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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',:;,::::]
338                PERFORMANCE, STABfLITY, DYNAMICS, AND CONTROL
          C12 C
[{IlI2;] =  'g"'  -, 22  'g:] [::":]
          C32 C
(4.108)
where Cii, C12, C13 are the direction cosines of the urut vector 12 with respect to
the OxiYizi system and so on. We observe that
C21
C31
   :  :
- I2 . II
    :   >
- J2 '11
=k2 11
C12 - l2 . Ji
C22 : j2 . jl
C32 = k2 . 71
C13 - 22 . ki
C23 -  j2 . ki
C33 = 7C2 . ki
(4.109)
(4.110)
(4.111)
where the " - " denotes thescalar product  Thus,knowing allthe elements Czj, /,  j  =
1, 3 of the direction cosine matrix, we can transform a vector from the OxiYiZi to
the Ox2Y2z2 system as follows:
Let
so that
or
          C12 C
[X:2;l= Cg"' 22 ":l[X;
         C32 C
cf =
Xz =
[X;2;]
X, = [x:"]
C12
C22
C32
X2 = C21Xl
(4.112)
(4.113)
(4.114)
(4.115)
(4.116)
(4.117)
(4.118)
(4.119)
EQUATIONS OF MOTION AND ESTIMATION OF STABILITY DERIVATIVES 339
so that
C~C~ = /
(4.120)
(4.121)
where (C~)-i = (Cl)' = C~..
   Carrying out the matrix ~nultiplications and equating the corresponding terms
on the left- and right-hand sides of the above equation, we get a total of nine
equations. It can be easily ver:ified that three of these equations are redundant,
i.e., repeat themselves. In other words, we have only six equations relating nine
parameters C,j, /,  j -. 1, 3 as follows:
C~l + C2?1 + C321 =
C/2 + C:~2 + C322 = 1
C~3 + C~3 + C323 = 1
CIIC12 + C21C22 + C31C32 - 0
Cll C13 + C21 C23 + C31 C33  - 0
C12C13 + C22C23 + C32C33 = O
(4.122)
(4.123)
(4.124)
(4,125)
(4.126)
(4.127)
  The fact that the nine parameters Czj,/, j = 1,3 forming the elements of a
coordinate transformation matrix have to satisfy six constraint Eqs. (4.122-4.127)
implies that only three of them are free. This result should not be a surprise to us
because we know that the three Euler angles Vr, 0, and 4 are necessary and sufficient
　
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