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2
4
3
5
2
4
3
5
2
4
3
5
X 0
Y 0
Z 0
X
Y
Z
cosʵ Y cosʵ Z
cosʵ X sinʵ Z þsinʵ X sinʵ Y cosʵ Z
sin
cosʵ Y sinʵ Z
cosʵ X cosʵ Z þsinʵ X sinʵ Y sinʵ Z
sinʵ Y
sinʵ X cosʵ Y
cos
ʵ X sin
ʵ Z þ
cos
ʵ X sin
ʵ Y cos
ʵ Z
sin
ʵ X cos
ʵ Z þ
cos
ʵ X sin
ʵ Y sin
ʵ Z
ʵ X cos
ʵ Y
ð
7
:
6
Þ
When
ʵ X ,
ʵ Y , and
ʵ Z are very small, we neglect the second-order small quantities
to obtain:
9
=
; :
cos
ʵ X
cos
ʵ Y
cos
ʵ Z
1
sin
ʵ X ʵ X , sin
ʵ Y ʵ Y , sin
ʵ Z ʵ Z
sin
ʵ X sin
ʵ Y
sin
ʵ Y sin
ʵ Z
sin
ʵ Z sin
ʵ X
0
Then ( 7.6 ) can be written as:
2
4
3
5
2
4
3
5
2
4
3
5 ,
X 0
Y 0
Z 0
X
Y
Z
1
ʵ Z
ʵ Y
ʵ Z
1
ʵ X
ð
7
:
7
Þ
ʵ Y
ʵ X
1
where the coefficient matrix is also called the differential rotation matrix. Compar-
ing ( 7.2 ) and ( 7.6 ) gives:
9
=
; :
cos
ʳ 3
cos
ʵ X cos
ʵ Y
cos
ʲ 2
cos
ʵ X cos
ʵ Z þ
sin
ʵ X sin
ʵ Y sin
ʵ Z
ð
7
:
8
Þ
cos α 1 ᄐ cos ʵ Y cos ʵ Z
Omitting the small terms higher than third order created by mutual multiplica-
tion of
ʵ X ,
ʵ Y , and
ʵ Z in ( 7.8 ) produces:
p
ʵ
=
; :
ʳ 3
X þ ʵ
Y
p
ʵ
X
Z
ð
7
:
9
Þ
ʲ 2
þ ʵ
p
ʵ
2
2
Z
α 1
Y þ ʵ
Equation ( 7.7 ) shows that by neglecting the small quantities of the second-order,
the rotation matrices are commutative.
The three rotation angles
ʵ X are called the yaw, pitch, and roll,
respectively, in describing the vehicle's attitude and are used to represent the
precession, rotation (spin), and nutation, respectively while studying the rotation
of rigid bodies.
ʵ Z ,
ʵ Y , and
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