Avoiding numerical pitfalls in social force models

Abstract

The social force model of Helbing and Molnár is one of the best known approaches to simulate pedestrian motion, a collective phenomenon with nonlinear dynamics. It is based on the idea that the Newtonian laws of motion mostly carry over to pedestrian motion so that human trajectories can be computed by solving a set of ordinary differential equations for velocity and acceleration. The beauty and simplicity of this ansatz are strong reasons for its wide spread. However, the numerical implementation is not without pitfalls. Oscillations, collisions, and instabilities occur even for very small step sizes. Classic solution ideas from molecular dynamics do not apply to the problem because the system is not Hamiltonian despite its source of inspiration. Looking at the model through the eyes of a mathematician, however, we realize that the right hand side of the differential equation is nondifferentiable and even discontinuous at critical locations. This produces undesirable behavior in the exact solution and, at best, severe loss of accuracy in efficient numerical schemes even in short range simulations. We suggest a very simple mollified version of the social force model that conserves the desired dynamic properties of the original many-body system but elegantly and cost efficiently resolves several of the issues concerning stability and numerical resolution.

Vadere: An Open-Source Simulation Framework to Promote Interdisciplinary Understanding

“…The best known and analyzed force-based model is Helbing's and Molnár's social force model (SFM) from 1995 [34]. Extensions were proposed by Helbing and his group [40,55] as well as others [85,52,59,58,9,47]. Agents are treated like particles that are accelerated or decelerated by "social" forces taking Newton's second law of dynamics as a guiding principle.…”

“…The algorithmic formulation, on the other hand, is through a numerical solver, such as Euler's method, which may -or may not -converge to the solution of the ODE. See [47] for numerical issues. Another example is the use of different pseudo random number generators with different starting seeds.…”

The Superposition Principle: A Conceptual Perspective on Pedestrian Stream Simulations

“…As in the gradient navigation model, discretisation is an issue of numerically solving the ordinary differential equations resulting from the model's formulation (also see [45]). Since the velocity is only manipulated indirectly through the acceleration, force-based models can be seen as principally different to the other models.…”

“…Force-based models and the gradient navigation model build on ordinary differential equations since they are formulated as the first or second derivative of agents' positions, which can lead to numerical challenges that have to be dealt with [45]. In deterministic cellular automata and the optimal steps model, a numerical optimisation scheme has to be employed.…”

“…The introduction of psychological or physiological aspects, such as a field of vision, can lead to discontinuities in the scalar field over time. These discontinuities can be a problem for models based on differential equations and some optimisation schemes [44,45]. On the other hand, extensions developed for one model based on scalar fields can easily be adopted for other models within the superposition paradigm.…”