# An Adaptive RBF-NMPC Architecture for Trajectory Tracking Control of Underwater Vehicles

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Description

## 3. Controller Design

#### 3.1. RBFNN Training

#### 3.2. Objective Function and Constraints

#### 3.3. AGWO Algorithm

Algorithm 1 RBF-NMPC | |

1: | Develop RBFNN predictive model offline using offline data; |

2: | Initialize the parameters of RBF-NMPC; |

3: | For k = 1 to N do |

4: | Sample the plant output $y(k)$; |

5: | Update the parameters of RBFNN to adapt the real environment; |

6: | Calculate the prediction outputs $\widehat{y}(k+p)$; |

7: | While current iteration times $t<{t}_{\mathrm{max}}$; |

8: | Compute the control signal by AGWO; |

9: | $t++$; |

10: |
End while |

11: | Sent the control signal to the underwater vehicle; |

12: | end for |

## 4. Simulation Results

#### 4.1. Model Identification Results

_{u}is set as [−2000 N, 2000 N], the thrust range of F

_{v}is set as [−2000 N, 2000 N], and the range of F

_{r}is set as [−900 Nm, 900 Nm]. The input of the RBFNN is $x(k)=[{F}_{u}(k-1),{F}_{v}(k-1),{F}_{r}(k-1),u(k-1),$ $v(k-1),r(k-1)]$, and the actual output of the RBFNN is $y(k)=[u(k),v(k),r(k)]$. Figure 3 shows 2000 sets of input and output data for underwater vehicles dynamic model identification, where 1900 groups are used as training data and 100 groups are used as test data.

#### 4.2. Optimization Results of AGWO

#### 4.3. Trajectory Tracking Control Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Leon, A.Z.; Huvenne, V.A.I.; Benoist, N.; Ferguson, M. Assessing the repeatability of automated seafloor classification algorithms, with application in marine protected area monitoring. Remote Sens.
**2020**, 12, 1572. [Google Scholar] [CrossRef] - Jalal, F.; Nasir, F. Underwater Navigation, Localization and Path Planning for Autonomous Vehicles: A Review. In Proceedings of the 2021 International Bhurban Conference on Applied Sciences and Technologies (IBCAST), Islamabad, Pakistan, 12–16 January 2021. [Google Scholar]
- Liu, H.; Liu, F.W.; Shi, H.J. Design of an AUV Heading Attitude Controller Based on a Disturbance Observer. In Proceedings of the 2020 International Conference on Electrical Engineering and Control Technologies (CEECT), Melbourne, VIC, Australia, 10–13 December 2020. [Google Scholar]
- Elmokadem, T.; Zribi, M.; Youcef-Toumi, K. Terminal sliding mode control for the trajectory tracking of underactuated autonomous underwater vehicles. Ocean. Eng.
**2017**, 129, 613–625. [Google Scholar] [CrossRef] - Zhang, H.; Zhang, Y.; Xu, Y.; Zhou, J.; Guo, Y. Research on Backstepping Tracking Control of Deep-diving AUV Based on Biological Inspiration. In Proceedings of the 2020 Chinese Control and Decision Conference (CCDC), Hefei, China, 22–24 August 2020. [Google Scholar]
- Raju, S.S.; Swamy, G.N.; Bharath, Y.; Nandini, C.N. Simulation and Performance Analysis of Autonomous Underwater Vehicle using Advanced Control Algorithms. In Proceedings of the 2020 International Conference on Communication and Signal Processing (ICCSP), Chennai, India, 28–30 July 2020. [Google Scholar]
- Jiang, C.M.; Wan, L.; Sun, Y.S.; Li, Y.M. Design of novel sliding-mode controller for high-velocity AUV with consideration of residual dead load. J. Cent. South. Univ.
**2018**, 25, 121–130. [Google Scholar] [CrossRef] - Aras, M.S.M.; Abdullah, S.S.; Rahman, A.F.N.A.; Hasim, N. Depth control of an underwater remotely operated vehicle using neural networkpredictive control. J. Bacteriol.
**2015**, 74, 85–93. [Google Scholar] - Mao, R.Q.; Cui, R.X.; Yan, W.S.; Chen, L.P. Neural Learning-Based Integrated Guidance and Control Algorithm of Multiple Underactuated AUVs. In Proceedings of the 2018 IEEE 8th International Conference on Underwater System Technology: Theory and Applications (USYS), Wuhan, China, 1–3 December 2018. [Google Scholar]
- Li, J.; Du, J. Command filtered adaptive NN trajectory tracking control of underactuated autonomous underwater vehicles. In Proceedings of the 2019 19th International Conference on Control, Automation and Systems (ICCAS), Jeju, Korea, 15–18 October 2019. [Google Scholar]
- Jia, L.Y.; Zhu, Z.Y. Improved fractional-order integral sliding mode control for AUV based on RBF neural network. In Proceedings of the 2019 Chinese Automation Congress (CAC), Hangzhou, China, 1–3 December 2018. [Google Scholar]
- Guo, L.W.; Liu, W.D.; Li, Z.Y.; Li, L.F. An adaptive sliding mode control strategy for the heading control of autonomous underwater vehicles. In Proceedings of the Global Oceans 2020: Singapore—U.S. Gulf Coast, Biloxi, MS, USA, 5–30 October 2020. [Google Scholar]
- Dutta, D.; Upreti, S.R. An optimal feedback control strategy for nonlinear, distributed-parameter processes. Processes
**2019**, 7, 758. [Google Scholar] [CrossRef] [Green Version] - Wei, Y.; Zhu, D.; Chu, Z. Underwater Dynamic Target Tracking of Autonomous Underwater Vehicle Based on MPC Algorithm. In Proceedings of the 2018 IEEE 8th International Conference on Underwater System Technology: Theory and Applications (USYS), Wuhan, China, 1–3 December 2018. [Google Scholar]
- Nabi, G.; Affan, M.; Khan, R.; Hameed, M.; Ali, Z. Adaptive Tracking Controller Design for the Horizontal Planner Motion of an Autonomous Underwater Vehicle. In Proceedings of the 2019 16th International Bhurban Conference on Applied Sciences and Technology (IBCAST), Islamabad, Pakistan, 8–12 January 2019. [Google Scholar]
- Zhang, J.; Sun, T.; Liu, Z. Robust model predictive control for path-following of underactuated surface vessels with roll constraints. Ocean. Eng.
**2017**, 143, 125–132. [Google Scholar] [CrossRef] - Bai, G.X.; Liu, L.; Meng, Y.; Luo, W.D. Path tracking of mining vehicles based on nonlinear model predictive control. Appl. Sci.
**2019**, 9, 1372. [Google Scholar] [CrossRef] [Green Version] - McCue, L. Handbook of marine craft hydrodynamics and motion control (Bookshelf). IEEE Control. Syst. Mag.
**2016**, 36, 78–79. [Google Scholar] - Shen, C.; Shi, Y.; Buckham, B. Trajectory tracking control of an autonomous underwater vehicle using Lyapunov-based model predictive control. IEEE Trans. Ind. Electron.
**2018**, 65, 5796–5805. [Google Scholar] [CrossRef] - Wilamowski, B.M.; Yu, H. Improved computation for levenberg–marquardt training. IEEE Trans. Neural Networ.
**2010**, 21, 930–937. [Google Scholar] [CrossRef] [PubMed] - Qiao, J.F.; Zhou, H.B. Modeling of energy consumption and effluent quality using density peaks-based adaptive fuzzy neural network. IEEE/Caa. J. Autom. Sin.
**2018**, 5, 968–976. [Google Scholar] [CrossRef] - Zhou, H.B.; Qiao, J.F. Soft sensing of effluent ammonia nitrogen using rule automatic formation-based adaptive fuzzy neural network. Desalination Water Treat.
**2019**, 140, 132–142. [Google Scholar] [CrossRef] - Gan, W.; Zhu, D.; Hu, Z.; Shi, X.; Chen, Y. Model predictive adaptive constraint tracking control for underwater vehicles. IEEE Trans. Ind. Electron.
**2019**, 67, 7829–7840. [Google Scholar] [CrossRef] - Mozaffari, A.; Azad, N.L. Empirical investigation and analysis of the computational potentials of bio-inspired nonlinear model predictive controllers: Success and challenges. Int. J. Bio-Inspired Comput.
**2017**, 9, 19–34. [Google Scholar] [CrossRef] - Kar, A.K. Bio inspired computing—A review of algorithms and scope of applications. Expert Syst. Appl.
**2016**, 59, 20–32. [Google Scholar] [CrossRef] - De Mendonça Mesquita, E.; Sampaio, R.C.; Ayala, H.V.H.; Llanos, C.H. Recent meta-heuristics improved by self-adaptation applied to nonlinear model-based predictive control. IEEE Access
**2020**, 8, 118841–118852. [Google Scholar] [CrossRef] - Mirjalili, S.; Mirjalili, S.M.; Lewis, A. Grey wolf optimizer. Adv. Eng. Softw.
**2014**, 69, 46–61. [Google Scholar] [CrossRef] [Green Version] - Gu, W.; Zhou, B.K. Improved grey wolf optimization based on the Quantum-behaved mechanism. In Proceedings of the 2019 IEEE 4th Advanced Information Technology, Electronic and Automation Control Conference (IAEAC), Chengdu, China, 20–22 December 2019. [Google Scholar]

**Figure 3.**The sets of input and output data for dynamic system identification: (

**a**) The excitation signal of longitudinal thrust; (

**b**) The response signal of surge velocity; (

**c**) The excitation signal of lateral thrust; (

**d**) The response signal of sway velocity;(

**e**) The excitation signal of heading moment; (

**f**) The response signal of yaw angular velocity.

**Figure 6.**Comparison chart of actual value and predicted value of test set: (

**a**) The actual and predicted values of surge velocity; (

**b**) The actual and predicted values of sway velocity; (

**c**) The actual and predicted values of yaw angular velocity.

**Figure 9.**Tracking control comparison results: (

**a**) Tracking effect in xoy plane; (

**b**) Tracking effect of each state.

**Figure 10.**Computation time of the RBF-NMPC and LMPC: (

**a**) Computation time of the RBF-NMPC; (

**b**) Computation time of the LMPC.

**Figure 11.**Tracking control comparison results under interference environment: (

**a**) Tracking effect in xoy plane; (

**b**) Tracking effect of each state.

**Figure 12.**Computation time of the RBF-NMPC and LMPC under interference environment: (

**a**) Computation time of the RBF-NMPC; (

**b**) Computation time of the LMPC.

MSE | PSO | DE | GWO | AGWO | Improvement (AGWO to GWO) |
---|---|---|---|---|---|

$x[{m}^{2}]$ | 0.1342 | 0.0026 | 0.0028 | 0.0012 | 57% |

$y[{m}^{2}]$ | 0.0703 | 0.0160 | 0.0158 | 0.0111 | 30% |

$\psi [ra{d}^{2}]$ | 0.1312 | 0.0059 | 0.0059 | 0.0047 | 20% |

Environment | MSE | LMPC | RBF-NMPC | Improvement |
---|---|---|---|---|

No interference | $x[{m}^{2}]$ | 0.0095 | 0.0012 | 87% |

$y[{m}^{2}]$ | 0.0196 | 0.0111 | 43% | |

$\psi [ra{d}^{2}]$ | 0.0096 | 0.0047 | 51% | |

Interference | $x[{m}^{2}]$ | 0.0106 | 0.0026 | 75% |

$y[{m}^{2}]$ | 0.0205 | 0.0154 | 25% | |

$\psi [ra{d}^{2}]$ | 0.0115 | 0.0051 | 56% |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Chu, Z.; Wang, D.; Meng, F.
An Adaptive RBF-NMPC Architecture for Trajectory Tracking Control of Underwater Vehicles. *Machines* **2021**, *9*, 105.
https://doi.org/10.3390/machines9050105

**AMA Style**

Chu Z, Wang D, Meng F.
An Adaptive RBF-NMPC Architecture for Trajectory Tracking Control of Underwater Vehicles. *Machines*. 2021; 9(5):105.
https://doi.org/10.3390/machines9050105

**Chicago/Turabian Style**

Chu, Zhenzhong, Da Wang, and Fei Meng.
2021. "An Adaptive RBF-NMPC Architecture for Trajectory Tracking Control of Underwater Vehicles" *Machines* 9, no. 5: 105.
https://doi.org/10.3390/machines9050105