In this paper, we present a novel theoretical framework for online adaptation of Control Barrier Function (CBF) parameters, i.e., of the class K functions included in the CBF condition, under input constraints. We introduce the concept of locally validated CBF parameters, which are adapted online to guarantee finite-horizon safety, based on conditions derived from Nagumo's theorem and tangent cone analysis. To identify these parameters online, we integrate a learning-based approach with an uncertainty-aware verification process that account for both epistemic and aleatoric uncertainties inherent in neural network predictions. Our method is demonstrated on a VTOL quadplane model during challenging transition and landing maneuvers, showcasing enhanced performance while maintaining safety.

Control barrier functions (CBFs) have seen widespread success in providing forward invariance and safety guarantees for dynamical control systems. A crucial limitation of discrete-time formulations is that CBFs that are nonconcave in their argument require the solution of nonconvex optimization problems to compute safety-preserving control inputs, which inhibits real-time computation of control inputs guaranteeing forward invariance. This paper presents a novel method for computing safety-preserving control inputs for discrete-time systems with nonconvex safety sets, utilizing convex optimization and the recently developed class of matrix control barrier function techniques. The efficacy of our methods is demonstrated through numerical simulations on a bicopter system.

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.

Safety filters, particularly those based on control barrier functions, have gained increased interest as effective tools for safe control of dynamical systems. Existing correct-by-construction synthesis algorithms for such filters, however, suffer from the curse-of-dimensionality. Deep learning approaches have been proposed in recent years to address this challenge. In this paper, we add to this set of approaches an algorithm for training neural control barrier functions from offline datasets. Such functions can be used to design constraints for quadratic programs that are then used as safety filters. Our algorithm trains these functions so that the system is not only prevented from reaching unsafe states but is also disincentivized from reaching out-of-distribution ones, at which they would be less reliable. It is inspired by Conservative Q-learning, an offline reinforcement learning algorithm. We call its outputs Conservative Control Barrier Functions (CCBFs). Our empirical results demonstrate that CCBFs outperform existing methods in maintaining safety while minimally affecting task performance. Source code is available at https://github.com/tabz23/CCBF.

-- Control barrier functions (CBFs) are a powerful tool for the constrained control of nonlinear systems; however, the majority of results in the literature focus on systems subject to a single CBF constraint, making it challenging to synthesize provably safe controllers that handle multiple state constraints. This paper presents a framework for constrained control of nonlinear systems subject to box constraints on the systems' vector-valued outputs using multiple CBFs. Our results illustrate that when the output has a vector relative degree, the CBF constraints encoding these box constraints are compatible, and the resulting optimization-based controller is locally Lipschitz continuous and admits a closed-form expression. Additional results are presented to characterize the degradation of nominal tracking objectives in the presence of safety constraints. Simulations of a planar quadrotor are presented to demonstrate the efficacy of the proposed framework.

Synthesizing safe sets for robotic systems operating in complex and dynamically changing environments is a challenging problem. Solving this problem can enable the construction of safety filters that guarantee safe control actions -- most notably by employing Control Barrier Functions (CBFs). This paper presents an algorithm for generating safe sets from perception data by leveraging elliptic partial differential equations, specifically Poisson's equation. Given a local occupancy map, we solve Poisson's equation subject to Dirichlet boundary conditions, with a novel forcing function. Specifically, we design a smooth guidance vector field, which encodes gradient information required for safety. The result is a variational problem for which the unique minimizer -- a safety function -- characterizes the safe set. After establishing our theoretical result, we illustrate how safety functions can be used in CBF-based safety filtering. The real-time utility of our synthesis method is highlighted through hardware demonstrations on quadruped and humanoid robots navigating dynamically changing obstacle-filled environments.

Contraction metrics are crucial in control theory because they provide a powerful framework for analyzing stability, robustness, and convergence of various dynamical systems. However, identifying these metrics for complex nonlinear systems remains an open challenge due to the lack of scalable and effective tools. This paper explores the approach of learning verifiable contraction metrics parametrized as neural networks (NNs) for discrete-time nonlinear dynamical systems. While prior works on formal verification of contraction metrics for general nonlinear systems have focused on convex optimization methods (e.g. linear matrix inequalities, etc) under the assumption of continuously differentiable dynamics, the growing prevalence of NN-based controllers, often utilizing ReLU activations, introduces challenges due to the non-smooth nature of the resulting closed-loop dynamics. To bridge this gap, we establish a new sufficient condition for establishing formal neural contraction metrics for general discrete-time nonlinear systems assuming only the continuity of the dynamics. We show that from a computational perspective, our sufficient condition can be efficiently verified using the state-of-the-art neural network verifier $α,\!β$-CROWN, which scales up non-convex neural network verification via novel integration of symbolic linear bound propagation and branch-and-bound. Built upon our analysis tool, we further develop a learning method for synthesizing neural contraction metrics from sampled data. Finally, our approach is validated through the successful synthesis and verification of NN contraction metrics for various nonlinear examples.

-- Certifying safety in dynamical systems is crucial, but barrier certificates -- widely used to verify that system trajectories remain within a safe region -- typically require explicit system models. When dynamics are unknown, data-driven methods can be used instead, yet obtaining a valid certificate requires rigorous uncertainty quantification. For this purpose, existing methods usually rely on full-state measurements, limiting their applicability. This paper proposes a novel approach for synthesizing barrier certificates for unknown systems with latent states and polynomial dynamics. A Bayesian framework is employed, where a prior in state-space representation is updated using input-output data via a targeted marginal Metropolis-Hastings sampler . The resulting samples are used to construct a candidate barrier certificate through a sum-of-squares program. It is shown that if the candidate satisfies the required conditions on a test set of additional samples, it is also valid for the true, unknown system with high probability. The approach and its probabilistic guarantees are illustrated through a numerical simulation. Ensuring the safety of dynamical systems is a critical concern in applications such as human-robot interaction, autonomous driving, and medical devices, where failures can lead to severe consequences. In such scenarios, safety constraints typically mandate that the system state remains within a predefined allowable region. Barrier certificates [1] provide a systematic framework for verifying safety by establishing mathematical conditions that guarantee that system trajectories remain within these regions.

This work proposes a hybrid framework for car-like robots with obstacle avoidance, global convergence, and safety, where safety is interpreted as path invariance, namely, once the robot converges to the path, it never leaves the path. Given a priori obstacle-free feasible path where obstacles can be around the path, the task is to avoid obstacles while reaching the path and then staying on the path without leaving it. The problem is solved in two stages. Firstly, we define a ``tight'' obstacle-free neighborhood along the path and design a local controller to ensure convergence to the path and path invariance. The control barrier function technology is involved in the control design to steer the system away from its singularity points, where the local path invariant controller is not defined. Secondly, we design a hybrid control framework that integrates this local path-invariant controller with any global tracking controller from the existing literature without path invariance guarantee, ensuring convergence from any position to the desired path, namely, global convergence. This framework guarantees path invariance and robustness to sensor noise. Detailed simulation results affirm the effectiveness of the proposed scheme.

Merely pursuing performance may adversely affect the safety, while a conservative policy for safe exploration will degrade the performance. How to balance the safety and performance in learning-based control problems is an interesting yet challenging issue. This paper aims to enhance system performance with safety guarantee in solving the reinforcement learning (RL)-based optimal control problems of nonlinear systems subject to high-relative-degree state constraints and unknown time-varying disturbance/actuator faults. First, to combine control barrier functions (CBFs) with RL, a new type of CBFs, termed high-order reciprocal control barrier function (HO-RCBF) is proposed to deal with high-relative-degree constraints during the learning process. Then, the concept of gradient similarity is proposed to quantify the relationship between the gradient of safety and the gradient of performance. Finally, gradient manipulation and adaptive mechanisms are introduced in the safe RL framework to enhance the performance with a safety guarantee. Two simulation examples illustrate that the proposed safe RL framework can address high-relative-degree constraint, enhance safety robustness and improve system performance.