Sensorless Tilt Compensation for a Three-Axis
Optical Pickup Using a Sliding-Mode Controller
Equipped With a Sliding-Mode Observer
Paul C.-P. Chao and Chien-Yu Shen
Abstract—A mode controller equipped with a sliding-mode observer is synthesized and applied to a novel three-axis four-wire optical pickup for the purpose of sensorless tilt compensation. The three-axis pickup owns the capability to move the lens holder in three directions of focusing, tracking and tilting, which is re-quired particularly for higher data-density optical disks and pre-cision measuring instruments to annihilate nonzero lens tiltings. To achieve the sensorless compensation, Lagrange’s equations are first employed to derive equations of motion for the lens holder. A sliding-mode controller is then designed to perform dynamic de-coupling and forge control efforts toward the goals of precision tracking, focusing, and zero tilting. Along with the controller, a sliding-mode observer is designed to perform the online tilt esti-mation of the lens holder. This estimated tilt allows the previously designed sliding-mode controller to be implemented in most ex-isting commercial pickups without additional photodiodes to de-tect the tilting motion of the lens holder. A full-order high-gain ob-server is next forged to estimate the moving velocities of the lens holder in order to provide low-noised feedback velocity signals for the designed sliding-mode controller. Simulations are carried out to choose appropriate controller and observer gains. Finally, exper-iments are conducted to validate the effectiveness of the controller for annihilating lens tilting and the capability of the tilt observer for performing sensorless tilt compensation.
Index Terms—High-gain observer, sensorless compensation, sliding-mode controller, sliding-mode observer, three-axis optical pickup.
I. INTRODUCTION
F
OR OPTICAL disk drives (ODDs) and some surface-pro-filing instruments in micro- or nano-precision [1]–[3], the key component determining the performance is the optical pickup, which conducts data-reading via a well-designed optical system installed inside the pickup. Fig. 1 shows a photograph of a three-axis four-wire-type pickup actuator, which is designed and manufactured by the Industrial Technology and Research Institute (ITRI) of Taiwan. This pickup consists mainly ofManuscript received September 03, 2006; revised November 04, 2007. Man-uscript received in final form April 03, 2008. First published June 13, 2008; cur-rent version published February 25, 2009. Recommended by Associate Editor S. Devasia. This work was supported in part by the National Science Council of ROC through Contract 96-2221-E-009-069 and Contract 96-2622-E-009-010-CC3 and by the NCTU “Building Foundation” Project.
P. C.-P. Chao is with the Department of Electrical and Control Engineering National Chiao Tung University, Hsinchu 300, Taiwan, R.O.C. (e-mail: [email protected]).
C.-Y. Shen is with the Department of Mechanical Engineering Chung-Yuan Christian University, Chung-Li 32023, Taiwan, R.O.C. (e-mail: [email protected]).
Digital Object Identifier 10.1109/TCST.2008.924560
Fig. 1. Structure of the three-axis four-wire-type optical pickup by ITRI.
an objective lens, a lens holder (often called “bobbin”), wire springs, sets of wound coils, and permanent magnets. Thanks to the flexibility of the wire springs, the bobbin could easily be in motion as the forces acting on the bobbin are generated by the electromagnetic interactions between the magnetic fields in-duced by permanent magnets and the currents conducted in sets of coils. A conventional pickup actuator (not the three-axis one shown in Fig. 1) often owns only two sets of wound coils—the focusing and tracking coils and the associated magnets. In this way, two independent actuating forces are generated in the directions of focusing (vertical to the disk) and tracking (horizontal to the disk) to perform precision positioning of the lens. This type of actuator is commonly named a “two-axis actuator.”
High numerical apertures (NAs) and short-wavelength laser diodes (like violet diodes) have recently been employed for ob-jective lens designs in pickups in order to produce a smaller op-tical detecting spot on an opop-tical disk for better data-reading resolution. This aims at increasing detectable data density via decreasing the circular radius of the aberration region of the op-tical spot, which is the main factor limiting resolution of data storage in disks for ODDs or surface profiling for measuring in-struments. With the size of the optical spot decreased, original electromechanical designs of the pickup structure might become obsolete. One of the critical challenges arises from the unavoid-able tilting of the bobbin during its motion since the resulted coma aberration increases in proportion to NA , where is
the wavelength of the laser diode [4]. This tilting arises from two possible factors. The first factor is an uneven magnetic field due to manufacturing tolerance and/or the mispass of the net electro-magnetic 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 unparallelism in practice between the lens and the optical disc in high-speed ro-tations or the surface to profile. To restrain the bobbin tilting to a small level for more accurate, faster data-reading, some research works [5]–[10] have successfully developed the tilt servo sys-tems for the bobbin, in addition to original focusing and tacking ones. The tilt servo makes possible the capability of suppressing the unavoidable bobbin tiltings. The new pickups with tilt servo are often called the “three-axis” optical pickups. Fig. 1 shows one that was designed and fabricated by ITRI.
Among the aforementioned studies [5]–[10], Chao et al. [9], [11], [12] had established the equations governing the dynamics of the bobbin and designed a robust sliding-mode and intelligent controllers and a high-gain observer that is able to perform simultaneous positionings in three degrees of freedom (DOFs) of tracking, focusing, and tilting. The sliding-mode controller (SMC) [13], [14] was chosen due to its advantages (when compared with well-known controllers) of the robustness against parameter uncertainties and the capability to tackle system couplings and nonlinearities. However, the SMC in practice needs to acquire online feedbacks of bobbin motions not only in DOFs of focusing and tracking but also tilting. For conventional pickups, only motions of tracking and fo-cusing can be detected by specifically designed optical systems and several patches of photo-detectors [15]. Fig. 2 illustrates a typical optical/sensing system for measuring the motions of the bobbin, where Fig. 2(a) illustrates the overall optical system, Fig. 2(b) shows the basic principle of quad-detectors for detecting focusing motions, and Fig. 2(c) demonstrates the three-beam method for tracking motions. It is can easily be seen from Fig. 2 that the conventional optical system needs additional patches of photo-detectors or apply the recently proposed methods of signal analysis [16]–[18] on the reflected light intensity to equip itself with the capability of tilt detection. The former approach requires significant hardware modifica-tion, while the latter arrives reportedly at moderate accuracies due to sensor sensitivity variation and/or geometry of the disk surface. To circumvent the aforementioned shortcomings, a sliding-mode observer [19]–[26] is synthesized in this study to perform the online estimation of the tilting motions of the bobbin based on the motions in focusing and tracking detected by the original commercial optical systems, as shown in Fig. 2. The estimated tilting motion allows the predesigned SMC to be implemented in all existing commercial pickups without any hardware modification on the conventional optical disk drives. Along with the sliding-mode controller and observer is a full-order high-gain observer [27]–[33] forged in this study to estimate bobbin velocities in practice, which is aimed to pro-vide low-noise feedback velocity signals for the designed SMC. Experiments are finally conducted to verify the effectiveness of the designed SMC scheme for annihilating bobbin tilting and
Fig. 2. (a) Optical system of the pickup including photodiodes. (b) Method of quad-detectors measuring focusing error. (c) Three-beam method for measuring tracking error [15].
the capability of the observers for estimating the bobbin tilting motions and reducing the effects of digital noise while pickup in positioning.
This paper is organized as follows. Section II presents the mathematical modeling of the three-axis four-wire-type lens actuator. Section III presents the design of the SMC and the high-gain observer, while Section IV offers the synthesis of the sliding-mode tilt observer. The numerical and experimental re-sults are presented in Sections V and VI, respectively, to predict and verify the performance of the proposed controller/observer scheme. Finally, conclusions are given in Section VII.
II. MATHEMATICALMODELING A. Dynamic Modeling of the Bobbin
A typical three-axis pickup actuator designed and fabricated by ITRI as shown in Fig. 1 is considered in this study. The conventional bobbin, due to its specially designed supporting structure of four parallel wires, exhibits motions mainly in the DOFs of tracking -axis and focusing -axis . In addition to the motions in the - and -directions, small tilting often occurs about the -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. It is assumed that the pickup assembly can be simply modeled as a lumped mass-spring-damper system due to
Fig. 3. Planar dynamic model of the bobbin from the side view in Fig. 1.
bobbin’s high material rigidity compared with the flexibility of the suspending wires. Fig. 3 shows the schematic on the bobbin from the planar side view of Fig. 1 (from the viewpoint toward the – plane) and accompanying coordinates/notations defined for capturing the bobbin motion. Shown in Figs. 1 and 3 are coordinates defined as the body-fixed ones to the moving bobbin, while coordinates are global ground coordinates. also serves as a dynamic variable, capturing the horizontal, tracking motion; does the vertical focusing motion; does the rotating angle of the bobbin about , i.e., the tilting angle. The displacement vector w for a given point of the bobbin can be captured by
(1)
where is the position vector of bobbin cen-troid , measured from the origin of the ground coordinates
. Also
is the transformation matrix due to and is the position vector of a differential mass (dm) in the bobbin, as shown in Fig. 3. Differentiating (1) with respect to time and
putting into kinetic energy, the kinetic energy of the bobbin can be obtained as
(2) where is the mass moment of inertia of the bobbin about its centroid along the -axis, while dm and
are first mass moments of inertia with respect to the -and -axes, respectively. The potential energy of the pickup is next expressed as
(3) where and are the equivalent spring stiffnesses in tracking, focusing, and tilting directions, is the mass of bobbin, and is the gravitation. Finally, the nonconservative virtual work can be derived as
(4) where denotes virtual work while and represent the actuation forces acting on the centroid, respectively, in the tracking and focusing directions. denotes the torsional mo-ment about , and is the virtual bobbin displacement due to the applied force . Substituting (2)–(4) into Lagrange’s equa-tion [34], the equaequa-tions of moequa-tion can be readily obtained as
(5) where contains the generalized coordinates for describing the motion of the bobbin, and are overall mass and stiffness matrices, contains the centrifugal and Coriolis force terms, captures the gravitational effect, and captures the actuator forces. Their expressions are given here, with defined at the bottom of the page:
Fig. 4. Moment generation of the four-wire-type optical pickup. (a) 3-D view. (b) Side view of the bobbin from the positive direction.
The stiffness coefficients in the above comply with (6) where is the elastic modulus, is the area moment of inertia about the - or -axis for the wire, and is the length of each wire. The expression of is next due to be derived. To this end, Fig. 4 illustrates how to derive the moment responsible for the tilting of the bobbin. In Fig. 4(b), represents the combined electromagnetic force in focusing and tracking directions, which is generated by the current carried by a wire at some instant. is the angle between and the -axis. Assuming an even magnetic field, the electromagnetic forces induced by other three wires are identical and can also then be denoted by . Then, the net moment acting on the bobbin is
where , as shown in Fig. 4(b), is the distance between the bobbin center and the wire. The angular deflection is next de-rived for calculating the equivalent rotational (tilting) stiffness , which is started with expressing the translational deflections in and directions due to the total electromagnetic force as
(7)
Fig. 5. Circuit model of the VCM.
The net deflection along is
(8) Assuming small motions of the bobbin, thus, . Hence-forth
(9) where and are, respectively, as shown in Fig. 4(b), the distances in the - and -directions between the bobbin center and each wire.
B. Modeling of Voice Coil Motors
The electromagnetic forces acting on the bobbin in the direc-tions of focusing, tracking, and tilting are parameterized in this section. The actuators composed of sets of coils in the pickup are namely voice coil motors (VCMs), which are the electromag-netic dynamic balances of which, in pickup operation, as equiv-alently shown in Fig. 5, can be derived based on the Kirchhoff’s law, yielding
(10) where are the independent VCM input voltages in three directions, is the back electromotive force (EMF), and and represent the resistance, current, in-ductance, and back EMF constant of the VCMs, respectively. With the electrical dynamics in (10) derived, the Fleming’s left-hand rule [35] is then employed to derive the electromagnetic forces for actuation, which is
(11) where is the number of coil loops, is the magnetic flux density within the air gap between the bobbin and magnets, and is the total effective coil length for a single coil loop. Based
on the fact that the electrical dynamics of the conducted current are much faster than those mechanical ones in the directions of focusing, tracking, and tilting, the term in (11) can be neglected. Incorporating further the simplified in (11) into the system model in (5) arrives at a net system model with additional consideration of wire damping as
(12)
where is of the form which contains
the input voltages into the VCMs in three DOFs of tracking, fo-cusing, and tilting, respectively. The remaining two expressions in (12), , and , are given as shown at the bottom of the page.
C. Modeling of Uncertainties and Disturbances
There often exist an uneven magnetic filed generated by mag-nets and manufacturing tolerances with respect to various cru-cial dimensions of the pickup structure, which are the factors, other than the movement of the bobbin, causing the mispasses of the electromagnetic forces on the mass center of the bobbin; as a result, this leads to a nonzero tilting. For later control de-sign, the uncertainties due to the uneven magnetic field and man-ufacturing tolerance are formulated into the dynamical model in (12) as structured parametric uncertainties [36]. The formu-lation process is similar to that provided in [9], the details of which are omitted for the sake of simplicity. The formulation finally yields the system equations as
(13)
where , and .
and are the modeled uncertainties associated with their nominal valuess , and , respectively. The de-tailed expressions of and are provided in [9]. On the other hand, in (13), is in fact , where and are the disturbance resulting from the mea-sured radial vibrations in the - and -directions, respectively. The measured radial vibrations are shown in time and frequency
Fig. 6. Time trace and fast Fourier transform of measured runouts.
domains in Fig. 6, where it is seen that the practical radial vi-bratory disturbance consists of a primary harmonic of 1 m at 7850 rpm, which is exactly the disk rotational frequency.
III. DESIGN OF THESLIDING-MODECONTROLLER
An SMC is synthesized here for precision positioning of the pickup actuator in three DOFs. This well-designed SMC is ex-pected to accept the estimated states of and from the high-gain observer and the sliding-mode observer designed later in this section and then to calculate required control efforts to be fed to the optical pickup. Fig. 7 illustrates the system in a block diagram.
A. SMC
The design of the SMC starts with rewriting (13) as
Fig. 7. Block diagram of the controlled system.
where . The error vector of the system is defined as
(15)
where are targeted focusing, tracking and tilting positions. Note that is set as zero to eliminate non-zero tilting. A switching vector function containing the integrals of the positioning errors is next defined as
(16)
where . Note that is a matrix with
positive diagonal elements to be determined. The determina-tion of the elements in decides relative convergence speeds among three DOFs. Taking time derivatives of in (16) and incorporating (14) and (15) leads to
(17) To find an appropriate control law, the reaching law of the states with proportional plus constant power rates in the form of
(18) is first set to be achieved. Note that
with
being the components of the given sliding-mode matrix . Furthermore, and are positive weighting coefficients to be designed, while the choice of also allows one to adjust the convergence speeds. By selecting appropriate values of and in (18), the convergence of state trajectories to
the switching surface can be guaranteed since the reaching law (18) directly leads to
(19) To make possible the reaching law (18), the control efforts in (17) can be designed as follows, based on theory of sliding-mode control:
(20) By selecting large values of and for the control effort in (20), one is not only able to reach the convergence in (19), but also retain the robustness against the uncertainties of
and and the disturbance . Note that the necessary con-dition for reaching the robustness against the uncertainties and disturbance by the voltage input in (20) is the satisfaction of the input matching condition [37]. This condition can, in fact, be easily proven satisfied, since the four-wire optical pickup is close to a typical spring-mass-damper system, as derived in (13), thanks to the smallness of nonlinearity and gravity . Finally, in order to reduce the known chattering phenomenon near the switching surface, the function preproposed in (20) is replaced by a saturation function inside the predesignated boundary layer [13]. The saturation function is of the form
if
if (21)
where is the boundary layer width of the switching surface.
B. High-Gain Observer
A high-gain observer is next synthesized and augmented into the controlled system in practice, as shown in Fig. 7, with the
aim to estimate the moving velocities of the bobbin in three DOFs of tracking, focusing, and tilting. The estimated veloc-ities would be provided to the controller as the feedback sig-nals in places of those digitally computed time derivatives from the measured displacements of the moving lens/bobbin. The re-placements are motivated by the fact that the computed deriva-tives are often contaminated by the noises caused by sensor limitation and magnified by consequent digital discretization in practice.
The design process of the high-gain observer follows the pro-cedure similar to that in [24] and [31], the details of which are not presented herein. This process finally yield an observer in the form of
(22) where and denote, respectively, the estimates of the actual displacements and velocities of the bobbin/lens, respectively. On the other hand, and are designated as the diagonal matrices with constant elements, i.e., and , with the aim to achieve an estimation con-vergence for and . Convergence of the estimation can be guaranteed, even against the uncertainties of , and , by choosing the values of ’s and ’s such that the spectra (roots) of the characteristic polynomials
(23) are all in the left half plane. Finally, the value of in the ob-server in (22) is chosen as a small positive parameter, serving as a detuning parameter to adjust the convergence speed of the designed observer.
IV. DESIGN OF THESLIDING-MODETILTOBSERVER
A sliding-mode observer is synthesized in this section for es-timating the tilting motion of the bobbin. A typical design proce-dure of a sliding-mode observer consists of two steps [24]. This starts with first arranging the dimensional controlled system (13) in state-space form as
(24) where
(25) (26) Note in (24) that captures the measurable states, while does the states of bobbin tilting angle and velocity to be estimated by
the observer. The sliding surface is next defined as the estima-tion errors between measurable states and the estimates by the observer , leading to
(27) The siding-mode observer is then designed to comprise
(28a) (28b) where ’s are the estimated model of the ’s given in (24), ’s and ’s are gain matrices to be designed, and
(29) The terms and in (28) provide linear error convergence outside the sliding surface, while and gen-erate switching error-convergence efforts inside the sliding sur-face. Note that, from the term “ ” in (28b) and (27), it is conceived that estimation of the nonmeasurable states are driven by the estimation error of the measurable states . The dynamics of the error between the system and the estimates by the observer can be captured by subtracting (28) from (24), yielding
(30a) (30b) where
Following the standard procedure of sliding-mode observer de-sign, one can use the simple forms of and as
(31)
with large and to reach the convergence of . How-ever, due to the complex nonlinearity involved in the term in (30a), which lays great difficulty in determining values of and , the standard procedure of sliding-mode observer design cannot be employed herein to determine the observer gains of and . To solve this problem, the error (30a)–(b) are simpli-fied by only considering the first linear terms of and , yielding the approximate linearized model of
(32a) (32b)
where
(33) The entries in the above own particular structures due to the characteristics of the system state form in (24), which yield
(33a)
(33b)
where zero entries of ’s in (33) result from the reduced-order state-space representation for the original system in (24). Note that the expressions of ’s, the entries in the above ’s, can be easily obtained by a basic standard computation procedure of linearization; therefore, they are omitted herein for the sake of concise presentation. Considering the linearized model in (32), the convergence condition for based on the theory of the sliding-mode observer can be derived as
(34) From the structure of the above condition, one can set the gains of sufficiently large to stabilize the eigenvalues of , while also set the gains of sufficiently large to let the term suppress the coupling term . The corresponding lower limits of gains in and for satisfying condition (34) can be determined with known and reasonable assumptions on the ranges of states during pickup operations. Since the process of deriving these limits is straight-forward and tedious, they are not shown herein. Note that the gains of and are often and should be over-designed to achieve robustness for the observer. The next step is to determine the gain matrices associated with the unavailable states to estimate, i.e., and . As approaching zeros to satisfy condition (34), combination of (32a)–(32b) leads to
(35) As long as the matrix in (35) is Hurwitz, the estimation error on the bobbin tilting and associated deriva-tive in go to zero eventually. This can be accomplished by designing appropriate gains in . A special design procedure
TABLE I
GEOMETRIC ANDMAGNETICPARAMETERS OF THEVCM ACTUATOR
TABLE II
CASES FORCHOOSINGWEIGHTINGSPANDW
for determining is proposed in the following to ensure con-vergence of and further an easy control on the conver-gence rate. The design procedure for determining starts with forcing to be in the form of
(36) where ’s satisfies
(37) and ’s are to be assigned. Note that, with the special nature of —zero entries in the first and third rows, ’s in (37) can be easily solved for given ; then can be determined based
Fig. 8. Simulated step responses, control efforts, and errors of the pickup system controlled by the designed SMC in three directions with control parameters of Case 2 given in Table II. (a) Tracking position. (b) Focusing position. (c) Tilting angle. (d) Control effort. (e) Control effort. (f) Control effort. (g) Tracking position error. (h) Focusing position error. (i) Tilting angle error.
on (36) with predesigned . Incorporating (36) into system (35), one yields
(38) Based on the particular structure of the matrix in (38), the eigen-values of the system in (35) are and . One can assign the speed of convergence for error estimation of the balancing ball by designating the values of and to enforce eigenvalues of to be the desired ones. With designated and can be determined using (36) and (37) with predesigned .
It is pertinent to note that the proposed observer is forged based upon the linearized part of the subsystem due to the fact that the system possesses highly nonlinearity that makes the ob-server design based on the original nonlinear system almost im-possible. With sufficiently large gains inside the matrices and , even though the observer is designed from the linear part of the system, it still stands a fair chance to have a robust convergence of estimation.
V. NUMERICALSIMULATIONS
Numerical simulations are conducted here to find suitable controller parameters and observer gains. A practical three-axis
pickup designed and manufactured by ITRI, as shown in Fig. 1, is used for the ensuing numerical simulations in this section and further experimental validation in the next section. All param-eters of the considered three-axis four-wire-type optical pickup actuator are calibrated or obtained from documented properties. They are kg m m Pa m m m s kg m kg m kg m (39)
while the parameter values of the VCMs are given in Table I. Note that all of the physical meanings of the notations in (39) were previously given in Section II.
Fig. 9. Simulated step responses, control efforts, and errors of the pickup system controlled by the designed SMC in three directions with control parameters of Case 4 given in Table II. (a) Tracking. (b) Focusing. (c) Tilting. (d) Control effort. (e) Control effort. (f) Control effort. (g) Tracking error. (h) Focusing error. (i) Tilting error.
A. Choosing Parameters of the SMC
Numerical simulations on the controlled system are carried out here to choose appropriate parameter values for the SMC and the high-gain observer designed in Section III. The ap-propriateness can be ensured if the error convergence in (19) is reached in the presences of the disturbances due to radial burnouts, as specified by in (13), and the parametric uncer-tainties, also in (13), which are prescribed as 5% of respective dimensions of the pickup bobbin. It can be obtained from the SMC design process proposed in Section III that the main con-trol parameters affecting the concon-troller performance are and . Among these parameters, one can only consider and
as the main parameters to be tuned for convergence and ro-bustness, since the effects of are only on the convergence which can also be tuned by and . On the other hand, de-termines only the relative convergence speeds among the states. Therefore, in the following process of controller tuning, is set to be unity to leave the job of convergence tuning to and
, while is designated as for
re-flecting desired relative convergence speeds among the states. Four cases of varied and as listed in Table II are con-sidered herein for simulations to obtain satisfactory controller
performance with appropriately chosen values of and . The first, second, and third cases are integrally set to consider rela-tive large, medium, and small levels of and a fixed , with the aim to find the most suitable value of for better control convergence and robustness. With the desired trajectories in the and directions set to be step functions of 10 m and the de-sired tilting in direction to be zero, simulations are conducted for the aforementioned four cases and the associated root mean squares of errors (RMSEs) of the bobbin positioning in three di-rections of tracking, focusing, and tiling are given in Table II. It can be seen from the RMSEs corresponding to Cases 1–3 in Table II that the case with the prechosen medium value of provides the best pickup actuation performance since it renders the smallest RMSE in all three directions. Fig. 8 presents the simulated time-domain responses for Case 2, where it is seen that the bobbin positionings in directions of focusing and tracking are achieved within 70 s, while the positioning in the tilting direction to zero is accomplished in a shorter pe-riod of 30 s. Of most importance is that the designed decou-pling SMC is shown capable of containing an almost zero tilting angle to deg as the bobbin is positioned to 10 m in both focusing and tracking directions. In the next step, the simulations with the parameter values considered in Case 4 in
Fig. 10. Simulated step responses, control efforts, and errors of the pickup system controlled by the designed SMC assisted by the high-gain observer in three directions. The employed controller and observer parameters are those of Case 2 in Table III. (a) Tracking. (b) Focusing. (c) Tilting. (d) Control effort. (e) Control effort. (f) Control effort. (g) Tracking error. (h) Focusing error. (i) Tilting error.
Table II are conducted to find suitable value of for a satis-factory controller performance. In Case 4, with fixed to the pre-chosen medium value of is increased from the value of 10 in Cases 1–3 to 1000 for a chance to speed up the convergence near the sliding surface. The corresponding results are shown in Fig. 9. A general comparison between Figs. 8 and 9 renders that an increased does not significantly shorten re-sponse time especially in the transient stage, while brings small levels of fluctuation due to the higher control gain of near the sliding surface. The fluctuation is perceivable since dic-tates level of the switching effort near the sliding surface. To avoid the fluctuation, the suitable value of remains as the prechosen 10.
Efforts are next devoted to find suitable high-gain observer gains also based on simulations. In the first step, the values of pa-rameters ’s and ’s to form the characteristic polynomial (23) are designated to lead to
and with the aim to
ensure robust stability of the observer. Second, the small param-eter is first chosen as for a relative fast convergence on state velocity estimations. The resulted responses are plotted in Fig. 10, where the responses of the controlled system with the feedback velocities estimated from the high-gain observer are represented by dot-dashed curves, while those with those
ac-TABLE III
CASES FORCHOOSINGDETUNINGPARAMETER"
tual velocities are represented by solid curves for comparison. It is seen from the errors in Fig. 10(g)–(i) that the high-gain observer-based controller achieves precision positionings in the three DOFs of focusing, tracking, and tilting. Also, the RMSE values of the controlled system with the high-gain observer em-ployed are close to those of actual responses in all three DOFs, indicating a satisfactory convergence in velocity estimation by the high-gain observer. is next detuned to a smaller value, , to aim for a better observer convergence. The cor-responding RMSEs of the simulated response are given in the second row of Table III. It is seen from in Table III that all of the RMSEs for Case 2 are slightly smaller than their counterparts for Case 1. This indicates that a smaller is a better
TABLE IV
CASES FORCHOOSINGOBSERVERGAINSk ; a ;ANDa
choice than . Note that the previous finding in fact reflects the theoretical core of the high-gain observer, that is, in each design of the observer, one ought to continuously detune the value of until the required computation load is bearable for the DSP module.
B. Choosing Parameters of the Sliding-Mode Observer
Further numerical simulations with varied gain values of the sliding-mode observer are carried out in this section to choose the appropriate ones to achieve fast observer error convergence on the tilting motion of the optical pickup. Based on the design procedure of the sliding-mode observer developed in the end of Section IV, the observer performance is dominated by the designed observer gains and . Among these gains, and are diagonal entries of the gain matrices as previ-ously defined in (31) and they rule the convergence speed of the observation for available states, i.e., focusing and tracking motions, while and , as previously defined in (36) and appearing in (38), dictate the convergence speed of unavail-able states, i.e., tilting angle and the associated velocity of the moving bobbin, respectively. The process of determining all ob-server gains starts with and . It is found based on theoret-ical ground and a number of simulations which are not shown for the sake of brevity that with varied and the conver-gence speed for measurable focusing and tracking motions can easily be tuned to desireds by simply increasing one of and to certain level, since both gains of or affects the con-vergence of available states simultaneously. Therefore, the fol-lowing gain design process would only consider as the tuning gain for available states with fixed to 10, while and for unavailable states.
Six different combinations of , and listed in Table IV are considered to carry out simulations for the prac-tical three-axis pickup designed and manufactured by ITRI, as shown in Fig. 1. The resulted performances of the designed controller and observer for the aforementioned six combina-tions are evaluated in terms of RMSEs for positioning and observation errors, also listed in respective cases in Table IV. Note that the positioning errors refers to the difference between the actual responses and the commands in three DOFs of the
bobbin, while the positioning errors does the difference between the actual responses and the estimated counterparts provided by the predesigned sliding-mode observer. With a well-designed controller and observer, the actual responses ought to be settled at the commands with small RMSEs and close to the estimated counterparts provided by the sliding-mode observer at steady state. From the results of the first four cases in Table IV where only the gain of is varied to choose its best value, it is seen that, as is decreased from 50 of Case 1 to 0.1 of Case 3, the convergence is enhanced with evidence of RMSE for the tilting of actual responses deceased to around deg, which is apparently a satisfied performance for tilt compensation. As further decreasing to 0.01 of Case 4, the tilting RMSE remains around deg, which indicates that, as determining , one can choose 0.1 or smaller for maximum performance. Fig. 11 depict the simulated system responses, control and observation errors for Case 3. It is seen from Fig. 11(g)–(i) that the precision focusing and tracking are well performed while, most importantly, the tilting of the pickup is totally annihilated with small observation errors within a short period of time, which is also clearly seen from the enlarged figure beside Fig. 11(c). It is also seen from Fig. 11(j)–(i) for observation errors that the sliding-mode observer has achieved a fast observation in tilting and satisfactory observation after the controller/observer scheme is activated around 0.3 s. With gain determined, the second step is to determine gains of and . Gains of and lower and higher than those in Cases 1–4 are considered in Cases 5 and 6, respectively, for investigating their effects on the observation convergence. Recall that affects the convergence speed of the bobbin tilt angle, while
does the tilt velocity. Therefore, based on the theory of the sliding-mode observer, smaller gains of and as given in Case 6 stand a stronger risk losing observation convergence. A general comparison among Cases 3, 5, and 6 confirms the risk. As increased from 10 (Case 3) to 100 (Case 5) and from 100 (Case 3) to 1000 (Case 5), the observation convergence holds as evidenced from the unchanged resulted RMSEs given in Table IV. On the contrary, as decreased from 10 (Case 3) to 0.1 (Case 6) and from 100 (Case 3) to 1 (Case 6), the observation collapses as evidenced from a larger tilting RMSE
Fig. 11. Simulated step responses, control efforts, and errors of the pickup system controlled by the designed SMC assisted by the sliding mode observer in three directions. The employed controller and observer parameters are those of Case 3 in Table IV. (a) Tracking. (b) Focusing. (c) Tilting. (d) Control effort. (e) Control effort. (f) Control effort. (g) Tracking error. (h) Focusing error. (i) Tilting error. (j) Tracking observed error. (k) Focusing observed error. (l) Tilting observed error.
of the actual responses, . Also seen from the results of Case 6 in this table is a large tilting RMSE of the observation error, , indicating that the designed sliding-mode observer fail to estimate the tilting angle of the optical pickup within a required time frame. Fig. 12 depicts the simulated system responses, control, and observation errors for Case 6. It is seen from Fig. 12(i) and (l) that the observation errors appear unacceptably large at transient and steady states, which indicates the failure of the sliding-mode observer with overdecreased and .
VI. EXPERIMENTALVALIDATION
Experiments are conducted to verify the expected effective-ness of the designed SMC and the accompanied sliding-mode tilt observer with the parameters and gains chosen in Section V. Fig. 13 illustrates the experimental setup employed, which consists of a laser displacement sensor (MTI 250), two optical fiber displacement sensors (MTI KD-300), and a three-axis pickup provided by the Industrial Technology and Research Institute (ITRI), Taiwan. The implementation of the previously designed controller/observer algorithms are accomplished by
a DSP module (dSPACE1103) with sampling time interval of 100 s. This DSP accepts the measurements of the bobbin motions from the laser and optical sensors, based on which calculations are conducted following the previously designed controller/observer algorithms to forge required output control efforts. The efforts are further amplified by a high-speed linear and bipolar amplifier (NF-HSA4051) before they are fed into the pickup for generating bobbin motions. The tracking motions of the bobbin (horizontal, along the direction) in the pickup are measured directly by the laser displacement sensor, while the focusing motions (vertical, along the direction) are ob-tained by averaging two optical sensor signals. The averaging is conducted by the employed DSP module. The resolution of the laser displacement is up to 0.1 m, while those of the optical sensors are around 0.3 m. It should be also noted at this point that the laser and optical sensors used herein play integrally the role of the optical detection system employed in commercial pickups. As mentioned in the Introduction, several mature detection methods are available to provide the estimation on the bobbin motions. Fig. 2 illustrates some of commonly used systems and their working principles.
Fig. 12. Simulated step responses, control efforts, and errors of the pickup system controlled by the designed SMC assisted by the sliding mode observer in three directions. The employed controller and observer parameters are those of Case 6 in Table IV. (a) Tracking position. (b) Focusing position. (c) Tilting angle. (d) Control effort. (e) Control effort. (f) Control effort. (g) Positioning error. (h) Positioning error. (i) Angle error. (j) Error of observer. (k) Error of observer. (l) Error of observer.
Fig. 13. Schematic diagram of the experimental setup.
Experiments with and without tilt compensation are carried out to verify the expected performance of the proposed
con-troller and observers. The employed concon-troller and observer gains were determined previously based on the simulations in Sections V-A and V-B, respectively. They corresponding cases are denoted as “for experiments” in Tables II–IV. Fig. 14 plots the experimental responses, control efforts, time histories of control, and observation errors with the theoretical counterparts for comparison. Also shown in Fig. 14(c), (i), and (l) are the tilt responses without the tilt compensation in dashed curves, com-pensation error, and observation error, respectively, for demon-strating the importance of tilt compensation. It is generically seen from Fig. 14(a)–(i) that the synthesized controller and ob-server are able to perform precision positionings simultaneously in the directions of tracking and focusing, and tilting compensa-tion, rendering steady-state RMSEs of 0.115 m, 0.258 m and 0.0016 , respectively. Note that the RMSEs obtained herein for tracking and focusing are slightly below the resolutions of the used laser and optical sensors, respectively, thanks to the nu-merical integrations performed by the high-gain observer to re-duce noises by the sensors and environment. Other than the
posi-Fig. 14. Experimental and simulated step responses, control efforts, and errors of the pickup system controlled by the designed SMC assisted by the sliding mode tilt observer in three directions. (a) Tracking. (b) Focusing. (c) Tilting. (d) Control effort. (e) Control effort. (f) Control effort. (g) Tracking error. (h) Focusing error. (i) Tilt compensation error. (j) Tracking observation error. (k) Focusing observation error. (l) Tilting observation error.
tioning errors, small steady-state observation errors provided by the designed sliding-mode observer in three DOFs are also seen from Fig. 14(j)–(l), respectively. A general closeness is also present between the simulated curves and experimental counter-parts. It should be noted from Fig. 14(c) and (i) that, without the tilt compensation implemented (simply switching off the con-trol effort in the tilt direction), the pickup would not be able to contain the tilt at steady state. Finally, it should also be noted that the large fluctuations in the control efforts in Fig. 14(d) and (e) prior to the initiation of the step commands are due to the sub-stantial estimation errors of the bobbin velocities induced by the high-gain observer. Based on theory of the high-gain observer convergence [31], the aforementioned observer estimation er-rors can easily be shown inevitable as the positioning erer-rors are close to zeros—as before the step commands are initiated.
VII. CONCLUSION
This study is dedicated to perform sensorless tilt compensa-tion and simultaneous precision posicompensa-tionings in direccompensa-tions of
focusing and tracking for a three-axis optical pickup via de-sign and experimental validation of an SMC equipped with a sliding-mode tilt observer. Based on the simulated and experi-mental results, the following conclusions are drawn.
1) Simulations and experimental studies have demonstrated the capability of the synthesized SMC and observer in performing simultaneous positioning of the bobbin in the pickup in the directions of focusing, tracking, and reducing tilting to acceptable, small levels without tilt sensor. 2) The levels of the tilting in the cases without tilt
compen-sation activated are found to be substantially larger than those with the titling compensated by the designed SMC accompanied by a sliding-mode observer, showing the im-portance of employing a three-axis controller and the effec-tiveness of the accompanied controller/observer algorithm proposed in this study to suppress nonzero tilting. 3) The design and application of the sliding mode observer
allows the SMC to perform tasks of decoupling and preci-sion positioning of the moving bobbin without a tilt sensor
required, making possible an easy implementation of sen-sorless tilt compensation in commercial pickups.
ACKNOWLEDGMENT
The authors would like to thank the staff of the Industrial Technology and Research institute (ITRI), Taiwan, R.O.C., for their help with the experimental hardware and setup.
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Paul C.-P. Chao received the B.S. degree from National Cheng-Kung
Uni-versity, Tainan, Taiwan, R.O.C., in 1989 and the M.S. and Ph.D. degrees from Michigan State University, Lansing, in 1993 and 1997, respectively.
After graduation, he worked for the CAE Department, Chrysler Corporation, Auburn Hill, MI, for two years. He is currently a Faculty Member of the Elec-trical and Control Engineering Department, National Chiao Tung University, Hsinchu, Taiwan. In recent years, his research interests focus on the electronics for optical systems; micro-mechatronics, control technology, micro- sensors, and actuators.
Prof. Chao was the recipient of the 1999 Arch T. Colwell Merit Best Paper Award from the Society of Automotive Engineering, the 2004 Long-Wen Tsai Best Paper Award from the National Society of Machine Theory and Mech-anism, the 2005 Best Paper Award from the National Society of Engineers, Taiwan, the 2002/2003/2004 CYCU Innovative Research Award, the AUO Award in 2006, and the Acer Long-Term Second-Prize Award in 2007.
Chien-Yu Shen received the B.S.E. degree in civil engineering and the M.S.
degree in mechanical engineering from Chung Yuan Christian University, Chung-Li, Taiwan, R.O.C., in 2002 and 2004, respectively. He is currently working toward the Ph.D. degree in mechanical engineering from Chung Yuan Christian University.
He specializes in mechanics of materials, mechanical dynamics, and system control theories include PID compensation, lead-lag compensation, fuzzy neural network control, and sliding-mode control theory. In order to practice novel op-tical data-reading systems, while pursuing his master’s and doctoral degrees, he has inclined his efforts toward researching dynamic modeling of novel axis optical pickup head and design of controllers and observers for novel three-axis optical pickup heads. Furthermore, he also performed a cooperative plan of The Industrial Technology Research Institute in 2004, which is about develop-ment in auto-tuning algorithm of optical data-reading servo.
