Third-Order, Minimal-Parameter Solution of the Orthogonal Matrix Differential Equation

The problem of minimal-parameter solution of the orthogonal matrix differential equation is addressed. This well-knownequationarisesnaturallyinthree-dimensionalattitudedeterminationproblems (inaircraftandsatellite navigation systems ), as well as in the square-root solution 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 determined by only n(n i 1)/2 independent (albeit nonunique ) parameters, a much more efe cient solution may, conceivably, be obtained by a parametrization of the problem in terms of such parameters. A new, third-order minimal parametrization, which is motivated by the Peano ‐Baker solution of linear matrix differential equations, isintroduced. Theparametersand theircorresponding differential equation arevery simple and natural. The proposed method is used to provide a new derivation of a closed-form third-order quaternion propagationalgorithm, which iscommonly usedin strapdown inertialnavigation systemsutilizing rate-integrating gyros. A numerical example is used to demonstrate the viability and high efe ciency of the new algorithm.