Control methods for drive mode of MEMS vibratory gyroscope with spring hardening nonlinearity

Control methods for drive mode of MEMS vibratory gyroscope with spring hardening nonlinearity

In driving mode of MEMS gyroscope， the spring hardening nonlinearity would become significant when with large vibration amplitude. To solve this problem， two control methods， i.e. phase-locked loop(PLL) and self-oscillation loop， are compared in the presence of such a nonlinearity. The comparison results show that the driving method of PLL would fail to track the resonant frequency of the nonlinear mode due to the phase-response hysteresis in frequency domain， while the self-oscillation loop can drive the nonlinear mode to stabilize at a resonance frequency thanks to the working principle of the self-oscillation loop. A modified digital PLL drive method is presented， which can significantly improve the stability at the cost of a large driving force. Experiment results show that they are in agreement with those of the simulations， verifying the feasibility of the proposed driving method.

MEMS gyroscope； spring hardening nonlinearity； phase-locked loop； self-oscillation loop

Increasing the vibration amplitude of the drive mode is one of effective means to improve the sensitivity of MEMS vibratory gyroscope. However， the spring nonlinearity in the drive mode becomes significant when the vibration amplitude gets large. This nonlinearity in MEMS comb-drive resonators has been well described and analyzed in plenty of literatures[1-2]. In paper [3]， the nonlinear behaviors of the drive mode of microgyroscope are discussed in detail， including the external resonance and non-resonant hard excitation. The nonlinear dynamics of the gyroscope designed by our research group are presented in [4]. Although the source and the behavior of the nonlinear mode have been specifically studied， few of the literatures concentrate on close-loop control methods for the nonlinear mode.

In this paper， the phase-locked loop (PLL)[5]and the self-oscillation loop[6]， which are generally used in drive mode of MEMS gyroscope， are compared in the presenceof the nonlinearity. In Section 1， the effect of the spring hardening nonlinearity is briefly introduced， as well as the approximate analytical frequency response of the nonlinear drive mode. Section 2 gives the compare of the two control methods by using numerical simulations. A modified PLL drive method， together with its simulation，is presented in Section 3. Section 4 shows the results of the experiments carried out， and some conclusions are drawn in Section 5.

1 Spring hardening nonlinearity in drive mode

In the drive mode of MEMS vibratory gyroscope，the spring nonlinearity， generally referring to the cubic stiffness term， derives from the nonlinear deformation of the suspension beams. The folded beams operate out of the linear range under large deformation and introduce the cubic stiffness. The nonlinear drive mode of the gyroscope， consisting of the drive frame， proof mass，together with the actuating and sensing combs， can be modeled as

Where x， ξx，ωnx，γ，F(xiàn)，ωdand mxare displacement，damping ratio， natural frequency， cubic stiffness parameter，amplitude and frequency of drive force and effective mass of the drive mode， respectively. Equation (1) describes a classical Duffing oscillator which has no closed-form solution. However， by using the method of multiple scales with the assumptions of weak damping and weak nonlinearity， which are always satisfied in MEMS gyroscopes， an analytical approximation for the forced response can be obtained as

where Mis the magnification factor， aand?are the amplitude and the phase of the solution respectively for Equation (1). The amplitude-response curve leans toward the higher frequencies for a positive value of γ， resulting in spring hardening nonlinearity. The phase-response curve is also distorted， as seen in Fig.1. The peak value of the magnification factor can be found from Equation (2) as

at the frequency of

Equation (4) indicates that the maximum value of the magnification factor is independent of the cubic stiffness parameter， while Equation (5) manifests that the resonant frequency depends on the strength of the nonlinearity as well as the vibration amplitude.

A profound difference between the response of the linear system and that of the nonlinear system is that the response of the latter is multivalued. For a fixed value of the drive frequency， there can be three different amplitudes and phases in the hysteresis region， as exhibited in Fig.1. The parameter values used to plot the curves are ξx=10-5， ωnx=104×2π rad/s， F=0.5 μN(yùn)， mx=3.33×10-7kg and the cubic spring force equals 5% of the linear spring force. The middle value of the three values appeared in the hysteresis region is unstable and will not be observed in an experiment. A jump down (up) phenomenon takes place at the upper (lower) boundary of the hysteresis region as the drive frequency gradually sweeps up (down). The influence of the hysteresis and the jump phenomenon on control methods for the gyroscope will be studied in the next section.

Fig.1 Frequency response of nonlinear resonator

2 Drive methods of PLL and self-oscillation

The control system for the drive mode of MEMS vibratory gyroscope has two basic requirements. The system should: a) maintain constant amplitude of vibration velocity of the proof mass or， approximately， constant amplitude of vibration displacement； b) provide a demodulation reference signal in-phase with the velocity of the proof mass to eliminate the zero-rate output introduced by quadrature error. Generally to follow these requirements，MEMS vibratory gyroscopes can be driven by either a PLL or a self-oscillation loop. The typical block diagrams of the two control methods are shown in Fig.2， wherepreKis the gain of the pre-amplifier which transfers a displacement of the structure into a voltage within a reasonable range，andvfKis the gain of drive voltage to drive force.

For a linear system， which means γ=0 in Equation (1)， both the two control methods can stably drive the gyroscope at resonance， although through different approaches.In the drive method of PLL， the resonator is driven by the VCO whose frequency is controlled by a PID-type controller fed with the phase error. The drive frequency is stable around the frequency point where the phase lag of the drive mode is 90° with small ripples. In the self-oscillation loop， there is no such a component like VCO who outputs the drive signal continuously. However，when satisfying the oscillation conditions， widely known as Barkhausen criterion， the system maintains a stable vibration whose frequency is determined by the phase response of the resonator and that of the phase shifter.

Fig.2 Typical control block diagrams for the drive mode

In the nonlinear case， the governing equations of the control loop presented in (2a) are described as follows:

where A is the amplitude of the VCO output，1T is the time constant of the low pass filter (LPF)， k is the sensitivity of the VCO，0ωis the initial angular frequency of the VCO，1pKand1iKare the proportional parameter and integral parameter of the controller， respectively. Considering that Equations (6)-(9) cannot be solved analytically， and even an approximate solution will be rather complicated， the system is numerically simulated with the parameter values listed in Tab.1. The frequency of the VCO output fails to track the natural frequency of the drive mode of the gyroscope due to the hysteresis in the phase-response. The jump phenomenon occurring whenever the frequency sweeps up across the upper boundary or down across the lower boundary degrades the stability of the control system and makes the frequency widely fluctuate， as shown in Fig.3a.

Tab.1 Parameter values used in the simulations

Similarly， the governing equations of the control loop presented in Fig.2b are summarized as

Equations (10)-(13) are numerically simulated with the parameter values listed in Tab.1. The simulations show that the self-oscillation loop stably drives the gyroscope. The responses of the self-oscillation loop for different cubic stiffness parameters are very close except with different vibrating frequencies. The effect that the resonant frequency is higher than the natural frequency is verified in the spectrum of the displacement signal plotted in Fig.3b.

Fig.3 Simulations for drive methods of PLL and self-oscillation

Different from the drive method of PLL， the drive method of self-oscillation evolves a quasi-constant drive frequency which satisfies the phase condition of Barkhausen criterion. The frequency variation caused by the change of the amplitude is very small. Therefore， the hysteresis in the phase-response does not affect the stability of the system significantly. The essence which makes the two control methods different in stability is that the phaseresponse curve is multivalued with respect to frequency，but single valued with respect to phase.

3 The modified drive method of digital PLL

Considering the failure of the PLL drive method and the difficulty in implementing a precise phase-shifter within a wide frequency range in digital circuits， a modified digital PLL solution is presented in Fig.4a. The digital controlled oscillator (DCO) offers two groups of orthogonal signals with the same frequency. As the same with the original PLL solution， the frequency control loop regulates the phase difference between the displacement and the drive force to a desired value which is set by the parameter d with the relationship of πdφ=??. This modified drive method differs from the original one in an extra close-loop， called delay control loop. One of the two groups of signals from the DCO is delayed from the other group for a particular phase which is determined by an independent parameter in the delay control loop. The delayed group of signals is used for synchronizing the displacement signal of the drive mode. Once the delay control loop locks the displacement signal， which is orthogonal to the Coriolis force， the delayed group of signals can be used as the demodulation references to extract Coriolis signal and quadrature signal.

The special configuration of the modified drive method of PLL allows setting the desired value d other than 0.5. Thus， the nonlinear mode can be driven away from the exact resonant point to avoid severe hysteresis region. The stability of the system is significantly improved by the non-resonant vibration. However， the drive force is no longer in-phase with the Coriolis force and cannot be used as demodulation reference. Nevertheless， with the help of the delay control loop， the Coriolis signal and the quadrature signal can be easily distinguished. The validity of the modified PLL solution is verified through the simulation of the frequency tracking，illustrated in (4b) where the desired value d in the frequency control loop is set as 0.25.

Fig.4 The modified drive method of PLL

4 Experiments on the nonlinear drive mode

The structure of the studied gyroscope is designed to be driven up to a displacement of 20 μm which makes the cubic nonlinearity remarkable. Fig.5a plots the measured amplitude-response and phase-response of the nonlinear drive mode. The jump phenomenon was clearly observed at the frequencies of 7617.8 Hz and 7615.4 Hz which constrain the boundaries of the hysteresis region.

The drive methods based on PLL， including the original method and the modified one， were implemented in a field-programmable gate array (FPGA) with necessary front-end devices， such as amplifies， AD convertors and DA convertors. As predicted by the simulations carriedout， the original PLL method failed to stably drive the gyroscope. The frequency of the drive signal， illustrated in Fig.5b， widely fluctuated by more than 3 Hz.

Fig. 5 The nonlinear drive mode of the studied gyroscope

Taking the advantage of the flexibility of FPGA， the control algorithm presented in Fig.4a was easily implemented without any modification of hardware. Fig.6a shows the power-on response of the error between the regulated phase delay and the desired phase delay in the frequency control loop. Fig.6b presents the controlled phase relationships among the drive signal， the displacement signal and the reference signal for demodulation. The displacement signal lagged behind the drive force signal for about 45°. The reference signal for demodulation was orthogonal to the displacement signal and the quadrature signal， which implied that the reference signal was in-phase with the Coriolis force and could be used to eliminate the zero-rate output introduced by the quadrature error.

Fig.6 Experiment result of the modified drive method of PLL

The drive method of self-oscillation based on analog circuits was also tested. The vibration amplitude of the studied gyroscope was increased step by step in the self-oscillation loop. Tab.2 lists the different operating points of the drive mode with various drive amplitudes. It can be revealed from Tab.2 that the resonant frequency was increased when the drive amplitude increased， as predicted by Equation (5). The adaptability of the selfoscillation loop was verified through several gyroscopes. The vibration parameters of three gyroscopes with the same self-oscillation drive circuit are listed in Tab.3. The vibration amplitudes were set as 12 μm. For different gyroscopes， the variations of their natural frequencies ofdrive modes vary between tens to hundreds of Hz due to manufacturing tolerance. The self-oscillation loop adapted to the various gyroscopes without any adjustment of circuit parameter.

Tab.2 Operating points of the drive mode in self-oscillation drive circuit with various drive amplitudes

Tab.3 Vibration parameters of three gyroscopes with the same self-oscillation drive circuit

5 Conclusion

The drive methods of PLL and self-oscillation loop for the nonlinear drive mode of the MEMS gyroscope are studied in this paper. The PLL drive method fails to stably drive the nonlinear mode due to the hysteresis effect in the frequency domain， while the self-oscillation drive method shows good stability and adaptability. Considering the failure of the PLL method， a modified digital PLL method is presented. This modified drive method can stably drive the nonlinear mode by avoiding the fierce hysteresis region in the frequency response. The experiments are carried out， which verify the analysis results of the two original control methods and demonstrate the feasibility of the modified drive scheme.

[1] Braghin F， Resta F， Leo E， Spinola G. Nonlinear dynamics of vibrating MEMS[J]. Sensors and Actuators， A: Physical， 2007， 134: 98-108.

[2] Elshurafa A M， Khirallah K， Tawfik H H， Emira A， Abdel Aziz A K S， Sedky S M. Nonlinear dynamics of spring softening and hardening in folded-MEMS comb drive resonators[J]. Journal of Microelectromechanical Systems，2011， 20: 943-958.

[3] Tsai N C， Sue C Y. Stability and resonance of micromachined gyroscope under nonlinearity effects[J]. Nonlinear Dynamics 2009， 56: 369-379.

[4] Xu L， Li H， Liu J， Ni Y， Huang L. Research on nonlinear dynamics of drive mode in z-axis silicon microgyroscope [J]. Journal of Sensors， 2014.

[5] Sun X， Horowitz R， Komvopoulos K. Stability and resolution analysis of a phase-locked loop natural frequency tracking system for MEMS fatigue testing[J]. Journal of Dynamic Systems， Measurement and Control，Transactions of the ASME， 2002， 124: 599-605.

[6] Cui J， Chi X Z， Ding H T， Lin L T， Yang Z C， Yan G Z. Transient response and stability of the AGC-PI closedloop controlled MEMS vibratory gyroscopes[J]. Journal of Micromechanics and Microengineering， 2009.

DING Xu-kai， LI Hong-sheng， NI Yun-fang， SHAO An-cheng
(Key Laboratory of Micro-Inertial Instrument and Advanced Navigation Technology of Ministry of Education， School of Instrument Science and Engineering， Southeast University， Nanjing 210096， China)

1005-6734(2015)03-0379-06

具有硬彈簧非線性的MEMS振動(dòng)式陀螺儀驅(qū)動(dòng)模態(tài)控制方法

丁徐鍇，李宏生，倪云舫，邵安成
(東南大學(xué) 儀器科學(xué)與工程學(xué)院 微慣性儀表與先進(jìn)導(dǎo)航技術(shù)教育部重點(diǎn)實(shí)驗(yàn)室，南京 210096)

當(dāng)振動(dòng)式MEMS陀螺儀的驅(qū)動(dòng)模態(tài)的振幅較大時(shí)，驅(qū)動(dòng)模態(tài)中的硬彈簧非線性將變得顯著。在驅(qū)動(dòng)模態(tài)具有此非線性的情況下，比較了MEMS陀螺儀中常用的兩種控制方法，即鎖相環(huán)驅(qū)動(dòng)和自激驅(qū)動(dòng)。由于非線性模態(tài)在頻域內(nèi)的相位響應(yīng)有遲滯效應(yīng)，鎖相環(huán)驅(qū)動(dòng)方式不能穩(wěn)定地鎖定非線性模態(tài)的諧振頻率。然而得益于自激驅(qū)動(dòng)方式的工作原理，自激方式可以將非線性模態(tài)驅(qū)動(dòng)在諧振點(diǎn)上。提出了一種改進(jìn)的數(shù)字鎖相環(huán)驅(qū)動(dòng)方式。該改進(jìn)的驅(qū)動(dòng)方式以較大的驅(qū)動(dòng)力為代價(jià)，提高了控制回路的穩(wěn)定性。實(shí)驗(yàn)結(jié)果與仿真結(jié)果相一致，并且驗(yàn)證了所提出的驅(qū)動(dòng)方式的可行性。

MEMS陀螺儀;硬彈簧非線性;鎖相環(huán);自激回路

TH824+.3

A

2015-01-27;

2015-04-20

江蘇省科技支撐計(jì)劃資助項(xiàng)目(BE2014003-3)

丁徐楷(1988—)，男，博士研究生，從事MEMS慣性儀表研究。E-mail:dingxukai@126.com

聯(lián) 系 人:李宏生(1964—)，男，教授，博士生導(dǎo)師。E-mail:hsli@seu.edu.cn

10.13695/j.cnki.12-1222/o3.2015.03.018