Abstract

In this paper, a compact four-degree-of-freedom (4-DOF) measurement system is presented. With a special optical configuration, the pitch error, yaw error, and two straightness errors of the moving target are able to be detected by only a single laser beam from a collimated laser diode. A 2D hybrid mirror angle steering mount is designed to perform the large angle turning for the axis alignment and very fine angle tuning by PZT actuators for real-time beam drift compensation. A series of calibration and comparison experiments have been carried out to verify the performance of the proposed system. The developed active compensation system could effectively suppress the beam’s angular drift to within ± 0.01 arc-sec in both of yaw and pitch directions. The developed 4-DOF measuring system is compact, low cost, and suitable for long distance geometric error measurement of linear stages.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
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References

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  18. Y. Zhao, C. Lu, L. Qiu, L. Zou, and J. Cui, “Enhancing laser beam directional stability by single-mode optical fiber and feedback control of drifts,” Rev. Sci. Instrum. 86(3), 036101 (2015).
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2017 (1)

U. Mutilba, E. Gomez-Acedo, G. Kortaberria, A. Olarra, and J. A. Yagüe-Fabra, “Traceability on machine tool metrology: a review,” Sensors (Basel) 17(7), 1605 (2017).
[Crossref] [PubMed]

2016 (2)

X. Yu, S. R. Gillmer, S. C. Woody, and J. D. Ellis, “Development of a compact, fiber-coupled, six degree-of-freedom measurement system for precision linear stage metrology,” Rev. Sci. Instrum. 87(6), 065109 (2016).
[Crossref] [PubMed]

C. Cui, Q. Feng, B. Zhang, and Y. Zhao, “System for simultaneously measuring 6DOF geometric motion errors using a polarization maintaining fiber-coupled dual-frequency laser,” Opt. Express 24(6), 6735–6748 (2016).
[Crossref] [PubMed]

2015 (3)

P. Hu, S. Mao, and J. B. Tan, “Compensation of errors due to incident beam drift in a 3 DOF measurement system for linear guide motion,” Opt. Express 23(22), 28389–28401 (2015).
[Crossref] [PubMed]

X. Yu, S. R. Gillmer, and J. D. Ellis, “Beam geometry, alignment, and wavefront aberration effects on interferometric differential wavefront sensing,” Meas. Sci. Technol. 26(12), 125203 (2015).
[Crossref]

Y. Zhao, C. Lu, L. Qiu, L. Zou, and J. Cui, “Enhancing laser beam directional stability by single-mode optical fiber and feedback control of drifts,” Rev. Sci. Instrum. 86(3), 036101 (2015).

2014 (1)

K. C. Fan, H. Y. Wang, H. W. Yang, and L. M. Chen, “Techniques of multi-degree-of-freedom measurement on the linear motion errors of precision machines,” Adv. Opt. Technol. 3(4), 375–386 (2014).

2013 (3)

2008 (1)

H. Schwenke, W. Knapp, H. Haitjema, A. Weckenmann, R. Schmitt, and F. Delbressine, “Geometric error measurement and compensation of machines—an update,” CIRP Ann. 57(2), 660–675 (2008).
[Crossref]

2006 (1)

W. Zhao, L. Qiu, Z. Feng, and C. Li, “Laser beam alignment by fast feedback control of both linear and angular drifts,” Optik (Stuttg.) 117(11), 505–510 (2006).
[Crossref]

2005 (1)

C. Kuang, Q. Feng, B. Zhang, B. Liu, S. Chen, and Z. Zhang, “A four-degree-of-freedom laser measurement system (FDMS) using a single-mode fiber-coupled laser module,” Sens. Actuators A Phys. 125(1), 100–108 (2005).
[Crossref]

2000 (2)

K. C. Fan and M. J. Chen, “A 6-degree-of-freedom measurement system for the accuracy of XY stages,” Precis. Eng. 24(1), 15–23 (2000).
[Crossref]

R. Ramesh, M. A. Mannan, and A. N. Poo, “Error compensation in machine tools—a review,” Int. J. Mach. Tools Manuf. 40(9), 1235–1256 (2000).
[Crossref]

1998 (1)

K. C. Fan, M. J. Chen, and W. M. Huang, “A six-degree-of-freedom measurement system for the motion accuracy of linear stages,” Int. J. Mach. Tools Manuf. 38(3), 155–164 (1998).
[Crossref]

1995 (1)

P. S. Huang and J. Ni, “On-line error compensation of coordinate measuring machines,” Int. J. Mach. Tools Manuf. 35(5), 725–738 (1995).
[Crossref]

1993 (1)

J. Ni and S. M. Wu, “An on-line measurement technique for machine volumetric error compensation,” J. Eng. Ind. 115(1), 85–92 (1993).
[Crossref]

1979 (1)

J. B. Bryan, “The Abbé principle revisit: An updated interpretation,” Precis. Eng. 1(3), 129–132 (1979).
[Crossref]

Bin, Z.

Bryan, J. B.

J. B. Bryan, “The Abbé principle revisit: An updated interpretation,” Precis. Eng. 1(3), 129–132 (1979).
[Crossref]

Chen, L. M.

K. C. Fan, H. Y. Wang, H. W. Yang, and L. M. Chen, “Techniques of multi-degree-of-freedom measurement on the linear motion errors of precision machines,” Adv. Opt. Technol. 3(4), 375–386 (2014).

Chen, M. J.

K. C. Fan and M. J. Chen, “A 6-degree-of-freedom measurement system for the accuracy of XY stages,” Precis. Eng. 24(1), 15–23 (2000).
[Crossref]

K. C. Fan, M. J. Chen, and W. M. Huang, “A six-degree-of-freedom measurement system for the motion accuracy of linear stages,” Int. J. Mach. Tools Manuf. 38(3), 155–164 (1998).
[Crossref]

Chen, S.

C. Kuang, Q. Feng, B. Zhang, B. Liu, S. Chen, and Z. Zhang, “A four-degree-of-freedom laser measurement system (FDMS) using a single-mode fiber-coupled laser module,” Sens. Actuators A Phys. 125(1), 100–108 (2005).
[Crossref]

Cui, C.

Cui, J.

Y. Zhao, C. Lu, L. Qiu, L. Zou, and J. Cui, “Enhancing laser beam directional stability by single-mode optical fiber and feedback control of drifts,” Rev. Sci. Instrum. 86(3), 036101 (2015).

F. Zhu, J. Tan, and J. Cui, “Common-path design criteria for laser datum based measurement of small angle deviations and laser autocollimation method in compliance with the criteria with high accuracy and stability,” Opt. Express 21(9), 11391–11403 (2013).
[Crossref] [PubMed]

Cuifang, K.

Cunxing, C.

Delbressine, F.

H. Schwenke, W. Knapp, H. Haitjema, A. Weckenmann, R. Schmitt, and F. Delbressine, “Geometric error measurement and compensation of machines—an update,” CIRP Ann. 57(2), 660–675 (2008).
[Crossref]

Ellis, J. D.

X. Yu, S. R. Gillmer, S. C. Woody, and J. D. Ellis, “Development of a compact, fiber-coupled, six degree-of-freedom measurement system for precision linear stage metrology,” Rev. Sci. Instrum. 87(6), 065109 (2016).
[Crossref] [PubMed]

X. Yu, S. R. Gillmer, and J. D. Ellis, “Beam geometry, alignment, and wavefront aberration effects on interferometric differential wavefront sensing,” Meas. Sci. Technol. 26(12), 125203 (2015).
[Crossref]

Fan, K. C.

K. C. Fan, H. Y. Wang, H. W. Yang, and L. M. Chen, “Techniques of multi-degree-of-freedom measurement on the linear motion errors of precision machines,” Adv. Opt. Technol. 3(4), 375–386 (2014).

K. C. Fan and M. J. Chen, “A 6-degree-of-freedom measurement system for the accuracy of XY stages,” Precis. Eng. 24(1), 15–23 (2000).
[Crossref]

K. C. Fan, M. J. Chen, and W. M. Huang, “A six-degree-of-freedom measurement system for the motion accuracy of linear stages,” Int. J. Mach. Tools Manuf. 38(3), 155–164 (1998).
[Crossref]

Feng, Q.

C. Cui, Q. Feng, B. Zhang, and Y. Zhao, “System for simultaneously measuring 6DOF geometric motion errors using a polarization maintaining fiber-coupled dual-frequency laser,” Opt. Express 24(6), 6735–6748 (2016).
[Crossref] [PubMed]

C. Kuang, Q. Feng, B. Zhang, B. Liu, S. Chen, and Z. Zhang, “A four-degree-of-freedom laser measurement system (FDMS) using a single-mode fiber-coupled laser module,” Sens. Actuators A Phys. 125(1), 100–108 (2005).
[Crossref]

Feng, Z.

W. Zhao, L. Qiu, Z. Feng, and C. Li, “Laser beam alignment by fast feedback control of both linear and angular drifts,” Optik (Stuttg.) 117(11), 505–510 (2006).
[Crossref]

Fenglin, Y.

Gillmer, S. R.

X. Yu, S. R. Gillmer, S. C. Woody, and J. D. Ellis, “Development of a compact, fiber-coupled, six degree-of-freedom measurement system for precision linear stage metrology,” Rev. Sci. Instrum. 87(6), 065109 (2016).
[Crossref] [PubMed]

X. Yu, S. R. Gillmer, and J. D. Ellis, “Beam geometry, alignment, and wavefront aberration effects on interferometric differential wavefront sensing,” Meas. Sci. Technol. 26(12), 125203 (2015).
[Crossref]

Gomez-Acedo, E.

U. Mutilba, E. Gomez-Acedo, G. Kortaberria, A. Olarra, and J. A. Yagüe-Fabra, “Traceability on machine tool metrology: a review,” Sensors (Basel) 17(7), 1605 (2017).
[Crossref] [PubMed]

Haitjema, H.

H. Schwenke, W. Knapp, H. Haitjema, A. Weckenmann, R. Schmitt, and F. Delbressine, “Geometric error measurement and compensation of machines—an update,” CIRP Ann. 57(2), 660–675 (2008).
[Crossref]

Hu, P.

Huang, P.

Huang, P. S.

P. S. Huang and J. Ni, “On-line error compensation of coordinate measuring machines,” Int. J. Mach. Tools Manuf. 35(5), 725–738 (1995).
[Crossref]

Huang, W. M.

K. C. Fan, M. J. Chen, and W. M. Huang, “A six-degree-of-freedom measurement system for the motion accuracy of linear stages,” Int. J. Mach. Tools Manuf. 38(3), 155–164 (1998).
[Crossref]

Knapp, W.

H. Schwenke, W. Knapp, H. Haitjema, A. Weckenmann, R. Schmitt, and F. Delbressine, “Geometric error measurement and compensation of machines—an update,” CIRP Ann. 57(2), 660–675 (2008).
[Crossref]

Kortaberria, G.

U. Mutilba, E. Gomez-Acedo, G. Kortaberria, A. Olarra, and J. A. Yagüe-Fabra, “Traceability on machine tool metrology: a review,” Sensors (Basel) 17(7), 1605 (2017).
[Crossref] [PubMed]

Kuang, C.

C. Kuang, Q. Feng, B. Zhang, B. Liu, S. Chen, and Z. Zhang, “A four-degree-of-freedom laser measurement system (FDMS) using a single-mode fiber-coupled laser module,” Sens. Actuators A Phys. 125(1), 100–108 (2005).
[Crossref]

Li, C.

W. Zhao, L. Qiu, Z. Feng, and C. Li, “Laser beam alignment by fast feedback control of both linear and angular drifts,” Optik (Stuttg.) 117(11), 505–510 (2006).
[Crossref]

Li, Y.

Liu, B.

C. Kuang, Q. Feng, B. Zhang, B. Liu, S. Chen, and Z. Zhang, “A four-degree-of-freedom laser measurement system (FDMS) using a single-mode fiber-coupled laser module,” Sens. Actuators A Phys. 125(1), 100–108 (2005).
[Crossref]

Lu, C.

Y. Zhao, C. Lu, L. Qiu, L. Zou, and J. Cui, “Enhancing laser beam directional stability by single-mode optical fiber and feedback control of drifts,” Rev. Sci. Instrum. 86(3), 036101 (2015).

Mannan, M. A.

R. Ramesh, M. A. Mannan, and A. N. Poo, “Error compensation in machine tools—a review,” Int. J. Mach. Tools Manuf. 40(9), 1235–1256 (2000).
[Crossref]

Mao, S.

Mutilba, U.

U. Mutilba, E. Gomez-Acedo, G. Kortaberria, A. Olarra, and J. A. Yagüe-Fabra, “Traceability on machine tool metrology: a review,” Sensors (Basel) 17(7), 1605 (2017).
[Crossref] [PubMed]

Ni, J.

P. S. Huang and J. Ni, “On-line error compensation of coordinate measuring machines,” Int. J. Mach. Tools Manuf. 35(5), 725–738 (1995).
[Crossref]

J. Ni and S. M. Wu, “An on-line measurement technique for machine volumetric error compensation,” J. Eng. Ind. 115(1), 85–92 (1993).
[Crossref]

Olarra, A.

U. Mutilba, E. Gomez-Acedo, G. Kortaberria, A. Olarra, and J. A. Yagüe-Fabra, “Traceability on machine tool metrology: a review,” Sensors (Basel) 17(7), 1605 (2017).
[Crossref] [PubMed]

Poo, A. N.

R. Ramesh, M. A. Mannan, and A. N. Poo, “Error compensation in machine tools—a review,” Int. J. Mach. Tools Manuf. 40(9), 1235–1256 (2000).
[Crossref]

Qibo, F.

Qiu, L.

Y. Zhao, C. Lu, L. Qiu, L. Zou, and J. Cui, “Enhancing laser beam directional stability by single-mode optical fiber and feedback control of drifts,” Rev. Sci. Instrum. 86(3), 036101 (2015).

W. Zhao, L. Qiu, Z. Feng, and C. Li, “Laser beam alignment by fast feedback control of both linear and angular drifts,” Optik (Stuttg.) 117(11), 505–510 (2006).
[Crossref]

Ramesh, R.

R. Ramesh, M. A. Mannan, and A. N. Poo, “Error compensation in machine tools—a review,” Int. J. Mach. Tools Manuf. 40(9), 1235–1256 (2000).
[Crossref]

Ren, L.

Schmitt, R.

H. Schwenke, W. Knapp, H. Haitjema, A. Weckenmann, R. Schmitt, and F. Delbressine, “Geometric error measurement and compensation of machines—an update,” CIRP Ann. 57(2), 660–675 (2008).
[Crossref]

Schwenke, H.

H. Schwenke, W. Knapp, H. Haitjema, A. Weckenmann, R. Schmitt, and F. Delbressine, “Geometric error measurement and compensation of machines—an update,” CIRP Ann. 57(2), 660–675 (2008).
[Crossref]

Tan, J.

Tan, J. B.

Wang, H. Y.

K. C. Fan, H. Y. Wang, H. W. Yang, and L. M. Chen, “Techniques of multi-degree-of-freedom measurement on the linear motion errors of precision machines,” Adv. Opt. Technol. 3(4), 375–386 (2014).

Weckenmann, A.

H. Schwenke, W. Knapp, H. Haitjema, A. Weckenmann, R. Schmitt, and F. Delbressine, “Geometric error measurement and compensation of machines—an update,” CIRP Ann. 57(2), 660–675 (2008).
[Crossref]

Wei, H.

Woody, S. C.

X. Yu, S. R. Gillmer, S. C. Woody, and J. D. Ellis, “Development of a compact, fiber-coupled, six degree-of-freedom measurement system for precision linear stage metrology,” Rev. Sci. Instrum. 87(6), 065109 (2016).
[Crossref] [PubMed]

Wu, S. M.

J. Ni and S. M. Wu, “An on-line measurement technique for machine volumetric error compensation,” J. Eng. Ind. 115(1), 85–92 (1993).
[Crossref]

Yagüe-Fabra, J. A.

U. Mutilba, E. Gomez-Acedo, G. Kortaberria, A. Olarra, and J. A. Yagüe-Fabra, “Traceability on machine tool metrology: a review,” Sensors (Basel) 17(7), 1605 (2017).
[Crossref] [PubMed]

Yang, H. W.

K. C. Fan, H. Y. Wang, H. W. Yang, and L. M. Chen, “Techniques of multi-degree-of-freedom measurement on the linear motion errors of precision machines,” Adv. Opt. Technol. 3(4), 375–386 (2014).

Yu, X.

X. Yu, S. R. Gillmer, S. C. Woody, and J. D. Ellis, “Development of a compact, fiber-coupled, six degree-of-freedom measurement system for precision linear stage metrology,” Rev. Sci. Instrum. 87(6), 065109 (2016).
[Crossref] [PubMed]

X. Yu, S. R. Gillmer, and J. D. Ellis, “Beam geometry, alignment, and wavefront aberration effects on interferometric differential wavefront sensing,” Meas. Sci. Technol. 26(12), 125203 (2015).
[Crossref]

Yusheng, Z.

Zhang, B.

C. Cui, Q. Feng, B. Zhang, and Y. Zhao, “System for simultaneously measuring 6DOF geometric motion errors using a polarization maintaining fiber-coupled dual-frequency laser,” Opt. Express 24(6), 6735–6748 (2016).
[Crossref] [PubMed]

C. Kuang, Q. Feng, B. Zhang, B. Liu, S. Chen, and Z. Zhang, “A four-degree-of-freedom laser measurement system (FDMS) using a single-mode fiber-coupled laser module,” Sens. Actuators A Phys. 125(1), 100–108 (2005).
[Crossref]

Zhang, Z.

C. Kuang, Q. Feng, B. Zhang, B. Liu, S. Chen, and Z. Zhang, “A four-degree-of-freedom laser measurement system (FDMS) using a single-mode fiber-coupled laser module,” Sens. Actuators A Phys. 125(1), 100–108 (2005).
[Crossref]

Zhao, S.

Zhao, W.

W. Zhao, L. Qiu, Z. Feng, and C. Li, “Laser beam alignment by fast feedback control of both linear and angular drifts,” Optik (Stuttg.) 117(11), 505–510 (2006).
[Crossref]

Zhao, Y.

C. Cui, Q. Feng, B. Zhang, and Y. Zhao, “System for simultaneously measuring 6DOF geometric motion errors using a polarization maintaining fiber-coupled dual-frequency laser,” Opt. Express 24(6), 6735–6748 (2016).
[Crossref] [PubMed]

Y. Zhao, C. Lu, L. Qiu, L. Zou, and J. Cui, “Enhancing laser beam directional stability by single-mode optical fiber and feedback control of drifts,” Rev. Sci. Instrum. 86(3), 036101 (2015).

Zhu, F.

Zou, L.

Y. Zhao, C. Lu, L. Qiu, L. Zou, and J. Cui, “Enhancing laser beam directional stability by single-mode optical fiber and feedback control of drifts,” Rev. Sci. Instrum. 86(3), 036101 (2015).

Adv. Opt. Technol. (1)

K. C. Fan, H. Y. Wang, H. W. Yang, and L. M. Chen, “Techniques of multi-degree-of-freedom measurement on the linear motion errors of precision machines,” Adv. Opt. Technol. 3(4), 375–386 (2014).

Appl. Opt. (1)

CIRP Ann. (1)

H. Schwenke, W. Knapp, H. Haitjema, A. Weckenmann, R. Schmitt, and F. Delbressine, “Geometric error measurement and compensation of machines—an update,” CIRP Ann. 57(2), 660–675 (2008).
[Crossref]

Int. J. Mach. Tools Manuf. (3)

R. Ramesh, M. A. Mannan, and A. N. Poo, “Error compensation in machine tools—a review,” Int. J. Mach. Tools Manuf. 40(9), 1235–1256 (2000).
[Crossref]

P. S. Huang and J. Ni, “On-line error compensation of coordinate measuring machines,” Int. J. Mach. Tools Manuf. 35(5), 725–738 (1995).
[Crossref]

K. C. Fan, M. J. Chen, and W. M. Huang, “A six-degree-of-freedom measurement system for the motion accuracy of linear stages,” Int. J. Mach. Tools Manuf. 38(3), 155–164 (1998).
[Crossref]

J. Eng. Ind. (1)

J. Ni and S. M. Wu, “An on-line measurement technique for machine volumetric error compensation,” J. Eng. Ind. 115(1), 85–92 (1993).
[Crossref]

Meas. Sci. Technol. (1)

X. Yu, S. R. Gillmer, and J. D. Ellis, “Beam geometry, alignment, and wavefront aberration effects on interferometric differential wavefront sensing,” Meas. Sci. Technol. 26(12), 125203 (2015).
[Crossref]

Opt. Express (4)

Optik (Stuttg.) (1)

W. Zhao, L. Qiu, Z. Feng, and C. Li, “Laser beam alignment by fast feedback control of both linear and angular drifts,” Optik (Stuttg.) 117(11), 505–510 (2006).
[Crossref]

Precis. Eng. (2)

K. C. Fan and M. J. Chen, “A 6-degree-of-freedom measurement system for the accuracy of XY stages,” Precis. Eng. 24(1), 15–23 (2000).
[Crossref]

J. B. Bryan, “The Abbé principle revisit: An updated interpretation,” Precis. Eng. 1(3), 129–132 (1979).
[Crossref]

Rev. Sci. Instrum. (2)

Y. Zhao, C. Lu, L. Qiu, L. Zou, and J. Cui, “Enhancing laser beam directional stability by single-mode optical fiber and feedback control of drifts,” Rev. Sci. Instrum. 86(3), 036101 (2015).

X. Yu, S. R. Gillmer, S. C. Woody, and J. D. Ellis, “Development of a compact, fiber-coupled, six degree-of-freedom measurement system for precision linear stage metrology,” Rev. Sci. Instrum. 87(6), 065109 (2016).
[Crossref] [PubMed]

Sens. Actuators A Phys. (1)

C. Kuang, Q. Feng, B. Zhang, B. Liu, S. Chen, and Z. Zhang, “A four-degree-of-freedom laser measurement system (FDMS) using a single-mode fiber-coupled laser module,” Sens. Actuators A Phys. 125(1), 100–108 (2005).
[Crossref]

Sensors (Basel) (1)

U. Mutilba, E. Gomez-Acedo, G. Kortaberria, A. Olarra, and J. A. Yagüe-Fabra, “Traceability on machine tool metrology: a review,” Sensors (Basel) 17(7), 1605 (2017).
[Crossref] [PubMed]

Other (2)

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Figures (12)

Fig. 1
Fig. 1 Optical configuration of the 4-DOF measuring system.
Fig. 2
Fig. 2 Principle of straightness error measurement: (a) optical path, (b) spot position measured by QPD3
Fig. 3
Fig. 3 Principle of angular error measurement: (a) optical path, (b) autocollimation.
Fig. 4
Fig. 4 Working principle of beam drift active compensation, (a) optical paths, (b) control loop.
Fig. 5
Fig. 5 (a) Design of a PZT embedded steering mirror mechanism, (b) photo of the prototype.
Fig. 6
Fig. 6 Experimental setup for 4-DOF geometric error measurement of a moving carrier.
Fig. 7
Fig. 7 Calibration setup: (a) straightness, (b) yaw and pitch.
Fig. 8
Fig. 8 Results of calibration: (a) horizontal straightness, (b) vertical straightness, (c) yaw (d) pitch.
Fig. 9
Fig. 9 Comparison of straightness error measurement (a) experimental setup, (b) Bryan principle.
Fig. 10
Fig. 10 Comparison results of measured horizontal straightness by (a) 4-DOF system, (b) converted LSMS, (c) averaged error and residual; measured vertical straightness by (d) 4-DOF system, (e) LSMS, (f) averaged error and residual.
Fig. 11
Fig. 11 Comparison results of measured yaw errors by (a) 4-DOF system, (b) AutoMAT 5000, (c) average and residuals; measured pitch errors by (d) 4-DOF system, (e) AutoMAT 5000, (f) average and residuals.
Fig. 12
Fig. 12 Stability results: (a) beam drift in yaw (b) beam drift in pitch (c) horizontal straightness (d) vertical straightness (e) measured yaw error (f) measured pitch error.

Tables (1)

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Table 1 Performance of the 4-DOF system

Equations (21)

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δ x = Δ x Q P D 3 = k δ x ( i 1 + i 4 ) ( i 2 + i 3 ) ( i 1 + i 2 + i 3 + i 4 ) δ y = Δ y Q P D 3 = k δ y ( i 1 + i 2 ) ( i 3 + i 4 ) ( i 1 + i 2 + i 3 + i 4 )
ε x = Δ x Q P D 2 f 2 and ε y = Δ y Q P D 2 f 2
n m 1 = [ N m 1 x N m 1 y N m 1 z ] T = [ 2 2 0 2 2 ] T
R m 1 = [ 1 2 N m 1 x N m 1 x 2 N m 1 x N m 1 y 2 N m 1 x N m 1 z 2 N m 1 x N m 1 y 1 2 N m 1 y N m 1 y 2 N m 1 y N m 1 z 2 N m 1 x N m 1 z 2 N m 1 y N m 1 z 1 2 N m 1 z N m 1 z ] = [ 0 0 1 0 1 0 1 0 0 ] .
n r m 1 = R m 1 n L D = [ 1 0 0 ]
n P B S = [ N p x N p y N p z ] T = [ 2 2 0 2 2 ] n m 2 = [ N m 2 x N m 2 y N m 2 y ] T = [ 2 2 0 2 2 ]
R p b s = [ 0 0 1 0 1 0 1 0 0 ] and R m 2 = [ 0 0 1 0 1 0 1 0 0 ]
n A C 1 = R P B S n r m 1 = [ 0 0 1 ] T n r m 2 = R P B S n r m 1 = [ 0 0 1 ] T
n ' L D = [ ε d y ε d p 1 ] T
n ' A C 1 = R p b s R m 1 n ' L D = [ ε d y ε d p 1 ] T
n ' r m 2 = R m 2 R m 1 n ' L D = [ ε d y ε d p 1 ] T .
ε d y = Δ x Q P D 1 f 1 and ε d p = Δ y Q P D 1 f 1
n ' m 1 = T m 1 n m 1 = [ 1 0 β m 1 0 1 α m 1 β m 1 α m 1 1 ] 2 2 [ 1 0 1 ] = 2 2 [ 1 + β m 1 α m 1 1 β m 1 ]
R ' m 1 = [ β m 1 2 2 β m 1 α m 1 + α m 1 β m 1 β m 1 2 1 α m 1 + α m 1 β m 1 1 α m 1 2 α m 1 α m 1 β m 1 β m 1 2 1 α m 1 α m 1 β m 1 β m 1 2 + 2 β m 1 ] .
n " A C 1 = R p b s R ' m 1 n ' L D [ 2 β m 1 ε d y ε d p + α m 1 1 ] T
n ' ' r m 2 = R m 2 R ' m 1 n ' ' L D [ ε d y 2 β m 1 ε d p + α m 1 1 ] T
β m 1 = ε d y 2 and α m 1 = ε d p .
n " r m 2 = [ 0 0 1 ] T = n r m 2
d x ε d p L x and d y 1 2 ε d y L y
δ A y ( z ) = δ B y ( z ) + ε z ( z ) L B A x ( z )
δ A x ( z ) = δ B x ( z ) ε z ( z ) L B A y ( z )

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