Optimized support vector regression algorithm-based modeling of ship dynamics

Applied Ocean Research 90 (2019) 101842

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Applied Ocean Research journal homepage: www.elsevier.com/locate/apor

Optimized support vector regression algorithm-based modeling of ship dynamics

T

Man Zhua,b, , Axel Hahnb, Yuan-Qiao Wena,c,d, Wu-Qiang Sune ⁎

a

Intelligent Transportation Systems Research Center, Wuhan University of Technology, 430063 Hubei, China Computer Science, Carl-von-Ossietzky University of Oldenburg, 26121 Oldenburg, Germany c National Engineering Research Center for Water Transport Safety, 430063 Hubei, China d Hubei Key Laboratory of Inland Shipping Technology, 430063 Hubei, China e Sunwin Intelligent Co., Ltd., 230000 Hefei, China b

ARTICLE INFO

ABSTRACT

Keywords: Parameter estimation Simplified ship dynamic model Different types of ships Support vector regression algorithm Artificial bee colony algorithm Maritime traffic simulation

This study contributes to solving the problem of how to derive a simplistic model feasible for describing dynamics of different types of ships for maneuvering simulation employed to study maritime traffic and furthermore to provide ship models for simulation-based engineering test-beds. The problem is first addressed with the modification and simplification of a complicated and nonlinearly coupling vectorial representation in 6 degrees of freedom (DOF) to a 3 DOF model in a simple form for simultaneously capturing surge motions and steering motions based on several pieces of reasonable assumptions. The created simple dynamic model is aiming to be useful for different types of ships only with minor modifications on the experiment setup. Another issue concerning the proposed problem is the estimation of parameters in the model through a suitable technique, which is investigated by using the system identification in combination with full-scale ship trail tests, e.g., standard zigzag maneuvers. To improve the global optimization ability of support vector regression algorithm (SVR) based identification method, the artificial bee colony algorithm (ABC) presenting superior optimization performance with the advantage of few control parameters is used to optimize and assign the particular settings for structural parameters of SVR. Afterward, the simulation study on identifying a simplified dynamic model for a large container ship verifies the effectiveness of the optimized identification method at the same time inspires special considerations on further simplification of the initially simplified dynamic model. Finally, the further simplified dynamic model is validated through not only the simulation study on a container ship but also the experimental study on an unmanned surface vessel so-called I-Nav-II vessel. Either simulation study results or experimental study results demonstrate a valid model in a simple form for describing the dynamics of different types’ ships and also validate the performance of the proposed parameter estimation method.

1. Introduction In the past few decades, massive digitization of machinery and equipment have been used in shipping industry to make navigation convenient and precise, but they add a burden of work to mariners which means too much information available [1]. In order to reduce the human error caused by the burden of too much information available and integrate navigational tools effectively, International Maritime Organization (IMO) proposed the e-Navigation strategy [2]. One issue concerning the robust e-Navigation technology is the verification and validation. As the maritime transportation can be understood as a sociotechnical system, this issue can be studied through simulation-based test beds [3], such as a modeling and simulation toolset named HAGGIS

⁎

(Hybrid Architecture for Granularly, Generic and Interoperable Simulations) [4]. Within components of HAGGIS, each one has its functionality and works interactively. Instance for better understanding is maritime traffic simulator (MTS) which is a flexible and usable maritime traffic simulation for implementing, executing and observing the behaviours of multiple vessels in a realistic context [4]. For every vessel involved in MTS, its behaviours comprising traffic information are animated by simultaneously using dynamic model, kinematic model under environmental conditions. Due to the kinematic model is functional in combination with outputs of the dynamic model, more attention should be paid on the ship dynamic model. Besides, another notable point is the requirement of dynamic models for ships that implies every ship needs a compatible dynamic model applicable for

Corresponding author at: Intelligent Transportation Systems Research Center, Wuhan University of Technology, 430063 Hubei, China. E-mail address: [email protected] (M. Zhu).

https://doi.org/10.1016/j.apor.2019.05.027 Received 6 August 2018; Received in revised form 3 February 2019; Accepted 29 May 2019 0141-1187/ © 2019 Elsevier Ltd. All rights reserved.

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capturing its unique motion characteristics. For this point, it would be beneficial if one model can be applicable for describing ships’ motions with minor modification such as changing parameters but retaining the model structure. Generally, the ship is reasonably assumed as a rigid-body object and then its dynamic model is derived based on Newton's second law of motion [5]. Typical and widely-used examples are Abkowitz model built in Taylor series expansion [6], Maneuvering Modeling Group (MMG) model also called modulus model [7], Fossen's vectorial representation model additionally regarded as modulus model in vector matrix form [5], and response model also so-called Nomoto model [8]. The former three models with the ability of completely capturing ship's dynamics is complicated and nonlinearly coupling in the use of a number of parameters. Therefore, major modification consuming much time and finance is required to adopt and switch one of these models among different types of ships due to unignorable differences existing in hull, propeller and rudder among various ships. Recall the above mentioned noticeable point about the requirement of dynamic models for ships, the former three models are not desirable fit. Trough the studies on the Nomoto model, it is reported to be applicable for control design especially the autopilot design in maritime engineering due to its simple structure and easy comprehension [9] but its ability of capturing 1 DOF motions in yaw direction is not so expected to describe ship dynamics in particular used in ship's motion simulation such as the MTS. Therefore, the task of how to match the requirement on establishment of one dynamic model for different types of ships is deserved considerable study to be resolved. This task envelops two sub-tasks, i.e., the determination of model structure, and the estimation of parameters in the determined model. Taking Newton's second law of motion and relatively derived models into account, the structure of ship dynamic model is basically deterministic but requiring additional modification to get a appropriate model for ships. For another task in estimating parameters involved in the determined model, an adequate solution is required. Even though four core methods reported in studies are extensively used for estimating ship dynamic models, including captive model test with planar motion mechanism (PMM) [10], estimation with empirical formulas [11], numerical calculation based on computational fluid dynamics (CFD) [12], and system identification in combination with free-running model test or full-scale trial [13], they are still underdeveloped due to their respective weakness such as scaling effects, demand of extreme computation, etc. explained as follows. The captive model test with PMM is applicable to obtain most parameters, but it has a remaining problem of scaling effects aroused by the difference of Reynolds’ number between the real ship and the scaled model ship which makes the measured value of parameters not wholly reliable [10]. The estimation with empirical formulas is practical and straightforward. The formulas are built based on statistical analysis of a set of ships, so it is not suitable for the latest type of ships, and the estimation results are not precise. The rest two estimation methods are both powerful methods, but the numerical calculation based on CFD always requires extreme computing power, and its validation dramatically depends on quality and amount of experimental data. Comparatively, system identification in combination with free-running model test or full-scale trial avoids the scaling effect and is easy to be undertook, but parameter drift unpredictably existing among parameters if there were too many parameters in a model can compromise the accuracy of identification results. Other than scaling effects, demand of extreme computational power and uninvolved empirical calculation, the influence of parameter drift on estimation accuracy of ship dynamic model is tricky to be overcome. Hence, the derived ship dynamic model should be not only in appropriate form but also in parameter-reduced version for mitigating even avoiding parameter drift influence and improving the accuracy of simulation of ship motions. The challenges required to be overcome can be summarized as follows. The ship dynamic model is paramount for the MTS but not

available to capture dynamics of different types’ ships with minor modification. The estimation methods are unexpectedly associated with weaknesses negatively influencing the accuracy and reliability of the estimated model. Therefore, this study proposes to develop an appropriately simple ship dynamic model with a small amount of terms to overcome the described challenges. The complicated dynamic model is simplified to get term-reduced model through a series of reasonable assumptions. Afterwards, the parameters involved in the simplified model are estimated by using the artificial bee colony algorithm incorporated with support vector regression algorithm. To validate the feasibility of the simplified model and effectiveness of the optimized estimation method, simulated data which are generated by a high realistic but complicated dynamic model of a large container ship with predefined parameters are employed. Through the evaluation results of simulation study, special considerations on further simplification of the simplified dynamic model with eight terms are inspired. The further simplified model with five terms is validated through not only the simulation study on the container ship but also the experimental study on a small unmanned surface vessel, so-called I-Nav-II vessel. The main contribution of this study is developing an approach for modeling dynamics of different types’ ships by firstly using the optimized identification method, which is described in detail as follows. The modeling of ship dynamics is achieved through simplifying a complicated and nonlinearly coupling dynamic models into a reducedterm model with relatively low complexity and acceptable accuracy based on pieces of reasonable assumptions and practical case studies on a large container ship as well as a Unmanned Surface Vessel. The identified model is feasible for capturing different types of ships’ dynamics. The performance and accuracy of the estimation method is improved by using a intelligent algorithm, which is the first time adopted in maritime domain. This paper is organized as follows. Section 2 reviews the theory related to the ship dynamic model and parameter estimation method to provide the foundations for the remainder of this study. In Section 3, the solution associated with modeling of ship dynamics is explained from three aspects including the simplification of ship dynamic model, the determination of optimized estimation method, and its implementation. The simulated maneuvers of the large container ship are used as training samples and validation samples for parameter estimation of the simplified ship dynamic model in Section 4 where comparisons among three steering models are carried out which inspires special considerations on further simplification of the nonlinear steering model. Furthermore, experimental study on a real unmanned surface vessel is undertook in Section 5 to validate the special considerations. Finally, concluding remarks are summarized in Section 6. 2. Related work 2.1. Ship dynamic models Since over one century, Davidson and Schiff [14] started working with the mathematical modeling of ship dynamics. This model describes linear steering dynamics especially the interactions between sway and yaw. Based on this model, Nomoto [15] derived a first or second order transfer function used for only analyzing yaw mode by eliminating the equation describing sway mode from the model proposed by Davison and Schiff. By considering the nonlinearity of ship dynamics in the operation of large rudder angle, Norrbin [16] adjusted the form of Nomoto's first order transfer function by adding a nonlinear component. Analogously, Nomoto's second order transfer function was also modified by adding a nonlinear component by Bech and Wagner Smith [17], in which they additionally proposed the well-known Bech's reverse spiral maneuver and introduced the approach on applying such maneuver to estimate parameters in the nonlinear component. Either Nomoto's first or second linear transfer function or the corresponding 2

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modified first or second order nonlinear transfer function is still recognizable today and is still widely used for guidance and control design due to its simple and effective structure and easy-understanding. A contribution to the work of modeling ship dynamics by Abkowitz [18] should be highlighted. In his work, the forces (surge and sway) and moments (yaw) acting on a ship in 3 DOF is expressed by Taylor-series expansions at forward cruise speed and have an arbitrary degree of precision. The model involves a large number of coefficients, which further lead to a tough problem about how to estimate all these coefficients. Some works caring this problem will be introduced later in section on system identification of the ship dynamic model. Later, Son and Nomoto [19] expanded the application of Abkowitz model to more ships, e.g., the large container ship, by analyzing and accounting the roll motions. As said previously, many coefficients are involved in the Taylor-series expansion models, and some terms have no physical interpretation, which leads to the estimation of coefficients difficult and furthermore makes the application of models troublesome. Therefore, simplifications to the models are meaningful as reported in some papers [20–23], where the sensitivity study was commonly used to analyze how sensitive is the model response to the variation of a model coefficient. According to the results of sensitivity study, the model was then simplified by omitting the coefficients with very low sensitivity degree. Different from Abkowitz model, i.e., the Taylor-series expansion model, another new type model called modulus model was developed in 1963 by Fedayevsky and Sobolev [24] who used the quadratic modulus equations with low order to present hydrodynamic forces and moments. In 1970, Norrbin [25] made some modifications of the former modulus model in modeling the hydrodynamic effects by using nonlinear quadratic and quartic terms for even functions and quadratic modulus terms for odd functions. This model is a little complicated for application, so Blanke [26] simplified it by removing some unwieldy and unphysical terms at same time maintained the significant parts. To comprehensively study ship dynamics including roll motions, Christensen and Blanke [27] developed a linear-time varying model with roll equation but without surge equation. Even though the surge equation is ignored, the forward speed generally resulted by the surge and sway speed is regarded as a varying parameter to be involved in sway, yaw and roll equations. Such linear-time varying model is used to discuss the problems such as rudder roll damping problem and fin roll damping problem. In the findings by Hooft [28], the cross-flow drag formulations were applied to analyze the lateral forces on a ship during maneuvering through tests performed as a function of ship's lateral velocity and as a function of ship's yaw rate. The key advantage of cross-flow models is that it is dependent on few hydrodynamic parameters, typically the drag coefficient for transverse flow and the hull friction coefficient [29]. With the development of such approach within years, some studies make efforts to propose some comprehensive model, see the corresponding work found in [30]. For the modeling of ship dynamics, the work of Mathematical Modeling Group makes the prominent influence and provides the fundaments on deriving models. In their works, the effects of hull hydrodynamic, propeller, rudder and their interactions are dealt with separately. The recommended references are [31,32]. The work by Fossen, especially in [5], demonstrates a comprehensive and extensive study of the latest research in navigation, guidance, control systems for marine vehicles. The detailed information about the implementation of these techniques is presented in a systematic way. The equations of motion are expressed in the form of matrix-vector, since nonlinear system properties such as symmetry, skew-symmetry, and positiveness of matrices can be exploited in the passivity or stability analysis, and such form of the model makes it easy to simulate the model in either Matlab or C++.

Fig. 1. Basic framework of identification of the ship dynamic model.

2.2. Parameter estimation methods The application of system identification to identify the ship dynamic model can be briefly described. The parameters of ship dynamic model are estimated which is guided by a hypothesized model structure and the collected observed data set. If the validation test on the identified model is ideal, the identified dynamic model is feasible to describe and capture ship dynamics. Basically, the framework of identification of ship dynamic model according to the system identification procedure is illustrated in Fig. 1. The execution of identification of ship dynamic models is trickily dependent on the procedure of system identification technique including core four steps, i.e. acquisition of observed data, determination of the set of model structure, decision of identification method, and validation of identified model. The data extracted from maneuvres carried out under the condition of the input satisfying persistence of excitation should be informative. The ship dynamic model studied in this study is classified as the grey-box model. The final step in validating identified model can be directly carried out by judging the degree of agreement between the compatible outputs of identified model and original manoeuvres. Unlike these three steps, decision of identification method is conducted effectively according to the comprehensive survey on the applications of various identification methods in maritime domain. For the identification method, many choices exist for performing system identification on modeling ship dynamics. Åström and Källström [33] estimated the parameters of the ship steering model by applying the measurements from full-scale tests to the maximum likelihood method (ML). The model of the Esso Osaka tanker was identified using the ridge regression method by Yoon and Rhee [34]. In Taylor's work [35], EKF was applied to estimate coefficients of a dynamic model for an Arleigh Burke Class Destroyer for the model-based control design. Caccia, et al. [36] used LS and data from sea trails to estimate parameters of the proposed practical dynamic model for an autonomous catamaran prototype Charlie2005. Additionally, the modified or improved LS is also studied to work in identifying ship dynamic models, see [37][38]. Apart from the traditional methods, many intelligent algorithms have been developed for identifying ship dynamic models in recent years. The genetic algorithm (GA) was adopted by Sutulo et al. [39] to identify a 3 DOF moderately complicated manoeuvring model. Besides, Erunsal [40] utilized GA and global search algorithm (GS) together to identify a model which was further applied to design controller. Ferri, et al. [41] adopted the simulated annealing algorithm (SA) to identify the dynamic model of an autonomous surface vehicle (ASV) taking into account the performance of sensors used for measurements record. Bhattacharyya and Haddara [42] employed the neural network method (NN) to estimate the hydrodynamic derivatives embedded in the nonlinear steering model. SVMs is firstly successively applied in 2009 by Luo and Zou [13] in maritime domain to estimate hydrodynamic derivatives of Abkowitz model taking into account the free-running tests. 3

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On the basis of those identification methods, some studies focus on simultaneously cooperating two approaches and successively applying them to identify ship dynamic models. The examples are a recursive least square method combined with a recursive prediction error method for identifying ship maneuvering dynamics [43], an EKF with constrained least square method used to estimate maneuvering parameters [44], a nonlinear prediction error method and a unscented Kalman filtering method for identifying dynamic models of a high-speed trimaran ferry [45]. By analyzing the performance of these identification methods on estimating parameters in the ship dynamic model, it is easy to reveal some noticeable points. The traditional identification methods such as LS and ML present some inherent drawbacks, i.e., the identification results rely on either the mathematical model of the research object or the initial values of parameters needed to be estimated firstly. The intelligent algorithm-based identification methods can overcome the drawbacks of traditional identification methods while used for identification. But some methods involving GS, GA, SA, and NN cannot always guarantee the generalization performance and optimal solution and avoid the curse of dimensionality. Differently, SVMs based identification method can prevent these deficiencies and guarantee satisfying performance. That is because of the unreplaceable merits of SVMs. Firstly, SVMs can work well with finite samples. Then, its optimization problem is a convex typically solved by using the quadratic program (QP), from which the global and unique optimal solutions can be guaranteed [46]. Thirdly, the simultaneous minimization of the empirical risk and structural risk promotes its generalization capability, and the curse of dimensionality can be omitted due to the computational complexity is only related to the number of support vectors instead of the dimension of inputs. The performance of SVM based methods for either offline or online identification can benefit from these natural merits. Besides these attractive merits, one point of SVM based identification method are ought to be noticed, which is the stability problem associated with dealing with a large amount of data. The small-scale training set would result in local optimal solutions. But the large-scale training set could increase the computation burden due to a large size of kernel matrix. From the suggestion proposed by Suykens et al. [46], the length of the training set is to be less than 2000. In addition, a robust method (i.e., optimal truncated SVM) with the ability of diminishing parameter uncertainty on demand of a low computational cost by reducing the dimensionality of the kernel matrix can be applied to solve this stability problem. In this study, attention to applying a small amount of training data less than 2000 to the offline identification of the simplified ship dynamic model is paid. One noticeable deficiency of SVM is that its identification results are sensitive to the structural parameters in SVM such as the insensitivity factor, the regularization parameter, and kernel parameters. For instance, the regularization parameter controls the trade-off between empirical risk and confidence interval, or in other words, the trade-off between achieving a low error on the training data and minimising the norm of the weights. Thus, the selection of these parameters for SVM is one of the most critical steps when using it as an identifier. To remedy this deficiency of SVM, Frohlich et al. [47] used the grid search on the log-scale of the parameter in combination with cross-validation technique to tune parameters in SVM. Even though this technique is simple, it is often time-consuming and at average accuracy [48]. From the extensive literature reviews, many different approaches have been proposed for tuning parameters in SVM. One widely-used technique is the swarm intelligence, for instance, the PSO [49], the GA [50], the firefly algorithm [51], ABC [48], etc. Apparently, the design of the developed hybrid model is different regarding the various problems needed to be solved in related domains. For the problem of parameter estimation of the ship dynamic model, the hybrid approach of ABC with SVM has not been applied, which will be studied and contributed in this study.

3. Solution for modeling ship dynamics 3.1. Simplified ship dynamic model As elaborated in related work on ship dynamic model, each typical model, e.g., Abkowitz model, MMG model, or vectorial representation model, can completely describe ship dynamics, but a number of parameters involved in the model makes challenges in its application. This is due to the tough task on estimating the unknown parameters. To estimate all these parameters, optimal experiments are required to be designed to acquire informative observations, which is of much difficulty including but not limited to the weather condition, the equipment, the financial support. Additionally, the problem of parameter identifiability is hard to be avoided due to the multicollinearity happens much frequently for the complicated model with series parameters. Furthermore, the ship dynamic model studied is specific to a ship. Switching the application of the model between different types of ship is on demand of re-setup experiments which is rarely found. Therefore, the vectorial model is selected as the fundamental form for simplification and modification to derive a model in a simple form. Its order reduced version expressed in 3 DOF investigated by our previous study [52] is employed to obtain the simplified ship dynamic model containing the decoupled surge and steering models, for which the forces and moments acting on the ship is calculated by the function of rudder deflection and propeller shaft speed. 3.1.1. 3 DOF ship dynamic model The simplified 3DOF dynamic model based on a series of reasonable assumptions is extracted from [52] and illustrated as (1)

M + C ( ) + D( ) + g( ) = where the expression of each symbol is shown as g ( ) = [0, 0, 0]T , = [ X , Y , N ]T

m 0 0 0 m mx g 0 mx g Iz

M = MRB + MA = m

Xu

0

0 0

=

m mx g

+

0 Yr Nr

(2)

0 0 m (x g r + v )

0

0

0

0

0

0

0 0 mu

Yv v +

0 0 1 (Yr + Nv ) r Xu u 2

Yv v

0 Yv Nv

0 Yv mx g Yr , Nv Iz Nr

C ( ) = CRB ( ) + CA ( ) =

+

Xu 0 0

= [u , v , r ]T ,

m (x g r + v ) mu 0

1 (Yr + Nv ) r 2 Xu u 0

(Yv

1 1 m) v + ( Yr + Nv 2 2 mx g ) r

=

(Yv

m) v 1 ( Yr 2 1 + Nv 2 mx g ) r

4

(m

(m Xu ) u

Xu ) u 0

,

(3)

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M. Zhu, et al.

D ( ) = DL + DNL ( ) = X|u | u |u| 0 0

+

Xu

Xu 0 0

0 Yv Nv

moments can be described as

0 Yr Nr

= [ X,

0 Y|v | v |v| N|v | v |v|

X|u | u |u|

Y|r | v |r| N|r | v |r|

Y|v | r |v| N|v | r |v|

0 Yv

0

0

Y|v | v |v|

Yr

0

Nv

Y|v | r |v| Y|r | r |r|

N|v | v |v|

Nr

N|r | v |r|

,

N|v | r |v| N|r | r |r|

= (m

Y=

N=

Xu ) u + (Yv

m )vr + (Yr

mx g ) r 2

Xu u

(m Yv ) v + (mx g Yr ) r + (m Xu )ur Y|r | v |r| v Yr r Y|v | r |v| r Y|r | r |r| r

Yv v

(mx g

(Nv

Nv ) v + (Iz

Nr ) r

(Yv

(m Xu ) u0 v Nv v N|v | v |v| v N|v | r |v| r N|r | r |r| r

m)uv

X|u | u |u| u Y|v | v |v| v

mx g )ur Nr r

(7)

which will be used as the fundamental 3 DOF model for the following simplification to obtain initial version of simplified dynamic model. 3.1.2. Actuator forces and moments The actuator forces and moments are generated by a set of thrusters with revolutions per second n = [n1, n2, …, np1 ]T and a set of control surfaces (or propeller blade pitch) with angles = [ 1, 2, …, p2 ]T . They are related to the vector τ through the mapping

= [ X,

T Y, N ]

= Bf (vr , n, )

(m

(9)

Let the force attack points of {T } located at coordinate (xn, yn, zn) in the body-fixed frame, and likewise (xδ, yδ, zδ) for the rudders. Then the actuator configuration matrix is

1 0 0 1 yn x

B=

u=

According to Ref. [53], the thrust force can be expressed as

T|n | n > 0

(11)

where n represents the propeller revolutions, T|n | n is a parameter depending on the propeller's diameter D and the density of the water ρ. From foil theory [54], the lift forces are modeled as

L=

2

AR CL ( )|uR | uR

v=

(12)

with

CL ( ) = c1

c2 | |

(16)

N ]

(17)

Xu ) u = Xu u + X|u | u |u| u + Xuuu u3 + T|n | n |n| n

Xu m

Xu

u+

X|u | u Xuuu 3 T|n | n |n| n |u| u + u + m Xu m Xu m Xu

(18)

Iz

Nr

(( Yv

Y|v | v |v|

Y|r | v |r|) v + ((m mx g

Y|v | r |v|

Y|r | r |r|) r + Y )

N|v | v |v|

N|r | v |r|) v + ( Nr

Yr

N|v | r |v|

Xu ) u 0

(( (m

Yr

Xu ) u 0

Nv

N|r | r |r|) r + N ) (19)

(13)

r=

where AR is the effective rudder area, uR is the relative velocity of the fluid at the rudder surface, CL(δ) is the non-dimensional lift coefficient. In order to simplify the expression of L which is expressed as

L = k2

(15)

3.1.4. Steering model The remaining steering model describing sway and yaw motions is obtained from (1) and the resultant forward speed is calculated by U = u 02 + r 2 based on the constant surge speed (u0). Hence, the decoupled steering model yields

(10)

T = T|n | n |n| n ,

T|n| n |n| n k2 yn T|n | n |n| n + x k2

=

The model is a second order modulus model with an extra third order term in the surge. The reason for adding the third order term is that it can show better fit to the experimental data. Note that the surge model (17) is still a valid representation of (5) if the sway velocity (v ) and yaw rate (r) are small. Considering the identification of the surge model, the four inputs (u, |u| u, u3and |n| n) in such model are not enough to identify all five parameters (Xu, X|u | u , Xuuu, T|n | n and Xu ). In other words, all hydrodynamic derivatives which define ship characteristics and its complete behaviors cannot be entirely estimated. Hence, Eq. (17) is reorganized into

(8)

where B is an actuator configuration matrix, f (vr , n, ) is a function that for each velocity vr relates the actuator set-point (n, δ) to a vector of forces. The propeller provides the thrust forces {T } while the rudders generate lift forces {L} and drag forces {D} . Here disregarding the drag forces, the force vector can be described as

f (vr , n, ) = [T , L]T

|n| n

3.1.3. Surge model The simplified model obtained finally should be applicable for the description of dynamics of multiple vessels varying from small to large vessels including the small unmanned surface vessel. At slow speeds small vessels act similarly to large vessels that drag themselves through the water, completely replying upon buoyancy to stay afloat. At higher speeds the small to medium vessels are pushed up out of the water with the hull of the vessel skidding across the water. To add to the complexity of these vessels’ dynamics, their steady state pitch angle changes based upon speed changing the planfrom exposed to the water and consequently changing the added mass terms. With these added complexities the simplified surge and steering models involving a speed scheduled set of parameters is sought as an alternative to the nonlinear horizontal 3 DOF model expressed in (5)–(7). The simplified surge model retains drag terms but excludes terms that couple surge velocity with sway velocity and yaw rate, which is given as

(5) (6)

N|r | v |r| v

T|n | n 0 0 k2

= [T|n | n |n| n Y

(4)

Eventually, the expressions of the surge, sway, and yaw motions are X

= Bf (vr , n, ) = Bku

For some kinds of ships, the force attack point of the thruster is located along with the longitudinal direction of the ship, which means yn = 0. Then the actuator forces and moments can be presented with the Yδ replacing k2 and Nδ for xδk2.

0

Y|r | v |r|

=

1 0 0 1 yn x

=

Y|r | r |r| N|r | r |r|

T N]

Y,

(14)

Nv

mx g

(( Yv

Y|v | v |v|

Y|r | v |r|) v + ((m

Y|v | r |v|

Y|r | r |r|) r + Y ) +

N|v | v |v|

N|r | v |r|) v + ( Nr

m

Yv

Xu ) u 0

(( (m

N|v | r |v|

Xu ) u 0

Yr Nv

N|r | r |r|) r + N ) (20)

with k2 = 2 AR (c1 c2 | |)|uR | uR . Then k = diag{ T|n| n, k2} acquired is regarded as the force coefficient matrix. Hence, the actuator forces and

with ∇=(m 5

Yv )(Iz

Nr )

(mx g

Yr )(mx g

Nv ) .

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3.1.5. The Nomoto models The first order linear Nomoto model is the most straightforward set describing the relationship between turn rate and rudder angle, and its nonlinear extension appeared later as well. The first order linear Nomoto model can be derived from the linearized simplified steering model. Linearize the simplified nonlinear steering model by removing the sway–yaw coupling terms from (19) and (20)

m mx g

Yv mx g Yr Nv Iz Nr

(m

Yv Xu ) u0 + Nv

matrix are expressed

A=[a1 a2 a3]1 × 3 , B=[b1 b2 b3 b4 b5 b6 b7]1 × 7 , C=[c1 c2 c3 c4 c5 c6 c7]1 × 7 , with

a1 = 1 +

v = r (m

Xu ) u0 + Yr Nr

hXuuu , m Xu 2hT|n | n |n 0 | , a3 = m Xu (Nr Iz ) Yv h + ((m b1 = 1 + a2 =

Y N

v r +

(21)

Subsequently, a 1 DOF model so-called the second order Nomoto model is obtained by eliminating the sway speed from (21). This model is widely applied to design ship autopilot because of its compromise between simplicity and accuracy. Its time domain function is

b2 =

(22)

T1 T2 r¨ + (T1 + T2 ) r + r = K + KT3

b3 =

where the parameters are calculated using hydrodynamic coefficients as

T1 T2 =

(m

Yv )(Iz

Yv Nr + (m

T1 + T2 = +

Nr ) Xu ) u 0

Yv (Iz

Nr )

(mx g

Yr )(mx g

Yr )((m

Nr (m

Nv )

((m

Xu ) u 0

b5 =

Xu ) u 0 + Yr )(mx g

Yv Nr + (m Xu ) u0 Yr )((m ((m Xu ) u 0 + Nv )(mx g Yr ) Yv Nr + (m

b4 =

Xu ) u0 + Nv

Yv)

Yr )((m

Nv )

b6 =

Xu ) u 0 + Nv )

b7 =

Xu ) u 0 + Nv ) c1 =

((m Xu ) u 0 + Nv ) Y K= Yv Nr + (m Xu ) u 0 Yr )((m Xu ) u 0 + Nv ) T3 =

(mx g Yv Nr + (m

Nv ) Y Xu ) u 0

(m Yr )((m

c2 =

Yv ) N

c3 =

Xu ) u 0 + Nv )

Eq. (22) is further converted into the first order Nomoto model by setting the effective time constant satisfy the relationship T = T1 + T2 − T3. The time domain function can present the first order linear Nomoto model which is illustrated as

Tr + r = K

c5 =

(23)

c6 =

Tr + n1 r + n3 r 3 = K

c7 =

(24)

where n3 is the coefficient of the nonlinear item, n1 is a coefficient determined according to ship's stability of course, e.g., n1 = 1 for a course-stable ship and n1 =−1 for a course-unstable ship.

Nr ) Y|v | v h + (mx g

Yr ) N|v | v h

(Iz

Nr ) Y|r | v h + (mx g

Yr ) N|r | v h

(Iz

Nr )((m

Xu ) u 0

Nr ) Y|v | r h + (mx g

Yr ) N|v | r h

(Iz

Nr ) Y|r | r h + (mx g

Yr ) N|r | r h

(Iz

Nr ) Y h

(mx g

Yr ) N h

,

,

Yv )((m

(Nv

mx g ) Y|v | v h

(m

Yv ) N|v | v h

(Nv

mx g ) Y|r | v h

(m

Yv ) N|r | v h

Xu ) u 0

Xu ) u 0 + Nv ) h

mx g ) Y|v | r h

(m

Yv ) N|v | r h

(N v

mx g ) Y|r | r h

(m

Yv ) N|r | r h

mx g ) Y h + (m

Yv ) N h

,

, ,

Yr ) h

(Nv

(Nv

,

,

mx g ) Yv h

mx g )((m

Yr ) Nr h

,

(Nv

(Nv

(m

,

,

Yr ) h + (mx g

(Iz

Yr ) h

(m

Yv ) Nr h

,

, ,

,

in which h is the time sample, k + 1 and k are two successive data, = (m Yv )(Iz Nr ) (mx g Yr )(mx g Nv ) . By similarly processing the Nomoto models, one also can get the input–output samples. For the first order linear Nomoto model, inputs are [r (k ), (k )]T2 × 1, output is r(k + 1), and parameters are h Kh [(1 T ) T ]1 × 2 . For the first order nonlinear Nomoto model, inputs are

3.1.6. Constructure of samples for identification In order to construct samples for parameter estimation according to the characteristics of the proposed identification method, the forwarddifference approximation of Eulers stepping method is employed to discrete the perturbation surge dynamic model and the simplified steering model. The perturbation dynamic model presented in (25) is obtained under the initial condition of state and input with the setting of u = u0, v = 0 , r = 0, δ = 0, and n = n0.

Xu + 2X|u | u |u0 | Xuuu 2T |n | u+ u3 + |n| n 0 n , m Xu m Xu m Xu

Xu ) u0 + Nv )(mx g

(Iz

c4 = 1 +

for which the nonlinear version is described as

u=

h (Xu + 2X|u | u |u0|) , m Xu

[r (k ), r 3 (k ), (k )]T3 × 1, output hn1 hn3 Kh [(1 ) ] . T T T 1×3

is

r(k + 1),

and

parameters

are

3.2. Optimized estimation method Due to the ship dynamic models are linear with respect to the parameters required to be estimated, one modified version of SVM, i.e., LS-SVR with the linear kernel function, is determined as the estimation methods. To tune particular setting for the regularization parameter of LS-SVR, the artificial bee colony algorithm (ABC) is selected because since ABC was proposed by Karaboga in 2005 based on foraging behaviors of bees it has been proven to be a robust and efficient algorithm for solving global optimization problems over continuous space [55]. In ABC system, artificial bees fly around in a multidimensional search space and some (employed and onlooker bees) choose food sources depending on the experience of themselves and their nest

(25)

where Δu = u − u0, Δn = n − n0. Then, the input-output samples can be given as

X=[ u (k ), u3 (k ), n (k )]T3× 1 , Y=[v (k ), |v (k )| v (k ), |r (k )| v (k ), r (k ), |v (k )| r (k ), |r (k )| r (k ), (k )]T7 × 1 , Z=[v (k ), |v (k )| v (k ), |r (k )| v (k ), r (k ), |v (k )| r (k ), |r (k )| r (k ), (k )]T7 × 1 . u(k + 1) = AX, v (k + 1) =BY, r(k + 1) = CZ where the parameter 6

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Fig. 2. Graphical representation of ABC-LSSVR.

The prototype is implemented through a series of files and functions which are limited to important core functionalities. Two classes of data including simulated data and experimental data are stored as the observed data. The simulated data are generated by using a high reality model with predetermined parameters of a full-scale ship, which is implemented directly in MATLAB. The experimental data stored in Excel spreadsheet file are extracted from measurements recorded by sensors. The observed data are divided into two groups, i.e., training data and validation data. The training data are then read through the built-in function xlsread() and transferred to the function of ship dynamic model. Meanwhile, with the application of the function of identification method, the parameters involved in the model are estimated. Before determining the values of these parameters, the validity of the identification results shall be verified through validation process. In this process, the inputs in the validation data, e.g., a time series of commanded rudder angle, is delivered to the identified model to generate the relevant output which is compared with the actual output stemmed from the validation data to get its MSE. If the MSE is not greater than the pre-set number, the identified model is accepted, and the values of parameters will be shown in the command window of MATLAB, otherwise, ‘failed’ will appear.

mates, and adjust their positions. Some (scouts) fly and choose the food sources randomly without using experience. If the nectar amount of a new source is higher than that of the previous one in their memory, they memorize the new position and forget the previous one. Thus, ABC system combines local search methods, carried out by employed and onlooker bees, with global search methods, managed by onlookers and scouts, attempting to balance exploration and exploitation process. The procedure of ABC-LSSVR for identification is briefly introduced as follows with a graphic abstract in Fig. 2. For more details regarding LS-SVR, ABC, and ABC-LSSVR, please refer to our previous work [52]. 3.3. Implementation To evaluate the proposed solution in modelling ship dynamics, a software prototype so-called SIMSD (System Identification-based Modeling of Ship Dynamics) is created based on LS-SVMlab1.8 Toolbox [56]. Such prototype is shown in Fig. 3 where the primary components and interaction between them are presented. The components including observed data, ship dynamic model, identification method, and validation are determined according to the procedure of system identification in identify ship dynamic model. Each component is implemented by using relevant functions and files built in MATLAB. This way leads to a modular design of the implementation of the estimation of the ship dynamic model based on system identification technique. It is additionally convenient to change every modular according to the requirement.

4. Simulation study In application of the previously created implementation, the proposed solution on optimized identification method-based modeling of 7

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Fig. 3. Component diagram of the SIMSD system.

ship dynamics can be evaluated in this section through the simulation study on a large container ship.

4.2. Identified model using ABC-LSSVR After setting the properties of ABC as NP = 20, S = 10, D = 1, x jmin = 10 2 , Limit = 20, T = 30, the best regularization parameters for identifying speed and steering models are calculated effectively, which are listed in Table 2. Meanwhile, by comparing the regression results of the optimized LS-SVR with the training data shown in Fig. 5, we can see that the approximations of LS-SVR agree well with the simulations with almost ignorable MSEs (mean squared errors between prediction and simulation results). Afterwards, the parameters of the simplified dynamic model are estimated by the optimized LS-SVR, which are recorded in Table 3.

4.1. Data processing As suggested by Ljung [57], the input used to generate training maneuvres to supply sufficiently informative data for identification should satisfy the persistence of excitation condition for guaranteeing convergence and ensuring robustness properties of identification method. Ideally, the phase trajectories resulting from such maneuvres are ought to fill some sufficiently large domain in the state space [39]. For instance, the pseudo-random binary sequences (PRBS) are rich enough and desirable choice to guarantee realiable estimation of all parameters involved in the model [34,39]. But PRBS requires very long tests to achieve good loop filling, which is not beneficial for the proposed ABC-LSSVR identification method dealing with a finite set of data. What is more, the desirable maneuvres are in fact not possible to be achieved and not simple to be executed with the standard means of control. From the studies many researchers have done on the identification of ship mathematical models, the standard zigzag maneuvres can be alternatives [34,39,58]. Comparing the 3 DOF maneuvring mathematical model investigated in [39] with our simplified model, we can obviously observe that our model has lower complexity. Therefore, the standard zigzag maneuvres are used for identification in this study. The well-studied model of a large container ship with predefined parameter values extracted from the study in [59] is used to simulate special maneuvers for the estimation of surge and steering model parameters, respectively. Straight lines maneuvers generated by varying propeller shaft speed and keeping rudder angle constant are used to identify the simplified surge model. 10°/10° and 20°/20° zigzag tests are carried out under the condition of varying rudder deflection but constant propeller shaft speed for the steering model identification. These maneuvers are undertook under the same initial conditions of U0 = u0 = 8 m/s, v0 = 0 , r0 = 0, δ0 = 0, ψ0 = 0, n0 = 80 rpm, t = 900 (the total sample time), and h = 0.5 s. Concrete and visualiable illustration of these maneuvers can refer to Table 1 and Fig. 4.

4.3. Validation The second straight line and 20°/20° maneuvers are predicted by using the identified models, respectively. As presented in Fig. 6, the trend of the predicted data is very similar to the original simulation in sway and yaw motion, which to some degree indicates that the identified steering model shows agreement with the ones of the complex nonlinear large container ship. However, the big deviation between the original data and predicted data was observed on the moment when the rudder command is executed. The predicted speed is a little higher than the simulated speed, but it can still illustrate that ABC-LSSVR performs good generalization because the small errors in surge speed predictions are not significant for practical aspects or to affect the applicability of the ABC-LSSVR method. It is worth to be noticed that the errors existing in predictions make few influences on the applicability and effectiveness of the proposed ABC-LSSVR method to identify the simplified ship dynamic model. The aspects for arising these errors in predictions is that the underlying physical characteristics of this container ship and the ignorance of the inter-coupling effect existing between the surge, steering and roll motions. 4.4. Comparison of steering models Nomoto models are widely used to describe ship's heading reaction 8

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Table 1 The scheme of ship maneuvers. Maneuver

Commanded rudder angle (°)

Propeller shaft speed (rpm)

Purpose

Straight line Straight line 10°/10° zigzag 20°/20° zigzag

0 0 ± 10 ± 20

Vary in [120, 160] Vary in [100, 160] 80 80

Identify the surge model Validate the identified surge model Identify the steering model Validate the identified steering model

to the commanded rudder order due to its simplicity and easy understanding. The first order Nomoto models are compared with the proposed nonlinear steering model to present the performance of the nonlinear model. The data extracted from the above 10°/10° zigzag maneuver is also used to identify these Nomoto models. For the comprehensive evaluation of the performance of the proposed identification method on identifying the first order linear Nomoto model, the analytical solution reported in [60,61] and data stemmed from the training zigzag maneuvre are used simultaneously to calculate K and T. From the Table 4 listing the identification results, calculation results and MSEs of predictions of the 20/20 zigzag maneuvre, we can observe that the performance of these two identified Nomoto models is almost the

same on account of the MSE and the estimated maneuvring indices. The calculation results from analytical solution are a little different from the identification results but very close to each other. From the prediction results, it can be found that all predictions have no so expected agreements with the simulated training maneuvre but have very similar trend. Besides, the nonlinear steering model does not show expected better performance than these Nomoto models in terms of MSE. From the practical engineering point of view, this difference makes few influences on deciding which model is to describe yaw motion, but the first order linear/nonlinear Nomoto model is suggested as the better choice for control design because of its simple construction. This difference is probably due to the parameter drift, which is a common issue

Fig. 4. Simulated data for optimization and identification of simplified dynamic models: the top sub-figure presents the surge speed of the first straight line maneuver as described in Table 1; the middle sub-figure donates the heading and rudder angles of the 10°/10° zigzag maneuver; the bottom sub-figure is the sway speed and yaw rate of the 10°/10° zigzag maneuver. 9

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Table 2 The best regularization parameter and MSE. Method

ABC

Best regularization value

MSE

SM

SS

SY

SM (m2/s2)

SS (m2/s2)

SY (°2/s2)

4.3313 × 107

8.9486 × 109

19.9067

4.0346 × 10−6

3.5005 × 10−6

4.3924 × 10−6

Note: SM, surge model; SS, steering model (sway motion); SY, steering model (yaw motion); ABC, ABC-LSSVR.

in the statistical regression analysis, involved in the ABC-LSSVR based identification. In the estimation of ship dynamic models, two types of parameter drift including the drift of linear hydrodynamic coefficients and the drift of the nonlinear hydrodynamic coefficients affect, which can be explained respectively from the dynamic cancellation and statistical regression analysis points of view. As suggested by Luo et al. [58], three kinds of measures containing the model simplification, difference method and additional excitation method can be employed to diminish the parameter drift effectively. Using the correlation analysis, the number of parameters involved in the model is reduced firstly to alleviate the multicollinearity. Afterwards, the samples with respect to the explanatory variables in the simplified model are modified by means of the difference method to change values of all explanatory variables as well as the additional excitation method to alter the number of explanatory variables. Considering the research goal, this study adopts the first measure with comparatively high priority to further simply the previously simplified nonlinear steering model. Unlike the Abkowitz model studied in [58], the nonlinear steering model ignores any surge velocity related coupling terms. So the trial and error incorporated with knowledge instead of correlation analysis will be used to carry out further simplification of the nonlinear steering model (Fig. 7).

Table 3 Estimated parameters by ABC-LSSVR. A

a1 9942

a2 −1.603

a3 85.962

B

b1 0.9743

b2 0.0043

b3 1.5306

b4 −1.6533

b5 3.5656

b6 −140.312

b7 −0.0278

C

c1 −0.0092

c2 0.0002

c3 −0.0010

c4 0.0223

c5 0.0024

c6 0.0001

c7 0.0153

4.5. Special considerations on further simplification of dynamic model To intuitive present the further simplification process, the simplified nonlinear steering model is rewritten into the following form

v = b v v + b|v | v |v| v + b|v | v |r| v + br r + b|v | r |v| r + b|r | r |r| r + b r = c v v + c|v | v |v| v + c|v | v |r| v + cr r + c|v | r |v| r + c|r | r |r| r + c (26) where

bv =

br =

(Iz

Nr ) Yv + ((m

(Iz

Nr ) Y|v | v

(mx g

Xu ) u0 + Nv )(Yr

Yr ) N|v | v

mx g)

,

,

Fig. 5. ABC-LSSVR regression: the upper sub-figure is the surge speed of the first straight line maneuver; the middle one is the sway speed of the 10°/10° zigzag maneuver; the lower one is the yaw rate of the 10°/10° zigzag maneuver. 10

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Fig. 6. Prediction of 20°/20° zigzag maneuver on the basis of the identified ship dynamic models. The upper sub-figure is the surge speed of the second straight line maneuver; the middle one is the sway speed of the 20°/20° zigzag maneuver; the lower one is the yaw rate of the 20°/20° zigzag maneuver. Table 4 Identification and calculated results of the three steering models. Model

K (1/s)

NSM LNM(identified) LNM(calculated) NNM

T (s)

0.0310 0.0579 0.0310

30.2487 26.3158 30.2487

n3 (×10( − 11))

MSE [SY (°2/s2)]

7.0853

0.1196 0.0548 0.0697 0.0548

cr = c|v | v =

c|r | v =

Note: NSM, nonlinear steering model; LNM, the first order linear Nomoto model; NNM, the first order nonlinear Nomoto model.

b|v | v =

(Iz

brv =

b|v | r =

b|r | r =

b =

cv =

(Iz

Nr ) Y|r | v

Nr )(Yr

(mx g

(m

Yr ) N|r | v

Xu ) u 0 )

Nr ) Y|v | r

(mx g

Yr ) N|v | r

(Iz

Nr ) Y|r | r

(mx g

Yr ) N|r | r

(Iz

(Nv

Nr ) Y

(mx g

mx g ) Yv + (m

Yr ) N

,

(mx g

(Iz

c|v | r = c|r | r =

Yr ) Nr

c =

,

,

Xu ) u0 + Nv )

mx g ) Y|v | v

(m

Yv ) N|v | v

(Nv

mx g ) Y|v | v + (m

Yv ) N|v | v

(m

Yv ) Nr

(N v

mx g ) Y|v | r + (m

Yv ) N|v | r

(N v

mx g ) Y|r | r + (m

Yv ) N|r | r

(N v

(Nv

mx g ) Y + (m

,

,

mx g)((m

Yv ) N

Xu ) u 0

Yr )

,

,

,

,

and = (m Yv )(Iz Nr ) (mx g Yr )( mx g Nv ) . Several groups of simplification of the nonlinear steering model are carried out, the corresponding estimated sway and yaw parameters are listed in Tables 5 and 6 where the MSE of predicted 20°/20° zigzag maneuver is also included. The identified values of linear terms such as bv , br, bδ, c v , cr and cδ are physically consistent. The MSE of re-simplified models in the first and fourth row is not obtained, for which the inconsistent estimated parameters should be responsible. By comparing the MES in tables, the combination of sway model in the sixth row of Table 5 and yaw model in the same row of Table 6 is indicated as the best one describing the steering motion. Thus, the suitable simplified dynamic model for the container ship is given as

,

,

Yv )((m

(Nv

,

11

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Fig. 7. Predictions of 20°/20° zigzag maneuver using different identified steering models. Table 5 Sway parameters. Model 1 2 3 4 5 6 7 8 9 10

bv

br

b|v | v

brv

b|v | r

b|r | r

bδ

MSE [SS (m2/s2)]

−0.0419 −0.0400

−3.3219 −3.5520

−10.102

−0.8184

−704.84 −105.78

−0.0481 −0.0476 −0.0532 −0.0025

−3.7079 −3.7649 −3.6720

0.0264 −0.0280 −0.0338 0.0264

−0.0769 −0.0542 −0.0568 −0.0770 −0.0637 −0.0637 −0.0606 −0.1153 −0.0610 −0.1179

NaN 0.0604 0.0495 NaN 0.0472 0.0472 0.0411 0.0413 0.0462 0.0730

−0.0094

−0.0183 −0.0184

−3.6746

−10.101 −0.7618

−1.2368 −0.8215

−704.62

0.9208

−0.6068

Table 6 Yaw parameters. Model 1 2 3 4 5 6 7 8 9 10

cv

cr

c|v | v

crv

c|v | r

c|r | r

cδ

MSE [SS (°2/s2)]

−0.0007 −0.0007

−0.0799 −0.0738

−0.0026

0.0015

0.0004 2.4079

−0.0006 −0.0005 −0.0004

−0.0706 −0.0691 −0.0720 −0.0436

−0.0300 0.0007 0.0008 −0.0016

0.0300 0.0026 0.0027 0.0023 0.0029 0.0029 0.0028 0.0023 0.0028 0.0022

NaN 0.0688 0.0646 NaN 0.0169 0.0162 0.0684 0.1346 0.0407 0.0447

−0.0004

u

−0.4861

−0.0335

=

1.0059u

4

=

The study undertook in this part are additional tests on verifying if the further simplified ship dynamic model obtained through simulation study is also useful for describing the dynamics of a real unmanned surface vessel, so-called I-Nav-II vessel. The optimized LS-SVR based on ABC is used to identify the model of I-Nav-II vessel by employing the experimental data stemmed from a set of full-scale trails.

3.7649r

+ 0.9208 |v| r r=

0.0005v

−0.0285

5. Experimental study

|n| n

1.0059 when u0 = 8m / s ) 0.0476v

17.1446

0.0069

Xu + X|u | u |u0 | m Xu

v=

0.0198 −0.1546

0.6847

3.321 × 10 5u3

+ 0.1072 × 10 (

0.1267 0.0280

0.064

0.0691r

0.0285 |v| r + 0.0029

5.1. Data acquisition and processing

The prediction results of 20°/20° zigzag maneuver with the comparison of the first order Nomoto model in yaw motion are depicted in Fig. 8

The tests were carried out on the I-Nav-II vessel in relatively calm water where is the East Lake in China. The vessel is presented in Fig. 9, 12

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Fig. 8. Predictions of 20°/20° zigzag maneuver using the further simplified steering model.

position, yaw rate, heading angle and actual rudder angle can be measured by a global positioning system (R93T GPS), an inertial measurement unit (Mit-G-700 IMU), a compass (HCM356B compass) and a rudder indicator. The RPM of the propeller is impossible to be measured. Consequently, the surge model of the I-Nav-II vessel is unable to be determined. The available data are extracted from a set of zigzag maneuvers carried out from a steady forward speed at around 3 m/s. The median filter is utilized to eliminate disturbances and noise corrupted in the measured data. After the filtering, the data is restored with the interval of 1 s. The velocities containing surge speed and sway speed are unavailable but calculated by taking the derivatives of the measured positions. 142 samples extracted from 10°/10° zigzag maneuver are regarded as the training data to identify the simplified steering model of the I-Nav-II vessel. The training data including yaw rate, actual rudder angle, heading angle and sway speed are illustrated in Fig. 10. The rest experimental data stemmed from 20°/20° and 25°/ 25° zigzag maneuvers are treated as validation data which are then used to validate the identified model.

Fig. 9. The I-Nav-II vessel.

5.2. Identification results and validation

which is powered by two batteries with the maximum voltage of 48 V. The particulars of this vessel are the length of 4.15 m, the breath of 1.6 m, the draught varying from 0.3 to 0.5 m, and the rudder deflection set at ± 25. The data obtained is less desirable than the ones acquired in the previous section. Two reasons can be explained for this. Firstly, the tests were carried out in relatively calm water, but there still noticeable environmental disturbances induced by wind. Secondly, the measurement devices used to record experimental data are limited. Only the

After processing the proposed modeling approach, the identification results for the steering model of the I-Nav-II vessel are obtained. The model expressed with estimated parameters is given as

v= r=

0.5706v 0.1386v

1.1529r + 1.4889 |v| r 0.0775 0.5113r 1.4883 |v| r + 0.2839

Then the identified steering model is validated by comparing experimental maneuvers with predictions of 20°/20° and 25°/25° zigzag 13

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Fig. 10. Experimental data of 10°/10° zigzag maneuver on the I-Nav-II vessel.

Fig. 11. Predictions of 20°/20° zigzag maneuver.

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Fig. 12. Predictions of 25°/25° zigzag maneuver.

Fig. 13. Errors of 20°/20° zigzag maneuver. 15

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Fig. 14. Errors of 25°/25° zigzag maneuver.

and experimental data conducted by a real ship, devices are needed to be mounted on the ship to collect benign experimental data used to verify and validate the simplified surge model. Besides, full-scale trails should be carried out in a calmer environment where the disturbances induced by wave, current and wind are ignorable so that the tests on proposed identification method and simplified ship dynamic model would be more effective. (2) For LS-SVR based identification method, this study applies the linear kernel function due to the ship dynamic model is linear with respect to the parameters, the future point worthy of attention can be on investigating the performance of other kernel functions for the identification of more complex model. (3) The application of either offline identification or online identification has existed in marine engineering. Thus, the extensive study on performing online identification by using the modified LS-SVR in conjunction with ABC could be contemplated, which would be no easy task. Noticeably, realtime data acquisition and data preprocessing are certainly necessitated.

maneuvers respectively shown in Figs. 11 and 12 . The trend of the experimental simulation and prediction fits quite well. Although the error can be observed in the simulation as shown in Figs. 13 and 14 , it attributes negligible errors in estimating the parameters of the steering model. The identified steering model is still ideal to describe steering motions for the I-Nav-II vessel. 6. Conclusions A model with relatively low complexity and high accuracy is desirable for describing characteristics of ship dynamics, which is attenuating the difficulty in identifying the model at the same time increasing the flexibility in simulating different types of ships’ dynamics. This study makes great attempts to derive a simplistic model feasible for describing dynamics of different types of ships. The vectorial model is selected as the fundamental model for modification and simplification due to the effects of ship hull, rudder, propeller and their interferences are completely expressed in vector form which eases the local changes and model simulation and is intuitive for analyzing properties of a ship using passivity and stability analysis. The simplistic 3 DOF model with eight terms is tested with the use of simulated data stemmed from a container ship and ABC-LSSVR identification method, through which the ABC-LSSVR method is proved to be a effective identification method but the nonlinear steering model does not preform so well as Nomoto models in terms of MSE. The differences between the nonlinear model and Nomoto models reasons to the influence of parameter drift, which is mitigated by further simplifying the nonlinear model to be with five terms. The measure is verified effectively through the simulation study and experimental study which demonstrate that the further simplified model is not only applicable for accurately capturing ship dynamics but also useful for describing dynamics of different types’ ships. To well enhance more contributions based on present achievements, the forthcoming points deserved to be studied are presented as follows. (1) Even though the further simplified steering model has been validated through the simulated data generated by a large container ship

Acknowledgements The work described in this paper was supported by the Ministry of Science and Culture of Lower Saxony for funding the Graduate school of Safe Automation of Maritime Systems (SAMS); the National Science Foundation of China (NSFC) through Grant No. 51579204; the National Key R&D Program of China through Grant Nos. 2018YFC1407405 and 2018YFC0213904. References [1] C. Denker, Assessing the spatio-temporal fitness of information supply and demand on an adaptive ship bridge, International Conference on Knowledge Engineering and Knowledge Management (2014) 185–192 Springer. [2] IMO, Msc85/annex 20 – Strategy for the Development and Implementation of eNavigation. [3] A. Hahn, V. Goll“ucke, C. Buschmann, S. Schweigert, Virtual test bed for maritime safety assessment, Zeszyty Naukowe Akademii Morskiej w Szczecinie (2015). [4] A. Hahn, Test bed for safety assessment of new e-navigation systems, Int. J. e-

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