For high-speed robotics dynamic (shaking) force and (shaking) moment balance are important properties to obtain low base vibrations, high precision, and short cycle times, a.o., while for large moving structures such as movable bridges and movable roofs force balance is important for safety and low energy usage. To find applicable solutions, dynamic balance can be considered in the very beginning of the design process by synthesis from principal vector linkages: fundamental kinematic architectures with inherent force balance.

Succeeding the already studied open-chain principal vector linkages, this paper introduces the design and analysis of closed-chain principal vector linkages with two approaches. With the proposed open-chain method solutions are obtained by simply closing the chain of an open-chain principal vector linkage. With the proposed method of the mass equivalent principal chain less complex solutions are obtained with consideration of the loop closure relations. Then an open-chain principal vector linkage is closed with an additional element which, with a general mass distribution, is modeled mass equivalently in a new way with two real equivalent masses and one virtual equivalent mass. The results are validated with a dynamic simulation and a grasper mechanism with payload balanced together is shown as application example.