BubbleCBF: Constructive Safety for Second-Order Systems and Application to Autonomous Robot Navigation in Unknown Environments

This tutorial provides a critical review of the practical application of Control Barrier Functions (CBFs) in robotic safety. While the theoretical foundations of CBFs are well-established, I identify a recurring gap between the mathematical assumption of a safe controller's existence and its constructive realization in systems with input constraints. I highlight the distinction between candidate and valid CBFs by analyzing the interplay of system dynamics, actuation limits, and class-K functions. I further show that some purported demonstrations of safe robot policies or controllers are limited to passively safe systems, such as single integrators or kinematic manipulators, where safety is already inherited from the underlying physics and even naive geometric hard constraints suffice to prevent collisions. By revisiting simple low-dimensional examples, I show when CBF formulations provide valid safety guarantees and when they fail due to common misuses. I then provide practical guidelines for constructing realizable safety arguments for systems without such passive safety. A crowd-navigation simulation study further illustrates that CBF-derived reward shaping in reinforcement learning can improve empirical behavior without establishing formal safety. The goal of this tutorial is to bridge the gap between theoretical guarantees and actual implementation, supported by an open-source interactive web demonstration that visualizes these concepts intuitively.

Safe navigation of autonomous robots remains one of the core challenges in the field, especially in dynamic and uncertain environments. One prevalent approach is safety filtering based on control barrier functions (CBFs), which are easy to deploy but difficult to design. Motivated by the shortcomings of existing learning- and model-based methods, we propose a simple yet effective neural CBF design method for safe robot navigation in dynamic environments. We employ the idea of a composite CBF, where multiple neural CBFs are combined into a single CBF. Individual CBFs are trained using data generated offline via the Hamilton-Jacobi reachability framework to approximate the optimal safe set for single moving obstacles. Additionally, we use a residual neural architecture, ensuring that the estimated safe set does not intersect with the corresponding failure set. The method is extensively evaluated in simulation experiments for a ground robot and a quadrotor, comparing it against several baseline methods. The proposed method improves success rates by up to 18% over the strongest baseline, while maintaining comparable or lower path lengths and motion times. The method is also demonstrated in hardware experiments for both types of robots.

Safe navigation often relies on well-defined conditions based on the shape of robots and obstacles, and can be challenging when they have irregular geometries. While Control Barrier Functions (CBFs) offer an efficient mechanism to enforce safe set forward invariance, common shape surrogates (e.g., spheres or super-ellipsoids) are either overly conservative in unstructured scenes or require many local primitives, which inflates constraint counts and degrades real-time performance. In this paper, we introduce a novel geometry-aware Control Barrier Function (CBF) based on Bernstein-Polynomial Signed Distance Fields (BP-SDFs). It provides a unified way to represent the obstacles and robots, so as to represent the barrier function with a unified minimum distance. Benefiting from the differentiability of the Bernstein polynomials, one can easily enforce the control constraints in a closed loop. We validate the method's efficiency and performance to guarantee safety in single-robot navigation and heterogeneous multi-robot collision avoidance via simulations under different environments.