\section{Related Work}
\label{sec:relatedWork}


Like mentioned before, most state-of-the-art systems use recursive state estimators like Kalman- and particle filters.
They differ mainly by the used sensors, their probabilistic models and how environmental information is incorporated.
For example \cite{Li2015} recently presented an approach combining methods of pedestrian dead reckoning (PDR), \docWIFI{}
fingerprinting and magnetic matching using a Kalman filter. While providing good results, fingerprinting methods
require an extensive offline calibration phase. Therefore, many other systems like \cite{Fang09} or \cite{Ebner-15}
use signal strength prediction models like the log-distance or wall-attenuation-factor model.
Additionally, the sensors noise is not always Gaussian or satisfies the central limit theorem. Using
Kalman filters is therefore problematic \cite{sarkka2013bayesian, Nurminen2014}.
All this shows, that sensor models differ in many ways and are a subject in itself.
A good discussion on different sensor models can be found in \cite{Yang2015}, \cite{Gu2009} or \cite{Khaleghi2013}. 

However, within this work, we use simple models, configured using a handful of empirically chosen parameters and
address their inaccuracies by harnessing prior information like the pedestrian's desired destination. Therefore,
instead of examining different sensors and their contribution to the localisation process, we will focus
on the state transition and how to incorporate environmental and navigational knowledge.

A widely used and easy method for modelling the movement of a pedestrian, is the prediction of a new position
using both, a walking direction and a to-be-walked distance, starting from the previous position.
If the line-of-sight between the new and the old position intersects a wall, the probability for this
transition is set to zero \cite{Woodman08-PLF, Blanchert09-IFF, Koeping14-ILU}.
However, as \cite{Nurminen13-PSI} already stated, it "gives more probability to a short step".
An additional drawback of these approaches is that for every transition an intersection-test
must be executed and thus often yields a high computational complexity.

These disadvantages can be avoided by using spatial models like indoor graphs. 
Here, two main classes can be distinguished: symbolic and geometric spatial models \cite{Afyouni2012}.
Especially geometric spatial models (coordinate-based approaches) are very popular, since they integrate metric properties to provide highly accurate location and distance information.
One of the most common environmental representations in indoor localization literature is the Voronoi diagram \cite{Liao2003}. 
It represents the topological skeleton of the building's floorplan as an irregular tessellation of space. 
This drastically removes degrees of freedom from the map, and results in a low complexity.

In the work of \cite{Nurminen2014} a Voronoi diagram is used to approximate the human movement. 
It is assumed that the pedestrian can be anywhere on the topological links.
The probabilities of changing to the next link are proportional to the total link lengths.
However, for highly accurate localisation in large-scale buildings, this network of one-dimensional
curves is not suitable \cite{Afyouni2012}.
Therefore, \cite{Hilsenbeck2014} searches for large open spaces (e.g. a lobby) and extends the Voronoi diagram
by adding those two-dimensional areas.
The final graph is then created by sampling nodes in regular intervals across the links and filling up the open
spaces in a tessellated manner.  Similar to \cite{Ebner-15}, they provide a state transition model that selects
an edge and a node from the graph according to a sampled distance and heading. 

Nevertheless, most corridors are still represented by just one topological link. 
While the complexity is reduced, it does not allow arbitrary movements and leads to suboptimal trajectories. 
Far more flexible and variable geometric spatial models are regularly tessellated approaches e.g. based on grids. 
Those techniques are trivially implemented, but yet very powerful. 
In \cite{Afyouni2012}, a square-shaped or hexagonal grid covers the entire map.
Especially in the area of simultaneous localisation and mapping (SLAM), so-called occupancy-grid approaches are
very popular \cite{elfes1989using, Thrun2003}. 
Occupancy grids assign a high probability to cells within the accessible space.
Likewise, cells occupied by obstacles or walls are less likely. 
Additionally, every grid cell is able to hold some context information about the environment (e.g. elevators or stairs)
or the behaviour of a pedestrian at this particular position (e.g. jumping or running). 

A similar approach, presented in \cite{Li2010}, \cite{Ebner-15}, is also used within this work. 
Assuming the floorplan is given beforehand, occupied cells can be removed. 
The remaining cells are described by their centre/bounding-box and represent free spaces within the environment. 
A graph is defined by using the centres as nodes and connecting direct neighbours with edges. 

In order to enable floor changes, some approaches suggest to simply connect the nodes at staircases \cite{Ebner-15, Hilsenbeck2014}. 
However, as mentioned before changing the floor in a discrete manner does not resemble real-world conditions. 
Therefore, \cite{GarciaPuyol2014} presented a stepwise floor change based on a hexagonal gridded-graph. 
We introduce a similar approach for square-shaped grids.

All this allows a wide range of possibilities for modelling the pedestrian's movement, while only sampling valid locations. 
In virtual environments like video games and simulations, the human motion is often modelled using graphs and path finding techniques. 
Here, the goal is not only to provide a shortest path, but also the least-cost path, most natural path or least-dangerous path. 
For example, \cite{Bandi2000} uses an A*-algorithm to search a 3D gridded environment for the shortest path to a goal. 
An additional smoothing procedure is performed to make the path more natural. 
They are considering foot span, body dimensions and obstacle dimensions when determining whether an obstacle is surmountable. 
However, many of this information is difficult to ascertain in real-time or imply additional effort in real-world environments. 
Therefore, more realistic simulation models, mainly for evacuation simulation, are just using a simple shortest path on regularly
tessellated graphs \cite{Sun2011, tan2014agent}. A more costly, yet promising approach is shown by \cite{Brogan2003}. They use a
data set of previously recorded walks to create a model of realistic human walking paths. 

Finally, it seems that currently none of the localisation system approaches are using realistic walking paths as additional
source of information to provide a more targeted and robust movement. Most common systems are sampling a new state only in
regard of the user's heading and speed using one of the above mentioned indoor graphs.




