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Formation Control via Rotation Symmetry Constraints

Zamir Martinez, Daniel Zelazo

TL;DR

This work addresses distributed formation control using rotation symmetry constraints, introducing a symmetry-forcing potential $F(p)$ and a gradient-based controller that drives agents to a $\mathcal{C}_n$-symmetric planar formation, requiring only $n-1$ communication links via a spanning-tree. It leverages a matrix-weighted Laplacian $Q = E(\Gamma_r) E(\Gamma_r)^T$, proving that $\mathrm{Null}(Q)$ corresponds to the symmetry manifold and guaranteeing convergence to the target formation. An augmentation with a time-varying virtual trajectory allows maneuvering (translation, rotation, scaling) along a predefined path, and a 3D extension is demonstrated numerically (cube-like symmetry). Collectively, the approach offers a scalable, flexible alternative to rigidity-based methods with reduced communication while enabling coordinated motion.

Abstract

We present a distributed formation control strategy for multi-agent systems based only on rotation symmetry constraints. We propose a potential function that enforces inter-agent \textbf{rotational} symmetries, with its gradient defining the control law driving the agents toward a desired symmetric and planar configuration. We show that only $(n-1)$ edges, the minimal connectivity requirement, are sufficient to implement the control strategy, where $n$ is the number of agents. We further augment the design to address the \textbf{maneuvering problem}, enabling the formation to undergo coordinated translations, rotations, and scalings along a predefined virtual trajectory. Numerical simulations demonstrate the effectiveness and flexibility of the proposed method.

Formation Control via Rotation Symmetry Constraints

TL;DR

This work addresses distributed formation control using rotation symmetry constraints, introducing a symmetry-forcing potential and a gradient-based controller that drives agents to a -symmetric planar formation, requiring only communication links via a spanning-tree. It leverages a matrix-weighted Laplacian , proving that corresponds to the symmetry manifold and guaranteeing convergence to the target formation. An augmentation with a time-varying virtual trajectory allows maneuvering (translation, rotation, scaling) along a predefined path, and a 3D extension is demonstrated numerically (cube-like symmetry). Collectively, the approach offers a scalable, flexible alternative to rigidity-based methods with reduced communication while enabling coordinated motion.

Abstract

We present a distributed formation control strategy for multi-agent systems based only on rotation symmetry constraints. We propose a potential function that enforces inter-agent \textbf{rotational} symmetries, with its gradient defining the control law driving the agents toward a desired symmetric and planar configuration. We show that only edges, the minimal connectivity requirement, are sufficient to implement the control strategy, where is the number of agents. We further augment the design to address the \textbf{maneuvering problem}, enabling the formation to undergo coordinated translations, rotations, and scalings along a predefined virtual trajectory. Numerical simulations demonstrate the effectiveness and flexibility of the proposed method.

Paper Structure

This paper contains 9 sections, 3 theorems, 36 equations, 10 figures.

Table of Contents

  1. INTRODUCTION
  2. Symmetry in Graphs and Frameworks
  3. Symmetry in Graphs
  4. Symmetry in frameworks
  5. Symmetry-based Formation Control
  6. Symmetry-based Control Law
  7. Formation Maneuvering
  8. Extension to $\mathbb{R}^3$
  9. Concluding Remarks

Key Result

Proposition 1

Let $Q$ be the symmetry-constraining matrix-weighted Laplacian associated with the spanning tree graph $\mathcal{G}_I$. Then:

Figures (10)

  • Figure 1: Cycle graph $C_3$, with $6$ automorphisms in $\mathrm{Aut}(\mathcal{G})$.
  • Figure 2: Symmetric frameworks with $C_n$ as the underlying graph. (a) and (b) are $\mathcal{C}_4$-symmetric, and (c) is $\mathcal{C}_6$-symmetric.
  • Figure 3: The contribution $\tau(\gamma_{ji})p_j-p_i$ in the control law \ref{['ea_dyn']} drives $p_i$ to a symmetric position of $p_j$.
  • Figure 4: Underlying graph $\mathcal{G}$.
  • Figure 5: Trajectories generated from \ref{['ctrl_1']}.
  • ...and 5 more figures

Theorems & Definitions (14)

  • Definition 1
  • Definition 2
  • Definition 3
  • Example 1
  • Definition 4
  • Example 2
  • Proposition 1
  • proof
  • Theorem 1
  • proof
  • ...and 4 more