Painlevé paradox and dynamic self-locking during passive walking of bipedal robot

When analyzing the passive walking of bipedal robot using the traditional methods, it usually assumes the contact foot and the slope maintain stick state. However, in the actual stable walking, the sliding contact caused by structural vibration is inevitable. Some special configuration of bipedal robot can result in dynamic self-locking or "jam" at the sliding contact if the coefficient of friction (COF) is sufficiently large; this has been termed, Painlevé paradox in rigid body dynamics. The objective of this paper is to propose a rigid body dynamic combined with Linear Complementary Problem to calculate the occurrence conditions of Painlevé paradox, and develop a completely flexible body model (CFBM) to analyze its bouncing motion induced by dynamic self-locking during passive walking of bipedal robots. The structural deformation field and inertial field of CFBM are discretized by finite element method. The contact forces and transient waves propagating in the CFBM are calculated. In the range of configurations and COF where Painlevé paradox occurs, analysis based on rigid body dynamics gives results indicating that either there are multiple solutions or the solution doesn't exist; analysis using CFBM finds that the bouncing motion will cause walking unstable and the normal contact force increases abruptly. In addition, increasing structural compliance will decrease the critical COF of dynamic self-locking roughly by 46%.