Control Barrier Functions (CBF) are a powerful tool for designing safety-critical controllers and motion planners. The safety requirements are encoded as a continuously differentiable function that maps from state variables to a real value, in which the sign of its output determines whether safety is violated. In practice, the CBFs can be used to enforce safety by imposing itself as a constraint in a Quadratic Program (QP) solved point-wise in time. However, this approach costs computational resources and could lead to infeasibility in solving the QP. In this paper, we propose a novel motion planning framework that combines sampling-based methods with Linear Quadratic Regulator (LQR) and CBFs. Our approach does not require solving the QPs for control synthesis and avoids explicit collision checking during samplings. Instead, it uses LQR to generate optimal controls and CBF to reject unsafe trajectories. To improve sampling efficiency, we employ the Cross-Entropy Method (CEM) for importance sampling (IS) to sample configurations that will enhance the path with higher probability and store computed optimal gain matrices in a hash table to avoid re-computation during rewiring procedure. We demonstrate the effectiveness of our method on nonlinear control affine systems in simulation.

翻译：控制屏障函数（CBF）是设计安全关键型控制器与运动规划器的有力工具。安全需求被编码为一个连续可微函数，该函数将状态变量映射为实数值，其输出的符号决定是否违反安全条件。在实践中，CBF可通过将其作为二次规划（QP）中的逐点求解约束来强制执行安全性。然而，这种方法耗费计算资源，且可能导致求解QP时出现不可行问题。本文提出一种新颖的运动规划框架，将基于采样的方法、线性二次型调节器（LQR）与CBF相结合。本方法无需为控制综合求解QP，并在采样过程中避免显式碰撞检测，而是利用LQR生成最优控制，并借助CBF剔除不安全轨迹。为提高采样效率，我们采用交叉熵方法（CEM）进行重要性采样（IS），以更高概率采样有助于优化路径的构型，并将计算得到的最优增益矩阵存储于哈希表中，避免在重连程序中进行重复计算。我们通过仿真在非线性控制仿射系统上验证了该方法的有效性。