# Design and Analysis of a Novel Piezo-ActuatedXYθz Micropositioning Mechanism with Large Travel and Kinematic Decoupling

Authors: Ruijiang Xiao; Shubao Shao; Minglong Xu; Zijian Jing
Journal: Advances in Materials Science and Engineering
 (2019)

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## Abstract

A novel hybrid-typeXYθz micropositioning mechanism driven by piezoelectric actuators is proposed in this paper. With the purpose of realizing a large motion range and 3-DoF independent motion within a compact size, the mechanism is designed using a symmetric translational part and a rotational part that are linked serially. The translational part is based on a double-amplification mechanism incorporating a guidance mechanism for decoupling; the rotational part uses a nonuniform beam with an amplification mechanism to translate the linear output displacement of piezoelectric actuators into a large rotational angle around the Z axis. To precisely predict the output displacements and implement dimensional design, electromechanical models of the translational mechanism and rotational mechanism are established. According to the theoretical model, dimensional optimization is carried out to achieve large motion ranges within a compact size. A prototype of the proposed mechanism is fabricated according to the optimized results, and the performance of the mechanism is validated by experiment. The experimental results show that translational travel in X and Y directions of 204.2 μm and 212.8 μm, respectively, and travel of 8.7 mrad in the θz direction can be realized in a small size of 106 mm × 106 mm × 23 mm. And, the output coupling was evaluated to be below 3%, indicating an excellent decoupling performance.


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## Body

## 1. Introduction

Three-degree-of-freedom (3-DoF)XYθz micropositioning mechanisms based on flexure hinges are widely applied in various fields; e.g., such mechanisms are used for atomic force microscopes, micromanipulation, scanning systems, high-precision image stabilization systems, and biomedical applications [1–7]. The mechanisms can improve the repeatability and reliability of scanning and image resolution of an optical-image stabilization system. Following the rapid development of optoelectronic technology and the expanding fields of implementation, it is necessary to correct the optical axis shift of space camera in real time and compensate image motion caused by low-frequency vibration via the design of servo mechanism in the optical path. Therefore, the XYθz micropositioning mechanism with large travel, high displacement resolution, and compact structure that can be applied for the optical-image stabilization system is urgently required.Generally, a piezoelectric actuator is commonly used in these micropositioning mechanisms because of its advantages of high positioning resolution, fast response, high stiffness, and no electromagnetic interference issues [8, 9]. However, the small deformation range of piezoelectric materials, which is merely about 1 μm/mm, restricts engineering applications. Existing piezo-actuated XYθz micropositioning mechanisms typically therefore have limited motion travel [10–12]. Hwang et al., for example, developed an in-plane XYθz positioning mechanism with dimensions of 240 mm × 240 mm × 25 mm and translational and rotational motion ranges of 58 μm and 1.05 mrad, respectively [11]. Zhu et al. presented a redundantly piezo-driven XYθz compliant mechanism with high dynamic characteristics but working ranges in X, Y, and θz directions of merely 12.42 μm, 24.82 μm, and 1.75 mrad, respectively [12]. To increase motion ranges, bridge-type amplification mechanisms are used for amplification of the deformation of piezoelectric materials because of their compact structure and high displacement amplifying ratio [13–17]. Although an existing XYθz micropositioning system having an amplification mechanism has larger working ranges, it employs larger stack-type actuators, which increase the system size [18]. Additionally, in the design of the above parallel-type XYθz mechanisms employing four or three actuators, the actuation of rotational motion shares the same piezoelectric stacks with the actuation of translational motion. This means that the strokes of rotation and translation cannot reach their expected maximum values simultaneously in reality [7, 18]. And also, the actuators of rotational part are often driven by high voltage for achieving large rotational angle, which limits practical application in the design of miniaturized instrument.To overcome the aforementioned problems and satisfy the application in the optical-image stabilization system, a novel hybrid-typeXYθz micropositioning mechanism featuring an enlarged workspace, compact size, and 3-DoF independent motion is proposed and developed in this study. The mechanism mainly comprises a translational part and rotational part, which are linked serially for kinematic decoupling. The parallel translational mechanism characterized by high precision and compactness uses a double-amplification mechanism incorporating a guidance mechanism for decoupling, aiming to realize large translational travel and guide the moving platform in the desired direction by preventing the platform from moving in a parasitic manner [19–21]. The rotational mechanism consists of a nonuniform beam and two piezoelectric actuators arranged in opposite directions. To translate the linear output displacement of the piezoelectric actuators into a larger output angle around the Z-axis at a lower voltage, a nonuniform beam with lever amplification is designed using flexure hinges. Generally, to estimate the output displacements of the XYθz micropositioning mechanism, electromechanical models of the translational mechanism and rotational mechanism are established employing a pseudo-rigid-body model and beam theory. Because of the finite stiffness of the piezo stack, the deformations affected by the input voltage and counterforce are simultaneously considered in this analysis, which improve the accuracy of the theoretical model. Using the established model, the mechanism is optimized to achieve larger motion ranges within a compact size. Finite element analysis (FEA) is then carried out to verify the output displacements and to investigate the dynamic performance of the mechanism. Finally, a prototype of the proposed micropositioning mechanism is fabricated according to the optimized parameters and the performance of the mechanism is experimentally validated.


## 2. Structure Design and Operating Principle

Figure1 shows the structure of the proposed XYθz micropositioning mechanism, which mainly comprises a translational part, rotational part, and the cruciform connecting structure illustrated in Figure 2(a). The parallel translational mechanism connected with the rotational part serially by the cruciform connecting structure employs a double-amplification mechanism including a rhombic piezoelectric actuator (RPA) and two levers that amplify the deformation of the piezo stack. On the basis of the right circular flexure hinges, the guidance mechanism is designed to deliver the platform high-resolution motion in desired (X and Y) directions and to restrict the platform in undesired directions. In the proposed mechanism, the rotational part consists of two piezoelectric actuators arranged in opposite directions, a nonuniform beam and a fixture. Figure 2(b) shows the design of the nonuniform beam, the two ends of which are fixed by flexure hinges. The beam consists of five segments. One rigid segment in the center works as an output port connected with the cruciform connecting structure by a groove and screw; two rigid arms are used to drive the output port; and two right circular flexure hinges are used to connect the above three rigid segments. Because of the symmetry of the translational part and the characteristic of hybrid-type design, independent motions of the platform along the X, Y, and θz directions can be achieved.Figure 1
Model of the proposedXYθz micropositioning mechanism: (a) 3D model; (b) translational part; and (c) rotational part.
(a)
(b)
(c)Figure 2
(a) Cruciform connecting structure and (b) nonuniform beam.
(a)
(b)To actuate 3-DoF in-plane motion independently and achieve large motion ranges, the platform requires three pairs of piezoelectric actuators. The independentX-directional motion of the platform can be implemented by the coordinated work of piezoelectric actuators PZT1 and PZT3. Similarly, the Y-directional motion can be realized by actuating PZT2 and PZT4. When one pair of differential voltages is applied to the pair of piezoelectric actuators located on one axis (X or Y), the corresponding pair of RPAs applies forces and displacements to one end of the levers in a push-and-pull driving mode. In this mode, the RPAs produce displacements and forces in two opposing directions. In the first direction, as shown in Figure 3(a), the RPA produces a downward force or displacement, which is induced by an increase in the voltage input to the piezo stack, resulting in transverse elongation of the mechanism. In the second direction, the RPA produces an upward output, as shown in Figure 3(b), which is induced by a decrease in the input voltage, generating elastic recovery of the compliant mechanism. Therefore, when the voltage applied to the RPA of one axis (X or Y) is increased and the voltage applied to the RPA opposing that axis is decreased, the platform produces translational motion along that axis via the flexure guidance mechanism.Figure 3
Schematics of the RPA: (a) pull driving mode; (b) push driving mode.
(a)
(b)The independentθz-directional motion of the platform can be obtained by simultaneously driving PZT5 and PZT6. When a voltage pair is applied to the pair of oppositely located piezoelectric actuators in the rotational part, each piezoelectric actuator applies an actuation force on the rigid arm of the nonuniform beam in either the positive or negative Y direction (F and −F) through flexure hinges, generating a torque around the Z-axis, which imposes a tilt angle of the output port that allows the whole translational mechanism to generate θz-directional motion. Accordingly, 3-DoF independent in-plane motion of the platform can be achieved using the three pairs of piezoelectric actuators.In the described micropositioning mechanism, the parallel-type translational motion part and the rotational motion part are decoupled and linked serially. The two parts are therefore modeled, respectively. The static analysis of the proposed mechanism is further discussed in Section3.


## 3. Kinematic Analysis

### 3.1. Analysis of the Translational Mechanism

The translational mechanism can be abstracted as a serial mechanical network, as shown in Figure4. To clarify the performance of the mechanism, it is necessary to establish an accurate theoretical model to comprehensively determine the translational motion transmission from the piezo stacks to the platform. A governing equation of the RPA is therefore derived as a starting point for calculating the relationship of input-output static displacements and forces.Figure 4
Schematic diagram of translational motion transmission.Because of the symmetry of the rhombic mechanism, we consider only a quarter of the structure as the study object. The mechanical model based on a pseudo-rigid-body model and Euler–Bernoulli beam theory is shown in Figure5. In this analysis, the rhombic mechanism is considered to be constructed of rigid bodies interconnected by flexure arms in which the mechanism compliance is lumped. According to the energy principle, the input and output displacements of the structure can be expressed as(1)Δin2=∫0lFNxEA⋅2dFNxdFindx+∫0lMxEI⋅2dMxdFindx,(2)ΔP2=∫0lFNxEA⋅2dFNxdFPdx+∫0lMxEI⋅2dMxdFPdx.Figure 5
Model of a quarter of the rhombic mechanism.Here,Fin is the driving force provided by the piezo stack, which produces input displacement Δin for the rhombic mechanism, Fp is the output force, and Δp is the output displacement of the RPA; E is the elastic modulus; A and I are, respectively, the area and moment of inertia of the corresponding cross section of the beam; and FNx and Mx are, respectively, the axial tension and moment along the neutral axis. According to the static analysis shown in Figure 5, we have(3)FNx=12FPsinα−Fincosα,Mx=12FPcosα+Finsinαx−l2.By substituting equation (3) into equations (1) and (2), the relationship between input and output forces and thus the displacements can be obtained:(4)Fincosα−FPsinα=EAlΔincosα−ΔPsinα,Finsinα+FPcosα=2clΔinsinα+ΔPcosα.Here,(5)c=6EIl2.As mentioned above, the input displacement and force acting on the rhombic mechanism are provided by the piezo stack (P885.91, PI). When the hysteresis nonlinearity of the piezoelectric material is ignored, the linear deformation of the piezo stack can be written as(6)δ=UinCV+1KPfP,where δ is the output displacement of the piezo stack, CV is the piezoelectric constant and can be calculated as CV = nd33 (with n being the number of layers of the piezo stack), Uin is the change in the voltage applied to the piezo stack, Kp is the stiffness of the piezo stack, fp is the external force imposed on the piezo stack, and fp = −Fin. Thus,(7)Δin=UinCV−1KPFin.Substituting (7) into (4), the governing equation of the RPA can be expressed in matrix form as(8)FPΔP=1sinαcosα2c−EA2ccos2α+EAsin2α+2EAcKpl2EAcll+1KpEAcos2α+2csin2αEAcos2α+2csin2αFinCVUin=f11f12f21f22FinCVUin.To further characterize the output displacement of the double-amplification mechanism, the mechanical model of the lever mechanism is shown in Figure6. The right circular flexure hinges sharing the same geometry configuration are simplified as rotational springs attached to the lever. Each flexure hinge therefore imposes an angular moment on the lever:(9)Mk=Kr1θk,k=C,O,D,where θk is the rotational angle of the hinges and equals the tilt angle of the rigid lever. θk can be expressed as θk=ΔP/L2 owing to the small value of θk. Kr1 is the rotation stiffness of right circular flexure hinge 1, which can be calculated using the equations presented by Zhu et al. [22]. The force equilibrium equation and geometrical equation of the rigid lever are then established:(10)MC+M0+MD+FCyL1−FDyL2=0,Δout=L1L2ΔP,where FCy and FDy are vertical forces of nodes C and D, and FDy = −FP/2.Figure 6
Mechanical model of the lever mechanism.In the analysis of the parallel configuration, the reaction of the guidance mechanism can be considered equivalent to an elastic load with port stiffness ofKload. According to Newton’s third law of motion, the output force of the double-amplification mechanism is then(11)Fout=−FCy=−Kloadxout4.The mechanical model of the guidance mechanism with four rods is shown in Figure7. The equivalent stiffness Kload can be expressed as(12)Kload=4kx,where kx is the stiffness in the x direction of each guidance rod and be calculated as(13)kx=Kr1+Kr2L32,with Kr2 being the rotation stiffness of right circular flexure hinge 2.Figure 7
Mechanical model of the guidance mechanism.Owing to the symmetry of the translational mechanism, the output displacements alongX and Y axes are equal. The electromechanical model of the translational mechanism, which relates the output displacement of the translational mechanism to the voltage applied to the piezo stack, can therefore be obtained from equations (8)–(11) as(14)xout=2L1L212Kr1+KloadL12f2212Kr1+KloadL12+2f12L22f2112Kr1+KloadL12+2f11L22−f12CVUin.


### 3.2. Analysis of the Rotational Mechanism

In this subsection, to accurately establish a theoretical model of the rotational mechanism, the nonuniform beam is modeled using the stiffness equation of the flexible segments. It is assumed that the compliance of the rotational part is lumped in the flexure hinges and the flexible segments of the beam. The modeling of the mechanism therefore begins with the analysis of the flexure connectors.The flexible segment, namely, the right circular hinge, is used to connect the rigid segments and to deliver high-resolution motion to the output port. Figure8 shows the degrees of freedom of the two nodes and the nodal loads of a right circular hinge in the coordinate system originating at node i. The stiffness equation of the right circular hinge can be referred to the literature [22].Figure 8
Schematic of the right circular flexure hinge.In the nonuniform beam, the other flexure joints between rigid bodies also employ right circular flexure hinges to fix the two ends of the beam and connect the beam with two piezoelectric actuators. These flexure hinges have the same geometrical configuration. A mechanical model of the beam is established to characterize the static response of the structure. The right circular flexure hinges are simplified as rotational springs. The flexure hinges then impose an angular momentMi (i = A1, B1, C1, D1) on the rigid beams, as shown in Figure 9. We thus have(15)Mi=Kr3θi,i=A1,B1,C1,D1,where θi is the rotational angle of the hinges and equals the tilt angle of the rigid arms. Kr3 is the rotation stiffness of flexure hinge 3.Figure 9
Simplified model of the nonuniform beam.The force equilibrium equations of the beam can then be derived:(16)FA1+FC1=FB1+FD1,MA1+MB1+MC1+MD1−FB1x+FC1L−x=FD1L,where FB1 and FC1 are the actuation forces provided by two piezoelectric actuators and FB1 = FC1 = F.From equation (16), the nodal forces of the right circular flexure hinge used to construct a flexible segment at point E can be expressed as(17)FE=1L2Fx−4Kr3θi,ME=1−2laLFx−2Kr3θi.The axial loads are ignored in this analysis owing to the small deformation of the beam. The stiffness equation of the right circular flexure hinge can therefore be derived to characterize the deflection of the beam:(18)FR=KRXR,where(19)FR=FEMEFFMFT,XR=ΔyEΔθEΔyFΔθFT=θilaθiθolo2−θoT,KR=1C1C3−C22C1C2−C1C2C2C3−C22RC2−C3−C1−C2C1−C2C22RC2−C3−C2C3,with Cm (m = 1 – 4) denoting the compliance, which refers to the displacements produced by the corresponding applied forces, and R being the cutting radius.Then, from (17) and (18), the defined equivalent stiffnesses Ko and Ki can be obtained:(20)Ko=Fxθo=pLC1la+C2+4Kr3f−pLC1lo/2+C22,Ki=Fxθi=Kof,where(21)p=1C1C3−C22,f=C2la+22RC2−C3−L−2laC1lo/2+C22C2la+C3+2/p1−2la/LKr3−L−2laC1la+C2+4Kr3/pL.According to (20), the output angle θo depends on the moment M = Fx that is derived from the actuation force generated by one piezoelectric actuator. It is therefore necessary to obtain the force F of the piezoelectric actuator that consists of a preloading mechanism and a piezo stack. A simplified model of the piezoelectric actuator is thus built as shown in Figure 10. In this model, the piezo stack is equivalent to a source of displacement connecting in series with a spring, which has stiffness Kp. The preloading mechanism, which has stiffness Kc, is incorporated with the piezo stack to output the actuation force F. The force equilibrium equation can be established according to the deformation equation of the piezo stack given as equation (6):(22)F=CVUin−δKp−δKc,where δ is given by δ = xθi owing to the small value of θi. By substituting equation (22) into (20), the electromechanical model of the rotational part, which relates the output angle around the Z axis of the mechanism to the voltage applied to the piezo stack, is obtained:(23)θo=FxKo=xKpCVfx2Kp+Kc+KiUin.Figure 10
Model of the piezoelectric actuator.


## 3.1. Analysis of the Translational Mechanism

The translational mechanism can be abstracted as a serial mechanical network, as shown in Figure4. To clarify the performance of the mechanism, it is necessary to establish an accurate theoretical model to comprehensively determine the translational motion transmission from the piezo stacks to the platform. A governing equation of the RPA is therefore derived as a starting point for calculating the relationship of input-output static displacements and forces.Figure 4
Schematic diagram of translational motion transmission.Because of the symmetry of the rhombic mechanism, we consider only a quarter of the structure as the study object. The mechanical model based on a pseudo-rigid-body model and Euler–Bernoulli beam theory is shown in Figure5. In this analysis, the rhombic mechanism is considered to be constructed of rigid bodies interconnected by flexure arms in which the mechanism compliance is lumped. According to the energy principle, the input and output displacements of the structure can be expressed as(1)Δin2=∫0lFNxEA⋅2dFNxdFindx+∫0lMxEI⋅2dMxdFindx,(2)ΔP2=∫0lFNxEA⋅2dFNxdFPdx+∫0lMxEI⋅2dMxdFPdx.Figure 5
Model of a quarter of the rhombic mechanism.Here,Fin is the driving force provided by the piezo stack, which produces input displacement Δin for the rhombic mechanism, Fp is the output force, and Δp is the output displacement of the RPA; E is the elastic modulus; A and I are, respectively, the area and moment of inertia of the corresponding cross section of the beam; and FNx and Mx are, respectively, the axial tension and moment along the neutral axis. According to the static analysis shown in Figure 5, we have(3)FNx=12FPsinα−Fincosα,Mx=12FPcosα+Finsinαx−l2.By substituting equation (3) into equations (1) and (2), the relationship between input and output forces and thus the displacements can be obtained:(4)Fincosα−FPsinα=EAlΔincosα−ΔPsinα,Finsinα+FPcosα=2clΔinsinα+ΔPcosα.Here,(5)c=6EIl2.As mentioned above, the input displacement and force acting on the rhombic mechanism are provided by the piezo stack (P885.91, PI). When the hysteresis nonlinearity of the piezoelectric material is ignored, the linear deformation of the piezo stack can be written as(6)δ=UinCV+1KPfP,where δ is the output displacement of the piezo stack, CV is the piezoelectric constant and can be calculated as CV = nd33 (with n being the number of layers of the piezo stack), Uin is the change in the voltage applied to the piezo stack, Kp is the stiffness of the piezo stack, fp is the external force imposed on the piezo stack, and fp = −Fin. Thus,(7)Δin=UinCV−1KPFin.Substituting (7) into (4), the governing equation of the RPA can be expressed in matrix form as(8)FPΔP=1sinαcosα2c−EA2ccos2α+EAsin2α+2EAcKpl2EAcll+1KpEAcos2α+2csin2αEAcos2α+2csin2αFinCVUin=f11f12f21f22FinCVUin.To further characterize the output displacement of the double-amplification mechanism, the mechanical model of the lever mechanism is shown in Figure6. The right circular flexure hinges sharing the same geometry configuration are simplified as rotational springs attached to the lever. Each flexure hinge therefore imposes an angular moment on the lever:(9)Mk=Kr1θk,k=C,O,D,where θk is the rotational angle of the hinges and equals the tilt angle of the rigid lever. θk can be expressed as θk=ΔP/L2 owing to the small value of θk. Kr1 is the rotation stiffness of right circular flexure hinge 1, which can be calculated using the equations presented by Zhu et al. [22]. The force equilibrium equation and geometrical equation of the rigid lever are then established:(10)MC+M0+MD+FCyL1−FDyL2=0,Δout=L1L2ΔP,where FCy and FDy are vertical forces of nodes C and D, and FDy = −FP/2.Figure 6
Mechanical model of the lever mechanism.In the analysis of the parallel configuration, the reaction of the guidance mechanism can be considered equivalent to an elastic load with port stiffness ofKload. According to Newton’s third law of motion, the output force of the double-amplification mechanism is then(11)Fout=−FCy=−Kloadxout4.The mechanical model of the guidance mechanism with four rods is shown in Figure7. The equivalent stiffness Kload can be expressed as(12)Kload=4kx,where kx is the stiffness in the x direction of each guidance rod and be calculated as(13)kx=Kr1+Kr2L32,with Kr2 being the rotation stiffness of right circular flexure hinge 2.Figure 7
Mechanical model of the guidance mechanism.Owing to the symmetry of the translational mechanism, the output displacements alongX and Y axes are equal. The electromechanical model of the translational mechanism, which relates the output displacement of the translational mechanism to the voltage applied to the piezo stack, can therefore be obtained from equations (8)–(11) as(14)xout=2L1L212Kr1+KloadL12f2212Kr1+KloadL12+2f12L22f2112Kr1+KloadL12+2f11L22−f12CVUin.


## 3.2. Analysis of the Rotational Mechanism

In this subsection, to accurately establish a theoretical model of the rotational mechanism, the nonuniform beam is modeled using the stiffness equation of the flexible segments. It is assumed that the compliance of the rotational part is lumped in the flexure hinges and the flexible segments of the beam. The modeling of the mechanism therefore begins with the analysis of the flexure connectors.The flexible segment, namely, the right circular hinge, is used to connect the rigid segments and to deliver high-resolution motion to the output port. Figure8 shows the degrees of freedom of the two nodes and the nodal loads of a right circular hinge in the coordinate system originating at node i. The stiffness equation of the right circular hinge can be referred to the literature [22].Figure 8
Schematic of the right circular flexure hinge.In the nonuniform beam, the other flexure joints between rigid bodies also employ right circular flexure hinges to fix the two ends of the beam and connect the beam with two piezoelectric actuators. These flexure hinges have the same geometrical configuration. A mechanical model of the beam is established to characterize the static response of the structure. The right circular flexure hinges are simplified as rotational springs. The flexure hinges then impose an angular momentMi (i = A1, B1, C1, D1) on the rigid beams, as shown in Figure 9. We thus have(15)Mi=Kr3θi,i=A1,B1,C1,D1,where θi is the rotational angle of the hinges and equals the tilt angle of the rigid arms. Kr3 is the rotation stiffness of flexure hinge 3.Figure 9
Simplified model of the nonuniform beam.The force equilibrium equations of the beam can then be derived:(16)FA1+FC1=FB1+FD1,MA1+MB1+MC1+MD1−FB1x+FC1L−x=FD1L,where FB1 and FC1 are the actuation forces provided by two piezoelectric actuators and FB1 = FC1 = F.From equation (16), the nodal forces of the right circular flexure hinge used to construct a flexible segment at point E can be expressed as(17)FE=1L2Fx−4Kr3θi,ME=1−2laLFx−2Kr3θi.The axial loads are ignored in this analysis owing to the small deformation of the beam. The stiffness equation of the right circular flexure hinge can therefore be derived to characterize the deflection of the beam:(18)FR=KRXR,where(19)FR=FEMEFFMFT,XR=ΔyEΔθEΔyFΔθFT=θilaθiθolo2−θoT,KR=1C1C3−C22C1C2−C1C2C2C3−C22RC2−C3−C1−C2C1−C2C22RC2−C3−C2C3,with Cm (m = 1 – 4) denoting the compliance, which refers to the displacements produced by the corresponding applied forces, and R being the cutting radius.Then, from (17) and (18), the defined equivalent stiffnesses Ko and Ki can be obtained:(20)Ko=Fxθo=pLC1la+C2+4Kr3f−pLC1lo/2+C22,Ki=Fxθi=Kof,where(21)p=1C1C3−C22,f=C2la+22RC2−C3−L−2laC1lo/2+C22C2la+C3+2/p1−2la/LKr3−L−2laC1la+C2+4Kr3/pL.According to (20), the output angle θo depends on the moment M = Fx that is derived from the actuation force generated by one piezoelectric actuator. It is therefore necessary to obtain the force F of the piezoelectric actuator that consists of a preloading mechanism and a piezo stack. A simplified model of the piezoelectric actuator is thus built as shown in Figure 10. In this model, the piezo stack is equivalent to a source of displacement connecting in series with a spring, which has stiffness Kp. The preloading mechanism, which has stiffness Kc, is incorporated with the piezo stack to output the actuation force F. The force equilibrium equation can be established according to the deformation equation of the piezo stack given as equation (6):(22)F=CVUin−δKp−δKc,where δ is given by δ = xθi owing to the small value of θi. By substituting equation (22) into (20), the electromechanical model of the rotational part, which relates the output angle around the Z axis of the mechanism to the voltage applied to the piezo stack, is obtained:(23)θo=FxKo=xKpCVfx2Kp+Kc+KiUin.Figure 10
Model of the piezoelectric actuator.


## 4. Dimensional Optimization

According to the above theoretical analysis, it is obvious that the dimensions are critical to the actuation performance of theXYθz micropositioning mechanism. Hence, to confirm the dimensions of the X/Y-directional part and θz-directional part, design optimization is performed in this section with the goal of achieving large motion travel within a compact size.According to (14) and (23), the output displacements of the mechanism rely on parameters of the geometric configuration, the performance of the piezoelectric actuators, machining feasibility, and material properties. In this optimization analysis, aluminum alloy AL7075-T6 is used because it has a high ratio of the yield strength to Young’s modulus. To reduce the rotation stiffness and increase the rotation precision of the right circular flexure hinges, the hinge thickness and cutting radius should be designed as small as possible. However, the minimum of the two parameters has to be larger than 0.3 mm to avoid the melting of the material during wire electro-discharge machining. In addition, to achieve a low-voltage design in specific cases, the range of the operating voltage in the rotational part is set from 0 to 50 V. All these fixed parameters are given in Table 1.Table 1
Fixed parameters of the mechanism.
Piezoelectric actuator (P885.91, Physik Instrumente, Germany)
Stiffness of the piezo stack
25 N/μm
Stiffness of the preloading mechanism
5.89 N/μm
Size of piezo stack
5 mm × 5 mm × 36 mm
Operating voltage
−20 V–120 V
Piezoelectric constant
0.316μm/V
Material (Al 7075)
Elastic modulus
72 GPa
Shear modulus
27 GPa
Density
2750 kg/m3
Yield strength
500 MPa
Right circular flexure hinge 1
Hinge thickness,t1
0.4 mm
Cutting radius,R1
1.2 mm
Right circular flexure hinge 2
Hinge thickness,t2
0.5 mm
Cutting radius,R2
1.0 mm
Right circular flexure hinge 3
Hinge thickness,t3
0.4 mm
Cutting radius,R3
0.5 mmThe output displacements in theX/Y direction and θz direction are therefore functions formulated by equations (14) and (23), respectively, in terms of the remaining design parameters. Then, equations of the objective functions are determined as follows:(24)Maximumxout=f1α,w,L1,Maximumθo=f2la,lo,R,x.The constraints are described as follows:(1)
To satisfy several specifications, these design parameters are constrained. To achieve a compact size, the following geometric constraints are imposed for the optimal design:(25)L1+L2=21.8,10≤L1≤20,5≤α≤40,1≤w≤4,2la+4R+lo≤44.2,6≤x≤21,5≤lo<48,0.3≤R≤3.(2)
In order to guarantee design reliability, the following maximum stress at each outer surface of the thinnest part of the flexure hinge needs to be treated [19]:(26)σmax=E1+β9/20β2fβθ≤σ=σyn,where β = t/R, θ is an angular displacement, σy is the yield stress of the material, and n is the safety factor.The optimization process can be carried out in MATLAB software to investigate how the maximum output displacement depends on each parameter in a given range; for an arbitrary value of a parameter, there is a corresponding maximum of the output displacement in the parameter space formed by the given domain of the remaining parameters. The optimization results are shown in Figures11–13.Figure 11
Dependence of the maximum translational displacement on the designed parameters.
(a)
(b)Figure 12
Maximum translational displacement plot in terms ofL1 and α.Figure 13
Dependence of the angle travelθz on the design parameters: (a) la; (b) lo; (c) R; (d) x.
(a)
(b)
(c)
(d)Figures11(a) and 11(b) show how the translational displacement changes with beam angle α, thickness w, and length L1. It can be seen from Figure 11(a) that when the angle α is constant, the output displacement increases as the thickness w decreases. Thus, the thickness w of the beam should be the minimum value in its range. Angle α and length L1 reach their respective thresholds when the translational displacement is a maximum, and when α or L1 increases continuously, the maximum translational displacement decreases. Figure 12 shows the dependence of the maximum translational travel on α and L1 when thickness w is taken as constant as discussed above. The constraint for α and L1 is shown as a plane in this 3D plot. It is seen that the translational travel has a maximum value. The thresholds of α and L1 at the maximum displacement can therefore be calculated as(27)∂f1α,L1∂L1=0,∂f1α,L1∂α=0.Figure13(a)–13(d) shows the dependence of the maximum angle travel θz on the design parameters of the θz-directional part. These plots clearly show that la should take a maximum value while lo, R, and x should take minimum values in their ranges.The optimization results of the design parameters of theX/Y-directional part and θz-directional part are summarized in Table 2.Table 2
Optimization parameters and results.
Parameters
X/Y-directional part
θ
z-directional part
w
α
L
1
l
a
l
o
R
x
Parameter limit
Lower
1
5
10
0
5
0.3
6
Upper
4
40
20
19
48
3
21
Optimization results
1
17
17.8
19
5
0.3
6
Unit
mm
deg
mm
mm
mm
mm
mm


## 5. FEA

In this section, FEA of theXYθz micromechanism is carried out using ANSYS software according to the optimized parameters. The 3D model is generated by Solidworks and imported to ANSYS. A multiphysics analysis is then conducted to estimate the performance of the mechanism.To begin with, a static analysis via FEA numerical simulation is conducted to validate the output displacements of the mechanism. In order to calculate the output displacements accurately, piezoelectric stacks are defined as piezoelectric bodies in ANSYS software. And, voltage boundary conditions are exerted to two surfaces of corresponding piezoelectric stack. The results are shown in Figure14. Figure 14(a) shows the motion of the platform along the positive X-axis with a driving voltage of 70 V. And, the maximum bidirectional translational travel can be obtained; the output displacement in the X/Y direction is 240.0 μm. In addition, when the platform is driven in the X direction, the output displacement in the Y direction is almost zero, which shows excellent output decoupling performance of the micropositioning mechanism. Therefore, when the two pairs of PZTs in the translational part are given the same voltages simultaneously, the platform has motion in a direction 45 degrees to the X-axis as seen in Figure 14(b). The motion in the θz direction is shown in Figure 14(c), and the rotational angle around the Z-axis is determined as 10.0 mrad. The maximum stress at maximum deformation is 345.9 MPa and occurs in the flexure hinge. It is lower than the yield stress of the material which is 500 MPa. Meantime, the rotational motion of the whole structure around Z-axis is shown in Figure 14(d). It can be seen that the torsional motion does not cause the translational displacement in X/Y direction because of the excellent decoupling characteristics of the mechanism.Figure 14
Results of static analysis: (a) motion along the positiveX-axis; (b) motion in a direction 45 degrees to the X-axis; (c) motion of rotational part; (d) motion of the entire structure in the θz direction.
(a)
(b)
(c)
(d)The results of theoretical analysis and FEA are compared in Table3. The corresponding relative errors of output displacements in the X/Y direction and θz direction are about 8.2% and 13.0%, respectively. The output displacements obtained from FEA are smaller than the theoretical values. This is mainly due to the rigid levers and rigid arms having elastic deformation in the FEA.Table 3
Comparison of outputs obtained employing the theoretical model and FEA.
X/Y (μm)
θ
z (mrad)
Theoretical modal
259.7
11.3
Finite element analysis
240.0
10.0
Error (%)
8.2
13.0Moreover, the modal analysis of the mechanism with consideration of piezoelectric actuators is also carried out through FEA to investigate the dynamic performance. The modal analysis results in Figure15(a)–15(c) show that natural frequencies along the θz, Y, and X directions are about 105, 505, and 518 Hz.Figure 15
Modal shapes of the structure in (a)θz direction, (b) Y direction, and (c) X direction.
(a)
(b)
(c)A prototype of theXYθz micropositioning mechanism was fabricated according to the results of optimization. The translational part and rotational part were monolithically machined from AL-7075 using wire electrodischarge machining technology for high precision. As shown in Figure 16, the overall dimensions of the prototype were 106 mm × 106 mm × 23 mm. Three pairs of piezo stacks (P885.91, PI) were used to drive the platform.Figure 16
Prototype of theXYθz micropositioning mechanism.


## 6. Experiment

The basic characteristics of the mechanism are experimentally investigated in this section. The principle experimental components are pictured in Figure17. A LabVIEW system (LabVIEW 8.5 with NI PXI-1050) was used to generate the driving signals that were sent to the power amplifier (PA-12D developed at Xi’an Jiaotong University) and to acquire the signal of output displacements from the laser sensor. To measure the output displacements, two pairs of differential voltages were applied to the two pairs of piezoelectric actuators located in the translational part and one pair of the same voltage was applied to the actuators located in the rotational part. Figure 18 shows the relationship of output displacements in X, Y, and θz directions with input voltages. The input voltage of each axis is described by the one input voltage of the voltage pair to the corresponding axis. To achieve a low-voltage design in specific cases and guarantee the security of the structure, the range of the operating voltage in the rotational part is set from 0 to 50 V.Figure 17
Principle and experimental setup of the mechanism.Figure 18
Tested output displacements in (a)X and Y directions and (b) θz direction.
(a)
(b)Figure18(a) shows that the output displacement along the X-axis reached 204.2 μm, which is slightly shorter than the output displacement along the Y-axis (212.8 μm) for a voltage of 120 V. These errors seem to originate from the difference in piezo stacks, machining errors, and installation errors of the mechanical structures. Figure 18(b) shows that the output angle around Z-axis reached 8.7 mrad with a driving voltage of 50 V.Owing to the design of the guidance mechanism and the symmetrical design of the translational part, there is no coupling displacement between theX-axis and Y-axis in theory. However, it is virtually impossible to achieve absolute symmetry of the mechanism owing to the machining and installation errors of piezoelectric actuators. Therefore, the coupling displacement of the two axes was also tested by using the sinusoidal input signal in this work. Figure 19 shows the output displacements when only the X-axis was actuated. It is seen that the maximum coupling error along the Y-axis was about 2.5%. Similarly, when only the Y-axis was actuated, the corresponding maximum coupling error along the X-axis was about 2.9%.Figure 19
Coupling error of the translational part.Figure20 compares the experimental results with the results of FEA. There are two curves for each direction of the experimental result because of the hysteresis of piezo stacks. The experimental results indicate that the actual translational travel and rotational angle around the Z axis were smaller than the results obtained using the finite element model, and the corresponding maximum relative differences (i.e., errors) between the experimental results and FEA were about 12.7% and 14.9%, respectively. These errors mainly originated from machining errors and installation errors of the mechanical structures.Figure 20
Comparison of the finite element model and experiment results: (a) translational travel; (b) output angle around theZ-axis.
(a)
(b)To determine the first natural frequency of the platform, the frequency response was measured. Sweeping sinusoidal signals were used to excite the PZTs inθz direction, and frequency response signal was measured by using laser sensor. Figure 21 shows that the first natural frequency is 87.8 Hz, which is slightly smaller than the FEA result. The deviation may be due to machining error and installation error of the structure. The additional mass caused by the mounting screws and the wires of the piezoelectric stacks can also reduce the natural frequency of the actual structure.Figure 21
Tested first natural frequency of the mechanism.In addition, the motion resolution is tested. As can be seen in Figure22, the minimum resolutions are clearly resolved from the multistep response experiment, which are 52 nm in the X/Y-axis and 9 μrad in the θz-axis, respectively. In this test, the movement of the platform is measured by KEYENCE laser displacement sensor. The resolution of laser sensor is lower than the capacity position sensor. If the resolution of the displacement sensor can be improved, a higher resolution of the developed structure can be achieved.Figure 22
Multistep response for the resolution test: (a)X/Y-axis; (b) θz-axis.
(a)
(b)


## 7. Conclusions

This paper presents the design, modeling, optimization, and experimental testing of a novel piezo-actuatedXYθz micropositioning mechanism featuring large travel, compact size, and 3-DoF independent motion. This mechanism consists of a translational motion part and a rotational motion part, which are linked serially. To achieve large motion ranges, electromechanical models of the translational mechanism and rotational mechanism were established to describe the relationship between the input voltages and the output displacements. The mechanism was then optimized using the theoretical model, realizing large output displacements within a compact size. On the basis of optimized parameters, the FEA results of the mechanism confirmed that the parallel translational part was well decoupled. Finally, a prototype of the mechanism with dimensions of 106 mm × 106 mm × 23 mm was fabricated. Experimental results show that translational travel in X and Y directions was, respectively, 204.2 and 212.8 μm, while travel of 8.7 mrad in the θz direction could be realized. And, the minimum resolutions are 52 nm in the X/Y-axis and 9 μrad in the θz-axis. In addition, the output coupling was evaluated to be below 3%, indicating an excellent decoupling performance. These results show that the developed XYθz micropositioning mechanism can be used in several industrial applications that require large motion ranges within small spaces, especially the optical image stabilization system. The testing of closed-loop control of the micropositioning mechanism will be presented in future work.



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*Source: 5461725-2019-07-01.xml*