Design Actuator of Piezoelectric Positioning System Based on The Feedback of Strain Gauge

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1 Design Actuator of Piezoelectric Positioning System Based on The Feedback of Strain Gauge Wei Wang 1,2, Yu Bao 2, Li Niu 2, Guoyu Zhang 1, * 1. The School of Photo-Electronic Engineering, hangchun University of Science and Technology, hangchun, hina 2. hangchun Institute of Applied hemistry, hinese Academy of Sciences, hangchun, hina *orresponding author( zh_guoyu@13.com) Abstract Nano positioning technology is one of the key technologies of nanometer-measurement and nano-machining. In the field of nano-positioning technology, a piezoelectric ceramic positioning system is the most widely used positioning element. The principle was analyzed of output displacement measuring for piezoelectric ceramic by strain gauge sensor. According to the load characteristics of the piezoelectric ceramic and the output signal properties of the strain gauge, the error amplification high-voltage driving circuit and the strain gauge signal detection circuit were designed. It should be pointed out that a PID controller for precise positioning control of piezoelectric positioning system was designed and improved. Under the open loop control, the average hysteresis of the piezoelectric ceramic positioning system is 8.5%, the maximum absolute error of the sine wave tracking is over 1μm, and the positioning error exceeds.5μm. Moreover, the value of above three parameter were lower than.5%,.3 μm, and 5 nm by improving the PID control algorithm. Therefore, the designed actuator can be used to eliminate the effect on positioning accuracy caused by the nonlinearity, hysteresis and creep of the piezoelectric ceramics, exhibiting good trajectory tracking performance. Key words: Nano-Positioning, Strain Gauge Sensors, Piezoelectric eramics, PID ontrol. 1. INTODUTION Nano positioning technology plays an important role in nanotechnology, which is not only the origin and foundation of nanotechnology, but also one of the main difficulties in developing nanotechnology. In the nano positioning technology, the piezoelectric ceramic positioning system is most widely used. Piezoelectric ceramic is a kind of piezoelectric poly-crystal, which is named for its producing process is similar to the ceramic. Because of the converse piezoelectric effect, the voltage applied to piezoelectric ceramics can be transformed into tiny mechanical displacement with the advantages of high positioning accuracy, high displacement resolution, fast response and low heating, thus it has been widely used in nano positioning even sub nanometer positioning, becoming the very ideal driving element in nano positioning technology at present(xu, 215; Mohammadzaheri and Alqallaf, 217; hen and Su, 21). But in practical applications, the piezoelectric ceramic output is not ideal with the disadvantages of nonlinearity, hysteresis and creep properties for the influence of polarization mechanism and the electromechanical coupled effect. Non-linearity refers to the relationship between voltages applied to both ends of the piezoelectric ceramic and its displacement is not strictly linear; hysteresis refers to the displacement difference between the step-up and step-down curves of piezoelectric ceramic, namely, there is a significant difference in the displacement between the step-up curve and the step-down curve at the same voltage, which varies with the driving voltage and generally can reach more than 1% of the whole displacement; creep properties indicate the output displacement of piezoelectric ceramics is not fixed at a certain value, but changes slowly with the time to become stable after a certain time when the voltages applied to both ends of the piezoelectric ceramic don t change(parashar and Kumar, 215; Kang and Park, 215; Yong and Fleming, 217). In order to eliminate the disadvantages of piezoelectric ceramic, it is necessary to eliminate the positioning error of the piezoelectric ceramic positioning system by precise positioning control. The control methods generally can be divided into open-loop control and closed loop control. The open-loop control method mainly describes the hysteresis behavior of the piezoelectric positioning system by establishing mathematical models of the hysteresis non-linearity such as PI model, Maxwell model, Bouc-Wen model and so on, which uses the inverse model feed-forward control to compensate for the voltage to reduce the hysteresis of piezoelectric positioning system(nasir and Mukhtar, 21; Habineza and akotondrabe, 215). The open loop control is of fast response speed, but the control accuracy is low for the limitation of model and the accuracy of parameter identification. In order to improve the positioning accuracy of the piezoelectric positioning system, closed loop control method is used to achieve the precise positioning control. Fuzzy control, PID control, adaptive control, robust control and neural network control are mainly used closed loop control methods, among which the PID 27

2 control is of simple algorithm, mature technology and easy implement in engineering and widely used in practical engineering application(maithripala and Berg, 215). 2. EXPEIMENTAL POEDUE 2.1. Measuring output displacement of piezoelectric ceramic by strain gauge sensor apacitive sensors and strain gauge sensors (SGS) are two main kinds of micro displacement sensors used commonly. apacitance sensor is of high precision, high resolution, good linearity, whose resolution can even reach tens of picometers, while it is of expensive price, high requirement for the acquisition circuit and strict control of noise, and it is easy to be affected by changes in the external environment and parasitic capacitance parameters in application. Strain gauge is a sensing element used to measure mechanical deformation of the object. Because the strain gauge is very thin, it can be affixed directly to the surface of the object by adhesive, thus the strain gauge deforms with the mechanical deformation of the measured object, which results in resistance changes, then the deformation of the measured object can be obtained by measuring changes in the resistance. The resistance strain gauge is widely used to measure the output displacement of the piezoelectric ceramic in many occasions due to its mature processing technology, good stability, fast response speed and high resolution. The resistance value of the strain gauge is not measured directly for its small span, and the Whiston bridge structure is used to enlarge its dynamic range(almir and Samir, 215; Meng and Fan, 217)., as shown in Figure 1, where V EF is the voltage reference provide the bridge circuit with a high precision reference voltage and low temperature drift. In the bridge circuit structure, if there are 2 strain gauges and 2 fixed resistors among 1 to 4, it will be the half bridge structure; if all 4 are resistance strain gauges, it will be a full bridge structure, in which changes of temperature characteristics of strain gauges will offset each other, and it s a major advantage of the full bridge structure. Figure 1. Measurement of output displacement of piezoelectric ceramic by SGS Full bridge structure is composed of four strain gauges to measure the single direction output displacement of the piezoelectric ceramic, as shown in Figure 1. Four strain gauges are pasted on the two non-working surfaces of the symmetrical piezoelectric ceramic in two groups with 2 in each group; two strain gauges pasted on the same surface and a sensitive grid are parallel to the direction of deformation of piezoelectric ceramics, another is vertical to the direction of deformation. From Figure 1, the output voltage of the bridge circuit can be calculated as, 3 2 u VEF VEF ( 3 4 ) ( 1 2 ) When the piezoelectric ceramic positioning system deforms in the direction shown in Figure 1, the resistance values of 1, 2, 3, and 4 will change, among which 1 1, 2 2, 3 3, 4 4. It can be concluded that for the parameters of the 4 strain gauges are basically the same, thus formula (1) can be further deduced: u. u VEF (2) Thus, the output displacement of the piezoelectric ceramic is proportional to the output of the bridge circuit (1) 277

3 2.2. Design of driver hardware circuit and PID controller High voltage amplifier circuit design The performance of the high voltage amplifier circuit determines the resolution and stability of the output displacement of the piezoelectric ceramic, which is the key part of accurate positioning in piezoelectric positioning system. When the piezoelectric ceramic working under the inverse piezoelectric effect, its electrical load characteristics are similar to load capacitance; as the leakage current is very small in the voltage range, the load of the driving circuit can be regarded as a pure capacitance. Based on this, this paper designs an error amplifier circuit structure, as shown in Figure 2. The advantage of this driving circuit structure is the feedback voltage taken directly from the load of piezoelectric ceramics, which enhances the ability to control the output power and makes the linearity and ripple of the output voltage more ideal. Error amplifier stage Power amplifier stage 2V Q 2 DA Q 1 1 D PZT Figure 2. Scheme of the error amplifier circuit In order to achieve high precision linear and power amplification of the voltage, the two-stage amplifier circuit is in need. In the front stage, the error amplifying circuit composed of low-voltage precision operational amplifier is used to obtain the smaller input offset voltage and higher bandwidth to achieve high precision linear amplification. In the latter stage, a power amplifying circuit composed of discrete components is used to obtain higher output voltage and larger output current, which enhances the load capacity. The advantages of the twostage amplifier circuit structure are high accuracy achieved by the output voltage sampled and fed back without error; the input offset voltage of the whole amplification circuit is determined by a preamplifier with small input offset voltage, which is very suitable for the design of piezoelectric ceramic driving circuit with single power supply Strain gauge signal detection circuit design Since the output voltage signal of strain gauge in full bridge circuit Δu is relatively weak, generally at the mv level, it should be amplified before the A/D conversion, which can significantly improve the signal-to-noise ratio and reduce the noise jamming of a weak signal before the A/D conversion. After being amplified, Δu needs to transform signal to reach or be close to the full-scale input range of AD, finally outputs to the A/D converter through low pass filter. It is used to eliminate the high frequency signal leads to the aliasing of A/D converter and filter peak noise in analog signal to improve the resolution of displacement detection, as the schematic diagram of the detection circuit shown in Figure 3. VIN VIN Signal amplification Signal transformation EF Signal filtering VOUT Figure 3. Schematic diagram of strain gauge signal detection circuit PID controller design PID is one of the most widely used control method in engineering, and it can obtain good control effect without knowing the mathematical model of the controlled object, which is of mature technology, high reliability, simple programming and convenient parameter adjustment. The discrete PID controller can be expressed as: k 1 u( k) KP e( k) e( i) TD e( k) e( k 1) T (3) I i 278

4 Where e(k) is the deviation between the sampling value and the a given value at the kth sampling time, u(k) is the output of the PID controller at the kth sampling time, K P, T I and T D are proportional coefficient, integral and differential time constant respectively(maithripala and Berg, 215; Lee and ho, 21). The PID controller needs to be improved accordingly in order to make it a good control effect with fast response, small overshoot and high steady state precision, which can make the piezoelectric positioning system of good static and dynamic characteristics. First, in order to improve the steady-state positioning precision of piezoelectric positioning system, the discrete rectangular integral of PID controller should be changed into parabola integral more close to the true value. According to Simpson parabola integral formula, the integral of the parabola integral element can be obtained, t 3 t3 t1 ( k 1) ( k 1) ( ) ( 1) 4 ( 2) ( 3 e t dt ) ( 1) 4 ( ) ( 1) t1 e t e t e t e k e k e k 1 e ( k 1) 4 e ( k ) e ( k 1) 3 Where t 1 and t 3 is the (k-1)th and the (k1)th sampling time respectively. From formula (3) and (4), the integral for improved PID controller is, K K 1 ui( k) e( i) e( k 1) 4 e( k) e( k 1) K e( k 1) 4 e( k) e( k 1) T k k k P P I I i TI i1 3 i1 (5) Where K I=K P/(3T I) Second, improve the differential term in PID controller. In order to achieve fast and stable positioning, the whole control process requires no vibration and overshoot due to the given value of the piezoelectric positioning system may change frequently during the work. In the formula (3), the differential term is given by the rate of variation of the deviation. In order to make the differential link accelerate the response of the system without being too sensitive to the interference, differential separation method is adopted to improve the differential term in the PID controller. When the deviation is larger, it s necessary to reserve the differential term to accelerate the response of the system, while it s necessary to remove the differential term when the deviation is smaller, in order to prevent the differential link from being too sensitive to the deviation and reduce the overshoot so as to avoid the system instability. The differential term for improved PID controller is, ud( k) KD e( k) e( k 1) () Where K D=K PT D; is the coefficient for the separation of differential, and (4) e( k) 1 e( k) (7) Where is the threshold for deviation. The improved PID controller can be expressed as, k u( k) KPe( k) K e( k 1) 4 e( k) e( k 1) KD e( k) e( k 1) (8) I i esults and discussion Block diagram of the control system is shown in Figure 4, and the PID control algorithm is written in language into AM controller to implement. In this paper, the experimental object is the three-dimensional piezoelectric ceramic positioning system type XP-11.XYZ, produced by Harbin core tomorrow technology o., Ltd. Photo of the piezoelectric ceramic positioning system and actuator is shown in Figure 5. Figure 4. Block diagram of the control system 279

5 Figure 5. Photo of three dimensional piezoelectric ceramic positioning system and actuator Step response test onduct step response test on the piezoelectric ceramic positioning system in open-loop control and closed-loop control respectively, with the step amplitude is 5 μm, and the test curves shown in Figure (a) and (b). It can be seen that there is overshoot and and steady-state error about.2 μm in the piezoelectric positioning system under open-loop control. The step response of the piezoelectric positioning system is slow under closed loop control without overshoot, and its steady-state error is essentially zero if not considering the noise effect (a) Open loop control (b) losed loop control Figure. Step response curve onstant displacement output test The constant displacement output curves of the piezoelectric positioning system under open-loop control and closed-loop control are shown in Figure 7 (a) and 7 (b) respectively. Under open-loop control, the displacement of the piezoelectric positioning system changes about.5 μm within 2s due to the existence of piezoelectric ceramic creep, and the output displacement tends to be stable as time increases. Under the closedloop control, the output displacement of piezoelectric positioning system is always constant at about 5 μm with the positioning error less than 5nm, which is achieved by adjusting the driving voltage at both ends of the piezoelectric ceramics in real time by the PID control algorithm according to the collected displacement signal Time/s Time/s (a) Open loop control (b) losed loop control Figure 7. onstant displacement output curve Step-up and step-down curve test The step-up and step-down curves of the piezoelectric positioning system under open-loop control and closed-loop control are shown in Figure 8 (a) and 7 (b) respectively. It can be seen from Figure 8 (a) that the 28

6 relationship between voltages applied to both ends of the piezoelectric ceramic and the output displacement of piezoelectric positioning system is not strictly linear in the open-loop control; there is a displacement difference between the step-up and step-down curves with the maximum near 1 μm and the average hysteresis is 8.5%. The step-up and step-down curves of the piezoelectric positioning system coincide basically under closed loop control, and the average hysteresis is less than.5%. 1 Step-up curve Step-down curve Step-up curve Step-down curve Voltage/V Voltage/V (a) Open loop control (b) losed loop control Figure 8. Step-up and step-down curve Sine wave signal tracking test In order to test the tracking performance of the piezoelectric positioning system, the sinusoidal signal with frequency of 5Hz, amplitude of 1 μm and offset of 4 μm is selected as the target signal, which is expressed as y (4 1sin(1 t)) μm. The sine wave tracking test curves under open-loop control and closed-loop control are shown in Figure 9 and 1 respectively. It can be seen from Figure 9 (a) that there is a small phase shift between the response signal of piezoelectric positioning system and the target signal under the open loop control, which is caused by the creep and hysteresis of the piezoelectric ceramics. The maximum tracking error can reach ±1μm in the open-loop control, and the tracking error tends to decrease due to the creep characteristics of the piezoelectric ceramics. Under closed loop control, the response signal of piezoelectric positioning system is capable of tracking the target signal with no phase shift between them, and the tracking error is within.3 μm (a) Target signal and output response (b) Tracking error Figure 9. Sine wave tracking test under open loop control (a) Target signal and output response (b) Tracking error Figure 1. Sine wave tracking test under closed loop control 281

7 3. ONLUSIONS In order to satisfy the the positioning demand with high precision of piezoelectric positioning system in engineering application, the high-voltage driving error amplifier circuit and the strain gauge signal detection circuit are designed according to the load characteristic of piezoelectric ceramics and the output signal characteristics of strain gauge, and the PID controller is designed and improved for precision positioning in piezoelectric positioning system. The results show that the piezoelectric positioning system is of serious nonlinearity, hysteresis and creep under the open loop control, which will cause great error in applying the precision positioning of piezoelectric positioning system. The above problems.can be effectively solved by using the improved PID controller for precision positioning of piezoelectric ceramic positioning control system. In the improved PID control algorithm, the steady-state error is zero basically, the positioning system hysteresis is less than.5%, the positioning error is less than 5nm, the measurement results of the positioning accuracy and the parameters of the repetitive positioning precision of piezoelectric ceramic are consistent, and the maximum of the sine wave tracking absolute error is less than.3 μm with good trajectory tracking performance. Acknowledgements This work was supported by the National Natural Science Foundation of hina (Grant: ). EFEENES Xu, Q.S. (215) Digital Sliding Mode Prediction ontrol of Piezoelectric Micro/Nanopositioning System, IEEE Transactions on ontrol Systems Technology, 23(1), pp Mohammadzaheri, M., Alqallaf, A. (217) Nanopositioning Systems with Piezoelectric Actuators, urrent State and Future Perspective, Science of Advanced Materials, 9(7), pp hen, X.K., Su,.Y., Li, Z., Yang, F. (21) Design of Implementable Adaptive ontrol for Micro/Nano Positioning System Driven by Piezoelectric Actuator, IEEE Transactions on Industrial Electronics, 3(1), pp Parashar, S.K., Kumar, K. (215) Three-dimensional analytical modeling of vibration behavior of piezoceramic cylindrical shells, Archive of Applied Mechanics, 85(5), pp.41-5 Kang, Y.K., Park, H.., Agrawal, B. (215) Optimization of Piezoceramic Sensor/Actuator Placement for Vibration ontrol of Laminated Plates, Aiaa Journal, 3(9), pp Yong, Y.K., Fleming, A.J. (217) Note: An improved low-frequency correction technique for piezoelectric force sensors in high-speed nanopositioning systems, eview of Scientific Instruments, 88(4), pp.5-5 Nasir,., Mukhtar, H., Man, Z. (21) Prediction of gas transport across amine mixed matrix membranes with ideal morphologies based on the Maxwell model, sc Advance, (3), pp Habineza, D., akotondrabe, M., Gorrec, Y.L. (215) Bouc Wen Modeling and Feedforward ontrol of Multivariable Hysteresis in Piezoelectric Systems: Application to a 3-DoF Piezotube Scanner, IEEE Transactions on ontrol Systems Technology, 23(5), pp Almir, T., Samir,., oman, B., Franz, K., Thilo, S., Franz, K. (215) MEMS Flow Sensors Based on Self-Heated age-thermistors in a Wheatstone Bridge, 15(5), pp Meng, L.J., Fan, S.., Mahpeykar, S.M., Wang, X.H. (217) Digital microelectromechanical sensor with an engineered polydimethylsiloxane (PDMS) bridge structure, Nanoscale, 9(3), pp Maithripala, D.H.S., Berg, J.M. (215) An intrinsic PID controller for mechanical systems on Lie groups, Automatica, 54(), pp Lee, J., ho, W., Edgar T.F. (21) An improved technique for PID controller tuning from closed-loop tests, Aiche Journal, 3(12), pp