Global Positioning System Reference
In-Depth Information
datum offsets, GEOID03 was constructed to accommodate these origin differences
and directly convert between NAD 83 and NAVD 88, rather than express a region
of an idealized global geoid. In addition, offsets of 0.5m or more in national height
datums are common, as tabulated in [11]. For these reasons, (2.2) is valid as a con-
ceptual model but may be problematic in actual precision applications. Detailed
treatment of height systems is beyond the scope of this text. However, more
information may be found in [12, 13].
2.3
Fundamentals of Satellite Orbits
2.3.1 Orbital Mechanics
As described in Section 2.1, a GPS user needs accurate information about the posi-
tions of the GPS satellites in order to determine his or her position. Therefore, it is
important to understand how the GPS orbits are characterized. We begin by describ-
ing the forces acting on a satellite, the most significant of which is the Earth's gravi-
tation. If the Earth were perfectly spherical and of uniform density, then the Earth's
gravitation would behave as if the Earth were a point mass. Let an object of mass m
be located at position vector r in an ECI coordinate system. If G is the universal grav-
itational constant, M is the mass of the Earth, and the Earth's gravitation acts as a
point mass, then, according to Newton's laws, the force, F , acting on the object
would be given by
mM
r 3
Fa
==−
m
r
(2.3)
where a is the acceleration of the object, and r
. The minus sign on the right-hand
side of (2.3) results from the fact that gravitational forces are always attractive.
Since acceleration is the second time derivative of position, (2.3) can be rewritten as
follows:
=|
r
|
2
d
dt
r
=− µ
r
(2.4)
2
3
r
where
is the product of the universal gravitation constant and the mass of the
Earth. In WGS 84, the original value of
µ
10 8 m 3 /s 2 . Subsequently,
µ
was 3986005
×
the value of
10 8 m 3 /s 2 , but to maintain
backward compatibility of the GPS navigation message, the original value of
3986005
µ
in WGS 84 was updated to 3986004.418
×
10 8 m 3 /s 2 is still used. Equation (2.4) is the expression of so-called
two-body or Keplerian satellite motion, in which the only force acting on the satel-
lite is the point-mass Earth. Because the Earth is not spherical and has an uneven dis-
tribution of mass, (2.4) does not model the true acceleration due to the Earth's
gravitation. If the function V measures the true gravitational potential of the Earth
at an arbitrary point in space, then (2.4) may be rewritten as follows:
×
2
d
dt
r
=∇
V
(2.5)
2
 
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