Novel mismatch compensation methods for

rate-integrating gyroscopes

J. A. Gregory, J. Cho, and K. Najafi Center for Wireless Integrated Microsensing and Systems (WIMS2), University of Michigan Ann Arbor, USA

Abstract-We propose two new controls to improve the per­formance of rate-integrating MEMS gyroscopes (RIGs). The first control loop dynamically determines and compensates for damping mismatch by creating a force on the gyroscope to oppose the apparent force due to damping mismatch. The second control loop extends this concept to frequency mismatch to eliminate residual quadrature which would otherwise accumulate due to lag in the control system. The proposed control loops have been investigated using simulation results from a non-ideal gyroscope including frequency and damping mismatch as well as control delay and capacitive feed-through. Experimental results of the controls being applied to a MEMS rate-integrating gyroscope are also presented. In simulation, the controls can reduce the drift in a gyroscope by better than two orders of magnitude. In a physical gyroscope, the mismatch parameters do not match the normal first order model, however RMS drift is reduced by 25%.

I. INTRODUCTION

We propose novel compensation methods for frequency and damping mismatch to reduce drift in rate-integrating gyro­scopes (RIGs). RIGs offer three primary advantages over rate gyroscopes: mechanically unlimited bandwidth, sensitivity that is purely a function of geometry, and wide dynamic range. The primary disadvantages of RIGs are the increased complexity of the controls and increased drift. While an ideal RIG will have zero drift, in practice even relatively small mismatch in the parameters of the degenerate modes will produce significant drift. We propose two novel control mechanisms to reduce the drift in RIGs so that relatively inexpensive MEMS RIGs can achieve good performance. Published MEMS RIGs [I] [2] are 1 to 1.5cm across and less than a millimeter thick, and there are several efforts to reduce the device size while improving performance. Potential applications are miniature flying vehicles and personal navigation systems which could benefit from the very large bandwidth and dynamic range of RIGs.

The principal of a vibratory RIG is that the orientation of a standing wave on a structure will lag behind the rotation of that structure. The absolute rotation of the structure is measured by tracking the orientation of the standing wave with respect to the structure. Currently, RIGs rely on extreme manufacturing tolerance to achieve rate-integration. Traditional controls for a RIG track the oscillation phase, maintain total oscillation energy and compensate for quadrature error. An optional ori­entation control loop can set the standing wave orientation, and is typically used for rate-mode operation. We have designed additional controls based on feedback loops to dynamically determine the relative frequency and damping mismatch and

y �

�� .......... ���/ X Q .. '

Fig. I. A single mass attached to a frame with springs aligned with the X and Y axis can be used as a generic model of all Coriolis vibrator gyroscopes.

compensate for them without interfering with the gyroscope operation.

In section II we present the operation of RIGs, in sec­tion III we describe the compensation algorithms in detail, in section IV we present the modeling and implementation of the controls, and in section V we present some experimental results which show the utility of the proposed methods while indicating that the error models may need to be improved for MEMS RIGs.

II. GYROSCOPE MODEL

A simple conceptual vibratory gyroscope is illustrated in Fig. I. A mass is vibrated with with amplitude a and resonance frequency w along an angle B with respect to the X axis. Due to non-idealities in the gyroscope, some quadrature motion q develops, which leads to drift in the orientation B. Sensors are placed to measure the motion along the X and Y axes and the motion of the mass projected onto these axes is sensed as [3]

x = acos(B)cos(wt) + qsin(B)sin(wt) y = asin(B)cos(wt) - qsin(B)sin(wt). (1)

Instead of a single mass, it is also possible to use any other structure with degenerate orthogonal modes such as the wineglass mode of a ring or cylindrical resonator [4].

The sensed gyroscope motion can be IQ demodulated as

[�: ��] =

[:�:: :�::] [�;] where ¢ = wt + ¢' is the phase of the demodulating signal which may have some error ¢'. The parameters cx, cy, sx, Sy

978-1-4673-0387-3/12/$31.00 ©2012 IEEE 252

Scale, Modulate and Sum

Ampl Cont.

Quad Cont. Demod,

Parameter 14-------- Extraction

Fig. 2. A RIG control system must implement blocks to maintain the total oscillation energy, cancel quadrature error, and proportion drive signals according to the oscillation orientation.

can be used to calculate the control 'slow' mode gyroscope

parameters

E 2 2 2 2 2 2 = Cx + Sx + Cy + Sy = a + q Q = 2(cxSy - cysx) = 2aq R = c2 + S

2 - c

2 - S2 = (a2 - q2) cos 28 (2) x x y y S = 2(cxcy + SxSy) = (a2 - q2) sin 28

Li = 2( CxSx + CySy) = (a2 - q2) sin 28¢

where E is the oscillation energy, Q is the quadrature error, R and S are the X, Y projections used to calculate the oscillation

angle, 8¢ is the phase error between the gyroscope oscillation

and the reference signal, and Li is the the phase error in the

demodulation signal.

The parameters E,Q,R,S, and Li form the basis of the

controls used in this work. A basic RIG control system is

illustrated in Fig. 2. There are four controls loops: amplitude,

quadrature, orientation, and phase. In an ideal gyroscope, only

the amplitude and phase control are necessary. The quadrature

control cancels the quadrature error to reduce the drift due to

variation in the resonance frequency and damping with the

oscillation orientation. The orientation control can steer the

oscillation to a desired orientation and the output of the control

is proportional to the rate of rotation plus any intrinsic drift

at that orientation. The amplitude, quadrature and orientation

control values are proportioned between the X and Y axes

based on the measured value of the oscillation orientation 8.

III. NOVEL COMPENSATION CONTROLS

The ideal MEMS RIG can be modeled as

1 x + -± + 2Agf2y + w2x = 0

7

.. 1. 2A r.' 2 0 Y + -y - gHX + w y =

7

(3)

where 7 is the damping time constant, Ag is the mechanical

angular gain, w is the resonance frequency and f2 is the rate

of rotation. When modeling the gyroscope, it can be useful to

think of on-axis velocity terms as damping forces, cross-axis

velocity terms as drift forces, and position terms as quadrature

forces. Due to material properties and fabrication uncertainty,

the damping and frequency will depend on the orientation of

the oscillation. A commonly used model of the damping and

frequency anisotropy is

1 1 1 - = -+ � -cos 2(8 - 8r) 7 70 7

w2 =w5+ w�wcos2(8 - 8w)

(4)

(5)

where 8r is the orientation where the damping is minimum,

8w the orientation where the resonance frequency is maximum,

A l I I . . d f th d . . h u - = - - - IS the magmtu e 0 e ampmg mlsmatc

7 71 72 w2

_ w2

and wt::.w = 1 2 the resonance frequency mismatch.

The terms 71. 72, 21 and W2 are the damping and resonance

frequency parameters at 8r,w and 8r,w + 7r /2 respectively.

By combining (3) with (4) and (5), a model of the gyroscope

which includes the mismatch errors can be derived [3]

x - 2Agf2y + �± + t::. (�) (±cos28r + ysin28r)

+ w2x - wt::.w (X cos 28w + y sin 28w) = 0

ii + 2Agf2± + �y + � (� ) (-± sin 28r + Y cos 28r)

+w2y - w�w(-xsin28w + ycos 28w) = O.

A. Damping Mismatch Compensation Loop

(6)

In [5], a method is described for dynamically determining

the magnitude and orientation of the damping mismatch in a

RIG and adjusting the amplitude control to account for the

variation in a gyroscope which has both a ring drive electrode

and drive electrodes on the X and Y axes. The drift due to

damping is not directly addressed and our own simulations

show that although amplitude variation is removed, the drift

is not. The method presented here does not use a ring

electrode, which is not always available in MEMS RIGs, and

directly addresses the drift due to damping mismatch. Also,

the derivation presented here comes more intuitively from the

gyroscope model than that presented in [5]. To maintain the gyroscope oscillation, it is necessary to

drive the X and Y axes with a force in phase with the

oscillation velocity. Examining (6) with only the damping

mismatch terms and including forces to maintain steady-state

behavior

1 1 1 x + -± + � -±cos28r + � -ysin28r + w2x = Fex + Fsx

7 7 7

.. 1. AI. 28 AI. . 28 2 F + F y+-y-u -ycos r+u-xsm r+WX= ey sy 7 7 7

(7)

where Fex and Fey maintains the oscillation amplitude and

Fsx and Fsy compensates for any drift terms. By assigning the

damping terms to Fex,Fey and the drift terms to Fsx,Fsy, the

steady-state values of these forces to maintain the oscillation

253

and compensate any drift are

Fex = �x + ��xcos20 T T

Fsx = ��y sin 20 T

1 1 Fey = Y + -y - �-ycos20 T T

Fsy = ��xsin20 T

(8)

If the quadrature compensation loop maintains a > > q, then using (1) the velocities can be written as

x = wa cos 0 sinwt y = wasinOsinwt

(9)

and by inserting (9) in to (8), the forces can be simplified to

Fex = ('fJ cos 0 + 'fJe cos 0) sinwt Fey = ('fJ sin 0 - 'TJc sin 0) sin wt Fsx = ('fJs sinO) sinwt Fsy = ('fJs cosO) sinwt

where the terms 'fJ, 'TJc and 'fJs are

1 'fJ =wa-

T 1

'fJe = wa� -cos 207 T

'fJs = wa�� sin 207 T

The basic amplitude control will have a mean output of 'fJ. Returning to the analysis in [5], the terms 'fJe and 'fJs can be found by implementing controls on the amplitude modulated by cos 20 and sin 20 leading to the control equations

�E=E- Eo

'fJ = Ap�E + L [Ai�E + Ad(�E - �E-l)]

'TJc = L cos 20 [Ai�E + Ad(�E - �E-l)]

'fJs = L sin 20 [Ai�E + Ad(�E - �E-l)]

(10)

where Ap, Ai, Ad are proportional, integral and derivative gains which can be chosen to optimize the response of the gyroscope, and the subscript -1 indicates the previous value.

B. Frequency Mismatch Compensation Loop

A control loop to reduce the quadrature error is required for non-ideal RIGs. Since the rate of growth of the quadrature is angle dependent, the control loop will tend to over or under compensate at various orientations and depending on the rotation rate. The gyroscope equations with only the frequency mismatch non-ideal terms and forces to cancel the quadrature growth are

x + w2x - w�w (x cos 20w + y sin 20w) = Fqx Y + w2y + w�w (-xsin20w + y cos 20w) = Fqy (11)

where Fqx, Fqy cancel growth in quadrature error from the frequency. There are other sources of quadrature error such as

rotational acceleration and damping mismatch, however the frequency mismatch terms dominate for even small values of �w. Since the form of (11) is similar to (7), it is proposed that a similar control scheme can be used to improve the response to resonance frequency variation with O.

Setting the cancellation forces equal to the frequency mis­match terms gives

Fqx = -w�w (x cos 20w + y sin 20w) Fqy = +w�w(-xsin20w + ycos 20w)

which, using the relationship (9) and assuming a > > q, can be rewritten as

Fqx = -(vesinO+ vscosO)coswt Fqy = + (ve cos 0 - Vs sin 0) coswt

where the terms Ve and Vs are

Ve = w�wacos20w Vs = w�wasin20w

By an analysis identical to that for damping mismatch compensation, the following control laws are proposed

v = ApQ + L [AiQ + Ad(Q - Q -l)]

Ve = L cos 20 [AiQ + Ad(Q - Q -d] (12)

Vs = L Sin20 [AiQ + Ad(Q - Q -l)]

where Ve and Vs will compensate the quadrature growth due frequency mismatch and v will control the quadrature growth from other sources. The final quadrature cancellation signals including v are then

Fqx = -((v + ve ) sinO + Vs cosO) coswt Fqy = + ((v + ve ) cosO - Vs sinO) coswt.

If the frequency mismatch is large, the required Fqx,Fqy may be larger than can be supported by the control hardware. Also, large quadrature cancellation signals are undesirable since they can saturate the sense circuits due to capacitive feed-through or cause error in the amplitude detection if not perfectly removed by feed-through cancellation. To reduce the level of the quadrature drive signals, Ve and Vs can be used to adjust DC bias voltages which reduce the mismatch terms by electrostatic spring softening. Ve and Vs can be transformed into effective stiffness mismatch and cross coupling parameters by

. Iv2 + v2 �k = skw o27r v e s

'fJT

{-I ifvs > 0 Sk = 1 if Vs :::; 0

Ve kxy = -2�karctan-. Vs

An automatic mode matching routine such as that described in [6] can then be used to determine the bias voltages, keeping the mismatch within a range that can be compensated for by the dynamic technique above.

254

Simulation Module

Fixed Point PLL

Control Software

Fig. 3. The simulation architecture used to test the proposed control functions. The simulation block replicates the FPGA firmware and gyroscope. The controls use the same code whether the software is being run in simulation mode or with physical hardware. Simulation mode is selected using a command line argument to the control interface software.

IV. SIMUL ATION

Simulation with a custom simulator has been used to verify that the controls successfully compensate for the modeled errors. The next sections describe the simulator used and results from the simulations.

A. Simulation Method

The simulation architecture illustrated in Fig. 3 is built upon the control software we previously presented in [7],[2],[6]. On a fast laptop, the following simulation runs at about Y2 real time (10 seconds of simulated gyroscope dynamics and control requires roughly 20 seconds of real time). The USRP FPGA hardware is replaced with a software model of the PLL implemented in the FPGA, including the use of a first order IIR filter, with precision limited to 16bits. The simulated PLL and the actual implementation in the FPGA are critical to the control performance. The reference phase is accumulated as a 32bit integer at the same sampling frequency as used in the FPGA, 500kS/s for these simulations, according to

<P = LWO + <Ppll. Capacitive feed-through is modeled by taking the derivative

of the drive signals and adding them to the output of the gyroscope model according to

[Xft] = [Cxx

Yft Cxy where Xft, Yft are the feed through signals, Xd, Yd are the x and y axis drive signals, and Cuu are coefficients chosen to match the measured feed through of MEMS gyroscopes tested in our lab. For the low frequency 3kHz CING, the feed-through is small enough to be ignored. It is more significant for the 20kHz CING.

The drive amplitudes provided by the control software are modulated by the output of the PLL as

The modulated drive signals are used as inputs to the gyroscope simulation. The gyroscope simulation is a C++ implementation using the GSL [8] ODE solver of the equation 6 from [3]

x - 2Agf2y - Agf2y + �X + � (�) (X cos 28r + y sin 28r)

+ (w2 - A�f22) X - w�w (X cos 28w + ysin28w) = Xd ii + 2Agf2x + Agf2x + �y - � (�) (X sin 28r - y cos 28r)

+ (w2 - A�f22) Y + w�w (X sin 28w - Y cos 28w) = Yd which includes angular acceleration and centripetal force as additional terms for quadrature error generation. The constant A� is a function of the gyroscope geometry. The velocity vari­ables x,y are used as the output of the gyroscope simulation.

The position variables can also be used as the output to model different sense amplifiers. The gyroscope output is demodulated to create the gyroscope parameters used by the controls according to

[cx cy] = [cos <P + <Pdly cos <P + <PdlY] [Xs]

Sx Sy sin <P + <Pdly sin <P + <Pdly Ys where <Ppll is the output of the phase accumulator and <Pdly is

<Ppll plus a constant to compensate or model any phase shift in the circuits.

The control parameter Li is calculated within the FPGA simulation and used as the error signal for the PLL PIO. The demodulated sense signals are transferred to the control software where they are filtered and decimated and then used to calculate the control variables in (2). The gyroscope orientation is often given as

however, this does not capture the full range of 8 and it is not immediately clear how to extend this to 28 E [-271",271"] required for the control laws (10), (12). The method used in our control is

1 0

,

-271" 28 = tan-1(SjR) + �::

71",

if R > 0 and Cx > 0 if R > 0 and S > 0 if R > 0 and S < 0 if R :::; 0 and 8_1 < 0 if R :::; 0 and 8_1 > 0

where 8_1 is the previous value of 8. Since R switches from

positive to negative when 8 is ± �, 8_1 will not change sign

when R is less than O. 2

255

-O.211z -O.lllz -O.ll1z

Fig. 4. Simulated drift versus oscillation orientation for a gyroscopes with a nominal frequency of 5kHz and Q of 25,000 (7 = 1.6sec) with various amounts of mismatch with only basic quadrature compensation enabled.

B. Simulation Results

The above compensation schemes have been tested over a range of gyroscope parameters. The results presented below are for a gyroscope with a resonance frequency of 5kHz, Q of 25,000 (T = 1.6sec) and Ag of 0.3. This Ag is predicted for a ring or hemispherical resonator from FEM simulation and is larger than the measured Ag for the CING resonator used in the experimental results section [2].

To determine the drift versus angle in each simulation, a constant rotation rate is applied to the gyroscope (60°/ s or 90° / s for the largest � T) and then the measured rate of rotation is calculated. The drift is the measured rate minus the applied rate. Because of delay in the control, which is modeled to represent delay in the real system, the drive signals lag behind actual rotation and shift the average rate away from the applied rate, resulting in a slight positive bias in the drift plots.

The drift for several values of �� and w�w without the T

proposed mismatch controls are are plotted in Fig. 4. By the standard analysis, reducing the quadrature error to zero will reduce the corresponding drift to zero, however there is a residual drift proportional to the frequency mismatch that is an orders of magnitude larger than predicted by the residual quadrature error. This drift may be due to the effective damping mismatch introduced by the frequency mismatch since damping is proportional to w(B)a/T.

With the frequency mismatch control enabled, this effective damping mismatch is evident in modeled gyroscopes with no damping time constant variation. By enabling both the frequency and damping control, the residual drift is reduced to nearly zero. The drift for the same gyroscopes as Fig. 4 but with the damping mismatch control enabled are plotted in Fig. 5. The frequency mismatch control loop settles within 9 sec­onds with non-optimized control values when a 90° / s rotating drive signal is applied, however the damping mismatch takes much longer to settle, over 266 seconds as shown in Fig. 6. The damping mismatch takes a long time to settle because the deviations of E from the set value with the amplitude control enabled are very small. An alternate method of generating the

0.4

� 0.1

(§

-0.1

-OllsO -135

6.T � 0.68, 6.[ � -0.211 Z

6.T � 0.68,6.[ � -O.ll1z

6.T � 0.38,6.[ � -O.ll1z 90 135 ISO

Fig. 5. Simulated drift versus oscillation orientation for a gyroscopes with a nominal frequency of 5kHz and Q of 25,000 (7 = 1.6sec) with various amounts of mismatch with both the proposed damping and mismatch com­pensation enabled. The ripples in the remaining drift are due to the settling of the PLL after enabling the compensation controls.

2.5

2.0 t .... � \ . "

� , " i "Tj

1.5 , , 0 "1", 0 , , ." 1l � E " z

-0.50

Fig. 6. In this simulation, the frequency mismatch loop, Ve settles to 10/0 in 9 seconds and the damping mismatch loop, T]e, in 266 seconds with 90° Is applied rate of rotation.

control values based on the derivative of the amplitude control

�E=E-Eo 'f} = Ap�E + L [Ai�E + Ad(�E -�E-l)l

'f}e = sin 2B L [Ai�E + Ad(�E -�E-l)l 'f}s = -cos2BL [Ai�E+Ad(�E-�E_l)1

(13)

settles much faster (under 14 seconds with only 60° / s applied rate), as in Fig. 7 but this method is unstable at low rates of rotation. In the future, the control will automatically switch between the two methods based on whether the rotation rate is above some threshold.

In the above plots, simulated rates of rotation were applied to the gyroscopes in order to move the wave orientation. It is also possible to use electric signals to move the wave orientation or create a simulated rotation. However, the lack of centripetal force and angular acceleration mean the electronic

ISO 135

g 90 c -0 45 0 c ." 0

1l .� � -45 Q)

§ 8 z -90

-135

10 15 20 25 30 35 -ISO

Time (s) Fig. 7. With the modified control, (B), where the damping mismatch is extracted from the derivative of the damping variation, settling of the damping control is much faster. Here, Ve settles to 10/0 in 13.5 seconds and the damping mismatch loop, T]e, in 13.8 seconds with 60° Is applied rate of rotation.

256

rotation is not the same as physical rotation. Also, in physical systems, various out of plane vibrations can be coupled in to the sense axis vibrations by rotation which is not the case with electrical rotation. Finally, the drive signals for electrical rotation may be large and due to electrical feed-through create apparent error signals causing the compensation loops to over or under compensate.

V. EXPERIMENTAL RESULTS

The system was originally designed and used to test the University of Michigan CING gyroscope [2]. The Single

Crystal-Si Cylindrical Rate-Integrating Gyro (CING) is a capacitive vibratory MEMS gyro. The CING is made of a Si wafer aligned on the (111) plane and operates at the fundamental flexural (wineglass) mode. The two versions of

CING operate at 18kHz (radius: 2.5mm, height:300um, Q up to �60,OOO) and 3kHz (radius:6mm, height:300um, Q up to �100,OOO) respectively. The mechanics of the CING are similar to other micro-ring gyroscopes, except that a large out of plane component of the wineglass mode reduces the rotation sensitivity [2].

At the beginning of each test, the mode frequencies are matched within a fraction of a Hertz with electrostatic tuning using tuning electrodes under the gyroscope. The bias voltage for these gyroscopes is IOV, and the tuning voltages are ap­proximately -3V (Vbias - Vxtune = 13V) for cross-axis tuning and -1 V (Vbias -Vjtune = 11 V) for on-axis tuning. Simulation of the tuning mechanisms does not show any interference with the control system, however there is some concern that the large tuning voltages might stress or unbalance the gyroscope in unknown ways.

The damping mismatch control loop is shown to settle in 3 seconds when driven with an rotating drive signal in Fig. 8. The frequency mismatch control loop settles in 7 seconds in Fig. 9. The tests were made without optimizing the control gains. The inability of the control loops to fully settle is due to the measured errors not matching the model in (4) and (5). The frequency and damping mismatch as measured with aCING gyroscope are plotted in Fig. 10. The cos 20 dependence is clear, but there are distortions which may be due to higher order variation in the parameters or multiple sources of error. While the results look noisy, in Fig. 11 the same sweep is made three times with nearly identical results indicating the the jaggedness is not a random noise.

Despite the mismatch between the model and the gyroscope errors, the controls are still able to reduce the drift of aCING gyroscope as shown in Fig. 12. The RMS drift is reduced by 25%, and the remaining drift is dominated by the regions where the model and measured error are the most different. The source of this difference could be particular to the CING, the hardware implementation, or a more general problem with MEMS RIG. The error model can also be updated to better approximate the measured error and further improve performance.

"r----,------OT-----.------,,-----, I M 1.2

135

0.6

0.'1

0.2

0.0

-0.20 10 Time (5)

Fig. 8. Enabling the damping mismatch control with CING gyroscope reduces the amplitude control variation and removes some of the angle dependency of the rate scale factor. The control is enabled at 10 seconds and has mostly settled within 3 seconds.

� '!l � � o o � -0.5 <ii E 0-1.0 Z

135

� o

� -45 .�

v,li} -90 0

Fig. 9. Enabling the frequency mismatch control with CING gyroscope reduces the quadrature control variation and the residual quadrature signa­ture.The control is enabled at 9 seconds and is mostly settled within 7 seconds.

� 0.5 o o 1fi 0.0 .............. ; .. . .� <ii E (5 -0.5 z

-1.0 .

-00 -45 0 Orientation (0)

Fig. 10. The frequency and damping variation measured using the quadrature and amplitude controls of a CING gyroscope. The wave orientation is controlled with electronic steering.

4s

1.20r-r---.----,----...,----.-----,----.----,----..-, 1.15 1.10 1.05

Z 1.00 "" 0.95

0.00

0.85

Fig. 11. The measured damping mismatch in a CING gyroscope derived from the variation in the amplitude control over 3 full rotations of the gyroscope wave.

257

-2�I�W���--�--��--�O������--�--�180 Orientation (0)

Fig. 12. The drift as measured from the variation of the measured rotation rate from the average rotation rate of a CING gyroscope under electronic rotation with and without the proposed compensation enabled. With this level of mismatch, no rate less than nearly 2000/ s would fully rotate the gyroscope orientation.

VI. CONCLUSION

We have described two novel control loops to characterize and compensate for damping and frequency mismatch in RIG, which we expect to be critical for high-performance operation of MEMS RIG. The loops use the residual error in the traditional control loops to determine orientation dependent error and applies a force to cancel the drift induced by the mismatch. The controls have been tested in simulation and implemented in a functional control system. In simulation, this work is capable of reducing the drift due to fabrication error by better than two orders of magnitude. This can either turn a meticulously manufactured (and therefor expensive) MEMS gyroscope into a high-performance RIG, or it can enable simpler, less expensive gyroscopes to operate as RIG for applications where bandwidth and dynamic range are more critical than drift. Disparity between the modeled errors and measured error reduce the effectiveness with a CING

gyroscope and more work is required to find the source of the disparity and to further improve the error model to reduce the residual drift.

ACKNOWLEDGMENT

The authors would like to thank Dr. R. Peterson for her help in reviewing this paper. This research is supported by DARPA HERMIT (W31P4Q-04-1-ROOl) and MRIG (W31P4Q-ll-l-0002) projects.

REFERENCES

[1] I. P. Prikhodko, S. A. Zotov, A. A. Trusov, and A. M. Shkel, "Foucault pendulum on a chip: angle measuring silicon MEMS gyroscope," in 2011 IEEE 24th International Conference on Micro Electro Mechanical Systems (MEMS). IEEE, Jan. 2011, pp. 161-164.

[2] J. Cho, J. Gregory, and K. Najafi, "High-Q, 3kHz Single-Crystal-Silicon cylindrical Rate-Integrating gyro (CING)," in Micro Electro Mechanical Systems (MEMS), 2012 IEEE 25th International Conference on, Paris, France, Feb. 2012, pp. 172-175.

[3] D. D. Lynch, "Vibratory gyro analysis by the method of averaging," in 2nd St. Petersburg Int. Con! on Gyroscopic Technology and Navigation, St. Petersburg, Russia, 1995.

(4) V. Zhuravlev, ''Theoretical foundations of solid-state wave gyroscopes," Mechanics of Solids, vol. 28, no. 3, pp. 3-15, 1993.

(5) Y. K. Zhbanov, "Amplitude control contour in a hemispherical resonator gyro with automatic compensation for difference in q -factors," Mechanics of Solids, vol. 43, no. 3, pp. 328-332, Jun. 2008.

[6] J. Gregory, J. Cho, and K. Najafi, "Characterization and control of a high­Q MEMS inertial sensor using low-cost hardware," in IEEEIlON PLANS 2012. Myrtle Beack, CA: IEEE, Apr. 2012.

(7) --, "MEMS rate and rate-integrating gyroscope control with commer­cial software defined radio hardware," in Solid-State Sensors, Actuators and Microsystems Conference (TRANSDUCERS), 2011 16th Interna­tional, Jun. 2011, pp. 2394-2397.

[8] Free Software Foundation, "GSL - GNU scientific library - GNU project," Mar. 2012. [Online). Available: http://www.gnu.orglsoftware/gsV

258

本文献由“学霸图书馆-文献云下载”收集自网络，仅供学习交流使用。

学霸图书馆（www.xuebalib.com）是一个“整合众多图书馆数据库资源，

提供一站式文献检索和下载服务”的24 小时在线不限IP

图书馆。

图书馆致力于便利、促进学习与科研，提供最强文献下载服务。

图书馆导航：

图书馆首页 文献云下载 图书馆入口 外文数据库大全 疑难文献辅助工具