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\documentclass[11pt]{article} |
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\usepackage{fullpage} |
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\usepackage{latexsym} |
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\usepackage{amsmath} |
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\begin{document} |
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notation: |
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\begin{itemize} |
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\item $x$ - true x position |
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\item $\tilde{x}$ - sensed x |
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\item $\hat{x}$ - desired |
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\end{itemize} |
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\textbf{motion model} |
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\[ |
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\bigl [ \omega_l[k] \;,\; \omega_r[k] \bigr] = |
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\max \left \{ \min \left \{ |
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\left[\begin{array}{cc} \frac{R}{2} & \frac{R}{2}\\ -\frac{R}{L} & \frac{R}{L} \end{array}\right] |
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^ {-1} |
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\left[\begin{array}{c} v[k] \\ \omega[k] \end{array}\right] |
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\;, \quad \tt{minSpeed} \right \} \;, \quad \tt{maxSpeed} \right \} |
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+ \mathcal{N} |
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\] |
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Where $\mathcal{N}$ is the Gaussian noise model |
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|
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\[ |
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x[k] = k[k-1] + \cos ( \theta[k-1] ) v[k] dt |
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\] |
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\[ |
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y[k] = y[k-1] + \sin ( \theta[k-1] ) v[k] dt |
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\] |
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\[ |
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\theta[k] = \theta[k-1] + \omega[k]dt |
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\] |
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\textbf{sensor model} |
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$d$ is the sensor delay, in timesteps |
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\begin{align*} |
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\tilde{\phi}[k] &= \left \lfloor (\phi[k-d] + \mathcal{N}) \frac{8}{\pi} \right \rfloor \frac{\pi}{8} \\ |
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\tilde{x}[k] &= x[k-d] \\ |
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\tilde{y}[k] &= y[k-d] |
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\end{align*} |
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\end{document} |