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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
system to the Ox2Y2z2 system. Performing the indicated matrix multiplications,
::l:o]
O
1
0
[:,::/
1
y;
a)
::g0']
il:%]
c)
326 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
we obtain
Then,
X2 = T12Xi
(4.22)
(4.23)
where X denotes a vector expressed in matrix form and its suffix indicates the
coordinate system in which ffie 'vector X has its components. For example, Xi is
a vector having its components in the OxiYizi axes system.
Each transformation matrix A, B, or C is an orthogonal matrix. An important
property of an orthogonal matrix is that its transpose is equal to its inverse. The
transformation matrix T12, which is the product of the three orthogonal matrices,
is also orthogonal. Therefore,
T21 = (Ti2)-1 = (Ti2)t
sin0 sin 4 cos yr - sin ~ cos 4
sin Vr sin 0 sin 4 +cos yr COs 4
sin 4 cos 0
(4.24)
(4.25)
where superscripts -1 and ' denote the inverse and transpose of a matrix, respec-
tively.
Transformation of vectors. Let us consider the transformation of vectors
between various coordinate systems as follows.
Inertial to Earth:[rxed system. Here we are considering transformation be-
tween the Oxryizt and OxEyEzE systems. Let the OxEyEzE system coincide
with Oxt yrz, system at t - 0. For t > 0, ltr - fZet, 0 - ~ - 0, where S-2e iS the
angular velocity of the Earth about Ozi or OzE axis. Then using Eq. (4.22),
XE = riExi
cos S2et SiD CZet 0
-. -sinS'2et COSS2et O Xi
OO1
The reverse transformation is given by
where
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