Skip to main content
Log in

Controlling flat-foot limit cycle walkers with compliant joints based on local stability variation

  • Original Article
  • Published:
Biological Cybernetics Aims and scope Submit manuscript

Abstract

This study investigates local stability of a four-link limit cycle walking biped with flat feet and compliant ankle joints. Local stability represents the behavior along the solution trajectory between Poincare sections, which can provide detailed information about the evolution of disturbances. The effects of ankle stiffness and foot structure on local stability are studied. In addition, we apply a control strategy based on local stability analysis to the limit cycle walker. Control is applied only in the phases with poor local stability. Simulation results show that the energy consumption is reduced without sacrificing disturbance rejection ability. This study may be helpful in motion control of limit cycle bipedal walking robots with flat feet and ankle stiffness and understanding of human walking principles.

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

Access this article

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

Instant access to the full article PDF.

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

Similar content being viewed by others

References

  • Collins S, Wisse M, Ruina A (2001) A three-dimensional passive-dynamic walking robot with two legs and knees. Int J Robot Res 20:607–615

    Article  Google Scholar 

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

    Article  CAS  PubMed  Google Scholar 

  • Frigo C, Crenna P, Jensen LM (1996) Moment-angle relationship at lower limb joints during human walking at different velocities. J Electromyogr Kinesiol 6(3):177–190

    Article  CAS  PubMed  Google Scholar 

  • Goswami A, Thuilot B, Espiau B (1998) A study of the passive gait of a compass-like biped robot: symmetry and chaos. Int J Robot Res 17:1282–1301

    Article  Google Scholar 

  • Goswami A, Espiau B, Keramane A (1996) Limit cycles and their stability in a passive bipedal gait. In: Proceedings of IEEE International Conference on Robotics and Automation, Minneapolis, USA

  • Hirai K, Hirose M, Haikawa Y, Takenaka T (1998) The development of the Honda Humanoid robot. In: Proceedings of the IEEE International Conference on Robotics and Automation, Leuven, Belgium, pp 1321–1326

  • Hobbelen DGE, Wisse M (2007) Limit cycle walking, humanoid robots: human-like machines, pp 277–294

  • Hobbelen DGE, Wisse M (2008) Ankle actuation for limit cycle walkers. Int J Robot Res 27(6):709–736

    Article  Google Scholar 

  • Hosoda K, Takuma T, Nakamoto A, Hayashi S (2008) Biped robot design powered by antagonistic pneumatic actuators for multi-modal locomotion. Robot Auton Syst 56:46–53

    Article  Google Scholar 

  • Huang Y, Wang Q (2016) Torque-stiffness-controlled dynamic walking: analysis of the behaviors of bipeds with both adaptable joint torque and joint stiffness. IEEE Robot Autom Mag 23(1):71–82

    Article  Google Scholar 

  • Huang Q, Yokoi K, Kajita S, Kaneko K et al (2001) Planning walking patterns for a biped robot. IEEE Trans Robot Autom 17(3):280–289

    Article  Google Scholar 

  • Huang Y, Vanderborght B, Van Ham R, Wang Q et al (2013) Step length and velocity control of a dynamic bipedal walking robot with adaptable compliant joints. IEEE/ASME Trans Mechatron 18:598–611

    Article  Google Scholar 

  • Huang Y, Vanderborght B, Van Ham R, Wang Q (2014) Torque-stiffness-controlled dynamic walking with central pattern generators. Biol Cybern 108:803–823

    Article  PubMed  Google Scholar 

  • Huang Y, Huang Q, Wang Q (2017) Chaos and bifurcation control of torque-stiffness-controlled dynamic bipedal walking. IEEE Trans Syst Man Cybern Syst 47(7):1229–1240

    Article  Google Scholar 

  • Huang Y, Wang Q, Chen B, Xie G, Wang L (2012) Modeling and gait selection of passivity-based seven-link bipeds with dynamic series of walking phases. Robotica, vol. 30, pp 39–51

  • Hubicki C, Grimes J, Jones M et al (2016) ATRIAS: design and validation of a tether-free 3D-capable spring-mass bipedal robot. Int J Robot Res 35(12):1497–1521

    Article  Google Scholar 

  • Hurmuzlu Y, Moskowitz GD (1986) Role of impact in the stability of bipedal locomotion. Int J Dyn Stab Syst 1(3):217–234

    Google Scholar 

  • Hyon SH, Suewaka D, Torii Y, Oku N (2017) Design and experimental evaluation of a fast torque-controlled hydraulic humanoid robot. IEEE/ASME Trans Mechatron 22(2):623–634

    Article  Google Scholar 

  • Iida F, Rummel J, Seyfarth A (2008) Bipedal walking and running with spring-like biarticular muscles. J Biomech 41:656–667

    Article  PubMed  Google Scholar 

  • Kaneko K, Kanehiro F, Kajita S (2004) Humanoid robot HRP-2. In: Proceedings of the IEEE International Conference on Robotics and Automation, New Orleans, LA, USA, pp 1083–1090

  • Kuo AD (2002) Energetics of actively powered locomotion using the simplest walking model. J Biomech Eng 124:113–120

    Article  PubMed  Google Scholar 

  • Kuo AD (2005) Harvesting energy by improving the economy of human walking. Science 309:1686–1687

    Article  CAS  PubMed  Google Scholar 

  • Kwan M, Hubbard M (2007) Optimal foot shape for a passive dynamic biped. J Theor Biol 248:331–339

    Article  PubMed  Google Scholar 

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

    Article  Google Scholar 

  • Meng L, Macleod CA, Porr B, Gollee H (2018) Bipedal robotic walking control derived from analysis of human locomotion. Biol Cybern 112:277–290

    Article  PubMed  PubMed Central  Google Scholar 

  • Mombaur KD (2001) Stability optimization of open-loop controlled walking robots, PhD thesis, Heidelberg University, Ruperto Carola

  • Norris JA, Marsh AP, Granata KP, Ross SD (2008) Revisiting stability of 2D passive biped walking: local behavior. Physica D 237(23):3038–3045

    Article  Google Scholar 

  • Reher J, Ames AD (2021) Inverse dynamics control of compliant hybrid zero dynamic walking, In: Proceedings of IEEE International Conference on Robotics and Automation, Xian, China, pp 2040–2047

  • Saccon A, van de Wouw N, Nijmeijer H (2014) Sensitivity analysis of hybrid systems with state jumps with application to trajectory tracking. In: Proceedings IEEE Conference on Decision and Control (CDC), pp 3065–3070

  • Sharbafi MA, Seyfarth A (2015) FMCH: a new model for human-like postural control in walking, In: Proceedings of IEEE international conference on intelligent robots and systems, Hamburg, Germany, pp 5742–5747

  • Shiriaev AS, Freidovich LB, Gusev SV (2010) Transverse linearization for controlled mechanical systems with several passive degrees of freedom. IEEE Trans Automat Contr 55(4):893–906

    Article  Google Scholar 

  • Spong MW, Holm JK, Lee D (2007) Passivity-based control of bipedal locomotion. IEEE Robot Autom Mag 14(2):30–40

    Article  Google Scholar 

  • Tsagarakis NG, Morfey S, Medrano-Cerda GA et al (2013) Compliant humanoid COMAN: optimal joint stiffness tuning for modal frequency control. IEEE International Conference on Robotics Automation, Karlsruhe, Germany, pp 673–678

  • Van Ham R, Vanderborght B, Van Damme M, Verrelst B, Lefeber D (2007) MACCEPA, the mechanically adjustable compliance and controllable equilibrium position actuator: design and implementation in a biped robot. Robot Auton Sys 55:761–768

    Article  Google Scholar 

  • Vanderborght B, Verrelst B, Van Ham R, Van Damme M et al (2006) Exploiting natural dynamics to reduce energy consumption by controlling the compliance of soft actuators. Int J Robot Res 25:343–358

    Article  Google Scholar 

  • Wang Q, Huang Y, Wang L (2010) Passive dynamic walking with flat feet and ankle compliance. Robotica 28:413–425

    Article  CAS  Google Scholar 

  • Wisse M, Schwab AL, van der Helm FCT (2004) Passive dynamic walking model with upper body. Robotica 22:681–688

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Nos. 62073038 and 51922015) and also supported by the Beijing Institute of Technology Research Funds for High-Level Talents.

Author information

Authors and Affiliations

Authors

Contributions

Y.H. and Q.W. wrote the main manuscript text. Y.H. and Y.G. prepared the simulations and experiments. Q.W. and Q.H. supervised the study. All authors reviewed the manuscript.

Corresponding author

Correspondence to Qining Wang.

Ethics declarations

Conflict of interest

The authors declare no conflicts of interest.

Additional information

Communicated by Jordi Garcia-Ojalvo.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huang, Y., Gao, Y., Huang, Q. et al. Controlling flat-foot limit cycle walkers with compliant joints based on local stability variation. Biol Cybern 118, 111–126 (2024). https://doi.org/10.1007/s00422-024-00987-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00422-024-00987-y

Keywords

Navigation