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.