Global Positioning System Reference
In-Depth Information
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8.2 TRANSFORMING NEARLY ALIGNED COORDINATE SYSTEMS
The transformation of three-dimensional coordinate systems has been given much
attention ever since geodetic satellite techniques made it possible to relate local and
geocentric geodetic datums. Some of the pertinent works are Veis (1960), Moloden-
skii et al. (1962), Badekas (1969), Vanicek and and Wells (1974), Leick and van
Gelder (1975), and Soler and van Gelder (1987). We assume that the Cartesian coor-
dinates of points on the earth's surface are available in two systems. Often it might be
difficult to obtain the Cartesian coordinates in the local geodetic datum because the
geoid undulations with respect to the local datum might not be accurately known.
Figure 8.1 shows the coordinate system
(x)
=
(x,y,z)
, which is related to the
z
]
T
coordinate system
(u)
=
(u, v, w)
by the translation vector
t
=
[
∆
x
∆
y
∆
between the origins of the two coordinate systems and the small rotations
(ε,
)
around the
(u, v, w)
axes, respectively. The transformation equation expressed in the
(x)
coordinate system can be seen from Figure 8.1:
ψ
,
ω
[30
Lin
—
2.5
——
Lon
PgE
t
+
(
1
+
s)
Ru
−
x
=
o
(8.8)
where 1
s
denotes the scale factor between the systems and
R
is the product of
three consecutive orthogonal rotations around the axes of (u):
+
R
=
R
3
(
ω
)
R
2
(
ψ
)
R
1
(ε)
(8.9)
The symbol
R
i
denotes the rotation matrix for a rotation around axis
i
(see Section
A.2). The angles
(ε,
ψ
ω
[30
)
are positive for counterclockwise rotations about the re-
spective
(u, v, w)
axes, as viewed from the end of the positive axis. For nearly aligned
coordinate systems these rotation angles are differentially small, allowing the follow-
ing simplification
,
0
ω
−ψ
R
=
I
+
Q
=
I
+
−ω
0
ε
(8.10)
ψ
−
ε
0
z
w
T
v
y
Figure 8.1 Differential transformation be-
tween Cartesian coordinate systems.
x
u
