Laser Beam Stabilization in the CNC Machines Investigation and Modeling Yuri V. Fedosov, Maxim Ya. Afanasev, and Alexander I. Trifanov ITMO University, St. Petersburg, Russian Federation
[email protected],
[email protected] Abstract Lasers became a principal part of various technological equipment. In recent years they are widely used in numerous machine tools to cut, weld or cure polymers. During the manufacturing processes various vibrations may occur, which can influence the shape or quality of produced parts. Such vibrations can be caused by the equipment design, units location, and various external vibration sources. Despite their minor values these vibrations can cause manufacturing errors when objects to be produced are measured in microns. To prevent possible errors laser beam stabilization should be used. Optical stabilizing systems are widely used in various devices such as optical mounts, optical terminators, scanning devices and many others. Several manufacturers for instance, Canon, Nikon and Sony disclose optical stabilizing and focusing schemes, based on mechanical systems utilizing stepper motors and piezomotors.
chine to stabilize the laser beam. Possible design is pictured on the Figure 5. This system was modeled. Specific parameters were calculated for given angular ranges and positioning coordinates. Also specifications for sliding shafts lengths was stated.
Let us analyze a few of the optical systems. Among others, one system includes lens mount, where a moving lens is placed for focus adjustment. The lens is mounted on a frame moving along two cylindrical guides (Figure 1), and is moved by stepper motor and screw. Also a gear box can be added. Unfortunately, this system is open for frame play within the degree of accuracy and its use in technological equipment may require principal design improvements. Another system is presented in Figure 2. Frame with lens is moved in three guides by three linear drives to compensate possible lens shake. Frame is positioned on the optical axis with three spring parts, mounted axially in three points. Moreover, image stabilizing systems can utilize various carriers, in which compensating prism or lens can be mounted, moved by coils and magnets. Similar systems are used in various scanning and reading systems. Such system can be exemplified by CD-drive carriage (Figure 3). The system includes spring elements and coils, and allows to change focus and adjustment of the lens’ position according to track position on the laser disk. This system is intended to perform speFigure 1: Focusing mechanism cial task, has specific working zones and is not rigid enough to be used as a part of precise technological equipment. Another design of focusing system includes lens frame (or whole optical system) mounted in moving barrel, which can move back and Figure 2: Shake compensating forth driven by stepper motor, system or piezomotor. Consider for instance the system described in the patent of laser beam control for CNC machine tool. During machine work heating or cooling parts of laser beam Figure 3: Lens positioning system deflection system may occur, what leads to their initial positions change, changing the laser beam trajectory. To compensate the changes, special optical scheme is presented, which utilizes complicate tracking element and moving mirror.
3-2 1-1
2-1
6 3-1
5-1
5-4
7
4-1
Y
Let us treat the optical system mounted on modified Stewart platform, which is moved by four drives (Figure 4). Presence of four drives providing the linear motion decreases number of freedom degrees to three: two tilts around OX and OY and linear movement along OZ axis. This design sustain center of the moving platform on the optical axis of the system. Use of such architecture and presence of three freedom degrees allows to compensate its occasional deviations from the fixed optical axis by tilting the lens around the two axes. Focal setting changing by moving the platform along the optical axis is also possible. Linear drives, stepper motors with ballscrews or piezomotors can be used to vary sliding shafts lengths. The compensating element lens or prism can be mounted on moving platform. To make the corrections in real-time mode closed loop must be organized. Thereto gyroscope and three or more accelerometers must be installed to track vibrations. Data from sensors will be processed using Kalman filter. According to processed data proportional plus reset controller parameters will be adjusted. As a result of its rigidity, compensated positioning error and definite positioning this system can be integrated in CNC ma-
H +∆
α a+C
B β a+C
H −∆
A
H +∆
H
α
B β a+C
a+C
Figure 10: Correction definition
4-4
2-3
5-3
H
Figure 9: Two opposite drives
3-4
Z O
1-4
2-4
5-2
4-2
H +∆
(2)
S
∆>0 S = const A
2-2
Direction cosines (Figure 12) will be determined as eq. (3): 1-3
X
ax ex = ; |¯a|
3-3
Figure 4: Modified Stewart platform kinematic scheme. 1 – motors, 2 – ballscrews, 3 – sliding shafts, 4 – hinges, 5 – link ball joints, 6 – moving platform, 7 – lens
sin
ay ey = |¯a|
(3)
Z Y
S/2 S
γ
cos
ey ~e
C X
ex
Figure 11: Platform rotation Figure 5: Modified Stewart platform. 1 – motors, 2 –sliding shafts, 3 – hinges, 4 – link ball joints, 5 – moving platform, 6 – lens
Mathematical Model To solve the problem Z of automatic vibration compensation and adjustment of position of the laser beam during its move, optimal trajectories of the platform orientation changing were chosen. FigX Y ure 6 shows the calculated platform moveFigure 6: Platform movement range ment range. A platform mock-up was constructed to match optimal moving joints parameters and couplings with positioning elements (Figure 7).
Figure 12: Direction cosines
Euler’s unitary vector is eq. (4): θex θx ~ Θ= , = Θ~e = θ ey θy
(4)
where ex, ey is direction cosines of the unitary vector ~e in moving frame (OZ axis excluded, due to no rotation around it). Rotation vector is eq. (5): θ θx 2ex tan θ/2 Θ= = 2~e tan = θy 2ey tan θ/2 2
(5)
Quaternion with zero component is eq. (6): θ θ ~ ~ ~ Λ = Λ0 + Λ = λ0 + λ1i + λ2j = cos + ~e sin , 2 2 and quaternion scalar part is eq. (7) θ Λ0 = λ0 = cos , 2 quaternion vectorial part is eq. (8)
(6)
(7)
~Λ = λ1~i + λ2~j = ~e sin θ , (8) 2 where ~i, ~j are unit vectors of the moving frame. Rodriguez parameters is eq. (9) ˙ 2λ0 = −λ1ωx − λ2ωy 2λ˙ 1 = λ0ωx ˙ 2λ2 = λ0ωy
(9)
For Rodriguez–Hamilton parameters and Euler’s unitary vector components orthogonal transformation matrix is as follows (10): " 2 # 2 2 λ0 + λ1 − λ2 2λ1λ2 x,y Ci,j = 2λ1λ2 λ20 + λ22 − λ21
Figure 7: Platform mock-up
A Proposed Mechanism
S
1-2
4-3
Introduction
q q A2−(a+S/2−S/2 cos γ)2+ B 2−(a+S/2−S/2 cos γ)2 H= 2
All the parameters of the mock-up model are listed by Table 1 and showed in Figure 8. Geometrically two opposite drives can be presented as shown in Figure 9. Correction will be as follows in Figure 10. H Hmax = 78.5
Parameter
Value
Hmax 100.5 mm Hmin 80.5 mm Max steps 6500 Correction 22 mm Steps per mm 365.17 Resolution 2.7 µm Table 1: Mock-up’s parameters
A -10,5° max
X’ Hmin = 58.5 +11,4° max
Z +10,1° max
-11,4° max
Y’
(10)
Conclusions • Mathematical model of the beam stabilizing system for CNC machine is studied. • Several designs of the optical stabilizing and focusing systems were rewieved. • The optical beam stabilizing system based on a modified Stewart platform is proposed. • Stabilization system mock-up was built, and sliding shafts acceptable region was determined.
References Figure 8: Mock-up’s ranges
When S rotates by the angle (Figure 11). The equation to define correction is (1): S S C = − · cos γ, (1) 2 2 where C is correction, and S is moving platform length. Height will be defined as eq. (2):
[1] Joachim Mayer. Laser beam position control apparatus for a cnc laser equipped machine tool, patent US 6528762, published 04.03.2003. [2] David Sachs, Steven Nasiri, and Daniel Goehl. Image stabilization technology overview. [3] D. Stewart. A platform with six degrees of freedom. In Proc. Inst. Mech. Eng., volume 180, pages 371–386, 1965/1966.
