On global convergence of area-constrained formations of hierarchical multi-agent systems

TL;DR

This paper establishes conditions for the global convergence of area-constrained formation control in multi-agent systems, specifically for isosceles triangles and certain arbitrary formations, improving understanding of gain parameters.

Contribution

It provides necessary and sufficient conditions for global convergence in area-constrained formations, clarifying gain ranges for specific triangle configurations.

Findings

Global convergence conditions are derived for isosceles triangle formations.

High gain on the signed area is admissible for certain arbitrary triangle formations.

The results eliminate reflection ambiguities in formation control.

Abstract

This paper is concerned with a formation shaping problem for point agents in a two-dimensional space, where control avoids the possibility of reflection ambiguities. One solution for this type of problems was given first for three or four agents by considering a potential function which consists of both the distance error and the signed area terms. Then, by exploiting a hierarchical control strategy with such potential functions, the method was extended to any number of agents recently. However, a specific gain on the signed area term must be employed there, and it does not guarantee the global convergence. To overcome this issue, this paper provides a necessary and sufficient condition for the global convergence, subject to the constraint that the desired formation consists of isosceles triangles only. This clarifies the admissible range of the gain on the signed area for this case. In addition, as for formations consisting of arbitrary triangles, it is shown when high gain on the signed area is admissible for global convergence.