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CSE 6730 Project 1 Report

February 1, 2014

Contents

1   Notes on   Simulation of pedestrian dynamics

using a two-dimensional cellular automaton.

1.1 General

•   Long-range interactions between the pedestrians are mediated by a so-called floor field  which modifies the transition rates to neighboring cells.

•  This field, which can be discrete or continuous, is subject to diffusion and

decay.•  Can be modified by the motion of pedestrians.

•  The pedestrian leaves a “trace” by modifying the underground on his path.

•   Purpose is to transform effects of long-ranged interactions (e.g. follow-ing people walking some distance ahead) into a local interaction (with the“trace”). This allows for an efficient simulation on a computer.

•  To keep the model simple, we provide the particles with as little intelligenceas possible and achieve the formation of complex structures and collective

effects by means of self-organization. In contrast to older approaches, wedo not make detailed assumptions about the human behavior. Nevertheless,the model is able to reproduce many of the basic phenomena.

•  In the floor-field model, each cell carries a quantity (field) which be eitherdiscrete or continuous.

•  The field can have its own dynamics given by diffusion and decay coefficients.

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•  For long distances, the interaction is often attractive. For example, whenwalking in a crowded area, it is usually advantageous to follow directly behindthe predecessor. Large crowds may also be attractive due to curiosity.

•  We can consider adding stress to people such that people with higher stresswill move in a more random (or less focused) path.

1.2 Model

•  A two-dimensional grid, where each cell can either be empty or occupied byexactly one person.

•  Size of cell corresponds to 40x40 cm2.

•  It might be desirable to use a finer discretization, e.g. such that each pedes-trian occupies four cells instead of one.

•   Empirically, the average velocity of a pedestrian is about 1.3 m/s. This givesapproximately 0.3 s per transition.

1.2.1 Basic rules

•  All cells are updated in parallel.

•  For each particle, a desired move is chosen. If the target cell is occupied, theparticle does not move. If it is not occupied and no other particle targets the

same cell, the move is executed. If more than one particle share the sametarget cell, one is chosen according to the relative probabilities with whicheach particle chose their target. This particle moves while its rivals for thesame target keep their position.

1.2.2 Floor field

•   In order to reproduce certain collective phenomena, it is necessary to intro-duce longer-ranged interactions.

•  To keep model simple, we avoid modeling long-range interactions explicitly.

Instead, we use a floor field which implicitly maps these long-range interac-tions into local transition probabilities.

•  Two types of floor fields:   static  and  dynamic .

•   Static S : does not evolve with time and is not changed by presence of pedes-trians. Such a field can be used to specify regions of space which are moreattractive (e.g. emergency exits or windows).

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•   Dynamic D : modified by the presence of pedestrians and has its own dynam-ics, i.e., diffusion and decay. This is used to model long-ranged attractiveinteractions between pedestrians. Since the total transition probability is

proportional to the dynamic floor field, it becomes more attractive to followin the footsteps of other pedestrians.

•  In general, the transition probability  pij   in direction (i, j) is give by

 pij  = NM ijDijS ij(1 − nij) (1)

where

–   nij: occupation number of target cell (i.e.   nij  ∈ {0, 1}). In other words,transitions to occupied cells have probability zero.

–   N : normalization factor to insure 

i,j pij  = 1, where the sum is overthe nine possible target cells (which include current cell).

–   D: is modified according to its diffusion and decay rules.

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