# Improving Matching Accuracy of Underwater Gravity Matching Navigation Based on Iterative Optimal Annulus Point Method with a Novel Grid Topology

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## Abstract

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## 1. Introduction

## 2. Construction of Iterative Optimal Annulus Point (IOAP) Method with a Novel Grid Topology

#### 2.1. Matching Positioning Strategy of the Tracking Starting Point in Small Annulus (SPMP)

^{−3}m/s

^{2}, so the standard deviation of the INS drift error $E$ is about 1.8 km/h). ${\beta}_{j}=j\cdot {\theta}_{0}$ represents the rotated angle of the $j$th grid point on each ring in the small annulus, where $j=1,2,\cdots ,{j}_{\mathrm{max}}=\u230a\raisebox{1ex}{$360$}\!\left/ \!\raisebox{-1ex}{${\theta}_{0}$}\right.\u230b$.

#### 2.2. Matching Positioning Mechanism of the Tracking Ending Point in Three-Layer Annulus (EPMP)

#### 2.3. Implementation Process of Iterative Optimal Annulus Point Algorithm for New Grid Topology

## 3. Verification and Application of the Proposed IOAP Algorithm

#### 3.1. Verification on the Matched Performance Difference of IOAP with Different $\sigma $ Criteria

^{−3}m/s

^{2}, the sailing speed 10 m·s

^{−1}, the sailing direction northeast 70°, the initial position error 0 m, the velocity error 0.04 m·s

^{−1}and the sailing direction error 0.05°. In addition, the real-time measured data of the gravimeter are the sum of the sampled value in the gravity anomaly reference map and random noise with a standard deviation of 1 mGal, while the number of sampling points and the sampling period are set as 110 and 20 s, respectively. In this paper, suppose that the matching positioning accuracy is defined as $l$/m; the effective matching means that the absolute value difference between the matching position and the real position is located in the closed interval $\left[0,l\right]$. Thus, if the number of the effective matching of algorithms is $n$ in the $N$ experimental tests, the matching success rate can be shown as $\xi =\frac{n}{l}\times 100\%$. Meanwhile, the average value (mean), standard deviation (std), worst value (max) of the matching positioning accuracy in $N$ times tests and the average matching time T (excluding the simulation environment configuration time) are calculated and recorded as the performance evaluation index of the gravity matching algorithm.

#### 3.2. Verification of the Difference Influence on Matching Performance of IOAPs with Different Ring Radius Reference Angle

#### 3.3. Verification of the Good Matching Performance of IOAP for Different Tracking Starting Points

## 4. Conclusions

- (1)
- The construction of an iterative optimal annulus point model with a novel grid topology. On the basis of breaking out from the traditional square-shaped grid topology of the TERCOM, the annulus-shaped topology of the matching grid points was constructed. Then, the Matching Positioning strategy of the tracking Starting Point in small annulus (SPMP) and the Matching Positioning mechanism of the tracking Ending Point in three-layer annulus (EPMP) were proposed by employing the INS sailing direction and distance information. Furthermore, an iterative optimal annulus point model with a novel grid topology was constructed by coupling the SPMP and EPMP.
- (2)
- The optimization selection of the $\sigma $ criterion and the reference angular ring radius $R$ for the optimal matching navigation in the large annulus at the tracking ending point. In this paper, the matching evaluation indexes, such as the average matching accuracy, matching standard deviation, average matching time and matching success rate, were comprehensively compared. These evaluation indexes were taken as the selection basis of the model parameters, which resulted in verifying the different influences of the parameters on the matching performances of the proposed iterative optimal annulus point algorithm and realizing the selection of good parameters to improve the underwater gravity matching accuracy.
- (3)
- The improvement of the underwater matching navigation accuracy. For the matching performance testing in three regions, the results showed that the iterative optimal annulus point model with a novel grid topology, compared with the TECOM, had little difference of the average matching time. The worst matching accuracies of the proposed model were also improved by up to 47.24%, 63.96% and 72.16%. Simultaneously, the average matching accuracies of the iterative optimal annulus point model were increased by up to 20.37%, 40.39% and 13.88%, respectively. In summary, the iterative optimal annulus point model with a novel grid topology contributed to enhancing the matching accuracy of underwater vehicle gravity matching navigation.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Matched points distribution in the small annulus based on the SPMP (${\theta}_{0}=45,\lambda =1/3$).

**Figure 2.**Distribution diagram of the matched grid points in a large annulus by the EPMP ($\phi =\pi /6,M=9$).

**Figure 3.**Schematic diagram of the satellite remote sensing and the gravity reference map in the study area. (

**a**) Satellite remote sensing map and (

**b**) gravity reference map.

**Figure 4.**Comparison of the matching test results of gravity matching algorithms under different A criteria.

**Figure 5.**Comparison histogram of the successful matching probability for four algorithms under different positioning accuracies.

**Figure 9.**Histogram comparison on the matching success probability of IOAPs with different ring radius reference angles.

Algorithm | Mean/M | Std/M | Max/M | T/S | $\mathit{\xi}\left(\mathit{l}=100\right)$ | $\mathit{\xi}\left(\mathit{l}=100\sqrt{2}\right)$ |
---|---|---|---|---|---|---|

TERCOM | 64.67 | 45.41 | 415.70 | 2.28 × 10^{−2} | 82 | 99 |

1$\sigma $-IOAP | 394.95 | 518.07 | 2642.07 | 3.90 × 10^{−3} | 48 | 52 |

2$\sigma $-IOAP | 99.23 | 191.91 | 1009.29 | 1.21 × 10^{−2} | 88 | 92 |

3$\sigma $-IOAP | 49.39 | 40.68 | 370.01 | 2.41 × 10^{−2} | 95 | 99 |

**Table 2.**Comparison results of the successful matching probability $\xi $/% of four algorithms under different positioning accuracies.

Algorithm | $\mathit{l}\mathbf{\u2a7d}20$ | $\mathit{l}\mathbf{\u2a7d}40$ | $\mathit{l}\mathbf{\u2a7d}60$ | $\mathit{l}\mathbf{\u2a7d}80$ | $\mathit{l}\mathbf{\u2a7d}100$ | $\mathit{l}\mathbf{\u2a7d}120$ | $\mathit{l}\mathbf{\u2a7d}100\sqrt{2}$ |
---|---|---|---|---|---|---|---|

TERCOM | 0 | 0 | 82 | 82 | 82 | 82 | 99 |

1$\sigma $-IOAP | 9 | 24 | 41 | 45 | 48 | 50 | 52 |

2$\sigma $-IOAP | 17 | 47 | 61 | 79 | 88 | 92 | 92 |

3$\sigma $-IOAP | 10 | 51 | 75 | 90 | 95 | 98 | 99 |

**Table 3.**Comparison of the matched performances of IOAPs with different ring radius reference angles.

Algorithm | Mean/M | Std/M | Max/M | T/S | $\mathit{\xi}\left(\mathit{l}=100\right)$ | $\mathit{\xi}\left(\mathit{l}=100\sqrt{2}\right)$ |
---|---|---|---|---|---|---|

TERCOM | 64.67 | 45.41 | 415.70 | 2.28 × 10^{−2} | 82 | 99 |

1-IOAP | 53.93 | 32.85 | 156.43 | 1.77 × 10^{−2} | 88 | 98 |

1.5-IOAP | 49.39 | 40.68 | 370.01 | 2.41 × 10^{−2} | 95 | 99 |

2-IOAP | 46.82 | 27.53 | 145.58 | 3.11 × 10^{−2} | 95 | 99 |

2.5-IOAP | 45.74 | 24.52 | 156.94 | 4.07 × 10^{−2} | 97 | 99 |

**Table 4.**Comparison of the matching success probability of IOAPs with different location accuracies (%).

Algorithm | $\mathit{l}\mathbf{\u2a7d}20$ | $\mathit{l}\mathbf{\u2a7d}40$ | $\mathit{l}\mathbf{\u2a7d}60$ | $\mathit{l}\mathbf{\u2a7d}80$ | $\mathit{l}\mathbf{\u2a7d}100$ | $\mathit{l}\mathbf{\u2a7d}120$ | $\mathit{l}\mathbf{\u2a7d}100\sqrt{2}$ |
---|---|---|---|---|---|---|---|

TERCOM | 0 | 0 | 82 | 82 | 82 | 82 | 99 |

1-IOAP | 15 | 44 | 64 | 83 | 88 | 95 | 98 |

1.5-IOAP | 10 | 51 | 75 | 90 | 95 | 98 | 99 |

2-IOAP | 15 | 46 | 77 | 88 | 95 | 98 | 99 |

2.5-IOAP | 14 | 44 | 76 | 92 | 97 | 98 | 99 |

Starting Point | Algorithm | MEAN/M | STD/M | MAX/M | T/S | $\mathit{\xi}\left(\mathit{l}\right)$ | |||||
---|---|---|---|---|---|---|---|---|---|---|---|

40 | 60 | 80 | 100 | 120 | $100\sqrt{2}$ | ||||||

A | TERCOM | 78.16 | 41.03 | 269.87 | 2.07 × 10^{−2} | 0 | 62 | 62 | 62 | 62 | 99 |

1.5IOAP | 62.24 | 31.15 | 142.37 | 2.36 × 10^{−2} | 30 | 47 | 69 | 87 | 97 | 99 | |

B | TERCOM | 96.82 | 47.78 | 415.70 | 2.03 × 10^{−2} | 0 | 36 | 36 | 36 | 45 | 99 |

1.5IOAP | 57.71 | 34.97 | 149.82 | 2.35 × 10^{−2} | 33 | 58 | 73 | 84 | 96 | 99 | |

C | TERCOM | 83.48 | 55.58 | 534.45 | 2.04 × 10^{−2} | 0 | 66 | 66 | 66 | 66 | 99 |

1.5IOAP | 71.89 | 30.07 | 148.78 | 2.39 × 10^{−2} | 14 | 33 | 60 | 82 | 94 | 99 |

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**MDPI and ACS Style**

Zhao, S.; Zheng, W.; Li, Z.; Xu, A.; Zhu, H.
Improving Matching Accuracy of Underwater Gravity Matching Navigation Based on Iterative Optimal Annulus Point Method with a Novel Grid Topology. *Remote Sens.* **2021**, *13*, 4616.
https://doi.org/10.3390/rs13224616

**AMA Style**

Zhao S, Zheng W, Li Z, Xu A, Zhu H.
Improving Matching Accuracy of Underwater Gravity Matching Navigation Based on Iterative Optimal Annulus Point Method with a Novel Grid Topology. *Remote Sensing*. 2021; 13(22):4616.
https://doi.org/10.3390/rs13224616

**Chicago/Turabian Style**

Zhao, Shijie, Wei Zheng, Zhaowei Li, Aigong Xu, and Huizhong Zhu.
2021. "Improving Matching Accuracy of Underwater Gravity Matching Navigation Based on Iterative Optimal Annulus Point Method with a Novel Grid Topology" *Remote Sensing* 13, no. 22: 4616.
https://doi.org/10.3390/rs13224616