Regions of Attraction for Hybrid Limit Cycles of Walking Robots

Manchester, Ian R.; Tobenkin, Mark M.; Levashov, Michael; Tedrake, Russ

doi:10.48550/arxiv.1010.2247

Cited by 2 publications

(3 citation statements)

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“…Further, this method simplifies the search for stability certificate compared with previous Lyapunov-based method in [4], which requires a separate search for transversal surfaces and valid Lyapunov functions on those surfaces. By encapsulating the direction of transversal surfaces with the definition of orthogonality, this method allows the search for stability certificate by a single convex optimization problem -a search for a valid transverse contraction metric M (x).…”

Section: Application Examplementioning

confidence: 99%

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Transverse contraction criteria for stability of nonlinear hybrid limit cycles

Tang

Manchester

2014

53rd IEEE Conference on Decision and Control

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In this paper, we derive differential conditions guaranteeing the orbital stability of nonlinear hybrid limit cycles. These conditions are represented as a series of pointwise linear matrix inequalities (LMI), enabling the search for stability certificates via convex optimization tools such as sumof-squares programming. Unlike traditional Lyapunov-based methods, the transverse contraction framework developed in this paper enables proof of stability for hybrid systems, without prior knowledge of the exact location of the stable limit cycle in state space. This methodology is illustrated on a dynamic walking example.

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Section: Application Examplementioning

confidence: 99%

“…A major motivation of this work is the study of underactuated bipedal locomotion [1], which can be represented as limit cycles in the state space [2]. The control design and stability analysis of these "dynamic walkers" are difficult since their dynamics are inherently hybrid and highly nonlinear [3], [4].…”

Section: Introductionmentioning

confidence: 99%

Transverse contraction criteria for stability of nonlinear hybrid limit cycles

Tang

Manchester

2014

53rd IEEE Conference on Decision and Control

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“…The authors have also adapted the procedure proposed in this paper to verify regions of attraction for limit cycles of hybrid systems (see Manchester (2010); Manchester et al (2010)).…”

Section: Stability Of Limit Cyclesmentioning

confidence: 99%

Invariant Funnels around Trajectories using Sum-of-Squares Programming

Tobenkin

Manchester

Tedrake

2011

IFAC Proceedings Volumes

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This paper presents numerical methods for computing regions of finite-time invariance (funnels) around solutions of polynomial differential equations. First, we present a method which exactly certifies sufficient conditions for invariance despite relying on approximate trajectories from numerical integration. Our second method relaxes the constraints of the first by sampling in time. In applications, this can recover almost identical funnels but is much faster to compute. In both cases, funnels are verified using Sum-of-Squares programming to search over a family of time-varying polynomial Lyapunov functions. Initial candidate Lyapunov functions are constructed using the linearization about the trajectory, and associated time-varying Lyapunov and Riccati differential equations. The methods are compared on stabilized trajectories of a six-state model of a satellite.

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Regions of Attraction for Hybrid Limit Cycles of Walking Robots

Cited by 2 publications

References 0 publications

Transverse contraction criteria for stability of nonlinear hybrid limit cycles

Transverse contraction criteria for stability of nonlinear hybrid limit cycles

Invariant Funnels around Trajectories using Sum-of-Squares Programming

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