2424 IEEE TRANSACTIONS ON MAGNETICS, VOL. 33, NO. 3, MAY 1997

Bias in Disk Drive Rotary Actuators:Characterization, Prediction, and Compensation

Kyle Eddy, Member, IEEE,John Steele,Student Member, IEEE,and William Messner,Member, IEEE

Abstract—This paper examines bias, which is the averagetorque (or current) required to maintain a set actuator position ina disk drive. In this paper, we show that in a disk drive the valueof bias depends on several factors including the actuator position,the length of time at the set position, the direction of the actuatormotion, the distance traveled by the actuator, the proximity of theset position to locations at which the actuator changed direction,and the proximity of the set position to locations at which theactuator rested. With an adequate prediction of bias, the settlingtime of a seek can be reduced by properly selecting the initialconditions of the track following controller at the handoff betweentrack seek and track settle. Two prediction algorithms for biasare shown: one based on a series of calibration tests and onebased on Dahl’s friction model. A method for choosing the initialcondition of the integrator of an integral action controller tominimize settling time based on the bias prediction is developed.

Index Terms—Disk drives, hysteresis, mechanical factors, mod-eling, nonlinearities, servo systems.

I. INTRODUCTION

DEMANDS to increase the capacity of disk drives haveled to such narrow track pitches that nonlinear dynamic

effects, which are often grouped under the topic of “friction,”have become impossible to neglect if performance is to bemaintained and improved. This paper examines the nonlin-ear dynamics associated with “bias,” the average torque (orcurrent) needed to maintain a set actuator or head position.The bias is needed to cancel the effects of forces from theflex cable, gravity, and friction at the head/disk interface,and aerodynamic forces. Another component of bias is theresisting torque of the pivot ball bearing of the rotary actuatorof the disk drive. This resisting torque occurs even when thepivot is not moving and, therefore, is sometimes called the“static friction” of bearing. This paper shows the rich varietyof dynamic behavior of bias, demonstrates algorithms forpredicting bias, and develops methods for using bias predictioninformation to improve disk drive performance.

The servo systems used for head positioning in hard diskdrives typically consist of two distinct controllers. Theseek

Manuscript received October 27, 1995; revised October 18, 1996. Thiswork was supported in part by the National Science Foundation under GrantsECD-890768 and CMS-9409413 and a National Science Foundation GraduateFellowship.

K. Eddy was with the Department of Electrical and Computer Engineering,Data Storage Systems Center, Carnegie Mellon University, Pittsburgh, PA15213 USA. He is now with Seagate Technology, Inc., Bloomington, MN55420 USA (e-mail: kyleeddy@notes.seagate.com).

J. Steele and W. Messner are with the Department of Mechanical Engineer-ing, Data Storage Systems Center, Carnegie Mellon University, Pittsburgh, PA15213 USA (e-mail: jsteele@andrew.cmu.edu; bmessner@andrew.cmu.edu).

Publisher Item Identifier S 0018-9464(97)00947-3.

controller is used to manage gross movement of the headfrom track to track, and thetracking controller is used forposition regulation on a set track. During the track-followmode, the motion of the actuator has very small amplitude( 0.003 ball rotation and 50 nm ball displacement at theball radius) and high frequency. The largest oscillations occurat frequencies which are harmonics of the spindle rotationfrequency, which is typically 60–120 Hz. The topic of frictionin small oscillations of ball bearings has been addressedextensively in the literature, [1]–[6]. The broader topic offriction modeling and compensation has also been extensivelystudied, and it is currently an active area of research [7]–[11].

The seek mode of the controller contrasts markedly with thetrack following mode. For a 3.5 in (90 mm) form factor diskdrive, seek motion ranges from a single track (about 0.01and 1.5 m motion at the ball radius in the pivot bearing) tofull actuator stroke (about 30and 5 mm motion at the ballradius). A typical seek of 9occurs in 9 ms. The seek motionis characterized by high accelerations (110 s and 10g’s acceleration at the ball radius in the pivot bearing) andhigh maximum velocities (1000 s or 8 cm s at theball radius). Clearly, lubrication regimes in the bearing willchange rapidly in track-seek as bearings move from essentiallyzero velocity to high velocity and back to zero again. Thefriction dynamics of ball bearings associated with the types ofmotion required in track seeks have not been addressed in theliterature. In this paper, as part of the examination of bias, wehave examined the “static friction” of the pivot bearing.

Fig. 1 shows a schematic of the typical current profile inthe voice coil of a disk drive during seek and tracking. Duringthe seek mode, the voice coil is energized with current inone direction to accelerate the actuator. The current is reducedto zero during the constant velocity portion of the seek, andthen the voice coil is energized in the opposite direction todecelerate the actuator. When the actuator has deceleratedsufficiently and the head is very nearly at the destinationtrack, control is passed from the seek controller to the trackingcontroller. The point of this transition is calledhandoff,and itis the main concern of the control portion of this paper.

The seek controller is optimized for minimum time motionbetween tracks, subject to constraints due to dynamics of thevoice coil and resonances in the mechanical structure of thehead-actuator assembly. The tracking controller (regulator) isoptimized for disturbance rejection during tracking. It is notoptimized for transient response. The tracking controller willgenerally have a form similar to that in Fig. 2. Integral actionis used to compensate for bias. Without altering the structure

0018–9464/97$10.00 1997 IEEE

EDDY et al.: DISK DRIVE ROTARY ACTUATORS 2425

Fig. 1. Typical actuator current profile for a seek.

of the controller, its transient response can be improved bychoosing an optimal value of the integrator initial condition athandoff. The optimal value of the integrator initial conditionis based on the closed loop dynamics of the controller, otherinitial conditions of the controller (e.g., velocity and position),and the bias at the destination position. To determine theoptimal initial condition, accurate predictions of bias arenecessary.

This paper is structured as follows. Section II describesan experimental setup used for making measurements of biasdynamics and presents the results showing the dependence ofbias on several variables associated with seek history. SectionIII describes two methods for making predictions of bias. Onemethod is empirical and is based on a series of calibration tests.The other method is based on a state space model derived fromthe Dahl model of friction. The effectiveness of the predictionmethods are shown. In Section IV, a method is derived forminimizing the transient response of the tracking controllerby using the bias estimate to set the initial condition of thetracking controller integrator at handoff. Experimental resultsare shown. Section V contains concluding remarks.

II. BIAS CHARACTERIZATION

Experiments to characterize the various dynamics affectingbias were performed with a commercial 3.5 in (90 mm), 2 Gbhard disk drive, with dedicated servo platter. The spindle speedwas 5400 rpm. Two discrete 0.5 in (12.5 mm) ball bearingsmounted in a cylindrical housing of the swing arm comprisedthe pivot bearing of the rotary actuator. The drive was con-nected to a Pentium based workstation using a modified SCSIcard which allowed the operator to directly issue commandsto the DSP controller and to monitor certain control variables.The workstation was equipped with disk controller software,which is invoked from test programs written in MATLAB.The software and hardware configuration provides for rapiddevelopment and automation of test algorithms and for the

Fig. 2. Block diagram of a disk drive tracking controller.

Fig. 3. Bias as a function of track position showing direction dependence.

analysis of resulting data. Tests consisted of issuing seekcommands to the disk drive and measuring the bias at thetarget track of the seek. Bias was taken to be the averageof the integrated position error over one full revolution ofthe spindle after the head settled onto the track. Using theaveraged integrator value reduced the effects of repetitive andnonrepetitive runout. The units of bias are given as normalizedcurrent units, which are related to the actual bias torquethrough the gain of the digital-to-analog converter (D/A) andthe amplifier and torque constant of the voice coil motor. (Weare not authorized to disclose the conversions by agreementwith the corporate sponsor.) A movement across one trackcorresponds to roughly 0.01of rotation of the actuator.

The experiments indicated that the bias measurement at agiven track is affected by the position of the track, the amountof time spent on the track, the length of the seek to the track,the direction of the seek to the track, and the recent historyof the seeks leading to the given track. The remainder of thissection shows results of the experiments which examined thephenomena associated with the factors listed above. Much ofthe data presented here has also been collected by industry,but it has not been published.

2426 IEEE TRANSACTIONS ON MAGNETICS, VOL. 33, NO. 3, MAY 1997

Fig. 4. Bias measurements as a function of time at a single track.

TABLE ITEST FOR POSITION AND DIRECTION DEPENDENCE

A. Position and Direction Dependence

Table I describes the sequence of steps used to collect datato show the position and direction dependence of the bias.The results are shown in Fig. 3. The position dependence ofbias is not surprising. The origins of the phenomenon are theaerodynamic effects from air circulation from the spinningdisks and from the elastic deformation of the flex cablecarrying conductors to the voice coil motor andREAD/WRITE

heads. The direction dependence is the result of hysteresis inthe pivot bearing and in the flex cable. The effect of the flexcable may be minimal because direction dependence effectsstill appear when most of the flex cable is cut away. The twodirection dependent curves are accurately represented by twosecond-order polynomials that differ only in the constant term.The constant terms do exhibit some slow drift over the courseof many minutes. This appears to be due to DC bias drift inthe current driver electronics, rather than mechanical changesin the bias phenomena.

B. Time Dependence

When the head remains at a single track for an extendedlength of time, the measured bias asymptotically approaches a

constant value over a period of several seconds, as shown inFig. 4. The scatter in the data is probably due to aerodynamicturbulence in the drive and to contact or near-contact dynamicsat head/disk interface. The cause of the time dependencephenomenon may be due to the flow of lubricant in the pivotbearing or to the erosion of lubricant at the head/disk interface.For the disk drive in normal operation, this phenomenon isirrelevant for compensation because the actuator remains atone position for only a small fraction of a second, and theintegral action of the tracking controller compensates for slowchanges in bias.

This phenomenon may be more relevant as a caution toengineers doing research on compensation for friction in diskdrive actuators. One commonly used test is to observe theresponse of the actuator to a sinusoidal input over severalseconds, which obtains some measure of steady state be-havior. Yet, it is clear that during normal operation, thedisk drive tracking controller operates in a transient regime.Consequently, these tests may be less useful for developingcompensation techniques than formerly believed.

C. Seek Length Effects

The bias phenomena associated with small motions of theactuator are quite different from those observed with large mo-tions. Fig. 5 shows the bias as a function of position for singletrack seeks following a direction change in a test describedin Table II. The solid curves correspond to the inbound andoutbound curves of Fig. 3, which we call “position/direction”curves. The single track seeks trace a hysteresis loop, ifdirection is reversed again. The bias measurements for thesingle track seeks overshoot the outbound bias curve and thenexhibit undershoot before converging to the outbound curve.

EDDY et al.: DISK DRIVE ROTARY ACTUATORS 2427

(a)

(b)

Fig. 5. Transition between seek position/direction curves. (a) Schematic ofseek sequence. (b) Data from disk drive.

The overshoot and undershoot phenomena are more easilyseen in the results from the test described in Table III. Thedata from the test with the position dependent trend removedare shown in Fig. 6. While there is a fair amount of scatterin the data, seeks longer than 200 tracks show the bias valueassociated with very large movements of the actuator seen inFig. 3. For movements of approximately 200 to 100 tracks, asmall amount of undershoot is observed, and for movementsof approximately 100 to 50 tracks, overshoot is observed. Thepart of Fig. 6 from position 50 to 50 tracks illustrates thesame hysteresis effect as shown in Fig. 5 except that the seeklength is varied rather than the position.

The origin of the overshoot and undershoot phenomenais not well understood. The effect may be related to thetransition from prerolling displacement to pure rolling inthe ball bearing. The phenomena are frequently observedin friction measurement of ball bearings undergoing slowoscillations (e.g., [12] and [13]).

D. Effects of Direction Changes

The next set of experiments examined the effect of directionchanges at the source track on the bias measurement at thetarget track. Two tests were devised. In the first test, describedin Table IV, a sequence of motions was chosen so that themotion to the source position and the motion from the sourceposition to the target position were in the same direction. Thetop of Fig. 7 shows a schematic of the sequence of seeks used

TABLE IITEST SHOWING EFFECT OF SEEK LENGTH

TABLE IIITEST FOR SHOWING OVERSHOOT AND UNDERSHOOT PHENOMENA

TABLE IVTEST FORSHOWING THE EFFECT OFNO DIRECTION CHANGE AT SOURCE TRACK

when the source position is on the outside diameter (OD) sideof the target. When the source is on the inside diameter (ID)side of the target, the seeks to the ID and OD are reversed.For seeks of length 200 or less, a small amount of overshoot isobserved—much less than in the test data in Fig. 6 for whicha direction change always occurred at the source track.

In the second test, described in Table V, a sequence of seekswas chosen so that the motion to the source position and themotion from the source position to the target position werein the oppositedirection. Fig. 8(a) shows a schematic of thesequence of seeks used when the source position is on theOD position of the target. As in the previous test, when thesource is on the ID side of the target, the seeks to the ID andOD are reversed. For this disk drive, the magnitude of theovershoot in the opposite direction case is approximately thesame as that in the same direction case. The range of sourcepositions for which overshoot is observed is only about 200to 50 tracks, so the overshoot appears to be more pronounced.Determining if there is any undershoot is difficult because ofthe scatter in the data.

The bias measurement for source positions near the targeton the OD side of the target is nearly the asymptotic value forlong seeks in the ID direction, because the movement of theactuator was a very long outbound seek followed by a veryshort outbound seek. The bias measurement transition fromsource 50 to 0 tracks can then be explained by a continuityargument. Similar reasoning holds for source positions on theOD side of the target in this test.

2428 IEEE TRANSACTIONS ON MAGNETICS, VOL. 33, NO. 3, MAY 1997

(a)

(b)

Fig. 6. Bias measurements at a fixed target position as a function of the source position. (a) Schematic of seek sequence: 1) Seek to target from sourceposition and 2) seek to new source position. (b) Experimental data.

TABLE VTEST FORSHOWING THE EFFECT OFDIRECTION CHANGE AT SOURCE TRACK

E. Effects of Proximity to Past Seek Targets

The actuator assembly exhibits a curious memory property.The bias measurement at a particular target position is affectedby the past positioning of the actuator at nearby locations. Fig.9 shows a schematic of the seek sequences used to demonstratethis effect. The sequence of seeks is described in Table VI. Thepeaking around zero is actually adecreasein the magnitudeof the bias after removing the position-dependent trend, whichis given by the average of the curves in Fig. 3.

The phenomenon may be due to the buildup of lubricanton one side of the ball bearings in the pivot and/or to theerosion of lubricant at the head/disk interface. This explanationaccounts for the effect appearing (essentially) for intermediatepositions only on one side of the target track. When theintermediate position is between the OD and the target, anybuildup of lubricant in front of the balls of the bearing isplowed away when the motion from the OD to the targetoccurs. When the intermediate position is on the OD sideof the target, a buildup of lubricant in front of the ballsof the bearing remains when the actuator moves from theintermediate position to the OD and then to the target. It isnot clear, however, how the buildup of lubricant could resultin a relative decrease in the bias.

III. B IAS PREDICTION

The goal of the bias study is to devise an accurate easilycomputable prediction algorithm so that settling time can bereduced with some feedforward compensation method. Twoprediction algorithms were developed. The first algorithm used

EDDY et al.: DISK DRIVE ROTARY ACTUATORS 2429

(a)

(b)

Fig. 7. Fixed target position with no direction change at the source. (a)Schematic of seek sequences for source track on OD side of the target. (b)Experimental data.

several calibration tests similar to the ones shown in SectionII to derive piecewise linear curve fits to the data. The totalbias prediction was derived from the sum of the predictionsof the different phenomena. The second algorithm used a statespace model based on the Dahl friction model [13] with twotunable parameters.

A. Bias Prediction Using Calibration Curves

This algorithm gives the bias prediction as the sum of threeparts

bias (1)

The first term accounts for the position dependence of the bias.It could be derived from a lookup table. Instead, a second orderpolynomial least squares fit to the average of the two positiondependent curves in Fig. 3 was found to be adequate for thisdisk drive

(2)

where is the head position in tracks. The constant termin (2) is continually updated to account for offset drift in thecurrent driver electronics. The second term in (1),, accountsfor the length of the seek from the source to the target andaccounts for any direction change at the source. It is derivedfrom piecewise linear least squares fits to the curves in Figs.7 and 8 in which the bias at the target has been normalized

(a)

(b)

Fig. 8. Fixed target position with direction changes at the source. (a)Schematic of seek sequences for source track on OD side of the target. (b)Experimental data.

to zero. The third term in (1) attempts to account for thememory effects in the system. It is derived from a piecewiselinear least squares fit to the data in Fig. 9 with the bias atthe target normalized to zero. A similar curve is obtained formotions in the OD direction. In obtaining the value of the thirdterm, it is necessary to keep track of all positions at which theactuator stopped that have not been crossed by subsequentmotions. This system memory adds considerable complexityto the prediction algorithm.

The results of the algorithm are shown in Figs. 10 and 11.Fig. 10 shows the results for random seeks in 250 track rangewhen position and direction dependence are accounted for, butother bias dynamics effects are ignored. This corresponds tothe method used in the current industrial state of the art. Notethat a systematic trend is discernible in Fig. 10(a) and thatthe histogram of prediction errors is bimodal. Fig. 11 showsthe prediction power of the full algorithm. Fig. 11(a) shows nosystematic trend, and the histogram appears to have a Gaussianshape. The standard deviation in the prediction error has beenreduced from about 19 in Fig. 10 to just under eight in Fig.11, a reduction of more than 50%.

B. State-Space Bias Prediction Model

The prediction algorithm described in the previous sectionshowed good results for random seeks but does not accountfor hysteresis occurring during short seeks. Furthermore, the

2430 IEEE TRANSACTIONS ON MAGNETICS, VOL. 33, NO. 3, MAY 1997

(a)

(b)

Fig. 9. Effect of target position proximity to other seek locations. (a) Seeksequence. (b) Experimental data.

TABLE VITEST SHOWING THE EFFECT OFPROXIMITY TO PREVIOUS SEEK LOCATIONS

algorithm requires considerable memory and calibration over-head. A practical prediction algorithm based on a state spacemodel of bias is derived in this section. The state space modelis derived from the hysteresis model for friction proposed byDahl in [14]. Dahl’s model is used to predict dynamic friction,while the model here is used for the static resisting force orbias.

The bias model begins with the following differential equa-tions used by Dahl to model dry friction hysteresis:

sgn

sgn sgn (3)

is the hysteresis portion of the bias. The variablerepresents the position location in tracks. The variables, ,

(a)

(b)

Fig. 10. Prediction results using only position and direction. (a) Biasprediction error as a function of actuator position for random motions. (b)Histogram of prediction errors.

and are parameters. The first relation in (3) is derivedfrom calculus. The second relation is an empirical relationdeveloped by Dahl. For the sake of simplicity, we follow theprecedent in the literature [9] by assuming that . Bycombining the two parts of (3) and multiplying by thefollowing equation results:

sgn (4)

The seek motions are in discrete steps, and so a differenceequation approximation to (4) is made by the followingdefinitions:

(5)

EDDY et al.: DISK DRIVE ROTARY ACTUATORS 2431

(a)

(b)

Fig. 11. Prediction results using position/direction, seek length, directionchange, and position proximity information. (a) Bias prediction error as afunction of actuator position for random motions. (b) Histogram of predictionerrors.

The variable is the signed seek length, andis the numberof the seek. Making the assumption that and solvingfor using (4) and (5) results in

for

for

(6)

The solution to the difference equation forsingle track seeksin the same direction above is

forfor

(7)

(a)

(b)

Fig. 12. Single seek data. (a) Seek sequence. (b) Experimental data asfunction of track position.

TABLE VIITEST SHOWING THAT A SINGLE N -TRACK SEEK HAS

THE SAME EFFECT ON BIAS AS N SINGLE-TRACK SEEKS

Making the final assumption that a single seek of lengthgives the same bias measurement as moving

successive single track seeks of length, the bias estimateequation is

forfor

(8)

The last assumption is justified by similarity of the incrementalstep data shown in Fig. 5 and the longer single step data shownin Fig. 12 and described in Table VII.

The complete bias prediction is given by

(9)

2432 IEEE TRANSACTIONS ON MAGNETICS, VOL. 33, NO. 3, MAY 1997

Fig. 13. Interpretation of bias model parameters.

Fig. 14. Performance of state space bias model. Solid curves are predictedbias values, and dots are measured bias values.

where is from (2). An interpretation of the parametersof the state space model is shown in Fig. 13. The parameter

is one-half the separation between the two curves of Fig.3. The parameter is the initial slope of the hysteresis curve.These parameters can be derived from a calibration curve fit.

The performance of the prediction algorithm is shown inFig. 14. This state space model does a good job of accountingfor most of the hysteresis. It does not account for the overshootand undershoot phenomena. For short seeks, the standarddeviation of the prediction error is about 11.5, a reduction ofabout 40% from the standard deviation in the error in Fig. 10.While this standard deviation value is larger than the valueobtained with the prediction algorithm in Section III-A, thestate space algorithm is easier to calibrate, and it will havesuperior performance for repeated short seeks occurring in thesame direction. The algorithm of Section III-A is better forrandom seeks of arbitrary length.

IV. BIAS COMPENSATION

The simplest and most intuitive use of the bias estimateis to attempt to exactly cancel the constant disturbance byadding the negative of the bias estimate to the control signalof the track settle controller. This is equivalent to settingthe initial condition of the integrator of the controller toits predicted converged value. This fact can be shown bya straightforward change of variables, but the derivation isnot presented here for the sake of brevity. Due to integratorwind-up during the step response, however, initializing theintegrator at handoff to its converged value will result in large

overshoots in response relative to the tracking error at handoff.This section shows a derivation of an optimal integratorinitial condition (feedforward value) which minimizes settlingtime by reducing overshoot caused by the integrator windup.Furthermore, for small seeks of about five tracks or less, theoverhead of implementing a trajectory following track seekcontroller, which hands off to a tracking regulator, can begreater than simply sending a step reference change to thetracking regulator. Data from simulations and experimentswhich demonstrate the effectiveness of using the optimal valueare presented.

A. Derivation of Optimal Integrator Initial Condition

To derive equations for minimization of the settle time givenan accurate estimate of the bias, a simplified discrete-time statespace model of the actuator assembly is used. Following arestate definitions:

integrated position errorposition errorvelocityprevious control input

control input

constant disturbance input (bias) (10)

Without loss of generality, the output of the plant is assumedto be the position error, rather than the absolute position. Theprevious control input must be included as a state if thereis any computation delay in the controller [15]. The bias isconsidered to be a constant for the range of motion over whichsettle occurs. In this derivation the dynamics of the velocityestimator are ignored.

The state dynamic equations are

(11)

is the system dynamics matrix, is the input matrix, isthe output matrix, and is the state feedback matrix whichis chosen such that , the closed loop dynamics matrix, isstable. The solution to the difference equation above in (11) is

(12)

Assuming that the tracking controller is turned on at time, the head is defined to be settled on track at time

if

for all (13)

EDDY et al.: DISK DRIVE ROTARY ACTUATORS 2433

Substituting in the solution for of (12) into (13) andsolving for the integrator initial condition , an equivalentcondition for settle on track is found in terms of the integratorinitial condition. If the following two inequalities are satisfiedfor all , then the head has settled on track at time

(14)

The vector is the th standard basis vector. The aboveinequalities assume that . The inequality signs arereversed when . These two inequalities indicatethat settle has occurred at time when the constant value ofthe integrator initial condition remains between time varyingupper and lower bounds for all time . This definitionof settling contrasts with inequality 13 which represents thesettle condition in terms of a time varying quantity, the positionerror, staying between constant upper and lower bounds for all

. The two conditions are equivalent because inequality13 is true if and only if inequality 14 is true for all .

The optimal value of the integrator initial condition forminimum settling time is the one for which inequality 14 issatisfied for all for the smallest possible . There areseveral ways to find this optimal value. The easiest to presentis a backward search in time from infinity (in practice, fromsome sufficiently large maximum time, ). Defining

(15)

Note that and.

Equation (15) provides a methodology to locate the min-imum settling time. is the minimum of the upperbound curve between time and infinity. Likewise,is the maximum of the lower bound curve between timeand infinity. When , then there existsa constant such that for all .If the integrator initial condition is chosen to be, then thesystem has settled by time . The smallest possible settlingtime is the time such thatand . The correspondingoptimal initial condition is

(16)

The advantage of this approach over other initial value com-pensation methods is that it explicitly minimizes the settlingtime. This type of optimization is not developed in othermethods. For example, in [16] the authors design initialvalue compensators by minimizing a quadratic performanceindex or by achieving pole-zero cancellation. The authorsdemonstrate improved settling time but cannot show that theirmethods achieve an optimal minimum settling time. The otheradvantage of the method described in the present paper isthat once the optimal integrator initial condition is computed,additional computations are not needed, since the feedforwardsignal is constant. Other methods require computations at eachtime step after the handoff.

B. Experimental Verification

The algorithm was demonstrated on a commercial 3.5 in(90 mm) form factor hard disk drive actuator test bed. Thissystem includes a commercial 4 Gb hard drive without plattersand a current mode driver circuit. The position and velocity ofthe actuator E block of the drive were measured using a laserDoppler vibrometer. A tracking regulator was designed usinglinear quadratic regulator (LQR) methods to meet modestperformance specifications of a digital settle controller, i.e.,one track seek settle time less than 3 ms with less than30% overshoot. Note that in hard disk drives, the allowabletracking error is a fixed constant, so the settling time varieswith changes in step reference magnitude. Both the regulatorand the initial value (feedforward) compensation algorithmswere implemented on a 40 MHz TMS320C30 DSP with adual channel analog I/O board used to sample the positionand velocity measurements and output the control signal. Thecontrol loop runs at 10 KHz and has a computational delayof four tenths of a cycle. Since the DSP code was not fullyoptimized for speed, the computational delay for the initialvalue optimization computations is not reported here prior toinitiating the step command. However, initial results show thatless than 100 floating point operations would be necessaryfor these computations for the typical conditions reported.Track seeks of 15 m with a 0.5- m settling envelope wereperformed from rest following a 15-m track seek in theopposite direction to simulate the tracking requirements for a5- m track pitch (5080 tpi) drive with a 10% off-track (read)tolerance.

For the simulation, the initial position error wasm, and the velocity was . The value of

the constant bias was not needed for the simulation, sincethe linearity of the system allowed the optimal integratorinitial condition to be computed in the absence of bias effectsand then added to the integrator value necessary to hold theactuator steady in the presence of bias. Therefore, to avoidconfusion, all results are presented in terms of the feedforwardconstants used.

The upper and lower bounds of (14) for the simulationof the testbed are shown in Fig. 15. The solid lines are theupper and lower bound curves [ and ] of (14).By selecting any constant feedforward value in Fig. 15 andstepping through time, the reader can determine the points in

2434 IEEE TRANSACTIONS ON MAGNETICS, VOL. 33, NO. 3, MAY 1997

Fig. 15. Prediction of time to settle on track in terms of bounds on constant feedforward signals.

time at which the system response given will intersect theon-track bounds. Points above the upper bound correspondto overshoot, and points below the lower bound correspondto undershoot. Since a different testbed was necessary forthe controller implementation, the units of bias presentedhere differ from those used in characterization efforts earlier.Note that the optimal feedforward constant (19.5) producesa response which asymptotically avoids crossing the upperbound curve into the region of overshoot. It is unreasonable toexpect the performance of the experimental system to exactlymatch the model. For this reason, a more robust feedforwardconstant ( 21.7) was used (see Fig. 15) which was the averageof the value of the upper and lower bounds at the point of thelocal minimum of the upper bound.

As shown in Fig. 15, using zero feedforward (i.e., initial-izing the integrator to simply counteract the bias), settle ispredicted to occur in 2.6 ms. Using the robust choice of theoptimal value for feedforward, settle is predicted to occur in1.4 ms. The output of the discrete-time simulation of thissystem along with typical experimental results, shown in Fig.16, confirms these predictions. For both the simulation andexperimental results, using the robust optimal initial conditionfor the integrator reduces the settle time by more than 46%.

C. Comments on Using the Compensation Method

Several concerns must be considered regarding this com-pensation method. One, since finding the optimal feedforwardconstant relies on accurate prediction of the system response,having both a good estimate of the bias and a good statespace plant model is essential. Two, the algorithm assumesthat the bias is constant over the range of motion of theactuator, which is not an unreasonable assumption for small

motions such as those seen in the settle portion of a disk drivetrack seeking algorithm. Three, more rigorous definitions of arobust choice of feedforward signals could be developed byconsidering the sensitivity of the settling time to variations inthe feedforward signal but were not explored here since theywould have increased the real-time computational demandsof the algorithm. Four, because the feedforward signal is aconstant, this algorithm does not affect the stability of thesystem.

The biggest impediment to implementation is the compu-tation required to obtain the optimal feedforward constant.Fortunately, part of the computation can be done “off-line.”If the bias is constant, it simply adds a DC componentequal to the converged value of the integrator to both thebounds in (14). With this simplification, the bounds in (14)are linear functions only of , , , and . Thevectors which these parameters multiply can be computedahead of time and stored in arrays. The upper and lowerbounds on the initial condition for convergence can be quicklycomputed at any time step by scalar multiplication of thesearrays followed by array addition. The optimal value is foundby a search of the result. Although not implemented heredue to computational limitations, it is also feasible that theoptimization method could be extended to nonconstant biasdisturbance.

V. CONCLUSION

In this paper, we have examined three aspects of bias indisk drives. First, we examined the rich dynamic behavior ofthe phenomena of bias. We showed that bias in the actuatordepended on the position of the actuator, the amount of timespent at a position, the length of the seek to the position, the

EDDY et al.: DISK DRIVE ROTARY ACTUATORS 2435

Fig. 16. Evaluation of constant feedforward algorithm using 15-�m step response with 0.5-�m settling window.

direction of the seek to the position, and the recent time historyof the seeks leading to the given position. Next, we showedtwo algorithms for predicting bias, one of which was based ona state-space hysteresis friction model. Both were shown tobe effective at reducing prediction errors. Finally, we deriveda method for using the bias prediction to compute a constantfeedforward value for an integral action tracking controller athandoff to minimize the settle time.

There are several areas for continuing research on bias and,by extension, “friction” in bearings. The origin of the differ-ent observed behavior is unclear. Experiments with differentbearing configurations, such as lubricant, preload, surfacetreatments, and geometry are needed to determine from wherethese dynamics originate. Operating disk drive actuators with-out spinning disks would determine if some of the dynamicsarise from the head disk interface. Such experiments are inprogress. The results may be used to improve the design ofdisk drive actuators and possibly the bearings themselves.

Better and computationally efficient prediction models ofbias phenomena are needed for compensation algorithms. Theovershoot and undershoot phenomena in bias are not currentlyaccounted for in the state-space model. These models might beextended to improve the performance of the tracking controllerduring track follow mode by including nonlinear feedbackdynamics.

Efficient means for using the bias predictions in an optimalor near optimal manner must be developed. Theory andexperimental data have demonstrated that sophisticated useof the bias prediction can significantly reduce settle time. Thecomputations associated with using the bias prediction may

be significant and should be reduced to make these optimalmethods practical for disk drive applications.

REFERENCES

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[2] D. Abramovitch, G. Franklin, T. Hurst, and F. Wang, “Disk drive pivotnonlinearity modeling, Part II: Time domain,” inProc. American ControlConf., vol. 2, Baltimore, MD, June 1994, pp. 2604–2608.

[3] M. R. Lovell, M. M. Khonsari, and R. D. Marangoni, “Calculation ofultra-low-speed jitter in rolling balls,”Trans. ASME, J. Tribol.,vol. 114,pp. 589–594, 1992.

[4] D. Henze, T. Hurst, and F. Wang, “Understanding ball bearing pre-rolling behavior using the restoring force surface method,” presented atWinter Annual Meeting ASME, ISPS-7,Chicago, IL, 1994.

[5] K. L. Johnson and M. J. Todd, “A model for Coulomb torque hysteresisin ball bearings,”Int. J. Mech. Sci.,vol. 29, no. 5, pp. 339–354, 1987.

[6] N. A. Osborne and D. L. Rittenhouse, “The modeling of friction and itseffects on fine pointing control,” presented atAIAA Mech. Cont. FlightCont. Conf.,Anaheim, CA, Aug. 1974, no. 74-875.

[7] B. Armstrong-Helouvry, C. Canudas de Wit, and P. Dupont, “A surveyof models, analysis tools, and compensation methods for the controlof machines with friction,”Automatica,vol. 30, no. 7, pp. 1083–1138,July 1994.

[8] P.-A. J. Bliman, “Mathematical study of the Dahl’s friction model,”Eur.J. Mech. Solids,vol. 11, no. 6, pp. 835–848, 1992.

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2436 IEEE TRANSACTIONS ON MAGNETICS, VOL. 33, NO. 3, MAY 1997

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[16] T. Yamaguchi, Y. Soyama, H. Hosokawa, K. Tsuneta, and H. Hirai,“Mode switching control design with initial value compensation and itsapplication to head positioning on magnetic disk drives,”IEEE Trans.Ind. Electron.,vol. 43, no. 1, pp. 65–73, Feb. 1996.

Kyle Eddy (S’93–M’95) received the B.A. degree in Chinese and the B.A.degree in physics from Connecticut College, New London, in 1992 and theB.S. degree in electrical engineering from Washington University, St. Louis,MO, in 1993. He received the M.S. degree in electrical engineering from theData Storage System Center, Carnegie Mellon University, Pittsburgh, PA, in1994.

While at Washington University, he spent three semesters as a co-op studentworking in data storage at IBM, Rochester, MN. In November 1995, he joinedSeagate Technology, Inc., Bloomington MN.

John Steele(S’97) received the B.S. degree in mechanical engineering fromthe University of California, Berkeley, in 1995. He is currently working towardthe M.S. degree at Carnegie Mellon University, Pittsburgh, PA, where he isa member of the Data Storage Systems Center.

Mr. Steele was awarded a National Science Foundation Graduate Fellow-ship in 1996.

William Messner (S’89–M’91) received the B.A. degree in mathematicsfrom the Massachusetts Institute of Technology, Cambridge, in 1985 and theM.S. and Ph.D. degrees in mechanical engineering from the University ofCalifornia, Berkeley, in 1989 and 1992, respectively.

From 1985 to 1987, he worked for BBN Laboratories, Newport, RI. In1993, he joined the Department of Mechanical Engineering at CarnegieMellon University, Pittsburgh, PA, as an Assistant Professor. He conductsresearch in the areas of adaptive, learning, nonlinear, and distributed controlalong with research on the dynamics of sensors and actuators. Primaryapplications of his work are directed to robotics, information storage systems,and microelectromechanical systems.

Dr. Messner is a member of ASME.