Rank deficiency and superstability of hybrid systems

Wendel, Eric; Ames, Aaron D.

doi:10.1016/j.nahs.2011.09.002

Cited by 18 publications

(21 citation statements)

References 27 publications

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“…On the other hand, if one chooses a Poincaré section at pre-impact states, where y = 0 and ẏ < 0, other constraints are imposed by the chosen control law at flight phase (12), which forces convergence of the foot angle θ (t) towards its desired value of θ 0 prior to touchdown, as seen in Figure 5(b). Moreover, the leg's damping attenuates the two-mass oscillations and the distance r(t) also converges towards its free length l 0 (normalized value of 1) as also seen in Figure 5 [62], the effective rank of the Poincaré map drops from 7 to 3, as corroborated by the numerical results of the present study.…”

Section: Discussionsupporting

confidence: 87%

Analysis of Foot Slippage Effects on an Actuated Spring-Mass Model of Dynamic Legged Locomotion

Moravia

2016

International Journal of Advanced Robotic Systems

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The classical model of spring-loaded inverted pendulum (SLIP) and its extensions have been widely accepted as a simple description of dynamic legged locomotion at various scales in humans, legged robots and animals. Similar to the majority of models in the literature, the SLIP model assumes ideal sticking contact of the foot. However, there are practical scenarios of low ground friction that causes foot slippage, which can have a significant influence on dynamic behaviour. In this work, an extension of the SLIP model with two masses and torque actuation is considered, which accounts for possible slippage under Coulomb's friction law. The hybrid dynamics of this model is formulated and numerical simulations under representative parameter values reveal several types of stable periodic solutions with stick-slip transitions. Remarkably, it is found that slippage due to low friction can sometimes increase average speed and improve energetic efficiency by significantly reducing the mechanical cost of transport.

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Section: Discussionsupporting

confidence: 87%

Analysis of Foot Slippage Effects on an Actuated Spring-Mass Model of Dynamic Legged Locomotion

Moravia

2016

International Journal of Advanced Robotic Systems

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“…(Note that here we are assuming that the fixed point is in the switching surface S; more general definitions are possible [14], [46], but the one Fig. 1: Trajectories of the hybrid system (52) lying in the zero dynamics, where Z is the zero dynamics surface and S is the switching surface (50).…”

Section: B Solutions Periodic Orbits and The Poincaré Mapmentioning

confidence: 99%

Rapidly Exponentially Stabilizing Control Lyapunov Functions and Hybrid Zero Dynamics

Ames

Galloway

Sreenath

et al. 2014

IEEE Trans. Automat. Contr.

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381

367

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Abstract-This paper addresses the problem of exponentially stabilizing periodic orbits in a special class of hybrid modelssystems with impulse effects-through control Lyapunov functions. The periodic orbit is assumed to lie in a C 1 submanifold Z that is contained in the zero set of an output function and is invariant under both the continuous and discrete dynamics; the associated restriction dynamics are termed the hybrid zero dynamics. The orbit is furthermore assumed to be exponentially stable within the hybrid zero dynamics. Prior results on the stabilization of such periodic orbits with respect to the fullorder dynamics of the system with impulse effects have relied on input-output linearization of the dynamics transverse to the zero dynamics manifold. The principal result of this paper demonstrates that a variant of control Lyapunov functions that enforce rapid exponential convergence to the zero dynamics surface, Z, can be used to achieve exponential stability of the periodic orbit in the full-order dynamics, thereby significantly extending the class of stabilizing controllers. The main result is illustrated on a hybrid model of a bipedal walking robot through simulations and is utilized to experimentally achieve bipedal locomotion via control Lyapunov functions.

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“…Each element of this union, i.e. V α , is called a chart The dimension of the charts can typically depend on α [5,6,20]. A state of the overall hybrid dynamical system consists of an index α together with a point in the chart V α .…”

Section: Hybrid Dynamical System Formulation With Exogenous Inputmentioning

confidence: 99%

“…It is obvious that for hybrid systems in which the dimension of charts (dim (V α )) changes with α, the linearized Poincaré return map in (7) is always rank-deficient for some i ∈ {0, · · · , N − 1} [20]. Even with rank deficiencies and the dimension of x[n] is time varying, the liner time varying dynamics in (6) is suitable for deriving harmonic transfer functions using impulse response representation.…”

Section: And Likewise For B C D)mentioning

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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Rank deficiency and superstability of hybrid systems

Cited by 18 publications

References 27 publications

Analysis of Foot Slippage Effects on an Actuated Spring-Mass Model of Dynamic Legged Locomotion

Analysis of Foot Slippage Effects on an Actuated Spring-Mass Model of Dynamic Legged Locomotion

Rapidly Exponentially Stabilizing Control Lyapunov Functions and Hybrid Zero Dynamics

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

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