Online Learning-Enhanced High Order Adaptive Safety Control

1. Problem Statement

Control Barrier Functions (CBFs) are widely used to formally guarantee the safety of robotic systems by ensuring the system state remains within a safe set. However, standard CBFs rely heavily on the accuracy of the system's dynamic model. In real-world applications, model inaccuracies caused by unmodeled dynamics, payload changes, or external disturbances (e.g., wind) can invalidate these safety guarantees, leading to collisions or unsafe behavior.

Existing solutions face significant limitations:

• Robust CBFs: Require known bounds on uncertainty, often resulting in overly conservative controllers.
• Adaptive CBFs (aCBFs): Typically assume residuals are linear combinations of known basis functions or require extensive offline training. They struggle with time-varying, nonlinear residuals and often fail when residuals fall outside the training distribution.
• Pure Learning Approaches: Often lack safety guarantees during the learning phase or require large amounts of offline data.

The core challenge is to develop a controller that can adapt online to complex, time-varying, and nonlinear model perturbations while maintaining formal safety guarantees without requiring prior knowledge of the disturbance structure.

2. Methodology

The authors propose NODE-HO-aCBF, a hybrid framework combining a nominal model with a data-driven residual model using Knowledge-based Neural ODEs (KNODE).

A. Hybrid Model Architecture

Instead of learning the entire dynamics from scratch, the system uses a hybrid model:
$$\dot{x} = \underbrace{f(x) + g(x)u}_{\text{Nominal Model}} + \underbrace{\hat{d}_\theta(x)}_{\text{Neural Residual}}$$

• Nominal Model: A simplified, analytically tractable model (e.g., double integrator).
• Residual Model ($\hat{d}_\theta$): A Neural Network parameterized by $\theta$ that learns the unmodeled dynamics and disturbances directly from state-control data.
• KNODE: Utilizes Neural ODEs to learn the dynamical system directly, allowing for data sampling at irregular time intervals (crucial for real-time physical systems).

B. High-Order Adaptive CBF (HO-aCBF)

Since many mechanical systems have safety constraints with a relative degree greater than one, the authors extend standard adaptive CBFs to High-Order CBFs.

• The standard HOCBF constraint is modified to include the Lie derivative of the learned residual term ($L_{\hat{d}_\theta}\psi_{r-1}$).
• The control input $u$ is computed by solving a Quadratic Program (QP) at 100Hz:
  $$\min_u \|u - u_{des}\| \quad \text{s.t.} \quad \sup_{u \in U} [\dots + L_{\hat{d}_\theta}\psi_{r-1}(x) + \alpha_r(\psi_{r-1}(x))] \geq 0$$
• This ensures that even with the estimated residual, the safety set remains forward invariant.

C. Online Learning Loop

The system operates in a continuous loop:

1. Control Execution: The QP solves for a safe control input using the most recent hybrid model.
2. Data Collection: State-control pairs ($z = [x, u]$) are stored in a First-In-First-Out (FIFO) queue.
3. Online Training: A Neural ODE is trained on the queue data to update the residual model $\hat{d}_\theta$.
4. Adaptation: The updated model is immediately loaded into the QP solver, allowing the controller to adapt to changing dynamics (e.g., wind gusts) in real-time.

3. Key Contributions

1. Learning-Enhanced HO-aCBF: A novel framework that integrates online learning directly into high-order CBF constraints, enabling safety guarantees under complex, time-varying model perturbations.
2. Hybrid KNODE Approach: A data-efficient method that leverages a nominal model for structure while using Neural ODEs to capture complex nonlinear residuals, avoiding the need for extensive offline training.
3. Real-Time Performance: The algorithm runs at 100Hz on a real-time system, solving the QP and training the model simultaneously without compromising control frequency.
4. Robustness to Tuning: Unlike previous adaptive methods that require precise tuning of gain matrices or knowledge of basis functions, this approach is robust and requires no prior assumptions about the residual structure.

4. Experimental Results

A. Simulations

The method was tested on a double integrator system with three types of residuals: "Attractive," "Repulsive," and "Time-varying."

• Baselines: Compared against standard HOCBFs, tuned HO-aCBFs (requiring known basis matrices), and offline-trained models.
• Findings:
  • Standard HOCBFs failed to maintain safety under perturbations.
  • Tuned HO-aCBFs were fragile; they failed when the assumed basis matrix ($Y(x)$) was inaccurate or when residuals were time-varying.
  • NODE-HO-aCBF successfully adapted to all residual types. The offline version achieved near-perfect safety (0% violation time), while the online version adapted quickly, eventually matching the safety performance of the offline model.
  • The online method demonstrated superior data efficiency and stability compared to baselines that required large history stacks.

B. Physical Experiments

• Platform: A 38g nano quadrotor (Crazyflie 2.1).
• Scenario: Tracking a circular trajectory while avoiding a central obstacle in the presence of 18 km/h turbulent wind generated by a fan.
• Results:
  • Baseline controllers (HOCBF, HO-aCBF) converged to unsafe trajectories and violated safety constraints significantly (31%–47% violation time).
  • Online NODE-HO-aCBF: Initially experienced a "learning delay" but rapidly adapted to the wind, reducing safety violations to 35.3% and settling into a stable trajectory closer to the safety boundary.
  • Offline NODE-HO-aCBF: Achieved the best performance with only 15.9% violation time, demonstrating that pre-training on similar data further enhances safety.
  • The system successfully maintained a safe distance from the obstacle despite severe wind disturbances, a feat the baselines could not achieve.

5. Significance

This work bridges the gap between formal safety guarantees and the adaptability required for real-world robotics. By embedding Neural ODEs directly into the CBF optimization, the authors create a controller that is:

• Safe: Provides formal guarantees even with model errors.
• Adaptive: Learns complex, time-varying disturbances online without offline pre-training.
• Efficient: Operates at high frequencies (100Hz) suitable for agile robots like nano-drones.

The ability to deploy this on a 38g nano quadrotor in turbulent wind demonstrates its potential for swarm robotics and close-proximity flight, where traditional robust controllers are too conservative and pure learning methods are too slow or unsafe. This paves the way for safer, more autonomous operation in unstructured environments.