A Screw Theory Approach for the Conceptual Design of Flexible Joints for Compliant Mechanisms

This paper presents a screw theory based approach for the analysis and synthesis of flexible joints using wire and sheet flexures. The focus is on designing flexure systems that have a simple geometry, i.e. a parallel constraint pat-tern. We provide a systematic formulation of the constraint-based approach which has been mainly developed by pre-cision engineering experts in designing precision machines. The two fundamental concepts in the constraint-based ap-proach, constraint and freedom, can be represented math-ematically by a wrench and a twist in screw theory. For example, an ideal wire flexure applies a translational con-straint which can be described by a wrench of pure force. As a result, the design rules of the constraint-based approach can be systematically formulated in the format of screws and screw systems. Two major problems in compliant mecha-nism design, constraint pattern analysis and constraint pat-tern design are discussed with examples in details. At last a case study is provided to demonstrate the application of this approach to the design of compliant prismatic joints. This innovative method paves the way for introducing computa-tional techniques into the constraint-based approach for the synthesis and analysis of compliant mechanisms. Nomenclature Ω A 3D vector presenting the angular velocity of a twist V A 3D vector presenting the linear velocity of a twist p A scalar representing the pitch of a twist T ̂ A 6D general twist representing an allowable motion ∏T A twist matrix representing the allowable motion space F A 3D vector presenting the force part of a wrench ∗Address all correspondence to this author. M A 3D vector presenting the couple part of a wrench q A scalar representing the pitch of a wrench W ̂ A 6D general wrench representing a constraint ∏W A wrench matrix representing the constraint space