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Research Papers

Experimental Investigation of the Painlevé Paradox in a Robotic System

[+] Author and Article Information
Zhen Zhao, Wei Ma, Bin Chen

State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, Beijing 100871, P.R.C.

Caishan Liu

State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, Beijing 100871, P.R.C.lcs@mech.pku.edu.cn

J. Appl. Mech. 75(4), 041006 (May 13, 2008) (11 pages) doi:10.1115/1.2910825 History: Received November 26, 2006; Revised December 02, 2007; Published May 13, 2008

This paper aims at experimentally investigating the dynamical behaviors when a system of rigid bodies undergoes so-called paradoxical situations. An experimental setup corresponding to the analytical model presented in our prior work Liu [2007, “The Bouncing Motion Appearing in a Robotic System With Unilateral Constraint  ,” Nonlinear Dyn., 49(1–2), 217–232] is developed, in which a two-link robotic system comes into contact with a moving rail. The experimental results show that a tangential impact exists at the contact point and takes a peculiar property that well coincides with the maximum dissipation principle stated in the work of Moreau [1988, “Unilateral Contact and Dry Friction in Finite Freedom Dynamics  ,” Nonsmooth Mechanics and Applications, Springer-Verlag, Vienna, pp. 1–82] the relative tangential velocity of the contact point must immediately approach zero once a Painlevé paradox occurs. After the tangential impact, a bouncing motion may be excited and is influenced by the speed of the moving rail. We adopt the tangential impact rule presented by Liu to determine the postimpact velocities of the system, and use an event-driven algorithm to perform numerical simulations. The qualitative comparisons between the numerical and experimental results are carried out and show good agreements. This study not only presents an experimental support for the shock assumption related to the problem of the Painlevé paradox, but can also find its applications in better understanding the instability phenomena appearing in robotic systems.

Copyright © 2008 by American Society of Mechanical Engineers
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Figures

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Figure 1

Two-link manipulator contacting with a constantly moving belt

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Figure 2

The physical model of the experimental setup

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Figure 3

The sketch of the experimental system

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Figure 4

The relative velocity of the contact point in tangential direction (H=0.3775m, θ1=32deg, and vt=−0.16m∕s)

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Figure 5

The tangential velocity of the contact point (H=0.3775m, θ1=32deg, and vt=−0.075m∕s)

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Figure 6

The tangential velocity of the contact point (H=0.3775m, θ1=32deg, and vt=−0.075m∕s)

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Figure 7

The tangential velocity of the tip with H=0.3775m and θ1=25deg

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Figure 8

The tangential velocity of the tip with H=0.3775m and θ1=−15deg

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Figure 9

The tangential velocity of the tip with H=0.3775m and θ1=7deg

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Figure 10

The tangential velocity of the tip with H=0.3775m and θ1=15deg

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Figure 11

The tangential velocity of the tip with H=0.3775m and θ1=21deg

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Figure 12

The tangential velocity of the tip without the Painlevé paradox (H=0.25m and θ1=69.4deg)

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Figure 13

The stick-slip phenomenon in the configuration (H=0.314m and θ1=50deg)

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Figure 14

A single pendulum composed by a link and a torsional spring

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Figure 15

Experimental and numerical results for the tangential speed (H=0.3775m, θ1=32deg, and vt=−0.2m∕s)

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Figure 16

Experimental and numerical results for the normal velocity (H=0.3775m, θ1=32deg, and vt=−0.24m∕s)

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Figure 17

The comparison of the tangential speed between experimental and numerical results (H=0.3775m, θ1=30.5deg, and vt=−0.2m∕s)

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Figure 18

Comparison between the experimental and numerical results for the stick-slip phenomena under the configuration taken as H=0.314m and θ1=55deg

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