Bridging the gap: From cellular automata to differential equation models for pedestrian dynamics
Introduction
Several approaches for modeling microscopic pedestrian dynamics have been developed in the last decades [37], [5]. Among these, two compete for supremacy: cellular automata (CA) and models based on ordinary differential equations (ODE). In typical formulations, both model types use the idea that pedestrians are driven by repulsive and attractive forces. Other pedestrians and obstacles repel, targets attract. However, the mathematical formulations differ fundamentally, especially as far as treatment of space and navigation is concerned.
In CA [3], [16], [14], [9], [36], the given area is divided into cells of equal shape and area that are either empty or occupied by a pedestrian, a target or an obstacle. This status is updated at each time step, that is, virtual pedestrians move from cell to cell according to certain rules. Typically, the pedestrians navigate along a floor field that expresses attraction and repulsion acting on the pedestrians [11], [17]. In most CA-models, acceleration to each pedestrian's free-flow velocity is achieved instantaneously if there is space to move. The coarse discretization of space limits the choice of direction and influences space requirements and the handling of speed [17]. Advantages are high computational speed, simplicity, as well as easy and intuitive integration of rules governing pedestrian behavior.
ODE-models for pedestrian motion are usually inspired by Newtonian mechanics. They also consider attractive and repulsive forces but operate in continuous space and time. Navigation is realized indirectly by computing an acceleration vector from a superposition of forces. Acceleration is delayed by a friction term. The best known ODE-model is the Social Force Model (SFM) introduced by Dirk Helbing and Péter Molnár in 1995 [13]. Problems of this ansatz include a treatment of inertia that is often inappropriate for pedestrian dynamics [4] as well as numerical pitfalls [19]. Furthermore, typical or specific behavior of pedestrians, even prevention of overlapping, can only be achieved by introducing extra complexity [6], [5].
In this paper, we present and compare two new models that remove the major differences between the CA and ODE models: First, the Optimal Steps Model [27], [32], which remains rule based as a CA but allows movement in continuous space. Second, the Gradient Navigation Model [8], which uses ordinary differential equations like the SFM but computes velocity, and hence direction of movement, directly from the forces. See Fig. 1 for a schematic representation of how to bridge the gap between CA and ODE models. Both approaches maintain the advantages of the models they were inspired by, but do not suffer from the main disadvantages. They are robust and successfully validated according to the guidelines given in [25]. Both can be calibrated to a given fundamental diagram [8], [32]. Numerical experiments yield very similar trajectories showing that they are indeed more alike than the original CA and ODE models. See Section 4.
The paper is structured as follows: in Sections 2 Optimal Steps Model, 3 Gradient Navigation Model we briefly introduce the two new models stating their main ideas and underline where they deviate from the CA and social force approaches. For detailed descriptions we refer to the original publications [27], [32], [8]. Then we show the results of numerical experiments for all four model types to support our hypothesis that the two new models do not only perform better but produce similar results (Section 4). In Section 5 we discuss which differences remain. In the conclusion section (Section 6) we propose desirable next steps.
Section snippets
Optimal Steps Model
In this section we give an outline of the Optimal Steps Model (OSM) as it is described in [27] and enhanced in [32]. The model is inspired by the idea that pedestrians try to optimize their position in space according to a balance of goals: reaching the target and avoiding obstacles and other pedestrians. This approach is also used for cellular automata models of pedestrian movement [18] and the discrete choice framework described in [1]. Virtual pedestrians move by locally minimizing a scalar
Gradient Navigation Model
The Gradient Navigation Model (GNM) is composed of a set of ordinary differential equations to determine the position of each pedestrian xi in two dimensional space as well as the scalar speed and navigational direction [8]. The idea is to compute the velocity vector as the gradient of several distance dependent, scalar functions similar to Pi in the OSM (see Eq. (2)). This constitutes a strong deviation from Newtonian dynamics and hence from the Social Force Model [13], where the acceleration
Results
Grid restrictions in CA models and indirect navigation through acceleration in the ODE models constitute the major differences in the model formulations. In three computer experiments described below, we demonstrate how pedestrian trajectories become more similar when these differences are removed with the OSM and the GNM. In the first scenario, a single pedestrian leans against a wall from where he or she moves to a target located to the right (see Fig. 5). The CA and SFM produce unnatural
Remaining differences
The OSM with its quasi-continuous dynamic discretization of space has drawn nearer to continuous models, while the GNM has adopted the navigational ideas of successful CA-models and the OSM and integrated them in an ODE context. Nevertheless, some differences remain. In the OSM, there is no acceleration phase. Pedestrians reach their desired free flow velocity instantaneously if they are not hampered by preceding pedestrians or obstacles. In the GNM, the velocity is relaxed through an
Conclusion and future work
Two new models for pedestrian motion were presented, one emanating from rule based CA models, but with continuous treatment of space and one based on ODEs using navigation on geodesics as in CA models. The trajectories of the new models are very similar, thus bridging the gap between the traditional modeling approaches. Closing the gap entirely may well be possible, but entails more stringent mathematical formulations and studies of the OSM to complement the present algorithmic formulation and
Acknowledgments
This work was funded by the German Federal Ministry of Education and Research through the project MEPKA on mathematical characteristics of pedestrian stream models (17PNT028). Support from the TopMath Graduate Center of TUM Graduate School at Technische Universität München, Germany, and from the TopMath Program at the Elite Network of Bavaria is gratefully acknowledged.
Felix Dietrich (B.Sc Scientific Computing, B.Sc Mathematics) is a researcher at the Munich University of Applied Sciences. He focuses on numerical analysis of multiple scale systems, specifically pedestrian flows.
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Felix Dietrich (B.Sc Scientific Computing, B.Sc Mathematics) is a researcher at the Munich University of Applied Sciences. He focuses on numerical analysis of multiple scale systems, specifically pedestrian flows.
Prof. Dr. Gerta Köster teaches mathematics and computer science at the Munich University of Applied Sciences. She leads the research group for modelling and simulation of the behavior of people in crowds.
Michael Seitz (B.Sc Computer Science, M.Sc. Statistics) is a researcher at the Munich University of Applied Sciences. He focuses on the cognitive heuristics used by pedestrians in crowds.
Isabella von Sivers (B.Sc Scientific Computing, M.Sc. Mathematics) is a researcher at the Munich University of Applied Sciences. She focuses on models for affiliative behavior and social identity.