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.
Highlights
The behavior of humans in a crowd is generally complex and gives rise to numerous phenomena
In the Optimal Velocity model (5), from the time-to-collision point of view, all pedestrians assume that the preceding pedestrian will stop at any time
The approximations used have a natural interpretation as different types of anticipation used by the pedestrians. Since both approximations correspond to bad or unrealistic anticipation strategies, one should look for better strategies to estimate the time-to-collision
Summary
The behavior of humans in a crowd is generally complex and gives rise to numerous (collective) phenomena. The pair distribution of the distances, i.e. the probability of two pedestrians to have a certain spatial distance, calculated from a large collection of data, was found to strongly depend on the relative velocity while the time-to-collision accurately parametrizes its pair distribution [6]. In [10] the underlying problems of diverging times-tocollision and unrealistic high flows are briefly discussed In this contribution, it is investigated how the TTC can be used as a fundamental quantity in (minimal) pedestrian models. The time-to-collision is defined and calculated using different approximations These give rise to different models, the well-known Optimal-Velocity models and a new class of Time-to-Collision models. Fundamental problems of these are discussed and related to the implicitly assumed anticipation strategies.
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