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A different switching surface stabilizing an existing unstable periodic gait: an analysis based on perturbation theory

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Abstract

Limit cycle walkers are known as a class of walking robots capable of presenting periodic repetitive gaits without having local controllability at all times during motion. A well-known subclass of these robots is McGeer’s passive dynamic walkers solely activated by the gravity field. The mathematics governing this style of walking is hybrid and described by a set of nonlinear differential equations along with impulses. In this paper, by applying perturbation method to a simple model of these machines, we analytically prove that for this type of nonlinear impulsive system, there exist different switching surfaces, leading to the same equilibrium points (periodic solutions) with different stabilities. Furthermore, it has been shown that the number of existing periodic solutions depends on the characteristics of the switching surface.

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References

  1. An, K., Chen, Q.: A passive dynamic walking model based on knee-bend behaviour: stability and adaptability for walking down steep slopes. Int. J. Adv. Robot. Syst. 10, 1–11 (2013)

  2. Bhounsule, P.A., Cortell, J., Grewal, A., Hendriksen, B., Karssen, J.D., Paul, C., Ruina, A.: Low-bandwidth reflex-based control for lower power walking: 65 km on a single battery charge. Int. J. Robot. Res. 33(10), 1305–1321 (2014)

  3. Borzova, E., Hurmuzlu, Y.: Passively walking five-link robot. Automatica 40(4), 621–629 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Byl, K., Tedrake, R.: Stability of passive dynamic walking on uneven terrain. In: Proceedings of Dynamic Walking (2006)

  5. Byl, K., Tedrake, R.: Metastable walking machines. Int. J. Robot. Res. 28(8), 1040–1064 (2009)

    Article  Google Scholar 

  6. Collins, S., Ruina, A., Tedrake, R., Wisse, M.: Efficient bipedal robots based on passive-dynamic walkers. Science 307(5712), 1082–1085 (2005)

    Article  Google Scholar 

  7. Das, S., Chatterjee, A.: An alternative stability analysis technique for the simplest walker. Nonlinear Dyn. 28(3–4), 273–284 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Garcia, M., Chatterjee, A., Ruina, A., Coleman, M.: The simplest walking model: stability, complexity, and scaling. J. Biomech. Eng. 120(2), 281–288 (1998)

    Article  Google Scholar 

  9. Goswami, A., Thuilot, B., Espiau, B.: A study of the passive gait of a compass-like biped robot symmetry and chaos. Int. J. Robot. Res. 17(12), 1282–1301 (1998)

    Article  Google Scholar 

  10. Gritli, H., Khraief, N., Belghith, S.: Period-three route to chaos induced by a cyclic-fold bifurcation in passive dynamic walking of a compass-gait biped robot. Commun. Nonlinear Sci. Numer. Simul. 17(11), 4356–4372 (2012)

    Article  MathSciNet  Google Scholar 

  11. Hass, J., Herrmann, J.M., Geisel, T.: Optimal mass distribution for passivity-based bipedal robots. Int. J. Robot. Res. 25(11), 1087–1098 (2006)

    Article  Google Scholar 

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

    Article  Google Scholar 

  13. Hobbelen, D.G., Wisse, M.: Swing-leg retraction for limit cycle walkers improves disturbance rejection. IEEE Trans. Robot. 24(2), 377–389 (2008)

    Article  Google Scholar 

  14. Huang, Y., Wang, Q., Xie, G., Wang, L.: Optimal mass distribution for a passive dynamic biped with upper body considering speed, efficiency and stability. In: 8th IEEE-RAS International Conference on Humanoid Robots, 2008. Humanoids 2008, pp. 515–520. IEEE (2008)

  15. Hurmuzlu, Y., Moskowitz, G.D.: The role of impact in the stability of bipedal locomotion. Dyn. Stab. Syst. 1(3), 217–234 (1986)

    Article  Google Scholar 

  16. Hurmuzlu, Y., Moskowitz, G.D.: Bipedal locomotion stabilized by impact and switching: I. two-and three-dimensional, three-element models. Dyn. Stab. Syst. 2(2), 73–96 (1987)

    Article  MATH  Google Scholar 

  17. Hurmuzlu, Y., Moskowitz, G.D.: Bipedal locomotion stabilized by impact and switching: Il. structural stability analysis of a four-element bipedal locomotion model. Dyn. Stab. Syst. 2(2), 97–112 (1987)

  18. Iqbal, S., Zang, X., Zhu, Y., Zhao, J.: Bifurcations and chaos in passive dynamic walking: a review. Robot. Auton. Syst. 62(6), 889–909 (2014)

    Article  Google Scholar 

  19. Kwan, M., Hubbard, M.: Optimal foot shape for a passive dynamic biped. J. Theor. Biol. 248(2), 331–339 (2007)

    Article  Google Scholar 

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

    Article  Google Scholar 

  21. McGeer, T.: Passive walking with knees. In: 1990 IEEE International Conference on Robotics and Automation, 1990. Proceedings, pp. 1640–1645. IEEE (1990)

  22. Ning, L., Junfeng, L., Tianshu, W.: The effects of parameter variation on the gaits of passive walking models: simulations and experiments. Robotica 27(04), 511–528 (2009)

    Article  Google Scholar 

  23. Qi, F., Wang, T., Li, J.: The elastic contact influences on passive walking gaits. Robotica 29(05), 787–796 (2011)

    Article  MATH  Google Scholar 

  24. Safa, A.T., Naraghi, M.: The role of walking surface in enhancing the stability of the simplest passive dynamic biped. Robotica 33(01), 195–207 (2015)

    Article  Google Scholar 

  25. Schwab, A., Wisse, M.: Basin of attraction of the simplest walking model. In: Proceedings of the ASME Design Engineering Technical Conference, vol. 6, pp. 531–539 (2001)

  26. Wang, Q., Huang, Y., Zhu, J., Wang, L., Lv, D.: Effects of foot shape on energetic efficiency and dynamic stability of passive dynamic biped with upper body. Int. J. Humanoid Robot. 7(02), 295–313 (2010)

    Article  MATH  Google Scholar 

  27. Wisse, M., Atkeson, C.G., Kloimwieder, D.K.: Swing leg retraction helps biped walking stability. In: 2005 5th IEEE-RAS International Conference on Humanoid Robots, pp. 295–300. IEEE (2005)

  28. Wisse, M., Schwab, A.L., van der Helm, F.C.: Passive dynamic walking model with upper body. Robotica 22(06), 681–688 (2004)

    Article  Google Scholar 

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Authors and Affiliations

  1. School of Mechanical Engineering, Amirkabir University of Technology, No. 424, Hafez Avenue, 15914, Tehran, Iran

    Ali Tehrani Safa & Mahyar Naraghi

  2. School of Mechanical Engineering, Sharif University of Technology, Azadi Avenue, 14588, Tehran, Iran

    Aria Alasty

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  1. Ali Tehrani Safa
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  2. Aria Alasty
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  3. Mahyar Naraghi
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Correspondence to Mahyar Naraghi.

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Safa, A.T., Alasty, A. & Naraghi, M. A different switching surface stabilizing an existing unstable periodic gait: an analysis based on perturbation theory. Nonlinear Dyn 81, 2127–2140 (2015). https://doi.org/10.1007/s11071-015-2130-1

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  • DOI: https://doi.org/10.1007/s11071-015-2130-1

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