Wahba's problem

About: Wahba's problem is a research topic. Over the lifetime, 37 publications have been published within this topic receiving 2511 citations.

Three-axis attitude determination from vector observations

Abstract: Two computationally efficient algorithms are presented for determining three-axis attitude from two or more vector observations. The first of these, the TRIAD algorithm, provides a deterministic (i.e., nonoptimal) solution for the attitude based on two vector observations. The second, the QUEST algorithm, is an optimal algorithm which determines the attitude that achieves the best weighted overlap of an arbitrary number of reference and observation vectors. Analytical expressions are given for the covariance matrices for the two algorithms using a fairly realistic model for the measurement errors. The mathematical relationship of the two algorithms and their relative merits are discussed and numerical examples are given. The advantage of computing the covariance matrix in the body frame rather than in the inertial frame (e.g., in terms of Euler angles) is emphasized. These results are valuable when a single-frame attitude must be computed frequently. They will also be useful to the mission analyst or spacecraft engineer for the evaluation of launch-window constraints or of attitude accuracies for different attitude sensor configurations.

Quaternion Attitude Estimation Using Vector Observations

Abstract: This paper contains a critical comparison of estimators minimizing Wahba’s loss function Some new results are presented for the QUaternion ESTimator (QUEST) and Estimators of the Optimal Quaternion (ESOQ and ESOQ2) to avoid the computational burden of sequential rotations in these algorithms None of these methods is as robust in principle as Davenport’s q method or the Singular Value Decomposition (SVD) method, which are significantly slower Robustness is only an issue for measurements with widely differing accuracies, so the fastest estimators, the modified ESOQ and ESOQ2, are well suited to sensors that track multiple stars with comparable accuracies More robust forms of ESOQ and ESOQ2 are developed that are intermediate in speed

ESOQ: A Closed-Form Solution to the Wahba Problem

Abstract: A closed-form solution to the problem of optimal spacecraft attitude estimation based on vector observation, known as the Wahba problem, is presented. The algorithm first provides the closed-form expressions of a 4 × 4 matrix eigenvalues and then computes the eigenvector associated with the greatest of them (representing the optimal quaternion) using two different methods. The first method uses a vector cross-product in a four-dimensional space, while the second uses an equivalent technique requiring a 3 × 3 nonsingular matrix inversion. The resulting “Estimator of the Optimal Quaternion” (ESOQ) algorithm does not present any singularities and allows an easy identification of the approaching of the unresolvable condition of quasi-parallel observed vectors. Numerical accuracy tests, showing the average and the variance of the maximum attitude errors, are presented. Speed numerical tests, which demonstrate ESOQ as the fastest optimal attitude estimation algorithm to date, validate ESOQ as the most suitable algorithm when a fast and optimal attitude determination is required.

Fast Linear Quaternion Attitude Estimator Using Vector Observations

Abstract: As a key problem for multisensor attitude determination, Wahba’s problem has been studied for almost 50 years. Different from existing methods, this paper presents a novel linear approach to solve this problem. We name the proposed method the fast linear attitude estimator (FLAE) because it is faster than known representative algorithms. The original Wahba’s problem is extracted to several 1-D equations based on quaternions. They are then investigated with pseudoinverse matrices establishing a linear solution to $n$ -D equations, which are equivalent to the conventional Wahba’s problem. To obtain the attitude quaternion in a robust manner, an eigenvalue-based solution is proposed. Symbolic solutions to the corresponding characteristic polynomial are derived, showing higher computation speed. Simulations are designed and conducted using test cases evaluated by several classical methods, e.g., Shuster’s quaternion estimator, Markley’s singular value decomposition method, Mortari’s second estimator of the optimal quaternion, and some recent representative methods, e.g., Yang’s analytical method and Riemannian manifold method. The results show that FLAE generates attitude estimates as accurate as that of several existing methods, but consumes much less computation time (about 50% of the known fastest algorithm). Also, to verify the feasibility in embedded application, an experiment on the accelerometer–magnetometer combination is carried out where the algorithms are compared via C++ programming language. An extreme case is finally studied, revealing a minor improvement that adds robustness to FLAE, inspired by Cheng et al. Note to Practitioners —Attitude determination using vector observations can be applied in many areas. The most frequently involved are the accelerometer–magnetometer combination and star tracker array. Based on the proposed efficient fast linear attitude estimator algorithm, the time consumption of the sensor fusion can be significantly reduced, saving the execution time for fault detection, fail safe, and so on.