Methods for modeling and simulation of multi-destination pedestrian crowds

Introduction

Rapid growth in the volume of public transport and the need for its reasonable, efficient planning has made the description and modeling of transport and pedestrian behaviors an important research topic in the last twenty years. In the study of pedestrian behavior, evacuation scenarios (in which pedestrians all target a definite destination) and multi-agent systems (in which pedestrians are treated as heterogeneous individuals) have attracted much attention as two specific problems.
Comparatively little attention has been paid to the problem of pedestrian crowd behaviors in geometries with multiple destinations: each of the possibly many pedestrians moves to one out of a number of destinations. The objective of the present study is to investigate pedestrian behavior in such a context. The central problem is the modeling of crossing pedestrian streams. In view of a desirable practical relevance, realistic, i.e. rather complex geometries are studied in this context. 

Experiments

In order to obtain reliable empirical data of multi-directional, intersecting pedestrian flows for the evaluation of different simulation models, in 2010 we conducted human crowd experiments at the Technische Universität Berlin. One particular experimental setup, for example, arranged for two pedestrian flows (142 and 83 subjects) to intersect at an angle of 90 degrees for one minute in a region of about 25 square meters, resulting in peak densities of about four pedestrians per square meter.

The pedestrians' spatio-temporal positions were obtained via photogrammetric means from video data recorded with multiple temporally synchronized network surveillance video cameras. Tracking of the pedestrians was facilitated with the standard Lucas-Kanade algorithm. In our case, a particular challenge was presented by the fact that due to constructional limitations the scene could not be captured from a bird's eye view. Thus the effect of the pedestrians' different heights could not be assumed as negligible - and since the pedestrians' heights were not known beforehand, we needed to devise a method for extracting the pedestrians' positions reliably without this information.

Smooth trajectories were then obtained via approximation with cubic B-splines, and the combinatorial assignment of trajectories obtained from different camera perspectives was supported with the Kuhn-Munkres algorithm.  We then computed a local density field via nearest-neighbor kernel density estimation. Compared to the fixed bandwidth estimator commonly used in the literature, our approach yields a density field which provides spatially averaged values across mesoscopic regions that are more faithful to the standard method of counting pedestrians in that region.

We argue that the spatio-temporal density configuration as a representation of pedestrian dynamics is particularly suited for the calibration and validation of a variety of models: macroscopic simulations already produce density fields, and data obtained from experiments or microscopic simulations may be easily converted.

Simulation models

In the last years, several methodical approaches have been investigated for the modeling and the simulation of traffic problems. In the present context, it seems adequate to develop both microscopic approaches in which the pedestrian is considered as an individual interacting with other pedestrians, and macroscopic models in which pedestrian behavior is analyzed in terms of more global properties of a continuous stream. 
In the present project two microscopic models are developed. The first one is a grid-based approach rooted in cellular automata (CA) models. The second one is a combination of a force based and graph based approach. 
In the traditional CA model and its various extensions, the state change of the cell (i.e. position) is applied to describe the system dynamics of the simulation.  Due to this conceptual limit, to our knowledge, the simulation participants (i.e. pedestrians) are all associated with a fixed spacial size, defined by the size of the grid cell in the CA model.  Consequently the pedestrians in the simulation have a fixed exclusive personal space which differs from empirical observations.  In our model this exclusive personal space is given additional attention.  The effect of a modifiable exclusive personal space is achieved by defining the inaccessibility of the surrounding cell position of an arbitrary pedestrian.  By this means it is possible to describe simple group behaviors, i.e. pedestrians which belong to the same group may require a smaller exclusive personal space, while toward other pedestrians in the simulation environment, the nearby cell positions are declared as inaccessible and thus keep the latter at a relatively larger distance as what we would imagine in real-world situations.  Our model also presents an advanced local step calculation to enable the so-called multi-cell-step, i.e. the transition from a start position to a destination with a distance larger than one grid cell.  In the step calculation the execution sequence of the simulation participants is affected, in addition to the participant's own characteristics, by the actual system dynamics as well. This enables a substantial reduction of the ``deadlock'' phenomenon.  This model is applied with some simple configurations with which the pedestrians are given pre-defined start positions and destinations.  With the notion of the modifiable personal space, the simulation with advanced step calculation can be realized in combination with pedestrian density control, if necessary.

The combined force- and graph-based approach treats every participant as an agent, who makes her own decisions. While the graph is only needed to find a path from the origin to the desired destination, the agents’ actual locomotion is driven by a force-base model. The model not only encompasses simple repelling and attracting forces but also more complex forces for explicit collision avoidance. 

The macroscopic approach is based on a set of pedestrian-specific coupled partial differential equations (PDEs) The equations are not derived from the Euler-/Navier-Stokes-PDEs known from fluid and gas dynamics. The specific situation of multi-destination pedestrian crowds with crossing streams requires the development of appropriately adapted methods. This has been targeted by the use of simple heuristics.  

Typical applications of these approaches include real-world scenarios like airports, shopping malls, buildings of middle- to large size etc., where the participants (i.e. the pedestrians) do not exhibit an overall unanimity and (may) have different and multiple destinations. 

Beyond the modeling of the above-mentioned problems, a particular aim of this project will be the development, implementation and test of appropriate computer-based simulation models. The reliability of these models will be illustrated by a comparison with real data obtained from crossing pedestrian streams experiments.