On the Computation of Sensitivity Tubes

Abstract

Achieving robust robot control requires explicit treatment of model uncertainties. Closed-loop sensitivity has emerged as a powerful tool to analyze how parameter errors map into state and input deviations through so-called “sensitivity tubes”, traditionally built from <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ellipsoidal</i> uncertainty sets and used to robustify system constraints. These ellipsoids, however, are themselves smooth approximations of underlying <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">hyperboxes</i> in the parameter space, leading to an inaccurate estimation of the parameter set. This paper extends that framework by proposing two new formulations that more precisely represent the real closed-loop behavior of the system through improved computation of the sensitivity tubes. The first constructs tubes directly from <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">hyperboxes</i>, exactly preserving the original parameter bounds but producing a non-differentiable description. The second employs <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">superquadrics</i>, which smoothly approximate the hyperbox with user-tunable fidelity while preserving differentiability, as in the ellipsoidal case. Both methods are validated through an extensive simulation campaign, where the resulting input tubes ensure actuator constraints are respected. The results demonstrate that the new tubes better enclose the perturbed trajectories with respect to ellipsoidal ones, enhancing robustness for both online and offline trajectory planning.