一種具有低遲滯效應(yīng)的三自由度夾心壓電機械臂-設(shè)計、建模和實驗外文文獻翻譯、中英文翻譯、外文翻譯,一種,具有,遲滯,效應(yīng),自由度,夾心,壓電,機械,設(shè)計,建模,實驗,外文,文獻,翻譯,中英文 A 3-DOF sandwich piezoelectric manipulator with low hysteresis effect: Design, modeling and experimental ABSTRACT A novel 3-DOF piezoelectric manipulator was modeled, fabricated and tested in this work. The longitudinal deformation and the orthogonal bending deformations were used to achieve the motions along the Z, X and Y axes, respectively. A novel kinematic model was developed and verified by FEM simulation. A prototype was fabricated, the output dis-placement test, frequency response test, hysteresis and creep characters were investigated in sequence. The tested results stated that the maximum displacements of the manipulator were 13.4 lm, 13.3 lm and 3.1 lm in X, Y, Z direction, respectively. The resonant frequen-cies of the X, Y and Z axis were tested as 2.734 kHz, 2.758 kHz and 16.462 kHz, respec-tively. The hysteresis effect of the manipulator was just 1.53% under the voltage of 300 V and the frequency of 1 Hz. The tests of two-dimensional and three-dimensional circular trajectories were implemented to study the dynamic performances. The results showed that the maximum open-loop working frequency was up to 50 Hz with an error less than 5%. Prospectively, the manipulator could be applied to the precision positioning system with open-loop working and nano-positioning demands by virtue of its compact size, high resolution, high resonant frequency and low hysteresis effect. Keywords：Piezoelectric manipulator；Hysteresis effect；Three-degree of freedom；Fast positioning；Kinematic model I.INTRODUCTION Nanopositioning technology has attracted more and more scholars and it has made rapid advancements in the last few decades, especially for semiconductor fabricating, miniature motion robots, scanning probe microscopy and biological manipulation [1–9]. At present, there are two major ways to implement the nanopositioning function: the nanopositioning manipulator and the nanopositioning stage. In recent years, various nanopositioning manipulators or stages have been pro-posed, they can be mainly divided into the voice coil motor driving type and the piezoelectric driving type [10–13]. For voice coil motor driving stages, they show the merits of large traveling range up to millimeter level. Zhang et al. developed a 3-DOF nanopositioning stage with the symmetric structure using voice coil motors as driving unit, which achieved a cross-coupling effect of 1.7% among the three motion axes and obtained the workspace of 2.22 mm 2.22 mm 1.81 mm with an overall size of 176 mm 176 mm 198 mm [14]. This stage shows the merit of a large workspace. However, its complex structure makes the fabrication expensive and time-consuming. For most of the existing voice coil type stages, a complex and preci-sion control system is needed to achieve high positioning accuracy. Another drawback is that multiple voice coil motors are needed to achieve multi-degree-of-freedom motion, the usage of multiple voice coil motors increase the complexity of the structure and make it difficult to assemble and maintain. Many nano-manipulators or stages are proposed utilizing the piezoelectric stack as the driving unit, considering their superiorities of compact size, quick response, high resonant frequency, free of electromagnetic interference and high reso-lution at the level of nanometer or subnanometer [15–19]. In general, the existing piezoelectric stages can be mainly clas-sified into two categories: the piezoelectric tube-type and the flexure-based type [20–26]. Kwon et al. developed a piezoelectric tube scanner with a scanning range of 100 lm 100 lm and the first major resonant frequency of 500 Hz [27]. This piezoelectric tube manipulator shows the merits of compact size and low cost. However, this manipulator shows relatively low mechanical resonant frequency when it is fixed at one end due to its large length-to-diameter ratios. For most of the existing tube-type manipulators, their first order resonant frequencies are always lower than 1 kHz. Their low reso-nant frequencies limit their applications in the systems that require higher working frequency, especially limit the scanning speed in the atomic force microscopy system [28,29]. What’s more, one limitation for piezoelectric tube manipulator is the vibration that occurs on the device structure will reduce the positioning performance [30,31], another limitation is the dynamics-coupling that leads to positioning error [32]. The flexure-guided nano manipulator, followed with great interest, has made great progress in the last twenty years due to the merits of high mechanical resonant frequency and low cross-coupling effect [33–38]. Leang et al. proposed a flexure-guided manipulator driven by piezoelectric stacks, which achieved the working range of 10 lm 10 lm with the first major resonant frequency in the slow and fast axis at 1.5 kHz and 29 kHz, respectively [39]. However, the main drawback of the flexure-guided nano-manipulator is that the flexures are needed to be designed accurately enough to offer motion guiding and reduce coupling. The accurate flexures are at the cost of structural complexity. Furthermore, for flexure-guided nano-manipulator, multiple piezoelectric stacks are always required if the multi-degree-freedom movement is needed. In this study, a 3-DOF sandwich piezoelectric manipulator is modeled and tested. This study aims at designing a compact nano-manipulator to realize 3-DOF motion by using longitudinal and bending-bending deformations, as well as makes it possible for the manipulator to be used in fields of precision engineering, medical biotechnology, micro/nano processing and scanning probe system. The structure of the manipulator is illustrated in section 2. The kinematic model of the manip-ulator is performed in section 3. Then, the mechanical output characteristics of the prototype are tested and discussed. A conclusion is given in the last section. 2. Structure and operating principle The three-dimensional model of the manipulator is shown in Fig. 1(a), the manipulator is composed of a driving tip, a horn, a four partition PZT stack, a screw bolt, copper electrodes and an end cap. The ring-shaped PZT ceramics and electrode slices are glued piece by piece, and they are clamped between the end cap and horn by the screw bolt. The total length of the positioning tip and the horn is 48 mm. The inner, outer diameters and the thickness of the ceramic rings are 14 mm, 30 mm and 1 mm. Furthermore, 24 pieces of bronze electrodes with a thickness of 0.2 mm are employed to apply voltages to the ceramic rings. The overall size of the manipulator is designed as 60 mm 60 mm 92 mm. All the PZT plates are polarized in four separate regions as shown in Fig. 1(b) and (c): two positive regions and two neg-ative regions, which are marked with symbols ‘‘+” and ‘‘–”, respectively. The four separate regions are named as region 1 to region 4, respectively. The advantage of the manipulator is that only one four-zone piezoelectric stack is used but it can achieve three-dimensional motion, which simplifies the assembly procedure and makes the structure compact. The X-Y-Z motion can be achieved when four input voltage signals V1 to V4 are applied to region 1 to region 4. A detailed illustration of how to achieve three-dimensional motion by changing the exciting voltage signal is listed in Table 1. 3. Kinematic modeling and finite element analysis of the manipulator In this section, a new kinematic modeling method for the manipulator is developed first to establish the relationship between the exciting voltage signal and the displacement response of the manipulator. As mentioned before, the four-zone piezoelectric stack can be considered as four independent longitudinal vibration piezoelectric stacks. The structure of the manipulator can be seen as a special parallel mechanism as shown in Fig. 2, the equivalent model consists of a fixed plane, four connection rods, an upper plane and a position tip. The end cap is equivalent to the fixed plane while the four-zone piezoelectric stack is equivalent to four connecting rods. Furthermore, the contact surface between the piezoelectric stack and horn is equivalent to the upper plane. The key structural parameters of the manipulator used in the kinematic model are given in Table 2. Finally, the static analysis is carried out to verify the accuracy of the kinematic model, and some important arbitrary constants in the kinematic model are calculated by static analysis. 3.1. The two-dimensional motion modeling According to the driving principle described in section 2, the X direction motion and the Y direction motion are indepen-dent of each other, so the planar motion can be regarded as the combination of X direction motion and Y direction motion. The two-dimensional planar motion can be achieved by applying V1 to region 1 and region 3 while applying V2 to region 2 and region 4. The realizations of the X and Y motion are presented in Fig3. Fig. 3. The diagram of how to achieve the two-dimensional planar motion: (a) Y direction motion display in space, (b) Y direction motion display in plane,(c) X direction motion display in space, (d) X direction motion display in plane. 3.2. Finite element analysis The static analysis was carried out by the finite element method (FEM) in ANSYS software to verify the accuracy of the kinematic model and to acquire the value of the arbitrary constants k and b in Eq. (2). The boundary condition is that the end cap is fixed. The element types of piezoelectric ceramics and metals are SOLID227 (piezoelectric coupled-field element) and SOLID186 (structural element). The materials of screw bolt and end cap are stainless steel (mass density: 7850 kg/m3, Young’s modulus: 2.10 1011N/m2, Poisson’s ratio: 0.3) while the material of horn is aluminum alloy (mass density: 2810 kg/m3, Young’s modulus: 7.17 1010N/m2, Poisson’s ratio: 0.33). The material of the piezo plate is PZT-4 (mass density: 7600 kg/m3) considering its high resistance to depolarization and low dielectric losses under high electric drive. The piezo-electric constant matrix d, stiffness matrix at constant electric field cE and dielectric matrix at constant stress eT are listed as follows. The static analysis results are shown in Fig. 6. When the amplitude of the applied voltage signal is 600 V, the maximum response displacement in the X, Y and Z directions are calculated as 13.67 lm, 13.62 lm and 3.24 lm, respectively. From the finite element analysis, a series of voltage signals are applied to the piezoelectric stack to acquire the relationship between the deformation of piezoelectric ceramics and the excitation voltage signal. The obtained results are polynomial fitted, and the fitting equation can be expressed as: Therefore, the arbitrary constants k and b are calculated as 0.0054 and 0.00143. After getting the value of k and b, the output displacements in X and Y direction are all calculated as 13.26 lm by the developed kinematic model. These results calculated by the kinematic model match well with the output displacements of simulation results. The little difference of about 3% between them is reasonable due to the model simplification. When applied constant force F = 1 N on the surfaces of the position tip in three directions, the stiffnesses in the X, Y and Z directions are calculated as 19.06 N/lm, 18.69 N/lm and 306.27 N/lm, respectively. In summary, the developed kinematic model provides a simple expression to evaluate the relationship between applied voltages and response displacements. The exactness of the kinematic model is proved by static analysis. This section pro- vides a simple modeling method for the piezoelectric manipulator with similar structure, which has guiding significance for further studies of the manipulator’s trajectory planning and control. 4. Experiments A prototype was fabricated to validate the feasibility of the piezoelectric manipulator, whose weight was 0.61 kg. An experiment system was set up, as shown in Fig. 7, which consisted of the manipulator, capacitance displacement sensors (D-E20.050, PI, Germany), a computer, a multi-channel arbitrary wave generator (DG4162, RIGOL, China) and a power ampli-fier (E01.A4, Coremorrow, China). The multi-channel arbitrary wave generator is used to generate driving signals. These sig-nals are amplified by a power amplifier, and then they are applied to the manipulator. The capacitance displacement sensor is employed to measure the output displacement. Then, the primary issue was to consider the cross-coupling effect of the manipulator. The test point was set as position tip, and the tested cross-coupling effect could be seen from Fig. 8. The voltage signal of 300 Vp-p with the frequency of 1 Hz was applied to the manipulator in X direction, and the response displacement of the manipulator was 6.7 lm. The measured X/Y cross-coupling effect was 196 nm (2.93%) while Y/X cross-coupling effect was 213 nm (3.18%), and the measured X/Z cross-coupling effect was about 46 nm (0.687%). The main reason which causes the coupling-effect is the individual differ-ences for PZT plates. The individual differences are caused by inhomogeneous polarization. Because the PZT plates work in d33 mode, so the inhomogeneous polarization means the d33 parameters may be different between each PZT plates, and the d33 parameters may be different between each polarization zone in one PZT plates. Then, the longitudinal deformation along Z direction and the bending deformation along X and Y directions of the manip-ulator were tested versus voltages, as shown in Fig. 9. The response displacements in X, Y and Z directions are approximately linear related to the applied voltages. Furthermore, the maximum displacements in X, Y and Z direction are obtained as 13.4 lm, 13.3 lm and 3.1 lm under the exciting voltage of 600 Vp-p, and the frequency of excitation voltages in this test is 1 Hz. where t is the time, L(t) is the displacement of PZT ceramic, L (0.1) is the displacement at 0.1 s after applying the step signal, and c is the creep factor. From the experimental tests, the creep factor c takes 0.028. The origin data of creep and theoretical creep curves are presented in Fig. 10, and it can be seen that the tested creep curve and theoretical creep curve fit well with others. Then the hysteresis tests were carried out. The test of hysteresis effects under different voltage amplitudes and different voltage frequencies were carried out in sequence. The hysteresis effect of a single piezoelectric plate was about 8% according to the supplier. The special triangular wave signal shown in Fig. 11(a) was applied to the manipulator to study the hysteresis effect under different voltage amplitudes. The response displacement and hysteresis curves are shown in Fig. 11(b), where T1 is the ascendant curve while T2 is the descent curve. The hysteresis curve is shown in Fig. 11(c), while the amplification of the hysteresis curve is shown in Fig. 11(d). When the amplitude of exciting voltages are 300 Vp-p; 240 Vp-p; 180 Vp-p and 120 Vp-p, the hysteresis coefficient are 1.76%; 1.45%; 1.36% and 1.27%, respectively. From the experiment results, one of the reasons for the relatively low hysteresis effect is the low electric field intensity (E), because higher electric field intensity (E) means higher hysteresis effect. The maximum electric field intensity (E) of the commercial piezoelectric stack is about 1000 to 1500 V/mm while the maximum electric field intensity (E) of this work is only 600 V/mm. To study the hysteresis effect of different voltage frequencies, the sine signal was applied to the manipulator, and the hys-teresis curves were shown in Fig. 12. Considering the first bending resonant frequency was 2.734 kHz, the applied voltages frequencies are up to 250 Hz (about 1/10 of the first resonance). When the exciting voltage is 300 Vp-p with the voltage fre-quencies of 1 Hz to 250 Hz, the hysteresis effects are shown in Table 3. It can be seen that the hysteresis effects in X and Y directions exhibit a small difference, while the hysteresis effects in Z direction are greater than those in X (Y) direction. The reason for this phenomenon is that the movement in the X (Y) direction is caused by the deformation of the ceramics in the two regions while the movement in the Z direction is caused by the deformation of the ceramics in the four regions. The increase of the ceramic area involved in deformation will weaken the inhibition effect of the whole structure on the hystere-sis effect. Therefore, the hysteresis effects in Z direction are significantly greater than those in X (Y) direction. What’s more, the maximum hysteresis effect is 4.26% when the voltage frequency is 250 Hz. Combing the results of Figs. 11 and 12, the more important factors that may contribute to the low hysteresis effect of the manipulator is the structure itself. The usage of high rigidity bolts in the manipulator can be equivalent to a spring with a high stiffness coefficient. This spring has high linearity, therefore, it will restrain the nonlinearity of the piezoelectric stack. It is also verified by the experiment as we finally find that the hysteresis effect is reduced from 8% to 1.53%. In summary, there are two main reasons for the low hys-teresis effect, one is the low electric field intensity (E) and another is the high linearity of the structure. The response displacement of the position tip in X direction under stepwise excitation signal was tested to study the res-olution of the manipulator. The value of each step is set as 0.2 V with a time of 1.5 s. The stepwise excitation signal is shown in Fig. 13(a), while the response displacement is exhibited in Fig. 13(b). Based on the step-by-step voltage signal, the reso-lution of 4.8 nm is achieved. The bandwidth of the sensor is 1.24 kHz. The resolution of 4.8 nm is the corresponding mini-mum test displacement of the manipulator, and it can be seen as the relative resolution of the manipulator. The frequency response tests were carried out. The scanning laser doppler vibrometer (PSV-400-M2, Polytec, Germany) is utilized to measure the frequency response, and the test result is shown in Fig. 14. Small inputs voltage signal (25 V) is applied to the manipulator during the test to minimize the effect of nonlinearity such as hysteresis. The first-order resonant frequency of the X, Y and Z axis is tested as 2.734 kHz, 2.758 kHz and 16.462 kHz, respectively. Then, the two-dimensional trajectory experiments were performed. The applied signals to the four regions are illustrated in Eq. (37). The experiment results and calculated by the kinematic model are shown in Fig. 15(a). It can be seen that the maximum displacement of the manipulator is tested about 6.7 lm under voltage of 300 Vp-p, the trajectory curve of the manipulator matches well with the calculated curve, and the errors between them are shown in Fig. 15(b). The maximum errors of X and Y direction are calculated as 104 nm and 110 nm, respectively. The maximum errors of X and Y direction are just 3.11% and 3.28%, respectively. Therefore, the experiment results also prove the accuracy of the developed kinematic model. Then, the measured trajectories under different exciting frequencies were illustrated in Fig. 16. From the figure, it shows that the errors between the tested trajectory and the calculated trajectory increase with the increase of the frequency. The errors in X and Y direction are listed in Table 4. The results show that the open-loop frequency is up to 50 Hz with an error less than 5%. Similar to two-dimensional trajectory experiments, the three-dimension trajectory experiments were carried out. The calculated results of the response displacement in Y direction and Z direction are shown in Fig. 17(a). The exciting signal in X direction is calculated by Eq. (32). To form the s