In this paper, we investigate resource allocation algorithm design for multiuser unmanned aerial vehicle (UAV) communication systems in the presence of UAV jittering and user location uncertainty. In particular, we jointly optimize the two-dimensional position and the downlink beamformer of a fixed-altitude UAV for minimization of the total UAV transmit power. The problem formulation takes into account the quality-of-service requirements of the users, the imperfect knowledge of the antenna array response (AAR) caused by UAV jittering, and the user location uncertainty. Despite the non-convexity of the resulting problem, we solve the problem optimally employing a series of transformations and semidefinite programming relaxation. Our simulation results reveal the dramatic power savings enabled by the proposed robust scheme compared to two baseline schemes. Besides, the robustness of the proposed scheme with respect to imperfect AAR knowledge and user location uncertainty at the UAV is also confirmed.