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COLIN, Tony, 03/19/2016 09:12 PM


PART 4 : Position Estimation.

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.


1 - Ephemeris.

GPS satellite ephemeris data.

GPS satellite position calculation algorithm.

from GPS Interface Control Document


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.


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