The configuration space of a robotic arm over a graph

Abstract

In this paper, we investigate the configuration space [Formula: see text] associated with the movement of a robotic arm of length [Formula: see text] on a grid over an underlying graph [Formula: see text], anchored at a vertex [Formula: see text]. We study an associated poset with inconsistent pairs (PIP) [Formula: see text] consisting of indexed paths on [Formula: see text]. This PIP acts as a combinatorial model for the robotic arm, and we use [Formula: see text] to show that the space [Formula: see text] is a CAT(0) cubical complex, generalizing work of Ardila, Bastidas, Ceballos, and Guo. This establishes that geodesics exist within the configuration space, and yields explicit algorithms for moving the robotic arm between different configurations in an optimal fashion. We also give a tight bound on the diameter of the robotic arm transition graph — the maximal number of moves necessary to change from one configuration to another — and compute this diameter for a large family of underlying graphs [Formula: see text].