Undirected Rigid Formations are Problematic

TL;DR

This paper reveals that undirected rigid formations of autonomous agents, when faced with small discrepancies in distance understanding, tend to rapidly stabilize into a predictable circular orbit rather than maintaining the intended shape.

Contribution

It demonstrates that slight differences in distance estimates among agents cause the formation to converge to a stable circular orbit, highlighting a fundamental problem in undirected rigid formations.

Findings

Discrepancies lead to formation distortion.

Formations converge to a circular orbit.

Unexpected exponential convergence observed.

Abstract

By an undirected rigid formation of mobile autonomous agents is meant a formation based on graph rigidity in which each pair of "neighboring" agents is responsible for maintaining a prescribed target distance between them. In a recent paper a systematic method was proposed for devising gradient control laws for asymptotically stabilizing a large class of rigid, undirected formations in two dimensional space assuming all agents are described by kinematic point models. The aim of this paper is to explain what happens to such formations if neighboring agents have slightly different understandings of what the desired distance between them is supposed to be or equivalently if neighboring agents have differing estimates of what the actual distance between them is. In either case, what one would expect would be a gradual distortion of the formation from its target shape as discrepancies in desired or sensed distances increase. While this is observed for the gradient laws in question, something else quite unexpected happens at the same time. It is shown that for any rigidity-based, undirected formation of this type which is comprised of three or more agents, that if some neighboring agents have slightly different understandings of what the desired distances between them are suppose to be, then almost for certain, the trajectory of the resulting distorted but rigid formation will converge exponentially fast to a closed circular orbit in two-dimensional space which is traversed periodically at a constant angular speed.