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In this paper, control of fully-constrained parallel cable robots with elastic cables is studied in detail. In the modeling process, longitudinal vibration of cables is considered as their dominant dynamics, and the governing equations of motion are rewritten to the standard form of singular perturbation. The proposed composite controller consists of two main components. A rigid controller is designed based on the slow or rigid model of the system and a corrective term is added to guarantee asymptotic stability of the fast dynamics. Then, by using Tikhonov theorem, slow and fast variables are separated and incorporated into the stability analysis of the overall closed-loop system, and a set of sufficient conditions for the stability of the total system is derived. Finally, the effectiveness of the proposed control law is verified through simulations.