The configuration space of a robotic arm over a graph

TL;DR

This paper models the configuration space of a robotic arm on a graph as a CAT(0) cubical complex, providing algorithms for optimal movement and bounds on transition steps.

Contribution

It introduces a combinatorial model using PIP to analyze the configuration space, proving it is a CAT(0) complex and deriving algorithms and diameter bounds.

Findings

Configuration space is a CAT(0) cubical complex.

Algorithms for optimal robotic arm movement are provided.

A tight bound on the transition graph diameter is established.

Abstract

We investigate the configuration space $\mathcal{S}_{G,b,\ell}$ associated with the movement of a robotic arm of length $\ell$ on a grid over an underlying graph $G$, anchored at a vertex $b \in G$. We study an associated PIP (poset with inconsistent pairs) $\text{IP}_{G,b,\ell}$ consisting of indexed paths on $G$. This PIP acts as a combinatorial model for the robotic arm, and we use $\text{IP}_{G,b,\ell}$ to show that the space $\mathcal{S}_{G,b,\ell}$ 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 $G$.