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Self-stability of a simple walking model driven by a rhythmic signal

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Abstract

In this paper, we analyzed the dynamic properties of a simple walking model of a biped robot driven by a rhythmic signal from an oscillator. The oscillator receives no sensory feedback and the rhythmic signal is an open loop. The simple model consists of a hip and two legs that are connected at the hip. The leg motion is generated by a rhythmic signal. In particular, we analytically examined the stability of a periodic walking motion. We obtained approximate periodic solutions and the Jacobian matrix of a Poincaré map by the power-series expansion using a small parameter. Although the analysis was inconclusive when we used only the first order expansion, by employing the second order expansion it clarified the stability, revealing that the periodic walking motion is asymptotically stable and the simple model possesses self-stability as an inherent dynamic characteristic in walking. We also clarified the stability region with respect to model parameters such as mass ratio and walking speed.

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References

  1. Adolfsson, J., Dankowicz, H., Nordmark, A.: 3D passive walkers: finding periodic gaits in the presence of discontinuities. Nonlinear Dyn. 24, 205–229 (2001)

    Article  Google Scholar 

  2. Altendorfer, R., Moore, N., Komsuoglu, H., Buehler, M., Brown Jr., H.B., McMordie, D., Saranli, U., Full, R., Koditschek, D.E.: RHex: A biologically inspired hexapod runner, Auton. Robots 11, 207–213 (2001)

    Article  Google Scholar 

  3. Altendorfer, R., Koditschek, D.E., Holmes, P.: Stability analysis of legged locomotion models by symmetry-factored return maps. Int. J. Robot. Res. 23(10–11), 979–999 (2004)

    Article  Google Scholar 

  4. Altendorfer, R., Koditschek, D.E., Holmes, P.: Stability analysis of a clock-driven rigid-body SLIP model for RHex. Int. J. Robot. Res. 23(10–11), 1001–1012 (2004)

    Article  Google Scholar 

  5. Aoi, S., Tsuchiya, K.: Locomotion control of a biped robot using nonlinear oscillators. Auton. Robots 19(3), 219–232 (2005)

    Article  Google Scholar 

  6. Aoi, S., Tsuchiya, K., Tsujita, K.: Turning control of a biped locomotion robot using nonlinear oscillators. In: Proceedings of the IEEE International Conference on Robotics and Automation, pp. 3043–3048 (2004)

  7. Aoi, S., Tsuchiya, K.: Bifurcation and chaos of a simple walking model driven by a rhythmic signal. Int. J. Non-linear Mech. 41(3), 438–446 (2006)

    Article  Google Scholar 

  8. Asano, F., Yamakita, M.: Virtual gravity and coupling control for robotic gait synthesis. IEEE Trans. System, Man, and Cybernetics – Part A: System and Humans 31(6), 737–745 (2001)

    Google Scholar 

  9. Coleman, M.J., Chatterjee, A., Ruina, A.: Motions of a rimless spoked wheel: a simple three-dimensional system with impacts. Dyn. Stab. Syst. 12(3), 139–160 (1997)

    MathSciNet  Google Scholar 

  10. Coleman, M.J., Ruina, A.: An uncontrolled walking toy that cannot stand still. Phys. Rev. Lett. 80, 3658–3661 (1998)

    Article  Google Scholar 

  11. Collins, S.H., Wisse, M., Ruina, A.: A three-dimensional passive-dynamic walking robot with two legs and knees. Int. J. Robot. Res. 20(7), 607–615 (2001)

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Google Scholar 

  14. Geyer, H., Seyfarth, A., Blickhan, R.: Spring-mass running: simple approximate solution and application to gait stability. J. Theor. Biol. 232(3), 315–328 (2005)

    MathSciNet  Google Scholar 

  15. Ghigliazza, R., Altendorfer, R., Holmes, P., Koditschek, D.E.: A simply stabilized running model. SIAM J. Appl. Dyn. Syst. 2(2), 187–218 (2003)

    Article  MathSciNet  Google Scholar 

  16. Goswami, A., Espiau, B., Keramane, A.: Limit cycles in a passive compass gait biped and passivity-mimicking control laws. Auton. Robots 4, 273–286 (1997)

    Article  Google Scholar 

  17. 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)

    Google Scholar 

  18. Grillner, S.: Control of Locomotion in Bipeds, Tetrapods and Fish, pp. 1179–1236. Handbook of Physiology, American Physiological Society, Bethesda, MD (1981)

  19. Grillner, S.: Neurobiological bases of rhythmic motor acts in vertebrates. Science 228, 143–149 (1985)

    Article  Google Scholar 

  20. Grizzle, J.W., Plestan, F., Abba, G.: Poincaré’s method for systems with impulse effects: application to mechanical biped locomotion. In: Proceedings of the IEEE International Conference on Decision and Control, pp. 3869–3876 (1999)

  21. Grizzle, J.W., Abba, G., Plestan, F.: Asymptotically stable walking for biped robots: analysis via systems with impulse effects. IEEE Trans. Autom. Control 46(1), 51–64 (2001)

    Article  MathSciNet  Google Scholar 

  22. Hase, K., Miyashita, K., Ok, S., Arakawa, Y.: Human gait simulation with a neuro-musculo-skeletal model and evolutionary computation. J. Vis. Comput. Anim. 14, 73–92 (2003)

    Article  Google Scholar 

  23. Hiskens, I.A.: Stability of hybrid system limit cycles: Application to the compass gait biped robot. In: Proceedings of the IEEE International Conference on Decision and Control, pp. 774–779 (2001)

  24. Kuo, A.D.: Stabilization of lateral motion in passive dynamic walking. Int. J. Robot. Res. 18(9), 917–930 (1999)

    Article  Google Scholar 

  25. Kuo, A.D., Energetics of actively powered locomotion using the simplest walking model. ASME J. Biomech. Eng. 124, 113–120 (2002)

    Article  Google Scholar 

  26. Kuo, A.D.: The relative roles of feedforward and feedback in the control of rhythmic movements. Motor Control 6(2), 129–145.

  27. Lewis, M.A., Etienne-Cummings, R., Hartmann, M.J., Xu, Z.R., Cohen, A.H.: An in silico central pattern generator: silicon oscillator, coupling, entrainment, and physical computation. Biol. Cybern. 88, 137–151 (2003)

    Article  Google Scholar 

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

    Google Scholar 

  29. Mochon, S., McMahon, T.: Ballistic walking: an improved model. Math. Biosci. 52, 241–260 (1980)

    Article  MathSciNet  Google Scholar 

  30. Mombaur, K.D., Bock, H.G., Schlöder, J.P., Winckler, M.J., Longman, R.W.: Open-loop stable control of running robots–a numerical method for studying stability in the context of optimal control problems. In: Proceedings of the First International Conference on Climbing and Walking Robots, pp. 89–94 (1998)

  31. Mombaur, K.D., Bock, H.G., Longman, R.W.: Human-like actuated walking that is asymptotically stable without feedback. In: Proceedings of the IEEE International Conference on Robotics and Automation, pp. 4128–4133 (2001)

  32. Nakanishi, J., Morimoto, J., Endo, G., Cheng, G., Schaal, S., Kawato, M.: Learning from demonstration and adaptation of biped locomotion. Robot. Auton. Syst. 47(2–3), 79–91 (2004)

    Article  Google Scholar 

  33. Orlovsky, G.N., Deliagina, T., Grillner, S.: Neuronal Control of Locomotion: From Mollusc to Man. Oxford University Press, Oxford (1999)

  34. Osuka, K., Kirihara, K.: Motion analysis and experiments of passive walking robot QUARTET II. In: Proceedings of the IEEE International Conference on Robotics and Automation, pp. 3052–3056 (2000)

  35. Piiroinen, P.T., Dankowicz, H.J., Nordmark, A.B.: On a normal-formal analysis for a class of passive bipedal walkers. Int. J. Bif. Chaos 11(9), 2411–2425 (2001)

    Article  Google Scholar 

  36. Piiroinen, P.T., Dankowicz, H.J., Nordmark, A.B.: Breaking symmetries and constraints: transitions from 2D to 3D in passive walkers. Multibody Syst. Dyn. 10, 147–176 (2003)

    Article  MathSciNet  Google Scholar 

  37. Saranli, U., Buehler, M., Koditschek, D.E.: RHex: a simple and highly mobile hexapod robot. Int. J. Robot. Res. 20(7), 616–631 (2001)

    Article  Google Scholar 

  38. Schmitt, J. and Holmes, P., Mechanical models for insect locomotion: dynamics and stability in the horizontal plane I. Theory. Biol. Cybern. 83, 501–515 (2000)

    Article  Google Scholar 

  39. Schmitt, J., Holmes, P.: Mechanical models for insect locomotion: dynamics and stability in the horizontal plane—II. Application. Biol. Cybern. 83, 517–527 (2000)

    Article  Google Scholar 

  40. Schmitt, J., Holmes, P.: Mechanical models for insect locomotion: stability and parameter studies. Physica D 156, 139–168 (2001)

    Article  MathSciNet  Google Scholar 

  41. Schwab, A.L., Wisse, M.: Basin of attraction of the simplest walking model. In: Proceedings of the ASME International Conference on Noise and Vibration (2001)

  42. Spong, M.W., Bullo, F.: Controlled symmetries and passive walking. IEEE Trans. Autom. Control 50(7), 1025–1031 (2005)

    Article  MathSciNet  Google Scholar 

  43. Sugimoto, Y., Osuka, K.: Walking control of quasi-passive-dynamic-walking robot Quartet III based on delayed feedback control. In: Proceedings of the 5th International Conference on Climbing and Walking Robots, pp. 123–130 (2002)

  44. Sugimoto, Y., Osuka, K.: Stability analysis of passive-dynamic-walking focusing on the inner structure of poincaré map. In: Proceedings of the 12th International Conference on Advanced Robotics, pp. 236–241 (2005)

  45. Taga, G., Yamaguchi, Y., Shimizu, H.: Self-organized control of bipedal locomotion by neural oscillators. Biol. Cybern. 65, 147–159 (1991)

    Article  Google Scholar 

  46. Taga, G.: A model of the neuro-musculo-skeletal system for human locomotion I. Emergence of basic gait. Biol. Cybern. 73, 97–111 (1995)

    Google Scholar 

  47. Taga, G., A model of the neuro-musculo-skeletal system for human locomotion II. Real-time adaptability under various constraints. Biol. Cybern. 73, 113–121 (1995)

    Google Scholar 

  48. Wagner, H., Blickhan, R.: Stabilizing function of skeletal muscles: An analytical investigation. J. Theor. Biol. 199, 163–179 (1999)

    Article  Google Scholar 

  49. Wisse, M., Schwab, A.L., van der Linde, R.Q.: A 3D passive dynamic biped with yaw and roll compensation. Robotica 19, 275–284 (2001)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Aeronautics and Astronautics, Graduate School of Engineering, Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto, 606-8501, Japan

    Shinya Aoi & Kazuo Tsuchiya

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  1. Shinya Aoi
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  2. Kazuo Tsuchiya
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Correspondence to Shinya Aoi.

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Aoi, S., Tsuchiya, K. Self-stability of a simple walking model driven by a rhythmic signal. Nonlinear Dyn 48, 1–16 (2007). https://doi.org/10.1007/s11071-006-9030-3

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  • DOI: https://doi.org/10.1007/s11071-006-9030-3

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