two sensors with the same y-coordinate. This will cause a nonlinearity in the computation of the sigma means and the residuals."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "np.random.seed(12330)\n",
+      "target_pos = [0, 0]\n",
+      "f = moving_target_filter(std_noise)\n",
+      "plot_straight_line_target(f, std_noise)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x7fa2f0e09240>"
+       ]
+      }
+     ],
+     "prompt_number": 19
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "If we seed the random number generator just right we can get the filter to eventually lock onto the signal and converge to a solution, but of course this is untenable behavior for a real world filter. I will deal with this nonlinearity later; I bring it up now in case you are experimenting with the code above and you start to see this sort of behavior."
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "###### Exercise: Track a target moving in a circle\n",
       "\n",
-      "Q = Q_discrete_white_noise(2, dt, 1.1)\n",
-      "f.Q[0:2,0:2] = Q\n",
-      "f.Q[2:4, 2:4] = Q\n",
+      "Change the simulated target movement to move in a circle. Avoid angular nonlinearities by putting the sensors well outside the movement range of the target, and avoid the angles $0^\\circ$ and $180^\\circ$."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "# your solution here"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 20
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "###### Solution\n",
       "\n",
-      "f.R *= std_noise**2\n",
-      "f.P *= 1000\n",
+      "We have a few choices here. First, if we know the movement of the target we can design our filter's state transition function so that it correctly predicts the circular movement. For example, suppose we were tracking a boat race optically - we would want to take track shape into account with our filter. However, in this chapter we have not been talking about such constrained behavior, so I will not build knowledge of the movement into the filter. So my implementation looks like this."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "def plot_circular_target(f, std_noise, target_pos):\n",
+      "    xs = []\n",
+      "    txs = []\n",
+      "    radius = 100\n",
+      "    for i in range(300):\n",
+      "        target_pos[0] = math.cos(i/10)*radius + randn()*0.0001\n",
+      "        target_pos[1] = math.sin(i/10)*radius + randn()*0.0001\n",
+      "        txs.append(target_pos.copy())\n",
       "\n",
-      "xs = []\n",
-      "txs = []\n",
+      "        z = measurement(sa_pos, sb_pos, target_pos)\n",
+      "        z[0] += randn() * std_noise\n",
+      "        z[1] += randn() * std_noise\n",
       "\n",
-      "for i in range(300):\n",
-      "    target_pos[0] += 1 + randn()*0.0001\n",
-      "    target_pos[1] += 1 + randn()*0.0001\n",
-      "    txs.append(target_pos.copy())\n",
+      "        f.predict()\n",
+      "        f.update(z)\n",
+      "        xs.append(f.x)\n",
+      "\n",
+      "    xs = np.asarray(xs)\n",
+      "    txs = np.asarray(txs)\n",
+      "\n",
+      "    plt.plot(xs[:,0], xs[:,2])\n",
+      "    plt.plot(txs[:,0], txs[:,1], linestyle='-.')\n",
+      "    plt.axis('equal')\n",
+      "    plt.show()\n",
+      "\n",
+      "sa_pos = [-240, 200]\n",
+      "sb_pos = [240, 200]\n",
+      "\n",
+      "np.random.seed(12283)\n",
+      "std_noise = math.radians(0.5)\n",
+      "target_pos = [0, 0]\n",
+      "f = moving_target_filter(std_noise)\n",
+      "plot_circular_target(f, std_noise, target_pos)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x7fa2f52d3048>"
+       ]
+      }
+     ],
+     "prompt_number": 21
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "###### Discussion\n",
+      "\n",
+      "The filter tracks the movement of the target, but never really converges on the track.  This is because the filter is modeling a constant velocity target, but the target is anything but constant velocity. As mentioned above we could model the circular behavior by defining the `fx()` function, but then we would have problems when the target is not moving in a circle. Instead, let's tell the filter we are fare less sure about our process model by making $\\mathbf{Q}$ much larger."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "def circular_target_filter(std_noise):\n",
+      "    f = UKF(dim_x=4, dim_z=2, dt=0.1, hx=hx, fx=fx, kappa=2.)\n",
+      "    f.x = np.array([target_pos[0], 1., target_pos[1], 1.])\n",
+      "\n",
+      "    Q = Q_discrete_white_noise(2, dt, 10.1)\n",
+      "    f.Q[0:2,0:2] = Q\n",
+      "    f.Q[2:4, 2:4] = Q\n",
+      "\n",
+      "    f.R *= std_noise**2\n",
+      "    f.P *= 1000\n",
       "    \n",
-      "    z = measurement(sa_pos, sb_pos, target_pos)\n",
-      "    z[0] += randn() * std_noise\n",
-      "    z[1] += randn() * std_noise\n",
+      "    return f\n",
       "\n",
-      "    f.predict()\n",
-      "    f.update(z)\n",
-      "    xs.append(f.x)\n",
+      "std_noise = math.radians(0.5)\n",
+      "cf = circular_target_filter(std_noise)\n",
+      "target_pos = [0, 0]\n",
+      "plot_circular_target(cf, std_noise, target_pos)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": "iVBORw0KGgoAAAANSUhEUgAAAmUAAAJVCAYAAACf9YKKAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x7fa2f525a6d8>"
+       ]
+      }
+     ],
+     "prompt_number": 22
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "The convergence is not perfect, but it is far better. \n",
       "\n",
-      "xs = np.asarray(xs)\n",
-      "txs = np.asarray(txs)\n",
+      "###### Problem: Sensor Position Effects\n",
       "\n",
-      "plt.plot(xs[:,0], xs[:,2])\n",
-      "plt.plot(txs[:,0], txs[:,1])\n",
+      "Is the behavior of the filter invariant for any sensor position? Find a sensor position that produces bad filter behavior."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "# your answer here"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 23
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "###### Solution\n",
+      "\n",
+      "We have already discussed the problem of angles being modal, so causing that problem by putting the sensors at `y=0` would be a trivial solution. However, let's be more subtle than that. We want to create a situation where there are an infinite number of solutions for the sensor readings. We can achieve that whenever the target lies on the straight line between the two sensors. In that case there is no triangulation possible and there is no unique solution. My solution is as follows."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "std_noise = math.radians(0.5)\n",
+      "sa_pos = [-200,200]\n",
+      "sb_pos = [200,-200]\n",
+      "plt.scatter(*sa_pos, s=200)\n",
+      "plt.scatter(*sb_pos, s=200)\n",
+      "cf = circular_target_filter(std_noise)\n",
+      "target_pos = [0, 0]\n",
+      "plot_circular_target(cf, std_noise, target_pos)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x7fa2f5340780>"
+       ]
+      }
+     ],
+     "prompt_number": 24
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "I put the sensors at the upper left hand side and lower right hand side of the target's movement. We can see how the filter diverges when the target enters the region directly between the two sensors. The state transition always predicts that the target is moving in a straight line. When the target is between the two sensors this straight line movement is well described the bearing measurements from the two sensors so the filter estimate starts to approximate a straight line. \n",
+      "\n",
+      "\n",
+      "###### Exercise: Compute Position Errors\n",
+      "\n",
+      "The position errors of the filter vary depending on how far away the target is from a sensor. Write a function that computes the distance error due to a bearing error. "
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "# your solution here"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 25
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "###### Solution\n",
+      "\n",
+      "Basic trigonometry gives us this answer."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "def distance_error(target_distance, angle_error):\n",
+      "    x = 1-math.cos(angle_error)\n",
+      "    y = math.sin(angle_error)\n",
+      "    return target_distance*(x**2 + y**2)**.5\n",
+      "\n",
+      "d = distance_error(100, math.radians(1.))\n",
+      "print('\\n\\nError of 1 degree at 100km is {:.3}km'.format(d))"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "output_type": "stream",
+       "stream": "stdout",
+       "text": [
+        "\n",
+        "\n",
+        "Error of 1 degree at 100km is 1.75km\n"
+       ]
+      }
+     ],
+     "prompt_number": 26
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "###### Discussion\n",
+      "\n",
+      "This is an inherent physical limitation that is extremely difficult to deal with when designing filters. We can see that a very small angular error translates into a very large distance error. What is worse, this behavior is nonlinear - the error in the *x-axis* vs the *y-axis* will vary depending on the actual bearing. For example, here is a scatter plot that shows the error distibution for a $1^\\circ$ standard deviation in bearing for a $30^\\circ$ bearing."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "d = 100\n",
+      "xs = []\n",
+      "ys = []\n",
+      "\n",
+      "for i in range (3000):\n",
+      "    a = math.radians(30) + randn() * math.radians(1)\n",
+      "    \n",
+      "    xs.append(d*math.cos(a))\n",
+      "    ys.append(d*math.sin(a))\n",
+      "    \n",
+      "plt.scatter(xs, ys)\n",
+      "plt.show() "
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x7fa2f5204c50>"
+       ]
+      }
+     ],
+     "prompt_number": 27
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "###### Exercise: Explain Filter Performance\n",
+      "\n",
+      "We can see that for even very small angular errors the (x, y) positional errors are very large. Explain how we got such relatively good performance out of the UKF in the target tracking problems above. Answer this for the one sensor problem as well as the multiple sensor problem. "
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "###### Solution\n",
+      "\n",
+      "> This is **very** important to understand. Try very hard to answer this before reading the answer below. If you cannot answer this you really need to revisit some of the earlier Kalman filter material in the Multidimensional Kalman filter chapter.\n",
+      "\n",
+      "There are several factors contributing to our success. First, let's consider the case of having only one sensor. Any single measurement has an extreme range of possible psoitions. But, our target is moving, and the UKF is taking that into account. Let's plot the results of several measurements taken in a row for a moving target to form an intuition."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "pos = [5, 5]\n",
+      "for i in range(5):\n",
+      "    pos[0] += 0.5\n",
+      "    pos[1] += 1  \n",
+      "    \n",
+      "    actual_angle = math.atan2(pos[1], pos[0])\n",
+      "    d = math.sqrt(pos[0]**2 + pos[1]**2)\n",
+      "\n",
+      "    xs = []\n",
+      "    ys = []\n",
+      "\n",
+      "    for i in range (100):\n",
+      "        a = actual_angle + randn() * math.radians(1)\n",
+      "        xs.append(d*math.cos(a))\n",
+      "        ys.append(d*math.sin(a))\n",
+      "    plt.scatter(xs, ys)\n",
+      "\n",
+      "plt.axis('equal')\n",
+      "plt.plot([5.5, pos[0]], [6, pos[1]], c='g', linestyle='--')\n",
       "plt.show()"
      ],
      "language": "python",
@@ -1361,13 +1727,149 @@
       {
        "metadata": {},
        "output_type": "display_data",
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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f47d4127630>"
+        "<matplotlib.figure.Figure at 0x7fa2f0cf3e10>"
        ]
       }
      ],
-     "prompt_number": 98
+     "prompt_number": 28
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "We can see that each individual measurement has a very large position error. However, when we plot successive measurements we suddenly have a clear trend - the target is obviously moving towards the upper right. When the Kalman filter (whether a linear KF, an EKF, or UKF) computes the Kalman gain it takes the distribution of errors into account by using the measurement function. In this example the error lies on an approximately $45^\\circ$ degree line, so the filter will discount errors in that direction. On the other hand, there is almost no error in measurement orthogonal to that, and again the Kalman gain will be taking that into account. The end result is the track can be computed even with this sort of noise distribution. Of course, this graph makes it look easy because we have computed 100 possible measurements for each position update and plotted them all. This makes it easy for us to see where the target is moving. In contrast, the Kalman filter only gets 1 measurement per update. Therefore the filter will not be able to generate as good a fit as the dotted green line implies. \n",
+      "\n",
+      "The next interaction we must think about is the fact that the bearing gives us no distance information. Suppose in the example above we told the filter that the initial position of the target was 1000km away from the sensor (vs the actual distance of 7.07 km), and that we were highly confident in that guess by setting $\\mathbf{P}$ very small. At that distance a $1^\\circ$ error translates into a positional error of 17.5 km. The KF would never be able to converge onto the actual target position because the filter is incorrectly very certain about its position estimates and because there is no distance information provided in the measurements."
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Now let's consider the effect of adding a second sensor. Again, I will start by plotting the measurements so we can form an intuition about the problem."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "from math import sin, cos, atan2, pi\n",
+      "\n",
+      "def isct(pa, pb, alpha, beta):\n",
+      "    \"\"\" Returns the (x, y) intersections of points pa and pb\n",
+      "    given the bearing ba for point pa and bearing bb for\n",
+      "    point pb.\n",
+      "    \"\"\"\n",
+      "    \n",
+      "    B = [pb[0] - pa[0], pb[1]- pa[1]]\n",
+      "\n",
+      "    AB = math.sqrt((pa[0]-pb[0])**2 + (pa[1] - pb[1])**2) \n",
+      "    \n",
+      "    ab = atan2(B[1], B[0])\n",
+      "    a = alpha - ab    \n",
+      "    b = pi - beta - ab\n",
+      "    p = pi - b - a\n",
+      "    \n",
+      "    AP = (sin(b) / sin(p)) * AB\n",
+      "    \n",
+      "    x = cos(alpha) * AP + pa[0]\n",
+      "    y = sin(alpha) * AP + pa[1]\n",
+      "    return x, y\n",
+      "\n",
+      "\n",
+      "def plot_iscts(N=4):\n",
+      "    for i in range(N):\n",
+      "        pos[0] += 0.5\n",
+      "        pos[1] += 1  \n",
+      "\n",
+      "        actual_angle_a = math.atan2(pos[1]-sa[1], pos[0] - sa[0])\n",
+      "        actual_angle_b = math.atan2(pos[1]-sb[1], pos[0] - sb[0])\n",
+      "\n",
+      "        da = math.sqrt((sa[0]-pos[0])**2 + (sa[1]-pos[1])**2)\n",
+      "        db = math.sqrt((sb[0]-pos[0])**2 + (sb[1]-pos[1])**2)\n",
+      "\n",
+      "        xs, ys, xs_a, xs_b, ys_a, ys_b = [], [], [], [], [], []\n",
+      "\n",
+      "        for i in range (300):\n",
+      "            a_a = actual_angle_a + randn() * math.radians(1)\n",
+      "            a_b = actual_angle_b + randn() * math.radians(1)\n",
+      "\n",
+      "            x,y = isct(sa, sb, a_a, a_b)\n",
+      "            xs.append(x)\n",
+      "            ys.append(y)\n",
+      "\n",
+      "            xs_a.append(da*math.cos(a_a)+sa[0])\n",
+      "            ys_a.append(da*math.sin(a_a)+sa[1])\n",
+      "\n",
+      "            xs_b.append(db*math.cos(a_b)+sb[0])\n",
+      "            ys_b.append(db*math.sin(a_b)+sb[1])\n",
+      "\n",
+      "        plt.scatter(xs, ys, c='r', marker='.')\n",
+      "        plt.scatter(xs_a, ys_a)\n",
+      "        plt.scatter(xs_b, ys_b)\n",
+      "\n",
+      "    #plt.plot([5.5, pos[0]], [6, pos[1]], c='g', linestyle='--')\n",
+      "    plt.axis('equal')\n",
+      "    plt.show()\n",
+      "\n",
+      "pos = [4, 4]\n",
+      "sa = [0, 2]\n",
+      "sb = [8, 2]\n",
+      "\n",
+      "plt.scatter(*sa, s=100)\n",
+      "plt.scatter(*sb, s=100)\n",
+      "plot_iscts(N=4)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x7fa2f0dffac8>"
+       ]
+      }
+     ],
+     "prompt_number": 29
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "I placed the sensors nearly orthogonal to the target's initial position so we get theese lovely 'x' shape intersections. I then computed the (x, y) coordinate corresponding to the two noisy bearing measurements and plotted them with red dots to show the distribution of the noisy measurements in x and y. We can see how the errors in x and y change as the target moves by the shape the scattered red dots make - as the target gets further away from the sensors, but nearer the y coordinate of sensor B the shape becomes strongly elliptical.\n",
+      "\n",
+      "Next I will alter the starting positions and rerun the simulation. Here the shape of the errors in x and y changes radically as the target position changes relative to the two sensors. "
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "sa = [3, 4]\n",
+      "sb = [3, 9]\n",
+      "pos= [4, 4]\n",
+      "\n",
+      "plt.scatter(*sa, s=100)\n",
+      "plt.scatter(*sb, s=100)\n",
+      "plot_iscts(N=5)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": "iVBORw0KGgoAAAANSUhEUgAAAkkAAAJPCAYAAABo2UApAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x7fa2f5302358>"
+       ]
+      }
+     ],
+     "prompt_number": 30
     },
     {
      "cell_type": "markdown",

code/book_format.py

@@ -1,6 +1,7 @@
 # -*- coding: utf-8 -*-
 from IPython.core.display import HTML
 import matplotlib.pylab as pylab
+import matplotlib.pyplot as plt
 
 def load_style(name='../styles/custom2.css'):
     styles = open(name, 'r').read()
@@ -30,7 +31,7 @@ pylab.rcParams['xtick.minor.pad'] = 6
 pylab.rcParams['xtick.color'] = '555555'
 pylab.rcParams['ytick.direction'] = 'in'
 pylab.rcParams['legend.fancybox'] = True
-pylab.rcParams['figure.facecolor'] = '0.85'
+pylab.rcParams['figure.facecolor'] = '1.0'
 pylab.rcParams['figure.edgecolor'] = '0.50'
 pylab.rcParams['figure.subplot.hspace'] = 0.5
 pylab.rcParams['figure.figsize'] = 12,6
@@ -39,3 +40,10 @@ pylab.rcParams['grid.linestyle'] = 'solid'
 pylab.rcParams['grid.linewidth'] = 1.5
 
 
+def equal_axis():
+    pylab.rcParams['figure.figsize'] = 10,10
+    plt.axis('equal')
+
+def reset_axis():
+    pylab.rcParams['figure.figsize'] = 12, 6
+