A novel vibration measurement and active control method for a hinged flexible two-connected piezoelectric plate

Mechanical Systems and Signal Processing 107 (2018) 357–395

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A novel vibration measurement and active control method for a hinged flexible two-connected piezoelectric plate Zhi-cheng Qiu a,b,⇑, Xian-feng Wang a, Xian-Min Zhang a, Jin-guo Liu b a Guangdong Province Key Laboratory of Precision and Manufacturing Technology, School of Mechanical and Automotive Engineering, South China University of Technology, Guangzhou 510641, PR China b State Key Laboratory of Robotics, Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang 110016, PR China

a r t i c l e

i n f o

Article history: Received 25 September 2017 Received in revised form 22 January 2018 Accepted 25 January 2018

Keywords: Piezoelectric flexible hinged plate Active vibration control Binocular vision measurement Image processing RBFNNC algorithm Decoupling on measurement of the bending and torsional vibration

a b s t r a c t A novel non-contact vibration measurement method using binocular vision sensors is proposed for piezoelectric flexible hinged plate. Decoupling methods of the bending and torsional low frequency vibration on measurement and driving control are investigated, using binocular vision sensors and piezoelectric actuators. A radial basis function neural network controller (RBFNNC) is designed to suppress both the larger and the smaller amplitude vibrations. To verify the non-contact measurement method and the designed controller, an experimental setup of the flexible hinged plate with binocular vision is constructed. Experiments on vibration measurement and control are conducted by using binocular vision sensors and the designed RBFNNC controllers, compared with the classical proportional and derivative (PD) control algorithm. The experimental measurement results demonstrate that the binocular vision sensors can detect the low-frequency bending and torsional vibration effectively. Furthermore, the designed RBF can suppress the bending vibration more quickly than the designed PD controller owing to the adjustment of the RBF control, especially for the small amplitude residual vibrations. Ó 2018 Elsevier Ltd. All rights reserved.

1. Introduction A solar array system is one of the important components of spacecrafts. The solar array system is in folded state during spacecraft launch and ascent. After the spacecraft and launch vehicle are separated and the spacecraft is turned into the free flying orbit, the solar array system will be deployed by driving mechanism [1]. The large-scale flexible solar panel structures consist of several plates which are connected together by hinges. Due to the influence of orbit control or external disturbances, the vibration problem will be inevitably caused during operation. The vibration will last a long time without active control in space enviornment. Unexpected vibration will seriously affect the normal operation of spacecraft equipment, reduce its accuracy, accelerate structural fatigue damage, and even cause the tumble of the spacecraft [2]. Therefore, research on vibration measurement and suppression of flexible hinged plate structures is realistic and prospective. For vibration testing, sensors can be divided into contact type and noncontact type. Contact type sensors include piezoelectric patches (PZT, lead zirconate titanate), accelerometers, etc. PZT as a kind of smart material was effectively implemented to control the mechanical vibration of flexible structures [3]. Accelerometers are by far the most traditional and ⇑ Corresponding author at: Guangdong Province Key Laboratory of Precision and Manufacturing Technology, School of Mechanical and Automotive Engineering, South China University of Technology, Guangzhou 510641, PR China. E-mail address: [email protected] (Z.-c. Qiu). https://doi.org/10.1016/j.ymssp.2018.01.037 0888-3270/Ó 2018 Elsevier Ltd. All rights reserved.

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widely used sensors employed in modal testing due to broad band. However, the effects of mass-loading can corrupt a measurement, especially at higher frequency ranges or for lightweight structures. Furthermore, accelerometers typically measure motion at a limited number of discrete points [4]. Non-contact type sensors include vision camera, laser displacement sensor, scanning laser vibrometer, etc. Laser displacement sensors can measure motion at a limited number of discrete points, and each point needs a laser displacement sensor. Image-based stereo-photogrammetry techniques provide additional measurement capabilities for high-displacement and low-frequency vibrations, typically difficult to be measured with accelerometers and laser vibrometers. Furthermore, it can measure many points without mass-loading [4]. As a non-contact method, significant advantages of the noncontact digital video cameras are relatively low-cost, agile, ease of operation, high spatial resolution and flexibility to extract structural displacement responses at multiple points, a non-contact, global measurement technique [5]. Photogrammetric methods are particularly useful when the object to be measured is inaccessible or difficult to access, when the object moves and deforms, and when its contour and surface information is required [6]. Video camera based measurements have been successfully used for vibration measurements and modal analysis, based on techniques such as the digital image correlation (DIC) and the point-tracking [7]. Digital video cameras vibration measurement technology and the 3-dimensional point-tracking techniques have been used widely [8,9]. The photogrammetry technique is an image based measurement system and uses displacements of the points or surfaces on a structure to monitor the dynamics of the structure. Photogrammetry provides non-contacting full-field measurement capabilities and can be an alternative to the conventional point-wise measurement approaches [10]. Video camera based measurements have been used for vibration measurement of various types of structures successfully, such as civil engineering structures, helicopter blade beam structures, and bridge dynamic response displacements [11–13]. Avitable et al. [14] presented a method to measure the structural vibration using machine vision. Teyssieux et al. [15] uses a CCD camera to obtain the in-plane vibration displacement of a cantilever beam. Wang et al. [16] investigated a non-contact measurement method by using vision sensors, and they studied the fuzzy image by employing the method of image moment. Dubus et al. [17,18] investigated the relationship between the feature points of multiple images to estimate the vibration signal of the of a flexible arm’s tip. As for measurement on flexible plate structures, Purohit et al. [19] conducted experimental investigation on flow induced vibration of an externally excited flexible plate. A laser vibrometer, pressure microphone and a high-speed camera are employed to measure the plate vibration, pressure signal, and instantaneous images of the plate motion, respectively. Sabatini et al. [20] investigated an image-based technique for the vibrations data acquisition. In the experiments, the camera was used to identify the eigenfrequencies of the vibrating structure, and aluminum thin plates simulate very flexible solar panels. The intrinsic parameters of a camera should be calibrated before vision measurement. Zhang [21] proposed a technique to easily calibrate a camera. It only requires the camera to observe a planar pattern shown at a few different orientations. The extrinsic parameters for the binocular vision system should also be calibrated. The resolution of CCD cameras may be not adequate, or not fast enough frame rate, which resulted in long processing time for the images, thus, significant lag and low update rates will be caused [22]. Therefore, the important problem of vision sensing is the delay between image capturing and image processing. In addition, visual or optical devices require a free line of sight between object and camera. To overcome these problems, control algorithms should be investigated to compensate for time delay and improve the control performance for vision-based control. Obtaining the model of the system is the premise of the design and implementation of the model based controller. Among the modeling methods of the flexible structure, finite element method (FEM) is one of the most widely used methods for distributed parameter system. The solar panels were simplified as a single flexible plate, and the effect of the connecting hinges were neglected in the most previous works [23]. Xu and Li [24] obtained the model of the solar panel by using FEM, neglecting the effect of the connecting hinges. Zhang et al. [25] used 8-node Mindlin plate element and virtual work principle to obtain the dynamic equations of a piezoelectric cantilever plate. The control algorithms are vital for the control performance of the controlled objects. The design of controller must be ensured that the control system has a certain stability, adaptability and autonomy. With the development of control theory, a large number of active vibration control algorithms are investigated and utilized. To improve the control performance, intelligence algorithms have been employed, such as fuzzy control [26] and artificial neural networks algorithm. Fei and Zhou [27] proposed a robust adaptive control strategy using a fuzzy compensator for MEMS triaxial gyroscope, which has system nonlinearities, including model uncertainties and external disturbances. Artificial neural networks have been applied to control complex and nonlinear systems by choosing neural network structures and training the weights properly. A radial basis function neural network (RBF-NN) has many attractive features such as the ability of non-linear mapping and the ability of learning, as well as the ability to globally stabilize [28]. A RBF-NN is employed to approximate the nonlinearities and uncertainties. Fei and Ding [29] used RBF-NN in a class of time varying system in presence of uncertainties and external disturbance. A fully tuned RBF neural network is presented for a general class of strict-feedback nonlinear systems [30]. Fang et al. [31] combined sliding mode control and RBF neural network control algorithms for active power filter. For a class of nonlinear dynamic systems, Fei and Lu [32] proposed an adaptive sliding mode control system using a double loop recurrent neural network (DLRNN) structure. Abdeljaber et al. [33] introduced a new intelligent neural network controller to mitigate the vibration response of flexible cantilever plates by using the piezoelectric sensor/actuator pairs.

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For vibration suppression on flexible plate structures, the contact type sensors (for example, accelerometer, PZT sensor) will cause mass-loading effect and corrupt a measurement. Furthermore, they only measure motion at a limited number of discrete points. Thus, non-contact type sensors should be considered. Laser displacement sensors can measure motion at a limited number of discrete points. Therefore, the binocular vision measurement is utilized due to its low cost, and flexibility to extract structural displacement responses at multiple points, a non-contact global full field measurement technique. Compared with the previously published work, the major contribution of this work: (a) A kind of non-contact vibration measurement and control apparatus for piezoelectric flexible hinged plate is established with binocular vision; (b) Decoupling of the bending and torsional low-frequency vibration of the hinged flexible plate on measurement is realized by using binocular vision sensors and the decoupled detection method; (c) A nonlinear RBF neural network controller is designed to suppressed the bending and torsional vibration of the flexible plate. Simulation and experiments are conducted to verify the measurement method and the designed controller. The rest of this article is organized as follows. In Section 2, the vibration control system based on binocular vision is described, and the model is obtained by the finite element method (FEM). In Section 3, the processing method of the vibration image and the decoupling method of the bending and torsional vibration on measurement are investigated. Section 4 introduces the designing of RBF-NNC, and numerical simulations are carried out. Section 5 presents experiments using the designed control algorithms and the binocular vision measurement. Finally, a concluding remark is given in Section 6. 2. System description and finite element modeling 2.1. Description of piezoelectric flexible hinged plate system with binocular vision Fig. 1 shows the schematic diagram of the piezoelectric flexible hinged plate system with binocular vision sensors. The piezoelectric flexible hinged plate consists of two flexible plates, namely plate I and II, which are connected by connecting plate I and II. Several PZT patches are bonded on both surfaces of plate I for the purpose of driving the bending and torsional modes vibration. As detecting marks, several circle marks are bonded on one surface of plate II, used as image characteristic features. Two cameras are installed on a base, composed a binocular vision measurement system. The distance of the two cameras can be adjusted, to meet the requirement of stereo vision. Ten characteristic circles are arranged on plate II and the vibration image is acquired using two cameras. Images of these characteristic circles are processed to obtain the vibration information of the piezoelectric flexible hinged plate. Fig. 2 shows the locations of the bonded PZT actuators. According to previous works of Qiu et al. [23], the clamped side of plate I is the reasonable placement position for the bending mode actuators and the location orientation angle is 0 degree. PZT patches numbered 1–8 are bonded symmetrically on both surfaces of the plate close to the clamped side. Furthermore, the layout locations of the four PZT patches on each surface are symmetrical along the centerline in width direction. They are connected together on the circuit and used as one channel bending mode actuator. PZT patches numbered 9–12 are connected together on the circuit and used as one channel torsional mode actuator. The four PZT patches are bonded antisymmetrically in the middle of the longitudinal direction of the flexible plate I and close to sides of the widthwise edge. 2.2. Modeling for the two-connected flexible plate using FEM Four-node rectangular plate element is adopted to analyze the model of piezoelectric flexible hinged plate, as shown in Fig. 3. The element meshing and node numbering of the piezoelectric flexible hinged plate are illustrated in Fig. 4. Plate I and plate II are both divided into 429 elements and there are 14 nodes in the length direction and 34 nodes in the width direction. Furthermore, the connecting plate I and connecting plate II are both divided into 3 elements. The two-connected flexible plate has 864 elements in total. Since the two-connected flexible plate is treated as a one side clamped plate, in order to apply the boundary condition to the plate system, one should constrain all the degrees of freedom of nodes from 1 to 34 which are on the left edge of the plate I along the Y direction. Ten circle marks locate at the corresponding nodes, used as visual detection marks. According to Ref. [23], after assembling all the elements under the consideration of the boundary condition and the damping effect, one can obtain the equation of motion of the hinged flexible plate as

M €d þ C d_ þ Kd ¼ F P þ F

ð1Þ

where d ¼ ½w35 h35x h35y . . . w952 h952x h952y T is the nodal displacement vector; M and K are the global mass matrix and stiffness matrix, respectively; F P ¼ K p V a is the control force vector, and K p is coefficient matrix; V a ¼ f V a1 V a2 . . . V ana gT is the control voltage vector, and na denotes the number of PZT actuators; F P and F are the control force and external force vectors, respectively; C ¼ aM þ bK is the damping matrix, and a, b are the Rayleigh mass and stiffness damping constants. After assembling, the output displacement vector of the binocular vision measurement can be expressed as

YðtÞ ¼ S v d;

ð2Þ

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flexible f exible plate fl plate II

flexible f exible plate fl plate II II circle circle mark mark

camera camera base base

the motion control card piezoelectric amplifier

computer

terminal board

Fig. 1. The schematic diagram of the piezoelectric flexible hinged plate with binocular vision sensors.

Patch 1,2

Circle marks

Patch 3,4 Patch 9,10

Patch 5,6 Patch 7,8

Patch 11,12

Fig. 2. The schematic diagram of the piezoelectric flexible hinged two-connected plate.

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Fig. 3. The four-node rectangular plate element.

Fig. 4. Element mesh and node numbering for the piezoelectric flexible hinged two-connected plate.

where YðtÞ ¼ f Y 1 ðtÞ Y 2 ðtÞ . . . Y 10 ðtÞ gT represents the displacement vector of the ten circle marks; S v is the coefficient matrix. Using a transformation of d ¼ Wn, the equation of motion and output displacement vector are

(

 1 Kn  þM  1 F P þ M  1 F  1 C n_  M €n ¼ M YðtÞ ¼ S v Wn

;

ð3Þ

 ¼ WT CW; K  ¼ WT MW; C  ¼ WT KW; F p ¼ WT F p ; F ¼ WT F; W is the modal matrix; n is the modal displacement vector. where M Eq. (3) can be written as the state space representation

(

X_ ¼ AX þ Bu U a þ Bf f YðtÞ ¼ C l X

;

ð4Þ

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    0 I 0 _ T is the state vector; A ¼ where X ¼ fx1 ; x2 gT ¼ fn; ng ¼ is the system matrix; B u T 1 1 1   K  M  C  W K v is the conM M   0 trol force input matrix; Bf ¼  1 T is the external force input matrix; C l ¼ ½ S v W 0  is the output matrix. M W 3. Image processing The acquired vibration images are processed by using the line correction method, edge extraction and image moment. Then, the vibration information is obtained. The decoupling method on vibration measurement of the first bending and torsional mode is investigated. 3.1. Parallel binocular vision mode The left and right cameras are parallel, and the origin of the left and the right camera coordinate system are in their light heart, respectively. The origin of the left camera coordinate system is described as Ol and that of the right camera coordinate system is Or . The horizontal distance b between the origin of the left and right camera coordinate system is called the baseline. The horizontal lines are the direction of X c axis; the vertical direction is Z c axis; Y c axis is perpendicular to the plane X c Z c , as shown in Fig. 5. P l and P r are the projection points in the left and right camera imaging plane of point P. The triangular geometric relationship is expressed as

8 xl ¼ f xzww > > < xr ¼ f ðxwzwbÞ ; > > : y ¼ y ¼ f yw l

r

ð5Þ

zw

where ðxl ; yl Þ is the coordinates of Pl ; ðxr ; yr Þ is the coordinates of P r ; ðxw ; yw ; zw Þ is the three-dimensional coordinates of P; f is the focal length of the camera. The variables xw , yw and zw are obtained as

8 x ¼ bðulduo Þ > > < w yw ¼ bfx ðvf yl d v o Þ ; > > : zw ¼ bfdx

ð6Þ

where ðuo ; v o Þ is the coordinates of center of the image; ðul ; v l Þ is the pixel coordinates of Pl ; ur is the value of the horizontal pixel coordinates of Pr ; f x ¼ f =dx, f y ¼ f =dy; where dx and dy denote the physical dimensions of each pixel cell in the X axis and Y axis, respectively; d ¼ ul  ur is the parallax of point P in the image coordinate. From Eq. (6), one knows that the three-dimensional coordinates of the point P can be uniquely determined by knowing the camera’s intrinsic parameters uo , v o , f x , f y and parallax value d. 3.2. Calibration of the binocular vision measurement system In order to obtain the relationship between the object of the objective world and the images, one should establish the geometric model between the object and images. One should accurately know the camera’s intrinsic parameters uo , v o , f x , f y and camera extrinsic parameters. The process of getting intrinsic and extrinsic parameters of the camera is called camera calibration [34].

Fig. 5. The parallel binocular vision of three-dimensional mathematical model.

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The binocular vision measurement system should be calibrated before the three-dimensional coordinates of the spatial point, calculated by using Eq. (6). Before calibrating the binocular vision measurement system, the individual camera is calibrated to obtain the intrinsic parameters. Then, the obtained left and right camera parameters are used to complete the binocular vision system calibration, to obtain its structural parameters, including rotation matrix R and translation vector T. In the process of calibration, ðxw ; yw ; zw Þ is specified as the coordinates of P point in the world coordinate system; ðxl ; yl ; zl Þ and ðxr ; yr ; zr Þ are the coordinates of P point in the left and right camera coordinates system, respectively. The projection relationship can be derived as

8 9 xl > > > > " > = Rl R1 l r ¼ T > > z l 0 > > > : > ; 1

8 9 xr > > > #> > >  1 < yr = R T l  Rl Rr T r ¼ > > 0T 1 > > zr > > : ; 1

8 9 > xr > > > > > T
1 > zr > > > > > : ; 1

ð7Þ

where Rl and Rr are the rotation matrices of the left camera and the right camera, respectively; T l and T r are the translation vectors of the left camera and the right camera, respectively. From Eq. (7), the structural parameters of binocular vision measurement system R and T are obtained as

(

R ¼ Rl R1 r T ¼ T l  Rl R1 r Tr

:

ð8Þ

Zhang Zhengyou’s camera plane calibration method [21] is used to calibrate the binocular vision measurement system. MATLAB toolbox is used for calibration. The calibration flow chart is shown in Fig. 6. 3.3. Line correction Due to the installation and manufacture errors, it is difficult to meet the ideal model of parallel binocular vision. Therefore, the images are processed by using the line correction method. The aim is to keep that the conjugate line of the images is parallel, and the actual model of the parallel binocular vision is in line with the ideal model of parallel binocular vision. In addition, the polar line correction can be used to eliminate the vertical parallax of the corresponding feature points, and it also reduces the error for the subsequent vibration information extraction from the images. The line correction is a planar projection transformation of the two images. The corresponding poles of images are mapped to infinity so that the conjugate lines of images are on the same horizontal line. This is a process of transforming the imaging plane from Pl0 and Pr0 to Pl1 and Pr1 , where the conjugate lines of images are parallel to each other, as shown in Fig. 7. 3.4. Feature extraction and matching method of circle marks Open source computer vision library (Open CV) is a cross-platform, open source and free computer vision library of machine vision. It provides a variety of application programming interface (API) function of image processing, such as filtering bath, feature point detection and contour extraction. The OpenCV function library is utilized to complete the image processing. The geometrical moments of image reflect the image information. The centroid coordinates ðx; yÞ of the image shape can be obtained by the image geometric moments. The centroid coordinates is expressed as

8 1;0
: y ¼ M0;1

;

ð9Þ

M0;0

where M 0;0 is the zero-order image geometric moment; M 1;0 and M 0;1 are the first-order image geometric moments. Many disturbance factors exist in the environment. The image and the real picture are different. Therefore, one needs to do some pre-processing before extracting the feature of image. To eliminate the noise effect, the first step is to filter the noise. Then, the image is split to the background (flexible plate) and the foreground (circle marks). The circle coordinates of circular marks are extracted as feature points to complete the matching of the characteristic circles. Canny operator is used to eliminate the influence of noise, and the edge of these circles can be extracted well in the experiment. Canny operator is the best balance between image fuzzification and noise elimination. It is also the optimal approximation operator between signal-to-noise ratio and edge location. The Gaussian filter function is used to filter the image. Then, non-maximum point suppression method is used to locate the edge, and the edge becomes thinner. Using this property, the pixel gradient value is compared in the neighborhood and the non-maximum point is abandoned. Furthermore, the double threshold method is used to remove some false edge points. The contour of characteristic circle images is selected by the area method after that the edge of characteristic circle is extracted by using Canny operator. Then the center coordinates of these characteristic circles are obtained by using Eq. (9). The center coordinates of circular marks are extracted as the feature points, which provide the conditions of characteristic circle matching. Parallax and projection model can be combined to restore the three-dimensional information of object.

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start calibration

no

read the left image

read the right image

extract the corner

extract the corner

the calibration of internal parameter

the calibration of internal parameter

the analysis of error

the analysis of error

re-extraction corner

re-extraction corner

the optimization of internal parameter

the optimization of internal parameter

the processing is complete?

the processing is complete? yes

yes the calibration of external parameter

the calibration of external parameter

the internal and external parameters of left camera

the internal and external parameters of right camera

the internal and external parameters of binocular vision

save the results

end calibration Fig. 6. Calibration flow chart of the binocular vision system.

no

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365

Fig. 7. The schematic diagram of the line correction.

Firstly, the image must be processed by using line correction, so that the conjugate line was horizontal state. According to the constraint of pole line, the center of peer characteristics circle in the left and right images is at the same level of conjugate line. According to the order consistency constraint, the peer characteristic circle of left images and the peer characteristic circle of right images are arranged one by one correspondence, and the match of peer characteristic circle is completed in the left and right images. The verification is divided into two steps. The first step is that a characteristic circle is selected in the left image, and then it is found in the right image correspondingly. The second step is that the corresponding characteristic circle of the first step is found in the left image. The center coordinates are subtracted to obtain the parallax value after completing the characteristic circle matching and extracting the center coordinates. The three-dimensional coordinates of characteristic circle are obtained by using Eq. (6). 3.5. Decoupling method on measurement of bending and torsional vibration After using the line correction, Canny operator edge detection the characteristic circle contour extraction and image moment, the center coordinates of these characteristic circles are obtained. Assuming that the center of characteristic circle is point P 0 , and the initial moment is T 0 . The coordinate of point P 0 is ðx0 ; y0 ; z0 Þ when the flexible plate is stationary. Supposing that the vibration moment is T i , the coordinate of point P0 is ðxi ; yi ; zi Þ when the low frequency vibration is excited. One suppose that the out-of-plane displacement of point P 0 is Sz , the displacement in the x-axis direction is Sx , and the displacement in the y-axis direction is Sy . The expression of in-plane and outer displacement is

8 > < Sx ¼ xi  x0 Sy ¼ y i  y 0 : > : Sz ¼ zi  z0

ð10Þ

The camera’s optical axis and the vibration direction of flexible plate are same. Therefore, the change of z-axis coordinate value of point P 0 reflects the vibration information. The vibration displacement yðtÞ is

yðtÞ ¼ Sz ¼ zi  z0 :

ð11Þ

The numbering layout diagram of circular mark points on the flexible hinged plate is shown in Fig. 8. Among ten circles, the centers of the circles numbered 5 and 6 are the center line in the width direction of the flexible hinged plate. For the

Fig. 8. The numbering layout diagram of circular mark points.

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torsional vibration, the displacement is zero of the centerline in the width direction. And the size of torsional is equal and the direction is opposite on up and down side. They are symmetry of the center line of the flexible hinged plate, when the torsional vibration of the piezoelectric flexible hinged plate is excited. Let Sz ðiÞ; ði ¼ 1; 2; . . . ; 10Þ be the vibration displacement of the ith circle; then the decoupling method of bending and torsional vibration on measurement is accomplished. The vibration displacement of the center of the same column are added and then divided by 5. The bending vibration information is obtained as,

Sw ðtÞ ¼

Sz ðiÞ þ Sz ði þ 2Þ þ Sz ði þ 4Þ þ Sz ði þ 6Þ þ Sz ði þ 8Þ ; ði ¼ 1; 2Þ: 5

ð12Þ

The torsional vibration information is extracted by using the vibration displacements of these centers (such as No. 1 and No. 9), which are symmetrically located on the center line of the piezoelectric flexible hinged plate. The torsional vibration information is

Sn ðtÞ ¼

Sz ðiÞ  Sz ði þ 8Þ ; ði ¼ 1; 2Þ: 2

ð13Þ

The flow chart of the binocular vision measurement system is shown in Fig. 9. The flow chart comprises the calibration of the binocular vision measurement system, image acquisition, image processing, the extraction of vibration information, and

Fig. 9. The flow chart of vibration measurement using binocular vision sensors.

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the decoupling on measurement of bending and torsional vibration. In the image processing module, the image is processed by line correction so that the conjugate line of the images is parallel. Filter method is utilized to eliminate the noise. Then, Canny operator is used extracts the edge. The contour of images is extracted, and the contours of characteristic circles are selected by the area method after the edges of these characteristic circles are extracted by the Canny operator. Then, the center coordinates of these characteristic circles are obtained by using image moment of Eq. (9), and then the center coordinates are subtracted to obtain the parallax value after completing the matching of the characteristic circle. The three-dimensional coordinates of characteristic circles are obtained by Eq. (6). Finally, the vibration displacement of the piezoelectric flexible plate is obtained by Eq. (11). The bending and torsional vibration information is extracted by employing Eqs. (12) and (13). 4. Controller design The bending and torsional low-frequency vibration information obtained by using the binocular vision measure system can be used as feedback signal to suppress the vibrations. An RBF-NN control algorithm is designed. 4.1. The design of RBF-NN control algorithm RBF-NN is a feedforward network with three layers of neural cells, namely input layer, hidden layer and output layer, as shown in Fig. 10. x ¼ ½ x1 x2 . . . xn  is the input vector of the input layer, and the input layer has n neuron. u ¼ ½ u1 u2 . . . um  is the activation function of the hidden layer, and the hidden layer has m neuron. f is the output of the output layer. The input layer to the hidden layer ui ði ¼ 1; 2; . . . ; mÞ is nonlinear function. The hidden layer to the output layer is linear transformation. The connection weight vector is w ¼ ½ w1 w2 . . . wm . The whole RBF neural network output is

f ¼

m X

ui  wi :

ð14Þ

i¼1

The Gaussian kernel function is used as the activation function ui , and it is expressed as



ui ðxÞ ¼ exp 

 2 ; i ¼ 1; 2; . . . ; m; kx  c k i 2

1

ri

ð15Þ

where c i ¼ ½ci1 ; ci2 ; . . . ; cin  is the center vector of the i hidden layer neuron; ri is the width value of the i neuron. The RBF-NN is adopted to adjust the weight parameters of a single neuron. The block diagram is shown in Fig. 11. NNC stands for single neuron controller and NNI is the RBF neural network regulator. The inputs of the single neuron controller are x1 ¼ eðkÞ and x2 ¼ eðkÞeðk1Þ , where eðkÞ is the deviation signal. The control u is T

u¼

2 X xi wi ¼ x1  w1 þ x2  w2 :

ð16Þ

i¼1

The delta learning rule is adopted to update the weight of a single neuron controller, and it is expressed as

@y  xi ; @u

ð17Þ

wi ðkÞ ¼ wi ðk  1Þ þ Dwi ðkÞ þ aDwi ðkÞ;

ð18Þ

Dwi ðkÞ ¼ geðkÞ 

where i ¼ 1; 2; g is the learning rate of the single neuron controller; a is the momentum factor of the single neuron con@y troller; @u is the sensitivity information, also called the Jacobian information. The structure of RBF_NN is 3-9-1 structure. The kth time input of the RBF neural network is xðkÞ ¼ ½uðkÞ; yðkÞ; yðk  1Þ, the output of the kth time is

Fig. 10. The structure diagram of an RBF neural network.

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Fig. 11. The block diagram of RBF control algorithm.

yn ðkÞ ¼

9 X

ui  wi ;

ð19Þ

i¼1

where ui is the Gaussian kernel function. Jacobian information is calculated as 9 @yðkÞ @yn ðkÞ @yn ðkÞ X ci1  x1 wi  ui  ;  ¼ ¼ @uðkÞ @uðkÞ @x1 ðkÞ r2i i¼1

ð20Þ

where x1 ðkÞ is the first component of the input vector, namely x1 ðkÞ ¼ uðkÞ; ci1 is the center vector of the first input layer for the ith hidden layer neuron. From the gradient descent method, one can obtain that

8 @J Dwi ðkÞ ¼ g1 @w ¼ gen ui > > i > < 2 @J ik Dri ðkÞ ¼ g @ ri ¼ g1 en wi ui kxc r3i ; > > > : Dc ðkÞ ¼ g @J ¼ ge w u xj cij n i i r2 ij 1 @cij

ð21Þ

i

where i ¼ 1; 2; . . . ; 9; j ¼ 1; 2; 3, g1 is the learning rate. The update rules of the weight, width values and center vectors are based on the gradient descent method as

8 > < wi ðkÞ ¼ wi ðk  1Þ þ Dwi ðkÞ þ a1 ðwi ðk  1Þ  wi ðk  2ÞÞ ri ðkÞ ¼ ri ðk  1Þ þ Dri ðkÞ þ a1 ðri ðk  1Þ  ri ðk  2ÞÞ ; > : cij ðkÞ ¼ cij ðk  1Þ þ Dcij ðkÞ þ a1 ðcij ðk  1Þ  cij ðk  2ÞÞ

ð22Þ

where i ¼ 1; 2; . . . ; 9; j ¼ 1; 2; 3, a1 is the momentum factor of the RBF neural network. 4.2. Simulation results Simulations are carried out by using a PD control algorithm and the designed RBF control algorithms on bending and torsional vibration control for the piezoelectric flexible hinged plate, based on the mathematical model obtained by using FEM. In simulations, the geometric characteristics of circle marks are listed in Table 1. The flexible plates and connecting plates are made of same epoxy resin material. The structural sizes of the flexible plate and PZT patches are listed in Table 2. The geometric dimensions of the plates, connecting plate and PZT patches are listed in Table 2 and their material properties are listed in Table 3. Since the low-frequency modes of the piezoelectric flexible hinged plate are dominant in the vibration modes, the simulations of the first bending and the first torsional mode of the piezoelectric flexible hinged plate are carried out. The control voltages are applied to the piezoelectric flexible hinged plate when the vibration starts 1 s.

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The description of parameter

Shape The number of mark Diameter Arrangement Space

Circle 10 20 mm Rows: 5 rows, columns: 2 columns Space of rows: 110 mm, space of columns: 150 mm

Table 2 The geometric dimensions of the flexible plate, connecting plate and PZT patches. Geometric dimensions

Plate I

Plate II

Connecting plate

PZT patches

Length (mm)  Width (mm)  Thickness (mm)

539  500  2

500  500  2

40  50  2

50  15  1

Table 3 Material properties of the plate and PZT patches. Parameters

Symbols

Unit

Plate

PZT patches

Young’s modulus Poisson ratio Density PZT strain constant

E

GPa — kg/m3 m/V

34.64 0.33 1840 –

63 0.3 7650 166  1012

l qm d31

4.2.1. Simulations on the vibration of the first bending mode The time domain free vibration response of the first bending mode is shown Fig. 12(a). Fig. 12(b) shows that of Fig. 12(a) which zoom in the time axis. From Fig. 12(a), it can be seen that the free vibration without active control of the first bending mode will last for a long period of time, and it still has larger amplitude at 20 s. The PD control algorithm is applied for vibration control in the simulation. The time domain control response of the first bending mode using the PD controller is shown in Fig. 13(a). Fig. 13(b) shows the corresponding control voltage. Compared Fig. 13(a) with Fig. 12, one can see that the vibration of the first bending mode reduce too much smaller amplitude about 20 s using the PD controller. The larger amplitude bending vibration is suppressed quickly. However, the small amplitude vibration will last for a longer time. For bending vibration control in the simulation, the learning rate of the RBF-NN is g1 ¼ 0:1; the momentum factor is a1 ¼ 0:05, the initial input is x1 ¼ ½0; 0; 0. The initial value of the Gaussian function width value is z ¼ ½1; 1; 1; 1; 1; 1; 1; 1; 1T , where zi ði ¼ 1; 2; . . . ; 9Þ is the ith parameter of z. The initial value of the Gaussian function weight vector is given as w ¼ ½0:92; 0:99; 0:54; 0:63; 0:73; 0:83; 0:20; 0:48; 0:60T , where wi ði ¼ 1; 2; . . . ; 9Þ is the ith parameter of w. The initial value of the center vector of the Gaussian function is specified as

Fig. 12. Free vibration response of the first bending mode.

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Fig. 13. Vibration control results of the first bending mode under PD control.

2

3 5:00; 3:75; 2:50; 1:25; 0:00; 1:25; 2:50; 3:75; 5:00 4 c ¼ 5:00; 3:75; 2:50; 1:25; 0:00; 1:25; 2:50; 3:75; 5:00 5. The learning rate is chosen as g ¼ 0:0025; momentum factor 5:00; 3:75; 2:50; 1:25; 0:00; 1:25; 2:50; 3:75; 5:00 is a ¼ 0:05; the initial input is x ¼ ½0; 0; and the initial value of the weight of the single neuron controller is w ¼ ½0:82; 0:14T . The time domain control response of the first bending mode using the RBF controller is shown in Fig. 14(a). Fig. 14(b) shows the corresponding control voltage. The updating process curves z4 and z6 among z are illustrated in Fig. 15(a) and (b), respectively. The updating process curves w4 and w6 among w are shown in Fig. 16(a) and (b), respectively. The updating process curves c14 , c16 , c24 , c26 , c34 and c36 among c are the 4th and the 6th parameters of each row, as shown in Fig. 17(a), (b), (c), (d), (e) and (f), respectively. The adaptive adjustment process curves w0 and w1 among the single neuron controller w are shown in Fig. 18(a) and (b), respectively. From Fig. 14(a), one knows that the small amplitude vibration can be suppressed more effectively and the vibration amplitude can be suppressed approach zero in about 15 s under the RBF controller. Therefore, the PD and RBF control algorithm are have obvious suppression effect on the first bending mode vibration of the piezoelectric flexible hinged plate. The control effect of RBF algorithm is better than the PD algorithm in the small residual vibration stage. The reason is that the designed RBF control algorithm can adjust the control parameters. RBF algorithm still has a large amount of control volume in the small amplitude residual vibration stage, as shown in Fig. 14(b) compared with Fig. 13(b). Therefore, it can suppress small amplitude residual vibration quickly, as shown in Fig. 14(a).

Fig. 14. Vibration control results of the first bending mode under RBF control.

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Fig. 15. The updating curves of the width value.

1

-0.6 -0.65

0.8

-0.7

w

w

6

4

0.6 -0.75

0.4 -0.8 0.2

-0.85

0

-0.9 0

5

10

15

20

0

5

10

15

t/s

t/s

(a) The adjustment of w4

(b) The adjustment of w6

20

Fig. 16. The updating curves of the weight.

4.2.2. Simulations on the vibration of the first torsional mode Simulation is carried out for torsional vibration. Free torsional vibration and vibration control responses using PD control and the designed RBF-NN control algorithm are provided. Fig. 19 show the time domain free vibration response of the first torsional mode of the piezoelectric flexible hinged plate. Fig. 19(b) shows the vibration response of Fig. 19(a) which zoom in the time axis. From Fig. 19(a) and (b), one can see that the vibration of the first torsional mode will last longer than 20 s without active control, and still have larger amplitude at 15 s. The time domain vibration control response of the first torsional mode using the PD controller is shown in Fig. 20(a) and (b) shows the corresponding control voltage. For torsional vibration control using the designed RBF-NN control algorithm, the learning rate of a single neuron controller is g ¼ 0:4; the momentum factor is a ¼ 0:05; the initial input is x ¼ ½0; 0. The initial value of the weight of the single neuron controller is w ¼ ½14:5; 0:25T . The learning rate of the designed RBF-NN algorithm is g1 ¼ 0:65; the momentum factor is a1 ¼ 0:05; the initial input vector is x1 ¼ ½0; 0; 0. The Gaussian function center initial value vector is 2 3 5:00; 3:75; 2:50; 1:25; 0:00; 1:25; 2:50; 3:75; 5:00 The weight initial weight vector value is c ¼ 4 2:00; 1:50; 1:00; 0:50; 0:00; 0:50; 1:00; 1:50; 2:00 5. 2:00; 1:50; 1:00; 0:50; 0:00; 0:50; 1:00; 1:50; 2:00 w ¼ ½0:59; 0:071; 0:75; 0:15; 0; 0:15; 0:74; 0:45; 0:58T , where wi ði ¼ 1; 2; . . . ; 9Þ is the ith parameter of w. The initial Gaussian function width value vector is z ¼ ½3:9; 3:9; 3:9; 3:9; 3:9; 3:9; 3:9; 3:9; 3:9T , where zi ði ¼ 1; 2; . . . ; 9Þ is the ith parameter of z.

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-1.25

1.25

-1.3

c

c

16

1.3

14

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1.2

1.15

1.1 0

5

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t/s

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t/s

(b) The adjustment of c16

(a) The adjustment of c14 1.35

-1.2

1.3

c

c

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26

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1.25

-1.3

1.2 0

5

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t/s

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t/s

(c) The adjustment of c24

(d) The adjustment of c26

-1.2

1.3

c

c

36

1.35

34

-1.15

-1.25

1.25

-1.3

1.2 0

5

10

15

20

0

t/s

(e) The adjustment of c34 Fig. 17. The updating curves of the center vector.

5

10 t/s

15

(f) The adjustment of c36

20

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2.5

-0.12 -0.14

2

-0.18

1

1.5

w

w

0

-0.16

-0.2 1 -0.22

0.5

-0.24 0

5

10

15

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0

5

10

15

t/s

t/s

(a) The adaptive adjustment of w0

(b) The adaptive adjustment of w1

20

6

4

4

2

2

Displacement /mm

6

0 -2 -4

0 -2 -4

-6

-6 0

5

10

15

20

25

30

0

5

10

t/s

15

20

t/s

(a) Time domain response

(b) The 0 ~20 s of the free vibration

Fig. 19. Free vibration response of the first torsional mode.

6

300

4

200

2

100

Voltage /V

Displacement /mm

Displacement /mm

Fig. 18. The updating weight curves of the single neuron controller for the bending mode.

0

0

-2

-100

-4

-200

-6

-300

0

5

10

15

t/s

(a) Time domain response.

20

0

5

10 t/s

(b) Control voltage.

Fig. 20. Vibration control results of the first torsional mode under PD control.

15

20

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300

4

200

2

100

Voltage /V

Displacement /mm

The time domain vibration control response on the first torsional mode using the RBF controller is shown in Fig. 21(a) and (b) shows the corresponding control voltage. From Fig. 21(a) with Fig. 20(a), one can see that PD and RBF control algorithm are have obvious suppression effect on the first torsional mode vibration of the piezoelectric flexible hinged plate. Compared Fig. 20(a) with Fig. 21(a), it can be seen that the control effect of RBF algorithm and the PD algorithm is almost in the large amplitude vibration stage; the control effect of RBF-NN control algorithm is better than the PD algorithm in the small residual vibration stage. The reason is that the RBF-NN control algorithm can adjust the parameter on-line, as shown in Fig. 21(b) compared with Fig. 20(b). The updating weight curves of the single neuron controller w are shown in Fig. 22. The updating process curves among c are illustrated by the 1th, 3th, 7th, and 9th parameters of each row, as shown in Fig. 23. The updating process of weights w1 ; w3 ; w7 , and w9 among vector w are illustrated in Fig. 24. The updating process curves width value z1 , z2 ; z7 and z9 among z are shown in Fig. 25.

0

0

-2

-100

-4

-200

-6

-300 0

5

10

15

20

0

5

t/s

10

15

20

15

20

t/s

(a) Time domain response.

(b) Control voltage.

Fig. 21. Vibration control results of the first torsional mode under RBF control.

-10

0.5

-15

1

w

w

0

-20 0

-25

-30

-35

-0.5 0

5

10

15

20

0

5

10

t/s

t/s

(a) The adaptive adjustment of w0

(b) The adaptive adjustment of w1

Fig. 22. The updating weight curves of the single neuron controller for the torsional mode.

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-4.9

7

-5 6.5 -5.1 6

c

c

11

19

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5.5

-5.4 5 -5.5 4.5 0

5

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0

5

t/s

10

15

20

t/s

(a) The adjustment of c11

(b) The adjustment of c19

-1

3.2

3 -1.5

17

-2

c

c

13

2.8

2.6 -2.5 2.4

-3

2.2 0

5

10

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0

5

t/s

10

15

20

t/s

(c) The adjustment of c13

(d) The adjustment of c17

-1.5

4.5 4

-2

3.5 29

c

c

21

-2.5 3

-3 2.5 -3.5

2

-4

1.5 0

5

10

15

20

0

5

t/s

10

15

t/s

(e) The adjustment of c21

(f) The adjustment of c29

Fig. 23. The updating curves of the center vector.

20

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-0.5

3

2.5 -1

-1.5

c

c

23

27

2

1.5 -2 1

-2.5

0.5 0

5

10

15

20

0

5

10

15

t/s

t/s

(g) The adjustment of c23

(h) The adjustment of c27

-1.5

20

4

3.5 -2

39

-2.5

c

c

31

3

2.5 -3 2

-3.5

1.5 0

5

10

15

20

0

5

t/s

10

15

20

t/s

(i) The adjustment of c31

(j) The adjustment of c39

-0.8

2

-0.9 -1 1.5

-1.2

c

c

33

37

-1.1

-1.3

1

-1.4 -1.5 -1.6

0.5 0

5

10

15

20

t/s

0

5

10

15

t/s

(k) The adjustment of c33

(l) The adjustment of c37 Fig. 23 (continued)

20

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0

4.5 4

-0.5

3.5 3 9

-1.5

w

w

1

-1

2.5 2

-2

1.5 -2.5

1

-3

0.5 0

5

10

15

20

0

5

t/s

10

15

20

t/s

(b) The adjustment of w9

(a) The adjustment of w1 0.2

3

0 2.5 -0.2 -0.4

2

7

w

w

3

-0.6 -0.8 -1

1.5 1

-1.2 0.5 -1.4 -1.6

0 0

5

10

15

20

0

t/s

5

10

15

20

t/s

(c) The adjustment of w3

(d) The adjustment of w7

Fig. 24. The updating curves of the weight.

5. Experiments 5.1. Introducing of the measurement and control experimental system The vision-based piezoelectric flexible hinged plate experimental setup is constructed, as shown in Fig. 26. It is composed of a piezoelectric flexible hinged plate, a control computer, a binocular vision measurement system, a two-channel piezoelectric amplifier, a data acquisition and control board. The two-channel piezoelectric amplifier (using 4 APEX PA240CX chips) can amplify voltages from 5 V to 5 V to 260 V to 260 V. The data acquisition and control board (GTS-400-PV-PCI) is made by Googoltech Corporation, with eight channels 12 bits of A/D (analogue-to-digital) conversion and four channels of D/A (digital-to analogue) conversion, through the matching terminal board (GT2-400-ACC2-V-AD16). The binocular vision measurement system composes two German Basler acA1600-60gc industrial cameras, two lenses, and light sources. The industrial camera’s performance parameters are listed in Table 4. The industrial lens is Japan’s Computar lens, and its main performance parameters are listed in Table 5. The guide rail (second generation F1 SLR slide of FAMOUS Fermans) is installed on the floor of the camera base, and the guide rail comes with a slider to facilitate the link PTZ (Pan/Tilt/Zoom). It contains a 360 degree rotating sphere to achieve any attitude of the camera and contains the left and right balls locking knob to convenient left and right hand fast locking ball. The industrial camera is connected to the spherical of PTZ via the interface (3/8 interface of international standard), as shown in Fig. 26. The height of cameras can be adjusted so that the characteristic circle marks of the piezoelectric flexible hinged plate are within the camera field of view. The white striped of LED light source is used in the experiments, and the light irradiation the

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6

4 3.5

5.5

3

3

z

z

1

5 2.5

4.5 2 4

1.5

3.5

1 0

5

10

15

20

0

t/s

5

10

15

20

t/s

(a) The adjustment of z1

(b) The adjustment of z 2

4.5

5.5

4

5

z

z

9

6

7

5

3.5

4.5

3

4

2.5

3.5 0

5

10

15

20

0

5

10

15

t/s

t/s

(c) The adjustment of z7

(d) The adjustment of z9

20

Fig. 25. The updating curves of the width value.

Fig. 26. Photo of the piezoelectric flexible hinged plate experimental system with binocular vision.

circular mark area by 45 degrees from the left and right sides of the piezoelectric flexible board so that the whole area is illuminated uniformly. The illuminated of the circular mark area can improve the quality of images, which is conducive to extraction of image features. In experiments, the binocular vision system is adopted to detect the vibration of the piezoelectric flexible hinged plate. The frame rate of the camera is specified as 50 frames per second, and the sampling frequency is 50 Hz.

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Table 4 Parameters of the basler acA1600-60gc industrial camera. Parameter type

The description of parameter

The output format of video Maximum frame rate The type of sensor chip The size of sensor chip Format Resolution (pixels) The size of pixel Pixel depth Transmission interface Power

Bayer RG8, Bayer RG12, Bayer RG12 Packed, YUV 4:2:2 Packed 60 fps COMS Horizontal: 7.20 mm, vertical: 5.4 mm 1/1.800 Horizontal: 1600, vertical: 1200 Horizontal: 4.5 lm, vertical: 4.5 lm 12 bits GIGE PoE or 12VDC

Table 5 Parameters of computar M1214-MP2 Industrial Lens. Parameter type

The description of parameter

Lens Focus The maximum value of lens diameter to focus ratio The maximum size of image The size of object Working range of aperture Working range of the focus Resolution Back focus The length of flange lapel The size of lens The rate of strain Mounting frame Temperature of work

M1214-MP2 12 mm 1:1.4 8.8 mm  6.6 mm 2/300 , 11.0(H) cm  8.3(V) cm F1.4–F16C 0.15 m –ln f At the center and the edge more than 100 pairs/mm 13.1 mm 17.526 mm U33.5 mm  28.2 mm 2/300 0.1% (y = 5.5), 1/200 0.35% (y = 4.0) C holder 20  C  þ50  C

5.2. Calibration of binocular vision measurement system The designed black and white checkerboard calibration plate is shown in Fig. 27. Fig. 27(a) is the design drawing of black and white checkerboard calibration plate. Fig. 27(b) is the physical photograph of the black and white checkerboard calibration plate. The parameters of the calibration plate are listed in Table 6. The images with different gestures and angles of the black and white checkerboard calibration plate are acquired by using a parallel binocular vision measurement system during calibration experiments, and the number of the images 26.

(a) The design diagram

(b) The physical photograph

Fig. 27. Black and white checkerboard calibration plate.

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Z.-c. Qiu et al. / Mechanical Systems and Signal Processing 107 (2018) 357–395 Table 6 The parameters of black and white checkerboard calibration plate. Parameter type

The description of parameter

Calibration plate’s size Grid size Ranks number Material Accuracy

Length: 300 mm, width: 260 mm, thickness: 5 mm Horizontal: 25 mm, vertical: 25 mm Rows: 10 rows, columns: 11 columns Aluminum plate 0.005 mm

Fig. 28. The extraction corner of the black and white checkerboard calibration plate.

Table 7 The calibration results of the left and right camera. Camera

fx

fy

u0

v0

Left Right

2695.17156 2698.14329

2695.11034 2698.07241

788.30452 793.86947

571.53661 583.36301

In order to reduce the calibration error and improve the calibration accuracy, the outermost layer corners of calibration plate are not processed, which means only the corners of the second to tenth rows, and the corners of the second and ninth columns are extracted. In total, 72 corners are processed in the calibration experiment, as shown in Fig. 28. The extracted corners are processed by the calculation and optimization algorithm, and these intrinsic parameters of the left and right cameras are obtained, are listed in Table 7. In order to evaluate the camera calibration accuracy, the corner has been re-projected in the calibration experiment. The left camera is considered as an example. Regard the upper left corner as the origin; the horizontal direction has 1600 pixels and the vertical direction has 1200 pixels; the whole image size is 1600  1200. As shown in Fig. 29, the deviation of corners coordinates extracted in the calibration experiment are mostly between 0.2 and 0.2, and the small part is distributed between 0.3 and +0.3, which ensures the calibration accuracy of camera, as shown in Fig. 30. The structural parameters of binocular vision measurement system are obtained as

2

3 0:989909312889330 0:046486936811925 0:133860064846911 6 7 R ¼ 4 0:046957290459838 0:9988968341788640 0:00035712758837 5; 0:13369579323092 0:0066392298716090 0:991000179363791 and

T ¼ ½135:94714

 0:79435 3:59681:

Fig. 31 is the space position relationship diagram between the left camera and the calibration plate. Fig. 32 is the space position relationship diagram between the right camera and the calibration plate. Fig. 33 is the space position relationship diagram between the cameras and the calibration plate. It is obtained by the calibration process of binocular vision measurement system. Fig. 33 shows the relative position relationship between the left and right cameras.

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Fig. 29. A corner image re-projected of the camera.

Fig. 30. The error image of corner.

Fig. 31. Between the left camera and the calibration plate.

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Fig. 32. Between the right camera and the calibration plate.

17 18 11133025 27 1 2 20 12 16 26 10 48 29 75 19 21 24 236 15 28 22 3

100

0

-100

Z Left Camera X Z Right Camera X Y Y

1200 1000

-200

800 600

-200 400 0

200 200

0

Fig. 33. Position relationship between the cameras and the calibration plate.

5.3. Active vibration control experiments The active vibration control experiments of the first bending and the first torsional mode of the piezoelectric flexible hinged plate are conducted, to validate the binocular vision measurement methods and the investigated control algorithms. The control software system is developed by using VS 2010 software, including the OpenCV function library. The interactive interface is programed for easy operation. The flow diagram of the vibration control software system is shown in Fig. 34. 5.3.1. Vibration control experiments of the first bending mode Active vibration control experiments on the first bending mode are conducted by using the designed RBF control algorithms and PD controller. Fig. 35 shows the time domain free vibration response of the first bending mode. It can be seen that the vibration of the first bending mode will last much longer than 20 s without active control. The corresponding frequency response obtained by employing Power Spectral Density (PSD) is shown in Fig. 35(b). From Fig. 35(b), one knows that the natural frequency of the first bending mode is 0.90 Hz, and the amplitude of the first bending mode is about 37.80 dB. The designed RBF control algorithm is applied for vibration control in the experiments. The time domain control response and the frequency response obtained by employing PSD of the first bending mode using the RBF controller are shown in Fig. 36(a) and (b), respectively. Fig. 37 shows the corresponding control voltage. The learning rate of RB-NN is g1 ¼ 0:1; momentum factor is a1 ¼ 0:05, and the initial input is x1 ¼ ½0; 0; 0; the Gaussian 2 3 5:00; 3:75; 2:50; 1:25; 0:00; 1:25; 2:50; 3:75; 5:00 function initial value vector is c ¼ 4 5:00; 3:75; 2:50; 1:25; 0:00; 1:25; 2:50; 3:75; 5:00 5; the updating process of c is 5:00; 3:75; 2:50; 1:25; 0:00; 1:25; 2:50; 3:75; 5:00 illustrated by the 4th and 6th parameters of each row, as shown in Fig. 38.

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start

the initialization of main program

the initialization and configuration of the motion control card

the initialization and configuration of the camera

timer setting

the acquisition of image data

draw the curve of bending and torsional vibration

the process of image

control ?

no

yes the filtering and phase shifting of signal

active control algorithm

outputs the control volume

no

draw the curve of control volume

the end of experiment ? yes save the experimental data

end Fig. 34. The flow diagram of the designed vibration control program.

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40

15

30

5

Gain /dB

Displacement /mm

10

0

20

10

-5 0

-10

-15

-10

0

20

40

60

80

100

0.2

1

t/s

f/Hz

(a) Time domain free vibration response

(b) frequency response

4.2

15

Displacement /mm

10 5 0 -5 -10

-15 0

5

10

15

20

t/s

(c) The 0 ~20 s of the free vibration response Fig. 35. Free vibration of the first bending mode of the piezoelectric flexible hinged plate.

20

15 10

Gain /dB

Displacement /mm

10 5 0

0

-5

-10 -10

-15 0

5

10

15

t/s

(a) Time domain response

20

-20 0.2

1 f/Hz

(b) Frequency response

Fig. 36. Vibration control results of the first bending mode under RBF control.

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385

300 200

Voltage /V

100

0 -100 -200

-300 0

5

10

15

20

t/s Fig. 37. The control voltage of RBF algorithms.

The initial value of the Gaussian function weight is expressed as a vector w ¼ ½0; 0; 0; 0; 0; 0; 0; 0; 0T , where wi ði ¼ 1; 2; . . . ; 9Þ is the ith parameter of w; the updating process of w is illustrated by w4 and w6 , as shown in Fig. 39. The initial value of the Gaussian function width value is z ¼ ½1; 1; 1; 1; 1; 1; 1; 1; 1T , where zi ði ¼ 1; 2; . . . ; 9Þ is the ith parameter of z; and the updating process of z is illustrated by z4 and z6 , as shown in Fig. 40. The learning rate of a single neuron controller is g ¼ 0:0025; momentum factor is a ¼ 0:05; the initial input is x ¼ ½0; 0; the initial value of the weight of the single neuron controller is w ¼ ½0:700; 0:008T ; the adaptive adjustment process of w is shown in Fig. 41. The parameters of the PD controller are chosen as kp ¼ 0:8 and kd ¼ 0:02. The time domain control effect and the frequency response obtained by employing PSD of the first bending mode using the PD controller is shown in Fig. 42(a) and (b), respectively. Fig. 43 shows the corresponding control voltage. From Fig. 35(a), it can be seen that the vibration response of the first bending mode will last much longer than 25 s without active control. From Fig. 42(a), one knows that it takes about 15 s to suppress the vibration using the PD controller. The larger amplitude vibration is suppressed quickly, while the small amplitude vibration lasts for a longer time. From Fig. 36(a), one knows that the small amplitude vibration can be suppressed more effectively and the vibration amplitude can be suppressed approach zero in about 15 s under the RBF controller. Therefore, the PD and RBF control algorithm have obvious suppression effect on the first bending mode vibration of the piezoelectric flexible hinged plate. The control effect of the designed RBF algorithm is better than that of the PD controller in the small residual vibration suppression stage. From Fig. 35(b), it can be seen that the vibration amplitude of the first bending mode is about 37.80 dB. From Fig. 36(b), one knows that the vibration amplitude of the first bending mode about 17.68 dB under the RBF controller, reduced by about 20.12 dB. From Fig. 42(b), it can be seen that the vibration amplitude of the first bending mode about 19.21 dB using the PD controller, reduced by about 18.59 dB. Therefore, the PD and RBF control algorithm are have obvious suppression effect on the vibration of the first bending mode. The reason is that the designed RBF-NN control algorithm can regulate the corresponding control parameters. The RBF-NN algorithm still has a large amount of control volume in the small amplitude residual vibration stage. Therefore, it can suppress the small amplitude residual vibration quickly, as shown in Fig. 36(a). From the experimental results, it can be concluded as follows: (1) It is feasible to use binocular vision to measure the first bending mode vibration of the piezoelectric flexible hinged plate. (2) The designed RBF control algorithm and the PD algorithm are can suppress the first bending mode vibration of the piezoelectric flexible hinged plate.

5.3.2. Active vibration control experiments of the first torsional mode Active control vibration experiments of the first torsional mode of piezoelectric flexible hinged plate are conducted by the investigated RBFcontrol algorithms and the PD control algorithms. The torsional information is a displacement signal generated by twisting the piezoelectric flexible hinged plate, which is positively correlated with the twist angle. The time domain free vibration and the frequency response obtained by employing PSD of the first torsional mode are shown in Fig. 44(a) and (b), respectively. One knows that the vibration of the first torsional mode will last over 20 s without active control; the natural frequency is 3.55 Hz, and the amplitude is about 23.87 dB. The designed RBF control algorithm is applied for vibration control for the piezoelectric flexible hinged plate during the experiments. The time domain control response and the frequency response obtained by employing PSD of the first

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torsional mode using the RBF controller is shown in Fig. 45(a) and (b), respectively. Fig. 46 shows the corresponding control voltage. The learning rate of the designed RBF-NN control algorithm is g1 ¼ 0:35; the momentum factor is a1 ¼ 0:05; the initial input is x1 ¼ ½0; 0; 0; the Gaussian function center initial value vector is 2 3 5:00; 3:75; 2:50; 1:25; 0:00; 1:25; 2:50; 3:75; 5:00 c ¼ 4 2:00; 1:50; 1:00; 0:50; 0:00; 0:50; 1:00; 1:50; 2:00 5, and the update process of c is illustrated by the 1th, 3th, 2:00; 1:50; 1:00; 0:50; 0:00; 0:50; 1:00; 1:50; 2:00 7th, and 9th parameters of each row, as shown in Fig. 47. The Gaussian function initial value of the weight is w ¼ ½0; 0; 0; 0; 0; 0; 0; 0; 0T , where wi ði ¼ 1; 2; . . . ; 9Þ is the ith parameter of w, the updating process of w is illustrated by w1 , w3 ; w7 , and w9 , as shown in Fig. 48. The initial value of the Gaussian function width value is z ¼ ½0:5; 0:5; 0:5; 0:5; 0:5; 0:5; 0:5; 0:5; 0:5T , where zi ði ¼ 1; 2; . . . ; 9Þ is the ith parameter of z. The updating process of z is illustrated by z1 , z2 , z7 and z9 , as shown in Fig. 49. The learning rate of a single neuron controller is g ¼ 0:05; the momentum factor is a ¼ 0:05; the initial input is x ¼ ½0; 0; the initial value of the weight of the single neuron controller is w ¼ ½4:50; 0:02T , and the adaptive adjustment process of w is shown in Fig. 50.

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The designed PD controller is applied for vibration control for the piezoelectric flexible hinged plate during the experiments. The parameters of the PD controller are chosen as kp ¼ 1:60 and kd ¼ 0:01. The time domain control response and the frequency response of the first torsional mode using the PD controller are shown in Fig. 51(a) and (b), respectively. Fig. 52 shows the corresponding control voltage. From Fig. 44(a), it can be seen that the vibration of the first torsional mode will last longer than 15 s without active control, and still have a larger amplitude. From Fig. 51(a), one can see that the vibration of the first torsional mode is reduced to small amplitude about 15 s using the PD controller. From Fig. 45(a), one knows that the vibration of the first torsional mode is reduced to small amplitude about 15 s under the RBF controller. From Fig. 44(b), one can see that the vibration amplitude of the first torsional mode about 23.87 dB. From Fig. 45(b), one knows that the vibration amplitude of the first torsional mode about 19.45 dB under the RBF controller, reduced by about 4.42 dB. From Fig. 51(b), one can see that the vibration amplitude of the first torsional mode about 17.00 dB using the PD controller, reduced by about 6.87 dB. Therefore, it is feasible to use binocular vision to measure the first torsional mode vibration of piezoelectric flexible hinged plate. The designed PD and RBF control algorithm have suppression effect on the first torsional mode vibration of the piezoelectric flexible hinged plate. However, the large measurement noises by using the binocular vision system in the torsional measurement will destroy the control effect to some extent; this is the reason why the designed RBF-NN algorithm will adjust control parameters.

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6. Conclusions A new non-contact binocular vision vibration measurement method for the piezoelectric flexible hinged plate is proposed, including both the bending and torsional vibrations. The model is obtained by using FEM. A RBF-NN control algorithm is designed and applied to control the vibration. Numerical simulations are carried out. Experimental setup is constructed and experiments are conducted. Experimental results demonstrate that the binocular vision measurement method is feasible on the low frequency vibration of the piezoelectric flexible hinged plate; and the decoupling of the bending and torsional vibration signals by binocular vision measurement is validated. Furthermore, the designed RBF-NN and PD control methods can suppress both the bending and torsional vibrations more effectively. Acknowledgements This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 51775190, U1501247), partially Supported by State Key Laboratory of Robotics Foundation – China (Grant No. 2017-O012). The authors gratefully acknowledge these support agencies. References [1] H.Q. Li, X.F. Liu, S.J. Guo, G.P. Cai, Deployment dynamics and control of large-scale flexible solar array system with deployable mast, Adv. Space Res. 58 (2016) 1288–1302.

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