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Top1. Introduction
The objective of this paper is to solve the dynamic plane covering problem by the movement of multiple robots, which is typically appeared in sprinkling water to a large field by several vehicles or aircrafts, and all of the points in the field should be covered with an almost equal density.
In the conventional studies, the static plane covering problem is often considered, which is to find a set of static positions of the robots to cover the field, and the robots do not move once the positions are determined. The good reviews in the context of the sensor networks can be found in Martinez, Cortes, and Bullo (2007) and Howard, Mataric, and Sukhatme (2002). However, in this paper, we extend the static problem into the above dynamic one, in which robots should move around a given field because the field is much larger than the area the robots can cover in static positions.
One of the ways to solve the problem is the swarm leading control method, in which one robot, called a target, moves along a path in the field, and all the other robots moves around the leader with a fixed distance. Then, we should 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.