Stabilization of the passive walking dynamics of the compass-gait biped robot by developing the analytical expression of the controlled Poincare map

摘要

The compass-gait biped robot is a two-DoF legged mechanical system that has been known by its passive dynamic walking. This kind of passive biped robot is modeled by an impulsive hybrid nonlinear system that exhibits complex behaviors. This paper is concerned with the stabilization of the passive dynamic walking of the compass-gait biped robot by designing an analytical expression of the controlled Poincare map. The suggested analytical method starts with the time-piecewise linearization of the impulsive system augmented with the controller, around a desired one-periodic passive hybrid limit cycle. By virtue of the first-order Taylor series, we design an explicit expression of the controlled Poincare map. We present also a simplified expression of the controlled Poincare map having a reduced dimension. In order to accomplish our goal concerned with the stabilization of the passive walking dynamics of the compass-gait biped robot, we develop first the linearized Poincare map around the period-1 fixed point of the Poincare map and we adopt a state-feedback control law to stabilize it. Furthermore, in order to enhance the effectiveness and robustness of the stabilization process, we adopt an LMI-based optimization approach by considering the controlled Poincare map to design the feedback gain. In the end of this work, we provide some numerical and graphical simulation results, to show the validity of the designed control law by means of the controlled Poincare map in the stabilization of the passive dynamic walking of the compass-gait biped robot.