Fundamentals of Satellite Navigation - Understanding GPS: Principles and Applications - page 37

the GPS ephemeris message, which includes not only six integrals of two-body
motion, but also the time of their applicability (reference time) and a characteriza-
tion of how those parameters change over time. With this information, a GPS
receiver can compute the “corrected” integrals of motion for a GPS satellite at the
time when it is solving the navigation problem. From the corrected integrals, the
position vector of the satellite can be computed, as we will show. First, we present
the definitions of the six integrals of two-body motion used in the GPS system.
There are many possible formulations of the solution to the two-body problem,
and GPS adopts the notation of the classical solution, which uses a particular set of
six integrals of motion known as the Keplerian orbital elements. These Keplerian
elements depend on the fact that for any initial conditions r 0 and v 0 at time t 0 ,the
solution to (2.4) (i.e., the orbit), will be a planar conic section. The first three
Keplerian orbital elements, illustrated in Figure 2.9, define the shape of the orbit.
Figure 2.9 shows an elliptical orbit that has semimajor axis a and eccentricity e .
(Hyperbolic and parabolic trajectories are possible but not relevant for Earth-orbit-
ing satellites, such as in GPS.) In Figure 2.9, the elliptical orbit has a focus at point
F , which corresponds to the center of the mass of the Earth (and hence the origin
of an ECI or ECEF coordinate system). The time t 0 at which the satellite is at some
reference point A in its orbit is known as the epoch . As part of the GPS ephemeris
message, where the epoch corresponds to the time at which the Keplerian ele-
ments define the actual location of the satellite, the epoch is called reference time of
ephemeris . The point, P , where the satellite is closest to the center of the Earth
is known as perigee, and the time at which the satellite passes perigee, , is another
Keplerian orbital parameter. In summary, the three Keplerian orbital elements
that define the shape of the elliptical orbit and time relative to perigee are as
follows:

a = semimajor axis of the ellipse

e = eccentricity of the ellipse

= time of perigee passage

A

a

r

Direction
of perigee

ν

P

F

t = τ

ae

Figure 2.9 The three Keplerian orbital elements defining the shape of the satellite's orbit.