[en] consensus algorithms ; decentralized control ; swarm control ; synchronization ; differential geometry ; mean on manifolds ; special orthogonal group ; Grassmann manifold
[en] The present paper considers distributed consensus algorithms that involve N agents evolving on a connected compact homogeneous manifold. The agents track no external reference and communicate their relative state according to a communication graph. The consensus problem is formulated in terms of the extrema of a cost function. This leads to eﬃcient gradient algorithms to synchronize (i.e., maximizing the consensus) or balance (i.e., minimizing the consensus) the agents; a convenient adaptation of the gradient algorithms is used when the communication graph is directed and time-varying. The cost function is linked to a speciﬁc centroid deﬁnition on manifolds, introduced here as the induced arithmetic mean, that is easily computable in closed form and may be of independent interest for a number of manifolds. The special orthogonal group SO(n) and the Grassmann manifold Grass(p,n) are treated as original examples. A link is also drawn with the many existing results on the circle.
Fonds de la Recherche Scientifique (Communauté française de Belgique) - F.R.S.-FNRS