Path homotopy invariants and their application to optimal trajectory planning

• Published in: Annals of mathematics and artificial intelligence Vol. 84; no. 3-4; pp. 139 - 160
• Main Authors: Bhattacharya, Subhrajit, Ghrist, Robert
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.12.2018; Springer; Springer Nature B.V
• Subjects: Algorithms; Artificial Intelligence; Automation; Complex Systems; Computer Science; Configuration space path planning; Coordination; Euclidean geometry; Graph representations; Mathematics; Robotics; Robots; Trajectory optimization; Trajectory planning; Mathematical robotics; 55-04; 90C35; 55Q99; Path homotopy invariants; Graph search; Topological path planning; Knot and link complements; Coordination spaces
• ISSN: 1012-2443; 1573-7470
• DOI: 10.1007/s10472-018-9596-8

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 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.