# AUV Trajectory Tracking Models and Control Strategies: A Review

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## Abstract

**:**

## 1. Introduction

## 2. AUV Trajectory Tracking Model

#### 2.1. Analytical Model

#### 2.1.1. Model Building

**Definition**

**1.**

**v**).

**Definition**

**2.**

**v**).

**Definition**

**3.**

**v**) $-\left\{R\left(\mathit{v}\right)\right\}$.

#### 2.1.2. System Coupling Factors

#### 2.1.3. System Nonlinearity Factors

#### 2.1.4. Environmental Disturbance Factors

#### 2.2. System Identification

#### 2.2.1. Offline Identification

#### 2.2.2. Offline Identification

## 3. AUV Trajectory Tracking Control

#### 3.1. Significance of Control Strategy in AUV Trajectory Tracking

#### 3.2. Methodologies of Control Strategies Applied in AUV Trajectory Tracking

#### 3.2.1. Mathematical Modeling Study

- Optimal Control
- (1)
- Linear quadratic regulator (LQR)

- (2)
- State-dependent Riccati equation (SDRE) control

**p**, $\mathit{B}\left(\mathit{P}\right)={\left[{\mathit{B}}_{\mathbf{1}}{}^{T}\left(P\right),{0}^{T}\right]}^{T}\in {R}^{4}$ and $\mathit{d}={\left[{d}_{1}{}^{T},{0}^{T}\right]}^{T}\in {R}^{4}$, ${Z}_{r}$ is prescribed depth, ${x}_{s1}$ is the integral of the depth trajectory tracking error, ${\mathit{x}}_{\mathit{a}\mathbf{1}}$ is the state, ${\delta}_{s}$ is input, $C=\left[0,0,1,0\right]$, $\mathit{v}=\left(\begin{array}{c}{d}_{1}\\ {Z}_{r}\end{array}\right)\in {\Omega}_{v}\subset {R}^{3}$, $E=\left[\begin{array}{cc}\begin{array}{cc}1& 0\\ 0& 1\end{array}& \begin{array}{c}0\\ 0\end{array}\\ \begin{array}{cc}0& 0\\ 0& 0\\ 0& 0\end{array}& \begin{array}{c}0\\ 0\\ -1\end{array}\end{array}\right]$.

- (3)
- Model Predictive Control (MPC)

- Nonlinear Time-Invariant Control
- (1)
- Sliding mode control (SMC)

**i**= 1, 2, 3, ${\mathit{\beta}}_{\mathit{i}}$ determines the rate of decay for ${\mathit{s}}_{\mathit{i}}\left(t\right)$, the results show that the second-order sliding mode controller can compensate for the uncertainty of fluid dynamics and hydraulic parameters and eliminate external disturbances during movements.

- (2)
- Backstepping control

- Adaptive Control

- Robust Control

- Intelligent Control
- (1)
- Fuzzy control

- (2)
- Neural network (NN) control

- (3)
- Reinforcement learning (RL)

- Others
- (1)
- Cascade systems

- (2)
- Bio-inspired control

#### 3.2.2. Physical Experimental Study

#### 3.3. Control Performance

## 4. Conclusions and Future Perspectives

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**The schematic diagram of AUV system identification [62].

**Figure 4.**Nonlinear model predictive control system with disturbance observer (Reprinted from Reference [39] with permission from IJMS, copyright 2017).

**Figure 5.**ANITSMC control block diagram (Reprinted from reference [91] with permission from the Institution of Engineering and Technology, copyright 2017).

**Figure 6.**Block diagram of AFPISMC (Reprinted from reference [95] with permission from Springer-Verlag Berlin Heidelberg, copyright 2017).

**Figure 7.**Block diagram for the double-loop chattering-free adaptive integral sliding mode control scheme (Reprinted from reference [97] with permission from Elsevier, copyright 2017).

**Figure 9.**Block diagram of the three proposed exponentially convergent robust controllers (Reprinted from reference [23] with permission from Elsevier, copyright 2017).

**Figure 10.**Block diagram of self-tuning nonlinear fuzzy PID controller (Reprinted from reference [119] with permission from Hammad, copyright 2017).

Country | AUV Name | Research Institute | Working Depth (m) | Research Purpose |
---|---|---|---|---|

USA | REMUS-6000 | Woods Hole Oceanography Institute | 6000 | Offshore exploration, survey, and automatic sampling [4] |

Odyssey | Massachusetts Institute of Technology | 3000 | Scientific investigation and ocean automatic sampling network research [5] | |

CETUS | Massachusetts Institute of Technology | 4000 | Military torpedo detection search and danger elimination [6] | |

SAUVIM | University of Hawaii | 6000 | Cable laying and demining | |

China | CR-02 | Shenyang Automation Institute | 6000 | Mineral resources survey and development [7] |

CR-01 | Shenyang Automation Institute | 6000 | Pacific Polymetallic Nodules Survey [8] | |

“Explorer” | Shenyang Automation Institute | 4500 | Marine search and rescue, undersea resource survey [9] | |

Germany | DeePC | STN Company | 4000 | The ice survey [10] |

UK | AUTOSUB | Southampton Oceanographic Centre | 1600 | Multi-purpose marine survey and surveillance platform [11] |

France | ALIVE | Cybernextix Company | 3000 | Equipment maintenance and investigation, archaeology and dangerous goods collection [12] |

Portugal | Delfim | Dynamical systems and ocean roboticsLAB | 4000 | Collection and transmission of marine data [13] |

Norway | HUGIN1000 | Konsberg | 1000 | Mine search mission [14] |

HUGIN3000 | Konsberg | 3000 | Application of fuel cell to AUV [15] | |

Japan | Tri-TON 2 | University of Tokyo, Japan | 2000 | Detect underwater mineral storage [16] |

Canada | Theseus AUV | ISE research | 2000 | Ice cable laying [17] |

Classification | Authors | Influencing Factors | Method of Modeling | Important Finding |
---|---|---|---|---|

Analytical model | [43] | The external environment disturbance and its coupling factors. | The Newton–Euler equation and the Quasi-Lagrange equation. | Not only reduces the number of model parameters, but also helps to analyze the impact of dynamic equations on the vehicle motion. |

[46] | The external environment disturbance and its coupling factors. | Quasi-Lagrangian equation. | Divided the model into a series of interrelated subsystems. | |

[47] | The external environment disturbance and its coupling factors. | Introduces additional items $\Delta \left(\upsilon ,\eta \right).$ | Avoid nonlinear modeling errors. | |

[51] | System coupling. | The optimization problem can be broken down into three smaller sub-problems. | This sub-problem should be solved in parallel so the calculation time can be greatly reduced. | |

[77] | System coupling. | Ignore the coupling between the rolling surface motion and the two plane motions. | Shorten the calculation time required to determine each controller. | |

[53] | System coupling. | Affine nonlinear systems. | Improved trajectory tracking accuracy | |

[54] | Complexity and the nonlinearity of the model. | Linearizing an operating forward speed. | The high nonlinearity of the model is handled. | |

[56] | Complexity and the nonlinearity of the model. | SDRE | Offer great design flexibility systematic and effective means for the design of control systems for nonlinear dynamical systems. | |

[1] | Complexity and the nonlinearity of the model | local-CFDL | Avoid errors caused by the traditional linearization process. | |

[57] | The significant interference caused by shallow water waves is introduced into the translational motion of the AUV. | A superposition of multiple regular blogs to implement the description of random waves. | Perform mathematical model of these disturbances to facilitate exercise planning and control purposes. | |

System identification | [64] | N/A | least squares (LS) | Offline system identification method. |

[69] | Measurement noise and mild nonlinearity. | Observer Kalman filter. | Remove noise and nonlinearity to make the model more accurate. | |

[42] | The AUV system is influenced by nonlinear factors and require nonlinear filtering methods. | Extended Kalman filter (EKF). | Use the instantaneous linearization of each time step to approximate nonlinearity. | |

[73] | The external environment disturbance and its coupling factors. | NN approximator and adaptive technology. | Estimate the uncertainty of the model due to unknown vehicle parameters, unmodeled dynamics, and constant or time-varying disturbances caused by waves and ocean currents. |

Control Strategy | Classification | Improvement | Control Object | Control Effect | Ref. |
---|---|---|---|---|---|

LQR | LQR | — | Track the reference trajectory. | Accurate tracking of spiral, sawtooth paths and 3D Dubin paths. | [80] |

SDRE | SDRE | A hyperbolic tangent sigmoid function is introduced to equivalently replace the rudder angle variable. | Achieve the rudder saturation constraint problem. | The error of the trajectory tracking to converge smoothly to the steady state value. | [56] |

The quadratic performance index. | Deal with suboptimal underwater surface control problems in AUV. | Depth control is achieved with actuator saturation and parameter uncertainty. | [83] | ||

MPC | MPC | Genetic algorithm. | Track the reference trajectory. | Track the given nonlinear path with satisfactory accuracy. | [87] |

Recurrent neural network. | Control of AUVs in a vertical plane. | Track the given nonlinear path with satisfactory accuracy. | [88] | ||

LMPC | C/GMRES algorithm. | Handling the actual constraints of the AUV thruster. | Improve the algorithmic efficiency of the NMPC algorithm. | [85] | |

DMPC | Subproblems and the warm start strategy. | Solve AUV control problems to track time-varying trajectories. | Reduce controller runtime and achieve good control. | [51] |

Control Algorithm | Research Purposes | Improvement | Control Effect | Ref. |
---|---|---|---|---|

Sliding mode control | Improve control accuracy | Robust sliding mode controller | Successfully controls the AUV roll angle, pitch angle and yaw angle within 10 s. | [106] |

Sliding mode variable structure control | The AUV can accurately reach the termination point from the starting point. | [107] | ||

Fuzzy logic | The trajectory tracking accuracy of the vehicle in all directions is very high. | [46] | ||

Solve the jitter problem | Bounded adaptive estimation | Solve the problem of speed jump due to initial error in conventional backstepping method, avoiding thruster saturation and satisfying control input and speed constraint conditions. | [96] | |

Dual closed-loop adaptive integral sliding mode controller | The designed controller can effectively eliminate the flutter effect. | [97] | ||

Self-adaptive fuzzy PI sliding mode control | PISMC has less oscillator response and the shortest delay time. | [95] | ||

Adaptive fuzzy sliding mode with PID sliding surface | Avoid response oscillating and reduce arrival time. | [94] | ||

Achieve the finite-time convergence of the system dynamics | The terminal sliding mode | Force the AUV’s position to track the desired time-varying trajectory. | [54] | |

Adaptive nonsingular integral terminal sliding mode control | Better robustness and faster convergence. | [91] | ||

Solve the limitations of actuators | The second-order sliding mode controlled | Effectively compensate for the uncertainties of the hydrodynamic and hydrostatic parameters of the vehicle and can eliminate unpredictable disturbance effects. | [108] | |

A second-order sliding mode controller using PID sliding surface | 2-SMC with switching controller showed smaller rms error in steady state than 2-SMC without switching controller. | [98] | ||

Realize horizontal trajectory tracking | Line-of-sight method | The underwater vehicle can be accurately set according to the preset, except for deviations at the starting point and the turning point. | [109] | |

Combination of the lateral trajectory error method and the line-of-sight method | Guarantees global κ-exponential stability of the cross-track error to straight line trajectories in three-dimensional space. | [110] | ||

Combining the cross-tracking error method and the line-of-sight method | The sliding mode controller has good tracking performance for time-varying depth signals. | [111] | ||

Backstepping control | Estimate faster convergence of parameters and tracking errors | Adaptive control scheme | Realize three-dimensional track precise tracking control. | [104,112] |

AUV’s virtual speed control and trajectory tracking enable asymptotic stability | Hierarchical control | The robustness of the system under environmental disturbance is guaranteed. | [105,113] |

Control Algorithm | Classification | Improvement | Control Object | Control Effect | Ref. |
---|---|---|---|---|---|

Fuzzy control | Fuzzy PID | Self-tuning nonlinear fuzzy PID controller | Control position and speed to follow desired trajectories. | Compared with traditional PID, the response speed is faster and the minimum error time is reduced. | [119] |

Hierarchical closed-loop fuzzy control | Closed loop planar trajectory tracking. | Motion and velocity errors are bounded and fast converging, showing the robustness of the control algorithm for external disturbances. | [121] | ||

Direct adaptive control | Compensate for the effect of actuator saturation. | System stability for trajectory tracking in the presence of actuator saturation. | [20] | ||

Neural networkcontrol | Adaptive neural network | Unscented Kalman filter | Pose estimation. | Ensure the accuracy and certainty of the estimate, as well as the feasibility of trajectory tracking control. | [125] |

Filtered technique | Trajectory tracking of AUV with model errors and external disturbances. | Avoided “explosion of complexity”. | [126] | ||

Linearly parameterized radial basis function | Estimate unknown terms. | Removing the inherent error. | [123,127] | ||

Nonlinear adaptive controller | Precise trajectory tracking. | A satisfactory approximation capacity and clearly result in superior tracking performance. | [74] | ||

Online neural network controller | Dynamic linear compensator | Compensating model error. | Extend the operating range of the AUV beyond the capacity of the linear controller. | [128] | |

Reinforcement learning | Address unknown disturbances, parameter uncertainties and control input nonlinearities. | Obtain the optimal tracking performance. | [124,136] | ||

Hybrid control | Dynamic surface control | Tracking curve or straight line. | Reduces controller complexity. | [37,47] | |

Reinforcementlearning | Reinforcement learning | Designed the reward function | Precise trajectory tracking. | The thrusters were 11.14% less solicited by the latter controller. | [132] |

Deep reinforcement learning | A reward function for deep RL | Improve AUV trajectory tracking precise. | Effectively improve reliability and stability, reduce energy consumption, and restrain the vectored thruster sudden change. | [133] | |

Interactive reinforcement learning | Learns from both human rewards and environmental rewards at the same time | Improve rewards and learning efficiency. | AUV can converge faster than a DQN learner from only environmental reward. | [134] | |

Model-free goal-driven deep RL | Based on the DDPG algorithm | Self-tuning of the low-level PID controllers of mobile robots. | Improved adaptability, making the agent able to adapt to different operative conditions | [135] |

Control Types | Strength | Weakness | Future Improvement | Whether Based on Model |
---|---|---|---|---|

PID | Flexibility, simplicity, and good performance. | Poor resistance to external interference. | Developed by combining other control algorithms. | Y |

LQR | Can set the unstable system, and the method is simple and easy to implement. | Lacks the characteristics of robustness. | Online iterative learning linear quadratic regulator (OILLQR). | Y |

SDRE | Ensure a wide range of progressive stability | Only applicable to the affine nonlinear system in the form of state correlation coefficient (SDC). Can only guarantee the local asymptotic stability of the closed-loop system. | Each mathematical model of each cycle can be processed into a form similar to a linear system, and the feedforward method is used to compensate for the nonlinear redundancy terms. | Y |

MPC | Effectively overcome the uncertainty of controlled objects, the dynamic effects of lag and time-varying factors. | Heavy online computational burden. | Offline precomputation, delay compensation, event triggering strategies, and digital continuations. | Y |

SMC | Has a certain resistance to modeling errors, time-varying parameters, and external environment interference. | Jitter problem. | The filtering or fuzzy sliding mode control method. | Y |

Backstepping control | The actuator’s control output is continuous and the control system does not experience jitter. | Vehicle tracking speed jump problem.Inherent disadvantage of “explosion of complexity”. | The “dynamic surface control” technique. | N |

Adaptive control | Ability to re-adjust controller parameters online | Asymptotic convergence under ideal conditions of time infinity. | Combined with other control methods. | N |

Robustness control | Ensure a certain level of dynamic performance while maintaining stability. | Cannot counter the complex control system in actual engineering | The combination of different methods can make the control scheme more effective. | Y |

Fuzzy Control | The control system can maintain good performance even when the characteristics of the controlled object change or perturb. | When establishing methods of fuzzification and inverse fuzzification, there is a lack of systematic methods. | The neural network can dynamically adjust the membership function and the fuzzy rule according to the system information. | N |

NN | Greater degree of fault tolerance and strong data processing capabilities. | Exists a limitation referred to as “the curse of dimensionality”. | Use the “Minimum Learning Parameter (MLP)” algorithm to reduce the computational burden of the algorithm. | N |

Cascaded system | Simplifying the controller design, the expression control law is not complicated. | _ | _ | N |

Bio-inspired | Eliminate the speed jump problem. | _ | _ | Y |

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Li, D.; Du, L.
AUV Trajectory Tracking Models and Control Strategies: A Review. *J. Mar. Sci. Eng.* **2021**, *9*, 1020.
https://doi.org/10.3390/jmse9091020

**AMA Style**

Li D, Du L.
AUV Trajectory Tracking Models and Control Strategies: A Review. *Journal of Marine Science and Engineering*. 2021; 9(9):1020.
https://doi.org/10.3390/jmse9091020

**Chicago/Turabian Style**

Li, Daoliang, and Ling Du.
2021. "AUV Trajectory Tracking Models and Control Strategies: A Review" *Journal of Marine Science and Engineering* 9, no. 9: 1020.
https://doi.org/10.3390/jmse9091020