# Positioning Error Analysis and Control of a Piezo-Driven 6-DOF Micro-Positioner

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Positioning Error Analysis

#### 2.1. Mechanism Description

^{3}. The micro-positioner is driven by PZTs and measured by capacitive displacement sensors to ensure fast response and high precision. It consists of a working platform and a compliant mechanism that is used to amplify and transmit the motion of PZTs via elastic deformations of the right angle flexure hinges. The compliant mechanism is composed of three parts, the top platform, the middle platform and the bottom platform, which are shown in Figure 1b–d, respectively.

#### 2.2. Modeling Positioning Error

## 3. Positioning Error Compensation Design

#### 3.1. System Identification

#### 3.2. Feedforward Compensation

#### 3.3. Feedforward Plus Feedback Compensation

## 4. Simulation, Experiment and Discussion

#### 4.1. Experimental Setup

_{2}Mn, SHANGHAI DIHUA METAL MATERIAL, Inc., Shanghai, China) by Wire cut Electric Discharge Machining (WEDM) technology. PZTs (model Pst-40VS15 produced by COREMORROW, Inc., Harbin, China) are adopted to drive the micro-positioner, and PZTs are actuated with a voltage of 0–120 V through the PZT controller (XE-500/501 series modular controller, AYP Nano Solutions, Inc., Anaheim, CA, USA), deriving the maximum displacement of 40 μm. The output displacement of micro-positioner is measured by capacitive displacement sensors (model CS5, from MICRO-EPSILION, Inc., Raleigh, NC, USA) with a 100 nm resolution and a 5 mm measurement range, and the signals of the sensors are sent to the host computer through the sensor controller (model RS6500, MICRO-EPSILION, Inc., Raleigh, NC, USA). The hardware connection between the instruments is illustrated in Figure 5b.

#### 4.2. Experiments of Positioning Error

#### 4.3. Simulated Testing of Error Compensation

#### 4.4. Positioning Error Compensation Experiment

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Notations

Notation | Explanation |

$\mathsf{\delta}{\Delta}_{out}$ | Positioning error of the micro-positioner |

${\mathsf{\epsilon}}_{p}$ | Positioning error of the driving stage |

${\mathsf{\epsilon}}_{s}$ | Positioning error of the machining stage |

${\mathsf{\epsilon}}_{c}$ | Positioning error of the measuring stage |

$w\left(=x,y,z,\alpha ,\beta ,\gamma \right)$ | 6-DOF of the micro-positioner |

$v(=b,m,t)$ | Top, middle and bottom platform of the compliant mechanism |

${f}_{w}$ | Implicit functions of motion |

${w}_{out}$ | Output displacement of micro-positioner |

${w}_{pzt}$ | Load output displacement of PZTs |

${w}_{pzt0}$ | No-load output displacement of PZTs |

$\delta {w}_{pzt}$ | Driving error of PZTs |

$\delta {w}_{cp}$ | Measuring error of the micro-positioner |

$\delta {w}_{out}$ | Positioning error of the micro-positioner |

$\delta {d}_{wi}$ | Machining error of the geometric parameters |

$\varsigma $ | Ideal linear ratio of the output displacement to the input voltage of PZTs |

$H\left({u}_{pzt}^{w}\right)$ | Hysteresis nonlinear model of PZTs |

${p}_{pzt}^{w}\left(\right)$ | Preisach function |

$\mu \left(\right)$ | Weighting function |

${\widehat{\gamma}}_{\alpha \beta}$ | Hysteresis operator |

$\widehat{\alpha}$,$\widehat{\beta}$ | Up and down switching value of the input voltage |

${u}_{pzt}^{w}$ | Input voltage of PZTs |

${u}_{in}^{w}$ | External voltage supply of PZT controller |

${m}_{w}$ | Equivalent mass of compliant mechanism |

$c$ | Equivalent damping coefficient of compliant mechanism |

${k}_{w}$ | Equivalent stiffness of compliant mechanism |

${k}_{in}^{v}$ | Input stiffness of compliant mechanism |

${k}_{pzt}$ | Stiffness of PZT |

${F}_{pzt}^{w}$ | Driving force of PZTs |

$q$ | Electric charge of PZTs |

${F}_{in}^{w}$ | Input driving force of compliant mechanism |

$n$ | Number of piezoelectric ceramics of PZTs |

${h}_{c}$ | Amplification ratio of PZT controller |

${R}_{c}$ | Equivalent resistance of PZT controller |

${C}_{c}$ | Equivalent capacitance of PZT controller |

${G}_{pzt}^{w}\left(s\right)$ | Transfer function of PZT |

${G}_{stage}^{w}\left(s\right)$ | Transfer function of the stage |

${G}_{w}\left(s\right)$ | Transfer function of the entire system |

${w}_{\mathrm{pzt}}^{\mathrm{d}}$ | Expected output displacement |

$\lambda $ | Displacement amplification ratio of the compliant mechanism |

${w}_{pzt}^{p}$ | Load output displacement of PZTs of PID controller |

$t$ | Time variable |

${K}_{p}$ | Proportional gain of PID controller |

${T}_{i}$ | Integral time of PID controller |

${T}_{d}$ | Derivative time of PID controller |

$\mathsf{\theta}$ | Input matrix of BP-PID controller |

$f\left(\mathsf{\theta}\right)$ | Output matrix of BP-PID controller |

${W}^{(1)}$ | Weight matrix of the hidden layer |

${W}^{(2)}$ | Weight matrix of the output layer |

${\mathsf{\rho}}^{(1)}$ | Threshold matrix of the hidden layer |

${\mathsf{\rho}}^{(2)}$ | Threshold matrix of the output layer |

$\phi \left(w\right)$ | Activation functions of the hidden layer |

$g\left(w\right)$ | Activation functions of the output layer |

${a}_{i}$ | Binary selection vector of CMAC neural network |

${w}_{pzt}^{n}$ | Load output displacement of PZTs of CMAC neural network |

$\chi $ | Generalization parameter of CMAC neural network |

$E$ | Performance indicator function |

${e}_{loop}$ | The maximum hysteresis error |

${e}_{\mathit{max}}$ | The maximum positioning error |

${e}_{rms}$ | The root-mean-square error |

## References

- Aktakka, E.E.; Woo, J.K.; Egert, D.; Gordenker, R.J.M.; Najafi, K. A microactuation and sensing platform with active lockdown for in situ calibration of scale factor drifts in dual-axis gyroscopes. IEEE/ASME Trans. Mechatron.
**2014**, 20, 934–943. [Google Scholar] [CrossRef] - Thoma, F.; Goldschmidtböing, F.; Woias, P. A new concept of a drug delivery system with improved precision and patient safety features. Micromachines
**2015**, 6, 80–95. [Google Scholar] [CrossRef] - Putra, A.S.; Huang, S.; Tan, K.K.; Panda, S.K.; Lee, T.H. Design, modeling, and control of piezoelectric actuators for intracytoplasmic sperm injection. IEEE Trans. Control Syst. Technol.
**2007**, 15, 879–890. [Google Scholar] [CrossRef] - Habibullah, H.; Pota, H.R.; Petersen, I.R. A novel control approach for high-precision positioning of a piezoelectric tube scanner. IEEE Trans. Autom. Sci. Eng.
**2017**, 14, 325–336. [Google Scholar] [CrossRef] - Linß, S.; Schorr, P.; Zentner, L. General design equations for the rotational stiffness, maximal angular deflection and rotational precision of various notch flexure hinges. Mech. Sci.
**2017**, 8, 29–49. [Google Scholar] [Green Version] - Tseytlin, Y.M. Notch flexure hinges: An effective theory. Rev. Sci. Instrum.
**2002**, 73, 3363–3368. [Google Scholar] [CrossRef] - Zelenika, S.; Munteanu, M.G.; De Bona, F. Optimized flexural hinge shapes for microsystems and high-precision applications. Mech. Mach. Theory
**2009**, 44, 1826–1839. [Google Scholar] [CrossRef] - Valentini, P.P.; Pennestrì, E. Elasto-kinematic comparison of flexure hinges undergoing large displacement. Mech. Mach. Theory
**2017**, 110, 50–60. [Google Scholar] [CrossRef] - Zhu, W.L.; Zhu, Z.; Guo, P.; Ju, B.F. A novel hybrid actuation mechanism based XY nanopositioning stage with totally decoupled kinematics. Mech. Syst. Sig. Process.
**2018**, 99, 747–759. [Google Scholar] [CrossRef] - Kim, H.; Kim, J.; Ahn, D.; Gweon, D. Development of a nanoprecision 3-DOF vertical positioning system with a flexure hinge. IEEE Trans. Nanotechnol.
**2013**, 122, 234–245. [Google Scholar] [CrossRef] - Cai, K.; Tian, Y.; Liu, X.; Fatikow, S.; Wang, F.; Cui, L.; Zhang, D. Shirinzadeh, B. Modeling and controller design of a 6-DOF precision positioning system. Mech. Syst. Sig. Process.
**2018**, 104, 536–555. [Google Scholar] [CrossRef] - Lin, C.; Shen, Z.; Wu, Z.; Yu, J. Kinematic characteristic analysis of a micro-/nano positioning stage based on bridge-type amplifier. Sens. Actuators A
**2018**, 271, 230–242. [Google Scholar] [CrossRef] - Shinno, H.; Yoshioka, H.; Taniguchi, K. A newly developed linear motor-driven aerostatic x-y planar motion table system for nano-machining. CIRP Ann. Manuf. Technol.
**2007**, 56, 369–372. [Google Scholar] [CrossRef] - Zeng, Q.; Ehmann, K.F. Error modeling of a parallel wedge precision positioning stage. J. Manuf. Sci. Eng.
**2012**, 134, 061005. [Google Scholar] [CrossRef] - Kang, R.; Sheng, X.; Wang, K. Volumetric error modelling, measurement, and compensation for an integrated measurement-processing machine tool. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci.
**2010**, 224, 2477–2486. [Google Scholar] - Cui, H.; Zhu, Z.; Gan, Z.; Brogardh, T. Kinematic analysis and error modeling of TAU parallel robot. Rob. Comput. Integr. Manuf.
**2005**, 21, 497–505. [Google Scholar] [CrossRef] - Gatti, G.; Danieli, G. A practical approach to compensate for geometric errors in measuring arms: Application to a six-degree-of-freedom kinematic structure. Meas. Sci. Technol.
**2007**, 19, 015107. [Google Scholar] [CrossRef] - Cui, C.; Feng, Q.; Kuang, C.; Cuifang, K.; Yusheng, Z.; Fenglin, Y. Development of a simple system for simultaneously measuring 6dof geometric motion errors of a linear guide. Opt. Express
**2013**, 21, 25805–25819. [Google Scholar] - Zhang, B.; Cui, C.; Feng, Q.; Cui, C.; Chen, S.; Zhao, Y. Errors crosstalk analysis and compensation in the simultaneous measuring system for five-degree-of-freedom geometric error. Appl. Opt.
**2015**, 54, 458–466. [Google Scholar] - Lee, J.I.; Huang, X.; Chu, P.B. Nanoprecision mems capacitive sensor for linear and rotational positioning. J. Microelectromech. Syst.
**2009**, 18, 660–670. [Google Scholar] [CrossRef] - Ge, P.; Jouaneh, M. Generalized preisach model for hysteresis nonlinearity of piezoceramic actuators. Precis. Eng.
**1997**, 20, 99–111. [Google Scholar] [CrossRef] - Oh, J.H.; Bernstein, D.S. Semilinear duhem model for rate-independent and rate-dependent hysteresis. IEEE Trans. Autom. Control
**2005**, 50, 631–645. [Google Scholar] - Goldfarb, M.; Celanovic, N. Modeling piezoelectric stack actuators for control of micromanipulation. IEEE Control Syst. Mag.
**1997**, 17, 69–79. [Google Scholar] - Xu, Q.; Li, Y. Dahl model-based hysteresis compensation and precise positioning control of an XY parallel micromanipulator with piezoelectric actuation. J. Dyn. Syst. Meas. Contr.
**2010**, 132, 558–564. [Google Scholar] [CrossRef] - Gu, G.Y.; Yang, M.J.; Zhu, L.M. Real-time inverse hysteresis compensation of piezoelectric actuators with a modified prandtl-ishlinskii model. Rev. Sci. Instrum.
**2012**, 83, 65–83. [Google Scholar] [CrossRef] - Wang, X.; Pommier-Budinger, V.; Reysset, A.; Gourinata, Y. Simultaneous compensation of hysteresis and creep in a single piezoelectric actuator by open-loop control for quasi-static space active optics applications. Control Eng. Pract.
**2014**, 33, 48–62. [Google Scholar] [CrossRef] - Ru, C.; Sun, L. Hysteresis and creep compensation for piezoelectric actuator in open-loop operation. Sens. Actuators A
**2005**, 122, 124–130. [Google Scholar] - Yen, P.L.; Yan, M.T.; Chen, Y. Hysteresis compensation and adaptive controller design for a piezoceramic actuator system in atomic force microscopy. Asian J. Control
**2012**, 14, 1012–1027. [Google Scholar] [CrossRef] - Nguyen, P.B.; Sohn, J.W.; Choi, S.B. Position tracking control of a flexible beam using a piezoceramic actuator with a hysteretic compensator. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci.
**2010**, 224, 2141–2153. [Google Scholar] [CrossRef] - Sebastian, A.; Salapaka, S.M. Design methodologies for robust nano-positioning. IEEE Trans. Control Syst. Technol.
**2005**, 13, 868–876. [Google Scholar] [CrossRef] - Rakotondrabe, M.; Haddab, Y.; Lutz, P. Quadrilateral modelling and robust control of a nonlinear piezoelectric cantilever. IEEE Trans. Control Syst. Technol.
**2009**, 17, 528–539. [Google Scholar] [CrossRef] - Li, J.; Yang, L. Finite-time terminal sliding mode tracking control for piezoelectric actuators. Abst. Appl. Anal.
**2014**, 3, 760937. [Google Scholar] [CrossRef] - Xu, Q. Enhanced discrete-time sliding mode strategy with application to piezoelectric actuator control. IET Control Theory Appl.
**2013**, 7, 2153–2163. [Google Scholar] [CrossRef] - Gu, G.Y.; Zhu, L.M. An experimental comparison of proportional-integral, sliding mode, and robust adaptive control for piezo-actuated nanopositioning stages. Rev. Sci. Instrum.
**2014**, 85, 055112. [Google Scholar] [CrossRef] [Green Version] - Qi, K.; Xiang, Y.; Fang, C.; Zhang, Y.; Yua, C. Analysis of the displacement amplification ratio of bridge-type mechanism. Mech. Mach. Theory
**2015**, 87, 45–56. [Google Scholar] [CrossRef] - Wei, Y.; Tao, H. Preisach model research on hysteresis characteristics of piezoelectric actuators. Piezoelectrics Acorstooptics
**2004**, 5, 364–367. (In Chinese). Available online: http://www.cqvip.com/qk/92382x/200405/10536930.html (accessed on 10 August 2019).

**Figure 1.**Schematic diagram of the six-degree-of-freedom (6-DOF) micro-positioner. (

**a**) The overall structure of the micro-positioner, (

**b**) the structure of the middle platfoem, (

**c**) the structure of the top platfoem, (

**d**) the structure of the bottom platfoem.

**Figure 2.**Schematic diagram of the stage: (

**a**) schematic of the bridge-type amplifier, (

**b**) schematic of translation in the X direction, (

**c**) schematic of translation in the Z direction, (

**d**) schematic of rotation around the X axis and (

**e**) schematic of rotation around the Z axis.

**Figure 5.**Schematic diagram of the 6-DOF micro-positioner: (

**a**) experimental setup, (

**b**) hardware connection.

**Figure 6.**Comparison of the hysteresis loop between the experimental results and the simulated results: (

**a**) hysteresis loop, (

**b**) hysteresis error.

**Figure 7.**Machining error results for (

**a**) moving along X/Y axis, (

**b**) moving along Z axis, (

**c**) rotating along X/Y axis, and (

**d**) rotating along Z axis.

**Figure 9.**Schematic diagram of the inverse Preisach model compensation: (

**a**) the hysteresis loop, (

**b**) the inverse hysteresis loop, (

**c**) the compensated output, and (

**d**) the compensated output error.

**Figure 11.**Experimental motion tracking results of micro-positioner for the translational motion along X/Y axis: (

**a**,

**b**) feedforward compensation, (

**c**,

**d**) feedback compensation.

**Figure 12.**Experimental tracking results of (

**a**,

**b**) the translation along Z axis, (

**c**,

**d**) the rotation along X/Y axis, (

**e**,

**f**) the rotation along Z axis.

Top Platform | Middle Platform | Bottom Platform | |||
---|---|---|---|---|---|

l_{6} (mm) | 2 | l_{4} (mm) | 9 | l_{1} (mm) | 7 |

t_{6} (mm) | 0.8 | t_{4} (mm) | 0.8 | t_{1} (mm) | 0.8 |

l_{a3} (mm) | 14 | l_{a2} (mm) | 9 | l_{a1} (mm) | 9 |

α_{3} (rad) | 0.185 | α_{2} (rad) | 0.061 | α_{1} (rad) | 0.068 |

l_{7} (mm) | 2 | l_{5} (mm) | 9 | l_{2} (mm) | 10 |

t_{7} (mm) | 0.8 | t_{5} (mm) | 0.8 | t_{2} (mm) | 1 |

R (mm) | 4 | h (mm) | 5 | l_{3} (mm) | 28 |

J (mm) | 4 | d (mm) | 10 | _ |

Performance | HC | HC+PID | HC+BP-PID | HC+CMAC-PID |
---|---|---|---|---|

Settling time(sec) | 0.84 | 0.68 | 0.46 | 0.11 |

Overshoot (%) | 66.12 | 0 | 0 | 1.39 |

Settling value(μm) | 6.429 | 6.488 | 6.542 | 6.601 |

Positioning error (%) | 2.78 | 1.89 | 1.07 | 0.18 |

Performance | OPEN | HC | HC+PID | HC+BP-PID | HC+CMAC-PID |
---|---|---|---|---|---|

${e}_{\mathit{max}}$(%) | 18.22 | 5.46 | 2.91 | 1.07 | 0.63 |

${e}_{rms}$(%) | 6.70 | 2.61 | 1.38 | 0.62 | 0.23 |

Freedom | Translation Along Z Axis | Rotation Around X/Y Axis | Rotation Around Z Axis | ||||||
---|---|---|---|---|---|---|---|---|---|

Performance | OPEN | HC | HC+CMAC-PID | OPEN | HC | HC+CMAC-PID | OPEN | HC | HC+CMAC-PID |

e_{max} (%) | 11.62 | 6.79 | 1.77 | 30.77 | 22.16 | 6.97 | 54.17 | 31.53 | 19.02 |

e_{rms} (%) | 50 | 3.30 | 0.95 | 16.10 | 10.33 | 3.06 | 25.34 | 16.82 | 7.32 |

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## Share and Cite

**MDPI and ACS Style**

Lin, C.; Zheng, S.; Li, P.; Shen, Z.; Wang, S.
Positioning Error Analysis and Control of a Piezo-Driven 6-DOF Micro-Positioner. *Micromachines* **2019**, *10*, 542.
https://doi.org/10.3390/mi10080542

**AMA Style**

Lin C, Zheng S, Li P, Shen Z, Wang S.
Positioning Error Analysis and Control of a Piezo-Driven 6-DOF Micro-Positioner. *Micromachines*. 2019; 10(8):542.
https://doi.org/10.3390/mi10080542

**Chicago/Turabian Style**

Lin, Chao, Shan Zheng, Pingyang Li, Zhonglei Shen, and Shuang Wang.
2019. "Positioning Error Analysis and Control of a Piezo-Driven 6-DOF Micro-Positioner" *Micromachines* 10, no. 8: 542.
https://doi.org/10.3390/mi10080542