Using Fractional Order Elements for Haptic Rendering

  • Ozan TokatliEmail author
  • Volkan Patoglu
Part of the Springer Proceedings in Advanced Robotics book series (SPAR, volume 2)


Fractional order calculus—a generalization of the traditional calculus to arbitrary order differointegration—is an effective mathematical tool that broadens the modeling boundaries of the familiar integer order calculus. Fractional order models enable faithful representation of viscoelastic materials that exhibit frequency dependent stiffness and damping characteristics within a single mechanical element. We propose the use of fractional order models/controllers in haptic systems to significantly extend the type of impedances that can be rendered using the integer order models. We study the effect of fractional order elements on the coupled stability of the overall sampled-data system. We show that fractional calculus generalization provides an additional degree of freedom for adjusting the dissipation behavior of the closed-loop system and generalize the well-known passivity condition to include fractional order impedances. Our results demonstrate the effect of the order of differointegration on the passivity boundary. We also characterize the effective impedance of the fractional order elements as a function of frequency and differointegration order.


  1. 1.
    Adams, R.J., Hannaford, B.: Stable haptic interaction with virtual environments. IEEE Trans. Robot. Autom. 15(3), 465–474 (1999)CrossRefGoogle Scholar
  2. 2.
    Anderson, R., Spong, M.: Bilateral control of teleoperators with time delay. IEEE Trans. Autom. Control 34(5), 494–501 (1989)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bagley, R.L., Torvik, P.J.: Fractional calculus - a different approach to the anaylsis of viscoelastically damped structures. AIAA J. 21, 741–748 (1983)CrossRefzbMATHGoogle Scholar
  4. 4.
    Caponetta, R., Dongola, G., Fortuna, L., Petras, I.: Fractional Order Systems. World Scientific, Singapore (2010)CrossRefGoogle Scholar
  5. 5.
    Carpinteri, A., Mainardi, F.: Fractals and Fractional Calculus in Continuum Mechanics. Springer, Wien (1997)CrossRefzbMATHGoogle Scholar
  6. 6.
    Chen, Y.Q., Moore, K.: Discretization schemes for fractional-order differentiators and integrators. IEEE Trans. Circuits Syst. I: Fundam. Theory Appl. 49(3), 363–367 (2002)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chen, Y.Q., Petras, I., Xue, D.: Fractional order control - a tutorial. In: American Control Conference, pp. 1397–1411 (2009)Google Scholar
  8. 8.
    Colgate, J., Brown, J.: Factors affecting the z-width of a haptic display. In: IEEE International Conference on Robotics and Automation, vol. 4, pp. 3205–3210 (1994)Google Scholar
  9. 9.
    Colgate, J.E., Schenkel, G.G.: Passivity of a class of sampled-data systems: application to haptic interfaces. J. Robot. Syst. 14(1), 37–47 (1997)CrossRefGoogle Scholar
  10. 10.
    Colgate, J., Grafing, P., Stanley, M., Schenkel, G.: Implementation of stiff virtual walls in force-reflecting interfaces. In: Virtual Reality Annual International Symposium, pp. 202–208 (1993)Google Scholar
  11. 11.
    Colgate, J., Stanley, M., Brown, J.: Issues in the haptic display of tool use. In: IEEE/RSJ International Conference on Intelligent Robots and Systems. Human Robot Interaction and Cooperative Robots, vol. 3, pp. 140–145 (1995)Google Scholar
  12. 12.
    Colonnese, N., Sketch, S., Okamura, A.: Closed-loop stiffness and damping accuracy of impedance-type haptic displays. In: IEEE Haptics Symposium (HAPTICS), pp. 97–102 (2014)Google Scholar
  13. 13.
    Craiem, D., Magin, R.L.: Fractional order models of viscoelasticity as an alternative in the analysis of red blood cell (RBC) membrane mechanics. Phys. Biol. 7(1), 13001 (2010)CrossRefGoogle Scholar
  14. 14.
    Das, S., Pan, I.: Fractional Order Signal Processing: Introductory Concepts and Applications. Springer, Heidelberg (2012)CrossRefzbMATHGoogle Scholar
  15. 15.
    Diolaiti, N., Niemeyer, G., Barbagli, F., Salisbury, J.: Stability of haptic rendering: discretization, quantization, time delay, and coulomb effects. IEEE Trans. Robot. 22(2), 256–268 (2006)CrossRefGoogle Scholar
  16. 16.
    Efe, M.: Fractional order systems in industrial automation 2014; a survey. IEEE Trans. Ind. Inf. 7(4), 582–591 (2011)CrossRefGoogle Scholar
  17. 17.
    Ferreira, N.M.F., Machado, J.A.T.: Fractional-order hybrid control of robotic manipulators. In: International Conference on Advanced Robotics (2003)Google Scholar
  18. 18.
    Gil, J., Avello, A., Rubio, A., Florez, J.: Stability analysis of a 1 DOF haptic interface using the Routh–Hurwitz criterion. IEEE Trans. Control Syst. Technol. 12(4), 583–588 (2004)CrossRefGoogle Scholar
  19. 19.
    Gillespie, R.B., Cutkosky, M.R.: Stable user-specific haptic rendering of the virtual wall. In: Proceedings of The International Mechanical Engineering Congress and Exhibition (1995)Google Scholar
  20. 20.
    Haddadi, A., Hashtrudi-Zaad, K.: Bounded-impedance absolute stability of bilateral teleoperation control systems. IEEE Trans. Haptics 3(1), 15–27 (2010)CrossRefGoogle Scholar
  21. 21.
    Hannaford, B., Ryu, J.H.: Time-domain passivity control of haptic interfaces. IEEE Trans. Robot. Autom. 18(1), 1–10 (2002)CrossRefGoogle Scholar
  22. 22.
    Hogan, N.: Controlling impedance at the man/machine interface. In: IEEE International Conference on Robotics and Automation, pp. 1626–1631 (1989)Google Scholar
  23. 23.
    Hulin, T., Preusche, C., Hirzinger, G.: Stability boundary for haptic rendering: influence of physical damping. In: IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 1570–1575 (2006)Google Scholar
  24. 24.
    Kim, J.P., Ryu, J.: Robustly stable haptic interaction control using an energy-bounding algorithm. Int. J. Robot. Res. (2009)Google Scholar
  25. 25.
    Kobayashi, Y., Moreira, P., Liu, C., Poignet, P., Zemiti, N., Fujie, M.: Haptic feedback control in medical robots through fractional viscoelastic tissue model. In: Annual International Conference of the IEEE Engineering in Medicine and Biology Society, pp. 6704–6708 (2011)Google Scholar
  26. 26.
    Krishna, B.: Studies on fractional order differentiators and integrators: a survey. Signal Process. 91(3), 386–426 (2011)CrossRefzbMATHGoogle Scholar
  27. 27.
    Li, C., Zhang, F.: A survey on the stability of fractional differential equations. Eur. Phys. J. Spec. Top. 193(1), 27–47 (2011)CrossRefGoogle Scholar
  28. 28.
    Lorenzo, C.F., Hartley, T.T.: Energy considerations for mechanical fractional-order elements. J. Comput. Nonlinear Dyn. 10, (2015)Google Scholar
  29. 29.
    Luo, Y., Chen, Y.Q.: Fractional Order Motion Controls. Wiley, New Jersey (2012)CrossRefGoogle Scholar
  30. 30.
    Lurie, B.J.: Three-parameter tunable tilt-integral-derivative (TID) controller (1994)Google Scholar
  31. 31.
    Ma, C., Hori, Y.: Fractional-order control: theory and applications in motion control [past and present]. IEEE Ind. Electron. Mag. 1(4), 6–16 (2007)CrossRefGoogle Scholar
  32. 32.
    Machado, J.T., Kiryakova, V., Mainardi, F.: Recent history of fractional calculus. Commun. Nonlinear Sci. Numer. Simul. 16(3), 1140–1153 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Minsky, M., Ming, O.y., Steele, O., Brooks Jr., F.P., Behensky, M.: Feeling and seeing: issues in force display. In: Proceedings of the Symposium on Interactive 3D Graphics, pp. 235–241 (1990)Google Scholar
  34. 34.
    Monje, C.A., Chen, Y.Q., Vinagre, B.M., Xue, D., Feliu-Batlle, V.: Fractional-order systems and controls: fundamentals and applications. Springer, London (2010)CrossRefzbMATHGoogle Scholar
  35. 35.
    Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, Cambridge (1974)zbMATHGoogle Scholar
  36. 36.
    Ortigueira, M.D.: Fractional Calculus for Scientists and Engineers. Springer, Netherlands (2011)CrossRefzbMATHGoogle Scholar
  37. 37.
    Oustaloup, A., Mathieu, B., Lanusse, P.: The crone control of resonant plants: application to a flexible transmission. Eur. J. Control 1(2), 113–121 (1995)CrossRefGoogle Scholar
  38. 38.
    Petras, I.: Fractional-Order Nonlinear Systems. Springer, Heidelberg (2011)CrossRefzbMATHGoogle Scholar
  39. 39.
    Podlubny, I.: Fractional-order systems and pi/sup /spl lambda//d/sup /spl mu//-controllers. IEEE Trans. Autom. Control 44(1), 208–214 (1999)CrossRefzbMATHGoogle Scholar
  40. 40.
    Ryu, J.H., Kwon, D.S., Hannaford, B.: Stable teleoperation with time-domain passivity control. IEEE Trans. Robot. Autom. 20(2), 365–373 (2004)CrossRefGoogle Scholar
  41. 41.
    Xue, D., Chen, Y.Q.: A comparative introduction of four fractional order controllers. In: 4th World Congress on Intelligent Control and Automation, vol. 4, pp. 3228–3235 (2002)Google Scholar
  42. 42.
    Zhang, M., Nigwekar, P., Castaneda, B., Hoyt, K., Joseph, J.V., di Sant’Agnese, A., Messing, E.M., Strang, J.G., Rubens, D.J., Parker, K.J.: Quantitative characterization of viscoelastic properties of human prostate correlated with histology. Ultrasound Med. Biol. 34(7), 1033–1042 (2008)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Sabanci UniversityOrhanli, TuzlaTurkey

Personalised recommendations