Closed-form Solution of Wahba's Problem for Pairwise Similar Quaternions

Abstract

Wahba's problem is fundamental to spacecraft attitude estimation, seeking the optimal rotation that minimizes the weighted misalignment between sets of vector observations. Traditional solvers, including Davenport's $q$-method, QUEST, and ESOQ, reformulate the problem as an eigenvalue task for a $4 \times 4$ symmetric matrix, a process that obscures the underlying algebraic structure of the solution. This paper presents a novel, entirely quaternion-based closed-form solution for the pairwise similar quaternions. By establishing a direct connection to the homogeneous singular Sylvester equation: (i) we derive the necessary and sufficient condition for the existence of a quaternion that achieves zero Wahba's cost; (ii) we provide a closed-form analytic expression for the corresponding solution set; and (iii) we propose the computationally efficient and numerically stable Minimal Analytic Rotation Algorithm (MARA). Computational complexity analysis demonstrates that MARA achieves a $35.11\%$ reduction in total floating-point operations (FLOPs) compared to the state-of-the-art ESOQ2 algorithm. Numerical validation via $10^6$ Monte Carlo trials confirms that MARA achieves higher accuracy than established optimal solvers under stochastic noise, offering a computationally more efficient and analytically transparent alternative for high-frequency attitude determination systems.