Delay-Dependent Stability Analysis in Haptic Rendering

  • Ahmad Mashayekhi
  • Saeed BehbahaniEmail author
  • Fanny Ficuciello
  • Bruno Siciliano


Nowadays haptic devices have lots of applications in virtual reality systems. While using a haptic device, one of the main requirements is the stable behavior of the system. An unstable behavior of a haptic device may damage itself and even may hurt its operator. Stability of haptic devices in the presence of inevitable time delay in addition to a suitable zero-order hold is studied in the presented paper, using two different methods. Both presented methods are based on Lyapunov-Krazuvskii functional. In the first method, a model transform is performed to determine the stability boundary, while the second approach is based on Free Weighing Matrices (FWMs). Delay-dependent stability criteria are determined by solving Linear Matrix Inequalities (LMIs). Results of these two methods are compared with each other and verified by simulations as well as experiments on a KUKA Light Weight Robot 4 (LWR4). It is concluded that using free weighing matrices leads to more unknown parameters and needs more calculation, but its results are less conservative.


Haptic devices Stability Linear matrix inequality 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



  1. 1.
    Dang, Q.V., Vermeiren, L., Dequidt, A., Dambrine, M.: Analyzing stability of haptic interface using linear matrix inequality approach. In: 2012 IEEE International Conference on Robotics and Biomimetics (ROBIO), pp. 1129–1134, IEEE (2012)Google Scholar
  2. 2.
    Mashayekhi, A., Nahvi, A., Yazdani, M., Mohammadi Moghadam, M., Arbabtafti, M., Norouzi, M.: Virsense: A novel haptic device with fixed-base motors and a gravity compensation system. Indust. Robot: Int. J. 41(1), 37–49 (2014)CrossRefGoogle Scholar
  3. 3.
    Grajewski, D., Górski, F., Hamrol, A., Zawadzki, P.: Immersive and haptic educational simulations of assembly workplace conditions. Procedia Comput. Sci. 75, 359–368 (2015)CrossRefGoogle Scholar
  4. 4.
    You, B., Li, J., Ding, L., Xu, J., Li, W., Li, K., Gao, H.: Semi-autonomous bilateral teleoperation of hexapod robot based on haptic force feedback. J. Intell. Robot. Syst. 91(3–4), 583–602 (2018)CrossRefGoogle Scholar
  5. 5.
    Saafi, H., Laribi, M.A., Zeghloul, S.: Optimal torque distribution for a redundant 3-rrr spherical parallel manipulator used as a haptic medical device. Robot. Auton. Syst. 89, 40–50 (2017)CrossRefGoogle Scholar
  6. 6.
    Yoon, H.U., Wang, R.F., Hutchinson, S.A., Hur, P.: Customizing haptic and visual feedback for assistive human–robot interface and the effects on performance improvement. Robot. Auton. Syst. 91, 258–269 (2017)CrossRefGoogle Scholar
  7. 7.
    Adams, R.J., Hannaford, B.: Stable haptic interaction with virtual environments. IEEE Trans. Robot. Autom. 15(3), 465–474 (1999)CrossRefGoogle Scholar
  8. 8.
    Hogan, N.: Controlling impedance at the man/machine interface. In: 1989 IEEE International Conference on Robotics and automation, 1989. Proceedings, pp. 1626–1631, IEEE (1989)Google Scholar
  9. 9.
    Diolaiti, N., Niemeyer, G., Barbagli, F., Salisbury, J.K.: Stability of haptic rendering: Discretization, quantization, time delay, and coulomb effects. IEEE Trans. Robot. 22(2), 256–268 (2006)CrossRefGoogle Scholar
  10. 10.
    Gil, J.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
  11. 11.
    Gil, J.J., Sánchez, E., Hulin, T., Preusche, C., Hirzinger, G.: Stability boundary for haptic rendering: Influence of damping and delay. J. Comput. Inf. Sci. Eng. 9(1), 011005 (2009)CrossRefGoogle Scholar
  12. 12.
    Hulin, T., Albu-Schaffer, A., Hirzinger, G.: Passivity and stability boundaries for haptic systems with time delay. IEEE Trans. Control Syst. Technol. 22(4), 1297–1309 (2014)CrossRefGoogle Scholar
  13. 13.
    Mashayekhi, A., Boozarjomehry, R.B., Nahvi, A., Meghdari, A., Asgari, P.: Improved passivity criterion in haptic rendering: influence of coulomb and viscous friction. Adv. Robot. 28(10), 695–706 (2014)CrossRefGoogle Scholar
  14. 14.
    Abbott, J.J., Okamura, A.M.: Effects of position quantization and sampling rate on virtual-wall passivity. IEEE Trans. Robot. 21(5), 952–964 (2005)CrossRefGoogle Scholar
  15. 15.
    van der Schaft, A.: L2-Gain and Passivity Techniques in Nonlinear Control. Springer (2017)Google Scholar
  16. 16.
    Ryu, J.-H., Kim, Y.S., Hannaford, B.: Sampled-and continuous-time passivity and stability of virtual environments. IEEE Trans. Robot. 20(4), 772–776 (2004)CrossRefGoogle Scholar
  17. 17.
    Ryu, J.-H., Preusche, C., Hannaford, B., Hirzinger, G.: Time domain passivity control with reference energy following. IEEE Trans. Control Syst. Technol. 13(5), 737–742 (2005)CrossRefGoogle Scholar
  18. 18.
    Minsky, M., Ming, O.-Y., Steele, O., Brooks, F.P. Jr., Behensky, M.: Feeling and seeing: Issues in force display. In: ACM SIGGRAPH Computer Graphics, vol. 24, pp. 235–241, ACM (1990)Google Scholar
  19. 19.
    Mehling, J.S., Colgate, J.E., Peshkin, M.A.: Increasing the impedance range of a haptic display by adding electrical damping. In: Eurohaptics Conference, 2005 and Symposium on Haptic Interfaces for Virtual Environment and Teleoperator Systems, 2005. World Haptics 2005. First Joint, pp. 257–262, IEEE (2005)Google Scholar
  20. 20.
    Tognetti, L.J., Book, W.J.: Effects of increased device dissipation on haptic two-port network performance. In: Proceedings 2006 IEEE International Conference on Robotics and Automation, 2006. ICRA 2006, pp. 3304–3311, IEEE (2006)Google Scholar
  21. 21.
    Gosline, A.H., Campion, G., Hayward, V.: On the use of eddy current brakes as tunable, fast turn-on viscous dampers for haptic rendering. In: Proceedings of Eurohaptics, pp. 229–234 (2006)Google Scholar
  22. 22.
    Colgate, J.E., Schenkel, G.: Passivity of a class of sampled-data systems: Application to haptic interfaces. In: American Control Conference, 1994, vol. 3, pp. 3236–3240. IEEE (1994)Google Scholar
  23. 23.
    Janabi-Sharifi, F., Hayward, V., Chen, C.-S.: Discrete-time adaptive windowing for velocity estimation. IEEE Trans. Control Syst. Technol. 8(6), 1003–1009 (2000)CrossRefGoogle Scholar
  24. 24.
    Mashayekhi, A., Behbahani, S., Ficuciello, F., Siciliano, B.: Analytical stability criterion in haptic rendering: The role of damping. IEEE/ASME Trans. Mechatron. 23(2), 596–603 (2018)CrossRefGoogle Scholar
  25. 25.
    Wu, M., He, Y., She, J.-H.: Stability Analysis and Robust Control of Time-Delay Systems. Springer (2010)Google Scholar
  26. 26.
    Tian, E., Peng, C.: Delay-dependent stability analysis and synthesis of uncertain t–s fuzzy systems with time-varying delay. Fuzzy Sets Syst. 157(4), 544–559 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Wu, H.-N.: Delay-dependent stability analysis and stabilization for discrete-time fuzzy systems with state delay: A fuzzy lyapunov-krasovskii functional approach. IEEE Trans. Syst. Man Cybern/ Part B (Cybernetics) 36(4), 954–962 (2006)CrossRefGoogle Scholar
  28. 28.
    Cao, J., Yuan, K., Li, H.-X.: Global asymptotical stability of recurrent neural networks with multiple discrete delays and distributed delays. IEEE Trans. Neural Netw. 17(6), 1646–1651 (2006)CrossRefGoogle Scholar
  29. 29.
    Fridman, E.: New Lyapunov–Krasovskii functionals for stability of linear retarded and neutral type systems. Syst. Control Lett. 43(4), 309–319 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    He, Y., Wu, M., She, J.-H.: Delay-dependent stability criteria for linear systems with multiple time delays. IEE Proc.-Control Theory Appl. 153(4), 447–452 (2006)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Craig, J.J.: Introduction to Robotics: Mechanics and Control, vol. 3. Pearson/Prentice, Hall Upper Saddle River (2005)Google Scholar
  32. 32.
    Hulin, T., Preusche, C., Hirzinger, G.: Stability boundary for haptic rendering: Influence of human operator. In: IEEE/RSJ International Conference on Intelligent Robots and Systems, 2008. IROS 2008, pp. 3483–3488. IEEE (2008)Google Scholar
  33. 33.
    Karami, A., Sadeghian, H., Keshmiri, M.: Novel approaches to control multiple tasks in redundant manipulators: stability analysis and performance evaluation. Adv. Robot., 1–12 (2018)Google Scholar
  34. 34.
    “Lightweight robot four documentation.” Accessed 2019-01-17
  35. 35.
    “Fast research interface library.” Accessed 2019-01-17
  36. 36.
    Mashayekhi, A., Behbahani, S., Ficuciello, F., Siciliano, B.: A closed-form stability criterion in haptic rendering. Submitted to Transactions on Haptics (2018)Google Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringIsfahan University of TechnologyIsfahanIran
  2. 2.CREATE Consortium, PRISMA Laboratory, Department of Electrical Engineering and Information TechnologyUniversity of Naples Federico IINaplesItaly

Personalised recommendations