Optimal control of variable stiffness actuators with nonlinear springs

Variable Stiffness Actuators (VSA) are currently being regarded as promising actuator types in robotics research especially due to their capability to store potential energy in their elastic elements and to control this energy by altering the elastic properties of these elements. The controllable potential energy enables these actuators to outperform their rigid counterparts especially when realizing fast explosive motions. Most of joint designs with VSA's, however, are described by nonlinear deflection-torque relations and consequently a thorough analysis regarding their maximum attainable performance is difficult. In this work, we tackle this problem by using Pontryagin's Minimum Principle and develop a general method to solve the optimal control problem of minimizing any given terminal cost for these joints. In other words, we show the optimal control strategies to alter VSA's elastic properties for various tasks such as maximization of the final link velocity or time-optimal tracking, which are all found to depend on the change of the system's potential and kinetic energy relative to its total energy. The application of the method is illustrated for VSA's with adjustable linear and cubic springs, where the potential energy stored in the springs is maximized at a given final time.