A General Safety Framework for Learning-Based Control in Uncertain Robotic Systems (1705.01292v3)

Abstract: The proven efficacy of learning-based control schemes strongly motivates their application to robotic systems operating in the physical world. However, guaranteeing correct operation during the learning process is currently an unresolved issue, which is of vital importance in safety-critical systems. We propose a general safety framework based on Hamilton-Jacobi reachability methods that can work in conjunction with an arbitrary learning algorithm. The method exploits approximate knowledge of the system dynamics to guarantee constraint satisfaction while minimally interfering with the learning process. We further introduce a Bayesian mechanism that refines the safety analysis as the system acquires new evidence, reducing initial conservativeness when appropriate while strengthening guarantees through real-time validation. The result is a least-restrictive, safety-preserving control law that intervenes only when (a) the computed safety guarantees require it, or (b) confidence in the computed guarantees decays in light of new observations. We prove theoretical safety guarantees combining probabilistic and worst-case analysis and demonstrate the proposed framework experimentally on a quadrotor vehicle. Even though safety analysis is based on a simple point-mass model, the quadrotor successfully arrives at a suitable controller by policy-gradient reinforcement learning without ever crashing, and safely retracts away from a strong external disturbance introduced during flight.

• The paper introduces a method to integrate state-dependent disturbance bounds via Gaussian process modeling within differential games theory.
• It establishes key assumptions like Lipschitz continuity and deformation retraction to ensure unique and continuous system solutions.
• The framework enhances control system reliability and adaptability, paving the way for advancements in autonomous vehicles and robotics.

Analyzing State-Dependent Control Inputs via Gaussian Processes

The paper under consideration explores a sophisticated approach to modeling control systems where the disturbance inputs depend on the state of the system. The research extends the classical framework of differential games, which typically operate under fixed control sets, and adapts it to systems with state-dependent disturbance bounds. The fundamental challenge is ensuring the existence and uniqueness of solutions to such systems, which are formalized as Caratheodory solutions.

Key Contributions

The primary focus of the paper is on accommodating dynamic, state-dependent control sets into the system dynamics by leveraging the properties of Gaussian processes. The authors introduce two technical assumptions to enable the transformation of these systems into a form that allows the application of existing differential games theory:

1. Deformation Retraction: For every state $x$ in $\mathbb{R}^n$, the disturbance set $D(x)$ must be a deformation retract of a fixed set $D$. This ensures that the set can be continuously transformed into $D(x)$.
2. Lipschitz Continuity: The mapping function $r$, which integrates the state-dependence, is required to be Lipschitz continuous in the state variable and uniformly continuous in the disturbance input.

Utilizing these assumptions, the paper demonstrates that the posterior mean and standard deviation of the disturbance, modeled using Gaussian processes, satisfy the necessary Lipschitz conditions.

Theoretical Implications

The authors rigorously prove several propositions, illustrating that disturbance bounds derived from a Gaussian process model meet the stated assumptions. This ensures well-defined dynamical systems with unique continuous solutions, thus facilitating the application of differential games methodologies:

• Proposition 8: Shows that the saturated system dynamics are bounded and uniformly continuous, with Lipschitz continuity in state variables.
• Proposition 9 and 10: Validate that the posterior mean and standard deviation are Lipschitz continuous, grounding their approach in solid mathematical footing.
• Proposition 11: Extends this to ensure that the set-valued map describing the disturbance bounds is Lipschitz continuous under the Hausdorff distance.

Numerical Results and Claims

The paper supports its theoretical propositions with a formal proof structure, although specific numerical simulations or experimental validations are not highlighted in the sections provided. The mathematical constructs employed imply potential for strong results in practical implementation, especially in systems where robustness and adaptation to variable states are critical.

Practical Implications and Future Directions

The framework developed could have significant implications for control systems engineering, particularly in areas requiring adaptive strategies, such as autonomous vehicles and robotics. By ensuring that the disturbance bounds are appropriately modeled, the approach could enhance system reliability and stability.

The research opens pathways for future exploration, including the integration with real-world applications where Gaussian process models are used for learning and adaptation. Furthermore, there is potential for extending the model to more complex systems with non-linear dynamics or higher-dimensional state spaces.

Conclusion

Overall, the paper presents a rigorous and innovative approach to dealing with state-dependent disturbances in control systems. By embedding the problem within the structure of differential games and utilizing Gaussian processes, the research contributes valuable insights into the conditions necessary for existence and uniqueness, paving the way for both theoretical advancements and practical applications in adaptive control systems.