Rank Properties of Poincare Maps for Hybrid Systems with Applications to Bipedal Walking

摘要

The equivalence of the stability of periodic orbits with the stability of fixed points of a Poincare map is a well-known fact for smooth dynamical systems. In particular, the eigenvalues of the linearization of a Poincare map can be used to determine the stability of periodic orbits. The main objective of this paper is to study the properties of Poincare maps for hybrid systems as they relate to the stability of hybrid periodic orbits. The main result is that the properties of Poincare maps for hybrid systems are fundamentally different from those for smooth systems, especially with respect to the linearization of the Poincare map and its eigenvalues. In particular, the linearization of any Poincare map for a smooth dynamical system will have one trivial eigenvalue equal to 1 that does not affect the stability of the orbit. For hybrid systems, the trivial eigenvalues are equal to 0 and the number of trivial eigenvalues is bounded above by dimensionality differences between the different discrete domains of the hybrid system and the rank of the reset maps. Specifically, if n is the minimum dimension of the domains of the hybrid system, then the Poincare map on a domain of dimension m > n results in at least m — n + 1 trivial 0 eigenvalues, with the remaining eigenvalues determining the stability of the hybrid periodic orbit. These results will be demonstrated on a nontrivial multi-domain hybrid system: a planar bipedal robot with knees.