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EQUATIONS OF MOTION AND ESTIMATION OF STABILITY DERIVATIVES 343
Here, S2lb iS called the skew-symmetric form of angular velocity vector COb,.b. Equa-
tion (4.166) is the required relation for propagating the direction cosine matrix C~
forward in time given its initial val.ue. Generally,. the initial value of C~ is not
directly given. Instead, the irutial values of the Euler angles are given. With this
information, the elements of C~ can be obtained using Eqs. (4.128-4.136).
The skew-symmetric form of a given vector has the property that premultiplying
it to another vector gives the vector cross product of the first 'vector with the second
vector. For example,
S2{.bA = ojb X A (4,168)
Another property of the skew-symmetric form of angular velocity matrix can
be illustrated as foLlows. Suppose we are given 02{. b and are asked to find qZ:,b, we
can do this as follows:
cLcib = I (4.169)
CbCb + Cb C1 = O (4.170)
Ci S2/;b Cib + cLcib S2;;,1 _ O (4.171)
C;1S2lbCl = _C;/CibSZZi (4.172)
We know that
~21b = -g22r. (4.173)
Therefore,
C~tlb Crb = C~CbQrb (4.174)
or
g?:b = Cj,S2lb Ctb (4_175)
Having found S2;b, we are now in a position to obtain Co;b using the definition
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