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Machine Theory
Mechanism and Machine Theory 43 (2008) 427–444
ate/mechmt
Symmetrical characteristics of the workspace for spatial
parallel mechanisms with symmetric structure
Jing-Shan Zhao *, Fulei Chu, Zhi-Jing Feng
Department of Precision Instruments and Mechanology, Tsinghua University, Beijing 100084, PR China
Received 12 June 2006; received in revised form 12 February 2007; accepted 10 April 2007
Available online 7 June 2007
Abstract
The fact that symmetry of a spatial parallel mechanism can be directly mapped into the workspace of the end-effector is
defined as a strengthened theorem and is proved by utilizing the group theory. If the order of the symmetry group of the
1
mechanism structure is n, then the searching range can be reduced to n of the initial one. This can lead to significant reduc-
tions of putation task and time because only a fraction of the workspace may need to be investigated for a sym-
metric mechanism, the merits of which are especially obvious if discretization algorithms are used. Firstly, the symmetry
mapping from the mechanism structure into the workspace is generalized in a much more extension by a group theoretical
proof. And then, examples are presented to illustrate the applications of this research result in reducing the searching tasks
of the reachable workspace of an end-effector. The strengthened theorem herein will be adapted to all symmetry cases and
will be most useful for the conceptual design of spatial parallel mechanisms, particularly for those plicated
symmetries.
Ó 2007 Elsevier Ltd. All rights reserved.
Keywords: Workspace; Symmetry; Theorem; Spatial parallel mechanism
1. Introduction
The reachable workspace of the end-effector of a mechanism, which will be denoted as ‘‘workspace’’ for
short in this paper, is the reachable domain of its end-effector and is bounded in the three dimensional space
[1]. The workspace of a mechanism has been defined in liter