Control Barrier Functions

Definition

A Control Barrier Function (CBF) encodes a safety requirement as forward invariance of a safe set
$\mathcal{S}=\{\mathbf{x}:h(\mathbf{x})\ge 0\}$, the superlevel set of a continuous function $h$
(its zero level set $\partial \mathcal{S}$ is the boundary, $h>0$ the interior). $h$ is a CBF if there
exists a control input keeping the system in $\mathcal{S}$ for all future time; constructed directly
from the safe-set definition, it sidesteps the polynomial-optimization cost of older barrier
certificates and serves either as a stand-alone safe controller or as a safety filter layered on an
existing controller (akella2024risk, after Ames 2016). Under stochastic
disturbances, almost-sure invariance may be infeasible, so the source’s central object is the
discrete-time Risk-Aware CBF (RCBF): $h$ is an RCBF for $\mathcal{S}$ if a dynamic coherent risk
measure $\rho$ of the next-step value dominates a discounted current value, whose existence implies
safety in the coherent-risk sense (Theorem 1). The source is regime-agnostic (a survey); its
demonstrations are terrestrial — bipeds, a quadruped, trucks, automotive lane-keeping — not a space base.

Key Equations

Symbols per notation.md.

Safe set as the superlevel set of the barrier function $h$ (state $\mathbf{x}\in \mathcal{X}\subset {\mathbb{R}}^n$):

\partial\mathcal{S}=\{\boldsymbol x \mid h(\boldsymbol x)=0\}.$$ Discrete-time Risk-Aware CBF condition (akella2024risk, Def. 9), for a convex class-$\mathcal K$ function $\alpha$ with $\alpha(r)<r\ \forall r>0$ and dynamic coherent risk measure $\rho$: $$\rho\!\left(h(\boldsymbol x(t{+}1))\right)\;\ge\;\alpha\!\left(h(\boldsymbol x(t))\right), \qquad \forall \boldsymbol x(t)\in\mathcal X.$$ > Notation note: $\alpha$ here is the class-$\mathcal K$ comparison function of the CBF condition, **not** > the CVaR confidence level $\alpha$ listed in [notation.md](../notation.md). The most common choice is a > constant $\alpha(r)=\alpha_0 r,\ \alpha_0\in(0,1)$. The risk measure $\rho$ is source-faithful (e.g. > $\rho=\mathrm{CVaR}_\alpha$ recovers a CVaR-CBF) and is not canonicalized in notation.md. ## Source Support - [akella2024risk](../sources/akella2024risk.md) — survey of risk-aware V&V and safety-critical control; gives the safe-set/superlevel-set CBF formulation (after Ames 2016), defines $\rho$-safety, $\rho$-reachability, the discrete-time RCBF (Def. 9) and finite-time RCBF (Def. 10), and proves that an RCBF implies $\rho$-safety (Thm. 1). Notes CBF construction for complex systems remains an open problem. ## Related Topics - [barrier_lyapunov_function](barrier_lyapunov_function.md) — companion certificate that bounds tracking error inside a tube; a CBF certifies *set invariance / safety* whereas a BLF certifies *constraint- respecting stability*. - [chance_constraints](chance_constraints.md) — an alternative way to encode safety under uncertainty (probabilistic constraint vs. risk-measure barrier); the RCBF replaces a hard barrier with a coherent risk on the barrier value. - [conditional_value_at_risk](conditional_value_at_risk.md) — the canonical coherent risk measure $\rho$; choosing $\rho=\mathrm{CVaR}_\alpha$ in the RCBF condition yields the CVaR-CBF the source builds on. - [coherent_risk_measures](coherent_risk_measures.md) — the general class of $\rho$ for which RCBF existence implies $\rho$-safety (Theorem 1); CVaR is the special case. - [model_predictive_control](model_predictive_control.md) — alternative feedback-safety layer; CBFs act pointwise as a filter, MPC enforces safety over a receding horizon. The source pairs both with risk. - [keep_out_zone](keep_out_zone.md) — the inspection-side application: an obstacle / forbidden region is exactly the complement of a safe set, i.e. a candidate $h$ for a CBF safety filter. ## Open Questions - The source is regime-agnostic and demonstrated only on terrestrial systems (legged robots, trucks, automotive). Does an RCBF/CBF safety filter transfer to our free-**flying** space manipulator, where the fully-actuated 6-DOF base plus redundant arm give a high-dimensional, dynamically-coupled state? - The source states that *constructing* CBFs — let alone risk-aware ones — remains an open problem for complex systems. What is a tractable $h$ for keep-out / collision-avoidance during proximity inspection? - The RCBF in Def. 9 is **discrete-time**. Our control runs in continuous time; does the discrete-time coherent-risk invariance guarantee survive a continuous-time implementation at our control rate?