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：控制屏障函数是一种设计安全关键型控制器与运动规划器的强大工具。安全要求被编码为一个从状态变量映射到实数的连续可微函数，其输出的正负性用于判定安全是否被违反。在实践中，控制屏障函数可作为二次规划（QP）中的约束条件，通过逐点求解该二次规划来强制执行安全性。然而，这种方法会消耗计算资源，并可能导致二次规划求解的不可行性。本文提出一种新颖的运动规划框架，该框架将基于采样的方法与线性二次型调节器（LQR）及控制屏障函数相结合。我们的方法无需为控制综合求解二次规划，也无需在采样过程中进行显式碰撞检测，而是利用LQR生成最优控制，并利用控制屏障函数拒斥不安全的轨迹。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），以更高概率采样能够优化路径的构型，并将计算得到的最优增益矩阵存储于哈希表中，以避免在重连过程中重复计算。我们在仿真中针对非线性控制仿射系统验证了本方法的有效性。