Minimal parameter solution of the orthogonal matrix differential equation

• Published in: IEEE transactions on automatic control 1990-03, Vol.35 (3), p.314-317
• Main Authors: Bar-Itzhack, I.Y., Markley, F.L.
• Format: Article
• Language: English
• Subjects: Aerodynamics; Aerospace engineering; Angular velocity; Applied sciences; Computer science; control theory; systems; Control theory. Systems; Councils; Differential equations; Eigenvalues and eigenfunctions; Exact sciences and technology; Numerical Analysis; Position measurement; Riccati equations; Space technology; Symmetric matrices
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/9.50344

The straightforward solution of the first-order differential equation satisfied by all nth-order orthogonal matrices requires n/sup 2/ integrations to obtain the matrix elements. 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 in the present work. Several possibilities which are based on attitude determination in three dimensions are examined. It is concluded that not all 3-D methods have useful extensions to other dimensions, and that the 3-D Gibbs vector (or Cayley parameters) provide the most useful extension. An algorithm is developed using 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.< >