# Effect of Uneven Electrostatic Forces on the Dynamic Characteristics of Capacitive Hemispherical Resonator Gyroscopes

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Gyroscope Description and Theoretical Analysis

#### 2.1. Gyroscope Structure

#### 2.2. Theoretical Analysis

#### 2.2.1. Uneven Electrostatic Forces

#### 2.2.2. Resonator Deformation

_{f}, the radius of the middle surface of the resonator R

_{d}, the half thickness of the shell $h/2$, and the uneven capacitance gap d.

_{d}. The point A on the middle surface is deformed and moved to the point ${A}^{\prime}$, which obtains the deformation equation on the radius vector (

**OA**), the radius vector (

**OA′**), and the displacement vector (

**W**) after deformation, shown as follows in Equation (7).

_{2}, V

_{2,}and W

_{2}are the Rayleigh–Ritz functions [25,27].

_{2}and ϑ are respectively the wave amplitude and vibration angle of the resonant vibration, a and b are the values related to the wave amplitude.

## 3. Dynamic Model

#### 3.1. Dynamic Characteristics of Uneven Electrostatic Forces

_{1}located at the 0° electrode axis and E

_{3}located at the 45° electrode axis. E

_{1}is used for driving the resonator and stabilizing the amplitude, and E

_{3}is used for force feedback.

#### 3.2. Output Error Model

_{m}and f

_{a}, that is, the force feedback electrode can hold the standing wave at the null position because the constant amplitude of the resonator signifies a constant electrostatic force. Thus, according to Equation (27), $\vartheta $ can be held at the null position to make b equal to zero, which results in Equation (24) becoming Equation (30).

## 4. Simulation Analysis

#### 4.1. Effect of the First Four Harmonics

^{−3}°/h. The zero biases caused by the second harmonic, third harmonic, and fourth harmonic respectively exhibit periodic changes of four-period oscillation, six-period oscillation, and eight-period oscillation. Their maxima are respectively 2.274 × 10

^{−5}°/h, 7.563 × 10

^{−5}°/h and 1.03 × 10

^{−4}°/h which are smaller than that of the first harmonic, which can be neglected. For the same amplitude, the higher the harmonic order in the uneven capacitance gap is, the more uneven the electrostatic forces are, and the larger the zero bias is.

^{−4}°/s, 7.318 × 10

^{−4}°/s and 7.06 × 10

^{−4}°/s that is an order of magnitude smaller than that of the first harmonic (7.659 × 10

^{−3}°/s), which indicates that the output error of HRG is mainly affected by the uneven electrostatic forces caused by the first harmonic. It is seen that the effect of uneven electrostatic forces on the output of the HRG cannot be neglected, in particular the uneven electrostatic forces caused by the first harmonic.

^{−3}°/h which is caused by the first four harmonics, which indicates that each harmonic component has a large degree of influence on the zero bias. However, it is seen from Figure 8b–d, that the output error of the angular rate is approximately periodically varied with the change of the phase, which is mainly dominated by the first harmonic component. In Figure 8d, when the angular rate is 1°/s, the maximum of output error is 0.0095°/s. In the single-electrode control mode, the tolerance range of excitation capacitance gap is given, which is, zero bias controlled within the range of 4.124 × 10

^{−4}°/h, and the output error controlled within the range of 0.2885%, while the amplitude of the uneven capacitance gap should be less than 0.33 μm.

#### 4.2. Effect of Uneven Electrostatic Forces

_{1}= 0.1 μm (as cyan triangle marker), the maximum of the nonlinearities is 0.8586 ppm (less than 1 ppm). When e

_{1}equals 0.3 μm (as red star marker), the maximum of nonlinearities is 4.062 ppm. The main conclusion drawn from this section studies is that the bigger the amplitude of uneven capacitance gap, the more uneven are the electrostatic forces, and the bigger is the scale factor nonlinearity. The amplitude and initial phase of uneven capacitance gap codetermine the differences of the uneven electrostatic forces as well as the influence on the scale factor nonlinearity.

## 5. Conclusions

^{−4}°/h, the modification rate of output error being controlled within 0.2885%, and the scale factor nonlinearity being controlled within 4 ppm, the amplitude of the uneven capacitive gap should be less than 0.3 µm. Therefore, it is advantageous to improve the machining and installation technology of the capacitive resonator for better performance of the HRG.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The basic structure of the HRG. (

**a**) Simple structure diagram of the HRG; (

**b**) The three main components (the resonator, inner base, and outer base).

**Figure 3.**The selected simulation results of resonator deformation caused by the different amplitudes of the first harmonic of uneven capacitance gap.

**Figure 4.**Structure diagram of the resonator: (

**a**) The deformation of the resonator; (

**b**) The excitation capacitance driving the resonator.

**Figure 6.**The zero biases caused by the first four harmonics of uneven capacitance gap are respectively: (

**a**) Zero bias caused by the first harmonic; (

**b**) Zero bias caused by the second harmonic; (

**c**) Zero bias caused by the third harmonic; (

**d**) Zero bias caused by the fourth harmonic.

**Figure 7.**The output errors caused by the first four harmonics of uneven capacitance gap under 1°/s are respectively: (

**a**) Output error caused by the first harmonic; (

**b**) Output error caused by the second harmonic; (

**c**) Output error caused by the third harmonic; (

**d**) Output error caused by the fourth harmonic.

**Figure 8.**The HRG output error caused by the first four harmonics of uneven capacitance gap under the different input angular rates are respectively: (

**a**) Zero bias caused by the first four harmonics; (

**b**) Output error (Input: 10

^{−3}°/s); (

**c**) Output error (Input: 0.1°/s); (

**d**) Output error (Input: 1°/s).

**Figure 9.**The HRG output error caused by 1.3 μm–270° uneven capacitance gap in the angular rate 1°/s: (

**a**) Harmonic analysis of the HRG output; (

**b**) The modification rate of output error of the HRG output.

**Figure 10.**The fitting curves of the HRG output under the same harmonic amplitude (1 μm) and different initial phases (0°, 90°, 180°, and 270°) of uneven capacitance gap are respectively: (

**a**) Initial phase 0°; (

**b**) Initial phase 90°; (

**c**) Initial phase 180°; (

**d**) Initial phase 270°.

**Figure 11.**Fitting residuals of angular rate under 1 μm–0°, 90°, 180°, and 270° uneven capacitance gap.

**Figure 12.**Scale factor nonlinearity: (

**a**) Scale factor nonlinearities caused by different uneven capacitance gaps; (

**b**) Effect of amplitudes of uneven capacitance gap on the average nonlinearity.

Uneven Capacitance Gap | First Harmonic | Second Harmonic | Third Harmonic | Fourth Harmonic |
---|---|---|---|---|

Amplitude | 0.0078°/s | 0.0015°/s | 0.0021°/s | 0.0026°/s |

Initial phase | 123.82° | 67.30° | 11.39° | 314.82° |

Input Angular Rate (°/s) | Output Angular Rate (°/s) | Input Angular Rate (°/s) | Output Angular Rate (°/s) | Input Angular Rate (°/s) | Output Angular Rate (°/s) | Input Angular Rate (°/s) | Output Angular Rate (°/s) |
---|---|---|---|---|---|---|---|

−3 | −2.9808 | −0.07 | −0.0696 | 0.003 | 0.0030 | 0.5 | 0.4968 |

−2.5 | −2.4840 | −0.05 | −0.0497 | 0.005 | 0.0050 | 0.7 | 0.6955 |

−2 | −1.9872 | −0.03 | −0.0298 | 0.007 | 0.0070 | 1 | 0.9936 |

−1.5 | −1.4904 | −0.01 | −0.0099 | 0.01 | 0.0099 | 1.5 | 1.4904 |

−1 | −0.9936 | −0.007 | −0.0070 | 0.03 | 0.0298 | 2 | 1.9872 |

−0.7 | −0.6955 | −0.005 | −0.0050 | 0.05 | 0.0497 | 2.5 | 2.4840 |

−0.5 | −0.4968 | −0.003 | −0.0030 | 0.07 | 0.0696 | 3 | 2.9808 |

−0.3 | −0.2981 | −0.001 | −0.0010 | 0.1 | 0.0994 | ||

−0.1 | −0.0994 | 0.001 | 0.0010 | 0.3 | 0.2981 |

First Harmonic | 1 μm–0° | 1 μm–90° | 1 μm–180° | 1 μm–270° |
---|---|---|---|---|

K_{1} | 0.9936 | 0.9957 | 1.0065 | 1.0043 |

K_{2} | 0 | 2.7756 × 10^{−17} | 1.3878 × 10^{−17} | 2.7756 × 10^{−17} |

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**MDPI and ACS Style**

Xu, Z.; Yi, G.; Er, M.J.; Huang, C.
Effect of Uneven Electrostatic Forces on the Dynamic Characteristics of Capacitive Hemispherical Resonator Gyroscopes. *Sensors* **2019**, *19*, 1291.
https://doi.org/10.3390/s19061291

**AMA Style**

Xu Z, Yi G, Er MJ, Huang C.
Effect of Uneven Electrostatic Forces on the Dynamic Characteristics of Capacitive Hemispherical Resonator Gyroscopes. *Sensors*. 2019; 19(6):1291.
https://doi.org/10.3390/s19061291

**Chicago/Turabian Style**

Xu, Zeyuan, Guoxing Yi, Meng Joo Er, and Chao Huang.
2019. "Effect of Uneven Electrostatic Forces on the Dynamic Characteristics of Capacitive Hemispherical Resonator Gyroscopes" *Sensors* 19, no. 6: 1291.
https://doi.org/10.3390/s19061291