用连续介质力学统一建模多智能体编队控制，实现灵活几何约束。
Element-based Formation Control: a Unified Perspective from Continuum Mechanics
- 基于变形梯度构建能量模型，以元素为单位描述编队形变。
- 可统一实现平移、旋转、缩放等几何不变性，收敛性理论完备。
- 适用于需要精确几何保持的机器人编队场景，如无人机群组。
本文通过引入连续介质力学中的变形梯度概念，建立了一个统一的基于元素的编队控制框架。不同于依赖图边几何约束的传统方法，本研究将编队视为由单纯形元素构成的离散弹性体，定义基于局部变形梯度张量的广义畸变能量，并推导出一系列分布式控制律，可实现平移、旋转、缩放及仿射变换等几何不变性。详细分析了控制器的收敛性与特性，理论上证明该框架是现有刚性基与拉普拉斯基方法之间的桥梁：刚性基控制器在数学上等价于最小化变形能张量的特定投影；同时建立了能量最小化与拉普拉斯基控制的严格联系。2D与3D数值仿真验证了该框架的有效性与统一性。
原文摘要 · Abstract (English)
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.
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