SCIENCE CHINA Technological Sciences

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*Corresponding author (email: zhangxm@scut.edu.cn)

• Article • March 2014 Vol.57 No.3: 560–567

doi: 10.1007/s11431-013-5446-4

Design of single-axis flexure hinges using continuum topology optimization method

ZHU BenLiang1, ZHANG XianMin1* & FATIKOW Sergej1,2

1 Key laboratory of Precision Equipment and Manufacturing Technology of Guangdong Province, South China University of Technology, Guangzhou 510640, China;

2 Division Microrobotics Department of Computing Science, Uhlhornsweg 84, A1, 26111 Oldenburg, Germany

Received May 6, 2013; accepted December 7, 2013

The design of compliant hinges has been extensively studied in the size and shape level in the literature. This paper presents a method for designing the single-axis flexure hinges in the topology level. Two kinds of hinges, that is, the translational hinge and the revolute hinge, are studied. The basic optimization models are developed for topology optimization of the translational hinge and the revolute hinge, respectively. The objective for topology optimization of flexure hinges is to maximize the com-pliance in the desired direction meanwhile minimizing the compliances in the other directions. The constraints for accom-plishing the translational and revolute requirements are developed. The popular Solid Isotropic Material with Penalization method is used to find the optimal flexure hinge topology within a given design domain. Numerical results are performed to illustrate the validity of the proposed method.

topology optimization, translational hinge, SIMP, compliant mechanisms, revolute hinge

Citation: Zhu B L, Zhang X M, Fatikow S. Design of single-axis flexure hinges using continuum topology optimization method. Sci China Tech Sci, 2014, 57: 560567, doi: 10.1007/s11431-013-5446-4

1 Introduction

Compliant mechanisms offer several advantages compared with rigid body mechanisms, for example, fewer assembly process requirements, backlash elimination, and no need lubrication [1]. Therefore, compliant mechanisms are largely used in a variety of high precision applications such as 3-degrees of freedom (DOFs) precision positioning stage [2] and nano-handling devices [3]. Two principal approach-es, that is, the kinematic-based approach [1] and the struc-tural topology optimization-based approach [4–6], for the design of compliant mechanisms have been developed.

A large number of compliant mechanisms are construct-ed based on employment of flexure hinges [1,7], especially

synthesized by using the kinematic-based approach. Flexure hinges with specific profiles such as circular, corner-filleted, and elliptical shapes [8] are widely employed. Unlike rigid mechanical connections that allow different kinematic DOFs, a flexure hinge is a member that is able to provide the relative motion between two rigid parts through elastic deformations [7].

Ever since Paros and Weisbord [9] developed a thorough analytical presentation of conventional right circular flexure hinge, substantial research has been spent on deriving the theoretical equations of the flexure hinges with other shapes [10–20]. The design of new types of compliant hinges with a large displacement [21] is also an important direction be-cause of the limitation of range of motion of conventional compliant hinges.

In particular, Smith et al. [10] presented the compliance equation of elliptical flexure hinge and checked the model

Zhu B L, et al. Sci China Tech Sci March (2014) Vol.57 No.3 561

prediction using the finite element analysis and experi-mental data. Stress concentration factors for the elliptical hinge also are presented based on the finite element analysis results. Lobontiu et al. [11] presented the closed-form solu-tions of compliance of the elliptical hinge based on the Castigliano’s first theorem. Comparisons are also made between the elliptical flexure hinge and the right circular flexure hinge on bending compliant, stresses, and precise rotation. Lobontiu et al. developed compliance closed-form equations for conic-section (circular, elliptic, parabolic, and hyperbolic [12,13]) flexure hinges. The performance of the proposed conic-section flexure hinges is compared based on a non-dimensional analysis. Stiffness characterization of cornerfilleted flexure hinges has been developed theoreti-cally by Lobontiu et al. [14]. Tseytlin [15] presented effec-tive compliance equations of monolithic flexure hinges with different cross sections based on a conformal mapping the-ory. The accuracy therefore is improved compared with other theoretical equations. Awtar et al. [16] presented a general dimensionless analytical framework of beam-based flexure mechanisms and applied it to analyze several beam-based flexure modules.

A more reliable direct finite element analysis method has been developed by Smith [17]. Based on the finite element analysis, Schotborgh et al. [18] presented dimensionless design graphs for the circular, cross and beam hinges which can be used for determining the optimal geometry of the hinges. Bona and Munteanu [19] developed a method for an optimal design of flexure hinges by coupling a parametric finite element model. The method is approached for opti-mization of flexure hinges in the shape level, that is, the topology of the hinges can not be optimized. Later, Zelenika [20] presented a method for nonlinear parametric optimiza-tion of flexural hinge shapes based on the finite analysis. Based on the comparison results of performances with re-spect to hinges with conventional circular notches, it is con-cluded that optimized hinge shapes lead to better perfor-mances.

The aforementioned design or optimization procedures are developed based on either a known topology (e.g., in ref. [20]) or a known shape (e.g., in ref. [11]). Therefore, the design procedures depend on the designer’s experiences and inspirations to a large extent. For conceptually designing the flexible joints, screw theory-based methods have been de-veloped [22–24]. However, rigorous but abstruse mathe-matical knowledge is required which makes the methods difficult to extend.

This paper is devoted to present an alternative method for the design of flexure hinges in the topology level by using the continuum topology optimization method. The optimi-zation models for two specific flexure hinges, that is, a translational hinge and a revolute hinge, are developed. Numerical results are generated as well as different aspects of the proposed formulations. Conclusions are put forward finally.

The remainder of this paper is organized as follows: The goal of this work is introduced in Section 2. Section 3 pro-vides the basic concept of the solid isotropic material with penalization (SIMP) method-based topology optimization. The optimization models for two kinds of flexure hinges are developed as well in Section 3. The sensitivity analysis is proposed. Numerical results for topology optimization of two kinds of flexure hinges are provided in Section 4. Sec-tion 5 presents conclusions and future work.

2 Motivation

Once the topology of the flexure hinge is selected, the shape (geometric profile) of the hinge becomes the key factor in determining the performance of the flexure hinge. Two types of flexure hinges are considered in this paper: the translational hinge and the revolute hinge.

Translational and revolute hinges are two widely flexure hinges that are used in compliant mechanisms to provide translational and rotational DOFs. Translational hinges have been designed as two primitive types: leaf spring type (Fig-ure 1(a)) and notch type (Figure 1(b)). Both of them are analogous to a four-bar linkage. With a special topology, two well-studied and widely used revolute hinges with spe-cific geometric profiles are shown in Figure 2.

The flexure hinges are generally modeled and analyzed as small-displacement beams. However, even with this sim-plified problem, there are two basic questions that need to

Figure 1 Basic translational hinges with the same topology. (a) Leaf spring type and (b) notch type. In the figure, Γd is the Dirichlet (or dis-placement) boundary condition. When the hinge is loaded at point o with a force along the y-direction, a displacement will be produced along y-direction [1].

Figure 2 Basic revolute hinges with the same topology. (a) Circular flexure hinge and (b) constant rectangular cross-section flexure hinge. In the figure, Γd is the Dirichlet (or displacement) boundary condition. When the hinge is loaded at point o with a force along the y-direction, an angular displacement will be produced [12,13].

562 Zhu B L, et al. Sci China Tech Sci March (2014) Vol.57 No.3

be answered. (1) Is the pre-set topology of the flexure hinge the best? (2) How to find the best topology of the flexure hinge, that is, how to optimize a flexure hinge in the topol-ogy level?

The aim of this paper is to focus on topology optimiza-tion of flexure hinges. Two kinds of flexure hinges are con-sidered: the translational hinge and the revolute hinge. The following work is proposed based on several assumptions.

(1) The flexure hinges are designed to be used in two dimensional monolithic compliant mechanisms.

(2) Optimization models are formulated to characterize only in-plane motions.

(3) The flexure hinges are subjected to bending by force, while the shearing effects are not considered. The axis loading is considered to generate a well-posed design.

(4) The design domains of two proposed kinds of flexure hinges are symmetric with respect to the longitudinal axis. The design domains are considered to be fixed at one end and free at the other. The design domains of two proposed kinds of flexure hinges are both square.

3 Optimization problem setting

3.1 Topology optimization

The purpose of the topology optimization is to find the best layout of a structure. In many applications, the topology optimization method is used for the concept design in which the best layout is determined. During the process of topolo-gy optimization, it is capable for a slot to be removed or generated. The unique feature of topology optimization method is that optimal solution can be generated automati-cally even though the topology of the mechanism is not known in advance.

Topology optimization can be performed based on truss [25] or continuum discretization. Several methods have been developed for the topology optimization based on con-tinuum discretization [26–29]. Among them, the most pop-ular numerical topology optimization method is the SIMP method, and it has reached the stage of application in indus-trial software [30].

The proposed method in this paper is based on the con-cept of optimal determination of the isotropic material in a given design domain. The use of the SIMP method for to-pology optimization of flexure hinges allows us to convert the optimal topology problem into a sizing problem. It re-lates the Young’s modulus E to the density as

0( ) ,pE Eρ ρ (1)

where is the density vector consisting of the density of each finite element e which is treated as the optimization variable; E0 is the stiffness for the given isotropic material; and p is a free parameter to penalize intermediate densities and lead approximately to a 0−1 design. Usually, p3 is

required.

3.2 Optimization model

This section is devoted to develop the basic optimization models for the topology design of the translational hinge and the revolute hinge, respectively. A general design do-main for topology optimization of flexure hinges can be illustrated as in Figure 3 in which y denotes the desired di-rection and x denotes the undesired direction. D is the de-sign domain and R is the rigid domain. The design domain D is fixed at the boundary Γd.

An ideal flexure hinge should intend to be as compliant as possible in the direction DOFs while being as stiff as possible in other directions [31]. Therefore, ud due to Fd needs to be maximized meanwhile uud due to Fud needs to be minimized. An artificial spring ks is added to avoid the nu-merical instabilities. Besides this, other criterions, for ex-ample, axis drift, must be established as well for specific applications [7]. These performance criterions can be used for formulating the topology optimization model to ensure a well-posed design.

Recently developed topology optimization methods are strongly related to the finite element analysis [32] and the accuracy of the finite element analysis model is significant-ly influenced by the way that boundary conditions are as-signed on the design domain [33]. Hence, the width l of the rigid area R, is set to be at least equal to or greater than the width L of the design domain D.

Since the accuracy of the finite element analysis model is significantly influenced by the way the boundary conditions were assigned on the model [33], forces are applied at a point where at least l3L, as is shown in Figure 4. In this

Figure 3 The design domain of flexure hinge.

Figure 4 The design domain of the flexure hinge considered in this paper.

Zhu B L, et al. Sci China Tech Sci March (2014) Vol.57 No.3 563

paper, l=3L is used.

3.3 Translational hinge

For the designing of translational hinge, the goal is to find a constraint arrangement that provides approximate straight line motion. The popular hinge that can accomplish this function is the parallelogram flexure [16]. Here we try to develop the general optimization model for topology opti-mization of the translational hinge. For simplicity, Fx and Fy are set to be unity.

As shown in Figure 5(a), for designing a translational hinge, the compliance Cy due to a load Fy needs to be max-imized and the compliance Cx due to a load Fx needs to be minimized. Cx and Cy can be stated as follows:

d ( ) ( ) ,x ijkl ij x kl xD

C E e u e u (2)

d ( ) ( ) ,y ijkl ij y kl yD

C E e u e u (3)

where ux and uy are displacement fields due to Fx and Fy, respectively. Eijkl and ε are the stiffness and linearized strain tensors, respectively.

To accomplish the translational requirement, when a force is applied at point A along y-direction, the rigid part R should move along y-direction as shown in Figure 5(a). Therefore, the constraint on the translational requirement can be written as

21

2Tobj( ) ,y y

y y

A B

F Fu u T (4)

where y

y

A

Fu denotes the displacement of point A along

Figure 5 Fictitious deformations of the translational (a) and the revolute (b) hinges under Fy.

y-direction due to the load Fy and y

y

B

Fu denotes the dis-

placement of point B along y direction due to the load Fy. 0T

obj is a small constant.

In conclusion, the optimization model QT for topology optimization translational hinge can be defined as fol-lows:

Minimize

Subject to obj

T Tobj

: : ,

: ,

,

y xQ C C

V V

T

(5)

where V denotes the material usage and is constrained with upper limit Vobj.

3.4 Revolute hinge

For the design of revolute hinge, the main requirement is to fulfill a finite rotation around z-axis (perpendicular to the design domain).

The design condition is shown in Figure 5(b). The de-sired rotation along z-axis should be maximized. The com-pliance Cx due to load Fx (undesired) should be minimized. Similarly to the design of translational hinge, the compli-ance Cy due to a load Fy needs to be maximized. Therefore, a possible objective function for topology design of revolute hinge can be stated as

Minimize : ,C C y x (6)

Figure 5(b) shows a fictitious deformation of the design domain due to Fy. As the horizontal axis is symmetry axis, only half of the design domain is taken into consideration.

Generally, any hinges, which make u y

y

A

F not equal to

u y

y

B

F ( u y

y

A

F and u y

y

B

F are displacements at points A and B

due to the load Fy, respectively, as shown in Figure 5(b)), can be called revolute hinges. However, for real world ap-plications, an ideal revolute hinge can make the center of rotation fixed at point O [7].

To fulfill the rotation requirement, the should be as small as possible, which means the line determined by points A and B should cross the geometrical center of the flexure hinge, that is, point O (the motion of rigid area R is assumed to be the rigid motion). Therefore, for a small rota-tion angle, the following condition should be satisfied:

2

,u AO L l

LBOu

y

y

y

y

A

F

B

F

(7)

Let

2

1 2

2R .

u L l

Lu

y

y

y

y

A

F

B

F

(8)

564 Zhu B L, et al. Sci China Tech Sci March (2014) Vol.57 No.3

For topology optimization of revolute hinge, R needs to be as small as possible. In conclusion, the optimization model QR for topology optimization revolute hinge can be defined as follows:

Minimize

Subject to

R

obj

R Robj

: : ,

: ,

,

Q C C

V V

y x

(9)

where V denotes the material usage, and is constrained with upper limit Vobj. 0R

obj is a small constant.

3.5 Sensitivity analysis

Here, we use QT as an example. Incorporating with the SIMP method, the optimization problem QT can be rewrit-ten by using the FEM form as follows:

T1 2

1

Minimize

Subject to

0 1 1 2

( , ,... )

obj

T Tobj

min

: ( ) ( ) ( ),

: ,

( ) ,

,

,

, , , ..., ,

Ny x

N

x x

y y

e

J C C

v V

Ku = F

Ku = F

e N

e e

(10)

where N is the total number of finite elements and Fx and Fy are the load vectors. 0 001min . is introduced to avoid

singularity of the stiffness matrix. K is the stiffness matrix in the global sense which can be written as

1

,N

pe e sK K K (11)

where Ke is the (global level) element stiffness matrix and Ks is the stiffness matrix of the output spring ks in global level.

The SIMP approach leads to a sizing problem and the number of design variables is typically very large. This op-timization problem can be efficiently solved by the optimal-ity criteria when there is only one constraint [4]. For solving the topology optimization problem with multiple constraints, the method of moving asymptotes (MMA) [34] which be-longs to the class of sequential convex programming [35] might be the most computationally efficient approach. Since the optimization problem stated as in QT has two constraints in addition to a simple constraint giving upper and lower limits on the density variable, MMA is employed to solve the proposed optimization problem.

The sensitivity of the objective function can be written as

1 1T T .J

p u K u p u K u

p p

e y e y e x e xe

(12)

Since the material usage constraint involves only , the sensitivity can be easily obtained and stated as follows:

1 2, , , ... ,V

v e N

e

e

(13)

The sensitivity of T can be written as

T

( ) ,y y

y yy y

y y

A B

F FA B

F Fe e

u uu u

e

(14)

where

1 T ,y

y

A

F pe A e y

e

up u K u

(15)

1- T ,y

y

B

F pe B e y

e

up u K u

(16)

where uA and uB are displacement vectors obtained by the load vectors HA and HB, respectively. HA and HB are unit load vectors consisting of zero except for positions A and B (Figure 4) with an entry of HA=1 and HB=1, respectively.

The above-proposed sensitivities can be directly extend-ed for QR. To avoid reduplication, the details are no longer performed.

4 Numerical results

In this section, the numerical results for topology optimiza-tion of two kinds of flexure hinges using the proposed method are presented. For the following numerical cases, the artificial material properties are described as Young’s modulus for solid material is E=1 and Poisson’s ratio is 0 3 . . The power law mixing rule with p=3 is used in the computations.

4.1 Translational hinge design

In all following cases, four-node is oparametric full integra-tion elements with bilinear shape functions are used and the applied forces are assumed to be unity. Note that, for clearly plotting the optimal topologies, the whole rigid area R is not plotted even though different l/L might be used.

First, the ratio of l/L is set to be 3. The mesh size of D is set to be 100×100 and the mesh size of R is 100×300. The maximum material usage Vobj is set to be 0.2.

Figure 6 shows the obtained optimal configuration. Note that a typical four-bar linkage is obtained. The convergence histories of Cx and Cy are shown in Figure 7. The conver-gence history of T is shown in Figure 8. In this case, the

Tobj is set to be 0.1 and the resulting T is 0.0906. Note that

30 iterations are used for the optimization process. T

Zhu B L, et al. Sci China Tech Sci March (2014) Vol.57 No.3 565

Figure 6 The final design of the translational hinge with l/L=3.

Figure 7 The convergence histories of Cx and Cy with l/L=3.

Figure 8 The convergence history of T with l/L=3.

reached very large in the second iteration since no reasona-ble topology is obtained then.

Second, we examine when l/L is set to be small, for ex-ample, 0.1. The corresponding final design is shown in Fig-ure 9. Note that the optimal flexure hinge is not a simple four-bar linkage. However, if the area in the red circle is treated as a whole, the obtained translational hinge becomes a typical four-bar linkage. The reason for generating the to-pology is that the l/L is small and this makes the stiffness of the rigid domain R small. Therefore, a part of material is added on the rigid domain R to make it more stiffer. When the hinge is loaded, the R can therefore be moved without torsional deformation.

Figure 9 The final design of the translational hinge with l/L=0.1.

In case of l is small, the problem is actually considered as a compliant mechanism design problem. To categorize the entire design domain as one hinge, l generally must be large. Based on the results shown in Figures 6 and 9, one can con-firm that there is no topological change compared with the standard hinges as shown in Figure 1, which confirms that the existing designs are already optimal in the topology lev-el.

4.2 Revolute hinge design

4.2.1 Effect of the output spring ks

The relationship between the optimal configuration and the spring ks is examined in this section. The other conditions are fixed as follows: the mesh sizes of D and R are set to be 100×100 and 100×300, respectively.

The corresponding final designs of the four studied cases are shown in Figure 10. Note that the same topology is ob-tained, although the shape of the final design is slightly dif-ferent. It is shown that holes are generated inside the hinge no matter that ks is set to be large or small. This may be be-cause the design problem is modeled and analyzed as a small-displacement problem, that is, the linear finite element

Figure 10 The final designs of the revolute hinge with different ks, while other conditions are fixed. (a) ks=0.02; (b) ks=0.04; (c) ks=0.08; and (d) ks=0.1.

566 Zhu B L, et al. Sci China Tech Sci March (2014) Vol.57 No.3

analysis is used. In future, we hope to further investigate the design problem by using the nonlinear finite element analy-sis method.

The maximum material usage Vobj is set to be 0.3. Robj is

set to be 0.01. Four cases are studied in this section, which are ks=0.02, ks=0.04, ks=0.08 and ks=0.1.

The topologies that are shown in Figure 10 confirm that there should be a certain number of holes inside the layout of the revolute hinge. It also confirms that the topology of the hinges that is shown in Figure 2 is not the only choice and the best one.

The convergence histories of Cx and Cy are shown in Fig-ure 11. The convergence history of R is shown in Figure 12. Forty-two iterations are used for the optimization process. In case of ks=0.04, the resulting R is 0.00674.

4.2.2 Comparison with the constant rectangular cross- section flexure hinge

The resulting revolute hinges in this study are more like the constant rectangular cross-section flexure hinge. Next, the differences between these two kinds of revolute hinges, that is, the constant rectangular cross-section flexure hinge (Fig-ure 2(a)) and the revolute hinge obtained by using the

Figure 11 The convergence histories of Cx and Cy of the revolute hinge with ks=0.04.

Figure 12 The convergence history of R with ks=0.04.

proposed method (Figure 10(b)) are examined by using ANSYS quantitatively.

The main material used in this section is nylon (E=3 GPa; =0.4). The design domain D of the revolute hinge obtained by using the proposed method is set to be 10 mm×10 mm×1 mm. Since the maximum material usage Vobj is set to be 0.3, the size of the constant rectangular cross-section flexure is set to be 3 mm×10 mm×1 mm. The same input load Fy=1 mN is applied for the two studied hinges.

The displacements of points A and B along y-direction, R, and the maximum von-Mises stress of the two hinges are shown in Table 1, where traditional means the traditional constant rectangular cross-section flexure hinge and To-poDesign means the revolute hinge obtained in this study. θ is the angle due to the input force (as shown in figure 5(b)).

One can see that with the same maximum material usage, the obtained revolute hinge is more compliant than the tradi-tional one. The revolute hinge obtained in this study can undergo large rotor angle subjected to the same input load.

By using eq. (8) and the displacements of A and B, R of the two studied hinges can be directly obtained as shown in Table 1. Note that with the obtained revolute hinge, R is 0.0087, which means that the line determined by A and B can nearly cross point O, which confirms that the obtained revo-lute hinge can perform better precision of rotation than the traditional one.

The obtained revolute hinge in this study may suffer the stress concentration problem. The sharp corners have stress singularities and the maximum von-Mises stresses are very large (similar to the traditional constant rectangular cross-section flexure hinge). One possible way to handle this issue is to perform a shape optimization after the layout of the flexure hinge is determined.

We have explored the comparison study between the ob-tained revolute hinge in this study and the constant rectan-gular cross-section flexure hinge. Although the results per-formed here are preliminary, the design of flexure hinges using a topology optimization technique itself is a worthy research endeavor.

5 Conclusion

A method for topology optimization of single-axis flexure hinges based on continuum-type topology optimization techniques is proposed. The basic optimization models for topology optimization of translational hinge and revolute hinge are developed, respectively. Numerical results are

Table 1 Displacements (mm) along y-direction and von-Mises stresses (GPa) of the studied two revolute hinges due to the same input load

B A θ R Max stress

Traditional 0.2261 1.6700 2.7555 0.0750 0.0340

TopoDesign 0.2469 1.7610 2.9083 0.0087 0.0362

Zhu B L, et al. Sci China Tech Sci March (2014) Vol.57 No.3 567

performed to illustrate the validity of the proposed method. Conclusions can be summarized as follows.

1) The numerical results confirm that for designing the translational hinge, the existing designs are already optimal in the topology level.

2) For designing the revolute hinges subjected to a small displacement, a certain number of holes should be intro-duced into the layout of the revolute hinge.

3) It is important to design a flexure hinge starting from the topology level. After the layout of the flexure hinges is determined, the shape or sizing designs can be used to de-termine the detailed geometry of the flexure hinges.

Flexure hinges subjected to large deflections should be further investigated. Another issue that should be further concerned is stress concentration. How to model the stress concentration as a constraint will be investigated in future.

This research was supported by the National Natural Science Foundation of China (Grant No. 91223201), the Natural Science Foundation of Guangdong Province (Grant No. S2013030013355), Project GDUPS (2010), and the Fundamental Research Funds for the Central Universities (Grant No. 2012ZP0004).

1 Howell L L. Compliant Mechanisms. New York: John Wiley & Sons, 2001

2 Wang H, Zhang X. Input coupling analysis and optimal design of a 3-dof compliant micro-positioning stage. Mech Mach Theroy, 2008, 43: 400–410

3 Fatikow S. Automated Nanohandling by Microrobots. 1st ed. London: Springer, 2007

4 Bendsøe M P, Sigmund O. Topology Optimization: Theory, Methods and Applications. Berlin: Springer, 2003

5 Zhu B L, Zhang X M. A new level set method for topology optimiza-tion of distributed compliant mechanisms. Int J Numer Meth Eng, 2012, 91: 843–871

6 Zhu B L, Zhang X M, Wang N F. Topology optimization of hinge-free compliant mechanisms with multiple outputs using level set method. Struct Multidisc Optim, 2013, 47: 659–672

7 Lobontiu N. Compliant mechanisms: Design of flexure hinges. CRC Press, 2003

8 Tian Y, Shirinzadeh B, Zhang D, et al. Three flexure hinges for com-pliant mechanism designs based on dimensionless graph analysis. Precis Eng, 2010, 34: 92–100

9 Paros J M, Weisbord L. How to design flexure hinges. Mach Des, 1965, 37: 151–156

10 Smith S T, Badami V G, Dale J S, et al. Elliptical flexure hinges. Rev Sci Instrum, 1997, 68: 1474–1483

11 Lobontiu N, Paine J S N, Garcia E, et al. Corner-filleted flexure hinges. J Mech Design, 2001, 123: 346–352

12 Lobontiu N, Paine J S, Garcia E, et al. Design of symmetric con-ic-section flexure hinges based on closed-form compliance equations. Mech Mach Theory, 2002, 37: 477–498

13 Lobontiu N, Paine J S, O’Malley E, et al. Parabolic and hyperbolic

flexure hinges: Flexibility, motion precision and stress characteriza-tion based on compliance closed-form equations. Precis Eng, 2002, 26: 183–192

14 Lobontiu N, Garcia E, Hardau M, et al. Stiffness characterization of corner-filleted flexure hinges. Rev Sci Instrum, 2002, 75: 4896–4905

15 Tseytlin Y M. Notch flexure hinges: An effective theory. Rev Sci In-strum, 2002, 73: 3363–3368

16 Awtar S, Slocum A H, Sevincer E. Characteristics of beam-based flexure modules. J Mech Design, 2007, 129: 625–639

17 Smith S T. Flexures: Elements of Elastic Mechanisms. New York: Gordon and Breach, 2000

18 Schotborgh W O, Kokkeler F G, Tragter H, et al. Dimensionless de-sign graphs for flexure elements and a comparison between three flexure elements. Precis Eng, 2005, 29: 41–47

19 Bona F D, Munteanu M G. Optimized flexural hinges for compliant micromechanisms. Analog Integr Circ S, 2005, 44: 163–174

20 Zelenika S, Munteanu M G, Bona F D. Optimized flexural hinge shapes for microsystems and high-precision applications. Mech Mach Theory, 2009, 44: 1826–1839

21 Trease B P, Moon Y M, Kota S. Design of large-displacement com-pliant joints. J Mech Design, 2005, 127: 788–798

22 Yu J J, Li S Z, Su H J, et al. Screw theory based methodology for the deterministic type synthesis of flexure mechanisms. J Mech Robot, 2011, 3: 03100801–03100813

23 Su H J, Denis V, Dorozhkin et al. A screw theory approach for the conceptual design of flexible joints for compliant mechanisms. J Mech Robot, 2009, 1: 04100901–04100907

24 Hopkins J B, Culpepper M L. Synthesis of multi-degree of freedom, parallel flexure system concepts via freedom and constraint topology (FACT)-part I: Principles. Precis Eng, 2010, 34: 259–270

25 Frecker M I, Ananthasuresh G K, Nishiwaki S, et al. Topological synthesis of compliant mechanisms using multi-criteria optimization. J Mech Design, 1997, 119: 238–245

26 Sigmund O. On the desgn of compliant mechanisms using topology optimization. Mech Based Struct Mach, 1997, 25: 493–524

27 Nishiwaki S, Min S, Yoo J, et al. Optimal structural design consider-ing flexibility. Comput Methods Appl Mech Eng, 2001, 190: 4457–4504

28 Wang M, Wang X M, Guo D M. A level set method for structural topology optimization. Comput Methods Appl Mech Eng, 2003, 192: 227–246

29 Allaire G, Jouve F, Toader A M. Structural optimization using sensi-tivity analysis and a level set method. J Comput Phys, 2004, 194: 363–393

30 Rozvany G I N. A critical review of established methods of structural topology optimization. Struct Multidisc Optim, 2009, 37: 217–237

31 Yu J J, Li S Z, Su H J, et al. Screw theory based methodology for the deterministic type synthesis of flexure mechanisms. J Mech Robot, 2011, 3: 031008

32 Choi K K, Kim N H. Structural sensitivity analysis and optimization 1: Linear system. New York: Springer-Verlag, 2005

33 Yong Y K, Lu T F, Handley D C. Review of circular flexure hinge design equations and derivation of empirical formulations. Precis Eng, 2008, 32: 63–70

34 Svanberg K. The method of moving asymptotes-a new method for structural optimization. Int J Numer Methods Eng, 1987, 24: 359– 373

35 Barthelemy J F M, Haftka R T. Approximation concepts for optimum structural design–a review. Struct Multidisc Optim, 1993, 5: 129–144