A Digitalized Gyroscope System Based on a Modified Adaptive Control Method

Transcription

1 sensors Article A Digitalized Gyroscope System Based on a Modified Adaptive Control Method Dunzhu Xia *, Yiwei Hu and Peizhen Ni Key Laboratory of Micro-inertial Instrument and Advanced Navigation Technology, Ministry of Education, School of Instrument Science and Engineering, Souast University, Nanjing 2196, China; @seu.edu.cn (Y.H.); @seu.edu.cn (P.N.) * Correspondence: 11123@seu.edu.cn; Tel./Fax: Academic Editor: Stefano Mariani Received: 16 October 215; Accepted: 26 February 216; Published: 4 March 216 Abstract: In this work we investigate possibility of applying adaptive control algorithm to Micro-Electro-Mechanical System (MEMS) gyroscopes. Through comparing gyroscope working conditions with reference model, adaptive control method can provide online estimation of key parameters and proper control strategy for system. The digital second-order oscillators in reference model are substituted for two phase locked loops (PLLs) to achieve a more steady amplitude and frequency control. The adaptive law is modified to satisfy condition of unequal coupling stiffness and coupling damping coefficient. The rotation mode of gyroscope system is considered in our work and a rotation elimination section is added to digitalized system. Before implementing algorithm in hardware platform, different simulations are conducted to ensure algorithm can meet requirement of angular rate sensor, and some of key adaptive law coefficients are optimized. The coupling components are detected and suppressed respectively and Lyapunov criterion is applied to prove stability of system. The modified adaptive control algorithm is verified in a set of digitalized gyroscope system, control system is realized in digital domain, with application of Field Programmable Gate Array (FPGA). Key structure parameters are measured and compared with estimation results, which validated that algorithm is feasible in setup. Extra gyroscopes are used in repeated experiments to prove commonality of algorithm. Keywords: silicon microgyroscope; MEMS; adative control algorithm; rotation mode elimnation; parameter optimization; FPGA 1. Introduction MEMS gyroscopes are a kind of angular rate sensor widely used in fields of navigation, motor vehicles, and mobile devices. They provide inertial angular rate measurements with low cost and low power consumption [1,2]. Recent years have witnessed development of adaptive control methods for MEMS gyroscopes. The application of an adaptive control algorithm can provide on-line estimation of input angular rate and or parameters, thus it has fine robustness features against parameter variances and disturbances. Park has proposed an adaptive measurement mode for operating MEMS gyroscopes, and formulated a unified methodology for synsis and analysis for algorithm [3]. The algorithm can realize online estimation and compensation of varied defects and disturbances that influence behavior of a MEMS gyroscope. He also extended adaptive control method to mode-tuning and angle measurement applications, which can directly measure rotation angle without integration of angular rate [4,5]. John presented concept of an adaptively controlled single-mass tri-axial angular rate (AR) sensor [6]. The single mass is free to move in three directions, thus a tri-axial perfect Sensors 216, 16, 321; doi:1.339/s

2 Sensors 216, 16, of 22 oscillator model is chosen as a reference. According to control law, coefficients of damping, stiffness and input angular rate can be completely estimated. To ensure parameter convergence in continuous domain, a modified trajectory algorithm is presented with aim to reduce discretization errors. Fei adopted sliding mode control in MEMS gyroscopes [7 9]. A proportional and integral Sensors 216, sliding 16, 321 surface is defined and applied to calculate control forces of different 2 of 21 axes. In order to eliminate chattering, discontinuous control component is replaced by a smoothing oscillator model is chosen as a reference. According to control law, coefficients of damping, sliding mode stiffness component. and input angular The algorithm rate can be completely is also updated estimated. by To ensure robust sliding parameter mode convergence control method and applied in on continuous tri-axial domain, gyroscope a modified model. trajectory Leland algorithm developed is presented Lyapunov-based with aim adaptive to reduce controllers for MEMS discretization gyroscopes errors. to compensate Fei adopted sliding uncertainty mode control in in MEMS natural gyroscopes frequencies, [7 9]. A mode proportional coupling and damping. and The integral controllers sliding surface are tested is defined under and averaged applied to low calculate frequency control model forces and of different full gyroscope axes. In model order to eliminate chattering, discontinuous control component is replaced by a smoothing conditions, respectively [1,11]. Dong presented a new adaptive control method based on Active sliding mode component. The algorithm is also updated by robust sliding mode control method Disturbance and applied Rejection on Control tri-axial (ADRC), gyroscope whichmodel. can precisely Leland developed estimate and Lyapunov-based compensate adaptive disturbance on eachcontrollers axis, and for is applied MEMS gyroscopes on a vibrational to compensate beam gyroscope uncertainty [12]. in natural frequencies, mode Most coupling of and reported damping. control The controllers methods are tested remained under averaged at simulation low frequency stage, model and ors full are implemented gyroscope on model analogconditions, circuits. Thus, respectively for [1,11]. first time, Dong we presented look into a new adaptive possibilities control of method realizing based Active Disturbance Rejection Control (ADRC), which can precisely estimate and adaptive control algorithm on a digitalized gyroscope system. Once algorithm is implemented compensate disturbance on each axis, and is applied on a vibrational beam gyroscope [12]. in digitalized setup, measured angular rate will not be affected by variance of coupling Most of reported control methods remained at simulation stage, and ors are coefficients implemented or quality on analog factor circuits. of Thus, gyroscope, for first thus time, we look system into haspossibilities better adaptivity of realizing for different gyroscopes adaptive under control different algorithm conditions. a digitalized gyroscope system. Once algorithm is implemented in The algorithm digitalized presented setup, in measured this paper angular is based rate will on not a model be affected reference by variance adaptive of control coupling algorithm, and coefficients adaptive control or quality algorithm factor of is furr gyroscope, modified thus based system onhas previous better adaptivity work. The for adaptive different law is gyroscopes under different conditions. redesigned to estimate asymmetric coupling parameters and reference model is replaced with The algorithm presented in this paper is based on a model reference adaptive control algorithm, two PLLs and for FPGA adaptive implementation. control algorithm is Besides, furr modified rotation based on mode previous of work. gyroscope The adaptive is considered law is in this manuscript redesigned and to estimate corresponding asymmetric elimination coupling parameters sectionand is designed. reference This model paper is replaced is organized with as follows: two in PLLs Section for FPGA 2 implementation. structure diagram Besides, and rotation dynamic mode characteristic of gyroscope of is considered gyroscope in this system are firstmanuscript described, and and next corresponding Section elimination 3 adaptive section control is designed. method This with paper is robust organized resistance as to z-axis rotation follows: mode in Section is finally 2 structure presented. diagram Simulation and dynamic results and characteristic parameter of gyroscope estimation system process are are first described, and next in Section 3 adaptive control method with robust resistance to z- n discussed in Section 4, and finally algorithm implementation on FPGA-based digitalized axis rotation mode is finally presented. Simulation results and parameter estimation process are gyroscope n system discussed is in demonstrated Section 4, and finally in Section algorithm 5. implementation on FPGA-based digitalized gyroscope system is demonstrated in Section Gyroscope System Description 2. Gyroscope System Description A capacitive silicon micromachined gyroscope is used in this work. The designed z-axis gyroscope is a double decoupled A capacitive single silicon proof micromachined mass gyroscope. gyroscope It is is a two-layer used in this structure work. The which designed mainly z-axis consists of gyroscope is a double decoupled single proof mass gyroscope. It is a two-layer structure which drive mode components, sense mode components, proof mass, damping elements and mainly consists of drive mode components, sense mode components, proof mass, glass substrate, etc. The details of gyroscope including its packaging, comb fingers, coupling damping elements and glass substrate, etc. The details of gyroscope including its packaging, and decoupling comb fingers, springs coupling are and displayed decoupling Figure springs 1a. are displayed in Figure 1a. F YZ 1 y l m 2 z x F Y 2 h y1 F X 2 F XZ 2 m 1 oh x 2 m 1 F XZ 1 h F x1 X 1 m p h y 2 F Y 2 m 2 F YZ 2 Figure 1. Description of gyroscope used. SEM photo and packaging details of gyroscope; Figure 1. Description Schematic diagram of of gyroscope gyroscope used. rotation SEM mode. photo and packaging details of gyroscope; Schematic diagram of gyroscope rotation mode.

3 Sensors 216, 16, of 22 The gyroscope is fabricated by silicon-on-glass (SOG) process. The upper layer of gyroscope is silicon structure ICP etched through single crystal silicon bulk micromachining, while lower layer is glass substrate which is anodically bonded with silicon structure. The gyroscope used in this work has a resonant frequency of Hz in drive mode and Hz in sense mode. The symmetric structure of gyroscope is especially suitable for implementation of adaptive control algorithm proposed in [4]. The detailed structure parameters of gyroscope in this work are listed in Table 1. Table 1. The structure parameters of gyroscope. Parameter Value (unit) Structure thickness 65 (µm) Proof mass size 25 ˆ 25 (µm 2 ) Equivalent drive/sense mass 1.39 ˆ 1 6 (kg) U-shaped spring width 1 (µm) Drive/sense U-shaped spring length 52 (µm) Drive/sense beam size 25 ˆ 7 (µm 2 ) Comb finger width 4 (µm) Comb finger length 4 (µm) Comb finger gap 4 (µm) Comb finger overlap length 2 (µm) Drive/sense frequency 3194/315.6 (Hz) Q-factor (drive/sense mode) 8256/7839 From structure model it can be seen that ideal micromachined gyroscope is a mass-spring-damping system with two independent modes. However, due to inevitable fabrication errors of micromachined structure, actual gyroscope dynamic equation tends to be more complicated. Besides, fabrication imperfection of comb fingers and beams will contribute to stiffness coupling, stiffness asymmetry, drive force asymmetry and displacement detection asymmetry, etc. These structural errors will lead to undesired coupling of drive and sense modes, and even excite or sub-modes, for instance, differential mode drive force asymmetry will make proof mass to rotate around z axis. In such case, mechanical model of three freedom degrees system is considered in this paper to study dynamic characteristic, as illustrated in Equations (1) (3), and schematic diagram of gyroscope rotation mode, is also depicted in Figure 1b. The drive mode: The sense mode: m x.. x ` D xx ẋ ` D xy ẏ ` K xx x ` K xy y F dx ` 2m p Ω s ẏ (1) m y.. y ` D yy ẏ ` D yx ẋ ` K yy y ` K yx x F dy 2m p Ω s ẋ (2) The rotation mode:... I z θ ` D z θ ` Kz θ ΣF xi h xi ` ΣF yi h yi ` ΣF xzi l ` ΣF yzi l ` ΣF xci h xi ` ΣF yci h yi i 1, 2 (3) where x, y is displacement of drive axis and sense axis, m x and m y is vibration mass of se two modes, m P is value of proof mass, I z is rotary moment of inertia of proof mass around z axis, and θ is rotation angle of z axis. K xx, K yy are normal stiffness coefficients of drive and sense modes, and K xy, K yx are coupling stiffness coefficients. D xx, D yy are normal damping coefficients of drive and sense modes, and D xy, D yx are coupling damping coefficients. F dx and F dy are feedback forces that attached on drive mode and sense mode, respectively.

4 Sensors 216, 16, of 22 As illustrated in Figure 1b, D z and K z are damping and stiffness coefficient of rotation mode. F xi and F yi represent asymmetric drive forces of drive and sense axes, F xzi and F yzi are vibration components that couple from drive and sense modes to rotation mode. F xci and F yci are force that calculated to compensate asymmetric components, h xi and h yi are arms of differential drive forces, i 1, 2 indicates different sides of electrodes, l is dimension parameter of gyroscope, representing projection on se two axes of distance from beam bearing point to vibration mass centre: F xz `m p ` 2m 1 γx ω 2 xx (4) F yz `m p ` 2m 2 γy ω 2 yy (5) F x F dx λ x (6) F y F dy λ y (7) where γ x and γ y represent rotation mode coupling coefficients of drive and sense modes, λ x and λ y are drive force asymmetric coefficients of drive and sense modes. As depicted in Figure 1b, m p is proof mass, m 1 and m 2 are masses of fingers and frames that vibrate in two modes respectively. Thus m p ` 2m 1 and m p ` 2m 2 are equal to m x and m y in Equations (1) and (2) respectively. ω x and ω y are resonate frequencies of drive and sense mode respectively. From equations of dynamic characteristics of gyroscope, it can be concluded that drive and sense modes are coupled with each or, and z-axis mode is influenced by different kinds of disturbances and fabrication imperfections. Due to existence of rotation mode, measured displacements of drive and sense modes are contaminated by projection of rotation movement on or two modes. Assuming ˆx, ŷ to be displacements of x and y that influenced by rotation mode, δ x and δ y are beam fabrication asymmetric coefficients, n we get: ˆx cosθx δ x lsinθ (8) ŷ cosθy ` δ y lsinθ (9) Considering control algorithm formation of drive and sense modes, gyroscope used in our experiment is double decoupled dual-mode micromachined gyroscope. The drive and sense modes are designed symmetrically, thus adaptive control algorithm applied on two modes is also symmetric. The reference model of gyroscope is defined as two ideal oscillators, which will remain stable vibration at certain frequencies [4]. The dynamic equations of ideal reference model can be written as:.. q ` K ideal q (1) where stiffness matrix K ideal is: K ideal «K ideal_x K ideal_y ff (11) The elements on leading diagonal are stiffnesses of two modes, and vector q is: Equation (12) represent displacements of ideal gyroscope structure. q «x y ff (12)

5 Sensors 216, 16, of 22 Considering coupling terms and Coriolis force aroused from input angular rate, model of physical gyroscope mechanical structure can be expressed as: «m x m y ff `.. q ` D.q ` K q F d 2m p Ω. q (13) In Equation (13) D is damping matrix, K is stiffness matrix, and Ω is matrix constructed from input angular rate. F d is vector of drive force that attached to drive electrode fingers of gyroscope. D K Ω ««F d «D xx D yx K xx K yx «F x F y D xy D yy K xy K yy ff Ω z Ω z Define R to be estimation error between actually stiffness matrix and reference model: R K K ideal Due to fabrication imperfections, small cross-coupling terms still exist between two modes. In matrix, coupling coefficients D xy and D yx, K xy and K yx are not necessarily equal to each or, respectively, thus an algorithm that can deal with each components correspondingly is of great importance. 3. The Adaptive Control Algorithm By comparing output signal of reference model and physical gyroscope, first tracking error can be obtained. With proper adaptive law, estimated parameters are n corrected, thus parameters are estimated precisely and feedback control force can be generated through estimated parameters of gyroscope, and finally system is well controlled. The block diagram of whole system is shown in Figure 2, where differential subtract block compares velocity outputs with actual gyroscope and reference model, and generates an velocity error e, which contains information of parameter estimation error: «R xx R yx ff ff ff R xy R yy ff (14) (15) (16) (17) (18) e q q m (19) The reference model modification section calculates estimated stiffness matrix, which is used to control vibration frequency of reference model. The feedback control block furr generates a direct feedback signal, i.e., velocity error, which acts as first feedback signal.

6 generates an velocity error e, which contains information of parameter estimation error: = (19) The reference model modification section calculates estimated stiffness matrix, which is used to control vibration frequency of reference model. The feedback control block furr generates a direct feedback signal, i.e., velocity error, which acts as first feedback signal. Sensors 216, 16, of 22 Figure 2. The block diagram of adaptive control gyroscope system. Figure 2. The block diagram of adaptive control gyroscope system. The control law is designed to be implemented in discrete domain. Define sampling index k, and sampling time t, differential subtraction section provides estimation error e e k`1 e k, and multiplied directly by feedback coefficient γ, i.e., τ γ e (2) Meanwhile, velocity error coupling and displacement error coupling modules are multiplied by feedback components with corresponding velocity or displacement signals respectively, n to provide upgrade amount of different elements in parameter matrix. In feedforward control section, signal that can be used to compensate coupling and damping disturbances are finally generated, R, D, Ω are modulated with corresponding sinusoidal waves and sent to two modes. Assuming ˆR, ˆD, ˆΩ to be estimated value of matrix R, D, Ω, and rr, rd, r Ω to be estimation error of three matrixes, respectively. A positive definite Lyapunov function is chosen as shown in Appendix B. In practical conditions, coupling coefficients on anti-diagonal of parameter matrices are not necessarily equal to each or, and control law will decouple each of coupling coefficients, respectively. To make Lyapunov function semi-negative definite, adaptive control law can be chosen as: rr γ R q m τ T (21) rd γ D `qm pk ` 1q q m pkq τ T (22) r Ω γ Ω `qm pk ` 1q q m pkq τ T (23) The Lyapunov stability analysis of algorithm is illustrated in Appendix B. The final output angular velocity signal is derived by integral calculation: Ω ř r Ω elements on counter-diagonal of matrix Ω is estimated angular velocity. The reference model modification block estimates normal stiffness and coupling stiffness of two gyroscope modes, which will be utilized by reference model and feedforward control module. To maintain gyroscope working conditions stable, two phase-lock-loop sections are applied in each vibration mode. Each PLL contains a numerical control oscillator (NCO), which generates sinusoidal wave with calculated frequency and phase. Rar than two ideal oscillator models that are illustrated as in Equation (1), output signal of NCO has a constant amplitude, which will make algorithm easier for implementation without affecting Lyapunov stability.

7 Sensors 216, 16, of 22 Sensors 216, 16, of 21 The detailed flow chart of adaptive control algorithm of gyroscope system is shown in Figure In Equation 3. With(3) two assume PLLs acting =, as and to reference cancel off model, coupling control force error components informationin can different be transferred modes, to n estimate we get: different parameters through displacement signal channel and velocity signal channel. Additional differential detection section is attached at drive mode to collect rotation = /h (26) angular information so that least mean square demodulation (LMSD) section can generate signal τ r to eliminate such an undesirable rotation. = /h (27) Figure 3. Detailed flow chart of modified adaptive gyroscope control algorithm. Figure 3. Detailed flow chart of modified adaptive gyroscope control algorithm. Since frequencies of asymmetric drive force and stiffness coupling components are different, To fulfill gyroscope designed is adaptation forced to law, rotate with feedforward mixture control of module se two is designed frequencies. to update The demodulation parameter estimation algorithm and is applied generate in both final modes feedforward to figure signal. out The rotation final gyroscope components angular of each rate output frequency is also and generated generate from compensation module. The forces total drive and signal τ is of attached each frequency. on drive Substituting fingers of Equations two modes. (6) and It consists (7) into Equations of three parts (26) including and (27), n feedforward we get: signal, feedback signal τ, and rotation eliminate signal τ r, i.e., = /h (28) τ ˆD = q. m ` ˆRq m ` 2 ˆΩ q. m ` /h τ ` τ r (24) (29) By applying small-angle approximation in Equations (8) and (9), it can be written as: = h (3) =+h (31)

8 Sensors 216, 16, of 22 where τ r is defined as: τ r «F xc F yc ff (25) Components of drive signal have different frequency and phase, in such case add operation will not severely increase total drive signal amplitude or make digital amount overflow. To eliminate rotation components, rotation elimination circuit is specially designed. In Equation (3) assume θ, and to cancel off coupling force components in different modes, n we get: F xc F x F xz l{h x (26) F yc F y F yz l{h y (27) Since frequencies of asymmetric drive force and stiffness coupling components are different, gyroscope is forced to rotate with mixture of se two frequencies. The demodulation algorithm is applied in both modes to figure out rotation components of each frequency and generate compensation forces F xc and F yc of each frequency. Substituting Equations (6) and (7) into Equations (26) and (27), n we get: F xc F dx λ x m p γ x ω 2 xxl{h x (28) F yc F dy λ y m p γ y ω 2 yyl{h y (29) By applying small-angle approximation in Equations (8) and (9), it can be written as: ˆx x h x θ (3) ŷ y ` h y θ (31) The unmatched signals of drive modes and sense modes are measured from gyroscope detection pins by signal condition circuits and calculated in FPGA chip. Although frequencies of signals are predictable, phases of detected signals remain unknown. Due to fabrication variation, coefficients of gyroscope structure are not measureable, thus a demodulation method is applied to estimate amplitude of signal. Here LMSD algorithm is applied in both modes. The demodulation reference signals are provided by reference model, calculated results are multiplied with a pair of quadrature reference signals to generate appropriate rotation elimination signal τ r. The rotation angle θ contains frequency components of both modes oscillation frequencies. From detection output port of each mode, differential signal is demodulated with a pair of quadrature sinusoidal or cosine signals of a certain frequency. Assuming detected rotation signal is lsinθ, it contains different frequency components of ω and pω ` ωq, with corresponding magnitude of A, B, C, E, respectively. By applying LMSD algorithm, rotation signal can be easily decomposed and recombined with two quadrature reference sinusoidal signals: lsinθ Asin pωtq ` Bcos pωtq ` Csin pωt ` ωtq ` Ecos pωt ` ωtq pa ` Ccos ωt Esin ωtq sin pωtq ` pb ` Csinϕt ` Ecos ωtq cos pωtq (32) From Equation (32) it can be seen that, although original frequency components cannot be restored, force components that are acquired to eliminate rotation are calculated in real time and can be attached to drive fingers of corresponding mode. With utilization of rotation elimination circuit, unexpected rotation of proof mass can be effectively suppressed to be less than 2 ˆ 1 1 rad, compared with former angular output of 6 ˆ 1 1 rad. The rotation angular output and Lissajous trajectories for tri-modes are depicted in Figures 4 and 5 respectively.

9 From Equation (32) it can be seen that, although original frequency components cannot be restored, force components that are acquired to eliminate rotation are calculated in real time restored, force components that are acquired to eliminate rotation are calculated in real time and can be attached to drive fingers of corresponding mode. With utilization of rotation and can be attached to drive fingers of corresponding mode. With utilization of rotation elimination circuit, unexpected rotation of proof mass can be effectively suppressed to be less elimination circuit, than 2 1 unexpected rotation of proof mass can be effectively rad, compared with former angular output of 6 1 suppressed to be less rad. The rotation angular than Sensors2 1 output 216, and 16, 321rad, compared with former angular output of 6 1 Lissajous trajectories for tri-modes are depicted in Figures rad. The rotation angular 4 and 5, respectively. 9 of 22 output and Lissajous trajectories for tri-modes are depicted in Figures 4 and 5, respectively. 8.x1-1 Before elimination After 8.x1-1 Before elimination 6.x1-1 After elimination 6.x1-1 4.x1-1 4.x1-1 2.x1-1 2.x x x x x x x x x Figure 4. Rotation angular output before and after elimination circuit. circuit. Figure 4. Rotation angular output before and after elimination circuit. Rotation angle(rad) Rotation angle angle (deg) (deg) x 1-9 x Drive mode amplitude (V) Drive mode amplitude (V) 1 1 After elimination After Before elimination elimination Before elimination Sense mode amplitude(v) Sense mode amplitude(v) Figure 5. The Lissajous trajectories plotted with output of three modes. Figure Figure The The Lissajous Lissajous trajectories trajectories plotted plotted with with output output of of three three modes. modes. From simulation result illustrated in Figures 4 and 5 it can be concluded that algorithm based on LMSD can effectively suppress rotation mode of gyroscope. With differential movement of exited by rotation is suppressed, gyroscope works at a more stable condition and control and measurement precision is also enhanced. In terms of different modes displacement detection, differential mode signals are extracted in drive and sense modes, while common mode signal are picked out in rotation mode. In this case, asymmetric displacement of drive and sense modes are only generated by asymmetric drive force, i.e., ir rotation angle and linear displacement signals are totally decoupled. Therefore, rotation elimination circuits are relatively independent of adaptive control circuits for drive and sense modes. Theoretically, Lyapunov stability of adaptive algorithm controlled system is not affected by rotation section. The main parameters and coefficients of simulated system are listed as in Table 2. In implementation process of algorithm, apart from coefficients of mechanical-electrical interface and parameter of gyroscope dynamical model, all variables and coefficients are calculated or stored in FPGA chip.

10 Sensors 216, 16, of 22 Table 2. Coefficients and parameters of simulated system. Block Parameter/Coefficient Symbol Feedforward control Continuous Domain Value Digital Domain Value Angular rate integral coefficient γ Ω 87 8 Damping integral coefficient γ D 21 8 Stiffness error integral coefficient γ R /124 Reference model Feedback gain γ 1.9 ˆ Mechanical-electrical Interface Gyroscope dynamical model Analog-digital domain gain K ad 2.7 ˆ 1 5 LSB{V - Digital-analog domain gain K da 4.76 ˆ 1 6 V{LSB - Voltage-force coefficient K v f 2.8 ˆ 1 7 N{V - Displacement-capacitance-voltage coefficient K dcv ˆ 1 5 V{m - Gyroscope proof mass m 1.39 ˆ 1 6 kg - Drive mode quality factor Q x Sense mode quality factor Q y Drive mode resonating frequency ω x 3194 Hzˆ2π - Sense mode resonating frequency ω y Hzˆ2π - Sense-drive stiffness coefficient K xy.1557 N{m - Drive-sense stiffness coefficient K yx.1514 N{m - Sense-drive damping coefficient D xy 1.8 ˆ 1 7 N{m{s - Drive-sense damping coefficient D yx.9 ˆ 1 7 N{m{s - Compared with conventional adaptive control algorithm proposed in [3], modified adaptive control algorithm proposed in this work has following improvements: adaptive law is redesigned to suit asymmetric condition; rotation elimination section is added to suppress differential components in detection signal; reference model is replaced with two NCOs and modified adaptive law has a simpler arithmetic structure, algorithm is more suitable for FPGA implementation. 4. Algorithm Simulation Simulation of adaptive control based gyroscope system consists of two steps. The continuous domain simulation verifies adaptive control algorithm and digital domain simulation based on DSP builder tool provides convenient conversion from control algorithm simulation to hardware program language realization. Simulation system is built in Simulink to investigate performance of proposed algorithm. Changing of gyroscope parameters is simulated to verify adaptive characteristic of algorithm. The influences of coefficients on gyroscope system performance are also compared in this section. The value of γ R and γ D will also affect estimation convergence speed of R and D matrix, respectively, as shown in Figures 6 and 7. From curves that represent estimation processes with different γ R values, it can be inferred that larger γ R can provide faster stiffness estimation but has larger overshoot and may result in overflow of digital output signal. Like case of γ R value, γ D also has influences on overshoot and estimation time. Moreover, estimation result of damping components are multiplied by velocity of reference model, in such case, system vibration condition is sensitive to signal. When γ R value is too large, such as orange curve in Figure 7d, ripples will emerge in parameter estimation result, and amplitude control process is disturbed. From Figure 8 it can be concluded that γ D will influence maximum overshoot during start-up period and γ R value mainly relates with time consumption of amplitude control. The adaptive algorithm coefficientsω R, γ D will influence amplitude control speed of start-up process, thus a proper combination of se two values is necessary. After a trade-off is made between control time, overshoot and proper estimation of parameters that illustrated in Figures 6 and 7 coefficients are finally determined with a multi-object optimization function [12,13].

11 Sensors 216, 16, of 21 Sensors 216, 16, of 21 different Sensors 216, γ16, values, 321 it can be inferred that larger γ can provide faster stiffness estimation but 11 of has 22 different γ larger overshoot values, it can be inferred that larger γ and may result in overflow of digital can provide faster stiffness estimation but has output signal. larger overshoot and may result in overflow of digital output signal. Parameter estimation(lsb) Parameter estimation(lsb) Parameter estimation(lsb) Parameter estimation(lsb) x x1 6 R =1/ x1 6 R =1/124 R =1/ x1 6 R =1/512 R =1/ x1 6 R =1/ x x x x x x1 5 2.x x x x x1 5 5.x1 4 R =1/ x1 4 R =1/124 R =1/ x1 4 R =1/512 R =1/ x1 4 R =1/ x1 5-1.x (c) (d) (c) (d) Figure 6. Convergence process curves of stiffness estimation error matrix with different values. Figure Convergence process curves of of stiffness estimation error matrix R with different γ R,, (c), (d) values. R xx, R xy, (c) R yx,,(d) (d) R yy.. 1.x x x x x x x x x1-5 D = 4 1.x x1-5 8.x1 14 D = D = x1-5 8.x x1 14 D = 16 4.x1-5 D = 8 6.x x1 14 D = 16 4.x1-5 4.x x1-5 2.x x1-5 2.x1-2.x x x x x x x x x x x1-5 D =4 5.x1-5 4.x1-5 D =8 D =4 4.x1-5 D =16 3.x1-5 D =8 D =16 3.x1-5 2.x x1-5 1.x1-5 1.x1-5 Equivalent damping values(n/m/s) Equivalent damping values(n/m/s) Equivalent damping values(n/m/s) Equivalent damping values(n/m/s) Equivalent stiffness value(n/m) Equivalent stiffness value(n/m) Equivalent stiffness value(n/m) Equivalent stiffness value(n/m) 6.x1 5 6.x1 5 4.x1 5 4.x1 5 2.x1 5 2.x1 5-2.x x x1 5-4.x (c) (d) (c) (d) Figure 7. Convergence process curves of damping estimation matrix D with different γ values. Figure 7. Convergence process curves of damping estimation matrix with different,, (c), (d). values. Figure 7. Convergence process curves of damping estimation matrix D with different γ D values.,, (c), (d). D xx, D xy, (c) D yx, (d) D yy. Pamateter estimation(lsb) Pamateter estimation(lsb) Parameter estimation(lsb) Parameter estimation(lsb) R =1/248 R =1/124 R =1/248 R =1/512 R =1/124 R =1/ x x1 2.1x x1 1.8x1 6 6 R =1/ x1 1.5x1 6 6 R =1/124 R =1/ x1 1.2x1 6 6 R =1/512 R =1/ x1 9.x1 5 6 R =1/ x1 6.x x1.5 3.x x1 5-3.x1 5-3.x x x1 1.4x x1-5 1.x1 1.2x1 15 D = 4 8.x1-5 8.x1 1.x x1-5 D = D = x1-5 6.x1 8.x1 14 D = D = x1-5 4.x1 6.x1 14 D = 16 4.x1-5 4.x1 2.x x1-5 2.x1-5 2.x1-2.x x x1-2.x x x x x x x x x x x x1-5 D =4 5.x1-5 4.x1-5 D =8 D =4 4.x1-5 D =16 3.x1-5 D =8 D =16 3.x1-5 2.x x1-5 1.x1-5 1.x1-5 Equivalent damping values(n/m/s) Equivalent damping values(n/m/s) Equivalent damping values(n/m/s) Equivalent damping values(n/m/s) Equivalent stiffness value(n/m) Equivalent stiffness value(n/m) Equivalent stiffness value(n/m) Equivalent stiffness value(n/m)

12 estimation result, and amplitude control process is disturbed. From Figure 8 it can be concluded that γ will influence maximum overshoot during start-up period and γ value mainly relates with time consumption of amplitude control. The adaptive algorithm coefficients γ, γ will influence amplitude control speed of start-up process, thus a proper combination of se two values is necessary. After a trade-off is made between Sensors 216, control 16, 321time, overshoot and proper estimation of parameters that illustrated in Figures 6 and 12 of 22 7, coefficients are finally determined with a multi-object optimization function [12,13]. Figure 8. The amplitude control performance with different and values. Control time, Figure 8. The amplitude control performance with different γ Overshoot. R and γ D values. Control time, Overshoot. To furr investigate speed performance of input angular rate estimation, angular rate To estimation furrfunction investigate is tested speed with performance ±3 /s step-formed of input input angular signal. rate The estimation, angular rate estimation angular rate estimation parameter function can affect is tested slew with rate and 3 /s overshoot step-formed of angular input rate, signal. as depicted Thein angular Figure 9. rate estimation With integral effect, adaptive control method also has a fine performance against parameter Sensors 216, can 16, affect 321 slew rate and overshoot of angular rate, as depicted in Figure of 21 input noise, from Figure 1 it can be calculated that, white noise mixed with input signal are effectively suppressed to obtain clean output angular rate estimation signal, and specifically 3x SNR (Signal to noise ratio) has been greatly enhanced from 49 db to 9 db. 2x1 13 = 4 3 1x1 13 = 8 2 = 16 1 Angular velocity output(lsb) -1x1 13-2x1 13-3x Figure Figure 9. Angular 9. velocity output with step-formed input input signal. signal. Equivalent angular velocity(deg/s) With integral effect, adaptive control method also has a fine performance against 8 Input input noise, from Figure 1 it can be calculated that, whiteoutput noise2x1 mixed 13 with input signal are 6 effectively suppressed to obtain clean output angular rate estimation signal, and specifically Angular velocity with noise(rad/s) 4 SNR (Signal to noise ratio) has been greatly enhanced from 49 db to 9 1x1 db x1 13-2x1 13 Estimated angular velocity output(lsb

13 Angula -2x1 13-3x ity(deg/s) Sensors 216, 16, of 22 Figure 9. Angular velocity output with step-formed input signal. Angular velocity with noise(rad/s) 8 Input Output x1 13 1x1 13-1x1 13-2x1 13 Estimated angular velocity output(lsb) Figure Figure Noise suppression of of input input angular angular rate estimation. rate estimation. In or hand, is also simultaneously related with measurement precision and Inbandwidth or of hand, gyroscope γ Ω issystem. also simultaneously These two characteristics relatedare with contradict measurement against each or, precision thus and bandwidth a trade-off should gyroscope be made according system. to These expected two characteristics application of are gyroscope contradict system. against each or, thus a trade-off As shown should in Figure be made 11, to according meet gyroscope to expected application application requirement, of bandwidth gyroscopeshould system. As be shown more than in Figure 4 Hz, and 11, to meet measurement gyroscope precision application should be higher requirement, than.1 /s, finally bandwidth we choose should be γ value as 12. more than 4 Hz, and measurement precision should be higher than.1 /s, finally we choose With coefficients of algorithms are determined, performance against parameters γ Ω value as variation Sensors 12. are 216, also 16, 321 simulated. In practical circumstances, stiffness of working gyroscope 13 of may 21 change due to environmental factors, especially temperature. In practical condition, parameters are changed gradually, 2 but 4 to test 6 8 estimation 1 12 speed of algorithm, variations are simulated in form of 45step switches, as shown in Figures 12 and 13. In Figure 12, quality 4factors of gyroscope model vibration modes 5 are changed to 1/1 of ir original value, thus 35 damping coefficients are changed accordingly. In Figure 13, 3.1 resonating frequencies of drive mode and sense mode are both changed by 1 Hz at 5 ms, 25 i.e., from 3715 Hz to 3725 Hz in drive mode and from 3684 Hz to 3674 Hz.15 in sense mode. Sensors 216, 16, of Precision Bandwidth γ5 Ω 35 Figure 11. 3Measurement precision and bandwidth with different.1 values. Figure 11. Measurement 25 precision and bandwidth with different γ Ω values x x1.2 With coefficients of -5 8.x1 1 algorithms are determined, Precision performance against parameters 14 variation are also simulated. 6.x1 In5practical circumstances, Bandwidth stiffness of 14 4.x1.25 working -5 gyroscope may change due to environmental factors, 4.x1 14 especially temperature. In practical condition, parameters are γ2.x1-5 Ω changed gradually, but to test 2.x1 14 estimation speed of algorithm, variations are simulated in form of step switches, Figure as11. shown Measurement in Figures precision 12 and and bandwidth 13. with different values. Bandwidth(Hz) Bandwidth(Hz) -2.x x1 1.x x1 8.x x1 6.x x x x1-5 2.x1 Figure Damping coefficient estimation with switch change at t =.5. Figure 12. Damping coefficient estimation with switch change at t =.5. Figure 12. Damping coefficient estimation with switch change at t =.5. -1x1 4x x1 3x1 66 Rxx Rxy -2 2x1 6 Ryx Ryy 1 1x1 6 1 Figure 13. Stiffness coefficient estimation with switch change at t =.5 s. Precision(deg/s) Equivalent damping value(n/m/s) Equivalent damping value(n/m/s) Precision(deg/s) Equivalent stiffness value(n/m) Equivalent stiffness Dxx Dxy -2.x1-5 Dyx Dyy 6.x x x1 4x1 146 Dxx -2.x x1 14 Dxy 3x1 6 Rxx Dyx -6.x1 14 Dyy Rxy -4.x1 2x1 6 Ryx x1 14 Ryy x1 6 1 l value(lsb)

14 D -4.x x x1 14 Dxy Dyx Dyy -4.x Sensors 216, 16, of 22 Figure 12. Damping coefficient estimation with switch change at t =.5. e(n/m/s) 4x1 6 3x1 6 2x1 6 1x1 6-1x1 6-2x Figure Stiffness coefficient estimation with switch change at at t t =.5.5 s. s. From simulation curves it can be seen that algorithm can adapt to parameter In Figure 12, quality factors of gyroscope model vibration modes are changed to 1/1 of variations robustly and estimated results converge to real values after a short transition ir original value, thus damping coefficients are changed accordingly. In Figure 13, resonating period. frequencies of drive mode and sense mode are both changed by 1 Hz at 5 ms, i.e., from 3715 Hz to 3725 Hz in drive mode and from 3684 Hz to 3674 Hz in sense mode. 5. Experimental Validation From simulation curves it can be seen that algorithm can adapt to parameter variations robustly Based andon a estimated set of digitalized results converge gyroscope to system, real values modified after a short adaptive transition control period. algorithm is programmed in FPGA chip to control gyroscope. 5. Experimental Validation Based on a set of digitalized gyroscope system, modified adaptive control algorithm is programmed in FPGA chip to control gyroscope Open-Loop Measurement of Parameters To validate estimation performance of algorithm, gyroscope parameters are measured in open-loop testing in advance [14]. As illustrated in Section 2, parameters of gyroscope are contained in three modes that coupled with each or. Through frequency sweep and half-power measurement, resonance frequency ω and quality factor Q can be derived, along with stiffness and damping coefficient of each mode, i.e., K xx, K yy, D xx and D yy. The coupling coefficients can be calculated according to mode-coupling transfer function and gyroscope coupling output. However, mechanical and electrical coefficients cannot be measured directly, and thus drive mode force-voltage transfer functions are not available. To solve this dilemma, force-displacement transfer functions are transformed into voltage-voltage transfer function and measurement of unknown coefficients are avoided. The scale factor ratio of drive and sense modes is measured through exiting rotation mode of gyroscope when common-mode force is attached at drive mode. Assuming F x psq and F y psq to be resultant forces attached on drive mode and sense mode, dynamics equation of drive and sense modes can be expressed as: Rxx Rxy Ryx Ryy x psq F x psq 1 m `s 2 ` ω x {Q x s ` ωx 2 (33) y psq F y psq 1 (34) m s 2 ` ω y {Q y s ` ωy Equivalent stiffness value(n/m)

15 Sensors 216, 16, of 22 and under such drive force, displacement of sense mode can be described as: where: y psq s 2 ` c yy y psq s ` y psq y psq H y psq K xy x psq ` D xy s x psq (35) 1 H y psq ˆ (36) m s 2 ` ωy s ` ωy Q 2 y Considering scale factor ratio of drive and sense modes p, Equation (36) can be written as: y psq x psq p Uy psq U x psq K xy H y psq ` D xy s H y psq (37) By frequency sweeping around resonating frequency, coupling coefficients can be derived from least square method. The detailed procedure is described as: y psq m ` nj (38) x psq H y psq a ` bj (39) s H y psq c ` dj (4) where m, n are measured data, and a, b, c, d can be calculated through gyroscope transfer function. Assuming frequency sweeping points are ω 1, ω 2... ω n, corresponding identification equations can be written in matrix format that listed as in Equation (41).» m pω 1 q m pω 2 q. m pω n q fi» ffi ffi ffi ffi fl Equation (41) can be written in form of: a pω 1 q a pω 2 q. a pω n q c pω 1 q c pω 2 q. c pω n q fi ffi «ffi ffi ffi fl K xy D xy ff (41) M nˆ1 A nˆ2 P 2ˆ1 (42) By operating least square procedure, calculated parameters are obtained under meaning of least variance, i.e.: rp pa1 Aq 1 A1 M (43) To measure coupling coefficients that form sense mode to drive mode, open-loop drive signal is attached on sense mode and coupling signal is measured from drive mode. Conducting same process as illustrated above, K yx and D yx can be calculated. The schematic diagram of gyroscope system circuit is shown in Figure 14, when used in open-loop measurement, digital board is removed and external signals are attached at drive electrodes of two modes. A crystal oscillator is used to provide carrier wave of gyroscope, and capacitances variance signals of two modes are detected by two diode rings. Since double-sided drive method is applied in gyroscope, feedback signals are transferred through inverting amplifiers and attached on opposite drive electrodes.

16 diagram of gyroscope system circuit is shown in Figure 14, when used in open-loop measurement, digital board is removed and external signals are attached at drive electrodes of two modes. A crystal oscillator is used to provide carrier wave of gyroscope, and capacitances variance signals of two modes are detected by two diode rings. Since double-sided drive method is applied in gyroscope, feedback signals are transferred through inverting Sensors 216, 16, of 22 amplifiers and attached on opposite drive electrodes. Figure Figure Schematic Schematic diagram diagram of of gyroscope gyroscopesystem systemcircuit. circuit Experimental 5.2. Implementation Result and Comparison For For convenience of of algorithm implementation, simulation is is performed with with DSP DSP builder builder toolbox, toolbox, where where algorithms algorithms of of system are verifiedand and word word length length and and truncation truncation of of each section are determined. Most of sections in algorithm can be resolved into basic addition, each section are determined. Most of sections in algorithm can be resolved into basic addition, subtraction, multiplication and integral calculations. The most complex and resource consumption subtraction, multiplication and integral calculations. The most complex and resource consumption section is reference model with two NCOs, which are implemented using CORDIC algorithm [15]. section is reference model with two NCOs, which are implemented using CORDIC algorithm [15]. As for calculation of matrixes, different elements are calculated and transferred separately As for through calculation different branches. of matrixes, Since different usage of fixed elements point multiplier are calculated will consume and transferred large amount separately of through hardware different resources branches. in Since FPGA and usage add ofpower fixed point consumption multiplier of will digitalized consume system, largemost amount of of hardware constant resources coefficients in FPGA are determined and add as powers of consumption 2, thus multiplication of digitalized operation system, can be most of realized constant by coefficients shifting operation. are determined as powers of 2, thus multiplication operation can be realized by After shifting algorithm operation. implementation, values of parameters estimated program are output After from DA chips algorithm and recorded, implementation, as plotted values in Figure of 15. In parameters plots, right estimated vertical program axes are labeled are output with from physical amount equivalent to digital value system. The dash lines marked with DA chips and recorded, as plotted in Figure 15. In plots, right vertical axes are labeled with corresponding curves are measured values acquired from open-loop experiment. physical amount equivalent to digital value system. The dash lines marked with corresponding curves are measured values acquired from open-loop experiment. Sensors 216, 16, of 21 Parameter estimation(v) Dxx Dxy *Dash lines of corresponding Dyx colors are Dyy measured values Time(s) 1.2x1-6 1.x1-6 8.x1-7 6.x1-7 4.x1-7 2.x x1-7 Equivalent damping values(n/m/s) Parameter estimation(v) Rxx Rxy Ryx Ryy Time(s) *Dash lines of corresponding colors are measured values Equivalent stiffness value(n/m) Figure 15. Experimental parameter estimation. Damping values; Stiffness values. Figure 15. Experimental parameter estimation. Damping values; Stiffness values. By comparing measurement result with estimated value, performance of adaptive algorithm can be evaluated. Parameters Estimated in program and measured from open-loop By comparing measurement result with estimated value, performance of adaptive method are compared in Table 3. In Figure 15 measured values are marked with dash lines in algorithm can be evaluated. Parameters Estimated in program and measured from open-loop corresponding colors. From Figure 15 and Table 3 it can be concluded that estimation results of method are compared adaptive control in Table algorithm 3. Inhighly Figureconsistent 15 measured with open-loop values are measurement marked with results in dash lines in corresponding primary colors. values, From and less Figure estimation 15 andprecision Table 3 in it can weak be concluded coupling coefficients that estimation that has slight results of adaptiveinfluence control in algorithm control highly system, consistent like and with. open-loop measurement results in primary Table 3. Comparison between measurement value and estimated value. Measured Value Estimated Value Parameter Symbol (Open-Loop) (Modified Adaptive Control) Drive mode normal stiffness coefficient N/m N/m

17 The experimental setup established in our work is presented as in Figure 17. The gyroscope system consists of an analog front end, a gyroscope and a digital board based on an FPGA chip. The program is down loaded into an EPCS16 flash chip, so that once system is supplied with a constant voltage of ±5 V, algorithm will control two modes vibrate at constant frequency and amplitude. As can be measured from oscilloscope, drive mode frequency is Hz and Sensors 216, 16, of 22 values, and less estimation precision in weak coupling coefficients that has slight influence in control system, like D xy and D yx. Table 3. Comparison between measurement value and estimated value. Parameter Symbol Measured Value (Open-Loop) Estimated Value (Modified Adaptive Control) Drive mode normal stiffness coefficient K xx N{m N{m Stiffness coupling coefficient K xy 3 N{m 4 N{m K yx.155 N{m.17 N{m Sense mode normal stiffness coefficient K yy N{m N{m Drive mode normal damping coefficient D xx 4.1 ˆ 1 6 N{m{s 4.4 ˆ 1 6 N{m{s Damping coupling coefficient D xy 2 ˆ 1 7 N{m{s 1 ˆ 1 7 N{m{s D yx 1 ˆ 1 7 N{m{s 8 ˆ 1 8 N{m{s Sense mode normal damping coefficient D yy 3.92 ˆ 1 6 N{m{s 4.1 ˆ 1 6 N{m{s To validate commonality of algorithm, anor two gyroscopes in same batch are also picked for repeated test using same digitalized circuit. Parameters of corresponding gyroscopes are listed in Table 4, and a screenshot of vibration of two modes is shown in Figure 16. Table 4. Parameters of gyroscope 2 and gyroscope 3. Parameter Symbol Gyroscope 2 Gyroscope 3 Drive mode normal stiffness coefficient K xx N{m 57.6 N{m Stiffness coupling coefficient K xy 1.5 ˆ 1 3 N{m 1.13 ˆ 1 3 N{m K yx 1 ˆ 1 3 N{m 7.4 ˆ 1 4 N{m Sense mode normal stiffness coefficient K yy N{m N{m Drive mode resonating frequency f x Hz Hz Drive mode quality factor Q x Drive mode normal damping coefficient D xx 4.48 ˆ 1 6 N{m{s 1.15 ˆ 1 5 N{m{s Damping coupling coefficient D xy 1 ˆ 1 8 N{m{s 7.9 ˆ 1 8 N{m{s D yx 6 ˆ 1 9 N{m{s 7.1 ˆ 1 8 N{m{s Sense mode normal damping coefficient D yy 6.18 ˆ 1 6 N{m{s 1.18 ˆ 1 5 N{m{s Sense mode resonating frequency f y Hz Hz Sense mode quality factor Q y Sensors 216, 16, of 21 Figure 16. Drive and sense mode signal when controlled by adaptive algorithm. Gyroscope 2; Figure 16. Drive and sense mode signal when controlled by adaptive algorithm. Gyroscope 2; Gyroscope 3. Gyroscope 3.

18 Sensors 216, 16, of 21 SensorsFigure 216, 16, Drive and sense mode signal when controlled by adaptive algorithm. Gyroscope 18 2; of 22 Gyroscope 3. The The experimental experimental setup setup established established in in our our work work is is presented presented as as in in Figure Figure The The gyroscope gyroscope system consists of an analog front end, a gyroscope and a digital board based on an FPGA system consists of an analog front end, a gyroscope and a digital board based on an FPGA chip.chip. The The program is down loaded into an EPCS16 flash chip, so that once system is supplied with program is down loaded into an EPCS16 flash chip, so that once system is supplied with aa constant V, algorithm algorithm will will control control two two modes modes vibrate vibrate at constant voltage voltage of of 5 ±5 V, at constant constant frequency frequency and and amplitude. As can be measured from oscilloscope, drive mode frequency is amplitude. As can be measured from oscilloscope, drive mode frequency is Hz Hz and and sense mode frequency is Hz. sense mode frequency is Hz. Figure 16. Drive and sense mode signal when controlled by adaptive algorithm. Gyroscope 2; Gyroscope 3. The experimental setup established in our work is presented as in Figure 17. The gyroscope system consists of an analog front end, a gyroscope and a digital board based on an FPGA chip. The program is down loaded into an EPCS16 flash chip, so that once system is supplied with a constant voltage of ±5 V, algorithm will control two modes vibrate at constant frequency and frequency is Hz and amplitude. As can be measured from oscilloscope, drive mode sense Figure mode frequency is Hz. control system. Digitalized system with three gyroscopes used; 17. The modified adaptive Figure 17. The modified adaptive control system. Digitalized system with three gyroscopes used; Experimental Experimental setup. setup. 9 Original digital SMG system output Allan variance(deg/hr) /s, with a nonlinearity As shown in Figure 18, gyroscope system has scale factor of 136 LSB/contol Adaptive algorithm system output 8 Fitting result of original output of 339 ppm. According to Allan variance result, bias instability is 13.8 /h, better than 14.9 /h of Fitting result of adaptive contol algorithm 7 digitalized system without adaptive control algorithm [14]. Compared with output result of system output same gyroscope with conventional control algorithm, improvement of Allan variance can be 6 attributed to stability of reference model. The FPGA utilized in our work is Cyclone EP4CE4, 5 and a total of 8125 logic elements (LE) are consumed when whole modified adaptive control 4 algorithm and AD/DA convertor interface sections are programmed. It can be concluded from Bias Instability experimental results that digitalized adaptive3control algorithm is successfully 14.9 deg/hr applied on Bias FPGA chip17. and achieve fineadaptive performance. enhancement of with performance will be Instability carried out Figure The modified control Furr system. three gyroscopes used; 2Digitalized system 13.8 deg/hr in our research. future Experimental setup Allan variance(deg/hr) 8 Figure 18. Performance of system. Scale 7 factor; Allan system Bias Instability 14.9 deg/hr 3 Bias Instability 13.8 deg/hr Sampling time(s) Original digital SMG system output Adaptive contol algorithm system output Fitting result of original output Fitting result of adaptive contol algorithm variance compared with original system output Sampling time(s) Figure of system. Scale Allan comparedcompared with original Figure Performance Performance of system. factor; Scale factor; variance Allan variance with system. original system.

19 Sensors 216, 16, of Conclusions In this paper we present a digitalized micromachined gyroscope based on a modified adaptive control method. The algorithm can estimate some of gyroscope key coefficients in real time and make both drive and sense modes vibrate at constant amplitude, which will enhance system robustness against environmental variances and disturbances. The rotation mode is also considered in model, with corresponding parameters estimated and common mode vibration amplitude is well controlled, differential movement exited by rotation mode is also investigated. Differential detection and LMS demodulation methods are used in system to eliminate such rotation. Performance including parameter estimation and system dynamic characteristics are investigated to choose a set of optimized key coefficients. To validate commonality of algorithm, anor two gyroscopes are used for repeated test. The digitalized gyroscope system applied with adaptive control method achieves scale factor of 136 LSB/ /s, with a nonlinearity of 339 ppm, and bias instability of 13.8 /h by Allan variance. Comparison of various adaptive control methods are summarized in Table 5. In recent years, different adaptive control methods have been tried on different gyroscopes by some leading researchers. Due to algorithm complexity, some works still focused on model building, adaptive control algorithm deduction and simulation verification. Normally, adopted gyroscope should be designed symmetrically, where z-axis MEMS gyroscope and tri-axial gyroscope can be always utilized. Based on previous work, modified adaptive control method is experimentally verified and realized on our digital gyroscope system, which furr proves that adaptive control method is feasible in real application. Table 5. Comparison of different gyroscope control methods in recent years. Institute Year Control Method Gyroscope Used University of California at Berkeley [3] 2 RMIT University [6] 26 University of Louisiana at Lafayette [7] Cleveland State University [12] This work 215 Adaptive control strategy Tri-axial adaptively controlled algorithm Sliding mode control Active Disturbance Rejection Control Modified Adaptive control z-axis MEMS gyroscope Single-mass tri-axial AR sensor z-axis MEMS gyroscope Vibrational beam gyroscope z-axis MEMS gyroscope Simulation and Experiment Algorithm simulation Modeled fabrication and algorithm simulation Algorithm simulation Analog circuit implementation Simulation and FPGA digital circuit implementation Future research of adaptive method controlled gyroscope system may focus on furr enhancement of performance and application of adaptive control method on or types of gyroscopes for instance, ring, disk or hemispherical shell gyroscopes. Acknowledgments: This work was supported in part by National Natural Science Foundation (No , 61148), Key Laboratory of Micro-Inertial Instrument and Advanced Navigation Technology, Ministry of Education, China (Project No. KL2112), Major Project Guidance Foundation of Basic Scientific Research Operation Expenses, Souast University (No ), Natural Science Fund of Jiangsu Province (BK ,BK213636), Aeronautical Science Foundation of China (No ). Author Contributions: Dunzhu Xia convinced modified algorithm and realized programming; Yiwei Hu finalized simulation and performed experiments for validation; Peizhen Ni designed PCB board and debugged circuit system. Conflicts of Interest: The authors declare no conflict of interest.

20 Sensors 216, 16, of 22 Appendix A Nomenclature Symbol Nomenclature Symbol Nomenclature K ideal Ideal Stiffness matrix τ Feedback section output vector θ Z axis rotation angle τ Total drive signal matrix q Displacement vector τ r Rotation elimination signal vector x Drive mode displacement m p Gyroscope proof mass y Sense mode displacement m x Drive mode vibration mass F Feedback force matrix m y Sense mode vibration mass F x Drive mode feedback force Q x Drive mode quality factor F y Sense mode feedback force Q y Sense mode quality factor D Damping matrix γ Ω Angular rate integral coefficient K Stiffness matrix γ D Damping integral coefficient R Stiffness estimation error matrix γ R Stiffness error integral coefficient Ω Angular rate matrix γ Feedback gain e Displacement estimation error vector V Lyapunov function Appendix B The Lyapunov stability proof process is illustrated as follows: V 1 2 e T γ e ` e T γ Ke ` trtγ 1 R r RrR T ` γ 1 D r DrD T ` γ 1 Ω Ω r Ω r T u (B1) By taking differential operation on Lyapunov function, it can be derived that: V V k`1 V k γ e T p e k`1 e k q ` γ p e k`1 e k q T K m e! `tr γ 1 r R R k rr T ` γ 1 r D D k rd T ` γ 1 r Ω Ω k Ω r T) (B2) rr rr k`1 rr k rd rd k`1 rd k r Ω r Ω k`1 rω k (B3) (B4) (B5) Assuming gyroscope controlled by modified control law is under a stable condition, and control signal is composed of feedback and feed forward outputs, i.e.:.. q ` D. q ` Kq ˆD. q m ` ˆRq m ` 2 ˆΩ. q m γ ė 2Ω. q.. e ` K m e ˆD. q m D. q ` ˆRq m Rq ` 2 ˆΩ. q m 2Ω. q γ ė ˆD. q m D. q ` ˆRq m Rq ` 2 ˆΩ. q m 2Ω. q γ ė rd. q m ` rrq m ` 2 r Ω. q m γ ė (B6) (B7) After left multiplication by γ e T on both sides of Equation (B7), and written in discrete time domain, it can be derived that: rd γ e T p e k`1 e k q ` γ e T K m e γ e T pq pk ` 1q q pkqq ` rrq pkq ` 2Ωpq r pk ` 1q q pkq γ 2 et e (B8)

21 Sensors 216, 16, of 22 Substituting Equation (B8) into Equation (B2), differential Lyapunov function can be written as:! V γ e T p e k`1 e k q ` γ e T K m e ` tr γ 1 r R R k rr T ` γ 1 r D D k rd T ` γ 1 r Ω Ω k Ω r T) τ rd `q m pk ` 1q q m pkq ` rrq m pkq ` 2Ωpq r m pk ` 1q q m pkq! γ 2 et e ` tr γ 1 r R R k rr T ` γ 1 r D D k rd T ` γ 1 r Ω Ω k Ω r T) To judge wher Equation (B9) is negative semi-definite, first term should be combined with third term and convert into formula of matrix trace, taking τ rrq as example: τ rrq m ı «R11 r R12 r τ x τ y rr 21 R22 r ff «q mx q my #«R r 11 τ x q mx ` rr 21 τ y q mx ` rr 12 τ x q my ` rr 22 τ y q my. rr11 qmx ` rr. ff 12 qmy ı +! tr. rr 21 qmx ` rr. τ x τ y tr γ 1 r ) R Rγ R q m τ T 22 qmy ff (B9) (B1) It can be seen that under condition of asymmetric coupling coefficients, Equation (B1) still exists. Similarly following equations can also be proved: τ rd `q! m pk ` 1q q m pkq tr γ 1 r ) D Dγ D `qm pk ` 1q q m pkq τ T! τ Ω r `qm pk ` 1q q m pkq tr γ 1 r ) Ω Ωγ Ω `qm pk ` 1q q m pkq τ T Substituting Equations (B1) (B12) into Equation (B9): (B11) (B12) V γ 2 et e ` tr! R r γ 1 R rr T q m τ T ` rd γ 1 D rd T `q m pk ` 1q q m pkq τ T ` rω γ 1 Ω Ω r T `q ) (B13) m pk ` 1q q m pkq τ T By applying adaptive law: rr γ R q m τ T (B14) rd γ D `qm pk ` 1q q m pkq τ T r Ω γ Ω `qm pk ` 1q q m pkq τ T (B15) (B16) Thus: tr! R r γ 1 ` rd γ 1 D rd T `q m pk ` 1q q m pkq τ T γ 1 Ω Ω r T `q ) m pk ` 1q q m pkq τ T R rr T q m τ T ` rω V γ 2 et e ď, and system is proofed to be Lyapunov stable. References 1. Xia, D.; Yu, C.; Kong, L. The development of micromachined gyroscope structure and circuitry technology. Sensors 214, 14, [CrossRef] [PubMed] 2. Xia, D.; Yu, C.; Kong, L. A micro dynamically tuned gyroscope with adjustable static capacitance. Sensors 213, 13, [CrossRef] [PubMed] 3. Park, S. Adaptive Control Strategies for MEMS Gyroscopes. Ph.D. Thesis, University of California, Berkeley, CA, USA, Park, S.; Horowitz, R. Adaptive control for conventional mode of operation of MEMS gyroscopes. J. Microelectromech. Syst. 23, 12, [CrossRef] 5. Park, S.; Horowitz, R.; Tan, C.W. Dynamics and control of a MEMS angle measuring gyroscope. Sens. Actuators A Phys. 28, 144, [CrossRef]

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