Path Homotopy Invariants and their Application to Optimal Trajectory Planning
This addresses a challenging problem in robotics for planning optimal paths in complex environments, though it appears incremental by extending existing methods to more general configuration spaces.
The paper tackles the problem of finding optimal trajectories in different homotopy classes for path planning in robotics, proposing automated solutions by constructing fundamental group presentations and solving word problems, with explicit results for knot complements and coordination spaces.
We consider the problem of optimal path planning in different homotopy classes in a given environment. Though important in robotics applications, path-planning with reasoning about homotopy classes of trajectories has typically focused on subsets of the Euclidean plane in the robotics literature. 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 \emph{word problem} in such groups. We present explicit results that apply to knot and link complements in 3-space, discuss how to extend to cylindrically-deleted coordination spaces of arbitrary dimension, and also present results in the coordination space of robots navigating on an Euclidean plane.