# Estimation and Error Analysis for Optomechanical Inertial Sensors

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulation and Estimation Methods

#### 2.1. Sensor Model

#### 2.2. Calibration Phase: Estimating the Sensor Bias

#### 2.3. Operation Phase: Estimating the Forcing Acceleration

#### 2.3.1. Least Squares Formulation

#### 2.3.2. Kalman Filter Formulation

## 3. Numerical Simulation Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. State Transition Matrix

## Appendix B. Process Noise Covariance

## References

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**Figure 1.**Flowchart of the calibration and operation Kalman filters. As shown, the outputs of each filter are provided as inputs to the other. Equation (20) can easily be substituted into the calibration step to use the lerast squares solution in place of the Kalman filter.

**Figure 2.**Results for the bias estimate obtained from the simulated calibration phase. The top figure shows the true and estimated bias over the calibration time interval of 10 min. On the bottom, the error in the bias estimate is plotted in blue with the corresponding $3\sigma $ bounds in red.

**Figure 3.**Results for the forcing acceleration estimate error from the moving least squares procedure. Estimate error is plotted in blue with the associated $3\sigma $ bounds in red. This is accompanied by the true bias time history in yellow.

**Figure 4.**Results for the forcing acceleration estimate error from the Kalman filter. Estimate error is plotted in blue with the associated $3\sigma $ bounds in red. This is accompanied by the true bias time history in cyan.

**Figure 5.**Results for the forcing acceleration estimate error from the moving least squares procedure for 100 Monte Carlo iterations. Estimate error is plotted in blue with the associated $3\sigma $ bounds in red.

**Figure 6.**Results for the forcing acceleration estimate error from the Kalman filter procedure for 100 Monte Carlo iterations. Estimate error is plotted in blue with the associated $3\sigma $ bounds in red.

Term | Value | Units |
---|---|---|

$\omega $ | 3.76 | Hz |

Q | $1.14\times {10}^{5}$ | |

${f}_{s}$ | 30.5 | Hz |

Term | Value | Units |
---|---|---|

${\sigma}_{v}$ | $1\times {10}^{-9}$ | m/s $\sqrt{\mathrm{Hz}}$ |

${\sigma}_{u}$ | $1\times {10}^{-8}$ | m/s${}^{2}\sqrt{\mathrm{Hz}}$ |

${\sigma}_{m}$ | $1\times {10}^{-11}$ | m |

${\sigma}_{g}$ | $1.8107\times {10}^{-8}$ | m/s $\sqrt{\mathrm{Hz}}$ |

${\sigma}_{b0}$ | $1.3506\times {10}^{-8}$ | m/s${}^{2}$ |

**Table 3.**Key performance statistics from 10,000 Monte Carlo trials of both acceleration estimation procedures.

Term | Value | Units |
---|---|---|

Least Squares Error Mean | $7.7990\times {10}^{-5}$ | μg |

Least Squares Error Std Dev | $3.6446\times {10}^{-2}$ | μg |

Kalman Filter Error Mean | $3.7493\times {10}^{-5}$ | μg |

Kalman Filter Error Std Dev | $3.5476\times {10}^{-2}$ | μg |

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**MDPI and ACS Style**

Kelly, P.; Majji, M.; Guzmán, F.
Estimation and Error Analysis for Optomechanical Inertial Sensors. *Sensors* **2021**, *21*, 6101.
https://doi.org/10.3390/s21186101

**AMA Style**

Kelly P, Majji M, Guzmán F.
Estimation and Error Analysis for Optomechanical Inertial Sensors. *Sensors*. 2021; 21(18):6101.
https://doi.org/10.3390/s21186101

**Chicago/Turabian Style**

Kelly, Patrick, Manoranjan Majji, and Felipe Guzmán.
2021. "Estimation and Error Analysis for Optomechanical Inertial Sensors" *Sensors* 21, no. 18: 6101.
https://doi.org/10.3390/s21186101