Global Positioning System Reference
In-Depth Information
4.2 DIRECTION COSINE MATRIX
(
1-3
)
In this section, the direction cosine matrix will be introduced. A simple two-
dimensional example will be used to illustrate the idea, which will be extended
into a three-dimensional one without further proof. Figure 4.1 shows two two-
dimensional systems (
x
1
,
y
1
)and(
x
2
,
y
2
). The second coordinate system is
obtained from rotating the first system by a positive angle
α
. A point
p
is used
to find the relation between the two systems. The point
p
is located at (
X
1
,
Y
1
)
in the (
x
1
,
y
1
) system and at (
X
2
,
Y
2
)inthe(
x
2
,
y
2
) system. The relation between
(
X
2
,
Y
2
)and(
X
1
,
Y
1
) can be found from the following equations:
X
2
=
X
1
cos
α
+
Y
1
sin
α
=
X
1
cos
(X
1
on
X
2
)
+
Y
1
cos
(Y
1
on
X
2
)
Y
2
=−
X
1
sin
α
+
Y
1
cos
α
=
X
1
cos
(X
1
on
Y
2
)
+
Y
1
cos
(Y
1
on
Y
2
)
(4.1)
In matrix form this equation can be written as
X
2
Y
2
cos
(X
1
on
X
2
)
X
1
Y
1
cos
(Y
1
on
X
2
)
=
(
4
.
1
)
cos
(X
1
on
Y
2
)
cos
(Y
1
on
Y
2
)
FIGURE 4.1
Two coordinate systems.
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