Theory of Stiffness Control in Robotics Using the Conservative Congruence Transformation

Abstract

In this paper, we present a new theory of conservative congruence transformation (CCT) to replace the conventional congruence formulation, K θ = J θ T K p J θ , first derived by Salisbury in 1980. The conservative congruence transformation defines the correct and consistent mapping of the stiffness matrices between the joint and Cartesian spaces. We present the theory and simulation, and show that the conventional formulation is only valid when the manipulator is always maintained at unloaded position. Once the grasping and manipulation are deviated from the unloaded configuration by the application of conservative force, the CCT must be used. The CCT takes into consideration the changes in geometry through the differential Jacobian, or the Hessian, matrix of the robot manipulators. We also show that the omission of the changes in the Jacobian during grasping and manipulation would result in discrepancy of the work, and lead to contradiction to the fundamental physical properties of stiffness control. The CCT, however, preserves the conservative and consistent properties of stiffness control in robotics for the mapping between the joint and Cartesian spaces.