Dynamic controllers for column synchronization of rotation matrices: a QR-factorization approach

In the multi-agent systems setting, this paper addresses continuous-time\ndistributed synchronization of columns of rotation matrices. More precisely, k\nspecific columns shall be synchronized and only the corresponding k columns of\nthe relative rotations between the agents are assumed to be available for the\ncontrol design. When one specific column is considered, the problem is\nequivalent to synchronization on the (d-1)-dimensional unit sphere and when all\nthe columns are considered, the problem is equivalent to synchronization on\nSO(d). We design dynamic control laws for these synchronization problems. The\ncontrol laws are based on the introduction of auxiliary variables in\ncombination with a QR-factorization approach. The benefit of this\nQR-factorization approach is that we can decouple the dynamics for the k\ncolumns from the remaining d-k ones. Under the control scheme, the closed loop\nsystem achieves almost global convergence to synchronization for quasi-strong\ninteraction graph topologies.\n