On Fast Attitude Filtering Based on Matrix Fisher Distribution with Stability Guarantee

This paper addresses two interrelated problems of the nonlinear filtering mechanism and fast attitude filtering with the matrix Fisher distribution (MFD) on the special orthogonal group. By analyzing the distribution evolution along Bayes' rule, we reveal two essential properties that enhance the performance of Bayesian attitude filters with MFDs, particularly in challenging conditions, from a theoretical viewpoint. Benefiting from the new understanding of the filtering mechanism associated with MFDs, two closed-form filters with MFDs is then proposed. These filters avoid the burdensome computations in previous MFD-based filters by introducing linearized error systems with right-invariant errors but retaining the two advantageous properties. Moreover, we leverage the two properties and closed-form filtering iteration to prove the almost-global exponential stability of the proposed filter with right-invariant error for the single-axis rotation, which, to our knowledge, is not achieved by existing directional statistics-based filters. Numerical simulations demonstrate that the proposed filters are significantly more accurate than the classic invariant Kalman filter. Besides, they are also as accurate as recent MFD-based Bayesian filters in challenging circumstances with large initial error and measurement uncertainty but consumes far less computation time (about 1/5 to 1/100 of previous MFD-based attitude filters).