Kybernetika 49 no. 5, 792-808, 2013

Parameter influence on passive dynamic walking of a robot with flat feet

Xiangze Lin, Haibo Du and Shihua Li


The biped robot with flat feet and fixed ankles walking down a slope is a typical impulsive dynamic system. Steady passive gaits for such mechanism can be induced on certain shallow slopes without actuation. The steady gaits can be described by using stable non-smooth limit cycles in phase plane. In this paper, it is shown that the robot gaits are affected by three parameters, namely the ground slope, the length of the foot, and the mass ratio of the robot. As the ground slope is gradually increased, the gaits exhibit universal period doubling bifurcations leading to chaos. Meanwhile, the phenomena of period doubling bifurcations also occur by increasing either the foot length or the mass ratio of the robot. Theory analysis and numerical simulations are given to verify our conclusion.


chaos, biped robot, impulse dynamic systems, limit cycles, bifurcations


93A14, 93C10, 93D15, 93D21


  1. E. Borzova and Y. Hurmuzlu: Passively walking five-link robot. Automatica 40 (2004), 4, 621-629.   CrossRef
  2. S. Collins, A. Ruina, R. Tedrake and M. Wisse: Efficient bipedal robots based on passive dynamic walkers. Science 307 (2005), 1082-1085.   CrossRef
  3. M. Garcia, A. Chatterjee, A. Ruina and M. Coleman: The simplest walking model: Stability, complexity, and scaling. J. Biomech. Engrg. 120 (1998), 2, 281-288.   CrossRef
  4. A. Goswami, B Thuilot and B. Espiau: Compass-like biped robot Part I: Stability and bifurcation of passive gaits. IINRIA Res. Rep. No. 2996, 1996.   CrossRef
  5. A. Goswami, B. Thuilot and B. Espiau: A study of the passive gait of a compass-like biped robot: Symmetry and chaos. Internat. J. Robotics Res. 17 (1998), 12, 1282-1301.   CrossRef
  6. J. W. Grizzle, G. Abba and F. Plestan: Asymptotically stable walking for biped robots: Analysis via systems with impulse effects. IEEE Trans. Automat. Control 46 (2001), 1, 51-64.   CrossRef
  7. I. Hiskens: Stability of hybrid system limit cycles: Application to the compass gait biped robot. In: Proc. IEEE Conference on Decision and Control, Orlando 2001.   CrossRef
  8. J. K. Holm: Control of Passive Dynamic Robots Using Artificial Potential Energy Fields. M. S. Thesis, Univ. Illinois Urbana-Champaign 2005.   CrossRef
  9. J. K. Holm, D. J. Lee and M. W. Spong: Time scaling for speed regulation in bipedal locomotion. In: Proc. IEEE Conference Robotics Automation, Rome 2007, pp. 3603-3608.   CrossRef
  10. Y. Ikemata, A. Sano and H. Fusimoto: Analysis of stable limit cycle in passive walking. In: SICE Anual Conference, Fubukui 2003, pp. 117-122.   CrossRef
  11. J. Kim, C. Choi and M. W. Spong: Passive dynamic walking with symmetric fixed flat feet. In: Proc. IEEE International Conference on Control and Automation, Guangzhou 2007, pp. 24-30.   CrossRef
  12. A. D. Kuo: Stabilization of lateral motion in passive dynamic walking. Internat. J. Robotics Res. 18 (1999), 9, 917-930.   CrossRef
  13. M. J. Kurz, T. N. Judkins, C. Arellano and M. Scott-Pandorf: A passive dynamic walking robot that has a deterministic nonlinear gait. J. Biomech. 41 (2008), 6, 1310-1316.   CrossRef
  14. M. J. Kurz and N. Stergiou: An artificial neural network that utilizes hip joint actuations to control bifurcations and chaos in a passive dynamic bipedal walking model. Biolog. Cybernet. 93 (2005), 3, 213-221.   CrossRef
  15. M. J. Kurz and N. Stergiou: Hip actuations can be used to control bifurcations and chaos in a passive dynamic walking model. J. Biomech. Engrg. 129 (2007), 2, 216-222.   CrossRef
  16. X. Lin, Y. Ding, M. Shen and S. Li: Feedback stabilization of unstable periodic orbits for chaotic passive compass-like biped robot. In: Proc. 7th World Congress on Intelligent Control and Automation, Chongqing 2008, pp. 7285-7290.   CrossRef
  17. T. McGeer: Passive dynamic walking. Internat. J. Robotics Res. 9 (1990), 2, 62-68.   CrossRef
  18. T. McGeer: Passive walking with knees. In: Proc. IEEE Conference on Robotics and Automation, Cincinnati 1990, pp. 1640-1645.   CrossRef
  19. D. J. Miller, N. Stergiou and M. J. Kurz: An improved surrogate method for detecting the presence of chaos in gait. J. Biomech. 39 (2006), 15, 2873-2876.   CrossRef
  20. E. Ott: Chaos in Dynamical Systems. Cambridge University Press, Cambridge 1993.   CrossRef
  21. T. S. Parker and L. O. Chua: Practical Numerical Algorithms for Chaotic Systems. Springer-Verlag, New York 1989.   CrossRef
  22. R. Seydel: Practical Bifurcation and Stability Analysis. Second edition. Springer-Verlag, New York 1994.   CrossRef
  23. M. W. Spong: Passivity based control of the compass gait biped. In: Proc. World Congress of IFAC, Beijing 1999, pp. 19-24.   CrossRef
  24. M. W. Spong and G. Bhatia: Further results on control of the compass gait biped. In: Proc. IEEE International Conference on Intelligent Robots and Systems, Las Vegas 2003, pp. 1933-1938.   CrossRef
  25. M. W. Spong and F. Bullo: Controlled symmetries and passive walking. IEEE Trans. Automat. Control 50 (2005), 7, 1025-1031.   CrossRef
  26. N. Stergiou, U. H. Buzzi, M. J. Kurz and J. Heidel: Nonlinear tools in human movement. In: Innovative Analysis of Human Movement, Human Kinetics (N. Stergiou, ed.), Champaign 2004.   CrossRef
  27. M. Wisse, A. L. Schwab and F. C. T. Van der Helm: Passive dynamic walking model with upper body. Robotica 22 (2004), 6, 681-688.   CrossRef