Formation scaling control using the stress matrix

This paper investigates the formation scaling control problem for multi-agent systems. In the existing literature, it is known that utilizing the formation's stress matrix, the scaling of the whole formation in IR^d can be achieved by only controlling d pairs of agents whose position vectors span IR^d, under the assumption that each of the d pairs of agents has the exact knowledge of the formation scaling parameter. In this paper, this stringent assumption is relaxed and we require only one pair of agents share the scaling information. We design a new class of distributed control laws by employing stresses and orthogonal projections such that the agents are steered to prescribed relative positions with respect to their neighbors. We show that if the corresponding stress matrix admits a generic universally rigid framework, the equilibrium of the closed-loop system is constrained only to the translation and scaling of the given configuration among all the possible affine transformations. Simulations are provided to validate the theoretical results.