Attitude estimation via matrix Fisher distributions on SO(3) using non-unit vector measurements

This note presents a novel Bayesian attitude estimator with the matrix Fisher distribution on the special orthogonal group, which can smoothly accommodate both unit and non-unit vector measurements. The posterior attitude distribution is proven to be a matrix Fisher distribution with the assumption that non-unit vector measurement errors follow the isotropic Gaussian distributions and unit vector measurements follow the von-Mises Fisher distributions. Next, a global unscented transformation is proposed to approximate the full likelihood distribution with a matrix Fisher distribution for more generic cases of vector measurement errors following the non-isotropic Gaussian distributions. Following these, a Bayesian attitude estimator with the matrix Fisher distribution is constructed. Numerical examples are then presented. The proposed estimator exhibits advantageous performance compared to the previous Bayesian estimator with the matrix Fisher distributions and the classic multiplicative extended Kalman filter in the case of non-unit vector measurements.