This paper proposes a novel formation maneuver control method for both 2-D and 3-D space, which enables the formation to translate, scale, and rotate with arbitrary orientation. The core innovation is the novel design of weights in the proposed augmented Laplacian matrix. Instead of using scalars, we represent weights as matrices, which are designed based on a specified rotation axis and allow the formation to perform rotation in 3-D space. To further improve the flexibility and scalability of the formation, the rotational axis adjustment approach and dynamic agent reconfiguration method are developed, allowing formations to rotate around arbitrary axes in 3-D space and new agents to join the formation. Theoretical analysis is provided to show that the proposed approach preserves the original configuration of the formation. The proposed method maintains the advantages of the complex Laplacian-based method, including reduced neighbor requirements and no reliance on generic or convex nominal configurations, while achieving arbitrary orientation rotations via a more simplified implementation. Simulations in both 2-D and 3-D space validate the effectiveness of the proposed method.

翻译：本文提出了一种适用于二维与三维空间的新型编队机动控制方法，使编队能够实现任意方向的平移、缩放与旋转。其核心创新在于所提出的增广拉普拉斯矩阵中权重的全新设计。我们采用矩阵而非标量来表示权重，这些矩阵基于指定的旋转轴进行设计，使编队能够在三维空间执行旋转。为进一步提升编队的灵活性与可扩展性，本文提出了旋转轴调整方法与动态智能体重构方法，使编队能够绕三维空间中任意轴旋转，并允许新智能体加入编队。理论分析表明，所提方法能够保持编队的原始构型。该方法继承了基于复拉普拉斯方法的优势，包括减少邻居数量需求、不依赖通用或凸的名义构型，同时通过更简化的实现方式实现了任意方向的旋转。二维与三维空间的仿真验证了所提方法的有效性。