Cable-driven parallel robots distinguish themselves from other robot types through their large workspace and high dynamic capabilities. Investigations show that with purely kinematic control schemes, the theoretical workspace of such robots cannot be fully realized in practice. Current control methods usually rely on complex models whose parameters are difficult to determine with the required precision or feedback from expensive sensors that can measure the actual platform pose. This work presents a force control method that enables high dynamic capabilities within a large workspace for redundantly-constrained cable-driven parallel robots in practice, without the aforementioned drawbacks. The new method modifies the cable forces within the nullspace of the robot's structure matrix to keep them within their feasible limits and as close as possible to a desired level without changing the platform's pose. In simulation, it achieves similar performance as a state-of-the-art model predictive control method, with 79% less computational effort. Experiments show that it can quickly reject disturbances and is significantly better at keeping the cable forces within their limits on highly dynamic trajectories than a purely kinematic control scheme.