On the Computation of Sensitivity Tubes
Andrea Pupa, Tommaso Belvedere, Cristian Secchi, Paolo Robuffo Giordano
AI summary
Problem
Traditional sensitivity tubes approximate parameter uncertainty with ellipsoids, which become increasingly restrictive and inaccurate in high-dimensional spaces, leading to under-estimated perturbations and degraded planning robustness.
Approach
The authors derive exact tube radii from hyperboxes for gradient-free optimization and introduce superquadrics to smoothly approximate hyperboxes while maintaining differentiability for gradient-based optimization.
Key results
- Hyperbox-based tube formulation for gradient-free optimization
- Superquadric-based approximation preserving differentiability and tunable fidelity
- Formal proofs for both new methods and the existing ellipsoidal approach
- Simulation validation demonstrating tighter trajectory enclosure and improved constraint satisfaction
Why it matters
Provides robot planners and controllers with more accurate, flexible tools to guarantee safety and performance under model uncertainty.
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 ellipsoidal uncertainty sets and used to robustify system constraints. These ellipsoids, however, are themselves smooth approximations of underlying hyperboxes 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 hyperboxes, exactly preserving the original param- eter bounds but producing a non-differentiable description. The second employs superquadrics, which smoothly approximate the hyperbox with user-tunable fidelity while preserving differen- tiability, 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.