Chapter 1 Introduction
2.2 Piezoelectric Constitutive Equations
Piezoelectric Constitutive Equation is described mathematically as below：
d-form：
The four state variables (S, T, D, and E) can be rearranged arbitrarily to give an additional 3 forms for a piezoelectric constitutive equation by mathematics operation.
It is possible to transform piezoelectric constitutive data in one form to another form.
In addition to the coupling matrix d, they contain the other coupling matrices e, g, or h in another 3 forms. What follows another 3 piezoelectric constitutive equations and their mutual transformations：
e-form：
Generally speaking, common publish material data exhibit d and g, whereas certain software such as finite element codes requires piezoelectric data entered as e. So it is essential to be the other forms for our convenience. All the descriptions of matrix variables used in the piezoelectric constitutive equations are shown in Table 3 and Table 4.
The four forms described above could be combined with the matrix form as follows. Thereinafter, for example, there are the d-form LiNbO3 constitutive equations for a full matrix form. All values of the parameters in matrix could be found in Table 5 – Table 9.
⎥⎥
The meaning of the suffixes of the symbols for S and D are respectively expressed as below；all meanings of the suffixes of all the symbols is shown in Figure ：
S1：x strain；
D1：x electric charge density；
D2：y electric charge density；
D3：z electric charge density；
The other three forms (e, g, and h-form) could be done similarly, and the relationships of transformation for four forms are：
e-form：
E
E s
c = 1 、εS = εT
(
1 − k d2)
、 sEe= d (2.6)
g-form：sD =sE
(
1−kd2)
、T
g d
= ε (2.7)
h-form： E
(
d2)
D s 1 k
c 1
= − 、
(
d2)
2 d
k 1 d h k
= − (2.8)
The different relationships of transformation are applied for different actuators or sensors for convenience.
Chapter 3
Preliminary Design of Piezoelectric Gyroscope
The conceptual design of H-type piezoelectric gyroscope is presented in this chapter. As requested by the adaptive control algorithm, we designed a long upper arm and a short lower arm for our H-type vibratory gyroscopes. The insulation layers, which were utilized to minimize the electrostatic coupling, is also presented and discussed. Lastly, an approximated 2nd order model for our H-type gyroscope is obtained, based on the FEM simulation results, for the subsequent adaptive control strategy.
3.1 Design of Piezoelectric Gyroscope Configuration
As requested by the subsequent adaptive control strategy, the designed piezoelectric gyroscopes must be actuated and sensed in both fork-mode and H-mode simultaneously. The requirement defeats the existing H-type gyroscope designs proposed in [2, 3], which the Y (fork) mode existed only in actuation arms and the H mode in both the driving and detecting arms. Here we proposed two conceptual designs to meet this performance requirement. The case I design is a Y-type gyroscope and the case II is a H-type gyroscope with different arm lengths. Both designs based on the bimorph bending motion of piezoelectric materials to achieve its resonance.
The basic equations that described the bimorph motion is listed in APPENDIX I.
3.1.1 Case I: The Y-type piezoelectric gyroscopes
In order to reduce the electromechanical coupling effects between driving and
driving and detecting electrodes suppress coupling effects, and the intermediate isolation layer among driving and detecting electrodes prevents the voltage feedthrough from driving voltage electrode to sensing voltage electrodes. Most importantly, this structure can be actuated and their respectively vibrating motion can be measured in both Y-mode and H-mode. However, it is a debatable point whether the vibratory arm which driving and detecting electrodes are put together is still the pure bending motion. Moreover, the leakage electric coupling between the driving and detecting electrodes could also be the problem, even though the insulation layers are inserted.
3.1.2 Case II: The H-type piezoelectric gyroscopes
To benefit the advantages from the existing H-types gyroscopes such as less electrostatic coupling [2, 3], we designed a newly H-type gyroscope with the Y mode is advisedly arisen in both the driving and detecting arms so that the gyroscope have an ability to control and measure the motion in dual axes. This is done by deliberately using a thin connecting bar in between left and right arms and different arm lengths of upper arms and lower arms. After done so, the effect of the quadratic spring is worsen and is expected to be dealt with by the subsequent adaptive control strategies.
The amplitude of the vibratory mode changes with the size of a gyroscope, and the resonant frequencies of the two modes also depend on arm thickness, arm width and arm length. Figure 3.5 shows the size of the designed gyroscope, which is advisedly established the different size between driving and detecting arms. The deflection of upper arms will arise in our design configuration while applying force to the lower
designed gyroscope. Two vibratory modes simulated by FEM are shown in Figure 3.7 and Figure 3.8, and lateral views are shown in Figure 3.9 and Figure 3.10. Their resonant frequencies are respectively 214 kHz for Y mode and 220 kHz for H mode.
Based on the design with enough space divided the driving electrode from the detecting electrode, and which has the ability to control and measure the motion in dual axes. The case II proposes a more suitable model than the case I for the designed gyroscope to work with the adaptive control strategies.
3.2 Electrostatic Coupling Analysis and Reduction
With the permittivity property of piezoelectric materials, the output signal of the piezoelectric gyroscope contains unnecessary electrostatic leakage coupling from the driving electrode to the detection electrode as shown in Figure 3.11. That is the k1 term described in section 1.3.1. Because the leakage output passes through the intermediate layer of the H-gyro between driving and sensing electrode, it’s possible to set up an isolation layer to prevent the unnecessary leakage signal. Figure 3.12 shows the concept design with the intermediate layer GND. If the method works, the frequency response of the gyroscope will have a similar response and the leakage would be minimized.
Figure 3.13 and Figure 3.14 shows the FEM simulation results of cases with and without isolation layers. Figure 3.13 shows the frequency response of the designed gyroscope with setting up the isolating material in the intermediate layer, and there is no any leakage voltage passing through the intermediate layer of the H-gyro. Figure 3.14 shows the frequency response of the designed gyroscope without any restriction
the gyroscope pasting with the intermediate planar electrode GND. As the above Figures indicate, the frequency response of Figure 3.15 is almost the same with that of Figure 3.13, and different from that of Figure 3.14. That is to say, our design with the intermediate layer GND can effectively suppress the leakage electrostatic coupling effect of H-gyro.
3.3 Dynamics for the Designed Gyroscope
This section describes the dynamic motion and how we obtain an approximated 2nd order dynamic equation of the designed H-gyro. Then, we will model the mechanical coupling term of spring constant and use these parameters to be simulated in next chapter.
3.3.1 Dynamic Motion of the Designed Gyroscope
The relative deflection percentage between upper and lower arms in the vibratory mode is presented in this section. According to the configuration of the designed model in Figure 3.5, Figures 3.16 shows the relative displacements Uzn/Uzo at node n
= #1~17 for Y vibratory mode. Uzn is the displacement of every divided averagely node in the y direction, and Uzo is the maximum displacement at the short one of two arms. According to the result of Figure 3.16, we can confirm the amplification factor of detecting output voltage in the detecting arms because of the asymmetrical motion between upper and lower arms.
Besides, Figures 3.17 shows the relative displacements Uyn/Uyo at node n = #1~17 for H vibratory mode, Uyn is the displacement of every divided averagely node in the
to the result of Figure 3.16, the H vibrating mode also amplifies the displacement in the long arm.
According to the Figure 3.16 and Figure 3.17, we can know the deflection in the upper arms is amplified in the dynamic motion of the designed gyroscope, that is to say, we can detect larger output signal in the detecting arms than the existing H-gyro due to the design of long detecting arms. It will be helpful to measure the output signals of detecting arms.
3.3.2 Frequency Response & Approximate 2nd Order Model
With the resonant frequencies of the two axes as mentioned above, the frequency characteristics of the transfer function from the driving arm (short arm) to the detecting electrode (long arm) can be analyzed in the harmonic analysis simulated by FEM. In this section, the V /V voltage relationship between the driving and in out detecting electrode is obtained instead of the relationship between the force and displacement. The main reason is that the frequency characteristics of the V /V in out can directly show the relationship between driving electrode and detecting electrode without transferring the displacement signal and force signal to voltage signal. On the other hand, it is more convenient to deal with our data. Detailed account for the transferred relationship is given below.
Equation (3.1) shows the basic dynamic equations of the conventional two-axes vibratory gyroscope, 2M yΩ& and 2M xΩ& , are due to the Coriolis acceleration and the two terms are usually used to measure the angular velocity Ω .
xx xy xx xy x
Mx+d x+d y+k x+k y= +2MWy My+d x+d y+k x+k y= -2MWx
⎧ τ
⎨ τ
⎩
&& & & &
&& & & & (3.1)
M is a mass, d and k are damping and spring term respectively. To match up the adaptive control strategies, we first divide the equation (3.1) by the mass M and get：
xx xx x
And with properties of piezoelectric materials, there is constant relationship (K v K ) either between the deflection and sensing voltage in the detecting arms, or m
between the control input voltage and control input force.
x=Kv⋅ ；V τ =Km⋅Vin (3.3)
Then, we can obtain dynamic equations with Vin/Vout voltage relationship：
m in x
Now, equation (3.4) is model with Vin/Vout relationship which is suited for the compensation of adaptive control strategies. Frequency responses of Y and H modes are respectively simulated by FEM.
The adaptive strategies in section 4.1 are suitable for a 2nd order model. But the identity of the H-type model is a high order system. Here, we get an approximated 2nd order model by utilizing curve fitting method around the appropriate operation frequency. Figure 3.18 and Figure 3.19 show the conception of curve fitting method and red curve is a standard 2nd order system. Blue curve is real frequency response and red curve is an approximate 2nd order model by curve fitting. As the result of using the adaptive algorithm for the approximated 2nd order model, all the parameters
The approximated 2nd order model can be described as follows.
That is proven by the same way for the parameters of y axis and all the parameters are shown in Table 12.
3.3.3 Modeling the Electromechanical Coupling
Dynamic coupling effects are necessarily eliminated in a high-resolution gyroscope.
In practice, small fabrication imperfections usually occur and arise due to the asymmetric spring and damping terms, which results in dynamic coupling between two axes even under zero angular velocity. So these fabrication coupling effects are major factors limiting the performance of gyroscopes. Aside from fabrication imperfections, the essential electromechanical couplings of piezoelectric material are also factors limiting the performance.
Now, we try to model the unnecessary coupling terms [14]. Assume that the real spring constants are k and1 k , and the real spring axes are titled by angel2 α from the driving and detecting axis for ideal gyroscope because of unnecessary couplings.
Let [ x y, ] be the ideal axes and [ x y′ ′ ] be the real coordinates. So the real axes ,
'
Then, the relationship between forces and displacement in the real axes is given by
x 1
the relationship between forces and displacement in the ideal axes is therefore given by
Therefore, we get
'
Compare with the above two equations, we get
2 2 1 2 1 2
1cos 2sin cos 2
2 2
x
k k k k
k =k α+k α = + + − α (3.13)
2 2 1 2 2 1
1sin 2cos cos 2
2 2
y
k k k k
k =k α+k α = + + − α (3.14)
(
1 2)
sin cos 1 2sin 2xy 2
k k
k = k −k α α = − α (3.15)
Therefore, we can calculate the electromechanical coupling term Kxy by above
equations. And we will assume damping parameter D due to the fabrication xy imperfection in my simulation. All the estimations of the unknown parameters are shown in Table12.
3.3.4 Convergence Analysis of Result by ANSYS
Different mesh size controls of the ANSYS program adequate for the different model we are analyzing. In this case, we are going to specify the mesh size of the designed gyroscope. Figure 3.20 and Figure 3.21 show the simulative results in the different mesh density. As a result, we can find the suitable mesh size for our model.
Therefore, we can verify the validity of the result by the convergent analysis. As shown in Figure 3.20 and Figure 3.21, the reasonable mesh density is for our designed gyroscope is obtained.
Chapter 4
Adaptive Control of Piezoelectric Gyroscope
To verify the feasibility of the adaptive control strategies, the designed model presented in previous chapter will be employed to simulate in this chapter. After the configuration of the gyroscope is accomplished, we begin to deal with the coupling terms such as k2 and k3 described in section 1.3. They are able to be eliminated with H-type configuration by FEM. And the mechanical-thermal and electrical-thermal noises are also considered. We use the adaptive control method to identify the unknown coupling terms and we can obtain an accurate angular velocity at the same time.
4.1 Adaptive Control Strategies
Park [14] proposed adaptive control strategies for MEMS Gyroscopes. The basic idea of the adaptive control approach is to treat the angular rate, along with the unknown coupling terms due to fabrication defects, as the parameters which will be identified by using adaptation algorithms. First, the adaptive control problems of the gyroscope is described as follows： Given the gyroscope plant equation with unknown constant parameters D, K andΩ .
q 2 Kq
q D
q&&+ &+ =τ− Ω& (4.1)
where
⎥⎦
⎢ ⎤
⎣
=⎡ y
q x (4.2)
⎥⎦ In the adaptive control strategies by Park, the persistent excitation condition is an important factor to estimate the unknown parameters correctly. To match this condition, a trajectory following approach is used. By following the reference model, the persistent excitation condition is met and all unknown parameters converge. The reference model is generated by an ideal oscillator and the trajectory of the gyroscope is controlled to follow that of the reference model. The reference model is defined as
Here, what has to be noticed is that resonant frequencies of two axes of the reference model must be unmatched by persistent excitation condition [14]. τ is the
0
Now, the trajectory error dynamics becomes
( )
p p m m mp D 2 e Ke Dq Rq 2 q
e&& + γ+ + Ω & + = ~& +~ + Ω~ &
(4.14)
Then define the Lyapunov function
( )
The derivative of the Lyapunov function is
( )
definite such that V V⋅ <& 0 and Lyapunov function converges.(
m T0)
the parameter estimates of the gyroscope converge to their true values.4.2 Estimation of Noise
For gyroscopes designed for small-size applications, it is expectable that noise floor has a large influence on the resolution, so the question about the fundamental noise limit of the gyroscope is vital. In general, the two noise sources of piezoelectric transducer, the mechanical-thermal noise of the damped mechanical harmonic oscillator and the electrical-thermal noise of the piezoelectric material are often the limiting noise components for gyroscope. This subsection discusses the estimated noise of the designed gyroscope, and simulation results with adaptive control strategies are presented in next section.
4.2.1 Mechanical-thermal Noise
A brownian system resulting in the mechanical-thermal noise is an example of noise-related theory that is applied to various natural phenomena. Several studies have emphasized the features of this phenomenon and its complexity arising from sensitivity to structural parameters [26, 27]. PSD (power spectral density) is a general unit describing how the power (or variance) of a time series is distributed with frequency. Mathematically, a standard PSD of noise is white noise (it is independent
of ω and its plot against frequency is "flat") which has the same power at every frequency.
( ) ( )
m B
S f = 4K TR N/ Hz ,f ≥ 0 (4.20)
K is Boltzmann’s constant (B 1 38 10. × −23J K/ ) and T is the absolute temperature.
The PSD (power spectral density) of the fluctuating force related to any mechanical resistance.
In our designed H-gyro, some parameters are shown as follows and S of m mechanical-thermal noise can be obtained.
f =220kHz, Q 100= ,R=4.13 10 N s m× -4 ⋅ / ,T=300 Ko
Then, we obtainSm =2.62 10× -12N Hz.
4.2.2 Electrical-thermal Noise
The mechanism of the electrical-thermal noise is related with losses in the piezoelectric material and depends on its loss factor or dissipation factor, which is inverse of the PE material’s quality factor [28].
By definition[x], loss factor η equals 1
η = CR
ω (4.21)
The Power spectral density of the electrical-thermal noise is given
( )
e B n
S = 4K TR V Hz (4.22)
These equations can be used for the calculation of the fundamental noise limit of
K is Boltzmann’s constant (B 1 38 10. × −23J K/ ) and T is the absolute temperature.
Here, R is an equivalent noise resistor which is related with frequency (n ω = π ), 2 f capacitor (C), and loss factor (η).
n
R 1
C 1
= ⎛ ⎞
ω ⎜⎝η +η⎟⎠
(4.23)
In our designed H-gyro, some parameters are shown as follows and S of e electrical-thermal noise can be obtained.
f =220kHz,η =0 01. ,C=26 7 pF. T=300 Ko
Then, we obtain S =2.12 10 Ve × -8 Hz.
4.3 Simulation Results and Discussion
Some simulations are implemented by MATLAB software package. These simulation results are shown to verify the feasibility of the designed gyroscope associated with the adaptive control strategies.
In this case, comparing the mechanical-thermal noise with the electrical-thermal noise, the electrical-thermal noise has a more critical influence on the output signal of the detecting electrode than that of mechanical-thermal noise. Their PSD, position noises and velocity noises are shown in Table 10.
A simulation using the preliminary design parameters was conducted, and the noise of the gyroscope was included, too. All gyroscope parameters in the model are summarized in Table 11 and Table 12. The simulation results are also presented in this section. To verify the performance of the gyroscope, we calculate the standard
mode.
Before starting the simulation, an important issue has to be noted is how we specify the reference model. In adaptive control strategies, the gyroscope is designed to follow the reference model such that the parameters of the gyroscope are convergent.
On the other hand, the reference trajectory is generated by an ideal oscillator and the trajectory of the real gyroscope is controlled to follow that of the reference model.
Therefore, the setting value of the amplitude of vibration of the reference trajectory must meet certain physical behaviors of the real gyroscope. The reference gyroscope must be met with the gyroscope in real world while we set up the dynamic characteristics of the reference model in our simulation.
Such an Adaptive mode of operation, using reference model signals instead of the real measurement signals, has the advantages that the reference signals q 、m q& are m not contaminated by measurement noises. Therefore, the performance of the gyroscope is determined by measurement signals of control input τ. How we reduce the noises of the measurement signals in the sensing electrode is the key factor for improving the resolution of the gyroscope.
Some simulation results are shown. Figure 4.1 and Figure 4.2 respectively show the trajectories of the proof mass and the tracking errors of x and y axes. The various estimation errors of D, R in the gyroscope are shown in Figure 4.3 and Figure 4.4 respectively. The estimate of angular velocity Ω response to a constant input angular velocity is shown in Figure 4.5, and in this figure, the standard deviation of the angular velocity estimate error is obtained.
In our simulation, the adaptive control gains are as shown in Table 13.
Ω of the gyroscope