Stiffness control and transformation for robotic systems with coordinate and non-coordinate bases

In this paper, the application of the conservative congruence transformation (CCT) to the stiffness mapping between non-coordinate basis and coordinate basis systems is studied and presented. Through the stiffness transformation between the 2 degree-of-freedom cylindrical and joint spaces, we illustrate that the CCT can be applied either directly or indirectly to the stiffness transformation between any two systems with either coordinate basis or non-coordinate basis. It is found that the same stiffness control for a conservative system will render a symmetric stiffness matrix with respect to a coordinate basis, but an asymmetric matrix with respect to a non-coordinate basis. The direct and indirect CCT methods are presented, with the latter requiring an intermediate coordinate system with a generalized coordinate basis. The relationships of the effective K_g matrices between the direct and indirect CCT methods is found and validated.