Analysis and Control on a Specific Electromagnetic Micromirror

In this paper, 10 resonant modes were divided based on the structure of a specific two-dimensional electromagnetic micromirror from Professor Shen’s team and Finite Element Method (FEM), and using as many as 10 resonant modes to do such analysis was the first time according to the best of our knowledge. The results showed that the slow axis can participate in several resonant motions under the signals of resonant frequencies. In particular, participating in Mode 1 for slow-scan axis was the key reason to the instability of Vertical Refreshing Scanning (VRS) in raster scanning. In addition, a piecewise PID control based on filters design for this electromagnetic micromirror was proposed to suppress the unexpected resonant oscillation and to improve the angular positioning accuracy in slow-scan axis control. Finally, the proposed method was applied to electromagnetic micromirror stages, and the experimental results showed that the proposed approach was reliable.

Ji et al. [16] proposed a two-dimensional electromagnetic micromirror using single turn coil and radial magnetic field from concentric magnet assembly in 2007, which was the first paper reported on this topic. Design and fabrication of this electromagnetic micromirror and 6 vibration modes were focused on in their paper, and the control was located in open loop control. Chen et al. [5] reported a two-dimensional scanning micromirror electromagnetically driven for slow-scan axis and harmonically driven for the fast-scan axis with larger mechanical angles up to ±7.5 ∘ and ±12.5 ∘ , respectively. Structure and fabrication of the electromagnetic micromirror and the sensitivity of piezo-resistive sensor were emphasized to study, but control is still in open loop control. Steve et al. [17] focused on Newton's method to determine the harmonic coefficients of the two-dimensional electromagnetic micromirror, but controlling the mirror deflection to correctly produce the desired angular motion presented a significant engineering challenge. Han et al. [18] proposed a new electromagnetic micromirror of unique current paths and gimbal geometry to provide independent 2-DOF (degree-of-freedom) actuation capability and large mechanical half scan angle up to ±6.43 ∘ and ±4.2 ∘ , and PID control and a low-pass filter (LPF) were used in their research. However, the rings from driving coils of two axes were difficult to be eliminated. erefore, based on the structure of an electromagnetic micromirror supported by patents [19][20][21][22] from Prof. Wenjiang Shen's team, 10 resonant modes were divided to estimate the resonant frequencies for the first time in this paper. e results showed that the slow axis participated in several resonant motions under the signals of resonant frequencies, which were the key reasons for the instability in slow-scan axis control of this two-dimensional electromagnetic micromirror. Modal analysis of this electromagnetic micromirror was presented, and resonant frequencies were acquired. en, a piecewise PID controller with filters was designed to improve the performances of this instability. Finally, the proposed methods were applied to an electromagnetic micromirror platform to determine their reliability.

Modal Analysis of Electromagnetic Micromirror
Structure of this electromagnetic micromirror is shown in Figure 1. e inner frame is connected to the outer frame through slow beams (Beam 1 and Beam 2). Ellipse micromirror and ellipse reinforce ring are connected to the inner frame through quick beams (Beam 3 and Beam 4). Four-Terminal Piezo-resistive Sensors (FTPSs) [23] are designed at the end of Beam 2 and Beam 4, respectively. In the beginning, a magnetic flux density B is applied externally with 45°to both the quick beams and the slow beams. Meanwhile, the beams, outer frame, inner frame, ellipse reinforce ring, ellipse micromirror, and FTPSs are in the same plane. Quick beams are perpendicular to slow beams. Beam 1 and Beam 2, Beam 3 and Beam 4 are symmetrical, respectively, and they are not in torsion. m turns of DACs (Driving Au Coils) are formed elliptically on the inner frame to drive this electromagnetic micromirror by current flow. Since current flow in a wire results from motion of charges in this wire, a wire with current placed in a magnetic field where directions of current and electromagnetic cannot parallel exerts magnetic forces. For a differential length dl, of a wire of current I → placed in a magnetic field B → , the force d F → [24] is given by where × is a cross product. Combining with (1), an electromagnetic torque T e that DACs generate under B → is shown as follows: where r → j 1 is the distance between unit length of the coil and the rotation axis on the j 1 th coil (j 1 � 1, . . . , m), the innermost elliptical Au coil is the first coil, and the outermost elliptical Au coil is the m th coil. L j 1 is the closed path where current is running through on the j 1 th coil (j 1 � 1, . . . , m), and θ is the angle between I → and B → . Elliptical ring 1 and elliptical ring 2 are under ellipse micromirror and inner frame, respectively.
Based on this structure ( Figure 1) together with the main properties of this electromagnetic micromirror shown in Table 1, 10 resonant modes ( Figure 2) are divided through ANSYS Workbench [18] in FEM, and the characters of modal analysis are shown in Table 2, which shows a detailed description of the 10 kinds of movements corresponding to Figure 2.
Among motions in Figure 2, Mode 1 and Mode 6 are usually used in Lissajous scanning as main operations, but they are hard to be controlled [13]. Different from Lissajous scanning, in this paper, raster scanning [1] is applied (Figure 3), which can avoid rings caused by driving coils for the fast axis and the slow axis. Sine is a signal of single frequency. Any signal, such as triangular waves, step signals, and so on, can be thought of as a composite of many sine signals of different amplitudes and frequencies [25]. If these signals of different frequencies as the input signals inspired the resonant motions shown in Figure 2, the triggered ringing effects could also occur. To avoid this phenomenon, sine signal is chosen as the input voltage. Sine signal of a high frequency in the horizontal direction called Horizontal Fast Scanning (HFS) is given; meanwhile, sine signal of a low frequency in the vertical direction called Vertical Refreshing Scanning (VRS) is supplied. Mode 6 is used as the quick-scan axis and is called HFS in raster scanning; DACs mentioned in structure description of the electromagnetic micromirror is used as the slow-scan axis and is called VRS in raster scanning. e analysis results show that the slow axis participates in many resonant motions under the signals of resonant frequencies ( Figure 3 and Table 2). Combined with the FEM (Figure 2), in VRS control, when the slow-scan axis operates in a noisy environment with resonant frequencies shown in Table 2, ten resonant motions are inspired to destroy raster scanning, which means that the key to stable control of electromagnetic micromirror is the stability of   Discrete Dynamics in Nature and Society      controlling slow-scan axis. In this research, slow-scan axis control is focused on, and filters of the slow-scan axis are designed to give a solution to this issue.

Filters Design
Although there are many noise removal methods such as m-derived filtering, Butterworth filters, and elliptical filters [26], two issues of filtering for electromagnetic micromirror in this paper should be briefly discussed. One is that the designed filtering must effectively eliminate noise with frequencies close to that in ten modes to avoid vibrations of the electromagnetic micromirror. In particular, environment noise of frequencies near mode 1 should be filtered to ensure the reliability of slow axis control in laser scanning. Another is that the effectiveness of the signal must be ensured under no phase distortion to realize closed-loop control in slow-scan axis control for electromagnetic micromirror better. So, secondary filtering is developed. First filtering, equal-ripple low-pass filtering, would filter these noises with resonating frequencies effectively. Second filtering, zero-phase filtering, would preserve the features of the feedback signal at the exact location where they occur and would avoid a phase distortion.
In slow-scan axis control, noise with frequencies close to the resonant frequencies in ten resonant modes can inspire unexpected vibrations of the electromagnetic micromirror (Table 2).
e actual resonant frequency f ac1 of electromagnetic micromirror in Mode 1 is derived from the ideal resonant frequency f id1 � 770.97 Hz (see also Figure 2) as where the scaling factor (the ratio between the actual resonant frequency and the ideal resonant frequency) Q 1 � 0.8432 in Mode 1 in this paper. It can be derived from (3) as follows:

Equal-Ripple Low-Pass Filtering. Difference equation of
Finite Impulse Response (FIR) filtering was described by Charles and Shenoi [27,28]: where y(i) is the output, x(n − i) are the inputs, h(i) are the coefficients, and n is the filtering order. For slow-scan axis control in electromagnetic micromirror system, noise with frequencies close to resonant frequencies in ten modes can result in unexpected vibrations of the electromagnetic micromirror. Equal-ripple low-pass filtering, which has the advantage of controlling the boundary frequency strictly in the aiming frequency compared with other filters [29], is developed here.y f is the output of this equal-ripple low-pass filtering.
H d (ω) is defined as the ideal magnitude of equal-ripple low-pass filtering, H g (ω) is the actual magnitude, N is the length of filtering, which is usually chosen as an odd number, and n is the order of filtering; the weighted error function W(ω) is constant. en, the error E(ω) between H d (ω) and H g (ω) can be expressed as follows: with Combined with (A.14), the unit impulse response of the designed filtering satisfies When electromagnetic micromirror operates in slowscan axis control with equal-ripple low-pass filtering, the constrained inequality about passband cut-off frequency f p and stopband cut-off frequency f s should be satisfied: From (4) and (11), it becomes Urey [9] reported that the slow-scan axis with the signal of the frequency 60 Hz and the quick-scan axis with the signal of the frequency 20 kHz, respectively, could support SVGA (800 × 600 pixels) for display. In this paper, 60 Hz < 650 Hz.
So, sine signal with a frequency of 60 Hz is used to drive the slow-scan axis and noise with resonant frequencies in ten resonant modes should be filtered by equal-ripple low-pass filter. We set f p � 60 Hz, f s � 61 Hz. From (7), (10), and (A.15), we get the following: (2h(i)cos(ωi) − h(2)cos(2ω)) .

(14)
Combined with Chebyshev Best Consistent Approximation (CBCA) [29,30], E(ω) on equal-ripple low-pass filtering in domain F � [0, π] could be represented as In the domain F, coefficients are chosen to make the maximum of E(ω) minimum.
is is the first filtering.

Zero-Phase Filtering.
For slow-scan axis of the electromagnetic micromirror, in order to preserve the features of feedback signal avoiding a phase distortion, the second filtering, zero-phase filtering, is developed, which would help preserve features appearing at the exact location to avoid the phase distortion [31]. Two traditional basic algorithms can express zero-phase filtering. One is FRR (Forward filter, Reverse filter, Reverse output), and the other is RRF (Reverse filter, Reverse filter, Forward output) [32]. eir principles are similar and the principle of FRR is given here: In (17)- (20), y f (i) is the signal sequence from first filtering, * is the convolution algorithm, y i 1 (i), (i 1 � 1, 2, 3) are the output sequences after every transformation, h(i) is the digital impulse response sequence, and y s (i) is the final output sequence after the FRR. In the frequency domain, (21)-(24) are deduced: In (25), there is zero phase between Y s (e jω ) and Y f (e jω ). is is the second filtering.

Experiments on Filters.
Experiments and validations completed on platform (see Section 4) on these filters are shown in Figures 4 and 5. Figure 4 shows the slow-scan axis response from collecting circuit after equal-ripple low-pass filtering filtered by zero-phase filtering under external disturbance, especially under electromagnetic interference similar to cell phone calls. Equal-ripple low-pass filtering effectively eliminates the vibration caused by environmental noise with the frequency close to the resonant frequency 650 Hz in Mode 1 (see the third rows in Figure 4). After the equal-ripple lowpass filtering, only the signal with the frequency of 60 Hz is left, which is the same frequency of the input sine signal (see the first three rows in Figure 5), but this signal appears hairy (see the third row in Figure 4). Zero-phase filtering decreases this hairy appearance, especially in peaks and troughs (see the fourth row in Figure 4). Meanwhile, it preserves the frequency (see the fourth row in Figure 5). e forward coefficient In a word, the first filtering, equal-ripple low-pass filtering, succeeds in filtering the signal with resonating frequency for slow-scan axis control of electromagnetic micromirror effectively, and the second filtering, zerophase filtering, succeeds in preserving the features of feedback signal after equal-ripple low-pass filtering at the original location where they occurred as soon as possible.

Piecewise PID Control and Experiments
Feedback control of MEMS is complicated due to the limited availability of sensor and disturbing characteristics on MEMS device. In this paper, FTPS is used to acquire the output signal to indirectly detect the angle of slow-scan axis [1], and the resolution of FTPS on the end of slow axis in VRS control is 1V@3deg. Slow axis operates in open loop control with input sine signal voltage of a frequency 60 Hz and an amplitude 2V, respectively. e maximum absolute angle error 0.67 ∘ appears in sines peaks and troughs (shown in Figure 6). Discrete PID control with filters design is firstly considered to be utilized for angle error reduction in slow-scan axis control. Its algorithm is presented by with an integral gain K i : and a derivative gain K d : where T is a sampling time, e(k) is a current error at k point, e(k − 1) is a previous error at k − 1 point, K p is a proportional gain, and e(0) is set as 0.   Discrete Dynamics in Nature and Society Figure 7 shows its control results. Indirectly measured real angle from feedback voltage can track the desired angle to some extent (in the 2nd and 3rd row of Figure 7). e maximum absolute angle error in sines peaks and troughs is reduced, compared with that in open loop control (Figure 6). But this maximum absolute angle error is still big. Here, K p � 3.685, K i � 0.002144, and K d � 35.175, respectively.
Due to the accumulation in an integral part K i k j 2 �0 e(j 2 ), the angle error may be bigger for slow-scan axis control of electromagnetic micromirror in discrete PID control. erefore, a discrete piecewise PID control with filters design is developed and its algorithm can be denoted as follows: with a switching coefficient β: Figure 8 shows its structure diagram in slow-scan axis control. When |e(k)| > e 0 , discrete Proportional Derivative (PD) control is adopted to avoid the accumulation of integral part to reduce the angle error; when |e(k)| ≤ e 0 , discrete PID control is used to achieve the desired angle of slow-scan axis for the electromagnetic micromirror. Figure 9 shows that its control advantage is that the maximum absolute error is limited in 0.  Discrete Dynamics in Nature and Society removed in PID control as shown in Figure 10(b). Substantial amount of ringing effect was removed in discrete piecewise PID control (Figure 10(c)). Figure 11 shows the raster scanning realized by the single red laser beam projected perpendicularly to this electromagnetic micromirror of two dimensions and  Discrete Dynamics in Nature and Society reflected to the screen. In Figure 11, the slow axis operates in piecewise PID control; meanwhile, the quick axis is harmonically driven by sine signal with the resonant frequency 22.113 kHz and the amplitude 1.8 V supplied by the function waveform generator (Agilent 33250A). is raster scanning pixel resolution is SVGA (800 × 600 pixels).

Conclusions
In this paper, motions of electromagnetic micromirror succeed in being analyzed through FEM. Mode 1 is emphasized to be eliminated to improve the stability in the slow-scan axis control of raster scanning. e discrete piecewise PID control with secondary filters put forward here was efficiently verified by experiment to remove the ringing effect caused by the instability of slow-scan control. where f p is the passband cut-off frequency, f s is the stopband cut-off frequency, a p is the maximum passband attenuation, a s is the minimum stopband attenuation, ξ 1 is the ripple of the passband, ξ 2 is the ripple of the stopband, and F S is the sampling frequency. We set f p � 60 Hz, f s � 61 Hz, a p � 3.5, and a s � 12 and combining with equations (A.3)-(A.6) and F S � 5 kHz mentioned in experiments, we have  Figure 11: Raster scanning realized by the electromagnetic micromirror of two dimensions: 1. function waveform generator (Agilent 33250A); 2. electromagnetic micromirror with FTPS, driving circuits, and data acquisition circuits; 3. red laser; 4. screen; 5. dSPACE and PC; 6. V in 1; 7. V out 1; 8. power.

Data Availability
Partial properties of electromagnetic micromirror on structure used in analysis are protected by patient privacy and cannot be opened, while requests for further information if necessary will be considered by authors upon request (tany@shnu.edu.cn or wjshen2011@sinano.ac.cn).

Conflicts of Interest
e authors declare no conflicts of interest.