Data Processing for Positioning (GPS and Galileo Receiver) Part 2

Computation of Satellite Position

This section connects Earth-centered and Earth-fixed (ECEF) coordinates X, Y , Z to a satellite position described in space by Keplerian orbit elements. First we recall the orbit elementslisted in Figure 8.5. The six Keplerian orbit elements constitute an important description of the orbit, so they are repeated in schematic form in Table 8.4.

The rest of this section is unavoidably somewhat mathematical; many readers will proceed, assuming ECEF coordinates are found.

The X -axis points toward the intersection between equator and the Greenwich meridian. For our purpose this direction can be considered fixed. The Z-axis co-incides with the spin axis of the Earth. The Y-axis is orthogonal to these two directions and forms a right-handed coordinate system.

FIGURE 8.5. The Keplerian orbit elements: semi-major axis a, eccentricity e, inclination of orbit i, right ascension & of ascending node K, argument of perigeeand true anomaly f. Perigee is denoted P. The center of the Earth is denoted C.

The orbit plane intersects the Earth equator plane in the nodal line. The direction in which the satellite moves from south to north is called the ascending node K. The angle between the equator plane and the orbit plane is the inclination i. The angle at the Earth’s center C between the X-axis and the ascending node K is calledit is a right ascension. The angle at C between K and the perigee P is called argument of perigee m; it increases counterclockwise viewed from the positive Z-axis.

Figure 8.6 shows a coordinate system in the orbital plane with origin at the Earth’s center C. Thepoints to the perigee and the n-axis toward the descending node. Theis perpendicular to the orbit plane. From Figure 8.6 we E and the true anomaly f .

TABLE 8.4. Keplerian orbit elements: Satellite position

	semi-major axis	size and shape of orbit
	eccentricity
	argument of perigee	the orbital plane in the apparent system
	right ascension of ascending node
	inclination
	mean anomaly	position in the plane

 

FIGURE 8.6. The elliptic orbit withcoordinates. The true anomaly f at C.

Also, immediately we have

Hence the position vector r of the satellite with respect to the center of the Earth C is

Simple trigonometry leads to the following expression for the norm:

In general, E varies with time t while a and e are nearly constant. (There are long and short periodic perturbations to e, only short for a.) Recall thatis the geometric distance between satellite S and the Earth center C = (0, 0).

For later reference we introduce the mean motion n, which is the mean angular satellite velocity. If the period of one revolution of the satellite is T, we have

Letbe the time the satellite passes perigee, so thatKepler’s famous equation relates the mean anomalyand the eccentric anomaly E :

From Equation (8.1) we finally get

By this we have connected the true anomaly f, the eccentric anomaly E, and the mean anomalyThese relations are basic for every calculation of a satellite position.

It is important to realize that the orbital plane remains fairly stable in relation to the geocentric X, Y, Z-system. In other words, seen from space, the orbital plane remains fairly fixed in relation to the equator. The Greenwich meridian plane rotates around the Earth spin axis in accordance with Greenwich apparent sidereal time (GAST), that is, with a speed of approximately 24h/day. A GPS satellite performs two revolutions a day in its orbit having a speed of 3.87 km/s.

In the orbital plane the Cartesian coordinates of satellite S are given as

wherecomes from (8.2) with a, e, and E evaluated forrefer to Figure 8.5.

This vector is rotated into the X, Y, Z-coordinate system by the following sequence of 3D rotations:

The matrix that rotates the XY-plane byand leaves the Z-direction alone, is

and similarly for a rotation about the X-axis:

Finally, the geocentric coordinates of satellite k at timeare given as

However, GPS satellites do not follow the presented normal orbit theory. We have to use time-dependent, more accurate orbit values. They come to us as the socalled broadcast ephemerides; see Section 8.2.2. We insert those values in a procedure given below and finally we get a set of variables to be inserted into (8.8).

Obviously, the vector is time-dependent, and one speaks about the ephemeris (plural: ephemerides, emphasis on "phem") of the satellite. These are the parameter values at a specific time. Each satellite transmits its unique ephemeris data.

The parameters chosen for description of the actual orbit of a GPS satellite and its perturbations are similar to the Keplerian orbital elements. The broadcast ephemerides are calculated using the immediate previous part of the orbit and they predict the following part of the orbit. The broadcast ephemerides are accurate to 1-2 m. For geodetic applications, better accuracy is needed. One possibility is to obtain post-processed precise ephemerides, which are accurate at the dm-level.

An ephemeris is intended for use from the epoch toe of reference counted in seconds of the GPS week. It is nominally at the center of the interval over which the ephemeris is useful. The broadcast ephemerides are intended for use during this period. However, they describe the orbit to within the specified accuracy for 2 hours afterward. The broadcast ephemerides include the parameters in Table 8.2. The coefficientscorrect argument of perigee, orbit radius, and orbit inclination due to inevitable perturbations of the theoretical orbit caused by variations in the Earth’s gravity field, albedo and sun pressure, and attraction from sun and moon.

Given the transmit time t (in GPS time), the following procedure gives the necessary variables to use in (8.8):

The mean Earth rotation is denotedThis algorithm is coded as the M-file satpos. The function calculates the position of any GPS satellite at any time. It is fundamental to every position calculation.

FIGURE 8.7. Correlation of 12 s data with the preamble.

To avoid under- or overflow at the beginning or end of a week, we use the M-file check_t.