  In this paper, a new probability distribution, referred to as the matrix
Fisher-Gaussian (MFG) distribution, is proposed on the nonlinear manifold
$\mathrm{SO}(3)\times\mathbb{R}^n$. It is constructed by conditioning a
(9+n)-variate Gaussian distribution from the ambient Euclidean space into
$\mathrm{SO}(3)\times\mathbb{R}^n$, while imposing a certain geometric
interpretation of the correlation terms to avoid over-parameterization. The
unique feature is that it may represent large uncertainties in attitudes,
linear variables of an arbitrary dimension, and angular-linear correlations
between them in a global fashion without singularities associated with local
parameterizations. Various stochastic properties and an approximate maximum
likelihood estimator of MFG are developed. Furthermore, two methods are
developed to propagate uncertainties though a stochastic differential equation
representing attitude kinematics. Based on these, a Bayesian estimator is
proposed to estimate the attitude and time-varying gyro bias concurrently.
Numerical studies indicate that the proposed estimator exhibits a better
accuracy against the well-established multiplicative extended Kalman filter for
two challenging cases.
