Analyzing Symmetries of Swarms of Mobile Robots Using Equivariant Dynamical Systems

TL;DR

This work addresses how symmetry properties arise and evolve in swarms of autonomous, anonymous robots within the $OBLOT$ model under fully-synchronous $F$sync dynamics and $LCM$ cycles. It models the collective state as $\mathbf{z}^+ = F(\mathbf{z})$ and shows that $F$ commutes with the action of the group $O(2) \times S_n$, establishing a formal equivariant framework for swarm dynamics. Using equivariant dynamical systems theory, it derives a hierarchy of potential symmetry increases during evolution, which can be refined by decoupling the Look phase to graph automorphisms, and it characterizes all symmetry-increase types when the decoupled Compute/Move phase is invertible; the linear Compute/Move case yields a reduced linear analysis. These results provide a principled foundation for predicting and shaping swarm patterns, aiding protocol design to manage symmetry for robust pattern formation and collision avoidance.

Abstract

In this article, we investigate symmetry properties of distributed systems of mobile robots. We consider a swarm of $n\in\mathbb{N}$ robots in the $\mathcal{OBLOT}$ model and analyze their collective $\mathcal{F}$sync dynamics using of equivariant dynamical systems theory. To this end, we show that the corresponding evolution function commutes with rotational and reflective transformations of $\mathbb{R}^2$. These form a group that is isomorphic to $\mathbf{O}(2) \times S_n$, the product group of the orthogonal group and the permutation on $n$ elements. The theory of equivariant dynamical systems is used to deduce a hierarchy along which symmetries of a robot swarm can potentially increase following an arbitrary protocol. By decoupling the Look phase from the Compute and Move phases in the mathematical description of an LCM cycle, this hierarchy can be characterized in terms of automorphisms of connectivity graphs. In particular, we find all possible types of symmetry increase, if the decoupled Compute and Move phase is invertible. Finally, we apply our results to protocols which induce state-dependent linear dynamics, where the reduced system consisting of only the Compute and Move phase is linear.