New walking dynamics in the simplest passive bipedal walking model
Qingdu Li
a
, Xiao-Song Yang
b,
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a
Key Laboratory of Network Control and Intelligent Instrument, Ministry of Education, Chongqing University of Posts and Telecommunications,
Chongqing 400065, China
b
Department of Mathematics, Huazhong University of Science and Technology, Wuhan 430074, China
article info
Article history:
Received 18 February 2011
Received in revised form 16 December 2011
Accepted 21 December 2011
Available online 30 December 2011
Keywords:
Periodic gaits
Chaos
Topological horseshoe
Basin of attraction
Passive bipedal model
abstract
This paper revisits the simplest passive walking model by Garcia et al. which displays chaos
through period doubling from a stable period-1 gait. By carefully numerical studies, two
new gaits with period-3 and -4 are found, whose stability is verified by estimates of eigen-
values of the corresponding Jacobian matrices. A surprising phenomenon uncovered here is
that they both lead to higher periodic cycles and chaos via period doubling. To study the
three different types of chaotic gaits rigorously, the existence of horseshoes is verified
and estimates of the topological entropies are made by computer-assisted proofs in terms
of topological horseshoe theory.
Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction
Walking is an activity that most of us do on a daily basis, generally with little cognitive effort or even without care. How-
ever, walking is an extremely complicated task and not very well understood. Inspired by the success of legged animals,
walking research was carried out by people coming from very different backgrounds and currently is an extremely popular
area of scientific research [1–3].
With the belief that bipedal walking might be largely understood as a passive mechanical process, McGeer demonstrated
in 1990 [4], by both physical-model construction and computer simulation, that some anthropomorphic legged mechanisms
can exhibit stable, human-like walking on a range of shallow slopes with no actuation and no control (energy lost in friction
and collisions is recovered from gravity). Unlike control-based models of animal locomotion, where the controller tries to
force a motion on the system, gait cycles of McGeer’s models (sequences of exactly-repeated steps) are inherent consequence
of the models’ dynamics for the given parameters.
McGeer’s results on passive dynamic walking machines suggest that the mechanical parameters of the human body such
as lengths, mass distributions, have a greater effect on the existence and quality of gait. This implies that one needs to study
mechanics, not just activation and control, to fully understand walking. Thus studying dynamics of various passive walking
models is important to understanding the mechanism of walking of animals and human beings and control design of legged
robots.
To get a better sense of the role of passive dynamics, Garcia et al. considered a simpler model, i.e., a limiting case of the
straight-legged walker of McGeer [5]. Comparing many of its modifications, such as the double-pendulum (‘compass-gait’)
point-foot model studied by Goswami et al. [6,7], the model with upper body presented by Wisse et al. [8], the model replac-
ing ramps with stairs by Safa et al. [9], and the model with toed feet studied by Kumar et al. [10], etc., it is regarded as the
0307-904X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved.
doi:10.1016/j.apm.2011.12.049
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Corresponding author.
E-mail addresses: ql78@cornell.edu (Q. Li), yangxs@mail.hust.edu.cn (X.-S. Yang).
Applied Mathematical Modelling 36 (2012) 5262–5271
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Applied Mathematical Modelling
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