Arithmetic and geometric solutions for average rigid-body rotation

Several existing formulations for the rotation average are reviewed and classified into the Euclidean and Riemannian solutions. A novel, more efficient characterization of the Riemannian-based average is proposed. The discussion addresses the issue of bi-invariance of the underlying distance metrics, and how the different solutions are interrelated. A not bi-invariant arithmetic average of rotation vectors is considered and shown to be an approximate solution to both the Riemannian and Euclidean averages. Results for four numerical examples are presented demonstrating the closeness of all solutions in practical applications, but also their differences when the rotations to be averaged are orthogonal to each other.