# A Novel Calibration Method for Gyro-Accelerometer Asynchronous Time in Foot-Mounted Pedestrian Navigation System

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## Abstract

**:**

## 1. Introduction

## 2. The Reference Frame Definitions

## 3. Gyro-Accelerometer Asynchronous Time

#### 3.1. Error Model of Gyro-Accelerometer Asynchronous Time

#### 3.2. Effects of Gyro-Accelerometer Asynchronous Time on Pedestrian Navigation

#### 3.3. Simulation

## 4. A Calibration Method for Gyro-Accelerometer Asynchronous Time

#### 4.1. Error Model of Pedestrian Navigation System Based on Gyro-Accelerometer Asynchronous Time

#### 4.2. Zero-Velocity Detection

#### 4.3. Kalman Filter Design

## 5. Experiments and Analysis

## 6. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

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**Figure 1.**Dynamic inconsistent error of coordinate frame caused by gyro-accelerometer asynchronous time.

**Figure 3.**Velocity errors and position errors caused by gyro-accelerometer asynchronous time. (

**a**) Velocity errors, (

**b**) Position errors.

**Figure 5.**Attitudes before and after compensating for the installation angle errors. (

**a**) Yaw angle, (

**b**) Pitch angle, (

**c**) Roll angle.

**Figure 7.**The results of zero-velocity detection. (

**a**) Walking on flat ground, (

**b**) Going upstairs, (

**c**) Going downstairs.

**Figure 11.**Velocity errors and position errors caused by gyro-accelerometer asynchronous time in the experiment. (

**a**) Velocity errors, (

**b**) Position errors.

**Figure 16.**The position errors at three standard points. (

**a**) Standard point 1, (

**b**) Standard point 2, (

**c**) Standard point 3.

**Table 1.**The effect of gyro-accelerometer asynchronous time on pedestrian navigation under different motions.

- | Pitch | Roll | Yaw |
---|---|---|---|

$\delta {v}_{E}^{n}$ | $\tau sin{\psi}_{0}{\int}_{0}^{t}{\omega}_{\theta}{f}_{U}^{n}dt$ | $\tau cos{\psi}_{0}{\int}_{0}^{t}{\omega}_{\gamma}{f}_{U}^{n}dt$ | $-\tau {\int}_{0}^{t}{\omega}_{\psi}{f}_{N}^{n}dt$ |

$\delta {v}_{N}^{n}$ | $-\tau cos{\psi}_{0}{\int}_{0}^{t}{\omega}_{\theta}{f}_{U}^{n}dt$ | $\tau sin{\psi}_{0}{\int}_{0}^{t}{\omega}_{\gamma}{f}_{U}^{n}dt$ | $\tau {\int}_{0}^{t}{\omega}_{\psi}{f}_{E}^{n}dt$ |

$\delta {v}_{h}^{n}$ | $\tau {\int}_{0}^{t}{\omega}_{\theta}{f}_{U}^{n}dt$ | $\tau {\int}_{0}^{t}{\omega}_{\gamma}{f}_{U}^{n}dt$ | $\tau \sqrt{{\left({\int}_{0}^{t}{\omega}_{\psi}{f}_{N}^{n}dt\right)}^{2}+{\left({\int}_{0}^{t}{\omega}_{\psi}{f}_{E}^{n}dt\right)}^{2}}$ |

$\delta {v}_{U}^{n}$ | $\tau {\int}_{0}^{t}{\omega}_{\theta}{f}_{//}^{n}dt$ | $-\tau {\int}_{0}^{t}{\omega}_{\gamma}{f}_{\perp}^{n}dt$ | 0 |

$\delta L$ | $-\frac{\tau cos{\psi}_{0}}{{R}_{M}+h}{\int}_{0}^{t}{\int}_{0}^{t}{\omega}_{\theta}{f}_{U}^{n}dtdt$ | $\frac{\tau sin{\psi}_{0}}{{R}_{M}+h}{\int}_{0}^{t}{\int}_{0}^{t}{\omega}_{\gamma}{f}_{U}^{n}dtdt$ | $\frac{\tau}{{R}_{M}+h}{\int}_{0}^{t}{\int}_{0}^{t}{\omega}_{\psi}{f}_{E}^{n}dtdt$ |

$\delta \lambda $ | $\frac{\tau sin{\psi}_{0}secL}{{R}_{N}+h}{\int}_{0}^{t}{\int}_{0}^{t}{\omega}_{\theta}{f}_{U}^{n}dtdt$ | $\frac{\tau cos{\psi}_{0}secL}{{R}_{N}+h}{\int}_{0}^{t}{\int}_{0}^{t}{\omega}_{\gamma}{f}_{U}^{n}dtdt$ | $\frac{-\tau secL}{{R}_{N}+h}{\int}_{0}^{t}{\int}_{0}^{t}{\omega}_{\psi}{f}_{N}^{n}dtdt$ |

$\delta h$ | $\tau {\int}_{0}^{t}{\int}_{0}^{t}{\omega}_{\theta}{f}_{//}^{n}dtdt$ | $-\tau {\int}_{0}^{t}{\int}_{0}^{t}{\omega}_{\gamma}{f}_{\perp}^{n}dtdt$ | 0 |

Motion | Horizontal Velocity Error | Vertical Velocity Error |
---|---|---|

Pitch motion | The errors will cancel each other out in opposite directions | Increase |

Roll motion | The errors will cancel each other out in opposite directions | Increase |

Yaw motion | The errors will cancel each other out in opposite directions | Make no difference |

Walking on flat ground | The errors will cancel each other out when walking on a closed-loop trajectory | Increase |

Going upstairs | The errors will cancel each other out when walking on a closed-loop trajectory | Increase |

Going downstairs | The errors will cancel each other out when walking on a closed-loop trajectory | Increase |

Performance | Gyros | Accelerometers |
---|---|---|

In-run stability | ${10}^{\circ}$/h | 40 ug |

Random walk | $0.{4}^{\circ}$/$\sqrt{}h$ | 0.06 m/s /$\sqrt{}h$ |

Full range | $\pm {2000}^{\circ}$/h | ±40 g |

Method | Detail |
---|---|

Method 1 | ZUPT with gyro-accelerometer asynchronous time ignored |

Method 2 | ZUPT with gyro-accelerometer asynchronous time considered |

Method 3 | ZUPT/height constraint with gyro-accelerometer asynchronous time ignored |

Method 4 | ZUPT/height constraint with gyro-accelerometer asynchronous time considered |

Errors | Method 1 | Method 2 | Method 3 | Method 4 |
---|---|---|---|---|

North position error (m) | 0.38 | 0.35 | 0.17 | 0.18 |

East position error (m) | 0.36 | 0.23 | 0.27 | 0.15 |

Horizontal position error (m) | 0.52 | 0.42 | 0.32 | 0.23 |

Error percentage (%D) | 0.50 | 0.40 | 0.31 | 0.22 |

Height error (m) | 1.03 | 0.76 | 0.08 | 0.06 |

Errors | Method 1 | Method 2 | Method 3 | Method 4 |
---|---|---|---|---|

North position error (m) | 0.93 | 0.75 | 0.44 | 0.36 |

East position error (m) | 1.06 | 0.92 | 1.01 | 0.85 |

Horizontal position error (m) | 1.41 | 1.19 | 1.10 | 0.92 |

Error percentage (%D) | 0.70 | 0.59 | 0.54 | 0.46 |

Height error (m) | 2.48 | 1.36 | 0.20 | 0.18 |

Errors | Method 1 | Method 2 | Method 3 | Method 4 |
---|---|---|---|---|

North position error (m) | 1.70 | 1.38 | 0.89 | 0.66 |

East position error (m) | 1.84 | 1.60 | 1.66 | 1.43 |

Horizontal position error (m) | 2.51 | 2.11 | 1.88 | 1.58 |

Error percentage (%D) | 0.84 | 0.70 | 0.64 | 0.53 |

Height error (m) | 3.37 | 2.11 | 0.22 | 0.23 |

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**MDPI and ACS Style**

Chen, T.; Yang, G.; Cai, Q.; Wen, Z.; Zhang, W.
A Novel Calibration Method for Gyro-Accelerometer Asynchronous Time in Foot-Mounted Pedestrian Navigation System. *Sensors* **2022**, *22*, 209.
https://doi.org/10.3390/s22010209

**AMA Style**

Chen T, Yang G, Cai Q, Wen Z, Zhang W.
A Novel Calibration Method for Gyro-Accelerometer Asynchronous Time in Foot-Mounted Pedestrian Navigation System. *Sensors*. 2022; 22(1):209.
https://doi.org/10.3390/s22010209

**Chicago/Turabian Style**

Chen, Tianyu, Gongliu Yang, Qingzhong Cai, Zeyang Wen, and Wenlong Zhang.
2022. "A Novel Calibration Method for Gyro-Accelerometer Asynchronous Time in Foot-Mounted Pedestrian Navigation System" *Sensors* 22, no. 1: 209.
https://doi.org/10.3390/s22010209