Optimal Control of Variable Stiffness Policies: Dealing with Switching Dynamics and Model Mismatch

Controlling complex robotic platforms is a challenging task, especially in designs with high levels of kinematic redundancy. Novel variable stiffness actuators (VSAs) have recently demonstrated the possibility of achieving energetically more efficient and safer behaviour by allowing the ability to simultaneously modulate the output torque and stiffness while adding further levels of actuation redundancy. An optimal control approach has been demonstrated as an effective method for such a complex actuation mechanism in order to devise a control strategy that simultaneously provides optimal control commands and time-varying stiffness profiles. However, traditional optimal control formulations have typically focused on optimisation of the tasks over a predetermined time horizon with smooth, continuous plant dynamics. In this chapter, we address the optimal control problem of robotic systems with VSAs for the challenging domain of switching dynamics and discontinuous state transitions arising from interactions with an environment. First, we present a systematic methodology to simultaneously optimise control commands, time-varying stiffness profiles as well as the optimal switching instances and total movement duration based on a time-based switching hybrid dynamics formulation. We demonstrate the effectiveness of our approach on the control of a brachiating robot with a VSA considering multi-phase swing-up and locomotion tasks as an illustrative application of our proposed method in order to exploit the benefits of the VSA and intrinsic dynamics of the system. Then, to address the issue of model discrepancies in model-based optimal control, we extend the proposed framework by incorporating an adaptive learning algorithm. This performs continuous data-driven adjustments to the dynamics model while re-planning optimal policies that reflect this adaptation. We show that this augmented approach is able to handle a range of model discrepancies in both simulations and hardware experiments.