arXiv subjects

Calin A. Belta

Publications and source records attributed to Calin A. Belta.

At least 19 recordsLinked to original sources

arXiv2026

Iterative Convex Optimization with Control Barrier Functions for Obstacle Avoidance among Polytopes

Obstacle avoidance of polytopic obstacles by polytopic robots is a challenging problem in optimization-based control and trajectory planning. Many existing methods rely on smooth geometric approximations, such as hyperspheres or ellipsoids, which allow differentiable distance expressions but distort the true geometry and restrict the feasible set. Other approaches integrate exact polytope distances into nonlinear model predictive control (MPC), resulting in nonconvex programs that limit real-time performance. In this paper, we construct linear discrete-time control barrier function (DCBF) constraints by deriving supporting hyperplanes from exact closest-point computations between convex polytopes. We then propose a novel iterative convex MPC-DCBF framework, where local linearization of system dynamics and robot geometry ensures convexity of the finite-horizon optimization at each iteration. The resulting formulation reduces computational complexity and enables fast online implementation for safety-critical control and trajectory planning of general nonlinear dynamics. The framework extends to multi-robot and three-dimensional environments. Numerical experiments demonstrate collision-free navigation in cluttered maze scenarios with millisecond-level solve times.

cs.RO↗

arXiv2026

Sampling-Aware Control Barrier Functions for Safety-Critical and Finite-Time Constrained Control

In safety-critical control systems, ensuring both safety and feasibility under sampled-data implementations is crucial for practical deployment. Existing Control Barrier Function (CBF) frameworks, such as High-Order CBFs (HOCBFs), effectively guarantee safety in continuous time but may become unsafe when executed under zero-order-hold (ZOH) controllers due to inter-sampling effects. Moreover, they do not explicitly handle finite-time reach-and-remain requirements or multiple simultaneous constraints, which often lead to conflicts between safety and reach-and-remain objectives, resulting in feasibility issues during control synthesis. This paper introduces Sampling-Aware Control Barrier Functions (SACBFs), a unified framework that accounts for sampling effects and high relative-degree constraints by estimating and incorporating Taylor-based upper bounds on barrier evolution between sampling instants. The proposed method guarantees continuous-time forward invariance of safety and finite-time reach-and-remain sets under ZOH control. To further improve feasibility, a relaxed variant (r-SACBF) introduces slack variables for handling multiple constraints realized through time-varying CBFs. Simulation studies on a unicycle robot demonstrate that SACBFs achieve safe and feasible performance in scenarios where traditional HOCBF methods fail.

eess.SY↗

arXiv2026

Event-Triggered Adaptive Taylor-Lagrange Control for Safety-Critical Systems

This paper studies safety-critical control for nonlinear systems under sampled-data implementations of the controller. The recently proposed Taylor--Lagrange Control (TLC) method provides rigorous safety guarantees but relies on a fixed discretization-related parameter, which can lead to infeasibility or unsafety in the presence of input constraints and inter-sampling effects. To address these limitations, we propose an adaptive Taylor--Lagrange Control (aTLC) framework with an event-triggered implementation, where the discretization-related parameter defines the discretization time scale and is selected online as state-dependent rather than fixed. This enables the controller to dynamically balance feasibility and safety by adjusting the effective time scale of the Taylor expansion. The resulting controller is implemented as a sequence of Quadratic Programs (QPs) with input constraints. We further introduce a selection rule to choose the discretization-related parameter from a finite candidate set, favoring feasible inputs and improved safety. Simulation results on an adaptive cruise control (ACC) problem demonstrate that the proposed approach improves feasibility, guarantees safety, and achieves smoother control actions compared to TLC while requiring a single automatically tuned parameter.

eess.SY↗

arXiv2026

Safety-Critical Control with Guaranteed Lipschitz Continuity via Filtered Control Barrier Functions

In safety-critical control systems, ensuring both system safety and smooth control input is essential for practical deployment. Existing Control Barrier Function (CBF) frameworks, especially High-Order CBFs (HOCBFs), effectively enforce safety constraints, but also raise concerns about the smoothness of the resulting control inputs. While smoothness typically refers to continuity and differentiability, it does not by itself ensure bounded input variation. In contrast, Lipschitz continuity is a stronger form of continuity that not only is necessary for the theoretical guarantee of safety, but also bounds the rate of variation and eliminates abrupt changes in the control input. Such abrupt changes can degrade system performance or even violate actuator limitations, yet current CBF-based methods do not provide Lipschitz continuity guarantees. This paper introduces Filtered Control Barrier Functions (FCBFs), which extend HOCBFs by incorporating an auxiliary dynamic system-referred to as an input regularization filter-to produce Lipschitz continuous control inputs. The proposed framework ensures safety, control bounds, and Lipschitz continuity of the control inputs simultaneously by integrating FCBFs and HOCBFs within a unified quadratic program (QP). Theoretical guarantees are provided and simulations on a unicycle model demonstrate the effectiveness of the proposed method compared to standard and smoothness-penalized HOCBF approaches.

eess.SY↗

arXiv2026

Feasibility-aware Learning of Robust Temporal Logic Controllers using BarrierNet

Control Barrier Functions (CBFs) have been used to enforce safety and task specifications expressed in Signal Temporal Logic (STL). However, existing CBF-STL approaches typically rely on fixed hyperparameters and per-step optimization, which can lead to overly conservative behavior, infeasibility near tight input limits, and difficulty satisfying long-horizon STL tasks. To address these limitations, we propose a feasibility-aware learning framework that constructs trainable, time-varying High Order Control Barrier Function (HOCBF) constraints and hyperparameters that guarantee satisfaction of a given STL specification. We introduce a unified robustness measure that jointly captures STL satisfaction, constraint feasibility, and control-bound compliance, and propose a neural network architecture to generate control inputs that maximize this robustness. The resulting controller guarantees STL satisfaction with strictly feasible HOCBF constraints and requires no manual tuning. Simulation results demonstrate that the proposed framework maintains high STL robustness under tight input bounds and significantly outperforms fixed-parameter and non-adaptive baselines in complex environments.

eess.SY↗

arXiv2025

Risk-Aware Adaptive Control Barrier Functions for Safe Control of Nonlinear Systems under Stochastic Uncertainty

This paper addresses the challenge of ensuring safety in stochastic control systems with high-relative-degree constraints, while maintaining feasibility and mitigating conservatism in risk evaluation. Control Barrier Functions (CBFs) provide an effective framework for enforcing safety constraints in nonlinear systems. However, existing methods struggle with feasibility issues and multi-step uncertainties. To address these challenges, we introduce Risk-aware Adaptive CBFs (RACBFs), which integrate Discrete-time Auxiliary-Variable adaptive CBFs (DAVCBFs) with coherent risk measures. DAVCBFs introduce auxiliary variables to improve the feasibility of the optimal control problem, while RACBFs incorporate risk-aware formulations to balance safety and risk evaluation performance. By extending discrete-time high-order CBF constraints over multiple steps, RACBFs effectively handle multi-step uncertainties that propagate through the system dynamics. We demonstrate the effectiveness of our approach on a stochastic unicycle system, showing that RACBFs maintain safety and feasibility while reducing unnecessary conservatism compared to standard robust formulations of discrete-time CBF methods.

math.OC↗

arXiv2025

Learning Safety for Obstacle Avoidance via Control Barrier Functions

Obstacle avoidance is central to safe navigation, especially for robots with arbitrary and nonconvex geometries operating in cluttered environments. Existing Control Barrier Function (CBF) approaches often rely on analytic clearance computations, which are infeasible for complex geometries, or on polytopic approximations, which become intractable when robot configurations are unknown. To address these limitations, this paper trains a residual neural network on a large dataset of robot-obstacle configurations to enable fast and tractable clearance prediction, even at unseen configurations. The predicted clearance defines the radius of a Local Safety Ball (LSB), which ensures continuous-time collision-free navigation. The LSB boundary is encoded as a Discrete-Time High-Order CBF (DHOCBF), whose constraints are incorporated into a nonlinear optimization framework. To improve feasibility, a novel relaxation technique is applied. The resulting framework ensure that the robot's rigid-body motion between consecutive time steps remains collision-free, effectively bridging discrete-time control and continuous-time safety. We show that the proposed method handles arbitrary, including nonconvex, robot geometries and generates collision-free, dynamically feasible trajectories in cluttered environments. Experiments demonstrate millisecond-level solve times and high prediction accuracy, highlighting both safety and efficiency beyond existing CBF-based methods.

cs.RO↗

arXiv2025

Learning-Enabled Iterative Convex Optimization for Safety-Critical Model Predictive Control

Safety remains a central challenge in control of dynamical systems, particularly when the boundaries of unsafe sets are complex (e.g., nonconvex, nonsmooth) or unknown. This paper proposes a learning-enabled framework for safety-critical Model Predictive Control (MPC) that integrates Discrete-Time High-Order Control Barrier Functions (DHOCBFs) with iterative convex optimization. Unlike existing methods that primarily address CBFs of relative degree one with fully known unsafe set boundaries, our approach generalizes to arbitrary relative degrees and addresses scenarios where the unsafe set boundaries must be inferred. We extract pixel-based data specifically from unsafe set boundaries and train a neural network to approximate local linearizations of these boundaries. The learned models are incorporated into the linearized DHOCBF constraints at each time step, enabling real-time constraint satisfaction within the MPC framework. An iterative convex optimization procedure is developed to accelerate computation while maintaining formal safety guarantees. The benefits of computational performance and safe avoidance of obstacles with diverse shapes are examined and confirmed through numerical results. By bridging model-based control with learning-based environment modeling, this framework advances safe autonomy for discrete-time systems operating in complex and partially known settings.

eess.SY↗

arXiv2025

Auxiliary-Variable Adaptive Control Barrier Functions

This paper addresses the challenge of ensuring safety and feasibility in control systems using Control Barrier Functions (CBFs). Existing CBF-based Quadratic Programs (CBF-QPs) often encounter feasibility issues due to mixed relative degree constraints, input nullification problems, and the presence of tight or time-varying control bounds, which can lead to infeasible solutions and compromised safety. To address these challenges, we propose Auxiliary-Variable Adaptive Control Barrier Functions (AVCBFs), a novel framework that introduces auxiliary variables in auxiliary functions to dynamically adjust CBF constraints without the need of excessive additional constraints. The AVCBF method ensures that all components of the control input explicitly appear in the desired-order safety constraint, thereby improving feasibility while maintaining safety guarantees. Additionally, we introduce an automatic tuning method that iteratively adjusts AVCBF hyperparameters to ensure feasibility and safety with less conservatism. We demonstrate the effectiveness of the proposed approach in adaptive cruise control and obstacle avoidance scenarios, showing that AVCBFs outperform existing CBF methods by reducing infeasibility and enhancing adaptive safety control under tight or time-varying control bounds.

eess.SY↗

arXiv2025

Safety-Critical Planning and Control for Dynamic Obstacle Avoidance Using Control Barrier Functions

Dynamic obstacle avoidance is a challenging topic for optimal control and optimization-based trajectory planning problems. Many existing works use Control Barrier Functions (CBFs) to enforce safety constraints for control systems. CBFs are typically formulated based on the distance to obstacles, or integrated with path planning algorithms as a safety enhancement tool. However, these approaches usually require knowledge of the obstacle boundary equations or have very slow computational efficiency. In this paper, we propose a framework based on model predictive control (MPC) with discrete-time high-order CBFs (DHOCBFs) to generate a collision-free trajectory. The DHOCBFs are first obtained from convex polytopes generated through grid mapping, without the need to know the boundary equations of obstacles. Additionally, a path planning algorithm is incorporated into this framework to ensure the global optimality of the generated trajectory. We demonstrate through numerical examples that our framework allows a unicycle robot to safely and efficiently navigate tight, dynamically changing environments with both convex and nonconvex obstacles. By comparing our method to established CBF-based benchmarks, we demonstrate superior computing efficiency, length optimality, and feasibility in trajectory generation and obstacle avoidance.

cs.RO↗

arXiv2024

Feasibility-Guaranteed Safety-Critical Control with Applications to Heterogeneous Platoons

This paper studies safety and feasibility guarantees for systems with tight control bounds. It has been shown that stabilizing an affine control system while optimizing a quadratic cost and satisfying state and control constraints can be mapped to a sequence of Quadratic Programs (QPs) using Control Barrier Functions (CBF) and Control Lyapunov Functions (CLF). One of the main challenges in this method is that the QP could easily become infeasible under safety constraints of high relative degree, especially under tight control bounds. Recent work focused on deriving sufficient conditions for guaranteeing feasibility. The existing results are case-dependent. In this paper, we consider the general case. We define a feasibility constraint and propose a new type of CBF to enforce it. Our method guarantees the feasibility of the above mentioned QPs, while satisfying safety requirements. We demonstrate the proposed method on an Adaptive Cruise Control (ACC) problem for a heterogeneous platoon with tight control bounds, and compare our method to existing CBF-CLF approaches. The results show that our proposed approach can generate gradually transitioned control (without abrupt changes) with guaranteed feasibility and safety.

math.OC↗

arXiv2024

Auxiliary-Variable Adaptive Control Lyapunov Barrier Functions for Spatio-Temporally Constrained Safety-Critical Applications

Recent work has shown that stabilizing an affine control system while optimizing a quadratic cost subject to state and control constraints can be mapped to a sequence of Quadratic Programs (QPs) using Control Barrier Functions (CBFs) and Control Lyapunov Functions (CLFs). One of the main challenges in this method is that the QPs could easily become infeasible under safety and spatio-temporal constraints with tight control bounds. In our own recent work, we defined Auxiliary-Variable Adaptive CBFs (AVCBFs) to improve the feasibility of the CBF-based QP, while avoiding extensive parameter tuning. In this paper, we consider spatio-temporal constraints as finite-time reachability requirements. In order to satisfy these requirements, we generalize AVCBFs to Auxiliary-Variable Adaptive Control Lyapunov Barrier Functions (AVCLBFs) that work for systems and constraints with arbitrary relative degrees. We show that our method has fewer conflicts with safety and input constraints, and outperforms the state of the art in term of adaptivity and feasibility in solving the QP. We illustrate our approach on an optimal control problem for a unicycle.

math.OC↗

arXiv2024

Auxiliary-Variable Adaptive Control Barrier Functions for Safety Critical Systems

This paper studies safety guarantees for systems with time-varying control bounds. It has been shown that optimizing quadratic costs subject to state and control constraints can be reduced to a sequence of Quadratic Programs (QPs) using Control Barrier Functions (CBFs). One of the main challenges in this method is that the CBF-based QP could easily become infeasible under tight control bounds, especially when the control bounds are time-varying. The recently proposed adaptive CBFs have addressed such infeasibility issues, but require extensive and non-trivial hyperparameter tuning for the CBF-based QP and may introduce overshooting control near the boundaries of safe sets. To address these issues, we propose a new type of adaptive CBFs called Auxiliary-Variable Adaptive CBFs (AVCBFs). Specifically, we introduce an auxiliary variable that multiplies each CBF itself, and define dynamics for the auxiliary variable to adapt it in constructing the corresponding CBF constraint. In this way, we can improve the feasibility of the CBF-based QP while avoiding extensive parameter tuning with non-overshooting control since the formulation is identical to classical CBF methods. We demonstrate the advantages of using AVCBFs and compare them with existing techniques on an Adaptive Cruise Control (ACC) problem with time-varying control bounds.

math.OC↗

arXiv2023

Iterative Convex Optimization for Model Predictive Control with Discrete-Time High-Order Control Barrier Functions

Safety is one of the fundamental challenges in control theory. Recently, multi-step optimal control problems for discrete-time dynamical systems were formulated to enforce stability, while subject to input constraints as well as safety-critical requirements using discrete-time control barrier functions within a model predictive control (MPC) framework. Existing work usually focus on the feasibility or the safety for the optimization problem, and the majority of the existing work restrict the discussions to relative-degree one control barrier functions. Additionally, the real-time computation is challenging when a large horizon is considered in the MPC problem for relative-degree one or high-order control barrier functions. In this paper, we propose a framework that solves the safety-critical MPC problem in an iterative optimization, which is applicable for any relative-degree control barrier functions. In the proposed formulation, the nonlinear system dynamics as well as the safety constraints modeled as discrete-time high-order control barrier functions (DHOCBF) are linearized at each time step. Our formulation is generally valid for any control barrier function with an arbitrary relative-degree. The advantages of fast computational performance with safety guarantee are analyzed and validated with numerical results.

math.OC↗

arXiv2023

Learning Feasibility Constraints for Control Barrier Functions

It has been shown that optimizing quadratic costs while stabilizing affine control systems to desired (sets of) states subject to state and control constraints can be reduced to a sequence of Quadratic Programs (QPs) by using Control Barrier Functions (CBFs) and Control Lyapunov Functions (CLFs). In this paper, we employ machine learning techniques to ensure the feasibility of these QPs, which is a challenging problem, especially for high relative degree constraints where High Order CBFs (HOCBFs) are required. To this end, we propose a sampling-based learning approach to learn a new feasibility constraint for CBFs; this constraint is then enforced by another HOCBF added to the QPs. The accuracy of the learned feasibility constraint is recursively improved by a recurrent training algorithm. We demonstrate the advantages of the proposed learning approach to constrained optimal control problems with specific focus on a robot control problem and on autonomous driving in an unknown environment.

math.OC↗

arXiv2022

Control Barrier Functions for Systems with Multiple Control Inputs

Control Barrier Functions (CBFs) are becoming popular tools in guaranteeing safety for nonlinear systems and constraints, and they can reduce a constrained optimal control problem into a sequence of Quadratic Programs (QPs) for affine control systems. The recently proposed High Order Control Barrier Functions (HOCBFs) work for arbitrary relative degree constraints. One of the challenges in a HOCBF is to address the relative degree problem when a system has multiple control inputs, i.e., the relative degree could be defined with respect to different components of the control vector. This paper proposes two methods for HOCBFs to deal with systems with multiple control inputs: a general integral control method and a method which is simpler but limited to specific classes of physical systems. When control bounds are involved, the feasibility of the above mentioned QPs can also be significantly improved with the proposed methods. We illustrate our approaches on a unicyle model with two control inputs, and compare the two proposed methods to demonstrate their effectiveness and performance.

eess.SY↗

arXiv2021

High Order Control Lyapunov-Barrier Functions for Temporal Logic Specifications

Recent work has shown that stabilizing an affine control system to a desired state while optimizing a quadratic cost subject to state and control constraints can be reduced to a sequence of Quadratic Programs (QPs) by using Control Barrier Functions (CBFs) and Control Lyapunov Functions (CLFs). In our own recent work, we defined High Order CBFs (HOCBFs) for systems and constraints with arbitrary relative degrees. In this paper, in order to accommodate initial states that do not satisfy the state constraints and constraints with arbitrary relative degree, we generalize HOCBFs to High Order Control Lyapunov-Barrier Functions (HOCLBFs). We also show that the proposed HOCLBFs can be used to guarantee the Boolean satisfaction of Signal Temporal Logic (STL) formulae over the state of the system. We illustrate our approach on a safety-critical optimal control problem (OCP) for a unicycle.

eess.SY↗

arXiv2020

Bridging the Gap between Optimal Trajectory Planning and Safety-Critical Control with Applications to Autonomous Vehicles

We address the problem of optimizing the performance of a dynamic system while satisfying hard safety constraints at all times. Implementing an optimal control solution is limited by the computational cost required to derive it in real time, especially when constraints become active, as well as the need to rely on simple linear dynamics, simple objective functions, and ignoring noise. The recently proposed Control Barrier Function (CBF) method may be used for safety-critical control at the expense of sub-optimal performance. In this paper, we develop a real-time control framework that combines optimal trajectories generated through optimal control with the computationally efficient CBF method providing safety guarantees. We use Hamiltonian analysis to obtain a tractable optimal solution for a linear or linearized system, then employ High Order CBFs (HOCBFs) and Control Lyapunov Functions (CLFs) to account for constraints with arbitrary relative degrees and to track the optimal state, respectively. We further show how to deal with noise in arbitrary relative degree systems. The proposed framework is then applied to the optimal traffic merging problem for Connected and Automated Vehicles (CAVs) where the objective is to jointly minimize the travel time and energy consumption of each CAV subject to speed, acceleration, and speed-dependent safety constraints. In addition, when considering more complex objective functions, nonlinear dynamics and passenger comfort requirements for which analytical optimal control solutions are unavailable, we adapt the HOCBF method to such problems. Simulation examples are included to compare the performance of the proposed framework to optimal solutions (when available) and to a baseline provided by human-driven vehicles with results showing significant improvements in all metrics.

eess.SY↗