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Learning-Based Measurement-Robust Control Barrier Functions for Obstacle Avoidance under State Estimation Error

Nicholas Rober, Yixuan Jia, Jonathan P. How

arXiv:2608.20467v1eess.SYcs.RO

TL;DR

State-estimation errors can undermine safety filters through collisions or excessive conservativeness. The paper develops certifiable DMR-CBFs and learned NMR-CBFs for obstacle avoidance, with the latter reducing conservativeness and computational cost. Across simulations and hardware experiments, the proposed methods maintain strong safety under uncertainty, including successful quadruped navigation under odometry drift.

  • Problem

    Safety filters commonly assume perfect state information, while existing robust methods can violate safety or become overly conservative as estimation error grows.

  • Method

    The paper develops DMR-CBFs with worst-case drift optimization and NMR-CBFs that learn the robustness term through supervised pretraining and differentiable trajectory fine-tuning.

  • Results

    Both DMR-CBF and NMR-CBF achieve perfect safety records without excessive conservativeness across the planar double-integrator and quadrotor evaluations.

  • Takeaways & Limitations

    NMR-CBF provides empirical safety comparable to DMR-CBF while reducing conservativeness and computational cost, and it protects a quadruped against onboard odometry drift.

Abstract

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Safety filters are an effective tool for enforcing constraints in safety-critical systems, but most existing methods assume perfect state information, which is rarely available in practice. Recent work has begun to close this gap by developing filtering mechanisms that are robust to state estimation error, but these methods can still exhibit safety violations or overly conservative behavior as estimation error grows. Focusing on obstacle avoidance, we develop two new control barrier function (CBF) formulations: drift-measurement-robust (DMR)-CBFs and neural measurement-robust (NMR)-CBFs. The DMR-CBF augments the standard CBF condition with an inner optimization over the worst-case uncertainty in the drift dynamics, improving robustness to estimation error. This DMR-CBF then supervises a pretraining phase for the NMR-CBF, which replaces the inner optimization with a learned term. The NMR-CBF is subsequently finetuned through differentiable trajectory rollouts, yielding a filter that achieves empirical safety comparable to the DMR-CBF while reducing both conservativeness and computational cost. We provide theoretical analysis of the DMR-CBF along with numerical results on a planar double integrator and a 12D quadrotor, where both proposed approaches prevent collisions while other robust methods either fail or are overly conservative. Finally, we deployed the NMR-CBF on a Unitree Go2, enabling successful navigation of an obstacle field under odometry errors that caused a standard CBF to collide.

I. INTRODUCTION

Safety filters must account for state-estimation error in safety-critical obstacle avoidance. The paper proposes DMR-CBF and NMR-CBF formulations intended to preserve safety while reducing conservativeness and computational cost.

  • Most safety filters assume perfect state information, although practical systems rely on estimates that inevitably contain error.
  • Existing robust filters can become unsafe or overly conservative as state-estimation uncertainty increases.
  • DMR-CBF combines point- and set-based robustness in a certifiable formulation for systems subject to estimation error.
  • NMR-CBF learns a robustness term to reduce conservativeness and computational cost relative to DMR-CBF while maintaining comparable empirical safety.
  • Numerical experiments show both proposed methods achieve safety and performance, while baselines are either safe but conservative or performant but collision-prone.
  • Hardware experiments deployed NMR-CBF on a Unitree Go2 under onboard estimate drift that caused a standard CBF to collide.

A. System Dynamics

The paper models a control-affine nonlinear system with an unknown true state and a known estimate accompanied by an error bound. Safety is represented by a forward-invariant set enforced through a CBF-based quadratic program using the estimated state.

  • The system has control-affine dynamics, with locally Lipschitz drift f(x), input matrix g(x), and control input u.
  • The true state is unknown, but the available estimate has a known element-wise error bound.
  • The error bound defines an uncertainty set centered at the state estimate through a Minkowski sum.
  • Safety is defined by the set C = {x ∈ X | h(x) ≥ 0}, with safety requiring forward invariance of C.
  • When the state is known, a quadratic program minimally modifies a nominal controller while enforcing the CBF condition; here the estimate replaces the unknown true state.

C. Safety Filtering Under State Estimation Error

Existing safety filters under state estimation error use either point-based or set-based robustifications, but each has important limitations. The paper therefore seeks methods that retain safety under large errors without excessive conservativeness or poor scalability.

  • Existing robust strategies: Point-based robust CBFs modify constraints using state estimates and robustifying terms, while set-based methods enforce safety over an uncertainty set.MR-CBFs, R-CBFs, and R-CBF-QPs are point-based; Duality CBFs and GUARDIAN are set-based.
  • Point-based limitations: MR-CBFs may be difficult or impossible to construct as state-estimation error increases, despite strong guarantees for the original safe set.The existence of a standard CBF does not guarantee a valid MR-CBF.
  • Point-based limitations: R-CBFs use fixed hyper-parameters that do not adapt to varying estimation error, allowing possible safety violations.R-CBF-QPs adaptively tune these parameters but, beyond a certain error level, guarantee safety only for an inflated safe set.
  • Set-based limitations: Set-based methods can become overly conservative because one control must preserve safety across the entire uncertainty set.This issue is identified for Duality CBFs and GUARDIAN.
  • Set-based limitations: GUARDIAN additionally requires a scalar control input, excluding many obstacle-avoidance problems, while Duality CBF and GUARDIAN face scalability issues.These limitations restrict their practical application to smaller systems compared with point-based filters.
  • Proposed direction: The proposed DMR-CBF and NMR-CBF combine point- and set-based robustness for obstacle avoidance under estimation uncertainty.DMR-CBF is certifiable; NMR-CBF uses supervised pretraining and differentiable trajectory fine-tuning to reduce conservativeness and computational cost.

A. Drift–Measurement Robust CBFs

The DMR-CBF robustifies the uncontrolled drift component of the CBF condition over the state-uncertainty set while retaining point-based treatment of the control component. Its certificate identifies when the resulting control guarantees forward invariance of the true safe set.

  • DMR-CBF construction: The DMR-CBF modifies the standard CBF constraint by finding the worst-case drift term over the state-uncertainty set.The resulting control is computed using the modified constraint.
  • DMR-CBF construction: DMR-CBF robustness is set-based for the autonomous drift term but point-based for the control term, preserving nominal control authority.This balances estimation-error robustness against the conservativeness of fully set-based control.
  • Uncertainty model: The uncertainty set is constructed as a box around the estimated state, and the true state lies inside it by construction of the error bound.The box is convex and connected.
  • Safety analysis: The true CBF condition decomposes into nonnegative DMR slack, nonnegative drift gap, and a control-coefficient mismatch term.The decomposition isolates the control-coefficient mismatch as the only potentially negative term.
  • Safety analysis: If the control-coefficient mismatch is nonnegative, the true CBF condition holds and the true safe set is forward invariant.The DMR-CBF certificate formally captures this implication.
  • Safety certificate: A DMR-CBF solution is certified safe when its estimated-state margin exceeds the control-authority gap.The certificate is a cheap a posteriori scalar minimization evaluated at the solved control.

B. Neural Measurement-Robust CBFs

The NMR-CBF replaces the DMR-CBF’s analytic worst-case drift correction with a learned nonnegative residual. It conditions on estimated state and uncertainty radius to reduce the need for repeated inner optimization across noise levels.

  • NMR-CBF formulation: The NMR-CBF replaces the DMR-CBF inner minimization and analytic drift correction with a learned nonnegative residual ρξ.A softplus output head enforces nonnegativity for every constraint.
  • NMR-CBF formulation: The network takes the concatenated estimated state and per-dimension uncertainty radius as input.Conditioning on uncertainty allows one model to generalize across radii up to the training envelope.
  • NMR-CBF limitation: The NMR-CBF has no formal safety guarantee, although later experiments assess its empirical safety.Its learned term is intended to replace the analytic correction while reducing computational burden.
  • NMR-CBF training: The NMR-CBF is initialized by regression against the DMR-CBF oracle for each safety constraint.The oracle provides the target robustness term used during supervised pretraining.

1) Supervised Pretraining:

Supervised pretraining makes the NMR-CBF imitate the DMR-CBF’s robustness correction. The pretrained network then replaces the DMR-CBF inner minimization with one MLP evaluation per step.

  • Supervised Pretraining: The pretraining target is the DMR-CBF drift correction, computed using projected-gradient ascent with random restarts over the uncertainty set.Targets are clipped to [0, ϕmax] to guard against ill-conditioned outliers near CBF singularities.
  • Supervised Pretraining: Training samples pair uniformly sampled safe state estimates with per-dimension noise radii drawn from Uniform([0, e]).Samples inside obstacles are rejected before minimizing the per-constraint mean-squared error.
  • Supervised Pretraining: One MLP evaluation per step replaces the DMR-CBF inner minimization at reduced computational cost.The pretrained network functions as a drop-in replacement for that optimization.

2) Differentiable Fine-Tuning:

The NMR-CBF is fine-tuned on differentiable closed-loop rollouts so its learned robustness term can shrink where trajectories have safety slack while preserving safety-oriented behavior.

  • Supervised pretraining incorporates the DMR-CBF’s worst-case assumption into the NMR-CBF, initially preserving its conservativeness.
  • Differentiable closed-loop rollouts fine-tune the robustness term while propagating gradients through the projected control and system dynamics.The rollout uses sampled initial states and bounded estimation errors.
  • The per-episode loss penalizes buffered safety violations and deviation from the nominal policy.The safety term activates before trajectories cross the constraint boundary, while the control term supplies a dense shaping signal.
  • A regularizer drives the learned robustness term toward zero where safety and minimal-deviation pressures do not require additional conservatism.The nonnegative softplus structure is retained while nominal-CBF aggressiveness is recovered at low estimation error.

3) Properties:

The NMR-CBF replaces DMR-CBF’s inner optimization with a learned evaluation and trajectory-level fine-tuning, trading formal out-of-distribution guarantees for lower conservativeness and computation. Across planar obstacle-avoidance experiments, the proposed methods maintain safety across uncertainty levels while baselines become conservative or collision-prone.

  • The NMR-CBF replaces the DMR-CBF inner optimization with a single MLP evaluation and trajectory-level loss optimization.This reduces conservativeness and computational cost relative to pointwise worst-case optimization.
  • The learned robustness term has no formal guarantee away from the training distribution and may under-tighten the constraint relative to DMR-CBF.
  • 1,000 initial states per method and uncertainty level were evaluated using collision rate and time to goal as comparison metrics.Trajectories were categorized as Reached, Timeout, or Unsafe within a 20 s simulation limit.
  • Set-based Duality CBF and GUARDIAN exhibit slower time to goal and more timeouts, while GUARDIAN also incurs safety violations when ε > 0.Their difficulty selecting an obstacle-avoidance direction is attributed to optimizing over the entire uncertainty set.
  • Standard CBF becomes increasingly collision-prone as ε grows, whereas R-CBF (0.2) lacks robustness at high ε and R-CBF (1.0) is overly conservative.
  • DMR-CBF and NMR-CBF are the only approaches that avoid collisions for every ε value without being overly conservative.NMR-CBF matches R-CBF-QP time to goal for ε = 0.1 and 0.2 and is at least as fast as R-CBF (1.0) for ε = 0.3, 0.4, and 0.5.

B. Quadrotor

The quadrotor evaluation tests scalability in a 12D state space with multiple safety constraints. Across 1,000 trajectories, DMR-CBF and NMR-CBF retain perfect safety without excessive conservatism, while NMR-CBF adds little computational overhead.

  • B. Quadrotor: The quadrotor state has 12 dimensions covering position, velocity, attitude, and angular rates.
  • B. Quadrotor: 1,000 quadrotor trajectories were evaluated, with GUARDIAN omitted because its HJ formulation does not scale to the 12D state space.
  • B. Quadrotor: DMR-CBF and NMR-CBF are the only approaches with a perfect safety record without being overly conservative, and NMR-CBF matches or outperforms baselines in time to goal.
  • B. Quadrotor: 0.31 ms versus 0.26 ms per step for planar double integrator and 0.57 ms versus 0.47 ms for quadrotor are the NMR-CBF and standard CBF runtimes, respectively.
  • B. Quadrotor: 2.27 ms versus 45.28 ms per step shows DMR-CBF is substantially faster than Duality CBF on the 12D quadrotor.

V. HARDWARE EXPERIMENTS: QUADRUPED

Hardware experiments deployed the NMR-CBF on a Unitree Go2 navigating two obstacles using onboard state estimates affected by injected bias and odometry drift. Compared with a standard CBF, the NMR-CBF accounted for estimation error and kept the true robot state safe.

  • Experimental setup: The hardware platform was a Unitree Go2 equipped with onboard computing, motor encoders, and an IMU.The Go2 was modeled as a planar double integrator and controlled through its sportmode API.
  • Experimental setup: The Go2 navigated from an initial position to a goal through a space containing two obstacles.The robot used onboard proprioceptive odometry rather than external positioning during navigation.
  • Comparison: A standard CBF collided with an obstacle under injected bias and odometry drift, whereas the NMR-CBF avoided both obstacles from similar initial conditions.The standard CBF did not account for state estimation error; the NMR-CBF accounted for possible errors.
  • Comparison: The standard CBF could collide with either obstacle depending on the initial configuration because the biased estimate caused under-correction.For left-side initial conditions, the estimate appeared beyond the first obstacle from the true state, causing entry into the keep-out region.
  • NMR-CBF behavior: The NMR-CBF remained safe across all initial conditions by dynamically covering injected bias and growing drift in its learned robustness term.It kept the ground-truth state away from both obstacles despite the same estimation errors that caused the CBF to fail.

VI. CONCLUSION

The paper develops DMR-CBF and NMR-CBF formulations for obstacle avoidance under state estimation error. Numerical and hardware experiments show improved empirical safety, while the NMR-CBF reduces conservativeness and computational cost relative to the DMR-CBF.

  • Conclusion: The DMR-CBF provides an a posteriori safety certificate but requires an inner optimization and is more conservative than the NMR-CBF.The NMR-CBF is pretrained using DMR-CBF-generated data and fine-tuned with trajectory rollouts.
  • Conclusion: Both DMR-CBF and NMR-CBF offer significant empirical safety improvements over all baselines, especially in high-uncertainty regimes.The conclusion reports this pattern across the numerical results.
  • Conclusion: Hardware experiments on a quadruped demonstrate NMR-CBF utility against odometry drift in onboard state estimation.The experiments show protection against drift naturally present in the robot’s state estimation module.
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