Boundary Sampling to Learn Predictive Safety Filters via Pontryagin's Maximum Principle
Authors: James Dallas, Thomas Lew, John Talbot, Jonathan DeCastro, Somil Bansal, John Subosits
Organizations: Toyota Research Institute, Los Altos, CA 94022, USA. · Department of Aeronautics and Astronautics, Stanford University, Stanford, CA 94305, USA.
Safety filters provide a practical approach for enforcing safety constraints in autonomous systems. While learning-based tools scale to high-dimensional systems, their performance depends on informative data that includes states likely to lead to constraint violation, which can be difficult to efficiently sample in complex, high-dimensional systems. In this work, we characterize trajectories that barely avoid safety violations using the Pontryagin Maximum Principle. These boundary trajectories are used to guide data collection for learned Hamilton-Jacobi Reachability, concentrating learning efforts near safety-critical states to improve efficiency. The learned Control Barrier Value Function is then used directly for safety filtering. Simulations and experimental validation on a shared-control automotive racing application demonstrate PMP sampling improves learning efficiency, yielding faster convergence, reduced failure rates, and improved safe set reconstruction, with wall times around 3ms.
Figures & tables
Fig. 1 : Left : Extremal trajectory that barely stays in the safe set S and reachable sets from x0 . Right : Generation of boundary trajectories to learn a safety filter.
Fig. 3 : Experimental platform.
Fig. 4 : Failure rate between PMP and uniform sampling. Performance vs training epochs and prediction horizon (top). Performance vs dataset size (bottom). PMP consistently improves failure rate. Shaded regions denote ±1 S.D. Significance shown with ∗p<0.05 and ∗∗p<0.01 .
Fig. 5 : Value function at a speed of 10.5 m/s for 50k epochs and 30k samples for PMP sampling (center) and uniform sampling (right). Ground truth (Sec. V-A ) is shown on left. Brown solid line is the 0-level set of the value function and black dashed lines are track edges and centerline.
Layers
Setting
PMP
Uniform
2
30k epochs
0.7±0.01
0.40±0.01
2
50k epochs
0.73±0.01
0.46±0.04
2
70k epochs
0.74±0.01
0.57±0.08
2
45k samples
0.72±0.01
0.52±0.02
2
60k samples
0.73±0.01
0.51±0.04
3
30k epochs
0.6±0.03
0.51±0.05
TABLE I : IOU across model capacity and dataset size. Bold indicates where PMP sampling outperforms uniform sampling with statistical significance.
Fig. 6 : Shared control experiment with γ=0.1 for steering difference (left) torque difference (center), and velocity (right). Color depicts intervention (left, center) and velocity magnitude (right).
Fig. 7 : Segment of γ=0.1 experiment for steering (left) and position (right). CBVF-QP output is shown in blue and driver request in green). Track edges are shown in red.