5.7 Satellite Positioning Systems

5.7 Satellite Positioning Systems

There are two methods for positioning a satellite: distance measurement from the ground station, and GPS as shown in Figure 5.7.1. As GPS is explained in 6.8, the former method will be explained here. The measurement of distance for satellite positioning is called the range and range rate system, by which the time of a radio wave transmitted between the ground station and a transponder onboard the satellite is measured with the Doppler frequency. It enables the distance or range and the range rate to be measured. The accuracy of the range and range rate is a few meters/second, respectively. The accuracy depends on the parameters of frequency, signal, the location of the ground station, coordinate system, time measurement system, reflection in the troposphere and the ionosphere etc.

The position of a satellite by the range and range rate method is limited only near the ground station and is also discrete. In order to determine the satellite orbit in a time sequential function, it is necessary to construct a model.

A primitive model is a parabolic curve based on the third theory of Kepler that defines the motion of two bodies in space, termed the two body problem, under the law of universal gravitation. The parabolic curve can be expressed with the six elements of the Keplerian orbit.

However in reality there are influences from other planets, which results in departure from the parabolic curve. This is called the perturbation.

There are two methods to determine the precision for an body problem (n>2); the numerical integral calculation, which has high accuracy but is time consuming and the analytical method with a lower accuracy but faster calculation, as shown in Table 5.7.1.

The satellite position in an orbit will be determined at certain time intervals, from which the orbit at an arbitrary time can be interpolated by the least square method for high order polynomials or by Lagrange's interpolation method.