Minimal realization of an arbitrary spatial stiffness matrix with a parallel connection of simple and complex springs

This article presents a method for determining a minimal realization of an arbitrary spatial stiffness matrix K through the use of a mechanism constructed of a parallel connection of springs. Two types of springs are used: simple and complex. The term simple spring refers to a purely translational or purely rotational passive spring while a complex spring couples translational and rotational components. Any symmetric positive definite spatial stiffness matrix can be realized with a parallel connection of such springs. However, to reduce the compliance mechanism´s complexity, it is desirable to minimize the number of complex springs. This article presents a method for determining minimal realizations for any symmetric positive definite or semi-definite spatial stiffness matrix. These realizations are minimal in the sense that they minimize both the number of complex springs and the total number of springs