Interpolation of survey coordinate differences

• US 6,370,476 B1
• Filed: 03/05/1999
• Issued: 04/09/2002
• Est. Priority Date: 04/27/1998
• Status: Expired due to Term

• Current Assignee: Trimble Navigation Limited (Trimble Inc.)
• Original Assignee: Trimble Navigation Limited (Trimble Inc.)
• Inventors: McBride, Kenneth W.
• Primary Examiner(s): Hilten, John S.
• Assistant Examiner(s): Vo, Hein
• Application Number:
  US09/263,960
• Time in Patent Office: 1,131 Days
• Field of Search: 702/14, 702/94, 702/95, 702/150, 342/352, 342/357, 342/463, 701/207, 701/213, 701/300, 333/20
• US Class Current: 702/14
• CPC Class Codes: G01S 1/04 Details

Abstract

A system for improving the accuracy of location coordinate determined in a survey of a chosen region. A grid of spaced apart points is imposed on the region, and a set of survey control points is provided. A “near set” of nearest survey control points is associated with each grid point, and the number of elements in this new set may vary with the grid point. Definition of a near set can vary from one grid point to the next. For each grid point, a transformation T from a first coordinate system to a second coordinate system is determined that minimizes a collective difference between coordinates of each survey control point in the near set and the corresponding coordinates of that survey control point under the transformation T. For one, two or three coordinates of each grid point, the difference(s) between the coordinate(s) of the grid point in the near set and the corresponding coordinate(s) of that grid point under the transformation T are computed. An interpolation function is determined that approximately matches the coordinate difference(s) at each grid point. The interpolation function provides a continuous datum-to-datum mapping between all points of the first and the second coordinate system. Determination of location of each survey point may use GPS, GLONASS, modified LEO or any other suitable location determination system.

Claims

1. A method for improving the accuracy of a survey of a region, the method comprising the steps of:

determining, a grid of at least three survey reference points in a selected first coordinate system for a selected region;

obtaining a set of N survey control locations associated with the region;

providing a computer that is programmed;

to identify, for each grid point in the grid, an associated survey control set of a selected number K (2≦

K≦

N) of survey control locations that are associated with the grid point, and determining a transformation T that has at least one adjustable parameter and that maps location coordinates in the first coordinate system into location coordinates in a selected second coordinate system;

for each grid point in the grid, to determining a coordinate difference between at least a first coordinate of the grid point and a corresponding coordinate of the image of the grid point under the transformation T; and

to determine a difference interpolation function, defined on and continuous at all points in the selected region, which is approximately equal to the coordinate difference for each point on the grid.

2. The method of claim 1, further comprising the step of selecting said number K of said survey control locations to be different for at least two of said survey reference points in said grid.

3. The method of claim 1, further comprising the step of determining said selected set of K survey control points to be said survey control points, each of whose generalized distance to said grid point is no greater than a generalized distance between said grid point and any survey control point that is not in said selected set of said K survey control points.

4. The method of claim 3, further comprising the step of selecting said generalized distance between said grid point, having location coordinates (x_m,G,y_m,G,z_m,G), and said survey control point, having location coordinates (x_SC-n,y_SC-n,z_SC-n, to have the form

5. The method of claim 4, further comprising the step of selecting said coefficients c_x, c_y and c_z to each be equal to a selected positive number and selecting q=2.

6. The method of claim 1, further comprising the step of choosing said at least one adjustable parameter so that a generalized sum of distances between each survey control location in said selected set of K survey control locations and the corresponding image under said transformation T, is minimized.

7. The method of claim 4, further comprising the steps of:

determining, for each said grid point in said grid, a second coordinate difference between at least a second coordinate of said grid point and a corresponding second coordinate of the image of said grid point under said transformation T; and

determining a second difference interpolation function, defined on and continuous at all points in the selected region, which is substantially equal to the second coordinate difference for each point on the grid.

8. The method of claim 7, further comprising the steps of:

determining, for each said grid point in said grid, a third coordinate difference between at least a third coordinate of said grid point and a corresponding third coordinate of the image of said grid point under said transformation T; and

determining a third difference interpolation function, defined on and continuous at all points in the selected region, which is substantially equal to the third coordinate difference for each point on the grid.

9. The method of claim 1, further comprising the step of selecting said first coordinate system and said second coordinate system to have a spatial dimension 2, with said location coordinates (x, y).

10. The method of claim 9, wherein said step of choosing said at least one adjustable parameter comprises the steps of:

obtaining a first set of K coordinate pairs {(x_k, y_k)} (k=1 . . . , K;

K≧

2;

K≧

2) for said survey control locations in said first coordinate system and applying said transformation T to each coordinate pair in the first set to determine an image under T of the coordinate pair in said second coordinate system

11. The method of claim 10, further comprising the step of choosing said transformation T the linear transformation defined by

12. The method of claim 11, wherein said step of choosing said at least one adjustable parameter further comprises the step of selecting at least one of said adjustable coefficients h_ij to minimize said error sum ε

_p.

13. The method of claim 10, further comprising the step of choosing as said transformation T the transformation defined by (1) a rotation of the first coordinate system around a selected rotation axis by a selected angle θ

1, (2) multiplication of said coordinates resulting from the rotation by a selected real number scale factor L, and (3) translation of the resulting first and second location coordinates by selected translation distances Δ

x and Δ

y, respectively, where at least one of the coefficients θ

1, L, Δ

x and Δ

y is an adjustable parameter.

14. The method of claim 10, further comprising the step of selecting said positive number p to be equal to 2 and choosing each of said non-negative numbers e_k and f_k to be equal to a selected positive number.

15. The method of claim 1, further comprising the step of selecting said first coordinate system and said second coordinate system to have a spatial dimension 3, with said location coordinates (x, y, z).

16. The method of claim 15, wherein said step of choosing said at least one adjustable parameter comprises the steps of:

obtaining a first set of K coordinate triples {(x_k, y_k, z_k)} (k=1, . . . , K;

K≧

2) for said survey control locations in said first coordinate system and applying said transformation T to each coordinate triple in the first set to determine an image under T of the coordinate triple in said second coordinate system

17. The method of claim 16, further comprising the step of choosing as said transformation T the linear transformation defined by

18. The method of claim 17, wherein said step of choosing said at least one adjustable parameter further comprises the step of selecting at least one of said adjustable coefficients h_ij to minimize said error sum ε

_p.

19. The method of claim 16, further comprising the step of choosing as said transformation T the transformation defined by (1) a first rotation of the first coordinate system around a selected first rotation axis by a selected first angle θ

1, (2) a second rotation of the first coordinate system around a selected second rotation axis, which differs from the selected first rotation axis, by a selected second angle θ

2, (3) a third rotation of the first coordinate system around a selected third rotation axis, which differs from the selected second rotation axis, by a selected second angle θ

3, (4) multiplication of said coordinates resulting from the first, second and third rotations by a selected real number scale factor L, and (5) translation of the resulting first, second and third location coordinates by selected translation distances Δ

x, Δ

y and Δ

z, respectively, where at least one of the coefficients Δ

1, θ

2, θ

3, L, Δ

x, Δ

y and Δ

z is an adjustable parameter.

20. The method of claim 16, further comprising the step of selecting said positive number p to be equal to 2 and choosing each of said non-negative numbers e_k, f_k and g_k to be equal to a selected positive number.

21. Apparatus for transforming coordinates for at least one location, surveyed in a first coordinate system, into location coordinates in a second coordinate system , the apparatus comprising a computer that is programmed

(1) to obtain location coordinates for a selected set of N survey control locations in a first coordinate system, and to obtain a subset of K selected survey control locations (N≧

K≧

2) that are closer to a selected location than any other survey control location in the selected set;

(2) to obtain location coordinates for the set of survey control locations, expressed in a selected second coordinate system;

(3) to determine a coordinate transformation T, having, at least one adjustable parameter, of the first coordinate system onto the second coordinate system to minimize a selected generalized sum of distances between each survey control location in the selected subset of K survey control locations and the corresponding image under the transformation T, by choice of at least one adjustable parameter; and

(4) to determine an interpolation function that estimates the difference between at least one coordinate of a selected location in the first coordinate system and the corresponding coordinate of the image under T of the selected location.

22. The apparatus of claim 21, wherein said computer is further programmed to select said number K of said survey control locations to be different for at least two of said survey reference points in said grid.

23. The apparatus of claim 21, wherein said computer is further programmed to apply said transformation T to said coordinates of at least one location in said first coordinate system that is not a survey control location.

24. The apparatus of claim 23, wherein said computer is programmed to determine said transformation T at approximately a time that a location is determined for said at least one location that is not a survey control location in said first coordinate system.

25. The apparatus of claim 24, wherein said computer is handheld.

26. The apparatus of claim 21, further comprising a location determination (LD) system that receives LD signals from a plurality of LD signal sources and that provides present location information for said computer from which said computer determines said coordinates for each of said plurality of survey control locations in at least one of said first coordinate system and said second coordinate system.

27. The apparatus of claim 26, further comprising a display that, in a selected mode of operation, automatically displays, in at least one of visually perceptible and audibly perceptible form, a selected number of survey control locations that are closest to said present location of said LD system.

28. The apparatus of claim 26, wherein said LD signals are drawn from a class of timed signals consisting of Global Positioning System signals, Global Orbiting Navigational Satellite System signals and Low Earth Orbit system signals.

29. The apparatus of claim 21, further comprising a display that displays, in at least one of visually perceptible and audibly perceptible form, at least one of (1) a selected number of survey control locations that are closest to a selected location;

(2) at least one transformation parameter that describes said transformation T;

(3) said difference between said at least one coordinate of said selected location in said first coordinate system and said corresponding coordinate of said image under T of said selected location; and

(4) an estimate, using, said interpolation function, of said difference between said at least one coordinate of said selected location in said first coordinate system and said corresponding coordinate of said image under T of said selected location.

30. The system of claim 21, wherein said computer is further programmed to select said number K of said survey control locations to be different for at least two of said survey reference points in said grid.

31. The apparatus of claim 21, wherein said computer is further programmed to determine said selected set of K survey control points to be 5said survey control points, each of whose general distance to said grid point is no greater than a general distance between said grid point and any survey control point that is not in said selected set of said K survey control points.

32. The apparatus of claim 31, wherein said computer is further programmed to select said general distance between said grid point, having location coordinates (x_m,G,y_m,G,z_m,G), and said survey control point, having location coordinates (x_SC-n,y_SC-n,z_SC-n), to have the form

33. The apparatus of claim 32, wherein said coefficients c_x, c_y and c_z are selected to each be equal to a selected positive number and said number q=2.

34. The apparatus of claim 21, wherein said computer is further programmed to choosing said at least one adjustable parameter so that a generalized sum of distances between each survey control location in said selected set of K survey control locations and the corresponding image under said transformation T, is minimized.

35. The apparatus of claim 21, wherein said computer is further programmed:

to determine, for each said grid point in said grid, a second coordinate difference between at least a second coordinate of said grid point and a corresponding second coordinate of the image of said grid point under said transformation T; and

to determine a second difference interpolation function, defined on and continuous at all points in the selected region, which is substantially equal to the second coordinate difference for each point on the grid.

36. The apparatus of claim 21, wherein said computer is further programmed:

to determine, for each said grid point in said arid, a third coordinate difference between at least a third coordinate of said grid point and a corresponding third coordinate of the image of said grid point under said transformation T; and

to determine a third difference interpolation function, defined on and continuous at all points in the selected region, which is substantially equal to the third coordinate difference for each point on the grid.

37. The apparatus of claim 21, wherein said computer is further programmed to select said first coordinate system and said second coordinate system to have a spatial dimension 2, with said location coordinates (x, y).

38. The apparatus of claim 37, wherein said computer is further programmed to choose said at least one adjustable parameter by a process comprising the steps of:

obtaining a first set of K coordinate pairs {(x_k, y_k)} (k=1, . . . , K;

K≧

2) for said survey control locations in said first coordinate system and applying said transformation T to each coordinate pair in the first set to determine an image under T of the coordinate pair in said second coordinate system

39. The apparatus of claim 38, wherein said computer is further programmed to choose as said transformation T the linear transformation defined by

40. The apparatus of claim 39, wherein said computer is further programmed to choose said at least one adjustable parameter further comprises the step of selecting at least one of said adjustable coefficients h_ij to minimize said error sum ε

_p.

41. The apparatus of claim 38, wherein said computer is further programmed to choose as said transformation T the transformation defined by (1) a rotation of the first coordinate system around a selected rotation axis by a selected angle θ

1, (2) multiplication of said coordinates resulting from the rotation by a selected real number scale factor L, and (3) translation of the resulting first and second location coordinates by selected translation distances Δ

x and Δ

y, respectively, where at least one of the coefficients θ

1, L Δ

x and Δ

y is an adjustable parameter.

42. The apparatus of claim 38, wherein said computer is further programmed to select said positive number p to be equal to 2 and choosing each of said non-negative numbers e_k, and f_k to be equal to a selected positive number.

43. The apparatus of claim 21, wherein said computer is further programmed to select said first coordinate system and said second coordinate system to have a spatial dimension 3, with said location coordinates (x, y, z).

44. The apparatus of claim 43, where in said computer is further programmed to choose said at least one adjustable parameter by a process comprising the steps of:

obtaining a first set of K coordinate triples {(x_k, y_k, z_k)} (k=1, . . . , K;

K≧

2) for said survey control locations in said first coordinate system and applying said transformation T to each coordinate triple in the first set to determine an image under T of the coordinate triple in said second coordinate system

45. The apparatus of claim 44, wherein said computer is further programmed to choose as said transformation T the linear transformation defined by

 

x′

=h₁₁x+h₁₂y+h₁₃z+h₁₄,

46. The apparatus of claim 45, wherein said computer is further programmed to choose said at least one adjustable parameter further comprises the step of selecting at least one of said adjustable coefficients h_ij to minimize said error sum ε

_p.

47. The apparatus of claim 44, wherein said computer is further programmed to choose as said transformation T the transformation defined by (1) a first rotation of the first coordinate system around a selected first rotation axis by a selected first angle θ

1, (2) a second rotation of the first coordinate system around a selected second rotation axis, which differs from the selected first rotation axis, by a selected second angle θ

2, (3) a third rotation of the first coordinate system around a selected third rotation axis, which differs from the selected second rotation axis, by a selected second angle θ

3, (4) multiplication of said coordinates resulting from the first, second and third rotations by a selected real number scale factor L, and (5) translation of the resulting first, second and third location coordinates by selected translation distances Δ

x, Δ

y and Δ

z, respectively, where at least one of the coefficients θ

1, θ

2, θ

3, L, Δ

x, Δ

y and Δ

z is an adjustable parameter.

48. The apparatus of claim 47, wherein said computer is further programmed to select said positive number p to be equal to 2 and choosing, each of said non-negative numbers e_k, f_k and g_k to be equal to a selected positive number.