The operational space control (OSC) of multilink cable-driven hyper-redundant robots (MCDHRs) is required to perform tasks in many applications. As a new coupled active-passive (CAP) MCDHR system, due to the multiple couplings between the active cables, the passive cables, the joints, and the end-effector, the OSC becomes more and more complicated. However, there is currently no robust and effective control method to solve the OSC problem of such types MCDHRs. In this paper, an OSC framework of CAP-MCDHRs using a dynamics-based iterative-learning-control (ILC) method is proposed, considering multivariate optimization. First, the multi-coupling kinematics and the series-parallel coupling dynamics equation (i.e., cable-joint-end) of the CAP-MCDHR are derived. Then, a dynamics-based trajectory tracking framework was constructed. Moreover, an OSC accuracy evaluation model based on a high-precision laser tracker was also designed. The framework allows the tracking of operational space trajectories (OSTs) online with the feasible cable tension and the joint angle. It is also shown that the tracking performance can be improved through the ILC when the desired trajectory is repeatedly performed. Finally, a simulation and an experimental hardware system are built. The results show that the proposed control framework can be easily and effectively applied to the CAP-MCDHR used in real-time.