Technical Summary: Online Adaptive Control Barrier Functions Under Epistemic and Aleatoric Uncertainty

1. Problem Statement

Control Barrier Functions (CBFs) are a standard tool for enforcing safety constraints in robotic systems by synthesizing safety filters that minimally modify a nominal controller. However, the practical performance of CBF-based controllers is heavily dependent on the choice of class-$\mathcal{K}$ function parameters (decay rates). This creates a fundamental tradeoff:

• Conservative parameters ensure feasibility and safety but often result in slow, circuitous motion.
• Aggressive parameters improve progress but can render the safety filter infeasible or unsafe, particularly under input constraints or near obstacles.

Existing learning-based approaches that attempt to adapt these parameters online often treat the learned model as a direct generator of optimal parameters. This introduces a critical risk: if the model is overconfident on out-of-distribution (OOD) states or obstacle configurations, it may output invalid parameters that directly compromise the safety filter's feasibility. Furthermore, many existing methods lack explicit mechanisms to assess predictive uncertainty or validate parameters against input constraints before application.

This paper addresses the challenge of online CBF parameter adaptation under input constraints, aiming to reduce conservatism while guaranteeing safety through a rigorous, uncertainty-aware verification process.

2. Methodology: Online Adaptive CBF (OA-CBF)

The proposed framework, OA-CBF, shifts the paradigm from direct parameter prediction to uncertainty-aware evaluation of candidate parameters. Instead of predicting a single optimal parameter, the system samples candidate parameters, evaluates them against safety and performance metrics, and selects the best verified candidate.

A. Locally Validated CBF Parameters

Rather than requiring a single parameter to be globally valid for all future states (which is often intractable and overly conservative), OA-CBF introduces the concept of locally validated CBF parameters. A parameter is considered locally valid if it satisfies the Input-Constrained CBF (ICCBF) condition over a finite prediction horizon $[t, t+T]$.

• The framework certifies that the system trajectory remains within an "inner safe set" $C^\star(\alpha)$ during this horizon.
• By repeatedly updating the parameter at intervals shorter than the horizon, safety is preserved via a receding-horizon validation mechanism.

B. Prediction Model Design

To evaluate candidates efficiently, OA-CBF employs a Probabilistic Ensemble Neural Network (PENN) trained offline.

• Graph-Attention Encoder: To handle environments with a variable number of obstacles, the model uses a graph neural network (GNN) with attention mechanisms. It aggregates information from all observed obstacles (robot, goal, and obstacles as nodes) without padding or heuristic selection, producing a context vector.
• Input Structure: The candidate CBF parameter $\tilde{\alpha}$ is explicitly concatenated with the environmental context vector. The network predicts distributions over two metrics:
  1. Safety Metric ($\phi$): A risk level derived from a safety-loss density function that penalizes ICCBF constraint violations and proximity to obstacles.
  2. Performance Metric ($\delta$): A measure of progress efficiency (e.g., time to goal).
• Ensemble Architecture: The PENN consists of multiple independent neural networks to capture both epistemic uncertainty (model uncertainty due to limited data) and aleatoric uncertainty (inherent noise in the system).

C. Two-Stage Uncertainty-Aware Verification

Before a candidate parameter is applied, it must pass two rigorous gates:

1. Epistemic Uncertainty Gate (Conformal Prediction):

   • The system quantifies model disagreement using the Jensen-Rényi Divergence (JRD) across the ensemble.
   • A threshold $D_{thr}$ is calibrated using conformal prediction on a held-out dataset to guarantee a specific in-distribution recall rate.
   • If the JRD exceeds the threshold, the input is deemed OOD, and the prediction is rejected to prevent overconfident extrapolation.

2. Aleatoric Risk Screening (Distributionally Robust CVaR):

   • For candidates passing the epistemic gate, the system evaluates the Conditional Value at Risk (CVaR) of the predicted safety metric.
   • A distributionally robust condition ensures that even the most conservative ensemble member predicts a risk level below the safety threshold with high probability. This screens for aleatoric risks (noise/variability) that could lead to safety violations.

D. Selection and Control

Among the candidates that pass both verification stages, the system selects the parameter that minimizes the predicted performance metric $\delta$. This selected parameter is then applied to the controller, which can be realized as either a CBF-QP or a sampled-data MPC-CBF.

3. Key Contributions

The paper makes four primary contributions:

1. Locally Validated CBF Formulation: A novel problem formulation that adapts CBF parameters by verifying them over a finite horizon rather than seeking a global solution, proving that repeated local validation preserves safety.
2. Uncertainty-Aware Verification: A two-stage procedure combining conformal prediction for epistemic gating and distributionally robust CVaR for aleatoric risk screening, ensuring that only reliable predictions influence the controller.
3. Graph-Attention Prediction Architecture: A scalable encoder that processes variable-size obstacle environments without information loss, allowing the model to evaluate arbitrary candidate parameters rather than outputting a single deterministic value.
4. Broad Applicability: Demonstration that the framework works with both MPC-CBF and CBF-QP filters and is applicable to both distance-based (ICCBF) and non-distance-based (Dynamic Parabolic CBF) safety constraints.

4. Experimental Results

The OA-CBF framework was evaluated on five nonlinear, input-constrained systems: Dynamic Unicycle, Planar Quadrotor (Quad2D), 3D Quadrotor (Quad3D), Kinematic Bicycle, and VTOL Quadplane.

• Safety vs. Conservatism: Compared to fixed-parameter baselines, OA-CBF significantly reduced conservatism. For instance, on the Dynamic Unicycle, it achieved a 0% safety-failure rate with an average reach time of 10.8s, whereas a conservative fixed parameter took 31.0s (0% failure), and an aggressive fixed parameter had a 39% failure rate.
• Comparison to Learning Baselines: OA-CBF outperformed learning-based baselines like BarrierNet and optimal-decay methods. While some baselines achieved fast times on successful trials, they suffered from high failure rates (e.g., BarrierNet had a 60% failure rate on the Unicycle). OA-CBF maintained 0% failure rates across all distance-based benchmarks while achieving the fastest reach times among safe methods.
• Graph-Attention Impact: Replacing the proposed graph-attention encoder with a standard fully-connected "closest-obstacle" encoder increased safety-failure rates (e.g., from 0% to 11% on Quad2D), demonstrating the importance of processing the full obstacle environment.
• Non-Distance-Based CBFs: On the Kinematic Bicycle using a Dynamic Parabolic CBF (defined in relative velocity space), OA-CBF reduced the safety-failure rate from 84% (aggressive fixed) to 1%, proving the method's applicability beyond signed-distance functions.

5. Significance and Claims

The paper claims that OA-CBF provides a tractable mechanism to resolve the tradeoff between safety and performance in CBF-based control. By treating the learned model as an uncertainty-aware evaluator rather than a direct generator, the framework explicitly rejects unsafe or unreliable updates before they are applied.

The authors emphasize that the method:

• Reduces Conservatism: It allows the system to operate closer to the boundaries of the safe set when conditions permit, unlike fixed conservative parameters.
• Ensures Robustness: Through conformal calibration and CVaR screening, it mitigates the risks associated with neural network overconfidence and distribution shifts.
• Generalizes: It is effective across diverse dynamics (from unicycles to VTOL aircraft) and different CBF formulations (distance-based and relative-velocity based).

The paper concludes modestly, acknowledging that the adaptation quality depends on the coverage of the offline training data and that hardware deployment will require further work to address sensing errors, unmodeled dynamics, and real-time optimization latency.