# A Distributed Formation Joint Network Navigation and Positioning Algorithm

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## Abstract

**:**

## 1. Introduction

## 2. Distributed Joint Navigation Method

#### 2.1. Principles of Distributed Joint Navigation

#### 2.2. Self-Positioning Process of Distributed Joint Navigation and Positioning and Construction of Relative Navigation Information

- (1)
- Distributed joint navigation and positioning node self-positioning process [24]:

- (2)
- Relative Navigation Information Construction of Distributed Joint Navigation and Positioning:

#### 2.3. Platform Composition and Overall Architecture of Distributed Joint Navigation and Positioning System

#### 2.4. Ranging and Time Synchronization of Distributed Joint Navigation and Positioning

#### 2.4.1. Ranging Scheme

#### 2.4.2. Time Synchronization

- (1)
- Time synchronization between the onboard data link clocks of each node:

- (2)
- Time synchronization between the INS of node i and the airborne data link device:

- (3)
- Time synchronization between nodes (including INS) and satellites:

## 3. Establishment of Distributed Joint Navigation and Positioning Model

#### 3.1. State Model

#### 3.2. Measurement Model

## 4. Simulation Verification

#### 4.1. Simulation Parameter Configuration

#### 4.2. Simulation Results

#### 4.2.1. Joint Navigation and Positioning Scenarios without an Altimeter Assistance

- (1)
- For the position error curve, the position errors of Node 1 and Node 2 in the north (N), east (E), and down (D) directions are not very different, and the mean error is in the order of 1 m in both the N and E directions, while the error in the D direction is relatively large, reaching the order of 10 m, mainly because the GNSS elevation accuracy makes it difficult to distinguish the height of the moving node [32], which results in a larger error compared to the N and E directions. Correspondingly, the position error accuracy (STD) of two nodes has a similar behavior. In addition, the mean value of the pure INS autonomous navigation error of the two nodes is basically greater than 100 m, and the accuracy is also higher than 100 m. It can be seen that even without the aid of an altimeter, the algorithm can significantly suppress the problem of pure INS position divergence.
- (2)
- For the velocity error curve, the velocity errors of Node 1 and Node 2 in the N, E, and D directions are also not much different. The accuracy is in the order of 0.1 m/s; in comparison, the mean of the respective pure INS navigation errors of the two nodes is about 1 m/s, and the accuracy is also close to 1 m/s. Similarly, the algorithm can also obviously suppress the problem of pure INS navigation velocity divergence.

#### 4.2.2. Altimeter-Assisted Joint Navigation and Positioning Scenarios

- (1)
- For the position error curve, with the assistance of an unbiased altimeter, the mean errors of Node 1 and Node 2 in the N and E directions are also in the order of 1 m, and compared with the altimeter-free scene, the error was significantly improved, especially in the D direction, which has an error of the order of 0.1 m showing relative improvement of two orders of magnitude. For accuracy, the N and E directions were also significantly improved, and the D direction accuracy also improved by two orders of magnitude. When the altimeter deviation is 10 m and 30 m, the mean values of the position errors of Node 1 and Node 2 in the N, E, and D directions also gradually increase, and the accuracy also gradually deteriorates. However, we deduct the fixed error accuracy, and it can be found that the corresponding position accuracy is also very impressive.
- (2)
- For the velocity error curve, compared with the scene without altitude assistance, although the addition of the altimeter does not significantly improve the velocity error in the N and E directions, the velocity error improvement in the D direction is very significant. The accuracy is improved by one order of magnitude. In addition, as the altimeter deviation increases, the corresponding joint navigation and positioning velocity indicators also gradually deteriorate. Similarly, when we deduct the fixed deviation, we can also obtain a good joint navigation and positioning effect, which is also in line with expectations.

#### 4.2.3. Influence of Formation Baseline on Joint Navigation and Positioning Performance

- (1)
- For the N direction error: regardless of Node 1 or Node 2, the position error finally converges between 0 m~10 m, the maximum fluctuation is about 50 m, the velocity error almost converges to zero, and the maximum fluctuation is not more than 1 m/s;
- (2)
- For the E direction error: the position error finally converges between 0 m and 20 m, and the maximum fluctuation is about 50 m. For the velocity error, the error of node 1 and Node 2 increases with the increase of the baseline interval, and the final convergence error also increased, respectively, but ultimately did not exceed 1 m/s, and the maximum fluctuation did not exceed 2 m/s;
- (3)
- For the D direction error: whether it is the position error or the velocity error, the final result converges to zero, and the maximum fluctuation is less than 3 m and 0.2 m/s, respectively.

- (1)
- When the baseline interval is between 10 m and 1 km, the error is relatively small, which is very suitable for small UAVs formation flying situation;
- (2)
- When the baseline interval increases to more than 1 km, at this time, the error is relatively large, but the final error curves all have zero-crossing points, which means that the error is convergent, and this situation is suitable for the formation of large- and medium-sized UAVs.

## 5. Algorithm Comparison

#### 5.1. Comparison of Different LEO Systems

#### 5.2. Comparison with MEO Constellation Algorithm

#### 5.3. Comparison with Other Algorithms

## 6. Conclusions

- (1)
- Compared with the traditional leader-fellow collaborative navigation structure that relies on the leader node, our scheme is a distributed collaborative navigation and positioning scheme, which, without the distinction between leader and follower, is a flexible formation collaboration scheme; when performing special tasks, it will gain huge formation reconfiguration advantages;
- (2)
- Even without the aid of an altimeter, our algorithm can well suppress the divergence of the pure INS collaborative navigation scheme. With the aid of an altimeter, the collaborative navigation performance is further improved since the altimeter has the advantage of low cost compared with other expensive sensors; thus, it has great practical value;
- (3)
- Even if the node baseline interval gradually increases, our algorithm position can converge to zero with or without altimeter assistance, which has a certain robustness and can meet the needs of joint navigation and positioning location services in challenging environments. It is suitable for formation flying and other application scenarios that have high requirements for the accuracy and robustness of moving target cooperative navigation.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 5.**Positioning error curve of joint navigation without an altimeter assistance. (

**a**) Position error curve, (

**b**) velocity error curve, (

**c**) 3D trajectory error curve, and (

**d**) 3D projection error curve.

**Figure 6.**Altimeter-assisted joint navigation positioning error curve. (

**a**) Position error curve, (

**b**) velocity error curve, (

**c**) 3D trajectory error curve, and (

**d**) 3D projection error curve.

**Figure 7.**Joint navigation and positioning error curves at different baseline intervals. (

**a**) Position travel curve, (

**b**) velocity error curve, (

**c**) 3D trajectory error curve, and (

**d**) 3D projection error curve.

**Figure 8.**Comparison curve of joint navigation and positioning errors for different LEO systems. (

**a**) Position error curve, (

**b**) velocity error curve, (

**c**) 3D trajectory error curve, and (

**d**) 3D projection error curve.

**Figure 9.**Comparison results with the four major GNSSs: GPS, GLONASS, Galileo, and BDS. (

**a**) Position travel curve, (

**b**) velocity error curve, (

**c**) 3D trajectory error curve, and (

**d**) 3D projection error curve.

Parameter | Node 1 | Node 2 |
---|---|---|

Gyroscope random walk ($deg/\sqrt{h}$) | 0.0005 | 0.0005 |

Accelerometer random walk ($m/s/\sqrt{h}$) | 0.003 | 0.003 |

Gyroscope first-order Markov noise RMS ($deg/\sqrt{h}$) | 0.002 | 0.002 |

Accelerometer first-order Markov noise RMS ($\mu {g}_{0}$) | 10 | 10 |

Position noise (m) | 0.1 | 0.1 |

Velocity noise (m/s) | 0.01 | 0.01 |

Data link ranging error (m) | 10 | 10 |

Flight velocity (m/s) | 200 | 200 |

Flight duration (s) | 420 | 420 |

Parameter | Value |
---|---|

Constellation configuration type | Walker [36] |

Track height (km) | 1250 |

Orbital inclination (deg) | 99 |

Number of orbital faces | 20 |

Number of satellites per orbit (total/orbit) | 50 |

Number of satellites per orbit | 1000 |

Indicators | Node 1 | Node 2 | ||||
---|---|---|---|---|---|---|

N | E | D | N | E | D | |

Mean (m) | −130.71 | 97.47 | −91.31 | −147.07 | −154.98 | −91.26 |

STD (m) | 119.91 | 87.73 | 86.10 | 124.78 | 119.79 | 86.05 |

Indicators | Node 1 | Node 2 | ||||
---|---|---|---|---|---|---|

N | E | D | N | E | D | |

Mean (m) | −3.23 | 6.50 | −13.8 | −8.30 | 5.11 | −16.98 |

STD (m) | 6.80 | 5.89 | 14.11 | 5.02 | 7.55 | 11.79 |

Indicators | Node 1 | Node 2 | ||||
---|---|---|---|---|---|---|

N | E | D | N | E | D | |

Mean (m/s) | −1.02 | 0.51 | −0.69 | −1.03 | 0.49 | −0.69 |

STD (m/s) | 0.63 | 0.43 | 0.45 | 0.63 | 0.41 | 0.45 |

Indicators | Node 1 | Node 2 | ||||
---|---|---|---|---|---|---|

N | E | D | N | E | D | |

Mean (m/s) | 0.01 | 0.04 | −0.01 | −0.02 | −0.09 | −0.06 |

STD (m/s) | 0.22 | 0.19 | 0.20 | 0.21 | 0.39 | 0.37 |

**Table 7.**Statistics of joint navigation and positioning position error in the altimeter presence scenario.

Node | Alt (m) | Mean (m) | STD (m) | ||||
---|---|---|---|---|---|---|---|

N | E | D | N | E | D | ||

0 | −1.45 | 3.49 | −0.41 | 5.73 | 7.22 | 0.64 | |

Node 1 | 10 | −1.74 | 10.58 | 9.37 | 8.59 | 11.83 | 1.15 |

30 | −2.33 | 24.78 | 28.96 | 15.56 | 24.37 | 2.98 | |

0 | −4.54 | −4.39 | −0.39 | 15.54 | 19.47 | 0.76 | |

Node 2 | 10 | −3.46 | 0.32 | 9.40 | 16.11 | 21.08 | 1.23 |

30 | −1.29 | 9.75 | 28.99 | 16.17 | 27.90 | 3.02 |

**Table 8.**Statistics of joint navigation positioning velocity error in the altimeter presence scenario.

Node | Alt (m) | Mean (m/s) | STD (m/s) | ||||
---|---|---|---|---|---|---|---|

N | E | D | N | E | D | ||

0 | −0.008 | −0.42 | −0.01 | 0.11 | 0.21 | 0.03 | |

Node 1 | 10 | −0.11 | −0.53 | −0.01 | 0.11 | 0.22 | 0.03 |

30 | −0.16 | −0.59 | −0.02 | 0.16 | 0.24 | 0.02 | |

0 | −0.09 | −1.05 | −0.02 | 0.28 | 0.45 | 0.03 | |

Node 2 | 10 | −0.10 | −1.00 | −0.02 | 0.29 | 0.45 | 0.04 |

30 | −0.12 | −0.91 | −0.02 | 0.30 | 0.46 | 0.04 |

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**MDPI and ACS Style**

Ye, L.; Yang, Y.; Ma, J.; Deng, L.; Li, H.
A Distributed Formation Joint Network Navigation and Positioning Algorithm. *Mathematics* **2022**, *10*, 1627.
https://doi.org/10.3390/math10101627

**AMA Style**

Ye L, Yang Y, Ma J, Deng L, Li H.
A Distributed Formation Joint Network Navigation and Positioning Algorithm. *Mathematics*. 2022; 10(10):1627.
https://doi.org/10.3390/math10101627

**Chicago/Turabian Style**

Ye, Lvyang, Yikang Yang, Jiangang Ma, Lingyu Deng, and Hengnian Li.
2022. "A Distributed Formation Joint Network Navigation and Positioning Algorithm" *Mathematics* 10, no. 10: 1627.
https://doi.org/10.3390/math10101627