520 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 14, NO. 3, JUNE 2005
An Approach for Increasing Drive-Mode Bandwidth
of MEMS Vibratory Gyroscopes
Cenk Acar and Andrei M. Shkel, Associate Member, IEEE, Associate Member, ASME
Abstract—The limitations of the photolithography-based
micromachining technologies defines the upper-bound on the
performance and robustness of micromachined gyroscopes. Con-
ventional gyroscope designs based on matching (or near-matching)
the drive and sense modes are extremely sensitive to variations
in oscillatory system parameters that shift the natural frequen-
cies and introduce quadrature errors. Nonconventional design
concepts have been reported that increase bandwidth to improve
robustness, but with the expense of response gain reduction. This
paper presents a new approach that may yield robust vibratory
MEMS gyroscopes with better gain characteristics while retaining
the wide bandwidth. The approach is based on utilizing multiple
drive-mode oscillators with incrementally spaced resonance fre-
quencies to achieve wide-bandwidth response in the drive-mode,
leading to improved robustness to structural and thermal pa-
rameter fluctuations. Enhanced mode-decoupling is achieved
by distributing the linear drive-mode oscillators radially and
symmetrically, to form a multidirectional linear drive-mode and a
torsional sense-mode; minimizing quadrature error and zero-rate
output. The approach has been implemented on bulk-microma-
chined prototypes fabricated in a silicon-on-insulator (SOI)-based
process, and experimentally demonstrated. [1285]
Index Terms—Inertial sensors, micromachined gyroscopes,
MEMS, rate sensors.
I. INTRODUCTION
T
HE tolerancing capabilities of the current photolithog-
raphy processes and microfabrication techniques are
inadequate compared to the requirements for production of
high-performance inertial sensors. The resulting inherent im-
perfections in the mechanical structure significantly limits
the performance, stability, and robustness of MEMS gyro-
scopes [3], [6]. Thus, fabrication and commercialization of
high-performance and reliable MEMS gyroscopes that require
picometer-scale displacement measurements of a vibratory
mass have proven to be extremely challenging [1], [2].
The operation principle of the vast majority of all existing
micromachined vibratory gyroscopes relies on the generation
of a sinusoidal Coriolis force due to the combination of vibra-
tion of a proof-mass and an orthogonal angular-rate input. The
proof mass is generally suspended above the substrate by a sus-
pension system consisting of flexible beams. The overall dy-
namical system is typically a two degrees-of-freedom (2-DOF)
Manuscript received February 28, 2004; revised July 21, 2004. Subject Editor
R. R. A. Syms.
The authors are with the University of California, Irvine, Mechan-
ical and Aerospace Engineering Department, MicroSystems Laboratory
EG2110, Irvine, CA 92697 USA (e-mail: cacar@uci.edu; ashkel@uci.edu;
http://mems.eng.uci.edu).
Digital Object Identifier 10.1109/JMEMS.2005.844801
mass-spring-damper system, where the rotation-induced Cori-
olis force causes energy transfer to the sense-mode proportional
to the angular rate input. In most of the reported micromachined
vibratory rate gyroscopes, the proof mass is driven into res-
onance in the drive direction by an external sinusoidal elec-
trostatic or electromagnetic force. When the gyroscope is sub-
jected to an angular rotation, a sinusoidal Coriolis force is in-
duced in the direction orthogonal to the drive-mode oscillation
at the driving frequency. Ideally, it is desired to utilize resonance
in both the drive and the sense modes, to attain the maximum
possible response gain, and hence sensitivity. This is typically
achieved by designing and electrostatically tuning the drive and
sense resonant frequencies to match. Alternatively, the sense-
mode is designed to be slightly shifted from the drive-mode
to improve robustness and thermal stability, while intentionally
sacrificing gain and sensitivity [7].
The drive and sense mode matching (or near-matching) re-
quirement in vibratory gyroscopes renders the system response
very sensitive to variations in system parameters, e.g., due to
fabrication imperfections and fluctuations in operating condi-
tions, which shift the drive or sense resonant frequencies [6]. For
the devices packaged in vacuum to enhance the sensitivity by in-
creasing the drive and sense mode Q-factors, the bandwidths of
the drive and sense frequency responses are extremely narrow;
leading to much tighter mode-matching requirements and lim-
ited bandwidth of angular-rate detection. Extensive research has
focused on design of symmetric drive and sense-mode suspen-
sions for mode-matching and minimizing temperature depen-
dence, [18]. However, especially for lightly-damped devices,
it is recognized by many authors that the mode-matching re-
quirement is well beyond fabrication tolerances; and none of
the symmetric designs can provide the required degree of mode-
matching without feedback control [4], [5].
Furthermore, extremely small imbalances in the gyroscope
suspension due to fabrication imperfections introduce anisoe-
lasticities, which result in undesired mode coupling often larger
than the Coriolis motion. In order to suppress coupled oscilla-
tion and drift and to minimize the resulting zero-rate drift, var-
ious devices have been reported employing decoupled modes or
independent suspension systems for the drive and sense modes
[12]–[15]. The approach of structurally decoupling drive and
sense modes led to the first integrated commercial MEMS gy-
roscopes mass-produced by Analog Devices [17].
The mode-matching problem and the quadrature error due to
inherent fabrication imperfections are the two major challenges
in MEMS gyroscope design. We have previously reported gy-
roscope systems that offer improved robustness by increasing
the degree-of-freedom of the dynamical system [8], [9]. Even
1057-7157/$20.00 © 2005 IEEE
ACAR AND SHKEL: AN APPROACH FOR INCREASING DRIVE-MODE BANDWIDTH OF MEMS VIBRATORY GYROSCOPES 521
Fig. 1. Scanning electron microscope micrograph of a distributed-mass
micromachined gyroscope prototype, utilizing multiple drive-mode oscillators
with incrementally spaced resonance frequencies.
though increased-DOF gyroscope systems provide significantly
increased bandwidth (over 1 kHz), this is achieved with the ex-
pense of sacrificing response gain. This paper presents a novel
approach that may provide wider drive-mode bandwidth than
conventional MEMS gyroscopes, with less sacrifice in response
gain compared to previously reported wide-bandwidth devices.
The concept based on utilizing multiple drive-mode oscillators
with incrementally spaced resonance frequencies (see Fig. 1)
was introduced in [10] by these authors, with the preliminary ex-
perimental results on the first generation prototypes presented in
[11]. In this paper we generalize the approach in Section II, the-
oretically and experimentally explore the involved design trade-
offs to achieve a wide drive-mode bandwidth in Sections III and
IV, and present the experimental characterization results that
demonstrate the feasibility of the design concept in Section IV.
II. T
HE APPROACH
Since the Coriolis force, and the sense-mode response is di-
rectly proportional to the drive-mode oscillation amplitude, it
is desired to enhance the drive-mode amplitude by increasing
the Q factor with vacuum packaging and operating at the peak
of the drive-mode resonance curve. However, large drive-mode
amplitude and bandwidth cannot be achieved with a 1-DOF
drive system at the same time. The proposed approach explores
the possibility of increasing the drive-mode response bandwidth
of micromachined gyroscopes, by utilizing multiple resonators
with incrementally spaced resonant frequencies in the drive-
mode. The drive and sense modes are effectively decoupled by
forming a multidirectional linear drive-mode that transmits the
Coriolis force into a torsional sense-mode.
The design concept is based on forming multiple drive-mode
oscillators, distributed symmetrically around the center of a sup-
porting frame. The distributed drive-mode oscillators are driven
in-phase toward the center of symmetry, and are structurally
constrained in the tangential direction with respect to the sup-
porting frame. Each oscillator is driven at the same drive fre-
quency. In the presence of an angular rotation rate about the
z-axis, a sinusoidal Coriolis force at the drive frequency is in-
duced on each proof mass in the direction orthogonal to each
drive-mode oscillation directions (see Fig. 2). Thus, each of
the induced Coriolis force vectors lie in the tangential direc-
tion, combining to generate a resultant torque on the supporting
frame. The net Coriolis torque excites the supporting frame into
torsional oscillations about the z-axis, which are detected by
sense capacitors for angular rate measurement.
The multidirectional and axisymmetric nature of the drive-
mode oscillators offers several structural benefits over a con-
ventional gyroscope design.
• Instability and drift due to mechanical coupling be-
tween the drive and sense modes is minimized, since
the structure is designed to completely decouple the
multidirectional linear drive-mode and the rotational
sense-mode. Thus, zero-rate-output and quadrature
error are significantly reduced in the presence of
structural imperfections.
• The sensing electrodes are attached to the supporting
frame, and do not respond to the drive-mode vibrations
owing to the structural decoupling. This minimizes the
noise in the response induced by the drive-mode oscil-
lations.
• The torsional sense mode rejects external linear accel-
erations and vibrations.
• Since the drive forces applied to the drive-mode os-
cillators cancel out in all directions due to the radial
symmetry, the net force on the structure is effectively
suppressed. This results in near-zero reaction force in-
duced on the anchor, thus minimizing energy emission
to the substrate.
• The central single anchor structure minimizes the ef-
fects of packaging stresses and thermal gradients.
• The symmetry of the drive-mode oscillator structure
about several axes also cancels the effects of direc-
tional residual stresses, and elastic anisotropy of the
structural material.
A. The Coriolis Response
In the proposed approach, the distributed drive-mode oscilla-
tors are driven in-phase toward the center, and constrained in the
tangential direction with respect to the supporting frame. The
constrained dynamics of each proof-mass along the associated
drive axis with respect to the supporting frame reduces to
where is the th proof-mass, and is the drive-mode re-
sponse of the
th mass. Thus, in the presence of an angular rota-
tion rate about the z-axis, the Coriolis forces, which are propor-
tional to drive direction oscillation amplitudes, induced on each
proof mass are
The rotation-induced Coriolis forces are orthogonal to each of
the drive-mode oscillation directions. Thus, each of the induced
Coriolis force vectors lie in the tangential direction, combining
522 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 14, NO. 3, JUNE 2005
Fig. 2. Conceptual illustration of the distributed-mass gyroscope with eight symmetric drive-mode oscillators.
Fig. 3. (a) The frequency responses of the distributed drive-mode oscillators. (b) The frequency spectrum of the total Coriolis torque generated by the distributed
drive-mode oscillators.
to form a resultant torque on the supporting frame. The net Cori-
olis torque generated as the combination of each Coriolis force
becomes
where is the position vector of the oscillator center-of-mass,
and
is the unit vector in the -direction. The Coriolis torque
excites the supporting frame into torsional oscillations about
the z-axis, which is detected by the sense capacitors, providing
measurement of angular rate. Assuming the rate input is con-
stant and smaller compared to the driving frequency, the simpli-
fied equation of motion of the supporting frame in the sense-di-
rection is
where is the torsional deflection of the supporting frame,
denotes the moment of inertia of the supporting frame com-
bined with the proof masses,
is the sense-mode torsional
damping ratio, and
is the torsional stiffness of the suspen-
sion structure.
B. Wide Bandwidth Operation for Improving Robustness
In the presented design concept, a wide-bandwidth opera-
tion region is achieved in the drive-mode frequency response,
by designing or actively tuning the resonance frequency of each
drive-mode oscillator to be incrementally spaced [see Fig. 3(a)].
Since the tangential Coriolis forces induced on each proof mass
jointly generate a resultant torque on the supporting frame, a
“levelled” total Coriolis torque is achieved over a wide range of
ACAR AND SHKEL: AN APPROACH FOR INCREASING DRIVE-MODE BANDWIDTH OF MEMS VIBRATORY GYROSCOPES 523
Fig. 4. (a) The effect of damping and resonance frequency separation on the drive-mode response. (b) The effect of frequency separation on the response gain
and bandwidth (effecting sensitivity and robustness, respectively). The gain is maximized for zero frequency separation, and the overall bandwidth increases
proportionally to spacing.
driving frequencies [see Fig. 3(b)]. The device is nominally op-
erated in this levelled region of the Coriolis torque frequency
response, so that fluctuations in system parameters that shift
oscillator resonance frequencies will not result in a significant
change in the total Coriolis torque. If the sense-mode resonance
frequency is designed to be accommodated in the same fre-
quency band [see Fig. 3(b)], the requirement on the degree of
mode-matching is relaxed, and robustness against structural and
thermal parameter fluctuations is achieved.
1) Driving Scheme: The drive-mode oscillators are driven
at the same frequency inside the levelled frequency region. This
assures that the sinusoidal Coriolis forces induced on each drive-
mode oscillator are at the same driving frequency. Thus, the si-
nusoidal Coriolis forces are superposed, and generate a resul-
tant moment that excites the torsional sense-mode at the driving
frequency.
During operation of the device, the forced oscillation ampli-
tude of each oscillator will be different depending on the loca-
tion of the drive frequency within the operation region, but the
total drive-mode response will be constant at a known value.
Thus, constant-amplitude control is not implemented on the os-
cillators, and the same signal is used to drive all of the os-
cillators for the purposes of demonstration of the design con-
cept. In future implementations, a control architecture could be
adapted that identifies the drive-mode parameters of each oscil-
lator during calibration, and applies the appropriate drive signal
to each oscillator so that the resonance amplitude of each is
equal to a preset value.
2) Frequency Spacing Design: It should be noticed that the
resonance frequency separation of the oscillators are dictated
by the bandwidth of the response, and thus by damping. In
order to obtain a levelled operation region in the drive-mode,
the frequency separation should be less than the bandwidth of
a single oscillator. If the separation of frequencies is large for
low damping resonators, the resonance peaks become signifi-
cant [see Fig. 4(a)], and the levelled operation region will not
be achieved in the response. On the contrary, the total response
will converge to a 1-DOF resonance peak as the frequency sep-
aration approaches zero, where the highest possible gain is at-
tained with the narrowest bandwidth.
III. T
HEORETICAL ANALYSIS OF THE TRADEOFFS
The proposed design approach allows to widen the operation
frequency range of the gyroscope drive-mode to achieve im-
proved robustness, with the expense of sacrifice in the response
amplitude. The optimal compromise between amplitude of the
response and bandwidth (effecting sensitivity and robustness,
respectively) can be obtained by selecting the frequency incre-
ments of the drive-mode oscillators.
As a numerical example, the response of a device consisting
of eight drive-mode oscillators with resonance frequencies from
6.895 to 7 kHz and a frequency spacing of 15 Hz will be con-
sidered. For
input angular rate and a Q factor of 100 in the
drive and sense modes, the supporting frame of the distributed-
mass gyroscope will have an angular amplitude of response
equal to
, which is equivalent to
displacement at the sensing electrodes. If the frequency spacing
of the drive-mode oscillators is decreased from 15 Hz to 10
Hz, the amplitude of the response in the sense direction will
increase from
to ; while the re-
sponse bandwidth will decrease from 200 Hz to 140 Hz, which
is still over an order of magnitude larger than the bandwidth
of a single-mass conventional gyroscope. The bandwidth can
be further widened by increasing the number of oscillators. In
Fig. 4(b), the response of a gyroscope with 10 oscillators is
524 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 14, NO. 3, JUNE 2005
Fig. 5. SEM images of the characterized two different prototype structures: (a) the structure employing comb-drive actuation for large drive amplit
udes and (b)
the structure employing parallel-plate actuation for a wide electrostatic tuning range.
modeled along with 8-oscillator systems with 0, 10, and 15 Hz
spacing; illustrating the effect of frequency separation and the
number of oscillators. If the frequency separation is set to zero,
the response gain will be at its maximum of
,
with a bandwidth of 100 Hz. The tradeoffs between gain of
the response (higher sensitivity) and the system bandwidth (in-
creased robustness) are typically guided by application require-
ments.
IV. E
XPERIMENTAL ANALYSIS OF THE TRADEOFFS
A. Fabrication of Prototypes
The wide-bandwidth design concept was analyzed experi-
mentally on the bulk-micromachined prototype structures, fab-
ricated in the UCI Integrated Nano-Systems Research Facility
(see Fig. 5). Two different prototype gyroscope structures uti-
lizing the wide-bandwidth design concept were designed: one
structure employing comb-drive actuation to achieve large drive
amplitudes, and one structure employing parallel-plate actua-
tion for a wide electrostatic tuning range.
For the fabrication of prototypes, a one-mask process based
on SOI (Silicon-on-Insulator) wafers was developed and opti-
mized for high-aspect ratio structures. The developed process
relies on deep-reactive ion etching (DRIE) through the 100-
-device layer, and front-side release of the structures by etching
the Oxide layer in HF solution. The process and the device de-
sign was optimized to minimize notching at the oxide interface
and excessive undercutting. The DRIE process was performed
in an STS ICP, using 8 s etch step cycle with 130 sccm
and
13 sccm
, 600 W coil power and 15 W platen power; and
5 s passivation step cycle time with 85 sccm
, 600 W coil
power, and 0 W platen power. In the device,
holes were used to perforate the suspended structures, and 10
gaps were used in the sensing and actuation electrodes. The
lowest etch rates were observed for the
holes, at
approximately
, and 85 min DRIE time was used
to assure complete through-etch while minimizing excessive un-
dercutting in larger areas. The anchors were designed as unper-
forated areas larger than
for 25 min release
in 49%HF solution. Each drive-mode oscillator was designed
identically, although it will be shown in the next section that the
natural frequency of each oscillator will be shifted due to fabri-
cation imperfections. This phenomenon is exploited to naturally
provide the required frequency spacing for this demonstration.
B. Finite Element Analysis Results
In order to optimize the system parameters and verify the va-
lidity of the theoretical analysis assumptions, the operational
modes of the system were simulated using the Finite Element
Analysis package MSC Nastran/Patran. Each drive-mode mass
of the analyzed prototype system is
, sus-
pended by four
folded springs; yielding a reso-
nance frequency estimation of 7.15 kHz with an elastic mod-
ulus of 130 GPa for single-crystal Silicon in (100)-direction.
Through FEA simulations, the resonance frequency of the drive-
mode oscillators were obtained at 6.98 kHz. The torsional sense
mode resonance frequency of the structure about the sense axis
was then located at
with four
torsional suspension beams, by iteratively optimizing the beam
length.
C. Experimental Characterization Results
The dynamic response of the linear drive-mode oscillators
and the torsional sense-mode of the prototype gyroscope were
characterized in an MMR Vacuum Probe Station. The frequency
response of the prototype devices were acquired under varying
pressure values and at room temperature, using off-chip tran-
simpedance amplifiers with a feedback resistor of
connected to an HP Signal Analyzer in sine-sweep mode. The
drive-mode frequency responses were acquired utilizing one-
port actuation and detection, where a single electrode was used
for both driving and sensing at the same time. The driving ac
signal plus the dc bias voltage was imposed on the gyroscope
structure through the anchor, and the actuation and detection
port was directly connected to the transimpedance amplifier.
The resonance frequencies of the drive-mode resonators were
observed to be scattered between 4.546 kHz and 5.355 kHz
