Quaternion-based Optimal Interpolation of Similarity Transformations for Multi-agent Formation

This paper addresses the challenge of optimal motion interpolation in multi-agent formation control. The primary goal is to generate trajectories of similarity transformations that minimize various metrics, such as distance traveled, kinetic energy consumption, and overall smoothness. The quaternion-based representation is introduced for similarity transformations, providing an innovative solution to bypass the intricate mapping between Lie group elements and Lie algebra elements. The quaternion-based approach enables the calculation of time derivatives of agents through finite algebraic operations. This feature is pivotal for efficient motion interpolation. This simplification significantly enhances the efficiency and effectiveness of the interpolation process, making it a valuable tool for multi-agent formation control. To address the optimal motion interpolation problem, we formulate it as a variational problem and derive necessary conditions for both two-point and multiple-point optimal interpolations. To further improve computational efficiency while preserving the optimal solution's structure, we propose a polynomial approximation method. Finally, we illustrate the effectiveness of the proposed method through examples of ellipsoid formation. Comparative analyses demonstrate the optimality of our approach.