Vibration Measurement in High Precision for Flexible Structure ... - MDPI

robotics Article

Vibration Measurement in High Precision for Flexible Structure Based on Microscopic Vision Xian Tao 1, *, De Xu 1 , Zhengtao Zhang 1 , Kai Wang 2 and Xiaobo Qi 2 1 2

*

Research Center of Precision Sensing and Control, Institute of Automation, Chinese Academy of Sciences, Beijing 100190, China; [email protected] (D.X.); [email protected] (Z.Z.) Research Center of Laser Fusion, Chinese Academy of Engineering Physics, Mianyang 621900, China; [email protected] (K.W.); [email protected] (X.Q.) Correspondence: [email protected]; Tel.: +86-10-8254-4535

Academic Editor: Huosheng Hu Received: 13 January 2016; Accepted: 28 March 2016; Published: 30 March 2016

Abstract: Vibration measurement for flexible structures is widely used in various kinds of precision engineering fields. However, it is a challenge to measure vibration in special applications, such as cryogenic, dangerous and magnetic interference. In this paper, a high-precision vibration measurement system based on machine vision is designed. The circle center on the target is employed as the image feature. The circle feature is extracted using the improved algorithm based on gradient Hough transform. Then the image Jacobian matrix is used to compute the vibrations in Cartesian space from the image feature changes. Experiments verify the effectiveness of the proposed methods. Keywords: flexible structure; vibration measurement; machine vision

1. Introduction Flexible structures are widely used in various kinds of precision engineering fields such as the cables of cable-stayed bridges [1,2], flexible inspection arms in international thermonuclear experimental reactors (ITER) [3], cryogenic targets used in inertial confinement fusion (ICF) [4], and spacecraft with solar panels [5]. As a type of fusion energy research, ICF attempts to initiate nuclear fusion reactions by heating and compressing a fuel target. The vibration of a cryogenic target causes many problems, such as the effectiveness of laser aiming and the target’s displacement. So the vibration measurement of a cryogenic target is an important research topic. The modal damping of those flexible structures is small. Once subjected to some kind of exciting force, its substantial vibration could continue for a long time. This will affect the equipment and also result in damage to the structure. So the vibration measurements of flexible structures play a crucial role in the prediction of potential disasters. A quite large number of monitoring techniques are available for structure vibration measurement. Among them, the most popular are probably the accelerometer-based measurements, the laser and the strain gages [6–8]. Although these techniques fit well for most applications, they present some limitations in special applications, such as cryogenic, strong electrical and magnetic interference. Vision-based vibration measurements have been studied by many researchers [9–11]. They can measure vibration displacements at multiple points simultaneously. These contactless vibration measuring techniques would be preferred in the special applications above. In this paper, a vision system is presented to measure the vibration of a cryogenic target. The cryogenic target is mounted on a 3-DOF manipulator which is sealed in a closed cryogenic container. The developed system consists of two 1-DOF adjusting platforms, two microscopic cameras, and a computer. The techniques including the calibration of the image Jacobian matrix with active movements of the cryogenic target and the image feature extraction are presented. An improved adaptive Hough transform method is proposed to achieve the image feature extraction. The process Robotics 2016, 5, 9; doi:10.3390/robotics5020009

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with active movements of the cryogenic target and the image feature extraction are presented. An 2 of 12 improved adaptive Hough transform method is proposed to achieve the image feature extraction. The process of vibration measurement consists of three steps: (1) the image Jacobian matrix is of vibration measurement of three(2) steps: (1) the image calibrated before calibrated before vibration consists measurement; the coordinates of Jacobian the targetmatrix centerisare calculated via vibration measurement; coordinates the target center are calculated image feature extraction; (2) (3) the vibrations in of Cartesian space are computed fromvia the image image feature extraction; (3) the in Cartesian space arevibration computedcurve from the image feature changes using changes using thevibrations image Jacobian matrix. The of the cryogenic target in the the image Jacobian matrix. The vibrationobtained curve of in thereal cryogenic developed system can be automatically time. target in the developed system can be automatically obtained in real time. The rest of this paper is organized as follows. Section 2 gives a brief introduction of the system The of this is organized follows. 2 gives a brief introduction of the system setup. Inrest Section 3,paper the calibration of as the image Section Jacobian matrix is provided. In Section 4, the setup. In Section 3, the calibrationThe of the image Jacobian matrixbased is provided. In Section 4, the autofocusing method is described. image feature extraction on an improved gradient autofocusing method is described. The vibration image feature extraction based ontoo. an improved gradient Hough transform method is given. The calculation is presented, Section 5 provides Hough transformand method is given. is presented, the experiments results. Finally,The thisvibration paper is calculation concluded in Section 6. too. Section 5 provides the experiments and results. Finally, this paper is concluded in Section 6. 2. System Setup 2. System Setup In the process of keeping a container cryogenic, the working of the cryocooler will lead to the In the processThe of vibration keeping ameasurement container cryogenic, the the cryocooler lead to target’s vibration. system for theworking target isof designed as givenwill in Figure 1. the target’s vibration. The vibration measurement system for the target is designed as given It consists of two 1-DOF motion platforms, microscopic cameras, a light system and a computer. in A Figure 1. It camera consistsand of two 1-DOFplatform motion platforms, microscopic cameras, system microscopic a motion with a support mechanism form aa light vision unit. and The a computer. A microscopic camera and a motion platform with a support mechanism form a vision microscopic camera is formed with a microscope lens and a Charge-coupled Device (CCD) camera. unit.motion The microscopic is formed with a microscope and in a Charge-coupled Device (CCD) The platform iscamera employed to move the microscopiclens camera order to adjust its distance to camera. motion platform employed to move in order to adjust its the objectThe to be viewed. It has isone translation DOF. the Themicroscopic vision unit camera is mounted on the vibration distance to the object to computer be viewed.isItused has one translation DOF.via The vision unit is mounted on the isolation platform. The to capture images the microscopic camera and to vibration isolation platform. computer is used to capture images via the microscopic camera and calculate the vibration of the The cryogenic target. to calculate the vibration cryogenic target. The measured objectofisthe shown in Figure 1. It is a flexible structure object sealed in a cryogenic The measured object is shown in Figure 1. It a flexible structure object sealed in a cryogenic system. There is a circle at the end surface ofisthe measured object, which is etched by the system. Therelaser. is a circle at the measured object, is etched by the femtosecond There are end alsosurface circles of atthe other surfaces. Thewhich external diameter of femtosecond the circle is laser.μm. There also circles at other of thealong circle the is 500 measured 500 Theare measured object can surfaces. be movedThe byexternal a 3-DOFdiameter manipulator Xwµm. -, YwThe - and Zw-axis, object can be moved by a 3-DOF manipulator along the Xw -, Yw - and Zw -axis, respectively. respectively. Robotics 2016, 5, 9

Figure 1. System configuration scheme: (a) The system setup; (b) The measured object. Figure 1. System configuration scheme: (a) The system setup; (b) The measured object.

3. Calibration of Image Jacobian Matrix 3. Calibration of Image Jacobian Matrix In the microscopic vision system, the mapping from the plane of the clear view area to the In the microscopic vision system, the mapping from the plane of the clear view area to the image image plane is linear because of the negligible lens distortion. The image Jacobian matrix can plane is linear because of the negligible lens distortion. The image Jacobian matrix can describe describe this mapping, which is the transformation matrix from the increments of the feature’s this mapping, which is the transformation matrix from the increments of the feature’s motion in the motion in the Cartesian space to the image feature changes [12,13]. The feature’s increments in the Cartesian space to the image feature changes [12,13]. The feature’s increments in the Cartesian space Cartesian space are the vibration to be measured. The image Jacobian matrix should be calibrated in are the vibration to be measured. The image Jacobian matrix should be calibrated in advance in order advance in order to obtain the accurate vibration from the image feature changes. In this work, the to obtain the accurate vibration from the image feature changes. In this work, the target’s position is target’s position is adjusted by the 3-DOF manipulator. Hence, the image Jacobian matrix can be adjusted by the 3-DOF manipulator. Hence, the image Jacobian matrix can be calibrated via the active calibrated via the active motion of the target. motion of the target.

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Each microscopic camera is sensitive to 2-DOF translation. For example, microscopic camera 1 is sensitive to the translations along the Zw - and Yw -axis. Microscopic camera 2 is sensitive to the translations along the Xw - and Yw -axis. The relation between the translation of the cryogenic target in Cartesian space and the feature changes on the images of two microscopic cameras can be expressed in Equation (1) if each feature point is chosen on the image of each microscopic camera. » — — — –

∆uc1 ∆vc1 ∆uc2 ∆vc2

fi

»

ffi — ffi — ffi “ — fl –

J11 J21 J31 J41

J12 J22 J32 J42

J13 J23 J33 J43

fi » fi » fi ∆xm ffi ∆xm ffi — ffi — ffi ffi – ∆ym fl “ J – ∆ym fl fl ∆zm ∆zm

(1)

where (∆uc1 , ∆vc1 ) and (∆uc2 , ∆vc2 ) are the two feature changes on the images of the two microscopic cameras, (∆xm , ∆ym , ∆zm ) is the incremental translation of the cryogenic target moved by the manipulator. The relative positions of the cryogenic target in Cartesian and image space can be obtained for several active movements. Applying these data to the measuring model in Equation (1), an equation is formed. » fi » fi fi ∆uc11 ∆uc12 . . . ∆uc1n J11 J12 J13 » ∆x1 ∆x2 . . . ∆xn — ∆v ffi — ffi — — J21 J22 J23 ffi — ffi c11 ∆vc12 . . . ∆vc1n ffi (2) — ffi “ — ffi – ∆y1 ∆y2 . . . ∆yn fl – ∆uc21 ∆uc22 . . . ∆uc2n fl – J31 J32 J33 fl ∆z1 ∆z2 . . . ∆zn ∆vc21 ∆vc22 . . . ∆vc2n J41 J42 J43 where [∆xi ∆yi ∆zi ]T is the relative position vector related to the reference in Cartesian space, [∆ucji ∆vcji ]T is the relative image position vector related to the reference on the images of the j-th microscopic camera at i-th step active movement, and n is the step number of active motions, j = 1, 2. Then the image Jacobian matrix can be solved from Equation (2) using the least squares method, as given in Equation (3). ´1

J “ AB T pBB T q

(3)

where » — — J“— –

J11 J21 J31 J41

J12 J22 J32 J42

J13 J23 J33 J43

fi

»

ffi — ffi — ffi , A “ — fl –

∆uc11 ∆vc11 ∆uc21 ∆vc21

∆uc12 ∆vc12 ∆uc22 ∆vc22

... ... ... ...

∆uc1n ∆vc1n ∆uc2n ∆vc2n

fi » ∆x1 ffi ffi — ffi B “ – ∆y1 fl ∆z1

∆x2 ∆y2 ∆z2

... ... ...

fi ∆xn ffi ∆yn fl ∆zn

Therefore, once the feature changes on the images and the translation increments in Cartesian space with n steps of active movements are obtained, the image Jacobian matrix can be obtained with Equation (3). 4. Automated Measurement The vibration measurement process mainly includes three parts: target autofocusing, feature extraction, and vibration measurement. The image Jacobian matrix is calibrated in the initialization through several steps of active movements. The measuring apparatus is initialized by opening lighting sources and cameras. In the vibration measurement process, the clear images of the measured object can be captured by the two microscopic cameras through focus movement. In this work, there is a circle at the end surface of the measured object. When applied to other measurement objects, a specific shape target needs to be fixed at the surface of the measured object. Then, high-speed microscopic cameras record the video on the target vibration. The center coordinates of the target are extracted from the video captured by the two microscopic cameras. An improved gradient Hough transform algorithm is used to extract the

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the measured object. When applied to other measurement objects, a specific shape target needs to be fixed at the surface of the measured object. Then, high-speed microscopic cameras record the Robotics on 2016,the 5, 9 target vibration. The center coordinates of the target are extracted from the video 4 of 12 video captured by the two microscopic cameras. An improved gradient Hough transform algorithm is used to extract the circle center. Due to the large amount of image data acquisition, the image circle center. Due to the large amount of image data acquisition, the image processing will take a long processing will take a long time which leads to a slow vibration measurement. In order to speed up time which leads to a slow vibration measurement. In order to speed up the feature extraction, the the feature extraction, the method of the thread pool is used to synchronously calculate all the method of the thread pool is used to synchronously calculate all the images in the captured video. images in the captured video. The coordinate changes of feature points in adjacent frames are The coordinate changes of feature points in adjacent frames are obtained, and then the actual amount obtained, and then the actual amount of vibration can be calculated by the image Jacobian matrix. of vibration can be calculated by the image Jacobian matrix. Finally, the displacement curve and the Finally, the displacement curve and the spectrum curve can be obtained, which are used to spectrum curve can be obtained, which are used to calculate the failure analysis information of the calculate the failure analysis information of the flexible structure. flexible structure. 4.1. 4.1. Autofocus Autofocus Strategy Strategy During thevibration vibrationmeasurement measurement process, measured be clear on the clear During the process, the the measured objectobject shouldshould be on the imaging imaging plane. Manual focusing is laborious and slow, and the repeatability precision is not plane. Manual focusing is laborious and slow, and the repeatability precision is not ensured. ensured. So far, various autofocus techniques including focus evaluation functions [14,15] and focus So far, various autofocus techniques including focus evaluation functions [14,15] and focus searching searching [16] have been developed. Autofocus target by is achieved by microscopic moving the algorithmsalgorithms [16] have been developed. Autofocus of the targetofisthe achieved moving the microscopic camera along its optical axis. After autofocus of the target, the microscopic camera camera along its optical axis. After autofocus of the target, the microscopic camera should stay fixed should stay fixed during the rest of the process. The focus evaluation function used in this work is during the rest of the process. The focus evaluation function used in this work is the Sobel operator to the Sobel operator to enhance the importance of the edges. enhance the importance of the edges. JJ “ 

M N M ÿ N ÿ

(i, jq j)  SSpi,

(4) (4)

11 jj“ 11 ii“

where S(i, j) is the Sobel Sobel value value of of pixel pixel (i, (i, j). j). The size of the the autofocus autofocus area area is is M Mˆ × N. The focusing process is finished when the maximum value of Equation (4) is achieved in the predefined focusing range. When the image is clearest, the target is focused. 4.2. Feature Feature Extraction Extraction 4.2. After the thethe target video needsneeds to be captured. All images the video to After theautofocus, autofocus, target video to be captured. Allinimages inare theprocessed video are get the coordinates of the feature point. However, in our work, under different ultra-low temperature processed to get the coordinates of the feature point. However, in our work, under different environments, the image of the targetthe is inconsistent Figure 2). This is due impure ultra-low temperature environments, image of the(see target is inconsistent (seeto Figure 2). helium This is in the cryogenic vessel. It can cause the impurities attached to the target in different temperatures. due to impure helium in the cryogenic vessel. It can cause the impurities attached to the target in This requires the algorithm to requires have good forhave different environments. The image of different temperatures. This theadaptability algorithm to goodlighting adaptability for different lighting the target has two (see 2). The points(see are the centers of thefeature two circles. environments. Thecircles image of Figure the target hasfeature two circles Figure 2). The pointsThe aremost the popularly used algorithms for detecting circular patterns in an image are based on Hough transform. centers of the two circles. The most popularly used algorithms for detecting circular patterns in an In this are work, the on gradient Hough transform algorithm is used to detect the center of the[17,18] target. image based Hough transform. In this work, the[17,18] gradient Hough transform algorithm The detection process of theoftarget’s centers as follows. is used to detect the center the target. Theisdetection process of the target’s centers is as follows.

(a)

(b)

Figure (b) at at normal normal temperature. temperature. Figure 2. 2. Target Target images images at at different different temperatures: temperatures: (a) (a) at at low low temperature; temperature; (b)

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(1) The gradient field of image intensity is computed using the following Equation (5). #

Gx pi, jq “ p f pi ` 1, jq ´ f pi, jq ` f pi ` 1, j ` 1q ´ f pi, j ` 1qq{2 Gy pi, jq “ p f pi, jq ´ f pi, j ` 1q ` f pi ` 1, jq ´ f pi ` 1, j ` 1qq{2

(5)

where (i, j) are the pixel indices, Gx (i, j) is the gradient vector in x direction at the pixel (i, j), Gy (i, j) is the gradient vector in y direction at the pixel (i, j), f (i, j) is the image intensity at the pixel (i, j). The gradient directions of points on the circular pattern are either pointing towards the center of a circle or away from it. Based on this rule, the corresponding center coordinates of the circle can be obtained by the coordinates of the point on the circle and its gradient vector when the radius of the circle is fixed. The parameter space in Hough transform consists of the circle’s center and radius. So each point with a non-zero gradient vector is corresponding to a series of centers and radii. All points with a non-zero gradient vector are transformed to the center and radius space described by an accumulation array containing the different circle centers and radii. So, the elements’ values of the accumulation array are evaluated by a voting process similar to the standard Hough transform. The maximum value in the accumulation array gives the center and radius of a circle. The point whose gradient magnitude is less than the gradient filtering threshold GT is not transformed in order to reduce the noise and improve the real-time performance. The values of GT for different brightness images are manually given. (2) In order to pinpoint the circle’s center, the centers corresponding to the elements with a larger value than the threshold in the accumulation array need to be filtered. The Laplacian of Gaussian (LoG) filter [19] is applied to give the area contour of the centers. (3) The centroid of the area contour is calculated, which is taken as the accurate center of the circle. 4.3. Vibration Calculation When the centers of the target’s circles in each image are extracted, the coordinate changes of center points in adjacent frames can be obtained. The actual amount of vibration can be calculated as » »

fi

∆xmk — ´1 — ffi — – ∆ymk fl “ pJ T Jq J T — – ∆zmk

∆uc1k ∆vc1k ∆uc2k ∆vc2k

fi ffi ffi ffi fl

(6)

where (∆uc1k , ∆vc1k ) and (∆uc2k , ∆vc2k ) are the k-th increments of the two feature points on the images of the two microscopic cameras, and (∆xmk , ∆ymk , ∆zmk ) is the k-th translation increment of the cryogenic target. Finally, the displacement curve and the spectrum curve can be obtained. 5. Experiments and Results 5.1. Experiment System An experiment system was established according to the scheme given in Section 2, as shown in Figure 3. In this experiment system, both cameras are Basler cameras. Both cameras are equipped with Nikon zoom lenses with magnification of 10ˆ, and they captured images at 90 frames per second with an image size of 2048 ˆ 2048 pixels. The field-of-view dimensions of the two cameras were 1 mm ˆ 1 mm. The two microscopic cameras were placed on Sigma SGSP26-50 to move along their optical axes. The translation resolution was 1 µm. The CPU of the host computer was Intel Core™2 DUO with a frequency of 2.8 GHz. The measurement accuracy in this system was 0.5 µm and the measurement vibration frequency range was 0–40 Hz.

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1Microscopic Camera 2

red object

2Microscopic Camera 1

PlatformAdjusting Platform

Light Controller

Software Interface

Figure 3. Experiment system. Figure 3. Experiment system.

5.2. Image Jacobian Matrixes Calibration 5.2. Image Jacobian Matrixes Calibration The manipulator actively translated the target with different distances as shown in matrix Mt, The manipulator actively translated the target with different distances as shown in matrix whose column was the increments of the target along the Xw-, Yw- and Zw-axis respectively. The Mt, whose column was the increments of the target along the Xw -, Yw - and Zw -axis respectively. increments were reasonably distributed around the origin. The extracted feature increments were The increments were reasonably distributed around the origin. The extracted feature increments recorded in matrix Mtc, whose column was the feature increments on images captured by the two were recorded in matrix M , whose column was the feature increments on images captured by microscopic cameras. Then tcthe image Jacobian matrix J was computed in Equation (7). The the two microscopic cameras. Then the image Jacobian matrix J was computed in Equation (7). pseudo-inverse of the Jacobian matrix was given in Equation (8). The pseudo-inverse of the Jacobian matrix was given in Equation (8). 0 0 45 13 fi »  20 0   μm 0 1045 25´13 M t  200 020 00 40  ffi — Mt “ – 00 ´20 0 ´40  fl µm 0 40 0 1710 2525 0 0 40 0 ´17 ´25 2.01 71.20 4.82 -25.13 -37.20   0.01 »  fi  1.02  32.50 0.02 67.60 15.98 0.01 2.01 71.20 4.82 ´25.13 40.60 ´37.20  pixel M tc—  ffi  33.16 ´32.50 -4.87 0.81 ´67.60 -6.43 78.89 0.02 15.98 -25.48 40.60 — 1.02 ffi Mtc “ —  ffi pixel  – 33.16 fl ´6.43 15.84 78.89 40.65 ´25.48 -32.47 0.81 2.97 -67.51  1.09 ´4.87  1.09 ´32.47 2.97 ´67.51 15.84 40.65 J  M tc M tT ( M t M tT ) 1 T T ´1 J “»M -0.0339 1.6682  fi tc Mt pM t Mt q  0.0351  0.0351  ´0.0339 1.6682 (7)  0.0041 1.6665 0.0163  ffi —  (7) 0.0041 0.1290 1.6665 0.0815 0.0163 — 1.7597 ffi  “— ffi  fl –  1.7597 0.1290 0.0815 0.0102 1.6801 0.0689  0.0102 1.6801 0.0689 »-0.0267 -0.0225 0.5690 -0.0219 fi ´0.0267 ´0.0225 0.5690 ´0.0219 ffi ( J T J ´)11 J T —-0.0152 0.2977 -0.0021 0.2998  (8)(8) pJ T Jq J T “ – ´0.0152 0.2977 ´0.0021 0.2998 fl  0.5994  0.5994 -0.0029 ´0.0029-0.0120 ´0.01200.0159 0.0159 where M Mt t was wasthe the matrix matrix formed formedby by the the active active translation translation of of the the manipulator, manipulator, M Mtctc was wasthe thematrix matrix where formedby bythe thefeature featureincrements. increments. formed

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5.3. 5.3.Feature FeatureExtraction Extraction Robotics 2016, 5, 9

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In Inthe theexperiment experiment of offeature featureextraction, extraction, Figure Figure 4a 4awas wasthe thethree-dimensional three-dimensional (3D) (3D) view view of ofthe the accumulation array according to step (1) presented in Section 4.2. Figure 4b showed the detected result 5.3. Feature array Extraction accumulation according to step (1) presented in Section 4.2. Figure 4b showed the detected of the target. The point marked with “+”with was the circle center. The circle The marked with blue color was result of In thethe target. The point marked was center. circle marked experiment of feature extraction, “+” Figure 4athe wascircle the three-dimensional (3D) view ofwith the blue the contour of the target. color was the contour of the target. accumulation array according to step (1) presented in Section 4.2. Figure 4b showed the detected result of the target. The point marked with “+” was the circle center. The circle marked with blue color was the contour of the target.

(a)

(a)

(b)

(b)

Figure Feature extraction: (a) The accumulation array; (b) The result offeature feature extraction. Figure 4. Feature extraction: (a)The Theaccumulation accumulation array; (b) The result of feature extraction. Figure 4.4.Feature extraction: (a) array; (b) The result of extraction.

5.4.Offline Offline Vibration Experiment 5.4. Offline Vibration Experiment 5.4. Vibration Experiment In the offline experiment,the thevibration vibration of the target generated by abypiezoelectric ceramics Inthe the offline experiment, ofthe thetarget targetwas was generated piezoelectric ceramics In offline experiment, the vibration of was generated by aapiezoelectric ceramics motion platform. An experiment system system was was established, which was shown in Figure 5. The motion platform. An experiment established, which was shown in Figure 5. The motion platform. An experiment system was established, which was shown infor Figure 5. The sinusoidal sinusoidal signal was inputted in the piezoelectric ceramics motion platform the vibration. The sinusoidal signal was inputted in the ceramics piezoelectric ceramics motion forThe theinput vibration. The signal was vibration inputted in the piezoelectric motion platform forThe theplatform vibration. vibration input frequency was 1 Hz and the magnitude was 5 μm. software interface was built input vibration frequency was 1 Hz and the magnitude was 5 μm. The software interface was built frequency 1 Hzinstrument and the magnitude was µm. The software interface by the virtual by thewas virtual Labview and an5oscilloscope was used to verifywas thebuilt motion of the by the virtual instrument Labview and an oscilloscope was used to verify the motion the instrument Labview and an was used to verify along the motion theZplatform. In this of work, platform. In this work, theoscilloscope platform motion was controlled the Xw-ofand w-axis respectively. platform. In motion this work, the platform motion was controlled along the Xwand - and w-axis respectively. The frame rates ofwas the cameras were set to 15 actual measurements theZmeasured timeof the the platform controlled along the Xfps and Zw -axis respectively. The frame rates w - in The frame rates of the cameras were set to 15 fps in actual measurements and measured time was were five seconds. cameras set to 15 fps in actual measurements and the measured time was fivethe seconds. was five seconds. OscilloscopeMeasu

Motion Controller

OscilloscopeMeasu

Microscopic

Microscopic

Microscopic

Microscopic

Software Interface

Motion Controller

Camera

Piezoelectric

Camera

Camera

Software Interface

Ceramics

Piezoelectric

Measured Target

Ceramics

Motion

Motion

Isolation Platform

setupTarget for offline experiment. Camera Figure 5. System Measured

Figure 5. System setup for offline experiment.

Figure 5. System setup for offline experiment.

Isolation Platform

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The measurement It can can be seen that the output of the measurement results were were shown shown in in Figure Figure 6. 6. It the displacement curves curves in inthe theXXwwand and directions was the sinusoidal signal. The amplitude was ZwZdirections was the sinusoidal signal. The amplitude was 5 μm w 5and µmthe and the frequency Hz.output The output was consistent with the input vibration frequency was 1was Hz. 1The resultresult was consistent with the input vibration signal.signal. Since Since was no input Yw direction, the maximum amount of vibration in Ythe Yw direction there there was no input in theinYthe w direction, the maximum amount of vibration in the w direction was was less than 1 µm with disturbance from the external environment. The mean value of the vibration less than 1 μm with disturbance from the external environment. The mean value of the vibration in 17 µm. The variance of the vibration in the −17´μm. in directionwas was−1.24 ´1.24× ˆ thethe YwYw direction 1010 The variance of the vibration in the Yww direction was was ´ 2 4.48 ˆ 10−2. So, . So,the theproposed proposedmethod methodwas wasable abletotomeet meetthe themeasurement measurementrequirement. requirement. × 10 6

0.6 0.4

4 0.2 0 Amplitude/um

Amplitude/um

2

0

-2

-0.2 -0.4 -0.6 -0.8

-4 -1

-6

0

0.5

1

1.5

2

2.5 time/s

3

3.5

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4.5

-1.2

5

0

0.5

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1.5

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(a)

2.5 time/s

3

3.5

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4.5

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(b) 6

4

Amplitude/um

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-4

-6

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(c) Figure 6. 6. The The vibration vibration curves curves of of offline offlinevibration vibrationexperiment: experiment:(a) (a)in inXXw direction; direction; (b) (b) in in YYw direction; Figure w w direction; (c) in Z w direction. (c) in Zw direction.

5.5. Vibration Measurement 5.5. Vibration Measurement In the vibration measurement of the cryogenic target, the experiment system was established In the vibration measurement of the cryogenic target, the experiment system was established as as shown in Figure 3. The vibration frequency we focused on was 0–40 Hz. Since the camera’s frame shown in Figure 3. The vibration frequency we focused on was 0–40 Hz. Since the camera’s frame rate rate was 90 fps, the system was fully able to meet the measurement requirement according to the was 90 fps, the system was fully able to meet the measurement requirement according to the sampling sampling theorem. However, when the displacement of the target in the camera exposure time was theorem. However, when the displacement of the target in the camera exposure time was greater than greater than one pixel equivalent in the high frequency vibration, it could result in blurred images one pixel equivalent in the high frequency vibration, it could result in blurred images of the target. of the target. In this case, we can use the information of the blurred images such as geometric In this case, we can use the information of the blurred images such as geometric moments to directly moments to directly calculate the vibration amplitude [20–24]. calculate the vibration amplitude [20–24]. The measurement results were shown in Figure 7. The approximation of the μm/pixel scaling The measurement results were shown in Figure 7. The approximation of the µm/pixel scaling obtained by the pseudo-inverse of the Jacobian matrix in Equation (8) was 0.5 μm/pixel. As can be obtained by the pseudo-inverse of the Jacobian matrix in Equation (8) was 0.5 µm/pixel. As can be seen from Figure 7, the vibration amplitude in the Zw and Yw directions exceeded the amplitude in seen from Figure 7, the vibration amplitude in the Zw and Yw directions exceeded the amplitude in the the Xw direction. The maximum amount of vibration in the Yw direction reached 13 μm. Xw direction. The maximum amount of vibration in the Yw direction reached 13 µm.

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The Butterworth low-pass filter was used in this work to filter the high frequency vibration. Butterworth filter was used in this work to filter the high frequency vibration. The The Butterworth filter low-pass was The Butterworth filter was a  a z 1  a2 z 2  a3 z 3  a4 z 4 G( za) ` a0 z´11 ` ´22 ` a z3´3 ` a4 z´4 0 b01 b1 z 1 a2bz  b3 z3  b4 z 4 2z Gpzq “ ´ 1 ´ b0 ` b1 z ` b2 z 2 ` b3 z´3 ` b4 z´4

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where a0 = 9.3980851433795 × 10−2, a1 = 3.75923405735178 × 10−2, a2 = 0.563885108602767, where a0 = 9.3980851433795 ˆ 10´2 , a1 = 3.75923405735178 a2 = 0.563885108602767, a3 = 0.375923405735178, a4 = 9.3980851433795 × 10−2, b0 = 1, b1 =ˆ0,10 b2´=2 , 0.486028822068271, b3 = 0, ´ 2 ab34 == 0.375923405735178, 1.7664800872442 × 10a4−2.= 9.3980851433795 ˆ 10 , b0 = 1, b1 = 0, b2 = 0.486028822068271, b3 = 0, ´2 . b4 = 1.7664800872442 ˆ 10 The measurement results after filtering were shown in Figure 8. The spectrum of the vibration The measurement after9. filtering shown in Figure vibration experiment was shownresults in Figure As canwere be seen from Figure 8. 9, The the spectrum frequencyofofthe 1 Hz in the experiment in Figure 9. frequency. As can be The seentarget from reached Figure 9, frequency of 1 Hz in Yw directionwas wasshown the main vibration thethe ultra-low temperature inthe the Ycryocooler. was the vibration main vibration frequency. The the target reachedThe the operating ultra-low frequency temperature in main of the target was from cryocooler. of the w directionThe the cryocooler. main vibration of the frequency target wascomponents from the cryocooler. The in operating frequency cryocooler wasThe 1 Hz. In Figure 9, many were visible the spectra. Those of the cryocooler Hz. In Figure 9, many frequency components were visible vibration signals was were1 from mechanical excitation generated by the control systeminofthe thespectra. target. Those signals from mechanical excitation generated the control system of the target. Thesevibration vibrations can bewere eliminated by the suppressing vibration by bases. That vibration information These be eliminated by the suppressing bases. That vibrationofinformation can can bevibrations used to can calculate the vibration modes and vibration failure analysis information the measured be used to calculate the vibration modes and failure analysis information of the measured object. object.

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6. Conclusions Our main contribution is that a precision vibration measurement instrument is designed for flexible structures. For high precision measurement, the image Jacobian matrix is applied for transforming the image feature changes to the vibrations in Cartesian space. The improved algorithm based on the gradient Hough transform algorithm is used for the feature extraction. Actual measurements verify the effectiveness of the proposed methods. Acknowledgments: This work was supported by the National Natural Science Foundation of China under Grant 61473293 and Youth Innovation Promotion Association, CAS (2013097). Author Contributions: Xian Tao and Zhengtao Zhang directed and guided this research. De Xu supervised the research. Kai Wang and Xiaobo Qi provided suggestions on building the experiment platform. All authors discussed the results and agreed about the conclusions. Conflicts of Interest: The authors declare no conflict of interest.

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