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Belief Space Control of Safety-Critical Systems Under State-Dependent Measurement Noise

Rohan Walia, Mitchell Black, Andrew Schoer, Kevin Leahy

TL;DR

The paper tackles safety-critical control when measurements are corrupted by state-dependent noise, arguing that fixed additive-noise models are insufficient for learning-based sensing modalities. It extends Belief Control Barrier Functions (BCBF) by integrating the Generalized Extended Kalman Filter (GEKF) to explicitly model multiplicative measurement noise $\aleph(x) = p\ell(x) + v$ and to provide probabilistic safety guarantees in belief space via CVaR constraints. The approach yields a state-dependent innovation covariance and reduced Kalman gains, leading to less conservative safety enforcement while maintaining probabilistic safety; the key theoretical contributions re-derive safety guarantees under the state-dependent noise model and GEKF updates. Demonstrations on a 1D setpoint-tracking problem and a 2D unicycle trajectory-tracking task show that BCBF-GEKF achieves closer nominal performance with zero safety violations, whereas BCBF-EKF can be overly conservative or fail to stay safe due to misestimated beliefs. Overall, the work provides a principled framework for incorporating complex sensor noise into safe control, with implications for real-world autonomous systems that rely on perception-based measurements.

Abstract

Safety-critical control is imperative for deploying autonomous systems in the real world. Control Barrier Functions (CBFs) offer strong safety guarantees when accurate system and sensor models are available. However, widely used additive, fixed-noise models are not representative of complex sensor modalities with state-dependent error characteristics. Although CBFs have been designed to mitigate uncertainty using fixed worst-case bounds on measurement noise, this approach can lead to overly-conservative control. To solve this problem, we extend the Belief Control Barrier Function (BCBF) framework to accommodate state-dependent measurement noise via the Generalized Extended Kalman Filter (GEKF) algorithm, which models measurement noise as a linear function of the state. Using the original BCBF framework as baseline, we demonstrate the performance of the BCBF-GEKF approach through simulation results on a 1D single integrator setpoint tracking scenario and 2D unicycle kinematics trajectory tracking scenario. Our results confirm that the BCBF-GEKF approach offers less conservative control with greater safety.

Belief Space Control of Safety-Critical Systems Under State-Dependent Measurement Noise

TL;DR

The paper tackles safety-critical control when measurements are corrupted by state-dependent noise, arguing that fixed additive-noise models are insufficient for learning-based sensing modalities. It extends Belief Control Barrier Functions (BCBF) by integrating the Generalized Extended Kalman Filter (GEKF) to explicitly model multiplicative measurement noise and to provide probabilistic safety guarantees in belief space via CVaR constraints. The approach yields a state-dependent innovation covariance and reduced Kalman gains, leading to less conservative safety enforcement while maintaining probabilistic safety; the key theoretical contributions re-derive safety guarantees under the state-dependent noise model and GEKF updates. Demonstrations on a 1D setpoint-tracking problem and a 2D unicycle trajectory-tracking task show that BCBF-GEKF achieves closer nominal performance with zero safety violations, whereas BCBF-EKF can be overly conservative or fail to stay safe due to misestimated beliefs. Overall, the work provides a principled framework for incorporating complex sensor noise into safe control, with implications for real-world autonomous systems that rely on perception-based measurements.

Abstract

Safety-critical control is imperative for deploying autonomous systems in the real world. Control Barrier Functions (CBFs) offer strong safety guarantees when accurate system and sensor models are available. However, widely used additive, fixed-noise models are not representative of complex sensor modalities with state-dependent error characteristics. Although CBFs have been designed to mitigate uncertainty using fixed worst-case bounds on measurement noise, this approach can lead to overly-conservative control. To solve this problem, we extend the Belief Control Barrier Function (BCBF) framework to accommodate state-dependent measurement noise via the Generalized Extended Kalman Filter (GEKF) algorithm, which models measurement noise as a linear function of the state. Using the original BCBF framework as baseline, we demonstrate the performance of the BCBF-GEKF approach through simulation results on a 1D single integrator setpoint tracking scenario and 2D unicycle kinematics trajectory tracking scenario. Our results confirm that the BCBF-GEKF approach offers less conservative control with greater safety.
Paper Structure (18 sections, 5 theorems, 52 equations, 3 figures, 2 tables)

This paper contains 18 sections, 5 theorems, 52 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Deterministic Forward Invariance
  4. Probabilistic Forward Invariance
  5. State-dependent measurement noise
  6. Technical Approach
  7. Measurement and state estimation model
  8. Probabilistic Safety Guarantees under multiplicative state-dependent noise
  9. Innovation covariance under a multiplicative noise model
  10. Experimental Results
  11. Setpoint tracking for a single integrator system
  12. 2D Sinusoidal Trajectory Tracking
  13. System Model
  14. Target trajectory
  15. Gain-scheduled trajectory tracking controller
  16. ...and 3 more sections

Key Result

Theorem 1

Given a set $\mathcal{C}$ as defined in eq:safe_set for a continuously differentiable function $h: \mathbb R^n \mapsto \mathbb R$, any locally Lipschitz continuous controller $u$ that satisfies eq:cbf_condition will render $\mathcal{C}$ forward invariant.

Figures (3)

  • Figure 1: Estimated and true trajectories for the 1D nonlinear single-integrator setpoint tracking task under BCBF-GEKF and BCBF-EKF control. The inset highlights that the EKF estimate crosses the safety boundary due to large estimation error, even though the true system trajectory remains safe. In contrast, the GEKF estimate respects the safety boundary while enabling the true trajectory to operate closer to it.
  • Figure 2: Comparison of trajectories, measurements, and estimates for sinusoidal tracking with BCBF-GEKF vs. BCBF-EKF. GEKF achieves closer nominal tracking while preserving safety comparable to the EKF.
  • Figure 3: Evolution of trace of covariance under EKF and GEKF. Lower covariance trace of the EKF can be attributed to overconfidence in measurements due to a higher Kalman gain as compared to the GEKF.

Theorems & Definitions (9)

  • Definition 1: Forward Invariance
  • Theorem 1: cbf_theory_and_applications, Thm. 2
  • Definition 3: BCBF belief_cbf
  • Theorem 2: belief_cbf, Thm. 2
  • Theorem 3: belief_cbf, Thm. 3
  • Proposition 1
  • proof
  • Proposition 2
  • proof