Abstract

A multi-beam bilateral teleoperation system of holographic optical tweezers accelerated by a graphics processing unit is proposed and evaluated. This double-arm teleoperation system is composed of two haptic devices and two laser-trapped micro-beads. Each micro-bead is trapped and moved following the trajectory of each haptic device, and the forces to which the micro-beads are subjected, which are generated by Stokes drag, are measured and fed back to an operator via the haptic devices. This real-time telexistence was quantitatively evaluated based on the time response of the trapped beads and the fed-back forces. And the demonstration of touching red blood cells shows the effectiveness of this system for biomedical application.

© 2012 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11(5), 288–290 (1986).
    [CrossRef] [PubMed]
  2. D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
    [CrossRef] [PubMed]
  3. A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330(6150), 769–771 (1987).
    [CrossRef] [PubMed]
  4. K. Svoboda, C. F. Schmidt, D. Branton, and S. M. Block, “Conformation and elasticity of the isolated red blood cell membrane skeleton,” Biophys. J. 63(3), 784–793 (1992).
    [CrossRef] [PubMed]
  5. P. J. H. Bronkhorst, G. J. Streekstra, J. Grimbergen, E. J. Nijhof, J. J. Sixma, and G. J. Brakenhoff, “A new method to study shape recovery of red blood cells using multiple optical trapping,” Biophys. J. 69(5), 1666–1673 (1995).
    [CrossRef] [PubMed]
  6. K. Sasaki, M. Koshioka, H. Misawa, N. Kitamura, and H. Masuhara, “Pattern formation and flow control of fine particles by laser-scanning micromanipulation,” Opt. Lett. 16(19), 1463–1465 (1991).
    [CrossRef] [PubMed]
  7. R. L. Eriksen, P. C. Mogensen, and J. Glückstad, “Multiple-beam optical tweezers generated by the generalized phase-contrast method,” Opt. Lett. 27(4), 267–269 (2002).
    [CrossRef] [PubMed]
  8. D. G. Grier, J. E. Curtis, and B. A. Koss, “Dynamic holographic optical tweezers,” Opt. Commun. 207(1-6), 169–175 (2002).
    [CrossRef]
  9. N. Masuda, T. Ito, T. Tanaka, A. Shiraki, and T. Sugie, “Computer generated holography using a graphics processing unit,” Opt. Express 14(2), 603–608 (2006).
    [CrossRef] [PubMed]
  10. L. Ahrenberg, P. Benzie, M. Magnor, and J. Watson, “Computer generated holography using parallel commodity graphics hardware,” Opt. Express 14(17), 7636–7641 (2006).
    [CrossRef] [PubMed]
  11. M. Reicherter, S. Zwick, T. Haist, C. Kohler, H. Tiziani, and W. Osten, “Fast digital hologram generation and adaptive force measurement in liquid-crystal-display-based holographic tweezers,” Appl. Opt. 45(5), 888–896 (2006).
    [CrossRef] [PubMed]
  12. J. Leach, K. Wulff, G. Sinclair, P. Jordan, J. Courtial, L. Thomson, G. Gibson, K. Karunwi, J. Cooper, Z. J. Laczik, and M. Padgett, “Interactive approach to optical tweezers control,” Appl. Opt. 45(5), 897–903 (2006).
    [CrossRef] [PubMed]
  13. J. A. Grieve, A. Ulcinas, S. Subramanian, G. M. Gibson, M. J. Padgett, D. M. Carberry, and M. J. Miles, “Hands-on with optical tweezers: a multitouch interface for holographic optical trapping,” Opt. Express 17(5), 3595–3602 (2009).
    [CrossRef] [PubMed]
  14. R. W. Bowman, G. Gibson, D. Carberry, L. Picco, M. Miles, and M. J. Padgett, “iTweezers: optical micromanipulation controlled by an Apple iPad,” J. Opt. 13(4), 044002–044004 (2011).
    [CrossRef]
  15. K. Onda and F. Arai, “Robotic approach to multi-beam optical tweezers with computer generated hologram,” in Robotics and Automation (ICRA), 2011 IEEE International Conference (2011), pp. 1825–1830.
  16. F. Arai, M. Ogawa, and T. Fukuda, “Indirect manipulation and bilateral control of the microbe by the laser manipulated microtools,” in Intelligent Robots and Systems, 2000. (IROS 2000). Proceedings. 2000 IEEE/RSJ International Conference (2000), pp. 665–670.
  17. C. Pacoret, R. Bowman, G. Gibson, S. Haliyo, D. Carberry, A. Bergander, S. Régnier, and M. Padgett, “Touching the microworld with force-feedback optical tweezers,” Opt. Express 17(12), 10259–10264 (2009).
    [CrossRef] [PubMed]
  18. R. J. Anderson and M. W. Spong, “Bilateral control of teleoperators with time-delay,” IEEE Trans. Automat. Contr. 34(5), 494–501 (1989).
    [CrossRef]
  19. N. Mukohzaka, N. Yoshida, H. Toyoda, Y. Kobayashi, and T. Hara, “Diffraction Efficiency Analysis of a Parallel-Aligned Nematic-Liquid-Crystal Spatial Light Modulator,” Appl. Opt. 33(14), 2804–2811 (1994).
    [CrossRef] [PubMed]
  20. M. Polin, K. Ladavac, S. H. Lee, Y. Roichman, and D. G. Grier, “Optimized holographic optical traps,” Opt. Express 13(15), 5831–5845 (2005).
    [CrossRef] [PubMed]
  21. J. A. Davis, D. M. Cottrell, R. A. Lilly, and S. W. Connely, “Multiplexed Phase-Encoded Lenses Written on Spatial Light Modulators,” Opt. Lett. 14(9), 420–422 (1989).
    [CrossRef] [PubMed]
  22. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 237–246 (1972).
  23. A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61(2), 569–582 (1992).
    [CrossRef] [PubMed]
  24. K. Visscher, S. P. Gross, and S. M. Block, “Construction of multiple-beam optical traps with nanometer-resolution position sensing,” IEEE J. Sel. Top. Quantum Electron. 2(4), 1066–1076 (1996).
    [CrossRef]
  25. J. E. Colgate and J. M. Brown, “Factors affecting the Z-Width of a haptic display,” in Robotics and Automation, 1994. Proceedings., 1994 IEEE International Conference (1994), pp. 3205–3210.
  26. O. Bryngdahl, “Geometrical transformations in optics,” J. Opt. Soc. Am. 64(8), 1092–1099 (1974).
    [CrossRef]

2011

R. W. Bowman, G. Gibson, D. Carberry, L. Picco, M. Miles, and M. J. Padgett, “iTweezers: optical micromanipulation controlled by an Apple iPad,” J. Opt. 13(4), 044002–044004 (2011).
[CrossRef]

2009

2006

2005

2003

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[CrossRef] [PubMed]

2002

1996

K. Visscher, S. P. Gross, and S. M. Block, “Construction of multiple-beam optical traps with nanometer-resolution position sensing,” IEEE J. Sel. Top. Quantum Electron. 2(4), 1066–1076 (1996).
[CrossRef]

1995

P. J. H. Bronkhorst, G. J. Streekstra, J. Grimbergen, E. J. Nijhof, J. J. Sixma, and G. J. Brakenhoff, “A new method to study shape recovery of red blood cells using multiple optical trapping,” Biophys. J. 69(5), 1666–1673 (1995).
[CrossRef] [PubMed]

1994

1992

K. Svoboda, C. F. Schmidt, D. Branton, and S. M. Block, “Conformation and elasticity of the isolated red blood cell membrane skeleton,” Biophys. J. 63(3), 784–793 (1992).
[CrossRef] [PubMed]

A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61(2), 569–582 (1992).
[CrossRef] [PubMed]

1991

1989

R. J. Anderson and M. W. Spong, “Bilateral control of teleoperators with time-delay,” IEEE Trans. Automat. Contr. 34(5), 494–501 (1989).
[CrossRef]

J. A. Davis, D. M. Cottrell, R. A. Lilly, and S. W. Connely, “Multiplexed Phase-Encoded Lenses Written on Spatial Light Modulators,” Opt. Lett. 14(9), 420–422 (1989).
[CrossRef] [PubMed]

1987

A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330(6150), 769–771 (1987).
[CrossRef] [PubMed]

1986

1974

1972

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 237–246 (1972).

Ahrenberg, L.

Anderson, R. J.

R. J. Anderson and M. W. Spong, “Bilateral control of teleoperators with time-delay,” IEEE Trans. Automat. Contr. 34(5), 494–501 (1989).
[CrossRef]

Ashkin, A.

A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61(2), 569–582 (1992).
[CrossRef] [PubMed]

A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330(6150), 769–771 (1987).
[CrossRef] [PubMed]

A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11(5), 288–290 (1986).
[CrossRef] [PubMed]

Benzie, P.

Bergander, A.

Bjorkholm, J. E.

Block, S. M.

K. Visscher, S. P. Gross, and S. M. Block, “Construction of multiple-beam optical traps with nanometer-resolution position sensing,” IEEE J. Sel. Top. Quantum Electron. 2(4), 1066–1076 (1996).
[CrossRef]

K. Svoboda, C. F. Schmidt, D. Branton, and S. M. Block, “Conformation and elasticity of the isolated red blood cell membrane skeleton,” Biophys. J. 63(3), 784–793 (1992).
[CrossRef] [PubMed]

Bowman, R.

Bowman, R. W.

R. W. Bowman, G. Gibson, D. Carberry, L. Picco, M. Miles, and M. J. Padgett, “iTweezers: optical micromanipulation controlled by an Apple iPad,” J. Opt. 13(4), 044002–044004 (2011).
[CrossRef]

Brakenhoff, G. J.

P. J. H. Bronkhorst, G. J. Streekstra, J. Grimbergen, E. J. Nijhof, J. J. Sixma, and G. J. Brakenhoff, “A new method to study shape recovery of red blood cells using multiple optical trapping,” Biophys. J. 69(5), 1666–1673 (1995).
[CrossRef] [PubMed]

Branton, D.

K. Svoboda, C. F. Schmidt, D. Branton, and S. M. Block, “Conformation and elasticity of the isolated red blood cell membrane skeleton,” Biophys. J. 63(3), 784–793 (1992).
[CrossRef] [PubMed]

Bronkhorst, P. J. H.

P. J. H. Bronkhorst, G. J. Streekstra, J. Grimbergen, E. J. Nijhof, J. J. Sixma, and G. J. Brakenhoff, “A new method to study shape recovery of red blood cells using multiple optical trapping,” Biophys. J. 69(5), 1666–1673 (1995).
[CrossRef] [PubMed]

Bryngdahl, O.

Carberry, D.

R. W. Bowman, G. Gibson, D. Carberry, L. Picco, M. Miles, and M. J. Padgett, “iTweezers: optical micromanipulation controlled by an Apple iPad,” J. Opt. 13(4), 044002–044004 (2011).
[CrossRef]

C. Pacoret, R. Bowman, G. Gibson, S. Haliyo, D. Carberry, A. Bergander, S. Régnier, and M. Padgett, “Touching the microworld with force-feedback optical tweezers,” Opt. Express 17(12), 10259–10264 (2009).
[CrossRef] [PubMed]

Carberry, D. M.

Chu, S.

Connely, S. W.

Cooper, J.

Cottrell, D. M.

Courtial, J.

Curtis, J. E.

D. G. Grier, J. E. Curtis, and B. A. Koss, “Dynamic holographic optical tweezers,” Opt. Commun. 207(1-6), 169–175 (2002).
[CrossRef]

Davis, J. A.

Dziedzic, J. M.

A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330(6150), 769–771 (1987).
[CrossRef] [PubMed]

A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11(5), 288–290 (1986).
[CrossRef] [PubMed]

Eriksen, R. L.

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 237–246 (1972).

Gibson, G.

Gibson, G. M.

Glückstad, J.

Grier, D. G.

M. Polin, K. Ladavac, S. H. Lee, Y. Roichman, and D. G. Grier, “Optimized holographic optical traps,” Opt. Express 13(15), 5831–5845 (2005).
[CrossRef] [PubMed]

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[CrossRef] [PubMed]

D. G. Grier, J. E. Curtis, and B. A. Koss, “Dynamic holographic optical tweezers,” Opt. Commun. 207(1-6), 169–175 (2002).
[CrossRef]

Grieve, J. A.

Grimbergen, J.

P. J. H. Bronkhorst, G. J. Streekstra, J. Grimbergen, E. J. Nijhof, J. J. Sixma, and G. J. Brakenhoff, “A new method to study shape recovery of red blood cells using multiple optical trapping,” Biophys. J. 69(5), 1666–1673 (1995).
[CrossRef] [PubMed]

Gross, S. P.

K. Visscher, S. P. Gross, and S. M. Block, “Construction of multiple-beam optical traps with nanometer-resolution position sensing,” IEEE J. Sel. Top. Quantum Electron. 2(4), 1066–1076 (1996).
[CrossRef]

Haist, T.

Haliyo, S.

Hara, T.

Ito, T.

Jordan, P.

Karunwi, K.

Kitamura, N.

Kobayashi, Y.

Kohler, C.

Koshioka, M.

Koss, B. A.

D. G. Grier, J. E. Curtis, and B. A. Koss, “Dynamic holographic optical tweezers,” Opt. Commun. 207(1-6), 169–175 (2002).
[CrossRef]

Laczik, Z. J.

Ladavac, K.

Leach, J.

Lee, S. H.

Lilly, R. A.

Magnor, M.

Masuda, N.

Masuhara, H.

Miles, M.

R. W. Bowman, G. Gibson, D. Carberry, L. Picco, M. Miles, and M. J. Padgett, “iTweezers: optical micromanipulation controlled by an Apple iPad,” J. Opt. 13(4), 044002–044004 (2011).
[CrossRef]

Miles, M. J.

Misawa, H.

Mogensen, P. C.

Mukohzaka, N.

Nijhof, E. J.

P. J. H. Bronkhorst, G. J. Streekstra, J. Grimbergen, E. J. Nijhof, J. J. Sixma, and G. J. Brakenhoff, “A new method to study shape recovery of red blood cells using multiple optical trapping,” Biophys. J. 69(5), 1666–1673 (1995).
[CrossRef] [PubMed]

Osten, W.

Pacoret, C.

Padgett, M.

Padgett, M. J.

R. W. Bowman, G. Gibson, D. Carberry, L. Picco, M. Miles, and M. J. Padgett, “iTweezers: optical micromanipulation controlled by an Apple iPad,” J. Opt. 13(4), 044002–044004 (2011).
[CrossRef]

J. A. Grieve, A. Ulcinas, S. Subramanian, G. M. Gibson, M. J. Padgett, D. M. Carberry, and M. J. Miles, “Hands-on with optical tweezers: a multitouch interface for holographic optical trapping,” Opt. Express 17(5), 3595–3602 (2009).
[CrossRef] [PubMed]

Picco, L.

R. W. Bowman, G. Gibson, D. Carberry, L. Picco, M. Miles, and M. J. Padgett, “iTweezers: optical micromanipulation controlled by an Apple iPad,” J. Opt. 13(4), 044002–044004 (2011).
[CrossRef]

Polin, M.

Régnier, S.

Reicherter, M.

Roichman, Y.

Sasaki, K.

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 237–246 (1972).

Schmidt, C. F.

K. Svoboda, C. F. Schmidt, D. Branton, and S. M. Block, “Conformation and elasticity of the isolated red blood cell membrane skeleton,” Biophys. J. 63(3), 784–793 (1992).
[CrossRef] [PubMed]

Shiraki, A.

Sinclair, G.

Sixma, J. J.

P. J. H. Bronkhorst, G. J. Streekstra, J. Grimbergen, E. J. Nijhof, J. J. Sixma, and G. J. Brakenhoff, “A new method to study shape recovery of red blood cells using multiple optical trapping,” Biophys. J. 69(5), 1666–1673 (1995).
[CrossRef] [PubMed]

Spong, M. W.

R. J. Anderson and M. W. Spong, “Bilateral control of teleoperators with time-delay,” IEEE Trans. Automat. Contr. 34(5), 494–501 (1989).
[CrossRef]

Streekstra, G. J.

P. J. H. Bronkhorst, G. J. Streekstra, J. Grimbergen, E. J. Nijhof, J. J. Sixma, and G. J. Brakenhoff, “A new method to study shape recovery of red blood cells using multiple optical trapping,” Biophys. J. 69(5), 1666–1673 (1995).
[CrossRef] [PubMed]

Subramanian, S.

Sugie, T.

Svoboda, K.

K. Svoboda, C. F. Schmidt, D. Branton, and S. M. Block, “Conformation and elasticity of the isolated red blood cell membrane skeleton,” Biophys. J. 63(3), 784–793 (1992).
[CrossRef] [PubMed]

Tanaka, T.

Thomson, L.

Tiziani, H.

Toyoda, H.

Ulcinas, A.

Visscher, K.

K. Visscher, S. P. Gross, and S. M. Block, “Construction of multiple-beam optical traps with nanometer-resolution position sensing,” IEEE J. Sel. Top. Quantum Electron. 2(4), 1066–1076 (1996).
[CrossRef]

Watson, J.

Wulff, K.

Yamane, T.

A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330(6150), 769–771 (1987).
[CrossRef] [PubMed]

Yoshida, N.

Zwick, S.

Appl. Opt.

Biophys. J.

K. Svoboda, C. F. Schmidt, D. Branton, and S. M. Block, “Conformation and elasticity of the isolated red blood cell membrane skeleton,” Biophys. J. 63(3), 784–793 (1992).
[CrossRef] [PubMed]

P. J. H. Bronkhorst, G. J. Streekstra, J. Grimbergen, E. J. Nijhof, J. J. Sixma, and G. J. Brakenhoff, “A new method to study shape recovery of red blood cells using multiple optical trapping,” Biophys. J. 69(5), 1666–1673 (1995).
[CrossRef] [PubMed]

A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61(2), 569–582 (1992).
[CrossRef] [PubMed]

IEEE J. Sel. Top. Quantum Electron.

K. Visscher, S. P. Gross, and S. M. Block, “Construction of multiple-beam optical traps with nanometer-resolution position sensing,” IEEE J. Sel. Top. Quantum Electron. 2(4), 1066–1076 (1996).
[CrossRef]

IEEE Trans. Automat. Contr.

R. J. Anderson and M. W. Spong, “Bilateral control of teleoperators with time-delay,” IEEE Trans. Automat. Contr. 34(5), 494–501 (1989).
[CrossRef]

J. Opt.

R. W. Bowman, G. Gibson, D. Carberry, L. Picco, M. Miles, and M. J. Padgett, “iTweezers: optical micromanipulation controlled by an Apple iPad,” J. Opt. 13(4), 044002–044004 (2011).
[CrossRef]

J. Opt. Soc. Am.

Nature

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[CrossRef] [PubMed]

A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330(6150), 769–771 (1987).
[CrossRef] [PubMed]

Opt. Commun.

D. G. Grier, J. E. Curtis, and B. A. Koss, “Dynamic holographic optical tweezers,” Opt. Commun. 207(1-6), 169–175 (2002).
[CrossRef]

Opt. Express

Opt. Lett.

Optik (Stuttg.)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 237–246 (1972).

Other

K. Onda and F. Arai, “Robotic approach to multi-beam optical tweezers with computer generated hologram,” in Robotics and Automation (ICRA), 2011 IEEE International Conference (2011), pp. 1825–1830.

F. Arai, M. Ogawa, and T. Fukuda, “Indirect manipulation and bilateral control of the microbe by the laser manipulated microtools,” in Intelligent Robots and Systems, 2000. (IROS 2000). Proceedings. 2000 IEEE/RSJ International Conference (2000), pp. 665–670.

J. E. Colgate and J. M. Brown, “Factors affecting the Z-Width of a haptic display,” in Robotics and Automation, 1994. Proceedings., 1994 IEEE International Conference (1994), pp. 3205–3210.

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 (9)

Fig. 1
Fig. 1

Schematic diagram of the teleoperation system using holographic optical tweezers. The coordinate systems of the master, slave, camera, and hologram are denoted as Σm, Σs, Σc, and Σh, respectively.

Fig. 2
Fig. 2

Optical system of holographic optical tweezers.

Fig. 3
Fig. 3

The phase distribution of centered hologram is an example of composite hologram by using a Fresnel phase lens shown in left side and a Fourier hologram shown in right side for separating a zero-order beam and a first-order beam. This operation contributes to eliminate the disturbance caused by a zero-order beam.

Fig. 4
Fig. 4

Block diagram of bilateral control system.

Fig. 5
Fig. 5

Schematic diagram of the force measurement by using the image analysis of microscopic camera images.

Fig. 6
Fig. 6

Images of the teleoperation of clockwise rotation. Upper images show the operation of masters and lower show the movement of slaves which moves along the trajectory of masters.

Fig. 7
Fig. 7

Results of bilateral teleoperation of the left-side master-slave. The upper one graph shows the position and the lower two graphs show the force. -X or -Y indicates the position or force of x-axis or y-axis. In the force graphs, right-side y-axis shows estimated Stokes drag calculated from the slave velocity. (a) Position response. (b) Left-side y-axis is slave force Fs and right-side is Stokes drag. (c) Left-side y-axis is modified slave force fs and right-side is Stokes drag.

Fig. 8
Fig. 8

Ramp response of a laser beam and a slave against position instruction of a ramp input in trajectory control. Xcsi was measured by observing a laser beam which was reflected by a mirror set up on the plane of OL.

Fig. 9
Fig. 9

Demonstration of bilateral teleoperation of touching red blood cells. Upper images show slave movements and lower graph shows the magnitude of modified slave forces.

Tables (1)

Tables Icon

Table 1 Experimental condition for the evaluation of bilateral teleoperation

Equations (8)

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

ϕ h =( ϕ h + ϕ lens )mod2π
ϕ lens = π r 2 λf mod2π
F s = K t ( X s X si )
F s = K t K pcs ( X cs X cm )
K s = K t K pcs
X cs = T psc T pcs X cm
T pcs = H trans H rot s pcs
f s (t)= K s ( X cs (t) X cm (t T d ) )

Metrics