Survey coordinate transformation optimization
First Claim
1. A method for transforming coordinates for at least one location, surveyed in a first coordinate system, into location coordinates in a second coordinate system, the method comprising the steps of:
- determining coordinates for each of a plurality of already-surveyed locations in a first coordinate system;
obtaining coordinates for the already-surveyed locations in a second coordinate system, where each already-surveyed location in the second coordinate system corresponds to an already-surveyed location in the first coordinate system;
computing a coordinate transformation T, having at least one adjustable parameter, of the first coordinate system onto the second coordinate system having the at least one adjustable parameter;
forming a selected sum of magnitudes of difference between the coordinates of at least one already-surveyed location in the second coordinate system and the result of applying the transformation T to the corresponding at least one already-surveved location in the first coordinate system; and
choosing the at least one adjustable parameter to minimize the selected sum.
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Abstract
A system for selecting an optimal transformation T(G2;G1) between a first ellipsoid E1 (e.g., WGS 84) in a first global coordinate system G1, relative to which the survey measurements are made, and a second ellipsoid E2 (e.g., NAD 27) in a second global coordinate system G2. Location coordinates (x'"'"'m,2,y'"'"'m,2,z'"'"'m,2) for M previously-surveyed locations, numbered m=1, . . . , M (M>1) in the second system, and location coordinates (x'"'"'n,1,y'"'"'n,1,z'"'"'n,1) for N presently-surveyed locations, numbered n=1, . . . , N (M<N), in the first system are provided, where M presently-surveyed locations coincide with the M previously-surveyed locations. The transformation is chosen so that the images of previously-surveyed locations in the first system under the transformation T are as close as possible to the corresponding [previously-surveyed] locations in the second system. Given an ellipsoid and a selected survey plane τ0 that is tangent to the ellipsoid, a set of selected locations can be surveyed with reference to the ellipsoid, and the location coordinates of each such surveyed location can be mapped into a corresponding "survey location," defined by reference to a local coordinate system that uses the survey plane τ0 as its base. These transformations and projections are determined and optimized in real time, at the time a surveyor is measuring the presently-surveyed locations in the field. The instruments used for surveying may be part of a location determination system, such as GPS, GLONASS or Loran.
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Citations
20 Claims
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1. A method for transforming coordinates for at least one location, surveyed in a first coordinate system, into location coordinates in a second coordinate system, the method comprising the steps of:
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determining coordinates for each of a plurality of already-surveyed locations in a first coordinate system; obtaining coordinates for the already-surveyed locations in a second coordinate system, where each already-surveyed location in the second coordinate system corresponds to an already-surveyed location in the first coordinate system; computing a coordinate transformation T, having at least one adjustable parameter, of the first coordinate system onto the second coordinate system having the at least one adjustable parameter; forming a selected sum of magnitudes of difference between the coordinates of at least one already-surveyed location in the second coordinate system and the result of applying the transformation T to the corresponding at least one already-surveved location in the first coordinate system; and choosing the at least one adjustable parameter to minimize the selected sum. - View Dependent Claims (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15)
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7. The method of claim 6, wherein said step of choosing said at least one adjustable parameter further comprises the step of selecting at least one of said adjustable coefficients hij to minimize said error sum ε
- P.
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8. The method of claim 5, further comprising the step of choosing as said transformation T the transformation defmed 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 K, 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, K, Δ
x and Δ
y is an adjustable parameter.
- 1, (2) multiplication of said coordinates resulting from the rotation by a selected real number scale factor K, and (3) translation of the resulting first and second location coordinates by selected translation distances Δ
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9. The method of claim 5, further comprising the step of selecting said positive number p to be equal to 2 and selecting each of said non-negative numbers en and fn to be equal a selected positive number.
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10. 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).
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11. The method of claim 10, wherein said step of choosing said at least one adjustable parameter comprises the steps of:
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obtaining a first set of N coordinate triples {(x'"'"'n,1,y'"'"'n,1,z'"'"'n,1)} (n=1, . . . , N) for said already-surveyed locations in said first coordinate system, where N is a selected integer that is at least 2, 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
space="preserve" listing-type="equation">T (x'"'"'.sub.n,1,y'"'"'.sub.n,1,z'"'"'.sub.n,1)=(x'"'"'.sub.n,2,y'"'"'.sub.n,2,z'"'"'.sub.n,2);choosing said at least one adjustable parameter for said transformation T so that an error sum ε
P defined by ##EQU21## is minimized, where p is a selected positive number and en, fn and gn are selected non-negative numbers.
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- 12. The method of claim 11, further comprising the step of choosing as said transformation T the linear transformation defined by
- space="preserve" listing-type="equation">x'"'"'.sub.n,2 =h.sub.11 x'"'"'.sub.n,1 +h.sub.12 y'"'"'.sub.n,1 +h.sub.13 z'"'"'.sub.n,1 +h.sub.14,
space="preserve" listing-type="equation">y'"'"'.sub.n,2 =h.sub.21 x'"'"'.sub.n,1 +h.sub.22 y'"'"'.sub.n,1 +h.sub.23 z'"'"'.sub.n,1 +h.sub.14,
space="preserve" listing-type="equation">z'"'"'.sub.n,2 =h.sub.31 x'"'"'.sub.n,1 +h.sub.32 y'"'"'.sub.n,1 +h.sub.33 z'"'"'.sub.n,1 +h.sub.14,where at least one of the coefficients hij (i=1,2,3;
j=1,2,3,4) is an adjustable parameter.
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13. The method of claim 12, wherein said step of choosing said at least one adjustable parameter further comprises the step of selecting at least one of said adjustable coefficients hij to minimize said error sum ε
- P.
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14. The method of claim 11, 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 K, 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, K, Δ
x, Δ
y and Δ
z is an adjustable parameter.
- 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
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15. The method of claim 11, further comprising the step of selecting said positive number p to be equal to 2 and selecting each of said non-negative numbers en, fn and gn to be equal a selected positive number.
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16. 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 determine coordinates for each of a plurality of already-surveyed locations in a first coordinate system; (2) to obtain coordinate for the already-surveyed locations in a second coordinate system, where each already-surveyed location in the second coordinate system corresponds to an already-surveyed location in the first 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 having at least one adjustable parameter; (4) to form a selected sum of magnitudes of difference between the coordinates of at least one already-surveyed location in the second coordinate system and the result of applying the transformation T to the corresponding at least one already-surveyed location in the first coordinate system; and (5) to choose the at least one adjustable parameter to minimize the selected sum. - View Dependent Claims (17, 18, 19, 20)
Specification