Control barrier functions coupled with an ensemble transform Kalman filter for calibrated chance-constrained safety filtering

Abstract

Control barrier functions (CBFs) certify safety for the true state, yet controllers act on estimates. We propose the Ensemble-Transform Control Barrier Function (ET-CBF), a safety filter that evaluates the discrete-time barrier condition on every member of an ensemble transform Kalman filter (ETKF) analysis and enforces a one-sided chance constraint on the resulting ensemble distribution. For control-affine systems the filter is a second-order cone program whose size is set by the ensemble size rather than the state dimension, and it needs no Jacobians of the dynamics, the measurement model or the barrier. We prove that ET-CBF coincides with the Kalman-filter chance-constrained CBF for affine barriers under linear-Gaussian models, show that it retains the curvature (Jensen) term that linearised belief CBFs omit, with a sign rule that predicts when linearisation is optimistic, and give a probabilistic safety certificate under ensemble consistency. We also identify a failure mode specific to ensemble beliefs, miscalibration at small ensemble sizes, and correct it with a Student-t predictive gain. In 5,600 closed-loop Monte Carlo runs of a unicycle with nonlinear range-bearing sensing, certainty-equivalent CBFs on EKF or ETKF means entered the obstacle in 69–79% of runs, while ET-CBF limited this to 0–4.5% with one-step failure rates of 3.0–3.6% against a 5% design level, on par with a linearised EKF chance constraint. With four members, the corrected gain reduced one-step failure from 14.7% to 5.1%. In a static test on a keep-in barrier, linearisation let the realised risk reach 10.5% against a 5% target, while a 200-member ensemble held it to 7.0%.