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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 - GPS satellite ephemeris data.
p=. !Eph12min.png!
h3. b - GPS satellite position calculation algorithm.
p=. !Alg12min.png!
These tables are extracted from GPS Interface Control Document *[2]*
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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 4.1 :* Pseudo-range measurement extracted from *[3]*
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) 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._
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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]* GPS Interface Control Document under http://www.gps.gov/technical/icwg/IS-GPS-200H.pdf
*[3]* 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 - GPS satellite ephemeris data.
p=. !Eph12min.png!
h3. b - GPS satellite position calculation algorithm.
p=. !Alg12min.png!
These tables are extracted from GPS Interface Control Document *[2]*
---
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 4.1 :* Pseudo-range measurement extracted from *[3]*
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) 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._
---
*References :*
*[1]* K. Borre, D. M. Akos, N. Bertelsen, P. Rinder, S. H. Jensen, A software-defined GPS and GALILEO receiver
*[2]* GPS Interface Control Document under http://www.gps.gov/technical/icwg/IS-GPS-200H.pdf
*[3]* Position Estimation Workshop, March 2016