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
C.C. Swan and S.F. Rahmatalla
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.
Topological Design and Nonlinear Control of Path-Following Compliant Mechanisms
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
C.C. Swan and S.F. Rahmatalla
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 .
Topological Design and Nonlinear Control of Path-Following Compliant Mechanisms
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).
C.C. Swan and S.F. Rahmatalla
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:
Topological Design and Nonlinear Control of Path-Following Compliant Mechanisms
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
C.C. Swan and S.F. Rahmatalla
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.
Topological Design and Nonlinear Control of Path-Following Compliant Mechanisms
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
C.C. Swan and S.F. Rahmatalla
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
Topological Design and Nonlinear Control of Path-Following Compliant Mechanisms
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:
C.C. Swan and S.F. Rahmatalla
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:
Topological Design and Nonlinear Control of Path-Following Compliant Mechanisms
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
C.C. Swan and S.F. Rahmatalla
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
Topological Design and Nonlinear Control of Path-Following Compliant Mechanisms
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
C.C. Swan and S.F. Rahmatalla
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
Topological Design and Nonlinear Control of Path-Following Compliant Mechanisms
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.
C.C. Swan and S.F. Rahmatalla
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
Topological Design and Nonlinear Control of Path-Following Compliant Mechanisms
20
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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)
C.C. Swan and S.F. Rahmatalla
21
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Topological Design and Nonlinear Control of Path-Following Compliant Mechanisms
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 +=
C.C. Swan and S.F. Rahmatalla
23
Figure 5 Algorithm for control problem interleaved with nonlinear analysis problem for the (n+1)th time-step of the problem.
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Topological Design and Nonlinear Control of Path-Following Compliant Mechanisms
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.
C.C. Swan and S.F. Rahmatalla
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.
Topological Design and Nonlinear Control of Path-Following Compliant Mechanisms
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
C.C. Swan and S.F. Rahmatalla
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
Topological Design and Nonlinear Control of Path-Following Compliant Mechanisms
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)(+
C.C. Swan and S.F. Rahmatalla
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.
Topological Design and Nonlinear Control of Path-Following Compliant Mechanisms
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
C.C. Swan and S.F. Rahmatalla
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.