\section{Introduction}

Since the advent of smartphones, location aware apps and services are ubiquitous and have become a natural part of our lives. Whether driving a car, jogging or shopping in the streets, GNSS-based applications are making orientation easier, point the way and even track our fitness achievements. But as soon as we drive into an underground car park or visit a shopping mall, they perform poorly. That is because satellite signals are to weak to pass through obstacles like buildings' walls. Moreover, their accuracy is not sufficient for individual parking spaces or rooms. Therefore, many different solutions for localizing a moving object within buildings have been developed in recent years \cite{}. Especially the hard problem of pedestrian localization and navigation has lately attracted a lot of interest. 

Most modern indoor localisation systems primarily use smartphones for determining the position of a pedestrian. Especially the phone's inertial measurement unit (IMU) as well as external information like Wi-Fi or Bluetooth are used for collecting the necessary data. Additionally, environmental knowledge is often incorporated by using floor maps. This combination of highly different sensor types is also known as sensor fusion. Here probabilistic methods like particle filters or Kalman filters or often used to approximate a probability distribution describing the uncertainties of the system. This procedure can be separated into two probabilistic models: The transition model represents the dynamics of the system and predicts the next accessible locations, while the evaluation model estimates a probability that the position also corresponds to the current sensor measurement.
 %Therefore, the most accurate position is represented by a peak of the probability distribution. 
In our previous work we were able to present such a localisation system based on all the above mentioned sensors including the phone's barometer \cite{Ebner-15}. 

In pedestrian navigation, the human movement underlies the characteristics of walking speed and walking direction. Additionally, environmental restrictions need to be considered as well, for example, walking through walls is in most cases impossible. Therefore, incorporating environmental knowledge is a necessary and gainful step. Like other systems, we are using a graph-based approach for this. The main advantage of such an approach is that the graph only samples valid locations. The unique feature of our approach is the way in how we model the human movement. This is done by using random walks on graphs, which are based upon the heading of the pedestrian. However, the system presented in \cite{Ebner-15} suffers from two major drawbacks, we want to solve within this work. 

Firstly, the transition model of our past approach uses discrete floors. Changing the floor on a discrete basis is like jumping down the staircase. This does not resemble real world floor changes and it could be shown that a correct estimation strongly depends on the quality of $z$-transitions. To address this problem we extended the graph by realistically shaped stairs, allowing a step-wise transition in the $z$-direction.

Secondly, another drawback is the way in how the pedestrian's walking behaviour is modelled. At the moment the heading is only calculated between two adjacent nodes. That means, we are only able to perform \SI{45}{\degree} turns. \commentByToni{Ich denke hier kann Frank E. mehr zu schreiben. Bin mir nicht sicher wie ich das Problem gut schildern kann.} blumenverteilung, kurven laufen fällt schwer... bessers ziehen.

The problem of localization can be simplified by assuming a person navigation. Such applications are used to navigate a pedestrian to a given target destination. So, based on this assumption the starting point, which is the current position of the pedestrian, as well as the destination are known beforehand. Regarding a graph-based transition model, one could suggest to calculate the shortest path between start and destination. However, this often leads to paths running very unnatural alongside walls. Additionally, the human walking behaviour is highly affected by visual distractions, comfort, disorientation and many other factors. Therefore, we present a novel method for pedestrian navigation by using XXX methods to achieve a preferably realistic path, areas near a wall are less likely to be choosen for the path then a door or a small hallway. ... probability map/graph ... 
\commentByToni{Wissen ja noch nicht was wir hier genau nehmen, deswegen erstmal leer}

\begin{itemize}
	\item Hinführen zum Thema 1/4 + Abstract (haben so wenig platz nur 8 seiten...)
	\item "In unseren vergangenen Arbeiten konnten wir zeigen ... aber ..."
	\item Probleme ganz konkret aufzählen und gleich die lösungen zu den problemen
	  \subitem diskrete stockwerke, senkrecht 
	  \subitem kurven laufen bereitet große probleme
	  \subitem lokalisationsergebnisse sind zwar gut, aber vor allem an treppen instabil und räume werden schwer erkannt.
	\item Was machen wir jetzt besser / anders. Hinführen zum Thema
	    \subitem Kontinuierliche Stockwerke
	    \subitem Bessers ziehen in der Transition 
	    \subitem zusätzliche Gewichtung der Knoten anstatt nur annahme des geradeaus laufens mit geringer chance auf turns. 
	    \subitem Annahme der Navigation
	\item Aufbau der Arbeit (falls platz, haben nur 8 Seiten)
\end{itemize}