root / branches / 16299_s10 / latex / math.tex @ 1821
History | View | Annotate | Download (1.75 KB)
1 |
\documentclass[11pt]{article} |
---|---|
2 |
\usepackage{fullpage} |
3 |
\usepackage{latexsym} |
4 |
\usepackage{amsmath} |
5 |
\begin{document} |
6 |
|
7 |
|
8 |
notation: |
9 |
\begin{itemize} |
10 |
\item $x$ - true x position |
11 |
\item $\tilde{x}$ - sensed x |
12 |
\item $\hat{x}$ - desired |
13 |
\item $r$ - true r |
14 |
\item $\tilde{r}$ - sensed r |
15 |
\item $\hat{r}$ - desired |
16 |
\item $\phi$ - true phi position |
17 |
\item $\tilde{\phi}$ - sensed phi |
18 |
\item $\hat{\phi}$ - desired |
19 |
\item $y$ - true y position |
20 |
\item $\tilde{y}$ - sensed y |
21 |
\item $\hat{y}$ - desired |
22 |
|
23 |
\item $(\hat{r}, \hat{\phi}) ^ n$ |
24 |
\item $(\hat{x}, \hat{y}) ^ n$ |
25 |
\item $(\hat{v}, \hat{\omega}) ^ n$ |
26 |
\item $(\tilde{v}, \tilde{\omega}) ^ n$ |
27 |
\item $(\tilde{x}, \tilde{y}, \tilde{\Theta}, \tilde{\phi}) ^ n$ |
28 |
\item $(\hat{\omega_l}, \hat{\omega_r}) ^ n$ |
29 |
\item $({\omega_l}, {\omega_r}) ^ n$ |
30 |
\item $(\tilde{\omega_l}, \tilde{\omega_r}) ^ n$ |
31 |
\item $(\tilde{x}, \tilde{y}, \tilde{\Theta}) ^ n$ |
32 |
|
33 |
\item $(\tilde{x}, \tilde{y}, \tilde{\phi}) ^ n$ |
34 |
\item $\phi ^ n$ |
35 |
\item $\tilde{\phi} ^ n$ |
36 |
\end{itemize} |
37 |
|
38 |
|
39 |
|
40 |
|
41 |
\textbf{motion model} |
42 |
|
43 |
\[ |
44 |
\bigl [ \omega_l[k] \;,\; \omega_r[k] \bigr] = |
45 |
\max \left \{ \min \left \{ |
46 |
\left[\begin{array}{cc} \frac{R}{2} & \frac{R}{2}\\ -\frac{R}{L} & \frac{R}{L} \end{array}\right] |
47 |
^ {-1} |
48 |
\left[\begin{array}{c} v[k] \\ \omega[k] \end{array}\right] |
49 |
\;, \quad \tt{minSpeed} \right \} \;, \quad \tt{maxSpeed} \right \} |
50 |
+ \mathcal{N} |
51 |
\] |
52 |
|
53 |
Where $\mathcal{N}$ is the Gaussian noise model |
54 |
|
55 |
\[ |
56 |
x[k] = k[k-1] + \cos ( \theta[k-1] ) v[k] dt |
57 |
\] |
58 |
|
59 |
\[ |
60 |
y[k] = y[k-1] + \sin ( \theta[k-1] ) v[k] dt |
61 |
\] |
62 |
|
63 |
\[ |
64 |
\theta[k] = \theta[k-1] + \omega[k]dt |
65 |
\] |
66 |
|
67 |
|
68 |
\textbf{sensor model} |
69 |
|
70 |
$d$ is the sensor delay, in timesteps |
71 |
|
72 |
\begin{align*} |
73 |
\tilde{\phi}[k] &= \left \lfloor (\phi[k-d] + \mathcal{N}) \frac{8}{\pi} \right \rfloor \frac{\pi}{8} \\ |
74 |
\tilde{x}[k] &= x[k-d] \\ |
75 |
\tilde{y}[k] &= y[k-d] |
76 |
\end{align*} |
77 |
|
78 |
|
79 |
\end{document} |