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Xi = T~XE
rE = (jriE)-l =
GrE)'
(4.26)
(4.27)
(4.28)
(4.29)
EQUATIONS OF MOTION AND ES11MATION OF STABILITY DERIVATIVES 327
n
Zi
Fig. 4.5 Inertial and nav:igational coordinate system,s.
Inertial system to navigational system Assume that the navigational system
OxeYeze is located in the northern hemisphere as shown in Fig. 4.5. Let / denote
the longitude and A the latitude of the origin of the navigational system. The
first step is to do a translation of the inertial axes system so that the origin of
the inertial axes system coincides with that of the navigational system. Then we
have to perform two Euler rotations, r/r = L and then 0 - -(90 + A,). With this.
the inertial axes system Oxiytzi coincides with the navigational system OxeYeze.
The third rotation involving 4 is not necessary. Therefore, Sb = 0. Substituting
these values of the three Euler angles in Eq. (4.22), we obtain
(4.30)
Let the vector Xi denote the position ofa particle P in the inertial frame ofreference
Oxiy,z, as shown in Fig. 4.5. We have
Xi - Xoi + Xei
(4.31)
Note that both vectors Xoi and Xet are having components in the inertial frame of
reference Ox,yizi. Then,
Xei = Xr ~ Xt,/
Xe = TreXer
= ir/e(Xr - Xoi)
Now vector Xe has components in the OxeYeze system and Tet = ~e)'
(4.32)
(4.33)
(4.34)
cos O cosy cos 0 sin p -sin O
r12 ~ sine sin4 cos p - sinVr cos 4 sin ~ sin0 sin 4 +cos p cos4 sin ~ cos 0
sin O cos 4 cos p + sin p sin 4 sin ~ sin0 cos 4 - cos p sin0 cos 4 cos0
sin0 cos 4 cos p + sin ~ sin 4]
sin p sin 9 cos ~ - cos p sin 0
cos 4 coso -l
rcos O cos Vr
= l-"o-::,~
-sin A cos L -sin A sin/
-[ie= -sinL cosL _ssl.A,]
-cos A cos/ -cos A sin / -
328 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
Inertial system to body axes system In general, a transformation of this na-
ture will involve all three Euler angles V , O, and 4 Equation (4-22) gives this
transformation matrix T,b as
cos8cosqr cosOsinp -sin 6
T/b= sinOsin4cosV,-sinl/cos4 sinysin9sin4+cospcos4 si.,l2:s%]
sm0 cos4 cos Vr +sin yr sin 4 sin ~ sin0 cos 4 - cos ~ sin 0 cos 4 co,
(4.35)
The transformation matrix from body to inertial system is given by
Tj = (Tib)'
coso cos .V sin0 sin 4 cos ~ - sin p cos 4 sin 0 cos 4 cos k + sin p su
_ cosOsinp sinpsinOsin~+cospcos4 sinVsinOcos4-cos,~,,%l
-sin0 sin4cos0 cos~cos8
(4.36)
Wind axes to body axes system. This transformation involves first rotation
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