IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS Robust Safety Critical Control Under Multiple State and Input Constraints: Volume Control Barrier Function Method Jinyang Dong, Shizhen Wu, Rui Liu, Xiao Liang, Senior Member, IEEE, Biao Lu, Member, IEEE, and Y ongchun Fang, Senior Member, IEEE Abstract --In this paper, the safety-critical control problem for uncertain systems under multiple control barrier function (CBF) constraints and input constraints is investigated. A novel framework is proposed to generate a safety filter that minimizes changes to reference inputs when safety risks arise, ensuring a balance between safety and performance. A nonlinear disturbance observer (DOB) based on the robust integral of the sign of the error (RISE) is used to estimate system uncertainties, ensuring that the estimation error converges to zero exponentially. This error bound is integrated into the safety-critical controller to reduce conservativeness while ensuring safety. To further address the challenges arising from multiple CBF and input constraints, a novel Volume CBF (VCBF) is proposed by analyzing the feasible space of the quadratic programming (QP) problem. To ensure that the feasible space does not vanish under disturbances, a DOB-VCBF-based method is introduced, ensuring system safety while maintaining the feasibility of the resulting QP . Subsequently, several groups of simulation and experimental results are provided to validate the effectiveness of the proposed controller. I NTRODUCTION A S automation systems have become integral to our daily lives, the development of safe and high-performance controllers for these systems is of paramount importance. To meet this need, the Control Barrier Function (CBF) is a powerful tool to ensure the safety of control systems [1].

--Polygonal collision avoidance (PCA) is short for the problem of collision avoidance between two polygons (i.e., polytopes in planar) that own their dynamic equations. This problem suffers the inherent difficulty in dealing with non-smooth boundaries and recently optimization-defined metrics, such as signed distance field (SDF) and its variants, have been proposed as control barrier functions (CBFs) to tackle PCA problems. In contrast, we propose an optimization-free smooth CBF method in this paper, which is computationally efficient and proved to be nonconservative. It is achieved by three main steps: a lower bound of SDF is expressed as a nested Boolean logic composition first, then its smooth approximation is established by applying the latest log-sum-exp method, after which a specified CBF-based safety filter is proposed to address this class of problems. T o illustrate its wide applications, the optimization-free smooth CBF method is extended to solve distributed collision avoidance of two underactuated nonholonomic vehicles and drive an underactuated container crane to avoid a moving obstacle respectively, for which numerical simulations are also performed. The control barrier function-based quadratic programming (CBF-QP) control method is popular for safe robotic control [1]-[5]. The CBF-based control can provide a simple and computationally efficient way for safe control synthesis [1], [2], and it has been gradually extended to higher-order systems [3]-[5]. Collision avoidance, i.e., driving the robot away from the obstacle and keeping a distance, is a common goal in reactive control of multi-agent robots such as [6], [7]. And recently, CBFs have been gradually used to achieve more complex collision avoidance [8]-[13]. When the shapes of robots and obstacles are complicated (instead of points or spheres), collision detection is not an obvious problem. Since polytopes can non-conservatively approximate any convex shapes, compared with collision avoidance between ellipsoids or generic convex sets [8]-[10], references [11]-[13] are particularly interested in developing CBFs for collision avoidance between polytopes/polygons, where polygon refers particularly to a polytope in planar. Recently, related works about developing CBF methods for avoiding obstacles with irregular shapes can also be found in [14].