Control Barrier Function Based Quadratic Programs for Safety Critical Systems

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Ames, Xu, Grizzle and Tabuada define safety as forward invariance of a set and certify it with barrier functions instead of computing reachable sets. They introduce two kinds: reciprocal barriers that blow up at the boundary (Ḃ ≤ α(1/B)) and zeroing barriers that vanish there (ḣ ≥ −α(h)), and show these conditions are necessary and sufficient for invariance under mild assumptions. For control-affine systems the conditions are affine in the input, so a quadratic program can enforce a control barrier function (CBF) as a hard constraint while a control Lyapunov function (CLF) for the task is relaxed with a slack variable. Safety always wins, and progress happens when the two do not conflict.

Findings

  • The CLF-CBF QP gives a controller that is locally Lipschitz under stated conditions, which is needed for the existence of solutions.
  • Adaptive cruise control with wheel-force limits: the time-headway constraint held throughout, force limits held when a force-based barrier was used, and the car braked earlier and approached its desired speed more slowly.
  • A less conservative barrier let the car follow more closely while still satisfying the headway constraint.
  • The authors cite earlier real-time QP controllers at 200 Hz to 1 kHz on a bipedal robot and on scale cars.

Limits

  • The guarantee is about the model’s state. A wrong state estimate (spoofed GPS, drifting IMU) means the QP keeps the estimate safe, which may not be where the robot is (my reading).
  • With input bounds, the barrier and the actuator limits can conflict. The authors build problem-specific force-based barriers and leave a general construction open.
  • The CLF slack means nothing forces progress: the controller can stay safe and never reach its goal.

Relevance

The base layer for safe control in robot teams: wang-2016-safety-barrier-certificates turns it into a multi-robot collision filter, and qin-2021-neural-barrier-certificates and bousias-2025-equivariant-cbf learn the barrier instead of writing it. The relaxed-progress design is also where safe-but-stuck behaviour starts, the topic of Safe but Stuck.