On the stability and applications of distance-based flexible formations

TL;DR

This paper analyzes the stability of flexible formations in the plane, showing how small disturbances can reduce their degrees of freedom and proposing methods to exploit these effects for shape control.

Contribution

It provides a stability analysis of flexible formations under disturbances and introduces design strategies leveraging these disturbances for shape control.

Findings

Disturbed flexible formations can lose their flexibility and become rigid.

Eigenvalue analysis characterizes stability of disturbed equilibria.

Design methods exploit disturbances to achieve desired shapes with fewer edges.

Abstract

This paper investigates the stability of distance-based \textit{flexible} undirected formations in the plane. Without rigidity, there exists a set of connected shapes for given distance constraints, which is called the ambit. We show that a flexible formation can lose its flexibility, or equivalently may reduce the degrees of freedom of its ambit, if a small disturbance is introduced in the range sensor of the agents. The stability of the disturbed equilibrium can be characterized by analyzing the eigenvalues of the linearized augmented error system. Unlike infinitesimally rigid formations, the disturbed desired equilibrium can be turned unstable regardless of how small the disturbance is. We finally present two examples of how to exploit these disturbances as design parameters. The first example shows how to combine rigid and flexible formations such that some of the agents can move freely in the desired and locally stable ambit. The second example shows how to achieve a specific shape with fewer edges than the necessary for the standard controller in rigid formations.