Fundamentals of Satellite Navigation - Understanding GPS: Principles and Applications - page 43

able is measured at time t k , the time (in seconds) from epoch to the GPS system time
of signal transmission.
There are a few additional subtleties in the computations described in Table 2.3.
First, computation (5), which is Kepler's equation, (2.9), is transcendental in the
desired parameter, E k . Therefore, the solution must be carried out numerically. Kep-
ler's equation is readily solved either by iteration or Newton's method. A second
subtlety is that computation (6) must produce the true anomaly in the correct quad-
rant. Therefore, it is necessary either to use both the sine and the cosine or to use a
smart arcsine function. Also, to carry out computation (14), it is also necessary to
know the rotation rate of the Earth. According to IS-GPS-200, this rotation rate is
&

Ω ε = 7 2921151467 10 5 rad/s, which is consistent with the WGS 84 value to be

used for navigation, though WGS 84 also provides a slightly different value in defin-
ing the ellipsoid. Finally, IS-GPS-200 defines the value of

. ×

−

to be used by GPS user

equipment as exactly 3.1415926535898.
As can be seen from the computations in Table 2.3, the variations in time of the
orbital parameters are modeled differently for particular parameters. For example,
mean motion is given a constant correction in computation (2), which effectively
corrects the mean anomaly computed in (4). On the other hand, latitude, radius,
and inclination are corrected by truncated harmonic series in computations (8), (9),
and (10), respectively. Eccentricity is given no correction. Finally, longitude of the
node is corrected linearly in time in computation (14). It is a misnomer of GPS sys-
tem terminology, as in Table 2.2, that the longitude of the node, Ω 0 , is given “at

weekly epoch.” In reality, Ω 0 is given at the reference time of ephemeris, t 0 e , the same

as the other GPS parameters. This can be verified by inspection of computation (14)
from Table 2.3. Reference [15] provides an excellent description of the tradeoffs
that resulted in the use of ephemeris message parameters and computations
described in Tables 2.2 and 2.3.

2.3.2 Constellation Design
A satellite “constellation” is characterized by the set of orbital parameters for the
individual satellites in that constellation. The design of a satellite constellation
entails the selection of those orbital parameters to optimize some objective function
of the constellation (typically to maximize some set of performance parameters at
minimum cost—i.e., with the fewest satellites). The design of satellite constellations
has been the subject of numerous studies and publications. Our purpose here is to
provide a general overview of satellite constellation design to summarize the salient
considerations in the design of constellations for satellite navigation, to provide
some perspective on the selection of the original 24-satellite GPS constellation, and
to set the ground work for discussions of future satellite navigation constellations
such as GALILEO.

2.3.2.1 Overview of Constellation Design
Given innumerable combinations of orbital parameters for each satellite in a con-
stellation, it is convenient to segregate orbits into categories. One categorization of
orbits is by eccentricity: