Determination of spring installation configuration on statically balanced planar articulated manipulators

This paper presents a design methodology to determine spring configuration on statically balanced planar articulated manipulator. From the energy perspective, the summation of potential energies remains constant at any configuration. The gravitational potential energy changes due to the movement of linkages, and the elastic potential energy corresponds to the spring configuration. By formulating similar representation of matrix form, the equilibrium equation is simplified as the summation of gravitational stiffness block matrix and elastic stiffness block matrices remain unchanged. This paper discusses the distribution features of entries in gravitational stiffness block matrix, the characteristic of elastic stiffness block matrix associated with the attachment angles and attachment points of the spring, and the equivalent spring installations that produce same elastic potential effect but with different design parameters or configurations. According to the interrelation between gravitational and elastic stiffness block matrix, the general criteria of the admissible spring configuration are obtained. For minimum number of springs and minimum total number of articulated joints that springs span over, the additional criteria and admissible spring configurations are derived. A three-degree-of-freedom planar articulated manipulator is shown as an illustrative example.