The bouncing motion appearing in a robotic system with unilateral constraint

Abstract

The hopping or bouncing motion can be observed when robotic manipulators are sliding on a rough surface. Making clear the reason of generating such phenomenon is important for the control and dynamical analysis for mechanical systems. In particular, such phenomenon may be related to the problem of Painlevé paradox. By using LCP theory, a general criterion for identifying the bouncing motion appearing in a planar multibody system subject to single unilateral constraint is established, and found its application to a two-link robotic manipulator that comes in contact with a rough constantly moving belt. The admissible set in state space that can assure the manipulator keeping contact with the rough surface is investigated, and found which is influenced by the value of the friction coefficient and the configuration of the system. Painlevé paradox can cause either multiple solutions or non-existence of solutions in calculating contact force. Developing some methods to fill in the flaw is also important for perfecting the theory of rigid-body dynamics. The properties of the tangential impact relating to the inconsistent case of Painlevé paradox have been discovered in this paper, and a jump rule for determining the post-states after the tangential impact finishes is developed. Finally, the comprehensively numerical simulation for the two-link robotic manipulator is

Experimental Investigation of the Painlevé Paradox in a Robotic System

“…Recent studies have indicated that the Painlevé paradox may be a common phenomenon and could be found in a variety of different applications such as robotic manipulation, legged locomotion, and vehicle braking systems [15,26,27,28]. Especially, according to our recent work for a robotic system that comes into contact with a moving belt [28], the Painlevé paradox will appear even though the coefficient of friction takes a very small value.…”

“…An experimental setup corresponding to the analytical model presented in our prior work [28] is developed, in which a two-link robotic system comes into contact with a moving rail. The experimental results clearly show that a tangential impact does exist at the contact point and takes a peculiar property well coinciding with the maximum dissipation principle stated in [11] by Moreau; the relative tangential velocity of the contact point must immediately approach zero once a Painlevé paradox occurs.…”

“…Therefore, it is crucial to search for a new example that can involve the paradox and can be easily implemented by experiments. Such examples have been presented in [15,28], where it has been shown that the Painlevé paradoxes may occur for arbitrarily small values of the coefficient of friction.…”

“…Especially, according to our recent work for a robotic system that comes into contact with a moving belt [28], the Painlevé paradox will appear even though the coefficient of friction takes a very small value. In this paper, we will use the analytical model presented in [28] to develop an experimental setup for the demonstration of the dynamical behaviors associated with the Painlevé paradox.…”

“…According to the theoretical analysis shown in [28], the Painlevé paradox can be found by setting the robotic system within a paradoxical configuration, which depends on the practical value of the coefficient of friction. By conducting the shock assumption into the paradoxical situation of the robotic system, and using the Darboux-Keller's method [21,[32][33][34][35][36][37], it is also shown that the tangential impact will take a peculiar property; the relative motion of the contact point must be immediately brought into a stick in tangential direction.…”