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Minimal parameter solution of the orthogonal matrix differential equationAs demonstrated in this work, all orthogonal matrices solve a first order differential equation. The straightforward solution of this equation requires n sup 2 integrations to obtain the element of the nth order matrix. There are, however, only n(n-1)/2 independent parameters which determine an orthogonal matrix. The questions of choosing them, finding their differential equation and expressing the orthogonal matrix in terms of these parameters are considered. Several possibilities which are based on attitude determination in three dimensions are examined. It is shown that not all 3-D methods have useful extensions to higher dimensions. It is also shown why the rate of change of the matrix elements, which are the elements of the angular rate vector in 3-D, are the elements of a tensor of the second rank (dyadic) in spaces other than three dimensional. It is proven that the 3-D Gibbs vector (or Cayley Parameters) are extendable to other dimensions. An algorithm is developed emplying the resulting parameters, which are termed Extended Rodrigues Parameters, and numerical results are presented of the application of the algorithm to a fourth order matrix.
Document ID
Document Type
Reprint (Version printed in journal)
External Source(s)
Bar-Itzhack, Itzhack Y.
(NASA Goddard Space Flight Center Greenbelt, MD, United States)
Markley, F. Landis
(NASA Goddard Space Flight Center Greenbelt, MD, United States)
Date Acquired
August 14, 2013
Publication Date
March 1, 1990
Publication Information
Publication: IEEE Transactions on Automatic Control
Volume: 35
ISSN: 0018-9286
Subject Category
Numerical Analysis
Accession Number
Distribution Limits

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