Two Novel Measurements for the Drive-Mode Resonant Frequency of a Micromachined Vibratory Gyroscope

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1 Sensos 2013, 13, ; doi: /s Aticle OPEN ACCESS sensos ISSN Two Novel Measuements fo the Dive-Mode Resonant Fequency of a Micomachined Vibatoy Gyoscope Ancheng Wang, Xiaoping Hu *, Bing Luo, Mingming Jiang, Xiaofeng He and Kanghua Tang College of Mechatonics and Automation, National Univesity of Defense Technology, Changsha , China; s: wangancheng@ymail.com (A.W.); uobing@nudt.edu.cn (B.L.); toppo@yeah.net (M.J.); hexiaofeng@nudt.edu.cn (X.H.); tt_kanghua@hotmail.com (K.T.) * Autho to whom coespondence should be addessed; xphu@nudt.edu.cn; Tel./Fax: Received: 5 Septembe 2013; in evised fom: 1 Novembe 2013 / Accepted: 1 Novembe 2013 / Published: 19 Novembe 2013 Abstact: To investigate the dive-mode esonance fequency of a micomachined vibatoy gyoscope (MVG), one needs to measue it accuately and efficiently. The conventional appoach to measue the esonant fequency is by pefoming a sweep fequency test and spectum analysis. The method is time-consuming and inconvenient because of the equiements of many test points, a lot of data stoage and off-line analyses. In this pape, we popose two novel measuement methods, the seach method and tack method, espectively. The fome is based on the magnitude-fequency chaacteistics of the dive mode, utilizing a one-dimensional seach technique. The latte is based on the phase-fequency chaacteistics, applying a feedback contol loop. Thei pefomances in pecision, noise esistivity and efficiency ae analyzed though detailed simulations. A test system is implemented based on a field pogammable gate aay (FPGA) and expeiments ae caied out. By compaing with the common appoach, feasibility and supeioities of the poposed methods ae validated. In paticula, significant efficiency impovements ae achieved wheeby the conventional fequency method consumes nealy 5,000 s to finish a measuement, while only 5 s is needed fo the tack method and 1 s fo the seach method. Keywods: micomachined vibatoy gyoscope; dive mode; esonant fequency; one-dimensional seach; golden section

2 Sensos 2013, Intoduction With thei advantages of low powe dissipation, compact bulk, and low weight, MVGs have boad applications in militay and commecial aeas, such as navigation assistance, vehicle platfom stabilization, consume electonics and automation. The opeation of an MVG based on the Coiolis foce. Fist, a vibation is geneated and maintained along the dive diection. If thee exists an angula movement aound the input axis, a Coiolis foce will be fomed. Then, the foce causes a vibation in the sense diection. One can obtain the input angula ate by detecting the vibation along the sense axis [1 3]. Commonly, the dive mode of an MVG is excited at esonance with constant amplitude. In this case, the sense-mode vibation fequency equals the dive-mode esonant fequency [4,5]. Ideally, the dive-mode esonant fequency is a constant depending on the designed stuctue paametes. In pactice, it is unpedictable and vaiable due to unavoidable fabication inaccuacies, tempeatue vaiations o stiffness aging [6 11]. When the fequency vaies, the vibation signal of the sense mode changes, and then the output of the gyoscope will dift, so the esonant fequency becomes a cucial paamete affecting the pefomance of the gyoscope. In ode to investigate its chaacteistics, it is necessay to focus on the measuement of the fequency. The conventional appoach to measue the esonant fequency is by pefoming sweep fequency tests and spectum analyses [12 15], which in this pape is called the sweep fequency method. Geneally, to cay out a sweep fequency test, special equipment such as a fequency esponse analyze [13] o dynamic signal analyze [14], a lot of data stoage, and off-line data analysis ae equied. The whole test pocess is time-consuming and inconvenient. In paticula, the sweep fequency method becomes unwokable when the esonant fequency changes apidly. The eason is that the dynamics of the system may have changed substantially duing the couse of measuement [11]. Theefoe, moe efficient methods should be exploed. In this pape, two novel esonance fequency measuement methods fo the dive mode of MVGs ae intoduced. One, called the seach method, is based on the magnitude-fequency chaacteistics, and utilizes a one-dimensional seach technique. The othe, called the tack method, is based on the phase-fequency chaacteistics, and applies a feedback contol loop. The poposed measuements can be un on-line and pefom efficiently and accuately. In the next section, we pesent and analyze dynamics of the MVG. Section 3 intoduces basic fundamentals of the poposed methods. Section 4 discusses the pefomance in simulation. Section 5 pesents the implemented system and expeimental esults. Finally, conclusions ae povided in Section Dynamics of the MVG A typical MVG includes a vibation stuctue suppoted by suspensions and some electodes. Nomally, the stuctue oscillates feely along two othogonal axes: the dive and sense axis. The oveall system can be modeled as a mass-sping-dampe stuctue having two degees of feedom (2-DOF), as shown in Figue 1 [16].

3 Sensos 2013, Figue 1. A simplified model of a MVG. Dampe Rotation ate Sense axis Sping Mass Dive axis In the dive diection, a contolled sinusoidal foce is geneated to make the mass vibate at the dive-mode esonant fequency and achieve stable amplitude by the use of the automatic gain contol (AGC) method [17]. When an angula otation aound the input axis exists, a Coiolis foce will be fomed. The foce causes a vibation in the sense diection. Ideally, the dynamics of MVG can be descibed as follows [16]: m 0 x Dx 0 x kx 0 x Fd 0 m y 0 D y y 0 k y y 2mx whee m denotes the mass of the vibation stuctue, x and y ae the oscillation displacements in the dive and sense axes. The paametes k x, k y epesent the stiffness, and D x, D y ae the damping paametes. The expession 2mx epesents the Coiolis foce and Ω is the input angula ate aound the otation axis. F d denotes the extenal foce that can be an electostatic, piezoelectic o electomagnetic foce. Fom Equation (1), the tansfe function fom the foce F d to the dive-mode vibation displacement x can be witten as: x 1 1/ m Hs () F ms D s k s Q s d x x ( / ) whee k / m is the esonant fequency and Q k m / D is the quality facto of the dive mode. x Accoding to the tansfe function in Equation (2), the magnitude and the phase as a function of fequency can be esolved by substituting s = j ω. They ae: A( ) H( j) x 1/ m Q ( ) / x (1) (2) (3) And: ( ) H( j) actan ( 2 2 ) Q (4) As an example, an MVG with the paametes listed in Table 1 is consideed. With these values, the magnitude A(ω) and phase φ(ω) ae as dawn in Figue 2. Fom this figue, the magnitude of the

4 () () A() (m/n) Sensos 2013, tansfe function inceases as the esonant fequency is appoached, and eaches a maximum nea the esonant fequency [5]. The phase stats at 0, eaches 90 at the esonant fequency, and tends to 180 at high fequencies. Futhe analysis will be pesented in Section 3. Table 1. Example paametes fo an MVG. Paametes m f / (2 ) Q Value 5 μg 4KHz 2,000 Figue 2. The magnitude and phase tansfe function plot, with the inset showing a detailed view aound the esonant fequency Fequency (Hz) Fequency (Hz) 3. Fundamentals of Dive-mode Resonant Fequency Measuements Taditionally, the sweep fequency method is adopted to measue the esonant fequency of the MVG. In this method, the fequency of the input signal inceases o deceases by a set step (the stepped fequency) in a given ange (sweep ange), and all of the esponse signals should be collected and ecoded. In ode to get a moe accuate esult, the stepped fequency should be small enough, and the dwell time at each fequency point should be long enough to avoid the influence of the tansient. Geneally, a sweep fequency test equies moe than a few minutes. A poblem aises that the obtained esonant fequency may not be the eal fequency because the dynamics of the system may have changed substantially duing the couse of the measuement. In this section, two efficient methods, the seach method and the tack method, ae intoduced and thei fundamentals ae pesented below The Seach Method Diffeentiating Equation (3) with espect to ω yields:

5 Sensos 2013, [ (1 ) 2 ] 2Q A'( ) m( ) / Q 3/2 (5) By setting A (ω) = 0, and consideing ω > 0, the only fequency whee the magnitude eaches its maximum value is solved to be: 2 m Q 1 1/ (2 ) (6) In pactice, the MVG is nomally packaged in specific vacuum level and Q is moe than 1,000 [18]. Thus, ω m is vey close to ω. Fo example, if Q = 1,000, (ω ω m ) ω =2.5e-7, which implies that the magnitude-fequency function A(ω) is unimodal and appoximately eaches the maximum value at the esonant fequency. Thus, to measue the esonant fequency is equivalent to finding maximum point of A(ω), and futhe to minimize A(ω). Accoding to the optimization theoy, a one-dimensional seach technique is a good candidate to solve the unconstained minimization poblem of a one-dimensional unimodal function [19]. Theefoe, we apply it to measue the esonant fequency of the dive mode. In this pape, we name the measuement as the seach method, whose pinciple diagam is depicted in Figue 3. The main modules and thei functions ae descibed below: Figue 3. The block diagam of the poposed seach method. A Seach contolle Oscillato sin( t) A 2 I Q 2 2 I Q Synchonous demodulation Asin( t) Dive mode of MVG Figue 4. The pinciple diagam of the synchonous demodulation technique. Asin( t) sin( t) Low pass filte I Low pass filte Q cos( t) The synchonous demodulation module is adopted to obtain the amplitude of the esponse signal, whose pinciple diagam is pesented in Figue 4. Obviously: 1 I LPF{sin( t ) sin( t)} cos 2 1 Q LPF{sin( t ) cos( t)} sin 2 (7)

6 Sensos 2013, Whee LPF epesents low-pass-filteing, I, Q denote the in-phase component and the quadatue component. Then the amplitude can be calculated as: (8) A I Q A seach contolle is used to contol seach pocess. It is ealized based on the one-dimensional seach technique, whose flowchat is shown in Figue 5. In the flowchat, y(f) = A(ω) = A(2π f) epesents function value, and f l, f h epesent the stat and end fequency of the seach ange, espectively. ɛ denotes eo toleance detemining whethe to end the seach o not. λ is a constant to contol the stepped fequency. Typically, when λ = 0.618, the algoithm is also called golden section algoithm. The oscillato is used to geneate sine signal fo a given fequency. Diect Digital Synthesize (DDS) is a candidate to achieve the function [20]. Figue 5. The flowchat of the applied one dimension seach technique. Stat Set f, f,, l h f1 fh ( fl fl ) f2 fl ( fh fl ) y1 y( f1) y y( f ) 2 2 Y y y 1 2 N fl f1 f f, y y f2 fl ( fh fl ) y y( f ) 2 2 fh f2 f f, y y f1 fh ( fh fl ) y y( f ) 1 1 f h fl f h Y 1 f ( fh fl ) 2 N End 3.2. The Tack Method Accoding to Equation (4): 2 2 ( ) Q cot (9) Defining a paamete ω to epesent the diffeence between the cuent excitation signal fequency and the esonance fequency, as: Equation (9) can be ewitten as: (10)

7 Sensos 2013, (2 ) Q cot ( ) (11) Geneally, ω is moe than 10,000 and ω is less than 200, so ω << ω, and: 2Q cot (12) It is implied that cotφ is in diect popotion to ω. So the value of cotφ can be thought as an indicato fo the diffeence of the cuent excitation signal fequency and the esonance fequency, that is to say, the fequency ω satisfied with cot[φ(ω)] = 0 equals the esonance fequency ω. Accoding to Equation (7), cotφ can be calculated as: cot I / Q (13) Based on the above analyses, a feedback contol system is poposed to measue the esonance fequency ω, which is shown in Figue 6. The main modules and thei functions ae intoduced as follows: Figue 6. The block diagam of the tack method. cot PID contolle Tack contolle Fequency adjusting Oscillato sin( t) cot I / Q I Q Synchonous demodulation sin( t ) Dive mode of MVG The synchonous demodulation module hee is the same as that in the seach method. The tack contolle is a key module in the system, which is used to smooth the value of cot and tune the cuent excitation signal fequency. It should be caefully designed accoding to the instantaneous esponse and steady-state pefomance of the whole contol system. The oscillato module hee is the same as that in Section 3.1, which is also used to geneate sine signals. The closed-loop contol system is simila to the phase locked loop (PLL) [21]. The main diffeence between the loop and conventional PLL used in dive-mode fequency keeping is the descibe. In a PLL, a phase detecto is equied and ealized accoding to Equation (4), so it needs an actangent calculation pocess, but in the poposed contol loop, the descibe can be ealized accoding to Equation (13), so the phase calculation is unneeded. Thus, as with a PLL, this loop has fewe calculations. Additionally, the descibe cotφ has lage lineaity ange than φ used in PLL, as seen in Figue 7 which shows the compaison of the poposed descibe and conventional PLL descibe. In the simulation, ω = 2π 4,000 ad, Q = 2,000. When the loop eaches stability, the fequency ω is locked to the esonant fequency ω. If the esonant fequency changes, the loop will tune cuent fequency ω to tack it. So the measuement is named as tack method in this pape.

8 Seach Result /Hz The output(nomalized) of descibe Sensos 2013, Figue 7. The compaison of the poposed descibe (cotφ) and conventional PLL descibe ( φ) cot (ad) 4. Simulation Analyses In this section, simulation systems ae built using Simulink to investigate the pefomance of the poposed methods, including the efficiency, pecision, and noise esistivity. MVG paametes used hee ae same as those in Table 1. Fo the seach method, we set the seach ange as 3,000~5,000 Hz, eo toleance ɛ as , and λ as Fo the tack method, we use PID contolle with P = 32, I = 256 and D = 0, and the contol peiod is chosen as 50 ms Simulations fo the Seach Method Fist, the pecision and efficiency ae studied by assigning six diffeent esonant fequencies in the ange of 4 5 khz. The esults ae pesented in Figue 8. The eo and seach steps ae listed in Table 2. Clealy, the elative eo of the seach method is on the ode of 10 8 and a measuement pocess equies only 44 seach steps. Figue 8. Measuement esults fo diffeent esonant fequencies using the seach method f=4000 f=4200 f=4400 f=4600 f=4800 f= Steps

9 Seach Results /Hz Sensos 2013, Table 2. Measuement eos and seach steps fo diffeent esonant fequencies using the seach method. Real Value(Hz) Measuement value(hz) Relative Eo e e e e e e-8 Seach steps Then, the pefomance compaison is simulated at diffeent noise levels. In the simulation, the esonant fequency to be measued is set as 4,000 Hz and andom noise with diffeent signal-noise-atio (SNR) is injected into the detected amplitude of the esponse signal. Simulation esults ae plotted in Figue 9 and listed in Table 3. Obviously, the effect of the noise to the measuement pecision is significant, while its effect on measuement efficiency is slight. When the SNR is 50 db, the elative eo of the seach measuement is on the ode of Figue 9. Measuement esults of the seach method unde diffeent level noise without noise with noise: SNR=70dB with noise: SNR=50dB with noise: SNR=30dB with noise: SNR=10dB Steps Table 3. Measuement esults statistics of the seach method unde diffeent level noise. Noise Level Without Noise SNR = 70 db SNR = 50 db SNR = 30 db SNR = 10 db Measuement value (Hz) Relative eo 6.248e e e e Seach steps Simulations fo the Tack Method Thee simulations ae caied out to investigate the pefomance of the tack method. In the fist simulation, the esonant fequency is set as a constant (f = 4,000 Hz). The tack esult and tack eo gaphs ae plotted in Figue 10. As shown in this figue, the tack eo appoaches 3.5e-7 Hz afte 5 s.

10 Fequency /Hz Relative Eo Tack Result /Hz Tack Eo /Hz Sensos 2013, Figue 10. Measuement value (left) and eo (ight) fo a constant esonant fequency (f = 4,000 Hz) using the tack method Tack esult Tack eo x Time /Second The second simulation is caied out to evaluate the pefomance fo tacking vaying fequency, whee the esonant fequency changes linealy with a 10 Hz/s slope fom 4,000 Hz to 4,050 Hz. The esults ae plotted in Figue 11, which shows the elative eo is about 5e-5. Figue 11. Tack esults when esonant fequency changes linealy fom 4,000 Hz to 4,050 Hz using the tack method x Refeence esonant fequency Tack esult Relative eo Time /Second Finally, the pefomance of noise esistivity is simulated. The esonant fequency hee is still set as a constant (f = 4,000 Hz), and andom noise detemined by the SNR is added to the detected phase. Tack eos unde diffeent level noise ae pesented in Figue 12. It is obvious that the tack method behaves well but its measuement pecision deceases. When SNR = 50 db, the measuement eo is less than 0.01 Hz.

11 Tack Eo /Hz Sensos 2013, Figue 12. Eos of the tack method unde diffeent level noise (f = 4,000 Hz) without noise with noise:snr=70db with noise:snr=50db with noise:snr=30db with noise:snr=10db Time /Second 5. Implementation and Expeiments In ode to study the pesented methods expeimentally, a measuement system is designed and implemented, as shown in Figue 13. The system includes two pats: the analog pat and digital pat. The analog pat is mainly used fo detecting the dive-mode vibation signal and filteing the noise. The digital pat is implemented on an FPGA chip, including synchonous demodulation module, contolle module, oscillato module and seial pot module. Contolles of these two methods, seach contolle and tack contolle, ae sepaate and pogammable to switch. The measuement esults ae tansfeed to PC though a RS232 cable. As a compaison, the sweep fequency method using fequency esponse analysis FRA5087 is also employed, labeled in dashed lines in Figue 12b. Figue 13. The implemented system fo expeiments. (a) The PCB. (b) The block diagam. Digital Analog To PC Seial Pot FPGA MVG Synchonous demodulation Seach contolle Tack contolle Oscillato A D C Filte Digital Pat Analog Pat Readout electonics Out FRA5087 In MVG I D A C (a) (b) Two expeiments ae caied out to assess the actual pefomance of the poposed measuement methods. One is fo nomal tempeatue test whee the esonant fequency vaies slightly. The othe is unde diffeent tempeatue conditions whee the esonant fequency changes significantly. In each

12 Measuement esult /Hz Sensos 2013, expeiment, thee methods ae used in tun to facilitate compaison. The common paametes in the expeiments ae pesented in Table 4. Paametes Table 4. Paametes in the expeiments. Seach Method Tack Method Sweep Fequency Method Range Stepped time P I D Contol peiod Range Stepped fequency dwell time Value KHz ms ms KHz 0.01 Hz 50 ms In nomal tempeatue expeiment, five epeated tests ae caied out. Figue 14 shows measuement esults and consumed-time compaison of thee methods. The statistics ae pesented in Table 5. Because the tue value of the dive-mode esonant fequency is unknown and vaiable, we cannot evaluate the measuement pecision diectly, but the measued aveage values of these methods ae close (the maximum diffeence is about 0.1 Hz), which denotes the feasibility of the poposed measuements. Besides, the tack method exhibits smalle measuement vaiance (about Hz), than the seach method (about Hz). As fo measuement efficiency, significant diffeences ae obseved that the sweep fequency method consumes nealy 5,000 s to finish one test, but the tack method only 5 s and the seach method 1 s. Thus, the poposed methods yield a big efficiency impovement. Figue 14. Expeimental esults of thee methods in nomal tempeatue. (a) The measuement esults. (b) Consumed-time compaison Sweep fequency method Tack method Seach method Sweep fequency method nealy 5000 s Consumed time (a) (b) Tack method 5 s Seach method 1 s Table 5. Statistics of expeimental esults in nomal tempeatue Aveage Standad vaiance Sweep fequency Method(Hz) Tack method(hz) Seach method(hz)

13 Measuement esult /Hz Sensos 2013, The second expeiment is caied out to measue the vaying esonant fequency. The system is placed in a the tempeatue chambe. We make the tempeatue change fom 5 C to 50 C with each tempeatue point kept fo 1 h. The measuement esults ae plotted in Figue 15. Clealy, the measuements using the thee diffeent methods ae appoximately equivalent, which again validates the feasibility of the poposed methods. Figue 15. Expeimental esults of thee methods fo vaying esonant fequency in diffeent tempeatue Sweep fequency method Tack method Seach method Tempeatue/ C It should be noticed that these expeimental esults ae inadequate to evaluate the measuement pecision diectly. The eason is twofold. Fist, the tue value of the dive-mode esonant fequency is unknown and vaiable. Second, fo each expeiment, thee methods ae conducted at diffeent time, meaning that thei tue values may be diffeent. 6. Conclusions This pape focuses on the measuement fo the dive-mode esonant fequency of a MVG. Two novel methods, the seach method and tack method, ae poposed. The seach method is based on the magnitude-fequency chaacteistics of the dive mode, and utilizes a one-dimensional seach technique, while the tack method is based on the phase-fequency chaacteistics, and applies a closed-loop contol technique. The feasibilities of the two measuement methods ae validated by simulations and expeiments. The simulations esults show that they behave well in both measuement accuacy and noise esistivity. When the SNR of a detected signal is 50 db, the elative eo of the measuement value using the seach method is only on the ode of 10 5, and 10 6 fo the tack method. Significant impovements in measuement efficiency ae achieved by the poposed methods. Expeimental esults show that the taditional sweep fequency method consumes nealy 5,000 s to finish one test, while only 5 s ae needed fo the tack method and 1 s fo the seach method.

14 Sensos 2013, Additionally, the poposed methods ae easy to implement on-line because they equie only a small amount of data stoage. They ae also applicable fo esonatos simila to MVGs. It should be noticed that these two novel methods fail to povide infomation about othe paametes, fo example, the Q-facto, so they ae suitable fo uses who only want to obtain the esonant fequency of a MVG. Acknowledgments This wok was suppoted by the National Natual Science Foundation of China (Gant No ) and National Univesity of Defense Technology Innovation Foundation Fo Postgaduate (Gant No N). Conflicts of Inteest The authos declae no conflict of inteest. Refeences 1. Yazdi, N.; Ayazi, F.; Najafi, K. Micomachined inetial sensos. Poc. IEEE 1998, 86, Baibou, N.; Schmidt, G. Inetial senso technology tends. Sens. J. IEEE 2001, 86, Liu, K.; Zhang, W.; Chen, W.; Li, K.; Dai, F.; Cui, F.; Wu, X.; Ma, G.; Xiao, Q. The development of mico-gyoscope technology. J. Micomech. Micoeng. 2009, 19, Pak, S.; Tan, C.-W.; Kim, H.; Hong, S.K. Oscillation contol algoithms fo esonant sensos with applications to vibatoy gyoscopes. Sensos 2009, 9, Saukoski, M. System and Cicuit Design fo a Capacitive MEMS Gyoscope; Helsinki Univesity of Technology: Espoo, Finland, Poeddy, S.R. Design and Dynamic Analysis of MEMS Gyoscopes; Univesity of Missoui-Columbia: Columbia, MO, USA, Lobu, M.; Holovatyy, A. Reseach of Influence of Mateials Popeties on Resonance Fequencies of Micomechanical Tuning Fok Gyoscope. In Poceedings of the Intenational Confeence on Pespective Technologies and Methods in MEMS Design (MEMSTECH 2007), Lviv-Polyana, Ukaine, May 2007; pp Zhang, L.M.; Uttamchandani, D.; Culshaw, B.; Dobson, P. Measuement of Young s modulus and intenal stess in silicon micoesonatos using a esonant fequency technique. Meas. Sci. Technol. 1990, 1, Walsh, D.; Culshaw, B. Tempeatue dependence of esonant fequency in all-fibe optically addessed silicon micoesonato sensos. Sens. Actuatos A 1991, 25 27, Hou, Z.; Xiao, D.; Wu, X.; Dong, P.; Chen, Z.; Niu, Z.; Zhang, X. Effect of axial foce on the pefomance of micomachined vibatoy ate gyoscopes. Sensos 2011, 11, M Closkey, R.; Gibson, J.S.; Hui, J. System identification of a MEMS gyoscope. J. Dyn. Sys. Meas. Cont. 2001, 123, Hill, J.J.; Jewson, P.M. Digital method of sweep fequency geneation. Electon. Lett. 1982, 18,

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