A vision-based model of Pedestrian Dynamics

Since the pioneering work of Helbing and collaborators on the Social Force Model [1,2] it has been customary in the field of Pedestrians Dynamics to represent a crowd as a system of many classical particles interacting by means of real forces of contact type and fictitious «social forces» acting at a distance.

There are, however, certain conceptual shortcomings connected with a force-based model, the most obvious being that social forces do not exist in reality. In addition, there is no a-priori reason to assume that social interactions between pedestrians can be combined as vector-like quantities.

For these reasons, an alternative approach based on cognitive heuristics has been proposed in the past decade by Moussaïd and collaborators [3,4]. From this «behavioural» perspective, the questions to be answered at the modelling stage are:

  • What are the goals of a pedestrian?
  • What is the nature of the information used by pedestrians?
  • How is this information processed and translated into actions?

Following this latter approach I developed a model of Pedestrian Dynamics based on three fundamental assumptions:

  • The ideal goal for a pedestrian is to reach his destination in the least possible time without experiencing intrusions of his own personal space.
  • Humans rely primarily on their vision in order to obtain information concerning the surrounding environment.
  • This information is employed in order to anticipate the actions of other pedestrians with the purpose of avoiding potentially dangerous collisions.

Outline of the model

In abridged notation, the equations of motion for a single pedestrian read:

                                                                  

The two terms on the right-hand side correspond respectively to social and physical interactions. The social acceleration as tends to adapt the instantaneous velocity after a desired velocity vd, while the term Fc takes contact forces into account.

Social interactions

Social interactions between pedestrians are treated by means of an explicit collision avoidance mechanism. As shown in Figure 1, each pedestrian extrapolates the future position of its visible neighbours at a suitably defined interaction time.

The predicted positions are employed to compute a danger function θ↦D(θ) which encodes the two competing tendency of pedestrians to avoid collisions and to minimise detours. In a sense, the value D(θ) represents the likelihood of an incoming collision in the direction specified by the angle θ in the visual field.

The new desired direction of motion is then determined by a minimisation of the danger function. The new desired walking speed depends instead on the distance to the next incoming collision.

Physical interactions

Pedestrians experience strong repulsion forces whenever they collide against each other or against a wall. In the present model contact forces are decomposed into a normal conservative reaction, a normal dissipation and a tangential friction.

Altogether, these physical forces preclude the existence of any meaningful integral of the motion such as those of energy and momentum.

Integration of the equations of motion

The equations of motion have been integrated using a modified multi-step Velocity Verlet algorithm which takes advantage of the large difference between the characteristic time scales of social and physical interactions.

Figure 1L - Extrapolation of future positions. Black arrows represent the current velocities of pedestrians. The blue arrow represents the new desired direction of motion.

Figure 1R - Danger function corresponding to the blue pedestrian.

Results

The validity of the model has been tested in a series of simulations ranging over a variety of scenarios and configurations. The results show that the model is able to reproduce the main phenomena observed in real crowds at moderate and high densities. These phenomena may be individual, such as the ability of a single pedestrian to find a path in a crowded environment, or collective, such as the formation of lanes in bidirectional and intersecting flows of pedestrians. However, the behaviour of the model is not entirely satisfactory at very high densities, and further work is needed in this direction.

Example 1: Lane formation in bidirectional flows

It is well-known from everyday experience that two groups of pedestrians walking in opposite directions tend to form segregated lanes in order to minimise the likelihood of head-on collisions [1,5].

Figure 2 shows an enlargement from a simulation performed in a 4 m wide corridor at an average density of 1 pedestrian per square metre. The scalar field displayed in the figure is the component of the local flux of pedestrian along the corridor main axis. It is possible to observe a clear separation between regions of positive and negative flux, corresponding respectively to pedestrians walking in the rightward and leftward directions.

Figure 2 - Lane formation in a 4 m wide corridor. The local flux of pedestrians (see Appendix) is measured in pedestrian per metre per second.

Example 2: Intersection of two groups

The formation of lanes is not unique to bidirectional flows in corridors. A similar phenomenon is also observed between two groups of pedestrians intersecting at right angles. A simple argument based on Galilean relativity shows that in the present case the resulting lanes are inclined at an angle of 45° with respect to the direction of motion, as illustrated in Figure 3.

Figure 3 - Intersection of two groups of pedestrians at right angles.

Example 3: Intersection of four groups

The intersection of four groups of pedestrians at right angles is significantly more complicated than the case considered in the previous example. In this scenario no lanes can be observed, and the resulting flow appears to be quite heterogeneous.

On the other hand, the field lines of the local flux of pedestrians show characteristic vortex-like structures which are continuously created and dissipated, as illustrated in Figure 4. This means that the members of the four groups are temporarily able to self-organize their motion into an evanescent «roundabout» pattern, thus increasing the efficiency of the flow [5].

Figure 4 - Intersection of four groups of pedestrians at right angles. The blue arrows represent the local flux of pedestrians (see Appendix).

Example 4: Bottleneck flow

Another familiar phenomenon observed in everyday life is the formation of clogs in the proximity of entrances and exits. Still, the flow of pedestrians through a bottleneck appears to be quite a divisive topic within the research community.

For example, it has been common wisdom during the last two decades to assume that an obstacle placed in front of an exit door has the effect of increasing the flux rate of pedestrians [2]. However, the existence of this peculiar effect has recently been questioned [6].

A further topic of debate is whether the flux rate through a bottleneck increases with its width in a stepwise or in a linear fashion [7,8].

As illustrated in Figures 5-7 the results of my simulations indicate that the flux rate increases linearly with the bottleneck width and does not depend on the presence of an obstacle. On the other hand, an obstacle placed in front of an exit door helps to reduce the density at the centre of the jam.

Figure 5 - Flow of pedestrian through a 1 m wide bottleneck, with and without a circular obstacle of radius 0.5 m.

Figure 6 - Flux rate through a bottleneck (in pedestrians per second) as a function of the bottleneck width. Blue line: simulations without obstacle. Yellow line: simulations with obstacle. Orange line: experimental data of Seyfried et al. [8] without obstacle.

Figure 7 - Voronoi density of pedestrians at a bottleneck (in pedestrians per square metre) with and without obstacle. The density inside the clog appears to be reduced in the presence of the obstacle.

Appendix

The local density and the local flux of pedestrians employed in the previous examples have been defined in the style of Smoothed-Particle Hydrodynamics as a sum of averaged individual contributions:

                                                       

Here t↦rn(t) is the law of motion of the n-th pedestrian and ψϵ:ℝ+→ℝ+ is a bell-shaped smoothing function with compact support in the interval [0,ϵ]. Therefore a pedestrian does not contribute to the local density and the local flux outside the points of a circle of radius ϵ around its current position.

The physical meaning of these two quantities is apparent:

  • The integral of the local density ρ over a plane region Σ is equal to the number of pedestrians inside that region.
  • The integral of the local flux J across a line Γ is equal to the rate at which pedestrians cross that line in the direction specified by its normal.

In addition, it is not difficult to show that the two definitions are consistent in the following sense: if the smoothing function ψϵ is regular then ρ and J satisfy the continuity equation:

                                                                  

References

[1] - D. Helbing, P. Molnár, Social force model for pedestrian dynamics, Phys. Rev. E 51, 4282 (1995)

[2] - D. Helbing, I. Farkas and T. Vicsek, Simulating dynamical features of escape panic, Nature 407, 487–490 (2000)

[3] - M. Moussaïd, D. Helbing and G. Theraulaz, How simple rules determine pedestrian behavior and crowd disasters, PNAS 108 (17) 6884–6888 (2011)

[4] - M. Moussaïd and J. D. Nelson, Simple Heuristics and the Modelling of Crowd Behaviours, in Pedestrian and Evacuation Dynamics 2012

[5] - D. Helbing, Traffic and related self-driven many-particle systems, Rev. Mod. Phys. 73, 1067 (2001)

[6] - A. Garcimartín, D. Maza, J. M. Pastor et al., Redefining the role of obstacles in pedestrian evacuation, New J. Phys., 123025 (2018)

[7] - S.P. Hoogendoorn and W. Daamen, Pedestrian Behavior at Bottlenecks, Transport. Sci. 39 (2), 147–159 (2005)

[8] - A. Seyfried, O. Passon, B. Steffen et al., New Insights into Pedestrian Flow Through Bottlenecks, Transport Sci. 43 (3) 395–406 (2009)