Boundary Sampling to Learn Predictive Safety Filters via Pontryagin's Maximum Principle

1. Problem Statement

Autonomous systems require safety filters to enforce constraints and prevent failures. While Control Barrier Functions (CBFs) are widely used, they suffer from two main limitations:

1. Myopic Behavior: Standard CBFs rely only on instantaneous state information, often leading to abrupt or suboptimal interventions because they lack a prediction horizon.
2. Design Difficulty: Constructing valid CBFs for complex, high-dimensional systems is difficult and often requires expert intuition.

Hamilton-Jacobi (HJ) Reachability offers a solution by encoding trajectory lookahead (predictive safety) through Backward Reachable Tubes (BRT). However, exact HJ methods suffer from the curse of dimensionality, making them intractable for high-dimensional systems.

Learning-based HJ Reachability (e.g., using neural networks like DeepReach) addresses scalability but remains data-inefficient. Uniform sampling of the state space fails to capture the critical "boundary" trajectories that define the limits of safety. Consequently, learned safety filters often fail to accurately reconstruct the safe set, leading to high failure rates or overly conservative behavior.

The Core Challenge: How to efficiently generate informative training data for learned HJ reachability that specifically targets the safety-critical boundary states in high-dimensional, complex systems without relying on expensive optimization solvers (like MPC) for every sample.

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2. Methodology

The authors propose a novel data generation strategy called Boundary Sampling via Pontryagin's Maximum Principle (PMP).

A. Theoretical Foundation: PMP Characterization

The paper establishes a theoretical link between abnormal extremals in optimal control and trajectories that "barely" avoid safety violations.

• Minimum Time Problem: The authors formulate an optimal control problem (OCP) to find the trajectory that reaches the safety boundary ($\partial S$) in minimum time.
• PMP Conditions: Using the Pontryagin Maximum Principle, they derive necessary optimality conditions. They prove a key lemma: Trajectories that stay within the safe set $S$ but can barely remain there (touching the boundary and staying tangent) correspond to abnormal extremals (where the cost multiplier $p_0 = 0$).
• Corollary: These boundary-defining trajectories satisfy specific conditions where the adjoint vector $p_T$ at the boundary is normal to the boundary surface ($\nabla h(x_T)$), and the velocity is tangent to the boundary.

B. Boundary Sampling Algorithm

Instead of uniform random sampling, the authors use the derived PMP conditions to generate targeted training data:

1. Initialization: Randomly sample initial states within the safe set.
2. Boundary Identification: Use gradient descent to find terminal points $(x_T, p_T)$ on the safety boundary that satisfy the PMP terminal conditions (specifically, $h(x_T)=0$ and $p_T \propto \nabla h(x_T)$).
3. Backward Integration: Integrate the PMP state-costate dynamics backward in time from these boundary points.
4. Data Augmentation: The resulting trajectories lie exactly on the boundary of the Backward Reachable Tube. These samples are added to the training dataset of the neural network (DeepReach), concentrating learning capacity on the most safety-critical regions.

C. Safety Filter Implementation

The learned value function is used to construct a Control Barrier Value Function (CBVF). This is integrated into a Quadratic Program (QP) safety filter:
$$u_{QP} = \arg \min \|u - u_d\|^2 \quad \text{s.t.} \quad \nabla_t V_\gamma + L_f V_\gamma + L_g V_\gamma \cdot u \geq -\gamma V_\gamma$$
This filter minimally modifies a nominal controller ($u_d$) to ensure the system remains within the learned safe set, providing predictive safety guarantees.

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3. Key Contributions

1. Theoretical Derivation: The authors derived PMP-based optimality conditions proving that trajectories barely avoiding constraint violations are characterized by abnormal extremals. This provides a principled mechanism for identifying safety boundaries.
2. Data-Efficient Sampling: They developed an algorithm to generate boundary-focused training samples using PMP, significantly improving the data efficiency of learned HJ reachability in high-dimensional systems.
3. Predictive Safety Filter: They constructed a learned CBVF that encodes future trajectory information, demonstrating its effectiveness as a minimally invasive, predictive safety filter for human-driven vehicles operating at the limits of handling.

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4. Results

The approach was validated through simulations and real-world experiments on a shared-control automotive racing application (a 3-DOF single-track vehicle model with 8 states).

• Learning Efficiency & Failure Rate:
  • PMP sampling significantly reduced failure rates compared to uniform sampling.
  • With limited training epochs (30k), failure rates dropped from 0.66 to 0.30 (2-layer network) and 0.50 to 0.45 (3-layer network).
  • PMP sampling showed statistically significant improvements across varying dataset sizes and prediction horizons.
• Safe Set Reconstruction:
  • Visualizations of the learned value functions showed that PMP sampling accurately captured the shape of the safe set (matching a high-fidelity proxy ground truth), whereas uniform sampling produced overly optimistic (unsafe) sets.
  • Intersection-over-Union (IOU): PMP sampling consistently achieved higher IOU scores with the ground truth compared to uniform sampling.
• Real-World Validation:
  • Experiments were conducted on a custom drive-by-wire vehicle on a closed oval track.
  • The filter demonstrated proactive behavior: it preemptively reduced speed before entering turns (slowing from 13.5 m/s to 9.7 m/s) rather than reacting myopically at the track edge.
  • Minimal Intervention: The filter respected driver inputs on straights but intervened aggressively (up to 0.6 rad steering correction, >1200 Nm torque reduction) only when necessary to prevent track departure.
  • Real-Time Performance: The safety filter ran with wall times of approximately 3 ms, suitable for real-time control.

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5. Significance

This work addresses a critical bottleneck in deploying learned safety filters for complex autonomous systems: data inefficiency.

• Scalability: By focusing learning on the "hard" parts of the state space (the boundaries) using PMP, the method makes learned HJ reachability viable for high-dimensional systems where uniform sampling fails.
• Predictive Safety: It successfully bridges the gap between the theoretical optimality of HJ reachability and practical implementation, enabling safety filters that "look ahead" and act proactively rather than reactively.
• Real-World Applicability: The successful deployment on a racing vehicle demonstrates that these theoretically grounded methods can handle nonlinear dynamics, complex constraints, and real-time requirements in safety-critical scenarios.

In summary, the paper provides a robust framework for training predictive safety filters that are both computationally efficient and highly reliable, leveraging optimal control theory to guide the learning process.