Quaternion estimation using Kalman filtering of the vectorized K-matrix

Optimal-REQUEST is an optimal recursive time-varying estimator of the quaternion of rotation. It relies, however, on a conservative estimation performance index and on a scalar gain in order to estimate the so-called K-matrix. These two deficiencies are covered in the present work, where a Kalman filter of the K-matrix is developed. Rather than preserving the matrix nature of the K-matrix plant, the approach in this work consists in vectorizing the matrix state-space equations of the K-matrix, and truncating the resulted state vector using the linear dependence between the elements of the K-matrix. This leads to a linear reduced model on which a linear Kalman filter is applied. The special case of zero-mean white propagation noises is considered here. Additional parameters such as gyro biases can be easily incorporated to the estimation algorithm. The quaternion is extracted, whenever it is needed, from the updated K-matrix using a classical method. In adequation with the dynamics specifications of various operational missions, the present algorithm assumes that the same batch of at least two non-collinear vector measurements is acquired at each sampling time. The performance of the proposed algorithm is demonstrated by means of extensive Monte-Carlo simulations.