# Noise Reduction of MEMS Gyroscope Based on Direct Modeling for an Angular Rate Signal

^{1}

^{2}

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## Abstract

**:**

^{0.5}and a bias instability of 44.41°/h were reduced to 0.4°/h

^{0.5}and 4.13°/h by the KF under a given bandwidth (10 Hz), respectively. The 1σ estimated error was reduced from 1.9°/s to 0.14°/s and 1.7°/s to 0.5°/s in the constant rate test and swing rate test, respectively. It also showed that the filtered angular rate signal could well reflect the dynamic characteristic of the input rate signal in dynamic conditions. The presented algorithm is proved to be effective at improving the measurement precision of the MEMS gyroscope.

## 1. Introduction

## 2. Modeling of Stochastic Error for MEMS Gyroscope

_{b}, and n(t) is the ARW white noise.

Noise Terms | Allan Variance | Slope | Coefficient | Unit |
---|---|---|---|---|

ARW | N^{2}/τ | −1/2 | N = σ(1) | °/h^{0.5} |

Bias instability | (0.6643B)^{2} | 0 | B = σ(f_{0})/0.6643 | °/h |

RRW | K^{2}τ/3 | +1/2 | K = σ(3) | °/h^{1.5} |

## 3. Optimal KF Algorithm for Gyroscope Noise Reduction

#### 3.1. KF State and Measurement Equation

_{ω}[16]:

**X**(t) = [ω, b]

^{T}; thus, the KF state and measurement equation can be expressed as:

**w**(t) = [n

_{ω},w

_{b}]

^{T}and

**v**(t) = n(t) with E[

**w**(t)

**w**

^{T}(t + τ)] =

**Q**δ(τ) and E[

**v**(t)

**v**

^{T}(t + τ)] =

**R**δ(τ), representing system process noise and measurement noise. The KF coefficient matrix

**F**= 0

_{2×2}. The measurement matrix

**H**, covariance matrices

**Q**and

**R**can be written as:

_{b}is the variance of RRW noise, q

_{n}is the variance of ARW noise, and q

_{ω}is the variance of white noise n

_{ω}. The values of parameter q

_{ω}should be determined by the bandwidth requirement of the input signal in a dynamic condition. From the state and measurement Equation (4), the continuous-time KF for the true angular rate and bias drift estimate can be given as:

**P**(t) is the estimated covariance, and

**K**(t) is the filter gain. It indicates that the state vector

**X**(t) can be obtained by the

**K**(t); thus, there is a need to solve the differential Equation (8). In particular, as for a KF filtering system with constant coefficients, the implementation of KF can be simplified if the filter has a steady-state gain. Therefore, the properties of the

**K**(t) and

**P**(t) will be analyzed in the next section.

#### 3.2. Analysis of KF Observability

**F**and

**H**will be analyzed. From the definition of matrices

**F**and

**H**, the rank of system observability matrix for the KF state-space model of Equation (4) is equal to one, thus the KF system is not completely observable. Here, a basic discrete iterative KF method in the following is used to off-line analyze the property and characteristic of estimated covariance

**P**(t) and filter gain

**K**(t) [22].

^{0.5}and 1200°/h

^{1.5}, and the value of q

_{ω}is set as 1000(°/h)

^{2}, where the filtering period is set as 0.005 s. By using Equations (9)–(11), the plot of changing trend for

**P**(t) and

**K**(t) are shown in Figure 1.

**Figure 1.**Plot of changing trend of the estimated covariance

**P**(t) and filter gain

**K**(t). (

**a**) Kalman filter (KF) covariance

**P**(t). (

**b**) KF gain

**K**(t).

**P**(t) will be increased with increasing iteration times, and the component values of the matrix

**P**(t) are diverged without steady-state values. However, fortunately the component values of the filter gain

**K**(t) approaches a steady-state value. In particular, multiple analyses have demonstrated that different KF parameters will not affect the convergent property of the filter gain, but only change the steady-state value and convergence time. In this paper, an off-line approach will be used to get the steady-state filter gain; thus, the state vector

**X**(t) can be estimated and obtained by using of a steady-state filter gain based on the Equation (6).

#### 3.3. Discrete-Time KF for True Rate Signal Estimate

**K**

_{s}to construct a KF will simplify the implementation of system. Here, we will establish a discrete-time equation for true angular rate estimate through solving the continuous-time Equation (6). The reason is that the inherent stable property of KF can be revealed and explained. In particular, the KF bandwidth can be easily analyzed through the KF frequency response, providing a basis for determining the q

_{ω}in a dynamic condition.

**K**s obtained by the off-line approach of Equations (9)–(11) is defined as ${\mathit{K}}_{s}={[{k}_{1},{k}_{2}]}^{T}$, the extract vector for the true angular rate and bias drift are defined as e

_{1}= [1, 0] and e

_{2}= [0, 1], respectively. Using of filter gain

**K**

_{s}, the estimation of true angular rate can be obtained by a continuous-time KF:

**S**is a full matrix whose columns are the corresponding eigenvectors, and

**Λ**is a diagonal matrix of eigenvalues. With the definition of matrix

**m**, the two eigenvalues of

**m**can be deduced as λ

_{1}= (k

_{1}+ k

_{2}) and λ

_{2}= 0, then we get:

- Step 1: Form the covariance matrices
**Q**and**R**by the ARW and RRW noise variance and variance q_{ω}; - Step 2: Analyze the steady-state filtering gain
**K**s off-line by using of Equations (9)–(11), ${\mathit{K}}_{s}={[{k}_{1},{k}_{2}]}^{T}$; - Step 3: Perform the eigenvalue decomposition of matrix
**m**, $\mathit{m}=\mathit{S}\text{\Lambda}{\mathit{S}}^{-1}$; - Step 4: Extract the eigenvectors matrix
**S**and eigenvalues λ_{1}and λ_{2}; - Step 5: Calculate the matrices
**A**and**B**,$$\mathit{A}=\mathit{S}\left[\begin{array}{cc}{e}^{-{\text{\lambda}}_{1}T}& 0\\ 0& 1\end{array}\right]{\mathit{S}}^{-1},\mathit{B}=\mathit{S}\left[\begin{array}{cc}-{\text{\lambda}}_{1}^{-1}({e}^{-{\text{\lambda}}_{1}T}-1)& 0\\ 0& T\end{array}\right]{\mathit{S}}^{-1}$$ - Step 6: Perform the discrete-time KF equation,$$\{\begin{array}{l}{\widehat{\mathit{X}}}_{k+1}=\mathit{A}\cdot {\widehat{\mathit{X}}}_{k}+\mathit{B}\cdot {\mathit{K}}_{s}\cdot {Z}_{k+1}\\ {\widehat{\omega}}_{k+1}=e\cdot _{1}{\widehat{\mathit{X}}}_{k+1}\\ {\widehat{b}}_{k+1}=e\cdot _{2}{\widehat{\mathit{X}}}_{k+1}\end{array}$$

#### 3.4. Analysis of KF Bandwidth

_{ω}even though it can also be affected by other parameters such as noise statistics of the gyroscope. Consequently, selecting a suitable value of q

_{ω}is prerequisite for implementation of a KF.

_{ω}in the implementation of KF. A linear fit Equation (18) is obtained to describe the relationship between the −3 dB bandwidth of KF and parameter $\sqrt{{q}_{\omega}}$ based on the multiple analyses of KF frequency response with choosing different values of $\sqrt{{q}_{\omega}}$ (see in Figure 3). Thus, a suitable value of parameter q

_{ω}for the KF implementation can be easily determined by the Equation (18):

## 4. Experiment and Discussion

**A**and

**B**are determined. In this work, the components values of the matrices

**A**and

**B**are calculated off-line in advance, and then these matrices are written into the signal processing program. The hardware system is designed by the digital signal processing (DSP) technique. Therefore, the gyroscope’s outputs signal can be collected by the DSP unit for real-time processing by the filtering program which solidified in the DSP processor, and then providing an optimal estimation of input rate signal having a higher accuracy.

#### 4.1. Static Drift Test Result

_{ω}to implement KF. The comparison of noise density and root Allan variance are shown in Figure 4 and Figure 5. The results are demonstrated in Table 2.

^{−0.5}for the original rate signal and 0.013°/s·Hz

^{−0.5}for the filtered rate signal with bandwidth of 10 Hz, resulting in a reduction factor of about 11. From Table 2, it can be found that the values of the noise density will decrease with a decrease in bandwidth, i.e., the noise density is about 0.061°/s·Hz

^{−0.5}for the 30 Hz bandwidth, which is higher than that of 10 Hz.

^{0.5}to 0.40°/h

^{0.5}, making a noise reduction factor of about 12 and 10 for the ARW and bias drift, respectively.

Terms | Noise (°/s/Hz^{0.5}) | ARW (°/h^{0.5}) | Bias drift (°/h) |
---|---|---|---|

Original gyro | 0.150 | 4.8668 | 44.4129 |

BW = 10 Hz | 0.013 | 0.4006 | 4.1344 |

BW = 20 Hz | 0.040 | 1.2037 | 12.1383 |

BW = 30 Hz | 0.061 | 1.8879 | 19.7388 |

**Figure 6.**Auto-correlation functions of the filtered rate signal and original rate signal under different KF bandwidths. (

**a**) Filtered rate signal. (

**b**) Original rate signal.

_{ω}. Therefore, in a practical application we should trade off bandwidth for accuracy, and then using Equation (18) to select a suitable value of q

_{ω}to set KF bandwidth in a dynamic condition.

#### 4.2. Constant Rate Test Result

Rate (°/s) | Mean of Estimated Error (°/s) | STD of Estimated Error (°/s) | ||
---|---|---|---|---|

Original Gyro | After Filtering | Original Gyro | After Filtering | |

10 | 0.0413 | −0.0421 | 1.9378 | 0.1120 |

30 | 0.0197 | −0.0209 | 2.0195 | 0.1075 |

50 | 0.0582 | −0.0607 | 1.9751 | 0.1069 |

80 | 0.0776 | −0.0812 | 2.6189 | 0.1407 |

#### 4.3. Swing Rate Test Result

_{0})°/s with three different frequencies f = 0.1, 0.3 and 0.5 Hz, and initial phase φ

_{0}= 0. The estimated rate signal and errors by KF filtering are shown in Figure 8. The detailed results are illustrated in Table 4, where the KF bandwidth was set as about 20 Hz.

Frequency f (Hz) | Amplitude (°/s) | STD Error (°/s) | ||
---|---|---|---|---|

Original Gyro | After Filtering | Original Gyro | After Filtering | |

0.1 | 22.4481 | 20.0973 | 1.6749 | 0.3836 |

0.3 | 22.6760 | 20.1926 | 1.6242 | 0.5510 |

0.5 | 23.1104 | 20.7806 | 1.7234 | 0.6866 |

## 5. Conclusions

^{0.5}to 0.4°/h

^{0.5}and 44.41°/h to 4.13°/h by the KF, respectively. The dynamic test results demonstrated that KF can effectively reduce the measurement noise and accurately reflect the dynamic characteristic of the input signal.

_{ω}to ensure the dynamic characteristic of the input rate signal without attenuation after KF filtering.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Xue, L.; Jiang, C.; Wang, L.; Liu, J.; Yuan, W.
Noise Reduction of MEMS Gyroscope Based on Direct Modeling for an Angular Rate Signal. *Micromachines* **2015**, *6*, 266-280.
https://doi.org/10.3390/mi6020266

**AMA Style**

Xue L, Jiang C, Wang L, Liu J, Yuan W.
Noise Reduction of MEMS Gyroscope Based on Direct Modeling for an Angular Rate Signal. *Micromachines*. 2015; 6(2):266-280.
https://doi.org/10.3390/mi6020266

**Chicago/Turabian Style**

Xue, Liang, Chengyu Jiang, Lixin Wang, Jieyu Liu, and Weizheng Yuan.
2015. "Noise Reduction of MEMS Gyroscope Based on Direct Modeling for an Angular Rate Signal" *Micromachines* 6, no. 2: 266-280.
https://doi.org/10.3390/mi6020266