Towards Constructing Finer then Homotopy Path Classes

TL;DR

This paper introduces a new path classification method that distinguishes paths based on geometric and topological features, providing a finer classification than homotopy, and facilitates obstacle avoidance and planning in complex environments.

Contribution

It proposes a novel, computationally efficient path classification criterion that captures finer distinctions than homotopy, leveraging workspace geometry and robot topology.

Findings

Path classes are equivalent to and finer than homotopy classes.

The method enables easier comparison and planning of paths.

It supports obstacle avoidance and navigation through narrow passages.

Abstract

This work presents a new path classification criterion to distinguish paths geometrically and topologically from the workspace, which is divided through cell decomposition, generating a medial-axis-like skeleton structure. We use this information as well as the topology of the robot to bound and classify different paths in the configuration space. We show that the path class found by the proposed method is equivalent to and finer than the path class defined by the homotopy of paths. The proposed path classes are easy to compute, compare, and can be used for various planning purposes. The classification builds heavily upon the topology of the robot and the geometry of the workspace, leading to an alternative fiber-bundle-based description of the configuration space. We introduce a planning framework to overcome obstacles and narrow passages using the proposed path classification method and the resulting path classes.