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LEAP-CBF: A Safety Filter for Uncertain Systems with Least-Effort Adversarial Potentials

Oswin So, Eric Yu, Chuchu Fan

arXiv:2609.28364v1cs.ROcs.LGmath.OC

TL;DR

Safety certificates may measure clearance without quantifying how much disturbance effort would cause failure. LEAP measures that effort and supports safety filtering under a cumulative disturbance budget; experiments report improved safety–task tradeoffs against evaluated HJ and CBF baselines.

  • Problem

    Existing safety values can measure clearance rather than the disturbance effort required to cause failure, leaving a gap in how certificate magnitude represents disturbance tolerance.

    Source: §I · p. 1

  • Method

    LEAP defines the minimum cumulative disturbance effort needed to cause failure under a fixed fallback and uses it in safety filters, with an on-policy reinforcement learning procedure to approximate the potential and improve the fallback.

    Source: §I · pp. 1–2, §V · p. 5

  • Results

    Experiments demonstrate improved safety–task performance tradeoffs under control noise, adversarial interactions, and tracking latency, including relative to evaluated HJ and CBF baselines.

    Source: §I · p. 2, §VIII · p. 7

  • Takeaways & Limitations

    The LEAP value gives robustness an operational meaning and supports safety guarantees when cumulative disturbance effort is bounded.

    Source: §I · pp. 1–2, §IV · p. 4

Abstract

from arXiv · show

Control barrier functions (CBF) are a popular safety filter to ensure safety for nonlinear dynamical systems. However, when the system is subject to uncertainties and disturbances, this requires the use of robust variants of CBFs, which can be difficult to construct and can be overly conservative, especially for high-dimensional systems under input constraints. In this work, we propose a new approach to solve these challenges by introducing Least-Effort Adversarial Potentials (LEAP), a certificate that quantifies the robustness of a given state against disturbances in terms of the effort required by the disturbance to cause failure. We show that LEAP is a CBF for the undisturbed system, but can also be used to construct a safety filter that is robust to disturbances whose cumulative effort is bounded. We propose a method for constructing LEAPs with on-policy deep reinforcement learning. Next, we demonstrate LEAPs in simulation on a variety of multi-agent systems with disturbances and uncertainties. Finally, hardware experiments on a quadruped and quadrotors validate that LEAPs are well suited to tackle the disturbances and uncertainties from real-world robotic systems.

I. INTRODUCTION

Robust safety filters need certificates that reflect disturbance tolerance, not only geometric clearance. LEAP measures effort-to-failure under a fixed fallback and connects that value to nominal CBFs, cumulative-budget filtering, and learned approximations.

I. INTRODUCTION

  • Brief but large disturbances can make pointwise-robust filters overly conservative, motivating certificates tied to cumulative disturbance effort.Pointwise bounds permit persistent disturbances at the bound, while LEAP makes effort-to-failure explicit for runtime filtering.Source: §I · p. 1
  • LEAP is the minimum cumulative disturbance effort required to cause failure under a fixed fallback policy.Its value gives robustness a meaning tied to system dynamics and the disturbance model.Source: §I · pp. 1–2
  • Under suitable regularity assumptions, positive-threshold shifts of exact LEAP values are nominal-system CBFs, and LEAP-based filters guarantee safety under a cumulative disturbance budget.The contribution is the connection to nominal CBFs and disturbance-budget safety, not running-cost reachability itself.Source: §I · pp. 1–2
  • An on-policy reinforcement-learning procedure alternates adversary training and fallback improvement, then estimates effort values with the fallback fixed.The introduction also describes learning the fallback policy and approximating the potential.Source: §I · pp. 1–2
  • LEAP-CBF improves safety–task performance tradeoffs over evaluated HJ and CBF baselines across navigation simulations and quadruped and quadrotor hardware experiments.The listed settings include noisy Dubins navigation, adversarial crowd navigation, and multi-agent position swapping with tracking latency.Source: §I · p. 2

II. PROBLEM SETUP

The setup models a controlled system subject to disturbances and defines safety as avoiding a failure set. For the nominal system, CBF conditions support invariant safe sets and a control-affine safety filter.

II. PROBLEM SETUP

  • The problem is to avoid a failure set despite unknown disturbances, with state, control, and measurable disturbance variables.The disturbance may be unbounded, and the setup assumes zero is an available disturbance when defining nominal dynamics.Source: §II · p. 2
  • The nominal system is defined under the assumption that zero belongs to the disturbance set.This isolates the dynamics used for nominal CBF conditions.Source: §II · p. 2
  • For control-affine nominal dynamics, the CBF condition is linear in the control, and a sufficiently regular policy satisfying it makes the certified set forward invariant.This yields a CBF-QP safety filter for the nominal command.Source: §II · p. 2

III. SAFETY FOR UNDISTURBED SYSTEMS

The section establishes the LEAP value as a control barrier function for nominal dynamics and interprets its magnitude in terms of disturbance effort. Its nonnegative set can be made forward invariant using a nominal CBF-QP safety filter.

III. SAFETY FOR UNDISTURBED SYSTEMS

  • LEAP is a running-cost reachability value that minimizes cumulative disturbance effort to failure while holding the fallback policy fixed.The value is nonnegative, equals zero on the failure set, and is infinite when no disturbance can cause failure.Source: §III · pp. 2–3
  • The LEAP value is a CBF for the nominal dynamics.The proof verifies the failure-set condition and uses the value function’s stationary HJB equation to establish the other CBF condition.Source: §III · p. 3
  • Using the LEAP value in a CBF-QP filter makes its nonnegative set forward invariant within the failure-free region.For control-affine dynamics, the CBF condition is linear in the control input.Source: §III · p. 3
  • Unlike signed-distance HJ values, LEAP’s magnitude directly represents the minimum cumulative disturbance effort needed for failure under the fallback.The HJ value’s magnitude depends on the chosen surface function and, with signed distance, measures clearance rather than disturbance tolerance.Source: §III · p. 3

IV. SAFETY FOR DISTURBED SYSTEMS

LEAP expresses robustness as the disturbance effort needed to cause failure, enabling filters that certify safety under a cumulative disturbance budget. The section develops switching and QP filters and compares their safety and intervention behavior with HJI.

  • A_t is accumulated disturbance effort; with quadratic effort the budget measures disturbance energy, while absolute effort measures cumulative magnitude.Source: §IV · p. 3
  • LEAP matches the observed safety of the safest evaluated HJI configurations with less intervention, while remaining safe in displayed rollouts where disturbances exceed HJI’s pointwise bounds.The effort budget permits short, large disturbance peaks while limiting their cumulative effect.Source: Figure 4 · p. 4, §IV · p. 4
  • For initial states with LEAP at least D, the switching filter guarantees safety against disturbances whose cumulative effort is bounded by D.The filter switches to the fixed fallback when LEAP reaches or falls below D.Source: §IV · p. 4
  • Combining a nominal CBF-QP with fallback switching also guarantees disturbance-budget safety for initial states in the certified set.The fallback is used when the QP is infeasible or LEAP is at or below D.Source: §IV · p. 4
  • The budget-aware CBF-QP preserves J(x_t) ≥ D − A_t while accumulated effort A_t remains below D, thereby guaranteeing disturbance-budget safety.Its feasibility requires H ≥ 0 and A < D.Source: §IV · pp. 4–5
  • For control-affine dynamics with additive disturbances and quadratic effort, disturbance minimization yields a linear constraint in the control and a QP when the input set is a polytope.Source: §IV · p. 5

V. SOLVING LEAP WITH DEEP REINFORCEMENT LEARNING

The paper approximates LEAP with cumulative-return reinforcement learning and an alternating on-policy actor-critic procedure. It refines the learned value for fallback-unsafe and adversary-unreachable states, while leaving formal convergence analysis for future work.

  • LEAP is approximated with standard cumulative-return reinforcement learning after discretizing its sum-of-costs objective.The paper contrasts this with trajectory-extremum HJ objectives, which use Bellman updates or PDE solvers.Source: §V · p. 5
  • An alternating on-policy actor-critic procedure trains an adversary to reach failure with least disturbance cost against a fixed fallback policy.The stochastic adversary is trained to explore nonzero disturbances and favor least-effort attacks rather than maximal disturbances at every step.Source: §V · p. 5
  • The fallback is trained on nominal, undisturbed rollouts to maximize the minimum disturbance effort required for failure.The training objective uses the adversary critic as an estimate of negative LEAP.Source: §V · p. 5
  • The deployed value combines a critic with classifiers for fallback safety and adversary reachability, assigning zero effort to predicted fallback-unsafe states and a cost cap to predicted unreachable states.Source: §V · p. 5
  • Formal convergence analysis of the coupled training procedure is left to future work, although one alternating iteration sufficed empirically across the experiments.Source: §V · p. 5

VI. SIMULATION EXPERIMENTS

Across three simulation tasks, the evaluation tests safety filters under disturbances and uncertainties. In noisy Dubins Reach-Avoid, LEAP reaches the goal faster than baselines while meeting a 95% safety threshold as noise increases.

Evaluation setup

  • The simulations test time-correlated steering noise, adversarial agent motion, and tracking latency across three tasks.The evaluation uses task-agnostic training environments and perturbed task environments at test time.Source: §VI · pp. 5–6

Baselines

  • Baselines include learned HJ and HJI safety filters and task-specific handcrafted CBFs, with a robust handcrafted variant.HJ learns a CBF; HJI is its robust counterpart under bounded disturbances; R-HCBF uses a worst-case counterpart over box-bounded disturbances.Source: §VI · p. 6

Noisy Dubins Reach-Avoid

  • With correlation time τ = 0.5 s, LEAP reaches the goal faster than baselines at the 95% safety threshold as noise increases.The comparison uses the fastest goal-reaching time among filter configurations achieving at least 95% safety, versus noise level σ.Source: §VI · p. 6, Figure 7 · p. 6

B. Adversarial Crowd Navigation

In Crowd Navigation, LEAP achieves a better safety–completion tradeoff than the baselines, including evaluated HJI disturbance bounds. The task tests collision avoidance while a faster crowd moves around a slower ego agent.

Task setup

  • A unicycle ego moves through five agents traveling at a nominal 0.9 m/s, faster than the ego’s maximum speed of 0.6 m/s.At evaluation, an adversary can briefly perturb an agent toward the ego after it enters the ego’s sensing radius.Source: B. Adversarial Crowd Navigation · p. 6

Metrics

  • Safety rate is the fraction of episodes without collision, and completion rate is the fraction in which the ego crosses the finish line.The finish line is at y = 12 m, with the ego starting at y = 0 m.Source: B. Adversarial Crowd Navigation · p. 6

Performance

  • LEAP achieves a better safety–completion tradeoff than the baselines, including all evaluated HJI disturbance bounds.Small HJI bounds provide insufficient robustness, whereas large bounds restrict progress; LEAP uses a single dmax = ∞.Source: B. Adversarial Crowd Navigation · p. 6, Figure 8 · p. 6

C. Tracking Latency in Position Swapping

In Position Swapping, LEAP maintains the best safety–reach tradeoff as tracking latency increases, although latency is absent during training. HJI is competitive at moderate latency but degrades sharply at larger values.

Task setup

  • Six double-integrator agents start at rest and must swap positions without colliding, with each agent’s goal set to another’s starting position.Robustness to tracking latency is evaluated at a fixed team size.Source: C. Tracking Latency in Position Swapping · p. 7

Latency model

Results

VII. HARDWARE EXPERIMENTS

In hardware trials, LEAP performs best, followed by HJI, on safety rate and final-waypoint distance after 30 seconds. When the drone turns unexpectedly into the quadruped’s path, HJ collides while LEAP avoids collision.

Hardware setup

  • A Unitree Go2 quadruped follows four waypoints while avoiding a Crazyflie drone whose heading can change toward the Go2.The safety filter controls the Go2; both robots are localized with the Crazyflie lighthouse system, and commands come from a centralized computer.Source: §VII · p. 7

Training and deployment

  • The drone’s evaluation behavior and the nominal waypoint controller are unavailable to baseline methods during training; the waypoint controller is introduced at deployment.The task tests robustness to departures from modeled drone motion while preserving waypoint progress.Source: §VII · p. 7

Hardware results

VIII. CONCLUSION

LEAP measures robustness as the minimum cumulative disturbance effort required to cause failure under a fixed fallback policy, supporting nominal CBF guarantees and disturbance-budget safety filters. Simulations and hardware experiments show improved safety–task tradeoffs, while learned-certificate guarantees and training convergence remain unresolved.

  • Simulations and hardware experiments show improved safety–task performance tradeoffs under control noise, adversarial interactions, and tracking latency.Source: §VIII · p. 7
  • Under suitable regularity assumptions, positive-threshold shifts of exact LEAP values are nominal-system CBFs, and LEAP-based filters guarantee safety under cumulative disturbance budgets.Source: §VIII · p. 7
  • LEAP assigns robustness an operational meaning: the minimum cumulative disturbance effort required to cause failure under a fixed fallback policy.Source: §VIII · p. 7
  • An alternating reinforcement learning procedure approximates LEAP while improving its fallback policy.Source: §VIII · p. 7
  • Classifier-based LEAP approximation is heuristic, lacks guaranteed convergence, and does not automatically inherit theoretical safety guarantees; fallback-training convergence is also unproven.Source: §VIII · p. 7

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