SOURCEFORGE » RALF » 2015 Software defined GPS receiver

h1. PART 4 : Position Estimation.

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p(. 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.

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h2. 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.

h3. a - Introduction of satellite orbit.

p=. !OrbitalPlanePositioningMin.png!

*Figure 4.1 :* Orbital plane positioning.

Orbital plane positioning parameters :

p=. !Parameters1.PNG!

p=. !OrbitPositioningInTheOrbitalPlaneMin.png!

*Figure 4.2 :* Orbit positioning in the orbital plane.

Orbit positioning in the orbital plane :

p=. !Parameters2.PNG!

p=. !SatellitePositioningMin.png!

*Figure 4.3 :* Orbital plane positioning.

Shape of the orbit :

p=. !Parameters3.PNG!

Positioning of the satellite on the orbit :

p=. !Parameters4.PNG!

Induced parameters :

p=. !Parameters5.PNG!

h3. b - GPS satellite ephemeris data.

p=. !Eph12min.png!

*Figure 4.4 :* List of ephemeris parameters included in GPS frames.

h3. c - GPS satellite position calculation algorithm.

p=. !Alg12min.png!

*Figure 4.5 :* Description of the algorithm step by step.

These tables are extracted from GPS Interface Control Document *[3]*

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h2. 2 - Navigation computation.

h3. 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 :

p=. !003.PNG!

*Figure 4.6 :* Pseudo-range measurement extracted from *[4]*

h3. 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 : 

p=. !Pos1.png!

*Equation 1*

or put in another way,

p=. !Pos2.png!

*Equation 2*

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 :

p=. !Pos3.png!

*Equation 3*

Hence,

p=. !Pos4.png!

*Equation 4*

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 :

p=. !Pos5.png!

*Equation 5*

which can be expressed as :

p=. !Pos6.png!

*Equation 6*

with the Least Square solution :

p=. !Pos7.png!

*Equation 7*

Thus, it is possible to retrieve the receiver position.

_Note that all unknowns are depicted in red color._

h3. 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.

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*References :* 

*[1]* K. Borre, D. M. Akos, N. Bertelsen, P. Rinder, S. H. Jensen, A software-defined GPS and GALILEO receiver

*[2]* M. Bousquet, Orbits and Satellite Platforms lecture script, January 2016

*[3]* GPS Interface Control Document under http://www.gps.gov/technical/icwg/IS-GPS-200H.pdf

*[4]* Position Estimation Workshop, March 2016