  This paper focuses on a stochastic formulation of Bayesian attitude
estimation on the special orthogonal group. In particular, an exponential
probability density model for random matrices, referred to as the matrix Fisher
distribution is used to represent the uncertainties of attitude estimates and
measurements in a global fashion. Various stochastic properties of the matrix
Fisher distribution are derived on the special orthogonal group, and based on
these, two types of intrinsic frameworks for Bayesian attitude estimation are
constructed. These avoid complexities or singularities of the attitude
estimators developed in terms of quaternions. The proposed approaches are
particularly useful to deal with large estimation errors or large uncertainties
for complex maneuvers to obtain accurate estimates of the attitude.
