  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.
