Translational and Rotational Invariance in Networked Dynamical Systems

In this paper, we study the translational and rotational ( SE(N)) invariance properties of locally interacting multiagent systems. We focus on a class of networked dynamical systems, in which the agents have local pairwise interactions, and the overall effect of the interaction on each agent is the sum of the interactions with other agents. We show that such systems are SE(N)-invariant if and only if they have a special quasilinear form. The SE(N) -invariance property, sometimes referred to as left invariance, is central to a large class of kinematic and robotic systems. When satisfied, it ensures independence to global reference frames. In an alternate interpretation, it allows for integration of dynamics and computation of control laws in the agents' own reference frames. Such a property is essential in a large spectrum of applications, for example, navigation in global positioning system (GPS)-denied environments. Because of the simplicity of the quasilinear form, this result can impact ongoing research on the design of local interaction laws. It also gives a quick test to check if a given networked system is SE(N)-invariant.