DEVELOPMENT OF A MAGNETIC SUSPENSION STAGE AND ITS APPLICATIONS IN NANO-IMPRINTING AND NANO-METROLOGY DISSERTATION

Transcription

1 DEVELOPMENT OF A MAGNETIC SUSPENSION STAGE AND ITS APPLICATIONS IN NANO-IMPRINTING AND NANO-METROLOGY DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Shih-Kang Kuo, M.S. * * * * * The Ohio State University 2003 Dissertation Committee: Professor Chia-Hsiang Menq, Adviser Professor Donald R Houser Professor James L Lee Professor Vadim Utkin Approved by Adviser Department of Mechanical Engineering

2 ABSTRACT The objective of this research is to develop an improved magnetic suspension system for ultra precision motion control and to demonstrate its feasibility in the area of nanotechnology via two applications, namely nano-imprinting and nano-metrology. In order to increase the travel range in horizontal axes and to reduce mechanical vibrations in the formerly developed magnetic suspension system, both actuation scheme and laser measurement systems are modified. Adaptive control and disturbance observer were employed to compensate model variations due to varying airgaps in DC electromagnets and to achieve uniformly high precision motion control throughout the workspace of the system. In the first application, a nano-metrology system consisted of an optical pick up head from a compact disc drive and the magnetic suspension system was developed. As the optical pick up head is a 1D probe, the magnetic suspension system was employed as a precision motion control scanning stage to obtain surface geometry of miniature objects having 2½ D features. Experiment results demonstrated that the developed metrology system can provide nano-meter measurement resolution over several millimeters range. In the second application, a nano-imprinting procedure was employed to fabricate miniature pores on a polymer membrane. In order to produce high aspect ratio, nano- II

3 sized pores over a large area, a nano-imprinting unit was developed to facilitate the process, along with the developed magnetic suspension system incorporated into the procedure to provide precise position and orientation control. System integration has been performed and preliminary experiments have been performed to show the feasibility of the process. III

4 VITA April 1, 1974 Born Chia-Yi, Taiwan B.S. Power Mechanical Engineering, National Tsing-Hua University, Hsin-Chu, Taiwan M.S. Power Mechanical Engineering, National Tsing-Hua University, Hsin-Chu, Taiwan Research Engineer, Opto-Electronis and System Laboratories, Industrial Technology Research Institute, Hsin-Chu, Taiwan present.graduate Research Associate, The Ohio State University Major Field : Mechanical Engineering FIELD OF STUDY IV

5 TABLE OF CONTENTS ABSTRACT...II VITA...IV LIST OF FIGURES...VII LIST OF TABLES...X CHAPTER 1. INTRODUCTION Background and motivation Objectives and Specific Aims: Dissertation Overview LITERATURE REVIEW Development of Magnetic Suspension Stage Control of Magnetic Suspension System Applications of Magnetic Suspension System Nano-metrology Nano-imprinting DEVELOPMENT OF AN IMPROVED MAGNETIC SUSPENSION SYSTEM Introduction Actuation System Design Improved laser interferometry system Vertical axis laser interferometer system Horizontal laser interferometer system Optical Path Analysis Mechanical design and force distribution Force distribution Digital Control System SYSTEM MODELING AND IDENTIFICATION Introduction System model identification for horizontal axes motion...50 V

6 4.2.1 On-line system identification algorithm System model identification for vertical axes motion control Summary LARGE TRAVEL PRECISION MOTION CONTROL IN MULTIPLE DEGREES OF FREEDOM Introduction Indirec adaptive controller design An overview of previously developed MSS Modeling and System Identification Disturbance observer Nominal Linear Control Disturbance Observer Design Experiment results APPLICATION(I) : DEVELOPMENT OF A NANO-METROLOGY SYSTEM Introduction A Laser Pick-up head APC circuit System Configuration A Calibration Result System integration Experiment Results Summary APPLICATION (II) : TOOLING AND MOTION CONTROL FOR NANO- IMPRINTING Introduction Nano-imprinting process Design and development of a Nano-imprinting Unit Capacitive sensor Mechanical Design of Nano-imprinting Unit System Integration Real-time measurement of target position Real-time control of the target position and orientation Experiment result Summary and Future Work CONCLUSION References VI

7 LIST OF FIGURES Figure 3.1 Subsystems in a magnetic suspension system...21 Figure 3.2 Configuration of DC electromagnet...24 Figure 3.3 Configuration of single phase linear motor...24 Figure 3.4 FEM analysis using ANSOFT 27 Figure 3.5 FEM simulation result of a DC linear motor...28 Figure 3.6 Optical configuration of vertical axes motion measurement...30 Figure 3.7 Resolution test of the improved laser measurement system...30 Figure 3.8 Optical configuration of horizontal axis measurement system...31 Figure 3.9 OPC analysis of horizontal measurement system...36 Figure 3.10 Geometric model of the measurement system...37 Figure 3.11 An exploded view of the Magnetic Suspension Stage...39 Figure 3.12 Bottom view of the floator...39 Figure 3.13 Available working area on a retroreflector...41 Figure 3.14 Free body diagram of MSS in x y θ motion...43 Figure 3.15 Control hardware integration...46 Figure 3.16 Software structure of the control system...46 Figure 3.17 Flow charts of the control software (a): main routine; (b) interrupt service routine...47 Figure 4.1 System identification process...56 Figure 4.2 Position and control effort response during system identification process...57 Figure 4.3 Signal conditioning before system identification...57 Figure 4.4 Auto-correlation test of system identification...58 Figure 4.5 Pole-zero map of the identified system...59 Figure 4.6 Parameter variation of x-axis model...60 Figure 4.7 Control block diagram of vertical axis motion control...64 Figure 4.8 Position and control effort response of z-axis during system identification process...64 Figure 4.9 Pole-zero map of the identified system...65 Figure 5.1 Basic components of previously developed MSS...72 Figure 5.2 Actuation scheme of the previously developed MSS...73 Figure 5.3 Free body diagram of the MSS in x-y-theta motion...75 Figure 5.4 Feedback linearized model of the x-axis motion...82 Figure 5.5 Parameter identification under two different initial conditions...82 Figure 5.6 Estimated αˆ x versus operating positions along the x-axis...84 Figure 5.7 Closed-loop control block diagram for the x-axis...88 VII

8 Figure 5.8 A general H control design block diagram with a loop-shaping procedure92 Figure 5.9 Trajectory of contouring a 0.6mm diameter circle...95 Figure 5.10 Tracking and contouring error of a 0.6mm diameter circle: (a) with indirect adaptive control (b) with robust H control...95 Figure 5.11 Trajectory of contouring a 2.0mm diameter circle...96 Figure 5.12 Tracking and contouring error of a 2.0mm diameter circle: (a) with indirect adaptive control (b) with robust H control...96 Figure 5.13 Tracking control trajectory in the θ-axis...97 Figure 5.14 Tracking error of the θ-axis : (a) with indirect adaptive control (b) with robust H control...97 Figure 5.15 On-line system identification results in indirect adaptive control (a) Circle contouring (b) theta axis tracking...98 Figure 5.16 Stabilized system model after nominal linear control Figure 5.17 Control block diagram of disturbance observer Figure 5.18 A time-varying disturbance Figure 5.19 A levitation process Figure 5.20 Positioning stability of the MSS Figure nm steps in Y direction Figure 5.22 Disturbance rejection of a time-varying disturbance Figure 5.23 Circular contouring and tracking errors Figure 5.24 A spiral curve trajectory Figure 5.25 Disturbance prediction for z-axis motion control Figure 5.26 Tracking error of 3D contouring using SODP Figure 5.27 Tracking error of 3D contouring using ZODP Figure 6.1 Sony SF-P151Z laser pick-up head Figure 6.2 Configuration of a laser pick-up head Figure 6.3 Focus Error signal Figure 6.4 APC circuit Figure 6.5 Feedback control of APC circuit Figure 6.6 BJT current drive Figure 6.7 System configuration of nano-metrology system Figure 6.8 Z-axis motion of the floator during calibration process Figure 6.9 A calibration process Figure 6.10 Curve fitting of FE signal using a 10 th order polynomial Figure 6.11 Control system of the nano-metrology system Figure 6.12 Picture of the developed system Figure 6.13 A focus-locking process Figure 6.14 Focus-locking process using different controller gains Figure 6.15 Experiment setup of gauge block measurement Figure 6.16 Experiment result of gauge block height difference measurement Figure 6.17 Experiment setup for mirror surface measurement Figure 6.18 X-axis motion of the floator during the measuring process Figure 6.19 Procedure of signal process for form error measurement Figure 6.20 Measurement result of a mirror surface Figure 6.21 Surface profile of a CD-R disk VIII

9 Figure 7.1 A nano-imprinting process Figure 7.2 Alignment between imprinting master and polymer Figure 7.3 Integration of magnetic suspension system and nano-imprinting unit Figure 7.4 The developed nano-imprinting unit Figure 7.5 Top view of nano-imprinting unit Figure 7.6 Picture of a nano-imprinting prototype system Figure 7.7 System integration scheme Figure 7.8 Relation between capacitive sensor coordinate and target Figure 7.9 Control system design for nano-imprinting system Figure 7.10 Z-axis reference trajectory for the imprinting process Figure 7.11 Capacitive sensor readings during the alignment process Figure 7.12 Target orientation adjustment during the alignment process Figure 7.13 Compensation for angle misalignment Figure 7.14 Three capacitive sensors readings of a nano-imprinting process Figure 7.15 SEM photo of nano-imprinting result (I) : top view Figure 7.16 SEM photo of nano-imprinting result (II) : side view IX

10 LIST OF TABLES Table 2.1 Specifications of three typical magnetic suspension stages...14 Table 3.1 Parameters of single phase linear motor...25 Table 4.1 System identification results at different location along x-axis...58 Table 4.2. x-axis identification at different locations along z-axis...59 Table 4.3 System identification results at different location along z-axis...65 Table 4.4 Current-position system model of z-axis...66 Table 6.1 Specifications of a laser pick-up head Table 7.1 Specifications of capacitor sensor X

11 CHAPTER 1 INTRODUCTION 1.1 Background and motivation Advanced tool actuation technologies that deliver ultra precision positioning accuracy with high bandwidth are critically important to modern fabrication processes such as micro/nano fabrication and ultra precision machining. Following one of the most important principles in traditional precision engineering, current precision machinery are designed to operate with respect to a number of kinematic constraints, including linear guide ways and rotating axes, that ensure the necessary precision. These kinematic references are often assemblages of mechanical members and need be compounded or cascaded in order to achieve multiple degrees of freedom motion, thus hindering the implementation of an actuation scheme with high bandwidth. In addition, the limitation of friction losses due to mechanical contacts is an engineering challenge when ultra precision actuation is desired. It is therefore very difficult to develop, based on current technology, high performance actuation systems capable of large travel in multiple degrees of freedom with nanometer tracking accuracy at elevated bandwidths. As magnetic suspension is contact-free, there is virtually no friction and wear, thereby enabling ultra precision positioning accuracy, and multiple degrees of freedom actuation can be realized without mechanical compounding or cascading, thus providing high 1

12 bandwidth precision tracking. Therefore, magnetic suspension is perceived to be a feasible alternative in the development of high performance actuation systems and is at the forefront of technologies resulting in technically advanced products that deliver superior performance with longer life. A magnetic suspension stage (MSS) was developed at CMML to achieve high precision six-degree-of-freedom (DOF) motion control [Shan et al 2002]. The MSS utilizes ten electromagnets, therefore ten power amplifiers, to suspend and servo the moving stage. A six-axis laser interferometer system, with a resolution of 1.24 nm, in conjunction with advanced computational algorithms was developed and employed for feedback control. The control architecture takes the six control variables provided by the laser measuring system and intends to control the six-dof motion through regulating the current in the ten electromagnets. The control architecture consists of three components: a) two-degree-of-freedom linear controllers, b) force distribution, and c) feedback linearization. Although the moving stage can be actively regulated in all six DOF, control of rotational motion was not attempted. It is due to the fact that modeling error of the electromagnetic force tends to increase rapidly as the target plate of an electromagnet deviates from its nominal orientation. In other words, the current MSS is actually a three-axis motion stage. Two experiments were conducted to demonstrate the performance of the developed MSS. The control algorithm was carried out at a sampling rate of 1000 Hz. In the first experiment, the moving stage rested on the stator initially and was levitated in the vertical direction to a height of 1 mm in 1.0 second. It was shown that the z-axis motion followed precisely the specified trajectory, while small accompanying motions in the other five 2

13 degrees of freedom were observed during transient response. It demonstrated that the levitated stage could be stabilized using current control architecture. It also showed that the MSS could achieve ±10 nm horizontal positioning stability and ±15 micro degrees orientation stability. However, the best vertical positioning stability achieved was ±20 nm and could not be easily repeated due to several reasons, including vibration of the supporting structure of the laser measuring system, modeling errors of the system, and limitations of the control algorithms employed. In the second experiment, the stage was commanded to contour a circle with a diameter of 0.6 mm on the horizontal plane. The experimental result showed that the stage precisely tracked the circle and that the contouring error and the tracking errors of both axes were within ±10 nm. However, the tracking accuracy degrades rapidly when increasing the travel range. The problem becomes worse when traveling along the vertical direction. In other words, the current MSS has very small workspace in three translation directions. There are at least three reasons inhibiting large travel in all six DOF for a magnetic suspension stage, whose moving stage has invariant configuration. The first reason is related to the actuation scheme employed. In a magnetic suspension system, large rotation results in large variations in air gaps of at least some, if not all, magnetic actuators. The second reason is related to the measurement of six DOF motion. In a laser interferometer system, a plane mirror interferometer has very narrow tolerance on the ranges of the pitch and yaw angles. In order to have larger tolerance on rotation, the plane mirror reflector can be replaced by a combination of a retroreflector and a plane mirror. The only drawback of employing a retrorefector is that the transverse range is limited by the size of the retroreflector. The size of a retroreflector can be up to 100 mm 3

14 in diameter. However, increasing rotational range severely limits the size of and twists the shape of the translation workspace of the MSS. The third reason is related to the control algorithms needed to achieve high precision motion control through a large travel range. In order to meet the demand for modern micro/nano fabrication, developing an improved magnetic suspension stage capable of large travel in multiple axes with nanometer tracking accuracy is highly desirable. At the present time, CMML has an ongoing project to develop a five-axis ultra precision motion control stage. The moving stage of the proposed five-axis system is separated into two parts, namely translation stage and rotation stage. The translation stage is suspended in six DOF and actuated to move along three translation axes within a workspace of 4mm 4mm 2mm. It is evident that although the translation stage can also rotate its main purpose is to achieve large translation. The rotation stage is a two-degree-of-freedom rotary actuator, the stator of which is cascaded to and thus move along with the translation stage, and the rotor of which is able to orientate a precision platen over a range of ±30 ±30. The five-axis stage in conjunction with the precision platen can have many applications. It can be the motion stage of a coordinate measuring machine for 3D microscale structures, components, and systems. It can also be an ultra precision motion stage facilitating various high precision micro/nano scale manufacturing processes, such as micro EDM and nano imprinting. The design of the rotary actuator is similar to that of a motorized probe head in CMM measurement. However, rather than being a sphere its rotor is a hemisphere. The purpose is to minimize the linear movement of the micro part when orientating the platen. In the proposed five-axis ultra precision motion control stage, 4

15 before the micro part is brought to interact with a micro probe or tool by the translation stage the rotation stage orientates the precision platen, and thus the micro part, locking into one of a series of repeatable locations around each of the two rotation axis. In other words, the proposed rotary actuator is employed to change the geometric configuration of the moving stage and precision platen assembly. Its accuracy is not necessarily high. The precision of the platen s position and orientation is guaranteed by a real-time laser interferometer measurement system and the translation stage. In order to realize the proposed system, four basic components of the project are identified. They are: 1) development of an improved translation stage; 2) development of an improved laser measuring system; 3) design and construction of a two-axis rotation stage; and 4) calibration and error mapping. This researchj focuses on the first two components of the project. In addition, two applications, namely nano-imprinting and nano-metrology, are proposed to demonstrate the feasibility of using the developed ultra precision motion control stage for micro/nano scale manufacturing and metrology. 1.2 Objectives and Specific Aims: The proposed research has two major objectives. The first one is to develop an improved magnetic suspension stage, which can provide ultra precision motion control throughout its 4mm 4mm 2mm workspace in three translation directions. In order to reduce mechanical vibrations and to increase travel ranges, both actuation schemes and laser measurement system are redesigned. For the purpose of achieving large travel motion control, advanced control algorithm will be developed and implemented so as to 5

16 compensate variations in system models induced by time-varying airgaps. In order to accomplish this objective, two specific aims are identified: Specific Aim #1: The first specific aim is to redesign the actuation scheme employed to suspend and servo the moving stage, as well as the laser measuring system for feedback control. In the proposed actuation scheme, DC electromagnets are used for vertical axes motion control as large levitation force can be generated by reluctant force. The actuation scheme for horizontal axes is similar to the one that employed by DC linear motors. In order to reduce the mechanical vibrations coming from the supporting structure of the laser interferometer system, the configuration of the measurement system and the arrangement of optical components have been improved such that the improved laser measurement system is more compact in physical size and stiffer in structure. Specific Aim #2: The second specific aim is to design advanced control schemes for large travel motion control. For the improved MSS, large travel results in large variations in airgaps of the actuators, which lead to significant variations in system models. To compensate model variations and to achieve uniformly ultra precision motion control throughout the workspace, adaptive control and disturbance observer control will be investigated and implemented. The second objective is to propose two applications, nano-metrology and nanoimprinting, to demonstrate the feasibility of using the developed magnetic suspension 6

17 stage for micro/nano scale manufacturing and metrology. Two specific aims are proposed. Specific Aim #3: The third specific aim is to illustrate the feasibility of the improved stage in the area of nano-metrology. A nano-coordinate measuring system will be developed. The coordinate measuring system consists of an optical pick-up head as the measurement probe and the developed MSS as the scanning stage. Using the astigmatic method and a quadrant photo diode, the laser emits from the optical head is able to convert the depth of a spot on a surface into electric signals. The scanning stage can be commanded to move precisely within the whole working space such that the probing depth is kept constant. From the laser readings of the six-axis measurement system, the translation motion of the moving stage, and thus the 3D coordinates of the micro part s surface point, can be calculated. The developed coordinate measuring system will be applicable for coordinate metrology of miniature objects having 2½ D features. Specific Aim #4: The fourth specific aim is to utilize the developed MSS as a motion controlled stage so as to facilitate a nano-fabrication procedure namely nanoimprinting. The procedure will be used to fabricate nano-pores on a polymer layer using a mask made of two-dimension array of optical fibers. The stage will be utilized as a motion control platform, providing precise position and angular alignment between the surface of the polymer and the tips of optical fiber array. The relative position and orientation between these two objects can be characterized by three parameters, i.e. one translation and two angular displacements. Taking these parameters as control variables, 7

18 the motions of the stage can be controlled such that the pattern formed by the optical fibers can be precisely transferred to the polymer layer placed on the stage. Since the laser measurement system provides translation motion of the moving stage, the absolute position of the moving stage is not known directly. In order to provide information of these three parameters, capacitive sensors will be employed to provide the three parameters that characterize the relative position and orientation between the surface of the polymer and the tips of optical fiber array. 1.3 Dissertation Overview This dissertation presents the development and control algorithm design of an improved magnetic suspension stage, as well as its applications in nano-metrology and nano-imprinting. A literature review will be presented in chapter 2. In chapter 3, development of the improved magnetic suspension system, including revised actuation and measurement system are discussed. System identification is investigated in chapter 4 for characterizing model variations due to changing airgaps when traveling inside the working space. To reach uniform performance within a three dimension working space, control algorithms for large travel motion control are addressed in Chapter 5 in which two approaches, adaptive control and disturbance observer, are attempted to compensate model variations during large travel motion control. In order to demonstrate the feasibility of employing the developed magnetic suspension system in the field of nano-technology, two researches, nano-metrology and nano-imprinting, are investigated. Chapter 6 presents the investigation of nano-metrology 8

19 in which a metrology system was developed for surface geometry measurement. A laser pick-up head embedded in a Compact Disc drive is utilized as a single axis probe for surface depth inspection. The magnetic suspension system is served as a precision moving stage for the scanning process. In Chapter 7, the developed magnetic suspension stage was incorporated with a nano-imprinting procedure for fabricating nano-pores on a thin polymer membrane. Magnetic suspension system is utilized for making alignment as well as precision motion control in the process. An imprinting unit, which is used to help performing alignment during the process, is developed and preliminary imprinting experiment results are shown. Conclusions are drawn and future research issues are recommended in Chapter 8. 9

20 CHAPTER 2 LITERATURE REVIEW This research focuses on development of a magnetic suspension stage along with its applications in nano-technology. In the development of a magnetic suspension system, mechanical hardware construction, including design of actuation scheme and measurement system, is important since it determines system characteristics, including available travel range and vulnerability to external disturbances such as mechanical vibrations. Since the magnetic suspension system is inherently an unstable system, control system design is another critical issue which influences greatly on system stability and positioning performance. Many studies regarding these two issues have been investigated by other researchers and this chapter will give a brief review on these investigations. Since the magnetic suspension system developed in this research provides a platform that delivers precision motion control with multiple degrees of freedom, which is needed for many applications in nano/micro technology, two applications in nano-technology, namely nano-metrology and nano-impriniting, were examined in this research. Related investigations in these two areas will also be reviewed. 10

21 2.1 Development of Magnetic Suspension Stage Many magnetic suspension systems have been developed to provide multiple degrees of freedom motion control. In order to achieve long-range scanning, a long-range scanning stage (LORS) was constructed [Holmes et al 2000]. The LORS stage utilizes four six-phase linear motors, therefore twenty four power amplifiers, to suspend and servo a moving platen. The suspended platen is floated in oil to enhance the mechanical damping of the system. The vertical position of the moving platen is measured by three capacitance probes, with a resolution of about 0.1 nm, provided by ADE Technologies, the lateral position by three heterodyne interferometers, with a resolution of better than 0.1 nm, from Zygo Corporation. The LORS stage achieves a horizontal positioning noise of less than 0.6 nm three sigma and a vertical positioning noise of less than 2.2 nm three sigma, and is consider to be one of the world s most accurate positioning stages. While positioning noise is very small, the positioning accuracy is limited by many factors, including the wavelength instability of the HeNe light source and the waviness of optical mirrors. The LORS stage has been designed to achieve a positioning accuracy of 10 nm. Since the actuation scheme is like a planner linear motor, the workspace in three translation directions is 25mm 25mm 0.1 mm. In other words, although the platen is suspended in six-dof the system is more like a two-axis motion control stage due to two reasons associated with the actuation scheme and measurement method employed. Firstly, linear motors are essentially narrow gap electromagnetic devices. The suspension force in the vertical direction decreases rapidly as the gap grows. Secondly, a plane mirror interferometer has very narrow tolerance on the ranges of the pitch and yaw angles, which are typically to only a few hundreds of arcseconds. 11

22 Another long stroke planar magnetic bearing actuator was developed [Molenaar et al 1998]. The vertical motion of the magnetic bearing is driven by magnetic reluctance force. Lorentz force is employed for providing actuation forces for horizontal motion control. Although the magnetic bearing was designed to achieve linear travel range of 160 mm for both x and y axes, the measurement system is incapable of covering the whole range. In addition, disturbing forces and heat generated by the coils on the moving stage create a potential limitation for the performance of the system [Jabben 1998]. The resulting steady state positioning error of the z-axis is 220 nm peak to peak. These investigations demonstrate the viability of large planar motion control by using electromagnetic forces. However, in many applications it requires not only long-range horizontal motions but also large travel in vertical axis. Therefore, these magnetic suspension systems have limited capability to cope with applications that requires large travel in vertical axis. A magnetic suspension stage (MSS) was developed at CMML to achieve high precision six-degree-of-freedom (DOF) motion control [Shan 2002]. The MSS utilizes ten electromagnets, therefore ten power amplifiers, to suspend and servo the moving stage. A six-axis laser interferometer system, with a resolution of 1.24 nm, in conjunction with advanced computational algorithms was developed and employed for feedback control. The control architecture takes the six control variables provided by the laser measuring system and intends to control the six-dof motion through regulating the current in the ten electromagnets. The control architecture consists of three components: 12

23 a) two-degree-of-freedom linear controllers, b) force distribution, and c) feedback linearization. Although the desired traveling volume is 2mm 2mm 2mm, the performance of the system is not consistent within the traveling volume and hence it has small working space. The non-uniform performance is due to several reasons. First, the force generated by a electromagnet decays rapidly as the floator moves away from it when multiple degrees of motion is desired. Therefore, it is very inefficient to use electromagnets to servo the magnetic suspension system in situations where large travel is needed. Second, because of the mechanical vibrations amplified by the structure which supports the laser interferometer measurement system, the performance is not satisfactory. The third reason is related to controller design. To compensate the force variation, feedback linearization is performed by using a lumped parameter model [Trumper 1997] and advanced algorithms such as robust H control is employed [Shan 2002]. From the experiment results it seems that these controllers are capable of rejecting model variations and servoing the MSS in large horizontal motion very well. However, These control algorithms are not capable of controlling vertical axes motions actuated by four narrow-gap electromagnets, in which model variations are more significant than horizontal axes. Table 2.1 lists the specifications of these three magnetic suspension stages to give a comparison of the performance. Strictly speaking, the first two stages can be classified as a two axis stage since they all operate at very narrow vertical gaps. As for the third stage, due to limited travel range and less impressive positioning stability caused by mechanical 13

24 vibrations and model variations, improvements are needed in order to achieve ultraprecision three-axis motion control. Actuation Measurement Travel Range (mm) Positioning Stability (Amplitude) Long Range Scanning Stage By Trumper et al 4 AC Linear motors Reluctance force Lorentzforce Capacitive sensors and interferometers 50 x 50 x nm 1nm Long Stroke Planar Magnetic Bearing By Molenaar et al Reluctance force Lorentz force Eddy current sensor 130 x 130 x nm Not reported Multiple Degrees of Freedom Magnetic Suspension Stage at CMML 10 electromagnets Reluctance force for both vertical and horizontal axes Interferometers 2 x 2 x 2 20 nm 10 nm Table 2.1 Specifications of three typical magnetic suspension stages 2.2 Control of Magnetic Suspension System Since magnetic suspension system is inherently an unstable system, control system design becomes an important issue in the development of a magnetic suspension system. Many researches have been focused on control algorithm design, especially for magnetic bearing systems throughout years. Because the electromagnetic force generated by DC electromagnets is in highly nonlinear relationship with both airgap and current, in order to have very large levitation force, magnetic suspension systems are often operated at very narrow, constant airgap. In these systems, however, changing positions of the floator corresponds to changing system 14

25 models, which degrades positioning performance when large travel motion control is desired. In order to minimize the model variations, feedback linearization was employed [Mittal et al 1997, Trumper et al 1997 and Shan et al 2002]. However, due to modeling errors perfect feedback linearization is not possible, and parameters of the feedback linearized system vary during the course of motion control. When parameter variations were moderate, by employing a linear robust control algorithm, it was demonstrated experimentally that the controller regulated the closed-loop system dynamics such that the transient response of the system was independent of the operating points [Shan 2002]. However, when large travel motion control or rotational motion control is desired, parameter variations can be significant even after employing feedback linearization and constant gain robust controllers may not be adequate for realizing ultra precision motion control. 2.3 Applications of Magnetic Suspension System Magnetic suspension system is considered to be a feasible tool for delivering ultraprecision motion control in recent years. It has many advantages over conventional positioning stages like x-y tables and piezo stages. The travel range of magnetic suspension can be as large as several millimeters while only several microns for a piezo stage. Because the magnetic suspension stage is suspended in the air and is directly driven by electromagnetic forces, there no need to use mechanical cascaded structures, which can be usually seen in conventional x-y tables. In spite of these advantages, there are only few applications of magnetic suspension systems appeared in the literature. Generally, the magnetic suspension stage can be applied to three fields in the area of 15

26 micro/nano technology in which precision motion control is strongly desired. These fields are metrology, fabrication and assembly. In this research, two applications of the developed magnetic suspension stage are demonstrated. In the first application, a nanometrology system was developed by integrating the magnetic suspension stage with an optical pick-up head. While in the second application, the magnetic suspension stage was combined with a nano-imprinting process to fabricate a thin layer with nano-size pores on it. A brief literature review of these two areas will provide context and motivations for developing these applications. 2.4 Nano-metrology Because of rapid growth in ultra-precision technology, development of the associated metrology systems received great attention in recent years. A metrology system usually consists of a probe and a moving mechanism. The probe is used for providing dimensional information of the measured object while the moving mechanism is used for either locating the probe or positioning the object such that the object remains within detectable range of the probe. Various probes have been developed and they fall into two categories, namely contact probe and non-contact probe. Classified by the measurable dimensions, the probes can be divided into 1D, 2D and 3D probes. In conventional CMM measurement, 3D touch-trigger probes are very commonly used. However, accuracy and tip size limit their application to measure miniature parts. In the current technology, the smallest probe size that can be fabricated is 100 µm in diameter and trigger force as small as 1 µn. The difficulty of probe assembly limits further downsizing of the probe. Optical microscopes are 2D probes providing two-dimensional 16

27 information on the focal plane [Kim et al 1999]. A moving mechanism provides 1D motion control is employed for constructing geometry of the feature. Scanning probe microscopes such as AFM are known to have very high resolutions. In a scanning atomic force microscope(afm), a sub-micron silicon probe tip is brought into interact with the surface to be measured [Karulkar et al 1993]. AFM systems employ various kinds of methods to detect the probes deformation, such as light leverage method and piezoelectric method, based on which the gap between the probe tip and the surface can be calculated. AFM probes can have nanometer resolution but provide only 1D measurement. Although these probes provide satisfactory accuracy and precision, they are usually very expensive and complicated. Because of its low cost and attainability, using a CD pickup head as an optical probe in a metrology system has been investigated recently [Fan et al 2000, Zhang et al 1997 and Armstrong et al 1992]. A CD pickup head is used to restore data recorded on a CD disk by using optical method. Therefore, it is capable of submicron resolution in vertical axis and the lateral resolution is around 2 micron. Because the optical head is a 1D probe, a moving mechanism is necessary when measuring a surface geometry over a certain area. Generally, two kinds of method are employed for surface geometry measurement in previous studies, i.e. focus error signal method and auto-focusing method. Focus error method directly measures the focus error signal and then converts it into the surface profile. The measurement range is limited by the linear range of the focus error signal and is very small (~10 µm). Because of mechanical misalignment, the motion of the moving mechanism cannot be perfectly perpendicular to the measurement axis of the optical 17

28 head. Usually the surface depth variation is larger than 10µm over several millimeters range and hence it is not eligible for measurement over a large area. The auto-focusing method is preferred because the measurement range is not restricted by the linear range of the focus error signal. In auto-focusing method, another moving mechanism is employed for adjusting relative distance between the optical head and the measured object. The motion of either optical head or object is controlled such that the measured surface always stays on the focal plane during the measuring process. Conventional actuators such as motorized x-y table and piezo actuators are employed as moving mechanism for motion control. However, motorized table suffers from nonlinearities like backlash, friction, hence the achieved precision is pretty low. In order to achieve ultra precision measurement, piezo actuators are employed for motion control in some researches. However, the travel range of piezo actuator is very small (~30µm) and hence the measurement range is restricted. 2.5 Nano-imprinting The major difficulty for mass-production of nano or micro-scale devices is the lack of a reliable and repeatable procedure. Nano-imprinting is a single mask-procedure and is one of the most promising technology for mass-production of nano-pattern. Only a few research results in nano-imprinting have been reported to demonstrate the viability of generating nano-scale size pattern over a large area with high throughput and low cost. Krauss transfer a strip pattern from a mold made of silicon dioxide to an Aluminum film [Krauss et al 1996]. The resulting strips are 70 nm wide and 200 nm tall giving a high aspect ratio. Another similar procedure for manufacturing dot pattern was done by Y. 18

29 Hirai, demonstrated the practicability of mass-production of large scaled quantum dots memory [Hirai 2000]. In these investigations, a commercial hydraulic press was used for alignment between SiO 2 mold and aluminum film in the nano-imprinting procedure. A compact imprinting system using driving power of stepping motors was also developed [Igaku 2000]. The z-axis of the mold and x and y-axis of the stage on which the aluminum film was placed were controlled by three stepping motors. Because of the trend to minimize sensors such that measurements can be acquired from small volume samples, such as biological cells and tissue, nano-tips arrays are fabricated on the distal faces of coherent fiber-optic bundles [Dam et al 1999 and Liu et al 2000]. The nano-tips arrays not only provide chemical properties of a localized area, such as Ph sensitivity, but also serve as an manufacturing tool for fabricating nano/microstructure with high aspect ratio. Utilizing the nano-tips arrays, a nano-imprinting procedure that can fabricate polymer was designed and proposed by L.J.Lee [Lee et al 2000]. In the process a mask made of optical fibers is contacted with polymer layer solution, the indented polymer was then cured to gelation by a ultra-violet light source before the mask is lifted. Finally a thin polymer membrane with nano-pores on it is fabricated. In such a process, both linear displacement and angular orientation needed be precisely regulated so that the orientation placement error between the mask and the polymer layer can be compensated and the array of holes produced can have precise uniform size. Therefore, a precision motion controlled stage with multiple degrees of freedom is highly desired. 19

30 CHAPTER 3 DEVELOPMENT OF AN IMPROVED MAGNETIC SUSPENSION SYSTEM 3.1 Introduction A magnetic suspension stage is an electro-mechanical system consists of three major components: actuation, measurement and control systems. As shown in Figure 3.1, these three components are cascaded in a feedback control loop and the performance of a system is determined by each component in the loop. In order to deliver high positioning stability as well as large travel range motion control, each sub-system must be designed properly. In a magnetic suspension system previously developed at CMML, ten DC electromagnets are used for controlling six degrees-of-freedom motions. It is known that the electromagnetic force generated by an electromagnet is in highly nonlinear relationship with current and airgap. Even though feedback linearization is applied to minimize nonlinearities, it still suffers from significant model variations and hence both travel range and positioning accuracy are limited. Mechanical vibration is an undesired disturbance that must be isolated from a precision motion control system. In the previous design of the magnetic suspension stage, in order to arrange the laser interferometer system for vertical axis measurement, a large levitation table was employed to support the whole system and the space under the table was used to accommodate interferometers 20

31 as well as related optical components. However, the levitated structure amplifies the vibration coming from the optical table and hence degrades the performance. Control system Actuation system Measurement system Figure 3.1 Subsystems in a magnetic suspension system In this chapter, the design and development of an improved magnetic suspension stage is presented. The objective of this improvement is to achieve uniform positioning stability within the entire three-dimensional working space. In order to accomplish this, both actuation and measurement systems are re-designed. Section 3.1 presents the principle and design of the actuation system. Improved laser interferometer measurement system is presented in section 3.2. Optical path analysis along with its inverse solution is also addressed. Control system provides integration between measurement and actuation system. Section 3.3 presents the digital control system for the improved magnetic suspension stage. 21

32 3.1 Actuation System Design In current technology, reluctance force and Lorentz force are two kinds of electromagnetic forces most frequently employed by electromechanical devices such as servo motors and magnetic bearings. Reluctance force comes from gradient of energy variation due to changing magnetic reluctance. It can be expressed explicitly as W ( x) F( x) =, where W(x) is the electromagnetic energy stored in the system. In most x cases the force is in nonlinear relationship with position. Usually a DC electromagnet is employed for generating reluctance force in a MSS. Figure 3.2 shows the configuration of a DC electromagnet. An iron plate, serving as a target, is placed facing normally to the pole face of the DC electromagnet. The attraction force generated by a DC electromagnet 2 C i F z ( g + a) can be expressed as =, where C is a constant related to two factors, geometry 2 of the electromagnet and number of coils, g is the airgap between electromagnet and iron target and a is a constant parameter that can be determined by calibration. It is seen that the reluctance force is in highly nonlinear relationship with both current and airgap, which makes it difficult to provide constant servo force when large travel is desired. Another electromagnetic force that has been commonly used in electromechanical system is Lorentz force. Lorentz force is generated by conducting a current through a magnetic field. The force direction is perpendicular to the field orientation and current direction and can be expressed as F = IL B, in which I is the magnitude of the current, L is the length of the conductor immersed in magnetic field and B is the 22

33 magnetic field density. As long as magnetic field is constant, Lorentz force is in linear relationship with the applied current, which makes it suitable for large travel planar motion control. Lorentzc force can be used to generate large force so as to drive the motor. However, there are some issues to be considered when developing an actuation system for a magnetic suspension stage. First, magnetic suspension stage is suspended in the air, in order to keep disturbance force as small as possible, it is undesirable to attach wires on the moving stage. Secondly, in a rotary motor, to generate large magnetic field, permanent magnets are used as a reliable source for magnetic field B and use a ferromagnetic material (such as iron) as the core of the coil. The wire core made of ferromagnetic material provides small reluctance and hence the flux density in the airgap is very large. Because of the symmetric structure of the rotor in a rotary motor, the attractive forces generated between permanent magnets and wire core cancel out each other and the resulting net force is very small. However, in the case of a magnetic suspension system, it is highly desired to use non-ferromagnetic material as the core for the coil since the large attraction force will result in extremely large disturbance force such that the system becomes uncontrollable. A single phase linear motor is employed to generate driving forces for horizontal motion control. The configuration of a single phase linear motor is shown in Figure 3.3, which has the similar structure as a DC linear motor. The single phase linear motor consists of four permanent electromagnets attached on a moving stage and coils wounded on a wire core made by aluminum, which is mounted on the stator. In order to have larger actuation force, a modern permanent magnet material NdFeB is used for providing 23

34 strong magnetic field. Some specifications of the single phase linear motor is listed in Table 3.1. DC electromagnet Ferromagnetic iron core µ Z d 0 µ µ 0 Fz Ferromagnetic plate µ Figure 3.2 Configuration of a DC electromagnet Current Sheet Non-ferromagnetic material Attached to Stator x N S Fx Direction of force Experienced by floator Permanent Magnet Aarry gap Attached to floator Figure 3.3 Configuration of a single phase linear motor 24

35 Size of Permanent magnets Resistance Dia = 20mm Height = 5mm 7.5 Ohm Wire Size AWG 24 Inductance 0.17mH Table 3.1 Parameters of single phase linear motor The model of a DC single phase linear motor was constructed and simulated using the electromagnetic field software ANSOFT MAXWELL-3D. Figure 3.4 shows the magnetic field distribution of permanent magnets. ANSOFT uses Finite Element Method (FEM) to simulate the magnetic field distribution of the linear motor model and obtain the magnitude and direction of the Lorenz force. It is seen in Figure 3.5 when the gap between the permanent magnet and the wire core is constant, the variation of actuation force is not significant as the moving range in x-axis is within ± 5mm. As the airgap changes, the actuation force decrease linearly with the gap distance. This characteristic makes the force variation easier to be compensated by using control algorithms. However, compared to electromagnet, the force generated by single phase linear motor is 25

36 very small, which makes it inefficient for vertical axes motion control. Therefore, four DC electromagnets are still used to provide large levitation forces for vertical axes motion control. 26

37 Figure 3.4 FEM analysis using ANSOFT 27

38 Fx(N) Airgap=1.5mm Airgap=3.5mm Airgap=5.0mm x(mm) Figure 3.5 FEM simulation result of a DC linear motor 3.2 Improved laser interferometry system In the previous developed magnetic suspension system, in order to arrange the optical devices for measuring six degrees-of-freedom motion of the floator, a supporting table was employed to levitate the magnetic suspension stage. Three laser interferometers are placed under the supporting table for vertical axes motion measurement. However, the design not only made the optical arrangement complicated but also induced mechanical vibrations. A modified laser interferometry system is presented in this dissertation. The measuring system has been improved in two ways. First, to reduce the mechanical vibrations, the whole stage is lowered down such that all optical components are mounted onto the optical table. The air damper of the optical table can block most of the vibrations comes from the floor. To measure vertical axes motions, three large plane mirrors are used to change the directions of laser beams such that they are incident normally to the top surface of the retrorefletors which are used for vertical axes 28

39 measurement. Second, since a retroreflector (weight = kg) is heavier than a plane mirror (weight = 0.08 kg), the three retroreflectors for horizontal axes measurement are replaced by three plane mirrors. The retroreflectors along with smaller plane mirrors can be arranged on the optical table as the beam-return device. Such an arrangement not only reduces the weight of the floator but also increases the measurement resolution Vertical axis laser interferometer system The configuration of the vertical axis laser interferometer system is shown in Figure 3.6. Laser beam is split into measurement beam and reference beam inside the interferometer. The measurement beam passes through interferometer and first reaches mirror Mr. The purpose of Mr is to change the direction of laser beam from horizontal into vertical. The measurement beam was redirected by Mr and hit the retroreflector on the floator, then reaches the return mirror M 1 on the optical table. After that, the laser beam travels all the way back to the interferometer and combines with the reference beam at the receiver, in which the interference occurs. In order to demonstrate that the modified measurement system has reduced mechanical vibrations, two sets of single axis interferometry systems are setup up for comparison. One set of the interferometry system is mounted directly on the optical table while the other is mounted on the support table, which is placed on the optical table. The resolution tests in Figure 3.7 shows that the interferometer placed on the support table experience a vibration of 10 nm magnitude, while the one placed on the optical table has much smaller vibration. Consequently, vibrations coming from the floor and environment have smaller impacts on the improved measurement system. 29

40 Interferometer M ref R Floator Retro-reflectors λ/4 M 1 PBS λ/4 Laser Beam M r Receiver Figure 3.6 Optical configuration of vertical axes motion measurement 4 x 10-9 Resolution Test results of both setup 2 X(m) time(sec) 1 x (a) X(m) time(sec) Figure 3.7 Resolution test of the improved laser measurement system (b) 30

41 3.2.2 Horizontal laser interferometer system The configuration of the horizontal laser interferometer system is shown in Figure 3.8. The measurement beam passing through the interferometer reaches a large plane mirror placed on the floator as the moving target of the interferometry system. The mirror M reflects the beam to a fixed retroreflector R and the retroreflector R returns the beam back to a fixed return mirror. Since the return mirror is arranged such that its normal parallels the direction of the beam leaving from laser interferometer, the return mirror can reflect the beam back along the reverse of the coming path. φ = 15 Interferometer M ref Target mirror M λ/ 4 Return mirror PBS Receiver λ/ 4 Attached to moving stage P R1 Retroreflector R Figure 3.8 Optical configuration of horizontal axis measurement system This configuration allows large measurement range in both translation and rotation motions. Since the measurement beam hits the target mirror four times, the measurement resolution is twice as large as the vertical interferometer system. In the 31

42 system setup, a ZMI 1000 measurement board is used and therefore a sub-nanometer (0.62nm) resolution can be obtained. The measurement range is determined by the size of the optical devices as well as the angle φ. A small angle φ results in large translation motion range but makes arrangement difficult. The optical configuration and design parameters are fully discussed in [Menq et al, 2000] Optical Path Analysis Since a laser interferometer detects the optical path change (OPC) induced by the moving target, which is characterized by six displacement parameters, six interferometer readings with effective real time algorithm are required for solving six degrees of freedom motions. The algorithm cannot be too complicated such that it can be performed within a single sampling interval during real time motion control. The algorithm for real time measurement can be divided into two stages. In the first stage, translation motion along z axis and rotation motion α and β can be determined by three laser readings coming from optical path change, which are induced by three retroreflectors on the moving floator. The algorithm along with analysis has been thoroughly discussed in [Shan 2001]. In the second stage, with motions z, α and β determined along with optical path change induced by three plane mirrors on the floator, x, y and θ axes motion can be obtained. The optical path analysis can be illustrated using Figure 3.9. As shown in Figure 3.9, the horizontal interferometry system consists of an interferometer, a retroreflector and a small plane mirror as beam return device and a large plane mirror as the moving target. P R is the position of the retroreflector nodal point, 32

43 which is located on the surface of the retroreflector. P M is the position of the moving target and Mˆ is the unit vector normal to the surface of the moving target. For convenience in OPC derivation, the virtual image of the retroreflector generated by the * moving target is also shown and its nodal point is denoted as P R. target P * R T = P + 2( P P ) MM ˆ ˆ (3.1) R M R After translation and rotation of the target, new position and normal vector of the P M ' and ˆM ' is formed. The nodal point of the retroreflector image is moved to P * R T ' = P + 2( P ' P ) Mˆ ' Mˆ ' (3.2) R M R Under the assumption that optical path change inside the retroreflector can be neglected [Shan 2001], the optical path change can be expressed as T * * OPC = ρ ˆ ( P ' P ) (3.3) where ρˆ is the measurement axis of interferometer. R R * P R ' Six targets, including three large plane mirrors and three retroreflectors, are placed on the floator and six interferomters are employed to measure optical path changes induced by the targets. Real time motions of the floator can then be calculated from laser readings generated by interferometrs. The task of finding six DOF motions of the floator can be divided into two parts: (a) determination of vertical motions and (b) determination of horizontal motions. The first part can be solved independently and has been addressed in Shan s report while the solution of the second part depends on the results of the first part. In this dissertation, only the second part is explained. Figure 3.9 shows the configuration of the horizontal interferometry system. There are two coordinate systems to be defined, one is fixed reference coordinate system and the other is moving coordinate system that 33

44 moves with floator. As it is illustrated in Figure 3.10, the position and orientation of the moving target with respect to the fixed reference coordinate system can then be expressed with a translation vector o f T o = f f f, and a rotation matrix f R = R( θ ) R( β ) R( α). The t [ x y z ] position of the moving targets relative to the moving coordinate system are ( i = 1 ~ 3) ˆ = T and the normal vector of the target surface are M i ( i 1 ~ 3), where P x y z ] T M 2 = [ xm2 ym2 z, T m2 P M [ xm3 ym3 zm3] P ] 3 =, T M ˆ Mˆ 1 2 = [cosφ sinφ 0] P Mi M1 = [, m1 m1 m1 = and ˆ T 3 = [ sinφ cosφ 0. The location of the nodal point of the retroreflectors with M ] respect to reference coordinate are expressed as PR = PM + LSli, where l ˆ ˆ 1 = l2 = [ cos 2φ sin 2φ 0], l ˆ3 = [sin 2φ cos 2φ 0], S retroreflector nodal point and plane mirror surface. i L is the normal distance between i ˆ To convert horizontal motions x,y and θ from three laser readings, three OPCs has to be formulated first. Taking first measurement axis as an example, the virtual image of the nodal point * R 1 P is expressed as * R1 P = P + 2( P P Mˆ Mˆ = P + L lˆ + 2( P P L ˆ Mˆ Mˆ (3.4) R1 M1 T R1) 1 1 M 1 S 1 M1 M 1 T sl1) 1 1 After translation and rotation of the floator, a new image of the nodal point * R 1 P is expressed as *' R1 P = P + 2( P P Mˆ Mˆ = P + L lˆ + 2( RP + t P L ˆ RMˆ RMˆ (3.5) R1 ' M1 T R1) ' 1 ' 1 M1 S 1 M1 0 f M1 T sl1) 1 1 where P and M ' is the position and normal vector of the mirror after translation and ' M1 ˆ 1 rotation of the floator. The optical path change can be expressed as 34

45 T *' OPC = ρ ( P P ) (3.6) 1 ˆ x R1 R 1 In order to reduce complexity and cost of computation, first order approximation ( sin δ δ be expressed as ) has been made. With the help of MATLAB symbolic TOOLBOX, OPC 1 can OPC1 C xf 1x f + C yf 1 y f + C zf 1z f + C xm1x m1 + C ym1 ym1 + C zm1z m1 + C Ls1 = L (3.7) s where C xf1,c yf1,c zf1,c xm1,c ym1,c zm1 and C Ls1 are functions of α, β and θ. Using the same procedure, OPC 2 and OPC 3 can also be derived. By subtracting OPC 1 and OPC 2, θ can be obtained B = B 2 4AC 2A θ (3.8) where A = 2sin( φ )( cos( φ) + sin( φ) α β ) 2 2 B = 2 cos ( φ ) 2sin ( φ) α β + 4sin( φ) cos( φ) α β 2 2 OPC1 OPC2 C = ym x and y axes motion can then be solved from OPC1 and OPC3, x f Cyf 3N1 Cyf 1N3 =, y D f C 3 N1 + C D N xf xf 1 3 = (3.9) where N 1 = OPC ( C zf 1z f + C xm1x m + C ym1 ym + C zm1z m + C Ls1L 1 s N = OPC ( C z + C x + C y + C z + C L ), 3 3 zf 3 f xm3 m ym3 m zm3 m Ls3 s ), D = C C C C, xf 1 yf 3 xf 3 yf 1 35

46 and coefficients C * are listed below 2 C xf 1 = ( sin( φ ) θ cos( φ) + sin( φ) αβ ) C yf 1 = ( sin( φ ) θ cos( φ) + sin( φ) αβ ) ( cos( φ) θ + sin( φ) + sin( φ) αβθ) C zf 1 = (sin( φ) α + cos( φ) β ) (sin( φ) αβ sin( φ) θ cos( φ)) C xm1 = sin( φ) ( αβ θ ) (sin( φ) αβ sin( φ) θ cos( φ)) C ym1 = θ ( cos( φ) + sin( φ) αβ )(sin( φ) αβ sin( φ) θ cos( φ)) C zm1 = (sin( φ) α + cos( φ) β )(sin( φ) αβ sin( φ) θ cos( φ)) C LS 1 = sin( φ ) αβ (cos(2φ ) αβ + 2sin(2φ )) + 2sin( φ) αβθ(sin( φ)cos( φ)( θ + αβ ) 1) + sin( φ) θ 2 C xf 3 = (sin( φ ) θ + cos( φ) + cos( φ) αβ )(sin( φ) cos( φ) + cos( φ) αβ ) 2 C yf 3 = (cos( φ ) αβθ + sin( φ) θ + cos( φ) C zf 3 = ( cos( φ ) α + sin( φ) β )(cos( φ) αβθ + sin( φ) θ + cos( φ)) C xm3 = cos( φ )( αβ θ)(cos( φ) αβθ + sin( φ) θ + cos( φ)) C ym3 = θ (sin( φ) + cos( φ) αβ )(cos( φ) αβθ + sin( φ) θ + cos( φ)) C zm3 = ( cos( φ ) α + sin( φ) β )(cos( φ) αβθ + sin( φ) θ + cos( φ)) C LS 3 = 2sin( φ) cos( φ) αβθ + 2sin( φ)cos( φ) αβ (1 αβθ) + θ (cos( φ) cos(2φ ) α β sin( φ) ) Virtual image of R1 Y φ = 15 M 1 ˆρ 1 P M 1 L s θ X M 2 ˆl 1 P R1 R1 P M 2 P M 3 Floator ˆρ 2 M 3 ˆl 2 P R2 P R3 ˆl 3 ˆρ 3 Figure 3.9 OPC analysis of horizontal measurement system 36

47 retroreflector ẑ f ŷ f flotor ρˆ i r i xˆ f M i O ẑ o o f t ŷ o o xˆo Figure 3.10: Geometry model of the measurement system 3.3 Mechanical design and force distribution Mechanical design is the most tedious work in the development of a magnetic suspension system. In order to come up with a reasonable design, several factors must be compromised and iterations must be taken. Force distribution is another important issue that relates to actuator arrangement and geometrical dimension of the mechanical structure. Mechanical Design The mechanical parts of the stage are designed using Solidworks. Figure 3.11 show an exploded view of the developed magnetic suspension stage. The stage has a floator that can move freely inside the stator. To have minimal weight of the floator while 37

48 maintaining its mechanical strength, the shape of the floator is designed simply as a plate with actuation and measurement components hooked on the bottom surface of it. A small supporting table is fixed inside the stator to support the floator. Four DC electromagnets are placed on top of the floator to control the vertical motions. A large ferromagnetic plate is fixed on the floator as the target for electromagnets. Four single phase linear motors are used for horizontal actuation. Each linear motor uses four permanent magnets as a source of magnetic field, which are placed on the floator, while wire coils are wounded on the stator offering currents for force generation. Three retro-reflectors along with three large plane mirrors are placed on the floator as the moving targets for interferometry system. The location of these moving targets, including retroreflectors and plane mirrors, for interferometry system is shown in Figure The total weight of the floator is around 2.5 Kg and the inertial is I x = kg-m 2, I y = kg-m 2, I z = kg-m 2. 38

49 Figure 3.11 An exploded view of the Magnetic Suspension Stage X-axis mirror #2 Y-axis mirror Z-axis retroreflector #3 Z-axis retroreflector #1 Z-axis retroreflector #2 X-axis mirror #1 Figure 3.12 Bottom view of the floator 39

50 To have vertical travel range as large as 2.0mm, the initial airgaps of electromagnets are a little larger than 2.0mm to have more tolerance for mechanical design and machining. As it has been mentioned earlier, the actuation force generated by a electromagnet is in square inverse proportion with airgap, which makes the levitation force very small if large airgap, or large travel is desired. It is this nonlinear feature in reluctance force that restricts the vertical travel range. The horizontal axes travel range is around 4mm, which is not restricted by actuation characteristic but rather by the size of retroreflectors. Since vertical axes motions are measured by optical path changes induced by three retroreflectors, laser beams coming from interferometers scans across the top surface of retroreflectors when the floator moves in an x-y plane. Because a retroreflector is consisted of three plane mirrors facing perpendicularly to each other. Therefore, the available working area of a single retroreflector is only one sixth of the top surface as shown in Figure It is not allowed to have laser beam scan across the edges of the mirrors since large measurement error will occur. The travel area of the magnetic suspension stage is smaller than the available working area of a single retroreflector since it is very difficult to make perfect alignment for three laser interferometers. It is seen that the size of the retroreflector somehow determines the horizontal travel ranges. To have large travel range, it is better to have larger retroreflectors, but that also increases the weight of the floator. 40

51 Available working area of a retroreflector : one-sixth of the surface Travel area of magnetic suspension system 4mm x 4mm 38.1 mm Figure 3.13 Available working area on a retroreflector Force distribution According to the developed actuation scheme, six degrees of freedom motions are controlled by eight actuation forces. Obviously, force distribution is necessary for developing a suitable control strategy. The force distribution problem can be solved by dividing the actuation forces into two groups, i.e., the horizontal group that control x, y and θ motions and the vertical group that control α, β and z motions. The details of the force distribution of the vertical group is similar to the system developed by Shan et al [Shan et al 2001]. A free body diagram shown in Figure 3.12 is utilized for solving the horizontal group force distribution problem. It is evident that the system has 3 outputs, x, y and θ, and 4 inputs, F i (i = 1 ~ 4). In order to develop a suitable control strategy, three virtual terms, u x, u y and u θ are defined. u x and u y are defined as net forces of along x and 41

52 42 y-axes and u θ is defined as net torque around z-axis. The relationship between actuation forces F i and net forces u x, u y and u θ can be easily formulated as follows: = F F F F W W W W u u u y x θ where W is the half width of the floator. By using pseudo-inverse method, control forces F i can be expressed as combination of net forces, which are obtained through control algorithm during real time motion control. = u θ u u AA A F F F F y x T T ) (, where = W W W W A Finally, forces F i ( i = 1~4 ) can be expressed explicitly as = u θ u u W W W W F F F F y x The system dynamics of horizontal motion control can then be decoupled into three SISO systems.

53 F 3 u y F 4 u θ u x F 2 F 1 Figure 3.14 Free body diagram of MSS in x y θ motion 3.4 Digital Control System Since magnetic suspension system is highly nonlinear and open loop unstable, development of a real-time control system, including necessary instrumentation equipment and real-time control software, is necessary for ultra precision motion control. Figure 3.15 shows the block diagram of the currently integrated hardware system. The optical measurement signals from the laser interferometers are processed on six VME bus based measurement boards before being fetched by the digital controller through two VME to PCI interface cards. The electromagnets are powered by linear amplifiers, which are capable of delivering up to 6 amperes of current. The nominal bandwidth of the power amplifiers for resistive load is 10K Hz. The amplifiers are controlled by the digital controller through a digital-to-analog converter (DAC) board. 43

54 The current digital controller is a personal computer with an Intel Pentium Pro 200 CPU. Correspondingly, the interface cards, including the PCI to MXI-2 converter card PCI-MXI-2, made by National Instruments, and the analog output card CIO-DAC 16/16, which is a 16 bit resolution 16 channel ISA card made by Computerboards, Inc., are installed in the extension slots of this computer. The six measurement boards from Zygo are located in a cage with a VME bus back plane. The VME to MXI-2 converting board, VME-MXI-2 from National Instrument, also resides on this back plane as the slot 0 controller of VME bus, and is connected to PCI-MXI-2 board via a MXI-2 cable. MXI-2 is a bus in a cable bridging the gap between two different bus protocols PCI and VME. Figure 3.16 shows the integrated software architecture that carries out the tasks of control algorithm, measurement computation, data input/output and user interface. The present control software is an interrupt based DOS program. The kernel of the program, namely the real-time measurement computation and control algorithms, is carried out in the interrupt service routine of DOS interrupt 0x1c. This interrupt is generated by a system timer on the PC and originally used to provide real time clock for the PC system. The interrupt vector for this interrupt is redirected to the measurement and control routine, and the interrupt frequency is reprogrammed as the sampling frequency for real time control. By using current configuration the sampling can be as fast as 5K Hz on the Pentium Pro 200 computer. 44

55 The main routine outside the interrupt service routine, on the other hand, will take care of non time-critical work such as data logging and user command interpreting. If an interrupt is triggered, this routine will be suspended. After the interrupt service routine releases the CPU resource, this routine can be resumed. The flow charts of the main routine and the interrupt routine are illustrated in Figure

56 MSA 1.24 nm resolution DC Power Supplies Linear Power Amplifiers 0-6 amps 10 Khz bandwidth VME Interface Control Timing CPU Digital Computer (Controller) Analog Output Board Figure 3.15 Control hardware integration Control Parameters User Position Data User Interface Measurement Computation and Control Algorithm VME Interface D/A Interface Digital Controller Measurement Data Control output Measurement System Magnetic Suspension Actuator Figure 3.16 Software structure of the control system 46

57 begin begin reprogram timer read position data from sensors redirect interrupt vector measurement computation log data control algorithm keyboard input? No send out control efforts Yes interpret command buffer data for logging Exit? No end/return Yes post process data end (a) (b) Figure 3.17 Flow charts of the control software (a): main routine; (b) interrupt service routine 47

58 CHAPTER 4 SYSTEM MODELING AND IDENTIFICATION 4.1 Introduction The improved MSS has a traveling volume of 4mm 4mm 2mm, hence the floator can move freely within this three-dimensional space. One of the specific objectives in this research is to reach uniform positioning stability within the three dimension working space. However, as the large travel motion in x,y and z axis is needed, the performance degrades because of the changing airgaps. The objective of this research is to have a working space as large as the traveling volume, which means the performance must be uniform inside this three dimensional (3D) space. To accomplish this, system model variation due to changing airgaps must be studied and investigation of advanced control algorithm is necessary. As it has been discussed in previous section, the actuation scheme of the developed magnetic suspension stage can be divided into two parts, horizontal axes (x, y and θ) and vertical axes (α, β and z). To generate large levitation force, four DC electromagnets are employed for vertical axes motion control. For the purpose of canceling nonlinearities existing in the dynamic equations, a lumped parameter model of the force-current relationship of a DC electromagnet was employed by feedback linearization method. 48

59 However, it has been shown that even though the force-current relationship was calibrated, the nonlinearity cannot be eliminated completely due to unmodeled dynamics, disturbance and mechanical misalignment [Shan, 2002]. As for horizontal axes motion control, employment of four single phase linear motors greatly simplify the actuation scheme and reduce model nonlinearity for large planar motion. However, for 3D translation motion control it is required that the floator travels along z axis, hence the gap between the coil and permanent magnets in the single phase linear motor is not constant as in the case of planar motion. To develop a control method realizing large travel 3D motion, model variations within the working space must be studied through system identification. System identification is a well-developed technology to fit a physical system with a mathematical model, which indicates the relationship between input and output of the system. The system identification process is shown in Figure 4.1, in which input-output information are collected and analyzed by identification algorithms for obtaining the mathematical model. The accuracy of the identified model is affected by the quality of the measured signal and therefore signal conditioning must be performed in advance. There exist many methods for system identification, which can be classified into two categories, frequency domain and time-series domain. In frequency domain system identification, bode plots are obtained by input sine waves with different frequencies and measure the amplitude and phase shift of the output signal. Usually a spectrum analyzer is used for precise amplitude and phase measurement. A frequency curve fitting procedure is then used to fit the frequency domain response with a linear model [Young 49

60 et al 1993, Pintelon 1996]. Since the system is characterized by amplitude and phase information of input-output signal, it can only be used for stable, linear system. On the other hand, time-series system identification employs a pre-defined excitation signal and solves the parameters of a difference equation by input-output time sequence. The difference equation can be expressed by a z-domain transfer function, which is not necessarily stable. Therefore, time-series system identification provides more freedom in signal choosing and is applicable to unstable system. Furthermore, it is very convenient to be implemented in a computer control system especially when on-line system identification is desired. In this section, system model in both horizontal axes and vertical axes will be identified by time-series system identification and will be presented in section 4.2 and 4.3 respectively. Furthermore, model variations will be quantified through the results of system identification. 4.2 System model identification for horizontal axes motion In horizontal axis motion control, three degrees of freedom motions (x,y, and θ) are controlled by four actuation forces generated by single phase linear motors. From the point of view of control, it is a MIMO system (4 inputs and 3 outputs) and therefore not easy to apply normal controller design method on it. For the purpose of controller design, force distribution is accomplished by using pseudo inverse method. After force distribution, the MIMO system is then decoupled into three SISO systems. The real system is always nonlinear, however, for the purpose of controller design, an ARMA 50

61 51 (auto-regressive moving-average) model is used to fit the real system when operated at a specific position. An ARMA model for x-axis motion control can be expressed as N N N N x z a z a z a z b z b z b z u z x z G = = ) ( ) ( ) ( (4.1) Generally, for a mechanical system, a second order model is sufficient to capture the dominant dynamics of the system. However, in a real system there are always undesired high frequency dynamics such as mechanical vibrations and dynamics of electrical current drives. Discovering these dynamics not only provides more knowledge of the actual system but also gives more specific requirement in controller design procedure. When large travel motion control is desired, changing position of the moving stage corresponds to changing gaps in electromagnets and linear motors, thus parameters in (4.1) can be assumed to be dependent on position of the moving stage T f f f z y x p ] [ =, i.e. ) ( p a a i i =, ) ( p b b i i = for N i 1,2, =. To perform system identification, (4.1) is first expressed in time domain series as ) ( 2) ( 1) ( ) ( ) ( 2) ( ) ( 1) ( ) ( ) ( N k u b k u b k u p b N k x a k x p a k x p a k x x N x x N = θ φ T (k) = (4.2) where [ ] T x x x N k u k u k u N k x k x k x k ) ( 2) ( 1) ( ) ( 2) ( 1) ( ) ( = φ, T N N p b p b p b p a p a p a )] ( ) ( ) ( ) ( ) ( ) ( [ = θ. Since the magnetic suspension stage is an inherently unstable plant, identification of the magnetic suspension stage

62 cannot be obtained from open loop response. Therefore, PID controllers are employed for stabilizing the system and data sequences x (k) and u x (k) are acquired after the floator is stabilized around a specified position. In order to satisfy persistently excited condition, the floator is commanded to track a square wave as shown in Figure 4.2. The resulting control effort is similar to a periodic impulse signal, which is rich enough in frequency domain so that the parameters can be determined uniquely. There are two kinds of algorithms to perform system identification, gradient method and least square method. Because of its computation simplicity, gradient method is suitable for on-line system identification. However, the converged parameter values are highly dependent on initial values and parameter updating rate. Despite of its higher computation cost, least square method is able to locate the global minimum location of the cost function. Therefore, least square method is employed in this phase of the research. To perform system identification, signal conditioning must be performed. Offset values must be eliminated from z and v. If the offset value is time-varying as shown in Figure 4.3, a curve is used to fit the time-varying mean value and then remove the moving trend from the original data. By defining T ε ( k ) = x( k) φ( k) θ. The objective of least square method is to minimize cost function J = where N L is the total data length acquired. N L k= N (ε ( k)) (4.3) 52

63 T By defining the matrix Φ = [ φ( N ) φ( N 1) φ( N + 1) ] and L L T = [ x( N L ) x( N L 1) x( N + 1) the cost function can be expressed as Y ] J = ( Y Φθ ) ( Y Φθ ) T *. The optimum parameters θ that minimize (4.3) is * T 1 T θ = ( Φ Φ) Φ Y (4.4) On-line system identification algorithm In some situations where on-line system identification is required such as indirect adaptive control or the dimension of the matrix Φ is too massive to be stored in the limited memory of the hardware, on-line algorithm is desired. The algorithm obtains pseudo inverse solution (4.4) by iteration at each step. Therefore, instead of solving optimal * θ in single step, on-line algorithm updates parameter θ (k) at every iteration. Intermediate variables P(k) and K(k) are obtained at each iteration, and θ (k) are given by K( k) where I 2N is the 2N 2N identity matrix 1 + φ( k) P( k 1) φ( k) = (4.5a) T P( k 1) φ( k) T N φ (4.5b) P( k) = ( I 2 K( k) ( k) ) P( k 1) θ ( k) = θ ( k 1) + K( k)( x( k) φ( k) T θ ( k 1)) (4.5c) Choosing N = 7 and performed system identification procedure for x-axis motion control, parameters are obtained. Correlation test is used to validate the accuracy of the system identification. Auto-covariance of the residual error is given by 53

64 N L Cov( i) = e( k) e( k i) (4.5) k = N + 1 T * where e( k) = x( k) φ( k) θ is the residual error of the system identification. If the result of system identification is acceptable, the residual error should be un-correlated, i.e. N L 1 Cov( i) = e( k) e( k i) = = k N if if i = 0 i 0 (4.6) The auto-covariance Cov(i) is shown in Figure 4.4, which is very close to a delta function. Therefore, the 7 th order ARMA model is sufficient for obtaining input-output relation of the magnetic suspension stage. To interpret the physical meanings of the identified model, the discrete time model is then converted to continuous time model using zero order hold (ZOH) method. The system identification for x-axis is performed in two stages to inspect model variation within the working space. In the first stage, system identification is performed at different positions along x-axis to demonstrate model variation, while in the second stage, system identification is performed at different locations along z-axis. The identification results of the first stage are listed in Table 4.1, in which ARMA model (4.1) is expressed in term of partial fraction expansion. It is seen that the dynamics of the system model can be divided into four modes. The first mode is the slowest and very close to a double integrator, which is the dominant dynamics of the system to be regulated by controller. The second and third modes are mechanical vibrations of the 54

65 magnetic suspension system. The resonant frequencies of these two vibration modes are around 240 and 400 Hz. The fourth mode comes from the system dynamics of the current drives. When driving the current of the single phase linear motor by power amplifier, a bench test shows that the bandwidth of the current loop reaches over 5 khz. Since the bandwidth of the current loop is several times higher than the sampling rate, signal aliasing occurs and therefore the identified dynamics of current loop is not correct. Figure 4.5 shows the pole-zero map of the identified x-axis model. It is seen that poles and zeros of the second and third modes are close to each other, resulting in underdamped mechanical vibrations. The first mode dynamics is of great interest when designing controller since it represents nominal model of the rigid body motion of magnetic suspension stage. It is seen that the system model is almost invariant along x-axis, which is highly agree with the feature of the single phase linear motor. Because of the symmetric structure of single phase linear motor, the model variation of x-axis actuation along y-axis is even smaller. It is obvious that the utilization of single phase linear motors greatly reduce model variations when planar motion control is needed. In the second stage, system identification is performed at different position along z-axis. As it has been mentioned earlier, the gap of the single phase linear motor varies as floator moves vertically inside the working space. To verify the model variations of x- axis actuation along the vertical axis, same procedure was performed at different height 55

66 and the identified nominal model ( first mode dynamics ) is shown in Table 4.2. It is seen that model variation is larger, especially the numerator. Figure 4.6 show that the numerator of P(s) varies linearly with the airgap between the coil and permanent magnets, which is highly consistent with the FEM simulation results. The system identification results show that the x-axis model is almost invariant when large planar motion control is desired. As for large travel in vertical axis, the parameter variation is in more significant and needed be compensated by advanced controller design. INPUT SYSTEM TO BE IDENTIFIED OUTPUT MEASUREMENT NOISE MEASUREMENT NOISE OBSERVED INPUT u(k) IDENTIFICATION TECHNIQUE OBSERVED OUTPUT y(k) SYSTEM MODEL Figure 4.1 System identification process 56

67 2 X (micron) Time (sec) Control Effort (N) Time (sec) Figure 4.2 Position and control effort response during system identification process Time After trend removing 0 Time Figure 4.3 Signal conditioning before system identification 57

68 20 x Auto-covariance of residual error Index I Figure 4.4 Auto-correlation test of system identification 1 st mode 2 nd mode 3 rd mode 4 th mode ~ P( s) = s 5.35 ± j11.77 s + 97 ± j1454 s ± j2295 s ~ P( s) = s 5.95 ± j11.77 s ± j1502 s ± j2542 s ~ P( s) = s 5.99 ± j11.77 s ± j1603 s + 121± j2623 s Table 4.1 System identification results at different location along x-axis 58

69 Pole zero map rd mode 2nd mode th mode 1st mode Imag Axis Real Axis Figure 4.5 Pole-zero map of the identified system Z=0.5mm Z=1.0 mm Z=1.5 mm Z=2.0 mm X-axis system model P( s) = s 5.12 ± j P( s) = s 5.93± j P( s) = s 5.71± j P( s) = s 5.49 ± j11.76 Table 4.2. x-axis identification at different locations along z-axis 59

70 Numerator of P(s) Z(mm) Figure 4.6 Parameter variation of x-axis model 4.2 System model identification for vertical axes motion control For vertical axis motion control, feedback linearization is first carried out by employing a calibrated force-current model. Figure 4.7 shows the schematic diagram of vertical axis motion control. The equation for the model of the MSS can be expressed as C iz M z = Mg (4.7) 2 ( d z + a) 0 2 where i z is the current output by current drive d 0 is the nominal airgap, C and a are two parameters calibrated by force-current relationship. It is obviously seen that the force generated by a electromagnet is nonlinear with respect to current i z and position z. 60

71 The desired current is calculated by the computer and was sent to the current drive. The current drive receives the current command i z * and converted it into real current i z. If the dynamics of the current drive is fast enough, i.e. i z follows i * z quickly * such that i z = i z at the end of every sampling instant, we can make the system in a * linear form by introducing a virtual term v, which is related to i z as follows: i M = ( d 0 + a z) ( u z g) (4.8) C * z + The nonlinearity is eliminated and system dynamics becomes z = (4.9) u z and a linear controller can be designed for stabilizing and regulating position z. However, because of the significant inductance of the coil inside the electromagnet, the real current i z cannot follow the current command i * z quick enough, and hence the dynamics of the current drive must be considered in system modeling as shown in Figure 4.7. Furthermore, since parameters C and a of the electromagnet cannot be calibrated perfectly, the system is still a nonlinear system even though feedback linearization is performed, which makes controller design difficult. Following the same approach as performed for x-axis system identification, the floator is levitated to a certain height and start to track square wave reference for 20 seconds. The position response z and control effort u z is shown in Figure 4.8. Using the system identification algorithm for x-axis, the parameters of the z-axis model are identified at different height and the identification results are shown in Table

72 The system dynamics of z-axis motion control can be divided into four modes as shown in Figure 4.9. Similar to what was obtained in x-axis system identification, the first mode is a slow, unstable dynamics which must be stabilized by control system. The first mode is slow and very close to a double integrator, which is the dominant system to be controlled. The second and third modes are mechanical vibrations of the magnetic suspension system. The resonant frequencies of these two vibration modes are very close to the frequencies identified in x-axis system identification. The fourth mode comes from the system dynamics of the current drives. Because of large inductance, the power amplify cannot convert the current command into real current very quickly, and a bench test shows that the bandwidth of the current loop is around 120 Hz, which is highly consistent with the bandwidth of the fourth mode. The first mode is of great interest since it is the dominant dynamics of the system. From the system identification result, the parameter variation is more significant than the x-axis model. For example, the first parameter in the denominator obtained at z = 2.0mm is five times as large as which obtained at z = 0.5mm. From the results of system identification, it is obvious that each axis of the improved MSS can be modeled as a position dependent system in which parameters variation are significant when traveling along z-axis. Another system identification procedure was carried out for verifying effectiveness of feedback linearization. Instead of using virtual force u z, current command i * z is employed as system input in the identification algorithm. The identification result is shown in Table 62

73 4.4. It is shown that the transfer function has two real-axis poles, one is stable while the other is unstable. The result matches with the real physical model of magnetic suspension very well, which can be obtained by linearizing (4.7) at a specified position. It can also be observed that the unstable pole moves away from the imaginary axis as the airgap decreases. 63

74 System Identification z d e Linear Controller u z PC Feedback Linearization M + a z) ( uz C * z = ( d 0 + i i z * g) Current Drive i 1.0 * i i z C i M z z = 2 ( d z + a) W(rad/s) 0 2 Mg z Figure 4.7 Control block diagram of vertical axis motion control Z (m) x Control Effort (N) Figure 4.8 Position and control effort response of z-axis during system identification process 64

75 Z=0.5 mm 1 st mode 2 nd mode 3 rd mode 4 th mode Z=1.0 mm Z=1.5 mm Z=2.0 mm Table 4.3 System identification results at different location along z-axis 3000 Pole zero map 3rd mode th mode 2nd mode 1st mode Imag Axis Real Axis Figure 4.9 Pole-zero map of the identified system 65

76 Z=0.5mm Z=1.0 mm Z=1.5 mm Z=2.0 mm Z-axis system current-position model P( s) = 2 s s poles : , P( s) = 2 s s poles : , P( s) = s 2 P( s) = s poles : , s poles : , s Table 4.4 Current-position system model of Z-axis 4.3 Summary In this chapter, system identification was performed to obtain the system model of the developed magnetic suspension system. In order to investigate model variations inside the working space, system was identified at various positions and models were obtained. Because of different actuation methods employed for vertical and horizontal motion control, it is concluded that the vertical actuation system suffers much more model variation than horizontal ones. The identified model can be separated into two parts, nominal part and perturbed part, which were then interpreted by its physical meanings. 66

77 CHAPTER 5 LARGE TRAVEL PRECISION MOTION CONTROL IN MULTIPLE DEGREES OF FREEDOM 5.1 Introduction Since magnetic suspension systems are inherently nonlinear, unstable systems, control systems must be developed for real time control purpose to achieve stabilization. In the case that DC electromagnets are employed and the air-gap varies during the course of motion control, feedback linearization of the nonlinear force relationship in terms of the coil current and the air-gap can be implemented so that the nominal model of the feedback linearized system is a linear one. However, due to modeling errors perfect feedback linearization is not possible. When parameter variations were moderate, by employing a linear robust control algorithm, it was demonstrated experimentally that the controller regulated the closed-loop system dynamics such that the transient response of the system was independent of the operating point [Shan, 2002]. However, when large travel motion control or rotational motion control is desired, parameter variations can be significant even after employing feedback linearization and constant gain robust controllers may not be adequate for realizing ultra precision motion control. Therefore, advance control algorithms are necessary for dealing with time-varying parameters. 67

78 In Chapter 4, it was demonstrated from the results of system identification that the magnetic suspension system actuated by electromagnets (or reluctance forces) suffers from significant model variations within the desired working space. Many methods are proposed for compensating small model variations through out decades, such as H- infinity, adaptive control, gain-scheduling, sliding-mode control. H-infinity control is well known for its robustness for both parameter variations and disturbance rejection. Since the robustness is guaranteed by small gain theorem, which is a sufficient condition for achieving robust stability, it becomes too conservative when applied to systems with large parameter variations. Sliding mode control is another candidate for maintaining robustness under model variations, in which disturbance and modeling error can be compensated by high gain control. However, sliding mode is also infamous for its chattering effect and therefore performance would be seriously decayed. In this research, two kinds of controller design, indirect adaptive control and disturbance observer, are attempted for model regulation and disturbance rejection. In Section 5.2, an indirect adaptive controller is implemented for a magnetic suspension system previously developed in CMML. Section gives an overview of a magnetic suspension system actuated by 10 electromagnets, which is capable of six degrees of freedom motion control. In section 5.2.2, based on a lumped parameter force model of the electromagnet a nonlinear model for the x-y-θ motion of the MSS is derived. Feedback linearization is then implemented in order to cancel the nonlinearities existing in the dynamic equations. Due to modeling errors, feedback linearization is imperfect and the parameter values of the feedback linearized system vary as the operating point changes. Consequently, for the purpose of x-y-θ motion control, each 68

79 axis of the feedback linearized system is modeled as a double integrator having gain value depending on the position of the stage and subjected to a disturbance. In section 5.2.3, for the purpose of large travel x-y-θ motion control, an indirect adaptive control algorithm is designed and implemented for each axis of the feedback linearized system. Finally, experimental results are shown in Section to demonstrate that the indirect adaptive controllers have superior tracking ability when compared to constant gain robust linear H controllers. In Section 5.3 a disturbance observer is designed for the newly developed magnetic suspension system for large travel motion control in x-y-z axes. The objective is to achieve uniform performance within the three-dimensional working space 4mm 4mm 2mm. It is seen from system identification that each axis of the magnetic suspension system can be modeled as a second order system with position-dependent parameters. For controller design, a nominal model is used and the system is stabilized by nominal linear control in Section The external disturbance and nonlinear parts are lumped into a disturbance term satisfying matching condition. In Section 5.3.2, a disturbance prediction algorithm is incorporated into the disturbance observer design for disturbance rejection. Several experiments are performed and the results are shown in Section Indirec adaptive controller design In this section, an indirect adaptive controller is implemented for a magnetic suspension system previously developed at CMML. The actuation scheme uses 10 electromagnets for six degrees-of-freedom motion control. Because of the significant 69

80 nonlinearities induced by electromagnetic actuation, precise large travel motion is difficult to be achieved by constant gain linear control An overview of previously developed MSS Figure 5.1 depicts the magnetic suspension stage (MSS) designed and developed by Shan et al at CMML. It consists of two separate parts: a floator that can move freely within the working volume and a fixed frame out of the floator called stator. The electromagnets are mounted on the stator, while the floator contains the electromagnetic force targets (ferromagnetic plates) and sensor targets, namely, six retroreflectors, as part of the laser interferometric sensing system. This design can provide six-degree-offreedom actuation and measurement. The traveling volume of the MSS is 3mm 3mm 1.8mm in the three linear degrees of freedom. The mass of the floator is 2.4 kg. The working principle of the stage is illustrated in Figure 5.2. The four electromagnetic actuators in the vertical scheme provide the vertical suspension force. These four actuators also provide torques for rotations around the x-axis and the y-axis. The x-axis actuation force is provided by two electromagnets in the x-axis. The other four actuators in the horizontal scheme provide the z-axis torque in addition to the y-axis actuation force. A six-dof measurement system was developed and integrated with the MSS. The developed measurement system consists of a laser source and six interferometers with 70

81 retroreflectors mounted on the floator. The measurement system is based on six modified Michelson heterodyne interferometers [Zhang et al 1999]. Three laser beams are arranged along the vertical direction, and the other three are in the horizontal plane. This configuration is analytically and numerically verified to be free of singularity and possesses very high resolution in linear motion measurement. By properly arranging the six retroreflectors and the six interferometers high rotational measurement sensitivity can also be achieved. Since the combination of a corner cube retroreflector and a plane mirror is used to replace the plane mirror in the homemade interferometer, the developed measuring system has a much larger rotational range compared to plane mirror laser interferometer systems. The measurement volume is dependent on the dimension of the assembly and on the size of each retroreflector. Using six interferometers with specific designs in the developed system, the measured optical path changes can be related to physical quantities such as linear and/or angular displacements along the transducer axes. In the developed system, these measured displacements need be integrated so as to compute the six-dof parameters of the controlled motion. The relation between the controlled six-degrees-of-freedom motion and the six laser readings was derived [Zhang et al 2000]. Based on this relation, an advanced computational algorithm was developed to calculate in real time the six parameters of the resulting motion. The developed measuring system has a resolution of 1.24 nm in linear measurement, and in a very short period of time, the influences of environment such as temperature change and the low frequency mechanical vibration are small. 71

82 The closed-loop control system of the MSS consists of 10 linear power amplifiers as the current source of the 10 electromagnets, one Pentium 233 PC as the controller, and a 12-bit D/A card with 16 channels converting control effort signals to current commands of the 10 power amplifiers. The measurement results are generated by 6 counter boards, which calculate the phase difference from the laser interferometer and transfer the readings to computer through a VME bus. The maximum amplitude of the current is limited at 6 amps to protect the electromagnetic coils from overheating. Due to the complexity of the employed control algorithm as well as the needed computation time in converting six laser readings to six-degrees-of freedom motion, the sampling rate is set at 1 khz. Figure 5.1 Basic components of previously developed MSS 72

83 F9 F8 F10 F7 F2 F6 F3 xˆ ẑ ŷ F4 F5 F1 Floator Figure 5.2 Actuation scheme of the previously developed MSS Modeling and System Identification In this section, based on a lumped parameter force model of the electromagnet a nonlinear model for the x-y-θ motion of the MSS is derived. In order to cancel the nonlinearities existing in the dynamic equations, feedback linearization of the nonlinear force relationship in terms of the coil current and the air-gap is implemented. The parameters of the feedback linearized system are identified at different operating points by closed-loop system identification. Due to imperfect feedback linearization, the parameter values of the feedback linearized system vary as the operating point changes. Therefore, for the purpose of x-y-θ motion control, each axis of the system is modeled as 73

84 a double integrator having gain value depending on the position of the stage and subjected to a disturbance. Modeling of X-Y-q Motion Figure 5.3 shows the free body diagram of the x-y-θ motion of the magnetic suspension stage. Electromagnets EM1 and EM2 generate push-pull forces to control the x axis motion of the MSS, while EM3, EM4, EM5 and EM6 generate forces to control the y axis motion and the theta rotation. In many applications, the theta motion is used to compensate for orientation error that is normally less than 1. Assuming small θ rotation, each ferromagnetic plate is nearly parallel to the pole face of its corresponding electromagnet and the resulting magnetic force is dominated by the normal component, which is characterized by the air-gap and coil current. In this paper, a lumped parameter model [Shan et al 2000] of the normal force is employed: F i 2 i I = C (5.1) 2 ( a + g ) i where g i denotes the air-gap, I i is the control current, and C and a are two constant parameters. Assuming small θ rotation, the equations of motion are decoupled and can be derived as follows: M x = F 1 F 2 (5.2a) M y = F (5.2b) 3 + F4 F5 F6 74

85 I zz W θ = ( F4 + F6 F3 F5 2 ) (5.3c) where M and I zz are the mass and moment of inertia of the floator, and W is the width of the floator. F s ( i = 1 ~ 6 ) are normal forces produced by electromagnets EM1~EM6 i given by (5.1). EM3 EM4 Y EM2 θ X EM1 M EM6 EM5 Figure 5.3 Free body diagram of the MSS in x-y-theta motion In order to cancel the nonlinearities existing in the dynamic equations, feedback linearization of the nonlinear force relationship is implemented through the following formulation, I ui + f0 = ( gˆ aˆ) Cˆ i (5.3) i + 75

86 where Ĉ and â are calibrated values based on the curve fitting results in [Shan et al 2000], ĝ i is the air-gap estimated from position and orientation of the floator, u i is the desired control force as if perfect feedback linearization can be accomplished, f 0 is the biased force added to each electromagnets to enhance stiffness of the system. It is evident that the system has 3 outputs, x, y and θ, and 6 inputs, u i ( i = 1 ~ 6 ), with additional restrictions u i > f0. In order to develop a suitable control strategy, force distribution is necessary. For x - y -θ motion, the task can be divided into two components, the first one is for the x -axis motion, while the second component is for the y -θ motion. Taking the x axis as an example, (5.1) and (5.3) are substituted into (5.2a), the equation of motion is given by where M x = K ) (5.4) 1u1 K 2u2 + ( K1 K 2 f0 K i 2 C( gˆ i + aˆ) =. 2 Cˆ( g + a) i One possible solution for force distribution in the x -axis is to define a virtual term u x, acting as the net control effort for the x -axis motion in a common control problem, and the distribution between u 1 and u 2 can be formulated as follows [Shan et al 2000]: 76

87 u x if u x 0 u1 = 0 if u x < 0 (5a) u x if u x < 0 u2 = 0 if u x 0 (5b) Force distribution for the y - θ motion is more complicated than that for the x -axis motion. For a general approach, one can refer to [Shan et al 2000]. With feedback linearization and force distribution, the x y θ motion of the magnetic suspension stage can be modeled as follows: x = α u + d (6) x x x y α = θ α y θy α α yθ θ u u θ d + d y θ y Ideally, α x, α y and α θ take the value of 1, and α y θ and θy α are zeros. However, due to the discrepancy between the estimated values and the real values, α x, α y, α θ, α y θ and α θy are no longer constants and are functions of the position and orientation, x, y and θ, of the floator. Additional terms, d x, due to imperfect feedback linearization of x, y and θ axes. d y and d θ, are considered as input disturbances By assuming small θ rotation, α y θ and θy α are small when compared to diagonal terms α y and α θ. For simplicity, the 2 input-2 output system can be simplified as two SISO systems. 77

88 y = α u + d (5.7a) y α θ y θ y θ = u + d (5.7b) θ System Identification Equation (5.6) and (5.7) describe the feedback linearized system dynamics of the magnetic suspension stage in x-y-theta motion. Each axis of the feedback linearized system can be viewed as a double integrator subjected to an input disturbance with the gain value depending on the air-gaps, and thus on the position of the floator. The feedback linearized model of the x -axis motion is shown as a block diagram in Figure 5.4. In this section, system identification is performed to demonstrate the validity of the proposed mathematical model as well as parameter variations due to changing air-gaps, i.e. different position of the magnetic suspension stage along both the x axis and the y axis. Since the system identification procedure is executed in a digital computer, it is practical to express the plant model in a discrete time form. The continuous equations (5.6) and (5.7) can be transformed into difference equations using the forward Euler s method. For example, in the x-axis, the difference equation is expressed as 2 x K = 2xK 1 xk 2 + T [ α x ( xs ) uk 2 + d x ( xs )] (5.8) 78

89 where T is the sampling time, x S denotes the specific position where system identification is performed. By using ˆα x, K and d x, K ˆ as the estimation of α x ) and d x ) at time index K, x ( S x ( S the estimated value xˆ K can be generated as 2 x ˆ ( ˆ ˆ K = 2xK 1 xk 2 + T α x, KuK 2 + d x, K ) (9) In view of the fact that the MSS is an inherently unstable plant, data sequences x K and u K are acquired after the floator is stabilized around a specified position. Since there are two unknown parameters to be identified, in order to have two parameters uniquely determined, the signal u K needs to be persistently excited (PE) at least of order 2. In order to achieve this condition, the floator was stabilized at a specified position initially and then commanded to track a square wave trajectory at time equals to 5 second as shown in Figure 5.5. The resulting control effort u K is similar to a periodic impulse signal rich enough in frequency domain to determine parameters α x ) and d x ) x ( S uniquely. The basis of parameter estimation is to minimized the objective function x ( S 2 K K K 2 J = ε = ( x xˆ ) (5.10) 79

90 where K 2 2 ε is the estimation error. By defining ϕ = [ T u T ] T, ˆ = [ αˆ dˆ ] T x S x S K K 2 Θ, Θ = [ α ( x ) d ( x )] T ~, and ΘK = Θ Θˆ K, the estimation error can be expressed as follow. K K K ε K T K = Θ ~ ϕ (5.11) K A projection algorithm can be incorporated into the system identification procedure [Astrom et al 1989], given by ηε ϕ Θˆ K = Θˆ K 1 + (5.12) ρ + ϕ ϕ K K T K K with ρ > 0 and 0 < η < 2. Under the PE condition, ˆα x, K and d ˆ converge to ) x, K α x ( x S and d x ). x ( S Figure 5.5 shows an experimental result on parameter identification when the MSS was stabilized at the origin of the x -axis. The sampling rate T is 1ms. The same experiment was conducted twice, at each time different initial values were specified. As can be seen from Figure 5.5, since the PE condition was not met in the first 5 second the calculated parameter values very much depended on the initial values. Nevertheless, starting from t = 5 sec the PE condition had been met and the parameters converged to the same values in spite of different initial guesses. It is seen that while the stage is 80

91 located at the origin of the x -axis, or x = 0, the identified parameters are α ( 0) = 0 79 S x. and d x ( 0) = It is obvious that the feedback linearization cannot completely eliminate nonlinearity of the force model of DC electromagnet. 81

92 d x u x α x 1 2 s x Figure 5.4 Feedback linearized model of the x-axis motion αˆ x dˆx time (sec) Figure 5.5 Parameter identification under two different initial conditions 82

93 Parameter variations The estimated parameter values, α ( 0) = 0 79 and d x ( 0) = shown in Figure 5.5, x. also demonstrate the fact that the implemented feedback linearization cannot be perfect and the discrepancy between the estimated values and the ideal ones cannot be ignored. Moreover, the feedback linearized system is not really a linear one, consequently the parameter values of the feedback linearized system vary as the operating point changes. In order to illustrate parameter variations along the x -axis, the system identification approach was performed at x s = ± 0. 5mm and ± 1.0mm. Figure 5.6 shows the variations of α x versus operating positions along the x -axis. The maximum deviation of α x from the nominal position x S = 0 is about 16.7%. Similar result was observed for the y-axis. It is evident that when deviating from the nominal position x = 0, one of the two s air-gaps becomes smaller. When decreasing the air-gap, the inductance of the electromagnetic coil increases rapidly and modeling uncertainty can be greater. Therefore, parameter variations cannot be neglected when large travel, causing one of the air-gaps to become smaller, is desired. Particularly, in tracking control, the air-gap changes constantly, therefore, the parameters of the system model vary with time. There are at least three options in current control technology that may be able to deal with tracking system having time-varying parameters. First, gain scheduling can be used to adjust control parameters according to pre-determined models. However, stability cannot be easily proved in this approach. Moreover, it is very time consuming to perform off-line system identification and controller design for a large number of operating 83

94 points. Second, the variations of α x can be viewed as multiplicative uncertainties, consequently the system can be regulated by robust linear controllers such as H controllers. Since sufficient condition for stability in H control is guaranteed by small gain theorem, the resulting design is often too conservative to accommodate large multiplicative uncertainties. Third, to ensure superior tracking performance while maintaining stability of the system, adaptive control with adjustable controller parameters is another alternative αˆ x position (mm) Figure 5.6 Estimated αˆ x versus operating positions along the x-axis Indirect adaptive controller design Adaptive control can be divided into several categories over decades of evolution. Adaptive control combined with neuro-network or fuzzy logic, applied to nonlinear systems, has been studied extensively in recent years [Sun et al 2001 and Yoo et al 2000]. 84

95 However, it is not practical to implement these control techniques in the current system due to excessive computation time needed for those control algorithms. Nonlinear direct adaptive control [Slotine et al 1991 and Yeh et al 1995] is another option for the MSS since stability can be easily guaranteed by Lyapunov function approach. There are some inherent weaknesses such as over parameterization [Astrom et al 1989] when applied to tracking problem. In the remainder of this section an indirect adaptive controller design is presented and the stability as well as tracking performance of the controlled system is discussed. In indirect adaptive control, the system parameters are estimated on-line and used to calculate the controller parameters. Therefore, the control system for the MSS now consists of an on-line system identification mechanism and a feedback controller. The feedback controller has three components, feedback linearization, model cancellation, and nominal linear control. A closed-loop control block diagram for the x -axis of the controlled MSS is depicted in Figure 5.7, in which feedback linerization is not explicitly shown. The on-line system identification implemented here is different from system identification discussed in section in two ways. In tracking control, with prescribed input trajectory PE condition can hardly be met by on-line system identification. For the MSS, air-gaps constantly change during tracking control, therefore the system parameters vary with time. Consequently, the estimated parameters by on-line system identification do not necessarily converge to their true values as obtained in section Nevertheless, the objective of on-line system identification is still to make the estimation error ε, defined in (11), converge to zero, and for control design purposes the estimated 85

96 parameters are treated as true parameters and are used to calculate the controller parameters. This design approach is called certainty equivalence [Ioannou et al 1996 and Li et al 1988]. By treating the estimated parameters, αˆ x and dˆ x, as if they were true ones, α x and d x, the normalization is performed by model cancellation such that the resulting dynamics of the plant is a pure double integrator. According to the control block diagram depicted in Figure 5.7, the x -axis motion can be regulated to e = v + ( α αˆ ) u + ( d dˆ = v + ε (5.13) x x x x x ) under model cancellation u x 1 = ( v dˆ x + x d ) (5.14) α ˆ x where ε is the estimation error of system identification and v is the nominal linear control with constant parameters. Without loosing generality, v can be expressed as v = D( s) e (5.15) where e = x x, and D (s) is a realizable linear controller designed for the linearized d system, which is a double integrator. Substitute (5.15) into (5.13), we get 86

97 G( s) e = ε = H ( s)ε (5.16) 1 + G( s) D( s) 1 where G ( s) =. 2 s In (5.16), it is obvious that the tracking error is induced by the resulting estimation errors ε of the on-line system identification process through a stable transfer function H (s). When moving along the trajectory, parameter values of the system vary at the same frequency as that of the trajectory, and thus the resulting estimation errors ε. While the adaptive law is designed to minimize the estimation errors, D (s) must be designed such that the gain of H (s) is minimized over the frequency range of the desired tracking trajectory. In order to maintain stability and to prevent saturation of the control effort, be bounded away from zero in (5.14). One method to avoid αˆ x must αˆ x going through zero is to modify the adaptation law using a projection method. Such a modification does not affect the convergence of ε and is achieved by using a prior knowledge of α x, namely the lower bound α 0, such that α x α0, is known [Ioannou et al 1996]. Applying the projection method to the adaptive law, we obtain 87

98 αˆ x, K = αˆ x, K 1 + η αˆ T 2 ρ + T 4 x, K 1 ε K u (1 + u K 2 2 K 2 ) if αˆ x, K otherwise α 0 (5.17) at initial time, α ˆ x, 0 is chosen so that α ˆ,0 > α0 x. On-line system identification x d e 2 s D(s) x d v dˆx 1 αˆx u x α x d x G(s) 1 2 s x Nominal Linear Control Model Cancellation Feedback Linearized Plant Figure 5.7 Closed-loop control block diagram for the x-axis Experimental results Indirect adaptive controllers are designed and implemented for the magnetic suspension stage. Experiments are conducted. In order to demonstrate its superior tracking performance, the experimental results are compared with those of linear H controllers with fixed gains. 88

99 Adaptive Controller design Following the design procedure discussed in section 5.2.3, discrete time models of the magnetic suspension stage are employed in on-line system identification and the projection method is adopted as a parameter adjustment mechanism. The parameter adaptation rate η should be selected large enough so as to compensate fast parameters variations. However, when η > 2, the on-line system identification mechanism becomes unstable. Parameter ρ is chosen to avoid singularity in (5.17) when u K is small. For x- axis, η = 0.4, ρ = 1 are selected to meet the requirements above. The nominal linear controller is chosen to be as follow. D( s) = K s + K µ s + 1 K s I D P + + K s II 2 (18) where K, K, K and D P I K II are parameters to be tuned and µ is a small positive number such that D (s) is realizable. The controller D (s) is simply a PID controller plus a double integrator, which regulates the steady state error to zero when subjected to type I reference trajectory. Performance of the closed-loop system is regulated by tuning the parameters of D (s) such that the resulting H (s) is stable, with minimized gain over the frequency range of the desired tracking trajectory. This procedure is applied to the controller design for each individual axis. In this paper, only the design parameters for 89

100 the x -axis, 2 K D = 10, 4 K P = , 4 K I = 18 10, 4 K II = 60 10, 4 µ = , are given to illustrate typical design results. H Controller design H control has been widely accepted to be a powerful linear controller design method and has been extensively implemented on magnetic bearing systems and magnetic levitation systems based on linearized models. For the purposes of comparison, H controllers are designed for the magnetic suspension stage. Figure 5.8 shows a generalized control block diagram in the H controller design framework. In the figure, P stands for the linearized model, W ( ), W ( ) and W (s) s are the weighting functions 1 s 2 s 1 to be chosen, and Ps ( s) = P( s) Ws ( s). NM = Ps ( s) is the coprime factorization of P s (s). It has been shown from the result of system identification that the magnetic suspension stage with large travel motion can be modeled as a double integrator with slow time varying gain subjected to an input disturbance. Since the plant P now has two poles on the imaginary axis, the problem is singular as some of the invariant zeros appearing on the augmented plant are located on the imaginary axis. Consequently, the general H controller design procedure established in MATLAB cannot be applied directly. In this paper, a loop-shaping procedure [McFarlane et al 1992, Kang et al 1999 and Tsai et al 1991] is adopted to overcome this problem. The objective is to solve K (s) such that 90

101 W1 W 2K ( Ps K ) M 1 (19) The inequality (19) is equivalent to z z 2 2 r as shown in Figure 5.8. By using the fact that the poles of M (s) match the roots of the closed-loop characteristic equation, a portion of the system dynamics can be specified through assigning the dynamics of M (s) [Tsai et al 1991]. To have faster closed loop dynamics, M (s) is chosen as 4 M ( s) = s. Since K 4 ( s (s) has to be cascaded by W s (s) to form the controller, i.e. + 50) K( s) = K ( s) W ( s), some desirable dynamics in the controller K (s) can be attributed to s W s (s). In tracking controller design, integrators are often incorporated into the controller to eliminate steady state tracking error. In addition, two LHP zeros are added to W s such that the phase lag induced by the double integrator is reduced at cross over frequency. ( s + 12)( s + 80) Hence, the weighting function W s (s) is selected as W s ( s) =. 2 s To minimize the order of the controller so as to manage the computation time, the selected weighting functions W ( ) and W ( ) are constants. The resulting K (s ) for the x -axis is given below. 1 s 2 s s s s K ( s) = (5.20) s s s s

102 With weighting functions selected above, some specifications of the H control system can be accomplished by manipulating the inequality (5.19). It can be shown that the closed-loop system is capable of tracking a reference signal consisting of frequency component up to 3 Hz under the variation in α x up to ± 30%. The procedure is also applied to the controller design for other axes. W1 z1 W 2 z 2 r 1 M K Ws P y Controller P s Figure 5.8 A general H control design block diagram with a loop-shaping procedure Circular Contouring and Angular Tracking In order to illustrate the superior tracking ability of the designed indirect adaptive controllers, three experiments were conducted. The first experiment was to command the stage to contour a circle with a diameter of 0.6 mm in 10 seconds as shown in Figure 5.9, while the second one was to contour a larger circle with a diameter of 2.0 mm again in 10 seconds as shown in Figure According to the results from system identification in 5.2.2, parameter variation under the second experiment is greater than that under the first one. 92

103 The resulting stage trajectories when employing indirect adaptive controllers in both experiments are also shown in Figure 5.9 and Figure 5.11, respectively. It is seen that the stage precisely tracks the two circles. The contouring and tracking errors are within about ± 10 nm, as shown in Figure 5.10a and Figure 5.12a. The contouring and tracking errors when employing H controllers are shown in Figure 5.10b and Figure 5.12b. To eliminate low frequency error, the H controller is designed with high gain in low frequencies. To minimize amplitude of positioning noise, the cross over frequency must be high enough to give levitated bandwidth and suppress high frequency noises. It is seen that the performance of the H controller is as good as that of the adaptive controller if the plant variation is small. In the second experiment, as the system model variation becomes larger, high bandwidth structure of the H controller tends to excite unmodeled uncertainties in the system. Therefore, the performance degrades rapidly as shown in Figure 5.12b. On the other hand, the indirect adaptive controller can adjust its parameters according to system model variations. In the third experiment, the stage was commanded to track a sine wave trajectory with a magnitude of o 0.5 in the theta axis, as shown in Figure As shown in Figure 5.14a, the indirect adaptive controller can achieve tracking error as small as about 5 ± degrees, while the H controller can only reach 4 ± degrees as can be seen in Figure 5.14b. These experiments demonstrate that adaptive controllers have superior tracking ability when compared to controllers with constant gains. Estimated time varying parameters are shown in Figure Since the trajectories of the three experiments all vary smoothly, the resulting control efforts are not persistently 93

104 excited enough so as to distinguish αˆ and dˆ independently. Therefore, the identification results are different from those in and the estimated parameters do not converge to their true values. It is seen that in Figure 5.15a, variations of both parameters are not independent and are totally out of phase. The parameter variations under the large circle contouring are greater than that of the small circle, thus the controller parameters adaptation is increased to accommodate larger model variation encountered in the large circle. Parameters variation in Figure 5.15b shows that the variation of αˆ θ is larger when θ is varying in positive range. In this situation, the constant gain H controller is unable to compensate the varying parameter and shows large tracking error in Figure

105 4 x y(m) x(m) x 10-4 Figure 5.9 Trajectory of contouring a 0.6mm diameter circle Figure 5.10 Tracking and contouring error of a 0.6mm diameter circle: (a) with indirect adaptive control (b) with robust H control 95

106 Figure 5.11 Trajectory of contouring a 2.0mm diameter circle X-axis tracking error(m) Y-axis tracking error(m) Contouring error(m) X-axis tracking error(m) Y-axis tracking error(m) Contouring error(m) Time(sec) Time(sec) (b) Figure 5.12 Tracking and contouring error of a 2.0mm diameter circle: (a) with indirect adaptive control (b) with robust H control 96

107 Degree Time(sec) Figure 5.13 Tracking control trajectory in the θ-axis Tracking error(degree) 2 x Time(Sec) 4 x 10-4 (a) Tracking error(degree) Time(Sec) (b) Fig 5.14 Tracking error of the θ-axis : (a) with indirect adaptive control (b) with robust H control 97

108 0.82 αˆ x diameter = 2.0 mm 0.76 diameter = 0.6 mm dˆ x diameter = 0.6 mm diameter = 2.0 mm time(sec) (a) 0.9 αˆ θ dˆθ time(sec) (b) Figure 5.15 On-line system identification results in indirect adaptive control (a) Circle contouring (b) theta axis tracking 98

109 5.3 Disturbance observer It is shown in Section 5.2 that adaptive control is able to cope with large parameters variation. Performance can be regulated through adjusting controller parameters by adaptation mechanism. Although it demonstrates that performance is better than using constant gain H-infinity control in section 5.2, parameters adaptation should be carefully monitored so as to avoid instability especially for an unstable plant [Lawrence et al 1990 and Golden et al 1991]. To fulfill this, projection method is usually adopted to update the controller parameters such that they are bounded inside a predefined boundary. This would results in a trade-off condition between stability and performance. For instance, a small boundary would guarantee higher stability but results in lower performance. The approach of using disturbance observers has been widely recognized as a convenient method for disturbance rejection. In a disturbance observer, disturbance is estimated on-line from input-output signal at previous sampling instants to achieve compensation. It is very effective to compensate steady state or slow time-varying disturbance using a disturbance observer. In this dissertation, an approach that predicts the time-varying disturbance using disturbance prediction algorithm is proposed. A nominal linear control is first utilized to stabilize the magnetic suspension system after force-current feedback linearization. Parameter uncertainties along with external disturbance force are then lumped into a time-varying disturbance term satisfying matching condition. The disturbance is then estimated using a conventional disturbance observer configuration and an algorithm is employed for predicting current disturbance from the past history disturbance information. Since the disturbance observer is formulated in discrete time, the prediction algorithm can be easily implemented by 99

110 utilizing a FIR (finite impulse response) filter. Performance of the proposed method will be demonstrated by experimental results Nominal Linear Control As it is shown in Chapter 4, the system model of each axis can be formulated as a nonlinear system x 1 = x 2 x = a p)x + a ( p)x + b( p)u + h( p, ) (5.21) 2 1( t where p denotes position of the moving stage, parameters a ( p) = a ~ ( 0 a p), i i + ~ b ( p) = b0 + b( p), in which a i0, b 0 are the nominal parameter value, a~ ~ i ( p) and b ( p) are the position-dependent parameter variations. Moreover, h ( p, t) is the external disturbance force. The system dynamics in the second state equation contains a linear part and a nonlinear part. The strategy here is to collect all linear parts and all nonlinear parts respectively. By defining x d as the desired reference trajectory and selecting new state variable x 1 = x 1 x d, x = x 2 xd 2 as error, the error dynamics of the system can be expressed as x 1 = x 2 (5.22) x 2 = a1( p)x1 + a2 ( p)x2 + b( p)u + ( xd + a1( p) x d + a2 ( p) xd ) + h( p, t) ~ ~ = a x + a x + b u + F } + { a ~ ( p)x + a~ ( p)x + b ( p)u + F + h( p, )} { t = { a10 x1 + a20 x2 + b0u + F0 } + η( p,u, t) 100

111 where F = x + a x + a x ] and F = a x + a x ]. The nonlinear parts, including 0 [ d 10 d 20 d ~ [ ~ ~ 1 d 2 d parameter variation and external disturbance, are lumped into a single term ( p, u, t) which is treated as a disturbance of the system. η, Since the controller is implemented in a digital computer, to make controller design more practical, the continuous time system (5.22) is transformed into a discrete system G S (z) using the forward Euler s method, d / dt ( z 1) /( t) =, x 1 ( 1 2 k + 1) = x ( k) + x ( k) t x2 ( k + 1) = x2 ( k) + [ a10 ( p) x1( k) + a20 ( p) x2 ( k) + b0 ( p) u( k) + F0 ( k) + η ( p, u, k)] t (5.23) The controller design approach based on disturbance observer can be expressed as 1 u( k) = ( u N ( k) + ud ( k)) (5.24) b 0 where u N (k) is the nominal linear control which applies state feedback regulation and feedforward control to the system, while u D (k) is the disturbance observer for compensating time-varying disturbance term η (k). 1 b Applying nominal linear control = ( K x ( k) K x ( k) F ( k)), where state u N feedback gains 2 1 a10 wn K = +, K 2 a20 + 2εwn =, the system is regulated to x 1 ( 1 2 x k + 1) = x ( k) + x ( k) t 2 2 ( 2 n 2 n 1 D η k + 1) = x ( k) + [ 2 w x ( k) w x ( k) + u ( k) + ( p, u, k)] t ε (5.25) 101

112 (5.25) can be viewed as a stable, linear system G P (z) subjected to input disturbance η to be compensated by control u D. Figure 5.16 shows the control block diagram of the system after applying nominal linear control Disturbance Observer Design Although the system has been stabilized by nominal control u N, the performance of the system is not regulated well because the disturbance term η ( ) appears on the right hand side of the state equation gives undesired perturbation. As it has been mentioned earlier, disturbance observer offers attractive feature for regulating performance by observing disturbance in real time estimation. The control block diagram of the disturbance observer is presented in Figure In its ideal form, the disturbance observer inverts G P (z) to determine the effective net disturbance on the system from plant output x ( ) and input u D(k). An additional compensating component is then 1 k supplied to cancel this net disturbance. A feedback controller D (z) is implemented in the outer loop for performance regulation. Since G P (z) is strictly proper in most cases, inverse of G P (z), which equals to G 1 P ( z), is unrealizable. Additional time delays are added such that N 2 1 G is realizable, i.e. G ( z) = z G ( z). Researches have been N 1 P 1 P P investigated for designing filter Q (z). Generally, to regulate the model uncertainty as well as reject disturbance, Q (z) is a low pass filter with unity gain at low frequencies. The bandwidth of Q is a trade-off result between performance and robust stability. To reach better performance as well as maintaining robust stability, high order disturbance 102

113 observer design has been investigated [Yamada et al 1996]. In the simplest implementation where measurement noise and high frequency uncertainties are not an 1 issue, Q (z) is simply a time delay, z. This implementation is also termed as time delay control [Reddy et al 1992 and Mittal et al 1997], which is widely utilized in precision motion control applications, and it provides accurate compensation for disturbances which are steady state or slow time varying. When MSS is stabilized at a position or traveling in a small volume, system model variation is very small and hence disturbance is changing slowly with respect to sampling rate, employing time delay control is sufficient for obtaining satisfactory performance. However, as it has been shown in previous section, for the improved magnetic suspension system, parameters vary greatly due to different position within the working space. When 3D large travel motion is desired, the variation of η (k) due to time varying parameters is no longer assumed to be slow and hence the disturbance estimation method needs to be improved by using higher order estimator. In this research, an innovative disturbance estimation method is proposed for compensating time-varying disturbance by using both Q (z) filter and a disturbance prediction filter P (z). Suppose the sampling rate is fast enough, the disturbance η (k) changes smoothly with time as shown in Figure A localized figure in Figure 5.18 shows that the disturbance can be fitted by a n-th order polynomial, i.e. n m η ( k ) = 0a m k, m= where a m is the coefficient. It is obvious that the fitting result becomes more accurate as 103

114 n gets larger. In order to obtain this n-th order polynomial, n+1 unknown coefficient a m must be solved by n+1 equations, which means a sequence with n+1 elements ( k 1) η ( k 2) ( k n 1) η, η must be acquired through employing input-output relationship. Since the curve fitting process is tedious and requires high cost computation, another method of finding n-th order polynomial that fits η (k) is adopted [Abramowitz 1972]. + 1 d n η( k) d k Using the property = 0 dη( k) d k and finite difference operator = η( k) η( k 1), η (k) can be estimated as n + 1 m+ 1 n η ˆ( k) = ( 1) Cmη ( k m). (5.26) m= 1 Therefore, (5.26) implies that the present disturbance η (k) can be calculated directly from previous n+1 data ( k 1) η, η ( k 2) η ( k n 1) 104. Time delay control is simply an estimation of a zero order polynomial when n=0, i.e. η ˆ( k) = η( k 1). By taking large number of terms in expansions, one obtains the First Order Disturbance Prediction (FODP) as η ˆ( k ) = 2η ( k 1) η ( k 2) and Second Order Disturbance Prediction (SODP) as η ˆ( k) = 3η ( k 1) 3η ( k 2) + η( k 3). The disturbance prediction method is implemented in the feedback loop of the disturbance observer as shown in Figure A third order low pass filter is employed as the Q filter for noise filtering and model regulation. The prediction algorithm can be realized easily by a Finite Impulse Response (FIR) filter P (z). Sending estimated disturbance (k) η into P (z), predicted disturbance η ˆ( k) is obtained and then fed back to the system to eliminate the influence of the disturbance.

115 The controller design scheme presented in this section provides more freedom in parameter tuning for an unstable plant since the system is first stabilized by nominal linear control. For an unstable plant, G has right half plane (RHP) poles which are supposed to be canceled completely by the same RHP zeros of the nominal plant inverse N G 1 P. However, system model variation results in parameters drift and therefore complete pole-zero cancellation is impossible. The performance of a control system having a plant with RHP poles and zeros has limited performance and is highly unstable [Doyle et al 1992 and Zhou et al 1997]. It is therefore very difficult to stabilize it by just tuning bandwidth of Q filter in conventional disturbance observer design. On the other hand, the proposed controller design first stabilize the unstable plant using a nominal linear control and then handles time-varying disturbance by disturbance prediction filter. The separation design makes it having more freedom in adjusting system bandwidth and disturbance rejection capability. 105

116 u D (k) η(k) u(k) G P (z) G S (z) x1 ( k ), x 2 ( k ) u N (k) x d Feedforward Control F 0 State-feedback Control [ K 1 K 2 ] Figure 5.16 Stabilized system model after nominal linear control η(k) D(z) ηˆ ( k) u D (k) G P (z) x ( k 1 ) P(z) η(k) Q(z) G N P 1 ( z ) Figure 5.17 Control block diagram of disturbance observer 106

117 η(k) η(k) n=2 n=1 n=0 k-2 k-1 k Time index Figure 5.18 A time-varying disturbance Experiment results The proposed control algorithm has been implemented on the improved magnetic suspension system on each individual axis. The compensator D (z) is chosen as a PD controller z 1 D( z) = K P + KD. To verify the control algorithm and demonstrate the Tz performance of the magnetic suspension stage, several experiments are performed and can be categorized into three categories, Stabilization and positioning, disturbance rejection as well as tracking and contouring. 107

118 Stabilization and positioning: A levitation process was performed to demonstrate the stability performance of the developed MSS. The robust control algorithm was implemented and zeros-order disturbance prediction (ZODP) was employed for steady state disturbance rejection. In the experiment, the moving stage rested on the stator initially and was levitated in the vertical direction to a height of 1.0 mm in 0.4 seconds. The experiment result is shown in Figure The left bottom plot shows the z-axis motion trajectory, while the other plots show the accompanying motions in the other five degrees of freedom, two translations (x and y) and three rotations (α, β and γ). Figure 5.20 shows that the MSS achieves ±3 nm horizontal and ±10 nm vertical positioning stability, and ±3 micro degrees orientation stability. To further investigate the positioning resolution, the experimental result with multiple 10 nm steps in y-axis is shown in Figure Since the positioning stability is ±3 nm, the steps can be seen clearly. It is obvious that the system performs well by using ZODP in both positioning and stabilization process. Disturbance rejection: This experiment illustrates external disturbance rejection capability of the proposed disturbance observer. The floater is first levitated and stabilized at z = 1.0 mm. At time equals 8 second, the MSS suffers a time varying disturbance in x-axis as shown in Figure It is seen that FODP rejects the disturbance and regulates the positioning error very well. For comparison, the result using a ZODP is also shown. The effectiveness of the proposed disturbance estimation algorithm can clearly be seen. 108

119 Tracking and Contouring: Two experiments are performed to demonstrate the motion tracking performance of the developed system, namely circular contouring and 3D contouring. In circular contouring, the floator was first levitated and stabilized at z = 1.0 mm and was commanded to track a circle in x-y plane. From the experiment result shown in Figure 5.23, it is seen that by using ZODP, even though feedforward control has been implemented to eliminate tracking error, it is unable to track the circle precisely because of time-varying external disturbance. To improve the performance, FODP is employed and satisfactory results shows that the tracking error is around ±5 nm. The second experiment is a 3D contouring process. The stage is first levitated in z- axis to a height of 0.5 mm, and then commanded to follow a spiral curve with a 4.0 mm diameter and 1.5 mm pitch in 17 seconds as shown in Figure The trajectory almost covers the desired working space 4 mm 4 mm 2 mm. As the MSS moving along z- axis, system model suffers significant change, which can be treated as a time-varying disturbance of the system. To obtain satisfactory performance, the MSS not only have to follow the time-varying reference precisely, but also rejects fluctuating disturbance. In the experiment two kinds of disturbance observers, ZODP and SODP, are implemented as a comparison for demonstrating the proposed disturbance observer. In order to illustrate the efficiency of the proposed method, predicted z-axis disturbance η ˆ( k) in both cases are shown in Figure 5.25, which shows that SODP is more suitable for predicting time-varying disturbance than ZODP. The experiment result shown in Figure 5.26 shows that SODP accurately estimate and then compensate this time-varying disturbance during the course of contouring process, tracking errors in both x and y-axis are within ±5 nm 109

120 while about ±20 nm in z-axis. However, ZODP is unable to eliminate the tracking errors imposed by the disturbance as shown in Figure It is seen that the proposed higher order disturbance observer precisely estimate time varying uncertainties and regulates the performance of the system during large travel motion control. 5 x x 10-4 x(m) 0-5 α(rad ) x x 10-4 y(m) 0-5 β (rad ) 0-1 z(m) x time (sec) θ (rad ) x time (sec) Figure 5.19 A levitation process 110

121 5 0.2 x (nm) 0 (µrad) α y (nm) 0 (µrad) 0 z (nm) β time (sec) θ (µrad) time (sec) Figure 5.20 Positioning stability of the MSS Y (nm) time (sec) Figure nm steps in Y direction 111

122 0.25 Disturbance (N) X (nm) ZODP FOPD Time (sec) Figure 5.22 Disturbance rejection of a time-varying disturbance Y (mm) X (mm) (a) X-axis tracking error (nm) X-axis tracking error (nm) Tracking error of ZODP Tracking error of FODP Time (sec) Figure 5.23 Circular contouring and tracking errors Y-axis tracking error (nm) Y-axis tracking error (nm) (b)

123 Z (mm) X (mm) Y (mm) 2 3 Figure 5.24 A spiral curve trajectory η ˆ ( k ) ( N ) ZODP SODP time(sec) Figure 5.25 Disturbance prediction for z-axis motion control 113

124 X axis error (nm) Y axis error (nm) Z axis error (nm) Time (sec) Figure 5.26 Tracking error of 3D contouring using SODP X axis error (nm) Y axis error (nm) Z axis error (nm) Time (sec) Figure 5.27 Tracking error of 3D contouring using ZODP 114

125 CHAPTER 6 APPLICATION(I) : DEVELOPMENT OF A NANO-METROLOGY SYSTEM 6.1 Introduction A laser pick-up head is a core element in a CD drive developed by interdisciplinary technologies. A laser pick-up head consists of four subsystems, which are optical, mechanical, electrical and magnetic systems. Because of mass production, it becomes very affordable and is commercially available. The laser pick-up head measures the distance between the measured object, such as a CD-ROM disk, and the laser head itself. By integrating with a moving mechanism, many metrology systems have been developed by using the laser head for surface geometry measurement. In this research, a nanometrology system is developed by integrating both magnetic suspension system and laser pick-up unit. In a CD drive, the surface of the disk must be kept within the focal range of the optical system such that the information on the disk can be pick-up by the optical system. Since the focal range is so small, to obtain measurement result over a large area, a positioning stage capable of multiple degrees of freedom motion is highly desired. Since the MSS has nano-meter positioning stability within a working space as large as 4mm 4mm 2mm, it is capable of integrating with many processes in nano/micro technology such as metrology, fabrication and assembly where precise motion control over large travel range is extremely preferred. Under the principle of auto-focusing 115

126 method, a nano-metrology system using the optical pickup head as an optical probe and the MSS as a scanning stage is introduced. The remainder of this chapter is organized as follows. Section 6.2 gives an overview of the laser pick-up head. In section 6.3, to enlarge the measurement range in developing the proposed metrology system, a method is proposed for system integration in which two control loops are arranged in a hierarchical structure to provide real time information to the MSS control system from the optical pick-up head signals. Experimental results are shown in section 6.4 as a performance demonstration of the developed metrology system. 6.2 A Laser Pick-up head In this research, a commercially available pickup head, SANYO SF-P151Z1, was utilized. Figure 6.1 shows a picture of the laser pick-up unit, the objective lens located in the center of the pick-up head can be actuated by voice coils in two degrees of freedom motion. The principle and configuration of the optical pick-up head is shown in Figure 6.2. A polarized laser beam emitted from the laser diode passes through a polarized beam splitter (PBS) and a quarter wavelength plate. An objective lens and a cylindrical lens change the wave form of the laser beam from a plane wave to a elliptic wave, which has two different center of curvature along z-axis. The shape of the laser spot formed on the object surface changes if the object moves along z-axis. Consequently, the laser beam is then reflected by the surface of the object and returns to a quadrant photo diode through the quarter wave plate. If the tested surface lies on the focal point, the reflected beams will generate symmetrical pattern on the photo diode, which is shown in Figure 6.3(b). If the tested surface is near or far away from the focal point, the resulting pattern 116

127 will be Figure 6.3(a) or Figure 6.3(c). The photo diode detects light intensity of each quadrant and produce signals A, B, C and D using I/V (current-voltage) converters. A focus error signal defined by FE=(A+C)-(B+D) is used for indicating the distance of the tested surface from the focal point. As the tested surface is approaching near the optical head, an S-curve shown in Fig 6.3(d) can be observed. In this metrology system focal point is of interest since it is served as a reference point during the measurement procedure. In a CD drive, the motions of the lens are actuated by two voice coils such that the focal point of the lens tracks the data stored on the disc rotated by a spindle motor. In this research, since the vertical position of the object is controlled by magnetic suspension system, the voice coil is not employed. The lens is fixed by glue so as to prevent undesired motion during the measurement process. Since the laser intensity is sensitive to surrounding temperature, a monitor diode (MD), which measures the light intensity output, is embedded inside the laser pickup head as a feedback signal device for laser power control. Table 6.1 also lists some specifications that related to the optical components of the laser pick-up head. Objective Lens Connector Figure 6.1 Sony SF-P151Z laser pick-up head 117

128 Optical Laser Pick-up Head Laser Diode PBS ¼ Wave Plate Cylindrical Lens (c) (b) (a) A D B C Quadrant Photo Diode Figure 6.2 Configuration of a laser pick-up head z FE=(A+C)-(B+D) (c) A D B C 3 Zero cross point (b) D A C B FE 2 z (a) D A C B 1 (d) Linear range Figure 6.3 Focus Error signal 118

129 Laser wavelength 785nm Objective lens NA = 0.45 Focal distance=2.88mm Working Distance=1.61mm Laser diode Operation voltage Average: 1.9V Maximum: 2.5V Operation Current Average: 47mA Photo Diode Supplied voltage Range of operation supply voltage Bandwidth Power Dissipation Maximum: 65mA Maximum : 9V 4.5~5.5 V 100 MHz 100 mw Table 6.1 Specifications of a laser pick-up head 119

130 6.2.1 APC circuit APC (Auto-Power Control) circuit is an analog circuit for regulating laser diode power. It is known that the power emitted from a laser diode is very sensitive to ambient temperature variation, thus the laser power suffers low frequency drift if the temperature is not controlled. Unregulated laser power would cause signal drift and will degrade the accuracy of the measurement result. Normally in a CD drive, the APC circuit is integrated inside the DSP chip set, which is not commercially available. Therefore, the APC circuit must be developed to turn on the laser in this research. Figure 6.4 show the APC circuit diagram, which consists of two parts, a feedback loop for power regulation and a current drive. A PNP type BJT transistor (2SA1110-ND) is used as the current drive for the laser diode LD. A monitor diode MD is embedded inside the laser pick-up unit as a light intensity sensor for power feedback control. MD receives the laser light intensity and converts it into minute current and flows through a resistor VR. The resistor VR transform the current signal into voltage signal and send it into the non-inverting terminal of the operational amplifier LM324. The op amp receives the feedback signal and controls the current of the transistor by feedback resistor R f. The capacitor C 1 connected between base and emitter of the transistor has two functions. First, it bypasses noise from op amp output Vo. Second, it is served as a temporary voltage source when the power system suddenly shut down. The feedback loop can be further analysis by Figure 6.5. By using Thevenin equivalent, the resistor R 1, R 2 and supply voltage can be transformed into voltage source 120

131 V eq with internal resistance R eq. Suppose the laser light intensity measured by monitor diode is VMD, we have V eq V R eq MD V = MD V R f O which gives V O R = ) R f f ( 1+ VMD Veq. It is seen that Vo gets larger as V MD gets higher, Req Req since R f R f >> 1, the amplification gain is ( Req Req ). In order to understand how power of the laser diode is regulated by the feedback loop, Figure 6.6 is used to explain the current drive part. To obtain the driving current sent into the laser diode, the bias loop must be analyzed first. From the bias loop, we have the following equation, 5 R ( β + 1) I 0. 7 R I = V. e B b B O Where β is the current amplification rate, which equals 50 for 2SA1110ND. The driving current I c is obtained I C = β 4.3 VO Re( β + 1) + R b, which is inverse proportional with Vo. As V MD gets larger, which means the power of laser diode becomes higher, Vo also increases and therefore reduces the driving current I c and vice versa. The power can be regulated by this mechanism for compensating the power drift by time-varying ambient temperature. In this research, to have large emitted laser intensity, the voltage received from MD is around 0.37 V, which is about twice as large as the laser power in a CD drive, 0.18V. 121

132 In the APC circuit, the power level is set by variable resistors R 1 and R 2. Since R f is much larger than R 1 and R 2, the current flow through R f can be negligible. Therefore, during operation, the voltage level at the inverting terminal of the op amp is 5 R 2, which R1 + R2 must be equal to V MD, the light intensity receives by monitor diode. 122

133 Feedback loop +5V R1=24.3 k R2=2 k Rf=84.2 k LM324 Current drive +5V Re=8.6 Rb=4.2 k GND VR MD LD VR Laser Pick-up Head Figure 6.4 APC circuit R2 V eq = 5V R + R R eq = R 1 // R Rf=84.2 k LM324 Vo V O Bias loop I B Rb=4.2 k +5V Re=8.6 I E I C V MD LD Figure 6.5 Feedback control of APC circuit Figure 6.6 BJT current drive 123

134 6.3 System Configuration The configuration of the developed nano-metrology system is shown in Figure 6.7. The laser probe, consisted of a laser pick-up head and a APC circuit, is fixed on top of the magnetic suspension stage with its objective lens facing downward to the measuring surface of the object, which is placed on the working platform of the floator. Signals from the photo diode are then feedback to the computer and received by analog to digital converters (ADCs). The computer receives the signal and control the motion of the floator such that the laser beam can scan across the object surface and measure the geometry of the surface. A data acquisition board CIO-DAS16JR/16 was used for performing analog-digital signal conversion. It has 16 channel analog input, 16 bits resolution and maximum 100kHz sampling rate per channel. The acquisition board uses ISA bus to communicate with PC and therefore there are no difficulties in integration with MSS control program. One can easily control the AD conversion by accessing registers on the ISA bus address. Signals from the photo diodes are converted into digital form and focus error signals can be easily obtained. During the measurement process, the motion of the floator is controlled by the control system of the magnetic suspension system. Laser probe signals and magnetic suspension motions are stored in the computer for signal processing. The surface geometry of the measured object can then be constructed after signal process. 124

135 To ADC * i Signals Power Amps i Laser probe Laser head Floator APC circuit Object Fixture Computer Stator Figure 6.7 System configuration of nano-metrology system A Calibration Result To have accurate measurement result, S-curve calibration is very important since the height of the measurement point is determined by both the magnitude of FE signal and z- axis position of the floator. By using calibration result of S-curve, FE signal can be converted to real probe distance. A calibration result was obtained by placing a mirror on the floator with its surface face normally to the laser beam. The z-axis position of the floator is shown in Figure 6.8. The floator was levitated in z-axis and was stabilized at z = 0.5mm. At this time the mirror surface is still out of the focus range. At time equals third second, the floator started to move in vertical axis to a height of 0.7mm in 4 seconds, the mirror surface scans through the measurement range of the laser head and both the focus error and RF signal (RF=A+B+C+D) were obtained simultaneously. It is shown in Figure 125

136 6.9 that the S-curve has a linear range of 10µm. From the S-curve, it is seen that FE signal is also zero when the mirror is far away from the focal point, this characteristic makes the identification of the focal point difficult. Since RF signal measures the total laser intensity received by the quadrant photo diode, it shows in Figure 6.9 that the level of RF signal rises up after the mirror surface enters the measurement range. Therefore, recognition of focal point becomes: if RF is larger than some standby level while at the same time FE is zero, then the object surface reached focal point. This identification method is used in most of the compact disc drives. In measurement process, a high order polynomial is used to fit the linear region of S- curve as shown in Figure A 10 th order polynomial is used, z = a0 + a1fe + a2fe + a10fe (6.1) and least square method is employed for optimizing coefficients a0, a1, a10. Given a FE signal, the real distance between the focal plane and the mirror surface can be obtained from this calibration relationship. Since the surface reflectivity of different materials are not equal, in order to generate measurement with high resolution, it is necessary to perform calibration process before for each measurement. 126

137 0.8 Z-axis position (mm) µm 4 seconds Time (sec) Figure 6.8 Z-axis motion of the floator during calibration process FE(V) Linear range 10 µ m RF (V) z( µ m ) Figure 6.9 A calibration process 127

138 0.5 FE signal FE (V) A 10 th order polynomial µm Figure 6.10 Curve fitting of FE signal using a 10 th order polynomial System integration To measure surface geometry, the developed MSS is employed as a moving mechanism to move the object so that the laser beam is able to scan perpendicularly across the surface. As it has been mentioned, for measurement over a large area, 10µm linear range is not enough. To overcome this problem, it is highly desired that the probe depth of the laser head remains constant during the measurement process. In order to accomplish this, the object to be measured is placed on the floator and the motion of the floator must be controlled such that the area being covered by the laser spot always stays very close to focal point during the scanning process. Therefore, system integration of both the MSS 128

139 and the laser head must be done. During the scanning process, FE signal and vertical axis position of the floator are recorded during the course of scanning process. Afterwards, these signals are used for reconstructing the surface geometry. In order to keep probe depth constant during the measurement process, a system integration method shown in Figure 6.11 has been proposed and implemented. It consists of two control loops configured in a hierarchical structure. The inner control loop is the developed MSS, which is composed by floator, stator, measurement system and control system. A nonlinear controller stabilized the motions of the floator at reference trajectories given by outer control loop. The outer control loop has two tasks, one is to configure the laser scanning path by generate the reference trajectory for the floator in both x and y axes, while the other task is to regulate the height of the floator by a servo controller. Once the object surface pass through the focal point and identified by both FE and RF signals, the outer control loop is activated. Since the bandwidth of the MSS in z- axis motion is about 100Hz, with damping ratio around 0.22, a low pass filter is used to filter out high frequency noise from focus error signal so that the vibration around natural frequencies 100Hz would not be excited. However, the bandwidth of the low pass filter cannot be too low since the induced phase lag would cause instability. As for the servo controller design, some issues are needed to be concerned. In order to locate the z-axis position of the object precisely, it should consist at least one integrator. Since the MSS is a stable system, there is no need to compensate phase at cross over frequency. For current design, the bandwidth of the low pass filter is 5Hz, and a PI controller 129

140 D ( s ) = K P + K s I is served as servo controller. Figure 6.12 shows the real nano-metrology system developed. A focus-locking process shown in Figure 6.13 was performed for locating the focal point on the object surface. It is seen that after the focal point is identified from FE and RF signals, the outer control loop is enabled and hence z-axis position of the object is controlled and FE is regulated to zero. Figure 6.14 shows the response of the FE signal during the focus-locking process by using different sets of PI controller gains. In the first experiment K p =10-5, K I =10-4, while in the second experiment K p =10-4,K I = It is seen that in the second experiment, FE converges to zeros immediately after outer control loop turned on. However, large mechanical vibration from magnetic suspension is then coupled into the system during steady state, resulting larger measurement noise as shown in Figure Therefore, case II shows worse positioning stability, which decreases the measurement accuracy and is undesired during measurement process. 130

141 FE PI Controller Outer Control Loop Path Planning Surface Profile z r yr Nonlinear Controller z Inner Control Loop z x y y * i x MSS i Floator Stator Laser Interferometer System Figure 6.11 Control system of the nano-metrology system Fixure Lase Pick-up Head APC circuit Working platform Measured Object Stator Figure 6.12 Picture of the developed system 131

142 FE(V) Outer control loop switch on RF(V) 2 1 Standby Level L Figure 6.13 A focus-locking process FE (V) Case I : K = 1e 5, K = 1e 4 P I FE (V) Case II : K = 1e 4, K = 5e 4 P I Larger Noise Time (sec) Figure 6.14 Focus-locking process using different controller gains 132

143 6.4 Experiment Results Five experiments are performed for evaluating the performance of the developed nanometrology system. The object to be measured is placed on the working platform of the floator, the motion of which is controlled by the proposed integration system. Height difference of gauge blocks The second experiment is to measure the step height difference of two gauge blocks. It is shown in Figure 6.15 that two gauge blocks with nominal height of and inch are placed on another larger gauge blocks, which is served as a flat reference plane. Following the same process, the focal point was locked after the floator was levitated and then was commanded to move along x-axis back and forth two times with travel range of 3.0mm. A signal processing procedure that is similar to mirror surface measurement was performed and the measurement results associated with two straight lines departed by 2.54µm is shown in Figure It is seen that the measurement result has high accuracy and excellent repeatability. Because the edge of the gauge block is rounded, the laser light is deflected away and hence the FE signal is zero. Therefore, the servo controller input equals zero and z-axis motion of the floator is stopped when the laser beam scanned across the edge. This would explain the unrepeated portion of two different scans in Figure The round edge of the gauge block is estimated to be around 300µm. The similar experiment was done with a motorized table as the moving mechanism [Fan et al 2000]. However, due to waviness of the table, the measured surface profile exhibits an imposed periodic sine wave on it, which degrades the accuracy of the result. 133

144 Optical head z x µ 2.54 m GB #1 GB #3 GB #2 Figure 6.15 Experiment setup of gauge block measurement Edge Surface profile ( µ m ) Scanning direction : -X 10 Scanning direction : +X X (mm) Figure 6.16 Experiment result of gauge block height difference measurement 134

145 Surface profile of a mirror A mirror surface was measured by the developed system. Figure 6.17 shows the configuration of the experimental setup. A mirror was placed on the working platform of the floator and was levitated in z-axis until it reaches the focal point. At the same time the focus-locking process was performed and outer control loop is enabled such that the mirror surface is locked on the focal point. The floator was then commanded to move along x-axis back and forth for four times with a travel range equals to 2.0mm as shown in Figure Since the outer control loop is activated, z-axis motion of the floator was controlled by a servo controller such that the probe depth almost remains the same through out the measurement process. It is seen that from Figure 6.20 the FE is very close to zero during the process. Because the FE signal measures the distance between the mirror surface and the focal point, it can be inferred that FE is proportional to z-axis position of the floator plus surface profile of the mirror. As the Fe signal is close to zero, the surface profile of the mirror can be roughly seen from z-axis motion of the floator, which is a tilted plane. To investigate repeatability of the measurement system, the scanning rate of the fourth scan is reduced by 50%. Finally, the surface profile can be reconstructed from z-axis motion of the floator and the FE signal. It is seen that because of the misalignment, the mirror surface is tilted by 40µm over 2.0mm. A procedure for signal processing as shown in Figure 6.19 is performed to obtain form error of the mirror surface. Since there was larger measurement error induced by the positioning error in the second scanning process, the measured form error of first, third and fourth scan were shown in Figure It is seen clearly that the form error is around 150nm over 2.0mm range and that the measurement results are highly repeatable. The two sharp peaks may 135

146 due to small particles or tiny scratches on the mirror surface. A localized figure shows that the measurement resolution of the developed metrology system is around 20nm. It is demonstrated that by integrating the optical head with the magnetic suspension stage, the measurement range can be very large and no longer restricted by the linear range of FE signal. Optical head z Mirror Working Platform x Floator Floator Figure 6.17 Experiment setup for mirror surface measurement 136

147 2.5 2 X-axis motion ( mm ) Scanning rate 0.25 mm/s Scanning rate 0.5 mm/s Time (sec) Figure 6.18 x-axis motion of the floator during the measuring process Procedure of Signal Process for obtaining Form Error of a Mirror Surface Convert FE signal into distance z f Obtain Surface profile SF = z z f Extract surface profile SF i ( i=1~4 ) of each scan from SF Fit straight lines L i to each SF i Form Error = SF i -L i Figure 6.19 Procedure of signal process for form error measurement 137

148 ( m) z µ Form Error (nm) time(sec) FE(V) Form Error of a Mirror time(sec) X (mm) Form Error (nm) 20 nm Figure 6.20 Measurement result of a mirror surface 138

149 Surface profile of CD-R disk The third experiment is to measure the surface profile of a blank CD-R disk. Unlike CD-ROM disk, a blank CD-R disk has tracks on the surface of the disk to help CD-writer locate the write-able area. According to the specification, the track distance is around 1.6µm while the height of the track is around 0.2µm. The measurement result in Figure 6.21 shows that the measured track pitch is around 1.67µm and the height is around 0.22µm. Since the CD-R disk is measured four times and it shows the surface profiles created are highly repeated, the measurement result should be fairly reliable. The inconsistent amplitude of the track height is probably due to uneven etched depth during the manufacturing process. Surface Profile ( µ m ) CD-R x ( µ m) Figure 6.21 Surface profile of a CD-R disk 139

150 6.5 Summary In order to show the feasibility of the magnetic suspension stage in the area of metrology, a nano-metrology system is developed for surface geometry inspection. The system uses a laser pick-up head as a one dimension optical probe and magnetic suspension system as a moving stage. To enlarge the measurement range of the optical probe, a control system is developed to adjust the vertical position of the measured object during the measurement process. Several samples are measured by the developed metrology system and the results demonstrate that the developed metrology system can provide nano-meter measurement resolution over several millimeters range. 140

151 CHAPTER 7 APPLICATION (II) : TOOLING AND MOTION CONTROL FOR NANO-IMPRINTING 7.1 Introduction In recent years, manufacturing of miniature structures has become an important research area. Nano-imprinting, among many other innovative techniques, is a promising technique for fabricating high aspect ratio nanometer-sized structures with high efficiency. A nano-imprinting lithography (NIL) technique was proposed for manufacturing patterns with sub 100nm resolution on 6 in. Si wafer [Heidari 1999 and 2000]. The manufacturing process is very similar to that employed by Compact Disc (CD) production. A NIL machine was developed for achieving high degree of parallelism between stamp and substrate. The purpose of NIL is to maintain uniform press force between stamp and the substrates made of stiff materials such as aluminum. For many medical and chemical applications, fabricating polymer devices with a high aspect ratio and nano-sized features is critical. For example, polymer membranes with a certain pore size cut-off or track-etched holes are currently used in drug encapsulation, cell immunoprotection, and protein separation. A nano-imprinting procedure that could fabricate polymer was designed and proposed by L.J.Lee et al [Lee et al 2000]. A mask made of optical fibers is used for producing two dimensional array of holes with nano- 141

152 size on a thin layer of curable polymer solution. The polymer solution is then cured to gelation by a ultra-violet light source before the mask is removed. In such a process, both linear displacement and angular orientation needed be precisely regulated so that the orientation placement error between the mask and the polymer layer can be compensated and the array of holes produced can have precise uniform size. Therefore, a precision motion controlled stage with multiple degrees of freedom is highly desired. In an attempt to facilitate the nano-imprinting procedure, the developed magnetic suspension system will be integrated into the nano-imprinting process so as to achieve ultra-precision displacement and orientation motion control that are required in the process. A nano-imprinting unit has been designed and will be integrated with the magnetic suspension stage for the nano-imprinting process. The nano-imprinting unit is used to mount and adjust the capacitive sensors that are employed to measure the orientation and the height of the substrate on which polymer solution is placed during the imprinting process. Development of nano-imprinting unit and system integration will be presented in the remainder of this chapter. Section 7.2 gives a brief introduction of the nano-imprinting process. Design and development of the nano-imprinting unit is illustrated in Section 7.3 In order to make alignment precisely, capacitive sensor signals must be feedback to the magnetic suspension control system and a system integration scheme having two control loops is developed and will be explained in Section 7.4. Preliminary experiment results are shown in Section 7.5, while Summary and future work is drawn in Section

153 7.2 Nano-imprinting process The nano-lithography procedure is illustrated in Figure 7.1. A photo-curable liquid resin or a highly concentrated polymer solution can be spin-coated on an optically clear sacrifice substrate (PDMS), which is then placed on a motion controlled stage. An imprinting master made of optical fibers was fabricated by chemical etching process and nano-sized tips are formed on the end surface of fibers. Furthermore, by arranging a large amount of optical fibers in an area as large as 10mm 10mm, a nano-sized patterns over a large area can be manufactured. Since the nanotips are fragile, a soft polymer layer made of PDMS is used to protect the nano-tips during the imprinting process. By controlling the position and orientation of the stage, the master is then dipped into the polymer liquid and the substrate surface is slightly indented. Next, the liquid resin is cured to gelation by a light source from the bottom of the sacrifice substrate before the master is removed. Using the nano-lithography procedure, the nano-sized pattern can be transferred to the polymer. Finally, nano-pores over a large area on a thin membrane can be fabricated. 143

154 PDMS attached to the wafer to protect master Attached to floator Photo-Curable Liquid resin (Or Polymer Solution) Master with nano-tips UV Light Master with an Array of Nano-pores attached to stator Figure 7.1 A nano-imprinting process 144

155 7.3 Design and development of a Nano-imprinting Unit When producing a single pore, a single measurement probe with one-degree-of freedom motion control is sufficient. However, when a master consisting of an array of optical fibers is used to generate a large number of pores on the polymer, both linear displacement and angular orientation need to be precisely regulated so that the orientation displacement error between the master and the polymer layer can be compensated and the array of pores produced can have precise and uniform size. This situation can be illustrated by Figure 7.2. Since only one sensor is used for measuring relative displacement between master and sensor target, the orientation of the target cannot be determined, and thus the produced pores cannot have uniform size. As more sensors come into the setup, misalignment can be compensated and uniform nano-pores can be made. 145

156 displacement sensor Master d Displacement sensor #1 Photo-curable solution PDMS layer Target Displacement sensor #2 d d Figure 7.2 Alignment between imprinting master and polymer Capacitive sensor Displacement sensors employed in the nano-imprinting process should meet the following specifications: (1) the sensors must be able to provide nano-meter accuracy, (2) since any contact force would cause disturbance to the system, there should be no contact between target and sensor, (3) the operation time takes around 1 minute, the drift due to ambient variation must be small. 146

157 Three kinds of sensors had been considered when developing the process, they are laser interferometers, eddy current sensors and capacitive sensors. Eddy current sensors have the lowest accuracy while laser interferometer has the highest accuracy and largest measurement range. However, laser interferometers suffers from significant thermal drift if the ambient temperature is not controlled. Capacitive sensors have low thermal drift, high resolution and adequate bandwidth for the application and thus are chosen for the development. After surveying commercial capacitive sensors from manufacturers, capacitive sensors from ADE Technology are employed. A single set of capacitive sensor consists of a capacitor probe (Model 2810) and a electronic gaging unit (Model 4800). High frequency sinusoidal signal is sent from the gaging module to the capacitive sensors as excitation signal. Since the impedance of a capacitive sensor varies linearly with the gap size, the voltage across the capacitive sensor has linear output. Table 7.1 lists the specifications of the sensor. The resolution of the capacitive sensor depends largely on the A/D converter. If a 16 bit, ±10 V, A/D board is used, then the resolution would be 3nm. 147

158 Measurement range Measurement bandwidth Standoff range Probing area Active area Sensor Output Thermal Stability ±100µm 1kHz (maximum) 100µm 20 mm dia 10 mm dia ±10 V 200ppm=0.02%/ C Table 7.1 Specifications of capacitance sensor Mechanical Design of Nano-imprinting Unit A nano-imprinting unit has been developed to carry out alignment between nano-tips and photo-curable resin. The imprinting unit is used to mount and adjust the master and capacitive sensors that are employed to measure the orientation and the height of the substrate on the magnetic suspension stage during the imprinting process. The integrated imprinting unit and the magnetic suspension stage are shown in Figure 7.3. A substrate with flat surface is served as polymer holder and targets for capacitive sensors. The nanoimprinting unit is fixed on the top of the magnetic suspension stage to look at the substrate. 148

159 Nano-imprinting Unit Stator Substrate Floator Figure 7.3 Integration of magnetic suspension system and nano-imprinting unit The imprinting unit is composed of three capacitive sensors, imprinting master, adjustable mounting fixtures, and a capacitive sensor target. Figure 7.4 shows components of the imprinting unit assembly. The components of the imprinting unit are described as follows. (1) Capacitive sensor: ADE s 2810 capacitive sensor probe with the diameter of 20 mm is adopted in the design. The 2810 capacitive sensor probe is cylindrical and has three mounting holes that are 120 apart on a 15 mm diameter circle. Figure 7.4 shows the upper view of the arrangement of the sensors and the imprinting unit. 149

160 They are positioned such that the distances between sensors, and the distances between sensors and the imprinting master are the same. Although the axial signal cables of the capacitive sensors are not shown in Figures 7.3 and 7.4, the design has taken the cables into consideration to avoid interferences with other components. (2) Capacitive sensor target: According to the specification, a 25 mm minimum target size is recommended for 2810 probe. Therefore, a 6 silicon wafer coated with coated with gold is used as the capacitive sensor target. Since the flatness of the silicon wafer is pretty high, the wafer not only served as sensor target, but also as a working substrate in the imprinting process. Polymer solution with PDMS can be loaded on the wafer. Cap_mount Optical mount (Newport : P100-P) Z-stage Adapter miniature stage (Newport : M-MR1.4) Capacitive probe (ADE : 2810) Tips-adapter Base Nano-tips Support holder and post (Newport : VPH-1 & SP-1) Figure 7.4 The developed nano-imprinting unit 150

161 (3) Miniature angle mount: The model P-100P miniature angle mount from Newport is adopted to adjust the orientation of the capacitive sensor. The capacitive sensor is attached to the miniature angle mount. The miniature angle mounts are attached on the capacitive sensor mount (Cap_mount). The position of the miniature angle mount on Cap_mount is adjustable along z-direction. (4) Adapter: The adapter to mount the capacitive sensor (CS) on the miniature angle mount (P-100P). There are three thread holes to fix the capacitive sensor. (5) Cap_mount: The capacitive sensor mount (Cap_mount) is designed to fix capacitive sensor. CS_mount is attached to another adapter (CS_mount_2_P100P) and an optical mount (P100-P) that provides adjustment to the orientation of the three sensors. The adjustment is important to assure the parallelism between the sensor faces and the target surface. (6) Base: The base is to fix the capacitor sensors to the post SP-1. The center of the adapter has the same distance to all capacitive sensors. (7) Optical mount (P100-P): The optical mount provides orientation adjustment of the capacitive sensors. Since the total weight of the Cap_mount, miniature angle mounts, and capacitive sensors is heavy, an optical mount with larger load capacity may be necessary. (8) Support holder and post: The support holder (VPH-1) and post (SP-1) attach the optical mount to a Z-stage. It provides additional rotational and translation degrees of freedom to the sensor mount. (9) Z-stage (OptoSigma ): The Z-stage provides adjustment of the sensor mount along z-axis. 151

162 (10) Nano-tips: The nano-tips is made of a bunch of optical fibers. Each optical fiber has 200µm diameter. After etching process, each of the optical fiber would have ~1000 tips on its cutting surface. By arranging thee fibers together, a nano-size pattern over a large area can be formed. Z-stage 1 2 Capcitive sensor Target (6 wafer) Capacitive sensor, adapter and angle mount Miniature stage 3 Cap_mount Nano-tips and adapter Figure 7.5 Top view of nano-imprinting unit 152

163 Figure 7.6 Picture of a nano-imprinting prototype system 7.4 System Integration Figure 7.6 shows the picture of a nano-imprinting prototype system. During the process, three capacitive sensors measures the gaps of the target placed on the floator. The signals from capacitive probes are sent to gaging module 4800, which produces voltage output signal that is in proportional to gap distance between capacitive probes and target. The gap distance is then converted to real displacement of the target by the following equation g Vi + 10 µ m) = 200 m + 20V ( Stand-off-distance i µ where g i is the ith capacitive sensor gap distance, V i is the voltage output from gaging model and stand-off-distance is 100µm. The control system of the magnetic suspension stage takes the measurement result and moves the target by controlling the motion of the floator. 153

164 Interactions may be occurring when two or more capacitive sensors are used to look at the same target. To minimize measurement errors due to interference between probes, phase lock the excitation signal to the probes is necessary. To do this, the board synchronization signals must be bussed across all modules. When more than one unit is used, the phase of the probe driving signals must be set to an appropriate angles. In the case where three probes are used, the first board should be set to 0, the second set to 120, and the third set to 240. To ADC Output Signals 4800 Gaging module Capacitive sensor signals Nano-imprinting Unit * i Power Amps i Floator Computer Stator Figure 7.7 System integration scheme Real-time measurement of target position Since capacitive sensors constitutes a reference coordinate for nano-tips during the imprinting process, in order to make alignment between nano-tips and polymer, it is 154

165 necessary to provide real-time position of the target referred to capacitive sensors. Therefore, conversion from capacitive readings to target position is necessary. Figure 7.8 shows the geometric model of the capacitive probes and the target. The output from the 4800 gaging system gives three gap distance g 1, g 2 and g 3 of the probes. Usually the arrangement is aligned manually before the process, but small misalignment still exists and would cause the produced uneven sized nano-pores over a large area. To compensate the alignment error, vertical position z and orientation of the target α and β must be obtained. By using small angle approximation, one can obtain g + g2 + Z C = 3 1 g3 α C = g3 g D 23 2 β C = g2 g D 12 1 where D ij is the transverse distance between i-th and j-th probes. z y β 1 2 Capacitive probes g1 2 Target g 3 x g 3 α Figure 7.8 Relation between capacitive sensor coordinate and target 155

166 7.4.2 Real-time control of the target position and orientation To make perfect alignment during the imprinting process, the position of the target must be controlled precisely respect to the coordinate defined by the three capacitive sensors. Consequently, the motion of the floator must be controlled such that the gap distances are regulated. In order to accomplish this, both MSS control system and imprinting unit are integrated by a control scheme as shown in Figure 7.9. The integrated scheme is composed of two control loops, namely inner control loop and outer control loop, arranged in a hierarchical structure. The inner control loop stabilizes the motion of the floator at the reference position or trajectories determined by outer control loop while the outer control loop takes three capacitive sensor readings and convert them into the real position of the target, which is then regulated through a servo controller. To have accurate position control, a Proportion-Integral (PI) controller is served as the servo 6 controller in which = K I = 1.0 for regulation of C 5 K, = P K for regulation of Zc and = 0. 4 I K, α and β C. The system integration scheme is similar to the one that used for development of the nano-metrology system in Chapter 6. It provides a general solution for many applications of MSS. P An experiment was performed to demonstrate the feasibility of the proposed system. The objective of the demonstration is to control the position of the target such that gap distance g1, g2 and g3 were all 280µm. The floator was commanded to levitate to a height of 0.5mm initially as shown in Figure At time equals 3 second, the floator started to rise in vertical direction at a rising rate mm/sec. At the same time capacitive sensor readings are acquired by the AD coverter and z-axis position of the 156

167 target Zc was calculated. When Zc reaches 280µm, then Flag_Outer_Loop is set to 1 and outer control loop is closed. The regulation of the target position is shown in Figure It is seen that the capacitive sensor readings all converged to 280µm after 2 seconds. Figure 7.12 shows the response of target orientation during the regulation process. z Capacitive sensor readings Outer Control Loop β α Parameters computation Z C α C β C Trajectory planning Servo Controller z r α r β r Nonlinear Controller * i i Floator Inner Control Loop z α β Stator Laser Interferometer System MSS Figure 7.9 Control system design for nano-imprinting system 157

168 1.2 Z-axis reference of the floator (mm) Floator rise up Rising rate : mm/sec Outer loop turn on Return to ground Time (sec) Figure 7.10 Z-axis reference trajectory for the imprinting process Capcitive sensor readings ( µ m) Target position Zc ( µ m) Flag_outer_loop Time (sec) Figure 7.11 Capacitive sensor readings during the alignment process 158

169 target orientation (rad) 12 x α C β C Figure 7.12 Target orientation adjustment during the alignment process Since the relative position and orientation between the nano-tips and capacitive sensors are not calibrated, controlling the position of the target precisely does not mean that the nano-tips and the polymer are perfectly aligned. Therefore a few trials must be performed to calibrate the relations between sensors and the nano-tips. After every imprinting process, the produced sample must be examined under microscope to determine the desired position of the target at the next attempt. The process will be iterated until perfect sample has been made. Therefore, in the iteration process it is necessary to tilt the orientation of the target such that the tip orientation and sample are aligned. Another experiment was performed to show the feasibility of performing 159

170 regulation of orientation motion. In the experiment the orientation of the target is regulated such that the three capacitive sensor readings are separated by 10um as shown in Figure It is also seen that the orientation of the target is precisely controlled to some certain values. Capacitive sensor readings ( µ m) Target Orientation (rad) x Time (sec) Figure 7.13 Compensation for angle misalignment 160

171 7.5 Experiment result Working with Mr. Wang, Shengnian, a student of Professor Lee, preliminary nanoimprinting experiment was conducted to illustrate the feasibility of the developed system. The process can be illustrated by three capacitive sensor readings in Figure The floator is levitated to 0.5mm and start to rise in z-direction until the target reaches measurement range of the capacitive sensors. Once Zc reaches 280 µm, regulation begins and the target was stabilized at 290µm. And then the target was commanded to move in positive z-axis until the nano-tips contact the polymer sample. The curing light was turned-on and the polymer was cured to gelatin in 30 seconds. Finally the floator was commanded to touch down to the ground. Because of the material property of the polymer used in this experiment, it was unable to be separated from the nano-tips. The cured polymer sample along with the nano-tips was examined under Scanning Electron Microscope (SEM). The area examined is around 35µm 25µm. It is shown in Figure 7.15 from top view of the sample that some of the tips are still immersed inside the polymer while most tips penetrated through the polymer surface, which means the orientation of nano-tips and polymer surface is not aligned perfectly. The side view is shown in Figure Currently, method for detecting contact of nano-tips and polymer surface has not yet been established. To bring nano-tip into contact with polymer, experiments were repeated and the results were examined under microscope to compensate the reference height for the target in the next trial. The process is time consuming and further research must be conducted. 161

172 300 Nano-imprinting Process Cap readings ( µ m ) 260 Out of range Adjustment 200 Curing Time(sec) Figure 7.14 Three capacitive sensors readings of a nano-imprinting process. Figure 7.15 SEM photo of nano-imprinting result (I) : top view (Courtesy of Shengnian Wang) 162

173 Figure 7.16 SEM photo of nano-imprinting result (II) : side view (Courtesy of Shengnian Wang) 163