Simulation of conservative congruence transformation: conservative properties in the joint and Cartesian spaces

In this paper, the stiffness characteristics of robot systems via the conservative congruence of transformation (CCT) and the conventional congruence transformation (CT) between the joint and Cartesian spaces are investigated. A stiffness matrix is conservative if (1) the force resulting from the stiffness matrix is conservative, and (2) the work done by such force along a closed path is zero, i.e., independent of the path. The criteria result in the derivation of the conservative congruence transformation (CCT) between the joint and Cartesian spaces. Numerical simulation of a two-link planar manipulator, manipulating along various closed paths with no self-intersection, is implemented. The results verify that a stiffness matrix in R^3×3 Cartesian space or joint space will be conservative if it is symmetric and satisfies the exact differential criterion. Furthermore, we also illustrate the importance of the effect of changes in geometry in grasping and manipulation using stiffness control via CCT. The results show that the CCT is the correct mapping for stiffness matrices between the joint and Cartesian spaces.