Decentralized Consensus Protocols on SO(4)^N and TSO(4)^N with Reshaping

Consensus protocols for a multi-agent networked system consist of strategies that align the states of all agents that share information according to a given network topology, despite challenges such as communication limitations, time-varying networks, and communication delays. The special orthogonal group (Formula presented.) plays a key role in applications from rigid body attitude synchronization to machine learning on Lie groups, particularly in fields like physics-informed learning and geometric deep learning. In this paper, N-agent consensus protocols are proposed on the Lie group (Formula presented.) and the corresponding tangent bundle (Formula presented.), in which the state spaces are (Formula presented.) and (Formula presented.), respectively. In particular, when using communication topologies such as a ring graph for which the local stability of non-consensus equilibria is retained in the closed loop, a consensus protocol that leverages a reshaping strategy is proposed to destabilize non-consensus equilibria and produce consensus with almost global stability on (Formula presented.) or (Formula presented.). Lyapunov-based stability guarantees are obtained, and simulations are conducted to illustrate the advantages of these proposed consensus protocols.