Path Homotopy Invariants and their Application to Optimal Trajectory Planning

TL;DR

This paper introduces automated methods for optimal path planning across different homotopy classes in complex configuration spaces, leveraging fundamental group presentations and algorithms for the word problem, with applications in robotics and knot theory.

Contribution

It develops general algorithms for solving the homotopy class planning problem in diverse configuration spaces using fundamental group presentations.

Findings

Applicable to knot and link complements in 3-space

Extends to cylindrically-deleted coordination spaces of arbitrary dimension

Provides solutions for robots navigating on Euclidean planes

Abstract

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.