Skip to main content
SpringerLink
Log in

LCP method for a planar passive dynamic walker based on an event-driven scheme

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

Abstract

The main purpose of this paper is to present a linear complementarity problem (LCP) method for a planar passive dynamic walker with round feet based on an event-driven scheme. The passive dynamic walker is treated as a planar multi-rigid-body system. The dynamic equations of the passive dynamic walker are obtained by using Lagrange’s equations of the second kind. The normal forces and frictional forces acting on the feet of the passive walker are described based on a modified Hertz contact model and Coulomb’s law of dry friction. The state transition problem of stick-slip between feet and floor is formulated as an LCP, which is solved with an event-driven scheme. Finally, to validate the methodology, four gaits of the walker are simulated: the stance leg neither slips nor bounces; the stance leg slips without bouncing; the stance leg bounces without slipping; the walker stands after walking several steps.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. McGeer, T.: Passive dynamic walking. Int. J. Robot. Res. 9, 62–82 (1990)

    Article  Google Scholar 

  2. Garcia, M., Chatterjee, A., Ruina, A.: Efficiency, speed, and scaling of two-dimensional passive-dynamic walking. Dynam. Stab. Syst. 15, 75–99 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  3. Walsh, C.J., Paluska, D., Pasch, K., et al.: Development of a lightweight, underactuated exoskeleton for load-carrying augmentation. In: IEEE International Conference on Robotics and Automation, ICRA 2006, May 15–19, Orlando, Florida, USA, 3485–3491 (2006)

  4. Collins, S., Ruina, A., Tedrake, R., et al.: Efficient bipedal robots based on passive-dynamic walkers. Science 307, 1082–1085 (2005)

    Article  Google Scholar 

  5. Kuo, A.D.: Biophysics. Harvesting energy by improving the economy of human walking. Science 309, 1686–1687 (2005)

    Article  Google Scholar 

  6. Alexander, M.N.: Walking made simple. Science 308, 58–59 (2005)

    Google Scholar 

  7. Liu, N., Li, J., Wang, T.: Passive walker that can walk down steps: simulations and experiments. Acta Mech. Sin. 24, 569–573 (2008)

    Article  MATH  Google Scholar 

  8. Liu, N., Li, J., Wang, T.: The effects of parameter variation on the gaits of passive walking models: simulations and experiments. Robotica 27, 511–528 (2009)

    Article  Google Scholar 

  9. Gritli, H., Belghith, S.: Computation of the Lyapunov exponents in the compass-gait model under OGY control via a hybrid Poincaré map. Chaos Solitons Fractals 81, 172–183 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gritli, H., Belghith, S.: Bifurcations and chaos in the semi-passive bipedal dynamic walking model under a modified OGY-based control approach. Nonlinear Dyn. 83, 1955–1973 (2016)

    Article  MathSciNet  Google Scholar 

  11. Gritli, H., Belghith, S.: Walking dynamics of the passive compass-gait model under OGY-based control: emergence of bifurcations and chaos. Commun. Nonlinear Sci. Numer. Simul. 47, 308–327 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gritli, H., Belghith, S.: Walking dynamics of the passive compass-gait model under OGY-based state-feedback control: Analysis of local bifurcations via the hybrid Poincaré map. Chaos Solitons Fractals 98, 72–87 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  13. Qi, F., Wang, T., Li, J.: The elastic contact influences on passive walking gaits. Robotica 29, 787–796 (2011)

    Article  Google Scholar 

  14. Qi, F., Bi, L.Y., Wang, T.S., et al.: The experimental study on the contact process of passive walking. Acta Mech. Sin. 28, 1163–1168 (2012)

    Article  Google Scholar 

  15. Asano, F.: Stability analysis of underactuated compass gait based on linearization of motion. Multibody Syst. Dyn. 33, 93–111 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Asano, F., Saka, T., Fujimoto, T.: Passive dynamic walking of compass-like biped robot on slippery downhill. In: IEEE/RSJ International Conference on Intelligent Robots and Systems, Hamburg, Germany, 4113–4118 (2015)

  17. Asano, F., Saka, T., Harata, Y.: 3-DOF passive dynamic walking of compass-like biped robot with semicircular feet generated on slippery downhill. In: IEEE International Conference on Robotics and Automation (ICRA), Stockholm, Sweden, 3570–3575 (2016)

  18. Asano, F., Seino, T., Tokuda, I., et al.: A novel locomotion robot that slides and rotates on slippery downhill. In: IEEE International Conference on Advanced Intelligent Mechatronics (AIM), Banff, Alberta, Canada, 425–430 (2016)

  19. Pennestrì, E., Rossi, V., Salvini, P., et al.: Review and comparison of dry friction force models. Nonlinear Dyn. 83, 1785–1801 (2016)

    Article  MATH  Google Scholar 

  20. Gamus, B., Or, Y.: Dynamic bipedal walking under stick–slip transitions. SIAM J. Appl. Dyn. Syst. 14, 609–642 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Pfeiffer, F.: On non-smooth dynamics. Meccanica 43, 533–554 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Zhao, Z., Liu, C., Chen, B., et al.: Asymptotic analysis of Painlevé’s paradox. Multibody Syst. Dyn. 35, 299–319 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. Zhao, Z., Liu, C.: Contact constraints and dynamical equations in Lagrangian systems. Multibody Syst. Dyn. 38, 77–99 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wang, Q.T., Tian, Q., Hu, H.Y.: Contact dynamics of elasto-plastic thin beams simulated via absolute nodal coordinate formulation. Acta Mech. Sin. 32, 525–534 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  25. Wang, Q., Tian, Q., Hu, H.: Dynamic simulation of frictional multi-zone contacts of thin beams. Nonlinear Dyn. 83, 1919–1937 (2016)

    Article  Google Scholar 

  26. Zhuang, F., Wang, Q.: Modeling and simulation of the nonsmooth planar rigid multibody systems with frictional translational joints. Multibody Syst. Dyn. 29, 403–423 (2013)

    MathSciNet  Google Scholar 

  27. Zhuang, F.F., Wang, Q.: Modeling and analysis of rigid multibody systems with driving constraints and frictional translation joints. Acta Mech. Sin. 30, 437–446 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  28. Wang, X., Lv, J.: Modeling and simulation of dynamics of a planar-motion rigid body with friction and surface contact. Int. J. Mod. Phys. B 31, 1744021 (2017)

    Article  MATH  Google Scholar 

  29. Duan, W., Wang, Q., Wang, T.: Simulation research of a passive dynamic walker with round feet based on non-smooth method. Lixue Xuebao/Chin. J. Theor. Appl. Mech. 43, 765–774 (2011)

    Google Scholar 

  30. Johnson, K.L.: One hundred years of Hertz contact. Proc. Inst. Mech. Eng. 196, 363–378 (1982)

  31. Johnson, K.L.: Contact mechanics. J. Tribol. 108, 464 (1985)

    Google Scholar 

  32. Yigit, A.S., Ulsoy, A.G., Scott, R.A.: Spring-dashpot models for the dynamics of a radially rotating beam with impact. J. Sound Vib. 142, 515–525 (1990)

    Article  Google Scholar 

  33. Machado, M., Moreira, P., Flores, P., et al.: Compliant contact force models in multibody dynamics: evolution of the Hertz contact theory. Mech. Mach. Theory 53, 99–121 (2012)

    Article  Google Scholar 

  34. Koshy, C.S., Flores, P., Lankarani, H.M.: Study of the effect of contact force model on the dynamic response of mechanical systems with dry clearance joints: computational and experimental approaches. Nonlinear Dyn. 73, 325–338 (2013)

    Article  Google Scholar 

  35. Wang, Q., Peng, H., Zhuang, F.: A constraint-stabilized method for multibody dynamics with friction-affected translational joints based on HLCP. Discrete Cont. Dyn. B 2, 589–605 (2011)

    MathSciNet  MATH  Google Scholar 

  36. Flores, P., Leine, R., Glocker, C.: Modeling and analysis of planar rigid multibody systems with translational clearance joints based on the non-smooth dynamics approach. Multibody Syst. Dyn. 23, 165–190 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  37. Glocker, C.: Set-Valued Force Laws: Dynamics of Non-Smooth Systems. In: Lecture Notes in Applied & Computational Mechanics, Vol. 1, Springer-Verlag Berlin Heidelberg (2001)

  38. Cottle, R.W., Dantzig, G.B.: Complementary pivot theory of mathematical programming. Linear Algebra Appl. 1, 103–125 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  39. Leine, R.I., Campen, D.H.V., Glocker, C.H.: Nonlinear dynamics and modeling of various wooden toys with impact and friction. J. Vib. Control 9, 25–78 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  40. Mcgeer, T.: Passive bipedal running. Proc. R. Soc. Lond. 240, 107–134 (1990)

    Article  Google Scholar 

Download references

Acknowledgements

The project was supported by the National Natural Science Foundation of China (Grants 11372018, 11772021).

Author information

Authors and Affiliations

  1. School of Aeronautic Science and Engineering, Beihang University, Beijing, 100083, China

    Xu-Dong Zheng & Qi Wang

Authors
  1. Xu-Dong Zheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qi Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qi Wang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zheng, XD., Wang, Q. LCP method for a planar passive dynamic walker based on an event-driven scheme. Acta Mech. Sin. 34, 578–588 (2018). https://doi.org/10.1007/s10409-018-0749-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10409-018-0749-0

Keywords

Search

Navigation