Decentralised attitude synchronisation of multiple rigid bodies on lie group SO (3)

Here, the global and coordinate-free approach has been applied to study the leaderless attitude synchronisation problem of multiple rigid bodies with kinematic evolving on Lie group SO (3). A decentralised consensus law based on the gradient of a smooth cost function which depends on the relative attitude of neighbouring agents is proposed. The advantage of using this cost function lies in the fact that there is a relation between the Hessian matrix of this function in a consensus configuration and the Laplacian matrix of a graph. In addition, a new approach for analysing the attitude synchronisation problem was developed and proved that convergence to synchronisation can be achieved for any connected and undirected graph if the initial attitudes of neighbouring agents are sufficiently close. Using the critical points of the cost function, the effects of graph topology on synchronisation problem were investigated. The initial attitudes sets ensuring the convergence to synchronisation are obtained for the undirected tree, complete, and ring graphs. The initial set is more restrictive for ring graph owing to the existence of other critical points induced by graph topology. Eventually, using the appropriate Lyapunov function, sufficient condition that guarantees convergence to synchronisation for general connected graph was obtained, and numerical simulation carried out to illustrate the theoretical results.