Robust Optimal Design of Energy Efficient Series Elastic Actuators: Application to a Powered Prosthetic Ankle

Abstract

Design of rehabilitation and physical assistance robots that work safely and efficiently despite uncertain operational conditions remains an important challenge. Current methods for the design of energy efficient series elastic actuators use an optimization formulation that typically assumes known operational requirements. This approach could lead to actuators that cannot satisfy elongation, speed, or torque requirements when the operation deviates from nominal conditions. Addressing this gap, we propose a convex optimization formulation to design the stiffness of series elastic actuators to minimize energy consumption and satisfy actuator constraints despite uncertainty due to manufacturing of the spring, unmodeled dynamics, efficiency of the transmission, and the kinematics and kinetics of the load. To achieve convexity, we write energy consumption as a scalar convex-quadratic function of compliance. As actuator constraints, we consider peak motor torque, peak motor velocity, limitations due to the speed-torque relationship of DC motors, and peak elongation of the spring. We apply our formulation to the robust design of a series elastic actuator for a powered prosthetic ankle. Our simulation results indicate that a small trade-off between energy efficiency and robustness is justified to design actuators that can operate with uncertainty.

Fig. 1.

Energy flow of an SEA. Dashed lines indicate that the energy path may or may not exist depending on the construction of the device. For instance, energy flowing from the load to the battery requires that the load is high enough to backdrive the motor-transmission system.

Fig. 2.

Left: Energy consumption as a function of compliance, α, where the energy savings (E.S.) region 0 ≤ α ≤ − b/a provides E_m below the rigid level c. Right: Case of motor and load that would not benefit energetically from series elasticity.

Fig. 3.

Schematic SEA for powered prosthetic ankle.

Fig. 4.

Position (top) and torque (bottom) of the human ankle during level ground walking [31]. The solid line indicates the mean values for a 69.1 kg subject. The shaded region around the nominal trajectory illustrates the uncertainty in the position ε_ql = ±5° and the mass of the subject ε_m = ±8.8 kg (uncertainty based on the standard deviation in [31]).

Fig. 5.

Speed and torque requirements of different actuators for a powered prosthetic ankle. The region enclosed by the dotted line describes the speeds and torques that satisfy the specifications of the motor, i.e., feasible region. Figure shows three possible actuator designs: (a) rigid actuator Maxon EC-30 without series elasticity, (b) SEA using the same motor with optimal stiffness that satisfies constraints only for the nominal data, and (c) SEA with the same motor that satisfies actuator constraints despite uncertainty using our formulation. The robust design (c) is the only actuator that satisfies the actuator constraints for all possible values of uncertainty.