A randomized greedy algorithm for piecewise linear motion planning

TL;DR

This paper introduces a randomized algorithm that generates near-minimal piecewise linear motion planners for robots in polyhedral spaces, leveraging topological complexity techniques for robust motion planning.

Contribution

It presents the first automated, probabilistically near-minimal solution for robot motion planning using simplicial complexity, avoiding costly subdivisions of the space.

Findings

Algorithm produces motion planners close to minimal in size.

Reveals SC model can recast Farber's invariant efficiently.

Implementation discretizes homotopic distance for practical estimation.

Abstract

We describe and implement a randomized algorithm that inputs a polyhedron, thought of as the space of states of some automated guided vehicle $\mathcal{R}$, and outputs an explicit system of piecewise linear motion planners for $\mathcal{R}$. The algorithm is designed in such a way that the cardinality of the outputed system is probabilistically close (with parameters chosen by the user) to minimal possible. This yields the first automated solution for robust-to-noise robot motion planning in terms of simplicial complexity (SC) techniques, a discretization of Farber's topological complexity TC. Besides its relevance toward technological applications, our work revels that, unlike other discrete approaches to TC, the SC model can recast Farber's invariant without having to introduce costly subdivisions. We develop and implement our algorithm by actually discretizing Mac\'ias-Virg\'os and Mosquera-Lois' notion of homotopic distance, thus encompassing computer estimations of other sectional category invariants as well, such as the Lusternik-Schnirelmann category of polyhedra.