Abstract-This paper introduces several new algorithms for consensus over the special orthogonal group. By relying on a convex relaxation of the space of rotation matrices, consensus over rotation elements is reduced to solving a convex problem with a unique global solution. The consensus protocol is then implemented as a distributed optimization using (i) dual decomposition, and (ii) both semi and fully distributed variants of the alternating direction method of multipliers technique -all with strong convergence guarantees. The convex relaxation is shown to be exact at all iterations of the dual decomposition based method, and exact once consensus is reached in the case of the alternating direction method of multipliers. Further, analytic and/or efficient solutions are provided for each iteration of these distributed computation schemes, allowing consensus to be reached without any online optimization. Examples in satellite attitude alignment with up to 100 agents, an estimation problem from computer vision, and a rotation averaging problem on SO (6) Traditional consensus in Euclidean space has a rich history [9] , with results offering strong convergence guarantees under mild and realistic assumptions using purely local protocols. Further, with the resurfacing of distributed optimization methods such as dual decomposition [10] and the alternating direction method of multipliers [11] , these methods have been successfully applied to achieve so-called "fast consensus" in a distributed [12] and semi-distributed (i.e. sensor fusion) [11] setting. Unfortunately, generalizing these approaches to manifolds, even those with as much structure as the group of rotation matrices, has proven nontrivial. Fortunately, although the space of rotation matrices is highly non-linear, it is a Lie Group. This structure has allowed for sophisticated consensus methods to be created with so-called "almost-global" convergence -that is to say the only stable stationary points of the protocol are global minimizers of the underlying optimization. These consensus schemes fall under two broad categories, namely intrinsic and extrinsic approaches.
The techniques were also integrated with decentralization schemes in REF to create a distributed algorithm for consensus over SO(n).
A convex approach to consensus on SO(n)
2014 52nd Annual Allerton Conference on Communication, Control, and Computing (Allerton)
2014 52nd Annual Allerton Conference on Communication, Control, and Computing (Allerton)
Mathematics Computer Science