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                                                                              cos O
(4.69)
(4.70)
(4.71)
(4.72)
(4,73)
(4.74)
(4.75)
  0
_ 0                                  (4.76)
Slimilarly,
                                                        0 sin 0 cos 4 sin 4 - 0 sin 0 sin ~ cos 4
      (4)2-P+-        ---    . (4.77)
                                                                                  cos o
                     0
                           = p+ O                                                          (4.78)
Thus, we see that the Euler angle rates ,j/ and 4 assume an indeterminate (0l0)
form, which means that we have a singularity at O = rc/2. We can get around
this difficulty using L'Hospital Rule and obtain approximate expressions for Euler
angle rates that are valid around O = rrl2 as follows:2
                                                        go(q sin~ +r o
      (ddy,)t=h.m d -( ) 4))
Vrith
                                                      dd0 (-) -. dd,(-):l0
we obtain  .
               (-,, ),  =:   -:  [-+q cos ~4 + r cos ~ - r sin ~4
                                         ~s n
(4.79)
(4.80)
(4.81)
"~

cos4 tani
 -S,.r~:] [q-jj,]
sec 0 cos *
334                PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
Substituting for q and r from Eqs. (4.73) and (4.74), we obtain
                                      (  v,.)t- = -:(4 sin4 + 04 +r cos 4)                        (4.82)
We have from Eq. (4.72),
so that
4 = Jr+p
(4.83)
           (4) 2  =  p - :(q sin 4+04 +r cos 4)                           (4.84)
　
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