Generic method for deriving the general shaking force balance conditions of parallel manipulators with application to a redundant planar 4-RRR parallel manipulator

This paper proposes a generic method for deriving the general shaking force balance conditions of parallel manipulators. Instead of considering the balancing of a parallel manipulator link-by-link or leg-by-leg, the architecture is considered altogether. The first step is to write the linear momentum of each element. The second step is to substitute the derivatives of the loop equations, by which the general force balance conditions are obtained. Subsequently specific kinematic conditions are investigated in order to find advantageous, simple balance solutions. As an example, the method is applied to a planar 4-RRR parallel manipulator, for which the force balance conditions and solutions are discussed and illustrated for each step respectively. By including the loop equations, linear relations of the motion among mechanism elements lead to an increase of balance possibilities.For specific kinematic conditions, additional linear relations among the motion of mechanism elements may be obtained, resulting in another increase of balance possibilities. For the latter, symmetric motion is an important feature for which a 4-RRR manipulator is advantageous