The higher topological complexity of subcomplexes of products of spheres---and related polyhedral product spaces

TL;DR

This paper develops optimal higher motion planners for systems modeled by polyhedral product spaces, like constrained robot arms, and calculates their topological complexity to optimize motion planning strategies.

Contribution

It introduces a method to construct optimal higher motion planners for polyhedral product spaces and computes their topological complexity explicitly.

Findings

Constructed explicit higher motion planners for polyhedral product spaces.

Determined the higher topological complexity for various families of these spaces.

Proved the optimality of the constructed motion planners through cohomology calculations.

Abstract

We construct "higher" motion planners for automated systems whose space of states are homotopy equivalent to a polyhedral product space $Z(K,\{(S^{k_i},\star)\})$, e.g. robot arms with restrictions on the possible combinations of simultaneously moving nodes. Our construction is shown to be optimal by explicit cohomology calculations. The higher topological complexity of other families of polyhedral product spaces is also determined.