On the stability of distance-based formation control with minimally globally rigid graphs

Flip ambiguities are a notorious issue with distance-based formation control over minimally rigid graphs, owing to the presence of multiple equilibrium points in the closed-loop formation dynamics. This paper introduces a method for proving the stability of such controllers over minimally globally rigid graphs to avoid these ambiguities in 2D. The proposed method involves embedding the 2D formation in 3D and introducing virtual 3D body coordinates frames for the agents. For an N-agent system, we show that by translating N−3 virtual frames along the z-axis (perpendicular to the formation plane), the rigidity matrix has full row rank. As a result, a typical Lyapunov stability analysis can be used to prove exponential convergence to the desired formation shape for all generic initial positions as well as when N−3 agents are initially collocated or collinear. The paper concludes by experimentally demonstrating the benefit of using distance-based formation control with minimally globally rigid graphs over the standard minimally rigid graph case.