Element-based Formation Control: a Unified Perspective from Continuum Mechanics

Abstract

This paper establishes a unified element-based framework for formation control by introducing the concept of the deformation gradient from continuum mechanics. Unlike traditional methods that rely on geometric constraints defined on graph edges, we model the formation as a discrete elastic body composed of simplicial elements. By defining a generalized distortion energy based on the local deformation gradient tensor, we derive a family of distributed control laws that can enforce various geometric invariances, including translation, rotation, scaling, and affine transformations. The convergence properties and the features of the proposed controllers are analyzed in detail. Theoretically, we show that the proposed framework serves as a bridge between existing rigidity-based and Laplacian-based approaches. Specifically, we show that rigidity-based controllers are mathematically equivalent to minimizing specific projections of the deformation energy tensor. Furthermore, we establish a rigorous link between the proposed energy minimization and Laplacian-based formation control. Numerical simulations in 2D and 3D validate the effectiveness and the unified nature of the proposed framework.

Figures (4)

• Figure 1: (a) Simplicial complex representation of the multi-agent formation. The numbered circles and solid black lines denote the physical agents and the 1D communication topology, respectively. The macroscopic formation is tiled by five 2D simplicial elements (colored regions). The red triangular nodes ($\mathcal{V}_e$) and dashed lines illustrate the corresponding dual graph, which models the adjacency and interaction pathways among neighboring elements. (b) Illustration of the deformation gradient mapping a reference element to its current configuration in 2D space.
• Figure 2: Simulation results for 2D formation control ($N=6$). (a) The trajectories of agents under different control laws. The colored arrows at the final positions of the Translation-and-Rotation-invariant and Similarity-invariant modes indicate the local coordinate frames of the agents, demonstrating that the final formation orientation is achieved without a global reference frame. (b) The evolution of the potential energy $V(\mathbf{p})$ on a logarithmic scale, showing exponential convergence.
• Figure 3: Simulation results for 3D formation control ($N=7$). (a) 3D trajectories of the agents. The final configurations show the formation of a 3D heart-shaped structure (a pyramid with a heart base). (b) The energy evolution confirms the stability and fast convergence of the proposed methods in 3D space.
• Figure 4: Heatmap of $\mathrm{CoV}(V)$ for rigidity-based (distance-, bearing-, and RoD-based) and element-based (with energy functions $\Psi_{\mathrm{TR}}$, $\Psi_{\mathrm{TS}}$, and $\Psi_{\mathrm{TRS}}$) controllers with different perturbations to the top-right node of $\mathbf{q}$.

Theorems & Definitions (20)

• Lemma 1
• proof
• Theorem 1
• Remark 1
• Lemma 2: Centroid Invariance
• proof
• Lemma 3: Coordinate-Free Implementation
• proof
• Lemma 4
• proof