Safe Pontryagin Differentiable Programming (2105.14937v2)

Abstract: We propose a Safe Pontryagin Differentiable Programming (Safe PDP) methodology, which establishes a theoretical and algorithmic framework to solve a broad class of safety-critical learning and control tasks -- problems that require the guarantee of safety constraint satisfaction at any stage of the learning and control progress. In the spirit of interior-point methods, Safe PDP handles different types of system constraints on states and inputs by incorporating them into the cost or loss through barrier functions. We prove three fundamentals of the proposed Safe PDP: first, both the solution and its gradient in the backward pass can be approximated by solving their more efficient unconstrained counterparts; second, the approximation for both the solution and its gradient can be controlled for arbitrary accuracy by a barrier parameter; and third, importantly, all intermediate results throughout the approximation and optimization strictly respect the constraints, thus guaranteeing safety throughout the entire learning and control process. We demonstrate the capabilities of Safe PDP in solving various safety-critical tasks, including safe policy optimization, safe motion planning, and learning MPCs from demonstrations, on different challenging systems such as 6-DoF maneuvering quadrotor and 6-DoF rocket powered landing.

• The paper introduces the Safe PDP framework which integrates barrier functions to enforce safety constraints without compromising gradient accuracy.
• It demonstrates that both trajectories and gradients are accurately approximated while strictly maintaining system constraints, validated through simulations on quadrotors and rocket landing tasks.
• The framework offers computational efficiency via a linear scaling auxiliary control system, making it practical for high-dimensional, long-horizon control tasks.

Safe Pontryagin Differentiable Programming: A Framework for Safety-Critical Learning and Control

The paper introduces a methodological framework named Safe Pontryagin Differentiable Programming (Safe PDP), aimed at addressing safety-critical learning and control tasks. The framework ingeniously integrates system constraints into the optimization problem via barrier functions, akin to interior-point methods, offering systematic treatments for constraints involving system states and inputs. This work is particularly relevant for tasks requiring strict safety guarantees throughout the learning and optimization processes, which are often encountered in complex systems such as 6-DoF maneuvering quadrotors and rocket landing.

Methodology and Theoretical Results

Safe PDP employs barrier functions to incorporate constraints directly into the cost function of the optimal control problem, transforming a constrained problem into a sequence of unconstrained ones. The paper asserts three primary contributions:

1. Both the solution trajectory and its gradient can be approximated using their unconstrained counterparts.
2. These approximations are not only controllably accurate but also maintain all original constraints, hence guaranteeing safety throughout the learning process.
3. The auxiliary control system provides an efficient mechanism to compute the gradient with respect to parameters, leveraging the Pontryagin differentiable programming (PDP) approach.

Numerical Experiments

The paper backs its claims with empirical evidence. It demonstrates the efficacy of Safe PDP in a range of safety-critical applications, such as safe policy optimization, safe motion planning, and learning Model Predictive Controllers (MPCs) from demonstrations. Results are particularly promising for safety-critical systems where any constraint violation could result in catastrophic failures.

Numerical Results and Algorithmic Implications

One noteworthy result is the framework's ability to manage constraints without resorting to projection methods, thus potentially easing integration into differentiable programming frameworks and reducing computational burdens. Theoretical assertions concerning the accuracy of the barrier methods validate the empirical outcomes, which show not only convergence to near-optimal solutions but also strict adherence to constraints during intermediate stages.

The computational efficiency of solving the auxiliary control system scales linearly with the time horizon, rendering Safe PDP feasible for high-dimensional, long-horizon control tasks. This efficiency primarily stems from leveraging the sparse structure ubiquitously found in most control systems.

Future Directions and Applications

The framework's potential is broad, with applications spanning various domains such as autonomous vehicles, robotics, and other safety-critical AI systems. Future research may explore enhancing Safe PDP's robustness to model uncertainties, broadening its applicability to systems operating under unpredictable conditions. Moreover, expanding Safe PDP to integrate state-of-the-art machine learning models could lead to improvements in real-time adaptability while maintaining safety guarantees.

While current implementations require feasible initial solutions, relaxing these prerequisites through better heuristic initialization strategies could further democratize its use across tasks where feasible solutions are challenging to derive a priori. Additionally, investigating strategies to handle non-differentiable constraints might augment its usability in diverse practical scenarios.

In conclusion, Safe PDP provides a rigorous, theoretically-founded approach to handling high-stakes, constraint-laden optimization problems in machine learning and control. The framework's balance between maintaining theoretical guarantees and demonstrating empirical success makes it a promising candidate for future research and application in ensuring safe autonomy.