Stabilization of a class of underactuated systems

Rosas-Flores, J.A.; Álvarez-Gallegos, Jaime; Castro‐Linares, R.

doi:10.1109/cdc.2000.914116

Cited by 7 publications

(1 citation statement)

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“…While the zero dynamics for a system modeled by ordinary differential equations is a well-known [31] and increasingly used concept, [5], [33], [51], [59], the hybrid zero dynamics is a novel notion developed in this paper to deal with the impact map that is common in legged locomotion models. The hybrid zero dynamics may be defined analogously to the zero dynamics: the largest internal dynamics compatible with the output being identically zero.…”

Section: Introductionmentioning

confidence: 99%

Hybrid zero dynamics of planar biped walkers

Westervelt

Grizzle

Koditschek

2003

IEEE Trans. Automat. Contr.

758

721

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Planar, underactuated, biped walkers form an important domain of applications for hybrid dynamical systems. This paper presents the design of exponentially stable walking controllers for general planar bipedal systems that have one degree-of-freedom greater than the number of available actuators. The within-step control action creates an attracting invariant set-a two-dimensional zero dynamics submanifold of the full hybrid model-whose restriction dynamics admits a scalar linear time-invariant return map. Exponentially stable periodic orbits of the zero dynamics correspond to exponentially stabilizable orbits of the full model. A convenient parameterization of the hybrid zero dynamics is imposed through the choice of a class of output functions. Parameter optimization is used to tune the hybrid zero dynamics in order to achieve closed-loop, exponentially stable walking with low energy consumption, while meeting natural kinematic and dynamic constraints. The general theory developed in the paper is illustrated on a five link walker, consisting of a torso and two legs with knees. KeywordsBipeds, hybrid systems, Poincaré sections, underactuated system, zero dynamics This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of the University of Pennsylvania's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to pubs-permissions@ieee.org. By choosing to view this document, you agree to all provisions of the copyright laws protecting it. Abstract-Planar, underactuated, biped walkers form an important domain of applications for hybrid dynamical systems. This paper presents the design of exponentially stable walking controllers for general planar bipedal systems that have one degree-of-freedom greater than the number of available actuators. The within-step control action creates an attracting invariant set-a two-dimensional zero dynamics submanifold of the full hybrid model-whose restriction dynamics admits a scalar linear time-invariant return map. Exponentially stable periodic orbits of the zero dynamics correspond to exponentially stabilizable orbits of the full model. A convenient parameterization of the hybrid zero dynamics is imposed through the choice of a class of output functions. Parameter optimization is used to tune the hybrid zero dynamics in order to achieve closed-loop, exponentially stable walking with low energy consumption, while meeting natural kinematic and dynamic constraints. The general theory developed in the paper is illustrated on a five link walker, consisting of a torso and two legs with knees.

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