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(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
to perform such coordinate transformations. Therefore, if we introduce more than
three parameters, we will have to have that many extra constraint relations among
the parameters. These additional constraint relations are often called redundancy
relations and are useful to determine any of the missing elements (up to three)
from the given direction cosine matrix. We will illustrate this concept at the end
of this section with the help of an illustrative example.
    Re/ations between Euler angles and direction cosines.    Suppose we are
given the Euler angles yr, 0, and 49 which transform a vector given in the OxiYizi
to the Ox2Y222 system. The transformation matrix based on these Euler angles
is given by Tl- of Eq. (4,22). Similarly, if we use the method of direction cosine
matrix, the transformation matrix is given by C21 of Eq. (4.112). Equating the two
matrices, we get the following relations:
Cll = cos 0 cos yr
C12 - cos O sin /r
(4.128)
(4.129)
rCll C21 C31  :11 CJ2      1 0 01
"g:2 22 :lji:][Cgj::! :2; g3:]=[: 1 tl
Lc13 C23 C33  31 C32     0 O 1_]
Z:l]
['g:
Xl - (C/-:)-lX2
   = C;X2
   = C;-C~Xl
246            PERFORMANCE, STABILl-fY, DYNAMICS, AND CONTROL
and
                         K2 = Cho + Cha(aw.oL - iw + /,) + CM.e8eo   .
Now
                                          Kt - -G] Secerlt
                                                                         -.  -5.0 *  1.85 * 0.608 * 0.9
                           - -5.0616
                                                                            Cml  :  -a, V i r7r r
                                                         - -O.I * 0.6 * 0.9 * 0.5
                              ~
                                       -. -0.027
　
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