The configuration space of a robotic arm over a graph

2021/11/11 by Derric Denniston, Robert Muth, Denniston, Derric +3
Computer Science · Mathematics · #05C12 #05E99 #06A07 #68R05 #68U05 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2111.06252

openalex publication_date 2021/11/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We investigate the configuration space S_G,b,ℓ associated with the movement of a robotic arm of length ℓ on a grid over an underlying graph G, anchored at a vertex b ∈ G. We study an associated PIP (poset with inconsistent pairs) IP_G,b,ℓ consisting of indexed paths on G. This PIP acts as a combinatorial model for the robotic arm, and we use IP_G,b,ℓ to show that the space S_G,b,ℓ 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.

Related

• The configuration space of a robotic arm in a tunnel
• Finding topology in a factory: configuration spaces
• A Classification of Configuration Spaces of Planar Robot Arms with Application to a Continuous Inverse Kinematics Problem
• Constrained Motion Spaces of Robotic Arms
• CAT(0) geometry, robots, and society