# Random Error Analysis of MEMS Gyroscope Based on an Improved DAVAR Algorithm

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Allan Variance Analysis and Dynamic Allan Variance Analysis

#### 2.1. Allan Variance Analysis

#### 2.2. Allan Variance of Analog MEMS Gyroscope Signal

^{−3}°/h

^{1/2}. The least square fitting results are as shown in Figure 3.

#### 2.3. Dynamic Allan Variance (DAVAR) Analysis

- (1)
- t
_{1}is defined as a fixed analysis time t, t = t_{1}. - (2)
- x
_{1}(t) is obtained by capturing the original signal with a rectangular window whose width is T and center is on t_{1}. - (3)
- The Allan standard deviation $\sigma ({t}_{1},\tau )$ is obtained by calculating the clipped signal x
_{1}(t). - (4)
- t
_{2}is defined as another analysis time point t, t = t_{2}. The two interception signals overlap through repeat step (2), and then steps (3) and (4) are repeated, respectively. The Allan standard deviation sequence $\sigma ({t}_{1},\tau ),\sigma ({t}_{2},\tau )\cdots \sigma ({t}_{n},\tau )$ is finally obtained.

## 3. Kurtosis and Sliding Kurtosis Contribution Coefficient

_{ωp}is obtained by intercepting the original signal with a L length rectangular window at the p data point in time t

_{p}.

_{mp}is defined as the sequence in the original signal sequence excluding the x

_{ωp}sequence. Though calculation, the kurtosis of the x

_{mp}sequence is as follows.

_{mp}is the mean value of x

_{mp}and σ

_{mp}is the standard deviation of x

_{mp}. The sliding kurtosis contribution coefficient of sequence x

_{ωp}is as follows.

## 4. Improved DAVAR Algorithm

#### 4.1. Extension

#### 4.2. Improved Window Length Adjusting Adaptively Algorithm

_{1},K

_{2}], such that at the analysis moment t

_{p}, the kurtosis value of this sequence K

_{p}is calculated and the following relation is obtained.

_{mp}of the front L − l length signal in the L window intercepting signal, the sliding kurtosis contribution coefficient $S=\left|{K}_{p}-{K}_{mp}\right|/{K}_{p}$ of the l window is calculated. S reflects the stability of the signal at the end of the L window and achieves an accurate sensitivity to the end of the intercepted signal sequence. The sliding kurtosis contribution coefficient in [S

_{1},S

_{2}] is designed to represent the stable signal, which leads to the existence of Equation (17) on the t

_{p}moment.

_{2}reflects whether there is an impact component at the ${t}_{p}+\frac{L-l}{2}{\tau}_{0}$ moment. Where ${\tau}_{0}$ is the sampling interval.

_{p}) is 0, the characterization of the signal is stationary, and the end of the intercept signal contains no impact components. When f(t

_{p}) is 1, the signal is unstable, and the rising edge of f(t

_{p}) is collected, which is known as the impact component, exists at time ${t}_{p}+\frac{L-l}{2}{\tau}_{0}$.

_{max}for a steady signal and the minimum window length L

_{min}for an unstable signal are usually designed. When f(t

_{p}) is 1, the signal contains the impact component at the ${t}_{p}+\frac{L-l}{2}{\tau}_{0}$ moment, and the L window starts to capture the signal at time ${t}_{p}+\frac{L-l}{2}{\tau}_{0}$ with the minimum window length L

_{min}. Assuming that the initial window length of the L window is ${L}^{\prime}$, the maximum distance of the center about the L window moving distance is $\frac{{L}^{\prime}-{L}_{\mathrm{min}}}{2}$, and the design of the window length variation factor ∆L

_{1}is as follows.

_{p}) = 1, and at the next point t

_{p}

_{+1}, the L window’s window length L(t

_{p}

_{+1}) is as follows:

_{max}. The maximum distance of L window center point moving is $\frac{{L}_{\mathrm{max}}-{L}^{\prime}}{2}$, then the window length variation factor ∆L

_{0}is designed at this time.

_{p}

_{+1}, the L window’s window length L(t

_{p}

_{+1}) is as follows.

## 5. Simulation and Test

_{max}= 600, L

_{min}= 400; K

_{1}= 2.5, K

_{2}= 3.3; S

_{1}= 0, S

_{2}= 0.07; l = 100; ∆L

_{0}= ∆L

_{1}= 2. The l window’s window length and above parameters can be adjusted according to the sensitivity of the impact components in signal. The kurtosis value of the captured signal which is obtained by capturing original signal with rectangular windows with window length of 600, 400, or adaptive window length are obtained, respectively, as shown in Figure 9. It can be found that before the impact signal is intercepted, the kurtosis value of the adaptive window length rectangular window intercepting signal coincides with the kurtosis value of the 600 window length rectangular window intercepting signal. And after the impact signal, the kurtosis value of the adaptive window length rectangular window intercepting signal coincides with the kurtosis value of the 400 window length rectangular window intercepting signal.

_{1}, sliding kurtosis contribution coefficient, f

_{2}, f and adaptive window length is shown in Figure 10. It can be seen from Figure 10 that the adaptive window length becomes the minimum window length at the 2803 s, and the data of the 3003 s is intercepted which is at the end of the intercepted signal and which is the starting point of the shock signal. Compared with the sliding kurtosis contribution coefficient, kurtosis can characterize the signal stability better, but it is insensitive to the end position of the impact component in the signal. As shown in Figure 10, the interval [K

_{1},K

_{2}] is relatively small (Some slightly fluctuating signals are shown unstable and f

_{1}jumps to 1). In this case, the shock signal is only perceived when the L window reaches 3840 s. And at this time, the end of the L window intercepting signal is the data at 4140 s. So the design of adaptive window length simply according to kurtosis is difficult to meet the requirements. Compared to the sensitivity of the sliding kurtosis contribution coefficient, the L window is already sensitive to the end of the shock signal when it reaches 3743 s. And at this time, the end of the L window intercepting signal is the data at 4043 s. Therefore, the reduction of window length at this time can improve the sensitivity of the impact component.

^{1/2}.

_{min}. And when the beginning of the shock signal is collected by the end of the rectangular window, the length of the rectangular window increases gradually to the maximum length L

_{max}. In the interval of the end of the shock signal, there is a slight fluctuation in the signal for a short time due to temperature and electromagnetic interference which leads to reduce the length of the rectangular window in the improved DAVAR method. The sliding kurtosis contribution coefficient highlights the more sensitive ability, such as the peak in Figure 19. And the window length decreases in the growth interval after 4000 s, so the tracking ability of dynamic Allan variance has improved by the window length decreasing. The measured signals are analyzed with the improved DAVAR method, and the dynamic Allan variance analysis results of the measured signals are as shown in Figure 20.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 7.**Analysis results of kurtosis and sliding kurtosis contribution coefficient under the same window length.

**Figure 12.**The angle random walk analysis results of intercepted signal of a rectangular window with different window lengths.

**Figure 19.**The analysis result of kurtosis, sliding kurtosis contribution coefficient, and adaptive window length.

**Table 1.**The comparison of confidence level and dynamic tracking ability of the analysis results between A-DAVAR, DAVAR-400, and DAVAR-600.

Analysis Node | DAVAR-400 | DAVAR-600 | A-DAVAR |
---|---|---|---|

ARW (10^{−3}°/h^{1/2}) (0~2000 s) | 4.03 | 3.87 | 3.83 |

ARW (10^{−3}°/h^{1/2}) (3000~4000 s) | 10.97 | 10.32 | 9.45 |

Beginning (3000 s) | 2800 | 2702 | 2703 |

End (3000 s) | 3200 | 3300 | 3156 |

Beginning (4000 s) | 3800 | 3700 | 3740 |

End (4000 s) | 4200 | 4300 | 4200 |

**Table 2.**The comparison of confidence level and dynamic tracking ability of the analysis results between A-DAVAR, DAVAR-400 and DAVAR-600.

Analysis Node | DAVAR-400 | DAVAR-600 | A-DAVAR |
---|---|---|---|

Refer value (°/h^{1/2}) | 0.355 | 0.355 | 0.355 |

ARW (°/h^{1/2}) (0~2000 s) | 0.518 | 0.476 | 0.451 |

ARW (°/h^{1/2}) (3000~4000 s) | 1.083 | 1.032 | 1.025 |

Beginning (3000 s) | 2800 | 2702 | 2713 |

End (3000 s) | 3200 | 3300 | 3189 |

Beginning (4000 s) | 3800 | 3700 | 3723 |

End (4000 s) | 4200 | 4300 | 4176 |

**Table 3.**The comparison of confidence level and dynamic tracking ability of the analysis results between A-DAVAR, DAVAR-400, and DAVAR-600.

Analysis Node | DAVAR-400 | DAVAR-600 | DAVAR-K-3.3 | A-DAVAR |
---|---|---|---|---|

Refer value (°/h^{1/2}) | 0.355 | 0.355 | 0.355 | 0.355 |

ARW (°/h^{1/2}) (1000~2000 s) | 1.136 | 1.091 | 1.107 | 0.963 |

ARW (°/h^{1/2}) (2500~3500 s) | 0.423 | 0.385 | 0.389 | 0.383 |

ARW (°/h^{1/2}) (4000~5000 s) | 1.172 | 1.105 | 1.086 | 1.061 |

Beginning (1000 s) | 800 | 700 | 738 | 738 |

End (1000 s) | 1200 | 1300 | 1288 | 1209 |

Restore 600 (1000 s) | - | - | 1616 | 1388 |

Beginning (2000 s) | 1800 | 1700 | 1616 | 1697 |

End (2000 s) | 2200 | 2300 | 2297 | 2215 |

Beginning (4000 s) | 3800 | 3700 | 3760 | 3760 |

End (4000 s) | 4200 | 4300 | 4167 | 4128 |

Beginning (5000 s) | 4800 | 4700 | 4713 | 4739 |

End (5000 s) | 5200 | 5300 | 5301 | 5126 |

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**MDPI and ACS Style**

Song, J.; Shi, Z.; Wang, L.; Wang, H.
Random Error Analysis of MEMS Gyroscope Based on an Improved DAVAR Algorithm. *Micromachines* **2018**, *9*, 373.
https://doi.org/10.3390/mi9080373

**AMA Style**

Song J, Shi Z, Wang L, Wang H.
Random Error Analysis of MEMS Gyroscope Based on an Improved DAVAR Algorithm. *Micromachines*. 2018; 9(8):373.
https://doi.org/10.3390/mi9080373

**Chicago/Turabian Style**

Song, Jinlong, Zhiyong Shi, Lvhua Wang, and Hailiang Wang.
2018. "Random Error Analysis of MEMS Gyroscope Based on an Improved DAVAR Algorithm" *Micromachines* 9, no. 8: 373.
https://doi.org/10.3390/mi9080373