diff --git a/doc/argraf/thesis_ar_gr02a.tex b/doc/argraf/thesis_ar_gr02a.tex new file mode 100644 index 0000000000000000000000000000000000000000..fe2d55aa600e73399e37d6b2c899021764191ac4 --- /dev/null +++ b/doc/argraf/thesis_ar_gr02a.tex @@ -0,0 +1,168 @@ +%% LyX 2.0.6 created this file. For more info, see http://www.lyx.org/. +%% Do not edit unless you really know what you are doing. +\RequirePackage{fix-cm} +\documentclass[english]{article} +\usepackage[T1]{fontenc} +\usepackage[utf8]{luainputenc} +\setlength{\parskip}{\medskipamount} +\setlength{\parindent}{0pt} +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{fixltx2e} + +\makeatletter +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Textclass specific LaTeX commands. +\numberwithin{equation}{section} +\numberwithin{figure}{section} +\numberwithin{table}{section} + +\makeatother + +\usepackage{babel} +\begin{document} + +\title{Automated Routing in Pedestrian Dynamics} + + +\author{Arne Graf} + + +\date{2015-09-01} + +\maketitle + +\section{Introduction} + + +\section{ODE based Model (dep.? -> opt.?)} + + +\section{Modelling} + + +\subsection{Eikonal Equation} + +The Eikonal Equation'' in a domain $\Omega$, subset of $\mathbb{R}^{n}$, + +\begin{align*} +\vert\nabla u(x)\vert\quad= & \quad F(x),x\in\Omega,\\ +\mathrm{s.t.\qquad u|_{\partial\Omega}\quad=} & \quad0\\ +\end{align*} + + +yields first-arrival-times'' in a spacial domain provided a target +region within the domain. + + +\subsection{Safe Navigation using the Floorfield} + + +\subsection{Distances-Field} + + +\subsubsection{Cost of a full'' preprocessing step } + + +\subsubsection{Distances-Field and repulsive Wall-Forces} + + +\subsection{Variant Model} + +In the latter, a new approach in modeling is described, aiming for +the avoidance of faulty interaction of pedestrians and walls while +maintaining the positive characteristics of row-formation, stop-and-go +waves and such, like seen in experimental data. In many of the existing +models using mathematical formulations in continouus domain, agents +breach wall-surfaces and get stuck inside of walls or obstacles. + +There are two mechanics used to avoid clipping'': +\begin{enumerate} +\item The routing of pedestirans makes use of the eikonal-equation, computed +with an inhomogenious speed-function, $s(x)$, which favours keeping +a distance to obstacles, walls and corners. +\item The distance to the closest wall of each pedestrian affects the moving +speed if and only if the agent's moving vector includes a component +geared towards the wall. +\end{enumerate} +In order to keep the model simple, repulsive wall forces as seen in +Social Force Models are omitted. An analogy to repulsive pedestrian +forces though is used to keep agents from colliding with each other. +The model differs from SFMs, as other agents effect the desired moving +direction in full, the magnitude on the other hand is effected by +other agents only to a certain degree (as described in ). + +\begin{alignat*}{1} +\Delta\vec{x}\quad=\quad & \Delta t\cdot\vec{v}_{res}\\ +\vec{v}_{res}\quad=\quad & \left(1-\frac{1}{2}\left[(\vec{v}_{n}\cdot(-\nabla distances)_{n})+\vert(\vec{v}_{n}\cdot(-\nabla distances)_{n})\vert\right]\right)\cdot\vec{v}_{n}\\ +\vec{v}_{n}\quad=\quad & g(g(\vec{v}_{ff})+g(\sum\vec{v}_{repP,i}))\\ +\vec{v}_{ff}\quad=\quad & v_{ff}(\vec{x})\\ +\end{alignat*} + + +\begin{align*} +\Delta\vec{v}\quad= & \quad\Delta t\cdot\vec{v}_{res}\\ +\vec{v}_{res}\quad= & \quad\bigg(1-\frac{1}{2}\bigg[\langle\vec{v}_{n},(-\nabla distances)_{n}\rangle+\big\vert\langle\vec{v}_{n},(-\nabla distances)_{n}\rangle\big\vert\bigg]\bigg)\cdot\vec{v}_{n}\\ +\vec{v}_{n}\quad= & \quad g\big(\quad g(\vec{v}_{ff})+g(\underset{\small i}{\sum}\vec{v}_{repP,i})\quad\big)\\ +\vec{v}_{ff}\quad= & \quad v_{ff}(\vec{x}) +\end{align*} + + + +\subsection{Idea of Separation of a Moving-Vector into Direction and Magnitute} + + +\subsubsection{no clipping} + + +\subsubsection{Recycling the Distances Field (neg. Gradient must be saved)} + +$\vec{v}_{res}\quad=\quad\bigg(1-\frac{1}{2}\bigg[\langle\vec{v}_{n},(-\nabla distances)_{n}\rangle+\big\vert\langle\vec{v}_{n},(-\nabla distances)_{n}\rangle\big\vert\bigg]\bigg)\cdot\vec{v}_{n}$ + + +\section{Testing} + + +\section{Shortcomings} + + +\subsection{Floorfield} + + +\subsubsection{Multiple Goals } + +The floorfield is a usefull tool in routing of pedestirans through +any geometry. + + +\subsubsection{Multiple Floors} + + +\paragraph{Neighboring Relations} + + +\subsection{Avoid Clipping} + + +\section{Outlook} + + +\subsection{Usage in JuPedSim} + + +\subsection{Floorfields in Triangulated Domains} + + +\subsection{Parallelization} + + +\section{Appendices} + + +\subsection{Classes and their Relations} + + +\subsection{Code Snippets} + + +\section{Bibliography} +\end{document}