# Coupling Mechanism Analysis and Fabrication of Triaxial Gyroscopes in Monolithic MIMU

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Working Principle and Structure Design

_{z}), due to the Coriolis effect (Figure 2b). There are four Yaw Sense Frames distributed at the periphery of Big Frames.

_{y}) shown in Figure 2c. The Outer Pitch Frame has 2-DOF in the in-plane driving direction and out-of-plane z-sensing direction. Upon the Outer Pitch Frame is steadily driven, when there is an angular velocity Ω

_{y}applied on the gyroscope, the Pitch Sense Frame will move together with the Outer Pitch Frame in z direction under the Coriolis effect. The sensing electrodes in Pitch/Roll Mode are designed to be comb fingers, which have different thickness in z-axis, so that the capacitance change is proportional to the z-axis movement caused by Coriolis force.

_{x}) shown in Figure 2d. Its working principle is similar to the Pitch Mode.

## 3. Mechanical Coupling Stiffness Analysis

_{1}, D

_{2}, D

_{3}, D

_{4}, D

_{1}’, D

_{2}’, D

_{3}’, D

_{4}’), when a driving voltage is applied on the drive electrodes. Under the participation of yaw coupling beams (Y

_{3}, Y

_{4}, Y

_{5}, Y

_{6}, Y

_{11}, Y

_{12}), the Big Frame will move together with the Drive Frames. The driving beams (D

_{5}, D

_{6}, D

_{7}, D

_{8}) will bring the yaw sense electrodes to move together with the Big Frame. Yaw sensing beams (Y

_{1}, Y

_{2}) will limit the yaw sense electrodes to have 2-DOF in-plane movement, when an angular rate in z-direction is applied on the gyroscope.

_{9}, D

_{10}, D

_{11}, D

_{12}, D

_{13}, D

_{14}, D

_{15}, D

_{16}). When an angular rate in x/y-direction is applied on the gyroscope, under the participation of pitch/roll sensing beams (P

_{1}, P

_{2}, P

_{3}, P

_{4}), the Outer Frame in Pitch/Roll Modes has 2-DOF in the in-plane drive direction and out-of-plane sensing direction. The schematic diagrams for the gyroscope and various beams are shown in Figure 3 and a summary of all the driving and sensing beams and frames mentioned above is shown in Table 1.

#### 3.1. Mechanical Error Analysis

_{system}is the stiffness matrix of an ideal beam; k

_{system}’ is the stiffness matrix of beam with fabrication imperfection, which can be approximately expressed by diagonal terms.

_{6}, the gas pressure difference between two U-shaped coupling beam arms can have the different etching rates. For specific performance, one of the two arms that are close to the anchor will be etched slower than that of the other arm as shown in Figure 4a, which is equivalent to having an additional angle α on the whole U-shaped beam and greatly affects the terms of the coupling stiffness (k

_{xy}/k

_{yx}). Due to the terms of the coupling stiffness introduced by fabrication imperfection, the whole gyroscope unit will have an undesired in-plane movement when the Drive Frames are steadily driven without angular velocity (Ω

_{z}) input, which will mostly affect the quadrature error for Drive Mode to Yaw Mode.

_{y}/Ω

_{x}) input.

_{Rtot}’ is the current system stiffness matrix with rotation movement, T

_{R}is the transfer matrix and K

_{tot}is the ideal system stiffness matrix.

_{1}, D

_{2}, D

_{3}, D

_{4}, D

_{1}’, D

_{2}’, D

_{3}’, D

_{4}’), yaw coupling beams with Big Frame (D

_{5}, D

_{6}, D

_{7}, D

_{8}, Y

_{3}, Y

_{4}, Y

_{5}, Y

_{6}, Y

_{11}, Y

_{12}) and yaw sensing beams in Yaw Frame (Y

_{1}, Y

_{2}). Take all the beams above into consideration, the equivalent matrix k is obviously the summation of each matrix for the coupling beams, the equation is shown below:

_{xx}and k

_{yy}are the diagonal coefficients, while k

_{xy}and k

_{yx}are the off-diagonal coefficients in Equation (4).

_{9}, D

_{10}, D

_{11}, D

_{12}), Pitch/Roll coupling beams in Outer Pitch/Roll Frame (P

_{1}, P

_{2}) and pitch/roll sensing beams in Inner Pitch/Roll Frame (D

_{13}, D

_{14}, D

_{15}, D

_{16}, P

_{3}, P

_{4}). Table 2 shows the part of the structural dimensions with fabrication error, for example 0.1°.

#### 3.1.1. Dynamics Analysis for Drive Mode to Yaw Mode

_{1}and k

_{1}’ are the summation of stiffnesses of U-shaped coupling beams D

_{1}, D

_{2}, D

_{3}, D

_{4}and D

_{1}’, D

_{2}’, D

_{3}’, D

_{4}’; k

_{2}and k

_{2}’ are the stiffness of decoupling beams Y

_{3}and Y

_{4}; k

_{4}and k

_{4}’ are the stiffness of coupling beams Y

_{1}, Y

_{2}, Y

_{5}and Y

_{6}; k

_{3}is summation of the stiffness of coupling beams D

_{7}and D

_{8}between Yaw Sense Frame and Big Frame; k

_{5}and k

_{5}’ are the stiffness of crab-leg beams Y

_{11}and Y

_{12}; k

_{6}is the summation of the stiffness of coupling beams D

_{9}, D

_{10}, D

_{11}, D

_{12}; k

_{8}is the summation of the stiffness of decoupling beams P

_{1}and P

_{2}. All the labels of decoupling beams are shown in Figure 3.

_{1}and m

_{1}’ represent the masses of Drive Frame, m

_{3}represents the Yaw Sense Frame; m

_{2}’ and m

_{2}” represent the Big Frame and Inner Drive Frame respectively and m

_{4}is the Outer Plane Frame (Pitch/Roll Mode). To simplify the dynamic model and equations, the related masses are classified in degrees of freedom: Drive Frames (m

_{1}and m

_{1}’) and Inner Drive Frame (m

_{2}”) have one degree of freedom in direction x; Yaw Sense Frame (m

_{3}) has one degree of freedom in direction y. In particular, Big Frame (m

_{2}′) has two degrees of freedom in direction x and y, while Outer Plane Frame (m

_{2}”) has two degrees of freedom in direction x and z. Since m

_{2}’, m

_{2}” and m

_{4}are driven together in direction x, when driving force is applied on the MIMU, so the three masses can be treated as a whole mass.

_{2}’), Inner Drive Frame (m

_{2}”) and Outer Pitch/Roll Frame (m

_{4}) move together by the driving force. The simplified dynamic equation in driving direction is shown below:

_{2}’) only with two degrees of freedom in direction x and y. So insulating the inner masses of the Big Frame, including the Inner Drive Frame (m

_{2}”) and Pitch/Roll Frames, the simplified dynamic model for Yaw Mode is shown in Figure 6.

_{10}represents the summation of U-shaped decoupling beams Y

_{7}, Y

_{8}, Y

_{9}and Y

_{10}(Figure 3). The other springs are the same in Figure 5. So the simplified dynamic equation in Yaw Sense direction is shown below:

_{z}= 0, owning to the existence of coupling stiffness k

_{yx}, coupling force k

_{yx}x generated by driving force in direction x will attach to the MIMU in direction y, and make the Big frame vibrate in the sense direction.

_{5xy}+ k

_{6xy}+ k

_{8xy})y can be neglected, and the dynamic equations of the gyroscope in drive direction x and sense direction y can be rewritten as follow:

_{d}= F

_{e}sinω

_{d}t, the equation of driving displacement can be expressed below:

_{e}is the amplitude of the electrostatic force, the k

_{xx}is the summation of 2k

_{1xx}, k

_{3xx}, 2k

_{5xx}, k

_{6xx}, k

_{8xx}. ω

_{d}is the frequency of driving voltage and ω

_{x}is the natural resonant frequency of Drive Mode, Q

_{drive}is the quality factor of the Drive Mode. A

_{x,drive}below is the amplitude of driving displacement when ω

_{d}≈ ω

_{x}.

_{z}= 0), the yaw sensing electrodes will be driven in direction y under the coupling term (k

_{5yx}

^{α}+ k

_{10yx}

^{α})x, and quadrature error y

_{Qerror,yaw}between Drive Mode to Yaw Mode can be expressed below:

_{y}is the natural resonant frequency of Yaw Mode. Q

_{yaw}is the quality factor of the Yaw Mode. k

_{5yx}

^{α}and k

_{10yx}

^{α}are the coupling stiffness terms of decoupling beams Y

_{11}, Y

_{12}and Y

_{7}, Y

_{8}, Y

_{9}, Y

_{10}correspondingly. From the equation above, by eliminating the coupling stiffness terms 2k

_{5yx}

^{α}+ k

_{10yx}

^{α}, the quadrature error y

_{Qerror,yaw}can simultaneously be reduced.

_{xy}

^{α}(k

_{yx}

^{α}) introduced by fabrication imperfection can be expressed as (k

_{xx}− k

_{yy})α. By reducing the equivalent angle of etching error α or conducting pre-compensation in the structural design on the key decoupling beams (Y

_{7}, Y

_{8}, Y

_{9}, Y

_{10}and Y

_{11}, Y

_{12}), the quadrature error between Drive Mode to Yaw Mode can be reduced effectively.

_{z}, to simplify calculation, the coupling term (2k

_{5yx}+ k

_{10yx})x which leads to the quadrature error of Yaw Mode can be neglected in Equation (8). When the Drive Mode is in resonant state (ω

_{x}≈ ω

_{d}), and the mechanical sensitivity S

_{yaw}of Drive Mode to Yaw Mode can be easily calculated and expressed below:

#### 3.1.2. Dynamics Analysis for Drive Mode to Pitch/Roll Mode

_{y}applied on the MIMU. The simplified dynamic model of Pitch Mode is shown in Figure 7.

_{2}”) is driven together with the Big Frame (m

_{2}’) via decoupling beams Y

_{7}, Y

_{8}, Y

_{9}and Y

_{10}in Figure 3, so Pitch mode shares the same driving equation (Equation (6)) with the other sense mode as that in Yaw Mode.

_{6}represents the summation of U-shaped driving beams D

_{9}, D

_{10}, D

_{11}, D

_{12}. Besides, k

_{7}is the summation of Trampoline beams P

_{1}and P

_{2}, and k

_{9}is the summation of Trampoline beams P

_{3}and P

_{4}. k

_{8}represents the summation of four double U-shaped beams D

_{13}, D

_{14}, D

_{15}and D

_{16}. The Outer Pitch Frame (m

_{4}) has two degrees of freedom and takes the Inner Pitch Frame (m

_{5}) move in direction z via double U-shaped beams (D

_{13}, D

_{14}, D

_{15}, D

_{16}). The dynamic equation of Pitch Mode in drive direction x and sense direction z can be written below:

_{pitch}of Pitch Mode can be expressed below:

_{z}is the natural resonant frequency of Pitch/Roll Mode, Q

_{pitch}is the quality factor of the Pitch Mode. When the frequency of driving force ω

_{d}is equal to the natural resonant frequency of Drive Mode ω

_{d}, the mechanical sensitivity of Pitch/Roll Mode can be expressed in Equation (14).

_{Qerror,pitch}of Drive Mode to Pitch/Roll Mode is led by the coupling term k

_{zx}

^{θ}= (k

_{7zx}+ k

_{9zx})θ, where θ is the equivalent offset angle by etching error of decoupling beams P

_{1}, P

_{2}, P

_{3}and P

_{4}in Figure 4. The calculation process is the same as that in Equation (11) and expressed below.

_{1}, P

_{2}, P

_{3}and P

_{4}), the quadrature error from Drive Mode to Pitch/Roll Mode can be reduced effectively.

#### 3.2. Cross-Axis Error Analysis between Sense Modes

#### 3.2.1. Cross-Axis Error from Yaw Mode to Pitch Mode

_{z}, the Inner Pitch Frame (m

_{5}) will be motionless in theory. Due to the existence of fabrication imperfection of the coupling beams (D

_{7}, D

_{8}, Y

_{7}, Y

_{8}, Y

_{9}, Y

_{10}, Y

_{11}, Y

_{12}), however, the Big Frame (m

_{2}’) together with the Inner Drive Frame (m

_{2}”) will be driven in turn by Yaw Frame, and further leads to the movement of Pitch Frame (m

_{4}and m

_{5}), which will have an extra output signal called cross-axis error [23]. To analyze the cross-axis error, the dynamic model is established in Figure 8.

_{z}only, and Yaw Sense Frame (m

_{3}) and Big Frame (m

_{2}’) move together in direction y, the displacement can be expressed by Equation (12) with mechanical sensitivity S

_{yaw}. Due to the existence of coupling terms k

_{5xy}

^{α}, Big Frame (m

_{2}’) as well as two Drive Frame (m

_{1}/m

_{1}’) have an extra displacement in direction x. By the transmission of decoupling beams Y

_{11}, Y

_{12}and Y

_{7}, Y

_{8}, Y

_{9}, Y

_{10}(k

_{5}/k

_{5}’ and k

_{10}), the extra displacement will be delivered to Inner Drive Frame (m

_{2}”), and further leads to the movement of Outer Pitch Frame (m

_{4}) and Inner Pitch Frame (m

_{5}).

_{2}”). The dynamic equations are written below:

_{x}is the equivalent resonant frequency for Inner Drive Mode. Q

_{drive}is the equivalent natural quality factors for Inner Drive Mode. A

_{sx}is the amplitude of the drive displacement for Inner Drive Frame (m

_{2}”). Actually, Inner Drive Mode shares the same resonant frequency and quality factor with Drive Mode. If Yaw Mode works in driving frequency ω

_{d}, Equation (16) can be simplified as above.

_{z}, the output of Pitch Mode is only from fabrication error of Pitch Mode, more specifically, from the terms of the coupling stiffness k

_{7zx}

^{θ}+ k

_{9zx}

^{θ}, and Equation (14) can be taken for referenced. So the cross-axis error from Yaw Mode to Pitch Mode can be expressed below:

_{5xy}

^{α}+ k

_{9zx}

^{α}and k

_{7zx}

^{θ}+ k

_{9zx}

^{θ}, more specifically, the coupling stiffness terms of crab-leg beams Y

_{11}, Y

_{12}, U-shaped beams Y

_{7}, Y

_{8}, Y

_{9}, Y

_{10}, trampoline beams P

_{1}, P

_{2}, P

_{3}and P

_{4}, that the equivalent fabrication angle α and θ matters most (Figure 4), should be decreased as much as possible to reduce the value of cross-axis error S

_{yaw}

_{2pitch}.

#### 3.2.2. Cross-Axis Error from Pitch Mode to Yaw Mode

_{5}) is driven in direction z by Coriolis force under the angular velocity input Ω

_{y}, the Yaw Sense Frame (m

_{3}) will be motionless in theory. Fabrication imperfection of the coupling beams (D

_{9}, D

_{10}, D

_{11}, D

_{12}, P

_{1}, P

_{2}) will lead to the movement of Outer Pitch Frame (m

_{4}) in direction z, together with the Inner Drive Frame (m

_{2}”) and Big Frame (m

_{2}’) in direction x, and further lead to the movement of Yaw Sense Frame (m

_{3}). The extra output signal of Yaw Frame is the cross-axis error from Pitch Mode to Yaw Mode.

_{7zx}

^{θ}+ k

_{9zx}

^{θ}, Inner Drive Frame (m

_{2}”) will have an extra movement, and further leads to the movement of Big Frame (m

_{2}’), under the decoupling beams Y

_{7}, Y

_{8}, Y

_{9}, Y

_{10}(k

_{10}). Then, Yaw Sense Frame (m

_{3}) will further be driven by Y

_{11}, Y

_{12}(k

_{5yx}

^{α}/k

_{5yx}

^{α}’). Dynamic model in Figure 5 can be taken for reference and the dynamic equations are written below, from which the equivalent drive displacement for Inner Drive frame together with the Big Frame can be calculated.

_{x}, Q

_{sx}, A

_{sx}and ω

_{d}are the same parameters as those in Equation (16). Therefore, Equation (18) can be simplified as above.

_{y}, the output of Yaw Mode is only from fabrication error of decoupling beams Y

_{11}, Y

_{12}and Y

_{7}, Y

_{8}, Y

_{9}, Y

_{10}, more specifically, from the terms of the coupling stiffness 2k

_{5yx}

^{α}+ k

_{10yx}

^{α}, and Equation (11) can be taken for reference. So the cross-axis error from Pitch Mode to Yaw Mode can be expressed below:

_{5yx}

^{α}+ k

_{10yx}

^{α}and k

_{7zx}

^{θ}+ k

_{9zx}

^{θ}, more specifically, the coupling stiffness terms of the crab-leg beams Y

_{11}/Y

_{12}and Y

_{7}, Y

_{8}, Y

_{9}, Y

_{10}, linked with the Big Frame (m

_{2}’), and trampoline beams P

_{1}, P

_{2}, P

_{3}and P

_{4}. The equivalent fabrication angle α, and θ (Figure 4), should be decreased as much as possible to reduce the value of cross-axis error S

_{pitch2yaw}.

_{yaw}, S

_{pitch}, S

_{roll}represent the mechanical sensitivity of three sense modes; The S

_{yaw}

_{2pitch}(S

_{yaw}

_{2roll}), S

_{pitch}

_{2yaw}(S

_{roll}

_{2yaw}), S

_{roll}

_{2pitch}(S

_{pitch}

_{2roll}) represent the cross-axis error from one sense mode to the other mode, respectively. In addition, the y

_{Qerror,yaw}, z

_{Qerror,pitch}, z

_{Qerror,roll}are the quadrature error corresponding to the sense modes.

^{−6}μm/°/s, and the mechanical sensitivity of Yaw Mode is about 1.59 × 10

^{−4}μm/°/s, with quality factor (Q

_{yaw}) of in-plane movement in vacuum about 300. Quadrature error of Yaw Mode is 1.849 × 10

^{−3}μm with fabrication error α = 0.1° of decoupling beams k

_{5}/k

_{5}’ (Y

_{11}/Y

_{12}) and k

_{10}(Y

_{7}, Y

_{8}, Y

_{9}, Y

_{10}). Likewise, cross-axis error from Yaw Mode to Pitch/Roll Mode is about 2.4 × 10

^{−6}μm/°/s, and the mechanical sensitivity of Pitch/Roll Mode is about 1.42 × 10

^{−4}μm/°/s, with quality factor (Q

_{pitch}/Q

_{roll}) of out-of-plane movement in vacuum about 1000. Quadrature error of Pitch/Roll Mode is 4.397 × 10

^{−3}μm with fabrication error θ = 0.1° of decoupling beams k

_{7}(P

_{1}/P

_{2}) and k

_{9}(P

_{3}/P

_{4}). Summary of all the parameters above is shown in Table 4.

## 4. Fabrication Process

_{2}is worked as sacrificial layer. Reactive Ion Etching (RIE) is adopted to etch the SiO

_{2}and exposes the underneath silicon for the MIMU structure, including decoupling beams, comb fingers and proof masses etc. (Figure 10a).

_{2}covering on the anchors for anodic bonding should be removed, with wet etching in buffered oxide etch (BOE) solution (49%HF: 40%NH

_{4}F = 1:5).

_{2}deposited with plasma enhanced chemical vapour deposition (PECVD) for sacrificial layer for next step, the thickness of SiO

_{2}can be chosen from 500 nm to 600 nm.

_{2}should be removed by BOE solution (49% HF: 40% NH

_{4}F = 1:5) for several hours.

_{7}, Y

_{8}, Y

_{9}and Y

_{10}), which is the main source of the quadrature error of different sense modes. By measuring the size of the MIMU structure in microscope, fabrication quality and fabrication error can be acquired. Figure 13 shows the measurement for some vital structural parts of the MIMU. In order to improve the contrast of the structure and make it clear to express the vital structure parts of the MIMU in the photograph by microscope, the SiO

_{2}covered on the structure with pink color is not disposed by BOE. Table 5 shows both the designed and actual parameters of the vital structure of the MIMU.

## 5. Experimental Results

_{drive}about 455; resonant frequency of the Yaw Mode is 7054.4 Hz with the quality factor Q

_{yaw}about 66, whose frequency split is about 104.2 Hz relative to the Drive Mode; the Pitch Mode and Roll Mode resonant frequencies are 7034.2 Hz and 7040.5 Hz with quality factor Q

_{pitch}for 109 and Q

_{roll}for 107, whose frequency split are about 84 Hz and 90.3 Hz. Although the Pitch Frame and Roll Frame of the MIMU are totally symmetrical in design, there still exists mismatching in actual fabrication process, whereas the difference is in an acceptable range. Vacuum sealed packaging and temperature prediction and controlling of circuit [31] can enhance the quality factor and further improve the performance of the whole device, which is the research emphasis in future work.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Cui, J.; Chi, X.Z.; Ding, H.T.; Lin, L.T.; Yang, Z.C.; Yan, G.Z. Transient response and stability of the AGC-PI closed-loop controlled MEMS vibratory gyroscopes. J. Micromech. Microeng.
**2009**, 19, 125015. [Google Scholar] [CrossRef] - Yoon, S.; Park, U.; Rhim, J.; Yang, S.S. Tactical grade MEMS vibrating ring gyroscope with high shock reliability. Microelectron. Eng.
**2015**, 142, 22–29. [Google Scholar] [CrossRef] - Ayazi, F. Multi-DOF inertial MEMS: From gaming to dead reckoning. In Proceedings of the IEEE Solid-State Sensors, Actuators and Microsystems Conference, Beijing, China, 5–9 June 2011; pp. 2805–2808. [Google Scholar]
- Braghin, F.; Resta, F.; Leo, E.; Spinola, G. Nonlinear dynamics of vibrating MEMS. Sens. Actuators A Phys.
**2007**, 134, 98–108. [Google Scholar] [CrossRef] - Li, Z.; Yang, Z.; Xiao, Z.; Hao, Y.; Li, T.; Wu, G. A bulk micromachined vibratory lateral gyroscope fabricated with wafer bonding and deep trench etching. Sens. Actuators A Phys.
**2000**, 83, 24–29. [Google Scholar] [CrossRef] - Liu, J.; Tang, J.; Shi, Y. The design and test of the single chip integration accelerometer gyroscope. In Proceedings of the 5th WSEAS International Conference on Instrumentation, Measurement, Circuits and Systems, Hangzhou, China, 16–18 April 2006; pp. 333–339. [Google Scholar]
- Boysel, R.M.; Fiscus, T.E.; Ross, L.J. Development of a single chip 6 DOF MEMS IMU for robotic and UV navigation. In Proceedings of the International Technical Meeting of the Satellite Division of the Institute of Navigation, Portland, OR, USA, 20–23 September 2011; pp. 3930–3936. [Google Scholar]
- Geen, J.A.; Sherman, S.J.; Chang, J.F.; Lewis, S.R. Single-chip surface micromachined integrated gyroscope with 50°/h Allan deviation. IEEE J. Solid-State Circuits
**2002**, 37, 1860–1866. [Google Scholar] [CrossRef] - Kim, J.; Park, S.; Kwak, D.; Ko, H.; Cho, D.I.D. An x-axis single-crystalline silicon microgyroscope fabricated by the extended SBM process. J. Microelectromech. Syst.
**2005**, 14, 444–455. [Google Scholar] - Ding, H.; Cui, J.; Liu, X.; Chi, X.; Yang, Z.; Yan, G. A highly double-decoupled self-oscillation gyroscope operating at atmospheric pressure. In Proceedings of the 7th IEEE Sensors Conference, Lecce, Italy, 27–29 October 2008; pp. 674–677. [Google Scholar]
- Ding, H.; Liu, X.; Lin, L.; Chi, X.; Cui, J.; Kraft, M.; Yang, Z.; Yan, G. A high-resolution silicon-on-glass, z axis gyroscope operating at atmospheric pressure. IEEE Sens. J.
**2010**, 10, 1066–1074. [Google Scholar] [CrossRef] - Guo, Z.Y.; Lin, L.T.; Zhao, Q.C.; Cui, J. An Electrically Decoupled Lateral-Axis Tuning Fork Gyroscope Operating at Atmospheric Pressure. In Proceedings of the International Conference on MICRO Electro Mechanical Systems, Sorrento, Italy, 25–29 January 2009; pp. 104–107. [Google Scholar]
- Guo, Z.Y.; Lin, L.T.; Zhao, Q.C.; Yang, Z.C.; Xie, H.; Yan, G.Z. A lateral-axis microelectromechanical tuning-fork gyroscope with decoupled comb drive operating at atmospheric pressure. J. Microelectromech. Syst.
**2010**, 19, 458–468. [Google Scholar] - Wang, M.C.; Jiao, J.W.; Yan, P.L.; Mi, B.W.; Qin, S. A novel tri-axis MEMS gyroscope with in-plane tetra-pendulum proof masses and enhanced sensitive springs. J. Micromech. Microeng.
**2014**, 24, 325–332. [Google Scholar] [CrossRef] - Zhao, Q.C.; Liu, X.S.; Lin, L.T.; Guo, Z.Y.; Cui, J.; Chi, X.Z. A doubly decoupled micromachined vibrating wheel gyroscope. In Proceedings of the TRANSDUCERS 2009-2009 International Solid-State Sensors, Actuators and Microsystems Conference, Denver, CO, USA, 21–25 June 2009; pp. 296–299. [Google Scholar]
- Chouvion, B.; Fox, C.H.J.; Mcwilliam, S.; Popov, A.A. In-plane free vibration analysis of combined ring-beam structural systems by wave propagation. J. Sound Vib.
**2010**, 329, 5087–5104. [Google Scholar] [CrossRef] - Iyer, S.; Zhou, Y.; Mukherjee, T. Analytical modeling of cross-axis coupling in micromechanical springs. Model. Simul. Microsyst.
**1999**, 632–635. [Google Scholar] - Tatar, E.; Alper, S.E.; Akin, T. Quadrature-error compensation and corresponding effects on the performance of fully decoupled mems gyroscopes. J. Microelectromech. Syst.
**2012**, 21, 656–667. [Google Scholar] [CrossRef] - Riaz, K.; Bazaz, S.A.; Saleem, M.M.; Shakoor, R.I. Design, damping estimation and experimental characterization of decoupled 3-dof robust mems gyroscope. Sens. Actuators A Phys.
**2011**, 172, 523–532. [Google Scholar] [CrossRef] - Sonmezoglu, S.; Taheri-Tehrani, P.; Valzasina, C.; Falorni, L.G.; Zerbini, S.; Nitzan, S. Single-structure micromachined three-axis gyroscope with reduced drive-force coupling. IEEE Electron Device Lett.
**2015**, 36, 953–956. [Google Scholar] [CrossRef] - Mochida, Y.; Tamura, M.; Ohwada, K. A micromachined vibrating rate gyroscope with independent beams for the drive and detection modes. Sens. Actuators A Phys.
**2000**, 80, 170–178. [Google Scholar] [CrossRef] - Vigna, B. Tri-axial MEMS gyroscopes and six degree-of-freedom motion sensors. In Proceedings of the 2011 IEEE Electron Devices Meeting (IEDM), Washington, DC, USA, 5–7 December 2011; pp. 29–31. [Google Scholar]
- Xia, D.; Kong, L.; Gao, H. Design and analysis of a novel fully decoupled tri-axis linear vibratory gyroscope with matched modes. Sensors
**2015**, 15, 16929–16955. [Google Scholar] [CrossRef] [PubMed] - Xia, D.; Kong, L.; Gao, H. A mode matched triaxia lvibratory wheel gyroscope with fully decoupled structure. Sensors
**2015**, 15, 28979–29002. [Google Scholar] [CrossRef] [PubMed] - Ni, Y.; Li, H.; Huang, L.; Ding, X.; Wang, H. On bandwidth characteristics of tuning fork micro-gyroscope with mechanically coupled sense mode. Sensors
**2014**, 14, 13024–13045. [Google Scholar] [CrossRef] [PubMed] - Painter, C. Micromachined Vibratory Gyroscopes with Imperfections. Ph.D. Thesis, University of California, Irvine, CA, USA, 2005. [Google Scholar]
- Wang, R.; Cheng, P.; Xie, F.; Young, D.; Hao, Z. A multiple-beam tuning-fork gyroscope with high quality factors. Sens. Actuators A Phys.
**2011**, 166, 22–33. [Google Scholar] [CrossRef] - Wisher, S.; Shao, P.; Norouzpour-Shirazi, A.; Yang, Y. A high-frequency epitaxially encapsulated single-drive quad-mass tri-axial resonant tuning fork gyroscope. In Proceedings of the IEEE International Conference on MICRO Electro Mechanical Systems, Shanghai, China, 24–28 January 2016; pp. 930–933. [Google Scholar]
- Nguyen, M.N.; Ha, N.S.; Nguyen, L.Q.; Chu, L.Q.; Vu, H.N. Z-Axis Micromachined Tuning Fork Gyroscope with Low Air Damping. Micromachines
**2017**, 8, 42. [Google Scholar] [CrossRef] - Chen, X.G.; Li, Y.M. Design and Analysis of a New High Precision Decoupled XY Compact Parallel Micromanipulator. Micromachines
**2017**, 8, 82. [Google Scholar] [CrossRef] - Xia, D.; Kong, L.; Hu, Y.; Ni, P. Silicon microgyroscope temperature prediction and control system based on BP neural network and Fuzzy-PID control method. Measurement Sci. Technol.
**2015**, 26, 025101. [Google Scholar] [CrossRef]

**Figure 2.**The modes of the gyroscope unit for the MIMU: (

**a**) The Drive Mode; (

**b**) The Yaw Mode (Ω

_{z}); (

**c**) The Pitch Mode (Ω

_{y}); (

**d**) The Roll Mode (Ω

_{x}).

**Figure 4.**(

**a**) In-plane fabrication error α of quadrature error for Drive Mode to Yaw Mode; (

**b**) Out-of-plane fabrication error θ of quadrature error for Drive Mode to Pitch Mode.

**Figure 9.**Simulation of mechanical output for sense modes: (

**a**) Yaw Mode with angular rate input Ω

_{z}; (

**b**) Pitch (Roll) Mode with angular rate input Ω

_{y}(Ω

_{x}).

**Figure 11.**Photographs of the MIMU structure: (

**a**) Whole structure of the MIMU; (

**b**) Detailed comb fingers of the driving electrodes in Drive Frame; (

**c**) Detailed comb fingers of the pitch electrodes in Pitch Frame; (

**d**) The anchors and U-shaped coupling beams of Drive Frame; (

**e**) Trampoline beams in Inner-Roll Frame; (

**f**) Trampoline beams in Outer-Roll Frame; (

**g**) Detailed decoupling beams between Big Frame and Outer-Roll Frame.

**Figure 12.**(

**a**) Bottom trenches first and anchors afterwards in ICP etching process for smooth edge; (

**b**) Anchors first, and bottom trenches afterwards with the protection of photoresist in ICP etching process for rough edge.

**Figure 13.**Measurement for vital structure parts of the MIMU. (

**a**) Crab-leg beams and U-shaped beams linking the Big Frame; (

**b**) Trampoline beams in Outer-Pitch Frame; (

**c**) Double U-shaped beams in Inner-Roll Frame; (

**d**) U-shaped beams linking Drive Frame and Big Frame.

Function Description | Part Number | ||
---|---|---|---|

Gyroscope Unit | Drive Mode | U-shaped beam | D_{1}, D_{2}, D_{3}, D_{4} |

D_{1}’, D_{2}’, D_{3}’, D_{4}’ | |||

D_{5}, D_{6}, D_{7}, D_{8} | |||

Drive electrodes | D-a, D-b | ||

Yaw Mode | U-shaped beam | Y_{1}, Y_{2}, Y_{3}, Y_{4}, Y_{5}, Y_{6} | |

Crab-leg beam | Y_{11}, Y_{12} | ||

Yaw electrodes | Y-a | ||

Pitch/Roll Mode | U-shaped beam | D_{9}, D_{10}, D_{11}, D_{12} | |

Double U-shaped beam | D_{13}, D_{14}, D_{15}, D_{16} | ||

Trampoline beam | P_{1}, P_{2}, P_{3}, P_{4} | ||

Pitch electrodes | P-a, P-b | ||

Roll electrodes | R-a, R-b | ||

Accelerometer Unit | Acceleration for direction x/y | Acc-xy | |

Acceleration for direction z | Acc-z |

Tabs | Summation of Coupling Beams | K_{xx} [N/m] | k_{yy} [N/m] | Coupling Terms [N/m] |
---|---|---|---|---|

k_{1}/k_{1}’ | D_{1}, D_{2}, D_{3}, D_{4}/D_{1}’, D_{2}’, D_{3}’, D_{4}’ | 49.6576 | 1950 | 190.0342 |

k_{2}/k_{2}’ | Y_{3}/Y_{4} | 1860.7 | 43.2814 | 181.7419 |

k_{3} | D_{5}, D_{6}, D_{7}, D_{8} | 53.3203 | 1998 | 194.4680 |

k_{4} | Y_{1}, Y_{2}, Y_{5}, Y_{6} | 1603.6 | 27.9724 | 157.5628 |

k_{5}/k_{5}’ | Y_{11}/Y_{12} | 53.4492 | 96.8923 | 39.6541 |

k_{6} | D_{9}, D_{10}, D_{11}, D_{12} | 49.6576 | 1950 | 380.0684 |

k_{7} | P_{1}, P_{2} | 2165.9 | 101.0704 (k_{zz}) | 112.4212 |

k_{8} | D_{13}, D_{14}, D_{15}, D_{16} | 71.2309 | 3482.1 | 341.0869 |

k_{9} | P_{3}, P_{4} | 2163.5 (k_{yy}) | 203.9288 (k_{zz}) | 115.5196 |

k_{10} | Y_{7}, Y_{8}, Y_{9}, Y_{10} | 1669.5 | 31.4864 | 163.8014 |

Parameter and Variable Name | Symbol | Value [Unit] |
---|---|---|

Drive Frame | m_{1}/m_{1}’ | 1.04508 × 10^{−4} g |

Big Frame | m_{2}’ | 1.6699392 × 10^{−4} g |

Inner Drive Frame | m_{2}” | 3.456 × 10^{−5} g |

Mass in Yaw Mode | m_{3} | 1.154208 × 10^{−4} g/2.8224 × 10^{−5} g |

Outer Pitch/Roll Frame | m_{4} | 2.5531 × 10^{−4} g |

Inner Pitch/Roll Frame | m_{5} | 7.503 × 10^{−5} g |

Mechanical Sensitivity (μm/°/s) | Cross-Axis Error (μm/°/s) | Quadrature Error (μm) | Capacity Sensitivity (F/°/s) | |
---|---|---|---|---|

Yaw Mode | S_{yaw} = 1.59 × 10^{−4}Q _{yaw} ≈ 300 | S_{pitch2yaw}/S_{pitch2yaw} = 5.6 × 10^{−6} | y_{Qerror,yaw} = 1.849 × 10^{−3}(α = 0.1°) | S_{cyaw} = 8.56 × 10^{−17} |

Pitch/Roll Mode | S_{pitch}/S_{roll} = 1.42 × 10^{−4}Q _{pitch} ≈ 1000 | S_{yaw2pitch}/S_{roll2pitch} = 2.4 × 10^{−6} | z_{Qerror,pitch}/z_{Qerror,roll} = 4.397 × 10^{−3}(θ = 0.1°) | S_{cpitch}/S_{croll} = 3.43 × 10^{−17} |

Tabs | Dimensions | Design Size (μm) | Actual Measured Size (μm) | Fabrication Rotation Error (°) | |
---|---|---|---|---|---|

U-shaped beam (D _{5}–D_{8}/D_{1}–D_{4}/Y _{3}, Y_{4}/Y_{7}–Y_{10}/Y _{1}, Y_{2}, Y_{5}, Y_{6}) | L^{u} = 400/410/430/480 /500 w ^{u}= 10 | L^{u} = 400.21/409.58/430.54/478.98 /500.16 w ^{u} = 11.09 | α = 0.13–0.53 | ||

Crab-leg beam (Y _{11}, Y_{12}) | L^{c}_{1} = 2040L ^{c}_{2} = 1622.5w ^{c}_{1} = 45w ^{c}_{2} = 45 | L^{c}_{1} = 2038.37L ^{c}_{2} = 1623.29w ^{c}_{1} = 44.27w ^{c}_{2} = 44.34 | α = 0.15–0.55 | ||

Double U-shaped beam (D _{13}–D_{16}) | L^{d−u} = 460w ^{d−u} = 10 | L^{d−u} = 460.74w ^{d−u} = 10.32 | α = 0.11–0.36 | ||

Trampoline beams (P _{1}, P_{2}/P_{3}, P_{4}) | L^{t}_{1} = 176/250L ^{t}_{2} = 170w ^{t} = 8 | L^{t}_{1} = 175.32/248.87L ^{t}_{2} = 169.46w ^{t} = 7.83 | θ = 0.11–0.24 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Xia, D.; Xu, L.
Coupling Mechanism Analysis and Fabrication of Triaxial Gyroscopes in Monolithic MIMU. *Micromachines* **2017**, *8*, 310.
https://doi.org/10.3390/mi8100310

**AMA Style**

Xia D, Xu L.
Coupling Mechanism Analysis and Fabrication of Triaxial Gyroscopes in Monolithic MIMU. *Micromachines*. 2017; 8(10):310.
https://doi.org/10.3390/mi8100310

**Chicago/Turabian Style**

Xia, Dunzhu, and Lei Xu.
2017. "Coupling Mechanism Analysis and Fabrication of Triaxial Gyroscopes in Monolithic MIMU" *Micromachines* 8, no. 10: 310.
https://doi.org/10.3390/mi8100310