Control Barrier Functions (CBFs) are a popular approach for safe control of nonlinear systems. In CBF-based control, the desired safety properties of the system are mapped to nonnegativity of a CBF, and the control input is chosen to ensure that the CBF remains nonnegative for all time. Recently, machine learning methods that represent CBFs as neural networks (neural control barrier functions, or NCBFs) have shown great promise due to the universal representability of neural networks. However, verifying that a learned CBF guarantees safety remains a challenging research problem. This paper presents novel exact conditions and algorithms for verifying safety of feedforward NCBFs with ReLU activation functions.

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) are a popular tool for safety certification of nonlinear dynamical control systems. Recently, CBFs represented as neural networks have shown great promise due to their expressiveness and applicability to a broad class of dynamics and safety constraints. However, verifying that a trained neural network is indeed a valid CBF is a computational bottleneck that limits the size of the networks that can be used. To overcome this limitation, we present a novel framework for verifying neural CBFs based on piecewise linear upper and lower bounds on the conditions required for a neural network to be a CBF. Our approach is rooted in linear bound propagation (LBP) for neural networks, which we extend to compute bounds on the gradients of the network. Combined with McCormick relaxation, we derive linear upper and lower bounds on the CBF conditions, thereby eliminating the need for computationally expensive verification procedures. Our approach applies to arbitrary control-affine systems and a broad range of nonlinear activation functions. To reduce conservatism, we develop a parallelizable refinement strategy that adaptively refines the regions over which these bounds are computed. Our approach scales to larger neural networks than state-of-the-art verification procedures for CBFs, as demonstrated by our numerical experiments.

Control barrier functions (CBFs) have been demonstrated as an effective method for safety-critical control of autonomous systems. Although CBFs are simple to deploy, their design remains challenging, motivating the development of learning-based approaches. Yet, issues such as suboptimal safe sets, applicability in partially observable environments, and lack of rigorous safety guarantees persist. In this work, we propose observation-conditioned neural CBFs based on Hamilton-Jacobi (HJ) reachability analysis, which approximately recover the maximal safe sets. We exploit certain mathematical properties of the HJ value function, ensuring that the predicted safe set never intersects with the observed failure set. Moreover, we leverage a hypernetwork-based architecture that is particularly suitable for the design of observation-conditioned safety filters. The proposed method is examined both in simulation and hardware experiments for a ground robot and a quadcopter. The results show improved success rates and generalization to out-of-domain environments compared to the baselines.

-- 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.

The synthesis of Control Barrier Functions (CBFs) often involves demanding computations or a meticulous construction. However, structural properties of the system dynamics and constraints have the potential to mitigate these challenges. In this paper, we explore how equivariances in the dynamics, loosely speaking a form of symmetry, can be leveraged in the CBF synthesis. Although CBFs are generally not inherently symmetric, we show how equivariances in the dynamics and symmetries in the constraints induce symmetries in CBFs derived through reachability analysis. This insight allows us to infer their CBF values across the entire domain from their values on a subset, leading to significant computational savings. Interestingly, equivariances can be even leveraged to the CBF synthesis for non-symmetric constraints. Specifically, we show how a partially known CBF can be leveraged together with equivariances to construct a CBF for various new constraints. Throughout the paper, we provide examples illustrating the theoretical findings. Furthermore, a numerical study investigates the computational gains from invoking equivariances into the CBF synthesis.

-- Safe navigation in unknown and cluttered environments remains a challenging problem in robotics. Model Predictive Contour Control (MPCC) has shown promise for performant obstacle avoidance by enabling precise and agile trajectory tracking, however, existing methods lack formal safety assurances. T o address this issue, we propose a general Control Lyapunov Function (CLF) and Control Barrier Function (CBF) enabled MPCC framework that enforces safety constraints derived from a free-space corridor around the planned trajectory. T o enhance feasibility, we dynamically adapt the CBF parameters at runtime using a Soft Actor-Critic (SAC) policy. The approach is validated with extensive simulations and an experiment on mobile robot navigation in unknown cluttered environments.

We present a novel method for designing higher-order Control Barrier Functions (CBFs) that guarantee convergence to a safe set within a user-specified finite. Traditional Higher Order CBFs (HOCBFs) ensure asymptotic safety but lack mechanisms for fixed-time convergence, which is critical in time-sensitive and safety-critical applications such as autonomous navigation. In contrast, our approach imposes a structured differential constraint using repeated roots in the characteristic polynomial, enabling closed-form polynomial solutions with exact convergence at a prescribed time. We derive conditions on the barrier function and its derivatives that ensure forward invariance and fixed-time reachability, and we provide an explicit formulation for second-order systems. Our method is evaluated on three robotic systems - a point-mass model, a unicycle, and a bicycle model and benchmarked against existing HOCBF approaches. Results demonstrate that our formulation reliably enforces convergence within the desired time, even when traditional methods fail. This work provides a tractable and robust framework for real-time control with provable finite-time safety guarantees.

We propose a novel cooperative herding strategy through backstepping control barrier functions (CBFs), which coordinates multiple herders to herd a group of evaders safely towards a designated goal region. For the herding system with heterogeneous groups involving herders and evaders, the behavior of the evaders can only be influenced indirectly by the herders' motion, especially when the evaders follow an inverse dynamics model and respond solely to repulsive interactions from the herders. This indirect interaction mechanism inherently renders the overall system underactuated. To address this issue, we first construct separate CBFs for the dual objectives of goal reaching and collision avoidance, which ensure both herding completion and safety guarantees. Then, we reformulate the underactuated herding dynamics into a control-affine structure and employ a backstepping approach to recursively design control inputs for the hierarchical barrier functions, avoiding taking derivatives of the higher-order system. Finally, we present a cooperative herding strategy based on backstepping CBFs that allow herders to safely herd multiple evaders into the goal region. In addition, centralized and decentralized implementations of the proposed algorithm are developed, further enhancing its flexibility and applicability. Extensive simulations and real-world experiments validate the effectiveness and safety of the proposed strategy in multi-robot herding.

Safety stands as the primary obstacle preventing the widespread adoption of learning-based robotic systems in our daily lives. While reinforcement learning (RL) shows promise as an effective robot learning paradigm, conventional RL frameworks often model safety by using single scalar negative rewards with immediate episode termination, failing to capture the temporal consequences of unsafe actions (e.g., sustained collision damage). In this work, we introduce a novel approach that simulates these temporal effects by applying continuous negative rewards without episode termination. Our experiments reveal that standard RL methods struggle with this model, as the accumulated negative values in unsafe zones create learning barriers. To address this challenge, we demonstrate how Control Barrier Functions (CBFs), with their proven safety guarantees, effectively help robots avoid catastrophic regions while enhancing learning outcomes. We present three CBF-based approaches, each integrating traditional RL methods with Control Barrier Functions, guiding the agent to learn safe behavior. Our empirical analysis, conducted in both simulated environments and real-world settings using a four-wheel differential drive robot, explores the possibilities of employing these approaches for safe robotic learning.