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     The transformation matrix Lo  is not an orthogonal matrix because the vectors
lv, O, and 4 are not mutually perpendicular. Hence, the inverse is not equal to the
 transpose. Therefore, we have to compute L~l  in the usual way as follows:
                               ~:l = (A(1,))adj( ~)                      (4.60)
 The determinant A(L) of matrix Lcn is given by
                                               A(L) = COS2 ~ cos O + S1112 ~ COS 0                              (4.61)
                                                            N cos O                                                                       (4.62)
                N
 The adjoint of matrix L~ is given by
                            'cos0   sin0 sin~   cos@ sin
     adj(Lo,)= O cos~cos0 -s,.,'S,]   (4.63)
                 0    sin~     cos4
 so that
or
~:l = [:
tan 0  sin ~
    cos 4
sec 0  sin ~
Substituting in Eq, (4.59), we get
[,/i] = [:
tan 0  sin 4
     cos 4}
sec O  sin 4
cos ~ tan c
-s,.11:]  (4.64)
sec O cos ~
4 = p + tan O(q sin 4 + r cos 4)
0 =qcos4-rsin4
Vr = sec O(q sin 4 +r cos 4)
(4.65)
(4.66)
(4.67)
(4.68)
      Singularity in Euler angle rates.     Equations (4.66-4.68) for Euler angle rates
become singular at O  -. 7r]2. The existence of the singularity can be demonstrated
as follows.
EQUATIONS OF MO1-ION AND ESTIMATION OF STABILI-fY DERIVATIVES 333
4=P+
q sin 0  sin ~  +  r sin O cos4
cos 0
O -q cos4-r sin~
                           Vr = q sin~ + r cos4
                                                                           cOs 0
For O = 7t/2, from Eqs. (4.55-4.57),
                     . p-4-'rk .
                                    q -o coS4
                                      r - -0 sin~
    Substituting in Eq. (4.71), we have
                                          . (l/t) 2  =  0 sin 4 cos 4 - 0 sin4 cos c)b
　
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