Quaternion Information Filters With Inaccurate Measurement Noise Covariance: A Variational Bayesian Method

Quaternion Kalman filters (QKFs) are designed for state estimation in three-dimensional (3-D) space. To simplify initialization, this paper focuses on the quaternion information filter (QIF), which converts the information vector and matrix into quaternion form. While QIF demonstrates strong performance under the assumption of known quaternion measurement noise statistics, this assumption frequently does not hold in practical scenarios. To address this issue, a variational Bayesian adaptive QIF (VBAQIF) is proposed by modeling the inverse of the covariance matrix for the quaternion measurement noise as the quaternion Wishart distribution in this paper. First, the adaptive QIF is derived under the recursive Bayesian estimation framework to propagate the quaternoin information vector and information matrix. Then, the quaternion measurement noise covariance matrix together with the quaternion state is inferred using the variational Bayesian approach. Furthermore, a corresponding square root version, called variational Bayesian adaptive square-root QIF (VBASQIF), is developed to enhance numerical stability of VBAQIF, and this stability is analyzed from a theoretical perspective. Finally, a 3-D target tracking example is simulated to demonstrate that the proposed VBAQIF exhibits excellent performance even in the presence of uncertainties in the quaternion measurement noise covariance matrices.