Equation (12-15b) can be verified by direct substitution of Eq. (12-13) into Eq. (12-15a), but the algebra is exceedingly tedious. The relationship of Eq. (12-15b) to the quaternion product is given in Appendix D. Note that the evaluation of Eq. (12-15b) involves 16 multiplications and the computation of Eq. (I2-I5a) requires 27; this is another advantage of Euler symmetric parameters.

Gibbs Vector. The direction cosine matrix can also be parameterized by the Gibbs vector * which is defined by

The direction cosine matrix is given in terms of the Gibbs vector by l+gî-gl-gl 2(g,g2 + s3) 2(glg3-g2) 2(glg2~g3) ^(gigj + gi)

1+r where G is the skew-symmetric matrix

G =

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