  In many multirobot applications, planning trajectories in a way to guarantee
that the collective behavior of the robots satisfies a certain high-level
specification is crucial. Motivated by this problem, we introduce counting
temporal logics---formal languages that enable concise expression of multirobot
task specifications over possibly infinite horizons. We first introduce a
general logic called counting linear temporal logic plus (cLTL+), and propose
an optimization-based method that generates individual trajectories such that
satisfaction of a given cLTL+ formula is guaranteed when these trajectories are
synchronously executed. We then introduce a fragment of cLTL+, called counting
linear temporal logic (cLTL), and show that a solution to planning problem with
cLTL constraints can be obtained more efficiently if all robots have identical
dynamics. In the second part of the paper, we relax the synchrony assumption
and discuss how to generate trajectories that can be asynchronously executed,
while preserving the satisfaction of the desired cLTL+ specification. In
particular, we show that when the asynchrony between robots is bounded, the
method presented in this paper can be modified to generate robust trajectories.
We demonstrate these ideas with an experiment and provide numerical results
that showcase the scalability of the method.
