PART 4 : Position Estimation.¶

• Table of contents
• PART 4 : Position Estimation.
  • 1 - Ephemeris.
    • GPS satellite ephemeris data.
    • GPS satellite position calculation algorithm.
  • 2 - Navigation computation.
    • a - Reminder about the impairments.
    • b - Demonstration of the Pseudo-ranges with Least Square method.

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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1 - Ephemeris.¶

GPS satellite ephemeris data.¶

GPS satellite position calculation algorithm.¶

from GPS Interface Control Document

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2 - Navigation computation.¶

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 :

Figure 4.1 : Pseudo-range measurement extracted from [2]

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 :

Equation 1

or put in another way,

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 :

Equation 3

Hence,

Equation 4

or in matrix equation form,

Equation 5

which can be expressed as :

Equation 6

with the Least Square solution :

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] Position Estimation Workshop, March 2016