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                      p
                                   = 2 - /79(q sin4+t cos4)
From Eq. (4.70), we have
(O) z  = q cos 4 - r sin 4
(4.85)
(4.86)
    Equations (4.82), (4.85), and (4.86) give the values of Euler angle rates in the
neighborhood of 0 ~ 7r/2. Thus, during computations, use Eqs. (4.66-4.68), for
all values of O exceptin the neighborhood ofl01  ~  rr/2. When fol  -+  7r/2 , switch
over to Eqs. (4.82), (4.85), and (4.86).
     Angu/ar ve/ocity of a body in the navigational system.    Consider the mo-
tion of a spacecraft orbiting the Earth. We want to determine the angular velocity
of this orbiting spacecraft with respect to an inertial axe,s system fixed at the center
of the Earth and obtain its components in the navigational system xeyeze.
         Suppose the orbitis containedin the plane ofthe equator (see Fig.4.7a). Then, the
angular velocity of the vehicle with respect to the xiyizi  system is kr/ where L  -
 L(t) is thelongitude at any time t.Here, we assume that thelongitude L is measured
 in the plane of the equator and, from the Oxi axis, positive counterclockwise. Next,
 assume that the orbit is contained in a vertical plane L  -. const. Then, the angular
 velocit}r vector is contained in the plane of the equator and has the magrutude A.Its
components along    e Oxi and Oyi axes are z~sin L and - jiA cos"Le:'In general,
 when the orbital plane is inclined, both A and L vary with time t, and we have
7.O;.b =
(4.87)
Here, t7){ , is the angular velocity of the spacecraft with respect to the inertial
system and has components in theinertialsystem.Note that an arrow over a symbol
denotes a vector. T~e order of the subscripts and the superscript are as follows:
the subscripts /, b have the meaning of the body with respect to the inertial system
and the superscript / means that the vector has components in the inertial system.
EQUATIONS OF MOTION AND ESTIMATION OF STABILITY DERIVATIVES 335
n
zi , L
o
a)
  k--.           
┏━━━━━━━┓
┃       \      ┃
┃         \    ┃
┃            \ ┃
┗━━━━━━━┛
st
PLani,
Plane o. Rotation
Fig.4.7   Angular velocityin a navigationalsystem.
-iS,:os: ]
336            PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
We have
                                  wi,b = TteCO;,b
Using Eq. (4.30) for 7,e, we obtain
(4.88)
                  -sin A cos L     -sin A,sin L      cos A
Wf.b= -siriL  cosL  O,AAl[_:s;os:l ,489,
                -cos A,cos L    -cos A sin L    -sin A
                       =  [-,j-:}'o{~. : ]                                                                   (4.90)
Here, coZb is.the angular velocity of the spacecraft with respect to theinertial system
and having components in the navigational system.
[p7]=AB[_B]+A[2jl+[:;] (4.91)
Substituting for matrices A and B from Eqs. (4.14) and (4.20) and simplifying, we
obtain
[q:] = [:
From Eq. (4.92), we get
   0       ' +sin a
  cos 4      -sin 4 cos ct
-sin ~  -cos~cos ct
  0
 cos 4
-sin 4
p =4+Bsina
　
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