Kinematic Analysis of Parallel Mechanisms with a Constraint Chain of Motion

Abstract

A closed loop mechanism with plural active connecting chains and one passive constraint chain, which restricts the motion of an output link, can be considered as a parallel mechanism with a constrained connecting chain. In this paper, concise kinematic analysis formular for this type of mechanisms has been derived. By modeling the passive constraint chain as an active connecting chain with some fixed active joints, equations for analyzing a parallel mechanism can be directly applied. The basis of the motion space of the constraint chain is firstly determined and that of the connecting chain is determined such that it contains all the coordinate vectors of the constraint chains. The basis of wrench space is determined as the dual basis of that of a connecting chain. The determination of the basis in this way makes it possible to separate the elements of wrenches to elements involved in the motion and other elements. This separation makes it possible to reduce the dimension of the Jacobian of the output link to that of the motion space of the constrained chain, and the motion and force analysis can be expressed by this Jacobian. Developed method is examined by appling to a 2 DOF prismatic kinematics to derive torques of active pairs verses inertial force of links.