mechanical sciences - copernicus.org...2012/03/15 · 2mechanical engineering, bucknell university,...

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Mechanical Sciences

Open Access

Building block method: a bottom-up modular synthesismethodology for distributed compliant mechanisms

G. Krishnan1, C. Kim2, and S. Kota3

1Mechanical Engineering, University of Michigan, Ann Arbor, MI 48105, USA2Mechanical Engineering, Bucknell University, Lewisburg, PA 17837, USA

3Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48105, USA

Correspondence to:G. Krishnan ([email protected])

Received: 19 March 2011 – Revised: 31 January 2012 – Accepted: 2 March 2012 – Published: 28 March 2012

Abstract. Synthesizing topologies of compliant mechanisms are based on rigid-link kinematic designs orcompletely automated optimization techniques. These designs yield mechanisms that match the kinematicspecifications as a whole, but seldom yield user insight on how each constituent member contributes towardsthe overall mechanism performance. This paper reviews recent developments in building block based designof compliant mechanisms. A key aspect of such a methodology is formulating a representation of complianceat a (i) single unique point of interest in terms of geometric quantities such as ellipses and vectors, and (ii) rel-ative compliance between distinct input(s) and output(s) in terms of load flow. This geometric representationprovides a direct mapping between the mechanism geometry and their behavior, and is used to characterizesimple deformable members that form a library of building blocks. The design space spanned by the buildingblock library guides the decomposition of a given problem specification into tractable sub-problems that canbe each solved from an entry in the library. The effectiveness of this geometric representation aids user insightin design, and enables discovery of trends and guidelines to obtain practical conceptual designs.

1 Introduction

Kinematics of conventional rigid-link mechanisms and theirsystematic synthesis has been studied for almost a coupleof centuries (Erdman et al., 2001). This has resulted in adatabase of a number of tried and tested mechanisms, eachadept in a specific task (Sclater and Chironis, 2007). A de-signer can either choose an existing conceptual design fromthe database, or systematically combine a number of designsto meet a more complex specification. Once the conceptualdesign is chosen, it can be refined for the application at handby first analyzing the various geometrical configurations thatrigid links and kinematic pairs assume, and then analyzingforces in links to determine the amount of material requiredto maintain strength and rigidity. For most applications mul-tiple solutions can be generated, and secondary criteria suchas aesthetics and ergonomics can be used to determine thebest solution. The simplicity of this process is as a result ofthe decoupling between kinematic and structural aspects ofdesign.

In the past couple of decades researchers are interested inmonolithic mechanisms devoid of joints and links that de-rive their mobility on elastic deformation alone. They havesignificant advantages that are well documented in literaturesuch as elimination of friction, backlash and reduced manu-facturing costs by avoiding assembly (Howell, 2001). Fur-thermore, their versatility is increased by their scalabilityin various length scales. However, elastic deformation maylead to material failure at certain regions having high stressesthus limiting range of motion and the load carrying ability ofthese mechanisms. The main design challenge for compli-ant mechanisms is the intricate coupling between kinematicsand elasticity. In other words, the topology of a mechanismand the size of its individual elements together determinehow the mechanism moves and the internal forces within itsmembers, thus precluding synthesis akin to rigid-link mech-anisms.

As a first attempt at design, Howell and Midha usedconventional rigid-link topologies and the associated designmethodologies to first design a rigid-link mechanism that

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16 G. Krishnan et al.: Building block method

meets the kinematic specifications. Appropriate torsionalsprings were placed at the joints of this mechanism to meetthe stiffness specification. The rigid links were then replacedwith beams of equivalent length to match the required kine-matics. The cross-section of the beams were then deter-mined based on the stiffness of the torsional springs and thestress considerations. This technique where an equivalentrigid link and a torsional spring is used to analyze and de-sign monolithic elastic mechanisms is known as pseudo-rigidbody model. The designs that resulted from this methodol-ogy, though practical yield distinct areas where flexibility islumped. These are also the areas where stresses are concen-trated and thus limit lead bearing and large range of motions.

Topology optimization was developed byAnanthasuresh(1994) that aimed at a better utilization of the design spacewithout restricting to conventional topologies. This proce-dure initially lists all possible interconnections of beams ora planar finite element mesh of the design domain, whosewidths or thickness are the design variables for optimization.Elements with the values of design variables lower than aspecific cut-off will not be considered to contribute towardsthe final topology. The optimization algorithm determinesan optimal topology that maximizes an objective function.If the objective function defines the kinematic behavior ofthe mechanism alone, without incorporating strength consid-erations, the optimization yields mechanisms that are com-posed of rigid members and thin flexures (Saxena and Anan-thasuresh, 2003). Yin and Ananthasuresh(2003) andCan-field et al. (2007) have demonstrated the use of additionalconstraints that prevent large relative rotations between twofinite elements that constitute the topology in the design do-main. Nevertheless, there is considerable complexity in in-corporating stress considerations in topology optimization.Furthermore, this process yields little insight on the func-tional contribution of different elements in the topology. Fur-thermore, the optimum connectivity may not be present in thedesign parametrization, which leads to suboptimal or convo-luted designs.

Thus, there is a need to understand the contributions ofeach member that constitutes the design. This understandingcan yield insightful conceptual synthesis of compliant mech-anisms, where simple deformable members are systemati-cally combined to obtain the overall topology. Such a build-ing block method was proposed byKim (2005); Kim et al.(2008).

2 Overview of building block methods

Complex systems such as an automobile, aircraft and elec-tronic gadgets are always broken down into simpler subsys-tems that can be easily designed. For example, an automobileconsists of combustion, transmission, electrical subsystemsall working together. These subsystems are designed and fab-ricated separately and integrated during assembly. A similar

Problem Specification

Evaluation of Candidate Building Blocks

Problem Decomposition

Mechanism Assembly

Figure 1. Schematic of the building block method for design.

building block approach was proposed for mechanism designby Kota and Chiou(1992) for conventional rigid link mech-anism synthesis. The building block method is captured inthe flowchart shown in Fig.1 (Kim, 2005). Once kinematicspecifications were determined, they were compared with en-tries in the database of existing designs. If no entry is found,the problem specification is broken down to tractable sub-problems, whose solutions are found in the database. Thefinal design is an assembly of the individual subproblems.

Building block methods are common in conceptual de-signs where there is a functional independence between theconstituent building blocks. This means that when two build-ing blocks are combined, one does not change the inherentbehavior of the other. For example, a slider joint remainsa slider irrespective of the number of links attached to it.However, in domain specific design problems such as com-pliant mechanism synthesis, the deformation behavior of abuilding block is in general determined by rest of the topol-ogy. In other words the same building block may have twodifferent deformation behaviors based on its loading condi-tion. Consider a simple cantilever beam that is ubiquitousin most compliant mechanism topologies. Its deformationbehavior can range from fixed-free to fixed-guided based onloads acting on the free end as seen in Fig.2. In most prob-lems this deformation behavior cannot be determined beforehand. Is it then possible to use the elements of the buildingblock method as shown in Fig.1 for compliant mechanismsynthesis?

The answer for the above question lies in the representa-tion of compliance that favors building block method. Sucha representation must provide

1. Expression of compliance quantities from first princi-ples: the representation can aid visual insight if geo-metric quantities are used to express various aspects of

Mech. Sci., 3, 15–23, 2012 www.mech-sci.net/3/15/2012/

G. Krishnan et al.: Building block method 17

(a) (b)

Figure 2. A beam with an end load has different deformation pro-files when its end is(a) free, or (b) guided (constrained rotationand axial motion). Whether it behaves as(a) or (b), or between thetwo in a compliant mechanism depends on the subsequent membersattached to it.

compliance. Furthermore the representation must beintrinsic to the topology and independent of referenceframes used to evaluate it.

2. Parametric characterization: simple deformable build-ing blocks are characterized with changing geometricparameters to span the design space. Use of geomet-ric quantities enable visualization of the tractable designspace.

3. Systematic decomposition of problem into tractablesubproblems: the compliance representation must pro-vide user insight in problem decomposition.

4. Enabling seamless assembly: the physics must enableseamless integration of the subproblems into the finalsolution.

There have been efforts in the past decade to propose var-ious representations that permit a building block approach.At a single point of interest the compliance matrix has beencharacterized as a three dimensional ellipsoid (Kim et al.,2008; Kim, 2005). This representation enabled designingmechanisms with a given stiffness behavior by parallel com-bination of curved beams and dyads (series combination ofbeams). Design for single-input single output mechanismshave been similarly proposed based on combining a num-ber of single point mechanisms between an input and output.Though these techniques yield conceptual designs that meetkinematic specifications, the representation lacks mathemat-ical rigor and thus yields limited insight in the process ofcombining building blocks. This paper reviews recent ad-vances in proposing a physically insightful, mathematically

β+δ

ef1ef2 δrE

CoE

Input

fx fy

m

Figure 3. Eigen-twist and Eigen-wrench parameters for a particularbuilding block geometry.

robust compliance representation at a single point of interest(Krishnan et al., 2011) and relative compliance between twopoints (Krishnan et al., 2010).

3 Single port compliance: eigen-twist andeigen-wrench characterization

Single port compliance involves characterizing the force dis-placement relationship at a single point of interest. This is ofimportance in the design of constraints (Awtar et al., 2007),suspensions for microsystems, and elastic vision based sen-sors (Cappelleri, 2008). This force displacement relation-ship is given by the compliance matrix (or its inverse stiff-ness matrix). The dimensions of the compliance matrix inplanar two dimensional case is 3×3. The three degrees offreedom are two translations in the plane of the geometryand a rotation about an axis perpendicular to the plane. Theterms in the compliance matrix thus consist of both transla-tional and rotational terms having different dimensions. Toavoid dimensional inconsistencies, it is desired that transla-tions and rotations are dealt with separately. This is accom-plished by shifting the point of interest from the input to anew point where decoupling translational and rotation termsof the compliance is possible (Lipkin and Patterson, 1992).This point, known as the Center of Elasticity (CoE) and is al-ways unique for a planar geometryKim (2008) and is shownfor a compliant dyad in Fig.3. This point is similar to theremote center of compliance (RCC) in robotics, and the wellestablished concepts of center of stiffness or center of com-pliance (Ciblak and Lipkin, 2003) as defined for a planar ge-ometry. If a rigid connection is established between the CoEand the input, then any force applied at this point yields puretranslation, and any moment applied yields a pure rotation.Thus the translational compliance at this point can be repre-sented by an ellipse whose semi-major and semi-minor axes(af1 andaf2) denote the primary and secondary compliancedirections. In these directions (ef1 andef2), any force applied

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if any force is applied along the direction of the coupling vec-tor no rotation is observed. This vector is thus named as thecoupling vector. Similar to the compliance matrix, the termsof the stiffness matrix can be represented as a stiffness ellipseand a stiffness coupling vector as seen in Fig. 5. Further in-sight into this characterization can be obtained in Krishnanet al. [11].

δ

-1500 -500 0 1000 2000-1500

-500

0

500

1500

af1

af2

-100

0

100

β+δ

-150 -50 0 50 150

0β+δ

rE2/kg

rE/kg

C11 C12 C13C12 C22 C23C13 C23 C33

Fig. 4. Compliance ellipse and Compliance coupling vector (~cv).

-1000 -500 0 500 1000

-1000

-500

0

500

10001/af1

1/af2δ

-100 -50 0 50 100

-100

-50

0

50

100

β+δ+γ

scS11 S12 S13S12 S22 S23S13 S23 S33

Fig. 5. Stiffness ellipse and Stiffness coupling vector (~sc).

This representation of compliance easily sets stage fora systematic building block based synthesis method. Thefirst stage for the building block method after determiningthe problem specification is evaluation of candidate build-ing blocks, or developing a library of building blocks. Themost versatile building block for compliant mechanism syn-thesis is shown to be a series combination of two beams, asa number topologies are shown to be composed of them [8],[12]. The eigen-twist and the eigen-wrench parameters canbe evaluated by varying the angle between the two beamsthat make up the dyad and their relative lengths. Figure 6plots np = a f2/a f1 for varying length ratios (radius of the po-lar plot) and dyad angles α. This gives an indication of thedesign space spanned by the dyad for np. Similarly otherparameters (rE , β, a f1 ) are plotted in Kirshnan et al [11].

210

240270

300

3301

1.52

1

1.5

2

30

6090

120

150

180 0

0.01

0.1

0.2

0.4 0.50.6

0.5

α

l1

l2 = l2norm× l1

radius: l2normangle : α

1 1.5

2

np

Fig. 6. Parametric characterization of a compliant dyad for its eigen-twist and eigen-wrench parameters. The figure shows one such plotadapted from [11].

Consider an example where equal biaxial (X and Y )stiffness is required at a point without any coupling trans-lational and rotational terms. Such a specification is requiredfor a vision-based force sensor [18], where external appliedforce can be evaluated by measuring the deformation of apoint. Such a problem specification requires a circular com-pliance ellipse with zero coupling vector magnitude shown inFig. 7a. Comparing from the database of compliant dyads nodesign matches these specifications [11]. Figure 7b-c and 8illustrate achieving these specifications using series and par-allel combination of dyads. In series combination, the cou-pling vectors of individual building blocks add. Thus thezero coupling vector specification can be achieved by align-ing equal and opposite building block specifications. Sincethe degenerate shift ellipse depends on the coupling vectororientations alone, its magnitude can be evaluated and sub-tracted from the required ellipse to obtain a net ellipse (Fig.7b). Two dyads are then chosen from the building block li-brary to meet the ellipse specifications. The next step in-volves assembly of dyads between themselves, and betweenone of the dyads and the input using rigid connecters as theydonot change the compliance characteristics at the CoE of abuilding block. Thus all the steps illustrated in Fig. 1 areaccomplished with geometrically intutive quantities.

One of the limitations of series combination is that theCoE always lies within the footprint of the mechanism (forproof of this, please refer Kirshnan et al. [11]). This doesnot provide an easy access of the input for interacting withthe objects in the vision based force sensor application. Toovercome this, parallel combination of building blocks arerecommended. During parallel combination, the stiffness el-lipses and striffness coupling vectors of the building blocksadd. Two sub-mechanisms whose stiffness ellipses are cir-cular and whose stiffness coupling vectors are aligned equaland opposite to each other are connected together as shownin Fig. 8. A practical realization of this involves parallelcombination of symmetric halves with some accommodationfor a rigid probe as shown in Fig 8d. The resulting input hasequal biaxial compliance and decoupled translational and ro-tational compliance.

Figure 4. Compliance ellipse and Compliance coupling vector (cv).

yields translation along the same direction. The rotationalstiffness at this point can be represented by a scalar valuekg.The orientation of the CoE is given by an angleβ and theangleδ refers to the orientation of the geometry in the twodimensional plane. These quantities are illustrated in Fig.3.They are called eigen-twist and eigen-wrench characteriza-tion as the parameters can be obtained by eigen value analy-sis of the compliance matrix with selective normalization ofits twists and wrenches (Kim, 2008).

Though it is convenient to characterize compliance at thecenter of elasticity as seen above, it is required to relate thisto the compliance at the input point, as this was our originallocation of interest. The compliance at the input can be repre-sented by the very same terms that characterize it at the CoEwith some additional terms as seen in Fig.4. The translationellipse at the CoE (af1 andaf2) is supplemented with a degen-erate ellipse oriented perpendicular to the line jointing theinput and the CoE. Furthermore the coupling between trans-lational rotational compliance at the input is represented by acoupling vector (cv) whose magnitude denotes the amount oftranslation obtained due to a unit moment, or the amount ofrotation obtained due to a unit force. However, if any force isapplied along the direction of the coupling vector no rotationis observed. This vector is thus named as the coupling vector.Similar to the compliance matrix, the terms of the stiffnessmatrix can be represented as a stiffness ellipse and a stiffnesscoupling vector as seen in Fig.5. Further insight into thischaracterization can be obtained inKrishnan et al.(2011).

This representation of compliance easily sets stage fora systematic building block based synthesis method. Thefirst stage for the building block method after determiningthe problem specification is evaluation of candidate build-ing blocks, or developing a library of building blocks. Themost versatile building block for compliant mechanism syn-

-1000 -500 0 500 1000

-1000

-500

0

500

10001/af1

1/af2δ

-100 -50 0 50 100

-100

-50

0

50

100

β+δ+γ

scS11 S12 S13S12 S22 S23S13 S23 S33

Figure 5. Stiffness ellipse and Stiffness coupling vector (sc).

210

240270

300

3301

1.52

1

1.5

2

30

6090

120

150

180 0

0.01

0.1

0.2

0.4 0.50.6

0.5

α

l1

l2 = l2norm× l1

radius: l2normangle : α

1 1.5

2

np

Figure 6. Parametric characterization of a compliant dyad for itseigen-twist and eigen-wrench parameters. The figure shows onesuch plot adapted fromKrishnan et al.(2011).

thesis is shown to be a series combination of two beams, as anumber topologies are shown to be composed of them (Kim,2005; Krishnan et al., 2010). The eigen-twist and the eigen-wrench parameters can be evaluated by varying the angle be-tween the two beams that make up the dyad and their relativelengths. Figure6 plotsnp = af2/af1 for varying length ratios(radius of the polar plot) and dyad anglesα. This gives anindication of the design space spanned by the dyad fornp.Similarly other parameters (rE, β, af1) are plotted inKrish-nan et al.(2011).



Consider an example where equal biaxial (X andY) stiff-ness is required at a point without any coupling translationaland rotational terms. Such a specification is required for avision-based force sensor (Cappelleri et al., 2010), where ex-ternal applied force can be evaluated by measuring the de-formation of a point. Such a problem specification requiresa circular compliance ellipse with zero coupling vector mag-nitude shown in Fig.7a. Comparing from the database ofcompliant dyads no design matches these specifications (Kr-ishnan et al., 2011). Figures7b–c and8 illustrate achievingthese specifications using series and parallel combination ofdyads. In series combination, the coupling vectors of indi-vidual building blocks add. Thus the zero coupling vectorspecification can be achieved by aligning equal and oppositebuilding block specifications. Since the degenerate shift el-lipse depends on the coupling vector orientations alone, itsmagnitude can be evaluated and subtracted from the requiredellipse to obtain a net ellipse (Fig.7b). Two dyads are thenchosen from the building block library to meet the ellipsespecifications. The next step involves assembly of dyads be-tween themselves, and between one of the dyads and the in-put using rigid connecters as they donot change the compli-ance characteristics at the CoE of a building block. Thus allthe steps illustrated in Fig.1 are accomplished with geomet-rically intutive quantities.

One of the limitations of series combination is that theCoE always lies within the footprint of the mechanism (forproof of this, please referKrishnan et al., 2011). This doesnot provide an easy access of the input for interacting withthe objects in the vision based force sensor application. Toovercome this, parallel combination of building blocks arerecommended. During parallel combination, the stiffness el-lipses and striffness coupling vectors of the building blocksadd. Two sub-mechanisms whose stiffness ellipses are cir-cular and whose stiffness coupling vectors are aligned equaland opposite to each other are connected together as shownin Fig.8. A practical realization of this involves parallel com-bination of symmetric halves with some accommodation fora rigid probe as shown in Fig.8d. The resulting input hasequal biaxial compliance and decoupled translational and ro-tational compliance.

Thus, it is shown how compliance representation in theform of eigen-twist and eigen-wrench parameters enables asystematic and insightful building block method to synthe-size single-port compliant mechanisms.

4 Multi-port compliance: load flow analogy

In the previous section we dealt with compliance where de-formation was directly actuated by an applied force at therequired point. There are a number of transmission prob-lems such as grasping objects and amplifying, where forceapplied is spatially separated from the required output de-formation. The challenge for the building block method is

E2 E1Ipm

rE2→ rI

→SubtractShift ellipsefrom target

2(rI)2

kg

1

2

3

1

2

3

E2 E1Ipm

rI→rE2

→

Net Ellipse

Subdivide into two smaller ellipses

(a) Problem Specification

(b) Spatial orientation

(c) Net Ellipse Subdivision

Rigid Connector

Zero Coupling vector and Circular Compliance

af1= af2= 0.15 mm/N

= 0.09 mm/N

0.060.15

0.12

0.035

0.03

0.025

rI = rE2

(d) Building Block Geomtery

Figure 7. Guidelines with an example.(a) Problem Specificationin terms of Compliance Ellipse and Coupling vector(b) Choose E1,E2, Ipm and evaluate shift ellipse(c) Net ellipse evaluation and sub-division into smaller building block ellipses(d) Design geometry ofthe two building blocks and their orientation.

formulating a representation for relative compliance betweenany two points in a continuum that can facilitate a buildingblock method. In other words, it must enable representationof problem specification that can easily be decomposed intotractable subproblems. In this section, we present a load flowbased analogy of relative compliance representation.

Between two points in a continuum, the relation betweenthe applied forces and deformation is obtained from the ex-tended form of the compliance matrix as shown below.[

uin

uout

]=

[Cin Cin-out

CTin-out Cout

][f in

fout

](1)



CoE

θ θ

CoE of the combined mechanism

θθ

+ =

(d)

Left half sv Right half sv Net sv

+ =

Left half ellipse Right half ellipse Net ellipse

CoE

(a)

(b)

(c)

Figure 8. Parallel Combination(a) Two symmetric halves(b) Ad-dition of Stiffness Coupling Vectors(c) Addition of Stiffness el-lipses(d) Final mechanism with a rigid probe.

whereuin andf in are the displacements and applied load re-spectively at the input, anduout andfout are the displacementsand applied load respectively at the output. Load transfer be-tween the two ports is defined as an equivalent applied loadat the output that produces the same output displacement asa unit input load. Load transfer can be thus defined as theoutput load that would cause the same output deformationas an applied input load as shown in Fig.9. This is similarto the notion of transferred forces defined inHarasaki andArora (2001). The relation between this transferred load andthe applied load is given by the LT matrix given by Eq. (2).

f̃tr =C−1outC

Tin-outf in

TL =C−1outC

Tin-out (2)

whereTL is the Load Transfer (LT) matrix that relates thetransferred load and the input loadf in. The detailed deriva-tion of the Eq. (2) is shown inKrishnan et al.(2010). Further-more, it must be noted that the transferred force in Eq. (2) isnot the same as an applied output load in Eq. (1). The impli-cation of defining transferred load is that a two-port problembetween the input and the output is converted into a singleport problem where the transferred force acting at the outputproduces the required output displacement.

Input

Output

fin uin

uout

fin= 0

uoutfout= 0

fout

~

(a) (b)

Figure 9. Deriving the Load Transfer matrix for Complaint Mecha-nisms (Krishnan et al., 2010). (a) Output displacement is evaluatedfor an applied input load(b) Output reaction load is evaluated byenforcing the output displacement from(a) with no input load. Thisreaction load is the transferred load.

While this characterization captures the relative compli-ance between two ports, its usefulness for a systematic syn-thesis is captured through an important property that enablesmodularity. Consider the two geometries and their deformedprofiles in Fig.10a and b. These geometries are composed ofa beam that acts as an input constraint in series with a beamthat connects the input and output. The output in Fig.10a isconstrained by a third beam which is absent in Fig.10b. It isfound that the transferred load evaluated at the output for thetwo different geometries is exactly equal, implying its inde-pendence on the output constraint. This property is true forall geometries consisting of a general input constraint in se-ries with a general transmitter element between the input andthe output. The detailed proof of the property is presentedin Krishnan et al.(2010). Thus, the fundamental buildingblock for load transfer between two points is identified as aLoad-Transmitter Constraint (LTC) set. The implication ofthis property in a mechanism composed of a number of LTCsets is that the transferred load in each LTC set can be inde-pendently evaluated of succeeding ones. This enables mod-ularity in analysis and design of two-port compliant mecha-nisms. One other observation is that the transferred load isindependent of the deformation profile of the transmitter. Forexample, the transmitter in Fig.10a is almost fixed-guided(like Fig. 2b), while the transmitter in Fig.10b deforms as arigid body.

The simplest of the LTC sets is a compliant dyad as seen inFig.10b. It has a beam for a constraint and a beam for a trans-mitter. Irrespective of the direction of the input force appliedthe transferred force is always oriented along the axis of thetransmitter. If a unit input force is applied perpendicular tothe orientation of the input constraint,the output transferredforce is given by



12

3

12

3

4

αα

α

Input Constraint

TransmitterOutputConstraint

finfin

ftrftr

~ ~

(c)

(b)(a)l1

l2

b1

b2

br=b2/b1

l2norm=l2/l1

Figure 10. Comparison of two geometries containing(a) inputconstraint beam, transmitter beam and output constraint beam and,(b) input constraint beam and transmitter beam alone reveals thatthe transferred force at the output is independent of the output con-straint. (c) The output displacement depends on the orientation ofthe output constraint. In general any direction of the output dis-placement is permitted±90◦ with respect to the direction of trans-ferred force.

f̃tr =finy

sin(α)(3)

whereα is the inclination of the transmitter with respect tothe constraint. Applying an input moment changes the direc-tion of the transferred force along with inducing transferredmoment. The transferred force evaluated will then yield

f̃trx = cot(α) fin−3(l22normcos(α)+b3

r )mi

2l1(l2norm+b3r )l2normsin(α)

f̃try = fin−3l2normmi

2l1(b3r + l2norm)

m̃tr =−b3

r mi

2(b3r + l2norm)

(4)

wherel2norm is the ratio of the transmitter beam length to theinput constraint beam length,l1 is the length of input con-straint beam length,br is the ratio of the transmitter beamthickness (in-plane) to the input constraint beam thickness.The output transferred forces (f̃ox and f̃oy) depends upon theinput force and input moment. However, the output trans-ferred moment ( ˜mo) is dependent on the input moment aloneand its direction is opposite to the input moment. Further-more, from the above equation, the magnitude of the trans-ferred moment is lower than the applied moment.

Though output constraints do not affect the transferredforce, they determine the magnitude and direction of outputdisplacement. Shown in Fig.10c is a semicircular band±90◦

with respect to the transferred load. From the positive def-initeness of the stiffness or compliance matrix, the outputdisplacement is constrained within this band. Its actual di-rection is dependent on the output constraint. For example

1

2

3

4

5

67

They do notintersect

1

2

∩

They intersect

1

3

2

1

3

2

1

2

3

4

5

(a) (b)

(c) (d)

Figure 11. Steps involved in the load flow based conceptual syn-thesis of two-port compliant mechanisms.(a) Kinematic problemspecification and the inability of a single load path to solve the prob-lem, (b) two load paths with the appropriate load flow directionsthat meet the kinematic specifications,(c) constraints that enforcethe load flow directions must be oriented along a truncated band,and(d) final mechanism topology and the deformed configuration.

the output constraint beam in Fig.10a constrains the outputto move along the direction determined by the intersection ofits degree of freedom with the semicircular band.

So far, we have formulated a representation for relativecompliance between two distinct points, identified its uniqueproperties that permit modular analysis and characterized asimple building block, namely a dyad. We will present asimple example of how this technique can be used for sys-tematic synthesis. Consider a problem specification shownin Fig. 11a where force applied at the input is at point “1”in the y-direction and the required displacement is at point“2” at an angle of−45◦. This is a nontrivial problem be-cause no direct connection between the input and output willyield the required kinematic specification. This is apparentfrom the figure as the direction of load flow in the transmit-ter does not intersect with any of the semicircular band di-rections at the output. To enable intersection, the problemis decomposed into two load paths, and the direction of loadflow in the transmitters is determined. The mechanism topol-ogy (i.e. the constraints) will be designed such that the pre-determined load flow directions are imposed. The constraintsat input point “1” must enable y-direction displacement. Abeam shown in Fig.11c thus acts as the input constraint. Theconstraint at point “3” must be oriented such that load flowdirections are preserved. This can be ensured if the degree of



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(a)

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(c)

34

(d)

Figure 12. Steps involved in generating conceptual solutions hav-ing multiple load paths.(a) Kinematic problem specification andplanning two parallel load paths between input and output,(b) trun-cated bands at each node,(c) constraints that enforce the load flowdirections, and(d) final mechanism topology and the deformed con-figuration with widths of transmitters and constraints optimized sothat the required deformation is achieved.

freedom direction of the constraint is along a truncated bandobtained by the intersection of the bands at points “2” and“3” as shown in Fig.11c. Furthermore the output constraintat point “2” is a beam that constraints it to move at an angleof −45◦. The deformed configuration is shown in Fig.11d.

The example illustrates the generality of the methodologyinvolving a single load path from input to the output. Thesame example can be used to demonstrate the use of multipleload paths between the input and the output. This is illus-trated in Fig.12. The required direction of transferred loadat point 1 can be obtained by a combination of load paths1-3-2 and 1-4-2. The net load transferred due to each pathadd in proportion to the stiffness of the individual paths. Forexample, if stiffness of path 1-3-2 is greater than 1-4-2 thenthe former would dominate. If the stiffness of each load pathis tuned so that they are equal, then the net load transferred isa vector combination of the individual paths. The constraintsare designed in Fig.12c such that they correspond to one ofthe truncated band directions. The widths of the constraintsand transmitters are optimized such that the output in point 2moves along a 45◦ angle.

Thus, a number of conceptual solutions can be gener-ated by planning load paths and constraints using the abovemethod. Comparison of each conceptual solution in terms ofstress distribution, stiffness and mechanical efficiency is re-

quired to choose the best solution for a given problem speci-fication.

5 Contributions and future research directions

This article shows how building block methods can be usedto generate conceptual designs for compliant mechanisms.The ease with which kinematic specifications are met bysystematic combination of simple deformable members, andthe ability to quickly obtain alternate solutions highlights theusefulness of this method. Furthermore the lack of numeri-cal complexity and the emphasis on user insight makes thistechnique excellent for classroom education. It is the rep-resentation of compliance that enables this user insight intosystematic synthesis. In this article, such a representation andsynthesis methodology is reviewed for single port problems,where force displacement relationship is characterized at asingle point of interest, and for multi-port problems whererelative compliance between two or more ports are charac-terized.

5.1 Contributions

Representing compliance at a single point is accomplished bydecoupling translational and rotational terms thus preservingdimensional consistency. Ability to represent translationalcompliance as an ellipse and the coupling between transla-tions and rotations as vectors enables insightful quantifica-tion of the compliance characteristics. Series and parallelcombination of any deformable member is explained as a ge-ometric combination of their individual ellipses and vectors.Thus solving for a given compliance characteristics, whichwas so far remotely accomplished through optimization isnow possible through systematic, yet intuitive methods.

Synthesis of two-port mechanisms have always been con-sidered non-intuitive as the contributions of each member to-wards the overall mechanism behavior is hard to understand.This is the first attempt towards identifying the functions ofeach member as a transmitter and a constraint. Representingdeformation behavior as load flow enables identifying andthus determining feasible load paths that meets a given kine-matic specification. Such a representation enables load pathto act as a skeleton for the overall mechanism geometry. Asseen in the examples, it is possible to synthesize single con-tinuous load path and multiple parallel load paths for any ap-plication with relative ease. This insight and ability to obtainalternate solutions with ease highlights the usefulness of thismethod.

To summarize, building block method with geometricallyinsightful compliance representation is a novel synthesismethod from first principles. Though this article focused ondesigning for kinematic specifications alone, the versatilityof the method may show promise in designing for strengthbased considerations and manufacturing limitations.



5.2 Future research directions

The main challenge of compliant mechanism design is ob-taining distribution of stresses evenly within all its con-stituent members. While meeting kinematic specificationsalone leads to multiple solutions, it is necessary to evalu-ate which of these solutions lead towards distributed com-pliance. Secondly, the effects of large deformations on com-pliance representations must be studied. In single-port de-sign this translates towards determining the change of theeigen-twist and eigen-wrench parameters with deformation.In two-ports the changes of load flow direction and magni-tude must be determined for large deformations. Thus, thefuture directions in building block methods is towards under-standing what constitutes distributed compliance, and formu-lating strategies in achieving them.

While planar examples were alone presented in this paper,the ideas proposed are as relevant to spatial mechanisms.The development of screw theory based methods (Hopkinsand Culpepper, 2010a,b) in designing mechanisms withpredetermined degrees of freedom and constraint can beconsidered as a spatial extension of the single port problem.However in mechanisms with relative deformation betweentwo or more ports that are not in general connected by arigid body, a combination of screw theory and load flow canbe envisioned.

Edited by: J. HerderReviewed by: two anonymous referees

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