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Timing Error Compensator Design for Spiral Based Self-Servowriting in Disk Drives
5th IFAC Symposium on Mechatronic Systems Marriott Boston Cambridge Cambridge, MA, USA, Sept 13-15, 2010
Timing Error Compensator Design for Spiral Based Self-Servowriting in Disk Drives Feng Dan Dong, Masayoshi Tomizuka Mechanical Engineering Department, University of California, Berkeley Berkeley, CA, USA, 94720 (e-mail: [email protected], [email protected] ) Abstract: This paper considers the timing error compensation problem for spiral based Self-Servowriting (SSW) process in disk drives manufacturing, wherein the product servo patterns are written based on the prewritten spiral tracks. The disk eccentricity and position error in the spiral tracks cause Repeatable Timing Error (RTE) which deteriorates the product servo pattern format and, as a result, the servo sectors are not uniformly distributed along the circular tracks. The spindle speed variation and sensor noise induce Non-repeatable Timing Error (NRTE) which results in phase incoherency at the servo sectors between adjacent tracks. In this paper, a Recursive Least Square (RLS) based Parameter Adaptive Algorithm (PAA) is proposed to estimate and cancel the RTE. An error shaping filter is designed to improve the estimation performance of PAA and reduce the effect of NRTE in the system. Finally, simulation studies using industry supplied data show the effectiveness of the proposed control scheme. The RTE can be cancelled up to 90% and the quality of the written servo sectors is improved by 91%. Keywords: Spiral based Self-Servowriting, Recursive Least Square, Parameter Adaptive Algorithm. the write head in HDD unit writes the concentric product servo sectors onto the circular tracks.
1. INTRODUCTION In recent Hard Disk Drive (HDD) manufacturing industry, spiral based Self-Servowriting (SSW) [1] technique is widely used to write the product servo sectors 0 ( N 1) along the concentric tracks (as illustrated by the dashed lines in Fig. 1) by referring to the spiral tracks (as illustrated by the solid lines in Fig. 1) which are prewritten on the disk.
With the continuously increasing data density, HDD manufacturers write data to disks such that the data density is substantially uniform throughout the disk. Uniform density (as shown in Fig. 2(a)) requires an accurate clock while writing the spiral tracks. During the manufacturing, when the spiral tracks are written to the disk by an external spiral writer, the eccentricity of disk and position error of the sync marks as written in the spiral tracks introduce the written-in timing errors, which are periodic and synchronized to the disk rotation. These written-in timing errors shift the spiral tracks from the ideal locations, as illustrated by the dashed lines in Fig. 3. After the spiral tracks are written, the disk drive is operated to self-write the product servo patterns by referring to them, wherein the drifts of spiral tracks cause Repeatable Timing Error (RTE) which deteriorates the product servo pattern format and, as a result, the sector-tosector spacing is not even, as shown in Fig. 2(b).
Fig. 1 Illustration of a disk with spiral tracks and product servo tracks.
(a)
Fig. 2 (a) The sector-to-sector spacing is ideally even. (b) The sector-to-sector spacing is not even due to RTE.
In general, the spiral tracks are prewritten on the disk by using an external spiral writer. After that, the read head in HDD unit is used to read these spirals and generate a read signal. The Phase Lock Loop (PLL) serves the purpose of a servo write clock [2] generator. With the servo write clock,
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Mechatronics'10 Cambridge, MA, USA, Sept 13-15, 2010
ID
Spiral trajectories with drifts
ID : inner diameter
Ideal spiral trajectories
OD : outer diameter
k
k 1
k2
doesn’t deal with the repeatable timing error (i.e. the reference signal) appropriately.
k 3
In this paper, a Recursive Least Square (RLS) based Parameter Adaptive Algorithm (PAA) is proposed to estimate and cancel RTE from the reference. A comb filter is designed as the error shaping filter to improve the PAA estimation performance and reduce the contamination of non-repeatable timing distortions in SSW process.
track i 1
track i
OD
The remainder of this paper is organized as follows. In Section 2, the timing control system in spiral based SSW process is presented and the timing error control problem is formulated. A RLS based PAA is proposed for RTE estimation and cancellation in Section 3. For further enhancement of RTE cancellation and NRTE reduction, an error shaping filter is designed. Section 4 shows the simulation studies using the industry supplied data. Conclusions are given in Section 5.
Time
Fig. 3 Drifts of spiral tracks due to written-in timing errors. In spiral based SSW process, except for the RTE, there exits some Non-repeatable Timing Error (NRTE) induced by the spindle speed variation and sensor noise, which causes phase incoherency at the servo sectors between adjacent tracks, as shown in Fig. 4(b). Therefore, it is important to compensate the RTE and reduce NRTE when the product servo patterns are written based on the spiral tracks. Trk i 1
Trk i 1 Trk i
(a )
2. TIMING ERROR CONTROL PROBLEM FORMULATION IN SPIRAL BASED SSW 2.1 Timing Control System in Spiral based SSW Process
Trk i
The general linearized block diagram for spiral based SSW timing control system in HDD is shown in Fig. 5.
(b)
Fig. 4 (a) In ideal case, the phase between the sectors in adjacent tracks is coherent. (b) Due to NRTE, the phase is incoherent between the sectors in adjacent tracks.
rR i ( k )
A lot of research work has been done on the Repeatable Runout (RRO) compensation for the position control loop in SSW process. In [3] – [4], adaptive filters were designed for compensating the RRO to improve the track-following performance of disk drives. Their control objective is to minimize the off-track (i.e., the deviation of the read/write head from the centre of tracks). So the filter coefficients are intuitively updated track by track. These methods are not suitable for the timing control loop in spiral based SSW process because they cannot guarantee the servo sectors are uniformly written along each individual track. The uniform distribution of servo sectors along the circular tracks requires the timing error to be adaptively corrected sector by sector.
1 z 1
ei (k )
i (k )
Fig. 5 Linearized block diagram of PLL circuit in spiral based SSW process. 1
In the figure, the integrator z 1 is the mathematical model of the Voltage Controlled Oscillator (VCO) in PLL, C ( z ) is the feedback controller, rR i (k ) represents the reference signal which is read from the spiral tracks and appears as RTE during SSW process. The index i 1, 2,... is the track number; k 0,1, 2,...N 1 is the sector number where N denotes the number of servo sectors in one track, and the output signal i (k ) represents the write head timing signal, i.e. the timing deviation written to the concentric servo sectors.
Little research effort has been reported on Repeatable Timing Error compensation for spiral based SSW process. In [5], a feedforward compensation method was proposed. This method measures the RTE at all of the servo sectors in one whole track and the mean is used for generating the feedforward compensation signal. Thus, it spends much time to compute the feedforwad compensation value before writing the product servo patterns on the track. In [6], a novel Adaptive Feedforward Compensation (AFC) scheme was designed to contain the closure error in each individual track and attenuate the track-to-track timing error propagation for concentric SSW process, which is an alternative selfservowriting technique in HDD manufacturing. However, this approach is unfit for spiral based SSW process since it
2.2 Control Objectives In spiral based SSW process, the product servo patterns are written based on the prewritten spirals. This implies that any timing distortion in spiral tracks will degrade the quality of servowriting. Therefore, the first and foremost significant objective is to reduce the RTE rR i ( k ) which is the major portion of timing distortion in SSW process. In order to effectively compensate the RTE, it is necessary to estimate it first because it is not directly measurable during SSW process. 559
Mechatronics'10 Cambridge, MA, USA, Sept 13-15, 2010
The other equally important goal is to assure good quality of product servo patterns even with the presence of NRTE rNR i ( k ) as shown in Fig. 6. Namely, the timing signal from the write head is desired to be evenly distributed along the circular tracks, i.e. the written timing variation is as minimum as possible. Using the signal i (k ) in Fig. 5, the performance index can be written as m
m
i
i 1
i 1
1 N 1 2 i (k ) N k 0
In Fig. 6, the PAA output vi (k ) is the estimated RTE and written as n
vi (k ) aij (k 1) sin(2 kj / N ) bij (k 1) cos(2 kj / N ) T (k )i (k 1) j 1
(2)
where iT (k ) a i1 (k ), a i 2 (k ),, a in (k ), b i1 (k ), b i 2 (k ),, b in (k ) . Remarks:
m 1
(1) (k ) in Equation (1) and (2) is known for all of k .
Ideally, i (k ) 0 for all k . To achieve this target, NRTE should be dealt with as well as RTE.
(2) i in Equation (1) is the parameter to be estimated; and i (k ) in Equation (2) is the estimated parameter.
3. RLS BASED PAA AND ERROR SHAPING FILTER DESIGN
(3) If i i (k 1) , then vi (k ) will completely cancel rR i (k ) from the reference.
The proposed control scheme is shown in Fig. 6. The design includes two steps. In the first, a RLS based PAA is applied to estimate the RTE and the output vi (k ) is then used to cancel it. Secondly, an error shaping filter W ( z ) is added to enhance the performance of PAA and reduce the effect of NRTE in the system.
Now assume that the following (k 1) sets of data are known: rR i (0), rR i (1), ..., rR i (k ) (0), (1), ..., (k )
The design problem is to find the parameter i (k ) to minimize the following cost function,
3.1 Recursive Least Square (RLS) based Parameter Adaptive Algorithm (PAA)
i (k ) is obtained by solving
PAA refers to a problem of identifying or estimating unknown parameters of a system, the structure of which is assumed to be known and identical to that of the system. For this purpose, we use a mathematical model and adjust the model parameters so that the input-output behaviour of the model and that of the system become close to each other.
n
Then the RLS based
(4)
where Fi ( k ) is the RLS gain for the k th sector on the i th track. We note that in the timing control loop, only the system error ei (k ) is measurable, i.e. rR i ( k ) is not directly measurable and the a-priori prediction error ei0(k ) cannot be used in PAA. However, in Fig. 6, before adding the error shaping filter W ( z ) and assuming the system is not contaminated by NRTE rNR i ( k ) , the system error ei (k ) is represented as
(1)
j 1
T ( k ) sin 2 k / N sin 4 k / N sin 2 kn / N , , cos 2 k / N ,cos 4 k / N , ,cos 2 kn / N
ai1 , ai 2 , , ain , bi1 , bi 2 , , bin , and the Fourier coefficients aij and bij are not dependent on k . T i
ei (k ) rR i (k ) vi (k ) i ( k ) rR i (k ) vi (k ) P ( z )ei ( k )
(5)
S ( z ) T (k ) i i (k 1) S ( z )ei0 (k )
vi (k ) rNR i (k )
ui (k ) rR i (k )
dG i (k ) 0. d i ( k )
0 i (k ) i (k 1) F ( k ) (k )ei (k ) 0 e (k ) rR i (k ) T ( k )i (k 1) i F (k 1) (k ) T (k ) Fi ( k 1) Fi ( k ) Fi (k 1) i 1 T (k ) Fi (k 1) (k )
(k 0,1,..., N 1 and n N / 2)
where
(3)
PAA can be described by the following equations:
RTE rR i ( k ) in Fig. 5 can be written as a sum of n sinusoids of known frequencies since it is periodic and synchronized to the disk rotation. rR i ( k ) aij sin(2 kj / N ) bij cos(2 kj / N ) T (k )i
2 1 k G i (k ) rR i ( j ) T ( j )i (k ) 2 j 0
ei (k )
1 z 1
C ( z) is the open loop transfer function in PLL where P( z ) z 1
i (k )
circuit and S ( z ) 1 P1 ( z ) is the sensitivity function. Hence, ei0(k ) can be obtained from the measurable quality ei (k ) , i.e.
Fig. 6 Proposed control scheme with PAA and error shaping filter.
ei0(k ) S ( z ) ei (k ) s (k ) ei ( k ) 1
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where s (k ) is the impulse response of S ( z ) .
The filter W ( z ) is considered as a comb filter
1
The PAA in Equation (4) can be modified in various ways. For example, it is possible to introduce a forgetting factor 0 1 . In such case, the updating equation for Fi ( k ) becomes, Fi (k )
1 N 2
W ( z)
Remarks: (1) The filter is stable if and only if 1. The zeros are Z k e2 k j / N and poles are Pk e2 k j / N for k 0,1,..., N 1 .
Notice that Equation (4) is a standard least square algorithm. Hence, lim ei0 ( k ) 0 and lim i (k ) i can be easily proved, k k which guarantee the stability of the system.
(2) The frequency response of W ( z ) consists of a series of evenly-spaced notches, giving the appearance of a comb. Fig. 7 gives an example of comb filter with delay length N 50 and factor coefficient 0.69 .
3.2 Effect of NRTE on the Performance of PAA If NRTE rNR i ( k ) is present in the system, Equation (5) can be rewritten as
Comb filter (N = 50, a = 0.69) 0
Magnitude (dB)
ei (k ) rR i (k ) vi ( k ) i ( k ) S ( z ) rR i (k ) vi (k ) rNR i (k )
(7)
S ( z ) T (k ) i i (k 1) rNR i (k )
-50
-100
-150
Equation (7) shows that the performance of PAA is affected by the presence of NRTE rNR i (k ) since ei (k ) is used in PAA update equation. Hence the estimation accuracy of vi (k ) is degraded due to rNR i (k ) .
1500 2000 Frequency (Hz)
2500
3000
Closed - loop bode plots
Using Equation (8) and adding the filter W ( z ) , the system error ei (k ) is rewritten as
0
Magnitude (dB)
(8)
5
S ( z ) rR i (k ) vi ( k ) rNR i ( k ) W ( z )ei (k )
1000
(3) In Equation (10), the factor 1 N 2 is for setting the filter gain to unity in the passband (i.e., between the notches). Fig. 8 shows the closed-loop responses (i.e. from rRi (k ) to i (k ) ) before and after adding the filter W ( z ) , which have the same bandwidth except that the system with W ( z ) has the notches which are used to reduce the NRTE in the system.
In this subsection, an error shaping filter W ( z ) (as shown in Fig. 6) is proposed to enhance the estimation performance of PAA and reduce the NRTE in the system. The error shaping filter is designed by introducing the new correction signal,
ei (k ) S ( z ) rR i ( k ) ui (k ) rNR i (k )
500
Fig. 7 Frequency response of a comb filter given in Equation (10). ( N 50 and 0.69 ).
3.2 Error Shaping Filtering for Reducing the Effect of NRTE
ui (k ) vi (k ) W ( z )ei (k )
(10)
where is the scaling factor.
Fi ( k 1) (k ) T (k ) Fi (k 1) 1 Fi (k 1) T (k ) Fi (k 1) (k )
rR i (k ) vi (k ) P ( z )ei ( k ) rNR i ( k )
1 zN N N 1 z
(9)
-5 -10 -15
Analysis:
-20 10
original sys. original sys. with filter 100
1000
10000
Frequency (Hz)
rR i ( k ) vi ( k )
is repeatable (1) In Equation (9), the term residual error which is determined by the PAA in Equation (4).
Fig. 8 Closed-loop response of the systems with and without the comb filter.
(2) The term rNR i ( k ) W ( z )ei ( k ) is non-repeatable residual error; from which, it can be seen that if rNR i (k ) can be reduced by W ( z )ei (k ) , then the system error ei (k ) will also be reduced.
(4) The repeatable components of system error ei (k ) are eliminated by W ( z ) . In other words, the non-repeatable components of ei (k ) are isolated for the purpose of reducing NRTE rNR i (k ) .
Therefore, it is desired to design W ( z ) to reduce NRTE rNR i (k ) . For this purpose, W ( z ) can be designed to extract the non-repeatable components from ei (k ) , i.e., remove its repeatable components out.
4. SIMULATION STUDIES
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PAA (i.e. the system in Fig. 6 before adding the filter W ( z ) ). In the simulation study, the forgetting factor was 0.86 and n in Equation (1) was 1. Fig. 11 compares the write head performance i for 19 tracks. The solid line is the case with PAA, which achieves a better timing pattern with 77.7% smaller average timing variation than the original system as shown by the dotted line.
In the simulation studies, 19 tracks of RTE rR i (as shown in Fig. 9) and NRTE rNR i (as shown in Fig. 10) are collected from a real disk drive. The spindle speed is 7200 rpm and the servo sector number in one track is N 208 . The low pass filter in the PLL circuit is chosen as, C( z)
0.477 z 0.4725 z2 z
(11)
The sampling period is T 4.63 10 5 sec.
12
15
10
19 revolutions
Ts (4ns)
5 0
6
-5
4
-10
2
-15 0
20
40
60
80
100
120
140
160
180
0
200
Sector number
2
4
6
8
10
12
14
16
18
20
Track number
Fig. 11 Written timing variation for 19 tracks with PAA ( avg i 2.36Ts ) and without PAA ( avg i 10.56Ts ).
Fig. 9 RTE profiles (19 tracks). Remarks:
In the second study, the estimation performance of PAA are compared in the systems with and without NRTE rNR i . The estimation performance of PAA for track i is defined as
(1) In practical SSW process, the spindle speed is generally slower than the nominal disk drive speed. Hence the sampling period in the simulation is chosen as 5
T 4.63 10 sec which is larger than
Pi
60 4 105 sec . 208 7200
vi rR i 100% rR i
(12)
In Fig. 12, the solid line denotes the injected RTE source. The dashed line is the ideal case when rNR i does not exist in the system. However, when rNR i is present, the estimation performance of PAA is clearly degraded as shown in the dotted line. This implies that the presence of rNR i adversely affects PAA performance. The average Pi is compared in Fig. 13. It can be seen that the estimation performance is degraded from 18.0% to 28.8% if the system is contaminated by rNR i .
(2) RTE profile looks like sinusoidal because it is mainly contributed by the disk eccentricity due to non-centric alignment of disks during writing the spiral tracks by the external spiral writer. (3) NRTE profile is continuous since its main cause is the spindle speed variation. (4) In the simulations, all signals are normalized to Ts 4ns . 10
30
19 revolutions 5
RTE source
25
estimated RTE w/o NRTE
20
estimated RTE with NRTE
15
Ts (4ns)
Ts (4ns)
original system original system + PAA
8
[ i] (T s )
10
0
10 5 0
-5
-5
-10 0
-10
500
1000
1500
2000
2500
3000
3500
4000
-15
19 x 208 sectors
1500
1550
1600
1650
1700
1750
1800
1850
1900
19 x 208 sectors
Fig. 10 NRTE profiles (19 tracks). Fig. 12 RTE and estimated RTE profiles for 19 tracks. 4.1 PAA with and without NRTE rNR i In order to show the effectiveness of the proposed PAA, rR i and rNR i are injected to SSW system for 19 revolutions. Two cases are firstly studied and compared: without PAA (i.e. the original system in Fig. 5) and with 562
Mechatronics'10 Cambridge, MA, USA, Sept 13-15, 2010
100
100 original sysytem + PAA w NRTE original sysytem + PAA w/o NRTE
60
60
40
40
20
20 2
4
6
8
10 12 Track number
14
16
18
2
20
Fig. 13 Comparison of PAA performance with rNR i ( avg Pi 28.8% ) and without rNR i ( avg Pi 18.0% ).
6
8
10 12 Track number
14
16
18
In order to further show the performance of the error shaping filter W ( z ) , Fig. 17 compares i in both schemes with and without W ( z ) . The solid line denotes the system using PAA and the comb filter W ( z ) . The resulting average i is reduced from 2.36Ts (using PAA only as shown in the dotted line) to 0.93Ts , and it is much smoother since the effect of NRTE rNR i has been reduced by W ( z ) .
6 original sysytem + PAA w NRTE original sysytem + PAA w/o NRTE
5
4
Fig. 15 Feedforward signal ui using PAA with W ( z ) ( avg Pi 9.67% ) and without W ( z ) ( avg Pi 28.8% ).
Fig. 14 compares the average i in the systems with and without NRTE rNR i . In the figure, we note that the quality of written timing signal is 62.7% better if the system is not contaminated by NRTE rNR i .
[ i] (T s )
original system + PAA original system + PAA + comb filter
80
Pi (%)
Pi (%)
80
4 3
30
RTE source estimated RTE with PAA + comb filter estimated RTE with PAA
25
2
20 15
2
4
6
8
10 12 Track number
14
16
18
Ts (4ns)
1 20
10 5 0 -5
Fig. 14 Written timing variation using PAA with rNR i ( avg i 2.36 Ts ) and without rNR i ( avg i 0.88Ts ).
-10 -15 1400
1450
1500
1550
1600
1650
1700
1750
1800
19 x 208 sectors
5.2 PAA with Error Shaping Filtering
Fig. 16 RTE and Feedforward signal ui profiles for 19 tracks
An error shaping filter W ( z ) in Equation (10) is designed to reduce the effect of rNR i in the system and improve the servowriting quality. The scaling factor is 0.69 . Fig. 15 Fig. 17 show the performance of W ( z ) . Fig. 15 and Fig. 16 compare the performance of the feedforward correction signal ui for 19 tracks using PAA with and without the filter W ( z ) . The performance of ui is defined by Equation (13), which is similar to Equation (12), ui rR i Pi 100% rR i
6 original system + PAA original system + PAA + comb filter
[i] (T s )
5 4 3 2 1
(13)
2
It is shown that the system with PAA plus W ( z ) achieves 26.8% more accurate RTE estimation than that with PAA only.
4
6
8
10 12 Track number
14
16
18
20
Fig. 17 Written timing variation using PAA with W ( z ) ( avg i 0.93Ts ) and without W ( z ) ( avg i 2.36Ts ). 5. CONCLUSIONS This paper presented the timing error compensation algorithms for the spiral based Self Servowriting process in HDD manufacturing. A Recursive Lease Square based 563
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Parameter Adaptive Algorithm was applied to estimate and cancel the Repeatable Timing Error in the reference. An error shaping filter was designed to improve the performance of PAA and reduce the effect of Non-repeatable Timing Error in the system. The simulation results showed the effectiveness of the proposed control scheme. The Repeatable Timing Error can be cancelled up to 90% and the quality of the written servo pattern is improved by as much as 91% from the original system. ACKNOWLEDGEMENT This research was supported by the Computer Mechanics Laboratory (CML) in the Department of Mechanical Engineering, University of California, Berkeley. REFERENCES Swearingen P. and Shepherd S., System for self-servowriting a disk drive, US Patent, 5,668,679. M. D. Schultz, E. J. Yarmchuk, B. C. Webb and T. J. Chainer, A self-servowrite clocking process, IEEE Trans. Magn., vol. 37, no.4, pp. 1878-1880, Jul. 2001. A. Sacks, M. Bodson, and W. Messner, Adaptive methods for repeatable runout compensation, IEEE Trans. On Magn., vol. 31, no.2, 1031-1036, 1995. M. Kawafuku, M. Iwasaki, H. Hirai, M. Kobayashi, and A. Okuyama, Rejection of repeatable runout for HDDs using adaptive filters, Proc. Of International Workshop on Advanced Motion Control, 2004, pp.305-310. Y. M. Lifchifts, W. Ying, Y. Cai, and S. Weerasooriya, Servo Writing a Disk Drive by Synchronizing a Servo Write Clock to a Reference Pattern on the Disk and Compensating for Repeatable Phase Error, US Patent, 7,333,280. F. Dong and M. Tomizuka, Timing Error Compensator design for Self-Sevowriting System in Hard Disk Drives, Proc. Of DSC Conference, Oct. 2009.
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Timing Error Compensator Design for Spiral Based Self-Servowriting in Disk Drives Feng Dan Dong, Masayoshi Tomizuka Mechanical Engineering Department, University of California, Berkeley Berkeley, CA, USA, 94720 (e-mail: [email protected], [email protected] ) Abstract: This paper considers the timing error compensation problem for spiral based Self-Servowriting (SSW) process in disk drives manufacturing, wherein the product servo patterns are written based on the prewritten spiral tracks. The disk eccentricity and position error in the spiral tracks cause Repeatable Timing Error (RTE) which deteriorates the product servo pattern format and, as a result, the servo sectors are not uniformly distributed along the circular tracks. The spindle speed variation and sensor noise induce Non-repeatable Timing Error (NRTE) which results in phase incoherency at the servo sectors between adjacent tracks. In this paper, a Recursive Least Square (RLS) based Parameter Adaptive Algorithm (PAA) is proposed to estimate and cancel the RTE. An error shaping filter is designed to improve the estimation performance of PAA and reduce the effect of NRTE in the system. Finally, simulation studies using industry supplied data show the effectiveness of the proposed control scheme. The RTE can be cancelled up to 90% and the quality of the written servo sectors is improved by 91%. Keywords: Spiral based Self-Servowriting, Recursive Least Square, Parameter Adaptive Algorithm. the write head in HDD unit writes the concentric product servo sectors onto the circular tracks.
1. INTRODUCTION In recent Hard Disk Drive (HDD) manufacturing industry, spiral based Self-Servowriting (SSW) [1] technique is widely used to write the product servo sectors 0 ( N 1) along the concentric tracks (as illustrated by the dashed lines in Fig. 1) by referring to the spiral tracks (as illustrated by the solid lines in Fig. 1) which are prewritten on the disk.
With the continuously increasing data density, HDD manufacturers write data to disks such that the data density is substantially uniform throughout the disk. Uniform density (as shown in Fig. 2(a)) requires an accurate clock while writing the spiral tracks. During the manufacturing, when the spiral tracks are written to the disk by an external spiral writer, the eccentricity of disk and position error of the sync marks as written in the spiral tracks introduce the written-in timing errors, which are periodic and synchronized to the disk rotation. These written-in timing errors shift the spiral tracks from the ideal locations, as illustrated by the dashed lines in Fig. 3. After the spiral tracks are written, the disk drive is operated to self-write the product servo patterns by referring to them, wherein the drifts of spiral tracks cause Repeatable Timing Error (RTE) which deteriorates the product servo pattern format and, as a result, the sector-tosector spacing is not even, as shown in Fig. 2(b).
Fig. 1 Illustration of a disk with spiral tracks and product servo tracks.
(a)
Fig. 2 (a) The sector-to-sector spacing is ideally even. (b) The sector-to-sector spacing is not even due to RTE.
In general, the spiral tracks are prewritten on the disk by using an external spiral writer. After that, the read head in HDD unit is used to read these spirals and generate a read signal. The Phase Lock Loop (PLL) serves the purpose of a servo write clock [2] generator. With the servo write clock,
978-3-902661-76-0/10/$20.00 © 2010 IFAC
(b)
558
10.3182/20100913-3-US-2015.00131
Mechatronics'10 Cambridge, MA, USA, Sept 13-15, 2010
ID
Spiral trajectories with drifts
ID : inner diameter
Ideal spiral trajectories
OD : outer diameter
k
k 1
k2
doesn’t deal with the repeatable timing error (i.e. the reference signal) appropriately.
k 3
In this paper, a Recursive Least Square (RLS) based Parameter Adaptive Algorithm (PAA) is proposed to estimate and cancel RTE from the reference. A comb filter is designed as the error shaping filter to improve the PAA estimation performance and reduce the contamination of non-repeatable timing distortions in SSW process.
track i 1
track i
OD
The remainder of this paper is organized as follows. In Section 2, the timing control system in spiral based SSW process is presented and the timing error control problem is formulated. A RLS based PAA is proposed for RTE estimation and cancellation in Section 3. For further enhancement of RTE cancellation and NRTE reduction, an error shaping filter is designed. Section 4 shows the simulation studies using the industry supplied data. Conclusions are given in Section 5.
Time
Fig. 3 Drifts of spiral tracks due to written-in timing errors. In spiral based SSW process, except for the RTE, there exits some Non-repeatable Timing Error (NRTE) induced by the spindle speed variation and sensor noise, which causes phase incoherency at the servo sectors between adjacent tracks, as shown in Fig. 4(b). Therefore, it is important to compensate the RTE and reduce NRTE when the product servo patterns are written based on the spiral tracks. Trk i 1
Trk i 1 Trk i
(a )
2. TIMING ERROR CONTROL PROBLEM FORMULATION IN SPIRAL BASED SSW 2.1 Timing Control System in Spiral based SSW Process
Trk i
The general linearized block diagram for spiral based SSW timing control system in HDD is shown in Fig. 5.
(b)
Fig. 4 (a) In ideal case, the phase between the sectors in adjacent tracks is coherent. (b) Due to NRTE, the phase is incoherent between the sectors in adjacent tracks.
rR i ( k )
A lot of research work has been done on the Repeatable Runout (RRO) compensation for the position control loop in SSW process. In [3] – [4], adaptive filters were designed for compensating the RRO to improve the track-following performance of disk drives. Their control objective is to minimize the off-track (i.e., the deviation of the read/write head from the centre of tracks). So the filter coefficients are intuitively updated track by track. These methods are not suitable for the timing control loop in spiral based SSW process because they cannot guarantee the servo sectors are uniformly written along each individual track. The uniform distribution of servo sectors along the circular tracks requires the timing error to be adaptively corrected sector by sector.
1 z 1
ei (k )
i (k )
Fig. 5 Linearized block diagram of PLL circuit in spiral based SSW process. 1
In the figure, the integrator z 1 is the mathematical model of the Voltage Controlled Oscillator (VCO) in PLL, C ( z ) is the feedback controller, rR i (k ) represents the reference signal which is read from the spiral tracks and appears as RTE during SSW process. The index i 1, 2,... is the track number; k 0,1, 2,...N 1 is the sector number where N denotes the number of servo sectors in one track, and the output signal i (k ) represents the write head timing signal, i.e. the timing deviation written to the concentric servo sectors.
Little research effort has been reported on Repeatable Timing Error compensation for spiral based SSW process. In [5], a feedforward compensation method was proposed. This method measures the RTE at all of the servo sectors in one whole track and the mean is used for generating the feedforward compensation signal. Thus, it spends much time to compute the feedforwad compensation value before writing the product servo patterns on the track. In [6], a novel Adaptive Feedforward Compensation (AFC) scheme was designed to contain the closure error in each individual track and attenuate the track-to-track timing error propagation for concentric SSW process, which is an alternative selfservowriting technique in HDD manufacturing. However, this approach is unfit for spiral based SSW process since it
2.2 Control Objectives In spiral based SSW process, the product servo patterns are written based on the prewritten spirals. This implies that any timing distortion in spiral tracks will degrade the quality of servowriting. Therefore, the first and foremost significant objective is to reduce the RTE rR i ( k ) which is the major portion of timing distortion in SSW process. In order to effectively compensate the RTE, it is necessary to estimate it first because it is not directly measurable during SSW process. 559
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The other equally important goal is to assure good quality of product servo patterns even with the presence of NRTE rNR i ( k ) as shown in Fig. 6. Namely, the timing signal from the write head is desired to be evenly distributed along the circular tracks, i.e. the written timing variation is as minimum as possible. Using the signal i (k ) in Fig. 5, the performance index can be written as m
m
i
i 1
i 1
1 N 1 2 i (k ) N k 0
In Fig. 6, the PAA output vi (k ) is the estimated RTE and written as n
vi (k ) aij (k 1) sin(2 kj / N ) bij (k 1) cos(2 kj / N ) T (k )i (k 1) j 1
(2)
where iT (k ) a i1 (k ), a i 2 (k ),, a in (k ), b i1 (k ), b i 2 (k ),, b in (k ) . Remarks:
m 1
(1) (k ) in Equation (1) and (2) is known for all of k .
Ideally, i (k ) 0 for all k . To achieve this target, NRTE should be dealt with as well as RTE.
(2) i in Equation (1) is the parameter to be estimated; and i (k ) in Equation (2) is the estimated parameter.
3. RLS BASED PAA AND ERROR SHAPING FILTER DESIGN
(3) If i i (k 1) , then vi (k ) will completely cancel rR i (k ) from the reference.
The proposed control scheme is shown in Fig. 6. The design includes two steps. In the first, a RLS based PAA is applied to estimate the RTE and the output vi (k ) is then used to cancel it. Secondly, an error shaping filter W ( z ) is added to enhance the performance of PAA and reduce the effect of NRTE in the system.
Now assume that the following (k 1) sets of data are known: rR i (0), rR i (1), ..., rR i (k ) (0), (1), ..., (k )
The design problem is to find the parameter i (k ) to minimize the following cost function,
3.1 Recursive Least Square (RLS) based Parameter Adaptive Algorithm (PAA)
i (k ) is obtained by solving
PAA refers to a problem of identifying or estimating unknown parameters of a system, the structure of which is assumed to be known and identical to that of the system. For this purpose, we use a mathematical model and adjust the model parameters so that the input-output behaviour of the model and that of the system become close to each other.
n
Then the RLS based
(4)
where Fi ( k ) is the RLS gain for the k th sector on the i th track. We note that in the timing control loop, only the system error ei (k ) is measurable, i.e. rR i ( k ) is not directly measurable and the a-priori prediction error ei0(k ) cannot be used in PAA. However, in Fig. 6, before adding the error shaping filter W ( z ) and assuming the system is not contaminated by NRTE rNR i ( k ) , the system error ei (k ) is represented as
(1)
j 1
T ( k ) sin 2 k / N sin 4 k / N sin 2 kn / N , , cos 2 k / N ,cos 4 k / N , ,cos 2 kn / N
ai1 , ai 2 , , ain , bi1 , bi 2 , , bin , and the Fourier coefficients aij and bij are not dependent on k . T i
ei (k ) rR i (k ) vi (k ) i ( k ) rR i (k ) vi (k ) P ( z )ei ( k )
(5)
S ( z ) T (k ) i i (k 1) S ( z )ei0 (k )
vi (k ) rNR i (k )
ui (k ) rR i (k )
dG i (k ) 0. d i ( k )
0 i (k ) i (k 1) F ( k ) (k )ei (k ) 0 e (k ) rR i (k ) T ( k )i (k 1) i F (k 1) (k ) T (k ) Fi ( k 1) Fi ( k ) Fi (k 1) i 1 T (k ) Fi (k 1) (k )
(k 0,1,..., N 1 and n N / 2)
where
(3)
PAA can be described by the following equations:
RTE rR i ( k ) in Fig. 5 can be written as a sum of n sinusoids of known frequencies since it is periodic and synchronized to the disk rotation. rR i ( k ) aij sin(2 kj / N ) bij cos(2 kj / N ) T (k )i
2 1 k G i (k ) rR i ( j ) T ( j )i (k ) 2 j 0
ei (k )
1 z 1
C ( z) is the open loop transfer function in PLL where P( z ) z 1
i (k )
circuit and S ( z ) 1 P1 ( z ) is the sensitivity function. Hence, ei0(k ) can be obtained from the measurable quality ei (k ) , i.e.
Fig. 6 Proposed control scheme with PAA and error shaping filter.
ei0(k ) S ( z ) ei (k ) s (k ) ei ( k ) 1
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(6)
Mechatronics'10 Cambridge, MA, USA, Sept 13-15, 2010
where s (k ) is the impulse response of S ( z ) .
The filter W ( z ) is considered as a comb filter
1
The PAA in Equation (4) can be modified in various ways. For example, it is possible to introduce a forgetting factor 0 1 . In such case, the updating equation for Fi ( k ) becomes, Fi (k )
1 N 2
W ( z)
Remarks: (1) The filter is stable if and only if 1. The zeros are Z k e2 k j / N and poles are Pk e2 k j / N for k 0,1,..., N 1 .
Notice that Equation (4) is a standard least square algorithm. Hence, lim ei0 ( k ) 0 and lim i (k ) i can be easily proved, k k which guarantee the stability of the system.
(2) The frequency response of W ( z ) consists of a series of evenly-spaced notches, giving the appearance of a comb. Fig. 7 gives an example of comb filter with delay length N 50 and factor coefficient 0.69 .
3.2 Effect of NRTE on the Performance of PAA If NRTE rNR i ( k ) is present in the system, Equation (5) can be rewritten as
Comb filter (N = 50, a = 0.69) 0
Magnitude (dB)
ei (k ) rR i (k ) vi ( k ) i ( k ) S ( z ) rR i (k ) vi (k ) rNR i (k )
(7)
S ( z ) T (k ) i i (k 1) rNR i (k )
-50
-100
-150
Equation (7) shows that the performance of PAA is affected by the presence of NRTE rNR i (k ) since ei (k ) is used in PAA update equation. Hence the estimation accuracy of vi (k ) is degraded due to rNR i (k ) .
1500 2000 Frequency (Hz)
2500
3000
Closed - loop bode plots
Using Equation (8) and adding the filter W ( z ) , the system error ei (k ) is rewritten as
0
Magnitude (dB)
(8)
5
S ( z ) rR i (k ) vi ( k ) rNR i ( k ) W ( z )ei (k )
1000
(3) In Equation (10), the factor 1 N 2 is for setting the filter gain to unity in the passband (i.e., between the notches). Fig. 8 shows the closed-loop responses (i.e. from rRi (k ) to i (k ) ) before and after adding the filter W ( z ) , which have the same bandwidth except that the system with W ( z ) has the notches which are used to reduce the NRTE in the system.
In this subsection, an error shaping filter W ( z ) (as shown in Fig. 6) is proposed to enhance the estimation performance of PAA and reduce the NRTE in the system. The error shaping filter is designed by introducing the new correction signal,
ei (k ) S ( z ) rR i ( k ) ui (k ) rNR i (k )
500
Fig. 7 Frequency response of a comb filter given in Equation (10). ( N 50 and 0.69 ).
3.2 Error Shaping Filtering for Reducing the Effect of NRTE
ui (k ) vi (k ) W ( z )ei (k )
(10)
where is the scaling factor.
Fi ( k 1) (k ) T (k ) Fi (k 1) 1 Fi (k 1) T (k ) Fi (k 1) (k )
rR i (k ) vi (k ) P ( z )ei ( k ) rNR i ( k )
1 zN N N 1 z
(9)
-5 -10 -15
Analysis:
-20 10
original sys. original sys. with filter 100
1000
10000
Frequency (Hz)
rR i ( k ) vi ( k )
is repeatable (1) In Equation (9), the term residual error which is determined by the PAA in Equation (4).
Fig. 8 Closed-loop response of the systems with and without the comb filter.
(2) The term rNR i ( k ) W ( z )ei ( k ) is non-repeatable residual error; from which, it can be seen that if rNR i (k ) can be reduced by W ( z )ei (k ) , then the system error ei (k ) will also be reduced.
(4) The repeatable components of system error ei (k ) are eliminated by W ( z ) . In other words, the non-repeatable components of ei (k ) are isolated for the purpose of reducing NRTE rNR i (k ) .
Therefore, it is desired to design W ( z ) to reduce NRTE rNR i (k ) . For this purpose, W ( z ) can be designed to extract the non-repeatable components from ei (k ) , i.e., remove its repeatable components out.
4. SIMULATION STUDIES
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PAA (i.e. the system in Fig. 6 before adding the filter W ( z ) ). In the simulation study, the forgetting factor was 0.86 and n in Equation (1) was 1. Fig. 11 compares the write head performance i for 19 tracks. The solid line is the case with PAA, which achieves a better timing pattern with 77.7% smaller average timing variation than the original system as shown by the dotted line.
In the simulation studies, 19 tracks of RTE rR i (as shown in Fig. 9) and NRTE rNR i (as shown in Fig. 10) are collected from a real disk drive. The spindle speed is 7200 rpm and the servo sector number in one track is N 208 . The low pass filter in the PLL circuit is chosen as, C( z)
0.477 z 0.4725 z2 z
(11)
The sampling period is T 4.63 10 5 sec.
12
15
10
19 revolutions
Ts (4ns)
5 0
6
-5
4
-10
2
-15 0
20
40
60
80
100
120
140
160
180
0
200
Sector number
2
4
6
8
10
12
14
16
18
20
Track number
Fig. 11 Written timing variation for 19 tracks with PAA ( avg i 2.36Ts ) and without PAA ( avg i 10.56Ts ).
Fig. 9 RTE profiles (19 tracks). Remarks:
In the second study, the estimation performance of PAA are compared in the systems with and without NRTE rNR i . The estimation performance of PAA for track i is defined as
(1) In practical SSW process, the spindle speed is generally slower than the nominal disk drive speed. Hence the sampling period in the simulation is chosen as 5
T 4.63 10 sec which is larger than
Pi
60 4 105 sec . 208 7200
vi rR i 100% rR i
(12)
In Fig. 12, the solid line denotes the injected RTE source. The dashed line is the ideal case when rNR i does not exist in the system. However, when rNR i is present, the estimation performance of PAA is clearly degraded as shown in the dotted line. This implies that the presence of rNR i adversely affects PAA performance. The average Pi is compared in Fig. 13. It can be seen that the estimation performance is degraded from 18.0% to 28.8% if the system is contaminated by rNR i .
(2) RTE profile looks like sinusoidal because it is mainly contributed by the disk eccentricity due to non-centric alignment of disks during writing the spiral tracks by the external spiral writer. (3) NRTE profile is continuous since its main cause is the spindle speed variation. (4) In the simulations, all signals are normalized to Ts 4ns . 10
30
19 revolutions 5
RTE source
25
estimated RTE w/o NRTE
20
estimated RTE with NRTE
15
Ts (4ns)
Ts (4ns)
original system original system + PAA
8
[ i] (T s )
10
0
10 5 0
-5
-5
-10 0
-10
500
1000
1500
2000
2500
3000
3500
4000
-15
19 x 208 sectors
1500
1550
1600
1650
1700
1750
1800
1850
1900
19 x 208 sectors
Fig. 10 NRTE profiles (19 tracks). Fig. 12 RTE and estimated RTE profiles for 19 tracks. 4.1 PAA with and without NRTE rNR i In order to show the effectiveness of the proposed PAA, rR i and rNR i are injected to SSW system for 19 revolutions. Two cases are firstly studied and compared: without PAA (i.e. the original system in Fig. 5) and with 562
Mechatronics'10 Cambridge, MA, USA, Sept 13-15, 2010
100
100 original sysytem + PAA w NRTE original sysytem + PAA w/o NRTE
60
60
40
40
20
20 2
4
6
8
10 12 Track number
14
16
18
2
20
Fig. 13 Comparison of PAA performance with rNR i ( avg Pi 28.8% ) and without rNR i ( avg Pi 18.0% ).
6
8
10 12 Track number
14
16
18
In order to further show the performance of the error shaping filter W ( z ) , Fig. 17 compares i in both schemes with and without W ( z ) . The solid line denotes the system using PAA and the comb filter W ( z ) . The resulting average i is reduced from 2.36Ts (using PAA only as shown in the dotted line) to 0.93Ts , and it is much smoother since the effect of NRTE rNR i has been reduced by W ( z ) .
6 original sysytem + PAA w NRTE original sysytem + PAA w/o NRTE
5
4
Fig. 15 Feedforward signal ui using PAA with W ( z ) ( avg Pi 9.67% ) and without W ( z ) ( avg Pi 28.8% ).
Fig. 14 compares the average i in the systems with and without NRTE rNR i . In the figure, we note that the quality of written timing signal is 62.7% better if the system is not contaminated by NRTE rNR i .
[ i] (T s )
original system + PAA original system + PAA + comb filter
80
Pi (%)
Pi (%)
80
4 3
30
RTE source estimated RTE with PAA + comb filter estimated RTE with PAA
25
2
20 15
2
4
6
8
10 12 Track number
14
16
18
Ts (4ns)
1 20
10 5 0 -5
Fig. 14 Written timing variation using PAA with rNR i ( avg i 2.36 Ts ) and without rNR i ( avg i 0.88Ts ).
-10 -15 1400
1450
1500
1550
1600
1650
1700
1750
1800
19 x 208 sectors
5.2 PAA with Error Shaping Filtering
Fig. 16 RTE and Feedforward signal ui profiles for 19 tracks
An error shaping filter W ( z ) in Equation (10) is designed to reduce the effect of rNR i in the system and improve the servowriting quality. The scaling factor is 0.69 . Fig. 15 Fig. 17 show the performance of W ( z ) . Fig. 15 and Fig. 16 compare the performance of the feedforward correction signal ui for 19 tracks using PAA with and without the filter W ( z ) . The performance of ui is defined by Equation (13), which is similar to Equation (12), ui rR i Pi 100% rR i
6 original system + PAA original system + PAA + comb filter
[i] (T s )
5 4 3 2 1
(13)
2
It is shown that the system with PAA plus W ( z ) achieves 26.8% more accurate RTE estimation than that with PAA only.
4
6
8
10 12 Track number
14
16
18
20
Fig. 17 Written timing variation using PAA with W ( z ) ( avg i 0.93Ts ) and without W ( z ) ( avg i 2.36Ts ). 5. CONCLUSIONS This paper presented the timing error compensation algorithms for the spiral based Self Servowriting process in HDD manufacturing. A Recursive Lease Square based 563
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Parameter Adaptive Algorithm was applied to estimate and cancel the Repeatable Timing Error in the reference. An error shaping filter was designed to improve the performance of PAA and reduce the effect of Non-repeatable Timing Error in the system. The simulation results showed the effectiveness of the proposed control scheme. The Repeatable Timing Error can be cancelled up to 90% and the quality of the written servo pattern is improved by as much as 91% from the original system. ACKNOWLEDGEMENT This research was supported by the Computer Mechanics Laboratory (CML) in the Department of Mechanical Engineering, University of California, Berkeley. REFERENCES Swearingen P. and Shepherd S., System for self-servowriting a disk drive, US Patent, 5,668,679. M. D. Schultz, E. J. Yarmchuk, B. C. Webb and T. J. Chainer, A self-servowrite clocking process, IEEE Trans. Magn., vol. 37, no.4, pp. 1878-1880, Jul. 2001. A. Sacks, M. Bodson, and W. Messner, Adaptive methods for repeatable runout compensation, IEEE Trans. On Magn., vol. 31, no.2, 1031-1036, 1995. M. Kawafuku, M. Iwasaki, H. Hirai, M. Kobayashi, and A. Okuyama, Rejection of repeatable runout for HDDs using adaptive filters, Proc. Of International Workshop on Advanced Motion Control, 2004, pp.305-310. Y. M. Lifchifts, W. Ying, Y. Cai, and S. Weerasooriya, Servo Writing a Disk Drive by Synchronizing a Servo Write Clock to a Reference Pattern on the Disk and Compensating for Repeatable Phase Error, US Patent, 7,333,280. F. Dong and M. Tomizuka, Timing Error Compensator design for Self-Sevowriting System in Hard Disk Drives, Proc. Of DSC Conference, Oct. 2009.
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