\subsection{Kalman Filtear (KF)}
Bayes Filter for Gauss. dist of R.V. \& linear meas. model.
Initial state $\vec{x}_0 \sim \cN(\hat{\vec{x}}, \mat{P}_0)$, $\mat{P}_0$ previous covariance;

\bi{Prediction} With linear state transition model ($\vec{u}_k$ odometry, $\vec{w}_k$ noise (covariance $\mat{Q}_k$)):\\
$\vec{x}_k = \mat{F}\vec{x}_{k - 1} + \mat{G}\vec{u}_k + \mat{L}\vec{w}_k$ with $\vec{w}_k \sim \cN(\vec{0}, \mat{Q}_k)$:
\begin{itemize}
    \item \bi{Mean} $\hat{\vec{x}}_{k | k - 1} = \mat{F} \hat{\vec{x}}_{k - 1} + \mat{G} \vec{u}_k$
    \item \bi{Covariance} $\mat{P}_{k | k - 1} = \mat{F} \mat{P}_{k - 1 | k - 1} \mat{F}^\top + \mat{L} \mat{Q}_{k} \mat{L}^\top$
\end{itemize}
\bi{Update} Lin. meas.: $\tilde{\vec{z}}_k = \mat{H}\vec{x}_k + \vec{v}_k$ with $\vec{v_k} \sim \cN(\vec{0}, \mat{R}_k)$:
\newpage
\begin{itemize}
    \item \bi{Meas. residual}: $\vec{y}_k = \tilde{\vec{z}}_k - \mat{H} \hat{\vec{x}}_{k | k - 1}$
    \item \bi{Resid. Cov}: $\mat{S}_k = \mat{H}\mat{P}_{k | k - 1} \mat{H}^\top \mat{S}_k^{-1}$
    \item \bi{Kalman gain}: $\mat{K}_k = \mat{P}_{k | k - 1} \mat{H}^\top \mat{S}_k^{-1}$
    \item \bi{Updated mean}: $\hat{\vec{x}}_{k | k} = \hat{\vec{x}}_{k | k - 1} + \mat{K}_k \vec{y}_k$
    \item \bi{Updated Cov.}: $\mat{P}_{k | k} = (\mat{I} - \mat{K}_k \mat{H}) \mat{P}_{k | k - 1}$
\end{itemize}
