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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng (in press)

Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.1861

Synthesis of contact-aided compliant mechanismsfor non-smooth path generation

N. D. Mankame1,,, and G. K. Ananthasuresh2,,

1Vehicle Development Research Laboratory, General Motors Research & Development, Warren, MI, U.S.A.2Mechanical Engineering, Indian Institute of Science, Bangalore, India

SUMMARY

Topology optimization is used in this paper for the systematic synthesis of contact-aided compliantmechanisms that trace prescribed, non-smooth paths in response to a single, monotonically increasing inputforce. Intermittent contact interactions that enable these mechanisms to exhibit non-smooth responses alsolead to algorithmic difficulties when the techniques from the synthesis of ordinary compliant mechanismsare used to design contact-aided compliant mechanisms. A sequential optimization approach based ona regularized normal contact model for large displacements is used in this work to circumvent thesedifficulties and to enable the use of computationally efficient, gradient-based optimization methods. Weuse an objective function based on Fourier shape descriptors, which allows the designer to emphasizedifferent aspects of the design intent (such as the shape, the size and the orientation of the outputpath) separately. A variable-stiffness input spring is used to allow the synthesis procedure to choose theappropriate magnitude of the input force. An arc-length finite element solver and heuristic measures thatguard against local and global instabilities add to the robustness of the synthesis procedure as demonstratedby the two design examples presented in this paper. Copyright q 2006 John Wiley & Sons, Ltd.

Received 28 October 2005; Revised 15 May 2006; Accepted 10 July 2006

KEY WORDS: contact-aided compliant mechanisms; non-smooth optimization; topology optimization;path generation

Correspondence to: N. D. Mankame, Vehicle Development Research Laboratory, General Motors Research &

Development, 30500 Mound Road, MC 480-106-256, Warren, MI 48090, U.S.A.E-mail: [email protected]: [email protected] done as a graduate student at The University of Pennsylvania. Formerly at The University of Pennsylvania.

Contract/grant sponsor: National Science Foundation; contract/grant number: DMI-0200362

Copyright q 2006 John Wiley & Sons, Ltd.


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N. D. MANKAME AND G. K. ANANTHASURESH

1. INTRODUCTION

A compliant mechanism (CM) effectively uses its elastic deformation to transmit force and mo-

tion. Such mechanisms derive their mobility from the flexibility of their bodies unlike conven-

tional rigid-body mechanisms, which use only kinematic joints such as hinges for the samepurpose. The absence of joints in single-piece CMs makes it possible to fabricate them over

a wide range of length scales using techniques such as surface and bulk micromachining at

the micrometer scale, to stamping, extruding and injection moulding at the mm to m scale.

CMs also exhibit high repeatability of motion due to the absence of problems like backlash that

are associated with joint clearances in rigid-body mechanisms. Consequently, CMs have been

proposed for a number of applications ranging from disk loading mechanisms [1] to bicycle

clutches [2], surgical tools [3], micro-electromechanical relays [4] and fastener-free clamps [5]

amongst others.

The single-piece nature of some CMs also imposes certain limitations on their performance.

For example, CMs have a smaller range of motion than rigid-body mechanisms. Rigid-body

mechanisms, especially higher-pair mechanisms such as cams, gears, etc., are known for the

versatility of their force and motion transmission characteristics. However, like any stable elas-

tic body, CMs exhibit smooth motion and force transmission characteristics. The displacement

(or stress) field in a CM that is subjected to a gradually applied, monotonically increasing

force (or displacement) is continuous and has continuous first derivatives (i.e. C1 continuity) at

all points.

We [6] introduced the concept of contact-aided compliant mechanisms (CCM) to bridge the

gap between the functionality of CMs and rigid-body mechanisms. Intermittent contact interac-

tions allow CCMs to exhibit non-smooth motion or force transmission characteristics similar to

those exhibited by rigid-body mechanisms while retaining the advantages of scalability and ease

of manufacture that are associated with CMs. As an example of non-smooth response consider

the deformation of the body in Figure 1 that generates the output path P, which has C0 con-

tinuity. The qualifier intermittent is used to emphasize that the contacts used in CCMs are notclosed over the entire range of the mechanisms motion. This distinguishes CCMs from CMs

with a passive revolute joint which has a higher-pair kinematic joint involving a contact that is

always closed[7]. Caroll et al. [8] used contact interactions in a different way to assemble planar

structures into CMs that function in three dimensions. The contacts in these mechanisms are re-

tained after the assembly is completed, and hence are unlike the intermittent contacts that are used

in CCMs.

In earlier work we have shown that CCMs can be designed to exhibit repeatable, non-smooth

forcedeflection characteristics [9, 10], and to undergo large deflections repeatedly without com-

promising their ability to do useful work[11]. Building upon that work, this paper considers the

more general problem of systematically designing planar CCMs whose output port is required to

trace a prescribed non-smooth path in response to a single, gradually applied, increasing input

force. The restriction on the number (1) and nature of actuating forces allows the mechanism togenerate a complex motion with only a simple actuator.

Referring to Figure 1, it is required to design a CCM that fits inside the design domain (B).

The mechanism will be anchored to the ground over some part of its boundary (Sd), and will be

subjected to input forces(fe)over the part (Sf) of B. The output port (O) is required to trace the

path (P) in response to the input forces, while working against the load resistance (fres), which

acts at O. It is necessary that the output port of the CCM trace the path(P)as closely as possible;

Copyright q 2006 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (in press)

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SYNTHESIS OF CCMS FOR NON-SMOOTH PATH GENERATION

??

?

w

o

p

B

sc

sf

sd

fe

Figure 1. A schematic showing the CCM synthesis problem addressed in this work.

it is equally important that the CCM have sufficient stiffness to work against the output load while

tracing P.

The task of designing a mechanism such that a point on it traces a prescribed path when

the mechanism is actuated is known as path generation in kinematics [12]. A comprehensive

review of the developments in the design of rigid-body mechanisms for path generation is given

in Erdman[13]. CMs have also been designed for path generation. Howell [2, p. 277] describes a

compliant, approximate, straight line mechanism obtained by replacing the hinges in a rigid-body

straight line mechanism with flexures.Howell and Midha [14] designed CMs for non-linear path generation using the pseudo-rigid-

body methodapproach. Details of this method, which combines design techniques from rigid-body

kinematics with equations of static equilibrium for elastic bodies, can be found in [2].

Saxena and Ananthasuresh [15] and Pedersen et al. [16] were the first to use the topology

optimization approach to systematically design CMs for path generation. They noted that CMs

that trace a path of sizable length need to undergo large displacements, therefore, it is neces-

sary to use geometrically non-linear finite element analysis (FEA) in the synthesis procedure.

Recently, Saxena [17] used genetic algorithms that can work with discrete (01) and continu-

ous design variables to revisit the problem. The work reported in all of these references used

the structural error objective, which is common in optimization-based synthesis of rigid-body

mechanisms. This objective seeks to minimize the deviation of the CMs output path from the

desired path. The deviation is measured in terms of the Euclidean distance between the twopaths at the precision points, which are the points used to define the paths. Tai et al. [18]

used a morphological representation scheme in conjunction with genetic algorithms to design

path-generating CMs.

Unlike the design-centric approaches described in the above references, Swan and Rahmatalla

[19] take a control-centric approach to obtain a CM that can trace a family of non-linear paths in

the vicinity of a nominal prescribed path.


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Section 2 describes the motivation for and the development of a regularized normal contact

model that is used in this work for modelling contact interactions in CCMs undergoing large

displacements. The structural error objective function used in the above references, is replaced by

a more robust objective function that is based on Fourier shape descriptors. This objective function

emphasizes a global comparison of the shapes of the desired and actual paths. The new objectiveis an adaptation to CCMs of an objective proposed by Ullah and Kota [20] for the synthesis of

path-generating rigid-body mechanisms. Section 3 covers the formulation of the new objective. In

a CM, the force and displacement fields are coupled via the elastic constitutive law. In loss-less

rigid-body mechanisms, the input energy is equal to the output energy. In CMs some of the input

energy is stored in the elastically deformed body of the CM. These factors make it difficult for

a designer to specify the input force corresponding to a prescribed output path and output load

resistance a priori. Section 4 describes a way to alleviate this problem by treating the input force

as a design variable during the optimization process. CMs undergoing large displacements are

susceptible to local and global structural instabilities which make it difficult for the optimizer

to explore the entire design domain. Section 5 covers a geometrically non-linear FEA procedure

based on a co-rotational, total Lagrangian formulation (following[21, 22]) and an arc-length solver

(following [2325]). The task of synthesizing a CCM for path generation is posed as an opti-

mization problem in Section 6. This section covers the choice of design variables, and constraints,

and measures used to counteract the influence of structural instabilities. Two examples of CCMs

designed, using the approach described in this paper, to trace non-smooth paths are presented in

Section 7. Analysis of the designs shows that the designs are robust to moderate variations in man-

ufacturing and friction at the contact interface. Details of the work presented here can be found in

Mankame [26].

In this first effort to systematically synthesize CCMs, we make the following simplifying

assumptions. This work is restricted to frictionless and adhesionless contact. As the behaviour of

CMs is dominated by bending, we assume that the CCM will experience large deflections but the

associated strain in the body remains small enough to justify the use of a linear elastic constitutive

law. Finally, CCMs are designed for quasi-static behaviour in this work. These assumptions aremade only to simplify the implementation; the procedure presented in this work can be used even

if these assumptions are not made.

2. A REGULARIZED NORMAL CONTACT MODEL

The formally correct way of dealing with non-smoothness in an optimization problem is to use

non-smooth optimization methods [27]. However, it is desirable to use the smooth optimization

methods for the CCM design problem for reasons of computational efficiency and robustness. This

section describes a regularized normal contact model that smoothens the contact interaction, and

thereby enables us to use smooth optimization algorithms.

2.1. Motivation for a regularized contact model

Figures 24 show three stages in the deformation of an elastic body (B) that deforms un-

der the influence of an external force (f) and eventually contacts a rigid, frictionless, exter-

nal surface (W). The contact interaction is modelled by the following set of constraints on the


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B

W

P

fS

f

cS

dS

Figure 2. Complementarity between the contact reaction and the separation between two contacting bodiesleads to non-smoothness in their responsecase (a) no contact.

B

W

Pf

Figure 3. Complementarity between the contact reaction and the separation between two contacting bodiesleads to non-smoothness in their responsecase (b) degenerate contact.

displacement(u)and stress(r)at a point P on the part (Sc)of the boundary ofB which eventually

contacts W.

un g 0 on Sc (1)

n 0 on Sc (2)

n(un g) = 0 on Sc (3)


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t2

C

F

F

R

F

ww(b)

t1

Figure 5. Intermittent contact interactions give rise to non-smoothness in the CCM synthesis problem.

for the response at this point. Smooth optimization algorithms search for regular stationary points

that satisfy the KahrushKuhnTucker (KKT) constraint qualification conditions. The non-smooth

point described above does not satisfy these conditions. Therefore, smooth optimization algorithms

cannot be used directly for the CCM synthesis problem. Non-smooth points pose a challenge to

smooth optimization algorithms even when the optimal design does not lie at a non-smooth

point, because the large variations in the contact stresses across these points lead to convergence

difficulties for these algorithms[28].

Regularization or smoothening of the contact interaction removes the non-smoothness in the

response of a CCM by replacing a sudden change in the boundary conditions with a gradual one.Next, we develop a regularized normal contact model that is suitable for designing path-generating

CCMs.

2.2. A regularized contact model for large displacements

The regularized contact model for CCMs undergoing small displacements, which was presented in

Mankame and Ananthasuresh [9], uses a variable-stiffness, non-linear spring connected between

corresponding points on the potential contact boundary (Sc) and the rigid external surface (W).

This model cannot be used for modelling contact in CCMs that undergo large displacements for

the following reason.

The model assumes that the end of a contact spring attached to Wis fixed in space as shown in

Figure 6. The contact (-induced) force is a function of the deflection of the contact spring and isdirected along the axis of the spring. When a point on Sc undergoes large displacement, the line of

action of the contact force is no longer normal to the contact surface at B. The use of a potential

surface (also known as a level set) to represent an external contact surface eliminates this problem

while retaining the essential features of the earlier model. The revised model also allows a single

external contact surface to interact with many contact nodes. This leads to fewer design variables in

the CCM synthesis problem than if the contact model from Mankame and Ananthasuresh [9] is used.


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B

F

Figure 6. The regularized contact model for small displacements is not physically meaningfulif the elastic body undergoes large displacements.

F

B

~A

w = 0

w < 0

w > 0

n

A0

A

~A0

Figure 7. The regularized contact model for CCMs undergoing large displacements usespotential surfaces to model the external contact surfaces.

Consider a material point on the potential contact boundary (Sc) of an elastic body (B) that

moves from the position A0, when the body is undeformed, to the position A in the deformed

configuration of the body in Figure 7. Let Wbe a straight, planar, rigid, external surface represented


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by the following equation:

wc= wc0y+ wc1x+wc2= 0 (4)

where wci , i = 0, 1, 2 are constants. The line (wc= 0) partitions the plane of the page into theregionswc0. If the coefficients wci are chosen such that the body (B) lies in the

region wc>0 and |wc0| = 1 then, the shortest, directeddistance of a point (X, Y) from the line

(wc= 0) is given by

=wc(X,Y)

1+w2c1

(5)

The distance()is directedas it distinguishes between a point in the region wc>0 from its image,

under reflection by the external surface (wc= 0), in the region wc


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0 2020406080 40 60 80

0

10

20

30

40

50

60

70

80

fs/(

0)

/ 0

m= 0.1m= 1

m= 10

> 0, No Contact< 0, Resting Contact

Figure 8. Normalized contact reaction for the regularized normal contact model proposed forCCMs undergoing large displacements.

0 2020406080 40 60 80

0

0.2

0.4

0.6

0.8

1

ks

/

/ 0

m= 0.1

m= 1

m= 10

> 0, No Contact< 0, Resting Contact

Figure 9. Normalized contact stiffness for the regularized normal contact model proposedfor CCMs undergoing large displacements.


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where nwis the unit outward normal to the external surface Wat the point of contact. The use of the

normal to the external surface, rather than the normal to the contact boundary of the elastic body in

Equation (10) is an approximation made to simplify the contact analysis. This assumption does not

invalidate the synthesis procedure as contact occurs at a few isolated points in practical realizations

of CCMs. The geometry of the contact interface at these points can be modified to make the two nor-mals coincide, without altering the topology of the mechanism. The above model can also be used

to analyse contact with curved external surfaces by approximating these surfaces with polygons.

2.3. Use of the regularized contact model in CCM synthesis

The regularized contact model allows the non-smooth CCM synthesis problem to be decomposed

into a sequence of smooth, approximate problems. The members of this sequence are characterized

by increasing values of the sharpness parameter (m). This sequence converges to the actual non-

smooth problem as m [29].

A small value for m is used at the beginning of the optimization process. The resulting smooth

sub-problem is solved by a smooth optimization algorithm. The optimal design from this sub-

problem is then used as the starting guess for the next sub-problem that has a larger value for m .This process is continued until a reasonably high (e.g. 500) value ofm is reached. A high value of

m guarantees that the final design is consistent with the mechanics of contact for the problem. The

sequential optimization procedure, which is similar to continuation or homotopy methods [30],

also helps to widen the radius of convergence of the synthesis procedure by suppressing numerous,

local minima that correspond to poor conformance with the design intent.

3. A SHAPE-BASED OBJECTIVE FUNCTION

Shape is a fundamental characteristic of the desired output path. In this section, we develop an

objective function that emphasizes the shape of the output path in addition to its size and orientation.

3.1. Motivation for a shape-based objective

Most of the prior works (ref. Section 1) in the optimization-based synthesis of path-generating

mechanisms use the structural error objective function. The structural error between the actual

and desired output paths of a planar mechanism is given by the following expression:

SE=

iP

(xi xi )2 +(yi yi )

2 (12)

where (xi ,yi ) is the position of the output port of the mechanism at the i th precision point. The

corresponding point on the desired path (P) is ( xi , yi ). The correspondence is usually expressed

in terms of input load or displacement levels.

Although the structural error objective (SE)is simple to formulate, it does not perform well in

practice because it does not capture the design intent accurately. Its shortcomings as an objective

for path generation problems are summarized below.

1. The structural error objective is aimed at selecting a mechanism whose output path matches

the shape, size and orientation of the desired path simultaneously. This makes the problem

overly restrictive [20].


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3

3

2

1

2

1

(a)

3

2

31

2

1

3

2

31

2

1

(b) (c)

Figure 10. The structural error objective does not capture the design intent ofpath generation problems unambiguously.

2. SE implicitly imposes timing constraints, which require the mechanism to traverse a

specific precision point in P corresponding to a specified input force (or displacement)level [20].

3. The local or pointwise comparison between two curves at a finite number of precision points,

as performed by SE, does not unambiguously capture the differences in shape between the

two curves.

4. The SE objective is known to generate multiple, spurious, local minima that correspond

to designs which do not conform to the design intent. This results in poor designs from

optimization-based synthesis methods.

An example will serve to emphasize these shortcomings. The two curves in Figure 10(a) are

close in size and shape, and hence, the difference between them as measured by the SE objective

will be small. Panel (b) in the figure shows the same curves as in (a), but the corresponding

points on the two curves are different. Due to its implicit enforcement of timing constraints, SEwill indicate a significant difference between the two curves, although visually the curves are

similar. Timing constraints are not required in most applications of path-generating mechanisms;

only bounds on the permissible inputs (forces or displacements) have to be satisfied. By adding

constraints that are not required by the design intent, the SE objective makes the problem

specification overly restrictive[20]. The two curves in Figure 10(c) are very different. However, as

they happen to coincide at the precision points, the SE objective will consider them to be similar

curves. Pedersen et al. [16] worked around this problem by increasing the number of precision

points from three to five. Similarly, we [31] used 13 precision points to get a reasonable CCM

design. But at the outset, the number of precision points that adequately capture the design intent is

not clear.

The use of Fourier descriptor-based objective, which is discussed next, addresses these problems

and leads to a more robust objective that corresponds closely to the design intent.

3.2. Fourier shape descriptors

The curve describing the change in the slope of a closed planar curve as a function of its path

co-ordinate is periodic in nature, and hence, can be expanded as a Fourier series. Fourier shape

descriptors are functions of the amplitudes and phases of the harmonics in this expansion [32].


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l3

X

Y

0V = V

0 M

V1

V2

V3

l

2

3

M

ll

l2

Figure 11. A schematic showing the computation of Fourier shape descriptors for asimple, piecewise linear, planar curve.

These shape descriptors are a function of only the shape of the curve and are independent of its

size, position, and orientation.

The path traced by the output port of a CCM is defined by a piecewise linear curve that joins

the successive positions of the output port as the external forces are applied gradually. The desiredpaths for the CCM synthesis problem are expected to be simple, open curves. These are converted

into simple closed curves for the purpose of computing the shape descriptors by simply joining

the last point to the starting point.

Let Vi denote the i th vertex of a piecewise linear, simple, closed, planar curve as shown in

Figure 11. The vertices are labelled as indicated in the figure starting from V0. The change in the

angular direction at Vi is i and the length of the segment between the vertices V(i 1) and Viis i . If the co-ordinates of Vi are (xi ,yi ), then the following expressions are obtained for the

length and inclination with respect to the X-axis:

xi = xi x(i1) (13)

yi = yi y(i1) (14)

i =

(xi )2 +(yi )2 (15)

i = arctan 2

yi

xi

(16)


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and

i

=

(i+1) i , i (,)

(i+1)

i

2,

i>

(i+1) i +2, i


15/42


The last two measures are not Fourier descriptors. They are added to the objective to ensure

that in addition to the shape, the size and orientation of the actual and prescribed paths can also

be compared. The scaling factorsCA,C,CL andC are used to adjust the contributions from the

various error terms to correct for differences in their orders of magnitude and to assign relative

importance to these errors for a given application.The key advantages of using F in the CCM synthesis problem can be summarized as

Emphasizes shape

F allows the designer to emphasize a similarity in shape between the desired and actual

paths.

Compares shape, size and orientation independently

This gives the user the flexibility to assign different weights to the satisfaction of different

requirements.

Provides a global comparison

Unlike SE,Fcompares characteristics of two curves that are based on the complete curves.

Is convenient to use with the arc-length method as it does not impose timing constraints

The arc-length method, which is used in the FEA part of the CCM synthesis procedure(see Section 5), treats the level of applied load () as an independent variable along with

the displacement vector during the solution of the static equilibrium equations. Hence,

it is difficult to compute the state response for a specified level of the applied load.

As SE implicitly imposes timing constraints based on the input force (or displacement)

levels, it requires the output response to be computed for specific input force (or dis-

placement) levels. Therefore, the arc-length method cannot be used efficiently in

conjunction with SE. This is not a problem for F, which does not require the points

that define the output path to be determined at specific input load (or displacement)

levels.

The ability ofF to separate the shape, size and orientation requirements is important in the

design of CCMs. By focusing on the overall shape of the output path,

Freduces the occurrence ofspurious local minima that plague SE. Other advantages of choosing Fover SEare highlighted

by the following example.

Figure 12 shows six curves that represent the output paths of candidate CCM designs. The desired

path is shown in a solid (red) line with square markers, which correspond to the precision points

used for evaluating SE; the actual path is shown as a dashed (blue) line with circular markers;

the closing segment of both curves are shown by dotted lines; and the points on the actual curve

that correspond, in terms of timing, to the precision points are marked by (black) asterisks. The

numerical values of the objective functions are listed under each curve for the following choice

of coefficients in Equation (19): CA= 50, C= 0, C= 1000, CL= 1.5. The choice ofC= 0 is

motivated by the observation that err generates comparable errors for all curves, and thus, does

not provide a means to discriminate between them.

The desired and actual curves are similar in the case of C1. They also have similar timing.Both F and SE attain their lowest values for this case. Therefore, C1 is used as the reference

for comparing the relative performance ofF and SE. The relative error (ej ) for the curve

Cj , j= 1, . . . , 6 that is plotted in Figure 13 is given by ej= (Ej E1)/E1, where Ej is the

Colours refer to the electronic version of this document.


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1 1.5 2 2.5 3

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

1 1.5 2 2.5 3

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

1 1.5 2 2.5 3

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

1 1.5 2 2.5 3

11.2

1.4

1.6

1.8

2

2.2

2.4

2.6

C1

C4

C2

C3

1 1.5 2 2.5 3

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

C5

1 1.5 2 2.5 3

C6

SE F

Figure 12. Comparison of the SEand the F objectives for different output path types that are commonlyencountered in the optimization-based synthesis of CCMs.

1 2 3 4 5 6

0

10

20

Relative

30

40

50

60

70

80

Curve number

FSE

Figure 13. Comparison of the normalized values for the SE and the F objectives corresponding to theactual and desired output paths shown in Figure 12. The dashed portion of the curve for the structural

error corresponds to the case C4 for which the objective cannot be computed.


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numerical value of the error for curve Cj . This normalization allows for a fair comparison of the

two objectives by eliminating scale differences.

Curve C1: Similar curves

Curve C2: Influence of timing constraintsThe curves are the same as inC1, but their timing is different. As F is insensitive to timing

constraints, it has the same error value as in C1. But SE registers a substantial difference

between C1 and C2 because of the difference in timing.

In all subsequent cases, the timing of the actual path is chosen to be the same as in

C1, unless indicated otherwise, to de-emphasize the contribution of the timing-induced error

in SE.

Curve C3: Influence of scale

The actual path in this case is a 1:2 scaled reduction of the actual path inC1. Frecognizes the

similarity in shape, and registers a small difference between the two curves. This difference

is due only to the size term (Lerr) in Equation (19). The points on the actual curve that

correspond to the precision points are in the same relative position as in C1, but have been

shifted in space as a consequence of the scaling. As SE compares the curves only at thecorresponding precision points, it registers a substantial difference between them.

In some applications the overall size may not be important at the synthesis stage because

it may be feasible to simply scale up the design for getting the desired size. When using F,

the only change that is needed to accommodate this flexibility is to reduce the value ofCL.

As SE measures all these characteristics simultaneously, the actual objective would have to

be reformulated for a specificreduction in scale. This is not only tedious, but it also imposes

an unnecessary and arbitrary constraint on the synthesis procedure as it is difficult to specify

a fixed reduction in scale a priori.

Curve C4: Incomplete analyses

Structural instabilities cause analysis of candidate CM designs, that undergo large displace-

ments, to fail before the entire input load is applied. SE cannot be computed in this case

as a one-to-one comparison between all of the corresponding precision points cannot beestablished. Therefore, this common occurrence stalls the optimization procedure when SEis used as the objective function. In contrast, F can be evaluated consistently even when

the analysis fails to complete. A penalty approach is used in this work to guard against the

degenerate case when the actual path is defined by less than three points.

Curve C5: Smooth approximation of a non-smooth curve

The actual path in this case is a smooth approximation of the non-smooth, desired path. Our

initial experience with optimization-based synthesis of CCMs showed that the optimization

algorithms had a tendency to exploit structural instabilities instead of contact interactions

to satisfy the design requirements. Despite the significant visual difference between the two

curves, SE is not able to discriminate between them.

Curve C5: Random curve

The actual path in this case is a random curve that passes through the same three pointsthat correspond to the precision points as in C1. Once again, despite the significant visual

difference between the two curves, SE is unable to detect any difference between them.

The above examples show that SE does not capture the design intent unambiguously as it ex-

hibits false negatives and false positives. A false negative results when an objective flags curves

as being different when they are similar from the viewpoint of the functionality of a CCM,


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e.g. C2,C3. A false positive corresponds to the case when an objective identifies curves as being

similar when they are different, e.g. C5,C6. The objective F not only avoids these errors,

but also provides a consistent way to handle incomplete analyses (e.g. C4) that cannot be

analysed using SE.

4. INPUT FORCE AS A DESIGN VARIABLE

The transmission of force and motion in a CM are closely coupled because both are transmitted

via the deformation of the body of the CM. This makes it difficult to specify a priori, how much

input force needs to be applied to a CM for it to trace a given path while working against a given

output resistance. The difficulty is more pronounced in the case of CCMs as their elastic response

is highly non-linear. In this section, we show how this difficulty can be circumvented by treating

input force as a design variable in the synthesis procedure.

As discussed in Section 3, the design of a (C)CM is constrained by how much force the

driving actuator can provide. Typically, the (C)CM is designed with the assumption that the

0

12

3

4

00.5 1 1.5 2

2.53 3.5

0

0.5

1

fin

X

Ya

b

c

a

b

c

d

0 0.5 1 1.5 2 2.5 3 3.5

0

0.5

1

X

Y

a

b

ca

b

c

d

0 0.5 1 1.5 2 2.5 3 3.5 4

0

0.5

1

f in

Y

a

b

ca

b

c

d

(c)

(b)(a)

Figure 14. A priori specification of maximum input force in the CM synthesistask can lead to the rejection of useful designs.


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CS

B

W

F

IS

Figure 15. A variable input spring (IS) allows the maximum magnitude of theinput force to be a design variable.

entire force capacity of the actuator will be used. This assumption makes the problem easier to

pose, but it introduces an unnecessary constraint on the synthesis procedure. Figure 14 shows

three views of two curves: c1 (a-b-c) and c2 (a-b-c-d) drawn, respectively, in (red) solid lines

with square markers and (blue) dashed line with circle markers. The closing segments of both

curves are shown by dotted lines of the same colour. It is required to design a CCM that cantrace the curve c1 given an upper bound of F1= 4 N on the input force. Let D be a CCM

design that can trace the curve c2 which coincides with the curve c1 for an applied force level of

F2(= 2.5 N)


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framework of displacement-based FEA in a simple manner. As the input spring and the mechanism

are mechanically in parallel at the input port, the externally applied force F is effectively split

into two components: Fm, which is the input force actually experienced by the mechanism; and

Fs which is the part that stretches the input spring. For a fixed F, which is dictated by the upper

bound on the actuator force, the magnitude of the effective mechanism input force (Fm) can bevaried by varying the stiffness of the input spring. This also has physical significance in practice

because most actuators have stiffness of their own. This concept is simple to implement in the

synthesis procedure as any element type (e.g. same element type that is used to discretize the

design domain) can be used to model IS.

5. GEOMETRICALLY NON-LINEAR FINITE ELEMENT ANALYSIS

The success of an optimization-based synthesis scheme depends critically on its ability to explore

the entire design domain. In the context of the design of path-generating CMs, this requires us

to use a robust FEA procedure that can simulate highly non-linear elastic behaviour. This sectiondevelops a co-rotational, total Lagrangian formulation and an arc-length solver that are used for

geometrically non-linear FEA in this work.

5.1. Finite element formulation

Let the body (B)in Figure 1 deform under the action of the external surface tractions (fe)applied

over the part Sf of its boundary and body forces (fB), while it is anchored to the ground on the

part Sdof its boundary. The part Sc of the boundary of the body can potentially come into contact

with the rigid external surface (W) as the body deforms. Using the principle of virtual work,

the underlying boundary value problem can be represented in the total Lagrangian framework as

follows: B0

: dV=

B0

fBu dV+

Sf0

fe u dS+

Sc0

fcu dS (24)

where and are the stress and strain tensors at a point and u is a kinematically admissible

perturbation in the displacement field (u). The subscripts 0 in the domains of integration (e.g. in

Sc0) indicate referral to the initial (or undeformed) configuration of the body. The displacement

on Sc and contact-induced traction fc are constrained by Equations (1)(3). These constraints are

satisfied in an approximate manner by expressing fc as a function of the separation between B

and W according to the regularized contact model (ref. Equations (8) and (10)) developed in

Section 2.

The static equilibrium equation (24) can be rewritten in a discretized form in the following

manner:

fint(u)fext fc(u)= 0 (25)

where the nodal internal force vector (fint) contains the contributions from the internal stresses,

fext is the nodal external force vector and fc is the vector of contact forces exerted on the body

(B)by the external surface (W). The vectors fint and fextare constructed in the usual manner[21],

while fc is assembled by summing contributions (see Equation (10)) from all nodes on Sc.


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For the frame-like structures such as those used in this work, it is easier to assemble the

above equation in a co-rotational FE framework. In this framework, the displacement field of

an elastic body is decomposed into a rigid-body component, which does not induce strain, and a

deformational component, which induces strain. This separation allows use of the engineering strain

measure for geometrically non-linear analysis. We adopt the approach described in Crisfield [21]and Belytschko and Hsieh [22] for its simplicity.

5.2. A modified arc-length solver

The arc-length method proposed by Riks [33]and Wempner [34], and later modified by Riks [35],

Crisfield[23]and Ramm[36]among others, can simulate highly non-linear elastic responses such

as snap-through, snap-back, etc. This method uses a combined incrementaliterative approach to

solve Equation (25). Unlike the commonly used adaptive load incrementation method, which varies

only the applied load level()or displacement control method, which varies only the displacement

(u) at a selected subset of nodes in a controlled manner, the arc-length method adaptively varies

boththe load level () and the displacement (u)to trace the complete elastic response of the

body. In this work, we use the modified arc-length method based on the work of Crisfield [23],Hellwegg and Crisfield [25] and Kuo and Yang [24].

5.3. Sensitivity analysis

Analytical expressions for the computation of the sensitivity of the state solution (i.e. the dis-

placement field u) to perturbations in the design parameters of the CCM are obtained in this

section by direct differentiation of the static equilibrium equation (25). Analytical expressions al-

low the sensitivity of the objective (F) to be computed in an efficient manner. This significantly

reduces the computational burden of the synthesis procedure and its susceptibility to numerical

problems.

The vector (x)of design variables comprises the vector of material distribution parameters (w),

the stiffness of the input spring (wis), and the vector of variables (wci , i= 1, 2) that control the

position and orientation of the external surfaces (W) relative to the design domain (B). The state

sensitivity is obtained by differentiating equation (25) with respect to x.

(Kt Kc)du

dx=

fint

x+

fc

x(26)

where Kt (= fint/u) is the tangent stiffness matrix, and Kc (= fc/u) is the contact (-induced)

stiffness matrix. The computation of Kt and fint/x follows standard practice in geometrical

non-linear FEA (see, e.g. [21] for details).

The contact stiffness matrix is obtained by collecting the contact-induced stiffness contributions

from all nodes on Sc in the following manner:

Kc=

i Sck

ic= k

iswc0

n21 n1n2n1n2 n

22

(27)

where kic is the contact stiffness matrix at node number i on Sc, kic is the magnitude of the

corresponding contact spring stiffness (ref. Equation (9)), the components of the unit normal

to the external contact surface (nw) are (n1, n2)=

wc1

1+w2c1,wc0

1+w2c1

, and wci ,


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i= 0, 1, 2 are coefficients in the parametric equation (4) for the contact surface. The term fc/x

in Equation (26) can be obtained by differentiating equation (10) with respect to x as follows:

fc

x

=fcx n

w+ f

c

nw

x

(28)

The first partial derivatives in the above equation can be expressed in terms of du/dx using

Equation (10).

The FEA procedure described in the previous sub-section is used to trace the path of a candidate

CCM design until a specified load level or displacement magnitude is exceeded. The state sensitivity

is computed at all increments in the FEA solution by re-using the Kt and Kc matrices that were

computed for the state solution at that increment. These results are then used to compute the

sensitivity ofF in a straightforward manner. The FEA and the analytical sensitivity computation

procedures described above are validated by comparing them, respectively, with solutions from a

commercial FEA software ABAQUS [37] and the central finite difference method.

6. FORMULATION OF THE OPTIMIZATION PROBLEM

Based on the concepts presented above, the task of systematically synthesizing CCMs that can trace

specified, non-smooth paths can be posed as an optimization problem in the following manner:

Minx

F2= L1F +(1)

L2F (29)

subject to

K(i)eff(u)u

(i) +f(i)c (u) = (i)fext i= (1, 2, . . . , m) (30)

xL x xU (31)

1.0wc(xh,yh)

0 hCh (32)

JDwJ

W 1.0 (33)

The objective F2 comprises a weighted combination of two load cases: (L1)corresponds to the

absence of any resistance at the output port, and the second load case (L2) corresponds to the

presence of a specified resistance at the output port. Pedersen et al. [16] note that this approach

leads to designs for path-generating CMs that are not flimsy, and hence, are capable of driving

an useful output load in addition to tracing a specified path. If the exact nature of the output

resistance for the application, in which the CCM will be used, is known it is used in the loadcase (L2). For the load case L2 in the design examples in Section 7, we assume an arbitrary

output resistance that opposes the general direction of motion of the output port as it traces

the desired path. The weighting factor () is chosen such that a greater emphasis is placed on

the load case of interest. The effective stiffness matrix Keff (=Kt Kc) and the corresponding

displacement field u are evaluated at every increment (i= 1, . . . , m)in the history of deformation

of the body.


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All design variables are bound by side constraints (Equation (31)) to restrict them to physi-

cally meaningful values. Recall that the initial separation (controlled by wc2) and the orientation

(controlled by wc1) of the external contact surface (W) relative to the design domain (B) are

also design variables. Equation (32) imposes the constraint that only those combinations of these

two variables that lead to the external surface (W) being clear of the design domain (B), for agiven wc0, are meaningful in this problem. As B and Ware both piecewise linear, this constraint

needs to be imposed only on the convex hull (Ch) ofB. The resource constraint in Equation (33)

limits the amount of material that can be distributed in the design domain; varying the maximum

permissible material (W) allows the designer to explore different parts of the design domain.Additional functional constraints such as local bounds on the Von-Mises stress in elements or

local constraints to guard against buckling could also be used. But such local constraints are known

to result in a disconnected design domain, which makes it difficult for the optimization algorithm to

search the entire design domain. Constraint relaxation techniques that address these problems have

been reported in literature (e.g. see Saxena and Ananthasuresh [38] and the references therein).

Such constraints are not imposed in this work, but are proposed as future enhancements.The design vector (x) comprises variables that represent different physical quantities whose

ranges are very different. As an example, for the synthesis example 1 presented in the next section,

the widths (w) included in the stiffness design vector are of O(1), the input spring width wisand the variable representing the external surface slope wc1 are of O(0.1) , and the variable

representing the initial external surface separation(wc2) is of O(10); where O indicates order of

magnitude. It was observed that the performance of the optimization algorithm was enhanced if

all design variables and constraints were scaled linearly to be of O(1).

6.1. Numerical issues

Structural instability at the local and global levels is one of the major hurdles confronting the

optimization-based synthesis of CMs undergoing large displacements. There is no known system-

atic and computationally efficient approach to address this difficulty. We use different heuristic

approaches to address local and global instabilities as they arise from different causes.The process of redistributing material in the design domain to synthesize a CM leads to void

regions with little or no material. These regions make an insignificant contribution to the elastic

response of a CM, and hence, are considered to be absent from the final design. The low stiffness of

these regions leads to a local loss of structural stability, or equivalently of the positive-definiteness

ofKeff. This leads to numerical problems that stall the optimization process and the algorithm is

unable to explore the design domain completely.To avoid this problem, the lower bound on the design variables corresponding to the stiffness

distribution is fixed at a small positive value (wlb ) instead of zero. It is desirable to make wlb as

small as possible to ensure that elements, whose widths have reached this value do not contribute

significantly to the overall structural response. However, this desire needs to be balanced by

the requirement of avoiding the loss of structural stability. The sequential optimization approach

described in Section 2 is used to address these conflicting requirements in the following manner.A comparatively high value ofwlb is used for the first step in the sequential optimization procedure.

This value, which is dictated by the need to avoid the local buckling of elements, ensures a stable

structural response that by avoiding extremely small values for the widths. The value ofwlb is

decreased by a fixed factor (wlb ) at every step in the sequential procedure until wlb is reduced

to a value such that elements with this width make a negligible contribution to the overall elastic

response of the body.


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Isolatednodes are created in the void regions during the synthesis procedure when all elements

connected to these nodes reach their lower bounds. As these regions are excluded from the final

design, we follow Buhl et al.[39]in relaxing the convergence criterion at these nodes for the state

solution.

Global instabilities are simpler to address than local ones because they are not an inherent issuefor a well-posed problem. If the mechanism is required to drive a non-negligible output load while

it traces the desired path, the optimization algorithm should steer away from unstable designs. The

use of an arc-length solver also helps as it can analyse post-buckling responses. But some times,

especially during the determination of a search direction by the optimizer, problematic candidate

designs can arise. In this work, we penalize such designs with an artificially high objective value

(e.g. O(100)) and attempt to push the design away from this iterate by assigning an artificial

sensitivity to stabilize the design.

7. SYNTHESIS EXAMPLES

The synthesis procedure developed in this paper is used to design CCMs that can trace non-smoothpaths, whose lengths are of the same order of magnitude as the overall size of the mechanisms.

7.1. Example 1

The mechanical cycle multiplier [9, 10] was designed with the assumption that it would undergo

only small displacements. As the resulting design actually experienced large displacements, it had

to be refined on a trial-and-error basis after the synthesis procedure for it to meet the design

specifications. We revisit that problem by changing the prescribed specification from a function

generation task to a path generation task in keeping with the objective of this paper.

7.1.1. Example 1: problem specification. A schematic of the problem specification is shown in

Figure 16 using a ground structure discretization for the design domain. In this figure, the node

IS

EF

P

O

Fop

W

I

N2

N1

Figure 16. Schematic showing the problem specification for example 1.


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Table I. Parameters used in the synthesis example 1.

Parameter name Units Value

Initial width lower bound (w0

L

) mm 0.2

Final width lower bound (wL) mm 0.018

Reduction rate for the width lower bound

wL(i +1)

wL(i)

0.9

Width upper bound (wU) mm 4.0

Width starting guess

w0 wL

wU wL

0.2

Volume constraint

W

Wmax

0.40

Lower bound for external surface slope variable (wLc1) 5.67

Upper bound for external surface slope variable (wUc1) 11.43

Start guess for external surface slope variable (wc1) 7.6

Lower bound for external surface offset variable (wL

c2

) mm 22.68

Upper bound for external surface offset variable (wUc2) mm 91.44

Start guess for external surface offset variable (wc2) mm 34.0

Lower bound for input spring width variable (wLis) mm 0.5e6

Upper bound for input spring width variable (wUis) mm 2.0

Start guess for the input spring width variable (wis) mm 0.5e4

Push-off factor for the no-initial-penetration constraint

wU

0.5

Normalized weight for load case 1 in the objective () 0.85

Maximum magnitude of actuation force (| fext |) N 200.0

Maximum magnitude of perturbation force (| fop |) N 2.0

Coefficient of Fourier amplitude error measure (CA) 5.128

Coefficient of Fourier phase error measure (C) 0.4878Coefficient of path length error measure (CL) 0.0294

Coefficient of path orientation error measure (C) 384.26

Starting sharpness parameter (m0) 10.0Final sharpness parameter (mf) 200.0

Sharpness parameter incrementation factor

m(i +1)

m(i)

0.15

Post-contact stiffness () N mm1 5.0E+03Thickness (t) mm 3.175

Youngs Modulus (E) N mm2 1312.0

marked E is anchored to the ground. The actuation force (F) is applied at the input port (I).

The input spring (IS) is connected at the input port such that it experiences a tensile stress due

to F. One external contact surface (W) is positioned such that it does not intersect the design

domain. The nodes on the discretized contact boundary (N1 and N2 on Sc)are marked by boxes.

The output port (O) is required to move along the path indicated by P . The load (Fop) at O is

the perturbing load for the second load case in the objective evaluation (see Equation (29)). The


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010 10 20 30 40

0

5

5

10

15

20

25

30

35

40

45

[mm]

[mm]

Figure 17. Topology of the solution for synthesis example 1.

parameters used in this problem are given in Table I. In this table, the superscripts (i +1) and (i)

refer to successive steps in the sequential optimization procedure. The optimization problem

is solved using the fmincon function, which is an implementation of the Sequential

Quadratic Programming algorithm for constrained non-linear optimization, from the Matlab

Optimization Toolbox [40].

7.1.2. Example 1: solution. The topology of the CCM design obtained by solving the optimization

problem is shown in Figure 17, where the final position of the external contact surface is shown by

the dashed line. The widths of the various elements are approximately proportional in the figure,but are not to scale. Successive positions of the final design as it deforms due to the actuation

force are shown in Figure 18. The undeformed configuration is plotted in green and the deformed

configurations are shown in orange.

The actual output path for the above CCM design is plotted in the Figure 19 using a solid

dark (blue) line with square markers. For comparison, the prescribed output path is overlaid in

the figure in the dashed grey (red) line with circle markers. In the spirit of topology optimization,

the elements whose widths have reached their lower bounds, are removed from the design. The

output path for this refined design as obtained from a non-linear simulation using ABAQUS is

also overlaid in the figure using light grey, inverted triangles. Note that the desired and the actual

curves are comparable in shape, size and orientation.

7.1.3. Example 1: discussion. The performance of the solution and the synthesis procedure for

synthesis example 1 is analysed next.

The design undergoes large displacements

The amplitude of the output motion is approximately 3.5 mm for an input stroke of approx-

imately 7.0 mm. The total path length (11.3 mm) is approximately 28% of the characteristic

dimension (40 mm) of the design domain. Thus, the device experiences large displacements.


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Figure 18. Successive positions of the design as predicted by non-linear finiteelement simulation in ABAQUS.

32 33 34 35 36 37 38 39 40

16

17

18

19

20

21

[mm]

[mm]

Actual Path (C)

Desired Path

Actual Path (A)

Figure 19. Desired and actual output port paths for synthesis example 1. Results from the simulationusing the FEA procedure developed in this work (C) and ABAQUS (A).


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0 10105

20 30 40

0

5

10

15

20

25

30

35

40

45

[mm]

[mm]

Figure 20. The CCMs obtained from the synthesis procedure if all design elements with widths of 0.1 mmor less are discarded from the final design.

Constraint activity

Only two design elements in the ground structure reached the upper bound, and none reached

the lower bound. The Lagrange multipliers corresponding to the active upper bound constraints

were ofO(103). The small value indicates a very small improvement in the objective func-

tion corresponding to an increase in the upper bound. A number of elements with widths close

to the lower bound were obtained. Their influence on the design is discussed in the next topic.

The input spring width was also close to its prescribed lower bound. The volume constraintand the no-initial-penetration constraints were inactive. These results indicate that little or

no further gain in the objective was possible by modifying the constraint parameters from

Table I.

Interpretation of the topology

The final value of the lower bound on the design width variables is wL= 0.018 mm, which

yields the ratio wL/wU 0.004. Therefore, the contribution of the lower bound elements to

response of the final design is relatively small as is desirable in a topology optimization

procedure.

In the final design, there exist a number of elements that are near the lower bound, but

have not quite reached it. These elements are difficult to fabricate in a physical prototype.

Therefore, it is instructive to study the effect of discarding these elements on the overall

response of the design. In Figure 20 all elements below a width threshold of 0.1 mm arediscarded; in Figure 21 the threshold is 0.15 mm and in Figure 17 the threshold is 0.2 mm.

The threshold of 0.2 mm leaves a single element in the top right corner that is not connected

to the rest of the structure. This element does not contribute significantly to the behaviour of

this design and is considered to be absent from the final topology.

The output path of the design corresponding to the width threshold of 0.2 mm as obtained

from a non-linear simulation in ABAQUS is plotted in Figure 22 using a solid grey (magenta)


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0 1010

5

20 30 40

0

5

10

15

20

25

30

35

40

45

[mm]

[mm]

Figure 21. The CCMs obtained from the synthesis procedure if all design elements with widths of 0.15 mmor less are discarded from the final design.

32 33 34 35 36 37 38 39 40

16

17

18

19

20

21

[mm]

[mm]

Actual path (C), T=0.0177, =0Desired path

Actual path (A), T=0.0177, =0Actual path (A), T=0.0177, =0.35

Actual path (A), T=0.2, =0.35

Actual path (A), T=0.2, =0

Figure 22. Prescribed and actual output port paths for synthesis example 2 withdifferent width thresholds and friction coefficients.


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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.1

0.1

0.2

0.2

0.3

0.4

0.5

Load proportionality factor ()

NormalizedcontactforceatN

1(

c=

fc/fI)

= 0

= 0.35

Figure 23. The normalized contact force at the contact node for non-linear CCM synthesis example withdifferent width thresholds and friction coefficients.

line with asterisks () as markers. Note that despite the removal of several elements, this

design generates an output path that is close to the desired path because the contribution

of the removed elements to the elastic response of the design is not significant. In the next

synthesis example, we will use this observation to specify a less stringent wL. As mentioned

earlier, a less stringent lower bound on the design widths eliminates candidate designs withflimsy elements, thus leading to a more stable synthesis procedure.

Contact interaction and the influence of friction

The CCM design experiences contact only at the node N1 in Figure 16. The evolution of

the contact force as the input loads are applied gradually is shown in Figure 23 as a solid

black (blue) line with circle markers. The magnitude of contact force is normalized by the

magnitude of the maximum input force (200 N).

Although frictionless contact was assumed in the synthesis procedure, the physical proto-

types will experience friction. It is, therefore, of interest to study the influence of friction on

the behaviour of the final design. Plots of the output paths of the design for friction coefficients

of= 0.0 and 0.35 as obtained from non-linear finite element simulation in ABAQUS are

overlaid in the Figure 22 for two different width threshold values. The width threshold in mm

is denoted by Tin the caption. It can be noted that consideration of friction does not changethe qualitative or quantitative behaviour of the design significantly.

The evolution of the normalized contact reaction for the case = 0.35 is plotted in a solid

(grey) line with square markers in Figure 23. The normal reaction magnitude remains the

same, but as a consequence of friction, the actual maximum force needed to traverse the

complete path is increased. Hence, the normalized contact force for the frictional contact case

lies below the frictionless contact case in the figure.


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0 50 100 150 200 250

100

101

102

103

sharpness factor (m)

Normalizedobjectivefunction(F=f/f

0)

Figure 24. Normalized convergence history.

Convergence history and computational cost

Figure 24 shows the convergence history for synthesis example 1. The Y-axis is normalized

such that the value of the objective function for the initial guess is 1.0.

The number of iterations required for the different steps in the sequential optimization

procedure is a measure of the computational cost of that step and is plotted in Figure 25. This

example required approximately 72 hours on a 266 MHz, single processor, Sun Ultra Sparc5 workstation. It is noted that although the objective function decreases steadily as the se-

quential optimization progresses, the computational cost does not decrease continuously. This

is unlike the computational cost for the synthesis of CCMs undergoing small displacements

[9], which was the highest for the first step and decreased sharply subsequently. Hence, the

computational cost for the non-linear synthesis procedure is relatively much more than for

the linear synthesis procedure.

7.2. Example 2

A compliant gripper that can grasp a rigid work-piece and pull it back in a continuous motion was

designed in[31]using the structural error objective function. As noted in Section 3.1, 13 precision

points were required in the specification of the desired path to obtain a CCM that conformed to thedesign intent. In this section, we show that the revised objective function allows the same problem

to solved with only three points.

7.2.1. Example 2: problem specification. The problem specification is shown schematically in

Figure 26 following the same notation as in example 1. A design that is symmetric about the

centreline CL is desired, hence only one-half of the design domain is discretized for this problem


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0 50 100 150 200 2500

50

100

150

200

250

300

350

400

450

500

sharpness factor (m)

No.ofoptimizationiterationsforconvergence

Figure 25. The computational cost for the synthesis example.

N

E

F I

ISR P

CLWP

W

O

Fop

1N2

Figure 26. Schematic showing the problem specification for example 2.

and symmetry constraints are applied at the nodes (marked R in the figure) that are shared by

the two symmetric halves of the design. Contact with the rigid work-piece (WP ) is modelled by

contact with a hypothetical contact surface (W) that is tangential to WP. The parameters used in

this problem are given in Table II.

Unlike example 1, the lower bound (wL) on design widths is not reduced across steps in

the sequential optimization procedure in this example. It will be seen later that elements with


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Table II. Parameters used in the synthesis example 2.

Parameter name Units Value

Initial width lower bound (w0

L

) mm 0.23

Final width lower bound (wL) mm 0.23

Reduction rate for the width lower bound

w(i +1)L

w(i)L

1.0

Width upper bound (wU) mm 5.0

Width starting guess

w0 wL

wU wL

0.2

Volume constraint

W

Wmax

0.35

Lower bound for external surface slope variable (wLc1) 0.0083

Upper bound for external surface slope variable (wUc1) 0.0083

Start guess for external surface slope variable (wc1) 0.0

Lower bound for external surface offset variable (wLc2) 4.75

Upper bound for external surface offset variable (wUc2) 5.25

Start guess for external surface offset variable (wc2) 5.0

Lower bound for input spring width variable (wLis) mm 0.029

Upper bound for input spring width variable (wUis) mm 11.0

Start guess for the input spring width variable (wis) mm 0.4

Push-off factor for the no-initial-penetration constraint

wU

0.05

Normalized weight for load case 1 in the objective () 0.85

Maximum magnitude of actuation force (|fext|) N 2000.0

Maximum magnitude of perturbation force (|fop|) N 14.1

Coefficient of Fourier amplitude error measure (CA) 1.634Coefficient of Fourier phase error measure (C) 0.0

Coefficient of path length error measure (CL) 0.0367

Coefficient of path orientation error measure (C) 7.553

Starting sharpness parameter (m0) 8.4

Final sharpness parameter (mf) 200.0

Sharpness parameter incrementation factor

m(i +1)

m(i)

1.15

Post-contact stiffness () N mm1 5.0E+03

Thickness (t) mm 3.175

Youngs Modulus (E) N mm2 1312.0

widths less than the initial lower bound w0L= 0.23 mm do not affect the behaviour of the design

substantially. This is due to the fact that the elastic response of the design is dominated by bending

of the elements, and the ratio of the bending stiffnesses of elements with widths (wL= 0.23mm)

and (wU= 5.0 mm) is O(1e4). The external contact surface (W) has a fixed configuration


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(i.e. orientation and location) in the problem specification. However, it was observed that the

performance of the optimization algorithm was improved significantly when W was permitted

to move within a small range around its nominal configuration. It will be shown later that the

design obtained with this relaxation of the problem specification works well even for the nominal

configuration ofW.Non-linear FEA of the starting design (i.e. the initial guess for the optimization) is used to pick

some of the design parameters. The starting sharpness parameter (m0)is chosen such that the ratio

of the deflections at the output port due to the contact and input forces is 0.01. This ensures that there

is a small but non-zero initial contact force, and hence, the regularized contact model is realistic.

The width of the input spring is allowed to vary between limits([0.029, 11.0] mm)that correspond

to[5, 95]% of the input force being transferred to the CCM in the initial design. The initial width is

picked to be closer to the lower bound. This together with a high magnitude of the input force, ensure

that the input force acting on the CCM can be varied over a wide range during the synthesis pro-

cedure. The coefficientsCi , i= A, ,L , in Equation (19) are chosen such thatCAAerr :CLLerr :

Cerr= 0.5 : 0.25 : 0.25 and F= 1.0 for the starting guess. Note that C is chosen to be zero

for reasons discussed in Section 3.3.

7.2.2. Example 2: solution. The topology of one-half of the symmetric gripper design obtained

from the synthesis procedure is shown in Figure 27. The disappeared elements, i.e. those with

widthsw, wwL, are not shown.

The actual output path for the above CCM design is plotted in the Figure 28 using a solid

dark (blue) line with square markers. Note that the prescribed output path, which is overlaid in

the dashed grey (red) line with circle markers, is specified using only three points. All elements,

including the disappeared elements, are included in the FE simulation that generated the actual

0 10 20 30 40 50 60

0

5

5

10

10

15

20

25

30

35

40

[mm]

[mm]

Figure 27. Topology of the solution for synthesis example 2.


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50 51 52 53 54 55 56 57 58 59 60

6

5

4

3

2

1

0

1

X displacement [mm]

Ydisplacement[mm]

Desired path

Actual path (C), T=0

Actual path (A), T=0.23

Figure 28. Desired and actual output port paths for synthesis example 2. Results from the simulationusing the FEA procedure developed in this work (C) and ABAQUS (A).

output path of the CCM. If the disappeared elements are removed from the mesh and the resulting

design is analysed in ABAQUS, the output port of the CCM traces the path that is plotted shown

by a light grey line with triangle markers. As in example 1, it can be seen that the desired and the

actual curves are comparable in shape, size and orientation.

7.2.3. Example 2: discussion. The performance of the solution and the synthesis procedure for

synthesis example 2 is analysed next.

The design undergoes large displacements

The total length of the output path is 11.7 mm, which is approximately 20% of the character-

istic dimension of the design domain (60 mm). Thus, the device experiences large displace-

ments.

Constraint activity

No design element in the ground structure had reached either the upper or the lower bound.

As in example 1, a number of elements with widths close to the lower bound were obtained.

The input spring width was close to its prescribed lower bound. None of the design con-

straints were active. Thus, the design is a local, interior optimum point. The influence ofthe elements with widths close to the lower bound on the elastic response of the design is

discussed next.

Interpretation of the topology

The design shown in the Figure 27 comprises only those elements whose widths are greater

than the lower bound(wL= 0.23mm). Recall that although the configuration of the external

contact surface is fixed in the design specifications, it was allowed to vary within a small


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0 10 20 30 40 50 60

0

5

10

10

5

15

20

25

30

35

40

[mm]

[mm]

Figure 29. The design obtained from the synthesis procedure if all design elements with widths of 0.5 mmor less are discarded from the final design.

range in the optimization procedure to avoid an equality constraint in the formulation. As

a consequence, the configuration of the external contact surface in the final design does

not correspond to the line (x + 5 = 0) as shown in Figure 26. Instead it corresponds to the

line with a slope (1e4) passing through the point (30, 4.766). This interpretation of the

design will be referred to as D1. Figure 29 shows a schematic of the design comprisingonly elements with widths greater than a threshold of 0.5 mm; Figure 30 shows the design

drawn to scale; and Figure 31 shows successive positions of the design as it deforms.

In modelling this design, we approximate the external contact surface by a line passing

through the point (30,4.766), but with a slope of zero. This interpretation of the design is

labelled D2. The output path for D2 as obtained from ABAQUS is plotted in the Figure 32

in a grey (fawn) line with upright triangle markers. It is seen that this output path is similar

to both: the output path obtained for D1 (solid, light grey line with inverted triangle markers)

and the desired path (dashed red line with circle markers). From a functional viewpoint, any

width threshold which leads to a design that satisfies the design specification is acceptable.

In general, it is difficult to predict a priori up to what threshold the design will conform to

the specifications, but our experience suggests that a threshold ofwL= 0.1 wU works well

for bending dominated designs. Contact interaction and the influence of friction

The design experiences contact only at the node N1 in Figure 26. As the contact is

assumed to be frictionless in the synthesis procedure, the contact force at N1 is normal

to the contact surface. The evolution of this force as the input load is applied gradually for

the design D1 is plotted in Figure 33 in a solid grey line with inverted triangle markers

(N(A), T= 0.23,= 0). The magnitude of contact force is normalized by the magnitude


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Figure 30. A CAD scale model of the design for example 2.

Figure 31. Successive positions of this design as it deformspredicted by non-linear finite elementsimulation in ABAQUS (bottom).

of the maximum input force (2000 N). If the simulation is repeated for contact with mod-

erate friction (coefficient of friction, = 0.35), the normal and shear forces at the contact

interface vary according to the curves plotted in a solid, light grey (cyan) line with cross


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50 51 52 53 54 55 56 57 58 59 60

0

1

2

3

4

5

6

1

X displacement [mm]

Ydisplacement[mm]

Desired path

Actual path (C), T=0,=0

Actual path (A), T=0.23,=0

Actual path (A), T=0.23,=0.35

Actual path (A), T=0.5,=0Actual path (A), T=0.5,=0.35

Figure 32. The output paths for different interpretations of the design simulatedunder different contact conditions.

0 0.05 0.1 0.15 0.2 0.25 0.3

0

0.01

0.01

0.02

0.03

0.04

0.05

Load proportionality factor []

Normalizedcontactforce[f/fi,max

]

N (A), T=0.23, =0

N (A), T=0.23, =0.35

S (A), T=0.23, =0.35N (A), T=0.5, =0

N (A), T=0.5, =0.35

S (A), T=0.5, =0.35

Figure 33. Normal and shear forces at the contact interface for the synthesis example 2 as obtained fromnon-linear FE simulations in ABAQUS.


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0 50 100 150 200 250

100

101

102

103

Sharpness factor (m)

Normalizedobjectivefunction(f/f

0)

Figure 34. Normalized convergence history for synthesis example 2.

()markers (N(A), T= 0.23,= 0.35) and a dashed light grey (cyan) line with cross ()

markers (S(A), T= 0.23, = 0.35), respectively.

Repeating the above simulations for the design D2 yields the evolution of the normal

contact force (solid, grey (fawn) line with upright triangle markers, N(A), T= 0.5,= 0)for the frictionless case; and the normal contact force (solid, light grey (magenta) line with

upright circle markers, N(A), T= 0.5, = 0.35) and the shear contact force (dashed, lightgrey (magenta) line with circle markers, S(A), T= 0.5,= 0.35) for the frictional contact

case as shown in Figure 33.

The output paths for the frictionless and frictional contact cases are nearly coincident

for both D1 and D2 as shown in Figure 32. This is also true for synthesis example 1

(see Figure 22. Thus, we conclude that friction at the contact interface does not change the

kinematics of the mechanism design, but a larger input force is required to obtain the same

output displacement.

Convergence history

The normalized convergence history for this example is shown in Figure 34. The dashed line

connecting the first two points in the plot corresponds to the improvement in the objective

function in the first step of the sequential optimization procedure. A significant reduction in

the first step of the synthesis process underscores the importance of the starting value of thesharpness parameter (m0). The same trend was noted in example 1 as seen in Figure 24.

8. CONCLUSION

A topology optimization-based systematic procedure is developed for the synthesis of contact-aided

compliant mechanisms (CCM) that trace prescribed non-smooth paths.


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A regularized contact model, which was proposed in an earlier work[9], is extended in this paper

to account for CCMs that undergo large displacements. This model forms the basis of a sequential

optimization procedure that is used to circumvent the mathematical challenges arising from the

non-smoothness introduced by contact interactions. This approach allows efficient, gradient-based

optimization algorithms to be used for synthesizing CCMs. An objective function based on the useof Fourier shape descriptors is proposed for the synthesis of path-generating CCMs. This objective

captures the design intent more closely than the commonly used structural error objective, and

gives the designer the flexibility to emphasize different aspects (e.g. shape, size, orientation) of

the desired path independently. A variable-stiffness spring connected between the input port and

ground allows the effective input force for the CCM to be a design variable while the maximum

input force is held constant. This allows a variable input force to be included in the traditional

displacement-FE-based topology optimization framework in a straightforward manner.

An adaptive arc-length solver that is capable of simulating highly non-linear elastic responses

is used in this work. This enables the optimization algorithm to search a bigger part of the design

domain than was possible using algorithms that use only adaptive load or displacement control.

However, the optimization procedure still stalled occasionally due to problems associated with

local and global structural instabilities. These problems are addressed by heuristic means. Two

design examples are used to illustrate the capability of the synthesis procedure. An analysis of the

performance of the designs and the synthesis procedure is also included.

The synthesis procedure proposed in this work brings together a number of concepts that make

it easier to systematically synthesize CCMs for path-generation. Natural extensions of this work

include relaxation of the assumptions made in the implementation, reduction of the computational

cost, formulation of a more rigorous way to address local and global structural instabilities, etc.

ACKNOWLEDGEMENTS

We gratefully acknowledge the support of The National Science Foundation for this work via the gran

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