A stiffness matrix approach for the design of statically balanced planar articulated manipulators

A methodology is developed to determine the spring installation for the design of a statically balanced planar articulated manipulator without parallel auxiliary links. The spring installation is characterized by the connectivity of springs among links, the selection of spring constants, and the locations of spring attachment points. The static equilibrium analysis of the spring-loaded planar articulated manipulator is based on the energy approach, formulated by a constant stiffness block matrix and its associated configuration block matrices. The stiffness block matrix quantifies the resistance or assistance of a manipulator to the change of configuration due to the gravitational forces and the elastic spring forces. Such a matrix uniquely represents both the gravitational potential energy and the elastic potential energy of springs of the system at any configuration. By solving the isotropic condition of the stiffness block matrix, all design parameters of springs can be obtained for any given planar articulated manipulator with prescribed dimensions and inertia. Exact solutions for the locations of attachment points are given in detailed in the examples of a spring-loaded one-, two- and three- degrees of freedom articulated manipulators.