Distributed-Mass Micromachined Gyroscopes forEnhanced Mode-Decoupling

Cenk AcarMicrosystems Laboratory

Mechanical and Aerospace Engineering Dept.University of California at Irvine

Irvine, CA, USAcacar@uci.edu

Andrei M. ShkelMicrosystems Laboratory

Mechanical and Aerospace Engineering Dept.University of California at Irvine

Irvine, CA, USAashkel@uci.edu

AbstractA novel micromachined z-axis gyroscope with multi-directional drive-mode is presented. The design conceptaims to relax mode-matching requirement, eliminate ef-fects of directional residual stresses, and completely de-couple the drive and sense modes. The proposed ap-proach is based on de�ning multiple drive-mode oscil-lators, distributed symmetrically around the center ofa supporting frame. Quadrature error and zero-rate-output are minimized, since the instability and drift issuppressed, due to the enhanced decoupling of multi-directional linear drive-mode and the torsional sense-mode. By designing each drive-mode oscillator withincrementally spaced resonance frequencies, a constanttotal Coriolis torque is achieved over a wide range ofdriving frequencies, leading to relaxed mode-matchingrequirements; and thus, robustness and reduced sensi-tivity to structural and thermal parameter uctuationsare achieved. Bulk-micromachined prototypes have beenfabricated in a one-mask SOI-based process, and exper-imentally characterized.

KeywordsMEMS, inertial sensors, micromachined gyroscopes.

INTRODUCTIONMiniaturization of vibratory gyroscopes with continu-ously maturing micromachining technologies is expectedto become an attractive solution to current inertial sens-ing market needs [1]. However, the scarce capabilities ofphotolithography and micro-fabrication processes, andthe resulting inherent imperfections in the mechanicalstructure signi�cantly limits the performance, stability,and robustness of MEMS gyroscopes [3, 6]. Thus, fabri-cation and commercialization of high-performance andreliable MEMS gyroscopes that require picometer-scaledisplacement measurements of a vibratory mass haveproven to be extremely challenging.

The conventional vibratory micromachined rate gyro-scopes consist of a 2 degree-of-freedom system with asingle proof mass suspended by exures anchored tothe substrate, which render the mass free to oscillatein two orthogonal directions, namely the drive and the

Figure 1. Conceptual illustration of the Distributed-Mass Gy-roscope (DMG).

sense directions. The proof mass is sustained in reso-nance in the drive direction, and in the presence of anangular rotation, the Coriolis force proportional to theinput angular rate is induced, exciting the proof massin the sense direction. To achieve high sensitivity, thedrive and the sense resonant frequencies are typicallydesigned and tuned to match to a certain degree, andthe device is controlled to operate at or near the peakof the response curve [1, 4].

The drive and sense mode matching requirement in vi-bratory gyroscopes renders the system response verysensitive to variations in system parameters due to fab-rication imperfections and uctuations in operating con-ditions, which shift the drive or sense resonant frequen-cies [6]. For the devices packaged in vacuum to enhancethe sensitivity, the bandwidth of the resonance peaksis extremely narrow; leading to much tighter mode-matching requirements. Extensive research has beenfocused on design of symmetric drive and sense-modesuspensions for mode-matching and minimizing tem-perature dependence [13, 15]. However, especially forlightly-damped devices, the mode-matching requirementis well beyond fabrication tolerances; and none of thesymmetric designs can provide the required degree ofmode-matching without feedback control [4, 5].

Furthermore, as the modes are matched closer, the me-chanical interference between the modes becomes moresigni�cant, resulting in operation instability and drift.

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In order to suppress coupled oscillation and drift and tominimize the resulting zero-rate drift, various deviceshave been reported employing independent suspensionbeams for the drive and sense modes [8, 9, 10, 11, 13].The approach of decoupling drive and sense modes ledto the �rst integrated commercial MEMS gyroscopesproduced by Analog Devices [14]. The Distributed-Mass Gyroscope (DMG) concept also aims to decouplethe drive and sense modes, with the di�erence of em-ploying multiple linear drive-modes completely decou-pled from the rotational sense-mode; while providing awide-band robust drive-mode.

GYROSCOPE STRUCTURE ANDPRINCIPLE OF OPERATIONWe propose a novel micromachined gyroscope systemwith multi-directional linear drive-mode and a rotationalsense-mode to minimize quadrature error, eliminate ef-fects of directional residual stresses, and e�ectively de-couple the drive and sense modes.

The design concept is based on forming multiple drive-mode oscillators, distributed symmetrically around thecenter of a supporting frame. The distributed drive-mode oscillators are driven in-phase along the axes nor-mal to the tangents of the supporting frame, and con-strained in the tangential direction with respect to thesupporting frame. In the presence of an angular rota-tion rate about the z-axis, the Coriolis forces are in-duced on each proof mass orthogonal to each drive-mode oscillation direction (Figure 1). Thus, each ofthe induced Coriolis force vectors lie in the tangentialdirection, combining to generate a resultant torque onthe supporting frame. The net Coriolis torque excitesthe supporting frame into torsional oscillations aboutthe z-axis, which are detected by sensing capacitors forangular rate measurement.

The multi-directional and axi-symmetric nature of thedrive-mode oscillators has several bene�ts over a con-ventional gyroscope design. Firstly, since the drive forcesapplied to the drive-mode oscillators cancel out in alldirections due to the radial symmetry, the net driv-ing force on the structure reduces to zero. The sym-metry of the design is assured in the fabrication pro-cess, where the complete structure is de�ned lithograph-ically in one mask, eliminating the possibility of maskmisalignment. Furthermore, the sensing electrodes at-tached to the supporting frame do not de ect due to thedrive-mode oscillations. Secondly, instability and driftdue to mechanical coupling between the drive and sensemodes is minimized, since the structure allows completedecoupling of multi-directional linear drive-mode andthe rotational sense-mode. Thus, zero-rate-output andquadrature error are suppressed e�ectively in the pres-ence of structural imperfections. The symmetry of thedrive-mode oscillator structure about several axes also

(a)

(b)

Figure 2. (a) The frequency responses of the distributed drive-mode oscillators. (b) The frequency spectrum of the total Cori-olis torque generated by the distributed drive-mode oscillators.

cancels the e�ects of directional residual stresses, andelastic anisotropy of the structural material.

The Coriolis ResponseIn the proposed approach, the distributed drive-modeoscillators are driven in-phase towards the center, andconstrained in the tangential direction with respect tothe supporting frame. The constrained dynamics ofeach proof-mass along the associated drive axis withrespect to the supporting frame reduces to:

mi �xi + cx _xi + kxxi = Fd + 2miz _yi

where mi is the ith proof-mass, and xi it the drive-

mode oscillation amplitude of the ith mass. Thus, in

the presence of an angular rotation rate about the z-axis, the Coriolis forces, which are proportional to drive

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direction oscillation amplitudes, induced on each proofmass are:

Fci = 2miz _xi

The rotation-induced Coriolis forces are orthogonal toeach drive-mode oscillation direction. Thus, each of theinduced Coriolis force vectors lie in the tangential di-rection, combining to form a resultant torque on thesupporting frame. The net Coriolis torque generated asthe combination of each Coriolis force becomes:

�!

Mc =

nXi=1

�!r c ��!

F ci =

nXi=1

2rcmiz _xi � bkwhere rc is the position vector of the oscillator center-of-mass, and bk is the unit vector in the z-direction. TheCoriolis torque Mc excites the supporting frame intotorsional oscillations about the z-axis, which are de-tected by sensing capacitors for angular rate measure-ment. Thus, the equation of motion of the supportingframe in the sense-direction is

Iz��+Dz

_�+Kz� =Mc;

where � is the torsional de ection of the supportingframe, Iz denotes the moment of inertia of the sup-porting frame combined with the proof masses, Dz isthe sense-mode rotational damping ratio, and Kz is thetorsional sti�ness of the suspension structure.

WIDE-BANDWIDTH OPERATION FORIMPROVING ROBUSTNESSThe most prominent advantage of the design concept isthe capability to provide a wide-bandwidth operationregion in the drive-mode frequency response. If eachdrive-mode oscillator proof-mass is designed to incre-mentally increase by �m such that

mi = m0 + i�m;

then the resonance frequencies of each drive-mode oscil-lator will be incrementally spaced (Figure 2a). Since thetangential Coriolis forces induced on each proof massjointly generate a resultant torque on the supportingframe, a constant total Coriolis torque is achieved overa wide range of driving frequencies (Figure 2b). The de-vice is nominally operated in the at region of the Cori-olis torque frequency response. Thus, uctuation in sys-tem parameters that shift oscillator resonance frequen-cies will not result in a signi�cant change in the totalCoriolis torque through the operation frequency band.If the sense-mode resonance frequency is designed tobe accommodated in the same frequency band (Figure2b), the requirement on the degree of mode-matching isrelaxed, and robustness against structural and thermalparameter uctuations is achieved.It should be noticed that the resonance frequency sepa-ration of the oscillators are dictated by the bandwidthof the response, which is determined by damping. In or-der to obtain a at operation region in the drive-mode,

Figure 3. The effect of damping and resonance frequency sep-aration on the drive-mode response.

the frequency separation should be less that 10% of thebandwidth. If the separation of frequencies is large forlow damping resonators, spacing of the resonance peaksbecome signi�cant in the response (Figure 3), and the at operation region will not be achieved in the drive-mode response.

Sensitivity and Robustness AnalysesThe proposed design approach allows to widen the op-eration frequency range of the gyroscope drive-mode toachieve improved robustness, while sacri�cing the re-sponse amplitude. The optimal compromise betweenamplitude of the response and bandwidth (e�ecting sen-sitivity and robustness, respectively) can be obtained byselecting the frequency increments of the drive-mode os-cillators.

As a numerical example, the response of a device con-sisting of 8 drive-mode oscillators with resonance fre-quencies located at 6.87 to 6.99kHz and a frequencyspacing of 15Hz will be analyzed. For 1o=sec input an-gular rate and a Q factor of 100 in the drive and sensemodes, the supporting frame of the distributed-massgyroscope will have an amplitude of response equal to2:8�10�3�m at the sensing electrodes. If the frequencyspacing of the drive-mode oscillators is decreased from15Hz to 10Hz, the amplitude of the response in thesense direction will increase from 2:8�10�3�m to 3:9�10�3�m; while the response bandwidth will decreasefrom 200 Hz to 140 Hz, which is still over an order ofmagnitude larger than the bandwidth of a single-massconventional gyroscope. The bandwidth can be furtherwidened by increasing the number of oscillators (Figure4). Thus, the design concept provides more freedom in

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Figure 4. The effect frequency separation on the response gainand bandwidth (effecting sensitivity and robustness, respec-tively). The gain is maximized for zero frequency separation,and the overall bandwidth increases proportional to spacing.

de�ning trade-o�s between gain of the response (highersensitivity) and the system bandwidth (increased ro-bustness), typically guided by application requirements.

Figure 5. The results of frequency mismatch on the overallsystem response. A mismatch of 3% results in only 16% error,while mismatches less than 1% have no significant effect.

Figure 5 illustrates the case of a potential shift in thesense-mode resonance frequency, e.g. due to tempera-ture uctuations, residual stresses, or fabrication vari-ations. It is observed that the response amplitude issustained at a constant value to a great extent withoutrequiring feedback control or active tuning of resonancefrequencies. For example, a 3% shift in the sense-mode

resonance frequency results in only 16% error in the re-sponse amplitude, while mismatches less than 1% haveno signi�cant e�ect on the response. Without activecompensation, a conventional 2-DOF gyroscope can ex-hibit over 60% error for the same 1% frequency shiftunder the same operation conditions [7].

Finite Element Analysis ResultsIn order to optimize the system parameters and ver-ify the validity of the theoretical analysis assumptions,the operational modes of the system were simulatedusing the Finite Element Analysis package MSC Nas-tran/Patran. Each drive-mode mass of the analyzedprototype system is 1080�m � 510�m, suspended byfour 460�m� 5�m folded springs; yielding a resonancefrequency estimation of 3.28kHz with 159GPa elasticmodulus. Through FEA simulations, the drive-moderesonance frequency of the drive-mode oscillators wereobtained at 3.34kHz (Figure 6a). The torsional sensemode resonance frequency of the structure about thesense axis was then located at !z =3.41kHz (Figure 6b)with four 769�m � 10�m torsional suspension beams,by iteratively optimizing the beam length.

(a)

(b)

Figure 6. The FEA simulation results. (a) The linear in-planeresonance mode of the drive oscillators, obtained at 3.34kHz. (b)The torsional sensing mode of the complete structure, optimizedas !z =3.41kHz.

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FABRICATION OF A PROTOTYPEThe �rst generation bulk-micromachined prototype gy-roscopes were designed and fabricated in the UCI In-tegrated Nano-Research Facility. A one-mask processbased on SOI (Silicon on Insulator) wafers was devel-oped and optimized for high-aspect ratio structures.The developed process relies on deep-reactive ion etch-ing (DRIE) through the 100�m device layer, and front-side release of the structures by etching the Oxide layerin HF solution. The process and the device design wasoptimized to minimize notching at the Oxide interfaceand excessive undercutting. In the device, 10�m�10�mholes were used to perforate the suspended structures,and 10�m gaps were used in the sensing and actuationelectrodes. Each drive-mode oscillator was designedidentically in the �rst generation devices.

Figure 7. Scanning Electron Microscope (SEM) photograph ofthe characterized Distributed-Mass Micromachined Gyroscopeprototype.

Preliminary Experimental ResultsThe linear drive-mode resonance frequencies, and thetorsional sense-mode resonance frequency of the pro-totype gyroscope were measured in a cryogenic MMRProbe Station. The frequency response of the device

was acquired under atmospheric pressure and room tem-perature, using o�-chip transimpedance ampli�ers con-nected to an HP Signal Analyzer in sine-sweep mode.The drive-mode frequency responses were acquired uti-lizing one-port actuation and detection, where a singleelectrode was used for both driving and sensing at thesame time.

(a)

(b)

Figure 8. Experimental measurements of the frequency re-sponse: (a) The drive-mode oscillators, where each drive portis connected to two resonators; (b) The torsional sense-modefrequency response of the supporting frame.

The resonance frequencies of the drive-mode resonatorswere observed to be scattered between 1.324 kHz and1.354 kHz within a 30Hz frequency band (Figure 8a).The sense-mode resonance frequency of the frame wasmeasured at 835Hz (Figure 8b). The large deviation

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from the FEA results could be attributed to excessivelateral and vertical over-etching due to the mask resolu-tion, and the photolithography process. In atmosphericpressure, the Q factor in the drive and sense modes wereobserved as 34 and 28, respectively. The close spacingof the drive-mode resonance frequencies allowed all ofthe resonators to be driven at the same time, to jointlygenerate a resultant Coriolis torque. The total Corio-lis torque, which is estimated for 1o=sec input rate bysumming the measured response of the four ports, wasobserved to provide a at range of over 30Hz (Figure9). The next generation prototypes will be designed forhigher drive-mode damping for wider bandwidth of thedrive-mode response, and with separate ports for eachresonator for individually addressable characterization.

Figure 9. Experimental frequency response measurements ofthe total Coriolis force induced in the sense-mode for 1o=secinput rate, obtained by summing the measured drive-mode re-sponse of the four ports.

CONCLUSIONA novel distributed-mass micromachined gyroscope de-sign was proposed, that is based on de�ning multipledrive-mode oscillators, distributed symmetrically aroundthe center of a supporting frame. The design conceptcompletely decouples the drive and sense modes to min-imize quadrature error and zero-rate-output by forminga multi-directional linear drive-mode and a torsionalsense-mode, which prevents instability and drift due tomechanical coupling to a great extent. By designingeach drive-mode oscillator to have incrementally spacedresonance frequencies, a constant total Coriolis torque isachieved over a wide range of driving frequencies, lead-ing to relaxed mode-matching requirements. Thus, thedesign concept is expected to provide inherently robust,low-drift and reliable MEMS gyroscopes.

ACKNOWLEDGEMENTSThis work is supported by the National Science Foun-dation Grant CMS-0223050. The authors gratefully ac-knowledge the invaluable assistance of Hung-Pin (Har-rison) Chang, and Vu Phan in fabrication of prototypes.

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