Simulation of conservative properties of stiffness matrices in congruence transformation

In this paper, the conservative properties of stiffness matrices via the nonconservative congruence mapping between the joint and Cartesian spaces are investigated with simulation of two fingers manipulating an object. The properties of both constant and configuration dependent stiffness matrices are presented with integration of work when manipulating along a closed path with no self-intersection. A stiffness matrix is conservative if the force resulting from the stiffness matrix is conservative, and the work done by such force along a closed path is zero, i.e., independent of the path. Both theoretical derivation and numerical simulation show that a stiffness matrix in R^3×3 Cartesian space or joint space with n generalized coordinates will be conservative if it is symmetric and satisfies the exact differential criterion. Simulation of two fingers manipulating an object is implemented using OpenGL with both Cartesian-based and joint-based stiffness control scheme. The results show that the congruence transformation generally results in nonconservative stiffness matrix, except for a special group of configuration dependent solutions.