A Screw Theory Approach for the Conceptual Design of Flexible Joints for Compliant Mechanisms Hai-Jun Su University of Maryland, Baltimore County

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]

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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 mathematically 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 formulated 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 realization 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-

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, all the rules in constraint pattern qF analysis and design can be explained and mathematically repre- α sented using screw theory. The result is a powerful tool that is c capable of systematically finding intuitive design topologies. The a problem of analyzing and synthesizing spatial stiffness/ y compliance of a general elastic mechanism with spring elements ˆ was investigated extensively by Lipkin ͓26–28͔, Schimmels T ˆ r ͓29,30͔, and others. As far as the authors’ knowledge, applying z x W screw theory to the type synthesis of compliant mechanisms with flexure elements is a new contribution to the field. In some senses, ˆ this work provides a mathematical representation of a graphical Fig. 1 A general wrench W does work on a body with the mo- approach for synthesizing flexure systems presented by Hopkins tion defined by a general twist Tˆ and Culpepper ͓20͔. This linking of screw theory to constraint- based compliant mechanism design allows a wide variety of computational techniques developed in the kinematics field ͓36͔ to be combined together with the constraint-based design approach to a pitch q=m/ f. Similar to the case of twist, a pure force and a achieve design automation for compliant mechanisms. pure couple are represented by a wrench of zero pitch and infinite The rest of the paper is organized as follows. Section 2 provides pitch, respectively, written as a review of screw theory. Section 3 describes a screw representation of the basic concepts including constraint, constraint space, pure force ͑q =0͒: Wˆ = ͑fu͉r ϫ fu͒͑5͒ freedom and freedom space. Also in this section, the general steps for constraint pattern analysis and design are presented and illustrated with examples. Section 4 provides a case study of designing pure couple ͑q = ϱ͒: Wˆ = ͑0͉mu͒͑6͒ compliant prismatic joints with wire and sheet flexures. Section 5 lists the contribution of this work and future research. Section 6 Figure 1 illustrates a general twist Tˆ and a general wrench Wˆ in presents conclusions and discussion. space. 2.2 Reciprocity of Screws. Let us denote the normal distance 2 Screw Theory Review and skew angle of axes of a general wrench Wˆ and a general twist This section provides a concise review of general screw theory. Tˆ by a and ␣, respectively, ͑Fig. 1͒. The virtual power of wrench ˆ ˆ 2.1 Twists and Wrenches. It is well known that screw theory W acting on a moving body with motion W is given by the recip- underlies the foundation of both instantaneous kinematics and rocal product of the twist and the wrench, calculated as statics. In kinematics, a general spatial motion of a rigid body is a screw motion ͑a rotation and a translation͒ about a line in space Tˆ ؠ Wˆ = F · V + M · ⍀ called a screw axis. The rotation and translation is further coupled ͓͑ ␻͒͑ ͒ ␻͑ ͒ ͑ ϫ ͔͒ by a scalar quantity called a pitch. The screw in kinematics also = fv + m s · u + f c − r · s u called a twist is formed by a pair of three-dimensional vectors, = ͓͑fv + m␻͒cos ␣ − f␻a sin ␣͔͑7͒ namely, angular velocity ⍀ and linear velocity V, written as A twist is called reciprocal to a wrench when their reciprocal Twist: Tˆ = ͑⍀͉V͒ = ͑␻s͉c ϫ ␻s + vs͒͑1͒ product is zero. To find all possible reciprocal conditions, we con- ͑ ͒ Tˆ Wˆ ͑ ͒ where vectors s and c denote the direction of and a point on the sider nontrivial none zero twist and nontrivial wrench . 1 If ˆ ˆ twist axis, respectively, scalars ␻ and v are the magnitude of f =0͑q=ϱ͒ and ␻=0͑p=ϱ͒, T is always reciprocal to W. This says angular velocity and partial linear velocity along the axis, respec- that a pure couple is always reciprocal ͑does no work͒ to a pure tively. The pitch is defined as the ratio of the linear velocity to translation. ͑2͒ If f =0,␻0orf 0,␻=0, Tˆ ؠWˆ =0 is satisfied if angular velocity, i.e., p=v/w. As special cases, a pure rotation and and only if cos ␣=0, i.e., the twist axis and the wrench axis are a pure translation in space are represented by a twist of zero pitch perpendicular to each other. ͑3͒ If f 0͑qϱ͒ and ␻0͑pϱ͒, and infinite pitch, respectively, written as Eq. ͑7͒ is reduced to pure rotation ͑p =0͒: Tˆ = ͑␻s͉c ϫ ␻s͒͑2͒ Tˆ ؠ Wˆ = f␻͓͑p + q͒cos ␣ − a sin ␣͔͑8͒ pure translation ͑p = ϱ͒: Tˆ = ͑0͉vs͒͑3͒ where we have substituted the definition of pitches. By consider- ͑ ͒ Note that a translation can be also viewed as a rotation with an ing the values of the pitches p and q in Eq. 8 , we can obtain the axis at infinity. The screw axis of planar motion degenerates to a following observations that can be very useful as thumb rules in point on the plane called the instantaneous center ͓13͔, displace- design practice. ment pole, or virtual pivot. The pitch of a planar twist is always ͑a͒ If p+q=0 ͑including the case p=q=0͒, the two screws zero. are reciprocal if either a=0 or sin ␣=0. This situation Similarly in statics, a general screw or a wrench consists of two occurs when the twist axis and wrench axis are coplanar, ͑ ͒ vectors representing a force F and a couple moment M acting on i.e., intersecting or parallel to each other a rigid body, written as ͑b͒ If the two screw axes are perpendicular cos ␣=0, their Wrench: Wˆ = ͑F͉M͒ = ͑fu͉r ϫ fu + mu͒͑4͒ reciprocity is independent of their pitches. This can occur only when a=0, i.e., two axes intersect. where vectors u and r are the direction of and a point on the wrench axis, respectively, scalars f and m are the magnitude of Reciprocal product is considered a linear operation on either force and partial moment along the axis, respectively, coupled by twist or wrench separately. For instance, the reciprocal product of

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Downloaded From: http://mechanismsrobotics.asmedigitalcollection.asme.org/ on 02/24/2014 Terms of Use: http://asme.org/terms ˆ ˆ ˆ y a twist T with a linear combination of two wrenches W1 ,W2 can be expressed as the linear combination of the reciprocal product of wire flexure Wˆ x Tˆ with each of the two wrenches, that is o ؠ ͑ ˆ ˆ ͒ ˆ ؠ ˆ ˆ ؠ ˆ ͑ ͒ z ˆ T k1W1 + k2W2 = k1T W1 + k2T W2 9 rigid body reference body u = r = q = where the coefficients k1 and k2 are arbitrary constants. This lin- (1 0 0), (0 0 0), 0 earity property is very important in the freedom and constraint pattern analysis and synthesis. Fig. 3 An ideal wire flexure imposes on a rigid body an ideal constraint, which removes the translational freedom in the 3 The Screw Theory Representation of the Constraint- axial direction of the wire flexure Based Design Approach This section relates concepts and design rules in constraint- based design approach to screw theory. ˆ ͉ W = ͑F͉M͒ = ͑Fx Fy Fz Mx My Mz ͒ 3.1 Freedom and Constraint of a Rigid Body. Constraint 2 2 2 ͑ ͒ and freedom are key concepts in the constraint-based design ap- such that FxMx + FyMy + FzMz =0,Fx + Fy + Fz 0 11 proach. In screw theory, a DOF is represented by a general twist. In real designs, the ideal constraint can usually be approximated As special cases, a rotational freedom or a translational freedom by a rigid link with two ball joints at both ends or a compliant link can be represented by a twist of pure rotation or pure translation ͑wire flexure͒ that is much stiffer along the direction of its axis shown in Eqs. ͑2͒ and ͑3͒, respectively. than along the direction perpendicular to the axis ͓17͔. Figure 3͑a͒ In this paper, all vectors are considered row vectors. For ex- shows a rigid body constrained by an ideal wire flexure, which ⍀෈ 3 ⍀ ͑⍀ ⍀ ⍀ ͒ ample, R is denoted by = x y z . Also, a general does not allow compression or stretch in its axial direction but is twist is essentially a six-dimensional row vector, i.e., Tˆ fully compliant in the perpendicular directions. Without losing 6 x =͑⍀x ⍀y ⍀z ͉ Vx Vy Vz͒෈R . An unconstrained rigid body generality, let us assume the wire axis aligns along the -axis. The has six DOF in space, i.e., three rotations and three translations constraint representing the wire flexure is denoted by a wrench along three orthogonal axes ͑Fig. 2͒ denoted by principle twists, Wˆ =͑100͉ 000͒, shown in Fig. 3͑b͒. The freedoms subject written as to the constraint can be obtained by requiring a general twist Tˆ ˆ ͑ ͉ ͒ reciprocal to the wrench. This allows us to obtain from Eq. ͑7͒ TRx = 100 000 Tˆ ؠ Wˆ =0,⇒ V =0 ͑12͒ ͒ ͉ ͑ ˆ TRy = 010 000 x ˆ ͑ ͉ ͒ Clearly this constraint removes the translational freedom along the TRz = 001 000 ͑10͒ x-axis. ˆ ͑ ͉ ͒ TTx = 000 100 Even though it is quite straightforward to design a translational Ά · constraint, it is not trivial to design a rotational constraint ͑infinite ˆ ͑ ͉ ͒ TTy = 000 010 pitch͒ or a general constraint ͑finite pitch͒. Typically a complex Tˆ = ͑000͉ 001͒ structure formed through cascading intermediate bodies must be Tz used. Hunt ͓23͔ provided a “wrench support” that applies a gen- When a mechanical connection is built between the rigid body eral constraint to a body. Blanding ͓17͔ showed an example of and a reference base in such as way that the number of DOF of the rotational constraint realized by using two pulleys and a cable. rigid body is reduced, this body is said to be constrained. The Since building rotational and general constraints is relatively com- number of reduced DOF is defined as degree-of-constraint ͑DOC͒ plicated ͑arguably not cost effective͒, it is preferable to use trans- of the mechanical connection. In screw theory, a constraint can be lational ͑ideal͒ constraints when possible in compliant mechanism described by a general wrench. A constraint that eliminates trans- design. lation along a line is called a translational constraint, which can be represented by a wrench of pure force shown in Eq. ͑5͒.A 3.2 Freedom Space and Constraint Space. The freedom constraint that eliminates rotation about a line is called a rota- space ͑topology͒ of a rigid body represents all of its allowable tional constraint represented by a wrench of pure couple shown in motion in space. Mathematically, the freedom space can be de- ͟ Eq. ͑6͒. scribed by a twist matrix T formed by f independent twists ˆ In the constraint-based design approach, an ideal constraint is a Tj͑j=1,...,f͒, written as slender structural member that is infinitely stiff along its axis but ˆ ⍀ ͉ is infinitely compliant perpendicular to its axis. The ideal con- T1 1 V1 straint is essentially a nontrivial translational constraint repre- Tˆ ⍀ ͉ V sented by a wrench of pure force, expressed as ͟ = 2 = 2 2 ͑13͒ T ΄ ΅ ΄ ] ΅ ]]] ⍀ ͉ ˆ f Vf y Tf Tˆ ,Tˆ rotation (p=0) Ry Ty ˆ Tˆ translation (p=∞) where f is called the dimension of the freedom space. Tj are the p basis twists that span the freedom space. Any motion in the free- screw motion dom space can be denoted by a linear combination of the basis x twists f Tˆ ,Tˆ z Rx Tx ˆ ˆ ͑ ͒ T = ͚ kjTj 14 j=1 ˆ ˆ TRz ,TTz where kj are arbitrary constants and cannot be zero simulta- ˆ Fig. 2 An unconstrained rigid body has three translations and neously. If the rank of the twist matrix ͟T is less than f, twists Tj three rotations represented by six principle twists are redundant, meaning that some twists can be written as linear

Journal of Mechanisms and Robotics NOVEMBER 2009, Vol. 1 / 041009-3

Downloaded From: http://mechanismsrobotics.asmedigitalcollection.asme.org/ on 02/24/2014 Terms of Use: http://asme.org/terms ˆ ˆ T2 2 z T3 z Tˆ z y ˆ 1 W1 ˆ r ˆ T2 p ≠ 0 ˆ r 1 Tˆ T1 2 W r 3 2 1 c ˆ 2 ˆ y Wˆ y T1 c c W2 3 1 r Tˆ 2 Tˆ c c 1 1 rigid body 2 2 1 x Ideal sheet flexure (a) constraint space (b) freedom space (a) two parallel lines (b) two intersecting lines (c) two skew lines Fig. 5 Constraint and freedom space of a rigid body con- Fig. 4 Two-dimensional freedom space generated by „a… two strained by an ideal sheet ﬂexure parallel lines, „b… two intersecting lines, and „c… two skew lines

Wˆ F ͉ M combinations of others. 1 1 1 ͉ Freedom space sometimes can have a geometric representation Wˆ F2 M2 ͟ = 2 = ͑18͒ in space. For instance, a one-dimensional freedom space is a W ΄ ΅ ΄ ] ΅ ]] ] single line in space. A two-dimensional freedom space is a surface ͉ ˆ Fc Mc generated by two lines. A three-dimensional space is a volume, Wc e.g., a solid sphere generated by three rotational twists intersecting ˆ at the same point. Recently, Hopkins and Culpepper ͓20͔ system- where c is called the dimension of the constraint space. Wi are the ͟ atically enumerated the topologies of freedom and constraint basis wrenches that span the whole constraint space. If matrix W ˆ space and provided a graphical illustration for each case. does not have a full rank, the wrenches Wi are redundant, meaning Let us take a look at a two-dimensional freedom space spanned that removing some of the constraints does not affect the mobility by two pure rotations. Physically this freedom space can be gen- of the constrained body. Similar to freedom space, any constraint erated by a serial chain of two revolute joints. Depending on in the constraint space can be represented by a linear combination whether the two rotation axes are parallel, intersecting, or skew, of the basis wrenches the freedom space is a plane, a disk, or a cylindroid, respectively. c Figure 4͑a͒ shows a plane generated by two parallel rotational Wˆ = ͚ k Wˆ ͑19͒ ˆ ͑⍀͉ ϫ⍀͒ ˆ ͑⍀͉ ϫ⍀͒ i i twists T1 = c1 and T2 = c2 . Any parallel lines on i=1 the plane can be represented by Figure 5͑a͒ shows a rigid body constrained by an ideal sheet Tˆ = k Tˆ + k Tˆ = ͑͑k + k ͒⍀͉͑k c + k c ͒ ϫ ⍀͒͑15͒ flexure. Since its thickness is much smaller than its width and 1 1 2 2 1 2 1 1 2 2 length, an ideal sheet flexure allows the bending but not stretching where the coefficients k1 and k2 can be viewed as the angular or compressing in the sheet plane. In the constraint-based design speeds of the joints if the freedom space is generated by a serial approach ͓17͔, an ideal sheet flexure applies three ideal constraints chain of two revolute joints. If the angular speeds are constrained on the sheet plane. In our screw approach, these three ideal con- ͑ ͒ such that k1 +k2 =0, the twist in Eq. 15 shows a pure translation straints can be represented by a set of any three independent in the normal direction of the plane formed by the two parallel wrenches on the sheet plane. An example set of three constraints lines. This result tells us that a serial chain of two parallel revolute is shown in Fig. 5͑a͒. Wrenches Wˆ and Wˆ are parallel to the joints can generate an instantaneous translation by driving the two 1 2 y-axis and intersecting the x-axis at r =͑100͒ and r joints with same angular velocity but in the opposite direction. 1 2 ͑ ͒ ˆ Figure 4͑b͒ shows a disk generated by two rotational twists = −100, respectively. And the wrench W3 aligns with the intersecting at a point c. Any freedom in the space can be ex- x-axis. They are written as pressed by ˆ ͑ ͉ ͒ W1 = 010 001 ˆ ˆ ˆ ͑ ⍀ ⍀ ͉ ϫ ͑ ⍀ ⍀ ͒͒ ͑ ͒ T = k1T1 + k2T2 = k1 1 + k2 2 c k1 1 + k2 2 16 ˆ ͑ ͉ ͒ ͑ ͒ Clearly, every freedom in this space is still a pure rotation about a W2 = 010 00−1 20 ⍀ ⍀ line through the same point c and in the direction k1 1 +k2 2. ˆ ͑ ͉ ͒ Figure 4͑c͒ shows a cylindroid generated by two skew rota- W3 = 100 000 tional twists. Any freedom in the space can be described by a Any other ideal constraint on the sheet plane in the direction general twist F=͑Fx Fy 0͒ through a point r=͑rx ry 0͒ is written as ˆ ˆ ˆ ͑ ⍀ ⍀ ͉ ϫ ⍀ ϫ ⍀ ͒ T = k1T1 + k2T2 = k1 1 + k2 2 c1 k1 1 + c2 k2 2 ˆ ͉ W = ͑F͉r ϫ F͒ = ͑Fx Fy 0 00rxFy − ryFx ͒ ͑21͒ ͑17͒ which can be expressed a linear combination of Wˆ , Wˆ , and Wˆ ˆ ˆ 1 2 3 One can see that all motions in the space except T1 ,T2 are in the as form of screw motion since the pitch of twist in Eq. ͑15͒ is none zero in general. ˆ ͩ rxFy − ryFx + Fy ͪ ˆ ͩ Fy − rxFy + ryFx ͪ ˆ ˆ W = W1 + W2 + FxW3 The freedom space of four and five dimensions cannot be rep- 2 2 resented graphically in general. Only in some special cases could ͑22͒ we use the combination of lower dimensional spaces to describe a higher dimensional space. For these cases, the twist matrix is 3.3 Rule of Complementary Patterns. The design rules in more preferable to describe higher dimensional freedom space. the constraint-based design approach are mostly illustrated in the ͑ ͒ The constraint space topology of a rigid body represents all form of subjective statements. The most important one is the Rule the forbidden motions of the body subject to a constraint arrange- of Complementary Patterns, which states that, “when a pattern of ment. In screw theory, a constraint space can be represented by a c nonredundant constraints is applied between an object and a ˆ wrench matrix formed by c independent wrenches Wi͑i reference body, the object will have f =6−c independent degrees- =1,...,c͒, written as of-freedom.” This rule can be explained in screw theory as fol-

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Downloaded From: http://mechanismsrobotics.asmedigitalcollection.asme.org/ on 02/24/2014 Terms of Use: http://asme.org/terms ˆ ˆ y y lows. Let wrenches W ͑i=1,...,c͒ denote c nonredundant con- W2 ˆ i Tˆ W1 1 Tˆ ( p = 0) Tˆ ͑j f͒ f 1 straints and twists j =1,..., denote nonredundant degrees- Wˆ Wˆ of-freedom. And all wrenches are reciprocal to all twists 2 1 ˆ p = ˆ T2 ( 1) T2 ˆ ؠ ˆ Tj Wi =0, i =1,...,c, j =1,...,f, and c + f =6 ˆ x W3 ˆ x ͑23͒ W3 z Wˆ z Wˆ 4 There are two major categories of compliant mechanism design 4 using the constraint-based design approach. One is called con- (a) physical arrangement (b) constraint space straint pattern analysis, which studies the mobility of a rigid body subject to a pattern of constraints. The other is called constraint Fig. 6 A body constrained by a pattern of ideal constraints pattern design, which seeks for a pattern of constraints to achieve can have a screw motion in space a specified pattern of freedoms. We discuss each in the Secs 3.4 and 3.5, respectively. 3.4 Constraint Pattern Analysis. Here let us use the sheet ˆ ͑ ͉ ͒ flexure shown in Fig. 5͑a͒ as an example to demonstrate the steps W1 = 010 001 for constraint pattern analysis. Step 1. Write all constraints in the format of wrenches, which Wˆ = ͑010͉ 00−1͒ ͑ ͒ 2 are written in Eq. 20 . ͑25͒ ˆ ͑⍀ ⍀ ⍀ ͉ ͒ Step 2. Require a general twist T= x y z Vx Vy Vz Wˆ = ͑001͉ 000͒ reciprocal to all wrenches in Eq. ͑20͒ and yield a system of linear 3 equations ˆ ͑ ͉ ͒ W4 = 101 −101 ؠ ˆ ⇒ ⍀ ˆ T W1 =0 1Vy +1 z =0 Following the constraint pattern analysis steps yields two independent twists Tˆ ؠ Wˆ =0⇒ 1V −1⍀ =0 ͑24͒ 2 y z ˆ ͑ ͉ ͒ T1 = 010 000 p1 =0 ؠ ˆ ⇒ ͑26͒ ˆ T W3 =0 1Vx =0 Tˆ = ͑100͉ 100͒ p =1 where the reciprocal product is explicitly expressed to show the 2 2 ͑ ͒ consistence in units. They represent a pure rotation p1 =0 around y-axis and a screw ⍀ Step 3. Solve the above linear system to obtain Vx =Vy = z =0 motion around x-axis with pitch p2 =1, respectively. These two and write the complementary freedom space in the format of a independent twists form the basis of the two-dimensional freedom general twist space that defines the allowable motion of the constrained body. Let us have a look at the freedom space to see if there exists ˆ ͑⍀ ⍀ ͉ ͒ T = x y 0 00Vz ˆ any rotational freedom other than T1. According to the definition Step 4. Find independent basis twists from the above twist sys- of freedom space, any freedom Tˆ =͑⍀͉V͒ in the freedom space ⍀ ⍀ tem. For this example, simply setting one of x, y, and Vz to be ˆ ˆ nonzero and the other two to be zero yields three basis twists can be expressed as linear combination of T1 and T2, written as ˆ ͑ ͉ ͒ Tˆ = k Tˆ + k Tˆ = ͑k k 0 ͉ k 00͒ ͑27͒ T1 = 100 000 1 1 2 2 2 1 2 If Tˆ is a rotation, we must have Tˆ = ͑010͉ 000͒ 2 ⍀ ⇒ 2 ⇒ ͑ ͒ · V =0 k2 =0 k2 =0 28 ˆ ͑ ͉ ͒ T3 = 000 001 ˆ ˆ However, this means that T is simply a multiply of T1. Hence this Clearly they represent rotation around x-axis, rotation around proves that no other rotation line exists in this freedom space. In ͑ ͑ ͒͒ ˆ y-axis and translation along z-axis, respectively see Fig. 5 b . other words, all freedoms except the rotation T1 are in the form of Note the step of choosing basis twists is not unique. Any three screw motion. independent twists in the freedom space should span the same freedom space. 3.5 Constraint Pattern Design. The constraint pattern design It should be pointed out that a general freedom is usually in the starts with specifying a freedom space described by a set of f ͑ ͒ ˆ form of screw motion represented by a general twist finite pitch . twists Tj͑j=1,...,f͒. The objective is to find the complementary However, for the sake of intuition, we prefer to use rotational or constrained space, which is to find c=6−f independent constraints translational twists as basis twists whenever possible to represent denoted by wrenches Wˆ ͑i=1,...,c͒. the freedom space. An interesting question is, “Can we always i Here we use an example to demonstrate the constraint pattern find f =6−c rotational freedom for a pattern of c constraints?” design steps. Suppose we want to design a compliant mechanism Unfortunately the answer is NO even when all the constraints in with two allowable motions: a pure rotation around z-axis and a an arrangement are ideal, i.e., q=0. A counter example is shown ͑ ͒͑ ͒ in Fig. 6 where a rigid body is subject to four ideal constraints. pure translation along the direction of 011 see Fig. 7 .We The physical arrangement of this constraint pattern and corre- seek an arrangement of ideal wire flexures, which constrain a rigid sponding constraint space are shown in Figs. 6͑a͒ and 6͑b͒, re- body and allow the prescribed motion. The constraint space is ˆ ˆ found by the following steps. spectively. Wrenches W1 and W2 are parallel to the y-axis and Step 1. Denoting all specified freedoms in twists yields intersect with the x-axis at +1 and −1, respectively. Wrench Wˆ 3 ˆ ͑ ͉ ͒ ˆ T1 = 001 000 p1 =0 aligns the z-axis. And the last wrench W4 is a skew line on the plane y=−1 and has an angle of 45 deg with both x-axis and ͑29͒ ˆ ͑ ͉ ͒ ϱ z-axis. They are written as T2 = 000 011 p2 =

Journal of Mechanisms and Robotics NOVEMBER 2009, Vol. 1 / 041009-5

Downloaded From: http://mechanismsrobotics.asmedigitalcollection.asme.org/ on 02/24/2014 Terms of Use: http://asme.org/terms y Wˆ freedoms are translational, by going through the design steps, one ˆ 4 can ﬁnd that there do not exist 6− f complementary ideal con- W3 ˆ straints. Apparently, these cases are not rare. For these design T2 cases, one would have to use cascaded complex structures to provide rotational or general constraints. This is beyond the scope of ˆ this paper and shall be an interesting research topic in the future. W2

x 4 Case Study: Design of Compliant Prismatic Joints Wˆ In this section, we show how to apply the proposed approach to Tˆ 1 z 1 the design of a compliant prismatic joint using wire or sheet ﬂex- ures. Without losing generality, let us assume that the given single freedom is a translational motion along the x-axis, denoted by a Fig. 7 Four ideal constraints found for given a pattern of 2 twist Tˆ =͑000͉ 100͒. By following the constraint design DOF steps, we ﬁnd Fx =0, i.e., all wrenches representing ideal constraints in the constraint space must have the form

Wˆ = ͑0 F F ͉ M M M ͒ ͑36͒ Step 2. Requiring a general wrench Wˆ y z x y z To design the constraint pattern, we only need to find five in- =͑Fx Fy Fz ͉ Mx My Mz͒ reciprocal to both twists yields dependent constraints in the constraint space. Suppose we want to ͒ ͑ ͒ ͑ ⇒ ˆ ؠ ˆ T1 W =0 Mz =0 30 use five wire flexures ideal constraints to achieve the design. This means that Eq. ͑36͒ is subject to ͒ ͑ Tˆ ؠ Wˆ =0⇒ F + F =0⇒ F =−F ͑31͒ 2 2 2 y z z y FyMy + FzMz =0,Fy + Fz 0 37 Since we are only interested in ideal constraints ͑design with wire If Mx =0, we substitute either Fy =0 or Fz =0 ͑note Fy and Fz flexures͒, we also want cannot be zero simultaneously͒ into Eq. ͑37͒ and obtain the fol- 2 2 2 ͑ ͒ lowing four subcases FxMx + FyMy + FzMz =0,Fx + Fy + Fz 0 32 Step 3.Substituting Eqs. ͑30͒ and ͑31͒ into Eq. ͑32͒,wecan Fy 0,Fz =0,My =0,Mz =0 write a general wrench in the complementary constraint space as Fy 0,Fz =0,My =0,Mz 0 ͑38͒ ˆ ͑ ͉ ͒ Ά Fy =0,Fz 0,My =0,Mz =0· W = Fx Fy − Fy Mx My 0 Fy =0,Fz 0,My 0,Mz =0 such that F M + F M =0,F2 +2F2 0 ͑33͒ x x y y x y which lead us the following four independent wrenches Step 4. Categorize the condition Eq. ͑33͒ and find independent ˆ ͑ ͉ ͒ subcases. For the sake of intuitiveness, we prefer to find con- W1 = 010 000 straints parallel to coordinate axes or plane whenever possible. Wˆ = ͑010͉ 001͒ This can be done by assigning one or more force elements to be 2 ͑39͒ zero and solve the other elements by Eq. ͑33͒. For this example, ˆ ͑ ͉ ͒ ΆW3 = 001 000· the following subcases are obtained ˆ ͑ ͉ ͒ W4 = 001 010 F 0,F =0,M =0,M =0 x y x y They represent two constraint lines parallel to the y-axis and two Fx 0,Fy =0,Mx =0,My 0 ͑ ͒ ͑34͒ constraint lines parallel to the z-axis, respectively Fig. 8 . And if Ά Fx =0,Fy 0,Mx =0,My =0· Mx 0, setting Fz =My =Mz =0 yields the last ideal constraint Fx =0,Fy 0,Mx =0,My 0 ˆ ͑ ͉ ͒ ͑ ͒ W5 = 010 100 40 ͑ ͒ Step 5. Write the subcases in 32 in the form of wrenches ͑ ͒ This constraint is a line in the direction F5 = 010 and passing ˆ ͑ ͉ ͒ through a point r =͑001͒. By checking the rank of the wrench W1 = 100 000 5 matrix of the constraint space, we verify that the five wrenches are ˆ ͑ ͉ ͒ W2 = 100 010 not redundant. ͑35͒ ˆ ͑ ͉ ͒ The design of the compliant prismatic joint using five wire ΆW3 = 01−1 000· flexures is shown in Fig. 9͑a͒. The compliant prismatic joint al- ˆ ͑ ͉ ͒ W4 = 01−1 100 By checking the rank of the wrench matrix, we find that the four ͑ ͒ ˆ ˆ y wrenches in Eq. 35 are independent. Wrenches W1 and W2 are ˆ parallel to the x-axis and intersecting the twist T1. Both wrenches ˆ Wˆ Wˆ ˆ ˆ ˆ W 1 2 W3 and W4 are perpendicular with T2 and lie on the y-plane and 5 z-plane. See Fig. 7. In general the pitches of wrenches in the complementary con- ˆ straint space may be zero, infinite or nonzero finite, which corre- T x spond to translational ͑ideal͒, rotational and general constraint, respectively. Because of the relatively low cost of building ideal Wˆ constraints in compliant mechanism design, we would prefer to 3 find as many ideal constraints in the constraint space as possible. Wˆ z 4 As in constraint pattern analysis, one should be aware that it is NOT always possible to find 6− f independent ideal constraints for arbitrary f freedoms. For instance, if two or three of the given Fig. 8 The constraint space for a compliant prismatic joint

041009-6 / Vol. 1, NOVEMBER 2009 Transactions of the ASME

Downloaded From: http://mechanismsrobotics.asmedigitalcollection.asme.org/ on 02/24/2014 Terms of Use: http://asme.org/terms Wˆ Wˆ y 4 y 4 6 Conclusion rigid body rigid body x x A screw theory based approach for the design of flexure based z pure z pure joints for compliant mechanisms is introduced in this paper. This Wˆ translation translation 3 approach presents a mathematical representation of the constraint- Wˆ Wˆ Wˆ 1 2 sheet 2 based design approach, which is typically characterized by sub- flexure jective statements. A constraint and a freedom in the constraint- Wˆ 5 based design approach can be mathematically denoted by a reference reference body body wrench and a twist, respectively. The constraint topology or space (a) (b) or pattern formed by a system of constraints acting on a rigid body is essentially a linear space spanned by a system of independent Fig. 9 Design of a compliant prismatic joint using „a… five wire wrenches. Similarly, a freedom space that describes the possible flexures or „b… one flexure sheet and two wire flexures instantaneous motion of a rigid body is represented by a system of independent twists. These two spaces are complementary and can be found from each other using linear algebra. Two major com- y pliant mechanism design problems, constraint pattern analysis, rigid body y pure and constraint pattern design are elaborated with examples of ˆ ˆ ˆ translation compliant mechanisms and flexures. Lastly, the approach is illus- ˆ W1 W6 W2 W5 sheet sheet trated by a case study of the design of compliant prismatic joints flexure 1 flexure 2 Tˆ using wire flexures and sheet flexures. It should be pointed out x that the motion of compliant mechanisms studied in this paper Wˆ x 3 must be sufficiently small since screw theory deals with instanta- Wˆ reference z 4 neous motion only. Nevertheless, the proposed analysis/design z body (a)(b) framework may be beneficial to the type synthesis of compliant mechanisms. Fig. 10 Design of a compliant prismatic joint using two parallel flexure sheets Acknowledgment The authors acknowledge the funding provided by the National Science Foundation under Grant No. CMMI-0457041 for this re- lows the rigid body to translate along the x direction only. As search. Hai-Jun Su also acknowledges the support provided by the shown in Sec. 3.2, a single sheet flexure provides three ideal con- National Science Foundation CAREER under Grant No. CMMI- straints whose axes lie on the same plane. An alternative physical 0845793. arrangement is to use a single sheet flexure to replace the wire ˆ ˆ ˆ ͑ ͒ flexures W1, W3, and W5. This design is shown Fig. 9 b . Nomenclature ˆ ⍀ ϭ a 3D vector presenting the angular velocity of Furthermore let us manually add a sixth constraint W6, which is a twist parallel to Wˆ and intersects Wˆ at the point ͑101͒͑Fig. 10͑a͒͒. 2 4 V ϭ a 3D vector presenting the linear velocity of a ˆ ͑ Since W6 is redundant can be written as a linear combination of twist ˆ ϭ Wi͑i=1,...,5͒, it does not apply extra constraint to the rigid body. p a scalar representing the pitch of a twist However this allows us to use a second flexure sheet to apply the Tˆ ϭ a 6D general twist representing an allowable constraints Wˆ , Wˆ , and Wˆ . The result is the well known parallel motion 2 4 6 ͟ ϭ flexure design of compliant prismatic joint ͑see Fig. 10͑b͒͒. T a twist matrix representing the allowable mo- As one can see, the choice of the basis constraints is not unique, tion space which means that there are multiple constraint patterns that will F ϭ a 3D vector presenting the force part of a achieve a given freedom pattern. Even for the same constraint wrench pattern, there may be multiple physical arrangements ͑designs͒. M ϭ a 3D vector presenting the couple part of a Apparently this is beneficial to the designers as they have multiple wrench ways to achieve the same design goal. An interesting future re- q ϭ a scalar representing the pitch of a wrench search task is to systematically find all possible physical arrange- Wˆ ϭ a 6D general wrench representing a constraint ments for a given freedom pattern. ͟W ϭ a wrench matrix representing the constraint space 5 Future Work References The major contribution of this paper is a systematic formulation ͓1͔ Howell, L. L., 2001, Compliant Mechanisms, Wiley-Interscience, New York. of the constraint-based design approach from the precision engi- ͓2͔ Howell, L. L., and Midha, A., 1995, “Parametric Deflection Approximations neering community using the screw theory, which is well studied for End-Loaded, Large-Deflection Beams in Compliant Mechanisms,” ASME by the kinematics community. This formulation provides a rigor- J. Mech. Des., 117͑1͒, pp. 156–165. ͓ ͔ ous mathematical proof of the narrative ͑arguably vague͒ design 3 Su, H.-J., 2009, “A Pseudorigid-Body 3R Model for Determining Large De- flection of Cantilever Beams Subject to Tip Loads,” ASME J. Mech. Rob., rules of the constraint-based design approach. It enables a large 1͑2͒, p. 021008. body of future work. The presented work mainly focuses on the ͓4͔ Xu, P., Yu, J., Zong, G., Bi, S., and Hu, Y., 2009, “A Novel Family of Leaf- qualitative design, i.e., type synthesis, of flexural joints that are Type Compliant Joints: Combination of Two Isosceles-Trapezoidal Flexural ͑ ͒ the basic building blocks of compliant mechanisms. One interest- Pivots,” ASME J. Mech. Rob., 1 2 , p. 021005. ͓5͔ Ananthasuresh, G. K., and Kota, S., 1995, “Designing Compliant Mecha- ing future research topic is to incorporate quantitative design into nisms,” Mech. Eng. ͑Am. Soc. Mech. Eng.͒, 117͑11͒, pp. 93–96. the framework. Another future research topic would be expanding ͓6͔ Frecker, M. I., Ananthasuresh, G. K., Nishiwaki, S., Kikuchi, N., and Kota, S., the study of the wire/sheet flexure to other flexure elements. 1997, “Topological Synthesis of Compliant Mechanisms Using Multi-Criteria ͑ ͒ Lastly, through the presented mathematical formulation, many Optimization,” ASME J. Mech. Des., 119 2 , pp. 238–245. ͓7͔ Sigmund, O., 1997, “On the Design of Compliant Mechanisms Using Topol- computational techniques such as optimization and kinematic ogy Optimization,” Mech. Struct. Mach., 25͑4͒, pp. 493–524. solvers can be applied to the synthesis of compliant mechanisms. ͓8͔ Hull, P. V., and Canfield, S., 2006, “Optimal Synthesis of Compliant Mecha-

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Downloaded From: http://mechanismsrobotics.asmedigitalcollection.asme.org/ on 02/24/2014 Terms of Use: http://asme.org/terms nisms Using Subdivision and Commercial FEA,” ASME J. Mech. Des., ͓22͔ Ball, R. S., 1998, The Theory of Screws, Cambridge University Press, Cam- 128͑2͒, pp. 337–348. bridge, UK, Original Work Published in 1876. ͓9͔ Zhou, H., and Ting, K.-L., 2005, “Topological Synthesis of Compliant Mecha- ͓23͔ Hunt, K. H., 1978, Kinematic Geometry of Mechanisms, Oxford University nisms Using Spanning Tree Theory,” ASME J. Mech. Des., 127͑4͒, pp. 753– Press, New York. 759. ͓24͔ Phillips, J., 1984, Freedom in Machinery. Volume 1, Introducing Screw Theory, ͓10͔ Howell, L. L., and Midha, A., 1996, “A Loop-Closure Theory for the Analysis Cambridge University Press, Cambridge, UK. and Synthesis of Compliant Mechanisms,” ASME J. Mech. Des., 118͑1͒, pp. ͓25͔ Phillips, J., 1990, Freedom in Machinery. Volume 2, Screw Theory Exempli- 121–125. ﬁed, Cambridge University Press, Cambridge, UK. ͓ ͔ ͓11͔ Su, H.-J., and McCarthy, J. M., 2006, “A Polynomial Homotopy Formulation 26 Lipkin, H., and Patterson, T., 1992, “Geometric Properties of Modelled Robot of the Inverse Static Analysis of Planar Compliant Mechanisms,” ASME J. Elasticity—Part I: Decomposition,” Proceedings of ASME Design Technical Mech. Des., 128͑4͒, pp. 776–786. Conferences, DE-Vol. 45, pp. 187–193. ͓ ͔ ͓12͔ Wang, M. Y., Chen, S., Wang, X., and Mei, Y., 2005, “Design of Multimaterial 27 Lipkin, H., and Patterson, T., 1992, “Geometric Properties of Modelled Robot Compliant Mechanisms Using Level-Set Methods,” ASME J. Mech. Des., Elasticity—Part II: Decomposition,” Proceedings of ASME Design Technical Conferences 127͑5͒, pp. 941–956. , DE-Vol. 45, pp. 179–185. ͓28͔ Patterson, T., and Lipkin, H., 1993, “Structure of Robot Compliance,” ASME ͓13͔ Kim, C. J., Kota, S., and Moon, Y.-M., 2006, “An Instant Center Approach J. Mech. Des., 115͑3͒, pp. 576–580. Toward the Conceptual Design of Compliant Mechanisms,” ASME J. Mech. ͓29͔ Huang, S., and Schimmels, J. M., 1998, “The Bounds and Realization of Des., 128͑3͒, pp. 542–550. Spatial Stiffnesses Archieved With Simple Springs Connected in Parallel,” ͓14͔ Su, H.-J., and McCarthy, J. M., 2007, “Synthesis of Bistable Compliant Four- IEEE Trans. Robot. Autom., 14͑3͒, pp. 466–475. Bar Mechanisms Using Polynomial Homotopy,” ASME J. Mech. Des., ͓30͔ Huang, S., and Schimmels, J. M., 2000, “The Bounds and Realization of 129͑10͒, pp. 1094–1098. ͓ ͔ Spatial Compliance Archieved With Simple Serial Elastic Mechanisms,” IEEE 15 Wang, M. Y., 2009, “A Kinetoelastic Formulation of Compliant Mechanism Trans. Robot. Autom., 16͑1͒, pp. 99–103. ͑ ͒ Optimization,” ASME J. Mech. Rob., 1 2 , p. 021011. ͓31͔ Dai, J. S., Huang, Z., and Lipkin, H., 2006, “Mobility of Overconstrained ͓ ͔ 16 Maxwell, J. C., 1890, General Considerations Concerning Scientiﬁc Appara- Parallel Mechanisms,” ASME J. Mech. Des., 128͑1͒, pp. 220–229. tus, Dover, New York. ͓32͔ Adams, J. D., and Whitney, D. E., 2001, “Application of Screw Theory to ͓ ͔ 17 Blanding, D. L., 1999, Exact Constraint: Machine Design Using Kinematic Constraint Analysis of Mechanical Assemblies Joined by Features,” ASME J. Processing, ASME, New York. Mech. Des., 123͑1͒, pp. 26–32. ͓18͔ Hale, L. C., 1999, “Principles and Techniques for Designing Precision Ma- ͓33͔ Smith, D., 2003, “Constraint Analysis of Assemblies Using Screw Theory and chines,” Ph.D. thesis, MIT, Cambridge, MA. Tolerance Sensitivities,” MS thesis, Brigham Young University, Provo, UT. ͓19͔ Awtar, S., and Slocum, A. H., 2007, “Constraint-Based Design of Parallel ͓34͔ Kong, X., and Gosselin, C. M., 2004, “Type Synthesis of 3-DOF Translational Kinematic XY Flexure Mechanisms,” ASME J. Mech. Des., 129͑8͒, pp. 816– Parallel Manipulators Based on Screw Theory,” ASME J. Mech. Des., 126͑1͒, 830. pp. 83–92. ͓20͔ Hopkins, J. B., and Culpepper, M. L., 2009, “Synthesis of Multi-Degree of ͓35͔ Kim, C. J., 2008, “Functional Characterization of Compliant Building Blocks Freedom, Parallel Flexure System Concepts Via Freedom and Constraint To- Utilizing Eigentwists and Eigenwrenches,” Proceedings of ASME IDETC/CIE pology ͑FACT͒—Part I: Principles,” Precis. Eng., July. 2008, New York, August 3–6. ͓21͔ Shai, O., and Pennock, G. R., 2006, “A Study of the Duality Between Planar ͓36͔ McCarthy, J. M., 2000, Geometric Design of Linkages, Springer-Verlag, New Kinematics and Statics,” ASME J. Mech. Des., 128͑3͒, pp. 587–598. York.

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