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

Design, Analysis, and Testing of a Novel 5-DOF Flexure-Based Alignment Stage A high pattern resolution is critical for fabricating roll-to-roll printed electronics (R2RPE) products. For enhanced over- lay 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 align- ment 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 sub- structures. 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 pro- poses 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. Keywords Flexible mechanism, Micro-positioning, Multi-DOF, Alignment stage R2RPE products will not satisfy the requirements of the 1 Introduction high-end market. Therefore, high-precision multi-layer Roll-to-roll printed electronics (R2RPE) has been vali- alignment systems are urgently required. dated as a viable method for manufacturing a various The production process of the R2RPE primarily electronic devices [1–4]. Considering electronic devices includes gravure printing, flexographic printing, reverse produced by traditional methods, R2RPE products have offset printing, screen printing, and inkjet printing [9– the advantages of having a large area and being fast, 13]. Within the spectrum of manufacturing techniques, flexible, and inexpensive [5, 6]. For multi-layer printed inkjet printing distinguishes itself through non-contact electronic products, the accuracy of multi-layer print- deposition, maskless fabrication capabilities, and mate- ing is crucial, as different materials must be printed on rial-efficient operation while maintaining a high feature each layer to achieve different structures, which directly resolution critical for micro-scale patterning applica- affects the performance of multi-layer printed electronic tions [14, 15]. Multi-layer registration accuracy should products [7, 8]. If the accuracy of the multi-layer align- be guaranteed to obtain high-resolution inkjet printing ment cannot be guaranteed, the pattern resolution of patterns of R2RPE. However, the accuracy of multi-layer registration can be affected if a relative position error *Correspondence: occurs between the printer and the flexible web during Lei Wang [email protected] the printing process. School of Automation Science and Electrical Engineering, Beihang The precision of inkjet printing processes is critically University, Beijing 100191, China challenged by multi-source registration errors between School of Electrical Engineering and Automation, Anhui University, Hefei 230039, China the printhead assembly and advancing flexible web © The Author(s) 2025. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/. Yang et al. Chinese Journal of Mechanical Engineering (2025) 38:172 Page 2 of 15 substrates. For example, unstable web transport speed system with multiple-input multiple-output (MIMO) or uneven web stretching caused by nonuniform web closed-loop control that achieves submicron-level align- tension can result in web position errors [16, 17]. The ment precision for large-scale continuous printing pro- posture error of the support rollers can also affect web cesses. Chen et  al. [19] proposed a motion stage with a position [18]. Moreover, the posture error of the printer remote center of motion to adjust the roller posture. has a significant impact on the alignment accuracy [19, The proposed device supports rollers using air spherical 20]. Therefore, multi-layer alignment systems must be bearings with high stiffness and a flexible mechanism to adopted for high-resolution inkjet printing. generate accurate motion. Thus, the roller posture can be Numerous studies have been conducted to improve the adjusted, and the web position error can be compensated. accuracy of the multi-layer alignment, and certain results Many scholars have designed various types of alignment have been achieved. In the field of web tension control, mechanisms for error compensation in different scenar - Kang et  al. [21, 22] proposed a new theoretical model ios and have achieved good results [34–36]. that considers the lateral position errors of the substrate The literature review shows that many scholars have web and roller and provides compensation methods. Kim designed various types of alignment devices and achieved et al. [23, 24] proposed a new design of roll-to-roll (R2R) many research results. However, current alignment printing equipment for R2RPE production. The designed devices are primarily concerned with compensating for R2RPE system consists of tension control components the position error of the rollers. For inkjet printing, the such as feeders, load cells, and charge-coupled device positional error of the printer head should be adjusted. cameras to detect the relative position errors of patterns Based on the above considerations, a novel five-degree- for high-precision printing. Jeong et  al. [25] proposed a of-freedom (5-DOF) flexible alignment stage is proposed tension model for each section to successfully predict to adjust the posture of the printer head in real time. The the tension applied to such a system, the sagging of the proposed stage consists of a various flexible modules con - film according to tension, and deformation due to resid - nected in serial and parallel. The dynamic performance ual stress, and built an accurate R2R system to minimize of the proposed device has been significantly enhanced tension reduction. Lee et  al. [26] proposed an advanced through novel structural design. The remainder of this model to determine the tension disturbances caused by paper is organized as follows: Section 2 details the design run-out resulting from the axis mismatch, roundness considerations and architectural configuration of the error, imbalance, and velocity variation of the rollers in developed stage. Section 3 introduces a theoretical analy- an industrial-scale R2R printing process, and a high aver- sis of the kinematics, stiffness, and dynamics of the pro - age accuracy of 92.4% was achieved. For position error posed stage. Parametric optimization and finite element compensation based on the above methods, a traditional analysis (FEA) validation are systematically addressed in rigid hinge and motion stage are typically adopted to Section  4. Experimental verification through prototype adjust the posture of a roller or printer. The multi-layer implementation and performance characterization is alignment accuracy is limited to 40 µm resolution [27]. addressed in Section 5. The concluding remarks are given For the posture error of the printer or roller to be elimi- in Section 6. nated with high accuracy, a flexible mechanism can be used to design compensation mechanisms, leveraging 2 Mechanism Design inherent advantages such as frictionless operation, back-2.1 Design Consideration lash elimination, and lubrication-free monolithic design When processing of inkjet-printed electronics, multiple [28–30]. Baldesi et  al. [31] engineered an R2R-compati- layers of patterns should be printed on the flexible web. ble compliant-stage printing system, leveraging elastic Therefore, a high overlay accuracy should be achieved deformation principles to eliminate backlash and slid- to ensure printing resolution. Hence, the position error ing friction. The flexible mechanism is driven manually between the inkjet printer and existing web should be by micrometer heads to eliminate the position error of compensated. The position error is primarily caused by the roller. In addition, Zhou et  al. [32, 33] engineered a the position deviation between the different support roll - flexible R2R printing system for real-time adjustment of ers. In R2R inkjet printing, rotation about the x-axis is roller posture. The flexible mechanisms are driven by lin - inherently constrained by the web transport mechanism, ear stepper motors and voice coil motors (VCMs). In the making compensation of θ unnecessary for alignment. design of such flexible mechanisms, the output motion in u Th s, the 5-DOF design focuses on critical errors in θ , working directions are not decoupled. In addition, flex - θ , and translations [7]. As shown in Figure  1, the posi- ible mechanisms are often designed to withstand loads, tion error can be divided into three linear errors and two which is an important factor affecting the stability of rotational errors. Two methods can be used to compen- alignment stage. Li et al. [7] developed a multi-layer R2R sate for the position errors. One is to change the position Y ang et al. Chinese Journal of Mechanical Engineering (2025) 38:172 Page 3 of 15 Figure 1 Position errors between the inkjet printer and flexible web 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 com- pensated, a 5-DOF alignment stage should be adopted to adjust the posture of the inkjet printer. The three translational position errors can be com - pensated for by adopting a decoupled XYZ motion Figure 2 Alignment processes with (a) inner rotation center and (b) stage. The rotational position error around z- axis can external rotation center be compensated for by adopting a θ motion stage. Compensation for rotational errors around y-axis may introduce additional parasitic errors. As shown in Fig- in the design stage should be suppressed to ensure ure 2(a), if the rotation center O is inside the alignment the error compensation accuracy of the alignment stage, the inkjet printer, which is the output end of the mechanism and simplify the control complexity of the alignment stage, will generate lateral offsets along x - mechanism. In this design, the main measures to sup- and z-axes. To avoid this type of coupled motion, the press parasitic errors are as follows: (1) By designing rotation center should coincide with the center point symmetrical and decoupled mechanisms, the coupling of the underside of the inkjet printer, as shown in Fig- of parallel motion is reduced and parasitic errors are ure  2(b). Therefore, a flexible rotary-motion platform simulated; (2) through the design of the RCM mecha- should have the characteristics of a remote motion nism, the rotation center of the rotating mechanism is center (RCM) [37, 38]. Therefore, the parasitic motion aligned with the surface of the inkjet head. In addition, Yang et al. Chinese Journal of Mechanical Engineering (2025) 38:172 Page 4 of 15 the alignment stage must exhibit good dynamic perfor- can generate 5-DOF motions in the required directions mance to obtain the real-time and fast pose compensa- to achieve the alignment function. Moreover, the axes of tion capability of inkjet printers. the 5-DOF motion intersect at one point, which coincides with the center point of the underside of the inkjet printer. 2.2 Structure Description Therefore, the 5-DOF motions are decoupled from each Based on the above considerations, a decoupled 5-DOF other. flexible alignment stage for R2R inkjet printing is pro - Considering the symmetry of the structure while ensur- posed. The device primarily consists of a 1-DOF linear ing equivalent structural quality and not affecting the stage, two 2-DOF linear stages and two 3-DOF off-plane performance evaluation of the alignment mechanism, the stages, as shown in Figure 3. The 1-DOF linear stage can structure and layout were optimized by inverting the cross- actively generate translational motion along x-axis using beam used for installing the inkjet head in the middle to a linear motor. The two 2-DOF linear stages are driven facilitate the display of a finite element simulation analysis using four VCMs. The two 3-DOF off-plane stages are and performance evaluation results. designed to satisfy the requirements of motion output. Therefore, the 2-DOF linear and 3-DOF off-plane stages 3 Theoretical Analysis are combined to form a parallel 4-DOF stage to gener-3.1 Kinematic Analysis ate motions along y and z-axes and motions around y and Based on the introduction, flexible mechanisms are used to z-axes. The 1-DOF linear stage is connected to the 4-DOF adjust the pose of a printer. To evaluate the kinematics of motion stage in series. Therefore, the proposed device 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 sim- plified model of the 5-DOF flexible alignment stage can be given as shown in Figure 4. By combining the input forces F , F , F , F , and F , the output stage generates motion x y1 y2 z1 z2 along the required directions. Moreover, the output point O coincides with the underside center point of the inkjet printer, which is the RCM point. When applying input force/displacement ( F /δ ) to the x x 1-DOF linear stage, the output stage can generate output displacement along x-axis, which is given by out Figure 3 Mechanical design of the 5-DOF flexible alignment stage: δ = δ . (1) (a) Overall view, (b) 1-DOF linear stage, (c) 2-DOF linear stage, (d) 3-DOF off-plane stage The input force F can be calculated as Figure 4 Simplified model of the proposed device Y ang et al. Chinese Journal of Mechanical Engineering (2025) 38:172 Page 5 of 15 F = k δ , ∼ x x x (2) δ = L(1 − cosθ) = Lcosθ , (14) where k is the stiffness of the 1-DOF linear stage. where θ are θ are the two passive rotational motions, 1 2 When a couple of input forces/displacements ( F , y1 and δ is the passive linear motion. F /δ , δ ) are applied to the two 2-DOF linear stage with y2 y1 x For the rotational motion about z-axis, according to F = F = F and δ = δ = δ , the output stage can y1 y2 y y1 y2 y the energy equation, the following relationship can be generate output displacement along y-axis, which is given obtained: by 1 1 1 1 out 2 2 2 δ = δ . y (3) 2 F δ = 2 k δ + k θ + k θ . y (15) y y y 2 3 y 2 3 2 2 2 2 Using the same method, the output stage can generate u Th s, the input stiffness about z- axis can be repre- the output displacement along z-axis, which is given by sented as out δ = δ . z (4) 2 2 K = k L + k + k θ . (16) θz y 1 3 The input force F and F can be derived as y z By applying an analogous analytical methodology, the F = k δ , y y y (5) y-axis input stiffness is formulated through an energy- based derivation as follows: F = k δ , z z z (6) 2 2 K = k L + k + k θ . (17) where k , k are the stiffnesses of the 2-DOF linear stage θy z 2 3 y z y along y- and z-axes, respectively. Therefore, the input stiffness can be deduced as 3.2 Statics Analysis K = k , x x (7) To evaluate the static and provide principles for select- ing actuators, this section analyzes the stiffness analyses K = 2k , y y (8) 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 mod - K = 2k . (9) z z ules. The spatial structure and deformation character - For the output rotational motion around y-axis, the istics of the flexible module are shown in Figure  5. The input forces/displacements ( F , F /δ , δ ) are given by z1 z2 z1 z2 flexible beam is selected as the deformation element of F =−F and δ =−δ . Thus, the output rotational z1 z2 z1 z2 the flexible module because it has good flexibility and motion θ can be derived as can generate significant deformation. Each flexure has a secondary stage to reduce deformation of the flexure θ = arcsin , (10) y beam and enlarge the stroke of the primary stage. Thus, a 1-DOF linear stage can achieve a large stroke and good where L is the rotation radius of the stage. orientation [39]. For the output rotational motion about z-axis, the As shown in Figure  3(c), the 2-DOF linear stage three input forces/displacements ( F , F /δ , δ ) are given by flexure modules along each working direction. The y1 y2 y1 y2 F =−F and δ =−δ . Thus, the output rotational spatial structure and deformation characteristics of y1 y2 y1 y2 motion θ can be deduced as θ = arcsin . (11) Moreover, the deformations of the passive 3-DOF off- plane stage along three directions are θ = θ , 1 y (12) θ = θ , 2 z (13) Figure 5 Flexure module of the 1-DOF linear stage Yang et al. Chinese Journal of Mechanical Engineering (2025) 38:172 Page 6 of 15 Therefore, according to Eqs. (7)–(9), the input stiffness of the proposed device along x-, y-, and z-axes can be cal- culated as 8Eb t K = , (22) 24Eb t K = , y (23) Figure 6 Flexure module of the 2-DOF linear stage 24Eb t K = . z (24) According to Figure  4, when the output stage gener- ates 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 Figure 7 Schematic of force analysis for flexure beam are used to generate 3-DOF out of plane motions, as shown in Figure 8(a). Only the in-plane stiffness must be considered for the the flexible modules are shown in Figure  6. Each flex - 1-DOF and 2-DOF linear stages. However, to derive ible module of the linear platform consists of two sets of the stiffness of the 3-DOF off-plane stage, we require deformation units, one set containing two flexible beams. a space stiffness model. Therefore, a stiffness model Figure 7 shows the deformation characteristics of a single of the 3-DOF off-plane stage was established using beam. According to the theory of beam deformation and the spatial flexibility matrix method. According to the boundary conditions, the deformation state parameters force analysis of a quarter of the flexure module shown can be represented by the following equation: in Figure  8(b), the torque and bending moment at any position can be represented as 3 3 Fl Fl bt M = , δ = , I = , (18) M(θ ) = M sinθ + F Rcosθ − M cosθ, x z y 2 12EI 12 (25) where E is the elastic modulus of the material, I is the T (θ ) = M sinθ + F Rcosθ − M cosθ, x z y (26) moment of inertia of the cross section, δ is the vertical displacement of the end, and l, b, and t are the length, where M , M , and F represent the moments and exter- x y z width, and thickness, respectively. Consequently, the nal force applied to point B, and R = (R + R )/2 rep- 1 2 stiffness of the 1-DOF linear stage can be calculated as resents the average radius. The displacements at point follows: B under an external force or moment can be obtained. 3 u Th s, we can inferred that F 4F 8Eb t A 1 k = 4 = 4 = . x (19) δ 2δ l The stiffness along the two functional directions of the 2-DOF linear platform can be expressed as F 4F 12Eb t y 2 k = 3 = 3 = , y (20) δ δ l F 4F 12Eb t z 3 k = 3 = 3 = . z (21) Figure 8 (a) Flexure 3-DOF off-plane stage, (b) Force analysis δ δ z l of stage (quarter module) Y ang et al. Chinese Journal of Mechanical Engineering (2025) 38:172 Page 7 of 15 π π alignment stage. In this paper, the dynamic model of the 2 2 M(θ)M(θ) T (θ)T (θ) z z δ = ∫ Rdθ + ∫ Rdθ device in five working directions is used to obtain the EI GI 0 0 P first five natural frequencies. Using the PRBM method, 3 3 2 2 πR (3π − 8)R R R all linkages are considered to be rigid bodies, and only = + F + + M z x 4EI 4GI 2EI 2GI P P the deformation of flexible hinges is considered. There - 2 2 fore, the corresponding energy can be calculated using πR (4 − π)R + − + M the displacement and rotation angle of the connecting 4EI 4GI rod. The kinetic energy of the proposed device along the = c F + c M + c M , 11 z 12 x 13 y five functional directions can be calculated as (27) 2 2 where M(θ ) and T (θ ) represents the corresponding ˙ ˙ z z 1 1 δ 1 δ 1 1 1 2 2 ˙ ˙ T = m δ + 8 m + 32 m + 2 m δ 1 1 2 3 4 unit loads, and G is the shear modulus of the material. 1 1 2 2 2 2 4 2 The output angles about x- and y-axes can be represented 1 1 2 2 ˙ ˙ similarly: +2 m δ + + m δ , 5 6 1 1 2 2 (33) θ = c F + c M + c M , x 11 z 12 x 13 y (28) 1 1 1 1 2 2 2 2 ˙ ˙ ˙ ˙ T = 2 m δ + m δ + 2 m δ + 4 m δ 2 5 6 7 8 2 2 2 2 2 2 2 2 θ = c F + c M + c M . y 31 z 32 x 33 y (29) 2 (34) 1 δ Considering Eqs. (27)–(29), the expression between +24 m , 2 2 deformation and force of flexible components can be obtained: 1 1 1 1 2 2 2 2        ˙ ˙ ˙ ˙ T = 2 m δ + m δ + 2 m δ + 4 m δ 3 5 6 7 8 3 3 3 3 δ c c c F F z 11 12 13 z z 2 2 2 2        θx = c c c M = C M , 2 (35) 21 22 23 x 0 x 1 δ θ c c c M M y 31 32 33 y y +24 m , 2 2 (30) where C denotes the compliance matrix of the quarter 1 1 1 1 δ module. Because the stiffness matrix is the inverse matrix 4 2 2 2 ˙ ˙ ˙ T = 2 m δ + 4 m δ + 2 m δ + 24 m 4 7 8 10 9 4 4 4 of the flexibility matrix, it can be expressed as 2 2 2 2 2   c c c + I θ , 11 12 13 1 −1 2   c c c K = C = . 0 21 22 23 (31) (36) c c c 31 32 33 1 1 1 1 δ 2 2 2 ˙ ˙ ˙ T = 2 m δ + 4 m δ + 2 m δ + 24 m 5 7 8 10 9 Based on the method of spatial matrix transformation, 5 5 5 2 2 2 2 2 combined with the above derivation, the stiffness matrix of the 3-DOF off-plane stage can be calculated. Owing + I θ , to the structural symmetry of circular flexible hinges, we (37) can obtain the stiffness matrix in coordinate I : −xyz where m , m , and m are the masses of output stage, 1 2 3 motion stage, and flexure hinge, respectively, of the K = diag k k k , 1 2 3 (32) 1-DOF linear stage. m , m , and m are the masses of 4 5 6 where K = 4k denotes the lin- 1 11 the 2-DOF linear, 3-DOF off-plane, and output stages, ear stiffness along the z- axis, respectively. m , m , and m are the masses of the output 7 8 9 2 2 2 K = K = 4[k + (R k − 4Rk − 4Rk + 4k ) ] stage, input stage, and flexure hinge of the 2-DOF linear 2 3 11 31 13 33 denote the rotational stiffnesses about x-axes and y- axes. stage, respectively. m is the mass of half of the 3-DOF The input stiffness can be derived by substituting k , k , off-plane stage. I and I are the moments of inertia along 1 2 1 2 k into Eqs. (16) and (17). two rotational directions. Therefore, the kinetic energy of the proposed alignment mechanism can be calculated as 3.3 Dynamic Analysis T = T + T + T + T + T . (38) 1 2 3 4 5 The dynamic characteristics are important indicators of the response speed of error compensation in a reaction Moreover, the potential energy of the proposed device system. Consequently, a dynamic analysis is conducted along the five functional direction can be expressed as to evaluate the dynamic performance of the proposed Yang et al. Chinese Journal of Mechanical Engineering (2025) 38:172 Page 8 of 15 1 4 Parameter Selection and FEA Validation U = K δ , (39) 1 x 4.1 Parameter Selection For good dynamic characteristics of the proposed the 5-DOF flexible alignment stage, the first natural fre - U = K δ , (40) 2 y 2 quency 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 U = K δ , (41) constrained. The output motion range of the stage should 3 z 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 U = K δ , (42) 4 θ y 4 actuators. The structural size of the stage should ensure that it can achieve web printing of 150 mm. For the actual 2 printing process, the proposed stage is only responsible U = K δ , (43) 5 θ y 5 for compensating for posture errors. Large-scale print- ing is performed using a large-stroke XYZ linear motion U = U + U + U + U + U . stage. Therefore, according to error analysis, the output 1 2 3 4 5 (44) motion range of the proposed stage should be larger To obtain the dynamic equations of the system, this than 300 µm × 300 µm × 300 µm × 2.5 mrad × 2.5 mrad paper uses the Lagrange equation, which takes the follow- in five working directions according to error analysis. ing form: Moreover, the first natural frequency of the device should exceed 60 Hz for real-time compensation. Owing to its d ∂T ∂T ∂U − + = F , i = 1, 2, . . . , N , ( ) high strength, high elasticity, and low density, aluminum dx ∂q˙ ∂q˙ ∂q˙ i i i alloy (Al 7075-T6) is very suitable for processing flexible (45) mechanisms and was selected as the processing material where q denotes the vector of linearly independent gen- for this stage. The key dimensional parameters and inher - eralized coordinates. N corresponds to the dimensional- ent material properties are listed in Table 1. ity of the generalized coordinate space (specifically, N = 5 for the proposed 5-DOF alignment stage) and F denotes the externally applied force vector. Under free-vibration 4.2 FEA Validation conditions, the external force term F is nullified through To validate the performance of the proposed stage, we the boundary constraint enforcement. By substituting conducted FEA simulations using ANSYS Workbench Eqs. (38) and (44) into Eq. (45), the characteristic free- 16.0. A 3D model was constructed using the 3D software motion dynamic equation is derived as SolidWorks 2023. This section examines the deforma - tion, stiffness, center shift, and modal analysis. First, a Mq ¨ + Kq = 0, (46) deformation analysis was performed. With input forces where M and K represent the mass and stiffness matri - applied at the input position, the deformation results ces of the dynamic system, respectively, which can be along five directions are depicted in Figure  9. To observe expressed as the RCM characteristics more intuitively, we designed a conical cover with its tip of the conical cover coinciding M = diag M M M M M , (47) 11 22 33 44 55 with the RCM point at the output. According to the sim- ulation results, the output motion of the RCM platform K = diag K K K K K . 11 22 33 44 55 (48) matched the design requirements, verifying the effective - ness of the proposed device. Additionally, based on the Based on the above dynamic equations and vibration parameters of the linear motors and VCMs, the maxi- theory, the characteristic equation of the system can be mum stress was measured when the maximum output derived as force was applied at the input position. The FEA results showed that stress only occurred at the flexible hinge, K − ω M = 0, (49) with a maximum stress of 219.28 MPa, indicating that the safety factor of the material was at least 2.29. where ω (i = 1, 2, 3, 4, 5) represents the corresponding Moreover, when specified input forces were applied to natural cycle frequency of the system. Thus, the natural the input position of the flexible stage, the correspond - frequency can be obtained as f = (1/2π)ω . i j ing displacement or rotation angle of the platform output Y ang et al. Chinese Journal of Mechanical Engineering (2025) 38:172 Page 9 of 15 Table 1 Dimensional parameters and material properties of the proposed stage Parameters Values Flexure hinge of 1-DOF stage ( l × b × t ) 22 mm × 13 mm × 0.8 mm 1 1 1 Flexure hinge of 2-DOF stage ( l × b × t ) 18 mm × 8 mm × 0.4mm 2 2 2 Flexure hinge of 3-DOF mechanism ( R × b × t ) 11 mm × 6 mm × 0.6 mm 3 3 Rotation radius of the output stage L 118.2 mm Operating space of the output stage l 25 mm Density ρ 2810 kg/m Yield strength σ 503 MPa Young’s modulus E 71.7 GPa Poisson ratio v 0.33 Figure 9 Deformation of the proposed stage along (a) x direction, (b) y direction, (c) z direction, (d) θ direction, and (e) θ direction y z results of theoretical modeling analysis, the maximum Table 2 Stiffness performance of the proposed stage deviation of FEA results was 13.29%. This deviation was Stiffness Unit Theoretical FEA Deviation (%) primarily caused by the nonlinear characteristics of flex - ible components and model errors of the PRBM. How- K N/μm 0.359 0.401 10.47 ever, the stiffness performance of the proposed device K N/μm 0.151 0.167 9.58 achieved the design goals. K N/μm 0.151 0.171 11.7 A center shift was detected to evaluate the rotational θ N · m/rad 2.121 2.401 11.66 accuracy of the device. With the output stage rotating θ N · m/rad 2.121 2.446 13.29 from 0 to 5 mrad about y- and z-axes, respectively, the central shift of the conical cap vertex in the correspond- ing direction was measured. As shown in Figure  10, could be obtained, and the stiffness of the proposed plat - when a rotational motion of 5 mrad was generated about form in five directions could be calculated based on y-axis at the output end, the maximum offset value was these data. The stiffness results at this stage are listed in detected, which was less than 1 µm. Compared with Table 2. We observed that the linear stiffness along y- axis the rotational displacement output d by the functional and z-axis were equal, whereas the rotational stiffness end, which can be given by d = l θ , the maximum 0 0 out about y-axis and z-axis were equal. Compared with the center offset was less than 0.8%. This indicated that the Yang et al. Chinese Journal of Mechanical Engineering (2025) 38:172 Page 10 of 15 Figure 10 Center shift of the proposed stage for rotation about (a) y-axis and (b) z-axis alignment stage exhibited good motion-decoupling y- and z-axes. The first five corresponding resonances performance. calculated using the dynamic model were 106.5, 129.93, A modal analysis was conducted on the dynamic char- 129.93, 138.82, and 142.29 Hz, respectively. The maxi - acteristics of the proposed platform using FEA software. mum deviation between the theoretical model and FEA In the free vibration state (without actuating elements), results was 8.77%, which was within the allowable range. the first five vibration modes of the device are shown To assess the risk of the fatigue failure of flexible in Figure  11, and the first five corresponding resonance mechanisms, we conducted a fatigue analysis on the were 106.86, 125.63, 129.74, 151.22, and 154.77 Hz, platform. By applying an alternating force correspond- respectively. The first modal shape was a linear motion ing to the full stroke at the input end, we obtained the along x-axis. The second and third mode shapes were fatigue analysis results of the mechanism, as shown in linear motions along the y- and z-axes. The fourth and Figure  12. Under the action of alternating stress with a fifth mode shapes were the rotation motions around the maximum stress amplitude of 125.7 N, the minimum Figure 11 Modal analysis of the proposed stage Y ang et al. Chinese Journal of Mechanical Engineering (2025) 38:172 Page 11 of 15 Figure 12 Fatigue analysis of the proposed stage: (a) Stress distribution, (b) Fatigue life number of fatigue cycles of the mechanism was as the actuating components for the 2-DOF and 1-DOF 6 6 1.52×10 , exceeding the design requirement of 1×10 . linear stages, respectively. The maximum output force and stroke of the VCMs were 80 N and 6.3 mm, respec- 5 Experimental Results tively. The maximum output force and stroke of the linear 5.1 Prototype Development motor were 150 N and 40 mm, respectively. Owing to the The 5-DOF flexible alignment stage prototype is fabri - large displacement of the alignment stage, a laser sen- cated by machining. The experimental setup and sensor sor (LK-H020, KEYENCE, Inc.) was used to measure the arrangement are shown in Figure  13. The experimen - output motion along the working direction. Considering tal system consisted of the proposed alignment stage, the measurement accuracy, capacitive sensors (CPL190, capacitive sensor, laser sensor for actuator calibration, probe model: C8-2.0-2.0, from Lion Precision, Inc.) were output motion measurement, sensor controllers, VCM used to calibrate the output displacement and measure controller, linear motion driver, and DC power. In addi- the output coupling errors. For rotational motions, the tion, to facilitate measurement of the output motion, we measurement of the output angle were achieved using a installed a measurement block at the output end of the laser sensor or capacitive sensor, as shown in Figure  14, stage. Before the experiment, we tested the environmen- and calculated as tal noise values of the laser and capacitive sensors, which were 0.06 and 0.02 µm, respectively. θ = , m (50) Four VCMs (XVLC80-06-00A, XIVI, Inc.) num- bered from 1 to 4 in Figure  13(b) and a linear motor (DRS42SB2-04 KA, Oriental Motor, Inc.) were selected Figure 13 (a) Set up of the experimental system, (b) Sensor arrangement Yang et al. Chinese Journal of Mechanical Engineering (2025) 38:172 Page 12 of 15 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), Figure 14 Working principle of the output angle measuring 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 Figure 15 Relationship between input voltage and displacement FEA results primarily caused by the nonlinearity of the of VCMs VCM displacement output, which can be reduced using precise closed-loop control algorithms. where δ represents the output displacement measured Third, to verify the decoupling performance of the by sensor, and l represents the distance between the 2-DOF linear stage, we measured the coupling error rotation center and the measurement point of sensor. The between motions along y-axis and z-axis. A laser sen- sensor was installed through the designed components to sor was used to measure the output displacement in one achieve positioning with the measured part and obtain an direction and a capacitive sensor was used to measure accurate distance l . 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. 5.2 Experimental Tests The coupling error was largest at approximately 0.42 µm. Before the experiment, the displacement outputs of the Therefore, the coupling error of the stage was less than two actuators were calibrated. Linear motors controlled 0.11%, which confirmed that the 2-DOF linear stage had the displacement output through input pulse signals, good decoupling performance. whereas the VCM controlled the displacement output Finally, the natural frequency of the stage was tested. through the voltage. Laser and capacitive sensors were After the platform actuator was removed, an impulse used to calibrate the corresponding relationship between force was applied to the measuring block using a modal the input signal and output displacement of the linear hammer. The striking point and application direction motors and VCMs. Based on the calibration results, the of the modal hammer are indicated by red arrows in input–output corresponding relationship of the linear Figure  18(a). Laser sensors were used to measure the motor had good linearity, whereas the VCM was slightly amplitude of the mechanism after applying an impact nonlinear, as shown in Figure  15. Therefore, the testing force. The collected data were analyzed using fast Fou - components used in the experiment could accurately rier transform in MATLAB. The time and correspond - measure the relationship between the input and output ing frequency responses are shown in Figure  18(b) and displacements of the proposed stage, thereby verify (c), respectively. The analysis results indicated that the ing the performance of the alignment stage. Owing to first five resonance frequencies were 78.4, 90.2, 98.8, 138, the need to verify the performance of the mechanism, and 139.8 Hz. Compared with the theoretical and FEA including the stroke, input–output relationship, and results, the experimental results were lower, which was coupling error, open-loop control was adopted in all the primarily caused by the additional mass of the measuring experiments in this study. Y ang et al. Chinese Journal of Mechanical Engineering (2025) 38:172 Page 13 of 15 Figure 16 Workspace of the proposed stage along the different directions Figure 17 Coupling errors of the proposed stage along (a) y-axis and (b) z-axis blocks and bolts. However, the deviation was within a 6 Conclusions reasonable range. Table  3 provides a performance comparison of the (1) A novel 5-DOF flexure-based alignment stage developed alignment stage with several representative to adjust the posture of the inkjet printer head is alignment stages, focusing on key parameters such as dis- designed, which is composed of a 1-DOF linear placement and force resolutions. 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 Yang et al. Chinese Journal of Mechanical Engineering (2025) 38:172 Page 14 of 15 Figure 18 Frequency test: (a) Impact point and applying direction, (b) Time response, (c) Frequency response Table 3 Performance comparison with other alignment stages Reference DOF Workspace Coupling error (%) [7] 5-DOF 2000 μm × 2000 μm × 20000 μm × 2 mrad × 2 mrad 0.5 [34] 6-DOF 77.42 μm × 67.45 μm × 24.56 μm × 0.93 mrad × 0.93 mrad × 0.93 mrad – [35] 6-DOF 240 μm × 240 μm × 240 μm × 2.5 mrad × 2.5 mrad × 2.5 mrad 1.8 [36] 5-DOF 143 μm × 142 μm × 212 μm × 0.111 mrad × 0.109 mrad – This work 5-DOF 335.1 μm × 418.9 μm × 408.1 μm × 3.4 mrad × 3.29 mrad 0.11 gave some advice on the manuscript. All authors read and approved the final stiffness and accuracy by adopting a combination of manuscript. series and parallel designs and a decoupling design. Consequently, the proposed stage can achieve out- Funding Supported by Natural Science Research Project of Anhui Educational Commit- put motion in five working directions. tee (Grant No. 2024AH040010). (2) The PRBM method was used to model the flexible driving structure of the micro-positioning platform. Data availability Not applicable. A theoretical analysis of the kinematics, stiffness, and dynamics at this stage was conducted. The per - Competing Interests formance of the platform was verified using FEA. The authors declare no competing financial interests. (3) A prototype was developed for experimental research. The prototype test results show that the Received: 29 March 2025 Revised: 19 June 2025 Accepted: 21 July 2025 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 cou- pling of less than 0.11% along the y- and z-axes, References which satisfy the compensating requirements. 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International Journal of Precision Engineering and Manufactur- Lei Wang born in 1981, is currently a professor at School of Auto- ing, 2015, 16(12): 2569-2575. mation Science and Electrical Engineering, Beihang University, China. [25] J Jeong, A N Gafurov, P Park. Tension modeling and precise tension His research interests include automatic control theory, micro/nano control of roll-to-roll system for flexible electronics. Flexible and Printed manipulation, and computer vision. Electronics, 2021, 6(1): 015005. [26] J Lee, M Jo, C Lee. Advanced tension model for highly integrated flex - ible electronics in roll-to-roll manufacturing. IEEE/ASME Transactions on Xiantao Sun born in 1985, is currently an associate professor at Mechatronics, 2022, 27(5): 2908-2917. School of Electrical Engineering and Automation, Anhui University, [27] P F Moonen, I Yakimets, J Huskens. Fabrication of transistors on flexible China. He received PhD degree in mechanical engineering from Bei- substrates: from mass-printing to high-resolution alternative lithography hang University, China, in 2015. His research interests include adaptive strategies. Advanced Materials, 2012, 24(41): 5526-5541. gripper, flexure mechanism, precision manipulation, force/torque [28] L L Howell. Compliant mechanisms. New York: John Wiley & Sons, 2001. sensor, and variable stiffness design, etc. [29] J P Bacher, C Joseph, R Clavel. Flexures for high precision robotics. Indus- trial Robot, 2002, 29(4): 349-353. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Chinese Journal of Mechanical Engineering Springer Journals

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

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Springer Journals
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Copyright © The Author(s) 2025
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10.1186/s10033-025-01332-5
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Abstract

A high pattern resolution is critical for fabricating roll-to-roll printed electronics (R2RPE) products. For enhanced over- lay 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 align- ment 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 sub- structures. 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 pro- poses 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. Keywords Flexible mechanism, Micro-positioning, Multi-DOF, Alignment stage R2RPE products will not satisfy the requirements of the 1 Introduction high-end market. Therefore, high-precision multi-layer Roll-to-roll printed electronics (R2RPE) has been vali- alignment systems are urgently required. dated as a viable method for manufacturing a various The production process of the R2RPE primarily electronic devices [1–4]. Considering electronic devices includes gravure printing, flexographic printing, reverse produced by traditional methods, R2RPE products have offset printing, screen printing, and inkjet printing [9– the advantages of having a large area and being fast, 13]. Within the spectrum of manufacturing techniques, flexible, and inexpensive [5, 6]. For multi-layer printed inkjet printing distinguishes itself through non-contact electronic products, the accuracy of multi-layer print- deposition, maskless fabrication capabilities, and mate- ing is crucial, as different materials must be printed on rial-efficient operation while maintaining a high feature each layer to achieve different structures, which directly resolution critical for micro-scale patterning applica- affects the performance of multi-layer printed electronic tions [14, 15]. Multi-layer registration accuracy should products [7, 8]. If the accuracy of the multi-layer align- be guaranteed to obtain high-resolution inkjet printing ment cannot be guaranteed, the pattern resolution of patterns of R2RPE. However, the accuracy of multi-layer registration can be affected if a relative position error *Correspondence: occurs between the printer and the flexible web during Lei Wang [email protected] the printing process. School of Automation Science and Electrical Engineering, Beihang The precision of inkjet printing processes is critically University, Beijing 100191, China challenged by multi-source registration errors between School of Electrical Engineering and Automation, Anhui University, Hefei 230039, China the printhead assembly and advancing flexible web © The Author(s) 2025. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/. Yang et al. Chinese Journal of Mechanical Engineering (2025) 38:172 Page 2 of 15 substrates. For example, unstable web transport speed system with multiple-input multiple-output (MIMO) or uneven web stretching caused by nonuniform web closed-loop control that achieves submicron-level align- tension can result in web position errors [16, 17]. The ment precision for large-scale continuous printing pro- posture error of the support rollers can also affect web cesses. Chen et  al. [19] proposed a motion stage with a position [18]. Moreover, the posture error of the printer remote center of motion to adjust the roller posture. has a significant impact on the alignment accuracy [19, The proposed device supports rollers using air spherical 20]. Therefore, multi-layer alignment systems must be bearings with high stiffness and a flexible mechanism to adopted for high-resolution inkjet printing. generate accurate motion. Thus, the roller posture can be Numerous studies have been conducted to improve the adjusted, and the web position error can be compensated. accuracy of the multi-layer alignment, and certain results Many scholars have designed various types of alignment have been achieved. In the field of web tension control, mechanisms for error compensation in different scenar - Kang et  al. [21, 22] proposed a new theoretical model ios and have achieved good results [34–36]. that considers the lateral position errors of the substrate The literature review shows that many scholars have web and roller and provides compensation methods. Kim designed various types of alignment devices and achieved et al. [23, 24] proposed a new design of roll-to-roll (R2R) many research results. However, current alignment printing equipment for R2RPE production. The designed devices are primarily concerned with compensating for R2RPE system consists of tension control components the position error of the rollers. For inkjet printing, the such as feeders, load cells, and charge-coupled device positional error of the printer head should be adjusted. cameras to detect the relative position errors of patterns Based on the above considerations, a novel five-degree- for high-precision printing. Jeong et  al. [25] proposed a of-freedom (5-DOF) flexible alignment stage is proposed tension model for each section to successfully predict to adjust the posture of the printer head in real time. The the tension applied to such a system, the sagging of the proposed stage consists of a various flexible modules con - film according to tension, and deformation due to resid - nected in serial and parallel. The dynamic performance ual stress, and built an accurate R2R system to minimize of the proposed device has been significantly enhanced tension reduction. Lee et  al. [26] proposed an advanced through novel structural design. The remainder of this model to determine the tension disturbances caused by paper is organized as follows: Section 2 details the design run-out resulting from the axis mismatch, roundness considerations and architectural configuration of the error, imbalance, and velocity variation of the rollers in developed stage. Section 3 introduces a theoretical analy- an industrial-scale R2R printing process, and a high aver- sis of the kinematics, stiffness, and dynamics of the pro - age accuracy of 92.4% was achieved. For position error posed stage. Parametric optimization and finite element compensation based on the above methods, a traditional analysis (FEA) validation are systematically addressed in rigid hinge and motion stage are typically adopted to Section  4. Experimental verification through prototype adjust the posture of a roller or printer. The multi-layer implementation and performance characterization is alignment accuracy is limited to 40 µm resolution [27]. addressed in Section 5. The concluding remarks are given For the posture error of the printer or roller to be elimi- in Section 6. nated with high accuracy, a flexible mechanism can be used to design compensation mechanisms, leveraging 2 Mechanism Design inherent advantages such as frictionless operation, back-2.1 Design Consideration lash elimination, and lubrication-free monolithic design When processing of inkjet-printed electronics, multiple [28–30]. Baldesi et  al. [31] engineered an R2R-compati- layers of patterns should be printed on the flexible web. ble compliant-stage printing system, leveraging elastic Therefore, a high overlay accuracy should be achieved deformation principles to eliminate backlash and slid- to ensure printing resolution. Hence, the position error ing friction. The flexible mechanism is driven manually between the inkjet printer and existing web should be by micrometer heads to eliminate the position error of compensated. The position error is primarily caused by the roller. In addition, Zhou et  al. [32, 33] engineered a the position deviation between the different support roll - flexible R2R printing system for real-time adjustment of ers. In R2R inkjet printing, rotation about the x-axis is roller posture. The flexible mechanisms are driven by lin - inherently constrained by the web transport mechanism, ear stepper motors and voice coil motors (VCMs). In the making compensation of θ unnecessary for alignment. design of such flexible mechanisms, the output motion in u Th s, the 5-DOF design focuses on critical errors in θ , working directions are not decoupled. In addition, flex - θ , and translations [7]. As shown in Figure  1, the posi- ible mechanisms are often designed to withstand loads, tion error can be divided into three linear errors and two which is an important factor affecting the stability of rotational errors. Two methods can be used to compen- alignment stage. Li et al. [7] developed a multi-layer R2R sate for the position errors. One is to change the position Y ang et al. Chinese Journal of Mechanical Engineering (2025) 38:172 Page 3 of 15 Figure 1 Position errors between the inkjet printer and flexible web 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 com- pensated, a 5-DOF alignment stage should be adopted to adjust the posture of the inkjet printer. The three translational position errors can be com - pensated for by adopting a decoupled XYZ motion Figure 2 Alignment processes with (a) inner rotation center and (b) stage. The rotational position error around z- axis can external rotation center be compensated for by adopting a θ motion stage. Compensation for rotational errors around y-axis may introduce additional parasitic errors. As shown in Fig- in the design stage should be suppressed to ensure ure 2(a), if the rotation center O is inside the alignment the error compensation accuracy of the alignment stage, the inkjet printer, which is the output end of the mechanism and simplify the control complexity of the alignment stage, will generate lateral offsets along x - mechanism. In this design, the main measures to sup- and z-axes. To avoid this type of coupled motion, the press parasitic errors are as follows: (1) By designing rotation center should coincide with the center point symmetrical and decoupled mechanisms, the coupling of the underside of the inkjet printer, as shown in Fig- of parallel motion is reduced and parasitic errors are ure  2(b). Therefore, a flexible rotary-motion platform simulated; (2) through the design of the RCM mecha- should have the characteristics of a remote motion nism, the rotation center of the rotating mechanism is center (RCM) [37, 38]. Therefore, the parasitic motion aligned with the surface of the inkjet head. In addition, Yang et al. Chinese Journal of Mechanical Engineering (2025) 38:172 Page 4 of 15 the alignment stage must exhibit good dynamic perfor- can generate 5-DOF motions in the required directions mance to obtain the real-time and fast pose compensa- to achieve the alignment function. Moreover, the axes of tion capability of inkjet printers. the 5-DOF motion intersect at one point, which coincides with the center point of the underside of the inkjet printer. 2.2 Structure Description Therefore, the 5-DOF motions are decoupled from each Based on the above considerations, a decoupled 5-DOF other. flexible alignment stage for R2R inkjet printing is pro - Considering the symmetry of the structure while ensur- posed. The device primarily consists of a 1-DOF linear ing equivalent structural quality and not affecting the stage, two 2-DOF linear stages and two 3-DOF off-plane performance evaluation of the alignment mechanism, the stages, as shown in Figure 3. The 1-DOF linear stage can structure and layout were optimized by inverting the cross- actively generate translational motion along x-axis using beam used for installing the inkjet head in the middle to a linear motor. The two 2-DOF linear stages are driven facilitate the display of a finite element simulation analysis using four VCMs. The two 3-DOF off-plane stages are and performance evaluation results. designed to satisfy the requirements of motion output. Therefore, the 2-DOF linear and 3-DOF off-plane stages 3 Theoretical Analysis are combined to form a parallel 4-DOF stage to gener-3.1 Kinematic Analysis ate motions along y and z-axes and motions around y and Based on the introduction, flexible mechanisms are used to z-axes. The 1-DOF linear stage is connected to the 4-DOF adjust the pose of a printer. To evaluate the kinematics of motion stage in series. Therefore, the proposed device 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 sim- plified model of the 5-DOF flexible alignment stage can be given as shown in Figure 4. By combining the input forces F , F , F , F , and F , the output stage generates motion x y1 y2 z1 z2 along the required directions. Moreover, the output point O coincides with the underside center point of the inkjet printer, which is the RCM point. When applying input force/displacement ( F /δ ) to the x x 1-DOF linear stage, the output stage can generate output displacement along x-axis, which is given by out Figure 3 Mechanical design of the 5-DOF flexible alignment stage: δ = δ . (1) (a) Overall view, (b) 1-DOF linear stage, (c) 2-DOF linear stage, (d) 3-DOF off-plane stage The input force F can be calculated as Figure 4 Simplified model of the proposed device Y ang et al. Chinese Journal of Mechanical Engineering (2025) 38:172 Page 5 of 15 F = k δ , ∼ x x x (2) δ = L(1 − cosθ) = Lcosθ , (14) where k is the stiffness of the 1-DOF linear stage. where θ are θ are the two passive rotational motions, 1 2 When a couple of input forces/displacements ( F , y1 and δ is the passive linear motion. F /δ , δ ) are applied to the two 2-DOF linear stage with y2 y1 x For the rotational motion about z-axis, according to F = F = F and δ = δ = δ , the output stage can y1 y2 y y1 y2 y the energy equation, the following relationship can be generate output displacement along y-axis, which is given obtained: by 1 1 1 1 out 2 2 2 δ = δ . y (3) 2 F δ = 2 k δ + k θ + k θ . y (15) y y y 2 3 y 2 3 2 2 2 2 Using the same method, the output stage can generate u Th s, the input stiffness about z- axis can be repre- the output displacement along z-axis, which is given by sented as out δ = δ . z (4) 2 2 K = k L + k + k θ . (16) θz y 1 3 The input force F and F can be derived as y z By applying an analogous analytical methodology, the F = k δ , y y y (5) y-axis input stiffness is formulated through an energy- based derivation as follows: F = k δ , z z z (6) 2 2 K = k L + k + k θ . (17) where k , k are the stiffnesses of the 2-DOF linear stage θy z 2 3 y z y along y- and z-axes, respectively. Therefore, the input stiffness can be deduced as 3.2 Statics Analysis K = k , x x (7) To evaluate the static and provide principles for select- ing actuators, this section analyzes the stiffness analyses K = 2k , y y (8) 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 mod - K = 2k . (9) z z ules. The spatial structure and deformation character - For the output rotational motion around y-axis, the istics of the flexible module are shown in Figure  5. The input forces/displacements ( F , F /δ , δ ) are given by z1 z2 z1 z2 flexible beam is selected as the deformation element of F =−F and δ =−δ . Thus, the output rotational z1 z2 z1 z2 the flexible module because it has good flexibility and motion θ can be derived as can generate significant deformation. Each flexure has a secondary stage to reduce deformation of the flexure θ = arcsin , (10) y beam and enlarge the stroke of the primary stage. Thus, a 1-DOF linear stage can achieve a large stroke and good where L is the rotation radius of the stage. orientation [39]. For the output rotational motion about z-axis, the As shown in Figure  3(c), the 2-DOF linear stage three input forces/displacements ( F , F /δ , δ ) are given by flexure modules along each working direction. The y1 y2 y1 y2 F =−F and δ =−δ . Thus, the output rotational spatial structure and deformation characteristics of y1 y2 y1 y2 motion θ can be deduced as θ = arcsin . (11) Moreover, the deformations of the passive 3-DOF off- plane stage along three directions are θ = θ , 1 y (12) θ = θ , 2 z (13) Figure 5 Flexure module of the 1-DOF linear stage Yang et al. Chinese Journal of Mechanical Engineering (2025) 38:172 Page 6 of 15 Therefore, according to Eqs. (7)–(9), the input stiffness of the proposed device along x-, y-, and z-axes can be cal- culated as 8Eb t K = , (22) 24Eb t K = , y (23) Figure 6 Flexure module of the 2-DOF linear stage 24Eb t K = . z (24) According to Figure  4, when the output stage gener- ates 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 Figure 7 Schematic of force analysis for flexure beam are used to generate 3-DOF out of plane motions, as shown in Figure 8(a). Only the in-plane stiffness must be considered for the the flexible modules are shown in Figure  6. Each flex - 1-DOF and 2-DOF linear stages. However, to derive ible module of the linear platform consists of two sets of the stiffness of the 3-DOF off-plane stage, we require deformation units, one set containing two flexible beams. a space stiffness model. Therefore, a stiffness model Figure 7 shows the deformation characteristics of a single of the 3-DOF off-plane stage was established using beam. According to the theory of beam deformation and the spatial flexibility matrix method. According to the boundary conditions, the deformation state parameters force analysis of a quarter of the flexure module shown can be represented by the following equation: in Figure  8(b), the torque and bending moment at any position can be represented as 3 3 Fl Fl bt M = , δ = , I = , (18) M(θ ) = M sinθ + F Rcosθ − M cosθ, x z y 2 12EI 12 (25) where E is the elastic modulus of the material, I is the T (θ ) = M sinθ + F Rcosθ − M cosθ, x z y (26) moment of inertia of the cross section, δ is the vertical displacement of the end, and l, b, and t are the length, where M , M , and F represent the moments and exter- x y z width, and thickness, respectively. Consequently, the nal force applied to point B, and R = (R + R )/2 rep- 1 2 stiffness of the 1-DOF linear stage can be calculated as resents the average radius. The displacements at point follows: B under an external force or moment can be obtained. 3 u Th s, we can inferred that F 4F 8Eb t A 1 k = 4 = 4 = . x (19) δ 2δ l The stiffness along the two functional directions of the 2-DOF linear platform can be expressed as F 4F 12Eb t y 2 k = 3 = 3 = , y (20) δ δ l F 4F 12Eb t z 3 k = 3 = 3 = . z (21) Figure 8 (a) Flexure 3-DOF off-plane stage, (b) Force analysis δ δ z l of stage (quarter module) Y ang et al. Chinese Journal of Mechanical Engineering (2025) 38:172 Page 7 of 15 π π alignment stage. In this paper, the dynamic model of the 2 2 M(θ)M(θ) T (θ)T (θ) z z δ = ∫ Rdθ + ∫ Rdθ device in five working directions is used to obtain the EI GI 0 0 P first five natural frequencies. Using the PRBM method, 3 3 2 2 πR (3π − 8)R R R all linkages are considered to be rigid bodies, and only = + F + + M z x 4EI 4GI 2EI 2GI P P the deformation of flexible hinges is considered. There - 2 2 fore, the corresponding energy can be calculated using πR (4 − π)R + − + M the displacement and rotation angle of the connecting 4EI 4GI rod. The kinetic energy of the proposed device along the = c F + c M + c M , 11 z 12 x 13 y five functional directions can be calculated as (27) 2 2 where M(θ ) and T (θ ) represents the corresponding ˙ ˙ z z 1 1 δ 1 δ 1 1 1 2 2 ˙ ˙ T = m δ + 8 m + 32 m + 2 m δ 1 1 2 3 4 unit loads, and G is the shear modulus of the material. 1 1 2 2 2 2 4 2 The output angles about x- and y-axes can be represented 1 1 2 2 ˙ ˙ similarly: +2 m δ + + m δ , 5 6 1 1 2 2 (33) θ = c F + c M + c M , x 11 z 12 x 13 y (28) 1 1 1 1 2 2 2 2 ˙ ˙ ˙ ˙ T = 2 m δ + m δ + 2 m δ + 4 m δ 2 5 6 7 8 2 2 2 2 2 2 2 2 θ = c F + c M + c M . y 31 z 32 x 33 y (29) 2 (34) 1 δ Considering Eqs. (27)–(29), the expression between +24 m , 2 2 deformation and force of flexible components can be obtained: 1 1 1 1 2 2 2 2        ˙ ˙ ˙ ˙ T = 2 m δ + m δ + 2 m δ + 4 m δ 3 5 6 7 8 3 3 3 3 δ c c c F F z 11 12 13 z z 2 2 2 2        θx = c c c M = C M , 2 (35) 21 22 23 x 0 x 1 δ θ c c c M M y 31 32 33 y y +24 m , 2 2 (30) where C denotes the compliance matrix of the quarter 1 1 1 1 δ module. Because the stiffness matrix is the inverse matrix 4 2 2 2 ˙ ˙ ˙ T = 2 m δ + 4 m δ + 2 m δ + 24 m 4 7 8 10 9 4 4 4 of the flexibility matrix, it can be expressed as 2 2 2 2 2   c c c + I θ , 11 12 13 1 −1 2   c c c K = C = . 0 21 22 23 (31) (36) c c c 31 32 33 1 1 1 1 δ 2 2 2 ˙ ˙ ˙ T = 2 m δ + 4 m δ + 2 m δ + 24 m 5 7 8 10 9 Based on the method of spatial matrix transformation, 5 5 5 2 2 2 2 2 combined with the above derivation, the stiffness matrix of the 3-DOF off-plane stage can be calculated. Owing + I θ , to the structural symmetry of circular flexible hinges, we (37) can obtain the stiffness matrix in coordinate I : −xyz where m , m , and m are the masses of output stage, 1 2 3 motion stage, and flexure hinge, respectively, of the K = diag k k k , 1 2 3 (32) 1-DOF linear stage. m , m , and m are the masses of 4 5 6 where K = 4k denotes the lin- 1 11 the 2-DOF linear, 3-DOF off-plane, and output stages, ear stiffness along the z- axis, respectively. m , m , and m are the masses of the output 7 8 9 2 2 2 K = K = 4[k + (R k − 4Rk − 4Rk + 4k ) ] stage, input stage, and flexure hinge of the 2-DOF linear 2 3 11 31 13 33 denote the rotational stiffnesses about x-axes and y- axes. stage, respectively. m is the mass of half of the 3-DOF The input stiffness can be derived by substituting k , k , off-plane stage. I and I are the moments of inertia along 1 2 1 2 k into Eqs. (16) and (17). two rotational directions. Therefore, the kinetic energy of the proposed alignment mechanism can be calculated as 3.3 Dynamic Analysis T = T + T + T + T + T . (38) 1 2 3 4 5 The dynamic characteristics are important indicators of the response speed of error compensation in a reaction Moreover, the potential energy of the proposed device system. Consequently, a dynamic analysis is conducted along the five functional direction can be expressed as to evaluate the dynamic performance of the proposed Yang et al. Chinese Journal of Mechanical Engineering (2025) 38:172 Page 8 of 15 1 4 Parameter Selection and FEA Validation U = K δ , (39) 1 x 4.1 Parameter Selection For good dynamic characteristics of the proposed the 5-DOF flexible alignment stage, the first natural fre - U = K δ , (40) 2 y 2 quency 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 U = K δ , (41) constrained. The output motion range of the stage should 3 z 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 U = K δ , (42) 4 θ y 4 actuators. The structural size of the stage should ensure that it can achieve web printing of 150 mm. For the actual 2 printing process, the proposed stage is only responsible U = K δ , (43) 5 θ y 5 for compensating for posture errors. Large-scale print- ing is performed using a large-stroke XYZ linear motion U = U + U + U + U + U . stage. Therefore, according to error analysis, the output 1 2 3 4 5 (44) motion range of the proposed stage should be larger To obtain the dynamic equations of the system, this than 300 µm × 300 µm × 300 µm × 2.5 mrad × 2.5 mrad paper uses the Lagrange equation, which takes the follow- in five working directions according to error analysis. ing form: Moreover, the first natural frequency of the device should exceed 60 Hz for real-time compensation. Owing to its d ∂T ∂T ∂U − + = F , i = 1, 2, . . . , N , ( ) high strength, high elasticity, and low density, aluminum dx ∂q˙ ∂q˙ ∂q˙ i i i alloy (Al 7075-T6) is very suitable for processing flexible (45) mechanisms and was selected as the processing material where q denotes the vector of linearly independent gen- for this stage. The key dimensional parameters and inher - eralized coordinates. N corresponds to the dimensional- ent material properties are listed in Table 1. ity of the generalized coordinate space (specifically, N = 5 for the proposed 5-DOF alignment stage) and F denotes the externally applied force vector. Under free-vibration 4.2 FEA Validation conditions, the external force term F is nullified through To validate the performance of the proposed stage, we the boundary constraint enforcement. By substituting conducted FEA simulations using ANSYS Workbench Eqs. (38) and (44) into Eq. (45), the characteristic free- 16.0. A 3D model was constructed using the 3D software motion dynamic equation is derived as SolidWorks 2023. This section examines the deforma - tion, stiffness, center shift, and modal analysis. First, a Mq ¨ + Kq = 0, (46) deformation analysis was performed. With input forces where M and K represent the mass and stiffness matri - applied at the input position, the deformation results ces of the dynamic system, respectively, which can be along five directions are depicted in Figure  9. To observe expressed as the RCM characteristics more intuitively, we designed a conical cover with its tip of the conical cover coinciding M = diag M M M M M , (47) 11 22 33 44 55 with the RCM point at the output. According to the sim- ulation results, the output motion of the RCM platform K = diag K K K K K . 11 22 33 44 55 (48) matched the design requirements, verifying the effective - ness of the proposed device. Additionally, based on the Based on the above dynamic equations and vibration parameters of the linear motors and VCMs, the maxi- theory, the characteristic equation of the system can be mum stress was measured when the maximum output derived as force was applied at the input position. The FEA results showed that stress only occurred at the flexible hinge, K − ω M = 0, (49) with a maximum stress of 219.28 MPa, indicating that the safety factor of the material was at least 2.29. where ω (i = 1, 2, 3, 4, 5) represents the corresponding Moreover, when specified input forces were applied to natural cycle frequency of the system. Thus, the natural the input position of the flexible stage, the correspond - frequency can be obtained as f = (1/2π)ω . i j ing displacement or rotation angle of the platform output Y ang et al. Chinese Journal of Mechanical Engineering (2025) 38:172 Page 9 of 15 Table 1 Dimensional parameters and material properties of the proposed stage Parameters Values Flexure hinge of 1-DOF stage ( l × b × t ) 22 mm × 13 mm × 0.8 mm 1 1 1 Flexure hinge of 2-DOF stage ( l × b × t ) 18 mm × 8 mm × 0.4mm 2 2 2 Flexure hinge of 3-DOF mechanism ( R × b × t ) 11 mm × 6 mm × 0.6 mm 3 3 Rotation radius of the output stage L 118.2 mm Operating space of the output stage l 25 mm Density ρ 2810 kg/m Yield strength σ 503 MPa Young’s modulus E 71.7 GPa Poisson ratio v 0.33 Figure 9 Deformation of the proposed stage along (a) x direction, (b) y direction, (c) z direction, (d) θ direction, and (e) θ direction y z results of theoretical modeling analysis, the maximum Table 2 Stiffness performance of the proposed stage deviation of FEA results was 13.29%. This deviation was Stiffness Unit Theoretical FEA Deviation (%) primarily caused by the nonlinear characteristics of flex - ible components and model errors of the PRBM. How- K N/μm 0.359 0.401 10.47 ever, the stiffness performance of the proposed device K N/μm 0.151 0.167 9.58 achieved the design goals. K N/μm 0.151 0.171 11.7 A center shift was detected to evaluate the rotational θ N · m/rad 2.121 2.401 11.66 accuracy of the device. With the output stage rotating θ N · m/rad 2.121 2.446 13.29 from 0 to 5 mrad about y- and z-axes, respectively, the central shift of the conical cap vertex in the correspond- ing direction was measured. As shown in Figure  10, could be obtained, and the stiffness of the proposed plat - when a rotational motion of 5 mrad was generated about form in five directions could be calculated based on y-axis at the output end, the maximum offset value was these data. The stiffness results at this stage are listed in detected, which was less than 1 µm. Compared with Table 2. We observed that the linear stiffness along y- axis the rotational displacement output d by the functional and z-axis were equal, whereas the rotational stiffness end, which can be given by d = l θ , the maximum 0 0 out about y-axis and z-axis were equal. Compared with the center offset was less than 0.8%. This indicated that the Yang et al. Chinese Journal of Mechanical Engineering (2025) 38:172 Page 10 of 15 Figure 10 Center shift of the proposed stage for rotation about (a) y-axis and (b) z-axis alignment stage exhibited good motion-decoupling y- and z-axes. The first five corresponding resonances performance. calculated using the dynamic model were 106.5, 129.93, A modal analysis was conducted on the dynamic char- 129.93, 138.82, and 142.29 Hz, respectively. The maxi - acteristics of the proposed platform using FEA software. mum deviation between the theoretical model and FEA In the free vibration state (without actuating elements), results was 8.77%, which was within the allowable range. the first five vibration modes of the device are shown To assess the risk of the fatigue failure of flexible in Figure  11, and the first five corresponding resonance mechanisms, we conducted a fatigue analysis on the were 106.86, 125.63, 129.74, 151.22, and 154.77 Hz, platform. By applying an alternating force correspond- respectively. The first modal shape was a linear motion ing to the full stroke at the input end, we obtained the along x-axis. The second and third mode shapes were fatigue analysis results of the mechanism, as shown in linear motions along the y- and z-axes. The fourth and Figure  12. Under the action of alternating stress with a fifth mode shapes were the rotation motions around the maximum stress amplitude of 125.7 N, the minimum Figure 11 Modal analysis of the proposed stage Y ang et al. Chinese Journal of Mechanical Engineering (2025) 38:172 Page 11 of 15 Figure 12 Fatigue analysis of the proposed stage: (a) Stress distribution, (b) Fatigue life number of fatigue cycles of the mechanism was as the actuating components for the 2-DOF and 1-DOF 6 6 1.52×10 , exceeding the design requirement of 1×10 . linear stages, respectively. The maximum output force and stroke of the VCMs were 80 N and 6.3 mm, respec- 5 Experimental Results tively. The maximum output force and stroke of the linear 5.1 Prototype Development motor were 150 N and 40 mm, respectively. Owing to the The 5-DOF flexible alignment stage prototype is fabri - large displacement of the alignment stage, a laser sen- cated by machining. The experimental setup and sensor sor (LK-H020, KEYENCE, Inc.) was used to measure the arrangement are shown in Figure  13. The experimen - output motion along the working direction. Considering tal system consisted of the proposed alignment stage, the measurement accuracy, capacitive sensors (CPL190, capacitive sensor, laser sensor for actuator calibration, probe model: C8-2.0-2.0, from Lion Precision, Inc.) were output motion measurement, sensor controllers, VCM used to calibrate the output displacement and measure controller, linear motion driver, and DC power. In addi- the output coupling errors. For rotational motions, the tion, to facilitate measurement of the output motion, we measurement of the output angle were achieved using a installed a measurement block at the output end of the laser sensor or capacitive sensor, as shown in Figure  14, stage. Before the experiment, we tested the environmen- and calculated as tal noise values of the laser and capacitive sensors, which were 0.06 and 0.02 µm, respectively. θ = , m (50) Four VCMs (XVLC80-06-00A, XIVI, Inc.) num- bered from 1 to 4 in Figure  13(b) and a linear motor (DRS42SB2-04 KA, Oriental Motor, Inc.) were selected Figure 13 (a) Set up of the experimental system, (b) Sensor arrangement Yang et al. Chinese Journal of Mechanical Engineering (2025) 38:172 Page 12 of 15 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), Figure 14 Working principle of the output angle measuring 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 Figure 15 Relationship between input voltage and displacement FEA results primarily caused by the nonlinearity of the of VCMs VCM displacement output, which can be reduced using precise closed-loop control algorithms. where δ represents the output displacement measured Third, to verify the decoupling performance of the by sensor, and l represents the distance between the 2-DOF linear stage, we measured the coupling error rotation center and the measurement point of sensor. The between motions along y-axis and z-axis. A laser sen- sensor was installed through the designed components to sor was used to measure the output displacement in one achieve positioning with the measured part and obtain an direction and a capacitive sensor was used to measure accurate distance l . 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. 5.2 Experimental Tests The coupling error was largest at approximately 0.42 µm. Before the experiment, the displacement outputs of the Therefore, the coupling error of the stage was less than two actuators were calibrated. Linear motors controlled 0.11%, which confirmed that the 2-DOF linear stage had the displacement output through input pulse signals, good decoupling performance. whereas the VCM controlled the displacement output Finally, the natural frequency of the stage was tested. through the voltage. Laser and capacitive sensors were After the platform actuator was removed, an impulse used to calibrate the corresponding relationship between force was applied to the measuring block using a modal the input signal and output displacement of the linear hammer. The striking point and application direction motors and VCMs. Based on the calibration results, the of the modal hammer are indicated by red arrows in input–output corresponding relationship of the linear Figure  18(a). Laser sensors were used to measure the motor had good linearity, whereas the VCM was slightly amplitude of the mechanism after applying an impact nonlinear, as shown in Figure  15. Therefore, the testing force. The collected data were analyzed using fast Fou - components used in the experiment could accurately rier transform in MATLAB. The time and correspond - measure the relationship between the input and output ing frequency responses are shown in Figure  18(b) and displacements of the proposed stage, thereby verify (c), respectively. The analysis results indicated that the ing the performance of the alignment stage. Owing to first five resonance frequencies were 78.4, 90.2, 98.8, 138, the need to verify the performance of the mechanism, and 139.8 Hz. Compared with the theoretical and FEA including the stroke, input–output relationship, and results, the experimental results were lower, which was coupling error, open-loop control was adopted in all the primarily caused by the additional mass of the measuring experiments in this study. Y ang et al. Chinese Journal of Mechanical Engineering (2025) 38:172 Page 13 of 15 Figure 16 Workspace of the proposed stage along the different directions Figure 17 Coupling errors of the proposed stage along (a) y-axis and (b) z-axis blocks and bolts. However, the deviation was within a 6 Conclusions reasonable range. Table  3 provides a performance comparison of the (1) A novel 5-DOF flexure-based alignment stage developed alignment stage with several representative to adjust the posture of the inkjet printer head is alignment stages, focusing on key parameters such as dis- designed, which is composed of a 1-DOF linear placement and force resolutions. 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 Yang et al. Chinese Journal of Mechanical Engineering (2025) 38:172 Page 14 of 15 Figure 18 Frequency test: (a) Impact point and applying direction, (b) Time response, (c) Frequency response Table 3 Performance comparison with other alignment stages Reference DOF Workspace Coupling error (%) [7] 5-DOF 2000 μm × 2000 μm × 20000 μm × 2 mrad × 2 mrad 0.5 [34] 6-DOF 77.42 μm × 67.45 μm × 24.56 μm × 0.93 mrad × 0.93 mrad × 0.93 mrad – [35] 6-DOF 240 μm × 240 μm × 240 μm × 2.5 mrad × 2.5 mrad × 2.5 mrad 1.8 [36] 5-DOF 143 μm × 142 μm × 212 μm × 0.111 mrad × 0.109 mrad – This work 5-DOF 335.1 μm × 418.9 μm × 408.1 μm × 3.4 mrad × 3.29 mrad 0.11 gave some advice on the manuscript. All authors read and approved the final stiffness and accuracy by adopting a combination of manuscript. series and parallel designs and a decoupling design. Consequently, the proposed stage can achieve out- Funding Supported by Natural Science Research Project of Anhui Educational Commit- put motion in five working directions. tee (Grant No. 2024AH040010). (2) The PRBM method was used to model the flexible driving structure of the micro-positioning platform. Data availability Not applicable. A theoretical analysis of the kinematics, stiffness, and dynamics at this stage was conducted. The per - Competing Interests formance of the platform was verified using FEA. The authors declare no competing financial interests. (3) A prototype was developed for experimental research. The prototype test results show that the Received: 29 March 2025 Revised: 19 June 2025 Accepted: 21 July 2025 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 cou- pling of less than 0.11% along the y- and z-axes, References which satisfy the compensating requirements. 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Journal

Chinese Journal of Mechanical Engineering – Springer Journals

Published: Sep 8, 2025

Keywords: Flexible mechanism; Micro-positioning; Multi-DOF; Alignment stage

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