Painlevé paradox during passive dynamic walking of biped robots

Abstract

An analytical method is proposed to study the walking stability problem and Painlevé paradox during the passive walking of bipedal robots. The whole walking process can be seen as the repetition of the four states, i.e. left single-leg support state, double-legs support state, right single-leg support state and double-legs support state. Based on the multiple states, the hybrid rigid body frictional model is established. The contact constraint is treated by completely inelastic collision hypothesis, and the initial conditions are calculated by conservation of angular momentum. Dynamic equations for the four system states are derived by using Lagrange equation. The relationship between the ratio ft/fn and time t under initial condition of the stable walking is obtained, and by using this relationship, region of friction coefficient for the stable walking state is obtained. The possible singular phenomenon — Painlevé paradox problem is analyzed. The critical value of the coefficient of friction for Painlevé paradox is specified. In addition, the analytical results of robot's dynamic responses are compared with the finite element numerical simulation. It shows that the results from the flexible body model are totally different from solutions calculated by the rigid body model. The critical value of coefficient of friction is smaller than those of the rigid body model.