XII International Conference AVIA-2015, April 2015 ...

XII International Conference AVIA-2015, April 2015, NAU, Kyiv, Ukraine UDC: 629.7 V. V. Chikovani, Doctor of Engineering (Institute of Air Navigation National Aviation University) Low-Cost Multi-Functional Metallic Resonator Vibratory Gyroscope Abstract. Three of four modes of vibratory gyro operation, namely: rate, rate-integrating, and recently developed by Ukrainian specialist, differential mode are analyzed in this paper. Standing wave control system block diagram and angle rate measurement results for each mode of operation using metallic resonator gyro are presented. Possibility to combine these three modes of operation in one gyro is discussed.

Introduction Coriolis vibratory gyro (CVG) is one of the chronologically latest gyroscopic technology appeared in the world market in the 90-s of the previous century. This technology for the sufficiently short time spread out all over the world mainly due to its micro-miniature variant based on micro-electro-mechanical system, MEMS gyro. Four modes of operation can be realized in CVG: 1. Open loop mode, where elastic standing wave excited in resonator is not controlled by any forces, except drive one. At gyro rotation, Coriolis force diverts standing wave through the angle proportional to gyro rotation rate. 2. Closed loop mode or rate mode, where standing wave is held in the vicinity of the drive electrode by the control forces. At gyro rotation, Coriolis force is compensated for by applying the control force to keep the standing wave at the stable position in the vicinity of the drive electrode. Control force amplitude that compensates for the Coriolis force is proportional to angle rate, in this case. 3. Whole angle mode or rate-integrating mode, where at gyro rotation, instead of Coriolis, quadrature force is only compensated for. In this case, gyro rotation angle caused by Coriolis force is proportional to the standing wave rotation angle. 4. Differential mode of operation, where standing wave is held between the electrodes by the applied control forces. In this case, at gyro rotation, there arise two measurement channels with positive and negative angle rates, which difference increases angle rate signal and compensates for the errors equal in these two channels. First mode of operation is not almost used in practice because of large nonlinearity and too low accuracy. The second one is most popular because of lower influence of manufacturing imperfections, lower noise when measuring small angle rate and acceptable bandwidth for most applications. The third one can have extremely high dynamic range and high bandwidth, but very sensitive to manufacturing imperfections. The fourth one has been developing in Ukraine for ring-like resonators. It has higher resistance to external disturbances, ability to on-line compensate for the frequency mismatch (like third mode) and pre-start bias calibration, but it has 2 times higher noise, than the second mode of operation because of subtraction of two channels should be taken.

XII International Conference AVIA-2015, April 2015, NAU, Kyiv, Ukraine The second, third and fourth mode of operation on the example of digital metallic alloy cylindrical resonator CVG produced by Ukrainian manufacturer JPSC “Kyiv Automatic Plant named after G. Petrovsky” are discussed in this paper. Design features of metallic cylindrical resonator CVG are presented. Control system block diagrams for each of the considered modes of operation are also presented. Measurement results and advantages and disadvantages for each mode of CVG operation are discussed. Possibility to realize these three (2-nd, 3-d, 4-th) modes of operation in one gyro with automatic switch from one to another is also discussed. Metallic resonator CVG Design Fig.1 shows sensor design components. It consists of only three parts – base with stem for resonator, resonator in view of cylinder with holes on its wall and piezo-plates as electrodes glued on the spokes created after making the holes, and cover under which air is pumped out to create vacuum of about 10-4 atm. Piezo-electrode hole

Fig.1. Components of metallic resonator CVG

Sensor assembly is very simple and low-cost. Resonator is mounted on the stem and fixes with standard screw. Fig. 2 shows sensing element, circuit boards to provide controlling standing wave, and ready-to-operate CVG.

Fig.2. Metallic resonator CVG components.

Fig.3 shows two variants of 3-dimensional gyro units based on CVGs. Size of this units are 1005050 mm3 and 808040 mm3, respectively.

XII International Conference AVIA-2015, April 2015, NAU, Kyiv, Ukraine

Fig.3. 3D gyro units based on metallic resonator CVG.

Rate CVG To keep the standing wave in stationary position in the vicinity of drive electrode despite on action of Coriolis force, feedback control is realized to compensate for this Coriolis force. Rate CVG control system block diagram is illustrated in fig. 4 [1].

Fig.4. Rate CVG control system block diagram

The standing wave is excited at the resonant frequency r. Excitation control is based on phase lock loop (PLL) subsystem and gain control subsystem. These two subsystems generate excitation signal Xin(t) of proper amplitude, frequency and phase. At gyro rotation there appears at Yout electrode oscillation amplitude caused by Coriolis force together with error signal which is called quadrature. These two signal components are separated by demodulation processes using sine and cosine reference signals, sine component is proportional to angle rate SF*, where SF is coefficient of proportionality, called scale factor, and quadrature signal is an error caused by resonator manufacturing imperfections. Then, these two signals are re-

XII International Conference AVIA-2015, April 2015, NAU, Kyiv, Ukraine modulated, combined and sent to the Yin electrode as a compensation signal to keep the standing wave in stationary position. Fig.5 shows metallic resonator CVG bias instability at +500C constant temperature. As can be seen instability parameters is very promising for such a low cost gyro. Metallic resonator CVG output signal

Allan variance

deg/h

deg/h

Random walk = 0.0026 deg/h; Min.Allan variance = 0.01 deg/h; RMS (100 s average)=0.014 deg/h; RMS (15 min average)=0.011 deg/h; RMS (1 h average)=0.007 deg/h.

Time, s

Averaging time, s

Fig.5. Bias instability of metallic resonator CVG in rate mode of operation.

Fig.6 shows metallic resonator CVG bias instability and its day-to-day repeatability in the temperature range [-40 +50]0C with temperature ramp 1 0C/min.

deg/h

Metallic resonator CVG bias drift

Day-to-day bias repeatability: 1st day Bias=-8.42 deg/h; 2nd day Bias=-8.84 deg/h; 3rd day Bias=-9.7 deg/h Mean=-9 deg/h; RMS=0.5 deg/h

In run bias stability in the temperature range: 1st day RMS=4.7 deg/h; 2nd day RMS=6.1 deg/h; 3rd day RMS=4.6 deg/h; Mean=5.1 deg/h; RMS=0.84 deg/h

Time, s Fig.6. Metallic resonator CVG bias repeatability and instability in the temperature range

XII International Conference AVIA-2015, April 2015, NAU, Kyiv, Ukraine As can be seen from fig.6 a lot of work should be made to bring bias instability in the temperature range nearer to that of stable temperature. One of the ways to do this is on-line bias correction that in principle can be done for vibratory gyros [2]. Fig. 7 shows metallic resonator CVG scale factor temperature instability when operating in rate mode.

1/(deg/s)

Metallic resonator CVG scale factor temperature instability and repeatability

SF temperature instability ≈0.2% (RMS); SF day-to-day repeatability ≈ 0.17% (RMS); SF temperature sensitivity ≈ 17ppm/ 0C Temperature, 0C Fig.7. Metallic resonator CVG scale factor temperature instability and day-to-day repeatability

Rate integrating CVG Coriolis force caused by gyro rotation is not compensated for in rate integrating mode X of operation and this results in standing wave  a rotation through the angle proportional to gyro rotation angle. Coefficient proportionali- rt+' q ty between those two angles of rotation, i.e. Y rate integrating CVG scale factor, is called Brian coefficient k or angle gain coefficient. In absence of gyro rotation resonator elementary mass point motion trajectory is ellipse Fig.8. Point trajectory in oscillation. shown in fig. 8. Ellipse parameters are designated in the fig.8, a is a vibration amplitude; q is a quadrature amplitude; r is resonant frequency; ’ is a vibration phase;  is vibrating pattern orientation relative to X axis (drive electrode). At gyro rotation the ellipse turns in the direction of rotation with the lag coefficient k. Frequency mismatch and Q-factor mismatch resulting from resonator manufacturing imperfections and external forces result in that parameters a, q, ’ and  are changing versus time. These changing are much slower than vibration period T=r/2. In order to calculate parameters a, q, ’ and  demodulation should be used

XII International Conference AVIA-2015, April 2015, NAU, Kyiv, Ukraine based on mixing the signals Xout(t) and Yout(t) with reference signals sinrt and cosrt and low pass filtering to obtain four demodulated variables Cx, Sx, Cy, Sy. Parameters a, q, ’ and  are calculated using the following expressions [3]:







1 1 E  E 2  Q2 ; q  E  E 2  Q2 2 2 E  Cx2  S x2  C y2  S y2 ; Q  2(Cx S y  C y S x );

a

; (1)

2(Cx C y  S x S y ) 2(Cx S x  C y S y ) 1 1   arctan 2 ;   arctan 2 ; 2 2 Cx  S x2  C y2  S y2 Cx  S x2  C y2  S y2

The aim of the CVG control system in rate integrating mode of operation is to keep the standing wave parameters in process of operation at the following values: q=0, a2= E=const, =0;

(2)

Control system block diagram providing relationships (2) for the standing wave parameters is presented in fig.9. cos Demod-r Xin

Yout Xout

Yin

Demod-r sin cos

Cy

 Regul-r

Sy

Cx Demod-r Demod-r Sx sin

sin cos fxe

PLL cos

E Regul-r E0=a2   k  (t )dt

q Regul-r

fye sin cos fxq fyq sin

Fig.9. CVG control system block diagram in rate integrating mode of operation.

Fig.10 shows output signal of rate integrating CVG when input angle rate is constant.

=

Fig.10. CVG output signal in rate integrating mode of operation.

XII International Conference AVIA-2015, April 2015, NAU, Kyiv, Ukraine Wave rotation angle can be expressed through the resonator’s manufacturing imperfections as follows [4]:   k  

12  22 2aq 1 a2  q2  1  cos 2(   )     sin 2(   ) ; 2 2 2 a  q 2 a2  q2    2  22 1 1 1      ; 2  1 ;    2   1 2

(3)

Where 1, 2, 1, 2 are time constants along resonator damping pricipal axes and resonant frequencies along rigidity principal axes, located under angle  and  relative to direction of standing wave oscilation repectively. When q=0 in correspondance with (2), q regulator in fig.9 should make it, then (3) is simplified to: 1 1 (4)   k    sin 2(   ) ; 2

  



So, as follows from (4), when =, nonlinearity is zero and the measured  crosses the straight line with tilt angle -k shown in fig.10. Differential CVG In differential CVG standing wave is located between the electrodes so that wave angle 0, /4, that is standing wave oscillation direction is not coincident with any electrodes. In this case measurement equations can be written as [5]: 2k Dy tan 2  d xy Dy tan 2  Dx d xx  Z x ; 2k Dx cot 2  d xy Dx cot 2  Dy d yy  Z y ,

(5)

where 1 2  1 1 d xx    h cos 2(   )  ; h       ;      1 2 2 1 1 2    ; d yy    h cos 2(   )  ;  1 2   d xy  h sin 2(   ).

Dx and Dy are transformation coefficients of X and Y electrodes deformations into voltages. As can be seen from the first equation of (5) angle rate  has negative sign, and in the second one it has positive sign. Thus, control system that holds standing wave between electrodes realizes differential mode of operation for CVG. To effectively realize differential mode of operation it is necessary to align standing wave under angle * at which the following condition is valid: SFy D 1 1 Dy tan 2*  Dx cot 2* ,or *  arctan x  arctan . 2 Dy 2 SFx

(6)

In this case SFx = SFy, and when summing two channel measurements information about angle rates is eliminated, therefore, current information about error components can be obtained. Sum and difference measurement channels of differential CVG can be presented as follows [6]:

XII International Conference AVIA-2015, April 2015, NAU, Kyiv, Ukraine 2 4k Dy tan 2*  ( Dx  Dy )  ( Dx  Dy )h cos 2(*   )  Z x  Z y ;  2 2 Dy h sin 2(*   ) tan 2*  ( Dx  Dy )  ( Dx  Dy )h cos 2(*   )  Z x  Z y . 

(7)

There is no damping cross coupling term hsin2(-) in these measurements. Differential CVG control system block diagram is presented in fig.11 [7]. Demod1 Controller1 VCO X sense cosrt Demod2 Controller2 cosrt - Ax sinrt Y sense

Y drive

X drive

sinrt cosrt

zx-zy

Quadrature Control Demod3 cosrt Demod4 sinrt

Controller sinrt Controler5 - Ay

-zx+zy cosrt

Fig.11. Differential CVG control system block diagram

Fig. 12 shows two channel signals of metallic resonator differential CVG.

Fig.12. Differential CVG X and Y channel measurements of ± 40 deg/s angle rate.

Fig.13 represents frequency mismatch during angle rate measurement. As can be seen quadrature control unit reduces frequency mismatch to 10-3 Hz.

Fig.13. Differential CVG frequency mismatch when measuring angle rate

XII International Conference AVIA-2015, April 2015, NAU, Kyiv, Ukraine After calibration of two channels scale factors SFx and SFy, angle * can be determined. The standing wave then is located under *angle to obtain SFx= SFy that results in angle rate compensation in sum Zx+Zy of two channels. Fig.14 shows sum of two channels under measuring angle rate when standing wave is under * angle.

Fig.14. Sum of two channels under measuring angle rate by differential CVG.

The angle * has very low temperature sensitivity. Fig.15 shows angle * temperature sensitivity in the temperature range [+25 +75] 0C. As can be seen * temperature sensitivity for metallic resonator gyro is about 1ppm/ 0C (10-4 %/0C).

Fig.15. Angle * temperature sensitivity for metallic resonator CVG

Differential CVG has more external shock and vibration resistance than non differential CVG, because of external disturbances act on both channels equally and in output signal (difference of channels) are compensated for. Fig.16 shows computer simulation results on comparison of shock sensitivity for rate mode (curve 2) and differential mode (curve 3) of operations for metallic resonator CVG [6]. Curve 1 presents transformed into angle rate voltage pulse due to shock using scale factor of rate CVG. As can be seen differential CVG erroneous angle rate response on 100 g, 5 ms duration shock is almost 4 times less than that of non differential (rate) CVG.

Fig.16. Comparison of shock resistance of differential and rate CVGs

XII International Conference AVIA-2015, April 2015, NAU, Kyiv, Ukraine Multi-functional CVG All three considered in this paper CVG mode of operations can be realized in one CVG on the basis of modified rate integrating algorithm with ability to automatically (or manually if necessary) switch from one mode to another. The example of such CVG with ability to automatically switch from the second mode to the third one has been presented in [8]. In order to realize multi-functional CVG with modes switching, advantages and disadvantage of each mode of operation should be analyzed. Advantages and disadvantages of three mode of operation are presented in the table 1. Table1. Advantages and disadvantages of CVG modes of operation Rate CVG

Rate Integrating CVG

Differential CVG

Advantages • Low noise • Standing wave control simplicity • Easy to calibrate resonator’s imperfections

• High dynamic range • High bandwidth • High stability of scale factor

• Higher resistivity to external disturbances • Effective on-line compensation for the frequency mismatch • Possibility to on-line bias compensation.

Disadvantages • Higher sensitivity to external disturbances • Lower scale factor temperature stability • Limited bandwidth

• Higher sensitivity to external disturbances • Higher noise especially when measuring small rate • Higher sensitivity to resonator’s imperfections

• Limited bandwidth • 2 times higher noise than rate CVG • Lower scale factor temperature stability

Based on results of the above table the following switching logic can be proposed: yes

in>max

no Disturbances?

no

Rate integrating mode

yes

Disturbances?

yes

Differential mode

no Rate mode Fig.17. Example of modes switching logic

XII International Conference AVIA-2015, April 2015, NAU, Kyiv, Ukraine Conclusion • • •

•

CVG has four modes of operation, three of which can provide practical requirements on accuracy and environments. Each mode of operation has advantages on accuracy at different environments. CVG can be realized in multi functional modes of operation with automatic (or manual if necessary) switching from one mode to another in dependence of environmental conditions and value of angle rate to provide maximum accuracy in widest dynamic range and environmental conditions. New differential mode of CVG operation can provide lower sensitivity to external disturbances and lower drift than other rate and rate integrating modes. References

1.

2. 3.

4. 5.

6. 7.

8.

V.V. Chikovani “Trends of Ukrainian all digital Coriolis vibratory gyroscopes development” ”.- IEEE Proc. Intern. Conf. on Methods and Systems of Navigation and Motion Control (MSNMC), NAU, Kyiv, Ukraine, Oct.14-17, 2014, pp.25-28. Robert E. Stewart “Apparatus and method for self-calibration of Coriolis vibratory gyroscope”.- US Patent #8011246, G01P 9/00, G01P 21/00, Sept. 6, 2011. D. Lynch “Vibratory Gyro Analysis by the Method of Averaging”, Proc. 2nd St. Petersburg Conf. on Gyroscopic Technology and Navigation, St. Petersburg, Russia, May 24-25, 1995, pp.26-34. J.Y. Cho “High-performance micromachined vibratory rate- and rate- integrating gyroscopes”.- Ph.D. Dissertation, Electrical Engineering, Michigan Univ., 2012, p.265. V.V. Chikovani, O.A. Suschenko “Differential mode of operation for ring-like resonator CVG”.- IEEE Proc. Intern. Conf. on Electronics and Nanotechnology (ELNANO), NTUUKPI, Kyiv, Ukraine, 15-18 April, 2014, pp.451-455 V. V. Chikovani, G.V. Tsiruk “Bias Compensation in Differential Coriolis Vibratory Gyro”. - Electronics and control systems, №4 (37), 2013 р., pp.99-103. V.V. Chikovani , H.V. Tsiruk “Differential CVG shock damping capacity. Computer simulation results”.- IEEE Proc. Intern. Conf. on Methods and Systems of Navigation and Motion Control (MSNMC), NAU, Kyiv, Ukraine, Oct.14-17, 2014, pp.132-134. D.D. Lynch, A. Matthews “Dual Mode Hemispherical Resonator Gyro Operating Characteristics”.- 3-rd S. Petersburg Int. Conf. on Integrated Navigation Systems, part 1, pp. 3744, 1996.