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Topological Design and Control of Path-Following Compliant Mechanisms

Colby C. Swan * and Salam F. Rahmatalla.† Civil &Environmental Engineering, Center for Computer Aided Design, The University of Iowa, Iowa City, Iowa

52242, USA

A methodology for continuum topology design of continuous, monolithic, hinge-free compliant mechanisms is presented and mechanisms are designed to use finite elastic deformation such that an output port region moves in a desired direction when a specified force is applied at an input port region. To extend the functionality of the compliant mechanisms, the ability to make the output port region follow specified curvilinear trajectories is then investigated. Specifically, control algorithms are presented in which sequences of actuation forces to the mechanisms’ input port are found so that the output port follows the desired trajectory in an optimum sense. For each step of the mechanisms’ nonlinear load-deformation response analysis, a control problem is solved for the desired actuation force(s). The methodology is successfully tested in this work on a number of problems, so that the mechanisms do indeed follow their specified trajectories and nearby trajectories with high degree of accuracy even when confronted with varying degrees of resistance.

I. Introduction HILE rigid-body mechanisms (Fig1.a) can be optimal in innumerable macroscopic mechanical systems, they are generally less suited for micro-scale applications due to the fundamental difficulty of fabricating reliable

hinged-joints on such small scales. One potential answer to this problem is to employ compliant mechanisms1 in which force and motion are transmitted primarily via elastic deformation of the system. The elastic deformation of compliant mechanisms can be either concentrated in flexible hinge regions (Fig. 1b)2, or it can be more or less uniformly distributed throughout the mechanism (Fig. 1c)3. In the former case, an attempt is usually made to re-design the hinged joints of rigid-body mechanisms as flexible hinges in such a way that the performance of the resulting compliant mechanism is roughly comparable to that of the rigid-body mechanism. This is a nontrivial endeavor, however, as designing flexible hinges in a way that permits only rotation at the joint, and so that the material in the hinge is not overstressed or overstrained is very challenging. For these reasons, compliant mechanism designs that feature distributed elastic deformation may be more designable and also more durable.

* Associate Professor, Wheeler Faculty Fellow of Engineering, colby-swan@uiowa.edu, AIAA Member. † Post-doctoral Associate, srahmata@engineering.uiowa.edu.

W

Figure 1. Schematic drawing of generic mechanisms, a) pin-jointed rigid-link mechanism; b) pseudo-rigid link mechanism (compliant hinges substitute for pin-jointed hinges); and c) hinge-free distributed deformation compliant mechanism.

rigid link

flexible hinge

flexible link rigid linkpin jointed hinge

a) b) c)

10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference30 August - 1 September 2004, Albany, New York

AIAA 2004-4521

Copyright © 2004 by Colby C. Swan. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

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Continuum structural topology design methods have over the past decade been investigated quite actively in the design of compliant mechanisms, beginning with the seminal work of Ref. 4 and then others5. Since continuum topology design methods solve for the layout of structural material in continuum structures and mechanical systems, it is somewhat ironic that when applied to design of compliant mechanisms, they have a tendency to produce systems that function as pseudo-rigid-body mechanisms. Such pseudo-rigid-body mechanisms generally feature de facto hinge regions, which are artifacts of the numerical model that behave effectively as hinges. Achieving compliant mechanism designs free of de facto hinges within a continuum topology optimization framework has been addressed previously where it has been observed that such designs generally function as distributed deformation compliant mechanisms6,7,8,9.

Since the material comprising compliant mechanisms will generally undergo finite strains, displacements, and rotations when the mechanism functions under normal design actuation forces, the analysis and design framework must be general enough to treat finite deformation effects. One important class of applications among compliant mechanisms are so-called path-following mechanisms in which the output ports of the mechanisms follow a specified trajectory under the effect of a sequence of actuation (input) forces. Research on utilizing continuum topology optimization methods to achieve such path-following compliant mechanisms is still in a state of relative infancy and only a small number of papers have been published toward this end (e.g. 10,11.) The approach taken in the cited works is to perform variable topology material layout design of the mechanism such that under a sequence of actuation forces applied to the input port, and under a specified workpiece resistance supplied at the output port, the output port follows a target curvilinear trajectory. Assuming that one were completely successful with this approach it suffers a lack of robustness in that the performance of the mechanism can be highly sensitive to the output port resistance. Restated, even small changes to the output port resistance can result in the mechanism output port following a path significantly different than the target trajectory.

In response to the potential problem perceived with the preceding approach, an alternative path is proposed and investigated here. Specifically, hinge-free compliant mechanism designs that have both sufficient flexibility and strong sensitivity of output port response to input port actuation forces are first obtained utilizing a particular continuum topology optimization formulation9 that is presented herein. After such mechanism designs are obtained, a control methodology is proposed to solve for the sequence of actuation forces that, when applied to the mechanism’s input port, result in the mechanism’s output port following a specified trajectory in an error-minimizing sense. The attractive aspects of the methodology being proposed here are: (1) realization and usage of compliant mechanism designs that are free of de facto hinges; and (2) the capability to make the mechanism output port follow any realizable curvilinear path◊ with good precision even when working against varying workpiece resistances.

To present the proposed design and control methodology, the manuscript is organized in the following way. In Section II, some key generic aspects of the computational tools that will be used are presented. These include: details of the nonlinear finite deformation hyperelastic analysis of the compliant mechanisms for a fixed mechanism design and a fixed sequence of actuation forces; continuum topology design sensitivity analysis and optimization of the mechanism for fixed actuation forces but varying material layout design; and nonlinear, incremental control of the mechanism for fixed design but variable actuation forces. With these important yet generic details covered, a specific formulation for design of hinge-free compliant mechanisms is then presented, explained, and demonstrated in Section III. Then, in Section IV, some of the mechanisms designs of the preceding section are taken and controlled with the proposed generic control algorithms such that the output ports follow a variety of different trajectories under varying workpiece resistances. Discussion of the proposed analysis, design, and control methods are then presented in Section V of the manuscript.

II. Element of Formulation

A. Structural Analysis Model Since the compliant mechanisms being modeled and designed undergo finite displacements, rotations, and

strains the analysis framework should accommodate it. Accordingly, the strong form of the nonlinear elliptic boundary value problem to be solved for the structural displacement field is as follows:

Find 3]),0[(: ℜ×Ω aTSu , such that:

◊ Clearly, not all desired paths will be physically and mathematically realizable for a given mechanism design.

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0γρτ j0jij, =+ on SΩ ],,0[ Tt∈∀ (1a)

subject to the boundary conditions:

(t)g(t)u jj = on gjΓ for ,1,2,3j = T][0,t∈∀ (1b)

(t)hτn jiji = on hjΓ for ,1,2,3j = T][0,t∈∀ (1c)

Above, τ denotes the Kirchhoff stress tensor field which is related to the Cauchy stress tensor σ via the relation στ J= , where )det(F=J and F is the deformation gradient operator. As is customary, it is assumed that the

Lagrangian surface hjgj ΓΓΓ ∪= bounding the Lagrangian structural domain SΩ admits the decomposition

∅=Γ∩Γjj hg for 3,2,1=j For a given mesh discretization of sΩ whose complete set of nodes is denoted η,

the subsequent design formulation is facilitated by introducing a subset of nodes ηh at which non-vanishing external forces are applied, and a subset of nodes ηg at which non-vanishing prescribed displacements are applied. The nodes in the model at which the unknown displacements remain to be determined form the set denoted gηη − .

The particular isotropic hyperelastic strain energy function E used here is that of Ref. 12 wherein the volumetric )(U and deviatoric )(W strain energy functions are assumed to be decoupled and of the forms:

)()()( θWJUE +=F ; ]3)([21

−= θtrW µ ; ⎥⎦⎤

⎢⎣⎡ −−= )ln()1(21

21)( 2 JJKJU (2)

In the preceding expression, J is again the determinant of F ; K is a constant bulk modulus; µ is a constant

shear modulus; TFF=θ is the left Cauchy-Green deformation tensor; and θθ (2/3)−= J is its scaled counterpart having a determinant of unity. For this model, therefore, the Kirchhoff stress τ in a material is thus related to deformation quantities as follows:

( ) θI1Jτ 2 : 12 devK µ+−= (3)

where 1 is the rank-2 identity tensor, and ( )11II ⊗−= 31

dev is the rank-4 deviatoric tensor with I the rank-4 identity operator.

Using standard techniques, the virtual work equivalent of the original problem statement in Eqs. (1) can be obtained in the following form:

∫∫ ∫ +=hS S Γ

hjjΩ Ω

Sjj0Sijij dΓδuhdΩδuγρdΩδετ (4)

In the expression above, the quantity on the left represents the internal virtual work intWδ , and that on the right, the external virtual work extWδ .

Usage of a Galerkin formulation, in which the real and variational kinematic fields are expanded in terms of the same nodal basis functions, and discretization of the time domain into a finite number of discrete time points, leads to the following force balance equations at each unrestrained node A in the mesh as here at the thn )1( + time step:

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( )

whileEnd

while End

1 )( : vectorforce residual e Updat

:ntdisplaceme e Updat

:nt displaceme lincrementafor system linearized Solve

) and While(

)( vector force residual initial Compute

:force external fixed obtain

E :predictornt displaceme ;0 :Initialize

:(Newton) iterations solutionnt Displaceme ) and 1While(

; :timeIncrement obtained. were0 satisfying ntsdisplaceme nodal At

11

11

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11

1111

11

max

11

11

11

max1max

1

knn

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kn

kn

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tolk

1n

kn

kn

nextext

n

gnE

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n

nn

nnn

kk

kkr

t

k

ttnnttt

t

++

++

++

+++

++

++++

++

+

++

++

++

+

+

=

+=

∆+=

−=∆⋅

∆

≤≥

=

=

∈∀∆+=

=

≤≤+∆+=

=

dd

dr

udd

ruK

u

r

drr

ff

gdd

rd

η

Figure 2 Algorithm for Newton iterations during a representative time/load step (n+1)th of nonlinear structural analysis without control.

gA

1nextA

1nintA

1n -A )()( ηη∈∀=−= +++ 0ffr (5)

where

∫Ω

+++ =S

S1nT

1nAA

1nint dΩ:)()( τBf (6)

∫ ∫Ω Γ

+++ +=S h

h1nA

S1nA

0A

1next dΓNdΩNρ)( hγf (7)

In Eq. (6), A1n+B represents the spatial infinitesimal nodal strain displacement matrix ))(N( AA

1n 1xB s

xn+∇=+ , and AN

denotes the nodal basis function for the thA node. Under finite deformations, Eq. (5) represents a set of nonlinear algebraic equations that must be solved in an iterative fashion for the incremental nodal displacements

An

A1n

A uuu −=∆ ++1)( n for each time step of the analysis problem and

gA ηη −∈∀ . When external forces applied to a structure are independent of its response, the derivative of the thi residual force vector component at the thA node with respect to the thj displacement vector component of

the thB node is simply:

∫ ∫Ω Ω

Ω+Ω=S S

SilBkjk

AjS

Bkljk

Aji

ABil dNNdBcB δτ ,,K (8)

where jkc is the spatial elasticity tensor in condensed form. Assembly of this nodal stiffness operator for all unrestrained nodes A and B gives the structural tangent stiffness matrix. To solve for nonlinear deflection responses of hyperelastic structures, Newton iterations (Fig. 2) are usually performed at each load step of the analysis problem. These involve solving for the set of nodal displacements 1+nu that satisfy the force-balance equilibrium condition of Eq. (5).

B. Continuum Topology Optimization In continuum topology optimization, one frequently solves for the spatial distribution of a fixed volume of

structural material in SΩ such that the desired performance characteristics of the structure are optimized. In the

current framework, this is achieved by using the same 0C bi-linear nodal basis functions of the structural analysis

problem to interpolate nodal volumetric densities of structural material throughout the structural domain SΩ .

Specifically, in the infinitesimal neighborhood about a point SΩ∈X , the volumetric density of solid structural material )(Xφ is given by

∑∈

=η

φφA

AAN )()( XX (9)

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where the Aφ represent nodal volumetric densities of solid material. Since at each point SΩ∈X there is generally a mixture of a solid structural material and a void-like material with respective volume fractions )(Xφ and

)(1 Xφ− a methodology is generally needed to determine the effective stiffness properties of the solid-void mixture or composite. A number of different possibilities exist, and a fairly detailed review was presented in 15. Here, a simple iso-deformation powerlaw mixing rule is employed in which it is assumed that both the solid and void-like material at a point X undergo identical deformations. Accordingly, at each point X , both the solid and void-like materials are assumed to share the same deformation gradient:

XxXFXFXF

∂∂

≡== )()()( voidsolid (10)

Although the solid and void-like materials share the same state of deformation, the stress states in each are generally consistent with their own constitutive behaviors. Assuming that both the solid and void-like materials can be represented by the hyperelastic constitutive model of the preceding section it follows that a point X the stresses in the respective materials would be:

( )

( ) .: 12

;: 12

void

solid

θI1Jτ

θI1Jτ

2

2

devvoidvoid

devsolidsolid

K

K

µ

µ

+−=

+−= (11)

In accordance with the powerlaw mixing rule the average stress in the solid-void mixture at point X is simply the weighted sum as follows:

( ) ( ) ( )[ ] voidP

solidP τXτXXτ φφ −+= 1 (12)

To achieve the effect of a void-like material in this work, the bulk and shear moduli of the void material are taken to be 610− times those in the solid structural material. In the mixing rule of Eq. (12), the powerlaw exponent P is generally chosen larger than unity, but less than or equal to four. A value of unity yields the classical Voigt rule of mixtures, whereas a value of 4=P leads to a penalized mixture in which stiffness approaching that of the solid material is achieved only for values of φ very close to unity.

In continuum topology optimization, the layout of structural material within SΩ is iteratively varied, and for each variation, the structure is re-analyzed de novo. The design of such a structure can be represented by a finite dimensional vector Nℜ∈b wherein each component of the vector represents a nodal volume fraction of solid material, and N denotes the number of nodes in the analysis model at which the design can be varied. Since the nodal volume fractions are continuous on the interval [ ]1,0∈φ , and since the design of a structure is represented by N such variables, where N can easily be on the order of 310 or greater, gradient-based optimization methods are

typically most effective for solving continuum structural topology design problems. In general terms, a design problem is usually solved by specifying: a performance-based objective function ( )bℑ

for the structure; a set of 0≥m equality constraint functions ( ) m,0,1,2,k 0 K∈∀=bkh ; and a set of 0≥n

inequality constraint functions ( ) n,0,1,2,l 0 K∈∀=blg ; The design space Nℜ is then searched for the design *b that satisfies, at a minimum, the first-order optimality conditions. In gradient-based optimization, it is thus

necessary to compute the first-order design derivatives of the objective and constraint functionals. The first order derivative of a quantity such asℑ is usually computed as follows:

bu

ubb dd

dd

⋅∂∂ℑ

+∂∂ℑ

=ℑ (13)

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Figure 3 Algorithm for nonlinear structural analysis and sensitivity analysis embedded within design optimization problem.

if-End* tourn Ret

,,, :Compute

feasible)not isit or conditions optimalityorder first satisfy not does If(for-End

)(

)(

(..

()()(

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()()()(

while- End

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;0 Initialize

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⎟⎠⎞

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ℑ∆

=

∆ℑ∆+ℑ=ℑ∂ℑ∆

+ℑ

=ℑ

∂∂

∆∂ℑ∆∂

+∂ℑ∆∂

=

∆∆∂ℑ∆∂

+∂ℑ∆∂

=ℑ∆

−=∆

+=

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−=∆

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≤≥

=

+==

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ℑ

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dd

knn

n

n

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nnn

nnn

kn

kn

kn

kn

kn

kn

kn

k1n

kn

tolkn

kn

kn

nnkn

nn

o

o To facilitate the computation, the observation is usually

made that:

buK

br

bu

ur

br0

br

dd

dd

dd

⋅+∂∂

=⋅∂∂

+∂∂

== (14)

With re-arrangement, it follows that

brK

bu

∂∂⋅−= −1

dd . (15)

Substitution of this result into Eq. (13) yields:

bru

bbrK

ubb ∂∂⋅+

∂∂ℑ

=∂∂⋅⋅

∂∂ℑ

−∂∂ℑ

=ℑ − a

dd 1 (16)

where au is the so-called adjoint displacement vector13

that satisfies the linear adjoint problem:

uuK

∂∂ℑ

−=⋅ a (17)

In all of the equations above, K denotes the tangent stiffness matrix for the structural model at its current state as defined in Eq. (8). Fairly extensive details on computation of incremental design derivatives of performance functionals associated with hyperelastic structures at finite deformations were provided in14 and for materially nonlinear structures in15. For the sake of

brevity, these details are not reproduced here. Nevertheless, an algorithmic view of finite deformation structural analysis and design sensitivity analysis embedded within a continuum topology design optimization framework is shown in Fig. 3.

C. Control within A Nonlinear Analysis Framework For a given layout of material within the structural model, it is useful to be able to solve for a sequence of

actuation forces inN

inininin fffff , , , , , 3210 K that, when applied to the mechanism’s input port, will result in the mechanism’s output port moving along a desired trajectory specified by a corresponding sequence of output port

displacements: ( ) ( ) ( ) ( ) op**2

*1

*0 , , , , N

opopopuuuu K .

Within the standard nonlinear analysis load step (Fig. 2), the external load extn 1+f on the structure at time 1+nt is

prescribed, and one iterates simply for the unknown nodal displacements that satisfy the equilibrium state of Eq. (5). However, within a typical single incremental load- or time-step ( )1+n of the nonlinear structural analysis problem,

one solves a sequence of trial incremental analysis problems with trial actuation forces ( ) K0,1,2,j ,1 =+jin

nf until the

equilibrium output port displacement ( ) 11

11

++

++ ⊂ j

nopj

n uu associated with the trial actuation force is as close as possible

to the target value for that increment ( )opn*

1+u (Fig.4). In a formal sense, the following optimization problem is

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solved for the actuation force numnpndofinn

×++ ℜ∈⊂ ext

1n1 ff associated with each load step of the structural analysis problem:

For predetermined ninn df and find 11 and ++ n

inn df

such that

)(minin

1n

g +f

and 0dfr =++ ),( 1nin

1n (18)

where:

( )( )

21

2

1*

1in1n 1))((

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−= ∑

+

++

iop

ni

opni

u

ug fd (19)

is the objective function; numnpndof ×++ ℜ∈⊂ 1n

op1n du is

the resulting output port displacement due to the

actuation force in1n+f ; and ( )op*

1nu + is the target

output port displacement for the ( )thn 1+ load step. The control problem is solved as an

unconstrained optimization problem within each load step of the nonlinear analysis problem, using an iterative conjugate gradient algorithm. For a trial value of the actuation force ( ) j

nin

1+f an equilibrium problem 0r =+

jn 1 is first solved. Then,

at the trial equilibrium state jn 1+d the gradient of the

objective function with respect to the actuation force is computed as:

( ) ( ) in1n

11+

++ ⋅−=∇ffw

ddg

extjn

ajn

(20)

where Nℜ∈aw is a vector of adjoint displacements satisfying the following linear adjoint problem:

( )j

n

jn

a g

111n

j

+++ ⎟

⎠⎞

⎜⎝⎛∂∂

−=⋅u

wK (21)

in which jn 1+K is tangent stiffness operator at the

current trial equilibrium state of the model

associated with ( ) jinn 1+f . Once the gradient of the

objective function is obtained, a conjugate gradient algorithm is used to obtain the search direction for the new actuation force. A line search is then employed using a Golden search followed by

Figure 4. Schematic of iterative control problem for actuation forces that make mechanism output port follow a

input Γ output Γ

A * B

*C

Desired path

op 1 1 ) (

x

op 2 1 ) (

x op3

2 )(x

op1 2 )(

x

op22 )(

x

op 1 ) * (

x

op*2 )(

xTrial path

op 3 1 ) (

x

op42 )(

x

Desired displacement

Trial displacement

o0 uxx +=

inf

Figure 5 Algorithm for control problem interleaved with nonlinear analysis problem for the (n+1)th time-step of the problem.

( )( )

( )

while-End; 1; ;

if-End * toJump

))1

fitting). polynomial andsearch (Golden search line a from size step theFind

method.gradient conjugate using ),(direction search Find

** tojump gradgIf

,for solve

then)1 and ( If

1

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) : vectorforce residual initial Compute

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:(Newton) iterationssolution nt Displaceme*

force) control (trial )( ;0 :Initialize :iterations force Control

at nt displacemeport output target :)( Initialize

) and 1( While; : timeInitialize ;0counter step time:Initialize

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1

max1max

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+++

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+=

+=

∇∇

≤∇

∇

≤+≥

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−=

=

∆+=

−=∆⋅

∆

≤≥

=

∈∀∆+=

=

==

≤≤+∆+==

∑

nnnjnn

jn

jn

jinn

jinn

jn

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maxtolj

n

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opn

1jij

n

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opjn

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tolkn

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inn

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in

nop

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u

ug

kkr

k

j

t

ttnntttn

uu **

.v(f(f

v

(u(u

dr

udd

ruK

u

r

r(dr

gdd

ff

u

α

α

η

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polynomial fitting to find the scaling factor. An overview of the complete control algorithm within a fixed load- or time-step of the analysis problem is provided in Figure 5.

III. Continuum Topology Design of Hinge-Free Mechanisms

A. Design Problem Formulation The first objective in the proposed framework is to

achieve workable hinge-free compliant mechanism designs that can subsequently be controlled with actuation forces so that the output port follows specified trajectories. In the topology design formulation, usage is made of two distinctly different sets of springs having different purposes. The first set of springs are called artificial springs and they are chosen to be very stiff9. The second set of springs is called workpiece springs and these represent the much smaller resistance supplied by the workpiece when manipulated by the mechanism. To design a mechanism within the proposed framework, a mathematical mechanism model on a spatial region

3ℜ∈Ωs is first created and support conditions are prescribed (Fig. 6). An input port region

inΓ to which an input force inf will be applied is identified, as is an output port region outΓ at which op

out uu ≡ is monitored. Compliant mechanisms can be designed and fabricated with a wide variety of materials and here the material

considered is aluminum GPa; 73( =E .35)=ν . Typically layout optimization of material in compliant mechanisms utilizing continuum topology optimization is performed with a prescribed amount C of material specified as a fraction Mℑ of the mechanism’s envelope volume. For a given design, the ratio of structural material volume to the mechanism’s envelope volume V is computed as follows:

( )∫Ω

Ω=ℑb

VM d 1 Xφ (22)

To achieve mechanism designs free of de facto hinges, the material layout problem is solved to minimize the sign inverse of the elastic mutual potential energy )(MPE under a given actuation force inf while working against the stiff artificial springs attached to both the I/P and O/P of the mechanism.. Here, the MPE is defined as follows:

)1(out

vout ufMPE ⋅= (23)

where (1)outu is the displacement at the output port due to a load inf applied at the input port, and v

outf is a virtual force at the output port specifying the direction of the desired output port displacement. A solution of the following optimization problem P1 is obtained subject to a material usage constraint and existence of an equilibrium solution of the structural equilibrium problem:

P1: For fixed material usage constraint value C and artificial spring stiffnesses kb:

( ) ( )[ ]b

buru ),(min 1 ⋅+−ℑ+− aM CMPE λ (24)

where numnp x ndofℜ∈r is the residual force vector for the elastic structural model which vanishes when the structure is in equilibrium under the applied actuation forces and the spring reaction forces. Also in the above, 1λ is the

inΓ

outΓ

inf

voutf

ink

outk

Fig. 6. Schematic of compliant mechanism design problem using artificial springs attached to both input port and output port.

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9

nonnegative Lagrange multiplier associated with the material usage constraint, and numnp x ndofℜ∈au is a vector of nodal adjoint displacements that serve as Lagrange multipliers to the structural equilibrium equality constraint16.

It is emphasized that design solutions of P1, for a specified amount C of structural material, will generally be very stiff. To subsequently model how such mechanism designs function at finite deformations under real workpiece resistance, the stiff artificial springs are removed and the second set of workpiece springs are attached only to the O/P of the mechanism. A realistic goal in design of compliant mechanisms is to have the mechanism be free of de facto hinges, and to have a complimentary compliance CE at finite deformation that exceeds a certain threshold value *

CE when working against the workpiece resistance in response to a specified actuation force inf . Here the complimentary compliance CE of the mechanism is defined as:

( ) MPEEvout

inout

voutv

out

inC *)1(

f

fuf

f

f=⋅= (25)

Depending on the material usage constraint value C for which design problem P1 was solved, the resulting design solution might very well be too stiff with *

CC EE < . Nevertheless, design problem P1 can be re-solved with

progressively smaller values of the material usage constraint value C until *CC EE = . The objective is thus to find

the largest value of the material usage constraint value C for which *CC EE = . A concise mathematical statement

of the extended design problem P2 that corresponds to this procedure is as follows:

P2: For specified artificial springs ( ink , outk ) and workpiece springs workpiecek find:

( )1,0inf ∈C and ( )MPE-minNℜ∈b

(26a)

such that:

0M ≤−ℑ C ; material usage constraint (26b)

0bur =),,( )1( C ; Case 1 equilibrium (26c)

0bur =),,( )2( C ; Case 2 equilibrium (26d)

( ) 0,,)2(* ≤− CEE CC bu ; Case 2 compliance (26e)

In P2, the Case 1 analysis has stiff springs attached to both the I/P and O/P of the structural model, and the structure is analyzed using linear elastic analysis. The resulting displacement field in the structural model from which MPE is computed is denoted )1(u . In Case 2 analysis, the stiff springs are removed from the model’s I/P and O/P and moderate workpiece springs are attached to the O/P. The finite deformation hyperelastic response of the structure

)2(u to the actuation force inf is computed, from which the complimentary compliance CE is also computed. In Eq. (26e) of P2, *

CE is the target value for complimentary compliance when working against the workpiece springs

under actuation force inf . The approach taken herein to solve P2 is to first solve design problem P1 for numerous values of the material usage constraint C . Each of the designs for different C values is then analyzed at finite deformation under the actuation force inf and complimentary compliances ( )CEC ,,)2( bu are computed. The design

b associated with the material usage constraint value C that yields the target complimentary compliance *CE is

then selected.

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B. Design of Hinge-Free Inverter Mechanisms The function of this device is to have the output port displace in a direction opposite to that of an input force

applied at the input port. Fig. 7 shows the design domain sΩ of the inverter problem with partial fixed support boundaries at the left hand side. The domain, which is discretized with a minimum of 100 x 100 bilinear quadrilateral finite elements, is loaded with Nf in 100= applied to the input port. The deflection at the output port

in the direction of voutf is to be maximized. The large artificial spring stiffness values used on both the input and

output ports of the mechanism are -110 mN106.1 ⋅⋅=bk . To demonstrate the effects of material usage constraint on resulting compliant mechanism characteristics, the

design problem P1 was solved with: 30.0=C (Fig. 7b,c); 10.0=C (Fig. 7d,e); and 03.0=C (Fig. 7f,g). Each design functions without any de facto hinges, and the more sparse designs feature nicely distributed elastic deformation. If for an actuation force Nfin 100= a

threshold complimentary compliance JEC3* 102 −⋅= is

desired when then 05.0inf =C as is shown in Fig. 8.

IV. Inverter as a Path-Following Compliant Mechanism In the preceding section the inverter mechanisms (Fig. 7) were designed such that their output ports follow a

horizontal path under the effect of a horizontal actuation force. The next goal of this work is to test the ability of the proposed control algorithm to solve for actuation forces so that the mechanisms can follow paths close to the originally intended path or even paths not so close to the originally intended path. Unless noted otherwise, in all of the control problems solved below, the sparse inverter mechanism design shown in Fig. 7f was utilized. Before being utilized in the control problems, the design was finely re-meshed with a conforming mesh of quadratic triangular continuum elements using techniques similar to those described in Ref. 17.

The mechanism was first controlled so that the output port would follow the backward horizontal path for which it was originally designed, and then a forward horizontal path for which it was not designed. In this first case, there was no workpiece resistance applied to the mechanism model’s output port (Fig. 9). The required forces (Fig. 9c) to control the mechanism in the backward path are quite linear with output port displacement whereas those required to move the O/P forward increase nonlinearly with displacement. The physical explanation for this observation can be seen in the deformed shapes of the mechanism (Figs. 9a&b). As the input port moves backward and the output port

Fig. 8 Computed complimentary compliances of the aluminum inverter mechanism at finite deformation versus material usage factor C for different workpiece spring stiffnesses.

1.E-05

1.E-04

1.E-03

1.E-02

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35M aterial Usage C

Com

plim

enta

ry C

ompl

ianc

e (N

m)

k=0.1 MN/mk=1 MN/mk=10 MN/m

10-3

10-2

10-4

10-5

b)

c)

3cm

e) g)

f)

a)

f in

100 elements

3cm

100 elements out k

bk bkvoutf

d)

Fig. 7. a) Inverter mechanism design region and loading conditions; b) design solution of P1 with C=.30; c) deformed configuration; d) design solution for C=.10; e) deformed configuration; f) design solution with C=.03; g) deformed configuration.

mNkworkpiece /106=

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moves forward, the mechanism is flattening out which diminishes the tendency to produce further forward motion of the output port.

Next, the mechanism was controlled to follow a

backward inclined path making an angle of 45° with respect to the horizontal axis, and then a forward path at the same inclination. No workpiece resistance was applied during this motion (Fig. 10). It is interesting to note here that relatively modest vertical forces need to be applied to the input port of the mechanism to achieve significant vertical motion of the output port in the same direction. What is not shown in Fig. 10, but can instead be seen in video of the mechanism going through this motion18 is that geometric advantage of vertical motion at the output port relative to that at the input port is roughly only about 3

1 . This is in contrast to that of the horizontal motion, which has a geometric advantage approaching unity.

The final control test for the inverter

mechanism in the absence of workpiece resistance solved for the actuation forces required to move the O/P in a backward parabolic trajectory 2xy −=

(Fig. 11a) and then a forward parabolic trajectory 2xy = (Fig. 11b). This trajectory involves both horizontal and vertical motion and the control algorithm is clearly very effective in keeping the output port along the proper path (Fig. 11c).

Fig. 9 a) Mechanism is controlled to follow a backward horizontal path; and b) a forward horizontal path; c) graph showing the computed relation between the input force components and the output port displacements.

)a )b

Desired path

)(+x

)(+y

)(−x

Desired path

)c

100200

0

-400

-100-200-300

0 1-1-2 2-3ux(mm)

Fx(N)

400300

-500

xF)(+ xF)(−

Output displacement; TrajectoryOutput displacement; Trajectory

Fig. 10. a) Mechanism when controlled to follow backward inclined path; and b) a forward inclined path; c) graph showing computed relations between actuation force components and output port displacements.

)c

10.5

0-0.5-1

-1.5-2-2.5

100500

-50-100-150-200

-250

0 1-1-2 2 3-3

0 100 200 300-100-200-300

uy(mm)

ux(mm)

Fy(N)

Fx(N)

Input force; Output displacement; Trajectory

)(+x

)(+y

)(−x

Desired path)a

xF)(+

xy =yF)(−

xy =

)b

xF)(−

yF)(+

300

-3

-2

0

-1

2-200 -100 0 100 200-300

0

100

-100

-200

200

-400

32

1

0-1-2-3

-300

-4

1ux(mm)

uy(mm)

Fx(N)

Fx(N)

)a

Desired path

)b

2xy=

)(+x

)(+y xF)(−

yF)(+xF)(+

yF)(−

xF)(+

yF)(−

)(−x

2xy −=Desired path

Input force; Output displacement; TrajectoryInput force; Output displacement; Trajectory

Fig. 11. a) Mechanism when controlled to follow a backward parabolic path; and b) a forward parabolic path; c) graph showing computed relations between the input forces and output displacements.

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Against relatively light workpiece resistance ( )llk ⊗⋅= −1510 mN where yx eel += the mechanism was then

controlled to undergo a backward inclined motion ( )0 with x <= xy , and then a forward inclined motion ( )0 with x >= xy (Fig. 12). The computed control force versus displacement characteristics of the mechanism are similar to those computed in the absence of any resistance (Fig. 10), although the magnitudes of the necessary control forces are somewhat larger, as would be expected. When an attempt was made to control the sparse mechanism to follow the same path when working against a resistance level increased by a factor of 100, the mechanism was too compliant relative to the workpiece springs and buckled against them. In this case, a similar mechanism design but stiffer (see Fig. 7d and e) was then employed with very good success as shown in Figure 13. The actuation force levels required to work against such high resistance are at least an order of magnitude larger than those required in all of the preceding test problems.

V. Discussion and Conclusions In this work, a methodology for design of distributed compliance mechanisms, as opposed to compliant

mechanisms with de facto hinges was first presented and demonstrated in Section 3 on an inverter mechanism. The key rationale for using distributed compliance mechanisms is that their designs are very unambiguous, and they are perceived as more durable when used in practice, since they contain no regions of strongly concentrated deformation. Different hinge-free inverter mechanism designs were then taken and controlled with the algorithm proposed in Section 2.3 to find sequences of input actuation forces that result in the mechanism output port following a specified trajectory when working against a given level of workpiece resistance. With the proposed control algorithm embedded within incremental nonlinear analysis (Fig. 5), the optimal incremental actuation forces in a given load- or time-step of the analysis were typically found in five to ten conjugate gradient method iterations. Generally fewer control iterations are required when the mechanism model is closer to its unloaded state, and more iterations required as the deformations in the model increase in magnitude. When taking account of line search iterations employed in the optimal control problem, nonlinear control problems such as those solved herein can be

Fig. 12. a) Mechanism when controlled to follow forward inclined path against workpiece springs with stiffness 105 N/m; and b) when following a backward inclined; c) graph showing computed relations between input forces and output displacements.

Desired pathxy =

)b

Desired path

)c

10.50

-0.5-1

-1.5

-22.5

100

0

-400

-100

-300

-200

300

0 1-1-2 2

0 100 200 300-100-200-300

uy(mm)

ux(mm)

Fy(N)

Input force; Output displacement; TrajectoryInput force; Output displacement; Trajectory

200

1.5

Fx(N)400-400 500

0.5 1.5-0.5-1.5

)(+x

)(+y

)(−x

xF

yF

)a

xy =

xF

yF

)c

1

0.5

0

-0.5

-1

-1.5

1

0

-1

-3

-2

3

0 1-1-2-2.5

0 5 10 15-25 -15

uy(mm)

ux(mm)

Fy(kN)

Trajectory

2

1.5

Fx(kN)

-20

0.5 1.5-0.5-1.5

)(+x

)(+y

)(−x

Desired pathxy =

)a

xF

yF

)b

xF

yF

Desired pathxy =

-5-10

Input force; Output displacement;Input force; Output displacement;

Fig. 13. a) Mechanism when controlled to follow forward inclined path against workpiece springs with stiffness 107N/m; and b) when following a backward inclined path; c) graph showing computed relations between the input forces and output displacements.

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an order of magnitude more expensive than pure nonlinear analysis in which the actuation forces are all specified in advance.

In the introduction to this manuscript it was noted that other investigators in this field have proposed methods for continuum topology design of compliant mechanisms such that under the action of a sequence of specified input actuation forces, and specified levels of output port workpiece resistance, the output port will follow the desired curvilinear trajectory. It was pointed out that such an approach might be limited by even modest changes in workpiece resistance causing the output port to follow much different trajectories than those desired. It bears mentioning here that mechanisms so-designed would also be good candidates for application with the optimal control algorithm proposed herein (Fig. 5) so that they can be made to follow the desired trajectory (if possible) even under different workpiece resistance levels.

While the optimal control algorithms for compliant mechanisms presented herein are seen as very promising, there is still room for considerable improvement in the design and control methodology. Specifically, it is noted that the stress levels in the mechanism models, and their proximity to material failure and/or high fatigue levels were not considered, as the models underwent nonlinear optimal control. Ideally, the mechanism should be both designed and controlled such that in performing the desired functions under reasonably varying degrees of workpiece resistance, the capacity of the mechanism is not exceeded. For this reason, subsequent extensions of the current work might consider simultaneous design and control of the mechanism with stress constraints rather than sequential design followed by control as considered herein with no stress constraints.

VI. Acknowledgments This research was funded in part by a grant from the University of Iowa CIFRE program and in part by NSF

Grant No. DMS-9874015.

VII. References 1Howell, L.L, and Midha, A., “A Loop-Closer Theory for the Analysis and Synthesis of Compliant Mechanisms,” Journal of

Mechanical Design, Vol. 118, 1996, pp. 121-125. 2Smith, S. T., 2000, Flexures: Elements of Elastic Mechanisms, Gorden & Breach, Amsterdam. 3Howell, L.L., 2001, Compliant Mechanisms, John Wiley & Sons, Inc., New York. 4Ananthasuresh, G.K., Kota, S, and Gianchandani, Y., 1994, “A Methodical Approach to the Design of Compliant

Micromechanisms,” Solid-State Sensor, Actuator Workshop, South Carolina, June 13-16. 5Sigmund, O., 1997, “On the Design of Compliant Mechanisms Using Topology Optimization,” Mech. of Struct. and Mach.,

25 (4) , pp. 495-526. 6Poulsen TA. A simple scheme to prevent checkerboard pattern and one-node connected hinges in topology optimization.

Struct. Multidisc. Optim. 2002; 24: 396-399. 7Poulsen TA. A new scheme for imposing a minimum length scale in topology optimization. Int. J. Num. Meth. Engng. 2003;

57: 741-760. 8Yin L, Ananthasuresh GK. Design of Distributed Compliant Mechanisms. Mechanics Based Design of Structures and

Machines. 2003; 31 (2): 151-179. 9Rahmatalla, S.F., and Swan, C.C., 2004, “Sparse monolithic compliant mechanisms using continuum structural topology

optimization,” in review. 10Pedersen, C.B.W., Buhl, T., and Sigmund, O., 2001, “Topology synthesis of large-displacement compliant mechanisms,” Int.

J. Num. Meth. Engrg. 50, pp. 2683-2705. 11Saxena, A., and Ananthasuresh, G.K., 2001, “Topology synthesis of compliant Mechanisms for Nonlinear Force-Deflection

and Curved Path Specifications,” J. Mech. Design. ASME, 123, pp. 33-42. 12Ciarlet, P. G., 1988, “Mathematical Elasticity, Volume I: Three-Dimensional Elasticity”, Elsevier, Amsterdam. 13Cardoso J.B. and Arora, J.S., 1991, “Adjoint sensitivity analysis for nonlinear dynamic thermoelastic systems,” AIAA J. 29

(2), 253-263. 14Rahmatalla, S., and Swan, C.C., 2003, “Continuum topology optimization of buckling-sensitive structures,” AIAA J. 41(5),

1180-1189. 15Swan, C.C., and Kosaka, I., 1997, “Voigt-Reuss topology optimization for structures with nonlinear material behaviors,” Int.

J. Numer. Meth. Engrg. 40, 3785-3814. 16Arora, J.S., 1989, Introduction to Optimum Design, McGraw-Hill, Inc., New York. 17Kim, H.J. and Swan, C.C., 2003, “Algorithms for automated meshing and unit cell analysis of periodic composites with

hierarchical tri-quadratic tetrahedral elements,” Int. J. Numer. Meth. Engrg. 58 1683-1711. 18http://www.engineering.uiowa.edu/~swan/struct_opt/videos/INVERTER03_OUTW_INW_ICL.avi