- Email: [email protected]
A Twist-Roller Friction Drive for Nanometer Positioning – Simplified Design Using Ball Bearings
A Twist-Roller Friction Drive for Nanometer Positioning Simplified Design Using Ball Bearings
-
H. Mizumoto, S. Arii, A. Yoshimoto, Tottori University, Tottori; T. Shimizu, Nachi-Fujikoshi Corp., Toyama; N. lkawa (1). Osaka University, Osaka; Japan Received on January 3,1996
Abstract The authors have reported in a previous paper that the twist-roller friction drive can realize subnanometer positioning resolution. However, a structural difficulty of the device is that hydrostatic bearings are used to support the twist-rollers. The present paper describes how the structure of the twist-roller friction drive can be simplified by using ball bearings to support the twist-rollers. Even with this simplification, the experiments showed that the positioning resolution is less than one nanometer. The paper concludes that the twist-roller friction drive of ball bearing type facilitates the manufacture of a nanometer positioning system which can be used in a clean environment, and can take the place of the ball screw or capstan friction drive.
: nanotechnology, precision machine, position accuracy
1. introduction
The positioning system for ultra-precision machine tools and measuring machines requires a nanometer order of positioning resolution and several hundred millimeters of stroke. To realize such a nanometer positioning system, many types of feed drive devices have been proposed. These systems employ the ball screw, the capstan friction drive and the hydrostatic lead screw (1-51. Recently, we proposed a feed drive device employing the twistroller friction drive mechanism, whose main feature is its small lead [6]. The twist-roller ftjction drive has three cylindrical rollers which are pressed against a drive shaft with a twist angle between the axis of the shaft and that of each roller. We, therefore, call the rollers "twist-roller". As the drive shaft rotates, the twist-rollers rotate helically relative to the drive shaft. Then, the rollers are forced to move in the axial direction of the shaft. This axial movement can be used for feeding and positioning. By decreasing the twist angle, the axial movement of the rollers for one revolution of the drive shaft, namely the lead of this mechanism, can be reduced to less than 0.lmrn. Therefore, by using a driving motor with the resolution of one-millionth of a revolution, the nominal positioning resolution can be refined to less than O.lnm. In the twist-roller friction drive, while rotation of the twist roller is inevitable, the accurate supporting of the rollers with low friction is essential. Therefore, hydrostatic bearings were used to support the twistrollers, This twist-roller friction drive of hydrostatic bearing type was incorporated into a positioning system using a hydrostatic guideway, and it was
Annals of the ClRP Vol. 45/1/1996
proved that the positioning resolution of the system is 0.2nm [7]. Thus, the use of hydrostatic bearings is effective for obtaining high performance of positioning. However, complicated designing and difficult machining is needed for the roller-supporting hydrostatic bearings, and the hydraulic oil used in the bearings and the guideway may contaminate the environment. Thus, the twist-roller friction drive of hydrostatic bearing type becomes expensive and is not suitable for use in a clean environment. These disadvantages of the device can be overcome by using ball bearings to support the twistrollers. In the present paper, a twist-roller friction drive of ball bearing type is proposed, and incorporated into a table guided by an aerostatic guideway. Therefore, this positioning system can be used in a clean environment and the cost of the system can be reduced. The influence of the simplification of the roller-supporting bearing on the positioning performance of the device is analyzed, Then, the advantages and disadvantages of the device are discussed in comparison with the ball screw and the capstan friction drive. 2
.
..
.
i
Figure 1 is a schematic view of the twist-roller friction drive of ball bearing type. The internal mechanism of the device is shown by removing the front cover of the housing. There are three rollers, and the upper roller is shown in cross-section. Each roller is supported by four angular contact ball bearings (#7003). These three twist-rollers are pressed against a drive shaft. The surfaces of the twist-rollers and the drive shaft are plated by
50 1
chromium, then finished to sub-micrometer accuracy by grinding. To decrease the lead of the mechanism, a minute twist angle (about 0.001 rad.) between the axis of the drive shaft and that of the twist-roller is made by inclining the axis of the hole for the roller shaft in the housing. This twist-roller friction drive is mounted on the table of a positioning system shown in Fig. 2. The table is guided by an aerostatic guideway; both the table and the guideway are made of alumina ceramics. The drive shaft of the friction drive is supported by an aerostatic radial bearing at one end and by an aerostatic radial/thrust bearing at the other end, where an AC servo driving motor is connected. The resolution of this driving motor is ten-millionths of a revolution. Therefore, a simple calculation indicates that the nominal positioning resolution is 0.01nm. The mechanical stroke of the table is about 300mm. The stroke is restricted by the length of the drive shaft and the guideway. The driving motor is controlled by a closed-loop control system using a micro computer. Therefore, the resolution and the stroke of the positioning system are also restricted by the characteristics of the measuring device for the table movement. In the present system, the table movement is detected by a laser scale for full stroke with low resolution (1Onm) and a fiber optic sensor for short stroke with high resolution. The fiber optic sensor used in the present paper has a dual optical path: one is for measurement, and the other is for reference. Owing to the dual path, the resolution of the sensor can be refined to O.lnm within the stroke of 3pm (81. A block diagram of the control system for the positioning with high resolution is shown in Fig. 3. The control signal for the driving motor is output from the DIA converter in the micro computer, and input to the driving amplifier of the motor. The twist-roller friction drive converts the rotation of the motor into the linear motion of the table. The displacement of the table is measured by the fiber optic sensor. This analog signal of the table position is ND converted, then input into the computer. The control signal continues to be output from the computer until the table reaches the final position within the resolution of the control system.
3.
H c ~ isno
Tr i
‘4 Fig. 1 Mechanism of twist-roller friction drive
Fig. 2 Nanometer positioning system with the twistroller friction drive
I
+,
Micro computer
.. .
The twist-roller friction drive transmits driving force for the table by rolling friction between the drive shaft and the twist-roller. For positioning control, the thrust force to the table should be small enough so as not to cause axial macro slip between the drive shaft and the roller. However, the thrust force causes some axial elastic displacement of the device. By pulling the table, such axial displacement is measured. The maximum thrust force applied is 100N, while no macro slip occurs and the displacement is elastic. From the measurement, the thrust stiffness is calculated to be 32N/pm.
502
,
1
Driving
motor
sensor
friction drive
Fig. 3 Block diagram of closed-loop control system
The measurement of the lead of the system shows that this twist-roller friction drive is a righthanded screw with the lead of 66pm for both clockwise and counter clockwise revolutions. The lead of the twist-roller friction drive of hydrostatic type was 60pm [S].Therefore, the roller-supporting ball bearings in the present paper can maintain the twist angle as small as the hydrostatic bearings can. The dynamic response in the axial direction of the positioning system is shown in Fig. 4. The vertical axis of the upper diagram indicates the ratio of the table velocity to the input voltage for the driving motor, and that of the lower diagram indicates the phase angle. The rotational speed of the driving motor is proportional to the input voltage. Therefore, the sinusoidal voltage with various frequencies is input to the driving motor, and the response of the positioning system is evaluated at the table, while the closed-loop control is not used. Figure 4 indicates that the positioning system is controllable for dynamic input lower than 30Hz. Nanometer step positioning of the system is executed by using the control system shown in Fig. 3. The table displacement is measured by the fiber optic sensor, and the result of the measurement is shown in Fig. 5, where a 5HZ low-pass filter is used. The command signal for a step motion with the width of (a) lnm, (b) 0.5nm or (c) 0.2nm is repeated ten times in one direction, then the command of reverse motion is also repeated ten times. Owing to the influences of electrical noise and mechanical vibration induced from the environment, some fluctuation appears in each step, however, Fig. 5 shows that the 0.5nm step is clearly resolved. The 0.2nm step is also resolved, though, the noise level is larger than the step width. Therefore, we conclude that the positioning resolution of this twistroller frictional drive of ball bearing type is 0.5nm, while the influence of the substitution of the hydrostatic bearings by the ball bearings for support of the rollers on the positioning resolution seems to be negligible.
m
9
-
iE
1
:
I
,
, 1 1 1 1
Frequency
f
Hz
Fig. 4 Dynamic response of the positioning system
8 E
c
0
6
c
c i
4
Ir
(b) Step = 0 . 5 nm
A
m
4.
The positioning resolution and the stroke are contradictory requirements for a positioning system, and various feed drive devices have been proposed to satisfy these two requirements. The degree of satisfaction can be evaluated by the ratio of stroke/resolution of the positioning system. Figure 6 shows the positioning resolution 6x(nm) and the stroke(mm)/resolution(mm) ratio Sr of several numerical control positioning systems (1 -5,7, 9-1 21 and the results of the present paper. The ratio Sr is the measure of the maximum steps of the numerical control system. The nominal positioning resolution and stroke of commercially available ultra-precision machine tools of the type known as 'aspheric generators" are about lnm and 300mm, respectively [lo- 12). Therefore, the ratio Sr of the positioning
Fig. 5 Nanometer step positioning (with fiber optic sensor and 5Hz low-pass filter)
503
'W?
10-11
' """"
'
""".'
'
'"""I
capstan friction drive, the twist-roller friction drive of ball bearing type can be used in many nanometer positioning systems.
Acknowledaements X
Nanoforrn 300(b) ASP-G25(h)
%
c
.-2
iAHN10@),
:10'v)
2!
The authors wish to thank Nachi-Fujikoshi Co., Ltd. for their assistance. This research was financially supported by the Japanese Ministry of Education (Grant-in-Aid for Scientific Research, N0.07555375, 1995).
.
, ,
0
aA
Precision machine
Tools
Ordinary NC Machine tools ,,,,,I , , , , , , , , I
,,,,,,,,,
1o7 108 Strokelresolution ratio Sr
i1
1o9
Fig. 6 Resolution and stroke of positioning systems: o commercially available machines (except LODTM) 0 systems reported in Ref.[9] 0 system reported in the present paper (letters in parentheses indicate the feed drive devices used, b = ball screw, c = capstan friction drive, and h = hydrostatic lead screw)
system of these aspheric generators is about 3x1Oa, which may be the highest value at present. These positioning systems use the ball screw or capstan friction drive. As shown in Fig. 6 by the symbol 0 , higher positioning performance can be obtained by a positioning system employing the hydrostatic lead screw and the twist-roller friction drive of hydrostatic bearing type[9]. However, these feed drive devices are difficult to manufacture and hydraulic equipment is needed. The high cost of these devices may only be justified in restricted applications. The twist-roller friction drive of ball bearing type proposed in the present paper shows that the positioning resolution is 0.5nm and the stroke is 300mm. As shown in Fig. 6 by the symbol 0 , the ratio Sr is Sxl@. Therefore, the positioning performance of the twist-roller friction drive of ball bearing type is superior to that of the ball screw and the capstan friction drive. The manufacture of the twist-roller friction drive of ball bearing type is easier than that of the ball screw and almost as easy as that of the capstan friction drive. Recently, it has been pointed out that the use of the capstan friction drive should be reexamined because of the difficulty of tuning the loop gain of the controller for various loads[5]. Therefore, instead of the ball screw and
504
References Donaldson, R. R. and Maddux, A. S., 1984, 3esign of a High-Performance Slide and Drive System for a Small Precision Machining Research Lathe, Ann. of the CIRP, 3311, 243-249. -eadbeater P. B., Clarke. M., Wills-Moren, W. J. and Wilson, T. J., 1989, A unique machine lor grinding large off-axis optical components: the OAGM 2500, Precision Engineering, 1114, 191-196. Usuki, M. and Yabuya, M., 1989, Ultra precision aspheric generator, J. of Japan Society for Precision Engineering, 5516, 967-971. Carlisie, K. and Shore, P., 1991, Experiences in the development of ultra stiff CNC aspheric generating machine tools for ductile regime grinding of brittle materials, Proc. of the 6th IPES, 85-94. Youden, D. H., 1992, Capstans or lead screws?, Proceedings of the ASPE 1992 annual meeting, 116-1 21. Mizumoto, H., Nomura, K., Matsubara, T. and Shimizu, T., 1993, An ultra-precision positioning system using a twist-roller friction drive", J. of ASPE, 1513, 180-184 Mizumoto, H., Yabuya, M., Shimizu, T. and Kami, Y., 1995, An angstrom-positioning system using a twist-roller friction drive", J. of ASPE, 1711, 57-62 Ikawa, N., Shimada, S. and Morooka, H., 1987, Photoelectronic Displacement Sensor with Precision Engineering, Inanometer resolution, 912, 79-82 Wizumoto, H., Yabuya, M., Shimizu, T. and Kami, Y., 1996, Comparison of Positioning Resolution Iof Feed Drive Devices for Ultra-precision Machine Tools, J. of Japan Society for Precision Engineering, 6213, 361-365. "Nanoforrn 600," 1989, Catalogue of Rank Pneumo Inc. "Nanocentre," 1993, Catalogue of Cranfield Precision Engineering Ltd. "AHNGO," 1994, Catalogue of Toyoda Machine Works Ltd.
-
H. Mizumoto, S. Arii, A. Yoshimoto, Tottori University, Tottori; T. Shimizu, Nachi-Fujikoshi Corp., Toyama; N. lkawa (1). Osaka University, Osaka; Japan Received on January 3,1996
Abstract The authors have reported in a previous paper that the twist-roller friction drive can realize subnanometer positioning resolution. However, a structural difficulty of the device is that hydrostatic bearings are used to support the twist-rollers. The present paper describes how the structure of the twist-roller friction drive can be simplified by using ball bearings to support the twist-rollers. Even with this simplification, the experiments showed that the positioning resolution is less than one nanometer. The paper concludes that the twist-roller friction drive of ball bearing type facilitates the manufacture of a nanometer positioning system which can be used in a clean environment, and can take the place of the ball screw or capstan friction drive.
: nanotechnology, precision machine, position accuracy
1. introduction
The positioning system for ultra-precision machine tools and measuring machines requires a nanometer order of positioning resolution and several hundred millimeters of stroke. To realize such a nanometer positioning system, many types of feed drive devices have been proposed. These systems employ the ball screw, the capstan friction drive and the hydrostatic lead screw (1-51. Recently, we proposed a feed drive device employing the twistroller friction drive mechanism, whose main feature is its small lead [6]. The twist-roller ftjction drive has three cylindrical rollers which are pressed against a drive shaft with a twist angle between the axis of the shaft and that of each roller. We, therefore, call the rollers "twist-roller". As the drive shaft rotates, the twist-rollers rotate helically relative to the drive shaft. Then, the rollers are forced to move in the axial direction of the shaft. This axial movement can be used for feeding and positioning. By decreasing the twist angle, the axial movement of the rollers for one revolution of the drive shaft, namely the lead of this mechanism, can be reduced to less than 0.lmrn. Therefore, by using a driving motor with the resolution of one-millionth of a revolution, the nominal positioning resolution can be refined to less than O.lnm. In the twist-roller friction drive, while rotation of the twist roller is inevitable, the accurate supporting of the rollers with low friction is essential. Therefore, hydrostatic bearings were used to support the twistrollers, This twist-roller friction drive of hydrostatic bearing type was incorporated into a positioning system using a hydrostatic guideway, and it was
Annals of the ClRP Vol. 45/1/1996
proved that the positioning resolution of the system is 0.2nm [7]. Thus, the use of hydrostatic bearings is effective for obtaining high performance of positioning. However, complicated designing and difficult machining is needed for the roller-supporting hydrostatic bearings, and the hydraulic oil used in the bearings and the guideway may contaminate the environment. Thus, the twist-roller friction drive of hydrostatic bearing type becomes expensive and is not suitable for use in a clean environment. These disadvantages of the device can be overcome by using ball bearings to support the twistrollers. In the present paper, a twist-roller friction drive of ball bearing type is proposed, and incorporated into a table guided by an aerostatic guideway. Therefore, this positioning system can be used in a clean environment and the cost of the system can be reduced. The influence of the simplification of the roller-supporting bearing on the positioning performance of the device is analyzed, Then, the advantages and disadvantages of the device are discussed in comparison with the ball screw and the capstan friction drive. 2
.
..
.
i
Figure 1 is a schematic view of the twist-roller friction drive of ball bearing type. The internal mechanism of the device is shown by removing the front cover of the housing. There are three rollers, and the upper roller is shown in cross-section. Each roller is supported by four angular contact ball bearings (#7003). These three twist-rollers are pressed against a drive shaft. The surfaces of the twist-rollers and the drive shaft are plated by
50 1
chromium, then finished to sub-micrometer accuracy by grinding. To decrease the lead of the mechanism, a minute twist angle (about 0.001 rad.) between the axis of the drive shaft and that of the twist-roller is made by inclining the axis of the hole for the roller shaft in the housing. This twist-roller friction drive is mounted on the table of a positioning system shown in Fig. 2. The table is guided by an aerostatic guideway; both the table and the guideway are made of alumina ceramics. The drive shaft of the friction drive is supported by an aerostatic radial bearing at one end and by an aerostatic radial/thrust bearing at the other end, where an AC servo driving motor is connected. The resolution of this driving motor is ten-millionths of a revolution. Therefore, a simple calculation indicates that the nominal positioning resolution is 0.01nm. The mechanical stroke of the table is about 300mm. The stroke is restricted by the length of the drive shaft and the guideway. The driving motor is controlled by a closed-loop control system using a micro computer. Therefore, the resolution and the stroke of the positioning system are also restricted by the characteristics of the measuring device for the table movement. In the present system, the table movement is detected by a laser scale for full stroke with low resolution (1Onm) and a fiber optic sensor for short stroke with high resolution. The fiber optic sensor used in the present paper has a dual optical path: one is for measurement, and the other is for reference. Owing to the dual path, the resolution of the sensor can be refined to O.lnm within the stroke of 3pm (81. A block diagram of the control system for the positioning with high resolution is shown in Fig. 3. The control signal for the driving motor is output from the DIA converter in the micro computer, and input to the driving amplifier of the motor. The twist-roller friction drive converts the rotation of the motor into the linear motion of the table. The displacement of the table is measured by the fiber optic sensor. This analog signal of the table position is ND converted, then input into the computer. The control signal continues to be output from the computer until the table reaches the final position within the resolution of the control system.
3.
H c ~ isno
Tr i
‘4 Fig. 1 Mechanism of twist-roller friction drive
Fig. 2 Nanometer positioning system with the twistroller friction drive
I
+,
Micro computer
.. .
The twist-roller friction drive transmits driving force for the table by rolling friction between the drive shaft and the twist-roller. For positioning control, the thrust force to the table should be small enough so as not to cause axial macro slip between the drive shaft and the roller. However, the thrust force causes some axial elastic displacement of the device. By pulling the table, such axial displacement is measured. The maximum thrust force applied is 100N, while no macro slip occurs and the displacement is elastic. From the measurement, the thrust stiffness is calculated to be 32N/pm.
502
,
1
Driving
motor
sensor
friction drive
Fig. 3 Block diagram of closed-loop control system
The measurement of the lead of the system shows that this twist-roller friction drive is a righthanded screw with the lead of 66pm for both clockwise and counter clockwise revolutions. The lead of the twist-roller friction drive of hydrostatic type was 60pm [S].Therefore, the roller-supporting ball bearings in the present paper can maintain the twist angle as small as the hydrostatic bearings can. The dynamic response in the axial direction of the positioning system is shown in Fig. 4. The vertical axis of the upper diagram indicates the ratio of the table velocity to the input voltage for the driving motor, and that of the lower diagram indicates the phase angle. The rotational speed of the driving motor is proportional to the input voltage. Therefore, the sinusoidal voltage with various frequencies is input to the driving motor, and the response of the positioning system is evaluated at the table, while the closed-loop control is not used. Figure 4 indicates that the positioning system is controllable for dynamic input lower than 30Hz. Nanometer step positioning of the system is executed by using the control system shown in Fig. 3. The table displacement is measured by the fiber optic sensor, and the result of the measurement is shown in Fig. 5, where a 5HZ low-pass filter is used. The command signal for a step motion with the width of (a) lnm, (b) 0.5nm or (c) 0.2nm is repeated ten times in one direction, then the command of reverse motion is also repeated ten times. Owing to the influences of electrical noise and mechanical vibration induced from the environment, some fluctuation appears in each step, however, Fig. 5 shows that the 0.5nm step is clearly resolved. The 0.2nm step is also resolved, though, the noise level is larger than the step width. Therefore, we conclude that the positioning resolution of this twistroller frictional drive of ball bearing type is 0.5nm, while the influence of the substitution of the hydrostatic bearings by the ball bearings for support of the rollers on the positioning resolution seems to be negligible.
m
9
-
iE
1
:
I
,
, 1 1 1 1
Frequency
f
Hz
Fig. 4 Dynamic response of the positioning system
8 E
c
0
6
c
c i
4
Ir
(b) Step = 0 . 5 nm
A
m
4.
The positioning resolution and the stroke are contradictory requirements for a positioning system, and various feed drive devices have been proposed to satisfy these two requirements. The degree of satisfaction can be evaluated by the ratio of stroke/resolution of the positioning system. Figure 6 shows the positioning resolution 6x(nm) and the stroke(mm)/resolution(mm) ratio Sr of several numerical control positioning systems (1 -5,7, 9-1 21 and the results of the present paper. The ratio Sr is the measure of the maximum steps of the numerical control system. The nominal positioning resolution and stroke of commercially available ultra-precision machine tools of the type known as 'aspheric generators" are about lnm and 300mm, respectively [lo- 12). Therefore, the ratio Sr of the positioning
Fig. 5 Nanometer step positioning (with fiber optic sensor and 5Hz low-pass filter)
503
'W?
10-11
' """"
'
""".'
'
'"""I
capstan friction drive, the twist-roller friction drive of ball bearing type can be used in many nanometer positioning systems.
Acknowledaements X
Nanoforrn 300(b) ASP-G25(h)
%
c
.-2
iAHN10@),
:10'v)
2!
The authors wish to thank Nachi-Fujikoshi Co., Ltd. for their assistance. This research was financially supported by the Japanese Ministry of Education (Grant-in-Aid for Scientific Research, N0.07555375, 1995).
.
, ,
0
aA
Precision machine
Tools
Ordinary NC Machine tools ,,,,,I , , , , , , , , I
,,,,,,,,,
1o7 108 Strokelresolution ratio Sr
i1
1o9
Fig. 6 Resolution and stroke of positioning systems: o commercially available machines (except LODTM) 0 systems reported in Ref.[9] 0 system reported in the present paper (letters in parentheses indicate the feed drive devices used, b = ball screw, c = capstan friction drive, and h = hydrostatic lead screw)
system of these aspheric generators is about 3x1Oa, which may be the highest value at present. These positioning systems use the ball screw or capstan friction drive. As shown in Fig. 6 by the symbol 0 , higher positioning performance can be obtained by a positioning system employing the hydrostatic lead screw and the twist-roller friction drive of hydrostatic bearing type[9]. However, these feed drive devices are difficult to manufacture and hydraulic equipment is needed. The high cost of these devices may only be justified in restricted applications. The twist-roller friction drive of ball bearing type proposed in the present paper shows that the positioning resolution is 0.5nm and the stroke is 300mm. As shown in Fig. 6 by the symbol 0 , the ratio Sr is Sxl@. Therefore, the positioning performance of the twist-roller friction drive of ball bearing type is superior to that of the ball screw and the capstan friction drive. The manufacture of the twist-roller friction drive of ball bearing type is easier than that of the ball screw and almost as easy as that of the capstan friction drive. Recently, it has been pointed out that the use of the capstan friction drive should be reexamined because of the difficulty of tuning the loop gain of the controller for various loads[5]. Therefore, instead of the ball screw and
504
References Donaldson, R. R. and Maddux, A. S., 1984, 3esign of a High-Performance Slide and Drive System for a Small Precision Machining Research Lathe, Ann. of the CIRP, 3311, 243-249. -eadbeater P. B., Clarke. M., Wills-Moren, W. J. and Wilson, T. J., 1989, A unique machine lor grinding large off-axis optical components: the OAGM 2500, Precision Engineering, 1114, 191-196. Usuki, M. and Yabuya, M., 1989, Ultra precision aspheric generator, J. of Japan Society for Precision Engineering, 5516, 967-971. Carlisie, K. and Shore, P., 1991, Experiences in the development of ultra stiff CNC aspheric generating machine tools for ductile regime grinding of brittle materials, Proc. of the 6th IPES, 85-94. Youden, D. H., 1992, Capstans or lead screws?, Proceedings of the ASPE 1992 annual meeting, 116-1 21. Mizumoto, H., Nomura, K., Matsubara, T. and Shimizu, T., 1993, An ultra-precision positioning system using a twist-roller friction drive", J. of ASPE, 1513, 180-184 Mizumoto, H., Yabuya, M., Shimizu, T. and Kami, Y., 1995, An angstrom-positioning system using a twist-roller friction drive", J. of ASPE, 1711, 57-62 Ikawa, N., Shimada, S. and Morooka, H., 1987, Photoelectronic Displacement Sensor with Precision Engineering, Inanometer resolution, 912, 79-82 Wizumoto, H., Yabuya, M., Shimizu, T. and Kami, Y., 1996, Comparison of Positioning Resolution Iof Feed Drive Devices for Ultra-precision Machine Tools, J. of Japan Society for Precision Engineering, 6213, 361-365. "Nanoforrn 600," 1989, Catalogue of Rank Pneumo Inc. "Nanocentre," 1993, Catalogue of Cranfield Precision Engineering Ltd. "AHNGO," 1994, Catalogue of Toyoda Machine Works Ltd.