Global Positioning System Reference
In-Depth Information
The direction cosine matrix is defined as
cos
(X
1
on
X
2
)
cos
(Y
1
on
X
2
)
C
1
≡
(
4
.
2
)
cos
(X
1
on
Y
2
)
cos
(Y
1
on
Y
2
)
This represents that the coordinate system is transferred from system 1 to
system 2.
In a three-dimensional system, the directional cosine can be written as
cos
(X
1
on
X
2
)
cos
(Y
1
on
X
2
)
cos
(Z
1
on
X
2
)
C
1
≡
cos
(X
1
on
Y
2
)
cos
(Y
1
on
Y
2
)
cos
(Z
1
on
Y
2
)
(
4
.
3
)
cos
(X
1
on
Z
2
)
cos
(Y
1
on
Z
2
)
cos
(Z
1
on
Z
2
)
Sometimes it is difficult to make one single transform from one coordinate
to another one, but the transform can be achieved in a step-by-step manner. For
example, if the transform is to rotate angle
α
around the
z
-axis and rotate angle
β
around the
y
-axis, it is easier to perform the transform in two steps. In other
words, the directional cosine matrix can be used in a cascading manner. The first
step is to rotate a positive angle
α
around the
z
-axis. The corresponding direction
cosine matrix is
cos
α
sin
α
0
C
1
=
−
sin
α
cos
α
0
(
4
.
4
)
0
0
1
The second step is to rotate a positive angle
β
around the
x
-axis; the correspond-
ing direction cosine matrix is
1
0
0
C
2
=
0
β
sin
β
(
4
.
5
)
0
−
sin
β
cos
β
The overall transform can be written as
1
0
0
cos
α
sin
α
0
C
1
=
C
2
C
1
=
0
β
sin
β
−
sin
α
cos
α
0
0
−
sin
β
cos
β
0
0
1
cos
α
sin
α
0
=
−
sin
α
cos
β
cos
α
cos
β
sin
β
(4.6)
sin
α
sin
β
−
cos
α
sin
β
cos
β
It should be noted that the order of multiplication is very important; if the order
is reversed, the wrong result will be obtained.
Suppose one wants to transform from coordinate system 1 to system
n
through
system 2
,
3
,...n
−
1. The following relation can be used:
C
1
C
n
−
1
···
C
2
C
1
=
(
4
.
7
)
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