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A Closed-form Solution to the Wahba Problem for Pairwise Similar Quaternions

Hristina Radak, Christian Scheunert, Frank H. P. Fitzek

TL;DR

This work solves Wahba's problem in the quaternion domain for two vector observations, delivering a closed-form, analytic solution rather than relying on eigen-decomposition or iterative methods. It proves that zero cost is possible iff the two-quaternion pair (a1,a2) is pairwise similar to (b1,b2) and then constructs all solutions via a two-step factorization q = q1 q2, where each step uses a closed-form expression derived from quaternion similarity and quaternion square roots. The approach hinges on the connection between the homogeneous singular Sylvester equation aq − qb = 0 and Wahba's cost, providing necessary and sufficient solvability conditions and explicit parameterizations. The results yield a precise, algebraic understanding of the two-observation Wahba problem and pave the way for efficient, exact attitude estimation in this special case.

Abstract

We present a closed-form solution to Wahba's problem in the quaternion domain for the special case of two vector observations. Existing approaches, including Davenport's $q$-method, QUEST, Horn's method, and ESOQ algorithms, recover the optimal quaternion through the eigendecomposition of a $4\times4$ matrix or iterative numerical methods. Consequently, these methods do not reveal the analytic structure of the optimal quaternion. In this work, we derive an explicit analytical characterization of all quaternions that yield zero Wahba cost for the case $\ell=2$. Our approach builds on a connection between quaternion similarity, the singular Sylvester equation $aq=qb$, and quaternion square roots established in our previous work [1]. We provide (i) necessary and sufficient conditions under which the Wahba's cost function is zero and (ii) a closed-form parameterization of all such quaternions. This eliminates the need for eigenvalue computations and enables a direct algebraic understanding of the underlying geometry of Wahba's problem.

A Closed-form Solution to the Wahba Problem for Pairwise Similar Quaternions

TL;DR

This work solves Wahba's problem in the quaternion domain for two vector observations, delivering a closed-form, analytic solution rather than relying on eigen-decomposition or iterative methods. It proves that zero cost is possible iff the two-quaternion pair (a1,a2) is pairwise similar to (b1,b2) and then constructs all solutions via a two-step factorization q = q1 q2, where each step uses a closed-form expression derived from quaternion similarity and quaternion square roots. The approach hinges on the connection between the homogeneous singular Sylvester equation aq − qb = 0 and Wahba's cost, providing necessary and sufficient solvability conditions and explicit parameterizations. The results yield a precise, algebraic understanding of the two-observation Wahba problem and pave the way for efficient, exact attitude estimation in this special case.

Abstract

We present a closed-form solution to Wahba's problem in the quaternion domain for the special case of two vector observations. Existing approaches, including Davenport's -method, QUEST, Horn's method, and ESOQ algorithms, recover the optimal quaternion through the eigendecomposition of a matrix or iterative numerical methods. Consequently, these methods do not reveal the analytic structure of the optimal quaternion. In this work, we derive an explicit analytical characterization of all quaternions that yield zero Wahba cost for the case . Our approach builds on a connection between quaternion similarity, the singular Sylvester equation , and quaternion square roots established in our previous work [1]. We provide (i) necessary and sufficient conditions under which the Wahba's cost function is zero and (ii) a closed-form parameterization of all such quaternions. This eliminates the need for eigenvalue computations and enables a direct algebraic understanding of the underlying geometry of Wahba's problem.

Paper Structure

This paper contains 5 sections, 12 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Singular Sylvester equation in quaternions
  4. Solutions to Wahba's problem in quaternions
  5. Conclusion

Key Result

Lemma 2.1

Let $\qa,\qb$ be nonreal pure quaternions and define $\qc = \mathrm{Im}\,(\qa\qb) = \qa \times \qb$. Then, $\qa\qc = -\qc\qa$.

Theorems & Definitions (23)

  • Lemma 2.1
  • proof
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4: radak+scheunert++2025_on
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Lemma 3.1: radak+scheunert++2025_on
  • Lemma 3.2: radak+scheunert++2025_on
  • ...and 13 more