Fundamental diagram as a model input – direct movement equation of pedestrian dynamics

Publication Type:

Conference Paper


Pedestrian and Evacuation Dynamics PED 2012 (2012)


Leading by advantages of continuous and discrete approaches to model pedestrian dynamics we develop a new discrete-continuous model SIgMA.DC [5]. In this model people (particles) move in a continuous space in this sense model is continuous, but number of directions where particles may move is a model parameter (limited and predetermined by a user) in this sense model is discrete. Here we deal with an individual approach when models give coordinates of each person.

A space and an infrastructure (obstacles) are known. People may move to free space only. Shape of each particle is a disk with a diameter of di, [m], initial position of a particle i is given by the coordinate of the center of the disk xi(0) =(xi1(0), xi2(0)), i=1,...,N, N – number of particles. Each particle is assigned with a free movement velocity vi0, [m/s], square of projection, mobility group, age of each person. Let assume the nearest exit as a target point of each pedestrian. Intersections of particles, particles and obstacles are forbidden. To orient in the space particles use the static floor field S [8].
An idea of the approach proposed is come from the formula that connects path and velocity. It’s finite-difference expression in a vector way gives as an opportunity to present movement equation in a direct form, when current position of the particle is determined as a function of a previous position and local particle’s velocity. For each time t coordinate of each particle i are given by the following formula:
xi(t)= xi(t-dt)+ vi(t)ei(t)dt, i=1,...,N, (1)
where xi(t-dt), [m, m] – coordinates in previous time moment; vi(t), [m/s] – current velocity of the particle; dt, [s] – length of a time step (fixed). Unknown values in (1) are the direction ei(t) and the shift vi(t)dt. The direction is proposed to be random. Each time step t each particle i may move in one of the q predetermined direction, q – number of directions, model parameter. Choice of the direction where particles make a next step is stochastic and based on probabilities distribution. “Right” probabilities vary dynamically and are given by balancing of three contributions: a) the main driven force (given by destination point), b) interaction with other pedestrians, c) interaction with an infrastructure (non movable obstacles). Procedure of calculating probabilities to move to each of the directions is adopted from previously presented stochastic Cellular Automata (CA) floor field (FF) model [2, 3, 4, 6] and gives the highest probability to direction that has most preferable conditions for movement considering other particles and obstacles and strategy of the people movement (the shortest path and/or the shortest time). Directed movement is given by using the static floor field that shows a distance from each point of the space to the nearest exit.
At each time step current velocity vi(t) of the particle is determined by local density in accordance with the fundamental diagram [1, 7]. To be more exactly we use analytical expressions of velocity versus density that is given by formula in [1]. This formula was derived from previously presented results in the book by Predtechenskii and Milinskii [7]. Obviously one can use other data, (velocity vs. density) that is in table or formula form.
In the class of continuous model such approach has an advantage. Usually differential equation is used to give movement equation of the particle. To simulate movement of N particles a system of differential equations are solved. It worse to be mentioned that numerical solution of the problem in such statement is “a hard nut to crack”. Forces that act to the particle could not be adopted from physical laws directly and should satisfy at least one quality condition: model flow should correspond fundamental diagram. To describe forces in such way is difficult problem.
In the approach proposed we leaved apart this step. The movement equation is given in direct form, and velocity is controlled by local density in the direct form. In an assumption that movement direction is “right” such approach should give strong coincidence of the model flow and experimental flow that is used. So the success in the pedestrian movement modeling using such approach is to choose “right” directions for each particle in each time step. Approach of choosing directions that was created for CA model was adopted and adapted for continuous space.
Model dynamics was investigated for simple geometry, straight corridor b=2 meters in width. For periodic boundary conditions there was a control line and we measured time T that N=1000 people needed to pass this line. There was observed a perfect coincidence of the model flow J=1000/T/b, [pers/s/m], and real data [1] for low and middle densities. For high densities model is considerably slower than a real flow. It seems to be a computational impact of the model. The fact is the lower free movement velocity [1, 1] the better coincidence of the flows observed for high densities. Experiments on open boundary conditions showed a typical and qualitatively the same behavior that model should reproduce, and RIMEA collection presents. Experiments on other geometries and further development of the model are going on.