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

Abstract

We address several problems of coordination and consensus on $ SO(3) $ and $ S^3 $ 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 $ S^3 $ and map trajectories onto $ SO(3) $ via the double cover map $ S^3 \to SO(3) $ , instead of working directly on $ SO(3) $ .

Keywords

• geometric consensus theory
• lie group
• coordination
• formation flying
• attitude synchronization