Complete Jacobian matrix of a class of incompletely symmetrical parallel mechanisms with 4-DOF

Abstract

The complete Jacobian matrix of a lower-mobility parallel mechanism is a six by six matrix that consists of the actuation sub-matrix and the constraint sub-matrix. As the alge-braic feature of the former one cannot account for the related constraint characteristics, modeling complete six by six Jaco-bian matrix becomes necessary in the kinematic analysis and optimal design of geometric parameters of this kind of parallel mechanisms. A method for deducing the complete Jacobian matrix of an incompletely symmetrical lower-mobility parallel mechanism is illustrated by taking 2RPS-2UPS as an example based on the theory of reciprocal screw. Firstly, the systems of twists and reciprocal screws of the constraint limbs are estab-lished based on the screw theory, then the constraint sub-matrix is obtained through the orthogonal product. Secondly, by lock-ing active joints of each limb, the system of additional recipro-cal screws of both constraint and unconstraint limbs is estab-lished, then the actuation sub-matrix is also obtained through the orthogonal product. By integrating these two sub-matrices properly, the complete Jacobian matrix of an incompletely symmetrical lower-mobility parallel mechanism can be finally set up. In the end, the singular conditions of 2RPS-2UPS paral-lel mechanism are analyzed by investigating the ranks of the Jacobian matrices.