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Limit Cycle Gaits

  • Fumihiko Asano
Reference work entry

Abstract

Limit cycle walking is an approach to control of legged locomotion robots not as a robot arm fixed on the floor but as a limit cycle with impulse effects and is known as the most useful way to achieve energy-efficient robotic dynamic walking. This chapter mainly describes the mathematical basis for modeling of planar walkers, and introduces several major approaches to generation of limit cycle gaits on level ground. First, a model of a planar fully actuated compass-like biped robot with flat feet is introduced. The 2-DOF equation of motion for the single-support phase is developed according to the Lagrange’s method. The 6-DOF collision equation corresponding to the extended generalized coordinates is also developed on the assumption that the rear leg leaves the ground immediately after landing of the fore leg. Second, several methods for generating energy-efficient limit cycle gaits are introduced. Different biped models are considered depending on the methods, but the idea of restoring the lost kinetic energy at impact efficiently is common to all. With the introduction of these methods, energy efficiency calculation method and output-following control law are also explained. Third, the principles and analysis methods of limit cycle stability recently developed are described using passive and underactuated rimless wheel models. The limit cycle gait is treated as a linear time-invariant system with instantaneous state jumps, and the gait stability can be analytically determined without conducting numerical simulations.

References

  1. 1.
    M. Vukobratović, On the stability of anthropomorphic systems. Math. Biosci. 5(1–2), 1–37 (1972)CrossRefGoogle Scholar
  2. 2.
    T. McGeer, Passive dynamic walking. Int. J. Robot. Res. 9(2), 62–82 (1990)CrossRefGoogle Scholar
  3. 3.
    T. McGeer, Passive walking with knees, in Proceedings of the IEEE International Conference on Robotics and Automation, vol. 3, 1990, pp. 1640–1645Google Scholar
  4. 4.
    E.R. Westervelt, J.W. Grizzle, C. Chevallereau, J.H. Choi, B. Morris, Feedback Control of Dynamic Bipedal Robot Locomotion (CRC Press, New York, 2007)CrossRefGoogle Scholar
  5. 5.
    M. Wisse, R.Q. van der Linde, Delft Pneumatic Bipeds (Springer, Berlin, 2007)CrossRefGoogle Scholar
  6. 6.
    F. Asano, Z.-W. Luo, Energy-efficient and high-speed dynamic biped locomotion based on principle of parametric excitation. IEEE Trans. Robot. 24(6), 1289–1301 (2008)CrossRefGoogle Scholar
  7. 7.
    F. Asano, Z.-W. Luo, Efficient dynamic bipedal walking using effects of semicircular feet. Robotica 29(3), 351–365 (2011)CrossRefGoogle Scholar
  8. 8.
    A. Goswami, B. Espiau, A. Keramane, Limit cycles in a passive compass gait biped and passivity-mimicking control laws. Auton. Robot. 4(3), 273–286 (1997)Google Scholar
  9. 9.
    F. Asano, M. Yamakita, N. Kamamichi, Z.-W. Luo, A novel gait generation for biped walking robots based on mechanical energy constraint. IEEE Trans. Robot. Autom. 20(3), 565–573 (2004)CrossRefGoogle Scholar
  10. 10.
    F. Asano, Z.-W. Luo, M. Yamakita, Biped gait generation and control based on a unified property of passive dynamic walking. IEEE Trans. Robot. 21(4), 754–762 (2005)CrossRefGoogle Scholar
  11. 11.
    M.W. Spong, F. Bullo, Controlled symmetries and passive walking. IEEE Trans. Autom. Control 50(7), 1025–1031 (2005)MathSciNetCrossRefGoogle Scholar
  12. 12.
    A. Goswami, B. Thuilot, B. Espiau, A study of the passive gait of a compass-like biped robot: symmetry and chaos. Int. J. Robot. Res. 17(12), 1282–1301 (1998)CrossRefGoogle Scholar
  13. 13.
    F. Asano, T. Hayashi, Z.-W. Luo, S. Hirano, A. Kato, Parametric excitation approaches to efficient dynamic biped locomotion, in Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, 2007, pp. 2210–2216Google Scholar
  14. 14.
    T. Hayashi, K. Kaneko, F. Asano, Z.-W. Luo, Experimental study of dynamic bipedal walking based on the principle of parametric excitation with counterweights. Adv. Robot. 25(1–2), 273–287 (2011)Google Scholar
  15. 15.
    F. Asano, Z.-W. Luo, The effect of semicircular feet on energy dissipation by heel-strike in dynamic biped locomotion, in Proceedings of the IEEE International Conference on Robotics and Automation, 2007, pp. 3976–3981Google Scholar
  16. 16.
    F. Asano, Z.-W. Luo, Dynamic analyses of underactuated virtual passive dynamic walking, in Proceedings of the IEEE International Conference on Robotics and Automation, 2007, pp. 3210–3217Google Scholar
  17. 17.
    D.G.E. Hobbelen, M. Wisse, Swing-leg retraction for limit cycle walkers improves disturbance rejection. IEEE Trans. Robot. 24(2), 377–389 (2008)CrossRefGoogle Scholar
  18. 18.
    M.J. Coleman, A. Chatterjee, A. Ruina, Motions of a rimless spoked wheel: a simple three-dimensional system with impacts. Dyn. Stab. Syst. 12(3), 139–159 (1997)MathSciNetCrossRefGoogle Scholar
  19. 19.
    M.J. Coleman, Dynamics and stability of a rimless spoked wheel: a simple 2D system with impacts. Dyn. Syst. 25(2), 215–238 (2010)MathSciNetCrossRefGoogle Scholar
  20. 20.
    F. Asano, High-speed dynamic gait generation for limit cycle walkers based on forward-tilting impact posture. Multibody Syst. Dyn. 30(3), 287–310 (2013)MathSciNetCrossRefGoogle Scholar
  21. 21.
    J.W. Grizzle, G. Abba, F. Plestan, Asymptotically stable walking for biped robots: analysis via systems with impulse effects. IEEE Trans. Autom. Control 46(1), 51–64 (2001)MathSciNetCrossRefGoogle Scholar
  22. 22.
    E.R. Westervelt, J.W. Grizzle, D.E. Koditschek, Hybrid zero dynamics of planar biped walkers. IEEE Trans. Autom. Control 48(1), 42–56 (2003)MathSciNetCrossRefGoogle Scholar
  23. 23.
    F. Asano, 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 (2015)MathSciNetCrossRefGoogle Scholar
  24. 24.
    F. Asano, M. Suguro, Limit cycle walking, running, and skipping of telescopic-legged rimless wheel. Robotica 30(6), 989–1003 (2012)CrossRefGoogle Scholar
  25. 25.
    M. Garcia, A. Chatterjee, A. Ruina, Efficiency, speed, and scaling of two-dimensional passive-dynamic walking. Dyn. Stab. Syst. 15(2), 75–99 (2000)MathSciNetCrossRefGoogle Scholar
  26. 26.
    J.-S. Moon, M.W. Spong, Classification of periodic and chaotic passive limit cycles for a compass-gait biped with gait asymmetries. Robotica 29(7), 967–974 (2011)CrossRefGoogle Scholar
  27. 27.
    S. Iqbal, X. Zang, Y. Zhu, J. Zhao, Bifurcations and chaos in passive dynamic walking: a review. Robot. Auton. Syst. 62(6), 889–909 (2014)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.School of Information ScienceJapan Advanced Institute of Science and TechnologyIshikawaJapan

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