This paper explores a displacement-based formation control problem in the presence of misaligned orientations among different body coordinate frames. When agents sense and adjust their relative positions, these misalignments suggest the attitudes of the agents in the absence of an agreement, thereby distorting collective behavior. To mitigate this distortion, an angular velocity control protocol is provided and embedded into a positively invariant set, aiming to establish a predefined attitude agreement. Based on this agreement, a formation control protocol is then to achieve the desired formation shape. Unlike the original alignment case, not only the dynamics of the agents among different coordinate frames are considered, but also the connectivity on the underlying topologies is relaxed. It should be pointed out that constructing a common Lyapunov function for these systems is challenging, since typical conditions for its existence and solvability, e.g., the double stochasticity condition or the non-trivial eigenvalue assignment, may not be satisfied. To address these challenges, a geometric analysis framework, derived as several polytopes and associated properties, is extended to handle these systems without the requirement for such conditions. By utilizing these properties, the control objectives of attitude and formation control are achieved, indicating that the desired formation shape is guaranteed through the implementation procedure of the attitude consensus. Finally, an example is conducted to verify the main results.

中文翻译：

---

基于位移的编队控制，具有针对时变拓扑的预定义姿态


本文探讨了不同身体坐标系之间存在未对准方向时基于位移的编队控制问题。当代理人感知并调整他们的相对位置时，这些偏差表明了代理人在没有达成协议的情况下的态度，从而扭曲了集体行为。为了减轻这种失真，提供了角速度控制协议并将其嵌入到正不变集中，旨在建立预定义的姿态协议。基于该协议，编队控制协议将实现所需的编队形状。与原始对齐情况不同，不仅考虑了不同坐标系之间代理的动态，而且还放松了底层拓扑上的连接性。应该指出的是，为这些系统构造一个通用的李亚普诺夫函数是具有挑战性的，因为它的存在和可解性的典型条件，例如双随机条件或非平凡特征值分配，可能无法得到满足。为了应对这些挑战，我们扩展了由多个多面体和相关属性衍生的几何分析框架来处理这些系统，而无需此类条件。利用这些性质，实现了姿态和编队控制的控制目标，表明通过姿态共识的实施过程保证了期望的编队形态。最后通过算例验证了主要结果。