Optimization-free Smooth Control Barrier Function for Polygonal Collision Avoidance

--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].