Analytical Stiffness Formulation and Interpretation for Six-DOF Tensegrity Joints Using Screw Theory

Abstract

Compliant mechanisms, e.g., tensegrities, inherently exhibit nonlinear behavior, wherein the stiffness matrix, evaluated at a specific configuration, characterizes the instantaneous relationship between applied forces and resulting displacements. For traditional robot joints, the stiffness matrix is defined using Cartesian and Euler angle parameters. This representation is convenient when the joints display translation or single degree of rotation behavior. However, it faces parameterization issues in modeling higher degree of freedom joints due to singularities and lack of uniqueness. Lie groups and screw theory representations provide a minimal and intrinsic representation of the rigid body motion. This representation is well suited for tensegrity joints which combine tensile and compressive members and behave as six degree-of-freedom joints. A key challenge in this context is that computing the stiffness matrix necessitates differentiating the transformation matrix with respect to the screw, a task that is highly nontrivial. This work derives an analytical formulation of the stiffness matrix for six degree-of-freedom tensegrity joints using screw theory representation, including a closed-form expression for the derivative of the transformation matrix with respect to its exponential coordinates. The analytical results are validated against numerical differentiation while achieving approximately three times faster computation speeds. The paper further interprets the stiffness matrix through block-form, column-wise, and row-wise representations, providing additional physical insight into the translational, rotational, and coupled stiffness contributions. These contributions establish an efficient framework for the stiffness analysis and lay the foundation for future integration of screw theory methods into Euler-Lagrange dynamics for higher degree-of-freedom robot joints including tensegrity joints.