Research on a 3-DOF Motion Device Based on the Flexible ... - MDPI

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Research on a 3-DOF Motion Device Based on the Flexible Mechanism Driven by the Piezoelectric Actuators Bingrui Lv , Guilian Wang *, Bin Li, Haibo Zhou and Yahui Hu Tianjin Key Laboratory for Advanced Mechatronic System Design and Intelligent Control, Tianjin University of Technology, Tianjin 300384, China; [email protected] (B.L.); [email protected] (B.L.); [email protected] (H.Z.); [email protected] (Y.H.) * Correspondence: [email protected] Received: 5 September 2018; Accepted: 29 October 2018; Published: 6 November 2018

Abstract: This paper describes the innovative design of a three-dimensional (3D) motion device based on a flexible mechanism, which is used primarily to produce accurate and fast micro-displacement. For example, the rapid contact and separation of the tool and the workpiece are realized by the operation of the 3D motion device in the machining process. This paper mainly concerns the device performance. A theoretical model for the static performance of the device was established using the matrix-based compliance modeling (MCM) method, and the static characteristics of the device were numerically simulated by finite element analysis (FEA). The Lagrangian principle and the finite element analysis method for device dynamics are used for prediction to obtain the natural frequency of the device. Under no-load conditions, the dynamic response performance and linear motion performance of the three directions were tested and analyzed with different input signals, and three sets of vibration trajectories were obtained. Finally, the scratching experiment was carried out. The detection of the workpiece reveals a pronounced periodic texture on the surface, which verifies that the vibration device can generate an ideal 3D vibration trajectory. Keywords: flexible mechanism; three degrees of freedom; matrix-based compliance modeling; piezoelectric actuator

1. Introduction The flexible hinge motion platform has significant advantages such as high transmission efficiency, non-backlash, high resolution and high motion accuracy. It has important applications in electron microscopy, measurement and calibration, ultra-precision positioning platforms, micro force detection, micromanipulators, robots and other fields [1,2]. Therefore, a large number of micro–nano motion platforms based on flexible mechanisms have been designed in recent years [3–6]. Researchers are trying to use different strategies to design more practical vibration devices in certain fields. For example, ways to increase the size of the flexible hinge to increase the bandwidth of the device are used in designing nanopositioning modules for high-throughput nanomanufacturing applications [7]. The decoupled positioning platform with compound parallelogram flexures and a compound bridge-type displacement amplifier is more suitable for the field of microscopic operation [8]. The design of the 3-legged prismatic-prismatic-spherical (3PPS) parallel-kinematic configuration makes the working space of the high-payload flexible parallel robot reach a few millimeters and degrees, which has shown application value in the field of ultraviolet nanometer lithography [9]. The lever-based amplification is used to enhance the displacement of the mechanism of the parallel three degrees of freedom (3-DOF) positioning platform. The large displacement and high repeat positioning accuracy

Micromachines 2018, 9, 578; doi:10.3390/mi9110578

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mechanism is more suitable for the micro–nano operation field [10]. Through the analysis of the influence of the quality and quantity of hinges on the performance of fast tool servo systems, it has been found that the single-hinge fast tool servo structure is more suitable for precision machining of the roll mold [11]. These areas require flexible devices with a wide range, high precision, high resolution, high reliability, and multiple degrees of freedom [12,13]. The application of a flexible mechanism in ultra-precision positioning platforms, electron microscopes and other precision instruments, requires that the motion device has a wide operational range [14,15]. There are several types of displacement amplification mechanisms: the lever type, the bridge type and the Scott-Russell mechanism. These basic amplifying mechanisms achieve superior performance through different combinations in practical applications. The self-guided displacement amplifying mechanism is obtained by combining a double composite guide rail and a bridge amplifying mechanism and can improve the output precision and range of the mechanism [1]. The double-rod amplifying structure based on the lever principle exhibits superior effects over the single-rod mechanism and improves the working range and natural frequency of the overall mechanism [16]. A mechanism consisting of multiple amplifiers has become a common approach. Lin and Zhang, respectively, designed a sub-millimeter working range of motion mechanisms. One motion mechanism is composed of multiple bridge amplifier structures [17], and the other motion mechanism is composed of a bridge amplifier and a lever compound amplifier [18]. The combination of two Scott–Russell and a half-bridge mechanism for double-stage displacement amplification provides a larger magnification ratio. The amplification ratios for the x and y directional motions can reach 5.2 and 5.4, respectively [19]. In addition, the flexible hinge mechanism requires the motion device to have high operational accuracy in applications such as micro gripper, micro-electro-mechanical system MEMS assembly, cell injection and other precision instruments. In general, most of high-precision positioning stages consist of two parts; actuators and guiding mechanisms. There are also some devices that use position feedback sensors to improve accuracy [20]. These devices can achieve an accuracy of approximately 30 nm using intuitive environmental force feedback during low-speed interactions [21]. The closed-loop positioning accuracy of the device can reach 600 nm, using a new type of repetitive compensation proportional integral differential (PID) controller combined with the inverted Prandtl–Ishlinskii model [22]. In order to increase processing efficiency, mechanisms are often required to have a higher natural frequency, based on the application of the flexible hinge mechanism, in the field of diamond turning and polishing. The 2-DOF motion device reduces mass through a compact design and its natural frequency reaches approximately 2900 Hz [23]. High-frequency motion devices also exhibit excellent performance in vibration-assisted turning [24]. Qu et al. designed a 2-DOF motion device. In order to have remote-center-of-motion (RCM) characteristics, the mass of the device is increased, and the natural frequency of the device is also reduced to 280.3 Hz [25]. The three-dimensional vibrating device for elliptical vibratory turning has a large motion range but its natural frequency is low [26]. When the motion device is designed, the input stiffness is usually determined due to the limitation of piezoelectric performance. Therefore, a simple structure and a small number of moving members are common choices for raising the natural frequency of the mechanism. This means that other performance may be degraded. For example, the addition of amplifiers and decoupling components increases the quality of the structure and causes the natural frequency to drop. In addition, as the input frequency increases, the output displacement of the general mechanism will also decay. These issues require a lot of research. For some other indicators, such as decoupling, compact structure, high-stability and multiple-degrees-of-freedom, some scholars have undertaken deep research. Zhou et al. established a 2-DOF flexible mechanism, using the parallel kinematic constraint map method, to achieve totally geometric decoupling and actuator isolation [27]. Guo et al. realized the coupling compensation of the motion platform through the decoupling feedforward/feedback controller [28] and applied it to micro/nano positioning control [29]. Cai et al. improved the stability and plane motion capability of


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the mechanism through the design of the “T-shaped” flexible hinge mechanism [30]. Lee et al. used closed-loop feedback to deal with the decoupling problem, so that the operating bandwidth of the device reached 100 Hz [3]. Li et al. invented a compact series piezoelectric drive platform through a “Z-shaped” flexible hinge design [31]. In addition, Cai et al. designed a six-degree-of-freedom precision positioning of the motion platform [32]. Chen et al. designed a large-range compliant remote center of motion RCM stage with input/output decoupling [33]. In general, for motion platforms based on the flexible mechanism, a lot of research and exploration has been done on the range of motion, motion accuracy and natural frequency. Meanwhile, various motion platforms have been developed to suit different applications, but there are still some problems Micromachines 2018, 9, x FOR PEER REVIEW 3 of 34 that remain unsolved. For example, because of the limitation of the spatial structure, the output positioning control [29]. Cai et been al. improved the stability and planeIn motion capability stiffnessmicro/nano problem of the output terminal has given less consideration. this case, someofdevices the mechanism through design of the “T-shaped” hinge mechanism [30].The Leeoutput et al. used may become inefficient when the a motion platform is usedflexible for precision machining. reliability, feedback to deal with the decoupling problem, so that the operating of the cutting stabilityclosed-loop and operational accuracy of these devices are relatively decreased due bandwidth to the complex device reached 100 Hz [3]. Li et al. invented a compact series piezoelectric drive platform through a force. In addition, most of the existing devices only achieve open-loop control; however, the actual “Z-shaped” flexible hinge design [31]. In addition, Cai et al. designed a six-degree-of-freedom displacement detection andofcontrol at the output terminal difficult to accomplish. precision positioning the motion platform [32]. Chen etare al. more designed a large-range compliant In this paper, aremote new type 3-DOFRCM motion that can realize closed centerof of motion stagedevice with input/output decoupling [33]. loop control with large output Inageneral, for motion platformsofbased on the is flexible mechanism, a lotcontents of research stiffness, and self-guided characteristic the output designed. The main are and as follows: has been analysis done on of thethe range of is motion, accuracy and natural of frequency. first, theexploration static and dynamic device carriedmotion out based on the analysis the mechanism Meanwhile, various motion platforms have been developed to suit different applications, but there and characteristics of the device; second, the dynamic performance, axial linear motion performance are still some problems that remain unsolved. For example, because of the limitation of the spatial and motion trajectory detection of the device are tested. structure, the output stiffness problem of the output terminal has been given less consideration. In this case, some devices may become inefficient when a motion platform is used for precision

2. Overall DesignThe of the System machining. output reliability, stability and operational accuracy of these devices are relatively

decreased due to the complex cutting force. In addition, most of the existing devices only achieve

2.1. Summary open-loop control; however, the actual displacement detection and control at the output terminal are

more difficult to device accomplish. paper, a new type of 3-DOF motion was device that can realize A 3-DOF motion withInx this y and z three direction translations designed in this study. closed loop control with large output stiffness, and a self-guided characteristic of the output is As shown in Figure 1a, the 3-DOF motion device has the characteristics of a simple control, large designed. The main contents are as follows: first, the static and dynamic analysis of the device is output stiffness, self-guided output. Meanwhile it and alsocharacteristics can realize closed control. carried outand based on the analysis of the mechanism of the loop device; second,The the device is mainly a seriesperformance, connected axial by the two-dimensional (2D)and motion platform x the anddevice y directions, dynamic linear motion performance motion trajectoryincluding detection of tested. and theare independent movement structure with z direction. The 2D motion platform is shown as Figure 1b, and the independent movement structure is shown as Figure 1c.

(a)

(b)

(c)

Figure 1. Schematic a three degrees of of freedom motion device. (a) Overall of the Figure 1. Schematic of aofthree degrees freedom(3-DOF) (3-DOF) motion device. (a) design Overall design of system; two-dimensional (2D) platform (x and y directions); (c) flexible-hinge with z with the system; (b)(b) two-dimensional (2D)motion motion platform (x and y directions); (c) flexible-hinge direction. z direction.

As shown in Figure 2, the 2D motion platform consists of a drive element, a force-removing element (consisting of four force parts), and x and y output decoupling guide elements. The driving element principle of the motion platform is shown in Figure 3. The principle of a double parallelogram is applied to ensure that the driving block moves steadily along the y axis under the

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(b)

(c)

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(b) degrees of freedom (3-DOF) motion device. (c) (a) Overall design of the Figure 1. Schematic of a three system; (b)Schematic two-dimensional motion platform(3-DOF) (x and motion y directions); z Figure 1. of a three(2D) degrees of freedom device.(c) (a)flexible-hinge Overall designwith of the direction. system; (b) two-dimensional (2D) motion platform (x and y directions); (c) flexible-hinge with z


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direction.

As shown in Figure 2, the 2D motion platform consists of a drive element, a force-removing element of four2,force parts), and platform xplatform and y output decoupling elements. The driving As (consisting shown in Figure the 2D motion consists of a force-removing motion consists of a driveguide element, element principle of the motion platform is shown in Figure 3. The principle of a double element (consisting of four force parts), and x and yy output output decoupling decoupling guide guide elements. elements. The driving parallelogram is applied to ensure that the driving block moves steadily along the y axis under the element principle principleof of motion platform is shown 3. Theof principle of a double thethe motion platform is shown in Figurein3. Figure The principle a double parallelogram action of the piezoelectric driving force, which can reduce the deflection error of the driving block parallelogram is applied that the driving block moves along the the y axis under is applied to ensure that to theensure driving block moves steadily alongsteadily the y axis under action of the caused the driving piezoelectric installation error. action by of the piezoelectric force, which reduce the deflection error of the caused drivingby block piezoelectric force,driving which can reduce thecan deflection error of the driving block the caused by theinstallation piezoelectric installation error. piezoelectric error.

Figure 2. 2D motion platform structure composition. Figure 2. 2. 2D 2D motion motion platform platform structure structure composition. composition. Figure

Figure3.3.Principle Principleofofthe thedriving drivingelement. element. Figure

3. Principle of the driving element. The force decomposition Figure element of the three-dimensional (3D) motion device can separate The force decomposition element of the three-dimensional (3D) motion device can separate the the force of driving element into x and y directions so as to realize the x and y movement of the force The of driving element into x element and y directions so as to realize the x and y movement ofseparate the output force decomposition of the three-dimensional (3D) motion device output platform by the different feeding strategies of the two driving blocks, blockcan A and blockthe B. platform by the different feeding strategies of the two driving blocks, block A and block B. The forcex-direction of driving motion elementprinciple into x and directions as to realize the xinand y movement of the The of ya 2D motionso platform is shown Figure 4. When the No.output 1 and x-direction motion principlefeeding of a 2D strategies motion platform is shown in Figure 4. WhenA theand No.block 1 andB. No. 2 platform by the different the two driving blocks, The No. 2 piezoelectric elements are extended or of shortened at the same time block at the same speed, the 2D piezoelectric elements are extended shortened at the time at the same speed, No. the 2D motion x-direction motion principle of a 2Dor motion is same shown in Figure 4. When 1 and No. 2 motion platform produces displacement inplatform the x direction. For example, blockthe A and block B are platform produces displacement in the x direction. For example, block A and block B are piezoelectric elements are extended or shortened at the same time at the same speed, the 2D motion respectively moved by distances y1 and y2 . When y1 = y2 , the output block of the 2D motion platform respectively moved bydisplacement distances y1 and 2. When y1 = y2, the output block of the 2D motion platform platform in ythe xIndirection. For example, A and blockblock B are generates produces a displacement in the x direction. fact, the equivalent rod ofblock the force-dividing is generates a displacement in the x direction. In fact, the equivalent rod of the force-dividing block is respectively moved by the distances y1 and y2. When y1 = y2, the output block of the 2D motion platform always inclined within operating range of the device. When the piezoelectric element is extended, always inclined within the operating range of the When piezoelectric element generates a displacement in the direction. In fact, thedevice. equivalent rod the of the force-dividing blockisis the 2D motion output terminal is xalways subjected to the component force along the x-positive direction, always inclined within the along operating range of Owing the device. Whenrebound, the piezoelectric element is and it can only move upward the x-direction. to material when the piezoelectric element is shortened, the 2D output terminal moves down along the x-negative direction.


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extended, the 2D motion output terminal is always subjected to the component force along the x-positive and it can onlyterminal move upward along the x-direction. Owing to material rebound, extended, direction, the 2D motion output is always subjected to the component force along the when the piezoelectric element is shortened, the 2D output terminal moves down along the 5 of 33 Micromachines 2018, 9, 578 x-positive direction, and it can only move upward along the x-direction. Owing to material rebound, x-negative when the direction. piezoelectric element is shortened, the 2D output terminal moves down along the x-negative direction.

Figure 4. Principle ofofx-direction motionofof2D2D motion platform. Figure 4. Principle x-direction motion motion platform. Figure 4. Principle of x-direction motion of 2D motion platform.

The y-direction motion principleofofaa2D 2D motion is shown in Figure 5. When5.the No. 1 the No. The y-direction motion principle motionplatform platform is shown in Figure When piezoelectric element is extended or shortened, shortened, the No. shortened The element y-direction principle of a 2D motion platform shown in element Figure 5. is When theshortened No.or1 1 piezoelectric ismotion extended or the No.2 piezoelectric 2ispiezoelectric element is or extended at the same speed at the same time. For example, the driving block A and block B move yor4 piezoelectric element is extended or shortened, the No. 2 piezoelectric element is shortened extended at the same speed at the same time. For example, the driving block A and block B move y4 and y5 to the right, respectively. When y4 time. = y5, the block the 2D block motion extended at the same speed at the Foroutput example, theofdriving A platform and blockgenerates B move ya4 and y5 to the right, respectively. Whensame y4 = y5 , the output block of the 2D motion platform generates a displacement in the y-direction. and y5 to the right, respectively. When y4 = y5, the output block of the 2D motion platform generates a displacement in the y-direction. displacement in the y-direction.

Figure 5. Principle of y-direction motion of 2D motion platform.

Figure 5. Principle of of y-direction 2D motion platform. Figure 5. Principle y-direction motion motion ofof2D motion platform.

2.2. x and y Output Decoupling Guide Element

2.2. x and Decoupling Element 2.2.y xOutput and y Output Decoupling Guide Element As shown in Figure 6,Guide since the 2D motion platform is often subjected to the movements Mx and y during vibration cutting or polishing, this requires a large torsional stiffness at the output As shown in Figure 6, since motionplatform platform isisoften subjected to the Mx andMx and AsM shown in Figure 6, since thethe 2D2D motion often subjected to movements the movements terminal of the 2D motion platform. In order to further enhance the device’s operational stability, M y during vibration cutting or polishing, this requires a large torsional stiffness at the output My during vibration cutting or polishing, this requiresdisplacement a large torsional stiffness at the output terminal reliability, and so that the actual in the x andoperational y directionsstability, can be terminal ofoutput the 2Daccuracy, motion platform. In order to further enhance the device’s of the 2D motion platform. In order to further enhance the device’s operational stability, reliability, detected any time, a fullyand decoupled guidance structure for the 2Dxmotion platform output reliability,atoutput accuracy, so that the actual displacement in the and y directions can be output accuracy, and so that the actual displacement in the x and y directions can be detected terminal was designed. detected at any time, a fully decoupled guidance structure for the 2D motion platform output at any time, a fully decoupled guidance structure for the 2D motion platform output terminal was designed. terminal was2018, designed. Micromachines 9, x FOR PEER REVIEW 6 of 34

Figure 6. Commonforce forceconditions conditions during work. Figure 6. Common during work.

The principle the x-guidance elementprinciple principle isisshown in in Figure 7. On it can it can The principle of theofx-guidance element shown Figure 7.the Onone thehand, one hand, reduce the influence of force or moment on the output terminal and improve the output precision reduce the influence of force or moment on the output terminal and improve the output precision and and output stiffness of the 2D motion platform, so as to improve the reliability of the device. On the output stiffness of the 2D motion platform, so as toofimprove the reliability thereflect device. the other other hand, detecting the displacement value the x-guidance structure of can the On actual hand, detecting the displacement value the x-guidance can reflect actual displacement of the x direction of theof2D motion platformstructure output terminal in realthe time or thedisplacement strain gauge can be mounted on the beam flexible hinge to calculate its actual displacement in the x direction. Because the beam flexure hinge will reduce the rigidity of the mechanism, a circular flexure hinge is adopted in this paper.

Figure 6. Common force conditions during work.

of the x-guidance element principle is shown in Figure 7. On the one hand, it can Micromachines The 2018,principle 9, 578

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reduce the influence of force or moment on the output terminal and improve the output precision and output stiffness of the 2D motion platform, so as to improve the reliability of the device. On the hand, of detecting displacement value of the x-guidance structure reflect the gauge actual can be of the xother direction the 2D the motion platform output terminal in real time can or the strain displacement of the x direction thecalculate 2D motion output terminalin inthe realxtime or the strain mounted on the beam flexible hingeofto itsplatform actual displacement direction. Because the gauge can be mounted on the beam flexible hinge to calculate its actual displacement in the x beam flexure hinge will reduce the rigidity of the mechanism, a circular flexure hinge is adopted in direction. Because the beam flexure hinge will reduce the rigidity of the mechanism, a circular this paper. flexure hinge is adopted in this paper.

Figure Thex-guidance x-guidance element principle. Figure 7. 7. The element principle.

The principle of y-guidance the y-guidancestructure structure is is shown shown ininFigure 8. Similar to thetox-guidance The principle of the Figure 8. Similar the x-guidance structure function, it not only improves the y-direction output accuracy, but also improves output structure function, it not only improves the y-direction output accuracy, but also improves output force conditions. In addition, it can realize the displacement detection of the 2D motion platform force conditions. In addition, it can The realize the displacement of x-direction. the 2D motion platform output terminal in the y direction. detection method is similardetection to that of the In order output terminal inthe theforce y direction. Theofdetection method similar to possible, that of the Inaorder to to improve performance the output terminalisas much as thisx-direction. design adopts improvestraight the force performance of theit output terminal as that much possible, this design adopts a straight round hinge. However, should be explained theas power element mechanics midpoint of the device is not exactly the same as the mechanical midpoint the x, y output decoupling guide of the round hinge. However, it should be explained that the powerofelement mechanics midpoint mechanics midpoint. This may resultmidpoint in slight deflection the output block of guide the 2Delement device iselement not exactly the same as the mechanical of the x, yofoutput decoupling mobile platform. mechanics midpoint. This may result in slight deflection of the output block of the 2D mobile platform. Micromachines 2018, 9, x FOR PEER REVIEW 7 of 34

Figure 8. 8. The element principle. Figure They-guidance y-guidance element principle. this paper, the power 3Dmotion motion device thethe principle of serial-to-hybrid In this In paper, the power partpart of of thethe3D deviceadopts adopts principle of serial-to-hybrid mixing, which has the advantages of both series and parallel structures. The parallel structure avoids mixing, which has the advantages of both series and parallel structures. The parallel structure avoids the coupling of the piezoelectric actuator holder and the moving part, which further the coupling of the piezoelectric actuator holder and the moving part, which further enhances the enhances the natural frequency of the device. In addition, the 2D motion platform adopts a parallel natural structure. frequency the1 device. addition, the 2D motion platform parallel TheofNo. and No. In 2 piezoelectric actuators are arranged in a adopts straight aline, which structure. is The No.advantageous 1 and No. 2forpiezoelectric actuators arranged a straightguide line,structure which is advantageous the structure and the forceare symmetry. Theindecoupling adopts the for the structure the force symmetry. decoupling structure adopts thenot principle of a principle ofand a parallelogram to ensure theThe parallel movementguide of the output terminal, which only improves the output stiffness, but also improves the running accuracy of the device. The parallelogram to ensure the parallel movement of the output terminal, which not only improves the element decomposes displacement of device. the outputThe terminal into the xelement output decoupling stiffness, but alsocharacteristic improves the runningthe accuracy of the decoupling and y directions. The automatic displacement decomposition inside the device provides an aid for characteristic decomposes the displacement of the output terminal into the x and y directions. the detection and control of the output displacement of the device. The automatic displacement decomposition inside the device provides an aid for the detection and control of the output displacement of the device. 3. Analysis of Mechanism Characteristics 3.1. Static Analysis 3.1.1. Calculation of Output Flexibility Using the MCM Method (1) Output Flexibility in the x and y Directions


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3. Analysis of Mechanism Characteristics 3.1. Static Analysis 3.1.1. Calculation of Output Flexibility Using the MCM Method (1) Output Flexibility in the x and y Directions The matrix-based compliance modeling (MCM) method is given in the Supplementary Materials [34]. The driving element and force-dividing element of the 2D platform provide impetus for the output platform, and the x and y output decoupling guide element improves the rigidity of the output platform. On the premise that the output center point of the power part is consistent with the mechanics center of the x and y output decoupling guide element, in order to calculate the whole output terminal flexibility, 2D motion platform is divided into two parts: the power part and the decoupling detection part. The power part of x and y is shown in Figure 9. In Figure 9, the red marked Micromachines 2018, 9, x FOR PEER REVIEW 8 of 34 area has little influence on its static performance. Therefore, this part is considered to be rigid.

Figure 9. x and y power part. Figure 9. x and y power part.

Some symbols are defined for convenient description. Some symbols are defined for convenient description. R is the flexibility The Cm matrix, which corresponds to the flexibility of the circular flexure hinge The 𝐶 is the flexibility matrix, which corresponds to the flexibility of the circular flexure hinge with the output terminal point m. It can be obtained according to Supplementary Materials Equation with the output terminal point m. It can be obtained according to Supplementary Materials Equation (S5). In this paper, the possible values of m include: A, C, A1 , C1 , A0 , C10 , A2 , C2 , A3 , C3 , A4 , C4 . (S5). In thisLpaper, the possible values of m include: 𝐴, 𝐶, 𝐴 , 𝐶 , A ,1C ,𝐴 ,𝐶 ,𝐴 ,𝐶 ,𝐴 ,𝐶 . The C is the flexibility matrix, which corresponds to the flexibility of the straight beam flexure The 𝐶n is the flexibility matrix, which corresponds to the flexibility of the straight beam flexure hinge with the output terminal point n. It can be obtained according to Supplementary Materials hinge with the output terminal point n. It can be obtained according to Supplementary Materials Equation (S6). In this paper, the possible values of n include: B, D, B1 , B2 , D1 , B10 , B3 , B4 . Equation (S6). In this paper, the possible values of n include: 𝐵, 𝐷, 𝐵 , 𝐵 , 𝐷 , B ,𝐵 , 𝐵 . The T V is a position matrix generated by moving the coordinate system with N as the coordinate The 𝑇N is a position matrix generated by moving the coordinate system with N as the origin to the coordinate system with V as the origin. It can be obtained according to Supplementary coordinate origin to the coordinate system with V as the origin. It can be obtained according to Materials Equations (S7)–(S9). In this paper, the possible values of V include: OK1 , OK2 , O1 , O3 , O4 . Supplementary Materials Equations (S7)–(S9). In this paper,0 the possible values of V The possible values of N include: A, B, C, D, A1 , B1 , C1 , D1 , A1 , B10 , C10 , A2 , B2 , C2 , A3 , B3 , C3 , A4 , include: 0 0𝑂 0, 𝑂 , 𝑂 ,𝑂 ,𝑂 . The possible values of N include: B4 , C4 , A , B , C , OK1 , OK2 . 𝐴, 𝐵, 𝐶, 𝐷,4𝐴 ,4𝐵 ,4𝐶 , 𝐷 , 𝐴 , 𝐵 , 𝐶 , 𝐴 , 𝐵 , 𝐶 , 𝐴 , 𝐵 , 𝐶 , 𝐴 , R 𝐵 , 𝐶 , 𝐴 , 𝐵 , 𝐶 , 𝑂 , 𝑂 . For example, in Figure 10, the flexibility matrix C of the straight circular hinge at point A1 can For example, in Figure 10, the flexibility matrix AC1 of the straight circular O hinge at point 𝐴 be obtained by the Supplementary Materials Equation (S5). The position matrix T AK1 from point A1 to from can be obtained by the Supplementary Materials Equation (S5). The position 1matrix T point OK1 can be obtained by the Supplementary Materials Equations (S7)–(S9). The flexibility of the point 𝐴 to point 𝑂 can be obtained by the Supplementary MaterialsO Equations (S7)–(S9). The T O R ∗ ( T K1 ) . straight circular hinge at the point OK1 can be obtained by TA K1 ∗ C A 1 byA1 𝑇 ∗ 𝐶 ∗ (𝑇 ) . flexibility of the straight circular hinge at the point 𝑂 can be1 obtained In addition, if two hinges are connected in series, it can be expressed as Cm = Cm1 + Cm2 . If two In addition, if two hinges are connected in series, it can be expressed as 𝐶 = 𝐶 + 𝐶 . If two −1 −1 −1 hinges are are connected connected in Cm1 + where the the flexibility flexibility of of hinges in parallel, parallel, it it can can be be expressed expressed as as C𝐶m = = ((𝐶 +C 𝐶m2 )) ,, where the two two hinges hinges is is C𝐶m1 and Cm𝐶is the total flexibility of the twotwo flexible hinges. the andCm2 𝐶 respectively, respectively, is the total flexibility of the flexible hinges. A static analysis of the power portions of x and y is shown in Figure 10. The half-output flexibility A static analysis of the power portions of x and y is shown in Figure 10. The half-output of the power part can be simplified as a parallel connection of two flexible rods with stiffness K1with and flexibility of the power part can be simplified as a parallel connection of two flexible rods then in series with the with stiffness 2 *with K2 . The K1 model be𝐾expressed as rod A1 B1 C1 and stiffness K1 and then inrod series with the rod stiffness 2 * K2can . The model can be expressed as rod 𝐴 𝐵 𝐶 and rod 𝐴 𝐵 𝐶 in parallel and then in series with the rod 𝐷 . Where the stiffness of rod A B C can be expressed as C , C and C in series; the stiffness of rod 𝐴 𝐵 𝐶 can be expressed as 𝐶 , 𝐶 and 𝐶 in series. The 𝐾 model can be expressed as 𝐶 , 𝐶 and 𝐶 in

series. Therefore, the flexibility matrix of K1 and K2 can be expressed as:


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rod A10 B10 C10 in parallel and then in series with the rod D1 . Where the stiffness of rod A1 B1 C1 can be expressed as C A1 , CB1 and CC1 in series; the stiffness of rod A10 B10 C10 can be expressed as CA0 , CB0 and 1 1 CC0 in series. The K2 model can be expressed as C A1 , CB1 and CC1 in series. Therefore, the flexibility 1 matrix of K1 and K2 can be expressed as:  T T T −1 O  R ∗ ( T OK1 ) + T OK1 ∗ C L ∗ ( T OK1 ) + T OK1 ∗ C R ∗ ( T O1 ) )  CK1 = (( TA K1 ∗ C A  B C B B A C C  1 1 1 1 1 1 1 1 1  T T T −1 T R ∗ ( T OK1 ) + T OK1 ∗ C L ∗ ( T OK1 ) + T OK1 ∗ C R ∗ ( T O1 ) ) )−1 + T OK1 ∗ C L ∗ ( T O1 ) +( TAO0K1 ∗ C A 0 0 0 0 0 0 0 0 D1 D1 D1 B C A B B C C  1 1 1 1 1 1 1 1 1   T T T  OK  R ∗ ( T O K2 ) + T O K2 ∗ C L ∗ ( T O K2 ) + T O K2 ∗ C R ∗ ( T O K2 ) ) ∗ T CK2 = TZ2 ∗ ( TA2 2 ∗ C A Z2 B2 C2 B2 B2 A2 C2 C2 2

(1)

where TZ2 is the coordinate transformation matrix of 90◦ around z-axis. The half-output flexibility of the power part of a 2D motion platform CDH can be expressed as: CDH = TO K2 ∗ (((CK1 )−1 + ( TY ∗ CK1 ∗ ( TY ) T ) O

−1

K1

T

K2 ) ∗ ( TOOK1 ) + 2 ∗ CK2

(2)

where TY is the coordinate transformation matrix of 180◦ around y-axis. The overall flexibility of the power part of a2018, 2D 9,motion platform as: Micromachines x FOR PEER REVIEWis symmetric about the x-axis. Therefore, CD can be expressed 9 of 34 −1 −1

T ）−11 ++ ((TTXX ∗∗CCDH =C （ (DH C DH ∗∗(T(XT)XT )− 1))− 1 ) CDC= (( )− D DH

(3)

(3)

where TX is the coordinate transformation matrix of 180° around the x-axis.

where TX is the coordinate transformation matrix of 180◦ around the x-axis.

Figure 10. Static analysis of x and y power part. Figure 10. Static analysis of x and y power part.

In order to simplify the calculation, when the x-direction statics analysis is performed, it can In order to simplify the calculation, when the x-direction statics analysis is performed, it can be be considered thatthe they-direction y-direction hinge is a rigid When the deformation of the y-direction considered that hinge is a rigid body. body. When the deformation of the y-direction hinge hinge is not considered inx-direction, the x-direction, the x-direction flexibility of the x anddecoupling y output decoupling is not considered in the the x-direction flexibility of the x and y output guide guide element considered as two hinges connected in series as shown in Figure 11. Where element cancan be be considered as two hinges connected in series as shown in Figure 11. Where the the flexibility of one of the hinges can be expressed as: flexibility of one of the hinges can be expressed as:

C

= ((C )−1 + (T * C

* (T )T )−1−)1

1X −1 X CXX = ((C1X ) + ( TX X ∗ C11X ∗ ( TXX )T )

)

C 1X = ((TAOO 4 * CRAR * (TOAO4 )TT + TBOO 4 * C LBL * (TOBO4 )TT + TCOO 4 * CRCR * (TOCO4 )TT)−−1 1 4

4

4

4

4

4

C1X = (( TA4 ∗ C A44 ∗ ( TA4 ) + TB44 ∗ CB44 ∗ ( TB44 ) + TC4 ∗ CC44 ∗ ( TC4 ) ) 4

O4

R CAR4' 0

O4 T T ' 4 ( TAO 4 0 )

4

O4

L CBL4' 0

+(TTA'O4 *∗ C *∗ (T ) ++TTB 'O4*∗C +( 4 0 4 0 A A A B 4

4

4

4

B4

O4 T T ( TBO4'04 )

4

O4

R CCR4'0

4

O 4 T T −1−1−1 ( TCO4' 04 ) ) )−1

*∗ (T ) ++TTC 'O4 *∗ C *∗ (T ) ) ) 4 0 C B C C 4

4

4

x-axis. where TX is the coordinate transformation matrix of 180° around ◦

where TX is the coordinate transformation matrix of 180 around x-axis.

Figure 11. Analysis of x-direction hinge.

4

(4)

(4)

X

1X

O4 A4

C 1X = ((T

X

1X

X

O4 T A4

* C AR * (T ) + TBO4 * C BL * (TBO 4 )T + TCO4 * C CR * (TCO4 )T )−1 4

O4

O4 T A4'

4

4

O4

4

O4 T B 4'

4

4

O4

4

(4)

O 4 T −1 −1 C 4'

+(TA' * C AR' * (T ) + TB ' * C BL' * (T ) + TC ' * C CR' * (T ) ) ) 4

4

4

4

4

4


9 of 33

where TX is the coordinate transformation matrix of 180° around x-axis.

Figure Figure 11. 11. Analysis Analysis of of x-direction x-direction hinge. hinge.

Similarly, when the y-direction statics analysis is performed, it can be considered that the Similarly, when the y-direction statics analysis is performed, it can be considered that the x-direction hinge is a rigid body. When the deformation of the x-direction hinge is not considered in the x-direction hinge is a rigid body. When the deformation of the x-direction hinge is not considered in y-direction, the y-direction flexibility of x and y output decoupling guide element can be considered the y-direction, the y-direction flexibility of x and y output decoupling guide element can be as four hinges connected in series, as shown in Figure 12. The flexibility of one of the hinges can be considered as four hinges connected in series, as shown in Figure 12. The flexibility of one of the expressed as: hinges can be expressed as: −1

T CCYY = = (( ((TTOAO3 ∗*CCRAR ∗*( T (TOA3O ))TT ++TTOB3O ∗ *C RC BR∗ (*T(OT3B)OT)+ +OT3CO∗ C*R C∗CR ( T*O(3T)CTO ))T )−1 T B3 A3 C3 B3 B3 A3 A3 C3 C3 3

3

3

3

3

3

3

T

3

3

3

3

3

3

3

T

3

T −1

O 33 ++TTY Y *∗ ((TTAOAO3 3 *∗ C C ARAR33 ∗* ((TTAOAO3333 ))T + + TTBO * CR R * (TO3O 3 )T + TO3O 3 * RC R * O (T3 O 3 )T )−∗1 T*YT)Y−)1−1 B 33 ∗ CB3B 3∗ ( TB3B )3 + TC3C 3∗ CC3 C∗3 ( TC3 C) 3 ) 33

Micromachines 9, x FOR PEERtransformation REVIEW the coordinate matrix of 180 180°◦ around the y-axis. where TTY is2018, where Y

(5)(5) 10 of 34

Figure Figure 12. 12. Analysis Analysis of of y-direction y-direction hinge. hinge.

output flexibility of theofflexible hinge device x and yindirections be expressed as: In summary, summary,the the output flexibility the flexible hinge in device x and y can directions can be expressed as:  −1  C )−1 + (2 ∗ CX )−1 WX = (CD -1 -1 -1 (6) CWX =（（C D） −+1（2*C X））−1 −1 C = (( C ) + 4 ∗ C ) ) ( WY D Y (6)  -1 −1 −1 C WY = ((C D ) + (4*C Y ) )

(2) Output flexibility in the z direction (2) Output flexibility in the z direction In order to simplify the calculation, when the z-direction statics analysis is performed, it can be In orderthat to simplify calculation, when z-direction staticsthe analysis is performed, it can be considered the x andthe y directions hinge is the a rigid body. When z-direction statics analysis is considered that the x and y directions hinge is a rigid body. When the z-direction statics analysis is performed, the influence of 2D motion platforms flexibility in the z direction is not considered, so only performed, theofinfluence of 2D hinge motion platforms flexibility the zhinge direction is not considered, the flexibility the z-direction is considered. Since theinz-axis is symmetric, only half so of only the flexibility of the z-direction hinge is considered. Since the z-axis hinge is symmetric, only the structure needs to be studied. The z-direction hinge can be considered as a series connection of half of the structure needs be beam studied. Thehinges. z-direction hinge can of bea considered as ahinge series two circular flexure hinges andtotwo flexure A static analysis half z-direction is connection of two circular flexure hinges and two beam flexure hinges. A static analysis of a half shown in Figure 13. The output flexibility of the z-direction hinge can be expressed as: z-direction hinge is shown in Figure 13. The output flexibility of the z-direction hinge can be  −1 −1 expressed as:  CZ = ((CO1 )−1 + ( TZ ∗ CO1 ∗ ( TZ ) T ) ) (7) O∗1 (T )TL)−1 )−1 O1 T O1 O1 T O1 O1 T  C = C Z O=1 ∗((CCRO )∗−1( T+O(1T)Z T ∗+C T R L O1 o1  T A1 B ∗ ZCB ∗ ( TB ) + TC ∗ CC ∗ ( TC ) + TD ∗ CD ∗ ( TD ) A A (7)  O O T O O T O O T O O T R L R L C o1 = TA 1 * C A * (TA 1 ) + TB 1 * C B * (TB 1 ) + T◦C 1 * C C * (TC 1 ) + TD 1 * C D * (TD 1 ) where Tz is the coordinate transformation matrix of 180 around z-axis.

where Tz is the coordinate transformation matrix of 180° around z-axis.

expressed as: C Z = ((C O )−1 + (TZ ∗ C O ∗ (TZ )T )−1 )−1  1 1  O1 O1 T R = + TBO1 * C BL * (TBO1 )T + TCO1 * C CR * (TCO1 )T + TDO1 * C DL * (TDO1 )T C T * C * ( T )  o1 A A A Micromachines 2018, 9, 578

where Tz is the coordinate transformation matrix of 180° around z-axis.

(7) 10 of 33

Figure 13. 13. Analysis Analysisof ofz-direction z-direction hinge. hinge. Figure

When the flexibility of the z-direction hinge mechanism is calculated, the influence of 2D motion When the flexibility of the z-direction hinge mechanism is calculated, the influence of 2D platform on the z-direction flexibility can be ignored. Therefore, the flexibility of the z-direction hinge motion platform on the z-direction flexibility can be ignored. Therefore, the flexibility of the device can be expressed as: z-direction hinge device can be expressed as: Cwz = Cz (8) C wz = C z (8) 3.1.2. Finite Element Analysis (FEA) Micromachines 2018, 9, x FOR PEER REVIEW

11 of 34

structure parameters: t = 0.7 mm, r = 2 mm, h = 5 mm, b = 10 mm, w = 10 mm. As shown in 3.1.2. The Finite (FEA) TheElement structureAnalysis parameters: t = 0.7 mm, r = 2 mm, h = 5 mm, b = 10 mm, w = 10 mm. As shown in Figure 14, full constraints are applied to all faces in the model that are attached to the fixture. Solid Figure 14, full constraints are applied to all faces in the model that are attached to the fixture. Solid model 10 node 187 elements are used in the finite element model. The material is an aluminum alloy model 10 node 187 elements are used in the finite element model. The material is an aluminum withalloy elasticity, linearitylinearity and isotropy. The elastic modulus is 70isMPa. Poisson’s with elasticity, and isotropy. The elastic modulus 70 MPa. Poisson’sratio ratioisis0.3. 0.3. A A force of 100 N is applied in three directions, x y and z, respectively. The displacement nephograms force of 100 N is applied in three directions, x y and z, respectively. The displacement nephogramsfor all directions are shown Figure The14. output compliance of the MCM method results and for all directions areinshown in 14. Figure The output compliance of the MCM methodanalysis analysis results FEAare results arein given in 1. Table 1. FEA and results given Table

(a) Figure 14. Cont.


11 of 33


12 of 34

(b)

(c) Figure 14. Finite element analysis outputcompliance. compliance. (a) nephogram in x in direction; Figure 14. Finite element analysis ofofoutput (a)Displacement Displacement nephogram x direction; (b) displacement nephogram y direction;(c) (c)displacement displacement nephogram inin z direction. (b) displacement nephogram in in y direction; nephogram z direction.


12 of 33


13 of 34

Table 1. The comparison of output compliance. Table 1. The comparison of output compliance. Methods δX Methods FX F(N) X (N) δX(µm) (µm) FEA 100100 84.87 FEA 84.87 MCM 100100 87.94 MCM 87.94

Error Error(%) (%)

FYFY(N) (N)

3.612 3.612

100 100 100 100

δY δY(µm) (µm) 133.17 133.17 131.63 131.63

Error Error (%) (%) 1.156 1.156

FFZZ (N) (N) 100 100 100 100

δZ(µm) (µm) δZ 151.86 151.86 139.52 139.52

Error(%) (%) Error 8.126 8.126

3.2. Dynamic Analysis 3.2.1. Theoretical Model The FEA given later in in this paper. According to FEA was wasperformed performedprior priortototheoretical theoreticalanalysis, analysis,but butis is given later this paper. According the FEA, since the the firstfirst andand second natural frequencies of theof3-DOF motion devicedevice occur occur in the in plane to the FEA, since second natural frequencies the 3-DOF motion the xoy, it is only to analyze naturalthe frequency device when the 2Dwhen motion has plane xoy, it necessary is only necessary tothe analyze natural of frequency of device theplatform 2D motion input signal. energy method to establish the dynamic model the flexible hinge platform hasUsing inputthe signal. Using the energy method to establish theofdynamic model of mechanism, the flexible the natural frequency be obtained according the stiffness matrixtoKthe andstiffness quality matrix matrix K M.and As hinge mechanism, thecan natural frequency can be to obtained according shown in Figure 15, the 3D motion device can be divided into six units of A ~G . Where x and y quality matrix M. As shown in Figure 15, the 3D motion device can be divided into sixDunits of D D D are the.generalized displacements the two drive motions respectively. energyrespectively. is generated A ~G Where x and y are theofgeneralized displacements of the twoThus, drivethe motions in eachthe unit of the is three-dimensional motion device. The energy of eachmotion unit was calculated to obtain Thus, energy generated in each unit of the three-dimensional device. The energy of the eachtotal unitenergy. was calculated to obtain the total energy.

Figure 15. Three-dimensional motion device structure.

As bebe divided into five parts of A kinetic energy of D can D5 . The As shown shown in inFigure Figure16, 16,unit unitA𝐴 can divided into five parts ofD1𝐴~A~𝐴 . The kinetic energy the unit A can be expressed as: D1 of the unit 𝐴 can be expressed as: . 2 1 TAD1 = 1m AD1 x D (9) 2 (9) 𝑇 = 𝑚 𝑥 . 2 where x D is the speed of the part A D1 , m AD1 is the mass of the part A D1 . is the of the is the mass of the part 𝐴 . where m A𝑥D1 can be speed expressed as:part 𝐴 , 𝑚 𝑚 can be expressed as: h i m𝑚 )w AD1 bρ 2w AD1𝑅 R− −𝜋𝑅 πR2]𝑏𝜌 bρ (10) A D1 ==( L (𝐿AD1 −−2R 2𝑅)𝑤 𝑏𝜌 + + 33[2𝑤 (10)

where ρ𝜌isisthe thematerial materialdensity, density, b is the thickness the motion mechanism. where b is the thickness ofof the 2D2D motion mechanism. can be expressed as: The kinetic energy of the unit 𝐴 The kinetic energy of the unit A D2 can be expressed as: 1 𝑥 𝑇 =1 𝐼 ( x. ) 2 2 𝐿 TAD2 = I AD2 ( D ) 2 L AD2 is the rotational inertia of the part 𝐴 . where 𝐼 be expressed 𝐼 I AD2can where is the rotational as: inertia of the part A D2 . 𝐼 where 𝑚

=𝑚

1 [ (𝐿 3

is the mass of the part 𝐴 .

− 2𝑅) + (𝐿

− 2𝑅)]𝑅 + 𝑅 ]

(11) (11)

(12)


13 of 33

I AD2 can be expressed as: i 1 I AD2 = m AD2 [ ( L AD2 − 2R)2 + ( L AD2 − 2R) ] R + R2 3 Micromachines 2018, 9, x FOR PEER REVIEW where m A D2 is the mass of the part A D2 . m AD2 can expressed as: as: 𝑚 becan be expressed m AD2 = ( L AD2 w AD2 − πR2 )bρ

𝑚

= (𝐿

𝑤

− 𝜋𝑅 )𝑏𝜌


𝑚

(12) 14 of 34

(13)

(13)

14 of 34

can be expressed as: 𝑚

= (𝐿

𝑤


(13)

Figure structure. Figure16. 16. Unit Unit 𝐴A Dstructure. As shown in Figure B point pointalong along x-axis x-axis and motion displacement of the local As shown in Figure 17, 17, thethe BD11 andy-axis y-axis motion displacement of the local coordinate system can be expressed as: coordinate system can be expressed as: 𝑥+𝑦

(16.𝑥Unit Figure = x𝐴 + structure. 2 y x0 = 𝑥 − (14) 2𝑦 x − y and y-axis motion displacement of the local = x-axis point along y𝑦0 = 22

(14)

As shown in Figure 17, the B coordinate system can be expressed as: Since the motion of the unit 𝐵 is extremely complicated, this paper only considers the 𝑥+𝑦 Since the motion ofTherefore, the unittheBkinetic complicated, this paper only considers the D is extremely translational motion. energy 𝑥 =of the unit 𝐵 can be expressed as: 2the unit B translational motion. Therefore, the kinetic energy of D1 can be expressed as: (14) 𝑥−𝑦 1 𝑥 −𝑦 𝑦𝑥 =+ 𝑦 (15) 𝑇 = 𝑚 𝜆 + 2. 2 " . 2 .2 2 . 2 # 1 x − y x + y D thisD paper only considers the D D complicated, Since the motion the unit is extremely TBof = part m B 𝐵𝐵 λ (15) is the mass of the , 1𝜆 is the2 rotation+ compensation where 𝑚 D1 2 the D1 2 be coefficient. translational motion. Therefore, kinetic energy of the unit 𝐵 can expressed as: 𝑚 can be expressed as: 1 𝑥 +𝑦 𝑥 −𝑦 = + [2(2𝑅 +compensation 𝑚𝑇 BD1 where m BD1 is the mass of the part (15) = 𝐿, λ 𝑚 𝑤is the 𝜆𝑏𝜌 rotation +𝑡)𝑅 − 𝜋𝑅 ]𝑏𝜌 coefficient. 2 1 2 2

m BD1 can be expressed as:

is the mass of the part 𝐵 , 𝜆 is the rotation compensation coefficient. where 𝑚 h i 𝑚 can be expressed as: 2

m BD1 = L BD1 w BD1 bρ + 2(2R + t) R − πR bρ 𝑚

=𝐿

𝑤

𝑏𝜌 + [2(2𝑅 + 𝑡)𝑅 − 𝜋𝑅 ]𝑏𝜌

Figure 17. Unit 𝐵 structure.

As shown in Figure 15, the kinetic energy of the unit 𝐶 𝑇 where 𝑚

can be expressed as:

1 𝑥 +𝑦 𝑥 −𝑦 = 𝑚 + 2 2 2 Figure17. 17. Unit Unit B 𝐵 D structure. Figure structure.

(16)

is the mass of the unit 𝐶 .

As shown in Figure 15, the kinetic energyof of the unit 𝐶 can bebe expressed as: as: As shown in Figure 15, the energy expressed D can As shown in Figure 18,kinetic since the motion ofthe theunit unitC𝐷 is also extremely complicated, this 𝑥 𝑥 + 𝑦 − 𝑦 paper only considers the translational1motion. Therefore, the kinetic energy of the unit 𝐷 can be " . . 2 + . . 2 # (16) 𝑇 = 𝑚 expressed as: 2 1 x D + 2y D x2D − y D TCD1 = mCD1 + 2 𝐶 . 2 2 is the mass of the unit where 𝑚 As shown in Figure 18, since the motion of the unit 𝐷 is also extremely complicated, this paper only considers the translational motion. Therefore, the kinetic energy of the unit 𝐷 can be expressed as:

(16)


14 of 33

where mCD1 is the mass of the unit CD1 . As shown in Figure 18, since the motion of the unit DD is also extremely complicated, this paper only considers the translational motion. Therefore, the kinetic energy of the unit DD1 can be expressed as: " . . . 2 . 2 # xD + yD 1 xD − yD Micromachines 2018, 9, x FOR 15 of 34 m DD1 λ2 TDPEER =REVIEW + (17) D1 2 2 2 1 λ2 is the𝑥rotation 𝑥 −𝑦 +𝑦 where m DD1 is the mass of the part D1 ,𝑚 𝑇 D= 𝜆 +compensation coefficient. 2 2 2 m DD1 can be expressed as:

(17)

is the mass of the part 𝐷 , 𝜆 is thehrotation compensationicoefficient. where 𝑚 𝑚 can be expressed m D as:= L D w D bρ + 2(2R + t) R − πR2 bρ D1

D1

𝑚

=𝐿

(18)

D1

𝑤

𝑏𝜌 + [2(2𝑅 + 𝑡)𝑅 − 𝜋𝑅 ]𝑏𝜌

(18)

Figure18. 18. Unit Unit D 𝐷 Dstructure. Figure structure.

As shown in Figure 𝐸 can be divided into five parts of 𝐸 ~𝐸 . As shown in Figure 19, E19, D can be divided into five parts of ED1 ~ED5 . The kinetic energy of the expressed as: The kinetic energy of the unitunit ED1𝐸 cancan bebeexpressed as: 1 = 𝑚 12

𝑇

TED1 =

where 𝑦 is the speed of the unit 𝐸 , 𝑚 𝑚 can be expressed as: .

𝑦

. 2

(19)

2 mass of the unit 𝐸 . is the

where y D is the speed of the unit ED1 , m ED1 is the mass of the unit ED1 . = (𝐿 − 2𝑅)𝑤 𝑏𝜌 + 3[2𝑤 𝑅 − 𝜋𝑅 ]𝑏𝜌 𝑚 m ED1 can be expressed as: The kinetic energy of the part 𝐸

(19)

m ED1 y D

(20)

can be expressed has:

i m ED1 = ( L ED1 − 2R)wE1D1 bρ + 𝑦3 2wED2 R − πR2 bρ 𝑇

= 𝐼 2

𝐿

(21)

(20)

The kinetic energy of the part ED2 can be expressed as: is the rotational inertia of the unit 𝐸 . where 𝐼 𝐼 can be expressed as: 𝐼

=𝑚

. 2 y 1 TE1D2 = IED2 [ (𝐿 − 2 2𝑅) +L(𝐿ED2 − 2𝑅)]𝑅 + 𝑅 ] 3

is the mass of the part 𝐸 . 𝑚 where Iwhere ED2 is the rotational inertia of the unit ED2 . 𝑚 can be expressed as: IED2 can be expressed as: 𝑚

IED2

= (𝐿

𝑤


1 = m ED2 [ ( L ED2 − 2R)2 + ( L ED2 − 2R)] R + R2 ] 3

where m ED2 is the mass of the part ED2 . m ED2 can be expressed as: m ED2 = ( L ED2 wED2 − πR2 )bρ

(22)

(21)

(23)

(22)

(23)

𝐼

can be expressed as: 1 [ (𝐿 3

=𝑚

𝐼

− 2𝑅) + (𝐿

− 2𝑅)]𝑅 + 𝑅 ]

(22)

is the mass of the part 𝐸 . where 𝑚 𝑚 can be expressed as: Micromachines 2018, 9, 578

= (𝐿

𝑚

𝑤


(23)

19. Unit ED structure.

Micromachines 2018, 9, x FOR PEER REVIEWFigure

15 of 33

16 of 34

As shown in Figure 20, FD can be divided into 𝐸eight parts of FD1 ~FD8 . The kinetic energy of the Figure 19. Unit structure. part FD1 can be expressed as: As shown in Figure 20, 𝐹 can be divided into eight parts of 𝐹 ~𝐹 . The kinetic energy of . . . 2 . 2 the part 𝐹 can be expressed as:

1 xD − yD m FD1 1 𝑥2 − 𝑦 2 = 𝑚 𝑇

TFD1 =

2

2

1 x + yD + IFD1 D 12 𝑥 +2L 𝑦F D1 + 𝐼 2

(24) (24)

2𝐿

where mwhere the mass of the part FD1 , 𝐹IFD1 is the rotational inertia of the unit F . FD1 is 𝑚 is the mass of the part , 𝐼 is the rotational inertia of the unit 𝐹 .D1 m FD1 can be expressed as: can be expressed as: 𝑚 mFD1 = ( L FD1 w FD1 − πR2 )bρ = (𝐿

m

𝑤


Figure20. 20. Unit Unit 𝐹 Figure FD structure. structure.

As shown in Figure 21, can 𝐺 can dividedinto into ten ten parts ~𝐺~G. The. kinetic energy of the As shown in Figure 21, G be be divided partsofof𝐺GD1 D D10 The kinetic energy of the part 𝐺 can be expressed as: part GD1 can be expressed as: 𝑥 . − 𝑦 . 2 1 1𝑚 xD − yD (25) 𝑇 = TGD1 = m (25) 2 2 2 GD1 2 where 𝑚

is the mass of the part 𝐺 .

where mGD1 𝑚is the of the part canmass be expressed as: GD1 . mGD1 can be expressed as: =𝐿

m

𝑤

𝑏𝜌 + 𝐿

𝑤

𝑏𝜌 + 3(2𝑤

𝑅 − 𝜋𝑅 )𝑏𝜌

(26)

2 The kineticm energy=ofLthe part 𝐺 can be expressed as: GD1 GD1 w GD1 bρ + L GD2 w GD1 bρ + 3 2w GD1 R − πR bρ 1 𝑥 −𝑦 = 𝐼 𝑇 The kinetic energy of the part GD2 can be 2expressed 2𝐿 as:

is the rotational inertia of the unit 𝐺 . . where 𝐼 xD 1 𝐼 can be expressed as: T = I GD2

𝐼

=𝑚

1 [ (𝐿 3

2

GD2

− 2𝑅) + (𝐿

where IGD2 is the rotational inertia of the unit GD2 . 𝑚 can be expressed as: IGD2 can be expressed as: 𝑚

IGD2

= (𝐿

𝑤

.

− yD 2LGD3

(26) (27)

2

− 2𝑅)]𝑅 + 𝑅 ]


2 1 = mGD2 [ ( LGD3 − 2R + ( LGD3 − 2R)] R + R2 ] 3

(27) (28)

(29)

(28)


16 of 33

mGD2 can be expressed as:

mGD2 = ( LGD3 wGD2 − πR2 )bρ

(29)


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Figure21. 21. Unit Unit 𝐺GDstructure. Figure structure.

According to Equations (9)–(29), thetotal totalkinetic kinetic energy thethe device can be expressed as According to Equations (9)–(29), the energyofof device can be expressed as 𝑇=𝑇

+ 4𝑇

+ 2𝑇

+𝑇

+ 2𝑇

+𝑇

+ 4𝑇

+ 8𝑇

+ 2𝑇

+ 8𝑇

(30)

T = TAD1 + 4TAD2 + 2TBD1 + TCD1 + 2TDD1 + TED1 + 4TED2 + 8TFD1 + 2TGD1 + 8TGD2 Equation (30) can be described as:

Equation (30) can be described as: 𝑇 = 0.5𝑀 𝑥

+ 0.5𝑀 𝑦

According to Equation (31), the quality matrix is . 2

(31) . 2

T = 0.5M1 x D𝑀 + 00.5M2 y D 𝑀=

0

(30)

𝑀

(32)

(31)

According Equation (31), the quality where M1 to is the equivalent quality matrix ofmatrix x, M2 isis the equivalent quality matrix of y. " # The stiffness matrices can be obtained according to the results of static investigation, and can be M1 0 expressed as: M=

𝐾=

0𝐾 0

0M2 𝐾

(32)

(33)

where M equivalent quality of x, M2 isinput the equivalent matrixof ofthe y. No. 2 1 is the the stiffness of the matrix No. 1 piezoelectric terminal, K4quality is the stiffness where K3 is The stiffness matrices can be obtained according to the results of static investigation, and can be piezoelectric input terminal. Then the first natural frequency and the second natural frequency can be calculated. expressed as: " # K3 0 1 𝐾 K= (33) (34) 𝑓= 0 2π 𝑀K4 where K3 isUsing the stiffness ofmethod, the No.the1 piezoelectric input terminal, K4 and is the of the No. 2 the energy first natural frequency is 439.59 Hz thestiffness second natural piezoelectric input terminal. frequency is 690.29 Hz. Then the first natural frequency and the second natural frequency can be calculated. 3.2.2. Finite Element Analysis (FEA)

r

1 Kby the energy method, the FEA method was In order to verify the natural frequency calculated f = applied to the four boundaries in the model. Solid (34) used for modal analysis. The full constraint was2π M model 10 node 187 elements were used in finite element model. The material is aluminum alloy Using energylinearity method,and theisotropy. first natural frequency is 439.59 and the second frequency withthe elasticity, The elastic modulus is 70 Hz MPa. Poisson’s ratio natural is 0.3. The is 690.29material Hz. density is 2.78 g/cm3. Figure 22 gives the vibrational mode shapes from mode 1 to mode 4. The natural frequencies of the first to fourth are 474.82 Hz, 656.98 Hz, 755.78 Hz and 800.61 Hz, respectively. ForAnalysis the first and second natural frequencies, the relative deviations of FEA and energy 3.2.2. Finite Element (FEA) method are 7.4196% and 4.8255% respectively. This shows that the results of the FEA method are In more orderortoless verify the natural by the energy method, the FEA method was consistent with the frequency results of thecalculated energy method. used for modal analysis. The full constraint was applied to the four boundaries in the model. Solid model 10 node 187 elements were used in finite element model. The material is aluminum alloy with


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elasticity, linearity and isotropy. The elastic modulus is 70 MPa. Poisson’s ratio is 0.3. The material density is 2.78 g/cm3 . Figure 22 gives the vibrational mode shapes from mode 1 to mode 4. The natural frequencies of the first to fourth are 474.82 Hz, 656.98 Hz, 755.78 Hz and 800.61 Hz, respectively. For the first and second natural frequencies, the relative deviations of FEA and energy method are 7.4196% and 4.8255% respectively. This shows that the results of the FEA method are more or less consistent with the results method. Micromachines 2018, of 9, xthe FORenergy PEER REVIEW 18 of 34

(a)

(b)

(c)

(d)

Figure 22. Modal Modal analysis analysis of of 3-DOF 3-DOF device. (a) (a) Mode Mode 1; 1; (b) (b) Mode Mode 2; 2; (c) (c) Mode 3; (d) Mode 4. Figure

4. Device Performance Test 4. Device Performance Test 4.1. Natural Frequency 4.1. Natural Frequency In this paper, a non-resonant 3-DOF motion device is designed. In order to maintain the stability In this paper, a non-resonant 3-DOF motion device is designed. In order to maintain the of the device and avoid damage, its operating frequency should effectively avoid the device’s natural stability of the device and avoid damage, its operating frequency should effectively avoid the frequency. In this paper, the natural frequency of the device was detected by the sweeping frequency device’s natural frequency. In this paper, the natural frequency of the device was detected by the method. The test system is shown in Figure 23. FEA shows that the direction of vibration of the first sweeping frequency method. The test system is shown in Figure 23. FEA shows that the direction of natural frequency is the y direction. A sine sweep signal (voltage range: 0.5–2 V, frequency range: vibration of the first natural frequency is the y direction. A sine sweep signal (voltage range: 0.5–2 V, 0–1000 Hz) is generated by a signal generator (Brand: Siglent Model: Sdg805, Shanghai, China.) and frequency range: 0–1000 Hz) is generated by a signal generator (Brand: Siglent Model: Sdg805, applied to a No. 1 piezoelectric actuator (Brand: Physik Instrumente (PI) GmbH Model: P-820.20, Shanghai, China.) and applied to a No. 1 piezoelectric actuator (Brand: Physik Instrumente (PI) Karlsruhe, Germany.) of a 3D motion device via a power amplifier (Brand: Physik Instrumente (PI) GmbH Model: P-820.20, Karlsruhe, Germany.) of a 3D motion device via a power amplifier (Brand: GmbH Model: E-663.00, Karlsruhe, Germany.). The capacitive sensor (Brand: Physik Instrumente Physik Instrumente (PI) GmbH Model: E-663.00, Karlsruhe, Germany.). The capacitive sensor (PI) GmbH Model: D-100.00, Karlsruhe, Germany.) is used to measure the output response in each (Brand: Physik Instrumente (PI) GmbH Model: D-100.00, Karlsruhe, Germany.) is used to measure direction, and the capacitance detection module (Brand: Physik Instrumente (PI) GmbH Model: PI the output response in each direction, and the capacitance detection module (Brand: Physik E509, Karlsruhe, Germany.) and the communication module (Brand: Physik Instrumente (PI) GmbH Instrumente (PI) GmbH Model: PI E509, Karlsruhe, Germany.) and the communication module Model: PI E516, Karlsruhe, Germany.) are used to achieve the output signal acquisition in each (Brand: Physik Instrumente (PI) GmbH Model: PI E516, Karlsruhe, Germany.) are used to achieve the output signal acquisition in each direction. The y-direction output response curve obtained by Fourier transform is shown in Figure 24. For the resonance frequency curve, the highest point of the curve is the first natural frequency of the motion device. Observing the response curve, the first natural frequency of the experiment is 414 Hz. The natural frequencies of the energy method and the


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direction. The y-direction output response curve obtained by Fourier transform is shown in Figure 24. For the resonance frequency curve, the highest point of the curve is the first natural frequency of the motion device. Observing the response curve, the first natural frequency of the experiment is 414 Hz. The natural frequencies of the energy method and the FEA are 439.59 Hz and 474.82 Hz, respectively. The experimental frequency is Micromachines 2018, 9, natural x FOR PEER REVIEW is slightly lower than the theoretical analysis. The main reason 19 of 34 that the assembly of the device is non-rigid and the capacitive sensor is mounted at the output, which improves the motion quality of the device. Micromachines 2018, 9, x FOR PEER REVIEW

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Figure Figure 23. Natural frequency detection (1) signal signalgenerator; generator; (2) power amplifier; (3) 23. Natural frequency detectionsystem. system. (1) (2) power amplifier; (3) Figure 23. Natural frequency detection system. (1) signal generator; (2) power amplifier; (3) three-dimensional motion device; capacitance sensor; capacitance detection channel. three-dimensional (3D) (3D) motion device; (4)(4)capacitance sensor;(5)(5) capacitance detection channel. three-dimensional (3D) motion device; (4) capacitance sensor; (5) capacitance detection channel. 180

180

160

X: 414 Y: 163

160

X: 414 Y: 163

140

140

120

120

100

100

80

80

60 40

60

20

40 0

20 0

0

100

200

300

400

500

600

700

800

900

1000

Frequency (Hz) 0

Figure 24. 100 200 Amplitude-frequency 300 400

curve 3-DOF motion device. 500 of a 600 700 800 Frequency (Hz)

900

1000

4.2. Dynamic Performance

Figure Figure 24. 24. Amplitude-frequency Amplitude-frequency curve of a 3-DOF motion device.

In general, if the mechanism can adapt to the stair step signal, the mechanism can meet the dynamic performance requirements of other signals. In order to simplify the experimental test, a 4.2. Dynamic Performance 4.2. Dynamic Performance square wave signal is used instead of a stair step signal. Sinusoidal signal is the most basic input In signal general, if the themotion mechanism canisadapt adapt toused theinstair stair step the the mechanism can meet the of 3-DOF device and widelyto engineering. However, mechanismcan has meet a In general, if mechanism can the step signal, signal, the mechanism the return trip effect. In order to better detect the output response of the device, triangle wave and sine dynamic performance requirements of other signals. In order to simplify the experimental test, dynamic performance requirements of other signals. In order to simplify the experimental test, aa wave are selected as testinstead signals inofthis paper. The flexible mechanism performance test is shown in input square wave wave signal is used used aa stair Sinusoidal signal is is the the most basic square signal is instead of stair step step signal. signal. Sinusoidal signal most basic input Figures 25 and 26. In Figure 25, ①, ②, ③ are power amplifiers connected to the No. 1, No. 2 and signal of 3-DOF motion device and is widely used in engineering. However, the mechanism has aa signal ofNo. 3-DOF motion device and is widely④, used However, mechanism has 3 piezoelectric actuators, respectively; ⑤,in ⑥engineering. are No. 1, No. 2 and No.the 3 piezoelectric return trip effect. In order to better detect the output response of the device, triangle wave and sine return trip effect.respectively. In order toThe better detect the output of the device, istriangle wave and sine actuators, capacitive sensor installedresponse in the x, y and z directions shown in Figure wave are are selected as test testsignals signals thispaper. paper.moving Theflexible flexible mechanism performance test is shown wave as ininthis The performance test shown in 26.selected In the experiment, the capacitance sensor plate ismechanism installed at the output terminal ofisthe in Figures 25 and 26. In Figure 25, , , are power amplifiers connected to the No. 1, No. 1 2 3 three-dimensional motion device, and the fixed plate is installed on the fixed frame. When the Figures 25 and 26. In Figure 25, ①, ②, ③ are power amplifiers connected to the No. 1, No. 2 and2 signal generatoractuators, output signal isrespectively; unchanged,④, the 4 x, ,y, 5and directions tested to obtain and No. 3 piezoelectric actuators, , 6zare are No.1,are 1,No. No. and No.a 3distance 3 piezoelectric No. 3 piezoelectric respectively; ⑤, ⑥ No. 2 2and No. piezoelectric curve in each direction. (The distance data measured by the experimental system begin withinthe actuators, respectively. y and z directions is shown Figure 26. actuators, respectively.The Thecapacitive capacitivesensor sensorinstalled installedininthe thex,x, y and z directions is shown in Figure same phase of the input signal.) The performance curves of the output terminal of the device are In the experiment, the capacitance sensor moving plate is installed at the output terminal of the 26. In the experiment, the capacitance obtained by comprehensive analysis.sensor moving plate is installed at the output terminal of the

three-dimensional motion fixed plate is installed on the frame.frame. WhenWhen the signal three-dimensional motion device, device,and andthe the fixed plate is installed on fixed the fixed the generator output signal is unchanged, the x, y, and z directions are tested to obtain a distance curve in signal generator output signal is unchanged, the x, y, and z directions are tested to obtain a distance each direction. (The distance data measured by the experimental system begin with the same phase curve in each direction. (The distance data measured by the experimental system begin with the same phase of the input signal.) The performance curves of the output terminal of the device are obtained by comprehensive analysis.


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of the input signal.) The performance curves of the output terminal of the device are obtained by comprehensive analysis. Micromachines 2018, 9, x FOR PEER REVIEW 20 of 34 Micromachines 2018, 9, x FOR PEER REVIEW

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Figure 25. Experiment of device response performance test. Figure 25. 25. Experiment response performance Figure Experimentof ofdevice device response performance test. test.

(a) (a)

(b) (b)

(c) (c)

Figure 26. Capacitive sensor installation in all directions. (a) x-direction test; (b) y-direction test; (c)

Figure 26. Capacitive sensor installationininall all directions. directions. (a)(a) x-direction test; test; (b) y-direction test; (c) test; (c) Figure 26. Capacitive sensor installation x-direction (b) y-direction z-direction test. z-direction z-direction test. test.

(1) x-direction (1) x-direction The square wave signals with the same frequency and phase were applied to the No. 1 and No. The square wave signals with the same frequency and phase were applied to the No. 1 and No. 2 piezoelectric actuators. Theoretically, square wave signals of the same period should be produced actuators. square wave signals of the same should be produced The2 piezoelectric square wave signals Theoretically, with the same frequency and phase wereperiod applied to the No. 1 and No. 2 in the x direction. The response curve of the output terminal in the x direction is shown in Figure in the xactuators. direction. The response curve of the output terminal in the x direction isshould shown in piezoelectric Theoretically, square wave signals of the same period beFigure produced in 27a. Observing the test curve for a half cycle, the response displacement is exponentially decayed. 27a. Observing the test curve for a half cycle, the response displacement is exponentially decayed. the x direction. The response of thetooutput inreturn the x direction shown in Figure 27a. After 0.025 s, the output curve signal tends be flat terminal because the response ofisthe piezoelectric After 0.025 s, the output signal tends to be flat because the return response of the piezoelectric Observing theand testmechanism curve forisalagging. half cycle, the response is exponentially decayed. system The triangular wave displacement and sine wave signals of the same phase and After system and mechanism is lagging. The triangular wave and sine wave signals of the same phase and the same frequency were applied toflat thebecause No. 1 piezoelectric actuator and No. 2 piezoelectric actuator, 0.025 s, the thesame output signal tends to be the return response of the piezoelectric system and frequency were applied to the No. 1 piezoelectric actuator and No. 2 piezoelectric actuator, respectively. The output curves of the system are shown in Figure 27b,c. The sinusoidal response of mechanism is lagging. The triangular wave and wave signals ofThe thesinusoidal same phase and of the same respectively. The output curves of the system aresine shown in Figure 27b,c. response the system is better, while the square wave and triangle wave response is poor. the were system is better,towhile the square wave and actuator triangle wave is poor. frequency applied the No. 1 piezoelectric andresponse No. 2 piezoelectric actuator, respectively. (2) y-direction The output curves of the system are shown in Figure 27b,c. The sinusoidal response of the system is (2) y-direction better, whileThe thesquare square wave and triangle wave response is poor. wave, triangle wave, and sinusoidal signals with the same frequency and a phase The square wave, triangle wave, and sinusoidal signals with the same frequency and a phase difference of 180° were applied to the No. 1 piezoelectric actuator and No. 2 piezoelectric actuator. difference of 180° were applied to the No. 1 piezoelectric actuator and No. 2 piezoelectric actuator. (2) y-direction The response curve of the output terminal in the y direction is shown in Figure 28. The output The response curve of the output terminal in the y direction is shown in Figure 28. The output response curve in the y direction was observed and it was found that: (1) When the rectangular response curve in the y direction was observed and it was found that: (1) When the rectangular The square wave, and sinusoidal with the same frequency and square wave signaltriangle is used aswave, the input signal, at 0.025 signals s, the output terminal responds to 90% of thea phase square wave signal is used as the input signal, at 0.025 s, the output terminal responds to 90% of the ◦ theoretical output response.;(2) when the triangular wave signal is used as the input signal, the difference of 180 output were response.;(2) applied to the No. piezoelectric and No. piezoelectric theoretical when the1triangular wave actuator signal is used as the2 input signal, theactuator. output signal amplitude fluctuates around the theoretical output signal; (3) when the sine wave The response the output terminal in the y direction shown in Figure 28. The output response outputcurve signal of amplitude fluctuates around the theoretical isoutput signal; (3) when the sine wave signal is used as the input signal, the output signal is basically the same as the theoretical output signal is used as the input signal, the output signal is basically the same as the theoretical output curve in the y direction was observed and it was found that: (1) When the rectangular square wave signal. signal issignal. used as the input signal, at 0.025 s, the output terminal responds to 90% of the theoretical (3) z-direction output (3) response; (2) when the triangular wave signal is used as the input signal, the output signal z-direction

(1) x-direction

amplitude fluctuates around the theoretical output signal; (3) when the sine wave signal is used as the input signal, the output signal is basically the same as the theoretical output signal. (3) z-direction

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The square square rectangular, rectangular, triangular, triangular, and and sinusoidal sinusoidal signals signals are are applied applied to to the the No. No. 3 piezoelectric piezoelectric The The square rectangular, triangular, and sinusoidal signals are applied to the No. 33 piezoelectric actuator. The The response response curve of of the output output terminal in in the the zz direction direction is is shown shown in in Figure Figure 29. 29. The The actuator. actuator. The response curve curve of the the output terminal terminal in the z direction is shown in Figure 29. The output response curve in the z direction was observed and found that: (1) when the rectangular output response responsecurve curveininthethe z direction observed found that: (1) when the rectangular output z direction waswas observed and and found that: (1) when the rectangular square square wave signal is an input signal, at 0.027 s, the output terminal responds to 90% 90% of of the the squaresignal waveis an signal is signal, an input signal, atoutput 0.027 terminal s, the output terminal to wave input at 0.027 s, the responds to 90%responds of the theoretical output theoretical output output response; (2) (2) when the the triangular triangular wave wave signal signal is is an an input input signal, signal, the the output output signal signal theoretical response; (2) when response; the triangularwhen wave signal is an input signal, the output signal amplitude fluctuates amplitude fluctuates around the theoretical output signal; (3) when the sine wave signal is an input amplitude the theoretical signal; when theinput sine signal, wave signal is an signal input around the fluctuates theoreticalaround output signal; (3) whenoutput the sine wave (3) signal is an the output signal, the the output output signal signal is is more more or or less less the the same same as as the the theoretical theoretical output output signal. signal. signal, is more or less the same as the theoretical output signal.

(a) (a)

(b) (b)

(c) (c)

Figure 27. 27. Response Response curve curve in in xx direction. direction. (a) (a) Square wave wave signal; signal; (b) (b) triangle triangle wave wave signal; signal; (c) (c) sine sine Figure (a) Square wave signal. signal. wave

(a) (a) Figure 28. Cont.

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Amplitude (um) Amplitude (um)

of 34 33 2221of 22 of 34

(b) (b)

(c) (c)

Figure 28. Response curve in y direction. (a) Square wave signal; (b) triangle wave signal; (c) sine 28. Response curve in y direction. direction. (a) Square wave signal; (b) triangle wave signal; signal; (c) sine sine Figure 28. wave signal. wave signal. signal. wave

(a) (a)

71.8 71.8 71.6 71.6 71.4 71.4 71.2 71.2 71 71 70.8 70.8 70.6 70.6 70.4 70.4 70.2 70.20 0

0.005 0.005

0.01 0.01

0.015 0.02 0.025 0.015 Time 0.02(s) 0.025

Time (s)

(b) (b)

0.03 0.03

0.035 0.035

0.04 0.04

(c) (c)

Figure 29. Response Response curve curve in z direction. (a) (a) Square Square wave signal; (b) triangle wave signal; (c) sine Figure 29. Response curve in z direction. (a) Square wave signal; (b) triangle wave signal; (c) sine wave signal. wave signal.

4.3. Linear Motion Performance in Each Direction 4.3. Linear Motion Performance in Each Direction 4.3. Linear Motion Performance in Each Direction Although the 3-DOF device can theoretically actualize independent movements in the x, y Although the 3-DOF device can theoretically actualize independent movements in the x, y and z thethere 3-DOF device theoretically actualize movements thetheoretical x, y and z and zAlthough directions, may be acan discrepancy between the independent actual displacement and in the directions, there may be a discrepancy between the actual displacement and the theoretical directions, there maytobe a discrepancy between thethree actual displacement and the theoretical displacement. In order verify the linear motion ability, input signals that theoretically produce displacement. In order to verify the linear motion ability, three input signals that theoretically displacement. In order to verify theare linear motion only x, y and z directions movements applied to theability, device.three input signals that theoretically produce only x, y and z directions movements are applied to the device. produce only x, y and z directions movements are applied to the device.

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The sinusoidal signals with the same frequency and phase were applied to the No. 1 and No. 2 The sinusoidal signals with the same frequency phasex-direction were applied to the No. 1The andoutput No. 2 piezoelectric actuators. Theoretically, the device only and produces displacement. piezoelectric actuators. Theoretically, the device only produces x-direction displacement. The output displacement of the three directions, x, y and z, were detected, respectively. The relationship displacement of the threeand directions, x, ythree and z,directions were detected, respectively. The relationship between between displacement time in is shown in Figure 30a. The following displacement anddrawn: time inThe three directions shown in Figure 30a. larger The following were conclusions were amplitude of is the x direction is much than the yconclusions and z directions, drawn: The amplitude of the x direction is much larger than the y and z directions, the y direction the y direction response is sinusoidal, and the amplitude is significantly larger than the z direction response sinusoidal, and the amplitude is significantly larger than the z direction response amplitude. response isamplitude. ◦ The sinusoidal signals with the same frequency and a phase difference of 180 180° were applied to the No. and No.No. 2 piezoelectric actuator, respectively. Theoretically, the device No.11piezoelectric piezoelectricactuator actuator and 2 piezoelectric actuator, respectively. Theoretically, the only produces y direction displacement. The output displacement of the three x, y and device only produces y direction displacement. The output displacement of thedirections, three directions, x, z, y was detected respectively, as shown in Figure 30b. The sinusoidal signals were applied to the No. 3 and z, was detected respectively, as shown in Figure 30b. The sinusoidal signals were applied to the piezoelectric actuator and theand output of the three directions, x y and z,x were No. 3 piezoelectric actuator the displacements output displacements of the three directions, y anddetected, z, were as shown as in shown Figure 30c. The experimental curve shows that the device hasdevice large has deviations from the detected, in Figure 30c. The experimental curve shows that the large deviations actual operation and theoretical operationoperation in the x direction and the yand direction. Figures 31–33 show from the actual operation and theoretical in the x direction the y direction. Figures 31– the projected trajectories of linear motion thealong x, y and 33 show the projected trajectories of linearalong motion thezx,axes. y and z axes.

(b)

Amplitude (um)

(a)

(c) Figure 30. Time-amplitude inin x Time-amplitudediagram diagramof oflinear linearmotion motionperformance performancetest testinineach eachdirection. direction.(a)(a)Driven Driven direction; (b)(b) driven inin y direction; (c)(c) driven in in z direction. x direction; driven y direction; driven z direction.


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24 of of 34 24 24 of 34 34

z(um) z(um) z(um)

Micromachines 2018, 2018, 9, xx FOR FOR PEER REVIEW REVIEW Micromachines Micromachines 2018, 9, 9, x FOR PEER PEER REVIEW

(a) (a)

(b) (b)

Figure 31. Linear motion performance test in x direction. (a) Projection on the xoy plane; (b)

Figure 31. motion performance test in (a) on the xoy (b) Figure 31. Linear Linear motion performance test in xx direction. direction. (a) Projection Projection onxoy theplane; xoy plane; plane; (b) Figure 31. Linear motion performance test in x direction. (a) Projection on the (b) projection projection on on the the xoz plane. plane. projection projection on the xoz xoz plane. on the xoz plane. 77.8 77.8 77.8 77.6 77.6 77.6 77.4 77.4 77.4 77.2 77.2 77.2 77 77 77 76.8 76.8 76.8 76.6 76.6 76.6 76.4 76.4 76.4 76.2 76.2 76.255 55 55

(a) (a)

55.2 55.2 55.2

55.4 55.4 55.4

55.6 55.6 55.6

55.8 55.8

55.8 y(um) y(um) y(um)

56 56 56

56.2 56.2 56.2

56.4 56.4 56.4

56.6 56.6 56.6

(b) (b)

z(um) z(um) z(um)

Figure 32. Linear Linear motion performance testy direction. in yy direction. direction. (a) Projection Projection onxoy theplane; xoy plane; plane; (b) Figure 32. motion performance test (a) on (b) Figure 32. Linear motion performance test in (a) Projection on the (b) projection Figure 32. Linear motion performance test in in y direction. (a) Projection on the the xoy xoy plane; (b) projection on the yoz plane. projection on projection on the the yoz yoz plane. plane. on the yoz plane.

(a) (a)

(b) (b)

Figure 33. Linear Linear motion performance testz direction. in zz direction. direction. (a) Projection Projection onxoz theplane; xoz plane; plane; (b) Figure 33. Linear motion performance test in (a) Projection on the (b) projection Figure 33. motion performance test (a) on (b) Figure 33. Linear motion performance test in in z direction. (a) Projection on the the xoz xoz plane; (b) projection on the yoz plane. on the yoz plane. projection on projection on the the yoz yoz plane. plane.

Table provides some data from the linear linear motion performance test. The maximum linear TableTable 2 provides somesome data data fromfrom the linear motion performance test.test. TheThe maximum linear motion 22 provides the motion performance maximum linear motion deviations in the x, y and z directions were approximately 6.67%, 5.71%, and 3.03%, motionindeviations in zthe x, y andwere z directions were approximately 5.71%,respectively. and 3.03%, The deviations the x, y and directions approximately 6.67%, 5.71%,6.67%, and 3.03%, respectively. The The reasons for for this error error are are mainly mainly because because of of the the following following aspects: aspects: (1) (1) since since the the respectively. reasons for this error reasons are mainly this because of the following aspects: (1) since the mechanical midpoint mechanical midpoint of the power component of the device is not exactly the same as the mechanical midpoint ofofthe power component of the device is not the same asmidpoint the mechanical of themechanical power component the device is not exactly the same asexactly the mechanical of the x, y output decoupling guiding element, the output block is slightly deflected; (2) the installation error has a great influence on the experimental results; (3) the inconsistent piezoelectric preload may cause the linear motion performance to deteriorate in x and y directions.


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midpoint of the x, y output decoupling guiding element, the output block is slightly deflected; (2) the installation2018, error has a great influence on the experimental results; (3) the inconsistent piezoelectric Micromachines 9, 578 24 of 33 preload may cause the linear motion performance to deteriorate in x and y directions. Table Table2.2.Experimental Experimentaldata dataanalysis analysisofoflinear linearmotion motionperformance. performance. Maximum Displacement Maximum Displacement

Project

Project

x direction input signal x direction input signal y direction input signal y direction input signal z direction input signal z direction input signal

xx Direction Direction(µm) (µm) 1.2 1.2 0.08 0.08 0.01 0.01

y Direction (µm) (µm) y Direction 0.08 0.08 1.4 1.4 0.01

0.01

z Direction (µm) z Direction (µm) 0.03 0.05 1.65

0.03 0.05 1.65

4.4. 4.4.3D 3DVibration VibrationTrajectory Trajectory 3D parallel movement in the x, yx,and z directions. It is necessary that 3Dvibration vibrationdevice devicecan canrealize realize parallel movement in the y and z directions. It is necessary the directions of x y and to realize 3D movement of the output against thatthree the three directions of x zy cooperate and z cooperate to the realize the 3D movement of theterminal output terminal actual application. In order to study the actual movement of the output terminal, different signals against actual application. In order to study the actual movement of the output terminal, different are applied the No. No. 2 and No.23and piezoelectric actuators, actuators, respectively. The motionThe trajectory signals are to applied to1,the No. 1, No. No. 3 piezoelectric respectively. motion of the device in all directions is detected using a capacitive sensor. The actual running trajectory is trajectory of the device in all directions is detected using a capacitive sensor. The actual running synthesized in a 3D space by detecting the displacement of x, y, z to the output during one cycle. The trajectory is synthesized in a 3D space by detecting the displacement of x, y, z to the output during three experimental schemes are shown in Figures 34–36inrespectively, and the runningand trajectories are one cycle. The three experimental schemes are shown Figures 34–36 respectively, the running shown in Figures 34d, 35d and 36d. In addition, the actual operational trajectory of the device is shown trajectories are shown in Figures 34d, 35d and 36d. In addition, the actual operational trajectory of in Figures 38, in when more37complex are used as inputs. the device37 is and shown Figures and 38, signals when more complex signals are used as inputs.

(a)

(b)

(c) Figure 34. Cont.

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(d) (d) 34. Experimental scenario (a)x-direction x-direction input input signal; input signal; (c) (c) 34. Experimental Experimental scenario 1. 1. (a) x-direction input signal; signal;(b) (b)y-direction y-direction input signal; (c) FigureFigure scenario 1. (a) (b) y-direction input signal; z-direction input signal; (d) output trajectory. trajectory. z-direction input signal; (d) output trajectory.

(a)

(a)

(b)

(b)

(c) Figure 35. Cont.

(c)


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(d)

(d) Figure 35. Experimental scenario 2. (a) x-direction inputsignal; signal; (b) y-direction input signal; (c) Figure 35. Experimental scenario2.2. (a) (a) x-direction x-direction input y-direction input signal; (c) (c) Figure 35. Experimental scenario input signal;(b)(b) y-direction input signal; z-direction input signal; (d) output trajectory. z-direction input signal; output trajectory. z-direction input signal; (d) (d) output trajectory.

(a)

(a)

(b)

(b) Figure 36. Cont.

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(c) (c)

(d) (d) Figure 36. Figure 36. 36. Experimental Experimental scenario scenario 3. 3. (a) (a) x-direction x-direction input input signal; signal; (b) (b) y-direction y-direction input input signal; signal; (c) (c) Figure Experimental scenario 3. signal; y-direction input signal; (c) z-direction input signal; (d) output trajectory. z-directioninput inputsignal; signal;(d) (d)output outputtrajectory. trajectory. z-direction

Figure Figure37. 37.Output Outputtrace traceNo. No.1.1.

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Figure Figure 38. 38. Output trace trace No. No. 2. 2.

5. Application Test 5. Application Test The scratching experiment system is shown in Figure 39 and the experimental tool is shown in The scratching experiment system is shown in Figure 39 and the experimental tool is shown in Figure 40. Since the output response of the device under no-load conditions is significantly different Figure 40. Since the output response of the device under no-load conditions is significantly different from the output response of the load, this still needs to be tested. There are two reasons that account from the output response of the load, this still needs to be tested. There are two reasons that account for this. On the one hand, the piezoelectric actuators will have a smaller output displacement when the for this. On the one hand, the piezoelectric actuators will have a smaller output displacement when output terminal is subjected to external loads compared to no-load. On the other hand, it is possible the output terminal is subjected to external loads compared to no-load. On the other hand, it is to cause internal deformation without generating an output displacement in a complex mechanical possible to cause internal deformation without generating an output displacement in a complex environment, if the design of the device is unreasonable. Therefore, two experiments are described in mechanical environment, if the design of the device is unreasonable. Therefore, two experiments the following section to observe the actual effect of the device under complex working conditions. The are described in the following section to observe the actual effect of the device under complex flexible mechanism device was installed on the spindle. The y-feed of the machine tool and the 3D working conditions. The flexible mechanism device was installed on the spindle. The y-feed of the vibrations jointly complete the scratching experiment. Table 3 gives the signal parameters applied to machine tool and the 3D vibrations jointly complete the scratching experiment. Table 3 gives the the motion device. The signal in x and y directions is generated by the piezoelectric driving control signal parameters applied to the motion device. The signal in x and y directions is generated by the system. The signal in z direction is generated by the signal generator. When the three directions’ piezoelectric driving control system. The signal in z direction is generated by the signal generator. signals are amplified by the power amplifier and applied to the No. 1, No. 2 and No. 3 piezoelectric When the three directions’ signals are amplified by the power amplifier and applied to the No. 1, No. actuators, respectively, flexible mechanism devices can produce 3D movement in space. Two groups of 2 and No. 3 piezoelectric actuators, respectively, flexible mechanism devices can produce 3D scratch experiments were carried out. A workpiece (material: aluminum alloy, AISI6061) is fixed to the movement in space. Two groups of scratch experiments were carried out. A workpiece (material: machine tool with a clamp and the tool is fixed at the 3D motion device output terminal. The y-feed aluminum alloy, AISI6061) is fixed to the machine tool with a clamp and the tool is fixed at the 3D rate of the machine table is 2 mm/min. The force measuring system (Brand: kistler Model: 9129AA, motion device output terminal. The y-feed rate of the machine table is 2 mm/min. The force Winterthur, Switzerland.) is installed on the bottom of the workpiece holder in order to detect the three measuring system (Brand: kistler Model: 9129AA, Winterthur, Switzerland.) is installed on the directions’ polishing force of the workpiece. The surface topography of the workpiece was observed bottom of the workpiece holder in order to detect the three directions’ polishing force of the using a microscope (Brand: Olympus Model: BX-15, Tokyo, Japan.) after the scratch experiment. workpiece. The surface topography of the workpiece was observed using a microscope (Brand: Olympus Model: BX-15, Tokyo, Japan.) after the scratch experiment. Table 3. Experimental signal parameters.

Project

Project Experiment No. 1

Tool Green rubber tool

Experiment Experiment No. 1 No. 2

Table 3. Experimental signal parameters. Sinusoidal Low High Signal Cycle (s) Voltage (V) Voltage Sinusoidal Low High(V) Direction

Tool

Green

rubber tool Diamond tool

Experiment No. 2

Diamond tool

x Signal yDirection z x x y z

y z x y z

0Voltag 0 e (V) 0 0 0 0 0

0 0 0 0 0

Voltag 8 e (V) 8 88 8 88 8 8 8 8

Cycle0.006 (s) 0.006 0.0060.3

0.006 0.006 0.30.006 1.2 0.006 0.006 1.2

Frequency Difference between Frequency x and y (s)

Difference between x and y0.001 (s) 0.001 0.001

0.001

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Figure 39. 3D vibration scratching experiment system (1) flexible mechanism device; (2) 39.3D3D vibration scratching experiment system (1)mechanism flexible mechanism device; (2) Figure 39. vibration scratching experiment system (1) flexible device; (2) experimental experimental tool; (3) workpiece holder; (4) workpiece. experimental tool; (3) workpiece holder; (4) workpiece. tool; (3) workpiece holder; (4) workpiece.

Figure Figure 40. 40. Experimental Experimental tool. tool. Figure 40. Experimental tool.

The The green green rubber rubber tool tool was was used used in in Experiment Experiment No.1. No.1. When When the the sine sine voltage voltage signal signal is is applied applied to to The green rubber toolmechanism was used in Experiment No.1. When theworkpiece sine voltage signal in is Figure applied41a to three directions of flexible device, the force curve of the is shown three directions of flexible mechanism device, the force curve of the workpiece is shown in Figure 41a threethe directions of flexible mechanism device, the force curve of theFigure workpiece The is shown in Figure 41a and and the workpiece workpiece surface surface obtained obtained by by the the experiment experiment is is shown shown in in Figure 42a. 42a. The diamond diamond tool tool was was and the workpiece surface obtained by the experiment is shown in Figure 42a. The diamond tool was used 2. When a sinusoidal voltage signalsignal of 2, 4,of 6, and V is applied to z-direction used ininExperiment ExperimentNo. No. 2. When a sinusoidal voltage 2, 4,8 6, and 8 V is applied to used in Experiment No. 2. When a sinusoidal voltage signal of 2, 4, 6, is and 8 V in is Figure applied41b. to of flexible mechanism device respectively, the force curve of the workpiece shown z-direction of flexible mechanism device respectively, the force curve of the workpiece is shown in z-direction ofvoltage flexiblesignal mechanism device respectively, the forcesurface curve of the workpiece is shown in When an 8 V was used as an input, the workpiece obtained by the experiment is Figure 41b. When an 8 V voltage signal was used as an input, the workpiece surface obtained by the Figure 41b. When an 8 VBased voltage signal was used as an input, the workpiece surface has obtained by the as shown in Figure 42b. on Figure 41, it is concluded that z-direction vibration the greatest experiment is as shown in Figure 42b. Based on Figure 41, it is concluded that z-direction vibration experiment as force shown 42b. Basedreason on Figure 41,the it ispolishing concluded that z-direction vibration influence onisthe of in theFigure workpiece. is that is perpendicular to the has the greatest influence on the force The of the workpiece. The reason istool that the polishing tool is has the greatest influence on the force of the workpiece. The reason is that the polishing tool is surface of the workpiece in the z direction, so the z-direction force changes very significantly; the perpendicular to the surface of the workpiece in the z direction, so the z-direction force changes very perpendicular to the surface the workpiece in the z direction, so actual the z-direction force changes very x-direction force is of second. While is the y-direction is the the tool is feed which significantly; the changes x-direction force changes second. While y-direction thedirection, actual tool feed significantly; the x-direction force changes is amplitude second. While the y-direction is theforce actual tool feed is affected by the feed, its force change curve is small; the z-direction increases as direction, which is affected by the feed, its force change curve amplitude is small; the z-direction direction, which is affected by the increases; feed, its force change curve amplitude is small; the z-direction the amplitude of the voltage signal Figure 42 shows the surface micro-morphology of the force increases as the amplitude of the voltage signal increases; Figure 42 shows the surface force increases as the amplitude of that the the voltage signalsurface increases; 42 shows the surface scratched workpiece was found workpiece aFigure clear, periodic microstructure. micro-morphology ofand theit scratched workpiece and it was foundhas that the workpiece surface has a micro-morphology ofoperating the scratched workpiece and itdevice was found that the workpiece surface has a When z-direction cycle of the motion is increased by the four times, the number clear, the periodic microstructure. When the 3D z-direction operating cycle of 3D motion device of is clear, periodic microstructure. When the z-direction operating cycle of the 3D motion device is microstructures is reduced to number 1/4. Thisofismicrostructures an effective basisisfor verifying the This practical of increased by four times, the reduced to 1/4. is anperformance effective basis increased by four times, the number of microstructures is reduced to 1/4. This is an effective basis the for device. verifying the practical performance of the device. for verifying the practical performance of the device.

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(a) (a)

(b) (b) Figure 41. Force ofof z-direction vibration on on initial force; (b) (b) Force curve during the Forcecurve. curve.(a)(a)Effect Effect z-direction vibration initial force; Force curve during Figure 41. Force curve. (a) Effect of z-direction vibration on initial force; (b) Force curve during the experiment. the experiment. experiment.

(a) (a)

(b) (b)

Figure 42. Workpiece topography. (a) Experiment Experiment No. No. 1; (b) Experiment No. 2. Workpiece surface topography. Figure 42. Workpiece surface topography. (a) Experiment No. 1; (b) Experiment No. 2.

6. Conclusions 6. Conclusions 6. Conclusions In this paper, paper, aa3-DOF 3-DOFmotion motiondevice device based a piezoelectric actuated flexible mechanism In this based onon a piezoelectric actuated flexible mechanism was In this paper, a 3-DOF motion device based on a piezoelectric actuated flexible mechanism was was designed, which has the characteristics of simple control, high output stiffness and self-guided designed, which has the characteristics of simple control, high output stiffness and self-guided designed, which has the characteristics of simple control, high output stiffness and self-guided output. Theoretical analysis, analysis, FEA, FEA, performance performance testing testing and and experiments experiments were were carried output. Theoretical carried out. out. The The main main output. Theoretical analysis, FEA, performance testing and experiments were carried out. The main conclusions are as follows: conclusions are as follows: conclusions are as follows: 1. The The device device is is mainly mainly series series connected connected by by aa 2D 2D motion motion platform platform including including the the xx and and yy directions, directions, 1. The device is mainly series connected by a 2D motion platform including the x and y directions, and the independent movement structure with z direction. The 3-DOF device adopts a circular and the independent movement structure with z The and the independent movement structure with z direction. The 3-DOF device adopts a circular flexure hinge, hinge, in in which which the 2D motion platform consists of three driving element, element, flexure three parts: parts: driving flexure hinge, in which the 2D motion platform consists of three parts: driving element, force-dividing element (composed of four force-dividing blocks), x and y output decoupling force-dividing element (composed of four force-dividing blocks), x force-dividing element (composed of four force-dividing blocks), x and y output decoupling guide element. element. The The driving driving element element is is designed designed using using the the principle principle of of double double parallelograms. parallelograms. guide guide element. The driving element is designed using the principle of double parallelograms. The function function of the force-dividing The force-dividing element elementisisto todecompose decomposethe theforce forceinto intoxxand andy ydirections; directions;The Thex The function of the force-dividing element is to decompose the force into x and y directions; The x and y output decoupling guide element can improve the output stiffness, and running x and y output decoupling guide element can improve the output stiffness, and running accuracy, and detect the actual displacement in the x and y directions. accuracy, and detect the actual displacement in the x and y directions.


2.

3.

4.

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and y output decoupling guide element can improve the output stiffness, and running accuracy, and detect the actual displacement in the x and y directions. The statics and dynamics of the 3-DOF motion device were analyzed. Using the MCM method, the output stiffness in the x, y and z directions are 1.137 N/µm, 0.760 N/µm and 0.717 N/µm. Using the FEA method, the output stiffness in the x, y and z directions are 1.178 N/µm, 0.751 N/µm, and 0.659 N/µm. Using energy method, the first natural frequency is 439.59 Hz and the second natural frequency is 690.29 Hz. Using FEA method, the first natural frequency is 474.82 Hz and the second natural frequency is 656.98 Hz. The first natural frequency of the experiment is 414 Hz. The device was tested for dynamic performance, linear motion, and vibration trace during no-load. Square wave, triangle wave and sine wave signals were applied in three directions to test dynamic response performance. When the square wave is input, the x-direction return response basically reaches the theoretical operating position at 0.025 s; the y direction and z direction reach 90% of the theoretical operating position at 0.025 s and 0.027 s, respectively. When the triangular wave signal is an input signal, the output signal amplitude fluctuates around the theoretical output signal; when the sine wave signal is an input signal, the output signal is basically the same as the theoretical output signal. When a signal that can only generate unidirectional motion is applied, the maximum linear motion errors in the x, y and z directions are approximately 6.67%, 5.71%, and 3.03%. Different signals were applied to the No. 1, No. 2 and No. 3 piezoelectric actuators to carry out five sets of vibration track test experiments. These results show that the device has excellent dynamic performance and can achieve 3D spatial trajectory. A three-dimensional vibration scratch experiment was carried out using a 3D motion device. When the vibration signal is only applied to the 3-DOF motion device, since the polishing tool is perpendicular to the surface of the workpiece in the z direction, the z-direction exhibits a significant periodic force change and coincides with the z-direction output signal cycle. The period and shape of the force curve are similar when the table is feeding or stationary. The workpiece surface scratched presents a pronounced periodic structure. These results proved the 3-DOF motion device has better reliability.

Supplementary Materials: The following are available online at http://www.mdpi.com/2072-666X/9/11/578/s1, Figure S1: Structure and driving principle of force-dividing block, Figure S2: Force-dividing block input-output response, Figure S3: The displacement nephogram of FEA, Figure S4: 3D motion device structure. (a) 3D motion device structural solid model; (b) Structural combination model, Figure S5: Schematic diagram of output center error position, Figure S6: Circular flexure hinge, Figure S7: Beam flexure hinge, Table S1: The structure parameters and FEA results, Table S2: Correction coefficient k. Author Contributions: Conceptualization, G.W.; Methodology, G.W.; Software, B.L. (Bingrui Lv); Validation, B.L. (Bin Li), H.Z. and Y.H.; Formal analysis, B.L. (Bingrui Lv); Investigation, B.L. (Bin Li); Resources, H.Z.; Writing—original draft preparation, B.L. (Bingrui Lv); Writing—review and editing, G.W.; Supervision, Y.H.; Project administration, G.W.; Funding acquisition, G.W. Funding: This work was supported by the National Natural Science Foundation of China (51405196) and the Natural Science Foundation of Tianjin (18JCYBJC20100, 17JCZDJC30400). Conflicts of Interest: The authors declare no conflicts of interest.

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