Identification, stability and stabilization of limit cycles in a compass-gait biped model via a hybrid poincaré map

Abstract

This chapter focuses on identification, stability analysis and stabilization of hybrid limit cycle in the passive dynamic walking of the compass-gait biped robot as it goes down an inclined surface. The walking dynamics of such biped is described by an impulsive hybrid nonlinear system, which is composed of a nonlinear differential equation and a nonlinear algebraic equation. Under variation of the slope parameter, the passive biped robot displays symmetric (stable one-periodic) and asymmetric (unstable one-periodic and chaotic) behaviors. Then, the main objective of this chapter is to stabilize a desired asymmetric gait into a symmetric one by means of a control input, the hip torque. Nevertheless, the design of such control input using the impulsive hybrid dynamics is a complicated task. Then, to overcome this problem, we constructed a hybrid Poincaré map by linearizing the impulsive hybrid nonlinear dynamics around a desired unstable one-periodic hybrid limit cycle for some desired slope parameter. We stress that both the differential equation and the algebraic equation are linearized. The desired hybrid limit cycle is identified and analyzed first through the impulsive hybrid nonlinear dynamics via the fundamental solution matrix. We demonstrate that identification of the one-periodic fixed point of the designed hybrid Poincaré map and its stability depend only upon the nominal impact instant. We introduce a state-feedback controller in order to stabilize the linearized Poincaré map around the one-periodic fixed point. We show that the developed strategy for the design of the OGY-based controller has achieved the stabilization of the desired one-periodic hybrid limit cycle of the compass-gait biped robot.