  The majority of existing Linear Temporal Logic (LTL) planning methods rely on
the construction of a discrete product automaton, that combines a discrete
abstraction of robot mobility and a B$\ddot{\text{u}}$chi automaton that
captures the LTL specification. Representing this product automaton as a graph
and using graph search techniques, optimal plans that satisfy the LTL task can
be synthesized. However, constructing expressive discrete abstractions makes
the synthesis problem computationally intractable. In this paper, we propose a
new sampling-based LTL planning algorithm that does not require any discrete
abstraction of robot mobility. Instead, it incrementally builds trees that
explore the product state-space, until a maximum number of iterations is
reached or a feasible plan is found. The use of trees makes data storage and
graph search tractable, which significantly increases the scalability of our
algorithm. To accelerate the construction of feasible plans, we introduce bias
in the sampling process which is guided by transitions in the
B$\ddot{\text{u}}$chi automaton that belong to the shortest path to the
accepting states. We show that our planning algorithm, with and without bias,
is probabilistically complete and asymptotically optimal. Finally, we present
numerical experiments showing that our method outperforms relevant temporal
logic planning methods.
