Learning Stability Certificate for Robotics in Real-World Environments (2510.03123v1)

Abstract: Stability certificates play a critical role in ensuring the safety and reliability of robotic systems. However, deriving these certificates for complex, unknown systems has traditionally required explicit knowledge of system dynamics, often making it a daunting task. This work introduces a novel framework that learns a Lyapunov function directly from trajectory data, enabling the certification of stability for autonomous systems without needing detailed system models. By parameterizing the Lyapunov candidate using a neural network and ensuring positive definiteness through Cholesky factorization, our approach automatically identifies whether the system is stable under the given trajectory. To address the challenges posed by noisy, real-world data, we allow for controlled violations of the stability condition, focusing on maintaining high confidence in the stability certification process. Our results demonstrate that this framework can provide data-driven stability guarantees, offering a robust method for certifying the safety of robotic systems in dynamic, real-world environments. This approach works without access to the internal control algorithms, making it applicable even in situations where system behavior is opaque or proprietary. The tool for learning the stability proof is open-sourced by this research: https://github.com/HansOersted/stability.

• The paper presents a data-driven method to learn Lyapunov functions from trajectory data for certifying robotic stability.
• It employs a novel neural network architecture with Cholesky factorization to ensure positive-definiteness and enforce ISS criteria.
• Demonstrated on noisy systems, the approach reliably certifies stability for black-box controllers in safety-critical applications.

Data-Driven Stability Certification for Robotics via Neural Lyapunov Functions

Introduction

The paper "Learning Stability Certificate for Robotics in Real-World Environments" (2510.03123) presents a framework for certifying the stability of robotic systems using data-driven methods, specifically by learning Lyapunov functions directly from trajectory data. The approach is designed to address the limitations of traditional stability analysis, which often requires explicit knowledge of system dynamics and is not feasible for complex, black-box, or proprietary controllers. The framework leverages neural networks to parameterize Lyapunov candidates, enabling stability certification in real-world, noisy environments without access to internal control algorithms.

Theoretical Framework

The stability notion adopted is Input-to-State Stability (ISS), which generalizes classical Lyapunov stability to systems with external inputs. The Lyapunov candidate $V$ is constructed as $V(e) = e^T Q e$, where $e$ is the tracking error and $Q$ is a symmetric positive-definite matrix. To guarantee positive definiteness, $Q$ is parameterized via its Cholesky factorization, $Q = LL^T$, with $L$ a lower-triangular matrix with strictly positive diagonal entries. The neural network outputs the elements of $L$, applying a Softplus activation to the diagonal to enforce positivity.

The ISS condition is enforced by ensuring $V(e) \leq \epsilon$ for some positive constant $\epsilon$, with $V(0) = 0$. The loss function penalizes violations of this condition, allowing for controlled relaxation to accommodate measurement noise and modeling uncertainties inherent in real-world data.

Neural Network Architecture and Training

The architecture ingests the tracking error and its derivatives as input features. The network consists of several hidden layers, culminating in outputs that are partitioned into off-diagonal and diagonal elements of $L$. The diagonal elements are passed through a Softplus transformation to ensure strict positivity, while the off-diagonal elements remain unconstrained. This design guarantees that the resulting $Q$ is positive-definite, a necessary property for Lyapunov functions.

Training is performed using gradient-based optimization, minimizing the loss function defined as $v(e) = \max\{0, h(e)\}$, where $h(e) = V(e) + \gamma$ and $\gamma$ is a small positive constant. The framework is robust to noise and does not require explicit system models, making it suitable for real-time verification of black-box controllers.

Implementation and Practical Considerations

The open-source implementation (https://github.com/HansOersted/stability) provides a practical tool for roboticists to certify stability from trajectory data. The framework is agnostic to the underlying control algorithm, supporting both classical and data-driven controllers, including those based on neural networks and reinforcement learning. The method tolerates measurement noise and can be deployed in real-world environments where system identification is infeasible.

Resource requirements are modest, as the neural network is relatively lightweight and the training process is efficient for moderate-dimensional systems. For high-dimensional systems, scalability may be limited by the complexity of the Cholesky factorization and the size of the neural network required to capture the relevant dynamics. The approach is particularly well-suited for safety-critical applications where formal stability guarantees are required but analytical methods are impractical.

Numerical Results and Claims

The paper demonstrates that the proposed framework can reliably learn Lyapunov functions that certify ISS for a variety of robotic systems, including those with unknown or complex dynamics. The method is shown to provide stability guarantees even in the presence of significant measurement noise, with high confidence in the certification process. The authors claim that the approach fills a critical gap in the field by enabling stability verification for black-box and data-driven controllers, a capability not previously available in open-source tools.

Implications and Future Directions

The proposed framework has significant implications for the deployment of autonomous robotic systems in safety-critical environments. By enabling data-driven stability certification, the method reduces reliance on expert knowledge and analytical modeling, democratizing access to formal verification tools. This is particularly relevant for systems controlled by neural networks or reinforcement learning agents, where traditional stability analysis is intractable.

Future developments may focus on scaling the approach to higher-dimensional systems, integrating uncertainty quantification, and extending the framework to certify other properties such as robustness and safety under adversarial conditions. The methodology could also be adapted to online learning scenarios, enabling continuous verification as new data becomes available.

Conclusion

This paper introduces a robust, data-driven framework for certifying the stability of robotic systems using neural Lyapunov functions learned from trajectory data. The approach is model-free, noise-tolerant, and applicable to black-box controllers, providing a practical tool for real-world stability assurance. The open-source implementation facilitates adoption and further research, with potential extensions to broader verification tasks in autonomous systems.