Design of a Novel MEMS Gyroscope Array.

0.2 MB·English

by  Wang, Wei

Sensors2013,13,1651-1663;doi:10.3390/s130201651
OPENACCESS
sensors
ISSN1424-8220
www.mdpi.com/journal/sensors
Article
Design of a Novel MEMS Gyroscope Array
WeiWang*,XiaoyongLvandFengSun
CollegeofAutomation,HarbinEngineeringUniversity,No. 145NantongStreet,Harbin150001,
Heilongjiang,China;E-Mails: [email protected](X.L.);Sunfeng [email protected](F.S.)
(cid:0)
*Authortowhomcorrespondenceshouldbeaddressed;E-Mail: [email protected];
Tel.: [phone];Fax: [phone].
Received: 23November2012;inrevisedform: 18January2013/Accepted: 21January2013/
Published: 28January2013
Abstract: This paper reports a novel four degree-of-freedom (DOF) MEMS vibratory
gyroscope. A MEMS gyroscope array is then presented using the novel gyroscope unit.
In the design of the proposed 4-DOF MEMS vibratory gyroscope, the elements of the
drive-modearesetinsidethewholegyroscopearchitecture,andtheelementsofsense-mode
are set around the drive-mode, which thus makes it possible to combine several gyroscope
units into a gyroscope array through sense-modes of all the units. The complete 2-DOF
vibratory structure is utilized in both the drive-mode and sense-mode of the gyroscope unit,
thereby providing the desired bandwidth and inherent robustness. The gyroscope array
combines several gyroscope units by using the unique detection mass, which will increase
the gain of sense-mode and improve the sensitivity of the system. The simulation results
demonstrate that, compared to a single gyroscope unit, the gain of gyroscope array (n = 6)
is increased by about 8 dB; a 3 dB bandwidth of 100 Hz in sense-mode and 190 Hz in
drive-mode are also provided. The bandwidths of both modes are highly matched with each
other, providing a bandwidth of 100 Hz for the entire system, thus illustrating that it could
satisfytherequirementsinpracticalapplications.
Keywords: structuraldesign;MEMS;gyroscopearray;multi-DOF;widebandwidth
Classification: PACS 85.85.+j
Sensors2013,13 1652
1.Introduction
Since the first micromachined tuning-fork gyroscope was reported by Draper laboratory in 1991 [1],
many types of MEMS gyroscopes have been designed and manufactured. Due to advantages such as
small size, low cost, reduced power consumption, and so on, micromachined vibratory gyroscopes have
received a great deal of attention in the past several decades. Currently, some micromachined vibratory
gyroscopes have been utilized in a variety of practical applications, such as in the automotive industry,
electronics,andINS(inertialnavigationsystem)fortacticalweaponsandvehicles,etc.
Conventional micromachined vibratory gyroscopes mainly employ the architecture with 1-DOF
drive-mode and sense-mode, which increases the mechanical gain by matching the natural frequencies
of two modes and allows enhancing the sensitivity [2–4]. These designs often utilize the design of
flexuralbeamsortuningelectronicstoadjustthestructuralfrequencyforthepurposeofmode-matching,
whichcouldincreasethesensitivityofthesystem. However,thehighsensitivityofthesemode-matched
gyroscopes is always at the cost of bandwidth or robustness and is susceptible to structural and
environmental parameter variations. Meanwhile, the structural coupling between two modes is also
adrawback.
To resolve the problems in the single-DOF MEMS vibratory gyroscopes, some multi-DOF
micromachined vibratory gyroscopes have been presented by increasing DOF of the drive-mode or
sense-mode. For the purpose of improving the robustness of MEMS vibratory gyroscopes, the 2-DOF
vibratory structure has been utilized in drive-mode or sense-mode to increase the bandwidth of the
respectivemode. A3-DOFMEMSvibratorygyroscopewith2-DOFdrive-modeand1-DOFsense-mode
is described [5], which uses 2-DOF dynamic vibration absorber(DVA) structure in drive-mode.
Thisdesignillustratedthewidebandwidthofdrive-mode,butthereisanarrowbandwidthofsense-mode.
Another type of 3-DOF micromachined vibratory gyroscope, with 1-DOF drive-mode and 2-DOF DVA
structureinsense-mode,isdescribed[6],whichshowedthewidebandwidthofsense-mode.
However, there is the limitation of structural design space in the DVA architecture. Moreover, the
tradeoffsamongoperationalfrequency,bandwidthanddiesize,aswellasdetectioncapacitancemustbe
considered[7]. ToovercomethelimitationintheDVAstructure,acomplete2-DOFarchitecturewithtwo
massesandthreespringshasbeenemployedinthesense-modeofsome3-DOFmicromachinedvibratory
gyroscopes [8–10]. This design demonstrated the advantage of gain and bandwidth in sense-mode.
However, these designs merely improve the bandwidth of drive-mode or sense-mode, respectively, and
intheothermode,therobustnessalsorequiresimprovement.
Recently,aMEMSgyroscopewith2-DOFdrive-modeandsense-modehasbeenproposedtoimprove
the robustness of both drive-mode and sense-mode [11], which utilizes the complete 2-DOF vibratory
structure to provide a robust gain-bandwidth frequency region and inherently improves the robustness
of drive-mode and sense-mode at the same time. The device decouples drive-mode and sense-mode
by adding decoupled mass and proof mass and, concurrently, a complete 2-DOF structure is formed in
bothmodes. Comparedtotheprevious1-DOFstructure,oneoftheadvantagesofutilizingthecomplete
2-DOF structure is its robust gain-bandwidth frequency region. Meanwhile, the implementation of the
complete 2-DOF system will reduce the complexity of electronic circuits and eliminate the electronic
noise due to the complex control circuit in drive-mode [12,13]. However, the gain of sense-mode also
Sensors2013,13 1653
needs to increase in order to improve the sensitivity of the device. In this paper, another novel 4-DOF
MEMS vibratory gyroscope structure is proposed. Based on this 4-DOF MEMS vibratory gyroscope
structure, a novel gyroscope array structure is presented. The proposed gyroscope array is different
from the previous gyroscope structure, which combines several gyroscope units into a single gyroscope
structure in a special way. The proposed gyroscope array could improve the gain of sense-mode
comparedtotheprevious4-DOFgyroscope[11].
The rest of this paper is organized as follows. Section 2 introduces the design of 4-DOF gyroscope
unit. Section 3 presents the structural design of the proposed gyroscope array based on the proposed
gyroscopeunit. Section4givesatheoreticaldynamicsanalysisandthesolutionsofthegyroscopearray.
Section5reportsthecomparativesimulationresults,andSection6concludesthepaperwithasummary
ofresults.
2.GyroscopeUnit
TheconceptualschematicofproposedgyroscopeunitisshowninFigure1. Similartothemulti-DOF
gyroscope [11], the complete 2-DOF vibratory structure is implemented in both drive-mode and
sense-mode,whichprovidesbothmodeswithalargebandwidth. Thedrive-combsaresetonthedriving
mass to drive drive-mode to oscillate along with x axis. However, unlike the previous 4-DOF MEMS
vibratory gyroscope [11], the elements of drive-mode are inside the schematic and the sense-mode
is around the drive-mode. The sensing-capacitors are set on the detection mass used to sense the
Coriolis-induced changing capacitance. The detection mass is set at the edge of the schematic, which
willbesuitableforcombiningseveralgyroscopeunitsintoagyroscopearray.
Figure1. Theschematicofproposedgyroscopeunit.
y
W x
z
The lump model of proposed 4-DOF MEMS gyroscope is simplified, as shown in Figure 2.
The proposed gyroscope consists of driving mass, decoupled frame, proof mass and detection mass
and springs. The driving mass, decoupled frame and springs, k , k , k , form the complete 2-DOF
1 2 3
drive-mode and the proof mass, detection mass and springs, k , k , k , form the complete 2-DOF
4 5 6
sense-mode. Drive-mode and sense-mode are linked to each other through decoupled frame and proof
mass,whichisolatevibrationbetweenbothmodes.
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Figure2. Modeloftheproposed4-DOFMEMSvibratorygyroscope.
F
d
k k k
1 2 3 k c
5 5
y
k c
c c c
k c 6 6
1 2 3
4 4
W x
z
When the gyroscope unit is operated, the driving force is applied to driving mass and drives driving
mass and decoupled frame to vibrate in drive direction (x). Moreover, proof mass could oscillate with
decoupledframeindrivedirection(x). IfanangularrateΩ isinputalongthedirection(z)perpendicular
z
to x − y plane, the Coriolis-induced force will excite proof mass and detection mass to move in sense
direction (y). Then the capacitors on detection mass will change, which helps pick up the motion and
calculatetheinputangularrateΩ .
z
3.GyroscopeArray
Inthissection,agyroscopearraywillbedesignedbasedonthegyroscopeunitpresentedinSection2.
ThelumpmodelofgyroscopearrayisshowninFigure3. Thereareagroupofgyroscopeunitsforming
agyroscopearray. Alltheunitsutilizetheidenticalelementsofdrive-modeandproofmassrespectively,
butthereisjustadetectionmass,whichisutilizedbyallthegyroscopeunits. Theproofmassesofevery
gyroscopeunitareconnectedtotheuniquedetectionmassthroughabeam.
Figure3. Lumpmodelofthe4-DOFMEMSvibratorygyroscopearray.
F
d
k k k
1 2 3 k c
5 5
y
k c
c c c
k c 6 6
1 2 3
4 4
W x
z
n
(cid:1)
F
d
k k k
1 2 3 k c
5 5
y
c c c
k c
1 2 3
4 4
W x
z
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In the design of gyroscope array, similar to gyroscope unit, the complete 2-DOF vibratory structure
is implemented in drive-mode of every gyroscope unit. In addition, the complete 2-DOF drive-mode
couldensurethedesiredbandwidththesameasagyroscopeunit. Insense-mode,however,thecomplete
2-DOF sense-mode of a gyroscope unit is changed into another model as shown in Figures 3 and 4.
The proof masses also belong to the respective gyroscope unit, but the detection masses of all the units
in gyroscope array are combined to a common detection mass, which is used by all the units. Then
the sense-mode of every gyroscope unit is also a complete 2-DOF vibratory structure providing a wide
bandwidthtosense-modeofgyroscopearray.
Figure4. Combinationof4-DOFMEMSvibratorygyroscopes.
y
W x
z
When the gyroscope array is operated, the driving masses of all the gyroscope units are driven along
x axis by the sinusoidal forces with the same frequency, amplitude and phase, which ensure that the
entiredrive-modewillworksynchronously. WhenthereisanangularrateΩ perpendiculartothex−y
z
plane, the Coriolis-induced force is applied to the respective proof mass. Then all the proof masses are
driven to oscillate along y axis and detection mass is excited by the force transmitted from all the proof
masses. The Coriolis-induced motion is picked up by the detection mass, which can help calculate the
inputangularrate Ω .
z
Compared with a single gyroscope unit, the gyroscope array combines several (i.e., n) gyroscope
unitsandpossessesauniquedetectionmass,whichwillenlargetheCoriolis-inducedforceappliedtothe
detectionmassandincreasetheamplitudeofthedetectionmass. Theenlargedamplitudeofthedetection
mass will increase the changing capacitance and sensitivity of the device. As shown in Figures 3 and 4,
there are several independent drive-mode elements and proof masses, which form a special drive-mode
and could improve insensitivity to structural parameters variations. The detailed analysis will be
givenlater.
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4.TheoreticalAnalysis
4.1.Drive-Mode
As the structural schematic of the micromachined vibratory gyroscope is shown in Figures 2 and 3,
all the gyroscope units use the identical drive-mode elements and the dynamic equations of drive-mode
canberepresentedbysimplifiedequations,
m x¨ +(c +c )x˙ −c x˙ +(k +k )x −k x = F (1)
1 1 1 2 1 2 2 1 2 1 2 2 d
(m +m )x¨ −c x˙ +(c +c )x˙ −k x +(k +k )x = 0 (2)
f 2 2 2 1 2 3 2 2 1 2 3 2
wherex andx representthemotionofdrivingmassm anddecoupledframem (withproofmassm )
1 2 1 f 2
alongwithxaxis. ThedrivingforceF issinusoidalappliedtothedrivingmassanddrivesdrivingmass
d
m anddecoupledframem (withproofmassm )tooscillateinthedrivedirection(x). Theproofmass
1 f 2
m provides the link between two modes and without input angular rate, it only oscillates in the drive
2
direction (x) with the decoupled frame m . When there is rotation about the z axis perpendicular to the
f
x − y plane, the Coriolis-induced force is applied to the proof mass m and drives proof mass m and
2 2
detectionmassm tomoveinthesensedirection(y).
3
Because the decoupled frame m and proof mass m provide the link between two vibratory modes
f 2
andtheirvibrationalongxaxiswillbetransferredtoy axisduetoCorioliseffect,thedynamicresponse
(x )ofdecoupledframem (withproofmassm )alongxaxiswilloccur. Assumingthesolutionx are
2 f 2 2
sinusoidal,accordingtoLaplacetransformthesteadysolutionfor x isgivenby,
2
(k +j!c )F
2 2 d
x = (3)
2
∆ (!)
d
where∆ (!) = (k +k −m !2+j!(c +c ))(k +k −(m +m )!2+j!(c +c ))−(k +j!c )2.
d 1 2 1 1 2 3 2 f 2 2 3 2 2
The Equation (2) provides theoretical argument for performance analysis. The structural frequency of
thecomplete2-DOFdynamicsystemindrive-modecanbedefinedby,
k +k k +k
!2 = 1 2;!2 = 2 3 (4)
d1 m d2 m +m
1 f 2
according to vibration dynamics, the anti-resonant frequency ! of the driving mass m is always
d0 1
between the two resonant frequencies of drive-mode. For obtaining the precise position of the robust
operationalregionindrive-mode,thestructuralfrequencies ! and! aresetequalto ! .
d1 d2 d0
Assuming zero-damping system and solving the eigenvalue equation ∆ (!) = 0, the resonant
d
frequenciesofdrive-modecanbeobtained,
v
u v
u u
u u k2
! = t!2 ±t 2 (5)
dH;L d0 m (m +m )
1 2 f
where ! are the high and low resonant frequencies of drive-mode. Define the resonant peak
dH;L
spacing∆ as,
d
∆ = ! −! (6)
d dH dL
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And by using Equations (4), (5) and (6), the design equations of the stiffnesses in drive-mode are
givenby,  √ √
 k2 = ∆d m1(m2 +mf) !d20 −0:25∆2d
k = m !2 −k (7)
 k1 = (m1 d+0 m )2!2 −k
3 f 2 d0 2
It is obvious that the resonant peak spacing ∆ of drive-mode can be set independently of the
d
operational frequency ! , which eliminates the limitation of DVA structure [7]. After the layout of
d0
drivingmassm anddecoupledframem (withproofmassm ),theoperationalfrequencycanbesetby
1 f 2
adjusting the stiffnesses. In the proposed gyroscope array, its drive-mode includes several independent
samedrivingelementsbelongingtorespectivegyroscopeunitsandtheirdynamicsequationsanddesign
equationsarethesameasEquations(1),(2)and(7),whichisadifferentpointincontrasttoothers.
4.2.Sense-Mode
As is shown in Figures 2 and 3, sense-mode elements of the gyroscope unit and gyroscope array are
differentfromeachother. Becausethereisproofmassm insense-modeofallgyroscopeunitsandthey
2
are identical, the motion of proof mass m in sense-mode of every gyroscope unit can be described by
2
thesameequation,
m y¨ +(c +c )y˙ −c y˙ +(k +k )y −k y = F (8)
2 1 4 5 1 5 2 4 5 1 5 2 C
where y represents the coordinate of proof mass m in all gyroscope units and y represents the
1 2 2
coordinate of detection mass m in gyroscope array. F is the Coriolis-induced force applied to proof
3 C
mass m . In gyroscope array, the unique detection mass m is utilized by all the gyroscope units of
2 3
gyroscopearrayanditsmotioncanbegivenby,
m y¨ −nc y˙ +(nc +c )y˙ −nk y +(nk +k )y = 0 (9)
3 2 5 1 5 6 2 5 1 5 6 2
wherenisthenumberofgyroscopeunitsinproposedgyroscopearray.
Because the detection mass m is used to pick up the Coriolis-induced motion through the
3
sensing-capacitor in it, the motion y of detection mass m is mainly focused on. Using
2 3
Equations(8)and(9),thesolutionofthedetectionmasscanbeobtained,
n(k +j!c )F
5 5 C
y = (10)
2
∆ (!)
s
where∆ (!) = (k +k −m !2 +j!(c +c ))(nk +k −m !2 +j!(nc +c ))−n(k +j!c )2.
s 4 5 2 4 5 5 6 3 5 6 5 5
After the design and layout of the masses, the stiffnesses of all the springs must be calculated.
According to vibration dynamics, the anti-resonant frequency ! of the proof mass m is
s0 2
always between the two resonant frequencies of sense-mode. Define the structural frequencies of
sense-modeby,
k +k nk +k
!2 = 4 5;!2 = 5 6 (11)
s1 m s2 m
2 3
For obtaining the precise position of the robust operational region in sense-mode, we set
! = ! = ! (! is the anti-resonant frequency of proof mass). Then assuming zero-damping
s1 s2 s0 s0
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system and solving the eigenvalue equation (∆ (!) = 0) of sense-mode, the resonant frequencies of
s
sense-modecanbeobtained,
v
u √
u
t nk2
! = !2 ± 5 (12)
sH;L s0 m m
2 3
where! arethehighandlowresonantfrequenciesofsense-mode. Theresonantpeakspacing∆ of
sH;L s
sense-modearedefinedby,
∆ = ! −! (13)
s sH sL
UsingEquations(11),(12)and(13),thedesiredstiffnessesk ,k ,k ,canbegivenby,
4 5 6
 √ √
 k5 = p1n∆s m2m3 !s20 −0:25∆2s
k = m !2 −k (14)
 k4 = m2!s20 −n5k
6 3 s0 5
which are the design equations of sense-mode in gyroscope array. The design equations of sense-mode
are different from those of drive-mode due to the different structures of both modes. Compared with
other gyroscope structures, another different point is that it utilizes several proof masses to enlarge the
CoriolisforceandtheirdynamicsequationsarethesameasEquation(8).
BothEquations(7)and(14)arethedesignequationsofproposedgyroscopearrayandthestiffnesses
k , k , k , k , k , k , can be gained in terms of the design equations. Because the resonant peak spacing
1 2 3 4 5 6
∆ and∆ determinethebandwidthsofdrive-modeandsense-mode,andtherobustregionisuptothe
d0 s0
central operational frequency, ∆ and ∆ should be set appropriately. Meanwhile, the anti-resonant
d0 s0
frequencies! equalto! isthecentraloperationalfrequency. Thenthebandwidthofdrive-modewill
d0 s0
behighlymatchedwiththatofsense-mode,whichprovidesthedesiredbandwidthofthesystem. Atthe
same time, the number n of gyroscope units in the gyroscope array should be selected appropriately to
ensurethatthecombinationofgyroscopeunitsincreasesthesensitivityofthesystemwithoutdecreasing
thedesiredbandwidthofthesystem.
5.SimulationResults
According to the theoretical analysis above, the solutions of both modes and design equations of the
systemaregivenbyEquations(3),(7),(10)and(14). Thevaluesofthemainparametersinthegyroscope
unit and proposed gyroscope array are given in Table 1. The stiffness could be calculated by the design
equations. Thenthesimulationwillbeshownintermsoftheseanalysis.
Since the drive-modes of both gyroscope unit and gyroscope array are identical, their
amplitude-frequency responses are the same. Figure 5 shows the frequency response of drive-mode.
It clear illustrates a 3 dB bandwidth of about 190 Hz around the central operational frequency 5 kHz.
Thus,thegyroscopeunitandgyroscopearrayhavethesamebandwidthandgaininthedrive-mode.
Sensors2013,13 1659
Table1. Parametersofthegyroscopeunitandproposedgyroscopearray.
Parameters Valuesofgyroscopeunit Valuesofgyroscopearray
m 2.36e(cid:0)7 kg 2.36e(cid:0)7 kg
1
m 5.23e(cid:0)8 kg 5.23e(cid:0)8 kg
f
m 2.547e(cid:0)7 kg 2.547e(cid:0)7 kg
2
m 1.35e(cid:0)7 kg 8.1e(cid:0)7 kg
3
! = ! 5.0kHz 5.0kHz
s0 d0
∆ 150Hz 150Hz
d
∆ 250Hz 100Hz
s
c 1e(cid:0)4 Ns/m 1e(cid:0)4 Ns/m
1
c 5e(cid:0)6 Ns/m 5e(cid:0)6 Ns/m
2
c 2e(cid:0)4 Ns/m 2e(cid:0)4 Ns/m
3
c 1e(cid:0)4 Ns/m 1e(cid:0)4 Ns/m
4
c 5e(cid:0)6 Ns/m 5e(cid:0)6 Ns/m
5
c 2e(cid:0)4 Ns/m 2e(cid:0)4 Ns/m
6
Figure5. Frequencyresponseofdrive-mode.
−18
−20
−22
−24
B)
d
n(−26
ai
G
−28
−30
−32
−34
4.8 4.85 4.9 4.95 5 5.05 5.1 5.15 5.2
Frequency(ω,kHz)
In sense-mode of the gyroscope array, the number n of gyroscope units needs to be set. Because
the number n has an effect on the frequency response of sense-mode, it must be set appropriately to
satisfythedesiredbandwidthandgain. Figure6showsfrequencyresponseofsense-modewithdifferent
n. When the number is 6, the frequency response has a high gain and also changes softly, which is
better than others. Figure 7 demonstrates the frequency response of gyroscope unit and gyroscope array
in sense-mode. In gyroscope unit, the 3 dB bandwidth is about 300 Hz around the central operational
frequency5kHz,andthegainisfrom–23dBto–20dB;ingyroscopearray,the3dBbandwidthisabout
100 Hz around the central operational frequency 5 kHz, and the gain is increased by 8 dB, from –15 dB
to–12dB.
Sensors2013,13 1660
Figure 6. Frequency response of gyroscope array with different number of gyroscope units
insense-mode.
−10
n=4
n=6
−15 n=8
−20
B)
d
n(−25
ai
G
−30
−35
−40
4.8 4.85 4.9 4.95 5 5.05 5.1 5.15 5.2
Frequency(ω,kHz)
Figure7. Frequencyresponseofgyroscopeunitandgyroscopearray(n = 6)insense-mode.
−10
gyroscope unit
gyroscope array
−15
−20
B)
d
n(−25
ai
G
−30
−35
−40
4.8 4.85 4.9 4.95 5 5.05 5.1 5.15 5.2
Frequency(ω,kHz)
Through combining gyroscope units into a gyroscope array, the gain of sense-mode is increased and
the sensitivity is improved inherently. On the other hand, the value of detection mass is also increased
by 6 times, as compared to the gyroscope unit shown in Table 1. Therefore, the changing capacitance is
6 times that of the gyroscope unit, which increases the output signal and also improves the sensitivity.
Although the bandwidth of gyroscope array is decreased to 100 Hz, it satisfies the need in practical
applications. Meanwhile, the simulation results show that the bandwidth and central frequency of
drive-mode are highly matched with those of sense-mode, which provides the system with a bandwidth
of100Hzandimprovestheinherentrobustnessofproposedgyroscopearray.
Thereissometoleranceinpracticalfabricationandthespringsaresusceptibletostructuralparameter
variations. The key spring k will be considered because the springs k provide the link between
5 5
detectionmass,andalltheproofmassesofgyroscopeunitsintheproposedgyroscopearrayasshownin