Abstract

White-light scanning interferometry is widely used for precision metrology of engineering surfaces. It needs a mechanical scanning for capturing an interferogram that determines where the surface of a measured sample is located. The residual vibration during the scanning procedure distorts the interferogram and it reduces the accuracy and the precision of the system. The residual vibration becomes bigger as the proportional gain gets higher for the fast response. So it is hard to achieve the fast and precise measurement simultaneously. In this study, input shaping which convolves a reference signal with the input shaper is investigated to reduce the residual vibration of the scanning system. The step response data is analyzed using Continuous Wavelet Transform (CWT) to design the input shaper. Using proposed method, the residual vibration of the white light scanning interferometry is reduced and it achieved both faster measurement speed and more accurate measurement.

© 2015 Optical Society of America

Full Article  |  PDF Article
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References

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  1. L. Deck and P. de Groot, “High-speed noncontact profiler based on scanning white-light interferometry,” Appl. Opt. 33(31), 7334–7338 (1994).
    [Crossref] [PubMed]
  2. T. Jo, K. Kim, S. Kim, and H. Pahk, “Thickness and Surface Measurement of Transparent Thin-Film Layers using White Light Scanning Interferometry Combined with Reflectometry,” Journal of the Optical Society of Korea 18(3), 236–243 (2014).
    [Crossref]
  3. T. Jo, S. Kim, and H. Pahk, “3D Measurement of TSVs Using Low Numerical Aperture White-Light Scanning Interferometry,” Journal of the Optical Society of Korea 17(4), 317–322 (2013).
    [Crossref]
  4. J. Kim, T. Jo, N. Kim, and D. Kwon, “Measurement of 3D Printed Structure Using a Peak Detection Method in Dispersive Interferometry,” J. Electron. Mater. 2014, 1–5 (2014).
  5. J. Schmit and A. G. Olszak, “Challenges in white-light phase-shifting interferometry,” in International Symposium on Optical Science and Technology(International Society for Optics and Photonics, 2002), pp. 118–127.
  6. L. L. Deck, “Suppressing vibration errors in phase-shifting interferometry,” in Optical Engineering + Applications(International Society for Optics and Photonics, 2007), pp. 670402–670402–670407.
  7. P. J. Groot, “Vibration in phase-shifting interferometry,” JOSA A 12(2), 354–365 (1995).
    [Crossref]
  8. W. Singhose, “Command shaping for flexible systems: A review of the first 50 years,” International Journal of Precision Engineering and Manufacturing 10(4), 153–168 (2009).
    [Crossref]
  9. J. R. Huey, “The intelligent combination of input shaping and PID feedback control,” (2006).
  10. N. C. Singer and W. P. Seering, “Preshaping command inputs to reduce system vibration,” J. Dyn. Syst. Meas. Control 112(1), 76–82 (1990).
    [Crossref]
  11. W. Singhose, W. Seering, and N. Singer, “Residual vibration reduction using vector diagrams to generate shaped inputs,” J. Mech. Des. 116(2), 654–659 (1994).
    [Crossref]
  12. W. Singhose, W. Seering, and N. C. Singer, “Time-optimal negative input shapers,” J. Dyn. Syst. Meas. Control 119(2), 198–205 (1997).
    [Crossref]
  13. D. Blackburn, W. Singhose, J. Kitchen, V. Patrangenaru, J. Lawrence, T. Kamoi, and A. Taura, “Command shaping for nonlinear crane dynamics,” Journal of Vibration and Control 16(4), 477–501 (2010).
    [Crossref]
  14. M. O. Cole, and T. Wongratanaphisan, “A direct method of adaptive FIR input shaping for motion control with zero residual vibration,” Mechatronics, IEEE/ASME Transactions on 18, 316–327 (2013).
    [Crossref]
  15. K.-S. Ou, K.-S. Chen, T.-S. Yang, and S.-Y. Lee, “Fast positioning and impact minimizing of MEMS devices by suppression of motion-induced vibration by command-shaping method,” Microelectromechanical Systems, Journalism 20, 128–139 (2011).
  16. J. Malzahn, M. Ruderman, A. Phung, F. Hoffmann, and T. Bertram, “Input shaping and strain gauge feedback vibration control of an elastic robotic arm,” in Control and Fault-Tolerant Systems (SysTol), 2010 Conference on(IEEE, 2010), pp. 672–677.
    [Crossref]
  17. M. Rakotondrabe, K. Rabenorosoa, J. Agnus, and N. Chaillet, “Robust feedforward-feedback control of a nonlinear and oscillating 2-dof piezocantilever,” Automation Science and Engineering, IEEE Transactions on 8(3), 506–519 (2011).
    [Crossref]
  18. A. Kamel, F. Lange, and G. Hirzinger, “New aspects of input shaping control to damp oscillations of a compliant force sensor,” in Robotics and Automation, 2008. ICRA 2008. IEEE International Conference on(IEEE, 2008), pp. 2629–2635.
    [Crossref]
  19. I. Daubechies, “The wavelet transform, time-frequency localization and signal analysis,” Information Theory, IEEE Transactions on 36(5), 961–1005 (1990).
    [Crossref]
  20. W. Staszewski, “Identification of damping in MDOF systems using time-scale decomposition,” J. Sound Vibrat. 203(2), 283–305 (1997).
    [Crossref]
  21. R. A. Carmona, W. L. Hwang, and B. Torresani, “Characterization of signals by the ridges of their wavelet transforms,” Signal Processing, IEEE Transactions on 45(10), 2586–2590 (1997).
    [Crossref]
  22. W. Singhose, E. Crain, and W. Seering, “Convolved and simultaneous two-mode input shapers,” in Control Theory and Applications, IEEE Proceedings-(IET, 1997), pp. 515–520.
    [Crossref]

2014 (2)

T. Jo, K. Kim, S. Kim, and H. Pahk, “Thickness and Surface Measurement of Transparent Thin-Film Layers using White Light Scanning Interferometry Combined with Reflectometry,” Journal of the Optical Society of Korea 18(3), 236–243 (2014).
[Crossref]

J. Kim, T. Jo, N. Kim, and D. Kwon, “Measurement of 3D Printed Structure Using a Peak Detection Method in Dispersive Interferometry,” J. Electron. Mater. 2014, 1–5 (2014).

2013 (1)

T. Jo, S. Kim, and H. Pahk, “3D Measurement of TSVs Using Low Numerical Aperture White-Light Scanning Interferometry,” Journal of the Optical Society of Korea 17(4), 317–322 (2013).
[Crossref]

2011 (2)

K.-S. Ou, K.-S. Chen, T.-S. Yang, and S.-Y. Lee, “Fast positioning and impact minimizing of MEMS devices by suppression of motion-induced vibration by command-shaping method,” Microelectromechanical Systems, Journalism 20, 128–139 (2011).

M. Rakotondrabe, K. Rabenorosoa, J. Agnus, and N. Chaillet, “Robust feedforward-feedback control of a nonlinear and oscillating 2-dof piezocantilever,” Automation Science and Engineering, IEEE Transactions on 8(3), 506–519 (2011).
[Crossref]

2010 (1)

D. Blackburn, W. Singhose, J. Kitchen, V. Patrangenaru, J. Lawrence, T. Kamoi, and A. Taura, “Command shaping for nonlinear crane dynamics,” Journal of Vibration and Control 16(4), 477–501 (2010).
[Crossref]

2009 (1)

W. Singhose, “Command shaping for flexible systems: A review of the first 50 years,” International Journal of Precision Engineering and Manufacturing 10(4), 153–168 (2009).
[Crossref]

1997 (3)

W. Singhose, W. Seering, and N. C. Singer, “Time-optimal negative input shapers,” J. Dyn. Syst. Meas. Control 119(2), 198–205 (1997).
[Crossref]

W. Staszewski, “Identification of damping in MDOF systems using time-scale decomposition,” J. Sound Vibrat. 203(2), 283–305 (1997).
[Crossref]

R. A. Carmona, W. L. Hwang, and B. Torresani, “Characterization of signals by the ridges of their wavelet transforms,” Signal Processing, IEEE Transactions on 45(10), 2586–2590 (1997).
[Crossref]

1995 (1)

P. J. Groot, “Vibration in phase-shifting interferometry,” JOSA A 12(2), 354–365 (1995).
[Crossref]

1994 (2)

L. Deck and P. de Groot, “High-speed noncontact profiler based on scanning white-light interferometry,” Appl. Opt. 33(31), 7334–7338 (1994).
[Crossref] [PubMed]

W. Singhose, W. Seering, and N. Singer, “Residual vibration reduction using vector diagrams to generate shaped inputs,” J. Mech. Des. 116(2), 654–659 (1994).
[Crossref]

1990 (2)

N. C. Singer and W. P. Seering, “Preshaping command inputs to reduce system vibration,” J. Dyn. Syst. Meas. Control 112(1), 76–82 (1990).
[Crossref]

I. Daubechies, “The wavelet transform, time-frequency localization and signal analysis,” Information Theory, IEEE Transactions on 36(5), 961–1005 (1990).
[Crossref]

Agnus, J.

M. Rakotondrabe, K. Rabenorosoa, J. Agnus, and N. Chaillet, “Robust feedforward-feedback control of a nonlinear and oscillating 2-dof piezocantilever,” Automation Science and Engineering, IEEE Transactions on 8(3), 506–519 (2011).
[Crossref]

Blackburn, D.

D. Blackburn, W. Singhose, J. Kitchen, V. Patrangenaru, J. Lawrence, T. Kamoi, and A. Taura, “Command shaping for nonlinear crane dynamics,” Journal of Vibration and Control 16(4), 477–501 (2010).
[Crossref]

Carmona, R. A.

R. A. Carmona, W. L. Hwang, and B. Torresani, “Characterization of signals by the ridges of their wavelet transforms,” Signal Processing, IEEE Transactions on 45(10), 2586–2590 (1997).
[Crossref]

Chaillet, N.

M. Rakotondrabe, K. Rabenorosoa, J. Agnus, and N. Chaillet, “Robust feedforward-feedback control of a nonlinear and oscillating 2-dof piezocantilever,” Automation Science and Engineering, IEEE Transactions on 8(3), 506–519 (2011).
[Crossref]

Chen, K.-S.

K.-S. Ou, K.-S. Chen, T.-S. Yang, and S.-Y. Lee, “Fast positioning and impact minimizing of MEMS devices by suppression of motion-induced vibration by command-shaping method,” Microelectromechanical Systems, Journalism 20, 128–139 (2011).

Daubechies, I.

I. Daubechies, “The wavelet transform, time-frequency localization and signal analysis,” Information Theory, IEEE Transactions on 36(5), 961–1005 (1990).
[Crossref]

de Groot, P.

Deck, L.

Groot, P. J.

P. J. Groot, “Vibration in phase-shifting interferometry,” JOSA A 12(2), 354–365 (1995).
[Crossref]

Huey, J. R.

J. R. Huey, “The intelligent combination of input shaping and PID feedback control,” (2006).

Hwang, W. L.

R. A. Carmona, W. L. Hwang, and B. Torresani, “Characterization of signals by the ridges of their wavelet transforms,” Signal Processing, IEEE Transactions on 45(10), 2586–2590 (1997).
[Crossref]

Jo, T.

T. Jo, K. Kim, S. Kim, and H. Pahk, “Thickness and Surface Measurement of Transparent Thin-Film Layers using White Light Scanning Interferometry Combined with Reflectometry,” Journal of the Optical Society of Korea 18(3), 236–243 (2014).
[Crossref]

J. Kim, T. Jo, N. Kim, and D. Kwon, “Measurement of 3D Printed Structure Using a Peak Detection Method in Dispersive Interferometry,” J. Electron. Mater. 2014, 1–5 (2014).

T. Jo, S. Kim, and H. Pahk, “3D Measurement of TSVs Using Low Numerical Aperture White-Light Scanning Interferometry,” Journal of the Optical Society of Korea 17(4), 317–322 (2013).
[Crossref]

Kamoi, T.

D. Blackburn, W. Singhose, J. Kitchen, V. Patrangenaru, J. Lawrence, T. Kamoi, and A. Taura, “Command shaping for nonlinear crane dynamics,” Journal of Vibration and Control 16(4), 477–501 (2010).
[Crossref]

Kim, J.

J. Kim, T. Jo, N. Kim, and D. Kwon, “Measurement of 3D Printed Structure Using a Peak Detection Method in Dispersive Interferometry,” J. Electron. Mater. 2014, 1–5 (2014).

Kim, K.

T. Jo, K. Kim, S. Kim, and H. Pahk, “Thickness and Surface Measurement of Transparent Thin-Film Layers using White Light Scanning Interferometry Combined with Reflectometry,” Journal of the Optical Society of Korea 18(3), 236–243 (2014).
[Crossref]

Kim, N.

J. Kim, T. Jo, N. Kim, and D. Kwon, “Measurement of 3D Printed Structure Using a Peak Detection Method in Dispersive Interferometry,” J. Electron. Mater. 2014, 1–5 (2014).

Kim, S.

T. Jo, K. Kim, S. Kim, and H. Pahk, “Thickness and Surface Measurement of Transparent Thin-Film Layers using White Light Scanning Interferometry Combined with Reflectometry,” Journal of the Optical Society of Korea 18(3), 236–243 (2014).
[Crossref]

T. Jo, S. Kim, and H. Pahk, “3D Measurement of TSVs Using Low Numerical Aperture White-Light Scanning Interferometry,” Journal of the Optical Society of Korea 17(4), 317–322 (2013).
[Crossref]

Kitchen, J.

D. Blackburn, W. Singhose, J. Kitchen, V. Patrangenaru, J. Lawrence, T. Kamoi, and A. Taura, “Command shaping for nonlinear crane dynamics,” Journal of Vibration and Control 16(4), 477–501 (2010).
[Crossref]

Kwon, D.

J. Kim, T. Jo, N. Kim, and D. Kwon, “Measurement of 3D Printed Structure Using a Peak Detection Method in Dispersive Interferometry,” J. Electron. Mater. 2014, 1–5 (2014).

Lawrence, J.

D. Blackburn, W. Singhose, J. Kitchen, V. Patrangenaru, J. Lawrence, T. Kamoi, and A. Taura, “Command shaping for nonlinear crane dynamics,” Journal of Vibration and Control 16(4), 477–501 (2010).
[Crossref]

Lee, S.-Y.

K.-S. Ou, K.-S. Chen, T.-S. Yang, and S.-Y. Lee, “Fast positioning and impact minimizing of MEMS devices by suppression of motion-induced vibration by command-shaping method,” Microelectromechanical Systems, Journalism 20, 128–139 (2011).

Ou, K.-S.

K.-S. Ou, K.-S. Chen, T.-S. Yang, and S.-Y. Lee, “Fast positioning and impact minimizing of MEMS devices by suppression of motion-induced vibration by command-shaping method,” Microelectromechanical Systems, Journalism 20, 128–139 (2011).

Pahk, H.

T. Jo, K. Kim, S. Kim, and H. Pahk, “Thickness and Surface Measurement of Transparent Thin-Film Layers using White Light Scanning Interferometry Combined with Reflectometry,” Journal of the Optical Society of Korea 18(3), 236–243 (2014).
[Crossref]

T. Jo, S. Kim, and H. Pahk, “3D Measurement of TSVs Using Low Numerical Aperture White-Light Scanning Interferometry,” Journal of the Optical Society of Korea 17(4), 317–322 (2013).
[Crossref]

Patrangenaru, V.

D. Blackburn, W. Singhose, J. Kitchen, V. Patrangenaru, J. Lawrence, T. Kamoi, and A. Taura, “Command shaping for nonlinear crane dynamics,” Journal of Vibration and Control 16(4), 477–501 (2010).
[Crossref]

Rabenorosoa, K.

M. Rakotondrabe, K. Rabenorosoa, J. Agnus, and N. Chaillet, “Robust feedforward-feedback control of a nonlinear and oscillating 2-dof piezocantilever,” Automation Science and Engineering, IEEE Transactions on 8(3), 506–519 (2011).
[Crossref]

Rakotondrabe, M.

M. Rakotondrabe, K. Rabenorosoa, J. Agnus, and N. Chaillet, “Robust feedforward-feedback control of a nonlinear and oscillating 2-dof piezocantilever,” Automation Science and Engineering, IEEE Transactions on 8(3), 506–519 (2011).
[Crossref]

Seering, W.

W. Singhose, W. Seering, and N. C. Singer, “Time-optimal negative input shapers,” J. Dyn. Syst. Meas. Control 119(2), 198–205 (1997).
[Crossref]

W. Singhose, W. Seering, and N. Singer, “Residual vibration reduction using vector diagrams to generate shaped inputs,” J. Mech. Des. 116(2), 654–659 (1994).
[Crossref]

Seering, W. P.

N. C. Singer and W. P. Seering, “Preshaping command inputs to reduce system vibration,” J. Dyn. Syst. Meas. Control 112(1), 76–82 (1990).
[Crossref]

Singer, N.

W. Singhose, W. Seering, and N. Singer, “Residual vibration reduction using vector diagrams to generate shaped inputs,” J. Mech. Des. 116(2), 654–659 (1994).
[Crossref]

Singer, N. C.

W. Singhose, W. Seering, and N. C. Singer, “Time-optimal negative input shapers,” J. Dyn. Syst. Meas. Control 119(2), 198–205 (1997).
[Crossref]

N. C. Singer and W. P. Seering, “Preshaping command inputs to reduce system vibration,” J. Dyn. Syst. Meas. Control 112(1), 76–82 (1990).
[Crossref]

Singhose, W.

D. Blackburn, W. Singhose, J. Kitchen, V. Patrangenaru, J. Lawrence, T. Kamoi, and A. Taura, “Command shaping for nonlinear crane dynamics,” Journal of Vibration and Control 16(4), 477–501 (2010).
[Crossref]

W. Singhose, “Command shaping for flexible systems: A review of the first 50 years,” International Journal of Precision Engineering and Manufacturing 10(4), 153–168 (2009).
[Crossref]

W. Singhose, W. Seering, and N. C. Singer, “Time-optimal negative input shapers,” J. Dyn. Syst. Meas. Control 119(2), 198–205 (1997).
[Crossref]

W. Singhose, W. Seering, and N. Singer, “Residual vibration reduction using vector diagrams to generate shaped inputs,” J. Mech. Des. 116(2), 654–659 (1994).
[Crossref]

Staszewski, W.

W. Staszewski, “Identification of damping in MDOF systems using time-scale decomposition,” J. Sound Vibrat. 203(2), 283–305 (1997).
[Crossref]

Taura, A.

D. Blackburn, W. Singhose, J. Kitchen, V. Patrangenaru, J. Lawrence, T. Kamoi, and A. Taura, “Command shaping for nonlinear crane dynamics,” Journal of Vibration and Control 16(4), 477–501 (2010).
[Crossref]

Torresani, B.

R. A. Carmona, W. L. Hwang, and B. Torresani, “Characterization of signals by the ridges of their wavelet transforms,” Signal Processing, IEEE Transactions on 45(10), 2586–2590 (1997).
[Crossref]

Yang, T.-S.

K.-S. Ou, K.-S. Chen, T.-S. Yang, and S.-Y. Lee, “Fast positioning and impact minimizing of MEMS devices by suppression of motion-induced vibration by command-shaping method,” Microelectromechanical Systems, Journalism 20, 128–139 (2011).

Appl. Opt. (1)

Automation Science and Engineering, IEEE Transactions on (1)

M. Rakotondrabe, K. Rabenorosoa, J. Agnus, and N. Chaillet, “Robust feedforward-feedback control of a nonlinear and oscillating 2-dof piezocantilever,” Automation Science and Engineering, IEEE Transactions on 8(3), 506–519 (2011).
[Crossref]

Information Theory, IEEE Transactions on (1)

I. Daubechies, “The wavelet transform, time-frequency localization and signal analysis,” Information Theory, IEEE Transactions on 36(5), 961–1005 (1990).
[Crossref]

International Journal of Precision Engineering and Manufacturing (1)

W. Singhose, “Command shaping for flexible systems: A review of the first 50 years,” International Journal of Precision Engineering and Manufacturing 10(4), 153–168 (2009).
[Crossref]

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

N. C. Singer and W. P. Seering, “Preshaping command inputs to reduce system vibration,” J. Dyn. Syst. Meas. Control 112(1), 76–82 (1990).
[Crossref]

W. Singhose, W. Seering, and N. C. Singer, “Time-optimal negative input shapers,” J. Dyn. Syst. Meas. Control 119(2), 198–205 (1997).
[Crossref]

J. Electron. Mater. (1)

J. Kim, T. Jo, N. Kim, and D. Kwon, “Measurement of 3D Printed Structure Using a Peak Detection Method in Dispersive Interferometry,” J. Electron. Mater. 2014, 1–5 (2014).

J. Mech. Des. (1)

W. Singhose, W. Seering, and N. Singer, “Residual vibration reduction using vector diagrams to generate shaped inputs,” J. Mech. Des. 116(2), 654–659 (1994).
[Crossref]

J. Sound Vibrat. (1)

W. Staszewski, “Identification of damping in MDOF systems using time-scale decomposition,” J. Sound Vibrat. 203(2), 283–305 (1997).
[Crossref]

JOSA A (1)

P. J. Groot, “Vibration in phase-shifting interferometry,” JOSA A 12(2), 354–365 (1995).
[Crossref]

Journal of the Optical Society of Korea (2)

T. Jo, K. Kim, S. Kim, and H. Pahk, “Thickness and Surface Measurement of Transparent Thin-Film Layers using White Light Scanning Interferometry Combined with Reflectometry,” Journal of the Optical Society of Korea 18(3), 236–243 (2014).
[Crossref]

T. Jo, S. Kim, and H. Pahk, “3D Measurement of TSVs Using Low Numerical Aperture White-Light Scanning Interferometry,” Journal of the Optical Society of Korea 17(4), 317–322 (2013).
[Crossref]

Journal of Vibration and Control (1)

D. Blackburn, W. Singhose, J. Kitchen, V. Patrangenaru, J. Lawrence, T. Kamoi, and A. Taura, “Command shaping for nonlinear crane dynamics,” Journal of Vibration and Control 16(4), 477–501 (2010).
[Crossref]

Microelectromechanical Systems, Journalism (1)

K.-S. Ou, K.-S. Chen, T.-S. Yang, and S.-Y. Lee, “Fast positioning and impact minimizing of MEMS devices by suppression of motion-induced vibration by command-shaping method,” Microelectromechanical Systems, Journalism 20, 128–139 (2011).

Signal Processing, IEEE Transactions on (1)

R. A. Carmona, W. L. Hwang, and B. Torresani, “Characterization of signals by the ridges of their wavelet transforms,” Signal Processing, IEEE Transactions on 45(10), 2586–2590 (1997).
[Crossref]

Other (7)

W. Singhose, E. Crain, and W. Seering, “Convolved and simultaneous two-mode input shapers,” in Control Theory and Applications, IEEE Proceedings-(IET, 1997), pp. 515–520.
[Crossref]

J. Malzahn, M. Ruderman, A. Phung, F. Hoffmann, and T. Bertram, “Input shaping and strain gauge feedback vibration control of an elastic robotic arm,” in Control and Fault-Tolerant Systems (SysTol), 2010 Conference on(IEEE, 2010), pp. 672–677.
[Crossref]

M. O. Cole, and T. Wongratanaphisan, “A direct method of adaptive FIR input shaping for motion control with zero residual vibration,” Mechatronics, IEEE/ASME Transactions on 18, 316–327 (2013).
[Crossref]

J. Schmit and A. G. Olszak, “Challenges in white-light phase-shifting interferometry,” in International Symposium on Optical Science and Technology(International Society for Optics and Photonics, 2002), pp. 118–127.

L. L. Deck, “Suppressing vibration errors in phase-shifting interferometry,” in Optical Engineering + Applications(International Society for Optics and Photonics, 2007), pp. 670402–670402–670407.

J. R. Huey, “The intelligent combination of input shaping and PID feedback control,” (2006).

A. Kamel, F. Lange, and G. Hirzinger, “New aspects of input shaping control to damp oscillations of a compliant force sensor,” in Robotics and Automation, 2008. ICRA 2008. IEEE International Conference on(IEEE, 2008), pp. 2629–2635.
[Crossref]

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Figures (8)

Fig. 1
Fig. 1 Experimental Setup.
Fig. 2
Fig. 2 Step responses (a) Low gains (b) High gains.
Fig. 3
Fig. 3 (a) 15um step response (b) CWT result of step response.
Fig. 4
Fig. 4 Extraction of damping ratios from modulus of CWT.
Fig. 5
Fig. 5 Step response before and after input shaping (a) time domain (b) frequency domain.
Fig. 6
Fig. 6 Scanning command.
Fig. 7
Fig. 7 Measured image of tilted sample (a) before input shaping (b) after input shaping.
Fig. 8
Fig. 8 Profile comparison along the horizontal direction at the center.

Tables (1)

Tables Icon

Table 1 Modal parameters and parameters of input shaper

Equations (6)

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

U c (s)=( k p + k i s )E(s)
 T(b,a)= 1 a + + u(t) ψ( tb a )dt
ψ= e t 2 /2 α 2 e iβt
 f(t)= i=1 n A i δ(t t i )
 [ t 1 t 2 A 1 A 2 ]=[ 0 π w d γ 1+γ 1 1+γ ]   ,     w d = w n 1 ζ 2 ,     γ= e ζπ 1 ζ 2
  f total (t)= f mode1 (t) f mode2 (t) f mod e n (t)

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