Affine connections for the Cartesian stiffness matrix

We study the 6/spl times/6 Cartesian stiffness matrix. We show that the stiffness of a rigid body subjected to conservative forces and moments is described by a (0,2) tensor which is the Hessian of the potential function. The key observation of the paper is that since the Hessian depends on the choice of an affine connection in the task space, so will the Cartesian stiffness matrix. Further, the symmetry of the Hessian and thus of the stiffness matrix depends on the symmetry of the connection. The connection that is implicit in the definition of the Cartesian stiffness matrix through the joint stiffness matrix (Salisbury, 1980) is made explicit and shown to be symmetric. In contrast, the direct definition of the Cartesian stiffness matrix in Griffis (1993), Ciblak and Lipkin (1994) and Howard et al. (1996) is shown to be derived from an asymmetric connection. A numerical example is provided to illustrate the main ideas of the paper.