Safe Navigation under State Uncertainty: Online Adaptation for Robust Control Barrier Functions

TL;DR

This work proposes a systematic method to improve R-CBFs, and demonstrates its advantages on a tracked vehicle that navigates among multiple obstacles through a new optimization-based online parameter adaptation scheme that reduces the conservativeness of existing R-CBFs.

Abstract

Measurements and state estimates are often imperfect in control practice, posing challenges for safety-critical applications, where safety guarantees rely on accurate state information. In the presence of estimation errors, several prior robust control barrier function (R-CBF) formulations have imposed strict conditions on the input. These methods can be overly conservative and can introduce issues such as infeasibility, high control effort, etc. This work proposes a systematic method to improve R-CBFs, and demonstrates its advantages on a tracked vehicle that navigates among multiple obstacles. A primary contribution is a new optimization-based online parameter adaptation scheme that reduces the conservativeness of existing R-CBFs. In order to reduce the complexity of the parameter optimization, we merge several safety constraints into one unified numerical CBF via Poisson's equation. We further address the dual relative degree issue that typically causes difficulty in vehicle tracking. Experimental trials demonstrate the overall performance improvement of our approach over existing formulations.

Figures (5)

• Figure 1: The proposed robust CBF (R-CBF) method in an outdoor safety-critical navigation scenario with a tracked mobile robot. Our R-CBF-QP-based safety filter maintains safety under state estimation uncertainty.
• Figure 2: Poisson safety function. The function $h_0$ describes the spatial safety specification, and it can be used to generate a CBF, as in \ref{['eq:modified_CBF']}.
• Figure 3: Nominal (no state uncertainty): COCP, vanilla CBF-QP, and our R-CBF-QP with online adaptation. Top: Robot trajectories with the safe set and the reference. The brown dot ($\bullet$) marks a vanilla CBF-QP deadlock: the robot reaches a point where no admissible input both preserves safety and moves toward the reference. The baseline controller $\mathbf{k_d}$ follows the sinusoidal reference with negligible steady-state error. Bottom: Time histories of $v$, $\omega$, ${\gamma_1, \gamma_2}$, and $h$. Our method maintains ${h \!\geq\! 0}$ and closely tracks the reference, similar to COCP. The tuned parameters are obtained as ${\Bar{\gamma}_1(\hat{\mathbf{ x }}) \!\approx\! 0}$, ${\Bar{\gamma}_2(\hat{\mathbf{ x }}) \!\approx\! 0}$ with the proposed online tuning method, as ${\mathbf{ e } \!\equiv\! 0}$.
• Figure 4: Robust (bounded state uncertainty): R-COCP, R-CBF-QP with fixed ${(\gamma_1 \!=\! 1.4, \gamma_2 \!=\! 0.3)}$, and tunable R-CBF-QP with ${\gamma_{1}(h), \gamma_{2}(h)}$ given in \ref{['eq:tun_par']}, and our R-CBF-QP with online adaptation. Top: Trajectories under the measurement uncertainty. Shaded tubes show the uncertainty envelope from bounded measurement error. Our R-CBF-QP and tunable R-CBF-QP track the reference more closely and are comparable to R-COCP near obstacles, while R-CBF-QP with fixed parameters is more conservative. Bottom: Time histories of $v$, $\omega$, ${\gamma_1, \gamma_2}$, and $h$. All methods satisfy ${h \!\geq\! 0}$. Our online tuning method increases ${\Bar{\gamma}_1, \Bar{\gamma}_2}$ only when needed, such as near the boundary.
• Figure 5: Experimental comparison on tracked robot. Pose is measured at the robot’s geometric center; therefore, each circular obstacle boundary is inflated by half the robot’s length. Dashed red circles denote the true safe-set boundaries, while solid red circles indicate the inflated boundaries used in the controller synthesis. Gray tubes visualize the 95% confidence envelope of the $(x, y)$ covariance along the executed paths. (Left) (Top) R-CBF-QP with online adaptation of parameters ${\Bar{\gamma}_{1}, \Bar{\gamma}_{2}}$, the proposed method. (Left) (Middle) R-CBF-QP with tunable robustness parameters, i.e., CBF-dependent ${\gamma_{1}(h), \gamma_{2}(h)}$. Tunable R-CBF-QP violated the safety constraint in the presence of time-varying state uncertainty, while its performance was similar to R-CBF-QP with online adaptation in simulation. (Left) (Bottom) Robustness parameters for the adaptive and tunable R-CBF-QPs. (Right) (Middle) R-CBF-QP without robustness parameters, i.e., ${\gamma_{1} \!=\! 0, \gamma_{2} \!=\! 0}$. This non-robust controller violates constraints, and the robot falls from the platform and tips over at 24 sec. The experiment was terminated after this unsafe behavior. This result illustrates the necessity of robustification in practice. The CBF-QP–controlled robot encounters a deadlock at the boundary of the first obstacle, marked by the purple cross ($\times$), since the CBF-QP only uses $v$. And, R-CBF-QP with fixed robustness parameters, i.e., ${\gamma_{1} \!=\! 1.0, \gamma_{2} \!=\! 0.2}$, this controller is more conservative. (Right) (Bottom) The value of CBF $h$ versus time.

Theorems & Definitions (7)

• Definition 1: Control Barrier Function, ames2017control
• Definition 2
• Definition 3: Robust Control Barrier Function, rahal_cdc_2025
• Theorem 1
• proof
• Remark 1
• Remark 2