and <o<=(oe is the angular velocity vector. Then d q .. q(t + At)-q(t)

If 0 is constant, we can formally integrate Eq. (16-3) to obtain v(0 = exp(«//2)v(0) (l6-4a)

In the weaker case that e is constant but u varies, the integration can still be carried out to yield.

The meaning of exponential functions of matrices and the relation of Eq. (16-4) to Eq. (16-1) are discussed in Appendix C.

The time dependence of the direction cosine matrix, A, can be similarly derived. We have

where A' is given by Eq. (12-7) with rotation angle Afl> and with e„ e2, e3 replaced by eu, eB, ew, as discussed above. If At is infinitesimal, small-angle approximations can be used for cos A$ and sinA$, yielding

A'= l + Q' At where 1 is the 3 x 3 identity matrix and

0 0

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