On the normal form of a spatial stiffness matrix

A key result in the study of spatial stiffness matrices is Loncaric´s normal form. When a spatial stiffness matrix is described in an appropriate coordinate frame, it will have a particularly simple structure. In this form, the 3×3 off-diagonal blocks of the stiffness matrix are diagonal. It has been shown that generically, a spatial stiffness matrix can be written in a normal form. For example, it is fairly well known that this is true for any positive definite spatial stiffness matrix. In this article, it is shown that any symmetric positive semi-definite matrix can be written in a normal form. Also, the conditions under which the spatial stiffness matrix can be diagonalized are identified. These results are used to design a compact parallel compliance mechanism with a prescribed positive semi-definite spatial stiffness matrix.