Optimal Energy Tank Initialization for Minimum Sensitivity to Model Uncertainties

Energy tanks have gained popularity inside the robotics and control communities over the last years, since they represent a formidable tool to enforce passivity (and, thus, input/output stability) of a controlled robot, possibly interacting with uncertain environments. One weak point of passification strategies based on energy tanks concerns, however, their initialization. Indeed, a too large initial energy can cause practical unstable behaviors, while a too low initial energy level can prevent the correct execution of the task. This shortcoming becomes even more relevant in presence of uncertainties in the robot model and/or environment, since it may be hard to predict in advance the correct (safe) amount of initial tank energy for a successful task execution. In this paper we then propose a new strategy for addressing this issue. The recent notion of closed-loop state sensitivity is exploited to derive precise bounds (tubes) on the tank energy behavior by assuming parametric uncertainty in the robot model. These tubes are then exploited in a novel nonlinear optimization problem aiming at finding both the best trajectory and the minimal initial tank energy that allow executing a positioning task for any value of the uncertain parameters in a given range. The approach is finally validated via a statistical analysis in simulation and experiments on real robot hardware.