In this paper, we propose a visual servoing control for a quadrotor unmanned aerial vehicle (UAV) which is based on a state transformation technique. The UAV is equipped with a single downwards facing camera, and the motion control objective is the regulation of relative displacement and yaw to a stationary visual target located on the ground. The state transformation is defined by a system of partial differential equations (PDEs) which eliminate roll and pitch rate dependence in the transformed image feature kinematics. A method for computing the general solutions of these PDEs is given, and we show a particular solution reduces to an established virtual camera approach. We treat point and line cases and introduce image moment features defined in the virtual camera image plane. Robustness of the control design is improved by accounting for attitude measurement bias, and uncertainty in thrust gain, mass, and image feature depth. The asymptotic stability of the closed-loop is proven. The method is based on a simple proportional-integral-derivative (PID) structure which can be readily implemented on-board. Experimental results show improved performance relative to previous work.