Methods for the determination of accessible workspaces of planar Stewart platforms of general design

Abstract

In recent years Stewart platforms have been increasingly studied and developed. These parallel manipulators offer a number of advantages over traditional serial manipulators including high rigidity, good positioning accuracy and high load to weight ratio. The main disadvantage associated with parallel manipulators is that they have relatively limited workspaces. Numerous researchers have thus emphasized the need to develop refined methods for the determination of workspaces of such manipulators. This study is primarily concerned with extensions to a novel optimization approach for the determination of manipulator accessible output sets. The optimization approach provides a general method for the determination of workspaces of both serial and parallel manipulators and has the considerable advantage that it may easily be automated. Furthermore, the approach allows for the easy and systematic implementation of various physical constraints acting on manipulators. Established methods for workspace determination are reviewed and illustrated by application to a simple two degree of freedom example. The original optimization approach is extended and generalize< to enable the determination of non-convex workspaces. Simply stated, the approach consists of finding the points of intersection of the workspace boundary with a number of successive search elements. The points of intersection are determined by means of optimization techniques in which a dynamic constrained optimization algorithm is used. Two new methodologies, the modified ray method and the chord method, are proposed. Differences between these methods are illustrated using a simple example. The optimization approach, embodied in the proposed methodologies, is applied to the determination of workspaces of planar Stewart platforms of varied designs. A formulation for all constraints acting on planar Stewart platforms is introduced and implemented in the optimization approach. A special case of manipulator geometry, where the orientation of the platform is effectively redundant in determining the extreme reach of the manipulat'1r, is identified and studied. A slight modification to the optimization methodologies is introduced to allow for the determination of workspaces of such redundant manipulators. The modified ray and chord methods proposed in this study have proven capable of determining convex and non-convex manipulator workspaces. Of the two new methods, the chord approach is the most reliable in determining non-convex workspaces. Both optimization methodologies have been implemented in practical interactive computer systems, which allow for the easy determination of workspaces of planar Stewart pla1 forms of arbitrary geometry.