Design, Modeling, and Optimization of a Nearly Constant Displacement Reducer with Completely Distributed Compliance

Abstract

This article proposes a displacement reducer based on distributed compliant mechanisms to improve the motion resolution of actuators commonly used in precision operation systems that require high-precision control and positioning, such as micro-grippers, biological manipulation, and micro-alignment mechanisms. Distributed compliance significantly diminishes its effective moving lumped mass, endowing the structure with advantages such as reduced stress concentration and an expansive range of motion. Additionally, the design incorporates an over-constraint structure through a dual-layer displacement reducer, ensuring that the reducer achieves a nearly constant reduction ratio. According to the compliance matrix method, the analytical model of the reducer is established to predict the input–output behaviors, which are verified by finite element simulations. On the basis of sensitivity analysis to structure parameters, including node positions and beam parameters, the Particle Swarm Optimization (PSO) algorithm is used to optimize the displacement reduction performance. Through finite element analysis and experimental results on the prototype, the proposed displacement reducer demonstrates a large reduction ratio of 11.03, an energy transfer efficiency of 39.6%, and a nearly constant reduction ratio with an input displacement range of 0 to 2000 µm.

1. Introduction

In applications requiring high-precision control and positioning, including micro-grippers, biological manipulation, and micro-alignment mechanisms, there is a growing demand for actuators with high resolution [1,2,3,4,5,6,7,8]. Compliant mechanisms, known for their advantages in operational accuracy and compactness, can be employed to design displacement reducers that enhance the motion resolution of actuators [9,10,11]. Compliant mechanisms leverage the elastic deformation and potential energy of materials to convert force and energy [12,13,14]. Compared to traditional rigid structures, compliant mechanisms offer benefits such as ease of fabrication, reduced assembly errors, improved operational precision, and a more compact structure [15,16,17,18].

In the previous literature, numerous designs, analyses, and optimization methodologies for the compliant-based displacement amplification mechanism have been presented [19,20,21]. In theory, resolution can be improved by swapping the input and output positions of a compliant amplification mechanism. However, this simple swap overlooks different standards in the design of amplifiers and reducers, such as drive compatibility, structural design specifications, and variations in mechanical performance [19,22,23,24,25]. Sakuma and Arai [26] designed a reducer that achieves the precise positioning of probe tips by combining springs of different stiffness levels. Guimin Chen [19] proposed a series design method for reducers, integrating two bridge amplification mechanisms to design the reducer. However, the use of flexible hinge connections in these designs may lead to stress concentration issues. Yunzhuang Chen [18] designed a distributed compliant displacement reducer that replaces the flexible hinge of the bridge mechanism with distributed compliant beams. However, this will lead to a decrease in output stroke. The design of the reducer necessitates the careful consideration of the constant reduction ratio range of the driving components [27,28,29]. A narrow linear interval between input and output strokes can lead to nonlinear issues, especially when subjected to significant input displacements, resulting in output deviations from expected performance standards and presenting challenges for practical implementation.

This study references existing research on displacement reducers and defines the reduction ratio as the ratio of input displacement to output displacement [18,19], facilitating theoretical calculations and performance evaluation. This study incorporates the large constant reduction ratio into the design of a compliant displacement reducer, employing an over-constrained structure to ensure the constant reduction ratio. Compared to the previous literature on reducers, this compliant mechanism maintains a nearly constant reduction ratio. Additionally, it features a small moving lumped mass, easing manufacturing at both macro- and micro-scales.

The main contributions of this paper can be encapsulated as follows:

• An over-constrained distributed compliant reducer is proposed, which provides a nearly constant reduction ratio across a wide range of input displacement. Simultaneously, this mechanism significantly reduces the moving lumped mass;
• The mathematical model of the displacement reducer is established based on the compliance matrix method, to predict the input–output relationship and to perform a detailed analysis of the structural sensitivity;
• A general optimization method for distributed compliance mechanisms based on the PSO algorithm is proposed. This optimization approach is applicable to the sizing optimization of most distributed compliant mechanisms. The optimization objectives under three different scenarios were considered separately.

The organization of this paper is as follows: Section 2 outlines the design concept of the mechanism. Section 3 establishes the analytical model of the mechanism. Finite element analysis (FEA) is employed to validate the analytical model. A sensitivity analysis of the structure is also conducted. Section 4 applies the PSO algorithm to optimize the designed mechanism. Subsequently, Section 5 conducts experimental testing on the optimized compliant mechanism. Finally, Section 6 presents conclusions and future prospects for compliant reducers.

2. Mechanism Design

In general, the reduction of displacement and the amplification of force in displacement reduction mechanisms can be achieved through centralized compliant lever mechanisms. However, large input displacements may lead to issues such as excessive deformation of the hinges and stress concentration. Therefore, this paper adopts a distributed compliant mechanism design. Additionally, an over-constrained design approach is employed to enhance the structural linear displacement range and output stiffness.

This study is based on the design concept of distributed compliant mechanisms and utilizes the deformation behavior of compliant beams for structural design. Inspired by the theoretical findings on the input–output relationship of compliant mechanisms from [28,30], a single-degree-of-freedom compliant displacement reducer with stable input–output characteristics has been successfully developed, as shown in Figure 1. This design adopts a double-layer over-constraint design method to enhance the range of the nearly constant reduction ratio and the load capacity of the displacement reduction mechanism.

Figure 1a shows that $DA{D}^{\prime }$ is a triangular mechanical amplification mechanism. Providing input displacement at position D and $D‘$ can achieve a stable output displacement at point A, where the amplification ratio increases as the angle ${\alpha }_1$ decreases. Conversely, when ${\alpha }_1$ is sufficiently large, displacement reduction can be achieved [28]. After swapping the input and output of two identical triangular lever amplification mechanisms (triangles $DEG$ and $D’E’G’$), they are connected to the triangular mechanical amplification mechanism to achieve an external secondary reduction mechanism, as shown in Figure 1b. Based on the principles shown in the previous two steps, an internal secondary reduction mechanism has been designed, where $CA{C}^{\prime }$ represents the triangular mechanical reduction mechanism and the reduction ratio is determined by ${\alpha }_2$. Due to the adoption of a symmetric design concept, the external secondary reduction mechanism and the internal secondary reduction mechanism can ensure that, when input displacement is provided at points E and $E^{\prime }$, a stable output displacement can be achieved at point A [28]. Additionally, to ensure an efficient use of space, the fixed constraints of the triangular lever mechanism (triangles $CAB$ and $C’A’B’$) in the internal secondary reduction mechanism are replaced by a symmetric structure, achieving effective spatial utilization, as shown in Figure 1c. The input and output positions of the internal and external secondary reduction mechanisms are connected. Additionally, to further enhance the reduction ratio and decrease the moving mass, a ground-mounted drive is implemented, and a third triangular mechanical reduction mechanism, $EF{E}^{\prime }$ (with the reduction angle determined by ${\alpha }_3$), is added. Furthermore, due to geometric relationships, the deflection of $Beam-DE$ and $Beam-D’E’$ is excessive, resulting in nonlinear deformation, which causes inconsistencies in the linear range of the internal and external secondary reduction mechanisms. To improve the linear range, $Beam-DE$ and $Beam-D’E’$ are replaced with slightly curved $Beam-DHE$ and $Beam-D’H’E’$, thereby increasing the range of the constant reduction ratio. The final structural design is shown in Figure 1d. In this figure, ${\alpha }_1$ represents the angle of triangle $DA{D}^{\prime }$, ${\alpha }_2$ represents the angle of triangle $CA{C}^{\prime }$, and ${\alpha }_3$ represents the angle of triangle $EF{E}^{\prime }$.

3. Kinetostatic Modeling and FEA

This section introduces the displacement analysis model of reducers based on the compliance matrix method. Using this method, we provide an analytical expression for the reduction ratio of the displacement reducer.

3.1. Model of Distributed Compliant Beams

Figure 2 illustrates the compliant beam between nodes j and k, denoted as $Beam-jk$. Node k represents the initial input position of the current compliant beam, which is subjected to the input force and displacement from the compliant beam or actuator in the previous stage at the same node. Node j represents the final input position of the compliant beam, which is also its output position. The force and displacement generated at this point will serve as the input conditions for the next stage of the compliant beam at node j or the final output position. Each node has three degrees of freedom: axial displacements $u_j$, $u_k$; lateral displacements $w_j$, $w_k$; and rotations ${\theta }_j$, ${\theta }_k$. Beam motion or deformation in the plane can be described using coordinate systems.

$O_j^{jk}-X_j^{jk}{Y}_j^{jk}$ and $O_k^{jk}-X_k^{jk}{Y}_k^{jk}$ represent local coordinate systems. The X-$axis$ of both coordinate systems is aligned with the beam, with its positive direction pointing from the input to the output positions. The above-mentioned forces and displacements can also be represented in any reference coordinate frames. The positive direction of the y-coordinate is defined as the vertical direction, and the positive direction of the x-coordinate is defined as the horizontal direction. $O_j^{t,jk}-X_j^{t,jk}{Y}_j^{t,jk}$ and $O_k^{t,jk}-X_k^{t,jk}{Y}_k^{t,jk}$ are established. The statics model of general small deformation compliant beams can be uniformly described using the concept of a stiffness matrix. This matrix exhibits a form similar to Hooke’s law [31]. Consequently, the force–position equilibrium relationship for distributed compliant beams can be established as Equation (1).

$$F_{jk}=K_{jk}{U}_{jk},$$

(1)

where $F_{jk}$ represents the resultant force acting on the beam at points j and k, $U_{jk}$ represents the resultant displacement generated at points j and k, and $K_{jk}$ represents the stiffness matrix that maps displacement to force. The specific parameters of $K_{jk}$ are calculated by considering the mechanical properties, geometry, and applied loads of the compliant beam. The stiffness matrix used in this paper is referenced from [27,28]. Vectors $F_{jk}$, $U_{jk}$, and $K_{jk}$ can be represented, respectively, by formulas Equations (2)–(4).

$$F_{jk}=\left[f_{j,x}^{jk}{f}_{j,y}^{jk}{m}_{j,z}^{jk}{f}_{k,x}^{jk}{f}_{k,y}^{jk}{m}_{k,z}^{jk}\right],$$

(2)

$$U_{jk}=\left[u_{j,x}^{jk}{u}_{j,y}^{jk}{u}_{j,\theta }^{jk}{u}_{k,x}^{jk}{u}_{k,y}^{jk}{u}_{k,\theta }^{jk}\right],$$

(3)

$${\displaystyle K_{jk}=\left[\begin{array}{cccccc}\frac{E{A}_{jk}}{L_{jk}}& 0& 0& -\frac{E{A}_{jk}}{L_{jk}}& 0& 0\\ 0& \frac{12E{I}_{jk}}{L_{jk}^3}& \frac{6E{I}_{jk}}{L_{jk}^2}& 0& -\frac{12E{I}_{jk}}{L_{jk}^3}& \frac{6E{I}_{jk}}{L_{jk}^2}\\ 0& \frac{6E{I}_{jk}}{L_{jk}^2}& \frac{4E{I}_{jk}}{L_{jk}}& 0& -\frac{6E{I}_{jk}}{L_{jk}^2}& \frac{2E{I}_{jk}}{L_{jk}}\\ -\frac{E{A}_{jk}}{L_{jk}}& 0& 0& \frac{E{A}_{jk}}{L_{jk}}& 0& 0\\ 0& -\frac{12E{I}_{jk}}{L_{jk}^3}& -\frac{6E{I}_{jk}}{L_{jk}^2}& 0& \frac{12E{I}_{jk}}{L_{jk}^3}& -\frac{6E{I}_{jk}}{L_{jk}^2}\\ 0& \frac{6E{I}_{jk}}{L_{jk}^2}& \frac{2E{I}_{jk}}{L_{jk}}& 0& -\frac{6E{I}_{jk}}{L_{jk}^2}& \frac{4E{I}_{jk}}{L_{jk}}\end{array}\right],}$$

(4)

where E represents the elastic modulus of the beam, I represents the moment of inertia of the cross-section, A represents the cross-sectional area, and L denotes the length of the beam.

Since discussions on the compliance (or stiffness) matrix are meaningful only within the same coordinate system, to establish the compliance (or stiffness) matrix for the entire compliant system, it is necessary to transform the coordinates of the output end to the input end coordinate system using a rotation matrix. Based on the established global coordinate system, this rotation matrix is established as shown in Equation (5):

$$R=\left[\begin{array}{cccccc}cos\alpha & sin\alpha & 0& 0& 0& 0\\ -sin\alpha & cos\alpha & 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 0& 0& cos\alpha & sin\alpha & 0\\ 0& 0& 0& -sin\alpha & cos\alpha & 0\\ 0& 0& 0& 0& 0& 1\end{array}\right],$$

(5)

According to the aforementioned Equation (1), we can derive Equation (6):

$$F_{jk}=R_{jk}^T{K}_{jk}{R}_{jk}{U}_{jk},$$

(6)

3.2. Modeling of the Compliant Reducer

The compliant reducer is symmetrical at its midpoint; thus, it can be statically modeled using only on one half, as shown in Figure 3. The figure retains only the geometric relationships for the sake of analysis convenience. This structure is decomposed into ten compliant beams, namely, $Beam$−$AC$, $Beam$−$CB$, $Beam$−$CE$, $Beam$−$BE$, $Beam$−$AD$, $Beam$−$DH$, $Beam$−$HE$, $Beam$−$DG$, and $Beam$−$GE$, and $Beam$−$EF$. The coordinate system $O-XY$ serves as the global reference coordinate system.

Given the input displacement $F_{in}$ and output displacement $F_{out}$, the coordinates of each node are defined as shown in Figure 3. The coordinates of nodes $A,B,C,D,E,F,G$ are in the coordinate system $O-XY$.

Assuming different widths for each beam, represented as $w_A$, $w_B$, $w_C$, $w_D$, $w_E$, $w_F$, $w_G$, $w_H$, and assuming the same material for each beam, we can derive the length of each beam, the angle of orientation, and the cross-sectional area.

This half configuration can be decomposed into ten straight strip beams, as shown in Figure 4. Based on the geometric relationships, the rotation angles of each beam can be obtained, as listed in

Table 1. Length, orientation angle, and cross−sectional area of the displacement reducer.

Beam	Length	Orientation Angle	Cross-Sectional Area
$BEAM-AC$	$L_{CA}=\sqrt{{(x_A-x_C)}^2+{(y_A-y_C)}^2}$	${\alpha }_{CA}=arctan\left({\displaystyle \frac{y_A-y_C}{x_A-x_C}}\right)$	$A_{CA}=w_{CA}·t_{CA}$
$BEAM-CB$	$L_{CB}=\sqrt{{(x_C-x_B)}^2+{(y_C-y_B)}^2}$	${\alpha }_{CB}=arctan\left({\displaystyle \frac{y_B-y_C}{x_B-x_C}}\right)$	$A_{CB}=w_{CB}·t_{CB}$
$BEAM-EC$	$L_{EC}=\sqrt{{(x_C-x_E)}^2+{(y_C-y_E)}^2}$	${\alpha }_{CE}=arctan\left({\displaystyle \frac{y_C-y_E}{x_C-x_E}}\right)$	$A_{EC}=w_{EC}·t_{EC}$
$BEAM-EB$	$L_{EB}=\sqrt{{(x_B-x_E)}^2+{(y_B-y_E)}^2}$	${\alpha }_{EB}=arctan\left({\displaystyle \frac{y_B-y_E}{x_B-x_E}}\right)$	$A_{EB}=w_{EB}·t_{EB}$
$BEAM-DA$	$L_{DA}=\sqrt{{(x_A-x_D)}^2+{(y_A-y_D)}^2}$	${\alpha }_{DA}=arctan\left({\displaystyle \frac{y_A-y_D}{x_A-x_D}}\right)$	$A_{DA}=w_{DA}·t_{DA}$
$BEAM-HD$	$L_{HD}=\sqrt{{(x_D-x_H)}^2+{(y_D-y_H)}^2}$	${\alpha }_{HD}=arctan\left({\displaystyle \frac{y_D-y_H}{x_H-x_D}}\right)$	$A_{HD}=w_{HD}·t_{HD}$
$BEAM-HE$	$L_{EH}=\sqrt{{(x_H-x_E)}^2+{(y_H-y_E)}^2}$	${\alpha }_{EH}=arctan\left({\displaystyle \frac{x_H-x_E}{y_E-y_H}}\right)$	$A_{EH}=w_{EH}·t_{EH}$
$BEAM-DG$	$L_{GD}=\sqrt{{(x_D-x_G)}^2+{(y_D-y_G)}^2}$	${\alpha }_{GD}=arctan\left({\displaystyle \frac{x_G-x_D}{y_D-y_G}}\right)$	$A_{GD}=w_{GD}·t_{GD}$
$BEAM-GE$	$L_{GE}=\sqrt{{(x_E-x_G)}^2+{(y_E-y_G)}^2}$	${\alpha }_{GE}=arctan\left({\displaystyle \frac{y_E-y_G}{x_E-x_G}}\right)$	$A_{GE}=w_{GE}·t_{GE}$

. The point e is taken as the coordinate origin.

In the local coordinate system, the global stiffness matrices of each beam are derived using geometric parameters from the previous section. These matrices incorporate beam orientation angles calculated according to Table 1 and Equation (6). The derived analytical model above can analyze half of the symmetric compliant displacement reducer. Hence, the rotations of the input and output rigid stages are constrained. Consequently, the static equilibrium equations for the seven unfixed nodes $A,B,C,D,E,F$, and H under the action of input force $F_{in}$ and output displacement $F_{out}$ can be obtained, as shown in Equation (7).

$$\left\{\begin{array}{c}{F}_D^{t,DA}+F_D^{t,HD}+F_D^{t,DG}=0\hfill \\ F_H^{t,HD}+F_H^{t,EH}=0\hfill \\ F_E^{t,EH}+F_E^{t,EG}+F_E^{t,EB}+F_E^{t,FE}=0\hfill \\ F_C^{t,CA}+F_C^{t,CB}+F_C^{t,EC}=0\hfill \\ F_B^{t,CB}+F_B^{t,EB}=0\hfill \\ f_F^{t,FE}={\displaystyle \frac{f_{in}}{2}}\hfill \\ f_{A,y}^{t,DA}+f_{A,y}^{t,CA}={\displaystyle \frac{f_{out}}{2}}\hfill \end{array}\right.,$$

(7)

Further derivation yields Equation (8):

$$\left\{\begin{array}{c}{K}_{DA,1}^t{U}_D^t+K_{DA,2}^t\langle 13,2\rangle U_A^t+K_{HD,4}^t{U}_D^t+K_{HD,3}^t{U}_H^t\hfill \\ +K_{DG,1}^t{U}_D^t+K_{DG,2}^t{U}_G^t=0\hfill \\ K_{HD,1}^t{U}_H^t+K_{HD,2}^t{U}_D^t+K_{EH,4}^t{U}_H^t+K_{EH,3}^t{U}_E^t=0\hfill \\ K_{EH,1}^t{U}_E^t+K_{EH,2}^t{U}_H^t+K_{EG,1}^t{U}_E^t+K_{EG,2}^t{U}_G^t+K_{EB,1}^t{U}_E^t\hfill \\ +K_{EB,2}^t\langle 13,2\rangle U_B^t+K_{FE,1}^t{U}_E^t+K_{FE,2}^t\langle 13,2\rangle U_E^t=0\hfill \\ K_{CA,1}^t{U}_A^t+K_{CA,2}^t\langle 13,2\rangle U_C^t+K_{CB,1}^t{U}_C^t+K_{CB,2}^t\langle 13,2\rangle U_B^t\hfill \\ +K_{EC,4}^t{U}_C^t+K_{EC,3}^t{U}_C^t=0\hfill \\ K_{CB,3}^t\langle 2,13\rangle U_C^t+K_{CB,4}^t\langle 2,2\rangle u_{b,y}^t+K_{EB,3}^t\langle 2,13\rangle U_E^t\hfill \\ +K_{EB,4}^t\langle 2,2\rangle u_{B,y}^t=0\hfill \\ K_{FE,3}^t\langle 2,13\rangle U_E^t+K_{FE,4}^t\langle 2,2\rangle u_{f,y}^t={\displaystyle \frac{f_{in}}{2}}\hfill \\ K_{DA,3}^t\langle 2,13\rangle U_D^t+K_{DA,4}^t\langle 2,2\rangle u_{a,y}^t+K_{CA,3}^t\langle 2,13\rangle U_C^t\hfill \\ +K_{CA,4}^t\langle 2,2\rangle u_{A,y}^t={\displaystyle \frac{f_{out}}{2}}\hfill \end{array}\right.,$$

(8)

From Equation (8), it is evident that, for a given input force $F_{in}$, the node displacements can be determined through analysis. Consequently, the displacement reducer’s reduction ratio can be determined as

$$r={\displaystyle \frac{u_{A,y}^t}{u_{F,y}^t}},$$

(9)

3.3. FEA Verification

To validate the analytical model from Section 3.2, FEA analyses are executed. Taking point E as the origin, the coordinates and beam width are defined as listed in

Table 2. Original width parameters and constraints.

Parameters	Before Optimization (mm)	Lower and Upper Limits of Width (mm)
$w_{AD}$	1.5	[1,2]
$w_{DG}$	1.5	[1,2]
$w_{AC}$	1.5	[1,2]
$w_{CB}$	1.5	[1,2]
$w_{CE}$	1.5	[1,2]
$w_{BE}$	1.5	[1,2]
$w_{DH}$	1.5	[1,2]
$w_{HE}$	1.5	[1,2]
$w_{EG}$	1.5	[1,2]
$w_{EF}$	1.5	[1,2]

and

Table 3. Original geometric parameters and constraints.

	x−Coordinates	y−Coordinates	Lower and Upper	Lower and Upper
Node	Before Optimization	Before Optimization	Limits of the x−Coordinates	Limits of the y−Coordinates
A	24	60	[20,30]	[60,70]
B	24	44	[20,30]	[40,50]
C	0	32	[0,10]	[30,40]
D	−30	30	[−30,−20]	[20,30]
E	0	0	0	0
F	24	−21	[20,30]	[−20,−10]
G	−24	−33	[−30,−20]	[−40,−30]
H	−15	−8	[−20,−10]	[−10,0]

.

In this study, the FEA simulations were conducted using ANSYS Workbench 2021 R1. In the simulation, the entire mechanism was modeled with large deflection. A tetrahedron elements method was employed, generating a mesh with an element size of 1 mm. The meshing is shown in Figure 5. Boundary conditions and input displacement were set by applying fixed support and displacement in the static structural analysis. Assuming that the reducer is made using $3D$ printing technology with PLA material, the Young’s modulus E is 3.5 GPa, Poisson’s ratio v is 0.33, and the density is 1.25 g/cm³. Through finite element analysis, the input–output relationship of the displacement reducer was obtained, as shown in Figure 6a. Figure 6b depicts the overlay of the analytical model and the finite element simulation model. The finite element model predicts an average reduction ratio of 4.5, whereas the analytical model calculates an average reduction ratio of 4.45, resulting in a discrepancy of 1.1%. Figure 6b illustrates the relationship between input displacement and input force, with an error of 3%. Figure 7 shows the stress distribution and deformation results of the model under ANSYS Workbench 2021 R1 simulation; the scale of deformation results is 1:1, with the dashed line representing the structure before deformation. The research results indicate that the model exhibits uniform stress distribution and deformation, thereby alleviating the phenomenon of stress concentration.

3.4. Sensitivity Analysis: Geometric Positions and Beam Width

In the design variables mentioned above, the geometric positions of the displacement reducer are given in the form of coordinates. The performance indicators of the displacement reducer are calculated using the analytical model presented in Section 3. Subsequently, the impact of different length and thickness parameters, as well as their relative coordinates, on the reduction ratio and output stiffness is discussed.

The effects of the variation in parameters of the displacement reducer within the specified range are shown in Figure 8 and Figure 9. In the figures, $x_i$ represents the horizontal coordinate variation of point i, $y_i$ represents the vertical coordinate variation, and $x_{ij}$ and $y_{ij}$ represent the relative position variations between points i and D, where $i,j=A,B,C,D,E,F,G,H$. Figure 8 shows that, as the horizontal distance between points A, B, and F increases, the reduction ratio of the displacement reducer gradually decreases, with similar variations. However, increasing the horizontal distance between points $C,G,F$ and H leads to an increase in the reduction ratio. Among these, the change at point G has the most significant impact on the structure, causing the reduction ratio to first increase and then decrease. The reduction ratio reaches its maximum when $x_D=-32$. Furthermore, increasing the relative distance between points C and D also results in an increase in the reduction ratio.

As shown in Figure 9, the vertical direction changes for each point do not significantly impact the displacement reducer. Only point b leads to a substantial increase in the reduction ratio. Similarly, changes in the relative position between points A and B also have a significant effect on the mechanism.

Figure 10 shows the sensitivity of the displacement reducer to different structural parameters. Figure 10a illustrates the impact of the coordinates of each point on the output characteristics of the displacement reducer. It can be seen that the displacement reducer is significantly affected by the geometric position changes of the structure, with the variation in the x−direction having a greater impact than the variation in the y−direction. Similarly, as shown in Figure 10b, the length of the beam also has a substantial effect on the reduction ratio of the structure, especially the changes in $Beam$−$BE$ and $Beam$−$HE$, which result in nearly 80%.

4. Structural Optimization

The displacement reducer proposed in this paper can be optimized by adjusting its geometric parameters. The objective function and parameter variables are determined using the analytical model calculated in Section 3.

This paper utilizes the PSO algorithm for further structural optimization. The PSO algorithm is a heuristic optimization algorithm based on swarm intelligence. Its principle originates from simulating the behavior of biological groups, such as bird flocking foraging or fish schooling. The algorithm efficiently searches for the global optimal solution in the solution space by simulating the information sharing and collaboration mechanisms among individuals in the swarm. The core principle of PSO can be summarized as follows:

Particle representation: In the PSO algorithm, each particle represents a candidate solution in the solution space. Each particle has two key attributes:

• Position vector $x_i\left(t\right)$: Represents the current position of the particle in the solution space, i.e., the current candidate solution;
• Velocity vector $v_i\left(t\right)$: Represents the search direction and step size of the particle in the solution space that determine the direction and speed of the particle’s next move.

Velocity update mechanism: The velocity update of the particle is the core part of the PSO algorithm. The velocity update depends on the following three factors:

• Individual best solution $p_{\mathrm{best}}$: The best solution found by each particle during the search process, denoted as $p_{\mathrm{best}}$;
• Global best solution $g_{\mathrm{best}}$: The best solution found by the entire particle swarm during the search process, denoted as $g_{\mathrm{best}}$;
• Inertia weight w: Used to balance global exploration and local exploitation. A larger inertia weight favors global exploration, while a smaller inertia weight helps local exploitation.

The velocity update formula is as follows:

$$v_i(t+1)=w{v}_i\left(t\right)+c_1{r}_1(p_{\mathrm{best}}-x_i\left(t\right))+c_2{r}_2(g_{\mathrm{best}}-x_i\left(t\right))$$

(10)

where w is the inertia weight, which controls the inertial part of the particle’s velocity. $c_1$ and $c_2$ are cognitive and social factors, respectively, used to adjust the particle’s dependence on individual experience and group collaboration. Typically, $c_1$ and $c_2$ are set around 2. $r_1$ and $r_2$ are random numbers uniformly distributed in the range $[0,1]$, introduced to add randomness and avoid early convergence to local optima.

Position update mechanism: The particle’s position is adjusted according to the updated velocity. The position update formula is as follows:

$$x_i(t+1)=x_i\left(t\right)+v_i(t+1)$$

(11)

Through this formula, the particle moves to a new position in the solution space and continues to search for a better solution.

Iterative optimization process: Through iterative updates of the particles’ velocity and position, the swarm gradually converges towards the global optimal solution, ultimately obtaining the optimal solution that satisfies the optimization objectives.

The PSO algorithm, with its simple structure, few parameters, fast convergence speed, and ease of implementation, has been widely applied in function optimization, engineering design, machine learning, and multi-objective optimization, among other fields. Its swarm collaboration mechanism and efficient global search capability make it one of the most effective tools for solving complex optimization problems.

The optimization process can be summarized as the following steps, serving as a general optimization method for distributed compliant mechanisms:

• Determine the structural parameter requirements and functional parameter indicators of the distributed compliant mechanism based on functional requirements;
• Choose an appropriate modeling method based on the functional parameter indicators and geometric parameter requirements for modeling;
• Perform a comparative optimization of structures under different optimization objectives;
• Based on the optimization results from step 3, perform weighting or constraint processing to ensure that other optimization objectives remain within a reasonable range and determine the optimal objective function.

In the design and optimization of displacement reducers, in addition to improving the reduction ratio, it is also necessary to maintain constant stiffness within a certain stroke range, which helps to enhance the load-bearing capacity of the displacement reducer. Furthermore, energy transfer efficiency should also be considered to prevent energy wastage. Therefore, in this section, based on the analytical model in Section 3, the constant stiffness range, reduction ratio, and energy transfer efficiency of the displacement reducer are set as the optimization objective functions. The PSO algorithm, implemented in Python 3.10 using PyCharm Community Edition 2020.3.5 x64, is applied. Each of the three objective functions is optimized separately, with the maximum value represented in reciprocal form. The constraint conditions for the design variables are set according to Table 2 and Table 3. When the results approach zero and converge, the function reaches its optimal value. The design variables obtained from the optimization are then used to establish the model in SOLIDWORKS, and the optimization results are presented in each subsection. These three optimization objectives are then combined for further integrated optimization.

4.1. Optimization of Constant Output Stiffness

Under load conditions, constant stiffness in the direction of motion is one of the essential characteristics required for the displacement reducer. To achieve good load-bearing capacity, the primary motion stiffness should remain constant or increase within the range of motion. Therefore, in this section, constant stiffness along the Y-axis is achieved by optimizing both the geometric and beam parameters.

The output displacement is divided into n segments. The maximum stiffness value of each segment is selected, and the difference between this value and the stiffness of the other segments is calculated. The smaller the difference, the closer the stiffness values between the segments, demonstrating that the goal of optimizing the constant stiffness range has been achieved, while also maximizing the overall stiffness. The displacement interval is divided into n segments, with each segment having a length

$$u_j={\displaystyle \frac{j{u}_{\max }}{n}},j=0,1,2,\cdots ,n,$$

(12)

where $u_j$ represents the displacement in each segment and $u_{\mathrm{out}}$ is the total displacement. According to Newton’s difference formula, the output stiffness in the y-direction is as defined below:

$$K_{\mathrm{out}}={\displaystyle \frac{f_j-f_{j-1}}{u_j-u_{j-1}}},j=0,1,2,\cdots ,n,$$

(13)

and $f_i$ represents the output force in each segment. The objective function can be defined as

$$\begin{array}{cc}\hfill \mathrm{Min}\rho \left(a_x,\cdots ,h_x,h_y,w_{\mathrm{ad}},\cdots ,w_{\mathrm{ef}}\right)& =\sqrt{{\displaystyle \frac{1}{n}}{\sum }_{j=1}^n{\left(K_{\mathrm{out}}-{\displaystyle \frac{1}{n}}{\sum }_{k=1}^n{K}_{\mathrm{out}}\right)}^2}\hfill \\ \hfill & j=0,1,2,\cdots ,n,\hfill \end{array}$$

(14)

Set the expression above as the objective function, with the constraints given in Table 2 and Table 3.

After 1000 generations of iteration, the fitness values are shown in Figure 11a. The results tend to converge after about 650 iterations, demonstrating the feasibility of the optimization algorithm. Figure 11b shows the structure with a constant stiffness in the Y-axis direction after optimization. The range of stiffness changed from 0.6–0.72 N/µm to 0.83–0.85 N/µm. It can be observed that some of the beam lengths have significantly increased, which helps to improve the load-bearing capacity of the mechanism. However, excessively thick beams may cause stress concentration, which could negatively affect the displacement reduction ratio of the structure. Figure 11c shows the stiffness characteristics of the displacement reducer before and after optimization. It can be seen that the stiffness has significantly increased after optimization and remains nearly constant from 0 to 2000 µm, which greatly enhances the overall load-bearing capacity of the structure.

The maximum stiffness is 0.85 N/µm, which is an improvement of 15.29% compared to the pre-optimization result of 0.6 N/µm. The range of variation decreased to 0.17% of its previous value.

4.2. Optimization of Reduction Ratio

Based on the design principle of the compliant displacement reducer, the displacement in the output direction should be minimized as much as possible to achieve the maximum reduction ratio. The objective function is set as

$$\mathrm{Max}R\left(a_x,\cdots ,h_x,h_y,w_{\mathrm{ad}},\cdots ,w_{\mathrm{ef}}\right)={\displaystyle \frac{u_{a,y}^t}{u_{f,y}^t}},$$

(15)

The constraints are given in Table 2 and Table 3.

After 1000 generations of iteration, the fitness values are as shown in Figure 12a. The results tend to converge after about 830 iterations, demonstrating the feasibility of the optimization algorithm. Figure 12b shows the structure with constant stiffness in the Y-axis direction after optimization. It can be observed that the slope of the end beams of the overall structure has increased and shows an elongation trend, which is beneficial for displacement reduction. However, excessively long beams may cause large deflection deformations, leading to nonlinear characteristics in the output displacement and output force. This would be detrimental to the load-bearing capacity of the structure and could also reduce the linear output range of the structure. Figure 12c shows a comparison of the output displacement before and after optimization. The reduction ratio prior to optimization was 4.23, whereas, post-optimization, it increased to 16, reflecting an improvement of 378.25%.

4.3. Optimization of Energy Transfer Efficiency

In the structural design of the displacement reducer, to avoid energy loss within the structure, the output efficiency is further set as the desired optimization objective. According to the principle of energy conservation, when the output displacement decreases, the output force should increase as much as possible. The energy transfer efficiency is defined as [32]

$$\mathrm{Min}\eta \left(a_x,\cdots ,h_x,h_y,w_{\mathrm{ad}},\cdots ,w_{\mathrm{ef}}\right)={\displaystyle \frac{F_{\mathrm{out}}·u_{\mathrm{out}}}{F_{\mathrm{in}}·u_{\mathrm{in}}}},i=0,1,2,\cdots ,n,$$

(16)

When $\eta$ approaches 1, the energy loss within the structure is minimized. The constraints are given in Table 2 and Table 3.

After 1000 generations of iteration, the fitness values are as shown in Figure 13a. The results tend to converge after about 820 iterations, demonstrating the feasibility of the optimization algorithm. The optimized structure is shown in Figure 13b, where the overall structure has a flattened shape. Figure 13c shows the comparison of energy transfer efficiency before and after optimization. Compared to the previous design, the energy transfer efficiency has increased by approximately 50%.

4.4. Comprehensive Discussion of Optimization Model

From the data analysis and modeling results of the above three sections, the coupling relationships between the three optimization objectives can be observed. The optimization of a single objective may lead to performance losses in other areas. Therefore, when several single-objective functions to be optimized are incompatible, multi-objective optimization (MOO) arises, which simultaneously considers multiple interrelated or conflicting optimization objectives. In this case, there is usually no single optimal solution, and the optimal solution must be obtained through weighting or by applying additional constraints to the functions. Thus, under the various optimization objectives described in this section, the trade-offs between the three objective functions are made based on the rate at which different design variables influence the objective function trends and the actual application of the displacement reducer. While ensuring a certain reduction ratio and output efficiency, the goal is to maximize the maximum constant stiffness range. The objective function is set as

$$\begin{array}{cc}\hfill \mathrm{Min}R\left(a_x,\cdots ,h_x,h_y,w_{\mathrm{ad}},\cdots ,w_{\mathrm{ef}}\right)& ={\omega }_1\rho +{\omega }_{2}R+{\omega }_3\eta ,i=0,1,2,\cdots ,n,\hfill \\ \hfill {\omega }_1+{\omega }_2+{\omega }_3& =1,\hfill \end{array}$$

(17)

$\omega$ is a weighted variable. In addition to the constraints shown in the table, the following additional constraints are specified: the reduction ratio $R>8$ and the energy transfer efficiency $\eta >20\%$.

After 1000 generations of iteration, the fitness values are as shown in Figure 14. The results tend to converge after about 580 iterations, demonstrating the feasibility of the optimization algorithm. Compared to Section 3, the reduction ratio increased by 149.5%, which increased from 4.5 prior to optimization to 11.23 following optimization.The maximum stiffness is 0.85 N/µm. Figure 15 shows a comparison of the structure before and after optimization, highlighting the effects of changes in geometric position. The dashed line represents the results before optimization, while the solid line represents the results after optimization. Figure 16 illustrates the relationship between input and output displacements before and after optimization.

Additionally, as the stroke gradually increases, the accuracy of the analytical model diminishes due to significant beam deformation. These issues will be addressed in our future work, where we plan to conduct more in-depth research.

Based on the final optimization results, modal simulations were conducted. Figure 17 shows the first four modal shapes of the reducer obtained from the finite element model, and the corresponding frequencies are listed in

Table 4. First 4 natural frequencies of the displacement reducer.

Modes	1	2	3	4
FEA results (Hz)	256.89	504.66	605.4	695.36

. It can be observed that the first mode corresponds to irregular lateral oscillations of the reducer in the X–Z plane. The second mode corresponds to an out-of-plane deflection mode. The third mode corresponds to the working mode of the reducer. The fourth mode corresponds to regular lateral oscillations in the X–Z plane.

5. Experimentation

Using the final design results obtained from Section 4, an experimental model of the displacement reducer was fabricated using additive manufacturing technology. The final reducer is shown in Figure 18a. The experimental setup is shown in Figure 18b. In the experimental tests, the FSL40 ball-screw sliding table (Chengdu Fuyu Technology Co., Ltd., Chengdu, China) is used as the actuator for the mechanism to generate input force and displacement. The sliding table provides a continuous thrust of 200 N, with an effective stroke range of 0–500 mm and a repeatability accuracy of 0.05 mm. A laser interferometer (IDS3005, Attocube, Munich, Germany), with a resolution of 1 pm, is employed to measure the displacement of the output mechanism. The force sensor used is the SF-50 digital push–pull force gauge (Airuipu Technology Co., Ltd., Quzhou, China), with a maximum measurement capacity of 50 N and an accuracy of 0.25 N.

Figure 19a,b shows the variation in input–output displacement and reduction ratio when the input displacement is increased to 4000 µm. From Figure 19a, it can be seen that the experimental results of the displacement reducer at 2000 µm matches the linear relationship calculated by the analytical model. After this point, the slope gradually decreases, reaching the maximum output value around 3000 µm. This indicates that, in the later stages of the stroke, due to the influence of the geometric relationship, the continuous increase in input displacement causes significant nonlinear changes in the structure, resulting in discrepancies between the modeling results and experimental data.

In Figure 19c, in the range of 0 to 2000 µm, the experimental results are superimposed on the analytical model curve, and a consistency is observed. Figure 19d compares the experimental results of the displacement reducer’s reduction ratio with the analytical model results. The constant reduction ratio of the displacement reducer is approximately 11.03, slightly higher than the predicted values from the dynamic–static model and the finite element model. After 2000 µm, nonlinear deformation occurs as the input displacement continues to increase. Therefore, the range of constant reduction ratio is from 0 to 2000 µm.

The experimental results were conducted within the load range of −10 N to 10 N at the output end, as shown in Figure 20. The results generally show a constant value with small fluctuations. It can be observed that, within the range of −10 N to 10 N, the variation in load has little effect on the displacement reducer. Increasing the load does not affect the reduction ratio of the displacement reducer. Even at the maximum load of 10 N, the change in reduction ratio does not exceed 16%. This indicates that the displacement reducer maintains a constant reduction ratio over a long stroke and under external load conditions. Additionally, the output stiffness calculated from Figure 21a is 0.7 N/µm.

Figure 21b shows the relationship between input displacement and input force obtained from the experiment. The input stiffness of the mechanism is approximately 0.0146 N/µm. Between 0 and 2000 µm, the input stiffness of the mechanism remains approximately constant, which is consistent with the previous optimization objectives. The ratio of output stiffness to input stiffness is approximately 47.94.

This study compares the reduction ratio, structural type, constant reduction ratio range, lumped mass, and energy transfer efficiency of the displacement reducers shown in

Table 5. Original width parameters and constraints.

	Structure	Reduction Ratio	Structural Type	Constant Reduction Ratio Range (µm)	Moving Lumped Mass	Energy Transfer Efficiency
This paper		11.03	Distributed compliance	2000	One input stage One output stage	39.6%
Ref. [19]		100	Lumped compliance	500	Two input stage One output stage Other rigid bodies	3.6%
Ref. [18]		7.19	Distributed compliance	1400	One input stage One output stage Other rigid bodies	10%
Ref. [26]		7.69	Distributed compliance	300	One input stage One output stage	-
Ref. [1]		8	Lumped compliance	200	One input stage One output stage Other rigid bodies	-

. Among the selected mechanisms, the proposed displacement reducer demonstrates the largest constant reduction ratio range and energy transfer efficiency. Simultaneously, this mechanism significantly reduces the moving lumped mass. The reducers in the second structure [19] and the fifth structure [1] in Table 5 adopt lumped compliance, which concentrates the load on a few compliant hinges, leading to higher stress concentration factors. In contrast, the other reducers employ distributed compliance, where the load is spread across multiple compliant beams, reducing the stress on each unit and minimizing the likelihood of stress concentration by ensuring a more even distribution of the load [28]. Compared to other reducers in Table 5, the proposed reducer provides a larger constant reduction ratio range and higher energy transfer efficiency. Overall, the displacement reducer presented in this paper exhibits the largest constant reduction ratio range, alleviates the concentrated stress problem, and achieves higher energy transmission efficiency, avoiding energy loss.

6. Conclusions

The distributed compliant mechanism-based displacement reducer with over-constrained structure is designed to allow a uniform output motion resolution/precision. This mechanism is capable of providing a nearly constant reduction ratio under a wide range. The ratio of output stiffness to input stiffness of the structure reached 47.94, and the energy transmission efficiency improved to 39.6%. Additionally, by establishing the mathematical model, the input–output relationship is derived, which facilitated a sensitivity analysis of the structural parameters. The mathematical model and finite element simulation results exhibit differences of less than 1.8%. Subsequently, the PSO algorithm is employed for the further optimization of the structure. Under the imposed constraints, the optimization of geometric relationships and beam width resulted in a 149.5% increase in the reduction ratio. This optimization approach is also applicable to other structural designs. The experimental reduction ratio was measured to be 11.03, while the constant reduction ratio ranges from 0 to 2000 µm. The main innovations of this study are as follows:

• Grounded in the principles of over-constrained design, it presents a distributed compliant reducer capable of maintaining a nearly constant reduction ratio over a wide range of input displacements, simultaneously minimizing the moving lumped mass;
• The analytical model of the mechanism is developed using the compliance matrix method, which is employed to conduct a detailed structural sensitivity analysis;
• An integrated optimization approach combining the PSO algorithm with the analytical model is applied to achieve a comprehensive optimization of the structure. This methodology is applicable for the structure optimization of most distributed compliant mechanisms;
• In the field of precision operations, such as micro-grippers, biological manipulation, and micro-alignment mechanisms, closed-loop systems with stringent environmental requirements and complex structures are commonly used to achieve high-precision positioning and operation. In contrast, the mechanism proposed in this study features a simple structure and easy fabrication, capable of achieving stable, high-precision output displacements under complex operating conditions, thus providing a more straightforward and reliable solution for precision operations.

Our subsequent main task is to design a compliant mechanism with high output stiffness and resolution, while also considering both input and output strokes. Building on the foundation of this study, further structural design and optimization enable it to be suitable for micro-/nano-operations. Additionally, the design content of this stage will serve as the first phase for future adjustable-resolution compliant mechanism designs.