How to Adapt Control Barrier Functions? A Learning-Based Approach with Applications to a VTOL Quadplane

TL;DR

Safety of autonomous systems under complex, nonlinear dynamics with input constraints is challenged by fixed CBF parameters that are difficult to tune. The authors propose online adaptation of CBF parameters via locally validated parameters, validated over finite horizons using tangent-cone/Nagumo theory, combined with a learning-based uncertainty-aware verifier. They integrate Probabilistic Ensemble Neural Networks (PENN) to predict safety margins and performance, with a two-stage verification that handles both epistemic and aleatoric uncertainty, ensuring local safety. The method is demonstrated on a VTOL quadplane during transitions and landing, showing reduced conservatism and improved performance while maintaining safety.

Abstract

In this paper, we present a novel theoretical framework for online adaptation of Control Barrier Function (CBF) parameters, i.e., of the class K functions included in the CBF condition, under input constraints. We introduce the concept of locally validated CBF parameters, which are adapted online to guarantee finite-horizon safety, based on conditions derived from Nagumo's theorem and tangent cone analysis. To identify these parameters online, we integrate a learning-based approach with an uncertainty-aware verification process that account for both epistemic and aleatoric uncertainties inherent in neural network predictions. Our method is demonstrated on a VTOL quadplane model during challenging transition and landing maneuvers, showcasing enhanced performance while maintaining safety.

Figures (4)

• Figure 1: Conceptual illustration of the inner safe set and locally validated CBF parameter. (a) Candidate inner safe set $\mathcal{C}^{*}(\tilde{\bm{\alpha}})$ defined via an ICCBF with a CBF parameter $\tilde{\bm{\alpha}}$. At boundary point ${\boldsymbol x}_1$, the CBF-constrained dynamics $G({\boldsymbol x}_1; \tilde{\bm{\alpha}})$ is non-empty, whereas at ${\boldsymbol x}_2$ it is empty, indicating that the set cannot be rendered forward invariant. (b) With locally validated CBF parameters, the trajectory remains within the inner safe set shown in (a) over the finite-horizon, ensuring safety for that interval. By adapting the CBF parameters, the corresponding inner safe set is reshaped dynamically, alleviating conservatism by allowing the trajectory to extend beyond a fixed, globally verified inner safe set $\mathcal{C}^{*}(\bm{\alpha})$.
• Figure 2: Neural network configurations for CBF parameter adaptation. (a) Prior approaches: A deterministic neural network directly outputs the optimal CBF parameter $\tilde{\bm{\alpha}}^{*}$. (b) Proposed method: A PENN model augments the input with the CBF parameter $\tilde{\bm{\alpha}}$ of interest, and outputs a GMM distribution representing the predicted characteristics when that CBF parameter is chosen, thereby enabling uncertainty-aware verification.
• Figure 3: Illustration of a VTOL quadplane aircraft.
• Figure 4: We visualize the aircraft trajectory along with its speed profile, rigid-body pose, and elevator angle $\delta_e$, with obstacles depicted in gray. (a) With fixed low parameters, the trajectory shows a significant altitude detour as the ICCBF constraints force the aircraft to pitch up to decelerate in response to the obstacle. (b) With fixed high parameters, the controller becomes infeasible, ultimately resulting in a collision. (c) Our adaptive approach dynamically adjusts the CBF parameters based on the aircraft’s speed and position. Initially, due to the high speed, it maintains low CBF parameters, prompting the elevator to pitch up and generate additional drag. As the aircraft slows down, the parameters increase to enhance performance, enabling the aircraft to fly safely beneath the obstacle and executing a smooth transition afterwards. Inset: Evolution of the adapted CBF parameters over time.

Theorems & Definitions (15)

• Definition 1: Tangent Cone blanchini_set-theoretic_2008
• Definition 2: Forward Invariance
• Theorem 1: Nagumo's Theorem nagumo_uber_1942
• Definition 3: Candidate CBF
• Definition 4: CBF ames_control_2019
• Lemma 1
• proof
• Definition 5: Inner Safe Set kim_learning_2025
• Definition 6: ICCBF agrawal_safe_2021
• Lemma 2