| The traditional method to investigate the walking of bipedal passive robots is usually based on a completely rigid body model incorporating with completely inelastic impact hypothesis.However,it neglects the compliant effects of local contact zone and whole structure.This will lead to a large error between the actual motion and the calculation results of the flexible biped robot.In addition,when analyzing the passive walking of the bipedal robot,in traditional methods we usually assume the contact foot and the slope maintain stick state.However,in the actual stable walking process,the sliding contact is inevitable.Some special configuration of mechanisms can result in dynamic self-locking or“jam”at the sliding contact if the coefficient of friction is sufficiently large and this has been termed as Painlevéparadox in rigid body dynamics.There are few researches on sliding friction problem of robots.Due to the strong non-smooth property of the system and the singularity of the dynamic equations,it is especially lack of researches on the Painlevéparadox of the robot.In this paper,the passive dynamic walking and Painlevéparadox are analyzed based on the rigid body theory incorporating with coulomb law of friction and linear complementary method,as well as the flexible body theory incorporating with nonlinear finite element technique.The main contents are as follows:(1)The completely rigid body model of the bipedal passive dynamic walking robot is proposed and the dynamic equations in single-leg support phase and double-leg support phase are derivate based on the Lagrange equation and completely inelastic impact hypothesis.The Newton-Raphson iterative method and linearization solution method are used to get the stable fixed point.Though inputting the parameters of the system into the algorithm programs,the initial motion state of the stable passive walking is obtained successfully.(2)A new completely flexible body model of the bipedal passive dynamic walking robot is proposed by considering the compliant effects of local contact zone and whole structure.The structural deformation field and inertial field are discretized by finite element theory,and the mesh convergence of the model is validated.By comparing with the rigid body model solutions and the flexible body model solutions,it can be found that the first step of walking of two models are in good agreement,the maximum error of angular displacement of subsequent motion can reach 10%and the error of the stride of each step is 0.1%~5.87%.It is also found that there exists friction energy loss in the flexible body model,which is different from the rigid body model.(3)Fine contact modes between foot and slope of a flexible biped robot in the stable walking is analyzed.It is found that the normal contact process of a single leg consists of three sub-phases cycles and they are macroscopic oblique impact sub-phase(duration time is 0.1~0.2s),rolling sub-phase(duration time is 0.6~0.7s)and long term separation sub-phase.Macroscopic oblique impact sub-phase consists of a series of fast normal contact-separation switches.It also can be found that the tangential contact process of a single leg consists of three sub-phases cycles,which are the sub-phase of stick-slip repeated switch,stick rolling sub-phase and long term separation sub-phase.The sub-phase of stick-slip repeated switch consists of a series of fast tangential stick-slip switches.In addition,a'dynamic self-locking'phenomenon is found as?=0.9,which will lead to the walking unstable.(4)Based on the linear complementarity theory,the criterion expression of the Painlevéparadox of the rigid body model is derived,and the Painlevéparadox is analyzed.The results show that some configurations in the stable walking when the coefficient of friction is large and the foot is sliding,it will appear the Painlevéparadox.And the critical coefficient of friction for the initial state configuration?_c~R=0.6630.The study also found that the region of Painlevéparadox will become larger as the coefficient of friction increases,moment of inertia decreases,the mass of the leg decreases or the mass of the hip increases.However,the length of the leg has no effect on the region of Painlevéparadox.(5)Based on the flexible body model,the dynamic self-locking phenomenon(closely related to the Painlevéparadox of the rigid body model)with no initial normal relative velocity is analyzed.It can be found that the dynamic self-locking phenomenon become more remarkable as the coefficient of friction increases,Young's modulus of the leg E_l decreases,the tangential relative velocity increases or the normal relative velocity increases.For the initial configuration,the critical coefficients of friction are?_c~F=0.50 and?_c~F=0.48 when Young's moduli of the leg are E_l=200GPa and E_l=70GPa,respectively.It also can be found that the region of dynamic self-locking is larger than the region of rigid body's Painlevéparadox.In addition,there exists the transmission,reflection and superposition of the transient stress waves which are generated by dynamic self-locking exist. |
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