This work presents the nonlinear dynamical model and motion controller of a system consisting of an unmanned aerial vehicle (UAV) that is tethered to a floating buoy in the three-dimensional (3D) space. Detailed models of the UAV, buoy, and the coupled tethered system dynamics are presented in a marine environment that includes surface-water currents and oscillating gravity waves, in addition to wind gusts. This work extends the previously modeled planar (vertical) motion of this novel robotic system to allow its free motion in all three dimensions. Furthermore, a Directional Surge Velocity Control System (DSVCS) is hereby proposed to allow both the free movement of the UAV around the buoy when the cable is slack, and the manipulation of the buoy’s surge velocity when the cable is taut. Using a spherical coordinate system centered at the buoy, the control system commands the UAV to apply forces on the buoy at specific azimuth and elevation angles via the tether, which yields a more appropriate realization of the control problem as compared to the Cartesian coordinates where the traditional x- , y- , and z -coordinates do not intuitively describe the tether’s tension and orientation. The proposed robotic system and controller offer a new method of interaction and collaboration between UAVs and marine systems from a locomotion perspective. The system is validated in a virtual high-fidelity simulation environment, which was specifically developed for this purpose, while considering various settings and wave scenarios.