Skip to main content
SpringerLink
Log in

Stability and oscillations in a slow-fast flexible joint system with transformation delay

  • Research Paper
  • Published:
Acta Mechanica Sinica Aims and scope Submit manuscript

Abstract

Flexible joints are usually used to transfer velocities in robot systems and may lead to delays in motion transformation due to joint flexibility. In this paper, a link-rotor structure connected by a flexible joint or shaft is firstly modeled to be a slow-fast delayed system when moment of inertia of the lightweight link is far less than that of the heavy rotor. To analyze the stability and oscillations of the slow-fast system, the geometric singular perturbation method is extended, with both slow and fast manifolds expressed analytically. The stability of the slow manifold is investigated and critical boundaries are obtained to divide the stable and the unstable regions. To study effects of the transformation delay on the stability and oscillations of the link, two quantitatively different driving forces derived from the negative feedback of the link are considered. The results show that one of these two typical driving forces may drive the link to exhibit a stable state and the other kind of driving force may induce a relaxation oscillation for a very small delay. However, the link loses stability and undergoes regular periodic and bursting oscillation when the transformation delay is large. Basically, a very small delay does not affect the stability of the slow manifold but a large delay affects substantially.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Zheng, Y.G., Wang, Z.H.: Advances in dynamics of slow-fast systems with time delay. Advances in Mechanics 41, 400–410 (2011) (in Chinese)

    Google Scholar 

  2. Erzgräber, H., Lenstra, D., Krauskopf, B., et al.: Mutually delay-coupled semiconductor lasers: Mode bifurcation scenarios. Optics Communications 255, 286–296 (2005)

    Article  Google Scholar 

  3. Boudjellaba, H., Sari, T.: Dynamic transcritical bifurcations in a class of slow-fast predator-prey models. Journal of Differential Equations 246, 2205–2225 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Fasano, A., Herrero, M.A., Rodrigo, M.R.: Slow and fast invasion waves in a model of acid-mediated tumour growth. Mathematical Biosciences 220, 45–56 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ji, Y., Bi, Q.S.: Bifurcation analysis of slow-fast behavior in modified Chua’s circuit. Acta Physica Sinica 61, 010202-1–010202-6 (2012)

    Google Scholar 

  6. Spong, M.W.: Modeling and control of elastic joint robots. Journal of Dynamic Systems, Measurement, and Control 109, 310–319 (1987)

    Article  MATH  Google Scholar 

  7. Fateh, M.M., Babaghasabha, R.: Impedance control of robots using voltage control strategy. Nonlinear Dynamics 74, 277–286 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Consolini, L., Gerelli, O., Bianco, C.G.L., et al.: Flexible joint control: A minimum-time feed-forward technique. Mechatronics 19, 348–356 (2009)

    Article  Google Scholar 

  9. Ott, C., Albu-Schaffer, A., Kugi, A., et al.: On the passivitybased impedance control of flexible joint robots. IEEE Transactions on Robotics 24, 416–429 (2008)

    Article  Google Scholar 

  10. Brogardh, T.: Present and future robot control development-An industrial perspective. Annual Reviews in Control 31, 69–79 (2007)

    Article  Google Scholar 

  11. Marc, J.R., Li, C.: Controlling robot manipulators by dynamic programming. Acta Mechanica Sinica 11, 20–33 (1995)

    Article  MATH  Google Scholar 

  12. Subudhi, B., Morris, A.S.: Dynamic modeling, simulation and control of a manipulator with flexible links and joints. Robotics and Autonomous Systems 41, 257–270 (2002)

    Article  MATH  Google Scholar 

  13. Dwivedy, S.K., Eberhard, P.: Dynamic analysis of flexible manipulators, a literature review. A Mechanism and Machine Theory 41, 749–777 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chen, G.R., Shieh, L.S.: On trajectory tracking of time-delayed systems with an application to flexible-joint robot arm. In: Proceedings of International Conference on Control and Information, Chinese University Press, Hong Kong (1995)

    Google Scholar 

  15. Chen, G.R., Desages, A., Julian, P.: Trajectory tracking and robust stability for a class of time-delayed flexible joint. International Journal of Control 68, 259–276 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ailon, A.: Asymptotic stability in a flexible-joint robot with model uncertainty and multiple time delays in feedback. Journal of the Franklin Institute 341, 519–531 (2004)

    Article  MathSciNet  Google Scholar 

  17. Valenzuela, M.A., Bentley, J.M., Lorenz, R.D.: Evaluation of torsional oscillations in paper machine sections. IEEE Transactions on Industry Applications 41, 493–501 (2005)

    Article  Google Scholar 

  18. Guckenheimer, J., Hoffman, K., Weckesserx, W.: Bifurcations of relaxation oscillations near folded saddles. International Journal of Bifurcation and Chaos 15, 3411–3421 (2005)

    Article  MATH  Google Scholar 

  19. Zheng, Y.G., Wang, Z.H.: Time-delay effect on the bursting of the synchronized state of coupled Hindmarsh-Rose neurons. Chaos 22, 043127 (2012)

    Article  Google Scholar 

  20. Lu, Q.S., Gu, H.G., Yang, Z.Q., et al.: Dynamics of firing patterns, synchronization and resonances in neuronal electrical activities: Experiments and analysis. Acta Mechanica Sinica 24, 593–628 (2008)

    Article  MATH  Google Scholar 

  21. Nayfeh, A.H.: Perturbation Methods. John Wiley & Sons, Inc, New York (1973)

    MATH  Google Scholar 

  22. Tikhonov, A.N.: Systems of differential equations containing a small parameter multiplying the derivative. Matematicheskii sbornik 31, 575–586 (1952)

    Google Scholar 

  23. Fenichel, N.: Geometric singular perturbation theory for ordinary differential equations. Journal of Differential Equations 31, 53–98 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  24. Jones, C.: Geometric singular perturbation theory. Lecture Notes in Math 1609, 45–118 (1994)

    Google Scholar 

  25. Rinaldi, S., Scheffer, M.: Geometric analysis of ecological models with slow and fast process. Ecosystems 3, 507–521 (2000)

    Article  Google Scholar 

  26. Izhikevich, E.M.: Neural excitability, spiking and bursting. International Journal of Bifurcation and Chaos 10, 1171–1266 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  27. Han, X., Jiang, B., Bi, Q.S.: Symmetric bursting of focus-focus type in the controlled Lorenz system with two time scales. Physics Letters A 373, 3643–3649 (2009)

    Article  MATH  Google Scholar 

  28. Ghorbel, F., Spong, M.W.: Integral manifolds of singularly perturbed systems with application to rigid-link flexible-joint multibody systems. International Journal of Non-Linear Mechanics 35, 133–155 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  29. Hori, Y., Iseki, H., Sugiura, K.: Basic consideration of vibration suppression and disturbance rejection control of multiinertia system using SFLAC (State feedback and load acceleration control). IEEE Transactions on Industry Applications 30, 889–896 (1994)

    Article  Google Scholar 

  30. Nordin, M., Gutman, P.: Controlling mechanical systems with backlash-a survey. Automatica 38, 1633–1649 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  31. Naidu, D.S.: Singular perturbations and time scales in control theory and applications: An overview. Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications & Algorithms 9, 233–278 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  32. Xu, J., Chung, K.W., Chan, C.L.: An efficient method for studying weak resonant double hopf bifurcation in nonlinear systems with delayed feedbacks. SIAM J. Applied Mathematics 6, 29–60 (2007)

    MathSciNet  MATH  Google Scholar 

  33. Xu, J., Chung, K.W.: Effects of time delayed position feedback on a van der Pol-Duffing oscillator. Physica D 180, 17–39 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  34. Wang, L.L.: Foundation of Stress Waves. (2nd edn.) National Defense Industry Press, Beijing (2005) (in Chinese)

    Google Scholar 

  35. Xu, J., Lu, Q.S., Huang, K.L.: Controlling erosion of safe basin in nonlinear parametrically excited systems. Acta Mechanica Sinica 12, 281–288 (1996)

    Article  MATH  Google Scholar 

  36. Wang, W.Y., Pei, L.J.: Stability and Hopf bifurcation of a delayed ratio-dependent predator-prey system. Acta Mechanica Sinica 27, 285–296 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  37. Wang, H.L., Wang, Z.H., Hu, H.Y.: Hopf bifurcation of an oscillator with quadratic and cubic nonlinearities and with delayed velocity feedback. Acta Mechanica Sinica 20, 426–434 (2004)

    Article  Google Scholar 

  38. Zhang, S., Xu, J.: Bursting-like motion induced by timevarying delay in an internet congestion control model. Acta Mechanica Sinica 28, 1169–1179 (2012)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Aerospace Engineering and Applied Mechanics, Tongji University, 200092, Shanghai, China

    Shan-Ying Jiang, Jian Xu & Yao Yan

Authors
  1. Shan-Ying Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jian Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yao Yan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jian Xu.

Additional information

The project was supported by the National Natural Science Foundation of China (11032009 and 11272236).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jiang, SY., Xu, J. & Yan, Y. Stability and oscillations in a slow-fast flexible joint system with transformation delay. Acta Mech Sin 30, 727–738 (2014). https://doi.org/10.1007/s10409-014-0064-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10409-014-0064-3

Keywords

Search

Navigation