﻿ 一种引入Hurst指数的MEMS陀螺仪去噪模型
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 大地测量与地球动力学  2022, Vol. 42 Issue (5): 457-461  DOI: 10.14075/j.jgg.2022.05.003

### 引用本文

GONG Yun, XIN Jie, NAN Shoujin. A Denoising Model of MEMS Gyroscope with Hurst Exponent[J]. Journal of Geodesy and Geodynamics, 2022, 42(5): 457-461.

### Foundation support

National Natural Science Foundation of China, No. 51674159.

### Corresponding author

XIN Jie, postgraduate, majors in inertial navigation data processing, E-mail: 2044922927@qq.com.

### About the first author

GONG Yun, PhD, associate professor, majors in simultaneous localization and mapping based on inertial navigation, E-mail: hbgongyun@xust.edu.cn.

### 文章历史

1. 西安科技大学测绘科学与技术学院，西安市雁塔中路58号，710054

1 去噪模型

CEEMDAN算法具体步骤参考文献[9]，本文不再赘述。

1.1 Hurst指数

 $\mathrm{IMF}_{1 \tau}=\frac{1}{\tau} \sum\limits_{t=1}^{\tau} \operatorname{IMF}_{1}(t)$ (1)

 $X(t, \tau)=\sum\limits_{u=1}^{t}\left(\mathrm{IMF}_{1}(u)-\mathrm{IMF}_{1 \tau}\right)$ (2)

 $R(\tau)=\max X(t, \tau)-\min X(t, \tau)$ (3)

Hurst标准偏差可表示为：

 $S(\tau)=\left[\frac{1}{\tau} \sum\limits_{t=1}^{\tau}\left(\operatorname{IMF}_{1}(t)-\operatorname{IMF}_{1 \tau}\right)^{2}\right]^{1 / 2}$ (4)

 $R / S=(\tau / 2)^{H}$ (5)

1.2 AKF

 $\left\{\begin{array}{l} \boldsymbol{X}(k)=\boldsymbol{F} \boldsymbol{X}(k-1)+\boldsymbol{W}(k) \\ \boldsymbol{Z}(k)=\boldsymbol{H} \boldsymbol{X}(k)+\boldsymbol{V}(k) \end{array}\right.$ (6)

 $\left\{\begin{array}{l} \hat{\boldsymbol{X}}_{k / k-1}=\boldsymbol{F}_{k / k-1} \hat{\boldsymbol{X}}_{k-1} \\ \boldsymbol{P}_{k / k-1}=\boldsymbol{F}_{k / k-1} \boldsymbol{P}_{k-1} \boldsymbol{F}_{k / k-1}^{\mathrm{T}}+\boldsymbol{W} \\ \hat{\boldsymbol{X}}_{k}=\hat{\boldsymbol{X}}_{k / k-1}+\boldsymbol{K}_{k}\left(\boldsymbol{Z}_{k}+\boldsymbol{H}_{k} \hat{\boldsymbol{X}}_{k / k-1}\right) \\ \boldsymbol{K}_{k}=\boldsymbol{P}_{k / k-1} \boldsymbol{H}_{k}^{\mathrm{T}}\left(\boldsymbol{H}_{k} \boldsymbol{P}_{k / k-1}+\boldsymbol{V}\right)^{-1} \\ \boldsymbol{P}_{k}=\left(\boldsymbol{I}-\boldsymbol{K}_{k} \boldsymbol{H}_{k}\right) \boldsymbol{P}_{k / k-1} \end{array}\right.$ (7)

 $\boldsymbol{P}_{k / k-1}=\lambda_{k+1} \boldsymbol{F}_{k / k-1} \boldsymbol{P}_{k-1} \boldsymbol{F}_{k / k-1}^{\mathrm{T}}+\boldsymbol{W}_{k-1}$ (8)

 $\lambda_{k}=\max \left\{1, \operatorname{tr}\left(\boldsymbol{N}_{k}\right) / \operatorname{tr}\left(\boldsymbol{M}_{k}\right)\right\}$ (9)

 $\boldsymbol{M}_{k} =\boldsymbol{H}_{k} \boldsymbol{F}_{k / k-1} \boldsymbol{P}_{k-1} \boldsymbol{F}_{k / k-1}^{\mathrm{T}} \boldsymbol{H}_{k}^{\mathrm{T}}$ (10)
 $\boldsymbol{N}_{k} =\boldsymbol{C}_{k}-\boldsymbol{H}_{k} \boldsymbol{W}_{k-1} \boldsymbol{H}_{k}^{\mathrm{T}}-\boldsymbol{V}_{k}$ (11)

1.3 引入Hurst指数的去噪模型

 图 1 去噪方法流程 Fig. 1 Flow chart of denoising method

1) 对原始陀螺信号s(t)进行CEEMDAN处理，得到n阶IMF和1个残差余量；

2) 计算各IMF分量的Hurst值，确定不同模态分量边界；

3) 采用AKF对混合模态进行滤波处理；

4) 将滤波后的IMF分量及保留的信息模态进行重构，得到去噪后的信号：

 $s^{\prime}(t)=\sum\limits_{k=M_{1}}^{M_{2}} \operatorname{IMF}_{k}(t)+\sum\limits_{k=M_{2}}^{n} \operatorname{IMF}_{k}(t)+r(t)$ (12)

5) 用信噪比(SNR)和均方根误差(RMSE)评价降噪效果。

2 实验分析 2.1 信号分解

 图 2 陀螺仪原始信号 Fig. 2 Gyroscope original signals

 图 3 IMF分量 Fig. 3 IMF components
2.2 模态筛选

 图 4 相关性分布 Fig. 4 Correlation distribution

2.3 去噪对比

 图 5 不同滤波器重构效果 Fig. 5 Reconstruction effect of different filters

3 结语

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A Denoising Model of MEMS Gyroscope with Hurst Exponent
GONG Yun1     XIN Jie1     NAN Shoujin1
1. College of Geomatics, Xi'an University of Science and Technology, 58 Mid-Yanta Road, Xi'an 710054, China
Abstract: Aiming at the errors caused by random drift of micro-electro-mechanical system(MEMS) gyroscope, we propose a complete ensemble empirical mode decomposition with adaptive noise(CEEMDAN) denoising model with Hurst exponent. Firstly, we decompose the original signal of gyroscope by CEEMDAN to obtain a series of intrinsic mode function (IMF) with high to low frequencies and a residual margin. Secondly, we introduce the Hurst exponential modal screening mechanism, and IMF components are divided into noise IMF, mixed IMF and information IMF. Finally, we filter the mixed modal components by the adaptive Kalman filter and reconstruct the signals. The results show that CEEMDAN has higher decomposition accuracy than EMD and EEMD.Using AKF to deal with mixed mode, the signal-to-noise of reconstructed signals through the Hurst exponential screening mechanism increases by about 12% and 36% compared with permutation entropy and correlation coefficient method. Using Hurst exponential screening mechanism, the RMSE of reconstructed signals of AKF is about 23% lower than that of wavelet threshold filtering.
Key words: CEEMDAN; intrinsic mode function; Hurst exponent; adaptive Kalman filter