INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 13, No. 3, pp. 401-406
MARCH 2012 / 401
DOI: 10.1007/s12541-012-0051-1
A Three-axis Translation Stage Using Opposing
Wedges with Error Compensation
Gyungho Khim1,#, Seung Kook Ro1, Jong Kweon Park1 and Kornel Ehmann2
1 Department of Ultra Precision Machines and Systems, Korea Institute of Machinery and Materials, 156, Gajeongbuk-ro, Yuseong-gu, Daejeon, Korea, 305-343
2 Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208-3111, USA
# Corresponding Author / E-mail: gyungho@kimm.re.kr, TEL: +82-42-868-7105, FAX: +82-42-868-7180
KEYWORDS: Wedge stage, Air bearings, Straightness error, Positioning error, Compensation
We describe the development of a three-axis translation stage using wedges and its motion error compensation. The threeaxis stage uses three wedges and is capable of translation in the vertical and longitudinal directions by controlling the
horizontal separation of two opposing wedges. An independent linear translation stage is used to achieve displacement
along the third axis. Compensation for straightness and positioning errors is achieved using the combined motion of the two
opposing wedges and the motion of an independent linear stage. The straightness error in the vertical and horizontal
directions through the longitudinal axis was 0.83 μm and 1.65 μm, respectively, before compensation and 0.22 μm and
0.29 μm with error compensation. The positioning error through the longitudinal axis was reduced from 8.96 μm to 0.85 μm.
Manuscript received: February 7, 2011 / Accepted: September 4, 2011
1. Introduction
A micro-factory is a miniature manufacturing system that
produces small products for IT (Information Technology), BT (Bio
Technology) and NT (Nano Technology) applications. It provides
space and energy savings as well as improved flexibility because of
the reduced dimensions.1,2 Micro-factory systems for recently
developed high-quality products must satisfy strict accuracy and
size requirements. Therefore, the multi-axis translation stage for
such a system also requires precise and compact mechanisms.
Conventional three-axis (x, y, z) translation stage systems
typically employ a stacked configuration.3,4 This is relatively simple
and easy to manipulate; however, it makes the system bulky, and
accumulated motion error can occur. If two or more stages are
stacked, the total stiffness will decrease because they can be
regarded as serially connected springs. Such a configuration also
has weak moment stiffness. It is especially difficult to realize
precise motion along the vertical axis because the feeding unit and
bearings should be installed perpendicular to the horizontal plane.
When a short stroke is required in the vertical direction, a wedgetype stage is sometimes used for the vertical lift motion. However,
as it produces only one-directional motion, it must be used in a
stacked configuration to realize motions along three axes. Moreover,
most commercialized wedge stages are composed of ballscrews and
circulating ball bearings, which are not suitable for precise
motion.5,6 Thus, it is desirable to employ non-stacked stages with
© KSPE and Springer 2012
low-profile designs and low friction bearings to manufacture
precise micro/meso-sized components in micro-factory systems.
To achieve this, we propose a compact three-axis translation
stage using opposing wedges. The three-axis stage uses three
wedges and is capable of translation in the vertical and longitudinal
directions by the combined simultaneous motion of two opposing
wedges. An independent linear translation stage is installed on the
center wedge to achieve displacement along the third axis. The
stage employs air bearings and linear motors to achieve precise
motion. This type of wedge-based stage is very compact and has a
low-profile design compared with conventional stacked
configurations. The other advantage of the developed stage is its
compensation for straightness and positioning errors. The vertical
straightness and positioning errors of the stage through the
longitudinal axis can be compensated for by using the combined
motion of the two opposing wedges. The horizontal straightness
error through the longitudinal axis can be also compensated for by
using an independent linear stage. We present an overview of the
design of the wedge stage and then examine the motion-error
compensation.
2. Design of the three-axis wedge stage
2.1 Operating principle of the wedge stage
Figure 1 shows the principle of operation of the wedge stage. If
402 / MARCH 2012
INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 13, No. 3
vx
y-axis slide
x2-slide
Center wedge
Center wedge
z
θ
Side wedge
vx
x
x1-slide
vx
Counterbalance air cylinder
(a) Motion along the x-axis
Side wedge
vz = vx tan θ
z
y
x
Center wedge
z
(a) Overall view
vx
vx
x
Side wedge
Center wedge
θ
Side wedge
Steel plate for magnetic preload
y-motor
Counterbalance air cylinder
x2-motor
(b) Motion along the z-axis
Fig. 1 Principle of operation of the wedge stage
the two side wedges move with the same velocity, vx, the center
wedge also moves with velocity of vx, as shown in Fig. 1(a).
However, if the direction of one of the wedges is reversed, the
center wedge only moves along the z-axis with velocity vz given by
vz = vx tan(θ),
x2-encoder
Magnetic preload
Air bearing
Counterbalance air cylinder
(1)
where θ is the angle defining the profile of the wedge.
2.2 Design of the three-axis wedge stage
The structure of the three-axis wedge stage is shown in Fig. 2(a).
The x-axis translation of the two side wedges produces motion
along the x- and z-axes. An independent linear stage is used to
produce motion along the y-axis and is installed on the center
wedge.
Orifice air bearings were used for all slide surfaces, including
the vertical and horizontal slide surfaces of the two side wedges and
the y-axis slide, as well as the inclined wedge surfaces and the
horizontal surfaces of the center wedge. The air bearings for the
bottom slides of the two side wedges and the bearings in the
inclined wedge slides were magnetically preloaded using circular
permanent magnets, whereas the other air bearings were preloaded
by opposing pads, as shown in Fig. 2(b) and (c). Rectangular steel
plates were attached on the aluminum slide surface for magnetic
preload. A counterbalance air cylinder (Airpel E16D4.0S), which
had anti-stiction and ultra-low-friction air bearings, was placed
under the two side wedges to support the mass of the center wedge
and the y-axis slide to improve control over the motion along the xand z-axes (see Fig. 2(b)). Pressurized air was supplied to the
pneumatic cylinder; the pressure was controlled to support the mass
of the center wedge and the y-axis slide.
Coreless linear motors (Trilogy, 110-1S) and linear scales
(Heidenhain, LIDA47) with a resolution of 50 nm were used for all
axes. A linear motor track and two motor coils (x1-motor, x2-motor)
x1-encoder
x1-motor
(b) Counterbalance air cylinder and magnetic preload
Air bearing
Magnetic preload
Steel plate for magnetic preload
(c) Side wedge slide
Fig. 2 Structure of the three-axis wedge stage
were installed in the two side wedges, and a linear scale and two
scale heads (x1-encoder, x2-encoder) were also installed under the
two side wedges. Therefore, each side wedge could move
independently and translate in the vertical and longitudinal
directions. The resolution of the x- and y-axis movements was 50
nm; however, the z-axis had a resolution of 28.9 (= 50 × tan 30°)
nm according to the wedge angle. The stroke in the z-direction was
limited by the wedge size, but had better resolution than motion
along the x-axis had. Therefore, this type of wedge stage is suitable
for a system that requires small strokes and high resolution in the
INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 13, No. 3
MARCH 2012 / 403
z
Side wedge
x1-motor
x
− δ x (x )
Actuator
Feedback
Bearing
Controller
Drive
Air cylinder
Stroke
Wedge angle
Material
Linear motor (Trilogy 110-1S)
Linear scale (Heidenhain LIDA 47, 50 nm)
Orifice air bearings with magnetic preload
UMAC (DeltaTau)
PWM drive (Copley Xenus XSJ-230-10)
Airpel E16D4.0S (Airpot)
x: 180 mm, y: 25 mm, z: 25 mm
30°
Anodized aluminum
vertical direction. The control system employed a UMAC
(DeltaTau) controller and Xenus PWM digital amplifiers (Copley
Controls). The stiffness of the x- and z-axes in this type of wedgebased stage is affected by the stiffness of the servo as well as the air
bearings, so the control gains were carefully determined.
Figure 3 shows a photograph of the translation stage, and
Table 1 summarizes the specifications of the stage. The stage was
fabricated from anodized aluminum. The overall stage size was
L584 × W203 × H101 mm, with a working volume of 180 × 25 ×
25 mm.
δ z (x) / tan θ
Fig. 4 Principle of compensation for the vertical straightness and
positioning errors
Z-axis displacement (mm)
Table 1 Specifications of the three-axis translation stage
x2-motor
− δ x (x )
− δ z (x) / tan θ
Fig. 3 Photograph of the stage
Side wedge
θ
Positioning error
δ x (x)
Center wedge
Vertical straightness error
δ z (x)
0.6 Slope=0.581
0.4
0.2
0.0
0.0
0.5
1.0
1.5
0.5
X-axis displacement (mm)
2.0
0.0
Fig. 5 Relationship between the x- and z-axis displacements
3. Experiments
Fig. 6 Measurement of the straightness error
3.1 Compensation for the vertical straightness error
displacement. If the positioning error is positive, then the two side
wedges must move in the negative x-axis direction at the same
velocity; if the error is negative, the two side wedges must move in
the positive x-axis direction at the same velocity.
Before compensating for the vertical straightness error, the
relationship between the x- and z-axis displacements must be
characterized; even though the angle of the wedge was designed to
be θ = 30°, improved control over the z-axis displacement was
anticipated by measuring this rather than simply using Eq. (1). The
measurement used to compensate for the vertical straightness error
was performed with a laser interferometer system (HP5529A).
Therefore, the z-axis displacement was measured as a function of
the separation of the two side wedges using the laser interferometer.
Following this, the angle of the wedge was calculated using Eq. (1).
Figure 5 shows the measured relationship between the x- and z-axis
displacements. The gradient of the curve was 0.581, which
corresponds to a wedge angle of θ = 30.16°.
The vertical straightness error along the x-axis was
One of the main advantages of the wedge stage is that it is
possible to compensate for the vertical straightness error by moving
the two side wedges. In a conventional linear translation stage, the
rail form error must be corrected to improve the straightness
error.7,8
Figure 4 illustrates the principle of compensation for the
vertical straightness error, δz(x), and positioning error, δx(x), of the
center wedge through the x-axis. The errors depended on the
relative motion of the two side wedges. When compensating for the
vertical straightness error, the two side wedges must move at
opposing velocities so that they do not change the x-axis
displacement. If we assume that the upward displacement of the
vertical straightness error is positive, then the two side wedges must
move apart with opposing velocities; if the error is negative, the two
side wedges must move together with opposing velocities. When
compensating for the positioning error, the two side wedges must
move with the same velocity so that they do not change the vertical
404 / MARCH 2012
Horizontal straightness error
Before compensation : 0.83 μm
0.4
δ y (x)
0.0
y-axis slide
-0.4
-0.8
0.8
0
30
60
90
120
150
After compensation : 0.22 μm
0.4
x
-0.4
Fig. 9 Principle of the horizontal straightness error compensation
0
30
60
90
120
150
180
Position, x (mm)
Fig. 7 Compensation for the vertical straightness error
10
8
Before compensation : 8.96 μm
6
4
Forward mean
Forward mean-2σ
Backward mean
Backward mean+2σ
2
0
10
8
− δ y (x )
y-motor
y
0.0
-0.8
Positioning error, δx(x) (μm)
Center
wedge
180
0
30
60
90
120
150
180
After compensation : 0.85 μm
Forward mean
Forward mean-2σ
Backward mean
Backward mean+2σ
6
4
2
2
Horizontal straightness error, δy(x) (μm)
Vertical straightness error, δz(x) (μm)
0.8
INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 13, No. 3
Before compensation : 1.65 μm
1
0
-1
2
0
30
60
90
120
150
180
150
180
After compensation : 0.29 μm
1
0
-1
0
30
60
90
120
Position, x (mm)
Fig. 10 Compensation for the horizontal straightness error
0
0
30
60
90
120
150
180
Position, x (mm)
Fig. 8 Positioning error in the x-axis with and without
compensation
compensated for by means of the relative motion of the two side
wedges. First, we measured the vertical straightness error using the
laser interferometer, as shown in Fig. 6. We then calculated the
change in the separation of the two side wedges, as shown in Fig. 4.
By using the UMAC controller compensation table, the vertical
straightness error was compensated for automatically while the two
side wedges were moving to achieve an x-axis displacement.
Figure 7 shows the measured straightness error in the vertical
direction as a function of the translation of the center wedge
through the x-axis with and without error compensation. The errorcompensation algorithm reduced the vertical straightness error from
0.83 μm to 0.22 μm.
3.2 Compensation for the positioning error
We followed the same procedure to compensate for the
positioning error. Thus, the vertical straightness and positioning
errors could be compensated for simultaneously by using a
compensation table. Figure 8 shows the x-axis positioning error as a
function of the x-axis translation, measured using a laser
interferometer that complied with ISO 230-2 standards.9 The error
was compensated for by translating the two side wedges
simultaneously using the UMAC controller compensation table.
Before compensation, the error was 8.96 μm (±2σ, where σ is the
standard deviation). By employing the error-compensation
algorithm, it was reduced to 0.85 μm. The experiment was carried
out in an environment where the temperature, humidity, and
fluctuation of air could not be controlled, which has implications
for the repeatability error.
3.3 Compensation for the horizontal straightness error
The straightness error in the horizontal direction, δy(x), was
compensated for by using the linear translation stage installed on
the center wedge, which could move independently of the x- and zaxes coupled-wedge system. The horizontal straightness error of the
center wedge along the x-axis was compensated for by using the yaxis linear motor while the center wedge was moving through the xaxis. Figure 9 shows the principle of the horizontal straightnesserror compensation.
Figure 10 shows the measured straightness error in the
horizontal direction with and without error compensation as a
function of the translation of the center wedge along the x-axis. The
error compensation algorithm resulted in a reduction of the
horizontal straightness error from 1.65 μm to 0.29 μm.
Displacement (μm)
INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 13, No. 3
without direct actuation or feedback.
0.3
1 step=50 nm
0.2
4. Conclusions
0.1
0.0
0
10
20
30
40
50
60
40
50
60
Displacement (μm)
Time (s)
(a) The x-axis
0.3
1 step=50 nm
0.2
0.1
0.0
0
10
20
30
Displacement (μm)
Time (s)
(b) The y-axis
0.20
0.15
We have described a novel three-axis translation stage
employing an opposing-wedge mechanism and its motion-error
compensation. The vertical (z-axis) displacement was achieved by
controlling the x-axis separation of the two side wedges. An
independent linear translation stage installed on the center wedge
was used to achieve y-axis displacement. The vertical straightness
and positioning errors through the x-axis were simultaneously
compensated for by controlling the displacement of the two side
wedges. The horizontal straightness error was compensated for by
controlling the displacement of the independent y-axis slide. The
vertical straightness error was reduced from 0.83 μm to 0.22 μm,
the horizontal straightness error was reduced from 1.65 μm to
0.29 μm, and the positioning error was reduced from 8.96 μm to
0.85 μm.
step=50 tan(
tan(θθ) )nm
nm
11 step=50
ACKNOWLEDGEMENT
0.10
0.05
0.00
0
Displacement (μm)
MARCH 2012 / 405
0.4
10
20
30
40
50
60
1 step=100 tan(θ ) nm
REFERENCES
0.3
0.2
0.1
0.0
0
10
20
30
This research was supported by the Industrial Source
Technology Development Programs funded by the Ministry of the
Knowledge Economy (MKE), Korea.
40
50
60
Time (s)
(c) The z-axis
Fig. 11 Measured response to x-, y-, and z-axis displacements
The straightness errors in the developed wedge stage were
effectively compensated for without directly correcting for the rail
form errors.
3.4 Fine motion response
Figure 11(a), (b), and (c) show the measured response to x-, y-,
and z-axis displacements, respectively. The step response was
measured using a capacitive sensor (ADE 6810, 0.25-nm resolution)
and a 24-bit A/D board (Dewetron, DEWE-43) with a low-pass
filter (Stanford Research System, SR640) to eliminate noise.
Control over the displacement to a resolution of 50 nm was
achieved along the x- and y-axes. The z-axis resolution was 29.2 nm
with each 50-nm movement of the two side wedges; however,
control over the z-axis displacement was relatively poor compared
with that over the x- or y-axis because the z-axis displacement was
achieved only by the combined motion of two opposing wedges
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INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 13, No. 3