Implementation and experiment of an active vibration reduction strategy for macro-micro positioning system

Accepted Manuscript Title: Implementation and Experiment of an Active Vibration Reduction Strategy for Macro-Micro Positioning System Authors: Lanyu Zhang, Jian Gao, Xin Chen, Yun Chen, Yunbo He, Yu Zhang, Hui Tang, Zhijun Yang PII: DOI: Reference:

S0141-6359(17)30170-8 http://dx.doi.org/10.1016/j.precisioneng.2017.09.002 PRE 6652

To appear in:

Precision Engineering

Received date: Revised date: Accepted date:

22-3-2017 17-8-2017 4-9-2017

Please cite this article as: Zhang Lanyu, Gao Jian, Chen Xin, Chen Yun, He Yunbo, Zhang Yu, Tang Hui, Yang Zhijun.Implementation and Experiment of an Active Vibration Reduction Strategy for Macro-Micro Positioning System.Precision Engineering http://dx.doi.org/10.1016/j.precisioneng.2017.09.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Implementation and Experiment of an Active Vibration Reduction Strategy for Macro-Micro Positioning System Lanyu Zhang, Jian Gao＊, Xin Chen, Yun Chen, Yunbo He, Yu Zhang, Hui Tang, Zhijun Yang Key Laboratory of Precision Microelectronic Manufacturing Technology & Equipment of Ministry of Education, Guangdong University of Technology, Guangzhou, Guangdong 510006, China

Corresponding author.

E-mail address: [email protected]. Highlights  We focus on the stage with high-speed, large-stroke with high positioning accuracy.  An adaptive vibration reduction action method for a macro-micro stage is proposed.  The start-up condition and extension of the piezoelectric device are finalized.  The stage settling time is greatly reduced without sacrificing positioning accuracy. Abstract—In microelectronics manufacturing, the macro-micro precision positioning system is often utilized to achieve a high-precision positioning motion with high-velocity and large-stroke. The working efficiency of the stage is affected by the inertial vibration of the macromotion which will cause more time to settle. This paper adopts an active vibration reduction method through a piezoelectric (PZT) device to quickly reduce the designed vibration amplitude of the macro-micro positioning stage. In the paper, we propose a dynamic adaptive vibration reduction strategy to work against the motion of the macro-stage through the PZT to settle the stage with much less time. The start-up condition and extension principle of the piezoelectric element used in the vibration reduction action are designed and finalized. The method can therefore dynamically start up the actuation of the PZT to act on the macromotion at right moment and determine the effect of the vibration reduction through the extension length of the PZT. A dynamic model, a force analysis, and an amplitude reduction analysis are performed to understand the vibration reduction action applied in the macro-micro stage and a dynamic simulation is conducted to examine the effect of the method. The experimental tests are carried out on the stage through the exploration of different extensions and waveform types of the piezoelectric device. The experimental results indicate that the settling time of the macromotion can be greatly reduced with the proposed method without decreasing its positioning accuracy for motion with high velocity and large stroke. This novel method is useful for applications of the electronic manufacturing equipment with high-velocity and high-accuracy requirements. Index Terms—Dynamic model, Force analysis, Precision positioning stage, Piezoelectric element, Settling time reduction.

1. Introduction The macro-micro precision positioning concept has become popular in precision engineering, microelectronic manufacturing engineering, semiconductor-based manufacturing industries, micro-nano manufacturing, and optical engineering, among others [1–4]. A macro-micro positioning system has been developed to combine the micromotion actuator with linear motor to realize large-stroke, high-velocity and high precision positioning [5, 6]. Specifically, a micromotion actuator, such as piezoelectric element (PZT), possesses the advantages of high-precision, large-thrust and high-frequency response, but it usually has difficulty achieving large stroke movement [7]. A linear motor, or a voice coil motor (VCM), has large-stroke and high-velocity characteristics, but its positioning accuracy is still limited [8]. Combining of the characteristics of a PZT actuator and a linear motor in macro-micro composite actuation can help solve the contradiction problem of high-velocity, large-stroke motion with high-precision positioning, which currently has been applied in related industrial fields [9, 10]. In the hard disk drives (HDD) 1

Corresponding author. E-mail address: [email protected].

application, a PZT actuator was used for precision positioning to increase the storage density, and a VCM is utilized to achieve a high-velocity and large-stroke motion [11]. In precision manufacturing fields, high-velocity and high-precision composite motion are normally designed through a macro-micro dual-stage [12]. Efficient positioning equipment with precision can be achieved by the macro-micro positioning stage with high-velocity and high-acceleration motion [13]. However, the inertial vibration resulted from a high velocity and high acceleration motion will obviously enlarge the settling time and thus affect the working efficiency of the positioning stage [14]. As indicated by the literature [15], the inertial vibration induced by the decelerating positioning motion cannot be quickly cancel out to a large extent and will influence the production efficiency of manufacturing equipment. To resolve this vibration problem, a number of studies have been conducted on achieving a vibration reduction [16, 17]. Pang et al. [18] proposed a singular perturbation control for a VCM-PZT system of a HDD device to suppress the actuators’ vibration via an inner loop. The experimental results showed that the vibration of the VCM and the PZT were effectively damped with a reduced relative degree. Chen et al. [19] developed a vibration damping method for reducing the overshoot induced by the servo motion part of the hard disk drive devices. This method was involved in limiting the vibration to the setting levels by changing the damping ratio of the closed-loop system. The settling time can be reduced significantly for this positioning system. Devasia et al. [20] developed a condition setting method to optimize the control for the VCM and the PZT in a dual-stage for improving the settling performance, and a fast settling was obtained with simulation verification. Numasato et al. [21] designed a minimum jerk trajectory for a VCM-PZT dual-stage to decrease the settling period and the inertial vibration of the stage. The experimental results showed that the settling responses of the stage were reduced with the designed control trajectory for a movement of 0.06 m/s velocity and 20 μm stroke. Li et al. [22] designed a residual vibration reduction method through motion profile planning for a precision positioning machine with an acceleration about 6 m/s2. This method utilized a sinusoidal-shaped waveform to control the changing rate of acceleration, and the residual vibration and settling time were reduced significantly. Salton et al. [23, 24] utilized a preview control strategy to adjust the overshoot of the macromotion part for a dual-stage used in the hard disk drive devices, and the control trajectory of this strategy was optimized with a designed constraint criterion to achieve an optimal motion control. The results showed the inertial vibration and settling time can be reduced with the designed control method. As we can see, through various control strategies, these methods can achieve the reduction of vibration and settling time for the positioning stage [25, 26]. Differently, there is a structure design method to quickly reduce the vibration of the positioning stage. Liu et al. [27] proposed a positioning table by utilizing a frictional sliding structure design to realize rapid and stable high-precision positioning, which is driven by a collision force of a piezo-VCM actuator. This system took 1.253 s for the stage stoke of 450 μm with high positioning accuracy. Zhang et al. [15, 28] and Wang et al. [29] adopted a damper device to reduce the vibration of the macro-micro motion stage, and the structure optimization design was conducted for the stage to achieve an effective performance by avoiding the resonance in the implementation [28]. Through a floating stator installed between the base and the actuator, the method can isolate the impact force vibration from the driving part of the macro-micro positioning system. Henmi et al. [30] designed a novel impact damper approach for a fine-motion flexure mechanism. Through the collision with the impact damper, the end of the flexure mechanism was settled with much less settling time. Through the clearance adjustment design, this method can achieve a good performance for flexure mechanisms. Considering the application of electronic manufacturing equipment, such as a wire bonding machine, the macro-micro motion stage usually has a feature of high-velocity high-acceleration motion with high positioning accuracy. The inertial vibration of this positioning system has been a serious problem for improvement of performance and efficiency of these machines. To solve the problem, we proposed an active vibration reduction method to quickly reduce the vibration amplitude through a PZT device installed on the macro-micro stage. The feasibility study of the proposed vibration reduction method was performed and introduced in [31]. Based on this study, we present a further work on the implementing mechanism of the reduction method and improve the vibration reduction effect with optimal actuation. In this paper, the PZT element is designed to be actuated dynamically and adaptively to act against the inertial vibration of the macromotion. The start-up condition of the actuation is detailed and implemented through the real-time position feedback. The PZT extension length is analysed and finalized, which determines the effect of the vibration reduction. Theoretical modelling and analysis including dynamic modeling, force analysis and amplitude reduction analysis are performed. Simulation and experimental work are carried out to examine the performance of the proposed method. The results show that the method can effectively reduce the inertial vibration and settling time for the macro-micro stage with a high-velocity and high-acceleration motion without decreasing its positioning accuracy. The rest of this paper is organized as follows. In Section 2, the construction and constituent design of the positioning stage is described, and the methodology including the adaptive extension principle and start-up condition of the PZT element are designed to implement the vibration reduction method. In Sections 3 and 4, the dynamic model and a force analysis of the vibration reduction method are performed, respectively. In Section 5, an amplitude reduction analysis is presented. The control strategies are then presented in Section 6. In Section 7, dynamic simulation of the proposed method is examined. The experimental setup of the stage is described in Section 8. Experimental results are then shown and discussed in Section 9. Finally, the achievements of this study are summarized in Section 10, along with some concluding remarks and an indication of areas of future work. 2. Methodology of the proposed adaptive vibration Reduction Method 2.1 Description of the macro-micro positioning stage To realize a vibration reduction for high-velocity and large-stroke motion with high positioning accuracy, a macro-micro stage

was designed in [31], and adopted for this study. The macromotion is implemented using a VCM motor, and the micromotion is designed with a special PZT-spring structure. The construction and components of the designed positioning stage are shown in Fig. 1. The micromotion is driven by the PZT1 actuator to achieve a precise positioning within its stoke. The vibration reduction mechanism consists of a sliding device and a PZT2 actuator placed near the target position.

2.2 Extension principle of the PZT2 in the vibration reduction method With the designed macro-micro stage, the proposed vibration reduction method is based on a collision mechanism to work against the oscillating motion of the stage for rapidly reducing the amplitude of the inertial vibration and the settling time of the macromotion. A dynamic adaptive extension principle of the PZT2 element in the vibration reduction method is illustrated in Fig. 2. In Fig. 2, L is the length between the PZT 2 position and the positioning target, and it is smaller than or equal to the limit stroke range Sp of the PZT2. Δi is the ith vibration amplitude with the vibration reduction method, Ai is the ith maximum inertial vibration amplitude of the macromotion without using the vibration reduction method, and Lpi is the ith extension displacement of the PZT2 for working against of the stage. The extension value of Lpi is set to satisfy the following condition:  L - A i  L pi  L   L pi  L pi  1 L  S p 

(1)

In equation (1), the extension Lpi of the PZT2 will be adaptively adjusted for each vibrating cycle based on the vibration condition. When the inertial vibration amplitude of the macromotion is decreased, the length Lpi of the PZT2 will be increased to work against its motion. The PZT 2 extension length for each actuation is determined on the basis of the energy conversation law. The calculation of Lpi is expressed as L pi = L -C e

M evi

2

(2) where, vi is the measured maximum velocity value of the stage for each cycle of the inertial vibration, and Me and Ke are the equivalent mass and stiffness of the stage during an inertial vibration, respectively. In addition, Ce (0
coefficient Ce can be used to adjust the PZT 2 extension length to determine how much vibration reduction can be achieved through the interactive process. 2.3 Start-up condition of the PZT2 for the vibration reduction Based on the extending principle of the PZT 2 in the vibration reduction method, the start-up moment of the PZT2 is designed and detailed. To work against the vibration of the macromotion during the decelerating positioning, the actuating moment of the PZT2 is determined by the vibration phase and position of the stage. The PZT2 is actuated against the vibration movement of the stage when the vibration direction is forward and the stage displacement is bigger than the position target. Under these conditions, the PZT2 action will not influence the positioning process, and can effectively suppress the vibration amplitude with the right moment. A diagram of the PZT2 start-up conditions is illustrated in Fig. 3. The start-up condition of the PZT 2 used for vibration reduction can be expressed as  d i  Pi  Pt  0   D i  Pi  Pi  1  0 ,

(3) ( i  1 , 2 , 3 ...)

where, di is the error between the current position value Pi of the stage and the target position Pt. Di is the difference between the current position value Pi of the stage and the previous position value Pi-1. If the start-up condition is satisfied, then the PZT2 is actuated to act on the macromotion opposite to the movement direction. Through the interactive action, the vibration of the stage can be reduced quickly and the stage can be settled with much less time. Then, the entire vibration reduction procedure including the dynamic start-up and the adaptive extension of the PZT2 is illustrated in the flowchart shown in Fig. 4. The procedure is described as follows. Step 1: The position signal of the stage is continually collected through the grating encoder and data acquisition card. Step 2: When the macromotion of the stage starts to decelerate the macromotion, the system begins to judge the right moment to actuate the PZT2 based on the real-time data feedback from the stage. Step 3: Once the start-up condition is met, the PZT2 is actuated to act on the oscillating stage with the extension length adaptively determined by the formula (1) to reduce the amplitude of the stage. Step 4: After the PZT2 action was taken place, the system will repeat this process to dynamically actuate the PZT2 if the condition is met. When the vibration amplitude is reduced to the ideal amplitude (|A|i <α), then the system will switch to the micromotion for the fine positioning with high accuracy. Otherwise, the system will continue the macromotion procedure and go back to Step 1.

Through the determination of the start-up condition and the extension length of the PZT 2, the proposed vibration reduction method can dynamically actuate the PZT2 to act on the macromotion at appropriate time with right force, and thus rapidly reduce the inertial vibration amplitude and settling time of the macromotion for a fast positioning with the required accuracy. 3. Dynamic modeling According to the methodology of the vibration reduction method and the design of the stage, the dynamic model of the stage with vibration reduction can be depicted simply as shown in Fig. 5. Based on Newton’s second law of motion, the differential equations of the stage with vibration reduction can be obtained as follows:   M 1 x 1  F1  K 1 x 1  C 1 x 1  K 2 ( x 1  x 2 )  C 2 ( x 1  x 2 )   M 2 x 2  K 2 ( x1  x 2 )  C 2 ( x1  x 2 )   x 2  F 2 

(4)

where C1, M1, and K1 are the damping, mass, and stiffness coefficients of the VCM axis, respectively; M2 is the collective mass of the specialized structure, including the stage and the PZT1 in the sleeve; C2 and K2 are the damping and stiffness coefficients of the spring element, respectively; and μ is the viscosity coefficient between the stage and the guide. In addition, F1 is designed as the driving force of the macromotion, which depends on the parameters of the VCM, and F2 is designed as the vibration reduction force of the PZT2 acting on the stage. When the actuating force is applied, the VCM axis and the stage displacements are X1 and X2, respectively. Based on (4), the matrix form of the dynamic model can be expressed as follows: K x  C x  M x  f  f'

where, K =

K1  K   K2

damping matrix,  0  f'      F2 

M

K2  K2 

2

= M 1   0

(5)

is the equivalent stiffness matrix of the dynamic model,

0   M 2

 x1 ( t )    x 2 (t ) 

is the equivalent mass matrix, x  

C

= C  

1

 C2

C2

C 2   (C 2   ) 

is the equivalent

is the displacement vector, and

 F1  f     0 

and

are the equivalent force vectors.

4. Force analysis To implement the proposed adaptive vibration reduction actions of the PZT 2, the forces of the PZT2 acting on the stage are investigated and analysed. The inertial vibration of the stage will be interrupted and altered when a collision from the PZT2 occurs. The force analysis model of the vibration reduction method is described in Fig. 6. When a vibration reducing collision occurs, the contact force Fc acts simultaneously on both the stage and the PZT2 actuator with the same values but in the opposite directions. The other forces acting on the stage are the spring and damping forces, which can be expressed in the same way as through a dynamic model analysis. By drawing a free-body PZT2 contact head in the collision model, there are two forces acting on it. One of them is contact force Fc acting on the contact interface of the PZT2 with the stage, and the other is actuating force Fp, which is generated by the rapid extension value of the the PZT2 actuator. The collision behaviour can be depicted simply through the following differential equation:   m 2 x 2 '  Fc  K 2 x 2  C  ..  m p x p  F p  Fc  ..

.

2

.

x2'   x2'

(6) where mp and xp are the mass and extension of the PZT2, respectively. On the basis of the Newton’s second law of motion, the driving force Fp of the PZT2 can be expressed as follows: F

 K

 x

(7) where Kp is the stiffness of the PZT2 element. With the above analysis, the contact force can be determined through a number of methods reported in [32-35] have studied this content. Through a stiffness-damper analysis [33], the contact force is expressed as follows: p

p

p

.

Fc  K c x c  C c x c

,

(8) where xc is the indentation of two contacting bodies and is equal to xp-x2, Kc is the stiffness coefficient for expressing the ratio of the indentation of the contact force, and Cc is the damping coefficient for expressing the energy loss of the collision. The force analysis is used to determine the appropriate active force to apply to the stage when implementing the vibration reduction actions through the PZT 2. 5. Amplitude reduction analysis In the proposed vibration reduction methodology, the PZT2 action is used to actively collide with the stage and thereby reduce the inertial vibration amplitude. To identify the amplitude reduction amount by using the PZT 2, the modal superposition method

is used in analysis of the vibration reduction procedure for the macromotion positioning process. The amplitude reduction analysis is described in this section. According to the above dynamic model, we can obtain a differential equation for the moment a collision occurs on the stage, which can be expressed as follow. M

x 2 + K 2 x 2  C 2 x 2 +  x 2  F2

(9) Based on the momentum conversation law, the impulse Ii of the stage during a collision can be obtained by Ii =



2

 i +  ti i

F2 i d t  M



2



( x2i  x2i )  M

2

( x2i )

(10)

where, the i is the number of collisions, and the τi and Δti are the collision moment and duration of the ith collision, respectively. In addition, x 2i and x 2- i are the velocity before and after the ith collision, and  x 2 i is the change in velocity of the ith collision. The average impact force F 2 i during the collision can therefore be expressed as F2 i 

Ii



 ti

M

2

( x2i )  ti

(11) Because the Δti is a tiny amount, we can use the Dirac function to express equation (11). F2 i  M

(  x 2 i ) ( t   )

(12) Based on equation (12) and the modal superposition method, we can transform the vibration differential equation of the stage n

through

x (t ) 



 i pi

2

. The transformed equation can then be expressed as

i 1

p i  2  i i p i   i p i   i (  x 2 i ) ( t   ) 2

T

(13) where, φi is the natural vibration vector, pi is the canonical coordinate, ωi =

K 2 /M

2

is the natural frequency of the system, and

ξi= (C2+μ)/(2M2ωi) is the modal damping ratio. According to the proposed vibration reduction method, the initial conditions of the inertial vibration motion of the stage are confirmed through the macromotion of the VCM actuator, which can be expressed as pi  x2 (0 )

pi  x2 (0 )

(14) Thus, based on equation (13) and the initial conditions, the solution to the inertial vibration displacement coordinate of the stage prior to a vibration reduction action can be solved through p i (t )  e

(   i i t )

[

x 2 ( 0 )   i i x 2 ( 0 )

 i'

 x 2 ( 0 ) c o s  i't ]

s i n  i't

(15)

( i  1, 2 .. n )

where ωi’ = 1 -   . The solution to the inertial vibration displacement coordinate of the stage after the vibration reduction action can be obtained as follows. 2

i

p i (t )  e

(   i i t )

[

 i ( x2i )

i

x 2 ( 0 )   i i x 2 ( 0 )

 i'

s i n  i't  x 2 ( 0 ) c o s  i't ]

(16)

T

+

 i'

e

(   i i t )

s i n  i' ( t   )

( i  1, 2 .. n )

Based on the above vibration equation in (16), we can obtain the inertial vibration amplitude after applying the vibration reduction collision. From this vibration analysis, it can be seen that the acting moment and the impact force are the key factors in achieving an effective reduction of the inertial vibration amplitude. It also provides a reference for the output control of the PZT 2 in the vibration reduction procedure. 6. Implementation of the designed vibration reduction method Based on the force and amplitude reduction analysis, the vibration reduction method for the stage is implemented through a control scheme. Fig. 7 shows the specific control scheme about the macromotion, the micromotio, and the dynamic and adaptive extension of the PZT2. The motion coupling of the macromotion and the micromotion is analysed. Based on the differential equation (5) with the Laplace transform, the transfer function matrix can be expressed as

1   M s 2  (C  C ) s  K  K  X 1(s)  1 1 2 1       X 2 (s) 0  

1 C2s  K

2

2

1 M 2 s  (C 2   ) s  K 2 2

   F (s)   1    F2 ( s )   

(17) Then, the coupling relation of macro-micro motion is obtained. X 1(s) 

1 C2s  K 2

F2 ( s )

(18) By applying the inverse Laplace transform, the equation (18) is presented as x1 ( t ) 

1 C2

-

e

K

2

C2

t

F2 (t )

(19) The adopted parameter values of the dynamic model are listed in Table 1. It can be seen that the value K2 is far larger than the C2. Thus, the x1(t) ≈ 0, the motion coupling relation of the macro-micro motion can be neglected for the stage. For the specific micromotion of the stage, the vibration reduction force of the PZT2 is controlled through the input voltage and extension control. The extension of the PZT element is proportional to the input voltage, which is based on the inverse piezoelectric effect and Kirchhoff’s law. Although the extension of the PZT is proportional to the input voltage, an open-loop control does not work sufficiently well because of the hysteresis and nonlinearity of the PZT elements. Thus, based on the tuned PID control parameters, a closed-loop PID controller using a packaged resistive strain position sensor is applied in the PZT control to achieve an accurate extension. In order to achieve a stable and precise macromotion positioning, a closed-loop nonlinear PID controller [36, 37] is implemented for the macromotion control. The closed-loop position feedback utilizes an incremental grating element in the VCM. Considering the problem of the nonlinearity and disturbance of the motion positioning, an adjustable mechanism with the proper PID parameters was utilized, which is presented as follows. Combining the error value, the output of the nonlinear PID controller can be expressed as

t

u (t )  P  e (t ,  ,  , )  I   e (t ,  ,  , ) d t  D 

d e (t ,  ,  , )

dt (20) where, P, I, and D are the PID control coefficients, ε is the deviation error value, γ and η are the control constants. This nonlinear control function e(t,ε,γ,η) can be expressed as follows: 0



  s ig n ( ),  e (t ,  ,  , ) =  1 , 1    

   

(21)

The control constants γ and η can be selected based on the studied guidelines, which are γ = 0.1~0.2 and η = 0.5~1, respectively. 7. Simulation of dynamic modeling To validate the theoretical analysis and examine the effectiveness of the proposed adaptive vibration reduction method, a dynamic simulation was implemented through different movement parameters. Depending on the dynamic vibration reduction model and the control scheme, the dynamic system was simulated with the MATLAB analysis software. Based on the proposed vibration reduction principle in Section II. B, the coefficient Ce is applied to reflect the amount of the vibration reduction extension of the PZT 2. Fig. 8 shows the positioning motion of the stage with different vibration reduction extensions of the PZT2. The simulation results show that the proposed vibration reduction method can effectively reduce the inertial vibration of the stage with the extension of the PZT 2. When using a large extension of the PZT 2 (a small Ce), the amplitude and settling time can be reduced more effectively. Based on the simulation performed, the settling time for the macromotion can be reduced from 0.0689 s to 0.0334 s for the movement of 40 mm stroke, 0.2 m/s velocity and 40 m/s2 acceleration with the Ce of 0.2, which is a reduction of 51.5%. The effects of the vibration reduction of the method with different waveform of the PZT 2 (as shown in Fig. 9) are also examined and simulated, and the results are shown in Fig. 10. It can be seen that the effectiveness of the vibration reduction varies with the used waveform, and the settling time can be reduced to 0.0316 s with the rectangle waveform for the motion of 40 mm stroke with 0.2 m/s velocity and 40 m/s2 acceleration. Based on the simulation results, the optimal setting of the PZT2 extension and the rectangle waveform are applied for the further work. Fig. 11 shows the settling time reduction of the stage for different strokes with and without the reduction method. It can be seen that the settling time of the macromotion can be reduced by 52.1% from 0.0652 s to 0.0312 s for the motion of 40 mm stroke with 0.2 m/s velocity and 20 m/s2 acceleration. Fig. 12 shows the simulated results of the settling time reduction for different velocities with and without the reduction method. It is found that the settling time of the macromotion can be reduced by 53.7% from 0.0668 s to 0.0309 s for the motion of 40 mm stroke with 0.2 m/s velocity and 40 m/s2 acceleration. 8. Experimental setup

The experiment tests were carried out on a macro-micro stage. The experimental setup of the stage was established and shown in Fig. 13. The parameters of the stage and the PZT elements were selected and determined based on the theoretical analysis, which are shown in Table 2. The applied voltages for the PZT elements were simulated through a Windows-sustained Virtual Studio (VS) programming. A absolute grating encoder (HEIDENHAIN LIC4015) with an input frequency of at most 500 kHz and a resolution of 1 nm was used to record the position of the stage and establish the closed-loop control between the PZT elements, VCM, and stage. A data acquisition card (NI PCI-6289) was utilized to collect the position data from the grating encoder, and transform the control signal into a power-amplified signal, and then into a PZT signal. The vibration of the stage was detected through a Doppler laser vibrometer (Polytec PSV-400), which has a resolution of 1 nm and a bandwidth of 1.5 MHz. A Renishaw laser interferometer was used to examine the positioning accuracy of the stage and the accuracy of the grating encoder. With the reference of the laser interferometer measurement, the mean error of 11 nm was achieved by the grating device measurement. 9. Experimental analysis and results 9.1 Effects of the PZT2 extensions To effectively verify the effectiveness of the PZT2 element acting against the inertial vibration, the vibration of the stage and the vibration reduction with different extensions of the PZT2 were examined while keeping other conditions constant. Based on the proposed methodology of the PZT2 action in Section II.B, the extensions of the PZT2 is determined by the extension coefficients Ce. Fig. 14 shows the results of the vibration waveform of the stage with and without the vibration reduction method. With different PZT2 extensions, the vibration amplitude can be reduced with different levels. In the experiments, the vibration reduction coefficient Ce is set to 0.2, 0.4, 0.6 and 0.8, then the extension of the PZT 2 is changed from large to small, the reduction of the vibration amplitude as well as the settling time can be reduced accordingly. Table 3 presents the settling time with and without the vibration reduction method. We can find from the table that the results are very close by comparing them from the experimental tests and simulations. From the experimental results, the settling time can be reduced effectively, and a maximum reduction in settling time was achieved by 51.5% from 66.2 ms to 32.1 ms with the extension coefficient of 0.2. Fig. 15 presents the relationship of the settling time with the coefficient Ce. The figure shows that the settling time of the stage decreases as the coefficient is set to a small value. The smallest coefficient can be determined by the stage designed, which is 0.075 for the stage with the 3 μm positioning accuracy of the macromotion and 40 μm PZT2 stroke. 9.2 Effects of the PZT2 input waveform Considering the effect of the PZT 2 waveform to the vibration reduction, four types of the PZT 2 input waveform are examined in the experimental tests. They are applied to actuate the PZT2 with a voltage width of tp. The other conditions, i.e., the motion parameters of the stage, and extension setting of the PZT2, remained constant in the experiments. The test results are shown in Fig. 16. It can be seen that a rectangular waveform can produce better results in vibration reduction than the other type of waveform. The settling time of the macromotion was affected by the waveform type as well. The reductions in settling time with different waveform for the PZT 2 actuation are presented in Table 4. With the rectangle waveform actuation, the settling time of the macromotion can be reduced by 54.4% from 66.2 ms to 30.2 ms, which is corresponding with the simulation. 9.3 Effects of different strokes To examine the effectiveness of the adaptive vibration reduction method used for large-stroke and high-velocity of the stage, several tests are carried out on the stage with different strokes and velocities. Fig. 17 shows the settling time reduction comparison with and without the PZT 2 based on the vibration reduction method when the stage stroke is set from 10 mm to 40 mm, while keeping the other motion parameters unchanged, such as the acceleration, the PZT 2 waveform and extension setting. The waveform and extension setting of the PZT 2 are selected with tested optimal parameters. The results of the settling time reduction for different strokes are presented in the Table 5. It can be seen from the results that the vibration reduction method can effectively reduce the vibration amplitude and settling time for the motion with different strokes, which are corresponding with the results of the simulation. The results indicate that the settling time for the macromotion can be effectively reduced at a level of 52~55.1% by using the selected optimal PZT 2 output setting, which is better than the previous effects of the vibration reduction in [31]. This will obviously help the stage to realize a fast positioning and thus improve the working efficiency of the stage. 9.4 Effects of different velocities Similarly, the experimental tests were carried out on the stage motion with different velocities. In the experiments, the stage motion was set with a stroke of 40 mm, an acceleration of 40 m/s2, and the velocity of the motion is set from 0.05 m/s to 0.2 m/s. Fig. 18 shows the results of the vibration reduction and settling time reduction with and without the PZT 2 vibration reduction method applied to the macromotion of the stage at different velocities. The results of the settling time reduction to the motion at different velocities are listed and compared in Table 6. The results show that the vibration reduction method can achieve an effective reduction in maximum amplitude and settling time of the motion. The results are also corresponding with the

examination of the simulation. The settling time of the macromotion can be reduced by 54.4% at a velocity of 0.2 m/s by implementing the selected optimal PZT 2 output setting, which is better than the previous effects of the vibration reduction in [31]. 10. Conclusions In this study, an adaptive vibration reduction method was proposed and implemented in a large-stroke and high-velocity macro-micro stage. The start-up condition and the extension principle of the vibration reduction element (PZT2) used in the method were defined and finalized. With these definitions, the proposed method can dynamically actuate the PZT 2 at right time and with appropriate extension by the realtime feedback of the stage. Based on the theoretical analysis of dynamic modeling, force analysis and amplitude reduction analysis, the control strategy of the macro-micro positioning stage was presented. The simulation results showed that the settling time of the macromotion can be effectively reduced by 54.7% (when Ce = 0.2). The experimental results showed that the settling time of the macromotion was reduced by 54.4% for a motion of 40 mm stroke, 0.2 m/s velocity and 40 m/s2 acceleration with a ±3 μm positioning accuracy. Through the simulation analysis and the experimental tests performed in the paper, we can see that the proposed method can effectively reduce the vibration of the macromotion and settle down the stage with much less time without decreasing its positioning accuracy. This method could be appropriate for the applications of large-stroke, high-velocity, high-acceleration motion with high positioning accuracy. Further work will be carried out on the system development of the vibration reduction device as a compact unit.

Acknowledgment This work was supported in part by the National Natural Science Foundation of China and the Guangdong Provincial Natural Science Foundation under Grant No. 51675106, No. U1601202 and No. 2015A030312008, and the Guangdong Provincial R&D Key Projects (No. 2015B010104008, No.2016A030308016 and No.2015A010104009).

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Fig. 1. Components of the macro-micro positioning stage.

Fig. 2. Extending principle of the PZT2 in the proposed vibration reduction method.

Fig. 3. PZT2 start-up conditions of the vibration reduction method.

Fig. 4. Vibration reduction procedure of the vibration reduction method.

Fig. 5. Dynamic model of the proposed vibration reduction method.

Fig. 6. Force analysis model of the proposed vibration reduction method.

Fig. 7. Control scheme of the positioning stage with the PZT2 vibration reduction (where up is the input displacement of the PZT2 element, and u is the input displacement of the positioning stage).

Fig. 8. Simulation results with different extensions of the PZT2.

Fig. 9. Types of the input waveform for the PZT2 action.

Fig. 10. Simulation results with different waveform of the PZT2. (Ce = 0.2)

Fig. 11. Simulation effects of the vibration reduction method for different strokes. (Velocity = 0.2 m/s; Acceleration = 20 m/s2; Ce = 0.2)

Fig. 12. Simulation effects of the vibration reduction method for different velocities. (Stroke = 40 mm; Acceleration = 40 m/s2; Ce = 0.2)

Fig. 13. Experimental setup of the designed macro-micro precision positioning stage.

Fig. 14. Vibration reduction effects with different PZT2 extensions.

Fig. 15. Relationship of the settling time with the vibration reduction coefficient Ce of the PZT2.

Fig. 16. Vibration reduction effects with different PZT2 waveform.

Fig. 17. Effects of the vibration reduction for different strokes. (Velocity = 0.2 m/s; Acceleration = 20 m/s2)

Fig. 18. Effects of the vibration reduction for different velocities. (Stroke = 40 mm; Acceleration = 40 m/s2)

Table 1 Parameter values of the dynamic model. Item

M1 (kg)

K1 (N/μm)

C1 (Ns/m)

M2 (kg)

K2 (N/μm)

C2 (Ns/m)

Value

0.8

65

0.2

0.6

150

0.2

Table 2 Parameters of the platform and the PZT elements. Item

Value/Material

Mass of voice coil motor axis (kg) Collective mass of specialized structure (kg) Resolution of VCM (μm) Repeatability positioning accuracy of VCM (μm) Size of platform (length × width × height) (mm3) Size of spring (wire × diameter × length) (mm3) Stiffness of spring (N/μm) Material of spring Material of platform Maximum/nominal stroke of PZT1 (μm) Stiffness of PZT1 (N/μm) Resonance frequency of PZT1 (kHz) Length/diameter of PZT1 (mm) Maximum/nominal stroke of PZT2 (μm) Stiffness of PZT2 (N/μm) Nominal pull/trust force of PZT2 (N) Electrostatic of PZT2 (μF) Resonance frequency of PZT2 (kHz) Length/diameter of PZT2 (mm) Mass of PZT2 (kg) Linearity of PZT2 (%) Maximum/nominal voltage of PZT2 (V)

0.8 0.6 0.1 ±3 120 × 70 × 8 2 × 17 × 25 150 Spring steel Iron 15/9 120 ± 20% 40 19/12 55/40 60 ± 20% 400/3500 7.2 ± 20% 20 46/15 0.21 0.1 -30-150/0-150

Table 3 Settling time reduction by different PZT2 output extensions. Settling time (mean value) PZT2 extension coefficient Ce

Without vibration reduction (ms)

With vibration reduction (ms)

Improvement (%)

Experiment

Simulation

Experiment

Simulation

Experiment

Simulation

0.8 0.6 0.4

66.2 66.2 66.2

68.9 68.9 68.9

44.2 41.3 38.6

44.9 43.7 38.8

33.2 37.6 41.7

34.8 36.6 43.7

0.2

66.2

68.9

32.1

33.4

51.5

51.5

Table 4 Settling time reduction by different PZT2 output waveform. PZT2 waveform

Settling time (Mean value) Without vibration reduction (ms)

With vibration reduction (ms)

Improvement (%)

Zigzag Triangle Sin

66.2 66.2 66.2

44.1 42.3 32.1

33.4 36.1 51.5

Rectangle

66.2

30.2

54.4

Table 5 Settling time reduction at different strokes. Strokes of the stage （ mm） 10 20 30

Settling time (mean value)

40

Without vibration reduction (ms)

With vibration reduction (ms)

Improvement (%)

65.4 66.3 63.9

30.5 29.8 30.6

53.4 55.1 52.1

Previous performance In Ref [31] (%) 46.6 46.8 48.1

64.3

30.9

52

48.2

Table 6 Settling time reduction at different velocities. Velocity Settling time (mean value) of the stage （ Without vibration With vibration reduction (ms) reduction (ms) m/s） 0.05 44.4 23.9 0.1 50.6 30.2 0.15 51.3 25.2

Improvement (%)

Previous performance In Ref [31] (%)

41.7 40.3 51

35.1 13.1 9.1

0.2

54.4

46.5

66.2

30.2