This paper proposes an energy-efficient adaptive robust tracking control method for a class of fully actuated, thrust vectoring unmanned aerial vehicles (UAVs) with parametric uncertainties including unknown moment of inertia, mass and center of mass, which would occur in aerial maneuvering and manipulation. We consider a novel vector thrust UAV with all propellers able to tilt about two perpendicular axes, so that the thrust force generated by each propeller is a fully controllable vector in 3D space, based on which an adaptive robust control is designed for accurate trajectory tracking in the presence of inertial parametric uncertainties and uncertain nonlinearities. Theoretically, the resulting controller achieves a guaranteed transient performance and final tracking accuracy in the presence of both parametric uncertainties and uncertain nonlinearities. In addition, in the presence of only parametric uncertainties, the controller achieves asymptotic output tracking. To resolve the redundancy in actuation, a thrust force optimization problem minimizing power consumption while achieving the desired body force wrench is formulated, and is shown to be convex with linear equality constraints. Simulation results are also presented to verify the proposed solution.