  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.
