Minimal realization of a spatial stiffness matrix with simple springs connected in parallel

This article presents a method for determining a minimal realization of an arbitrarily specified stiffness dominated spatial impedance through the use of a mechanism constructed of passive springs. This mechanism consists of simple springs connected in parallel, where the term simple spring refers to a purely translational or purely rotational passive spring. Not every stiffness matrix K can be realized with a parallel connection of simple springs. The characterizing condition is that the zipper right 3×3 submatrix of K has zero trace. Using this condition, the author shows how one can always synthesize any realizable spatial stiffness matrix with r parallel simple springs where r is the rank of K. These results are also applicable to realizing a spatial damping matrix