Satellite Coordinate Systems

Introduction

Satellite position usually described in orbital system as a function of time. Need to relate this system to a terrestrial system.

Perigee - point of closest approach to earth
Apogee - point farthest from earth
Line of apsides - line connecting perigee and apogee, major axis of orbital ellipse
Anomaly - the angular distance from perigee to the satellite, S

true anomaly, f - the angle between the line of apsides and the line joining the center of the earth with the satellite determined in a counter-clockwise direction
eccentric anomaly, E - the angle between the line of apsides and the line joining the geometrical center of the ellipse with the projection of the satellite S' on a concentric circle of radius a.
mean anomaly, m - is the true anomaly corresponding to the motion of an imaginary satellite having uniform angular velocity w; m = 0 at perigee and then increases linearly with time at a rate of 2p per revolution.

The relation between the eccentric and mean anomalies is: E = m + e sin E

Relationship be true and eccentric anomalies:

ORBITAL COORDINATE SYSTEM (OR)

Origin is at the earth's center of mass,
x^OR coincides with the line of apsides
y^OR corresponds to f = p/2
z^OR completes the right-handed coordinate system

The instantaneous position vector for the satellite is

where a and e are ellipsoid constants and r, f and E vary with time.

The orientation of the OR system has to be fixed in space with three more parameters. The orbital plane is extended to intersect the celestial sphere, and its trace, the projected orbit, intersects the celestial equator at the ascending node; that is, the point where the satellite crosses the celestial equator when going from the southern hemisphere to the northern hemisphere. Vice-versa, the descending node is the point where the satellite crosses the celestial equator from north to south. The angle between the orbital half-plane and the celestial equator is known the angle of inclination, i. The angle between the ascending nodal line and the line of apsides (counter-clockwise) is known as the argument of perigee, v. The angle between x^AP (true vernal equinox) and the ascending nodal line measured counter-clockwise from +Z^OR axis in the equatorial plane is the right ascension of the ascending node, W.

Combining these three quantities, i, v, and W with those describing the orbital ellipse and the motion of the satellite in the orbit constitute the six Keplarian orbital elements.

SIX KEPLARIAN ORBITAL PARAMETERS

Parameter	Name	Rational
a	semi-major axis	Size and shape of orbit
e	eccentricity	Size and shape of orbit
v	argument of perigee	Position of the orbit in the AP system
W	right ascension of the ascending node
i	inclination of orbit
m	mean (or other) anomaly	Position of satellite in orbit

An alternative system uses a geocentric Cartesian representation of (x, y, z, ) where (x, y, z) describe the position of the vector, and

is its velocity vector at given epoch t.

Point positioning requires the transformation of positions of the satellite (usually predicted in the broadcast ephemeris) from OR system to the CT system where they are used to compute the position of the observing station. This transformation proceeds in three steps: OR to AP, AP to IT, and IT to CT. The complete transformation is

The time-varying position of the satellite is known as its ephemeris. Since the ephemeris for a GPS satellite has the Earth's polar motion included in the six Keplarian parameters, the preceding equation can be simplified as

As we can see in this simpler equation, the last two rotations are both around the Z^CT axis. These two rotations can be combined into one rotation of l = GAST − W further simplifying the equation to

View animation

Directions for Viewing 3d Animation

You will need a VRML 2.0 viewer to see the animation to the left. Download and install the free Cortona vrml plug-in.
Select the graphic in the left screen or the link below it.
Please press the cube labeled 1 to view the first rotation of −(ν + ω) about the z satellite coordinate system axis. In the animation the satellite coordinate system is shown on the satellite as a reminder that we are rotating the (X, Y, Z)^T satellite coordinates.
The cube labeled 2 will animate the second rotation of −i about the once-rotated x axis. This rotation brings the twice-rotated satellite coordinate system's equatorial plane in coincident with the CTS equatorial plane.
The cube labeled 3 will animate the third rotation of −W about the twice rotated z axis of the satellite coordinate system. placing the satellite coordinates in the celestial reference frame.
Finally, the cube labeled 4 will rotate the satellite's thrice-rotated coordinates by an amount of GAST about the z axis to bring the satellite coordinates into the conventional terrestrial reference frame. The last two rotations can be combined into a single rotation about the z axis of l (GAST − W).
The cube labeled "R" will reset it to the original rotation so that you can view the rotations another time.
While viewing the animation, you can select the "study" and "turn" buttons on the left panel to roll the entire image so that the rotations can be viewed from any perspective. the "restore" button in the lower panel will reset the image to its original orientation.

The range mathematical model can be written as

where is the measured range from the station P_i to satellite position S_j, at time t_j; and = (x_i, y_i, z_i) are the unknown Cartesian coordinates for the receiver in the CT system. For each range observation, one range equation can be written, and four such equations are sufficient to uniquely solve for the receiver position and the unknown timing error in the receiver. This discussion is completed in Chapter 13 of Elementary Surveying: An Introduction in Geomatics, 11^th Ed. This

Transformations of Terrestrial Positions

It is often necessary to transform the position of a point, typically located on the surface of the earth from one coordinate system to another. The most typical transformations are:

The transformation of the geodetic curvilinear coordinates (f, l, h)^G into their representative Cartesian coordinates (x, y, z)^G and vice versa. (See Ellipse).
The transformation of the CT coordinates (x, y, z)^CT into non-geocentric geodetic coordinates (f, l, h) and vice versa.
The transformation of astronomical coordinates (F, L) into geodetic curvilinear coordinates (f, l) along with the transformation of the astronomical and geodetic azimuths (A and a), and the orthometric height (H) into the geodetic height (h), and vice versa.
The transformation of horizontal, geodetic curvilinear coordinates (f, l, h)₁ referred to ellipsoid₁ into another triplet (f, l, h)₂ referred to ellipsoid₂.
The transformation of horizontal, geodetic curvilinear coordinates (f, l) into map coordinates (x, y). (This topic of map projections and their use in surveying was extensively covered in SUR 262)

These transformations can be separated into two classes. The first consists of the transformation within one family of coordinate systems (such as in 1). The second class is between families of coordinate systems with different locations and orientations, and will be discussed herein.

2

This second transformation requires knowledge about the position and orientation of the reference ellipsoid. This task of positioning and orienting the reference ellipsoid is known as the establishment of a horizontal geodetic datum. The positioning of the reference ellipsoid requires the parameters a and f, six additional parameters for a total of eight parameters.

The six geocentric set of datum parameters are:

Datum translation components: T_x,T_y, T_z required to translate the origin
Datum misalignment angles: e_x, e_y, e_z required to align the axes.

Thus the second transformation which goes from a geocentric system(f, l, h)^G into (x, y, z)^CT is done in two steps.

(f, l, h)^G into (x, y, z)^G using standard geocentric coordinates
(x, y, z)^G into (x, y, z)^CT employing the formula

(1)

Obviously, if the two coordinate systems are parallel, i.e., e_x = e_y = e_z = 0, the above equations simplifies to

The accuracy of this transformation is typically about �2 m, but is actually dependent on the accuracy of the transformation parameters. These transformations are typically used with GPS.

3

The third transformation allows us to go from an astronomical system to a geodetic system is not performed today since astronomical positioning is no longer a major positioning technology. These transformations were covered in the lesson on Coordinate Systems, or can be easily accomplished by applying geodetic corrections to conventional instrument observations. This topic is included in a future lesson on Relative Positioning.

4

The fourth transformation is used to transform coordinates between two different datums such as NAD27 and NAD83. In this transformation it is not only necessary to account for differences in position and orientation of the geometric centers of the ellipsoids, but also the differing sizes of the ellipsoid. Consider two ellipsoids with defining parameters of (a₁, f₁) and (a₂, f₂); their geometrical centers with respect to the Earth's center of mass being and ; and their misalignment angles with respect to the CT system being (e_x₁, e_y₁, e_z₁) and (e_x₂, e_y₂, e_z₂). Furthermore, let us denote the coordinates of a point in the first datum as (f₁, l₁, h₁) and in the second system as (f₂, l₂, h₂).

There are two techniques for obtaining (f₂, l₂, h₂) as functions of (f₁, l₁, h₁). The first direct approach is to find the CT coordinates (x, y, z) from (f₁, l₁, h₁) and then transform back into (f₂, l₂, h₂) using the inverse operation. This method requires the transformations listed in 2. The second method requires knowledge of the differences da = a₂ - a_1;df = f₂ - f₁; dx_E = x_E₂ - x_E₁; ... ; de_z = de_z₂ - de_z₁ which must be sufficiently small.