Bounded Collision Force by the Sobolev Norm

TL;DR

This paper introduces a novel application of the Sobolev norm as a system norm to provide rigorous bounds on maximum collision force in robots, enabling analysis of complex models and control strategies for safety.

Contribution

The paper adapts the Sobolev norm to be a system norm for collision force analysis, allowing accurate bounds and controller synthesis for general dynamic and compliant robots.

Findings

Validated Sobolev norm experimentally and in simulation.

Demonstrated impact of control and compliance on collision force.

Showed trade-off between collision safety and environmental estimation.

Abstract

A robot making contact with an environment or human presents potential safety risks, including excessive collision force. While experiments on the effect of robot inertia, relative velocity, and interface stiffness on collision are in literature, analytical models for maximum collision force are limited to a simplified mass-spring robot model. This simplified model limits the analysis of control (force/torque, impedance, or admittance) or compliant robots (joint and end-effector compliance). Here, the Sobolev norm is adapted to be a system norm, giving rigorous bounds on the maximum force on a stiffness element in a general dynamic system, allowing the study of collision with more accurate models and feedback control. The Sobolev norm can be found through the $\mathcal{H}_2$ norm of a transformed system, allowing efficient computation, connection with existing control theory, and controller synthesis to minimize collision force. The Sobolev norm is validated, first experimentally with an admittance-controlled robot, then in simulation with a linear flexible-joint robot. It is then used to investigate the impact of control, joint flexibility and end-effector compliance on collision, and a trade-off between collision performance and environmental estimation uncertainty is shown.