Soft-actuators in cyclic motion: Analytical optimization of stiffness and pre-load

In this paper, we study the role of soft actuation in the reduction of the energy cost for mechanical systems that perform cyclic tasks. The objective is to determine the optimal stiffness value and spring pre-load such that a given cost functional is minimized. For the analysis, we consider both fully actuated and underactuated mechanical systems using elastic actuators which, depending on how and where the springs are placed w.r.t. the actuator and the load, can be Series Elastic Actuators (SEAs) or Parallel Elastic Actuators (PEAs). The energy consumption depends not only on the actuation parameters but also on the trajectories followed to perform a given cyclic task. We show that the general problem in which both joint trajectories and actuation parameters are the optimization variables, can be cast as a simpler problem in which optimization regards only joint trajectories. Simulations of fully actuated and underactuted compliant robots are reported to demonstrate the effectiveness of the method. Although the stiffness optimization method is analytical in nature, it is directly applicable to existing systems whose model is unknown. A model-free experimental application on a prototype of a hopping robot with SEA is presented.