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Static Attitude Determination Methods

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Fundamentals of Spacecraft Attitude Determination and Control

Part of the book series: Space Technology Library ((SPTL,volume 33))

Abstract

Attitude determination typically requires finding three independent quantities, such as any minimal parameterization of the attitude matrix. The mathematics behind attitude determination can be broadly characterized into approaches that use stochastic analysis and approaches that do not. We restrict the term “estimation” to approaches that explicitly account for stochastic variables in the mathematical formulation, such as a Kalman filter or a maximum likelihood approach [29]. Black’s 1964 TRIAD algorithm was the first published method for determining the attitude of a spacecraft using body and reference observations, but his method could only combine the information from two measurements [2]. One year later, Wahba formulated a general criterion for attitude determination using two or more vector measurements [36]. However, explicit relations to stochastic errors in the body measurements are not shown in these formulations. The connection to the stochastic nature associated with random measurement noise was first made by Farrell in a Kalman filtering application that appeared in a NASA report in 1964 [11], but was not published in the archival literature until 1970 [12]. Farrell’s filter did not account for errors in the system dynamics, which were first accounted for in a Kalman filter developed by Potter and Vander Velde in 1968 [27].

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Notes

  1. 1.

    We are indebted to Yang Cheng for providing the basis of the discussion in this paragraph.

  2. 2.

    Let M denote the 3 × 3 matrix defined by the right side of Eq. (5.49). The 4 − 4 component of Eq. (5.48) means that trB = trM. Then the upper left 3 × 3 submatrix of Eq. (5.48) says that B + BT = M + MT. Finally, the remaining 3 × 1 and 1 × 3 submatrices tell us that B − BT = M − MT, establishing Eq. (5.49).

  3. 3.

    We employ the convention that x1:0 or x5:4 is an empty vector, with no components.

  4. 4.

    One of the eigenvalues of the singular matrix M is zero, so its eigenvalue decomposition is M = μ1vvT + μ2wwT. Then mi ×mj = μ1μ2(v ×w)k(v ×w), where {i, j, k} is a cyclic permutation of {1, 2, 3}. The optimal indices are thus those with maximum |(mi ×mj)k|.

  5. 5.

    Paul Davenport discovered these relations, but did not publish them.

  6. 6.

    Much more important, in Paul Davenport’s opinion.

  7. 7.

    We can show that Eq. (5.112) gives an extremum of P𝜗𝜗 by differentiating Eq. (5.110) with respect to any ak, then substituting Eqs. (5.112) and (5.113) after differentiating, which gives zero for the derivative.

  8. 8.

    The really basic measurements are the electron counts in the individual pixels of the star tracker’s focal plane, but these are invariably reduced to centroids before being communicated to the attitude control system.

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Authors and Affiliations

  1. Attitude Control Systems Engineering Branch, NASA Goddard Space Flight Center, Greenbelt, MA, USA

    F. Landis Markley

  2. Mechanical and Aerospace Engineering, University at Buffalo State University of New York, Amherst, NY, USA

    John L. Crassidis

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Markley, F.L., Crassidis, J.L. (2014). Static Attitude Determination Methods. In: Fundamentals of Spacecraft Attitude Determination and Control. Space Technology Library, vol 33. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0802-8_5

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  • DOI: https://doi.org/10.1007/978-1-4939-0802-8_5

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