Control lyapunov functions and hybrid zero dynamics

Abstract

Abstract-Hybrid zero dynamics extends the Byrnes-Isidori notion of zero dynamics to a class of hybrid models called systems with impulse effects. Specifically, given a smooth submanifold that is contained in the zero set of an output function and is invariant under both the continuous flow of the system with impulse effects as well as its reset map, the restriction dynamics is called the hybrid zero dynamics. Prior results on the stabilization of periodic orbits of the hybrid zero dynamics have relied on input-output linearization of the transverse variables. The principal result of this paper shows how control Lyapunov functions can be used to exponentially stabilize periodic orbits of the hybrid zero dynamics, thereby significantly extending the class of stabilizing controllers. An illustration of this result on a model of a bipedal walking robot is provided.

Rapidly Exponentially Stabilizing Control Lyapunov Functions and Hybrid Zero Dynamics

“…The results presented in this paper were initially reported in [4]. The present paper adds to the original contribution in the following important ways: the non-hybrid case is fully developed through Theorem 1; the proof of the main resultTheorem 2-is carried out in full while it was omitted from [4]; and the theoretical results are verified experimentally.…”

“…The present paper adds to the original contribution in the following important ways: the non-hybrid case is fully developed through Theorem 1; the proof of the main resultTheorem 2-is carried out in full while it was omitted from [4]; and the theoretical results are verified experimentally. These factors combine to create a complete exposition on the application of control Lyapunov functions to hybrid zero dynamics with application to bipedal robotic locomotion.…”

Optimization-based locomotion planning, estimation, and control design for the atlas humanoid robot

“…Ames et al [4,3] used control Lyapunov functions for walking by solving QPs that minimize the input norm, ||u||, while satisfying constraints on the negativity of (x, u, t). In the discrete time setting, Wang and Boyd [66] describe an approach to quickly evaluating control Lyapunov policies using explicit enumeration of active sets in cases where the number of states is roughly equal to the square of the number of inputs.…”

Técnicas de control para el balance de un robot bípedo: un estado del arte

“…Las funciones de control de Lyapunov las utilizan Ames, Galloway y Grizzle (2012). Estos autores muestran cómo dichas funciones se pueden utilizar para estabilizar las órbitas periódicas de la dinámica híbrida cero de manera exponencial.…”