Added section on sensor fusion.

08_Designing_Kalman_Filters/Designing_Kalman_Filters.ipynb

@@ -1,7 +1,7 @@
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-  "signature": "sha256:f4bf9b95f2646bd4f54205934aa979f8de905052aad5861873b0283a625be1e9"
+  "signature": "sha256:99ce1d42001f74032bcccc6919adcf581e234259ab9db1c2aa8da6551ff2c728"
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  "nbformat": 3,
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@@ -261,7 +261,7 @@
        "output_type": "pyout",
        "prompt_number": 1,
        "text": [
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+        "<IPython.core.display.HTML at 0x7f7a9b19c5c0>"
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       }
      ],
@@ -354,9 +354,9 @@
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        "metadata": {},
        "output_type": "display_data",
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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fe9e05cd358>"
+        "<matplotlib.figure.Figure at 0x7f7a7e5d63c8>"
        ]
       }
      ],
@@ -698,26 +698,18 @@
       "\n",
       "p1, = plt.plot (xs, ys, 'r--')\n",
       "p2, = plt.plot (pxs, pys)\n",
-      "plt.legend([p1,p2], ['filter', 'measurement'], 2)\n",
+      "plt.legend([p1,p2], ['filter', 'measurement'], loc=2)\n",
       "plt.show()"
      ],
      "language": "python",
      "metadata": {},
      "outputs": [
-      {
-       "output_type": "stream",
-       "stream": "stderr",
-       "text": [
-        "/home/rlabbe/anaconda/lib/python3.4/site-packages/matplotlib/cbook.py:133: MatplotlibDeprecationWarning: The \"loc\" positional argument to legend is deprecated. Please use the \"loc\" keyword instead.\n",
-        "  warnings.warn(message, mplDeprecation, stacklevel=1)\n"
-       ]
-      },
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fe9e05cd3c8>"
+        "<matplotlib.figure.Figure at 0x7f7a7e21df98>"
        ]
       }
      ],
@@ -794,12 +786,20 @@
      "language": "python",
      "metadata": {},
      "outputs": [
+      {
+       "output_type": "stream",
+       "stream": "stderr",
+       "text": [
+        "/home/rlabbe/anaconda/lib/python3.4/site-packages/matplotlib/cbook.py:133: MatplotlibDeprecationWarning: The \"loc\" positional argument to legend is deprecated. Please use the \"loc\" keyword instead.\n",
+        "  warnings.warn(message, mplDeprecation, stacklevel=1)\n"
+       ]
+      },
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fe9e01f4a90>"
+        "<matplotlib.figure.Figure at 0x7f7a7df5ffd0>"
        ]
       },
       {
@@ -884,7 +884,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fe9e02b0cf8>"
+        "<matplotlib.figure.Figure at 0x7f7a7e207908>"
        ]
       }
      ],
@@ -899,6 +899,232 @@
       "Finally, let's look at the lower left side of $\\mathbf{P}$, which is all 0s. Why 0s? Consider $\\mathbf{P}_{3,0}$. That stores the term $p\\sigma_3\\sigma_0$, which is the covariance between $\\dot{y}$ and $x$. These are independent, so the term will be 0. The rest of the terms are for similarly independent variables."
      ]
     },
+    {
+     "cell_type": "heading",
+     "level": 2,
+     "metadata": {},
+     "source": [
+      "Sensor Fusion"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Early in the g-h filter chapter we discussed designing a filter for two scales, one that was accurate and one that was very inaccurate. We determined that we should always include the information from the inaccurate filter - we should never discard any information that is available to us. If you do not recall this discussion, consider this edge case. You have two scales, one accurate to 1 lb, the second to 10 lbs. You way yourself on each scale, and the first reads 170 lbs, and the second reads 181. If we only used the data from the first scale we would only know that our weight is in the range of 169 to 171 lbs. However, when we incorporate the measurement from the second scale we can decide that the correct weight can only be 171 lbs, since 171 lbs is exactly at the outside range of the second measurement, which would be 171 to 191 lbs. Of course most measurements don't offer such a exact answer, but the principle applies to any measurement.\n",
+      "\n",
+      "So consider a situation where we have two sensors measuring the same value at each time. How shall we incorporate that into our Kalman filter? \n",
+      "\n",
+      "Suppose we have a train or cart on a railway. It has a sensor attached to the wheels counting revolutions, which can be converted to a distance along the track. Then, suppose we have a GPS mounted to the train which reports position. Thus, we have two measurements, both reporting position along the track. Suppose further that the accuracy of the wheel sensor is 1m, and the accuracy of the GPS is 10m. How do we combine these two measurements into one filter? This may seem quite contrived, but aircraft use sensor fusion to fuse the measurements from sensors such as a GPS, INS, doppler radar, VOR, the airspeed indicator, and more.\n",
+      "\n",
+      "It is actually quite easy. The relevant matrix is the measurement matrix $\\mathbf{H}$. Recall that this matrix tells us how to convert from the Kalman filter's state $\\mathbf{X}$ to a measurement ${Z}$. Suppose that we decide that our Kalman filter state should contain the position and velocity of the train, so that\n",
+      "\n",
+      "$$ \\mathbf{X} = \\begin{bmatrix}x \\\\ \\dot{x}\\end{bmatrix}$$\n",
+      "\n",
+      "We have two measurements for position, so we will define the measurement vector to be a vector of the measurements from the wheel and the GPS.\n",
+      "\n",
+      "$$ \\mathbf{Z} = \\begin{bmatrix}x_{wheel} \\\\ x_{gps}\\end{bmatrix}$$\n",
+      "\n",
+      "So we have to design the matrix $\\mathbf{H}$ to convert $\\mathbf{X}$ to $\\mathbf{Z}$ . They are both positions, so the conversion is nothing more than multiplying by one:\n",
+      "\n",
+      "$$ \\begin{bmatrix}x_{wheel} \\\\ x_{gps}\\end{bmatrix} = \\begin{bmatrix}1 &0 \\\\ 1& 0\\end{bmatrix} \\begin{bmatrix}x \\\\ \\dot{x}\\end{bmatrix}$$\n",
+      "\n",
+      "To make it clearer, suppose that the wheel reports not position but the number of rotations of the wheels, where 1 revolution yields 2m of travel. In that case we would write\n",
+      "\n",
+      "$$ \\begin{bmatrix}x_{wheel} \\\\ x_{gps}\\end{bmatrix} = \\begin{bmatrix}0.5 &0 \\\\ 1& 0\\end{bmatrix} \\begin{bmatrix}x \\\\ \\dot{x}\\end{bmatrix}$$\n",
+      "\n",
+      "Now we have to design the measurement noise matrix $\\mathbf{R}$. Suppose that the measurement variance for the GPS is twice the variance of the wheel, and the standard deviation of the wheel is 1.5m. That gives us\n",
+      "\n",
+      "$$\\sigma_{wheel} =  1.5\\\\\n",
+      "\\sigma^2_{wheel} = 2.25 \\\\ \n",
+      "\\sigma_{gps} = 1.5*2 = 3 \\\\\n",
+      "\\sigma^2_{gps} = 9.\n",
+      "$$\n",
+      "\n",
+      "That is pretty much our Kalman filter design. We need to design for Q, but that is invariant to whether we are doing sensor fusion or not, so I will just choose some arbitrary value for Q.\n",
+      "\n",
+      "So let's run a simulation of this design. I will assume a velocity of 10m/s with an update rate of 0.1 seconds."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "import numpy.random as random\n",
+      "from numpy.random import randn\n",
+      "import numpy as np\n",
+      "import matplotlib.pyplot as plt\n",
+      "from filterpy.kalman import KalmanFilter\n",
+      "from numpy import array, asarray\n",
+      "\n",
+      "dt = 0.1\n",
+      "wheel_sigma = 1.5\n",
+      "gps_sigma = wheel_sigma*2\n",
+      "kf = KalmanFilter(dim_x=2, dim_z=2)\n",
+      "\n",
+      "kf.F = array ([[1., dt], [0., 1.]])\n",
+      "kf.H = array ([[1., 0.], [1., 0.]])\n",
+      "kf.X = array ([[0.], [1.]])\n",
+      "kf.Q *= 0.01\n",
+      "kf.P *= 100\n",
+      "kf.R[0,0] = wheel_sigma**2\n",
+      "kf.R[1,1] = gps_sigma**2 \n",
+      "\n",
+      "random.seed(1123)\n",
+      "xs = []\n",
+      "zs = []\n",
+      "nom = []\n",
+      "for i in range(1, 100):\n",
+      "    m0 = i + randn()*wheel_sigma\n",
+      "    m1 = i + randn()*gps_sigma\n",
+      "    z = array([[m0], [m1]])\n",
+      "\n",
+      "    kf.predict()\n",
+      "    kf.update(z)\n",
+      "\n",
+      "    xs.append(kf.X.T[0])\n",
+      "    zs.append(z.T[0])\n",
+      "    nom.append(i)\n",
+      "\n",
+      "xs = asarray(xs)\n",
+      "zs = asarray(zs)\n",
+      "nom = asarray(nom)\n",
+      "\n",
+      "\n",
+      "res = nom-xs[:,0]\n",
+      "print('fusion std: {:.3f}'.format (np.std(res)))\n",
+      "\n",
+      "plt.plot(xs[:,0], label='Kalman filter')\n",
+      "plt.plot(zs[:,0],linestyle=':', label='wheel')\n",
+      "plt.plot(zs[:,1],linestyle=':', label='GPS')\n",
+      "plt.legend(loc=4)\n",
+      "plt.show()"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "output_type": "stream",
+       "stream": "stdout",
+       "text": [
+        "fusion std: 0.422\n"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x7f7a769411d0>"
+       ]
+      }
+     ],
+     "prompt_number": 14
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "We can see the result of the Kalman filter in blue.\n",
+      "\n",
+      "##### Excercise: Prove that the GPS improves the filter\n",
+      "\n",
+      "Devise a way to prove that fusing the GPS and wheel measurements yields a better result than using the wheel alone.\n",
+      "\n",
+      "##### Solution 1\n",
+      "\n",
+      "Force the Kalman filter to disregard the GPS measurement by setting the measurement noise for the GPS to a near infinite value. Re-run the filter and observe the standard deviation of the residual. \n",
+      "\n",
+      "For example, if I edit the cell above with\n",
+      "\n",
+      "    kf.R[1,1] = 1.e80\n",
+      "\n",
+      "and run the cell again I get a standard deviation of 0.494, vs the standard deviation of 0.422 when the GPS measurement is used. \n",
+      "\n",
+      "\n",
+      "##### Solution 2\n",
+      "\n",
+      "This is more work, but we can write a Kalman filter that only takes one measurement."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "import numpy.random as random\n",
+      "from numpy.random import randn\n",
+      "import numpy as np\n",
+      "import matplotlib.pyplot as plt\n",
+      "from filterpy.kalman import KalmanFilter\n",
+      "from numpy import array, asarray\n",
+      "\n",
+      "dt = 0.1\n",
+      "wheel_sigma = 1.5\n",
+      "kf = KalmanFilter(dim_x=2, dim_z=1)\n",
+      "\n",
+      "kf.F = array ([[1., dt], [0., 1.]])\n",
+      "kf.H = array ([[1., 0.]])\n",
+      "kf.X = array ([[0.], [1.]])\n",
+      "kf.Q *= 0.01\n",
+      "kf.P *= 100\n",
+      "kf.R[0,0] = wheel_sigma**2\n",
+      "\n",
+      "random.seed(1123)\n",
+      "xs = []\n",
+      "zs = []\n",
+      "nom = []\n",
+      "for i in range(1, 100):\n",
+      "    m0 = i + randn()*wheel_sigma\n",
+      "    z = array([[m0]])\n",
+      "\n",
+      "    kf.predict()\n",
+      "    kf.update(z)\n",
+      "\n",
+      "    xs.append(kf.X.T[0])\n",
+      "    zs.append(z.T[0])\n",
+      "    nom.append(i)\n",
+      "\n",
+      "xs = asarray(xs)\n",
+      "zs = asarray(zs)\n",
+      "nom = asarray(nom)\n",
+      "\n",
+      "\n",
+      "res = nom-xs[:,0]\n",
+      "print('std: {:.3f}'.format (np.std(res)))\n",
+      "\n",
+      "plt.plot(xs[:,0], label='Kalman filter')\n",
+      "plt.plot(zs[:,0],linestyle=':', label='wheel')\n",
+      "plt.legend(loc=4)\n",
+      "plt.show()"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "output_type": "stream",
+       "stream": "stdout",
+       "text": [
+        "std: 0.523\n"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x7f7a768a9198>"
+       ]
+      }
+     ],
+     "prompt_number": 15
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "On this run I got a standard deviation of 0.523 vs the value of 0.422 for the fused measurements. "
+     ]
+    },
     {
      "cell_type": "heading",
      "level": 2,
@@ -991,7 +1217,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 14
+     "prompt_number": 16
     },
     {
      "cell_type": "markdown",
@@ -1034,19 +1260,19 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fe9d49847b8>"
+        "<matplotlib.figure.Figure at 0x7f7a768f84a8>"
        ]
       },
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fe9d4a5ecf8>"
+        "<matplotlib.figure.Figure at 0x7f7a76900978>"
        ]
       }
      ],
-     "prompt_number": 15
+     "prompt_number": 17
     },
     {
      "cell_type": "markdown",
@@ -1184,7 +1410,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 16
+     "prompt_number": 18
     },
     {
      "cell_type": "markdown",
@@ -1224,11 +1450,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fe9d4763e48>"
+        "<matplotlib.figure.Figure at 0x7f7a76784fd0>"
        ]
       }
      ],
-     "prompt_number": 17
+     "prompt_number": 19
     },
     {
      "cell_type": "markdown",
@@ -1369,7 +1595,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 18
+     "prompt_number": 20
     },
     {
      "cell_type": "markdown",
@@ -1417,13 +1643,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fe9d4579400>"
+        "<matplotlib.figure.Figure at 0x7f7a7e3109e8>"
        ]
       }
      ],
-     "prompt_number": 19
+     "prompt_number": 21
     },
     {
      "cell_type": "markdown",
@@ -1560,11 +1786,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fe9d49e2358>"
+        "<matplotlib.figure.Figure at 0x7f7a767aa588>"
        ]
       }
      ],
-     "prompt_number": 20
+     "prompt_number": 22
     },
     {
      "cell_type": "markdown",
@@ -1634,7 +1860,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 21
+     "prompt_number": 23
     },
     {
      "cell_type": "markdown",
@@ -1691,13 +1917,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x7fe9d457ee10>"
+        "<matplotlib.figure.Figure at 0x7f7a768d3668>"
        ]
       }
      ],
-     "prompt_number": 22
+     "prompt_number": 24
     },
     {
      "cell_type": "markdown",
@@ -1759,21 +1985,21 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x7fe9d49e26d8>"
+        "<matplotlib.figure.Figure at 0x7f7a88145f98>"
        ]
       },
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fe9e02a3fd0>"
+        "<matplotlib.figure.Figure at 0x7f7a768e9cc0>"
        ]
       }
      ],
-     "prompt_number": 23
+     "prompt_number": 25
     },
     {
      "cell_type": "markdown",
@@ -1797,21 +2023,21 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fe9d4763198>"
+        "<matplotlib.figure.Figure at 0x7f7a76796668>"
        ]
       },
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fe9d45cbe48>"
+        "<matplotlib.figure.Figure at 0x7f7a767ecef0>"
        ]
       }
      ],
-     "prompt_number": 24
+     "prompt_number": 26
     },
     {
      "cell_type": "markdown",