SICE-ICASE International Joint Conference 2006 Oct. 18-2 1, 2006 in Bexco, Busan, Korea Time Complexity of Motion Planning Algorithm for Homogeneous Combinatorial Robots Hong-Fa Ho1 and Hong-Shuo Tai2 Department of Industrial Education and Institute of Applied Electronic Technology, National Taiwan Normal University, Taipei, Taiwan, R.O.C. (Tel: +886-2-2321-9997; E-mail: jackhogntnu.edu.tw) 2 Institute of Applied Electronic Technology, National Taiwan Normal University, Taipei, Taiwan, R.O.C. (Tel: +886-2-2321-9997; E-mail: giorgmsI l.url.com.tw) Abstract: Some properties and an algorithm of the motion planning problem of homogeneous combinatorial robots are presented. Homogeneous combinatorial robots can be combined and separated freely in the process of moving. The motion planning problem of homogeneous combinatorial robots in a static discrete environment is proven to be compliant to the principle of optimality. A backward dynamic motion planning algorithm is used to find the optimal motion plans. Suppose that |VI is the number of all vertices in the graph, n is the number of robots, and k is the number of stages robots passing the graph. The time complexity without considering the limitation function of this problem is O(|V| ) . Furthermore, the time complexity with the limitation function of the problem is presented. 1. INTRODUCTION Motion planning problems have been studied widely in the artificial intelligence, control theory, robotics communities, computer algorithms, and so on [1]. Many different motion planning problems have been proposed already. Taxonomy of motion planning was proposed in [2]. The type of environment proposed includes stationary, time-varying, constrained, and movable-object environments. The type of robot proposed includes rigid robot, point, manipulator, and multiple robots. The motion planning approaches include skeleton [3-6], cell decomposition [7-10], potential field [11-12], algorithms [13-14], and so on. Configuration spaces [15] were proposed to simulate motion planning problems. The situations of configuration space are used to represent the states of the robots, obstacles, and the environment. A new motion planning problem is proposed. Considering that there is a graph in a static discrete environment. Some robots, located at the start point initially, move toward the end point. Robots are point robots. These point robots, referred to as combinatorial robots, can be combined and separated freely in the process of moving. In the process of moving of robots in the environment, the risk factor is considered. More robots combined, not only the risk but also the cost could be lower. Combinatorial robots could be either homogeneous or non-homogeneous. For homogeneous combinatorial robots, every robot is the same. For non-homogeneous ones, they could be not the same. In this paper, the study is only focused on homogeneous ones. Some properties of combinatorial robots of the motion planning problem make the problem more complicated. One of the properties is that all robots in the environment move concurrently. Many well-known algorithms, for examples, the shortest path algorithm, Dijkstra's algorithm, the breadth first search, the depth first search, A* search, the best first search, and the iterative deepening algorithm, are not suitable for this problem. The principle of optimality was proven in [16]. Based on the principle, a backward dynamic motion planning algorithm is used to find the optimal motion plans. The complexity is analyzed without considering the limitation of the number of robots entering the vertex. For further analysis, the limitation of the number of robots entering the vertex is considered. Section 2 introduces the environment, some properties, and the motion planning problem of homogeneous point robots. In Section 3, a backward dynamic motion planning algorithm for optimal plans is presented. And the time complexity without considering the limitation of the number of robots entering the vertex is obtained. The time complexity analysis with the limitation function is presented in Section 4. The concluding remarks are in Section 5. 2. COMBINATORIAL ROBOTS A motion planning problem of numerous homogeneous combinatorial robots in a static discrete environment is considered. Homogeneous combinatorial robots can be combined and separated freely. Let the graph, G= < V,E >, denote the environment. V is the set of vertices and E is the set of edges in G. Let the goal vertex of all robots be g E V and the start vertex be s E V. All robots are located at s, initially. Let Rx denote the combined robot of x robots. Initially, Rn is located at s. Finally, Rn is located at g. In the real world, there are limitations on each road for many applications. Suppose that each vertex, v E V, has limitation function about the number of robots which can go-thru it. Let the limitation function of a vertex v be denoted as L(v) = /ow, tLhigh v E V}. Suppose L(s) (1, n) and L(g)=(1, n). Rx is able to go-thou v only if Llow < X < Lhigh- The status of robots in the environment is referred to as the state. Combined robots and/or robots move from 89-950038-5-5 98560/06/$10 © 2006 ICASE 4280