Cable-driven parallel robots are well suited for applications that require a very large workspace. Thanks to their lightweight moving parts, they can achieve high dynamics while remaining pretty safe for nearby human workers. Furthermore, their size depends only on the length of the cables; thus, their scale is almost totally decoupled from their cost. However, due to the cables, the stiffness is very low with respect to rigid link robots, inducing slowly damped oscillations of the end effector. Previous works have shown that those vibrations can be effectively damped by the winch actuators thanks to active vibration damping techniques. In this paper, a gain scheduling approach is proposed based on a linearized model of the robot dynamics. This model is projected in the modal space yielding six decoupled transfer functions for six degrees-of-freedom (DoFs) of a cable-driven parallel robot using thin cables. The stability of the proposed control law is analyzed for a static and a moving end effector. The proposed control algorithm is validated experimentally on an eight-cable suspended robot prototype.