- Email: [email protected]
FBG sensor network for pressure localization of spacecraft structure based on distance discriminant analysis
Optik 125 (2014) 404–408
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
FBG sensor network for pressure localization of spacecraft structure based on distance discriminant analysis Jing Jin, Song Lin ∗ , Xiangyu Ye School of Instrument Science and Optic-electronics Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100191, China
a r t i c l e
i n f o
Article history: Received 18 February 2013 Accepted 30 June 2013
Keywords: Fiber Bragg grating Pressure localization Distance discriminant analysis
a b s t r a c t Structure used in aircrafts and spacecrafts often undergo high pressure loading that can produce various types of damage, thus prediction of existence and pressure position is important to damage localization. In this paper, the structure type investigated is a 50 mm thick sandwich panel with aluminum face-sheets and aluminum honeycomb core. One pressure test was performed on the panels, which was instrumented with 9 FBG sensors. The signals recorded at the various sensor locations varied with regard to Bragg wavelength shift of FBGs. Using this information and combining it with a localization algorithm based on distance discriminant analysis, the pressure location could be successfully determined. A description of the FBG sensors network and the mathematical model to determine the pressure location is provided. The pressure tests on the spacecraft structure, the response of the FBG sensors network and the analysis performed to determine the pressure location are described. © 2013 Elsevier GmbH. All rights reserved.
1. Introduction Onboard satellites, spacecraft and aircraft, a variety of sensors monitor the subsystem function and their health status. The sensors applied measure e.g. temperature, pressure, strain, etc. The acquired data are downlinked to the ground station as standard housekeeping data. Typically, the amount of data gathered is sufficient for controlling the function and the health of spacecraft subsystems. However, there are various situations when an aircraft is in flight where the availability of additional information would be desirable, especially when there is an anomaly or a subsystem failure. A variety of anomalies and subsystem failures can be caused by space debris impacts on the structure of spacecraft [1]. Meanwhile, pressure loading is a potential damaging event that may occur anytime during the service life of many spacecraft structures. High strength pressure may cause materials to sustain significant internal damage that can grow with time, leading to catastrophic failure of the spacecraft structure [2,3]. In order to detect these damaging events, information of the existence and position of a pressure loading on the structure are needed. As the position of the pressure is known with a certain precision, the inspection of the damage can be limited to this region and may reduce the complexity of the damage identification problem significantly.
∗ Corresponding author. E-mail address: [email protected] (S. Lin). 0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.06.087
Recently, fiber Bragg grating (FBG) sensors are proven to have many advantages in damage detection area [4–10], such as the high resistance to corrosion and fatigue, the wide bandwidth operation, and the immunity to electromagnetic interference. In addition, they are compatible with the most part of composite materials and concrete, safe to handle, and there is little detrimental effect on the mechanical properties of the host structure. Among fiber-optic sensors, FBG plays a dominant role due to their low costs and extreme versatility, in conjunction with a good multiplexing capability. A lot of work has been done to detect impact damage using FBG sensors [11–15], and most of the localization methods are based on the concept of time-of-flight of ultrasonic waves. Jeannot Frieden presents a method for the localization of an impact based on the arrival time of waves at the FBG sensors, which allows to predict the impact position with interpolation of a reference data set [16]. Cristobal Hiche proposed an impact localization method based on the relative placement of all sensors and the maximum strain amplitude measured by each sensor [17]. However, little work for pressure localization has been done. In this paper, 9 FBG sensors are surface mounted on a 50 mm thick sandwich panel with aluminum face-sheets and aluminum honeycomb core in different directions, and the sandwich plate is divided into 4 areas by the FBGs. Meanwhile, the distance discriminant analysis (DDA) method, which is mostly used in statistics, pattern recognition and machine learning, is utilized to determine which area pressure is loaded on. The Bragg wavelength shift values of FBG sensors are collected when pressure is loaded on different positions of the 4 areas to build the training set, and then the training data are normalized as unit vectors. A localization algorithm
J. Jin et al. / Optik 125 (2014) 404–408
405
is proposed to further prediction of the pressure position based on the concept of DDA method, and the results are validated to localize with a good accuracy.
as uj = 1j , 2j , , ij
2. Localization principle
dj = d x, uj =
2.1. Method to distinguish area
The algorithm for further localization is based on the assumption that Euclidian distance dj has a linear relationship with actual distance dj ’. In this algorithm, 3 reference points A, B and C, the Euclidian distances of which are da , db , dc are minimum among all the reference points, are picked up and predicted point I = (X,Y) can be calculated with the formula (5):
d2 (X, G) = (X − )
−1
(X − ), i = 1, 2, , k
(1)
where = E(X) = (1 , 2 , , n ) ’ and is the covariance matrix of the distribution. Consider that there are k different classes Gi , then the Mahalanobis distance between X and Gi can be expressed as Eq. (2): d2 (X, Gi ) = (X − i )
−1 i
(X − i ), i = 1, 2, , k
(2)
where i is the covariance matrix and can be estimated with Eq. (3):
ˆ
1 (xij − x¯ i )(xij − x¯ i ) , i = 1, 2, , k ni − 1 ni
i
=
(3)
i=1
The sample X can be assigned to class i if d2 (x, Gj ) = min d2 (x, Gi ). 1≤i≤k
2.2. Extrapolation of pressure position In order to localize pressure loading on a large structure, an amount of FBG sensor is needed due to the sensitivity range of the sensor, which increased the complexity of localization and the amount of data processing. In this paper, the whole structure is divided into 4 areas that are delimited by 9 FBG sensors, and the distance discriminant analysis method is utilized to determine which area the pressure is loaded on with the training set. After that, a localization algorithm is proposed for further prediction of the pressure position. Firstly, the Bragg wavelength shift value of every FBG sensor is normalized as a unit vector, the observation X = (1 , 2 , , n ) / ˙2n , where n is the Bragg wavelength shift value of the nth sensor. Then, 25 reference points are picked up in each area to load pressure, and the Bragg wavelength shift data of FBGs constructed the class Gi , namely the training set. Finally, the area where pressure is loaded can be determined by the DDA method. Assuming that the area determined previously is delimited by i sensors, the Bragg wavelength shift value of each sensor when pressure is loaded can be normalized as a unit vector x = (1 , 2 , , i ) / ˙2i . Therefore, the unit vector at refer-
ence points Ij = Xj , Yj which belong to this area can be recorded
˙2ij . Then, the Euclidian dis-
tance between x and uj can be computed with the formula (4):
Distance discriminant analysis is used to determine variables that discriminate between two or more naturally occurring groups, and the main classification rule is simply to assign an observation X to the class the mean of which is closer to x in the sense of the Mahalanobis distance. The “Mahalanobis distance” is a metric that is better adapted than the usual “Euclidian distance” to settings involving nonspherically symmetric distributions. Particularly, it is more useful when multinormal distributions are involved, although its definition does not require the distributions to be normal. Assuming that G = (X1 , X2 , , Xn ) ’ is an n elements class, and X = (x1 , x1 , , xn ) ’ is an observation, the Mahalanobis distance can be calculated with Eq. (1):
/
x − uj
(Xa − X)2 + (Ya − Y )2 = da =
x − uj
(4)
(Xb − X)2 + (Yb − Y )2 db
(Xc − X)2 + (Yc − Y )2 dc
(5)
There are two solutions O1, O2 for formula (5). Without taking account of the solutions are complex numbers, the point which is more close to the circumcenter of triangle constructed by points A, B and C is picked up as the predicted point. 3. Experiment setup The pressure localization method is validated by the static pressure loaded on a 50 mm thick sandwich panel with aluminum face-sheets and aluminum honeycomb core with a size of 1000 × 1000 mm; its surface is pretreated by sanding and cleaning to maximize adhesive bonding between the surface and the sensors are clamped on all sides by the fixture to impose a fixed–fixed boundary condition. 9 FBG strain sensors are surface mounted on the plate to acquire strain information at different positions of the plate, and a distance of 10 cm has been kept between FBG sensors and the edge of the plate. The experimental setup is shown in Fig. 1. Cyanoacrylate adhesive is used to mount the FBG sensors on the plate to minimize both air pockets and the adhesive layer thickness between the plates and the FBG sensors. In order to improve the precision of the experiment, a FBG temperature sensor is mounted on the plate to acquire the Bragg wavelength shift of FBGs induced by temperature change. Static pressure loading is provided by iron with a small and even underside. The FBG interrogation system is sm125 from Micron Optics with a working wavelength range from 1510 to 1590 nm, wavelength resolution is 1 pm, wavelength stability is 1 pm, wavelength repeatability is 0.5 pm at 1 Hz. To reduce the number of measurement channels, spectrally separated Bragg gratings were spliced in
Fig. 1. Experimental setup.
406
J. Jin et al. / Optik 125 (2014) 404–408 cm
FBG1
Reference data point
FBG5
FBG4
60 55
1
21
26
46
50
Area 1
45
Area 2
40 5
25
51
71
35
30
50
76
96
30 FBG2 25
FBG6
FBG3
20
Area 4
Area 3
15 10 5
80
75
55 FBG7
100 FBG9
FBG8
0 0
5
10
15
20
25
30
35
40
45
50
55
60 cm
Fig. 2. Reference points selected in the area and the location of FBG sensors.
chains with a distance of ∼0.3 m between the gratings. The spectral separation of the gratings in a chain was ∼10 nm with the Bragg wavelength in a range 1530 ∼ 1550 nm. 4. Result and discussion 4.1. Construction of training set The whole plate is divided into 4 areas by 9 FBGs; these areas are marked as Area 1, Area 2, Area 3 and Area 4, as is shown in Fig. 2. In order to build the training set, 100 reference points, marked as position 1 to position 100, are selected in the domain delimited by FBG sensors. Pressures are loaded on these points with strengths of 50 N, respectively. The Bragg wavelength shift values of FBG sensors are acquired by interrogation system sm125 (Fig. 3). During the experiment, pressure loading is repeated 3–5 times at each point to check the repeatability. The training sets are normalized as unit vectors, and the 100 samples are utilized in the discriminant rule to determine which area these corresponding points belong to. 4.2. Result of area classification and localization
Fig. 4. The error of exact location and predict location on the plate.
plate. The Bragg wavelength shift data is collected and processed by the interrogation system sm125 in the meantime. It turns out that all of the points are assigned to the correct area with the distance discriminant method. With the further localization algorithm, predicted position is calculated and compared to the exact position. The error of predict locations are listed in Table 1 and shown in Fig. 4. It can be seen that these verified points in each area are predicted well with a maximum error of 1.64 cm and an average error of 1.12 cm. The result turns out that the discriminant modal has a consistently successful prediction, and indicated the feasibility of the distance discriminant analysis method. Though all the points are predicted in the correct area in this experiment, there is a possibility of misjudgment when pressure is loaded on the edge of each area. Considering the situation that a point located is in Area 2, the coordinate of which is 47.5, 31, it might probably be wrongly predicted in Area 4. According to the reference data points in Area 4 and the calculations, there are two solutions of formula (5). The coordinates of solution 1 are 48.8, 23.7, and the coordinates of solution 2 are 47.4, 29.3. The predict error of the two solutions are 7.41 cm and 1.7 cm, respectively. The sketch map is shown in Fig. 5. It can be seen that the error is obviously
In order to validate the distance discriminant analysis method, 32 verified points, which are selected in different regions delimited by FBG sensors, are loaded with a static pressure of 50 N on the cm
FBG1
Exact location
FBG5
Predict location
FBG4
60 55 50
Area 2
Area 1
45 40
Exact location
35
Solution 2
30 FBG2
FBG3
25
86
20
Area 3
15
FBG6
91
Reference data point selected
Solution 1 92
Area 4
10 5
FBG7
FBG9
FBG8
0 0
Fig. 3. Wavelength shift of FBGs when pressure is loading at 25 reference points in area 1 (a) FBG1; (b) FBG2; (c) FBG3; (d) FBG4.
5
10
15
20
25
30
35
40
45
50
55
60 cm
Fig. 5. The misjudgment of predicted point on the edge of area 2.
J. Jin et al. / Optik 125 (2014) 404–408
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Table 1 Error of predict location. Number
Exact area
Predict area
Exact location (X/cm, Y/cm)
Predict location (X/cm, Y/cm)
Error (cm)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4
1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4
(22.6, 36.2) (19.4, 55.6) (14.5, 38) (5.7, 42.5) (7.3, 54.1) (17, 47.7) (13.3, 43.3) (26.5, 43.8) (41.5, 55.8) (53.5, 50.5) (38.8, 43.7) (42.3, 46.7) (47.6, 43.5) (34, 37.8) (50.9, 37) (36.6, 52) (6.3, 21.8) (17.5, 17) (21.9, 5.8) (23.6, 19.4) (13, 7.5) (17.4, 24.3) (6.2, 10.8) (12.5, 16.5) (39, 24.7) (37.2, 17.6) (44, 16.7) (7.8, 9.2) (48.4, 52.5) (52.7, 17.5) (47.1, 12.2) (52.7, 6.2)
(23, 37.5) (18.3, 55.8) (13.5, 38.4) (6.5, 41.8) (8.5, 53.5) (17.3, 47.3) (12.5, 43.5) (25.5, 42.5) (41.9, 57) (52.3, 50.7) (37.8, 44.1) (42.5, 46.2) (48.3, 43) (34.3, 36.5) (51.3, 38.2) (37, 53.1) (7.2, 22.3) (18.3, 17.2) (23, 6.5) (23.7, 18.3) (13.6, 8.6) (18.3, 23.6) (6.3, 12) (13.2, 17) (38.2, 23.7) (36.1, 18.1) (43.5, 16.2) (8.4, 8.2) (49, 53.8) (53.7, 18.2) (46.5, 11.2) (54, 6)
1.36 1.12 1.08 1.06 1.34 0.5 0.82 1.64 1.26 1.22 1.08 0.54 0.86 1.33 1.26 1.17 1.03 0.82 1.3 1.1 1.25 1.14 1.2 0.86 1.28 1.21 0.71 1.17 1.43 1.22 1.08 1.32
increasing and solution 1 with a bigger error is picked up as the predicted point as the reference points selected are all in Area 4. In order to solve this problem, more reference points are selected in the predict algorithm. Similarly, the point located in Area 2, the coordinates of which are 47.5, 31, is wrongly predicted in Area 4. Not only the reference data points in Area 4 but also the reference points in Area 2 and Area 3, which are near Area 4, are selected to calculate the location. The selection of reference points is shown in Fig. 6. Three reference points (40, 45 and 86) are calculated to further predict the position, and the coordinates of the reasonable solution are 47.3, 32, with an error of 1.02 cm.
cm
FBG1
FBG4
Exact location
FBG5
Predict location
60 55 50
Area 2
Area 1
45 40
40
35
Exact location 45
Reference data point selected
FBG6
30 FBG2
FBG3
25
Acknowledgment This research is supported by the National Natural Science Foundation of China (Grant No. 61007040).
20
Area 4
Area 3
References
10 5
Bragg wavelength of FBG sensors will shift when pressure is loaded on the structure, and the shift values of FBG sensors are applied in this paper to predict the position of pressure with the distance discriminant analysis method. The construction of the training set is important as it contributes a lot to the accuracy of the predict result. In this paper, the distance discriminant method is verified feasible with a consistent homology of prognosis for the 100 samples. During the pressure experiment on sandwich aluminum plate, 32 different points are randomly selected and all the points are predicted to the correct area. In the meantime, a maximum error of 1.64 cm and an average error of 1.12 cm are acquired with the algorithm for further localization. Regarding the situation that points are misjudged to incorrect area, more reference points are selected in the predict algorithm to solve this problem. The experimental results prove that the localization algorithm can predict pressure location successfully with a good accuracy, and error of prediction is smaller than 2 cm.
Solution 1 86 Solution 2
15
5. Conclusion
FBG7
FBG9
FBG8
0 0
5
10
15
20
25
30
35
40
45
50
55
60 cm
Fig. 6. The selection of reference points for solving misjudgment of predicted location.
[1] F. Schafer, The threat of space debris and micrometeoroids to spacecraft operations, ERCIM News 65 (April) (2006) 27–29. [2] C. Boller, Structural health monitoring in aerospace, in: Advance Course on SHM, Barcelona, Spain, 2009. [3] N. Takeda, Characterization of microscopic damage in composite laminates and real-time monitoring by embedded optical fiber sensors, Int. J. Fatigue 24 (2002) 281–289.
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[4] J.M. López-Higuera, L.R. Cobo, A.Q. Incera, A. Cobo, Fiber optic sensors in structural health monitoring, J. Lightwave Technol. 29 (4) (2011) 587–608. [5] A. Cusano, P. Capoluongo, S. Campopiano, A. Cutolo, M. Giordano, F. Felli, A. Paolozzi, M. Caponero, Experimental modal analysis of an aircraft model wing by embedded fiber Bragg grating sensors, IEEE Sensor. J. 6 (1) (2006) 67–77. [6] H.F. Lima, R. da Silva Vicente, R.N. Nogueira, I. Abe, P.S. de Brito André, C. Fernandes, H. Rodrigues, H. Varum, H.J. Kalinowski, A. Costa, J. de Lemos Pinto, Structure health monitoring of the Church of Santa Casa da Misericordia of Aveiro using FBG sensors, IEEE Sensor. J. 8 (7) (2008) 1236–1242. [7] C. Baldwin, J. Kiddy, T. Salter, P. Chen, J. Niemczuk, Fiber optic structural health monitoring system: rough sea trials of the RV Triton, in: Proceedings of Oceans Conference Record, IEEE, 2002, pp. 1807–1814. [8] N. Hirano, H. Tsutsui, J. Kimoto, T. Akatsuka, H. Sashikuma, N. Takeda, Y. Koshioka, Verification of the impact damage detection system for airframe structures using optical fiber sensors, in: Proc SPIE 7293, 72930Q, 2009. [9] S. Lin, N. Song, J. Jin, X. Wang, G. Yang, Effect of grating fabrication on radiation sensitivity of fiber Bragg gratings in gamma radiation field, IEEE Trans. Nucl. Sci. 58 (4, August) (2011). [10] J. Jing, L. Song, S. Ning-fang, Experimental investigation of pre-irradiation effect on radiation sensitivity of temperature sensing fiber Bragg grating, Chin. Phys. B 21 (6) (2012).
[11] S. Minakuchi, Y. Okabe, T. Mizutaniand, N. Takeda, Barely visible impact damage detection for composite sandwich structures by optical-fiber-based distributed strain measurement, Smart Mater. Struct. 18 (8) (2009) 085018. [12] S. Minakuchi, H. Tsukamoto, H. Banshoya, H. Takeda, Hierarchical fiberoptic-based sensing system: impact damage monitoring of large-scale CFRP structures, Smart Mater. Struct. 20 (8) (2011) 085029. [13] A.K. Mal, F.J. Shih, F. Ricci, S. Banerjee, Impact damage detection in composite structures using Lamb waves, Proc. SPIE 5768 (2005) 295–303. [14] Z. Djinovic, M. Scheererb, M. Tomicc, Impact damage detection of composite materials by fiber Bragg gratings, Proc. SPIE 8066 (2011) 80660T. [15] H. Tsutsui, A. Kawamata, J. Kimoto, A. Isoe, Y. Hirose, T. Sanda, N. Takeda, Impact damage detection system using small-diameter optical fiber sensors wavily embedded in CFRP laminate structures, Proc. SPIE 5054 (2003) 184–191. [16] J. Frieden, J. Cugnoni, J. Botsis, T. Gmür, Low energy impact damage monitoring of composites using dynamic strain signals from FBG sensors – Part I: Impact detection and localization, Compos. Struct. 94 (2) (2012) 438–445. [17] C. Hiche, C.K. Coelho, A. Chattopadhyay, M. Seaver, Impact localization on complex structures using FBG strain amplitude information, Proc. SPIE 7649 (2010) 764903.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
FBG sensor network for pressure localization of spacecraft structure based on distance discriminant analysis Jing Jin, Song Lin ∗ , Xiangyu Ye School of Instrument Science and Optic-electronics Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100191, China
a r t i c l e
i n f o
Article history: Received 18 February 2013 Accepted 30 June 2013
Keywords: Fiber Bragg grating Pressure localization Distance discriminant analysis
a b s t r a c t Structure used in aircrafts and spacecrafts often undergo high pressure loading that can produce various types of damage, thus prediction of existence and pressure position is important to damage localization. In this paper, the structure type investigated is a 50 mm thick sandwich panel with aluminum face-sheets and aluminum honeycomb core. One pressure test was performed on the panels, which was instrumented with 9 FBG sensors. The signals recorded at the various sensor locations varied with regard to Bragg wavelength shift of FBGs. Using this information and combining it with a localization algorithm based on distance discriminant analysis, the pressure location could be successfully determined. A description of the FBG sensors network and the mathematical model to determine the pressure location is provided. The pressure tests on the spacecraft structure, the response of the FBG sensors network and the analysis performed to determine the pressure location are described. © 2013 Elsevier GmbH. All rights reserved.
1. Introduction Onboard satellites, spacecraft and aircraft, a variety of sensors monitor the subsystem function and their health status. The sensors applied measure e.g. temperature, pressure, strain, etc. The acquired data are downlinked to the ground station as standard housekeeping data. Typically, the amount of data gathered is sufficient for controlling the function and the health of spacecraft subsystems. However, there are various situations when an aircraft is in flight where the availability of additional information would be desirable, especially when there is an anomaly or a subsystem failure. A variety of anomalies and subsystem failures can be caused by space debris impacts on the structure of spacecraft [1]. Meanwhile, pressure loading is a potential damaging event that may occur anytime during the service life of many spacecraft structures. High strength pressure may cause materials to sustain significant internal damage that can grow with time, leading to catastrophic failure of the spacecraft structure [2,3]. In order to detect these damaging events, information of the existence and position of a pressure loading on the structure are needed. As the position of the pressure is known with a certain precision, the inspection of the damage can be limited to this region and may reduce the complexity of the damage identification problem significantly.
∗ Corresponding author. E-mail address: [email protected] (S. Lin). 0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.06.087
Recently, fiber Bragg grating (FBG) sensors are proven to have many advantages in damage detection area [4–10], such as the high resistance to corrosion and fatigue, the wide bandwidth operation, and the immunity to electromagnetic interference. In addition, they are compatible with the most part of composite materials and concrete, safe to handle, and there is little detrimental effect on the mechanical properties of the host structure. Among fiber-optic sensors, FBG plays a dominant role due to their low costs and extreme versatility, in conjunction with a good multiplexing capability. A lot of work has been done to detect impact damage using FBG sensors [11–15], and most of the localization methods are based on the concept of time-of-flight of ultrasonic waves. Jeannot Frieden presents a method for the localization of an impact based on the arrival time of waves at the FBG sensors, which allows to predict the impact position with interpolation of a reference data set [16]. Cristobal Hiche proposed an impact localization method based on the relative placement of all sensors and the maximum strain amplitude measured by each sensor [17]. However, little work for pressure localization has been done. In this paper, 9 FBG sensors are surface mounted on a 50 mm thick sandwich panel with aluminum face-sheets and aluminum honeycomb core in different directions, and the sandwich plate is divided into 4 areas by the FBGs. Meanwhile, the distance discriminant analysis (DDA) method, which is mostly used in statistics, pattern recognition and machine learning, is utilized to determine which area pressure is loaded on. The Bragg wavelength shift values of FBG sensors are collected when pressure is loaded on different positions of the 4 areas to build the training set, and then the training data are normalized as unit vectors. A localization algorithm
J. Jin et al. / Optik 125 (2014) 404–408
405
is proposed to further prediction of the pressure position based on the concept of DDA method, and the results are validated to localize with a good accuracy.
as uj = 1j , 2j , , ij
2. Localization principle
dj = d x, uj =
2.1. Method to distinguish area
The algorithm for further localization is based on the assumption that Euclidian distance dj has a linear relationship with actual distance dj ’. In this algorithm, 3 reference points A, B and C, the Euclidian distances of which are da , db , dc are minimum among all the reference points, are picked up and predicted point I = (X,Y) can be calculated with the formula (5):
d2 (X, G) = (X − )
−1
(X − ), i = 1, 2, , k
(1)
where = E(X) = (1 , 2 , , n ) ’ and is the covariance matrix of the distribution. Consider that there are k different classes Gi , then the Mahalanobis distance between X and Gi can be expressed as Eq. (2): d2 (X, Gi ) = (X − i )
−1 i
(X − i ), i = 1, 2, , k
(2)
where i is the covariance matrix and can be estimated with Eq. (3):
ˆ
1 (xij − x¯ i )(xij − x¯ i ) , i = 1, 2, , k ni − 1 ni
i
=
(3)
i=1
The sample X can be assigned to class i if d2 (x, Gj ) = min d2 (x, Gi ). 1≤i≤k
2.2. Extrapolation of pressure position In order to localize pressure loading on a large structure, an amount of FBG sensor is needed due to the sensitivity range of the sensor, which increased the complexity of localization and the amount of data processing. In this paper, the whole structure is divided into 4 areas that are delimited by 9 FBG sensors, and the distance discriminant analysis method is utilized to determine which area the pressure is loaded on with the training set. After that, a localization algorithm is proposed for further prediction of the pressure position. Firstly, the Bragg wavelength shift value of every FBG sensor is normalized as a unit vector, the observation X = (1 , 2 , , n ) / ˙2n , where n is the Bragg wavelength shift value of the nth sensor. Then, 25 reference points are picked up in each area to load pressure, and the Bragg wavelength shift data of FBGs constructed the class Gi , namely the training set. Finally, the area where pressure is loaded can be determined by the DDA method. Assuming that the area determined previously is delimited by i sensors, the Bragg wavelength shift value of each sensor when pressure is loaded can be normalized as a unit vector x = (1 , 2 , , i ) / ˙2i . Therefore, the unit vector at refer-
ence points Ij = Xj , Yj which belong to this area can be recorded
˙2ij . Then, the Euclidian dis-
tance between x and uj can be computed with the formula (4):
Distance discriminant analysis is used to determine variables that discriminate between two or more naturally occurring groups, and the main classification rule is simply to assign an observation X to the class the mean of which is closer to x in the sense of the Mahalanobis distance. The “Mahalanobis distance” is a metric that is better adapted than the usual “Euclidian distance” to settings involving nonspherically symmetric distributions. Particularly, it is more useful when multinormal distributions are involved, although its definition does not require the distributions to be normal. Assuming that G = (X1 , X2 , , Xn ) ’ is an n elements class, and X = (x1 , x1 , , xn ) ’ is an observation, the Mahalanobis distance can be calculated with Eq. (1):
/
x − uj
(Xa − X)2 + (Ya − Y )2 = da =
x − uj
(4)
(Xb − X)2 + (Yb − Y )2 db
(Xc − X)2 + (Yc − Y )2 dc
(5)
There are two solutions O1, O2 for formula (5). Without taking account of the solutions are complex numbers, the point which is more close to the circumcenter of triangle constructed by points A, B and C is picked up as the predicted point. 3. Experiment setup The pressure localization method is validated by the static pressure loaded on a 50 mm thick sandwich panel with aluminum face-sheets and aluminum honeycomb core with a size of 1000 × 1000 mm; its surface is pretreated by sanding and cleaning to maximize adhesive bonding between the surface and the sensors are clamped on all sides by the fixture to impose a fixed–fixed boundary condition. 9 FBG strain sensors are surface mounted on the plate to acquire strain information at different positions of the plate, and a distance of 10 cm has been kept between FBG sensors and the edge of the plate. The experimental setup is shown in Fig. 1. Cyanoacrylate adhesive is used to mount the FBG sensors on the plate to minimize both air pockets and the adhesive layer thickness between the plates and the FBG sensors. In order to improve the precision of the experiment, a FBG temperature sensor is mounted on the plate to acquire the Bragg wavelength shift of FBGs induced by temperature change. Static pressure loading is provided by iron with a small and even underside. The FBG interrogation system is sm125 from Micron Optics with a working wavelength range from 1510 to 1590 nm, wavelength resolution is 1 pm, wavelength stability is 1 pm, wavelength repeatability is 0.5 pm at 1 Hz. To reduce the number of measurement channels, spectrally separated Bragg gratings were spliced in
Fig. 1. Experimental setup.
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Fig. 2. Reference points selected in the area and the location of FBG sensors.
chains with a distance of ∼0.3 m between the gratings. The spectral separation of the gratings in a chain was ∼10 nm with the Bragg wavelength in a range 1530 ∼ 1550 nm. 4. Result and discussion 4.1. Construction of training set The whole plate is divided into 4 areas by 9 FBGs; these areas are marked as Area 1, Area 2, Area 3 and Area 4, as is shown in Fig. 2. In order to build the training set, 100 reference points, marked as position 1 to position 100, are selected in the domain delimited by FBG sensors. Pressures are loaded on these points with strengths of 50 N, respectively. The Bragg wavelength shift values of FBG sensors are acquired by interrogation system sm125 (Fig. 3). During the experiment, pressure loading is repeated 3–5 times at each point to check the repeatability. The training sets are normalized as unit vectors, and the 100 samples are utilized in the discriminant rule to determine which area these corresponding points belong to. 4.2. Result of area classification and localization
Fig. 4. The error of exact location and predict location on the plate.
plate. The Bragg wavelength shift data is collected and processed by the interrogation system sm125 in the meantime. It turns out that all of the points are assigned to the correct area with the distance discriminant method. With the further localization algorithm, predicted position is calculated and compared to the exact position. The error of predict locations are listed in Table 1 and shown in Fig. 4. It can be seen that these verified points in each area are predicted well with a maximum error of 1.64 cm and an average error of 1.12 cm. The result turns out that the discriminant modal has a consistently successful prediction, and indicated the feasibility of the distance discriminant analysis method. Though all the points are predicted in the correct area in this experiment, there is a possibility of misjudgment when pressure is loaded on the edge of each area. Considering the situation that a point located is in Area 2, the coordinate of which is 47.5, 31, it might probably be wrongly predicted in Area 4. According to the reference data points in Area 4 and the calculations, there are two solutions of formula (5). The coordinates of solution 1 are 48.8, 23.7, and the coordinates of solution 2 are 47.4, 29.3. The predict error of the two solutions are 7.41 cm and 1.7 cm, respectively. The sketch map is shown in Fig. 5. It can be seen that the error is obviously
In order to validate the distance discriminant analysis method, 32 verified points, which are selected in different regions delimited by FBG sensors, are loaded with a static pressure of 50 N on the cm
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Fig. 3. Wavelength shift of FBGs when pressure is loading at 25 reference points in area 1 (a) FBG1; (b) FBG2; (c) FBG3; (d) FBG4.
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Fig. 5. The misjudgment of predicted point on the edge of area 2.
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Table 1 Error of predict location. Number
Exact area
Predict area
Exact location (X/cm, Y/cm)
Predict location (X/cm, Y/cm)
Error (cm)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4
1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4
(22.6, 36.2) (19.4, 55.6) (14.5, 38) (5.7, 42.5) (7.3, 54.1) (17, 47.7) (13.3, 43.3) (26.5, 43.8) (41.5, 55.8) (53.5, 50.5) (38.8, 43.7) (42.3, 46.7) (47.6, 43.5) (34, 37.8) (50.9, 37) (36.6, 52) (6.3, 21.8) (17.5, 17) (21.9, 5.8) (23.6, 19.4) (13, 7.5) (17.4, 24.3) (6.2, 10.8) (12.5, 16.5) (39, 24.7) (37.2, 17.6) (44, 16.7) (7.8, 9.2) (48.4, 52.5) (52.7, 17.5) (47.1, 12.2) (52.7, 6.2)
(23, 37.5) (18.3, 55.8) (13.5, 38.4) (6.5, 41.8) (8.5, 53.5) (17.3, 47.3) (12.5, 43.5) (25.5, 42.5) (41.9, 57) (52.3, 50.7) (37.8, 44.1) (42.5, 46.2) (48.3, 43) (34.3, 36.5) (51.3, 38.2) (37, 53.1) (7.2, 22.3) (18.3, 17.2) (23, 6.5) (23.7, 18.3) (13.6, 8.6) (18.3, 23.6) (6.3, 12) (13.2, 17) (38.2, 23.7) (36.1, 18.1) (43.5, 16.2) (8.4, 8.2) (49, 53.8) (53.7, 18.2) (46.5, 11.2) (54, 6)
1.36 1.12 1.08 1.06 1.34 0.5 0.82 1.64 1.26 1.22 1.08 0.54 0.86 1.33 1.26 1.17 1.03 0.82 1.3 1.1 1.25 1.14 1.2 0.86 1.28 1.21 0.71 1.17 1.43 1.22 1.08 1.32
increasing and solution 1 with a bigger error is picked up as the predicted point as the reference points selected are all in Area 4. In order to solve this problem, more reference points are selected in the predict algorithm. Similarly, the point located in Area 2, the coordinates of which are 47.5, 31, is wrongly predicted in Area 4. Not only the reference data points in Area 4 but also the reference points in Area 2 and Area 3, which are near Area 4, are selected to calculate the location. The selection of reference points is shown in Fig. 6. Three reference points (40, 45 and 86) are calculated to further predict the position, and the coordinates of the reasonable solution are 47.3, 32, with an error of 1.02 cm.
cm
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Reference data point selected
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Acknowledgment This research is supported by the National Natural Science Foundation of China (Grant No. 61007040).
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Area 3
References
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Bragg wavelength of FBG sensors will shift when pressure is loaded on the structure, and the shift values of FBG sensors are applied in this paper to predict the position of pressure with the distance discriminant analysis method. The construction of the training set is important as it contributes a lot to the accuracy of the predict result. In this paper, the distance discriminant method is verified feasible with a consistent homology of prognosis for the 100 samples. During the pressure experiment on sandwich aluminum plate, 32 different points are randomly selected and all the points are predicted to the correct area. In the meantime, a maximum error of 1.64 cm and an average error of 1.12 cm are acquired with the algorithm for further localization. Regarding the situation that points are misjudged to incorrect area, more reference points are selected in the predict algorithm to solve this problem. The experimental results prove that the localization algorithm can predict pressure location successfully with a good accuracy, and error of prediction is smaller than 2 cm.
Solution 1 86 Solution 2
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5. Conclusion
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Fig. 6. The selection of reference points for solving misjudgment of predicted location.
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