Structural Optimization of a Spherical Parallel Manipulator Using a Two-Level Approach

The Agile Wrist, a spherical wrist with a parallel, isotropic architecture for highest orientational accuracy, is being designed as a module of an 11-degree-of-freedom (dof) robot. The wrist consists of two main elements, the base and the moving plates. The two plates are coupled by means of three identical legs, each of these composed of two links, proximal and distal, coupled to each other by a revolute joint. Each leg, in turn, is coupled to its proximal plate via revolute joints. Moreover, the three axes of the leg-revolute joints are concurrent at the center of the wrist, each axis making an angle of 90 with its neighbor. Direct-drive DC motors are used to rotate the wrist proximal links and electrical brakes and optical encoders are located on each of the motor shafts for control purposes. In this paper we introduce a two-level approach to the optimum design of the proximal link of the Agile Wrist. First, the shape of the midcurve producing minimum stress concentrations is obtained by means of the concept of curve synthesis using cubic splines. At the second level, the optimum cross-section along the midcurve producing a link of minimum weight is determined. INTRODUCTION The Agile Wrist (AW), shown in Figure 1, is the threedof orienting module of the redundant, 11-axis Multi-Modular Manipulator (M3) currently under design at McGill University’s 1 Figure 1. A RENDERING OF THE AGILE WRIST Centre for Intelligent Machines (Angeles et al, 1999). It owes its name to the Agile Eye (AE) developed by Gosselin and his team at Université Laval, Quebec, Canada (Gosselin and Lavoie, 1993; Gosselin and Hamel, 1994; Gosselin and Gagné, 1995). However, while keeping the advantages of a parallel, isotropic architecture for highest orientation accuracy, the AW has a lighter Copyright  2001 by ASME weight as compared to the AE; moreover, the AW avoids collisions between the tool installed on top of the mobile platform and the moving links. A robot architecture is said to be isotropic if it allows for robot postures whereby the Jacobian matrix has all its singular values identical and nonzero. The AW consists of two plates coupled by three identical legs. Each leg is composed of two links, the proximal and the distal, coupled to each other by means of a revolute joint. The shapes of the midcurves of the distal links are restricted to be circular arcs, whereas those of the proximal links are only shapeconstrained at their two ends. Since the design of the distal link is rather simple, we focus in this paper on the optimum design of the midcurve and the cross-section of the proximal link. In the design of the proximal link, we aim at minimizing the weight while avoiding stress concentrations. The design consists of two levels: (i) finding the shape of the midcurve which renders minimum the stress concentrations and (ii) determining the crosssection which minimizes the overall weight subject to a maximum allowable stress. The midcurve is obtained by means of the concept of curve synthesis (Angeles, 1983) using cubic splines; the optimization problem thus resulting is then solved with the aid of the ODA package (Teng, and Angeles, 2000). Next, we resort to finite element analysis (FEA) and structural optimization, by means of ProMechanica to solve the problem of determining the cross-section along the midcurve that produces a link of minimum weight subject to the maximum allowable stress. KINETOSTATIC ANALYSIS While the kinematic analysis of parallel spherical wrists has been the subject of many publications, the static analysis has received less attention. Yi, Freeman and Tesar (1992) proposed a force and stiffness transmission analysis for a spherical shoulder mechanism by means of the Jacobian matrix. This kind of analysis provides the designer with useful information to select the appropriate actuators, but in the case of spherical wrists, only moments are taken into account by the analysis, the forces at the joints remaining indeterminate. While designing the links of the Agile Wrist, we had to be sure that they can withstand the load of the tool, but no moments are created if this load is directed along a line passing through the center of the wrist, which is the case when the Agile Wrist is in its nominal posture, with its base oriented horizontally. In our case, the base of the Agile Wrist can be oriented vertically; in this orientation, it is then possible to determine the moment exerted by the weight of the tool at the centre of the wrist by using a Jacobian approach and then evaluating the loading moment that acts on the most heavily loaded leg. The ensuing kinetostatic analysis enables us to determine which maximum moment magnitude the Agile Wrist can withstand using the prescribed actuators. 2 Kinematics of the Agile Wrist and of the ith Leg