A Common Framework for Complete and Incomplete Attitude Synchronization in Networks With Switching Topology

In this paper, we study attitude synchronization for elements in the unit sphere in <inline-formula><tex-math notation="LaTeX">$\mathbb {R}^{\scriptscriptstyle {\scriptscriptstyle {3}}}$</tex-math></inline-formula> and for elements in the three-dimensional (3-D) rotation group, for a network with switching topology. The agents’ angular velocities are assumed to be the control inputs, and a switching control law for each agent is devised that guarantees synchronization, provided that all elements are initially contained in a region, which we identify later in the paper. The control law is decentralized and it does not require a common orientation frame among all agents. We refer to synchronization of unit vectors in <inline-formula><tex-math notation="LaTeX">$\mathbb {R}^{\scriptscriptstyle {3}}$</tex-math></inline-formula> as incomplete synchronization, and of 3-D rotation matrices as complete synchronization. Our main contribution lies in showing that these two problems can be analyzed under a common framework, where all agents’ dynamics are transformed into unit vectors dynamics on a sphere of appropriate dimension.