Gerta Köster

On modelling the influence of group formations in a crowd

Inspired by doubts from social scientists on the validity of computer models that see a crowd as a pure aggregation of individuals, we develop a mathematical model for group formation within crowds. It is based on a few simple characteristics. Most importantly, small groups stick together as they thread their way through a crowd. Additionally, groups have a tendency to walk abreast to ease communication. Through simulation, we establish that the occurrence of groups significantly impacts crowd movement, namely evacuation times. Further, we complement and validate the simulations by a small experiment: a classroom egress. The simulation results match the measurements qualitatively. We get a good quantitative match after calibrating the supposed desire to communicate while walking—and hence to walk abreast. We conclude that it is one of the crucial parameters to calibrate the group model against reality. While working on a mathematically complete model, a new gap between the mathematical modelling and the social sciences emerged: some model assumptions are based on the modellers intuition rather than on sociological or psychological insight validated by the scientific community. We hope the findings—and resulting suggestions—will in return inspire new cooperation between the disciplines.

Dynamic stride length adaptation according to utility and personal space

Pedestrians adjust both speed and stride length when they navigate difficult situations such as tight corners or dense crowds. They try to avoid collisions and to preserve their personal space. State-of-the-art pedestrian motion models automatically reduce speed in dense crowds simply because there is no space where the pedestrians could go. The stride length and its correct adaptation, however, are rarely considered. This leads to artefacts that impact macroscopic observation parameters such as densities in front of bottlenecks and, through this, flow. Hence modelling stride adaptation is important to increase the predictive power of pedestrian models. To achieve this we reformulate the problem as an optimisation problem on a disk around the pedestrian. Each pedestrian seeks the position that is most attractive in a sense of balanced goals between the search for targets, the need for individual space and the need to keep a distance from obstacles. The need for space is modelled according to findings from psychology defining zones around a person that, when invaded, cause unease. The result is a fully automatic adjustment that allows calibration through meaningful social parameters and that gives visually natural results with an excellent fit to measured experimental data.

Microscopic Pedestrian Simulations: From Passenger Exchange Times to Regional Evacuation

Pedestrian dynamics play an important role in diverse fields of application such as optimizing traffic logistics, e.g. the optimization of passenger exchange times, or egress planning of buildings, infrastructures and even whole regions. Quantitative predictions of pedestrian dynamics, namely of egress times, is an essential part of optimizing pedestrian flows. To obtain quantitative results simulations must be as realistic as possible. Here, we present a new microscopic pedestrian simulator based on a cellular automaton. It reduces discretization artifacts as typically observed in cellular automaton models to a minimum without loosing their efficiency. It reliably captures typical crowd phenomena and can simulate up to 50000 pedestrians in real time.

Gradient navigation model for pedestrian dynamics

We present a microscopic ordinary differential equation (ODE)-based model for pedestrian dynamics: the gradient navigation model. The model uses a superposition of gradients of distance functions to directly change the direction of the velocity vector. The velocity is then integrated to obtain the location. The approach differs fundamentally from force-based models needing only three equations to derive the ODE system, as opposed to four in, e.g., the social force model. Also, as a result, pedestrians are no longer subject to inertia. Several other advantages ensue: Model-induced oscillations are avoided completely since no actual forces are present. The derivatives in the equations of motion are smooth and therefore allow the use of fast and accurate high-order numerical integrators. At the same time, the existence and uniqueness of the solution to the ODE system follow almost directly from the smoothness properties. In addition, we introduce a method to calibrate parameters by theoretical arguments based on empirically validated assumptions rather than by numerical tests. These parameters, combined with the accurate integration, yield simulation results with no collisions of pedestrians. Several empirically observed system phenomena emerge without the need to recalibrate the parameter set for each scenario: obstacle avoidance, lane formation, stop-and-go waves, and congestion at bottlenecks. The density evolution in the latter is shown to be quantitatively close to controlled experiments. Likewise, we observe a dependence of the crowd velocity on the local density that compares well with benchmark fundamental diagrams.

A Sandwich Approach for Evacuation Time Bounds

In this article, we propose a novel modeling approach – the sandwich approach – to deal with evacuation time bounds (ETB) - in which lower and upper bounds for the evacuation time are calculated. A provable lower bound is achieved by computing a quickest flow, using a dynamic network flow model, an upper bound is obtained via simulation using a cellular automaton model. Coherence between the macroscopic network flow and the microscopic simulation model will be discussed. In order to validate our theoretical results, we report on our practical experiences with the Betzenberg, the region containing the Fritz-Walter soccer stadium in Kaiserslautern, Germany.

Towards the calibration of pedestrian stream models

Every year people die at mass events when the crowd gets out of control. Urbanization and the increasing popularity of mass events, from soccer games to religious celebrations, enforce this trend. Thus, there is a strong need to gain better control over crowd behavior. Simulation of pedestrian streams can help to achieve this goal. In order to be useful, crowd simulations must correctly reproduce real crowd behavior. This usually depends on the actual situation and a number of socio-cultural parameters. In other words, what ever model we come up with, it must be calibrated. Fundamental diagrams capture a large number of the socio-cultural characteristics in a very simple concept. In this paper we represent a method to calibrate a pedestrian stream simulation tool so that it can reproduce arbitrary fundamental diagram with high accuracy. That is, it correctly reproduces a given dependency of pedestrian speed on the crowd density. We demonstrate the functionality of the method with a cellular automaton model.

Validation of Crowd Models Including Social Groups

The development of group models within models of pedestrian motion has recently become a new focus of research. This interest was triggered by insight from the social sciences: Small groups often dominate the crowd at large events and the need to associate with family and friends may dominate over flight instincts. It is therefore desirable that crowd simulators adopt the new group models to better mitigate risks for example at large events or at public infrastructures. However, to make this feasible reliable validation tests must be made available. Developers and users alike should be able to check whether the adopted model indeed captures the essential characteristics of a crowd composed of subgroups. As a desirable side effect, common validation tests would make simulation tools easier to compare and their range of application easier to assess. This can help to ensure a minimum quality standard and thus to further mitigate risks. In this paper we suggest basic visual tests and some quantitative test were data is available.

How update schemes influence crowd simulations

Time discretization is a key modeling aspect of dynamic computer simulations. In current pedestrian motion models based on discrete events, e.g. cellular automata and the Optimal Steps Model, fixed-order sequential updates and shuffle updates are prevalent. We propose to use event-driven updates that process events in the order they occur, and thus better match natural movement. In addition, we present a parallel update with collision detection and resolution for situations where computational speed is crucial. Two simulation studies serve to demonstrate the practical impact of the choice of update scheme. Not only do density-speed relations differ, but there is a statistically significant effect on evacuation times. Fixed-order sequential and random shuffle updates with a short update period come close to event-driven updates. The parallel update scheme overestimates evacuation times. All schemes can be employed for arbitrary simulation models with discrete events, such as car traffic or animal behavior.

Bridging the gap: From cellular automata to differential equation models for pedestrian dynamics

Abstract Cellular automata (CA) and ordinary differential equation (ODE) models compete for dominance in microscopic pedestrian dynamics. There are two major differences: movement in a CA is restricted to a grid and navigation is achieved by moving directly in the desired direction. Force based ODE models operate in continuous space and navigation is computed indirectly through the acceleration vector. We present the Optimal Steps Model and the Gradient Navigation Model, which produce trajectories similar to each other. Both are grid-free and free of oscillations, leading to the conclusion that the two major differences are also the two major weaknesses of the older models.

Bidirectional Coupling of Macroscopic and Microscopic Approaches for Pedestrian Behavior Prediction

We combine a macroscopic and a microscopic model of pedestrian dynamics with a bidirectional coupling technique to obtain realistic predictions for evacuation times. While the macroscopic model is derived from dynamic network flow theory, the microscopic model is based on a cellular automaton. Output from each model is fed into the other, thus establishing a control cycle. As a result, the gap between the evacuation times computed by both models is narrowed down: the microscopic approach benefits from route optimization resulting in lower evacuation times. The network flow approach is enriched by including data of microscopic pedestrian behavior, thus reducing the underestimation of evacuation times.