Modeling and Simulation of a Swarm of Robots for Box ... - CiteSeerX

Modeling and Simulation of a Swarm of Robots for Box-pushing Task Yangmin Li and Xin Chen Department of Electromechanical Engineering, Faculty of Science and Technology University of Macau, Av.Padre Tomás Pereira S.J., Taipa, Macau SAR, P.R.China, Email: {[email protected] | [email protected]}

Abstract This paper investigates the application of swarm intelligence principles for box-pushing task, and proposes the mathematical model of the system. The paper analyses the system in two scopes: microscope and macroscope. Firstly, the structure of individual robots is described briefly. Secondly, in macroscope, based on the dynamic process theory, the paper proposes the dynamic equations of the system. The solution of the equations reveals the mechanism of cooperation among robots. And based on the solution and simulations, the paper discusses how to realize obstacle avoidance during the process of box-pushing.

1. Introduction There has been increasing research interest in swarm behavior. Two researching fields are regarded as the sources of swarm behavior: social insects in mathematical biology, and groups of interacting autonomous robots in engineering. Different from traditional multi-agent paradigm, which is based on deliberative agents and central control, swarm paradigm need no central controller to direct the behavior of the system. In other words, swarm systems are self-organizing Since it is social insects that inspire the research on swarm behavior, “understanding the nature of coordination in groups of simple agents is a first step toward implementing useful multi-robot system.”[1] Many researchers concentrate upon exploring the underlying mechanism of social insects [2]. Based on researching of biology systems, the community introduces swarm behavior into engineering, and has achieved many useful conclusions. In general, it’s believed that swarm systems are more flexible, more robust, and, more economical than traditional multi-robot systems. These advantages result from characteristics of swam systems: 1) Decentralization; 2) Homogeneity; 3) self-organization; 4) simplicity.

Now swarm intelligence has become an alternative approach to classical artificial intelligence. The paradigm of swarm systems is complete distributed control. There is no central controller directing the behaviors of swarm systems. So, the group behavior ‘emerges’ from individual behaviors of agents, or, robots in Robotics. It’s called as ‘self-organization’. Normally, agents in swarm systems need no explicit communication. Instead of it, stigmergy communication is an alternative way. That means all agents sense each other through the environment. Box-pushing has been an important benchmark for testing swarm architectures in swarm community. describes An ant-like transport system was proposed [3], in which robots cooperate with each other with local information. A transportation model was proposed on the basis of social insects, which can handle the uncertainty in transportation [4]. We think that the essential question of swarm design is to find out a mechanism that relates individual characteristics to the collective behavior of the entire system. So, it’s required to analyze swarm systems by mathematic model. It has been a challenge in swarm community. Microscopic and macroscopic methodologies were presented in [5], which based on Markov models for predicting the dynamics of swarm system. The paper also presented a physical system and simulations to illustrate their viewpoints. In addition, A mathematical model was constructed for asynchronous swarm with a fixed communication topology [6]. A new methodology was introduced to model coalition formation of electronic markets [7]. In this paper, the goal is to analyze Markov properties of the system and figure out the mathematic model of the box-pushing task by swarm paradigm, and discuss the superiority of swarm intelligence.

2. Behavior-based Robots Fig. 1 shows the simulation interface of the box-pushing. The center disk represents the box. And the

small disks represent the robots. It’s assumed that few robots can not move the box. So, the aim of the task is to push the box to the goal area marked as a sphere, shown on the right bottom corner.

Figure 2.

Figure 1. Simulation interface. The swarm is composed of a group of homogeneous robots. All robots are controlled by behavior-based control method [8]. The behavior layers include: 1) Wandering behavior: It is the lowest layer that makes robot wander randomly in the arena. In every certain period, it will send random speed command to two motors. 2) Chasing behavior: When robot wanders around the arena, the chasing behavior perceives the arena by using three photoelectric sensors. The one is for perceiving the light that denotes the goal area, and the other two are for perceiving the box. If the robot is behind the box and can push the box to the goal area, the behavior will suppress the output of the wandering behavior, and makes robot move to the box directly. 3) Pushing behavior: The behavior always checks the outputs of two bump switchs installed in the front of the robot. If the robot touches something and judges the object is the box through optical sensors, the robot will adjust its pose and push the box. 4) Stagnancy behavior: The behavior check robot’s movement. If robot is stagnant, the behavior will count the stagnancy time. Once the stagnant duration is greater than a certain threshold, the behavior will compel the robot to obey wandering behavior for a period. It should be emphasized that pushing is also regarded as a kind of stagnancy. That means even a robot is pushing the box, it will quit the pushing action after certain time and wander randomly for another period, this important for the system. In the control architecture, the higher layer behaviors can suppress the output the lower layer behaviors. Then the whole control architecture is shown in Fig.2.

The behavior-base architecture

Obviously, if there are not enough robots pushing the box together, the box could not be moved. So, robots need to cooperate with each other. But from the description of four-layer microscopic construction of the system, there is no direct behavior for cooperation. Even there is no communication behavior to coordinate their actions. In the following analysis of macroscopic model of system, we will explain that the model of cooperation is the property of system’s structure, and need no explicit cooperation indention at all.

3. Macroscopic model of the system Normally, the swarm behaviors can be regarded as a kind of dynamic process. So, a mathematical model is an idealized representation of a process. A mathematical model can describe a swarm system at two levels. One is microscopic model, which describes the agent’s interaction with other agents and the environment. The description of individual architecture refers to describe the microscopic model. The other is macroscopic model, which describes the collective group behavior of a multi-agent system. The swarm robotic system is composed of a group of homogeneous robots. Individual’s behavior is determined by itself. And among the robots, there is no explicit communication. Now, we explain how coordination behavior ‘emerges’ from interaction among robots. To simplify the analysis, we assume that obstacle avoidance can be ignored while wandering. And pushing behavior can count the pushing duration. Once the count exceeds a threshold, robots quit pushing, and wander randomly.Then the behavior of stagnancy can be canceled. The behaviors can be reduced to three ones: Wandering, Pushing, and Chasing. Firstly, we define the states of the system S i as: S1 : The average number of robots in state of Pushing; S 2 : The average number of robots in state of Chasing;

S 3 : The average number of robots in state of Wandering. φ is defined as the total number of robots, then:

φ=

∑S . 3

(1)

i

i =1




S=

∑ S qˆ 3

i =1

i

i

,

(2)

where vector qˆ i is a unit vector to represent each state.

Figure 3.

State diagram.

Fig. 3 shows the system’s states transition. In fact, it’s also individual robot’s state transition. Briefly, the transition can be described as: when a wandering robot finds the box, it changes from wandering state into chasing state to move to the box. And once it touches the box, it adjusts its pose to change into pushing state. But instead of pushing the box until reaching the goal area, it just pushes the box for a certain of period. And then it will give up pushing and wander again. In the latter section, we will explain why we adopt such pushing duration. Because of such individual state transition, the number of robots in each state also increases or decreases 1. From the description, it can be concluded that the system obeys Markov property. More accurately, the mathematic model analyzed in the paper is a states-discrete, parameters-continuous Markov chains. That means the future motion of the system is determined by the present states, not by the past. And the dynamics of system is time continuous. If the probability of state S
is defined as P (S , t ) , then,


P (S , t + ∆t ) =

∑ P(S , t + ∆t S ′, t)P(S ′, t) .











(3)

S′

For the continuous system, the transition rate is defined as the limit of transition probability:

W (S S ′; t ) = lim




P( S , t + ∆t S ′, t )

∆t → 0

∆t

.

(4)

So, the normal dynamic equations of the system is [9]: ∂S i = ∂t

So, the configuration of system can be expressed as:








∑w j




ji

(S )S j + S i

∑ w (S ) .


ij

(5)

j

To get the dynamic equation of the system, the following assumptions are proposed: 1) Once a robot perceived the box, it can adjust its direction to the box instantly. That means, the time for pose adjustment can be ignored. And robot moves to the box at a constant speed of v . 2) Comparing with the size of box, the size of robots can be ignored. That means there’s no density limit. And the quantity of robots is large enough. Then, we need not consider the limit of pushing robots. 3) Because the box’s shape is cylinder, the force acting on the box is through its center of mass, which is also the geometry center of box, and parallel with horizon. 4) The origin of reference frame is on the center of the box. And it’s assumed that, comparing with the velocity of robots, the speed of the box can be ignored. So, no matter the box is static or dynamic, robots will move to the box at a speed of v . Secondly, there are some definitions: 1) R : The radius of the box. 2) T P : The average pushing time. i.e., the period in which the robot pushes the box. 3) L : The boundary of a valid perceiving area. Every robot has identical perceiving ability, i.e. the individual perceives bound. Consequently, from the view of the box, there exists an area around it. If robot is in this area, it can perceive the box and move to the box directly. This area is called as the box’s valid perceiving area. And its boundary is described as L . The max distance between the boundary and center of box is L0 . Fig.4 shows a kind of simple perceiving area, which is the sector marked by dashed line. And the maximum of the boundary is 50 units. 4) H : The template of the attraction. The action of moving to the box can be regarded as a kind of attraction, therefore we need a template to describe such attraction field. According to the assumption 2), robot chases the box at constant velocity of v . It can be regarded as the product of certain constant and gradient of the attraction potential field. Because the velocity is constant, the gradient of the template should point to the center of the box, and the intensity of the gradient should be 1. So, the template is: H = L0 − x 2 + y 2 , v = A ⋅ ∇ R H = 1 ,

(6)

where R represents the position of the robot. In Fig. 4, circle lines denote the template. Of course, the template’s effect range should be equal to or larger than the valid perceiving area. Obviously, outside of the valid perceiving area, this attraction is of no effect.

Figure 4. The template and the valid perceiving area of the box. The radius of the box is 10cm . L0 = 50cm .

(8)

where S1 = C1 dθ , S = C dxdy . 2 ∫ ∫∫ 2 R







The dynamic equations can be obtained according to general expression Eq.(5). But Eq.(5) only reflects temporal change of the system, we should also find out special distribution of robots to reveal how to push the box. For example, if all pushing robots distribute around the box averagely, the composition of forces acting on the box is zero. And the box would not be moved. So, what we should concern is the density change of robots around the box. Of course, the density is differential of states on space, S i . Then, based on these assumptions and definitions, the dynamic equation of the system can be expressed as following: ∂C1 1 (7) = − C1 + Rv C 2 R , ∂t TP ∂C2 = f (φ − S1 − S2 ) − v∇ ⋅ (C2∇H ) , ∂t

robots attach to the box is T P . Averagely, there is 1 TP robots depart from the box per unit of time. At the same time, vC2 |B represents the rate of chasing robots that touches the box and pushes it. For unit coordination, vC2 |R multiplies with the radius of the box, R . It must be figured out that we can prove Eq. (7) is valid only when the template is in the form of Eq. (6). The second equation describes the dynamics of robots moving to the box, or the arrow from wandering state to chasing state. f (φ − S1 − S 2 ) represents the distribution density of robots that changes from wandering behavior into chasing behavior per unit of time. v∇ ⋅ (C2∇H ) describes the interaction between robots and the template, i.e. the attractiveness of the template gradient. Obviously, we take the action of chasing as a kind of diffusion. There exists a trend that robots move along the gradient of attraction field. Obviously, the box can move only under the condition that the composition of forces acting on the box is big than the static friction. If we assumed that every robot can provide the same pushing force, D f , the condition is expressed as:

S

C1 represents the density of robots for pushing the box. Because we ignore robots’ size, C1 is one dimensional distribution of number of pushing robots around the box. C 2 represents the density of robots in the valid perceiving area, i.e. the density of robots moving to the box. It’s two dimensional distribution of number of chasing robots. The first equation describes the dynamics of robotic density around the box, or, the arrow from state ‘chasing’ to ‘pushing’ in Fig. 3. Because the average time that

2π

F > Fstatic , where F = D f ∫ e iθ C1 dθ , 0

(9)




and the direction of F points to the goal area. So, the sufficient condition that the system will push the box to the goal area is that Eq. (9) is satisfied when ∂C1 ∂t = 0 , and ∂C2 ∂t = 0 , or, −

1 C1 + RvC 2 TP

R

= 0,

(10)

(11) f (φ − S1 − S 2 ) − v∇ ⋅ (C2∇H ) = 0 . Solving Eq.(10) and Eq.(11), and expressing the solution in polar coordinates: T f (φ − S 1 − S 2 ) 2 (12) (L − R 2 ) , C1 = P 2 C2 =

. f (φ − S1 − S 2 )  L2    r − r 2v  

(13)

In fact, from the state paradigm and Eq.(7) and Eq.(8), we know that the states of the system form a limit event set. And the Markov chain is an irreducible chain. If we assume that f (φ − S1 − S 2 ) is time-invariant. Say, it only relates to the number of robots in three states. And in the beginning, the box is away from the goal area far enough, there must be a stationary distribution of Markov chain during the process of moving the box. Obviously, this stationary distribution is expressed as Eq.(12) and Eq.(13).

4. Analysis on solutions From Eq.(12) and Eq.(13), it can be observed that if the

distribution density of robots changing into chasing behavior is known, the density of robots pushing the box is affected by T P , and L . So, proper design of average pushing time and perceiving area of a single robot can make robots work together! The cooperation among robots is a property of the system’s structure. An example is illustrated with following assumptions: After robots depart from the box, firstly they will wander randomly for a period without perceiving the arena. The average duration is TW . Then robots perceive the arena for a short time. If nothing found, they will change into random wandering state again. Or, they will move to the box. TW is long enough that position at which a robot suspends its perceiving ability has no relationship with the position at which the robot resumes perceiving ability. Consequently, γ (14) (φ − S1 − S 2 ) , f (φ − S1 − S 2 ) = TW S where γ represents the probability that robot is in the valid perceiving area when it resumes perceiving ability. S represents the area of the valid perceiving area. So, Eq.(14) means that the robots which are changing from wandering into chasing behavior ordinarily distribute in the valid perceiving area. Rewriting the expression of C1 as Tγ (15) C1 = P (φ − S1 − S 2 )(L2 − R 2 ). 2TW S If it is assumed that, T P = 3S , TW = 3S , γ = 0.4 , v = 40cm / S , the template and the valid perceiving area are as the same as Fig.3 shown, when Eq.(10) and Eq.(11) are satisfied, the average density of robots pushing the box is C1 = 6.27rad −1 . If D f = 10 N , the average composition of forces acting on the box is FMax = 108.6 N . And the force points to negative Y-axis. So, if FMax > Fstatic , the box will be moved. The pushing time T P plays a very important role in the system. Normally, all robots are required to get together to push the box in proper direction. Once robots attached to the box, they should not depart from it until the box arrives at goal area. Why do robots depart from the box after T P ? T P describes the property, homeostasis, of the system. It makes the system be more flexible. It’s also one of properties of swarm systems. If the pushing duration is infinite, all robots would attach to the box unavoidably. Then the dynamic process falls into a static state, Eq. (7) and Eq.(8) would be useless any more. Maybe collecting all robots’ power is a high efficient way to accomplish the task, but it will deduce the flexibility of the system. For example, if there is an obstacle on the way of the box, how do robots avoid it? Traditional methods require robots abandon pushing and perceive the obstacle’s position to reform the group. This will increase the cost of

decision-making. But if the system is homeostasis, there are always a part of robots wandering around the box. When they are near the obstacle, they can perceive the obstacle. At the same time, if they also find that the box is near them, they will move to it and work with other pushing robots to drive the box away from the obstacle in spite of whether they are in the valid perceiving area of the box or not. Or, another reasonable explanation is that, the obstacle changes the valid perceiving area of the box. Fig. 5 shows a simple example of this change. The red disk represents the box, the blue disk represents the obstacle, and the gray area is the valid perceiving area of the box. When the box is closed to the obstacle, just as 5(b) shows, the valid perceiving area is changed, because some robots around the box will move to the box in spit of the perceiving original area shown in 5(a). The arrow indicates the direction change of the composition of forces in two figures. Fig.6 is the simulation result. Fig. 6(a) shows the traces of the box in ten simulations. There are 25 robots in simulation. The box’s radius is 10 cm . The obstacle’s radius is 18cm . In each simulation, the box was required to be pushed from the origin to the goal area, (80, 80)cm. From the simulation, we can observe all traces of the box avoiding the obstacle. Figure 6(b) shows the relationship between average quantity of robots pushing the box and distance form the obstacle to the box. Then, without explicit indention for obstacle avoidance, the system can form a trajectory that rounds the obstacle. That means, without modifying anything about the individual strategy or adding any complex coordination methodology, the swarm can accomplish more complex behavior.

(a)

(b)

Figure 5. The change of the valid

perceiving area of the box.

Now review the assumptions mentioned above. Assumption 1) affects the rate of distribution density of robots changing into chasing behavior f (φ − S1 − S 2 ) . Assumption 2) means the density of robots around the box should not be limited, if the robots’ size and quantity can not be ignored, there would be a limit of C1 . Assumption 3) ensures that Eq.(9) is satisfied. The last assumption

ensures that the relative velocity of robot moving to the box would be constant. Obviously, these assumptions have no effects on the structure of dynamic equations Eq.(7) and Eq.(8), hence the model is valid.

large scale of robots to work together. At this stage, we have constructed a swarm of robots, and experimental study will be carried out later.

6. Acknowledgements This work was funded by the Research Committee of University of Macau under grant no.: RG024/03-04S/LYM/FST.

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7. References

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[1] C. R. Kube and E. Bonabeau, “Cooperative Transport by 40

Obstacle

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[2]

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[3]

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(a)

[4]

Averrage quantity of pushing robots

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[5]

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[6]

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[7]

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1.5 25

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Distance from the obstacle to the box

(b)

Figure 6. Simulation results of obstacle avoidance.

5. Conclusion Cooperation is a base property of multi-robot systems. The paper provides a paradigm of swarm intelligence to solve box-pushing issue and proposes the sufficient condition for achieving the task. It can be concluded from above analysis that cooperation can be achieved by proper design of the system’s structure. The dynamic equilibrium of system design brings about high flexibility. Because the requirement for individual robot’s intelligence and communication is very simple, it is feasible to construct a

[8] [9]

Ants and Robots”, Robotics and Autonomous System, Vol. 30, No.1, 1999, pp. 85-101. C. R. Kube and H. Zhang, “Collective Robotics: From Social Insects to Robots”, Adaptive Behavior, 2 (2), 1993, pp. 189-218. D. J. Stilwell and J. S. Bay, “Toward the Development of a Material Transport System using Swarms of Ant-like Robots”. Proceedings of IEEE International Conference on Robotics and Automation, Atlanta, USA, 1993, pp. 766-771. P. Lucic and D.Teodorovic, “Transportation modeling: an artificial life approach”, Proceedings of 14th IEEE International Conference on Tools with Artificial Intelligence, Nov. 4-6, 2002 , pp. 216 –223. A. Martinoli and K. Easton, “Modeling Swarm Robotic System”, Proceedings of the Eight Int. Symp. on Experimental Robotics, 2002, Sant'Angelo d'Ischia, Italy, pp. 297-306. Y. Liu, K. M. Passino, and M. M. Polycarpou, “Stability Analysis of M-Dimensional Asynchronous Swarms With a Fixed Communication Topology”, IEEE Transactions on Automatic Control, Vol. 48, No. 1, 2003, pp. 76 –95. K. Lerman and O. Shehory, “Coalition formation for large-scale electronic markets”, Proceedings of the 4th International Conference on Multiagent Systems, July 2000, pp. 167 – 174. R. A. Brooks, “A Robust Layered Control System for a Mobile Robot”, IEEE Journal of Robotics and Automation, Vol.2, No.1, 1986, pp. 14-23. C. W. Garnier, Handbook of Stochastic Methods, Springer, New York, NY. 1983.