Symmetry-Based Formation Control on Cycle Graphs Using Dihedral Point Groups

TL;DR

This paper introduces a symmetry-based formation control method for agents on cycle graphs, leveraging dihedral group constraints to achieve symmetric configurations with minimal communication, ensuring exponential convergence.

Contribution

It presents a novel framework using dihedral symmetries for formation control on cycle graphs, reducing communication links and guaranteeing convergence.

Findings

Achieves symmetric formations with only n-1 links

Guarantees exponential convergence to desired configurations

Extends to coordinated maneuvers along time-varying trajectories

Abstract

This work develops a symmetry-based framework for formation control on cycle graphs using Dihedral point-group constraints. We show that enforcing inter-agent reflection symmetries, together with anchoring a single designated agent to its prescribed mirror axis, is sufficient to realize every $\mathcal{C}_{nv}$-symmetric configuration using only $n-1$ communication links. The resulting control laws have a matrix-weighted Laplacian structure and guarantee exponential convergence to the desired symmetric configuration. Furthermore, we extend the method to enable coordinated maneuvers along a time-varying reference trajectory. Simulation results are provided to support the theoretical analysis.