Geometry-Aware Control Barrier Functions for Collision Avoidance via Bernstein Polynomial Approximations
Siwon Jo, Yanze Zhang, Yupeng Yang, Wenhao Luo
AI summary
Problem
Traditional Control Barrier Functions rely on overly conservative geometric surrogates like spheres, which fail to accurately represent irregular robot and obstacle shapes, leading to computational inefficiency or unsafe behavior in unstructured environments.
Approach
The method approximates both robots and obstacles with a unified, differentiable Bernstein polynomial Signed Distance Field, then derives explicit control constraints via KKT conditions to enforce forward-invariant safety in a quadratic program.
Key results
- Unified Bernstein polynomial Signed Distance Field representation for arbitrary robot and obstacle geometries
- Explicit derivation of geometry-aware Control Barrier Functions with closed-loop safety certificates
- C-space inflation technique to resolve non-smoothness in composite obstacle sets
- Simulation-validated single-robot navigation and heterogeneous multi-robot collision avoidance
Why it matters
Provides a computationally efficient, geometry-exact safety framework for autonomous robots navigating cluttered or multi-agent environments without relying on conservative shape approximations.
Abstract
Safe navigation often relies on well-defined condi- tions 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) either are 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 rep- resent 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.