  For a nonlinear system (e.g. a robot) with its continuous state space
trajectories constrained by a linear temporal logic specification, the
synthesis of a low-level controller for mission execution often results in a
non-convex optimization problem. We devise a new algorithm to solve this type
of non-convex problems by formulating a rapidly-exploring random tree of
barrier pairs, with each barrier pair composed of a quadratic barrier function
and a full state feedback controller. The proposed method employs a
rapid-exploring random tree to deal with the non-convex constraints and uses
barrier pairs to fulfill the local convex constraints. As such, the method
solves control problems fulfilling the required transitions of an automaton in
order to satisfy given linear temporal logic constraints. At the same time it
synthesizes locally optimal controllers in order to transition between the
regions corresponding to the alphabet of the automaton. We demonstrate this new
algorithm on a simulation of a two linkage manipulator robot.
