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Cable-actuated parallel manipulators (CPMs) rely on cables instead of rigid links to manipulate the moving platform in the taskspace. Upper and lower bounds imposed on the cable tensions limit the force capability in CPMs and render certain forces infeasible at the end effector. This paper presents a geometrical analysis of the problems to 1) determine whether a CPM is capable of balancing a given wrench within the cable tension limits (feasibility check); 2) minimize the 2-norm of the cable tensions that balance feasible wrenches; and 3) check for the existence of an all-positive nullspace vector, which is a necessary condition to have a wrench-closure configuration in CPMs. The unified approach used in this analysis is systematic and geometrically intuitive that is based on the formulation of the static force equilibrium problem as an intersection between two convex sets and the application of Dykstra's alternating projection algorithm to find the projection of a point onto that intersection. In the case of infeasible wrenches, the algorithm can determine whether the infeasibility is because of the cable tension limits or the non-wrench-closure configuration. For the former case, a method was developed by which this algorithm can be used to extend the cable tension limits to balance infeasible wrenches. In addition, the performance of the algorithm is explained in the case of incompletely restrained cable-driven manipulators and the case of manipulators at singular poses. This paper also discusses the algorithm convergence and termination rule. This geometrical and systematic approach is intended for use as a convenient tool for cable tension analysis during design.