Colony

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notation:

\begin{itemize}

\item $x$ - true x position

\item $\tilde{x}$ - sensed x

\item $\hat{x}$ - desired

\end{itemize}

\textbf{motion model}

\[

\bigl [ \omega_l[k] \;,\; \omega_r[k] \bigr] = 

\max \left \{ \min \left \{ 

\left[\begin{array}{cc} \frac{R}{2} & \frac{R}{2}\\ -\frac{R}{L} & \frac{R}{L} \end{array}\right]

^ {-1} 

\left[\begin{array}{c} v[k] \\ \omega[k] \end{array}\right]

\;, \quad \tt{minSpeed} \right \} \;, \quad \tt{maxSpeed} \right \}

+ \mathcal{N}

\]

Where $\mathcal{N}$ is the Gaussian noise model

\[

x[k] = k[k-1] + \cos ( \theta[k-1] ) v[k] dt

\]

\[

y[k] = y[k-1] + \sin ( \theta[k-1] ) v[k] dt

\]

\[

\theta[k] = \theta[k-1] + \omega[k]dt

\]

\textbf{sensor model}

$d$ is the sensor delay, in timesteps

\begin{align*}

  \tilde{\phi}[k] &= \left \lfloor (\phi[k-d] + \mathcal{N})  \frac{8}{\pi} \right \rfloor \frac{\pi}{8} \\

  \tilde{x}[k] &= x[k-d] \\

  \tilde{y}[k] &= y[k-d]

\end{align*}

\end{document}