# Micromachined Rate-Integrating Gyroscopes: Concept, Asymmetry Error Sources and Phenomena

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## Abstract

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## 1. Introduction

- (1)
- We applied a separation principle method to deeply characterize the error sources of RIG operation. A real MEMS gyro has a variety of non-ideal factors, and it is difficult to quantify their influences. Separated error study can simplify the analysis and better characterize the consequences.
- (2)
- We abridged the ZRO drift phenomena and scientific principles. The error factors of frequency split, damping and quality factor mismatch are added to the ideal gyro model and analyzed. Their errors, as time domain phenomena, are also explained to help engineers locate the major drifting source in practice.
- (3)
- We verified the theoretical study through a case to case comparison. A series of tests were utilized to demonstrate the influences of the major error sources with visualized trajectories.

## 2. Theoretical Analysis of RIG Errors and Phenomena

#### 2.1. Ideal Symmetric RIG Principle

#### 2.2. RIG with Anisoelasticity

- (1)
- The anisotropic properties of the material. MEMS gyros are typically made by single crystal silicon, whose mechanical properties are different along different orientations because of its crystal structures. Even assuming that the mechanical structures of each vibration mode are fabricated as perfectly symmetric, the equivalent stiffness would be different.
- (2)
- Fabrication imperfection. The micro gyros are fabricated through a series of processes, including photolithography, etching and deposition. Though these processes are considered mature enough to withhold the mechanical tolerance, it is very common to yield a stiffness difference level of 5% between the two modes.
- (3)
- Thermal mechanical deformations. As the temperature changes, the dimensions of the MEMS gyro also expand/shrink to result in equivalent stiffness variation. The variations of the two modes will not be uniform which will lead to further splitting.

#### 2.3. RIG with Damping Error Factors

## 3. Validation and Discussions

#### 3.1. Simulation Configuration

#### 3.2. Stiffness Mismatch and Coupling Results

#### 3.3. Damping Coupling and Quality Factor Mismatch Results

#### 3.4. Discussions and Suggestions

- (1)
- The periodic oscillation motions at the angle output are caused by the frequency split or stiffness coupling.
- (2)
- The exponential drift toward one major axis/mode is induced by the quality factor mismatch and damping coupling term.

- (1)
- When giving the symmetry the highest priority, e.g., designing a gyro in a converse way to reduce the frequency split/damping mismatch/mode coupling, even conducting a lower quality factor is acceptable. Developing a gyro using isotropic materials (e.g., poly silicon) is another solution.
- (2)
- Evaluating the RIG behavior, locating the major error source and modifying the design according to the analysis in this work. If the symmetry is already reaching the limits, it can be further optimized. If the ZRO pattern is dominated by drifting behavior, a higher Q and lower resonant frequency can improve the ZRO stability; if the ZRO behaves in an oscillation manner, higher resonant frequencies will reduce this error.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ZRO | Zero rate output |

MEMS | Micro-electro-mechanical systems |

Q | Quality factor |

RIG | Rate-integrating gyroscopes |

WA | Whole-angle |

## Appendix A. Detailed Derivation of Conventional Rate Gyro System

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**MDPI and ACS Style**

Qi, Z.; Wu, J.; Li, C.; Zhao, W.; Cheng, Y.
Micromachined Rate-Integrating Gyroscopes: Concept, Asymmetry Error Sources and Phenomena. *Symmetry* **2020**, *12*, 801.
https://doi.org/10.3390/sym12050801

**AMA Style**

Qi Z, Wu J, Li C, Zhao W, Cheng Y.
Micromachined Rate-Integrating Gyroscopes: Concept, Asymmetry Error Sources and Phenomena. *Symmetry*. 2020; 12(5):801.
https://doi.org/10.3390/sym12050801

**Chicago/Turabian Style**

Qi, Zhenxiang, Jin Wu, Chong Li, Wanliang Zhao, and Yuxiang Cheng.
2020. "Micromachined Rate-Integrating Gyroscopes: Concept, Asymmetry Error Sources and Phenomena" *Symmetry* 12, no. 5: 801.
https://doi.org/10.3390/sym12050801