Attitude determination based on global navigation satellite system information

1. A method for determining a 3-dimensional (3D) attitude of a platform, the method comprising:

receiving satellite relayed information regarding an ambiguous phase single-difference measurement (φ);

resolving a phase ambiguity of the ambiguous phase single-difference measurement (φ) to determine an unambiguous phase single-difference estimate (ϕ);

calculating coarse direction vectors x cor and y cor based on the unambiguous phase single-difference estimate (ϕ);

estimating improved direction vectors x and y based on the coarse direction vectors x cor and y cor and by imposing constraints on (1) a norm of each of the improved direction vectors x and y, and (2) an angle between the improved direction vectors x and y; and

calculating the 3D attitude of the platform from the improved direction vectors x and y,

wherein the 3D attitude indicates an orientation of the platform relative to a reference.

2. The method of claim 1 , wherein the platform has first to fifth receivers for receiving the satellite relayed information and the relayed information has a single frequency.

3. The method of claim 2 , wherein the first to third receivers are arranged along a first baseline, and the first, fourth and fifth receivers are arranged along a second baseline.

4. The method of claim 3 , wherein the first and second baselines are not collinear and the first and second baselines are not parallel.

5. The method of claim 4 , wherein the step of calculating coarse direction vectors x cor and y cor uses the unambiguous phase single-difference estimate (ϕ) calculated with measurements only from the first, second and fourth receivers.

6. The method of claim 5 , further comprising:

applying a Lagrange multiplier method to an objective function f that uses the unambiguous phase single-difference estimate (ϕ) calculated with measurements from the first, third, and fifth receivers,

wherein the objective function includes the improved direction vectors x and y.

7. The method of claim 6 , wherein a norm of the direction vectors x and y is constrained to be unity and an angle between the direction vectors x and y is constrained.

8. The method of claim 7 , wherein the direction vectors x and y are expressed in spherical coordinates.

9. The method of claim 1 , wherein the platform has first to third receivers for receiving the satellite relayed information and there are two carriers that relay the information, having a first frequency and a second frequency.

10. The method of claim 9 , wherein the first and second receivers are arranged along a first baseline, and the first and third receivers are arranged along a second baseline.

11. The method of claim 10 , wherein the first and second baselines are not collinear and the first and second baselines are not parallel.

12. The method of claim 11 , wherein the step of calculating coarse direction vectors x cor and y cor uses the unambiguous phase single-difference estimate (ϕ) calculated with measurements from the first frequency only.

13. The method of claim 12 , further comprising:

applying a Lagrange multiplier method to an objective function f that uses the unambiguous phase single-difference estimate (ϕ) calculated with measurements from the second frequency,

wherein the objective function includes the improved direction vectors x and y, a norm of each of the direction vectors x and y is constrained to be unity, and an angle between the direction vectors x and y is constrained.

14. A system for determining a 3-dimensional (3D) attitude of a platform, the system comprising:

plural receivers configured to receive satellite relayed information regarding an ambiguous phase single-difference measurement (φ); and

a controller connected to the plural receivers and configured to,

resolve a phase ambiguity of the ambiguous phase single-difference measurement (φ) to determine an unambiguous phase single-difference estimate (ϕ);

calculate coarse direction vectors x cor and y cor based on the unambiguous phase single-difference estimate (ϕ);

estimate improved direction vectors x and y based on the coarse direction vectors x cor and y cor and by imposing constraints on (1) a norm of each of the improved direction vectors x and y, and (2) an angle between the improved direction vectors x and y; and

calculate the 3D attitude of the platform from the improved direction vectors x and y,

wherein the 3D attitude indicates an orientation of the platform relative to a reference.

15. The system of claim 14 , wherein the platform has first to fifth receivers for receiving the satellite relayed information and the relayed information has a single frequency.

16. The system of claim 15 , wherein the first to third receivers are arranged along a first baseline, and the first, fourth and fifth receivers are arranged along a second baseline.

17. The system of claim 16 , wherein the first and second baselines are not collinear and the first and second baselines are not parallel.

18. The system of claim 14 , wherein the platform has first to third receivers for receiving the satellite relayed information and there are two carriers that relay the information, having a first frequency and a second frequency.

19. The system of claim 18 , wherein the first and second receivers are arranged along a first baseline, and the first and third receivers are arranged along a second baseline, and the first and second baselines are not collinear and the first and second baselines are not parallel.

20. The system of claim 19 , wherein the step of calculating coarse direction vectors x cor and y cor uses the unambiguous phase single-difference estimate (ϕ) calculated with measurements from the first frequency only.

21. A non-transitory computer readable medium including computer executable instructions, wherein the instructions, when executed by a processor, implement instructions for determining a 3-dimensional (3D) attitude of a platform, the instructions comprising:

receiving satellite relayed information regarding an ambiguous phase single-difference measurement (φ);

resolving a phase ambiguity of the ambiguous phase single-difference measurement (φ) to determine an unambiguous phase single-difference estimate (ϕ);

calculating coarse direction vectors x cor and y cor based on the unambiguous phase single-difference estimate (ϕ);

estimating improved direction vectors x and y based on the coarse direction vectors x cor and y cor and by imposing constraints on (1) a norm of each of the improved direction vectors x and y, and (2) an angle between the improved direction vectors x and y; and

calculating the 3D attitude of the platform from the improved direction vectors x and y,

wherein the 3D attitude indicates an orientation of the platform relative to a reference.