arXiv:2610.08332v1 Announce Type: cross Abstract: The collision cone used by velocity-space control barrier functions is conservative: it rejects every relative velocity aimed into an obstacle, however slow. We propose the \emph{shifted collision-cone CBF} (SC3BF), which adds a state-dependent \emph{allowance} to the cone condition, so the robot may approach the obstacle at a rate that grows with distance and with its own speed. SC3BF is enforced by an ordinary quadratic program, and its safe set is forward invariant under bounded inputs without a minimum forward speed or a clearance margin. We prove that a nonzero allowance preserving safety always exists, and derive one in closed form. Against three velocity-space baselines on a kinematic bicycle among up to $100$ moving obstacles, SC3BF reaches the goal more often and modifies the nominal input less than half as much.