Reachability Barrier Networks: Learning Hamilton-Jacobi Solutions for Smooth and Flexible Control Barrier Functions

TL;DR

This work tackles safety guarantees for high-dimensional autonomous systems by marrying Hamilton–Jacobi reachability with physics-informed neural networks to produce Reachability Barrier Networks (RBNs). RBNs yield differentiable, gamma-parameterized, CBF-like value functions learned offline via a PINN-based residual of the HJI/VI and augmented with conformal-prediction-based probabilistic safety guarantees, enabling online safety filtering through a CBF-QP with tunable conservativeness. The approach scales to 9D multi-vehicle collision avoidance, outperforming neural CBFs in safety and goal achievement, and is validated in hardware on TurtleBots. Together, RBNs offer a scalable framework for synthesizing safe controllers for general nonlinear systems with formal probabilistic assurances and controllable aggressiveness.

Abstract

Recent developments in autonomous driving and robotics underscore the necessity of safety-critical controllers. Control barrier functions (CBFs) are a popular method for appending safety guarantees to a general control framework, but they are notoriously difficult to generate beyond low dimensions. Existing methods often yield non-differentiable or inaccurate approximations that lack integrity, and thus fail to ensure safety. In this work, we use physics-informed neural networks (PINNs) to generate smooth approximations of CBFs by computing Hamilton-Jacobi (HJ) optimal control solutions. These reachability barrier networks (RBNs) avoid traditional dimensionality constraints and support the tuning of their conservativeness post-training through a parameterized discount term. To ensure robustness of the discounted solutions, we leverage conformal prediction methods to derive probabilistic safety guarantees for RBNs. We demonstrate that RBNs are highly accurate in low dimensions, and safer than the standard neural CBF approach in high dimensions. Namely, we showcase the RBNs in a 9D multi-vehicle collision avoidance problem where it empirically proves to be 5.5x safer and 1.9x less conservative than the neural CBFs, offering a promising method to synthesize CBFs for general nonlinear autonomous systems.

Figures (7)

• Figure 1: Sample experiment setup. Agents, endowed with a joint safety filter to avoid collisions, are independently attempting to reach their goals. The blue contours indicate the unsafe region for the blue agent at its heading.
• Figure 2: Low-dimensional ground truth comparison. The level set of the CBVF (left) and the RBN (right) are shown for $\theta=\pi$. Sets are plotted for $\gamma=\{0.0, 0.3, 0.5, 1.0\}$, shown in blue, orange, purple, and black, respectively. The insets (green) show the change in value over $\theta$, which exhibits the differentiability of the value function.
• Figure 3: Conformal expansions of the 9D learned value functions. (Left) Relationship between $\delta$ and probabilistic guarantee of safety ($\epsilon$) for varying $\gamma$. (Middle) Comparison of conformal expansions for $\gamma = {0, 0.5, 1}$. The uncalibrated zero-level set is included in black. (Right) Trajectory rollouts of the 9D multi-agent collision avoidance problem. Increasing $\gamma$ makes the control more aggressive and decreases the distance between agents.
• Figure 4: Rollouts and contours. (Left) Sample rollouts and their corresponding initial condition level sets (Right) for neural CBFs (Top) and our method (Bottom).
• Figure 5: Comparison of the controllers in hardware (the robots become less transparent as time progresses). A) Nominal controller. B) neural CBF controller. C) HJ cooperative pairwise controller. D) RBNs (our method). Our method is the only one capable of safely navigating this scenario.