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This paper deals with the wrench-feasible workspace (WFW) of n-degree-of-freedom parallel robots driven by n or more than n cables. The WFW is the set of mobile platform poses for which the cables can balance any wrench of a given set of wrenches, such that the tension in each cable remains within a prescribed range. Requirements of nonnegative cable tensions, as well as maximum admissible tensions, are thus satisfied. The determination of the WFW is an important issue since its size and shape are highly dependent on the geometry of the robot and on the ranges of allowed cable tensions. The approach proposed in this paper is mainly based on interval analysis. Two sufficient conditions are presented, namely, a sufficient condition for a box of poses to be fully inside the WFW and a sufficient condition for a box of poses to be fully outside the WFW. These sufficient conditions are relevant since they can be tested, with the means to test them being discussed in the paper. Used within usual branch-and-prune algorithms, these tests enable WFW determinations in which full-dimensional sets of poses (volumes) are found to lie within or, on the contrary, to lie outside the WFW. This provides a useful alternative to a basic discretization, the latter consisting of testing a discrete (zero-dimensional) finite set of poses. In order to improve the efficiency of the computations, a means to mitigate the undesirable effects of the so-called wrapping effect is introduced. The paper also illustrates how the proposed approach is capable of dealing with small uncertainties on the geometric design parameters of a parallel cable-driven robot.