Bayesian Attitude Estimation with Approximate Matrix Fisher Distributions on SO(3)

A matrix Fisher distribution on the special orthogonal group is a compact, global form of attitude uncertainty distribution that has been successfully utilized for Bayesian attitude estimations in an intrinsic fashion. This paper addresses two computational issues in implementing matrix Fisher distributions, namely numerical stability and computational efficiency. More precisely, an exponentially scaled normalizing constant of the matrix Fisher distribution and its mathematical properties are introduced for robust numerical implementations. Next, two approximate matrix Fisher distributions are formulated for a highly concentrated case and an almost uniformly distributed case respectively. These approximate forms yield an explicit form of attitude estimation schemes for the considered cases, and it also illustrate the similarity between the Gaussian distribution and the matrix Fisher distribution in the highly concentrated cases.