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New bifurcations in the simplest passive walking model.

Qingdu Li1, Song Tang1, Xiao-Song Yang1

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This study reveals new stable periodic gaits in a simple passive walking robot model, expanding beyond previously known gaits. These new gaits exhibit complex behaviors, including period-doubling leading to chaos and cyclic-fold bifurcations.

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Area of Science:

  • Robotics
  • Dynamical Systems Theory
  • Bipedal Locomotion

Background:

  • The simplest passive walking bipedal model is a foundational tool for studying robotic locomotion.
  • Previous research identified period-1 gaits in this model.
  • Understanding the full range of dynamic behaviors is crucial for advancing legged robotics.

Purpose of the Study:

  • To uncover new stable periodic gaits in the simplest passive walking bipedal model.
  • To investigate the stability and bifurcation phenomena associated with these newly discovered gaits.
  • To provide rigorous mathematical confirmation of the observed gaits and their associated bifurcations.

Main Methods:

  • Computer simulations to identify and observe new periodic gaits.
  • Analysis of period-doubling and cyclic-fold bifurcations.
  • Computer-assisted proofs using the interval Newton method.
  • Topological horseshoes theory for chaos verification.

Main Results:

  • Discovery of stable periodic gaits with periods 3 to 7, in addition to period-1 gaits.
  • Demonstration that new gaits lead to chaos via period-doubling bifurcation.
  • Observation of gaits losing stability through cyclic-fold bifurcation.
  • Identification of novel bifurcation scenarios.

Conclusions:

  • The simplest passive walking bipedal model exhibits a richer set of stable periodic gaits than previously known.
  • These gaits undergo complex bifurcations, including period-doubling to chaos and cyclic-fold bifurcations.
  • Computer-assisted proofs confirm the existence and stability of these gaits and their associated complex dynamics.