Neighbor Topology for Dynamic Plane Coverage in Swarm Leading Control

The objective of this paper is to solve the dynamic plane coverage problem by the movement of multiple robots, for example, sprinkling water to a large field by several vehicles or aircrafts, in which all of the points in the field should be covered by the robots in an almost equal density. One of the ways to solve it is the swarm leading control method, in which one of the robots, called a target, moves along a path in the field, and all the other robots move around the target with a fixed distance. In the process, the topology of the robots affects to the efficiency of the dynamic plane coverage problem. If the topology is a tight one, the swarm can be stable but the coverage area can be limited in a small area. On the other hand, if it is a loose one, an opposite thing can be happened. In this paper, the relation between the topology and the efficiency is discussed numerically. DOI: 10.4018/jalr.2012010106 60 International Journal of Artificial Life Research, 3(1), 59-75, January-March 2012 Copyright © 2012, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited. discuss the stability of the swarm formation to solve the dynamic plane covering problem. In the field of the swarm robotics (Ren, Beard, & Atkins, 2005) the formation and the stability of a swarm are of the central issues. Let us review the development of the swarm robotics briefly. The first research on an artificial swarm is the boid model proposed by Reynolds (1987). Each of the robots can know the relative speed and position of neighboring robots and decides its motion using them. Reynolds showed that a natural looking swarm can be emerged only by the three local interaction rules of separation, cohesion, and alignment. However, the boid model is a rule based system and the mathematical properties such as the stability of the swarm were not discussed. Tanner developed the dynamical system of mobile robots based on the idea of the boid model, particularly the alignment rule (Tanner, Jadbabaie, & Pappas, 2003, 2007). He showed and proved the stability of the system in which global communication between agents is allowed. Olfati-Saber (2008) gave the mathematical framework to the swarm formation. He formalized the motion of the robots as a dynamical system of particles, and showed several stability theorem of a swarm. For example, he proved that a swarm made by the boid model is easy to be fragmented, and that a swarm is stable if all of the robots know a destination. In the model, it is assumed that the entire agent knows a goal point. However, in natural swarms, it often happens that only a small number of the members know the goal, and the other members just follow them to reach to the goal. Couzin, Krause, Frank, and Levin (2005) discussed the relation between the number of agents which know the destination and the swarm navigation accuracy. They showed that most of the agents in the swarm can be navigated by a small number of the agents. Su, Wang, and Lin (2009) investigated the relation between the swarm stability and the number of robots which knows the destination. They showed numerically that a small number of the robots can navigate a swarm to the destination. All the models aim at the development and investigation of flocking mechanisms. On the other hand, in this paper, we apply the flocking mechanism to problem solving, and discuss the stability of the swarm from the viewpoint of the efficiency of the problem solving. We solve the dynamic plane coverage by the swarm leading control method, in which one of the robots, called a target or leader, moves along a path in the field, and all the other robots, sometime called followers, move around the target with a fixed distance (Naruse, Suenaga, & Fukui, 2010; Naruse, Fukui, & Luo, 2011). In the process, the topology of the robots affects to the efficiency of the dynamic plane coverage problem. If the topology is a tight one, the swarm can be more stable but the coverage area can be limited in a smaller area. On the other hand, if it is a loose one, an opposite thing can be happened. In this paper, we develop the dynamical system for the swarm leading control model, and investigate the relation between the robot neighbor topology and the efficiency of the dynamic plane coverage. More concretely, we represent the topology by the eigenvalues of the Laplacian matrix of the robot network, and investigate the relation between the eigenvalues and the coverage area. Since the topology can be changed during the motion of the swarm due to the flocking dynamics, we discuss the changing patterns of the eigenvalues, as well as the energy consumption for the robot control. 2. DYNAMICS OF SWARM LEADING CONTROL A. Single Agent Dynamics In the swarm leading control model, we have two kinds of robots: a leader and followers. Only a single leader exists in the system, and only the leader knows a destination of a swarm or determines a motion of the swarm. The motion of the leader is not influenced by the followers. On the other hand, all the other robots are called the follower, and they try to catch up the leader in a fixed distance, meanwhile the followers 15 more pages are available in the full version of this document, which may be purchased using the "Add to Cart" button on the product's webpage: www.igi-global.com/article/neighbor-topology-dynamic-planecoverage/65076?camid=4v1 This title is available in InfoSci-Journals, InfoSci-Journal Disciplines Medicine, Healthcare, and Life Science. Recommend this product to your librarian: www.igi-global.com/e-resources/libraryrecommendation/?id=2