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Keywords = thruster allocation

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19 pages, 1359 KB  
Article
ESO-Enhanced Actor–Critic Reinforcement Learning-Optimised Trajectory Tracking Control for 3-DOF Marine Vessels
by Xiaoling Liang and Jiajian Li
Mathematics 2026, 14(5), 867; https://doi.org/10.3390/math14050867 - 4 Mar 2026
Abstract
This paper develops an extended-state-observer (ESO)-enhanced actor–critic reinforcement learning (RL) scheme for the trajectory tracking control of 3-DOF marine vessels subject to uncertain hydrodynamics and environmental disturbances. A coordinate-consistent error construction is provided to obtain an exact strict-feedback second-order uncertain template. On this [...] Read more.
This paper develops an extended-state-observer (ESO)-enhanced actor–critic reinforcement learning (RL) scheme for the trajectory tracking control of 3-DOF marine vessels subject to uncertain hydrodynamics and environmental disturbances. A coordinate-consistent error construction is provided to obtain an exact strict-feedback second-order uncertain template. On this basis, an Hamilton–Jacobi–Bellman (HJB)-inspired optimised control structure is implemented: the critic approximates the optimal value-gradient and the actor generates the optimised control law. A key simplification is employed: rather than minimising the squared Bellman residual via complex gradients, we introduce an HJB-inspired actor–critic consistency regularisation through a weight-matching coupling. This yields computationally light online update laws and enables transparent Lyapunov-based stability analysis while not claiming exact HJB satisfaction or policy optimality. The ESO estimates lumped uncertainty and provides feedforward compensation, so the RL module learns only the observer residual. A composite Lyapunov analysis establishes the semi-global uniform ultimate boundedness of tracking errors and boundedness of all observer signals. Practical implementation with thruster allocation, explicit wind–wave–current disturbance shaping filters, and a theory-aligned ablation protocol are provided for reproducibility. Full article
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26 pages, 5247 KB  
Article
A Control-Oriented Thruster Management Framework for Fault-Tolerant Propulsion of Remotely Operated Vehicles
by Lu Wang, Yi Wu, Chao Fang, Jie Gao, Yonggang Gu, Chao Zhai and Zhen Zhang
J. Mar. Sci. Eng. 2026, 14(4), 352; https://doi.org/10.3390/jmse14040352 - 12 Feb 2026
Viewed by 230
Abstract
Remotely operated vehicles (ROVs) typically rely on over-actuated propulsion systems to achieve precise dynamic positioning and maneuvering in complex underwater environments. In practice, however, conventional propulsion management based on thrust allocation is often challenged by non-ideal actuator behaviors, such as cavitation-induced thrust degradation, [...] Read more.
Remotely operated vehicles (ROVs) typically rely on over-actuated propulsion systems to achieve precise dynamic positioning and maneuvering in complex underwater environments. In practice, however, conventional propulsion management based on thrust allocation is often challenged by non-ideal actuator behaviors, such as cavitation-induced thrust degradation, low-speed dead-zone effects, inter-thruster coupling, and partial actuator failures. Most existing approaches treat propulsion management as a static force distribution problem and implicitly assume ideal or fast thrust execution, which limits performance when actuator dynamics and execution uncertainty become significant. To address these limitations, this paper proposes a control-oriented thruster management framework that reformulates propulsion management as a feedback regulation problem rather than a static allocation task. In the proposed framework, actuator dynamics and thrust execution uncertainty are explicitly incorporated into the control loop. At the actuator level, thrust degradation and low-speed operation are compensated through disturbance-aware feedback control, while at the system level an LQI-based controller with thrust response compensation is employed to coordinate multi-degree-of-freedom (DOF) force and moment regulation and suppress cross-axis coupling. Fault tolerance is achieved inherently through feedback regulation without relying on explicit fault detection or reallocation. Experimental results obtained from an ROV propulsion platform, including single-thruster tests, coupled multi-DOF control, and a thruster shutdown scenario, demonstrate improved thrust executability, reduced coupling-induced disturbances, and enhanced fault-tolerant performance compared with conventional direct thrust allocation strategies. Full article
(This article belongs to the Section Ocean Engineering)
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22 pages, 4949 KB  
Article
The Effect of Wind–Wave Correlations on the Optimal Thruster Location for Offshore Vessels
by Francesco Mauro and Giada Kyaw’oo D’Amore
J. Mar. Sci. Eng. 2025, 13(11), 2025; https://doi.org/10.3390/jmse13112025 - 22 Oct 2025
Viewed by 516
Abstract
Offshore vessels are nowadays equipped with dynamic positioning systems, meaning they have additional thrusters dedicated to the station keeping of the unit. However, there is no rational criterion on the placement of these devices to increment station keeping capabilities. This is true both [...] Read more.
Offshore vessels are nowadays equipped with dynamic positioning systems, meaning they have additional thrusters dedicated to the station keeping of the unit. However, there is no rational criterion on the placement of these devices to increment station keeping capabilities. This is true both in case of a vessel retrofitting or for the design of a new unit. The present work proposes investigating a methodology for the optimal placement of thrusters along the hull of an offshore unit. This implies the adoption of a suitable optimisation algorithm capable of handling all the constraints of the optimisation problem. As the target is the optimal capability, the optimisation should handle multiple dynamic positioning capability calculations, meaning (in a quasi-static approach) that it is capable of solving multiple thrust allocation problems at each optimisation step. As thruster allocation is another optimisation problem, the process should handle two nested optimisations. Here, the global location problem is solved with a differential evolution algorithm, while the thrust allocation employs non-linear programming. As the capability calculations imply the adoption of a specific wind–wave correlation, the present work compares the effect of different correlations on the optimised location of the thrusters. The results presented on a reference Pipe Lay Crane Vessel highlight the differences in the final optimum as a function of the environmental modelling. Full article
(This article belongs to the Special Issue Design Optimisation in Marine Engineering)
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20 pages, 6462 KB  
Article
Thrust Allocation Control of an Underwater Vehicle with a Redundant Thruster Configuration
by Liping Deng and Jianguo Tao
Mathematics 2025, 13(11), 1766; https://doi.org/10.3390/math13111766 - 26 May 2025
Cited by 4 | Viewed by 2813
Abstract
This paper presents a fault-tolerant thruster configuration scheme and a thrust control allocation strategy for an underwater vehicle. First, to accommodate the vehicle’s flexible spatial motion capabilities and address potential thruster failures, an 8-thruster vector arrangement is designed, and the impact of thruster [...] Read more.
This paper presents a fault-tolerant thruster configuration scheme and a thrust control allocation strategy for an underwater vehicle. First, to accommodate the vehicle’s flexible spatial motion capabilities and address potential thruster failures, an 8-thruster vector arrangement is designed, and the impact of thruster failures on vehicle maneuverability is analyzed. Based on this configuration, a mathematical model of the vector propulsion system is then developed, establishing the relationship between the thrust generated by the individual thrusters and the virtual control forces applied to the vehicle’s 6 degrees of freedom (DOF). Subsequently, a thrust allocation strategy based on quadratic programming (QP) is proposed to optimize thrust allocation, enhancing energy efficiency while satisfying thrust saturation constraints. Finally, simulation results demonstrate that the proposed thruster configuration exhibits strong fault-tolerance. Moreover, compared to the least squares (LS) method based on the pseudo-inverse of the configuration matrix, the QP-based thrust allocation strategy achieves significantly better energy-saving performance. Full article
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19 pages, 6806 KB  
Article
An Energy-Efficient Thrust Allocation Based on the Improved Dung Beetle Optimizer for the Dynamic Positioning System of Vessels
by Yulong Tuo, Jianlong Lin, Zhouhua Peng, Yuanhui Wang and Shasha Wang
J. Mar. Sci. Eng. 2025, 13(6), 1041; https://doi.org/10.3390/jmse13061041 - 26 May 2025
Cited by 4 | Viewed by 1327
Abstract
This paper investigates the constrained nonlinear thrust allocation problem for the dynamic positioning system of vessels. Considering the wear, energy consumption, and allocation error of thrusters, a constrained nonlinear mathematical optimization model of thrust allocation is established based on the “Hai Yang Shi [...] Read more.
This paper investigates the constrained nonlinear thrust allocation problem for the dynamic positioning system of vessels. Considering the wear, energy consumption, and allocation error of thrusters, a constrained nonlinear mathematical optimization model of thrust allocation is established based on the “Hai Yang Shi You 201”. Based on the dung beetle optimizer (DBO) algorithm, a hybrid Osprey adaptive t-distribution DBO (HOATDBO) algorithm is presented to achieve the thrust allocation. The HOATDBO algorithm introduces the global exploration strategy of the Osprey algorithm, with the addition of the initialization of the good point set and adaptive t-distribution perturbations. The proposed HOATDBO algorithm has perfect global and local optimization capabilities, which can quickly and reliably obtain the optimal thrust solution, improve the thrust allocation accuracy of vessels, and reduce energy consumption. Finally, the simulation and comparison results are presented to verify the superiority of the proposed HOATDBO algorithm. Full article
(This article belongs to the Special Issue Control and Optimization of Ship Propulsion System)
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28 pages, 9566 KB  
Article
The Design of a New Type of Remotely Operated Vehicle System and the Realization of a Thrust Distribution Method
by Fushen Ren, Xin Guo, Xin Deng, Baojin Wang and Zhongyang Wang
Appl. Sci. 2025, 15(8), 4199; https://doi.org/10.3390/app15084199 - 10 Apr 2025
Cited by 2 | Viewed by 1266
Abstract
In order to realize the detection of marine engineering facilities, the hardware system of a new type of remotely operated vehicle (ROV) is designed independently, and the control system, including the lower computer program and the upper computer software, is developed. At the [...] Read more.
In order to realize the detection of marine engineering facilities, the hardware system of a new type of remotely operated vehicle (ROV) is designed independently, and the control system, including the lower computer program and the upper computer software, is developed. At the same time, in order to explore the thrust distribution of the thruster and realize the optimization of the thrust distribution under the installation position and installation angle of the designed thruster, the mathematical model of the ROV propulsion system is established. The simulation models of ROV motion control and thrust distribution are established in MATLAB R2022a and Unity 3D, respectively. Given the thrust input of the compound motion, the sequential quadratic programming (SQP) method and the direct logic method are used to compare the simulation results of thrust distribution. Finally, the underwater attitude control experiment and the application experiment of the actual scene are carried out. Combined with the simulation and experimental results, the feasibility of using the sequential quadratic programming method to optimize the thrust allocation is verified, and it is shown that the new ROV system can basically meet the expected design requirements. Full article
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23 pages, 8791 KB  
Article
Event-Triggered Neural Adaptive Distributed Cooperative Control for the Multi-Tug Towing of Unactuated Offshore Platform with Uncertainties and Unknown Disturbances
by Shaolong Geng, Yulong Tuo, Yuanhui Wang, Zhouhua Peng and Shasha Wang
J. Mar. Sci. Eng. 2024, 12(8), 1242; https://doi.org/10.3390/jmse12081242 - 23 Jul 2024
Cited by 3 | Viewed by 1532
Abstract
An event-triggered neural adaptive cooperative control is proposed for the towing system (TS) with model parameter uncertainties and unknown disturbances. Different from ordinary multi-vessel formation control, the tugs and unactuated offshore platform in the TS are connected together by towlines, and the resultant [...] Read more.
An event-triggered neural adaptive cooperative control is proposed for the towing system (TS) with model parameter uncertainties and unknown disturbances. Different from ordinary multi-vessel formation control, the tugs and unactuated offshore platform in the TS are connected together by towlines, and the resultant tension of the towlines serves as the actual drag force for the platform. Initially, based on the radial basis function neural network (RBFNN), an adaptive RBFNN is designed to compensate unknown disturbances and model parameter uncertainties of the TS, and we use minimal learning parameter (MLP) algorithm to reduce the online learning parameters of adaptive RBFNN. Combined with dynamic surface technology and event-triggered control (ETC) mechanism, an event-triggered neural adaptive virtual controller is designed to obtain the desired drag force of the platform. According to the quadratic programming algorithm, the desired drag force is allocated as the desired tensions of towlines. Subsequently, the desired towline length and the desired position information of the tugs are obtained sequentially through the towline model and the position relationship between the tugs and the platform. Then, according to the desired positions of tugs, an event-triggered neural adaptive distributed cooperative controller is designed for achieving the multi-tug towing of the offshore platform. The ETC mechanism is introduced to reduce the communication burden within the TS and the execution frequency of the tugs’ thrusters. Finally, the stability of the closed-loop system is proven using the Lyapunov theory, and the ETC mechanism proves that no Zeno behavior occurs. The effectiveness of the ETC mechanism and the MLP-based adaptive RBFNN on the controllers of TS is verified through simulations and comparison analysis. Full article
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22 pages, 2624 KB  
Article
Multi-Objective Optimization for Thrust Allocation of Dynamic Positioning Ship
by Qiang Ding, Fang Deng, Shuai Zhang, Zhiyu Du and Hualin Yang
J. Mar. Sci. Eng. 2024, 12(7), 1118; https://doi.org/10.3390/jmse12071118 - 3 Jul 2024
Cited by 1 | Viewed by 2888
Abstract
Thrust allocation (TA) plays a critical role in the dynamic positioning system (DPS). The task of TA is to allocate the rotational speed and angle of each thruster to generate the generalized control forces. Most studies take TA as a single-objective optimization problem; [...] Read more.
Thrust allocation (TA) plays a critical role in the dynamic positioning system (DPS). The task of TA is to allocate the rotational speed and angle of each thruster to generate the generalized control forces. Most studies take TA as a single-objective optimization problem; however, TA is a multi-objective optimization problem (MOP), which needs to satisfy multiple conflicting allocation objectives simultaneously. This study proposes an improved multi-objective particle swarm optimization (IMOPSO) method to deal with the non-convex MOP of TA. The objective functions of reducing the allocation error, and minimizing the power consumption and the tear-and-wear of thrusters under physical constraints, are established and solved via MOPSO. To enhance the global seeking ability, the improved mutation strategy combined with the roulette wheel mechanism is adopted. It is shown through test data that IMOPSO converges better than multi-objective algorithms such as MOPSO and nondominated sorting genetic algorithm II (NSGA-II). Simulations are conducted for a DP ship with two propeller–rudder combinations. The simulation results with the single-objective PSO algorithm show that the proposed IMOPSO algorithm reduces thrust allocation errors in the three directions of surge, sway, and yaw by 48.48%, 39.64%, and 15.02%, respectively, and reduces power consumption by 44.53%, which demonstrates the feasibility and effectiveness of the proposed method. Full article
(This article belongs to the Section Ocean Engineering)
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25 pages, 8357 KB  
Article
Experimental Study on Adaptive Backstepping Synchronous following Control and Thrust Allocation for a Dynamic Positioning Vessel
by Changde Liu, Yufang Zhang, Min Gu, Longhui Zhang, Yanbin Teng and Fang Tian
J. Mar. Sci. Eng. 2024, 12(2), 203; https://doi.org/10.3390/jmse12020203 - 23 Jan 2024
Cited by 6 | Viewed by 1908
Abstract
Cargo transfer vessels (CTVs) are designed to transfer cargo from a floating production storage and offloading (FPSO) unit into conventional tankers. The dynamic positioning system allows the CTV to maintain a safe position relative to the FPSO unit using a flexible cargo transmission [...] Read more.
Cargo transfer vessels (CTVs) are designed to transfer cargo from a floating production storage and offloading (FPSO) unit into conventional tankers. The dynamic positioning system allows the CTV to maintain a safe position relative to the FPSO unit using a flexible cargo transmission pipe, and the CTV tows the tanker during operating conditions. The operation mode can be considered a synchronization tracking control problem. In this paper, a synchronization control strategy is presented based on the virtual leader–follower configuration and an adaptive backstepping control method. The position and heading of the following vessel are proven to be able to globally exponentially converge to the virtual ship by the contraction theorem. Then, the optimization problem of the desired thrust command from the controller is solved through an improved firefly algorithm, which fully considers the physical characteristics of the azimuth thruster and the thrust forbidden zone caused by hydrodynamic interference. To validate the effectiveness of the presented synchronous following strategy and thrust allocation algorithm, a scale model experiment is carried out under a sea state of 4 in a seakeeping basin. The experimental results show that the CTV can effectively maintain a safe distance of 100 m with a maximum deviation of 3.78 m and an average deviation of only 0.99 m in the wave heading 180°, which effectively verifies that the control strategy proposed in this paper can achieve safe and cooperative operation between the CTV and the FPSO unit. To verify the advantages of the SAF algorithm in the thrust allocation, the SQP algorithm and PSO algorithm are used to compare the experimental results. The SAF algorithm outperforms the SQP and PSO algorithms in longitudinal and lateral forces, with the R-squared (R2) values of 0.9996 (yaw moment), 0.9878 (sway force), and 0.9596 (surge force) for the actual thrusts and control commands in the wave heading 180°. The experimental results can provide technical support to improve the safe operation of CTVs. Full article
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20 pages, 5811 KB  
Article
Planning Allocation for GTO-GEO Transfer Spacecraft with Triple Orthogonal Gimbaled Thruster Boom
by Guangfu Ma and Xianglong Kong
Mathematics 2023, 11(13), 2844; https://doi.org/10.3390/math11132844 - 25 Jun 2023
Cited by 2 | Viewed by 1899
Abstract
This paper proposes an electric propulsion platform based on a triple orthogonal gimbaled thruster boom to realize the GTO-GEO transfer process. The adjustment mechanism of the gimbaled thruster boom significantly improves the range of thrust vector variation enhances the efficiency of thrust vector [...] Read more.
This paper proposes an electric propulsion platform based on a triple orthogonal gimbaled thruster boom to realize the GTO-GEO transfer process. The adjustment mechanism of the gimbaled thruster boom significantly improves the range of thrust vector variation enhances the efficiency of thrust vector adjustment, and reduces the spacecraft burn-up. Additionally, to achieve the application performance, a planning allocation method based on the model prediction algorithm is proposed and verified through numerical simulation. Full article
(This article belongs to the Special Issue Mathematical Methods in Spacecraft Relative Motion)
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14 pages, 4664 KB  
Article
Dynamic Response Characteristics of the Hydraulic Rotary System of an Azimuth Thruster for a Dynamic Positioning Vessel
by Ruiqi Liu, Zhongyu Jin, Xinfei Li and Lihao Yuan
J. Mar. Sci. Eng. 2023, 11(2), 399; https://doi.org/10.3390/jmse11020399 - 11 Feb 2023
Cited by 2 | Viewed by 5622
Abstract
The composition characteristics and working principle of the hydraulic rotary system of the azimuth thruster were analyzed. The mathematical model of the rotary dynamic system of the pump-controlled hydraulic motor driving the gear reduction mechanism was established. Additionally, a fast tracking method for [...] Read more.
The composition characteristics and working principle of the hydraulic rotary system of the azimuth thruster were analyzed. The mathematical model of the rotary dynamic system of the pump-controlled hydraulic motor driving the gear reduction mechanism was established. Additionally, a fast tracking method for the azimuth angle of the azimuth thruster was proposed to analyze the rotary azimuth angle, angular velocity and dynamic response characteristics of the hydraulic system for different desired azimuth angles. The simulation results show that the established dynamic model can simulate the rotary motion response process of the real thruster, and can realize the rapid and accurate tracking of the azimuth angle. At the same time, the physical constraints of the rotary dynamic response were established. It provides an important reference for research on the motion control methods of dynamic positioning vessels. Full article
(This article belongs to the Special Issue Advances in Marine Propulsion)
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16 pages, 5715 KB  
Article
Fault-Tolerant Thrust Allocation with Thruster Dynamics for a Twin-Waterjet Propelled Vessel
by Zijing Xu, Roberto Galeazzi and Jingqi Yuan
J. Mar. Sci. Eng. 2022, 10(12), 1983; https://doi.org/10.3390/jmse10121983 - 13 Dec 2022
Cited by 3 | Viewed by 2691
Abstract
The availability of the propulsion system is of primary importance to ensure safe and stable operations of marine crafts, both during transit and station keeping. Diminished propulsion efficiency could impair the ability of a vessel to maintain speed and course and possibly lead [...] Read more.
The availability of the propulsion system is of primary importance to ensure safe and stable operations of marine crafts, both during transit and station keeping. Diminished propulsion efficiency could impair the ability of a vessel to maintain speed and course and possibly lead to a drifting craft. The waterjet’s propulsion efficiency is affected by several factors such as cavitation, erosion, vibration and noise emission. This paper addresses the design of a fault-tolerant thrust allocation algorithm able to maintain the seaworthiness of a twin-waterjet marine craft in the presence of a severe power loss in one of the waterjets. The proposed solution combines a load torque estimator with an optimization routine that accounts for the power limits when a waterjet is subject to a power loss. This prevents faults from quickly escalating into a complete failure of the waterjet due to excessive power demands. Two simulated case studies including zig-zag path following and sideways movements are presented to demonstrate the effectiveness of the fault tolerant control thrust allocation strategy. Full article
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28 pages, 13394 KB  
Article
Finite-Time Controller Design for the Dynamic Positioning of Ships Considering Disturbances and Actuator Constraints
by Yufang Zhang, Changde Liu, Nan Zhang, Qian Ye and Weifeng Su
J. Mar. Sci. Eng. 2022, 10(8), 1034; https://doi.org/10.3390/jmse10081034 - 27 Jul 2022
Cited by 4 | Viewed by 2451
Abstract
Focusing on dynamic positioning (DP) systems for ships, which are subject to environmental disturbances and actuator constraints, this paper presents a finite-time controller that uses a disturbance observer with the aid of a backstepping technique. First, to estimate the time-varying and unknown environmental [...] Read more.
Focusing on dynamic positioning (DP) systems for ships, which are subject to environmental disturbances and actuator constraints, this paper presents a finite-time controller that uses a disturbance observer with the aid of a backstepping technique. First, to estimate the time-varying and unknown environmental disturbances in finite time, two sliding-mode disturbance observers are constructed. Specifically, an adaptive disturbance observer (ADO) effectively decreases undesired chattering without the need for prior information on environmental disturbances. Then, to handle the actuator constraints, the designed control forces are distributed into multiple actuators using a control allocation algorithm to obtain the actual forces. Next, an auxiliary dynamic system is built to compensate for velocity tracking errors induced by the mismatch of the DP control law and thruster forces. Then, with the designed ADO and the auxiliary dynamic system, a finite-time controller with a fast exponential-reaching law is designed; this ensures that the positioning errors and the sliding surface converge to zero at a fast convergence rate. Finally, numerical simulations are presented: these present a cable-laying ship experiencing wind, currents, and waves in different sea states. The results show the effectiveness of the presented control scheme. Full article
(This article belongs to the Special Issue Marine Dynamic Positioning System)
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23 pages, 28765 KB  
Article
On the Adaptation of an AUV into a Dedicated Platform for Close Range Imaging Survey Missions
by Yevgeni Gutnik, Aviad Avni, Tali Treibitz and Morel Groper
J. Mar. Sci. Eng. 2022, 10(7), 974; https://doi.org/10.3390/jmse10070974 - 15 Jul 2022
Cited by 20 | Viewed by 6186
Abstract
This study presents the redesign of an existing autonomous underwater vehicle (AUV) with limited maneuverability, transforming it into a platform optimized for autonomous, near-seabed visual imaging missions. This work describes the enhancement of the AUV’s maneuverability through the addition of thrusters, the leveraging [...] Read more.
This study presents the redesign of an existing autonomous underwater vehicle (AUV) with limited maneuverability, transforming it into a platform optimized for autonomous, near-seabed visual imaging missions. This work describes the enhancement of the AUV’s maneuverability through the addition of thrusters, the leveraging of a state-of-the-art thrust allocation algorithm, and the development of both a path-following controller and a dedicated imaging system. The performance of the optimized platform is demonstrated in a simulation and in actual real sea visual survey missions. Full article
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24 pages, 7792 KB  
Article
Control Allocation Design for Torpedo-Like Underwater Vehicles with Multiple Actuators
by Yung-Yue Chen, Chun-Yen Lee, Ya-Xuan Huang and Tsung-Tso Yu
Actuators 2022, 11(4), 104; https://doi.org/10.3390/act11040104 - 29 Mar 2022
Cited by 6 | Viewed by 5581
Abstract
For solving the transformation problem between the desired nonlinear control laws and installed actuators’ input commands of torpedo-like underwater vehicles, one closed-form control allocation method is proposed in this article. The goal of this study is to optimally distribute the desired nonlinear control [...] Read more.
For solving the transformation problem between the desired nonlinear control laws and installed actuators’ input commands of torpedo-like underwater vehicles, one closed-form control allocation method is proposed in this article. The goal of this study is to optimally distribute the desired nonlinear control law to each single actuator installed on the torpedo-like underwater vehicle. The first step of this proposed control allocation method is to arrange the required types, numbers, and positions of the installed actuators and then build up the thrust configuration matrix for the developed torpedo-like underwater vehicle. In this step, the desired nonlinear control law can be optimally distributed to output commands of installed actuators based on the optimization method. Next, through collecting the input and output data of each installed actuator by practical experiments, the mathematical transformation of input and output commands of each installed actuator can be found. For verifying performance of this proposed control allocation method, simulations with the robust trajectory tracking design of a torpedo-like underwater vehicle with four fins, four rudders, and one thruster are executed in this investigation. Full article
(This article belongs to the Special Issue Design and Application of Actuators with Multi-DOF Movement)
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