A new Poincare map for investigating the complex walking behavior of the compass-gait biped robot

摘要

The planar compass-gait biped robot is recognized by its simple passive morphological structure and by a complex dynamic walking system modeled by an impulsive hybrid nonlinear dynamics. Such complexity inhibits investigating the biped locomotive mechanism by means of the Poincare map. Nevertheless, it is very difficult (and even impossible) to establish the exact closed form of the Poincare mapping. This paper is concerned with the development of an improved closed-form analytical expression of the impact-to-impact Poincare map for analyzing the complex walking behavior of the passive-dynamics biped walker and its stability. Our methodology consists in linearizing and then approximating the hybrid dynamics of the compass-gait biped robot around a predefined hybrid limit cycle. Based on the second-order Taylor series expansion, we design first an enhanced closed-form expression of the Poincare map as well as an analytical expression for the computation of the step period of the bipedal locomotion. Furthermore, a simplification of these developed expressions is presented by decreasing the dimension. We provide also an expression of the Jacobian matrix for investigating the stability of the period-1 fixed point of the designed simplified Poincare map. At the end of this work, we illustrate some simulation results in order to evaluate the validity of the new developed expression of the impact-to-impact Poincare map in analyzing the stability and the complex walking behavior of passive-dynamics compass biped robot.