Matrix Fisher-Gaussian Distribution on SO(3) ×ℝn for Attitude Estimation with a Gyro Bias

In this paper, we propose a new probability density function, referred to as the matrix Fisher-Gaussian (MFG) distribution, on the product of the special orthogonal group and the Euclidean space. MFG is constructed by conditioning a multivariate Gaussian distribution from the ambient Euclidean space, such that the correlation between attitudes and linear components is formulated at the tangent space of the mean attitude. The desirable feature is that MFG can globally represent large uncertainties in the attitude of a rigid body correlated with any variable in the Euclidean space, thereby eliminating singularities and complexities inherent to local coordinates. Several stochastic properties and an approximate maximum likelihood estimation are derived for MFG, and it is further utilized for unscented attitude estimation with a gyro bias. It is illustrated that the proposed attitude estimation scheme with MFG exhibits more accurate estimates than the multiplicative extended Kalman filter for a challenging case of large initial estimation errors.