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COLIN, Tony, 03/20/2016 04:21 PM

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h1. PART 4 : Position Estimation.
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{{toc}}
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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.
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GPS uses a particular algorithm in order to characterise satellite position. In comparison with GLONASS, this method requires more parameters, but less complexity.
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h3. a - Introduction of satellite orbit.
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p=. !OrbitalPlanePositioningMin.png!
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*Figure 4.1 :* Orbital plane positioning.
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Orbital plane positioning parameters :
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p=. !Parameters1.PNG!
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p=. !OrbitPositioningInTheOrbitalPlaneMin.png!
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*Figure 4.2 :* Orbit positioning in the orbital plane.
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Orbit positioning in the orbital plane :
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p=. !Parameters2.PNG!
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p=. !SatellitePositioningMin.png!
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*Figure 4.3 :* Orbital plane positioning.
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Shape of the orbit :
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p=. !Parameters3.PNG!
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Positioning of the satellite on the orbit :
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p=. !Parameters4.PNG!
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Induced parameters :
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p=. !Parameters5.PNG!
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h3. b - GPS satellite ephemeris data.
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p=. !Eph12min.png!
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*Figure 4.4 :* List of ephemeris parameters included in GPS frames.
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h3. c - GPS satellite position calculation algorithm.
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p=. !Alg12min.png!
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*Figure 4.5 :* Description of the algorithm step by step.
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These tables are extracted from GPS Interface Control Document *[3]*
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h2. 2 - Navigation computation.
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h3. a - Reminder about the range impairments.
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The following figure gives the impairments affecting the range in case of the GPS system as well as the correction process :
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p=. !003.PNG!
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*Figure 4.6 :* Pseudo-range measurement extracted from *[4]*
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h3. b - Demonstration of the Pseudo-ranges with Least Square method.
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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 : 
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p=. !Pos1.png!
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*Equation 1*
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or put in another way,
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p=. !Pos2.png!
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*Equation 2*
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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 :
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p=. !Pos3.png!
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*Equation 3*
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Hence,
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p=. !Pos4.png!
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*Equation 4*
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In practise, for a receiver located e.g. in France PR (t_0) can be described by Paris location as initialization for the algorithm.
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In vectorial form the equation becomes :
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p=. !Pos5.png!
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*Equation 5*
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which can be expressed as :
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p=. !Pos6.png!
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*Equation 6*
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with the Least Square solution :
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p=. !Pos7.png!
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*Equation 7*
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Thus, it is possible to retrieve the receiver position.
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_Note that all unknowns are depicted in red color._
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h3. c - Kalman filter.
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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.
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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 :* 
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*[1]* K. Borre, D. M. Akos, N. Bertelsen, P. Rinder, S. H. Jensen, A software-defined GPS and GALILEO receiver
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*[2]* M. Bousquet, Orbits and Satellite Platforms lecture script, January 2016
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*[3]* GPS Interface Control Document under http://www.gps.gov/technical/icwg/IS-GPS-200H.pdf
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*[4]* Position Estimation Workshop, March 2016