Theoretical Analysis Of The Painlevé Paradox In Frictional Contact Of A Flexible Manipulator And Its Dynamic Self-locking Experiment

When rigid body or multi-rigid-body systems come into contact with a rough surface in a specific configuration,combined with Coulomb law of friction,there will be no solution or multiple solutions when solving the dynamic equations of a mechanical system with a unilateral constraint,which is termed as Painlevéparadox.Paradox phenomenon are widespread in many mechanical systems that involve contact issues.In the study of various classic rigid body dynamic problems,the mechanism of the Painlevéparadox has not been thoroughly explained,and the research on this problem focuses more on avoiding the occurrence of paradox phenomena from the theoretical aspect,but less on the research related to mechanical systems such as the robotic manipulator.Experimental research about Painlevéparadox are rarely seen in detail and in-depth.Aiming at the model of the two-link manipulator system,the dynamic equations of the rigid body friction dynamics theory is used to derive the paradox region of the rigid body model.A dynamic self-locking experimental platform capable of realizing automatic control of the initial configuration was set up,and the experimental study of paradox-related phenomenon in almost all configuration region were completed for the first time.Finally,the experimental results were verified by LS-DYNA,and the parameters were discussed.The specific research contents and conclusions are as follows:(1)A complete rigid body model of the two-link manipulator is established,and the paradox region of the rigid body model is obtained.Combining Coulomb law of friction with the unilateral constrained motion hypothesis,the Lagrange equation of the second kind is used to derive the system dynamics equations of the two-link manipulator in contact with rough surfaces.According to the linear complementary problem between the contact force and the acceleration of the contact point,the paradox configuration region is obtained.The conclusion that the paradox region is related to the friction coefficient?and the initial configuration?₁ and?₂ is given.Paradox region under different?is obtained.(2)A dynamic self-locking experimental platform capable of realizing automatic and precise control of the initial configuration is established,and the relationship between the mathematical Painlevéparadox and the dynamic self-locking phenomenon in the real world is clarified.Frictional contact experiments were performed in the whole configuration region(i.e.,different ?₁ and ?₂)when?=0.8,and the configuration region when dynamic self-locking occurred was obtained.In the experiment,pure aluminum with low elastic modulus and large friction coefficient was used as the material of the robotic manipulator and sliding platform.In order to fix the initial configuration of the experiment,an Arduino development board,sensors,and actuators were used to design and manufacture the initial configuration controller,and recorded the motion and stress data.The experimental observations found that the frictional phenomenon occurring in the whole configuration region can be divided into three groups:continuous sticking,stick-bouncing,and continuous sliding.Among them,stick-bouncing is a specific feature of the dynamic self-locking phenomenon.Further research found that the initial configuration where the stick-bouncing occurred was basically located in the theoretical Painlevéparadox region,which indicates that the dynamic self-locking phenomenon is a manifestation of the Painlevéparadox in the real world.(3)A finite element dynamics numerical simulation of the experimental dynamic self-locking phenomenon was carried out to verify the experimental results.Based on the system parameters of the dynamic self-locking phenomenon in the experiment,a corresponding 3D flexible finite element model was established,and a numerical solution of the transient response of the system was obtained by LS-DYNA calculation.The results show that under the initial configuration of H=0.36m and?₁=28°,the experimental and simulation results agree well,and the displacement of contact point differ by 11.3%.The maximum normal contact force reaches 2.05k N.During the movement,there is a conversion of energy from kinetic energy to internal energy and then to kinetic energy again.The stress of contact point far exceeds the yield stress,reaching a maximum of 529 MPa,which results in a plastic strain of 0.0188 at the end of link and a plastic deformation of 4.93?m at the contact point on the sliding platform.(4)The influence of different system parameters on the dynamic self-locking phenomenon is discussed,and the influence of structural flexibility on the dynamic self-locking region is obtained.For the same initial configuration,changing the friction coefficient?,the material elastic modulus E,and the sliding platform speed Vp,frictional contact transient response under different parameters were calculated,and the influence of these three parameters on the paradox phenomenon was discussed.