spell check

11_Extended_Kalman_Filters.ipynb

@@ -1,7 +1,7 @@
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-     "cell_type": "heading",
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+     "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "The Extended Kalman Filter"
+      "#The Extended Kalman Filter"
      ]
     },
     {
@@ -257,13 +256,13 @@
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@@ -271,7 +270,7 @@
      "source": [
       "**Author's note: this is still being heavily edited and added to - there is a lot of duplicate material, incorrect code, and so on. The examples are just a brain dump, not fully formulated and not ready to be used.**\n",
       "\n",
-      "**The Uncented Kalman filter (UKF) chapter is much further along. The UKF is almost always better performing than the Extended Kalman filter, and is much easier to implement, so if you have an urgenet need for nonlinear Kalman filter I'll point you towards that chapter for now.**\n",
+      "**The Unscented Kalman filter (UKF) chapter is much further along. The UKF is almost always better performing than the Extended Kalman filter, and is much easier to implement, so if you have an urgent need for nonlinear Kalman filter I'll point you towards that chapter for now.**\n",
       "\n",
       "The Kalman filter that we have developed to this point is extremely good, but it is also limited. Its derivation is in the linear space, and hence it only works for linear problems. Let's be a bit more rigorous here. You can, and we have in this book, apply the Kalman filter to nonlinear problems. For example, in the g-h filter chapter we explored using a g-h filter in a problem with constant acceleration. It 'worked', in that it remained numerically stable and the filtered output did track the input, but there was always a lag. It is easy to prove that there will always be a lag when $\\mathbf{\\ddot{x}}>0$. The filter no longer produces an optimal result. If we make our time step arbitrarily small we can still handle many problems, but typically we are using Kalman filters with physical sensors and solving real-time problems. Either fast enough sensors do not exist, are prohibitively expensive, or the computation time required is excessive. It is not a workable solution.\n",
       "\n",
@@ -284,11 +283,10 @@
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     },
     {
-     "cell_type": "heading",
-     "level": 2,
+     "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "The Problem with Nonlinearity"
+      "##The Problem with Nonlinearity"
      ]
     },
     {
@@ -304,7 +302,7 @@
       "\n",
       "That's nonlinear. But it is also a very special form. You spent a lot of time, probably at least a year, learning how to integrate various terms, and you still can not integrate some arbitrary equation - no one can. We don't know how. If you took freshman Physics you perhaps remember homework involving sliding frictionless blocks on a plane and other toy problems. At the end of the course you were almost entirely unequipped to solve real world problems because the real world is nonlinear, and you were taught linear, closed forms of equations. It made the math tractable, but mostly useless. \n",
       "\n",
-      "The mathematics of the Kalman filter is beautiful in part due to the Gaussian equation being so special. It is nonlinear, but when we add and multipy it using linear algebra we get another Gaussian equation as a result. That is very rare. $\\sin{x}*\\sin{y}$ does not yield a $\\sin(\\cdot)$ as an output.\n",
+      "The mathematics of the Kalman filter is beautiful in part due to the Gaussian equation being so special. It is nonlinear, but when we add and multiply it using linear algebra we get another Gaussian equation as a result. That is very rare. $\\sin{x}*\\sin{y}$ does not yield a $\\sin(\\cdot)$ as an output.\n",
       "\n",
       "> If you are not well versed in signals and systems there is a perhaps startling fact that you should be aware of. A linear system is defined as a system whose output is linearly proportional to the sum of all its inputs. A consequence of this is that to be linear if the input is zero than the output must also be zero. Consider an audio amp - if a sing into a microphone, and you start talking, the output should be the sum of our voices (input) scaled by the amplifier gain. But if amplifier outputs a nonzero signal for a zero input the additive relationship no longer holds. This is because you can say $amp(roger) = amp(roger + 0)$ This clearly should give the same output, but if amp(0) is nonzero, then\n",
       "\n",
@@ -323,11 +321,10 @@
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     {
-     "cell_type": "heading",
-     "level": 2,
+     "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "The Effect of Nonlinear Transfer Functions on Gaussians"
+      "##The Effect of Nonlinear Transfer Functions on Gaussians"
      ]
     },
     {
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truncated
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"iVBORw0KGgoAAAANSUhEUgAAAysAAAGNCAYAAAARje6GAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f914ad4bcf8>"
+        "<matplotlib.figure.Figure at 0x7f1aa26a6780>"
        ]
       }
      ],
-     "prompt_number": 2
+     "prompt_number": 3
     },
     {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "This is an unsuprising result. The result of passing the Gaussian through $f(x)=2x+1$ is another Gaussian centered around 1. Let's look at the input, transfer function, and output at once."
+      "This is an unsurprising result. The result of passing the Gaussian through $f(x)=2x+1$ is another Gaussian centered around 1. Let's look at the input, transfer function, and output at once."
      ]
     },
     {
@@ -388,19 +385,19 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x7f914ad4bc18>"
+        "<matplotlib.figure.Figure at 0x7f1aa26a6668>"
        ]
       }
      ],
-     "prompt_number": 3
+     "prompt_number": 4
     },
     {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "The plot labelled 'input' is the histogram of the original data. This is passed through the transfer function $f(x)=2x+1$ which is displayed in the chart to the upper right. The red lines shows how one value, $x=0$ is passed through the function. Each value from input is passed through in the same way to the output function on the left. The output looks like a Gaussian, and is in fact a Gaussian. We can see that it is altered -the variance in the output is larger than the variance in the input, and the mean has been shifted from 0 to 1, which is what we would expect given the transfer function $f(x)=2x+1$ The $2x$ affects the variance, and the $+1$ shifts the mean.\n",
+      "The plot labeled 'input' is the histogram of the original data. This is passed through the transfer function $f(x)=2x+1$ which is displayed in the chart to the upper right. The red lines shows how one value, $x=0$ is passed through the function. Each value from input is passed through in the same way to the output function on the left. The output looks like a Gaussian, and is in fact a Gaussian. We can see that it is altered -the variance in the output is larger than the variance in the input, and the mean has been shifted from 0 to 1, which is what we would expect given the transfer function $f(x)=2x+1$ The $2x$ affects the variance, and the $+1$ shifts the mean.\n",
       "\n",
       "Now let's look at a nonlinear function and see how it affects the probability distribution."
      ]
@@ -422,19 +419,19 @@
       {
        "metadata": {},
        "output_type": "display_data",
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+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f914bb762e8>"
+        "<matplotlib.figure.Figure at 0x7f1aa2298ba8>"
        ]
       }
      ],
-     "prompt_number": 4
+     "prompt_number": 5
     },
     {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "This result may be somewhat suprising to you. The transfer function looks \"fairly\" linear - it is pretty close to a straight line, but the probability distribution of the output is completely different from a Gaussian.  Recall the equations for multiplying two univariate Gaussians:\n",
+      "This result may be somewhat surprising to you. The transfer function looks \"fairly\" linear - it is pretty close to a straight line, but the probability distribution of the output is completely different from a Gaussian.  Recall the equations for multiplying two univariate Gaussians:\n",
       "$$\\begin{aligned}\n",
       "\\mu =\\frac{\\sigma_1^2 \\mu_2 + \\sigma_2^2 \\mu_1} {\\sigma_1^2 + \\sigma_2^2}\\mbox{, } \n",
       "\\sigma = \\frac{1}{\\frac{1}{\\sigma_1^2} + \\frac{1}{\\sigma_2^2}}\n",
@@ -458,19 +455,19 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f9167bfa160>"
+        "<matplotlib.figure.Figure at 0x7f1aa0fb4b38>"
        ]
       }
      ],
-     "prompt_number": 5
+     "prompt_number": 6
     },
     {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "As you can see the probability function is futher distorted from the original Gaussian. However, the graph is still somewhat symmetric around $0$, let's see what the mean is."
+      "As you can see the probability function is further distorted from the original Gaussian. However, the graph is still somewhat symmetric around $0$, let's see what the mean is."
      ]
     },
     {
@@ -487,12 +484,12 @@
        "output_type": "stream",
        "stream": "stdout",
        "text": [
-        "input  mean, variance: 0.0002, 1.0015\n",
-        "output mean, variance: -0.0276, 2.2445\n"
+        "input  mean, variance: -0.0001, 0.9995\n",
+        "output mean, variance: -0.0275, 2.2404\n"
        ]
       }
      ],
-     "prompt_number": 6
+     "prompt_number": 7
     },
     {
      "cell_type": "markdown",
@@ -517,20 +514,20 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": "iVBORw0KGgoAAAANSUhEUgAAAxgAAAGaCAYAAACMmuWeAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f914a743b38>"
+        "<matplotlib.figure.Figure at 0x7f1aa2298160>"
        ]
       },
       {
        "output_type": "stream",
        "stream": "stdout",
        "text": [
-        "output mean, variance: -0.0003, 2.2534\n"
+        "output mean, variance: 0.0001, 2.2489\n"
        ]
       }
      ],
-     "prompt_number": 7
+     "prompt_number": 8
     },
     {
      "cell_type": "markdown",
@@ -561,12 +558,12 @@
        "output_type": "stream",
        "stream": "stdout",
        "text": [
-        "linear    output mean, variance: -0.0171, 7493.0617\n",
-        "nonlinear output mean, variance: -1.8226, 26276.2379\n"
+        "linear    output mean, variance: 0.0057, 7478.2709\n",
+        "nonlinear output mean, variance: -1.8465, 26169.1410\n"
        ]
       }
      ],
-     "prompt_number": 8
+     "prompt_number": 9
     },
     {
      "cell_type": "markdown",
@@ -576,11 +573,10 @@
      ]
     },
     {
-     "cell_type": "heading",
-     "level": 2,
+     "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "The Extended Kalman Filter"
+      "##The Extended Kalman Filter"
      ]
     },
     {
@@ -611,17 +607,17 @@
        "output_type": "display_data",
        "png": "iVBORw0KGgoAAAANSUhEUgAAAysAAAGNCAYAAAARje6GAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f914a84d198>"
+        "<matplotlib.figure.Figure at 0x7f1aa216d6a0>"
        ]
       }
      ],
-     "prompt_number": 9
+     "prompt_number": 10
     },
     {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "We want a linear appoximation of this function so that we can use it in the Kalman filter. We will see how it is used in the Kalman filter in the next section, so don't worry about that yet. We can see that there is no single linear function (line) that gives a close approximation of this function. However, during each innovation (update) of the Kalman filter we know its current state, so if we linearize the function at that value we will have a close approximation. For example, suppose our current state is $x=1.5$. What would be a good linearization for this function?\n",
+      "We want a linear approximation of this function so that we can use it in the Kalman filter. We will see how it is used in the Kalman filter in the next section, so don't worry about that yet. We can see that there is no single linear function (line) that gives a close approximation of this function. However, during each innovation (update) of the Kalman filter we know its current state, so if we linearize the function at that value we will have a close approximation. For example, suppose our current state is $x=1.5$. What would be a good linearization for this function?\n",
       "\n",
       "We can use any linear function that passes through the curve at (1.5,-0.75). For example, consider using f(x)=8x\u221212.75 as the linearization, as in the plot below."
      ]
@@ -646,11 +642,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f9149830b00>"
+        "<matplotlib.figure.Figure at 0x7f1aa1179f28>"
        ]
       }
      ],
-     "prompt_number": 10
+     "prompt_number": 11
     },
     {
      "cell_type": "markdown",
@@ -687,11 +683,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f9149fb0668>"
+        "<matplotlib.figure.Figure at 0x7f1aa22421d0>"
        ]
       }
      ],
-     "prompt_number": 11
+     "prompt_number": 12
     },
     {
      "cell_type": "markdown",
@@ -723,15 +719,14 @@
       "\n",
       "In the next chapter we will spend a lot of time on a new development, the unscented Kalman filter(UKF) which avoids many of these problems. I think that as it becomes better known it will supplant the EKF in most applications, though that is still an open question. Certainly research has shown that the UKF performs at least as well as, and often much better than the EKF. \n",
       "\n",
-      "I think the easiest way to understand the EKF is to just start off with an example. Perhaps the reason for some of my mathmatical choices will not be clear, but trust that the end result will be an EKF."
+      "I think the easiest way to understand the EKF is to just start off with an example. Perhaps the reason for some of my mathematical choices will not be clear, but trust that the end result will be an EKF."
      ]
     },
     {
-     "cell_type": "heading",
-     "level": 2,
+     "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "Example: Tracking a Flying Airplane"
+      "##Example: Tracking a Flying Airplane"
      ]
     },
     {
@@ -758,11 +753,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f914a78f278>"
+        "<matplotlib.figure.Figure at 0x7f1aa0fabac8>"
        ]
       }
      ],
-     "prompt_number": 12
+     "prompt_number": 13
     },
     {
      "cell_type": "markdown",
@@ -818,7 +813,13 @@
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "### Design the System Model\n",
+      "### Design the System Model"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
       "We will model this as a set of differential equations. So we need an equation in the form \n",
       "$$\\dot{\\mathbf{x}} = \\mathbf{Ax} + \\mathbf{w}$$\n",
       "\n",
@@ -868,16 +869,21 @@
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "$\\Phi$ is our system matrix. We cannot use greek symbols in Python, so the code uses the symbol `F` for $\\Phi$. This is admittedly confusing. In the math above $\\mathbf{F}$ represents the system of partial differential equations, and $\\Phi$ is the system matrix. In the Python the partial differential equations are not represented in the code, and the system matrix is `F`."
+      "$\\Phi$ is our system matrix. We cannot use Greek symbols in Python, so the code uses the symbol `F` for $\\Phi$. This is admittedly confusing. In the math above $\\mathbf{F}$ represents the system of partial differential equations, and $\\Phi$ is the system matrix. In the Python the partial differential equations are not represented in the code, and the system matrix is `F`."
      ]
     },
     {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "### Design the Measurement Model\n",
-      "\n",
-      "The measurement function for our filter needs to take the filter state $\\mathbf{x}$ and turn it into a slant range distance. This is nothing more than the pythagorean theorem.\n",
+      "### Design the Measurement Model"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "The measurement function for our filter needs to take the filter state $\\mathbf{x}$ and turn it into a slant range distance. This is nothing more than the Pythagorean theorem.\n",
       "\n",
       "$$h(\\mathbf{x}) = \\sqrt{x_{pos}^2 + x_{alt}^2}$$\n",
       "\n",
@@ -955,7 +961,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 13
+     "prompt_number": 14
     },
     {
      "cell_type": "markdown",
@@ -978,7 +984,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 14
+     "prompt_number": 15
     },
     {
      "cell_type": "markdown",
@@ -1015,7 +1021,7 @@
       "        self.alt = self.alt + .1*randn()\n",
       "        self.pos = self.pos + self.vel*self.dt\n",
       "    \n",
-      "        # add measurment noise\n",
+      "        # add measurement noise\n",
       "        err = self.pos * 0.05*randn()\n",
       "        slant_dist = math.sqrt(self.pos**2 + self.alt**2)\n",
       "        \n",
@@ -1024,7 +1030,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 15
+     "prompt_number": 16
     },
     {
      "cell_type": "markdown",
@@ -1052,7 +1058,7 @@
       "                            [0, 0, 0],\n",
       "                            [0, 0, 0]])*dt\n",
       "                            \n",
-      "After assigning reasonble values to $\\mathbf{R}$, $\\mathbf{Q}$, and $\\mathbf{P}$ we can run the filter with a simple loop\n",
+      "After assigning reasonable values to $\\mathbf{R}$, $\\mathbf{Q}$, and $\\mathbf{P}$ we can run the filter with a simple loop\n",
       "\n",
       "    for i in range(int(20/dt)):\n",
       "        z = radar.get_range()\n",
@@ -1140,36 +1146,35 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x7f914a8899e8>"
+        "<matplotlib.figure.Figure at 0x7f1aa218ccc0>"
        ]
       },
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f914a048198>"
+        "<matplotlib.figure.Figure at 0x7f1aa1935278>"
        ]
       },
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": "iVBORw0KGgoAAAANSUhEUgAAA0EAAAGkCAYAAAD36y7BAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f914a015c88>"
+        "<matplotlib.figure.Figure at 0x7f1aa10ccd68>"
        ]
       }
      ],
-     "prompt_number": 16
+     "prompt_number": 17
     },
     {
-     "cell_type": "heading",
-     "level": 2,
+     "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "Using SymPy to compute Jacobians"
+      "##Using SymPy to compute Jacobians"
      ]
     },
     {
@@ -1204,7 +1209,7 @@
        ],
        "metadata": {},
        "output_type": "pyout",
-       "prompt_number": 17,
+       "prompt_number": 18,
        "text": [
         "\u23a1       x_pos                    x_alt        \u23a4\n",
         "\u23a2\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500  0  \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u23a5\n",
@@ -1214,7 +1219,7 @@
        ]
       }
      ],
-     "prompt_number": 17
+     "prompt_number": 18
     },
     {
      "cell_type": "markdown",
@@ -1224,11 +1229,10 @@
      ]
     },
     {
-     "cell_type": "heading",
-     "level": 3,
+     "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "Designing Q"
+      "##Designing Q"
      ]
     },
     {
@@ -1286,7 +1290,7 @@
        ],
        "metadata": {},
        "output_type": "pyout",
-       "prompt_number": 18,
+       "prompt_number": 19,
        "text": [
         "\u23a1           3      2                2      2             2            \u23a4\n",
         "\u23a2  \\Delta{t} \u22c5w_vel        \\Delta{t} \u22c5w_vel     \\Delta{t} \u22c5w_alt\u22c5w_vel\u23a5\n",
@@ -1305,14 +1309,13 @@
        ]
       }
      ],
-     "prompt_number": 18
+     "prompt_number": 19
     },
     {
-     "cell_type": "heading",
-     "level": 2,
+     "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "Example: A falling Ball"
+      "##Example: A falling Ball"
      ]
     },
     {
@@ -1335,19 +1338,19 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f914a703f28>"
+        "<matplotlib.figure.Figure at 0x7f1aa11ae828>"
        ]
       }
      ],
-     "prompt_number": 19
+     "prompt_number": 20
     },
     {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "We can artifically force the Kalman filter to track the ball by making $Q$ large. That would cause the filter to mistrust its prediction, and scale the kalman gain $K$ to strongly favor the measurments. However, this is not a valid approach. If the Kalman filter is correctly predicting the process we should not 'lie' to the filter by telling it there are process errors that do not exist. We may get away with that for some problems, in some conditions, but in general the Kalman filter's performance will be substandard.\n",
+      "We can artificially force the Kalman filter to track the ball by making $Q$ large. That would cause the filter to mistrust its prediction, and scale the kalman gain $K$ to strongly favor the measurments. However, this is not a valid approach. If the Kalman filter is correctly predicting the process we should not 'lie' to the filter by telling it there are process errors that do not exist. We may get away with that for some problems, in some conditions, but in general the Kalman filter's performance will be substandard.\n",
       "\n",
       "Recall from the **Designing Kalman Filters** chapter that the acceleration is\n",
       "\n",
@@ -1362,15 +1365,6 @@
       "\n",
       "Here $\\beta$ is the ballistic coefficient, where a high number indicates a low drag."
      ]
-    },
-    {
-     "cell_type": "code",
-     "collapsed": false,
-     "input": [],
-     "language": "python",
-     "metadata": {},
-     "outputs": [],
-     "prompt_number": 19
     }
    ],
    "metadata": {}