Attitude Estimation via Matrix Fisher Distributions on SO(3) Using Non-Unit Vector Measurements

TL;DR

This work develops a Bayesian attitude estimator on $SO(3)$ using the matrix Fisher distribution that accommodates both unit and non-unit vector measurements without normalization. A key result shows that under isotropic Gaussian noise for non-unit measurements and von Mises–Fisher unit measurements, the posterior remains matrix Fisher with an updated parameter $F'$, extending prior results to more general sensor models. To address non-isotropic noise, a global unscented transformation is proposed, propagating uncertainty between $\mathbb{R}^3$ and $SO(3)$ while preserving a matrix Fisher form; this yields a practical estimator that achieves accurate, fast convergence. Numerical experiments demonstrate the method’s advantages over the MEKF and unit-vector MF-based estimators, particularly for non-unit measurements, highlighting its potential for robust attitude estimation in realistic sensing scenarios.

Abstract

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 with the previous attitude estimator with matrix Fisher distributions and the classic multiplicative extended Kalman filter in the case of non-unit vector measurements.

Figures (4)

• Figure 1: Case $I$: Isotropic Measurement Noises
• Figure 2: Case $II$: Non-isotropic Measurement Noises
• Figure 3: Case $III$: Large Measurement Uncertainty
• Figure 4: Case $III$: Steady-State Errors