  Motion planning under uncertainty for an autonomous system can be formulated
as a Markov Decision Process with a continuous state space. In this paper, we
propose a novel solution to this decision-theoretic planning problem that
directly obtains the continuous value function with only the first and second
moments of the transition probabilities, alleviating the requirement for an
explicit transition model in the literature. We achieve this by expressing the
value function as a linear combination of basis functions and approximating the
Bellman equation by a partial differential equation, where the value function
can be naturally constructed using a finite element method. We have validated
our approach via extensive simulations, and the evaluations reveal that to
baseline methods, our solution leads to in terms of path smoothness, travel
distance, and time costs.
