\subsection{Fixed-lag smoothing}

Within \cite{fetzer-16} we added an additional smoothing step to the localisation procedure. 
In contrast to normal filtering, smoothing methods are able to incorporate future measurements instead of just using current and past data.
Therefore, they are able to compute probability distributions in the form of $p(\mStateVec_t \mid \mObsVec_{1:T})$. 
Especially interesting for real-time applications is the so-called fixed-lag smoothing. 
In fixed-lag smoothing, one tries to estimate the current state, given measurements up to a time $t + \tau$, where $\tau$ is a predefined lag. 
By running backwards in time, they are able to remove multimodalities and improve the overall localisation result.
We can distinguish between two different smoothing algorithms: Forward-backward smoothing \cite{doucet2000} and backward simulation \cite{Godsill04:MCS}.  
Both perform very similar and are reweighting possible states based on a smoothing transition model.

The smoothing transition model calculates the probability of being in a state $\vec{q}_{t+1}$ in regard to previous states and the pedestrian's walking behaviour. 
Therefore, we compare the distance, angle and height between $\vec{q}_{t+1}$ and $\vec{q}_{t}$ in regard to the measurements gettered at time $t$. 
The resulting likelihood is then used for reweighting. 

%By writing
%\begin{equation}
%p(\vec{q}_{t+1} \mid \vec{q}_t, \mObsVec_t)_{\text{step}} = \mathcal{N}(\Delta d_t \mid \mu_{\text{step}}, \sigma_{\text{step}}^2)
%\label{eq:smoothingTransDistance}
%\end{equation}
%we receive a statement about how likely it is to cover a distance $\Delta d_t$ between two states $\vec{q}_{t+1}$ and $\vec{q}_{t}$.
%In the easiest case, $\Delta d_t$ is the euclidean distance between two states. 
%The average step length $\mu_{\text{step}}$ is based on the pedestrian's walking speed and $\sigma_{\text{step}}^2$ denotes the step length's variance. 
