# An Improved Positioning Method for Two Base Stations in AIS

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Displacement Correction Position Estimation Method

^{k}, λ

^{k}). Then (ψ

^{k}, λ

^{k}) can be converted into the horizontal and vertical coordinates (φ

^{k}, ω

^{k}) of the earth’s surface according to the Equation (1).

^{k}, Δω

^{k}) of the vessel after the time interval ΔT is given by

^{n}is the speed over ground and α

^{k}is the heading of the vessel at time k, which can be both obtained according to real-time output of the auxiliary sensors on the vessel, such as compasses and log indicators.

^{k}

^{+1}, ω

^{k}

^{+1}), which can be calculated as Equation (3).

^{k}

^{+1}, λ

^{k}

^{+1}) of the vessel can be calculated by (φ

^{k}

^{+1}, ω

^{k}

^{+1}).

_{i}. At time n, the position equation is linearized using Taylor-series keeping only terms below the second order.

^{k}, ω

^{k}) of the vessel can be calculated after several iterations. According to Equation (1), the geodetic coordinate (ψ

^{k}, λ

^{k}) of the vessel at time k can be estimated.

## 3. Improved Displacement Correction Position Estimation Method

#### 3.1. Improved Method for Accelerated Motion

^{k}, ω

^{k}) at any time k. During the time interval ΔT, displacement vector of the vessel is $\Delta {p}^{k}=\left[\begin{array}{cc}\Delta {\phi}^{k}& \Delta {\omega}^{k}\end{array}\right]$, which can be calculated as Equation (7).

**v**

^{k}and

**a**

^{k}are the speed over ground vector and the acceleration vector of the vessel at time k respectively.

**v**

^{k}can be divided into two components, ${v}_{H}^{k}$ and ${v}_{V}^{k}$. ${v}_{H}^{k}$ is the speed over ground in the direction of the horizontal, and ${v}_{V}^{k}$ is in the vertical direction.

^{k}is the heading angle in radian at time k. Similarly, the acceleration vector

**a**

^{k}can also be denoted like ${a}^{k}=[\begin{array}{cc}{a}_{H}^{k}& {a}_{V}^{k}\end{array}]$ as Equation (9).

^{k}, Δω

^{k}) after the time interval ΔT can be calculated by

^{k}

^{+1}, ω

^{k}

^{+1}) can be obtained. Equation (3) is improved by

^{k+}

^{1}, λ

^{k+}

^{1}) of the vessel can be calculated using Equation (1) according to the proposed improved method in the condition of accelerated mode. Equation (6) can be simplified by

**x**can be obtained by the least square method.

**x**is added to initial estimated position at each iteration to get the more accurate coordinate, which is also served as the initial estimated position of the next iteration. After a certain number of Newton iterations, the estimated position of the vessel would approach the real position at time k [23,24].

**v**is provided by the auxiliary sensor on the vessel, such as a log indicator [25,26]. The acceleration vector

**a**could be calculated by Equation (17).

#### 3.2. Improved Method for Turning Motion

^{k}, ω

^{k}) and the initial speed over ground

**v**

^{k}at time k. During time interval ΔT, the turning angle of the vessel is θ, if the counter clockwise direction is positive.

**v**

^{k}

^{+1}during the turning. Similar to the speed over ground vector

**v**

^{k}at time k, it can also be divided into components in the horizontal and vertical direction. During the turning process, the relationship between

**v**

^{k}and

**v**

^{k}

^{+1}is

^{k}and the time interval ΔT numerically.

**v**

^{k}and

**v**

^{k}

^{+1}both can be measured by the auxiliary sensor on the vessel, θ can be calculated according to Equation (18). Dividing the time interval ΔT, the angular velocity ω

^{k}can be obtained then. The radius of the turning r can be calculated by

_{H}, O

_{V}), which can be calculated as Equation (21).

^{k}

^{+1}, ω

^{k}

^{+1}) of the vessel after the time interval ΔT during the turning can be calculated according to the center of the turning as Equation (22).

^{k+}

^{1}, λ

^{k+}

^{1}) of the vessel can be calculated using Equation (1) according to the proposed improved method in the condition of turning mode. Then ${\widehat{L}}_{A}^{k+1}$ and ${\widehat{L}}_{B}^{k+1}$ in

**b**defined in Equation (15) can be calculated. Then, the position of the vessel can be obtained using the least square method and the Newton iteration.

^{k+}

^{1}, ω

^{k+}

^{1}) after the time interval ΔT using the above Equations (18)–(22), which is based on the geometric relationship in the model of turning motion. In contrast, the turning motion is converted into much a little rectilinear motion in the prior DCPE method, assuming that the trajectory of the vessel is approximated by a straight line in a very short time interval. So, it simply uses Equation (3) to estimate (φ

^{k+}

^{1}, ω

^{k+}

^{1}). Therefore, the proposed method is more accurate than the prior DCPE method.

#### 3.3. Motion Model Adaptation Selection

^{k}, λ

^{k}) can be obtained according to Equation (1).

## 4. Simulation Results and Analysis

#### 4.1. Simulation Scenario

^{2}. After that, the vessel moves in uniform rectilinear motion for 200 s. Then, the vessel moves in uniformly accelerated rectilinear motion for 150 s with the acceleration of −0.2 m/s

^{2}. After that, the vessel turns a corner with the centripetal acceleration of 0.156 m/s

^{2}, and the turning duration time is 50 s, corresponding to the upper right corner in Figure 5. Then, the vessel continually moves clockwise, and the motion of the vessel is similar to the above-mentioned process. Eventually the vessel’s trajectory is a red parallelogram, as shown in Figure 5. The destination is (38°34.421′ N, 121°38.193′ E), which is marked by a bigger yellow diamond. It can be seen in Figure 5 that the initial position and the destination could almost coincide.

#### 4.2. Comparison and Analysis

#### 4.2.1. Horizontal and Vertical Errors

_{i}(t) indicates the deviation, including the horizontal errors δ

_{1}(t) and vertical errors δ

_{2}(t). The mean of the deviation can be calculated by

_{1}indicates the starting moment and N is the total number of positioning. The standard deviation of errors is

^{2}during the time period of 1~150 s. From the simulation results, the standard deviations of the horizontal and vertical errors using the DCPE method are 0.0560 cm and 0.0596 cm, respectively. While using the proposed method, the standard deviation of horizontal errors is 0.0514 cm, and the standard deviation of vertical errors is 0.0492 cm. Similarly, the vessel makes a uniformly accelerated rectilinear motion with acceleration of −0.2 m/s

^{2}during the time period of 350~500 s. From the simulation results, the standard deviations of horizontal and vertical errors using the DCPE method are 0.0543 cm and 0.0605 cm, respectively. The standard deviations of horizontal and vertical errors using the proposed method are 0.0538 cm and 0.0489 cm.

#### 4.2.2. Position Errors

#### 4.2.3. Errors Analysis

_{A}and β

_{B}are the azimuth angles Laotieshan and Huangbaizui in Figure 5. Data in Table 3 and Table 4 corresponds to a part of the first turning process in Figure 5 during the 505th to 521st second. Horizontal error and vertical error represent positioning errors in the horizontal and vertical direction when the vessel is in turning motion.

_{A}and β

_{B}change in the same trend. At the third point, β

_{A}gradually increases, while β

_{B}continually decreases. In this process, β

_{A}and β

_{B}change in the opposite trend, which corresponds to data in blue font in the first two columns. After 15 points, β

_{A}and β

_{B}change in the same trend again.

_{A}and β

_{B}change in the opposite trend, which corresponding to the area of the upper right corner or the lower left corner in Figure 5. This directly leads to a large value of (

**A**

^{T}

**A**)

^{−1}

**A**

^{T}in the Equation (16), which results in a larger Δx in each iteration.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Del Peral-Rosado, J.A.; Castillo, R.E.I.; Mıguez-Sánchez, J.; Navarro-Gallardo, M.; Garcıa-Molina, J.A.; López-Salcedo, J.A.; Seco-Granados, G.; Zanier, F.; Crisci, M. Performance analysis of hybrid GNSS and LTE Localization in urban scenarios. In Proceedings of the 8th ESA Workshop on Satellite Navigation Technologies and European Workshop on GNSS Signals and Signal Processing (NAVITEC), Noordwijk, The Netherlands, 14–16 December 2016; pp. 1–8. [Google Scholar]
- Bayat, M.; Madani, M.H. Loran phase code revisited for continuous wave interference cancellation. IET Sci. Meas. Technol.
**2017**, 11, 322–330. [Google Scholar] [CrossRef] - Song, S.P.; Choi, H.H.; Kim, Y.B.; Lee, S.J.; Park, C. Verification of GPS aided error compensation method for eLoran using raw TOA measurements. In Proceedings of the 11th International Conference on Control, Automation and Systems, Gyeonggi-do, Korea, 26–29 October 2011; pp. 1620–1624. [Google Scholar]
- Johnson, G.; Shalaev, R.; Hartnett, R.; Swaszek, P.; Narins, M. Can LORAN meet GPS backup requirements? IEEE Aerosp. Electron. Syst. Mag.
**2005**, 20, 3–12. [Google Scholar] [CrossRef] - Helfrick, A. Question: Alternate position, navigation timing, APNT? Answer: ELORAN. In Proceedings of the IEEE/AIAA 33rd Digital Avionics Systems Conference (DASC), Colorado Springs, CO, USA, 5–9 October 2014; pp. 3C3-1–3C3-9. [Google Scholar]
- Wang, L.; Li, K.; Zhang, J.; Ding, Z. Soft Fault Diagnosis and Recovery Method Based on Model Identification in Rotation FOG Inertial Navigation System. IEEE Sens. J.
**2017**, 17, 5705–5716. [Google Scholar] [CrossRef] - Du, H.; Sun, Q.; Zhang, Y.; Qi, Z.; Wen, Y. Inertial navigation system positioning assisted by star sensor. In Proceedings of the International Conference on Control, Automation and Information Sciences (ICCAIS), Ansan, Korea, 27–29 October 2016; pp. 183–187. [Google Scholar]
- Sineglazov, V. Main features of terrain-aided navigation systems. In Proceedings of the IEEE 3rd International Conference on Methods and Systems of Navigation and Motion Control (MSNMC), Kiev, Ukraine, 14–17 October 2014; pp. 49–52. [Google Scholar]
- Mistary, P.V.; Chile, R.H. Real time Vehicle tracking system based on ARM7 GPS and GSM technology. In Proceedings of the Annual IEEE India Conference (INDICON), New Delhi, India, 17–20 December 2015; pp. 1–6. [Google Scholar]
- Essaadali, R.; Jebali, C.; Grati, K.; Kouki, A. AIS data exchange protocol study and embedded software development for maritime navigation. In Proceedings of the IEEE 28th Canadian Conference on Electrical and Computer Engineering (CCECE), Halifax, NS, Canada, 3–6 May 2015; pp. 1594–1599. [Google Scholar]
- Johnson, G.; Swaszek, P. Feasibility Study of R-Mode Combining MF DGNSS, AIS, and eLoran Transmissions. 2014. Available online: http://www.accseas.eu/publications/r-mode-feasibility-study/ (accessed on 21 February 2018).
- Johnson, G.; Swaszek, P.; Alberding, J.; Hoppe, M.; Oltmann, J.H. The Feasibility of R-Mode to Meet Resilient PNT Requirements for E-Navigation. In Proceedings of the 27th International Technical Meeting of the Satellite Division of the Institute of Navigation, Tampa, FL, USA, 8–12 September 2014; pp. 3076–3100. [Google Scholar]
- Julia, H.; Stefan, G. Launch of R-Mode Baltic Project—An Alternative Navigation System at Sea. 2017. Available online: http://www.dlr.de/dlr/en/desktopdefault.aspx/tabid-10260/370_read-24695#/gallery/28882 (accessed on 21 February 2018).
- Zhang, S.F.; Hu, Q. Ship Autonomous Positioning System Based on Automatic Identification System (AIS). CN Patent 102305936B, 17 July 2013. [Google Scholar]
- Zhang, J.B.; Zhang, S.F.; Wang, J.P. Pseudorange Measurement Method Based on AIS Signals. Sensors
**2017**, 17, 1183. [Google Scholar] [CrossRef] [PubMed] - Wang, X.Y.; Zhang, S.F.; Sun, X.W. The Additional Secondary Phase Correction System for AIS Signals. Sensors
**2017**, 17, 736. [Google Scholar] [CrossRef] [PubMed] - Hu, Q.; Jiang, Y.; Zhang, J.B.; Sun, X.W.; Zhang, S.F. Development of an Automatic Identification System Autonomous Positioning System. Sensors
**2015**, 15, 28574–28591. [Google Scholar] [CrossRef] [PubMed] - Jiang, Y.; Zhang, S.F.; Yang, D.K. A novel position estimation method based on displacement correction in AIS. Sensors
**2014**, 14, 17376–17389. [Google Scholar] [CrossRef] [PubMed] - Batista, P.; Silvestre, C.; Oliveira, P. Pseudo-range navigation with clock offset and propagation speed estimation. In Proceedings of the 54th IEEE Conference on Decision and Control (CDC), Osaka, Japan, 15–18 December 2015; pp. 7636–7641. [Google Scholar]
- Feng, K.L.; Li, A.; Qin, F.J. Integrated navigation method based on pseudo-range/pseudo-range rate/dual-differenced carrier phase. In Proceedings of the Chinese Automation Congress (CAC), Wuhan, China, 27–29 November 2015; pp. 1778–1782. [Google Scholar]
- Tanaka, Y.; Okuno, J.; Okubo, S. A new method for the computation of global viscoelastic post-seismic deformation in a realistic earth model (I)—Vertical displacement and gravity variation. Geophys. J. Int.
**2006**, 164, 273–289. [Google Scholar] [CrossRef] - Tanaka, Y.; Okuno, J.; Okubo, S. A new method for the computation of global viscoelastic post-seismic deformation in a realistic earth model (II)-horizontal displacement. Geophys. J. Int.
**2007**, 170, 1031–1052. [Google Scholar] [CrossRef] - Gao, L.P.; Sun, H.; Liu, M.N.; Jiang, Y. TDOA collaborative localization algorithm based on PSO and Newton iteration in WGS-84 coordinate system. In Proceedings of the IEEE 13th International Conference on Signal Processing (ICSP), Chengdu, China, 6–10 November 2016; pp. 1571–1575. [Google Scholar]
- Kutil, R.; Flatz, M.; Vajtersic, M. Improvements in Approximation Performance and Parallelization of Nonnegative Matrix Factorization with Newton Iteration. In Proceedings of the International Conference on High Performance Computing & Simulation (HPCS), Genoa, Italy, 17–21 July 2017; pp. 887–888. [Google Scholar]
- Tordal, S.S.; Løvsland, P.O.; Hovland, G. Testing of wireless sensor performance in Vessel-to-Vessel Motion Compensation. In Proceedings of the IECON 2016—42nd Annual Conference of the IEEE Industrial Electronics Society, Florence, Italy, 23–26 October 2016; pp. 654–659. [Google Scholar]
- Xu, B.; Bai, J.; Wang, G.; Zhang, Z.; Huang, W. Cooperative navigation and localization for unmanned surface vessel with low-cost sensors. In Proceedings of the DGON Inertial Sensors and Systems (ISS), Karlsruhe, Germany, 16–17 September 2014; pp. 1–14. [Google Scholar]
- Fillmore, J.P. A Note on Rotation Matrices. IEEE Comput. Graph. App.
**1984**, 4, 30–33. [Google Scholar] [CrossRef] - Jang, J.T.; Gong, H.C.; Lyou, J. Computed Torque Control of an aerospace craft using nonlinear inverse model and rotation matrix. In Proceedings of the 15th International Conference on Control, Automation and Systems (ICCAS), Busan, Korea, 13–16 October 2015; pp. 1743–1746. [Google Scholar]
- Ahmad, F.; Khan, R.A. Eigenvectors of a rotation matrix. Q. J. Mech. Appl. Math.
**2009**, 62, 297–310. [Google Scholar] [CrossRef] - Wang, Y.; Wu, Y. Direction and polarization estimation using quaternion and polarization rotation matrix. In Proceedings of the IEEE Symposium on Electrical & Electronics Engineering (EEESYM), Kuala Lumpur, Malaysia, 24–27 June 2012; pp. 319–322. [Google Scholar]
- Wang, S.; Gao, S.; Yang, W. Ship route extraction and clustering analysis based on automatic identification system data. In Proceedings of the Eighth International Conference on Intelligent Control and Information Processing (ICICIP), Hangzhou, China, 3–5 November 2017; pp. 33–38. [Google Scholar]
- Ren, W.; Li, G.; Lv, J. Analysis of Adopting Mobile Position System Based on GPSOne Technique in VTS System. In Proceedings of the International Conference on Intelligent System Design and Engineering Application, Changsha, China, 13–14 October 2010; pp. 281–284. [Google Scholar]
- Liu, C.Q. Vibratory source’s search and localization algorithm of Newton iteration based on method of weighted least squares. In Proceedings of the International Conference on Measurement, Information and Control, Harbin, China, 18–20 May 2012; pp. 988–991. [Google Scholar]
- Wei, Y. A revision of the Newton iteration format for nonlinear equation. In Proceedings of the Fourth International Workshop on Advanced Computational Intelligence, Wuhan, China, 19–21 October 2011; pp. 540–541. [Google Scholar]

**Figure 1.**Scene of the automatic identification system (AIS) ranging mode (R-Mode) with two base stations.

Name | MMSI | Latitude | Longitude |
---|---|---|---|

Laotieshan | 4,131,101 | $38\xb0{43.6420}^{\prime}$ N | $121\xb0{08.1330}^{\prime}$ E |

Huangbaizui | 4,131,104 | $38\xb0{54.2850}^{\prime}$ N | $121\xb0{42.9500}^{\prime}$ E |

Method | Error | 1~500 s | 501~550 s | 551~1050 s | 1051~1100 s | 1101~1600 s | 1601~1650 s | 1651~2150 s | 2151~2200 s |
---|---|---|---|---|---|---|---|---|---|

DCPE Method | η_{1} | 0.3826 | 0.9627 | 0.1457 | 0.2402 | 0.2710 | 0.7601 | 0.1382 | 0.1689 |

η_{2} | 0.3934 | 0.3165 | 0.1948 | 0.1868 | 0.2575 | 1.0614 | 0.2291 | 0.1959 | |

μ_{1} | −0.0010 | 0.0276 | −0.0007 | 0.0120 | 0.0009 | 0.0249 | −0.0010 | −0.0262 | |

μ_{2} | 0.0014 | −0.0724 | 0.0019 | −0.0195 | −0.0025 | 0.0524 | 0.0003 | −0.0175 | |

σ_{1} | 0.0444 | 0.2673 | 0.0251 | 0.1024 | 0.0459 | 0.2624 | 0.0271 | 0.0674 | |

σ_{2} | 0.0445 | 0.2358 | 0.0329 | 0.1107 | 0.0534 | 0.3429 | 0.0353 | 0.0808 | |

Improved Method | η_{1} | 0.2180 | 0.2043 | 0.1048 | 0.1996 | 0.1462 | 0.2633 | 0.1057 | 0.0766 |

η_{2} | 0.3174 | 0.1674 | 0.1805 | 0.1200 | 0.2280 | 0.3222 | 0.1816 | 0.0780 | |

μ_{1} | −0.00001 | −0.0019 | 0.0001 | 0.0023 | −0.0008 | 0.0128 | −0.0003 | −0.0035 | |

μ_{2} | 0.0004 | 0.0177 | 0.0016 | 0.0018 | −0.0006 | 0.0229 | −0.0007 | 0.0001 | |

σ_{1} | 0.0426 | 0.0800 | 0.0244 | 0.0494 | 0.0420 | 0.0952 | 0.0259 | 0.0390 | |

σ_{2} | 0.0442 | 0.0834 | 0.0302 | 0.0612 | 0.0466 | 0.1236 | 0.0347 | 0.0415 |

No. | ${\mathit{\beta}}_{\mathit{A}}^{\mathit{k}}$ | ${\mathit{\beta}}_{\mathit{B}}^{\mathit{k}}$ | Horizontal Error | Vertical Error |
---|---|---|---|---|

1 | 5.009371978 | 6.116692350 | 0.136755441 | −0.033688209 |

2 | 5.009364103 | 6.116582698 | −0.044890233 | 0.178701572 |

3 | 5.009360177 | 6.116480498 | 0.133527424 | −0.354018592 |

4 | 5.009360530 | 6.116386250 | −0.613144241 | −0.014801477 |

5 | 5.009364834 | 6.116299943 | 0.038769839 | −0.400999096 |

6 | 5.009373314 | 6.116221335 | −0.481755011 | −0.151736104 |

7 | 5.009385773 | 6.116150846 | −0.083722842 | −0.476183031 |

8 | 5.009402483 | 6.116088651 | −0.233052817 | 0.274446059 |

9 | 5.009422998 | 6.116034472 | 0.458114784 | −0.422002996 |

10 | 5.009447685 | 6.115988481 | 0.589510173 | −0.008883503 |

11 | 5.009476324 | 6.115950728 | 0.115658450 | −0.192622662 |

12 | 5.009508806 | 6.115921259 | 0.962727494 | −0.793874836 |

13 | 5.009545403 | 6.115900175 | −0.075081556 | −0.693991153 |

14 | 5.009585874 | 6.115887465 | −0.175334603 | −0.333361476 |

15 | 5.009630160 | 6.115883378 | 0.669634066 | −0.225495902 |

16 | 5.009678183 | 6.115887176 | 0.202885538 | −0.298708880 |

17 | 5.009730051 | 6.115899711 | 0.259243149 | −0.218549897 |

No. | ${\mathit{\beta}}_{\mathit{A}}^{\mathit{k}}$ | ${\mathit{\beta}}_{\mathit{B}}^{\mathit{k}}$ | Horizontal Error | Vertical Error |
---|---|---|---|---|

1 | 5.009371995 | 6.116692369 | 0.052833868 | 0.038729669 |

2 | 5.009364061 | 6.116582636 | 0.072796110 | −0.032886895 |

3 | 5.009360236 | 6.116480577 | 0.059390886 | −0.007025953 |

4 | 5.009360511 | 6.116386303 | 0.036391146 | 0.029988286 |

5 | 5.009364878 | 6.116299822 | −0.137737964 | 0.088876636 |

6 | 5.009373319 | 6.116221349 | 0.006360655 | 0.085991809 |

7 | 5.009385838 | 6.116150860 | 0.015130796 | 0.111074183 |

8 | 5.009402406 | 6.116088433 | −0.190604914 | −0.026010386 |

9 | 5.009423040 | 6.116034233 | −0.068020249 | 0.087595125 |

10 | 5.009447679 | 6.115988256 | −0.033166051 | 0.009970006 |

11 | 5.009476329 | 6.115950615 | 0.097515659 | 0.020200008 |

12 | 5.009508939 | 6.115921190 | 0.003789785 | 0.041866405 |

13 | 5.009545475 | 6.115900117 | 0.017615842 | 0.063760294 |

14 | 5.009585895 | 6.115887434 | 0.056106653 | 0.006849625 |

15 | 5.009630184 | 6.115883145 | 0.102804730 | 0.105033462 |

16 | 5.009678249 | 6.115887169 | 0.038196622 | 0.049230991 |

17 | 5.009730064 | 6.115899593 | 0.051263916 | −0.011125038 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Jiang, Y.; Wu, J.; Zhang, S.
An Improved Positioning Method for Two Base Stations in AIS. *Sensors* **2018**, *18*, 991.
https://doi.org/10.3390/s18040991

**AMA Style**

Jiang Y, Wu J, Zhang S.
An Improved Positioning Method for Two Base Stations in AIS. *Sensors*. 2018; 18(4):991.
https://doi.org/10.3390/s18040991

**Chicago/Turabian Style**

Jiang, Yi, Jiani Wu, and Shufang Zhang.
2018. "An Improved Positioning Method for Two Base Stations in AIS" *Sensors* 18, no. 4: 991.
https://doi.org/10.3390/s18040991