Zhao Hongzhee-mail: hongzhezhao@gmail.com

Bi Shushenge-mail: bishusheng@gmail.com

Yu Jingjune-mail: jjyu@buaa.edu.cn

Robotics Institute,

Beihang University,

Beijing 100191, P.R. China

Guo JunSchool of Astronautics,

Beihang University,

Beijing 100191, P.R. China

e-mail: guojunbh@buaa.edu.cn

Design of a Family ofUltra-Precision Linear MotionMechanismsThe parasitic motion of a parallel four-bar mechanism (PFBM) is undesirable for design-ers. In this paper, the rigid joints in PFBM are replaced with their flexural counterparts,and the center shift of rotational flexural pivots can be made full use of in order to com-pensate for this parasitic motion. First, three schemes are proposed to design a family ofultraprecision linear-motion mechanisms. Therefore, the generalized cross-spring pivotsare utilized as joints, and six configurations are obtained. Then, for parasitic motion ofthese configurations, the compensation condition is presented, and the design space ofgeometric parameters is given. Moreover, the characteristic evaluation of these configu-rations is implemented, and an approach to improve their performances is further pro-posed. In addition, a model is developed to parametrically predict the parasitic motionand primary motion. Finally, the analytic model is verified by finite element analysis(FEA), so these linear-motion mechanisms can be employed in precision engineering.[DOI: 10.1115/1.4007491]

1 Introduction

In the last fifty years, flexural pivots and compliant mechanismshave become an active area of research, because some substantialimprovements in performance can be achieved, such as ease of as-sembly, maintenance-free, no backlash, diminished friction, infin-itesimal resolution, and simplified manufacturing process [1,2].However, in order to overcome some inherent drawbacks, likesmall motion range and undesirable parasitic motion, challengesfor the researchers are posed [3,4]. So the complex flexural pivotswere investigated for the purpose that the parasitic error can bereduced by utilizing the building block approach [5].As the compliant linear-motion mechanisms (i.e., translational

flexural pivots) are considered, some novel designs were devel-oped deriving from rigid mechanisms, such as Roberts mechanism[68], Scott-Russel mechanism [9], and orthoplanar linear-motionspring [10]. Nevertheless, the prior literatures show that the vastmajority of linear-motion mechanisms are derived from thePFBM. A typical configuration is constructed by two leaf-springsin parallel, as illustrated in Fig. 1(a). The distributed compliancewas exploited so as to achieve large motion range [11,12]. Then,several mechanisms of this kind were combined in series or paral-lel, and the double and quaternary PFBM were developed toimprove the performances [1216]. Meanwhile, a ratio-control le-ver was used in such mechanism to reduce the errors originatingfrom manufacturing and assembly tolerances [17,18].In this paper, the aim is to diminish the parasitic motion arising

from PFBM, but not double or quaternary PFBM. Because the par-asitic motion for PFBM is not tractable, as illustrated in Fig. 1(a).In addition, the intermediate stage of double or quaternary PFBMpossesses an unwanted translational degrees of freedom (DOF)when the ground and final stage are held fixed. This issue wouldbe a problem for dynamic actuation and ambient vibrations. Underthis consideration, the conventional joints in PFBM were replacedwith the cartwheel flexural pivots (Fig. 1(b)) by some researchers[19,20]. However, the center shift of the cartwheel flexural pivotmakes the distance between its moving stage and fixed stage short-ened, so the application of this rotational flexural pivot (Fig. 1(b))will worsen the parasitic motion of PFBM. Even so, some helpful

information can be achieved from Fig. 1(b), according to the quali-tative analysis of this mechanism [21]. To be specific, if a flexuralpivot is used, whose center shift makes the distance between thetwo stages increase, the parasitic motion of the compliant PFBMwill be eliminated. For example, the mechanism as shown inFig. 1(c) can achieve desired performance.So the objective of this research is to synthesize the compliant

PFBM, and compensate for the parasitic motion by taking advant-age of some rotational flexural pivots. As a typical flexural build-ing block, the generalized cross-spring pivot [22,23] is chosen toimplement a quantitative analysis. Then, six configurations arepresented and the compensation condition of parasitic motion isdeveloped. Meanwhile, the characteristics of these configurations,such as the range of motion and manufacturing performance, areevaluated. Furthermore, taking one configuration as an example,the model is developed to parametrically predict the parasiticmotion and primary motion. Finally, FEA is carried out, in orderto verify the validity of the developed model.

2 The Compensation Principle of Parasitic Motion

It is universally accepted that the parasitic motion in PFBM isundesirable but unavoidable, if the mechanism is utilized for recti-linear motion. However, the center shift of the rotational flexuralpivot can be taken advantage of, in order to compensate for the in-herent parasitic errors of this PFBM. It is a promising method,especially for the leaf-type flexural pivots, because large range ofmotion and high precision may be achieved simultaneously.Moreover, since the leaf with distributed compliance plays bothroles of pivot and rigid link, a compact mechanism can bedesigned. In order to clarify the compensation principle of para-sitic motion, a single kinematic chain of the compliant PFBM istaken into account, and an equivalent rigid body model is given.As illustrated in Fig. 2, there are three schemes to accomplish

the requirement of error compensation. The link length is the ini-tial distance between the two pivot points, which is depicted byLl. In the y direction, the projection of the link is Lyh. Accordingly,normalized by a characteristic length of the mechanism L (L willbe defined in Sec. 3), the shortening of the projection dyl can beexpressed as

dyl Ll LyhL

LlL

1 cos h ll 1 cos h ll h2

2(1)

Contributed by the Mechanisms and Robotics Committee of ASME for publica-tion in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received February 29,2012; final manuscript received June 27, 2012; published online September 17,2012. Assoc. Editor: Yuefa Fang.

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On the other hand, the center shift of a flexural pivot can betreated as two prismatic joints (the prismatic joint in the x direc-tion is not illustrated in Fig. 2). Thus, the center shift in the ydirection (dyno) can be taken advantage of to compensate for theshortening of the projection, according to the topological relation-ship between the link and constraint boundary.First of all, in order to diminish the parasitic error of the

linear-motion mechanism, the center shift must be in the arrowdirection (as displayed in Fig. 2). More specifically, the instan-taneous center of the flexural pivot in scheme I (Fig. 2(a)) mustmove towards the moving stage; but in schemes II and III(Figs. 2(b) and 2(c)), the instantaneous center of the flexuralpivot must move to the fixed stage. Furthermore, when thedirection is satisfied, the magnitude of the center shift can betaken into consideration. As a result, the dominant term of theparasitic motion will be diminished if the center shift of a pivotis the following form.For scheme I

dyno dyl2 llh

2

4(2a)

For schemes II and III

dyno dyl2 llh

2

4(2b)

So it is a primary task to choose the suitable rotational flexuralpivot to satisfy condition (2a) and (2b). Note also that ll is a nor-malized parameter, and the dimensions of the link and the pivotwill be limited by each other. Consequently, one of the objectivesfor this paper is to harmonize the geometric parameters of the link

and the pivot, so as to compensate for the parasitic error of thePFBM.

3 Synthesis of the Compliant Linear-MotionMechanism

Based on the compensation principle proposed in Sec. 2, thesynthesis of the compliant linear-motion mechanism can beimplemented. First, according to a quantitative analysis, thesuitable flexural pivots with desirable parasitic motion aregiven. Then, employing the prescribed flexural pivot, someconfigurations are proposed based on the three schemes. More-over, the compensation condition of parasitic motion isdeveloped.

3.1 Choosing of the Rotational Flexural Pivot. Althoughthere are too many types of rotational flexural pivot, as summar-ized by Trease and the co-workers [12], the performances of someflexural pivots are not good and they are unsuitable to be used incompliant PFBM. However, as a typical rotational flexural pivot,the cross-spring pivot [1,2428] is an exception. In order to fur-ther improve the performance and expand the application domain,the generalized cross-spring pivot (Fig. 3), whose intersectionpoint is at arbitrary position of the leaf-springs [22,23], can beadopted as the rotational flexural module. Moreover, as a primi-tive joint, the generalized cross-spring pivot can be combined toform a complex flexural pivot, especially for the monolithicarrangement (as shown in Fig. 3(b), the two leaf-springs can bemanufactured in one plate).Accordingly, for the generalized cross-spring pivot, the charac-

teristics of center shift must be reviewed [23]. First of all, the non-dimensional parameters are defined as the following lower caseletters:

Fig. 2 Three schemes is clarified by the equivalent rigid body model of a single kinematic chain (pris-matic joints in the x direction are not shown). The dotted line is the initial position, and Ll is referring tothe dotted line: (a) scheme I, (b) scheme II, and (c) scheme III.

Fig. 1 The compliant parallel four-bar mechanism combined by flexural building blocks: (a) leaf spring buildingblock, (b) cartwheel flexural building block, and (c) generalized cross-spring building block

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mp MpLpEI

; fp FpL

2p

EI; pp

PpL2p

EI

dyp DYLp

; dxp DXLp

; dp 12 LpTp

2

where Mp, Fp, and Pp denote bending moment, horizontal force,and vertical force applied on the moving stage as the externalloads, and their directions are invariable in the space; dxp and dypare the center shift of this pivot (DX and DY are depicted in Fig.4); Lp, Wp, and Tp are the length, width, and thickness of thebeam, respectively, and they are named as shape parameters, I isthe moment of inertia, dp is the square ratio of Lp to Lp; E isYoungs modulus of the material.The geometric parameters k and a associated with the intersec-

tion point are described in Fig. 4. When the free end and fixed endof the leaf are A and B, respectively, and the two leaves (or theirextension lines) cross at the point O, the ratio between the directedline segment AO and AB is define as k. Thus, the center shift canbe expressed as

dxp Cxh ph3p Hxh p Cxf pfp Cxp ppp (3a)dyp Cyh ph2p Hyh p Cyf pfp Cyp ppp (3b)

where Cxh_p and Cyh_p are the coefficients of the dominant terms;Hxh_p and Hyh_p are the higher order terms; Cxf_p and Cyf_p are thecompliances for the horizontal force fp in the x and y directions,

respectively; Cxp_p and Cyp_p are the compliances for the verticalforce pp in the x and y directions, respectively. All these coeffi-cients can be obtained according to Refs. [21,23]. Each coefficienthas two values when the geometric parameter k is in the differentrange, (1, 0.5) and (0.5, 1).In terms of the center shift dyp, the dominant term of dyp,d can

be expressed as the following form, irrespective of the range ofgeometric parameter k.

dyp;d 115 cos a

9k2 9k 1 h2 (4)3.2 Six Configurations of the Compliant Linear-Motion

Mechanisms. According to the model of the generalized cross-spring pivot [23], the dominant term of the center shift dyp,d willbe positive, if the geometric parameter k is in the range(0.127322, 0.872678); but it will be negative when the geometricparameter k is in the range (1, 0.127322) or (0.872678, 1).Here, the two types of the pivots are named as I type pivot and IItype pivot, respectively. So the former satisfies the desirable direc-tion for scheme I, but the latter satisfies schemes II and III. Inaddition, the magnitude of dyp is of the order of h

2. Hence, thesynthesis of the compliant PFBM can be carried out, by employ-ing the generalized cross-spring pivot.As a result, six configurations are proposed, as schematic illus-

trated in Fig. 5. The blue lines refer to the leave-springs, and theblack lines denote the rigid bodies. Configuration 1 is developedfrom scheme I, and I type pivot is utilized as the flexural buildingblock. Utilizing II type pivot, configurations 24 are derived fromscheme II, and Configurations 5 and 6 are derived from schemeIII. Although some configurations are essentially identical, theyare still treated separately in consideration of arrangement. Forexample, both configurations 5 and 6 are obtained from schemesIII by II type pivot, but they are out-plane and in-plane arrange-ment, respectively.Moreover, the geometric parameters l, a, and a (a is shown in

Fig. 4) are defined to describe the configuration of these compliantlinear-motion mechanisms. And the width of these mechanisms Lis chosen as the characteristic length to normalize the parameters.So the compensation condition of parasitic motion that the geo-metric parameters must satisfy can be developed.

3.3 The Compensation Condition of Parasitic Motion. Thegeneralized cross-spring pivot is normalized by the leaf length.But for the compliant linear-motion mechanism, the nondimen-sional metrics are defined by its characteristic length L. So thetranslation of the normalized parameters needs to be implemented.First of all, the leaf length for the generalized cross-spring pivot is

Lp L=n (5)

Fig. 3 The generalized cross-spring pivot: (a) nonmonolithic arrangement and (b)monolithic arrangement

Fig. 4 Deflected configuration of the generalized cross-springpivot

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where n cosa/l. So the center shift of the flexural pivot can beexpressed as

dyp ndyno; dxp ndxno (6)where the suffixes no refers to the number of the flexural modules,and it corresponds to the parameters normalized by L but not Lp.After the translation of the parameters, dyno,d and dyl,d, which

denote the dominant terms of dyno and dyl, respectively, are for-mulated and listed in Table 1. In addition, according to Figs. 4and 5, the relationship of the geometric parameters and the con-straint condition are also summarized in Table 1.Then, the dominant terms of the parasitic motion can be com-

pensated for, in terms of the compensation principle. Configura-tion 1 complies with Eq. (2a) to diminish the parasitic error, butthe others achieve high precision by making use of Eq. (2b). Sothe compensation equations are yielded for all of sixconfigurations.Configuration 1

430cos2 al2 15cos2 a2a136a l36a2 0 (7a)

Configuration 2

430cos2 al2 15cos2 a2a136a l36a2 0 (7b)Configuration 3

4l2 36a 15 cos2 a1 2a l 36a2 0 (7c)Configuration 4

430cos2 al2 36a15cos2 a2a1 l36a2 0 (7d)Configuration 5

4 30cos2 al2 15cos2 a2a 1 36a l 36a2 0 (7e)Configuration 6

4 30cos2 al2 36a 15cos2 a1 2a l 36a2 0 (7f )

Fig. 5 Six configurations of the compliant linear-motion mechanisms: (a) configuration 1, (b) configuration 2, (c) configuration3, (d) configuration 4, (e) configuration 5, and (f) configuration 6

Table 1 The expression and constraint condition for the parameters

No. k dyno,d dy1,d Constraint conditiona

1 k 1 a=l 9a2 9al l2h215l cos2 a

1 2l a2

h2b1 < k < b20 l a < 1=2

c

2b k 1 a=l 9a2 9al l2h215l cos2 a

2a l 12

h2k 1a l > 1=2

3b k 1 a=l 9a2 9al l2h215l cos2 a

1 2a2

h2k 1

4 k a=l 9a2 9al l2h215l cos2 a

1 2a l2

h2k 0a l < 1=2

5 k 1 a=l 9a2 9al l2h215l cos2 a

2l a 12

h20 k < b1 or b2 < k 1l a > 1=2

6 k a=l 9a2 9al l2h215l cos2 a

1 2l a2

h20 k < b1 or b2 < k 10 l a < 1=2

aThe geometric parameters for all six configurations subject to the condition: 0 < l < 1, a 0.bIf monolithic manufacturing is considered, the condition becomes stricter: 0 < l < 1=2.cb1 3

5

p =6, b2 35

p =6.

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It is a remarkable fact that some compensation equations are thesame in the form, for example, Eqs. (7a) and (7e). However, theyare treated as two distinct configurations, because the ranges ofthe geometric parameters are different, and the arrangements ofthe leaf-springs are not identical.Therefore, the compensation condition of the parasitic motion

can be developed by considering the constraint condition (listed inTable 1) and the compensation equations (Eqs. (7a) and (7b))simultaneously. Furthermore, the design space for the geometricparameters l, a, and a are shown in Fig. 6. So these figures can beutilized as a design tool to determine the suitable geometricparameters.

4 Discussion for these Compliant Linear-MotionMechanisms

4.1 Characteristics of the Six Configurations. All of thesix configurations have achieved a high precision linear-motion interms of the design space as illustrated in Fig. 6. Meanwhile, theother characteristics also need to be evaluated. The range ofmotion and the manufacturing performance are two importantmetrics for the designers, so they are taken into account.First, in the x direction, the dominant term of the displacement is

llh, so the motion range of the mechanism is determined by thelength of the link and the angular range of the flexural pivot. Theformer is the initial distance between the intersection points, whichcan be readily obtained from dyl,d in Table 1. The latter can be ana-lyzed according to the stress characteristics of the generalized cross-spring pivot [22]. Comparing with a monolithic one (as shown in

Fig. 3), the nonmonolithic arrangement has a slightly larger angularrange if the sizes of the pivots (determined by the geometric param-eter l) are the same. Taking configuration 4 for example, the size ofthe monolithic pivot is too small, which will lead to a small range ofmotion for the linear-motion mechanism. Furthermore, the charac-teristics for other configurations are listed in Table 2.Second, as the manufacturing performance is considered, the

monolithic arrangement is undoubtedly desirable in a practicalapplication. It seems that configurations 3, 4, and 6 can bemachined form a metal plate. However, in light of the geometricparameter l (l> 0.5), configuration 6 must be assembled by morethan two layers of plate. Similarly, the manufacturing performancesfor all these configurations are evaluated and depicted in Table 2.Finally, according to the performance comparison in Table 2,

the performances of configuration 3 are best; however, the motionrange for configuration 4 is not satisfying.

4.2 Constructed With the Complex Flexural Pivot. Asmentioned in Sec. 3.1, only the primitive flexural pivot was utilizedin the three schemes. In this section, the complex flexural pivot willbe further exploited, in order to improve the performance of thelinear-motion mechanism. Configuration 4 is taken as an example,and the complex flexural pivot combined by mirroring two general-ized cross-spring pivots is employed. The new compliant linear-motion mechanism is named as configuration 7, and the geometricparameters are defined as illustrated in Fig. 7.According to Table 1, the link length in configuration 7 is

the same as that in configuration 4. But if the complex flexuralpivot rotates an angle h, the angular displacement for a single gen-eralized cross-spring pivot is about h/2. Thus, for the complexflexural pivot, the dominant term of the center shift dyno,d can bederived as

dyno;d 2 9a2 9al l2 15l cos2 a

h2

2 9a

2 9al l2 30l cos2 a

(8)

The relationship between the geometric parameters is

k 1 al

(9)

So the compensation equation can be yielded easily, accordingto scheme II in Sec. 2.

Fig. 6 Design space for the geometric parameters l, a, and a: (a) for configuration 1, (b) for configuration 2, (c) for configuration3, (d) for configuration 4, (e) for configuration 5, and (f) for configuration 6

Table 2 Evaluation of all these linear-motion mechanismsa

ConfigurationNo.

Rangeof pivot

Lengthof link

Range ofmechanism

Manufacturingperformace

1 0 02 0 0 0 03 0 4 5 0 0 6 0 a: poor; 0: normal; : good.

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2 30 cos2 al2 18a 15 cos2 a2a 1 l 18a2 0(10)

The constraint condition for the geometric parameters is

a l < 1=2 a > 0; l > 0 (11)

If the monolithic manufacturing (all the leaf-springs can bemanufactured in one plate) is required, the constraint conditionbecomes more severe as

a l < 1=4 a > 0; l > 0 (12)

Then, the design space for the geometric parameters is obtained,as illustrated in Fig. 8. When a is zero, the pivot becomes the cart-wheel flexural pivot. Under this scenario, the compliant linear-motion mechanism can not be manufactured in one plate, becausel is greater than 1/4 and condition (12) is not satisfied. Being dis-tinct from the design in literature [19], this configuration cannotachieve monolithic arrangement if the cartwheel flexural pivot isemployed to compensate for the parasitic motion.As a result, comparing with configuration 4, the new compliant

linear-motion mechanism (configuration 7) has some remarkableadvantages. First, the complemented component is in the blank ofthe plate, so the new configuration is more compact. Second, therotational angle of the generalized cross-spring pivot will reduceto a half, if the motion range of the mechanism is identical.Finally, but most importantly, the drive force is moved away fromthe intersection point, which leads to the decreased stiffness andlower stress level.Finally, it is worth noting that configuration 7 is only an exam-

ple to display the method of using the complex flexural pivot.Some other configurations that are not mentioned in this section

can also be synthesized. However, the difficulties need to be indi-cated: if the pivots with different geometric parameters and shapeparameters are utilized to combine the complex flexural pivot, therotational angles for these flexural building blocks will not beidentical, and Eq. (8) will not be valid.

5 Case Study

Because configuration 3 has some good performances, itsmodel will be developed to quantitative predict the characteristicsof the mechanism in this section. On the basis of a single kine-matic chain and the whole mechanism, the building block methodis taken advantage of to simplify the derivation.

5.1 The Model of a Single Kinematic Chain. First of all,similarly with Eq. (6), the other nondimensional parameters in thegeneralized cross-spring pivot need to be normalized again interms of the characteristic length L

mp mnon ; fp fno

n2; pp pno

n2; dp dno

n2 d

n2

Hence, the center shift of the generalized cross-spring pivot areexpressed as the following form on purpose that the physicalattributes are revealed clearly

dxno Cxh noh3no Hxh no Cxf nofno Cxp nopno (13a)dyno Cyh noh2no Hyh no Cyf nofno Cyp nopno (13b)

The physical meanings of these coefficients are similar withthose in Eqs. (3a) and (3b). But these parameters and coefficientsare normalized by L but not Lp.The linear-motion mechanism is derived from the PFBM, so a

single kinematic chain is modeled first. In terms of the buildingblock method, an equivalent model is proposed in order to utilizethe model of the generalized cross-spring pivot. Taking advantageof this equivalent model, the boundary condition is transformedinto an initial condition and another relaxed boundary condition.Thus, the first kinematic chain is taken as an example to explainthis manipulation. As shown in Fig. 9, the following conditionsneed to be satisfied to fulfill the equivalence of the transformation.

Fig. 7 New configuration is design by using complex flexural pivot: (a) configura-tion 4 is constructed by primitive flexural pivot and (b) configuration 7 is con-structed by primitive flexural pivot

Fig. 8 Design space for the geometric parameters of configu-ration 7 Fig. 9 An equivalent model for a single kinematic chain

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5.1.1 Boundary Condition. Both of the two flexural buildingblocks are fixed on a rotational base. A local coordinate systemthat defines loads and displacements will rotate with the base.This condition defines a virtual base.

5.1.2 Initial Condition. The moving stage of pivot 12 (asillustrated in Fig. 5(c)) must be in parallel with the x axis of theglobal coordinate system at any final time; at the initial time itmust be in parallel with the x12 axis of the local coordinate sys-tem. This condition defines a virtual initial position.According to this equivalent model, the single kinematic chain

is divided into two flexural modules, and the model of the general-ized cross-spring pivot can be exploited. Hence, the external loadson both pivots can be obtained in the local coordinate systems. Itis important to note that the rotational angle h12 is negative.

f11 f1 cosh12 p1 sinh12p11 p1 cosh12 f1 sinh12

m11 m1

8