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

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.