Salam F. Rahmatalla, - User page server for...

C.C. Swan and S.F. Rahmatalla

1

TOPOLOGICAL DESIGN AND NONLINEAR CONTROL OF PATH-FOLLOWING COMPLIANT MECHANISMS

Colby C. Swan∇, Ph.D

Wheeler Faculty Fellow of Engineering Civil & Environmental Engineering Center for Computer-Aided Design

The University of Iowa, Iowa City, Iowa 52242, USA

Salam F. Rahmatalla, Ph.D. Civil & Environmental Engineering Center for Computer-Aided Design

The University of Iowa, Iowa City, Iowa 52242, USA

∇ Corresponding author.

ABSTRACT

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.

Keywords: topology optimization; nonlinear control; MEMS; hyperelastic analysis.

Topological Design and Nonlinear Control of Path-Following Compliant Mechanisms

2

1. INTRODUCTION

While 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 mechanisms [1] 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) [15], or it can be more or less uniformly distributed throughout the mechanism

(Fig. 1c) [6]. 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.

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 Ananthasuresh et al [1] and then others such as [14]. 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 mechanisms [9][10][12] [17].

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


3

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. [8][13].) 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 potential 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 formulation [12] 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 2, 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

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


4

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 3.

Then, in Section 4, 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 and closure

on the proposed analysis, design, and control methods are then presented in Section 5.

2. ELEMENTS OF FORMULATION

2.1 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:

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


5

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 Ciarlet

[3] wherein the volumetric )(U and deviatoric )(W strain energy functions are assumed to

be decoupled and of the forms:

)()()( θWJUE +=F (2a)

⎥⎦⎤

⎢⎣⎡ −−= )ln()1(21

21)( 2 JJKJU (2b)

]3)([21

−= θtrW µ (2c)

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 .


6

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:

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( AA1n 1

xB sxn+

∇=+ , 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).


7

2.2 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)

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 [16]. 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:


8

( )

( ) .: 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


9

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)

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

b

brK

ubb

∂∂⋅+

∂∂ℑ

=

∂∂⋅⋅

∂∂ℑ

−∂∂ℑ

=ℑ −

a

dd

1

(16)

where au is the so-called adjoint displacement vector [3] 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 in

[11] and for materially nonlinear structures in [16]. 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.


10

2.3 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 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))((

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−= ∑

+

++

iopni

opni

u

ug fd (19)

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

op1n du is the resulting output port displacement


11

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 ( ) jn

in1+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 neqℜ∈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 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.

3. CONTINUUM TOPOLOGY DESIGN OF HINGE-FREE MECHANISMS

3.1 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 stiff [12]. The second set


12

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 opout 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(MPE out

vout uf ⋅= (23)

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

and voutf 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:


13

( )[ ]),( MPE- min M1 burub

⋅+−ℑ+ aCλ (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 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

constraint [2].

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:


14

( )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.

3.2 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


15

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 mNkworkpiece /106= then 05.0inf =C as is shown in Fig. 8.

4. 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 [7].

The mechanism was first controlled to 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 moves forward, the

mechanism is flattening out which diminishes the tendency to produce further forward


16

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

motion [18] is that geometric advantage of vertical motion at the output port relative to that at

the input port is roughly only about 31 . 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).

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.

5. DISCUSSION AND CONCLUSIONS


17

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 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


18

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.

6. ACKNOWLEDGEMENTS

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

7. REFERENCES

[1] G.K. Ananthasuresh, S. Kota, and Y. Gianchandani, A Methodical Approach to the Design of Compliant Micromechanisms in Proc. Solid-State Sensor, Actuator Workshop, South Carolina, (1994).

[2] J.S. Arora, Introduction to Optimum Design (McGraw-Hill, Inc., New York 1989).

[3] J.B. Cardoso and J.S. Arora, Adjoint sensitivity analysis for nonlinear dynamic thermoelastic systems,” AIAA J. 29 (2) (1991) 253-63.

[4] P.G. Ciarlet, Mathematical Elasticity, Volume I: Three-Dimensional Elasticity, (Elsevier,

Amsterdam, 1988). [5] L.L. Howell, and A. Midha, A Loop-Closer Theory for the Analysis and Synthesis of

Compliant Mechanisms, J. Mechanical Design, 118 (1996) 121-25.

[6] L.L. Howell, Compliant Mechanisms, (John Wiley & Sons, Inc., New York, 2001).

[7] H.J. Kim and C.C. Swan, Algorithms for automated meshing and unit cell analysis of periodic composites with hierarchical tri-quadratic tetrahedral elements, Int. J. Numer. Meth. Engrg. 58 (2001) 1683-711.

[8] C.B.W. Pedersen, T. Buhl and O. Sigmund, Topology synthesis of large-displacement compliant mechanisms, Int. J. Num. Meth. Engrg. 50 (2001) 2683-705.

[9] T.A. Poulsen, A simple scheme to prevent checkerboard pattern and one-node connected hinges in topology optimization. Struct. Multidisc. Optim. 24 (2002) 396-9.

[10] T.A. Poulsen, A new scheme for imposing a minimum length scale in topology

optimization, Int. J. Num. Meth. Engng. 57 (2003) 741-60. [11] S. Rahmatalla and C.C. Swan, Continuum topology optimization of buckling-sensitive

structures,” AIAA J. 41(5) (2003) 1180-9. [12] S.F. Rahmatalla, and C.C. Swan, Sparse monolithic compliant mechanisms using

continuum structural topology optimization, (2004) in review.


19

[13] A. Saxena and G.K. Ananthasuresh, Topology synthesis of compliant mechanisms for nonlinear force-deflection and curved path specifications, J. Mech. Design. 123 (2001) 33-42.

[14] O. Sigmund, On the design of compliant mechanisms using topology optimization,”

Mech. of Struct. and Mach., 25(4) (1997) 495-526. [15] S.T. Smith, Flexures: Elements of Elastic Mechanisms, (Gorden & Breach, Amsterdam,

2000). [16] C.C. Swan, and I. Kosaka, Voigt-Reuss topology optimization for structures with

nonlinear material behaviors, Int. J. Numer. Meth. Engrg. 40 (1997) 3785-814. [17] Y. Yin, and G.K. Ananthasuresh, Design of distributed compliant mechanisms,

Mechanics Based Design of Structures and Machines. 31(2) (2003) 151-79. [18] www.engineering.uiowa.edu/~swan/struct_opt/videos/INVERTER03_y=x.avi


20

( )

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

11

111

11

1111

11

max

11

11

11

max1max

1

knn

kn

kn

kn

kn

kn

kn

kn

kn

kn

tolk

1n

kn

kn

nextext

n

gnE

nkn

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.

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 linkrigid link pin jointed

hinge

a) b) c)


21

if-End* tourn Ret

,,, :Compute

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

)(

)(

(..

()()(

(.

()()()(

while- End

1 )( : vectorforce residual Update

:ntdisplaceme Update

.

nt displaceme lincrementafor system linearized Solve

) and ( While

) : vectorforce residual initial Compute

;0 Initialize

:timeIncrement );;0For(

,, tInitialize*

layout material startinga Given

11

1

1

11

111

11

11

11

111

11

111

11

max1

11

11

1

max

o

bbb bb

b

b

uu

ubbb

br)

Ku)b

bu)

u)bb

uuu

ur

uuu

ruK

u

r

r(ur

euu

bu

b

∆+=

⎟⎠⎞

⎜⎝⎛ ℵ

ℵℑ

ℑ∆

=

∆ℑ∆+ℑ=ℑ∂ℑ∆

+ℑ

=ℑ

∂∂

∆∂ℑ∆∂

+∂ℑ∆∂

=

∆∆∂ℑ∆∂

+∂ℑ∆∂

=ℑ∆

−=∆

+=

∆+=

−=∆

∆

≤≥

=

+==

∆+=++≤=

ℑ

++

+

+

++

+

++

+

+

+++

++

++

++

+++

++

++++

++

+

++

++

+

dd

dd

ddd

dd

dd

dd

kk

kkr

k

tttnnnn

dd

knn

n

n

1nk1n

1n

nn

1n

1n

nnn

nnn

kn

kn

kn

kn

kn

kn

kn

k1n

kn

tolkn

kn

kn

nnkn

nn

o

o

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


22

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

inputΓ outputΓA

inf

*B

*C

Desired path

Actuation input forces

inverter

op11)(x

op21 )(x

op32 )(x

op12 )(x

op22 )(x

op1 )*(x

op*2 )(xTrial path

Trial displacement

Desired displacement

op31 )(x

op42 )(x

op0 uxx +=


23

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

))while-End

)( : vectorforce residual Update

:ntdisplaceme Update

:nt displaceme lincrementafor system Solve

) and ( While

) : vectorforce residual initial Compute

E :predictornt displaceme ;0 :Initialize

:(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

111

1111

1

11

111

11

tol1j1n

11

11

21

2

1*

111

11

11

11

11

111

11

1111

11

max1

11

11

11

1*

1

max1max

1

+++

++++

+

++

+++

++

++

++

++

+

++

++

++

++

++

++

+++

++

++++

++

+

++

++

++

++

+

+

∆+=+==

+=

+=

∇∇

≤∇

∇

≤+≥

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−=

=

∆+=

−=∆⋅

∆

≤≥

=

∈∀∆+=

=

==

≤≤+∆+==

∑

nnnjnn

jn

jn

jinn

jinn

jn

jn

jn

jn

jn

maxtolj

n

iop

ni

opn

1jij

n

opkn

opjn

kn

kn

kn

kn

kn

kn

kn

kn

kn

tolkn

kn

kn

gnE

nkn

inn

jn

in

nop

n

n

nn

tttnn

jj

gg

g

jjgg

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


24

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.


25

b)

c) e) g)

f)

a)

f in

100 elements

3cm

3cm

100 elements outk

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.


26

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


27

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


28

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)(+


29

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.


30

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


31

)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.

Salam F. Rahmatalla, - User page server for...

Documents

Transcript of Salam F. Rahmatalla, - User page server for...