Decentralized Event-Based Controllers for Robust Stabilization of Hybrid Periodic Orbits: Application to Underactuated 3-D Bipedal Walking

Abstract

Models of bipedal walking are hybrid, with continuous-time dynamics representing the swing phases and discrete-time dynamics representing the impact events. The feedback controllers for these systems can be two-level, including both continuousand discrete-time (event-based) actions. This paper presents a systematic framework to design decentralized event-based controllers for robust stabilization of hybrid periodic orbits against possible disturbances in discrete-time phases. The properties of the Poincare return map are investigated to study the orbital input-to-state stability for the closed-loop system with respect to disturbance inputs. An optimization problem involving bilinear matrix inequalities is then presented to design ℌ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> - and ℌ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> -optimal decentralized event-based controllers. The power of the proposed framework is finally demonstrated through designing a set of decentralized two-level controllers for underactuated walking of a threedimensional autonomous bipedal robot with nine degrees of freedom and a decentralization scheme motivated by amputee locomotion with a transpelvic prosthetic leg.