Conservative congruence transformation for joint and Cartesian stiffness matrices of robotic hands and fingers

In this paper, we develop the theoretical work on the properties and mapping of stiffness matrices between joint and Cartesian spaces of robotic hands and fingers, and propose the conservative congruence transformation (CCT). In this paper, we show that the conventional formulation between the joint and Cartesian spaces, K_θ = J_θ^TK_pJ_θ, first derived by Salisbury in 1980, is only valid at the unloaded equilibrium configuration. Once the grasping configuration is deviated from its unloaded configuration (for example, by the application of an external force), the conservative congruence transformation should be used. Theoretical development and numerical simulation are presented. The conservative congruence transformation accounts for the change in geometry via the differential Jacobian (Hessian matrix) of the robot manipulators when an external force is applied. The effect is captured in an effective stiffness matrix, K_g, of the conservative congruence transformation. The results of this paper also indicate that the omission of the changes in Jacobian in the presence of external force would result in discrepancy of the work and lead to contradiction to the fundamental conservative properties of stiffness matrices. Through conservative congruence transformation, conservative and consistent physical properties of stiffness matrices can be preserved during mapping regardless of the usage of coordinate frames and the existence of external force.