Topological Complexity of Motion Planning

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 the 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: for spheres, two-dimensional surfaces, for 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.