where ee is the outer product (see Appendix Q and E is the skew-symmetric matrix

This representation of the spacecraft orientation is called the Euler axis and angle parameterization. It appears to depend on four parameters, but only three are independent because |e| = 1. It is a straightforward exercise to show that A defined by Eq. (12-7) is a proper real orthogonal matrix and that fe is the axis of rotation, that is, /4e=e. The rotation angle is known to be $ because the trace of A satisfies Eq. (12-6d).

It is also easy to see that Eq. (12-7) reduces to the appropriate one of Eqs. (12-6) when e lies along one of the reference axes. The Euler rotation angle, can be expressed in terms of direction cosine matrix elements by cos$=y[tr(^)-l]

If sin 4> ^ 0, the components of ê are given by e2°(An-Al3)/(2sm<b) =

Equation (12-9) has two solutions for <i>, which differ only in sign. The two solutions have axis vectors e in opposite directions, according to Eq. (12-10). This expresses the fact that a rotation about e by an angle $ is equivalent to a rotation about -e by

Euler Symmetric Parameters. A parameterization of the direction cosine matrix in terms of Euler symmetric parameters qlt q2, q3, qA has proved to be quite useful in spacecraft work. These parameters are not found in many modern dynamics textbooks, although Whittaker [1937] does introduce them and they are discussed by Sabroff, et a!., [1965]. They are defined by

Hie four Euler symmetric parameters are not independent, but satisfy the constraint equation q\+q\+ql + ql=\ (12-12a)

These four parameters can be regarded as the components of a quaternion,

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