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IEEE TRANSACTIONS ON MAGNETICS, VOL. 38, NO. 5, SEPTEMBER 2002
Modeling and Cancellation of Pivot
Nonlinearity in Hard Disk Drives
Jian Qin Gong, Lin Guo, Ho Seong Lee, Member, IEEE, and Bin Yao, Member, IEEE
Abstract—This paper considers the issue of modeling the pivot
nonlinearity in a typical hard disk drive, taking into account viscous friction, Coulomb friction and hysteresis effects. It presents
a nonlinear compensator design, based on a proposed model, to
make the compensated system behave like a double integrator regardless of the magnitude of the input current. It describes extensive experiments, carried out in both time domain and frequency
domain, to show the effects of pivot nonlinearity and the effectiveness of the compensator.
Index Terms—Compensator, friction, hard disk drive (HDD)
system, hysteresis, modeling.
I. INTRODUCTION
W
ITH the dramatically increasing density and decreasing
size in the hard disk drive (HDD) system, there are
always challenges in the servo control design for HDD [1].
Today’s information storage industry is forced to move toward
micro hard disk drives (microdrives) for laptop computers
and smaller hand-held devices as well as high performance
drives with extreme large storage capacity and fast data
retrieval/storage time. Instead of increasing the physical dimension of a hard disk to satisfy this requirement, a hard disk with
higher track density is in order. The requirement of higher track
density demands more precise and smaller motion of the actuator arm, which exerts more difficulties on track following and
single-track seeking. Consequently, it has posted a great challenge to the design of servo controllers for the moving parts of
the information storage devices, as the current industrial servo
design techniques may no longer be able to deliver the required
accuracy and speed, which is due to either the fundamental
performance limitations coming from the artificial constraint
of using only linear controller structures [2]–[5] or the inability
of the controller in dealing with the ever-increasing noticeable
effect of nonlinear disturbance forces. These nonlinearities
may include pivot bearing friction [6] and hysteresis [7], the
effects of internal/external vibrations [8]–[10], the eccentricity
caused by the discrepancy between the true center of the disk
and the center of the spindle motor [11], the influences of the
aeroelastic forces [12]–[15], and the uncertainties to the position error signal (PES) [16]–[19]. In order to capture the effects
of these nonlinearities, certain nonlinear models are needed.
However, it is usually very difficult, if it is not impossible, to
obtain those models which are physically meaningful and yet
simple enough for control design. In order to achieve a high
performance, all these undesirable nonlinear factors need to
be taken into account in the design of servo controller, which
must have a nonlinear structure. An alternative way to improve
the performance of HDD is the use of a dual-stage controller,
which may bypass the problems related to these nonlinearities.
As a matter of fact, the application of dual-stage actuator to
HDD does reduces the PES significantly [20]. However, it
also increases the cost of HDD. So, it is still meaningful to
investigate these nonlinearities in HDD.
In this paper, the main focus is put on the pivot nonlinearity
including pivot friction and pivot hysteresis, which has been
extensively studied in [21]–[23]. An excellent survey work
about friction has been presented in [24]. In [21], the fuzzy
logic model was obtained to approximate the pivot hysteresis.
Although the modeling results match the measurement data
very well, it may be too complicated to be used in industry.
Different models, such as preload plus two-slope model, hysteretic two-slope model, etc., were investigated in [22] and [23],
where both time domain and frequency domain performance
were considered. Although the models proposed in [22], [23]
may match the measurement data either in time domain or in
frequency domain, none of them can fit the measurement data
both in time domain and in frequency domain very well. In
this paper, based on measurement data, a nonlinear model is
developed. Based on the nonlinear model, a nonlinear compensator is then designed. In both time domain and frequency
domain, extensive experimental results are presented to verify
how well the nonlinear model can match the measurement
data. It is shown, both theoretically and experimentally, that the
compensated system behaves like a linear system, specifically,
a double integrator in the ideal case. In this way, it is confirmed
that the proposed model can match the pivot nonlinearities.
The rest of paper is organized as follows. The modeling of
the pivot nonlinearity of HDD system is discussed in Section II.
Based on the model developed in Section II, a nonlinear compensator is designed in Section III. With and without model
compensation, experimental comparison studies are carried out
in Section IV to show the effectiveness of the proposed compensator, followed by conclusions drawn in Section V.
II. MODELING OF PIVOT NONLINEARITY OF HDD SYSTEM
Manuscript received November 2, 2001; revised April 26, 2002.
J. Q. Gong and B. Yao are with the School of Mechanical Engineering, Purdue
University, West Lafayette, IN 47907 USA (e-mail: gong2@ecn.purdue.edu;
byao@ecn.purdue.edu).
L. Guo and H. S. Lee are with Maxtor Corporation, Milpitas, CA 95035 USA
(e-mail: lin_guo@maxtor.com; ho_lee@maxtor.com).
Digital Object Identifier 10.1109/TMAG.2002.802745.
The experimental setup is given in Fig. 1, where laser Doppler
vibrometer (LDV) is used to measure the displacement and velocity of the tip of the actuator arm, a floating point digital signal
processing (DSP) system is used to collect the data and execute
the modeling algorithm, and a dynamic signal analyzer (DSA)
0018-9464/02$17.00 © 2002 IEEE
GONG et al.: MODELING AND CANCELLATION OF PIVOT NONLINEARITY
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By applying the standard least-square method, the estimates of
parameter vector can be given as follows:
(5)
where
represents
the
value of at time instant , and
of a matrix .
estimate
of
,
with
being the
represents the inverse
B. Estimate of Hysteresis Loop
Fig. 1.
Experimental setup.
is used to obtain the performance of the system in frequency domain.
In the following sections, the pivot nonlinearity will be categorized into viscous friction, Coulomb friction, and hysteresis,
respectively.
When the actuator arm rotates in a small range with a low
speed, the approximation given in (2) is no longer precise
enough to describe the effect of hysteresis, and more detailed
modeling process is in order. In this paper, the effect of hysteresis is estimated by a simplified Dahl model [25], [26]. At
is estimated by the following differential
the starting point,
equation:
A. Estimate of Pivot Friction
(6)
The dynamic model of the HDD system is assumed to have
the following form:
(1)
where is the normalized1 moment of inertia of the system,
is the pivot angle, is the coefficient of the normalized viscous
friction, is the coefficient of the normalized Coulomb friction,
represents the effect of hysteresis loop, and is the input
current to the voice coil motor (VCM).
In this section, when the actuator arm rotates in a large range
with a high speed, the effect of hysteresis can be approximated
with an offset described by the
by a bearing stiffness term
following:
(2)
which will be experimentally verified in Section IV-A. Consequently, the following dynamic equation of the system is
obtained:
(3)
Alternatively, by considering as the output and as the input,
(3) can be rewritten in the following regressor form:
is the estimate of
, is the slope at
, is
where
, and is
the angular velocity, is the asymptotic value of
the exponential index.
When the angular velocity changes the direction, the value of
at that instant is called the turnaround value and is denoted
and
is then estimated by
by
(7)
In this paper, in order to save the memory of the computer and
to simplify the algorithm, only the latest turnaround value is
memorized. Other turnaround values are discarded. After the
is obtained either by (6) or by (7), it is saturated
estimate of
by the asymptotic value .
III. CANCELLATION OF PIVOT NONLINEARITY
In the last section, the estimation method of the friction, and
hysteresis is developed. Based on these estimates, the following
compensator is proposed:
(8)
The input current is composed of two components,
and is described as follows:
(4)
and
,
(9)
where is the extra efforts needed to compensate for the nonis the nominal input by assuming that there
linearities, and
is no nonlinearity at all and the system is a double integrator.
,
,
, and
In the ideal case that
, substituting (9) into (1) yields
1Normalized
with respect to the unit input current.
(10)
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IEEE TRANSACTIONS ON MAGNETICS, VOL. 38, NO. 5, SEPTEMBER 2002
Fig. 2. Schematic structure of measuring hysteresis loop.
Fig. 4.
Hysteresis loop in large range.
Fig. 5.
Estimate of angular acceleration and its true value.
Fig. 3. Modeling of hysteresis loop: experiment versus simulation.
which is in the form of the double integrator. In this sense,
the proposed compensator achieves the goal of linearizing the
system.
IV. EXPERIMENTAL COMPARISON STUDY
In this section, first, several experiments are carried out to
show the effects of the pivot nonlinearity; second, the coefficients of friction are estimated; third, experiments are carried
out to show how well the proposed compensator can eliminate
those nonlinearities; and fourth, the proposed compensator is
further applied to another HDD to show the effectiveness of this
approach.
A. Measurement of Hysteresis Loop
In order to prevent the actuator arm from drifting from the
center point, a controller is added to regulate the position of the
arm. The structure of system is shown in Fig. 2. Applying an
amplitude-modulated sinusoid wave with constant frequency to
the VCM, the hysteresis loop is obtained, which is presented by
the solid line in Fig. 3. By applying the simplified Dahl model
(6) and (7), the hysteresis loop can be estimated, which is shown
by the dotted line in Fig. 3. it can be found that the estimated
current can match the experimental data very well in the time
domain.
Furthermore, if the actuator arm rotates in a large range, the
corresponding hysteresis loop can be obtained as in Fig. 4. From
the result in Fig. 4, it can be easily found that the average effect
can be modeled as a bearing stiffness as follows:
(11)
where is the slope of the nominal shape. It is in accordance
with bearing stiffness term in (3).
Remark 1: The above result reveals that hysteresis can be approximated by a term proportional to the angular position with
an offset when actuator rotates in a large range. It also explains
and in model (3). In the following
the existence of terms
discussion, two models will be switched according to the situation of rotation of actuator arm. When fast rotation occurs,
model (3) will be employed while hysteresis model given in (6)
and (7) will be used in case of rotation with a low speed.
B. Estimates of Viscous Friction and Coulomb Friction
A square wave with frequency 6000 Hz and peak-to-peak
value 1 V is applied to VCM. The signals of displacement, velocity are measured by LDV, and the acceleration is calculated
by the backward difference method. The total displacement is
within 1 m, which is much smaller than the focusing range
of LDV, 10 m. By applying the least-square method stated in
Section II-A, the estimates of , , , and are obtained.
Fig. 5 shows that the estimate of acceleration matches the corresponding true value very well.
GONG et al.: MODELING AND CANCELLATION OF PIVOT NONLINEARITY
Fig. 6.
Schematic system structure of experiments.
Fig. 7.
Frequency responses of the plant with varying input current level.
C. Comparison Studies
In this subsection, the compensator proposed in Section III
will be used to cancel those pivot nonlinearities observed in the
last two subsections.
The schematic structure of the system is shown in Fig. 6. The
swept sine method is used to get the bode plots of the original
system and the compensated system. In order to hold the arm
around one fixed position, a simple PI control is added. In Fig. 7,
the bode plots without compensator are given, where the input
is the current applied to VCM, and the output is the displacement of the tip of actuator arm. It can be seen that those bode
plots change with the change of input current, which basically
means that system is nonlinear. It is also found that the resonant
frequency decreases with the increase of magnitude of the input,
and the damping factor increases with the increase of magnitude
of the input. Both are typical phenomena caused by pivot hysteresis.
In order to verify how well the compensator works, the same
set of input currents are applied to the VCM to carry out the
comparison study. When the model compensation part is added
to the system, the bode plots are shown in Fig. 8. In the first
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Fig. 8.
Bode plots: after compensation.
plot in Fig. 8, it can be seen that all of magnitude-frequency
relationships are almost the same regardless of the change of
the magnitude of the input current. To a certain degree, it shows
that the compensated system behaves like a linear system and
verifies that the proposed compensator does cancel the pivot
nonlinearity. In a way, it also confirms that proposed model can
match the pivot nonlinearities very well. A detailed observation
reveals that all the bode plots in the first plot in Fig. 8 have slopes
of approximate 40 dB/decade regardless of the change of input
current, and the phases shown in the second plot in Fig. 8 are
all around 180 . All these facts verify that the compensated
system does behave like a double integrator, just as expected in
Section III.
D. Validity for Other HDD
In order to further verify whether it can be used broadly, the
proposed compensator is applied to another HDD without any
modification. Both cases of inner diameter (ID) and middle diameter (MD) are investigated. Fig. 9 presents the experimental
results of MD. The effect of hysteresis clearly shows up before
compensation and almost disappears after compensation. The
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IEEE TRANSACTIONS ON MAGNETICS, VOL. 38, NO. 5, SEPTEMBER 2002
ACKNOWLEDGMENT
J. Q. Gong would like to express his sincere thanks to
Dr. M. Hatch of Maxtor Corporation for his tutorial talk about
friction.
REFERENCES
Fig. 9. Bode plots: comparison for another HDD (MD).
Fig. 10.
Bode plots: after compensation for another HDD.
performance of the compensated HDD at ID is given in Fig. 10.
It also clearly shows that the proposed compensator works well.
It is also reasonable to find out in Fig. 9 that the compensated
HDD cannot exactly behave like a double integrator since different HDDs may have different parameters. However, Figs. 9
and 10 do reveal that the proposed compensator almost cancels
the effects of pivot nonlinearity.
V. CONCLUSION
In this paper, by investigating the pivot nonlinearity of an
HDD in depth, a nonlinear model has been proposed. Based
on this model, a nonlinear compensator has been developed to
cancel the nonlinear effects to make the compensated system behave like a double integrator, which may make it easy to design
controllers. In both time domain and frequency domain, extensive experiments have been carried out to show the effectiveness
of the proposed approach.
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Jian Qin Gong received the B.Eng. degree in electrical engineering and the
M.Eng. degree in systems engineering from Xi’an Jiaotong University, Xi’an,
China, in 1993 and 1996, respectively, and the M.Eng. degree in electrical engineering from the National University of Singapore, Singapore, in 1999. He is
currently working toward the Ph.D. degree at the School of Mechanical Engineering, Purdue University, West Lafayette, IN.
His research interests include nonlinear modeling, control of hard disk drives,
control of rigid/flexible smart material robots, control of linear motor drive
system, adaptive control, robust control, and neural network control.
Lin Guo received the B.Eng degree in mechanical engineering from Shanghai
Jiao Tong University, Shanghai, China, in 1984, the M.Eng. degree in mechanical engineering from McGill University, Montreal, QC, Canada, in 1987, and
the Ph.D. degree in mechanical engineering from the University of California,
Berkeley, in 1996.
Between 1988 and 1993, he was an Experimental Scientist at the Commonwealth Scientific and Industrial Research Organization, Australia. Since 1996,
he has been with Maxtor Corporation, Milpitas, CA, where he is Senior Member
of Technical Staff and Director of Advanced Servo Technology Group. His research interests include digital motion control, optimal control, system identification, mechatronics, and disk drive dynamics.
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Ho Seong Lee (S’93–M’95) received the B.S. and M.S. degrees in mechanical design and production engineering from Seoul National University, Seoul,
Korea, the Eng. degree in mechanical engineering from Stanford University,
Stanford, CA, and the Ph.D. degree in mechanical engineering from the University of California, Berkeley, in 1979, 1981, 1987, and 1994, respectively. His
research at Stanford University focused on kinematics, rehabilitative robotics,
and sensor-integrated robotic grasping.
From 1988 to 1993, he was an Equipment Engineer with IBM Corporation,
East Fishkill, NY. Then, he was with Seagate Technology, Scotts Valley, CA,
for two years. Currently, he works in the Advanced Technology Group, Maxtor
Corporation, Milpitas, CA. His research interests include head-dish assembly,
mechanics, development of advanced actuators, and high-speed servo control of
heads for data storage systems.
Dr. Lee is a member of the IEEE Control Systems and Industrial Electronics
Societies.
Bin Yao (S’92–M’96) received the Ph.D. degree in mechanical engineering
from the University of California, Berkeley, in 1996.
Since 1996, he has been an Assistant Professor in the School of Mechanical
Engineering, Purdue University, West Lafayette, IN, where his research focuses
on the development of a general framework for the design of high-performance
adaptive robust control algorithms. Integration of the approach with actual
sensor properties, physical properties, and control-oriented modeling has been
carried out for different applications such as the control of electrohydraulic systems, linear motors, machine tools, and robot manipulators. He experimentally
verified the approach on the precision control of various electromechanical/hydraulic systems. His current research interests include design and coordinated
control of intelligent high-performance electromechanical/electrohydraulic
systems, modeling of hard disk drive systems, optimal adaptive and robust
control, nonlinear observer design and neural networks for virtual sensing,
modeling, fault detection, diagnostics, adaptive fault-tolerant control, and
data fusion.
Dr. Yao is the recipient of the 1997 Caterpillar Engineering Young Faculty
Development Fund for his work on electrohydraulic control and the 1998 National Science Foundation CAREER Award for his work on the engineering
synthesis of high-performance adaptive robust controllers for mechanical systems and manufacturing processes.