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Sensors and Actuators A 144 (2008) 56–63
Dynamics and control of a MEMS angle measuring gyroscope
Sungsu Park
a,∗
, Roberto Horowitz
b
, Chin-Woo Tan
c
a
Department of Aerospace Engineering, Sejong University, 98 Gunja-dong, Kwangjin-gu, Seoul, Republic of Korea
b
Department of Mechanical Engineering, University of California at Berkeley, Berkeley, CA 94720, United States
c
PATH, University of California at Berkeley, Richmond, CA 94804, United States
Re

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Available online at www.sciencedirect.com
Sensors and Actuators A 144 (2008) 56–63
Dynamics and control of a MEMS angle measuring gyroscope
Sungsu Park
a
,
∗
, Roberto Horowitz
b
, Chin-Woo Tan
c
a
Department of Aerospace Engineering, Sejong University, 98 Gunja-dong, Kwangjin-gu, Seoul, Republic of Korea
b
Department of Mechanical Engineering, University of California at Berkeley, Berkeley, CA 94720, United States
c
PATH, University of California at Berkeley, Richmond, CA 94804, United States
Received 17 March 2007; received in revised form 14 November 2007; accepted 31 December 2007Available online 17 January 2008
Abstract
This paper presents an algorithm for controlling vibratory MEMS gyroscopes so that they can directly measure the rotation angle withoutintegration of the angular rate, thus eliminating the accumulation of numerical integration errors incurred in obtaining the angle from the angularrate. The proposed control algorithm consists of a weighted energy control and a mode tuning control. The weighed energy control compensatesunequal damping terms and keeps the amplitude of oscillation constant in an inertial frame by maintaining the prescribed total energy. The modetuningcontrolcontinuouslytunesmismatchesinspringstiffnessinordertomaintainastraightlineofoscillationfortheproofmass.Thesimulationresults demonstrate the feasibility of the control algorithm and the viability of the concept of using a vibratory gyroscope to directly measurerotation angle.© 2008 Elsevier B.V. All rights reserved.
Keywords:
Angle measurement; MEMS gyroscope; Energy control; Mode tuning
1. Introduction
MEMSgyroscopesaretypicallyangularrategyroscopesthatare designed to measure the angular rate [1]. In order to obtain
the rotation angle using a MEMS rate gyroscope, it is requiredto integrate the measured angular rate with respect to time. Theintegration process, however, causes the rotation angle to driftover time and therefore the angle error to diverge quickly due tothe presence of bias and noise in the angular rate signal. Theseeffects are more severe for low cost MEMS rate gyroscopes.Several techniques have been proposed and commercializedto bound the error divergence resulted from the integration of gyroscope angular rate signal. The most common technique isto fuse rate gyroscopes with accelerometers and magnetometersbased on the fact that steady-state pitch and roll angles can beobtained using accelerometers, and yaw angles can be obtainedusing magnetometers. This technique, however, has a few draw-backs. The magnetometer signals can be severely distorted by
∗
Corresponding author. Tel.: +82 2 3408 3769; fax: +82 2 3408 3333.
E-mail addresses:
sungsu@sejong.ac.kr (S. Park),horowitz@me.berkeley.edu (R. Horowitz), tan@eecs.berkeley.edu(C.-W. Tan).
unwanted magnetic ﬁelds in the vicinity of the sensors. Therotation angles can be correctly obtained from accelerometermeasurements only when the moving object is in steady state.Moreover, yaw angle cannot be obtained using accelerometers,although there are a number of applications where yaw anglemust be measured correctly such as automobile and home robotnavigation [2].MEMS gyroscopes can conceptually operate in the rota-tion angle measurement mode. When an isotropic oscillator isallowed to freely oscillate, the precession of the straight lineof oscillation provides a measure of the angle of rotation. Forfreely oscillating, the natural frequencies of oscillation of thetwo vibrating modes must be the same and the modes are un-damped. Ideally, the vibrating modes of a MEMS gyroscope aresupposed to remain mechanically decoupled, their natural fre-quencies should be matched, and the output of the gyroscopeshould be sensitive to only rotation. In practice, however, fab-rication defects and environment variations are always present,resulting in a mismatch of the frequencies of oscillation for thetwovibratingmodesandthepresenceoflineardissipativeforceswith damping coefﬁcients [3]. These fabrication imperfections
are major factors that limit realization of an angle measuringgyroscope. Although most published control algorithms dealwith rate gyroscope [4–6], a few control algorithms for real-
0924-4247/$ – see front matter © 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.sna.2007.12.033
S. Park et al. / Sensors and Actuators A 144 (2008) 56–63
57
izing angle measuring gyroscopes have been presented in Refs.[7–12].FriedlandandHutton[7]suggestedtheuseofavibratory
gyroscope for measuring rotation angle. A composite nonlinearfeedback control is reported in Refs. [8–11], where the energy
control and angular momentum control are developed basedon the analytic results of Ref. [7]. However, their energy con-
trol relies on the equal damping assumption, and the angularmomentum control is vulnerable to interference with the Corio-lis acceleration. Another composite nonlinear feedback controlis proposed in Ref. [12], where the stability of the controlled
system is not proven.In this paper, we present a new control algorithm forrealizing angle measuring gyroscopes. The developed con-trol algorithm maintains the prescribed total energy level, andcompensates for mismatched stiffness and damping, so as toensure that the proof mass maintains a straight line of oscil-lation and keeps the magnitude of amplitude in the inertialframe.
2. Dynamics of a vibratory angle measuring gyroscope
The equation of motion of a mass freely oscillating in twodegrees-of-freedom(2-DOF)atfrequency
ω
0
inaninertialframeis given by¨
q
i
+
ω
20
q
i
=
0 (1)where
q
i
=
[
x
i
y
i
]
T
is displacement of mass along the ˆ
e
1
and ˆ
e
2
axisoftheinertialframe.Todescribethemotionofamassfreelyoscillating in the gyro frame, which rotates about the ˆ
e
3
axisof the inertial frame, a coordinate transformation is performedusing the relation:
q
i
=
C
ig
q
(2)where
C
ig
=
cos
ψ
−
sin
ψ
sin
ψ
cos
ψ
is the direction cosine matrix,
ψ
istherotationangle,and
q
=[
xy
]
T
isdisplacementvectoralongthe ˆ
g
1
and ˆ
g
2
axes of the gyro frame.If Eq. (2) is substituted to Eq. (1), then we get
¨
q
+
[˙
ω
gig
×
]
q
+
2[
ω
gig
×
]˙
q
+
(
ω
20
−
˙
ψ
2
)
q
=
0 (3)where [
ω
gig
×
]
=
0
−
˙
ψ
˙
ψ
0
is the angular rate matrix of thegyro frame with respect to the inertial frame.If the line of oscillation of the mass with amplitude
M
isaligned with the ˆ
e
1
axis, then the solution of Eq. (3) is given by
x
=
M
cos(˙
ψt
)sin(
ω
0
t
)
y
=−
M
sin(˙
ψt
)sin(
ω
0
t
)(4)Eq. (3) is approximated as following equation with the assump-
tion that
ω
0
>>
˙
ψ
and ¨
ψ
≈
0.¨
x
+
ω
20
x
=
2 ˙
ψ
˙
y
¨
y
+
ω
20
y
=−
2 ˙
ψ
˙
x
(5)
Fig. 1. MEMS gyroscope: (a) model and (b) gyroscope fabricated by SejongUniversity.
Eq. (5) describes the motion of 2-DOF freely oscillating masswith frequency
ω
0
in the gyro frame. The rotation angle
ψ
canbe calculated with Eq. (4) by measuring displacement
x
and
y
inthe gyro frame. Therefore, Eq. (5) is referred to as the dynamics
of an ideal vibratory angle measuring gyroscope.A physical angle measuring gyroscope can be implementedby the 2-DOF mass-spring-damper system whose proof massis suspended by spring ﬂexure anchored at the gyro frame, asshown in Fig. 1. A vibratory angle measuring gyroscope has
the same structure as a vibratory rate gyroscope, and there arereports of various types of rate gyroscopes in the literature andindustry.Considering fabrication imperfections and damping, a real-istic model of a
z
-axis gyroscope is described as follows:¨
x
+
d
x
˙
x
+
ω
2
x
x
+
ω
xy
y
=
f
x
+
2 ˙
ψ
˙
y
¨
y
+
d
y
˙
y
+
ω
2
y
y
+
ω
xy
x
=
f
y
−
2 ˙
ψ
˙
x
(6)where
d
x
and
d
y
are damping,
ω
x
and
ω
y
are natural frequenciesofthe
x
-and
y
-axis,
ω
xy
isacoupledfrequencyterm,and
f
x
and
f
y
arethespeciﬁccontrolforcesappliedtotheproofmassin ˆ
g
1
andˆ
g
2
axis of the gyro frame, respectively. The coupled frequencyterm, called quadrature error, comes mainly from asymmetriesin suspension structure and misalignment of sensors and actu-ators. Recently, a mechanically decoupled gyroscope structurehas been proposed in the literature and it is shown that twoaxes can be mechanically decoupled to a great extent by usinga unidirectional frame structure [13,14].
58
S. Park et al. / Sensors and Actuators A 144 (2008) 56–63
3. Design of control algorithm
The control problem of angle measuring gyroscope is for-malized as follows; given the realistic gyroscope model,¨
q
+
ω
20
q
=
f
−
D
˙
q
−
Rq
−
2[
ω
gig
×
]
q
(7)where
f
=
[
f
x
f
y
]
T
,
D
=
d
x
00
d
y
,
R
=
ω
x
00
ω
y
and
ω
x
=
ω
2
x
−
ω
20
,
ω
y
=
ω
2
y
−
ω
20
, determine the control lawsfor
f
x
and
f
y
, such that the damping terms,
d
x
and
d
y
, and mis-matches in natural frequencies,
ω
x
and
ω
y
, are correctlycompensated for the realistic gyroscope to be operated as anideal angle measuring gyroscope. Note that the gyroscope oper-ates at a ﬁxed frequency,
ω
0
, which is chosen by the designer.In such a way, natural frequencies of both axes can be activelytuned to be matched, and the associated signal processing canbe simpliﬁed as well.In this section, we propose an adaptive controller to compen-sate for damping terms and mismatches in natural frequenciesby performing two tasks: (a) initiating oscillation and maintain-ingtotalenergylevel,and(b)tuninganymismatchinthenaturalfrequencies of both axes.
3.1. Weighted energy control
When the gyroscope rotates, the line of oscillation precessesbecause the Coriolis acceleration transfers energy between thetwo axes of the gyroscope, while conserving the total energy of the gyroscope. This can be shown by deﬁning the instantaneoustotal mechanical energy
E
as
E
=
12(˙
q
T
˙
q
+
ω
20
q
T
q
) (8)and differentiating it along the trajectory of an ideal gyroscope(5) as follows.˙
E
=
˙
q
T
¨
q
+
ω
20
˙
q
T
q
=
˙
q
T
(
−
ω
20
q
−
2[
ω
gig
×
]˙
q
)
+
ω
20
˙
q
T
q
=
0(9)FromEq.(9),itisclearthattheangularratetermdoesnotchange
the total energy. However, in case of a non-ideal gyroscope, thetotal energy is not conserved because of the damping terms.Therefore, the purpose of an energy control should be tomaintaintheprescribedenergylevelsothatthedampingiscom-pensated without interference with the angular rate, and also toexcite the proof mass into oscillation. If the prescribed energylevel is larger than the current energy level, then the magnitudeof energy control is chosen to be positive for growing the oscil-lation, and conversely negative for damping the oscillation. Insuch a way, the magnitude of energy control effectively com-pensates the damping terms and sustains free oscillation of thesystem.The deviation of actual energy level of the system from theprescribed one is deﬁned by˜
E
=
E
0
−
12(˙
q
T
˙
q
+
ω
20
q
T
q
) (10)where
E
0
denotes the prescribed energy level. Note that totalenergy is computed based on the designed reference frequency
ω
0
. Now, consider the following positive deﬁnite function(PDF).
V
=
12
˜
E
2
+
1
K
I
tr
{
˜
D
˜
D
T
}
(11)where
K
I
is a positive constant, ˜
D
=
ˆ
D
−
D
where ˆ
D
is theestimate of
D
, and tr
{·}
denotes the trace of the matrix. Thederivative of the PDF
V
along the trajectory of Eq. (7) is
˙
V
=
˜
E
˙˜
E
+
1
K
I
tr
{
˜
D
˙˜
D
T
}
(12)If the energy control law
f
E
is chosen to be
f
E
=
ˆ
D
˙
q
+
f
1
(13)where
f
1
isanauxiliarycontrolactionthatwillbedeﬁnedsubse-quently,thenthederivativeofthePDF
V
iscomputedasfollows,assuming that the natural frequencies are compensated to be thereference natural frequency.˙
V
=
˜
E
(
−
˙
q
T
f
1
−
˙
q
T
˜
D
˙
q
)
+
1
K
I
tr
{
˜
D
˙˜
D
T
}
(14)If
f
1
is chosen to be
f
1
=
K
P
˜
E
˙
q
(15)where
K
P
is a positive constant, then Eq. (14) becomes:
˙
V
=−
K
P
˜
E
2
˙
q
T
˙
q
+
tr
1
K
I
˜
D
˙˜
D
T
−
˜
E
˜
D
˙
q
˙
q
T
(16)Eq. (16) suggests the following adaptation law for:
˙ˆ
D
=
K
I
˜
E
˙
q
˙
q
T
(17)leads to ˙
V
=−
K
P
˜
E
2
˙
q
T
˙
q
≤
0.
Theorem 1.
With the control laws (13) and (15), and damping
adaptation law (17), the following results hold.
(a) The total energy error ˜
E
and its time-derivative both con-verge to zero as
t
→∞
.(b) The convergence of the damping estimate, ˆ
D
, to its truevalue is guaranteed only when equal damping of both axesis assumed.According to Theorem 1, the energy control can compensate
the damping terms only when both axes have the same dampingvalues.Sinceunequaldampingtermscausedifferentdissipationof energy, different weightings on the total energy control arerequired. This fact suggests a modiﬁcation of the energy controllaw which we summarize in the following theorem.
Theorem 2.
If the damping ratio of both axes is known, thenthe total energy error and damping estimate error converge tozeroas
t
→∞
whenthefollowingcontrollaw(18)anddamping
adaptation law (19) are applied.
S. Park et al. / Sensors and Actuators A 144 (2008) 56–63
59
f
E
=
K
P
˜
E
˙
q
+
ˆ
αΛ
˙
q
(18)˙ˆ
α
=
K
I
˜
E
(˙
q
T
Λ
˙
q
) (19)where
Λ
is a damping ratio matrix to satisfy
D
=
αΛ
,
α
is anassociated scalar value, ˆ
α
is the estimate of
α
, and ˜
α
=
ˆ
α
−
α
.The srcinal energy control is modiﬁed using the dampingratio matrix and therefore compensates different dissipation of two axes. The damping ratio can be obtained at an initializationstage which we will explain later.
3.2. Mode tuning control
Thenaturalfrequenciesofthetwoaxesmustbematchedpre-cisely, but the accuracy required is beyond the manufacturingtolerance. Since the natural frequency changes with tempera-ture and other environment factors, we propose that the ﬁxedreference frequency,
ω
0
, is speciﬁed by the designer. There-fore, the purpose of mode tuning control is to track referencefrequency by compensating for
x
- and
y
-axis natural frequencydeviationsfromthereferencefrequency.Theintroductionoftheconcept of reference frequency is very useful since it can sim-plify signal processing needed for calculating the total energyand demodulating the output signals.The mode tuning control for both axes is given by
f
M
=
ˆ
Rq
=
ˆ
ω
x
x
ˆ
ω
y
y
(20)where the frequency deviations are estimated by a frequencydeviationestimator.Sincebothaxessharethesamemodetuningcontrol scheme, we will explain a frequency deviation estimatorfor
x
-axis only for simplicity. A frequency deviation estimatorconsists of two function blocks: phase detector and controller.The phase detector compares the phase difference betweenthe driving signal and the output. Consider an ideal gyroscopebehavior and a velocity feedback energy control, the drivingsignal can be assumed to be cos
ω
0
t
multiplied by a constant.Therefore, the output of phase detector is the product of themeasured position signal and the driving signal, cos
ω
0
t
, ﬁlteredby a low-pass ﬁlter, i.e.˜
θ
=
LPF(
x
sgn(
X
)cos
ω
0
t
) (21)where ˜
θ
is the phase difference,
x
is the measured position sig-nal, LPF denotes a low-pass ﬁlter, and sgn(
X
) is the sign of themeasuredvelocitysignal ˙
x
comparedtoreferencedrivingsignalcos
ω
0
t
, i.e.
X
=
LPF(cos
ω
0
t
˙
x
) (22)The
x
-axis frequency deviation,
ˆ
ω
x
, is calculated from thephase difference, ˜
θ
, by using an integral controller,
ˆ
ω
x
=
K
IM
s
˜
θ
(23)where
K
IM
is the integral control gain. Stability analysis of thismodetuningcontrolschemecanbeobtainedinasimilarfashionas that in the literature for PLL [16].
3.3. Initialization
AsmentionedinTheorem2,theenergycontrolneedsadamp-
ing ratio matrix
to compensate for the different dissipationsof two axes. There may be two approaches in identifying thisdamping ratio matrix. One approach is to drive both axes withthe same control such as
f
x,y
=
A
cos
ω
0
t
(24)where
A
is the ﬁxed amplitude of the control. The damping ratiois identiﬁed by calculating the energy ratio using the fact thatthe damping ratio is inversely proportional to the square-root of the energy ratio, i.e.
d
y
d
x
=
E
X
E
Y
(25)where
E
X
=
12(˙
x
2
+
ω
20
x
2
)
, E
Y
=
12(˙
y
2
+
ω
20
y
2
) (26)are the calculated energies of the
x
- and
y
-axis, respectively.The other approach is to implement energy control schemeandestimatethedampingtermsofbothaxesindependently.Theenergy control can be the same as that in Eq. (24), however, the
amplitudeoftheenergycontrol
A
isadjusteduntiltheprescribedenergy level is reached at both axes. Scalar versioned controllaws of (13), (15) and (17) can be used to estimate
d
x
and
d
y
,thus damping ratio
d
y
/
d
x
.Theseapproachesareusedattheinitialcalibrationstagewiththeassumptionofzeroangularratewhenthegyroscopeisturnedon, or at regular calibration sessions which may be performedperiodically to identify the ratio. Once the ratio is identiﬁed, itsvalue can be frozen until the next calibration session, becausethe variation of damping ratio is negligibly slow compared withdamping itself.Althougharecentlydevelopedmechanicallydecoupledgyro-scopestructurehasshownthatthetwoaxescanbemechanicallydecoupled to a great extent, there may be still a coupled stiff-ness term which comes mainly from misalignment of sensorsand actuators. In this case, a coupling compensator is neededto compensate for the coupling effect in stiffness between thetwoaxesataninitialcalibrationstage.Aforcebalancingcontrol[17] is used to compensate the coupling term so that it drivesthe
y
-axis output to and holds it at zero. It is given as a PI-typecontroller as follows.ˆ
ω
xy
=
K
1
+
K
2
s
LPF(
y
cos
ω
0
t
)
f
xb
=
ˆ
ω
xy
y
(27)where
f
xb
is a force-balancing control, and
K
1
and
K
2
are theproportional and integral gains.
3.4. Rotation angle calculation
When the gyroscope is allowed to freely oscillate andcontrolled to compensate for damping, mismatched natural fre-

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