A Simple Approach to the Multi-Predator Multi-Prey Pursuit Domain

Transcription

1 A Simple Approach to the Multi-Predator Multi-Prey Pursuit Domain Javier A. Alcazar Sibley School of Mechanical and Aerospace Engineering Cornell University 1. Abstract We present a different approach to a class of pursuit games: the Multi-Predator Multi- Prey domain. In the typical game, a group of predators tries to capture a group of preys, and all the agents have perfect knowledge of the prey and predator positions. In our problem definition the prey-agent and the predator-agent have only local information provided by its vision range, each predator independently tries to capture a prey in a one-predator-one-prey-pair way. The predator-prey-pair capture is not known in advance and both the predators and the preys are moving in the environment. We show that a simple greedy local predator rules are enough to capture all the preys. 2. Introduction and Previous Work This class of pursuit game has become a popular domain for the study of cooperative behavior in Distributed Artificial Intelligence (DAI). The pursuit domain was introduced by Brenda et. al. [Brenda 1986]. In his formulation, the prey moves randomly, and the predators can occupy the same position. In his work he made used of the center of gravity of the agents. Stephens and Merx [Stephens 1989], [Stephens 1990] experimented in a domain that did not allow two agents to occupy the same position, the predators alternated moves and the prey moved randomly. Their successful way to capture the prey made used of a centralized control mechanism. Korf [Korf 1992] introduced a simple solution to the pursuit domain. His simple algorithm made used of attractive forces between the predators and the prey, and repulsive forces between the predators. In his approach the predators had knowledge of the existence of all the other predators and the prey, then every single

2 2 predator compute the resultant force to choose their next move. All the approaches mentioned above share, among other things, two important properties: (1) the predators had knowledge of the existence of all the other predators and the prey, and (2) The predator s goal is to capture only one prey. L. E. Parker [Parker 2002] has studied a more complete problem, the one that she named Cooperative Multi-Robot Observation of Multiple Moving Targets or CMOMMT for short. In her domain the goal of the robots is to maximize the average number of targets that are being observed by at least one robot. The problem presented in this paper has the goal of maximize the number of capture preys by the predators. Observation in not enough, the predator will need to move in a greedy way to maximize the number of captured preys. We look at the multi-predator multi-prey domain, where the property (1) is dropped it and replaced with limiting sensing capabilities by the predators and the preys. This assumption is made because property (1) is inconsistent with biological systems since no creature has infinite sensing range. We also extended the problem by having several preys, this assumption introduce the problem of predator-prey assignation. 3. Multi-Predator Multi-Prey Problem Description The Predator-Prey pursuit domain has many different instantiations that can be used to illustrate different multi agent scenarios. The Multi-Predator Multi-Prey scenario used in the present paper has the following characteristics: 1. The arena or world is two-dimensional, bounded and continuous with a square shape. 2. We define a prey capture when the predator is on top of the prey. 3. Predators can move linearly choosing on any angle from 0 to 360 degrees. 4. Predators can occupy the same position. 5. There are n predators and m preys on the arena. 6. Predators and preys have a 360 degree field of view observation of limited range. 7. Predators and preys will move simultaneously. 8. Prey movement will be evasive. 9. No predator communication. 10. Predators and preys will bounce off the walls using the reflexion angle law: angle of incidence is equal to angle of reflexion. 11. Predators can move faster than the preys. 12. Preys can see farter than the predators.

3 3 Goal: To capture as many preys as possible, i.e. if m>n we can not capture more than n preys, and if m<=n we will capture all m-preys. Measure of performance: The total time to intercept (TTI) and capture as many preys as possible. 4. The Approach The way we propose to capture all the preys is by asking the predators to sweep the terrain by changing the direction its heading. The predators will make a left every time they travel the maximum distance in the terrain. This will be more precisely defined once we define the metrics of the multi-predator multi-prey domain The Multi-Predator Multi-Prey Metric We believe that the first step towards given a solution to this domain would be defining in numerical terms the characteristics listed in part 3 of this paper. For this purpose we will define: D = Diagonal distance across the arena. r = Robot (predator) radius, i.e. physical radius dimension. R = Robot (predator) vision radius, i.e. how far they can see. rp = Robot (prey) radius, i.e. physical radius dimension. Rp = Robot (prey) vision radius, i.e. how far they can see. v = Velocity (magnitude) of the predators. vp = Velocity (magnitude) of the preys. Figure. 1. Robot representation. Figure. 2. World representation.

4 4 Define the following ratios: C1 = 2r/D C2 = 2R/D C3 = Rp/R C4 = D/v C5 = vp/v In this paper we consider homogeneous predators and homogeneous preys, i.e. the parameters r and v are the same for all the predators, and rp and vp are the same for all the preys. Therefore we will have same size predators with same magnitude velocity, and same size preys with same magnitude prey velocity. Ratio s interpretation: C1 is a measure of the physical space used by the predator s dimension. C2 is a measure of the predator s vision coverage. C3 is the prey to predator vision ratio. C4 is the time T it takes for a predator to travel the maximum distance D. C5 is the prey to predator velocity ratio. More formally, the approach is to ask each predator to sweep the terrain changing the direction its heading, by -π/2, at every multiple of C4. In the rest of the paper we will consider that prey movement will always be evasive, and that the predator will either (a) sweep the terrain without changing the direction its heading, or (b) sweep the terrain changing the direction its heading, by minus 90 degrees, at every multiple of C4. Therefore the quantities (D, C1, C2, C3, C4, C5, n, m) will perfectly define a homogeneous multi-predator multi-prey pursuit domain, and we will say that (D, C1, C2, C3, C4, C5, n, m) will constitute the metrics of the pursuit domain Simulation of the Multi-Predator Multi-Prey domain. To evaluate the effectiveness of simple predator, as described in 4.1 above, we conducted several simulations where the Predators: (a) sweep the terrain without changing the direction its heading, (b) sweep the terrain changing the direction its heading, by minus 90 degrees, at every multiple of C4, or (c) sweep the terrain in spiral patterns. We are interested in the following question: How a change in C5 will affect the time to intercept the preys (TTI)? Like in [Parker 2002] we will assume that C5 < 1. This assumption allows predators an opportunity to capture the preys, if the preys could always move faster, then they could always evade the predators and the problem becomes trivially impossible for the predator team (i.e., assuming intelligent preys ) [Parker 2002]. For predator evasion to take place, the preys were privileged with a larger range of view, i.e. C3 >1. Case (a) is being used as a baseline for comparison with cases (b) and (c). In the following computer simulations the

5 5 preys will move linearly evasively, i.e. they try to avoid detection by nearby predators and if the preys did not see a predator within its vision range, it moved linearly along its current direction of movement. At the beginning of each experiment, predators and preys are randomly positioned and oriented in the world. We ran several simulations for the three cases: (a) sweep the terrain without changing the direction its heading Simple sweep, (b) sweep the terrain changing the direction its heading, by minus 90 degrees, at every multiple of C4 Smart sweep and (c) sweep the terrain in spiral patterns Spiral sweep. The idea behind the second and third approaches is to apply a simple local rule that will produce a good emergent behavior. In the context presented in [Hackwood 1991] SWARM Intelligence will emerge, i.e. in [Hackwood 1991] Swarm intelligence is defined as a property of systems of non-intelligent robots exhibiting collectively intelligent behavior. We will see that asking that the predators to sweep the terrain changing the direction its heading, by minus 90 degrees, at every multiple of C4 will give such an intelligent collective behavior. In some other context the second approach is referred as emerging complex behaviors from the bottom up. In the following simulations the predators are colored blue and the preys are colored red. One run of the simulations is shown in figure 3, where we have 10 predators and 4 preys that have been caught in TTI = T units. The metrics for figure 3 was (80 2, 0.02, 0.1, 1.2, 0.5, 0.9, 10, 4) Figure. 3. Simulation explanation.

6 6 5. Results The following results were obtained using the following metrics: (80 2, 0.02, 0.1, 1.2, 0.5, C5, 10, 4) 2 average plot (num predators, vision prey / predator)= (10,1.2) 1.8 Time To Intercept (TTI) [sec] C5= Vprey / Vpredator 2 Figure. 4. Simple sweep. average plot (num predators, vision prey / predator)= (10,1.2) 1.8 Time To Intercept (TTI) [sec] C5= Vprey / Vpredator Figure. 5. Smart sweep.

7 7 Figure. 6 Spiral sweep. Every vertical line on the plot represents 100 runs of the domain with the mean and one standard deviation. We can see from the figures 4, 5 and 6 that the smart sweep performed better than the simple sweep and the spiral sweep in average. The intuition will tell us that searching in a new direction every time we had travel the maximum distance in the terrain would result in less time to capture all the preys. 6. Conclusions and future work The presented domain has a highly dynamic environment. We assumed that no prior knowledge of the terrain was given and the main question to be answered is: using reactive predators, can we come up with intelligent ways to minimize the time to intercept all the preys? A reactive predator is such that the decision of action or behavior in the terrain will take almost no computational time. Therefore its actions will be like reflexes.

8 8 Self-organization in the predator team can be claimed on the smart sweep, because the predators have the ability to distribute itself optimally for the particular task of capturing as many preys as they can. The area of multi agent and multi robot systems is experiencing an explosive research interest. Of particular interest are the systems composed of multiple autonomous mobile robots exhibiting collective behavior. Behavioral-based approaches have been used to implement formation control [Balch 1998]. We believe that a predator formation will not reduce the time to intercept in the presented domain. It seems that forming clusters will not be a good searching strategy; on the other hand, a predator spreading strategy seems to be closer to a better search. We have introduced a simple attempt to solve real time problems of the multi-predator multi-prey pursuit domain. This constitutes a basic step towards more advance and fast ways to implement algorithms that can be use in real terrains. As yet, few applications of collective robotics have been reported, and supporting theory is still in its formative stages. References [Balch 1998] Balch, Tucker and Arkin, Ronald C. December 1998, Behavior-Based Formation Control for Multirobot Teams. IEEE Transactions on Robotics and Automation, Vol. 14, No. 6 [Brenda 1986] M. Brenda, B. Jagannathan, and R. Dodhiawala, July 1986, On optimal cooperation of knowledge sources. Technical Report BCS-G2010, Boeing AI Center, Boeing Computer Services, Seattle, Wa. [Hackwood 1991] Hackwood, S. and Beni, G. 1991, Self-Organizing sensors by deterministic annealing, IEEE IROS, [Korf 1992] Korf Richard E., February 1992, A Simple Solution to Pursuit Games, Proceedings of the 11th International Workshop on Distributed Artificial Intelligence, Glen Arbor, Michigan. [Parker 2002] Parker, Lynne E. 2002, Distributed Algorithms for Multi-Robot Observation of Multiple Moving Targets, Autonomous Robots, 12, 3, 2002 [Stephens 1989] Stephens, L. and M. Merx, Sept. 1989, Agent organization as an effector of DAI system performance, Proceedings of the Ninth Workshop on Distributed Artificial Intelligence, Eastsound, Washington, pp [Stephens 1990] Stephens, L. and M. Merx, October 1990, The effect on agent control strategy on the performance of a DAI pursuit problem, Proceedings of the 10th International Workshop on Distributed Artificial Intelligence, Bandera, Texas.