# An Improved Fault Detection and Isolation Method for Airborne Inertial Navigation System/Attitude and Heading Reference System Redundant System

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Traditional Generalized Likelihood Test Fault Detection for Redundant Systems

- (1)
- The three sets of subsystems are installed in the same direction and are parallel, and the inertial devices can be unified to the same coordinate origin;
- (2)
- The inertial devices for each subsystem are mounted orthogonally in three axes.

_{0}, and the fault hypothesis, H

_{1}, are as follows:

_{D}is the detection threshold. When a false alarm rate is given, it can be known by checking the Chi-square distribution table.

_{i}, the statistical characteristic is:

_{i}is column i of parity matrix V, and the maximum likelihood estimate for fault size f is:

## 3. Principal Component Parity Vector-Based Sequential Weighted Generalized Likelihood Ratio Test Fault Detection for Inertial Navigation System/Attitude and Heading Referential System Redundant System

#### 3.1. Principal Component Parity Vector Method

_{r}can be obtained by probability statistics of the retained principal component X

_{r}:

_{i}, the statistical characteristic is:

_{i}is column i of parity matrix V, the maximum likelihood estimate for fault size f

_{s}is:

#### 3.2. Principal Component Parity Vector-Based Sequential Weighted Generalized Likelihood Ratio Test Fault Detection

_{i}is the fault vector, f

_{s}is the fault amplitude, the ith element is one, and the rest is zero. The estimated error covariance PK

_{f}of ${\widehat{X}}_{f}$ is:

_{i}is a diagonal matrix, D

_{i}(i,i) = 0, and the other elements on the diagonal are one.

_{h}is:

_{h}can be converted to:

_{t}is:

_{r}, under the fault-free hypothesis H

_{0}and fault hypothesis H

_{1}, is as follows:

_{t}, the parity vector is P

_{t}and the fault function detection value is P

_{t}, the adaptive fault detection threshold is:

- Step 1 is the sequential parity vector sampling matrix initialization.
- Step 2 is the PCA analysis, and the construction of principal component parity vector P
_{r.} - Step 3 is the SWGLT fault detection function calculation; if $F{D}_{SWGLT}>{T}_{df}$, we proceed to the next step; otherwise, we return to step 2.
- Step 4 is the SWGLT fault isolation function calculation, where the failed subsystem is isolated and the fault alarm is reported.

## 4. Experimental Setup

## 5. Results

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

RNP | Required navigation performance |

INS | Inertial navigation system |

GLT | Generalized likelihood ratio |

AHRS | Attitude and heading reference system |

IMA | Integrated modular avionics |

FMS | Flight management system |

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**Figure 5.**Fault injection of INS2 X-axis accelerometer. (

**a**) Fault size of accelerometer. (

**b**) Output of accelerometer.

**Figure 6.**Soft fault injection of INS1 Z-axis gyro. (

**a**) Soft fault size of gyro. (

**b**) Output of gyro.

**Figure 7.**Comparison results for gyro faults detection. (

**a**) FD function result of GLT. (

**b**) FD function result of WGLT.

**Figure 8.**Comparison results for accelerometer faults detection. (

**a**) FD function result of GLT. (

**b**) FD function result of WGLT.

**Figure 9.**Comparison results for gyro soft faults detection. (

**a**) FD function result of GLT. (

**b**) FD function result of WGLT.

Subsystems | Bias Instability of Gyro (°/h) | Bias Instability of Acc (m/s^{2}) |
---|---|---|

INS1 | 0.01 | 1 × 10^{−4} g |

INS2 | 0.01 | 1 × 10^{−4} g |

AHRS | 0.1 | 5 × 10^{−3} g |

Statistical Items | Fault Detection Rate (FDR) | False Alarm Rate (FAR) | Accuracy |
---|---|---|---|

Traditional GLT | 68.33% | 67.15% | 50.59% |

WGLT | 66.66% | 0% | 83.33% |

PPV-aided SWGLT | 99.27% | 0% | 99.64% |

Statistical Items | Fault Detection Rate (FDR) | False Alarm Rate (FAR) | Accuracy |
---|---|---|---|

Traditional GLT | 65.40% | 59.63% | 52.89% |

WGLT | 67.74% | 0% | 83.87% |

PPV-aided SWGLT | 99.38% | 0% | 99.69% |

Statistical Items | Fault Detection Rate (FDR) | False Alarm Rate (FAR) | Accuracy | Detection Delay |
---|---|---|---|---|

Traditional GLT | 33.0% | 78.32% | 27.34% | 33.5 s |

WGLT | 31.4% | 0% | 65.70% | 34.3 s |

PPV-aided SWGLT | 74.2% | 0% | 87.10% | 12.9 s |

Statistical Items | Fault Detection Rate (FDR) | False Alarm Rate (FAR) | Accuracy | Detection Delay |
---|---|---|---|---|

Soft fault rate with 0.03°/h/s | ||||

Traditional GLT | 40.2% | 76.18% | 32.01% | 29.9 s |

WGLT | 34.6% | 0% | 67.3% | 32.7 s |

PPV-aided SWGLT | 76.4% | 0% | 88.2% | 11.8 s |

Soft fault rate with 0.05°/h/s | ||||

Traditional GLT | 49.4% | 68.98% | 40.21% | 25.3 s |

WGLT | 50.2% | 0% | 75.1% | 24.9 s |

PPV-aided SWGLT | 83.74% | 0% | 91.48% | 8.13 s |

Soft fault rate with 0.1°/h/s | ||||

Traditional GLT | 85.98% | 69.22% | 58.38% | 7.01 s |

WGLT | 85.1% | 0% | 92.55% | 7.45 s |

PPV-aided SWGLT | 91.84% | 0% | 95.92% | 4.08 s |

Soft fault rate with 0.2°/h/s | ||||

Traditional GLT | 96.52% | 69.58% | 63.47% | 1.74 s |

WGLT | 96.04% | 0% | 98.02% | 1.98 s |

PPV-aided SWGLT | 98.48% | 0% | 99.24% | 0.48 s |

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**MDPI and ACS Style**

Dai, Y.; Lai, J.; Zhang, Q.; Li, Z.; Shen, Y.
An Improved Fault Detection and Isolation Method for Airborne Inertial Navigation System/Attitude and Heading Reference System Redundant System. *Aerospace* **2023**, *10*, 1024.
https://doi.org/10.3390/aerospace10121024

**AMA Style**

Dai Y, Lai J, Zhang Q, Li Z, Shen Y.
An Improved Fault Detection and Isolation Method for Airborne Inertial Navigation System/Attitude and Heading Reference System Redundant System. *Aerospace*. 2023; 10(12):1024.
https://doi.org/10.3390/aerospace10121024

**Chicago/Turabian Style**

Dai, Yuting, Jizhou Lai, Qieqie Zhang, Zhimin Li, and Yugui Shen.
2023. "An Improved Fault Detection and Isolation Method for Airborne Inertial Navigation System/Attitude and Heading Reference System Redundant System" *Aerospace* 10, no. 12: 1024.
https://doi.org/10.3390/aerospace10121024