  This paper proposes a new optimal control synthesis algorithm for multi-robot
systems under global temporal logic tasks. Existing planning approaches under
global temporal goals rely on graph search techniques applied to a product
automaton constructed among the robots. In this paper, we propose a new
sampling-based algorithm that builds incrementally trees that approximate the
state-space and transitions of the synchronous product automaton. By
approximating the product automaton by a tree rather than representing it
explicitly, we require much fewer memory resources to store it and motion plans
can be found by tracing sequences of parent nodes without the need for
sophisticated graph search methods. This significantly increases the
scalability of our algorithm compared to existing optimal control synthesis
methods. We also show that the proposed algorithm is probabilistically complete
and asymptotically optimal. Finally, we present numerical experiments showing
that our approach can synthesize optimal plans from product automata with
billions of states, which is not possible using standard optimal control
synthesis algorithms or off-the-shelf model checkers.
