Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-17T20:13:37.992Z Has data issue: false hasContentIssue false
  • English
  • Français

Classification of periodic and chaotic passive limit cycles for a compass-gait biped with gait asymmetries

Published online by Cambridge University Press:  23 March 2011

Jae-Sung Moon  and
Mark W. Spong
Jae-Sung Moon
Affiliation:
Department of Industrial and Enterprise Systems Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
Mark W. Spong*
Affiliation:
Department of Electrical Engineering, University of Texas at Dallas, Richardson, TX 75080, USA
*
*Corresponding author. E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we study the problem of passive walking for a compass-gait biped with gait asymmetries. In particular, we identify and classify bifurcations leading to chaos caused by the gait asymmetries because of unequal leg masses. We present bifurcation diagrams showing step period versus the ratio of leg masses at various walking slopes. The cell mapping method is used to find stable limit cycles as the parameters are varied. It is found that a variety of bifurcation diagrams can be grouped into six stages that consist of three expanding and three contracting stages. The analysis of each stage shows that marginally stable limit cycles exhibit period-doubling, period-remerging, and saddle-node bifurcations. We also show qualitative changes regarding chaos, i.e., generation and extinction of chaos follow cyclic patterns in passive dynamic walking.

Keywords

Passive dynamic walkingBifurcations and chaosGait asymmetriesSix stages of bifurcationsComposite hybrid systems
Type
Articles
Information
Robotica , Volume 29 , Issue 7 , December 2011 , pp. 967 - 974
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Aoi, S. and 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).CrossRefGoogle Scholar
2.Asano, F. and Luo, Z.-W., “Pseudo Virtual Passive Dynamic Walking and Effect of Upper Body as Counterweight,” Proceedings of IEEE/RSJ International Conference on Intelligent Robots Systems, Nice, France (2008) pp. 2934–2939.Google Scholar
3.Asano, F. and Luo, Z.-W., “On Efficiency and Optimality of Asymmetric Dynamic Bipedal Gait,” Proceedings of IEEE International Conference on Robotics Automation, Kobe, Japan (2009) pp. 1972–1977.Google Scholar
4.Bier, M. and Bountis, T. C., “Remerging feigenbaum trees in dynamical systems,” Phys. Lett. A 104 (5), 239–244 (1984).CrossRefGoogle Scholar
5.Ephanov, A. and Hurmuzlu, Y., “Generating pathological gait patterns via the use of robotic locomotion models,” J. Technol. Health Care 10, 135–146 (2002).CrossRefGoogle ScholarPubMed
6.Feigenbaum, M. J., “Quantitative universality for a class of nonlinear transformations,” J. Stat. Phys. 19 (1), 25–52 (1978).CrossRefGoogle Scholar
7.Garcia, M., Chatterjee, A., Ruina, A. and Coleman, M., “The simplest walking model: Stability, complexity and scaling,” ASME J. Biomech. Eng. 120, 281–288 (1998).CrossRefGoogle ScholarPubMed
8.Goswami, A., Espiau, B. and Keramane, A., “Limit Cycles and Their Stability in a Passive Bipedal Gait,” Proceedings of IEEE International Conference on Robotics Automation, Vol. 1, Minneapolis, MN (1996) pp. 246–251.CrossRefGoogle Scholar
9.Goswami, A., Thuilot, B. and Espiau, B., “Compass-Like Biped Robot. Part I: Stability and Bifurcation of Passive Gaits,” INRIA Technical Report no. 2996 (INRIA, Grenoble, France. 1996).Google Scholar
10.Goswami, A., Thuilot, B. and 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).CrossRefGoogle Scholar
11.Guckenheimer, J. and Holmes, P., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, New York, 1986).Google Scholar
12.Harata, Y., Asano, F., Taji, K. and Uno, Y., “Efficient Parametric Excitation Walking with Delayed Feedback Control,” Proceedings of IEEE/RSJ International Conference on Intelligent Robots Systems, St. Louis, MO (2009) pp. 2934–2939.Google Scholar
13.Hilborn, R. C., Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers (Oxford University Press, New York, 2000).CrossRefGoogle Scholar
14.Howell, G. W. and Baillieul, J., “Simple Controllable Walking Mechanisms which Exhibit Bifurcations,” Proceedings of IEEE Conference on Decision Control, Tampa, FL (1998) pp. 3027–3032.Google Scholar
15.Hsu, C. S., Cell-to-Cell Mapping : A Method of Global Analysis for Nonlinear Systems (Springer-Verlag, New York, 1987).CrossRefGoogle Scholar
16.Hurmuzlu, Y. and Moskowitz, G., “The role of impact in the stability of bipedal locomotion,” Dyn. Stab. Syst. 1 (3), 217–234 (1986).Google Scholar
17.Kurz, M. J. and Stergiou, N., “Hip actuations can be used to control bifurcations and chaos in a passive dynamic walking model,” ASME J. Biomech. Eng. 129 (2), 216–222 (2007).CrossRefGoogle Scholar
18.McGeer, T., “Passive dynamic walking,” Int. J. Robot. Res. 9 (2), 62–82 (1990).CrossRefGoogle Scholar
19.Mochon, S. and McMahon, T. A., “Ballistic walking,” J. Biomech. 13 (1), 49–57 (1980).CrossRefGoogle ScholarPubMed
20.Mochon, S. and McMahon, T. A., “Ballistic walking: An improved model,” Math. Biosci. 52 (3–4), 241–260 (1980).CrossRefGoogle Scholar
21.Moon, J.-S. and Spong, M. W., “Bifurcations and Chaos in Passive Walking of a Compass-Gait Biped with Asymmetries,” Proceedings of IEEE International Conference on Robotics Automation, Anchorage, AK, (2010) pp. 1721–1726.Google Scholar
22.Ott, E., Chaos in Dynamical Systems (Cambridge University Press, New York, 1993).Google Scholar
23.Ott, E., Grebogi, C. and Yorke, J. A., “Controlling chaos,” Phys. Rev. Lett. 64 (11), 1196–1199 (1990).CrossRefGoogle ScholarPubMed
24.Parker, T. S. and Chua, L. O., Practical Numerical Algorithms for Chaotic Systems (Springer-Verlag New York, NY, 1989).CrossRefGoogle Scholar
25.Shinbrot, T., Grebogi, C., Ott, E. and Yorke, J. A., “Using small perturbations to control chaos,” Nature 363, 411–417 (1993).CrossRefGoogle Scholar
26.Spong, M. W., Hutchinson, S. and Vidyasagar, M., Robot Modeling and Control (Wiley, Hoboken, NJ, 2006).Google Scholar
27.Strogatz, S. H., Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (Addison-Wesley, Reading, MA, 1994).Google Scholar
28.Suzuki, S. and Furuta, K., “Enhancement of Stabilization for Passive Walking by Chaos Control Approach,” Proceedings of IFAC Triennial World Congress, Vol. B, Barcelona, Spain (2002) pp. 133–138.Google Scholar