Implementation Issues of Force Based Pedestrian Motion Models

Forced based models in the form of ordinary differential equations (ODE), such as the social force model, are among the best known approaches to simulating pedestrian flow. They adopt 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 ODEs for velocity and acceleration. The models are widely spread in science and application. Nevertheless, oscillations, collisions, and instabilities occur even for small step sizes. We identify some mathematical properties at the root of the problem: The right hand side of the differential equation may be non-differentiable and discontinuous at target locations. This produces undesirable behavior in the solution and severe loss of accuracy in efficient numerical schemes. Using the social force model as an example, we propose a very simple mollification so that the dynamic properties of the original many-body system are conserved. This elegantly and cost-efficiently resolves several of the issues concerning stability and numerical resolution. On the other hand, we show that it is insufficient to remove the typical but undesirable circular movement of pedestrians moving towards a target.