This study presents a cable-driven parallel robot (CDPR) that shows flexible motion on 3D curved surfaces. Because the actuators of a CDPR are placed outside the robot and the actuation torque is transmitted through a set of cables, the CDPR has a large workspace and highly dynamic properties, in contrast with conventional robot manipulators that have rigid links or joints. To implement the kinematic control of the CDPR, we set a kinematic subspace in which the cable tensions were always greater than or equal to zero. In this subspace, controlling the CDPR by manipulating the cable length instead of the cable tension is possible. The CDPR can be used as a 3D printer to manufacture an object or as a 3D plotter to draw textures on a complex curved surface. Experimental results showed that the developed CDPR was capable of flexible motion on a 3D curved surface. A simple and intuitive kinematics control algorithm, which does not require the measurement of cable tension to achieve flexible motion, was implemented.