On the Computation of Sensitivity Tubes

Abstract

Achieving robust and reliable control in robotic systems is crucial, especially in presence of model uncertainties. Over the last years the notion of closed-loop sensitivity has emerged as an effective tool for analyzing and quantifying how uncertainties in the model parameters affect the closed-loop system behavior. In particular, several previous works have shown how the sensitivity matrixes can be used to map uncertainty ellipsoids in the parameter space to the corresponding state/input ellipsoids (the so-called sensitivity tubes) that can be leveraged to robustify any system constraint. This paper extends these previous works based on an ellipsoidal modeling of the parametric uncertainty by proposing two new approaches that significantly improve the computation of the sensitivity tubes. The first method replaces the ellipsoids with hyperboxes for constructing the tubes: this solution avoids any approximation of the parameter set but yields a non-differentiable formulation of the resulting tube. The second method, instead, leverages superquadrics that can approximate a hyperbox with a tunable precision while retaining differentiability of the resulting tubes (as in the ellipsoid case). Both methods have been validated via a simulation campaign and compared with the previous approaches based on an ellipsoidal modeling of the uncertainties. The results confirm the effectiveness of the proposed techniques in producing state/input tubes that accurately envelope the perturbed system behavior, which is an important asset for providing a robustness layer to any online/offline trajectory generation algorithm.