Skip to main content
SpringerLink
Log in

Analysis of steady and target walking speeds in limit cycle walking

  • Published:
International Journal of Dynamics and Control Aims and scope Submit manuscript

Abstract

This paper presents the mathematical analysis of steady walking speed and target walking speed generation in 1-DOF limit cycle walking driven by time-settling control inputs. An actuated combined rimless wheel (CRW) model is introduced to analyse the steady walking state when the CRW is walking on level ground. First the initial and terminal boundary conditions driven by discrete stepwise control systems are analysed. The steady step period can be calculated and the target walking period can be generated by the formulas. Second we extend the mathematical analysis to \((n+1)\)-period stepwise control system and derive the general formula of the steady step period. Finally, the continuous piecewise control systems are analysed mathematically by discretizing the control input, and thus the boundary conditions can be analysed by the formula of \((n+1)\)-period stepwise control system. As a result, if the generated walking gait is single-step-cycle, the steady step period can be calculated and target walking steady speeds can be generated by our general formulas in the time-settling control systems.

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
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

References

  1. McGeer T (1990) Passive dynamic walking. Int J Rob Res 9(2):62–82

    Article  Google Scholar 

  2. Grizzle JW, Abba G, Plestan F (2001) Asymptotically stable walking for biped robots: analysis via systems with impulse effects. IEEE Trans Automat Contr 46(1):51–64

    Article  MathSciNet  MATH  Google Scholar 

  3. Asano F (2015) Stability analysis of underactuated compass gait based on linearization of motion. Multibody Syst Dyn 33(1):93–111

    Article  MathSciNet  MATH  Google Scholar 

  4. Asano F (2015) Fully analytical solution to discrete behavior of hybrid zero dynamics in limit cycle walking with constraint on impact posture. Multibody Syst Dyn 35(2):191–213

    Article  MathSciNet  MATH  Google Scholar 

  5. Nanayakkara T, Byl K, Liu H, et al (2012) Dominant sources of variability in passive walking. In: Proceedings of the IEEE international conference on robotics and automation, pp 1003–1010

  6. Hobbelen DGE (2008) Controlling the walking speed in limit cycle walking. Int J Rob Res 27(9):989–1005

    Article  Google Scholar 

  7. Hobbelen DGE, Wisse M (2007) A disturbance rejection measure for limit cycle walkers: the gait sensitivity norm. IEEE Trans Rob 23(6):1213–1224

    Article  Google Scholar 

  8. Zhou K, Kemin J, Doyle C, Glover K (1996) Robust and optimal control, vol 40. Prentice Hall, New Jersey

    MATH  Google Scholar 

  9. Coleman MJ, Chatterjee A, Ruina A (1997) Motions of a rimless spoked wheel: a simple three-dimensional system with impacts. Dyn Stab Syst 12(3):139–159

    Article  MathSciNet  MATH  Google Scholar 

  10. Coleman MJ (2010) Dynamics and stability of a rimless spoked wheel: a simple 2D system with impacts. Dyn Syst 25(2):215–238

    Article  MathSciNet  MATH  Google Scholar 

  11. Asano F (2011) Stability analysis of passive compass gait using linearized model. In: Proceedings of the IEEE international conference on robotics and automation, pp 557–562

  12. Asano F, Xiao X (2013) Role of deceleration effect in efficient and fast convergent gait generation. In: Proceedings of the IEEE international conference on robotics and automation, pp 5649–5654

  13. Xiao X, Asano F (2014) Analytical solution of steady step period in 1-DOF limit cycle walking driven by stepwise control inputs. In: Proceedings of the IEEE international conference on mechatronics and automation (ICMA), pp 245–250

  14. Coleman MJ, Garcia M, Ruina A et al (1999) Stability and chaos in passive-dynamic locomotion. In: IUTAM symposium on new applications of nonlinear and chaotic dynamics in mechanics. Springer, Netherlands

  15. Xiao X, Asano F (2015) Analytical solution of target steady walking speed in 1-DOF limit cycle walking. In: Proceedings of the 2015 IEEE international conference on robotics and automation, pp 4525–4531

  16. Ridders C (1979) A new algorithm for computing a single root of a real continuous function. IEEE Trans Circuits Syst 26(11):979–980

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. The School of Information Science, Japan Advanced Institute of Science and Technology, 1-1 Asahidai, Nomi, Ishikawa, 923-1292, Japan

    Xuan Xiao & Fumihiko Asano

Authors
  1. Xuan Xiao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fumihiko Asano
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xuan Xiao.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xiao, X., Asano, F. Analysis of steady and target walking speeds in limit cycle walking. Int. J. Dynam. Control 5, 454–465 (2017). https://doi.org/10.1007/s40435-015-0212-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40435-015-0212-z

Keywords

Search

Navigation