Renamed Chapters

Renamed chapters to make them easier to work with from the command line
using tab completion.

07_Kalman_Filter_Math/Kalman_Filter_Math.ipynb

@@ -1,7 +1,7 @@
 {
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-  "signature": "sha256:01ef2af3b68e3abffe35ee916e0df666540b2491ee054656e7641112e4934357"
+  "signature": "sha256:b7f65a1b7640cd43f95156970a5c73ee923fb430d06acbc4debe6508e5327549"
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@@ -284,7 +284,7 @@
      "level": 2,
      "metadata": {},
      "source": [
-      "Modeling a Dynmaic System that Has Noise"
+      "Modeling a Dynamic System that Has Noise"
      ]
     },
     {
@@ -293,7 +293,7 @@
      "source": [
       "We need to start by understanding the underlying equations and assumptions that the Kalman filter uses. We are trying to model real world phenomena, so what do we have to consider?\n",
       "\n",
-      "First, each physical system has a process. For example, a car traveling at a certain velocity goes so far in a fixed amount of time, and it's velocity varies as a function of it's acceleration. We describe that behavior with the well known Newtonian equations we learned in high school.\n",
+      "First, each physical system has a process. For example, a car traveling at a certain velocity goes so far in a fixed amount of time, and it's velocity varies as a function of its acceleration. We describe that behavior with the well known Newtonian equations we learned in high school.\n",
       "\n",
       "\n",
       "$$\n",
@@ -306,12 +306,16 @@
       "And once we learned calculus we saw them in this form:\n",
       "\n",
       "$$\n",
-      " \\mathbf{v} = \\frac{d \\mathbf{x}}{d t}\\\\ \\quad \\mathbf{a} = \\frac{d \\mathbf{v}}{d t} = \\frac{d^2 \\mathbf{x}}{d t^2} \\,\\!\n",
+      "\\begin{aligned}\n",
+      " \\mathbf{v} &= \\frac{d \\mathbf{x}}{d t}\\\\ \n",
+      " \\quad \\mathbf{a} &= \\frac{d \\mathbf{v}}{d t}\\\\\n",
+      " &= \\frac{d^2 \\mathbf{x}}{d t^2} \\,\\!\n",
+      "\\end{aligned}\n",
       " $$\n",
       " \n",
       "A typical problem would have you compute the distance travelled given a constant velocity or acceleration. But, of course we know this is not all that is happening. First, we do not have perfect measures of things like the velocity and acceleration - there is always noise in the measurements, and we have to model that. Second, no car travels on a perfect road. There are bumps that cause the car to slow down, there is wind drag, there are hills that raise and lower the speed. If we do not have explicit knowledge of these factors we lump them all together under the term \"process noise\".\n",
       "\n",
-      "Trying to model all of those factors explicitly and exactly is impossible for anything but the most trivial problem. I could try to include equations for things like bumps in the road, the behavior of the car's suspension system, heck, the effects of hitting bugs with the windshield, but the job would never be done - there would always be more effects to add. What is worse, each of those models would in themselves be a simplification - do I assume the wind is constant, that the drag of the car is the same for all angles of the wind, that the suspension act as perfect springs, that the suspension for each wheel acts identically, and so on.\n",
+      "Trying to model all of those factors explicitly and exactly is impossible for anything but the most trivial problem. I could try to include equations for things like bumps in the road, the behavior of the car's suspension system, even the effects of hitting bugs with the windshield, but the job would never be done - there would always be more effects to add and limits to our knowledge (how many bugs do we hit in an hour, for example). What is worse, each of those models would in themselves be a simplification - do I assume the wind is constant, that the drag of the car is the same for all angles of the wind, that the suspension act as perfect springs, that the suspension for each wheel acts identically, and so on.\n",
       "\n",
       "So control theory makes a mathematically correct simplification. We acknowledge that there are many factors that influence the system that we either do not know or that we don't want to have to model. At any time $t$ we say that the actual value (say, the position of our car) is the predicted value plus some unknown process noise:\n",
       "\n",
@@ -321,12 +325,12 @@
       "\n",
       "This is not meant to imply that $noise(t)$ is a function that we can derive analytically or that it is well behaved. If there is a bump in the road at $t=10$ then the noise factor will just incorporate that effect. Again, this is not implying that we model, compute, or even know the value of *noise(t)*, it is merely a statement of fact - we can *always* describe the actual value as the predicted value from our idealized model plus some other value. \n",
       "\n",
-      "Let's express this in linear algebra. Using the same notation from previous chapters, we can say that our model of the system (without noise) is:\n",
+      "Let's express this with linear algebra. Using the same notation from previous chapters, we can say that our model of the system (without noise) is:\n",
       "\n",
       "$$ f(\\mathbf{x}) = \\mathbf{Fx}$$\n",
       "\n",
       "That is, we have a set of linear equations that describe our system. For our car, \n",
-      "$mathbf{F}$ will be the coefficients for Newton's equations of motion. \n",
+      "$\\mathbf{F}$ will be the coefficients for Newton's equations of motion. \n",
       "\n",
       "Now we need to model the noise. We will just call that *w*, and add it to the equation.\n",
       "\n",
@@ -338,7 +342,7 @@
       "\n",
       "And that's it. That is the equation that Kalman set out to solve, and he found a way to compute an optimal solution if we assume certain properties of $w$.\n",
       "\n",
-      "However, we took advantage of something I left mostly unstated in the last chapter. We were able to provide a definition for $\\mathbf{F}$ because we were able to take advantage of the exact solution that Newtonian equations provide us. However, if you have an engineering background you will realize what a small class of problems that covers.If you don't, I will explain it next, and provide you with several ways to compute $\\mathbf{F}$."
+      "However, we took advantage of something I left mostly unstated in the last chapter. We were able to provide a definition for $\\mathbf{F}$ because we were able to take advantage of the exact solution that Newtonian equations provide us. However, if you have an engineering background you will realize what a small class of problems that covers.If you don't, I will explain it next, and provide you with several ways to compute $\\mathbf{F} for arbitrary systems$."
      ]
     },
     {

README.md

@@ -79,63 +79,63 @@ Contents
 Motivation for the book. Where to download, how to use.
 
 
-* [**Chapter 1: The g-h Filter**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/Chapter01_gh_filter/g-h_filter.ipynb)
+* [**Chapter 1: The g-h Filter**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/01_gh_filter/g-h_filter.ipynb)
 
 Intuitive introduction to the g-h filter, which is a family of filters that includes the Kalman filter. Not filler - once you understand this chapter you will understand the concepts behind the Kalman filter. 
 
 
-* [**Chapter 2: The Discrete Bayes Filter**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/Chapter02_Discrete_Bayes/discrete_bayes.ipynb)
+* [**Chapter 2: The Discrete Bayes Filter**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/02_Discrete_Bayes/discrete_bayes.ipynb)
 
 Introduces the Discrete Bayes Filter. From this you will learn the probabilistic reasoning that underpins the Kalman filter in an easy to digest form.
 
 
-* [**Chapter 3: Least Squares Filter**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/Chapter03_Least_Squares/Least_Squares_Filters.ipynb)
+* [**Chapter 3: Least Squares Filter**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/03_Least_Squares/Least_Squares_Filters.ipynb)
 
 Introduces the least squares filter in batch and recursive forms.
 
 
-* [**Chapter 4: Gaussian Probabilities**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/Chapter04_Gaussians/Gaussians.ipynb)
+* [**Chapter 4: Gaussian Probabilities**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/04_Gaussians/Gaussians.ipynb)
 
 Introduces using Gaussians to represent beliefs. Gaussians allow us to implement the algorithms used in the Discrete Bayes Filter to work in continuous domains.
 
 
-* [**Chapter 5: One Dimensional Kalman Filters**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/Chapter05_Kalman_Filters/Kalman_Filters.ipynb)
+* [**Chapter 5: One Dimensional Kalman Filters**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/05_Kalman_Filters/Kalman_Filters.ipynb)
 
 Implements a Kalman filter by modifying the Discrete Bayesian Filter to use Gaussians. This is a full featured Kalman filter, albeit only useful for 1D problems. 
 
 
-* [**Chapter 6: Multivariate Kalman Filter**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/Chapter06_Multivariate_Kalman_Filter/Multivariate_Kalman_Filters.ipynb)
+* [**Chapter 6: Multivariate Kalman Filter**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/06_Multivariate_Kalman_Filter/Multivariate_Kalman_Filters.ipynb)
 
 We extend the Kalman filter developed in the previous chapter to the full, generalized filter. 
 
 
-* [**Chapter 7: Kalman Filter Math**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/Chapter07_Kalman_Filter_Math/Kalman_Filter_Math.ipynb)
+* [**Chapter 7: Kalman Filter Math**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/07_Kalman_Filter_Math/Kalman_Filter_Math.ipynb)
 
 We gotten about as far as we can without forming a strong mathematical foundation. This chapter is optional, especially the first time, but if you intend to write robust, numerically stable filters, or to read the literature, you will need to know this.
 
 
-* [**Chapter 8: Designing Kalman Filters**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/Chapter08_Designing_Kalman_Filters/Designing_Kalman_Filters.ipynb)
+* [**Chapter 8: Designing Kalman Filters**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/08_Designing_Kalman_Filters/Designing_Kalman_Filters.ipynb)
 
 Building on material in Chapter 6, walks you through the design of several Kalman filters. Discusses, but does not solve issues like numerical stability.
 
 
-* [**Chapter 9: Extended Kalman Filters**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/Chapter09_Extended_Kalman_Filters/Extended_Kalman_Filters.ipynb)
+* [**Chapter 9: Extended Kalman Filters**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/09_Extended_Kalman_Filters/Extended_Kalman_Filters.ipynb)
 
 Kalman filter as covered only work for linear problems. Extended Kalman filters (EKF) are the most common approach to linearizing non-linear problems.
 
 
-* [**Chapter 10: Unscented Kalman Filters**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/Chapter10_Unscented_Kalman_Filters/Unscented_Kalman_Filter.ipynb)
+* [**Chapter 10: Unscented Kalman Filters**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/10_Unscented_Kalman_Filters/Unscented_Kalman_Filter.ipynb)
 
 
 Unscented Kalman filters (UKF) are a recent development in Kalman filter theory. They allow you to filter nonlinear problems without requiring a closed form solution like the Extended Kalman filter requires.
 
 
-* [**Chapter 11: Designing Nonlinear Kalman Filters**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/Chapter11_Designing_Nonlinear_Kalman_Filters/Designing_Nonlinear_Kalman_Filters.ipynb)
+* [**Chapter 11: Designing Nonlinear Kalman Filters**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/11_Designing_Nonlinear_Kalman_Filters/Designing_Nonlinear_Kalman_Filters.ipynb)
 
 EKF and UKF are linear approximations of nonlinear problems. Unless programmed carefully, they are not numerically stable. We discuss some common approaches to this problem.
 
 
-* [**Chapter 12: H-Infinity Filters**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/Chapter12_HInfinity_Filters/HInfinity_Filter.ipynb)
+* [**Chapter 12: H-Infinity Filters**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/12_HInfinity_Filters/HInfinity_Filter.ipynb)
     
 
 * [**Chapter 13: Numerical Stability**](not implemented)