Consensus and coordination on groups SO(3) and S3 over constant and state-dependent communication graphs

We address several problems of coordination and consensus on SO(3) and S3 that can be formulated as minimization problems on these Lie groups. Then, gradient descent methods for minimization of the corresponding functions provide distributed algorithms for coordination and consensus in a multi-agent system. We point out main differences in convergence of algorithms on the two groups. We discuss advantages and effects of representing 3D rotations by quaternions and applications to the coordinated motion in space. In some situations (and depending on the concrete problem and goals) it is advantageous to run algorithms on S3 and map trajectories onto SO(3) via the double cover map S3 → SO(3), instead of working directly on SO(3)