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.

翻译：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单元的编队控制框架。与依赖图边上几何约束的传统方法不同，我们将编队建模为由单纯形单元组成的离散弹性体。通过基于局部变形梯度张量定义广义畸变能量，我们推导出一系列分布式控制律，这些控制律能够强制执行各种几何不变性，包括平移、旋转、缩放和仿射变换。对提出的控制器的收敛特性和特征进行了详细分析。理论上，我们证明该框架充当了现有基于刚性和基于拉普拉斯方法之间的桥梁。具体来说，我们证明基于刚性的控制器在数学上等价于最小化变形能量张量的特定投影。此外，我们在提出的能量最小化与基于拉普拉斯的编队控制之间建立了严格联系。二维和三维数值仿真验证了所提框架的有效性和统一性。