# Vertical Profile Diving and Floating Motion Control of the Underwater Glider Based on Fuzzy Adaptive LADRC Algorithm

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Dynamic Model of the Underwater Glider

## 3. Design of the Fuzzy Adaptive LADRC Controller

#### 3.1. Fuzzy Adaptive LADRC Control Block Diagram

#### 3.2. Fuzzy Adaptive LADRC Controller

- Construction of the control structure;
- Estimating the value of $\mathrm{b}$ and set other LADRC parameters;
- Finding out the variation laws of e, ec and ${\alpha}_{1}$, ${\alpha}_{2}$ according to the engineering practice, where ec is the differential value of the diving depth error e of the underwater glider;
- Design the fuzzy membership function and establish the fuzzy law.

#### 3.2.1. LADRC Controller

#### 3.2.2. Design of Fuzzy Controller

- When the deviation |e| is large, the system is in the rising stage, and in order to improve the system response speed, it should take a larger ${\alpha}_{1}$. Meanwhile, |e| of the instantaneously large may lead to the differential oversaturation and make the control effect beyond the permitted range, so take a smaller ${\alpha}_{2}$;
- When the control system is in normal operation, |e| and |ec| are medium, and in order to make the depth with a small overshoot, ${\alpha}_{1}$ should be taken smaller. At this time, the value of the ${\alpha}_{2}$ impact on the system is larger, should take a smaller value;
- When |e| is small, ${\alpha}_{1}$ should be increased appropriately so that the system has good steady-state performance. In order to prevent the system from oscillation near the set value, while taking into account the performance of the system against interference, the value ${\alpha}_{2}$ must be properly selected, as ${\alpha}_{2}$ is mainly based on |ec| to regulate; when |ec| larger, choose a smaller ${\alpha}_{2}$, and vice versa to take a larger ${\alpha}_{2}$.

## 4. Simulation and Results Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Abbreviations and Variable | Definition |
---|---|

PID | Proportion integral differential |

ADRC | Active disturbance rejection control |

LADRC | Linear active disturbance rejection control |

FLADRC | Fuzzy adaptive linear active disturbance rejection control |

TD | Tracking differentiator |

LSEF | Linear state error feedback |

LESO | Linear extended state observer |

LQR | Linear quadratic regulator |

DSMC | Dynamic sliding mode control |

$b={[x,y,z]}^{T}$ | Position of the origin of the body coordinate system |

$\varphi $ | Cross-roll angle |

$\theta $ | Pitch angle |

$\psi $ | Yaw angle |

$V={[u,v,w]}^{T}$ | Linear velocity in the body coordinate system |

$p$, $q$, $r$ | Angular velocity in the body coordinate system |

$\alpha $ | Attack angle |

β | Sideslip angle |

${M}_{{f}_{1}}$, ${M}_{{f}_{2}}$, ${M}_{{f}_{3}}$ | Added mass |

${I}_{{f}_{1}}$, ${I}_{{f}_{2}}$, ${I}_{{f}_{3}}$ | Added moment of inertia |

${K}_{MR},{K}_{M0},{K}_{p},{K}_{M},{K}_{q},{K}_{MY},{K}_{r}$,${K}_{{D}_{0}},{K}_{{L}_{0}},{K}_{\beta},$ ${K}_{L}$ | Hydrodynamic coefficients |

${m}_{b}$ | Mass of the adjustable net buoyancy |

${m}_{p}$ | Mass of the movable block |

${m}_{rb}$ | Mass of the underwater glider shell |

${R}_{p}$ | Offset of the movable block |

$\gamma $ | Rotation angle of the movable block |

${r}_{{b}_{1}}$ | The position of the variable ballast mass on the ${e}_{1}$-axis of the body coordinate system |

${r}_{{p}_{1}}$ | Position of the movable block in the body coordinate system |

${U}_{1}$ | Mass of pump oil to adjust the net buoyancy |

${U}_{2}$ | Position of the moving mass |

## References

- Yu, C.; Liu, C.; Lian, L.; Xiang, X.; Zeng, Z. ELOS-based path following control for underactuated surface vehicles with actuator dynamics. Ocean Eng.
**2019**, 187, 106139. [Google Scholar] [CrossRef] - Jawhar, I.; Mohamed, N.; Al-Jaroodi, J.; Zhang, S. An Architecture for Using Autonomous Underwater Vehicles in Wireless Sensor Networks for Underwater Pipeline Monitoring. IEEE Trans. Ind. Inform.
**2019**, 15, 1329–1340. [Google Scholar] [CrossRef] - Cho, H.; Jeong, S.-K.; Ji, D.-H.; Tran, N.-H.; Vu, M.T.; Choi, H.-S. Study on Control System of Integrated Unmanned Surface Vehicle and Underwater Vehicle. Sensors
**2020**, 20, 2633. [Google Scholar] [CrossRef] [PubMed] - Davis, R.E.; Eriksen, C.C.; Jones, C.P. Autonomous buoyancy-driven underwater gliders. In Technology and Applications of Autonomous Underwater Vehicles; Taylor and Francis: London, UK, 2002; pp. 37–58. [Google Scholar]
- Tian, X.; Zhang, H.; Zhang, L.; Wang, Y.; Yang, Y. Research on positive buoyancy underwater glider and its sailing efficiency. Appl. Ocean Res.
**2021**, 110, 102592. [Google Scholar] [CrossRef] - Edwards, D.; Arnold, N.; Heinzen, S.; Strem, C.; Young, T. Flying emplacement of an underwater glider. In Proceedings of the OCEANS 2017-Anchorage, Anchorage, AK, USA, 18–21 September 2017; pp. 1–6. [Google Scholar]
- Imlach, J.; Mahr, R. Modification of a military grade glider for coastal scientific applications. In Proceedings of the 2012 Oceans, Hampton Roads, VA, USA, 14–19 October 2012; pp. 1–6. [Google Scholar]
- Castelao, R.; Glenn, S.; Schofield, O.; Chant, R.; Wilkin, J.; Kohut, J. Seasonal evolution of hydrographic fields in the central middle atlantic bight from glider observations. Geophys. Res. Lett.
**2008**, 35, 183–199. [Google Scholar] [CrossRef] - Daniel, L.R.; Sylvia, T.C. On sampling the ocean using underwater gliders. J. Geophys. Res. Ocean.
**2011**, 116, C08010. [Google Scholar] - Webb, D.; Simonetti, P.; Jones, C. SLOCUM: An underwater glider propelled by environmental energy. IEEE J. Ocean. Eng.
**2001**, 26, 447–452. [Google Scholar] [CrossRef] - Sherman, J.; Davis, R.; Owens, W.; Valdes, J. The autonomous underwater glider “spray”. IEEE J. Ocean. Eng.
**2001**, 26, 437–446. [Google Scholar] [CrossRef] [Green Version] - Nakamura, M.; Asakawa, K.; Hyakudome, T.; Kishima, S.; Matsuoka, H.; Minami, T. Hydrodynamic Coefficients and Motion Simulations of Underwater Glider for Virtual Mooring. IEEE J. Ocean. Eng.
**2013**, 38, 581–597. [Google Scholar] [CrossRef] [Green Version] - Yu, J.; Zhang, A.; Jin, W.; Chen, Q.; Tian, Y.; Liu, C. Development and Experiments of the Sea-Wing Underwater Glider. China Ocean Eng.
**2011**, 25, 721–736. [Google Scholar] [CrossRef] [Green Version] - Liu, F.; Wang, Y.; Wu, Z.; Wang, S. Motion analysis and trials of the deep sea hybrid underwater glider Petrel-II. China Ocean. Eng.
**2017**, 31, 55–62. [Google Scholar] [CrossRef] - Leonard, N.E.; Graver, J.G. Model-based feedback control of autonomous underwater gliders. Ocean. Eng.
**2001**, 26, 633–645. [Google Scholar] [CrossRef] [Green Version] - Fan, S. Dynamics Modeling, Motion Analysis and Controller Design of Underwater Gliders under the Influence of Ocean Currents; Zhejiang University: Hangzhou, China, 2013. [Google Scholar]
- Huang, Z.; Zheng, H.; Wang, S.; Ma, J.; Liu, Y. A self-searching optimal ADRC for the pitch angle control of an underwater thermal glider in the vertical plane motion. Ocean Eng.
**2018**, 159, 98–111. [Google Scholar] [CrossRef] - Zhou, H.; Wei, Z.; Zeng, Z.; Yu, C.; Yao, B.; Lian, L. Adaptive robust sliding mode control of autonomous underwater glider with input constraints for persistent virtual mooring. Appl. Ocean Res.
**2020**, 95, 102027. [Google Scholar] [CrossRef] - Vu, M.T.; Le, T.-H.; Thanh, H.L.N.N.; Huynh, T.-T.; Van, M.; Hoang, Q.-D.; Do, T.D. Robust Position Control of an Over-actuated Underwater Vehicle under Model Uncertainties and Ocean Current Effects Using Dynamic Sliding Mode Surface and Optimal Allocation Control. Sensors
**2021**, 21, 747. [Google Scholar] [CrossRef] - Xiang, X.; Yu, C.; Lapierre, L.; Zhang, J.; Zhang, Q. Survey on Fuzzy-Logic-Based Guidance and Control of Marine Surface Vehicles and Underwater Vehicles. Int. J. Fuzzy Syst.
**2018**, 20, 572–586. [Google Scholar] [CrossRef] - Cao, J.; Cao, J.; Zeng, Z.; Lian, L. Nonlinear multiple-input-multiple-output adaptive backstepping control of underwater glider systems. Int. J. Adv. Robot. Syst.
**2016**, 13, 1729881416669484. [Google Scholar] [CrossRef] [Green Version] - Xu, H.; Oliveira, P.; Soares, C.G. L1 adaptive backstepping control for path-following of underactuated marine surface ships. Eur. J. Control
**2021**, 58, 357–372. [Google Scholar] [CrossRef] - Isa, K.; Arshad, M. Neural network control of buoyancy-driven autonomous underwater glider. In Recent Advances in Robotics and Automation; Springer: Berlin/Heidelberg, Germany, 2013. [Google Scholar]
- Sands, T. Development of Deterministic Artificial Intelligence for Unmanned Underwater Vehicles (UUV). J. Mar. Sci. Eng.
**2020**, 8, 578. [Google Scholar] [CrossRef] - Zhang, S.; Yu, J.; Zhang, A.; Zhang, F. Spiraling motion of underwater gliders: Modeling, analysis, and experimental results. Ocean Eng.
**2013**, 60, 1–13. [Google Scholar] [CrossRef] - Vu, M.T.; Van, M.; Bui, D.H.P.; Do, Q.T.; Huynh, T.-T.; Lee, S.-D.; Choi, H.-S. Study on Dynamic Behavior of Unmanned Surface Vehicle-Linked Unmanned Underwater Vehicle System for Underwater Exploration. Sensors
**2020**, 20, 1329. [Google Scholar] [CrossRef] [Green Version] - Han, J. From PID to Active Disturbance Rejection Control. IEEE Trans. Ind. Electron.
**2009**, 56, 900–906. [Google Scholar] [CrossRef] - Gao, Z.; Hu, S.; Jiang, F. A novel motion control design approach based on active disturbance rejection. In Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, FL, USA, 4–7 December 2001; pp. 1547–1552. [Google Scholar]
- Gao, Z. Scaling and bandwidth-parameterization based controller tuning. In Proceedings of the 2003 American Control Conference, Denver, Colorado, 4–6 June 2003; pp. 4989–4996. [Google Scholar]
- Wang, Y.; Zhang, W.; Dong, H.; Yu, L. A LADRC based fuzzy PID approach to contour error control of networked motion control system with time arying delays. Asian J. Control
**2019**, 22, 1973–1985. [Google Scholar] [CrossRef] - Li, H.; Liu, X.; Li, J. The research of fuzzy immune linear active disturbance rejection control strategy for three-motor synchronous system. Control Eng. Appl. Inform.
**2015**, 14, 50–58. [Google Scholar]

**Figure 10.**Underwater glider diving to a fixed depth of 100 m in the vertical profile under strict input constraint.

**Figure 12.**Control input for diving 100 m fixed depth in vertical profile under strict input constraint.

**Figure 13.**Underwater glider diving to a fixed depth of 100 m in the vertical profile under external disturbances and strict input constraints.

**Figure 14.**Depth following errors of PID, FLADRC, and LADRC under external disturbances and strict input constraints.

**Figure 15.**Control input for a 100 m dive in the vertical profile under external disturbances and strict input constraints.

**Figure 16.**The diving velocity of the glider in the vertical profile under external disturbances and strict input constraints.

α_{1} | ec | NB | NM | NS | ZO | PS | PM | PB |
---|---|---|---|---|---|---|---|---|

e | ||||||||

NB | PB | PB | PM | PM | PS | ZO | ZO | |

NM | PB | PB | PM | PS | PS | ZO | PS | |

NS | PM | PM | PM | PS | ZO | PS | PS | |

ZO | PM | PM | PS | ZO | PS | PM | PM | |

PS | PS | PS | ZO | PS | PS | PM | PM | |

PM | PS | ZO | PS | PM | PM | PM | PB | |

PB | ZO | ZO | PM | PM | PM | PB | PB |

α_{2} | ec | NB | NM | NS | ZO | PS | PM | PB |
---|---|---|---|---|---|---|---|---|

e | ||||||||

NB | PS | PS | PB | PB | PB | PM | PS | |

NM | PS | PS | PB | PM | PM | PS | ZO | |

NS | ZO | PS | PM | PM | PS | PS | ZO | |

ZO | ZO | PS | PS | PS | PS | PS | ZO | |

PS | ZO | PS | PS | ZO | PS | PS | ZO | |

PM | PS | PM | PS | PS | PS | PM | PS | |

PB | PB | PM | PM | PM | PS | PS | PB |

Parameters | Value |
---|---|

Shell static mass | ${m}_{h}$ = 54.28 kg |

Moving mass block | ${m}_{p}$ = 11 kg |

Buoyancy adjustment mass | −0.5 kg ≤ ${m}_{b}$ ≤ 0.5 kg |

Overall drainage mass | $m$ = 65.28 kg |

Additional mass factor | ${M}_{f}$ = diag [1.48, 49.58, 65.92] |

Additional inertia term | ${I}_{f}$ = diag [0.53, 7.88, 10.18] |

Resistance factor | ${K}_{D}$ = 386.29, ${K}_{D0}$ = 7.19 |

Lift force factor | ${K}_{L0}$ = −0.36, ${K}_{L}$ = 440.99 |

Lateral force coefficient | ${K}_{\beta}$ = −115.65 |

Transverse rocking moment coefficient | ${K}_{MR}$ = −58.27, ${K}_{P}$ = −19.83 |

Pitch moment coefficient | ${K}_{M0}$ = 0.28, ${K}_{q}$ = −205.64, ${K}_{M}$ = −65.84 |

Depth Controller | Parameter | Value |
---|---|---|

TD | r | 6000 |

h | 0.01 | |

LSEF | ${\alpha}_{1}$ | 0.25 (initial) |

${\alpha}_{2}$ | 0.75 (initial) | |

b | 0.5 | |

LESO | ${\beta}_{1}$ | 160 |

${\beta}_{2}$ | 1820 | |

${\beta}_{3}$ | 0.069 |

Controller | Maximum Overshoot | FLADRC Relatively Reduction |
---|---|---|

PID | 1.73 m | 75.1% |

LADRC | 0.99 m | 56.6% |

FLADRC | 0.43 m | 0 |

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**

Wang, Z.; Yu, C.; Li, M.; Yao, B.; Lian, L.
Vertical Profile Diving and Floating Motion Control of the Underwater Glider Based on Fuzzy Adaptive LADRC Algorithm. *J. Mar. Sci. Eng.* **2021**, *9*, 698.
https://doi.org/10.3390/jmse9070698

**AMA Style**

Wang Z, Yu C, Li M, Yao B, Lian L.
Vertical Profile Diving and Floating Motion Control of the Underwater Glider Based on Fuzzy Adaptive LADRC Algorithm. *Journal of Marine Science and Engineering*. 2021; 9(7):698.
https://doi.org/10.3390/jmse9070698

**Chicago/Turabian Style**

Wang, Zhiguang, Caoyang Yu, Mingjie Li, Baoheng Yao, and Lian Lian.
2021. "Vertical Profile Diving and Floating Motion Control of the Underwater Glider Based on Fuzzy Adaptive LADRC Algorithm" *Journal of Marine Science and Engineering* 9, no. 7: 698.
https://doi.org/10.3390/jmse9070698