Sequential parametrized motion planning and its complexity, II

TL;DR

This paper extends the theory of sequential parametrized motion planning, providing a complete topological complexity analysis for even-dimensional Euclidean spaces and presenting an explicit minimal-complexity motion planning algorithm for multiple robots and obstacles.

Contribution

It offers a full topological complexity characterization for arbitrary even dimensions and introduces a practical motion planning algorithm applicable to multiple robots and obstacles.

Findings

Complete topological complexity for even-dimensional Euclidean spaces.

Explicit motion planning algorithm with minimal topological complexity.

Applicable to any number of robots and obstacles.

Abstract

This is a continuation of our recent paper in which we developed the theory of sequential parametrized motion planning. A sequential parametrized motion planning algorithm produced a motion of the system which is required to visit a prescribed sequence of states, in a certain order, at specified moments of time. In the previous publication we analysed the sequential parametrized topological complexity of the Fadell - Neuwirth fibration which in relevant to the problem of moving multiple robots avoiding collisions with other robots and with obstacles in the Euclidean space. Besides, in the preceeding paper we found the sequential parametrised topological complexity of the Fadell - Neuwirth bundle for the case of the Euclidean space $\Bbb R^d$ of odd dimension as well as the case $d=2$. In the present paper we give the complete answer for an arbitrary $d\ge 2$ even. Moreover, we present an explicit motion planning algorithm for controlling multiple robots in $\Bbb R^d$ having the minimal possible topological complexity; this algorithm is applicable to any number $n$ of robots and any number $m\ge 2$ of obstacles.