December 15, 2014 / Vol. 39, No. 24 / OPTICS LETTERS
6997
Frequency-scanning interferometry for dynamic absolute distance measurement using Kalman filter Long Tao, Zhigang Liu,* Weibo Zhang, and Yangli Zhou State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an 710049, China *Corresponding author: [email protected] Received September 4, 2014; accepted November 5, 2014; posted November 20, 2014 (Doc. ID 220936); published December 15, 2014 We propose a frequency-scanning interferometry using the Kalman filtering technique for dynamic absolute distance measurement. Frequency-scanning interferometry only uses a single tunable laser driven by a triangle waveform signal for forward and backward optical frequency scanning. The absolute distance and moving speed of a target can be estimated by the present input measurement of frequency-scanning interferometry and the previously calculated state based on the Kalman filter algorithm. This method not only compensates for movement errors in conventional frequency-scanning interferometry, but also achieves high-precision and low-complexity dynamic measurements. Experimental results of dynamic measurements under static state, vibration and one-dimensional movement are presented. © 2014 Optical Society of America OCIS codes: (120.3180) Interferometry; (120.3930) Metrological instrumentation; (120.7280) Vibration analysis; (140.3600) Lasers, tunable; (280.3400) Laser range finder. http://dx.doi.org/10.1364/OL.39.006997
Frequency-scanning interferometry (FSI) is a promising method for absolute distance measurement [1–6]. For a static target, the absolute distance is determined by counting the interference fringes produced by laser frequency scanning without a priori knowledge of the distance [3]. However, a major disadvantage of this method is that it is very sensitive to the movement of the target. A movement of the target in one optical wavelength is interpreted as a movement in a synthetic wavelength, causing the measurement error to be amplified [4]. A dual-laser scanning system has been devised to solve this problem [3,5]. The measurement errors are eliminated by averaging the two phase shifts produced by the frequency scanning of two lasers in opposite directions. Nevertheless, using two lasers will increase the complexity and cost of the measurement systems. In addition, some algorithms combining several successive measurements have been developed for single laser systems; they are mainly applied in situations involving slow movements with constant velocity [4,6]. For a practical application in an industrial environment, the ability to perform dynamic measurements with high precision and low complexity is highly desirable, while maintaining a fast system response. The Kalman filtering (KF) technique is an optimal estimation algorithm based on a linear unbiased minimalvariance estimate principle using discrete observation data with noise in real time [7]. It has been widely applied in guidance, navigation and real-time tracking [8,9]. In this report, we propose a dynamic absolute distance-measurement method by FSI employing the KF technique. The laser consecutively scans forward and backward during the measurement. Then the current absolute distance and moving speed of the target can be estimated by using the current measurement of FSI and the previously state calculated by the KF algorithm. This method not only solves the problem mentioned above, but also achieves dynamic tracking of a moving target. To explain our method in detail, we first analyze the influence of movement on FSI. For FSI with a fixed path, 0146-9592/14/246997-04$15.00/0
the phase change Δφ resulting from the optical frequency scanning from vs to ve is Δφ
4πn LΔv; c
(1)
where c is the speed of light in vacuum, n denotes the refractive index of air, L denotes the absolute distance between the two arms of the interferometer, and Δv denotes the mode-hop-free frequency scanning range Δv ve − vs . Rearranging Eq. (1), L can be determined by counting the interference fringes and measuring the frequency scanning range. If the drift of absolute distance is defined as ΔL during the frequency scanning, the total phase difference will be Δφ0
4πn L ΔLve − Lvs : c
(2)
Assuming ve ∼ vs v~ , we can obtain the measured length LM by LM
v~ cΔφ0 L ΔL: 4πnΔv jΔvj
(3)
Notice that the second term ΔL~v∕jΔvj in Eq. (3) denotes the movement error, where the plus-minus sign is related to the direction of the frequency scanning (+ for a forward and − for a backward frequency scan). As shown in Eq. (3), the drift ΔL is amplified by a factor v~ ∕Δv. For a mode-hop-free scanning range of 96 GHz in a tunable laser with center wavelength λ 1064 nm, the amplification factor is approximately 2937. That is, a 5-nm drift of the target would introduce an error of 14.68 μm in the measurement. Therefore, it is very important to reduce the influence of the movement error on FSI. In this study, we establish a dynamic model for FSI and employ KF to compensate for the errors caused by the target movement. Assuming that the absolute distance L can be regarded as a continuous function of time, © 2014 Optical Society of America
6998
OPTICS LETTERS / Vol. 39, No. 24 / December 15, 2014
denoted as Lt, and is differentiable at the time t, it can be expressed by a second-order Taylor expansion as 1 Lt ΔT Lt stΔT atΔT 2 ; 2
(5)
where 2
1 A 40 0
ΔT 1 0
3 ΔT 2 ∕2 ΔT 5; 1
(6)
and wk represents the process noise. In addition, from Eq. (4), the target movement ΔL during frequency scanning can be rewritten as 1 ΔL sΔt aΔt2 ; 2
(7)
where Δt is the duration of the optical frequency scanning for one measurement. Therefore, at the time step k, the observation vector zk of the state-space dynamic model can be expressed as follows: zk Hxk vk ;
(8)
where
v~ Δt H 1 jΔvj
v~ Δt2 ; 2jΔvj
(9)
zk is a 1 × 1 vector whose element is the measured length LM defined in Eq. (3), and vk represents the measurement noise. Equations (5) and (8) constitute the state and observation equations, respectively, for the state-space dynamic model. Note that the random variables wk and vk corresponding covariances of σ 2w and σ 2v , respectively, are assumed to be independent and white, with normal probability distributions. The covariance matrices of wk and vk are 2
0 Qk 4 0 0
3 0 0 0 0 5; 2 0 σ w ∕ΔT
Rk σ 2v ∕ΔT;
(11)
P −k1 AP k AT Q;
(12)
K k1 P −k1 H T HP −k1 H T R−1 ;
(13)
xˆ k1 xˆ −k1 K k1 zk1 − H xˆ −k1 ;
(14)
P k1 I − K k1 HP −k1 ;
(15)
(4)
where ΔT is the time interval, st is the derivative of Lt with respect to time, and at denotes the second derivative of Lt. Let us assume that the state vector of the state-space dynamic model of KF is x L; s; aT . If the absolute distance of the target is measured at every ΔT, then according to Eq. (4), the true state vector xk1 at time step k 1 can be evolved from the value xk given at step k using a linear stochastic difference model xk1 Axk wk ;
xˆ −k1 Axˆ k ;
(10)
respectively. With the above state-space dynamic model, the KF time update equations and measurement update equations are developed to estimate the absolute distance and velocity of the target step-by-step with consecutive measurements, which is described by the following iterations [10]:
where xˆ k1 is the a posteriori state estimate, xˆ −k1 is the a priori state estimate; P −k1 and P k1 denote the a priori estimate error covariance and the a posteriori estimate error covariance, respectively; K k1 is the Kalman gain matrix. The KF process can be described as follows: First, the time update Eqs. (11) and (12) project the current state and covariance estimates, respectively. Once the outcome of the measurement is obtained, these estimates are updated through measurement update Eqs. (13)–(15). Because of the algorithm’s recursive nature, it can run in real time only using the present input measurements and the previously calculated state, no additional past information is required [11]. The consecutive measurements of FSI can be achieved by consecutive forward and backward optical frequency scans. By scanning in the opposite direction, the movement error caused by motion will have an opposite sign. Therefore, in the process of KF recursive calculation, the signs of the second and third elements of vector H in Eq. (9) will change according to the direction of the frequency scan. In order to reduce the system error, the time interval ΔT between measurements should be as short as possible. While ΔT is related to the performance of the tunable laser, the minimum of ΔT is limited by the maximum laser tuning rate. Therefore, the vibration frequency of the target should not be more than half of the measuring frequency 0.5∕ΔT, otherwise the results of KF will deviate from the true state. The KF is an optimal recursive data processing algorithm that minimizes the variance of the estimates. With the performing of measurement, the estimate error covariance will tend to be zero or a steady-state value, and the results of KF will be closer to the true state. So, a judgment criterion is introduced to determine whether the state estimate reaches a sufficient accuracy. The judgment criterion is based on the norm of the estimate error covariance ‖P k ‖, and ‖P k ‖ is compared with a certain threshold value ε. The judgment criterion can be expressed as ‖P k ‖ < ε.
(16)
To verify the effectiveness of the dynamic absolute distance measurement, an FSI was constructed, as shown in Fig. 1. The system utilized an external cavity diode laser (ECDL) (New Focus, TLB-7021) with a mode-hop-free scan range of 96 GHz and a center wavelength λ 1064 nm, and a high-finesse (>200)
December 15, 2014 / Vol. 39, No. 24 / OPTICS LETTERS
6999
Fig. 1. Schematic of the FSI.
Fabry–Perot (FP) etalon (Thorlabs, SA200 with FSR of 1.5 GHz) for measuring Δv by counting the resonances in the cavity of the FP etalon. The laser frequency scanning was achieved by a piezo actuator inside the laser head of the ECDL. It is well known that piezo actuators do not respond absolutely linearly to an applied voltage. Thus, a triangle waveform signal with nonlinear voltage ramps [12], of 20 Hz, was generated to drive the piezo actuator. By using this triangle waveform signal, we can realize forward and backward linear frequency scans. The signals of the FP and interference fringes were detected by photodetectors PD1 and PD2 respectively, and were sampled by a data acquisition card simultaneously at a rate of 2 MS∕s with an accuracy of 14 bits. The triangular wave signal was also collected simultaneously so as to determine the sign of H by software. The KF algorithm is initialized with the initial state vector xˆ and the initial error covariance matrix P. In this report, the first element of xˆ was set to the first measured distance of FSI. The second and third elements were set to be zero. P was set to a three-order identity matrix. The parameter σ 2v depends on the precision of the measurement instrument, so σ 2v can be determined by the variance deduced from the first 10 raw FSI measurements in static state. Since we cannot obtain the accurate value of σ 2w directly, the determination of σ 2w is quite subjective. The numerical simulations of KF aiming at different test environments can be carried out to determine the reasonable value of σ 2w according to the simulation results. In order to evaluate the performance of the dynamic absolute distance measurements in static state, the target RR2 was fixed with an absolute distance of approximately 660 mm. Figure 2 shows the measurement results of FSI with and without KF. During the measurements, the target cannot maintain a completely static state, and the measured distance was always influenced by the vibration and noise from environment. In Fig. 2, the measurement results of FSI, marked in red, fluctuates obviously. The fluctuation is mainly caused by the movement errors and noise. The continuous line corresponds to the measured absolute distances based on KF. As can be seen in Fig. 2, the results of KF have obviously large errors at the beginning of filtering. However, with the dynamic measurements, the measured absolute distance curve converges quickly from the initial value of 660.400 mm to the final value. Notice that there are still optical-path difference (OPD) drifts in static state, which are mainly caused by air turbulence and temperature fluctuation. When the estimate error covariance reaches
Fig. 2. Performance of FSI with and without Kalman filter as the target being in a static state.
the judgment criterion, the subsequent measurements have a standard deviation (SD) of 0.23 μm. However, for the traditional FSI, this value is 3.12 μm. It clearly demonstrates the working of the KF. In addition, the performance of dynamic absolute distance measurement in different positions, ranging from 27 to 62 cm, was tested under on-site environment. Two hundred sequential scans were performed in each position. The measurement results are listed in Table 1. The SDs of the remaining measured distances are presented when the estimate error covariance reaches the judgment criterion. In Table 1, the measured SD of FSI with KF are significantly lower than that without KF. Note that, in positions 3, 5 and 6, the SDs of FSI without KF are significantly larger than in other positions. It was because the measurements were influenced by the air disturbance. To test the vibration measurement performance, a piezoelectric stage (Thorlabs, NF15AP25 with a resolution of 25 nm) was employed to produce a sinusoidal vibration to the target RR2 with an amplitude of 4.5 μm and a period of 0.5 s. Figure 3 shows the vibration measurement results. The estimates of absolute distance converge to a sinusoidal movement after 1 s. The measured and expected vibrations agree very well, within the 0.25%–5.86% for amplitude, and 0.004%–0.014% for frequency. We also tested the dynamic tracking measurement of one-dimensional movement. The target RR2 was fixed on a linear stage (Newport, M-IMS400PP with a minimum incremental motion 1.25 μm). Furthermore, the laser beam falling on the target was adjusted parallel to the rails, so that the one-dimensional movement of the target could be performed with the movement of the stage.
Table 1. Distance Measurement Precision in Different Locations with and without Kalman Filter FSI
With Kalman Filter
Position
Mean (mm)
SD (μm)
Mean (mm)
SD (μm)
1 2 3 4 5 6
270.396630 340.348829 410.293685 480.221637 550.166741 620.101830
5.195 8.225 13.334 8.928 13.769 10.325
270.396369 340.348482 410.293487 480.220752 550.165543 620.101193
0.271 0.280 0.365 0.231 0.591 0.489
7000
OPTICS LETTERS / Vol. 39, No. 24 / December 15, 2014
Fig. 3. Performance of dynamic absolute distance measurements in sinusoidal vibration.
Fig. 5.
Measured speed errors spread versus time.
In conclusion, a dynamic FSI technique is proposed in this report. We demonstrated the potential of Kalman filter in FSI for dynamic absolute distance measurement, and its high precision under different practical conditions. Using the dynamic model of KF, the movement error in traditional FSI can be effectively compensated. Moreover, only one ECDL laser is needed in this method, greatly simplifying the complexity of the structure and reducing the costs compared with the dual-laser scanning system. The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (Grant No. 51375376).
Fig. 4. Dynamic tracking measurement test with measured absolute distance (a) and speed (b).
Figure 4 shows an example of the measured results for the tracking movement of the target. The moving speed of the linear stage was set to be 1 mm∕s in the direction of OPD reduction. Figure 4(a) represents the measured absolute distance and Fig. 4(b) represents the measured movement speed. As can be seen in Fig. 4(b), the speed falls from 0 mm∕s to −1 mm∕s when the target starts to move, then fluctuates round −1 mm∕s with the measurements. The curves indicate that the performance of the dynamic tracking is good. Figure 5 shows the measured speed errors by dynamic measurement method. The errors decrease with successive measurements obviously. The upper and lower deviations are 0.05 mm∕s, respectively.
References 1. J. A. Stone, A. Stejskal, and L. Howard, Appl. Opt. 38, 5981 (1999). 2. P. A. Coe, D. F. Howell, and R. B. Nickerson, Meas. Sci. Technol. 15, 2175(2004). 3. H. J. Yang, S. Nyberg, and K. Riles, Nucl. Instrum. Meth. Phys. Res. Sec. A 575, 395 (2007). 4. B. L. Swinkels, N. Bhattacharya, and J. J. M. Braat, Opt. Lett. 30, 2242 (2005). 5. S. Kakuma and Y. Katase, Opt. Rev. 19, 376 (2012). 6. A. Cabral and J. Rebordao, Opt. Eng. 46, 073602 (2007). 7. C. K. Chui and G. Chen, Kalman Filtering with Real-Time Applications (Springer-Verlag, 1987). 8. M. Cen and D. Luo, Appl. Opt. 49, 5384 (2010). 9. T. E. Zander, V. Madyastha, A. Patil, P. Rastogi, and L. M. Reindl, Opt. Lett. 34, 1396 (2009). 10. G. Welch and G. Bishop, http://clubs.ens‑cachan.fr/krobot/ old/data/positionnement/kalman.pdf. 11. T. N. Mamidwar, N. P. Bandal, and K. U. Kanhere, Int. J. Adv. Res. Comput. Sci. Manage. Studies 2, 64 (2014). 12. T. Führer and T. Walther, Opt. Lett. 33, 372 (2008).
6997
Frequency-scanning interferometry for dynamic absolute distance measurement using Kalman filter Long Tao, Zhigang Liu,* Weibo Zhang, and Yangli Zhou State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an 710049, China *Corresponding author: [email protected] Received September 4, 2014; accepted November 5, 2014; posted November 20, 2014 (Doc. ID 220936); published December 15, 2014 We propose a frequency-scanning interferometry using the Kalman filtering technique for dynamic absolute distance measurement. Frequency-scanning interferometry only uses a single tunable laser driven by a triangle waveform signal for forward and backward optical frequency scanning. The absolute distance and moving speed of a target can be estimated by the present input measurement of frequency-scanning interferometry and the previously calculated state based on the Kalman filter algorithm. This method not only compensates for movement errors in conventional frequency-scanning interferometry, but also achieves high-precision and low-complexity dynamic measurements. Experimental results of dynamic measurements under static state, vibration and one-dimensional movement are presented. © 2014 Optical Society of America OCIS codes: (120.3180) Interferometry; (120.3930) Metrological instrumentation; (120.7280) Vibration analysis; (140.3600) Lasers, tunable; (280.3400) Laser range finder. http://dx.doi.org/10.1364/OL.39.006997
Frequency-scanning interferometry (FSI) is a promising method for absolute distance measurement [1–6]. For a static target, the absolute distance is determined by counting the interference fringes produced by laser frequency scanning without a priori knowledge of the distance [3]. However, a major disadvantage of this method is that it is very sensitive to the movement of the target. A movement of the target in one optical wavelength is interpreted as a movement in a synthetic wavelength, causing the measurement error to be amplified [4]. A dual-laser scanning system has been devised to solve this problem [3,5]. The measurement errors are eliminated by averaging the two phase shifts produced by the frequency scanning of two lasers in opposite directions. Nevertheless, using two lasers will increase the complexity and cost of the measurement systems. In addition, some algorithms combining several successive measurements have been developed for single laser systems; they are mainly applied in situations involving slow movements with constant velocity [4,6]. For a practical application in an industrial environment, the ability to perform dynamic measurements with high precision and low complexity is highly desirable, while maintaining a fast system response. The Kalman filtering (KF) technique is an optimal estimation algorithm based on a linear unbiased minimalvariance estimate principle using discrete observation data with noise in real time [7]. It has been widely applied in guidance, navigation and real-time tracking [8,9]. In this report, we propose a dynamic absolute distance-measurement method by FSI employing the KF technique. The laser consecutively scans forward and backward during the measurement. Then the current absolute distance and moving speed of the target can be estimated by using the current measurement of FSI and the previously state calculated by the KF algorithm. This method not only solves the problem mentioned above, but also achieves dynamic tracking of a moving target. To explain our method in detail, we first analyze the influence of movement on FSI. For FSI with a fixed path, 0146-9592/14/246997-04$15.00/0
the phase change Δφ resulting from the optical frequency scanning from vs to ve is Δφ
4πn LΔv; c
(1)
where c is the speed of light in vacuum, n denotes the refractive index of air, L denotes the absolute distance between the two arms of the interferometer, and Δv denotes the mode-hop-free frequency scanning range Δv ve − vs . Rearranging Eq. (1), L can be determined by counting the interference fringes and measuring the frequency scanning range. If the drift of absolute distance is defined as ΔL during the frequency scanning, the total phase difference will be Δφ0
4πn L ΔLve − Lvs : c
(2)
Assuming ve ∼ vs v~ , we can obtain the measured length LM by LM
v~ cΔφ0 L ΔL: 4πnΔv jΔvj
(3)
Notice that the second term ΔL~v∕jΔvj in Eq. (3) denotes the movement error, where the plus-minus sign is related to the direction of the frequency scanning (+ for a forward and − for a backward frequency scan). As shown in Eq. (3), the drift ΔL is amplified by a factor v~ ∕Δv. For a mode-hop-free scanning range of 96 GHz in a tunable laser with center wavelength λ 1064 nm, the amplification factor is approximately 2937. That is, a 5-nm drift of the target would introduce an error of 14.68 μm in the measurement. Therefore, it is very important to reduce the influence of the movement error on FSI. In this study, we establish a dynamic model for FSI and employ KF to compensate for the errors caused by the target movement. Assuming that the absolute distance L can be regarded as a continuous function of time, © 2014 Optical Society of America
6998
OPTICS LETTERS / Vol. 39, No. 24 / December 15, 2014
denoted as Lt, and is differentiable at the time t, it can be expressed by a second-order Taylor expansion as 1 Lt ΔT Lt stΔT atΔT 2 ; 2
(5)
where 2
1 A 40 0
ΔT 1 0
3 ΔT 2 ∕2 ΔT 5; 1
(6)
and wk represents the process noise. In addition, from Eq. (4), the target movement ΔL during frequency scanning can be rewritten as 1 ΔL sΔt aΔt2 ; 2
(7)
where Δt is the duration of the optical frequency scanning for one measurement. Therefore, at the time step k, the observation vector zk of the state-space dynamic model can be expressed as follows: zk Hxk vk ;
(8)
where
v~ Δt H 1 jΔvj
v~ Δt2 ; 2jΔvj
(9)
zk is a 1 × 1 vector whose element is the measured length LM defined in Eq. (3), and vk represents the measurement noise. Equations (5) and (8) constitute the state and observation equations, respectively, for the state-space dynamic model. Note that the random variables wk and vk corresponding covariances of σ 2w and σ 2v , respectively, are assumed to be independent and white, with normal probability distributions. The covariance matrices of wk and vk are 2
0 Qk 4 0 0
3 0 0 0 0 5; 2 0 σ w ∕ΔT
Rk σ 2v ∕ΔT;
(11)
P −k1 AP k AT Q;
(12)
K k1 P −k1 H T HP −k1 H T R−1 ;
(13)
xˆ k1 xˆ −k1 K k1 zk1 − H xˆ −k1 ;
(14)
P k1 I − K k1 HP −k1 ;
(15)
(4)
where ΔT is the time interval, st is the derivative of Lt with respect to time, and at denotes the second derivative of Lt. Let us assume that the state vector of the state-space dynamic model of KF is x L; s; aT . If the absolute distance of the target is measured at every ΔT, then according to Eq. (4), the true state vector xk1 at time step k 1 can be evolved from the value xk given at step k using a linear stochastic difference model xk1 Axk wk ;
xˆ −k1 Axˆ k ;
(10)
respectively. With the above state-space dynamic model, the KF time update equations and measurement update equations are developed to estimate the absolute distance and velocity of the target step-by-step with consecutive measurements, which is described by the following iterations [10]:
where xˆ k1 is the a posteriori state estimate, xˆ −k1 is the a priori state estimate; P −k1 and P k1 denote the a priori estimate error covariance and the a posteriori estimate error covariance, respectively; K k1 is the Kalman gain matrix. The KF process can be described as follows: First, the time update Eqs. (11) and (12) project the current state and covariance estimates, respectively. Once the outcome of the measurement is obtained, these estimates are updated through measurement update Eqs. (13)–(15). Because of the algorithm’s recursive nature, it can run in real time only using the present input measurements and the previously calculated state, no additional past information is required [11]. The consecutive measurements of FSI can be achieved by consecutive forward and backward optical frequency scans. By scanning in the opposite direction, the movement error caused by motion will have an opposite sign. Therefore, in the process of KF recursive calculation, the signs of the second and third elements of vector H in Eq. (9) will change according to the direction of the frequency scan. In order to reduce the system error, the time interval ΔT between measurements should be as short as possible. While ΔT is related to the performance of the tunable laser, the minimum of ΔT is limited by the maximum laser tuning rate. Therefore, the vibration frequency of the target should not be more than half of the measuring frequency 0.5∕ΔT, otherwise the results of KF will deviate from the true state. The KF is an optimal recursive data processing algorithm that minimizes the variance of the estimates. With the performing of measurement, the estimate error covariance will tend to be zero or a steady-state value, and the results of KF will be closer to the true state. So, a judgment criterion is introduced to determine whether the state estimate reaches a sufficient accuracy. The judgment criterion is based on the norm of the estimate error covariance ‖P k ‖, and ‖P k ‖ is compared with a certain threshold value ε. The judgment criterion can be expressed as ‖P k ‖ < ε.
(16)
To verify the effectiveness of the dynamic absolute distance measurement, an FSI was constructed, as shown in Fig. 1. The system utilized an external cavity diode laser (ECDL) (New Focus, TLB-7021) with a mode-hop-free scan range of 96 GHz and a center wavelength λ 1064 nm, and a high-finesse (>200)
December 15, 2014 / Vol. 39, No. 24 / OPTICS LETTERS
6999
Fig. 1. Schematic of the FSI.
Fabry–Perot (FP) etalon (Thorlabs, SA200 with FSR of 1.5 GHz) for measuring Δv by counting the resonances in the cavity of the FP etalon. The laser frequency scanning was achieved by a piezo actuator inside the laser head of the ECDL. It is well known that piezo actuators do not respond absolutely linearly to an applied voltage. Thus, a triangle waveform signal with nonlinear voltage ramps [12], of 20 Hz, was generated to drive the piezo actuator. By using this triangle waveform signal, we can realize forward and backward linear frequency scans. The signals of the FP and interference fringes were detected by photodetectors PD1 and PD2 respectively, and were sampled by a data acquisition card simultaneously at a rate of 2 MS∕s with an accuracy of 14 bits. The triangular wave signal was also collected simultaneously so as to determine the sign of H by software. The KF algorithm is initialized with the initial state vector xˆ and the initial error covariance matrix P. In this report, the first element of xˆ was set to the first measured distance of FSI. The second and third elements were set to be zero. P was set to a three-order identity matrix. The parameter σ 2v depends on the precision of the measurement instrument, so σ 2v can be determined by the variance deduced from the first 10 raw FSI measurements in static state. Since we cannot obtain the accurate value of σ 2w directly, the determination of σ 2w is quite subjective. The numerical simulations of KF aiming at different test environments can be carried out to determine the reasonable value of σ 2w according to the simulation results. In order to evaluate the performance of the dynamic absolute distance measurements in static state, the target RR2 was fixed with an absolute distance of approximately 660 mm. Figure 2 shows the measurement results of FSI with and without KF. During the measurements, the target cannot maintain a completely static state, and the measured distance was always influenced by the vibration and noise from environment. In Fig. 2, the measurement results of FSI, marked in red, fluctuates obviously. The fluctuation is mainly caused by the movement errors and noise. The continuous line corresponds to the measured absolute distances based on KF. As can be seen in Fig. 2, the results of KF have obviously large errors at the beginning of filtering. However, with the dynamic measurements, the measured absolute distance curve converges quickly from the initial value of 660.400 mm to the final value. Notice that there are still optical-path difference (OPD) drifts in static state, which are mainly caused by air turbulence and temperature fluctuation. When the estimate error covariance reaches
Fig. 2. Performance of FSI with and without Kalman filter as the target being in a static state.
the judgment criterion, the subsequent measurements have a standard deviation (SD) of 0.23 μm. However, for the traditional FSI, this value is 3.12 μm. It clearly demonstrates the working of the KF. In addition, the performance of dynamic absolute distance measurement in different positions, ranging from 27 to 62 cm, was tested under on-site environment. Two hundred sequential scans were performed in each position. The measurement results are listed in Table 1. The SDs of the remaining measured distances are presented when the estimate error covariance reaches the judgment criterion. In Table 1, the measured SD of FSI with KF are significantly lower than that without KF. Note that, in positions 3, 5 and 6, the SDs of FSI without KF are significantly larger than in other positions. It was because the measurements were influenced by the air disturbance. To test the vibration measurement performance, a piezoelectric stage (Thorlabs, NF15AP25 with a resolution of 25 nm) was employed to produce a sinusoidal vibration to the target RR2 with an amplitude of 4.5 μm and a period of 0.5 s. Figure 3 shows the vibration measurement results. The estimates of absolute distance converge to a sinusoidal movement after 1 s. The measured and expected vibrations agree very well, within the 0.25%–5.86% for amplitude, and 0.004%–0.014% for frequency. We also tested the dynamic tracking measurement of one-dimensional movement. The target RR2 was fixed on a linear stage (Newport, M-IMS400PP with a minimum incremental motion 1.25 μm). Furthermore, the laser beam falling on the target was adjusted parallel to the rails, so that the one-dimensional movement of the target could be performed with the movement of the stage.
Table 1. Distance Measurement Precision in Different Locations with and without Kalman Filter FSI
With Kalman Filter
Position
Mean (mm)
SD (μm)
Mean (mm)
SD (μm)
1 2 3 4 5 6
270.396630 340.348829 410.293685 480.221637 550.166741 620.101830
5.195 8.225 13.334 8.928 13.769 10.325
270.396369 340.348482 410.293487 480.220752 550.165543 620.101193
0.271 0.280 0.365 0.231 0.591 0.489
7000
OPTICS LETTERS / Vol. 39, No. 24 / December 15, 2014
Fig. 3. Performance of dynamic absolute distance measurements in sinusoidal vibration.
Fig. 5.
Measured speed errors spread versus time.
In conclusion, a dynamic FSI technique is proposed in this report. We demonstrated the potential of Kalman filter in FSI for dynamic absolute distance measurement, and its high precision under different practical conditions. Using the dynamic model of KF, the movement error in traditional FSI can be effectively compensated. Moreover, only one ECDL laser is needed in this method, greatly simplifying the complexity of the structure and reducing the costs compared with the dual-laser scanning system. The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (Grant No. 51375376).
Fig. 4. Dynamic tracking measurement test with measured absolute distance (a) and speed (b).
Figure 4 shows an example of the measured results for the tracking movement of the target. The moving speed of the linear stage was set to be 1 mm∕s in the direction of OPD reduction. Figure 4(a) represents the measured absolute distance and Fig. 4(b) represents the measured movement speed. As can be seen in Fig. 4(b), the speed falls from 0 mm∕s to −1 mm∕s when the target starts to move, then fluctuates round −1 mm∕s with the measurements. The curves indicate that the performance of the dynamic tracking is good. Figure 5 shows the measured speed errors by dynamic measurement method. The errors decrease with successive measurements obviously. The upper and lower deviations are 0.05 mm∕s, respectively.
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