Nonholonomic Hybrid Zero Dynamics for the Stabilization of Periodic Orbits: Application to Underactuated Robotic Walking

This brief addresses zero dynamics associated with relative degree one and two nonholonomic outputs for exponential stabilization of given periodic orbits for hybrid models of bipedal locomotion. Zero dynamics manifolds are constructed to contain the orbit while being invariant under both the contin...

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Bibliographic Details
Published in:IEEE transactions on control systems technology 2020-11, Vol.28 (6), p.2689-2696
Main Authors: Akbari Hamed, Kaveh, Ames, Aaron D.
Format: Article
Language:English
Subjects:
Control stability
Hybrid zero dynamics (HZD)
Legged locomotion
Locomotion
Manifolds
Mathematical model
Mathematical models
Model reduction
nonholonomic outputs
Numerical models
Orbital mechanics
Orbital stability
Orbits
periodic orbits
Robot kinematics
Stability analysis
underactuated bipedal robots
Walking
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Summary:This brief addresses zero dynamics associated with relative degree one and two nonholonomic outputs for exponential stabilization of given periodic orbits for hybrid models of bipedal locomotion. Zero dynamics manifolds are constructed to contain the orbit while being invariant under both the continuous- and discrete-time dynamics. The associated restriction dynamics are termed the hybrid zero dynamics (HZD). Prior results on the HZD have mainly relied on input-output linearization of holonomic outputs and are referred to as holonomic HZD (H-HZD). This brief presents reduced-order expressions for the HZD associated with nonholonomic output functions referred to as nonholonomic HZD (NH-HZD). This brief systematically synthesizes NH-HZD controllers to stabilize periodic orbits based on a reduced-order stability analysis. A comprehensive study of H-HZD and NH-HZD is presented. It is shown that NH-HZD can stabilize a broader range of walking gaits that are not stabilizable through traditional H-HZD. The power of the analytical results is finally illustrated on a hybrid model of a bipedal robot through numerical simulations.
ISSN:1063-6536
1558-0865
DOI:10.1109/TCST.2019.2947874