Infinitesimal Weak Rigidity and Stability Analysis on Three-Agent Formations

In this paper, new concepts of weak rigidity matrix and infinitesimal weak rigidity for planar frameworks are introduced. 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 propose applying the new concepts to a three-agent formation control problem with a gradient control law, and further we finally propose a modified Henneberg construction which is a technique to generate minimally rigid (or weakly rigid) graphs.