Control Barrier Functions

Shared autonomy blends operator intent with autonomous assistance. In cluttered environments, linear blending can produce unsafe commands even when each source is individually collision-free. Many existing approaches model obstacle avoidance through potentials or cost terms, which only enforce safety as a soft constraint. In contrast, safety-critical control requires hard guarantees. We investigate the use of control barrier functions (CBFs) at the inverse kinematics (IK) layer of shared autonomy, targeting post-blend safety while preserving task performance. Our approach is evaluated in simulation on representative cluttered environments and in a VR teleoperation study comparing pure teleoperation with shared autonomy. Across conditions, employing CBFs at the IK layer reduces violation time and increases minimum clearance while maintaining task performance. In the user study, participants reported higher perceived safety and trust, lower interference, and an overall preference for shared autonomy with our safety filter. Additional materials available at https://berkguler.github.io/barrierik.

This paper presents a general end-to-end framework for constructing robust and reliable layered safety filters that can be leveraged to perform dynamic collision avoidance over a broad range of applications using only local perception data. Given a robot-centric point cloud, we begin by constructing an occupancy map which is used to synthesize a Poisson safety function (PSF). The resultant PSF is employed as a control barrier function (CBF) within two distinct safety filtering stages. In the first stage, we propose a predictive safety filter to compute optimal safe trajectories based on nominal potentially-unsafe commands. The resultant short-term plans are constrained to satisfy the CBF condition along a finite prediction horizon. In the second stage, instantaneous velocity commands are further refined by a real-time CBF-based safety filter and tracked by the full-order low-level robot controller. Assuming accurate tracking of velocity commands, we obtain formal guarantees of safety for the full-order system. We validate the optimality and robustness of our multistage architecture, in comparison to traditional single-stage safety filters, via a detailed Pareto analysis. We further demonstrate the effectiveness and generality of our collision avoidance methodology on multiple legged robot platforms across a variety of real-world dynamic scenarios.

In UAV dynamic decision, complex and variable hazardous factors pose severe challenges to the generalization capability of algorithms. Despite offering semantic understanding and scene generalization, Large Language Models (LLM) lack domain-specific UAV control knowledge and formal safety assurances, restricting their direct applicability. To bridge this gap, this paper proposes a train-free two-layer decision architecture based on LLMs, integrating high-level safety planning with low-level precise control. The framework introduces three key contributions: 1) A fuzzy Control Barrier Function verification mechanism for semantically-augmented actions, providing provable safety certification for LLM outputs. 2) A star-hierarchical graph-based retrieval-augmented generation system, enabling efficient, elastic, and interpretable scene adaptation. 3) Systematic experimental validation in pursuit-evasion scenarios with unknown obstacles and emergent threats, demonstrating that our SAGE-LLM maintains performance while significantly enhancing safety and generalization without online training. The proposed framework demonstrates strong extensibility, suggesting its potential for generalization to broader embodied intelligence systems and safety-critical control domains.

Control Barrier Functions (CBFs) are a powerful tool for ensuring robotic safety, but designing or learning valid CBFs for complex systems is a significant challenge. While Hamilton-Jacobi Reachability provides a formal method for synthesizing safe value functions, it scales poorly and is typically performed offline, limiting its applicability in dynamic environments. This paper bridges the gap between offline synthesis and online adaptation. We introduce refineCBF for refining an approximate CBF - whether analytically derived, learned, or even unsafe - via warm-started HJ reachability. We then present its computationally efficient successor, HJ-Patch, which accelerates this process through localized updates. Both methods guarantee the recovery of a safe value function and can ensure monotonic safety improvements during adaptation. Our experiments validate our framework's primary contribution: in-the-loop, real-time adaptation, in simulation (with detailed value function analysis) and on physical hardware. Our experiments on ground vehicles and quadcopters show that our framework can successfully adapt to sudden environmental changes, such as new obstacles and unmodeled wind disturbances, providing a practical path toward deploying formally guaranteed safety in real-world settings.

Despite the strong performance of large language models (LLMs) across diverse tasks, their susceptibility to adversarial attacks and unsafe content generation remains a significant obstacle to deployment, particularly in high-stakes settings. Addressing this challenge requires safety mechanisms that are both practically effective and theoretically grounded. In this paper, we introduce BarrierSteer, a novel inference-time framework that improves response safety by embedding learned nonlinear safety constraints directly into the model's latent representation space. BarrierSteer treats hidden-state safety classifiers as Control Barrier Functions (CBFs), enabling constraint-guided steering of unsafe latent trajectories during generation. By composing multiple safety constraints through efficient constraint merging without modifying the underlying LLM parameters, BarrierSteer preserves model utility. We provide theoretical results showing that applying CBFs in the latent space yields a principled, modular, and computationally efficient approach for steering with respect to learned safety constraints, with guarantees conditional on the learned barriers capturing the intended safety property. Our extensive experimental results across multiple model families and datasets demonstrate that BarrierSteer substantially reduces adversarial attack success rates and unsafe generations, outperforming the existing method. The code is available in our GitHub repository.

Ensuring safe robot operation in cluttered and dynamic environments remains a fundamental challenge. While control barrier functions (CBFs) provide an effective framework for real-time safety filtering, their performance critically depends on the underlying geometric representation, which is often simplified, leading to either overly conservative behavior or insufficient collision coverage. Superquadrics (SQs) offer an expressive way to model complex shapes using a few primitives and are increasingly used for robot safety. To integrate this representation into CBF, most existing approaches directly use their implicit functions as barrier candidates. However, we identify a critical but overlooked issue in this practice: the gradients of the implicit SQ function can become severely ill-conditioned, potentially rendering the optimization infeasible and undermining reliable safety filtering. To address this issue, we formulate an SQ-based safety filter that uses signed distance functions (SDFs) as barrier candidates. Since analytical SDFs are unavailable for general SQs, we compute the distance using the efficient Gilbert-Johnson-Keerthi algorithm and derive its gradient in closed form. Extensive simulation and real-world experiments demonstrate consistent collision-free manipulation in cluttered scenes, showing robustness to challenging geometries, sensing noise, and dynamic disturbances, while improving task efficiency in teleoperation tasks. These results highlight a pathway toward safety filters that remain precise and reliable under the geometric complexity of real-world environments.

Latent safety filters extend Hamilton-Jacobi (HJ) reachability to operate on latent state representations and dynamics learned directly from high-dimensional observations, enabling safe visuomotor control under hard-to-model constraints. However, existing methods implement "least-restrictive" filtering that discretely switch between nominal and safety policies, potentially undermining the task performance that makes modern visuomotor policies valuable. While reachability value functions can, in principle, be adapted to be control barrier functions (CBFs) for smooth optimization-based filtering, we theoretically and empirically show that current latent-space learning methods produce fundamentally incompatible value functions. We identify two sources of incompatibility: First, in HJ reachability, failures are encoded via a "margin function" in latent space, whose sign indicates whether or not a latent is in the constraint set. However, representing the margin function as a classifier yields saturated value functions that exhibit discontinuous jumps. We prove that the value function's Lipschitz constant scales linearly with the margin function's Lipschitz constant, revealing that smooth CBFs require smooth margins. Second, reinforcement learning (RL) approximations trained solely on safety policy data yield inaccurate value estimates for nominal policy actions, precisely where CBF filtering needs them. We propose the LatentCBF, which addresses both challenges through gradient penalties that lead to smooth margin functions without additional labeling, and a value-training procedure that mixes data from both nominal and safety policy distributions. Experiments on simulated benchmarks and hardware with a vision-based manipulation policy demonstrate that LatentCBF enables smooth safety filtering while doubling the task-completion rate over prior switching methods.

Control Barrier Functions (CBFs) can provide provable safety guarantees for dynamic systems. However, finding a valid CBF for a system of interest is often non-trivial, especially for systems with low computational resources, higher-order dynamics, and moving close to obstacles of complex shape. A common solution to this problem is to use a purely distance-based CBF. In this paper, we study Hyperplane CBFs (H-CBFs), where a hyperplane separates the agent from the obstacle. First, we note that the common distance-based CBF is a special case of an H-CBF where the hyperplane is a supporting hyperplane of the obstacle that is orthogonal to a line between the agent and the closest point of the obstacle. We then show that a less conservative CBF can be found by optimising over the orientation of the supporting hyperplane, in order to find the Least Restrictive Hyperplane CBF. This enables us to maintain the safety guarantees while allowing controls that are closer to the desired ones, especially when moving fast and passing close to obstacles. We illustrate the approach on a double integrator dynamical system with acceleration constraints, moving through a group of arbitrarily shaped static and moving obstacles, and show that the proposed approach reduces the average gap between desired and safe controls by an order of magnitude.

Reinforcement learning (RL), while powerful and expressive, can often prioritize performance at the expense of safety. Yet safety violations can lead to catastrophic outcomes in real-world deployments. Control Barrier Functions (CBFs) offer a principled method to enforce dynamic safety -- traditionally deployed online via safety filters. While the result is safe behavior, the fact that the RL policy does not have knowledge of the CBF can lead to conservative behaviors. This paper proposes CBF-RL, a framework for generating safe behaviors with RL by enforcing CBFs in training. CBF-RL has two key attributes: (1) minimally modifying a nominal RL policy to encode safety constraints via a CBF term, (2) and safety filtering of the policy rollouts in training. Theoretically, we prove that continuous-time safety filters can be deployed via closed-form expressions on discrete-time roll-outs. Practically, we demonstrate that CBF-RL internalizes the safety constraints in the learned policy -- both enforcing safer actions and biasing towards safer rewards -- enabling safe deployment without the need for an online safety filter. We validate our framework through ablation studies on navigation tasks and on the Unitree G1 humanoid robot, where CBF-RL enables safer exploration, faster convergence, and robust performance under uncertainty, enabling the humanoid robot to avoid obstacles and climb stairs safely in real-world settings without a runtime safety filter.

Control Barrier Functions (CBFs) are utilized to ensure the safety of control systems. CBFs act as safety filters in order to provide safety guarantees without compromising system performance. These safety guarantees rely on the construction of valid CBFs. Due to their complexity, CBFs can be represented by neural networks, known as neural CBFs (NCBFs). Existing works on the verification of the NCBF focus on the synthesis and verification of NCBFs in deterministic settings, leaving the stochastic NCBFs (SNCBFs) less studied. In this work, we propose a verifiably safe synthesis for SNCBFs. We consider the cases of smooth SNCBFs with twice-differentiable activation functions and SNCBFs that utilize the Rectified Linear Unit or ReLU activation function. We propose a verification-free synthesis framework for smooth SNCBFs and a verification-in-the-loop synthesis framework for both smooth and ReLU SNCBFs. and we validate our frameworks in three cases, namely, the inverted pendulum, Darboux, and the unicycle model.

Model Predictive Path Integral (MPPI) control provides a sampling-based framework for optimal control, while Control Barrier Functions (CBFs) provide a principled means of enforcing safety constraints. We introduce BR-MPPI, which integrates CBF-like conditions into MPPI's control sampling procedure. CBFs impose inequality constraints that bound the rate of change of barrier functions using a class-K function of the barrier value. We instead impose the CBF condition as an equality constraint using a parametric linear class-K function and augment the system state with its parameter. The parameter's time derivative serves as an additional control input optimized by MPPI. We further design a cost function that promotes parameter values consistent with Nagumo's condition at the safe-set boundary, thereby encouraging safety. The resulting multiple state- and control-dependent equality constraints pose a challenge for random control sampling. We address this through state transformations and control projections inspired by manifold path planning that map sampled controls onto the constraint manifold. We also incorporate learned signed distance fields to represent robot geometry and reduce computation time. Simulations demonstrate improved sample efficiency over vanilla MPPI and higher navigation success rates across five robot models compared with safety-oriented MPPI variants. Hardware experiments on a quadrotor further demonstrate the method's ability to navigate constrained environments near safe-set boundaries.

Robot learning has produced remarkably effective ``black-box'' controllers for complex tasks such as dynamic locomotion on humanoids. Yet ensuring dynamic safety, i.e., constraint satisfaction, remains challenging for such policies. Reinforcement learning (RL) embeds constraints heuristically through reward engineering, and adding or modifying constraints requires retraining. Model-based approaches, like control barrier functions (CBFs), enable runtime constraint specification with formal guarantees but require accurate dynamics models. This paper presents SHIELD, a layered safety framework that bridges this gap by: (1) training a generative, stochastic dynamics residual model using real-world data from hardware rollouts of the nominal controller, capturing system behavior and uncertainties; and (2) adding a safety layer on top of the nominal (learned locomotion) controller that leverages this model via a stochastic discrete-time CBF formulation enforcing safety constraints in probability. The result is a minimally-invasive safety layer that can be added to the existing autonomy stack to give probabilistic guarantees of safety that balance risk and performance. In hardware experiments on an Unitree G1 humanoid, SHIELD enables safe navigation (obstacle avoidance) through varied indoor and outdoor environments using a nominal (unknown) RL controller and onboard perception.

Control barrier functions (CBFs) provide a principled framework for safety-critical control, but their construction typically requires an explicit and differentiable description of the safe or unsafe region. It becomes challenging for data-defined unsafe regions that may evolve over time. This paper proposes SafeLink, a data-driven CBF construction and adaptation method based on a cost-sensitive random vector functional link (RVFL) network. SafeLink introduces asymmetric misclassification costs to promote conservative unsafe-region representation while preserving a closed-form solution. We establish the Lipschitz continuity of the learned CBF and its derivatives, and derive sufficient conditions for conservative unsafe-region coverage and the corresponding interval-wise safety guarantees. Analytical updates are further developed for adjusting the misclassification cost and for incrementally adding or decrementally removing samples, avoiding full retraining when the unsafe region changes. Experiments on a two-link manipulator demonstrate that SafeLink rapidly adapts to evolving unsafe regions, enables collision-free target reaching, and achieves substantially lower update runtimes than baselines.

Control Barrier Function Quadratic Programs (CBF-QPs) provide formal safety guarantees for general control-affine systems and accommodate actuation constraints, but do not inherently guarantee liveness, i.e., sustained progress toward and convergence to a goal. They may admit undesirable equilibria or dysfunctional circulation that impede goal convergence. In contrast, Modulation of Dynamical Systems (Mod-DS) promotes liveness through geometric reshaping of nominal vector fields, but is primarily suited to fully actuated systems and does not directly accommodate general actuation constraints. We establish theoretical connections between these paradigms, showing that normal Mod-DS is a special case of CBF-QP and that reference Mod-DS is linked to CBF-QP through a shared formulation. These connections motivate the Modulated CBF-QP (MCBF-QP) framework, which incorporates geometric guidance from Mod-DS into CBF-QPs to enhance liveness while retaining formal safety and control-affine generality. We develop reference MCBF-QP, which reduces undesirable equilibria and dysfunctional circulation for star-shaped obstacles, and on-manifold MCBF-QP, which eliminates undesirable equilibria and reduces dysfunctional circulation for arbitrarily shaped concave obstacles. Although no Lyapunov function is explicitly constructed, we prove that the tangent guidance constraint introduced into the CBF-QP formulation implicitly enforces the decrease condition of a hybrid Lyapunov function, yielding asymptotic goal convergence and thus liveness under the stated assumptions. Extensive simulation and hardware experiments demonstrate safe and live obstacle avoidance with the proposed MCBF-QP framework in static and dynamic environments.

Merely pursuing performance may adversely affect safety, while a conservative policy for safe exploration will degrade the performance. How to guarantee both safety and performance in learning-based control problems is an interesting yet challenging issue. This paper aims to enhance system performance with a safety guarantee by solving reinforcement learning (RL)-based optimal control problems for nonlinear systems subject to high-relative-degree state constraints and unknown time-varying disturbance/actuator faults. A new type of control barrier functions (CBFs), termed high-order reciprocal-based control barrier function, is proposed to handle high-relative-degree constraints, which extends the design of CBFs to enforce robust safety without knowing the disturbance bound. The concept of gradient similarity is proposed to quantify the relationship between safety and performance. Finally, gradient manipulation and adaptive mechanisms are introduced in the model-based safe RL framework to enhance the performance with a safety guarantee. Two simulation examples illustrate the efficacy of the proposed algorithms.

Safe control in unknown environments is a key challenge in mobile robotics. Control Barrier Functions (CBFs) provide a principled framework for guaranteeing safety constraint satisfaction. State-of-the-art CBF approaches assume either known environments with predefined obstacles, or rely only on obstacles currently within the robot's Field of View (FoV). However, practical robots in a priori unknown environments can observe their surroundings only partially, and therefore can violate safety due to limited FoV, sensor range, or occlusion. This paper incorporates the memory of previously observed obstacles of arbitrary shape that have left the robot's FoV into CBF-based safe control. In particular, we couple the Signed Distance Function (SDF)-based CBF formulation to an occupancy grid map built online during the system's operation. Furthermore, the lack of steering authority induced by the SDF gradient degeneracy when facing obstacles head-on is addressed by constructing a Gaussian pyramid of the SDF, yielding a multi-level CBF. The efficacy of the proposed approach is evaluated against memory unaware baselines in the CARLA simulator. Moreover, we demonstrate the generalizability of the proposed approach in real deployments on a small warehouse robot and a large, articulated frame steering autonomous wheel loader.