Quaternion Attitude Estimation Using Vector Observations
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Quaternion Attitude Estimation Using Vector Observations F. Landis Markley ' and Daniele Mortarf'
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.
Introduction In many spacecraft attitude systems, the attitude observations are naturally represented as unit vectors. Typical examples are the unit vectors giving the direction to the Sun or a star and the unit vector in the direction of the Earth's magnetic field. This paper will consider algorithms for estimating spacecraft attitude from vector measurements taken at a single time, which are known as "single-frame" methods or "point" methods, instead of filtering methods that employ information about spacecraft dynamics. Almost all single-frame algorithms are based on a problem proposed in 1965 by Grace Wahba [1]. Wahba's problem is to find the orthogonal matrix A with determinant + 1 that minimizes the loss function (1)
where {hi} is a set of unit vectors measured in a spacecraft's body frame, {r.} are the corresponding unit vectors in a reference frame, and {ai} are non-negative 'Guidance, Navigation, and Control Center, NASA's Goddard Space Flight Center, Greenbelt, MD, [email protected]. 2Aerospace Engineering School, University of Rome, "La Sapienza," Rome, Italy, daniele@psm. uniromal.it, Visiting Associate Professor, Texas A&M University, College Station, TX.
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weights. In this paper we choose the weights to be inverse variances, a, = (J"i- 2 , in order to relate Wahba's problem to Maximum Likelihood Estimation [2]. This choice differs from that of Wahba and many other authors, who assumed the weights normalized to unity. It is possible and has proven very convenient to write the loss function as L(A)
=
Ao - tr(AB T )
(2)
with (3)
and
(4) Now it is clear that L(A) is minimized when the trace, tr(AB T ) , is maximized. This has a close relation to the orthogonal Procrustes problem, which is to find the orthogonal matrix A that is closest to B in the Frobenius norm (also known as the Euclidean, Schur, or Hilbert-Schmidt norm) [3]
IIMII} == 2: Ml =
tr(MM
T
)
(5)
i,j
Now
so Wahba's problem is equivalent to the orthogonal Procrustes problem with the proviso that the determinant of A must be
+ 1.
The purpose of this paper is to give an overview in a unified notation of algorithms for solving Wahba's problem, to pro
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