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Cable-suspended robots are structurally similar to parallel-actuated robots, but with the fundamental difference that cables can only pull the end-effector, but not push it. These input constraints make feedback control of cable-suspended robots a lot more challenging than their counterpart parallel-actuated robots. In this paper, we present a computationally efficient control design procedure for a cable robot with six cables, which is kinematically determined as long as all cables are in tension. The control strategy is based on dynamic aspects of statically feasible workspace. The basic idea suggested in this paper is to represent the reachable domain in terms of achievable set points under a specified control law that respects the input constraints. This computational framework is recursively used to find a set of reachable domains, using which, we are able to expand the region of feasibility by connecting adjacent domains through common points. The salient feature of the technique is that it is computationally efficient, or online implementable, for the control of a cable robot with positive input constraints. However, due to the complexity of the dynamics of general motion of a cable robot, we consider only translations. No cable interference is considered in this paper. Finally, the effectiveness of the proposed method is illustrated by numerical simulations and laboratory experiments on a six-degree-of-freedom cable-suspended robot.