Learning for Layered Safety-Critical Control with Predictive Control Barrier Functions

TL;DR

The paper tackles safety in complex nonlinear control where a reduced-order model (RoM) safety filter may fail to ensure full-order model (FoM) safety due to RoM-FoM discrepancies. It introduces Predictive CBFs that augment the RoM CBF condition with a horizon-based robustness term $\delta$, computed via FoM rollouts, to guarantee safety on both models. The authors prove existence and safety guarantees under mild tracking assumptions and develop a learning framework to estimate $\delta$ from massively parallel simulations with domain randomization, enabling practical deployment. They validate the approach in simulation and on the ARCHER 3D hopping robot, achieving safe navigation in cluttered environments while reducing conservatism relative to prior RoM-based safety filters.

Abstract

Safety filters leveraging control barrier functions (CBFs) are highly effective for enforcing safe behavior on complex systems. It is often easier to synthesize CBFs for a Reduced order Model (RoM), and track the resulting safe behavior on the Full order Model (FoM) -- yet gaps between the RoM and FoM can result in safety violations. This paper introduces \emph{predictive CBFs} to address this gap by leveraging rollouts of the FoM to define a predictive robustness term added to the RoM CBF condition. Theoretically, we prove that this guarantees safety in a layered control implementation. Practically, we learn the predictive robustness term through massive parallel simulation with domain randomization. We demonstrate in simulation that this yields safe FoM behavior with minimal conservatism, and experimentally realize predictive CBFs on a 3D hopping robot.

Figures (5)

• Figure 1: First, a CBF is synthesized on a RoM; tracking errors from the FoM can lead to unsafe behavior. We learn a structured robustification term $\delta( \mathbf{x} )$ from massively parallel simulation, and deploy the robustified CBF on hardware to obtain safe behaviors.
• Figure 2: Layered architecture of a RoM and FoM interacting through a Predictive CBF.
• Figure 3: Top: Comparison of the set $\mathcal{S}_{\delta_0}$ from Theorem \ref{['thm:predcbfs-exists']} for different values of $\delta_0$ and trajectories of the closed-loop FoM in Example \ref{['example:scalar']}. Bottom: Comparison between the optimized (left) and learned (middle) values of $\delta( \mathbf{x} )$, as well as the corresponding $\mathcal{C}_{\mathrm{FoM}}^\delta$. (Right) compares the performance of these predictive CBFs to the nominal CBF without $\delta$.
• Figure 4: Comparison between the performance of the Predictive CBF (both Learned and Optimized) to the nominal RoM CBF Safety Filter. Both versions of the Predictive CBF ensure safety for the FoM, while the nominal safety filter violates the safety constraint.
• Figure 5: The 3D hopping robot ARCHER (left) navigates a cluttered environment using each of the nominal safety filter (SF), predictive safety filter (PSF) and learned PSF. (Middle) The trajectories of the hopper in space; (Right) values of $h$ and $\delta$ plotted over time for each approach. The nominal SF is unsafe, but both the PSF and LPSF maintain safety.

Theorems & Definitions (10)

• remark 1
• theorem 1
• definition 1
• remark 2
• theorem 2: Predictive CBFs $\implies$ Safety
• theorem 3: Existence of Predictive CBFs
• Theorem
• proof
• Theorem
• proof