Forced Symmetric Formation Control

TL;DR

This work addresses distance-constrained formations under explicit spatial symmetry by leveraging symmetry-forced rigidity. It develops a gradient-based control framework that combines a standard distance-measurement potential with symmetry-enforcing terms, enabling formations to achieve prescribed symmetric configurations with significantly fewer edges than classical rigidity bounds. Key contributions include the orbit rigidity matrix formulation, edge-count reductions to at most $(1+1/|\Gamma|)|\mathcal{V}|$, and a centroid-consensus augmentation to target symmetries about a moving origin. The results establish local exponential stability for the symmetric formations under isostatic, free-symmetry assumptions and provide practical guidance for implementing symmetry-constrained multi-agent coordination with reduced communication. The approach broadens formation control design by integrating group-theoretic symmetry into the rigidity framework, offering a scalable path for symmetric pattern generation in multi-robot systems.

Abstract

This work considers the distance constrained formation control problem with an additional constraint requiring that the formation exhibits a specified spatial symmetry. We employ recent results from the theory of symmetry-forced rigidity to construct an appropriate potential function that leads to a gradient dynamical system driving the agents to the desired formation. We show that only $(1+1/|Γ|)n$ edges are sufficient to implement the control strategy when there are $n$ agents and the underlying symmetry group is $Γ$. This number is considerably smaller than what is typically required from classic rigidity-theory based strategies ($2n-3$ edges). We also provide an augmented control strategy that ensures the agents can converge to a formation with respect to an arbitrary centroid. Numerous numerical examples are provided to illustrate the main results.

Figures (6)

• Figure 1: The framework in (a) is infinitesimally rigid, whereas the framework with dihedral symmetry in (b) is flexible, as shown in (c).
• Figure 2: The cycle graph $C_4$ has $8$ automorphisms in $\mathrm{Aut}(\mathcal{G})$.
• Figure 3: Symmetric frameworks with $C_4$ as underlying graph. (a) is $\mathcal{C}_{4v}$-symmetric (and hence $\tau(\Gamma)$-symmetric for any subgroup $\tau(\Gamma)$ of $\mathcal{C}_{4v}$) and (b) and (c) are $\mathcal{C}_s$-symmetric (with respect to the reflection $\sigma$) and $\mathcal{C}_2$-symmetric, respectively. The framework in (c) has a non-trivial $\mathcal{C}_2$-symmetric infinitesimal motion, which extends to a continuous symmetry-preserving motion.
• Figure 4: The quotient $\Gamma$-gain graphs of the graphs of the frameworks in Fig. \ref{['symfwks']}.
• Figure 5: A $\tau(\Gamma)$-symmetric graph with $y$-axis symmetry not satisfying Assumption \ref{['vertexorbit_subgraph']}.

Theorems & Definitions (32)

• Definition 1
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• Definition 4
• Example 1
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• Example 2
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• Example 3