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White-light scanning interferometry is widely used for precision metrology of engineering surfaces. It needs a mechanical scanning for capturing an interferogram that determines where the surface of a measured sample is located. The residual vibration during the scanning procedure distorts the interferogram and it reduces the accuracy and the precision of the system. The residual vibration becomes bigger as the proportional gain gets higher for the fast response. So it is hard to achieve the fast and precise measurement simultaneously. In this study, input shaping which convolves a reference signal with the input shaper is investigated to reduce the residual vibration of the scanning system. The step response data is analyzed using Continuous Wavelet Transform (CWT) to design the input shaper. Using proposed method, the residual vibration of the white light scanning interferometry is reduced and it achieved both faster measurement speed and more accurate measurement.
© 2015 Optical Society of America
1. Introduction
During past two decades White-Light Scanning Interferometry (WLSI) was widely used for precision profile metrology of engineering surfaces [1–4]. WLSI make it possible to profile surfaces that have uncertain fringe order by the fringe localization. For the localization, WLSI requires scanning for capturing the white-light interferogram which determines the location of the surface. Usually, piezoelectric transducer (PZT) moves the objective lens along the z direction for obtaining the interferogram. During the scanning procedure, the vibrations from the circumstance distort the interferogram and make phase errors [5–7]. One source of the vibration which reduces the performance of the system is scanning itself. Scanning time is required to became more faster because of the mass production issues. So, it requires high proportional feedback gain for faster scanning. However, high proportional gain induces the residual vibration to the WLSI system. It is hard to achieve the fast and precise measurement simultaneously especially for the multi modal system.
The input shaping such as Zero Vibration (ZV), Zero Vibration Derivative (ZVD), Unity Magnitude Zero Vibration Shaper (UMZV), Extra Insensitive (EI) [8–12] have been applied to robots, cranes, MEMS [12–17] for reducing residual vibration. And it is already shown that the identical input shaper can be applied to the step and ramp input [18]. In this paper, input shaping method is applied to multi modal WLSI system for reduction of residual vibrations. For designing input shaper, the scanning system is analyzed by Continuous Wavelet Transform (CWT) [19] method which is convenient to extract the natural frequencies and damping ratios. After that, the input shaper is designed on the basis of the CWT analysis result. And the step response after input shaping was compared with the conventional scanning system. Finally, to verify the performance improvement of WLSI, an image of tilted specimen is compared with conventional scanning system.
2. Experimental setup
The basic setup for the white-light scanning interferometry (WLSI) measurement system using a mirau interferometer is shown in Fig. 1 The WLSI system is consist of the PZT actuated scanner, a feedback controller, interferometric objectives and flexures. The PZT controlled by a feedback controller translates the mirau objective along the vertical axis. The camera captures the interference images. A CCD camera of 640 x 480 pixel resolutions is used to capture the specimen image for the measurement. The CCD pixel size is 7.4 by 7.4 square microns, and it makes spatial resolution as 0.148 μm per pixel. It provides satisfactory resolution for the measurement. The light source has a wavelength range of 420 nm to 700 nm. Captured images from the CCD camera are transferred to a computer for image processing to detect sub-pixel edges.
The analog feedback control is used to achieve the precise motion of the scanning. The Integral feedback gain Ki with proportional gain Kp are set together for controlling the settling time and rise time
The step responses with the high gain and the low gain were examined to analyze how the residual vibration and the response speed change as the feedback gains change. With the low gain, the response speed is so slow that it cannot achieve required scanning speed as Fig. 2(a). On the other way, the high feedback gain for the fast response induces bigger residual vibrations as Fig. 2(b). These two figures show that the WLSI system cannot achieve the fast movement and small residual vibration simultaneously. In this paper, the input shaping method is applied to overcome these limitations.3. Design of the input shaper
Input shaping is a control technique for reducing vibrations. It works by creating a command signal that cancels its own vibration. Vibration caused by the part of the first command signal is canceled by vibration caused by the part of the second command. Input shaping is performed by convolving an input shaper, a sequence of impulses with a desired command. The convolved command is then used to drive the system. If the impulses in the shaper are chosen correctly, the system will respond without vibration to the unshaped command. The amplitudes and time locations of the impulses are obtained from the system's natural frequencies and damping ratios. For combining the feedback control and the input shaper, the feedback gains that satisfy the response speed are set first and the input shaper is designed on the basis of the closed-loop dynamics. The closed-loop dynamics is analyzed by Continues Wavelet Transform (CWT) as Eq. (2) for calculating natural frequencies and damping ratios of the system [20].
where ψ is mother function, a defines the scale and b defines the time shift. It transforms the time domain function u(t) into frequency-time domain function T(b,a). CWT shows the amplitude and frequency at a local time, which are convenient for analyzing the free damped multimodal system.Figure 3(b) shows the CWT result of the step response when the Morlet function in Eq. (3) is used as a mother function
where the parameter α allows trade between time and frequency resolutions. And β is frequency of morlet function. To extract the natural frequencies and damping ratios from the CWT result, a direct search algorithm [21] is applied. Two solid lines in Fig. 4 indicate two significant vibration frequencies of the WLSI system and the damping ratios are estimated at each vibration frequency.Input shaper f(t) is composed of the amplitudes Ai and the time delays ti as Eq. (4). It can be made from natural frequencies wn and damping ratios which were extracted from CWT result. In this paper, ZV input shaper in Eq. (5), which has short time delay, is used to reduce the vibration of the WLSI system.
where,The WLSI system has two residual vibration frequencies as in Table 1. Final input shaper is made by convolving all shapers of each mode Eq. (6) [22].
Table 1. Modal parameters and parameters of input shaper
4. Result
15 µm step responses before and after input shaping were compared in Figs. 5(a)-5(b). It shows that the residual vibration is significantly reduced by applying the proposed method to the WLSI scanning system. The frequency domain analysis result also shows the reduction of the residual vibration as Fig. 5(b).
To verify the effect of the input shaping in the WLSI, a tilted surface sample is measured and leveled before and after input shaping. The applied scanning command, which is used for translating the inteferometric objective, is composed of step and ramp as Fig. 6.
Before the input shaping, the residual vibration of the WLSI scanner causes the wrinkles in the measured image as Fig. 7(a). On the contrary, applying proposed technique, we can see that the wrinkles in the measured image are reduced by 87%.
The comparison of the profile along the horizontal direction in the measured image clearly shows that the performance of SWLI is improved as Fig. 8.
5. Conclusion
This article concerns with the performance improvement of WLSI using input shaping in order to obtain fast and precise measurement simultaneously. To solve this problem, input shaping is applied to the WLSI. The CWT analysis of the step response is performed to extract the natural frequencies and the damping ratios of the scanning system for designing the input shaper. Applying the input shaping to the WLSI system, scanning system achieved the fast scanning speed and the reduction of the residual vibration at the same time. The measured image of the tilted sample shows the performance improvement distinctly. After applying the input shaping, the wrinkle caused by residual vibration is dramatically reduced. Finally the WLSI system shows better performance of fast and precise measurement by input shaping.
Acknowledgments
This work was supported by the SNUPrecision and also supported in part by the Institute of Advanced Machinery and Design at Seoul National University.
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