A constrained quaternion estimator for single-frame attitude determination

This paper presents a novel algorithm for quaternion estimation from single-frame vector measurements, which is developed in the realm of deterministic constrained least-squares estimation. Hinging on the interpretation of quaternion measurements errors as angular errors in the four-dimensional Euclidean space, a novel cost function is developed and a minimization problem is formulated under the quaternion unitnorm constraint. This approach sheds a new light on the Wahba problem and on the qmethod. The optimal estimate can be interpreted as achieving the least angular distance among a collection of planes in ℝ⁴ that are constructed from the vector observations. The resulting batch algorithm is mathematically equivalent to the q-method. Yet, taking advantage of the gained geometric insight, a recursive algorithm is developed, where the update stage consists of a rotation in the four-dimensional Euclidean space. The rotation is performed in a plane that is generated by the a priori quaternion estimate and by a measurement-related quaternion. The rotation angle is empirically designed as a fading memory factor. The main highlights of this novel algorithm are that the quaternion update stage is multiplicative such as to preserve the estimated quaternion unit-norm, and that no iterative search for eigenvalues is required. Extensive Monte- Carlo simulations showed that the novel recursive algorithm asymptotically converges to the q-method solution. Beyond the framework of single-frame quaternion estimation, this approach appears as a promising tool to embed norm-preserving quaternion update stages in augmented state estimators. More generally, the proposed approach lends itself to norm-preserving estimation algorithms in higher dimensions.