This article proposes a nonlinear model predictive control (NMPC) strategy for the position tracking of cable-driven parallel robots (CDPRs). The NMPC formulation handles explicitly the cable tensions and their limits. Accordingly, the cable tension distribution is performed as an integral part of the NMPC feedback control strategy, which notably allows the CDPR to operate on the wrench-feasible workspace boundaries without failure. In order to integrate the cable tension minimization within the NMPC formulation, the concept of wrench equivalent optimality (WEO) is introduced. The WEO is a nonnegative measure able to evaluate if the wrench generated by a given cable tension vector can be generated by an alternative tension vector with smaller 2-norm. The redundancy resolution performed by means of the minimization of the WEO enables the stability of the closed-loop system to be proved. More precisely, sufficient conditions for the uniform asymptotic stability are deduced using results from the analysis of NMPC schemes without terminal constraints and costs. Furthermore, the proposed NMPC strategy is validated experimentally on a fully constrained 6 degree-of-freedom CDPR.