  Sampling-based algorithms, such as Rapidly Exploring Random Trees (RRT) and
its variants, have been used extensively for motion planning. Control barrier
functions (CBFs) have been recently proposed to synthesize controllers for
safety-critical systems. In this paper, we combine the effectiveness of
RRT-based algorithms with the safety guarantees provided by CBFs in a method
called CBF-RRT$^\ast$. CBFs are used for local trajectory planning for
RRT$^\ast$, avoiding explicit collision checking of the extended paths. We
prove that CBF-RRT$^\ast$ preserves the probabilistic completeness of
RRT$^\ast$. Furthermore, in order to improve the sampling efficiency of the
algorithm, we equip the algorithm with an adaptive sampling procedure, which is
based on the cross-entropy method (CEM) for importance sampling (IS). The
procedure exploits the tree of samples to focus the sampling in promising
regions of the configuration space. We demonstrate the efficacy of the proposed
algorithms through simulation examples.
