A Convex Approach to Consensus on SO(n)

TL;DR

This paper presents a convex relaxation-based distributed consensus algorithm for rotation matrices on SO(n), providing strong convergence guarantees and exact solutions at all iterations, validated through multiple practical examples.

Contribution

Introduces a convex relaxation approach for consensus on SO(n), with distributed algorithms that guarantee convergence and exact solutions at each iteration.

Findings

Convex relaxation yields exact solutions at all iterations.

Distributed algorithms converge with strong guarantees.

Validated on satellite, vision, and rotation averaging problems.

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 $SO(6)$ validate the approach.