Lyapunov based estimation of the basin of attraction of Poincare maps with applications to limit cycle walking

Limit cycles occur in a wide range of nonlinear and hybrid systems including dynamic walking. While methods for determining local stability of limit cycles are available, they are relevant only for small perturbations. Characterizing the robustness of limit cycles to larger perturbations is an important, yet mostly open, challenge. Robustness can be characterized by the basin of attraction (BA), but standard techniques for computing BAs are computationally intensive. Here we present an algorithm to estimate inner bounds of BAs of fixed points of polynomial discrete maps. The algorithm is based on a convex optimization problem known as Sum of Squares programming, and results in BA estimates which are expressed as sublevel sets of polynomial Lyapunov functions. The method can be applied to estimate BAs of fixed points of discrete Poincaré Maps and thus the BAs of limit cycles on selected Poincaré sections, as demonstrated on a simple biped walking model under different modes of actuation. For each mode of actuation, the Poincaré map was fitted using a small number of simulations and an inner bound for the BA was estimated. Comparing the BAs achieved by the different modes of actuation demonstrates that the BA of a passive biped model can be enlarged using minimalistic event-driven actuation, and that this relationship is also apparent from the estimated inner bounds of the BAs.