By fully exploiting the mobility of unmanned aerial vehicles (UAVs), UAV-based aerial base stations (BSs) can move closer to ground users to achieve better communication conditions. In this paper, we consider a scenario where an aerial BS is dispatched for covering a maximum number of ground users before exhausting its on-board energy resources. The resulting trajectory optimization problem is a mixed integer non-linear problem (MINLP) which is non-convex and is challenging to solve. As such, we propose an iterative algorithm which decomposes the problem into two sub-problems by applying both successive convex optimization and block coordinate descent techniques to solve it. To be specific, the trajectory of the aerial BS and the user scheduling and association are alternately optimized within each iteration. In addition, to achieve better coverage performance and speed up convergence, the problem of designing the initial trajectory of the UAV is also considered. Finally, to address the unavailability of accurate user location information (ULI) in practice, two different robust techniques are proposed to compensate the performance loss in the existence of inaccurate ULI. Simulation results show both energy and coverage performance gains for the proposed schemes compared to the benchmark techniques, with an up to 50% increase in coverage probability and an up to 20% reduction in energy.