Determination of Stability by Linear Approximation of a Periodic Solution of a System of Differential Equations With Discontinuous Right-Hand Sides

Айзерман, М.А.; Gantmacher, Felix R.

doi:10.1093/qjmam/11.4.385

Cited by 28 publications

(26 citation statements)

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“…3, previous work has shown that, when the contact mode sequence is fixed, trajectory outcomes vary continuously [Bal00, Thm. 20] and differentiably [AG58] with respect to variations in initial conditions (i.e. initial states and parameters).…”

Section: Differentiability Through Contactmentioning

confidence: 99%

Decoupled limbs yield differentiable trajectory outcomes through intermittent contact in locomotion and manipulation

Pace

Burden

2017

2017 IEEE International Conference on Robotics and Automation (ICRA)

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When limbs are decoupled, we find that trajectory outcomes in mechanical systems subject to unilateral constraints vary differentiably with respect to initial conditions, even as the contact mode sequence varies. OrganizationWe begin in Sec. 2 by specifying the class of dynamical systems under consideration, namely, mechanical systems subject to unilateral constraints. Sec. 3 imposes conditions on the system dynamics and trajectories that enable us in Sec. 4 to report that trajectories vary differentiably with respect to initial conditions, even as the contact mode sequence varies. d k=1 (D k M m + D m M k − D M km ) [JK13, Eqn. 30]. 3 We let 2 n = {J ⊂ {1, . . . , n}} denote the power set (i.e. the set containing all subsets) of {1, . . . , n}.

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Section: Differentiability Through Contactmentioning

confidence: 99%

Decoupled limbs yield differentiable trajectory outcomes through intermittent contact in locomotion and manipulation

Pace

Burden

2017

2017 IEEE International Conference on Robotics and Automation (ICRA)

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“…The submanifold Σ is referred to as a Poincaré section. It is known that this procedure yields a map that is well-defined and smooth near the fixed point ξ = P (ξ) [13], [28], [32], [33]. Unlike the classical case, Poincaré maps in hybrid systems need not be full rank.…”

Section: B Hybrid Periodic Orbits and Hybrid Poincaré Mapsmentioning

confidence: 99%

Model Reduction Near Periodic Orbits of Hybrid Dynamical Systems

Burden

Revzen

Sastry

2015

IEEE Trans. Automat. Contr.

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We show that, near periodic orbits, a class of hybrid models can be reduced to or approximated by smooth continuous-time dynamical systems. Specifically, near an exponentially stable periodic orbit undergoing isolated transitions in a hybrid dynamical system, nearby executions generically contract superexponentially to a constantdimensional subsystem. Under a non-degeneracy condition on the rank deficiency of the associated Poincaré map, the contraction occurs in finite time regardless of the stability properties of the orbit. Hybrid transitions may be removed from the resulting subsystem via a topological quotient that admits a smooth structure to yield an equivalent smooth dynamical system. We demonstrate reduction of a high-dimensional underactuated mechanical model for terrestrial locomotion, assess structural stability of deadbeat controllers for rhythmic locomotion and manipulation, and derive a normal form for the stability basin of a hybrid oscillator. These applications illustrate the utility of our theoretical results for synthesis and analysis of feedback control laws for rhythmic hybrid behavior. I. INTRODUCTION Rhythmic phenomena are pervasive, appearing in physical situations as diverse as legged locomotion [1], dexterous manipulation [2], gene regulation [3], and electrical power generation [4]. The most natural dynamical models for these systems are piecewise-defined or discontinuous owing to intermittent changes in the mechanical contact state of a locomotor or manipulator, or to rapid switches in protein synthesis or constraint activation in a gene or power network. Such hybrid systems generally exhibit dynamical behaviors that are distinct from those of smooth systems [5]. Restricting our attention to the dynamics near periodic orbits in hybrid dynamical systems, we demonstrate that a class of hybrid models for rhythmic phenomena reduce to classical (smooth) dynamical systems.Although the results of this paper do not depend on the phenomenology of the physical system under investigation, a principal application domain for this work is terrestrial locomotion. Numerous architectures have been proposed to explain how animals control their limbs; for steady-state locomotion, most posit a principle of coordination, synergy, symmetry or synchronization, and there is a surfeit of neurophysiological data to support these hypotheses [6]- [10]. Taken together, the empirical evidence suggests that the large number of degrees-of-freedom (DOF) available to a locomotor can collapse during regular motion to a low-dimensional dynamical attractor (a template) embedded within a higher-dimensional model (an anchor) that respects the locomotor's physiology [1], [11]. We provide a mathematical framework to model this empirically observed dimensionality reduction in the deterministic setting.A stable hybrid periodic orbit provides a natural abstraction for the dynamics of steady-state legged locomotion. This widely-adopted approach has generated models of bipedal [12] . In certain cases, it has been possibl...

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“…Since the solution intersects the surface of discontinuity without tangency and the initial time does not correspond to a crossing of the surface of discontinuity, the monodromy matrix can still be constructed from Eq. (15) [30]. For simplicity, an explicit forward Euler method is applied here to approximate the resulting matrix, so that it is not necessary to know the switching time in advance.…”

Section: Stability Of Non-linear Normal Modesmentioning

confidence: 99%

Large-amplitude non-linear normal modes of piecewise linear systems

Jiang

Pierre

Shaw

2004

Journal of Sound and Vibration

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A numerical method for constructing non-linear normal modes (NNMs) for piecewise linear autonomous systems is presented. These NNMs are based on the concept of invariant manifolds, and are obtained using a Galerkin-based solution of the invariant manifold's non-linear partial differential equations. The accuracy of the constructed non-linear modes is checked by the comparison of the motion on the invariant manifold to the exact solution, in both time and frequency domains. It is found that thisconstruction approach can accurately capture the NNMs over a wide range of amplitudes, including those with strong non-linear effects. Several interesting dynamic characteristics of the non-linear modal motion are found and compared to those of linear modes. A two-degree-offreedom example is used to illustrate the technique. The existence, stability and bifurcations of the NNMs for this example are investigated.

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Determination of Stability by Linear Approximation of a Periodic Solution of a System of Differential Equations With Discontinuous Right-Hand Sides

Cited by 28 publications

References 0 publications

Decoupled limbs yield differentiable trajectory outcomes through intermittent contact in locomotion and manipulation

Decoupled limbs yield differentiable trajectory outcomes through intermittent contact in locomotion and manipulation

Model Reduction Near Periodic Orbits of Hybrid Dynamical Systems

Large-amplitude non-linear normal modes of piecewise linear systems

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