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


  • Jakob Cordes Civil Safety Research, Forschungszentrum Jülich, Jülich, Germany and Institute for Theoretical Physics, University of Cologne, Cologne, Germany
  • Mohcine Chraibi Civil Safety Research, Forschungszentrum Jülich, Jülich, Germany
  • Antoine Tordeux Institute for Security Technology, University of Wuppertal, Wuppertal, Germany
  • Andreas Schadschneider Institute for Theoretical Physics, University of Cologne, Cologne, Germany




pedestrian dynamics, optimal velocity models, fundamental diagram, single-file motion, time-to-collision


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.


Nicolas, A., Kuperman, M., Ibañez, S., Bouzat, S., Appert-Rolland, C.: Mechanical response of dense pedestrian crowds to the crossing of intruders. Scientific Reports 9(1), 105 (2019). doi:10.1038/s41598-018-36711-7

Helbing, D., Molnár, P.: Social force model for pedestrian dynamics. Phys. Rev. E 51, 4282-4286 (1995). doi:10.1103/PhysRevE.51.4282

Burstedde, C., Klauck, K., Schadschneider, A., Zittartz, J.: Simulation of pedestrian dynamics using a two-dimensional cellular automaton. Physica A 295, 507-525 (2001). doi:10.1016/S0378-4371(01)00141-8

Bando, M., Hasebe, K., Nakayama, A., Shibata, A., Sugiyama, Y.: Dynamical model of traffic congestion and numerical simulation. Phys. Rev. E 51(2), 1035-1042 (1995). doi:10.1103/PhysRevE.51.1035

Nakayama, A., Hasebe, K., Sugiyama, Y.: Instability of pedestrian flow and phase structure in a two-dimensional optimal velocity model. Physical Review E 71(3), 036121 (2005). doi:10.1103/PhysRevE.71.036121

Karamouzas, I., Skinner, B., Guy, S.J.: A universal power law governing pedestrian interactions. Phys. Rev. Lett. 113(5), 238701 (2014). doi:10.1103/PhysRevLett.113.238701

Moussaïd, M., Helbing, D., Theraulaz, G.: How simple rules determine pedestrian behavior and crowd disasters. PNAS 108(17), 6884-6888 (2011). doi:10.1073/pnas.1016507108

Xiao, Y., Gao, Z., Qu, Y., Li, X.: A pedestrian flow model considering the impact of local density: Voronoi diagram based heuristics approach. Transportation Research Part C: Emerging Technologies 68, 566-580 (2016). doi:10.1016/j.trc.2016.05.012

Asano, M., Iryo, T., Kuwahara, M.: Microscopic pedestrian simulation model combined with a tactical model for route choice behaviour. Transport Res. C–Emer. 18(6), 842-855 (2010). doi:10.1016/j.trc.2010.01.005

von Krüchten, C.: Development of a cognitive and decision-based model for pedestrian dynamics. Ph.D. thesis, Universität zu Köln, Köln (2019)

Pipes, L.A.: An operational analysis of traffic dynamics. J. Appl. Phys. 24(3), 274 - 281 (1953). doi:10.1063/1.1721265

Newell, G.F.: Nonlinear Effects in the Dynamics of Car Following. Operations Research 9(2), 209-229 (1961). doi:10.1287/opre.9.2.209

Jiang, R., Wu, Q., Zhu, Z.: Full velocity difference model for a car-following theory. Physical Review E 64(1), 017101 (2001). doi:10.1103/PhysRevE.64.017101

Lassarre, S., Roussignol, M., Tordeux, A.: Linear stability analysis of first-order delayed car-following models on a ring. Physical Review E 86, 036207 (2012). doi:10.1103/PhysRevE.86.036207

Steffen, B., Seyfried, A.: Methods for measuring pedestrian density, flow, speed and direction with minimal scatter. Physica A 389(9), 1902-1910 (2010). doi:10.1016/j.physa.2009.12.015

Tordeux, A., Chraibi, M., Schadschneider, A., Seyfried, A.: Influence of the number of predecessors in interaction within acceleration-based flow models. Journal of Physics A: Mathematical and Theoretical 50(34), 345102 (2017). doi:10.1088/1751-8121/aa7fca

Seyfried, A., Portz, A., Schadschneider, A.: Phase coexistence in congested states of pedestrian dynamics. In: Bandini, S., Manzoni, S., Umeo, H., Vizzari, G. (eds.) Cellular Automata, Lecture Notes in Computer Science, vol. 6350, pp. 496-505. Springer-Verlag Berlin Heidelberg (2010). doi:10.1007/978-3-642-15979-4_53




How to Cite

Cordes, J., Chraibi, M., Tordeux, A., & Schadschneider, A. (2022). Time-To-Collision Models for Single-File Pedestrian Motion. Collective Dynamics, 6, 1–10. https://doi.org/10.17815/CD.2021.133



Pedestrian and Evacuation Dynamics 2021