Parameter fine-tuning method for MMG model using real-scale ship data Rin Suyamaa,∗ , Rintaro Matsushitab , Ryo Kakutab , Kouki Wakitaa and Atsuo Makia,∗ a Department of Naval Architecture and Ocean Engineering, Graduate School of Engineering, Osaka University, 2-1 Yamadaoka, Suita, Osaka 565-0971, arXiv:2312.04224v1 [eess.SY] 7 Dec 2023 Japan b Monohakobi Technology Institute Co., Ltd., Yusen Building, 3-2 Marunouchi, 2-Chome, Chiyoda-ku, Tokyo 100-0005, Japan ARTICLE INFO ABSTRACT Keywords: Autonomous Vessel System Identification Fine-Tuning Real-scale Ship Data CMA-ES 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. 1. Introduction The development of maneuvering models is essential to the research and the development of autonomous navigation algorithms. Autonomous navigation algorithms, studied intensively by institutions in many countries in recent years, should be validated by maneuvering simulators. The reliability of the algorithm can be improved by testing it under maneuvering simulators with accurate maneuvering models. Research on modern maneuvering models has a long history, with literature dating back to the 1940s, such as Davidson and Schiff (1946). The maneuvering models have been studied in terms of hydrodynamics by Motora (1959); Inoue et al. (1981) and control engineering by Motora (1955); Nomoto et al. (1957). In addition, researches have been conducted on models of the hydrodynamic forces of individual components, such as hull, propeller, and rudder. For instance, Abkowitz (1964); Åström and Källström (1976) utilized Taylor expansion with respect to state variables to derive a maneuvering model. The MMG model (Ogawa and Kasai, 1978; Yasukawa and Yoshimura, 2015) was groundbreaking in that it could represent each individual component and their interaction. Fossen (2011) formulated a maneuvering model in vector form inspired by mechanical systems, and this formulation is often used in model-based control design. In addition, there exist studies on models expressed with polynomials of motion states and actuator states (Miyauchi et al., 2023; Dong et al., 2023). In contrast to these white box models, some studies have addressed the ∗ Corresponding author Email addresses: suyama_rin@naoe.eng.osaka-u.ac.jp (R. Suyama); rintaro_matsushita@monohakobi.com (R. Matsushita); ryo_kakuta@monohakobi.com (R. Kakuta); kouki_wakita@naoe.eng.osaka-u.ac.jp (K. Wakita); maki@naoe.eng.osaka-u.ac.jp (A. Maki) approximation of maneuvering models using black box models based on machine learning. Some examples of the black box models are artificial neural networks (ANN) (Moreira and Soares, 2003; Rajesh and Bhattacharyya, 2008; Zhang and Zou, 2013; Oskin et al., 2013; Luo and Zhang, 2016; Woo et al., 2018; He et al., 2022; Wakita et al., 2022; Wang et al., 2023), support vector machine (Luo and Zou, 2009; Wang et al., 2020), and Gaussian process (Ramirez et al., 2018; Xue et al., 2020). Among various maneuvering models, the MMG model has high accuracy in simulations of maneuvering motion. This is explained by the fact that it is formulated based on detailed hydrodynamics. Due to its high performance, the MMG model is widely used for the design and validation of autonomous navigation algorithms. Li et al. (2013) validated the path following control law based on sliding mode control by numerical simulations following the MMG model. Zhang et al. (2017) validated the course keeping control law with nonlinear feedback in a simulation based on the MMG model. Zheng et al. (2022) designed a path following control of an unmanned surface vessel by applying the framework of reinforcement learning with the state transition model based on the MMG model. There are also examples of simulations based on the MMG model that solve the autonomous berthing/unberthing problems in the framework of off-line trajectory planning (Maki et al., 2020; Miyauchi et al., 2022b; Suyama et al., 2022). The MMG model contains constant parameters to be identified. There have been various methods for the identification of parameters in the MMG model for model-scale ships. Classically, captive tests were conducted to identify the parameters (Ogawa and Kasai, 1978; Yasukawa and Yoshimura, 2015). Computational fluid dynamics (CFD) is also used for the identification of the parameters in the MMG ORCID (s): Suyama et al.: Preprint submitted to Elsevier Page 1 of 13 Parameter fine-tuning method for MMG model using real-scale ship data model (Zhang et al., 2019). Sakamoto et al. (2019) identified the values of the parameters in the MMG model based on the captive model tests simulated with CFD. So far in most cases, the parameters for the real-scale ship MMG model were identified based on the results of model-scale captive tests. One of the differences between the maneuvering motion of model-scale ships and real-scale ships is the difference in Reynolds numbers. To correct the effect of the difference in Reynolds numbers, for instance, for limited parameters such as the frictional resistance coefficient for an equivalent flat plate and the wake coefficient in straight moving, there are widely used methods such as the three-dimensional extrapolation method (e.g. (Yasukawa and Yoshimura, 2015)). However, at this time, the methodology for the identification of the parameters in the real-scale ship MMG model using captive tests is not established. Some parameters, such as the flow straightening coefficient, are sometimes used without correcting for the effect of the difference in Reynolds numbers, although the effects have been noted (Aoki et al., 2006). In addition, the effects of the difference in Reynolds numbers are not completely clarified for all parameters in the MMG model. If captive tests on real-scale ships were conducted, the parameters could be identified without being affected by the difference in Reynolds numbers, but captive tests on large ships, such as containers, are not possible. Another method for identification of the parameters in the real-scale ship MMG model is a heuristic tuning of previously determined values by referring to the maneuvering motion data of the real-scale ship. In most cases, this tuning is implemented by experienced engineers using their own heuristic methods based on their senses (Sutulo and Soares, 2014). This heuristic tuning directly refers to real-scale ship maneuvering motion data, so the accuracy of the MMG model with the tuned parameters can be steadily improved. However, heuristic tuning of a large number of parameters is not optimal, so an automatic tuning method needs to be established. System identification (SI) is applicable for the determination of the values of parameters in the real-scale ship MMG model. SI is an identification framework for system parameters using input/output data of the target system. Therefore, SI for the parameters in the MMG model of real-scale ships can be implemented only with time series data of the maneuvering motion of the subject real-scale ship. SI has been applied to parameter identification of ship maneuvering models. Kalman filter is one of the popular methods for SI, and Åström and Källström (1976); Abkowitz (1980) applied Kalman filter to the Abkowitz model. A combination of state estimation and SI was proposed by (Yoon and Rhee, 2003) for parameter identification. Araki et al. (2012) validated the SI method using the maneuvering motion data simulated using CFD. Sutulo and Soares (2014) adopted the genetic algorithm as the exploration method for the optimal values of parameters in SI. This study was extended to SI for the maneuvering model with a modular structure (Sutulo and Soares, 2023). Miyauchi et al. (2022a) Suyama et al.: Preprint submitted to Elsevier proposed a parameter identification method for the MMG model without the initial guess of the parameters using the SI framework. This study was extended to the parameter identification of the Abkowitz model with more than 200 parameters and validated its effectiveness (Miyauchi et al., 2023). However, in the method proposed by Miyauchi et al. (2022a, 2023), maneuvering motion data with random inputs, which is difficult to obtain in the operation of realscale ships, were used, and the validation of the method was limited to model-scale ship cases. Several studies on SI utilize time series data of real-scale ship maneuvering motion (Abkowitz, 1980; Kim, 2018; Meng et al., 2022; Kambara et al., 2022; Wang et al., 2023), but these studies have focused on certain components of the maneuvering model, linear maneuvering models, or ANN models. To the best of the authors’ knowledge, the SI method for the whole MMG model using real-scale ship data is not discussed. In this study, the authors propose a fine-tuning method for all of the arbitrarily indicated target parameters of the MMG model using the framework of SI. The proposed method is designed to automatically fine-tune the parameter values previously determined based on hydrodynamics, captive model tests, and CFD to the ones for the real-scale MMG model. The proposed method directly refers to the time series data of real-scale ship maneuvering motion and steadily improves the performance of the MMG model with the tuned parameter in terms of the accuracy of the simulated maneuvering motion. The parameter tuning is formulated as a constrained minimization problem, and the solution is explored using Covariance Matrix Adaption Evolutionary Computation (CMA-ES) (Hansen, 2007; Hansen and Auger, 2014). To obtain realistic values for the MMG parameters, the ranges of available values for the target parameters in the fine-tuning problem are constrained to the neighborhood of the previously determined values. The proposed parameter fine-tuning method is applied to a container ship with 𝐿pp = 83m and validated. The objective of this study is not to clarify the effect of the difference in Reynolds numbers between real-scale ships and model-scale ships, but to establish a practical method for fine-tuning MMG parameters. The rest of the manuscript is organized as follows. Sec.2 describes the notation used in this manuscript. The MMG model of the subject ship of this study is detailed in Sec.3. The proposed parameter fine-tuning method is described in Sec.4. Sec.5 details the CMA-ES. Sec.6 shows the results of parameter tuning for a container ship. The discussion on the proposed method is presented in Sec.7. Sec.8 concludes this manuscript. 2. Notation ℝ represents the set of all real numbers. |𝑥| represents the absolute value of 𝑥 ∈ ℝ. The overdot “ ̇ ” represents the derivative with respect to time 𝑡. 𝐴⊤ represents the transposed array of the array 𝐴. Page 2 of 13 Parameter fine-tuning method for MMG model using real-scale ship data Table 2 Explanation of symbols. Vector 𝑝 𝜉 𝜏 Elements 𝑝1 𝑝2 𝜓 𝑢 𝑣m 𝑟 𝑛P 𝛿 Description Position on the Earth-fixed frame. Position on the Earth-fixed frame. Heading angle. Surge velocity. Sway velocity at the midship. Yaw angular velocity. Propeller revolution number. Rudder angle. Fig. 1: Coordinate systems. (𝑡), which indicates the dependence of variables on time, is omitted to simplify the description. States 𝑝 and 𝜉 follow the kinematics equation: Table 1 Principal particulars of the subject ship. Item 𝐿pp [m] 𝐵 [m] 𝑑 [m] 𝐶b 𝑥G [m] 𝐷P [m] 𝐻R [m] 𝐴R [m2 ] 𝑝̇ = 𝑅(𝑝)𝜉 , Value 83.0 13.5 3.8 0.737 0.93 2.80 3.49 6.282 where ⎛cos 𝜓 𝑅(𝑝) ∶= ⎜ sin 𝜓 ⎜ ⎝ 0 First, coordinate systems are defined as Fig. 1 shows. O − 𝑥𝑦 in Fig. 1 represents a body-fixed coordinate system with the origin fixed at the midship. The ship is assumed to move on an Earth-fixed coordinate system OE −𝑥E 𝑦E . In this study, the subject ship was a container ship M. V. SUZAKU. The principal particulars of the subject ship are summarized in Tab. 1. The maneuvering motion of the ship is represented with state variables: 𝑝(𝑡) ∶= (𝑝1 (𝑡) 𝑝2 (𝑡) 𝜓(𝑡))⊤ , (1) 𝜉(𝑡) ∶= (𝑢(𝑡) 𝑣m (𝑡) 𝑟(𝑡))⊤ . (2) The augmented variable: (3) is defined. The control input is defined as 𝜏(𝑡) ∶= (𝑛P (𝑡) 𝛿(𝑡))⊤ . (4) The physical meaning of each component of these variables is summarized in Tab. 2. Each component of these variables follows the coordinate systems in Fig. 1. In this study, the authors only consider forward motion with forward propeller revolution; 𝑢(𝑡) > 0 and 𝑛P (𝑡) > 0. In addition, current and wind are ignored as they are assumed to have little effect on the maneuvering motion in this study. In the following, Suyama et al.: Preprint submitted to Elsevier − sin 𝜓 cos 𝜓 0 0⎞ 0⎟ . ⎟ 1⎠ (6) The equation of motion of ship maneuvering is formulated as 3. MMG model 𝜁 (𝑡) ∶= (𝑝(𝑡)⊤ 𝜉(𝑡)⊤ )⊤ ∈ ℝ6 (5) ⎧(𝑚 + 𝑚𝑥 )𝑢̇ − (𝑚 + 𝑚𝑦 )𝑣m 𝑟 − 𝑥G 𝑚𝑟2 = 𝑋 ⎪ ⎨ (𝑚 + 𝑚𝑦 )𝑣̇ m + (𝑚 + 𝑚𝑥 )𝑢𝑟 + 𝑥G 𝑚𝑟̇ = 𝑌 ⎪ 2 ⎩ (𝐼g + 𝑥G 𝑚 + 𝐽𝑧𝑧 )𝑟̇ + 𝑥G 𝑚(𝑣̇ m + 𝑢𝑟) = 𝑁 (7) where 𝑚, 𝑚𝑥 , 𝑚𝑦 , 𝐼g , 𝐽𝑧𝑧 , 𝑥G are the displacement, the added mass for surge motion, the added mass for sway motion, the moment of inertia for yaw motion along the vertical line passing the midship, the added moment of inertia for yaw motion, 𝑥 coordinate of the center of gravity of the ship, respectively. 𝑋, 𝑌 , and 𝑁 in Eq. (7) represent the forces and moments, excluding centrifugal forces, on the body fixed coordinate system O − 𝑥𝑦. The MMG model (Yasukawa and Yoshimura, 2015) represents 𝑋, 𝑌 , and 𝑁 as the sum of the effects of the hull force, propeller force, and rudder force as follows: ⎧ 𝑋(𝜃 t ) = 𝑋H + 𝑋P + 𝑋R ⎪ t ⎨ 𝑌 (𝜃 ) = 𝑌H + 𝑌P + 𝑌R ⎪𝑁(𝜃 t ) = 𝑁 + 𝑁 + 𝑁 ⎩ H P R (8) where the argument 𝜃 t represents the target parameter to be tuned. The target parameter 𝜃 t is detailed in Sec.4.1. The MMG model’s representation of hull force, propeller force, and rudder force is unique in that it even takes into account the interference and interaction of each component, as detailed below. By reshaping Eq. (7), the system for 𝜉 is obtained as 𝜉̇ = 𝑓MMG (𝜉, 𝜏; 𝜃 t ) , (9) Page 3 of 13 Parameter fine-tuning method for MMG model using real-scale ship data 3.2. Propeller force (Okuda et al., 2023) where The propeller force for 𝑢 > 0 and 𝑛P > 0 was modeled t 𝑓MMG (𝜉, 𝜏; 𝜃 ) ∶= 𝑀 ⎛𝑋(𝜃 t ) + (𝑚 + 𝑚𝑦 )𝑣m 𝑟 + 𝑥G 𝑚𝑟2 ⎞ (10) ⎟ , 𝑌 (𝜃 t ) − (𝑚 + 𝑚𝑥 )𝑢𝑟 ⎜ ⎟ t 𝑁(𝜃 ) − 𝑥G 𝑚𝑢𝑟 ⎝ ⎠ as −1 ⎜ ⎛𝑚 + 𝑚𝑥 𝑀 ∶= ⎜ 0 ⎜ ⎝ 0 0 𝑚 + 𝑚𝑦 𝑥G 𝑚 0 ⎞ ⎟ . (11) 𝑥G 𝑚 ⎟ 2 𝐼g + 𝑥G 𝑚 + 𝐽𝑧𝑧 ⎠ Augmenting Eq. (5) and Eq. (9), the system for 𝜁 is obtained as 𝜁̇ = 𝑓𝜁 (𝜁 , 𝜏; 𝜃 t ) , (12) where ( )⊤ 𝑓𝜁 (𝜁 , 𝜏; 𝜃 t ) ∶= (𝑅(𝑝)𝜉)⊤ (𝑓MMG (𝜉, 𝜏; 𝜃 t ))⊤ . (13) The following sections detail the hydrodynamic force model for each component. The MMG model used in this study is based on the one proposed by Okuda et al. (2023). However, the model for hull force was the widely used polynomial model. In this manuscript, characters with the prime symbol represent nondimensionalized quantities. The nondimensionalized values of physical quantities with unit [m], [kg], [kgm2 ], and [m∕s] are calculated by dividing with 𝐿pp , 0.5𝜌𝐿2pp 𝑑, 0.5𝜌𝐿4pp 𝑑, and 𝑈 , respectively, where 𝑈 ∶= √ 𝑢2 + 𝑣2m is the ship speed. In the following, 𝛽 ∶= arctan(−𝑣m ∕𝑢) represents the drift angle. ⎧ 𝑋P = 1 𝜌𝑆P 𝑉 2 (1 − 𝑡P )𝐾T (𝜙P ) 𝑟 2 ⎪ ⎪ 1 2 ⎨ 𝑌P = 𝜌𝑆P 𝑉𝑟 𝐶P𝑌 (𝜙P ) 2 ⎪ ⎪𝑁 = 1 𝜌𝑆 𝐿 𝑉 2 𝐶 (𝜙 ) . ⎩ P 2 P pp 𝑟 P𝑁 P Here 𝑡P is a constant parameter called the thrust deduction factor. 𝜙P is defined as ( ) 𝑢P 180 𝜙P ∶= arctan (17) 𝜋 0.7𝜋𝑛P 𝐷P where 𝑢P ∶= (1 − 𝑤P )𝑢. The function 𝐾T (𝜙P ) was modeled as 𝐾T (𝜙P ) =3.31 × 10−6 𝜙3P − 3.72 × 10−4 𝜙2P − 2.60 × 10−3 𝜙P + 0.167 , ⎧ ′ ′ ′ ′ ′ ′2 ′ ′ ′ ′ ′2 ⎪ 𝑋H (𝑣m , 𝑟 ) = −𝑅0 + 𝑋𝑣𝑣 𝑣m + 𝑋𝑣𝑟 𝑣m 𝑟 + 𝑋𝑟𝑟 𝑟 ′ ′ ′ ′4 ⎪ + 𝑋𝑣𝑣𝑣𝑟 𝑣′3 m 𝑟 + 𝑋𝑣𝑣𝑣𝑣 𝑣m ⎪ ⎪ 𝑌H′ (𝑣′m , 𝑟′ ) = 𝑌𝑣′ 𝑣′m + 𝑌𝑟′ 𝑟′ ⎪ ′ ′ ′2 ′ + 𝑌𝑣𝑣𝑣 𝑣′3 ⎪ m + 𝑌𝑣𝑣𝑟 𝑣m 𝑟 (15) ⎨ ′ ′ ′2 ′ ′3 + 𝑌𝑣𝑟𝑟 𝑣m 𝑟 + 𝑌𝑟𝑟𝑟 𝑟 ⎪ ⎪𝑁 ′ (𝑣′ , 𝑟′ ) = 𝑁 ′ 𝑣′ + 𝑁 ′ 𝑟′ 𝑣 m 𝑟 ⎪ H m ′ ′ ′2 ′ ⎪ + 𝑁𝑣𝑣𝑣 𝑣′3 m + 𝑁𝑣𝑣𝑟 𝑣m 𝑟 ⎪ ′ ′ ′3 ⎪ + 𝑁𝑣𝑟𝑟 𝑣′m 𝑟′2 + 𝑁𝑟𝑟𝑟 𝑟 ⎩ where the coefficients are constant parameters which are called the hydrodynamic derivatives. Suyama et al.: Preprint submitted to Elsevier (18) by fitting measured data in the propeller open test shown in Fig.6 in the original paper (Okuda et al., 2023). The function 𝐶P𝑌 (𝜙P ), 𝐶P𝑁 (𝜙P ) were modeled as 𝐶P𝑌 (𝜙P ) = − 2.83 × 10−5 𝜙2P + 6.04 × 10−4 𝜙P − 1.28 × 10−2 , ⎧ − 2.48 × 10−4 𝜙 − 1.70 × 10−3 P ⎪ for 𝜙P < 20 , ⎪ 𝐶P𝑁 (𝜙P ) = ⎨ −4 −3 ⎪ − 1.86 × 10 𝜙P + 4.03 × 10 ⎪ for 𝜙P ≥ 20 , ⎩ 3.1. Hull force (Okuda et al., 2023) The surge force, the sway force, and the yaw moment induced by the interaction between the ship hull and the fluid were modeled as follows: ⎧ 𝑋H = 1 𝜌𝐿pp 𝑑𝑈 2 𝑋 ′ (𝑣′ , 𝑟′ ) H m 2 ⎪ ⎪ 1 ′ 2 ′ ′ (14) ⎨ 𝑌H = 𝜌𝐿pp 𝑑𝑈 𝑌H (𝑣m , 𝑟 ) 2 ⎪ ⎪𝑁 = 1 𝜌𝐿2 𝑑𝑈 2 𝑁 ′ (𝑣′ , 𝑟′ ) ⎩ H 2 pp H m The nondimensionalized forces and moment were modeled as (16) (19) (20) by fitting measured data shown in Fig.13 in the original paper (Okuda et al., 2023). In addition, 𝑆P ∶= 𝜋𝐷P2 4 is the propeller disk area, and √ 𝑉𝑟 ∶= 𝑢2P + (0.7𝜋𝑛P 𝐷P )2 (21) (22) is the apparent inflow velocity into the propeller. Here the wake fraction factor 𝑤P was modeled as 𝑤P = 𝑤P0 exp(𝐶𝑤 𝛽P2 ) (23) 𝛽P ∶= 𝛽 − 𝑙P′ 𝑟′ (24) where is the geometrical inflow angle of the propeller with 𝑙P′ = −0.5, 𝑤P0 is the value of 𝑤P at 𝛽P = 0. Page 4 of 13 Parameter fine-tuning method for MMG model using real-scale ship data 3.3. Rudder force (Okuda et al., 2023) The surge force, sway force, and yaw moment induced by the rudder were modeled as follows: ⎧ 𝑋R = −(1 − 𝑡R )𝐹N sin 𝛿 ⎪ ⎨ 𝑌R = −(1 + 𝑎H )𝐹N cos 𝛿 ⎪𝑁 = −(𝑥 + 𝑎 𝑥 )𝐹 cos 𝛿 R H H N ⎩ R Table 3 Pre-determined values of the added masses and added moment of inertia (Okuda et al., 2023). Item 𝑚′𝑥 𝑚′𝑦 ′ 𝐽𝑧𝑧 (25) Value 0.010 0.168 0.010 The rudder normal force 𝐹N was model as 1 𝜌𝐴 𝑈 2 {𝑓 (𝛿 ) sin 𝛼R + 𝐶l0 (𝛿f )} 2 R R 𝛼 f where 𝐴R is the rudder area, √ 𝑈R ∶= 𝑢2R + 𝑣2R 𝐹N = (26) (27) is the resultant inflow velocity of the rudder, and 𝛼R ∶= 𝛿 − 𝚊𝚝𝚊𝚗𝟸(𝑣R , 𝑢R ) (28) is the effective inflow angle of the rudder. The longitudinal inflow velocity component of the rudder 𝑢R was modeled as 𝑢R = max{𝑢∗R , 𝑢∗∗ R } (29) where Table 4 Pre-determined values of the parameters in the model of 𝑋H , 𝑌H , and 𝑁H (Okuda et al., 2023). Item 𝑅′0 ′ 𝑋𝑣𝑣 ′ 𝑋𝑣𝑟 + 𝑚′𝑦 𝑋𝑟𝑟′ ′ 𝑋𝑣𝑣𝑣𝑟 ′ 𝑋𝑣𝑣𝑣𝑣 Value 0.017 0.009 0.160 −0.0164 −0.824 −0.114 is the ratio of the diameter of the propeller to the rudder height. The lateral inflow velocity component of the rudder 𝑣R was modeled as 𝑣R = 𝛾R (𝑈 sin 𝛽 − 𝑙R 𝑟) (32) where 𝑙R is an experimental constant, and 𝛾R is a constant determined based on { 𝛾Rp for 𝛽R > 0 𝛾R = (33) 𝛾Rn for 𝛽R < 0 with effective inflow angle 𝛽R ∶= 𝛽 − 𝑙R′ 𝑟′ . (34) 𝑓𝛼 (⋅) and 𝐶l0 (⋅) in Eq. (26) was modeled as functions of the flap angle 𝛿f with 𝑓𝛼 (𝛿f ) = 𝑓𝛼0 + 𝑓𝛼2 𝛿f2 (35) 𝐶l0 (𝛿f ) = 𝐶l01 𝛿f + 𝐶l03 𝛿f3 (36) Value −0.329 0.090 −0.787 −0.022 −0.206 0.001 Item 𝑁𝑣′ 𝑁𝑟′ ′ 𝑁𝑣𝑣𝑣 ′ 𝑁𝑣𝑣𝑟 ′ 𝑁𝑣𝑟𝑟 ′ 𝑁𝑟𝑟𝑟 Value −0.106 −0.057 −0.037 −0.105 0.012 −0.008 Table 5 Pre-determined values of the parameters in the model of 𝑋P , 𝑌P , and 𝑁P (Okuda et al., 2023). { (√ ) } ⎧ ∗ 8𝐾T (𝜙P ) 1+ −1 +1 , ⎪ 𝑢R = 𝑢P 𝜖 𝜂R 𝜅 (30) 𝜋𝐽P2 ⎨ ⎪ ∗∗ ⎩𝑢R = 0.7𝜋𝑛P 𝐷P 𝑢R0 . Here, 𝜖 is the ratio of the wake fraction factors at the propeller and rudder positions, 𝜅 is an experimental constant, and 𝐷 𝜂R ∶= P (31) 𝐻R Item 𝑌𝑣′ 𝑌𝑟′ − 𝑚′𝑥 ′ 𝑌𝑣𝑣𝑣 ′ 𝑌𝑣𝑣𝑟 ′ 𝑌𝑣𝑟𝑟 ′ 𝑌𝑟𝑟𝑟 Item 𝑡P 𝑤P0 𝐶𝑤 Value 0.080 0.422 −2.0 Table 6 Pre-determined values of the parameters in the model of 𝑋R , 𝑌R , and 𝑁R (Okuda et al., 2023). Item 𝑡R 𝑎H 𝑥′H 𝜖 𝜅 𝑢R0 𝑙R′ Value −0.058 0.158 −0.605 1.27 0.5 0.14 −0.888 Item 𝛾Rp 𝛾Rn 𝑓𝛼0 𝑓𝛼2 𝐶l01 𝐶l03 Value 0.483 0.172 2.411 −0.381 1.164 −0.381 3.4. Pre-determined parameter values The parameters included in the MMG model described in the previous sections have values determined based on hydrodynamics, captive model tests, and CFD, without using the time series of real-scale ship trials. In this manuscript, these values are referred to as pre-determined values. The pre-determined values of the parameters used in this study are those given in the original paper (Okuda et al., 2023). The pre-determined values of added masses and added moment of inertia are shown in Tab. 3. Moreover, the predetermined values of the parameters used in the models of hydrodynamic forces and moments are shown in Tabs. 4 to 6. where 𝑓𝛼0 , 𝑓𝛼2 , 𝐶l01 , and 𝐶l03 are constants. The flap angle 𝛿f is determined depending upon the rudder angle 𝛿 in the manipulation system of the subject ship. The relationship between 𝛿 and 𝛿f is shown in Fig.5 in the original paper (Okuda et al., 2023). Suyama et al.: Preprint submitted to Elsevier Page 5 of 13 Parameter fine-tuning method for MMG model using real-scale ship data 4.3. Real-scale ship trials data 4. Parameter tuning method In this section, the target parameter to be tuned, the exploration range for the target parameter, the real-scale ship trials used in this study, and the mathematical formulation of the parameter tuning problem are described. In this study, the time series of real-scale ship trials were utilized. One time series was treated as matrix 𝐷 ∈ ℝ8×𝑇 with ⊤ In this study, among the parameters included in the MMG model shown in Sec.3, the following 12 parameters were set as the target parameter 𝜃 t to be tuned. 𝜃 t ∶= (𝑅′0 𝑡P 𝑤P0 𝐶𝑤 𝑡R 𝑎H 𝑥′H 𝜖 𝜅 𝑙R′ 𝛾Rp 𝛾Rn )⊤ (37) The target parameters were limited to the ones that have a big influence on the simulated forces and moment. However, the hydrodynamic derivatives were excluded from the target for the simplification of the problem. On the other hand, 𝑅′0 was included in the target of the tuning without applying the extrapolation method. The objective for this is to obtain an average value of 𝑅′0 concerning the variation of ship speed due to maneuvering motion. The applicability of the proposed method is not limited to the case with the target parameter Eq. (37). The target parameter can be selected arbitrarily by the users. 4.2. Exploration range In the parameter tuning, the exploration ranges for all elements of 𝜃 t were limited. In this study, it is assumed that all of the parameters in the MMG model of the modelscale ship of the subject ship have their values which have been determined based on hydrodynamics, captive model tests, and CFD. The exploration ranges were set to be the neighborhood of those values. By constraining the ranges of available values for the target parameters in the fine-tuning problem to the neighborhood of the pre-determined values, the proposed method limits the output in realistic ranges. In the following, the candidate value and the predetermined value of the target parameter 𝜃 t ∈ ℝ12 are described as 𝜃̂ ∈ ℝ12 and 𝜃 pre ∈ ℝ12 , respectively. The exploration range for the element of 𝜃̂𝑖 (𝑖 = 1, ⋯ , 12) was formulated as 12 ∏ (40) 𝑇 is the number of time steps included in the time series data with the time step size: Δ𝑡 = 1.0 s. Elements 𝑝𝑖 , 𝜉 𝑖 , and 𝜏 𝑖 represent 𝑝, 𝜉, and 𝜏 at the time step 𝑖, respectively. In this study, eight turning tests of the subject ship with 𝛿 = ±10, ± 20, ± 35, ± 40 deg. were prepared and are described as 𝐷±10 , 𝐷±20 , 𝐷±35 , 𝐷±40 , respectively. Fig. 2 exemplifies 𝐷+10 . These time series data were utilized both for the parameter tuning and the test of the tuned parameter. In the parameter tuning phase, the parameter that minimizes the deviation between numerically simulated time series and the real-scale ship data was explored. In the test phase, the tuned parameter was applied to the simulation of real-scale ship data which was not utilized in the tuning phase and evaluated. The sets of time series data for the parameter tuning and the performance test are defined as tune ∶= {𝐷+10 , 𝐷−20 , 𝐷+35 , 𝐷−40 } (41) test ∶= {𝐷−10 , 𝐷+20 , 𝐷−35 , 𝐷+40 } , (42) and respectively. 4.4. Problem formulation In this study, the performance of the maneuvering model ̂ was defined based on the with the given parameter 𝜃̂ ∈ Θ comparison of the simulated time series of maneuvering motion and the real-scale ship trial data. Here, the deviation of the simulated time series following Eq. (12) with the MMG model and the time series of real-scale ship trial was computed, and the candidate parameter 𝜃̂ which minimizes this deviation was output as the optimal value in the parameter tuning problem. This minimization problem is summarized as follows. (38) with 𝑎r > 0. The present study considered the hydrodynamics underlying the maneuvering motion to some extent by defining the exploration range with reference to the predetermined values. Further investigation of the exploration range for parameters based on the theoretical background is one of the issues for future research. The exploration range of 𝜃 t ∈ ℝ12 is described as ̂ ∶= Θ ⊤ 𝑖 = 1, ⋯ , 𝑇 . 4.1. Target parameter ̂ 𝑖 ∶= [ 𝜃 pre − 𝑎r |𝜃 pre |, 𝜃 pre + 𝑎r |𝜃 pre | ] Θ 𝑖 𝑖 𝑖 𝑖 ⊤ ( 𝐷1𝑖 ⋯ 𝐷8𝑖 )⊤ ∶=( 𝑝𝑖 𝜉 𝑖 𝜏 𝑖 )⊤ , ̂ tune ) ∶= minimize 𝐽 (𝜃; ̂ ̂ Θ 𝜃∈ 𝑇 ∑ ∑ 𝑝̃𝑖⊤ 𝑄𝑝̃𝑖 𝐷∈tune 𝑖=2 ⎧ 𝑝̃ ∶= 𝑝̂ − 𝑝 ⎪ 𝑖 𝑖 𝑖 𝑖 ⊤ ⎪ 𝑝̂ ∶= (𝜁̂1 𝜁̂2 𝜁̂3 ) s.t. ⎨ 𝑖 𝑖−1 𝑖−1 𝑖−1 ̂ ⎪ 𝜁̂ = 𝜁̂ + 𝑓𝜁 (𝜁̂ , 𝜏 ; 𝜃)Δ𝑡 ⎪ ̂1 1 ⎩𝜁 = 𝜁 𝑖 𝑖 𝑖 (43) for 𝑖 = 2, ⋯ , 𝑇 ̂𝑖 . Θ (39) 𝑖=1 Five cases of parameter fine-tuning with 𝑎r = 0.2, 0.3, 0.4, 0.5, 0.6 were conducted for the subject ship. Suyama et al.: Preprint submitted to Elsevier Here 𝑄 ∈ ℝ3×3 is a weight matrix for deviation 𝑝. ̃ The solution of the minimization problem Eq. (43) was explored using CMA-ES. This exploration algorithm is detailed in Sec.5. In the following, the output of CMA-ES for the problem Eq. (43) is described as 𝜃 ∗ . Page 6 of 13 3 2 600 200 250 500 100 0 750 0 250 0 250 500 750 500 750 50 0.0 0.5 0 0 250 500 750 0 50 t [s] 0 250 500 750 y E [m] r [deg. /s] 200 0 δ [deg. ] vm [m/s] x E [m] 400 nP [rpm] u [m/s] Parameter fine-tuning method for MMG model using real-scale ship data 0.4 0.2 0 250 500 750 t [s] Fig. 2: An example of time series data of real-scale ship trial prepared in this study: turning test with 𝛿 = 10 deg.. 5. Covariance Matrix Adaption Evolutionary Computation (CMA-ES) CMA-ES (Hansen, 2007; Hansen and Auger, 2014) is a numerical solver for optimization problems and is known to have high efficiency in the exploration of the optimal solution. In the field of ship navigation, CMA-ES has been applied, for instance, to the optimal control problem for autonomous berthing (Maki et al., 2020) and the parameter exploration problem for the MMG model based on SI (Miyauchi et al., 2022a). In general, evolutionary computation does not need the gradient of the objective function. Therefore, users do not have to consider the differentiability of the designed objective function. Although the minimization problem treated in this study includes the computation of maneuvering motion based on the nonlinear and uncontinuous MMG model, the exploration of the optimal solution with CMA-ES is not inappropriate, as the problem is similar to the one in the paper by Miyauchi et al. (2022a) where CMA-ES succeeded in the exploration. In the implementation of CMA-ES in this study, the box constraint was handled based on the method proposed by Sakamoto and Akimoto (2017). The algorithm of CMA-ES is summarized as follows. The values of the objective function for candidate solutions which are generated following the normal distribution defined with a given mean and a given covariance matrix are iteratively calculated. At each step, the algorithm updates the mean and the covariance matrix of the distribution based on the values of the objective function. This iterative update is continued until the convergence of the distribution, and the resultant value of the mean is output as the optimal solution. Suyama et al.: Preprint submitted to Elsevier In this research, CMA-ES with the restart strategy (Auger and Hansen, 2005) was applied. 6. Results 6.1. Verification of MMG model with pre-determined parameters For the verification of the MMG model, the simulation of maneuvering motion was conducted with pre-determined parameters shown in Tabs. 3 to 6. This simulation was calculated by applying Euler method to Eq. (12) with time step size: Δ𝑡 = 1.0 s. The initial state was 𝑝(0) = (0 0 0)⊤ , 𝑢(0) = 6 knots = 3.086 m∕s, 𝑣m (0) = 𝑟(0) = 0. The control input was fixed as 𝑛P = 106 rpm, 𝛿 = ±35 deg.. These conditions were equivalent to the ones set in the simulation shown in Fig.17 (Section 5.3.1) in the original paper (Okuda et al., 2023). The simulated time series is shown in Fig. 3. Fig. 3 shows that the simulated turning motion matches the one shown in the original paper (Okuda et al., 2023) in that the diameter of the right turning circle is larger than that of the left turning circle. This result verifies the MMG model set in this study. 6.2. Parameter tuning The weight matrix was set as 𝑄 = diag(𝐿pp , 𝐿pp , 0.25𝜋). The initial population size was set 𝜆 = 12, and the population size was doubled at the restart of exploration with the upper limit 𝜆 ≤ 𝜆̄ = 128. First, the process of the exploration by CMA-ES is explained. The case of exploration with 𝑎r = 0.2 is shown in Fig. 4. The horizontal axis of Fig. 4 shows the number of iterations, The vertical axis of the upper figure of Fig. 4 shows the smallest value of the objective function calculated Page 7 of 13 Parameter fine-tuning method for MMG model using real-scale ship data 250 1200 1000 J(θ ∗ ; D tune ) x E [m] 200 150 800 600 100 400 50 200 0 0 50 200 100 0 100 200 0.2 0.3 0.4 ar 0.5 0.6 Fig. 5: Values of the objective function on the tuned parameter for tuning data tune . y E [m] Fig. 3: The trajectory of turning motion simulated based on the MMG model with pre-determined values (𝛿 = ±35 deg.). J(θ ∗ ; D test ) 2000 J 15000 1500 10000 1000 5000 500 0 1000 2000 3000 4000 5000 log 10 (J − Jmin ) 0 0.2 0.3 0 0.4 ar 0.5 0.6 5 Fig. 6: Values of the objective function on the tuned parameter for test data test . 10 0 1000 2000 3000 Iteration 4000 5000 Fig. 4: Exploring process of CMA-ES (𝑎r = 0.2). for each of the candidate solutions at the iteration. The vertical axis of the lower figure of Fig. 4 shows the value plotted in the upper figure minus the smallest value of the objective function recorded in all iterations. The iteration at which the value of 𝐽 increases abruptly represents the timing when the convergence of the distribution is confirmed and the exploration restarts with randomly initialized candidate solutions. From Fig. 4, it can be confirmed that the value of 𝐽 has repeatedly converged to the same value. Not only in the case of 𝑎r = 0.2, but also in all the cases conducted in this study, it was confirmed that the optimal solution was explored with a process similar to Fig. 4. Since multiple restarts were observed in a single trial of the algorithm, and convergence to the same value was observed at each restart, the exploration algorithm for each case was run only once in this study. Second, the performance of the tuned parameter is shown. For each 𝜃 ∗ explored in each 𝑎r , the values of the objective function 𝐽 (⋅) were computed for the data set tune and test . These values are shown in Figs. 5 and 6. It can be confirmed that the value of 𝐽 (𝜃 ∗ ; tune ) decreases Suyama et al.: Preprint submitted to Elsevier as the value of 𝑎r increases. This means that the accuracy of the simulated maneuvering motion for the tuning data was improved by expanding the exploration range of the target parameter. The value of 𝐽 (𝜃 ∗ ; test ) has the minimum and the maximum at 𝑎r = 0.5 and 𝑎r = 0.6, respectively. In the range from 𝑎r = 0.2 to 𝑎r = 0.5, the wider the exploration range, the more accurate the MMG model with the tuned parameter applied. However, in the case 𝑎r = 0.6, while showing the best accuracy for tuning data, the value of 𝐽 (𝜃 ∗ ; test ) is larger than the other cases. In the case 𝑎r = 0.6, it is considered that the exploration of the parameter that can simulate the maneuvering motion more consistent with the tuning data was performed by expanding the exploration range, resulting in overfitting biased toward the tuning data. The output values of the target parameters in the exploration with 𝑎r = 0.2 ∼ 0.6 are shown in Tab. 7. In addition, the distribution of each element of the output parameter in its exploration range is shown in Fig. 7. In Fig. 7, the ̂ 𝑖 is offset to [−1, 1], and the values of exploration range Θ each parameter are also offset. From Fig. 7, it is observed that, in all cases, 𝑤P0 , 𝐶𝑤 , 𝑡R , 𝑎H , 𝑙R′ and 𝛾Rn converged to boundary values on the same side of the exploration range, respectively. Finally, the time series data of the four real-scale ship trials included in test were compared with the time series simulated based on the MMG model with the pre-determined and tuned parameter. The comparisons are shown in Figs. 8 Page 8 of 13 Parameter fine-tuning method for MMG model using real-scale ship data Table 7 Explored values of target parameters. 𝑎r 0.2 0.3 0.4 0.5 0.6 𝑅′0 0.0174 0.0179 0.0188 0.0207 0.0204 𝑡P 0.0960 0.1040 0.1019 0.0400 0.0320 𝑤P0 0.5064 0.5486 0.5908 0.6330 0.6752 𝐶𝑤 −2.4000 −2.6000 −2.8000 −3.0000 −3.2000 𝑡R −0.0464 −0.0406 −0.0348 −0.0290 −0.0232 𝑎H 0.1896 0.2054 0.2212 0.2370 0.2528 𝑥′H −0.7260 −0.7865 −0.8470 −0.9075 −0.9680 𝜖 1.0438 1.0248 1.0049 1.1872 1.3736 𝜅 0.6000 0.6500 0.7000 0.4014 0.2000 𝑙R′ −1.0656 −1.1544 −1.2432 −1.3320 −1.4208 𝛾Rp 0.4866 0.4531 0.4262 0.3655 0.3017 𝛾Rn 0.2064 0.2236 0.2408 0.2580 0.2752 Explored value (offset) 1.0 0.5 Limit of exploring range (offset) Tuned with ar = 0.2 Tuned with ar = 0.3 Tuned with ar = 0.4 Tuned with ar = 0.5 Tuned with ar = 0.6 0.0 0.5 1.0 R 0 0 tP wP0 Cw tR aH x 0 H ² Target parameter l 0 R γRp γRn Fig. 7: Distribution of the explored values of target parameters in their exploration ranges. to 11, respectively. All of the real-scale ship trials shown in Figs. 8 to 11 were not used in the parameter tuning phase. By comparing these with the simulated time series using the MMG model with the tuned parameters, the estimation performance of the tuned parameter on the real-scale ship maneuvering motion can be evaluated. From trajectories in the left figure of each of Figs. 8 to 11, in all cases of 𝑎r = 0.2 ∼ 0.6, the MMG models with the tuned parameter simulate time series closer to the maneuvering motion of the real-scale ship than the one with the pre-determined values. Comparing the best-performing case 𝑎r = 0.5 (blue-colored) and the worst-performing case 𝑎r = 0.6 (coral pink-colored), although the turning circle diameter differs by about 1.5𝐿pp for 𝐷−10 , not much difference in simulated maneuvering motion is observed for 𝐷+20 , 𝐷−35 , and 𝐷+40 . 7. Discussion From Figs. 8 to 11, it is confirmed that the parameters were tuned to simulate the turning motions with larger diameters than pre-determined parameters in all cases. Here, Fig. 7 shows that 𝑎H > 0 is tuned to be large and 𝑥′H < 0 is tuned to be small in all cases, suggesting that 𝑁R is computed to be large. On the other hand, 𝑤P0 and 𝑡R were tuned to be large so that the propeller inflow velocity 𝑢P and the fraction of rudder (1 − 𝑡R )𝐹N sin 𝛿 are computed to be small, respectively. It is analyzed that these effects are Suyama et al.: Preprint submitted to Elsevier dominant compared to the effect of largely estimated rudder moment 𝑁R . Among the parameter tunings conducted in this study, the parameter with the best performance on the test data was obtained with the exploration range defined by 𝑎r = 0.5. However, the proposed tuning method depends on the time series data set of the real-scale ship trial used for fine-tuning. When tuning the parameters of a maneuvering model, unexpected problems such as overfitting may occur depending on the time series data used for tuning, the target parameters to be tuned, and the exploration range of each parameter, as observed in Fig. 6. Therefore, for parameters explored under different conditions, it is not possible to determine which output is the best for the simulation of the real-scale ship maneuvering motion without at least checking the accuracy of simulated maneuvering motion for time series data other than tune , such as test in this study. Thus, if there is a sufficient amount of time series data available for parameter tuning, it is desirable to prepare test data separately from the tuning data and compare the performance against validation data to determine the best parameter. However, the amount of available time series data of real-scale ship maneuvering motion is limited. Therefore, there may be cases where it is not possible to prepare validation data. In such cases, parameters tuned in a certain condition would have to be used in practice without performance validation. In this case, one should be careful Page 9 of 13 Parameter fine-tuning method for MMG model using real-scale ship data nP [rpm] 3 u [m/s] Model-scale parameter Tuned with ar = 0.2 Tuned with ar = 0.3 Tuned with ar = 0.4 Tuned with ar = 0.5 Tuned with ar = 0.6 Real-scale ship trial 2 50 0 0 200 400 600 400 0 200 0 200 400 600 400 600 50 200 0 25 0.6 δ [deg. ] vm [m/s] x E [m] 100 0.4 0 25 0.2 0 200 200 400 600 50 t [s] 600 400 200 0 y E [m] r [deg. /s] 0.2 0.4 0.6 0 200 400 600 t [s] Fig. 8: Comparison of seven time series data of turning test (𝛿 = −10 deg.) which is not utilized in parameter tuning: real-scale ship trial 𝐷−10 , simulation by the MMG model with untuned parameters, and simulation by the MMG model with tuned parameters (𝑎r = 0.2 ∼ 0.6). not to set an unnecessarily wide exploration range, since parameters may be explored biased toward the tuning data. 8. Concluding remarks An automatic fine-tuning method for all of the arbitrarily indicated target parameters of the MMG model was proposed. The proposed method tunes the parameter values which are previously determined based on the hydrodynamics, captive model tests, and CFD to the ones for the MMG model of the real-scale ship using the framework of SI. The previously determined parameter values are utilized to constrain the tuned parameter values to the realistic ranges. By directly referring to the time series data of real-scale ship maneuvering motion, the proposed method can steadily improve the performance of the MMG model with the tuned parameter in terms of the accuracy of the simulated maneuvering motion. The proposed fine-tuning method was applied to a container ship and validated with 12 target parameters which is highly influential in the MMG model. In all cases of parameter fine-tuning conducted with different widths of exploration range, better parameters were obtained compared to untuned parameters in terms of the accuracy of the simulated real-scale trajectories. Suyama et al.: Preprint submitted to Elsevier CRediT authorship contribution statement Rin Suyama: Conceptualization, Formal analysis, Methodology, Software, Visualization, Writing - Original Draft . Rintaro Matsushita: Data Curation, Project administration, Writing - Review & Editing . Ryo Kakuta: Data Curation, Project administration, Writing - Review & Editing . Kouki Wakita: Methodology, Writing - Review & Editing . Atsuo Maki: Funding acquisition, Project administration, Supervision, Writing - Review & Editing . Acknowledgment This study was conducted as a part of the Nippon Foundation to Support Fully Autonomous Ship Development Project “MEGURI2040”. The authors are thankful to Mr. Takuya Taniguchi (Osaka University) for a helpful discussion. Moreover, this study was supported by a Grantin-Aid for Scientific Research from the Japan Society for Promotion of Science (JSPS KAKENHI Grant #22H01701) and by the Fundamental Research Developing Association for Shipbuilding and Offshore (REDAS23-5(18A)). Page 10 of 13 2 0 200 0.6 0 200 0 200 400 0 25 0.8 0 200 25 0.4 0 0 50 δ [deg. ] vm [m/s] 100 50 400 0.2 200 100 0 1 300 x E [m] nP [rpm] Model-scale parameter Tuned with ar = 0.2 Tuned with ar = 0.3 Tuned with ar = 0.4 Tuned with ar = 0.5 Tuned with ar = 0.6 Real-scale ship trial u [m/s] Parameter fine-tuning method for MMG model using real-scale ship data 200 400 50 400 t [s] 400 0.8 r [deg. /s] y E [m] 0.6 0.4 0 200 400 t [s] Fig. 9: Comparison of seven time series data of turning test (𝛿 = +20 deg.) which is not utilized in parameter tuning: real-scale ship trial 𝐷+20 , simulation by the MMG model with untuned parameters, and simulation by the MMG model with tuned parameters (𝑎r = 0.2 ∼ 0.6). References Abkowitz, M.A., 1964. 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