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Optimal Control Barrier Functions: Maximizing the Action Space Subject to Control Bounds

Logan E. Beaver

TL;DR

The paper addresses the constraint compatibility problem for control barrier functions (CBFs) under bounded actuation by constructing an optimal CBF that maximizes the feasible action space for both first- and second-order constraints. It proves that the optimal CBF induces a switching dynamic and defines a dynamical motion primitive on the safe-set boundary, which implicitly encodes a model of future trajectories for time-varying components. The authors provide constructive proofs (including Theorems for first- and second-order cases), a switching-system interpretation, and a definition of the motion primitive, along with a practical ACC simulation showing reduced conservatism compared with a linear CBF. The work demonstrates that optimal CBFs can maintain safety while expanding the allowable control actions, with potential implications for more realistic, reactive safety in autonomous systems and for developing data-driven or higher-order barrier formulations.

Abstract

This letter addresses the constraint compatibility problem of control barrier functions (CBFs), which occurs when a safety-critical CBF requires a system to apply more control effort than it is capable of generating. This inevitably leads to a safety violation, which transitions the system to an unsafe (and possibly dangerous) trajectory. We resolve the constraint compatibility problem by constructing a control barrier function that maximizes the feasible action space for first and second-order constraints, and we prove that the optimal CBF encodes a dynamical motion primitive. Furthermore, we show that this dynamical motion primitive contains an implicit model for the future trajectory for time-varying components of the system. We validate our optimal CBF in simulation, and compare its behavior with a linear CBF.

Optimal Control Barrier Functions: Maximizing the Action Space Subject to Control Bounds

TL;DR

The paper addresses the constraint compatibility problem for control barrier functions (CBFs) under bounded actuation by constructing an optimal CBF that maximizes the feasible action space for both first- and second-order constraints. It proves that the optimal CBF induces a switching dynamic and defines a dynamical motion primitive on the safe-set boundary, which implicitly encodes a model of future trajectories for time-varying components. The authors provide constructive proofs (including Theorems for first- and second-order cases), a switching-system interpretation, and a definition of the motion primitive, along with a practical ACC simulation showing reduced conservatism compared with a linear CBF. The work demonstrates that optimal CBFs can maintain safety while expanding the allowable control actions, with potential implications for more realistic, reactive safety in autonomous systems and for developing data-driven or higher-order barrier formulations.

Abstract

This letter addresses the constraint compatibility problem of control barrier functions (CBFs), which occurs when a safety-critical CBF requires a system to apply more control effort than it is capable of generating. This inevitably leads to a safety violation, which transitions the system to an unsafe (and possibly dangerous) trajectory. We resolve the constraint compatibility problem by constructing a control barrier function that maximizes the feasible action space for first and second-order constraints, and we prove that the optimal CBF encodes a dynamical motion primitive. Furthermore, we show that this dynamical motion primitive contains an implicit model for the future trajectory for time-varying components of the system. We validate our optimal CBF in simulation, and compare its behavior with a linear CBF.
Paper Structure (7 sections, 9 theorems, 46 equations, 2 figures, 1 table)

This paper contains 7 sections, 9 theorems, 46 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notation
  3. Control Barrier Function Preliminaries
  4. First Order Systems
  5. Second Order ZBFs
  6. Simulation Results
  7. Conclusions

Key Result

Theorem 1

The optimal ZBF eq:fo-opt-zbf can be approximated arbitrarily well by jointly imposing a linear CBF and control bounds.

Figures (2)

  • Figure 1: Switching description of the optimal behavior; the system operates within the control bounds until it reaches the safe set boundary.
  • Figure 2: Behavior of the optimal CBF (blue and orange) and the conservative linear CBF (purple and yellow) as the system approaches the boundary of the safe set at $b(\bm{x}) =0$.

Theorems & Definitions (22)

  • proof
  • Theorem 1
  • proof
  • Lemma 1
  • Definition 1
  • Lemma 2
  • proof
  • Remark 1
  • Lemma 3
  • proof
  • ...and 12 more