fractal, a microscopic crowd model

1st Reading September 17, 2008

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Fractals, Vol. 16, No. 4 (2008) 1–16 c World Scientific Publishing Company

FRACTAL, A MICROSCOPIC CROWD MODEL SETYAWAN WIDYARTO Department of Computer and Information Management State Polytechnics of Lampung Jalan Soekarno-Hatta No. 10 Rajabasa Bandar Lampung, 3514 2 Indonesia Faculty of Industrial Information Technology Bestari Jaya Campus, Jln Timur Tambahan 45600 Berjuntai Bestari Selangor, Darul Ehsan, Malaysia [email protected] M. S. ABD. LATIFF Department of Computer System and Communication Faculty of Computer Science and Information System Universiti Teknologi Malaysia 81310 UTM Skudai, Johor, Malaysia Received November 13, 2007 Accepted June 6, 2008

Abstract The core of this research is related to the human crowd problem. Some major problems are congestion, emergency evacuation, and fatal catastrophe. In fact, it has been realized that many crowd related problems can be resolved by influencing (controlling) human flow with providing various control measures. Thus, the problem itself becomes the motivation for this research and the solution is approached through model and simulation within the virtual environment. The relationship between fractal pattern and crowd behavior is produced with respect to the crowd behavior model. It exposes a comparison between the crowd paths resulting from the model developed and the fractal formation created to imitate the crowd paths. A new approach for crowd behavior modeling is proposed based on fractal features and, thus, gives new understanding about a relationship between fractal pattern and crowd behavior. Several innovations of fractal patterns that can be used in crowd behavior model applications adds to the novelty of the research contribution. Ultimately, the new method of modeling the crowd can be obtained 1


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S. Widyarto & M. S. Abd. Latiff

and at the same time the new direction of crowd modeling research can be directed. A new approach for crowd behavior modeling is proposed based on fractal features and, thus, gives new understanding about a relationship between fractal pattern and crowd behavior. Several innovations of fractal patterns that can be used in crowd behavior model applications adds to the novelty of the research contribution. Ultimately, the new method of modeling the crowd can be obtained and at the same time the new direction of crowd modeling research can be directed. Keywords:

1. INTRODUCTION The focus of this research is the crowd microscopic model simulation for large agents by considering the interaction forces among the crowd members but the research is extended to fractal investigation. The investigation also compares the characteristics of fractal formation and agent paths. The agent paths are used to analyze crowd behavior. The crowd behavior is approached with fractal patterns. The crowd behavior is a collective behavior from many agents. In the literatures, the collective behavior is a typical feature of living systems consisting of many similar units and it is suggested that the pattern would be fractal.However, fractal determination is a conjecture that needs convincing criteria. The overall of this research is in exposing discussion of the relationship between the crowd microscopic model and fractal. Some simulation results of fractal patterns are also supplied to support the conjecture relationship. Thus, the research objective is to study the relationship between fractal pattern and crowd behavior with respect to producing the crowd behavior model. The crowd behavior model is constructed with forces inclusion (crowd dynamics). One of the research novelties is some findings of fractal patterns that can be used in crowd behavior model applications.

2. A MICROSCOPIC SWARM MODEL SIMULATION This research differentiates between crowd and swarm on the reason behind the usage. When crowd is used it has danger behind but when swarm is used it is just a moving large number of agents and does not highlight any danger. This section uses both crowd and swarm. However swarm is used to refer to multi-agent simulation in neutral observation without any prejudice of potential fatal catastrophe.

The microscopic model simulation of agents is a computer simulation model of agent movement where every agent member in the model is treated as an individual agent. The microscopic model is also known as a particle-based model. An example of such a microscopic model is reported in robotics.1 The microscopic model is used to study collective behavior of a swarm of robots engaged in object aggregation and collaborative pulling. The stickpulling experiment was carried out to the study dynamics of collaboration of robots. Venutia et al.2 suggested that further research should be addressed to the determination of the values of the parameters involved by means of ad hoc experimental tests and microscopic models of the human agent flow could be included in the framework of crowd-structure interaction. However, this microscopic model and simulation in this research is somewhat particle system- or physical force-based, but in this research it is termed as “agent-based”. The model is termed as agent-based because this model is based on identifying each member of the crowd as an object or an agent. Moreover, it is a simulation of each agent motion with respect to all other agents and the environment in which it moves. This style of model depends on simulating the behavior of an object or agents individually where every agent member in the model is treated as an individual agent. The model has three variables of force types. The forces will be further explained in Sec. 2.1. Customization of physical-based variables as inputs of the model is designed to get autonomous agents. The autonomous agents’ paths will be different in every repeating simulation. It means the simulation will be given predefined inputs and the agent motion is free from a user. Agents in the microscopic simulation model are modeled as non-player agents (NPAs). NPAs are the autonomous agents that are free from the user’s


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Fractal, A Microscopic Crowd Model

control but initial conditions are keyed by the users. NPAs are seen from above the facilities (top view). An agent is modeled as a circle with a certain radius. Each agent’s initial conditions includes initial location, initial time, initial velocity, and predetermined target location (opposite to the initial location). These inputs can be determined by the user as a design experiment and can be specified randomly. NPAs will interpret an action and this interpretation process is important for the view in behavioral animation. This will lead to further autonomous actions in the virtual environment as well as intelligent responses to the action being carried out. Thus motion or path planning becomes much more complicated when an animation for large swarms must be made. The development in motion planning and in global techniques for improving the approach has been discussed3 but it focused on the probabilistic roadmap (PRM). Furthermore, a model to simulate the movement of virtual humans based on trajectories captured from filmed video sequences is investigated.4 It used a physically-based simulator to animate virtual humans and to reproduce the trajectories. Whereas the improvement for path planning techniques used for large swarms is very few. Some available video materials have been observed (for example Thawaf), Fig. 1. The figure shows crowd movement circling (determined with blurred area) the Kaa’bah (black box in the middle) and at the farther distance from the Kaa’bah crowd keeps at their place. Briefly, the summary of the characteristic features of bio-creatures crowd movement could be served as follows: (1) Crowd flow is further slowed by fallen or injured or stopped agent acting as

Fig. 1

Pilgrim’s circumbulating around Kaa’bah.

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“obstacles”. (2) The physical interactions in the jammed swarm add up. (3) People in special case show a tendency toward mass behavior, that is, to do what other people do.5 (4) Alternative exits are often overlooked or not efficiently used in escape situations.6 These observations have encouraged us to model the collective phenomenon of swarm flow in the framework of emerging swarm behavior, NPAs. The developed computer simulations of the swarm dynamics of agents are modeled as physical-based with explicit visual interaction. The explicit visual interaction might represent a generalized force model inside the collective behavior of the agents. The collective behavior in a crowd is dominantly influenced by socio-psychological force.7

2.1. Modeling Agent Movements Kirchner and Schadschneide8 and Henein and White9 with their field-based model represent forces in such a model that two individual factors — desire to move toward an exit and desire to follow others — within a physical space laid out in a grid pattern. Force in models of crowds is a basic element that must be represented for three main reasons. First, force has a direct effect on movement. Second, force is a perceptible input to the cognitive system and a major source of information in an informationstarved situation. Third, force carries the consequences of dangerous crowd scenarios: injuries. The model developed applied those three reasons. Modeling the agent crowd movement begins with an assumption that each agent is subjected to motivation to move ahead toward the target point or destination. The motivation is an analogy of the force that characterizes the internal driving force. Borrowing the general behavior rules in Literature Review from Thalmann et al.,10 the driving force is assumed proportional with the difference between the intended velocities and the current velocity. Briefly, Fig. 2 shows the method used in microscopic crowd modeling behavior. From Fig. 2, calibration refers to real world features observation, whereas, verifying and validating mean comparing observation with related existing works. The microscopic swarm model is a mathematical model that every agent has three variables of forces. The overall model is a mathematical model that is combination and extension of escape panic model,7,11,12 , bird flocking behavioral model13 and pedestrian traffic model.14 The development of the


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is applied toward the three forces together (global parameter) while the other three parameters are applied only for the particular force. The three forces are Eqs. (5)–(7). vˆf α ´ − v α α vâ χ ´ − v maa = χ χ

maf =

Fig. 2

The method used.

mar = model and simulation is designed to capture the individual swarm member and able to record the characteristics of its movements. The calibration of the simulation is concerned with the determination of the numerical value of the parameters and the results of the simulation. Whereas, validation and verification does not mean validating with real-life movement but it is rather an exposure to show that the model does work. However, this research extends the method with comparison between the characteristics of the microscopic model and fractal experiment result. The basic dynamical model will be xt+1 = xt + vt at = vt+1 − vt

(1) (2)

where xt , vt , at denotes the matrix or vector of current location, velocity and acceleration, respectively. The force (f ) is proportional with mass (m) and acceleration (a). The acceleration is a discrepancy between two velocities, and the model in this research is the summation of desired velocities, vˆ(t) and the actual current velocity, v(t). Thus, it is written as Eq. (2). Therefore, the force becomes Eq. (4). f (t) = m.a (3) dv(t) = vˆ(t) − v(t). (4) m dt The model developed comprises three variables of force types, i.e. the force to move ahead (i.e. a desired velocity), the force to prevent collision (i.e. acceleration or deceleration) forward, and the force to move away (i.e. change the direction) that is also to avoid collision. Each of the forces contains different parameters and is modeled obeying the Newton law. The developed model has four parameters. They are the mass, m, alpha, α, beta, β and chi, χ. Alpha influences the force to move ahead, beta for collision avoidance and chi for move away. The mass

(5) (6)

β´ vˆr − v. β β

(7)

Resultant of the three forces (addition of vectors) yields Eq. (8): m (af + aa + ar ) v´f vá v´r + + − = α χ β Say a = af +aa +ar and c = becomes Eq. (9) ma + cv =

α ´ χ ´ β´ + + α χ β

χ ´ β´ α ´ α + χ+ β,

v´f vá v´r + + . α χ β

v.

(8)

then Eq. (8)

(9)

The first force is the main force and starting force. It means the other two forces will not work if the main force does not exist. This force is the forward moving force and does move ahead. The forward force will drive the agents from any initial or current positions into target positions or destination. Generally, the movement of an agent is from the current location, p(t) toward the destination point, e(t). The magnitude of this force is intended to make the speed of the agents (crowd members) within the range of the human agent walking speed, which is from zero to maximum of the walking speed, µmax or at the mean human walking speed as shown in Table 1. This force will make an agent reach the desired velocity. Let F , a force, is applied to move ahead that directs the agent to move. The F force makes the agent path almost in a straight line and its direction is from the current location toward the destination. The gradient (direction) of the F is given by g(t), g(t) =

e(t) − p(t) . e(t) − p(t)

(10)

If there is no obstruction, the agent’s intended velocity reaches the maximum walking speed, µmax or smaller (0 ≤ vˆ(t) ≤ µmax . The existence of other agents or obstructions will reduce the walking


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Fractal, A Microscopic Crowd Model Table 1 Observed Walking Speed in Uncongested Corridors. Mean Speed (m/s)

Ref. 15 16 17 18 19 20 21

1.4 1.47 1.44 1.45 1.08 1.19 1.25 1.4 1.32 1.46 1.23 1.22

22 23 24 25 Estimated overall average

1.34

Standard Deviation (m/s)

0.23

0.26

1.0 0.63

Location Netherlands United Kingdom Australia Australia Saudi Arabia Hong Kong Sri Lanka Canada United Kingdom India Singapore Thailand

0.37

speed. Thus, the intended velocity for the F force is given by vˆf (t) =

µmax e(t) − p(t) µmax g(t) = . F F e(t) − p(t)

(11)

The norm in the denominator of the equation above represents the distance between the current position and the destination. Generally, the movement of an agent is from the current location, p(t) towards the destination point, e(t). Alpha force is applied as the force to move ahead that directs the agent to move. The alpha force makes the agent path almost in a straight line during absence of the other two forces. On the other hand, the absence of alpha force will make an agent immobilized in sense of no destination. The direction of the alpha force is from the current location towards the destination. However, the detail of the model is somewhat customized with physical-based variables that can be measured as inputs of the model. The model developed is different from the work by Helbing et al.11,26 where agents are influenced via velocities adjustment by assuming the “psychological” repulsive force between people in a crowd is exponentially increasing with distance reaching the force upon contact. The developed model eliminates the exponential increment and applies distance proportion to adjust the velocity because the model is not aimed for escape panic situation. The second force is the force to prevent collision but it does not change the direction. It means an

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agent will be patient by decreasing the speed with deceleration or may quickly respond by increasing the speed with acceleration. Thus, in both cases the agent keeps moving forward. However, the developed model applies another repulsive force of multi-agent reactions by assigning additional radii to guarantee collision avoidance. Thus, the developed microscopic agent simulation model is made based on the existing models to improve the deficiency of the existing models and keeps their main advantages. The model with the second force may avoid collision by keeping the safe distance between two agents. When two agents nearly collide (Fig. 3) they usually move away from each other within a certain distance but do not change the path. In others words, an agent decreases the speed along (dotted line in the picture) the same axe, e.g. X. They may not wait until their distance becomes too close to move away unless there are no spaces surrounding them. A similar behavior happens when an agent is following another slower agent. If d(t),y and r are, respectively, representing the distance between the agents, interference of the closest agent in the area in front of the actor and the influence radius of agent, the intended velocity of agent ith, vâi (t) due to F force to move away, is given by vâi (t) =

Fig. 3 plane.

µmax (2r − y(t)) µmax (2r − y(t)) = . F d(t) F pk (t) − pi (t) (12)

F Force to move away within sight distance in x-y


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The third force is is also to avoid collision but the force will make the agents move away and change the direction. The expected velocity is a kind of predicted velocity on which way the agent is going to move in the next time ahead. The expected velocity directs the acceleration and the forward force toward the target point. The direction of the expected velocity must be the same as the force and the acceleration. Based on the Newton law, the acceleration is proportional to the force with a constant proportion called mass, m. Since the acceleration has the same direction as the force, it is also the direction of the acceleration. When the expected velocity is equal to the current velocity, the force (and the acceleration) has zero value and the agent may be stopped or walking with a constant velocity. The model with the third force may avoid collision by changing the direction of the movement, thus the path will be curved. When two agents nearly collide, they usually move (repulse) away from each other by changing positions such that different force vectors occur (Fig. 4). They may not wait until their distance becomes too close to repulse away (three agents in the right in the picture), unless there are no spaces surrounding them. A similar behavior happens when an agent is following another slower agent (three agents on the left of the picture). They have to adjust their velocity direction so that any proper alignment can be reached matching velocity with a nearby agent. Align an actor’s velocity vector with that of the local flock. This move (repulse) away direction causes the path bend and the overall swarm path may also create an emerging behavior of curve path. A repulsive force is generated if the influence radius does overlap each other. The repulsive force considers all agents surrounding and the forces are summed up linearly. The force depends on the distance between the agent and other agents surrounding it. The repulsive force model is

Fig. 4

Repulse away.

given by

µmax 2r − dij(t) Xi(t) − Xj(t) − Vt . fi,r (t) = β dij(t) j

(13) The two forces are then totaled together with a weighing factor, the parameter m’ to define the acceleration. 1 (14) at = (ff (t) + fi,r(t) ). m It is different from the two models, this research mixes socio-psychological and physical force. The psychological force is an influencing force of the behavior in a swarm and the force is a moving force. In other words, agents never move if no force moves ahead (alpha = 0), even though other parameters are not zero. It is assumed that each agent is subject to “mixed forces” that represent motivation to move ahead toward the target location. The force here is the analogy of the force that characterizes the internal driving force or motivation of the agent.

2.2. Collision-Free Multi-Agents (Swarm) Motion Planning In practice, the repulse away force is not always sufficient to prevent a collision between swarm members and it is necessary to implement a skirting rule. The repulse away rule is an extremely local version of the separation rule. In other words, it only reckons the nearest swarm member ignoring multi-surrounding agents. To overcome this drawback and to guarantee collision free, some collision avoidance research can found from literatures and will be revealed in the following paragraphs. Provided with a situation that involves a large number of agents, for example an egress during an emergency, an agent is supposed to act and produce movement. Every single robot in a swarm is expected to move away from the threat but there must not be any movement that causes jam, obstruction or other non-adaptive directions. Canter,27 and Still,28–30 modeled the problem by producing movement in either predictable directions: towards the threat or away from the threat or no movement. This problem was reduced to three variables that interact — Objective, Motility and Constraint — and one parameter which represents the reaction time; Assimilation. The Objective is to reach the expected point. The Motility is the speed of each agent. The Constraint is the geometric size


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of each agent. The interaction between these three variables were plotted against each other and the results are fractal in nature.28 Tzafestas et al.31 studied collision avoidance for motion planning and Bruzzone and Signorile32 used repulsion among the entities to avoid collision with other entities. Furthermore, Helbing et al.12 was able to represent the collective phenomenon of escape panic. Nevertheless, all the above research have successfully exposed the model and simulation of agent movement with each of their own highlighted objectives. This research developed the agents movement model that is different from the above multi-agent models in respect of collision avoidance method. The collision method that is used in this research has characteristics that agents have sight distance and influence diameter (Fig. 5). Snapshot of simulation agents with sight distance is the left picture, whilst the right picture is with the circle barrier. If d(t), y and r are, respectively, representing the distance between the agents, interference of the closest agent in the area in front of the actor and the influence radius of agent, the expected velocity of agent ith, vâi (t) due to chi force to move away, is given by Eq. (15). vâi (t) =

µmax (2r − y(t)) µmax (2r − y(t)) = . ξd(t) ξ pk (t) − pi (t) (15)

By the alpha and chi forces, the agents can adjust the distance between two agents and are able to move away from each other. However, those two forces may not be able to prevent a collision when there are many agents in the arena. To further prevent a collision, a force that considers all surrounding agents is needed. For this purpose, it is assumed that each agent has an influence radius that represents his or her security awareness. By giving influenced radius they are able to keep a certain distance away from the nearest agent and to avoid collisions

Fig. 5

Collision avoidance characteristics.

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with a nearby agent. The force must be generated when at least two agents’ influencing radii partly cover each other as has been shown in Fig. 5. No repulsive force is generated if the influence radius does not overlap each other. Instead of considering the closest agent as the first repulsive force, the second repulsive force considers all surrounding agents and the forces are summed up linearly. Similar to the first repulsive force, the second repulsive force depends on the distance between the actor and other agents surrounding it. Since v(t) = 2 dp(t) and a(t) = dv(t) = d dtp(t) 2 , the formulation dt dt can be put together in terms of the current position of agent i, p(t) as a second order differential equation. Equation (16) is a nonlinear second order differential equation of agent positions that depend on each agent’s positions, speeds and accelerations. The analytical solution of the differential equation is very difficult and not practical since it is also dependent on the number of agents and the sight distance. Numerical method through simulation is more favorable and it has the benefit to visualize the movement of each agent in a plan as an animation. m

d2 pi (t) dpi (t) + dt2 dt 2r − y(t) e(t) − pi (t) + = µmax αe(t) − pi (t) ξpξ (t) − pi (t) 2r −1 + pi (t) − pi (t)

j

pj (t) − pi (t) × βpj (t) − pi (t)

.

(16)

The differential Eq. (17) vˆf (t) =

µmax e(t) − p(t) µmax g(t) = α α e(t) − p(t)

(17)

is solved numerically by the divide and conquer algorithm using Euler method, which provides adequate results while keeping the computational speed reasonable. Each equation is computed one by one, as each agent is assumed an autonomous agent. An agent has his own internal forces and influences other agents only through his position. The agent movement is based on the resultant forces that act upon him. Other numerical methods to solve the differential equation such as Runge Kutta may produce a better approach to the differential equation


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but it decreases the computational speed significantly if the number of agents is more than 100, thus, it is recommended for further study.

2.3. Modeling Fractal Patterns Defining the fractal equation or pattern that may have potential sharing features to crowd features needs many experiments. It is decided to find the pattern in the Holy Qur’an. Surah Yaasiin verses 36 says “Glory to Allah, Who created in pairs all things that the earth produces, as well as their own (human) kind and (other) things of which they have no knowledge.” Based on the knowledge from the Holy Qur’an, a fractal pattern with a pair of arms is created. A keyword from this verse is pair. Without trying to interpret it, a fractal pattern is able to create based on pairing. It is like a tree with a certain growing pattern. It starts with the first branch and grows up two branches and the pattern is iterated. The pattern is one branch generated toward left and right, as shown in Fig. 6, and some experiments using this fractal pattern have been conducted. The fractal model, which is created, starts with two branches, i.e. one branch generated “symmetrical”, toward left and right. Each branch has two variables, i.e. length and angle. In order to make the experiments more practical, a simple interface is created. The interface can be used in constructing a variant form of the pattern using a different length and angle of arm. Figure 7 shows the interface to make a series of experiments. By manipulating the branch length and angle, many formations can be created.

Fig. 6

Simple fractal.

Fig. 7

Simple fractal toolbox.

3. SIMULATION RESULTS: THE CROWD BEHAVIOR MODEL AND FRACTAL STUDY The simulation results are grouped into two subsections. The path resulted from the model is served before the path resulted from the fractal pattern.

3.1. Swarm Path Resulting from the Model The path graphs served in this section are extracted from the system created. The input snapshots of the system are shown in Fig. 8 whereas the output snapshots are shown in Fig. 9. In case of microscopic characteristics of agents’ movements, the effects of forces applied are explained from Fig. 10. It can be seen from Fig. 10

Fig. 8

Snapshot of inputs.


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Fig. 11 The effects of forward, and move away forces to agents’ paths.

Fig. 9

Snapshots of outputs.

Fig. 12 Forward and avoid collision. The effects of forces to agents’ paths.

Fig. 10

The effects of forward forces to agents’ paths.

that the forward force causes the paths to be a linear line. The effects of forces to agents’ paths means only alpha force applied, Eq. (5). It is shown the paths are linear, the agents may decelerate or accelerate but they do not change direction or repulse away. If repulse away forces are also applied as an effort to avoid collision, the agents’ path direction will be changed. Thus, repulse away forces direct the overtaking behavior, Fig. 11. The force Eqs. (5) and (6) were applied. It is clearly shown, some paths in Fig. 11 on the low lines are still linear because the density is not high enough or distance amongst agents is not close enough to stimulate the repulse away force. However, if avoid obstruction or collision force applied instead of repulse away force the agents’ path keeps linear, Fig. 12. It means the effects forward Eq. (5) and avoid collision forces [Eq. (7)] are applied. It is also noted that the paths are more distributed than the paths of forward forces application only, Fig. 10.

Fig. 13 The effects of forward, repulse away and avoid collision forces to agents’ paths.

It could be predicted that the paths are curving and the forces drive the overtaking behavior when the repulse away forces are applied again, Fig. 13. It means all three forces are applied, [Eqs. (5) to (7)]. In all cases of the experiment results (Figs. 10 to 13), it would be said that the paths have the same pattern but they are not self-similar. However, each figure has the same rules for every member of swarm, i.e. they move based on force(s) applied. In other words it is up repetition of the rules but they are not literally sequential. Therefore, the paths formed do not construct a tree as a fractal does.


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Fig. 14

The Sierpinski triangle, the game of chaos based on Barnsley.

Somewhat, the procedure of making a rule be applied in itself innumerable times, which is called recursion or recursive process, is iterated indeed. That means, each time this rule is applied a new result or a new path will be obtained. The difference between the swarms model and fractal path results would happen because of random generation inclusion. Further discussion of path resulted from fractal will be discussed in the following subsection.

3.2. Fractals Study and Agro-Biotechnology Results Here, some of the significant results produced using Matlab are presented. This is based on Michael F. Barnsley’s simulation of the game of chaos. The result is the Sierpinski triangle. The simulations are shown with various STEPS. The STEPS is the number of iterations thus the number of points in the image resulted. The following figures (Fig. 14) are simulation results with STEPS 100, 1000, and 10,000, respectively. Furthermore, the simulation of fractal application is applied in shaping both in 2D and 3D using Matlab and WRL software. Fractal 2D shape in Fig. 15 produces a remarkable shape of any biological structure. Meanwhile, the fractal 3D

Fig. 15

Two-dimensional fractal shaping.

Fig. 16

Three-dimensional fractal shaping.

shape in Fig. 16 produces an attractive shape of plantation. IFS (iterated function system) shaping simulation is exposed in another experimentation. The self-similarity dimension (also known as boxcounting or Minkowski dimension, Fig. 17) provides a measure of the degree of space-filling exhibited by a particular fractal curve.33 This has been increasingly applied as a means of characterizing data texture and shape in a large number of physical and biological sciences. On the other hand, Figs. 18 and 19 show the simulation results of half-fern fractal. Two basic variables are applied here, i.e. a number of points and a number of iterations. The result in Fig. 18 uses iteration as a variable with a fixed number of points. It is shown that the increased number of iterations does not significantly construct the desirable


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Fig. 17

The box-counting Minkowski dimension.

results. On the other hand, Fig. 19 gives a remarkable result by varying the number of points in fractal construction. Marana et al.34 used density estimation based on Minkowski fractal dimension. Fractal dimension has been widely used to characterize data texture in physical and biological sciences. The results of their experiments show that fractal dimension can also be

Fig. 18

used to characterize levels of people congestion in images of crowds. Some experiments, which manipulates branch lengths and angle of the fractal pattern with one branch generated symmetrical towards left and right (see Fig. 6), have been conducted. Figure 20 shows some results with the variant of symmetrical angle of branch but both asymmetrical and symmetrical length of branches. Both figures in Fig. 20 show that any curve path has not been formed yet if the fractal has symmetric angle of branch. Analogically, any scalar parameters [e.g. mass] in a physical-based model may not significantly influence the path direction, but any vector parameters [e.g. velocity] may cause bending of path direction. Furthermore, Fig. 21 produces a very significant pattern of fractal that can contribute towards a very complicated structure of swarm research. Figure 21, i.e. asymmetric angle of arm and the arm length both symmetric and asymmetric have shared characteristics of curve emerging behavior with the agent microscopic model developed that applied repulse/move away forces (see Figs. 11 and 13). However, these shared characteristics have not been mathematically proven except from the curve paths seen.

A hundred-points half-fern with various iterations.

Fig. 19

11

Fifty iterations half-fern with various points.


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Fig. 20 Left: Symmetric angle and symmetric length of branch. Right: Symmetric angle and asymmetric length of branch. Fig. 23

Fig. 21 Left: Asymmetric angle and asymmetric length of branch. Right: Asymmetric angle and symmetric length of branch.

The most important advantage of fractal usage to model swarm is that the whole members are relatively more uniformly distributed than any other models. Uniform distribution will achieve uniform swarm density. The occupied density is an influenced characteristic to control any swarm, crowd, flock or other large massive group of agents. The same experiment with equal arm’s length and 60◦ equal arm’s angle resulted in an emerging pattern of a beehive-like formation (Fig. 22). It is common knowledge that a bee is a grouping creature. Are there any relationships between group behavior and fractal characteristics? This question needs further research. Predictably, a straight line formation can be built by equal arm’s length and 0◦ as shown in Fig. 23. The straight line formation corresponds to the straight path of agents (in Fig. 10) when only a forward force is applied to the agents. Let

Fig. 22 angle.

Result with equal arm’s length and 60◦ equal arm’s

Result with equal arm’s length and 0◦ .

straight lines represent swarm member lanes, ideally the more dense the members are the more lanes must be formed. Consequently, swarm should be composed of cooperative agents in such an adaptive behavior that encodes a collective pattern. This adaptive pattern can be modeled from a fractal. As the research focuses on the formula with the basic shape and form of a fractal, the fractal images created were not further processed by transformations that transform and warp the shape of a fractal and combine various transformations to create complex effects. The research also does not involve coloring algorithms to give beautiful and complex images. However, Fig. 21 produce a very significant pattern of fractal that can contribute towards a very complicated structure of the crowd path.

4. PATH COMPARISON AND VALIDATING OBSERVATION This section compares the characteristics of the microscopic model result and fractal. Some research about fractal and its application in topics of crowd also motivate the project. Paths resulting from experiment of crowd microscopic model with forward forces applied (Fig. 23) in a swarm has sharing characteristics with paths resulting from the fractal experiment with equal arm’s length and a small degree e.g. 5◦ ) of arm’s angle (Fig. 24). In addition, the exit model could be counterparted with fern fractal using transformation manipulation, Fig. 25. It is clearer than original when it is zoomed in, Fig. 26. Regarding the validation and verification, crowd compiler can be used to visually compare between the real world (Fig. 27) and and simulation snapshots. Results are not validated with real data but with other models studied from literature. One validation


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Fig. 26

Fig. 24

13

Zoom in fern fractal with manipulation.

Result with equal arm’s length and 5◦ arm’s angle

Fig. 27

Fig. 28

Fig. 25

Fern fractal with manipulation.

is briefly evaluated with cellular automata (CA). CA are models that are discrete in space, time and state variables. To describe the state of a lane using a CA, the lane is first divided into cells of agent length. This corresponds to the typical space (agent’s length + distance to the preceding agent) occupied by an agent in a dense jam. Each cell can now either be empty or occupied by exactly one agent. Each agent is characterized by its current velocity v which could take the values V = 0 to Vmax . Here Vmax corresponds, e.g. to a speed limit and is therefore the same for all agents. An example

Crowd compiler.

CA configuration for validation.

of simple cases, a configuration of the lane is shown in Fig. 28. Nagel and Schreckenberg35 have introduced a rule set, which led to a realistic behavior. It consists of four steps that have to be applied at the same time to all of the agents (parallel or synchronous dynamics). • Step 1: Acceleration. All agents that have not already reached the maximal velocity, vmax , acceleration by one unit: v → v + 1. • Step 2: Slowing (safety distance). If an agent has d empty cells in front of it and has its velocity v (after step 1) larger then d, then it reduces the velocity to d : v → min(d, v). • Step 3: Congested (randomization). With probability p, the velocity is reduced by one unit (if v after step 2): v → v − 1.


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Configuration at time t

(a) Acceleration (νmax = 2)

(b) Braking

Fig. 30

Toolbox crowd setting used for validation.

(c) Randomization (p = 1/3)

(d) Driving (= configuration at time t + 1)

Fig. 29

CA configuration used for validation.

• Step 4: Moving ahead (driving). After steps 1–3 the new velocity vn for each agent n has been determined forward by vn cells: xn → xn + vn . Figure 29 shows a configuration at time t + 1 which is updated step-by-step to obtain the new configuration at time t. Step 1 describes the desire of the agents to move ahead as fast as possible (or allowed). Step 2 encodes the interaction between the agents. In this simple model, interactions only occur to avoid accidents. Step 3, in a very simple way, corresponds to many complex effects that play an important role in real crowd lane. Usually, a single agent will not move with a constant speed, but there are always small fluctuations of the velocity. An important point is overreactions at braking. An agent that had to decelerate (slowing) in step 2 will, with some probability p, decelerate even further than necessary to avoid a collision. This kind of imperfect movement can lead to a chain reaction, if the density of agents is large enough. In the end, it might lead to the stopping of an agent which leads to the creation of a jam. This jams occurs without obvious external reason and is therefore called “jam out of nowhere,” “phantom jam” or spontaneous jam. It shows the extreme importance of step 3, which reflects the imperfect behavior of the agents.

Fig. 31

CA-trajectories used for validation.

Finally, in step 4, all agents move according to their new velocity. With the use of Visions Of Chaos,36 some simulations can be conducted within a toolbox, Fig. 30. Whereas the result is generally served as trajectories graph, Fig. 31. It is interesting to note that the gradient and the intercept of the graph resulting from the experiment are different from the traffic (Fig. 32). The developed model result is not consistently negative gradient. First of all, the approach is not a


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jammed. The starting time for agents to interact could heavily determine the shape of the graph.

5. CONCLUSIONS

Fig. 32

Fig. 33

Speed and density comparison.

Flow versus density in CA traffic flow model.

traffic approach with a fixed number of lanes, but it is an adaptive lane both in lane numbers and direction. Second, the modeled agent characteristics may contribute to a difference result from the prior results. Alternatively, the result may get similar if the influenced agent diameter and sight agent distance are adjusted according to the speed applied. However, a literature 37 also shows that a negative gradient occurred in cellular automata for traffic flow model, Fig. 33. The graph describes the connection between density and flow rate on the lane. When the density is low, that is, agents are far from each other, the flow increases linearly with increasing density. When the density reaches certain value, agents start to “interact” with each other, agents become cautious and lower their velocities to maintain a safe distance to the agent ahead. The lowering of velocities causes the flow to decrease. If the density still increases, agent velocities will be getting lower and finally the flow rate drops to zero when crowd is completely

The research revealed that the microscopic swarm model studies have been successfully applied to explore the behavior of microscopic agents flow by showing the three influenced forces representing forward velocity, repulsive away of single agent reaction, and repulsive away of multi-agents’ reactions. It is clearly shown that fractal could be used to approach the emerging swarm behavior because the research has produced the result that has sharing characteristics with crowd microscopic model. Fractal study presented in this research introduces a very significant result in producing a complicated pattern specifically for crowd behavior research. The crowd behavior research has been introduced and has opened a lot of opportunities for computer scientists to get involved. The result from the game of chaos suggests, the more STEPS results, the clearer the shape of the Sierpinski triangle. Simulation results of fractal shaping application whether in 2D or 3D would suggest that random choices will give a deterministic picture. Fractal image compression is recognized as one of the most effective image compressions. The microscopic agent simulation model is developed to determine the microscopic characteristics of swarm. From the simulation results, there is relationship between density and agent speed. The relationship is influenced by maximum speed and velocity distributions. The resulting agents paths are studied and compared to paths that are created from the fractal. The research demonstrated that fractal could be used to approach the emerging swarm behavior. Since the research has produced the result that has shared characteristics with crowd microscopic model, modeling swarm with fractal equation would be potentially conducted in the future.

ACKNOWLEDGMENT The authors would like to thank Kardi Teknomo, Tohoku University, Japan and the Research Management Centre, Universiti Teknologi Malaysia.

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