PART 4 : Position Estimation.¶
- Table of contents
- PART 4 : Position Estimation.
Once the navigation bits from at least 4 satellites have been retrieved from the acquisition/tracking part, it is possible to estimate the desired position of the receiver.
1 - Ephemeris.¶
GPS uses a particular algorithm in order to characterise satellite position. In comparison with GLONASS, this method requires more parameters, but less complexity.
a - Introduction of satellite orbit from .¶
Figure 4.1 : Orbital plane positioning.
Orbital plane positioning parameters :
Figure 4.2 : Orbit positioning in the orbital plane.
Orbit positioning in the orbital plane :
Figure 4.3 : Orbital plane positioning.
Shape of the orbit :
Positioning of the satellite on the orbit :
Induced parameters :
b - GPS satellite ephemeris data.¶
GPS uses previous classical ephemeris data for orbit and satellite position determination, and decompose them into elementary parameters to be implemented in the navigation frame :
Figure 4.4 : List of ephemeris parameters included in GPS frames.
c - GPS satellite position calculation algorithm.¶
Starting from the GPS ephemeris present in the navigation frame - subframes 2 and 3 - it is now possible to compute the satellite position via the following algorithm :
Figure 4.5 : Description of the algorithm step by step.
These tables are extracted from GPS Interface Control Document 
2 - Navigation computation.¶
a - Reminder about the range impairments.¶
The following figure gives the impairments affecting the range in case of the GPS system as well as the correction process :
Figure 4.6 : Pseudo-range measurement extracted from 
b - Demonstration of the Pseudo-ranges with Least Square method.¶
Starting from the fact that can determine most of the elements within the pseudo-range measurement PR_sat(i) from the information provided by each satellite, we have the equation :
or put in another way,
Indeed 4 measurements are needed, providing 4 equations with 4 unknows which are the receiver coordinates and the clock bias of the receiver. As the equation is highly non-linear, it is important to proceed to a linearization such as the Taylor expansion :
In practise, for a receiver located e.g. in France PR (t_0) can be described by Paris location as initialization for the algorithm.
In vectorial form the equation becomes :
which can be expressed as :
with the Least Square solution :
Thus, it is possible to retrieve the receiver position.
Note that all unknowns are depicted in red color.
c - Kalman filter.¶
Another position estimation method is Kalman filter i.e. an algorithm that uses a series of measurements observed over time, containing statistical noise and other inaccuracies, and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone.
In this project, a single measurement will be used for "simplicity" purposes, therefore, the Least Square method is more appropriate for this issue.
 K. Borre, D. M. Akos, N. Bertelsen, P. Rinder, S. H. Jensen, A software-defined GPS and GALILEO receiver
 M. Bousquet, Orbits and Satellite Platforms lecture script, January 2016
 GPS Interface Control Document under http://www.gps.gov/technical/icwg/IS-GPS-200H.pdf
 Position Estimation Workshop, March 2016