Method for predicting the orbit of a satellite and corresponding satellite signal receiver

1. A method, comprising:

acquiring satellite signals with a satellite signal receiver; and

predicting, by a processor, at least one of a state or an orbit ( r (t)) of a satellite, including using a model ( {umlaut over (r)} SRP (t)) of solar radiation pressure operating on the satellite, the predicting including:

expressing the model of solar radiation pressure ( {umlaut over (r)} SRP (t)) operating on the satellite as a first Fourier series ( ë c′(t)=[ë′ x s ë′ y s ë′ z s ]) having a frequency function of an angle defined by a solar position with respect to a body fixed frame of the satellite and having respective first Fourier coefficients of said first Fourier series;

calculating position approximation errors (a pae (t i,j )) by comparing true satellite positions r eph (t i,j ) corresponding to the satellite signals at given time points against predicted satellite positions ( r pred (t i,j )) at corresponding time points;

estimating said first Fourier coefficients of said first Fourier series as a function of said position approximation errors (a pae (t i,j )); and

using said estimated first Fourier coefficients of said first Fourier series in the model of solar radiation pressure operating on the satellite expressed as a second Fourier series that is used in the model of solar radiation pressure to predict the at least one of the state or the orbit of the satellite; and

determining, by the processor, a position fix of the satellite signal receiver using the predicted at least one state or orbit of the satellite.

2. The method according to claim 1 , wherein the satellite signal receiver is a Global Navigation Satellite System (GNSS) receiver.

3. The method according to claim 1 , wherein the angle defined by the solar position with respect to the body fixed frame of the satellite is a satellite-Earth-Sun angle (α).

4. The method according to claim 1 , wherein calculating position approximation errors (a pae (t i,j )) by comparing true satellite positions r eph (t i,j ) at given time points against predicted satellite positions ( r pred (t i,j )) at corresponding time points includes:

obtaining a verification set of ephemerides (eph v ) at given time instants;

obtaining time end points of time intervals around said given time instants;

obtaining a prediction set of ephemerides (eph p );

using said prediction set of ephemerides (eph p ) to obtain initial conditions;

generating a predicted orbit as a function of said initial conditions ( r 0 , v 0 );

calculating predicted positions ( r pred (t i,j )) of said predicted orbit of the satellite at said time end points;

obtaining the true positions ( r eph (t i,j )) of the satellite at said time end points from said verification set of ephemerides (eph v ); and

comparing said predicted positions ( r pred (t i,j )) of said predicted orbit of the satellite at said time end points and true positions ( r eph (t i,j )) of the satellite at said time end points to obtain position approximation errors (a pae (t i,j )) at said time end points.

5. The method according to claim 1 , comprising:

truncating, by the processor, the first Fourier series ( ë (t)=[ë′ x s ë y s ë′ z s ]) of the model of the solar radiation pressure ( {umlaut over (r)} SRP (t)) at a given element (l max ).

6. The method according to claim 1 , wherein estimating said first Fourier coefficients of said first Fourier series as a function of said position approximation errors (a pae (t i,j )) includes choosing values minimizing said position approximation errors (a pae (t i,j )).

7. The method according to claim 1 , wherein said estimating said first Fourier coefficients of said first Fourier series as a function of said position approximation errors (a pae (t i,j )) includes:

selecting a coefficient among said first Fourier coefficients of the first Fourier series and setting the coefficient to a first given value;

for a determined number of iterations, setting all other first Fourier coefficients to fixed values and calculating an estimator of the approximation errors;

setting the coefficient to a second given value;

calculating a second estimator of the approximation errors to obtain a respective sequence of set values and a sequence of estimator values;

fitting said sequence of set values and said sequence of estimator values for the coefficients to identify an estimation function (y est ); and

selecting a value associated with the minimum value of the estimation function as an estimated value for the coefficient.

8. The method according to claim 7 , comprising:

repeating, by the processor, the estimating of said first Fourier coefficients of said first Fourier series as a function of said position approximation errors a determined number of times.

9. The method according to claim 1 , wherein generating the predicted orbit as a function of said initial conditions ( r 0 , v 0 ) includes:

using an approximated second derivative of a position error ( ë ′(t)) between the orbit of the satellite that takes the model of solar radiation pressure ( {umlaut over (r)} SRP (t)) into account and the orbit of the satellite that does not take the model of solar radiation pressure ( {umlaut over (r)} SRP (t)) into account.

10. The method according to claim 1 , wherein calculating position approximation errors (a pae (t i,j )) by comparing the true satellite positions r eph (t i,j ) at given time points against the predicted satellite positions ( r pred (t i,j )) at corresponding time points includes:

calculating velocity approximation errors; and

calculating acceleration approximation errors.

11. A satellite signal receiver, comprising:

a front-end to acquire satellite signals; and

a processor configured to:

express the model of solar radiation pressure operating on the navigation satellite as a first Fourier series having a frequency function of an angle defined by a solar position with respect to a body fixed frame of the navigation satellite and having respective first Fourier coefficients of said first Fourier series;

calculate position approximation errors by comparing true satellite positions corresponding to the satellite signals at given time points against predicted satellite positions at corresponding time points;

estimate said first Fourier coefficients of said first Fourier series as a function of said position approximation errors;

use said estimated first Fourier coefficients of said first Fourier series in the model of solar radiation pressure operating on the navigation satellite expressed as a second Fourier series that is used in the model of solar radiation pressure to predict a state or an orbit of the navigation satellite; and

determine a position fix of the satellite signal receiver using the predicted at least one state or orbit of the satellite, the predicting of the state or the orbit of the navigation satellite.

12. A satellite signal receiver according to claim 11 , wherein the satellite signal receiver is a Global Navigation Satellite System (GNSS) receiver.

13. A satellite signal receiver according to claim 11 , wherein the processor is further configured to:

obtain a verification set of ephemerides at given time instants;

obtain time end points of time intervals around said given time instants;

obtain a prediction set of ephemerides;

use said prediction set of ephemerides to obtain initial conditions;

generate a predicted orbit as a function of said initial conditions;

calculate predicted positions of said predicted orbit of the satellite at said time end points;

obtain the true positions of the satellite at said time end points from said verification set of ephemerides; and

compare said predicted positions of said predicted orbit of the satellite at said time end points and true positions of the satellite at said time end points to obtain position approximation errors at said time end points.

14. A satellite signal receiver according to claim 11 , wherein the processor is further configured to:

select a coefficient among said first Fourier coefficients of the first Fourier series and setting the coefficient to a first given value;

for a determined number of iterations, set all other first Fourier coefficients to fixed values and calculating an estimator of the approximation errors;

set the coefficient to a second given value;

calculate a second estimator of the approximation errors to obtain a respective sequence of set values and a sequence of estimator values;

fit said sequence of set values and said sequence of estimator values for the coefficients to identify an estimation function (y est ); and

select a value associated with the minimum value of the estimation function as an estimated value for the coefficient.

15. A satellite signal receiver according to claim 11 , wherein the processor is further configured to:

calculate velocity approximation errors; and

calculate acceleration approximation errors.

16. A satellite navigation system, comprising:

a satellite signal receiver coupleable to a plurality of navigation satellites, the satellite signal receiver configured to:

predict a state or an orbit of a satellite of the plurality of navigation satellites; and

calculate a position solution of the satellite signal receiver using the predicted state or orbit of the satellite, predicting the state or the orbit of the satellite causing the satellite signal receiver to:

express the model of solar radiation pressure operating on the satellite as a first Fourier series having a frequency function of an angle defined by a solar position with respect to a body fixed frame of the satellite and having respective first Fourier coefficients of said first Fourier series;

calculate position approximation errors by comparing true satellite positions at given time points against predicted satellite positions at corresponding time points;

estimate said first Fourier coefficients of said first Fourier series as a function of said position approximation errors; and

use said estimated first Fourier coefficients of said first Fourier series in the model of solar radiation pressure operating on the satellite expressed as a second Fourier series that is used in the model of solar radiation pressure to predict the state or the orbit of the satellite.

17. A satellite navigation system according to claim 16 , wherein the satellite signal receiver is a Global Navigation Satellite System (GNSS) receiver.

18. A satellite navigation system according to claim 16 , wherein predicting the state or the orbit of the satellite further causing the processor to:

obtain a verification set of ephemerides at given time instants;

obtain time end points of time intervals around said given time instants;

obtain a prediction set of ephemerides;

use said prediction set of ephemerides to obtain initial conditions;

generate a predicted orbit as a function of said initial conditions;

calculate predicted positions of said predicted orbit of the satellite at said time end points;

obtain true positions of the satellite at said time end points from said verification set of ephemerides; and

compare said predicted positions of said predicted orbit of the satellite at said time end points and true positions of the satellite at said time end points to obtain position approximation errors at said time end points.

19. A satellite navigation system according to claim 16 , wherein predicting the state or the orbit of the satellite further causing the processor to:

select a coefficient among said first Fourier coefficients of the first Fourier series and setting the coefficient at a first given value;

for a determined number of iterations, set all other first Fourier coefficients to fixed values and calculating an estimator of the approximation errors;

set the coefficient at a second given value;

calculate a second estimator of the approximation errors to obtain a respective sequence of set values and a sequence of estimator values;

fit said sequence of set values and said sequence of estimator values for the coefficients to identify an estimation function (y est ); and

select a value associated with the minimum value of the estimation function as an estimated value for the coefficient.

20. A satellite navigation system according to claim 16 , wherein predicting the state or the orbit of the satellite further causing the processor to:

calculate velocity approximation errors; and

calculate acceleration approximation errors.