Reciprocal Barrier Functions in Safety Control

Updated 7 January 2026

Reciprocal Barrier Functions (RBFs) are mathematical constructs that enforce forward invariance by converting safe set definitions into reciprocal metrics, ensuring a strong repulsive effect near unsafe boundaries.
RBFs integrate with quadratic programming and control Lyapunov functions to balance safety and performance across deterministic, stochastic, and disturbance-affected systems.
Advanced RBF extensions—including high-order, disturbance-robust, and neural network-based approaches—broaden their applications in automotive safety, collision avoidance, and complex robotic systems.

A Reciprocal Barrier Function (RBF) is a mathematical construct used in safety-critical control to encode forward invariance of a prescribed safe set. The fundamental operation is the reciprocity transform: given a continuously differentiable function $h:\mathbb{R}^n\to\mathbb{R}$ defining the safe set $\mathcal{C} = \{x\mid h(x)\ge0\}$ , the reciprocal barrier is defined as $B(x) = 1/h(x)$ on $\mathrm{Int}(\mathcal{C}) = \{x\mid h(x)>0\}$ . RBFs—and their controlled, stochastic, and neural network-parameterized extensions—play a central role in real-time safety-critical control, including deterministic, stochastic, and disturbance-affected systems. RBF-based safety constraints are frequently integrated into quadratic program (QP) formulations, allowing joint mediation of safety and performance objectives (Ames et al., 2016, Clark, 2020, Nishimura et al., 2022, Agrawal et al., 15 Apr 2025, Wang et al., 25 Jul 2025, So et al., 2023).

1. Core Definition and Deterministic Invariance Guarantees

Let $h:\mathbb{R}^n \rightarrow \mathbb{R}$ be $C^1$ . The safe set is

$\mathcal{C} = \{x \in \mathbb{R}^n : h(x) \geq 0\}, \quad \mathrm{Int}(\mathcal{C}) = \{h(x) > 0\}.$

Define $B(x) = 1/h(x)$ on $\mathrm{Int}(\mathcal{C})$ .

For the control-affine dynamics

$\dot{x} = f(x) + g(x)u, \quad x \in \mathbb{R}^n, \ u \in U \subset \mathbb{R}^m,$

a reciprocal control barrier function satisfies that for all $x \in \mathrm{Int}(\mathcal{C})$ there exists $u \in U$ so that

$L_f B(x) + L_g B(x) u \leq \alpha B(x), \qquad \alpha > 0,$

where $L_f B = \nabla B(x)f(x)$ , $L_g B = \nabla B(x)g(x)$ . If $u$ can always be chosen to satisfy this constraint, $\mathrm{Int}(\mathcal{C})$ is forward invariant: trajectories initialized in $\mathcal{C}$ remain in $\mathcal{C}$ for all future time (Ames et al., 2016). As $x$ approaches the boundary $h(x) \to 0^+$ , $B(x) \to +\infty$ , and the constraint asymptotically relaxes, which ensures a strong "repulsive" behavior away from unsafe states.

2. Quadratic Program Synthesis and Integration with Control Lyapunov Functions

RBF constraints are linear (affine) in $u$ , allowing their seamless integration with other affine objectives, such as those derived from equilibrium-stabilizing Control Lyapunov Functions (CLFs), in pointwise quadratic programs (Ames et al., 2016, Agrawal et al., 15 Apr 2025). A typical QP-based control law is: $\begin{array}{rl} u^*(x) = \underset{u \in U}{\arg\min} & \|u - u_{\text{des}}\|^2 \ \text{s.t.} & L_f V(x) + L_g V(x) u \leq -\gamma V(x), \ & L_f B(x) + L_g B(x) u \leq \alpha B(x) \end{array}$ where $V$ is a CLF, and $B$ is a RBF. The safety constraint is enforced as a hard barrier, while the CLF may be relaxed to mediate safety and performance, as in adaptive cruise control and lane keeping (Ames et al., 2016).

3. Stochastic Reciprocal Barrier Functions (RCBFs)

Stochastic extensions of RBFs are critical for safety-critical systems subject to diffusion-type noise, often represented by Itô SDEs: $dX_t = [f(X_t) + g(X_t)u_t]\,dt + \sigma(X_t)\,dW_t$ with $h$ and $B$ as above. A stochastic reciprocal CBF $B$ satisfies for all $x$ in the interior of the safe set: $\mathcal{L}_{f,g,\sigma}(u_o(x),u,B(x)) \leq \gamma B(x)$ where $\mathcal{L}$ is the extended generator: $\mathcal{L}_{f,g,\sigma}(u_o,u,B) = \nabla B\,f + \nabla B\,g(u_o+u) + \frac{1}{2}\operatorname{tr}[\sigma\sigma^\top D^2B].$ Enforcing this constraint almost surely guarantees invariance of the safe set under stochastic trajectories (Clark, 2020, Nishimura et al., 2022).

A distinctive property of stochastic RCBFs, established via maximal inequality and Tanaka's formula, is that if the control $u$ satisfies the generator inequality at all times, then $\Pr[X_t\in \mathcal{C},\ \forall t\ge0]=1$ (So et al., 2023). The generator condition imposes that the ratio of drift to diffusion in the Itô dynamics must diverge faster than $1/h(x)$ as $h\to0$ , or safety fails with nonzero probability, regardless of unbounded drift (So et al., 2023).

4. Extensions: High-Order, Disturbance-Robust, and Observer-Based RBFs

Expansion of RBF theory encompasses several advanced formulations:

High-order RBFs address safe sets defined by functions with higher relative degree. They recursively define additional variables to reduce the effective relative degree to one, generalizing the invariance result with affine-in- $u$ constraints (Clark, 2020, Wang et al., 25 Jul 2025).
Reciprocal Resistance–based Barrier Functions (RRBFs) introduce a resistance-like correction, replacing standard ZBF conditions with

$\Lie_f h(x) + \alpha(h(x)) - \beta(1/h(x)) \geq 0$

and, for systems with control,

$\Lie_f h(x) + \Lie_g h(x) u + \alpha(h(x)) - \beta(1/h(x)) \geq 0$

Here, $\beta(1/h(x))$ diverges near the safe set boundary, robustly dominating disturbances. RRBFs naturally create an interior "robust safe set" $\mathcal{S} = \{h(x) \ge h_s\}$ , and exhibit superior resilience near $\partial\mathcal{C}$ compared to both classic RBFs and ZBFs (Wang et al., 25 Jul 2025).

Disturbance observer–based RRCBFs (DO-RRCBF) further admit additive unmodeled disturbance estimates $\hat{d}(x)$ . Their admissible input sets remain valid for any bounded disturbance, without explicit knowledge of its bound (Wang et al., 25 Jul 2025).
Observer-based RBF synthesis addresses uncertain state measurement, with RBF constraints designed on state estimates (e.g., via Extended Kalman Filtering), and additional terms to bound estimation error (Clark, 2020).

Zeroing Barrier Functions (ZBFs), which enforce the constraint

$L_f h(x) + L_g h(x) u \geq -\beta(h(x)),$

provide an alternative certificate for set invariance. Under suitable assumptions, RBF ( $B=1/h$ ') and ZBF ( $h$ ') formulations are dual: each can be constructed from the other and both guarantee forward invariance (Ames et al., 2016, Agrawal et al., 15 Apr 2025). However, in stochastic settings, sufficient conditions for ZBF almost-sure invariance were shown to be invalid in general, while RCBFs retain almost-sure guarantees if the aforementioned drift/diffusion ratio diverges sufficiently fast (So et al., 2023). RBF-based constraints are numerically convenient (affine in $u$ ), whereas ZBFs avoid the unboundedness of $B$ near the boundary, which can introduce ill-conditioning in computation (Ames et al., 2016).

RRBFs, by introducing a reciprocal resistance term $\beta(1/h(x))$ , claim both improved disturbance robustness and inner "buffer sets" that ZBF and standard RBF approaches lack (Wang et al., 25 Jul 2025).

6. Neural Synthesis of Reciprocal CBFs

Physics-informed neural networks (PINNs) enable the direct data-driven synthesis of neural RBFs for complex, high-dimensional, or hard-to-parameterize systems. The neural RBF is defined as $B(x)=1/h(x)$ , with a bounded transformation $W_N(x)=\beta(B(x))$ (e.g., $\tanh$ ), enforced via a loss function incorporating the Zubov PDE, boundary, and supervised-safe/unsafe sample sets. This approach enables scalable specification of flexible safe sets, which can be smoothly tuned in training via the choice of $h(x)$ and the sample set (Agrawal et al., 15 Apr 2025).

A case study on the inverted pendulum shows that trained neural RBFs provide forward-invariant sublevel sets, whose boundary can be adjusted post hoc, and which can be incorporated into QP-based controllers analogously to analytic RBFs (Agrawal et al., 15 Apr 2025).

7. Representative Applications, Limitations, and Research Directions

RBFs and their variants have been applied in diverse contexts:

Automotive Safety: Adaptive cruise control (ACC) and lane keeping, leveraging QP-based mediation of safety vs. performance—even under actuator constraints (Ames et al., 2016, Wang et al., 25 Jul 2025).
Stochastic Multi-agent Collision Avoidance: Ensuring almost-sure safety in noisy, decentralized robotic systems via RCBF-based safety constraints (Clark, 2020).
Disturbed and Unmodeled Systems: Reciprocal resistance-based RBFs enforce safety under unknown or time-varying disturbances, outperforming standard RBF and ZBF approaches in numerical studies (Wang et al., 25 Jul 2025).

Limitations include the potential for control input (and thus actuator commands) to diverge near the safe set boundary due to $B(x)\rightarrow\infty$ , as well as challenges in digital (sample-and-hold) realization. ZBF-based and RRBF-based formulations can mitigate certain numerical issues and further extend robustness. The stochastic RCBF framework in particular exposes subtle requirements on the drift/diffusion scaling for effectiveness; insufficient drift divergence will not prevent stochastic exit from the safe set.

Ongoing research areas include stochastic generalizations (e.g., relaxing constant gain to class- $\mathcal{K}_\infty$ ), unified stochastic QP integration of safety and performance, hybrid and time-varying systems, robustification to model uncertainties, and efficient data-driven neural synthesis for high-dimensional systems (Nishimura et al., 2022, So et al., 2023, Agrawal et al., 15 Apr 2025, Wang et al., 25 Jul 2025).