  This work presents an optimal sampling-based method to solve the real-time
motion planning problem in static and dynamic environments, exploiting the
Rapid-exploring Random Trees (RRT) algorithm and the Model Predictive Path
Integral (MPPI) algorithm. The RRT algorithm provides a nominal mean value of
the random control distribution in the MPPI algorithm, resulting in
satisfactory control performance in static and dynamic environments without a
need for fine parameter tuning. We also discuss the importance of choosing the
right mean of the MPPI algorithm, which balances exploration and optimality
gap, given a fixed sample size. In particular, a sufficiently large mean is
required to explore the state space enough, and a sufficiently small mean is
required to guarantee that the samples reconstruct the optimal controls. The
proposed methodology automates the procedure of choosing the right mean by
incorporating the RRT algorithm. The simulations demonstrate that the proposed
algorithm can solve the motion planning problem in real-time for static or
dynamic environments.
