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Structural design and experimental characterization of torsional micromachined gyroscopes

with non-resonant drive mode

View the table of contents for this issue, or go to the journal homepage for more

2004 J. Micromech. Microeng. 14 15

(http://iopscience.iop.org/0960-1317/14/1/303)

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INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF MICROMECHANICS AND MICROENGINEERING

J. Micromech. Microeng. 14 (2004) 15–25 PII: S0960-1317(04)59396-8

Structural design and experimentalcharacterization of torsionalmicromachined gyroscopes withnon-resonant drive modeCenk Acar and Andrei M Shkel

Department of Mechanical and Aerospace Engineering, Microsystems Laboratory,University of California at Irvine, Engineering Gateway 2110, Irvine, CA 92697, USA

E-mail: cacar@uci.edu and ashkel@uci.edu

Received 10 February 2003, in final form 10 July 2003Published 18 August 2003Online at stacks.iop.org/JMM/14/15 (DOI: 10.1088/0960-1317/14/1/303)

AbstractThis paper reports a novel gimbal-type torsional micromachined gyroscopewith a non-resonant actuation scheme. The design concept is based onemploying a 2 degrees-of-freedom (2-DOF) drive-mode oscillatorcomprising a sensing plate suspended inside two gimbals. By utilizingdynamic amplification of torsional oscillations in the drive mode instead ofresonance, large oscillation amplitudes of the sensing element are achievedwith small actuation amplitudes, providing improved linearity and stabilitydespite parallel-plate actuation. The device operates at resonance in thesense direction for improved sensitivity, while the drive direction amplitudeis inherently constant within the same frequency band. Thus, the necessityto match drive and sense resonance modes is eliminated, leading toimproved robustness against structural and thermal parameter fluctuations.In the paper, the structure, operation principle and a MEMS implementationof the design concept are presented. Detailed analysis of the mechanics anddynamics of the torsional system is covered, and the preliminaryexperimental results verifying the basic operational principles of the designconcept are reported.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

Many emerging angular rate measurement applicationsdictate orders of magnitude reduction in size, weight, costand power consumption of existing high-end gyroscopetechnologies, including spinning wheel, laser-ring and fiber-optic devices. Thus, miniaturization of vibratory gyroscopeswith micromachining technologies is expected to become anattractive solution to current inertial sensing market needs, aswell as open new market opportunities with an even widerapplication range. Innovative micro-fabrication processesand gyroscope designs suggest drastic improvement inperformance and functionality of micromachined gyroscopesin the near future. Due to their robustness against shock

and vibration, potentially increased reliability and theircompatibility to mass-production, solid-state sensors areprojected to become a crucial part of automotive industry,military equipment and consumer electronics [1, 2].

Batch-fabrication of micromachined gyroscopes in VLSIcompatible surface-micromachining technologies constitutesthe key factor in low-cost production and commercialization.The first integrated commercial MEMS gyroscopes producedby Analog Devices Inc. have been fabricated by utilizingsurface micromachining technology [3]. However, the limitedthickness of structural layers attained in current surface-micromachining processes results in very small sensingcapacitances and higher actuation voltages, restricting theperformance of the gyroscope. Various devices have been

0960-1317/04/010015+11$30.00 © 2004 IOP Publishing Ltd Printed in the UK 15

C Acar and A M Shkel

Figure 1. Conceptual schematic of the torsional micromachinedgyroscope with non-resonant drive.

proposed in the literature that employ out-of-plane actuationand detection, with large capacitive electrode plates [4, 5].However, the highly non-linear and unstable nature ofparallel-plate actuation limits the actuation amplitude ofthe gyroscope. In this paper, we propose a novelsurface-micromachined torsional gyroscope design utilizingdynamical amplification of rotational oscillations to achievelarge oscillation amplitudes about the drive axis withoutresonance (figures 1 and 2), thus addressing the issuesof electrostatic instability while providing large sensecapacitance. The approach suggests to employ a three-mass structure with two gimbals and a sensing plate. Largeoscillation amplitudes in the passive gimbal, which containsthe sensing plate, are achieved by amplifying the smalloscillation amplitude of the driven gimbal (active gimbal).Thus, the actuation range of the parallel-plate actuatorsattached to the active gimbal is small, minimizing the non-linear force profile and instability.

The proposed non-resonant micromachined gyroscopedesign concept also addresses one of the major MEMSgyroscope design challenges, which is the mode-matchingrequirement [6]. Conventional micromachined rategyroscopes operate on the vibratory principle of a 2-DOFsystem with a single proof mass suspended by flexures

Figure 2. SEM micrograph of the fabricated prototype torsional micromachined gyroscopes.

anchored to the substrate. The proof mass is sustained inresonance in the drive direction, and in the presence of anangular rotation, the Coriolis force proportional to the inputangular rate is induced in the orthogonal direction (sensedirection). To achieve high sensitivity, the drive and thesense resonant frequencies are typically designed and tunedto match, and the device is controlled to operate at or nearthe peak of the response curve [7]. To enhance the sensitivityfurther, the device is packaged in high vacuum, minimizingenergy dissipation due to viscous effects of air surrounding themechanical structure. Extensive research has been focused onthe design of symmetric suspensions and resonator systemsfor mode matching and minimizing temperature dependence[8, 9]. However, especially for lightly-damped devices, therequirement for mode matching is well beyond fabricationtolerances, and none of the symmetric designs can provide therequired degree of mode matching without feedback control[10, 11]. The proposed design eliminates the mode-matchingrequirement by utilizing dynamic amplification of rotationaloscillations instead of resonance in the drive direction. Thus,the design concept is expected to overcome the small actuationand sensing capacitance limitation of surface-micromachinedgyroscopes, while achieving improved excitation stability androbustness against fabrication imperfections and fluctuationsin operation conditions.

The paper is organized as follows: we first present, insection 2, the structure and the principle of operation ofthe proposed torsional system. The dynamics of the deviceis then analyzed in section 3. A MEMS implementationof the design concept is presented in section 4, coveringthe basics of the mechanics and dynamics of the torsionalsystem, along with an approach for determining optimalsystem parameters to maximize sensor performance. Finally,in section 5, preliminary experimental results verifying thedesign objectives and operational principles are presented.

2. The torsional gyroscope structure and principleof operation

The overall torsional gyroscope system is composed of threeinterconnected rotary masses: the active gimbal, the passive

16

Torsional MEMS gyroscopes with non-resonant drive

Figure 3. (a) The frequency responses of the 2-DOF drive and1-DOF sense-mode oscillators. (b) The response of the overall3-DOF gyroscope system. The drive-direction oscillation amplitudeis insensitive to parameter variations and damping fluctuations in theflat operating region, eliminating mode-matching requirement.

gimbal and the sensing plate (figure 1). The active gimbal andthe passive gimbal are free to oscillate only about the driveaxis x. The sensing plate oscillates together with thepassive gimbal about the drive axis, but is free to oscillateindependently about the sense axis y, which is the axis ofresponse when a rotation along z-axis is applied.

The active gimbal is driven about the x-axis by parallel-plate actuators formed by the electrode plates underneath.The combination of the passive gimbal and the sensing platecomprises the vibration absorber of the driven gimbal. Thus,a torsional 2-DOF oscillator is formed in the drive direction.The frequency response of the 2-DOF drive oscillator has tworesonant peaks and a flat region between the peaks, where theresponse amplitude is less sensitive to parameter variations(figure 3(a)). The sensing plate, which is the only mass freeto oscillate about the sense axis, forms the 1-DOF torsionalresonator in the sense direction.

In the presence of an input angular rate about thesensitive axis normal to the substrate (z-axis), only the sensingplate responds to the rotation-induced Coriolis torque. Theoscillations of the sense plate about the sense axis are detectedby the electrodes placed underneath the plate. Since thedynamical system is a 1-DOF resonator in the sense direction,the frequency response of the device has a single resonancepeak in the sense mode. To define the operation frequencyband of the system, sense-direction resonance frequency ofthe sensing plate is designed to coincide with the flat regionof the drive oscillator (figure 3(a)). This allows operationat resonance in the sense direction for improved sensitivity,while the drive direction amplitude is inherently constant inthe same frequency band, in spite of parameter variations orperturbations. Thus, the proposed design eliminates the mode-matching requirement by utilizing dynamic amplification ofrotational oscillations instead of resonance in drive direction,leading to reduced sensitivity to structural and thermalparameter fluctuations and damping variations, while attainingsufficient performance with resonance in the sense mode.

A prototype of the torsional gyroscope has beenfabricated in a standard surface-micromachining process(figure 2), and the design objectives have been verifiedexperimentally (presented in section 5). The basic operationalprinciples of the design concept have also been experimentallydemonstrated with linear prototype gyroscopes [14], includingthe flat driving frequency band within which the drive-modeamplitude varies insignificantly, and mechanical amplificationof active mass oscillation by the sensing element.

2.1. The Coriolis response

The design concept suggests to operate at the sense-directionresonance frequency of the 1-DOF sensing plate, in orderto attain the maximum possible oscillation amplitudes inresponse to the induced Coriolis torque. The frequencyresponse of the 2-DOF drive direction oscillator hastwo resonant peaks and a flat region between the peaks(figure 3(a)). When the active gimbal is excited inthe flat frequency band, amplitudes of the drive-directionoscillations are insensitive to parameter variations due to anypossible fluctuation in operation conditions of the device.Moreover, the maximum dynamic amplification of activegimbal oscillations by the passive gimbal occurs in this flatoperation region, at the anti-resonance frequency. Thus, inorder to operate the sense-direction resonator at resonance,while the 2-DOF drive direction oscillators operate in theflat-region frequency bands, the flat region of the drive-oscillator has to be designed to overlap with the sense-direction resonance peak (figure 3(a)). This can be achievedby matching the drive direction anti-resonance frequencywith the sense-direction resonance frequency, as will beexplained in section 4.4. However, in contrast to theconventional gyroscopes, the flat region with significantlywider bandwidth can be easily overlapped with the resonancepeak without feedback control with sufficient precision inspite of fabrication imperfections and variations in operationconditions (figure 3(b)).

By utilizing dynamical amplification in the 2-DOFdrive-oscillator instead of resonance, increased bandwidth

17

C Acar and A M Shkel

Figure 4. Illustration of the non-inertial coordinate frames attachedto the sensing plate, passive gimbal, active gimbal and the substrate.

and reduced sensitivity to structural and thermal parameterfluctuations and damping changes can be achieved, whilesense-direction resonance provides high sensitivity of thedevice. Consequently, the design concept allows us tobuild z-axis gyroscopes utilizing surface-micromachiningtechnology with large sense capacitances, while resulting inimproved robustness and long-term stability over the operatingtime of the device. Thus, the approach is expected torelax control requirements and tight fabrication and packagingtolerances.

3. Dynamics of the gyroscope

The dynamics of each rotary proof mass in the torsionalgyroscope system is best understood by attaching non-inertialcoordinate frames to the center-of-mass of each proof mass andthe substrate (figure 4). The angular momentum equation foreach mass will be expressed in the coordinate frame associatedwith that mass. This allows the inertia matrix of each massto be expressed in a diagonal and time-invariant form. Theabsolute angular velocity of each mass in the coordinateframe of that mass will be obtained using the appropriatetransformations. Thus, the dynamics of each mass reduces to

Is �̇ωss + �ωs

s × (Is �ωs

s

) = �τse + �τsd

Ip �̇ωpp + �ωp

p × (Ip �ωp

p

) = �τpe + �τpd

Ia �̇ωaa + �ωa

a × (Ia �ωa

a

) = �τae + �τad + Md

where Is , Ip and Ia denote the diagonal and time-invariantinertia matrices of the sensing plate, passive gimbal andactive gimbal, respectively, with respect to the associatedbody attached frames. Similarly, �ωs

s, �ωpp and �ωa

a denotethe absolute angular velocity of the sensing plate, passivegimbal and active gimbal, respectively, expressed in theassociated body frames. The external torques �τse, �τpe, �τae and�τsd , �τpd, �τad are the elastic and damping torques acting on theassociated mass, whereas Md is the driving electrostatic torqueapplied to the active gimbal.

If we denote the drive direction deflection angle of theactive gimbal by θa , the drive direction deflection angle ofthe passive gimbal by θp, the sense-direction deflection angleof the sensing plate by φ (with respect to the substrate) andthe absolute angular velocity of the substrate about the z-axis

by �z (figure 4), the homogeneous rotation matrices fromthe substrate to active gimbal (Rsub→a), from active gimbal topassive gimbal (Ra→p) and from passive gimbal to the sensingplate (Rp→s), respectively, become

Rsub→a =1 0 0

0 cos θa −sin θa

0 sin θa cos θa

Ra→p =1 0 0

0 cos(θp − θa) −sin(θp − θa)

0 sin(θp − θa) cos(θp − θa)

Rp→s = cos φ 0 sin φ

0 1 0−sin φ 0 cos φ

.

Using the obtained transformations, the total absoluteangular velocity of the sensing plate can be expressed in thenon-inertial sensing plate coordinate frame as

�ωss =

0

φ̇

0

+ Rp→s

θ̇ p

00

+ Rp→sRa→pRsub→a

0

0�z

.

The absolute angular velocities of the active and passivegimbals are obtained similarly, in the associated non-inertialbody frame. Substitution of the angular velocity vectors intothe derived angular momentum equations yields the dynamicsof the sensing plate about the sense axis (y-axis), and the activeand passive gimbal dynamics about the drive axis (x-axis)

I sy φ̈ + Ds

yφ̇ +[Ks

y +(�2

z − θ̇2p

) (I sz − I s

x

)]φ

= (I sz + I s

y − I sx

)θ̇ p�z + I s

y θp�̇z +(I sz − I s

x

)φ2θ̇ p�z(

Ipx + I s

x

)θ̈ p +

(Dp

x + Dsx

)θ̇ p

+[Kp

x +(Ipy − Ip

z + I sy − I s

z

)�2

z

]θp

= Kpx θa − (

I sz + I s

x − I sy

)φ̇�z − I s

xφ�̇z

I ax θ̈ a + Da

x θ̇a + Kax θa = Kp

x (θp − θa) + Md

where I sx , I s

y and I sz denote the moments of inertia of the

sensing plate; Ipx , I

py and I

pz are the moments of inertia of

the passive gimbal; I ax , I a

y and I az are the moments of inertia

of the active gimbal; Dsx , D

px and Da

x are the drive-directiondamping ratios and Ds

y is the sense-direction damping ratio ofthe sensing plate; Ks

y is the torsional stiffness of the suspensionbeam connecting the sensing plate to the passive gimbal, K

px

is the torsional stiffness of the suspension beam connectingthe passive gimbal to the active gimbal and Ka

x is the torsionalstiffness of the suspension beam connecting the active gimbalto the substrate.

With the assumptions that the angular rate input isconstant, i.e. �̇z = 0, and the oscillation angles are small,the rotational equations of motion can be further simplified,yielding:

I sy φ̈ + Ds

yφ̇ + Ksyφ = (

I sz + I s

y − I sx

)θ̇ p�z(

Ipx + I s

x

)θ̈ p +

(Dp

x + Dsx

)θ̇ p + Kp

x θp = Kpx θa

I ax θ̈ a + Da

x θ̇a + Kax θa = Kp

x (θp − θa) + Md.

It should be noted in the sense-direction dynamics that,the term

(I sz + I s

y − I sx

)θ̇ p(t)�z is the Coriolis torque that

excites the sensing plate about the sense axis, with φ beingthe detected deflection angle about the sense axis for angularrate measurement.

18

Torsional MEMS gyroscopes with non-resonant drive

3.1. Cross-axis sensitivity

The response of the sensing plate to the angular inputrates (�x and �y) orthogonal to the sensitive axis (z-axis)can be modeled similarly, using the derived homogeneoustransformation matrices Rp→s , Ra→p and Rsub→a , andexpressing the total absolute angular velocity of the sensingplate as

�ωss,xy =

0

φ̇

0

+ Rp→s

θ̇ p

00

+ Rp→sRa→pRsub→a

�x

�y

0

.

With the derived total absolute angular velocity in thepresence of cross-axis inputs

( �ωss,xy

)and the assumption that

the input rates are constant (�̇x = �̇y = 0), the equation ofmotion of the sensing plate about the sense axis becomes:

Is �̇ωss,xy + �ωs

s,xy × (Is �ωs

s,xy

) = �τse + �τsd

I sy φ̈ + Ds

yφ̇ +[Ks

y +(θ̇2

p + 2θ̇ p�x + �2x

) (I sx − I s

z

)]φ

= (I sx − I s

z

)(θpθ̇p�y + θp�x�y).

For small oscillation angles and small magnitudes of thecross-axis inputs �x and �y , the equation of motion reducesto

I sy φ̈ + Ds

yφ̇ + Ksyφ = (

I sz − I s

x

)(2φθ̇p�x − θpθ̇p�y).

When the excitation terms on the right-hand side ofthis equation are compared to the excitation component(I sz + I s

y − I sx

)θ̇ p�z due to �z, it is seen that the additional

factors φ and θp in these terms make them orders of magnitude(over 10−5 times) smaller than the Coriolis excitation. Thus,the cross-axis sensitivity of the ideal system is negligible,provided that the sensor is aligned perfectly within the sensorpackage.

4. A MEMS implementation of the design concept

This section describes the principle elements of a MEMSimplementation of the conceptual design presented insection 2. First, the suspension system design for the torsionalsystem is investigated with the derivation of the stiffnessvalues. The capacitive sensing and actuation details arefollowed by the discussion of achieving dynamic amplificationin the drive mode, along with an approach for determiningoptimal system parameters to maximize sensor performance.Finally, the sensitivity and robustness analyses of the systemare presented.

4.1. Suspension design

The suspension system of the device that supports the gimbalsand the sensing plate is composed of thin polysilicon beamswith rectangular cross-section functioning as torsional bars.The active gimbal is supported by two torsional beams oflength La

x anchored to the substrate, aligned with the driveaxis, so that the gimbal oscillates only about the drive axis.The passive gimbal is also attached to the active gimbal withtwo torsional beams of length L

px aligned with the drive

axis, forming the 2-DOF drive-direction oscillator. Finally,the sensing plate is connected to the passive gimbal usingtwo torsional beams of length Ls

y lying along the sense axis,

Figure 5. SEM micrograph of the torsional suspension beams in theprototype gyroscopes.

allowing it to oscillate about the sense axis independent fromthe gimbals (figure 5).

Assuming each torsional beam is straight with a uniformcross-section, and the structural material is homogeneous andisotropic; the torsional stiffness of each beam with a length ofL can be modeled as [15]

K = SG + σJ

L

where G = E2(1−ν)

is the shear modulus with the elasticmodulus E and Poisson’s ratio ν; σ is the residual stressand J = 1

12 (wt3 + tw3) is the polar moment of inertia ofthe rectangular beam cross-section with a thickness of t and awidth of w. The cross-sectional coefficient S can be expressedfor the same rectangular cross-section as [15]

S =(

t

2

)3w

2

[16

3− 3.36

t

w

(1 − t4

12w4

)].

Assuming the same thickness t and width w for each beam,the torsional stiffness values in the equations of motion of theideal gyroscope dynamical system model can be calculated asfollows:

Kax = 2

SG + σJ

Lax

Kpx = 2

SG + σJ

Lpx

Ksy = 2

SG + σJ

Lsy

.

For the presented prototype design (figure 2), thesuspension beam lengths are La

x = Lpx = Ls

y = 30 µm,with the width of 2 µm and a structural thickness of 2 µm,resulting in the stiffness values of Ka

x = Kpx = Ks

y = 1.04 ×10−18 kg m2 s−2.

4.2. Finite element analysis results

In order to verify the validity of the assumptions in thetheoretical analysis, the operational modes and the otherresonance modes of the system were simulated using finiteelement analysis (FEA) package MSC Nastran/Patran.

The geometry of the device was optimized to match thedrive-mode resonant frequency of the isolated passive mass-spring system ω

px =

√K

px

/(I ax + I s

x

)with the sense-mode

resonance frequency of the sensing plate ωy =√

Ksy

/I sy ,

as will be explained in section 4.4. Theoretical analysis ofthe device geometry, which is presented in detail in figure 2,

19

C Acar and A M Shkel

(a)

(b)

(c)

Figure 6. Finite element analysis simulation of the torsionalsystem: (a) Sense-mode resonance frequency of the sensing plate atωy = 7.457 kHz. (b) Drive-mode resonant frequency of the isolatedpassive mass-spring system at ωxp = 7.097 kHz. (c) The undesiredmode with the lowest frequency at 8735 kHz, being the linearout-of-plane mode.

and a structural thickness of 2 µm yields Kpx = Ks

y =1.04 × 10−18 kg m2 s−2,

(I ax + I s

x

) = 4.97 × 10−18 kg m2

and I sy = 4.94 × 10−18 kg m2, resulting in ω

px = 7.285 kHz

and ωy = 7.263 kHz.Through FEA simulations, the sense-mode resonance

frequency of the sensing plate about the sense axis wasobtained to be ωy = 7.457 kHz (figure 6(a)), with a 2.28%discrepancy from the theoretical calculations. The drive-moderesonant frequency of the isolated passive mass-spring systemwas obtained as the first mode (figure 6(b)) at ωp

x = 7.097 kHz.The 5.91% discrepancy from theoretical analysis is attributedin part to the reduced stiffness due to the compliance of theactive and passive gimbal frame structures. More importantly,the torsional beams exhibit linear deflections as well, and donot undergo purely torsional deflections, which also providesconsiderable additional compliance.

The undesired resonance modes of the structure werealso observed to be sufficiently separated from the operationalmodes. In the FEA simulations, the undesired mode with thelowest frequency was observed to be the linear out-of-planemode (figure 6(c)), at 8735 kHz.

4.3. Electrostatic actuation

The active gimbal is excited about the drive axis bythe electrodes symmetrically placed underneath its edges.

Figure 7. Cross-section of the torsional electrostatic parallel-plateactuation electrodes attached to the active gimbal.

Applying a voltage of V1 = Vdc + νac sin(ωdt) to electrode 1on one side of the gimbal, and V2 = Vdc − νac sin(ωdt) toelectrode 2 on the opposite side, a balanced actuation schemeis imposed.

The net moment Md that drives the active gimbal is thesum of the positive and negative resultant moments appliedby electrode 1 and electrode 2, respectively. These momentscan be expressed by integrating the moments generated by theinfinite number of infinitesimally-small capacitors of width dx,located by a distance of x from the center of rotation (figure 7).Neglecting the fringing field effects, with symmetricelectrodes of width (a2 −a1) and length L placed by a distanceof a1 from the center-line, the net actuation moment Md , whichis a function of the deflection angle θa , can be expressed as

Md = M1 − M2 =∫ a2

a1x dF1 −

∫ a2

a1x dF2

=∫ a2

a1x

ε0V2

1 L dx

2(h − x tan θa)2−

∫ a2

a1x

ε0V2

2 L dx

2(h + x tan θa)2

where h is the elevation of the structure from the substrate, andε0 = 8.85 × 10−12Fm−1 is the permittivity of air. Assumingsmall angles of actuation, which are achieved due to dynamicamplification of oscillations as will be explained in the nextsection, the net electrostatic moment reduces to the expressionused in the simulations of the dynamic system:

Md = ε0L

2θ2a

(1

1 − a2hθa

− 1

1 − a1hθa

+ lnh − a2θa

h − a1θa

)V 2

1

− ε0L

2θ2a

(1

1 + a2hθa

− 1

1 + a1hθa

+ lnh + a2θa

h + a1θa

)V 2

2 .

For the presented prototype design (figure 2), the drivemode electrodes underneath the active gimbal are 380 µm ×60 µm, resulting in a total of 0.252 µN force per each electrodewith a 1 V actuation voltage. The total moment applied byeach electrode at the deflection of θa = 0 is 5.29 × 10−13 Nm.

4.4. Optimization of system parameters

Since the foremost mechanical factor determining theperformance of the gyroscope is the angular deflection φ ofthe sensing plate about the sense axis due to the input rotation,the parameters of the dynamical system should be optimizedto maximize φ. However, the optimal compromise betweenamplitude of the response and bandwidth should be obtainedto maintain robustness against parameter variations, while theresponse amplitude is sufficient for required sensitivity. Thetrade-offs between gain of the response (for higher sensitivity)and the system bandwidth (for increased robustness) will betypically guided by application requirements.

20

Torsional MEMS gyroscopes with non-resonant drive

For a given input rotation rate �z, in order to maximize theCoriolis torque

(I sz + I s

y − I sx

)θ̇ p(t)�z that excites the sensing

plate about the sense axis, the oscillation amplitude of thepassive gimbal about the drive axis should be maximized. Inthe drive mode, the gyroscope is simply a 2-DOF torsionalsystem. The sinusoidal electrostatic drive moment Md isapplied to the active gimbal. The combination of the passivegimbal and the sensing plate comprises the vibration absorberof the 2-DOF oscillator, which mechanically amplifiesthe oscillations of the active gimbal. Approximating the2-DOF oscillator by a lumped mass-spring-damper model, theequations of motion about the drive axis can be expressed as

I ax θ̈ a + Da

x θ̇a + Kax θa = Kp

x (θp − θa) + Md(Ipx + I s

x

)θ̈ p +

(Dp

x + Dsx

)θ̇ p + Kp

x θp = Kpx θa

where I sx , I s

y and I sz are the moments of inertia of the sensing

plate; Ipx , I

py and I

pz are the moments of inertia of the passive

gimbal; I ax , I a

y and I az are the moments of inertia of the active

gimbal; Dsx,D

px and Da

x are the drive-direction damping ratios;K

px is the torsional stiffness of the suspension beam connecting

the passive gimbal to the active gimbal and Kax is the torsional

stiffness of the suspension beam connecting the active gimbalto the substrate.

When the driving frequency ωdrive is matched with theresonant frequency of the isolated passive mass-spring system(ω

px

), i.e. ωdrive = ω

px =

√K

px

(Iax +I s

x ), the passive gimbal and the

sensing plate move to exactly cancel out the driving momentMd applied to the active gimbal, and maximum dynamicamplification is achieved at this anti-resonance frequency[18]. Thus, if the drive direction anti-resonance frequency

ωpx and the sense-direction resonance frequency ωy =

√K

py

I sy

are designed to match, maximum dynamic amplification indrive mode is achieved, the Coriolis torque drives the sensingplate into resonance and the drive-mode oscillator is excitedin the flat frequency band. The optimal design condition canbe summarized as follows:√

Kpx(

I ax + I s

x

) =√

Kpy

I sy

= ωdrive.

In contrast to the conventional gyroscopes, where twosharp resonance peaks with very narrow bandwidth have tobe matched with a very precise and constant ratio, this designcondition can easily be met without feedback control withsufficient precision in spite of fabrication imperfections andoperation condition variations, thanks to the flat region inthe drive-mode frequency response with significantly widerbandwidth.

4.5. Sensitivity and robustness analyses

The response of the complete electro-mechanical system ofthe torsional gyroscope was simulated by incorporating thepresented electro-mechanical modeling. With the sense-direction resonance frequency of 7.457 kHz as obtained fromthe finite element analysis simulations, the effective sense-direction response amplitude of the sensing capacitors to a1◦/s input angular rate was found to be 1.6 × 10−5 µm. It isassumed that the gyroscope is vacuum packaged so that the

Figure 8. The response of the complete torsional gyroscope system.5% mismatch between the sense-mode resonance frequency (ωy)and the drive-mode anti-resonance frequency (ωp

x ) results in only2.5% error in the response amplitude.

pressure within the encapsulated cavity is equal to 100 mTorr(13.3 Pa), and that the passive gimbal oscillates in the whole2 µm gap. The response of a torsional gyroscope with aresonant drive mode and the same geometry to the same inputis 0.53 × 10−5 µm, since the stable drive-mode actuationrange is limited to 0.66 × 10−5 µm. However, the requiredactuation voltage amplitude for the anti-resonant mode is3.9 times larger than the resonant drive-mode approach. Itshould also be noted that, in the presented prototype design(figure 2) with the sensing electrode area of 200 µm ×130 µm (nominal capacitance of 11.51 pF); 1◦/s inputangular rate results in a total capacitance change of 29.2 fF,which is considerably larger compared to in-plane surface-micromachined gyroscope designs.

In the case of a potential shift in the sense-mode resonancefrequency, e.g., due to temperature fluctuations, residualstresses or fabrication variations, the response amplitudeis sustained at a constant value to a great extent withoutactive tuning of resonance frequencies. For example, a5% mismatch in the sense-mode resonance frequency ofthe sensing plate (ωy) and the drive-mode anti-resonancefrequency

(ω

px

)results in only 2.5% error in the response

amplitude (figure 8). Without active compensation, aconventional 2-DOF gyroscope can exhibit over 60% errorfor the same frequency mismatch under the same operationconditions [6]. Thus, the increased bandwidth of the 2-DOFdrive-oscillator achieved by utilizing dynamical amplificationprovides improved robustness against structural and thermalparameter fluctuations.

5. Experimental characterization

The response of the fabricated prototype gyroscopes hasbeen characterized electrostatically under vacuum, andoptically using a Sensofar PLµ confocal imaging profiler andPolytec scanning laser Doppler vibrometer under atmosphericpressure.

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C Acar and A M Shkel

Figure 9. (a) Sense mode, and (b) drive-mode frequency responseacquired using off-chip transimpedance amplifiers, yieldingωy = 8.725 kHz, and ωp

x = 8.687 kHz for 0.7 V dc bias.

The sense-mode resonance frequency and the drive-modeanti-resonance frequency of the prototype gyroscope weremeasured in a cryogenic MMR vacuum probe station. Thefrequency response of the device was acquired using off-chiptransimpedance amplifiers connected to a HP signal analyzerin sine-sweep mode. Due to the large actuation and sensingcapacitances, actuation voltages as low as 0.7 V–1.8 V dc bias,and 30 mV ac were used under 40 mTorr vacuum. For sense-mode resonance frequency detection, one-sided actuation wasutilized, where one sensing electrode was used for driving,and the other for detection. For detecting the drive-mode anti-resonant frequency, which is equal to the resonant frequencyof the isolated passive mass-spring system

(ω

px

), a separate

test structure that consists of the passive gimbal-sensing plateassembly was used, with the one-sided actuation scheme, andthe same actuation voltages.

The resonance peaks in the sense and drive modeswere observed very clearly for the 0.7 V–1.8 V dc biasrange. With a 0.7 V dc bias, the sense-mode resonancefrequency was measured to be 8.725 kHz (figure 9(a)), and

Figure 10. (a) Electrostatic tuning of the drive and sense-moderesonance frequencies by changing the dc bias. Note that ωp

x andωy are exactly matched for Vdc = 1.0 V. (b) Drive and sense modesobserved in the same sweep with 1.3 V dc bias.

the drive-mode anti-resonance frequency was measured to be8.687 kHz (figure 9(b)). We were also able to electrostaticallytune both the drive and sense-mode resonance frequencies byseveral hundred Hertz with only 1 V dc bias tuning range(figure 10(a)). It was observed that ω

px and ωy are exactly

matched for Vdc = 1.0 V. Furthermore, it was possible toobserve both the drive and sense resonance modes in the samesweep, by driving the structure with one drive electrode, andsensing the response with one sense electrode. Figure 10(b)

shows the frequency response with 1.3 V dc bias, where theprimary excited drive mode appears as the large resonancepeak, and the sense mode appears as the secondary small peak.

The experimentally measured resonance frequencies, andthe estimated values via theoretical analysis and finite elementanalysis are summarized in table 1. To investigate the originsof over 1 kHz discrepancy between the measurements andFEA results, the prototype was analyzed using a PLµ confocalimaging profiler, for obtaining the structural parameters, suchas layer thickness, elevation, suspension beam geometry andany possible curling in the structure (figure 11(a)). ThePLµ optical profiler acquires confocal images using a

22

Torsional MEMS gyroscopes with non-resonant drive

Table 1. Summary of the estimated and measured resonance frequencies.

Theoretical FEA FEA results Experimental Measured Qanalysis results with l = 27.1 µm Vdc = 0.7 V (40 mTorr)

Drive mode (ωx) 7.28 kHz 7.097 kHz 8.364 kHz 8.687 kHz 50Sense mode (ωy) 7.26 kHz 7.457 kHz 8.642 kHz 8.725 kHz 100

Figure 11. (a) Confocal imaging profiler scan of the structure, forobtaining layer thickness, elevation, suspension beam geometry andcurling. (b) Effective length of the torsional suspension beams wasobserved as 27.1 µm due to the rounding effects.

proprietary confocal arrangement, fast scanning devices andhigh contrast algorithms, and includes a motorized x–y stageand a PZT scanning device [19]. In the confocal microscope,all structures out of focus are suppressed by the detectionpinhole, employing an arrangement of diaphragms. Thus,light rays from outside the focal plane are rejected, and three-dimensional data sets of the devices are acquired by out-of-plane scanning.

Using the PLµ profiler, the structural layer thickness wasmeasured as 1.97 µm, elevated by 2.1 µm from the substrate,with extremely small curling (60 nm elevation differencebetween the middle section and the edges). However, due tothe corner rounding effects in photo-lithography, the effectivelength of the torsional suspension beams were observed as27.1 µm (figure 11(b)). When the FEA was repeated withthe 27.1 µm suspension length, the results agreed withexperimental measurements to a great extent, with 3.6%discrepancy in the drive mode, and 0.95% discrepancy in thesense mode (table 1).

In order to verify the mode-shapes of the structure atthe measured frequencies, a Polytec scanning laser Dopplervibrometer was used under atmospheric pressure for dynamic

(a)

(b)

Figure 12. (a) Sense-mode, and (b) drive-mode dynamic responsemeasurements using the LDV in the scanning mode.

Figure 13. Scanning mode LDV measurements at theanti-resonance frequency, demonstrating dynamic amplification ofthe active gimbal oscillations by the passive gimbal. The passivegimbal was observed to achieve over 1.7 times larger oscillationamplitudes than the driven active gimbal.

optical profiling. Laser Doppler vibrometry (LDV) is a non-contact vibration measurement technique using the Doppler

23

C Acar and A M Shkel

Figure 14. Drive-mode static response measurements using the LDV in the dual-beam mode, with static excitation of the active gimbal by aramping dc signal.

effect, based on the principle of the detection of the Dopplershift of coherent laser light that is scattered from a smallarea of the test object. Laser vibrometers are typically two-beam interferometric devices which detect the phase differencebetween an internal reference and the measurement beam,which is focused on the target and scattered back to theinterferometer [20].

Dynamic response of the device in the drive and sensemodes was characterized using the scanning laser Dopplervibrometer in scanning mode, which allows us to measure theresponse of a dense array of points on the whole gyroscopestructure. Dynamic excitation of the sensing plate aboutthe sense axis at the experimentally measured sense-moderesonance frequency (ωy = 8.725 kHz) revealed that onlythe sensing plate responds in the sense mode, verifying thatthe 1-DOF resonator formed in the sense mode is decoupledfrom the drive mode (figure 12(a)), in agreement withthe intended design and finite element analysis simulations.Dynamic excitation of the active gimbal about the drive axis atfrequencies away from the anti-resonance frequencies verifiedthat the active gimbal oscillates independent from the passivegimbal-sensing plate assembly (figure 12(b)), constituting theactive mass of the 2-DOF oscillator.

Most prominently, dynamic amplification of the activegimbal oscillations by the passive gimbal was successfullydemonstrated. At the drive-mode anti-resonance frequency,which was measured to be 8.687 kHz, the passive gimbalwas observed to achieve over 1.7 times larger oscillationamplitudes than the driven active gimbal (figure 13). Thistranslates into attaining over 2.4 times larger drive-modedeflection angles at the sensing plate than the active gimbal.

DC response of the structure in the drive mode wasmeasured with static excitation of the active gimbal by aramping dc signal (figure 14). To eliminate the low-frequencyvibrations, LDV was used in the dual-beam mode, where thereference beam was located on the substrate. The resultsindicated that the passive gimbal and the sensing plate deflectwith the same angle independent from the active gimbal,forming a rigid passive mass in the drive mode. This

verifies that the sensing plate achieves the desired drive-modeoscillation amplitude for generation of the Coriolis torque,confirming the design objective. It also indicates that a 2-DOFoscillator is formed in the drive mode.

6. Conclusion

In this paper, the design concept of a gimbal-type torsionalmicromachined gyroscope with non-resonant drive mode isreported. The analysis of the system dynamics and structuralmechanics of the torsional system are presented, along withthe preliminary experimental results verifying the designobjectives. The proposed approach is based on forming atorsional 2-DOF drive-mode oscillator with the use of oneactive and one passive gimbal, to achieve large oscillationamplitudes in the passive gimbal by amplifying the smalloscillation amplitude of the driven gimbal. This allowsminimization of the non-linear force profile and instabilitydue to parallel-plate actuation of the active gimbal, whileeliminating the mode-matching requirement by obtaining aflat operation frequency band in the drive mode. Withthe basic operational principles experimentally verified, thedesign concept is expected to overcome the small actuationand sensing capacitance limitation of surface-micromachinedgyroscopes, while achieving improved excitation stability androbustness against fabrication imperfections and fluctuationsin operation conditions. Thus, the proposed approach isprojected to lead to reduced cost and complexity in fabricationand packaging of MEMS-based multi-axis inertial sensors.

Acknowledgments

This work is supported in part by the NationalScience Foundation Grant CMS-0223050, program managerDr Shin-Chi Liu. The authors would also like to thankSensofar [19] for providing the optical profilometer, andPolytech PI [20] for the scanning laser Doppler vibrometerfor experimental characterization of the micro-devices.

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Torsional MEMS gyroscopes with non-resonant drive

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