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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
q = -p sin~cos a +cycos4
r = -p cos ~ cos ce - a sin ~
Ci
(4.92)
(4.93)
(4.94)
(4.95)
(4.96)
EQUATIONS OF MOTION AND ESTIMATION OF STABILITY DERIVATIVES 337
We note that (T3)-1 j/ (Tu~)' because the angular velocity 'vectors a, 8, and 4 are
not mutually perpendicular. Therefore, the matrix T~ is not an orthogonal matrix,
and the inverse is not equal to the transpose. Therefore, (T~)-l was computed in
the usual fashion using Eq, (4.60) and is given by
Then,
Tbw = (T~)-l = [:
sin 4 tan ct
    cos @
sin 4 sec ct
4 = p+tana(qsin~+r cos~)
a = q cos ~ -r sin ~
p = -seca(q sin~ +r cos~)
(4.97)
(4.98)
(4.99)
(4.100)
4.2.7   Method of Direction Cosine Matrix
      Consider a vector A - iAx +  jAy + 7cAz in the Oxyz system. Let 81, 82, and
 83 be the angles the vector A makes with the x, y, and z axes, respectrvely. Then,
 we have
so that
    Ax
cos 81 = IAj                                             (4.101)
(4.102)
(4.103)
COS2 8i  +COS2 82 + COS2 83  =  1                                    (4.104)
The numbers cos 81, cos 82, and cos 83 are called the direction cosines of the vector
A with respect to the Oxyz axes system.
   Now consider the transformation of a vector from one coordinate system to
another using the direction cosines. Suppose we want to transform a vector given
in the OxiYizi systeminto Ox2Y222.Let
l2 = Ciiii + C12/1 + Cl3kl
l/2  =  C2111  + C22 71  + C23ki
k2 = C3111 + C32]1 + C33kl
(4.105)
(4.1'06)
(4.107)
'0-4vl
cos 4 sec cej
,os 82 = IAAj
,os83 = fAAI

-:,js',:;,::::]
424             PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
a)
                                            Mach Number
　
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