Fixed some latex issues.
This commit is contained in:
Roger Labbe committed
1 parent
511a3d1012
commit
acdc04af34
1 file changed
+119
-235
@@ -16,19 +16,11 @@
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},
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{
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"cell_type": "code",
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"execution_count": 52,
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"execution_count": 1,
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"metadata": {
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"collapsed": false
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},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"The autoreload extension is already loaded. To reload it, use:\n",
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" %reload_ext autoreload\n"
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]
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},
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{
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"data": {
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"text/html": [
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@@ -262,7 +254,7 @@
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"<IPython.core.display.HTML object>"
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]
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},
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"execution_count": 52,
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"execution_count": 1,
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"metadata": {},
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"output_type": "execute_result"
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}
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@@ -431,7 +423,7 @@
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"data": {
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"image/png": 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truncated
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"text/plain": [
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"<matplotlib.figure.Figure at 0x5318748>"
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"<matplotlib.figure.Figure at 0x56887f0>"
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]
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},
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"metadata": {},
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@@ -839,7 +831,7 @@
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### **Step3**: Design the Process Noise Matrix"
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"### **Step 3**: Design the Process Noise Matrix"
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]
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},
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{
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@@ -892,7 +884,7 @@
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"data": {
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"image/png": 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|
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"text/plain": [
|
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"<matplotlib.figure.Figure at 0x7ac7160>"
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"<matplotlib.figure.Figure at 0x7f859e8>"
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]
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},
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"metadata": {},
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@@ -945,7 +937,7 @@
|
||||
},
|
||||
{
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"cell_type": "code",
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"execution_count": 16,
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"execution_count": 15,
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"metadata": {
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"collapsed": false,
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"scrolled": true
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@@ -995,7 +987,7 @@
|
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},
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{
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"cell_type": "code",
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"execution_count": 17,
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"execution_count": 16,
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"metadata": {
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"collapsed": false,
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"scrolled": true
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@@ -1008,7 +1000,7 @@
|
||||
" [ 0.001175 , 0.0235 ]])"
|
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]
|
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},
|
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"execution_count": 17,
|
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"execution_count": 16,
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"metadata": {},
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"output_type": "execute_result"
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}
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@@ -1022,7 +1014,7 @@
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"### **Step 3**: Design the Control Function\n",
|
||||
"### **Step 4**: Design the Control Function\n",
|
||||
"\n",
|
||||
"The Kalman filter does not just filter data, it allows us to incorporate control inputs for systems like robots and airplanes. Suppose we are controlling a robot. At each time step we would send control signals to the robot based on its current position vs desired position. Kalman filter equations incorporate that knowledge into the filter equations, creating a predicted position based both on current velocity *and* control inputs to the drive motors. Remember, we *never* throw information away.\n",
|
||||
"\n",
|
||||
@@ -1042,7 +1034,7 @@
|
||||
},
|
||||
{
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||||
"cell_type": "code",
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"execution_count": 18,
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"execution_count": 17,
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"metadata": {
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||||
"collapsed": false
|
||||
},
|
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@@ -1074,7 +1066,7 @@
|
||||
"\n",
|
||||
"Now we can implement the update step of the filter. The good news is that you only have to supply two more matrices, and they are easy to understand. The bad news is that the Kalman filter equations themselves are harder to understand. So for now I'll leave the equations unexplained so we can get the full Kalman filter working. After that we'll look at the equations. \n",
|
||||
"\n",
|
||||
"### **Step 4**: Design the Measurement Function\n",
|
||||
"### **Step 5**: Design the Measurement Function\n",
|
||||
"\n",
|
||||
"The Kalman filter computes the update step in what we call **measurement space**. We mostly ignored this issue in the univariate chapter because of the complication it adds. Ee tracked our dog's position using a sensor that reported his position. Computing the *residual* was easy - subtract the filter's predicted position from the measurement:\n",
|
||||
"\n",
|
||||
@@ -1108,8 +1100,8 @@
|
||||
"\n",
|
||||
"$$\n",
|
||||
"\\begin{aligned}\n",
|
||||
"&\\textbf{y} = &\\mathbf{z} &- &\\mathbf{H}&\\mathbf{x^-} \\\\\n",
|
||||
"&\\begin{bmatrix}y \\end{bmatrix} = &\\begin{bmatrix}z\\end{bmatrix} &- &\\begin{bmatrix}?&?\\end{bmatrix} &\\begin{bmatrix}x \\\\ \\dot{x}\\end{bmatrix}\n",
|
||||
"\\textbf{y} &= \\mathbf{z} - \\mathbf{H}\\mathbf{x^-} \\\\\n",
|
||||
"\\begin{bmatrix}y \\end{bmatrix} &= \\begin{bmatrix}z\\end{bmatrix} - \\begin{bmatrix}?&?\\end{bmatrix} \\begin{bmatrix}x \\\\ \\dot{x}\\end{bmatrix}\n",
|
||||
"\\end{aligned}\n",
|
||||
"$$\n",
|
||||
"\n",
|
||||
@@ -1128,7 +1120,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
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||||
"execution_count": 19,
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||||
"execution_count": 18,
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||||
"metadata": {
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||||
"collapsed": true
|
||||
},
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@@ -1148,9 +1140,9 @@
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"### **Step 5**: Design the Measurement Noise Matrix\n",
|
||||
"### **Step 6**: Design the Measurement Noise Matrix\n",
|
||||
"\n",
|
||||
"The **measurement noise matri*x* is a matrix that models the noise in our sensors as a covariance matrix. Admittedly this can be difficult to do in practice. A complicated system may have many sensors, the correlation between them might not be clear, and usually their noise is not a pure Gaussian. For example, a sensor might be biased to read high if the temperature is high, and so the noise is not distributed equally on both sides of the mean.\n",
|
||||
"The **measurement noise matrix** models the noise in our sensors as a covariance matrix. In practice this can be difficult. A complicated system may have many sensors, the correlation between them might not be clear, and usually their noise is not a pure Gaussian. For example, a sensor might be biased to read high if the temperature is high, and so the noise is not distributed equally on both sides of the mean.\n",
|
||||
"\n",
|
||||
"In the univariate chapter we used a variance of 5 meters squared for our position sensor. Let's use the same value here. The Kalman filter equations uses the symbol $\\mathbf{R}$ for this matrix. The matrix will have dimension $m{\\times}m$, where $m$ is the number of sensors. It is $m{\\times}m$ because it is a covariance matrix, as there may be correlations between the sensors. We have only 1 sensor here so we write:\n",
|
||||
"\n",
|
||||
@@ -1167,7 +1159,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 20,
|
||||
"execution_count": 19,
|
||||
"metadata": {
|
||||
"collapsed": true
|
||||
},
|
||||
@@ -1186,9 +1178,7 @@
|
||||
"\n",
|
||||
"Finally, since this is a 1x1 matrix you can just use a scalar.\n",
|
||||
"\n",
|
||||
" kf.R = 5.\n",
|
||||
"\n",
|
||||
"Use whatever is the most readable to you."
|
||||
" kf.R = 5."
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -1207,7 +1197,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 21,
|
||||
"execution_count": 20,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
@@ -1226,7 +1216,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 22,
|
||||
"execution_count": 21,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
@@ -1257,14 +1247,14 @@
|
||||
"\n",
|
||||
"**5** We use the `Q_discrete_white_noise()` method to set $\\mathbf{Q}$'s variance to 0.1.\n",
|
||||
"\n",
|
||||
"**5**: We set $\\mathbf{P} = \\begin{bmatrix} 500&0\\\\0&400\\end{bmatrix}$\n",
|
||||
"**6**: We set $\\mathbf{P} = \\begin{bmatrix} 500&0\\\\0&400\\end{bmatrix}$\n",
|
||||
"\n",
|
||||
"I've rewritten the creation of the Kalman filter to allow you to specify different initial values for `R`, `P`, and `Q` and put it in a helper function. We will be creating and running many of these filters, and this saves us a lot of headaches."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 23,
|
||||
"execution_count": 22,
|
||||
"metadata": {
|
||||
"collapsed": true
|
||||
},
|
||||
@@ -1294,7 +1284,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 24,
|
||||
"execution_count": 23,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
@@ -1310,14 +1300,14 @@
|
||||
" \"\"\"\n",
|
||||
" dog_filter = pos_vel_filter(initial_x, R=R, Q=Q, P=P)\n",
|
||||
" \n",
|
||||
" pos, cov = [], [] \n",
|
||||
" for z in data:\n",
|
||||
" pos, cov = [], [] \n",
|
||||
" for z in data: \n",
|
||||
" # perform the kalman filter steps\n",
|
||||
" dog_filter.predict()\n",
|
||||
" dog_filter.update(z)\n",
|
||||
"\n",
|
||||
" pos.append(dog_filter.x)\n",
|
||||
" cov.append(dog_filter.P)\n",
|
||||
" \n",
|
||||
" # perform the kalman filter steps\n",
|
||||
" dog_filter.update(z)\n",
|
||||
" dog_filter.predict()\n",
|
||||
" \n",
|
||||
" pos = np.array(pos)\n",
|
||||
" cov = np.array(cov)\n",
|
||||
@@ -1339,7 +1329,7 @@
|
||||
"\n",
|
||||
" dog_filter = dog_tracking_filter(R=R, Q=Q, cov=500.)\n",
|
||||
" \n",
|
||||
"We will want to plot the filtered position, the measurements, and the covariance, so we will need to store them in lists. The next three lines initialize empty lists of length *count* in a pythonic way.\n",
|
||||
"We will want to plot the filtered position, the measurements, and the covariance, so we will need to store them in lists. The next three lines initialize empty lists of length *count* in a Pythonic way.\n",
|
||||
"\n",
|
||||
" pos = [None] * count\n",
|
||||
" zs = [None] * count\n",
|
||||
@@ -1356,7 +1346,25 @@
|
||||
" \n",
|
||||
"It really cannot get much simpler than that. As we tackle more complicated problems this code will remain largely the same; all of the work goes into setting up the `KalmanFilter` variables; executing the filter is trivial.\n",
|
||||
"\n",
|
||||
"Now let's look at the result. Here is some code that calls `filter_track()` and then plots the result. It is fairly uninteresting code, so I will not walk through it. The `DogSimulation` class from the previous chapter has been placed in `DogSimulation.py`. I have also implemented a few plot routines to visualize the data in `mkf_internal.py`. Both of these are in the *code* subdirectory if you wish to read them.\n"
|
||||
"Now let's look at the result. Here is some code that calls `filter_track()` and then plots the result. It is fairly uninteresting code, so I will not walk through it. The `DogSimulation` class from the previous chapter has been placed in `DogSimulation.py`. I have also implemented a few plot routines to visualize the data in `mkf_internal.py`. Both of these are in the *code* subdirectory if you wish to read them.\n",
|
||||
"\n",
|
||||
"The Kalman filter is designed as a recursive algorithm - as new measurements come in we immediately create a new estimate. But it is very common to have a set of data that have been already collected which we want to filter. Kalman filters can always be run in a *batch* mode, where all of the measurements are filtered at once. We have implemented this in `KalmanFilter.batch_filter()`. Internally, all the function does is loop over the measurements and collect the resulting state and covariance estimates in arrays.\n",
|
||||
"\n",
|
||||
"Here is an alternative form of the last function which uses batch filtering. `batch_filter` returns four NumPy `Array` objects; the first two contain the filtered estimates and covariances (the posteriors), and the last two contain the predicted estimates and covariances (the priors). The priors are useful for smoothing algorithms which we will learn later, but for now they are not useful to us, so I disregard them."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 24,
|
||||
"metadata": {
|
||||
"collapsed": true
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"def filter_dog(data, R=0, Q=0, P=500., initial_x=(0., 0.)):\n",
|
||||
" dog_filter = pos_vel_filter(initial_x, R=R, Q=Q, P=P) \n",
|
||||
" pos, cov, _, _ = dog_filter.batch_filter(data)\n",
|
||||
" return (pos, cov)"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -1376,7 +1384,6 @@
|
||||
" zs = [dog.move_and_sense() for t in range(count)]\n",
|
||||
" else:\n",
|
||||
" zs = data\n",
|
||||
" \n",
|
||||
" ps, cov = filter_dog(data=zs, R=R, Q=Q, P=P, initial_x=initial_x)\n",
|
||||
" plot_track(ps[:, 0], zs, cov, std_scale, **kwargs)\n",
|
||||
" return ps, cov"
|
||||
@@ -1386,7 +1393,7 @@
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Finally, call it. We will start by filtering 50 measurements with a noise variance of 10 and a process variance of 0.01. "
|
||||
"Finally, call it. We will start by filtering 50 measurements with a noise variance of 10 and a process variance of 0.01."
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -1396,18 +1403,11 @@
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"1\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x7d2b4a8>"
|
||||
"<matplotlib.figure.Figure at 0x7fa5c88>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1415,9 +1415,9 @@
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x7d2b978>"
|
||||
"<matplotlib.figure.Figure at 0x7f6ceb8>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1464,9 +1464,9 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x9632278>"
|
||||
"<matplotlib.figure.Figure at 0x8144940>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1551,7 +1551,7 @@
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x7ceb438>"
|
||||
"<matplotlib.figure.Figure at 0x7f81eb8>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1590,7 +1590,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 58,
|
||||
"execution_count": 29,
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"scrolled": true
|
||||
@@ -1598,9 +1598,9 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x964dda0>"
|
||||
"<matplotlib.figure.Figure at 0x9d550b8>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1653,7 +1653,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 57,
|
||||
"execution_count": 30,
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"scrolled": true
|
||||
@@ -1663,7 +1663,7 @@
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x95bfda0>"
|
||||
"<matplotlib.figure.Figure at 0xadc37b8>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1824,7 +1824,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 30,
|
||||
"execution_count": 31,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
@@ -1844,16 +1844,16 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 31,
|
||||
"execution_count": 32,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x7b7a828>"
|
||||
"<matplotlib.figure.Figure at 0x998f518>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1946,7 +1946,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 32,
|
||||
"execution_count": 33,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
@@ -1955,32 +1955,24 @@
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"variance 240.45554425\n",
|
||||
"1\n"
|
||||
"variance 240.45554425\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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||||
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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x7c03ba8>"
|
||||
"<matplotlib.figure.Figure at 0x99713c8>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"output_type": "display_data"
|
||||
},
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"1\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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||||
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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x9733128>"
|
||||
"<matplotlib.figure.Figure at 0x8029e80>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -2029,24 +2021,17 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 33,
|
||||
"execution_count": 34,
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"scrolled": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"1.0\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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|
||||
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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x95ad128>"
|
||||
"<matplotlib.figure.Figure at 0x9971320>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -2067,23 +2052,16 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 34,
|
||||
"execution_count": 35,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"1.0\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x959d828>"
|
||||
"<matplotlib.figure.Figure at 0x80c5748>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -2105,23 +2083,16 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 35,
|
||||
"execution_count": 36,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"1.0\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x7c06f28>"
|
||||
"<matplotlib.figure.Figure at 0x8176ba8>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -2182,7 +2153,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 68,
|
||||
"execution_count": 37,
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"scrolled": true
|
||||
@@ -2192,7 +2163,7 @@
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0xa650d30>"
|
||||
"<matplotlib.figure.Figure at 0x7f6cd30>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -2217,7 +2188,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 69,
|
||||
"execution_count": 38,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
@@ -2226,7 +2197,7 @@
|
||||
"data": {
|
||||
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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x7d425f8>"
|
||||
"<matplotlib.figure.Figure at 0x9d75710>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -2280,23 +2251,16 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 38,
|
||||
"execution_count": 39,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"1\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x972a780>"
|
||||
"<matplotlib.figure.Figure at 0x9d7bac8>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -2304,9 +2268,9 @@
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x7c90fd0>"
|
||||
"<matplotlib.figure.Figure at 0x9a0ba90>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -2341,23 +2305,16 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 39,
|
||||
"execution_count": 40,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"1\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x9668198>"
|
||||
"<matplotlib.figure.Figure at 0x98e6978>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -2365,9 +2322,9 @@
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0xa55f2b0>"
|
||||
"<matplotlib.figure.Figure at 0x9898a58>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -2392,23 +2349,16 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 40,
|
||||
"execution_count": 41,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"1\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x96d5ba8>"
|
||||
"<matplotlib.figure.Figure at 0x9990c88>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -2433,23 +2383,16 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 41,
|
||||
"execution_count": 42,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"1\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x95bac50>"
|
||||
"<matplotlib.figure.Figure at 0xadb2c50>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -2479,16 +2422,16 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 63,
|
||||
"execution_count": 43,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0xa6d1828>"
|
||||
"<matplotlib.figure.Figure at 0x8016208>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -2496,9 +2439,9 @@
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x95c3a20>"
|
||||
"<matplotlib.figure.Figure at 0x8144710>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -2599,16 +2542,16 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 64,
|
||||
"execution_count": 44,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0xaaa0c88>"
|
||||
"<matplotlib.figure.Figure at 0x9d5a048>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -2659,7 +2602,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 44,
|
||||
"execution_count": 45,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
@@ -2671,7 +2614,7 @@
|
||||
" [ 0. ]])"
|
||||
]
|
||||
},
|
||||
"execution_count": 44,
|
||||
"execution_count": 45,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
@@ -2738,9 +2681,9 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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||||
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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x95cbc50>"
|
||||
"<matplotlib.figure.Figure at 0xadc4a58>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -2789,9 +2732,9 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x958c358>"
|
||||
"<matplotlib.figure.Figure at 0xadc3780>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -2830,9 +2773,9 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x958c278>"
|
||||
"<matplotlib.figure.Figure at 0x998f518>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -2896,9 +2839,9 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x7ab23c8>"
|
||||
"<matplotlib.figure.Figure at 0x98b9dd8>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -2947,65 +2890,6 @@
|
||||
"In later chapters we will deal with the problem of nonlinearity. For now I want you to understand that designing the matrices of a linear filter is an experimental procedure more than a mathematical one. Use math to establish the initial values, but then you need to experiment. If there is a lot of unaccounted noise in the world (wind, etc) you may have to make $\\mathbf{Q}$ larger. If you make it too large the filter fails to respond quickly to changes. In the **Adaptive Filters** chapter you will learn some alternative techniques that allow you to change the filter design in real time in response to the inputs and performance, but for now you need to find one set of values that works for the conditions your filter will encounter. Noise matrices for an acrobatic plane might be different if the pilot is a student than if the pilot is an expert as the dynamics will be quite different."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## The Equations\n",
|
||||
"\n",
|
||||
"The brilliance of the Kalman filter is taking the insights of the chapter up to this point and finding an optimal mathematical solution. The Kalman filter finds what is called a *least squared fit* to the set of measurements to produce an optimal output. We will not trouble ourselves with the derivation of these equations. It runs to several pages, and offers a lot less insight than the words above, in my opinion. Furthermore, to create a Kalman filter for your application you will not be manipulating these equations, but only specifying the matrices that are used by them. First, let's see the equations. \n",
|
||||
"> Kalman Filter Predict Step:\n",
|
||||
">\n",
|
||||
"> $$\n",
|
||||
"\\begin{aligned}\n",
|
||||
"\\hat{\\textbf{x}}^-_{k+1} &= \\mathbf{F}_{k}\\hat{\\textbf{x}}_{k} + \\mathbf{B}_k\\mathbf{u}_k\\;\\;\\;&(1) \\\\\n",
|
||||
"\\textbf{P}^-_{k+1} &= \\mathbf{F}_k \\textbf{P}_k\\mathbf{F}_k^\\mathsf{T} + \\textbf{Q}_{k}\\;\\;\\;&(2)\n",
|
||||
"\\end{aligned}\n",
|
||||
"$$\n",
|
||||
"\n",
|
||||
"> Kalman Filter Update Step:\n",
|
||||
">\n",
|
||||
">$$\n",
|
||||
"\\begin{aligned}\n",
|
||||
"\\mathbf{S}_k &= \\textbf{H}_k\\textbf{P}^-_k\\textbf{H}_k^\\mathsf{T} + \\textbf{R}_k \\;\\;\\;&(3) \\\\\n",
|
||||
"\\textbf{K}_k &= \\textbf{P}^-_k\\textbf{H}_k^\\mathsf{T}\\mathbf{S}_k^{-1}\\;\\;\\;&(4) \\\\\n",
|
||||
"\\textbf{y}_k &= \\textbf{z}_k - \\textbf{H}_k\\hat{\\textbf{}x}^-_k\\;\\;\\;&(5) \\\\\n",
|
||||
"\\hat{\\textbf{x}}_k &= \\hat{\\textbf{x}}^-_k +\\textbf{K}_k\\textbf{y} \\;\\;\\;&(6)\\\\\n",
|
||||
"\\mathbf{P}_k &= (\\mathbf{I}-\\mathbf{K}_k\\mathbf{H}_k)\\mathbf{P}^-_k\\;\\;\\;&(7)\n",
|
||||
"\\end{aligned}\n",
|
||||
"$$\n",
|
||||
"\n",
|
||||
"Dash off, wipe the blood out of your eyes, and we'll discuss what this means. \n",
|
||||
"\n",
|
||||
"These are nothing more than linear algebra equations that implement the algorithm we used in the last chapter, but using multidimensional Gaussians instead of univariate Gaussians, and optimized for a least squares fit.\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"Different texts use different notation and variable names for the Kalman filter. Later we will expose you to these different forms to prepare you for reading the original literature. In the equations above I have adopted the variable names used by the Wikipedia article[2] on Kalman filters. Each bold letter denotes a matrix or vector. The subscripts indicate which time step the data comes from; $k$ is now, $k+1$ is the next step. The caret (^) indicates that the value is an estimate. Finally, I particularly like how Brown [3] uses a raised $^-$ to denote a prediction (the prior), and so I have adopted that approach.\n",
|
||||
"\n",
|
||||
"For a matrix $\\mathbf{A}$, $\\mathbf{A}^\\mathsf{T}$ signifies its transpose, and $\\mathbf{A}^{-1}$ its inverse. So, taken together, $\\hat{\\mathbf{x}}^-_{k+1}$ represents the prediction for the estimate of $\\mathbf{x}$ at time step $k+1$, where $\\mathbf{x}$ is some vector in the form $\\mathbf{x} = \\begin{bmatrix}x_1&x_2&..&x_n\\end{bmatrix}^\\mathsf{T}.$ In short, it is the prior at step $k+1$.\n",
|
||||
"\n",
|
||||
"### Kalman Equations Expressed as an Algorithm\n",
|
||||
"\n",
|
||||
"I find this notation to be dense, and unnecessarily complicated for writing code. Even now I have read papers very slowly to ensure I am interpreting everything correctly. The subscripts indicate the time step, but when we write code it is very clear what is being calculated at each time step. For most of this book I'm going to use the following simplified equations, which express an algorithm to be executed in a linear order.\n",
|
||||
"\n",
|
||||
"$$\n",
|
||||
"\\begin{aligned}\n",
|
||||
"\\text{Predict Step}\\\\\n",
|
||||
"\\mathbf{x^-} &= \\mathbf{F x} + \\mathbf{B u}\\;\\;\\;&(1) \\\\\n",
|
||||
"\\mathbf{P^-} &= \\mathbf{FP{F}}^\\mathsf{T} + \\mathbf{Q}\\;\\;\\;&(2) \\\\\n",
|
||||
"\\\\\n",
|
||||
"\\text{Update Step}\\\\\n",
|
||||
"\\textbf{S} &= \\mathbf{HP^-H}^\\mathsf{T} + \\mathbf{R} \\;\\;\\;&(3)\\\\\n",
|
||||
"\\mathbf{K} &= \\mathbf{P^-H}^\\mathsf{T} \\mathbf{S}^{-1}\\;\\;\\;&(4) \\\\\n",
|
||||
"\\textbf{y} &= \\mathbf{z} - \\mathbf{H x^-} \\;\\;\\;&(5)\\\\\n",
|
||||
"\\mathbf{x} &=\\mathbf{x^-} +\\mathbf{K\\textbf{y}} \\;\\;\\;&(6)\\\\\n",
|
||||
"\\mathbf{P} &= (\\mathbf{I}-\\mathbf{KH})\\mathbf{P^-}\\;\\;\\;&(7)\n",
|
||||
"\\end{aligned}\n",
|
||||
"$$\n",
|
||||
"\n",
|
||||
"What do all of the variables mean? What is $\\mathbf{P}$, for example? Don't worry right now. Instead, I am going to design a Kalman filter, and introduce the names as we go. Then we will pass them into Python function that implement these equations, and we will have our solution. Later sections will then delve into more detail about each step and equation. I think learning by example and practice is far easier than trying to memorize a dozen abstract facts at once. "
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
|
||||
Reference in new issue
Block a user