Matrix Fisher–Gaussian Distribution on $\mathrm{SO}(3)\times \mathbb {R}^{ n }$ and Bayesian Attitude Estimation

In this paper, a new probability distribution, referred to as the matrix Fisher-Gaussian distribution, is proposed on the product manifold of three-dimensional special orthogonal group and Euclidean space. It is constructed by conditioning a multivariate Gaussian distribution from the ambient Euclidean space into the manifold, while imposing a certain geometric constraint on the correlation term 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. Various stochastic properties and an approximate maximum likelihood estimator are developed. Furthermore, two methods are developed to propagate uncertainties through 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 provides more accurate estimates against the multiplicative extended Kalman filter and unscented Kalman filter for challenging cases.