Article Preview
Top1. Introduction
Parallel Manipulators are closed-loop mechanical systems, used for many applications such as fast pick and place, flight simulators, etc. due to their good performances in terms of accuracy, rigidity, manipulation speed and ability to manipulate large loads (CM Gosselin, 1996; Merlet, 2006; Pierrot, Fournier, & Dauchex, 1991). One of the first parallel manipulator was developed by Stewart (Stewart, 1965). The Stewart platform (or the so called Gough-Stewart platform) is a 6 DOF manipulator that incorporates six prismatic actuators. These actuators are mounted in pairs to the mechanism’s fix base, crossing over to three mounting points on a top plate(manipulator). From its initial introduction many designs of parallel manipulators were proposed. But despite the many advantages of parallel mechanisms, they also suffer from several drawbacks. Pierrot et. al. demonstrated some of the main disadvantages that are shared by many parallel manipulators: They have a complex direct kinematics; the position and the orientation of the moving platform are coupled; their workspace is small; and they require expensive joints such as the spherical joint (Hunt, 1983; Pierrot et al., 1991). Other platforms were proposed to tackle some of these disadvantages. The DELTA platform designed by Clavel (Clavel, 1988) solves the first two items. Delta manipulators are popular manipulators that are mainly being used for pick and place applications. Their main advantage is their high moving speed within their workspace. Figure 1 shows a 3 DOF version of the Delta robot, which was constructed in our lab, the schematic of its kinematic chain and the parameters such as length of the links. Changing of these parameters has an effect on workspace and also the dexterity of the robot. This manipulator is a more general 3 DOF configuration compared to Delta, since the 2 parameters of “d” and “e” are not considered as zero. One of the main features of parallel manipulators, is that in contrast with serial articulated arms, where singularities are located on the border of the workspace, parallel manipulators suffer from singularities inside their workspace, and thus some areas in the workspace of the manipulator should be avoided (Agrawal, 1991).
Figure 1. Left: The spatial three degrees of freedom parallel manipulator developed in our lab, Right: The schematics of its kinematics chain
On the other hand the location of such singularities in the workspace changes with modifying length of the links. It is desired to choose the best design parameters in order to increase the dexterity of the manipulator. Dexterity of a manipulator is related with the ease and precision a end-effector of a manipulator can achieve a desired position, and with some discretion it can be used as a measure of distance to a singularity on the workspace. In the next section, we will describe the condition number, which is introduced as a measure for dexterity of the manipulator at each single point inside the workspace. Thus the optimization problem is to select the design parameters in order to maximize this number for the whole workspace of the robot.