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 1n 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.