⚡ electrical engineering

VertexCBF: Improving Neural Control Barrier Functions via Vertex-Restricted Control Search

This paper introduces VertexCBF, a scalable and explainable framework that learns neural control barrier functions by approximating the Hamilton-Jacobi value function through physics-informed training and GPU-parallel vertex-restricted control search, effectively recovering large safe sets and ensuring safety in both simulation and real-world robotic experiments.

Original authors: Bojan Derajić, Sebastian Bernhard, Wolfgang Hönig

Published 2026-09-14

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CC BY 4.0

Original authors: Bojan Derajić, Sebastian Bernhard, Wolfgang Hönig

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

As autonomous machines become more common in our daily lives, from delivery bots on sidewalks to drones monitoring forests, a fundamental question arises: how do we ensure they never hurt anyone or anything? The answer lies in a mathematical concept known as a safety filter. Imagine a robot receiving a command to move forward, but a pedestrian steps into its path. A safety filter acts as a vigilant guardian, instantly checking if that command is safe. If the path is clear, the robot proceeds; if not, the filter subtly adjusts the robot's motion just enough to avoid a collision while still trying to reach its goal. For years, engineers have used a specific type of safety filter based on control barrier functions, a method that provides a rigorous guarantee of safety. However, designing these filters for complex robots has been notoriously difficult. Traditional methods often result in safety zones that are far too small, forcing robots to move cautiously and inefficiently, or they fail entirely when the robot's movements become too complicated to map out by hand.

A team of researchers has developed a new approach called VertexCBF to solve this problem, allowing robots to learn much larger and more practical safety zones. Instead of trying to calculate every possible future path a robot could take—a task that is computationally impossible for complex machines—they focused on a specific mathematical shortcut. They realized that for many robots, the most critical decisions about safety happen at the very edges of what the robot's motors can do. By restricting their search to these extreme limits, or "vertices," of the robot's control capabilities, they could generate high-quality training data much faster and more accurately than before. They then used this data to teach a neural network, a type of artificial intelligence, to recognize safe and unsafe states. The result is a system that learns to keep a robot safe without being overly cautious, effectively recovering the largest possible area where the robot can operate freely.

The researchers tested this method on fifteen different robotic systems, ranging from simple pendulums and drones to complex underwater vehicles and walking robots. In every case, they compared their new learning method against existing techniques. The older methods often produced safety zones that were either empty, meaning the robot could not move at all, or so small that the robot was practically paralyzed. In contrast, the new method consistently found large, reliable safe zones. For example, on a system simulating a robot balancing a pole, the traditional approaches failed to find any safe path, while the new method successfully identified a vast area where the robot could operate safely. The team also validated these findings by running the learned safety filter on a real mobile robot in a hardware experiment. The robot was tasked with navigating through a space while avoiding pedestrians who were intentionally moving to intercept it. Using the safety filter trained with their new method, the robot successfully avoided collisions in all ten trials, smoothly adjusting its path to keep a safe distance while continuing toward its destination.

What makes this approach particularly powerful is how it balances two competing needs: strict mathematical safety and practical efficiency. The researchers combined a physics-based rule, which ensures the robot never violates the laws of motion, with a data-driven learning process that fills in the gaps. They designed the neural network with a specific structure that guarantees the learned safety zone can never be larger than the physical limits of the robot, preventing the AI from "hallucinating" a safe path where none exists. This ensures that the robot remains within its true physical constraints. The study shows that by focusing on the extreme limits of control rather than trying to sample every possible middle ground, the system can learn to navigate complex environments with a level of confidence that previous methods could not achieve. This work suggests a path forward for deploying autonomous robots in crowded, unpredictable human environments, where safety must be absolute, but efficiency cannot be sacrificed.

Technical Summary: VertexCBF

Problem Statement

As autonomous robotics applications expand, ensuring safe and reliable operation remains a critical challenge. While Control Barrier Functions (CBFs) offer a theoretically grounded framework for safety filtering, designing effective CBFs for systems with state and input constraints is notoriously difficult. Existing learning-based methods often struggle with scalability, interpretability, or result in overly conservative safe sets. Furthermore, approaches relying solely on physics-informed learning (solving Hamilton–Jacobi–Bellman variational inequalities) often converge to trivial or suboptimal solutions, while methods using full-control trajectory optimization for supervision can be computationally expensive and less informative.

Methodology: VertexCBF

The authors propose VertexCBF, a framework for learning neural CBFs that approximate the optimal safe set characterized by Hamilton–Jacobi (HJ) reachability. The method integrates physics-informed learning with sparse, high-quality supervision derived from a specific structural property of control-affine systems.

Core Components

Neural Approximation of the Value Function:
The method approximates the stationary HJ safety value function $V(x)$ using a neural network. To ensure the learned safe set never exceeds the original physical constraints, the network employs a residual architecture:
$V_\Theta(x) = c(x) - r_\Theta(x)$
where $c(x)$ is the constraint function and $r_\Theta(x) \geq 0$ is a non-negative residual output by a Multi-Layer Perceptron (MLP). This guarantees $V_\Theta(x) \leq c(x)$ , ensuring the learned safe set is a subset of the constraint set.
Vertex-Restricted Control Search:
The key innovation lies in generating supervision labels. For control-affine systems ( $\dot{x} = f(x) + g(x)u$ ) with a convex polytope control set $U$ , the Hamiltonian is maximized at the vertices of $U$ .
- Instead of solving complex partial differential equations (PDEs) or performing full-control trajectory optimization (e.g., MPPI), VertexCBF restricts the control search to the vertices of the control polytope.
- This transforms the infinite-horizon safety problem into a finite-horizon optimization over a discrete set of control sequences.
- Supervision labels are generated via GPU-parallel tree search (using Beam Search, Stochastic Beam Search, or Branch-and-Bound) over these vertex-restricted trajectories. This avoids the need for numerical PDE solvers and reduces the computational burden compared to full-control sampling.
Training Objective:
The neural network is trained by minimizing a convex combination of two loss terms:
- Physics-Informed Loss ( $L_{PDE}$ ): Enforces the stationary HJ variational inequality locally on densely sampled, unlabeled states.
- Data Loss ( $L_{data}$ ): Minimizes the mean-squared error between the network output and the labels generated by the vertex-restricted tree search at sparsely sampled states.
  This combination guides the optimization away from trivial constant solutions (common in PDE-only learning) while maintaining local consistency with the system dynamics.

Key Contributions

Systematic Learning Framework: The paper introduces a method for learning neural CBFs that approximates the maximal safe set by combining HJ reachability theory with supervised learning.
Vertex-Restricted Supervision: By exploiting control-affine dynamics and convex control constraints, the authors demonstrate that supervision can be efficiently generated via vertex-restricted tree search. This approach is more scalable than full-control sampling and more effective than PDE-only optimization.
Residual Parametrization: The use of a residual architecture ensures the learned CBF is upper-bounded by the constraint function, preventing the learning process from expanding the safe set beyond physical limits.
Comprehensive Evaluation: The method is evaluated on 15 diverse dynamical systems (ranging from 2 to 13 states and 1 to 6 controls) and compared against PDE-only baselines and full-control MPPI supervision.

Experimental Results

The evaluation demonstrates that VertexCBF consistently recovers larger and more reliable safe sets than baseline methods:

Reliability: PDE-only learning (ND) frequently fails, collapsing to trivial solutions with empty safe sets or high false-safe rates ( $\rho_{FS}$ ). VertexCBF maintains low false-safe rates across most systems.
Effectiveness: Compared to Full-Control Data (FCD) generated via MPPI, VertexCBF (using Vertex-Restricted Control Data, VRCD) achieves competitive or superior effective safe volumes ( $\eta_{eff}$ ) and lower false-unsafe rates ( $\rho_{FU}$ ). The full-control MPPI supervision often results in overly conservative safe sets.
Hardware Validation: A hardware experiment with a mobile robot navigating among pedestrians confirmed that the learned neural CBF can be deployed in real-time as a safety filter, successfully avoiding collisions in all 10 trials despite model mismatches (e.g., tire slip, delayed perception).

Significance and Claims

The paper claims that VertexCBF provides a practical and systematic design approach for learning neural CBFs. Its significance lies in:

Scalability: It avoids the intractability of grid-based PDE solvers and the high computational cost of full-control trajectory optimization by leveraging the vertex structure of control sets.
Robustness: It overcomes the instability and triviality issues of PDE-only learning by introducing high-quality, sparse supervision derived from vertex-restricted searches.
Safety Guarantees: The residual architecture provides a built-in guarantee that the learned safe set remains within the original state constraints.

The authors acknowledge limitations, noting that the method does not yet provide formal convergence guarantees or deterministic error bounds, relying instead on post-training validation via closed-loop rollouts. They also note that while the method scales well to systems with up to 13 states, very high-dimensional whole-body dynamics (e.g., full humanoid models) may still pose computational challenges due to the exponential growth of the tree search with the number of control vertices. Future work is suggested to address convergence guarantees, scalability for high-dimensional systems, and extensions to uncertain dynamics.