Often, we are given map \(m\) of the environment, giving us further information about the robot pose \(x_t\). In general,

\begin{equation} p\left(x_{t} | u_{t}, x_{t-1}\right) \neq p\left(x_{t} | u_{t}, x_{t-1}, m\right) \end{equation}

And the map-based motion model should give better results. Computing this motion model in closed form is difficult. An approximation via factorization works well where the distance \(x_{t-1}\) and \(x_t\) is small.

\begin{equation} p\left(x_{t} | u_{t}, x_{t-1}, m\right)=\eta p\left(x_{t} | u_{t}, x_{t-1}\right) p\left(x_{t} | m\right) \end{equation}