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Matrices, Vectors, Frames, Transforms

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Fundamentals of Spacecraft Attitude Determination and Control

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

Abstract

This chapter begins with an overview of matrices and vectors, which are used extensively in attitude analysis. We assume that the reader has some familiarity with this material, so the account is not completely self-contained. The principal objective of this section is to define our notation and conventions.

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Notes

  1. 1.

    One example is an incorrect sign for the velocity aberration correction for star tracker measurements on the WMAP spacecraft, which fortunately was easily corrected.

  2. 2.

    This is true in classical physics. Various contemporary physical theories indicate that we live in a space having anywhere from two to eleven dimensions.

  3. 3.

    A vector product of two vectors can be defined only in three dimensions because an n × n skew-symmetric matrix has exactly n independent parameters only for n = 3.

  4. 4.

    This is the geocentric nadir vector. Some spacecraft use the geodetic nadir vector, which is normal to the surface of the reference ellipsoid, but we will not consider this complication.

  5. 5.

    The notation \(\bar {\mathbf {q}} \otimes \mathbf { q}\) was introduced in [7], and the notation \(\bar {\mathbf {q}} \odot \mathbf { q}\) is a modification of notation introduced in [13].

  6. 6.

    Leonhard Euler (1707–1783) laid the foundations for the analysis of rotations, and his fingerprints are all over the subject. Thus, attaching his name to anything serves poorly for distinguishing it from other results also bearing his name.

  7. 7.

    Otherwise, we would have x = (λI − M)−10 = 0.

  8. 8.

    Note that these power series expansions assume that we measure angles in radians.

  9. 9.

    Olinde Rodrigues (1795–1851) obtained a doctorate in mathematics in 1815, with a thesis containing his well-known formula for the Legendre polynomials. He published nothing mathematical for the next 21 years, devoting himself to banking, the development of the French railways, utopian socialism, writing several pamphlets on banking, and editing an anthology of workers’ poetry. Then he published eight papers between 1838 and 1845, including his 1840 paper [14] greatly advancing the state of the art in attitude analysis.

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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

Authors
  1. F. Landis Markley
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  2. John L. Crassidis
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Markley, F.L., Crassidis, J.L. (2014). Matrices, Vectors, Frames, Transforms. 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_2

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  • DOI: https://doi.org/10.1007/978-1-4939-0802-8_2

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4939-0801-1

  • Online ISBN: 978-1-4939-0802-8

  • eBook Packages: EngineeringEngineering (R0)

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