Forced Symmetric Formation Control

This work considers the distance constrained formation control problem with an additional constraint requiring that the formation exhibits a specified spatial symmetry. We employ recent results from the theory of symmetry-forced rigidity to construct an appropriate potential function that leads to a gradient dynamical system driving the agents to the desired formation. We show that only $(1+1/|\Gamma |)n$ edges are sufficient to implement the control strategy when there are $n$ agents and the underlying symmetry group is $\Gamma$. This number is considerably smaller than what is typically required from classic rigidity-theory-based strategies ($2n-3$ edges). We also provide an augmented control strategy that ensures that the agents can converge to a formation with respect to an arbitrary centroid. Numerous numerical examples are provided to illustrate the main results.