# MEMS Accelerometer Noises Analysis Based on Triple Estimation Fractional Order Algorithm

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Fractional Calculus and Fractional Noises

#### Fractional Noise

## 3. Triple Estimation Algorithm

#### 3.1. Order Estimation Filter KFo

#### 3.2. State Estimation Filter KFx

#### 3.3. Parameters Estimation Filter KFw

^{o}(e.g., ${\tilde{P}}_{k}^{o}$, ${Q}_{k-1}^{o}$), parameters of KFw are denoted with upper index ${}^{w}$ (e.g., ${\tilde{P}}_{k}^{w}$, ${Q}_{k-1}^{w}$) and parameters of KFx are without upper index. A detailed description of the Triple Estimation Algorithm is presented in [40].

## 4. Identification and Analysis of Fractional Order Noise Parameters

#### 4.1. Analysis of Fractional Constant and Variable Order System with Input Signal Known

- Noises parameters$$\mathrm{E}\left[\omega {\omega}^{T}\right]={10}^{-4},$$$$\mathrm{E}\left[\nu {\nu}^{T}\right]={10}^{-3},$$
- Parameters of KFx filter$${P}_{0}=\left[\begin{array}{c}0.01\end{array}\right],\phantom{\rule{0.277778em}{0ex}}{Q}_{0}=\left[\begin{array}{c}{10}^{-4}\end{array}\right],$$$${x}_{0}=\left[\begin{array}{c}0\end{array}\right],\phantom{\rule{0.277778em}{0ex}}R=\left[\begin{array}{c}{10}^{-3}\end{array}\right],$$
- Parameters of KFo filter$${P}_{0}^{o}=\left[\begin{array}{c}0.01\end{array}\right],\phantom{\rule{0.277778em}{0ex}}{Q}_{0}^{o}=\left[\begin{array}{c}0.1\end{array}\right],$$$${\alpha}_{0}=\left[\begin{array}{c}1\end{array}\right],\phantom{\rule{0.277778em}{0ex}}{R}^{o}=\left[\begin{array}{c}{10}^{-3}\end{array}\right],\phantom{\rule{0.277778em}{0ex}}\mathfrak{A}=1,\phantom{\rule{0.277778em}{0ex}}\mathfrak{B}=2,\phantom{\rule{0.277778em}{0ex}}{\delta}^{o}=0.5,$$
- Parameters of KFw filter$${P}_{0}^{w}=\left[\begin{array}{c}0.01\end{array}\right],\phantom{\rule{0.277778em}{0ex}}{Q}_{0}^{w}=\left[\begin{array}{c}0.1\end{array}\right],$$$${w}_{0}=\left[\begin{array}{c}0\end{array}\right],\phantom{\rule{0.277778em}{0ex}}{R}^{w}=\left[\begin{array}{c}{10}^{-3}\end{array}\right],\phantom{\rule{0.277778em}{0ex}}\mathfrak{A}=1,\phantom{\rule{0.277778em}{0ex}}\mathfrak{B}=2,\phantom{\rule{0.277778em}{0ex}}{\delta}^{w}=0.5.$$

**Example**

**1.**

- Noises parameters$$\mathrm{E}\left[\omega {\omega}^{T}\right]={10}^{-4},$$$$\mathrm{E}\left[\nu {\nu}^{T}\right]={10}^{-5},$$
- Parameters of KFx filter$${P}_{0}=\left[\begin{array}{c}0.01\end{array}\right],\phantom{\rule{0.277778em}{0ex}}{Q}_{0}=\left[\begin{array}{c}{10}^{-4}\end{array}\right],$$$${x}_{0}=\left[\begin{array}{c}0\end{array}\right],\phantom{\rule{0.277778em}{0ex}}R=\left[\begin{array}{c}{10}^{-5}\end{array}\right],$$
- Parameters of KFo filter$${P}_{0}^{o}=\left[\begin{array}{c}0.1\end{array}\right],\phantom{\rule{0.277778em}{0ex}}{Q}_{0}^{o}=\left[\begin{array}{c}0.001\end{array}\right],$$$${\alpha}_{0}=\left[\begin{array}{c}0.2\end{array}\right],\phantom{\rule{0.277778em}{0ex}}{R}^{o}=\left[\begin{array}{c}{10}^{-5}\end{array}\right],\phantom{\rule{0.277778em}{0ex}}\mathfrak{A}=1,\phantom{\rule{0.277778em}{0ex}}\mathfrak{B}=2,\phantom{\rule{0.277778em}{0ex}}{\delta}^{o}=0.5,$$
- Parameters of KFw filter$${P}_{0}^{w}=\left[\begin{array}{c}0.01\end{array}\right],\phantom{\rule{0.277778em}{0ex}}{Q}_{0}^{w}=\left[\begin{array}{c}0.001\end{array}\right],$$$${w}_{0}=\left[\begin{array}{c}-0.5\end{array}\right],\phantom{\rule{0.277778em}{0ex}}{R}^{w}=\left[\begin{array}{c}{10}^{-5}\end{array}\right],\phantom{\rule{0.277778em}{0ex}}\mathfrak{A}=1,\phantom{\rule{0.277778em}{0ex}}\mathfrak{B}=2,\phantom{\rule{0.277778em}{0ex}}{\delta}^{w}=0.5.$$

**Example**

**2.**

**Example**

**3.**

#### 4.2. Identification without Input Signal Knowledge

## 5. Identification and Analysis of MEMS Accelerometer’s Noises

^{2}C and SPI communication protocols.

#### 5.1. Experimental Setup

^{2}C communication protocol was used for data transmission between them. Additionally, the accelerometer’s range was set to 8 g, and measurement’s data were collected in the Arduino IDE environment for stationary located IMU, with sampling time $h=0.01$ s. The lock accelerometer position allows gathering its noises corrupted by constant gravity components. In the second phase of the experiment, the calculated mean value of data for each axis was subtracted from its measurement to have the pure noise corresponding to a particular accelerometer axis. Then, post-processing data for each axis without mean value were adapted as x-axis, y-axis, and z-axis noises in the Matlab environment. Having such prepared noises data and due to the fact that their mathematical models are independent from each other, we decided to apply the TEA separately for each axis. In fact, the state, parameter, and order estimation results were achieved for each axis separately under the same TEA configuration parameters except covariances R, ${R}^{o}$, and ${R}^{w}$ adjusted according to noise measurement corresponding to each axis.

- Parameters of KFx filter$${P}_{0}=\left[\begin{array}{c}0.01\end{array}\right],\phantom{\rule{0.277778em}{0ex}}{Q}_{0}=\left[\begin{array}{c}0.01\end{array}\right],$$$${x}_{0}=\left[\begin{array}{c}0\end{array}\right],\phantom{\rule{0.277778em}{0ex}}R=\left\{\begin{array}{cc}4\xb7{10}^{-4}& \mathrm{for}x-\mathrm{axis}\mathrm{noise}\\ 3.9\xb7{10}^{-4}& \mathrm{for}y-\mathrm{axis}\mathrm{noise}\\ 11\xb7{10}^{-4}& \mathrm{for}z-\mathrm{axis}\mathrm{noise}\end{array}\right.$$
- Parameters of KFo filter$${P}_{0}^{o}=\left[\begin{array}{c}0.01\end{array}\right],\phantom{\rule{0.277778em}{0ex}}{Q}_{0}^{o}=\left[\begin{array}{c}0.1\end{array}\right],$$$${\alpha}_{0}=\left[\begin{array}{c}1\end{array}\right],\phantom{\rule{0.277778em}{0ex}}\mathfrak{A}=1,\phantom{\rule{0.277778em}{0ex}}\mathfrak{B}=2,\phantom{\rule{0.277778em}{0ex}}{\delta}^{o}=0.5,\phantom{\rule{0.277778em}{0ex}}{R}^{o}=\left\{\begin{array}{cc}4\xb7{10}^{-4}& \mathrm{for}x-\mathrm{axis}\mathrm{noise}\\ 3.9\xb7{10}^{-4}& \mathrm{for}y-\mathrm{axis}\mathrm{noise}\\ 11\xb7{10}^{-4}& \mathrm{for}z-\mathrm{axis}\mathrm{noise}\end{array}\right.$$
- Parameters of KFw filter$${P}_{0}^{w}=\left[\begin{array}{c}0.01\end{array}\right],\phantom{\rule{0.277778em}{0ex}}{Q}_{0}^{w}=\left[\begin{array}{c}0.1\end{array}\right],$$$${w}_{0}=\left[\begin{array}{c}0\end{array}\right],\phantom{\rule{0.277778em}{0ex}}\mathfrak{A}=1,\phantom{\rule{0.277778em}{0ex}}\mathfrak{B}=2,\phantom{\rule{0.277778em}{0ex}}{\delta}^{w}=0.5,\phantom{\rule{0.277778em}{0ex}}{R}^{w}=\left\{\begin{array}{cc}4\xb7{10}^{-4}& \mathrm{for}x-\mathrm{axis}\mathrm{noise}\\ 3.9\xb7{10}^{-4}& \mathrm{for}y-\mathrm{axis}\mathrm{noise}\\ 11\xb7{10}^{-4}& \mathrm{for}z-\mathrm{axis}\mathrm{noise}.\end{array}\right.$$

#### 5.2. Experimental Results

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 11.**The real view of experimental setup with Arduino Due development board and MPU9252 IMU mounted on a shaft of servo motor in lock position.

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**MDPI and ACS Style**

Macias, M.; Sierociuk, D.; Malesza, W.
MEMS Accelerometer Noises Analysis Based on Triple Estimation Fractional Order Algorithm. *Sensors* **2022**, *22*, 527.
https://doi.org/10.3390/s22020527

**AMA Style**

Macias M, Sierociuk D, Malesza W.
MEMS Accelerometer Noises Analysis Based on Triple Estimation Fractional Order Algorithm. *Sensors*. 2022; 22(2):527.
https://doi.org/10.3390/s22020527

**Chicago/Turabian Style**

Macias, Michal, Dominik Sierociuk, and Wiktor Malesza.
2022. "MEMS Accelerometer Noises Analysis Based on Triple Estimation Fractional Order Algorithm" *Sensors* 22, no. 2: 527.
https://doi.org/10.3390/s22020527