Pedestrians and crowds

Experiments

Interactions between pedestrians in crowds produce emerging self-organized structures. For ins

tance, flows of pedestrians moving in opposite direction spontaneously segregate into lanes (see top picture). This phenomenon increases the traffic efficiency with no need of external control. This is often given as an example of 'collective intelligence' (an other example is the spontaneous organization of ant traffic into trails). The emergent organization of crowds is scarcely documented and experimental measurements in controlled condition are rare. The behavioral mechanisms that underly the emergent self-organization in crowds are not completely understood.

In this work, we have studied this phenomenon under controlled laboratory conditions, in the framework of the 'PEDIGREE' ANR project. We have derived mathematical and simulation models which are carefully validated and calibrated against the experimental data. Experiments have been conducted at the INRIA-Rennes (see second from the top picture, an experiment in a ring-shaped arena). The tracking is performed on real-time using infra-red sensors positionned on the shoulders and forehead of the subjects (middle picture).

Sensors are recorded by means of computer animation equipment (see second from the bottom picture). Complex data processing is required to entangle sensor trajectories when occultations occur (see bottom picture an example of sensor assignment to one pedestrian). As a result, the trajectories of all pedestrians during the entire session is available and allows the exploration of interaction rules.

Below are two examples of the collected data (left: a two-way experiment in the ring-shpaed arena with 60 pedestrians labeled blue (clockwise) and red (anti-clockwise) ; right: a one-way experiment with pedestrians walking in line on a circle).

Lane formation

Clustering method.

We have explored the spontaneous segregation into lanes in the ring-shaped arena.

To identify lane formation, we have designed a clustering method. Two pedestrians belong to the same lane if one of them is following the other.

We assume that a pedestrian is following another pedestrian, if the trajectory of the former at some later time passes at a small distance from the location of the latter (see figure).

Evolution of the number of lanes

The figure below shows the number of clusters (left panel) and the lateral barycenter of the clockwise and anti-clockwise pedestrians (right) as functions of the time. As the number of clusters diminishes (after an initial transient), the two barycenters become widely separated, indicating a segregation into lanes. We have studied the intermittency of cluster formation (see publication and CNRS Research Highlight).

Following behavior

From the experiment of the pedestrians walking in circle, we have calibrated a Follow-the-Leader model for pedestrians. It relates the acceleration of a subject to the distance to his leader and to the difference of their velocity.

The major finding of this work is the existence of a time delay (of about 0.5 seconds) before the reaction of the subject. This is a major change compared to Follow-the-Leader models of car traffic where this time-delay has often been neglected.

The figures below show the data (top left) the model (top right). Results of the Social Force model of Helbing (bottom left) and the Computer Graphics model of Reynolds (bottom right) are given for comparison. The figures represent the positions of the pedestrians as a function of time. The color code corresponds to the local density (inverse of the distance to the leader). The initial stop-and-go waves (appearing as red strips) and their damping are fairly well reproduced by the model. By contrast, the social force model shows an excess of undamped stop-and-go waves, while the computer graphics model shows an excess of damping.

Macroscopic models of two-way flow in corridors

The model describes counter flows of pedestrians in corridors and walkways. It is one-dimensional and based on two-way extensions of the Lighthill-Whitham-Richards (LWR) and Aw-Rascle (AR) models of car traffic.

In the LWR model, the flux is a function of the density and is bell-shaped as plotted in the right figure (phi denotes the flux, rho the car density).

The increasing flux at low density describes free traffic and the decreasing part at high density, congested traffic. The slopes of the red and green lines illustrate the difference between the agents' velocity (in red) and the congestion velocity (in green).

In our pedestrian model, the pedestrian flux in one direction depends on the co-moving and counter-moving pedestrian densities. A typical flux function is plotted in the left picture (f denotes the flux, rho_+ and rho-_ the co-moving and counter-moving pedestrian densities respectively). The flux is bell-shaped as a function of the co-moving pedestrian density as in car-traffic But It is always decreasing as a function of the counter-moving pedestrian density to take into account the 'friction' induced by these pedestrians.

At the moderate densities at which the experiments have been conducted, this simple model reproduces surprisingly well the observed structures.

At high density, the model exhibits congestions consisting of two opposite fronts slowly diffusing into each other. These structures are a consequence of the loss of hyperbolicity of the model. The picture below shows a congestion situation (left: initial condition ; right: final time. The right moving pedestrian density is in blue ; the left moving one in green and they are plotted as a function of position). Although this behaviour is unlikely in normal conditions, it may be relevant in special conditions (such as aircraft passengers getting in by both ends of the aircraft). The singular pressure model has also been included in the model to describe the jamming transition when the crowd reaches the congestion density.

Game theoretic pedestrian models

The model of Moussaid, Helbing, Theraulaz can be seen as a differential game in which pedestrians are rational agents optimizing their walking direction to fulfill some objective. Each pedestrian scans the possible directions and determines his largest possible walking distance before colliding with another pedestrian. Then, he chooses the walking direction which brings him the closest to his target. This model reproduces the experimentally observed lane formation and dynamics fairly well. We have proposed a macroscopic formulation of this model based on our established parallel between Nash equilibria in game theory and thermodynamic equilibria in kinetic theory. A similar macroscopic model also follows from the vision-based model of pedestrian behavior of Ondrej et al.

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