Experimental Investigation of the Painlevé Paradox in a Robotic System

Zhao, Zhen; Liu, Caishan; Ma, Wei; Chen, Bin

doi:10.1115/1.2910825

Cited by 29 publications

(45 citation statements)

References 32 publications

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“…The process of impact is dominated by the contact point with maximal potential energy, which is linked with the 'time-like' independent variable Zhao et al 2008). Similar to the contact dynamics, the tangential constraint at the instant of the relative tangential velocity vanishing can also be identified by the correlative coefficient ðm st i Þ obtained from the local dynamical equations.…”

Section: Introductionmentioning

confidence: 99%

Planar dynamics of a rigid body system with frictional impacts. II. Qualitative analysis and numerical simulations

2009

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The objective of this paper is to implement and test the theory presented in a companion paper for the non-smooth dynamics exhibited in a bouncing dimer. Our approach revolves around the use of rigid body dynamics theory combined with constraint equations from the Coulomb's frictional law and the complementarity condition to identify the contact status of each contacting point. A set of impulsive differential equations based on Darboux-Keller shock dynamics is established that can deal with the complex behaviours involved in multiple collisions, such as the frictional effects, the local dissipation of energy at each contact point, and the dispersion of energy among various contact points. The paper will revisit the experimental phenomena found in Dorbolo et al. (Dorbolo et al. 2005 Phys. Rev. Lett. 95, 044101), and then present a qualitative analysis based on the theory proposed in part I. The value of the static coefficient of friction between the plate and the dimer is successfully estimated, and found to be responsible for the formation of the drift motion of the bouncing dimer. Plenty of numerical simulations are carried out, and precise agreements are obtained by the comparisons with the experimental results.

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Section: Introductionmentioning

confidence: 99%

Planar dynamics of a rigid body system with frictional impacts. II. Qualitative analysis and numerical simulations

2009

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“…The impact assumption has been supported by the numerical investigations fulfilled under a variety of compliance-based models [12,13]. In addition, experimental investigations [28] also showed that the unstable phenomenon of a robotic system touching a moving belt essentially originated from an event of a tangential impact.…”

Section: Introductionmentioning

confidence: 77%

“…If and only if A(θ, μ) > 0, the solution obtained from the constraint-based method is unique for all B(θ,θ) [18,28]. If A(θ, μ) < 0, the normal contact force F n is either in an inconsistent mode if B(θ,θ) < 0, or is in an indeterminate mode if B(θ,θ) > 0.…”

Section: Painlevé Paradoxmentioning

confidence: 99%

“…Based on an LCP-based method, the singularity can be easily identified according to system configurations and friction state on the contact surface. Besides the widely studied Painlevé's example [18,28], recent studies have revealed that many mechanical systems are intervened by the singularity. Examples are a robot system on a moving belt [28][29][30], an IPOS-like mechanism [31], and the passive walking systems [32].…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic analysis of Painlevé’s paradox

Zhao

Liu²,

Chen³

et al. 2015

Multibody Syst Dyn

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Painlevé's paradox is a well-known problem in rigid-body dynamics, of which the forward dynamics equations could have no solution. To handle this situation, an assumption of tangential impact is often introduced. Although the assumption seems to provide a good fix, it still needs to be mathematically examined via analyzing the asymptotic property of a compliance-based model at the limit of rigidity. In this paper, we revisit the paradox using the typical Painlevé's example of a rod sliding on a rough surface. For convenience, the interaction at the contact point of the rod is represented by a linear spring to scale the local normal compliance, coupled with Coulomb's law to reflect friction. We perform an asymptotic analysis using the spring stiffness as a perturbation parameter. The rod dynamics in the Painlevé's paradox, accompanying the variation of friction status, consists of three distinct phases as follows: An initial period of sliding which allows contact force to diverge with the increase of the spring stiffness, a period of sticking which ends at the occurrence of a reverse slip motion, and a reverse slip phase which causes the rod to be detached from the contact surface. As the stiffness goes to infinity, all the time intervals of the three phases converge to zero. This analysis theoretically confirms the assumption of the tangential impact in the paradox of sliding rod dynamics.

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“…Defining a Lagrangian function L = T − U and using Euler-Lagrange equations, the dynamics of the system is governed by 14) where…”

Section: The Dynamics Of a Disc-ball Systemmentioning

confidence: 99%

Impact–contact dynamics in a disc–ball system

et al. 2013

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This paper concerns a disc-ball system, in which a moving ball collides against a disc resting on a rough, fixed horizontal surface. The complexity in such a simple object is due to the presence of the line contact between the disc and the fixed plate, which significantly influences the impact-generated state of the disc. We deal with this problem in a uniform framework that encapsulates different structures of the mathematical model, including contacts, impacts, stick-slip in friction, as well as the transitions among different states of the variablestructure dynamics. We design specific experiments that provide useful information to help determine the macroscopic parameters in impact and friction. Other complicated cases concerned with the couplings between impacts and friction are theoretically and experimentally investigated. Excellent agreements between numerical and experimental results support our theoretical developments.

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Experimental Investigation of the Painlevé Paradox in a Robotic System

Cited by 29 publications

References 32 publications

Planar dynamics of a rigid body system with frictional impacts. II. Qualitative analysis and numerical simulations

Planar dynamics of a rigid body system with frictional impacts. II. Qualitative analysis and numerical simulations

Asymptotic analysis of Painlevé’s paradox

Impact–contact dynamics in a disc–ball system

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