A Reachability-based Safety Certificate for Dynamical System Motion Policies

2026-09-29

AI Summary

This study addresses the issue that local safety policies for dynamical systems in unknown environments are prone to inducing collisions or spurious attractors. To overcome this, we propose a method for constructing deterministic finite-horizon safety certificates based on backward reachability value functions. By reformulating the infinite-dimensional reachability problem as deterministic receding-horizon optimization, we rigorously prove that this function constitutes the maximal forward-invariant subset of the obstacle-free region. The research integrates five system construction techniques, including Neural ODEs, diffeomorphic latent spaces, and LPV-DS, with experimental validation conducted on a Franka robotic manipulator. Results demonstrate that the proposed approach completely overcomes the saddle-point traps inherent in conventional Control Barrier Functions (CBFs), significantly enhancing safety in complex environments.

Abstract

Dynamical Systems (DS) are reactive motion policies representing vector fields trained with theoretical guarantees of stability and convergence. To ensure safety during deployment in unknown environments they must be locally reshaped, either through modulation or geometric control barrier function strategies. However, depending on the geometry of the obstacles and the complexity of the DS, these local strategies can lead the system to unavoidable collisions or spurious attractors. In this work, we certify safety with a value function drawn from the notion of backward reachability tube, which measures the worst-case safety along a rollout trajectory of the nominal DS. Usually, such a value function is intractable for a controlled system due to curse of dimensionality. We show that in the DS-based learning-from-demonstration setting, the absence of a control input collapses the reachability problem to a deterministic rollout, and the presence of certain stability conditions truncates the infinite horizon to a finite one, resulting in a well-defined value function. We further show that the value function we devised is the maximal forward-invariant subset of the obstaclefree region for the nominal DS flow. The application of this certificate function is validated across five DS constructions - analytical, Neural ODE, diffeomorphic latent space, LPV-DS, SE(3)and validate it on a Franka manipulator. Modulation and geometric CBFs also suffer from saddle point in cases of headon approach towards an unsafe zone. We show that CBF-on-V avoids this pitfall entirely.

Problem

• Dynamical Systems
• Safety Certificate
• Reachability
• Motion Policies
• Spurious Attractors

Innovation

• Backward Reachability Tube
• Safety Certificate
• Dynamical Systems
• Control Barrier Function
• Value Function