  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.
