On the Geometric Reduction of Controlled Three-Dimensional Bipedal Robotic Walkers

Ames, Aaron D.; Gregg, Robert D.; Wendel, Eric; Sastry, S. Shankar

doi:10.1007/978-3-540-73890-9_14

Cited by 37 publications

(41 citation statements)

References 7 publications

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“…This definition is not meant to be all-inclusivemany different forms of "hybrid Lagrangians" have appeared in the literature (cf. [5], [6], [7], [9] to name a few), although not under this specific name-but rather concrete enough to allow for explicit constructions, while general enough to include an interesting class of systems (such as bipedal robotic walkers [2]). The definition of a hybrid Lagrangian motivates the definition of a Lagrangian hybrid system; we explicitly construct Lagrangian hybrid systems from hybrid Lagrangians.…”

Section: Hybrid Lagrangiansmentioning

confidence: 99%

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Hybrid Routhian reduction of Lagrangian hybrid systems

Ames¹,

Sastry²

2006

2006 American Control Conference

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Abstract-This paper extends Routhian reduction to a hybrid setting, i.e., to systems that display both continuous and discrete behavior. We begin by considering a Lagrangian together with a configuration space with unilateral constraints on the set of admissible configurations. This naturally yields the notion of a hybrid Lagrangian, from which we obtain a Lagrangian hybrid system in a way analogous to the association of a Lagrangian vector field to a Lagrangian. We first give general conditions on when it is possible to reduce a cyclic Lagrangian hybrid system, and explicitly compute the reduced Lagrangian hybrid system in the case when it is obtained from a cyclic hybrid Lagrangian.

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Section: Hybrid Lagrangiansmentioning

confidence: 99%

“…[8], [14], [15]). In fact, the results of this paper are used in [2] to reduce the dimensionality of bipedal walkers. It is then possible to use results relating to two-dimensional bipedal walkers to allow three-dimensional bipedal walkers to walk while stabilizing to the upright position.…”

Section: Special Lagrangian Hybrid Systemsmentioning

confidence: 99%

Hybrid Routhian reduction of Lagrangian hybrid systems

Ames¹,

Sastry²

2006

2006 American Control Conference

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“…The map ∆ "compresses" the flight phase into an "event," and can be thought of as a (generalized) "impact map" [12], or a "reset map" [5]. In this setting, the hybrid dynamics of the ASLIP becomes…”

Section: Aslip Hybrid Dynamics Of Runningmentioning

confidence: 99%

“…The same general idea, albeit in a fully actuated setting, has been employed in [5] and [4], where the method of controlled symmetries introduced in [45] together with a generalization of Routhian reduction for hybrid systems were combined to extend passive dynamic walking gaits, such as those obtained by McGeer's passive walker [28], in three-dimensions.…”

Section: Introductionmentioning

confidence: 99%

The Spring Loaded Inverted Pendulum as the Hybrid Zero Dynamics of an Asymmetric Hopper

Poulakakis

Grizzle

2009

IEEE Trans. Automat. Contr.

234

139

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Abstract-A hybrid controller that induces provably stable running gaits on an Asymmetric Spring Loaded Inverted Pendulum (ASLIP) is developed. The controller acts on two levels. On the first level, continuous within-stride control asymptotically imposes a (virtual) holonomic constraint corresponding to a desired torso posture, and creates an invariant surface on which the two-degree-of-freedom restriction dynamics of the closed-loop system (i.e., the hybrid zero dynamics) is diffeomorphic to the center-of-mass dynamics of a Spring Loaded Inverted Pendulum (SLIP). On the second level, event-based control stabilizes the closed-loop hybrid system along a periodic orbit of the SLIP dynamics. The controller's performance is discussed through comparison with a second control law that creates a one-degreeof-freedom non-compliant hybrid zero dynamics. Both controllers induce identical steady-state behaviors (i.e. periodic solutions). Under transient conditions, however, the controller inducing a compliant hybrid zero dynamics based on the SLIP accommodates significantly larger disturbances, with less actuator effort, and without violation of the unilateral ground force constraints.

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“…Powerful analytical and numerical tools have been developed in order to control and analyze such hybrid dynamical systems and behaviors [1,7,10]. However, these tools are limited to the case when we have a full (and preferably simple) mathematical model-typically derived from first principles-that can accurately describe the system dynamics, but such modeling requires many creative decisions about what to neglect.…”

Section: Introductionmentioning

confidence: 99%

System identification of rhythmic hybrid dynamical systems via discrete time harmonic transfer functions

Ankaralı

Cowan

2014

53rd IEEE Conference on Decision and Control

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Abstract-Few tools exist for identifying the dynamics of rhythmic systems from input-output data. This paper investigates the system identification of stable, rhythmic hybrid dynamical systems, i.e. systems possessing a stable limit cycle but that can be perturbed away from the limit cycle by a set of external inputs, and measured at a set of system outputs. By choosing a set of Poincaré sections, we show that such a system can be (locally) approximated as a linear discrete-time periodic system. To perform input-output system identification, we transform the system into the frequency domain using discretetime harmonic transfer functions. Using this formulation, we present a set of stimuli and analysis techniques to recover the components of the HTFs nonparametrically. We demonstrate the framework using a hybrid spring-mass hopper. Finally, we fit a parametric approximation to the fundamental harmonic transfer function and show that the poles coincide with the eigenvalues of the Poincaré return map.

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On the Geometric Reduction of Controlled Three-Dimensional Bipedal Robotic Walkers

Cited by 37 publications

References 7 publications

Hybrid Routhian reduction of Lagrangian hybrid systems

Hybrid Routhian reduction of Lagrangian hybrid systems

The Spring Loaded Inverted Pendulum as the Hybrid Zero Dynamics of an Asymmetric Hopper

System identification of rhythmic hybrid dynamical systems via discrete time harmonic transfer functions

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