A Screw Theory Approach for the Type Synthesis of Compliant Mechanisms With Flexures

Transcription

1 Mechanical Engineering Conference Presentations, Papers, and Proceedings Mechanical Engineering A Screw Theor Approach for the Tpe Snthesis of Compliant Mechanisms With Fleures Hai-Jun Su Universit of Marland, Baltimore Count Denis V. Dorohkin Iowa State Universit, dorodv@gmail.com Jud M. Vance Iowa State Universit, jmvance@iastate.edu Follow this and additional works at: Part of the Computer-Aided Engineering and Design Commons Recommended Citation Su, Hai-Jun; Dorohkin, Denis V.; and Vance, Jud M., "A Screw Theor Approach for the Tpe Snthesis of Compliant Mechanisms With Fleures" (2009). Mechanical Engineering Conference Presentations, Papers, and Proceedings This Conference Proceeding is brought to ou for free and open access b the Mechanical Engineering at Iowa State Universit Digital Repositor. It has been accepted for inclusion in Mechanical Engineering Conference Presentations, Papers, and Proceedings b an authoried administrator of Iowa State Universit Digital Repositor. For more information, please contact digirep@iastate.edu.

2 A Screw Theor Approach for the Tpe Snthesis of Compliant Mechanisms With Fleures Abstract This paper presents a screw theor based approach for the tpe snthesis of compliant mechanisms with fleures. We provide a sstematic formulation of the constraint-based approach which has been mainl developed b precision engineering eperts in designing precision machines. The two fundamental concepts in the constraint-based approach, constraint and freedom, can be represented mathematicall b a wrench and a twist in screw theor. For eample, an ideal wire fleure applies a translational constraint which can be described a wrench of pure force. As a result, the design rules of the constraint-based approach can be sstematicall formulated in the format of screws and screw sstems. Two major problems in compliant mechanism design, constraint pattern analsis and constraint pattern design are discussed with eamples in details. This innovative method paves the wa for introducing computational techniques into the constraintbased approach for the snthesis and analsis of compliant mechanisms. Kewords VRAC, Screws, Compliant mechanisms Disciplines Computer-Aided Engineering and Design Mechanical Engineering This conference proceeding is available at Iowa State Universit Digital Repositor:

3 Proceedings of the ASME 2009 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE 2009 Proceedings of IDETC/CIE 2009 ASME 2009 International August 30 Design - September Engineering 2, 2009, Technical San Diego, Conferences California, USA & Computers and Information in Engineering Conference August 30-September 2, 2009, San Diego, CA, USA DETC DETC A SCREW THEORY APPROACH FOR THE TYPE SYNTHESIS OF COMPLIANT MECHANISMS WITH FLEXURES Hai-Jun Su Department of Mechanical Engineering Universit of Marland, Baltimore Count Baltimore, Marland haijun@umbc.edu Denis V. Dorohkin, Jud M. Vance Department of Mechanical Engineering Iowa State Universit Ames, IA dorodv@iastate.edu, jmvance@iastate.edu ABSTRACT This paper presents a screw theor based approach for the tpe snthesis of compliant mechanisms with fleures. We provide a sstematic formulation of the constraint-based approach which has been mainl developed b precision engineering eperts in designing precision machines. The two fundamental concepts in the constraint-based approach, constraint and freedom, can be represented mathematicall b a wrench and a twist in screw theor. For eample, an ideal wire fleure applies a translational constraint which can be described a wrench of pure force. As a result, the design rules of the constraint-based approach can be sstematicall formulated in the format of screws and screw sstems. Two major problems in compliant mechanism design, constraint pattern analsis and constraint pattern design are discussed with eamples in details. This innovative method paves the wa for introducing computational techniques into the constraint-based approach for the snthesis and analsis of compliant mechanisms. NOMENCLATURE Ω A 3D vector presenting the angular velocit of a twist V A 3D vector presenting the linear velocit of a twist p A scalar representing the pitch of a twist ˆT A 6D general twist representing an allowable motion T A twist matri representing the allowable motion space F A 3D vector presenting the force part of a wrench Address all correspondence to this author. M q Ŵ W A 3D vector presenting the couple part of a wrench A scalar representing the pitch of a wrench A 6D general wrench representing a constraint A wrench matri representing the constraint space 1 Introduction Compared with traditional rigid bod mechanisms, compliant mechanisms [1] or fleures have man advantages, such as high precision and a simplified manufacturing and assembl process. However the design and analsis of compliant mechanisms is comple due to the nonlinearit of deformation of the fleible members. Researchers in two isolated fields, kinematics and mechanisms and precision engineering, have independentl made major contributions to compliant mechanism design. In the kinematics and mechanisms communit, research has focused on appling computational techniques to determine the dimensions and/or topologies of compliant mechanisms to achieve a pre-specified design objective. The two most often used approaches in this field are the Pseudo-Rigid-Bod-Model (PRBM) [2] approach and the topological snthesis approach [3 8]. The former approach models a compliant mechanism as a rigid bod with one or more springs. These springs impose approimated lumped compliance to the rigid link models of compliant mechanisms. This allows the theories and methodologies developed for rigid bod mechanisms [9, 10] to be used to design compliant mechanisms. Because of this simplification in the modeling process, it is necessar to evaluate the designs to ensure 1 Copright c 2009 b ASME

4 the validit of the PRBM. The topological snthesis approach models the compliant linkage as a network of link members of different sies which together achieve a specified objective function such as geometric advantage and mechanical advantage. The result is a compliant mechanism of comple topolog with distributed compliance. This compleit results in mechanisms that are difficult to manufacture and produce non-intuitive motions. In parallel, the precision engineering communit have been using the constraint-based approach for the design of compliant instruments with fleures. The foundations of the constraintbased method were developed b Mawell [11] in the 1880s. It was recentl revisited b Blanding [12] and several researchers at the MIT Precision Engineering Labs [13 15] for the design of fitures, rigid bod machines and fleure sstems. The fundamental premise of the constraint-based method is that all motions of a rigid bod are determined b the position and orientation of the constraints (constraint topolog) which are placed upon the bod. The method is attractive because it is based upon motion visualiation and is therefore well-suited to conceptual development; however the proficienc in using the constraint-based methods for designing compliant mechanisms requires commitment to a steep learning curve and development of hands-on eperience to understand the stiffness characteristics of alternate designs. Hence the design process is not sstematical and ma not necessaril lead to the optimal design, especiall when the designer is ineperienced. In this paper, a mathematical formulation of the constraintbased approach based upon screw theor is presented. A screw is the geometric entit that underlies the foundation of statics and instantaneous (first-order) kinematics. Man authors have made contributions to screw theor. The two fundamental concepts in screw theor are twist representing a general helical motion of a rigid bod about an instantaneous ais in space, and wrench representing a sstem of force and moment acting on a rigid bod. These two concepts are often called dualit [16] in kinematics and statics. Ball [17] was the first to establish a sstematical formulation for screw theor. Hunt [18] and Phillips [19, 20] further developed the mathematical and geometrical representation of screws and screw sstems. Their focus lies on the application of screw theor to the analsis and snthesis of mechanisms. Lipkin and Pattern [21 23] sstematicall investigated the screw theor and its applications to compliance or elasticit analsis of robot manipulators. Huang and Schimmels [24, 25] studied the realiation of a prescribed stiffness matri with serial or parallel elastic mechanisms. Other applications of screw theor include mobilit analsis [26], assembl analsis [27, 28] and topolog snthesis [29]. Recentl Kim [30] studied the characteriation of compliant building blocks b utiliing the concept of eigentwists and eigenwrenches based on screw theor. A constraint and a degree-of-freedom in the constraint-based approach can be described b a wrench and a twist in screw theor respectivel. Therefore, all the rules in constraint pattern analsis and design can be eplained and mathematicall represented using screw theor. The result is a powerful tool that is capable of sstematicall finding intuitive design topologies. The problem of analing and snthesiing spatial stiffness/compliance of a general elastic mechanism with spring elements has been investigated etensivel b Lipkin [21 23], Schimmels [24,25] and others. As far as the authors knowledge, appling screw theor to the tpe snthesis of compliant mechanisms with fleure elements is a new contribution to the field. This linking of screw theor to constraint-based compliant mechanism design allows a wide variet of computational techniques developed in the kinematics field [31] to be combined together with the constraint-based design approach to achieve design automation for compliant mechanisms. The rest of the paper is organied as follows. Section 2 provides a review of screw theor. Section 3 describes a screw representation of the basic concepts including constraint, constraint space, freedom and freedom space. Also in this section, the general steps for constraint pattern analsis and design are presented and illustrated with eamples. Section 5 presents conclusions and discussion. 2 Screw Theor Review This section provides a concise review of general screw theor. 2.1 Twists and Wrenches It is well known that screw theor underlies the foundation of both instantaneous kinematics and statics. In kinematics, a general spatial motion of a rigid bod is a screw motion (a rotation and a translation) about a line in space called screw ais. The rotation and translation is further coupled b a scalar quantit called pitch. The screw in kinematics also called a twist is formed b a pair of three dimensional vectors, namel angular velocit Ω and linear velocit V, written as Twist: ˆT = (Ω V) = (ωs c ωs + vs) (1) where vectors s and c denote the direction of and a point on the twist ais respectivel, scalars ω and v are the magnitude of angular velocit and partial linear velocit along the ais. The pitch is defined as the ratio of the linear velocit to angular velocit, i.e. p = v/ω. As special cases, a pure rotation and a pure translation in space are represented b a twist of ero pitch and infinite pitch respectivel, written as, Pure Rotation(p = 0) : ˆT = (ωs c ωs) (2) Pure Translation(p = ) : ˆT = (0 vs) (3) 2 Copright c 2009 b ASME

5 Note that a translation can be also viewed as a rotation with ais at infinit. The screw ais of planar motion degenerates to a point on the plane called the instantaneous center [32], displacement pole or virtual pivot. The pitch of a planar twist is alwas ero. Similarl in statics, a general screw or a wrench consists of two vectors representing a force F and a couple (moment) M acting on a rigid bod, written as, Wrench: Ŵ = (F M) = ( f u r f u + mu) (4) where vectors u and r are the direction of and a point on the wrench ais respectivel, scalars f and m are magnitude of the force and partial moment along the ais, coupled b a pitch q = m/ f. Similar to the case of twist, a pure force and a pure couple are represented b a wrench of ero pitch and infinite pitch respectivel, written as, Pure Force(q = 0) : Ŵ = ( f u r f u) (5) Pure Couple(q = ) : Ŵ = (0 mu) (6) Figure 1 illustrates a general twist ˆT and a general wrench Ŵ in space. p Ω c Ω=ωs a F=fu qf α Ŵ Figure 1. A general wrench Ŵ does work on a bod with the motion defined b a general twist ˆT 2.2 Reciprocit of Screws Let us denote the normal distance and skew angle of aes of a general wrench Ŵ and a general twist ˆT b a and α respectivel (Fig. 1). The virtual power of wrench Ŵ acting on a moving bod with motion Ŵ is given b the reciprocal product of the twist and r the wrench, calculated as, ˆT Ŵ = F V + M Ω = [( f v + mω)(s u) + f ω(c r) (s u)] = [( f v + mω)cosα f ωasinα] (7) A twist is called reciprocal to a wrench when their reciprocal product is ero. To find all possible reciprocal conditions, we consider nontrivial (nonero) twist ˆT and nontrivial wrench Ŵ. 1. If f = 0(q = ) and ω = 0(p = ), ˆT is alwas reciprocal to Ŵ. This sas that a pure couple is alwas reciprocal (does no work) to a pure translation. 2. If f = 0,ω 0 or f 0,ω = 0, ˆT Ŵ = 0 is satisfied if and onl if cosα = 0 i.e. the twist ais and the wrench ais are perpendicular to each other. 3. If f 0(q ) and ω 0(p ), Eq.(7) is reduced to ˆT Ŵ = f ω[(p + q)cosα asinα], (8) where we have substituted the definition of pitches. B considering the values of the pitches p and q in Eq.(8), we can obtain the following observations that can be ver useful as thumb rules in design practice. (a) If p + q = 0 (including the case p = q = 0), the two screws are reciprocal if either a = 0 or sinα = 0. This situation occurs when the twist ais and wrench ais are coplanar, i.e., intersecting or parallel to each other (b) If the two screw aes are perpendicular cosα = 0, their reciprocit is independent of their pitches. This can occur onl when a = 0, i.e. two aes intersect. Reciprocal product is considered a linear operation on either twist or wrench separatel. For instance, the reciprocal product of a twist ˆT with a linear combination of two wrenches Ŵ 1,Ŵ 2 can be epressed as the linear combination of the reciprocal product of ˆT with each of the two wrenches, that is, ˆT (k 1 Ŵ 1 + k 2 Ŵ 2 ) = k 1 ˆT Ŵ 1 + k 2 ˆT Ŵ 2 (9) where the coefficients k 1 and k 2 are arbitrar constants. This linearit propert is ver important in the freedom and constraint pattern analsis and snthesis. 3 The Screw Theor Representation of the Constraint-Based Design Approach This section relates concepts and design rules in constraintbased design approach to screw theor. 3 Copright c 2009 b ASME

6 3.1 Freedom and Constraint of a Rigid Bod Constraint and freedom are ke concepts in the constraintbased design approach. In screw theor, a degree-of-freedom (dof) is represented b a general twist. As special cases, a rotational freedom or a translational freedom can be represented b a twist of pure rotation or pure translation shown in Eq.(2) and Eq.(3) respectivel. In this paper, all vectors are considered row vectors. For eample, Ω R 3 is denoted b Ω = (Ω Ω Ω ). Also a general twist is essentiall a si dimensional row vector, i.e. ˆT = (Ω Ω Ω V V V ) R 6. An unconstrained rigid bod has si dof in space, i.e. three rotations and three translations along three orthogonal aes (Fig 2) denoted b principle twists, written as ˆT R = ( ) ˆT R = ( ) ˆT R = ( ) ˆT T = ( ) ˆT T = ( ) ˆT T = ( ) (10) In the constraint-based design approach, an ideal constraint is a slender structural member that is infinitel stiff along its ais but is infinitel compliant perpendicular to its ais. The ideal constraint is essentiall a nontrivial translational constraint represented b a wrench of pure force, epressed as Ŵ = (F M) = (F F F M M M ),s.t. F M + F M + F M = 0,F 2 + F 2 + F 2 0. (11) In real designs, the ideal constraint can usuall be approimated b a rigid link with two ball joints at both ends or a compliant link (wire fleure) that is much stiffer along the direction of its ais than along the direction perpendicular to the ais [12]. Figure 3(a) shows a rigid bod constrained b an ideal wire fleure which does not allow compression or stretch in its aial direction but is full compliant in the perpendicular directions. Without losing generalit, let us assume the wire ais aligns along the ais. The constraint representing the wire fleure is denoted b a wrench Ŵ = ( ), shown in Figure 3(b). The freedoms subject to the constraint can be obtained b requiring a general twist ˆT reciprocal to the wrench. This allows us to obtain from Eq.(7): ˆT Ŵ = 0, V = 0 (12) Figure 2. p R, T R, T R, T rotation (p=0) translation (p= ) screw motion An unconstrained rigid bod has three translations and three rotations represented b si principle twists. Clearl this constraint removes the translational freedom along the ais. rigid bod Figure 3. wire fleure reference bod o Ŵ u= ( 1 0 0), r= (0 0 0), q= 0 An ideal wire fleure imposes on a rigid bod an ideal constraint which removes the translational freedom in the aial direction of the wire fleure When a mechanical connection is built between the rigid bod and a reference base in such as wa that the number of dof of the rigid bod is reduced, this bod is said to be constrained. The number of reduced dof is defined as degree-ofconstraint (doc) of the mechanical connection. In screw theor, a constraint can be described b a general wrench. A constraint that eliminates translation along a line is called a translational constraint which can be represented b a wrench of pure force shown in Eq.(5). A constraint that eliminates rotation about a line is called a rotational constraint represented b a wrench of pure couple shown in Eq.(6). Even though it is quite straightforward to design a translational constraint, it is not trivial to design a rotational constraint (infinite pitch) or a general constraint (finite pitch). Tpicall a comple structure formed through cascading intermediate bodies must be used. Hunt [18] provided a wrench support that applies a general constraint to a bod. Blanding [12] showed an eample of rotational constraint realied b using two pulles and a cable. Since building rotational and general constraints is relativel complicated (arguabl not cost effective), it is preferable to use translational (ideal) constraints when possible in compliant mechanism design. 4 Copright c 2009 b ASME

7 3.2 Freedom Space The freedom space (topolog) of a rigid bod represents all of its allowable motion in space. Mathematicall the freedom space can be described b a twist matri T formed b f independent twists ˆT j ( j = 1,... f ), written as ˆT 1 Ω 1 V 1 ˆT 2 T =. = Ω 2 V 2... ˆT f Ω f V f (13) And f is called the dimension of the freedom space. ˆT j are the basis twists that span the freedom space. An motion in the freedom space can be denoted b a linear combination of the basis twists, ˆT = f j=1 k j ˆT j (14) where k j are arbitrar constants and cannot be ero simultaneousl. If the rank of the twist matri T is less than f, twists ˆT j are redundant, meaning that some twists can be written as linear combinations of others. Freedom space sometimes can have a geometric representation in space. For instance, a one dimensional freedom space is a single line in space. A two dimensional freedom space is a surface generated b two lines. And three dimensional space is a volume, e.g. a solid sphere generated b three rotational twists intersecting at the same point. Recentl Hopkins and Culpepper [15] sstematicall enumerated the topologies of freedom and constraint space and provided a graphical illustration for each case. Let us take a look at a two dimensional freedom space spanned b two pure rotations. Phsicall this freedom space can be generated b a serial chain of two revolute joints. Depending on whether the two rotation aes are parallel, intersecting or skew, the freedom space is a plane, a disk or a clindroid respectivel. Figure 4(a) shows a plane generated b two parallel rotational twists ˆT 1 = (Ω c 1 Ω) and ˆT 2 = (Ω c 2 Ω). An parallel lines on the plane can be represented b ˆT = k 1 ˆT 1 + k 2 ˆT 2 = ((k 1 + k 2 )Ω (k 1 c 1 + k 2 c 2 ) Ω), (15) where the coefficients k 1,k 2 can be viewed as the angular speeds of the joints if the freedom space is generated b a serial chain of two revolute joints. If the angular speeds are constrained such that k 1 +k 2 = 0, twist (15) shows a pure translation in the normal direction of the plane formed b the two parallel lines. This result tells us that a serial chain of two parallel revolute joints can generate an instantaneous translation b driving the two joints with same angular velocit but in the opposite direction. 2 1 (a) two parallel lines Figure 4. c 2 c 1 Ω Ω 2 Ω Ω c (b) two intersecting lines c 2 c 1 2 Ω 2 1 p 0 Ω 1 (c) two skew lines Two dimensional freedom space generated b (a) two parallel lines, (b) two intersecting lines and (c) two skew lines. Figure 4(b) shows a disk generated b two rotational twists intersecting at a point c. An freedom in the space can be epressed b ˆT = k 1 ˆT 1 + k 2 ˆT 2 = (k 1 Ω 1 + k 2 Ω 2 c (k 1 Ω 1 + k 2 Ω 2 )), (16) Clearl ever freedom in this space is still a pure rotation about a line through the same point c and in the direction k 1 Ω 1 +k 2 Ω 2. Figure 4(c) shows a clindroid generated b two skew rotational twists. An freedom in the space can be described b a general twist ˆT = k 1 ˆT 1 + k 2 ˆT 2 = (k 1 Ω 1 + k 2 Ω 2 c 1 k 1 Ω 1 + c 2 k 2 Ω 2 ), (17) One can see that all motions in the space ecept ˆT 1, ˆT 2 are in the form of screw motion since the pitch of twist in (15) is nonero in general. The freedom space of four and five dimensions cannot be represented graphicall in general. Onl in some special cases, could we use the combination of lower dimensional spaces to describe a higher dimensional space. For these cases, the twist matri is more preferable to describe higher dimensional freedom space. 3.3 Constraint Space The constraint space (topolog) of a rigid bod represents all the forbidden motions of the bod subject to a constraint ar- 5 Copright c 2009 b ASME

8 rangement. In screw theor, a constraint space can be represented b a wrench matri formed b c independent wrenches Ŵ i (i = 1,...,c), written as Ŵ 1 F 1 M 1 Ŵ 2 W =. = F 2 M 2... Ŵ c F c M c (18) where c is called the dimension of the constraint space. Ŵ i are the basis wrenches that span the whole constraint space. If matri W does not have a full rank, the wrenches Ŵ i are redundant, meaning that removing some of the constraints does not affect the mobilit of the constrained bod. Similar to freedom space, an constraint in the constraint space can be represented b a linear combination of the basis wrenches, Ŵ = c i=1 k i Ŵ i (19) Figure 5(a) shows a rigid bod constrained b an ideal sheet fleure. Since its thickness is much smaller than its width and length, an ideal sheet fleure allows rotations about an line on the sheet plane and translating along the normal direction. It prohibits the rotation about the normal direction and translations in the plane. In the constraint-based design approach [12], an ideal sheet fleure applies three ideal constraints on the sheet plane. In our screw approach, these three ideal constraints can be represented b a set of an three independent wrenches on the sheet plane. An eample set of three constraints is shown in Fig. 5(a). Wrenches Ŵ 1 and Ŵ 2 are parallel to the ais and intersecting the ais at r 1 = (1 0 0) and r 2 = ( 1 0 0) respectivel. And the wrench Ŵ 3 aligns with the ais. The are written as, Ŵ 1 = ( ) Ŵ 2 = ( ) Ŵ 3 = ( ) (20) An other ideal constraint on the sheet plane in the direction F = (F F 0) through a point r = (r r 0) is written as Ŵ = (F r F) = (F F r F r F ), (21) which can be epressed a linear combination of Ŵ 1,Ŵ 2,Ŵ 3 as ( ) r F r F + F Ŵ = Ŵ ( F r F + r F 2 ) Ŵ 2 + F Ŵ 3 (22) Figure 5. Ŵ 1 Ŵ r r r 1 3 Ŵ Ŵ 2 3 r 2 rigid bod Ideal sheet fleure (a) constraint space an ideal sheet fleure. 2 3 (b) freedom space Constraint and freedom space of a rigid bod constrained b 3.4 Rule of Complementar Patterns The design rules in the constraint-based design approach are mostl illustrated in the form of subjective statements. The most important one is Rule of Complementar Patterns which state: when a pattern of c nonredundant constraints is applied between an object and a reference bod, the object will have f = 6 c independent degrees-of-freedom. This rule can be eplained in screw theor as follows. Let wrenches Ŵ i (i = 1,...,c) denote c nonredundant constraints and twists ˆT j ( j = 1,..., f ) denote nonredundant f degrees-of-freedom. And all wrenches are reciprocal to all twists ˆT j Ŵ i = 0, i = 1,...,c, j = 1,..., f,c + f = 6 (23) There are two major categories of design problems of compliant mechanisms using the constraint-based design approach. One is called constraint pattern analsis which studies the mobilit of a rigid bod subject to a pattern of constraints. The other is called constraint pattern design which seeks for a pattern of constraints to achieve a specified pattern of freedoms. We discuss each in the following. Constraint Pattern Analsis Here let us use the sheet fleure shown in Fig 5(a) as an eample to demonstrate the steps for constraint pattern analsis. Step 1: write all constraints in the format of wrenches which are written in Eq.(20) Step 2: require a general twist ˆT = (Ω Ω Ω V V V ) reciprocal to all wrenches in Eq.(20) and ield a sstem of linear equations, ˆT Ŵ 1 = 0 1V + 1Ω = 0 ˆT Ŵ 2 = 0 1V 1Ω = 0 (24) ˆT Ŵ 3 = 0 1V = 0 where the reciprocal product is eplicitl epressed to show the consistence in units. 1 6 Copright c 2009 b ASME

9 Step 3: solve the above linear sstem to obtain V = V = Ω = 0 and write the complementar freedom space in the format of a general twist, ˆT = (Ω Ω V ) Step 4: find independent basis twists from the above twist sstem. For this eample, simpl setting one of Ω,Ω,V to be nonero and the other two to be ero ields three basis twists: ˆT 1 = ( ) ˆT 2 = ( ) ˆT 3 = ( ) Clearl the represent rotation around ais, rotation around ais and translation along ais respectivel. See Fig. 5(b). Note the step of choosing basis twists is not unique. An three independent twists in the freedom space should span the same freedom space. It should be pointed out that a general freedom is usuall in the form of screw motion represented b a general twist (finite pitch). However for the sake of intuition, we prefer to use rotational or translational twists as basis twists whenever possible to represent the freedom space. An interesting question is can we alwas find f = 6 c rotational freedom for a pattern of c constraints? Unfortunatel the answer is NO even when all the constraints in an arrangement are ideal, i.e. q = 0. A counter eample is shown in Fig. 6 where a rigid bod is subject to four ideal constraints. The phsical arrangement of this constraint pattern and corresponding constraint space are shown in Fig. 6(a) and Fig. 6(b) respectivel. Wrenches Ŵ 1 and Ŵ 2 are parallel to the ais and intersect with the ais at +1 and 1 respectivel. Wrench Ŵ 3 aligns the ais. And the last wrench is a skew line on the plane = 1 and has an angle of 45 with both and ais. The are written as Ŵ 1 = ( ) Ŵ 2 = ( ) Ŵ 3 = ( ) = ( ) (25) Following the constraint pattern analsis steps ields two independent twists ˆT 1 = ( ) p 1 = 0 ˆT 2 = ( ) p 2 = 1 (26) The represent a pure rotation (p 1 = 0) around ais and a screw motion around ais with pitch p 2 = 1 respectivel. These two independent twists form the basis of the two dimensional freedom space that defines the allowable motion of the constrained bod. Let us have a look at the freedom space to see if there eists an rotational freedom other than ˆT 1. According to the definition of freedom space, an freedom ˆT = (Ω V) in the freedom space can be epressed as linear combination of ˆT 1, ˆT 2, written as ˆT = k 1 ˆT 1 + k 2 ˆT 2 = (k 2 k 1 0 k 2 0 0) (27) If ˆT is a rotation, we must have Ω V = 0 k 2 2 = 0 k 2 = 0. (28) However this means that ˆT is simpl a multiple of ˆT 1. Hence this proves that no other rotation line eists in this freedom space. In other words, all freedoms ecept the rotation ˆT 1 are in the form of screw motion. Ŵ 3 Ŵ 2 1 Ŵ 1 2 (a) phsical arrangement Figure 6. Ŵ 2 Ŵ 3 1 ( p= 0) Ŵ 1 (b) constraint space 2 ( p= 1) A bod constrained b a pattern of ideal constraints can have a screw motion in space. Constraint Pattern Design The constraint pattern design starts with specifing a freedom space described b a set of f twists ˆT j ( j = 1,..., f ). The objective is to find the complementar constrained space, that is to find c = 6 f independent constraints denoted b wrenches Ŵ i (i = 1,...,c). Here we use an eample to demonstrate the constraint pattern design steps. Suppose we want to design a compliant mechanism with two allowable motions: a pure rotation around ais and a pure translation along the direction of (0 1 1). See Fig 7. We seek an arrangement of ideal wire fleures which constrain a rigid bod and allows the prescribed motion. The constraint space is found b the following steps. 7 Copright c 2009 b ASME

10 Step 1: denoting all specified freedoms in twists ields ˆT 1 = ( ) p 1 = 0, ˆT 2 = ( ) p 2 = (29) 2 Ŵ 3 Ŵ 2 Step 2: requiring a general wrench Ŵ = (F F F M M M ) reciprocal to both twists ields 1 Ŵ 1 ˆT 1 Ŵ = 0 M = 0 (30) ˆT 2 Ŵ = 0 F + F = 0 F = F (31) Since we are onl interested in ideal constraints (design with wire fleures), we also want F M + F M + F M = 0,F 2 + F 2 + F 2 0. (32) Step 3: substituting Eqs.(30,31) into Eq.(32), we can write a general wrench in the complementar constraint space as Ŵ = (F F F M M 0),s.t. F M + F M = 0,F 2 + 2F 2 0. (33) Step 4: categorie the condition Eq.(33) and find independent subcases. For the sake of intuitiveness, we prefer to find constraints parallel to coordinate aes or plane whenever possible. This can be done b assigning one or more force elements to be ero and solve the other elements b Eq.(33). For this eample, the following subcases are obtained, F 0,F = 0,M = 0,M = 0 F 0,F = 0,M = 0,M 0 F = 0,F 0,M = 0,M = 0 F = 0,F 0,M = 0,M 0 Step 5: write the subcases in (32) in the form of wrenches, Ŵ 1 = ( ) Ŵ 2 = ( ) Ŵ 3 = ( ) = ( ) (34) (35) B checking the rank of the wrench matri, we find that the four wrenches in Eq.(35) are independent. Wrenches Ŵ 1,Ŵ 2 are parallel to the ais and intersecting the twist ˆT 1. Both wrenches Ŵ 3, are perpendicular with ˆT 2 and lie on the plane. See Fig 7. Figure 7. Four ideal constraints found for given a pattern of two degreeof-freedom. In general the pitches of wrenches in the complementar constraint space ma be ero, infinite or nonero finite which correspond to translational (ideal), rotational and general constraint respectivel. Because of the relativel low cost of building ideal constraints in compliant mechanism design, we would prefer to find as man ideal constraints in the constraint space as possible. As in constraint pattern analsis, one should be aware that it is NOT alwas possible to find 6 f independent ideal constraints for arbitrar f freedoms. For instance, if two or three of the given freedoms are translational, b going through the design steps, one can find that there do not eist 6 f complementar ideal constraints. Apparentl these kind of cases are not rare. As a matter of fact, most of spatial freedom space cannot be realied without using non-ideal constraints. For these design cases, one has to use cascaded comple structures to provide rotational or general constraints. 4 Tpe Snthesis of Compliant Prismatic Joints In this section, we show how to appl the proposed approach to the design of a compliant prismatic joint using wire or sheet fleures. Without losing generalit, let us assume that the given single freedom is a translational motion along the ais, denoted b a twist ˆT = ( ). B following the constraint design steps, we find f = 0, i.e. all wrenches representing ideal constraints in the constraint space must have the form Ŵ = (0 f f m m m ) (36) To design the constraint pattern, we onl need to find five independent constraints in the constraint space. Suppose we want to use five wire fleures (ideal constraints) to achieve the design. This means that Eq.(36) is subject to f m + f m = 0, f 2 + f 2 0. (37) If m = 0, we substitute either f = 0 or f = 0 (Note f and f cannot be ero simultaneousl) into Eq.(37) and obtain the 8 Copright c 2009 b ASME

11 following four subcases: f 0, f = 0,m = 0,m = 0 f 0, f = 0,m = 0,m 0 f = 0, f 0,m = 0,m = 0 f = 0, f 0,m 0,m = 0 which lead us the following four independent wrenches, Ŵ 1 = ( ) Ŵ 2 = ( ) Ŵ 3 = ( ) = ( ) (38) (39) The represent two constraint lines parallel to the ais and two constraint lines parallel to the ais respectivel (Fig. 8). And if m 0, setting f = m = m = 0 ields the last ideal constraint, Ŵ 5 = ( ) (40) This constraint is a line in the direction F 5 = (0 1 0) and passing through a point r 5 = (0 0 1). B checking the rank of the wrench matri of the constraint space, we verif that the five wrenches are not redundant. Ŵ 3 Figure 9. Ŵ 5 rigid bod Ŵ 1 (a) pure translation Ŵ 2 reference bod sheet fleure rigid bod pure translation reference bod Design of a compliant prismatic joint using (a) five wire fleures or (b) one fleure sheet and two wire fleures. the rigid bod. However this allows us to use a second fleure sheet to appl the constraints Ŵ 2,,Ŵ 6. The result is the well known parallel fleure design of compliant prismatic joint. See Fig. 10(b). Ŵ 3 Ŵ Ŵ1 Ŵ6 Ŵ2 5 (a) rigid bod sheet fleure 1 (b) (b) Ŵ 2 reference bod pure translation sheet fleure 2 Figure 8. Ŵ 3 Ŵ 5 Ŵ1 Ŵ2 The constraint space for a compliant prismatic joint. Figure 10. sheets. Design of a compliant prismatic joint using two parallel fleure As one can see, the choice of the basis constraints is not unique which means that there are multiple constraint patterns which will achieve a given freedom pattern. Even for the same constraint pattern, there ma be multiple phsical arrangements (designs). Apparentl this is beneficial to the designers as the have multiple was to achieve the same design goal. An interesting future research task is to sstematicall find all possible phsical arrangements for a given freedom pattern. The design of the compliant prismatic joint using five wire fleures is shown in Fig. 9(a). The compliant prismatic joint allows the rigid bod translate along the direction onl. As shown in Section 3.2, a single sheet fleure provides three ideal constraints whose aes lie on the same plane. An alternative phsical arrangement is to use a single sheet fleure to replace the wire fleures Ŵ 1,Ŵ 3,Ŵ 5. This design is shown Fig. 9(b). Furthermore let us manuall add a sith constraint Ŵ 6 which is parallel to Ŵ 2 and intersects at the point (1 0 1) (Fig. 10(a)). Since Ŵ 6 is redundant (can be written as a linear combination of Ŵ i (i = 1,...,5), it does not appl etra constraint to 5 Conclusion A screw theor based approach for the design of compliant mechanisms is introduced in this paper. This approach presents a mathematical representation of the constraint-based design approach which is tpicall characteried b subjective statements. A constraint and a freedom in the constraint-based design approach can be mathematicall denoted b a wrench and a twist respectivel. The constraint topolog or space or pattern formed b a sstem of constraints acting on a rigid bod is essentiall a linear space spanned b a sstem of independent wrenches. Sim- 9 Copright c 2009 b ASME

12 ilarl a freedom space that describes the possible instantaneous motion of a rigid bod is represented b a sstem of independent twists. These two spaces are complementar and can be found from each other using linear algebra. Two major compliant mechanism design problems, constraint pattern analsis and constraint pattern design, are elaborated with eamples of compliant mechanisms and fleures. The proposed analsis/design framework is beneficial for the earl design stages, such as tpe snthesis of compliant mechanisms. Acknowledgment All authors acknowledge funding provided b the National Science Foundation grant CMMI for this research. The first author acknowledges support provided b National Science Foundation CAREER grant. REFERENCES [1] Howell, L. L., Compliant Mechanisms. Wile- Interscience, New York, NY. [2] Howell, L. L., and Midha, A., Parametric deflection approimations for end-loaded, large-deflection beams in compliant mechanisms. ASME Journal of Mechanical Design, 117(1), pp [3] Ananthasuresh, G. K., and Kota, S., Designing compliant mechanisms. Mechanical Engineering, 117(11), pp [4] Frecker, M. I., Ananthasuresh, G. K., Nishiwaki, S., Kikuchi, N., and Kota, S., Topological snthesis of compliant mechanisms using multi-criteria optimiation. ASME Journal of Mechanical Design, 119(2), pp [5] Sigmund, O., On the design of compliant mechanisms using topolog optimiation. Mechanics of Structures and Machines, 25(4), pp [6] Wang, M. Y., Chen, S., Wang, X., and Mei, Y., Design of multimaterial compliant mechanisms using level-set methods. ASME Journal of Mechanical Design, 127(5), pp [7] Hull, P. V., and Canfield, S., Optimal snthesis of compliant mechanisms using subdivision and commercial fea. ASME Journal of Mechanical Design, 128(2), pp [8] Zhou, H., and Ting, K.-L., Topological snthesis of compliant mechanisms using spanning tree theor. ASME Journal of Mechanical Design, 127(4), pp [9] Howell, L. L., and Midha, A., A loop-closure theor for the analsis and snthesis of compliant mechanisms. ASME Journal of Mechanical Design, 118(1), pp [10] Su, H.-J., and McCarth, J. M., A polnomial homotop formulation of the inverse static analsis of planar compliant mechanisms. ASME Journal of Mechanical Design, 128(4), pp [11] Mawell, J. C., General considerations concerning scientific apparatus. Dover Press. [12] Blanding, D. L., Eact Constraint: Machine Design Using Kinematic Processing. ASME Press, New York, NY. [13] Hale, L. C., Principles and Techniques for Designing Precision Machines. PhD Thesis, MIT, Cambridge, MA. [14] Awtar, S., and Slocum, A. H., Constraint-based design of parallel kinematic fleure mechanisms. ASME Journal of Mechanical Design, 129(8), pp [15] Hopkins, J. B., and Culpepper, M. L., Snthesis of Multi-degree of Freedom Fleure Sstem Concepts via Freedom and Constraint Topologies (FACT) (submitted to Journal of Precision Engineering). April. [16] Shai, O., and Pennock, G. R., A stud of the dualit between planar kinematics and statics. ASME Journal of Mechanical Design, 128(3), pp [17] Ball, R. S., The Theor of Screws. Cambridge Universit Press, Cambridge, England (Originall published in 1876 and revised b the author in 1900, now reprinted with an introduction b H. Lipkin and J. Duff). [18] Hunt, K. H., Kinematic Geometr of Mechanisms. Oford Universit Press, New York, NY. [19] Phillips, J., Freedom in Machiner. Volume 1, Introducing Screw Theor. Cambridge Universit Press, Cambridge, U.K. [20] Phillips, J., Freedom in Machiner. Volume 2, Screw Theor Eemplified. Cambridge Universit Press, Cambridge, U.K. [21] Lipkin, H., and Patterson, T., Geometric properties of modelled robot elasticit: part i - decomposition. DE- Vol. 45, pp [22] Lipkin, H., and Patterson, T., Geometric properties of modelled robot elasticit: part ii - decomposition. DE- Vol. 45, pp [23] Patterson, T., and Lipkin, H., Structure of robot compliance. ASME Journal of Mechanical Design, 115(3), pp [24] Huang, S., and Schimmels, J. M., The bounds and realiation of spatial stiffnesses archieved with simple springs connected in parallel. IEEE Transactions on Robotics and Automation, 14(3), pp [25] Huang, S., and Schimmels, J. M., The bounds and realiation of spatial compliance archieved with simple serial elastic mechanisms. IEEE Transactions on Robotics and Automation, 16(1), pp [26] Dai, J. S., Huang, Z., and Lipkin, H., Mobilit of overconstrained parallel mechanisms. ASME Journal of Mechanical Design, 128(1), pp [27] Adams, J. D., and Whitne, D. E., Application of 10 Copright c 2009 b ASME

13 screw theor to constraint analsis of mechanical assemblies joined b features. ASME Journal of Mechanical Design, 123(1), pp [28] Smith, D., Constraint Analsis of Assemblies Using Screw Theor and Tolerance Sensitivities. MS Thesis, Brigham Young Universit, Provo, UT. [29] Kong, X., and Gosselin, C. M., Tpe snthesis of 3- dof translational parallel manipulators based on screw theor. ASME Journal of Mechanical Design, 126(1), pp [30] Kim, C. J., Functional characteriation of compliant building blocks utiliing eigentwists and eigenwrenches. In Proceedings of ASME IDETC/CIE August 3-6, New York, USA. [31] McCarth, J. M., Geometric Design of Linkages. Springer-Verlag, New York, NY. [32] Kim, C. J., Kota, S., and Moon, Y.-M., An instant center approach toward the conceptual design of compliant mechanisms. ASME Journal of Mechanical Design, 128(3), pp Copright c 2009 b ASME