Enhanced Stiffness Modeling and Identification Method for a Cable-Driven Spherical Joint Module

A cable-driven manipulator (CDM) has low stiffness and its stiffness identification is a critical issue. This paper focuses on stiffness modeling and identification for a cable-driven spherical joint module (CSJM), whose trajectory is a curve on
$SO(3)$
. In order to obtain the stiffness of the CSJM, it requires to evaluate the variation of the load against the displacement. However, since the vectors of displacement and load at different poses of the CSJM belong to different vector spaces of
$SO(3)$
, the algebraic operations between them can not be performed directly. Hence, a Riemannian metric and the Levi-Civita connection are defined on
$SO(3)$
, so that vectors can be parallel transported from one vector space to another along the trajectory curve. Consequently, the covariant derivative of the load with respect to the displacement is defined on
$SO(3)$
to establish the stiffness model. The resultant stiffness matrix is proved to be symmetric for a conservative system. In this way, the stiffness model with the system parameters of the CSJM is derived based on the kinetostatic analysis. Due to a part of the system parameters can not be accurately known, a feasible stiffness identification method is proposed based on the approximation of the covariant derivative, which merely require to measure the poses and loads of the CSJM. The experiment on the actual testbed validates the practical appeals of the proposed stiffness model and associate identification method.