Minimal parameter solution of the orthogonal matrix differential equation

The straightforward solution of the first-order differential equation satisfied by all nth-order orthogonal matrices requires n² 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