Matrix Transformations for ' Spacecraft Attitude

A common problem for experimental space physicists is the determination of the "attitude matrix" T which transforms vectors between representations in X and Xl coordinate systems according to Vx = TXX' VXI. Solution of this problem using the standard Eulerian angle techniques is in general tedious and circuitous. A straightforward, simple, and efficient solution for the transformation matrix is a "double-cross" transformation, such as " " " " " " " " t TXX ' = [A, A x B, Ax (A x B)JX [A, A x B, Ax (A x B) JxI, " " which is calculated from any two directions A and B which are known in both X and X' coordinates. " The B direction need be known only well enough to " " define the plane in which A and B lie. The problem of the intersection of two cones as applicable to attitude solutions is also discussed. T. Introduction Experimental space physicists frequently encounter the problem of determining the attitude of their sensors on rockets or satellites. The purpose of this report is to present a simple, efficient approach to the mathematical aspects of the problem of vehicle and sensor orientation. It is well known that a (column) vector VX' defined in an orthogonal coordinate system X' may be transformed to another orthogonal coordinate system X having the same origin by a transformation of the form where T is an orthonormal rotation, or attitude, matrix. Due to the orthonormal property of T, the inverse transformation is performed by the transpose of T: A A The problem we shall address is to determine T(A, B) when any two A A directions, A and B, are known in both X and X' representations. (The mathematics are, of course, equally applicable in terms of the alternative A A physical interpretation, that the transformation rotates A and B to new positions in a fixed coordinate system.) The usual method of expressing T involves Eulerian angles. (See, for example, Mechanics texts such as Goldstein [1950J, Marion [1965J, and others, or applications-oriented volumes such as Thomson [196lJ, Singer [1964J or Greensite [1970J.) The approach using Eulerian angles is suitable for physical problems involving rotations through known angles; however, it is difficult to apply in the situation being considered, and is cumbersome and expensive for computer applications due to the numerous