Conditions for parallel realizable configurations in synthesis of constraint-based flexure mechanisms

In synthesis of flexure mechanism, parallel arrangement is paid more attention due to its advantages, such as compact structure and higher stiffness. Researchers have derived many parallel flexure mechanisms, but seldom discuss which kind of flexure mechanism can be realized via fully parallel arrangement. The realizable conditions proposed in current work are complicated to engineering applications. To solve two problems on how to judge whether a flexure mechanism can be realized via fully parallel arrangement and how to realize those flexure mechanisms which cannot be realized via fully parallel arrangement, the algebraic condition is derived to judge whether a freedom space is parallel realizable after introducing the definition of parallel realizable and some propositions, and the condition is there exist 6-n independent line constraints in constraint space reciprocal to dimensional freedom space. Then the realizable constraint spaces reciprocal to freedom spaces with 1-3 dimensions are provided. As a result, not all freedom spaces are parallel realizable. For freedom spaces that are not parallel realizable, the criterion of decomposing DOF is proposed to achieve all motion patterns via parallel or hybrid arrangement, that is a high dimensional freedom space can be realized via combining several low dimensional freedom spaces which are parallel realizable. Specific decomposing strategies for 4 and 5 DOF are provided and a complete flowchart is presented to guide designing flexure mechanisms, particularly those which are not parallel realizable. As case studies, synthesis processes of two helical and 3T1R motions are provided to illustrate the proposed approach. The proposed method provides a feasible approach to realize all motion patterns.