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Fig. 12-1. The fundamental problem of three-axis attitude parameterization is to specify the orientation of the spacecraft axes ù, v, w in the reference 1, 2, 3 frame.

example, is the cosine of the angle between ft and the reference I axis. For this reason, A is often referred to as the direction cosine matrix. The elements of the direction cosine matrix are not all independent. For example, the fact that û is a unit vector requires w?+uf+uj=l and the orthogonality of û and v means that u, v, + u2v2 + u jo j=0

These relationships can be summarized by the statement that the product of A and its transpose is the identity matrix

(See Appendix C for a review of matrix algebra.) This means that A is a real orthogonal matrix. Also, the definition of the determinant is equivalent to det/i =û-(v Xw)

so the fact that ù, v, w form a right-handed triad means that det A = 1. Thus, A is a proper real orthogonal matrix.

The direction cosine matrix is a coordinate transformation that maps vectors from the reference frame to the body frame. That is, if a is a vector with components a„ a2, a3 along the reference axes, then

The components of A a are the components of the vector a along the body triad û, v, w. As shown in Appendix C, a proper real orthogonal matrix transformation preserves the lengths of vectors and the angles between them, and thus represents a rotation. The product of two proper real orthogonal matrices A" = A'A represents the results of successive rotations by A and A', in that order. Because the transpose and inverse of an orthogonal matrix are identical, AT maps vectors from the body frame to the reference frame.

It is also shown in Appendix C that a proper real orthogonal 3x3 matrix has

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