Abstract

Small motion measurement systems are widely used in industry measurement fields to measure small positional/angular motions. These systems usually consist of two parts: a measuring assembly and a reference assembly. The position-sensing detectors (PSDs) are embedded in either measuring assembly or reference assembly to sense the variations of laser light incidence points when there are any small positional/angular motions. To use these systems, it is necessary to determine the linear equations of PSD readings, which relate the six-degrees-of-freedom small positional/angular motions and PSD readings. The purpose of this paper is to derive these equations based on the paraxial raytracing method. Two measurement systems are used as illustrative examples to validate the proposed methodology. The methodology of this study will be useful for system design of PSD-based measurement systems and their applications.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. E. H. Bokelberg, H. J. Sommer III, and M. W. Tretheway, “A six-degree-of-freedom laser vibrometer, part I: Theoretical development,” J. Sound Vib. 178, 643–654 (1994).
    [CrossRef]
  2. E. H. Bokelberg, H. J. Sommer III, and M. W. Trethewey, “A six-degree-of-freedom laser vibrometer, part II: Experimental validation,” J. Sound Vib. 178, 655–667 (1994).
    [CrossRef]
  3. H. M. Varma, A. K. Nandakumaran, and R. M. Vasu, “Study of turbid media with light: Recovery of mechanical and optical properties from boundary measurement of intensity autocorrelation of light,” J. Opt. Soc. Am. A 26, 1472–1483 (2009).
    [CrossRef]
  4. M. T. Tavassoly, M. Amiri, A. Darudi, R. Aalipour, A. Saber, and A. R. Moradi, “Optical diffractometry,” J. Opt. Soc. Am. A 26, 540–547 (2009).
    [CrossRef]
  5. D. Patrick, “Sandia lab’s laser tracking system: a single station solution for position and attitude data,” IEEE J. Rob. Autom. RA-3, 323–344 (1987).
  6. P. D. Lin and C. H. Lu, “Modeling and sensitivity analysis of laser tracking systems by skew-ray tracing method,” J. Manuf. Sci. Eng. 127, 654–662 (2005).
    [CrossRef]
  7. R. B. Holmes, “Scintillation-induced jitter of projected light with centroid trackers,” J. Opt. Soc. Am. A 26, 313–316 (2009).
    [CrossRef]
  8. J. Ni and S. M. Wu, “An on-line measurement technique for machine volumetric error compensation,” J. Eng. Ind. 115, 85–92 (1993).
  9. K. H. Kim, K. F. Eman, and S. M. Wu, “Analysis alignment errors in a laser-based in-process cylindricity measurement system,” J. Eng. Ind. 109, 321–329 (1987).
    [CrossRef]
  10. C. W. Park, K. F. Eman, and S. M. Wu, “An in-process flatness error measurement and compensatory control system,” J. Eng. Ind. 110, 263–270 (1988).
    [CrossRef]
  11. P. D. Lin and K. F. Ehmann, “Sensing of motion related errors in multi-axis machines,” J. Dyn. Syst., Meas., Control 118, 425–433 (1996).
    [CrossRef]
  12. S. W. Lee, R. Mayor, and J. Ni, “Development of a six-degree-of-freedom geometric error measurement system for a meso-scale machine tool,” J. Manuf. Sci. Eng. 127, 857–865 (2005).
    [CrossRef]
  13. P. D. Lin, W. Y. Jywe, and C. J. Chen, “General method for determining the sensor readings of motion measurement systems by using skew-ray tracing method,” Optik (Stuttgart) 120, 257–264 (2009).
    [CrossRef]
  14. P. D. Lin and C. Y. Tsai, “First order gradients of skew rays of axis-symmetrical optical systems,” J. Opt. Soc. Am. A 24, 776–784 (2007).
    [CrossRef]
  15. P. D. Lin and C. K. Sung, “Matrix-based paraxial skew ray-tracing in 3D systems with non-coplanar optical axis,” Optik (Stuttgart) 117, 329–340 (2006).
    [CrossRef]
  16. P. D. Lin and C. K. Sung, “Camera calibration based on Snell's law,” J. Dyn. Syst., Meas., Control 128, 548–557 (2006).
    [CrossRef]
  17. P. D. Lin and C. K. Sung, “Comparing two new camera calibration methods with traditional pinhole calibrations,” Opt. Express 15, 3012–3022 (2007).
    [CrossRef] [PubMed]
  18. R. P. Paul, Robot Manipulators—Mathematics, Programming and Control (MIT, 1982).
  19. C. J. Chen, P. D. Lin, and W. Y. Jywe, “An optoelectronic measurement system for measuring 6-degree-of-freedom motion error of rotary parts,” Opt. Express 15, 14601–14617 (2007).
    [CrossRef] [PubMed]

2009 (4)

2007 (3)

2006 (2)

P. D. Lin and C. K. Sung, “Matrix-based paraxial skew ray-tracing in 3D systems with non-coplanar optical axis,” Optik (Stuttgart) 117, 329–340 (2006).
[CrossRef]

P. D. Lin and C. K. Sung, “Camera calibration based on Snell's law,” J. Dyn. Syst., Meas., Control 128, 548–557 (2006).
[CrossRef]

2005 (2)

P. D. Lin and C. H. Lu, “Modeling and sensitivity analysis of laser tracking systems by skew-ray tracing method,” J. Manuf. Sci. Eng. 127, 654–662 (2005).
[CrossRef]

S. W. Lee, R. Mayor, and J. Ni, “Development of a six-degree-of-freedom geometric error measurement system for a meso-scale machine tool,” J. Manuf. Sci. Eng. 127, 857–865 (2005).
[CrossRef]

1996 (1)

P. D. Lin and K. F. Ehmann, “Sensing of motion related errors in multi-axis machines,” J. Dyn. Syst., Meas., Control 118, 425–433 (1996).
[CrossRef]

1994 (2)

E. H. Bokelberg, H. J. Sommer III, and M. W. Tretheway, “A six-degree-of-freedom laser vibrometer, part I: Theoretical development,” J. Sound Vib. 178, 643–654 (1994).
[CrossRef]

E. H. Bokelberg, H. J. Sommer III, and M. W. Trethewey, “A six-degree-of-freedom laser vibrometer, part II: Experimental validation,” J. Sound Vib. 178, 655–667 (1994).
[CrossRef]

1993 (1)

J. Ni and S. M. Wu, “An on-line measurement technique for machine volumetric error compensation,” J. Eng. Ind. 115, 85–92 (1993).

1988 (1)

C. W. Park, K. F. Eman, and S. M. Wu, “An in-process flatness error measurement and compensatory control system,” J. Eng. Ind. 110, 263–270 (1988).
[CrossRef]

1987 (2)

K. H. Kim, K. F. Eman, and S. M. Wu, “Analysis alignment errors in a laser-based in-process cylindricity measurement system,” J. Eng. Ind. 109, 321–329 (1987).
[CrossRef]

D. Patrick, “Sandia lab’s laser tracking system: a single station solution for position and attitude data,” IEEE J. Rob. Autom. RA-3, 323–344 (1987).

1982 (1)

R. P. Paul, Robot Manipulators—Mathematics, Programming and Control (MIT, 1982).

Aalipour, R.

Amiri, M.

Bokelberg, E. H.

E. H. Bokelberg, H. J. Sommer III, and M. W. Trethewey, “A six-degree-of-freedom laser vibrometer, part II: Experimental validation,” J. Sound Vib. 178, 655–667 (1994).
[CrossRef]

E. H. Bokelberg, H. J. Sommer III, and M. W. Tretheway, “A six-degree-of-freedom laser vibrometer, part I: Theoretical development,” J. Sound Vib. 178, 643–654 (1994).
[CrossRef]

Chen, C. J.

P. D. Lin, W. Y. Jywe, and C. J. Chen, “General method for determining the sensor readings of motion measurement systems by using skew-ray tracing method,” Optik (Stuttgart) 120, 257–264 (2009).
[CrossRef]

C. J. Chen, P. D. Lin, and W. Y. Jywe, “An optoelectronic measurement system for measuring 6-degree-of-freedom motion error of rotary parts,” Opt. Express 15, 14601–14617 (2007).
[CrossRef] [PubMed]

Darudi, A.

Ehmann, K. F.

P. D. Lin and K. F. Ehmann, “Sensing of motion related errors in multi-axis machines,” J. Dyn. Syst., Meas., Control 118, 425–433 (1996).
[CrossRef]

Eman, K. F.

C. W. Park, K. F. Eman, and S. M. Wu, “An in-process flatness error measurement and compensatory control system,” J. Eng. Ind. 110, 263–270 (1988).
[CrossRef]

K. H. Kim, K. F. Eman, and S. M. Wu, “Analysis alignment errors in a laser-based in-process cylindricity measurement system,” J. Eng. Ind. 109, 321–329 (1987).
[CrossRef]

Holmes, R. B.

Jywe, W. Y.

P. D. Lin, W. Y. Jywe, and C. J. Chen, “General method for determining the sensor readings of motion measurement systems by using skew-ray tracing method,” Optik (Stuttgart) 120, 257–264 (2009).
[CrossRef]

C. J. Chen, P. D. Lin, and W. Y. Jywe, “An optoelectronic measurement system for measuring 6-degree-of-freedom motion error of rotary parts,” Opt. Express 15, 14601–14617 (2007).
[CrossRef] [PubMed]

Kim, K. H.

K. H. Kim, K. F. Eman, and S. M. Wu, “Analysis alignment errors in a laser-based in-process cylindricity measurement system,” J. Eng. Ind. 109, 321–329 (1987).
[CrossRef]

Lee, S. W.

S. W. Lee, R. Mayor, and J. Ni, “Development of a six-degree-of-freedom geometric error measurement system for a meso-scale machine tool,” J. Manuf. Sci. Eng. 127, 857–865 (2005).
[CrossRef]

Lin, P. D.

P. D. Lin, W. Y. Jywe, and C. J. Chen, “General method for determining the sensor readings of motion measurement systems by using skew-ray tracing method,” Optik (Stuttgart) 120, 257–264 (2009).
[CrossRef]

P. D. Lin and C. Y. Tsai, “First order gradients of skew rays of axis-symmetrical optical systems,” J. Opt. Soc. Am. A 24, 776–784 (2007).
[CrossRef]

C. J. Chen, P. D. Lin, and W. Y. Jywe, “An optoelectronic measurement system for measuring 6-degree-of-freedom motion error of rotary parts,” Opt. Express 15, 14601–14617 (2007).
[CrossRef] [PubMed]

P. D. Lin and C. K. Sung, “Comparing two new camera calibration methods with traditional pinhole calibrations,” Opt. Express 15, 3012–3022 (2007).
[CrossRef] [PubMed]

P. D. Lin and C. K. Sung, “Matrix-based paraxial skew ray-tracing in 3D systems with non-coplanar optical axis,” Optik (Stuttgart) 117, 329–340 (2006).
[CrossRef]

P. D. Lin and C. K. Sung, “Camera calibration based on Snell's law,” J. Dyn. Syst., Meas., Control 128, 548–557 (2006).
[CrossRef]

P. D. Lin and C. H. Lu, “Modeling and sensitivity analysis of laser tracking systems by skew-ray tracing method,” J. Manuf. Sci. Eng. 127, 654–662 (2005).
[CrossRef]

P. D. Lin and K. F. Ehmann, “Sensing of motion related errors in multi-axis machines,” J. Dyn. Syst., Meas., Control 118, 425–433 (1996).
[CrossRef]

Lu, C. H.

P. D. Lin and C. H. Lu, “Modeling and sensitivity analysis of laser tracking systems by skew-ray tracing method,” J. Manuf. Sci. Eng. 127, 654–662 (2005).
[CrossRef]

Mayor, R.

S. W. Lee, R. Mayor, and J. Ni, “Development of a six-degree-of-freedom geometric error measurement system for a meso-scale machine tool,” J. Manuf. Sci. Eng. 127, 857–865 (2005).
[CrossRef]

Moradi, A. R.

Nandakumaran, A. K.

Ni, J.

S. W. Lee, R. Mayor, and J. Ni, “Development of a six-degree-of-freedom geometric error measurement system for a meso-scale machine tool,” J. Manuf. Sci. Eng. 127, 857–865 (2005).
[CrossRef]

J. Ni and S. M. Wu, “An on-line measurement technique for machine volumetric error compensation,” J. Eng. Ind. 115, 85–92 (1993).

Park, C. W.

C. W. Park, K. F. Eman, and S. M. Wu, “An in-process flatness error measurement and compensatory control system,” J. Eng. Ind. 110, 263–270 (1988).
[CrossRef]

Patrick, D.

D. Patrick, “Sandia lab’s laser tracking system: a single station solution for position and attitude data,” IEEE J. Rob. Autom. RA-3, 323–344 (1987).

Paul, R. P.

R. P. Paul, Robot Manipulators—Mathematics, Programming and Control (MIT, 1982).

Saber, A.

Sommer, H. J.

E. H. Bokelberg, H. J. Sommer III, and M. W. Trethewey, “A six-degree-of-freedom laser vibrometer, part II: Experimental validation,” J. Sound Vib. 178, 655–667 (1994).
[CrossRef]

E. H. Bokelberg, H. J. Sommer III, and M. W. Tretheway, “A six-degree-of-freedom laser vibrometer, part I: Theoretical development,” J. Sound Vib. 178, 643–654 (1994).
[CrossRef]

Sung, C. K.

P. D. Lin and C. K. Sung, “Comparing two new camera calibration methods with traditional pinhole calibrations,” Opt. Express 15, 3012–3022 (2007).
[CrossRef] [PubMed]

P. D. Lin and C. K. Sung, “Matrix-based paraxial skew ray-tracing in 3D systems with non-coplanar optical axis,” Optik (Stuttgart) 117, 329–340 (2006).
[CrossRef]

P. D. Lin and C. K. Sung, “Camera calibration based on Snell's law,” J. Dyn. Syst., Meas., Control 128, 548–557 (2006).
[CrossRef]

Tavassoly, M. T.

Tretheway, M. W.

E. H. Bokelberg, H. J. Sommer III, and M. W. Tretheway, “A six-degree-of-freedom laser vibrometer, part I: Theoretical development,” J. Sound Vib. 178, 643–654 (1994).
[CrossRef]

Trethewey, M. W.

E. H. Bokelberg, H. J. Sommer III, and M. W. Trethewey, “A six-degree-of-freedom laser vibrometer, part II: Experimental validation,” J. Sound Vib. 178, 655–667 (1994).
[CrossRef]

Tsai, C. Y.

Varma, H. M.

Vasu, R. M.

Wu, S. M.

J. Ni and S. M. Wu, “An on-line measurement technique for machine volumetric error compensation,” J. Eng. Ind. 115, 85–92 (1993).

C. W. Park, K. F. Eman, and S. M. Wu, “An in-process flatness error measurement and compensatory control system,” J. Eng. Ind. 110, 263–270 (1988).
[CrossRef]

K. H. Kim, K. F. Eman, and S. M. Wu, “Analysis alignment errors in a laser-based in-process cylindricity measurement system,” J. Eng. Ind. 109, 321–329 (1987).
[CrossRef]

IEEE J. Rob. Autom. (1)

D. Patrick, “Sandia lab’s laser tracking system: a single station solution for position and attitude data,” IEEE J. Rob. Autom. RA-3, 323–344 (1987).

J. Dyn. Syst., Meas., Control (2)

P. D. Lin and K. F. Ehmann, “Sensing of motion related errors in multi-axis machines,” J. Dyn. Syst., Meas., Control 118, 425–433 (1996).
[CrossRef]

P. D. Lin and C. K. Sung, “Camera calibration based on Snell's law,” J. Dyn. Syst., Meas., Control 128, 548–557 (2006).
[CrossRef]

J. Eng. Ind. (3)

J. Ni and S. M. Wu, “An on-line measurement technique for machine volumetric error compensation,” J. Eng. Ind. 115, 85–92 (1993).

K. H. Kim, K. F. Eman, and S. M. Wu, “Analysis alignment errors in a laser-based in-process cylindricity measurement system,” J. Eng. Ind. 109, 321–329 (1987).
[CrossRef]

C. W. Park, K. F. Eman, and S. M. Wu, “An in-process flatness error measurement and compensatory control system,” J. Eng. Ind. 110, 263–270 (1988).
[CrossRef]

J. Manuf. Sci. Eng. (2)

S. W. Lee, R. Mayor, and J. Ni, “Development of a six-degree-of-freedom geometric error measurement system for a meso-scale machine tool,” J. Manuf. Sci. Eng. 127, 857–865 (2005).
[CrossRef]

P. D. Lin and C. H. Lu, “Modeling and sensitivity analysis of laser tracking systems by skew-ray tracing method,” J. Manuf. Sci. Eng. 127, 654–662 (2005).
[CrossRef]

J. Opt. Soc. Am. A (4)

J. Sound Vib. (2)

E. H. Bokelberg, H. J. Sommer III, and M. W. Tretheway, “A six-degree-of-freedom laser vibrometer, part I: Theoretical development,” J. Sound Vib. 178, 643–654 (1994).
[CrossRef]

E. H. Bokelberg, H. J. Sommer III, and M. W. Trethewey, “A six-degree-of-freedom laser vibrometer, part II: Experimental validation,” J. Sound Vib. 178, 655–667 (1994).
[CrossRef]

Opt. Express (2)

Optik (Stuttgart) (2)

P. D. Lin, W. Y. Jywe, and C. J. Chen, “General method for determining the sensor readings of motion measurement systems by using skew-ray tracing method,” Optik (Stuttgart) 120, 257–264 (2009).
[CrossRef]

P. D. Lin and C. K. Sung, “Matrix-based paraxial skew ray-tracing in 3D systems with non-coplanar optical axis,” Optik (Stuttgart) 117, 329–340 (2006).
[CrossRef]

Other (1)

R. P. Paul, Robot Manipulators—Mathematics, Programming and Control (MIT, 1982).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

Transformation of a vector.

Fig. 2
Fig. 2

Paraxial raytracing in an optical system with non-coplanar optical axis.

Fig. 3
Fig. 3

Ray travels along a straight line in a homogeneous medium before the reflection process at the i th boundary surface.

Fig. 4
Fig. 4

Flat mirror reflects light ray [ P i i 1 i i 1 ] T into ray [ P i i i i ] T .

Fig. 5
Fig. 5

Schematic diagram of sensor b of system from [8].

Fig. 6
Fig. 6

Modeling of sensor a of the small motion measurement system presented by [19].

Fig. 7
Fig. 7

Schematic representation of the six small positional/angular motions.

Equations (25)

Equations on this page are rendered with MathJax. Learn more.

A h g = [ I h g x J h g x K h g x t h g x I h g y J h g y K h g y t h g y I h g z J h g z K h g z t h g z 0 0 0 1 ] .
A i + 1 i = [ I i + 1 i x J i + 1 i x K i + 1 i x t i + 1 i x I i + 1 i y J i + 1 i y K i + 1 i y t i + 1 i y I i + 1 i z J i + 1 i z K i + 1 i z t i + 1 i z 0 0 0 1 ] = [ R i + 1 i t i + 1 i 0 1 × 3 1 ] .
[ Δ P i + 1 i Δ i + 1 i ] = [ A i + 1 i P i i A i + 1 i P ̱ i i A i + 1 i i i A i + 1 i ̱ i i ] = [ A i + 1 i ( P i i P ̱ i i ) A i + 1 i ( i i ̱ i i ) ] = [ R i + 1 i Δ P i i R i + 1 i Δ i i ] = [ I i + 1 i x J i + 1 i x K i + 1 i x 0 0 0 I i + 1 i y J i + 1 i y K i + 1 i y 0 0 0 I i + 1 i z J i + 1 i z K i + 1 i z 0 0 0 0 0 0 I i + 1 i x J i + 1 i x K i + 1 i x 0 0 0 I i + 1 i y J i + 1 i y K i + 1 i y 0 0 0 I i + 1 i z J i + 1 i z K i + 1 i z ] [ Δ P i i x Δ P i i y Δ P i i z Δ i i x Δ i i y Δ i i z ] = B i + 1 i [ Δ P i i Δ i i ] .
[ Δ P i i Δ i i 1 ] = [ 1 ̱ i i 1 x ̱ i i 1 y 0 λ ̱ i ̱ i i 1 x ̱ i i 1 y λ ̱ i 0 0 0 0 0 0 0 0 ̱ i i 1 z ̱ i i 1 y 1 0 ̱ i i 1 z ̱ i i 1 y λ ̱ i λ ̱ i 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 ] [ Δ P i i 1 Δ i i 1 ] = T i ( λ ̱ i ) [ Δ P i i 1 Δ i i 1 ] ,
[ Δ P i i Δ i i ] = [ 1 0 0 0 0 0 0 0 0 0 0 0 0 ̱ i 1 z ̱ i 1 y 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 ] [ Δ P i i Δ i i 1 ] = M i [ Δ P i i Δ i i 1 ] .
[ Δ P i + 1 i Δ l i + 1 i ] = B i + 1 i [ Δ P i i Δ l i i ] = B i + 1 i M i T i [ Δ P i i 1 Δ l i i 1 ] .
[ Δ P n n Δ l n n ] = T n ( B n n 1 M n 1 T n 1 ) ( B i + 1 i M i T i ) ( B 3 2 M 2 T 2 ) ( B 2 1 M 1 T 1 ) [ A 1 0 P 0 0 A 1 0 l 0 0 ] .
A reference measuring = Trans ( δ x , δ y , δ z ) Rot ( z , η z ) Rot ( y , η y ) Rot ( x , η x ) ,
A i 0 = A i measuring A measuring reference A reference 0 ,
A measuring reference = [ 1 η z η y δ x η z 1 η x δ y η y η x 1 δ z 0 0 0 1 ] .
[ Δ P 3 3 Δ l 3 3 ] = T 3 ( B 3 2 M 2 T 2 ) ( B 2 1 M 1 T 1 ) [ A 1 0 P 0 0 A 1 0 l 0 0 ] .
A 1 0 = A 1 measuring A measuring reference A reference 0 = Rot ( z , 45 ° ) Trans ( f x h x , h y , 0 ) A measuring reference Trans ( h x + f x , q 0 , 0 ) ,
A 2 1 = A 2 0 A 0 1 = A 2 measuring A measuring 1 = Rot ( z , 45 ° ) Trans ( f x + h x + g x , 0 , 0 ) Rot ( z , 45 ° ) ,
A 3 2 = A 3 0 A 0 2 = Trans ( g x + h x + f x , g y , 0 ) Trans ( h x f x , q 0 , 0 ) A measuring reference Trans ( g x , h y , 0 ) Rot ( z , 45 ° ) = Trans ( g x , g y + q 0 , 0 ) A measuring reference Trans ( g x , h y , 0 ) Rot ( z , 45 ° ) .
[ Δ X b Δ Z b ] = [ Δ P 3 3 x Δ P 3 3 z ] = [ 2 δ x ( 2 h y + f x + h x + g x ) η z [ 2 ( g y + q 0 + h y ) + ( f x + h x + g x ) ] η x + ( f x + h x + g x ) η y ] .
[ Δ P 2 2 Δ l 2 2 ] = T 2 ( B 2 1 M 1 T 1 ) [ A 1 0 P 0 0 A 1 0 l 0 0 ] .
A reference 0 = I 4 × 4 ,
A 1 measuring = Rot ( y , 180 ° ) Rot ( x , ϕ ) Rot ( z , 90 ° ) Trans ( 2 h 3 6 , 0 , h 3 3 ) ,
A 2 0 = Trans ( 0 , d , 0 ) Rot ( z , 135 ° ) Rot ( y , 180 ° ) Rot ( x , ϕ ) Rot ( z , 90 ° ) Trans ( 2 h 3 6 , 0 , h 3 3 ) ,
A 2 1 = A 2 0 A 0 1 = A 2 0 A 1 0 1 = A 2 0 A 0 reference A reference measuring A measuring 1 = Trans ( 0 , d , 0 ) Rot ( z , 135 ° ) Rot ( y , 180 ° ) Rot ( x , ϕ ) Rot ( z , 90 ° ) Trans ( 2 h 3 6 , 0 , h 3 3 ) A reference measuring Trans ( 2 h 3 6 , 0 , h 3 3 ) Rot ( z , 90 ° ) Rot ( x , ϕ ) Rot ( y , 180 ° ) ,
[ Δ X a Δ Z a ] = [ Δ P 2 2 x Δ P 2 2 z ] = [ 2 δ x + ( 2 h 3 + 2 d ) ( η y η z ) d ( η y + η z ) ] .
Rot ( x , η x ) = [ 1 0 0 0 0 C η x S η x 0 0 S η x C η x 0 0 0 0 1 ] ,
Rot ( y , η y ) = [ C η y 0 S η y 0 0 1 0 0 S η y 0 C η y 0 0 0 0 1 ] ,
Rot ( z , η z ) = [ C η z S η z 0 0 S η z C η z 0 0 0 0 1 0 0 0 0 1 ] ,
Trans ( δ x , δ y , δ z ) = [ 1 0 0 δ x 0 1 0 δ y 0 0 1 δ z 0 0 0 1 ] ,

Metrics