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Proceedings of the ASME 2009 International Design Engineering Technical Conferences &Computers and Information in Engineering Conference

IDETC/CIE 2009August 30-September 2, 2009, San Diego, USA

DETC2009-86845

SYNTHESIS METHODS IN COMPLIANT MECHANISMS: AN OVERVIEW

Juan A. GallegoDept. of BioMechanical Engineering

3ME - BmechEDelft University of Technology

Delft, 2628 CDThe Netherlands

[email protected]

Just HerderDept. of BioMechanical Engineering

3ME - BmechEDelft University of Technology

Delft, 2628 CDThe Netherlands

[email protected]

ABSTRACT

Compliant mechanisms are rapidly gaining importance, yet

their design remains challenging. A great variety of methods are

being developed as it is reported in a growing stream of publi-

cations. However, so far no review of this body of literature is

available.

This paper provides a comprehensive and conceptual

overview of the main notions behind the most relevant design

methods for compliant mechanisms. Rigid-Body-Replacement

methods including the Pseudo-Rigid-Body model and the FACT

approach are covered, as well as Building Block approaches. In

addition an introduction and explanation on Topology Optimiza-

tion and Shape Optimization is provided, including their most

common parameterizations and formulations.

This work aims to serve as an introduction into compliant

mechanism design methods and as a reference work for more

experienced scholars and professionals. It is intended to be a

starting point for the exploration of the literature, as well as a

guide to specific papers about a particular design problem one

may have. For this reason, the paper presents the methods in a

wide perspective, emphasizing the conceptual ideas behind every

method and refers to literature for details and advanced features.

Address all correspondence to this author.

1 INTRODUCTION

Compliant mechanisms are those mechanisms that accom-

plish their function due to the deformation of one or more slen-der segments of their members; they do not rely exclusively on

the relative motion between joints and the rigid link. From the

referenced literature, advantages of compliant mechanisms were

collected, which may be summarized as follows.

Due to their monolithic nature compliant mechanisms pos-

sess two main benefits over conventional rigid-link mechanisms,

namely no relative motion among pieces and no overlapping

pieces.

The absence of relative motion implies the absence of slid-

ing friction, which eliminates wear, noise, vibration and the need

for lubrication. Consequently, less maintenance is required. Fur-

thermore, backlash is eliminated, which leads to reduced posi-tioning error and therefore increased precision.

The fact that there are no overlapping pieces allows fewer

parts and single piece production, which reduces the assembly

and weight. Therefore, compactness, miniaturization are en-

hanced while production costs are reduced.

All these benefits help to create more innovative designs and

actuation arrangements which increase the solution search space.

In the case of adaptive structures, compliant mechanisms mean

that fewer actuators are required.

Apart from the above advantages, the monolithic nature of

compliant mechanisms also gives rise to some drawbacks. Due

to potential energy storage in the compliant segments, the input-

1 Copyright c 2009 by ASME

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output relationship is affected. In particular, energy efficiency

is challenged. As a consequence, synthesis and analysis cannotbe done by separating kinematics and dynamics. The design pro-

cess is even more complicated if the compliant segments undergo

large deflections, in which case the governing stress and strain

equations become non-linear. In the design process of compliant

mechanisms the stress and strain relationships must be consid-

ered because they determine the deformed shape of the elements

and therefore the input-output behavior of the mechanism.

These disadvantages tend to turn the design into a trial and

error process highly dependent on the designers experience [1].

This prevents the wide use of compliant mechanisms and there-

fore few examples are available to be used as inspiration for new

developments [2]. Although compliant mechanisms have beenused for more than a century, the last 20 years have shown a

proliferation of new methods for analysis and synthesis of such

mechanisms.

Despite all the work that has been done in compliant mech-

anisms, there are only a few introductory documents and books

available [3] [4] [5], but none of them present an accessible and

comprehensive introduction to the synthesis methods.

For those with little or no experience in the field of com-

pliant mechanisms this has posed a problem: finding a starting

point from where they can be guided towards the solution of aspecific design problem. This problem is also enlarged by the

amount of knowledge areas that converge here: compliant mech-

anisms involve kinematics of mechanisms, multi-body dynam-

ics, non-linear mechanics of materials, numerical optimization

techniques, etc. This paper aims to present a comprehensive

overview of the most common synthesis approaches for compli-

ant mechanisms. The document is also intended to provide the

reader with a basic understanding of the methods in a way that

the reader acquires a sufficiently wide knowledge base to inves-

tigate particular methods or to find methods that are suitable for

particular application. Three main different design synthesis ap-

proaches for compliant mechanisms are distinguished; the kine-

matics based approaches, the building blocks approaches and the

structural optimization based approaches. The organization of

the individual methods within these main approaches is repre-

sented in Fig. 1.

In the kinematics based approaches two methods are dis-

cussed: the rigid-body-replacement method based on flexure

joints and pseudo-rigid-body models, and the freedom and con-

straints topologies. In the building blocks approaches two meth-

ods were identified: the instant center approach and the flexi-

ble building blocks. In the topology and shape optimization ap-

proach the optimization problem is presented as well as the most

common parameterizations and objective formulations.

2 KINEMATIC APPROACHES

Two main methods can be found here; the Freedom and Con-straints Topologies (FACT) and the Rigid-Body-Replacement

method. As the name suggest these methods aim to obtain de-

signs by focusing on kinematic requirements.

2.1 The FACT method

The FACT method [6] [7] is based on mapping a set of geo-

metric entities in the freedom space into a set of geometric enti-

ties in the constraint space, where the topology solutions for the

design problem can be found.

Basically the designer translates the required mechanisms

motion into degrees of freedom (DOF) which are used to find the

geometric entities that describe the required motion in the free-dom space. Knowing these geometric entities, it is possible to

find in the constraint space the topologies of the flexure elements

that provide the desired motion.

In the FACT method there are twelve sets of geometric en-

tities, but only eight have importance in flexure systems. The

other four are still in the process of being correlated with flexure

systems.

To give a clear idea of the method a simple example will be

given. Imagine that some device is needed having two DOFs; a

displacement in thez axis and a rotation about the x axis, Fig. 2.

Figure 2. Device with two DOFs.

The rotation can be represented by one of the twelve ge-

ometrical entities, the P-plane. The P-plane represents a plane

containing all co-planar and parallel rotation lines with a specific

orientation. In our case the P-plane is parallel to thexy plane,and the lines have the same orientation of the rotation axis, see

Fig. 3.

The displacement can be represented by a geometrical entity,

the Hoop, this entity represents the displacement as a rotation in

the infinite about any axis that lies in the xy plane, Fig. 4.

Figure 3. The P-plane entity.


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Figure 1. Synthesis of compliant mechanisms

Figure 4. The Hoop entity.

Now that the DOFs have been translated into a set of two

geometrical entities, a Hoop and a P-plane, the mapping from

the freedom space to the constraint space can be performed.

According to the FACT method the Hoop and P-plane to-gether in the freedom space are mapped in the constraint space

with two other geometrical entities, the Box and the A-plane1,

see Fig. 5. From these geometrical entities the constraints will

be obtained.

The Box represents the constraint lines inside this box,

which are parallel in the direction specified by the P-plane in the

freedom space. The A-plane represents any constraint line that

1In the FACT method the mapping is done by searching in a table the case

and the type that better suit the concerning problem. In the example, the problem

requires four non-redundant constraints which match with Case 4 and the geo-

metrical entities in the freedom space (hoop and P-plane) match with a Type 2

problem

Figure 5. mapping between the freedom space and the constraint

space.

lies on this plane, which is parallel or coincident with the lines in

the Box.Because the problem has two DOF it means that it needs four

constraints. The constraints can be thought of as truss elements

(a bar between base and object, connected by spherical joints).

A solution to the example can be seen in Fig. 6(a), where the

constraints were replaced by beam elements designed in such a

way that they provide the proper flexibility and stiffness in the

required directions to replace the constraints, see Fig. 6(b).

The FACT method is a type synthesis method that only pro-

vides topologies for the mechanisms; it requires the use of some

complementary dimensional synthesis to complete the design

process. The method not only considers the rotational and trans-

lational displacements but also the screw movement, which is a


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(a) (b)

Figure 6. Possible solution for the FACT example. (a) Selected con-

straints, (b) possible solution, here the compliant beam replaces three of

the constraints.

coupled rotation and translation movement.

2.2 The Rigid-Body-Replacement

The Rigid-Body-Replacement method [3] [1] makes exten-

sive use of the Pseudo-Rigid-Body (PRB) method [4].

Basically the Rigid-Body-Replacement consists of finding a

rigid body mechanism that accomplish the desired function and

then convert it into a compliant version. The conversion is per-

formed by replacing the joints using a Pseudo-Rigid-Body model

or beam deflection models, or by simply replacing the conven-

tional joints with flexure joints.

In the classification of compliant mechanisms two main

types of compliance can be distinguished, lumped complianceand distributed compliance. In the case of lumped compliant

mechanism the deformation takes place in a concentrated part of

the constitutive elements while in a distributed compliant mech-

anism the deformation occurs along a broader part on the consti-

tutive elements, Fig. 7.

(a) (b)

Figure 7. (a) Lumped compliance, (b) distributed compliance.

Based on the mentioned distinctions, the synthesis ap-

proaches in the rigid-body-replacement can be divided in design

based on flexure joints (either with lumped compliance or dis-

tributed compliance) and design based on PRB model (either for

lumped compliance or distributed compliance).

2.2.1 Flexure joints. A flexure joint is a region which

can undergo large deflections relative to stiffer adjacent regionsin the same element. Normally these stiffness differences are

attained through the geometrical characteristics of the deflection

regions. Depending on these characteristics, the flexure joint can

show single or multiple deflection axes which can be rotational

or translational axes.

Flexure joints can be categorized as primitive and complex

flexures [8]. Flexural joints with rotational axes are also known

as flexure hinges or flexure pivots.

Flexure joints for lumped compliance Among primitive flex-

ures there are the small length flexures and the notch-type flex-

ure hinges were the notch profile can be a rectangular section,

corner filleted, circular, parabolic, hyperbolic, elliptical, inverseparabolic, secant or hybrid sections [912], as shown in Fig. 8.

(a) (b)

(c) (d)

Figure 8. Notch-type flexure hinges. (a) Corner filet, (b) circular, (c)

parabolic, (d) hybrid.

Usually the design of the notch-type flexure hinges is con-

fined to small displacements but they can be designed to undergo

large deformations [13,14].

Flexure joints for distributed compliance Flexure joints for

distributed compliance can be both primitive and complex flex-

ures. The primitive flexures can be shaped as ellipses, four-bars,

chevron, etc. See Fig. 9.

(a) (b) (c)

Figure 9. Primitive flexures for distributed compliance. (a) ellipse, (b)

chevron, (c) four-bar.

Complex flexures are combinations of more simple flexures


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which can be design to act as revolute joints, prismatic joints or

as universal joints [15,16], Fig. 10.

(a) (b) (c) (d)

Figure 10. Complex flexural joints, reproduced from [16]. (a) universal

joint, (b) revolution joint, (c) and (d) prismatic joints.

Complex flexures can be combined to create even more com-

plex elements, i.e. [17] present the design of Compliant Paral-

lel Kinematic Machines (CPKMs) using a set constraining legs.

Their design is based on the previous set of three complex flexure

joints, see Fig. 11.

Figure 11. Flexural joints for the design of constraining legs for CPKMs,

reproduced from [17].

Examples of other complex flexures are the Compliant

Contact-Aided Revolute (CCAR) joint [18] and the CompliantRolling-contact Element (CORE) [19], which can act as a com-

bination of bearings and springs, see Fig. 12.

2.2.2 Pseudo-Rigid-Body model (PRB). The PRBmodel is an approach that allows to find a rigid-body mechanism

that emulates the behavior of a compliant member that under-

goes large, nonlinear deflections. The deflection path is given

by the kinematics of the rigid-body mechanism whilst the force-

deflection relation is approximated by springs that represent the

members stiffness.

During the design of a compliant mechanism, the PRB

model has its main role in the conceptual design stage in the

(a) (b)

Figure 12. Contact-based joints. (a) The CCAR joint [18], (b) The CORE

joint [19].

transition from the type synthesis to the dimensional synthesis.

Analyses based on kinematics are simpler, so the use of PRB

model provides with a quick way to test concepts and therefore

reduces efforts to obtain final concepts, just before to proceed

with the detailed design.

The PRB models vary depending on the boundary conditions

applied at both ends of the beam, these conditions can be fixed-

fixed, fixed-pinned and pinned-pinned; they determined how the

loads are applied.

When two compliant members interact it is important to de-

termine which condition better suits this interaction.

PRB model for lumped compliance The design of compliantmechanisms with lumped compliance using PRB models is based

on the PRB model for small-length flexural pivot [1, 4]. Here

the compliant member to be designed shows two segments, one

large and stiff and the other short and flexible. The short one and

flexible is known as the small-length flexural pivot.

The idea is to find the position of the characteristic pivot

and the characteristic stiffness for the spring in the PRB model.

Figure 13 shows a member with a small-length flexural pivot and

its PRB model.

(a) (b)

Figure 13. PRB for a beam with small-length flexural pivot, reproduced

from [4]. (a) Deflected beam, (b) equivalent PRB model.


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The pivot is placed in the middle of the short segment and

the stiffness constant is given by

K=(EI)l

l(1)

Wherelis the length of the small-length flexure pivot,Eand

Iare the Youngs modulus and cross-section second moment of

inertia respectively for this segment.

An accurate use of the PRB model requires that: L l (Lten times or more larger thenl), andEIL EIl , also the membermust be subjected to pure bending.

PRB model for distributed compliance In this approach thecompliant member is assume to have a constant cross-section.

The most important PRB model for distributed compliance

is the model for a cantilever beam (fixed-pinned condition2) with

a force acting at the free end [4,20], see Fig. 14.

(a) (b)

Figure 14. PRB for a cantilever beam with force at free end, reproduced

from [4]. (a) Deflected beam, (b) Equivalent PRB model

Here the position of the characteristic pivot is given by the

value of characteristic radius . The characteristic stiffness forthe spring is given by a stiffness coefficient k and , see eq. 2.

K= KEI

l(2)

Both, k and, are function of the n parameter which setsthe orientation of the applied force at the free end as a proportion

of its components, see Fig. 15.

The equation for(n)can be found in [4, 2022]. Equationfork(n)can be found in [4,21, 22].

When the applied force at the free end has an orientation

angle ranging 63.4 135 or 0.5 n 1, a constant value

2This means that no moments are created at the free end of the beam

Figure 15. Applied force at the free end of the beam.

of=0.85 and k=2.65 can be assumed for rough calculations,otherwise it must be used the equations for (n)andk(n).

These values provide an accuracy of 0.5% on the deflectionpath for deflection angles below 77.

[23] presents a PRB model for combined end loads with

positive end moments. [24] presents a PRB model for compliant

members that are initially-curved with pinned-pinned boundary

condition. [25] presents a PRB model for a cantilever beam with

an end moment acting opposite to an end force. [26] presents a

PRB model for a beam with boundary conditions fixed-fixed. Su

[27] presents a PRB model for deflection angles larger than 77,

where the beam is divided in four rigid segments with three joints

and their respective characteristic springs. This model allows

combined end force and moment.

3 BUILDING BLOCKS APPROACHES

In the building blocks approach the idea is to concatenate

multiple compliant mechanisms that perform simple functions

to create compliant mechanisms that can perform more complex

functions. There are two main building block approaches; one

based on instant centers and compliance ellipsoids and the other

one based on flexible building blocks and optimization.

3.1 Building blocks by instant centers

The building block approach based on instant centers [28

33] is a conceptual design procedure.The idea is to find a mechanism that provides for a given in-

put displacement an output with a desired displacement direction

and geometric advantage (GA). The mechanism is found by con-

catenating two different basic blocks, the compliant dyad build-

ing block(CDB) illustrated in Fig. 16(a) and the compliant four-

bar building block (C4B) illustrated in Fig. 16(b). These two

blocks can be used to form combinations, like those presented in

Fig. 17(a) and 17(b)

This synthesis process is called the dual stage synthesis. Be-

fore explaining how the dual stage synthesis is used, three con-

cepts are introduced: the principal compliance vector, the instant

center and the decomposition point.


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(a) (b)

Figure 16. Basic building blocks, reproduced from [31], (a)Compliant

dyad building block (CDB), (b) four-bar building block (C4B).

(a) (b)

Figure 17. Building blocks concatenation, reproduced from [31]. (a)

Combination of two C4B, (b) combination of C4B and CDB.

Theprincipal compliance vector(PCV) is a unit vector that

points in the direction of the major compliance at the output port

of a mechanism. By intuition it can be seen that a cantilever

beam like the one in Fig. 18(a) is more compliant in the vertical

direction than in the horizontal one, then the PCV for the tip

of the beam points in the vertical direction; another example is

presented on Fig. 18(b), at the input port the PCV shows that the

displacement will be vertical due to the input force, and at the

output port the PCV indicates the mechanism will move in the

horizontal direction.

(a) (b)

Figure 18. Principal compliance vector. (a) PVC of a Cantilever beam,

(b) PVC of a compliant mechanism at the input and output port.

The instant center is defined as the point around which a

rigid body with plane motion seems to rotate in a particular in-stant. For the case of a C4B the instant center can be identified

by projecting the perpendicular lines from the PVC at the input

and output port and finding the intersection point of these lines

as shown in Fig. 19.

Figure 19. Instant center of a C4B.

Thedecomposition point(DP) is the point inside the design

space where two building blocks are concatenated. At this point

the output port of the first building block coincides with input

port of the second one.

Now in the dual stage synthesis two building blocks are con-

catenated by finding a proper decomposition point and at this

point finding the direction of the principal compliance vector thatensures the desired GA (Fig. 20).

(a) (b)

Figure 20. (a) Decomposition point and instant centers, (b) Final design

using a C4B and CDB.

Basically any point in the design space can act as a decom-

position point and for any decomposition point there is a PCV

that ensures the geometrical advantage. Now, what distinguishes

one decomposition point from another one is the way that the to-

tal GA is generated by the two blocks, meaning that one building

block could contribute more to the total GA than the other one;

remembering the expression for the GA:


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GAtotal = GA1 GA2 (3)

whereGA1 andGA2 are the GAs generated by block 1 and

block 2 respectively. These geometrical advantages are equal to

the ratio of the length from the output port to the instant center

over the length from the input port to the instant center. The

fitness of the decomposition point that provides a proper GA is

measured by thegeometric advantage index (nGA).

nGA=logGAtarget(GA2) (4)

The GA index normalizes the GA for block 2 in the range

[0, 1]. Ideally it is searched for decomposition points with a GA

index of 0.5, which means that block 1 and 2 generate equal GA.

Another important issue in the design process is the selection

of the position for the moving junctions. Moving junction are

the elements that connects the floating link with the ground. A

proper selection gives less error between the desired and the real

GA. In Fig. 21 it can be seen how the same decomposition point

and PCVs have different arrangements for the moving junctions.

(a) (b)

Figure 21. Two designs with the same DP, PCV and desired GA but

different moving junctions, hence different GA error.

3.2 Flexible building blocks and optimization

In this design method, a compliant mechanism is considered

as an assembly of multiple flexible building blocks [34]. The

idea behind the method consists in searching for an optimal dis-

tribution of these flexible building blocks inside a mesh that acts

as the design domain, which is defined by the number of building

blocks and their size (height and width).

The building blocks are elementary units that are formed by

joining two, three or more nodes with beam elements inside a

mesh, see Fig. 22. Each building block has its own characteris-

tic stiffness matrix which is created by assembling the stiffnessmatrices of all beam elements that form the building block. The

stiffness matrix of the compliant mechanism is calculated by as-

sembling the stiffness matrixes of all the building blocks in the

design domain.

Figure 22. Building blocks as elementary units, reproduce from [34].

After the user manually defines blocks in the design domain

as well as inputs, outputs, ground ports, contacts and loading

conditions, a multi-objective genetic algorithm generates a set of

possible topologies by finding optimal distributions of the build-

ing blocks inside the design domain (see Fig. 23).

The solutions are found by optimizing the balance between

stiffness and compliance, and displacement and force, using dif-

ferent objective formulations, like mutual potential energy, strain

energy, geometrical advantage, the mechanical advantage, etc.

Grossard et al. [35] took the approach one step further by

adding blocks with integrated piezoelectric actuators. It is in-

troduced the finite element formulation for the active building

blocks as well as their implementation into the genetic algorithm.

4 STRUCTURAL OPTIMIZATION APPROACHES

The structural optimization approaches are based on the use

of optimization and search techniques to obtain the design of a

compliant mechanism that satisfies an objective function for a set

of parameters and constraints.

First it is essential to understand optimization and the con-

cepts behind topology, shape and size optimization, this will be

treated in section 4.1. The next issues to be considered are the

objective formulation, the design parametrization, and solution

methods. The first two are discussed in sections 4.2 and 4.3, re-

spectively, while the solution methods are outside the scope of

this paper.


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Figure 23. Building blocks assembling a compliant mechanisms in the

mesh design domain, reproduce from [34].

4.1 Optimization

In general, an optimization problem is a procedure to mini-

mize or maximize a function, while a set of constraint functions

and inequalities are satisfied. Maximizing a function is equal to

minimizing the same function multiplied by -1, for this reason

optimization problems are often set as minimization problems.

The function that is being minimized is called the objective

function, the variables in the function are called the design vari-

ables and the domain of the design variables is called the search

space.

The formulation of an optimization problem (see eq. 5) con-

tains the objective functionf(x), thepconstraint functionshi, the

m inequalities gj, the n design variables x and the search space

. Normally in literature for compliant mechanisms, the designvariablesx1,x2,...,xn are presented as the design vectorx.

minimizex

f(x)

subject to

hi(x) =0 i =1,2,...,pgj(x) 0 j=1,2,...,m

where

x=

x1 x2 . . . xn

(5)

Topology, shape and size Topology is a branch of mathemat-

ics that studies how the properties of a space are preserved or

change when this space is subjected to deformations.

If an object is imagined to be composed of small elements

connected to their neighbors, then this connectivity is said to de-

fine the topology. Deformation does not affect the topology as

long as the connections between the elements remain. However

if a hole is made, it requires some connections to be broken, inwhich case the topology is modified.

In the case of mechanisms the topology refers to the connec-

tions among the constitutive elements of the mechanism, even if

they are small discrete elements; constitutive elements also in-

cludes the input ports (where input loads and movements are ap-

plied), the output ports, ground ports, etc.

Topology optimization then refers to the process of finding

the optimal topology that satisfies in the best way the objective

function.

In the shape optimization problem, the word shape refers to

the shape of every constitutive element, if topology forms the

skeleton, then the shape is the contour appearance of every bone

in this skeleton. Figure 24 shows two examples (continuum anddiscrete representation) , with the same topology but with differ-

ent shape.

(a) (b)

(c) (d)

Figure 24. Example of same topology, different shape. (a) and (b) con-

tinuum structures, (c) and (d) Discrete structures.

Shape optimization then refers to the process of finding the

optimal shape of the contour or surface that satisfies in the best

way the objective function in a fixed topology.

In the size optimization problem the idea is to find the opti-

mal set of sizing variables that satisfies in the best way the objec-

tive function. The design variables are the sizing variables or in

other words the variables that define the dimensional properties

of a model, i.e. thickness, cross sections, diameters, radii, etc.

In a typical size optimization problem the shape and topol-

ogy of the model have been already defined, for example think


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on a plate like the one in Fig. 25, here the topology is defined

and also the shape of the contours, now the problem is to find theproper dimensions of the two semi-axis for the elliptical outer

contour and the radius for circular inner contour as well as the

thickness of the cross section.

Figure 25. Size optimization.

The last example highlight the fact that size variables can

define sometimes the shape and even the topology. Topology op-

timization using gradient based algorithms typically formulates

the problem in this way, using size variables as the design vari-

ables.

When a synthesis problem for compliant mechanisms is de-

cided to be solved by using topology, shape or size optimization,

three main aspects need to be considered: the objective func-

tion formulation, the design parameterization and the solution

method.

In the following are presented the most common objectivefunction formulations and design parameterizations in the opti-

mization of compliant mechanisms. The solution methods are

not presented in this document, they are out of the scope of this

work.

4.2 The objective function formulationDeepak et al. [36] present a comparative study about the five

dominant objective formulations found in the literature which

are:

- Mutual potential energy (MPE) and strain energy (SE)

- Mechanical and geometrical advantage

- Energy efficiency- Characteristic stiffness

- Artificial I/O spring formulation

Reference [36] provided a thorough overview; therefore they

are explained here only briefly.

4.2.1 Mutual Potetial Energy (MPE) and Strain En-ergy (SE). In this formulation the compliant mechanism isseen as a structure which is stiff enough to resist the applied

loads while at the same time it is compliant enough to allow the

desired deflection. The MPE and SE formulation seeks to con-

ciliate these two requirements.

The MPE accounts for the deflection requirements, while the

SE ensures the structural requirements. From this premise Anan-thasuresh [37] presents an objective function based on a weighted

linear combination of these two parameters.

minimize: wMPE+ (1w)SE, 0 w 1 (6)

Wherewis the control variable for the relation compliance-

stiffness and which is problem dependent. Frecker et al [38],

present an objective formulation that overcomes this dependency,

namely the ratio between these two quantities.

minimize: (MPE/SE) (7)

Saxena and Ananthasuresh [39,40] present a generalization

of this objective formulation as a power ratio of MPE and SE.

Frecker et al. [41] present the use of ratio between MPE and

SE for the design of compliant mechanisms with multiple out-

puts.

4.2.2 Mechanical Advantage (MA), GeometricalAdvantage (GA) and Mechanical Efficiency (ME).

These formulations are related with the functional specificationsfor the compliant mechanism rather than their structural require-

ments as in MPE and SE formulations. In other words, these

formulations are more focus on what the mechanism should

do than how it should do it.

The idea in these objective formulations is to maximize the

ratio of a parameter between the output and the input port. Table

1 shows the maximized ratio depending on which parameter is

considered between the input and output port.

Table 1. Objective formulation depending the parameter at I\O port

I\O Parameter Maximized Ratio Expression

Force MA Fo/Fi

Displacement GA o/i

Work ME Foo/Fii

The output force is modeled as the force exerted by a virtual

spring whose stiffness is the stiffness of the workpiece.

Fo=kso (8)


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Sigmund [42] presents an objective function for the maxi-

mization of the mechanisms MA considering an initial gap be-tween the mechanism and the workpiece.

minimize: R ()

Fin(9)

R() is the reaction force at the output port, is the vectorcontaining the design variables and F in is the force at the input

port.

Canfield and Frecker [43] compare the GA and the ME

through the design of an amplifier for piezoelectric actuators.

Frecker and Bharti [44] present a similar work using GA for a

given force and stroke in the actuator.Lau et al. [45] compile the objective formulations for MA,

GA and ME and show the result of two examples using the three

formulations.

Pedersen et al. [46] propose an objective formulation based

on [42] for large-displacements mechanisms by maximizing the

work of a virtual spring (a spring that simulates the workpieces

stiffness) at the output port, which allows to emphasize on force

generation or displacement generation.

Jung and Gea [47] present an objective formulation based on

GA for non-linear materials.

4.2.3 Characteristic stiffness. Chen and Wang [48]propose an objective formulation that combines strength and

functional requirements.

minimize: e(GAGA)kinkout (10)

The terme(GAGA) accounts for the strength requirements

by specifying a desiredGA. The termkin andkout account for

the strength requirements and represent the characteristic stiff-

ness at the input and output port, respectively.

The characteristic stiffness can be thought of as the stiffness

kp of an equivalent spring that allows the same deflection p ofa cantilever beam when force fis applied at point p, see Fig. 26.

Figure 26. Characteristic stiffness of a pointp on a cantilever beam.

4.2.4 Error formulations. Objective functions based

on error calculation are commonly used for designs where path,function or motion generation are required.

The main idea is that the behavior of the mechanism during

the optimization is fitted to a prescribed behavior. This is done

by minimizing the difference between the actual behavior and the

prescribed one.

The difference between behaviors is calculated by using

some form of error function, e.g., least squares error (LSE).

Larsen et al. [49] present an objective formulation to achieve

a desired MA combined with a desired GA using error calcula-

tion.

minimize:(MAMA)2

(MA)2 +

(GAGA)2

(GA)2 (11)

Examples developed by using error formulations can be

found in [46, 5054].

4.3 The design parameterizationThe design parameterization refers to the model that is used

to represent the topology, the shape or the size in order to create

a proper set of design variables.

In the case of topology some common design parameteri-

zation are the representation of the connectivity by using finiteelements like discrete ground structures or by means of graph

theory like the load-path and spanning tree representations.

In shape optimization the design parameterization uses two

main approaches [55]: The shape optimization based on finite

element (FE) models, and the shape optimization based on ge-

ometry models.

In the shape optimization based on FE, the design variables

are the coordinates of the nodes on the contours and surfaces on

2D and 3D models respectively; this can be seen in Fig. 27(a)

where the shape is given by the position of the outer and inner

contour nodes.

In the shape optimization based on geometry models, the ge-ometry of the model is parameterized and these parameters are

the design variables. For example, in Fig. 27(b), the shape is

given by the values of the parameters rands that represent the

radius of curvature and the arc length, respectively, as consec-

utive contour segments. The design parameterization based on

splines, Bezier curves and wide curves are good examples of this

kind of parameterization.

4.3.1 Spline, Bezier and wide curves.

Spline Parkinson et al. [56] present a design strategy in which

they combine optimization and analysis tools. Their method


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(a) (b)

Figure 27. Approaches for shape optimization. (a) Shape based on FE,

(b) Shape based on parameterization.

starts by creating from the design requirements an initial design,

which comprises two parametric models; a model for optimiza-

tion and a model for analysis.

The parameterization means that the mechanism is defined

by using parameters like width, thickness, material properties or

even the position of control points, in cases when shape is given

by spline curves, Bezier, etc.

After the two models are created, an iterative solution pro-

cess is followed. First an optimization over the optimization

model is performed and the new parameters values feed the anal-

ysis model which is used in a finite element analysis. If the finite

element analysis shows that the design behavior is the desired

one, the process ends, otherwise, a new design is selected and

the new optimization model is optimized and so on until the de-

sired behavior is achieved.

In [56] the authors develop two examples, one for a constant

force mechanism and another for a path generation mechanism

by using as the objective function an error formulation between

the desired and the actual design.

Vehar-Jutte and Kota [57] [58]proposed a method based on

optimization that provides the topology, shape and size of springs

that can achieve a desired nonlinear load-displacement function.

The method uses a parametric model that represents the spring

as a planar fractal-like network of splines. This design param-

eterization makes use of cubic B-splines of five control points.The nonlinear behavior is achieved by incorporating geometric

nonlinearities instead of material nonlinearities.

The objective function is to minimize the error relative to a

prescribed curve at some target points (see Fig. 28) while dis-

placement and buckling penalties are taken into account. The

search for the optimum is performed using genetic algorithms.

The splines form a ground topology created by three

branches joined at the input port and where each branch is

formed by one primary spline and two secondary splines (see

Fig. 29(a)). All the ends points are ground ports (the input port

is also the output port and is called the applied input).

The set of design variables is form by topology variables,

Figure 28. Relative error to the prescribed load-displacement curve.

(a) (b)

Figure 29. (a) Ground topology for the plane-fractal-like network of

splines, (b) Pictorial representation of a result using splines in [57] [58].

shape variables and size variables. Topology variables are the

existence or not of a spline, the connection points between pri-

mary and secondary splines and the boundary conditions at the

end points. The shape variables are the position of all the con-

trol points. Finally, size variables are the in-plane height of each

spline bounded between a lower and upper limit.

The designs are modeled with nonlinear finite elements us-

ing the beam element B21H from the ABAQUS software.

In Fig. 29(b) is an example of what could be a result after

the optimization, here the second branch has been completely

removed as well as one of the secondary splines in branch one

and three, also the out-of-plane height and the position of the

control points and ground ports have been changed.

Bezier Xu and Ananthasuresh proposed a method for the shape

optimization where the parameterization is based on cubic Bezier

curves to represent the compliant segments in a given topol-

ogy [59]. Here the design variables in the optimization process

are the coordinates of the Beziers control points, while the points

of the Bezier curve are used as nodes in the finite element beam

model. This simplifies the re-meshing process after every itera-

tion during the optimization.

Figure 30 shows a design parameterization example, where

the mechanism is composed of one rigid element and three

compliant segments which have been parameterized with cubic

Bezier curves where only the two middle control points are used


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Figure 30. Example of design parameterization of a compliant mecha-

nism with Bezier curves.

as design variables.

Wide curves Zhou and Ting present a design approach where

in a given topology the shape and size of the compliant seg-

ments are optimized by parameterizing these segments using

wide Bezier curves [60] [61]. A wide curve is a curve that pos-

sesses a variable width or cross section, see Fig. 31(a). The

resulting curve can be thought of as the trace that is left by a

moving circle of variable radius.

A wide Bezier curve is then a Bezier curve where its width

can vary; to do this the control points are replaced by control cir-

cles where the centers of these circles define the control polygonand their radii define the variation of the width along the curve,

see Fig. 31(b).

(a) (b)

Figure 31. (a) Wide curve, (b) Wide Bezier curve.

The design variables are the radii and the center positions of

the control circles.

4.3.2 Morphological representation of topology.Tai and Chee [6264] present an approach where the topology

and shape of a compliant mechanism are optimized by using a

parameterization that is inspired by the morphology of vertebrate

creatures.

(a) (b)

(c) (d)

Figure 32. Different topologies by adding flesh to same skeleton struc-

ture. (a) Skeleton following the contour of the Bezier curves, (b) thick-

ness=1, topology with two holes, (c) thickness=2 on curves 2 and 3, topol-

ogy with one hole, (d) thickness=2 on curve 1 and 2, topology with three

holes.

A valid structure for a mechanism is the one that presents

a connection among the input, output and ground ports. These

connections are made by Bezier curves, which are used to define

the structures skeleton.

The structure of the mechanism is formed by a skeleton

which is surrounded or covered by flesh. The skeleton is formed

by the elements in the design domain that follow the contour of

the Bezier curves as shown in Fig. 32(a).

The amount of flesh that is added to the skeleton defines the

shape, as well as potentially the topology. This amount of flesh

or thickness is added considering each skeleton element and is

not constant; it can vary along the curves segments. The seg-ments are defined every two control points, so if the curve has

four control points, it is divided in three segments.

In Fig. 32(b), 32(c) and 32(d) three examples are shown of

how different topologies can be created with the same skeleton

by changing the surrounding flesh.

Here the parameterization variables are the position of the

control points of the Bezier curve and the segments thicknesses

along the curves. The optimization procedure is based on a ge-

netic algorithm.

4.3.3 Intrinsic functions. Lan and Cheng [65] intro-

duce a parameterization for the shape optimization of a compli-


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ant link using intrinsic functions.

Intrinsic function parameterization means that the param-eterization is made by using essential functions; for instance

trigonometric functions.

Figure 33. Compliant link representation using intrinsic functions, taken

from [65].

In this parameterization the shape of a single compliant link

is given by an angle function (u) and the lateral thickness func-tion w(u), where u[0 1] is the non-dimensional length of thelink along the neutral axis (see Fig. 33); the link is assumed to

have constant out-of-plane thickness.

Both intrinsic functions are represented as polynomials

(equation 12) and their coefficients are the design variables for

the optimization procedure.

(u) =m

i=0

ciui

w(u) =k

j=0

djuj

(12)

In their work, the authors present the equations necessary

for the calculation of the Cartesian coordinates for the points on

the neutral axis and lateral surfaces as well as the constraints that

are required for a correct formulation and to avoid for example

self-intersections or loops in the compliant link.

4.3.4 Load-path representation and spanning treebased topologies. Load-path representation and the span-ning tree based topologies are methods based on graph theory to

represent the topology of a mechanism, but they differ on how

topologies are obtained during the optimization procedure.

Load-path representation The load-path representation [50,

6670] is a design method that integrates topology, size and ge-

ometry synthesis by implementing a design-space parameteriza-

tion that solves some of the ambiguities in the topology, i.e., gray

areas or disconnected structures.

The method treats the mechanisms topology as a graph,

where vertices in the graph represent rigid connections with no

degrees of freedom and the edges represent the beams where the

degrees of freedom occur.The basic requirement of a valid compliant mechanism is

that there must be a physical connection between the input port,

output port and the ground port; in other words, there must be

a path in the graph connecting every pair of these ports. Those

vertices that are not input, output or ground ports are called in-

termediate connection ports.

The design variables in the load-path representation method

are grouped in four sets: variables for the path sequence, vari-

ables for the presence of path, variables for the cross-section di-

mension of the segments and variables for the location of the

intermediate connection ports. The first two sets define the topol-

ogy, the third set defines the size and the fourth defines the shape.

Figure 34. Load-path representation.

The optimization is performed by a genetic algorithm, so

during the procedure the variables for path sequence and the

variables for presence of a path are modified, creating different

graphs which means different topologies.

Spanning tree based topologies In the spanning tree based

topologies [71] [72], the optimization is performed by a genetic

algorithm using as design variables the position of the interme-

diate nodes and the segments cross-section dimensions.

The topology is not defined by the design variables. The

topology is given by the presence or absence of every segment in

a structural universal.

A structural universal can be thought of as a ground structure

containing all possible connections among vertices. Here edges

represent segments (Fig. 35).

The presence or absence of a segment is given by the seg-

ments cross-sectional design variables. A zero value means ab-

sence of the segment. When the optimization algorithm finds

a candidate topology inside the structural universal, it must be

checked for being a spanning tree.


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Figure 35. Structural universal, reproduced from [72].

A spanning tree is tree on a graph that connects all vertices

without creating cycles or loops. A spanning tree guaranties the

connection among all vertices with a minimum number of edges.

If the candidate topology is a spanning tree, it is a valid

topology. Figure 36 shows two examples of spanning trees from

the structural universal on Fig. 35

(a) (b)

Figure 36. Two different spanning trees from the same structural univer-

sal. (b) reproduced from [72].

So in other words, every time that the optimization algorithm

propose a new set of design variables, this new set creates a newcandidate topology from the structural universal which must be

evaluated to be a valid topology using the spanning tree criterion.

4.3.5 Unit truss cell. The unit truss cell is a designapproach for cellular structures presented by Wang and Rosen

[73] [74]. A unit truss is a central node surrounded by n half-

struts. A common strut connects neighboring unit truss cells.

The idea is to create a starting structure topology of unit

truss cells and modify the strut diameters by optimization. When

a diameter becomes zero, that strut is removed.

The design variables are the struts diameters. From this

point of view, the parameterization works in a similar way as a

Figure 37. Neighboring unit truss cells.

ground truss structure but according to the author presents the

advantage of better accuracy due to the simultaneous analysis

of multiple-degree-of-freedom deformations and by considering

non-linearities.

In their work, the authors present the mathematical formula-

tion for the constitutive equations of a unit truss cell, which arederived from beam theory.

4.3.6 Discrete structure - Truss, frame, beamground structures. A ground structure is said to be discretewhen the design domain has been discretized using common one-

dimensional finite elements3 like truss or frames. This implies

that not the entire design domain space is represented or mapped

by structural finite elements.

The discrete ground structures can be divided in full ground

structures and partial or modular ground structures. Hetrick and

Kota present in [75] a comparison between the full ground struc-

ture and the modular ground structure by developing the designof a compliant gripper and a compliant wrench using both param-

eterizations. In addition they present a more extensive discussion

about both ground structures and their advantages and disadvan-

tages.

When discrete structures are used, the design variables for

the optimization procedure normally are variables that describe

the geometric characteristics of the finite elements like cross sec-

tional areas, node positions or elements lengths, out-of-plane

thickness, in-plane widths, slopes, etc.

The discrete ground structure can be modeled by using truss

or frame elements. Examples using frame elements can be found

in [43,44,54,7678]; examples using truss elements can be foundin [38] and [76].

Joo et al. [79] and Joo and Kota [80] present specifically the

development of discrete ground structures using linear and non-

linear beam elements, respectively. Ramrkahyani et al. [81] de-

velop a model for a hinged beam element that can have a pin con-

nection on one or both ends in contrast to a conventional beam

element where both ends are clamped.

Full ground structure In a full ground structure, each node in

the design domain is connected with all the other nodes through

one-dimensional finite elements as shown in Fig. 38.

3element where one of whose dimensions is larger than the other two15 Copyright c 2009 by ASME

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Figure 38. Full ground structure.

Dziedzic and Frecker [82] present two design examples of

a scissor-grasper mechanism by using three-dimensional full

ground structures. Frecker et al. [38] in the presentation of the

multi-criteria objective function make use of a two-dimensional

full ground structure of truss elements for the design of a com-

pliant gripper.

Partial or modular ground structure It is said that a ground

structure is modular when the nodes are connected just to neigh-

boring nodes instead of all the nodes in the design domain. An

example of this can be seen in Fig. 39.

Figure 39. Partial ground structure.

Frecker and Bharti [44] and Mankame and ananthasuresh

[83] make use of the modular ground structure in the develop-

ment of their examples.

4.3.7 Continuum structure. A ground structure issaid to be a continuum structure when the entire design do-

main has been discretized and every discrete sub-domain is

Figure 40. Continuum structure.

modeled by some mathematical representation of the macro-microstructure, see Fig. 40. Consequently, the entire design do-

main is mapped to some structural model representation.

The basic idea behind the use of continuum structures is to

start with a design domain which is full of elements while grad-

ually, during the optimization, those elements that are not effec-

tively used are removed, so that at the end, only the essential

elements remain to achieve the design requirements,

The ESO4 method [84] is representative of this approach.

Other approaches can remove or add elements during the opti-

mization like the BESO5 method.

Ansola et al. [85] show an additive version of the ESO

method applied to compliant mechanisms; Here the groundstructure is a fully compliant structure which is an empty design

domain and gradually, during the optimization, the elements are

added where required, instead of being removed.

When continuum structures are used in the design of compli-

ant mechanisms, two parameterization models are highlighted:

the SIMP method and artificial density, and the homogenization

method or hole-in-cell.

SIMP method and artificial density Normally when a topol-

ogy optimization is performed, it is desired that the design vari-

ables express the existence or inexistence of an element in the

design domain. This is done by assuming a value of either 1, the

element exists, or 0, the element is removed from the domain.When optimization algorithm based on gradients whose

variables can range in the [0, 1] interval, a problem arise, the al-

gorithm can not deal with Boolean variables. The SIMP6 method

helps to overcome this problem by penalizing the design vari-

ables, so intermediate values can be assumed as 0 or 1.

When SIMP is used, the design variables are the density of

the elements. Here density can be thought as the material den-

sity or as some kind of artificial density or cost variable. Sig-

4ESO which stands for Evolutionary structural optimization is also known as

SERA which stands for Sequential Element Rejections and Admissions5BESO stands for Bidirection ESO method6SIMP is an acronym for Solid Isotropic Material Penalization


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mund [42] presents an optimization procedure for maximizing

the mechanical advantage in compliant mechanisms using artifi-cial density by modifying the Youngs modulus for every element

in a continuum structure.

Ee = (xe)E0, e=1, . . . ,N (13)

HereE0 is the Youngs modulus of solid material,Ee is the

Youngs modulus of a single element, is a penalization factorandxe is the design variable for a single element.

When the optimization procedure ends, the design variables

modify the Youngs modulus in all the elements inside the design

domain. Ifxe is close to zero7 it means a low Youngs modulus

Ee thus low stiffness and therefore the element is removed. Con-

trarily, values close to one means Youngs modulus equal to the

modulus of solid materialE0 thus high stiffness and therefore the

element remains.

For a more extensive description of the SIMP method see

[84] [86]. Examples applied to the design of compliant mecha-

nisms can be found in [49] and [46].

Honeycomb representation Checkerboard patterns are a

common problem found in topology optimization due to the use

of squared elements for the structural representation of the design

domain. These patterns consist of a sequence of void elements

and solid elements where the solid elements are joined by theirnodes instead of their edges (see Fig. 41), therefore creating ar-

eas of zero stiffness.

Figure 41. checkerboard pattern.

To overcome this problem R.Saxena and A.Saxena [87] and

Mankame and A.Saxena [88] present a parameterization using

hexagonal elements which discretize the design domain into a

honeycomb pattern, see Fig. 42.

7the minimum value for the design variables normally is not zero to avoid

singularities on the stiffness matrix

Figure 42. Honeycomb parameterization.

This type of discretization avoids the zero stiffness areas by

ensuring edge-to-edge contact among the elements. The stiffnessmatrix for a hexagonal element is derived by considering it as the

junction of two four-node finite elements as shown in Fig. 43.

Figure 43. Hexagonal cell represented as two four-nodes finite element.

The design variables in this approach can be the artificial

densities used in the SIMP method. For each hexagonal element,

there is a design variable meaning that the two four-node ele-

ments share the design variable to define their material proper-

ties.

Homogenization method (Hole-in-cell) The homogenization

method allows to obtain a relation between stiffness and den-

sity by representing the microstructure as a unit cell where the

amount of material can change by modifying the geometry of a

hole that is inside the unit cell.

The unit cell is defined by the height (a) and width (b) of theinner hole and the rotation of the cell (), see Fig. 44. Dependingon the values for these variables the unit cell can be completely

void (a=b=1) which means no material; completely full (a=b=0)

which means solid structure or something in between.

The variables from all the unit cells in the design domain

are the design variables and when their optimal values are found,

they define the material distribution along the entire design do-

main.

More about the fundamentals of the homogenization method

can be found in [89] [90] and [86]. Frecker et al. [38] and Nishi-

waki et al. [91] apply the homogenization method to the design

of compliant mechanisms.


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Figure 44. Representation of the microstructure by homogenization.

4.3.8 Control meshes and subdivision. In the pa-rameterization based on control meshes and subdivision [92]

[93], the design variables are the existence state of the control

meshes. A single control mesh is a squared subregion in the de-

sign domain defined by four control points. This mesh must not

be confused with a mesh for an analysis with finite elements.

These meshes are just a geometrical division of the design do-

main.

During the optimization process, which is performed by a

genetic algorithm the state of the control mesh assumes values of

0 or 1 (existence or inexistence of the control mesh). After the

algorithm proposes a new arrangement for the control meshes, a

subdivision process is performed.

The subdivision provides a more detailed and smooth design

which is used for the meshing in the FEA and then for the eval-

uation of the objective function. An example of subdivision can

be seen in Fig. 45.

(a) (b)

Figure 45. Subdivision process. (a) Initial control meshes and control

points, (b) Refined geometry after subdivision.

Subdivision is a process that creates more control points

from the initial control points that define the control meshes, so

in this way the optimization variables are fewer, because they

do not describe the entire detail of the mechanisms shape and

topology but they can be used to recreate them.

An important feature of subdivision is the reduction on the

formation of lumped compliance.Subdivision relies in the fact that there is a unique mapping

from the initial model with few control points to the more dense

final model.

Figure 46, shows three steps during the optimization pro-

cedure: the initial control meshes, a proposed control meshes

arrangement after some iterations and the arrangement just after

subdivision and before meshing for FEA. Notice how using a few

design variables is possible to obtain a more complex geometry

by subdivision.

(a) (b)

(c)

Figure 46. Three steps during optimization, reproduced from [92].

(a) Initial control meshes, (b) proposed control meshes arrangement,

(c)proposed arrangement after subdivision.

5 CONCLUSIONS

This document presented for the first time a systematic com-

pendium for most of the actual research done in the development

of design methods for compliant mechanisms. Due to the rapid

development of the field it can not be claimed to be a complete

overview but it allows the inclusion of new techniques.

The document was presented as an introductory document

for those interested on the subject without a strong background

on the matter and as a reference document for experienced re-

searchers in the field.

Three main research directions were found in the design of

compliant mechanisms: Methods based on kinematics concepts,

methods based on building blocks and methods based on struc-

tural optimization and search techniques.

Two main methods based on kinematics were identified and

discussed: the Rigid-Body Replacement method and the Free-

dom and Constraints (FACT) method. It was shown how the

Rigid-Body Replacement method is applied using either design


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of flexure joint or Pseudo-Rigid-Body model to obtain compliant

mechanisms with lumped or distributed compliance. The FACTmethod was explained through the development of an example.

Two building block methods were identified: the building

blocks based on instant centers and the building blocks based on

optimization. Building blocks using compliant ellipsoids were

not presented due to its extension but it is a further development

on the instant centers based approach. Therefore it is left to the

interested reader to follow the references to this work.

On the structural optimization section, more attention was

paid to the design parameterizations. They constitute a big part

on the research contributions in this area but almost no litera-

ture is dedicated on this matter. Commonly, discussion is dedi-

cated in most articles to the objective function formulation and

the solution methods; for this reason few words were dedicated

to objective functions and no words on solution methods.

It was presented how almost any approach that is useful to

describe the geometrical shape or topology of a domain and its

association with a mechanical behavior, can be used as a design

parameterization for an optimization procedure.

It can be said that most of the design methods relies on three

basic ideas:

- Design of compliance mechanisms can be done using the

well known kinematics of Rigid-Body mechanisms. Find a

way to get rid of conventional joints and stay on the elastic

domain.- Design of compliance mechanisms can be done by au-

tomating the search of a solution that fulfils a desired func-

tion and constraints. Find the proper way to describe the

topology, shape and size, and what is need it to be fulfilled

to get the proper design.

- Design of compliance mechanisms can be done by the

premise divide and conquer, first design small pieces and

then join them; either by some automate process or using the

well known kinematics from Rigid-Body mechanisms.

- A combination of all the previous.

ACKNOWLEDGMENT

This research is part of the VIDI Innovational Research

Incentives Scheme grant for the project Statically Balanced

Compliant Mechanisms, NWO-STW 7583.

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