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

Abstract

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 projection that enforces the safety half-space constraint, eliminating the need for an online quadratic program (QP) solver. We provide the exact closed-form solution to the joint problem of enforcing both the ECBF constraint and the actuation bound, eliminating the gap between the relaxed projection and the full constrained problem. We also provide a one-step discretization bound that quantifies the maximum barrier decay within a sample interval. The approach is validated through Monte Carlo simulations and computational benchmarks demonstrating a 99.61% safety rate across 10,000 randomized trials, with the residual failure set shown to be invariant across three sampling rates, and a mean latency of 3.40 microseconds---approximately 464x faster than a standard QP solver (OSQP) on the relaxed half-space-only problem from scratch and 11.3x faster than an optimized warm-started implementation. Because the joint problem's ball constraint is a second-order cone constraint that OSQP cannot represent directly, this comparison is conservative and a like-for-like SOCP benchmark is left as future work.