Cellular Automata in Pedestrian Simulation and Modeling-A

Literature Review

Evarist Ruhazwe1, Deo Chimba

2

1Graduate Research Assistant, Department of Civil and Architectural Engineering, Tennessee State University,

2Assistant Professor, Department of Civil and Architectural Engineering, Tennessee State University

Abstract

Cellular Automaton simulation involves mathematical idealization in which space and time are

discrete and physical quantities takes on definite set of discrete values [1]. It finds a lot of

applications in modeling different systems ranging from biological systems to pedestrian

characteristics. A literature review was done to determine how cellular automata can be used in

simulation and modeling of pedestrian movement under different scenario. It was found that the

simulation can be done in determining the maximum evacuation times in buildings [2],

identification of possible conflict points such as bottlenecks in buildings and surroundings.

design of pedestrian control facilities like crosswalks and sidewalks [3] , [4]. On its application

to such modeling, the space had to be divided into grids (discretized) which size depends on the

pedestrian size. Wolfram (1983) [1] used 0.6mx0.6m by considering the dynamic size (as

provided by HCM) while [3] used 0.4mx0.4m considering the size of pedestrians compacted in a

building.

Background and Definition

A cellular automaton modeling involves mathematical idealizations in which space and time are

discrete and physical quantities take on a definite set of discrete values [1] . A cellular

automaton consists of a regular uniform lattice (or array) usually infinite in extent with a discrete

variable at each site/cell. It was first introduced by von Neumann and Ulam as an idealization of

biological systems with a purpose of modeling biological self-reproduction. [5]

Generally cellular automata find a lot of applications in different aspects. Many biological

systems have been modeled by cellular automata [1]. The simple rules governing structure and

pattern of living organism can be studied in this model. The application fields for Cellular

automata can be extended to simulation of sand and snow dunes [4]

Pedestrian characteristics can also be modeled using this methodology. Pedestrian modeling are

employed for applications like determination of maximum evacuation times [2], identification of

possible conflict points or bottlenecks in buildings and surroundings, determination of optimal

evacuation routes [6], design of pedestrian and control facilities like crosswalks [3] and

sidewalks(1)

Before modeling process general principles governing the model itself and several characteristics

of the pedestrians should be known [4]

Principles of cellular Automata

Several things have to be clearly understood if one is going to use Cellular Automata approach in

modeling. First Cellular automata are mathematical models with discrete values in time, space

and state (10). This means that the continuous variables like length, speed time has to be

discretized for the sake of effective modeling. Terminologies used in this are time, space, state

and environment.

The time and space unit should be set depending on the expected speed and cell size [7]. A new

unit discrete time is set to express the speed of changing cells as number of cells per that unit

time.

Space and environment

Cellular automata divide the simulation domain into cells. These cells have to be regular, so that

they form a regular grid. The grid structure can form different patterns depending on

performance of each to a particular simulation. The basic forms as expressed by [4] are triangular

grid, rectangular grid and hexagonal grid

Figure 1: Different possible cell grids

The use of any grid will depend upon the simulation to be done. Each grid type has its own

simplicity and complexity. Triangular and rectangular grids are simpler than hexagonal grids but

sometimes hexagonal grids represent the actual case more than two of them.

Cell state is also an important parameter. Every cell has one certain state value from a finite set

of possible states [4]. John Conway expresses his one of the simplest cellular automaton in the

“Game of life”. A rectangular grid is considered where the cells carry just two possible states,

dead or live.

A cell is assigned a state that will describe its condition at a certain time. The set of all possible

states depends upon a particular simulation being performed. The state of a cell automaton in a

particular generation depends upon the states of its neighboring cells and the state the cell had in

the previous generation.

Just saying neighboring cells leads us to another important terminology “cell neighborhood”.

Different considerations for what neighborhood of cells is, exist depending on the type of the

grids used and given consideration. The widely known neighborhoods are Von Newmann and

Moore Neighborhoods [4] as shown below.

Figure 2:Von-Neumann neighborhood:

Just the neighboring cells which share one side with the basic cell are taken into account

Figure 3:Moore neighborhood:

All cells sharing at least one corner with the basic cell are considered as neighbors:

General Modeling Concept for Pedestrians

Pedestrian movements can be experienced in many areas. Starting from buildings to the

crosswalks and sidewalks [3] different characteristics can be experienced. Greifswald et. al

(2013) [4] tried explained such characteristics.

Firstly, pedestrians are considered as individuals with different targets. If several pedestrians

have the same target they may try to reach it following different routes. Secondly, every

pedestrian has an individual velocity which is strongly influenced by interactions with other

individuals. The velocities in these interactions might lead to overall reduction velocity or even

stop completely the movement a situation occurring in jams.

Space and environment

As said earlier space is made of grids which have defined size. Cell size depends on the size of

the people under consideration and their situation. Also when they carry bags with them their

space becomes wider. According to HCM, static thickness for pedestrian is 0.3m and the

dynamic thickness is 0.6m – 0.8m. Feng et. al (2013) [3] used 0.6 m when studying pedestrian

dynamics at crosswalk. When people cross street, they will follow the one in the front of him

closely to keep conformity with others. Considering pedestrians in a building, [2] used 0.4m x

0.4m which is the typical space occupied by an occupant in a dense crowd. Feng et al, (2013) [3]

used the varying size of the crosswalk depending on the actual size of vehicle lane, so the

pedestrian flow was simulated according to several typical sizes of crosswalks(grid): 17 × 7, 17

× 10, 27 × 7 and 27 × 10.

Some of the considerations about the cells have to be defined. Those cells with initial assigning

of wall will not change throughout their simulation. The cells states can hold different values

depending on the study under consideration. Pengyuan (2009) [6] defines the possible cell state

as object, available, decision or held. S ∈ {Object, Available, Decision, Held}

Where "object" refers to a cell that is held by the objects and cannot be gone through, such as

barrier, shop and canyon; "available" refers to a cell that is available to get across; "Decision"

refers to a cell that will be held by pedestrians next time, because it has been “decided” by other

agents; "held" refers to that this cell is held by one pedestrian [6]. Greifswald et. al (2013) used

states of cell as empty cell, obstacle or cell occupied by a pedestrian [4]. Update of cell state

depends on its current state and condition of the surrounding cells [7].

Time and update Type

Time discreteness can be set depending on the expected pedestrian speed and cell size decided

[7]. On modeling pedestrians at crosswalk [3] used the average velocity of pedestrians in signal

control sections (1.2m/s) and cell size of 0.6m x 0.6m to get the speed of 2 steps per second and

discrete time 0.5 s. In every time step, one pedestrian can move at most one step.

Update rules

Update rules ensures time development from t to t+1. Parallel and sequential update rules exist as

the types of time update rules. Different models can use either of that depending on situation of

the involved simulation. On studying the pedestrian movement at crosswalks [3] used a front-to-

back sequential update rule to simulate the model. Sequential means pedestrians move one by

one, front-to-back means pedestrian who is in front of crosswalk by his/her direction moves first,

i.e., if pedestrian 1 is in the front of pedestrian 2, pedestrian 2 will not move until pedestrian 1

moves.

Parallel update has been used by [4] in the modeling of pedestrian movement in a building. By

using parallel update rules the moves of all pedestrians are done simultaneously. This raises

conflicts if two or more pedestrians want to enter the same target cell. Only the winner of the

conflict is allowed to move, the others have to stay in their cells. However, the writer preferred

this approach as the issue of pedestrian conflicts is of interest when considering emergency

evacuations [4].

Modeling

After considering above parameters which are, grid size and type, update rule to be used,

pedestrian speed and cell states, modeling can now be done. A key idea for the model is to

introduce complex interactions in the form of potential fields. The purpose of pedestrian

movement is to minimize the distance to their destination which can be an exit for building

evacuation [4] or the other side of street for the street pedestrian crosswalk [3]. This is

represented numerically in a potential field describing the metrics for the pedestrians [4].

Two classes of floor fields namely static and dynamic potential fields exists. The static floor field

does not evolve with time and it is not influenced by the presence of pedestrians. The dynamic

floor field is modified by the presence of pedestrians and it is updated using two procedures

called diffusion and decay [7]. Static floor fields have been widely used in much literature.

Static floor field S

First one defines a static potential field S to determine the pedestrian dynamics. For that many

definitions are possible. In this static field the influence of the ground, obstacles and the

attraction by doors are taken into account.

Greifswald (2013) [4] considered distances field values as follows

First the euclidic distance to the nearest exit is determined,

X El and YEl are the coordinates of the lth

exit. In the next step the maximum value of ~ Sij is

computed: In the next step the maximum value of Sij is computed:

The resulting static floor field S is now given through:

So the "field values increase with decreasing distance to an exit and is zero for the cell farthest

away from the door". In determination of the floor field values [2] Specifies the distances to the

nearest exits as the values of the fields themselves. The distance of any point to the exit can be

represented by the formula

The typical floor field values as determined by [5] is as follows

Figure 4: Floor field values for a two exit building

When using Van-Neumann neighborhood, and if no obstacles are involved, [4] suggested the use

of the Manhattan distance;

The transition probability

Transition probability depends on the floor field value and the state of the neighborhood cells.

Considering the static floor field and Van Newmann neighborhood; transition probability can be

expressed as:-

Where N is the normalization parameter to make sure Here, N is a normalization factor for

ensuring ∑i,jPij = 1. Sij is the Static floor field value and ks is the scaling factor (or coupling

factor) taking into account the possibility of the pedestrian to deviate from the planned route due

to some factors (14). For small kS the pedestrians perform a random walk while for big kS the

transition probability towards the cell next to the door is much bigger than the others, so the

pedestrians accumulate in front of the door [4].

[3] Considered Moore’s neighborhood and used three parameters to express the field potential of

the crosswalk. The parameters are benefit parameter, attraction parameter and occupy parameter.

The benefit parameter is used to describe the will that pedestrian want to cross a street. The

attraction parameter is used to show the conformity behavior of pedestrians. And the occupy

parameter is used to ensure that one grid can be occupied by only one cell. The probability of

pedestrians’ movement is the operating of these three parameters. Considering such parameters

an update rule was developed that will give transition probabilities. They combined the effects of

such parameters together in obtaining the full model.

where Vij is the value of the movement matrix, i, j = −1, 0, 1; Bij is the value of the benefit

matrix; Aij is the value of the attraction matrix; Oij is the value of the occupy value; β and α is the

weight of benefit parameter and attraction parameter, respectively, where β + α = 1, β ≥ α; Vij is

the normalization value of the movement matrix; Pij is the probability of pedestrian’ movement.

Rules governing motion

Each particle is given a direction of preference. From this direction, a 3×3matrix of preferences

is constructed which contains the probabilities for a move of the particle. The central element

describes the probability for the particle not to move at all, the remaining 8 correspond to a move

to the neighboring cells.

Pedestrians at crosswalk will make their movement at next step according to their surrounding

environment, and they can choose to wait or move to one of the eight cells around them (Moore

neighborhood is used as is shown in Fig. 3) [3].

Model Rules and Scenario

A cell without a person will have a person in the following cycle if and only if; a) at least one

adjacent cell have an individual b) the distance from the cell under consideration to an exit is less

than the distance from the cell occupied to the exit. In other case the cell doesn’t change the

state. In running a scenario in a building, pedestrians are assumed to be generated from one end

in which are assumed to spread. They are assumed to be generated for the possibility of reaching

the next exit.

Results and Discussions

The basic findings present in many studies is what is termed as faster is slower. Variation of

pedestrian speed with the evacuation time shows that beyond certain desired velocity evacuation

time is longer the same it is with velocity below such velocity. Such case is what is seen below

from the study done by [3].

The other study of interest was the variation of exit size with total evacuation time. The study

done by [5] indicates a decrease in evacuation time with increase in exit size, under the same

environment

In showing the variation of evacuation time with pedestrian density under conditions of fire or

not [8] showed the increase in evacuation time as the density increased as shown in the figure

below The evacuation time as function of pedestrian density for fire occurred or not [8].

Conclusion

Using Cellular Automata microsimulation allows for examination of conditions that are complex

and difficult to model [9]. In spite of the assumptions usually introduced for obtaining a simpler

model, the Cellular Automata results to be very suitable tools for modeling different problems

with results very similar to those expected to occur in a real evacuation situation [5]. However

[6] suggests that the use of trained neural network in agents makes the simulation more realistic.

To make the agents more intelligent and realistic other factors can also be included, such as

psychological and visional factors.

References

[1] S. Wolfram, "Statistical Mechanics of Cellular Automata," Reviews of Modern Physics, vol. 55, no. 3,

pp. 601-644, 1983.

[2] Z. Xiaoping, L. Wei and G. Chao, "Simulation of Evacuation Processes in a square with partition wall

using cellular Automaton Model for Pedestrian Dynamics," Physica A, no. 389, pp. 2177-2188, 2010.

[3] S. Feng, N. Ding, T. Chen and H. Zhang, "Simulation of Pedestrian Flow Based on Cellular Automata:

A case of pedestrian crossing street at section in China," Physica A, no. 393, pp. 2847-2859, 2013.

[4] C. N. Greifswald, R. Schneider and B. Bruhn, "Cellular Automata Modeling for Pedestrian Dynamics,"

2013.

[5] P. C. Tissera, M. Printista and M. L. Errecalde, "Evacuation Simulations using Cellular Automata,"

JCS&T, vol. 7, no. 1, pp. 14-20, 2007.

[6] S. Pengyuan, "A more realistic Simulation of Pedestrian based on Cellular Automata," Institute of

Electrical and Electronics Engineers, pp. 24-29, 2009.

[7] S. Bandini, F. Rubagotti, G. Vizzari and K. Shimura, "A cellular Automata Model for Pedestrian And

Group Dynamics".

[8] Y. Zheng, B. Jia, X.-G. Li and N. Zhu, "Evacuation dynamics with fire spreading based on cellular

automaton," Physica A, no. 390, pp. 3147-3156, 2011.

[9] V. J. Blue and J. L. Adler, "Cellular Automata Microsimulation For Modeling Bi-directional Pedestrian

Walkways," Transportation Research, vol. Part B, no. 35, pp. 293-312, 2001.

[10] C. Burstedde, K. Klauck, A. Schadschneider and J. Zittartz, "Simulation of Pedestrian Dynamics using

a two-dimensional Cellular Automaton," Physics A, no. 295, pp. 507-525, 2001.

[11] V. J. Blue and J. L. Adler, "Cellular Automata Microsimulation for Modelling bi-directional pedestrian

walkways," Transportation Research, vol. Part B, no. 35, pp. 293-312, 2001.

[12] L. Fu, J. Luo, M. Deng, L. Kong and H. Kuang, "Simulation of Evacuation Processes in a Large

Classroom Using an Improved Cellular Automaton Model for Pedestrian Dynamics," in International

Conference on Advances in Computational Modeling and Simulation, Guilin, 2012.

[13] Y.-C. Peng and C.-I. Chou, "Simulation of pedestrian flow through a “T” intersection: A multi-floor

field," Computer Physics Communications, no. 182, p. 205–208, 2011.