### abstract

- A novel two-stage quaternion estimator from vector observations that is a synthesis between Wahba's approach and the Kalman filtering approach is presented. The first stage features an optimal denoising procedure of the elements of a time-varying noisy K-matrix. The second stage produces a quaternion estimate from the filtered K-matrix via any eigenvalue-eigenvector solver. This work's contribution consists in performing the denoising via Kalman filtering. For that purpose, a matrix Kalman filter (MKF) is developed, which has the advantage of preserving the natural formulation of the matrix plant equations. As a result, two aspects of a previous algorithm, called Optimal-REQUEST (OPREQ), are improved: the K-matrix update estimation stage uses a matrix gain rather than a scalar gain, and that gain is optimized with respect to the classical minimum-variance cost. This work assumes that the sensed lines of sight (LOS) are time invariant as seen in the chosen reference frame. This assumption fits in various operational mission architectures. An exact Kalman filter is developed that accounts for the state-multiplicative noise in the process equation. A reduced estimator is also developed assuming simple expressions for the filter covariance matrices. A constrained estimator, which enforces the symmetry and null-trace of the estimated matrix, is designed using the pseudomeasurement (PM) technique. Extensive Monte-Carlo simulations illustrate the performance of the novel filters with a spinning and nutating spacecraft (SC) as a case study. Extensive Monte-Carlo simulations show that the proposed estimator outperforms OPREQ. As illustrated by additional Monte-Carlo simulations, the constrained MKF exhibits a better transient and a better steady-state accuracy than the unconstrained filter for large initial disturbances in the symmetry and null-trace properties.