An inverse Prandtl–Ishlinskii model based decoupling control methodology for a 3-DOF flexure-based mechanism

TLDR: In this article, a modified Prandtl-Ishlinskii (P-I) hysteresis model is developed to form the feedforward controller for a 3-DOF flexure-based mechanism.

Abstract: A modified Prandtl–Ishlinskii (P–I) hysteresis model is developed to form the feedforward controller for a 3-DOF flexure-based mechanism. To improve the control accuracy of the P–I hysteresis model, a hybrid structure that includes backlash operators, dead-zone operators and a cubic polynomial function is proposed. Both the rate-dependent hysteresis modeling and adaptive dead-zone thresholds selection method are investigated. System identification was used to obtain the parameters of the newly-developed hysteresis model. Closed-loop control was added to reduce the influence from external disturbances such as vibration and noise, leading to a combined feedforward/feedback control strategy. The cross-axis coupling motion of the 3-DOF flexure-based mechanism has been explored using the established controller. Accordingly, a decoupling feedforward/feedback controller is proposed and implemented to compensate the coupled motion of the moving platform. Experimental tests are reported to examine the tracking capability of the whole system and features of the controller. It is demonstrated that the proposed decoupling control methodology can distinctly reduce the coupling motion of the moving platform and thus improve the positioning accuracy and trajectory tracking capability.

Figures

Fig. 5. Block diagram of feedforwar/feedback controller in the y direction.

Fig. 6. The trajectory tracking of a sinusoidal signal in the y direction with feedforward controller.

Table 2. Parameter of decoupling algorithm

Fig. 10. The trajectory tracking of a sinusoidal signal and coupling error with decoupling feedforward/feedback controller: (a) trajectory tracking in the y direction and coupling error in x2 and x3; (b) trajectory tracking in the x direction and coupling error in y; (c) trajectory tracking in the θZ direction and coupling error in y.

Fig. 2. Block diagram of modified inverse P-I hysteresis model.

Fig. 4. Response time of three different inverse hysteresis models.

Full text

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Original citation:
Guo, Z., Tian, Y., Liu, Xianping, Shirinzadeh, B., Wang, F. and Zhang, D.. (2015) An
inverse Prandtl–Ishlinskii model based decoupling control methodology for a 3-DOF
flexure-based mechanism. Sensors and Actuators A : Physical, 230 . pp. 52-62
http://dx.doi.org/10.1016/j.sna.2015.04.018
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An inverse Prandtl-Ishlinskii model based decoupling control methodology for a
3-DOF flexure-based mechanism
Z. Guo
1
, Y. Tian
1
, C. Liu
1
, X. Liu
2
, B. Shirinzadeh
3
, F. Wang
1
, D. Zhang
1
1
Key Laboratory of Mechanism Theory and Equipment Design of Ministry of Education, Tianjin University,
Tianjin 300072, China
2
School of Engineering, University of Warwick, Coventry CV4 7AL, UK
3
Robotics and Mechatronics Research Laboratory, Department of Mechanical and Aerospace Engineering,
Monash University, VIC 3800, Australia
Abstract ： A modified Prandtl-Ishlinskii (P-I) hysteresis model is developed to form the
feedforward controller for a 3-DOF flexure-based mechanism. To improve the control accuracy of
the P-I hysteresis model, a hybrid structure that includes backlash operators, dead-zone operators
and a cubic polynomial function is proposed. Both the rate-dependent hysteresis modeling and
adaptive dead-zone thresholds selection method are investigated. System identification was used to
obtain the parameters of the newly-developed hysteresis model. Closed-loop control was added to
reduce the influence from external disturbances such as vibration and noise, leading to a combined
feedforward/feedback control strategy. The cross-axis coupling motion of the 3-DOF flexure-based
mechanism has been explored using the established controller. Accordingly, a decoupling
feedforward/feedback controller is proposed and implemented to compensate the coupled motion of
the moving platform. Experimental tests are reported to examine the tracking capability of the
whole system and features of the controller. It is demonstrated that the proposed decoupling control
methodology can distinctly reduce the coupling motion of the moving platform and thus improve
the positioning accuracy and trajectory tracking capability.
Keywords: decoupling control, flexure-based mechanism, P-I hysteresis model, piezoelectric
actuator.

1. Introduction
Micro/nano positioning is one of the key enabling techniques in the scientific and engineering
fields including AFM (Atomic Force Microscopy) [1], STM (Scanning Tunnel Microscopy) [2],
optical fiber alignment [3, 4], bio-micro-surgery [5, 6] and micro-assembly [7, 8]. In order to reach
the required high precision, the piezo-driven flexure-based mechanism has been widely used due to
the large output force and quick response [9-11]. However, the hysteresis of the piezoelectric
ceramic significantly reduces the positioning accuracy of the developed system, so a variety of
hysteresis models and control methodologies for the piezoelectric have been proposed. In addition,
electromagnetic-driven flexure stage starts to be designed in recent years [12-14], especially in the
requirement of large stroke, but electromagnetic drive such as voice coil motor usually owns small
output force, leading to a small stiffness of the stage and further a low resonant frequency. In
reference [15, 16], it is easy to find the mainstream modeling and control methods of piezo-actuated
nanopositioning stages, and in reference [17], a corresponding general skeleton on this is presented.
Among various hysteresis models such as Preisach model [18, 19], Bouc-Wen model [20, 21],
Prandtl-Ishlinskii (P-I) model [22, 23], the P-I hysteresis model attracts more attention for the
structural simplicity, easy model identification and analytical inverse. As a result, the P-I model is
selected to model the hysteresis and further form a feedforward-feedback controller to control the
motion of the platform.
Micro/nano planar positioning systems generally employ parallel kinematics mechanisms
because of their compact structure, high stiffness and ease of monolithic fabrication [24, 25].
However, a degree of motion coupling between axes is one disadvantage for many potential
applications. Although the coupling effects can be reduced to some extent through good mechanical
design and optimization, the influences of assembly and manufacturing tolerances on the different
axes will inevitably result in some residual coupling phenomena [26]. Further, the initial conditions
of the installed piezoelectric actuators will significantly affect the static and dynamic properties,
especially, the preload and the Hertzian contact between the piezoelectric actuator and the driving
point. Discrepancy between the kinematic chains will induce motion coupling in such mechanisms,
because the platform motion is determined by all of the kinematic chains in the parallel mechanism
and apparently single-axis motion may depend on several actuators. For example, the rotational
angle is implemented by driving three actuators simultaneously in the mechanisms described in [27,
28], two piezoelectric actuators are utilized to realize the rotational and translational displacement

in the mechanism described in [29]. Therefore, it remains important to develop a decoupling control
methodology for parallel planar positioning systems even if their mechanism is designed with
decoupling capability.
This paper proposes a modified inverse P-I hysteresis model, identified by a direct inverse
modelling approach [30], to form the feedforward controller and thus compensate the nonlinearity
of the piezoelectric actuators. In order to improve the accuracy of the hysteresis model without
increasing the structural complexity, a hybrid structure of the inverse P-I hysteresis model is
developed. It includes several serial connected backlash operators and dead-zone operators and a
parallel connected cubic polynomial input function. A proportional-integral controller is added to
form a feedforward/feedback controller to further improve the trajectory tracking performance of
the developed flexure-based mechanism. A decoupling methodology has been proposed and
constructed to reduce the coupling effects at the moving platform. The performance of the
developed controller has been validated using a number of experimental tests. The rest of this paper
is arranged as follows: Section II briefly describes the mechanical design and control system of the
developed 3-DOF flexure-based mechanism. The modified P-I hysteresis model and parameter
identification are provided in Section III. Section IV explores the decoupling feedforward/feedback
control method for the 3-DOF flexure-based mechanism. Section V then examines the trajectory
tracking capability of the developed controller. Finally, conclusions are drawn in Section VI.
2. 3-DOF flexure-based mechanism
A newly developed 3-DOF flexure-based micro/nano positioning mechanism is shown in Fig.
1(a). It is monolithically manufactured from an Aluminum Alloy T7075 plate using the Wire
(a) 3-DOF flexure-based mechanism
(b) Control system setup
Fig. 1. Developed 3-DOF flexure-based mechanism and control system.
PZT
1
PZT
2
PZT
3
Bolt

Electric Discharge Machining (WEDM). The moving platform is supported by flexure-hinge links
which are orthogonally arranged in the x and y directions. Two pairs of links are connected in
parallel in the x direction and one pair of links in the y direction. Thus, the platform can translate in
the x and y direction and rotate about the z axis. Three piezoelectric actuators are installed between
the base and the driving points, and ball contacts (modeled as Hertzian) are used to avoid imposing
bending force on the piezoelectric actuators (Model: AE 0505D18F, THORlabs Company USA).
The preload is set by adjusting the bolt behind the actuator. The nominal maximal displacement of
these specific actuators is 15 µm at the driving voltage of 100 V. After assemble the piezoelectric,
the plane displacement of the flexure-based mechanism can reach 12.74 µm, 12.22 µm in the x and
y direction, respectively, the maximal rotation angle in clockwise and anticlockwise are 0.0088° and
0.0103°, respectively, the first resonant frequency is 790 HZ in the θ
Z
direction. More detailed
information about the positioning mechanism can reference our previous work [31]. The actuators
are driven by a piezoelectric amplifier (E-505.00, PI, Germany) that receives command signals from
the I/O interface of a dSPACE DS1103 R&D control board, on which the newly developed control
methodology is implemented with a sampling rate of 10 kHz. Three laser displacement sensors
(LK-H050, Keyence, Japan) provide real time displacement sensing and measurement for the
moving platform. Two of them are arranged in parallel in the x direction, allowing to translation in
the x direction and rotation about the z axis be measured. The third one measures the platform
displacement in the y direction. In order to reduce the influences of external disturbances, the whole
system is mounted on a Newport RS-4000 optical table, as shown in Fig. 1(b).
3. The feedforward/feedback controller based on a modified inverse P-I hysteresis model
A new design of feedforward/feedback controller for the flexure-based mechanism, accounting
for hysteresis and external disturbances, has been established and implemented in the dSPACE
DS1103 R&D control board. The rate dependent hysteresis of the piezoelectric actuator is
compensated by exploiting a modified inverse P-I hysteresis model in a feedforward controller,
using the direct parameter identification technique. Then a closed-loop feedback controller is used
to eliminate the effects of external disturbances such as vibration and noise on the positioning
accuracy of the mechanism.
3.1. The modified inverse P-I hysteresis model
The traditional P-I hysteresis model is generally constructed from parallel-connected backlash
operators which can be expressed as

Frequently asked questions

Q1. What have the authors contributed in "An inverse prandtl-ishlinskii model based decoupling control methodology for a 3-dof flexure-based mechanism" ?

In this paper, a modified Prandtl-Ishlinskii ( P-I ) hysteresis model is developed to form the feedforward controller for a 3-DOF flexure-based mechanism. 

Q2. What future works have the authors mentioned in the paper "An inverse prandtl-ishlinskii model based decoupling control methodology for a 3-dof flexure-based mechanism" ?

In order to further improve the static and dynamic characteristics including the high frequency trajectory tracking performance, their future work will focus on the laser interferometry based sensing and measurement technique to improve the robustness and stability of the proposed novel feedforward/feedback control methodology. 

Q3. What is the purpose of the decoupling strategy?

when the platform is actuated, any coupled error motions will degrade its tracking property of a 2-D trajectory, so a decoupling strategy has been introduced as a necessary part of the controller. 

Q4. What is the effect of external disturbances on the positioning accuracy of the actuator?

Then a closed-loop feedback controller is used to eliminate the effects of external disturbances such as vibration and noise on the positioning accuracy of the mechanism. 

Q5. How many parameters are tuned by Ziegler-Nichols method?

The parameter of the feedback controller are tuned by the Ziegler-Nichols method, intranslational direction kp and ki are 0.05 and 50, respectively, while in the motion rotate about the z axis they are 0.06 and 70, respectively. 

Q6. What is the maximum resonant frequency of the flexure-based mechanism?

After assemble the piezoelectric, the plane displacement of the flexure-based mechanism can reach 12.74 µm, 12.22 µm in the x and y direction, respectively, the maximal rotation angle in clockwise and anticlockwise are 0.0088° and 0.0103°, respectively, the first resonant frequency is 790 HZ in the θZ direction. 

Q7. What is the effect of the installation of the piezoelectric actuators on the moving platform?

the installation of the piezoelectric actuators will introduce additional stiffness in that side of the moving platform and thus disturb the symmetric properties of the entire system. 

Q8. What is the hysteresis of the backlash operator?

the backlash operator has symmetrical shape about the central line and so the modeled hysteresis loop must also possess symmetrical characteristics. 

Q9. What is the common configuration of the dead-zone operators?

The common configuration is to connect the dead-zone operators or cubic polynomial input functions in series with the backlash operators. 

Q10. What is the inverse hysteresis of the piezoelectric actuator?

The rate dependent hysteresis of the piezoelectric actuator is compensated by exploiting a modified inverse P-I hysteresis model in a feedforward controller, using the direct parameter identification technique. 

Q11. What is the current design of the actuators?

The actuators are driven by a piezoelectric amplifier (E-505.00, PI, Germany) that receives command signals from the I/O interface of a dSPACE DS1103 R&D control board, on which the newly developed control methodology is implemented with a sampling rate of 10 kHz. 

Q12. What is the design of the dSPACE DS1103?

A new design of feedforward/feedback controller for the flexure-based mechanism, accountingfor hysteresis and external disturbances, has been established and implemented in the dSPACE DS1103 R&D control board. 

Q13. What is the way to describe the inverse hysteresis model?

The modified inverse P-I hysteresis model can be used to form the feedforward controller, butit cannot compensate the unmodeled errors, including creep of the piezoelectric actuator and external disturbances such as vibration and noise, which also influence the positioning accuracy of the mechanism. 

Q14. What is the way to describe the asymmetric hysteresis?

It is demonstrated that both the dead-zone operator and polynomial input function can be usedto describe the asymmetric hysteresis [32]. 

Q15. What is the decoupling compensation for the non-actuated directions?

The decoupling compensation signal for the non-actuated directions is calculated based on the experimental results from single axis actuation testing (Fig.7), and these signals are just compensated to the input command of the modified inverse P-I model, the tracking error e still determined by difference between the desired displacement and the measured displacement. 

Q16. What is the linear relationship between the weight and the input rate?

The linear relationship between the weight and the input rate is bkwh ty (4)where k=[k1, k2,…, kn] T and b=[b1, b2,…, bn] T are the slop and intercept vectors, respectively, which can be determined in the parameter identification. 

Q17. How can the inverse hysteresis model be used to improve the performance?

These errors can be effectively compensated by feedback methods, so in order to improve the static and dynamic performance of the entire system, a combined feedforward/feedback controller is introduced. 

Q18. How can the inverse hysteresis model be improved?

the modelling precision with only polynomial input function is hardly further improved, and the modelling precision can be improved by increasing the number of the dead-zone operators resulting in an obvious increase of the response time. 

Q19. How is the performance of the 3-DOF flexure-based mechanism examined?

The performance of the 3-DOF flexure-based mechanism with the newly-developedfeedforward/feedback controller has been examined through sinusoidal signal tracking. 

Q20. What is the reason why a real 3-DOF flexure-based mechanism will exhibit?

Even if ideally designed in principle, a real 3-DOF flexure-based mechanism will exhibit somecoupling motions between different motion directions because of inevitable manufacturing and assembly errors. 

Q21. How can the decoupling control algorithm be used to compensate for the coupling error?

This can be realized in practice by changing the input signal to the modified inverse P-I hysteresis model, and this signal can be achieved by adding the decoupling control algorithm onto the previous version of the controller.