Abstract— This paper proposes a novel design methodology

to synthesize flexure-based parallel manipulators (FPM) for

high precision micro/nano-scale manipulation. Unlike

traditional synthesis methods, the proposed method uses a

structural optimization algorithm that is independent of human

intuition, to synthesize compliant joints with optimal stiffness

characteristics. This algorithm is able to evolve the topology and

shape of the compliant joints. Based on finite element analysis,

the synthesized compliant joints are able to achieve better

stiffness characteristics than the traditional compliant joints.

This allows the synthesized joints to achieve a large deflection

range while maintaining their capabilities to resist external

wrenches in the non-actuating directions. A planar motion FPM

with a workspace of 4 mm2 × 2 is formed by assembling the

optimal compliant joints. The actuating compliance of the joints

and FPM are validated by experiments and their deviation

between the experimental results and the simulation prediction

are within 10% and 18% respectively.

I. INTRODUCTION

Compliant mechanisms are structures that have high compliance in the actuating directions while maintaining high stiffness in the non-actuating directions. As these structures achieve their motions via elastic deformation, they are able to eliminate backlash and mechanical play [1]. These unique characteristics allow the compliant mechanisms to achieve highly repetitive motions, making them the ideal candidates to perform micro/nano-scale manipulations [2-4].

The elasto-kinematics method is an established method to synthesize multiple degrees-of-freedom (DOF) compliant mechanisms [5, 6]. This method classifies the components of the compliant mechanism into groups of rigid-bodies (linkages) coupled together by the elastic-bodies (compliant joints). There are two basic types of compliant joints, the notch-type and beam-type, and their deformation characteristics resemble motion achieved by traditional joints that have springs attached to them. By integrating the compliant joints with rigid-bodies, motions of the compliant mechanisms and the traditional mechanisms resemble one another [5, 6]. Based on this concept, Ryu et al. and Lee et al.

have utilized notch-type compliant joints to develop X-Y-z planar motion compliant mechanisms [7, 8]. Ryu et al. is able to achieve 8 nm translational resolution and 0.057 arcsecond rotational resolution while Lee et al. is able to achieve 10 nm

1 School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore 639798. 2 Department of Mechanical Engineering and Robotics Institute, Carnegie

Mellon University, PA 15213. 3 Mechatronics Group, Singapore Institute of Manufacturing Technology,

Singapore 638075.

translational resolution and 0.2 arcsecond rotational resolution respectively. However, their workspace is small - ranging between hundreds of micrometers in the translational direction and hundreds of arcseconds in the rotational direction. This small workspace characteristic is due to the high actuating stiffness nature of the notch-type compliant joint. In order to increase the workspace, the notch-type compliant joints can be replaced by the more compliant beam-type compliant joints. This was demonstrated by Yang

et al. where a beam-type X-Y-z planar motion compliant mechanism which has a much larger workspace of

2.5mm×2.5mm×2.5 is developed [9]. Although the beam-type compliant mechanisms have larger workspace than their notch-type counterparts, their non-actuating stiffness are also lower. This reduces the beam-type compliant mechanisms' capability to resist external wrenches in the non-actuating directions and thus affecting its overall precision.

Despite having intensive research on the design of the notch-type and beam-type compliant joints, it should be noted that these joints are fundamentally created via human intuition. Thus, the stiffness characteristics of these compliant joints might not be optimized. This prevents the assembled compliant mechanisms from achieving their full potential of enhancing their ratio of non-actuating to actuating stiffness. A promising new compliant joint synthesis method that is not limited by human intuition was suggested by Wang [10]. By using a structural optimization algorithm, Wang was able to independently synthesize a prismatic compliant joint that resembles the linear spring design [10]. The design procedure is to first discretize the design domain into a mesh of quadrilateral finite elements. During the optimization, behavior of the compliant mechanism is determined by finite element analysis (FEA). Subsequently, based on the fitness function, the loading and boundary conditions of the design domain, the optimization algorithm will determine the final state of these elements. Ideally, all the elements should either be black (solid) or white (void) at the final state. Although this synthesis method had demonstrated great potential, the final design obtained by Wang contains ambiguous "grey" elements which have no physical meaning. Thus, it is still a challenging task to synthesize feasible compliant joints by using a structural optimization algorithm and eventually assemble them into a multi-DOF compliant mechanism.

This paper presents a novel hybrid topological and structural optimization method that is able to synthesize feasible compliant joints with optimal stiffness characteristics. The obtained compliant joints are able to achieve actuating stiffness that are in the same order as the beam-type joints but with higher non-actuating stiffness. The synthesized compliant joints are eventually assembled to form a 3-DOF

A Hybrid Topological and Structural Optimization Method to

Design a 3-DOF Planar Motion Compliant Mechanism

Guo Zhan Lum1,2,3

, Student Member; IEEE, Tat Joo Teo3, Member; IEEE, Guilin Yang

3, Member;

IEEE, Song Huat Yeo1 and Metin Sitti

2, Senior Member; IEEE

2013 IEEE/ASME International Conference onAdvanced Intelligent Mechatronics (AIM)Wollongong, Australia, July 9-12, 2013

978-1-4673-5320-5/13/$31.00 ©2013 IEEE 247

planar motion flexure-based parallel manipulator (FPM). The organization of the paper is as follows: the kinematics configuration of the proposed FPM is described in Section II. Sections III and IV will discuss about the development of the compliant joints and their assembly into the FPM respectively. The experimental results of the compliant joints and FPM prototype are shown in Section V. Lastly, the conclusion is provided in Section VI.

II. PROPOSED 3PPR FPM

The proposed 3-DOF compliant mechanism is articulated

by a parallel kinematics configuration. In order to realize a

X-Y-z planar motion compliant mechanism, there are three

possible combinations, 3-legged Revolute-Revolute-

Revolute (3RRR), 3-legged Prismatic-Revolute-Revolute

(3PRR), and the 3-legged Prismatic-Prismatic-Revolute

(3PPR) [9]. Here, the 3-legged prismatic-prismatic-revolute

3PPR is selected over the 3PRR and 3RRR because

compliant prismatic joints are generally stiffer and more

deterministic than the compliant revolute joints. The 3PPR

schematic is shown in Fig. 1 where the end-effector (centre

of the moving platform) is connected to the fixed base by

three identical parallel kinematics chains. Each chain

comprises of a serially-connected active prismatic joint (P)

and a passive prismatic-revolute (PR) joint.

Fig. 1. Schematic drawings of the 3PPR at home pose.

A global frame, {g}, is fixed at the center of the mechanism when it is at the home pose. Each kinematic chain

is assigned a corresponding local frame, {i}, {i +, 1 ≤ i ≤

3}, and their origins are fixed at the home position of their P joint. The orientation of the frames {1}, {2} and {3} have a

relative z-axis rotation angle of [1 2 3] =

]

from the global frame respectively. The superscripts of the vectors or matrices indicate the frame used to describe them. For example, a vector

{g}r means that this vector is viewed in

the global frame. These frame conventions will be used for the analyses in Section IV.

III. SYNTHESIZING COMPLIANT JOINTS WITH OPTIMAL

STIFFNESS CHARACTERISTICS

In order to synthesize a compliant mechanism which has

identical kinematic requirements as the 3PPR parallel

manipulator, the utilized conventional joints are converted to

corresponding compliant joints. Two compliant joints – P

and PR joints will be synthesized via the hybrid topology and

structural optimization.

A. The Hybrid Topological and Structural Optimization

Algorithm

There are various types of structural optimization algorithms in literature such as the SIMP [11], morphological representation [12], homogenous methods [13] and the mechanism approach [14]. In this paper, the mechanism approach is the selected algorithm to synthesize the compliant joints. This algorithm uses traditional mechanisms as seeds to represent the topology and shape of the compliant joints. A notable advantage for this algorithm is that it is able to eliminate infeasible solutions such as disconnected solid elements and ambiguous “gray” elements while exhibiting good convergence capability.

In order to implement the mechanism approach, an appropriate traditional mechanism that has the same DOF requirement as the compliant joint is selected as the seed. The seed selection can be achieved by using the well-established classical mechanism synthesis methods as a guideline. This seed will then be superimposed onto a design domain where all its finite elements are initially void. The pose of the seed can be varied by changing the link lengths and their orientations. Each link of the seed is represented by one cubic curve, one harmonic curve, and their reflected curves about the link. This is illustrated in Fig. 2 using the four-bar linkage as the seed. The four curves form the boundaries used in the selection of solid elements. Based on the value of m assigned to each link ( ), different combinations of solid elements can be generated. If m = 1, all the elements bounded between the original curves and the link are solid. When m = 2, all the elements bounded by the reflected curves and the link are solid. When m = 3, the solid elements will be the combined elements of the first two cases.

Fig. 2. Curves generated by the last link of the seed.

A corresponding structure is produced when all the links

follow the above-mentioned description as shown in Fig. 3.

By varying the design parameters such as the seed’s pose and

the curve parameters, different structures are generated.

Although the mechanism approach uses a seeding method,

the topologies are not limited by its seed. The seed’s

topology can be changed if any link’s length approaches zero

during the optimization. The fixed points and the coupler

point of the seed correspond to the fixed points and loading

points of the compliant joints. Note that there is only one

loading point for compliant joints.

Original

Reflected

x1

Cubic curves

Harmonic curves

y1 m = 1

m = 2

m = 3

Seed

Design Domain

{g}

x1 y1

x2

y2

x3

y3

Moving

platform

P

P

P

PR

PR

PR

248

Fig. 3. Example of a topology by the mechanism approach.

The mechanical behavior of the structures is evaluated via

FEA, where the stiffness of one finite element, Ke, and the

entire structure, Ks, are given as:

∭ ∑ ( ) (1)

The matrices B and D are the commonly used deformation

matrix in FEA and isotropic compliance matrix in solid

mechanics respectively. The variables V and represent the

volume and state of the finite element. If element i is solid, i

= 1; if it is void, i = 10-6

, to prevent numerical instabilities.

Thus, the governing equation for the FEA is:

Ksus = f (2)

where us and f represent the global nodal deformation vector

and global loading vector respectively.

In order to evaluate a structure’s stiffness characteristics,

its 6×6 compliance matrix, that is the inverse of the 6×6

stiffness matrix, has to be determined. This matrix is

obtained by solving Eq. (2) six times, each replacing f with

an orthogonal unit wrench at the loading point. The resultant

position and orientation deformation vector of the loading

point induced by a corresponding loading forms a column in

the 6×6 compliance matrix. The first three columns of the

matrix are induced by unit force loadings in the x, y and z-

axis respectively while the last three columns of the matrix

are induced by unit torque loadings in the x, y and z-axis

respectively. Note that the first three rows of the 6×6

compliance matrix represent the translational deformation of

the loading point in the x, y and z-axis respectively. The last

three rows represent the rotational deformation of the loading

point in the x, y and z-axis respectively.

By formulating a fitness function that minimizes the

actuating stiffness and maximizes the non-actuating stiffness,

the solver Genetic Algorithm (G.A.) will eventually obtain

an optimum compliant joint. Note that this is also equivalent

to maximizing the actuating compliance and minimizing the

non-actuating compliance in the 6×6 compliance matrix.

B. Synthesis of the Passive PR Joint

The synthesis of the PR joint is broken down into two sub-

optimization problems to reduce the computational time. The

first optimization is performed on a coarse mesh while the

second optimization refines the design on a fine mesh. For

both optimizations, a 20 mm × 50 mm × 10 mm aluminum

design domain is used and its Young’s Modulus and the

Poisson ratio are 71 GPa and 0.33 respectively. The design

domain is discretized into a mesh of 3-D 8-node bilinear finite

elements for both optimizations.

As the compliant PR joint has 2-DOF, a corresponding

2-DOF 5-bar linkage with a coupler point is selected as the

seed for the first optimization (shown in Figure 4(a)). The

coupler point of the seed (also the loading point) is

constrained to move along the top row elements while there

are two fixed points located at the base. Note that one of the

fixed points location is fixed while the other's location is

constrained to move along the base. Ideally, the PR joint has

two low actuating stiffness and they are represented by

having high C11 and C66 in the 6×6 compliance matrix. In

addition, in order to have high non-actuating stiffness for the

PR joint, other components in the 6×6 compliance matrix

should be low. Therefore, the optimization problem is

formulated as:

( ( ) |∏ ∏

|)

subjected to:

h1(x) ≤ 0 (3)

The C11 and C66 components are raised to the power of 19

as there are 19 non-actuating stiffness components. The

inequality constraints ensure that the pose of the seed

remains in the design domain and the equality constraint

represents the FEA governing equations. Figure 4(b) shows

the final result of the first optimization after G.A. evolves a

population of 500 chromosomes via 100 generations. At the

end of this optimization, it is found that two link lengths of

the seed are reduced to zero. This essentially changes the

topology of the original 5-bar linkage seed into a thin beam

supported by a structure. The second optimization uses Eq.

(3) again to refine the design. By using G.A. to evolve a

population of 200 chromosomes via 50 generations, the

optimal PR compliant joint shown in Fig. 4(c) is obtained.

The optimal PR joint resembles a non-uniform beam

supported by an “arch”. The high actuating compliance is

provided by the beam while the “arch” provides the high

stiffness in the non-actuating directions.

Fig. 4. (a) Using a 5-bar linkage seed to carry out the PR joint

synthesis. (b) The coarse mesh optimal solution. (c) The optimal PR

joint obtained after refining the coarse mesh solution.

As it might be easier to apply z-directional torques to the

compliant joint if the joint is symmetrical about its loading

fx

non-uniform Beam

x

y

Arch Supports

loading point

fx mz fx

mz

(a) 5-bar linkage

topology (b) Evolved into 3-

bar (c) Optimal PR joint

249

point, the finalized PR joint (Fig. 5(a)) is obtained by

reflecting the obtained structure about its loading point. The

6×6 stiffness matrix of the finalized design, KPR, is shown in

Eq. (4) and it is compared to the stiffness matrix of the

traditional beam-type design, KEK1 in Eq. (5). If any

components in KPR and KEK1 are smaller than 10-2

, a zero is

assigned to them as they are relatively insignificant. Note

that the traditional beam-type design is a thin beam that is

fixed on both ends (Fig. 5(b)). As it might be easier to

compare if both designs have at least one identical actuating

compliance, the beam thickness of the traditional design is

adjusted to 0.56 mm to match the rotary actuating stiffness,

K66, of the finalized PR joint design.

[

]

(4)

[

]

(5)

A matrix Ra is used to compare KPR and KEK1. Each

component in Ra, R

aij, represents the ratio of KPR,ij against

KEK1,ij if neither are zero. A zero will be assigned to Raij if

both KPR,ij and KEK1,ij are zero.

[

]

(6)

In order for the finalized PR joint to exhibit lower actuating stiffness and higher non-actuating stiffness, R

a11 and

Ra66 should be less than 1 while other non-zero R

aij should be

more than 1. Based on Eq. (6), the final design has a lower translational actuating stiffness than the traditional design as R

a11 is less than 1. Furthermore, except for R

a33, the finalized

PR joint has higher non-actuating stiffness than the traditional design too. This is especially true for the R

a55 and R

a42 where

the finalized design is more than 2 and 1.5 times stiffer than the traditional design respectively. Note that out of the seven non-zero stiffness components, five components have improved. Since the finalized PR joint is shown to have lower actuating stiffness and higher non-actuating stiffness, it can be concluded that the finalized PR joint design is better.

Fig. 5. (a) The finalized PR joint design that is obtained by reflecting Fig. 4(c) about its loading point. (b) The traditional PR joint design.

C. Synthesis of the Active P Joint

Similar to the synthesis of the PR joint, the P joint

synthesis is split into two optimization problems so as to

reduce computational time. The first optimization uses a

coarse mesh while the second optimization refines the design

with a fine mesh. For both optimizations, the design domain

is a 25 mm × 50 mm × 10 mm aluminum metal plate and its

Young’s Modulus and Poisson ratio are 71 GPa and 0.3

respectively. The design domain is discretized into a mesh of

identical 3-D 8-node bilinear finite elements.

Similar to a linear guide, the ideal characteristic of a

compliant P joint is that it has low stiffness in one actuating

translational direction and high stiffness for other non-

actuating directions. This infers that the P-joint is a 1-DOF

compliant joint and thus a simple 4-bar linkage with a coupler

point is selected as the seed for the first optimization. The

seed’s coupler point (loading point) is located at the top row's

central element. There are two fixed points and they are

located at the extreme ends of the bottom row. Note that the

pose of the seed used in the first optimization is random (as

shown in figure 6(a)) and it can be evolved.

The actuating stiffness is represented by the C11

component in the C6×6. In order to minimize this stiffness and

maximize the non-actuating stiffness, C11 and other

components in C6×6 have to be maximized and minimized

respectively. As C6×6 is a symmetry matrix, the optimization

problem is:

( ( ) |∏ ∏

|)

subjected to:

h2(x) ≤ 0 (7)

The C11 component is raised to the power of 20 as there

are 20 non-actuating stiffness components. The vector x

represents the design variables while the equality constraints

represent the FEA governing equation. The inequality

constraints are used to ensure that the seed's pose remains in

the design domain. The first optimization is carried out by

evolving a population of 400 chromosomes via 100

generations of evolution and Fig. 6(b) shows the final result.

Although the topology of the optimal seed remains as a four-

bar linkage after the first optimization, the optimal pose has

been determined. The links of the seed are placed either

vertically or horizontally (see the outlined yellow lines in Fig.

6(b)). The second optimization refines this design with a fine

mesh by using Eq. (7) and the optimal seed pose shown in

Fig. 6(b). The optimal P joint is obtained after G.A. evolves a

random initial population of 200 chromosomes via 50

generations and it is shown in figure 6(c).

The optimal P joint resembles a tapered-shape rigid-link

supported by two thin beams. A comparison between the

optimal P joint and the traditional beam-type linear-spring

design is made in Fig. 7. The 6×6 stiffness matrix for the

optimal P joint, KP, and the traditional beam-type design,

KEK2, are shown in Eq. (8) and (9) respectively. Note that any

x

y

fx

mz

a) Finalized PR joint design

fx

mz

b) Traditional design

10 mm

2 mm 0.4 mm

0.56 mm

250

components in KP and KEK2 that are smaller than 10-2

are

assigned with a zero because they are relatively insignificant.

Fig. 6. (a) Using a 4-bar linkage seed to carry out the P joint

synthesis. (b) The coarse mesh optimal solution. (c) The optimal P

joint obtained after refining the coarse mesh solution.

Fig. 7. (a) The traditional linear spring P joint. (b) The obtained

optimal P joint

[

]

(8)

[

]

(9)

A matrix Rb is used to compare KP and KEK2. Each

component in Rb, R

bij, represents the ratio of KP,ij against

KEK2,ij if neither are zero. A zero will be assigned to Rbij if

both KP,ij and KEK2,ij are zero.

[

]

(10)

In order for the optimal P joint to exhibit lower actuating

stiffness and higher non-actuating, Rb11 should be less than 1

while other non-zero components should be more than 1. As

all the non-zero components in Rb are either more than or

equal to 1, this means that the overall stiffness for the

optimal P joint is higher than the traditional design. Thus,

this also implies that the optimal P joint has higher non-

actuating stiffness too. This is especially true for the Rb

32 and

Rb22 where the optimal topology design is about 14% and 9%

stiffer respectively. Since both the designs have identical

actuating stiffness (Rb11 = 1), it can be concluded that the

optimal P joint has a better overall stiffness characteristics.

The optimal P joint is able to exhibit higher non-actuating

stiffness because its tapered-shape rigid link has more layers

of finite elements. However, as the compliance in the

actuating direction is largely attributed by the supporting thin

beams, when both designs have identical beams, their

actuating stiffness are identical too.

IV. ASSEMBLY OF THE PROPOSED 3PPR FPM

The optimal compliant joints from Section III are

combined together to form the 3PPR FPM as shown in Fig.

8. Note that the sharp edges which appeared in the compliant

joints have been smoothen out to prevent stress

concentration. In this work, electromagnetic voice-coil (VC)

actuators are used to drive this 3PPR FPM due to their ability

to provide millimeter range stroke. During the design stage,

it is estimated that each VC actuator should provide a

continuous force of at least 30 N. In addition, the minimum

dimension of such VC actuators is estimated to be at least

Ø60 × 60mm2. Thus, each kinematic chain is fixed at 90 mm

× 90 mm in order to encase each VC actuator within it. In

this work, the 3PPR FPM is monolithically cut from a

SUS316 stainless steel work-piece. Young's Modulus, E, and

Poisson ratio, , of SUS316 are 200 GPa and 0.33

respectively. Lastly, a standard 19-mm thick work-piece is

used to enhance the non-actuating stiffness along the z-axis.

Fig. 8. Schematic drawing for the 3PPR FPM.

A size optimization is required to determine the optimal

space distribution between the compliant joints so as to

achieve the best stiffness characteristics for the end-effector.

This is because increasing l3 (refer to Fig. 8) will increase the

non-actuating stiffness of the PR joints but it will also

decrease the actuating compliance of the P joint. In order to

retain the actuating compliance of the joints, the size

optimization does not alter the thickness of the beams. The

stiffness matrices for the compliant joints are determined via

FEA and are expressed in terms of their local chain frame.

The chain frames are the same as the ones shown in Section

II where {1}, {2} and {3} have a relative z-axis rotation

angle of [1 2 3] =

] from the global frame {g}

respectively. The 6×6 compliance matrix of kinematic chain

i, Cchain i,6×6, at the PR joint loading point can be obtained by:

l3

90- l3

P

Length (90 mm)

PR

ri

Smoothen

edges bi

x

y

x

y fx

(b) Parallel 4-bar (c) Optimal P joint (a) 4-bar linkage topology

Supports

Loading point

(a) Traditional Design

fx

25.2 mm

Thin

Beams

Rigid link 50

mm 0.4mm

2mm

mm 10 mm

(b) Optimal P joint

fx

Thin

Beams

25.2 mm

Tapered-shape

rigid link

251

[

] (11)

where [ ̂

] is the Jacobian matrix.

The 3x3 matrices I3x3, O3x3 and ̂ ( ) represent the identity

matrix, zero matrix and the skew-symmetry matrix form of

the position vector ri (shown in Fig. 8) respectively. Note

that ri represent the displacement vector from chain i's the

loading point to its P joint's loading point. Once the chain's

stiffness is identified, the 6×6 end-effector’s stiffness matrix,

Kee,6×6, can be obtained by:

∑ ]

[

]

] (12)

where [ ( ) ̂ ( )

( )] is the adjoint matrix.

The 3x3 matrices Rz and ̂ represent the standard z-axis

rotational matrix and the skew-symmetry matrix form of the

position vector bi (shown in Fig. 8) respectively. Note that bi

represents the displacement vector from the end-effector to

chain i's loading point. As the size optimization aims to

maximize the non-actuating diagonal stiffness while

minimizing the actuating stiffness of the end-effector, the

fitness function is formulated as:

(

) (13)

After Eq. (13) was solved by G.A using a population of

10 chromosomes which undergoes 10 generations of

evolution, the optimal solution l3 = 20 mm is obtained. Based

on the optimal space distribution, the 3PPR FPM prototype

shown in Fig. 9 has been developed.

Fig. 9. The 3PPR FPM prototype.

V. EXPERIMENTAL RESULTS

In this section, the actuating compliance of the synthesized joints and FPM will be investigated. The

actuating compliances can be determined by finding the linear relationship between the loadings and their corresponding deformations. The compliant P and PR joints are fabricated from a monolithic piece and they are shown in Fig. 10. Note that the sharp notches of the joints are not smoothen out yet as the authors would like to replicate similar design to evaluate the accuracy of the FEA. However, experiments for the FPM are performed on the FPM prototype (with smoothen edges) shown in Fig. 9. All experiments are performed on an anti-vibration table.

Fig. 10. Fabricated compliant joints.

A. Evaluation of Translational Compliance for Joints

In these experiments, the fixed points of the compliant joints are constrained by a fixed plate and their loading points are mounted by an electromagnetic nano-positioner, FELA[15]. Different magnitudes of force can be applied to the joints by varying the current supplied to the actuator. Upon loading, the applied force and linear deflections of the joints will be measured by a force sensor and a linear probe respectively. Fig. 11 shows the experimental set ups. Note that the force sensor is located between the joints and FELA.

Fig. 11. Experiment setups for force loadings.

Fig. 12. Experimental results for P joint's linear deflection where the input force, f, is plotted against the deflection, x. The best fit line is f = 1.42x.

For both experiments, three sets of data were collected; each set consists of 10 data points. The compiled data for the P and PR joints' experiments are shown via the scatter plots (which have a best fit line) in Fig. 12 and 13 respectively. Based on the gradient of the best fit lines, the P and PR joints have an average compliance of 7.04×10

-4 m/N and 6.00×10

-4

m/N respectively. These experimental results agree with the

Loading for PR joint

FELA

Linear

Probe

Force Sensor

PR joint

Loading for P joint

FELA Linear

Probe

P joint

Force

Sensor

Loading point

of PR joint

Loading point

of P joint

10 cm P joint

PR joint

Smoothen

edges

Metrology

frame 3PPR FPM

VC

actuator

Structure

frame

1 cm

252

FEA simulation where the predicted compliance for the P and PR joints should be 7.47×10

-4 m/N and 6.68×10

-4 m/N (based

on Eq. (8) and Eq.(4)) respectively. The deviation between the FEA and experiments for the P and PR joints are 6% and 10% respectively and they may be caused by manufacturing errors, such as having non-uniform beam widths. However, these deviations are negligible and this implies that the FEA translational compliance predictions have high credibility.

Fig. 13. Experimental results for PR joint's linear deflection where the input force, f, is plotted against the deflection, x. The obtained best fit line is f = 1.66x.

B. Evaluation of Angular Compliance for Joints

The actuating angular compliance of the PR joint is investigated in these experiments. Similar to the previous experiments, the fixed points of the PR joint are constrained by a fixed plate. In order to apply an external torque to the loading point of the PR joint, a stepper motor is used to replace the FELA. Likewise to FELA, different magnitudes of torques can be applied to the joints by varying the current supplied to the actuator. Upon loading, the linear deflections of a specific point (defined as point A) and applied torque will be measured by using the linear probe and torque sensor respectively. Note that the torque sensor is located between the joints and FELA. The angular deflection can be obtained by dividing the point A’s linear deflection by a prior known moment arm. Note that the moment arm is 20 mm as shown in the experimental setup in Fig. 14.

Fig. 14. Experiment setup for torque loading.

Similarly, three sets of 10 data points were collected. The compiled data is represented by the scatter plot (with a best fit line) shown in Fig. 15. Based on the gradient of the best fit line, the PR joint should have an angular compliance of 0.834 rad/(Nm). This result agrees with FEA where the predicted angular compliance is 0.909 rad/(Nm) (based on Eq. (4)). The deviation between the experiments and FEA simulation is 9% and this might be due to manufacturing errors, such as non-uniform beam widths. The small deviation

suggests high credibility in FEA predictions for the angular compliance.

Fig. 15. Experimental results for PR joint's angular deflection where

the input torque, , is plotted against the angular deflection, . The

obtained best fit line is = 1.19.

C. Evaluation of 3PPR FPM's Actuating Compliance

The actuating compliances of the 3PPR FPM are investigated in these experiments. As the FPM has 3-DOF, its end-effector has three actuating compliances - x-axis force loading Fx, y-axis force loading Fy and z-axis torque loading Mz. Due to the symmetrical configuration of the FPM, the Fx and Fy loadings of the FPM should be identical. Thus, in this section, experiments are only conducted to investigate the Fy and Mz loadings' actuating compliance.

In the Fy loading experiments, a picomotor is placed collinearly with the end-effector's y-axis. This picomotor will provide a range of y-axis force loadings to the end-effector. Similarly, in the Mz loading experiments, the picomotor is used to provide a range of z-axis torque loadings to the end-effector. This is achieved by placing the picomotor with an offset distance along the x-axis from its original position in the Fy loading experiments. Note that the angular z-axis deflections induced by Fy loadings are found to be negligible, thus these induced deflections are neglected in the Mz loading experiments. For both types of experiments, the deflections of the end-effector are measured by a 3-D measurement GOM camera. The force and torque loadings are measured by using a 6-axis force/torque sensor. Note that the force/torque sensor is placed between the end-effector and the picomotor for both types of experiments. The experimental setup for the y-axis loading experiments is shown in Fig. 16.

Fig. 16. Experimental setup for the Fy loading experiments

For both types of experiments, three sets of 10 data points

were collected. The compiled data for the Fy loading and Mz

loading experiments are represented by the scatter plots

shown in Fig. 17 and Fig. 18 respectively. Based on the

PR joint

Torque

sensor

Linear Probe Motor

Point A

20 mm

Picomotor

Force/torque

sensor

GOM Camera

3PPR FPM

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gradient of the best fit lines, the FPM's actuating compliance

in the Fy and Mz axes are 3.91×10-5

m/N and 0.0301 rad/(Nm)

respectively. The predicted actuating Fy and Mz compliance

by FEA (based on the final values of Eq. (13)) are

3.39 ×10-5

m/N and 0.0248 rad/(Nm) respectively. Thus, the

experimental results agree with the FEA prediction and the

deviation between the experiments and FEA for the Fy and Mz

loadings are 15% and 18% respectively. These deviations

might be caused by manufacturing errors such as non-uniform

beam widths or lengths. However, these deviations are

considered small and it suggests that the FEA for the FPM

stiffness characteristics have high credibility. Based on the

experimental stiffness characteristics, VC actuators with

26.35 N output forces are selected so as to enable the 3PPR

FPM to achieve a workspace of 4 mm2 × 2.

Fig. 17. Experimental results for FPM Fy loading where the input force, f, is plotted against the deflection, y. The best fit line is f = 25.6y.

Fig. 18. Experimental results for FPM Mz loading where the input

moment, Mz, is plotted against the deflection, . The obtained best fit

line is Mz= 33.2.

VI. CONCLUSION

This paper describes a novel design methodology to

synthesize multi-DOF compliant mechanisms for high

precision micro/nano-manipulation. The design of a 3PPR

FPM which comprises of two basic compliant joints - P and

PR joints, is used to illustrate the synthesis process. Unlike

traditional synthesis methods, a novel structural optimization

synthesis method which is not limited by human intuition has

been used to design compliant joints with optimal stiffness

characteristics. The proposed structural optimization

algorithm uses traditional mechanisms as seeds to represent

the compliant joints. During the optimization process, the

topology and shape of the compliant joints are evolved

gradually until an optimal solution is achieved. Based on

FEA, the optimal compliant joints have better overall

stiffness characteristics than their traditional counterparts.

For example, the P joint is able to increase its non-actuating

stiffness while achieving the same magnitude of actuating

compliance as the traditional design. Similarly, the PR joint

also has higher non-actuating and lower actuating stiffness

compared to the traditional design. When these compliant

joints are assembled, the optimal space distribution between

the joints are determined via another optimization process so

as to achieve the optimal stiffness characteristics for the

3PPR FPM. The actuating compliance for the joints and

FPM are validated by experiments and their deviation with

their FEA predications are within 10% and 18% respectively.

The developed 3PPR FPM has a workspace of 4 mm2 × 2.

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