control barrier functions

Designing high-order control barrier functions (CBFs) for complex nonlinear dynamical systems remains challenging, and conventional hyperplane-based approximations of unsafe regions often yield overly conservative control policies. To address this, we propose a unified optimization framework that jointly tunes CBF parameters and control inputs. Our key innovation is the “minimally restrictive hyperplane CBF,” which employs continuous parametrization and co-optimization to guarantee strict safety while maximizing control freedom. The method accommodates both static and dynamic obstacles and explicitly incorporates practical actuation constraints—such as acceleration limits. Evaluated on a double-integrator system, our approach significantly improves trajectory flexibility and obstacle avoidance robustness compared to baseline methods, achieving a superior trade-off between safety and performance.

Conventional control barrier functions (CBFs) suffer from spatially varying relative degrees in the state space, leading to regions of undefined control within the safe set—causing boundary chattering and safety violations. Method: This paper proposes a novel CBF synthesis paradigm based on boundary value problem (BVP) modeling and physics-informed neural network (PINN) solving. It formalizes CBF synthesis for the first time as a BVP enforcing constant relative degree and physical constraints, thereby eliminating control-undefined regions at their root; nonlinear control theory is integrated with PINN-based optimization to ensure strict first-order relative degree over the entire feasible domain. Results: Simulation and real-world quadrotor experiments demonstrate significant suppression of safety boundary chattering, rigorous invariance of the safe set, and an empirical constraint satisfaction rate of 99.7%.

Existing control barrier function (CBF) approaches for nonlinear systems often suffer from insufficient compatibility among multiple CBF constraints, hindering simultaneous enforcement of multiple safety objectives. Method: This paper proposes a unified multi-CBF synthesis framework based on vector relative degree. It rigorously characterizes compatibility conditions for multi-output CBF constraints, establishes—for the first time—the existence of locally Lipschitz continuous controllers satisfying all CBF constraints, and derives their analytical optimal solutions via an optimization formulation that explicitly trades off safety and tracking performance. Contribution/Results: The resulting controller provably ensures closed-loop stability while guaranteeing multi-state safety constraints. Theoretical analysis confirms the framework’s completeness and robustness. Extensive quadrotor simulations demonstrate its effectiveness in maintaining both safety and control accuracy under complex, nonconvex safety boundaries.

This work addresses the challenge of safely transferring safety guarantees between heterogeneous systems with mismatched dynamics by proposing a transfer Control Barrier Function (tCBF) framework. The approach systematically migrates safety constraints from a source system to a target system by integrating a simulation function with an explicit margin term, which compensates for model mismatch. Safety is enforced via a quadratic programming-based safety filter that minimally modifies the nominal control input. Notably, this method achieves cross-system safety certificate transfer without requiring assumptions on matching state dimensions or dynamical structures. The explicit margin ensures robustness against model discrepancies, thereby preserving safety in the target system. The efficacy of tCBF is demonstrated in a quadrotor obstacle avoidance task, where safety constraints are successfully transferred with negligible interference to the original controller, highlighting the framework’s generality and practical utility.

Ensuring safety for nonlinear affine systems under complex, uncertain disturbances remains challenging. Method: This paper proposes a robust safety controller that integrates uncertainty estimation with high-order control barrier functions (HOCBFs). It innovatively embeds bounds on estimation errors directly into the CBF constraints and extends the formulation to a second-order cone programming (SOCP) framework. The approach unifies elastic actuator modeling, HOCBF-based safety constraints, and quadratic-programming (QP) feedback control. Contribution/Results: It is the first method to provide rigorous robust safety guarantees against both matched and mismatched disturbances. Evaluations in simulation and on a tracked robot navigating inclined terrain demonstrate 100% safety constraint satisfaction, a 42% improvement in disturbance rejection, and significantly enhanced motion robustness and real-time safety under dynamic uncertainties.

Real-time computation of safety-preserving control inputs for discrete-time systems subject to nonconvex safe sets is challenging due to inherent nonconvexity in the underlying safety constraints. Method: This paper proposes a novel design framework integrating matrix control barrier functions (MCBFs) with convex optimization. We extend MCBFs—originally formulated for continuous-time systems—to discrete-time dynamics and construct an equivalent convex optimization problem via judicious convex relaxation, thereby circumventing direct solution of nonconvex programs while ensuring forward invariance of the safe set. Contribution/Results: The method unifies system dynamics, safety requirements, and convexification techniques to significantly improve computational efficiency and online implementability. Extensive simulations on a quadrotor platform demonstrate the approach’s superiority in safety enforcement, state convergence, and real-time performance. This work establishes a new paradigm for safety-critical control under nonconvex safety constraints.

Real-time safety-critical control of input-constrained systems—particularly for resource-limited aerospace platforms—remains challenging due to the high computational cost of conventional backup Control Barrier Functions (bCBFs), which require online solution of high-dimensional quadratic programs. Method: This paper proposes a closed-loop solvable bCBF framework. We derive, for the first time, an analytical closed-form solution to the bCBF optimization problem by optimally interpolating between a nominal controller and a backup controller, thereby jointly ensuring system safety and input boundedness without runtime numerical optimization. Contribution/Results: The proposed method drastically reduces computational complexity while rigorously enforcing nonlinear safety constraints. Experimental validation on a double-integrator system and a nonlinear fixed-wing aircraft model demonstrates both theoretical safety guarantees and real-time feasibility. This work provides a provably safe, lightweight control solution tailored for computationally constrained platforms.

This work addresses the myopic nature of traditional Control Barrier Functions (CBFs) and their difficulty in simultaneously satisfying safety requirements and control constraints. The authors propose Predictive Flow Control Barrier Functions (P-CBF), which extend safety verification over the entire predicted trajectory by integrating a terminal backup safe set with a planning time-shift mechanism to jointly optimize both the control policy parameters and real-time inputs. By employing an adjustable prediction horizon, the method enables end-to-end trajectory safety certification while unifying finite-horizon cost optimization with safety guarantees. Under convex polyhedral control constraints, the resulting problem reduces to a quadratic program (QP), amenable to efficient real-time solution. Experimental results on nonholonomic ground robots in dense navigation scenarios demonstrate that the proposed FlowBarrier approach achieves the highest goal-reaching success rate, zero safety violations, and the lowest computation time across 100 trials.

This work addresses the challenge of maintaining safety in complex dynamic environments where traditional control barrier functions (CBFs) struggle to adapt in real time to sudden obstacles or environmental disturbances. The authors propose an online optimization framework that integrates Hamilton-Jacobi (HJ) reachability analysis with a warm-start mechanism to locally and incrementally refine unsafe or approximate CBFs, thereby adaptively updating the safety value function. This approach achieves, for the first time, online local reconstruction of safety certificates grounded in HJ reachability, ensuring monotonic improvement of safety during adaptation while supporting seamless deployment from simulation to physical hardware. Experimental validation on ground vehicles and quadrotor drones demonstrates the framework’s effectiveness in handling unexpected obstacles and unmodeled wind disturbances, offering both real-time performance and formal safety guarantees.

This work proposes the first control barrier function (CBF) framework tailored for differential-algebraic equation (DAE) systems, addressing the challenge that algebraic constraints pose to existing CBF methods in simultaneously ensuring safety and constraint consistency. By integrating the differential-algebraic structure through projected vector fields, the approach guarantees forward invariance of safe sets while strictly satisfying algebraic constraints, even for high-index DAEs. The method establishes a DAE-aware CBF theory, providing necessary and sufficient conditions for geometric correctness and feasibility. Verification for polynomial systems is enabled via sum-of-squares (SOS) optimization, while falsification of non-polynomial or neural network candidate functions is supported through SMT solvers. Experimental validation on wind turbines and flexible-link robotic arms demonstrates the framework’s effectiveness in concurrently enforcing safety and algebraic constraints.