Symmetrical characteristics of the workspace for spatial parallel mechanisms with symmetric structure

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 mechanism structure is n, then the searching range can be reduced to image of the initial one. This can lead to significant reductions of the computation task and time because only a fraction of the workspace may need to be investigated for a symmetric 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 with complicated symmetries.