arXiv subjects

David E. J. van Wijk

Publications and source records attributed to David E. J. van Wijk.

10 recordsLinked to original sources

arXiv2026

Filtering for Liveness: Recurrence via Backup Reachability Control Barrier Functions

Liveness requires a dynamical system to repeatedly accomplish a desired objective over time. In this work, we consider a particular form of liveness known as recurrence, which requires the closed-loop trajectory to regularly revisit a prescribed set, with no more than a specified amount of time between visits. Inspired by the control barrier function (CBF) safety-filter paradigm, we develop a controller-filtering framework that makes adjustments to a nominal controller only when necessary to guarantee recurrence. Unlike conventional CBF methods, which enforce set forward invariance, our framework allows the trajectory to leave the set while guaranteeing its timely return. To achieve this, our framework combines two components: a backup reachability CBF that preserves the ability to return to the prescribed set within the allowable time, and a timing-based mechanism that invokes a reaching controller only when the nominal behavior would otherwise fail to meet the return-time requirement. The resulting framework guarantees recurrence while preserving the nominal behavior when possible, enabling control objectives that require temporary excursions from the set, such as traversal between disjoint regions. Our method is successfully utilized in a drone racing example with a recurrence constraint on beacon visibility.

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arXiv2026

Uncertainty Estimators for Robust Backup Control Barrier Functions

Designing safe controllers is crucial and notoriously challenging for input-constrained safety-critical control systems. Backup control barrier functions offer an approach for the construction of safe controllers online by considering the flow of the system under a backup controller. However, in the presence of model uncertainties, the flow cannot be accurately computed, making this method insufficient for safety assurance. To tackle this shortcoming, we integrate backup control barrier functions with uncertainty estimators and calculate the flow under a reconstruction of the model uncertainty while refining this estimate over time. We prove that the controllers resulting from the proposed Uncertainty Estimator Backup Control Barrier Function (UE-bCBF) approach guarantee safety, are robust to unknown disturbances, and satisfy input constraints.

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arXiv2026

Backup Control Barrier Function Synthesis using Sum-of-Squares Reachability

Backup control barrier functions (bCBFs) enforce safety for input-constrained nonlinear systems using a pre-certified backup set and controller, but their performance depends strongly on this prescribed pair. This letter develops a constructive method for synthesizing a less conservative backup pair via finite horizon sum-of-squares (SOS) backward reachability. Starting from an initial backup set, we compute an SOS-certified finite horizon backward reachable set and controller that satisfy safety and input constraints while steering trajectories to the original backup set. We then provide conditions under which this certified set becomes a valid backup set for a piecewise backup controller. The resulting backup pair is integrated into the bCBF framework to certify larger safe sets.

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arXiv2026

Robust Adaptive Backup Control Barrier Functions

We propose a notion of robust adaptive backup control barrier functions for nonlinear control affine systems with parametric uncertainty in both the drift dynamics and actuation matrix. Backup control barrier functions guarantee safety by predicting the system's trajectory under a pre-certified safe controller. However, these predictions rely on the model and can be inaccurate when the system contains unknown parameters. To address this issue, we estimate the unknown parameters using element-wise certified adaptive estimators that provide a parameter adaptation law and component-wise estimation error bounds. We compute the backup flow using the estimated model and tighten the safety conditions using these certified bounds. The resulting safety conditions account for the sensitivity of the predicted flow to parameter estimation errors. Moreover, to handle uncertainty in the actuation matrix, we use a duality-based reformulation that enables the use of a computationally efficient quadratic-program-based safety filter. We prove that controllers satisfying the proposed robust adaptive backup control barrier function constraints guarantee safety under parametric uncertainty and input constraints.

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arXiv2026

Output Feedback Backup Control Barrier Functions: Safety Guarantees Under Input Bounds and State Estimation Error

Guaranteeing the safety of controllers is vital for real-world applications, but is markedly difficult when the states are not perfectly known and when the control inputs are bounded. Backup control barrier functions (bCBFs) use predictions of the flow under a prescribed controller to achieve safety in the presence of bounded inputs and perfect state information. However, when only an estimate of the true state is known, this flow may not be precisely computed, as the initial condition is unknown. Furthermore, the true flow evolves using feedback from the estimated state, thus introducing coupling between known and unknown flows. To address these challenges, we propose a technique that leverages an uncertainty envelope centered around the estimated flow and show that ensuring the safety of this envelope guarantees that the true state satisfies the safety constraints. Additionally, we show that in the presence of state uncertainty, using the resulting Output Feedback Backup Control Barrier Functions (O-bCBFs), there always exists a feasible control input that can guarantee the safety of the true state, even in the presence of input constraints.

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arXiv2026

Probabilistic Control Barrier Functions for Systems with State Estimation Uncertainty using Sub-Gaussian Concentration

Safety-critical control systems, such as spacecraft performing proximity operations, must provide formal safety guarantees despite stochastic uncertainties from state estimation and unmodeled dynamics. Although Control Barrier Functions (CBFs) have been extended to stochastic systems, existing approaches typically face a trade-off between the tightness of probabilistic guarantees and computational tractability. This paper presents a particle-based probabilistic CBF framework that overcomes this limitation by exploiting the sub-Gaussian structure of the barrier function increment under Gaussian uncertainties. We establish that Gaussian uncertainties propagating through Lipschitz-continuous control-affine dynamics preserve sub-Gaussianity of the barrier function increment, with explicit tail bounds. Leveraging this structure, we derive finite-sample bounds on the approximation error between particle-based Conditional Value at Risk (CVaR) estimates and ground-truth probabilistic constraints; applying this yields a tractable optimization problem formulation with finite-sample safety certificates. We show through numerical experiments how the proposed approach provides tight yet provably valid probabilistic safety guarantees.

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arXiv2026

SafeSpace: Aggregating Safe Sets from Backup Control Barrier Functions under Input Constraints

Control barrier functions (CBFs) provide a principled framework for enforcing safety in control systems -- yet the certified safe operating region in practice is often conservative, especially under input bounds. In many applications, multiple smaller safe sets can be certified independently, e.g., around distinct equilibria with different stabilizing controllers. This paper proposes a framework for uniting such regions into a single certified safe set using \emph{combinatorial CBFs}. We refine the combinatorial CBF framework by introducing an auxiliary variable that enables logical compositions of individual CBFs. In the proposed framework, we show that such compositions yield a \emph{generalized combinatorial CBF} under a condition termed \emph{conjunctive compatibility}. Building on this result, we extend the framework to enable the aggregation of multiple implicit safe sets generated by the backup CBF framework. We show that the resulting CBF-based quadratic program yields a continuous safety filter over the aggregated safe region. The approach is demonstrated on two spacecraft safety problems, safe attitude control and safe station keeping, where multiple certified safe regions are combined to expand the operational envelope.

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arXiv2026

Generalizations of Backup Control Barrier Functions: Expansion and Adaptation for Input-Bounded Safety-Critical Control

Guaranteeing the safety of nonlinear systems with bounded inputs remains a key challenge in safe autonomy. Backup control barrier functions (bCBFs) provide a powerful mechanism for constructing controlled invariant sets by propagating trajectories under a pre-verified backup controller to a forward invariant backup set. While effective, the standard bCBF method utilizes the same backup controller for both set expansion and safety certification, which can restrict the expanded safe set and lead to conservative dynamic behavior. In this study, we generalize the bCBF framework by separating the set-expanding controller from the verified backup controller, thereby enabling a broader class of expansion strategies while preserving formal safety guarantees. We establish sufficient conditions for forward invariance of the resulting implicit safe set and show how the generalized construction recovers existing bCBF methods as special cases. Moreover, we extend the proposed framework to parameterized controller families, enabling online adaptation of the expansion controller while maintaining safety guarantees in the presence of input bounds.

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arXiv2025

Safety-Critical Control with Bounded Inputs: A Closed-Form Solution for Backup Control Barrier Functions

Verifying the safety of controllers is critical for many applications, but is especially challenging for systems with bounded inputs. Backup control barrier functions (bCBFs) offer a structured approach to synthesizing safe controllers that are guaranteed to satisfy input bounds by leveraging the knowledge of a backup controller. While powerful, bCBFs require solving a high-dimensional quadratic program at run-time, which may be too costly for computationally-constrained systems such as aerospace vehicles. We propose an approach that optimally interpolates between a nominal controller and the backup controller, and we derive the solution to this optimization problem in closed form. We prove that this closed-form controller is guaranteed to be safe while obeying input bounds. We demonstrate the effectiveness of the approach on a double integrator and a nonlinear fixed-wing aircraft example.

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arXiv2024

Disturbance-Robust Backup Control Barrier Functions: Safety Under Uncertain Dynamics

Obtaining a controlled invariant set is crucial for safety-critical control with control barrier functions (CBFs) but is non-trivial for complex nonlinear systems and constraints. Backup control barrier functions allow such sets to be constructed online in a computationally tractable manner by examining the evolution (or flow) of the system under a known backup control law. However, for systems with unmodeled disturbances, this flow cannot be directly computed, making the current methods inadequate for assuring safety in these scenarios. To address this gap, we leverage bounds on the nominal and disturbed flow to compute a forward invariant set online by ensuring safety of an expanding norm ball tube centered around the nominal system evolution. We prove that this set results in robust control constraints which guarantee safety of the disturbed system via our Disturbance-Robust Backup Control Barrier Function (DR-bCBF) solution. The efficacy of the proposed framework is demonstrated in simulation, applied to a double integrator problem and a rigid body spacecraft rotation problem with rate constraints.

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