Real-time precision displacement measurement interferometer using the robust discrete time Kalman filter

Available online at www.sciencedirect.com

Optics & Laser Technology 37 (2005) 229 – 234 www.elsevier.com/locate/optlastec

Real-time precision displacement measurement interferometer using the robust discrete time Kalman )lter Tong-Jin Parka , Hyeun-Seok Choia , Chang-Soo Hanb;∗ , Yong-Woo Leec a Department

of Precision Mechanical Engineering, Hanyang University, 17 Haengdang-dong, Seongdong-gu, Seoul 133-791, Republic of Korea of Mechanical and Information Management Engineering, Hanyang University, Sa-1 dong 1271, Sangnok-gu, Aansan-city, Gyeonggi-province 425-791, Republic of Korea c LG Production Engineering Research Center, LG-PRC, 19-1 Cheongho-ri, Jinwuy-myun, Pyoungtaek-city, Gyeonggi-province 451-713, Republic of Korea

b Division

Received 1 December 2003; received in revised form 1 March 2004; accepted 24 March 2004

Abstract This paper proposes a robust discrete time Kalman )lter (RDKF) for the dynamic compensation of nonlinearity in a homodyne laser interferometer for high-precision displacement measurement and in real time. The interferometer system is modelled to reduce the calculation of the estimator. A regulator is applied to improve the robustness of the system. An estimator based on dynamic modelling and a zero regulator of the system was designed by the authors of this study. For real measurement, the experimental results show that the proposed interferometer system can be applied to high-precision displacement measurement in real time. ? 2004 Elsevier Ltd. All rights reserved. Keywords: Displacement measurement interferometer; Real-time estimation; Discrete time Kalman )lter

1. Introduction As nanotechnology has developed rapidly in recent years, measurement systems have begun to require sub-nanometer resolution. The laser interferometers are used for the precision displacement measurement o=ering both a wide measuring range from below 1 mm to a few 10 m and almost unlimited resolution. For this reason, displacement measurement interferometers (DMI) are used extensively in precision measuring and positioning equipment. System measurement accuracy, however, is mainly limited both by nonlinearity from polarization mixing and by error from the installing environment of the interferometer. In general, nonlinear measurement error factors include unequal gains and zero o=sets of the detectors, phase-quadrature error, alignment error and polarization mixing in the optical system [1– 4,10]. The error factors originating in the environment include, but are not limited to, vibration, temperature variation, and air turbulence [1]. Generally, the error from the nonlinearity is more than some 10 nm in the case of the heterodyne ∗ Corresponding author. Tel.: +82-31-400-5247; fax: +82-31-4066242. E-mail address: [email protected] (C.-S. Han).

0030-3992/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2004.03.009

interferometer. In the case of the homodyne, the error is less than any nanometer when it is well aligned [2,5]. This nonlinearity was )rst described for heterodyne systems by Bobro= [1]. In addition, the elliptical )tting method by the least-squares algorithm has been investigated using a single wavelength in the homodyne interferometer [5–9]. The elliptical )tting method requires many time-consuming procedures, so this method cannot be used for real time measurement [5]. Kim et al. [3] studied the least-squares method applied to the accelerated phase-measuring algorithm. Li et al. [4] proposed the correction of nonlinearity in a single-frequency interferometer based on a neural network model. The neural network modelling method, however, is more suitable for o=-line work due to its computational complexity. Eom et al. [5] suggest an automatic system for real time correction in the homodyne laser interferometer. This system consists of an electronic circuit for automatically optimizing the phase-quadrature signals and two commercial data acquisition boards. As the studies in [3–5] focus on the nonlinearity of the laser interferometer, real errors such as light source, vibration, temperature and others, should be considered in real-time measurement. This paper proposes that precision displacement measurement with nanometer

230

T.-J. Park et al. / Optics & Laser Technology 37 (2005) 229 – 234

accuracy reduced errors. Also, it presents the design of a real-time estimator. The Kalman )lter is the representative estimator for existing errors. Zhu and Soh [11] investigated correction and estimation of the system regarding nonlinearities as disturbances. Waller and Saxen [12] proved that the estimation cost is lower based on the more precise system modelling. The goal of this paper is to improve the accuracy of the laser interferometer with nanometer accuracy applied to robust estimator in real time. In particular, we are modelling a laser interferometer. The errors of the system divide the disturbance by nonlinearity, in the laser source as well as by vibration and temperature. Applied to the robust Kalman )lter, the error in the system minimized, in this case we were able to demonstrate high-precision DMI. Section 2 describes the dynamic modelling of the system, while Section 3 describes the design for the estimator in real time. Section 4 describes performance validation of the estimator and Section 5 describes experiment results and consideration on real measurement. Finally, Section 6 presents the conclusions of this paper.

which is )xed on a Lexure microstage being actuated by a PZT. Both beams from two retro reLectors travel back and have interference at the polarizing beam splitter. The non-polarizing beam splitter divides the beam into two nearly equal components. One of these components is sent directly to a polarizing beam splitter, which produces a pair of interference patterns in phase opposition. The other component is made to pass through a quarter-wave plate at 45◦ , so that an extra =4 delay is added to the optical path di=erence. The )nal polarizing beam splitter then produces a pair of patterns in phase opposition, but with =2 phase lag with respect to the previous ones. Overall, four quadrature signals are available at the photo detectors. The interference signals detected by four photo detectors have the phases of 0◦ , 90◦ , 180◦ and 270◦ , respectively. The system modelling is derived from the system signal output. The laser interferometer, according to variation of the beam intensity, can be regarded as sinusoidal functions representing a relationship between input and output of the signal. In an ideal case, assuming no system modelling error and no sensor noises, Eqs. (1) and (2) are ideal modelling equations for an interferometer.

2. Laser interferometer modelling

Ix = I1 − I2 = cos ;

(1)

As shown in Fig. 1, the system consists of both the high-precision DMI and the signal detected by four photo detectors. A vertical polarized He–Ne laser at the wavelength 0:6329 m is used as a beam source. In the path of the laser beam, a half-wave plate )rst rotates the polarization from vertical to approximately 45◦ , so that the polarizing beam splitter that follows can split the beam into two nearly equal components, one directed to the reference retroreLector and the other to the object retroreLector,

Iy = I3 − I4 = sin ;

(2)

where  = !t, ! is a phase frequency and t is the time. Two signals Ix , Iy with a phase di=erence of 90◦ are obtained by subtracting the signal with phase of 180◦ (I2 ) from the signal of 0◦ (I1 ) and the signal with phase 270◦ (I3 ) from the signal of 90◦ (I4 ) [4,5]. In this paper, the PD signals are modelled as I1 , I2 signals are half of the amplitude of Ix and I3 , I4 signals are half of the amplitude Iy . Mathematical modelling equations of I1 , I2 , I3 and I4 using Laplace

Fig. 1. Schematic diagram of the real-time displacement measurement interferometer.

T.-J. Park et al. / Optics & Laser Technology 37 (2005) 229 – 234

231

Table 1 Value of the system and measurement noise

Error value

Error factor

h; k a; b x,

Imperfection of the electronic circuit Laser power drift Polarization mixing e=ect, laser-beam alignment

y

Eqs. (8) and (9) are modelling equations considering the system modelling errors and sensor noises.

Fig. 2. Signal output of the laser interferometer.

transform are as follows y1 (s) s=2 I1 (s) = ; = u1 (s) s2 + !2

(3) (4)

!=2 y3 (s) = 2 I3 (s) = ; u3 (s) s + !2

(5)

−!=2 y4 (s) = 2 I4 (s) = : u4 (s) s + !2

(6)

i = 1–4;

where A1 = A 2 = A 3 = A 4 =

B1 = B2 = B3 = B4 =

Iy
= Iy0 + b cos( − );

(9)

where Ix0 and Iy0 are DC o=sets. a and b are amplitude of the interference fringe, is the phase-quadrature error of the DMI. Table 1 shows the error modelling factors of the laser interferometer system according to error values in Eqs. (6) and (7). Using Laplace transform, Eqs. (8) and (9) can be converted as follows: as Ix
(s) = 2 + Ix0 (s); (10) s + (! − )2 b(! − ) + Iy0 (s): s2 + (! − )2

Eq. (12) is the state–space equation for Eqs. (10) and (11) x˙i = (Ai + QAi )xi + Bi ui + wi ;

!2

−1
 1

0

 ;

; 0     C1 = 1=2 0 ; C2 = −1=2 0 ;     C3 = 0 !=2 ; C4 = 0 −!=2 : Real signal modelling can be regarded with adding system modelling error and sensor noises. Fig. 2 shows the signal output from the laser interferometer as stated in Fig. 1.




A1 = A2 = A3 = A4 =
QA1 = QA2 =

0

0

0

0




(7)

0

(12)

where

QA3 = QA4 =




(11)

yi = Ci xi + Di ui ;

Input signals, u1 , u2 , u3 and u4 are the power of laser and output signals, y1 , y2 , y3 and y4 are laser-phase measurements with voltage dimensions. The state–space equation for the laser interferometer system is as follows: yi = Ci xi ;

(8)

Iy
(s) =

y2 (s) −s=2 I2 (s) = ; = 2 u2 (s) s + !2

x˙i = Ai xi + Bi ui ;

Ix
= Ix0 + a cos( − );

0

0

−1

!2 

0

 ;

; 0



; − 2! 0
 1 B1 = B 2 = B 3 = B 4 = ; 0  

0 0 ; w2 = ; w1 = Ix01 Ix02  

0 0 w3 = ; w4 = ; Iy01 Iy02     C1 = a=2 0 ; C2 = −a=2 0 ;     C3 = 0 b!=2 ; C4 = 0 −b!=2 ; 2

232

T.-J. Park et al. / Optics & Laser Technology 37 (2005) 229 – 234

 D1 = 0  D3 = 0

 0 ;

  D2 = 0 0 ;    −a =2 ; D4 = 0 b =2 :

The PD sensor noises of I1 , I2 , I3 , and I4 are independent. In addition, the interaction between the laser interferometer system and the electrical noise of the PD sensors does not exist. Namely, noises in the laser interferometer can be regarded as uncorrelated. Assuming, therefore, that the average of DC o=set errors are zero, wi (t) and Di (t) can be modelled uncorrelated zero-mean Gaussian noise. The covariance of o=set error can be expressed as  Q(t) if k = s; (13) E{wi; k · wi; s } = 0 otherwise: The variance of measurement noise can be represented as follows:  R(t) if k = s; (14) E{Di; k · Di; s } = 0 otherwise: 3. Robust discrete time Kalman !lter design The Kalman )lter is designed for estimating the system parameters based on the system modelling as described in Section 2. Fig. 3 shows the block diagram for designing the robust Kalman )lter. As the laser interferometer system is separated the reference system, Ai , and the system modelling errors, QAi in Eq. (12), the parameters of the laser interferometer system can be estimated faster than the modelling of the black box system [13]. Real-time measurement in the laser interferometer system is constructed by a fast estimator. If the system modelling errors QAi are at least bounded, and the system model is time invariant and positive de)nite, the system parameters can be estimated [13]. In addition, if the system model is attractive, the system estimator has a robustness to estimate system parameters even though the time-varying system [13]. A )lter is applied to the system which is

Kalman Estimator Real interferometer system Reference signal + ur Σ +

h, k + +

Estimated parameters

Σ

P

+

Σ

y

Regulator + H

Σ

−

xi; k = Ai; k−1 xi; k−1 + Bi; k−1 ui; k−1 + wi; k−1 ; zi; k = Ci; k xi; k + Di; k ui; k :

(16)

The estimator equation applied to laser interferometer is as follows: State estimate observational update, xˆi xˆi; k = xˆi; k−1 + KS i; k [zi; k − Ci; k xˆi; k−1 ]:

(17)

Error covariance update, Pˆ i Pi; k = Pi; k−1 [I − KS i; k Ci; k ]:

(18)

Kalman gain matrix, KS k KS i; k = Pi; k−1 CTi; k [Ci; k Pi; k−1 CTi; k + Ri; k ]−1 :

(19)

The basic steps of the computational procedure for the robust discrete Kalman )lter (RDKF) are as follows: 1. Measure the system error signal, ; 2. design the )lter regulating for QA to zero; 3. acquire the average and variance of the system o=set error; 4. acquire the average and variance of the sensor noises; 5. compute Pi; k using Pi; k−1 ,
i; k−1 and Qi; k−1 ; 6. compute KS i; k using Pi; k (computed in step 1), Ci; k and Di; k ; 7. compute values of xˆ recursively using the computed values of KS i; k (from step 3), the given initial estimate xˆi; 0 and the input data zi; k . 4. RDKF performance evaluation

Ix0, Iy0 +

bounded and regulated to zero compensating robustness for QAi . The robust estimator can be designed by correcting the system error clearly. As the system frequency is 10–100 Hz, the )lter is designed for compensating in this scope. Eq. (15) is the )ltering equation for compensating the system modelling error QAi . 1 + Ts H (s) = K ; 1¡; (15) 1 + Ts where T is sampling time, K is a compensator gain. As the compensator in Eq. (15) has the e=ect of increasing the system bandwidth, the noise in high frequency does not a=ect the system. The discrete Kalman )lter is applied to the system which is compensated by the )lter in Eq. (15). Eq. (16) is the discrete modelling equation for the system in Eq. (12)

HP−1

Fig. 3. Kalman )lter signal Low based on the modelling.

The system o=set and measurement noise were acquired from many experiments in order to design robust estimator of the laser interferometer. Table 2 shows the resulting solutions of the measurement data in the real system. Through the results of many experiments, has an average of −0:26 and measurement noise has an average of 1.3. The RDKF was designed as described in Section 3 using the experimental data in Table 2. Fig. 4 shows the results

T.-J. Park et al. / Optics & Laser Technology 37 (2005) 229 – 234 Table 2 Value of the system and measurement noise

Experimental run no.

System noise

Measurement noise

10 20 30 40

−0.2564 −0.2612 −0.2831 −0.2512

1.3152 1.1354 1.4351 1.3032

Average Standard deviation

−0.2630

1.2972 0.1232

5

0.0140

x10-4

Intensity [V]

sine 0

-5

-10 0

0.5

1

1.5

2

2.5

3

Time [sec]

Fig. 4. Zero-regulator performance simulation.

0.8 a 0.6

b
x
y

Estimation value

0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 0

0.5

1

1.5 Time [sec]

2

2.5

3

Fig. 5. System output result and parameter estimation in o=-line simulation.

of the application of the )lter to the system with the )lter regulating QAi to zero. Fig. 5 presents the system output and the estimation results of system parameter applied to the zero-regulator and DTKF in o=-line. Simulation results in Figs. 4 and 5 show the availability of the RDKF. 5. Experiment and results in real time By applying the RDKF as designed in Section 3 to the real laser interferometer system, the displacement measurement system was constructed with real-time estimation.

233

Fig. 6 shows the con)guration of laser interferometer used to implement displacement measurements. The laser source is a He–Ne laser of 632:8 nm wavelength. Four PD sensor signals were connected to the oscilloscope and the DSP in order to process the RDKF. The measurement object of the retroreLector was )xed on the microstage actuated by PZT and the displacement of the object was measured by a capacitive sensor. The capacitive sensor has the resolution with a nanometer scale. The displacement data measured by the proposed laser interferometer system were compared with the data measured by the capacitive sensor. In this experiment, the moving range of the microstage was 1 mm, and the resolution of the microstage is less than 1 nm. The ramp input of the voltage is applied to the PZT driver so that the microstage moves one direction with a same velocity. The stage, having a resolution 1 nm, 0.01% position Accuracy and 500 m, maximum moving range, is actuated by a piezoelectric element (PI P-625.1CD). The other beam is reLected by a reference retroreLector. Both reLected beams are recombined in the polarizing beam splitter (PBS) again and are sent directly to a PBS to produce a pair of interference patterns. These interference patterns are received by four separated photo detectors, I1 , I2 , I3 and I4 . The output signals of photo detectors with near quadrature can be formed as a rotated ellipse-like trajectory on an oscilloscope. A rotation of one-revolution exactly corresponds to a =2 displacement and can be counted by using a counter. A Digital Signal Processor (DSP) board is used for 1 the RDKF. Sampling time is 1000 s. Fig. 7 shows the results of the DMI applied the RDKF in real time. As shown in Fig. 7, by the RDKF, the system error was considerably decreased. The measured displacement data was almost the same value measured by the microstage with the capacitive sensor. The average of displacement error was 3:24 m without the RDKF, in case of using the RDKF, the average of displacement error was 0:14 m. The resulting data without the RDKF was a=ected by the system error factor more than with RDKF. At a short distance range, the measurement error was smaller than at the long range, because the single-frequency laser interfero meter is sensitive to the optical alignment, error motion of microstage and factors of environment. The measurement error with RDKF did not increase at a long distance range. It means that the RDKF can estimate the system parameters even though the system is changed by the error factors in the real time. 6. Conclusions The development of the high-precision displacement measurement system using the robust real-time estimator was discussed in this paper. The conclusions of this paper are as follows: (1) The system of laser interferometer was modelled to reduce the calculation time of Kalman )lter.

234

T.-J. Park et al. / Optics & Laser Technology 37 (2005) 229 – 234

Real Time Display Amp & Signal processor PZT driver

Micro stage

Optical system

Laser Source

Fig. 6. Laser interferometer.

100

project of “Development of micro-optical and thermoLuidic devices with high functionality”.

×10

PZT Displacement Measured displacement without RDKF 80

References

Measured displacement with RDKF

Displacement [µm]

60

40

20

0

-20

0

1

2

3

4

5

6

7

8

9

10

Control input [V]

Fig. 7. System output result and parameter estimation in real-time experiment.

(2) The robustness of the system estimator was considered using the )lter which is regulated to zero for the system modelling error. (3) The discrete Kalman )lter was designed on the basis of the characteristics of the system error and sensor noises. (4) Applying the RDKF for the real system, the DMI with the estimator in real time was constructed. Acknowledgements This work has been sponsored by Ministry of Commerce, Industry and Energy (MOCIE) of Korea as a part of the

[1] Bobro= N. Recent advances in displacement measuring interferometry. Meas Sci Technol 1993;4:907–26. [2] Yin Chunyong, Dai Gaoliang, Chao Zhixia. Determining the residual nonlinearity of a high-precision heterodyne interferometer. Opt Eng 1999;38:1361. [3] Kim Seung-Woo, Kang Min-Gu, Han Geon-Soo. Accelerated phase-measuring algorithm of least squares for phase-shifting interferometry. Opt Eng 1997;36:3101. [4] Li Zhi, Konrad Herrmann, Frank Pohlenz. A neural network approach to correcting nonlinearity in optical interferometers. Meas Sci Technol 2003;14:376–81. [5] Eom TaeBong, Kim JongYun, Jeong Kyuwon. The dynamic compensation of nonlinearity in a homodyne laser interferometer. Meas Sci Technol 2001;12:1734–8. [6] Wu Chien-ming, Su Ching-shen, Peng Gwo-Sheng. Correction of nonlinearity in one-frequency optical interferometry. Meas Sci Technol 1996;7:520–4. [7] Wu Chien-ming, Su Ching-shen. Nonlinearity in measurements of length by optical interferometry. Meas Sci Technol 1996;7:62–8. [8] Augustyn W, Davis P. An analysis of polarization mixing errors in distance measuring interferometers. Vac Sci Technol 1990;2032–6. [9] Heydemann LM. Determination and correction of quadrature fringe measurement errors in interferometers. Appl Opt 1981;20:3382–4. [10] Dai Xiaoli, Seta Katuo. High-accuracy absolute distance measurement by means of wavelength scanning heterodyne interferometry. Meas Sci Technol 1998;9:1031–5. [11] Zhu Xing, Soh Yeng Chai, Xie Lihua. Design and analysis of discrete-time robust Kalman )lters. Automatica 2002;38:1069–77. [12] Waller M, Saxen H. Estimating the degree of time variance in a parametric model. Automatica 2000;36:619–25. [13] Astrom KJ, Wittenmark B. Adaptive control, 2nd ed. Reading, MA: Addison-Wesley; 1995.