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 if the shape of the unsafe region is complex and the CBFs are of higher order. A common solution to this problem is to make a conservative approximation of the unsafe region in the form of a line/hyperplane, and use the corresponding conservative Hyperplane-CBF when deciding on safe control actions. In this letter, we note that conservative constraints are only a problem if they prevent us from doing what we want. Thus, instead of first choosing a CBF and then choosing a safe control with respect to the CBF, we optimize over a combination of CBFs and safe controls to get as close as possible to our desired control, while still having the safety guarantee provided by the CBF. We call the corresponding CBF the least restrictive Hyperplane-CBF. Finally, we also provide a way of creating a smooth parameterization of the CBF-family for the optimization, and illustrate the approach on a double integrator dynamical system with acceleration constraints, moving through a group of arbitrarily shaped static and moving obstacles.

翻译：控制屏障函数（CBFs）能够为动态系统提供可证明的安全性保证。然而，为特定系统寻找有效的CBF通常并非易事，尤其当不安全区域的形状复杂且CBF为高阶时。该问题的常见解决方案是采用直线/超平面形式对不安全区域进行保守近似，并在确定安全控制动作时使用相应的保守超平面CBF。本文指出，保守约束仅在其阻碍我们实现预期目标时才构成问题。因此，我们不再先选择CBF再根据CBF选择安全控制，而是通过联合优化CBF与安全控制的组合，在保持CBF所提供安全性保证的前提下，尽可能接近期望控制。我们将对应的CBF称为最小约束超平面CBF。最后，我们还提出了一种为优化过程构建CBF族平滑参数化的方法，并通过具有加速度约束的双积分器动态系统在任意形状静态与动态障碍物群中穿行的案例，阐明了该方法的有效性。