In underactuated robots with closed kinematic chains, the task space configuration variables are coupled through complex dynamics. In this article, the regulation control of a 3-DOF underactuated cable-driven parallel robot is investigated by using the interconnection and damping assignment passivity-based control approach, which is designed based on the solution of some challenging partial differential equations (PDEs). Additionally, as cables can only pull, positive tension in cables shall also be taken into account. Here, the corresponding PDE for the controller design is solved by transforming the PDE into some Pfaffian differential equations. Then, boundedness of control efforts are ensured via suitable modification of the gains. Furthermore, an adaptation law for the mass of the robot is designed and the system stability is investigated through the Lyapunov direct method. The efficiency of the proposed controller is confirmed and compared to another method through experiment.