Probabilistically safe controllers based on control barrier functions and scenario model predictive control

TL;DR

The paper addresses safe real-time control under uncertainty by integrating Control Barrier Functions (CBFs) with scenario-based Model Predictive Control (MPC). It replaces full-horizon constraint enforcement with a one-step-ahead probabilistic safety condition encoded via CBFs and derives finite-sample, distribution-free guarantees on the closed-loop safety violation frequency using scenario-based results, characterized by a bound involving the horizon, the number of scenarios $m$, and the safety parameter $\epsilon$. The approach is demonstrated on a multi-UAV collision-avoidance problem with additive disturbances, showing successful position swapping while keeping the average safety violation probability below $\epsilon$ (a priori) and validating theoretical bounds with empirical results. A comparative study against a recent stochastic CBF method indicates sharper one-step safety guarantees and competitive performance, highlighting the practicality of distribution-free, data-driven probabilistic safety in real-time robotic systems.

Abstract

Control barrier functions (CBFs) offer an efficient framework for designing real-time safe controllers. However, CBF-based controllers can be short-sighted, resulting in poor performance, a behaviour which is aggravated in uncertain conditions. This motivated research on safety filters based on model predictive control (MPC) and its stochastic variant. MPC deals with safety constraints in a direct manner, however, its computational demands grow with the prediction horizon length. We propose a safety formulation that solves a finite horizon optimization problem at each time instance like MPC, but rather than explicitly imposing constraints along the prediction horizon, we enforce probabilistic safety constraints by means of CBFs only at the first step of the horizon. The probabilistic CBF constraints are transformed in a finite number of deterministic CBF constraints via the scenario based methodology. Capitalizing on results on scenario based MPC, we provide distribution-free, \emph{a priori} guarantees on the system's closed loop expected safety violation frequency. We demonstrate our results through a case study on unmanned aerial vehicle collision-free position swapping, and provide a numerical comparison with recent stochastic CBF formulations.

Figures (4)

• Figure 1: UAV position swap maneuver: polygons show unsafe regions, triangles are UAVs, circles mark initial positions, crosses the targets, and lines depict the closed-loop UAV trajectories.
• Figure 2: Deterministic (blue) and robust (solution of the scenario based MPC) trajectories (red) for one UAV of Fig.\ref{['UAVsnapshot']}.
• Figure 3: Empirical distribution (calculated based on $100$ independent runs) of average safety violations for $T=90$ time steps.
• Figure 4: Comparison of one-step exit probability: cosner2023robust and ours (Lemma \ref{['nxt_stp_prob']}). The left boxplot, linked to the left verical axis represents the approach in cosner2023robust. The Right boxplot, linked to the right vertical axis corresponds to our proposed approach.

Theorems & Definitions (2)

• Lemma 1
• Theorem 1