Convex Optimization for Spring Design of Parallel Elastic Actuators

Elastic actuation can improve human-robot interaction and energy efficiency for wearable robots. Previous work showed that the energy consumption of series elastic actuators can be a convex function of the series spring compliance. This function is useful to optimally select the series spring compliance that reduces the motor energy consumption. However, series springs have limited influence on the motor torque, which is a major source of the energy losses due to the associated Joule heating. Springs in parallel to the motor can significantly modify the motor torque and therefore reduce Joule heating, but it is unknown how to design springs that globally minimize energy consumption for a given motion of the load. In this work, we introduce the stiffness design of linear and nonlinear parallel elastic actuators via convex optimization. We show that the energy consumption of parallel elastic actuators is a convex function of the spring stiffness and compare the energy savings with that of optimal series elastic actuators. We analyze robustness of the solution in simulation by adding uncertainty of 20% of the RMS load kinematics and kinetics for the ankle, knee, and hip movements for level-ground human walking. When the winding Joule heating losses are dominant with respect to the viscous losses, our optimal PEA designs outperform SEA designs by further reducing the motor energy consumption up to 63%. Comparing to the linear PEA designs, our nonlinear PEA designs further reduced the motor energy consumption up to 31%. From our convex formulation, our global optimal nonlinear parallel elastic actuator designs give two different elongation-torque curves for positive and negative elongation, suggesting a clutching mechanism for the final implementation. In addition, the different torque-elongation profiles for positive and negative elongation for nonlinear parallel elastic actuators can cause sensitivity of the energy consumption to changes in the nominal load trajectory.