Infinitesimal Weak Rigidity, Formation Control of Three Agents, and Extension to 3-dimensional Space
For researchers in multi-agent formation control, this provides a new theoretical tool for analyzing and designing control laws with guaranteed convergence.
This paper introduces weak rigidity matrix and infinitesimal weak rigidity for planar frameworks, enabling direct rigidity checks. It applies these to three-agent formation control, proving instability at incorrect equilibria and convergence to desired formations, and extends the concept to 3D.
In this paper, we introduce new concepts of weak rigidity matrix and infinitesimal weak rigidity for planar frameworks. The weak rigidity matrix is used to directly check if a framework is infinitesimally weakly rigid while previous work can check a weak rigidity of a framework indirectly. An infinitesimal weak rigidity framework can be uniquely determined up to a translation and a rotation (and a scaling also when the framework does not include any edge) by its inter-neighbor distances and angles. We apply the new concepts to a three-agent formation control problem with a gradient control law, and prove instability of the control system at any incorrect equilibrium point and convergence to a desired target formation. Also, we propose a modified Henneberg construction, which is a technique to generate minimally rigid (or weakly rigid) graphs. Finally, we extend the concept of the weak rigidity in R^2 to the concept in R^3.