In this paper we address a challenging version of the problem of controlling tethered aerial vehicles (also known as UAV, MAV, and UAS) by considering the aerial robot linked to a generic and independently moving platform. We solve the exact tracking control problem for both the 3D position of the robot (either absolute or with respect to the platform) and the tension along the link. To achieve this goal we prove some fundamental system properties, useful to design a nonlinear controller, such as differential flatness and dynamic feedback linearizability. To close the control loop a set of minimal and standard sensors is proposed. Then we show that it is possible to retrieve the full system state from those sensors by means of nonlinear measurements transformations and a bank of low-dimension estimators based on the nonlinear high gain observer. The ability of the proposed observer-controller method is tested by extensive numerical simulations spanning many non-ideal conditions.