Sensor fusion at different rates

Discussed how to perform sensor fusion when the sensors run at different rates.

08_Designing_Kalman_Filters/Designing_Kalman_Filters.ipynb

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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 0x7f89ce772dd8>"
+        "<matplotlib.figure.Figure at 0x7f7f48720828>"
        ]
       }
      ],
@@ -707,9 +707,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f89ce7820b8>"
+        "<matplotlib.figure.Figure at 0x7f7f48720940>"
        ]
       }
      ],
@@ -788,9 +788,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f89b1816eb8>"
+        "<matplotlib.figure.Figure at 0x7f7f2b8069e8>"
        ]
       },
       {
@@ -875,7 +875,7 @@
        "output_type": "display_data",
        "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x7f89a9ec8cc0>"
+        "<matplotlib.figure.Figure at 0x7f7f23ebfcc0>"
        ]
       }
      ],
@@ -997,7 +997,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f89b1821f28>"
+        "<matplotlib.figure.Figure at 0x7f7f23ebfba8>"
        ]
       }
      ],
@@ -1204,7 +1204,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 75
+     "prompt_number": 17
     },
     {
      "cell_type": "heading",
@@ -1247,7 +1247,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 76
+     "prompt_number": 18
     },
     {
      "cell_type": "markdown",
@@ -1280,7 +1280,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 77
+     "prompt_number": 19
     },
     {
      "cell_type": "markdown",
@@ -1305,7 +1305,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 78
+     "prompt_number": 20
     },
     {
      "cell_type": "markdown",
@@ -1334,13 +1334,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f89a8593898>"
+        "<matplotlib.figure.Figure at 0x7f7f2b542e80>"
        ]
       }
      ],
-     "prompt_number": 79
+     "prompt_number": 21
     },
     {
      "cell_type": "markdown",
@@ -1368,7 +1368,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 80
+     "prompt_number": 22
     },
     {
      "cell_type": "code",
@@ -1384,13 +1384,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f89b1848630>"
+        "<matplotlib.figure.Figure at 0x7f7f2b535630>"
        ]
       }
      ],
-     "prompt_number": 81
+     "prompt_number": 23
     },
     {
      "cell_type": "markdown",
@@ -1415,13 +1415,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f89a9ed5128>"
+        "<matplotlib.figure.Figure at 0x7f7f2c6ca320>"
        ]
       }
      ],
-     "prompt_number": 82
+     "prompt_number": 24
     },
     {
      "cell_type": "markdown",
@@ -1448,13 +1448,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f89a84faef0>"
+        "<matplotlib.figure.Figure at 0x7f7f2b7f05f8>"
        ]
       }
      ],
-     "prompt_number": 83
+     "prompt_number": 25
     },
     {
      "cell_type": "markdown",
@@ -1479,13 +1479,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f89a8504b70>"
+        "<matplotlib.figure.Figure at 0x7f7f2261d828>"
        ]
       }
      ],
-     "prompt_number": 84
+     "prompt_number": 26
     },
     {
      "cell_type": "markdown",
@@ -1518,13 +1518,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f89a84e3240>"
+        "<matplotlib.figure.Figure at 0x7f7f225906a0>"
        ]
       }
      ],
-     "prompt_number": 85
+     "prompt_number": 27
     },
     {
      "cell_type": "markdown",
@@ -1559,13 +1559,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x7f89a8358550>"
+        "<matplotlib.figure.Figure at 0x7f7f22617a90>"
        ]
       }
      ],
-     "prompt_number": 86
+     "prompt_number": 28
     },
     {
      "cell_type": "markdown",
@@ -1599,13 +1599,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f89b1518dd8>"
+        "<matplotlib.figure.Figure at 0x7f7f22620be0>"
        ]
       }
      ],
-     "prompt_number": 87
+     "prompt_number": 29
     },
     {
      "cell_type": "markdown",
@@ -1636,13 +1636,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f89a8358240>"
+        "<matplotlib.figure.Figure at 0x7f7f225904e0>"
        ]
       }
      ],
-     "prompt_number": 88
+     "prompt_number": 30
     },
     {
      "cell_type": "markdown",
@@ -1676,7 +1676,7 @@
        "output_type": "display_data",
        "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x7f89b15a4c18>"
+        "<matplotlib.figure.Figure at 0x7f7f225d2780>"
        ]
       },
       {
@@ -1684,11 +1684,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f89b15a4400>"
+        "<matplotlib.figure.Figure at 0x7f7f2b55ca90>"
        ]
       }
      ],
-     "prompt_number": 89
+     "prompt_number": 31
     },
     {
      "cell_type": "markdown",
@@ -1724,7 +1724,7 @@
        "output_type": "display_data",
        "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x7f89a8358f60>"
+        "<matplotlib.figure.Figure at 0x7f7f224f2668>"
        ]
       },
       {
@@ -1732,11 +1732,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f89a844a940>"
+        "<matplotlib.figure.Figure at 0x7f7f225fcd68>"
        ]
       }
      ],
-     "prompt_number": 96
+     "prompt_number": 32
     },
     {
      "cell_type": "markdown",
@@ -1765,11 +1765,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f89a8585128>"
+        "<matplotlib.figure.Figure at 0x7f7f2b893208>"
        ]
       }
      ],
-     "prompt_number": 71
+     "prompt_number": 33
     },
     {
      "cell_type": "markdown",
@@ -1833,11 +1833,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f89a8623908>"
+        "<matplotlib.figure.Figure at 0x7f7f2b887518>"
        ]
       }
      ],
-     "prompt_number": 97
+     "prompt_number": 34
     },
     {
      "cell_type": "markdown",
@@ -1866,7 +1866,7 @@
        "output_type": "display_data",
        "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x7f89b1656400>"
+        "<matplotlib.figure.Figure at 0x7f7f48714f60>"
        ]
       },
       {
@@ -1874,11 +1874,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f89a9ef39e8>"
+        "<matplotlib.figure.Figure at 0x7f7f2b5e39b0>"
        ]
       }
      ],
-     "prompt_number": 98
+     "prompt_number": 35
     },
     {
      "cell_type": "markdown",
@@ -1909,7 +1909,7 @@
        "output_type": "display_data",
        "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x7f89b15af6d8>"
+        "<matplotlib.figure.Figure at 0x7f7f2b505160>"
        ]
       },
       {
@@ -1917,11 +1917,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f89b1880160>"
+        "<matplotlib.figure.Figure at 0x7f7f225d9198>"
        ]
       }
      ],
-     "prompt_number": 100
+     "prompt_number": 36
     },
     {
      "cell_type": "markdown",
@@ -1950,7 +1950,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f89a835f438>"
+        "<matplotlib.figure.Figure at 0x7f7f224958d0>"
        ]
       },
       {
@@ -1958,11 +1958,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f89a835f400>"
+        "<matplotlib.figure.Figure at 0x7f7f225d94a8>"
        ]
       }
      ],
-     "prompt_number": 101
+     "prompt_number": 37
     },
     {
      "cell_type": "heading",
@@ -1980,30 +1980,30 @@
       "\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",
+      "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-like sensor which I'll call a 'PS' 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 PS 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",
+      "Kalman filters for inertial filters are very difficult, but fusing data from two or more sensors providing measurements of the same state variable (such as position) 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",
+      "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 PS.\n",
       "\n",
-      "$$ \\mathbf{Z} = \\begin{bmatrix}x_{wheel} \\\\ x_{gps}\\end{bmatrix}$$\n",
+      "$$ \\mathbf{Z} = \\begin{bmatrix}x_{wheel} \\\\ x_{ps}\\end{bmatrix}$$\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",
+      "$$ \\begin{bmatrix}x_{wheel} \\\\ x_{ps}\\end{bmatrix} = \\begin{bmatrix}1 &0 \\\\ 1& 0\\end{bmatrix} \\begin{bmatrix}x \\\\ \\dot{x}\\end{bmatrix}$$\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",
+      "$$ \\begin{bmatrix}x_{wheel} \\\\ x_{ps}\\end{bmatrix} = \\begin{bmatrix}0.5 &0 \\\\ 1& 0\\end{bmatrix} \\begin{bmatrix}x \\\\ \\dot{x}\\end{bmatrix}$$\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",
+      "Now we have to design the measurement noise matrix $\\mathbf{R}$. Suppose that the measurement variance for the PS is twice the variance of the wheel, and the standard deviation of the wheel is 1.5m. That gives us\n",
       "\n",
       "$$\\sigma_{wheel} =  1.5\\\\\n",
       "\\sigma^2_{wheel} = 2.25 \\\\ \n",
-      "\\sigma_{gps} = 1.5*2 = 3 \\\\\n",
-      "\\sigma^2_{gps} = 9.\n",
+      "\\sigma_{ps} = 1.5*2 = 3 \\\\\n",
+      "\\sigma^2_{ps} = 9.\n",
       "$$\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",
@@ -2022,7 +2022,7 @@
       "from filterpy.kalman import KalmanFilter\n",
       "from numpy import array, asarray\n",
       "\n",
-      "def fusion_test(wheel_sigma, gps_sigma, do_plot=True):\n",
+      "def fusion_test(wheel_sigma, ps_sigma, do_plot=True):\n",
       "    dt = 0.1\n",
       "    kf = KalmanFilter(dim_x=2, dim_z=2)\n",
       "\n",
@@ -2033,7 +2033,7 @@
       "                   [(dt**2)/2,  dt      ]]) * 0.02\n",
       "    kf.P *= 100\n",
       "    kf.R[0,0] = wheel_sigma**2\n",
-      "    kf.R[1,1] = gps_sigma**2 \n",
+      "    kf.R[1,1] = ps_sigma**2 \n",
       "\n",
       "    random.seed(1123)\n",
       "    xs = []\n",
@@ -2041,7 +2041,7 @@
       "    nom = []\n",
       "    for i in range(1, 100):\n",
       "        m0 = i + randn()*wheel_sigma\n",
-      "        m1 = i + randn()*gps_sigma\n",
+      "        m1 = i + randn()*ps_sigma\n",
       "        z = array([[m0], [m1]])\n",
       "\n",
       "        kf.predict()\n",
@@ -2061,7 +2061,7 @@
       "    if do_plot:\n",
       "        plt.plot(xs[:,0], label='Kalman filter')\n",
       "        plt.plot(zs[:,0],linestyle=':', label='wheel', lw=4)\n",
-      "        plt.plot(zs[:,1],linestyle='--', label='GPS')\n",
+      "        plt.plot(zs[:,1],linestyle='--', label='PS')\n",
       "        plt.legend(loc=4)\n",
       "        plt.ylim(0,100)\n",
       "        plt.show()\n",
@@ -2083,7 +2083,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f89b18a5780>"
+        "<matplotlib.figure.Figure at 0x7f7f2b827208>"
        ]
       }
      ],
@@ -2101,18 +2101,66 @@
      "level": 3,
      "metadata": {},
      "source": [
-      "Excercise: Prove that the GPS improves the filter"
+      "Exercise: Can you Kalman Filter GPS outputs?"
      ]
     },
     {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "Devise a way to prove that fusing the GPS and wheel measurements yields a better result than using the wheel alone.\n",
+      "In the section above I have you apply a Kalman filter to 'GPS-like' sensor. Can you apply a Kalman filter to the output of a commercial Kalman filter? In other words, will the output of your filter be better than, worse than, or equal to the GPS's output?"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "# Think your answer here"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": []
+    },
+    {
+     "cell_type": "heading",
+     "level": 4,
+     "metadata": {},
+     "source": [
+      "Solution"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Commercial GPSs have a Kalman filter built into them, and their output is the filtered estimate created by that filter. So, suppose you have a steady stream of output from the GPS consisting of a position and position error. Can you not pass those two pieces of data into your own filter?\n",
+      "\n",
+      "Well, what are the characteristics of that data stream, and more importantly, what are the fundamental requirements of the input to the Kalman filter?\n",
+      "\n",
+      "Inputs to the Kalman filter must be *Gaussian* and *time independent*. The output of the GPS is *time dependent* because the filter bases it's current estimate on the recusive estimates of all previous measurements. Hence, the signal is not white, it is not time independent, and if you pass that data into a Kalman filter you have violated the mathematical requirements of the filter. So, the answer is no, you cannot get better estimates by running a KF on the output of a commercial GPS. \n",
+      "\n",
+      "Another way to think of it. Kalman filters are optimal in a least squares sense. There is no way to take an optimal solution, pass it through a filter, any filter, and get a 'more optimal' answer because it is a logical impossibility. At best the signal will be unchanged, in which case it will still be optimal, or it will be changed, and hence no longer optimal.\n",
+      "\n",
+      "This is a difficult problem that hobbiests face when trying to integrate GPS, IMUs and other off the shelf sensors. It is too early in the book to try to address this problem."
+     ]
+    },
+    {
+     "cell_type": "heading",
+     "level": 3,
+     "metadata": {},
+     "source": [
+      "Excercise: Prove that the PS improves the filter"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Devise a way to prove that fusing the PS 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. "
+      "Force the Kalman filter to disregard the PS measurement by setting the measurement noise for the PS to a near infinite value. Re-run the filter and observe the standard deviation of the residual. "
      ]
     },
     {
@@ -2140,7 +2188,7 @@
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "Here we can see the error in the filter where the GPS measurement is almost entirely ignored is greater than that where it is used.\n",
+      "Here we can see the error in the filter where the PS measurement is almost entirely ignored is greater than that where it is used.\n",
       "\n",
       "\n",
       "##### Solution 2\n",
@@ -2213,7 +2261,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f89b1821518>"
+        "<matplotlib.figure.Figure at 0x7f7f2256bcc0>"
        ]
       }
      ],
@@ -2226,6 +2274,171 @@
       "On this run I got a standard deviation of 0.523 vs the value of 0.391 for the fused measurements. "
      ]
     },
+    {
+     "cell_type": "heading",
+     "level": 3,
+     "metadata": {},
+     "source": [
+      "Exercise: Different Data Rates"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "It is rare that two different sensor classes output data at the same rate. Assume that the PS produces an update at 1 Hz, and the wheel updates at 4 Hz. Design a filter that incorporates all of these measurements.\n",
+      "\n",
+      "**hint**: This is a difficult problem in that I have not explained how to do this. Think about which matrices incororate time, and which incorporate knowledge about the number and kind of measurements. All of these will have to be designed to work with this problem. If you can correctly enumerate those matrices you are most of the way to a solution."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "#your solution here!"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": []
+    },
+    {
+     "cell_type": "heading",
+     "level": 4,
+     "metadata": {},
+     "source": [
+      "Solution"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "We can do this by setting the data rate to 0.25 seconds, which is 4 Hz. As we loop, on every iteration we call `update()` with only the wheel measurement. Then, every fourth time we will call `update()` with both the wheel and PS measurements. \n",
+      "\n",
+      "This means that we vary the amount of data in the z parameter. The matrices associated with the measurement are $\\mathbf{H}$ and $\\mathbf{R}$. In the code above we designed H to be \n",
+      "\n",
+      "$$\\mathbf{H} = \\begin{bmatrix}1 &0 \\\\ 1& 0\\end{bmatrix}$$\n",
+      "\n",
+      "to account for the two measurements of position. When only the wheel reading is available, we must set\n",
+      "$$\\mathbf{H} = \\begin{bmatrix}1 &0\\end{bmatrix}.$$\n",
+      "\n",
+      "The matrix $\\mathbf{R}$ specifies the noise in each measurement. In the code above we set\n",
+      "\n",
+      "$$\\mathbf{R} = \\begin{bmatrix}\\sigma_{wheel}^2 &0 \\\\ 0 & \\sigma_{PS}^2\\end{bmatrix}$$\n",
+      "\n",
+      "When only the wheel measurement is available, we must set\n",
+      "\n",
+      "$$\\mathbf{R} = \\begin{bmatrix}\\sigma_{wheel}^2\\end{bmatrix}$$\n",
+      "\n",
+      "The two matrices that incorporate time are $\\mathbf{F}$ and $\\mathbf{Q}$. For example,\n",
+      "\n",
+      "$$\\mathbf{F} = \\begin{bmatrix}1 & \\Delta t \\\\ 0& 1\\end{bmatrix}.$$\n",
+      "\n",
+      "Since the wheel and PS reading coincide once every 4 readings we can just set $\\delta t =0.25$ and not modify it while filtering. If the readings did not coincide in each iteration you would have to calculate how much time has passed since the last predict, compute a new $\\mathbf{F}$ and $\\mathbf{Q}$, and then call `predict()` so the filter could make a correct prediction based on the time step required.\n",
+      "\n",
+      "So here is the code. "
+     ]
+    },
+    {
+     "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",
+      "def fusion_test(wheel_sigma, ps_sigma, do_plot=True):\n",
+      "    dt = 0.25\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 *= array([[(dt**3)/3, (dt**2)/2],\n",
+      "                   [(dt**2)/2,  dt      ]]) * 0.02\n",
+      "    kf.P *= 100\n",
+      "    kf.R[0,0] = wheel_sigma**2\n",
+      "    kf.R[1,1] = ps_sigma**2 \n",
+      "\n",
+      "    random.seed(1123)\n",
+      "    xs = []\n",
+      "    zs_wheel = []\n",
+      "    zs_ps = []\n",
+      "    nom = []\n",
+      "    for i in range(1, 101):\n",
+      "        \n",
+      "        if i % 4 == 0:\n",
+      "            m0 = i + randn()*wheel_sigma\n",
+      "            m1 = i + randn()*ps_sigma\n",
+      "            z = array([[m0], [m1]])\n",
+      "            kf.H = array([[1., 0.], [1., 0.]])\n",
+      "            R = np.eye(2)\n",
+      "            R[0,0] = wheel_sigma**2\n",
+      "            R[1,1] = ps_sigma**2      \n",
+      "            \n",
+      "            zs_wheel.append(m0)\n",
+      "            zs_ps.append((i,m1))\n",
+      "        else:  \n",
+      "            m0 = i + randn()*wheel_sigma\n",
+      "            z = array([m0])\n",
+      "            kf._H = array([[1., 0.]])\n",
+      "            R = np.eye(1) * wheel_sigma**2\n",
+      "            zs_wheel.append(m0)\n",
+      "\n",
+      "        kf.predict()\n",
+      "        kf.update(z, R)\n",
+      "\n",
+      "        xs.append(kf.x.T[0])\n",
+      "        nom.append(i)\n",
+      "\n",
+      "    xs = asarray(xs)\n",
+      "    nom = asarray(nom)\n",
+      "\n",
+      "    res = nom-xs[:,0]\n",
+      "    print('fusion std: {:.3f}'.format (np.std(res)))\n",
+      "    if do_plot:\n",
+      "        plt.plot(xs[:,0], label='Kalman filter')\n",
+      "        plt.plot(zs_wheel, linestyle=':', label='wheel', lw=4)\n",
+      "        plt.plot(*zip(*zs_ps), linestyle='--', label='PS')\n",
+      "        plt.legend(loc=4)\n",
+      "        plt.ylim(0,100)\n",
+      "        plt.show()\n",
+      "\n",
+      "fusion_test(1.5, 3.0)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "output_type": "stream",
+       "stream": "stdout",
+       "text": [
+        "fusion std: 0.452\n"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x7f7f221f1240>"
+       ]
+      }
+     ],
+     "prompt_number": 104
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "We can see from the standard deviation that the performance is a bit worse than when the PS and wheel were measured in every update, but better than the wheel alone.\n",
+      "\n",
+      "The code is fairly straighforward. The `update()` function optionally takes R as an argument, and I chose to do that rather than alter `KalmanFilter.R`, mostly to show that it is possible. Either way is fine. I modified `KalmanFilter.H` on each update depending on whether there are 1 or 2 measurements available. The only other difficulty was storing the wheel and PS measurements in two different arrays because there are a different number of measurements for each. "
+     ]
+    },
     {
      "cell_type": "heading",
      "level": 2,
@@ -2367,7 +2580,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f89b18db048>"
+        "<matplotlib.figure.Figure at 0x7f7f2247f278>"
        ]
       },
       {
@@ -2375,7 +2588,7 @@
        "output_type": "display_data",
        "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x7f89b18f4080>"
+        "<matplotlib.figure.Figure at 0x7f7f23ecf5f8>"
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       }
      ],
@@ -2560,7 +2773,7 @@
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        "text": [
-        "<matplotlib.figure.Figure at 0x7f89a859e390>"
+        "<matplotlib.figure.Figure at 0x7f7f2242eeb8>"
        ]
       }
      ],
@@ -2756,7 +2969,7 @@
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        "text": [
-        "<matplotlib.figure.Figure at 0x7f89a8358fd0>"
+        "<matplotlib.figure.Figure at 0x7f7f222e60b8>"
        ]
       }
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@@ -2899,7 +3112,7 @@
        "output_type": "display_data",
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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f89a84f2be0>"
+        "<matplotlib.figure.Figure at 0x7f7f2b51fc50>"
        ]
       }
      ],
@@ -3037,7 +3250,7 @@
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        "text": [
-        "<matplotlib.figure.Figure at 0x7f89b1852240>"
+        "<matplotlib.figure.Figure at 0x7f7f2b860550>"
        ]
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@@ -3105,7 +3318,7 @@
        "output_type": "display_data",
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        "text": [
-        "<matplotlib.figure.Figure at 0x7f89a84f2cc0>"
+        "<matplotlib.figure.Figure at 0x7f7f2b51f160>"
        ]
       },
       {
@@ -3113,7 +3326,7 @@
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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f89b272d940>"
+        "<matplotlib.figure.Figure at 0x7f7f2b824438>"
        ]
       }
      ],
@@ -3143,7 +3356,7 @@
        "output_type": "display_data",
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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f89a8593780>"
+        "<matplotlib.figure.Figure at 0x7f7f2242eba8>"
        ]
       },
       {
@@ -3151,7 +3364,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f89a84ad278>"
+        "<matplotlib.figure.Figure at 0x7f7f2b50ebe0>"
        ]
       }
      ],