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In this paper, a new probability distribution, referred to as the matrixFisher-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 geometricinterpretation of the correlation terms to avoid over-parameterization. Theunique feature is that it may represent large uncertainties in attitudes,linear variables of an arbitrary dimension, and angular-linear correlationsbetween them in a global fashion without singularities associated with localparameterizations. Various stochastic properties and an approximate maximumlikelihood estimator of MFG are developed. Furthermore, two methods aredeveloped to propagate uncertainties though a stochastic differential equationrepresenting attitude kinematics. Based on these, a Bayesian estimator isproposed to estimate the attitude and time-varying gyro bias concurrently.Numerical studies indicate that the proposed estimator exhibits a betteraccuracy against the well-established multiplicative extended Kalman filter fortwo challenging cases.

Matrix Fisher-Gaussian Distribution on $\mathrm{SO}(3)\times\mathbb{R}^n$ for Attitude Estimation with a Gyro Bias