Hybrid Invariant Manifolds in Systems With Impulse Effects With Application to Periodic Locomotion in Bipedal Robots

Abstract

Motivated by the problem of controlling walking in a biped with series compliant actuation, this paper develops two main theorems relating to the stabilization of periodic orbits in systems with impulse effects. The first main result shows that when a periodic orbit of a system with impulse effects lies within a hybrid invariant manifold, there exist local coordinate transforms under which the Jacobian linearization of the Poincare return map has a block upper triangular structure. One diagonal block is the linearization of the system as restricted to the hybrid invariant manifold, also called the hybrid zero dynamics. The other is the product of two sensitivity matrices related to the transverse dynamics-one pertaining to the impact map and the other pertaining to the closed-loop vector field. When either of these sensitivity matrices is sufficiently close to zero, the stability of the return map is determined solely by the stability of the hybrid zero dynamics. The second main result of the paper details the construction of a hybrid invariant manifold, such as that required by the first main theorem. Forward invariance follows from the methods of Byrnes and Isidori, and impact invariance is achieved by a novel construction of impact-updated control parameters. In addition to providing impact invariance, the construction allows entries of the impact sensitivity matrix of the transverse dynamics to be made arbitrarily small. A simulation example is provided where stable walking is achieved in a 5-link biped with series compliant actuation.