BROADCAST EPHEMERIS DATA
Forces of gravitational and non-gravitational origin perturb the motion of the GPS satellites, causing the orbits to deviate from a Keplerian ellipse in inertial space (Figure 1 below) -- defined by the six elements a (semi-major axis), e (eccentricity), i (inclination), (longitude of the ascending node), (angle of perigee), and f (true anomaly) or E (eccentric anomaly). The perturbations are characterised by periodic and secular components, and must be continually determined through the analysis of tracking data. In the case of the GPS broadcast ephemerides, this procedure is a three-step process:
Figure 1. The Keplerian ellipse in space.
In order to adequately describe the GPS orbits during the interval of time for which the ephemeris information is transmitted (at least an hour), a representation based on Keplerian elements plus perturbations is used -- see Table 1 and Figure 2 below.
Figure 2. Broadcast Ephemeris orbital representation.
|Mo||Mean anomaly at reference time|
|n||Mean motion difference from computed value|
|Square root of the semi-major axis|
|o||Right Ascension at reference time|
|io||Inclination angle at reference time|
|Rate of change of inclination angle|
|Argument of Perigee|
|Rate of change of Right Ascension angle|
|Cuc, Cus||Amplitude of the cosine and sine harmonic correction terms to the argument of latitude|
|Crc, Crs||Amplitude of the cosine and sine harmonic correction terms to the orbit radius|
|Cic, Cis||Amplitude of the cosine and sine harmonic correction terms to the angle of inclination|
|toe||Ephemeris reference time (seconds in the GPS week)|
Note that the satellite ephemerides are broadcast two hours in advance of the epoch for which they were calculated (and valid). Generally there is a daily update, though sometimes more frequent. Therefore the portions of ephemeris data for the second through fourteenth day are not normally transmitted, except when upload is not possible. At the same time each satellite's clock state is estimated, then extrapolated into the future, and the information is formatted into the Navigation Message.
The parameters are given in terms of the ephemeris reference time toe -- nominally the centre of the transmission period. (GPS system time, as derived from the coded signals, and toe is measured in seconds from the start of the GPS week, at sunday midnight.) Although the Keplerian representation has physical meaning, additional parameters are required to model the perturbations about the Keplerian orbit. The following parameters are introduced:
These parameters are obtained from a curve fit to the predicted satellite ephemeris over an interval of 4 to 6 hours. They are not true Keplerian elements as they only describe the ephemeris over the interval of applicability and not for the whole orbit. (Although only intended for use during the transmission period, they do, however, describe the orbit to the required accuracy over intervals of 1.5 to 5 or more hours.) While it is difficult to judge the accuracy of the broadcast ephemerides (the procedures are undergoing continual improvement), it is expected that it is on average better than 10m.
A sample of the broadcast Navigation Message for one satellite is presented in Table 2 below.
Table 2. Sample Broadcast Ephemeris message for satellite PRN26.
PRN Date/time of clock toc a0 (msec) a1 (msec/day) a2 (msec/day2) 26 92 07 26 08 51 44.0 1.635868102312D-06 -3.410605131648D-13 0.000000000000D+00 Age of ephemeris (sec) Crs (m) Dn (rad/sec) Mo (rads) 2.170000000000D+02 1.128125000000D+01 4.738768932810D-09 5.863412966114D-02 Cuc (rads) e Cus (rads) 4.898756742477D-07 8.141674217768D-03 9.039416909218D-06 5.153610269547D+03 toe (secs in GPS wk) Cic (rads) Wo (rads) Cis (rads) 3.190400000000D+04 -7.636845111847D-08 -1.371976967685D+00 -4.470348358154D-08 io (rads) Crc (m) w (rads) 9.603279283700D-01 2.067812500000D+02 -1.491259472097D+00 -8.059264366977D-09 GPS week number -4.760912775126D-10 1.000000000000D+00 6.550000000000D+02 0.000000000000D+00
The procedure set out in Table 3 allows the user to derive the earth-centred / earth-fixed Cartesian coordinates from the broadcast orbital parameters in Table 1. The earth-fixed reference system is presently based on the WGS84 (prior to January 1987 the system used was WGS72). The algorithm in Table 3 is implemented within every code-correlating GPS receiver, be it the "navigation" or "surveying" variety.
Table 3. Broadcast Ephemeris computational procedure.
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© Chris Rizos, SNAP-UNSW, 1999