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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
PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
C13 = -sin 0
C21 - sin0 sin~ cosl/  - sin rk cos~
C22  -. siri r/t siri0 sin 4 + cos Vr cos 4
                    C23  = sin ~ cos O
C31  -. sin O cos {b cos ~ + sin p sin ~
C32  - sin yr siri0 cos ~ -  cos lV sin O
                   C33 = COS ~ cos8
(4.130)
(4.131)
(4.132)
(4.133)
(4.134)
(4.135)
(4.136)
    Therefore, given the elements of a direction cosine matrix, we can obtain the
Euler angles using Eqs. (4.129), (4.130), and (4.133) as follows:
(4.137)
(4.13 8)
(4.139)
    Recall that we have imposed certain restrictions on Euler angles as given by
Eqs. (4.1-4.3). Subject to these restrictions, the quadrants in which the Euler
angles lie can be detemuned as follows.2
     Consider the pitch angle O. If C13 < O, then O lies in the first quadrant or O  <
0 <,rl2. On thevother hrand, if C13 > O, then O will lie in the fourth quadrant or
-rtl2 < O < 0. Thus, cos 0 will always be positive.
     Next consider the bank angle ~. We have C33  - COS ~ COS 0. Because cos 0 is
 always positive, the sign of C33  iS governed by the sign ofcos ~.Now check the sign
of C23 = sin 4 cos 0. Suppose C33 > 0 and 8u > o. Then ~ is in the first quadrant
On the other hand, if C33 > O and C23 < 0, then ~ is in the fourth quadrant  If
C33 <0 and C23 > 0, then ~ lies in the second quadrant. Kboth C33 and C23 are
negative, then 4 lies in the third quadrant.
     Similarly, examining the signs of Cii  and C12, the quadrant in which the yaw
angle ly falls can be determined.
   Updating direction cosine matrix.  During a continuous motion, the ele-
ments of the direction cosine matrices continuously vary. To determine their vari~
ation with time, we need to know the derivative of the direction cosine matrix with
respect to time, which can be obtained as follows.
EQUATIONS OF MOTION AND ESTIMATION OF STABILITY DERIVATIVES 341
   Consider the transformation of unit vectors from body-fixed axes system to
inertial axes system using the direction cosine matrix as given.by
:::l=[l"' g'2: g,3/l[:::]      (4*140)
[:"l] :,,[:l]
(4.141)
C. onsider the first equation,
                                                 lf  = Clllb + C12 jb + C132b .                               (4.142)
       dd, =CllZb+Ciidd +C127b+C12dttt+C13kb+C13d~ (4.143)
                                                                                                           (
　
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