These equations can be verified by substituting Gqs. (12-11) into them, using some trigonometric identities, and comparing them with Gq. (12-7).

The Euler symmetric parameters corresponding to a given direction cosine matrix, A, can be found from q4=±±(\+Au + A22+A„)1'2 (12-14a)

Note that there is a sign ambiguity in the calculation of these parameters. Inspection of Eq. (12-13) shows that changing the signs of all the Euler symmetric parameters simultaneously does not affect the direction cosine matrix. Equations (12-14) express one of four possible ways of computing the Euler symmetric parameters. We could also compute q^±\(\+An-An-A33)1'2

and so forth. All methods are mathematically equivalent, but numerical inaccuracy can be minimized by avoiding calculations in which the Euler symmetric parameter appearing in the denominator is close to zero. Other algorithms for computing Euler symmetric parameters from the direction cosine matrix are given by Klumpp [1976].

Euler symmetric parameters provide a very convenient parameterization of the attitude. They are more compact than the direction cosine matrix, because only four parameters, rather than nine, are needed. They are more convenient than the Euler axis and angle parameterization (and the Euler angle parameterizations to be considered below) because the expression for the direction cosine matrix in terms of Euler symmetric parameters does not involve trigonometric functions, which require time-consuming computer operations. Another advantage of Euler symmetric pa /neters is the relatively simple form for combining the parameters for two indi dual rotations to give the parameters for the product of the two rotations. Thus, if




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