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

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.

💡 Research Summary

The paper addresses Wahba’s problem—determining the optimal rotation quaternion that aligns two sets of vector observations—specifically for the case of exactly two vector pairs (n = 2). Traditional solutions such as Davenport’s q‑method, QUEST, Horn’s algorithm, and ESOQ rely on forming a 4 × 4 attitude matrix and extracting its dominant eigenvector, which hides the analytic relationship between the cost function and the optimal quaternion and incurs non‑trivial computational overhead.

The authors exploit the algebraic structure of quaternions to obtain a fully closed‑form solution. They introduce the notion of quaternion similarity (a ∼ b) defined by the existence of a non‑zero quaternion p satisfying p⁻¹ap = b, and show that for non‑real quaternions this is equivalent to equality of real parts and norms. This similarity is directly linked to the singular Sylvester equation a q = q b, whose zero‑cost condition is |q⁻¹aq − b|² = 0. Extending this to pairs, they define pairwise similarity (a₁,a₂) ∼ (b₁,b₂) if a single p simultaneously maps a₁→b₁ and a₂→b₂. Lemma 2.6 proves that pairwise similarity holds iff a₁ ∼ b₁, a₂ ∼ b₂, and Re(a₁a₂) = Re(b₁b₂).

The main theorem (Theorem 4.2) states that Wahba’s cost f(q) = ∑|q⁻¹aₗq − bₗ|² vanishes for some non‑zero quaternion q if and only if the two pairs are pairwise similar. The solution is constructed in two stages.

1. First stage (q₁): Solve a₁ q₁ = q₁ b₁, i.e., the singular Sylvester equation for the first vector pair. Using results from the authors’ earlier work, the general solution is
   q₁ = λ₁·(Im a₁)(Im b₁)⁎ + μ₁·Im(a₁ + b₁),
   where λ₁, μ₁ ∈ ℝ satisfy |λ₁| + |μ₁·Im(a₁ + b₁)| = 0, and an orthogonality condition Im a₁·Im q₁ = 0 when Im a₁ = −Im b₁.

2. Second stage (q₂): After mapping a₁ to b₁, the remaining constraint a₂→b₂ reduces to a Sylvester equation involving the cross products a₃ = q₁⁻¹(a₁×a₂)q₁ and b₃ = b₁×b₂, which are pure quaternions with equal norms and zero real parts. Solving a₃ q₂ = q₂ b₃ yields
   q₂ = λ₂·q₁⁎(Im a₁×Im a₂)q₁·(Im b₂×Im b₁) + μ₂·Im(a₃ + b₃),
   with λ₂, μ₂ ∈ ℝ satisfying |λ₂| + |μ₂·Im(a₃ + b₃)| = 0 and an analogous orthogonality condition when a₃ = −b₃.

The final quaternion is q = q₁q₂. This factorisation separates the problem into two independent similarity sub‑problems, each admitting a simple parametric family of solutions. Consequently, the existence of a zero‑cost solution can be checked by verifying pairwise similarity—a purely algebraic test—without any eigenvalue computation.

The authors emphasize that the closed‑form expressions involve only quaternion multiplication, scalar products, and cross products, making them highly suitable for real‑time embedded implementations where matrix decompositions are costly. Moreover, the parameterisation reveals that, when the similarity conditions hold, there is generally an infinite continuum of optimal quaternions, reflecting the geometric freedom inherent in aligning two vector pairs.

In summary, the paper provides (i) necessary and sufficient conditions for the Wahba cost to be zero with two observations, (ii) an explicit analytic parameterisation of all optimal quaternions, and (iii) a clear pathway to implement these results without resorting to numerical linear algebra. This contributes both theoretical insight into the structure of Wahba’s problem and practical tools for fast attitude determination in aerospace, robotics, and autonomous systems.