Closed-Form CBF Filtering with Input Saturation for Safe Point Robot Navigation

This paper presents a computationally efficient safety filter for a 2D point robot navigating around a circular obstacle. The proposed approach combines a simple input saturation pre-filter with a closed-form Exponential Control Barrier Function (ECBF) projection, eliminating the need for an online quadratic program (QP) solver. The filter enforces collision avoidance whenever the ECBF half-space projection is feasible, while achieving deterministic sub-microsecond execution times. We provide a rigorous derivation of the closed-form projection, including a numerical safeguard for degenerate cases and a feasibility analysis for the single-obstacle case. The approach is validated through Monte Carlo simulations and computational benchmarks demonstrating a 99.61% safety rate across 10,000 randomized trials and a mean latency of 3.34 μs with minimal jitter—approximately 500× faster than standard QP-based implementations