Coordinate-free formulation of the Cartesian stiffness matrix

In the paper we study the Cartesian stiffness matrix using methods of differential geometry. We show that the stiffness of a conservative mechanical system is described by a ((0)(2)) tensor and that components of the Cartesian stiffness matrix are given by evaluating this tensor on a pair of basis twists. Our formulation leads to three important results: (a) The stiffness matrix does not depend on the parameterization of the manifold; (b) The stiffness matrix depends on the choice of a connection on the manifold; and (c) The standard definition of the Cartesian stiffness matrix assumes an asymmetric connection and this is the reason that the matrix is, in general, asymmetric.