On averaging rotations

In this paper two common approaches to averaging rotations are compared to a more advanced approach based on a Riemannian metric. Very often the barycenter of the quaternions or matrices that represent the rotations are used as an estimate of the mean. These methods neglect that rotations belong to a non-linear manifold and re-normalization or orthogonalization must be applied to obtain proper rotations. These latter steps have been viewed as ad hoc corrections for the errors introduced by assuming a vector space. The article shows that the two approximative methods can be derived from natural approximations to the Riemannian metric, and that the subsequent corrections are inherent in the least squares estimation.