Control Barrier Function

Writing ‘don't cross this boundary’ as a constraint, nudging the control command only minimally, and only when safety is close to being violated.

A control barrier function defines a safe set with a function h(x): h(x) ≥ 0 counts as safe, where x is the system state (e.g., position and velocity). Its core condition is ḣ ≥ −α·h (α a positive constant): the closer to the boundary — the smaller h is — the more the allowed rate of decrease of h is slowed down, and right at the boundary it can no longer decrease at all, which keeps the state inside the safe set forever. Ames, Tabuada, and others combined this with quadratic programming (arXiv 2016, journal version in IEEE TAC): among all control values satisfying that condition, find the one closest to the original desired command u_des — this is the CBF-QP. Because it only adjusts the command when necessary, it's often used as a safety filter wrapped around teleoperation or a learned policy. It's the counterpart of the control Lyapunov function: a CLF guarantees ‘converge to the goal,’ while a CBF guarantees ‘never leave the safe region’; compared to artificial potential fields, it offers a formal safety guarantee.

ExampleA 1D cart moving as ẋ = u toward a wall at x = 5 meters: taking h = 5 − x and α = 2, the condition becomes u ≤ 2(5 − x). At x = 4, the speed limit is 2 m/s, so an original command of 1 m/s is unaffected; at x = 4.9, the limit drops to 0.2 m/s, and the CBF-QP clamps the command to 0.2 m/s. The closer the cart gets to the wall, the slower it's allowed to go, so it never hits it.

Also called

CBF, CBF-QP, Barrier Function

Related

Control Lyapunov Function · Safety Filter · Quadratic Programming · Hamilton-Jacobi Reachability Analysis · Safe Reinforcement Learning · Artificial Potential Field

Sources

Ames et al., Control Barrier Functions: Theory and Applications (arXiv:1903.11199)
Ames, Xu, Grizzle, Tabuada: Control Barrier Function Based Quadratic Programs for Safety Critical Systems (arXiv:1609.06408)