It is known that new power transmission line inspection techniques have been developed over the last few years with great potential to improve and, in some cases, even replace traditional inspection procedures such as using helicopters and cars. A series of Unmanned Aerial Vehicles (UAVs) such as fixed-wing or rotary-wing UAVs, and vehicles that climb on the power transmission line, promise to revolutionize the inspection market. In this light, at least 39 new research studies and/or products have been conducted and/or introduced to the market, respectively. However, in an incipient way, some works point to the fusion of some technologies: the development of multi-rotor UAVs and the ability to connect and move over the power transmission line. In line with this, the current work was proposed, with significant unprecedented advances (such as an over-actuated control capacity with tilt rotors, the capability of a displacement in the angle, and the maintenance of active motors on the power transmission line), and the design, modeling, and control of an over-actuated UAV able to move over the conductor cable without the need for a new locomotion system is presented. The aircraft allows for a greater response and the indispensable ability to approximate landing in a power transmission line arbitrary position rather than the catenary lowest point (due to its ability to forward/backward move using the tilting rotors). Its design is detailed, its subsystems are described, and its normal and coupled flight mode dynamics are modeled. The results show good stability and reliable maneuvers for the coupling-to-power-transmission-line flight mode, without any overshoots, and the ability to follow the entire catenary through different Real Control Action (RCA) sets.