Time-To-Collision Models for Single-File Pedestrian Motion

Abstract

We apply the concept of time-to-collision (TTC) to the modeling of pedestrian dynamics. The TTC combines the spatial distances with the velocities to quantify the 'distance' to a collision. Therefore, it is a promising candidate for modeling the interactions between pedestrians. Empirical studies also indicate that the interaction between pedestrians can be described by the TTC: While the pair distribution of the distances, i.e. the probability of two pedestrians to have a certain spatial distance, was found to strongly depend on the relative velocity, the TTC accurately parametrizes its pair distribution. However, there are still few pedestrian models that use the TTC. After giving a general definition of the TTC, we present the widely used approximations for its calculation, especially in a one-dimensional setting. Combined with a desired time-gap, these give rise to different models, namely an Optimal-Velocity model and a new Time-to-Collision model. The TTC model exhibits, however, generic inconsistencies which are related to the estimates we use to approximate the speed of the predecessor. The estimates have a large impact on the dynamics and must therefore be interpreted as reflecting the pedestrians behavior, i.e. as anticipation strategies. We propose new estimates for the predecessor's speed. These give rise to a rich family of models based on the TTC which are analyzed by means of linear stability analysis and simulations.