Abstract
Dual-mass MEMS gyroscope is one of the most popular inertial sensors. In this chapter, the structure design and electrostatic compensation technology for dual-mass MEMS gyroscope is introduced. Firstly, a classical dual-mass MEMS gyroscope structure is proposed, how it works as a tuning fork (drive anti-phase mode), and the structure dynamical model together with the monitoring system are presented. Secondly, the imperfect elements during the structure manufacture process are analyzed, and the quadrature error coupling stiffness model for dual-mass structure is proposed. After that, the quadrature error correction system based on coupling stiffness electrostatic compensation method is designed and evaluated. Thirdly, the dual-mass structure sensing mode modal is proposed, and the force rebalancing combs stimulation method is utilized to achieve sensing mode transform function precisely. The bandwidth of sensing open loop is calculated and experimentally proved as 0.54 times with the resonant frequency difference between sensing and drive modes. Then, proportional-integral-phase-leading controller is presented in sensing close loop to expand the bandwidth, and the experiment shows that the bandwidth is improved from 13 to 104 Hz. Finally, the results are concluded and summarized.
Keywords
- MEMS gyroscope
- dual-mass structure
- mode analysis
- quadrature error
- bandwidth expansion
- electrostatic compensation
1. Introduction
The precision of micro-electro-mechanical system (MEMS) gyroscope improves a lot in this decade, and achieves the tactical grade level. On the benefits of the small size, low costs, and light weight the MEMS gyro is applied in more and more areas, such as inertial navigation, roller detection, automotive safety, industrial controlling, railway siding detection, consumer electronics and stability controlling system [1, 2, 3, 4, 5]. During use, the acceleration along the sense axis causes great error in MEMS gyroscope output signal, and dual-mass gyroscope structure restrains this phenomenon well by employing differential detection technology; so, a lot of research institutes are interested in this structure [6, 7, 8].
1.1. Development of dual-mass MEMS gyroscope structure quadrature error compensation
Most of the literature informs that the dominate signal component in output signal is quadrature error, which is generated in the structure manufacture process, and brings over several 100° S−1 equivalent input angular [9, 10, 11, 12, 13, 14]. The original source of quadrature is the coupling stiffness, which is modulated by drive mode movement and generates quadrature error force. The force has same frequency but has a 90° phase difference with Coriolis force and stimulates sense mode [11]. Most previous works utilize phase-sensitivity demodulation method to pick Coriolis signal from sense channel [9, 13], which requires accurate phase information and long-term, full-temperature range stability. However, the demodulation phase error and noise usually exist (sometimes more than 1° [9, 12]), which bring undesirable bias. The coupling stiffness drift (the drive and sense modes’ equivalent stiffness vary with temperature and generate the drift of coupling stiffness [6, 11]) causes the quadrature error force drift, which is considered to be one of the most important reasons leading to bias long-term drift, and is proved by [11, 14] experimental work.
The previous works provide several effective ways to reduce quadrature error and are concluded into three aspects after the structure is manufactured [9, 11]: the quadrature signal compensation, the quadrature force correction and coupling stiffness correction. In work [9], the quadrature error is reduced by dc voltage based on synchronous demodulation and electrostatic quadrature compensation method, and the sigma-delta technology is employed in ADC and DAC. The research in Ref. [14] also employs coupling stiffness correction method to improve the performance of “butterfly” MEMS gyroscope. The bias stability and scale factor temperature stability enhance from 89°/h and 662 ppm/°C to 17°/h and 231 ppm/°C, respectively, which achieves the correction goal. The quadrature error correction in dual-mass tuning fork MEMS gyro structure is investigated in literature [11], and this work also proves the quadrature stiffness are different in left and right masses. A quadrature error correction closed loop is proposed in the work, and utilizing the coupling stiffness correction method, the masses are corrected separately. The stiffness correction combs utilize unequal gap method with dc voltages [15]. The bias stability improves from 2.06 to 0.64°/h with Allan Deviation analysis method, and the noise characteristic is also optimized [11]. Another coupling stiffness correction work is proposed in literature [7]; in this work, coupling stiffness correction controller uses PI technology, and the quadrature error equivalent input angular rate is measured as 450°/s. The experiment in the work shows that the bias stability and ARW improve from 7.1°/h and 0.36°/√h to 0.91°/h and 0.034°/√h, respectively. In Ref. [16], quadrature signal is compensated based on charge injecting technology in the sense loop, the compensation signal has same frequency, amplitude and anti-phase with quadrature error signal. The quadrature error correction method proposed in literature [8] employs both the quadrature force and stiffness correction methods, the modulation reference signal is generated by PLL technology and the correction loop uses PI regulator; the two masses are controlled together. A novel quadrature compensation method is proposed in literature [17] based on sigma-delta-modulators (ΣΔM), the quadrature error is detected by utilizing a pure digital pattern recognition algorithm and is compensated by using DC bias voltages, and the system works beyond the full-scale limits of the analog ΣΔM hardware. The quadrature error is compensated by open-loop charge injecting circuit in Ref. [18], the circuit is implemented on application specific integrated circuits (ASIC) and the experimental results show that the quadrature error component is effectively rejected.
1.2. Development of dual-mass MEMS gyroscope bandwidth expansion
High precision MEMS gyros are reported in literatures, and the bias drift parameters are even better than the tactical grade requirement. But the bandwidth performance always restrains the MEMS gyro application (100 Hz bandwidth is required in both Tactical and Inertial Grade) [4]. For most linear vibrating MEMS gyro, the mechanical sensitivity is determined by the difference between drive and sensing modes’ resonant frequencies
This chapter focuses on the investigation of the dual-mass MEMS gyroscope structure, and the electrostatic compensation method for the structure, including quadrature error correction and bandwidth expansion technology. Through these technologies, the static and dynamic performance of the MEMS gyroscope is improved.
2. Dual-mass MEMS gyroscope structure design and analysis
2.1. Dual-mass MEMS gyroscope structure design
The fully decoupled linear vibrating gyroscope structure’s ideal movement model can be found in many papers; the model can be described as two “spring-mass-damping” systems: drive mode and sense mode. In
The drive U-shaped springs’ stiffness coefficients are large along
The drive mode of the structure bases on tuning for k theory. The left and right masses are coupled by connect U-shaped spring, when two sensing masses are coupled by the
2.2. Dual-mass MEMS gyroscope structure working principle analysis
Due to a large difference (>1000 Hz) between the in-phase (the first mode) and anti-phase drive modes frequencies, the quality factor of drive anti-phase mode is
are the mass, displacement, stiffness, damping and external force matrix, respectively;
where
The abovementioned equations indicate that the movement of drive and sensing modes are the compound motion of stable vibration and attenuation vibration. Since drive mode closed loop is employed, drive mode is stimulated with stable amplitude (
The sensing in-phase and anti-phase modes movement equation can be got from (3):
Then, the mechanical sensitivity can be expressed as:
The mechanical sensitivity of dual-mass sensing mode coupled structure is determined by vibration amplitude of drive mode and frequency differences between drive working mode and sensing modes (including the second and third modes). Furthermore, the one near the fourth mode is the dominant element. Therefore, the sensing anti-phase mode determines the gyro structure mechanical sensitivity. The schematic diagram of gyro sensing mode is shown in Figure 4.
where
2.3. Dual-mass MEMS gyroscope monitoring system
The gyro control and detection system is shown in Figure 5. In drive loop, the drive frame displacement
3. Dual-mass MEMS gyroscope quadrature error compensation
3.1. Dual-mass MEMS gyroscope structure quadrature error model
The quadrature error is caused by coupling stiffness, which is generated in structure processing stage, and the stiffness elements in Eq. (1) can be calculated by:
where
The equivalent stiffness and masses system and structure motion of dual-mass gyro structure is shown in Figure 6. The design drive and sense stiffness axis are
3.2. Dual-mass MEMS gyroscope structure coupling stiffness compensation
The CSC method utilizes quadrature error correction combs to generate negative electrostatic stiffness and correct quadrature error coupling stiffness. This special comb is unequal gap style, and is introduced in [11]; its stiffness is expressed as:
where
We have
Then, combining (9), (11) and (12), we have:
When the system is under stable state, s = 0, and the above equation has:
Then:
The coupling stiffness is corrected. The CSC system is simulated and the curves are shown in Figure 8. The Pole-Zero Map is shown in Figure 8(a), no pole is in the positive real axis and Figure 8(b) is the Nyquist Diagram, the curve does not contain (−1,0j) point, which proves the system’s stability. The time-domain simulation curves are shown in Figure 8(c)–(e), and the curves indicate that the CSC system is under stable state after about 0.7 s. It is obvious that in start-up stage, the sense channel signal mainly consists of quadrature error signal. But, in stable state, the dominate element is Coriolis in-phase signal. Furthermore, the overall coupling stiffness
4. Dual-mass MEMS gyroscope bandwidth expansion
4.1. Dual-mass MEMS gyroscope structure sense mode model
The force rebalancing combs stimulation method (FRCSM) is employed to test the sense mode. Force rebalancing combs are slide-film form (does not vary
where
After Laplace transformation, we can get [35]:
where,
In Figure 10, point A (frequency is ∆
4.2. Dual-mass MEMS gyroscope bandwidth expanding
Sensing closed-loop control method provides electrostatic force to rebalance the Coriolis force applied on sensing mode, which is one of the most effective ways to improve gyro dynamic performance. When sensing closed-loop works, the sensing frame’s displacement is restricted and the Coriolis force is transformed into electronic signal directly. And it also avoids the nonlinearity of sensing mode displacement. Meanwhile, closed loop provides better anti-vibration and anti-shock characteristics to the gyroscope. The sensing closed-loop schematic is shown in Figure 9, where the real input angular rate
Combining Eq. (20) with (21), we have:
Because
The above equation means that the scale factor in closed loop is constant value and is not restricted by resonant peak (A point).
Generally speaking, the open-loop Bode diagram of sensing closed-loop is expected to have the following characteristics: (a) In low frequency range, a first-order pure integral element is configured to achieve enough gain and reduce the steady state error of the system, (b) In middle frequency range, the slope of magnitude line is designed to be −20 dB/dec at 0 dB crossing frequency point. The cut off frequency is
According to the analysis, we make
The open-loop Bode Diagram of the sensing closed-loop is simulated in Simulink software and shown in Figure 12(a). Figure 12 indicates that the minimum phase margin of the loop is 34.6° and the magnitude margin is 7.21 dB, which satisfy the design requests. The Pole-Zero Map and Nyquist Map of sensing closed-loop are shown in Figure 12(b) and (c). The poles distribute in negative side of real axis and the Nyquist curve does not contain (−1, 0j) point. These two criterions both illustrate the closed system is pretty stable. The Bode Diagram of sensing closed-loop simulation is shown in Figure 12(d), whose curves indicate that the bandwidth of the gyro is 100 Hz, the lowest point within the bandwidth range is −13.8 dB, the DC magnitude is −12.3 dB and the highest point is −10.4 dB. The resonant peak point A (shown in Figure 10) is compensated, and the new bandwidth bottleneck point is valley B. Therefore, one of the best methods to expand the bandwidth under this condition is to enlarge the frequency difference between
5. Conclusion
In this chapter, the recent achievements in our research group for dual-mass MEMS gyroscope are proposed. Three main parts are discussed: dual-mass gyroscope structure design, quadrature error electrostatic compensation and bandwidth expansion. First, dual-mass MEMS gyroscope structure is designed and simulated in ANSYS soft and dual-mass structure movement function is derived. Second, quadrature error is traced to the source, and the coupling stiffness electrostatic compensation method is employed to reduce the quadrature error. Finally, proportional-integral-phase-leading controller is presented in sensing close loop to expand the bandwidth from 13 to 104 Hz.
Acknowledgments
This work was supported by National Natural Science Foundation of China No.51705477. The research was also supported by Research Project Supported by Shanxi Scholarship Council of China No. 2016-083, Fund of North University of China, Science and Technology on Electronic Test and Measurement Laboratory No. ZDSYSJ2015004, and The Open Fund of State Key Laboratory of Deep Buried Target Damage No. DXMBJJ2017-15. The authors deliver their special gratitude to Prof. Hongsheng Li (Southeast University, Nanjing, China) and his group for the guidance, discussions and help.
References
- 1.
Zaman M, Sharma A, Hao Z, et al. A mode-matched silicon-yaw tuning-fork gyroscope with subdegree-per-hour Allan deviation bias instability. Journal of Microelectromechanical Systems. 2008;(6):1526-1536. DOI: 10.1109/jmems.2008.2004794 - 2.
Xu Y, Chen XY, Wang Y. Two-mode navigation method for low-cost inertial measurement unit-based indoor pedestrian navigation. Journal of Chemical Information & Computer Sciences. 2016; 44 (5):1840-1848. DOI: 10.1177/0037549716655220 - 3.
Huang H, Chen X, Zhang B, et al. High accuracy navigation information estimation for inertial system using the multi-model EKF fusing Adams explicit formula applied to underwater gliders. ISA TRANSACTIONS. 2017; 66 :414-424. DOI: 10.1016/j.isatra.2016.10.020 - 4.
Antonello R, Oboe R. Exploring the potential of MEMS gyroscopes. IEEE Industrial Electronics Magazine. 2012; 3 :14-24. DOI: 10.1109/mie.2012.2182832 - 5.
Xia DZ, Yu C, Kong L. The development of micromachined gyroscope structure and circuitry technology. Sensors. 2014; 14 :1394-1473. DOI: 10.3390/s140101394 - 6.
Cao HL, Li HS. Investigation of a vacuum packaged MEMS gyroscope structure’s temperature robustness. International Journal of Applied Electromagnetics & Mechanics. 2013; 4 :495-506. DOI: 10.3233/jae-131668 - 7.
Tatar E, Alper S, Akin T. Quadrature- error compensation and corresponding effects on the performance of fully decoupled MEMS gyroscopes. Journal of Microelectromechanical Systems. 2012;(3):656-667. DOI: 10.1109/jmems.2012.2189356 - 8.
Chaumet B, Leverrier B, Rougeot C, et al. A new silicon tuning fork gyroscope for aerospace applications. In: Proceedings of Symposium Gyro Technology. 2009, 1.1-1.13 - 9.
Saukski M, Aaltonen L, Halonen K. Zero-rate output and quadrature compensation in vibratory MEMS gyroscopes. IEEE Sensors Journal. 2007;(12):1639-1652. DOI: 10.1109/jsen.2007.908921 - 10.
Tally C, Waters R, Swanson P. Simulation of a MEMS Coriolis gyroscope with closed-loop control for arbitrary inertial force, angular rate, and quadrature inputs. Proceedings of IEEE Sensors. 2011:1681-1684 - 11.
Li HS, Cao HL, Ni YF. Electrostatic stiffness correction for quadrature error in decoupled dual-mass MEMS gyroscope. Journal of Micro-Nanolithography Mems And Moems. 2014; 13 (3): 033003. DOI: 10.1117/1.jmm.13.3.033003 - 12.
Walther A, Blanc CL, Delorme N, Deimerly Y, Anciant R, Willemin J. Bias contributions in a MEMS tuning fork gyroscope. Journal of Microelectromechanical Systems. 2013; 22 (2):303-308. DOI: 10.1109/jmems.2012.2221158 - 13.
Cao HL, Li HS, Liu J et al. An improved interface and noise analysis of a turning fork microgyroscope structure. Mechanical Systems and Signal Processing. 2016; s 70–71 :1209-1220. DOI: 10.1016/j.ymssp.2015.08.002 - 14.
Su J, Xiao D, Wu X, et al. Improvement of bias stability for a micromachined gyroscope based on dynamic electrical balancing of coupling stiffness. Journal of Micro-Nanolithography Mems And Moems. 2013; 12 (3):033008. DOI: 10.1117/1.jmm.12.3.033008 - 15.
Ni YF, Li HS, Huang LB. Design and application of quadrature compensation patterns in bulk silicon micro-gyroscopes. Sensors. 2014; 14 :20419-20438. DOI: 10.3390/s141120419 - 16.
Seeger J, Rastegar A, Tormey T. Method and apparatus for electronic cancellation of quadrature error : United States Patent. No. 7290435B2; 2007 - 17.
Maurer M, Northemann T, Manoli Y. Quadrature compensation for gyroscopes in electro-mechanical bandpass ΣΔ-modulators beyond full-scale limits using pattern recognition. Procedia Engineering. 2011; 25 :1589-1592. DOI: 10.1016/j.proeng.2011.12.393 - 18.
Antonello R, Oboe R, Prandi L, et al. Open loop compensation of the quadrature error in MEMS vibrating gyroscopes. In: Proceedings of the IEEE Industrial Electronics Society Conference; 2009. pp. 4034-4039 - 19.
Cao HL, Li HS, Sheng X, et al. A novel temperature compensation method for a MEMS gyroscope oriented on a periphery circuit. International Journal of Advanced Robotic System. 2013; 10 (5):1-10. DOI: 10.5772/56759 - 20.
M H Bao. Handbook of sensors and actuators. 1st ed. Amsterdam; The Netherlands,Elsevier; 2000. DOI: 10.1016/9780444505583 - 21.
Ezekwe CD, Boser BE. A mode-matching ΣΔ closed-loop vibratory gyroscope readout interface with a 0.004°/s/√Hz noise floor over a 50Hz band. IEEE Journal of Solid-State Circuits. 2008; 43 (12):3039-3048. DOI: 10.1109/jssc.2008.2006465 - 22.
Sonmezoglu S, Alper SE, Akin T. An automatically mode-matched MEMS gyroscope with wide and tunable bandwidth. Journal of Microelectromechanical Systems. 2014; 23 (2):284-297. DOI: 10.1109/jmems.2014.2299234 - 23.
Sung WT, Sung S, Lee JG, Kang T. Design and performance test of a MEMS vibratory gyroscope with a novel AGC force rebalance control. Journal of Micromechanics & Microengineering. 2007; 17 (10):1939. DOI: 10.1088/0960-1317/17/10/003 - 24.
Cui J, Guo Z, Zhao Q, et al. Force rebalance controller synthesis for a micromachined vibratory gyroscope based on sensitivity margin specifications. Journal of Microelectromechanical. 2011; 20 (6):1382-1394. DOI: 10.1109/jmems.2011.2167663 - 25.
He C, Zhao Q, Liu Y, et al. Closed loop control design for the sensing mode of micromachined vibratory gyroscope. Science China-Technological Sciences. 2013; 56 (5):1112-1118. DOI: 10.1007/s11431-013-5201-x - 26.
Alshehri A, Kraft M, Gardonio P. Two-mass MEMS velocity sensor: Internal feedback loop design. IEEE Sensors Journal. 2013; 13 (3):1003-1011. DOI: 10.1109/jsen.2012.2228849 - 27.
Trusov AA, Schofield AR, Shkel AM. Study of substrate energy dissipation mechanism in in-phase and anti-phase micromachined vibratory gyroscope. In: IEEE Sensors Conference; 2008. pp. 168-171 - 28.
Y Ni, H Li, L Huang, et al. On bandwidth characteristics of tuning fork micro-gyroscope with mechanically coupled sensing mode. Sensors 2014; 14 (7):13024-13045. DOI: 10.3390/s140713024 - 29.
Si C, Han G, Ning J, Yang F. Bandwidth optimization design of a multi degree of freedom MEMS gyroscope. Sensors. 2013; 13 (8):10550-10560. DOI: 10.3390/s130810550 - 30.
Acar C, Shkel AM. An approach for increasing drive-mode bandwidth of MEMS vibratory gyroscopes. Journal of Microelectromechanical Systems. 2005; 14 (3):520-528. DOI: 10.1109/jmems.2005.844801 - 31.
Feng ZC, Fan M, Chellaboina V. Adaptive input estimation methods for improving the bandwidth of microgyroscopes. IEEE Sensors Journal. 2007; 7 (4):562-567. DOI: 10.1109/jsen.2007.891992 - 32.
Cui J, He C, Yang Z, et al. Virtual rate-table method for characterization of microgyroscopes. IEEE Sensors Journal. 2012; 12 (6):2192-2198. DOI: 10.1109/jsen.2012.2185489 - 33.
Y Yin, S Wang, C Wang, et al. Structure-decoupled dual-mass MEMS gyroscope with self-adaptive closed-loop. In: Proceedings of the 2010 5th IEEE International Conference on Nano/Micro Engineered and Molecular Systems; Xiamen. China: 2010. p. 624-627 - 34.
Cao HL, Li HS, Shao XL, Liu ZY, Kou ZW, Shan YH, Shi YB, Shen C, Liu J. Sensing mode coupling analysis for dual-mass MEMS gyroscope and bandwidth expansion within wide-temperature range. Mechanical Systems & Signal Processing. 2018; 98 :448-464. DOI: 10.1016/j.ymssp.2017.05.003 - 35.
Cao HL, Li HS, Kou ZW, Shi YB, Tang J, Ma ZM, Shen C, Liu J. Optimization and experiment of dual-mass MEMS gyroscope quadrature error correction methods. Sensors. 2016; 16 (1):71. DOI: 10.3390/s16010071