Passivity of time-delayed whole-body operational space control with series elastic actuation

Whole-Body Control has been extensively used to achieve humanoid robot force and motion tasks simultaneously during recent years. However, most existing results have not incorporated low-level actuator dynamics and time delays yet. In this study, we propose a novel time-delayed Whole-Body Operational Space control (WBOSC) with series elastic actuator (SEA) dynamics. This type of controller generalizes our previously proposed distributed control structure to multi-input and multi-output free floating humanoid robotic systems. Namely, Cartesian stiffness control is adopted to design the WBOSC at the centralized level while motor damping control is implemented at the embedded level to remedy the stability deterioration caused by time delays. Additionally, embedded-level torque feedback control is formulated and physically interpreted as a shaping of the motor inertia. To ensure passivity, we separate the overall system into two subsystems interconnected in a feedback configuration. By the Lyapunov-Krasovskii functional technique, we propose a delay-dependent passivity criterion of the closed-loop system in the form of linear matrix inequalities (LMIs), and solve for the allowable maximum time delays via the passivity criterion. Numerical simulations of a dynamic locomotion process are used to validate the proposed passivity criterion and the WBOSC framework.