Path Homotopy Invariants and their Application to Optimal Trajectory Planning

We consider the problem of optimal path planning in different homotopy classes in a given environment. Though important in applications to robotics, homotopy path-planning in applications usually focuses on subsets of the Euclidean plane. The problem of finding optimal trajectories in different homotopy classes in more general configuration spaces (or even characterizing the homotopy classes of such trajectories) can be difficult. In this paper we propose automated solutions to this problem in several general classes of configuration spaces by constructing presentations of fundamental groups and giving algorithms for solving the word problem in such groups. We present explicit results that apply to knot and link complements in 3-space, and also discuss how to extend to cylindrically-deleted coordination spaces of arbitrary dimension.