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In this paper, we introduce new concepts of weak rigidity matrix andinfinitesimal weak rigidity for planar frameworks. The weak rigidity matrix isused to directly check if a framework is infinitesimally weakly rigid whileprevious work can check a weak rigidity of a framework indirectly. Aninfinitesimal weak rigidity framework can be uniquely determined up to atranslation and a rotation (and a scaling also when the framework does notinclude any edge) by its inter-neighbor distances and angles. We apply the newconcepts to a three-agent formation control problem with a gradient controllaw, and prove instability of the control system at any incorrect equilibriumpoint and convergence to a desired target formation. Also, we propose amodified Henneberg construction, which is a technique to generate minimallyrigid (or weakly rigid) graphs. Finally, we extend the concept of the weakrigidity in R^2 to the concept in R^3.

Infinitesimal Weak Rigidity, Formation Control of Three Agents, and Extension to 3-dimensional Space