Theoretical, Experimental and Numerical Analyses for Painlevé Paradox of Two-Link Robotic Manipulator System

When multi-rigid-body come into contact with a rough surface in certain configurations, multiple solutions or no solution would occur in the theoretical derivation of the dynamic equation, which is termed Painlevé paradox. In this paper, two-link robotic manipulator system as a kind of Painlevé paradox model is studied from theory, experiment and simulation. The theoretical solution is obtained by the linear complementary problem (LCP) method, which offers guidance to the experiment. Then the feasibility of experiment is validated by numerical simulation. For experiment, two-link robotic manipulator set-up is built. The apparatuses verify the continuity of two-link system motion as a function of initial configuration. The two-link robotic manipulator model is also built in LS-DYNA. The experiment and simulation results show that Painlevé paradox is always accompanied with dynamic jam. Meanwhile, there is no clear boundary between dynamic jam region and non-dynamic jam region derived as the LCP solution indicates. Instead, it tends to be a gradual change process with certain transformation law. Sticking-bounce motion is found in the experiment and simulation. Several different motion characteristics are concluded corresponding to the initial angles of the two links. The summary of the variation of dynamic responses is given for further studying the mechanism of tangential impact of similar robotic manipulator system, especially for guiding how to avoid such universal but unexpected action existing in robotic manipulator system.

Could Chalk Hopping Be Caused by Reverse Chatter?

Anyone who has ever used a chalkboard is probably familiar with the phenomenon of “chalk hopping,” where the chalk unexpectedly skips across the chalkboard, leaving a dotted line in its wake. Such behavior is ubiquitous to mechanical systems with moving parts in contact, where it is almost always undesirable. It is widely believed that hopping behavior is a physical manifestation of either the classical Painlevé paradox or a related phenomenon called dynamical jam. The present paper poses the question of whether chalk hopping might be caused by a different, and much more recently discovered, instability called “reverse chatter,” in which two bodies initially in sustained contact can lose contact through a sequence of impacts with increasing amplitude. Previous simulations of reverse chatter have considered only constant external loads, which do not adequately model the forces exerted on a piece of chalk. The current work presents simulation results for a model system in the presence of a control algorithm that mimics the human hand by attempting to keep the chalk in contact with the chalkboard. The simulations reveal that there exist physically realistic parameter values for which a loss of contact occurs that cannot be attributed to either the classical Painlevé paradox or dynamical jam, but which can only be attributed to reverse chatter. Furthermore, the subsequent motion of the system after losing contact is found to be strikingly similar to that of chalk hopping on a chalkboard, to a hitherto unparalleled degree. These results show that neither the classical Painlevé paradox nor dynamical jam is necessary for hopping behavior, and suggest that reverse chatter may be the most plausible explanation for chalk hopping.

Numerical Location of Painlevé Paradox-Associated Jam and Lift-Off in a Double-Pendulum Mechanism

In this article, we will introduce the phenomenon known as the Painlevé paradox and further discuss the associated coupled phenomena, jam and lift-off. We analyze under what conditions the Painlevé paradox can occur for a general two-body collision using a framework that can be easily used with a variety of impact laws, however, in order to visualize jam and lift-off in a numerical simulation, we choose to use a recently developed energetic impact law as it is capable of achieving a unique forward solution in time. Further, we will use this framework to derive the criteria under which the Painlevé paradox can occur in a forced double-pendulum mechanical system. First, using a graphical technique, we will show that it is possible to achieve the Painlevé paradox for relatively low coefficient of friction values, and second we will use the energetic impact law to numerically show the occurrence of the Painlevé paradox in the double-pendulum system.