Micromachined electrostatically suspended gyroscope with a spinning

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Micromachined electrostatically suspended gyroscope with a spinning ring-shaped rotor

This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2012 J. Micromech. Microeng. 22 105032 (http://iopscience.iop.org/0960-1317/22/10/105032) View the table of contents for this issue, or go to the journal homepage for more

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IOP PUBLISHING

JOURNAL OF MICROMECHANICS AND MICROENGINEERING

doi:10.1088/0960-1317/22/10/105032

J. Micromech. Microeng. 22 (2012) 105032 (9pp)

Micromachined electrostatically suspended gyroscope with a spinning ring-shaped rotor F T Han 1 , Y F Liu 1 , L Wang 1 and G Y Ma 2 1

Department of Precision Instruments and Mechanology, Tsinghua University, Beijing 100084, People’s Republic of China 2 No. 9 Academy, China Aerospace Science and Technology Corporation, Beijing 100854, People’s Republic of China E-mail: [email protected]

Received 19 June 2012, in final form 2 August 2012 Published 10 September 2012 Online at stacks.iop.org/JMM/22/105032 Abstract A micromachined electrostatically suspended gyroscope is described in this paper, in which a spinning ring-shaped rotor is suspended by an electric bearing in five degrees of freedom and driven by a three-phase variable-capacitance motor. The electric bearing provides contactless suspension of the spinning rotor, allowing the rotor through a torque-rebalance loop to precess about two input axes that are orthogonal to the spin axis. In this way, the micromachined spinning-rotor gyroscope can be used as a two-degree-of-freedom angular rate sensor by detecting the precession-induced torque. Design and simulation of the dual-axis torque-rebalance loop, by considering actual negative spring effect in rotor dynamics, are presented to investigate the loop stability and explain the experimental measurement. The prototype device has been fabricated by bulk micromachining technique and tested successfully with a suspended rotor spinning at a speed of 10 085 rpm. Initial measurements of the rate gyroscope shows that an input range of ± 100◦ s−1, a noise floor of 0.015◦ s−1 Hz−1/2, and a bias stability of 50.95◦ h−1 have been achieved. The detailed results of the prototype device, electric bearing and motor spin-up are also described. (Some figures may appear in colour only in the online journal)

1. Introduction

Likewise, the micromachined electrostatically suspended gyroscope (MESG) is one of the most promising candidates toward high performance, low cost and miniaturized inertial sensors [12, 13]. It usually consists of a spinning rotor suspended in a vacuum cavity and several sets of electrodes that are symmetrically arranged on the inner surface of the cavity. Electrostatic bearing is employed to suspend the rotor, eliminating the mechanical bearing and thus friction effects between the rotor and associated substrate. Such a device has considerable advantages over existing approaches, including increased independence of fabrication tolerances, online characteristics adjustment and in particular, compared to vibratory rate gyroscopes, the intrinsic absence of mode coupling error and the need for resonant mode tuning [14, 15]. Early development of a MEMS rate gyroscope employing electrostatic bearing was reported using a surface

In order to enhance the performance of MEMS gyroscopes, various gyroscope designs relying on structure optimization, improved microfabrication techniques and control electronics have been proposed [1], among which, a micro counterpart of the conventional electrostatically suspended gyroscope (ESG) has recently become of interest to MEMS gyro designers [2–8]. The ESG employs electrostatic forces to suspend a spherical rotor in order to isolate the spinning rotor from unwanted mechanical friction [9, 10], so that the long-term stability of the gyroscope is improved greatly with proven strategic-grade capability. However, such macro-scale ESGs are too expensive and bulky so that they are found merely in some highend applications with the most severe accuracy requirements such as submarines navigation and space missions [9, 11]. 0960-1317/12/105032+09$33.00

1

© 2012 IOP Publishing Ltd

Printed in the UK & the USA

J. Micromech. Microeng. 22 (2012) 105032

F T Han et al

micromachining technique by SatCon Technology Co. (USA) [2]; however, the sensor failed to operate due to charged induced adhesion. Recent reports from Tokimec, Inc. (Japan) have demonstrated the potential performance of a spinningrotor MESG [5, 12]. Such a device, fabricated using bulk micromachining technique and packaged in a vacuum cavity to prevent viscous damping, has improved accuracy compared to vibratory types and is coming into use in a compass system for small vessels [13]. Currently, most of the published work in the literature focuses on the gyro design [2, 15], fabrication techniques [5, 7, 8], capacitive sensing [16, 17], suspension and drive of the rotor [6, 7, 18]. However, very little work has been reported on the performance test of such a device as rate gyroscope due to the several combined techniques required, such as challenging microfabrication and vacuum package, sophisticated suspension of the rotor in five degrees of freedom (DOFs) and spin-up of the rotor over 104 rpm. Our initial measurements on the suspension and rotation of such a device for use as a micromotor have been reported with an attainable spin speed of only 73.3 rpm at atmospheric pressure [6, 18]. One of the most promising applications is a rate gyroscope where the sensitivity is proportional to the spin speed of the rotor [3]. Therefore, it is necessary to package the device in a vacuum so as to minimize the viscous drag effect and yield much higher spin speed. The purpose of this paper is to examine the design and performance of a prototype device operated inside a vacuum chamber where the rotor is spun at a much higher speed of 10 085 rpm. The experimental results of electrostatic bearing, motor drive and rate gyroscope will be presented. Design and simulation of the dual-axis torquerebalance loop, by considering actual negative spring effect in rotor dynamics, are also introduced to explain the experimental gyro performance.

H

O

Figure 1. Rotor spinning in an inertial space.

Based on the conservation of angular momentum, the MESG operates as a two-DOF angular rate sensor and can be expressed using the following basic equations [19]: ωy = Mx /H,

ωx = −My /H,

(2)

where Mx and My are the applied precession torques, ωx and ωy are two angular rate inputs that are orthogonal to the spin axis and can be measured simultaneously. 2.2. Overview of the MESG The device consists of a glass/silicon/glass triple stack structure with a complex electrode pattern depicted in figure 2. The essential geometry comprises a spinning ring-shaped rotor surrounded by sets of sense, suspension and spin electrodes [18]. The rotor was designed to have an outer radius of 2.0 mm, an inner radius of 1.73 mm and a thickness of 68 μm. An electric bearing system is employed to suspend the spinning rotor electrostatically in five DOFs so that a virtually frictionless spin bearing is provided to permit free rotation of the rotor about the spin axis of the gyroscope. Rotor spinning is based on the principle of a planar variable-capacitance motor [6] and controlled by the outermost spin electrodes located symmetrically above and underneath the rotor. The step angle in our design is π /21 (8.57◦ ). Bulk microfabrication-based glass/silicon/glass bonding was used to assemble this device [6]. Note that the device, here for use as a spinning-rotor gyroscope, must be evacuated to high degree of vacuum so as to eliminate the effect of air drag and permit the highest possible spin speed.

2. Description of the MESG 2.1. Principle of operation of the MESG The MESG consists of a spinning rotor and associated stators that maintain the rotor at its null position by electric bearing. The rotor is driven at a constant speed by an electrostatic motor which is attached to the gyro case. When an angular rotation orthogonal to the spin axis is applied, a precession torque generated by the rebalance loop returns the rotor to the null position. The magnitude and direction of the restraining torque would be a measure of the input rate to the gyroscope. A schematic diagram of a rotor spinning in an inertial space is shown in figure 1 where the rotor spins with a constant angular momentum, H. The precession of the rotor is governed by the equation: M = ω × H,

ω

M

3. Design of the rebalance loop This section introduces a dual-axis rebalance loop for the twoDOF MESG intended for a strapdown system. The torquerebalance loops control the rotor, which is an inertial sensing element, so that the device may be operated at pickoff null [19, 20]. It is used to control the gyroscope over its dynamic range and provide a precise output measurement of two-axis inertial angular rates applied to the gyro case.

(1)

where M is a torque applied to the rotor and ω is the precession rate of the rotor with respect to inertial space. If no torque is applied to the rotor, it will maintain its orientation in the inertial space. However, if a torque perpendicular to the spin axis is applied to the rotor, the rotor will precess at the rate ω about an axis perpendicular to the spin axis and the axis of the applied torque.

3.1. Motion equations of the gyro rotor The motion equation of the rotor is first introduced for the following analysis and design. Here, two coordinate frames are defined [19]. The c-frame is fixed in the gyro case. The 2


ϕy

ϕz Rotor spin electrodes

O

F T Han et al

zn

y

ϕx

x

θy

θx

zc

Electrodes for radial suspension

ϕz

Top glass wafer

O Silicon wafer

ϕx xc

Bottom glass wafer

Ring-shaped rotor

ϕy

θx

yn

yc

xn Figure 3. Orientation of the rotor-referenced frame Oxnynzn and the gyro case frame Oxcyczc.

Common excitation Electrodes for axial electrodes suspension

Further, the angular momentum of the spin rotor is ⎡ ⎤ Jrx ωnx ⎦, Jry ωny Hnr = ⎣ (5) Jrz (ωnz + ω0 ) where ω0 is the preselected angular velocity of the rotor relative to the gyro case, which is assumed to be held constant by the electrostatic motor and is much larger than ωnz , and Jrx , Jrx and Jrz are the rotor moments of inertia about the corresponding rotor axes. The moment on the rotor, written in the n-frame, is equal to the rate of change of its angular momentum: di n dn n H = H + ωnin × Hnr . (6) Mnr = dt r dt r Substituting (4) and (5) into (6) yields the following moments on⎡the rotor written in the n-frame: ⎤ Jrx ω˙ nx + (Jrz − Jry )ωny ωnz + Jrz ωny ω0 (7) Mnr = ⎣Jry ω˙ ny + (Jrx − Jrz )ωnx ωnz − Jrz ωnx ω0 ⎦ . Jrz ω˙ nz + (Jry − Jrx )ωnx ωny The moments on the rotor can be expressed in the case fixed coordinate frame as ⎡ ⎤ ⎡ ⎤ n Mrx + θy Mrzn Mx n ⎦ . (8) Mry − θx Mrzn Mcr = ⎣My ⎦ = Ccn Mnr = ⎣ n n n − θy Mrx + θx Mry + Mrz Mz Ideally, the motor can only provide moments to the rotor along the case fixed zc-axis. Moments applied to the rotor along the perpendicular axes, Mx and My , are provided mainly by torquers of the rebalance loop. Expanding these equations will yield the nonlinear motion equations of the rotor as described in [19]. However, these equations of motion can be reasonably linearized by assuming that the products including the pickoff angles,θx and θy , are small on the order of 10−3 rad and retaining only the linear terms of the derivatives of ϕ. Then the linearized equations can be reasonably derived as [21]

Jr θ¨x + H θ˙y = Mx − Jr ϕ¨x − H ϕ˙y , (9) Jr θ¨y − H θ˙x = My − Jr ϕ¨y + H ϕ˙x where H = Jrz ω0 is the angular momentum of the rotor about its zn-axis and assuming the rotor is symmetric so that Jrx = Jry = Jr .

(a)

(b)

Figure 2. Micromachined electrostatically suspended gyros with integrated electric bearing: (a) exploded view of the device and (b) the fabricated device.

n-frame is the rotor-referenced frame but does not spin with the rotor, and the zn-axis is aligned with the rotor spin axis. Figure 3 shows the orientation of the n-frame with respect to the c-frame. When external angular rate inputs ϕ˙x and ϕ˙y with respect to inertial space occur, the instantaneous attitude of the rotor with respect to the case frame is specified by the two pickoff angles θ x and θ y, as defined in figure 3. The angular velocity of the n-frame with respect to inertial space written in the n-frame is (3) ωnin = ωnic + ωncn = Cnc ωcic + ωncn , T T where ωcic = ϕ˙x ϕ˙y ϕ˙z , ωncn = θ˙x θ˙y 0 and the direction cosine matrix Cnc , for small θ x and θ y, can be simplified as ⎡ ⎤ 1 0 −θy 1 θx ⎦ . Cnc = ⎣ 0 θy −θx 1 Then, substituting the above quantities into (3) yields ⎤ ⎡ ⎤ ⎡ ϕ˙x + θ˙x − θy ϕ˙z ωnx ωnin = ⎣ωny ⎦ = ⎣ ϕ˙y + θ˙y + θy ϕ˙z ⎦ . ωnz ϕ˙z− θx ϕ˙y + θy ϕ˙x

θy

θy

θx

(4)

3


F T Han et al

The linearized equations of motion in (9) can be expressed in transfer function form by means of the Laplace transform: 1 ϕ˙x (s) θx (s) Mx (s) = G(s) − , θy (s) My (s) s ϕ˙y (s)

Gf ( s)

ϕx

(10)

1/ s

where G11 (s) G12 (s) G(s) = G11 (s) G12 (s) ⎡ ⎤ Jr −H ⎢ Jr2 s2 + H 2 s Jr2 s2 + H 2 ⎥ ⎥. =⎢ ⎣ ⎦ Jr H 2 2 2 2 2 2 s(Jr s + H ) Jr s + H

Ks

Gc ( s )

Vy

KT

−+

Mx

+

ϕy 1/ s

− +

θy

Ks

Gc ( s)

G11 ( s)

++

G12 ( s)

Ka

G21 ( s )

Vx

KT

Gf ( s)

+ − +

My M d, y

G22 ( s)

++

ux

Figure 4. Schematic diagram of the dual-axis gyro rebalance loop.

Examination of figure 4 reveals that the dual-axis rebalance loop is comprised of a pair of cross-axis loops and a pair of direct-axis loops [20]. A cross-axis loop is one which derives its input from a pickoff sensor which senses angular motion about one axis and delivers its output voltages to the suspension electrodes which apply a torque about the same axis. On the other hand, a direct-axis loop is one which derives its input from a pickoff sensor which senses angular motion about one axis and delivers its output voltages to the suspension electrodes which apply a torque about the other axis. In a gyroscope of the torque-rebalance type, the precession along the xc axis is produced by a torque along the yc axis as described in (11).

3.2. Design of the loop compensators The rebalance loop is mainly composed of capacitive pickoff sensors, loop compensators and electrostatic torquers. In the rate-sensing mode, when an angular rate about an input axis is applied to a moving vehicle where the gyro case is attached, the resulting angular offset between the rotor spin axis and the gyro case is measured by the capacitive sensing electronics and is called the pickoff angle. This signal is then fed to the servo controller which generates a control torque to be applied to the rotor in such a way as to precess the rotor at a rate equal to the inertial angular rate applied to the gyro case, thus driving the pickoff angle back to null. Thus, the torque-rebalance loop is used to keep the rotor spin axis perpendicular to the gyro input axes, keeping the pickoff angles nulled, and its output, the control torque, is used as a measurement of the inertial angular rates to which the gyro case is subjected. A governing equation for the electrostatic torquers, where small pickoff angles are considered, can be linearized as Mc,y = −KTVx + Ka θy ,

M d, x

θx

Ka

When the motor is spinning on the order of 104 rpm, H will play a dominant role in the dynamics of the spinning rotor. In this case, the examination of (10) reveals that two-axis coupling characteristics become dominant for the operation of such two-DOF rate gyroscopes.

Mc,x = −KTVy + Ka θx ,

+ −

uy

3.3. Stability analysis of the rebalance loop The rebalance loop must also have sufficient stability margins so that the loop will remain stable in the presence of modeling errors, pickoff and torquer misalignments, and changes in operating conditions. The complex method was applied to the stability analysis of the two-DOF MESG [19]. The primary advantage of the complex method is that it turns the two-input– two-output real gyro model into a single-input–single-output (SISO) system described by complex coefficient differential equations. This in turn allows classical SISO design tools to be used in the analysis and design of the two-DOF rebalance loop. The complex method is applied to the gyro equations of motion by defining the following complex variables:

(11)

where KT is the torquer gain and Ka the angular position stiffness, Vx and Vy are applied control voltages to generate controllable electrostatic torques. Further, let the unmodeled disturbance moments on actual rotor be Md,x and Md,y ; then, considering the rebalance loop of figure 4, we obtain the total moments applied to the rotor, i.e. Mx = Mc,x + Md,x and My = Mc,y + Md,y . In order to achieve the desired gyro performance, certain compensators are required to be included in the rebalance loop. A block diagram of the closed-loop torque-rebalance loop is shown in figure 4. The loop controller is denoted by Gc(s) and the pickoff sensor is simply modeled as the gain Ks. A pair of low-pass filters denoted by Gf(s) is placed before gyro outputs, ux and uy. The rebalance loop of the MESG is a type of servo control that is inherently unstable unless the loop contains appropriate phase compensators. The final controller design using classical lag–lead compensation [18] is given by (s + 50)(s + 2500) Gc (s) = 40 . (12) (s + 1)(s + 20000)

θc = θx + jθy ,

ϕ˙c = ϕ˙x + jϕ˙y ,

Mc = Mc,x + jMc, y . (13)

Expressing (9) in terms of the complex pickoff angle, complex inertial angular rate, and complex control torque, yields a complex coefficient transfer function description of the two-DOF gyro model: 1 θc (s) = 2 , (14) Mc (s) + Me (s) Jr s − jHs where Me (s) represents the error moments and Me (s) = Md (s) − Jr s2 ϕc (s) + jHsϕc (s). Further, by defining the complex variable Vc = Vx + jVy , the open loop transfer function of the dual-axis rebalance loop 4

J. Micromech. Microeng. 22 (2012) 105032 Me

Gc ( s )

KT Vθ

-5

1 Js 2 − jHs − K a

x10

θc

Ks

4 2

0

θy (rad)

+ −

-7

x10 4

θx (rad)

Vc

θ ref = 0

F T Han et al

-4

0 -2

Figure 5. Block diagram of the rebalance loop with complex models.

-8 0

2

1

-4

3

0

Time (s)

-40 10 -2

0

2

3

2 0

-0.01 -2

Go(-jω) 10 0

10 2

10

-0.02

4

0

1

2

Time (s)

Frequency (Hz) Phase (degree)

uy (V)

ux (V)

Magnitude (dB)

Go(jω) -20

3

4

0.01

20 0

2

1

Time (s)

40

-100

3

-4

0

1

Time (s)

Figure 7. Simulated responses of the rebalance loop to a sinusoidal angular rate 100o s−1 at 1 Hz, two-axis pickoff angles of θ x and θ y and gyro outputs of ux and uy are shown.

-150 -200

positive phase margin of 48.0o, a negative gain margin of 3.20 dB and a positive gain margin of 6.16 dB. These phase and gain margins can be increased by using a high gain controller at a cost of increased noise in gyro outputs.

-250 -300 10 -2

10 0

10 2

10

4

Frequency (Hz)

Figure 6. Open loop bode plot of the lag–lead compensated rebalance loop.

3.4. Simulation results The response of the closed-rebalance loop to a 100o s−1 sinusoidal input in the x-axis, ϕ˙x , is used as a measure of the gyro performance. The time responses of the pickoff angles and gyro output voltages are shown in figure 7. The simulated amplitudes of the sinusoidal pickoff angles, θ x and θ y, are 6.97 × 10−8 and 2.33 × 10−5 rad, respectively, and far below the allowable maximum on the order of 10−3 rad. The gyro output of the y-axis loop is 2.596 V corresponding to a scale factor of 25.96 mV−1 (o s−1). However, the x-axis output is not equal to an expected zero but at an amplitude of 7.78 mV, which corresponds to a cross-axis coupling effect and could be eliminated by decouple control [21]. It is also noted that the system behaves as a lightly damped sinusoidal oscillation at the nutation frequency of approximately 2ω0. It is shown that the nutation amplitudes at the gyro output, ux and uy, are on the order of several hundred microvolts. These small high-frequency components, originated from the gyro nutation dynamics, can be attenuated greatly by including high-Q nutation notch filters in the rebalance loop design [20].

Table 1. Parameters of the gyro rebalance loop. Parameters (symbol)

Value (unit)

Rotor radial moment of inertia (Jr) Sensitivity of angle sensor (Ks) Angular velocity of rotor about spin axis (ω0) Angular moment of rotor (H, about spin axis) Torquer gain (KT) Angular position stiffness (Ka) Gain of low pass filters at gyro outputs

7.17 × 10−13 kg m2 5.9 × 102 V rad−1 1056 (rad s−1) 1.51 × 10−9 kgm2 s−1 1.94 × 10−9 Nm V−1 3.35 × 10−6 Nm rad−1 4.80

can be transformed into a SISO system via a complex method and given is by 1 Go (s) = Ks KT Gc (s) 2 . (15) Jr s − jHs − Ka The resulting feedback loop is shown in figure 5. The signals Vc and Vθ are the control torque input and pickoff angle output of the gyroscope, respectively. The reference input signal, θref , is equal to zero. The transfer function from θref to θc is used as a measure of loop performance, while the control torque to the gyroscope is a measure of the inertial angular rate applied to the gyro case. A complex version of Bode plots of the compensated open loop system, Go(s), is used to determine the stability margins of the loop design with the parameters listed in table 1. The frequency response of the complex system is different for positive and negative frequencies and yields two different Bode plots (see figure 6). The stability margins obtained from the plots indicate a negative phase margin of 20.9o and a

4. Experimental results The experimental system that is used throughout the paper is a micromachined rate gyroscope with a ring-shaped rotor supported by a five-axis electric bearing. Aerodynamic drag from the surrounding air is believed to provide the main limitation of the rotor speed in atmospheric conditions. For example, using a drive voltage of 28.3 V, such a device achieved a maximum speed of only 73.3 rpm in air [6]. Therefore, for a spinning-rotor rate gyroscope where the sensitivity is proportional to the spin speed of the rotor 5


F T Han et al

Magnitude (dB)

10

ω0

0 -10 Free air -20 10 0

High vacuum 10 1

10 2 Frequency (Hz)

10 3

Figure 10. Closed-loop frequency responses of the z-axis electric bearing system. The measured bandwidths are 221.6 and 1820 Hz for the device operated at air and in vacuum, respectively. Figure 8. Vacuum set-up for the test of the MESG prototype. Table 2. Parameters of the electrostatic bearing system. Control unit of the rate table

Temperature test chamber

MESG electronics

Parameters (unit)

z-axis

x-, y-axes

Mass of the rotor (kg) Bias voltage (V) Sensitivity of position sensor (V m−1) Voltage-to-force coefficient (N V−1) Position stiffness (N m−1)

4.1 × 6.0 3.0 × 105 2.45 2.45

10−7 11.2 4.5 × 105 1.19 1.96

4.1. Electrostatic bearing

Vacuum chamber

Rate table

Vacuum pipe

Eliminating viscous drag by operating the device in a vacuum is expected to significantly increase the spinning speed of the rotor. However, operation in vacuum eliminates a convenient way to control the damping in a micromachined device, namely squeeze film damping. Therefore, all suspension compensators should include the phase-lead portion, compared with the devices at atmospheric pressure [18], in order to provide adequate damping force and thus stabilize the bearing system. Accordingly, the performance of the electric bearing for gyro application is very different compared with those micromotors operated in free air [6]. Figure 10 shows the measured frequency responses of the z-axis suspension loop with the bearing parameters listed in table 2. It is clearly shown that the closed-loop responses are quite different for the device operated in a vacuum and at atmospheric pressure. In contrast to the case in air, the rotor has virtually no natural damping in vacuum where the use of phase-lead compensation will extend the suspension bandwidth greatly [18]. The measured closed-loop bandwidths are 1.95, 1.81, 1.82, 1.62 and 1.70 kHz for five loops in the x-, y-, z-, ϕ x- and ϕ y-axes, respectively. Simulation results also show that these suspension loops must be designed to have high bandwidth on the order of kilohertz to satisfy the required suspension stiffness and overload capacity [18].

Vacuum system

Figure 9. Photograph of the MESG test. Note the rate table is mounted inside a temperature test chamber which does not work in our test.

typically over 104 rpm, the electrode cavity in which the rotor rotates must be evacuated to as high a degree of vacuum as possible to minimize viscous drag effect and yield high spin speed. In our test, both the device and the capacitive sensing electronics were fixed inside a small vacuum chamber, while the suspension and drive control electronics are mounted outside. The vacuum chamber can be maintained at high vacuum by a pumping station including a combination of turbo-molecular pump and back pump from Pfeiffer vacuum. An ionization vacuum gauge is mounted near the inlet tube and utilized as a vacuum indicator. The vacuum pipe used to connect the pumping station and the vacuum chamber is 2 m in length. An experimental set-up for testing the MESG prototype is shown in figure 8. The single-axis rate table used in the gyro test has a resolution of 0.0001o s−1 and its rotational axis is nominally vertical (see figure 9). Both the vacuum chamber and associated control electronics were mounted in the fixture on the rate table so that the gyro input axis is nominally parallel to the table rotational axis [22]. Note that both the output axis and spin axis are arranged horizontally. The gyro output signals were connected to a data acquisition unit (Agient 34401) to record the measured data.

4.2. Motor spin-up The spinning rotor is driven by a three-phase variablecapacitance motor. Since this motor type is synchronous, i.e. produces torque only when the rotation frequency and excitation frequency are the same, it requires a variablefrequency drive source to spin up. In our design, the motor is driven with six balanced bipolar voltages so that the rotor will 6


F T Han et al

Gyro output voltage (V)

5 4

Figure 11. Principle of high-speed three-phase motor drive.

3 2 1 0 -1 -2 -3 -4 -100

0 -50 50 Input angular rate (º/s)

100

PA+

Figure 13. Input-output characteristics of the rate gyro. PB+

at 10 085 rpm. The higher the vacuum level, the lower the drive voltage. An idea of measuring the drag effect of the residual gas in the vacuum chamber may be gained from the facts that at 0.6 Pa, a rotor spinning at 104 rpm will take about 10 min to slow to 20 rpm after removing the motor drive. Further analysis of the motor dynamics shows the damping coefficient due to the fact that viscous drag torque is about 1.53 × 10−14 N m s rad–1. For the device to be used as a rate gyroscope, higher vacuum level is preferred to lower drive voltage and resultant electrostatic disturbance torque on the rotor.

PC +

Vspin

Figure 12. Three-phase drive voltages and rotor speed at 10 085 rpm where the voltage is set at 7.8 V and the frequency at 2.353 kHz.

4.3. Angular rate gyroscope

remain near zero potential, as show in figure 11. Motor start-up will require that the drive frequency starts at several Hz and ramps up to the full-speed value over 2.33 kHz (104 rpm). This is accomplished by program control of three enhanced PWM modules in a digital signal processor (TMS320F28335). These PWM modules produce three square waves with 120◦ phase difference and a variable drive frequency. The drive generator takes the programmable output level of a D/A converter and produces the drive voltage Vd and its complementary signal −Vd for driving the output stages. The output circuit contains six analogue switches, one for each bipolar phase, and can deliver up to ± 15 V. For the test of motor spin-up, the pressure indicated by the vacuum gauge is set at about 0.6 Pa. Figure 12 shows three-phase voltages and rotor speed at 10 085 rpm where the drive voltage is set at 7.8 V and the frequency at 2.353 kHz. A periodic waveform, Vspin , at 14 cycles per revolution is available from sensing the capacitance variation between the rotor and two stators [6]. The spin speed is constrained by stress limits in the rotor, viscous drag, drive voltage and stiffness of the electric bearing [2]. Analysis of the motor dynamics shows that a rate up to 4.56 × 104 rpm can be attained with an attainable drive voltage of 15 V at a cost of much longer startup time. However, this is well below the ultimate spin rate (about 1.2 × 106 rpm) determined by material strength limits. Since the scale factor of the gyroscope is proportional to the spin speed, it is important to maintain speed constant during normal operation. In this case, once the preselected speed is reached, the motor is switched to phase-locked loopbased PD constant speed control. It was found that the drive voltage is about 7.02 V to maintain the rotor constant spinning

The performance of the micromachined gyroscope has been tested using the single-axis rate table. Once the preselected vacuum degree was reached, the rotor was spun up and maintained at 10 085 rpm. The response of the sensor to various input rates between −100o s−1 and +100o s−1 is shown in figure 13. In these tests, the vacuum valve was closed and the vacuum pipe was disconnected from the vacuum chamber to achieve unconstrained turn of the rate table. The experimental results show that the scale factor is 39.8 mV−1 (o s−1) to the angular rate input and linearity error is 0.58% in ± 100o s−1 input range. In addition, a higher scale factor of 47.4 mV−1 (o s−1) is attained at a higher rotor speed of 12 000 rpm. The experimental results verify that the gyro sensitivity is proportional to the spin speed of the rotor as predicated by (2). However, the measured scale factor seems to be 53.5% higher than the theoretical predication in section 3.4. The observed mismatch between the simulated and experimental results may be due to the fabrication imperfection of the device and simplicity in modeling the electrostatic torquer as given in (11). The measured performance of the rate gyroscope is summarized in table 3 where the gyro signal is low-pass filtered at a bandwidth of 10 Hz for fine resolution. Figure 14 shows the gyro output to step inputs of ± 0.5 o s−1. The measured noise floor and resolution of the gyro output were 0.015 o s−1 Hz−1/2 and 0.014o s−1, respectively. For testing the bias stability, 5 h measurement data were collected at room temperature and zero table rate. The measured gyro output is shown in figure 15, where the output curve descends continuously until the end of the first four hours. Thus, only 7


F T Han et al

spinning-rotor rate gyroscope in strapdown applications, the rate of case rotation is measured by detecting the precessioninduced torque of the spinning rotor using the dual-axis rebalance loop. Since the rebalance loop is of importance in determining dynamic range, sensitivity and accuracy of the MESG, simulation results show that the decouple control and nutation attenuation will be necessary in future loop design to further improve gyro performance. The detailed experimental results of electrostatic bearing, motor spin-up and rate gyro are presented using the prototype device. Some problems have been solved, but others remain. Calibration of the drift model by multiposition drift rate test will be one of the most promising means to improve the gyro accuracy as those classical two degree-of-freedom spinningrotor gyroscopes. Its reliable use is likely to require a vacuum package. Future work will focus on sealing the device wafer in a vacuum ceramic chip package and test of the gyro drift model.

Angular rate (º/s)

0.8 0.4

0 -0.4 -0.8

0

50

100

150

Time (s)

Figure 14. Response of the gyro to a step input of ± 0.5◦ s−1.

Angular rate (º/s)

0.6 0.4 0.2 0

Acknowledgments

-0.2 -0.4 0

1

2

3

4

This work was partly supported by the National Natural Science Foundation (grant no 41074049) and the Aviation Science Foundation of China (grant no 20100858005).

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Time (hour)

References

Figure 15. Stability of the gyro output with time. Table 3. Measured performance of the rate gyroscope. Performance

Value (unit)

Input range Scale factor Nonlinearity Resolution Noise floor Bias stability

± 100 ◦ s−1 39.8 mV−1 (◦ s−1) 0.58% 0.014 ◦ s−1 0.015 o s−1 Hz−1/2 50.95 o h−1

[1] Liu K, Zhang W P, Chen W Y, Li K, Dai F Y, Cui F, Wu X S, Ma G Y and Xiao Q J 2009 The development of micro-gyroscope technology J. Micromech. Microeng. 19 113001 [2] Torti R, Gondhalekar V, Tran H and Selfors B 1994 Electrostatically suspended and sensed micro-mechanical rate gyroscope Proc. SPIE 2220 27–38 [3] Murakoshi T, Endo Y, Fukatsu K, Nakamura S and Esashi M 2003 Electrostatically levitated ring-shaped rotational-gyro accelerometer Japan. J. Appl. Phys. 42 2468–72 [4] Nakamura S 2005 MEMS inertial sensor toward high accuracy and multi-axis sensing Proc. 4th IEEE Conf. on Sensors (Irvine, CA, Oct 31–Nov 3) pp 939–42 [5] Damrongsak B, Kraft M, Rajgopal S and Mehregany M 2008 Design and fabrication of a micromachined electrostatically suspended gyroscope Proc. Inst. Mech. Eng. C 222 53–63 [6] Han F T, Wu Q P and Wang L 2010 Experimental study of a variable-capacitance micromotor with electrostatic suspension J. Micromech. Microeng. 20 115034 [7] Cui F, Liu W, Chen W Y, Zhang W P and Wu X S 2011 Hybrid microfabrication and 5-DOF levitation of micromachined electrostatically suspended gyroscope Electron. Lett. 47 976–8 [8] Ellis C D and Wilamowski B M 2008 Fabrication and control of an electrostically levitated rotating gyro Proc. SPIE 6959 59590Q [9] Hill D A, Letendre T and Mills H A 2004 Embedded, real-time DSP control of an electrostically suspended gyroscope Proc. 2004 American Control Conf. (Boston, MA, June 30–July 2) pp 3321–6 [10] Han F T, Gao Z Y, Li D M and Wang Y L 2005 Nonlinear compensation of active electrostatic bearings supporting a spherical rotor Sensors Actuators A 113 177–86 [11] Bencze W J, Eglington M E, Brumley R W and Buchman S 2007 Precision electrostatic suspension system for the Gravity Probe B relativity mission’s science gyroscopes Adv. Space Res. 39 224–9

the data in the last 1 h are used for the standard deviation calculation of the drift rate, which yields a bias stability is 50.95 o h−1. Note that this stability level is simply tested without the identification and compensation of the gyro drift error. The bias stability of the MESG could be significantly improved by the calibration of the drift rate model as such macro-scale mechanical gyroscopes [22].

5. Conclusions A micromachined electrostatically suspended gyroscope with a spin speed of the rotor at 10 085 rpm was designed, fabricated and tested successfully. The spinningrotor gyroscope was implemented using a microfabrication process, which combines high-aspect-ratio dry etching and glass/silicon/glass bonding. The use of electrostatic bearing aims to eliminate mechanical friction, and hence enhance long-term stability of the MEMS inertial sensor. Experimental results show that the suspension loops must be designed to have high bandwidth on the order of kilohertz to satisfy the required suspension stiffness and overload capacity. For this 8


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[12] Esashi M 2009 Micro/nano electro mechanical systems for practical applications J. Phys.: Conf. Ser. 187 012001 [13] Fukuda G and Hayashi S 2010 The basic research for the new compass system using latest MEMS Int. J. Mar. Navig. Saf. Sea Transp. 4 317–22 [14] Kraft M and Damrongsak B 2010 Micromachined gyroscopes based on a rotating mechanically unconstrained proof mass IEEE Sensors 2010 Conf. pp 23–8 [15] Damrongsak B 2009 Development of a micromachined electrostatically suspended gyroscope PhD Thesis University of Southampton, UK [16] Giindila M V and Kraft M 2003 Electronic interface design for an electrically floating micro-disc J. Micromech. Microeng. 13 S11–6 [17] Han F T, Wu Q P, Zhang R and Dong J X 2009 Capacitive sensor interface for an electrostatically levitated micromotor IEEE Trans. Instrum. Meas. 58 3519–26

[18] Han F T, Wang L, Wu Q P and Liu Y F 2011 Performance of an active electric bearing for rotary micromotors J. Micromech. Microeng. 21 085027 [19] Coffee J R 1988 A comparison of multivariable design methodologies for a two-degree-of-freedom gyro torque-rebalance loop MS Thesis Massachusetts Institute of Technology, USA [20] Paquet F, Vinnins M F and Gallop L 1992 Analog torque rebalance loop for a tuned rotor gyroscope US Patent no. 5,138,883 [21] Ma G Y, Chen W Y, Zhang W P, Cui F and Li K 2010 Compact H∞ robust rebalance loop controller design for a micromachined electrostaticaly suspended gyroscope ISA Trans. 49 222–8 [22] The Gyro and Accelerometer Panel of the IEEE Aerospace and Electronic Systems Society 1989 IEEE specification format guide and test procedure for two-degree-of-freedom dynamically tuned gyros ANSI/IEEE Standard 813-1988

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