A Restricted Poincare Map for Determining Exponentially Stable Periodic Orbits in Systems with Impulse Effects: Application to Bipedal Robots

摘要

Systems with impulse effects form a special classof hybrid systems that consist of an ordinary, time-invariant differential equation (ODE), a co-dimension one switching surface, and a re-initialization rule. The exponential stability of a periodic orbit in a C~(1)-nonlinear system with impulse effects can be studied by linearizing the Poincare return map around a fixed point and evaluating its eigenvalues. However, in feedback design--where one may be employing an iterative technique to shape the periodic orbit subject to it being exponentially stable--recomputing and re-linearizing the Poincare return map at each iteration can be very cumbersome. For a nonlinear system with impulse effects that possesses an invariant hybrid subsystem and the transversal dynamics is sufficiently exponentially fast, it is shown that exponential stability of a periodic orbit can be determined on the basis of the restricted Poincare map, that is, the Poincare return map associated with the invariant subsystem. The result is illustrated on a walking gait for an underactuated planar bipedal robot.