Precision positioning of a three-axis optical pickup via a double phase-lead compensator equipped with auto-tuned parameters

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T E C H N I C A L P A P E R

Precision positioning of a three-axis optical pickup via a double

phase-lead compensator equipped with auto-tuned parameters

Paul C.-P. ChaoÆ Chi-Wei Chiu Æ Jackal C.-Y. Shen

Received: 3 August 2008 / Accepted: 10 December 2008 / Published online: 31 January 2009 Ó Springer-Verlag 2009

Abstract The objective of this study is to design an intelligent control servo scheme for the three-axis optical pickups employed in the next-generation optical disc drives. The three-axis pickup owns the capability to move the lens holder in three directions of focusing, tracking and tilting, which is required particularly for higher data-density optical disks and precision measuring instruments to annihilate non-zero lens tilting. The intelligent controller utilizes a commercially often-used double phase-lead compensator equipped with the capability of auto-tuning on control parameters. In this way, the model uncertainty of the pickups caused by manufacturing tolerance and the coupling between three different DOFs of the three-axis pickup can be overcome to render desired precision data-reading. In the initial stage of the study, Lagrange’s equations are employed to derive equations of motion for the lens holder. A double-lead controller equipped with a fuzzy logic parameter tuning algorithm is then designed to perform dynamic decoupling and forge control efforts toward the goals of precision tracking, focusing and zero tilting simultaneously. Along with the controller, a genetic algorithm is developed to search the optimal designed parameters of previously designed auto-tuning algorithm. Finally, the experiments are conducted to show the effec-tiveness of the controller. With validated performance, the

designed intelligent controller is ready to be employed for the next-generation optical disc drives.

1 Introduction

For optical disk drives (ODDs) and some surface-profiling instruments in micro- or nano-precisions (Zhang and Cai

1997; Fan et al. 2000, 2001), the key component deter-mining the performance is the optical pickup, which conducts data-reading via a well-designed optical system installed inside the pickup. Figure1 shows a photo of a three-axis four-wire type pickup actuator, which is designed and manufactured by the Industrial Technology and Research Institute (ITRI), Taiwan. This pickup consists mainly of an objective lens, a lens holder (often called ‘‘bobbin’’), wire springs, sets of wound coils and perma-nent magnets. Thanks to flexibility of wire springs, the bobbin could easily be in motions as the forces acting on the bobbin are generated by the electromagnetic interac-tions between the magnetic fields induced by permanent magnets and the currents conducted in sets of coils.

High-numerical apertures (NA) and short-wavelength laser diode (like violet diodes) are recently employed for objective lens designs in pickups in order to produce a smaller optical detecting spot on an optical disk for better data-reading resolution. This aims at increasing detectable data-density via decreasing the circular radius of the aberration region of the optical spot, the main factor lim-iting resolution of data storage in disks for ODDs or surface profiling for measuring instruments. With the size of optical spot decreased, original electro-mechanical designs of the pickup structure might become obsolete. One of critical challenges arises from the unavoidable

P. C.-P. Chao (&) C.-W. Chiu

Department of Electrical and Control Engineering, National Chiao Tung University, Hsinchu 300, Taiwan e-mail: pchao@mail.nctu.edu.tw

J. C.-Y. Shen

Department of Mechanical Engineering,

Chung-Yuan Christian University, Chung-Li 320, Taiwan DOI 10.1007/s00542-008-0772-3

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tilting of the bobbin during its motion since the resulted coma aberration increases in proportion to NA3/k, where k is the wavelength of the laser diode (Nagasato and Hoshino 1996). This tilting arises from two possible factors. The first factor is an uneven magnetic field due to manufacturing tolerance and/or the mis-pass of the net electromagnetic force in the directions of focusing and tracking to the mass center of the bobbin while the bobbin moves from its static position to desired vertical and radial positions. This factor leads to a tilting moment on the bobbin and then a nonzero bobbin tilt. The second factor is the unavoidable small un-parallelism in practice between the lens and the optical disc in high-speed rotations or the surface to profile. To restrain the bobbin tilting to a small level for a more accurate, faster data-reading, some research works (Choi et al. 2001; Kang and Yoon 1998; Rosmalen1987; Yamada et al.2000; Chao et al.2003; He et al. 2004) have successfully developed the tilt servo systems for the bobbin, in addition to original focusing and tacking ones. The tilt servo makes possible the capability

of suppressing the unavoidable bobbin tiltings. One of challenges for implementation, however, arises from the manufacturing tolerance existing in each commercial four-wire type pickup, which makes uncertain if the afore-designed controllers still works in the presence of tolerance and the dynamic coupling between three axes of the pickup. Table1 lists dynamic characteristics of four-four-wire type optical pickups of the same model no. and their variations tested in the laboratory. It is clear that the dynamics of the pickups are not the same to each other, and thus a controller designed based on one single pickup is not guaranteed to also work for others.

To solve above-mentioned problem, the available intelligent servo methods for the next-generation servo to perform precision positioning is developed in this study. The design concept of this intelligent control servo system relies on tuning the control parameters automatically using some intelligent frameworks. We use the well-known fuzzy logic controller (FLC) to change the parameters in double phase-lead compensators to save the response time and

Fig. 1 Structure of the three-axis four-wire type optical pickup by ITRI

Table 1 Specifications of two difference four-wire type optical pickups

Difference optical pickup Focusing Tracking Tilting

No. 1 No. 2 No. 1 No. 2 No. 1 No. 2

Resonance frequency (Hz) 53.3 54.5 77.68 79.22 145.86 143.62

Resonance peak (dB) 59.181 60.21 59.99 61.56 48.7 47.66

Magnitude (dB) 52.3 53.1 51.35 50.33 46.2 38.3

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attain better performance of controller. In addition, the membership functions are searched by genetic algorithm (GA) to perform dynamic decoupling and forge control efforts toward the goals of precision tracking, focusing and zero tilting. In Akgu¨l and Morgu¨l (1997), Hong et al. (1992), Visioli (1999), and Hsu and Tsai (1996), the studies describe the tunings on the parameters of the PID or lead– lag controller by FLC designs for improving the system responses. From above studies, the main advantage of the FLC is its inherent robustness and ability to handle any nonlinear behavior of the structure. Another advantage is that engineers can design a FLC easily because the prin-ciple of a FLC is simple to understand. Therefore, the parameter of the double-lead compensator is tuned by FLC in this study. However, there are some parameters needed to be searched in the FLC for attaining better performance. GA is chosen herein for searching for optimal memberships and associated peak gains in the FLC. In Hwang and

Thompson (1994), a similar GA is presented for the FLC. The domains of the membership functions of the FLC are searched. It is pertinent to note at this point that the intelligent servo in practice for this study needs to acquire on-line feedbacks of bobbin motions not only in DOFs of focusing and tracking but also tilting. For conventional pickups, only motions of tracking and focusing can be detected by specifically designed optical systems and sev-eral patches of photo-detectors (Marchant1990). Figure2

illustrates a typical optical/sensing system for measuring the motions of the bobbin, where Fig.2a presenting the overall optical system, Fig. 2b showing the principle of quad-detectors for detecting focusing motions, while Fig.2c the three-beam method for tracking motions. It is can easily be seen from Fig.2that the conventional optical system needs additional patches of photo-detectors or apply the recently proposed methods of signal analysis (Katayama et al. 2001; Miyano and Nagara 2004;

Fig. 2 aThe optical system of the pickup including photodiodes, b The method of quad-detectors measuring focusing error, c The three-beam method for measuring tracking error

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Yamasaki et al. 2006) on the reflected light intensity to equip itself with the capability of tilt detection.

In the next section, the mathematical modeling for the optical pickup will be presented. In Sect.3the double-lead compensator and auto-tuning algorithm are designed. Numerical and experimental results are subsequently pre-sented in Sects.4 and5, respectively. Finally conclusions and future works are stated in Sect. 6.

2 Mathematical modeling

A typical three-axis pickup actuator designed fabricated by ITRI, as shown in Fig.1, is considered in this study. This pickup mainly consists of a lens holder—bobbin, inner/ outer yokes, four-wire springs, coils for actuations in directions of tracking/focusing/tilting, four permanent magnets and a PCB holder. To actuate the pickup, three external voltages are applied independently across the respective spring wires to generate the wire-carried cur-rents through the magnetic fields posed by surrounding magnets. It generates independent actuation forces and moments to perform simultaneous by positioning and rotating in the directions of focusing/tracking and tilting, respectively. According to (Chao et al. 2003), the mathe-matical modeling of three-axis pickup was established accurately as following section.

2.1 Dynamic modeling of actuator

The conventional bobbin, due to its specially designed supporting structure of four parallel wires, exhibits motions mainly in the DOFs of tracking (X-axis) and focusing (Y-axis). In addition to the motions in X and Y directions, small tilting often occurs about h-axis, which is caused by manufacturing tolerance, uneven magnetic fields, and/or geometric mis-passes of the electro-magnetic forces acting line on the bobbin mass center. The objective of this study is to design a fuzzy controller that owns three independent actuating forces and moment in X, Y and h directions in order to perform precision focusing/tracking and to simultaneously achieve zero tilting to avoid any errors in optical reading signals. The design of such controller starts with an establishment of the system dynamic model. It is assumed that the pickup assembly can be simply modeled as a lumped mass-spring-damper system due to bobbin’s high material rigidity compared to the flexibility of the suspending wires. Figure3shows the schematic on the bobbin from the planar side view of Fig.1 (from the viewpoint toward the X–Y plane), and accompanying coordinates/notations defined for capturing the bobbin motion. As seen in Figs.1 and 3 are coordinates xyz defined as the body-fixed ones to the moving bobbin, while

coordinates XYZ are global, ground coordinates. X also serves as a dynamic variable, capturing the horizontal, tracking motion; Y does the vertical, focusing motion; h does the rotating angle of the bobbin about Z; i.e., the tilting angle. The displacement vector w for a given point of the bobbin can be represented by

w¼ R þ Tr ð1Þ

where R = [X Y O]T is the position vector of bobbin centroid, O, measured from the origin of the ground coordinates XYZ, O. Also,

T¼ cos h sin h 0 sin h cos h 0 0 0 1 2 6 4 3 7 5

is the transformation matrix due to h, and r = [X Y O] is the position vector of a differential mass dm in the bobbin as shown in Fig.3. Differentiating Eq.1 with respect to time and putting into kinetic energy, the kinetic energy of the bobbin can be obtained as

LT ¼ 1 2 Z m _ wTwdm_ ¼1 2m _X 2_{þ _}_Y2 þ1 2Ih_h 2 _X _h Ix sin hþ Iycos h þ _Y _h Ix cos h Iysin h; ð2Þ where Ihis the mass moment of inertia of the bobbin about

its centroid along z-axis, while Ix¼R_mxdm and Iy¼ R

mydm are first mass moments of inertia with respect to x and y axes, respectively. The potential energy of the pickup is next expressed as

V ¼1 2 kxX 2 þ kyY2þ khh2 þ mgY; ð3Þ

where kx, kyand khare the equivalent spring stiffnesses in

tracking, focusing and tilting directions; m is the mass of

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bobbin; g is the gravitation. Finally, the non-conservative virtual work can be derived as

dW¼ Z

A T F ð ÞTdwdA

¼ Fx cos h Fysin hdXþ Fx sin hþ Fycos hdY

þ Fhdh; ð4Þ

where dW enotes virtual work while, Fx and Fy represent

the actuation forces acting on the centroid, respectively, in the tracking and focusing directions. Fh denotes the

torsional moment about h. dW is the virtual bobbin displacement due to the applied force F. Substituting Eqs.2–4into Lagrange’s equation (Meirovitch 1967), the equations of motion can be readily obtained as

M€qþ Kq þ N þ G ¼ TF; ð5Þ

where q = [X Y h]T contains the generalized coordinates for describing the motion of the bobbin. M and K are overall mass and stiffness matrices. N contains the cen-trifugal and Coriolis force terms. G captures the gravitational effect. F captures the actuator forces. Their expressions are given in the followings,

The stiffness coefficients in the above K0, (kx, ky)

comply with kx¼ ky¼ 4

12EIw

L3 ; ð6Þ

where E is the elastic modulus, Iw is the area moment of

inertia about x- or y-axis for the wire, and L is the length of each wire. The expression of khis next due to be derived.

To this end, Fig.4 are first depicted to illustrate how to derive the moment M responsible for the tilting of the bobbin. In Fig.4, F represents the combined electro-magnetic force in focusing and tracking directions, which is generated by the current carried by a wire at some instant. / is the angle between F- and x-axis. Assuming an even magnetic field, the electro-magnetic forces induced by other three wires are identical and can also then be denoted by F. Then the net moment acting on the bobbin is

M¼ 4FD;

where D, as shown in Fig.4b, is the distance between the bobbin center and the wire. The angular deflection h is next derived for calculating the equivalent rotational (tilting) stiffness kh, which is started with expressing the

translational deflections in x and y directions due to the total electro-magnetic force F as

dx¼4F cos / kx ¼F cos /L 3 12EIw and dy¼4F sin / ky ¼ F sin /L3 12EIw ð7Þ

The net deflection along F is

d¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid2_xþ d2y q

: ð8Þ

Fig. 4 Moment generation of the four-wire-type optical pickup

M¼

m 0 Ix sin hþ Iycos h

0 m Ixcos h Iysin h

Ix sin hþ Iycos h Ixcos h Iysin h Ih 2 6 4 3 7 5; K0¼ diag kx; ky; kh F¼ FX½ FY Fh T ; G¼ 0 mg½ 0T; N¼ _h2Ixcos h Iysin h _h2Ixsin hþ Iycos h _

X _h Ix cos h Iysin hþ _Y _h Ix sin hþ Iycos h 2 6 4 3 7 5:

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Assuming small motions of the bobbin, thus, d = Dh. Henceforth, kh¼ M h ¼ 4FD d=D¼ 48EIw D2xþ D 2 y L3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dx=D ð Þ2þ Dx=Dð Þ2 q ; ð9Þ

where Dxand Dyare, respectively, as shown in Fig. 4b, the

distances in x and y directions between the bobbin center and each wire.

2.2 System identification

The prerequisite condition for designing a favorable FLC controller is to build a precise system dynamic model. This precise model is used for evaluating FLC performance before it is tested in the hardware system. The model can be derived by theoretical dynamic characteristics as stated in the previous section or through system identification in this section. The usage of the curve fitting method to identify system dynamic model is used herein for a single degree of freedom system. In general, this modeling method of single degree of freedom system could not sat-isfy the controller design of multi-axial system which has the coupling between different DOFs. Nevertheless, the intelligent servo controller—FLC, which will be proposed in next section, has capability to overcome the nonlinear coupling among three axes of the optical pickup, although the FLC is designed only based on three independent system dynamic models.

To identify the three independent models, a dynamic signal analyzer is used to obtain the frequency responses of the four-wire type optical pickup as subjected to a swept sine excitations in three different directions. The transfer functions of the real system can then be estimated. Figure5

shows the whole experiment system for identification, including the dynamic signal analyzer, three-axis optical

pickup, power amplifier and displacement sensor. The dynamic signal analyzer provides the input voltage signal, which is powered by an amplifier into the three-axis optical pickup. The optical fiber displacement sensor measures the displacement of the objective lens tip and feedbacks the signal to the signal analyzer. The frequency of the sine wave swept ranges from 5 Hz to 10 kHz. The frequency responses of the real system can be obtained. Figure6

shows the frequency responses long three directions. The forms of system transfer function to be identified is con-sidered as G sð Þ ¼ kx 2 n s2_{þ 2fxns}_{þ x}2 n : ð10Þ

where xnand f are the nature frequency and damping ratio,

respectively. The damping ratio f can be calculated by the following equation

G jxð Þ j jmax¼

1

2fpffiffiffiffiffiffiffiffiffiffiffiffiffi1 f2: ð11Þ

The gain k in Eq.10is computed by the DC gain of the real system by the following equation

DC gain¼ 20 log kx 2 n s2_{þ 2fxn}_{þ x}2 n _s¼jx;x¼5: ð12Þ Following the above basic identification procedure in Eqs.11, and 12, the system transfer function could be identified, yielding GFðsÞ ¼ 2:737 10 6 s2_{þ 53:48s þ 1:117 10}5ðlm=VÞ; ð13aÞ GTrðsÞ ¼ 3:583 10 6 s2_{þ 55:29s þ 1:194 10}5ðlm=VÞ; ð13bÞ GTiðsÞ ¼ 3:5631 10 6 s2_{þ 77:91s þ 9:485 10}5ðlm=VÞ; ð13cÞ along focusing, tracking and tilting, respectively. The identified system frequency responses are shown in Fig.6, where it is seen that the responses of the real system and the identified two-order transfer function of system are closely matched before 10 k (rad/s), which is normally beyond the actuation bandwidth of a optical pickup. However, beyond 10 k (rad/s) the responses are different between the real system and the identified model. This is due to noticeable noises from sensors in high frequency.

3 Auto-tuning scheme

The auto-tuning servo scheme is presented in this section. This intelligent control servo is designed based on a commercially often-used double-lead compensator, which

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is equipped with a fuzzy logic controller for real-time auto-tuning on control parameters. Finally, a genetic algorithm (GA) is used to optimize the auto-tuning process for better performance of pickup positioning.

3.1 Double-lead compensator

For an unstable system, a well-designed phase-lead com-pensator is employed to increase the phase margin of system for reaching desired stability. Since the desired compensated phase angle is more than 60°, the double-lead compensator is employed to provide enough phase margins for conducting better performance. The system block dia-gram is shown in Fig.7, which r is the reference signal; Gc1(s) is the first-lead compensator; Gc2(s) is the

second-lead compensator; K is the loop gain; 1/s is an integrator; y is the output signal. Note that in order to eliminate the steady-state error, the system with the double-lead com-pensator needs to be combined with an integrator. The transfer function of the double-lead compensator is Gcð Þ ¼ Ks

1þ a1T1s 1þ T1s

1þ a2T2s

1þ T2s : ð14Þ

where K [ 0, T1 & T2[ 0, a1 & a2[ 1. For simplicity,

among parameters K, T1, T2, a1, a2 of the phase-lead

compensators, only the parameter a1 of the first-lead

compensator is chosen in this study to be tunable. The design steps of the double-lead compensator in Eq.14are stated as below,

1. In order to annihilate the steady-state error, free integrators are added into individual transfer functions in Eqs. 13 for the dynamics along three axes of the pickup, resulting in GFðsÞ ¼ 2:737 10 6 s s_ð2_{þ 53:48s þ 1:117 10}5_Þ; ð15aÞ GTrðsÞ ¼ 3:583 10 6 s s_ð2_{þ 55:29s þ 1:194 10}5_Þ; ð15bÞ GTiðsÞ ¼ 3:5631 10 6 s s_ð 2_{þ 77:91s þ 9:485 10}5_Þ: ð15cÞ 102 103 -30 -20 -10 0 10 20 30 40 50 Frequence (rad/sec) magnitude (dB)

Focusing Bode Diagram

Real Model Identification Model 102 103 ₁₀4 -40 -30 -20 -10 0 10 20 30 Frequence (rad/sec) Magnitude (dB)

Tilting Bode Diagram

Real Model Identification Model 102 103 -10 0 10 20 30 40 50 Frequence (rad/sec) Magnitude (dB)

Tracking Bode Diagram

Real Model Identification Model Fig. 6 Frequency responses of

real and identified models

1 1 1 1 1 a T s T s + + 2 c G

K

1s r y plant 2 2 2 1 1 a T s T s + + 1 c G

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2. The frequency responses of the uncompensated open-loop system are shown in Fig.8. The designed double-lead compensator must satisfy the time domain specifications which are

Mp¼ 0:35 35%ð Þ; Ts¼ 0:1 secð Þ; 40\PM\60 ð16Þ where Mpis the maximum overshoot; Tsis the settling

time and PM is the phase margin. Following a standard procedure, for the systems in Eqs. 15, Mp, Ts and PM

can be derived by PM¼ tan1 2f _{ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi}1 4f4þ 1 2f2 p !1 2 8 < : 9 = ;; ð17Þ Mp¼ epf1 ffiffiffiffiffiffiffiffi 1f2 p Ts¼ 4 fx_n 8 > < > : ; ð18Þ xb ¼ xn ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2f2_þqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi₂_4f2_{þ 4f}4 r ; ð19Þ K kx 2 n s2_{þ 2fxn}_s_{þ x}2 n _s¼xb 2 ¼ 1; ð20Þ

where f is the equivalent damping ratio of the systems in Eqs. 15. The desired phase margins are chosen as 57°, 57° and 45° herein for satisfying performance specifications for tracking, focusing and tilting. The damping ratio f, the maximum overshoot Mp and the

natural frequency xncan be computed by Eqs.17and

18. The bandwidth of the system xbcan be solved by

substituting f and xninto Eq.19and the loop gain K is

derived by Eq. 20.

3. The loop gain K is next multiplied by the transfer function G(s) to produce a new transfer function Gu(s)

which is called ‘‘uncompensated system’’. The transfer functions of the uncompensated system are

GuFðsÞ ¼ 1:513 10 8 s s2_ð þ 53:48s þ 1:117 105_Þ; ð21aÞ GuTrðsÞ ¼ 1:196 10 8 s s_ð2_{þ 55:29s þ 1:194 10}5_Þ; ð21bÞ GuTiðsÞ ¼ 3:5631 10 8 s s_ð 2_{þ 77:91s þ 9:485 10}5_Þ: ð21cÞ The resulted phase margins of the dynamics along focusing, tracking and tilting are -81.9°, -81.3° and -74.1°, respectively, while the gain margins are -26.4, -25.2 and -13.7 dB, respectively. The gain-crossover frequencies xgare 572, 572 and 1,120 rad/s as shown in

Fig.7. According to the above computation results, the differences between the phase margin of Gu(s)’s in

Eqs.21and the desired phase margin is more than 90°, thus failing to stabilize the systems. To solve the problem, in the next step, the first-lead compensator is designed to improve the phase margin of the uncompensated systems.

4. Having added the first phase-lead compensator into the system, the phase margin of system could be improved to about -10°. With the first phase-lead compensator in hand, a newly computed gain-crossover frequency x

g1 of system which is larger than the original one, thus improving the phase margin. Note that the phase margin drops moderately as compared to theoretical expected value by the aforementioned computation process, since x_g1 is not located at the maximum phase. In order to achieve the desired phase margin, the estimated value e1for the difference between the

final desired phase margin and the originally uncom-pensated margin is necessarily computed. The correlative equation is then

/m1¼ /d1 /1þ e1; ð22Þ

where /d1is the desired phase margin, /1is the phase

margin of the uncompensated system, /m1is the phase

angle which is needed to be compensated. The com-pensation /m1 is accomplished by assigning a1 in

Eq. 14by

a1¼1þ sin /m1 1 sin /m1

: ð23Þ

Substituting the desired /m1 ‘s into Eq.23, the

parameter a1 of the first phase-lead controller can be

derived. The derived a1’s for each of three axes are

91.79, 101.51 and 787.8, respectively.

5. Define xm1 as the frequency of the maximum phase

margin. When xm1¼ xg1;, the first phase-lead com-pensator will provide 10 log a1dB for the gain at the

maximum phase margin. Thus x_g1 can be computed by 20 log Gu jxg1 ¼ 10 log a1: ð24Þ

The parameter T1of the first phase-lead compensator

in Eq.14 can be derived by substituting a1 and xm1

into T1¼ 1

xg1 ffiffiffiffiffia1

p : ð25Þ

Finally, the transfer function of the first phase-lead compensator is

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-100 -50 0 50 Magnitude (dB) -270 -225 -180 -135 -90 Phase (deg)

Focusing uncompensated Bode Diagram

Gm = -26.4 dB (at 334 rad/sec) , Pm = -81.9 deg (at 572 rad/sec)

Frequency (rad/sec) -80 -60 -40 -20 0 20 Magnitude (dB) 102 103 ₁₀4 -270 -225 -180 -135 -90 Phase (deg)

Tilting uncompensated Bode Diagram

Gm = -13.7 dB (at 974 rad/sec) , Pm = -74.1 deg (at 1.12e+003 rad/sec)

Frequency (rad/sec) -100 -50 0 50 Magnitude (dB) 101 -270 -225 -180 -135 -90 Phase (deg)

Tracking uncompensated Bode Diagram

Gm = -25.2 dB (at 346 rad/sec), Pm = -81.3 deg (at 572 rad/sec)

Frequency (rad/sec)

102 ₁₀3 ₁₀4

101 ₁₀2 ₁₀3 ₁₀4

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Gc1ðsÞ ¼1þ a1T1s

1þ T1s ð26Þ

The phase margins and gain margins of the compensated systems along focusing, tracking and tilting with the help from the first phase-lead compensator in Eq.26are -8.83° and -23.1 dB, -8.15° and -21.7 dB, -1.72° and -8.34 dB, respectively. The frequency response is depicted in Fig.9.

6. The design process of the second phase-lead compen-sator is the same as the first one. The transfer function of the second phase-lead compensator is

Gc2ðsÞ ¼1þ a2T2s

1þ T2s : ð27Þ

The transfer functions of the double-lead compensator are finally derived as

GcFðsÞ ¼ 55:3 1þ 0:008754s 1þ 0:00009537s 1þ 0:002413s 1þ 0:0006167s ð28aÞ GcTrðsÞ ¼ 33:48 1þ 0:009142s 1þ 0:00009006s 1þ 0:002352s 1þ 0:0006478s ð28bÞ GcTiðsÞ ¼ 100:0 1þ 0:0122s 1þ 0:00001549s 1þ 0:0007517s 1þ 0:0001067s ð28cÞ Figure10shows the phase margins and gain margins are 56.6° and 19.5 dB, 57 and 19.6 dB, 45.6° and 25.5 dB, respectively, for focusing, tracking and tilting, which satisfy the original time-domain specifications as listed in Table1. 3.2 Auto-tuning algorithm

Some parameters of the designed controllers are tuned in off-line fashion. The process consumes time and cost. The method developed herein tunes the parameters automati-cally in an on-line fashion to expedite the development of the pickup servo. First, the double-lead compensator is chosen to be the base controller which is designed in pre-vious sections. Second, the fuzzy logic controller is used to tune the parameter a1 of the double-lead compensator to

perform dynamic decoupling and forge the control efforts toward the goals of precision positioning. Finally, a genetic algorithm is employed off-line to search the optimal membership functions of the fuzzy logic controller (FLC) to render better performance.

3.2.1 Fuzzy logic controller

The structure of the double-lead compensator with FLC is called fuzzy double-lead controller. The structure is

illustrated in Fig. 11. Previous sections accomplish design of the double-lead compensator. The fuzzy logic controller is synthesized herein to tune the parameters of the double-lead compensator, the process of which consists of (1) defining inputs and outputs of FLC; (2) fuzzification; (3) rule table; (4) defuzzification.

3.2.1.1 Defining inputs and outputs of FLC The control output error e(t) and the increment of terror de(t) are chosen to be considered as the input signals to FLC. Therefore,

eðtÞ ¼ rðtÞ yðtÞ; ð29Þ

deðtÞ ¼ eðt þ 1Þ eðtÞ ð30Þ

where r(t) is the reference signal and y(t) is the output signal of the system. The FLC is intended to tune the performance of a standard phase-lead compensator C sð Þ ¼1þ aTs

1þ Ts T[ 0; a[ 1: ð31Þ

where a and T can be designated to move the zero and pole of the C(s) in the s-plane, as shown in Fig.12. Two dif-ferent kinds of movements for zeros and poles can be realized. They are

1) Moving -1/aT to the origin will improve the rising and settling times; however, the maximum overshoot will be increased with -1/aT closing the origin. 2) Moving -1/T far away the origin will decrease the

maximum overshoot; however, the rising and settling times will be increase as T becomes small.

According to above two points, the performance of the system can be decided by tuning a and T. In order to simplify the auto-tuning system and increase the effec-tiveness of computation, only one parameter a of a phase-lead compensator is tuned. The parameter a1 of the

first-lead compensator in Eq.14is chosen herein.

3.2.1.2 Fuzzification Fuzzification is the process of decomposing system input, output signals into one or more fuzzy sets. In order to fuzzify input/output signals, the membership functions are first defined. Many types of membership functions can be used, but the triangular or trapezoidal shaped ones are the most common because they are easier to be computed during fuzzification and later de-fuzzification. Figure 13 shows triangular membership functions used in this study for inputs and outputs of the system. The inputs are error and error increments defined in Eqs.29and30, while the output is the value of a1previously

given in Eq.14. Note that normalizations are applied to the error and error increment before fuzzification. The scaling factors for normalizations are chosen such that the error and

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-150 -100 -50 0 50 100 M agnitude (dB) 100 101 102 103 104 105 106 -270 -180 -90 0 Phase (deg)

Focusing first compensated Bode Diagram

Frequency (rad/sec) -150 -100 -50 0 50 100 M agnitude (dB) -270 -180 -90 0 Phase (deg)

Tracking first compensated Bode Diagram

Frequency (rad/sec) -200 -150 -100 -50 0 50 100 M a gnitude (dB) 107 -270 -180 -90 0 Phase (deg)

Tilting first compensated Bode Diagram

Gm = -8.34 dB (at 1.62e+003 rad/sec) , Pm = -1.72 deg (at 2.3e+003 rad/sec)

Frequency (rad/sec)

100 101 102 103 104 105 106

Fig. 9 The systems compensated by the first-lead compensator

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-150 -100 -50 0 50 100 M a gnitude (dB) -270 -180 -90 0 Phase (deg)

Tracking double-lead compensated Bode Diagram

Gm = 17.4 dB (at 6.55e+003 rad/sec) , Pm = 39.4 deg (at 1.76e+003 rad/sec)

Frequency (rad/sec) -150 -100 -50 0 50 100 M a gnitude (dB) 100 ₁₀1 ₁₀2 ₁₀3 ₁₀4 ₁₀5 ₁₀6 -270 -180 -90 0 Phase (deg)

Focusing double-lead compensated Bode Diagram

Gm = 17 dB (at 6.44e+003 rad/sec) , Pm = 39.4 deg (at 1.77e+003 rad/sec)

Frequency (rad/sec) -200 -150 -100 -50 0 50 100 M agnitude (dB) 100 ₁₀1 ₁₀2 ₁₀3 ₁₀4 ₁₀5 ₁₀6 ₁₀7 -270 -180 -90 0 90 Phase (deg)

Tilting double-lead compensated Bode Diagram

Gm = 25.5 dB (at 2e+004 rad/sec) , Pm = 40.8 deg (at 3.33e+003 rad/sec)

Frequency (rad/sec)

100 ₁₀1 ₁₀2 ₁₀3 ₁₀4 ₁₀5 ₁₀6

Fig. 10 The systems with the double-lead compensator

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error increment are large enough to span the interval [-1, 1]. Note in Fig.13 that the membership functions overlap to allow smooth mapping of fuzzification. The process of fuzzification allows the system inputs and outputs to be expressed in linguistic terms such that the control rules can be applied in a simple manner. The membership functions for parameter a1in the first phase-lead compensator in Eq.14

are determined such that different values of a1are expected

to achieve desired control goals.

3.2.1.3 Rule table Having completed fuzzification, efforts are paid next to establish linguistic rules regulating the input/output relations. This starts with understanding on the typical step responses for the focusing and tracking in Fig.14a and that for tilting in Fig. 14b, where it is seen that initial positions of the pickup along focusing and tracking direction are defined as negative and then to a zero value at steady state, while the initial tilting angle is zero since the pickup is initially in horizontal equilibrium. There are also corresponding histories of positioning error e(t) and increment of error de(t) depicted in Fig.14. As depicted in Fig.14, different time instants a - h are chosen for forging the aforementioned lin-guistic rules that regulates the relation from the inputs, e(t) and de(t) to the output, the parameter a1 in the

first-lead compensator in Eq.14. The forged linguistic rules are listed in the two rule tables in Fig.15 for focusing and tilting, respectively.

At varied time instants a-h denoted in Fig.14a, there are different combinations of magnitudes and signs of e(t) and de(t). The considered combinations are NB, NS, NM, ZE, PS, PM, PB, where S stands for Small, M for Medium, B for Big and ZE for Zero, while N for negative and P for positive. It is seen from Fig. 15a, b that a quasi-orbit is formed instants a to h, which correspond to zero e(t) and de(t), meaning that the control goal is achieved. For each combination in every square in both tables, a linguistic rule relating from e(t) and de(t) to a1needs to be proposed. The

rule is proposed based on the general effects of e(t) and de(t) that (1) when e(t) is large, a1 should be tuned to

decrease the maximum overshoot, following design guideline 2 in Sect.3.2.1.1for movement of zero and pole

1 1 1 1 1 a T s T s + + Focusing Fuzzy Logic Control 2 2 2 1 1 a T s T s + +

r

+−

y

e

de

auto-tuning

K

s

1 c G Optical Pickup

G u

2 c G 1 a Fuzzification Inference engine Defuzzification Fuzzy knowledge base Design input and output Design the membership function Design rule table input output

Fuzzy logic control

Fig. 11 The structure of the fuzzy double-lead controller

1

T

− −_aT1

jω

σ

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of the first-lead compensator in Eq.14; (2) when de(t) is large, a1 should be tuned to decrease rising and settling

times, following design guideline 1 in Sect.3.2.1.1. Based

on the above general guidelines, 49 rules can be forged. They are listed in Table 2. For example, near to the instant b in Fig. 15a, the error e(t) is close to zero while the increment of the error de(t) is negative and medium. At this point, in order to avoid a large overshoot, a1is tuned to a

small value, following guideline 2; thus, rule 23 is if ‘‘e(t) is ZE’’ and ‘‘de(t) is NB’’ then ‘‘a is SMALL’’.

3.2.1.4 Defuzzification Having forged fuzzy reasoning, linguistic output variables from applying Table2 need to be converted into numerical valued. The intension herein is to derive a single exact numeric value that best represents the inferred fuzzy values of the linguistic output variable. Defuzzification is such inverse transformation which maps the output from the fuzzy domain back into the numerical domain. The Center-of-Area (COA) method is chosen herein to complete the job, which is often referred to as the Center-of-Gravity (COG) method because it computes the centroid of the composite area representing the output fuzzy variables. -1 -0.5 0 0.5 1 0 0.2 0.4 0.6 0.8 1 e Degree of membership NB NM NS Z PS PM PB (a) -1 -0.5 0 0.5 1 0 0.2 0.4 0.6 0.8 1 de Degree of membership NB NM NS Z PS PM PB (b) 0 10 20 30 40 50 60 70 80 90 0 0.2 0.4 0.6 0.8 1 a Degree of membership B M S (c)

Fig. 13 Triangular membership functions

Fig. 14 Typical step response for a focusing and tracking directions; btilting direction

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3.2.2 Genetic algorithm

The main difficulty in FLC design and are the choices of proper membership functions and the associated adaptive gains. This subsection develops the application of a genetic algorithm (GA) technique in selecting the membership func-tions and the adaptive gain. A flow chart of a genetic algorithm is shown in Fig.16. First, choose the number of parameters need to be searched by GA. Then define the initial population. These parameters are coded to binary type and the fitness value is the defined and computed by GA to find the best parameters. Selection from the first generation of better parameters is reserved to next generation and compared with other parameters to search much better parameters in the second generation. Finally, the best parameters are derived and the maximum fitness value is reached.

3.2.2.1 Genetic algorithm for fuzzy double-lead control-ler A membership function in general has three important parameters to be specified: shape, distance and peak loca-tion, as described in Fig.17. Among those three values, the peak value of the triangle plays a key role in improving or

Rule1

…

PB PM PS ZE NS NM NB NB NM NS ZE PS PM PB a de e

…

_Rule7 Rule49

…

Rule43

…

a a b b cc d d ee ff g g h h (a) Rule1

…

PB PM PS ZE NS NM NB NB NM NS ZE PS PM PB a de e

…………

_Rule7 Rule49

…

Rule43

…

a a b b cc d d ee ff g g h h (b)

Fig. 15 Loci of time instants in Fig.14for a focusing and tracking; btilting directions

Table 2 Rule table

e(t) de(t)

NB NM NS ZE PS PM PB

Focusing rule table

NB B B B B B B B NM M M M B M S S NS S S M M S S S ZE S S S M S S S PS S S S M M S S PM S S M B M M M PB B B B B B B B

Tracking rule table

NB B B B B B B B NM M M M B M S S NS S S M M S S S ZE S S S M S S S PS S S S M M S S PM S S M B M M M PB B B B B B B B

Tilting rule table

NB M M B M S S S NM S M M S S S S NS S S M S S S S ZE S S M M S S S PS S M B M M M M PM B B B B B B M PB B B B B B B B

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degrading the performance of the controller. Different peak values generate different shapes of triangular membership functions. The four parameters to be specified in this study are those at points S1, S2, S3, S4, in Fig.17a; at points S5, S6, S7, S8, in Fig. 17b, while at point S19 in Fig.17c. The scaling factors Se and Sde are also considered to be the parameters for searching. Therefore, eleven parameters in total are searched by GA. The use of GA provides a sto-chastic optimization procedure to search for optimal parameter sets such that the controller performance can be improved. A population is initialized by setting up a ran-dom distribution of parameter vectors (incorporates the eleven parameters to optimize). The individual parameter

values are assumed with a uniform distribution across the allowable ranges.

GA works with a population of binary strings, not the parameters themselves. Before executing its algorithm, it is necessary to consider how a vector of values from the parameter set is converted to a binary string. The choice of coding length of each element in the vector is concerned with not only the resolution assigned by the designer in the corresponding search space, but also the type of spacings, such as logarithmic or linear spacings. In the binary coding method, the bit length 8 with linear spacing is adopted in this study for each element. As a result, the parameters of membership function can be transformed into a binary

(a)

(b)

(c)

Fig. 17 Membership functions for a the error; b the increment of error. c the parameter ‘‘a’’

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string with the bit length 104, and then the search space is formed. The decoding procedure is a reverse process of coding.

The cost function is chosen as a more general time-domain weighted error criterion and defined as

J¼X l t¼0 e tð Þ j j; ð32Þ

where e(t) = r(t) - y(t). Then the accumulated error is mapped into a fitness value to fit into the genetic algorithm. The fitness value can be regarded as how well a FLC can be tuned based on the string to actually minimize the error. The higher fitness value implies that the corresponding

Table 3 Parameters of genetic algorithm

Description Focusing Tracking Tilting

Initial population 60 60 60 Generations 100 100 100 Bit length 8 8 8 Reproduction Roulette wheel selection Roulette wheel selection Roulette wheel selection Crossover rate 0.7 0.7 0.8 Mutation rate 0.05 0.05 0.07 First-lead ‘‘a’’ range 0 ; 90 0 ; 90 0 ; 120 -1 -0.5 0 0.5 1 0 0.2 0.4 0.6 0.8 1 e NB NM NS Z PS PM PB (c1) Tilting -1 -0.5 0 0.5 1 0 0.2 0.4 0.6 0.8 1 de NB NM NS Z PS PM PB (c2) Tilting 0 20 40 60 80 100 120 0 0.2 0.4 0.6 0.8 1 a B M S (c3) Tilting -1 -0.5 0 0.5 1 0 0.2 0.4 0.6 0.8 1 e Degree of membership NB NM NS Z PS PPMB (a1) Focusing -1 -0.5 0 0.5 1 0 0.2 0.4 0.6 0.8 1 de Degree of membership NB NM NS Z PS PM PB (a2) Focusing 0 10 20 30 40 50 60 70 80 90 0 0.2 0.4 0.6 0.8 1 a Degree of membership B M S (a3) Focusing -1 -0.5 0 0.5 1 0 0.2 0.4 0.6 0.8 1 e NB NM NS Z PS PMPB (b1) Tracking -1 -0.5 0 0.5 1 0 0.2 0.4 0.6 0.8 1 de NB NM NS Z PS PM PB (b2) Tracking 0 10 20 30 40 50 60 70 80 90 0 0.2 0.4 0.6 0.8 1 a B M S (b3) Tracking

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string leads to a better solution. GA selects a parent with higher fitness values to generate better offspring. Therefore, a better FLC could be obtained by better fitness in GA. There are several methods to perform the mapping from a cost value to a fitness value. The windowing techniques using linear mapping is considered and the equation is given by

Fitness value¼Q

J; ð33Þ

where Q must be positive.

These GA operation are performed in a standard man-ner. Individuals are selected for breeding with a probability proportional to their fitness. For example, in the roulette wheel selection method, the ith string with high fitness value, Hi, is given a proportionately high probability of

reproduction, Ri, according to the distribution

Ri:¼PHi

Hi: ð34Þ

Once the strings are reproduced or copied for possible use in the next generation, they are put into a mating pool where they await further processing via crossover and mutation. After reproduction, simple crossover proceeds in three steps. First, two newly reproduced individuals from the mating pool are selected. Second, a cross point along the two strings is chosen uniformly at random. Third, the exchange of the characters following the crossover point is performed. Mutation is a rarely used random search operation, which increases the variability of the population in the mating pool and enhances GA performance to find a globally near-optimal solution.

According to above definition, the parameters of GA are listed in Table3. After the computation by GA, Fig.18

shows the optimized membership functions which are obviously different from Fig.13, while Fig.19 shows the corresponding convergent histories of fitness values for the dynamics in focusing, tracking and tilting. Based on the obtained results in Figs.18 and19, the best memberships of the FLC are successfully found by GA.

4 Numerical simulation and experimental validation Numerical simulations are conducted in this section to confirm the efficiency of the auto-tuning algorithm by checking if performance specifications are satisfies with expected counterparts. The designed auto-tuning algorithm is applied to three-axis optical pickups and the efficiency of this auto-tuning algorithm could be achieved via experi-mental validation. Figure20shows the implementation of the experimental system, which is accomplished mainly

by the dSPACE module. The output control signals are amplified by an OP-741 amplifier circuit to provide enough and safe input voltage to drive the pickup bobbin. The motions in the three directions of pickup are

0 20 40 60 80 100 30 35 40 45 50 55 (a) Gereration Best fitness 0 20 40 60 80 100 30 35 40 45 50 55 Generation Best fitness (b) 0 20 40 60 80 100 19 20 21 22 23 24 25 (c) Generation Best fitness

Fig. 19 Fitness value histories of the designed fuzzy double-lead controller in three directions; a focusing; b tracking; c tilting

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measured by a laser displacement sensor (MTI 250, MI-CROTRAK 7000) and two optical fiber displacement sensors, respectively. The sensor signals are feedbacked to dSPACE module that has the compiled auto-tuning algorithm to compute the control effort in real time fashion and then rendering precision positioning of the pickup. Note that the resolution of the laser displacement

is ±0.5–0.6 lm and the resolutions of the optical fiber displacement sensors are ±1–2 lm.

4.1 Fuzzy logic double-lead controller

Figure21 shows the numerical and experimental step responses of the closed-loop system with the fuzzy

Fig. 20 Experiment system

0 0.05 0.1 0.15 0.2 0.25 0.3 0 5 10 15 Focusing position Time(sec) Displacement( µ m) 0 0.05 0.1 0.15 0.2 0.25 0.3 0 5 10 15 Tracking position Time(sec) Displacement( µ m) 0 0.05 0.1 0.15 0.2 0.25 0.3 -10 -5 0 5x 10 -3 _{Tilting position} Time(sec) Angle(degree) Command Experiment Numerical Command Experiment Numerical Command Numerical

Experiment with tilting controller Experiment without tilting controller

Fig. 21 The actual responses of the three-axis optical pickup using the fuzzy double-lead controller

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double-lead controller. In this case, the designed position set as 10 lm in both focusing while in the tracking direc-tion, it is set as zero degree. It can be seen from this figure that the pickup reaches the expected position in three directions during a short settling time period simulta-neously. The numerical and experimental performances of the designed fuzzy double-lead controller are characterized in Table4 by different pre-defined performance indices. Based on these results, the auto-tuning algorithm has the capability to overcome the model uncertainties of the pickups caused by manufacturing tolerance and the

couplings among three different DOFs of the three-axis pickup, to reach zero tilting degree in a short period of settling time. This capability can be verified not only in numerical but also experimental data. Also seen from third subfigure of Fig. 21, the experimental responses without a tilting controller activated have much more error than those with a tilting controller. Moreover, it is also seen from Fig.21 that the experimental responses have slight time delays than the numerical counterparts. This is caused by the application of low-pass filter in experiments, which is intended to reduce the unavoidable sensor noises in experiment. The corresponding errors are shown in Fig. 22, while Fig.23shows control efforts.

4.2 Genetic algorithm applied to FLC memberships In order to attain better performance, the genetic algorithm was designed to seek optimum membership functions of the previously designed fuzzy double-lead controller in priori an off-line fashion. Figure 24show the responses in three dif-ferent directions of the controlled pickup. The corresponding time-domain performance of the fuzzy double-lead con-troller assisted with GA is summarized in Table5. Comparing Tables4and5, it can concluded that with well

Table 4 Time domain specifications of fuzzy double-lead controller

Direction Maximum overshoot (Mp) (%) Settling time (Ts) (s) Focusing Numerical 27.5 0.059 Experimental 27.5 0.062 Tracking Numerical 28 0.06 Experimental 28 0.062 Tilting Numerical 32 0.0645 Experimental 17 0.0645 0 0.05 0.1 0.15 0.2 0.25 0.3 -5 0 5 10 15 Focusing error Time(sec) Error value( µ m) 0 0.05 0.1 0.15 0.2 0.25 0.3 -5 0 5 10 15 Tracking error Time(sec) Error value( µ m) 0 0.05 0.1 0.15 0.2 0.25 0.3 -10 -5 0 5x 10 -3 Tilting position Time(sec) Angle(degree) Experiment Numerical Experiment Numerical

Experiment with tilting Experiment without tilting Numerical

Fig. 22 The position error by the fuzzy double-lead controller of three-axis optical pickup

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searched membership functions in hand, the less maximum overshoot and the shorter settling time than the case without genetic algorithm would be guaranteed, as shown in Fig.24

and Table 5. On the other hand, Fig.25 shows the corre-sponding error which is close to zero at steady state, while Fig.26shows the required control efforts.

0 0.05 0.1 0.15 0.2 0.25 0.3

-0.2 0 0.2 0.4

Control input focusing direct

Time(sec) Voltage(V) 0 0.05 0.1 0.15 0.2 0.25 0.3 -1 0 1

Control input tracking direct

Time(sec) Voltage(V) 0 0.05 0.1 0.15 0.2 0.25 0.3 -0.5 0 0.5

Control input tilting direct

Time(sec) Voltage(V) Experiment Numerical Experiment Numerical

Experiment with tilting controller Numerical

Fig. 23 The control effort by the fuzzy double-lead controller of three-axis optical pickup

0 0.05 0.1 0.15 0.2 0.25 0.3 0 5 10 15 Focusing position Time(sec) Displacement( µ m) 0 0.05 0.1 0.15 0.2 0.25 0.3 0 5 10 15 Tracking position Time(sec) Displacement( µ m) 0 0.05 0.1 0.15 0.2 0.25 0.3 -10 -5 0 x 10-3 Tilting position Time(sec) Angle(degree) Command Experiment Numerical Command Experiment Numerical Command Numerical

Experiment with tilting controller Experiment without tilting controller

Fig. 24 The actual responses of the three-axis optical pickup using the fuzzy double-lead controller equipped with a genetic algorithm

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4.3 Error analysis

In order to verify the expected performance of the fuzzy double-lead controller tuned by the genetic algorithm—the capability of overcoming the model uncertainties of the pickups caused by manufacturing tolerance and the cou-plings among the dynamics in three directions of the pickup, the positioning error analysis concerning two dif-ferent pickups is conducted herein with first defining the positioning error as Et¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Xn i¼1 P2 expðiÞ s ; ð35Þ

where n is the number of the experimental samples, Pexp

is the experimental positioning errors. The total error is defined as below the root mean square value of Et

thus

RMSE¼Et

n: ð36Þ

As shown in Figs. 27,28,29, although the fuzzy double-lead controller with the assistance from a genetic algorithm was applied for two different pickups, No. 1 and 2, the performances of these two pickups are still similar to each other. On the other hand, the error analysis for these cases, where 8,000 samples are taken from total time-domain responses in each case, results in that the total errors and the root mean square error (RMSE) of the pickup No. 1 are, respectively, 77.34 and 0.0097 lm for the focusing direc-tion, 74.69 and 0.0093 lm for the tracking direcdirec-tion, and 0.0193° and 0.00000024° for the tilting direction. The total errors and the root mean square error of the No. 2 optical pickup are, respectively, 78.64 and 0.0098 lm for the focusing direction, 67.67 and 0.0085 lm for the tracking direction, and 0.0618° and 0.00000077° for the tilting direction. The root mean square error of No. 1 and 2 optical pickups are both smaller than 0.1 lm, showing the satis-factory performance of the designed fuzzy controller equipped with tuning from the genetic algorithm.

Table 5 Time domain specifications of fuzzy double-lead controller with GA

Direction Maximum overshoot

(Mp) (%) Settling time (Ts) (s) Focusing Numerical 21 0.215 Experimental 20 0.0315 Tracking Numerical 20 0.0315 Experimental 18 0.0315 Tilting Numerical 17 0.0215 Experimental 16 0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 -5 0 5 10 Focusing error Time(sec) Error value( µ m) 0 0.05 0.1 0.15 0.2 0.25 0.3 -5 0 5 10 Tracking error Time(sec) Error value( µ m) 0 0.05 0.1 0.15 0.2 0.25 0.3 -10 -5 0 x 10-3 Tilting error Time(sec) Error value(degree) Experiment Numerical Experiment Numerical

Experiment with tilting Experiment without tilting Numerical

Fig. 25 The position error by the fuzzy double-lead controller with a genetic algorithm

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0 0.05 0.1 0.15 0.2 0.25 0.3 -0.1

0 0.1 0.2

Time(sec) Voltage(V) 0 0.05 0.1 0.15 0.2 0.25 0.3 -0.5 0 0.5 1 1.5

Time(sec) Voltage(V) 0 0.05 0.1 0.15 0.2 0.25 0.3 -1 0 1

Time(sec) Voltage(V) Experiment Numerical Experiment Numerical

Experiment with tilting controller Numerical

Fig. 26 The control effort by the fuzzy double-lead controller with a genetic algorithm

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 5 10 15 Focusing position Time(sec) Displacement( µ m) 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 5 10 15 Tracking position Time(sec) Displacement( µ m) 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 -5 0 5x 10 -3 _{Tilting position} Time(sec) Angle(degree) Command

Experiment first optical pickup Experiment second optical pickup

Command

Fig. 27 Experimental results of the the fuzzy double-lead controller with a genetic algorithm in two different three-axis optical pickups

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5 Conclusion

In order to overcome the model uncertainty of the pickups caused by manufacturing tolerance and the

coupling between three different DOFs of the three-axis pickup for rendering desired precision data-reading, the auto-tuning algorithm based on the fuzzy double-lead controller assisted by a genetic algorithm is proposed in 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 -10 0 10 Focusing error Time(sec) Error value( µ m) 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 -10 0 10 Tracking error Time(sec) Error value( µ m) 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 -5 0 5x 10 -3 _{Tilting error} Time(sec) Error value(degree)

Fig. 28 Experimental position error of the fuzzy double-lead controller with a genetic algorithm in two different three-axis optical pickups

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 -0.2

0 0.2 0.4

Time(sec) Voltage(V) 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 -1 0 1 2

Time(sec) Voltage(V) 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 -1 -0.5 0 0.5

Time(sec)

Voltage(V)

Fig. 29 Experimental control effort of the fuzzy double-lead controller with a genetic algorithm in two different three-axis optical pickups

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this study. The conclusive remarks are summarized as follows:

1. The design process of the auto-tuning algorithm has already been developed successfully. According to the numerical and experimental results, the focusing and tracking of the four-wire type optical pickup can be positioned at 10 lm and the tilting can be controlled to almost 0°.

2. As experimental result shows, the application of auto-tuning algorithm overcomes the couplings among three different DOFs of the three-axis pickup to render precise positioning successfully.

3. In order to attain better performance of the proposed fuzzy double-lead controller, the genetic algorithm is adopted to search optimum membership functions for the fuzzy logic controller was verified by experiments. Shorter settling time and less overshoot of experimen-tal responses are presented.

Acknowledgment The authors are greatly indebted to the National Science Council of ROC for the supports via the research contracts in nos. of NSC 96-2622-E-009-010-CC3 and NSC 97-2221-E-009-057-MY3.

References

Akgu¨l M, Morgu¨l O¨ (1997) Fuzzy controller design for parametric controllers. In: Proceedings of the 1997 IEEE international symposium on intelligent control, pp 67–72

Chao PCP, Huang JS, Lai CL (2003) Nonlinear dynamic analysis and actuation strategy for a three-DOF four-wire type optical pickup. Sens Actuator A Phys 105:171–182

Choi IH, Hong SN, Suh MS, Son DH, Kim YL, Park KW, Kim JY (2001) 3-axis actuator in slim optical pick-up for disc tilt compensation. In: Proceedings optical storage topical meeting, pp 178–180

Fan KC, Lin CY, Shyu LH (2000) Development of a low-cost focusing probe measurement. Meas Sci Technol 11:1–7

Fan KC, Lin CY, Shyu LH (2001) Development of a low-cost autofocusing probe measurement. Meas Sci Technol 12:2137– 2146

He N, Jia W, Yu D, Huang L, Gong M (2004) A novel type of 3-axis lens actuator with tilt compensation for high-density optical disc system. Sens Actuator A Phys 115:126–132

Hong HP, Park SJ, Han SJ, Cho KY, Lim YC, Park JK, Kim TG (1992) A design of auto-tuning PID controller using fuzzy logic. In: Proceedings of the 1992 international conference on indus-trial electronics, vol 2, pp 971–976

Hsu YP, Tsai CC (1996) Autotuning for fuzzy-PI control using genetic algorithm. In: Proceedings of the 1996 IEEE IECON 22nd international conference on industrial electronics, vol 1, pp 602–607

Hwang WR, Thompson WE (1994) Design of intelligent fuzzy logic controllers using genetic algorithms. In: Proceedings of the third IEEE conference on fuzzy systems , vol 2, pp 1383–1388 Kang JY, Yoon MG (1998) Robust control of an active tilting actuator

for high-density optical disk. In: Proceedings of the American control conference , vol 2, pp 861–865

Katayama R, Meguro S, Komatsu Y, Yamanaka Y (2001) Radial and tangential tilt detection for rewritable optical disks. Jpn J Appl Phys 40(3B):1684–1693

Marchant AB (1990) Optical recording. Addison–Wesley, New York Meirovitch L (1967) Analytical methods in vibrations. MacMillan,

London

Miyano K, Nagara T (2004) A new radial tilt detection method: differential wobble phase detection. In: Proceedings of optical data storage conferences, California, USA (5380), pp 189–196 Nagasato M, Hoshino I (1996) Development of two-axis actuator with

small tilt angles for one-piece optical head. Jpn J Appl Phys 1(35):392–397

Rosmalen GV (1987) A floating lens actuator. In: Proceedings of international symposium on optical memory

Visioli A (1999) Fuzzy logic based set-point weight tuning of PID controllers. IEEE Trans Syst Man Cybern A 29(6):587–592 Yamada S, Nishiwaki S, Nakamura A, Ishida T, Yamaguchi H (2000)

Track center servo and radial tilt servo system for digital versatile rewritable disc (DVD-RAM). Jpn J Appl Phys 39(12B):867–870

Yamasaki F, Arai A, Alkoh H (2006) Radial tilt detection using push-pull signals. Jpn J Appl Phys 45(2B):1158–1161

Zhang J, Cai L (1997) An autofocusing measurement system with a piezoelectric translator. IEEE ASME Trans Mechatron 2:213– 216