Design, Analysis, and Testing of a Novel 5-DOF Flexure-Based Alignment Stage

Abstract

A high pattern resolution is critical for fabricating roll-to-roll printed electronics (R2RPE) products. For enhanced overlay alignment accuracy, position errors between the printer and the substrate web must be eliminated, particularly in inkjet printing applications. This paper proposes a novel five-degree-of-freedom (5-DOF) flexure-based alignment stage to adjust the posture of an inkjet printer head. The stage effectively compensates for positioning errors between the actuation mechanism and manipulated objects through a series–parallel combination of compliant substructures. Voice coil motors (VCMs) and linear motors serve as actuators to achieve the required motion. Theoretical models were established using a pseudo-rigid-body model (PRBM) methodology and were validated through finite element analysis (FEA). Finally, an alignment stage prototype was fabricated for an experiment. The prototype test results showed that the developed positioning platform attains 5-DOF motion capabilities with 335.1 µm × 418.9 µm × 408.1 µm × 3.4 mrad × 3.29 mrad, with cross-axis coupling errors below 0.11% along y- and z-axes. This paper proposes a novel 5-DOF flexure-based alignment stage that can be used for error compensation in R2RPE and effectively improves the interlayer alignment accuracy of multi-layer printing.

Introduction

Roll-to-roll printed electronics (R2RPE) has been validated as a viable method for manufacturing a various electronic devices [1, 2, 3–4]. Considering electronic devices produced by traditional methods, R2RPE products have the advantages of having a large area and being fast, flexible, and inexpensive [5, 6]. For multi-layer printed electronic products, the accuracy of multi-layer printing is crucial, as different materials must be printed on each layer to achieve different structures, which directly affects the performance of multi-layer printed electronic products [7, 8]. If the accuracy of the multi-layer alignment cannot be guaranteed, the pattern resolution of R2RPE products will not satisfy the requirements of the high-end market. Therefore, high-precision multi-layer alignment systems are urgently required.

The production process of the R2RPE primarily includes gravure printing, flexographic printing, reverse offset printing, screen printing, and inkjet printing [9, 10, 11, 12–13]. Within the spectrum of manufacturing techniques, inkjet printing distinguishes itself through non-contact deposition, maskless fabrication capabilities, and material-efficient operation while maintaining a high feature resolution critical for micro-scale patterning applications [14, 15]. Multi-layer registration accuracy should be guaranteed to obtain high-resolution inkjet printing patterns of R2RPE. However, the accuracy of multi-layer registration can be affected if a relative position error occurs between the printer and the flexible web during the printing process.

The precision of inkjet printing processes is critically challenged by multi-source registration errors between the printhead assembly and advancing flexible web substrates. For example, unstable web transport speed or uneven web stretching caused by nonuniform web tension can result in web position errors [16, 17]. The posture error of the support rollers can also affect web position [18]. Moreover, the posture error of the printer has a significant impact on the alignment accuracy [19, 20]. Therefore, multi-layer alignment systems must be adopted for high-resolution inkjet printing.

Numerous studies have been conducted to improve the accuracy of the multi-layer alignment, and certain results have been achieved. In the field of web tension control, Kang et al. [21, 22] proposed a new theoretical model that considers the lateral position errors of the substrate web and roller and provides compensation methods. Kim et al. [23, 24] proposed a new design of roll-to-roll (R2R) printing equipment for R2RPE production. The designed R2RPE system consists of tension control components such as feeders, load cells, and charge-coupled device cameras to detect the relative position errors of patterns for high-precision printing. Jeong et al. [25] proposed a tension model for each section to successfully predict the tension applied to such a system, the sagging of the film according to tension, and deformation due to residual stress, and built an accurate R2R system to minimize tension reduction. Lee et al. [26] proposed an advanced model to determine the tension disturbances caused by run-out resulting from the axis mismatch, roundness error, imbalance, and velocity variation of the rollers in an industrial-scale R2R printing process, and a high average accuracy of 92.4% was achieved. For position error compensation based on the above methods, a traditional rigid hinge and motion stage are typically adopted to adjust the posture of a roller or printer. The multi-layer alignment accuracy is limited to 40 µm resolution [27].

For the posture error of the printer or roller to be eliminated with high accuracy, a flexible mechanism can be used to design compensation mechanisms, leveraging inherent advantages such as frictionless operation, backlash elimination, and lubrication-free monolithic design [28, 29–30]. Baldesi et al. [31] engineered an R2R-compatible compliant-stage printing system, leveraging elastic deformation principles to eliminate backlash and sliding friction. The flexible mechanism is driven manually by micrometer heads to eliminate the position error of the roller. In addition, Zhou et al. [32, 33] engineered a flexible R2R printing system for real-time adjustment of roller posture. The flexible mechanisms are driven by linear stepper motors and voice coil motors (VCMs). In the design of such flexible mechanisms, the output motion in working directions are not decoupled. In addition, flexible mechanisms are often designed to withstand loads, which is an important factor affecting the stability of alignment stage. Li et al. [7] developed a multi-layer R2R system with multiple-input multiple-output (MIMO) closed-loop control that achieves submicron-level alignment precision for large-scale continuous printing processes. Chen et al. [19] proposed a motion stage with a remote center of motion to adjust the roller posture. The proposed device supports rollers using air spherical bearings with high stiffness and a flexible mechanism to generate accurate motion. Thus, the roller posture can be adjusted, and the web position error can be compensated. Many scholars have designed various types of alignment mechanisms for error compensation in different scenarios and have achieved good results [34, 35–36].

The literature review shows that many scholars have designed various types of alignment devices and achieved many research results. However, current alignment devices are primarily concerned with compensating for the position error of the rollers. For inkjet printing, the positional error of the printer head should be adjusted.

Based on the above considerations, a novel five-degree-of-freedom (5-DOF) flexible alignment stage is proposed to adjust the posture of the printer head in real time. The proposed stage consists of a various flexible modules connected in serial and parallel. The dynamic performance of the proposed device has been significantly enhanced through novel structural design. The remainder of this paper is organized as follows: Section 2 details the design considerations and architectural configuration of the developed stage. Section 3 introduces a theoretical analysis of the kinematics, stiffness, and dynamics of the proposed stage. Parametric optimization and finite element analysis (FEA) validation are systematically addressed in Section 4. Experimental verification through prototype implementation and performance characterization is addressed in Section 5. The concluding remarks are given in Section 6.

Mechanism Design

Design Consideration

When processing of inkjet-printed electronics, multiple layers of patterns should be printed on the flexible web. Therefore, a high overlay accuracy should be achieved to ensure printing resolution. Hence, the position error between the inkjet printer and existing web should be compensated. The position error is primarily caused by the position deviation between the different support rollers. In R2R inkjet printing, rotation about the x-axis is inherently constrained by the web transport mechanism, making compensation of ${\theta }_x$ unnecessary for alignment. Thus, the 5-DOF design focuses on critical errors in ${\theta }_z$, ${\theta }_y$, and translations [7]. As shown in Figure 1, the position error can be divided into three linear errors and two rotational errors. Two methods can be used to compensate for the position errors. One is to change the position of the roller to ensure that the flexible roll is parallel to the output surface of the inkjet printer. However, the number of rollers is relatively large, and at least two roller postures should be adjusted, which would increase the real-time control difficulty. The other method involves adjusting the position of the inkjet printer, which is much easier to achieve. For all these position errors to be compensated, a 5-DOF alignment stage should be adopted to adjust the posture of the inkjet printer.

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Figure 1

Position errors between the inkjet printer and flexible web

The three translational position errors can be compensated for by adopting a decoupled XYZ motion stage. The rotational position error around z-axis can be compensated for by adopting a ${\theta }_z$ motion stage. Compensation for rotational errors around y-axis may introduce additional parasitic errors. As shown in Figure 2(a), if the rotation center $O_1$ is inside the alignment stage, the inkjet printer, which is the output end of the alignment stage, will generate lateral offsets along x- and z-axes. To avoid this type of coupled motion, the rotation center should coincide with the center point of the underside of the inkjet printer, as shown in Figure 2(b). Therefore, a flexible rotary-motion platform should have the characteristics of a remote motion center (RCM) [37, 38]. Therefore, the parasitic motion in the design stage should be suppressed to ensure the error compensation accuracy of the alignment mechanism and simplify the control complexity of the mechanism. In this design, the main measures to suppress parasitic errors are as follows: (1) By designing symmetrical and decoupled mechanisms, the coupling of parallel motion is reduced and parasitic errors are simulated; (2) through the design of the RCM mechanism, the rotation center of the rotating mechanism is aligned with the surface of the inkjet head. In addition, the alignment stage must exhibit good dynamic performance to obtain the real-time and fast pose compensation capability of inkjet printers.

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Figure 2

Alignment processes with (a) inner rotation center and (b) external rotation center

Structure Description

Based on the above considerations, a decoupled 5-DOF flexible alignment stage for R2R inkjet printing is proposed. The device primarily consists of a 1-DOF linear stage, two 2-DOF linear stages and two 3-DOF off-plane stages, as shown in Figure 3. The 1-DOF linear stage can actively generate translational motion along x-axis using a linear motor. The two 2-DOF linear stages are driven using four VCMs. The two 3-DOF off-plane stages are designed to satisfy the requirements of motion output.

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Figure 3

Mechanical design of the 5-DOF flexible alignment stage: (a) Overall view, (b) 1-DOF linear stage, (c) 2-DOF linear stage, (d) 3-DOF off-plane stage

Therefore, the 2-DOF linear and 3-DOF off-plane stages are combined to form a parallel 4-DOF stage to generate motions along y and z-axes and motions around y and z-axes. The 1-DOF linear stage is connected to the 4-DOF motion stage in series. Therefore, the proposed device can generate 5-DOF motions in the required directions to achieve the alignment function. Moreover, the axes of the 5-DOF motion intersect at one point, which coincides with the center point of the underside of the inkjet printer. Therefore, the 5-DOF motions are decoupled from each other.

Considering the symmetry of the structure while ensuring equivalent structural quality and not affecting the performance evaluation of the alignment mechanism, the structure and layout were optimized by inverting the crossbeam used for installing the inkjet head in the middle to facilitate the display of a finite element simulation analysis and performance evaluation results.

Theoretical Analysis

Kinematic Analysis

Based on the introduction, flexible mechanisms are used to adjust the pose of a printer. To evaluate the kinematics of the stage, this paper simplifies the theoretical model of the flexible stage using the pseudo-rigid-body model (PRBM) method. Thus, the 1-DOF linear stage is equivalent to a linear joint, and the 2-DOF linear stage is equivalent to an active 2-DOF linear joint. Moreover, the 3-DOF off-plane stage is equivalent to a combination of passive universal and passive linear joints. Based on this conversion, the simplified model of the 5-DOF flexible alignment stage can be given as shown in Figure 4. By combining the input forces $F_x$, $F_{y1}$, $F_{y2}$, $F_{z1}$, and $F_{z2}$, the output stage generates motion along the required directions. Moreover, the output point $O$ coincides with the underside center point of the inkjet printer, which is the RCM point.

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Figure 4

Simplified model of the proposed device

When applying input force/displacement ($F_x/{\delta }_x$) to the 1-DOF linear stage, the output stage can generate output displacement along x-axis, which is given by

1

$${\delta }_x^{\mathit{out}}={\delta }_x.$$

The input force $F_x$ can be calculated as

2

$$F_x=k_x{\delta }_x,$$where $k_x$ is the stiffness of the 1-DOF linear stage.

When a couple of input forces/displacements ($F_{y1}$, $F_{y2}/{\delta }_{y1}$, ${\delta }_x$) are applied to the two 2-DOF linear stage with $F_{y1}=F_{y2}=F_y$ and ${\delta }_{y1}={\delta }_{y2}={\delta }_y$, the output stage can generate output displacement along y-axis, which is given by

3

$${\delta }_y^{\mathit{out}}={\delta }_y.$$

Using the same method, the output stage can generate the output displacement along z-axis, which is given by

4

$${\delta }_z^{\mathit{out}}={\delta }_z.$$

The input force $F_y$ and $F_z$ can be derived as

5

$$F_y=k_y{\delta }_y,$$

6

$$F_z=k_z{\delta }_z,$$where $k_y$, $k_z$ are the stiffnesses of the 2-DOF linear stage along y- and z-axes, respectively. Therefore, the input stiffness can be deduced as

7

$$K_x=k_x,$$

8

$$K_y=2k_y,$$

9

$$K_z=2k_z.$$

For the output rotational motion around y-axis, the input forces/displacements ($F_{z1}$, $F_{z2}/{\delta }_{z1}$, ${\delta }_{z2}$) are given by $F_{z1}={-F}_{z2}$ and ${\delta }_{z1}={-\delta }_{z2}$. Thus, the output rotational motion ${\theta }_y$ can be derived as

10

$${\theta }_y=\text{arcsin}\frac{{\delta }_z}{L},$$where L is the rotation radius of the stage.

For the output rotational motion about z-axis, the input forces/displacements ($F_{y1}$, $F_{y2}/{\delta }_{y1}$, ${\delta }_{y2}$) are given by $F_{y1}={-F}_{y2}$ and ${\delta }_{y1}={-\delta }_{y2}.$ Thus, the output rotational motion ${\theta }_z$ can be deduced as

11

$${\theta }_z=\text{arcsin}\frac{{\delta }_y}{L}.$$

Moreover, the deformations of the passive 3-DOF off-plane stage along three directions are

12

$${\theta }_1={\theta }_y,$$

13

$${\theta }_2={\theta }_z,$$

14

$${\delta }_3=L\left(1-\text{cos}\theta \right)\cong \frac{1}{2}L\text{cos}{\theta }^2,$$where ${\theta }_1$ are ${\theta }_2$ are the two passive rotational motions, and ${\delta }_3$ is the passive linear motion.

For the rotational motion about z-axis, according to the energy equation, the following relationship can be obtained:

15

$$2\left(\frac{1}{2}{F}_y{\delta }_y\right)=2\left(\frac{1}{2}{k}_y{\delta }_y^2+\frac{1}{2}{k}_2{\theta }_2^2+\frac{1}{2}{k}_3{\theta }_3^2\right).$$

Thus, the input stiffness about z-axis can be represented as

16

$$K_{\theta z}=k_y{L}^2+k_1+k_3\frac{{\delta }_z}{L}{\theta }_z^2.$$

By applying an analogous analytical methodology, the y-axis input stiffness is formulated through an energy-based derivation as follows:

17

$$K_{\theta y}=k_z{L}^2+k_2+k_3\frac{{\delta }_z}{L}{\theta }_y^2.$$

Statics Analysis

To evaluate the static and provide principles for selecting actuators, this section analyzes the stiffness analyses of the three kinds of flexible stages.

In the proposed alignment platform design, given in Figure 3(b), the 1-DOF linear stage has four flexure modules. The spatial structure and deformation characteristics of the flexible module are shown in Figure 5. The flexible beam is selected as the deformation element of the flexible module because it has good flexibility and can generate significant deformation. Each flexure has a secondary stage to reduce deformation of the flexure beam and enlarge the stroke of the primary stage. Thus, a 1-DOF linear stage can achieve a large stroke and good orientation [39].

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Figure 5

Flexure module of the 1-DOF linear stage

As shown in Figure 3(c), the 2-DOF linear stage three flexure modules along each working direction. The spatial structure and deformation characteristics of the flexible modules are shown in Figure 6. Each flexible module of the linear platform consists of two sets of deformation units, one set containing two flexible beams. Figure 7 shows the deformation characteristics of a single beam. According to the theory of beam deformation and boundary conditions, the deformation state parameters can be represented by the following equation:

18

$$M=\frac{\mathit{Fl}}{2},{\delta }_1=\frac{F{l}^3}{12EI},I=\frac{b{t}^3}{12},$$where E is the elastic modulus of the material, I is the moment of inertia of the cross section, $\delta$ is the vertical displacement of the end, and l, b, and t are the length, width, and thickness, respectively. Consequently, the stiffness of the 1-DOF linear stage can be calculated as follows:

19

$$k_x=4\frac{F_A}{{\delta }_A}=4\frac{4F}{2\delta }=\frac{8E{b}_1{t}_1^3}{l_1^3}.$$

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Figure 6

Flexure module of the 2-DOF linear stage

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Figure 7

Schematic of force analysis for flexure beam

The stiffness along the two functional directions of the 2-DOF linear platform can be expressed as

20

$$k_y=3\frac{F_y}{{\delta }_y}=3\frac{4F}{\delta }=\frac{12E{b}_2{t}_2^3}{l_2^3},$$

21

$$k_z=3\frac{F_z}{{\delta }_z}=3\frac{4F}{\delta }=\frac{12E{b}_3{t}_3^3}{l_3^3}.$$

Therefore, according to Eqs. (7)–(9), the input stiffness of the proposed device along x-, y-, and z-axes can be calculated as

22

$$K_x=\frac{8E{b}_1{t}_1^3}{l_1^3},$$

23

$$K_y=\frac{24E{b}_2{t}_2^3}{l_2^3},$$

24

$$K_z=\frac{24E{b}_3{t}_3^3}{l_3^3}.$$

According to Figure 4, when the output stage generates rotational motion about y- or z-axis, the 3-DOF off-plane stage requires two rotational motions around the two axes and one linear motion. It has a compact structural design, as shown in Figure 3(d). It is a passive flexible 2-DOF Hooke hinge. Circular flexible hinges are used to generate 3-DOF out of plane motions, as shown in Figure 8(a).

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Figure 8

(a) Flexure 3-DOF off-plane stage, (b) Force analysis of stage (quarter module)

Only the in-plane stiffness must be considered for the 1-DOF and 2-DOF linear stages. However, to derive the stiffness of the 3-DOF off-plane stage, we require a space stiffness model. Therefore, a stiffness model of the 3-DOF off-plane stage was established using the spatial flexibility matrix method. According to the force analysis of a quarter of the flexure module shown in Figure 8(b), the torque and bending moment at any position can be represented as

25

$$M\left(\theta \right)=M_x\text{sin}\theta +F_{z}R\text{cos}\theta -M_y\text{cos}\theta ,$$

26

$$T\left(\theta \right)=M_x\text{sin}\theta +F_{z}R\text{cos}\theta -M_y\text{cos}\theta ,$$where $M_x$, $M_y$, and $F_z$ represent the moments and external force applied to point B, and $R=(R_1+R_2)/2$ represents the average radius. The displacements at point B under an external force or moment can be obtained. Thus, we can inferred that

27

$$\begin{array}{cc}\hfill {\delta }_z& =\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{M\left(\theta \right)\overline{M}{\left(\theta \right)}_z}{\mathit{EI}}R\text{d}\theta +\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{T\left(\theta \right)\overline{T}{\left(\theta \right)}_z}{G{I}_P}R\text{d}\theta \hfill \\ \hfill & =\left[\frac{\pi R^3}{4EI}+\frac{\left(3\pi -8\right)R^3}{4G{I}_P}\right]F_z+\left(\frac{R^2}{2EI}+\frac{R^2}{2G{I}_P}\right)M_x\hfill \\ \hfill & +\left[-\frac{\pi R^2}{4EI}+\frac{\left(4-\pi \right)R^2}{4G{I}_P}\right]M_y\hfill \\ \hfill & =c_{11}{F}_z+c_{12}{M}_x+c_{13}{M}_y,\hfill \\ \hfill \end{array}$$where $\overline{M}{(\theta )}_z$ and $\overline{T}{(\theta )}_z$ represents the corresponding unit loads, and G is the shear modulus of the material. The output angles about x- and y-axes can be represented similarly:

28

$${\theta }_x=c_{11}{F}_z+c_{12}{M}_x+c_{13}{M}_y,$$

29

$${\theta }_y=c_{31}{F}_z+c_{32}{M}_x+c_{33}{M}_y.$$

Considering Eqs. (27)–(29), the expression between deformation and force of flexible components can be obtained:

30

$$\left(\begin{array}{c}{\delta }_z\\ \theta x\\ {\theta }_y\end{array}\right)=\left(\begin{array}{ccc}{c}_{11}& c_{12}& c_{13}\\ c_{21}& c_{22}& c_{23}\\ c_{31}& c_{32}& c_{33}\end{array}\right)\left(\begin{array}{c}{F}_z\\ M_x\\ M_y\end{array}\right)={\mathit{C}}_0\left(\begin{array}{c}{F}_z\\ M_x\\ M_y\end{array}\right),$$where ${\mathit{C}}_0$ denotes the compliance matrix of the quarter module. Because the stiffness matrix is the inverse matrix of the flexibility matrix, it can be expressed as

31

$${\mathit{K}}_0={\mathit{C}}_0^{-1}=\left(\begin{array}{ccc}{c}_{11}& c_{12}& c_{13}\\ c_{21}& c_{22}& c_{23}\\ c_{31}& c_{32}& c_{33}\end{array}\right).$$

Based on the method of spatial matrix transformation, combined with the above derivation, the stiffness matrix of the 3-DOF off-plane stage can be calculated. Owing to the structural symmetry of circular flexible hinges, we can obtain the stiffness matrix in coordinate $I_{-xyz}$:

32

$$\mathit{K}=\text{diag}\left(\begin{array}{ccc}{k}_1& k_2& k_3\end{array}\right),$$where $K_1=4k_{11}$ denotes the linear stiffness along the z-axis, $K_2=K_3=4[k_{12}^2+(R^2{k}_{11}-4R{k}_{31}-4R{k}_{13}+4k_{33}{)}^2{]}^{\frac{1}{2}}$ denote the rotational stiffnesses about x-axes and y-axes. The input stiffness can be derived by substituting $k_1$, $k_2$, $k_3$ into Eqs. (16) and (17).

Dynamic Analysis

The dynamic characteristics are important indicators of the response speed of error compensation in a reaction system. Consequently, a dynamic analysis is conducted to evaluate the dynamic performance of the proposed alignment stage. In this paper, the dynamic model of the device in five working directions is used to obtain the first five natural frequencies. Using the PRBM method, all linkages are considered to be rigid bodies, and only the deformation of flexible hinges is considered. Therefore, the corresponding energy can be calculated using the displacement and rotation angle of the connecting rod. The kinetic energy of the proposed device along the five functional directions can be calculated as

33

$$\begin{array}{c}{T}_1=\frac{1}{2}{m}_1{\dot{\delta }}_1^2+8\frac{1}{2}{m}_2{\left(\frac{{\dot{\delta }}_1}{2}\right)}^2+32\frac{1}{2}{m}_3{\left(\frac{{\dot{\delta }}_1}{4}\right)}^2+2\frac{1}{2}{m}_4{\dot{\delta }}_1^2\\ +2\frac{1}{2}{m}_5{\dot{\delta }}_1^2++\frac{1}{2}{m}_6{\dot{\delta }}_1^2,\\ \end{array}$$

34

$$\begin{array}{c}{T}_2=2\frac{1}{2}{m}_5{\dot{\delta }}_2^2+\frac{1}{2}{m}_6{\dot{\delta }}_2^2+2\frac{1}{2}{m}_7{\dot{\delta }}_2^2+4\frac{1}{2}{m}_8{\dot{\delta }}_2^2\\ +24\frac{1}{2}{m}_9{\left(\frac{{\dot{\delta }}_2}{2}\right)}^2,\\ \end{array}$$

35

$$\begin{array}{c}{T}_3=2\frac{1}{2}{m}_5{\dot{\delta }}_3^2+\frac{1}{2}{m}_6{\dot{\delta }}_3^2+2\frac{1}{2}{m}_7{\dot{\delta }}_3^2+4\frac{1}{2}{m}_8{\dot{\delta }}_3^2\\ +24\frac{1}{2}{m}_9{\left(\frac{{\dot{\delta }}_3}{2}\right)}^2,\\ \end{array}$$

36

$$\begin{array}{c}{T}_4=2\frac{1}{2}{m}_7{\dot{\delta }}_4^2+4\frac{1}{2}{m}_8{\dot{\delta }}_4^2+2\frac{1}{2}{m}_{10}{\dot{\delta }}_4^2+24\frac{1}{2}{m}_9{\left(\frac{{\dot{\delta }}_4}{2}\right)}^2\\ +\frac{1}{2}{I}_1{\dot{\theta }}_4^2,\\ \end{array}$$

37

$$\begin{array}{c}{T}_5=2\frac{1}{2}{m}_7{\dot{\delta }}_5^2+4\frac{1}{2}{m}_8{\dot{\delta }}_5^2+2\frac{1}{2}{m}_{10}{\dot{\delta }}_5^2+24\frac{1}{2}{m}_9{\left(\frac{{\dot{\delta }}_5}{2}\right)}^2\\ +\frac{1}{2}{I}_2{\dot{\theta }}_5^2,\\ \end{array}$$where $m_1$, $m_2$, and $m_3$ are the masses of output stage, motion stage, and flexure hinge, respectively, of the 1-DOF linear stage. $m_4$, $m_5$, and $m_6$ are the masses of the 2-DOF linear, 3-DOF off-plane, and output stages, respectively. $m_7$, $m_8$, and $m_9$ are the masses of the output stage, input stage, and flexure hinge of the 2-DOF linear stage, respectively. $m_{10}$ is the mass of half of the 3-DOF off-plane stage. $I_1$ and $I_2$ are the moments of inertia along two rotational directions. Therefore, the kinetic energy of the proposed alignment mechanism can be calculated as

38

$$T=T_1+T_2+T_3+T_4+T_5.$$

Moreover, the potential energy of the proposed device along the five functional direction can be expressed as

39

$$U_1=\frac{1}{2}{K}_x{\delta }_1^2,$$

40

$$U_2=\frac{1}{2}{K}_y{\delta }_2^2,$$

41

$$U_3=\frac{1}{2}{K}_z{\delta }_3^2,$$

42

$$U_4=\frac{1}{2}{K}_{{\theta }_y}{\delta }_4^2,$$

43

$$U_5=\frac{1}{2}{K}_{{\theta }_y}{\delta }_5^2,$$

44

$$U=U_1+U_2+U_3+U_4+U_5.$$

To obtain the dynamic equations of the system, this paper uses the Lagrange equation, which takes the following form:

45

$$\frac{\text{d}}{\text{d}x}\left(\frac{\partial T}{\partial {\dot{q}}_i}\right)-\frac{\partial T}{\partial {\dot{q}}_i}+\frac{\partial U}{\partial {\dot{q}}_i}=F_i,\left(i=1,2,\dots ,N\right),$$where $q_i$ denotes the vector of linearly independent generalized coordinates. N corresponds to the dimensionality of the generalized coordinate space (specifically, N = 5 for the proposed 5-DOF alignment stage) and $F_i$ denotes the externally applied force vector. Under free-vibration conditions, the external force term $F_i$ is nullified through the boundary constraint enforcement. By substituting Eqs. (38) and (44) into Eq. (45), the characteristic free-motion dynamic equation is derived as

46

$$\mathit{M}\ddot{\mathit{q}}+\mathit{Kq}=\mathbf{0},$$where M and K represent the mass and stiffness matrices of the dynamic system, respectively, which can be expressed as

47

$$\mathit{M}=\text{diag}\left\{\begin{array}{ccc}{M}_{11}& M_{22}& M_{33}\end{array}\begin{array}{cc}{M}_{44}& M_{55}\end{array}\right\},$$

48

$$\mathit{K}=\text{diag}\left\{\begin{array}{ccc}{K}_{11}& K_{22}& K_{33}\end{array}\begin{array}{cc}{K}_{44}& K_{55}\end{array}\right\}.$$

Based on the above dynamic equations and vibration theory, the characteristic equation of the system can be derived as

49

$$\left|\mathit{K}-{\omega }_j^2\mathit{M}\right|=0,$$where ${\omega }_i(i=1,2,3,4,5)$ represents the corresponding natural cycle frequency of the system. Thus, the natural frequency can be obtained as $f_i=(1/2\pi ){\omega }_j$.

Parameter Selection and FEA Validation

Parameter Selection

For good dynamic characteristics of the proposed the 5-DOF flexible alignment stage, the first natural frequency should be as high as possible to guarantee the control bandwidth. Additionally, the output motion range, stiffness, and structural size of the device must be constrained. The output motion range of the stage should be sufficiently large to compensate for the printer posture errors. The input stiffness of the device in all the working directions should not exceed the output stiffness of the actuators. The structural size of the stage should ensure that it can achieve web printing of 150 mm. For the actual printing process, the proposed stage is only responsible for compensating for posture errors. Large-scale printing is performed using a large-stroke XYZ linear motion stage. Therefore, according to error analysis, the output motion range of the proposed stage should be larger than 300 µm × 300 µm × 300 µm × 2.5 mrad × 2.5 mrad in five working directions according to error analysis. Moreover, the first natural frequency of the device should exceed 60 Hz for real-time compensation. Owing to its high strength, high elasticity, and low density, aluminum alloy (Al 7075-T6) is very suitable for processing flexible mechanisms and was selected as the processing material for this stage. The key dimensional parameters and inherent material properties are listed in Table 1.

Table 1. Dimensional parameters and material properties of the proposed stage

Parameters		Values
Flexure hinge of 1-DOF stage ($l_1\times b_1\times t_1$)		$22\text{mm}\times 13\text{mm}\times 0.8\text{mm}$
Flexure hinge of 2-DOF stage ($l_2\times b_2\times t_2$)		$18\text{mm}\times 8\text{mm}\times 0.4\text{mm}$
Flexure hinge of 3-DOF mechanism ($R\times b_3\times t_3$)		$11\text{mm}\times 6\text{mm}\times 0.6\text{mm}$
Rotation radius of the output stage L		$118.2\text{mm}$
Operating space of the output stage $l_0$		$25\text{mm}$
Density $\rho$		2810 $\text{kg}/{\text{m}}^3$
Yield strength ${\sigma }_y$		503 MPa
Young’s modulus E		71.7 GPa
Poisson ratio $v$		0.33

FEA Validation

To validate the performance of the proposed stage, we conducted FEA simulations using ANSYS Workbench 16.0. A 3D model was constructed using the 3D software SolidWorks 2023. This section examines the deformation, stiffness, center shift, and modal analysis. First, a deformation analysis was performed. With input forces applied at the input position, the deformation results along five directions are depicted in Figure 9. To observe the RCM characteristics more intuitively, we designed a conical cover with its tip of the conical cover coinciding with the RCM point at the output. According to the simulation results, the output motion of the RCM platform matched the design requirements, verifying the effectiveness of the proposed device. Additionally, based on the parameters of the linear motors and VCMs, the maximum stress was measured when the maximum output force was applied at the input position. The FEA results showed that stress only occurred at the flexible hinge, with a maximum stress of 219.28 MPa, indicating that the safety factor of the material was at least 2.29.

[See PDF for image]

Figure 9

Deformation of the proposed stage along (a) x direction, (b) y direction, (c) z direction, (d) ${\theta }_y$ direction, and (e) ${\theta }_z$ direction

Moreover, when specified input forces were applied to the input position of the flexible stage, the corresponding displacement or rotation angle of the platform output could be obtained, and the stiffness of the proposed platform in five directions could be calculated based on these data. The stiffness results at this stage are listed in Table 2. We observed that the linear stiffness along y-axis and z-axis were equal, whereas the rotational stiffness about y-axis and z-axis were equal. Compared with the results of theoretical modeling analysis, the maximum deviation of FEA results was 13.29%. This deviation was primarily caused by the nonlinear characteristics of flexible components and model errors of the PRBM. However, the stiffness performance of the proposed device achieved the design goals.

Table 2. Stiffness performance of the proposed stage

Stiffness	Unit	Theoretical	FEA	Deviation (%)
$K_x$	$\text{N}/\mathrm{\mu }\text{m}$	0.359	0.401	10.47
$K_y$	$\text{N}/\mathrm{\mu }\text{m}$	0.151	0.167	9.58
$K_z$	$\text{N}/\mathrm{\mu }\text{m}$	0.151	0.171	11.7
${\theta }_y$	$\text{N}·\text{m}/\text{rad}$	2.121	2.401	11.66
${\theta }_z$	$\text{N}·\text{m}/\text{rad}$	2.121	2.446	13.29

A center shift was detected to evaluate the rotational accuracy of the device. With the output stage rotating from 0 to 5 mrad about y- and z-axes, respectively, the central shift of the conical cap vertex in the corresponding direction was measured. As shown in Figure 10, when a rotational motion of 5 mrad was generated about y-axis at the output end, the maximum offset value was detected, which was less than 1 µm. Compared with the rotational displacement output $d_0$ by the functional end, which can be given by $d_0=l_0{\theta }_{\mathit{out}}$, the maximum center offset was less than 0.8%. This indicated that the alignment stage exhibited good motion-decoupling performance.

[See PDF for image]

Figure 10

Center shift of the proposed stage for rotation about (a) y-axis and (b) z-axis

A modal analysis was conducted on the dynamic characteristics of the proposed platform using FEA software. In the free vibration state (without actuating elements), the first five vibration modes of the device are shown in Figure 11, and the first five corresponding resonance were 106.86, 125.63, 129.74, 151.22, and 154.77 Hz, respectively. The first modal shape was a linear motion along x-axis. The second and third mode shapes were linear motions along the y- and z-axes. The fourth and fifth mode shapes were the rotation motions around the y- and z-axes. The first five corresponding resonances calculated using the dynamic model were 106.5, 129.93, 129.93, 138.82, and 142.29 Hz, respectively. The maximum deviation between the theoretical model and FEA results was 8.77%, which was within the allowable range.

[See PDF for image]

Figure 11

Modal analysis of the proposed stage

To assess the risk of the fatigue failure of flexible mechanisms, we conducted a fatigue analysis on the platform. By applying an alternating force corresponding to the full stroke at the input end, we obtained the fatigue analysis results of the mechanism, as shown in Figure 12. Under the action of alternating stress with a maximum stress amplitude of 125.7 N, the minimum number of fatigue cycles of the mechanism was 1.52×10⁶, exceeding the design requirement of 1×10⁶.

[See PDF for image]

Figure 12

Fatigue analysis of the proposed stage: (a) Stress distribution, (b) Fatigue life

Experimental Results

Prototype Development

The 5-DOF flexible alignment stage prototype is fabricated by machining. The experimental setup and sensor arrangement are shown in Figure 13. The experimental system consisted of the proposed alignment stage, capacitive sensor, laser sensor for actuator calibration, output motion measurement, sensor controllers, VCM controller, linear motion driver, and DC power. In addition, to facilitate measurement of the output motion, we installed a measurement block at the output end of the stage. Before the experiment, we tested the environmental noise values of the laser and capacitive sensors, which were 0.06 and 0.02 µm, respectively.

[See PDF for image]

Figure 13

(a) Set up of the experimental system, (b) Sensor arrangement

Four VCMs (XVLC80-06-00A, XIVI, Inc.) numbered from 1 to 4 in Figure 13(b) and a linear motor (DRS42SB2-04 KA, Oriental Motor, Inc.) were selected as the actuating components for the 2-DOF and 1-DOF linear stages, respectively. The maximum output force and stroke of the VCMs were 80 N and 6.3 mm, respectively. The maximum output force and stroke of the linear motor were 150 N and 40 mm, respectively. Owing to the large displacement of the alignment stage, a laser sensor (LK-H020, KEYENCE, Inc.) was used to measure the output motion along the working direction. Considering the measurement accuracy, capacitive sensors (CPL190, probe model: C8-2.0-2.0, from Lion Precision, Inc.) were used to calibrate the output displacement and measure the output coupling errors. For rotational motions, the measurement of the output angle were achieved using a laser sensor or capacitive sensor, as shown in Figure 14, and calculated as

50

$${\theta }_m=\frac{{\delta }_m}{l_m},$$where ${\delta }_m$ represents the output displacement measured by sensor, and $l_m$ represents the distance between the rotation center and the measurement point of sensor. The sensor was installed through the designed components to achieve positioning with the measured part and obtain an accurate distance $l_m$.

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Figure 14

Working principle of the output angle measuring

Experimental Tests

Before the experiment, the displacement outputs of the two actuators were calibrated. Linear motors controlled the displacement output through input pulse signals, whereas the VCM controlled the displacement output through the voltage. Laser and capacitive sensors were used to calibrate the corresponding relationship between the input signal and output displacement of the linear motors and VCMs. Based on the calibration results, the input–output corresponding relationship of the linear motor had good linearity, whereas the VCM was slightly nonlinear, as shown in Figure 15. Therefore, the testing components used in the experiment could accurately measure the relationship between the input and output displacements of the proposed stage, thereby verifying the performance of the alignment stage. Owing to the need to verify the performance of the mechanism, including the stroke, input–output relationship, and coupling error, open-loop control was adopted in all the experiments in this study.

[See PDF for image]

Figure 15

Relationship between input voltage and displacement of VCMs

Second, the working space was tested along five directions. For the linear motor, given a group of input pulses, the test results for the output displacement along x-axis are shown in Figure 16(a). For the VCMs, given a group of input pulses of 1 V at each step, the test results of the output linear motions along y- and z-axes and the output rotational motions about y- and z-axes are shown in Figure 16(b), (c), (d), and (e), respectively. As shown in Figure 16(a), the maximum output range along x-axis was approximately 335.1 µm. Moreover, the input–output relationship exhibited good linearity. The deviation between the experimental and FEA results was primarily owing to the difference between the actual and theoretical outputs of the linear motor, as well as errors in the machining and assembly of the prototype. The maximum output ranges along y- and z-axes were 418.9 and 408.1 µm, respectively. The results indicated that when the input voltage increased, the linearity of the output displacement decreased. According to the previous calibration results, this was primarily caused by the nonlinear relationship between the input voltage and output displacement of the VCMs. The deviation between the experimental and FEA results primarily caused by the nonlinearity of the VCM displacement output, which can be reduced using precise closed-loop control algorithms.

[See PDF for image]

Figure 16

Workspace of the proposed stage along the different directions

Third, to verify the decoupling performance of the 2-DOF linear stage, we measured the coupling error between motions along y-axis and z-axis. A laser sensor was used to measure the output displacement in one direction and a capacitive sensor was used to measure the coupling error in the other direction. The coupling errors of the proposed device are shown in Figure 17. When the maxi-mum output displacement was 400 µm. The coupling error was largest at approximately 0.42 µm. Therefore, the coupling error of the stage was less than 0.11%, which confirmed that the 2-DOF linear stage had good decoupling performance.

[See PDF for image]

Figure 17

Coupling errors of the proposed stage along (a) y-axis and (b) z-axis

Finally, the natural frequency of the stage was tested. After the platform actuator was removed, an impulse force was applied to the measuring block using a modal hammer. The striking point and application direction of the modal hammer are indicated by red arrows in Figure 18(a). Laser sensors were used to measure the amplitude of the mechanism after applying an impact force. The collected data were analyzed using fast Fourier transform in MATLAB. The time and corresponding frequency responses are shown in Figure 18(b) and (c), respectively. The analysis results indicated that the first five resonance frequencies were 78.4, 90.2, 98.8, 138, and 139.8 Hz. Compared with the theoretical and FEA results, the experimental results were lower, which was primarily caused by the additional mass of the measuring blocks and bolts. However, the deviation was within a reasonable range.

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Figure 18

Frequency test: (a) Impact point and applying direction, (b) Time response, (c) Frequency response

Table 3 provides a performance comparison of the developed alignment stage with several representative alignment stages, focusing on key parameters such as displacement and force resolutions.

Table 3. Performance comparison with other alignment stages

Reference	DOF	Workspace	Coupling error (%)
[7]	5-DOF	$2000\mathrm{\mu }\text{m}\times 2000\mathrm{\mu }\text{m}\times 20000\mathrm{\mu }\text{m}\times 2\text{mrad}\times 2\text{mrad}$	0.5
[34]	6-DOF	$77.42\mathrm{\mu }\text{m}\times 67.45\mathrm{\mu }\text{m}\times 24.56\mathrm{\mu }\text{m}\times 0.93\text{mrad}\times 0.93\text{mrad}\times 0.93\text{mrad}$	–
[35]	6-DOF	$240\mathrm{\mu }\text{m}\times 240\mathrm{\mu }\text{m}\times 240\mathrm{\mu }\text{m}\times 2.5\text{mrad}\times 2.5\text{mrad}\times 2.5\text{mrad}$	1.8
[36]	5-DOF	$143\mathrm{\mu }\text{m}\times 142\mathrm{\mu }\text{m}\times 212\mathrm{\mu }\text{m}\times 0.111\text{mrad}\times 0.109\text{mrad}$	–
This work	5-DOF	$335.1\mathrm{\mu }\text{m}\times 418.9\mathrm{\mu }\text{m}\times 408.1\mathrm{\mu }\text{m}\times 3.4\text{mrad}\times 3.29\text{mrad}$	0.11

Conclusions

1. A novel 5-DOF flexure-based alignment stage to adjust the posture of the inkjet printer head is designed, which is composed of a 1-DOF linear stage, two 2-DOF linear stages and two 3-DOF off-plane stages. The parasitic errors of the mechanism are effectively reduced while ensuring structural stiffness and accuracy by adopting a combination of series and parallel designs and a decoupling design. Consequently, the proposed stage can achieve output motion in five working directions.

2. The PRBM method was used to model the flexible driving structure of the micro-positioning platform. A theoretical analysis of the kinematics, stiffness, and dynamics at this stage was conducted. The performance of the platform was verified using FEA.

3. A prototype was developed for experimental research. The prototype test results show that the developed positioning platform attains 5-DOF motion capabilities with 335.1 µm × 418.9 µm × 408.1 µm × 3.4 mrad × 3.29 mrad with output coupling of less than 0.11% along the y- and z-axes, which satisfy the compensating requirements. Moreover, a dynamic test validated that the proposed stage can compensate for position errors in real time.

Acknowledgements

Not applicable.

Authors' Contributions

Shang Yang wrote the manuscript; Lei Wang was in charge of the whole trial; Weihai Chen proposed ideas and edited revised the manuscript; Shang Yang, Yuxia Li and Xiantao Sun was responsible for prototype experiments and testing; Yuxia Li was responsible for experimental data processing; Shasha Chen gave some advice on the manuscript. All authors read and approved the final manuscript.

Funding

Supported by Natural Science Research Project of Anhui Educational Committee (Grant No. 2024AH040010).

Data availability

Not applicable.

Competing Interests

The authors declare no competing financial interests.

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