This paper uncovers several new stable periodic gaits in the simplest passive walking bipedal model proposed in the literature. It is demonstrated that the model has period-3 to period-7 gaits beside the period-1 gaits found by Garcia et al. By simulations, this paper shows that each of these new gaits leads to chaos via period-doubling bifurcation and loses its stability by cyclic-fold bifurcation. This interesting phenomenon suggests a series of new bifurcation scenarios that have not been observed before. To confirm the new gaits and their bifurcations, this paper presents computer assisted proofs on the existence and stability of each periodic gait and its period-doubling gaits with the interval Newton method. To verify that the routes indeed lead to chaos, computer-assisted proofs are also given by means of topological horseshoes theory.
Received 30 March 2013Accepted 30 September 2013Published online 25 October 2013
Lead Paragraph: This paper shows new bifurcation scenarios that have not been observed before in the simplest passive walking model proposed by Garcia et al. It demonstrated that there are many period-doubling routes to chaos in this model. To verify our finding rigorously, the paper presents computer assisted proofs about the existence and stability of each periodic gait, its first period-doubling gait and chaos.It expected that the new dynamics found in this paper could be of potential implication to dynamical walking, especially to experimental studies on passive dynamical walking. On the other hand, it is demonstrated that even in this simplest passive dynamic model more dynamic bifurcation scenarios can take place because of the hybrid nature of this model. Therefore, further investigation of the new bifurcations numerically found in the present paper should be of much interest from hybrid dynamical systems point of view.
We are very grateful to the reviewers for their valuable comments and suggestions. This work was supported in part by National Natural Science Foundation of China (61104150) and Science Fund for Distinguished Young Scholars of Chongqing (cstc2013jcyjjq40001).
Article outline: I. INTRODUCTION II. NEW BIFURCATIONS IN THE PASSIVE WALKING MODEL III. COMPUTER ASSISTED PROOFS OF THE STABLE PERIOD-n GAITS IV. THE BIFURCATION AND CONTINUATION OF PERIOD-n GAITS V. CONCLUSIONS
3.G. Berman and J. A. Ting, Exploring Passive-Dynamic Walking (Complex Systems Summer School, 2005).
4.B. W. Verdaasdonk, H. F. J. M. Koopman, and F. C. T. van der Helm, “Energy efficient walking with central pattern generators: From passive dynamic walking to biologically inspired control,” Biol. Cybern.101(1), 49–61 (2009).
11.A. Schwab and M. Wisse, “Basin of attraction of the simplest walking model,” in Proceedings of DETC'01 ASME 2001 Design Engineering Technical Conferences and Computers and Information in Engineering Conference Pittsburgh, Pennsylvania, September 9–12, 2001.
12.Q. Li, H. Zhou, and X.-S. Yang, “A study of basin of attraction of the simplest walking model based on heterogeneous computation,” Acta Phys. Sin.61(4), 040503 (2012).
13.Q. Li and X. S. Yang, “New walking dynamics in the simplest passive bipedal walking model,” Appl. Math. Model.36(11), 5262–5271 (2012).
16.G. Alefeld, “Inclusion methods for systems of nonlinear equations—The interval Newton method and modifications,” in Topics in Validated Computations, edited by J. Herzberger (Elsevier Science Publishers, Amsterdam, 1994), pp. 7–26.