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

arXiv:2512.07597v3 Announce Type: replace-cross 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.