Parametrized motion planning and topological complexity

TL;DR

This paper develops parametrized motion planning algorithms that adapt to external conditions, analyzing their topological complexity and providing explicit, optimal solutions for multi-robot systems with obstacles, including sphere bundles.

Contribution

It introduces a detailed analysis of parametrized topological complexity and presents explicit, optimal motion planning algorithms for robots in obstacle-rich environments.

Findings

Algorithm is optimal with minimal topological complexity for odd dimensions.

Modified algorithm is optimal for even dimensions.

Analyzes parametrized topological complexity of sphere bundles using characteristic classes.

Abstract

In this paper we study paramertized motion planning algorithms which provide universal and flexible solutions to diverse motion planning problems. Such algorithms are intended to function under a variety of external conditions which are viewed as parameters and serve as part of the input of the algorithm. Continuing a recent paper, we study further the concept of parametrized topological complexity. We analyse in full detail the problem of controlling a swarm of robots in the presence of multiple obstacles in Euclidean space which served for us a natural motivating example. We present an explicit parametrized motion planning algorithm solving the motion planning problem for any number of robots and obstacles.. This algorithm is optimal, it has minimal possible topological complexity for any d odd. Besides, we describe a modification of this algorithm which is optimal for d even. We also analyse the parametrized topological complexity of sphere bundles using the Stiefel - Whitney characteristic classes.