# Identification of Noise Covariance Matrices to Improve Orientation Estimation by Kalman Filter

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental Procedure

#### 2.2. Gold Standard Measure

^{−30}on the set square orientation estimate.

#### 2.3. MIMU Data Processing

^{2}/$\sqrt{\mathrm{Hz}}$ for the accelerometer, 2.2 mrad/s/$\sqrt{\mathrm{Hz}}\text{}\mathrm{for}\text{}\mathrm{the}\text{}\mathrm{gyroscope},$ and 160 nT/$\sqrt{\mathrm{Hz}}$ for the magnetometer.

#### 2.4. Identification of the Kalman Parameters

_{g}, σ

_{bg}, and σ

_{a}. At the end of this first step an “ideal” value of σ

_{bg}was obtained. In the second step, the complete error was analyzed by integrating the effect of the magnetometers. We then considered the three parameters σ

_{g}, σ

_{a}, and σ

_{m}. Indeed, these three parameters are very closely linked and must be computed together.

_{g}, σ

_{bg}, and σ

_{a}lasted around 27 × 103 × 30 = 810,000 s, which already corresponded to nine days.

#### 2.5. Optimality Criterion

#### 2.6. Validation Procedure

## 3. Results

#### 3.1. Identification of the Kalman Parameters

_{g}, σ

_{bg}, σ

_{a}, and σ

_{m}, the analysis of the results revealed interesting information concerning the behavior of the Kalman filter. For instance, Figure 4 represents, for the intermediate intensity movements, the RMSe translating orientation error between the orientation estimated with the optoelectronic system and that obtained with the Kalman filter, depending on the values of the Kalman parameters σ

_{g}and σ

_{a}. Most importantly, Figure 4 shows that a bad parameter choice can induce a huge error (up to 65° for this movement), which confirms that appropriate selection of the Kalman parameters is essential.

_{g}and low values assigned to σ

_{a}. In other words, these values force the Kalman filter to give maximum importance to the measurement resulting from the accelerometers, and to neglect the measurement from the gyroscopes. Conversely, the area on the far left corresponds to low values assigned to σ

_{g}and high values assigned to σ

_{a}. In other words, these values force the Kalman filter to give maximum importance to measurement from gyroscopes, and to neglect the measurement resulting from the accelerometers. These two areas usually lead to significant errors. Indeed, the error observed in the right-hand area translates into an error in orientation estimation due to external accelerations experienced by the MIMUs, while the error observed in the left-hand area corresponds to the drift phenomenon resulting from the numerical integration of noisy angular velocity. The zone containing the optimal values is therefore located between these two particular areas, since it reflects a compromise between the gyroscope measurement and the measurement resulting from the accelerometers.

_{a}is proportional to movement intensity (directly related with the external acceleration). These results also highlight the particular behavior of static condition because of which the Kalman filter must grant much less importance to gyroscopes (a hundred times less) in order to counteract bias instability.

#### 3.2. Results from the Validation Procedure

## 4. Discussion

_{a}/σ

_{b}function of movement intensity. A procedure that defines the evolution of this ratio according to the measured accelerations could then be proposed. We have in this study considered Kalman parameters common to all three Opal APDM MIMUs tested. However, each MIMU is slightly different, such that the assignment of specific parameters to each sensor could be advantageous. Such a choice would, however, require the reproduction of this identification process for all MIMUs.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Synchronization of the MIMUs and Optoelectronic Systems

#### Appendix A.1. Temporal Synchronization

**Figure A1.**Comparison of the angular velocity obtained from the MIMU (in red) with that obtained from the optoelectronic system (in black) at the beginning of the 10 min acquisition (at the top) and at end (at the bottom).

#### Appendix A.2. Identification of the Quaternion Representing the Transformation of the Optoelectronic System Coordinate System into the MIMU Coordinate System

## Appendix B. Indirect Kalman Filter Used in the Study

#### Appendix B.1. Attitude Prediction

#### Appendix B.2. Update

## Appendix C. Application of the Methodology to a MIMU of Tactical Grade

^{−4}rad/s, which is below 1.45 × 10

^{−4}rad/s, the threshold under which a MIMU is characterized as being “tactical grade” [30].

**Table A1.**Values of measurement noise identified by the method of the Allan variance and Kalman parameters identified for slow movements.

σ_{bg} (rad/s) | σ_{g} (rad/s) | σ_{a} (m/s^{2}) | σ_{m} (µT) | |
---|---|---|---|---|

Allan’s variance | 1 × 10^{−4} | 1 × 10^{−3} | 2 × 10^{−2} | 0.6 |

Identification | 0 | 2 × 10^{−4} | 1 | 4 |

_{a}/σ

_{b}.

**Figure A2.**RMSe between the orientation estimated with the optoelectronic system and the Kalman filter depending on the values of Kalman parameters σ

_{g}and σ

_{a}for the slow intensity movement.

RMSe (degree) | |
---|---|

Manufacturer algorithm | 0.8 |

Kalman filter set with identified parameters | 0.4 |

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**Figure 1.**Set square equipped with both an MIMU and five reflective markers. The reflective markers can be used to define local coordinate systems associated with the set square.

**Figure 3.**Identification process. MIMU: magneto-inertial measurement units; RMSe: root mean squared error.

**Figure 4.**RMSe and standard error (black dots) of the RMSe between the orientation estimated with the optoelectronic system and the Kalman filter, depending on the values of Kalman parameters σ

_{g}and σ

_{a}for the intermediate intensity movements.

**Figure 5.**Orientation errors computed during a 10 min movement for orientations obtained with four different settings of the Kalman filter, namely with adapted Kalman parameters and with Kalman parameters identified respectively on slow, intermediate, and fast movements. The orientation obtained with the optoelectronic system served as reference. The color bar at the top of the figure represents the intensity level recorded during the movement.

**Figure 6.**Orientation errors computed during the same 10 min movement for orientations obtained with adapted Kalman parameters and with the manufacturer algorithm. The orientation obtained with the optoelectronic system served as reference. The color bar at the top of the figure represents the intensity level recorded during the movement.

Intensity | Slow | Intermediate | Fast |
---|---|---|---|

Acceleration (g) | 0.03 $\pm $0.02 | 0.7 $\pm $0.5 | 4 $\pm $2 |

Angular velocity (°/s) | 40 $\pm $20 | 300 $\pm $150 | 700 $\pm $400 |

σ_{bg} | σ_{g} | σ_{a} | σ_{m} |
---|---|---|---|

3 × 10^{−3} rad/s | 1 × 10^{−3} rad/s | 4 × 10^{−3} m/s^{2} | 0.2 µT |

Static | Slow | Intermediate | Fast | |
---|---|---|---|---|

σ_{bg} (rad/s) | 10^{−4} | 10^{−4} | 10^{−3} | 3 × 10^{−5} |

σ_{g} (rad/s) | 0.1 | 10^{−3} | 10^{−2} | 6 × 10^{−3} |

σ_{a} (m/s^{2}) | 0.2 | 0.2 | 8 | 10 |

σ_{m} (µT) | 10 | 1.5 | 4 | 45 |

RMSe (degree) | 0.08 ± 0.01 | 1.5 ± 0.2 | 2.9 ± 0.3 | 13.7 ± 5.3 |

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**MDPI and ACS Style**

Nez, A.; Fradet, L.; Marin, F.; Monnet, T.; Lacouture, P.
Identification of Noise Covariance Matrices to Improve Orientation Estimation by Kalman Filter. *Sensors* **2018**, *18*, 3490.
https://doi.org/10.3390/s18103490

**AMA Style**

Nez A, Fradet L, Marin F, Monnet T, Lacouture P.
Identification of Noise Covariance Matrices to Improve Orientation Estimation by Kalman Filter. *Sensors*. 2018; 18(10):3490.
https://doi.org/10.3390/s18103490

**Chicago/Turabian Style**

Nez, Alexis, Laetitia Fradet, Frédéric Marin, Tony Monnet, and Patrick Lacouture.
2018. "Identification of Noise Covariance Matrices to Improve Orientation Estimation by Kalman Filter" *Sensors* 18, no. 10: 3490.
https://doi.org/10.3390/s18103490