Stabilization of Symmetric Formations

This work proposes a solution to the distance-constrained formation control problem to attain symmetric formations. Utilizing recent results from rigidity theory for symmetric frameworks, we design a gradient-based control strategy that simultaneously drives the agents to the desired inter-agent distances and also a special position characterizing additional symmetry constraints on the graph. We show that for graphs where there exist edges between agents in the same vertex orbit induced by the automorphism group, no additional information exchange links are required. Furthermore, leveraging the symmetry constraints of the system it is possible to solve the formation control problem with fewer edges than standard approaches. In particular, the framework is not required to be minimally infinitesimally rigid in this case. Our results are demonstrated with a numerical example.