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

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. The springs are of two types: simple and screw. The term simple spring refers to a purely translational or purely rotational passive spring, while a screw spring couples translational and rotational components. Any spatial stiffness matrix can be realized with a parallel connection of such springs. However, as the names suggest, simple springs are much easier to implement than screw springs. Thus, to reduce the compliance mechanism's complexity, it is desirable to minimize the number of screw springs. The article presents a method for determining minimal realizations for any symmetric positive definite or semidefinite spatial stiffness matrix. These realizations are minimal in the sense that they minimize both the number of screw springs and the total number of springs.