diff --git a/doc/mathesisargraf/masterthesisArGr.aux b/doc/mathesisargraf/masterthesisArGr.aux deleted file mode 100644 index f9ac8269d9f54d54298ade02bc25baaec891c49f..0000000000000000000000000000000000000000 --- a/doc/mathesisargraf/masterthesisArGr.aux +++ /dev/null @@ -1,38 +0,0 @@ -\relax -\select@language{english} -\@writefile{toc}{\select@language{english}} -\@writefile{lof}{\select@language{english}} -\@writefile{lot}{\select@language{english}} -\@writefile{toc}{\contentsline {section}{\numberline {1}Pedestrian Dynamics: Introduction}{5}} -\@writefile{toc}{\contentsline {section}{\numberline {2}ODE based Model}{5}} -\@writefile{toc}{\contentsline {section}{\numberline {3}Modelling}{5}} -\@writefile{toc}{\contentsline {subsection}{\numberline {3.1}Variant Model}{7}} -\@writefile{toc}{\contentsline {subsection}{\numberline {3.2}Eikonal Equation}{7}} -\newlabel{eikonalequation}{{3.2}{7}} -\@writefile{lof}{\contentsline {figure}{\numberline {3.1}{\ignorespaces Isolines of a floor-field.}}{7}} -\newlabel{fig:BottleneckObstaclePure}{{3.1}{7}} -\newlabel{replace with bibtex: http://roboticslab.uc3m.es/roboticslab/researchtopic/fast-marching}{{3.2}{8}} -\@writefile{toc}{\contentsline {subsection}{\numberline {3.3}Safe Navigation using the Floorfield}{8}} -\@writefile{toc}{\contentsline {subsection}{\numberline {3.4}Distances-Field}{9}} -\newlabel{fig:DistanceField2}{{3.4}{9}} -\newlabel{fig:BottleneckObstaclePure2}{{3.4}{9}} -\newlabel{fig:BottleneckObstacleEnhanced}{{3.4}{10}} -\@writefile{toc}{\contentsline {subsubsection}{\numberline {3.4.1}Cost of a ``full'' preprocessing step }{10}} -\@writefile{toc}{\contentsline {subsubsection}{\numberline {3.4.2}Distances-Field and repulsive Wall-Forces}{10}} -\@writefile{toc}{\contentsline {subsection}{\numberline {3.5}Idea of Separation of a Moving-Vector into Direction and Magnitute}{11}} -\@writefile{toc}{\contentsline {subsubsection}{\numberline {3.5.1}no clipping}{11}} -\@writefile{toc}{\contentsline {subsubsection}{\numberline {3.5.2}Recycling the Distances Field (neg. Gradient must be saved)}{11}} -\@writefile{toc}{\contentsline {section}{\numberline {4}Testing}{11}} -\@writefile{toc}{\contentsline {section}{\numberline {5}Outlook}{11}} -\@writefile{toc}{\contentsline {subsection}{\numberline {5.1}Floor-field}{11}} -\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.1.1}Multiple Goals }{11}} -\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.1.2}Multiple Floors}{11}} -\@writefile{toc}{\contentsline {paragraph}{Neighboring Relations}{11}} -\@writefile{toc}{\contentsline {subsection}{\numberline {5.2}Usage in JuPedSim}{12}} -\@writefile{toc}{\contentsline {subsection}{\numberline {5.3}Floor-fields in Triangulated Domains}{12}} -\@writefile{toc}{\contentsline {subsection}{\numberline {5.4}Parallelization}{12}} -\@writefile{toc}{\contentsline {section}{\numberline {6}Appendices}{12}} -\@writefile{toc}{\contentsline {subsection}{\numberline {6.1}Fast-Marching Algorithm}{12}} -\@writefile{toc}{\contentsline {subsection}{\numberline {6.2}Classes and their Relations}{12}} -\@writefile{toc}{\contentsline {subsection}{\numberline {6.3}Code Snippets}{12}} -\@writefile{toc}{\contentsline {section}{\numberline {7}Bibliography}{12}} diff --git a/doc/mathesisargraf/masterthesisArGr.bbl b/doc/mathesisargraf/masterthesisArGr.bbl deleted file mode 100644 index e69de29bb2d1d6434b8b29ae775ad8c2e48c5391..0000000000000000000000000000000000000000 diff --git a/doc/mathesisargraf/masterthesisArGr.bib b/doc/mathesisargraf/masterthesisArGr.bib index b3fe429dc0c2950463d01ca86e6b87852c5ec7e7..541af76392a6caca323623277e37749ef8f2117d 100644 --- a/doc/mathesisargraf/masterthesisArGr.bib +++ b/doc/mathesisargraf/masterthesisArGr.bib @@ -1,6 +1,143 @@ -@online{Universidad Carlos III de Madrid, RoboticsLab, -author = {L.Moreno et al.}, -title = {Fast Marching}, -date = {2014-12-19}, -url = {roboticslab.uc3m.es/roboticslab/research/fast-marching}, -} \ No newline at end of file +@online{Madrid, +AUTHOR = {Moreno}, +TITLE = {Fast Marching, roboticslab.uc3m.es/roboticslab/research/fast-marching}, +DATE = {2014-12-19}, +YEAR = {2014}, +URL = {roboticslab.uc3m.es/roboticslab/research/fast-marching}, +} + +@Article{Dietrich2014, + Title = {Gradient navigation model for pedestrian dynamics}, + Author = {Dietrich, Felix and K{\"o}ster, Gerta}, + Journal = {Arxiv e-prints}, + Year = {2014}, + + File = {:pdf\\Dietrich2014.pdf:PDF}, + Keywords = {Agent, Pedestrians, Modelling, Microscopic, Ordinary differential equations (ODE), Flow}, + Owner = {WeichenLiao}, + Timestamp = {2014.01.09} +} + +@PhdThesis{Chraibi2012, + Title = {Validated force-based modeling of pedestrian dynamics}, + Author = {Chraibi, Mohcine}, + School = {Universit{\"a}t zu K{\"o}ln}, + Year = {2012}, + Month = {March, 15}, + + File = {:pdf\\Chraibi2012.pdf:PDF}, + Owner = {Fred}, + Timestamp = {2012.07.18} +} + +@Article{KemlohWagoum2013, + Title = {Route choice modelling and runtime optimisation for simulation of building evacuation}, + Author = {Kemloh Wagoum, Armel Ulrich}, + Journal = {Schriften des Forschungszentrums J{\"u}lich}, + Year = {2013}, + Note = {pedestrian dynamics, route choice, evacuation, high performance computing}, + Volume = {17}, + + Abstract = {Increasing number of visitors at large-scale events combined with the increasing complexity of modern buildings set a major challenge for planners, operators and emergency services. Examples include multi-purpose arenas, large railway stations and airports. In this dissertation the use of modern parallel hardware in combination with optimised algorithms are for the first time used on site to speed up the simulation of large crowds. The aim is to perform real-time forecasts of pedestrian traffic. For this purpose, specialneighbourhood lists and a two-stage hybrid parallelisation are used. Thesecond part of this dissertation deals with route choice in complex structures, which plays an important role in achieving realistic computer simulations of pedestrian flows. The developed route choice process is based on visibility and perception of the local environment by the simulated agents. It has as basis a navigation graph. The generation of the graph, espe- cially in complex structures, has also been performed within the framework of this thesis. The work is closed with an empirical study in which the route choice profiles of spectators during various football games and concert performances are analysed and compared with the proposed model. The runtime optimisation strategies and route choice algorithms have been successfully tested in the ESPRIT arena in Düsseldorf (Germany), where they have been integrated in an evacuation assistant.}, + File = {:pdf/KemlohWagoum2013.pdf:PDF}, + ISBN = {978-3-89336-865-5}, + ISSN = {1868-8489}, + Keywords = {pedestrian dynamics, route choice, evacuation, high performance computing}, + Owner = {f}, + Timestamp = {2013.04.30} +} + +@Article{Nunez2011, + Title = {Parallel Implementation of Fast Marching Method, 18.337 final report}, + Author = {Leonardo Andrés Zepeda Núñez}, + Journal = {}, + Year = {2011}, +} + +@online{Emme, +AUTHOR = {INRO}, +TITLE = {http://www.inrosoftware.com/en/products/emme/}, +DATE = {}, +YEAR = {2015}, +URL = {http://www.inrosoftware.com/en/products/emme/}, +} + +@Article{Moussaid2011, + Title = {How simple rules determine pedestrian behavior and crowd disasters}, + Author = {Moussa\"id, Mehdi and Helbing, Dirk and Theraulaz, Guy}, + Journal = {PNAS}, + Year = {2011}, + + Abstract = {With the increasing size and frequency of mass events, the study of crowd disasters and the simulation of pedestrian flows have become important research areas. However, even successful modeling approaches such as those inspired by Newtonian force models are still not fully consistent with empirical observations and are sometimes hard to calibrate. Here, a cognitive science approach is proposed, which is based on behavioral heuristics. We suggest that, guided by visual information, namely the distance of obstructions in candidate lines of sight, pedestrians apply two simple cognitive procedures to adapt their walking speeds and directions. Although simpler than previous approaches, this model predicts individual trajectories and collective patterns of motion in good quantitative agreement with a large variety of empirical and experimental data. This model predicts the emergence of self-organization phenomena, such as the spontaneous formation of unidirectional lanes or stop-and-go waves. Moreover, the combination of pedestrian heuristics with body collisions generates crowd turbulence at extreme densities-a phenomenon that has been observed during recent crowd disasters. By proposing an integrated treatment of simultaneous interactions between multiple individuals, our approach overcomes limitations of current physics-inspired pair interaction models. Understanding crowd dynamics through cognitive heuristics is therefore not only crucial for a better preparation of safe mass events. It also clears the way for a more realistic modeling of collective social behaviors, in particular of human crowds and biological swarms. Furthermore, our behavioral heuristics may serve to improve the navigation of autonomous robots.}, + Comment = {Social Force Modell, microscopic, modelling, stop-and-go waves}, + File = {Moussaid2011.pdf:pdf/Moussaid2011.pdf:PDF}, + Owner = {portz}, + Timestamp = {2011.05.06} +} + +@Article{Chraibi2011, + Title = {Force-based models of pedestrian dynamics}, + Author = {Chraibi, M. and Kemloh, U., Seyfried, A. and Schadschneider, A.}, + Journal = {Networks and Heterogeneous Media}, + Year = {2011}, + Number = {3}, + Pages = {425--442}, + Volume = {6}, + + Abstract = {Force-based models describe the interactions of pedestrians in terms of physical and social forces. We discuss some intrinsic problems of this ap- proach, like penetration of particles, unrealistic oscillations and velocities as well as conceptual problems related to violations of Newton’s laws. We then present the generalized centrifugal force model which solves some of these prob- lems. Furthermore we discuss the problem of choosing a realistic driving force to an exit. We illustrate this problem by investigating the behaviour of pedes- trians at bottlenecks.}, + Doi = {10.3934/nhm.2011.6.425}, + File = {:pdf\\Chraibi2011.pdf:PDF}, + Keywords = {Pedestrian dynamics, flow, bottleneck, force-based models.}, + Owner = {mmueller}, + Timestamp = {2011.10.26}, + Url = {http://aimsciences.org/journals/displayPaper.jsp?paperID=6440} +} + +@Article{Hartmann2010, + Title = {Adaptive pedestrian dynamics based on geodesics}, + Author = {Hartmann, Dirk}, + Journal = {New Journal of Physics}, + Year = {2010}, + Pages = {043032}, + Volume = {12}, + + Abstract = {Here, we report on a new approach for adaptive path finding in microscopic simulations of pedestrian dynamics. The approach extends a widely used concept based on scalar navigation field -- the so-called floor field method. Adopting a continuum perspective, navigation fields used in our approach correspond to the shortest distances to the pedestrian's targets with respect to arbitrary metrics, e.g. metrics depending on the local terrain. If the metric correlates inversely with the expected speed, these distances could be interpreted as expected travel times. Following this approach, it is guaranteed that virtual pedestrians navigate along the steepest descent of the navigation field and thus follow geodesics. Using the Eikonal equation, i.e. a continuum model, navigation fields can be determined with respect to arbitrary metrics in an efficient manner. The fast marching method used in this work offers a fast method to solve the Eikonal equation (complexity N log N, where N is degree of freedom). Increasing computational efforts with respect to classical approaches only mildly, the consideration of locally varying metrics allows a realistic adaptive movement behavior like the avoidance of certain terrains. The method is outlined using a simple cellular automaton approach. Extensions to other microscopic models, e.g. cellular automata approaches or social force models, are possible.}, + Comment = {shortest path, quickest path, routing, re-routing, travel time}, + File = {Hartmann2010.pdf:pdf/Hartmann2010.pdf:PDF}, + Owner = {portz}, + Timestamp = {2010.10.22} +} + +@online{jupedsim, + Title = {JuPedSim. http://www.JuPedSim.org} + Author = {Forschungszentrum J{\"u}lich GmbH, JSC, CST} + Year = {2015} + URL = {http://www.jupedsim.org} +} + +@Book{Predtechenskii1971, + Title = {Personenstr\"ome in Geb\"auden - Berechnungsmethoden f\"ur die Projektierung}, + Author = {Predtechenskii, W. M. and Milinskii, A. I.}, + Publisher = {Verlagsgesellschaft Rudolf M{\"u}ller, K{\"o}ln-Braunsfeld}, + Year = {1971}, + Note = {Original in Russian, Stroiizdat Publishers, Moscow, 1969}, + + File = {Predtechenskii1971.pdf:pdf/Predtechenskii1971.pdf:PDF}, + Owner = {portz}, + Timestamp = {2008.01.09} +} + +@InProceedings{Hirai1975, + Title = {A simulation of the behavior of a crowd in panic}, + Author = {Hirai, K. and Tarui, K.}, + Booktitle = {Proc. of the 1975 International Conference on Cybernetics and Society}, + Year = {1975}, + + Address = {San Francisco}, + Pages = {409-411}, + + Abstract = {The purpose of this paper is to study the behavior of a crowd in panic by digital simulation. Psychologienl factors and effects of the environment such as the presence of sign or symbol to guide the crowd to exits in emergency, wall eonfigurations, and the location of emergency exits, were taken into consideration in constructing a mathematical model of a crowd. These simulation results ean be applied to the control of crowds in panic.}, + File = {Hirai1975.pdf:pdf/Hirai1975.pdf:PDF}, + Owner = {seyfried}, + Timestamp = {2012.10.26} +} diff --git a/doc/mathesisargraf/masterthesisArGr.blg b/doc/mathesisargraf/masterthesisArGr.blg deleted file mode 100644 index 89eeab06160ad8be2ef7d2f2c0bdd5173c23093e..0000000000000000000000000000000000000000 --- a/doc/mathesisargraf/masterthesisArGr.blg +++ /dev/null @@ -1,47 +0,0 @@ -This is BibTeX, Version 0.99d (TeX Live 2013/Debian) -Capacity: max_strings=35307, hash_size=35307, hash_prime=30011 -The top-level auxiliary file: masterthesisArGr.aux -I found no \citation commands---while reading file masterthesisArGr.aux -I found no \bibstyle command---while reading file masterthesisArGr.aux -You've used 0 entries, - 0 wiz_defined-function locations, - 83 strings with 506 characters, -and the built_in function-call counts, 0 in all, are: -= -- 0 -> -- 0 -< -- 0 -+ -- 0 -- -- 0 -* -- 0 -:= -- 0 -add.period$ -- 0 -call.type$ -- 0 -change.case$ -- 0 -chr.to.int$ -- 0 -cite$ -- 0 -duplicate$ -- 0 -empty$ -- 0 -format.name$ -- 0 -if$ -- 0 -int.to.chr$ -- 0 -int.to.str$ -- 0 -missing$ -- 0 -newline$ -- 0 -num.names$ -- 0 -pop$ -- 0 -preamble$ -- 0 -purify$ -- 0 -quote$ -- 0 -skip$ -- 0 -stack$ -- 0 -substring$ -- 0 -swap$ -- 0 -text.length$ -- 0 -text.prefix$ -- 0 -top$ -- 0 -type$ -- 0 -warning$ -- 0 -while$ -- 0 -width$ -- 0 -write$ -- 0 -(There were 2 error messages) diff --git a/doc/mathesisargraf/masterthesisArGr.log b/doc/mathesisargraf/masterthesisArGr.log deleted file mode 100644 index a857fd3f3b770eaa7d476ec45ae38e3f23203143..0000000000000000000000000000000000000000 --- a/doc/mathesisargraf/masterthesisArGr.log +++ /dev/null @@ -1,383 +0,0 @@ -This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/Debian) (format=pdflatex 2015.6.26) 9 SEP 2015 10:42 -entering extended mode - 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Durch die Betreuung von M.C. floss die Erfahrung zahlreicher Modelle ein und es gelang bei der Modelfindung ein geeignetes Testmodel neu zu beschreiben. -In this thesis, the effect of an alternate floor-field was analyzed, by using it in a newly composed test-model for pedestrian dynamics. In the simulation of pedestrian (crowd) movement, the routing of agents\footnote{An agent is the representation of a pedestrian in the simulation. Depending on the used model, an agent incorporates some kind of artificial intelligence or basic agent-attributes only (like size, speed attributes, etc.). In the latter case the model takes over the task of navigating agents.} is an integral part. Routing can be seen as the composition of two aspects: the global pathfinding through a geometry and the avoidance of static or dynamic obstacles\footnote{collision detection} (like walls or other agents) in a local\footnote{local in time and/or in space} situation. +In this thesis, the effect of an alternate floor-field is analyzed, by using it in a newly composed test-model for pedestrian dynamics. In simulations of pedestrian movement, the routing of agents\footnote{An agent is the representation of a pedestrian in the simulation. Depending on the used model, an agent incorporates some kind of artificial intelligence or basic agent-attributes only (like size, speed attributes, etc.). In the latter case the model takes over the task of navigating agents.} is an integral part. Routing can be seen as the composition of two aspects: the global pathfinding through a geometry and the avoidance of static or dynamic obstacles (like walls or other agents) in a local situation. -The history of pedestrian simulation shows various models with different answers to the question of navigation. Many of which make use of manually added elements\footnote{like some sort of domain-decomposition, e.g. through helplines} to solve the global pathfinding, which enable the user to simulate crowd movement in that very geometry. Other models use an algorithm, that will supply a navigation direction, calculated from the agent's current position, the destination and the geometry data. The model described by Dietrich \citep{Dietrich2014} is one of the later. It uses the solution of the Eikonal Equation (see chapter \ref{eikonalequation}), which describes a 2-D wave-propagation. The wave starts in the target region and propagates throughout the geometry. To navigate agents, they are directed in the opposite direction of the gradient of said solution of the Eikonal Equation. It is to be noted, that the solution of the Eikonal Equation can be calculated beforehand and does not contribute to the runtime of any given simulation scenario. The Routing using the plain floor-field will yield non-smooth pathways as described later. This could pose a problem for some models. Dietrich shows the existance and uniqueness of his problem-formulation by using the theorem of Picard-Lindelöf. -\footnote{Picard-Lindelöf theorem: Consider the initial value problem \newline \begin{center} -$y'(t) = f(t,y(t)), \quad y(t_0 ) = y_0, \quad t \in [t_0 - \epsilon, t_0 + \epsilon]$.\newline -\end{center} Suppose $f$ is uniformly Lipschitz continuous in y and continuous in $t$. Then, for some value $\epsilon > 0$, there exists a unique solution $y(t)$ to the initial value problem on the interval $[t_0 - \epsilon, t_0 + \epsilon]$.} -To apply this theorem, Lipschitz-continuous first derivatives of the input-functions must be given. Dietrich solves that problem by the use of a mollifier, which basically takes a locally integrable function and returns a smooth approximation. +Development of pedestrian simulation shows various models with different answers to the question of navigation. Many of which make use of manually added elements\footnote{like some sort of domain-decomposition, e.g. through helplines} to solve the global pathfinding, which enable the user to simulate crowd movement in that specific geometry. Other models use an algorithm, that will supply a navigation direction for any geometry. The \emph{Gradient Navigation Model}(GNM) described by Dietrich\citep{Dietrich2014} is one of the later. It uses the solution of the Eikonal Equation (see chapter \ref{eikonalequation}), which describes a 2-D wave-propagation. The wave starts in the target region and propagates throughout the geometry. Agents are directed in the opposite direction of the gradient of aforementioned solution of the Eikonal Equation. The Routing using the plain floor-field will yield non-smooth pathways. This could pose a problem for models, relying on a well-posed problem. Here, Dietrich shows the existance and uniqueness of a solution to his problem-formulation by using the theorem of Picard-Lindelöf. +%\footnote{Picard-Lindelöf theorem: Consider the initial value problem \newline \begin{center} +%$y'(t) = f(t,y(t)), \quad y(t_0 ) = y_0, \quad t \in [t_0 - \epsilon, t_0 + \epsilon]$.\newline +%\end{center} Suppose $f$ is uniformly Lipschitz continuous in y and continuous in $t$. Then, for some value $\epsilon > 0$, there exists a unique solution $y(t)$ to the initial value problem on the interval $[t_0 - \epsilon, t_0 + \epsilon]$.} +To apply this theorem, Lipschitz-continuous first derivatives of the input-functions must exist. Dietrich solves that problem by the use of a mollifier, which basically takes a locally integrable function and returns a smooth approximation. Thus he creates a well-posed problem. -In this thesis, a different floor-field is described, which solves above issue (non-smoothness) as a welcome side-effect. A research-group in Madrid \citep{Madrid}\footnote{Robotics Lab, Universidad Carlos III de Madrid} is working on the safe navigation of robots. They are not to follow paths, which cut corners (which come close to any obstacles). A so-called distance-field is created and used as we see later. The welcome side-effect is in \emph{smooth} pathways through the avoidance of walls and corners. The researchers take that approach even further, by reducing any geometry into a skeleton and thus having the domain in which the 2-D wave propagates reduced dramatically. Their intent is to re-calculate the floor-field in real-time using it for the reduced view-field of the robot's sensors. -Our interest in this sleight of hand is different. We welcome the smoothness of the resulting pathways and take special interest in the behavior of agents close to obstacles. The floor-field itself shows pathways, which show a wall-repulsive character in the negative gradient. This phenomenon enables us to formulate a new model, one that uses an altered floor-field. Thanks to the rich experience of Chraibi in creation and testing of pedestrian dynamics models, we follow his intuition to use that altered floor-field in a new model. The results seen in the simulations show remarkably good behavior. The model is easy to use, fast and shows superior characteristics in routing through complex geometries. The extent to which we alter the floor-field is subject to our analysis. +In this thesis, an enhanced floor-field is described, which addresses aforementioned issue (non-smoothness) as a welcome side-effect. A research-group at the Universidad Carlos III de Madrid\citep{Madrid} is working on safe navigation of robots. Since agents should not follow paths, which come close to any obstacle, a distance-field is created and used in the Fast-Marching algorithm, resulting in \emph{smooth} pathways, which favor a distance to walls. The researchers take that approach even further, by transforming any geometry into a skeleton (again using the distance-field) and thus having the domain in which the 2-D wave propagates reduced dramatically. Their intent is to re-calculate the floor-field in real-time using it for the reduced view-field of the robot's sensors. +Our interest in this sleight of hand is different. We take special interest in the behavior of agents close to obstacles. The floor-field itself yields pathways, which show a wall-repulsive character in the negative gradient. This component enables us to formulate a new model, one that uses an altered floor-field. The model is implemented in JuPedSim\cite{jupedsim}, a simulation suit for pedestrian simulation, developed at the Jülich Supercomputing Centre, Forschungszentrum Jülich GmbH. It is verified according to RiMEA tests and validated with respect to empirical data. The results seen in the simulations show remarkably good behavior. The model is easy to use, fast and shows an organic routing through complex geometries. The extent to which we alter the floor-field is subject to our analysis. \newpage \tableofcontents @@ -213,26 +213,27 @@ Our interest in this sleight of hand is different. We welcome the smoothness of \section{Pedestrian Dynamics: Introduction} %<< big picture: micor-/macroscopic models, cell automata/ODE-based, take a closer look in next chapter >> -Pedestrian Dynamics defines a field of research trying to understand the kinematic and mechanic of pedestrian crowd movement. Understanding, how crowds will react in different geometries under various circumstances, will lead to the ability to design our environment to best fit the needs of civil and security engineering. Results are applied to safely conduct large events, to create architecture (traffic infrastructure), through which large crowds can safely be moved and to optimize evacuation time in case of an emergency. -To simulate crowd behavior, many models exist with different characteristics. The history of the analysis reaches back to the 1970's to Predtetschenski and Milinkskii, as Kemloh\citep{KemlohWagoum2013} states in his dissertation. Since then, new models have been described throughout the decades. To maintain orientation, these models can be grouped into classes in the following manner, common in Pedestrian Dynamics (Chraibi, 2012)\citep{Chraibi2012}: +Pedestrian dynamics defines a field of research trying to understand the kinematic and mechanic of pedestrian crowd movement. Understanding, how crowds react in different geometries under various circumstances, enables a safe design of our environment to best fit the needs of civil and security engineering. Results are applied to safely conduct large events, to create architecture (traffic infrastructure), through which large crowds can safely be moved and to optimize evacuation time in case of an emergency. +To simulate crowd behavior, many models exist with different characteristics. The history of the analysis reaches back to the 1970's to Predtetschenski and Milinkskii\citep{Predtechenskii1971}. Since then, new models have been described throughout the decades. To maintain orientation, these models can be grouped into classes in the following manner: \begin{figure}[h!] \includegraphics[width=1.0\linewidth]{pics/organizationModels} -\caption{A possible hierarchical classification of models in pedestrian dynamics by Chraibi, 2012} +\caption{A possible hierarchical classification of models in pedestrian dynamics in \citep{Chraibi2012}} \end{figure} -\emph{Macroscopic} models describe crowd behavior without the need to characterize individuals, which make up the crowd. Aggregated values, like density or flow for instance, are used to describe the dynamic within the system. Imagine a model, which describes the change of the density over time throughout a geometry. Such a model can be mathematically captured by an PDE. The action of a single agent is neglected and it is assumed, that aggregated values are sufficient to describe the crowd behavior. Thus a crowd is seen as a continuous fluid, which can be modeled by the aggregated observables (density, speed, flow) only. No inter-particle relations are explicitly considered. Larger roadmap- and city-traffic-simulation are fields, where macroscopic models are widely spread and can supply travel times and point out bottlenecks.\citep{Emme} +\emph{Macroscopic} models tackle crowd behavior without the need to characterize individuals, which make up the crowd. The action of a single agent is neglected and it is assumed, that aggregated values are sufficient to describe the crowd behavior. Metrics, e.g. density or flow, are used to describe the dynamic within the system. Thus a crowd is seen as a continuous fluid, which can be modeled by these aggregated observables only. No inter-particle relations are explicitly considered. Given a model, which describes the change of the density throughout a geometry, it can be mathematically captured by an PDE. Larger roadmap- and city-traffic-simulation are fields, where macroscopic models are widely spread and can supply travel times and point out bottlenecks\citep{Emme}. Macroscopic models are fast but lack the ability to simulate heterogeneous groups. \emph{Microscopic} models consist of mathematical formulations describing the state and/or the interactions of every agent. Each agent has a position in the domain and interacts with its environment. It is assumed that the behavior of any crowd is only based on the sum of these individual actions. Within the model, these individual actions obviously must be far different from the attempt to model the complete, complex system of a person's psychology, which defines its motivation of movement inside a crowd. It is desirable to have few and simple equations to model the crowd's movement. Equations, that do scale as good as possible to achieve real-time capability for simulations. If one can not achieve real-time capability, at least reasonable computing time is a must-have-criteria for the model to be successful. A popular starting point origins in the modeling of the behavior of electrical charges in an electro-magnetic field. Charges of the same sign act on each other with a repelling force. This effect is used in the modeling of the natural collision-avoidance of a person to other persons, walls and obstacles in pedestrian dynamics. Agents can be modeled to react equally to charges in the electro-magnetic field of the surrounding environment. Let us add a driving force, that acts on the agent, forcing him towards its destination and we end up with a microscopic model called ``social force model''.\footnote{This simplified describtion shall be enough for this introduction. For further reference, please see corresponding literature. A comprehensive insight in SFMs is given by Chraibi's ``Validated force-based modeling of pedestrian dynamics''.\citep{Chraibi2012}} \begin{figure}[h!] \includegraphics[width=1.0\linewidth]{pics/forces_on_agent} \caption{Forces acting on agent A from: wall, obstacle and agent B} +\label{forcesOnAgent} \end{figure} We will leave the overview and set the stage for the ODE based models, for in this category our new model is to be found. -\section{ODE based, microscopic models} +\subsection{ODE based, microscopic models} %<< Explain different types (social force (2nd order), velocity based (1st order)) >> @@ -302,9 +303,33 @@ In order to keep the model simple, repulsive wall forces as seen in Social Force \subsection*{Definitions:} -The following functions will be used in the model-formulation and shall be introduced: +Let $\Omega$ be the discret set of gridpoints in the bounded domain, which holds the geometry of the simulation, a subset of $\mathbb{R}^{2}$.The following functions will be used in the model-formulation and shall be introduced: -\vspace*{1cm} +\[ d : \Omega \ni \vec{x} \longrightarrow d(\vec{x}) \in \mathbb{R}, \qquad \Omega \subseteq \mathbb{R}^{2} \] +The function $d$ assigns to each gridpoint in $\Omega$ the distance to the closest wall. It will be used to choose, in which mode the movement-vector candidate will be altered. + +\[ P : \mathbb{R}^2 \times \Omega \ni (\vec{v}, \vec{x}) \longrightarrow P(\vec{v}, \vec{x}) \in \mathbb{R}^2 \] +The funciton $P$ describes the orthogonal projection of a given orientation $\vec{v}$ onto the closest wall of $\vec{x}$. It yields an orientation parallel to that wall. + +\[ v_{ff} : \Omega \ni \vec{x} \longrightarrow v_{ff}(\vec{x}) = \vec{v}_{ff} \in \mathbb{R}^{2}|_{\left\lVert \cdot \right\rVert_2 = 1} \] +The vector $\vec{v}_{ff}$ is the (normalized) negative gradient of the floor-field at position $\vec{x}$. This vector describes the direction of the negative gradient only and always has unit length. It is used to contribute to an agent's moving vector. + +\[ g : \mathbb{R}^2 \ni \vec{v} \longrightarrow g(\vec{v}) \in \mathbb{R}^{2}|_{\left\lVert \cdot \right\rVert_2 \leq 1} \] +The function $g$ limits the length of any inputvector $\vec{v}$ to unit length. If $\left\lVert \vec{v} \right\rVert_2 \leq 1$, then $g$ is the identity of $\vec{v}$. + +\[ \sum_{i=1}^{n}\vec{v}_{repP,i} : \overset{n}{\underset{i=1}{\otimes}} \mathbb{R}^2_i \ni (\vec{v}_1, ... \vec{v}_n) \longrightarrow \sum_{i=1}^{n}\vec{v}_{repP,i} \in \mathbb{R}^{2} \] +The sum is adding the distance-dependand\footnote{The correlation of distance and force is the Gompertz-function. The function is smooth and adjustable in scale, range of influence and steepness.} repelling force of neighboring agents within a close vicinity. This sum is analog to the SFM model, yet the resulting direction is directly taken and not converted to an acceleration nor to a velocity. The letter $v$ is a hint to being a vector, \emph{not} to being a velocity. + +%\begin{tabular}{lllclll} +%$ d $ & : $\Omega $ & $ \ni \vec{x} $ & $ \longrightarrow $ & $ d(\vec{x}) $ & $\in \mathbb{R} \quad $ & $:= \quad $ distance to the closest wall \\ +%$ P $ & : $\mathbb{R}^2 \times \Omega $ & $ \ni (\vec{v}, \vec{x}) $ & $ \longrightarrow $ & $ P(\vec{v}, \vec{x}) $ & $\in \mathbb{R}^2 \quad $ & $ := \quad $ orth. proj. of $\vec{v}$ onto closest wall of $\vec{x}$ \\ +%$ v_{ff} $ & : $\Omega $ & $ \ni \vec{x} $ & $ \longrightarrow $ & $ v_{ff}(\vec{x}) = \vec{v}_{ff} $ & $ \in \mathbb{R}^2 \quad $ & $:= \quad $ floor-field at position $\vec{x}$ \\ +%$ g $ & : $\mathbb{R}^2 $ & $ \ni \vec{v} $ & $ \longrightarrow $ & $ g(\vec{v}) $ & $ \in \mathbb{S}^2 $ & $:= \quad $ proj. onto the unit-sphere in $\mathbb{R}^2$ \\ +%\end{tabular} + + +% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % +%\vspace*{1cm} \begin{align*} &\textbf{TEST Model:}\\ @@ -314,10 +339,10 @@ The following functions will be used in the model-formulation and shall be intro &\vec{v}_{n, res}&&= \left\{ \begin{aligned} %&\left(1-\frac{1}{2}\left[(\hat{v}\cdot(-\nabla \hat{d}))+\vert(\hat{v}\cdot(-\nabla \hat{d}))\vert\right]\right) & P(\vec{v}_{n}) \quad &: & d(\vec{x})&& < 0.1\\ - &\left(1-\frac{1}{2}\left[\langle\vec{v}_{n},-\nabla \hat{d}\rangle+\vert\langle\vec{v}_{n},-\nabla \hat{d}\rangle\vert\right]\right) & P(\vec{v}_{n}) \quad &: & d(\vec{x})&& < 0.1\\ + &\left(1-\langle\vec{v}_{n},-\nabla \hat{d}\rangle\right) & P(\vec{v}_{n}) \quad & \textbf{if} \quad && d(\vec{x}) < && 0.1; \quad \langle\vec{v}_{n},-\nabla \hat{d}\rangle \ge 0 \\ %&\left(1-\frac{1}{2}\left[(\hat{v}\cdot(-\nabla \hat{d}))+\vert(\hat{v}\cdot(-\nabla \hat{d}))\vert\right]\right) & \vec{v}_{n} \quad &: \quad 0.1 < & d(\vec{x})&& < 0.2\\ - &\left(1-\frac{1}{2}\left[\langle\vec{v}_{n},-\nabla \hat{d}\rangle+\vert\langle\vec{v}_{n},-\nabla \hat{d}\rangle\vert\right]\right) & \vec{v}_{n} \quad &: \quad 0.1 < & d(\vec{x})&& < 0.2\\ - & &\vec{v}_{n} \quad &: & d(\vec{x})&& > 0.2 + &\left(1-\langle\vec{v}_{n},-\nabla \hat{d}\rangle\right) & \vec{v}_{n} \quad & \textbf{if} \quad 0.1 < && d(\vec{x}) < && 0.2 ;\quad \langle\vec{v}_{n},-\nabla \hat{d}\rangle \ge 0\\ + & &\vec{v}_{n} \quad & \textbf{else} && &&%& d(\vec{x})&& > 0.2 \end{aligned} \right.\\ %&\vec{v}_{n, res}&&= \begin{cases} @@ -328,19 +353,9 @@ The following functions will be used in the model-formulation and shall be intro % \left(1-\frac{1}{2}\left[(\hat{v}\cdot(-\nabla \hat{d}))+\vert(\hat{v}\cdot(-\nabla \hat{d}))\vert\right]\right)\cdot\vec{v}_{n} & \quad : \quad d(\vec{x}) < 0.2,\\ % \vec{v}_{n} & \quad : \quad d(\vec{x}) > 0.2 % \end{cases}\\ -&\vec{v}_{n}&&= 0.8\cdot\vec{v}_{n-1, res} + 0.2\cdot g\left(g(\vec{v}_{ff})+g(\sum\vec{v}_{repP,i})\right) +&\vec{v}_{n}&&= 0.8\cdot\vec{v}_{n-1, res} + 0.2\cdot g\left(g(\vec{v}_{ff})+g(\sum_{i=1}^n\vec{v}_{repP,i})\right) \end{align*} -\begin{align*} -\textbf{Definitions:} \\ -\end{align*} -\begin{tabular}{lllclll} -$ d $ & : $\Omega $ & $ \ni \vec{x} $ & $ \longrightarrow $ & $ d(\vec{x}) $ & $\in \mathbb{R} \quad $ & $:= \quad $ distance to the closest wall \\ -$ P $ & : $\mathbb{R}^2 \times \Omega $ & $ \ni (\vec{v}, \vec{x}) $ & $ \longrightarrow $ & $ P(\vec{v}, \vec{x}) $ & $\in \mathbb{R}^2 \quad $ & $ := \quad $ orth. proj. of $\vec{v}$ onto closest wall of $\vec{x}$ \\ -$ v_{ff} $ & : $\Omega $ & $ \ni \vec{x} $ & $ \longrightarrow $ & $ v_{ff}(\vec{x}) = \vec{v}_{ff} $ & $ \in \mathbb{R}^2 \quad $ & $:= \quad $ floor-field at position $\vec{x}$ \\ -$ g $ & : $\mathbb{R}^2 $ & $ \ni \vec{v} $ & $ \longrightarrow $ & $ g(\vec{v}) $ & $ \in \mathbb{S}^2 $ & $:= \quad $ proj. onto the unit-sphere in $\mathbb{R}^2$ \\ -\end{tabular} - \vspace*{2cm} %\begin{alignat}{1} %\Delta\vec{x}\quad=\quad & \Delta t\cdot\vec{v}_{res}\\ @@ -509,11 +524,15 @@ The trajectories improved and look much more natural. This was achieved without %$\vec{v}_{res}\quad=\quad\bigg(1-\frac{1}{2}\bigg[\langle\vec{v}_{n},(-\nabla distances)_{n}\rangle+\big\vert\langle\vec{v}_{n},(-\nabla distances)_{n}\rangle\big\vert\bigg]\bigg)\cdot\vec{v}_{n}$ \newpage -\section{Validation} +\section{Verification and Validation} +In this chapter, we want to describe applicable tests to analyze the effect of the \emph{wall-avoid-distance} parameter to the output of the TEST-model. It shall be verified, that we see no overlapping and that the crowd shows typical phenomena, like row-formation. Following our set of tests, we will then validate the TEST-model to produce the pedestrian flow in bottleneck experiments, corresponding to empirical data. We will further compare the trajectories with a different model, one using automated triangulation to assist routing. The last test is a mere stress test to the model. We want to see how the model performes in an evacuation setting with a large, crowded geometry. + \subsection{Basic Tests} -The validation of a model is a complex task and makes up a separate research field. Researchers in the field of Civil Engineering are working on various approaches on how to validate a model. The research group, CST - Pedestrian Dynamics and Traffic Simulation, is developing a set of test-cases any serious model should aim to pass. Tests include the behavior of a single moving agent passing static objects like a dummy agent or an obstacle. (see figure \ref{test01}) +%The validation of a model is a complex task and makes up a separate research field. Researchers in the field of Civil Engineering are working on various approaches on how to validate a model. The research group, CST - Pedestrian Dynamics and Traffic Simulation, is developing a set of test-cases any serious model should aim to pass. Tests include the behavior of a single moving agent passing static objects like a dummy agent or an obstacle. (see figure \ref{test01}) + +%The TEST-model, implemented in its current state in the simulation-suit JuPedSim\citep{jupedsim}, passed applicable tests and it was shown, that the basic mechanics of the routing are working as specified. Yet to complete validation of the model, the floorfield needs to support multiple destinations. + -The TEST-model, implemented in the simulation-suit JuPedSim\citep{jupedsim}, passed these tests and it was shown, that the basic mechanics of the routing are working as specified. \begin{figure}[h!] \includegraphics[width=1.\textwidth]{pics/Testcases01} @@ -531,9 +550,11 @@ The TEST-model, implemented in the simulation-suit JuPedSim\citep{jupedsim}, pas \subsection{Variation of the Parameter} -In this thesis, we take special interest in the following testcase, as it validates the model to yield a fundamental diagram like seen in real world experiments: +We take special interest in the following test-case, as it validates the model to yield a fundamental diagram like seen in real world experiments: The tests demands to simulate a bottleneck experiment several times. This is to be repeated for various bottleneck-widths. The calculated flow through the bottleneck shall match the empirical data. -We have conducted this test for various \emph{wall-avoid-distance} parameter and want to analyze the change in the flow. It can be seen, that the more the agents will gear to avoid obstacles and walls, the lesser the flow will become (see \ref{funddiag}). This result is expectable. It was also shown, that without the enhancement of the floor-field, few agents did not pass the bottleneck, but got caught inside of walls. +We have conducted this test for various \emph{wall-avoid-distance} parameter and want to analyze the change in the flow. It can be seen, that the more the agents will gear to avoid obstacles and walls, the lesser the flow will become (see \ref{funddiag}). This result is expectable. It was also shown, that without the enhancement of the floor-field, few agents did \emph{not} pass the bottleneck, but got caught inside of walls. + + \begin{figure}[h!] \includegraphics[width=1.\textwidth]{pics/sim_flow_vs_experimental_data}