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Version 29 (COLIN, Tony, 03/20/2016 03:58 PM) → Version 30/35 (COLIN, Tony, 03/20/2016 04:18 PM)
h1. PART 4 : Position Estimation.
{{toc}}
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.
p=. !Parameters1.PNG!
p=. !OrbitPositioningInTheOrbitalPlaneMin.png!
*Figure 4.2 :* Orbital plane positioning.
p=. !Parameters2.PNG!
p=. !SatellitePositioningMin.png!
*Figure 4.3 :* Orbital plane positioning.
p=. !Parameters3.PNG!
p=. !Parameters4.PNG!
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.
---
*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
{{toc}}
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.
---
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.
p=. !Parameters1.PNG!
p=. !OrbitPositioningInTheOrbitalPlaneMin.png!
*Figure 4.2 :* Orbital plane positioning.
p=. !Parameters2.PNG!
p=. !SatellitePositioningMin.png!
*Figure 4.3 :* Orbital plane positioning.
p=. !Parameters3.PNG!
p=. !Parameters4.PNG!
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]*
---
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.
---
*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