Lambert W Function Controller Design for Teleoperation Systems

  • Soheil Ganjefar
  • Mohammad Hadi Sarajchi
  • Seyed Mahmoud Hoseini
  • Zhufeng ShaoEmail author
Regular Paper


Stability and transparency play key roles in a bilateral teleoperation system with communication latency. This study developed a new method of controller design, based on the Lambert W function for the bilateral teleoperation through the Internet. In spite of the time-delay in the communication channel, system disturbance, and modeling errors, this approach causes the slave manipulator tracks the master appropriately. Time-delay terms in the bilateral teleoperation systems result in an infinite number of characteristic equation roots making difficulty in the analysis of systems by traditional strategies. As delay differential equations have infinite eigenspectrums, it is not possible to locate all closed-loop eigenvalue in desired positions by using classical control methods. Therefore, this study suggested a new feedback controller for assignment of eigenvalues, in compliance with Lambert W function. Lambert W function causes the rightmost eigenvalues to locate exactly in desired possible positions in the stable left hand of the imaginary axis. This control method led to a reduction in the undesirable effect of time-delay on the communication channel. The simulation results showed great closed-loop performance and better tracking in case of different time-delay types.


Eigenvalue assignment Lambert W function Teleoperation systems Time-delay 



Funding was provided by National Natural Science Foundation of China (Grant No. 51575292).


  1. 1.
    Sheridan, T. B. (1995). Teleoperation, telerobotics and telepresence, a progress report. Control Engineering Practice, 3(2), 204–214.Google Scholar
  2. 2.
    Dinh, T. Q., Yoon, J. I., Marco, J., Jennings, P., Ahn, K. K., & Ha, C. (2017). Sensorless force feedback joystick control for teleoperation of construction equipment. International Journal of Precision Engineering and Manufacturing, 18(7), 955–969.Google Scholar
  3. 3.
    Truong, D. Q., Truong, B. N. M., Trung, N. T., Nahian, S. A., & Ahn, K. K. (2017). Force reflecting joystick control for applications to bilateral teleoperation in construction machinery. International Journal of Precision Engineering and Manufacturing, 18(3), 301–315.Google Scholar
  4. 4.
    Kim, J. Y., Jun, B. H., & Park, I. W. (2017). Six-legged walking of “Little Crabster” on uneven terrain. International Journal of Precision Engineering and Manufacturing, 18(4), 509–518.Google Scholar
  5. 5.
    Baek, S. Y., Park, S., & Ryu, J. (2017). An enhanced force bounding approach for stable haptic interaction by including friction. International Journal of Precision Engineering and Manufacturing, 18(6), 813–824.Google Scholar
  6. 6.
    Jung, K., Chu, B., Park, S., & Hong, D. (2013). An implementation of a teleoperation system for robotic beam assembly in construction. International Journal of Precision Engineering and Manufacturing, 14(3), 351–358.Google Scholar
  7. 7.
    Kim, T., Kim, H. S., & Kim, J. (2016). Position-based impedance control for force tracking of a wall-cleaning unit. International Journal of Precision Engineering and Manufacturing, 17(3), 323–329.Google Scholar
  8. 8.
    Liu, X., Tao, R., & Tavakoli, M. (2014). Adaptive control of uncertain nonlinear teleoperation systems. Mechatronics, 24(1), 66–78.Google Scholar
  9. 9.
    Li, L. S., & Li, J. N. (2017). Reliable control for bilateral teleoperation systems with actuator faults using fuzzy disturbance observer. IET Control Theory and Applications, 11(3), 446–455.MathSciNetGoogle Scholar
  10. 10.
    Ganjefar, S., Rezaei, S., & Hashemzadeh, F. (2017). “Position and force tracking in nonlinear teleoperation systems with sandwich linearity in actuators and time-varying delay. Mechanical Systems and Signal Processing, 86(Part A), 308–324.Google Scholar
  11. 11.
    Sheridan, T. B. (1992). Telerobotics, automation, and human supervisory control. Cambrige, MA: The MIT Press.Google Scholar
  12. 12.
    Lee, D., & Spong, M. W. (2006). Passive bilateral teleoperation with constant time delay. IEEE Robotics and Automation Society, 22(2), 269–281.Google Scholar
  13. 13.
    Craig, J. J. (1989). Introduction to robotics: Mechanics and control. London: Addison-Wesley Series.zbMATHGoogle Scholar
  14. 14.
    Janabi-Sharifi, F. (1995). Collision: Modelling, Simulation and identification of robotic manipulators interacting with environments. Journal of Intelligent and Robotic Systems, 13(1), 1–44.Google Scholar
  15. 15.
    Gu, K., & Silviu-Iulian, N. (2006). 4 Stability analysis of time-delay systems: A Lyapunov approach. Advanced Topics in Control Systems Theory, 328, 139–170.MathSciNetzbMATHGoogle Scholar
  16. 16.
    Liu, P. L. (2003). Exponential stability for linear time-delay systems with delay dependence. Journal of the Franklin Institute, 340(6–7), 481–488.MathSciNetzbMATHGoogle Scholar
  17. 17.
    Corless, R. M., Gonnet, G. H., Hare, D. E. G., Jeffrey, D. J., & Knuth, D. E. (1996). On the Lambert W function. Advances in Computational Mathematics, 5(1), 329–359.MathSciNetzbMATHGoogle Scholar
  18. 18.
    Yi, S., Nelson, P. W., & Ulsoy, A. G. (2010). “Time-delay systems: Analysis and control using the Lambert W function. Singapore: Word Scientific Publishing.zbMATHGoogle Scholar
  19. 19.
    Asl, F. M., & Ulsoy, A. G. (2003). Analysis of a system of linear delay differential equations. Journal of Dynamic Systems, Measurement, and Control, 12(5), 215–223.Google Scholar
  20. 20.
    Nelson, P. W., Ulsoy, A. G., & Yi, S. (2010). Eigenvalue assignment via the Lambert W function for control for time- delay systems. Journal of Vibration and Control, 16(7–8), 961–982.MathSciNetzbMATHGoogle Scholar
  21. 21.
    Yi, S., & Ulsoy, A. G. (2006). Solution of a system of linear delay differential equations using the matrix Lambert function. In Proceedings of American control conference (pp. 2433–2438).Google Scholar
  22. 22.
    Bellman, R. E., & Cooke, K. L. (1963). Differential-difference equations. New York: Academic Press.zbMATHGoogle Scholar
  23. 23.
    Nelson, P. W., Ulsoy, A. G., & Yi, S. (2007). Delay differential equations via the matrix Lambert W function and bifurcation analysis. Mathematical Biosciences and Engineering, 4(2), 355–368.MathSciNetzbMATHGoogle Scholar
  24. 24.
    Shinozaki, H., & Mori, T. (2006). Robust stability analysis of linear time-delay systems by Lambert W function: Some extreme point results. Automatica, 42(10), 1791–1799.MathSciNetzbMATHGoogle Scholar
  25. 25.
    Manitius, A., & Olbrot, A. W. (1979). Finite spectrum assignment problem for systems with delays. IEEE Transaction on Automatic Control, 24(4), 541–553.MathSciNetzbMATHGoogle Scholar
  26. 26.
    Chen, C. T. (1998). Linear system theory and design. New York: Oxford University Press.Google Scholar
  27. 27.
    Engelborghs, K., Vansevenant, P., Roose, D., & Michiels, W. (2002). Continuous Pole placement for delay equations. Automatica, 38(5), 747–761.MathSciNetzbMATHGoogle Scholar
  28. 28.
    Colgate, J. E. (1993). Robust impedance shaping telemanipulation. IEEE Transaction on Robotics and Automation, 9(4), 374–384.Google Scholar
  29. 29.
    Reinoso, O., Sabater, J. M., Perez, C., & Azorin, J. M. (2003). A new control method of teleoperators with time delay. In 11th International conference on advanced robotics, Coimbra, Portugal (pp. 100–105).Google Scholar

Copyright information

© Korean Society for Precision Engineering 2019

Authors and Affiliations

  1. 1.School of Electrical EngineeringBu-Ali Sina UniversityHamedanIran
  2. 2.State Key Laboratory of Tribology and Institute of Manufacturing Engineering, and Beijing Key Lab of Precision/Ultra-Precision Manufacturing Equipment and ControlTsinghua UniversityBeijing ShiChina

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