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Autonomous ships are essentially designed and equipped to perceive their internal and external environment and subsequently perform appropriate actions depending on the predetermined objective(s) without human intervention. Consequently, trajectory planning algorithms for autonomous berthing must consider factors such as system dynamics, ship actuators, environmental disturbances, and the safety of the ship, other ships, and port structures, among others. In this study, basing the ship dynamics on the low-speed MMG model, trajectory planning for an autonomous ship is modeled as an optimal control problem (OCP) that is transcribed into a nonlinear programming problem (NLP) using the direct multiple shooting technique. To enhance berthing safety, besides considering wind disturbances, speed control, actuators' limitations, and collision avoidance features are incorporated as constraints in the NLP, which is then solved using the Sequential Quadratic Programming (SQP) algorithm in MATLAB. Finally, the performance of the proposed planner is evaluated through (i) comparison with solutions obtained using CMA-ES for two different model ships, (ii) trajectory planning for different harbor entry and berth approach scenarios, and (iii) feasibility study using stochastically generated initial conditions and positions within the port boundaries. Simulation results indicate enhanced berthing safety as well as practical and computational feasibility making the planner suitable for real-time applications.

Many technologies need to be established to realize autonomous ships. In particular, accurate state estimation in real time is one of the most important technologies. In the ship and ocean engineering fields, there have been many studies on state estimation using nonlinear Kalman filters. Several methods have been proposed for nonlinear Kalman filters. However, there is insufficient verification on the selection of which filter should be applied among them. Therefore, this study aims to validate the filter selection to provide a guideline for filter selection. The effects of modeling error, observation noise, and type of maneuvers on the estimation accuracy of the unscented Kalman filter (UKF) and ensemble Kalman filter (EnKF) used in this study were investigated. In addition, it was verified whether filtering could be performed in real time. The results show that modeling error significantly impacts the estimation accuracy of the UKF and EnKF. However, the observation noise and types of maneuvers did not have an impact like the modeling error. Thus, we obtained the guideline that UKF and EnKF should be used differently depending on the required computation time. We also obtained that keeping the modeling error sufficiently small is essential to improving the estimation accuracy.

It is important to know the mean time until critical roll motion occurs in various operating and sea conditions, in order to determine and ensure the safety of ship designs and operating ships. Since typical ocean waves are irregular, the forcing and roll response of the ship is considered to be a stochastic process of colored noise type. However, the simulation of the corresponding first passage times is very time consuming. Therefore, an approach for the determination of mean first passage times of critical roll motion of ships is proposed in this paper which needs much less computation time. This approach is based on explicit formulas for the roll energy of the ship. These formulas are used to determine the mean first passage times based on integral expressions, which were previously obtained. The resulting integral expressions can be computed very fast using standard quadrature formulas. Moreover, the underlying model for ship dynamics is extended by introducing a new effective wave for short-crested sea states. This is an extension to the improved Grim's effective wave concept.

Among ship maneuvers, berthing/unberthing maneuvers are one of the most challenging and stressful phases for captains. Concerning burden reduction on ship operators and preventing accidents, several researches have been conducted on trajectory planning to automate berthing/unberthing. However, few studies have aimed at assisting captains in berthing/unberthing. The trajectory to be presented to the captain should be a maneuver that reproduces human captain's control characteristics. The previously proposed methods cannot explicitly reflect the motion and navigation, which human captains pay particular attention to reduce the mental burden in the trajectory planning. Herein, mild constraints to the trajectory planning method are introduced. The constraints impose certain states (position, bow heading angle, ship speed, and yaw angular velocity), to be taken approximately at any given time. The introduction of this new constraint allows imposing careful trajectory planning (e.g., in-situ turns at zero speed or a pause for safety before going astern), as if performed by a human during berthing/unberthing. The algorithm proposed herein was used to optimize the berthing/unberthing trajectories for a large car ferry. The results show that this method can generate the quantitatively equivalent trajectory recorded in the actual berthing/unberthing maneuver performed by a human captain., Comment: 16 pages, 12 figures, an accepted manuscript of a published paper on Ocean Engineering

A dynamic model for an automatic berthing and unberthing controller has to estimate harbor maneuvers, which include berthing, unberthing, approach maneuvers to berths, and entering and leaving the port. When the dynamic model is estimated by the system identification, a large number of tests or trials are required to measure the various motions of harbor maneuvers. However, the amount of data that can be obtained is limited due to the high costs and time-consuming nature of full-scale ship trials. In this paper, we improve the generalization performance of the dynamic model for the automatic berthing and unberthing controller by introducing data augmentation. This study used slicing and jittering as data augmentation methods and confirmed their effectiveness by numerical experiments using the free-running model tests. The dynamic model is represented by a neural network-based model in numerical experiments. Results of numerical experiments demonstrated that slicing and jittering are effective data augmentation methods but could not improve generalization performance for extrapolation states of the original dataset., Comment: 12 pages, 11 figures, Submitted to Journal of Marine Science and Technology

In this study, we consider a continuous min--max optimization problem $\min_{x \in \mathbb{X} \max_{y \in \mathbb{Y}}}f(x,y)$ whose objective function is a black-box. We propose a novel approach to minimize the worst-case objective function $F(x) = \max_{y} f(x,y)$ directly using a covariance matrix adaptation evolution strategy (CMA-ES) in which the rankings of solution candidates are approximated by our proposed worst-case ranking approximation (WRA) mechanism. We develop two variants of WRA combined with CMA-ES and approximate gradient ascent as numerical solvers for the inner maximization problem. Numerical experiments show that our proposed approach outperforms several existing approaches when the objective function is a smooth strongly convex--concave function and the interaction between $x$ and $y$ is strong. We investigate the advantages of the proposed approach for problems where the objective function is not limited to smooth strongly convex--concave functions. The effectiveness of the proposed approach is demonstrated in the robust berthing control problem with uncertainty.ngly convex--concave functions. The effectiveness of the proposed approach is demonstrated in the robust berthing control problem with uncertainty.

In this paper, a fine-tuning method of the parameters in the MMG model for the real-scale ship is proposed. In the proposed method, all of the arbitrarily indicated target parameters of the MMG model are tuned simultaneously in the framework of SI using time series data of real-sale ship maneuvering motion data to steadily improve the accuracy of the MMG model. Parameter tuning is formulated as a minimization problem of the deviation of the maneuvering motion simulated with given parameters and the real-scale ship trials, and the global solution is explored using CMA-ES. By constraining the exploration ranges to the neighborhood of the previously determined parameter values, the proposed method limits the output in a realistic range. The proposed method is applied to the tuning of 12 parameters for a container ship with five different widths of the exploration range. The results show that, in all cases, the accuracy of the maneuvering simulation is improved by applying the tuned parameters to the MMG model, and the validity of the proposed parameter fine-tuning method is confirmed., Comment: 13 pages, 11 figures, a preprint submitted to Ocean Engineering

A ship steering control is designed for a nonlinear maneuvering model whose rudder manipulation is constrained in both magnitude and rate. In our method, the tracking problem of the target heading angle with input constraints is converted into the tracking problem for a strict-feedback system without any input constraints. To derive this system, hyperbolic tangent ($\tanh$) function and auxiliary variables are introduced to deal with the input constraints. Furthermore, using the feature of the derivative of $\tanh$ function, auxiliary systems are successfully derived in the strict-feedback form. The backstepping method is utilized to construct the feedback control law for the resulting cascade system. The proposed steering control is verified in numerical experiments, and the result shows that the tracking of the target heading angle is successful using the proposed control law., Comment: 12 pages, 6 figures, a preprint submitted to the Journal of Marine Science and Technology

Capsizing is one of the worst scenarios in oceangoing vessels. It could lead to a high number of fatalities. A considerable number of studies have been conducted until the 1980s, and one of the discoveries is the weather criterion established by the International Maritime Organization (IMO). In the past, one of the biggest difficulties in revealing the behavior of ship-roll motion was the nonlinearity of the governing equation. On the other hand, after the mid-1980s, the complexity of the capsizing problem was uncovered with the aid of computers. In this study, we present the theoretical backgrounds of the capsizing problem from the viewpoint of nonlinear dynamics. Then, we discuss the theoretical conditions and mechanisms of the bifurcations of periodic solutions and numerical attempts for the bifurcations and capsizing.

Capsizing is one of the worst scenarios in oceangoing vessels. It could lead to a high number of fatalities. A considerable number of studies have been conducted until the 1980s, and one of the discoveries is the weather criterion established by the International Maritime Organization (IMO). In the past, one of the biggest difficulties in revealing the behavior of ship-roll motion was the nonlinearity of the governing equation. On the other hand, after the mid-1980s, the complexity of the capsizing problem was uncovered with the aid of computers. In this study, we present the theoretical backgrounds of the capsizing problem from the viewpoint of nonlinear dynamics. Then, we discuss the theoretical conditions and mechanisms of the bifurcations of periodic solutions and numerical attempts for the bifurcations and capsizing.

A VecTwin rudder system comprises twin fishtail rudders with reaction fins to increase its performance. With a constant propeller revolution number, the vessel can execute special low-speed maneuvers like hover, crabbing, reverse, and rotation. Such low-speed maneuvers are termed dynamic positioning (DP), and a DP vessel should be fully/overly actuated with several thrusters. This article introduces a novel and experimental VecTwin positioning system (VTPS) without making the ship fully/overly actuated. Unlike the usual dynamic positioning system (DPS), the VTPS is developed for low-speed operations in a calm harbor area. It is designed upon an assumption that the forces due to the interaction between the rudders, the propeller, and the hull are linear with the rudder angles within a range around the hover rudder angle. The linear relationship is obtained through linear regression of the results from several CFD simulations. The VTPS implements a PID controller that regulates the actuator forces to achieve the given low-speed positioning objective. It was tested in combined automatic docking and position-keeping experiments where disturbances from the environment exist. It shows promising potential for a practical application but with further improvements.

Automation of berthing maneuvers in shipping is a pressing issue as the berthing maneuver is one of the most stressful tasks seafarers undertake. Berthing control problems are often tackled via tracking a predefined trajectory or path. Maintaining a tracking error of zero under an uncertain environment is impossible; the tracking controller is nonetheless required to bring vessels close to desired berths. The tracking controller must prioritize the avoidance of tracking errors that may cause collisions with obstacles. This paper proposes a training method based on reinforcement learning for a trajectory tracking controller that reduces the probability of collisions with static obstacles. Via numerical simulations, we show that the proposed method reduces the probability of collisions during berthing maneuvers. Furthermore, this paper shows the tracking performance in a model experiment., Comment: 14 pages, 15 figures, Submitted to Journal of Marine Science and Technology

In the usual framework of control, a reference trajectory is needed as the set point for a feedback controller. This reference trajectory can be generated by solving a trajectory optimization problem. This problem is a continuous optimal control problem (OCP) that is transcribed into a finite-dimensional nonlinear optimization problem (NLP) and solved by SQP. For an underactuated conventional vessel, the mathematical model can be very intricate, hence the NLP itself. This causes significant computational time. This article demonstrates that the balance between the feasibility of the reference trajectory and the computational time can be achieved for an underactuated vessel in a disturbed and restricted environment. This is done by: (1) using an almost-globally optimal offline solution as a warm start in a semionline trajectory optimization to speed up the calculation, (2) including the prediction of wind dynamics, and (3) representing the ship as a rigid body and using a predefined boundary to generate the necessary spatial constraints via a point-in-polygon method that ensure a collision-free trajectory in a nonconvex region. Incorporation of these three things maintains a safe and dynamically feasible trajectory where the warm start gives a considerable computational speedup and better results than that without a warm start.

Accurate maneuvering estimation is essential to establish autonomous berthing control. The system-based mathematical model is widely used to estimate the ship's maneuver. Commonly, the system parameters of the mathematical model are obtained by the captive model test (CMT), which is time-consuming to construct an accurate model suitable for complex berthing maneuvers. System identification (SI) is an alternative to constructing the mathematical model. However, SI on the mathematical model of ship's maneuver has been only conducted on much simpler maneuver: turning and zig-zag. Therefore, this study investigates the SI on a mathematical model capable of berthing maneuver. The main contributions of this study are as follows: (i) construct the system-based mathematical model on berthing by optimizing system parameters with a reduced amount of model tests than the CMT-based scheme; (ii) Find the favorable choice of objective function and type of training data for optimization. Global optimization scheme CMA-ES explored the system parameters of the MMG model from the free-running model's trajectories. The berthing simulation with the parameters obtained by the proposed method showed better agreement with the free-running model test than parameters obtained by the CMT. Furthermore, the proposed method required fewer data amounts than a CMT-based scheme., Comment: 16 pages, 8 figures, preprint

Several studies on ship maneuvering models have been conducted using captive model tests or computational fluid dynamics (CFD) and physical models, such as the maneuvering modeling group (MMG) model. A new system identification method for generating a low-speed maneuvering model using recurrent neural networks (RNNs) and free running model tests is proposed in this study. We especially focus on a low-speed maneuver such as the final phase in berthing to achieve automatic berthing control. Accurate dynamic modeling with minimum modeling error is highly desired to establish a model-based control system. We propose a new loss function that reduces the effect of the noise included in the training data. Besides, we revealed the following facts - an RNN that ignores the memory before a certain time improved the prediction accuracy compared with the "standard" RNN, and the random maneuver test was effective in obtaining an accurate berthing maneuver model. In addition, several low-speed free running model tests were performed for the scale model of the M.V. Esso Osaka. As a result, this paper showed that the proposed method using a neural network model could accurately represent low-speed maneuvering motions., Comment: 13 pages, 7 figures, submitted to Journal of Marine Science and Technology for peer-review

We propose an approach to saddle point optimization relying only on oracles that solve minimization problems approximately. We analyze its convergence property on a strongly convex--concave problem and show its linear convergence toward the global min--max saddle point. Based on the convergence analysis, we develop a heuristic approach to adapt the learning rate. An implementation of the developed approach using the (1+1)-CMA-ES as the minimization oracle, namely Adversarial-CMA-ES, is shown to outperform several existing approaches on test problems. Numerical evaluation confirms the tightness of the theoretical convergence rate bound as well as the efficiency of the learning rate adaptation mechanism. As an example of real-world problems, the suggested optimization method is applied to automatic berthing control problems under model uncertainties, showing its usefulness in obtaining solutions robust to uncertainty., Comment: Accepted for ACM TELO

For the safe design and operation of high-speed craft it is important to predict their behaviour in waves. There still exists a concern, however, in the framework of the International Maritime Organization (IMO) with regards to the stability criteria. In particular, for high-speed craft, the higher limit of operational speed resulting in wave blocking as well as the lower limit known as the surf-riding threshold are important features. Therefore, by applying the polynomial approximation to wave induced surge force including the nonlinear surge equation, an analytical formula in order to predict the wave blocking and surf-riding thresholds is proposed. Comparative results of the surf-riding threshold and wave blocking threshold utilizing the proposed formula and the numerical bifurcation analysis indicate fairly good agreement. In addition, previously proposed analytical formulae are inclusively examined. It is concluded that the analytical formulae based on a continuous piecewise linear approximation and Melnikov’s method agrees well with the wave blocking threshold and the surf-riding threshold obtained by the numerical bifurcation analysis and the free-running model experiment. As a result, it is considered that these two calculation methods could be recommended for the early design stage tool for avoiding broaching and bow-diving.

In the research filed of nonlinear dynamical system theory it is well known that a homoclinic/heteroclinic point leads to unpredictable motions, such as chaos. Melnikov’s method enables us to judge whether the system has a homoclinic/heteroclinic orbit. Therefore, in order to assess a vessels safety against capsizing, Melnikov’s method has been applied for the investigations of chaos that appears in beam sea rolling. This is because chaos is closely related to capsizing incidents. In a previous paper 1), the formula to predict the capsizing boundary by applying Melnikov’s method to analytically obtain the non-Hamiltonian heteroclinic orbit, was proposed. However, in that paper, limited numerical investigation had been carried out. Therefore further comparative research between the analytical and numerical results is conducted, with the result being that the formula is validated.

Chaos appearing in a ship roll equation in beam seas, known as the escape equation, has been intensively investigated so far because it is closely related to capsizing accident. In particular, many applications of Melnikov integral formula have been reported in the existing literature. However, in all the analytical works concerning with the escape equation, Melnikov integral is formulated utilizing a separatrix for Hamiltonian part or a numerically obtained heteroclinic orbit for non-Hamiltonian part, of the original escape equation. To overcome such limitations, this paper attempts to utilise an analytical expression of the non-Hamiltonian part. As a result, an analytical procedure making use of a heteroclinic orbit of non-Hamiltonian part within the framework of Melnikov integral formula is provided.

Many technologies need to be established to realize autonomous ships. In particular, accurate state estimation in real time is one of the most important technologies. In the ship and ocean engineering fields, there have been many studies on state estimation using nonlinear Kalman filters. Several methods have been proposed for nonlinear Kalman filters. However, there is insufficient verification on the selection of which filter should be applied among them. Therefore, this study aims to validate the filter selection to provide a guideline for filter selection. The effects of modeling error, observation noise, and type of maneuvers on the estimation accuracy of the unscented Kalman filter (UKF) and ensemble Kalman filter (EnKF) used in this study were investigated. In addition, it was verified whether filtering could be performed in real time. The results show that modeling error significantly impacts the estimation accuracy of the UKF and EnKF. However, the observation noise and types of maneuvers did not have an impact like the modeling error. Thus, we obtained the guideline that UKF and EnKF should be used differently depending on the required computation time. We also obtained that keeping the modeling error sufficiently small is essential to improving the estimation accuracy., The version of record of this article, first published in Journal of Marine Science and Technology (Japan), is available online at Publisher’s website: https://doi.org/10.1007/s00773-023-00953-6

Although it is known from previous studies that porpoising in high-speed planing craft is a Hopf bifurcation, this study examined its occurrence and disappearance using the motion model identified from full-scale test data. Herein, we first analyzed the stability of the system linearized near the equilibrium point. From its results, we reconfirmed the knowledge that porpoising occurs when the system becomes unstable in the vicinity of the equilibrium point. We also found that the system became unstable as the thrust or trim angle of the outboard motor decreased. This finding was consistent with the results of a full-scale craft test performed in a previous study. Second, we confirmed that the limit cycle which is a result of the nonlinearity of the system was stable. The two analyses indicate that porpoising corresponds to the supercritical Hopf bifurcation. Furthermore, in the vicinity of the bifurcation point, it was found that stable equilibrium points and stable limit cycles can coexist. Finally, we confirmed this phenomenon in the full-scale test., This version of the article has been accepted for publication, after peer review (when applicable) and is subject to Springer Nature’s AM terms of use, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: https://doi.org/10.1007/s00773-023-00972-3, Hamada S., Maki A.. Linear and nonlinear analyses of the porpoising dynamics of high-speed planing craft using full-scale trial data. Journal of Marine Science and Technology (Japan) , (2023); https://doi.org/10.1007/s00773-023-00972-3.

The version of record of this article, first published in Journal of Marine Science and Technology (Japan), is available online at Publisher’s website: https://doi.org/10.1007/s00773-023-00962-5, Automation of berthing maneuvers in shipping is a pressing issue as the berthing maneuver is one of the most stressful tasks seafarers undertake. Berthing control problems are often tackled by tracking a predefined trajectory or path. Maintaining a tracking error of zero under an uncertain environment is impossible; the tracking controller is nonetheless required to bring vessels close to desired berths. The tracking controller must prioritize the avoidance of tracking errors that may cause collisions with obstacles. This paper proposes a training method based on reinforcement learning for a trajectory tracking controller that reduces the probability of collisions with static obstacles. Via numerical simulations, we show that the proposed method reduces the probability of collisions during berthing maneuvers. Furthermore, this paper shows the tracking performance in a model experiment.

The version of record of this article, first published in Journal of Marine Science and Technology (Japan), is available online at Publisher’s website: https://doi.org/10.1007/s00773-023-00953-6, Many technologies need to be established to realize autonomous ships. In particular, accurate state estimation in real time is one of the most important technologies. In the ship and ocean engineering fields, there have been many studies on state estimation using nonlinear Kalman filters. Several methods have been proposed for nonlinear Kalman filters. However, there is insufficient verification on the selection of which filter should be applied among them. Therefore, this study aims to validate the filter selection to provide a guideline for filter selection. The effects of modeling error, observation noise, and type of maneuvers on the estimation accuracy of the unscented Kalman filter (UKF) and ensemble Kalman filter (EnKF) used in this study were investigated. In addition, it was verified whether filtering could be performed in real time. The results show that modeling error significantly impacts the estimation accuracy of the UKF and EnKF. However, the observation noise and types of maneuvers did not have an impact like the modeling error. Thus, we obtained the guideline that UKF and EnKF should be used differently depending on the required computation time. We also obtained that keeping the modeling error sufficiently small is essential to improving the estimation accuracy.

This version of the article has been accepted for publication, after peer review (when applicable) and is subject to Springer Nature’s AM terms of use, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: https://doi.org/10.1007/s00773-023-00972-3, Hamada S., Maki A.. Linear and nonlinear analyses of the porpoising dynamics of high-speed planing craft using full-scale trial data. Journal of Marine Science and Technology (Japan) , (2023); https://doi.org/10.1007/s00773-023-00972-3., Although it is known from previous studies that porpoising in high-speed planing craft is a Hopf bifurcation, this study examined its occurrence and disappearance using the motion model identified from full-scale test data. Herein, we first analyzed the stability of the system linearized near the equilibrium point. From its results, we reconfirmed the knowledge that porpoising occurs when the system becomes unstable in the vicinity of the equilibrium point. We also found that the system became unstable as the thrust or trim angle of the outboard motor decreased. This finding was consistent with the results of a full-scale craft test performed in a previous study. Second, we confirmed that the limit cycle which is a result of the nonlinearity of the system was stable. The two analyses indicate that porpoising corresponds to the supercritical Hopf bifurcation. Furthermore, in the vicinity of the bifurcation point, it was found that stable equilibrium points and stable limit cycles can coexist. Finally, we confirmed this phenomenon in the full-scale test.