Penalty Approximation for Non-smooth Constraints in Vibroimpact

Paoli, Lætitia; Schatzman, Michelle

doi:10.1006/jdeq.2001.4027

Cited by 19 publications

(19 citation statements)

References 9 publications

(12 reference statements)

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“…In the case of a single impact, the result obtained for unilaterally constrained mechanical systems in [62], [63], [72] by penalization methods coincides with that presented previously. The latter, however, is not restricted to some special equation structure, as in the case of purely and partially elastic obstacles in impact mechanics, and is valid for general unilaterally constrained dynamic systems, encompassing mechanical systems as a particular case.…”

Section: Remarksupporting

confidence: 87%

“…Furthermore, the implications of applying the results obtained on the basis of such limit description to the actual system should be clarified, as well. The techniques currently available for modeling the limit behavior of systems with singular motions include quasi-differential equations [61] and differential inclusions [67], [55], [57], and complementarity [20] and penalty methods [63], [72], [66], with modeling reported in [64], [65]. The first two representations give system evolution not in terms of an isolated trajectory, but a set of trajectories, referred to as integral funnel.…”

Section: Introductionmentioning

confidence: 99%

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Dynamical Systems With Active Singularities of Elastic Type: A Modeling and Controller Synthesis Framework

Bentsman

Miller

2007

IEEE Trans. Automat. Contr.

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A new class of systems characterized by admitting impulsive control action within their singular motion phases, such as dimension changes, state discontinuities, and other irregularities, is introduced. This class, termed dynamical systems with active, or controlled, singularities, encompasses various applications with sensing and/or actuation ultra-fast in comparison with the natural system time scale, including mechanisms with impact-induced motion, power and sensor networks under faults, fast positioning devices, smart skins, and switching electronic circuits. The present work focuses on systems with impact-type singularities induced through system interaction with controlled, or active, state constraints. The latter are assumed to be characterized by parameter-dependent elastic-type constraint violation that vanishes in the limit as the parameter value tends to infinity. A physically well justified representation of this class of systems is proposed, but found to be not well suited for controller synthesis in the singular phase. To address this problem, two equations are derived -one describing controlled rigid impacts in auxiliary stretched time and compatible with the regular control design techniques, termed controlled infinitesimal dynamics equation, and another incorporating controlled impact dynamics in the infinitesimal form in terms of the shift operator along the trajectories of the first equation, termed the limit model. The latter is demonstrated to generate a unique isolated discontinuous system motion, i.e., to provide a tight and well-behaved description of the collision with rigid constraint. It is then shown that the corresponding paths generated by the original physically-based and the limit representations can be made arbitrarily close to each other uniformly except, possibly, in the vicinities of the jump points, by the appropriate choice of the value of the constraint violation parameter. This is shown to permit enforcing, for sufficiently large values of the parameter, the desired limit system behavior onto the original system by simply taking the control signals found through the limit representation, timerescaling them, and inserting the resulting signals directly into the original system. These features show that the procedure developed, in fact, provides a well-posed controller synthesis framework for the class of systems considered. Using this framework, a ball/racket rotationally controlled impact representation is developed, and on its basis, a singular phase control signal in the form of racket rotation velocity is designed, demonstrating that the soft racket provides the specified bounce-off angle increment under a noticeably lower racket rotation velocity and a longer rotation phase than the hard racket.Index Terms-Control of mechanical systems, controlled discrete transitions, generalized solutions, hybrid systems, impacts, impulsive control, nonsmooth analysis and synthesis, penalty method, singularities, systems with unilateral constraints.

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Section: Remarksupporting

confidence: 87%

Section: Introductionmentioning

confidence: 99%

Dynamical Systems With Active Singularities of Elastic Type: A Modeling and Controller Synthesis Framework

Bentsman

Miller

2007

IEEE Trans. Automat. Contr.

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“…Following Moreau [7,8] (see also [15,17] for a mathematical justification of this impact law by a penalty method) we assume inelastic impacts i.e.…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

An existence result for non-smooth vibro-impact problems

Paoli¹

2005

Journal of Differential Equations

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International audienceWe are interested in mechanical systems with a finite number of degrees of freedom submitted to frictionless unilateral constraints. We consider the case of a convex, non-smooth set of admissible positions given by $K =\{q ∈\mathbb R^d ; \phi_\alpha(q)\geq 0, 1\leq \alpha\leq v \}$, $v\geq 1$, and we assume inelastic shocks at impacts. We propose a time-discretization of the measure differential inclusion which describes the dynamics and we prove the convergence of the approximate solutions to a limit motion which satisfies the constraints. Moreover, if the geometric properties ensuring continuity on data hold at the limit, we show that the transmission of velocities at impacts follows the inelastic shocks rule

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“…Introducing the shift operator along the paths of (26) as in (21) and following the subsequent steps of the previous section, Eq. (26) yields the limit system corresponding to the original system (24) as µ → ∞.…”

Section: Multi-impact Controlled Discrete Transitionsmentioning

confidence: 99%

“…This approach indeed looks natural coming from Lagrangian and Hamiltonian mechanics for the systems with unilateral constraints and appears in the so-called measure-type reaction forces localized at the constraint boundary. Moreover, this approach is augmented by penalty approximation of dynamic equations [20][21][22] for simulation purposes. However, in Zeno-type problems the complementarity approach has very limited applicability, since the solutions of the complementary models in these cases are characterized by nonuniqueness beyond accumulation points without additional requirements of the analyticity of the input signals [2].…”

Section: Infinite Transition Sequences Characterized By Accumulation mentioning

confidence: 99%