Mechanical Engineering Publications Mechanical Engineering 9-28-2009 A Screw Theory Approach for the Conceptual Design of Flexible Joints for Compliant Mechanisms Hai-Jun Su University of Maryland, Baltimore County Denis V. Dorozhkin Iowa State University Judy M. Vance Iowa State University, [email protected] Follow this and additional works at: http://lib.dr.iastate.edu/me_pubs Part of the Mechanical Engineering Commons The ompc lete bibliographic information for this item can be found at http://lib.dr.iastate.edu/ me_pubs/27. For information on how to cite this item, please visit http://lib.dr.iastate.edu/ howtocite.html. This Article is brought to you for free and open access by the Mechanical Engineering at Iowa State University Digital Repository. It has been accepted for inclusion in Mechanical Engineering Publications by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. A Screw Theory Approach for the Conceptual Design of Flexible Joints for Compliant Mechanisms Abstract 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 pattern. We provide a systematic formulation of the constraint-based approach, which has been mainly developed by precision engineering experts in designing precision machines. The two fundamental concepts in the constraint-based approach, constraint and freedom, can be represented mathematically by a wrench and a twist in screw theory. For example, an ideal wire flexure applies a translational constraint, 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 mechanism design, constraint pattern analysis, and constraint pattern design are discussed with examples in details. Lastly, 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 computational techniques into the constraint-based approach for the synthesis and analysis of compliant mechanisms. Disciplines Mechanical Engineering Comments This article is from Journal of Mechanisms and Robotics (2009): 041009, doi:10.1115/1.3211024. Posted with permission. This article is available at Iowa State University Digital Repository: http://lib.dr.iastate.edu/me_pubs/27 A Screw Theory Approach for the Conceptual Design of Flexible Hai-Jun Su1 Department of Mechanical Engineering, University of Maryland, Joints for Compliant Mechanisms Baltimore County, Baltimore, MD 21250 This paper presents a screw theory based approach for the analysis and synthesis of e-mail: [email protected] flexible joints using wire and sheet flexures. The focus is on designing flexure systems that have a simple geometry, i.e., a parallel constraint pattern. We provide a systematic formulation of the constraint-based approach, which has been mainly developed by pre- Denis V. Dorozhkin cision engineering experts in designing precision machines. The two fundamental con- e-mail: [email protected] cepts in the constraint-based approach, constraint and freedom, can be represented math- ematically by a wrench and a twist in screw theory. For example, an ideal wire flexure Judy M. Vance applies a translational constraint, which can be described by a wrench of pure force. As e-mail: [email protected] a result, the design rules of the constraint-based approach can be systematically formu- lated in the format of screws and screw systems. Two major problems in compliant Department of Mechanical Engineering, mechanism design, constraint pattern analysis, and constraint pattern design are dis- Iowa State University, cussed with examples in details. Lastly, a case study is provided to demonstrate the Ames, IA 50011 application of this approach to the design of compliant prismatic joints. This innovative method paves the way for introducing computational techniques into the constraint-based approach for the synthesis and analysis of compliant mechanisms. ͓DOI: 10.1115/1.3211024͔ 1 Introduction the constraint-based approach for the design of compliant instru- ments with flexures. The foundations of the constraint-based Compared with traditional rigid body mechanisms, compliant method were developed by Maxwell ͓16͔ in the 1880s. It was mechanisms ͓1͔ or flexures have many advantages, such as high recently revisited by Blanding ͓17͔ and several researchers at the precision and simplified manufacturing and assembly process; MIT Precision Engineering Laboratories ͓18–20͔ for the design of however, the design and analysis of compliant mechanisms is fixtures, rigid body machines, and flexure systems. The fundamen- complex due to the nonlinearity of deformation of the flexible tal premise of the constraint-based method is that all motions of a members. Researchers in two isolated fields, kinematics and rigid body are determined by the position and orientation of the mechanisms and precision engineering, have independently made constraints ͑constraint topology͒, which are placed on the body. major contributions to compliant mechanism design. The method is attractive because it is based on motion visualiza- In the kinematics and mechanisms community, research has tion and is therefore well suited to conceptual development. How- focused on applying computational techniques to determine the ever, the proficiency in using the constraint-based methods for dimensions and/or topologies of compliant mechanisms to achieve designing compliant mechanisms requires commitment to a steep a prespecified design objective. The two most often used ap- learning curve and development of “hands-on” experience to un- proaches in this field are the pseudorigid body model ͑PRBM͒ derstand the stiffness characteristics of alternate designs. Hence ͓2–4͔ approach and the topological synthesis approach ͓5–9͔. The the design process is not systematical and may not necessarily former approach models a compliant mechanism as a rigid body lead to the optimal design, especially when the designer is inex- with one or more springs. These springs impose approximated perienced. lumped compliance to the rigid link models of compliant mecha- In this paper, a mathematical formulation of the constraint- nisms. This allows the theories and methodologies developed for based approach based on screw theory is presented. A screw is the rigid body mechanisms ͓10,11͔ to be used to design compliant geometric entity that underlies the foundation of statics and in- mechanisms. Because of this simplification in the modeling pro- stantaneous ͑first-order͒ kinematics. Many authors have made cess, it is necessary to evaluate the designs to ensure the validity contributions to screw theory. The two fundamental concepts in of the PRBM. The topological synthesis approach models the screw theory are twist representing a general helical motion of a compliant linkage as a network of link members of different sizes, rigid body about an instantaneous axis in space and wrench rep- which together achieve a specified objective function such as geo- resenting a system of force and moment acting on a rigid body. metric advantage and mechanical advantage. The result is a com- These two concepts are often called duality ͓21͔ in kinematics and pliant mechanism of complex topology with distributed compli- statics. Ball ͓22͔ was the first to establish a systematical formula- ance. This complexity results in mechanisms that are difficult to tion for screw theory. Hunt ͓23͔ and Phillips ͓24,25͔ further de- manufacture and produce nonintuitive motions. Recent techniques veloped the mathematical and geometrical representation of for designing compliant mechanisms include the level set method screws and screw systems. Their focus lies on the application of ͓12͔, the instant center approach ͓13͔, the polynomial homotopy screw theory to the analysis and synthesis of mechanisms. Lipkin method ͓14͔, and a kinetoelastic formulation ͓15͔. and Pattern ͓26–28͔ systematically investigated the screw theory In parallel, the precision engineering community has been using and its applications to compliance or elasticity analysis of robot manipulators. Huang and Schimmels ͓29,30͔ studied the realiza- tion of a prescribed stiffness matrix with serial or parallel elastic 1Corresponding author. mechanisms. Other applications of screw theory include mobility Contributed by the Mechanisms and Robotics Committee of ASME for publica- analysis ͓31͔, assembly analysis ͓32,33͔, and topology synthesis tionintheJOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received January 14, ͓ ͔ ͓ ͔ 2009; final manuscript received June 11, 2009; published online September 28, 2009. 34 . Recently, Kim 35 studied the characterization of compliant Review conducted by G. K. Ananthasuresh. building blocks by utilizing the concept of eigentwists and eigen- Journal of Mechanisms and Robotics Copyright © 2009 by ASME NOVEMBER 2009, Vol. 1 / 041009-1 Downloaded From: http://mechanismsrobotics.asmedigitalcollection.asme.org/ on 02/24/2014 Terms of Use: http://asme.org/terms wrenches based on screw theory for designing compliant mecha- Ω=ωs nisms. F=fu A constraint and a degree-of-freedom ͑DOF͒ in the constraint- p Ω based approach can be described by a wrench and a twist in screw theory, respectively. Therefore,