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 Ox{' coincides with Ox2 and Oz'i' falls in the OY222 plane.
        3) A third and finalr)otation 4 about Oxj' ( Ox;" or Ox2) talang O y;'to Oy;n (Oy2)
and Oz'i' to0z'i" (0z2).
EQUATIONS OF MOTION AND ES11MATION OF STABILITY DERIVATIVES 323
x.
Xl (xj"),x2(4)
Fig.4.3 Eulerangles.
To avoid ambiguities, the ranges of the Euler angles are limited as follows:
-1t < yr .< rr
{-0-~
-7T .< ~ < 7r
(4.1)
(4.2)
(4.3)
    Transformation matrices using Eu/er ang/es.   1) For first rotation t/J, we
have the following relation (see Fig. 4.4a):
                                           x; = XI COS Vr + yl sin ~                                  (4.4)
                                     .     y{ = -xi sin Vr + yi cos p                                (4.5)
                                       z; - Zi                                 (4.6)
Or, in matrix form,
           xi      rk  siny
    }l[x;i] (4.7)
           ZI          0
Let
                                                         cos l/J      sin .
          C= -sjnyr ,"~ 1]        (4.8)
              0   O
~"
             :(.4 1
 . Nr
   1:.
       In view ofthese restrictions, the Euler angles will have discontinuous (sawtooth)
variations for tvehicle motions involving continuous rotations. For example, in a
steady rolling maneuver, the bank angle 4 will have a sawtooth variation.
324              PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
so that
ri
z ;o
Z
Fig. 4.4    Orientation of various axes during transformation.
[xl,] =,[x;~]
(4.9)
    2) Next, we perform the rotation O as shown in Fig. 4.4b about the Oy{ axis.
Then we have
x{' = x{ cos8 - z'i sin O
          yi' N y{
zl  = x{ sin0 + z'i cos 0
(4.10)
(4.11)
(4.12)
EQUATIONS OF MOTION AND ESTIMATION OF STABILITY DERIVATIVES 325
Or, in matrix form,
so that
[x:}:'] =
     cos0 0
B-  0  1
     sin0 0
[x::~::] =B
[xl]' ,4.13,
[x:,] -.B.[x:~]
(4.14)
(4.15)
3) Finally, perform the rotation ~ about the Ox{u axis (see Fig. 4.4c) to obtain
Or, in matrix form,
Let
so that
           x;tl = x{t
  y{"  =  y{' cos 4 + z'i' sin ~
ziu = -y{'sin ~ + z7 cos 4
[x;,~},:,:] = [:
A= [:
(4.16)
(4.17)
(4.18)
  0
.OS,jl:%l[X:lj.:] (4.19)
-sin 4  ~
  O
cos*
-sin 4
(4.20)
[X:j}{:l-.[X:2;l=A[Xl'l=AB.[X;~] (4.21)
       Let T12  =  A BC.Here, T12 is the matrix that transforms a vector from the OxiYizi
　
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