Shape stabilization and flocking control for multi-agent systems with infinitesimal ratio-of-distance rigidity

Driven by the challenge of stabilizing formations in multi-agent systems under constraints of limited sensor measurements, this paper investigates a class of distributed formation control problems where each agent relies solely on its onboard sensing information, without access to global information. To address this, a novel distributed formation control approach is designed based on the theory of infinitesimal ratio-of-distance rigidity. This approach relies solely on ratio-of-distance and direction information, which can be derived from local sensor measurements, and achieves shape stabilization even when position, displacement, and aligned bearing information are unmeasurable. This allows the entire formation system to achieve stability, characterized by local exponential convergence while maintaining freedoms of translation, rotation, and scaling. Additionally, the concept of signed frameworks is introduced, which provides initial conditions that can avoid undesired equilibrium points and realize target shapes. As another significant contribution, this paper proves that triangular formations in 2D space and tetrahedral formations in 3D space can converge to the desired formation with almost global exponential stability. Finally, simulation examples and physical experiments are presented to validate the effectiveness of the proposed formation controller.