Digital simulation of the rotation of a moving triad with a known matrix spin vector

Abstract

This paper discusses the numerical evaluation of the elements of the transformation matrix of an orthogonal rectilinear triad moving with respect to a similar fixed system, when the former rotates with a known spin vector. Typical governing equations are formu lated in terms of Euler angles, in terms of quater nions, and in terms of direction cosines and their derivatives. Emphasis is placed on the latter formu lation for solution by digital computation.

The governing equations are reformulated into a homogeneous and linear matricial state variable equa tion of the first order, with a skew symmetric coefficient matrix called the rotation operator. The transition matrix of this state equation gives the required rotation matrix. The characteristics of the skew-symmetric matrices are used in the development of a suitable algorithm for the solution.

The exact rotation matrix can be obtained for cases of constant spin vector and for special cases of variable spin vectors. For the general case of a variable spin vector, a discrete-variable numerical solution based upon a mean coefficient value is used. The algorithm is based upon a recursive formula giving the elements of the rotation matrix at any arbitrary value of the independent variable. Errors resulting from the use of this method are compared (for a particular example) with those resulting from a fourth-order Runge-Kutta integration algorithm. The state equation for systems of higher degrees of free dom is discussed.