An Analysis of Stability of Systems with Impulse Effects: Application to Biped Robots

The approximate Jacobian matrix of the Poincaré return map at the fixed point is presented for the nonlinear system with impulse effects. And the sufficient condition to the existence of this approximate Jacobian matrix is given with the disturbance theory and linearization method. Since this approximate expression depends only on the configuration of the system with impulse effects, then the uniqueness of this approximate expression can be guaranteed and this approximate expression can be obtained precisely. In addition, this approximate Jacobian matrix can be used as a tool to study the asymptotical stability of the system with impulse effects. Since the biped robot gaits can be described by the nonlinear system with impulse effects, then the stability of the biped robot walking cycle can be studied with this tool. In order to study the stability, this approximate Jacobian matrix is applied to the compass-like passive biped robot gaits. It is shown that the approximate Jacobian matrix proposed in this paper is as useful as the ones proposed in the numerical methods. In the end this result is confirmed by simulations.