Optimal Control Strategies for Maximizing the Link Velocity of Elastic Joints with Nonlinear Impedance

Abstract

In this thesis, we investigate the problem of maximizing the link velocity of elastic joints using velocity-sourced elastic actuators. More specifically, focusing on joints with nonlinear series elastic actuators we derive motor control strategies such that the link velocity is maximized at a given time instant when the joint is initially at rest. Furthermore, we provide a physical interpretation for the derived strategies by exploiting their time optimality. The interpretation reveals the dependence of these strategies on periods of mass-spring systems which in turn explains how nonlinear torque-deflection profiles influence the maximal link velocity. In order to clearly illustrate this influence, we analyse in detail three different elastic joints with softening, linear and hardening springs. In particular, we compare their maximal link velocities as well as the corresponding control strategies and elaborate on the observed differences. Our theoretical results are experimentally validated on the DLR Floating Spring Joint where link velocities at least more than three times the maximally applied motor velocity are attained in less than a second. Several extensions are also provided which reveal the influence of damping and stiffness actuation on optimal control strategies. Finally, we give a proof of Pontryagin's Minimum Principle, the main theorem used in the thesis, by exploiting the properties of transition maps. Assuming an additional degree in the smoothness of the system dynamics and the cost functional, this leads to an extension of the principle, namely the Second Order Minimum Principle.