SE(N) Invariance in Networked Systems

In this paper, we study the translational and rotational ($SE(N)$) invariance properties of locally interacting multi-agent systems. We focus on a class of networked 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, quasi-linear 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, e.g., navigation in GPS-denied environments. Because of the simplicity of the quasi-linear form, this result can impact ongoing research on design of local interaction laws. It also gives a quick test to check if a given networked system is $SE(N)$-invariant.