sensors

Article Analysis and Compensation of Bias Drift for a Micromachined Spinning-Rotor Gyroscope with Electrostatic Suspension

Shunyue Wang and Fengtian Han *

Department of Precision Instrument, Tsinghua University, Beijing 100084, China; [email protected] * Correspondence: [email protected]; Tel.: +86-10-6279-8645



Received: 5 March 2020; Accepted: 22 March 2020; Published: 24 March 2020


Abstract: Bias stability is one of primary characteristics of precise gyroscopes for inertial navigation. Analysis of various sources of the bias drift in a micromachined electrostatically suspended gyroscope (MESG) indicates that the bias stability is dominated by the temperature-induced drift. The analytical results of temperature drift resulting from the rotor structure and capacitive position sensing electronics are modeled and analyzed to characterize the drift mechanism of the MESG. The experimental results indicate that the bias drift is mainly composed of two components, i.e., rapidly changing temperature drift and slowly changing time drift. Both the short-term and long-term bias drift of the MESG are tested and discussed to achieve online bias compensation. Finally, a neural network based-bias compensation scheme is presented and verified experimentally with improved bias stability of the MESG.

Keywords: micromachined electrostatically suspended gyroscope; temperature-induced drift; bias drift compensation; time drift; bias stability

1. Introduction Micromachined gyroscopes have attracted much attention in inertial measurements and navigation applications wherein lightweight, low-cost and high-performance sensors are crucial [1,2]. In order to minimize the mechanical friction and wear of micromachined spinning-rotor gyroscopes, various suspension schemes, including contactless gas-lubricated bearings [3], liquid-suspended bearings [4–6] and electrostatically and electromagnetically suspended bearings [7–9] have been studied and fabricated for better performance and longer lifetime. Among them, the micromachined electrostatically suspended gyroscope (MESG) is comparatively compatible with the existing micro-electromechanical system (MEMS) fabrication process [10,11]. Most of the published work in the literature focused on design [12–14], fabrication [11,15], suspension [7] and rotation [16] of the MESG. In this paper, the temperature characteristics of the MESG bias are studied first. Currently, most of these MESG devices are fabricated with glass/silicon/glass triple-wafer bonding and suspended electrostatically in five degrees of freedom (DOFs). Temperature-induced drift is one of dominant error sources for the MESG with its spinning rotor suspended by a frictionless electrostatic bearing, as the gyroscope output is sensitive to environmental temperature variation resulting from thermal impact on interface electronics and sensing structure [17]. The bias stability of gyroscopes is the most crucial performance gauge for inertial navigation applications [18]. Ambient temperature variation inevitably results in bias drift and performance degradation [19,20]. Constant temperature control and temperature drift compensation are the two usual approaches to reducing greatly the bias drift and enhancing the gyroscope performance. Generally, ultra-high precision gyroscopes are typically operated in a constant temperature but bulky

Sensors 2020, 20, 1799; doi:10.3390/s20061799 www.mdpi.com/journal/sensors Sensors 2020, 20, 1799 2 of 19 chamber to provide ultra-low bias drift [20–22]. By contrast, most of gyroscopes generally operate with temperature-based model compensation to attenuate the bias drift and thus enhance the gyroscope stability, which is an effective and inexpensive solution. a support vector machine (SVM) method was used to compensate the temperature drift of a dynamically tuned gyroscope, which reduces the average drift down to 0.003◦/h from a 735 daily averaged temperature drift data and real-time temperature data [17]. Recently, the modeling and compensation of temperature-induced drift have also been studied for high-performance MEMS gyroscopes. a drift self-compensation scheme for silicon quadruple mass gyroscopes was reported by using its resonant frequency measurement as an embedded thermometer, rather than placing a temperature sensor in close proximity to the gyroscope chip [23]. a back-propagation (BP) artificial neural network was used to compensate the bias drift of MEMS gyroscopes, which exhibits a non-linear model between the gyroscope output and ambient temperature [24]. The bias drift of a quadruple mass vibratory gyroscope was compensated by the mode-matching method, and the bias stability reaches as low as 0.2◦/h for a time duration of half a month [25]. However, the temperature drift compensation and the evaluation of the existing MEMS vibratory gyroscopes are mostly for short-time operation. Heretofore, very little research has been reported on long-term temperature drift characteristics of micromachined spinning-rotor gyroscopes, which have the potential to be used for high precision and long-term inertial measurements. Tokyo university has reported the static MESG data for 85 h with a sampling time of 2.2 ms, but only the most stable 4-h data are used to analysis the bias instability by Allan variance [26]. To achieve high performance of the MESG for long-term and wide-temperature operation, it is significant to analyze the mechanism and characteristics of long-term temperature drift to improve the design of MESG sensing structure and associated electronics. In addition, specific compensation methods can be utilized to suppress the temperature drift greatly and improve the bias stability of MESGs as well. This purpose of this paper is to examine the temperature-induced bias drift for a prototype MESG with a high-speed spinning rotor. The theoretical analysis and experimental evaluation of the thermal rotor deformation and sensing electronics drift are presented to characterize the temperature-induced bias drift of MESGs. The MESG measurements for up to 70 h in a temperature test chamber are presented and discussed to examine the long-term bias drift, which comprises rapidly changing temperature drift and slowly changing time drift. a BP neural network-based drift compensation scheme is experimentally validated to offer much improved long-term bias stability for a duration up to 70 h.

2. Device Description

2.1. Device Structure The MESG presented here consists of a glass-silicon-glass triple-wafer bonding structure with a complex electrode pattern, as shown in the exploded view and a fabricated die of the MESG in Figure1. a typical structure of the MESG consists of two Pyrex 7740 glass wafers, and one contactless ring-shaped rotor which is formed in the intermediate silicon wafer [8]. The sensing and actuation electrodes around the rotor are used to suspend the rotor electrostatically in DOFs and spin up the rotor at high speed so as to function as a two-DOF angular rate gyroscope. Planar electrodes symmetrically sputtered on the top and bottom glass wafers are used for axial suspension of the rotor in three DOFs: translation along the z-axis and rotation around the two in-plane axes. Moreover, the outmost electrodes on each glass wafer are used for rotation drive of the rotor which is operated as a planar variable-capacitance micro-motor. The silicon wafer is etched to form the ring-shaped rotor and truncated pie-shaped radial electrodes which are used for radial suspension of the rotor along the x- and y-axes. Main design parameters of the MESG structure are listed in Table1. Sensors 2020, 20, x FOR PEER REVIEW 3 of 19 Sensors 2020, 20, 1799 3 of 19

z ϕ y ω x o φ

Top glass wafer Top electrodes

Movable ring- Silicon wafer shaped rotor

Bottom glass wafer Bottom electrodes

(a) (b) Figure 1. Micromachined electrostatically suspended gyroscope: (a) an exploded view of the device Figure 1. Micromachined electrostatically suspended gyroscope: (a) an exploded view of the device and (b) the fabricated die. and (b) the fabricated die. Table 1. Main parameters of the micromachined electrostatically suspended gyroscope Table(MESG) 1. structure.Main parameters of the micromachined electrostatically suspended gyroscope (MESG) structure. Description Value Description Value Rotor outer radius r0 (µm) 2000 Rotor outer radius r0 (μm) 2000 Rotor inner radius ri (µm) 1730 RotorRotor thickness innerh (um)radius ri (μm) 1730 77 Mass ofRotor the rotor thicknessm (mg) h (um) 0.457 77 Moment of inertia of the rotor J (kg m2) 1.98 10 12 Mass of the rotor m· (mg) 0.457× − Axial gap d (µm) 5 Moment of inertiaz of the rotor J (kg·m2) 1.98 × 10−12 Radial gap dr (µm) 5 Axial suspension electrodeAxial outer gap radiusdz (μm)R o (µm) 18685 Axial suspension electrodeRadial inner gap radiusdr (μm)Ri (µm) 17385 AxialNumber suspension of stator electrode phases outerN radius Ro (μm) 18683 Number of stator electrodes Ns 24 Axial suspension electrode inner radius Ri (μm) 1738 Number of rotor poles Ns1 14 Number of stator phases N 3 Angle of rotor poles and stator electrodes θr π/14 Number of stator electrodes Ns 24 Number of rotor poles Ns1 14 The MEMS device was fabricated using bulk microfabrication based on silicon on glass Angle of rotor poles and stator electrodes θr π/14 (SOG) technique, which includes several key processes, such as high-aspect-ratio dry etching, twice glass-silicon anodic bonding, aluminum sacrificial layer removal and vacuum packaging. 2.2. Scale Factor and Full-Scale Range The size of each fabricated die is 6.5 6.5 1.1 mm, as depicted in Figure1b [27]. Basically, the MESG is a dual-axis× angular× rate sensor when the rotor is spinning at a constant and2.2. high Scale speed. Factor andIf no Full-Scale torque Rangeis applied on the rotor, it will maintain its orientation in the inertial space.Basically, However, the if MESG an angular is a dual-axis rotation angular orthogonal rate sensor to the when spin the axis rotor occu is spinningrs, a precession at a constant torque and generatedhigh speed. by Ifthe no rebalance torque is loop applied will onforce the the rotor, rotor it willto its maintain null position. its orientation The scalein factor theinertial is one of space. the mainHowever, design if parameters an angular rotationfor MESGs, orthogonal which describes to the spin the axis response occurs, of a the precession gyroscope torque output generated VΩ under by anthe angular-rate rebalance loop input will Ωx. force The precession the rotor to of its the null spinning position. rotor The is scalegoverned factor by is the one equation: of the main design Ω= HVKx Ω v , (1)

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parameters for MESGs, which describes the response of the gyroscope output VΩ under an angular-rate input Ωx. The precession of the spinning rotor is governed by the equation:

HΩx = V Kv, (1) | | Ω where H = Jz ω is the angular momentum of the rotor; Kv is the electrostatic torquer gain of the · 0 rebalance loop; ω0 and Jz are the rated spin rate and moment of inertia of the rotor spinning around the z-axis, respectively. 1 The scale factor Kf in V/(◦ s− ) can be expressed by

VΩ Jz ω0 Kf = = · . (2) Ωx Kv

Equation (2) indicates that the scale factor of MESGs is proportional to the rated spin rate ω0. Note that the scale factor stability is also closely related with the constant speed accuracy of the rotation control loop. To obtain high sensitivity and excellent stability, it is essential to have a high and precise spin rate of the rotor during design of the MESG structure, rotation electronics and vacuum packaging [8]. In principle, the full-scale range of the MESG in ◦/s can be given by

Vbr Ωr_max = , (3) Kf where Vbr is the bias voltage applied on the axial electrodes, which is also the allowable maximum voltage of the MESG output. According to Equation (2), the scale factor reaches 26.1 mV/(◦/s) at the rated spin rate of 1 104 rpm. When the bias voltage is set at 7.2 V, the full-scale measurement range × of the MESG is 275.86 /s. It is clear that the scale factor increases as the spin rate rises while the ± ◦ full-scale range will decrease as the scale factor becomes larger.

2.3. Vacuum Packaging As a spinning-rotor gyroscope, a vacuum environment is essential for operation of the MESG to reduce viscous drag dramatically and thus to enable ultra-high rotation of the rotor. High vacuum level as low as 0.01 Pa can minimize the effect of residual air-film damping and achieve a desirable spin rate over 1 104 rpm with a rotation voltage below 7.6 V, which is essential to dramatically enhance × the overall performance of the MESG. Figure2 shows a schematic illustration of the device-level vacuum packaging using a metallic chip package. To reduce the number of the package pins, a ceramic board was used to rearrange and combine part of the die’s pins, which have the same electrical function. Firstly, the 60-pin die was fixed to the ceramic circuit board using JM7000 adhesive (Henkel, Dusseldorf, Germany) baking at 175 ◦C for 2 h. Secondly, the MESG die was wire-bonded to the 55-pin ceramic circuit board. Then the ceramic board was further wire-bonded to the 55-pin metallic chip package. Thirdly, the parallel seam welding technique was used to seal the metallic package with the cover. When the two conical electrodes roll along the edge of the metal cover at a speed of 50 mm/s, a 1 kHz power pulse signal with a pulse width of 3 ms and a peak power of 2500 W was applied on the two electrodes to generate Joule heat. Then a partial molten state was formed between the cover and the package frame, and the sealing process was completed after solidification of the soldering seam. Fourthly, both the ceramic board and metallic package were high-temperature degassed in a vacuum chamber to bake out the water and gas that adhered in the surface. In order to minimize gas evolution from the material of die, the ceramic 5 board and metallic package, a 160 h, 10− Pa and 130 ◦C baking-out procedure was used to drive out chemically dissolved water, carbon dioxide, nitrogen and oxygen from the glass and hydrogen from the metal. Finally, a heating current of 3.3 a was applied for 10 min to activate the getter and thereafter maintain high vacuum of the MESG packaging. Sensors 2020, 20, x FOR PEER REVIEW 5 of 19 Sensors 2020, 20, 1799 5 of 19

Figure 2. Schematic diagram of the device-level vacuum packaging of the MESG. Figure 2. Schematic diagram of the device-level vacuum packaging of the MESG. It is essential for the vacuum packaging to provide a required initial vacuum as well as desirable It is essential for the vacuum packaging to provide a required initial vacuum as well as desirable duration in years after sealing [28]. The lifetime of device-level encapsulation is a key measure with duration in years after sealing [28]. The lifetime of device-level encapsulation is a key measure with which to evaluate the reliability of most sensor products in vacuum [29,30]. Any defects during the which to evaluate the reliability of most sensor products in vacuum [29,30]. Any defects during the sealing and packaging process could cause failure of MESGs or performance degradation over time. sealing and packaging process could cause failure of MESGs or performance degradation over time. Therefore, it is important to characterize the vacuum maintenance duration of the MESG which could Therefore, it is important to characterize the vacuum maintenance duration of the MESG which could provide valuable information for lifetime evaluation of the MESG. provide valuable information for lifetime evaluation of the MESG. The relationship between the viscous damping effect and speed decay duration of the rotor has The relationship between the viscous damping effect and speed decay duration of the rotor has been experimentally studied for MESGs [16]. When the rotation drive torque was removed, the been experimentally studied for MESGs [16]. When the rotation drive torque was removed, the rotation rotation speed would decrease due to weak damping effect of the residual gas inside the vacuum speed would decrease due to weak damping effect of the residual gas inside the vacuum package. package. Therefore, the speed decay time is a measure of the real vacuum pressure inside the Therefore, the speed decay time is a measure of the real vacuum pressure inside the package. Define the package. Define the decay time constant as τ = Jz /b where b is the viscous damping coefficient decay time constant as τ = Jz /b where b is the viscous damping coefficient associated with the vacuum associated with the vacuum pressure; then the instantaneous rotor speed ω (t) during the decay pressure; then the instantaneous rotor speed ω (t) during the decay process can be expressed as process can be expressed as t t ω(t) = ω(0) e−− τ , t 0. (4) ωω(t)=× (0)× eτ ,t ≥≥ 0 . (4)

SoSo the the gas gas pressure insideinside thethe vacuum vacuum package package can can be be estimated estimated by byτ, whichτ, which can can be obtained be obtained from fromspeed speed decay decay experiments. experiments. The stability The stability of vacuum of maintenancevacuum maintenance in one MESG in hasone beenMESG monitored has been by monitoredmeasuring by periodically measuring the periodically speed decay the time speed (τ) updecay to 20 time months. (τ) up The to experimental20 months. The results experimental show that resultsthe MESG show device that worksthe MESG stably devi duringce works 20 months stably andduring the estimated20 months vacuum and the pressure estimated is still vacuum below pressure0.1 Pa. As is shownstill below in Figure 0.1 Pa.3a, As the shown speed in decay Figure duration 3a, the decreases speed decay rapidly duration in the decreases first six months rapidly and in thestays first virtually six months constant andafter stays 18 virtually months. constant Figure3a after indicates 18 months. that the Figure speed decay3a indicates duration that was the reduced speed decayfrom 610duration s to 282 was s afterreduced a 20-month from 610 encapsulation. s to 282 s after a 20-month encapsulation.

600 0.015 ） s 550

（ 0.01

500 0.005

450 0

400 -0.005 350

Gyroscope Output (V) Output Gyroscope -0.01 300 -0.015 -2

Tim e durationthe of speed decayt [email protected]×10 Pa 250 -2 0 2 4 6 8 10 12 14 16 18 20 [email protected]×10 Pa -0.02 Packaging Time T（Month） 0 0.2 0.4 0.6 0.8 1 Time (h) (a) (b)

FigureFigure 3. 3. ((aa)) StabilityStability ofof vacuum vacuum tests tests up up to 20to months20 months and (andb) gyroscope (b) gyroscope noise atnoise different at different drive voltages. drive voltages.

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Equation (2) indicates that high spin rate would contribute to enhancing the scale factor of the gyroscope. The higher the vacuum level in the device package, the higher the spin rate that could be reached. Figure3b shows that at the same spin rate of 1.0 104 rpm, high vacuum level will contribute × to low rotation drive voltage and thus reduce gyroscope output noise. The gyroscope output noise is induced by a combination of the electrostatic suspension voltage and the rotation drive voltage. Note that the rotation drive torque is proportional to the square of the rotation voltage and the precession control torque varies linearly with the MESG output voltage. Then, by assuming these two noise sources are independent, the overall noise contributions to the gyroscope’s output from these two sources can be approximated by q 2 2 2 Vn = Vns + (knVr ) , (5) where Vn and Vns are the gyroscopic output noise and suspension voltage induced noise, Vr and kn are the rotation drive voltage and its noise coefficient, respectively. The experimental results in Figure3b 3 4 show that the root-mean-square values of Vns and kn are 1.8 10 V and 1.95 10 by data fitting in × − × − Equation (5), respectively. It is clear that the gyroscope output noise can be reduced greatly with higher vacuum to permit low rotation drive voltage. Equation (5) indicates that the gyroscopic noise will fall by 67% as the rotation voltage reduces from 5.98 V to 2.85 V. Therefore, a high vacuum packaging better than 4 10 2 Pa is the first prerequisite to realizing low-noise MESGs operating at the spin rate × − of 1.0 104 rpm. × 3. Temperature Dependent Characteristics of the MESG Theoretically, the variation of environment temperature and the heat generated by associated electronics of MESGs will lead to the change of the temperature field distribution, which is the dominant source of the temperature drift. The temperature-dependent output drift of the MESG could result from thermal variations in both the device structure and electronics.

3.1. Rotor Deformation-Induced Bias Drift

3.1.1. Theoretical Analysis of the Rotor Thermal’s Deformation The temperature drift of the gyroscope output comes from both the average temperature variation and internal temperature gradient fluctuation. The non-uniformly distributed or changing temperature field on the rotor could result in the variation of the angular momentum and the deviation of the center of mass from the center of support. Here, the temperature drift caused by thermal deformation of the rotor structure is evaluated approximately by an analytical model. Rotor deformation can be described by quasi-static thermoelastic theory, as the MESG works under ordinary heat exchange conditions. The displacement and temperature field are described by a simplified thermoelastic motion equation.

2 (1 2v∗) U + graddivU 2(1 + v∗)αTgradT = 0, (6) − ∇ − where v∗ is the poisson’s ratio; U is displacement vector; αT is the coefficient of thermal expansion; T is the temperature; and 2 is the Laplace operator. ∇ A uniform, thin rod is assumed to represent the ideal rotor suspended electrostatically, as shown in Figure4. Sensors 2020, 20, 1799 7 of 19 SensorsSensors 2020 2020, 20,, 20x FOR, x FOR PEER PEER REVIEW REVIEW 7 of7 of19 19

x x o -l-lo l l (a) (a)Uniform, Uniform, thin thin rod rod T T x x o -l-lo l l (b) (b)Temperature Temperature field field U U x x o -l -l o l l Uo1Uo1 (c) (c)Displacement Displacement Uo2Uo2 x x o -l-lo l l (d) (d)Deformation Deformation state state

Figure 4. A schematically thermoelastic deformation of the MESG rotor. FigureFigure 4. A4. schematicallyA schematically thermoelastic thermoelastic de formationdeformation of ofthe the MESG MESG rotor. rotor. Assuming that the temperature field varies along the rotor in accordance with linear temperature Assuming that the temperature field varies along the rotor in accordance with linear variation,Assuming then thethat expression the temperature of the temperature field varies variation along is the rotor in accordance with linear temperaturetemperature variation, variation, then then the the expres expressionsion of ofthe the temperature temperature variation variation is is T0 25−25 T = T−T−025x + 25,+ (7) Tx=25Tx=250 l + , , (7)(7) l l where l is the rotor diameter, T0 is the environmental temperature and x is the rotor deformation. where l is the rotor diameter, T0 is the environmental temperature and x is the rotor deformation. In whereIn a one-dimensional l is the rotor diameter, coordinate T0 is system, the environmental the expression temperature of the thermoelastic and x is the equation rotor deformation. is simplified In as a one-dimensionala one-dimensional coordinate coordinate system system, the, the expression expression of ofthe the thermoelas thermoelastictic equation equation is simplifiedis simplified as as dU(x) + * ()1 1v+∗* dUdU() x= x 1=+ v α vTTα(x). (8) = α* TTx(). (8) dx 1 v−∗* TTx(). (8) dxdx 1−−1v v Equation (9) gives the solution of Equation (8) with the boundary condition U(0) = 0. EquationEquation (9) (9) gives gives the the solution solution of ofEquation Equation (8) (8) with with the the boundary boundary condition condition U (0)U(0) = 0. = 0. * 1 + v∗+* T0 25− 2 +1 v T−0 25 2 U(x) = 1=+v αT[αT0 − 25 x2 + 25x]. (9) Ux()=+α* T [ x 25] x. (9) Ux()1 v∗−* T [2l x 25] x. (9) 1−−1v v 2l 2l Figure5 shows the analytical results of the rotor thermal deformation along the axial and radial FigureFigure 5 shows 5 shows the the analytical analytical results results of ofthe the roto rotor thermalr thermal deformation deformation along along the the axial axial and and radial radial axes. It is indicated that the coefficients of thermal deformation along the axial and radial directions axes.axes. It isIt indicatedis indicated11 that that the the coefficients coefficients9 of ofthermal thermal deformation deformation along along the the axial axial and and radial radial directions directions are 8.964 10− m/◦C and 4.426 10− m/◦C, respectively. areare 8.964 8.964 × 10× −1011 −m/11 m/°C°C and and 4.426 4.426 × × 10× −109 m/°C,−9 m/°C, respectively. respectively.

-9 -7 x-9 10 x-7 10 x 10 x 10 6 11 11 6

10 10

5 5

8 8 4 4 Thermal deformation(m) Thermal Thermal deformation(m) Thermal AxialAxial direction direction RadialRadial direction direction 3 6 6 3 45 4545 50 5050 55 55 60 6060 65 65 70 70 75 75 80 80 85 85 90 90 95 95 45 50 55 60 65 70 75 ℃ 80 85 90 95 TemTemperatureperature()℃()

FigureFigureFigure 5. 5. Theoretical5.Theoretical Theoretical results results results of of ofrotor rotor rotor thermal thermal thermal deform deformation deformationation along along along axial axial axial and and radial radial directions. directions. 3.1.2. Bias Drift Caused by Thermal Expansion of the Rotor Structure 3.1.2.3.1.2. Bias Bias Drift Drift Caused Caused by by Thermal Thermal Expansion Expansion of ofthe the Rotor Rotor Structure Structure Ideally, there is no alignment error between the rotor and the top/bottom electrodes. Ideally,Ideally, there there is nois no alignment alignment error error between between th eth rotore rotor and and the the top/bottom top/bottom electrodes. electrodes. However, However, However, the glass-silicon anodic bonding process inevitably leads to alignment error in the order of thethe glass-silicon glass-silicon anodic anodic bonding bonding process process inevitably inevitably leads leads to toalignment alignment error error in inthe the order order of ofmicrons. microns. microns. As shown in Figure6, there are constant alignment errors between the origins of top electrode AsAs shown shown in inFigure Figure 6, 6,there there are are constant constant alignment alignment errors errors between between the the origins origins of oftop top electrode electrode O1Ox11yx11zy11 zand1 and bottom bottom electrode electrode O 2Ox22yx22zy22 zwhen2 when they they are are projected projected onto onto the the O ’Ox’’yx’’zy’’ zcoordinates’ coordinates of ofthe the rotorrotor wafer. wafer. According According to tothe the experience experience in inbond bondinging alignment alignment process, process, th eth alignmente alignment precision precision can can

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O1x1y1z1 and bottom electrode O2x2y2z2 when they are projected onto the O’x’y’z’ coordinates of the rotorSensors wafer. 2020, 20 According, x FOR PEER to REVIEW the experience in bonding alignment process, the alignment precision can8 of be19 controlled within 2–5 µm. The O’x’y’z’ coordinates will move to Oxyz due to the thermal expansion be controlled within 2–5 μm. The O’x’y’z’ coordinates will move to Oxyz due to the thermal effect of the rotor. As shown in Figure6, the origin of the Ox1y1z1 coordinate is projected onto the expansion effect of the rotor. As shown in Figure 6, the origin of the Ox1y1z1 coordinate is projected silicon rotor as d = (∆x, ∆y, h/2)T, which denotes in the Oxyz coordinates. Similarly, the origin of the 1 T onto the silicon rotor as d1= (Δx, Δy, h/2) , which denotes in the Oxyz coordinates.T Similarly, the origin Ox2y2z2 coordinate is projected onto the silicon rotor as d2= (∆x, ∆y, h/2) . Therefore, the alignment of the Ox2y2z2 coordinate is projected onto the silicon rotor as d2=− (Δx, Δy, −h/2)T. Therefore, the error variation caused by thermal expansion introduces electrostatic disturbance torque, which affects alignment error variation caused by thermal expansion introduces electrostatic disturbance torque, the angular rate measurements along both the x and y directions. which affects the angular rate measurements along both the x and y directions.

y1

O1 x1 Top electrode

y y' O x Δy Rotor d1 O' x' d2 Δx

y2

O2 x2 Bottom electrode

Figure 6. Figure 6. Alignment error caused by thermal expansion of the rotor.rotor. In an electrostatically suspended system of the MESG, define the suspension stiffness along the x, In an electrostatically suspended system of the MESG, define the suspension stiffness along the y and z axes as [kx, ky, kz]. When the rotor suffers from a vector input acceleration [ax, ay, az], the origin x, y and z axes as [kx, ky, kz]. When the rotor suffers from a vector input acceleration [ax, ay, az], the will deviate to the coordinates ( max/ kx, may/ ky, maz/ kz). The moment arms l1 and l2 formed by origin will deviate to the coordinates− （−max/ kx, −may/ ky, −maz/ kz）. The moment arms l1 and l2 the top and bottom electrodes can be expressed as formed by the top and bottom electrodes can be expressed as

 TT  l1 =l (=Δ+∆(/,/,/2/)x + xma max/kx k, ∆ Δ+y y+ ma may/k ky, h h/2 + + mamaz/ kkz)   1 xx yy zz . (10)  l = ( x + ma k y + ma k h + ma k )T . (10) 2 ∆=Δ+(/,/,-/2/)x/ x, ∆ Δ+y/ y, /2 + z/ z T l2 x maxx k y ma yy k− h ma zz k When a small offset d’ due to thermal expansion occurs along the z axis of the frame, the electrostatic When a small offset d’ due to thermal expansion occurs along the z axis of the frame, the forces generated by the top and bottom electrodes on the rotor are as electrostatic forces generated by the top and bottom electrodes on the rotor are as

 222 2 2 2 εαεα(R0 −Ri ) (V⋅ bz Vfz−)  （）RRVV0bf()T T  F1=（= (0,， 0,− − izz· 2 − ) ） F1 0,0 ( ) 2  − d(-')z d0  2 dd2−z 2 , (11)  εα(R0 Ri ) (Vbz+Vfz) T , (11)  F2 = (0, 0, εα（ 22−− · ）⋅+2 ) 2  RRVV0bf(d +izzd ) ()T F =（0,0，− − z 0 ）  2 2  (+')ddz where α = π/3 is the arc angle of the axial electrodes; ε is the permittivity; Vbz and Vfz are the bias voltagewhere α = and π/3 feedbackis the arc angle voltage of the of theaxialz-axis, electrodes; respectively. ε is the permittivity; The electrostatic Vbz and moment Vfz are the caused bias voltage by the alignmentand feedback error voltage is of the z-axis, respectively. The electrostatic moment caused by the alignment error is 2 2 2 2εα(R0 Ri ) V M = l F + l F =εα 22−⋅− 2· bz [∆y, ∆x, 0]T. (12) f 1 1 2 2 2(RRV )2 =×× +×× 0 ibzdz ΔΔ T Mf1122lFlF = 2 [ y, x ,0] . (12) dz . The thermal expansion-induced moment produces an angular rate response ϕx of the MESG along ϕ the x Theaxis isthermal expansion-induced moment produces an angular rate response x of the MESG . H Mx along the x axis is ϕ = × . (13) x H2 HM× ϕ x . x = 2 (13) H According to the analytical results of thermal deformation in Section 3.1.1, we know that Δx/dT = Δy/dT = 4.43×10−9 m/°C. Then, the bias temperature drift of the MESG resulted from the rotor thermal expansion is 3.59 × 10−3 mV/°C based on Equation (13).

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According to the analytical results of thermal deformation in Section 3.1.1, we know that ∆x/dT = ∆y/dT = 4.43 10 9 m/ C. Then, the bias temperature drift of the MESG resulted from the rotor thermal × − ◦ expansion is 3.59 10 3 mV/ C based on Equation (13). × − ◦ 3.2. Radiative Thermal Transfer Model of the MESG As the rotor is suspended in high vacuum without any mechanical contact, the temperature stabilization or change on the rotor is mainly caused by radiative thermal transfer rather than solid thermal conduction. Therefore, it is necessary to calculate the thermal equilibrium time of the rotor under radiative thermal transfer only. In order to simplify the analysis, the thermal transfer process from the environment to the electrode chamber is omitted. Here we only discuss the most critical thermal transfer response between the electrode cavity and the contactless silicon rotor. Based on the energy balance relationship of microelement in the heat transfer theory, the governing equation for the MESG can be written as follows.

∂T(r, t) q(r, t) + g(r, t) = ρCp , (14) − ∇· ∂t where T is the temperature inside the microelement; Cp is the specific heat of the material; g is the heat energy generation rate per unit volume; q is the heat flux conducting along the boundary normal direction; q is the heat flux gradient conducting along the boundary normal direction; ρ is the ∇· material density; and r is the distance vector. Calculating the volume integral of the heat equilibrium equation (Equation (14)) yields Z Z Z ∂T(t) q(r, t)dV + g(t)dV = ρCp dV. (15) − ∇· ∂t V V V

The Gaussian integral formula is used on the first term on the left side of Equation (15), and the radiation law is used on the second term on the left side. Then Equation (15) can be rewritten as:

∂T(t) τ τ + T(t) = T + ∆g(t) + ∆Ta(t). (16) ∂t ∞ ρCp

The solution of Equation (16) can be expressed as

Zt t/τ 1 τ (t υ)/τ T(t) = T + (T0 T )e− + ( ∆g(υ) + ∆Ta(υ))e− − dv, (17) ∞ − ∞ τ ρCp 0

2 2 ρCpπ(ro r ) h − i · τ where τ = 2 2 3 and T = Ta0 + g0. Then Equation (18) is obtained based on [2 πr h+2π(ro r )] ε δTa ρCp 0 − i · 1 ∞ P∞ jnωt P∞ jnωt Fourier series ∆g(t) = ane− and ∆Ta(t) = bne− n= n= −∞ −∞ X t/τ ∞ τ 1 jnωt t/τ T(t) = T + (T0 T )e− + ( an + bn) (e e− ). (18) ∞ ∞ C + j n − n= ρ p 1 τ ω − −∞ The steady-state rotor temperature response can be obtained when t →∞ X ∞ τ 1 jnωt limT(t) = T + ( an + bn) e . (19) t ∞ ρC 1 + jτnω →∞ n= p −∞ As can be seen from Equations (18) and (19), the rotor temperature response changing from an initial state to its equilibrium is in an exponential decay process with a time constant τ = 711.5 s, Sensors 2020, 20, 1799 10 of 19

8 2 4 when the silicon emissivity is ε = 0.05 and Ta is 300 K. Here δ = 5.67 10 W/(m K ) is the blackbody 1 × − · radiation constant coefficient. The thermal equilibrium time of the suspended rotor is calculated as 3τ = 35.6 min, which roughly governs the temperature stabilization process of the MESG device.

3.3. Temperature Drift Analysis of the Position Sensing Circuit Six capacitive position sensing channels, as illustrated in Figure7, are used to detect the rotor Sensorsposition 2020 and, 20, thusx FOR realize PEER REVIEW electrostatic suspension of the rotor in five DOFs. a sinusoidal excitation10signal of 19 with an amplitude of 1.48 V and frequency of 1 MHz is applied to the common electrode Ccom. When the When the rotor deviates from the equilibrium position, a differential capacitance change is generated rotor deviates from the equilibrium position, a differential capacitance change is generated between the between the rotor and associate sensing electrodes; i.e., ΔCi =Ci-C’i (i=z1, z2, z3, z4, x, y). The rotor and associate sensing electrodes; i.e., ∆Ci =Ci-C’i (i=z1, z2, z3, z4, x, y). The unbalanced current is unbalanced current is then converted to the sensing voltage by a pair of differential charge amplifiers then converted to the sensing voltage by a pair of differential charge amplifiers with very high input with very high input impedance. The ensuing AC amplifier stage functions as a band-pass filter with impedance. The ensuing AC amplifier stage functions as a band-pass filter with a gain of up to 150. a gain of up to 150. The displacement sensing signal from the AC amplifier is a high frequency The displacement sensing signal from the AC amplifier is a high frequency amplitude modulation amplitude modulation wave and is fed to an analog multiplier for phase sensitive modulation. wave and is fed to an analog multiplier for phase sensitive modulation. Finally, the position sensing Finally, the position sensing output voltage is obtained by low-pass filtering at a cut-off frequency of output voltage is obtained by low-pass filtering at a cut-off frequency of 20 kHz. 20 kHz.

Phase shift Δϕ

Excitation signal Vsin Vz1 I/V V/V V/V

Cz1 Cz2 Cz3 Cz4

Ccom

Vz2 y I/V V/V V/V׳C Rotor

Cy … … … … com׳C z4׳z3 C׳z2 C׳z1 C׳C

Vy I/V V/V V/V

Charge Differential Analog Low pass amplifier amplifier AC amplifier multiplier filter (a) (b) (c) (d) (e)

FigureFigure 7. 7. AA schematic schematic of of the the rotor rotor displacement displacement sensing sensing circuit. circuit.

Since the series capacitors are used to couple the AC position sensing signal—see the capacitors Since the series capacitors are used to couple the AC position sensing signal—see the capacitors shown in Figure7a–c—the o ffset drifts of the charge amplifiers, differential amplifiers and AC amplifier shown in Figure 7a–c—the offset drifts of the charge amplifiers, differential amplifiers and AC caused by ambient temperature variation can be ignored. However, the temperature-induced offset amplifier caused by ambient temperature variation can be ignored. However, the temperature- voltage change of the analog multipliers and low-pass filters have to be considered to evaluate the induced offset voltage change of the analog multipliers and low-pass filters have to be considered to thermal stability of the position sensing outputs. evaluate the thermal stability of the position sensing outputs. According to datasheets of the amplifiers used in the MESG, the temperature drift coefficient of According to datasheets of the amplifiers used in the MESG, the temperature drift coefficient of input offset voltage for the low-pass filter is as low as 2 µV/ C. The analog multiplier is the most basic input offset voltage for the low-pass filter is as low as 2 μV/°C.◦ The analog multiplier is the most basic application of AD734AN (Analog Devices Inc., Norwood, MA, USA), where no external components application of AD734AN (Analog Devices Inc., Norwood, MA, USA), where no external components are needed to constitute the multiplier. The temperature drift range of the multiplier output, which is are needed to constitute the multiplier. The temperature drift range of the multiplier output, which mainly caused by the input offset voltage drift, is estimated from 0.9 mV/ C to 0.42 mV/ C according is mainly caused by the input offset voltage drift, is estimated− from −◦0.9 mV/°C to ◦0.42 mV/°C to AD734AN’s datasheet. according to AD734AN’s datasheet. To test the temperature drift of the capacitance sensing circuit, low temperature-drift capacitor To test the temperature drift of the capacitance sensing circuit, low temperature-drift capacitor pairs were used to replace the differential rotor-electrode capacitance of the MESG, as depicted in pairs were used to replace the differential rotor-electrode capacitance of the MESG, as depicted in Figure7. Then the entire sensing circuit was put into a temperature controlled chamber to evaluate Figure 7. Then the entire sensing circuit was put into a temperature controlled chamber to evaluate the temperature-induced output drift from 40 C to 60 C in a step of 10 C. The above-mentioned the temperature-induced output drift from 40 °C◦ to 60 °C◦ in a step of 10 °C.◦ The above-mentioned experiments were repeated three times to ensure the repeatability of the measured temperature drift experiments were repeated three times to ensure the repeatability of the measured temperature drift coefficient, which yielded 0.2895 mV/ C, 0.2848 mV/ C and 0.2571 mV/ C in three measurements, coefficient, which yielded −−0.2895 mV/°C,◦ −−0.2848 mV/°C◦ and −−0.2571 mV/°C◦ in three measurements, respectively. The temperature drift coefficient of the sensing circuit has a mean value of 0.2771 mV/ C, respectively. The temperature drift coefficient of the sensing circuit has a mean value− of −0.2771◦ which is within the temperature drift range of the analog multiplier. The experimental result agrees mV/°C, which is within the temperature drift range of the analog multiplier. The experimental result well with the predication that the analog multiplier dominates the temperature drift of the position agrees well with the predication that the analog multiplier dominates the temperature drift of the sensing circuit. position sensing circuit.

3.4. Gyroscopic Output Drift Caused by the Temperature Drift of Position Sensing Circuit A dual-axis rebalance loop for operation and measurement of the two-DOF MESG is shown in Figure 8. When an angular rate input is applied to the gyroscope, the dual-axis torque-rebalance loop stabilizes the rotor over its dynamic range and provides precise measurement signals Vx and Vy, as the Simulink simulation model shows in Figure 8.

Sensors 2020, 20, 1799 11 of 19

3.4. Gyroscopic Output Drift Caused by the Temperature Drift of Position Sensing Circuit A dual-axis rebalance loop for operation and measurement of the two-DOF MESG is shown in Figure8. When an angular rate input is applied to the gyroscope, the dual-axis torque-rebalance loop stabilizes the rotor over its dynamic range and provides precise measurement signals Vx and Vy, as the SimulinkSensors 2020 simulation, 20, x FOR PEER model REVIEW shows in Figure8. 11 of 19

Kf Vx ϕ + + x 1 - Mx - Ks Gc(s) Kv G11(s) s +

G21(s)

G12(s) ϕ y 1 + Ks Gc(s) Kv G (s) s - - 22 + y + M Vy

Kf

Figure 8. Block diagram of the dual-axis MESG rebalance loop.loop. In order to design the MESG rebalance loop with desirable performance, two lag-lead compensators In order to design the MESG rebalance loop with desirable performance, two lag-lead are used to provide appropriate phase compensation, as given by Equation (20), which stabilizes the compensators are used to provide appropriate phase compensation, as given by Equation (20), which open-loop unstable dual-axis system by feedback. stabilizes the open-loop unstable dual-axis system by feedback. (++s + 50)(s + 4000) ( )= = (s 50)(s 4000) Gc (s)s 150150 . . (20)(20) (s(++s 1)(s+ 1)( 40000)s + 40000)

Figure8 8indicates indicates that that the the dual-axis dual-axis rebalance rebalanc loope describesloop describes the relationship the relationship between between the angular the . . rate inputs ϕ , ϕ andϕϕ the gyroscopic outputs Vx, Vy, wherex y G(s) governs the dynamics of the spinning angular rate xinputsy xy, and the gyroscopic outputs V , V , where G(s) governs the dynamics of the rotor in the form of spinning rotor in the form of " #  Je H  G (s) G (s)J 2 2 2 -H 2−2 2  11 12 e Je s +H s(Je s +H )  G(s) = =22 2 22 2 ; (21) GGG(s)(s) G (s) (s) JHs+ H s( JH s+J )e  = 1121 12 22 ee2 2 2 2 2 2 G=(s) s(Je s +H ) Je s +H (21) GG21(s) 22 (s) H Je  G12(s) and G21(s), and G11(s) and G22(s)s(JH22 describe s + 2 ) the JH 22 s precession + 2 motion and rigid body ee; characteristics of the spinning-rotor gyro, respectively [14]. Main parameters of the rebalance loop are listedG12(s) in and Table G21(s),2. and G11(s) and G22(s) describe the precession motion and rigid body characteristics of the spinning-rotor gyro, respectively [14]. Main parameters of the rebalance loop are listed in Table 2. Table 2. Parameters of the dual-axis MESG rebalance loop. Description (Unit) Value Table 2. Parameters of the dual-axis MESG rebalance loop. 2 13 Rotor radial moment of inertia Je (kg m ) 7.86 10− · × 4 Spin rateDescription of rotor ω0 (rpm) (Unit) 1.0 Value10 2 1 × 9 Angular moment of rotor H (kg m s− )2 2.07 10− −13 Rotor radial moment of inertia· · 1Je (kg·m ) 7.86× × 109 Torque-voltage coefficient Kv (N m V− ) 4.31 10− Spin rate of rotor ɷ· 0 (rpm)· 1 1.0× × 105 4 Angular position stiffness Kf (N m rad− ) 2.7 10− · · 1 2 −1 × 2 −9 SensitivityAngular of position moment sensor of rotorKs (V Hrad (kg·m)·s ) 5.92.07 10× 10 · − × Torque-voltage coefficient Kv (N·m·V−1) 4.31 × 10−9 Angular position stiffness Kf (N·m·rad−1) 2.7 × 10−5 Simulink simulation results indicate that the temperature-induced bias drift rate of the MESG Sensitivity of position sensor Ks (V·rad−1) 5.9 × 102 output is 8.7 mV/◦C when the temperature drift coefficient of the capacitive position sensing is set at its measured mean of 0.28 mV/ C. It is clear that the bias drift induced from the position pick-off electronics Simulink simulation results◦ indicate that the temperature-induced bias drift rate of the MESG is virtually four orders of magnitude larger than that from thermal expansion of the rotor structure. output is 8.7 mV/°C when the temperature drift coefficient of the capacitive position sensing is set at its measured mean of 0.28 mV/°C. It is clear that the bias drift induced from the position pick-off electronics is virtually four orders of magnitude larger than that from thermal expansion of the rotor structure.

4. Experimental Results and Bias Drift Compensation

4.1. Experimental Set-Up for the MESG Figure 9a illustrates an experimental set-up of the MESG prototype comprising three printed circuit boards (PCBs) stacked by customized bus connectors. The MESG chip was mounted on the

Sensors 2020, 20, 1799 12 of 19 Sensors 2020, 20, x FOR PEER REVIEW 12 of 19

4.top Experimental PCB along with Results the and capacitive Bias Drift position Compensation sensing and associated suspension electronics. The rotation control electronics and the digital suspension controller were realized in the other two PCBs. 4.1. ExperimentalA temperature Set-Up controlled for the MESG chamber was used to set and maintain constant temperature of the MESGFigure at different9a illustrates settings an experimentalduring temperature set-up of drift the MESGmeasurements. prototype A comprising data acquisition three printedsystem circuit(KEITHLEY boards 2010 (PCBs) Multimeter, stacked by Keithley, customized Shanghai, bus connectors. China) was The used MESG to sample chip was and mounted record the on outputs the top PCBfrom alongthe MESG with and the capacitivethe temperature position sensor. sensing Additionally, and associated a control suspension panel programmed electronics.The by Labview rotation controlon a laptop electronics functioned and theas a digital man-machine suspension interface controller and used were to realized control in the the gyroscopic other two operation PCBs. and record complete experiment data.

Vacuum packaged MESG

Capacitive position sensing Rotor speed signal and suspension control Phase A active Rotation control circuit Phase B active DSP-based suspension Phase C active control circuit

（a） （b） Figure 9. (a) Experimental set-up of the MESG and (b) the rotor drive waveform. Figure 9. (a) Experimental set-up of the MESG and (b) the rotor drive waveform. A temperature controlled chamber was used to set and maintain constant temperature of the MESG The rotation of the suspended rotor is based on the principle of a three-phase variable- at different settings during temperature drift measurements. a data acquisition system (KEITHLEY capacitance electrostatic motor. The 14-pole ring-shaped rotor is misaligned to the 12-pole stator 2010 Multimeter, Keithley, Shanghai, China) was used to sample and record the outputs from the electrodes so that once a drive voltage is applied on the rotation electrodes of one phase, the rotor MESG and the temperature sensor. Additionally, a control panel programmed by Labview on a laptop will be forced to realign to the energized electrodes [16]. To rotate the rotor continuously, the functioned as a man-machine interface and used to control the gyroscopic operation and record controlled drive voltages are applied with a fixed duty ratio in a phase commutation order of Phases complete experiment data. A, B and C, as shown in Figure 9b. The rotation of the suspended rotor is based on the principle of a three-phase variable-capacitance In this section, the experimental results for the scale factor, temperature-induced bias drift, long- electrostatic motor. The 14-pole ring-shaped rotor is misaligned to the 12-pole stator electrodes so term bias stability and temperature drift compensation of the MESG prototype will be presented and that once a drive voltage is applied on the rotation electrodes of one phase, the rotor will be forced to discussed. realign to the energized electrodes [16]. To rotate the rotor continuously, the controlled drive voltages are applied with a fixed duty ratio in a phase commutation order of Phases A, B and C, as shown in 4.2. Scale Factor Figure9b. InEquation this section, (2) indicates the experimental that the scale results factor foris linearly the scale proportional factor, temperature-induced to the rotor speed. To bias test drift, the long-termscale factor, bias the stability MESG was and operated temperature at various drift compensation speed settings of of the 1.0 MESG × 104, prototype 1.3 × 104 and will 1.5 be × presented 104 rpm. andThe discussed.angular rate input was applied by a rate turntable while the MESG output was sampled by the data acquisition system at 1 Hz. 4.2. ScaleFigure Factor 10 indicates that the measured scale factor for the MESG operating at different rotor speedsEquation varying (2) from indicates 10,000 thatrpm theto 15,000 scale factorrpm. Theore is linearlyticalproportional and experimental to the results rotor speed.of the scale To test factor the scaleat different factor, speeds the MESG are waslisted operated in Table at 3 various for comparis speedon. settings It is clear of 1.0 that10 the4, 1.3experimental104 and 1.5 results10 4agreerpm. × × × Thewell angularwith the rate theoretical input was predication applied byby aEquation rate turntable (2). It is while interesting the MESG to calculate output wasthe nonlinearity sampled by and the dataasymmetry acquisition of the system scale atfactor 1 Hz. for performance evaluation of the MESG prototype. Figure 10 shows that theFigure measured 10 indicates nonlinearity that the measuredand asymmetry scale factor at 15,000 for the rpm MESG are 0.240% operating and at 0.586% different with rotor an speeds input varyingrange of from± 100°/s, 10,000 respectively. rpm to 15,000 rpm. Theoretical and experimental results of the scale factor at different speeds are listed in Table3 for comparison. It is clear that the experimental results agree well with the theoretical predication by Equation (2). It is interesting to calculate the nonlinearity and asymmetry of the scale factor for performance evaluation of the MESG prototype. Figure 10 shows that the measured nonlinearity and asymmetry at 15,000 rpm are 0.240% and 0.586% with an input range of 100 /s, respectively. ± ◦

SensorsSensors2020 2020, ,20 20,, 1799 x FOR PEER REVIEW 13 ofof 1919

4 10000RPM 3 13000RPM

2 15000RPM ） V

（ 1

0 4 -1 2

-2 0 Gyroscope Output Gyroscope 15000rpm Kf+ =39.10 mV/(°/s) -2 -3 Kf =39.23 mV/(°/s) Kf- =39.33 mV/(°/s) -4 -100 -50 0 50 100 -4 -100 -80 -60 -40 -20 0 20 40 60 80 100 Angular rate (°/s)

FigureFigure 10.10. Angular rate input-output characterizationcharacterization ofof MESGMESG withwith didifffferenterent rotorrotor speeds.speeds.

Table 3. Comparison between theoretical and experimental scale factors at different rotor speeds. Table 3. Comparison between theoretical and experimental scale factors at different rotor speeds. Scale Factor (mV/ /s) Rotor Speed Scale Factor (mV/◦ °/s) Rotor Speed Theoretical Experimental Nonlinearity Asymmetry Theoretical Experimental Nonlinearity Asymmetry 1.0 104 rpm 26.1 25.7 0.236% 1.980% 1.0 ×× 104 rpm 26.1 25.7 0.236% 1.980% 1.3 104 rpm 33.9 33.9 0.120% 0.678% × 1.31.5 × 101044 rpmrpm 33.939.0 39.2 33.9 0.240% 0.120% 0.586% 0.678% × 1.5 × 104 rpm 39.0 39.2 0.240% 0.586% 4.3. Bias Drift 4.3. Bias Drift The bias stability is the most crucial index for performance evaluation of such gyroscopes for inertialThe bias navigation. stability is The the biasmost drift crucial is highly index dependentfor performance on temperature evaluation variationof such gyroscopes of the MESG for operationinertial navigation. [31]. The bias drift is highly dependent on temperature variation of the MESG operation [31]. 4.3.1. Temperature-Dependent Bias Drift 4.3.1. Temperature-Dependent Bias Drift To test the temperature-dependent bias drift, the MESG prototype operated at 10,000 rpm was put intoTo thetest temperature-controlledthe temperature-dependent test chamber bias drift, with the aMESG working prototype temperature operated ranging at 10,000 from rpm 40 ◦C was to 60put◦C, into respectively, the temperature-controlled in 10 ◦C steps, as test shown chamber in Figure with a11 working. a platinum temperature resistance ranging thermometer from 40 °C was to embedded60 °C, respectively, inside the in MESG 10 °C package steps, as and shown utilized in Figure to sense 11. precisely A platinum the real-timeresistance temperature thermometer of was the MESGembedded device. inside The the measurement MESG package duration and utilized at each to temperature sense precisely point the was real-time lasted fortemperature nearly 2.5 of h the to ensureMESG thedevice. temperature The measurement sensor response duration would at each be close temperature to the rotor point temperature. was lasted Figure for nearly 12 shows 2.5 h the to biasensure voltage the temperature of the MESG sensor output response varies with would the be working close to temperature.the rotor temperature. At each temperature Figure 12 shows setting, the thebias MESG voltage output of the increases MESG output rapidly varies as the with temperature the working increases temperature. initially; theAt each output temperature voltage decreases setting, muchthe MESG more slowlyoutput after increases the temperature rapidly as becomes the temper nearlyature constant. increases initially; the output voltage decreases much more slowly after the temperature becomes nearly constant.

FigureFigure 11.11. Set-upSet-up ofof thethe temperaturetemperature driftdrift testing.testing.

Sensors 2020, 20, 1799 14 of 19 SensorsSensors 2020 2020, 20, 20, x, xFOR FOR PEER PEER REVIEW REVIEW 1414 of of 19 19 ） ） GyroscopeGyroscope Output Output varying varying with with Temperature Temperature V V （ （ 1.81.8

1.61.6 1.41.4

1.21.2 Gyroscope Output Gyroscope Output 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 TimeTime (h) (h) (a)(a) ℃ 60℃ 60 （） （）

4040

2020 Temperature Temperature 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 TimeTime (h) (h) (b)(b)

FigureFigure 12. 12. (a ()a Gyroscope ) Gyroscope output output along along h -axis-axish-axis andand and ((b ())b temperaturetemperature) temperature responseresponse response varyingvarying varying withwith with time.time. time. As can be seen from Figure 12, the angular rate measurement of the MESG is relatively sensitive AsAs can can be be seen seen from from Figure Figure 12, 12, the the angular angular rate rate measurement measurement of of the the MESG MESG is is relatively relatively sensitive sensitive to variation of the environmental temperature. It was found that the temperature-induced bias drift toto variation variation of of the the environmental environmental temperature. temperature. It It was was found found that that the the temperature-induced temperature-induced bias bias drift drift rate was 12.48 mV/ C (0.49 s 1/ C) from the data in Figure 12. raterate was was 12.48 12.48 mV/°C mV/°C◦ (0.49 (0.49 ◦°s °s−−1−/1°C)◦/°C) from from the the data data in in Figure Figure 12. 12. From the Simulink simulation results in Section 3.4, the temperature drift of gyroscope bias is FromFrom the the Simulink Simulink simulation simulation results results in in Section Section 3.4, 3.4, the the temperature temperature drift drift of of gyroscope gyroscope bias bias is is 8.7 mV/ C. The discrepancy between this experimental results and the simulation results is about 30%, 8.78.7 mV/°C. mV/°C.◦ The The discrepancy discrepancy between between this this experiment experimentalal results results and and the the simulation simulation results results is is about about which indicates the temperature drift of the MESG is mainly resulted from thermal stability of the 30%,30%, which which indicates indicates the the temperature temperature drift drift of of the the MESG MESG is is mainly mainly resulted resulted from from thermal thermal stability stability of of position sensing circuit as well. thethe position position sensing sensing circuit circuit as as well. well. 4.3.2. Long-Term Bias Stability of the MESG 4.3.2.4.3.2. Long-Term Long-Term Bias Bias Stability Stability of of the the MESG MESG To evaluate the long-term bias stability, the MESG prototype was also placed in the ToTo evaluate evaluate the the long-term long-term bias bias stability, stability, the the ME MESGSG prototype prototype was was also also placed placed in in the the temperature- temperature- temperature-controlled test chamber to maintain a constant temperature environment, which attenuated controlledcontrolled test test chamber chamber to to maintain maintain a aconstant constant temperature temperature environment, environment, which which attenuated attenuated greatly greatly greatly the drift from ambient temperature variations. thethe drift drift from from ambient ambient temperature temperature variations. variations. At a temperature setting of 51 ◦C, the outputs from the gyroscope and thermometer were sampled AtAt a atemperature temperature setting setting of of 51 51 °C, °C, the the outputs outputs from from the the gyroscope gyroscope and and thermometer thermometer were were by the data acquisition system over 70 h, as shown in Figure 13. sampledsampled by by the the data data acquisition acquisition system system over over 70 70 h, h, as as shown shown in in Figure Figure 13. 13. Gyroscope Output varying with Temperature C) Gyroscope Output varying with Temperature C) ° ° 51.151.1 51.0551.05 5151 50.9550.95 0 10 20 30 40 50 60 70 Temperature ( Temperature 0 10 20 30 40 50 60 70 Temperature ( Temperature TimeTime (h) (h) (a)(a) 1.621.62 1.61.6 1.581.58 1.561.56 0 0 1010 2020 3030 4040 5050 6060 7070 TimeTime (h) (h) (b)(b) 0.010.01 0 0 Gyroscope Output (V) Output Gyroscope Gyroscope Output (V) Output Gyroscope -0.01-0.01 0 0 1010 2020 3030 4040 5050 6060 7070 TimeTime (h) (h) (c)(c)

FigureFigureFigure 13. 13.13.Long-term Long-termLong-term bias bias stabilitybias stabilitystability in a temperature-controlled inin a a temptemperature-controllederature-controlled environment: environment:environment: (a) working ( temperature;a()a ) workingworking (temperature;btemperature;) raw drift of ( b the()b )raw MESGraw drift drift bias of of the and the MESG (cMESG) residual bias bias and drift and ( compensatedc()c )residual residual drift drift by compensated removing compensated the by slowly by removing removing changing the the timeslowlyslowly drift. changing changing time time drift. drift.

AfterAfterAfter the the temperature temperature responseresponse response insideinside inside thethe the chamberchamber chamber reachesreaches reaches itsits its steady steady state, state, the the overall overall gyroscope gyroscope outputoutputoutput drifts drifts down down slowly slowly slowly with withwith time. time.time. Considerin Considerin Consideringg gan an an extremely extremely extremely small small small temperature temperature temperature disturbance, disturbance, disturbance, as as asshownshown shown in in inFigure Figure Figure 13a, 13a,13 a,and and and the the temperature-dependent temperature-dependent bias bias drift driftdrift in inin Section SectionSection 4.3.1, 4.3.14.3.1, ,it itit is is is clear clear that that thethe the time-dependenttime-dependenttime-dependent drift driftdrift dominates dominatesdominates the long-term thethe long-termlong-term bias stability bibiasas stability instability the temperature inin thethe temperature controlledtemperature environment. controlledcontrolled Figureenvironment.environment. 13b indicates Figure Figure that13b 13b indicates the indicates MESG that that exhibits the the MESG MESG a time-dependent exhibits exhibits a atime-dependent time-dependent bias drift coe biasffi biascient drift drift of coefficient 0.737coefficient mV /ofh, of as0.7370.737 obtained mV/h, mV/h, byas as least-squareobtained obtained by by fittingleast-square least-square the measured fitting fitting the data.the measured measured data. data.

Sensors 2020, 20, x FOR PEER REVIEW 15 of 19 Sensors 2020, 20, 1799 15 of 19 After compensating the time-dependent drift of the MESG output, Figure 13c shows that he residual bias drift is closely correlated with the temperature variation of the MESG. After removing the timeAfter drift, compensating the correlation the coefficient time-dependent between drift the of temperature the MESG and output, gyroscope Figure output13c shows is 0.6596. that he residualFurthermore, bias drift is Figure closely 13b correlated shows that with a theslowly temperature changing variation time drift of can the last MESG. over After 70 h. removingBased on the analyticaltime drift, result the correlation of the thermal coefficient time betweenconstant, the 35.6 temperature min, given andin Section gyroscope 3.2, outputit seems is that 0.6596. this long- time Furthermore,drift source Figureexcludes 13 bthe shows possibility that a slowly of the changing rotor thermal time drift balance can last response. over 70 Therefore, h. Based on it the is worthwhileanalytical result to investigate of the thermal which time part constant, of the 35.6 MESG min, prototype given in Section is the 3.2dominant, it seems source that this of long-timethe time- dependentdrift source bias excludes drift. the possibility of the rotor thermal balance response. Therefore, it is worthwhile to investigateThis time which the part output of the time MESG drift prototype of the position is the dominant sensing sourcechannel of theitself time-dependent was tested firstly. bias drift.The positionThis sensing time the PCB output was time put driftinto the of the constant position temperature sensing channel test chamber itself was with tested a setting firstly. of The 50 position°C and testedsensing for PCB a duration was put of into 46 h. the The constant experimental temperature results test of two-channel chamber with position a setting sensing of 50 output◦C and for tested the hfor-axis a duration measurement of 46 h. are The shown experimental in Figure results 14. of two-channel position sensing output for the h-axis measurement are shown in Figure 14.

-1.36 -5.3

-1.365 -5.4 -1.37 Channel 1 Channel Channel 1 Voltage (V) Voltage (V) -5.5 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45

-1.325 -0.68 -1.33 -0.7 -1.335

Channel 2 Channel -0.72 Channel 2 Channel Voltage (V)

-1.34 Voltage (V) 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 C) C) ° ° 50.35 50.4 50.3 50.3 50.25 50.2 50.2

Temperature ( Temperature 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 Temperature ( Time (h) Time (h)

(a) (b)

Figure 14. Long-term output drift of position sensing circuit at 50 °C: (a) two-channel outputs with Figure 14. Long-term output drift of position sensing circuit at 50 ◦C: (a) two-channel outputs with similar magnitude and (b) twotwo channelschannels withwith veryvery didifferentfferentoutputs. outputs.

Figure 1414aa indicatesindicates that that the the drift drift rates rates of of Channel Channel 1 and 1 and Channel Channel 2 with 2 with time time are 55.2 are and55.2 75.8 andµ 75.8V/h, μrespectively.V/h, respectively. Moreover, Moreover, the simulation the simulation result result of the of rebalance the rebalance loop indicatesloop indicates that thethat gyroscope the gyroscope bias biasdrift drift of the ofh theaxis h generatedaxis generated by the by sensing the sensing circuit circuit of Channel of Channel 1 and 1 Channeland Channel 2 in Figure2 in Figure 14a is 14a only is only0.642 −0.642 mV/h. mV/h. The discrepancy The discrepancy between between the experimental the experimental drift rate drift0.737 rate mV −/0.737h and mV/h the simulation and the − − simulationresult is only result 12.89%. is only It is clear12.89%. that It the is long-termclear that biasthe driftlong-term of the MESGbias drift is mostly of the contributed MESG is mostly by the positioncontributed sensing by the circuit. position sensing circuit. To compare the eeffectsffects ofof didifferentfferent outputoutput voltagesvoltages onon timetime drift,drift, the amplitudeamplitude of the position sensing output is adjusted to to be larger and smaller relative relative to to its its operating operating voltage, voltage, respectively. respectively. Figure 1414bb indicates that the measured drift rates of the two adjusted channels with very different different output biases, Channel 1 and Channel 2, are 839 and 36 µμVV/h,/h, respectively. By comparing the drift rates ofof thesethese sensingsensing channels channels in in Figure Figure 14 14a,b,a,b, it canit can be be found found that that the the larger larger of the of sensingthe sensing circuit circuit bias biasis, the is, higher the higher the drift the ratedrift will rate be. will Therefore, be. Therefore, an eff ectivean effective method method to reduce to reduce the long-term the long-term time drift time of driftthe MESG of the biasMESG is to bias nullify is to thenullify bias the of thebias position of the position sensing sensing circuit by circuit adjusting by adjusting the capacitive the capacitive sensing sensingzero of thezero suspended of the suspended rotor. rotor.

4.4. Compensation of of Bias Bias Drift Drift Using Using a BP a NeuralBP Neural Network Network The experimental results show that the bias drift of the MESG is both temperature and time dependent. InIn orderorder to to attenuate attenuate the the drift drift caused caused by theby temperaturethe temperature and timeand concurrently,time concurrently, a BP neural a BP neuralnetwork network was used was to modelused andto model compensate and compensate for the bias driftfor the of thebias MESG. drift Bothof the the MESG. temperature Both andthe temperaturetime data were and directly time data fed intowere the directly input fed layer into of the input BP neural layer network, of the BP while neural the network, gyroscope while output the gyroscopedata were inputoutput into data the were output input layer. into the output layer.

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A single hidden layer neural network which includes the input layer, hidden layer and output A single hidden layer neural network which includes the input layer, hidden layer and output layer nodes was used in this work. In the MATLAB neural network toolbox, the neural net fitting layer nodes was used in this work. In the MATLAB neural network toolbox, the neural net fitting was used to create the BP neural network. The Levenberg–Marquardt algorithm was chosen for the was used to create the BP neural network. The Levenberg–Marquardt algorithm was chosen for the network training and prediction. In the following compensation experiment, the experimental drift network training and prediction. In the following compensation experiment, the experimental drift data were divided into a training set (the first 70% of the gyroscopic output data), a validation set data were divided into a training set (the first 70% of the gyroscopic output data), a validation set (the following 15% of the data) and a test set (the last 15% of the data). (the following 15% of the data) and a test set (the last 15% of the data). After compensation with the BP neural network model, Figure 15 shows that the bias stability After compensation with the BP neural network model, Figure 15 shows that the bias stability of of the MESG improves from 0.56 °/s to 0.027 °/s for 70 h and from 0.022 °/s to 0.018 °/s for 1 h data the MESG improves from 0.56 ◦/s to 0.027 ◦/s for 70 h and from 0.022 ◦/s to 0.018 ◦/s for 1 h data from from 23–24 h, respectively. Clearly, the compensation can attenuate the long-term time drift greatly 23–24 h, respectively. Clearly, the compensation can attenuate the long-term time drift greatly while the while the short-term 1 h drift is reduced by only 18% due to relatively large noise. In addition, the short-term 1 h drift is reduced by only 18% due to relatively large noise. In addition, the compensation compensation method is also evaluated in a temperature-uncontrolled environment. The method is also evaluated in a temperature-uncontrolled environment. The experimental result experimental result shows that the bias stability of the MESG improves from 1.2774 °/s to 0.0526 °/s shows that the bias stability of the MESG improves from 1.2774 ◦/s to 0.0526 ◦/s for 24 h. It is clear for 24 h. It is clear that the long-term bias drift can be modeled and compensated effectively to that the long-term bias drift can be modeled and compensated effectively to improve the MESG improve the MESG performance greatly. performance greatly.

Gyroscope Output before Compensation 1.62

1.6

1.58

0 10 20 30 40 50 60 70 Time（h）(a) Gyroscope Output after Compensation 0.01 Gyroscope Output (V） 0

-0.01 0 10 20 30 40 50 60 70 Time（h）(b) Figure 15. a b Figure 15.Comparison Comparison ofof thethe bias bias drift: drift: ((a)) biasbiasbefore before compensation compensation and and ( ()b) bias bias after after being being compensated by the BP neural network. compensated by the BP neural network. The Allan deviation analysis with 1-h MESG output data are shown in Figure 16, with and without The Allan deviation analysis with 1-h MESG output data are shown in Figure 16, with and drift compensation respectively. By comparing these results, it is indicated that the bias instability without drift compensation respectively. By comparing these results, it is indicated that the bias is improved from 29.5 /h to 26.5 /h after compensation. This implies that the compensation can instability is improved◦ from 29.5◦ °/h to 26.5 °/h after compensation. This implies that the improve the short-term bias performance, but not as much as the long-term result, as discussed above. compensation can improve the short-term bias performance, but not as much as the long-term result, In addition, the angle random walk is basically the same because it is a representation of the output as discussed above. In addition, the angle random walk is basically the same because it is a white noise which can not be reduced by the compensation method. As discussed in Section 2.3, representation of the output white noise which can not be reduced by the compensation method. As higher vacuum level is an effective solution to reduce the output noise of the MESG. discussed in Section 2.3, higher vacuum level is an effective solution to reduce the output noise of the MESG.

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3 10 before compensation after compensation

Angle random walk: 2.61°/√h 2 10 2.6207°/√h Allan deviation (°/h) Allan deviation Bias instability: 29.5°/h 26.5°/h 1 10 0 1 2 3 4 10 10 10 10 10 Time (s)

FigureFigure 16. 16.Allan Allan deviation deviation before before and and after after bias bias compensation. compensation.

5. Conclusions 5. Conclusions The temperature drift characteristics, long-term drift measurements and compensation of The temperature drift characteristics, long-term drift measurements and compensation of a a spinning-rotor MESG have been investigated in this paper. The experimental result indicates spinning-rotor MESG have been investigated in this paper. The experimental result indicates that the that the rotation drive voltage in a form of pulse waveform is a major noise source of the MESG, rotation drive voltage in a form of pulse waveform is a major noise source of the MESG, which can which can be attenuated by packaging the device under higher vacuum to permit ultra-low drive be attenuated by packaging the device under higher vacuum to permit ultra-low drive voltage voltage amplitude. The temperature-dependent characteristics of the rotor structure and capacitive amplitude. The temperature-dependent characteristics of the rotor structure and capacitive position position sensing circuit are analyzed to evaluate the bias drift of the MESG. It was found that the bias sensing circuit are analyzed to evaluate the bias drift of the MESG. It was found that the bias drift drift induced from the position pick-off electronics was virtually four orders of magnitude larger than induced from the position pick-off electronics was virtually four orders of magnitude larger than that that from thermal expansion of the rotor structure. The experimental result shows that the MESG bias from thermal expansion of the rotor structure. The experimental result shows that the MESG bias is is relatively sensitive to the temperature variation with a drift rate of 12.48 mV/ C, which is mainly relatively sensitive to the temperature variation with a drift rate of 12.48 mV/°C,◦ which is mainly a a result of the thermal stability of the position sensing circuit, as predicated theoretically. The long-term result of the thermal stability of the position sensing circuit, as predicated theoretically. The long- bias test results show that both the temperature and time dependent bias drifts exist in the MESG term bias test results show that both the temperature and time dependent bias drifts exist in the prototype.MESG prototype. The long-term The long-term bias drift bias has drift been has modeled been modeled and compensated and compensated effectively effectively by the BP by neural the BP networkneural network model tomodel improve to improve significantly significantly the bias the stability bias stability of the MESG. of the MESG. The future The workfuture will work focus will onfocus reduction on reduction of the outputof the output noise by noise packaging by packag theing device the device in higher in higher vacuum vacuum and improvement and improvement of the long-termof the long-term stability stability by optimizing by optimizing the capacitive the capacitive position position sensing sensing electronics. electronics. Author Contributions: S.W. contributed the modeling and simulation; S.W. and F.H. performed the experiments andAuthor data Contributions: analysis, and designed S.W. contributed the interface the modeling electronics. and S.W. simulation; and F.H. S.W. jointly and contributed F.H. performed to writing the experiments this paper. Alland authors data analysis, have read and and designed agreed the to theinte publishedrface electronics. version S.W. of the an manuscript.d F.H. jointly contributed to writing this paper. Funding:Funding:This This research research was was funded funded by by [National [National Natural Natural Science Science Foundation Foundation of of China] China] grant grant number number [41774189 [41774189 andand 61374207]. 61374207]. The The APC APC was was funded funded by by [41774189]. [41774189]. Conflicts of Interest: The authors declare no conflict of interest. Conflicts of Interest: The authors declare no conflict of interest.

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