A Randomized Greedy Algorithm for Piecewise Linear Motion Planning

We describe and implement a randomized algorithm that inputs a polyhedron, thought of as the space of states of some automated guided vehicle R, and outputs an explicit system of piecewise linear motion planners for R. The algorithm is designed in such a way that the cardinality of the output is probabilistically close (with parameters chosen by the user) to the minimal possible cardinality.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 reveals 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 Macias-Virgos 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.

Automated summary

The algorithm described and implemented in this paper produces an explicit system of piecewise linear motion planners for an automated guided vehicle by inputting a polyhedron. The cardinality of the output is probabilistically close to the minimal possible cardinality, providing an automated solution for robust robot motion planning. The implementation includes discretizing the homotopic distance concept and computer estimations of other invariants, such as the Lusternik-Schnirelmann category of polyhedra.