Geometrical method for modeling of asymmetric 6 × 6 Cartesian stiffness matrix

In this paper, we study the 6 × 6 Cartesian stiffness matrices of conservative systems using the method of changing basis in differential geometry of the motion of the rigid body. We show that the stiffness matrix is symmetric at the unloaded equilibrium configuration. When the system is subjected to external loads, the 6 × 6 Cartesian stiffness matrix becomes asymmetric. The skew-symmetric part of the stiffness matrix is equal to the negative one-half of the cross-product matrix formed by the externally applied load, referenced to the inertial frame. The method presented in this paper provides a systematic way of constructing 6 × 6 stiffness matrix in robotic grasping/manipulation and stiffness control.