3RD-ORDER, MINIMAL-PARAMETER SOLUTION OF THE ORTHOGONAL MATRIX DIFFERENTIAL-EQUATION

The problem of minimal-parameter solution of the orthogonal matrix dif ferential equation is addressed. This well-known equation arises natur ally in three-dimensional attitude determination problems (in aircraft and satellite navigation systems), as well as in the square-root solu tion of the matrix Riccati differential equation. A direct solution of this equation involves n(2) integrations for the elements of the nth- order solution matrix. However, since an orthogonal matrix is determin ed by only n(n - 1)/2 independent (albeit nonunique) parameters, a muc h more efficient solution may, conceivably, be obtained by a parametri zation of the problem in terms of such parameters. A new, third-order minimal parametrization, which is motivated by the Peano-Baker solutio n of linear matrix differential equations, is introduced. The paramete rs and their corresponding differential equation are very simple and n atural. The proposed method is used to provide a new derivation of a c losed-form third-order quaternion propagation algorithm, which is comm only used in strapdown inertial navigation systems utilizing rate-inte grating gyros. A numerical example is used to demonstrate the viabilit y and high efficiency of the new algorithm.