PART41 » History » Version 15
Version 14 (COLIN, Tony, 03/19/2016 09:13 PM) → Version 15/35 (COLIN, Tony, 03/19/2016 09:15 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.
---
h2. 1 - Ephemeris.
h3. a - GPS satellite ephemeris data.
p=. !Eph12min.png!
h3. b - GPS satellite position calculation algorithm.
p=. !Alg12min.png!
from GPS Interface Control Document
---
h2. 2 - Navigation computation.
h3. a - Reminder about the 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 *[2]*
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*
or in matrix equation form,
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]* 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.
h3. a - GPS satellite ephemeris data.
p=. !Eph12min.png!
h3. b - GPS satellite position calculation algorithm.
p=. !Alg12min.png!
from GPS Interface Control Document
---
h2. 2 - Navigation computation.
h3. a - Reminder about the 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 *[2]*
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*
or in matrix equation form,
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]* Position Estimation Workshop, March 2016