A Remark on Formation Control with Triangulated Laman Graphs: Genericity of Equivariant Morse Functions

TL;DR

This paper proves that for multi-agent systems with triangulated Laman graphs, the potential function is generically an equivariant Morse function, ensuring only finitely many nondegenerate critical orbits, which has broad implications.

Contribution

It establishes that the potential function in RMA systems with TLGs is generically an equivariant Morse function, confirming a key assumption in formation control theory.

Findings

Potential function is generically an equivariant Morse function for TLG-based RMA systems.

Finitely many nondegenerate critical orbits exist in these systems.

Implications for solving complex formation control problems.

Abstract

This paper, as a continuing work of [1], focus on establishing the fact that if we equip a reciprocal multi-agent (RMA) system with a triangulated Laman graph (TLG), then the associated potential function is generically an equivariant Morse function, i.e, there are only finitely many critical orbits each of which is nondegenerate. Though this assumption on the potential function of being an equivariant Morse function has been used, and in fact indispensable, in several occasions. But it is actually still an open question whether it is true for a given RMA system. Thus, in this paper we will provide a confirmative answer to the question for the class of RMA systems with TLGs. The main result, as well as the analysis of this paper, has many implications for other difficult problems.