Linearization of the scanning field for 2D torsional micromirror by RBF neural network

Sensors and Actuators A 121 (2005) 230–236

Linearization of the scanning field for 2D torsional micromirror by RBF neural network Yi Zhao∗ , Francis E.H. Tay, Fook Siong Chau, Guangya Zhou Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore Received 5 November 2004; received in revised form 11 January 2005; accepted 7 February 2005 Available online 25 March 2005

Abstract We report a radial basis function (RBF) neural network (NN) method to linearize the scanning field of a 2D torsional micromirror, which is distorted by the intrinsic nonlinearity of the electrostatic torques. The 2D micromirror system is modelled and the parameters are identified. The feasibility of the method is shown. The system model is coded in the analog hardware description language (AHDL) for system level simulation in SaberTM . The experiment is implemented in National Instrument’s LabVIEWTM with a real-time embedded controller and a data acquisition card. The NN algorithm is coded in C language as a Dynamic Link Library (DLL) to make it easily integrated into both Saber™ and LabVIEW™. The experimental results agree well with the simulation ones. The distortion rates are improved from 28% in simulation and 23% in experiment to a negligible level respectively. © 2005 Elsevier B.V. All rights reserved. Keywords: MEMS; 2D torsional micromirror; RBF neural network; Linearization

1. Introduction The 2D torsional micromirror based on micro-electromechanical systems (MEMS) technology has been investigated by several research groups for free-space fiber-optic switches [1], projection displays [2–4], and endoscopic optical coherence topography [5]. This system has three degrees of freedom (two orthogonal rotation movements and one normal direction movement). The displacement in the normal direction is small enough to be neglected. Thus the system can be modelled as a two inputs two outputs (TITO) system. Although some actuation methods such as magnetostrictive [6,7], thermal [8], piezoelectric [9] and electromagnetic [10] have been reported, the most widely used one is electrostatic actuation [11–13], which offers many advantages including low power consumption, fast response time and simple driving electronics. However, electrostatic actuation leads to the nonlinear relationship between the two tilt angles and the two input voltages. In [12,13], linearization of the scanning ∗

Corresponding author. Tel.: +65 9873 9457; fax: +65 6779 1450. E-mail address: [email protected](Y. Zhao).

0924-4247/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.sna.2005.02.004

field by differential driving scheme has been developed. This scheme can reduce the scanning distortion rate to 13%. It is well known that the neural network (NN) approach is suitable for nonlinear function approximation. Supplied with pairs of input and output samples, the NN is able to generate the correct output based on the derived training rules. First proposed by Broomhead and Lowe [14], the radial basis function (RBF) NN has become a popular technique for function approximation due to its simple structure and training scheme [15–18]. Compared with its main rival multi-layer perceptron (MLP), RBF NN is faster during training process and can approximate a nonlinear function more precisely [19]. In this paper, we apply the RBF NN technique to correct the scanning distortion, which is caused by the intrinsic nonlinearity and is worsened by the misalignment of the mirror plate. First, the system is modelled and its parameters are identified. We prove that if the approximation error is small enough, the scanning field can be linearized. Then the RBF NN is designed by selecting the centers, widths and hidden layer size (HLS). The centers of the RBF are selected by the k-means clustering algorithm while the widths are selected by the P-nearest algorithm. The HLS is selected by taking

Y. Zhao et al. / Sensors and Actuators A 121 (2005) 230–236

the mean squared error (MSE) into consideration. The RBF NN algorithm is coded into the Saber TM simulator for system level simulation. The method is also verified in experiment with the LabVIEWTM programming environment and a realtime embedded controller. Both the simulation results and the experimental results show that the scanning field can be well linearized by the proposed method.

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Table 1 System parameters Parameter

Value

Ix Iy (kg m2 ) Dx (N m/s) Dy (N m/s) Kx (N m) Ky (N m) xbias (␮m)

1.73 × 10−16 4.33 × 10−17 9.95 × 10−14 1.64 × 10−14 3.9 × 10−10 3.9×10−10 12

(kg m2 )

2. System modelling The device studied in this paper is fabricated by the multi-user MEMS process (MUMPs) poly-silicon surface micromachining process. The scanning electron microscopy (SEM) of the mirror is shown in Fig. 1(a). The mirror plate (400␮m × 400␮m) is made out of Poly1-Trapped OxidePoly2 sandwich structure with a total thickness of 4.25 ␮m. The double-gimballed structure is suspended by two pairs of torsional bars (180 ␮m long, 2 ␮m wide and 1.5 ␮m thick) for x- and y-axis scanning respectively. Fig. 1(b) shows the detailed dimensions of the structure. After fabrication, the micromirror is manually assembled and wires are bonded. The device is actuated by four electrodes, which lie underneath the mirror plate, as shown by the dashed gray squares in Fig. 1(b). The micromirror plate is raised up to a height of 70 ␮m from the substrate by pushing the four sliding plates towards the mirror center until they are locked by the mechanical latching structures. We model the mirror plate as a rigid body with two degrees of freedom and the torsional bars as linear elastic bodies. The system equations for the 2D torsional micromirror are given by

where 0 is the permittivity of air with the value of 8.85 × 10−12 F/m, Si (i = 1–4) and Vi are the integral range and voltage of the ith electrode respectively, g is the gap and α is the slope between the mirror plate and the substrate, given by the following equation:

Ix θ¨ + Dx θ˙ + Kx θ = Tθ

(1)

α(θ, φ) = cos−1 (cos θ cos φ)

Iy φ¨ + Dy φ˙ + Ky φ = Tφ

(2)

where Ix , Iy are the moments of inertia, Dx and Dy are the damping coefficients of air, Kx and Ky are the spring constants of the torsional bars for x-axis and y-axis respectively.

The electrostatic torques for x-axis and y-axis scanning are given by 1
2

0 Vi 2 4

Tθ =

 ×

i=1

1
2

0 Vi 2  ×

Si

(−y)

1 sin α α g − x cos θ sin φ + y sin θ 4

Tφ =



i=1

2 dx dy

(3)

dx dy

(4)

 Si

x

1 sin α α g − x cos θ sin φ + y sin θ

2

(5)

The system parameters are identified, as shown in Table 1. The parameter xbias comes from the misalignment of the electrodes and the mirror plate, which means the integral domains Si (i = 1–4) in Eqs. (3) and (4) have 12 ␮m bias along the x-axis.

Fig. 1. Micromirror system structure: (a) SEM picture; (b) detailed dimensions.

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The position of the light spot on the PSD screen can be obtained using the following equations:

Two independent voltages, Vx and Vy , combined with a bias voltage Vbias , is applied to obtain the four driving voltages. This is given in the following equations: −Vx + Vy 2

(6a)

−Vx − Vy V2 = Vbias + 2

(6b)

V3 = Vbias +

Vx − Vy 2

(6c)

V4 = Vbias +

Vx + Vy 2

(6d)

As for the modelling of the optical system, the relationship between the tilt angles and the position on the position sensitive detector (PSD) is derived. The optical system is illustrated in Fig. 2 and the parameters are shown in Table 2. The relationship between the incident light and the reflected light is [12] r = i − 2(i · m)m

(7)

After tilting angles θ and φ, the normal vector of the mirror plate m is    1 0 0 cos φ 0 sin φ    1 0  m0 (8) m =  0 cos θ − sin θ   0 0 sin θ

cos θ

i m0 n xPSD yPSD doo

xPSD = o s · xPSD

(10)

yPSD = o s · yPSD

(11)

xPSD = 2doo θ √ yPSD = 2doo φ

(12a) (12b)

3. Linearization solution by RBF NN The idea of linearizing the scanning field by RBF NN is illustrated in Fig. 3. The input pair (Vix , Viy ) is mapped to the expected PSD output pair (ˆxPSD , yˆ PSD ) linearly. The expected PSD output is then fed to the RBF NN to produce the actual voltage pair (Vx , Vy ) required to drive the 2D torsional micromirror to the position (xPSD , yPSD ). We define the micromirror system as the following mapping:  f (·) :

Vx Vy



 −→



xPSD yPSD

where f is a 2 × 2 matrix  f =

f11 (θ, φ) f12 (θ, φ) f21 (θ, φ) f22 (θ, φ)



The inverse of the above mapping is  f

−1

(·) :

xPSD yPSD



− sin φ 0 cos φ

 −→

Vx



Vy

The RBF NN is designed to approximate the inverse mapping. Since there always exists some approximation error, −1 the actual mapping of the RBF NN is f˜ . We have

Table 2 Optical system parameters Parameter

(9)

where xPSD and yPSD are unit vectors for xPSD -axis and yPSD axis respectively. Given θ and φ < 5◦ , the position of laser spot on the PSD can be obtained using the following equations:

Fig. 2. Optical system.

V1 = Vbias +

o s = doo n + dos r

Value 1 √ (0, −1, −1) 2 (0, 0, 1) 1 √ (0, 1, −1) 2 1 √ (0, −1, −1) 2 (1, 0, 0) 48 mm

−1 f˜ · f = I + ∆

where ∆ is the approximation error matrix. The position of the laser spot on the PSD is expressed by the following equation:    Vix kx 0 xPSD = (I + ∆) (13) 0 ky yPSD Viy where kx , ky are scale parameters specified by the user.

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Eq. (13) shows that if ∆∞ is small, the range of the PSD scanning field can be well linearized.

4. RBF neural network design The RBF NN takes on a structure with one input layer, one hidden layer and one output layer, as shown in Fig. 4. The RBF NN can be considered as a mapping in Euclidean space: T : Rr −→ Rs . Let xp ∈ Rr be the input vector, and ci ∈ Rr (i = 1, 2, . . . , k) be the centers. The output is then taken to be a linear combination of basis functions, given by yj (xp ) =

k


Fig. 4. Radial basis function neural network.

wji φi (xp − ci ),

j = 1, 2, . . . , s

(14)

j=1

where  ·  denotes the distance in Euclidean space, k is the HLS, φi is the basis function and wji is the output weight. Generally, the Gaussian function is used as the basis function due to its local property (φ( · ) −→ 0 as  ·  −→ ∞)[20]. The radial basis function is given by  p − ci  x φi (xp − ci ) = exp − , i = 1, 2, . . . , k σi2 (15) where ci and σi are the center and width of the ith neuron in the hidden layer respectively. The first stage of training the RBF NN is to get the centers ci and the widths σi . The k-means clustering method [21] is used to select the centers. This training method is unsupervised and the number of centers, k, is set in advance. k-means clustering initializes k centers randomly. Then each of the input data is reassigned to its nearest center in an iterative manner. At the end of each iteration, the centers are updated as follows: 1
p ci = x , i = 1, 2, . . . , k (16) Ni p∈Gi

where Ni is the number of samples in the clustering group Gi . The sum squared clustering function J is computed as J=

k



xp − ci 2

(17)

i=1 p∈Gi

The iteration terminates when the minimum of J is reached. The widths are computed by P-nearest neighbors method after all the centers have been finally obtained. Here P is set to

3. For each center, the P nearest centers are found. The width is obtained by

 P 1
σi =  (18) cm − ci 2 P m=1

The second stage is to obtain the output weights wji . Since the mapping of this stage is a linear superposition, the weights wji can be easily found by solving a set of linear equations. Eq. (14) can be written in matrix form by defining matrices with components (W)ji = wji , (Φ)pi = φi , Ypj = yj . This simplifies the equation set to ΦW T = Y and the formal solution for the weights is given by W T = Φ∆ Y

(19)

where Φ∆ ≡ (ΦT Φ)−1 ΦT is the pseudo inverse of Φ. W can be found by fast linear matrix inversion techniques, such as singular value decomposition (SVD) [22]. The last parameter that needs to be specified is the HLS. Too small an HLS leads to underfitting while too large an HLS results in overfitting and increases computing time [20]. In this paper, the cross validation method is applied to select the suitable HLS. Firstly, the whole data set D is randomly partitioned into an training subset T and a validation subset V. T is used to select the centers, widths and output weights and V is used to evaluate the RBF NN in terms of MSE. The HLS starts with a small number, such as 10. In each of the iteration, the RBF NN is trained by T and validated by V. The iteration terminates when MSE reaches its minimum.

5. System level simulation

Fig. 3. Scheme of the linearization by RBF NN.

The RBF NN method is simulated in Saber™. The layout of the simulation is shown in Fig. 5, where Ix , Iy , Dx , Dy , Kx , Ky have the same meaning as in Eqs. (1) and (2). The four capacitors between Si and the mirror plate are modelled by C1 to C4 . The input voltage pair (Vix , Viy ) is mapped to the expected position (ˆxPSD , yˆ PSD ) on the PSD

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Y. Zhao et al. / Sensors and Actuators A 121 (2005) 230–236

Fig. 5. Simulation layout in Saber™.

linearly. The signals pass through the RBF NN and the differential driving circuit to generate the actual four driving voltages V1 to V4 by Eqs. (6a)–(6d) to drive the mirror. The four voltages are then applied to the corresponding electrodes. When sampling the training data, the linear mapper and the RBF neural network are inactive. The bias voltage applied on the mirror plate Vbias is set to 50 V. Since the mirror will pull in if the applied voltages Vix and Viy are greater

than 2 V, we apply Vix and Viy from −2 to 2 V with a step of 0.2 V. Totally, 441 pairs of input and output data are sampled, as shown in Fig. 7. The distortion rate is 25% at the top, 28% at the two sides and 13% at the bottom, shown in Fig. 7(a). The difference in distortion rates between the top and bottom is due to the misalignment. The reason that the distortion rate of xPSD is bigger than that of yPSD is due to the scale difference from the scanning angles θ and φ to the position on the PSD, as shown in Eqs. (12a) and (12b). The linear mapper and the RBF neural network are activated to obtain the linearized scanning field, which is shown in Fig. 6(b). For training the RBF NN, the 441 pairs of data (xPSD , yPSD ) are taken as NN input and (Vx , Vy ) are taken as output. The whole data set is randomly partitioned into two parts, the training data set T (410 pairs of data) and the validation data set V (31 pairs of data). The relationship between the MSE and the HLS is shown in Fig. 7(a), where the MSE is normalized between 0 and 1. With the increment of the HLS, the MSE drops rapidly at the beginning. However, the MSE will increase steadily if HLS becomes larger. Fig.7(b) shows the simulation result and expected result for the validation data when HLS is selected as 200. This figure reveals that the simulation result is very close to the expected result.

Fig. 6. Simulated scanning field.

Fig. 7. Results of training RBF NN.

Y. Zhao et al. / Sensors and Actuators A 121 (2005) 230–236

235

Fig. 8. Experiment setup scheme.

Fig. 9. Experimental scanning field (Vbias = 50 V).

6. Experiment verification The RBF NN method for linearization has been implemented experimentally. The setup is shown in Fig. 8. A Hamamatsu™ C4674 2D PSD is used to detect the position of the laser spot on the screen. The PSD signals are fed into National Instrument™ PXI8175 Real-Time (RT) Embedded Controller with the data acquisition (DAQ) daughter card PXI6040E. The RT Controller can communicate with a host computer, where the LabVIEW™ programming environment is installed, via an Ethernet hub. Thus the position information on the PSD can be displayed and stored in the host computer in real-time. The experimental results are shown in Fig. 9. As in the simulation, 441 pairs of data (21 for xPSD and 21 for yPSD ) are sampled to train the RBF NN. The distortion rates at the left and right are larger than those at the top and bottom, as shown in Fig. 9(a). This is because of the different scale from the tilt angles to position on the PSD. The distortion rate at the top is larger than at the bottom and the scanning field is drifted along the y-axis. This is due to misalignment. The distortion can be well corrected by RBF NN, as shown in Fig. 9(b). The inner part of the scanning field is quite linear. Only the rim of the scanning field has been distorted a little. This can be explained by the fact that the RBF NN cannot

obtain sufficient information near the rim to approximate the nonlinearity.

7. Conclusion An RBF NN method to linearize the distorted scanning field of a 2D torsional micromirror has been presented and demonstrated. Both the simulation and experimental results show that the RBF NN can capture the nonlinearity and correct the distortion successfully. The distortion rates have been improved from 28% in simulation and 23% in experiment to a negligible level respectively.

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Biographies Yi Zhao received his BEng degree from Harbin Engineering University, MEng degree from Huazhong University of Science and Technology, PR China, in 1999 and 2002 respectively. Currently he is pursuing his PhD degree in National University of Singapore. His research interests are design, simulation and characterization of optical microsystems. Francis E.H. Tay is with the Mechanical Engineering Department in the National University of Singapore. He is an active researcher in the microsystems and some of his current research interests include biochip, wearable systems and microfluidics. He is concurrently a Group Leader in the Medical Devices Group in the Institute of Bioengineering and Nanotechnology. He has also been a Technical Manager of the Micro and Nano Systems Cluster in the Institute of Materials Research and Engineering. Fook Siong Chau is Head of Applied Mechanics Group in the Department of Mechanical Engineering, National University of Singapore, Singapore. His area of specialisation is in experimental stress analysis, particularly in the use of optical techniques for nondestructive evaluation and metrology. His current area of interest is in developing and characterizing micro-components and systems, particularly micro-optical devices. He is a member of the Institution of Engineers, Australia, the Institution of Engineers, Singapore. Guangya Zhou received the BEng and PhD degrees in optical engineering from Zhejiang Univeristy, Hangzhou, China, in 1992 and 1997 respectively. Currently, he is a research fellow in Microsystems Technology Initiative, National University of Singapore. His main research interests are in modelling and simulation of MEMS devices, micro-optics, diffractive optics, and MEMS devices for optical applications.