Global uniform symptotic attractive stability of the non-autonomous bouncing ball system

Leine, Remco I.; Heimsch, Thomas

doi:10.1016/j.physd.2011.04.013

Cited by 53 publications

(30 citation statements)

References 32 publications

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“…The first one consists of a mechanical system capturing the dynamics of a simple robotic manipulator that is required to interact physically with the environment through the effect of a control input affecting the continuous dynamics (see also [24], [25], [15,Section 7.3], [13, Section 6.5]) . The second application pertains to the bouncing ball system [26] with a control input affecting the impacts (see also [27], [28], [29] and [30], [31] [32] where stabilization and, respectively, trajectory tracking for the so-called juggling systems, namely mechanical systems controlled at impacts, has been addressed, and [33] where the stability of a controlled bouncing ball system is studied using Lyapunov-like techniques). Classical passivity-based control techniques such as passivation by feedback, energy shaping and damping injection are also applied to the two applications to illustrate their effectiveness in the hybrid systems setting.…”

Section: Contributionsmentioning

confidence: 99%

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Passivity-based control for hybrid systems with applications to mechanical systems exhibiting impacts

Naldi

Sanfelice

2013

Automatica

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Motivated by applications of systems interacting with their environments, we study the design of passivity-based controllers for a class of hybrid systems in which the energy dissipation may only happen along either the continuous or the discrete dynamics. A general definition of passivity, encompassing the said special cases, is introduced and, along with detectability and solution conditions, linked to stability and asymptotic stability of compact sets. The proposed results allow to take advantage of the passivity property of the system at flows or at jumps and are employed to design passivity-based controllers for the class of hybrid systems of interest. Two applications, one pertaining to a point mass physically interacting with a wall and another about controlling a ball bouncing on an actuated surface, illustrate the definitions and results throughout the paper.

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Section: Contributionsmentioning

confidence: 99%

“…x, 0)) = 0 (33) for some r ≥ 0. Due to detectability relative to the set (27), every solution starting from and staying in (33) converges to A. Moreover, since V is positive definite with respect to A, the only invariant set in (33) is for r = 0.…”

Section: Basic Propertiesmentioning

confidence: 99%

Passivity-based control for hybrid systems with applications to mechanical systems exhibiting impacts

Naldi

Sanfelice

2013

Automatica

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“…The corresponding ball motion is called the P 1 1 -orbit, where we adopt the nomenclature from [9]. The orbit P k l denotes a periodic ball motion where the ball impacts with the paddle l times per In order for a P k 1 -orbit to be stable, its nominal impact must occur during the interval where the paddle decelerates withp = −g/2, i.e.…”

Section: A Periodic Orbitsmentioning

confidence: 99%

“…Bouncing balls exhibit rich dynamical behavior, which has been well studied, see, for example, [8], [9]. One such behavior is that the ball trajectories are chaotic for a specific higher frequency paddle motion.…”

Section: Introductionmentioning

confidence: 99%

Control of nonlinear systems with symmetries using chaos

Reist

D’Andrea

2012

2012 IEEE 51st IEEE Conference on Decision and Control (CDC)

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We present a method that exploits chaos for the control of systems composed of subsystems with identical nonlinear dynamics and a shared, common control input. Due to symmetry, these systems are uncontrollable in a deterministic sense. However, the systems may be controllable in a stochastic sense when they are driven by process noise. We present a control strategy that exploits the sensitivity of chaotic motion to process noise. The chaotic subsystem trajectories evolve independently on the strange attractor, which enlarges the reachable set of the system. Specifically, we consider the control of a juggling machine bouncing multiple balls on a single, actuated paddle. The goal is to control the balls to a combination of stable periodic orbits. First, a paddle motion is applied that induces chaotic ball trajectories. Then, when the ball states reach the basins of attraction of the desired periodic orbits, the paddle motion is switched to the motion that stabilizes the orbits. Both simulation and preliminary experimental results are presented.

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“…2(a). This notation means that the ball motion has a period of 2 vibration cycles and describes one kind of trajectory during the orbit [27]. Out of this region, a chaotic dynamics prevails.…”

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confidence: 99%

Dynamics of a Grain-Filled Ball on a Vibrating Plate

2014

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We study experimentally how the bouncing dynamics of a hollow ball on a vibrating plate is modified when it is partially filled with liquid or grains. Whereas empty and liquid-filled balls display a dominant chaotic dynamics, a ball with grains exhibits a rich variety of stationary states, determined by the grain size and filling volume. In the collisional regime, i.e., when the energy injected to the system is mainly dissipated by interparticle collisions, an unexpected period-1 orbit appears independently of the vibration conditions, over a wide range. This is a self-regulated state driven by the formation and collapse of a granular gas within the ball during one cycle. In the frictional regime (dissipation dominated by friction), the grains move collectively and generate different patterns and steady modes: oscillons, waves, period doubling, etc. From a phase diagram and a geometrical analysis, we deduce that these modes are the result of a coupling (synchronization) between the vibrating plate frequency and the trajectory followed by the particles inside the cavity. A bouncing ball (BB) on a vibrated plate is a fundamental system used as a model in different physical and engineering problems [1]; for instance, it has been used to describe cosmic-ray particles in astrophysics [2], the dynamic stability in human performance [3], and the cantilever motion in atomic force microscopy [4]. The BB dynamics displays a period doubling route to chaos also found in biological, hydrodynamic, optical, and chemical systems [5]. More complex objects such as dimers [6], trimers [7], bouncing droplets [8], and quantum bouncers [9] also show nonlinear behaviors: bifurcations, rotations, etc. Dimers and trimers exhibit self-propulsion, providing models to study collective motions, such as fish schools [10], flocks [11], and bacteria colonies [12]. Clearly, the study of bouncing objects has been relevant to the description of a large variety of nonlinear behaviors.This Letter explores the bouncing dynamics of a hollow sphere partially filled with grains. We aim to understand how the classical BB dynamics (widely studied in the literature [1,5,13,14]) is affected by the fast energy dissipation typical of granular materials. We compared the dynamics of an empty, a liquid-filled, and a grain-filled sphere. The former two cases exhibit chaotic behaviors while the sphere with grains bounces primarily in steady states. Of particular interest was the observation of a period-1 orbit independent of the vibration conditions in a broad range. This state was observed at low filling ratios V f (< 12%) and using large beads (diameter d ¼ 2 mm). When tiny particles were used (300 μm), different patterns appeared depending on the vibration frequency f, for instance, stable oscillons and surface waves similar to those observed in vibrated granular beds [15][16][17][18][19]. The above grain size dependence reveals, as in the case of granular dampers [20][21][22][23][24][25], the relevance of the main dissipation mechanism: interparticle collisio...

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Global uniform symptotic attractive stability of the non-autonomous bouncing ball system

Cited by 53 publications

References 32 publications

Passivity-based control for hybrid systems with applications to mechanical systems exhibiting impacts

Passivity-based control for hybrid systems with applications to mechanical systems exhibiting impacts

Control of nonlinear systems with symmetries using chaos

Dynamics of a Grain-Filled Ball on a Vibrating Plate

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