1. INTRODUCTION
Emerging MEMS technologies enable us to have a gyroscope small enough that can be applied to many potential applications [1-8] such as virtual reality, platform stabilization, car navigation, inertial navigation, space avionics, etc. In order to achieve high performances for such applications, more improvements in performances are required. Feedback control scheme is one of the answers to that question [2]. Feedback controller enhances the sensor performances such as bandwidth, linearity, etc. When we apply feedback control scheme to gyroscopes, however, we should consider operating circumstances, e.g. vacuum condition of the sensor operation, since that is one of the main factors that influences the sensor performances.
When the vibratory gyroscope is operated, it usually needs the vacuum circumstances in order to get a large vibrating displacement at low air damping conditions [1-5], which leads to high sensitivity to the angular rate input. We usually call it as a ‘high-Q system’. The lower the vacuum level is, the higher the Q factor is. In open loop operation, however, those high-Q systems restrict the desirable sensor performance such as fast response [2,5] and linear output. Therefore, the vacuum level becomes a factor that draws the confliction between a large displacement of the proof mass (high mechanical sensitivity) and a large bandwidth (fast response). That is the main reason why we apply the feedback control to the gyroscope. Feedback control enables us to have a fast response without sacrificing other sensor performances much.
As a matter of fact, however, there needs some trade-off between bandwidth and sensitivity when we apply the feedback control at low vacuum level. In open loop, sensitivity – output voltage level to the angular rate input at a fixed noise level (We may use ‘SNR’ as another sensor performance instead of ‘sensitivity’ in this case.) – is directly proportional to the plant gain that is closely related to the vacuum level. In the closed loop, however, sensitivity is not directly proportional to the plant gain. Its increasing rate declines as the plant gain grows big. So the sensitivity of the feedback control system becomes lower than that of the open loop system at the high-Q (low vacuum) conditions.
In this paper, we investigate the operational principal and some features of the gyroscope in the case of open and closed loop system. And we analyze the quantitative performance
levels of the gyroscope such as sensitivity, linearity and bandwidth from the experiments when the vacuum level changes.
2. PRINCIPLE OF OPERATION
The SEM photograph of gyroscope structure is shown in Figure 1 and operation scheme is illustrated in the figure. The proof mass is driven along a driving axis (x-axis) at a resonant frequency of the driving mode by sinusoidal driving voltage.
When an angular rate input (z-axis) is applied to the sensor, the mass is oscillated along a sensing axis (y-axis), which is modulated signal by the driving frequency. The input angular rate is measured by detecting the capacitance change that is proportional to the modulated gap variation along the sensing axis. So the mechanical sensitivity to the rate input is basically influenced by these two modes of the gyroscope. Figure 2 shows a block diagram of the system. The rate input and two times of the mass of the structure are multiplied by the driving velocity of the moving mass, which induces Coriolis force that is an input to the sensing mode or plant of the system. Plant output is a voltage proportional to the capacitance change or displacement of the proof mass along sensing axis. Each mode has a low damping or high-Q characteristics to get a largest displacement by the smallest energy [1-8].
Fig. 1 SEM photograph of gyroscope and operation
Performance analysis of feedback controller for vibratory gyroscope at various vacuum levels
Woon Tahk Sung*, Jang Gyu Lee* and Taesam Kang**
*School of Electrical Engineering and Computer Science, Seoul National University, Seoul, Korea (Tel : +82-2-872-8190, +82-2-880-7308; E-mail: [email protected], [email protected])
**Department of Aerospace Engineering, Konkuk University, Seoul, Korea (Tel : +81-3-450-3544; E-mail: [email protected])
Abstract: In this paper, presented is a feedback control performance of vibratory gyroscope at various vacuum levels. Micro gyroscope, whose operation is based on the vibrating motion at the vacuum conditions, is highly influenced by the vacuum level of the operating circumstances. In general, we apply the feedback control scheme to the gyroscope in order to improve the performances of the sensor. And control performances of the gyroscope are related to those vacuum levels. So we need investigate the performances of the closed loop control at various vacuum conditions comparing with those of the open loop. The experimental results show that the sensitivity of the closed loop is less than that of the open loop especially in low vacuum conditions. Therefore, there should be trade-off between sensitivity and other sensor performances such as linearity, bandwidth when we apply feedback control to the gyroscope.
Keywords: Gyroscope, MEMS, Micromachining, Inertial sensor, Feedback control
2.1 Governing equations
The governing equations of the vibratory gyroscope can be expressed in equation (1)-(6). When the proof mass is driven by the electrical force or Fxe(t) along the driving axis, the equation of the dynamic motion of the mass can be simply expressed as (1), where M, Cx, Kx are mass, damping coefficient, spring constant, respectively, and x is variable that stands for the displacement along driving axis. The steady state output can be expressed as equation (2) with following assumptions:
1. Driving force is sinusoidal signal.
2. Driving frequency is same as the resonant frequency of the driving mode.
3. Quality factor of the driving mode is Qx.
where F0 is a magnitude of the signal and ωx is a resonant frequency of the driving mode.
) e(t Fx xx K xx C x
M&&+ &+ = (1)
xt F Kx Qx Kx
xe xF t Q
x()= = 0sinω (2)
If the external angular rate input is applied to the gyroscope, then the equation of induced Coriolis force is equal to (3) and the dynamic equation of the sensing mode can be expressed as (4) just like the above equation (1). We solve equation (4) to have a steady state output of y(t), displacement along the sensing axis, as expressed in (5) applying same conditions 1-3 at the above. The notations are analogous to the driving mode.
Finally, multiplying y(t) by the scale factor SFvy, displacement of the sensing mode to voltage, leads us to have equation (6).
xt F
x Qx
xt xF
Kx Qx x M
v Corilois M
F
ω ω
ω ω
0 cos 2
0cos 2
2
Ω
=
Ω
= Ω
=
(3)
xt F
x Qx Corilois F yy K yy C y M
ω 0 cosω
2 Ω
=
= + + &
&&
(4)
xt F
y M x
Qy Qx
xt F
Ky Qy x Qx t y
ω ω ω ω ω
0 sin 2 2
0 sin ) 2
(
Ω
=
Ω
=
(5)
xt F
y M x
SFvy Qy Qx
t vyy SF out t V
ω ω
ω 2 0 sin 2
) ( )
(
Ω
=
=
(6)
From the above results, in order to get a large output signal to the angular rate input or to get a high sensitivity, we need following conditions:
1. High quality factors of the driving and sensing mode 2. Resonant frequencies of two modes should be matched.
3. Large capacitance variation to sensing gap change 2.2 System illustration in frequency domain
Equation (6) is based on the several assumptions mentioned above for calculational convenience – steady state output, matched resonant frequency, driving at resonant frequency, but in practical case, those assumptions are not necessarily satisfied. Moreover, exact solution of the motion equation without such assumptions is complicated and we don’t need such a solution. We can rather deal with the system in a frequency domain and that helps us to have a useful insight of the principle of operation. In Figure 2, the driving and sensing mode in frequency domain are illustrated. How these modes are formed determines the performances of the gyroscope.
Fig. 3 Four configurations of two modes Fig. 2 System block diagram
y y
2 C s K
Ms + +
×
1s
x x
2 C s K
Ms + + 1
driving
F
Coriolis
F
x v
xy
SFyvV B P F
V (
(ra te ra te ) )
D riv in g m o d e S e n s in g m o d e
Ω
z 2MMa gn it ud e( dB )
F r e q u e n c y ( H z ) F r e q u e n c y ( H z ) Ma
gn it ud e( dB )
ω
xω
yy y
2 C s K
Ms + +
×
1s
x x
2 C s K
Ms + + 1
driving
F
Coriolis
F
x v
xy
SFyvV B P F
V (
(ra te ra te ) )
D riv in g m o d e S e n s in g m o d e
Ω
z 2MMa gn it ud e( dB )
F r e q u e n c y ( H z ) F r e q u e n c y ( H z ) Ma
gn it ud e( dB )
ω
xω
yFigure 3 shows the four forms of the driving and the sensing mode configuration. Four forms are: (a) high-Q mismatched, (b) high-Q matched, (c) low-Q mismatched and (d) low-Q matched. In Figure 3, (b) is best in sensitivity but worst in bandwidth and vice versa is (d). Mismatched (a) or (c) is the one that is a usual case in practice but is not desirable.
3. FEEDBACK CONTROL 3.1 Feedback loop configuration and sensitivity
Most desirable case is that the driving mode has a high-Q and the sensing mode has a low-Q with a high level of the magnitude, which is briefly shown in Figure 4. That cannot be achievable in the open loop system. Therefore we use a feedback control scheme for that purpose. Feedback control loop and mode shapes of the system are illustrated in the figure. The figure shows the relationship between the system parts (driving transfer function Gd, plant G, controller K and scale factor of the voltage-force transducer SFfv) and mode shapes. Plant G(sensing part transfer function), controller K and scale factor SFfv determines the sensing mode shape while Gd determines the driving mode shape. And the vacuum condition influences the all those shapes. Our mission is, therefore, to construct a feedback control loop in order to have best mode configuration with given vacuum conditions.
Fig. 4 Feedback control and mode shapes
In open loop, the output is clearly proportional to the plant gain, which is also proportional to the Q factor. But the closed loop is not. Since the closed loop output is the same as the controller output, the closed loop gain can be expressed as
) ( 1 GK SFfv
GK closed
G = + (7)
meaning that the sensitivity of the closed loop system increases slowly comparing with the open loop as the plant gain grows and is dependent on the value of SFfv.
Considering of SNR leads a similar result. Figure 5 shows a simple block diagram for calculating SNR in the open and the closed loop system. SNR in the open loop can be obtained as equation (8) and SNR in the closed loop as equation (9). Here we can see the term G/(1+GSFfv) for the closed loop case. If the noise at the controller output, n′, is negligible, the closed loop result is the same as the open loop. But that’s not applicable in practice. So we have to submit some degradation of the SNR at the high plant gain or high-Q system. The
example of SNR graph is sketched is Figure 6 (linear scale) with an assumption of r/n = 1. In the graph, SNR of the close loop is lower and grows slow comparing with the open loop as the gain of the plant increases.
(a) Open loop
(b) Closed loop
Fig. 5 Diagram for SNR of the open and closed loop system
n Gr yn
yr SNRopen
log10 10 20
log
20 =
= (8)
(
ifn n)
n r SF G
G n SF G n
Gr yn yn
yr closed
SNR
fv fv
′=
= +
+ ′
=
+ ′
=
) ( 1 log10 20
) 10 (
log 20
log10 20
(9)
Fig. 6 SNR of open and closed loop system
3.2 Feedback control with various Q values
Bode diagram of the system at various Q values are shown in Figure 7. Feedback controller is designed using the conventional PID control scheme [2]. The figure shows that the closed loop system is relatively insensitive to the Q values
whereas the open loop is greatly influenced by the Q values.
The bandwidth of the closed loop is guaranteed enough without much variation, but the open loop bandwidth is greatly decreased as Q increases. Nyquist plot of the feedback control system is shown in Figure 8. The graph shows that the stability of the control system is guaranteed and the Q factor of the system does not have much influence on the stability margin. We can consider that the feedback control system is pretty robust to the vacuum condition of the operation. If the Q values are too high, the mode shape of the closed loop can be somewhat distorted, which leads the system to have overshoot at the output. In such a case, reducing the derivative control gain gives us the desirable modification of the mode shape [2].
(a) Open loop system
(b) Closed loop system
Fig. 7 Bode diagram of open and closed loop system
Fig. 8 Nyquist plot of the closed loop system
4. EXPERIMENTS AND DISCUSSION In order to investigate the performances of the feedback controller in various vacuum circumstances, the experiments are accomplished. The experimental setup is sketched in Figure 9. A gyroscope with a fabricated circuit board is put in a vacuum chamber on a rate table and the output signal is measured in time and frequency domain.
At first, Q factors of the mode are measured and the mode shape is observed as the vacuum level varies (Figure 10). We can see the Q values and mode shapes are similar as the graph in Figure 7-(a). The Q factor is inversely proportional to the vacuum level.
Fig. 9 Experimental setup
Fig. 10 Sensing mode shapes and Q value vs. vacuum level Next, the dynamic tests are accomplished using the rate table. 10 deg/sec, 1Hz sinusoidal signal is applied to the circuit and the output signal is measured, maintaining out noise in constant level and the sensitivities of the open and closed loop system are calculated. The results are shown in Figure 11, showing that the sensitivity of the closed loop becomes lower than that of the open loop at the vacuum level of less than about 400mTorr.
Fig. 11 Sensitivity vs. vacuum level
Finally, bandwidths are measured at various vacuum levels.
The result shows that the bandwidth of the closed loop system is always larger that that of the open loop over whole vacuum levels as expected in design procedure. And the value of the bandwidth is nearly constant over whole vacuum levels. In addition to the bandwidth test, linearity test is accomplished.
The result is that the linearity of the closed loop is also improved over whole vacuum levels to nonlinearity of 0.4%
from the 1.1% of the open loop.
From the experimental results, feedback control improves the gyroscope performances such as bandwidth and linearity regardless of the vacuum level. But the sensitivity or SNR of the feedback loop grows worse as the vacuum level decreases.
Therefore we need trade-off between sensitivity and other performances when we apply feedback control to the gyroscope at low vacuum levels.
Fig. 12 Bandwidth vs. vacuum level
5. CONCLUSIONS
In this paper, feedback control performances of micro gyroscope at various vacuum levels are considered and experiments are accomplished.
Feedback controller generally improves the gyroscope performances such as bandwidth and linearity over whole vacuum conditions. But the sensitivity of the feedback loop is lower than that of the open loop. The experimental results show that the sensitivity of the closed loop grows worse than that of the open loop as the vacuum level of the circumstances decreases. Therefore, we need trade-off between sensitivity and other performances when we apply feedback control to the gyroscope at low vacuum levels.
ACKNOWLEDGMENTS
Authors gratefully are acknowledging the financial support by Agency for Defence Development and by ACRC (Automatic Control Research Center), Seoul National University.
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