Fundamentals of Satellite Navigation - Understanding GPS: Principles and Applications - page 41

GPS constellation. The remaining orbital parameters vary between satellites, so that
the constellation provides coverage of the entire Earth.
As previously indicated, the actual motion of a satellite is described by (2.7)
rather than (2.4). However, the Keplerian orbital elements may be computed for a
satellite at a particular instant in time from its true position and velocity vectors. In
this case, the orbital elements are known as osculating ; if all forces perturbing the
point-mass force of the Earth were to cease at the time of the osculating orbital ele-
ments, the satellite would follow the two-body orbit described by those osculating
elements. Because of the perturbing accelerations in (2.7), the osculating orbital ele-
ments of a satellite will change slowly over time. The osculating orbital elements do
not change quickly because the first term of the Earth's gravitational harmonic
series, (2.6), is still the dominant element in the force field acting on a satellite.
GPS almanac data and ephemeris data transmitted by the satellites include the
osculating Keplerian orbital elements, with the exception that the time of perigee
passage is converted to mean anomaly at epoch by (2.11). In order to be useful, it is
necessary for the osculating elements to include the reference time, known as the
time of epoch or time of ephemeris, at which the orbital elements were valid. Only
at epoch are the orbital elements exactly as described by the osculating values; at all
later times, the true orbital elements deviate slightly from the osculating values.
Because it is necessary for the GPS ephemeris message to contain very accurate
information about the satellite's position and velocity, it is insufficient to use only
the osculating Keplerian orbital elements for computing the position of a GPS satel-
lite, except very near the epoch of those elements. One solution to this problem
would be to update the GPS ephemeris messages very frequently. Another solution
would be for the GPS receiver to integrate the fully perturbed equation of motion,
(2.7), which would include a detailed force model, from epoch to the desired time.
Because both of these solutions are computationally intensive, they are impractical
for real-time operations. Therefore, the osculating Keplerian orbital elements in the
GPS ephemeris message are augmented by “correction” parameters that allow the
user to estimate the Keplerian elements fairly accurately during the periods of time
between updates of the satellite's ephemeris message. (Particulars on ephemeris
message updating are provided in Section 3.3.1.4.) Any time after the epoch of a
particular ephemeris message, the GPS receiver uses the correction parameters to
estimate the true orbital elements at the desired time.
Table 2.2 summarizes the parameters contained in the GPS ephemeris message.
These parameters are found in IS-GPS-200 [4], which is the interface specification
between the GPS space segment and GPS user segment. As can be seen, the first
seven parameters of the GPS ephemeris message are time of epoch and, essentially,
the osculating Keplerian orbital elements at the time of epoch, with the exceptions
that the semimajor axis is reported as its square root and that mean anomaly is used
instead of time of perigee passage. The next nine parameters allow for corrections to
the Keplerian elements as functions of time after epoch.
Table 2.3 provides the algorithm by which a GPS receiver computes the position
vector of a satellite ( x s , y s , z s ) in the ECEF coordinate system from the orbital ele-
ments in Table 2.2. For computation (3) in Table 2.3, t represents the GPS system
time at which the GPS signal was transmitted. In the notation of Table 2.3, the sub-
script k appearing in computation (3) and below means that the subscripted vari-