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

In this article, 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.