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    From Eq. (4.37), we have    .
-8
~
t;
c:l
6
r:
 o
6
€
j::,::~]
                                 cos a cos p          .
T3 =   sinasin4.cosp+sin-8cos4
                sin or cos4 cos f/ - sin  p sin ~
6
oI
u
a
u
a
0.2049 -1
-0.3882
0.8138 _1
3]
f
-=
c:
a
354                PERFORMANCE, STABILI-fY, DYNAMICS, AND CONTROL
Subsequently, the reference axes system xoYoZo remains fixed in space. This means
that we will have a different reference axes system for each value of a. With these
assumpOons,  k(o) = o(o) = 4(o) = 0 for each case. We have
p - s2 cos a
q-0
 r - 02 sinaf
Because cr is a held constant for each case, p = q = r = 0.
   1) Euler angles: the Euler angle rates are given by Eqs. (4.66-4.68). For this
case, these equations reduce to
4 = p+r tanOcos4
     0 = -r sin4
    j/ : r sec 0 cos4
The MATLABs code ODE45 was used for numericalintegration ofthe above Euler
angle rate equations. Whenever the value of any Euler angle exceeded 180 deg, it
was reset equal to -180 deg.
    For ct = 45 deg, the pitch angle 0 approaches -90 deg, and we encounter the
singularity present in Euler angle rate equations. To work around this difficulty,
when 89.5 < 0 < 90.5, we use Eqs. (4.82), (4.85), and (4.86), which assume the
following form for this case:
(4)-,=g/
                                                              (0).z =-r sin ~
                      (Vr),= 2p
2)  Method of direction cosines: the direction cosine matrix is given by
Xo = Cg Xb
   CI1 C12 CI3
- C21 C22 C23
  C31 C32 C33
where Cg = Tbo. Using Eq. (4.36), we get
                     cos o COS ip     sin0 sin 4 cos Vr - sin p cos 4     sin 0 cos ~ cos Vr + si
Tbo= cosOsiriV/ sinpsinO-sin4+cospcos4 sin sinocos4-cs~;":T]
     -sin0      sin4cos0         cos4cos0    _j
EQUATIONS OF MOTION AND ESTIMATION OF STABILITY DERIVATIVES 355
so that
C11 - cos o cos /r
C12 = siri0 sh ~ cc)s p -siri  vf cOs q)
C13 = sin 0 cos <b cos ~ + sin ~ sin ~
C21  - cos 0 sin V
C22  = sin 1/J siri0 sin 4 + cos y  cos ~
C23 = SiD VJ siri0 cos 4 - cos ~ sin 0
  C31 -. -sin 0
C32 - siri~ cos 0
C33 - COS ~ cos 0
     Substituting lt,(0) = 0(0) = ~(0) = 0. we get the initial values of the elements of
the direction cosines matrix as follows:
Cll-l    C12-0    C13 -0
C21-0     C22 -1     C23 -0
C31 - C32 - O      C33 - 1
The rate equations for updating the elements ofthe direction cosine matrix elements
are given by Eqs. (4.156-4.164), which are reproduced in the following:
Cll = C12r - C13q
C12 = C13 p - Ciir
C13 = Ciiq - C12P
C21 = C22r - C23q
C22 - C23 p - C21r
C23 = C21q - C22P
C31 = C32r - C33q
C32 = C33 p - C31T
C33 = C3iq - C32P
All these first-order, coupled ordinary differential equations along with the above
'g,
v.
;l/r
   ' .7 .
    . r.
:,.
:::
 H
 .:
 :.
  :'; :
356             PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
initial conditions were integrated using the MATLAB5 code ODE45. Then, know-
ing the values of Crj, /  = J  -  1, 3, the Euler angles were calculated using the
following relations:
　
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