Minimum-norm Solution for the Actuator Forces in Cable-based Parallel Manipulators based on Convex Optimization

Cable-based parallel manipulators (CPM) are light-weight manipulators that can reach high accelerations. The difference between the design of CPM and that of rigid-link parallel manipulators is that cables can only perform while under tension. Redundant limbs, such as extra cables, springs, or cylinders, can be used for applying forces on the mobile platform to generate cable tensions resulting in a redundantly actuated manipulator. To operate this manipulator, the actuator-force distribution amongst the cables and the redundant limbs needs to be determined. Actuator-force optimization techniques developed for rigid-link manipulators are unsuitable for CPM. In this study, a numerical procedure based on convex analysis and optimization is presented to calculate the minimum-norm solution that minimizes the 2-norm of actuator forces. The procedure is based on convex optimization that utilizes the Dykstra´s alternating projection algorithm to reach to the optimum solution. This numerical method is successfully applied to 3- and 6-degree-of-freedom (DOF) spatial CPMs to determine the optimum actuator forces for a given external load. This study addresses the static analysis in cable-based parallel manipulators in the language of convex analysis