We present LQR-CBF-RRT*, an incremental sampling-based algorithm for offline motion planning. Our framework leverages the strength of Control Barrier Functions (CBFs) and Linear Quadratic Regulators (LQR) to generate safety-critical and optimal trajectories for a robot with dynamics described by an affine control system. CBFs are used for safety guarantees, while LQRs are employed for optimal control synthesis during edge extensions. Popular CBF-based formulations for safety critical control require solving Quadratic Programs (QPs), which can be computationally expensive. Moreover, LQR-based controllers require repetitive applications of first-order Taylor approximations for nonlinear systems, which can also create an additional computational burden. To improve the motion planning efficiency, we verify the satisfaction of the CBF constraints directly in edge extension to avoid the burden of solving the QPs. We store computed optimal LQR gain matrices in a hash table to avoid re-computation during the local linearization of the rewiring procedure. Lastly, we utilize the Cross-Entropy Method for importance sampling to improve sampling efficiency. Our results show that the proposed planner surpasses its counterparts in computational efficiency and performs well in an experimental setup.

翻译：我们提出了LQR-CBF-RRT*，一种用于离线运动规划的增量式基于采样的算法。我们的框架利用控制障碍函数（CBF）和线性二次型调节器（LQR）的优势，为具有仿射控制系统描述的机器人生成安全关键且最优的轨迹。CBF用于保证安全性，而LQR则在边扩展过程中用于最优控制综合。基于CBF的安全关键控制公式通常需要求解二次规划（QP），这可能会带来较高的计算成本。此外，基于LQR的控制器需要对非线性系统重复应用一阶泰勒近似，这也可能增加额外的计算负担。为了提高运动规划效率，我们直接在边扩展中验证CBF约束的满足性，以避免求解QP的负担。我们将计算得到的最优LQR增益矩阵存储在哈希表中，以避免在重连过程的局部线性化中重复计算。最后，我们利用交叉熵方法进行重要性采样，以提高采样效率。实验结果表明，所提出的规划器在计算效率上优于同类方法，并在实验装置中表现良好。