Methods and apparatuses of estimating the position of a mobile user in a system of satellite differential navigation

• US 7,102,563 B2
• Filed: 02/26/2004
• Issued: 09/05/2006
• Est. Priority Date: 02/26/2004
• Status: Active Grant

• Current Assignee: Topcon GPS, LLC
• Original Assignee: Topcon GPS, LLC
• Inventors: Barabanov, Ivan, Khvalkov, Alexander, Rapoport, Lev Borisovich
• Primary Examiner(s): Issing, Gregory C.
• Application Number:
  US10/788,575
• Publication Number:
  US 20050190103A1
• Time in Patent Office: 922 Days
• Field of Search: 342/357.04
• US Class Current: 342/357.27
• CPC Class Codes: G01S 19/44 Carrier phase ambiguity resolution; Floating ambiguity; LAMBDA [Least-squares AMBiguity Decorrelation Adjustment] method

Abstract

Disclosed are methods and apparatuses for estimating the floating ambiguities associated with the measurement of the carrier signals of a plurality of global positioning satellites, such that the floating ambiguities are preferably consist for a plurality of different time moments. In one aspect of the invention, a real-time iterative matrix refactorization process is provided which reduces processor load and retains history of the measurements.

Claims

1. A method of estimating a set of floating ambiguities associated with a set of phase measurements of a plurality n of satellite carrier signals made by a first navigation receiver (B) and a second navigation receiver (R) separated by a distance, wherein a baseline vector (x^o,y^o,z^o) relates the position of the second receiver to the first receiver, each satellite carrier signal being transmitted by a satellite and having a wavelength, wherein each receiver has a time clock for referencing its measurements and wherein any difference between the time clocks may be represented by an offset, said method receiving, for a plurality of two or more time moments j, the following inputs:

a vector γ

_j^B representative of a plurality of pseudo-ranges measured by the first navigation receiver (B) and corresponding to the plurality of satellite carrier signals,

a vector γ

_j^R representative of a plurality of pseudo-ranges measured by the second navigation receiver (R) and corresponding to the plurality of satellite carrier signals,

a vector D_j^B representative of a plurality of estimated distances between the satellites and the first navigation receiver (B),

a vector D_j^R representative of a plurality of estimated distances between the satellites and the second navigation receiver (R),

a vector φ

_j^B representative of a plurality of full phase measurements of the satellite carrier signals measured by the first navigation receiver (B),

a vector φ

_j^R representative of a plurality of full phase measurements of the satellite carrier signals measured by the second navigation receiver (R),

a geometric Jacobian matrix H_j^γ

whose matrix elements are representative of the changes in the distances between the satellites and one of the receivers that would be caused by changes in that receiver'"'"'s position and time clock offset, said method comprising the steps of;

(a) generating, for each time moment j, a vector Δ

γ

_j of a plurality of range residuals of pseudo-range measurements made by the first and second navigation receivers in the form of;

Δ

γ

_j=(γ

_j^R−

γ

_j^B)−

(D_j^R−

D_j^B);

(b) generating, for each time moment j, a vector Δ

φ

_j of a plurality of phase residuals of full phase measurements made by the first and second navigation receivers in the form of;

Δ

φ

_j=(φ

_j^R−

φ

_j^B)−

Λ

<sup>−

1</sup>·

(D_j^R−

D_j^B),

where Λ

<sup>−

1</sup> is a diagonal matrix comprising the inverse wavelengths of the satellites;

(c) generating, for time moment j=1, an LU-factorization of a matrix M₁ or a matrix inverse of matrix M₁, the matrix M₁ being a function of at least Λ

<sup>−

1</sup> and H₁^γ

,

(d) generating, for time moment j=1, a vector N₁ as a function of at least Δ

γ

₁, Δ

φ

₁, and the LU-factorization of matrix M₁ or the matrix inverse of matrix M₁;

(e) generating, for an additional time moment j≠

1, an LU-factorization of a matrix M_j or a matrix inverse of matrix M_j, the matrix M_j being a function of at least Λ

<sup>−

1</sup>, H_j^γ

and an instance of matrix M generated for a different time moment; and

(f) generating, for an additional time moment j≠

1, a vector N_j as a function of at least Δ

γ

_j, Δ

φ

_j, and the LU-factorization or matrix M_j or the matrix inverse of matrix M_j, the vector N_j having estimates of the floating ambiguities.

2. The method of claim 1 wherein step (c) comprises generating an LU-factorization for a matrix comprising a form equivalent to (G^TP₁G), where:

the matrix G has 2n rows, n columns, an upper sub-matrix, and a lower sub-matrix, one of the sub-matrices comprises an n×

n zero matrix and the other sub-matrix comprising an n×

n identity matrix,

the matrix G^T comprises the transpose matrix of matrix G, and

the matrix P₁ has 2n rows, 2n columns, and a form which comprises a matrix equivalent to (R₁<sup>−

1</sup>−

R₁<sup>−

1</sup>Q₁(Q₁^TR₁<sup>−

1</sup>Q₁+qS₁)<sup>−

1</sup>Q₁^TR₁<sup>−

1</sup>) where the matrix R₁ is a weighting matrix, where the matrix R₁<sup>−

1</sup> comprises an inverse of matrix R₁, where the matrix Q₁ has an upper sub-matrix and a lower sub-matrix, one of the sub-matrices of Q₁ comprising matrix H₁^γ

and the other of the sub-matrices of Q₁ comprising the matrix product Λ

<sup>−

1</sup>H₁^γ

, and wherein the matrix Q₁^T comprises the transpose of matrix Q₁, and where the quantity qS_k is a zero matrix when the distance between the first and second navigation receivers is unconstrained and where q may be a non-zero weighting parameter and S_k may be a non-zero matrix when the distance between the first and second navigation receivers is constrained.

3. The method of claim 2 wherein step (d) comprises the step of generating vector N₁ to comprise a vector having a form equivalent to M₁⁻

1(G^TP₁μ

₁+qg₁), where the matrix M₁<sup>−

1</sup> comprises an inverse of matrix of matrix M₁, and where the vector μ

₁ comprises the vector [Δ

γ

₁, Δ

φ

₁]^T, and where the quantity qg_k is a zero vector when the distance between the first and second navigation receivers is unconstrained and where q may be non-zero and g_k may be a non-zero vector when the distance between the first and second navigation receivers is constrained.

4. The method of claim 1 wherein step (e) comprises generating an LU-factorization for a matrix comprising a form equivalent to M_j=M_j−

1+G^TP_jG, where;

M<sub>j−

1</sub> comprises the matrix M₁ of step (c) when j=2 and comprises the matrix M_j of step (e) for the j−

1 time moment when j>

2,

the matrix G has 2n rows, n columns, an upper sub-matrix, and a lower sub-matrix, one of the sub-matrices comprising an n×

n zero matrix and the other sub-matrix comprising an n×

n identity matrix,

the matrix G^T comprises the transpose matrix of matrix G, and

the matrix P_j has 2n rows, 2n columns, and a form which comprise a matrix equivalent to (R_j<sup>−

1</sup>−

R_j<sup>−

1</sup>Q_j(Q_j^TR_j<sup>−

1</sup>Q_j+qS_j)<sup>−

1</sup>Q_j^TR_j<sup>−

1</sup>) where the matrix R_j is a weighting matrix, where the matrix R_j<sup>−

1</sup> comprises an inverse of matrix R_j, where the matrix Q_j has an upper sub-matrix and a lower sub-matrix, one of the sub-matrices of Q_j comprising matrix H_j^γ

and the other of the sub-matrices of Q_j comprising the matrix product Λ

<sup>−

1</sup>H_j^γ

, and wherein the matrix Q_j^T comprises the transpose of matrix Q_j, and where the quantity qS_j is a zero matrix when the distance between the first and second navigation receivers is unconstrained and where q may be a non-zero weighting parameter and S_j may be a non-zero matrix when the distance between the first and second navigation receivers is constrained.

5. The method of claim 4 wherein step (f) comprises the step of generating vector N_j to comprise a vector having a form equivalent to N_j−

1+M_j<sup>−

1</sup>└

G^TP_j(μ

_j−

GN<sub>j−

1</sub>)+qg_j┘

, where the matrix M_j<sup>−

1</sup> comprises an inverse of matrix of matrix M_j, where the vector μ

_j comprises the vector [Δ

γ

_j, Δ

φ

_j]^T, and where the vector N<sub>j−

1</sub> comprises the vector N₁ generated by step (d) when j=2 and comprises the vector N<sub>j−

1</sub> generated by step (f) for the j−

1 time moment when j>

2, and where the quantity qg_j is a zero vector when the distance between the first and second navigation receivers is unconstrained and where q may be non-zero and g_j may be a non-zero vector when the distance between the first and second navigation receivers is constrained.

6. The method of claim 3 wherein the distance between the first and second navigation receivers is constrained to a distance L_RB, wherein step (c) generates matrix S₁ in a form equivalent to:

7. The method of claim 3 wherein weighting matrix R₁ comprises an identity matrix multiplied by a scalar quantity.

8. The method of claim 5 wherein the distance between the first and second navigation receivers is constrained to a distance L_RB, wherein step (e) generates matrix S_j in a form equivalent to:

9. The method of claim 5 wherein the weighting matrix R_j comprises an identity matrix multiplied by a scalar quantity for at least one time moment j.

10. The method of claim 4 wherein the generation of the LU-factorization in step (e) comprises the steps of:

(g) generating an LU-factorization of matrix M<sub>j−

1</sub> in a form equivalent to L<sub>j−

1</sub>L<sub>j−

1</sub>^T wherein L<sub>j−

1</sub> is a low-triangular matrix and L<sub>j−

1</sub>^T is the transpose of L<sub>j−

1</sub>;

(h) generating a factorization of G^TP_jG in a form equivalent to T_jT_j^T=G^TP_jG, where T_j^T is the transpose of T_j;

(i) generating an LU-factorization of matrix M_j in a form equivalent to L_jL_j^T from a plurality n of rank-one modifications of matrix L<sub>j−

1</sub>, each rank-one modification being based on a respective column of matrix T_j, where n is the number of rows in matrix M_j.

11. The method of claim 10 wherein step (h) generates matrix T_j from a Cholesky factorization of G^TP_jG.

12. The method of claim 10 wherein weighting matrix R_j has a form equivalent to:

13. The method of claim 10 wherein weighting matrix R_j is applied to a case where there is a first group of satellite signals having carrier signals in a first wavelength band and a second group of satellite signals having a carrier signals in a second wavelength band, the weighting frequency having a form equivalent to:

14. The method of claim 4 wherein step (d) comprises generating a vector B₁ to comprise a vector having a form equivalent to G^TP₁μ

₁+qg₁, where the vector μ

₁ comprises the vector [Δ

γ

₁, Δ

φ

₁]^T, and where the quantity qg_j is a zero vector when the distance between the first and second navigation receivers is unconstrained and where q may be non-zero and g_j may be a non-zero vector when the distance between the first and second navigation receivers is constrained; and

wherein step (f) further comprises generating, for each time moment j≠

1, a vector B_j to comprise a matrix having a form equivalent to B<sub>j−

1</sub>+G^TP_jμ

_j+qg_i, where the vector μ

_j comprises the vector [Δ

γ

_j, Δ

φ

_j]^T, and where the vector B<sub>j−

1</sub> is the vector B₁ generated by step (d) when j=2 and comprises the vector generated by step (f) for the for the j−

1 time moment when j>

2, and where the quantity qg_j is a zero vector when the distance between the first and second navigation receivers is unconstrained and where q may be non-zero and g_j may be a non-zero vector when the distance between the first and second navigation receivers is constrained; and

wherein step (f) further comprises generating vector N_j to comprise a vector having a form equivalent to N_j=[M_j]<sup>−

1</sup>B_j, where the matrix M_j<sup>−

1</sup> comprises an inverse of matrix of matrix M_j.

15. The method of claim 14 wherein the distance between the first and second navigation receivers is constrained to a distance L_RB, wherein step (c) generates matrix S₁ in a form equivalent to:

16. The method of claim 14 wherein the distance between the first and second navigation receivers is constrained to a distance L_RB, wherein step (e) generates matrix S_j in a form equivalent to:

17. The method of claim 14 wherein the weighting matrix R_j comprises an identity matrix multiplied by a scalar quantity for at least one time moment j.

18. A method of estimating a set of floating ambiguities associated with a set of phase measurements of a plurality n of satellite carrier signals made by a first navigation receiver (B) and a second navigation receiver (R), wherein a baseline vector (x^o,y^o,z^o) relates the position of the second receiver to the first receiver, each satellite carrier signal being transmitted by a satellite and having a wavelength, wherein each receiver has a time clock for referencing its measurements and wherein any difference between the time clocks may be represented by an offset, said method receiving, for a plurality of two or more time moments j, the following inputs for each time moment j:

a vector γ

_j^B representative of a plurality of pseudo-ranges measured by the first navigation receiver (B) and corresponding to the plurality of satellite carrier signals,

a vector γ

_j^R representative of a plurality of pseudo-ranges measured by the second navigation receiver (R) and corresponding to the plurality of satellite carrier signals,

a vector D_j^B representative of a plurality of estimated distances between the satellites and the first navigation receiver (B),

a vector D_j^R representative of a plurality of estimated distances between the satellites and the second navigation receiver (R),

a vector φ

_j^B representative of a plurality of full phase measurements of the satellite carrier signals measured by the first navigation receiver (B),

a vector φ

_j^R representative of a plurality of full phase measurements of the satellite carrier signals measured by the second navigation receiver (R),

a geometric Jacobian matrix H_j^γ

whose matrix elements are representative of the changes in the distances between the satellites and one of the receivers that would be caused by changes in that receiver'"'"'s position and time clock offset, said method comprising the steps of;

(a) generating, for each time moment j, a vector Δ

γ

_j of a plurality of range residuals of pseudo-range measurements made by the first and second navigation receivers in the form of;

Δ

γ

_j=(γ

_j^R−

γ

_j^B)−

(D_j^R−

D_j^B), said step generating a set of range residuals Δ

γ

_k, k=1, . . . , j;

(b) generating, for each time moment j, a vector Δ

φ

_j of a plurality of phase residuals of full phase measurements made by the first and second navigation receivers in the form of;

Δ

φ

_j=(φ

_j^R−

φ

_j^B)−

Λ

<sup>−

1</sup>·

(D_j^R−

D_j^B), where Λ

<sup>−

1</sup> is a diagonal matrix comprising the inverse wavelengths of the satellites, said step generating a set of phase residuals Δ

φ

_k, k=1, . . . , j;

(c) generating an LU-factorization of a matrix M or a matrix inverse of matrix M, the matrix M being a function of at least Λ

<sup>−

1</sup> and H_k^γ

, for index k of H_k^γ

covering at least two of the time moments j;

(d) generating a vector N of estimated floating ambiguities as a function of at least the set of range residuals Δ

γ

_k, the set of phase residuals Δ

φ

_k, and the LU-factorization of matrix M or the matrix inverse of matrix M.

19. The method of claim 18 wherein step (c) comprises generating matrix M in a form equivalent to the summation

20. The method of claim 19 wherein step (d) comprises generating matrix N in a form equivalent to:

21. The method of claim 20 wherein the distance between the first and second navigation receivers is constrained to a distance L_RB, wherein step (c) generates matrix S_k for the k-th time moment in a form equivalent to:

22. The method of claim 19 wherein at least one of the weighting matrices R_k comprises an identity matrix multiplied by a scalar quantity.

23. A computer program product for directing a data processor to estimate a set of floating ambiguities associated with a set of phase measurements of a plurality n of satellite carrier signals made by a first navigation receiver (B) and a second navigation receiver (R) separated by a distance, wherein a baseline vector (x^o,y^o,z^o) relates the position of the second receiver to the first receiver, each satellite carrier signal being transmitted by a satellite and having a wavelength, wherein each receiver has a time clock for referencing its measurements and wherein any difference between the time clocks may be represented by an offset, the process receiving, for a plurality of two or more time moments j, the following inputs:

a vector γ

_j^B representative of a plurality of pseudo-ranges measured by the first navigation receiver (B) and corresponding to the plurality of satellite carrier signals,

a vector γ

_j^R representative of a plurality of pseudo-ranges measured by the second navigation receiver (R) and corresponding to the plurality of satellite carrier signals,

a vector D_j^B representative of a plurality of estimated distances between the satellites and the first navigation receiver (B),

a vector D_j^R representative of a plurality of estimated distances between the satellites and the second navigation receiver (R),

a vector φ

_j^B representative of a plurality of full phase measurements of the satellite carrier signals measured by the first navigation receiver (B),

a vector φ

_j^R representative of a plurality of full phase measurements of the satellite carrier signals measured by the second navigation receiver (R),

a geometric Jacobian matrix H_j^γ

whose matrix elements are representative of the changes in the distances between the satellites and one of the receivers that would be caused by changes in that receiver'"'"'s position and time clock offset, the computer program product comprising;

a computer-readable medium;

a first set of instructions embodied on the computer-readable medium which directs the data processor to generate, for each time moment j, a vector Δ

γ

_j of a plurality of range residuals of pseudo-range measurements made by the first and second navigation receivers in the form of;

Δ

γ

_j=(γ

_j^R−

γ

_j^B)−

(D_j<sup>R−

D</sup>_j^B);

a second set of instructions embodied on the computer-readable medium which directs the data processor to generate, for each time moment j, a vector Δ

φ

_j of a plurality of phase residuals of full phase measurements made by the first and second navigation receivers in the form of;

Δ

φ

_j=(φ

_j^R−

φ

_j^B)−

Λ

<sup>−

1</sup>·

(D_j^R−

D_j^B), where Λ

<sup>−

1</sup> is a diagonal matrix comprising the inverse wavelengths of the satellites;

a third set of instructions embodied on the computer-readable medium which directs the data processor to generate, for time moment j=1, an LU-factorization of a matrix M₁ or a matrix inverse of matrix M₁, the matrix M₁ being a function of at least Λ

<sup>−

1</sup> and H₁^γ

;

a fourth set of instructions embodied on the computer-readable medium which directs the data processor to generate, for time moment j=1, a vector N₁ as a function of at least Δ

γ

₁, Δ

φ

₁, and the LU-factorization of matrix M₁ or the matrix inverse of matrix M₁;

a fifth set of instructions embodied on the computer-readable medium which directs the data processor to generate, for an additional time moment j≠

1, an LU-factorization of a matrix M_j or a matrix inverse of matrix M_j, the matrix M_j being a function of at least Λ

<sup>−

1</sup> and H_j^γ

; and

a sixth set of instructions embodied on the computer-readable medium which directs the data processor to generate, for an additional time moment j≠

1, a vector N_j as a function of at least Δ

γ

_j, Δ

φ

_j, and the LU-factorization or matrix M_j or the matrix inverse of matrix M_j, the vector N_j having estimates of the floating ambiguities.

24. The computer program product of claim 23 wherein the third set of instructions directs the data processor to generate an LU-factorization for a matrix comprising a form equivalent to (G^TP₁G), where:

the matrix G has 2n rows, n columns, an upper sub-matrix, and a lower sub-matrix, one of the sub-matrices comprises an n×

n zero matrix and the other sub-matrix comprising an n×

n identity matrix,

the matrix G^T comprises the transpose matrix of matrix G, and

the matrix P₁ has 2n rows, 2n columns, and a form which comprises a matrix equivalent to (R₁<sup>−

1</sup>−

R₁<sup>−

1</sup>Q₁(Q₁^TR₁<sup>−

1</sup>Q₁+qS₁)<sup>−

1</sup>Q₁^TR₁<sup>−

1</sup>) where the matrix R₁ is a weighting matrix, where the matrix R₁<sup>−

1</sup> comprises an inverse of matrix R₁, where the matrix Q₁ has an upper sub-matrix and a lower sub-matrix, one of the sub-matrices of Q₁ comprising matrix H₁^γ

and the other of the sub-matrices of Q₁ comprising the matrix product Λ

<sup>−

1</sup>H₁^γ

, and wherein the matrix Q₁^T comprises the transpose of matrix Q₁, and where the quantity qS_k is a zero matrix when the distance between the first and second navigation receivers is unconstrained and where q may be a non-zero weighting parameter and S_k may be a non-zero matrix when the distance between the first and second navigation receivers is constrained.

25. The computer program product of claim 24 wherein the fourth set of instructions directs the data processor to generate vector N₁ to comprise a vector having a form equivalent to M₁⁻

1(G^TP₁μ

₁+qg₁), where the matrix M₁¹ comprises an inverse of matrix of matrix M₁, and where the vector μ

₁ comprises the vector [Δ

γ

₁, Δ

φ

₁]^T, and where the quantity qg_k is a zero vector when the distance between the first and second navigation receivers is unconstrained and where q may be non-zero and g_k may be a non-zero vector when the distance between the first and second navigation receivers is constrained.

26. The computer program product of claim 23 wherein the fifth set of instructions directs the data processor to generate an LU-factorization for a matrix comprising a form equivalent to M_j=M_j−

1+G^TP_jG, where;

M<sub>j−

1</sub> comprises the matrix M₁ of step (c) when j=2 and comprises the matrix M_j of step (e) for the j−

1 time moment when j>

2,

the matrix G has 2n rows, n columns, an upper sub-matrix, and a lower sub-matrix, one of the sub-matrices comprising an n×

n zero matrix and the other sub-matrix comprising an n×

n identity matrix,

the matrix G^T comprises the transpose matrix of matrix G, and

the matrix P_j has 2n rows, 2n columns, and a form which comprise a matrix equivalent to (R_j<sup>−

1</sup>−

R_j<sup>−

1</sup>Q_j(Q_j^TR_j<sup>−

1</sup>Q_j+qS_j)<sup>−

1</sup>Q_j^TR_j<sup>−

1</sup>) where the matrix R_j is a weighting matrix, where the matrix R_j<sup>−

1</sup> comprises an inverse of matrix R_j, where the matrix Q_j has an upper sub-matrix and a lower sub-matrix, one of the sub-matrices of Q_j comprising matrix H_j^γ

and the other of the sub-matrices of Q_j comprising the matrix product Λ

<sup>−

1</sup>H_j^γ

, and wherein the matrix Q_j^T comprises the transpose of matrix Q_j, and where the quantity qS_j is a zero matrix when the distance between the first and second navigation receivers is unconstrained and where q may be a non-zero weighting parameter and S_j may be a non-zero matrix when the distance between the first and second navigation receivers is constrained.

27. The computer program product of claim 26 wherein the sixth set of instructions directs the data processor to generate a vector N_j to comprise a vector having a form equivalent to N_j−

1+M_j<sup>−

1</sup>└

G^TP_j(μ

_j−

GN<sub>j−

1</sub>)+qg_j┘

, where the matrix M_j<sup>−

1</sup> comprises an inverse of matrix of matrix M_j, where the vector μ

_j comprises the vector [Δ

γ

_j, Δ

φ

_j]^T, and where the vector N<sub>j−

1</sub> comprises the vector N₁ generated by step (d) when j=2 and comprises the vector N<sub>j−

1</sub> generated by step (f) for the j−

1 time moment when j>

2, and where the quantity qg_j is a zero vector when the distance between the first and second navigation receivers is unconstrained and where q may be non-zero and g_j may be a non-zero vector when the distance between the first and second navigation receivers is constrained.

28. The computer program product of claim 27 wherein the distance between the first and second navigation receivers is constrained to a distance L_RB, and wherein the fifth set of instructions directs the data processor to generate matrix S_j in a form equivalent to:

29. The computer program product of claim 28 wherein the fifth set of instructions comprises:

a seventh set of instructions that direct the data processor to generate an LU-factorization of matrix M<sub>j−

1</sub> in a form equivalent to L<sub>j−

1</sub>L<sub>j−

1</sub>^T wherein L<sub>j−

1</sub> is a low-triangular matrix and L<sub>j−

1</sub>^T is the transpose of L<sub>j−

1</sub>;

an eighth set of instructions that direct the data processor to generate a factorization of G^TP_jG in a form equivalent to T_jT_j^T=G^TP_jG, where T_j^T is the transpose of T_j; and

a ninth set of instructions that direct the data processor to generate an LU-factorization of matrix M_j in a form equivalent to L_jL_j^T from a plurality n of rank-one modifications of matrix L<sub>j−

1</sub>, each rank-one modification being based on a respective column of matrix T_j, where n is the number of rows in matrix M_j.

30. The computer program product of claim 29 wherein the eighth set of instructions directs the data processor to generate matrix T_j from a Cholesky factorization of G^TP_jG.

31. The computer program product of claim 29 wherein weighting matrix R_j has a form equivalent to:

32. The method of claim 29 wherein weighting matrix R_j is applied to a case where there is a first group of satellite signals having carrier signals in a first wavelength band and a second group of satellite signals having a carrier signals in a second wavelength band, the weighting frequency having a form equivalent to:

33. A computer program product for directing a data processor to estimate a set of floating ambiguities associated with a set of phase measurements of a plurality n of satellite carrier signals made by a first navigation receiver (B) and a second navigation receiver (R) separated by a distance, wherein a baseline vector (x^o,y^o,z^o) relates the position of the second receiver to the first receiver, each satellite carrier signal being transmitted by a satellite and having a wavelength, wherein each receiver has a time clock for referencing its measurements and wherein any difference between the time clocks may be represented by an offset, the process receiving, for a plurality of two or more time moments j, the following inputs:

a vector γ

_j^B representative of a plurality of pseudo-ranges measured by the first navigation receiver (B) and corresponding to the plurality of satellite carrier signals,

a vector γ

_j^R representative of a plurality of pseudo-ranges measured by the second navigation receiver (R) and corresponding to the plurality of satellite carrier signals,

a vector D_j^B representative of a plurality of estimated distances between the satellites and the first navigation receiver (B),

a vector D_j^R representative of a plurality of estimated distances between the satellites and the second navigation receiver (R),

a vector φ

_j^B representative of a plurality of full phase measurements of the satellite carrier signals measured by the first navigation receiver (B),

a vector φ

_j^R representative of a plurality of full phase measurements of the satellite carrier signals measured by the second navigation receiver (R),

a geometric Jacobian matrix H_j^γ

whose matrix elements are representative of the changes in the distances between the satellites and one of the receivers that would be caused by changes in that receiver'"'"'s position and time clock offset, the computer program product comprising;

a computer-readable medium;

a first set of instructions embodied on the computer-readable medium which directs the data processor to generate, for each time moment j, a vector Δ

γ

_j of a plurality of range residuals of pseudo-range measurements made by the first and second navigation receivers in the form of;

Δ

γ

_j=(γ

_j^R−

γ

_j^B)−

(D_j^R−

D_j^B);

a second set of instructions embodied on the computer-readable medium which directs the data processor to generate, for each time moment j, a vector Δ

φ

_j of a plurality of phase residuals of full phase measurements made by the first and second navigation receivers in the form of;

Δ

φ

_j=(φ

_j^R−

φ

_j^B)−

Λ

<sup>−

1</sup>·

(D_j^R−

D_j^B), where Λ

<sup>−

1</sup> is a diagonal matrix comprising the inverse wavelengths of the satellites;

a third set of instructions embodied on the computer-readable medium which directs the data processor to generate an LU-factorization of a matrix M or a matrix inverse of matrix M, the matrix M being a function of at least Λ

<sup>−

1</sup> and H_k^γ

, for index k of H_k^γ

covering at least two of the time moments j; and

a fourth set of instructions embodied on the computer-readable medium which directs the data processor to generate a vector N of estimated floating ambiguities as a function of at least the set of range residuals Δ

γ

_k, the set of phase residuals Δ

φ

_k, and the LU-factorization of matrix M or the matrix inverse of matrix M.

34. The computer program product of claim 33 wherein the third set of instructions directs the data processor to generate matrix M in a form equivalent to the summation

35. The computer program product of claim 34 wherein the fourth set of instructions directs the data processor to generate matrix N in a form equivalent to:

36. The computer program product of claim 35 wherein the distance between the first and second navigation receivers is constrained to a distance L_RB, wherein the third set of instructions directs the data processor to generate matrix S_k for the k-th time moment in a form equivalent to:

37. An apparatus for estimating a set of floating ambiguities associated with a set of phase measurements of a plurality n of satellite carrier signals made by a first navigation receiver (B) and a second navigation receiver (R) separated by a distance, wherein a baseline vector (x^o,y^o,z^o) relates the position of the second receiver to the first receiver, each satellite carrier signal being transmitted by a satellite and having a wavelength, wherein each receiver has a time clock for referencing its measurements and wherein any difference between the time clocks may be represented by an offset, said apparatus receiving, for a plurality of two or more time moments j, the following inputs:

a vector γ

_j^B representative of a plurality of pseudo-ranges measured by the first navigation receiver (B) and corresponding to the plurality of satellite carrier signals,

a vector γ

_j^R representative of a plurality of pseudo-ranges measured by the second navigation receiver (R) and corresponding to the plurality of satellite carrier signals,

a vector D_j^B representative of a plurality of estimated distances between the satellites and the first navigation receiver (B),

a vector D_j^R representative of a plurality of estimated distances between the satellites and the second navigation receiver (R),

a vector φ

_j^B representative of a plurality of full phase measurements of the satellite carrier signals measured by the first navigation receiver (B),

a vector φ

_j^R representative of a plurality of full phase measurements of the satellite carrier signals measured by the second navigation receiver (R),

a geometric Jacobian matrix H_j^γ

whose matrix elements are representative of the changes in the distances between the satellites and one of the receivers that would be caused by changes in that receiver'"'"'s position and time clock offset,

said apparatus comprising;

(a) means for generating, for each time moment j, a vector Δ

γ

_j of a plurality of range residuals of pseudo-range measurements made by the first and second navigation receivers in the form of;

Δ

γ

_j=(γ

_j^R−

γ

_j^B)−

(D_j^R−

D_j^B);

(b) means for generating, for each time moment j, a vector Δ

φ

_j of a plurality of phase residuals of full phase measurements made by the first and second navigation receivers in the form of;

Δ

φ

_j=(φ

_j^R−

φ

_j^B)−

Λ

<sup>−

1</sup>·

(D_j^R−

D_j^B), where Λ

<sup>−

1</sup> is a diagonal matrix comprising the inverse wavelengths of the satellites;

(c) means for generating, for time moment j=1, an LU-factorization of a matrix M₁ or a matrix inverse of matrix M₁, the matrix M₁ being a function of at least Λ

<sup>−

1</sup> and H₁^γ

;

(d) means for generating, for time moment j=1, a vector N₁ as a function of at least Δ

γ

₁, Δ

φ

₁, and the LU-factorization of matrix M₁ or the matrix inverse of matrix M₁;

(e) means for generating, for an additional time moment j≠

1, an LU-factorization of a matrix M_j or a matrix inverse of matrix M_j, the matrix M_j being a function of at least Λ

<sup>−

1</sup> and H_j^γ

; and

(f) means for generating, for an additional time moment j≠

1, a vector N_j as a function of at least Δ

γ

_j, Δ

φ

_j, and the LU-factorization or matrix M_j or the matrix inverse of matrix M_j, the vector N_j having estimates of the floating ambiguities.

38. The apparatus of claim 37 wherein means (c) comprises means for generating an LU-factorization for a matrix comprising a form equivalent to (G^TP₁G), where:

the matrix G has 2n rows, n columns, an upper sub-matrix, and a lower sub-matrix, one of the sub-matrices comprises an n×

n zero matrix and the other sub-matrix comprising an n×

n identity matrix,

the matrix G^T comprises the transpose matrix of matrix G, and

the matrix P₁ has 2n rows, 2n columns, and a form which comprises a matrix equivalent to (R₁<sup>−

1</sup>−

R₁<sup>−

1</sup>Q₁(Q₁^TR₁<sup>−

1</sup>Q₁+qS₁)<sup>−

1</sup>Q₁^TR₁<sup>−

1</sup>) where the matrix R₁ is a weighting matrix, where the matrix R₁<sup>−

1</sup> comprises an inverse of matrix R₁, where the matrix Q₁ has an upper sub-matrix and a lower sub-matrix, one of the sub-matrices of Q₁ comprising matrix H₁^γ

and the other of the sub-matrices of Q₁ comprising the matrix product Λ

<sup>−

1</sup>H₁^γ

, and wherein the matrix Q₁^T comprises the transpose of matrix Q₁, and where the quantity qS_k is a zero matrix when the distance between the first and second navigation receivers is unconstrained and where q may be a non-zero weighting parameter and S_k may be a non-zero matrix when the distance between the first and second navigation receivers is constrained.

39. The apparatus of claim 38 wherein means (d) comprises means for generating vector N₁ to comprise a vector having a form equivalent to M₁⁻

1(G^TP₁μ

₁+qg₁), where the matrix M₁<sup>−

1</sup> comprises an inverse of matrix of matrix M₁, and where the vector μ

₁ comprises the vector [Δ

γ

₁, Δ

φ

₁]^T, and where the quantity qg_k is a zero vector when the distance between the first and second navigation receivers is unconstrained and where q may be non-zero and g_k may be a non-zero vector when the distance between the first and second navigation receivers is constrained.

40. The apparatus of claim 37 wherein means (e) comprises means for generating an LU-factorization for a matrix comprising a form equivalent to

M_j=M_j−

1+G^TP_jG, where;

M<sub>j−

1</sub> comprises the matrix M₁ of generated by means (c) when j=2 and comprises the matrix M_j generated by means (e) for the j−

1 time moment when j>

2,

the matrix G has 2n rows, n columns, an upper sub-matrix, and a lower sub-matrix, one of the sub-matrices comprising an n×

n zero matrix and the other sub-matrix comprising an n×

n identity matrix,

the matrix G^T comprises the transpose matrix of matrix G, and

the matrix P_j has 2n rows, 2n columns, and a form which comprise a matrix equivalent to (R_j<sup>−

1</sup>−

R_j<sup>−

1</sup>Q_j(Q_j^TR_j<sup>−

1</sup>Q_j+qS_j)<sup>−

1</sup>Q_j^TR_j<sup>−

1</sup>) where the matrix R_j is a weighting matrix, where the matrix R_j<sup>−

1</sup> comprises an inverse of matrix R_j, where the matrix Q_j has an upper sub-matrix and a lower sub-matrix, one of the sub-matrices of Q_j comprising matrix H_j^γ

and the other of the sub-matrices of Q_j comprising the matrix product Λ

<sup>−

1</sup>H_j^γ

, and wherein the matrix Q_j^T comprises the transpose of matrix Q_j, and where the quantity qS_j is a zero matrix when the distance between the first and second navigation receivers is unconstrained and where q may be a non-zero weighting parameter and S_j may be a non-zero matrix when the distance between the first and second navigation receivers is constrained.

41. The apparatus of claim 40 wherein means (f) comprises means for generating vector N_j to comprise a vector having a form equivalent to N_j−

1+M_j<sup>−

1</sup>└

G^TP_j(μ

_j−

GN<sub>j−

1</sub>)+qg_j┘

, where the matrix M_j<sup>−

1</sup> comprises an inverse of matrix of matrix M_j, where the vector μ

_j comprises the vector [Δ

γ

_j, Δ

φ

_j]^T, and where the vector N<sub>j−

1</sub> comprises the vector N₁ generated by means (d) when j=2 and comprises the vector N<sub>j−

1</sub> generated by means (f) for the j−

1 time moment when j>

2, and where the quantity qg_j is a zero vector when the distance between the first and second navigation receivers is unconstrained and where q may be non-zero and g_j may be a non-zero vector when the distance between the first and second navigation receivers is constrained.

42. The apparatus of claim 39 wherein the distance between the first and second navigation receivers is constrained to a distance L_RB, wherein means (c) comprises means for generating matrix S₁ in a form equivalent to:

43. The apparatus of claim 41 wherein the distance between the first and second navigation receivers is constrained to a distance L_RB, wherein means (e) comprises means for generating matrix S_j in a form equivalent to:

44. The apparatus of claim 40 wherein means (e) for generating the LU-factorization of M_j comprises:

(g) means for generating an LU-factorization of matrix M<sub>j−

1</sub> in a form equivalent to L<sub>j−

1</sub>L<sub>j−

1</sub>^T wherein L<sub>j−

1</sub> is a low-triangular matrix and L<sub>j−

1</sub>^T is the transpose of L<sub>j−

1</sub>;

(h) means for generating a factorization of G^TP_jG in a form equivalent to T_jT_j^T=G^TP_jG, where T_j^T is the transpose of T_j; and

(i) means for generating an LU-factorization of matrix M_j in a form equivalent to L_jL_j^T from a plurality n of rank-one modifications of matrix L<sub>j−

1</sub>, each rank-one modification being based on a respective column of matrix T_j, where n is the number of rows in matrix M_j.

45. The apparatus of claim 44 wherein means (h) generates matrix T_j from a Cholesky factorization of G^TP_jG.

46. The apparatus of claim 44 wherein weighting matrix R_j has a form equivalent to:

47. The method of claim 44 wherein weighting matrix R_j is applied to a case where there is a first group of satellite signals having carrier signals in a first wavelength band and a second group of satellite signals having a carrier signals in a second wavelength band, the weighting frequency having a form equivalent to:

48. An apparatus for estimating a set of floating ambiguities associated with a set of phase measurements of a plurality n of satellite carrier signals made by a first navigation receiver (B) and a second navigation receiver (R) separated by a distance, wherein a baseline vector (x^o,y^o,z^o) relates the position of the second receiver to the first receiver, each satellite carrier signal being transmitted by a satellite and having a wavelength, wherein each receiver has a time clock for referencing its measurements and wherein any difference between the time clocks may be represented by an offset, said apparatus receiving, for a plurality of two or more time moments j, the following inputs:

a vector γ

_j^B representative of a plurality of pseudo-ranges measured by the first navigation receiver (B) and corresponding to the plurality of satellite carrier signals,

a vector γ

_j^R representative of a plurality of pseudo-ranges measured by the second navigation receiver (R) and corresponding to the plurality of satellite carrier signals,

a vector D_j^B representative of a plurality of estimated distances between the satellites and the first navigation receiver (B),

a vector D_j^R representative of a plurality of estimated distances between the satellites and the second navigation receiver (R),

a vector φ

_j^B representative of a plurality of full phase measurements of the satellite carrier signals measured by the first navigation receiver (B),

a vector φ

_j^R representative of a plurality of full phase measurements of the satellite carrier signals measured by the second navigation receiver (R),

a geometric Jacobian matrix H_j^γ

whose matrix elements are representative of the changes in the distances between the satellites and one of the receivers that would be caused by changes in that receiver'"'"'s position and time clock offset,

said apparatus comprising;

(a) means for generating, for each time moment j, a vector Δ

γ

_j of a plurality of range residuals of pseudo-range measurements made by the first and second navigation receivers in the form of;

Δ

γ

_j=(γ

_j^R−

γ

_j^B)−

(D_j^R−

D_j^B), said means generating a set of range residuals Δ

γ

_k, k=1, . . . , j;

(b) means for generating, for each time moment j, a vector Δ

φ

_j of a plurality of phase residuals of full phase measurements made by the first and second navigation receivers in the form of;

Δ

φ

_j=(φ

_j^R−

φ

_j^B)−

Λ

<sup>−

1</sup>·

(D_j^R−

D_j^B), where Λ

<sup>−

1</sup> is a diagonal matrix comprising the inverse wavelengths of the satellites, said means generating a set of phase residuals Δ

φ

_k, k=1, . . . , j;

(c) means for generating an LU-factorization of a matrix M or a matrix inverse of matrix M, the matrix M being a function of at least Λ

<sup>−

1</sup> and H_k^γ

, for index k of H_k^γ

covering at least two of the time moments j;

(d) means for generating a vector N of estimated floating ambiguities as a function of at least the set of range residuals Δ

γ

_k, the set of phase residuals Δ

φ

_k, and the LU-factorization of matrix M or the matrix inverse of matrix M.

49. The apparatus of claim 48 wherein means (c) comprises means for generating matrix M in a form equivalent to the summation

50. The apparatus of claim 49 wherein means (d) comprises means for generating matrix N in a form equivalent to:

51. The apparatus of claim 50 wherein the distance between the first and second navigation receivers is constrained to a distance L_RB, wherein means (c) comprises means for generating matrix S_k for the k-th time moment in a form equivalent to: