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                                                         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
of skew-matrix as given by the Eq. (4.167).
     Equation (4.175) states that if we have a skew-matrix of angular rates of the b
frame relative to the / frame with components in the b frame and we need to find
the components of the sc        rectorin the / frame, then we must premultiply it by
C; and postmult:iply it byamCb.:ec
    From Eq. (4.166), we have
Ct, = Cj,S?/~                              (4.176)
(Cl,)' = (C. bg2b:b)'
~2::b)'(C;))'
Cb, = _C2bb Cb
(4.177)
(4.17 8)
(4.179)
gjii]
-r _,P]
344              PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
where we have used the property (S?lb)'  = -S-2lb. Equation (4.179) is of the form
X - AX
The characteristic of Eq. (4.180) is given by
                                                        A(A/ - A) :0
(4.180)
(4.181)
where A (.) denotes the determinant of the argument matrix. Using this definition,
the characteristic of Eq. (4.179) is given by
so that
or
A(;k,2 + p2 +q2 +r2) = 0
A, - 0
A =+j,     -~-
+ j ltDlb I
　
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