Topological Complexity of Motion Planning

Abstract. In this paper we study a notion of topological complexity TC (X) for the motion planning problem. TC (X) is a number which measures discontinuity of the process of motion planning in the configuration space X . More precisely, TC (X) is the minimal number k such that there are k different "motion planning rules," each defined on an open subset of X× X , so that each rule is continuous in the source and target configurations. We use methods of algebraic topology (the Lusternik—Schnirelman theory) to study the topological complexity TC (X) . We give an upper bound for TC (X) (in terms of the dimension of the configuration space X ) and also a lower bound (in terms of the structure of the cohomology algebra of X ). We explicitly compute the topological complexity of motion planning for a number of configuration spaces: spheres, two-dimensional surfaces, products of spheres. In particular, we completely calculate the topological complexity of the problem of motion planning for a robot arm in the absence of obstacles.