Earth-Centered, Earth-Fixed Coordinate System - Fundamentals of Global Positioning System Receivers: A Software Approach - page 55

Substituting Equation (4.9) into Equation (4.8) the result can be simplified as

r = a s ( 1 − e s cos E) ( 4 . 10 )

The position of the satellite can be found as

x = r cos ν
y = r sin ν

z = 0 (4.11)

This equation does not reference any point on the surface of the earth but refer-
ences the center of the earth. It is desirable to reference to a user position that is
a point on or above the surface of the earth.
First a common point must be selected and this point must be on the surface
of the earth as well as on the satellite orbit. The satellite orbit plane intercepts the
earth equator plane to form a line. An ascending node is defined along this line
toward the point where the satellite crosses the equator in the north (ascending)
direction. The angle ω between the perigee and ascending node in the orbit plane
is referred to as the argument of the perigee. This angle information can be
obtained from the received satellite signal. Now let us change the x -axis from
the perigee direction to the ascending node. This transform can be accomplished
by keeping the z -axis unchanged and rotating the x -axis by the angle ω as
shown in Figure 4.3. In Figure 4.3 the y -axis is not shown. The x i -axis and the
z i -axis are perpendicular and the y i -axis is perpendicular to the x i z i plane. The
corresponding direction cosine matrix is

cos ω − sin ω 0

C 1 = sin ω cos ω 0 ( 4 . 12 )

0 0 1

In this equation the angle ω is in the negative direction; therefore the sin ω has a
different sign from Equation (4.4). This rotation changes the x 1 -axis to x 2 -axis.
The next step is to change from the orbit plane to the equator plane. This
transform can be accomplished by using the x 2 -axis as a pivot and rotate angle i .
This angle i is the angle between the satellite orbit plane and the equator plane
and is referred to as the inclination angle. This inclination angle is in the data
transmitted by the satellite. The corresponding direction cosine matrix is

10 0

C 2 = 0c s i − sin i ( 4 . 13 )

0 n i cos i

The angle i is also in the negative direction. After this transform, the z 3 -axis is
perpendicular to the equator plane rather than the orbit of the satellite and the
x 3 -axis is along the ascending point.