Time-Delay-Based Sliding Mode Tracking Control for Cooperative Dual Marine Lifting System Subject to Sea Wave Disturbances

Abstract

Dual marine lifting systems are complicated, fully actuated mechatronics systems with multi-input and multi-output capabilities. The anti-swing cooperative lifting control of dual marine lifting systems with dual ships’ sway, heave, and roll motions is still open. The uncertainty regarding system parameters makes the task of achieving stable performance more challenging. To adjust both the attitude and position of large distributed-mass payloads to their target positions, this paper presents a time-delay-based sliding mode-tracking controller for cooperative dual marine lifting systems impacted by sea wave disturbances. Firstly, a dynamic model of a dual marine lifting system is established by using Lagrange’s method. Then, a kinematic coupling-based cooperative trajectory planning strategy is proposed by analyzing the coupling relationship between the dual marine lifting system and dual ship motion. After that, an improved sliding mode tracking controller is proposed by using time-delay estimation technology, which estimates unknown system parameters online. The finite-time convergence of full-state variables is rigorously proven. Finally, the simulation results verify the designed controller in terms of anti-swing control performance. The hardware experiments revealed that the proposed controller significantly reduces the actuator positioning errors by 83.33% compared with existing control methods.

1. Introduction

Dual marine lifting systems (DMLSs) are commonly utilized for the transportation and installation of large-scale offshore platforms. As shown in Figure 1, large offshore platforms require lifting by two lifting booms that can move linearly along the horizontal mechanisms. The DMLS is installed on two ships to carry out offshore lifting operations. It is evident that the offshore platform’s swing is caused by inappropriate control commands for DMLS under sea wave disturbances. The undesirable swing of the offshore platform decreases the safety and productivity of operational tasks [1]. Therefore, it is crucial to design an efficient control approach for DMLSs subject to sea wave disturbances.

As a kind of multiple-input multiple-output fully actuated electromechanical system, the state variables of DMLSs (such as horizontal mechanism displacement and lifting boom displacement) are strongly nonlinear and highly coupled. Large offshore platforms cannot be considered as the mass point. The distributed-mass payloads make dynamic models of DMLSs complicated. DMLSs collaborate to move the offshore platform toward the destination. During the lifting process, it is crucial to maintain the balance of the offshore platform, despite the ship motion caused by sea wave disturbances [2]. Cooperative lifting for heavy and large offshore platforms is a highly complex control task.

The efficient and safe operation of marine lifting systems relies heavily on advanced control strategies. Over the past few decades, significant advancements have been made in the development of control approaches for single marine lifting systems, particularly for offshore cranes. These control methods fall into two main categories: linear control and nonlinear control [3]. Linear control strategies include linear quadratic regulator control [4], improved proportional–integral–derivative (PID) control [5,6,7], and linearized model-based control [8]. Nonlinear control strategies include trajectory planning [9,10,11,12], trajectory tracking control [13,14,15], active heave compensation control [16,17,18], fault-tolerant control [19], sliding mode control [20,21,22,23,24,25], output feedback regulation control [26,27,28], neural network control [29,30,31], and adaptive control [32,33,34]. Owing to its inability to adjust payload attitude and limited carrying capacity, a single marine lifting system struggles to accomplish high-precision heavy-lifting assignments. The above existing control methods for single marine lifting systems cannot be directly applied for DMLSs.

Compared with the single marine lifting system, the cooperative control of the DMLS is far more challenging due to the complicated system dynamics and geometric constraints. By considering the ship’s roll motion, a dynamics model for a fully actuated DMLS was established using Lagrange’s method [35]. An energy analysis-based (EAB) control method has been presented for a dual offshore crane lifting system, which considers the actuator saturation constraints [36]. An adaptive dynamic programming-based sliding mode controller was proposed for dual offshore crane lifting systems with ships’ roll motions caused by sea wave disturbances [37]. A neural network-based adaptive hierarchical sliding mode controller was investigated for application in dual offshore crane lifting systems [38]. The neural networks were designed to counteract the unmatched disturbance in ship roll movements caused by sea waves. In our previous work, a heave compensation control method based on the incremental model predictive approach was proposed for a DMLS with ship heave motion [39]. However, these existing control methods only consider the distances between the two ships as a constant. Owing to the ship sway motion caused by sea wave disturbances, the distance between the dual ships is usually a time-varying state parameter.

Based on our analysis of the existing research work, there are a number of open problems for DLMSs that need to be addressed:

• (1). None of the existing control methods for DMLSs simultaneously take into account the dual ships’ roll, heave, and sway motions caused by sea wave disturbances;

• (2). Most of the control methods need the exact system parameters of the DMLS. In practice, the system parameters (e.g., the mass and geometric parameters of both the actuators and payload, as well as friction coefficients of actuators) are difficult to measure accurately in different lifting tasks. System parameter uncertainty may increase the positioning errors of actuators.

To address the aforementioned issues, a sliding mode tracking control approach based on time delay estimation is proposed for cooperative DMLSs in this paper. Firstly, a DMLS dynamic model with dual ships’ sway, heave, and roll motions is established by using Lagrange’s method. Then, a cooperative trajectory planning method is designed based on the kinematic coupling relationship between the lifting booms and the horizontal mechanisms. After that, an improved sliding mode tracking control strategy by using the time-delay estimation technology is presented for DLMSs. Ultimately, the simulation and experiment validate the efficiency and stability of the proposed sliding mode tracking control approach, surpassing existing state-of-the-art control techniques.

In particular, the primary contributions of this research work are outlined via the following aspects:

• (1). Compared with the existing state-of-the-art control methods, the proposed controller considers more complicated dynamic behaviors and more degrees of freedom in dual ships’ sway, heave, and roll motions;

• (2). This paper presents a cooperative trajectory planning method based on the kinematic coupling relationship of the actuators. This innovative trajectory planning approach enables the steady adjustment of both the attitude and position of large distributed-mass payloads to their precise target positions, even amidst sea wave disturbances;

• (3). An adaptive sliding mode tracking controller with consideration of parameter uncertainty is realized by using the time-delay estimation technique, which effectively eliminates the positioning errors of the dual horizontal mechanisms and dual-lifting booms.

The organization of the remainder of this paper is as follows: Section 2 provides the DMLS dynamic model and control objectives of this paper. Section 3 presents the kinematic coupling-based trajectory planning method and the improved sliding mode tracking controller based on time delay estimation for the DMLS. Section 4 and Section 5 demonstrate simulations and experiments, respectively. Finally, Section 6 summarizes this paper.

2. Problem Statement and Preliminaries

2.1. Dynamics Modelling

An illustration of the DMLS model within the O–XY coordinate is shown in Figure 2. The parameters of the DMLS are provided in Table 1. The dynamic model equations of the DMLS were obtained by using Lagrange’s method as follows:

(1)$\begin{array}{l}\left(m_1+m_2+{\displaystyle \frac{1}{4}}{m}_p+{\displaystyle \frac{h^2}{d^2}}{m}_p\right){\ddot{a}}_1+\left(-{\displaystyle \frac{1}{4}}{m}_p\cos \left({\alpha }_1+{\alpha }_2\right)+m_p\sin \left({\alpha }_1+{\alpha }_2\right){\displaystyle \frac{h}{d}}+m_p\sin \left({\alpha }_1+{\alpha }_2\right){\displaystyle \frac{h^2}{d^2}}\right){\ddot{a}}_2\\ +\left(-{\displaystyle \frac{1}{4}}{m}_p\sin \left({\alpha }_1+{\alpha }_2\right)+m_p\cos \left({\alpha }_1+{\alpha }_2\right){\displaystyle \frac{h}{d}}-m_p\sin \left({\alpha }_1+{\alpha }_2\right){\displaystyle \frac{h^2}{d^2}}\right){\ddot{b}}_2\\ +\left(m_p{\displaystyle \frac{h^2}{d^2}}\left(l_b+l_c\right)+{\displaystyle \frac{1}{2}}{m}_2{b}_1+{\displaystyle \frac{1}{4}}{m}_p{b}_1+m_p{b}_1{\displaystyle \frac{h^2}{d^2}}+{\displaystyle \frac{1}{2}}{m}_2{l}_b+\left(m_1+m_2\right)l_c+{\displaystyle \frac{1}{4}}{m}_p\left(l_b+l_c\right)\right){\ddot{\alpha }}_1\\ +\left(\begin{array}{l}{m}_p{\displaystyle \frac{h^2}{d^2}}\left(l_b+l_c\right)\cos \left({\alpha }_1+{\alpha }_2\right)-{\displaystyle \frac{1}{4}}{m}_p\left(l_b+l_c\right)\cos \left({\alpha }_1+{\alpha }_2\right)+{\displaystyle \frac{1}{4}}{m}_p{l}_a\sin \left({\alpha }_1+{\alpha }_2\right)-{\displaystyle \frac{h^2}{d^2}}{m}_p{l}_a\sin \left({\alpha }_1+{\alpha }_2\right)\\ -{\displaystyle \frac{1}{4}}{m}_p\cos \left({\alpha }_1+{\alpha }_2\right)b_2+{\displaystyle \frac{h^2}{d^2}}{m}_p\cos \left({\alpha }_1+{\alpha }_2\right)b_2+{\displaystyle \frac{1}{4}}{m}_p\sin \left({\alpha }_1+{\alpha }_2\right)a_2-{\displaystyle \frac{h^2}{d^2}}{m}_p\sin \left({\alpha }_1+{\alpha }_2\right)a_2\\ +m_p{\displaystyle \frac{h}{d}}{l}_a\cos \left({\alpha }_1+{\alpha }_2\right)+m_p{\displaystyle \frac{h}{d}}\left(l_b+l_c\right)\sin \left({\alpha }_1+{\alpha }_2\right)+m_p{\displaystyle \frac{h}{d}}\cos \left({\alpha }_1+{\alpha }_2\right)a_2+m_p{\displaystyle \frac{h}{d}}\sin \left({\alpha }_1+{\alpha }_2\right)b_2\end{array}\right){\ddot{\alpha }}_2\\ -\left(\left(m_1+m_2\right)+{\displaystyle \frac{1}{4}}{m}_p+m_p{\displaystyle \frac{h^2}{d^2}}\right)\sin {\alpha }_1{\ddot{c}}_1-\left({\displaystyle \frac{1}{4}}{m}_p\sin {\alpha }_1+m_p{\displaystyle \frac{h}{d}}\cos {\alpha }_1+m_p{\displaystyle \frac{h^2}{d^2}}\sin {\alpha }_1\right){\ddot{c}}_2\\ -\left(\left(m_1+m_2\right)+{\displaystyle \frac{1}{4}}{m}_p+m_p{\displaystyle \frac{h^2}{d^2}}\right)\cos {\alpha }_1{\ddot{s}}_1-\left(-{\displaystyle \frac{1}{4}}{m}_p\cos {\alpha }_1+m_p{\displaystyle \frac{h}{d}}\sin {\alpha }_1+m_p{\displaystyle \frac{h^2}{d^2}}\cos {\alpha }_1\right){\ddot{s}}_2\\ +\left(m_2+{\displaystyle \frac{1}{2}}{m}_p+2m_p{\displaystyle \frac{h^2}{d^2}}\right){\dot{b}}_1{\dot{\alpha }}_1+\left({\displaystyle \frac{1}{2}}{m}_p\sin \left({\alpha }_1+{\alpha }_2\right)+2m_p{\displaystyle \frac{h}{d}}\cos \left({\alpha }_1+{\alpha }_2\right)-2m_p{\displaystyle \frac{h^2}{d^2}}\sin \left({\alpha }_1+{\alpha }_2\right)\right){\dot{a}}_2{\dot{\alpha }}_2\\ +\left(-{\displaystyle \frac{1}{2}}{m}_p\cos \left({\alpha }_1+{\alpha }_2\right)+2m_p{\displaystyle \frac{h}{d}}\sin \left({\alpha }_1+{\alpha }_2\right)+2m_p{\displaystyle \frac{h^2}{d^2}}\cos \left({\alpha }_1+{\alpha }_2\right)\right){\dot{b}}_2{\dot{\alpha }}_2\\ -\left(m_p{\displaystyle \frac{h^2}{d^2}}{l}_a+\left(m_1+m_2\right)a_1+{\displaystyle \frac{1}{4}}{m}_p{a}_1+m_p{\displaystyle \frac{h^2}{d^2}}{a}_1+{\displaystyle \frac{1}{2}}{m}_1{l}_a+m_2{l}_a+{\displaystyle \frac{1}{4}}{m}_p{l}_a\right){\dot{\alpha }}_1^2\\ +\left(\begin{array}{l}{\displaystyle \frac{1}{4}}{m}_p{l}_a\cos \left({\alpha }_1+{\alpha }_2\right)-m_p{l}_a{\displaystyle \frac{h^2}{d^2}}\cos \left({\alpha }_1+{\alpha }_2\right)+{\displaystyle \frac{1}{4}}{m}_p\left(l_b+l_c\right)\sin \left({\alpha }_1+{\alpha }_2\right)-m_p{\displaystyle \frac{h^2}{d^2}}\left(l_b+l_c\right)\sin \left({\alpha }_1+{\alpha }_2\right)\\ {\displaystyle \frac{1}{4}}{m}_p\cos \left({\alpha }_1+{\alpha }_2\right)a_2-m_p{\displaystyle \frac{h^2}{d^2}}\cos \left({\alpha }_1+{\alpha }_2\right)a_2+{\displaystyle \frac{1}{4}}{m}_p\sin \left({\alpha }_1+{\alpha }_2\right)b_2-m_p{\displaystyle \frac{h^2}{d^2}}\sin \left({\alpha }_1+{\alpha }_2\right)b_2\\ m_p{\displaystyle \frac{h}{d}}\left(l_b+l_c\right)\cos \left({\alpha }_1+{\alpha }_2\right)-m_p{\displaystyle \frac{h}{d}}{l}_a\sin \left({\alpha }_1+{\alpha }_2\right)+m_p{\displaystyle \frac{h}{d}}\cos \left({\alpha }_1+{\alpha }_2\right)b_2-m_p{\displaystyle \frac{h}{d}}\sin \left({\alpha }_1+{\alpha }_2\right)a_2\end{array}\right){\dot{\alpha }}_2^2\\ -\left(m_1+m_2+{\displaystyle \frac{1}{2}}{m}_p\right)g\sin {\alpha }_1-m_p{\displaystyle \frac{h}{d}}g\cos {\alpha }_1=F_1\end{array},$

(2)$\begin{array}{l}\left({\displaystyle \frac{1}{4}}{m}_2+{\displaystyle \frac{1}{4}}{m}_p+h^2{m}_p\right){\ddot{b}}_1-\left({\displaystyle \frac{1}{4}}{m}_p\sin \left({\alpha }_1+{\alpha }_2\right)+m_p\cos \left({\alpha }_1+{\alpha }_2\right){\displaystyle \frac{h}{d}}-m_p\sin \left({\alpha }_1+{\alpha }_2\right){\displaystyle \frac{h^2}{d^2}}\right){\ddot{a}}_2\\ -\left(-{\displaystyle \frac{1}{4}}{m}_p\cos \left({\alpha }_1+{\alpha }_2\right)+m_p\sin \left({\alpha }_1+{\alpha }_2\right){\displaystyle \frac{h}{d}}+m_p\sin \left({\alpha }_1+{\alpha }_2\right){\displaystyle \frac{h^2}{d^2}}\right){\ddot{b}}_2\\ -\left(m_p{\displaystyle \frac{h^2}{d^2}}{l}_a+{\displaystyle \frac{1}{2}}{m}_2{a}_1+{\displaystyle \frac{1}{4}}{m}_p{a}_1+m_p{a}_1{\displaystyle \frac{h^2}{d^2}}+{\displaystyle \frac{1}{2}}{m}_2{l}_a+{\displaystyle \frac{1}{4}}{m}_p{l}_a\right){\ddot{\alpha }}_1\\ -\left(\begin{array}{l}{\displaystyle \frac{1}{4}}{m}_p{l}_a\cos \left({\alpha }_1+{\alpha }_2\right)-m_p{\displaystyle \frac{h^2}{d^2}}{l}_a\cos \left({\alpha }_1+{\alpha }_2\right)+{\displaystyle \frac{1}{4}}{m}_p\left(l_b+l_c\right)\sin \left({\alpha }_1+{\alpha }_2\right)-{\displaystyle \frac{h^2}{d^2}}{m}_p\left(l_b+l_c\right)\sin \left({\alpha }_1+{\alpha }_2\right)\\ +{\displaystyle \frac{1}{4}}{m}_p\cos \left({\alpha }_1+{\alpha }_2\right)a_2-{\displaystyle \frac{h^2}{d^2}}{m}_p\cos \left({\alpha }_1+{\alpha }_2\right)a_2+{\displaystyle \frac{1}{4}}{m}_p\sin \left({\alpha }_1+{\alpha }_2\right)b_2-{\displaystyle \frac{h^2}{d^2}}{m}_p\sin \left({\alpha }_1+{\alpha }_2\right)b_2\\ +m_p{\displaystyle \frac{h}{d}}\left(l_b+l_c\right)\cos \left({\alpha }_1+{\alpha }_2\right)-m_p{\displaystyle \frac{h}{d}}{l}_a\sin \left({\alpha }_1+{\alpha }_2\right)+m_p{\displaystyle \frac{h}{d}}\cos \left({\alpha }_1+{\alpha }_2\right)b_2-m_p{\displaystyle \frac{h}{d}}\sin \left({\alpha }_1+{\alpha }_2\right)a_2\end{array}\right){\ddot{\alpha }}_2\\ +\left({\displaystyle \frac{1}{2}}{m}_2+{\displaystyle \frac{1}{4}}{m}_p+m_p{\displaystyle \frac{h^2}{d^2}}\right)\cos {\alpha }_1{\ddot{c}}_1-\left(-{\displaystyle \frac{1}{4}}{m}_p\cos {\alpha }_1+m_p{\displaystyle \frac{h}{d}}\sin {\alpha }_1+m_p{\displaystyle \frac{h^2}{d^2}}\cos {\alpha }_1\right){\ddot{c}}_2\\ -\left({\displaystyle \frac{1}{2}}{m}_2+{\displaystyle \frac{1}{4}}{m}_p+m_p{\displaystyle \frac{h^2}{d^2}}\right)\sin {\alpha }_1{\ddot{s}}_1+\left({\displaystyle \frac{1}{4}}{m}_p\sin {\alpha }_1+m_p{\displaystyle \frac{h}{d}}\cos {\alpha }_1-m_p{\displaystyle \frac{h^2}{d^2}}\sin {\alpha }_1\right){\ddot{s}}_2\\ -\left(m_2+{\displaystyle \frac{1}{2}}{m}_p+2m_p{\displaystyle \frac{h^2}{d^2}}\right){\dot{a}}_1{\dot{\alpha }}_1+\left(-{\displaystyle \frac{1}{2}}{m}_p\cos \left({\alpha }_1+{\alpha }_2\right)+2m_p{\displaystyle \frac{h}{d}}\sin \left({\alpha }_1+{\alpha }_2\right)+2m_p{\displaystyle \frac{h^2}{d^2}}\cos \left({\alpha }_1+{\alpha }_2\right)\right){\dot{a}}_2{\dot{\alpha }}_2\\ +\left(-{\displaystyle \frac{1}{2}}{m}_p\sin \left({\alpha }_1+{\alpha }_2\right)+2m_p{\displaystyle \frac{h}{d}}\cos \left({\alpha }_1+{\alpha }_2\right)-2m_p{\displaystyle \frac{h^2}{d^2}}\sin \left({\alpha }_1+{\alpha }_2\right)\right){\dot{b}}_2{\dot{\alpha }}_2\\ -\left(m_p{\displaystyle \frac{h^2}{d^2}}\left(l_b+l_c\right)+{\displaystyle \frac{1}{4}}\left(m_p+m_2\right)b_1+m_p{\displaystyle \frac{h^2}{d^2}}{b}_1+{\displaystyle \frac{1}{4}}{m}_2\left(l_b+2l_c\right)+{\displaystyle \frac{1}{4}}{m}_p\left(l_b+l_c\right)\right){\dot{\alpha }}_1^2\\ +\left(\begin{array}{l}{m}_p{l}_b{\displaystyle \frac{h^2}{d^2}}\cos \left({\alpha }_1+{\alpha }_2\right)-{\displaystyle \frac{1}{4}}{m}_p\left(l_b+l_c\right)\cos \left({\alpha }_1+{\alpha }_2\right)+m_p{\displaystyle \frac{h^2}{d^2}}{l}_c\cos \left({\alpha }_1+{\alpha }_2\right)\\ +{\displaystyle \frac{1}{4}}{m}_p{l}_a\sin \left({\alpha }_1+{\alpha }_2\right)-m_p{l}_a{\displaystyle \frac{h^2}{d^2}}\sin \left({\alpha }_1+{\alpha }_2\right)-{\displaystyle \frac{1}{4}}{m}_p\cos \left({\alpha }_1+{\alpha }_2\right)b_2+m_p{\displaystyle \frac{h^2}{d^2}}\cos \left({\alpha }_1+{\alpha }_2\right)b_2\\ +{\displaystyle \frac{1}{4}}{m}_p\sin \left({\alpha }_1+{\alpha }_2\right)a_2-m_p{\displaystyle \frac{h}{d}}\sin \left({\alpha }_1+{\alpha }_2\right)a_2+m_p{\displaystyle \frac{h}{d}}{l}_a\cos \left({\alpha }_1+{\alpha }_2\right)+m_p{\displaystyle \frac{h}{d}}\left(l_b+l_c\right)\sin \left({\alpha }_1+{\alpha }_2\right)\\ +m_p{\displaystyle \frac{h}{d}}\cos \left({\alpha }_1+{\alpha }_2\right)a_2+m_p{\displaystyle \frac{h}{d}}\sin \left({\alpha }_1+{\alpha }_2\right)b_2\end{array}\right){\dot{\alpha }}_2^2\\ +\left({\displaystyle \frac{1}{2}}{m}_2+{\displaystyle \frac{1}{2}}{m}_p\right)g\cos {\alpha }_1-m_p{\displaystyle \frac{h}{d}}g\sin {\alpha }_1=F_2\end{array},$

(3)$\begin{array}{l}\left(-{\displaystyle \frac{1}{4}}{m}_p\cos \left({\alpha }_1+{\alpha }_2\right)+m_p\sin \left({\alpha }_1+{\alpha }_2\right){\displaystyle \frac{h}{d}}+m_p\cos \left({\alpha }_1+{\alpha }_2\right){\displaystyle \frac{h^2}{d^2}}\right){\ddot{a}}_1\\ +\left(-{\displaystyle \frac{1}{4}}{m}_p\sin \left({\alpha }_1+{\alpha }_2\right)+m_p\cos \left({\alpha }_1+{\alpha }_2\right){\displaystyle \frac{h}{d}}-m_p\sin \left({\alpha }_1+{\alpha }_2\right){\displaystyle \frac{h^2}{d^2}}\right){\ddot{b}}_1+\left(m_1+m_2+{\displaystyle \frac{1}{4}}{m}_p+{\displaystyle \frac{h^2}{d^2}}{m}_p\right){\ddot{a}}_2\\ +\left(\begin{array}{l}{m}_p{\displaystyle \frac{h^2}{d^2}}{l}_b\cos \left({\alpha }_1+{\alpha }_2\right)-{\displaystyle \frac{1}{4}}{m}_p\left(l_b+l_c\right)\cos \left({\alpha }_1+{\alpha }_2\right)+{\displaystyle \frac{h^2}{d^2}}{m}_p{l}_c\cos \left({\alpha }_1+{\alpha }_2\right)+{\displaystyle \frac{1}{4}}{m}_p{l}_a\sin \left({\alpha }_1+{\alpha }_2\right)\\ -{\displaystyle \frac{h^2}{d^2}}{m}_p{l}_a\sin \left({\alpha }_1+{\alpha }_2\right)-{\displaystyle \frac{1}{4}}{m}_p\cos \left({\alpha }_1+{\alpha }_2\right)b_1+{\displaystyle \frac{h^2}{d^2}}{m}_p\cos \left({\alpha }_1+{\alpha }_2\right)b_1+{\displaystyle \frac{1}{4}}{m}_p\sin \left({\alpha }_1+{\alpha }_2\right)a_1\\ -{\displaystyle \frac{h^2}{d^2}}{m}_p\sin \left({\alpha }_1+{\alpha }_2\right)a_1+m_p{\displaystyle \frac{h}{d}}\left(l_b+l_c\right)\sin \left({\alpha }_1+{\alpha }_2\right)+m_p{\displaystyle \frac{h}{d}}{l}_a\cos \left({\alpha }_1+{\alpha }_2\right)\\ +m_p{\displaystyle \frac{h}{d}}\cos \left({\alpha }_1+{\alpha }_2\right)a_1+m_p{\displaystyle \frac{h}{d}}\sin \left({\alpha }_1+{\alpha }_2\right)b_1\end{array}\right){\ddot{\alpha }}_1\\ +\left(m_p{\displaystyle \frac{h^2}{d^2}}\left(l_b+l_c\right)+{\displaystyle \frac{1}{2}}{m}_2{b}_2+{\displaystyle \frac{1}{4}}{m}_p{b}_2+m_p{b}_2{\displaystyle \frac{h^2}{d^2}}+{\displaystyle \frac{1}{2}}{m}_2{l}_b+\left(m_1+m_2\right)l_c+{\displaystyle \frac{1}{4}}{m}_p\left(l_b+l_c\right)\right){\ddot{\alpha }}_2\\ -\left({\displaystyle \frac{1}{4}}{m}_p\sin {\alpha }_2+m_p{\displaystyle \frac{h}{d}}\cos {\alpha }_2+m_p{\displaystyle \frac{h^2}{d^2}}\sin {\alpha }_2\right){\ddot{c}}_1-\left(\left(m_1+m_2\right)+{\displaystyle \frac{1}{4}}{m}_p+m_p{\displaystyle \frac{h^2}{d^2}}\right)\sin {\alpha }_2{\ddot{c}}_2\\ -\left(-{\displaystyle \frac{1}{4}}{m}_p\cos {\alpha }_2+m_p{\displaystyle \frac{h}{d}}\sin {\alpha }_2+m_p{\displaystyle \frac{h^2}{d^2}}\cos {\alpha }_2\right){\ddot{s}}_1-\left(\left(m_1+m_2\right)+{\displaystyle \frac{1}{4}}{m}_p+m_p{\displaystyle \frac{h^2}{d^2}}\right)\cos {\alpha }_2{\ddot{s}}_2\\ +\left({\displaystyle \frac{1}{2}}{m}_p\sin \left({\alpha }_1+{\alpha }_2\right)+2m_p{\displaystyle \frac{h}{d}}\cos \left({\alpha }_1+{\alpha }_2\right)-2m_p{\displaystyle \frac{h^2}{d^2}}\sin \left({\alpha }_1+{\alpha }_2\right)\right){\dot{a}}_1{\dot{\alpha }}_1+\left(m_2+{\displaystyle \frac{1}{2}}{m}_p+2m_p{\displaystyle \frac{h^2}{d^2}}\right){\dot{b}}_2{\dot{\alpha }}_2\\ +\left(-{\displaystyle \frac{1}{2}}{m}_p\cos \left({\alpha }_1+{\alpha }_2\right)+2m_p{\displaystyle \frac{h}{d}}\sin \left({\alpha }_1+{\alpha }_2\right)+2m_p{\displaystyle \frac{h^2}{d^2}}\cos \left({\alpha }_1+{\alpha }_2\right)\right){\dot{b}}_1{\dot{\alpha }}_1\end{array}$

$\begin{array}{l}+\left(\begin{array}{l}{\displaystyle \frac{1}{4}}{m}_p{l}_a\cos \left({\alpha }_1+{\alpha }_2\right)-m_p{l}_a{\displaystyle \frac{h^2}{d^2}}\cos \left({\alpha }_1+{\alpha }_2\right)+{\displaystyle \frac{1}{4}}{m}_p\left(l_b+l_c\right)\sin \left({\alpha }_1+{\alpha }_2\right)-m_p{\displaystyle \frac{h^2}{d^2}}\left(l_b+l_c\right)\sin \left({\alpha }_1+{\alpha }_2\right)\\ +{\displaystyle \frac{1}{4}}{m}_p\cos \left({\alpha }_1+{\alpha }_2\right)a_1-m_p{\displaystyle \frac{h^2}{d^2}}\cos \left({\alpha }_1+{\alpha }_2\right)a_1+{\displaystyle \frac{1}{4}}{m}_p\sin \left({\alpha }_1+{\alpha }_2\right)b_1-m_p{\displaystyle \frac{h^2}{d^2}}\sin \left({\alpha }_1+{\alpha }_2\right)b_1\\ +m_p{\displaystyle \frac{h}{d}}\left(l_b+l_c\right)\cos \left({\alpha }_1+{\alpha }_2\right)-m_p{\displaystyle \frac{h}{d}}{l}_a\sin \left({\alpha }_1+{\alpha }_2\right)+m_p{\displaystyle \frac{h}{d}}\cos \left({\alpha }_1+{\alpha }_2\right)b_1-m_p{\displaystyle \frac{h}{d}}\sin \left({\alpha }_1+{\alpha }_2\right)a_1\end{array}\right){\dot{\alpha }}_1^2\\ -\left(m_p{\displaystyle \frac{h^2}{d^2}}{l}_a+\left(m_1+m_2\right)a_2+{\displaystyle \frac{1}{4}}{m}_p{a}_2+m_p{\displaystyle \frac{h^2}{d^2}}{a}_2+{\displaystyle \frac{1}{2}}{m}_1{l}_a+m_2{l}_a+{\displaystyle \frac{1}{4}}{m}_p{l}_a\right){\dot{\alpha }}_2^2\\ -\left(m_1+m_2+{\displaystyle \frac{1}{2}}{m}_p\right)g\sin {\alpha }_2+m_p{\displaystyle \frac{h}{d}}g\cos {\alpha }_2=F_3\end{array},$

(4)$\begin{array}{l}-\left({\displaystyle \frac{1}{4}}{m}_p\sin \left({\alpha }_1+{\alpha }_2\right)+m_p\cos \left({\alpha }_1+{\alpha }_2\right){\displaystyle \frac{h}{d}}-m_p\sin \left({\alpha }_1+{\alpha }_2\right){\displaystyle \frac{h^2}{d^2}}\right){\ddot{a}}_1\\ -\left(-{\displaystyle \frac{1}{4}}{m}_p\cos \left({\alpha }_1+{\alpha }_2\right)+m_p\sin \left({\alpha }_1+{\alpha }_2\right){\displaystyle \frac{h}{d}}+m_p\cos \left({\alpha }_1+{\alpha }_2\right){\displaystyle \frac{h^2}{d^2}}\right){\ddot{b}}_1+\left({\displaystyle \frac{1}{4}}\left(m_2+m_p\right)+{\displaystyle \frac{h^2}{d^2}}{m}_p\right){\ddot{b}}_2\\ -\left(\begin{array}{l}{\displaystyle \frac{1}{4}}{m}_p{l}_a\cos \left({\alpha }_1+{\alpha }_2\right)m_p-{\displaystyle \frac{h^2}{d^2}}{l}_a\cos \left({\alpha }_1+{\alpha }_2\right)+{\displaystyle \frac{1}{4}}{m}_p\left(l_b+l_c\right)\sin \left({\alpha }_1+{\alpha }_2\right)+m_p{\displaystyle \frac{h}{d}}\cos \left({\alpha }_1+{\alpha }_2\right)b_1\\ -{\displaystyle \frac{h^2}{d^2}}{m}_p\left(l_b+l_c\right)\sin \left({\alpha }_1+{\alpha }_2\right)+{\displaystyle \frac{1}{4}}{m}_p\cos \left({\alpha }_1+{\alpha }_2\right)a_1-{\displaystyle \frac{h^2}{d^2}}{m}_p\cos \left({\alpha }_1+{\alpha }_2\right)a_1+{\displaystyle \frac{1}{4}}{m}_p\sin \left({\alpha }_1+{\alpha }_2\right)b_1\\ -{\displaystyle \frac{h^2}{d^2}}{m}_p{l}_a\sin \left({\alpha }_1+{\alpha }_2\right)b_1+m_p{\displaystyle \frac{h}{d}}\left(l_b+l_c\right)\cos \left({\alpha }_1+{\alpha }_2\right)-m_p{\displaystyle \frac{h}{d}}{l}_a\sin \left({\alpha }_1+{\alpha }_2\right)-m_p{\displaystyle \frac{h}{d}}\sin \left({\alpha }_1+{\alpha }_2\right)a_1\end{array}\right){\ddot{\alpha }}_1\\ -\left(m_p{\displaystyle \frac{h^2}{d^2}}{l}_a+{\displaystyle \frac{1}{2}}{m}_2{a}_2+{\displaystyle \frac{1}{4}}{m}_p{a}_2+m_p{a}_2{\displaystyle \frac{h^2}{d^2}}+{\displaystyle \frac{1}{2}}{m}_2{l}_a+{\displaystyle \frac{1}{4}}{m}_p{l}_a\right){\ddot{\alpha }}_2\\ -\left(-{\displaystyle \frac{1}{4}}{m}_p\cos {\alpha }_2+m_p{\displaystyle \frac{h}{d}}\sin {\alpha }_2+m_p{\displaystyle \frac{h^2}{d^2}}\cos {\alpha }_2\right){\ddot{c}}_1+\left({\displaystyle \frac{1}{2}}{m}_2+{\displaystyle \frac{1}{4}}{m}_p+m_p{\displaystyle \frac{h^2}{d^2}}\right)\cos {\alpha }_2{\ddot{c}}_2\\ +\left({\displaystyle \frac{1}{4}}{m}_p\sin {\alpha }_2+m_p{\displaystyle \frac{h}{d}}\cos {\alpha }_2-m_p{\displaystyle \frac{h^2}{d^2}}\sin {\alpha }_2\right){\ddot{s}}_1-\left({\displaystyle \frac{1}{2}}{m}_2+{\displaystyle \frac{1}{4}}{m}_p+m_p{\displaystyle \frac{h^2}{d^2}}\right)\sin {\alpha }_2{\ddot{s}}_2\\ +\left(-{\displaystyle \frac{1}{2}}{m}_p\cos \left({\alpha }_1+{\alpha }_2\right)+2m_p{\displaystyle \frac{h}{d}}\sin \left({\alpha }_1+{\alpha }_2\right)+2m_p{\displaystyle \frac{h^2}{d^2}}\cos \left({\alpha }_1+{\alpha }_2\right)\right){\dot{a}}_1{\dot{\alpha }}_1\\ +\left(-{\displaystyle \frac{1}{2}}{m}_p\sin \left({\alpha }_1+{\alpha }_2\right)+2m_p{\displaystyle \frac{h}{d}}\cos \left({\alpha }_1+{\alpha }_2\right)-2m_p{\displaystyle \frac{h^2}{d^2}}\sin \left({\alpha }_1+{\alpha }_2\right)\right){\dot{b}}_1{\dot{\alpha }}_1-\left(m_2+{\displaystyle \frac{1}{2}}{m}_p+2m_p{\displaystyle \frac{h^2}{d^2}}\right){\dot{a}}_2{\dot{\alpha }}_2\\ +\left(\begin{array}{l}{m}_p{l}_b{\displaystyle \frac{h^2}{d^2}}\cos \left({\alpha }_1+{\alpha }_2\right)-{\displaystyle \frac{1}{4}}{m}_p\left(l_b+l_c\right)\cos \left({\alpha }_1+{\alpha }_2\right)+m_p{\displaystyle \frac{h^2}{d^2}}{l}_c\cos \left({\alpha }_1+{\alpha }_2\right)\\ +{\displaystyle \frac{1}{4}}{m}_p{l}_a\sin \left({\alpha }_1+{\alpha }_2\right)-m_p{\displaystyle \frac{h^2}{d^2}}\sin \left({\alpha }_1+{\alpha }_2\right)-{\displaystyle \frac{1}{4}}{m}_p\cos \left({\alpha }_1+{\alpha }_2\right)b_1+m_p{\displaystyle \frac{h^2}{d^2}}\cos \left({\alpha }_1+{\alpha }_2\right)b_1\\ +{\displaystyle \frac{1}{4}}{m}_p{\displaystyle \frac{h}{d}}\sin \left({\alpha }_1+{\alpha }_2\right)a_1-m_p{\displaystyle \frac{h}{d}}\sin \left({\alpha }_1+{\alpha }_2\right)a_1+m_p{\displaystyle \frac{h}{d}}{l}_a\cos \left({\alpha }_1+{\alpha }_2\right)+m_p{\displaystyle \frac{h}{d}}\left(l_b+l_c\right)\sin \left({\alpha }_1+{\alpha }_2\right)\\ +m_p{\displaystyle \frac{h}{d}}\cos \left({\alpha }_1+{\alpha }_2\right)a_1+m_p{\displaystyle \frac{h}{d}}\sin \left({\alpha }_1+{\alpha }_2\right)b_1\end{array}\right){\dot{\alpha }}_1^2\\ -\left(m_p{\displaystyle \frac{h^2}{d^2}}\left(l_b+l_c\right)+{\displaystyle \frac{1}{4}}{m}_2{b}_2+m_p{\displaystyle \frac{h^2}{d^2}}{b}_2+{\displaystyle \frac{1}{4}}{m}_2{l}_b+{\displaystyle \frac{1}{2}}{m}_2{l}_c+{\displaystyle \frac{1}{4}}{m}_p\left(l_b+l_c\right)\right){\dot{\alpha }}_2^2\\ +\left({\displaystyle \frac{1}{2}}{m}_2+{\displaystyle \frac{1}{2}}{m}_p\right)g\cos {\alpha }_2-m_p{\displaystyle \frac{h}{d}}g\sin {\alpha }_2=F_4\end{array},$

where $F_1\triangleq F_{a1}-F_{fa1},F_2\triangleq F_{b1}-F_{fb1},F_3\triangleq F_{a2}-F_{fa2},F_4\triangleq F_{b2}-F_{fb2}$. $F_{fa1},F_{fb1},F_{fa2}$ and $F_{fb2}$ are presented as [40,41,42]:

(5)$\left\{\begin{array}{c}{F}_{fa1}=f_1\tanh \left({\dot{a}}_1/\epsilon \right)-k_{r1}\left|{\dot{a}}_1\right|{\dot{a}}_1\\ F_{fb1}=f_2\tanh \left({\dot{b}}_1/\epsilon \right)-k_{r2}\left|{\dot{b}}_1\right|{\dot{b}}_1\\ F_{fa2}=f_3\tanh \left({\dot{a}}_2/\epsilon \right)-k_{r3}\left|{\dot{a}}_2\right|{\dot{a}}_2\\ F_{fb2}=f_4\tanh \left({\dot{b}}_2/\epsilon \right)-k_{r4}\left|{\dot{b}}_2\right|{\dot{b}}_2\end{array}\right.,$

where $f_1,f_2,f_3,f_3,\epsilon ,k_{r1},k_{r2},k_{r3},k_{r4}$ are coefficients of the friction forces.

In order to complete the controller design, dynamic model Equations (1)–(4) are rewritten as matrix-vector forms:

(6)$M\left(q\right)\ddot{q}+C\left(q,\dot{q}\right)\dot{q}+G\left(q\right)=U+D,$

where $q,M\left(q\right),C\left(q,\dot{q}\right),G\left(q\right),U,D$ are defined in Appendix A.

2.2. Control Objectives

The control objectives of the DMLS are described as follows.

• (1). The barycenter of payload $P\left(x_p,y_p\right)$ is lifted to the target position, in the sense that

  (7)$x_p\to 0,y_p\to y_{pd},$

  where $\left(0,y_{pd}\right)$ is the target barycenter of payload in the O-XY coordinate.

• (2). The payload swing angle $\theta$ is suppressed to zero, in the sense that

  (8)$\theta \to 0.$

• (3). The continuous external disturbance caused by the sea waves is eliminated for the DMLS.

3. Main Results

This section provides the kinematic coupling-based trajectory planning method and time-delay-based sliding mode tracking controller for the DMLS. The stability of the closed-loop system is demonstrated by utilizing the Lyapunov method.

3.1. Kinematic Coupling Based Trajectory Planning

This subsection presents the kinematic coupling-based trajectory planning method. In Figure 2, the top position of lifting boom 1 $B_1\left(x_{B1},y_{B1}\right)$ and top position of lifting boom 2 $B_2\left(x_{B2},y_{B2}\right)$ are calculated as follows

(9)$\left\{\begin{array}{c}{x}_{B1}=-s_1+\left(a_1+l_a\right)\cos {\alpha }_1+\left(l_c+{\displaystyle \frac{1}{2}}{b}_1+{\displaystyle \frac{1}{2}}{l}_b\right)\sin {\alpha }_1+{\displaystyle \frac{1}{2}}\left(b_1+l_b\right)\sin {\alpha }_1\\ y_{B1}=c_1-\left(a_1+l_a\right)\sin {\alpha }_1+\left(l_c+{\displaystyle \frac{1}{2}}{b}_1+{\displaystyle \frac{1}{2}}{l}_b\right)\cos {\alpha }_1+{\displaystyle \frac{1}{2}}\left(b_1+l_b\right)\cos {\alpha }_1\end{array}\right.,$

(10)$\left\{\begin{array}{c}{x}_{B2}=s_2-\left(a_2+l_a\right)\cos {\alpha }_2-\left(l_c+{\displaystyle \frac{1}{2}}{b}_2+{\displaystyle \frac{1}{2}}{l}_b\right)\sin {\alpha }_2-{\displaystyle \frac{1}{2}}\left(b_2+l_b\right)\sin {\alpha }_2\\ y_{B2}=c_2-\left(a_1+l_a\right)\sin {\alpha }_2+\left(l_c+{\displaystyle \frac{1}{2}}{b}_2+{\displaystyle \frac{1}{2}}{l}_b\right)\cos {\alpha }_2+{\displaystyle \frac{1}{2}}\left(b_2+l_b\right)\cos {\alpha }_2\end{array}\right..$

The center of mass of the payload $P\left(x_p,y_p\right)$ is calculated as follows

(11)$\left\{\begin{array}{c}{x}_p={\displaystyle \frac{x_{B1}+x_{B2}}{2}}-h\sin \theta \\ y_p={\displaystyle \frac{y_{B1}+y_{B2}}{2}}+h\cos \theta \end{array}\right..$

The full state variables have the following kinematic coupling relationship in the horizontal and vertical directions

(12)$\left\{\begin{array}{c}{x}_{B2}-x_{B1}=d\cos \theta \\ y_{B2}-y_{B1}=d\sin \theta \end{array}\right..$

To ensure the stability of the lifted payload, it is imperative that the payload’s swing angle remains at zero under sea wave disturbances. Moreover, the payload is needed to lift to its target position. Therefore, it can be determined that

(13)$\left\{\begin{array}{c}{x}_{B2}-x_{B1}=d\cos \theta \\ y_{B2}-y_{B1}=d\sin \theta \end{array}\right..$

By solving (11)–(13) together, the kinematic coupling-based trajectories can be obtained as follows

(14)$a_{1r}=s_1\cos {\alpha }_1+c_1\sin {\alpha }_1-l_a-{\displaystyle \frac{d}{2}}\cos {\alpha }_1-\left(y_{bd}-h\right)\sin {\alpha }_1,$

(15)$a_{2r}=s_2\cos {\alpha }_2+c_2\sin {\alpha }_2-l_a-{\displaystyle \frac{d}{2}}\cos {\alpha }_2-\left(y_{bd}-h\right)\sin {\alpha }_2,$

(16)$b_{1r}=s_1\sin {\alpha }_1+c_1\cos {\alpha }_1-\left(l_c+l_b\right)-{\displaystyle \frac{d}{2}}\sin {\alpha }_1+\left(y_{bd}-h\right)\cos {\alpha }_1,$

(17)$b_{2r}=s_2\sin {\alpha }_2+c_2\cos {\alpha }_2-\left(l_c+l_b\right)-{\displaystyle \frac{d}{2}}\sin {\alpha }_2+\left(y_{bd}-h\right)\cos {\alpha }_2.$

By applying the designed trajectories in (14)–(17), the large distributed-mass payloads can be adjusted to their precise target positions, even amidst sea wave disturbances.

3.2. Time-Delay-Based Sliding Mode Tracking Controller Designing

This subsection provides the improved sliding mode tracking controller based on time-delay estimation technology. Figure 3 presents the block diagram of the proposed improved sliding mode tracking controller.

To facilitate the controller design, the dynamic of DMLS (6) is rearranged into the following form as

(18)$\overline{M}\ddot{q}+N=U,$

where the gain matrix $\overline{M}\in {ℝ}^{4\times 4}$ and vector $N\in {ℝ}^{4\times 1}$ are expressed as follows

$\overline{M}\ddot{q}+N=U,$

$N=\left[\begin{array}{c}\left(m_{11}-{\overline{m}}_{11}\right){\ddot{a}}_1+m_{13}{\ddot{a}}_2+m_{14}{\ddot{b}}_2+c_{12}{\dot{b}}_1+c_{13}{\dot{a}}_2+c_{14}{\dot{b}}_2+g_1-d_1\\ \left(m_{22}-{\overline{m}}_{22}\right){\ddot{b}}_1+m_{23}{\ddot{a}}_2+m_{24}{\ddot{b}}_2+c_{21}{\dot{a}}_1+c_{23}{\dot{a}}_2+c_{24}{\dot{b}}_2+g_2-d_2\\ m_{13}{\ddot{a}}_1+m_{23}{\ddot{b}}_1+\left(m_{33}-{\overline{m}}_{33}\right){\ddot{a}}_2+c_{31}{\dot{a}}_1+c_{32}{\dot{b}}_1+c_{34}{\dot{b}}_2+g_3-d_3\\ m_{14}{\ddot{a}}_1+m_{24}{\ddot{b}}_1+\left(m_{44}-{\overline{m}}_{44}\right){\ddot{b}}_2+c_{41}{\dot{a}}_1+c_{42}{\dot{b}}_1+c_{43}{\dot{a}}_2+g_4-d_4\end{array}\right].$

A group of error signals of the actuators are expressed as

(19)$\begin{array}{cc}e& ={\left[\begin{array}{cccc}{e}_{a1}& e_{b1}& e_{a2}& e_{b2}\end{array}\right]}^T\hfill \\ & ={\left[\begin{array}{cccc}{a}_1-a_{1r}& b_1-b_{1r}& a_2-a_{2r}& b_2-b_{2r}\end{array}\right]}^T\hfill \end{array}.$

A sliding mode surface vector is defined as

(20)$s={\left[\begin{array}{cccc}{s}_{a1}& s_{b1}& s_{a2}& s_{b2}\end{array}\right]}^T=\dot{e}+\Lambda e,$

where gain matrix $\Lambda \in {ℝ}^{4\times 4}$ is expressed as $\Lambda =\mathrm{diag}\left\{\begin{array}{cccc}{k}_{a1}& k_{b1}& k_{a2}& k_{b2}\end{array}\right\}$. By taking the time derivative of (20), it can be obtained as

(21)$\dot{s}=\ddot{q}-q_a,$

where vector $q_a$ is expressed as $q_a={\left[\begin{array}{cccc}{\ddot{a}}_{1r}-k_{a1}{\dot{e}}_{a1}& {\ddot{b}}_{1r}-k_{b1}{\dot{e}}_{b1}& {\ddot{a}}_{2r}-k_{a2}{\dot{e}}_{a2}& {\ddot{b}}_{2r}-k_{b2}{\dot{e}}_{b2}\end{array}\right]}^T$.

Based on (18)–(21), an improved sliding mode tracking controller is designed for the DMLS as follows

(22)$U=\widehat{N}+\overline{M}\left[q_a-K\mathrm{sgn}\left(s\right)-Ps\right],$

where $K=\mathrm{diag}\left\{\begin{array}{cccc}{k}_1& k_2& k_3& k_4\end{array}\right\}$ and $P=\mathrm{diag}\left\{\begin{array}{cccc}{p}_1& p_2& p_3& p_4\end{array}\right\}$ are the control gain matrix. $\widehat{N}$ is the online estimation of vector $N$. $\widehat{N}$ can be designed by using the time-delay estimation method as follows

(23)$\widehat{N}=U_{t-L}-\overline{M}{\ddot{q}}_{t-L},$

where $L$ is the time delay. The estimation error of vector $N$ is calculated as

(24)$\epsilon =\widehat{N}-N={\left[\begin{array}{cccc}{\epsilon }_{a1}& {\epsilon }_{b1}& {\epsilon }_{a2}& {\epsilon }_{b2}\end{array}\right]}^T.$

Remark 1.

There are some comprehensive guidelines available for selecting the appropriate control gains. ${\overline{m}}_{11},{\overline{m}}_{22},{\overline{m}}_{33},{\overline{m}}_{44}$ are the positive gains, which should satisfy the stability condition $‖I-M^{-1}\overline{M}‖<1$ [43,44,45,46]. $k_{a1},k_{b1},k_{a2},k_{b2}$ are employed to adjust for the positioning errors of horizontal mechanisms and the lifting booms. The selection criteria for the traditional PID controller’s gains help to find the values of $k_{a1},k_{b1},k_{a2},k_{b2}$. The control gains $k_1,k_2,k_3,k_4$ are chosen by carefully considering $k_1>{\overline{\sigma }}_1,k_2>{\overline{\sigma }}_2,k_3>{\overline{\sigma }}_3,k_4>{\overline{\sigma }}_4$. ${\overline{\sigma }}_1,{\overline{\sigma }}_2,{\overline{\sigma }}_3,{\overline{\sigma }}_4$ are the upper bounds of ${\sigma }_1,{\sigma }_2,{\sigma }_3,{\sigma }_4$, which are expressed as ${\sigma }_1={\overline{m}}_{11}^{-1}{\epsilon }_{a1},{\sigma }_2={\overline{m}}_{22}^{-1}{\epsilon }_{b1},{\sigma }_3={\overline{m}}_{33}^{-1}{\epsilon }_{a2},{\sigma }_4={\overline{m}}_{44}^{-1}{\epsilon }_{b2}$. $p_1,p_2,p_3,p_4$ are used to adjust the convergence of the designed sliding mode surface $s$. It is not overly difficult to choose the values of $p_1,p_2,p_3,p_4$ by means of trial and error. $L$ represents the small time delay, which is usually selected as the sampling time period.

3.3. Closed-Loop System Stability Analysis

This subsection provides the closed-loop system stability analysis for the designed time-delay-based sliding mode tracking controller.

Theorem 1.

The full states of DLMS converge to their desired values in a finite time $t_f$, one can see that

(25)$\underset{t\to t_f}{\lim }{\left[\begin{array}{ccccc}{a}_1& b_1& a_2& b_2& \theta \end{array}\right]}^T={\left[\begin{array}{ccccc}{a}_{1r}& b_{1r}& a_{2r}& b_{2r}& 0\end{array}\right]}^T.$

Proof. 

Firstly, a Lyapunov function candidate $V\left(t\right)$ is defined as

(26)$V\left(t\right)={\displaystyle \frac{1}{2}}{s}^{T}s.$

Substituting (18) and (22)–(24) into (21), it can be obtained that

(27)$\dot{s}={\overline{M}}^{-1}\epsilon -K\mathrm{sgn}\left(s\right)-Ps=\left[\begin{array}{c}{\sigma }_1-k_1\mathrm{sgn}\left(s_{a1}\right)-p_1{s}_{a1}\\ {\sigma }_2-k_2\mathrm{sgn}\left(s_{b1}\right)-p_2{s}_{b1}\\ {\sigma }_3-k_3\mathrm{sgn}\left(s_{a2}\right)-p_3{s}_{a2}\\ {\sigma }_4-k_4\mathrm{sgn}\left(s_{b2}\right)-p_4{s}_{b2}\end{array}\right].$

Then, by differentiating (26) with respect to time and utilizing (27), it can be obtained that

(28)$\begin{array}{cc}\dot{V}\left(t\right)& =s^T{\overline{M}}^{-1}\epsilon -s^{T}K\mathrm{sgn}\left(s\right)-s^{T}Ps\hfill \\ & \le -\left(k_1-\left|{\sigma }_1\right|\right)\left|s_{a1}\right|-\left(k_2-\left|{\sigma }_2\right|\right)\left|s_{b1}\right|-\left(k_3-\left|{\sigma }_3\right|\right)\left|s_{a2}\right|-\left(k_4-\left|{\sigma }_4\right|\right)\left|s_{b2}\right|-s^{T}Ps\hfill \\ & \le -s^{T}Ps\le 0\hfill \end{array}.$

Based on $V\ge 0$ and $\dot{V}\le 0$, it is further obtained that

(29)$V\left(t\right)\in L_{\infty }\Rightarrow s_{a1},s_{b1},s_{a2},s_{b2}\in L_{\infty }.$

Substituting (26) into (28), it can be obtained that

(30)$\dot{V}\left(t\right)\le -2\min \left\{p_1,p_2,p_{3,}{p}_4\right\}V.$

By using (30) and Bellman–Gronwall inequality [47], it can be obtained that

(31)$V\left(t\right)\le V\left(0\right)e^{-2\min \left\{p_1,p_2,p_{3,}{p}_4\right\}t},\forall t\ge 0.$

Based on (31), it can further be obtained that

(32)$\underset{t\to \infty }{\lim }V\left(t\right)=0\Rightarrow \underset{t\to \infty }{\lim }s=0\Rightarrow \underset{t\to \infty }{\lim }{s}_{a1}=0,\underset{t\to \infty }{\lim }{s}_{b1}=0,\underset{t\to \infty }{\lim }{s}_{a2}=0,\underset{t\to \infty }{\lim }{s}_{b2}=0,$

which demonstrates that the designed sliding mode surface $s$ undergoes exponential convergence to 0.

Next, the proof of the finite-time convergence for the designed sliding mode surface $s$ needs to be completed. From (28), it can be obtained that

(33)$\begin{array}{cc}\dot{V}\left(t\right)& \le -\left(k_1-\left|{\sigma }_1\right|\right)\left|s_{a1}\right|-\left(k_2-\left|{\sigma }_2\right|\right)\left|s_{b1}\right|-\left(k_3-\left|{\sigma }_3\right|\right)\left|s_{a2}\right|-\left(k_4-\left|{\sigma }_4\right|\right)\left|s_{b2}\right|\hfill \\ & \le \varphi \sqrt{s_{a1}^2+s_{b1}^2+s_{a2}^2+s_{b2}^2}\hfill \end{array},$

where $\varphi =\min \left\{\left(k_1-\left|{\sigma }_1\right|\right),\left(k_2-\left|{\sigma }_2\right|\right),\left(k_3-\left|{\sigma }_3\right|\right),\left(k_4-\left|{\sigma }_4\right|\right)\right\}$.

Combined with (26) and (33), it can be further obtained that

(34)$\dot{V}\left(t\right)\le -\varphi \sqrt{2V\left(t\right)}\Rightarrow \sqrt{V\left(t\right)}-\sqrt{V\left(0\right)}\le {\displaystyle \frac{-\varphi }{\sqrt{2}}}t.$

When $s$ converges to 0 at time $t_f$, it can obtain $V\left(t_f\right)=0$. Therefore, it can be obtained from (34) that

(35)$t_f\le {\displaystyle \frac{\sqrt{2}}{\varphi }}\sqrt{V\left(0\right)}\in L_{\infty }.$

Combined with (28) and (35), it can be obtained that

(36)$\begin{array}{l}0\le V\left(t\right)\le V\left(t_f\right)=0,\mathrm{when}t>t_f\\ \Rightarrow V\left(t\right)=0,\forall t\ge t_f\Rightarrow s={\left[\begin{array}{cccc}{s}_{a1}& s_{b1}& s_{a2}& s_{b2}\end{array}\right]}^T={\left[\begin{array}{cccc}0& 0& 0& 0\end{array}\right]}^T,\forall t\ge t_f.\end{array}.$

By substituting (19) and (20) into (36), the resulting partial differential equations can be obtained

(37)$e={\left[\begin{array}{cccc}{e}_{a1}& e_{b1}& e_{a2}& e_{b2}\end{array}\right]}^T={\left[\begin{array}{cccc}0& 0& 0& 0\end{array}\right]}^T,\mathrm{when}t\ge t_f.$

Based on (19) and (37), it can easily get

(38)$\underset{t\to t_f}{\lim }{\left[\begin{array}{cccc}{a}_1& b_1& a_2& b_2\end{array}\right]}^T={\left[\begin{array}{cccc}{a}_{1r}& b_{1r}& a_{2r}& b_{2r}\end{array}\right]}^T.$

Combined with (12), (14)–(17), and (38), it can be obtained that

(39)$\underset{t\to t_f}{\lim }\theta =0.$

By substituting (38) and (39) into (9)–(11) and (14)–(17), it can be obtained that

(40)$\underset{t\to t_f}{\lim }{\left[\begin{array}{cc}{x}_p& y_p\end{array}\right]}^T={\left[\begin{array}{cc}0& y_d\end{array}\right]}^T.$

Till now, it is concluded that Theorem 1 is proven. □

4. Simulation Verifications

This section provides the simulation verifications of the proposed trajectory planning method and improved sliding mode tracking control method.

The system parameters of the DMLS simulation platform and control gains are given in Table 2. The sway, heave, and roll motions induced by sea wave disturbances in dual ships were calculated utilizing the Marine System Simulator (MSS) toolbox [48]. Figure 4 and Figure 5 show the sea state and the ship motions, respectively.

4.1. Simulation 1: Effectiveness Verification

This simulation compares the proposed controller with the EAB controller [36]. To enhance the ease of comparison, the proposed controller and EAB controller track the same reference trajectories (Ref. Tra.), which are designed in (14)–(17). The target position of the payload is set as P (0, 10).

Figure 6 presents the comparative simulation results of the proposed controller and EAB controller. When using the EAB controller, the payload has a maximum positioning error of 0.372 m in the X direction and 0.433 m in the Y direction. The maximum swing angle of the payload is 4.83 deg. There is residual swing for a long period of time. However, it is observed that the proposed controller drives the horizontal mechanisms and the lifting booms to reach reference trajectories accurately. Under the proposed controller, the positioning errors of the payload in the X and Y directions are only 0.036 m and 0.006 m, respectively. The swing angle of the payload is suppressed and converges to zero when using the proposed controller. Figure 7 depicts the payload position and attitude during the hoisting and transferring processes. Simulation 1’s results demonstrate that the proposed control method performs significantly better in payload positioning and eliminating payload swing compared with the existing EAB controller.

4.2. Simulation 2: Robustness Verification

This simulation carried out three cases of simulations to test the robustness of the proposed control method against different system parameters and different working requirements as follows:

Case 1: Changing the mass of the payload from $m_p=5000\mathrm{kg}$ to $m_p=8000\mathrm{kg}$, while the other system parameters remain the same.

Case 2: Changing the length of the payload from h = 1 m, $d$ = 3 m to h = 1.5 m, $d$ = 4 m, while other system parameters remain the same.

Case 3: Changing working requirements of the DMLS. The payload is first hoisted to $y_{pd}$ = 10 m and then lowered to $y_{pd}$ = 9.5 m.

Figure 8 and Figure 9 present the simulation results for changing payload mass and length, respectively. After changing the payload mass and length, the proposed controller retains its control performance under parameter uncertainty. Figure 8 and Figure 9 demonstrate that the actuator positioning errors and payload swing angles achieved through the proposed method are lesser compared with those attained using the EAB controller. The time-delay estimation technique in the proposed controller provides an effective adaptive solution for dealing with unknown system parameters. The simulation results for Case 1 and Case 2 verify that the proposed controller is robust to changing system parameters.

Figure 10 provides the simulation result of Case 3. Figure 11 shows the payload state under different working requirements. Although the working requirements varied, the proposed method demonstrated a rapid response to sea wave disturbances and effectively eliminated payload swing. The simulation results for Case 3 verified that the proposed control method was robust to different working requirements.

5. Experiment Verifications

Experimental studies were carried out to further validate the control performance of the proposed control method in this section. The marine lifting system experiment platform is shown in Figure 12. The system parameters of the marine lifting system experiment platform and control gains are provided in Table 3. The lifting boom and horizontal mechanism are actuated by two servo motors with encoder 1 and encoder 2. The lifting boom displacement and horizontal mechanism displacement were measured by using the embedded encoders with servo motors. The control command was calculated using a PC with MATLAB R2021a software. The transmission of the control commands and the collection of the encoder data rely on the data-acquisition board. The Stewart platform was used to simulate the ship’s sway, heave, and roll motions caused by sea wave disturbances. The Stewart platform engages in communication with the PC via Ethernet.

Figure 13 presents the hardware experimental results for the proposed controller and EAB controller. The target position of the payload is set as P (0, 1). When using the EAB controller, the payload had a maximum positioning error of 0.036 m in the X direction and 0.026 m in the Y direction. However, it was observed that the proposed controller drove the horizontal mechanisms and the lifting booms to reach reference trajectories accurately. Under the proposed controller, the positioning errors of the payload in the X and Y directions were only 0.006 m and 0.001 m, respectively. The positioning error of the actuator controlled by the proposed control method was smaller than that of the existing EAB controller. The experimental results demonstrate that the proposed control method performed significantly better in payload positioning compared with the existing EAB controller.

6. Conclusions

This article proposes a novel time-delay-based sliding mode tracking controller for DMLSs affected by sea wave disturbances. The main conclusions of this research are summarized as follows.

• (1). For the system modeling, the fully actuated DMLS dynamic model was established by considering dual ships’ sway, heave, and roll motions. The established dynamic model considered complex dynamic behaviors and more degrees of freedom than the existing model;

• (2). For trajectory planning, a cooperative trajectory planning method was designed based on the kinematic coupling relationship of the actuators. Even when sea waves disturb the ships, the designed trajectory planning method ensures that large distributed-mass payloads can accurately reach their intended target positions, both in terms of attitude and position;

• (3). For the controller design, a novel time-delay-based sliding mode tracking controller was proposed for the DMLS. The designed tracking controller based on the time-delay estimation technique had the adaptive scheme of real-time estimation of the unknown system parameters (e.g., the mass and geometric parameters of both the actuators and payload, as well as friction coefficients of actuators);

• (4). In the practical experiment verifications, the superiority of the proposed novel time-delay-based sliding mode tracking controller was confirmed, demonstrating a decrease of at least 83.33% in actuator positioning errors compared with state-of-the-art control methods.

Future work will consider six-degrees-of-freedom ship motion (including roll, heave, sway, pitch, surge, and yaw) in both dynamic analysis and control strategy design for the DMLS.

Author Contributions

Conceptualization, Y.C. and G.L.; methodology, J.L.; software, J.T.; validation, Y.C.; formal analysis, J.L.; investigation, Y.C.; resources, X.M.; data curation, G.L.; writing—original draft preparation, Y.C.; writing—review and editing, G.L. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

Data are contained within the article. No additional data.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1. Lifting operations using DMLS.

Figure 2. Model of the DMLS.

Figure 3. Block diagram of the proposed time-delay-based sliding mode tracking controller.

Figure 4. Waves and corresponding sea state.

Figure 5. Ship motion caused by sea wave disturbances.

Figure 6. Comparative simulation results.

Figure 7. Payload state of the comparative simulation: (a) Proposed controller, (b) EAB controller.

Figure 8. Simulation results with changing payload mass.

Figure 9. Simulation results with changing payload length.

Figure 10. Simulation results with changing working requirements.

Figure 11. Payload state of the robustness experiment: (a) Proposed controller, (b) EAB controller.

Figure 12. Marine lifting system experimental platform.

Figure 13. Experimental results.

Table 1

Parameters of the DMLS.

Parameter	Parameter Definition	Units
$m_1,m_2,m_p$	Horizontal mechanism, lifting boom, and payload masses	kg
$l_a,l_b$	Horizontal mechanism and lifting boom lengths	m
$l_c$	Distance between the center of ship and the horizontal mechanism	m
$a_i(i=1,2)$	Horizontal mechanism $i$ displacement	m
$b_i(i=1,2)$	Lifting boom $i$ displacement	m
$d$	Payload width	m
$h$	Half of the payload height	m
$\theta$	Payload swing angle	deg
${\alpha }_i\left(i=1,2\right)$	Ship $i$ roll angle	deg
$s_i\left(i=1,2\right)$	Ship $i$ sway displacement	m
$c_i\left(i=1,2\right)$	Ship $i$ heave displacement	m
$F_{ai}(i=1,2)$	Horizontal mechanism $i$ actuating force	N
$F_{bi}(i=1,2)$	Lifting boom $i$ actuating force	N
$F_{fai}(i=1,2)$	Horizontal mechanism $i$ friction force	N
$F_{fbi}(i=1,2)$	Lifting boom $i$ friction force	N

Table 2

Parameter values in the simulation.

Parameters	Values	Control Gains	Values
$$m_1$$	6000 kg	$$\overline{M}$$	diag{1000, 500, 1000, 500}
$$m_2$$	2000 kg	$$\Lambda$$	diag{10, 10, 10, 10}
$$m_p$$	5000 kg	$$K$$	diag{5, 5, 5, 5}
$$l_a$$	7.30 m	$$P$$	diag{5, 5, 5, 5}
$$l_b$$	4.00 m	$$L$$	0.005 s
$$l_c$$	2.70 m
$$d$$	3.00 m
$$h$$	1.00 m

Table 3

Parameter values in the experiment.

Parameters	Values	Control Gains	Values
$m_1$	6 kg	$\overline{M}$	diag{1, 0.5, 1, 0.5}
$m_2$	2 kg	$\Lambda$	diag{10, 10, 10, 10}
$m_p$	0.5 kg	$K$	diag{5, 5, 5, 5}
$l_a$	0.73 m	$P$	diag{5, 5, 5, 5}
$l_b$	0.40 m	$L$	0.005 s
$l_c$	0.27 m
$d$	0.03 m
$h$	0.05 m

Appendix A

The expressions of $q,M\left(q\right),C\left(q,\dot{q}\right),G\left(q\right),U,D$ are specified as follows $$M\left(q\right)=\left[\begin{array}{cccc}{m}_{11}& 0& m_{13}& m_{14}\\ 0& m_{22}& m_{23}& m_{24}\\ m_{13}& m_{23}& m_{33}& 0\\ m_{14}& m_{24}& 0& m_{44}\end{array}\right],C\left(q,\dot{q}\right)=\left[\begin{array}{cccc}0& c_{12}& c_{13}& c_{14}\\ c_{21}& 0& c_{23}& c_{24}\\ c_{31}& c_{32}& 0& c_{34}\\ c_{41}& c_{42}& c_{43}& 0\end{array}\right],$$ $$G\left(q\right)={\left[\begin{array}{cccc}{g}_1& g_2& g_3& g_4\end{array}\right]}^T,U={\left[\begin{array}{cccc}{F}_{a1}& F_{b1}& F_{a2}& F_{b2}\end{array}\right]}^T,q={\left[\begin{array}{cccc}{a}_1& b_1& a_2& b_2\end{array}\right]}^T,D={\left[\begin{array}{cccc}{d}_1& d_2& d_3& d_4\end{array}\right]}^T$$ $$\begin{array}{l}{m}_{11}=m_1+m_2+{\displaystyle \frac{1}{4}}{m}_p+{\displaystyle \frac{h^2}{d^2}}{m}_p,m_{13}=-{\displaystyle \frac{1}{4}}{m}_p\cos \left({\alpha }_1+{\alpha }_2\right)+m_p{\displaystyle \frac{h}{d}}\sin \left({\alpha }_1+{\alpha }_2\right)+m_p{\displaystyle \frac{h^2}{d^2}}\cos \left({\alpha }_1+{\alpha }_2\right),\\ m_{14}=-{\displaystyle \frac{1}{4}}{m}_p\sin \left({\alpha }_1+{\alpha }_2\right)-m_p{\displaystyle \frac{h}{d}}\cos \left({\alpha }_1+{\alpha }_2\right)+m_p{\displaystyle \frac{h^2}{d^2}}\sin \left({\alpha }_1+{\alpha }_2\right),m_{22}={\displaystyle \frac{1}{4}}\left(m_2+m_p\right)+m_p{\displaystyle \frac{h^2}{d^2}},\\ m_{23}=-{\displaystyle \frac{1}{4}}{m}_p\sin \left({\alpha }_1+{\alpha }_2\right)-m_p{\displaystyle \frac{h}{d}}\cos \left({\alpha }_1+{\alpha }_2\right)+m_p{\displaystyle \frac{h^2}{d^2}}\sin \left({\alpha }_1+{\alpha }_2\right),m_{33}=m_1+m_2+{\displaystyle \frac{1}{4}}{m}_p+{\displaystyle \frac{h^2}{d^2}}{m}_p,\\ m_{24}={\displaystyle \frac{1}{4}}{m}_p\cos \left({\alpha }_1+{\alpha }_2\right)-m_p{\displaystyle \frac{h}{d}}\sin \left({\alpha }_1+{\alpha }_2\right)-m_p{\displaystyle \frac{h^2}{d^2}}\cos \left({\alpha }_1+{\alpha }_2\right),m_{44}={\displaystyle \frac{1}{4}}\left(m_2+m_p\right)+m_p{\displaystyle \frac{h^2}{d^2}},\end{array}$$ $$\begin{array}{l}{c}_{12}=\left(m_2+{\displaystyle \frac{1}{2}}{m}_p+2m_p{\displaystyle \frac{h^2}{d^2}}\right){\dot{\alpha }}_1,c_{21}=-\left(m_2+{\displaystyle \frac{1}{2}}{m}_p+2m_p{\displaystyle \frac{h^2}{d^2}}\right){\dot{\alpha }}_1,\\ c_{13}=\left[{\displaystyle \frac{1}{2}}{m}_p\sin \left({\alpha }_1+{\alpha }_2\right)+2m_p{\displaystyle \frac{h}{d}}\cos \left({\alpha }_1+{\alpha }_2\right)-2m_p{\displaystyle \frac{h^2}{d^2}}\sin \left({\alpha }_1+{\alpha }_2\right)\right]{\dot{\alpha }}_2,\\ c_{14}=\left[-{\displaystyle \frac{1}{2}}{m}_p\cos \left({\alpha }_1+{\alpha }_2\right)+2m_p{\displaystyle \frac{h}{d}}\sin \left({\alpha }_1+{\alpha }_2\right)+2m_p{\displaystyle \frac{h^2}{d^2}}\sin \left({\alpha }_1+{\alpha }_2\right)\right]{\dot{\alpha }}_2,\\ c_{23}=\left[-{\displaystyle \frac{1}{2}}{m}_p\cos \left({\alpha }_1+{\alpha }_2\right)+2m_p{\displaystyle \frac{h}{d}}\sin \left({\alpha }_1+{\alpha }_2\right)+2m_p{\displaystyle \frac{h^2}{d^2}}\sin \left({\alpha }_1+{\alpha }_2\right)\right]{\dot{\alpha }}_2,\\ c_{24}=-\left[{\displaystyle \frac{1}{2}}{m}_p\sin \left({\alpha }_1+{\alpha }_2\right)+2m_p{\displaystyle \frac{h}{d}}\cos \left({\alpha }_1+{\alpha }_2\right)-2m_p{\displaystyle \frac{h^2}{d^2}}\sin \left({\alpha }_1+{\alpha }_2\right)\right]{\dot{\alpha }}_2,\\ c_{31}=\left[{\displaystyle \frac{1}{2}}{m}_p\sin \left({\alpha }_1+{\alpha }_2\right)+2m_p{\displaystyle \frac{h}{d}}\cos \left({\alpha }_1+{\alpha }_2\right)-2m_p{\displaystyle \frac{h^2}{d^2}}\sin \left({\alpha }_1+{\alpha }_2\right)\right]{\dot{\alpha }}_1,\\ c_{32}=\left[-{\displaystyle \frac{1}{2}}{m}_p\cos \left({\alpha }_1+{\alpha }_2\right)+2m_p{\displaystyle \frac{h}{d}}\sin \left({\alpha }_1+{\alpha }_2\right)+2m_p{\displaystyle \frac{h^2}{d^2}}\sin \left({\alpha }_1+{\alpha }_2\right)\right]{\dot{\alpha }}_1,\\ c_{34}=\left(m_2+{\displaystyle \frac{1}{2}}{m}_p+2m_p{\displaystyle \frac{h^2}{d^2}}\right){\dot{\alpha }}_2,c_{43}=-\left(m_2+{\displaystyle \frac{1}{2}}{m}_p+2m_p{\displaystyle \frac{h^2}{d^2}}\right){\dot{\alpha }}_2,\end{array}$$ $$\begin{array}{l}{c}_{41}=\left[-{\displaystyle \frac{1}{2}}{m}_p\cos \left({\alpha }_1+{\alpha }_2\right)+2m_p{\displaystyle \frac{h}{d}}\sin \left({\alpha }_1+{\alpha }_2\right)+2m_p{\displaystyle \frac{h^2}{d^2}}\sin \left({\alpha }_1+{\alpha }_2\right)\right]{\dot{\alpha }}_1,\\ c_{42}=-\left[{\displaystyle \frac{1}{2}}{m}_p\sin \left({\alpha }_1+{\alpha }_2\right)+2m_p{\displaystyle \frac{h}{d}}\cos \left({\alpha }_1+{\alpha }_2\right)-2m_p{\displaystyle \frac{h^2}{d^2}}\sin \left({\alpha }_1+{\alpha }_2\right)\right]{\dot{\alpha }}_1,\end{array}$$ $$\begin{array}{l}{g}_1=-\left(m_1+m_2+{\displaystyle \frac{1}{2}}{m}_p\right)g\sin {\alpha }_1-m_p{\displaystyle \frac{h}{d}}g\cos {\alpha }_1,\\ g_2=\left({\displaystyle \frac{1}{2}}{m}_2+{\displaystyle \frac{1}{2}}{m}_p\right)g\cos {\alpha }_1-m_p{\displaystyle \frac{h}{d}}g\sin {\alpha }_1,\\ g_3=-\left(m_1+m_2+{\displaystyle \frac{1}{2}}{m}_p\right)g\sin {\alpha }_2+m_p{\displaystyle \frac{h}{d}}g\cos {\alpha }_2,\\ g_4=\left({\displaystyle \frac{1}{2}}{m}_2+{\displaystyle \frac{1}{2}}{m}_p\right)g\cos {\alpha }_2-m_p{\displaystyle \frac{h}{d}}g\sin {\alpha }_2,\end{array}$$ $$\begin{array}{cc}{d}_1& =-F_{fa1}+\left(m_p{\displaystyle \frac{h^2}{d^2}}\left(l_b+l_c\right)+{\displaystyle \frac{1}{2}}{m}_2{b}_1+{\displaystyle \frac{1}{4}}{m}_p{b}_1+m_p{b}_1{\displaystyle \frac{h^2}{d^2}}+{\displaystyle \frac{1}{2}}{m}_2{l}_b+\left(m_1+m_2\right)l_c+{\displaystyle \frac{1}{4}}{m}_p\left(l_b+l_c\right)\right){\ddot{\alpha }}_1\hfill \\ & +\left(\begin{array}{l}{m}_p{\displaystyle \frac{h^2}{d^2}}\left(l_b+l_c\right)\cos \left({\alpha }_1+{\alpha }_2\right)-{\displaystyle \frac{1}{4}}{m}_p\left(l_b+l_c\right)\cos \left({\alpha }_1+{\alpha }_2\right)+{\displaystyle \frac{1}{4}}{m}_p{l}_a\sin \left({\alpha }_1+{\alpha }_2\right)-{\displaystyle \frac{h^2}{d^2}}{m}_p{l}_a\sin \left({\alpha }_1+{\alpha }_2\right)\\ -{\displaystyle \frac{1}{4}}{m}_p\cos \left({\alpha }_1+{\alpha }_2\right)b_2+{\displaystyle \frac{h^2}{d^2}}{m}_p\cos \left({\alpha }_1+{\alpha }_2\right)b_2+{\displaystyle \frac{1}{4}}{m}_p\sin \left({\alpha }_1+{\alpha }_2\right)a_2-{\displaystyle \frac{h^2}{d^2}}{m}_p\sin \left({\alpha }_1+{\alpha }_2\right)a_2\\ +m_p{\displaystyle \frac{h}{d}}{l}_a\cos \left({\alpha }_1+{\alpha }_2\right)+m_p{\displaystyle \frac{h}{d}}\left(l_b+l_c\right)\sin \left({\alpha }_1+{\alpha }_2\right)+m_p{\displaystyle \frac{h}{d}}\cos \left({\alpha }_1+{\alpha }_2\right)a_2+m_p{\displaystyle \frac{h}{d}}\sin \left({\alpha }_1+{\alpha }_2\right)b_2\end{array}\right){\ddot{\alpha }}_2\hfill \\ & -\left(\left(m_1+m_2\right)+{\displaystyle \frac{1}{4}}{m}_p+m_p{\displaystyle \frac{h^2}{d^2}}\right)\sin {\alpha }_1{\ddot{c}}_1-\left({\displaystyle \frac{1}{4}}{m}_p\sin {\alpha }_1+m_p{\displaystyle \frac{h}{d}}\cos {\alpha }_1+m_p{\displaystyle \frac{h^2}{d^2}}\sin {\alpha }_1\right){\ddot{c}}_2\hfill \\ & -\left(\left(m_1+m_2\right)+{\displaystyle \frac{1}{4}}{m}_p+m_p{\displaystyle \frac{h^2}{d^2}}\right)\cos {\alpha }_1{\ddot{s}}_1-\left(-{\displaystyle \frac{1}{4}}{m}_p\cos {\alpha }_1+m_p{\displaystyle \frac{h}{d}}\sin {\alpha }_1+m_p{\displaystyle \frac{h^2}{d^2}}\cos {\alpha }_1\right){\ddot{s}}_2\hfill \\ & -\left(m_p{\displaystyle \frac{h^2}{d^2}}{l}_a+\left(m_1+m_2\right)a_1+{\displaystyle \frac{1}{4}}{m}_p{a}_1+m_p{\displaystyle \frac{h^2}{d^2}}{a}_1+{\displaystyle \frac{1}{2}}{m}_1{l}_a+m_2{l}_a+{\displaystyle \frac{1}{4}}{m}_p{l}_a\right){\dot{\alpha }}_1^2\hfill \\ & +\left(\begin{array}{l}{\displaystyle \frac{1}{4}}{m}_p{l}_a\cos \left({\alpha }_1+{\alpha }_2\right)-m_p{l}_a{\displaystyle \frac{h^2}{d^2}}\cos \left({\alpha }_1+{\alpha }_2\right)+{\displaystyle \frac{1}{4}}{m}_p\left(l_b+l_c\right)\sin \left({\alpha }_1+{\alpha }_2\right)-m_p{\displaystyle \frac{h^2}{d^2}}\left(l_b+l_c\right)\sin \left({\alpha }_1+{\alpha }_2\right)\\ {\displaystyle \frac{1}{4}}{m}_p\cos \left({\alpha }_1+{\alpha }_2\right)a_2-m_p{\displaystyle \frac{h^2}{d^2}}\cos \left({\alpha }_1+{\alpha }_2\right)a_2+{\displaystyle \frac{1}{4}}{m}_p\sin \left({\alpha }_1+{\alpha }_2\right)b_2-m_p{\displaystyle \frac{h^2}{d^2}}\sin \left({\alpha }_1+{\alpha }_2\right)b_2\\ m_p{\displaystyle \frac{h}{d}}\left(l_b+l_c\right)\cos \left({\alpha }_1+{\alpha }_2\right)-m_p{\displaystyle \frac{h}{d}}{l}_a\sin \left({\alpha }_1+{\alpha }_2\right)+m_p{\displaystyle \frac{h}{d}}\cos \left({\alpha }_1+{\alpha }_2\right)b_2-m_p{\displaystyle \frac{h}{d}}\sin \left({\alpha }_1+{\alpha }_2\right)a_2\end{array}\right){\dot{\alpha }}_2^2\hfill \end{array}$$ $$\begin{array}{l}{d}_2=-F_{fb1}-\left(m_p{\displaystyle \frac{h^2}{d^2}}{l}_a+{\displaystyle \frac{1}{2}}{m}_2{a}_1+{\displaystyle \frac{1}{4}}{m}_p{a}_1+m_p{a}_1{\displaystyle \frac{h^2}{d^2}}+{\displaystyle \frac{1}{2}}{m}_2{l}_a+{\displaystyle \frac{1}{4}}{m}_p{l}_a\right){\ddot{\alpha }}_1\\ -\left(\begin{array}{l}{\displaystyle \frac{1}{4}}{m}_p{l}_a\cos \left({\alpha }_1+{\alpha }_2\right)-m_p{\displaystyle \frac{h^2}{d^2}}{l}_a\cos \left({\alpha }_1+{\alpha }_2\right)+{\displaystyle \frac{1}{4}}{m}_p\left(l_b+l_c\right)\sin \left({\alpha }_1+{\alpha }_2\right)-{\displaystyle \frac{h^2}{d^2}}{m}_p\left(l_b+l_c\right)\sin \left({\alpha }_1+{\alpha }_2\right)\\ +{\displaystyle \frac{1}{4}}{m}_p\cos \left({\alpha }_1+{\alpha }_2\right)a_2-{\displaystyle \frac{h^2}{d^2}}{m}_p\cos \left({\alpha }_1+{\alpha }_2\right)a_2+{\displaystyle \frac{1}{4}}{m}_p\sin \left({\alpha }_1+{\alpha }_2\right)b_2-{\displaystyle \frac{h^2}{d^2}}{m}_p\sin \left({\alpha }_1+{\alpha }_2\right)b_2\\ +m_p{\displaystyle \frac{h}{d}}\left(l_b+l_c\right)\cos \left({\alpha }_1+{\alpha }_2\right)-m_p{\displaystyle \frac{h}{d}}{l}_a\sin \left({\alpha }_1+{\alpha }_2\right)+m_p{\displaystyle \frac{h}{d}}\cos \left({\alpha }_1+{\alpha }_2\right)b_2-m_p{\displaystyle \frac{h}{d}}\sin \left({\alpha }_1+{\alpha }_2\right)a_2\end{array}\right){\ddot{\alpha }}_2\\ +\left({\displaystyle \frac{1}{2}}{m}_2+{\displaystyle \frac{1}{4}}{m}_p+m_p{\displaystyle \frac{h^2}{d^2}}\right)\cos {\alpha }_1{\ddot{c}}_1-\left(-{\displaystyle \frac{1}{4}}{m}_p\cos {\alpha }_1+m_p{\displaystyle \frac{h}{d}}\sin {\alpha }_1+m_p{\displaystyle \frac{h^2}{d^2}}\cos {\alpha }_1\right){\ddot{c}}_2\\ -\left({\displaystyle \frac{1}{2}}{m}_2+{\displaystyle \frac{1}{4}}{m}_p+m_p{\displaystyle \frac{h^2}{d^2}}\right)\sin {\alpha }_1{\ddot{s}}_1+\left({\displaystyle \frac{1}{4}}{m}_p\sin {\alpha }_1+m_p{\displaystyle \frac{h}{d}}\cos {\alpha }_1-m_p{\displaystyle \frac{h^2}{d^2}}\sin {\alpha }_1\right){\ddot{s}}_2\\ -\left(m_p{\displaystyle \frac{h^2}{d^2}}\left(l_b+l_c\right)+{\displaystyle \frac{1}{4}}\left(m_p+m_2\right)b_1+m_p{\displaystyle \frac{h^2}{d^2}}{b}_1+{\displaystyle \frac{1}{4}}{m}_2\left(l_b+2l_c\right)+{\displaystyle \frac{1}{4}}{m}_p\left(l_b+l_c\right)\right){\dot{\alpha }}_1^2\\ +\left(\begin{array}{l}{m}_p{l}_b{\displaystyle \frac{h^2}{d^2}}\cos \left({\alpha }_1+{\alpha }_2\right)-{\displaystyle \frac{1}{4}}{m}_p\left(l_b+l_c\right)\cos \left({\alpha }_1+{\alpha }_2\right)+m_p{\displaystyle \frac{h^2}{d^2}}{l}_c\cos \left({\alpha }_1+{\alpha }_2\right)\\ +{\displaystyle \frac{1}{4}}{m}_p{l}_a\sin \left({\alpha }_1+{\alpha }_2\right)-m_p{l}_a{\displaystyle \frac{h^2}{d^2}}\sin \left({\alpha }_1+{\alpha }_2\right)-{\displaystyle \frac{1}{4}}{m}_p\cos \left({\alpha }_1+{\alpha }_2\right)b_2+m_p{\displaystyle \frac{h^2}{d^2}}\cos \left({\alpha }_1+{\alpha }_2\right)b_2\\ +{\displaystyle \frac{1}{4}}{m}_p\sin \left({\alpha }_1+{\alpha }_2\right)a_2-m_p{\displaystyle \frac{h}{d}}\sin \left({\alpha }_1+{\alpha }_2\right)a_2+m_p{\displaystyle \frac{h}{d}}{l}_a\cos \left({\alpha }_1+{\alpha }_2\right)+m_p{\displaystyle \frac{h}{d}}\left(l_b+l_c\right)\sin \left({\alpha }_1+{\alpha }_2\right)\\ +m_p{\displaystyle \frac{h}{d}}\cos \left({\alpha }_1+{\alpha }_2\right)a_2+m_p{\displaystyle \frac{h}{d}}\sin \left({\alpha }_1+{\alpha }_2\right)b_2\end{array}\right){\dot{\alpha }}_2^2\end{array}$$ $$\begin{array}{cc}{d}_3& =-F_{fa2}+\left(m_p{\displaystyle \frac{h^2}{d^2}}\left(l_b+l_c\right)+{\displaystyle \frac{1}{2}}{m}_2{b}_2+{\displaystyle \frac{1}{4}}{m}_p{b}_2+m_p{b}_2{\displaystyle \frac{h^2}{d^2}}+{\displaystyle \frac{1}{2}}{m}_2{l}_b+\left(m_1+m_2\right)l_c+{\displaystyle \frac{1}{4}}{m}_p\left(l_b+l_c\right)\right){\ddot{\alpha }}_2\hfill \\ & +\left(\begin{array}{l}{m}_p{\displaystyle \frac{h^2}{d^2}}{l}_b\cos \left({\alpha }_1+{\alpha }_2\right)-{\displaystyle \frac{1}{4}}{m}_p\left(l_b+l_c\right)\cos \left({\alpha }_1+{\alpha }_2\right)+{\displaystyle \frac{h^2}{d^2}}{m}_p{l}_c\cos \left({\alpha }_1+{\alpha }_2\right)+{\displaystyle \frac{1}{4}}{m}_p{l}_a\sin \left({\alpha }_1+{\alpha }_2\right)\\ -{\displaystyle \frac{h^2}{d^2}}{m}_p{l}_a\sin \left({\alpha }_1+{\alpha }_2\right)-{\displaystyle \frac{1}{4}}{m}_p\cos \left({\alpha }_1+{\alpha }_2\right)b_1+{\displaystyle \frac{h^2}{d^2}}{m}_p\cos \left({\alpha }_1+{\alpha }_2\right)b_1+{\displaystyle \frac{1}{4}}{m}_p\sin \left({\alpha }_1+{\alpha }_2\right)a_1\\ -{\displaystyle \frac{h^2}{d^2}}{m}_p\sin \left({\alpha }_1+{\alpha }_2\right)a_1+m_p{\displaystyle \frac{h}{d}}\left(l_b+l_c\right)\sin \left({\alpha }_1+{\alpha }_2\right)+m_p{\displaystyle \frac{h}{d}}{l}_a\cos \left({\alpha }_1+{\alpha }_2\right)\\ +m_p{\displaystyle \frac{h}{d}}\cos \left({\alpha }_1+{\alpha }_2\right)a_1+m_p{\displaystyle \frac{h}{d}}\sin \left({\alpha }_1+{\alpha }_2\right)b_1\end{array}\right){\ddot{\alpha }}_1\hfill \\ & -\left({\displaystyle \frac{1}{4}}{m}_p\sin {\alpha }_2+m_p{\displaystyle \frac{h}{d}}\cos {\alpha }_2+m_p{\displaystyle \frac{h^2}{d^2}}\sin {\alpha }_2\right){\ddot{c}}_1-\left(\left(m_1+m_2\right)+{\displaystyle \frac{1}{4}}{m}_p+m_p{\displaystyle \frac{h^2}{d^2}}\right)\sin {\alpha }_2{\ddot{c}}_2\hfill \\ & -\left(-{\displaystyle \frac{1}{4}}{m}_p\cos {\alpha }_2+m_p{\displaystyle \frac{h}{d}}\sin {\alpha }_2+m_p{\displaystyle \frac{h^2}{d^2}}\cos {\alpha }_2\right){\ddot{s}}_1-\left(\left(m_1+m_2\right)+{\displaystyle \frac{1}{4}}{m}_p+m_p{\displaystyle \frac{h^2}{d^2}}\right)\cos {\alpha }_2{\ddot{s}}_2\hfill \\ & +\left(\begin{array}{l}{\displaystyle \frac{1}{4}}{m}_p{l}_a\cos \left({\alpha }_1+{\alpha }_2\right)-m_p{l}_a{\displaystyle \frac{h^2}{d^2}}\cos \left({\alpha }_1+{\alpha }_2\right)+{\displaystyle \frac{1}{4}}{m}_p\left(l_b+l_c\right)\sin \left({\alpha }_1+{\alpha }_2\right)-m_p{\displaystyle \frac{h^2}{d^2}}\left(l_b+l_c\right)\sin \left({\alpha }_1+{\alpha }_2\right)\\ +{\displaystyle \frac{1}{4}}{m}_p\cos \left({\alpha }_1+{\alpha }_2\right)a_1-m_p{\displaystyle \frac{h^2}{d^2}}\cos \left({\alpha }_1+{\alpha }_2\right)a_1+{\displaystyle \frac{1}{4}}{m}_p\sin \left({\alpha }_1+{\alpha }_2\right)b_1-m_p{\displaystyle \frac{h^2}{d^2}}\sin \left({\alpha }_1+{\alpha }_2\right)b_1\\ +m_p{\displaystyle \frac{h}{d}}\left(l_b+l_c\right)\cos \left({\alpha }_1+{\alpha }_2\right)-m_p{\displaystyle \frac{h}{d}}{l}_a\sin \left({\alpha }_1+{\alpha }_2\right)+m_p{\displaystyle \frac{h}{d}}\cos \left({\alpha }_1+{\alpha }_2\right)b_1-m_p{\displaystyle \frac{h}{d}}\sin \left({\alpha }_1+{\alpha }_2\right)a_1\end{array}\right){\dot{\alpha }}_1^2\hfill \\ & -\left(m_p{\displaystyle \frac{h^2}{d^2}}{l}_a+\left(m_1+m_2\right)a_2+{\displaystyle \frac{1}{4}}{m}_p{a}_2+m_p{\displaystyle \frac{h^2}{d^2}}{a}_2+{\displaystyle \frac{1}{2}}{m}_1{l}_a+m_2{l}_a+{\displaystyle \frac{1}{4}}{m}_p{l}_a\right){\dot{\alpha }}_2^2\hfill \end{array}$$ $$\begin{array}{cc}{d}_4& =-F_{fb2}-\left(m_p{\displaystyle \frac{h^2}{d^2}}{l}_a+{\displaystyle \frac{1}{2}}{m}_2{a}_2+{\displaystyle \frac{1}{4}}{m}_p{a}_2+m_p{a}_2{\displaystyle \frac{h^2}{d^2}}+{\displaystyle \frac{1}{2}}{m}_2{l}_a+{\displaystyle \frac{1}{4}}{m}_p{l}_a\right){\ddot{\alpha }}_2\hfill \\ & -\left(\begin{array}{l}{\displaystyle \frac{1}{4}}{m}_p{l}_a\cos \left({\alpha }_1+{\alpha }_2\right)m_p-{\displaystyle \frac{h^2}{d^2}}{l}_a\cos \left({\alpha }_1+{\alpha }_2\right)+{\displaystyle \frac{1}{4}}{m}_p\left(l_b+l_c\right)\sin \left({\alpha }_1+{\alpha }_2\right)+m_p{\displaystyle \frac{h}{d}}\cos \left({\alpha }_1+{\alpha }_2\right)b_1\\ -{\displaystyle \frac{h^2}{d^2}}{m}_p\left(l_b+l_c\right)\sin \left({\alpha }_1+{\alpha }_2\right)+{\displaystyle \frac{1}{4}}{m}_p\cos \left({\alpha }_1+{\alpha }_2\right)a_1-{\displaystyle \frac{h^2}{d^2}}{m}_p\cos \left({\alpha }_1+{\alpha }_2\right)a_1+{\displaystyle \frac{1}{4}}{m}_p\sin \left({\alpha }_1+{\alpha }_2\right)b_1\\ -{\displaystyle \frac{h^2}{d^2}}{m}_p{l}_a\sin \left({\alpha }_1+{\alpha }_2\right)b_1+m_p{\displaystyle \frac{h}{d}}\left(l_b+l_c\right)\cos \left({\alpha }_1+{\alpha }_2\right)-m_p{\displaystyle \frac{h}{d}}{l}_a\sin \left({\alpha }_1+{\alpha }_2\right)-m_p{\displaystyle \frac{h}{d}}\sin \left({\alpha }_1+{\alpha }_2\right)a_1\end{array}\right){\ddot{\alpha }}_1\hfill \\ & -\left(-{\displaystyle \frac{1}{4}}{m}_p\cos {\alpha }_2+m_p{\displaystyle \frac{h}{d}}\sin {\alpha }_2+m_p{\displaystyle \frac{h^2}{d^2}}\cos {\alpha }_2\right){\ddot{c}}_1+\left({\displaystyle \frac{1}{2}}{m}_2+{\displaystyle \frac{1}{4}}{m}_p+m_p{\displaystyle \frac{h^2}{d^2}}\right)\cos {\alpha }_2{\ddot{c}}_2\hfill \\ & +\left({\displaystyle \frac{1}{4}}{m}_p\sin {\alpha }_2+m_p{\displaystyle \frac{h}{d}}\cos {\alpha }_2-m_p{\displaystyle \frac{h^2}{d^2}}\sin {\alpha }_2\right){\ddot{s}}_1-\left({\displaystyle \frac{1}{2}}{m}_2+{\displaystyle \frac{1}{4}}{m}_p+m_p{\displaystyle \frac{h^2}{d^2}}\right)\sin {\alpha }_2{\ddot{s}}_2\hfill \\ & +\left(\begin{array}{l}{m}_p{l}_b{\displaystyle \frac{h^2}{d^2}}\cos \left({\alpha }_1+{\alpha }_2\right)-{\displaystyle \frac{1}{4}}{m}_p\left(l_b+l_c\right)\cos \left({\alpha }_1+{\alpha }_2\right)+m_p{\displaystyle \frac{h^2}{d^2}}{l}_c\cos \left({\alpha }_1+{\alpha }_2\right)\\ +{\displaystyle \frac{1}{4}}{m}_p{l}_a\sin \left({\alpha }_1+{\alpha }_2\right)-m_p{\displaystyle \frac{h^2}{d^2}}\sin \left({\alpha }_1+{\alpha }_2\right)-{\displaystyle \frac{1}{4}}{m}_p\cos \left({\alpha }_1+{\alpha }_2\right)b_1+m_p{\displaystyle \frac{h^2}{d^2}}\cos \left({\alpha }_1+{\alpha }_2\right)b_1\\ +{\displaystyle \frac{1}{4}}{m}_p{\displaystyle \frac{h}{d}}\sin \left({\alpha }_1+{\alpha }_2\right)a_1-m_p{\displaystyle \frac{h}{d}}\sin \left({\alpha }_1+{\alpha }_2\right)a_1+m_p{\displaystyle \frac{h}{d}}{l}_a\cos \left({\alpha }_1+{\alpha }_2\right)+m_p{\displaystyle \frac{h}{d}}\left(l_b+l_c\right)\sin \left({\alpha }_1+{\alpha }_2\right)\\ +m_p{\displaystyle \frac{h}{d}}\cos \left({\alpha }_1+{\alpha }_2\right)a_1+m_p{\displaystyle \frac{h}{d}}\sin \left({\alpha }_1+{\alpha }_2\right)b_1\end{array}\right){\dot{\alpha }}_1^2\hfill \\ & -\left(m_p{\displaystyle \frac{h^2}{d^2}}\left(l_b+l_c\right)+{\displaystyle \frac{1}{4}}{m}_2{b}_2+m_p{\displaystyle \frac{h^2}{d^2}}{b}_2+{\displaystyle \frac{1}{4}}{m}_2{l}_b+{\displaystyle \frac{1}{2}}{m}_2{l}_c+{\displaystyle \frac{1}{4}}{m}_p\left(l_b+l_c\right)\right){\dot{\alpha }}_2^2\hfill \end{array}$$

References

1. Pan, X.; Wang, K.; Jiang, W.; Zhang, L.; Yu, Y.; Ma, X. Soft actor-critic based controller for offshore lifting arm system of active 3-DOF wave compensation using gym environment with pyBullet. Proceedings of the 2024 14th Asian Control Conference (ASCC); Dalian, China, 5–8 July 2024; pp. 1650-1655.

2. Nam, M.; Kim, J.; Lee, J.; Kim, D.; Lee, D.; Lee, J. Cooperative control system of the floating cranes for the dual lifting. Int. J. Nav. Archit. Ocean Eng.; 2018; 10, pp. 95-102. [DOI: https://dx.doi.org/10.1016/j.ijnaoe.2017.03.003]

3. Cao, Y.; Li, T. Review of antiswing control of shipboard cranes. IEEE/CAA J. Autom. Sin.; 2020; 7, pp. 346-354. [DOI: https://dx.doi.org/10.1109/JAS.2020.1003024]

4. Sun, M.; Ji, C.; Luan, T.; Wang, N. LQR pendulation reduction control of ship-mounted crane based on improved grey wolf optimization algorithm. Int. J. Precis. Eng. Manuf.; 2023; 24, pp. 395-407. [DOI: https://dx.doi.org/10.1007/s12541-022-00763-7]

5. Chen, S.; Xie, P.; Liao, J.; Wu, S.; Su, Y. NMPC-PID control of secondary regulated active heave compensation system for offshore crane. Ocean Eng.; 2023; 287, 115902. [DOI: https://dx.doi.org/10.1016/j.oceaneng.2023.115902]

6. Chen, S.; Xie, P.; Liao, J. Cascade NMPC-PID control strategy of active heave compensation system for ship-mounted offshore crane. Ocean Eng.; 2024; 302, 117648. [DOI: https://dx.doi.org/10.1016/j.oceaneng.2024.117648]

7. Bozkurt, B.; Ertogan, M. Heave and horizontal displacement and anti-sway control of payload during ship-to-ship load transfer with an offshore crane on very rough sea conditions. Ocean Eng.; 2023; 267, 113309. [DOI: https://dx.doi.org/10.1016/j.oceaneng.2022.113309]

8. Kim, D.; Park, Y. Tracking control in x-y plane of an offshore container crane. J. Vib. Control; 2017; 23, pp. 469-483. [DOI: https://dx.doi.org/10.1177/1077546315581091]

9. Chen, H.; Zhang, R.; Liu, W.; Chen, H. A time optimal trajectory planning method for offshore cranes with ship roll motions. J. Frankl. Inst.; 2022; 359, pp. 6099-6122. [DOI: https://dx.doi.org/10.1016/j.jfranklin.2022.06.007]

10. Wu, Q.; Ouyang, H.; Xi, H. Real-time trajectory planning for ship-mounted rotary cranes considering continuous sea wave disturbances. Nonlinear Dyn.; 2023; 111, pp. 20959-20973. [DOI: https://dx.doi.org/10.1007/s11071-023-08953-2]

11. Li, S.; Zou, Y.; Lai, X.; Liu, Z.; Wang, X. Performance-maximum optimization of the intelligent lifting activities for a polar ship crane through trajectory planning. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci.; 2023; 237, pp. 765-781. [DOI: https://dx.doi.org/10.1177/09544062221111707]

12. Lu, B.; Lin, J.; Fang, Y.; Hao, Y.; Cao, H. Online trajectory planning for three-dimensional offshore boom cranes. Autom. Constr.; 2022; 140, 104372. [DOI: https://dx.doi.org/10.1016/j.autcon.2022.104372]

13. Martin, I.A.; Irani, R.A. A generalized approach to anti-sway control for shipboard cranes. Mech. Syst. Signal Process.; 2021; 148, 107168. [DOI: https://dx.doi.org/10.1016/j.ymssp.2020.107168]

14. Martin, I.A.; Irani, R.A. Dynamic modeling and self-tuning anti-sway control of a seven degree of freedom shipboard knuckle boom crane. Mech. Syst. Signal Process.; 2021; 153, 107441. [DOI: https://dx.doi.org/10.1016/j.ymssp.2020.107441]

15. Li, F.; Qian, Y.; Wu, S. A self-adapting trajectory tracking control for double-pendulum marine tower crane considering constraint dead zone against sea wave disturbances. Ocean Eng.; 2024; 309, 118358. [DOI: https://dx.doi.org/10.1016/j.oceaneng.2024.118358]

16. Chen, S.; Xie, P.; Liao, J.; Huang, Z. Model experimental studies on active heave compensation control strategy for electric-driven offshore cranes. Ocean Eng.; 2024; 311, 118987. [DOI: https://dx.doi.org/10.1016/j.oceaneng.2024.118987]

17. Li, Z.; Ma, X.; Li, Y.; Meng, Q.; Li, J. ADRC-ESMPC active heave compensation control strategy for offshore cranes. Ships Offshore Struct.; 2020; 15, pp. 1098-1106. [DOI: https://dx.doi.org/10.1080/17445302.2019.1703388]

18. Li, W.; Xu, C.; Lin, S.; Zhang, P.; Xu, H. Prediction and control strategy based on optimized active disturbance rejection control for AHC system. Ocean Eng.; 2023; 289, 116178. [DOI: https://dx.doi.org/10.1016/j.oceaneng.2023.116178]

19. Guo, B.; Chen, Y. Fuzzy robust fault-tolerant control for offshore ship-mounted crane system. Inf. Sci.; 2020; 526, pp. 119-132. [DOI: https://dx.doi.org/10.1016/j.ins.2020.03.068]

20. Kim, G.H.; Hong, K.S. Adaptive sliding-mode control of an offshore container crane with unknown disturbances. IEEE/ASME Trans. Mechatron.; 2019; 24, pp. 2850-2861. [DOI: https://dx.doi.org/10.1109/TMECH.2019.2946083]

21. Ngo, Q.H.; Nguyen, N.P.; Nguyen, C.N.; Tran, T.H.; Ha, Q.P. Fuzzy sliding mode control of an offshore container crane. Ocean Eng.; 2017; 140, pp. 125-134. [DOI: https://dx.doi.org/10.1016/j.oceaneng.2017.05.019]

22. Saghafi Zanjani, M.; Mobayen, S. Event-triggered global sliding mode controller design for anti-sway control of offshore container cranes. Ocean Eng.; 2023; 268, 113472. [DOI: https://dx.doi.org/10.1016/j.oceaneng.2022.113472]

23. Saghafi Zanjani, M.; Mobayen, S. Anti-sway control of offshore crane on surface vessel using global sliding mode control. Int. J. Control; 2022; 95, pp. 2267-2278. [DOI: https://dx.doi.org/10.1080/00207179.2021.1906447]

24. Yang, T.; Sun, N.; Chen, H.; Fang, Y. Swing suppression and accurate positioning control for underactuated offshore crane systems suffering from disturbances. IEEE/CAA J. Autom. Sin.; 2020; 7, pp. 892-900. [DOI: https://dx.doi.org/10.1109/JAS.2020.1003162]

25. Zhao, T.; Sun, M.; Wang, S.; Han, G.; Wang, H.; Chen, H.; Sun, Y. Dynamic analysis and robust control of ship-mounted crane with multi-cable anti-swing system. Ocean Eng.; 2024; 291, 116376. [DOI: https://dx.doi.org/10.1016/j.oceaneng.2023.116376]

26. Sun, N.; Wu, Y.; Liang, X.; Fang, Y. Nonlinear stable transportation control for double-pendulum shipboard cranes with ship-motion-induced disturbances. IEEE Trans. Ind. Electron.; 2019; 66, pp. 9467-9479. [DOI: https://dx.doi.org/10.1109/TIE.2019.2893855]

27. Chen, H.; Sun, N. An output feedback approach for regulation of 5-DOF offshore cranes with ship yaw and roll perturbations. IEEE Trans. Ind. Electron.; 2022; 69, pp. 1705-1716. [DOI: https://dx.doi.org/10.1109/TIE.2021.3055159]

28. Li, Z.; Ma, X.; Li, Y. Nonlinear partially saturated control of a double pendulum offshore crane based on fractional-order disturbance observer. Autom. Constr.; 2022; 137, 104212. [DOI: https://dx.doi.org/10.1016/j.autcon.2022.104212]

29. Tuan, L.A.; Cuong, H.M.; Trieu, P.V.; Nho, L.C.; Thuanb, V.D.; Anh, L.V. Adaptive neural network sliding mode control of shipboard container cranes considering actuator backlash. Mech. Syst. Signal Process.; 2018; 112, pp. 233-250. [DOI: https://dx.doi.org/10.1016/j.ymssp.2018.04.030]

30. Yang, T.; Sun, N.; Chen, H.; Fang, Y. Neural network-based adaptive antiswing control of an underactuated ship-mounted crane with roll motions and input dead zones. IEEE Trans. Neural Netw. Learn. Syst.; 2020; 31, pp. 901-914. [DOI: https://dx.doi.org/10.1109/TNNLS.2019.2910580]

31. Qian, Y.; Hu, D.; Chen, Y.; Fang, Y.; Hu, Y. Adaptive neural network-based tracking control of underactuated offshore ship-to-ship crane systems subject to unknown wave motions disturbances. IEEE Trans. Syst. Man Cybern. Syst.; 2022; 52, pp. 3626-3637. [DOI: https://dx.doi.org/10.1109/TSMC.2021.3071546]

32. Wu, Y.; Sun, N.; Chen, H.; Fang, Y. New adaptive dynamic output feedback control of double-pendulum ship-mounted cranes with accurate gravitational compensation and constrained inputs. IEEE Trans. Ind. Electron.; 2022; 69, pp. 9196-9205. [DOI: https://dx.doi.org/10.1109/TIE.2021.3112978]

33. Wu, Q.; Ouyang, H.; Xi, H. Adaptive nonlinear control for 4-DOF ship-mounted rotary cranes. Int. J. Robust Nonlinear Control; 2023; 33, pp. 1957-1972. [DOI: https://dx.doi.org/10.1002/rnc.6504]

34. Zhang, R.; Chen, H. An adaptive tracking control method for offshore cranes with unknown gravity parameters. Ocean Eng.; 2022; 260, 111809. [DOI: https://dx.doi.org/10.1016/j.oceaneng.2022.111809]

35. Wang, K.; Liu, X.; Guo, P.; Yu, Y.; Li, Z.; Ma, X. Dynamics modeling and analysis for twin marine lifting systems. Proceedings of the 2023 42nd Chinese Control Conference (CCC); Tianjin, China, 24–26 July 2023; pp. 635-640. [DOI: https://dx.doi.org/10.23919/CCC58697.2023.10240663]

36. Hu, D.; Qian, Y.; Fang, Y.; Chen, Y. Modeling and nonlinear energy-based anti-swing control of underactuated dual ship-mounted crane systems. Nonlinear Dyn.; 2021; 106, pp. 323-338. [DOI: https://dx.doi.org/10.1007/s11071-021-06837-x]

37. Qian, Y.; Hu, D.; Chen, Y.; Fang, Y. Programming-based optimal learning sliding mode control for cooperative dual ship-mounted cranes against unmatched external disturbances. IEEE Trans. Autom. Sci. Eng.; 2023; 20, pp. 969-980. [DOI: https://dx.doi.org/10.1109/TASE.2022.3182720]

38. Qian, Y.; Zhang, H.; Hu, D. Finite-time neural network-based hierarchical sliding mode antiswing control for underactuated dual ship-mounted cranes with unmatched sea wave disturbances suppression. IEEE Trans. Neural Netw. Learn. Syst.; 2024; 35, pp. 12396-12408. [DOI: https://dx.doi.org/10.1109/TNNLS.2023.3257508] [PubMed: https://www.ncbi.nlm.nih.gov/pubmed/37030785]

39. Wang, F.; Li, G.; Jiang, W.; Zhang, L.; Yu, Y.; Ma, X. Collaborative heave compensation control of dual ship-mounted lifting arm system based on incremental model predictive control. Proceedings of the 2024 14th Asian Control Conference (ASCC); Dalian, China, 5–8 July 2024; pp. 1790-1795.

40. Li, G.; Ma, X.; Li, Y. Dynamics analysis of the cooperative dual marine lifting systems subject to sea wave disturbances. Proceedings of the 2023 International Conference on Applied Nonlinear Dynamics, Vibration and Control; Hongkong, China, 4–6 December 2023; pp. 348-361. [DOI: https://dx.doi.org/10.1007/978-981-97-0554-2_27]

41. Li, G.; Ma, X.; Li, Y. Adaptive anti-swing control for 7-DOF overhead crane with double spherical pendulum and varying cable length. IEEE Trans. Autom. Sci. Eng.; 2024; 21, pp. 5240-5251. [DOI: https://dx.doi.org/10.1109/TASE.2023.3309937]

42. Li, G.; Ma, X.; Li, Y. Robust command shaped vibration control for stacker crane subject to parameter uncertainties and external disturbances. IEEE Trans. Ind. Electron.; 2024; 71, pp. 14740-14752. [DOI: https://dx.doi.org/10.1109/TIE.2024.3366197]

43. Boudjedir, C.E.; Bouri, M.; Boukhetala, D. An enhanced adaptive time delay control-based integral sliding mode for trajectory tracking of robot manipulators. IEEE Trans. Control. Syst. Technol.; 2023; 31, pp. 1042-1050. [DOI: https://dx.doi.org/10.1109/TCST.2022.3208491]

44. Ahmed, S.; Wang, H.; Tian, Y. Adaptive high-order terminal sliding mode control based on time delay estimation for the robotic manipulators with backlash hysteresis. IEEE Trans. Syst. Man Cybern. Syst.; 2021; 51, pp. 1128-1137. [DOI: https://dx.doi.org/10.1109/TSMC.2019.2895588]

45. Yang, Y.; Hui, W.; Li, J. Model-free composite sliding mode adaptive control for 4-DOF tower crane. Autom. Constr.; 2024; 167, 105673. [DOI: https://dx.doi.org/10.1016/j.autcon.2024.105673]

46. Yao, X.; Chen, H.; Liu, Y.; Dong, Y. Tracking approach of double pendulum cranes with variable rope lengths using sliding mode technique. ISA Trans.; 2023; 136, pp. 152-161. [DOI: https://dx.doi.org/10.1016/j.isatra.2022.11.019] [PubMed: https://www.ncbi.nlm.nih.gov/pubmed/36528393]

47. Khalil, H.K. Nonlinear Systems; 3rd ed. Prentice–Hall: Englewood Cliffs, NJ, USA, 2002.

48. Fossen, T.I.; Perez, T. Marine Systems Simulator (MSS). Available online: https://www.mathworks.com/matlabcentral/fileexchange/86393-marine-systems-simulator-mss (accessed on 25 November 2024).