\section{Related Work}

Like mentioned before, most state-of-the-art systems use recursive state estimators like Kalman- and particle filters.
They differ mainly by the sensors used, their probabilistic models and how the environmental information are incorporated.
For example \cite{Li2015} recently presented an approach combining methods of pedestrian dead reckoning (PDR), Wi-Fi
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}
are using signal strength prediction models like the log-distance model or wall-attenuation-factor model.
Additionally, the sensors noise is not always Gaussian or satisfies the central limit theorem, what makes the
usage of Kalman filters problematic \cite{sarkka2013bayesian, Nurminen2014}.

All this shows, that sensor models differ in many ways and are a subject in itself.
\commentByFrank{sagt man das so? meinst du: haben ihr eigenes forschungsgebiet?}
\commentByToni{"Sie sind ein Thema für sich". Glaub schon das man das so sagt. its own theme gibt es noch. find ich aber nicht so fresh}
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 parameters and address their inaccuracies by harnessing prior information like the pedestrian's desired destination. 
Therefore, we are not that interested in the different sensor representations but more in the state transition as well as incorporating environmental and navigational knowledge. 

A widely used and easy method for modelling the movement of a pedestrian, is the prediction of a new position
by adding an approximated covered distance to the current position. In most cases, a heading serves as
walking direction. If the connection line \commentByFrank{graph? oder generell?: line-of-sight?} \commentByToni{ganz generell. deshalb nur connection line. line of sight ist ja mehr blickachse oder sichtlinie}
between the new and the old position intersects a wall, the probability for the new position 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".
\commentByFrank{waende bevorzugen kurze schritte? wird das klar was hier gemeint ist?}
An additional drawback of these approaches is that for every transition an intersection-test
must be executed. This can result in a high computational complexity.

These disadvantages can be avoided by using spatial models like indoor graphs. 
Regarding modelling approaches, 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, what 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 and 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. 
The complexity is reduced but does not allow arbitrary movements and leads to suboptimal trajectories. 
Far more flexible and variable geometric spatial models are regular tessellated approaches like grid-based models. 
Those techniques are trivially implemented, but yet very powerful \cite{Afyouni2012}. 
Here, 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}. 
In an occupancy grid, a high probability is assigned to cells within accessible space, while 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 is presented in \cite{Li2010}, \cite{Ebner-15} and is also used within this work. 
By assuming that the floorplan is given beforehand, the occupied cells can be removed. 
The remaining cells are described by its centre and represent all free spaces in the indoor 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 in a discrete manner \cite{}. However, as mentioned before changing the floor in a discrete does not resemble real-world conditions. Therefore, \cite{} presented a stepwise floor change based on a hexagonal gridded-graph. A similar approach is presented in the here presented approach for a square-shaped grid.


All this allows a wide range of possibilities for modelling the pedestrian's movement, while only sampling valid locations. 

Computer Games

Evacuation Route Planning





In computer games like the sims or starcraft, intelligent npc movement is a key factor. hierbei geht es nicht nur um das umlaufen von hindernissen sondern auch um eine möglichst natürliche art der bewegung. 


ansätze die dijkstra einfach zum navigieren nutzen.

ansätze aus der robotic um einen roboter von a nach b zu schicken


the idea of using navigational knowledge to simulate the human movement 




