Fixed section numbering.
I had some sections at level 2 when they should have been level 3.
This commit is contained in:
Roger Labbe
parent
11f28beae6
commit
4f7c896507
@ -496,14 +496,14 @@
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Initialization - Choose State Variables and Set Initial Conditions"
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"### Initialization - Choose State Variables and Set Initial Conditions"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### **Step 1**: Design State Variable as a Multivariate Gaussian\n",
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"#### **Step 1**: Design State Variable as a Multivariate Gaussian\n",
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"\n",
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"In the univariate chapter we tracked a dog in one dimension by using a Gaussian. The mean $(\\mu)$ represented the most likely position, and the variance ($\\sigma^2$) represented the probability distribution of the position. In that problem the position is the *state* of the system, and we call $\\mu$ the *state variable*.\n",
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"\n",
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@ -702,12 +702,12 @@
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Predict Step\n",
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"### Predict Step\n",
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"\n",
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"The next step in designing a Kalman filter is telling it how to predict the state (mean and covariance) of the system for the next time step. We do this by providing it with equations that describe the physical model of the system.\n",
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"\n",
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"\n",
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"### **Step 2:** Design the State Transition Function\n",
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"#### **Step 2:** Design the State Transition Function\n",
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"\n",
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"In the univariate chapter we modeled the dog's motion with\n",
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"\n",
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@ -872,7 +872,7 @@
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### **Step 3**: 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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@ -957,7 +957,7 @@
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"<u>**Process Noise**</u>\n",
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"#### Process Noise\n",
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"\n",
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"A quick review on **process noise**. A car is driving along the road with the cruise control on; it should travel at a constant speed. We model this with $x=\\dot{x}\\Delta t + x_0$. However, it is affected by a number of unknown factors. The cruise control is not perfect, and cannot maintain a constant velocity. Winds affect the car, as do hills and potholes. Passengers roll down windows, changing the drag profile of the car. \n",
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"\n",
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@ -1016,7 +1016,7 @@
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### **Step 4**: Design the Control Function\n",
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"#### **Step 4**: Design the Control Function\n",
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"\n",
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"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",
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"\n",
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@ -1050,7 +1050,7 @@
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"metadata": {},
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"source": [
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"\n",
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"### Prediction: Summary\n",
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"#### Prediction: Summary\n",
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"\n",
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"Your job as a designer is to specify the matrices for\n",
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"\n",
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@ -1065,11 +1065,11 @@
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Update Step\n",
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"### Update Step\n",
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"\n",
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"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",
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"\n",
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"### **Step 5**: Design the Measurement Function\n",
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"#### **Step 5**: Design the Measurement Function\n",
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"\n",
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"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. We 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",
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"\n",
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@ -1145,7 +1145,7 @@
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### **Step 6**: Design the Measurement Noise Matrix\n",
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"#### **Step 6**: Design the Measurement Noise Matrix\n",
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"\n",
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"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",
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"\n",
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@ -1190,7 +1190,7 @@
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Implementing the Kalman Filter"
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"### Implementing the Kalman Filter"
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]
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},
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{
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@ -2932,7 +2932,7 @@
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"name": "python",
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@ -16,7 +16,7 @@
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@ -28,9 +28,10 @@
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@ -60,40 +61,44 @@
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@ -104,22 +109,19 @@
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@ -235,16 +242,20 @@
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@ -252,10 +263,13 @@
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"source": [
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"#format the book\n",
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"%matplotlib inline\n",
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"%load_ext autoreload\n",
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"%autoreload 2 \n",
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324
Appendix-I-Analytic-Evaluation-of-Performance.ipynb
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324
Appendix-I-Analytic-Evaluation-of-Performance.ipynb
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@ -0,0 +1,324 @@
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"[Table of Contents](http://nbviewer.ipython.org/github/rlabbe/Kalman-and-Bayesian-Filters-in-Python/blob/master/table_of_contents.ipynb)"
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"source": [
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"# Analytic Evaluation of Performance"
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" ol li{\n",
|
||||
" padding-left: 0.5em; \n",
|
||||
" margin-bottom: 0.5em; \n",
|
||||
" margin-top: 0.5em; \n",
|
||||
" }\n",
|
||||
" ul li{\n",
|
||||
" padding-left: 0.5em; \n",
|
||||
" margin-bottom: 0.5em; \n",
|
||||
" margin-top: 0.2em; \n",
|
||||
" }\n",
|
||||
" a:link{\n",
|
||||
" font-weight: bold;\n",
|
||||
" color:#447adb;\n",
|
||||
" }\n",
|
||||
" a:visited{\n",
|
||||
" font-weight: bold;\n",
|
||||
" color: #1d3b84;\n",
|
||||
" }\n",
|
||||
" a:hover{\n",
|
||||
" font-weight: bold;\n",
|
||||
" color: #1d3b84;\n",
|
||||
" }\n",
|
||||
" a:focus{\n",
|
||||
" font-weight: bold;\n",
|
||||
" color:#447adb;\n",
|
||||
" }\n",
|
||||
" a:active{\n",
|
||||
" font-weight: bold;\n",
|
||||
" color:#447adb;\n",
|
||||
" }\n",
|
||||
" .rendered_html :link {\n",
|
||||
" text-decoration: underline; \n",
|
||||
" }\n",
|
||||
" .rendered_html :hover {\n",
|
||||
" text-decoration: none; \n",
|
||||
" }\n",
|
||||
" .rendered_html :visited {\n",
|
||||
" text-decoration: none;\n",
|
||||
" }\n",
|
||||
" .rendered_html :focus {\n",
|
||||
" text-decoration: none;\n",
|
||||
" }\n",
|
||||
" .rendered_html :active {\n",
|
||||
" text-decoration: none;\n",
|
||||
" }\n",
|
||||
" .warning{\n",
|
||||
" color: rgb( 240, 20, 20 )\n",
|
||||
" } \n",
|
||||
" hr {\n",
|
||||
" color: #f3f3f3;\n",
|
||||
" background-color: #f3f3f3;\n",
|
||||
" height: 1px;\n",
|
||||
" }\n",
|
||||
" blockquote{\n",
|
||||
" display:block;\n",
|
||||
" background: #fcfcfc;\n",
|
||||
" border-left: 5px solid #c76c0c;\n",
|
||||
" font-family: 'Open sans',verdana,arial,sans-serif;\n",
|
||||
" width:680px;\n",
|
||||
" padding: 10px 10px 10px 10px;\n",
|
||||
" text-align:justify;\n",
|
||||
" text-justify:inter-word;\n",
|
||||
" }\n",
|
||||
" blockquote p {\n",
|
||||
" margin-bottom: 0;\n",
|
||||
" line-height: 125%;\n",
|
||||
" font-size: 100%;\n",
|
||||
" }\n",
|
||||
"</style>\n",
|
||||
"<script>\n",
|
||||
" MathJax.Hub.Config({\n",
|
||||
" TeX: {\n",
|
||||
" extensions: [\"AMSmath.js\"]\n",
|
||||
" },\n",
|
||||
" tex2jax: {\n",
|
||||
" inlineMath: [ ['$','$'], [\"\\\\(\",\"\\\\)\"] ],\n",
|
||||
" displayMath: [ ['$$','$$'], [\"\\\\[\",\"\\\\]\"] ]\n",
|
||||
" },\n",
|
||||
" displayAlign: 'center', // Change this to 'center' to center equations.\n",
|
||||
" \"HTML-CSS\": {\n",
|
||||
" scale:100,\n",
|
||||
" availableFonts: [],\n",
|
||||
" preferredFont:null,\n",
|
||||
" webFont: \"TeX\",\n",
|
||||
" styles: {'.MathJax_Display': {\"margin\": 4}}\n",
|
||||
" }\n",
|
||||
" });\n",
|
||||
"</script>\n"
|
||||
],
|
||||
"text/plain": [
|
||||
"<IPython.core.display.HTML object>"
|
||||
]
|
||||
},
|
||||
"execution_count": 2,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"#format the book\n",
|
||||
"%matplotlib inline\n",
|
||||
"%load_ext autoreload\n",
|
||||
"%autoreload 2 \n",
|
||||
"from __future__ import division, print_function\n",
|
||||
"import sys\n",
|
||||
"sys.path.insert(0,'./code')\n",
|
||||
"from book_format import load_style\n",
|
||||
"load_style()"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {
|
||||
"collapsed": true
|
||||
},
|
||||
"outputs": [],
|
||||
"source": []
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {
|
||||
"collapsed": true
|
||||
},
|
||||
"outputs": [],
|
||||
"source": []
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {
|
||||
"collapsed": true
|
||||
},
|
||||
"outputs": [],
|
||||
"source": []
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
"kernelspec": {
|
||||
"display_name": "Python 3",
|
||||
"language": "python",
|
||||
"name": "python3"
|
||||
},
|
||||
"language_info": {
|
||||
"codemirror_mode": {
|
||||
"name": "ipython",
|
||||
"version": 3
|
||||
},
|
||||
"file_extension": ".py",
|
||||
"mimetype": "text/x-python",
|
||||
"name": "python",
|
||||
"nbconvert_exporter": "python",
|
||||
"pygments_lexer": "ipython3",
|
||||
"version": "3.4.3"
|
||||
}
|
||||
},
|
||||
"nbformat": 4,
|
||||
"nbformat_minor": 0
|
||||
}
|
||||
@ -208,7 +208,7 @@
|
||||
<script>
|
||||
MathJax.Hub.Config({
|
||||
TeX: {
|
||||
extensions: ["AMSmath.js"]
|
||||
extensions: ["AMSmath.js", "autobold.js"]
|
||||
},
|
||||
tex2jax: {
|
||||
inlineMath: [ ['$','$'], ["\\(","\\)"] ],
|
||||
|
||||
Loading…
Reference in New Issue
Block a user