Optimal Design of Proportion Compliant Mechanisms with Corner-filleted

Flexure Hinges

Qiaoling Meng1, Jia Xu1 and Yangmin Li1,2,∗

Abstract— This paper presents a new analysis approach foroptimal design of proportional compliant mechanisms withcorner-filleted flexure hinges. It is the first time to use pseudorigid body model (PRBM) for cantilever beams to analyze theconnecting flexure hinges. The rotation center and rotationalangle of each flexure hinge are corrected by means of charac-teristics radius factor (γ) and parametric angle coefficient (Cθ

). The displacement proportion equation for such mechanismsis derived according to the new approach. Combining the newproposed design equation with the existed stiffness equation,a new proportion compliant mechanism with corner-filletedflexure hinges is designed by means of the least squaresoptimization. The designed models are verified by finite elementanalysis.

I. INTRODUCTION

Flexures can be connecting joints for micro-scale or nano-

scale motion of compliant mechanisms to implement high

precision function which the traditional rigid joints cannot

do. A mechanism which relies on the elastically deformation

of flexures or flexure hinges to realize its functions is

called flexure-based compliant mechanism. A flexure-based

compliant mechanism can be manufactured in monolith-

ic by using a block of material [1]. The characteristics

of such a mechanism, such as non-assembling and non-

lubrication, can improve its accuracy, reduce assembly time,

simplify manufacturing time, and reduce wear, weight and

cost. Therefore, flexure-based compliant mechanisms have

been utilized in ultra-precision positioners for microscope in

biological applications, amplifiers for piezoelectric actuators,

and ultra-precision scanning, aligning and machining, and so

on [2], [3], [20], [27].

However, designing and analyzing such a flexure-based com-

pliant mechanism is relative difficult [4]. Many researchers

put a lot of efforts onto this project. For instance, Tian et

al. designed a 3-DOF flexure-based parallel mechanism for

micro/nano manipulation and its corresponding kinematic

and dynamic modeling were analyzed by matrix method. Ex-

perimental results verified the two modeling for this parallel

mechanism [15]. Yong et al. proposed a flexure-based XY

stage which can scan over a relatively large range with high

This work was supported in part by the National Natural ScienceFoundation of China (Grant No. 61128008), Macao Science and Technolo-gy Development Fund (Grant No. 016/2008/A1), and Research Commit-tee of University of Macau (Grant no. MYRG203(Y1-L4)-FST11-LYM,MYRG183(Y1-L3)FST11-LYM).

1Q. Meng, J. Xu and Y. Li are with the Department of ElectromechanicalEngineering, Faculty of Science and Technology, University of Macau, Av.Padre Tomas Pereira, Taipa, Macao SAR, China

2Y. Li is also with School of Mechanical Engineering, Tianjin Universityof Technology, Tianjin 300384, China.

∗Corresponding author. E-mail: ymli@umac.mo

scanning speed for fast nanoscale positioning [17]. Choi et

al. designed a large displacement precision XY positioning

stage by using cross strip flexure joints. This mechanism is

over-constrained to incorporate symmetry in order to cancel

out the effects of center shifting when flexures were in

large deformation [21]. Liaw et al. developed a flexure-based

four-bar mechanism driven by piezo-actuator. The dynamic

equations for modeling of the mechanism were established

in this work[19], [23].

With the development of amplification mechanisms, com-

pliant amplifiers are becoming the focus in high-precision

positioning applications. There are three basic approaches

to design and analyze compliant amplification mechanisms.

These three approaches are matrix method, finite element

analysis (FEA) and topology design. The stiffness character-

istics responded by the load-deflection relationship become

evaluation criteria in analysis and synthesis of compliant

mechanisms [5], [6]. In 2003, Lobontiu et al. proposed an

analytical model for optimal design of a class of flexure-

based displacement amplifiers. Displacement and stiffness

equations were formulated based on the strain energy and

Castigliano’s displacement theorem. It is noted that the

displacement equations were derived by combining compli-

ance matrix and strain energy. This result obtained by this

approach is more complex [7]. Li et al. presented a total-

ly decoupled flexure-based XY parallel micromanipulator

which is consisted of four displacement amplifiers. Matrix

method was applied in this work in order to evaluate the

compliance of the proposed model and verified by means of

finite element analysis [8] [9]. Li et al. also designed a lever

displacement amplifier for a flexure-based dual-mode motion

stage by means of matrix method [24]. Tian et al. (2009)

proposed a flexure-based five-bar mechanism for micro/nano

manipulation which used lever mechanisms to increase the

working space in Cartesian. The kinematic modeling of the

proposed mechanism was established by means of matrix

method, then finite element analysis and experimental tests

verified the analytical results [10], [26]. They also presented

a methodology for modeling a high-precision flexure-based

mechanism driven by piezoelectric actuator [11]. Thereafter,

they modeled a flexure-based Scott-Russell mechanism for

nano-manipulation based on finite element analysis. This

Scott-Russell mechanism was monolithically constructed to

provide high rotational positioning accuracy and long-term

repeatability [18]. Zhang et al. presented a flexure parallel

micromanipulator based on multi-level displacement ampli-

fier in order to overcome the drawback of small stroke

happened in conventional flexure parallel micromanipulators

Proceedings of the 13thIEEE International Conference on NanotechnologyBeijing, China, August 5-8, 2013

978-1-4799-0676-5/13/$31.00 ©2013 IEEE 460

[14]. Concerning to FEA, Iula et al. designed and analyzed an

ultrasonic actuator based on a flexural mechanical amplifier

by means of finite analysis method in 2005 [12]. Based

on a symmetric five-bar topology, Ouyang et al. proposed

a compliant mechanical amplifier which can achieve large

amplification [13], [16]. Corner-filleted flexure hinges were

utilized in this work to be the connectors. Lever principle

was used to calculate the amplification ratio. Malosio et al.

proposed a design method for a multi-stage piezo stroke

amplifier which can be utilized as the actuated module by

means of topological approach for kinematic analysis of

rigid mechanism. Finite element analysis was employed to

verify their results [22]. Acob et al. optimized a compliant

mechanical amplifier with a symmetric five-bar topology

based on its design parameters together with goals of large

amplification ratio and high natural frequency [25]. Meng

et al. proposed a novel flexure-based proportion compliant

mechanism which can be used in high accuracy positioning

[28]. This work presented the proportion ratio equation and

closed-form stiffness equations for the proposed mechanism

based on the Catigliano’s displacement theorem.

To sum up, the amplification ratio for these existing flexure-

based compliant amplifiers or proportion mechanisms can be

calculated in two ways. One is the amplification including

stiffness matrix proposed by Lobontiu et al. [7]. This method

is more complex for obtaining the ratio but accurate. The

other one depends on the pseudo-rigid-body model (PRBM)

of the amplifier. This method is easier than the last one but

inaccurate. The reason is that the rotational center shifting

is ignored in PRBM. The target of this work is to correct

the proportion ratio equation presented by Meng et al. [28]

in order to optimize the design of a proportion mechanism.

Based on our previous experiences on optimal design of

compliant mechanism [29]-[33], this paper will optimize the

proposed proportion mechanism. The approach to correct the

proportion ratio equation can also be used in other flexure-

based compliant amplifiers.

The remaining of this paper is organized as follows.

PRBM in cantilever beam is recalled simply in Section II.

Section III introduces the approach to analyze a flexure hinge

by using the theory of PRBM for cantilever beams with

loads at the free end. A new corrected the proportion ratio

calculation is presented and empirical corrected coefficients

are proposed by using the new approach in Section IV. In

Section V, a compliant proportion mechanism is designed

by using the proposed coefficients. Finite element analysis

is employed to verify the results. Finally, the conclusions are

drawn in Section VI.

II. PRBM FOR CANTILEVER BEAM

Howell proposed PRBM to analyze the complex character-

istics of flexures and flexure-based compliant mechanisms.

Rigid link mechanism theory may be used in the model to

analyze compliant mechanisms [4].

Concerning a cantilever beam with a load at the end, Howell

has given the analysis details for some typical loading

situations, the rotational center point(or the rotational angle)

can be corrected. Therefore, PRBM for cantilever beam is

recalled simply here in order to implement the next step

work. Fig. 1 and Fig. 2 show a cantilever beam with loads

at the free-end and its PRB model.

We can figure out that the key parameter to the rotational

�0

F

b

a

P

nP

EI

Undeflection position

l

Path followed by beam end M

Fig. 1. A cantilever beam with loads at the free-end

Θ0

F

b

P

nP

Ma

l

l

Fig. 2. Its PRB model

center point is the characteristic radius, γ . However, picking

the value of γ relates with the loading approach at the free-

end. Three situations relating with three loading approaches

are presented as follows.

A. Cantilever Beam with a Force at the Free End

Considering only one force is applied at the free-end, the

vertical component of the force is P; the axial force is nP

(where the positive parameter n represents a force that will

produce compression in the un-deflected beam); therefore, F

can be written as:

F = P√

n2 +1 (1)

According to the work presented by Howell, the characteris-

tic radius factor γ can be decided thanks to the value of n. If

461

n is known, the characteristic radius factor γ can be chosen

in the literature [4].

Let us see the Figs. 1 and 2, the PRB rotational angle Θ

is not equaled to the real rotational angle θ0. The empirical

constant coefficient (or parametric angle coefficient), cθ ,

were found out in order to correct the PRB rotational angle.

While the real rotational angle can be written as:

θ0 = cθ Θ (2)

Where Θ can be written as:

Θ = atanb

1− l(1− γ)(3)

B. Cantilever Beam with a Pure Moment at the Free-End

Taking account into the cantilever beam with a pure

moment applied at the free-end as shown in Fig. 1. A method

similar to that used for the cantilever beam with a force at

the free-end can be used in finding a PRB model for this

situation. The results have already been proposed in [4] and

the characteristic radius factor γ is equal to 0.7346; and the

parametric angle coefficient is equal to 1.5164.

C. Cantilever Beam with Both a Moment and a Force at the

Free End

There are three loading cases were discussed by Howell:

case I is that force and moment applied in the same direc-

tion; case II is that force and moment applied in opposite

directions but no inflection point; and case III is that force

and moment applied in opposite direction with an inflection

point. This paper will analyze the conditions in detail in the

next section.

III. PRBM FOR FLEXURE HINGES

Consider a small-length flexural pivot (or it can be con-

sidered as a corner-filleted flexure hinge without filleted

corner.), the rotational center point is supposed at the middle

point of the small-length flexural pivot as shown in Fig. 3.

According to our knowledge, however, the rotational center

point is shifting with the deflection of the hinge. The errors

induced by the rotational center point for calculating the

proportion ratio can be eliminated by correcting the position

of rotation center point. It is the key point that finding the

coefficient is the PRB rotation (γP = 0.5) and the real value.

In the case of a cantilever beam with loads at the free-end,

however, the value of the characteristic radius factor, γ , is

more than 0.5 whatever the loaded approach is. While the

real rotation center point is approximate between the two

cases. Fig. 4 shows the distribution of the positions of the

three rotation center points. Θ is the PRB angle obtained

according to a cantilever beam with loads at the free-end; θ0

is the real rotation angle; and ΘP is the RBM angle obtained

according to pivot case, where the subscript P denotes pivot.

According Eqs. 2 and 3, the PRBM can be written as:

ΘP = cPθ0 (4)

where cp can be described as:

cP =cθ atan b

a−l(1−γ)

atan ba−l/2

(5)

ll/2

Fig. 3. A corner-filleted flexure hinge

ΘP

Θ

Path followed by beam end

(1-γ)l

l/2

b

a

FP

nP

EI

Undeflected position

θ0

θ0

M

Fig. 4. The distribution of center point of PRBM, real and flexure hinge

It is easy to figure out that the coefficient cP is relative to

the parameter angle coefficient (cθ ) and the characteristic

radius ratio(γ) which can be obtained by following the

descriptions aforementioned.

IV. DISPLACEMENT PROPORTION RATIO

EQUATION FOR FLEXURE-BASED PROPORTION

COMPLIANT MECHANISMS

A four crank-slider flexure-based proportion compliant

mechanism’s PRB model is recalled in this paper and shown

in Fig. 5. ri(i = 0,1,2,3, ...,9) is the length of the ith link.

While r5 is the initial horizontal distance between Point

A and Point C. r0 is the initial vertical distance between

Point A and Point B. Two links are connected by means of

a corner-filleted flexure hinge and its stiffness is described

as Ki(i = 1,2,3, ...,10). As shown in Fig. 5, half of the

mechanism is studied in this paper due to the mechanism’s

symmetric structure. Half of the model can be divided into

two crank-slide mechanisms. One is the above part consisted

of Link 1, Link 4 and slide B, the other one in the bottom

part consisted of Link 2, Link 3 and slider C.

The vertical input displacement of the model as shown in

Fig. 5 can be written as:

462

i

o

ɵ1 ɵ2

e1

r1

r2 r3

r4

e2

r0

r5

ɵ4ɵ3

A

B

C

i

ɵ5

ɵ6

ɵ7

ɵ8

C’

r6r7 r8

r9

Fig. 5. PRBM of a flexure-based proportion compliant mechanism

i = r0 − r1 sinθ1 − r4 sinθ4 (6)

The horizontal output displacement of the model is:

o = r5 − r2 cosθ2 − r3 cosθ3 (7)

Therefore, the displacement proportion ratio can be ex-

pressed simply as:

a =o

i=

r5 − r2 cosθ2 − r3 cosθ3

r0 − r1 sinθ1 − r4 sinθ4(8)

As what mentioned in Section II and Section III, the

force analysis for each corner-filleted flexure hinge is shown

as follows in order to obtain the choice conditions for the

corrected coefficient cP.

θ40

F

nP

P

l5 γl

5

(1-γ)l5

Fig. 6. The pseudo-rigid-body model of Hinge5

θ40

F

nP

P

l2 γl

2

(1-γ)l2

M2

Fig. 7. The pseudo-rigid-body model of Hinge2

1) Hinge 5: Fig. 6 shows the pseudo-rigid-body model

of Hinge5 with only forces at the free-end. Specifically, the

vertical component of the force is P; the axial force is nP ;

the total force is F .

Where the case of a cantilever beam with a force applied at

the free-end is used here to obtain the corrected coefficient.

The key parameter to define cθ and γ is n. From the force

analysis, we can obtain:

P = Fcosθ40 (9)

nP = F sinθ40 (10)

where θ40 is the initial angle of θ4. Therefore, the n can be

written as:

n =1

cotθ40(11)

2) Hinge 2: The static analysis of Hinge 2, as illustrated

in Fig. 7, is more complex than that of Hinge 5, where the

pseudo-rigid-body model combines end force and moment

at the free-end. And the force and moment cause deflection

in the same direction. The moment M0 creates a curvature in

the beam that is continuous throughout the entire beam. The

key parameter to define cP is the curvature, κ0 (described

in [4]).

From Fig. 5, we can obtain the moment applied at the end

of the hinge:

M2 = Fr4 cosθ40 (12)

The nondimensionalized parameter κ0 is:

κ2 =M2l2

EI2=

Fr4 cosθ40l4

EI2(13)

where E is the Young’s Modulus, I2 is the moment of inertia

about the vertical axis of Hinge 2, l2 is the length of Hinge

2.

463

3) Hinge 3: Fig. 8 shows the pseudo-rigid-body model

of Hinge 3 with combined end force and moment. However,

the force and moment load in opposite directions as shown

in Fig. 8. We use the method of inversion, which can make

the system to model but dose not change the relative motion

between the end. From Fig. 5, we obtain the moment applied

F

l3

γl3

(1-γ)l3

M3

2̟-θ30

nP

P

Fig. 8. The pseudo-rigid-body model of Hinge3

to Hinge 3:

M3 = F (e1 − r2 cosθ20) (14)

The nondimensionalized parameter κ0 is:

κ3 =M3l3

EI3=

F (e1 − r2 cosθ20) l3

EI3(15)

By using the similar approach aforementioned, we can

obtain the nondimensionalized parameter κ4 and κ1 for

Hinge 4 and Hinge 1. The values for κi(i = 1,2,3,4,5) are

listed in Table I.

TABLE I

THE VALUES FOR κi , AND Mi (i = 1,2,3,4) FOR DIFFERENT JOINTS

No. Mi κi

Joint1 F (r1 cosθ10 + r4 cosθ40)F(r1 cosθ10+r4 cosθ40)l1

EI1

Joint2 Fr4 cosθ40Fr4 cosθ40l2

EI2

Joint3 F (e1 − r2 cosθ2)F(e1−r2 cosθ20)l3

EI3

Joint4 F (r5 − r1 cosθ10 − r4 cosθ40)F(r5−r1 cosθ10−r4 cosθ40)l4

EI4

Such this, we can define the corrected coefficient cP by

choosing the parameters γ and cθ according to the curvature

κ0 [4].

Therefore, θ1 can be obtained as:

θ1c = θ10 +∆θ1

/

cP1 (16)

where the subscript c denotes the equation has been corrected

by the correcting coefficient cn.

Moreover, cP2 and ∆θ2 can be obtained by

cP1 = cP2 (17)

θ2c = θ20 +∆θ1 (18)

In consideration of the relationships (17) and (18), θ2 can

be written as follows:

θ2c = θ20 +∆θ1

/

cP1 (19)

and θ3 can be expressed as:

θ3c = θ30 +2π − arcsin

[

r2 sin(θ20+∆θ1/cP1)±e2

r3

]

−θ30

cP3(20)

where e2 is the vertical distance between point A and point

C, if point C is lower than the point A, the symbol is (20),

is positive +; if point C is above point A, the symbol in (20)

is negative -.

θ4 can be obtained by following the above procedure and

written as:

θ4c = θ40 +arccos

[

e1−r1 cos(θ10+∆θ1/cP1)r4

]

−θ40

cP4(21)

According to Eqs. 16),(19), (20), ( 8) can be corrected as:

a =o

i=

r5 − r2 cosθ2c − r3 cosθ3c

r0 − r1 sinθ1c − r4 sinθ4c

(22)

V. CASE STUDY: A FLEXURE-BASED

PROPORTION COMPLIANT MECHANISM DESIGN

Based on the proportion ratio equation presented above, a

new optimal design of a flexure-based proportion compliant

mechanism as shown in Fig. 5 was done in this work by

following the similar optimization procedure as aforemen-

tioned research work. The desired amplification ratio was

set as 5 in the paper. Therefore, the design variables, {X}={r1 r2 r3 r4 θ10 θ20 e1 e2} , can be optimized by

means of the commercial software MATLAB and the results

are shown in Table II. Five corner-filleted flexure hinges were

TABLE II

OPTIMIZED GEOMETRIC PARAMETERS

r1(mm) 11.1261 K1(Nmm) 45.851

r2(mm) 18.0224 K2(Nmm) 46.731

r3(mm) 14.5479 K3(Nmm) 47.096

r4(mm) 14.3358 K4(Nmm) 44.640

θ10(∘) 69.1044 K5(Nmm) 44.448

θ20(∘) 43.9974

e1(mm) 14.7144

e2(mm) 1.1815

designed by following [3] together with the stiffness shown

in Table II. Telfon (Young’s Moudulus E = 345MPa and

Yield strength Sy = 23MPa) is chosen as the material for the

designed mechanism. Based on these obtained parameters, a

flexure-based proportion compliant mechanism was modeled

by SOLIDWORKS, then the FEA model was analyzed by

the commercial FEA software COMSOL. Plane stress and

large deformation were set on for the finite element analysis

in order to verify the proposed equation in this paper can

also be used in nonlinear case. Refined meshing technique

was adopted here in order to insure the correctness of the

results. Finally, the data of input and output displacements

were read and their relationship is shown in Fig. 9.

From Fig. 9, we can observe that the FEA results using

the optimal designed flexure-based proportion compliant

mechanism are very close to the desired results. The error

induced by design is within 5.8%.

464

0 0.02 0.04 0.06 0.08 0.10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Input−displacement (mm)

Outp

ut−

dis

pla

cem

ent (m

m)

FEA Results

Desired Results

Fig. 9. Input-Output displacement

VI. CONCLUSIONS

This paper proposes a new approach to calculate the pro-

portion ratio for flexure-based proportion compliant mecha-

nisms. The method of PRBM in cantilever beam with loads

at the free-end is firstly utilized to analyze the position

of rotation center point for flexure hinges. Proportion ratio

equation for a four crank-slider proportion compliant mech-

anism is developed in this work based on the new analysis

method. At the end, a flexure-based proportion compliant

mechanism is designed by the proposed equation and verified

via finite element analysis. As a result, the error between the

proportion ratio obtained by the optimized mechanism and

the desired demand is no more than 5.8%.

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