Abstract

It is crucially important to establish an accurate model to represent the relationship between the actuator forces and the lap surface changes when polishing a large and highly aspheric optical surface. To facilitate a computer-controlled optical polishing process, a neural network based stressed lap surface shape model was developed. The developed model reflects the dynamic deformation of a stressed lap. The original data from the microdisplacement sensor matrix were used to train the neural network model. The experimental results show that the proposed model can represent the surface shape of the stressed lap accurately and provide an analytical model to be used to polish the stressed lap control system and the active support system for a large mirror.

© 2010 Optical Society of America

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  1. Y. Li and W. Jiang, “A study on computer stressed lap polishing technology for large aperture and highly aspherical optical surface,” J. Opto-electron. Eng. 26, 9-15 (1999).
  2. Y. Li, Y. Zheng, W. Jiang, J. Yuan, Z. Zeng, and F. Wu, “The flexible optical manufacturing technologies for large aspheric mirror,” J. Opt. Tech. 27, 490-492 (2001).
  3. Y. Li, Advanced Optics Manufacturing Technology (Science Press, Beijing, 2001).
  4. X. Cui. B. Gao, D. Wang, Y. Zhu, S. Yang, Z. Zhu, L. Wang, and Z. Xia, “A new polishing technology for large diameter and deep aspherical mirror,” J. Acta Optica Sin. 25, 402-407(2005).
  5. D. Wang, Y. Li, and S. Yang, “Study on control technology of active stressed lap polishing aspherical mirror,” J. Opt. Technique 31, 373-379 (2005).
  6. Z. Zheng, B. Gao, X. Li, and M. Liu, “Optical technology and testing method using stressed lap to polish asphere, ” J. Opt. Technique 31, 341-343 (2005).
  7. B. Fan, Y. Li, Z. Zeng, X. Li, and Y. Wan, “Method for representing the surface and testing technology of stressed-lap in CMAC controller,” J. Opt. Technique 31, 751-754(2005).
  8. B. Shan, S. Wang, H. Niu, and S. Liu, “Zernike polynomial fitting method and its application,” J. Opt. Precis. Eng. 10, 318-322 (2002).
  9. L. Xu, A. Kryzak, and A. Yuille, “On radial basis function nets and kernel regression: statistical consistency, convergence rates, and receptive field size,” J. Neural Net. 7, 609-628(1994).
    [CrossRef]
  10. M. Chen and D. A. Linkens, “A fast fuzzy modelling approach using clustering neural networks,” in World Congress on Computational Intelligence (IEEE, 1998), pp 1088-1093.
  11. M. Chen and D. A. Linkens, “A systematic neuro-fuzzy modeling framework with application to material property prediction,” IEEE Trans. Syst. Man Cybern. B 31, 781-790 (2001).
    [CrossRef]
  12. M. Chen and D. A. Linkenns, “Rule-base self-generation and simplification for data-driven fuzzy models,” J. Fuzzy Sets Syst. 142, 243-265 (2004).
    [CrossRef]

2005

X. Cui. B. Gao, D. Wang, Y. Zhu, S. Yang, Z. Zhu, L. Wang, and Z. Xia, “A new polishing technology for large diameter and deep aspherical mirror,” J. Acta Optica Sin. 25, 402-407(2005).

D. Wang, Y. Li, and S. Yang, “Study on control technology of active stressed lap polishing aspherical mirror,” J. Opt. Technique 31, 373-379 (2005).

Z. Zheng, B. Gao, X. Li, and M. Liu, “Optical technology and testing method using stressed lap to polish asphere, ” J. Opt. Technique 31, 341-343 (2005).

B. Fan, Y. Li, Z. Zeng, X. Li, and Y. Wan, “Method for representing the surface and testing technology of stressed-lap in CMAC controller,” J. Opt. Technique 31, 751-754(2005).

2004

M. Chen and D. A. Linkenns, “Rule-base self-generation and simplification for data-driven fuzzy models,” J. Fuzzy Sets Syst. 142, 243-265 (2004).
[CrossRef]

2002

B. Shan, S. Wang, H. Niu, and S. Liu, “Zernike polynomial fitting method and its application,” J. Opt. Precis. Eng. 10, 318-322 (2002).

2001

M. Chen and D. A. Linkens, “A systematic neuro-fuzzy modeling framework with application to material property prediction,” IEEE Trans. Syst. Man Cybern. B 31, 781-790 (2001).
[CrossRef]

Y. Li, Y. Zheng, W. Jiang, J. Yuan, Z. Zeng, and F. Wu, “The flexible optical manufacturing technologies for large aspheric mirror,” J. Opt. Tech. 27, 490-492 (2001).

1999

Y. Li and W. Jiang, “A study on computer stressed lap polishing technology for large aperture and highly aspherical optical surface,” J. Opto-electron. Eng. 26, 9-15 (1999).

1994

L. Xu, A. Kryzak, and A. Yuille, “On radial basis function nets and kernel regression: statistical consistency, convergence rates, and receptive field size,” J. Neural Net. 7, 609-628(1994).
[CrossRef]

Chen, M.

M. Chen and D. A. Linkens, “A systematic neuro-fuzzy modeling framework with application to material property prediction,” IEEE Trans. Syst. Man Cybern. B 31, 781-790 (2001).
[CrossRef]

M. Chen and D. A. Linkens, “A fast fuzzy modelling approach using clustering neural networks,” in World Congress on Computational Intelligence (IEEE, 1998), pp 1088-1093.

Jiang, W.

Y. Li, Y. Zheng, W. Jiang, J. Yuan, Z. Zeng, and F. Wu, “The flexible optical manufacturing technologies for large aspheric mirror,” J. Opt. Tech. 27, 490-492 (2001).

Y. Li and W. Jiang, “A study on computer stressed lap polishing technology for large aperture and highly aspherical optical surface,” J. Opto-electron. Eng. 26, 9-15 (1999).

Li, X.

Z. Zheng, B. Gao, X. Li, and M. Liu, “Optical technology and testing method using stressed lap to polish asphere, ” J. Opt. Technique 31, 341-343 (2005).

B. Fan, Y. Li, Z. Zeng, X. Li, and Y. Wan, “Method for representing the surface and testing technology of stressed-lap in CMAC controller,” J. Opt. Technique 31, 751-754(2005).

Li, Y.

B. Fan, Y. Li, Z. Zeng, X. Li, and Y. Wan, “Method for representing the surface and testing technology of stressed-lap in CMAC controller,” J. Opt. Technique 31, 751-754(2005).

D. Wang, Y. Li, and S. Yang, “Study on control technology of active stressed lap polishing aspherical mirror,” J. Opt. Technique 31, 373-379 (2005).

Li, Y.

Y. Li, Y. Zheng, W. Jiang, J. Yuan, Z. Zeng, and F. Wu, “The flexible optical manufacturing technologies for large aspheric mirror,” J. Opt. Tech. 27, 490-492 (2001).

Linkens, D. A.

M. Chen and D. A. Linkens, “A systematic neuro-fuzzy modeling framework with application to material property prediction,” IEEE Trans. Syst. Man Cybern. B 31, 781-790 (2001).
[CrossRef]

Liu, S.

B. Shan, S. Wang, H. Niu, and S. Liu, “Zernike polynomial fitting method and its application,” J. Opt. Precis. Eng. 10, 318-322 (2002).

Liu, M.

Z. Zheng, B. Gao, X. Li, and M. Liu, “Optical technology and testing method using stressed lap to polish asphere, ” J. Opt. Technique 31, 341-343 (2005).

Niu, H.

B. Shan, S. Wang, H. Niu, and S. Liu, “Zernike polynomial fitting method and its application,” J. Opt. Precis. Eng. 10, 318-322 (2002).

Wang, S.

B. Shan, S. Wang, H. Niu, and S. Liu, “Zernike polynomial fitting method and its application,” J. Opt. Precis. Eng. 10, 318-322 (2002).

Wang, D.

D. Wang, Y. Li, and S. Yang, “Study on control technology of active stressed lap polishing aspherical mirror,” J. Opt. Technique 31, 373-379 (2005).

Wu, F.

Y. Li, Y. Zheng, W. Jiang, J. Yuan, Z. Zeng, and F. Wu, “The flexible optical manufacturing technologies for large aspheric mirror,” J. Opt. Tech. 27, 490-492 (2001).

Yang, S.

D. Wang, Y. Li, and S. Yang, “Study on control technology of active stressed lap polishing aspherical mirror,” J. Opt. Technique 31, 373-379 (2005).

Yuan, J.

Y. Li, Y. Zheng, W. Jiang, J. Yuan, Z. Zeng, and F. Wu, “The flexible optical manufacturing technologies for large aspheric mirror,” J. Opt. Tech. 27, 490-492 (2001).

Yuille, A.

L. Xu, A. Kryzak, and A. Yuille, “On radial basis function nets and kernel regression: statistical consistency, convergence rates, and receptive field size,” J. Neural Net. 7, 609-628(1994).
[CrossRef]

Zeng, Z.

Y. Li, Y. Zheng, W. Jiang, J. Yuan, Z. Zeng, and F. Wu, “The flexible optical manufacturing technologies for large aspheric mirror,” J. Opt. Tech. 27, 490-492 (2001).

Zheng, Y.

Y. Li, Y. Zheng, W. Jiang, J. Yuan, Z. Zeng, and F. Wu, “The flexible optical manufacturing technologies for large aspheric mirror,” J. Opt. Tech. 27, 490-492 (2001).

Chen , M.

M. Chen and D. A. Linkenns, “Rule-base self-generation and simplification for data-driven fuzzy models,” J. Fuzzy Sets Syst. 142, 243-265 (2004).
[CrossRef]

Cui, X.

X. Cui. B. Gao, D. Wang, Y. Zhu, S. Yang, Z. Zhu, L. Wang, and Z. Xia, “A new polishing technology for large diameter and deep aspherical mirror,” J. Acta Optica Sin. 25, 402-407(2005).

Fan, B.

B. Fan, Y. Li, Z. Zeng, X. Li, and Y. Wan, “Method for representing the surface and testing technology of stressed-lap in CMAC controller,” J. Opt. Technique 31, 751-754(2005).

Gao, B.

Z. Zheng, B. Gao, X. Li, and M. Liu, “Optical technology and testing method using stressed lap to polish asphere, ” J. Opt. Technique 31, 341-343 (2005).

Gao, B.

X. Cui. B. Gao, D. Wang, Y. Zhu, S. Yang, Z. Zhu, L. Wang, and Z. Xia, “A new polishing technology for large diameter and deep aspherical mirror,” J. Acta Optica Sin. 25, 402-407(2005).

Kryzak, A.

L. Xu, A. Kryzak, and A. Yuille, “On radial basis function nets and kernel regression: statistical consistency, convergence rates, and receptive field size,” J. Neural Net. 7, 609-628(1994).
[CrossRef]

Li, Y.

Y. Li and W. Jiang, “A study on computer stressed lap polishing technology for large aperture and highly aspherical optical surface,” J. Opto-electron. Eng. 26, 9-15 (1999).

Y. Li, Advanced Optics Manufacturing Technology (Science Press, Beijing, 2001).

Linkenns, D. A.

M. Chen and D. A. Linkenns, “Rule-base self-generation and simplification for data-driven fuzzy models,” J. Fuzzy Sets Syst. 142, 243-265 (2004).
[CrossRef]

Linkens, D. A.

M. Chen and D. A. Linkens, “A fast fuzzy modelling approach using clustering neural networks,” in World Congress on Computational Intelligence (IEEE, 1998), pp 1088-1093.

Shan, B.

B. Shan, S. Wang, H. Niu, and S. Liu, “Zernike polynomial fitting method and its application,” J. Opt. Precis. Eng. 10, 318-322 (2002).

Wan, Y.

B. Fan, Y. Li, Z. Zeng, X. Li, and Y. Wan, “Method for representing the surface and testing technology of stressed-lap in CMAC controller,” J. Opt. Technique 31, 751-754(2005).

Wang, D.

X. Cui. B. Gao, D. Wang, Y. Zhu, S. Yang, Z. Zhu, L. Wang, and Z. Xia, “A new polishing technology for large diameter and deep aspherical mirror,” J. Acta Optica Sin. 25, 402-407(2005).

Wang, L.

X. Cui. B. Gao, D. Wang, Y. Zhu, S. Yang, Z. Zhu, L. Wang, and Z. Xia, “A new polishing technology for large diameter and deep aspherical mirror,” J. Acta Optica Sin. 25, 402-407(2005).

Xia, Z.

X. Cui. B. Gao, D. Wang, Y. Zhu, S. Yang, Z. Zhu, L. Wang, and Z. Xia, “A new polishing technology for large diameter and deep aspherical mirror,” J. Acta Optica Sin. 25, 402-407(2005).

Xu, L.

L. Xu, A. Kryzak, and A. Yuille, “On radial basis function nets and kernel regression: statistical consistency, convergence rates, and receptive field size,” J. Neural Net. 7, 609-628(1994).
[CrossRef]

Yang, S.

X. Cui. B. Gao, D. Wang, Y. Zhu, S. Yang, Z. Zhu, L. Wang, and Z. Xia, “A new polishing technology for large diameter and deep aspherical mirror,” J. Acta Optica Sin. 25, 402-407(2005).

Zeng, Z.

B. Fan, Y. Li, Z. Zeng, X. Li, and Y. Wan, “Method for representing the surface and testing technology of stressed-lap in CMAC controller,” J. Opt. Technique 31, 751-754(2005).

Zheng, Z.

Z. Zheng, B. Gao, X. Li, and M. Liu, “Optical technology and testing method using stressed lap to polish asphere, ” J. Opt. Technique 31, 341-343 (2005).

Zhu, Y.

X. Cui. B. Gao, D. Wang, Y. Zhu, S. Yang, Z. Zhu, L. Wang, and Z. Xia, “A new polishing technology for large diameter and deep aspherical mirror,” J. Acta Optica Sin. 25, 402-407(2005).

Zhu, Z.

X. Cui. B. Gao, D. Wang, Y. Zhu, S. Yang, Z. Zhu, L. Wang, and Z. Xia, “A new polishing technology for large diameter and deep aspherical mirror,” J. Acta Optica Sin. 25, 402-407(2005).

J. Neural Net.

L. Xu, A. Kryzak, and A. Yuille, “On radial basis function nets and kernel regression: statistical consistency, convergence rates, and receptive field size,” J. Neural Net. 7, 609-628(1994).
[CrossRef]

IEEE Trans. Syst. Man Cybern. B

M. Chen and D. A. Linkens, “A systematic neuro-fuzzy modeling framework with application to material property prediction,” IEEE Trans. Syst. Man Cybern. B 31, 781-790 (2001).
[CrossRef]

J. Acta Optica Sin.

X. Cui. B. Gao, D. Wang, Y. Zhu, S. Yang, Z. Zhu, L. Wang, and Z. Xia, “A new polishing technology for large diameter and deep aspherical mirror,” J. Acta Optica Sin. 25, 402-407(2005).

J. Fuzzy Sets Syst.

M. Chen and D. A. Linkenns, “Rule-base self-generation and simplification for data-driven fuzzy models,” J. Fuzzy Sets Syst. 142, 243-265 (2004).
[CrossRef]

J. Opt. Precis. Eng.

B. Shan, S. Wang, H. Niu, and S. Liu, “Zernike polynomial fitting method and its application,” J. Opt. Precis. Eng. 10, 318-322 (2002).

J. Opt. Tech.

Y. Li, Y. Zheng, W. Jiang, J. Yuan, Z. Zeng, and F. Wu, “The flexible optical manufacturing technologies for large aspheric mirror,” J. Opt. Tech. 27, 490-492 (2001).

J. Opt. Technique

D. Wang, Y. Li, and S. Yang, “Study on control technology of active stressed lap polishing aspherical mirror,” J. Opt. Technique 31, 373-379 (2005).

Z. Zheng, B. Gao, X. Li, and M. Liu, “Optical technology and testing method using stressed lap to polish asphere, ” J. Opt. Technique 31, 341-343 (2005).

B. Fan, Y. Li, Z. Zeng, X. Li, and Y. Wan, “Method for representing the surface and testing technology of stressed-lap in CMAC controller,” J. Opt. Technique 31, 751-754(2005).

J. Opto-electron. Eng.

Y. Li and W. Jiang, “A study on computer stressed lap polishing technology for large aperture and highly aspherical optical surface,” J. Opto-electron. Eng. 26, 9-15 (1999).

Other

Y. Li, Advanced Optics Manufacturing Technology (Science Press, Beijing, 2001).

M. Chen and D. A. Linkens, “A fast fuzzy modelling approach using clustering neural networks,” in World Congress on Computational Intelligence (IEEE, 1998), pp 1088-1093.

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Figures (11)

Fig. 1
Fig. 1

Basic structure of the stressed lap surface control system.

Fig. 2
Fig. 2

Top view of the actuators positioned on the stressed lap.

Fig. 3
Fig. 3

Computer-controlled activated stressed lap.

Fig. 4
Fig. 4

Displacement sensor array for measuring the shape changes of the lap surface.

Fig. 5
Fig. 5

Distribution of the stressed lap surface displacement detecting sensors.

Fig. 6
Fig. 6

Structure of the RBF network model.

Fig. 7
Fig. 7

Surface shape of the stressed lap with a one unit force produced by actuator 2.

Fig. 8
Fig. 8

Surface shape of the stressed lap with a one unit force produced by actuators 2 and 6.

Fig. 9
Fig. 9

Residual error distribution.

Fig. 10
Fig. 10

Surface shape of the stressed lap with a one unit force produced by actuators 1, 5, and 9.

Fig. 11
Fig. 11

Surface shape of the stressed lap with a one unit force produced by all 12 actuators.

Equations (9)

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

D = f ( F 1 , F 2 , , F m ) ,
D j = i = 1 S w i j h i ( F 1 , F 2 , , F m ) ,
h i = exp ( F c i 2 / σ i 2 ) / M i ,
M i = i = 1 S exp ( F c i 2 / σ i 2 ) ,
σ i = ( j = 1 m c i c j / m ) 1 2 .
E = 1 2 N j = 1 n ( D j d j ) 2 ,
Δ w i j ( t ) = β h i e j + γ Δ w i j ( t 1 ) ,
Δ c i k ( t ) = β h i j = 1 n e j ( F k c i k ) σ i k 2 ( w i j D j ) + γ Δ c i k ( t 1 ) ,
Δ σ i k ( t ) = β h i j = 1 n e j ( F k c i k ) σ i k 2 ( w i j D j ) + γ Δ σ i k ( t 1 ) ,

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