Weak rigidity theory and its application to formation stabilization

This paper introduces the notion of weak rigidity to characterize a framework by pairwise inner products of interagent displacements. Compared to distance-based rigidity, weak rigidity requires fewer constrained edges in the graph to determine a geometric shape in an arbitrarily dimensional space. A necessary and sufficient graphical condition for infinitesimal weak rigidity of planar frameworks is derived. As an application of the proposed weak rigidity theory, a gradient-based control law and a nongradient-based control law are designed for a group of single-integrator modeled agents to stabilize a desired formation shape, respectively. Using the gradient control law, we prove that an infinitesimally weakly rigid formation is locally exponentially stable. In particular, if the number of agents is one greater than the dimension of the space, a minimally infinitesimally weakly rigid formation is almost globally asymptotically stable. In the literature of rigid formation, the sensing graph is always required to be rigid. Using the nongradient control law based on weak rigidity theory, it is not necessary for the sensing graph to be rigid for local exponential stability of the formation. A numerical simulation is performed for illustrating the effectiveness of our main results.