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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:
                                         0 = sir1-1(-C31)
, = cos-, (
7 =,os-, (
C33
)sgn(C32,
)sgn(C2,
   3) Quaternions: the four Euler parameters are given by Eqs. (4.192-4.195),
which are reproduced in the following:
     Vr  0  4    rit . O . 4
eo = cos  ;~/ cos 2 coS ~+ sin ~ sm 2 sm 2
      q   o . ~  , Vr . 0  *
ei = cos ; cos 2 sm 2 -. sm 2 sm 2 cos 2
     /  r
            0.4
e2 =,os gsmg,os~+sm g,os;srn 2
      v/ . 0 . 4  . v   0  4
e3 = -cos 2 sm 2 sm g +sin 2 .os 2 ,Os 2
eo = -;:(el p + e2q + e3r) + Aee0
ei - }:(eoP + e2r - e3q) + A,€ei
e2 - /z(eoq + e3p - eir) + A,ee2
e3 - ;:(eor + eiq - e2P) + Aee3
where € is given by Eq. (4.215) and is reproduced in the following:
6 - 1 - (e% + ef + eg- + e~z)
We assume A - 0.0001 and integrate the above rate equations for four quaternion
parameters using MATLAB5 coE9e ODE45. Then, the Euler angles are obtained
using Eqs. (4.217), (4.221), and (4.222).
EQUATIONS OF MOTION AND ESTIMATION OF STABILITY DERIVATIVES 357
o
0
'o
f=
o
(,,
a)
-o
.:
rn
a
  DCM
N . N - -             quLtcrnims
Fig. 4.12    Euler angles (S z = 30 degts, a = 30 deg).
     The results are presented in Figs. 4.12-4.14. We observe that all three methods
　
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