Bounded Collision Force by the Sobolev Norm: Compliance and Control for Interactive Robots.

A robot making contact with a human or environment, both intended and unintended, is a major design consideration for interactive robots. Substantial experimental work has investigated the role of inertia, relative velocity, and interface stiffness in collision, but no analytical expression for maximum impact force of complex models is established. Here, the Sobolev norm is used to rigorously bound maximum impact force when both the force and its derivative are in $\mathcal{L}_p$, a condition which holds for a pure stiffness coupling damped inertias under impulsive excitation. The Sobolev norm allows unified consideration of contact with humans (free space and clamped) and pure stiffness environments, accommodating complex models without using elastic/inelastic collision assumptions. The Sobolev norm can be found through the $\mathcal{H}_2$ norm of a related system, allowing efficient computation and connection with existing control theory. 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 motor dynamics, control, joint and interface compliance on collision, and a trade-off between collision performance and environmental estimation is shown.