Analytical Modeling and Application for Semi-Circular Notch Flexure Hinges

Abstract

Flexure-based compliant mechanisms can be used to achieve bio-imitability and adaptability in the applications of biomedical engineering. However, a nonlinear load-displacement profile increases the design complexity of this type of compliant mechanism, especially when the cross-section of the flexure hinge is not constant. This paper proposes two general analytical models by analyzing the compliance and stress characteristics of the semi-circular notch flexure hinge undergoing large deflections, which is a typical variable cross-section of a flexure hinge, based on the Castigliano’s second theorem and the finite elements analysis method. As a case study for verification, three compliant four-bar linkage mechanisms are designed based on the proposed design approach, the design method proposed by Howell, and the equations proposed by Lobontiu, respectively. The results show that the design accuracy is improved 36% in comparison with designs from Howell and Lobontiu. Finally, a flexure-based artificial finger is designed and manufactured based on the proposed optimization approach. The performance test of the prototype shows that the artificial finger has good bio-imitability and adaptability with respect to joint movements.

1. Introduction

The flexure-based compliant mechanism (FCM) is a mechanism that uses flexure hinges to transmit motions and forces. Due to their advantages over conventional rigid mechanisms, such as minimal backlash, reduced weight, simplicity in construction, smooth and continuous displacement, and motion precision, FCMs are being employed more and more frequently in a variety of industries [1,2]. As the most important component of FCMs, the flexure hinge is a flexible connector that can provide a limited rotational motion between two rigid parts via deformation [3]. Nowadays, flexure hinges have been widely utilized in robots, biomedical engineering, and many other fields [4,5,6,7].

To our knowledge, the shape of the flexure hinge can guide its motion. There are a lot of FCMs designed by using the specified shape flexure hinges. Some researchers have proposed a variety of notch flexure hinges with different shapes. For instance, slender beam, corner-filleted, circular, elliptical, and V-shape [8,9,10,11,12]. These types of flexure hinges can implement rotation motion with the inner center due to their constant and symmetrical sections. However, such kinds of shapes in the cross-section cannot lead to an off-rotational center. The semi-circular notch flexure hinge, as shown in Figure 1, is a typically shaped flexure hinge that can guide a circle motion with a remote rotation center. It is widely utilized in exoskeletons, prosthetics, and so on [13,14].

There are two kinds of cross-section for the flexure hinges. One is the constant cross-section, and the other is the variable cross-section. The constant cross-section can be a special case occurring in the variable cross-sections. There is a lot of research on the compliance characteristics and the stress characteristics of the semi-circular flexure hinges within the constant cross-section. For instance, Howell et al. analyzed the stiffness characteristics of an initially curved beam based on the pseudo-rigid-body model (PRBM) [15]. The initially curved beam could be regarded as a semi-circular flexure hinge in a macro point. According to this viewpoint, some researchers built some multi-joint PRBMs of the initially circular hinges under different load conditions [16,17,18]. In addition to the PRBM analytical approach, there is much research on analytical stiffness models directly based on the Euler–Bernoulli beam theory. Ahuett-Garza et al. proposed a stiffness analytical model for the semi-circular hinge based on the Euler–Bernoulli beam theory with large deflection [19]. Additionally, Chen et al. proposed a method for the analysis of the initially curved hinge based on the elliptic integration method [20]. The beam was separated into a few circular elements and several methods for the discretization were presented. Unfortunately, the assumptions of the Euler–Bernoulli beam theory can limit the precise design of the semi-circular flexure hinges and the hinges-based compliant mechanisms. The nonlinear characteristics should be considered in the computation of the stiffness or the stress even though its cross-section is constant, for example the thickness–height ratio is not in the limited range. Currently, flexure hinges with non-constant cross-sections are extensively employed in the field of bio-medical engineering, such as in compliant artificial hands (Figure 1). The advantages of this type of flexure hinge lie in its ease of design and manufacturing. However, the aforementioned research findings are not applicable for analyzing the stiffness/compliance characteristics of semi-circular flexure hinges with constant or variable cross-sections.

Furthermore, there are other mathematical approaches to analyzing the properties of flexure hinges. Topology optimization is also an approach that is utilized in the analysis and design of compliant mechanisms and flexure hinges. Zhu et al. surveyed the continuum topology optimization method from three levels [21,22]. Cao et al. proposed a topology optimization framework in the design of the compliant mechanism composed of flexure hinges and beams. A super flexure hinge element was proposed in the study and the stiffness matrix was derived with consideration of the stress concentration effects [23,24]. However, the topology method is usually utilized in the design of a mechanism with an objective function, such as the requirement of compliance in the desired direction of the flexure hinges.

In addition, Castigliano’s second theorem is mainly utilized in the displacement calculation of an elastic body under strain energy. The theorem has been widely utilized in the stiffness calculation of the variable cross-section flexure hinge as it takes the effects of section structure and geometric dimension on stiffness into consideration. Lobontiu proposed several closed-form compliance equations based on Castigliano’s second theorem [3]. Of course, the accuracy of the proposed closed-form compliance equations is limited under the small deflections. With the development of finite element analysis tools, the finite element analysis approach is utilized to amplify the application of Castigliano’s second theorem. Xu et al. proposed the characteristics of stiffness and stress of the elliptical and corner-filleted flexure hinges through the finite element method. The results show the accuracy of their proposed equations [25]. Similarly, Smith et al. developed closed-form equations for symmetric single-axis semi-elliptical flexure hinges based on finite element analysis and experimental tests [26]. Qiu et al. proposed a multi-cavity flexure hinge with great compliance through the three-dimensional continuum topology optimization, the compliance equations of the hinge were obtained based on Castigliano’s theorem [27]. Yang et al. verified the stiffness of a variable thickness flexure pivot based on the FEA results [28]. Meng et al. presented several new empirical stiffness equations for corner-filleted flexure hinges based on FEA results and the accuracy of the equations under large deformation was greatly improved [29]. It shows that the FEA approach could be utilized together with Castigliano’s second theorem to analyze the characteristics of stiffness and stress of the flexure hinges with large deflection. Therefore, this paper analyzes the compliance characteristics and stress characteristics of the variable cross-section semi-circular notch flexure hinge under large deformation based on Castigliano’s second theorem and the FEA approach. The proposed design equations in this paper can also be analyzed in bionic properties.

The remainder of this paper is organized as follows: The model of the semi-circular flexure hinge is introduced, and the compliance and stress equations are presented in Section 2. Then, the finite element analysis is conducted, and the general design equations for the semi-circular notch flexure hinge are derived in Section 3. In Section 4, an improved optimization approach for the compliant mechanism is presented and a case study of compliant four-bar linkage mechanisms is conducted to verify the accuracy of the equations. In addition, as an application of the flexure hinge in the biomedical engineering field, a model of the flexure-based artificial finger is designed and evaluated in Section 5. Finally, the conclusion and discussion are shown in Section 6.

2. Compliance Modeling under Small Deformation

The structure model of the variable cross-section semi-circular notch flexure hinge is shown in Figure 2, while the radii of the two concentric circles are R and r, respectively. As is shown in Figure 1, L, w, and t denote the length, width, and minimum thickness of the hinge, respectively, and h represents the height of the rigid body. In general, the ratio of t/L is defined as the dimensionless design parameter to describe the geometry characteristics of the hinge. As a special variable cross-section flexure hinge, the inner circle of the hinge is always tangent to the two rigid bodies under the different value of t/L, and it can be deduced that L = 2r. Two fillets are designed at both sides of the hinge to reduce stress concentration. The fillet is tangent to the outer circle and rigid body with a radius of a. The origin of the coordinate system is the midpoint of the line that connects the two tangent points of the fillets and the inner circle. For the human finger, the joints are driven by the muscle and the moment is generated at each joint. Therefore, one end of the flexure hinge is fixed, and a moment is given at point ‘1’ of the other free end. Compared with many other semi-circular notch flexure hinges with a constant cross-section, both sides of the hinge proposed in this paper are thicker than the middle part. This kind of flexure hinge also shows good motion accuracy and better energy storage characteristics because of its geometrical characteristics.

As is shown in Figure 2, the thickness $t\left(x\right)$ of the variable cross-section semi-circular notch flexure hinge can be expressed as Equation (1):

$$t\left(x\right)=\left\{\begin{array}{l}\sqrt{(R+a{)}^2-(r-a{)}^2}-\sqrt{a^2-(x-a{)}^2}-\sqrt{r^2-(x-r{)}^2}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em} x\in \left(0,r-\frac{R\left(r - a\right)}{R + a}\right)\\ \sqrt{R^2-(x-r{)}^2}-\sqrt{r^2-(x-r{)}^2}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em} x\in \left(r-\frac{R\left(r - a\right)}{R + a},r+\frac{R\left(r - a\right)}{R + a}\right)\\ \sqrt{(R+a{)}^2-(r-a{)}^2}-\sqrt{a^2-(x+a-2r{)}^2}-\sqrt{r^2-(x-r{)}^2}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em} x\in \left(r+\frac{R\left(r - a\right)}{R + a},2r\right)\end{array}\right.$$

(1)

According to Castigliano’s second theorem, the relationship between load and the deformation of the hinge can be expressed as:

$$\left[\begin{array}{c}{\delta }_x\\ {\delta }_y\\ {\theta }_z\end{array}\right]=\left[\begin{array}{ccc}{C}_{1,x-F_x}& 0& 0\\ 0& C_{1,y-F_y}& C_{1,y-M_z}\\ 0& C_{1,{\theta }_z-F_y}& C_{1,{\theta }_z-M_{{}_z}}\end{array}\right]\left[\begin{array}{c}{F}_x\\ F_y\\ M_z\end{array}\right]$$

(2)

As the finger joint is mainly driven by the moment, for the moment $M_z$ applied at the point ‘1’ while the $F_x$ and $F_y$ are equal to zero, the Equation (2) can be simplified as:

$${\theta }_z=C_{1,{\theta }_z-M_z}{M}_z$$

(3)

The ${\theta }_Z$ is the rotation angle of the hinge. The theoretical compliance $C_{1,{\theta }_z-M_z}^T$ of the flexure hinge under small deformation can be calculated by Castigliano’s second theorem:

$$C_{1,{\theta }_z-M_z}^T=\frac{12}{Ew}{I}_1$$

(4)

where E is Young’s modulus of the material. Substitute Equation (1) and Equation (4) to Equation (3) and the ${\theta }_Z$ can be expressed as:

$${\theta }_z=C_{1,{\theta }_z-M_z}^T{M}_z=\frac{12}{Ew}{\displaystyle {\int }_0^L\frac{dx}{t{(x)}^3}}{M}_z$$

(5)

A given model is built to verify the accuracy of the compliance matrix based on Castigliano’s second theorem under large deformation, and the results obtained from Equation (5) were compared with the finite element analysis (FEA) results. The different rotation angles obtained from the FEA results and Equation (5) under the same moment are shown in Figure 3.

More specifically, the relative error between the FEA results ${\theta }^F$ and theoretical rotation angles ${\theta }^T$ are presented in Figure 4, while the error,

$$\mathrm{E}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r}=\left[\frac{{\theta }^F-{\theta }^T}{{\theta }^F}\right]\times 100\%$$

(6)

For different ratios of t/L (t/L = 0.10 and t/L = 0.25), the maximum errors reach over 35% and 25% because of the characteristics of the variable cross-section. Therefore, the compliance equations based on Castigliano’s second theorem cannot be utilized in the precise calculation under large deformation with the characteristics of the variable cross-section. The equations will be modified, and the general design equations will be deduced in Section 3.

3. General Analysis Modeling

3.1. FEA Modeling of Flexure Hinge and Modified Coefficient

FEA is an important approach to verify the nonlinear deformation and stress of a flexure hinge undergoing large deflection. The accuracy of these FEA models was verified by Lobontiu and Meng et. al., and was with a maximum 8% error compared to three experimental results [4,29]. This was completed in order to decrease the errors influenced by the materials and the geometrical parameters t/L and correct the theoretical compliance equations given by Equation (4). Eleven semi-circular beam flexure hinge models with different ratios of t/L are built in ANSYS 19.0 to obtain the reference principle compliance $C_{1,{\theta }_z-M_z}^F$ and the principle stress ${\mathsf{\sigma }}^{\mathrm{F}}$ of the hinges, the ratios of t/L are 0.025, 0.05, 0.1, 0.15, …, 0.5, respectively. It is noted that the model with the ratio t/L of 0.025 is chosen to improve the accuracy of the modified equations. In addition, the radius of the fillet is chosen as $a=0.5\hspace{0.33em}{t}_{\min }$ to decrease the concentration of stress. The geometry of one of these analysis models is shown in Figure 5. The analytical models had a depth of 10 mm. Triangle element type is more suitable to model irregular shapes and is chosen in this paper to generate the model mesh. The meshing technique was refined five times and used to produce automatically refined meshing at parts that had high stress concentrations. The size of the mesh is only approximately 1/50 of the thickness. In this way, the analysis of flexure hinges is always assumed to be like a cantilever beam. One end of the flexure hinge is fixed and the moment $M_z$ is applied at Point $P_1$ through the ANSYS 19.0, the triangles method and sizing of the mesh are utilized and set. In addition, the condition of the FEA is set as large deformations and the non-linear characteristic. In the meantime, the FEA investigation should follow the procedure described in the following to ensure the coincidence: The flexure hinges have the same deformation ${\theta }_z{}^F$ under the moment $M_z$. Therefore, the moment $M_z$ will increase with the ratio of t/L. Thus, the considered FEA simulations have been performed in ANSYS Workbench. In total, 220 data points are utilized to obtain modified equations.

The rotation angle ${\theta }_z{}^F$ can be obtained as described in Figure 5. The corresponding theoretical compliance $C_{1,{\theta }_z-M_z}^T$ under the same load condition can be calculated through the Equations (4) and (5). The modified coefficient $\mu$ is introduced in this paper while the $\mu =\frac{C_{1,{\theta }_z-M_z}^F}{C_{1,{\theta }_z-M_z}^T}$ is used to describe the error between the theoretical compliance and FEA compliance quantitatively. In all these cases, the resulting difference for the parameter $\mu$ is found to be always lower than 0.5 and 0.3%, respectively, allowing one to assume a negligible dependency on the material parameters.

The compliance and deformation are usually determined by the load, material properties (Young Modulus E), rotation angles, and geometrical parameters (t/L) [29]. Therefore, the modified coefficient $\mu$ and relative error may be determined by the Young Modulus E, rotation angles, and geometrical parameters (t/L).

In addition, the relative errors between the $C_{1,\theta z-Mz}^T$ and $C_{1,\theta z-Mz}^F$ with different parameters are obtained under the same FEA setting by following Equation (7):

$$\mathrm{E}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r}=\left[\frac{C_{1,\theta z-Mz}^F-C_{1,\theta z-Mz}^T}{C_{1,\theta z-Mz}^F}\right]\times 100\%$$

(7)

While the Young Modulus of ‘Material A’ is 345 MPa, and the Young Modulus of ‘Material B’ is 110 GA, the ratios of the hinge are chosen as 0.05, 0.1, and 0.25. In the results, the relative errors of the hinges of the same ratio of t/L with different materials are 0.09%, 0.25%, and 0.34%, respectively. In addition, with the increase of the rotation angle, the errors under the same material keep steady while the change ranges are approximately 0.16%, 0.36%, and 0.95%. Therefore, Young’s modulus is not a major influence parameter in the compliance characteristics.

On the contrary, the errors mainly depend on the geometrical parameters t/L. Therefore, the value of the modified coefficient $\mu$ can be simplified as the function of the ratio of t/L, the relationship between the ratio of t/L and modified coefficient $\mu$ will be discussed in the next part of the paper.

3.2. Modified Compliant Model

The modified compliance $C_{1,{\theta }_z-M_z}$ can be expressed in Equation (8), while the modified coefficient $\mu =\mu (t/L)$.

$$C_{1,{\theta }_z-M_z}=C_{1,{\theta }_z-M_z}^F=\mu C_{1,{\theta }_z-M_z}^T$$

(8)

Based on the output data obtained from the finite element simulation, the MATLAB curve fitting toolbox is utilized in the fitting of the coefficient for each single flexure hinge. The numerical curve is fitted by assuming $\mu =f \left(\frac{t}{L}\right)$ in the hypothesis. The fitting curve is chosen here via ordinary least squares techniques [30] in order to improve the calculation accuracy and get an accurate modified coefficient. The modified coefficient is shown in Equation (9) and the corresponding coefficients are shown in

Table 1. Coefficient for Equations (9) and (10).

Coefficients	Value
a0	63.91
a1	−85.78
a2	43.27
a3	−9.73
a4	2.205

. The maximum fitting error of the equations is 1.1499%. Substitute Equation (9) into Equation (8) and the modified compliance $C_{1,{\theta }_z-M_z}$ can be expressed as:

$$\mu =\sum _{i=0}^4{a}_i{\left(\raisebox{1ex}{$t$}\!\left/ \!\raisebox{-1ex}{$L$}\right.\right)}^i$$

(9)

$$C_{1,{\theta }_z-M_z}=\sum _{i=0}^4{a}_i{\left(\raisebox{1ex}{$t$}\!\left/ \!\raisebox{-1ex}{$L$}\right.\right)}^i{C}_{1,{\theta }_z-M_z}^T$$

(10)

3.3. Stress Characteristics Modeling

The stress and stress characteristics of the flexure hinge are the important characteristics in the design and practical application of the hinge. For the semi-circular flexure hinge, the strain is small considering the length and the structure. Therefore, the stress characteristics of the hinge are mainly analyzed. The stress of the flexure hinge under different rotation angles can be obtained from the FEA results. The relationship between the rotation angles and the corresponding stress for the flexure hinge with different ratios of t and L is presented in Equation (11):

$$\frac{\sigma }{E}=\sum _{i=0}^4{b}_{ij}(\raisebox{1ex}{$t$}\!\left/ \!\raisebox{-1ex}{$L$}\right.{)}^i{\theta }^j\hspace{0.33em}(i+j\le 4)$$

(11)

In particular, for a given maximum rotation angle ${\theta }_{\max }$, the $\sigma$ can be expressed as the ${\sigma }_s$ while the stress $\sigma$ is the yield strength ${\sigma }_s$ of the material. Equation (11) can be expressed as in Equation (12) which can be utilized in the selection of material in the design of flexure hinge for the given value of t/L and maximum rotation angle:

$$\frac{{\sigma }_s}{E}=\sum _{i=0}^4{b}_{ij}(\raisebox{1ex}{$t$}\!\left/ \!\raisebox{-1ex}{$L$}\right.{)}^i{\theta }^j\hspace{0.33em}\hspace{0.33em}(i+j\le 4)$$

(12)

The corresponding coefficients for Equations (11) and (12) are shown in

Table 2. Coefficient for Equations (11) and (12).

Coefficients	Value	Coefficients	Value	Coefficients	Value
$b_{00}$	0.0007568	$b_{02}$	0.008669	$b_{40}$	0.3867
$b_{10}$	−0.02399	$b_{30}$	−0.4565	$b_{31}$	0.6169
$b_{01}$	−0.01059	$b_{21}$	−0.6463	$b_{22}$	0.4631
$b_{20}$	0.1761	$b_{12}$	−0.2095	$b_{13}$	0.02927
$b_{11}$	0.8788	$b_{03}$	−0.0005916	$b_{04}$	−0.0003766

.

3.4. Optimization Design Approach of Compliant Mechanism with the Semi-Circular Notch Flexure Hinges

Based on the general design equations proposed in the Section 3.2, we present an enhanced optimization approach for compliant mechanisms with the semi-circular notch flexure hinges. This approach can be employed to design and select the actuator and input load for the given design conditions of the compliant mechanisms. The step-by-step design procedure is as follows:

Step 1:

Specify the geometric properties of the mechanism and desired motion, including maximum rotation angle;

Step 2:

Provide information about the material or value of t/L;

Step 3:

Optimize to determine either the value of t/L or the material of the hinge using Equations (5), (10), (11), and (12);

Step 4:

Calculate geometry dimensions of flexure hinge using the general design equations;

Step 5:

Obtain input moment through proposed compliance equation and virtual work principle.

4. Case Study: Compliant Four-Bar Linkage Mechanism

The compliant four-bar linkage mechanism is the most common compliant mechanism which can meet the accuracy requirements of motion in the general case. A compliant four-bar linkage mechanism based on the semi-circular notch flexure hinge is built to verify the accuracy of the general design equations in this part of the paper. In addition, the comparison of the given moment and the moment in the finite element analysis under the same desired rotation angle are conducted. The PRB model of the mechanism is shown in Figure 6. The revolute is simplified as the joint with a torsion spring.

Since the accuracy of the hinge is related to the motion direction for the hinge of transverse symmetry, the semi-circular flexure hinge can be applied in the unidirectional rotation. When a moment is loaded on the Rocker $r_2$ as described in Figure 5, the direction of rotation for $r_2$ will be clockwise. The flexure hinges 1 and 4 will also rotate in this same direction, while the flexure hinges 2 and 3 will rotate counterclockwise. Therefore, the model of the compliant mechanism based on the semi-circular notch flexure hinge is shown in Figure 7. The mechanism is composed of the rockers $r_2$ and $r_4$, coupler $r_3$, base $r_1$ and four semi-circular notch flexure hinges 1 to 4. A pure moment M is applied at the rocker $r_2$ and the corresponding angle of $r_2$ is ${\gamma }_2$ while the initial angle is ${\gamma }_{20}$.

As is listed in

Table 3. Parameter description of four-bar linkage mechanism.

Parameters Description	Symbol	Length (mm)
Base	$r_1$	400
Rocker	$r_2$	160
Coupler	$r_3$	240
Rocker	$r_4$	240

, the lengths of base $r_1$, rocker $r_2$, coupler $r_3$, rocker $r_4$, and the width of the flexure hinge are given in advance. For the given desired rotation angles ${\theta }_d$ of $r_2$, the corresponding rotation angles ${\theta }_2$, ${\theta }_3$, and ${\theta }_4$ of hinges 2 to 4 can be obtained from the closed-form equations of the compliant four-bar linkage mechanism while the initial angle ${\gamma }_{20}$ is 65 deg. For the maximum rotation angles ${\theta }_{\max }$ of ${\theta }_2$ to ${\theta }_4$, the ratio t/L can be figured out through the stress design equation in Equation (12) while the Teflon (E = 345 MPa, ${\sigma }_s$ = 23 MPa) is chosen as the material of the mechanism. The remaining dimensions of the mechanism are shown in

Table 4. Parameters of four-bar linkage mechanism designed by the general equations.

Dimension	Length $\mathit{L}$ (mm)	$\mathbf{Minimum} \mathbf{Thickness} {\mathit{t}}_{\mathbf{min}}$ (mm)	$\mathbf{Angle} {\mathit{\theta }}_{\mathbf{max}}$ (rad)	Width $\mathit{w}$ (mm)
Hinge#1	10	3.19	0.30	6
Hinge #2	10	2.07	0.47	6
Hinge #3	10	3.45	0.28	6
Hinge #4	10	3.57	0.11	6

considering the lengths of $r_1$, $r_4$, and the compliant equation.

For the given flexure hinges, the corresponding compliance $C_1$ to $C_4$ can be calculated by Equation (10) with the width of 6 mm. The theoretical moment $M_l^m$ can be figured out utilizing the principle of virtual work with the compliance and rotation angles. Furthermore, the finite element analysis with same desired rotation angles ${\theta }_d$ of hinge 2 of the mechanism is conducted and the moment $M_f^m$ obtained from the FEA is compared with the $M_l^m$. The error between the $M_f^m$ and $M_l^m$ can be utilized to measure the accuracy of the proposed design equations while:

$$\mathrm{E}\mathrm{r}\mathrm{r}\mathrm{o}{\mathrm{r}}_{\mathrm{m}}=\left[\right(M_l^m-M_f^m)/M_f^m]\times 100\%$$

(13)

As mentioned in Section 1, Howell et al. analyzed the characteristics of the initially curved beam through the method of the PRB model. In this paper, two compliant four-bar linkage mechanisms are built on the basic of the stress and compliance equations proposed by Howell and Lobontiu. The dimensions obtained from the two methods are shown in

Table 5. Parameter description of Hinges for the desired four-bar linkage mechanism.

Dimension	Length $\mathit{L}$ (mm)	$\mathbf{Minimum} \mathbf{Thickness} {\mathit{t}}_{\mathbf{min}}$ (mm)	$\mathbf{Angle} {\mathit{\theta }}_{\mathbf{max}}$ (rad)	Width $\mathit{w}$ (mm)
Hinge #1	10	3.32	0.30	6
Hinge #2	10	2.21	0.47	6
Hinge #3	10	3.56	0.28	6
Hinge #4	10	4.14	0.11	6

and

Table 6. Parameters of four-bar linkage mechanism designed by Lobontiu’s equations.

Dimension	Length $\mathit{L}$ (mm)	$\mathrm{Minimum} \mathrm{Thickness} {\mathit{t}}_{\min }$ (mm)	$\mathbf{Angle} {\mathit{\theta }}_{\mathbf{max}}$ (rad)	Width $\mathit{w}$ (mm)
Hinge #1	10	3.57	0.30	6
Hinge #2	10	2.38	0.47	6
Hinge #3	10	3.15	0.28	6
Hinge #4	10	4.66	0.11	6

, respectively.

$M_l^H$ and $M_l^L$ denote the theoretical moment obtained from Howell and Lobontiu’s equations, while $M_F^H$ and $M_F^L$ denote the corresponding FEA output results. The theoretical moment and the FEA moment are plotted in Figure 8. The stress distribution figure of the model obtained from the equations proposed by Howell is shown in Figure 9 while the maximum stress is 23.291 MPa. The corresponding stress of the three models is shown in Figure 10. The errors $\mathrm{E}\mathrm{r}\mathrm{r}\mathrm{o}{\mathrm{r}}_{\mathrm{H}}$ and $\mathrm{E}\mathrm{r}\mathrm{r}\mathrm{o}{\mathrm{r}}_{\mathrm{L}}$ between $M_F^{}$ and $M_L^{}$ of the two mechanisms are compared with the $\mathrm{E}\mathrm{r}\mathrm{r}\mathrm{o}{\mathrm{r}}_{\mathrm{m}}$. These errors are printed in the Figure 11.

As is shown in Figure 11, for the nonlinear characteristics of hinge and the axis drift under large deformation, the maximum error which happened in Lobontiu’s equations reached 41.29%. The maximum error that happened in the Howell’s equations is 25.42%. The emergence of large errors can be attributed to underlying assumptions. Howell et al. conducted an analysis on the stiffness characteristics of numerous flexure hinges and compliant mechanisms using the PRB model method, which assumes small deformation without considering non-linear deformation. However, in the case of semi-circular flexure hinges investigated in this study, even very small deformations, cannot neglect non-linear deformation. Additionally, Lobontiu analyzed the characteristics of variable cross-section flexure hinges based on Castigliano’s second theorem, which takes into account the influence of section structure and geometric dimensions on stiffness and proposes several equations. Nevertheless, it should be noted that these compliance equations have limited accuracy when applied to small deformations. Additionally, it can be figured out that with the increase of the rotation angle, the equations are more accurate. With the increase of the angle, the deformation is more concentrated, the cross-section of the deformed area is closer to the constant section, and the hinge is closer to the hinge with a constant cross-section.

Compared to Howell and Lobontiu’s equations [3,29], the errors of the general design equations keep steady and the equation is more accurate while the maximum error of the equation is 5.27%. Therefore, the results show that the accuracy of the equation can be improved nearly 36% and equations can be utilized in the accurate design of semi-circular flexure hinge under large deformation.

5. Example of Application: Flexure-Based Artificial Finger

5.1. Model and Characterization of Artificial Finger

The semi-circular notch flexure hinge can act as a joint that drives the finger to stretch. To illustrate the practicability of the semi-circular notch flexure hinge and general design equations, a finger of the artificial hand based on the flexure hinge is designed through the proposed optimization approach and equations. The PRB model of the finger is shown in Figure 12. The ${\theta }_1$, ${\theta }_2$, and ${\theta }_3$ in Figure 12 represent the rotation angles of metacarpophalangeal point (MCP) joint, proximal interphalangeal point (PIP) joint, and distal interphalangeal joint (DIP) joint, respectively.

Three semi-circular notch flexure hinges are utilized in the model acting as the three joints of the finger. Specifically, in this study, we determined the overall length and height of the artificial hand by considering the dimensions and grasping capabilities of the Chinese human hand. Additionally, we established the length of each hinge based on both its height and maximum rotation angel. The two-dimensional diagram is shown in Figure 13. The metacarpal is fixed, the geometric parameters of the finger and the semi-circular flexure hinge are determined by the physiological structure of the human finger. The rotation angle of the finger is given in advance. The parameters of the finger are shown in

Table 7. Parameters description of artificial finger.

Parameters Description	Symbol	Value	Parameters Description	Symbol	Value
Metacarpal length	$L_1$	30 mm	Flexure hinge #1 minimum thickness	$t_1$	1 mm
Proximal phalanx length	$L_2$	20 mm	Flexure hinge #2 minimum thickness	$t_2$	0.9 mm
Middle phalanx length	$L_3$	20 mm	Flexure hinge #3 minimum thickness	$t_3$	0.8 mm
Distal phalanx length	$L_4$	13 mm	Flexure hinge #1 maximum rotation angle	${\theta }_{1\max }$	0.7 rad
Flexure hinge #1 length	$L_5$	20 mm	Flexure hinge #2 maximum rotation angle	${\theta }_{2\max }$	0.8 rad
Flexure hinge #2 length	$L_6$	20 mm	Flexure hinge #3 maximum rotation angle	${\theta }_{3\max }$	0.9 rad
Flexure hinge #3 length	$L_7$	20 mm	Height of phalanges	$H$	10 mm

. The minimum thicknesses of the three flexure hinges are determined so that their rotation angles are consistent with those of the human fingers. In addition, the force is applied to the finger in order to bend the finger.

5.2. Prototype and evaluation

For the given maximum rotation angles and $\frac{\mathrm{t}}{\mathrm{L}}$ of the flexure hinge 1 to flexure hinge 3 listed in Table 7, the corresponding ratio of yield strength and the Young modulus can be figured out through Equation (10). The enhanced polylactic acid (PLA), a material designed for 3D printing (E = 2336 MPa, ${\mathsf{\sigma }}_s$ = 58 MPa, isotropic), is utilized in the development of the prototype. The prototype of the finger developed by 3D printing technology is shown in Figure 14.

Furthermore, an experiment is conducted and measuring equipment is utilized to evaluate the maximum rotation angle of the prototype. As is shown in Figure 14, the finger is actuated by the cable and four groups of circular marks are put on the rigid connectors of the prototype to measure the rotation angles of every hinge [30]. The measuring equipment can measure the coordinates of these marks in real time and the corresponding angles of each joint can be obtained. The measurement results are shown in Figure 15. It should be noted that the loads on each flexure hinge do not solely consist of moments. The cable is fixed at the artificial fingertip and can generate a moment at point 1 as shown in Figure 5. However, there is also an accompanying force exerted on this point. Consequently, these combined loading conditions may lead to increased motion errors in certain cases. Henceforth, it would be beneficial to investigate the entire stiffness characteristics of the semi-circular flexure hinge.

Figure 15 shows that with the increase of the force, the rotation angles reach can over a given angle. The results of the evaluation experiment in the section proved that the flexure hinge designed by the proposed design equations can be utilized in the design of the biomedical engineering field. The motion of the designed artificial finger is consistent with the motion of the human finger which illustrated the possible use of the hinge.

6. Conclusions

The present paper proposes an enhanced optimization design methodology for a compliant mechanism featuring semi-circular notch flexure hinges, aiming to enhance its bio-imitability and adaptability in the application of compliant mechanisms. We introduce a variable cross-section semi-circular notch flexure hinge that can serve as a joint for hand exoskeletons and propose a method for analyzing and optimizing stress and compliance characteristics of the variable cross-section flexure hinge under large deformation. Based on finite element analysis results and Castigliano’s second theorem, we put forward general equations for compliance and stress design of the hinge under large deformation. Furthermore, we propose three four-bar linkage mechanisms and conduct finite element analysis to validate the accuracy of our general design equations and approach. Compared to Lobontiu and Howell’s equations, our general design equations exhibit nearly 36% higher accuracy. Our proposed methodology can be easily applied in designing compliant mechanisms with semi-circular flexure hinges operating in moment-loaded environments, particularly in biomedical engineering applications such as soft prosthetic hands and exoskeleton robots. Finally, based on the proposed equations, we design and fabricate a flexure-based artificial finger whose performance test demonstrates excellent bio-imitability and adaptability in terms of joint movements.