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

Systems with impulse effects form a special class of 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 C¹-nonlinear systems with impulse effects can be studied by linearizing the Poincaré 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 Poincaré return map at each iteration can be very cumbersome. For a non- linear 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 Poincaré map, that is, the Poincaré return map associated with the invariant subsystem. The result is illustrated on a walking gait for an underactuated planar bipedal robot.