Modeling the Interactions of Pedestrians

When pedestrians are moving in a crowd they interact with each other. In order to simulate these intaractions models need to be developed.

Models in pedestrian dynamics describe the movement of single pedestrians, small groups, or the assembling and dispersal of large gatherings, which could happen, e.g., outdoor in a city or inside of buildings. With respect to their nature, models are often classified, in a generic way, as microscopic or macroscopic.

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Different possible combinations of characteristics of the microscopic pedestrian models. (macroscopic models present similar classification.)

Models are developed in natural science, computer science, engineering, social sciences, or psychology. The scope and the type of the models are closely related to the scientific area or the application and thus are diverse. For instance, in physics, pedestrians are treated as driven particles to study the emergence of collective phenomena like stop-and-go waves, lane formation, or clogging and intermittent flow at bottlenecks. The objective of minimal modelling approaches is to identify basic microscopic mechanisms and types of interaction that are relevant for the emergence of the observed collective phenomena.

In traffic engineering however, models focus on transport characteristics like flow, density, speed, or route choices. In fire safety science, the process of building evacuation including behavioral factors like the reaction time or the influence of smoke and fire is addressed. In all cases, it remains clear that models support planners in the design and dimensioning of pedestrian facilities in dense cities, high occupancy buildings, infrastructures for public transport, or large-scale events outside.

Acceleration-based Models

A popular class of microscopic models for pedestrian dynamics is defined by the so-called acceleration-based, also known as force-based models, where the movement of pedestrians is defined by a superposition of exterior forces. Given initial values of the state variables, the resulting second-order ordinary differential equations can be solved numerically to obtain the trajectories of pedestrians. In 1975 Hirai and Tarui proposed the first known microscopic force-based model to investigate the movement of pedestrians in a 2D space.

Velocity-Based Models

In contrast to acceleration-based models, the velocity in velocity-based models is instantaneously adjusted to the neighborhood and the environment with no inertia or an implicit implementation of a reaction time. Such a modelling approach is largely inspired by motion planning in robotics, since robots move with almost no inertia and react instantaneously to a command. Constraints on the velocity in case of contact allow to model hard-core body exclusion (volume exclusion) and to describe collision-free pedestrian dynamics with no overlapping.

This publication gives a general overview of mathematical modeling in pedestrian dynamics.

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Last Modified: 14.06.2022