Optimality principles in variable stiffness control: The VSA hammer

The control of a robot's mechanical impedance is attracting increasing attention of the robotics community. Recent research in Robotics has recognized the importance of Variable Stiffness Actuators (VSA) in safety and performance of robots. An important step in using VSA for safety has been to understand the optimality principles that regulate the synchronized variation of stiffness and velocity when moving in the shortest time while limiting possible impact forces (the safe brachistochrone problem). In this paper, we follow a similar program of understanding the use of VSA in performance enhancement, looking at very dynamic tasks where impacts are maximized. To this purpose we address a new optimization problem that consists in choosing the inputs for maximizing the velocity of a link at a given final position, such as, e.g., for maximizing the effect of a hammer impact. We first study the problem with fixed stiffness, and show that, under realistic modeling assumptions, there does exist an optimal linear spring for the given inertia and motor. We then study optimal control of VSA and show that varying the spring stiffness during the execution of the hammering task improves the final performance substantially. The optimal control law is obtained analytically, thus providing insight in the optimality principles underpinning general VSA control. Finally, we show the practicality of our theoretical results with experimental tests.