Energy-Based Adaptive Control for Variable Rope Length Double-Pendulum Ship-Borne Cranes: A Disturbance Rejection Stabilization Controller Without Overshoot

1. Introduction

Mast cranes are commonly applied industrial installation near modern ports. The ship-borne cranes are usually mounted on the ship, which may disturbed by throbbing waves, drifting wind, cable corrosion, etc. during cargo transportation process. Because the ship-borne crane works in a harsh marine environment, the current manual operation is dangerous, and the accuracy of control is easily influenced by environmental factors, so it is essential to design a controller to manage the operation of the crane. The mechanical structure and working principle of ship cranes are similar to those of typical land fixed cranes, however, they exhibit stronger state-variable coupling and more complex dynamic characteristics. Additionally, unlike land fixed cranes, there have hull motions with multiple degrees of freedom in marine environment, which increase the difficulty and challenges of controlling ship-borne cranes [1]. Nowadays the ship-borne crane with double pendulum takes more effects in the ocean engineering, the research of its advanced control method is very meaningful.

The problem of control of land fixed cranes has been studied extensively over the past few decades, such as optimal control [2,3,4], nonlinear control [5,6,7,8,9,10], and intelligent control [11,12,13,14,15,16,17,18]. Currently available methods include open-loop control methods, closed-loop control methods, and sliding mode control, etc. Compared to the single-pendulum crane, the double-pendulum crane system has more motion dimensions and stronger coupling, making it more difficult to eliminate load sway. In the case of the double pendulum effect, the traditional method applied to the single-pendulum crane system is difficult to achieve a good control effect. For this reason, in recent decades, the control problem of ship-borne cranes has attracted more and more attention, and many remarkable research results have been presented.

For ship-borne crane systems, in [19], Ren et al. propose a general model-free anti-swing control scheme that simplifies the control process by not requiring state-space equations. This approach effectively minimizes the motion of a pendular payload, regardless of the specific system configuration. Zhao et al. propose a sliding mode anti-swing controller in [20], and subsequently developed an enhanced adaptive sliding mode cooperative controller. In [21], Wu et al. introduce an adaptive controller capable of updating its adaptive laws in real time based on the system’s state and disturbances. Considering the issue of residual swing of the load, Chen et al. propose a time-optimal trajectory planning method for the offshore crane in [22]. Yang et al. present an adaptive control method utilizing neural networks to address the issue of nonlinear input dead zones [23]. In [24], Kuchler et al. propose an dynamic compensation system for the vertical vessel motion. Taking into account the nonlinear system of the offshore ship crane, along with actuator malfunctions and external interferences, Guo et al. propose an innovative event-triggered fuzzy robust fault-tolerant control approach [25]. In [26], Qian et al. develop a new adaptive robust coupling control method that integrates adaptive laws to handle unknown parameters.

Considering the control issues associated with dual ship-borne crane system, Hu et al. develope two energy-based nonlinear controllers, consisting of a full-state feedback control approach and an output feedback control approach [27]. Considering the scenario of jib rotation and horizontal transfer, Qian et al. present an innovative control scheme based on nonlinear switching logic for marine cranes [28]. Taking into account the complex disturbances caused by ship motions, Wu et al. introduce a novel adaptive dynamic output feedback control approach [29]. In [30], Kim et al. develop a tracking controller for cranes installed at mobile harbors, which features a dual-stage trolley system capable of dynamically positioning containers between the mobile harbor and the container ship, and vice versa. Considering the underwater working environment, Wang et al. develope a nonlinear control method based on a coupling characteristic indicator [31]. In [32], Chen et al. introduce an innovative hierarchical control strategy to address the challenge of active heave compensation for onboard offshore cranes during heavy lifting operations in harsh marine environments. In the context of ship-to-ship load transfer using offshore cranes, Bozkurt et al. propose an innovative control strategy that integrates a combined vertical, horizontal, and anti-swing control system [33].

In general, based on the analysis of the above literature, this paper proposes an energy-based adaptive controller for ship-borne cranes with double pendulum effect. Specific innovation points are summarized as follows:

1) For a ship-borne crane with double pendulum effect, an adaptive controller has been designed. Using the adaptive term, the uncertain friction of the actuator can be estimated online, effectively eliminating the positioning error of the jib and the load.

2) To address the issue of overshoot, we meticulously designed two overshoot limit terms. By introducing overshoot limit parameters, we constrained the overshoot of the rope length and the jib, ensuring stability and precision in the control process.

The structure of the rest of this paper is outlined as follows. Section II presents the dynamic model transformation of the ship-borne crane system. Section III describes the design process of the adaptive controller. Section IV analyzes the stability of the system. Section V presents the simulations. Section VI concludes the paper.

2. Dynamic Model Transformation

This section establishes the dynamic model of the ship-borne crane, with its schematic diagram in Figure 1. Where, $\left\{X_{g}O{Z}_g\right\}$ and $\left\{X_{s}O{Z}_s\right\}$ represent the earth-fixed and ship-fixed coordinates, respectively. $F_j$ and $F_l$ represent the driving force of the jib and rope respectively, $M,m_1,m_2$ respectively represent the jib mass, hook mass and load mass. $\rho$ represents the ship motion angle caused by sea wave. J is the jib rotational inertia. ${\zeta }_1$ is the jib luffing angle, ${\zeta }_2$, $L_j$ and $L_2$ represent the length of the rope, the jib and the distance between the hook and the center of gravity of the load respectively. ${\zeta }_3$ and ${\zeta }_4$ represent the swing angle of the hook and the load relative to the vertical direction. The dynamic equation of a ship-borne crane system with variable rope length is shown as follows[21]:

$$\begin{array}{cc}\hfill & \left[J+(m_1+m_2)L_j^2\right]{\ddot{\zeta }}_1-(m_1+m_2)L_j{\ddot{\zeta }}_{2}cos({\zeta }_3-{\zeta }_1)+(m_1+m_2)L_j{\zeta }_2{\ddot{\zeta }}_{3}sin({\zeta }_3-{\zeta }_1)\hfill \\ \hfill & +m_2{L}_2{L}_j{\ddot{\zeta }}_{4}sin({\zeta }_4-{\zeta }_3)+2(m_1+m_2)L_j{\dot{\zeta }}_3{\dot{\zeta }}_{2}sin({\zeta }_3-{\zeta }_1)+(m_1+m_2)L_j{\zeta }_2{\dot{\zeta }}_3^2\hfill \\ \hfill & \times cos({\zeta }_3-{\zeta }_1)-m_2{L}_2{L}_j{\dot{\zeta }}_4{\dot{\zeta }}_{3}cos({\zeta }_4-{\zeta }_3)+m_2{L}_2{L}_j{\dot{\zeta }}_4^{2}cos({\zeta }_3-{\zeta }_4)\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill & +\left({\displaystyle \frac{1}{2}}M+m_1+m_2\right)g{L}_{j}cos{\zeta }_1=F_j-f_j\hfill \\ \hfill & -(m_1+m_2)L_j{\ddot{\zeta }}_{1}cos({\zeta }_3-{\zeta }_1)+(m_1+m_2){\ddot{\zeta }}_2+m_2{L}_2{\ddot{\zeta }}_{4}sin({\zeta }_3-{\zeta }_4)-(m_1+m_2)\hfill \\ \hfill & \times \left[L_j{\dot{\zeta }}_1^{2}sin({\zeta }_3-{\zeta }_1)+{\zeta }_2{\dot{\zeta }}_3^2\right]-m_2{L}_2{\dot{\zeta }}_4^{2}cos({\zeta }_3-{\zeta }_4)\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill & -(m_1+m_2)gcos{\zeta }_3=F_l-f_l\hfill \\ \hfill & (m_1+m_2)L_j{\zeta }_2{\ddot{\zeta }}_{1}sin({\zeta }_3-{\zeta }_1)+(m_1+m_2){\zeta }_2^2{\ddot{\zeta }}_3+m_2{L}_2{\zeta }_2{\ddot{\zeta }}_{4}cos({\zeta }_3-{\zeta }_4)\hfill \\ \hfill & -(m_1+m_2)L_j{\zeta }_2{\dot{\zeta }}_1^{2}cos({\zeta }_3-{\zeta }_1)+m_2{L}_2{L}_j{\dot{\zeta }}_1{\dot{\zeta }}_{4}cos({\zeta }_4-{\zeta }_3)\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & +2(m_1+m_2){\zeta }_2{\dot{\zeta }}_2{\dot{\zeta }}_3+m_2{L}_2{\zeta }_2{\dot{\zeta }}_4^{2}sin({\zeta }_3-{\zeta }_4)+(m_1+m_2)g{\zeta }_{2}sin{\zeta }_3=0\hfill \\ \hfill & m_2{L}_2{L}_j{\ddot{\zeta }}_{1}sin({\zeta }_4-{\zeta }_3)+m_2{L}_2{\ddot{\zeta }}_{2}sin({\zeta }_3-{\zeta }_4)+m_2{L}_2{\zeta }_2{\ddot{\zeta }}_{3}cos({\zeta }_3-{\zeta }_4)+m_2{L}_2^2{\ddot{\zeta }}_4\hfill \\ \hfill & -m_2{L}_2{L}_j{\dot{\zeta }}_3{\dot{\zeta }}_{1}cos({\zeta }_4-{\zeta }_3)+2m_2{L}_2{\dot{\zeta }}_3{\dot{\zeta }}_{2}cos({\zeta }_3-{\zeta }_4)\hfill \\ \hfill & -m_2{L}_2{\zeta }_2{\dot{\zeta }}_3^{2}sin({\zeta }_3-{\zeta }_4)+m_{2}g{L}_{2}sin{\zeta }_4=0\hfill \end{array}$$

(4)

The friction force $f_j$ in (1) and $f_l$ in (2) are respectively expressed in the following forms:

$$\begin{array}{cc}\hfill & f_j=M_{f1}tanh\left({\displaystyle \frac{{\dot{\zeta }}_1}{d_1}}\right)+M_{f2}\left|{\dot{\zeta }}_1\right|{\dot{\zeta }}_1\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & f_l=d_2{\dot{\zeta }}_2\hfill \end{array}$$

(6)

where, $M_{f1},M_{f2},d_1,d_2\in R$ represent friction-related parameters. The model presented in (1)-(4) can alternatively be expressed in the following compact form:

$$\begin{array}{c}\hfill M_{\zeta }\left(\zeta \right)\ddot{\zeta }+C_{\zeta }\left(\zeta ,\dot{\zeta }\right)\dot{\zeta }+G_{\zeta }\left(\zeta \right)=U-F_f\end{array}$$

(7)

where, $\zeta ={\left[\begin{array}{cccc}{\zeta }_1& {\zeta }_2& {\zeta }_3& {\zeta }_4\end{array}\right]}^T$, $F_f={\left[\begin{array}{cccc}{f}_j& f_l& 0& 0\end{array}\right]}^T$. Under the premise of not losing generality, the following assumptions are considered reasonable and are commonly applied in the field of crane control:

Assumption 1 [34]: Considering the practice working condition, the hook and the load are always under the jib, so the swing angle of the hook and the load meet the following conditions:

$$\begin{array}{c}\hfill -{\displaystyle \frac{\pi }{2}}<{\zeta }_3\left(t\right)<{\displaystyle \frac{\pi }{2}},-{\displaystyle \frac{\pi }{2}}<{\zeta }_4\left(t\right)<{\displaystyle \frac{\pi }{2}}\end{array}$$

(8)

According to the model given in (1)-(4), a controller with $F_j$ and $F_l$ as driving forces will be designed to solve the control problem of double-pendulum ship-borne crane system through their joint action. Its control objectives are as follows:

$$\begin{array}{cc}\hfill & \underset{t\to \infty }{lim}{\zeta }_1\left(t\right)={\zeta }_{1d},\underset{t\to \infty }{lim}{\zeta }_2\left(t\right)={\zeta }_{2d},\hfill \\ \hfill & \underset{t\to \infty }{lim}{\zeta }_3\left(t\right)=0,\underset{t\to \infty }{lim}{\zeta }_4\left(t\right)=0\hfill \end{array}$$

(9)

where ${\xi }_{1d},{\xi }_{2d}$ are the corresponding desired values.

3. Controller Design

This section introduces the design procedure for the adaptive controller of double-pendulum ship-borne crane. To enable further analysis and the design of control laws, the following two error signals are introduced:

$$\begin{array}{cc}\hfill & e_1={\zeta }_1-{\zeta }_{1d},e_2={\zeta }_2-{\zeta }_{2d}\hfill \\ \hfill & e_3={\zeta }_3,e_4={\zeta }_4\hfill \\ \hfill & \Rightarrow {\dot{e}}_1={\dot{\zeta }}_1,{\dot{e}}_2={\dot{\zeta }}_2,{\dot{e}}_3={\dot{\zeta }}_3,{\dot{e}}_4={\dot{\zeta }}_4\hfill \end{array}$$

(10)

For convenience, let’s define $M_1={\displaystyle \frac{1}{2}}M+m_1+m_2,M_2=m_1+m_2$. Then construct a energy-like function as follows

$$\begin{array}{cc}\hfill E=& {\displaystyle \frac{1}{2}}{\dot{\zeta }}^T{M}_{\zeta }\left(\zeta \right)\dot{\zeta }+M_{2}g{\zeta }_2\left(1-cos{\zeta }_3\right)\hfill \\ \hfill & +m_{2}g{L}_2\left(1-cos{\zeta }_4\right)\hfill \end{array}$$

(11)

Derivation of Formula (11) gives:

$$\begin{array}{cc}\hfill \dot{E}=& {\dot{\zeta }}^T{M}_{\zeta }\left(\zeta \right)\ddot{\zeta }+{\displaystyle \frac{1}{2}}{\dot{\zeta }}^T{\dot{M}}_{\zeta }\left(\zeta \right)\dot{\zeta }+m_{2}g{L}_2{\dot{\zeta }}_{4}sin{\zeta }_4\hfill \\ \hfill & +M_{2}g{\dot{\zeta }}_2(1-cos{\zeta }_3)+M_{2}g{\zeta }_2{\dot{\zeta }}_{3}sin{\zeta }_3\hfill \end{array}$$

(12)

By substituting Formulas (1)–(4) into Formula (12), we can get

$$\begin{array}{cc}\hfill \dot{E}& =\left[\left(F_j-f_j\right)-M_{1}g{L}_{j}cos{\zeta }_1\right]{\dot{\zeta }}_1+\left[\left(F_l-f_l\right)+M_{2}g\right]{\dot{\zeta }}_2\hfill \\ \hfill & =\left[\left(F_j-{\gamma }_{\mathbf{j}}^{\mathbf{T}}{\omega }_{\mathbf{j}}\right)-M_{1}g{L}_{j}cos{\zeta }_1\right]{\dot{\zeta }}_1\hfill \\ \hfill & +\left[\left(F_l-{\gamma }_{\mathbf{l}}{\omega }_{\mathbf{l}}\right)+M_{2}g\right]{\dot{\zeta }}_2\hfill \end{array}$$

(13)

where,

$$\begin{array}{c}\hfill \begin{array}{c}{\gamma }_{\mathbf{j}}^{\mathbf{T}}={\left[\begin{array}{cc}tanh\left({\displaystyle \frac{{\dot{\zeta }}_1}{d_1}}\right)& \left|{\dot{\zeta }}_1\right|{\dot{\zeta }}_1\end{array}\right]}^T,{\omega }_{\mathbf{j}}={\left[\begin{array}{cc}{M}_{f1}& M_{f2}\end{array}\right]}^T\hfill \\ {\gamma }_{\mathbf{l}}={\dot{\zeta }}_2,{\omega }_{\mathbf{l}}=d_2\hfill \end{array}\end{array}$$

(14)

To integrate the control objective into the controller design, we further define the following continuously differentiable positive definite scalar function $W\left(t\right)$ based on Equation (11).

$$\begin{array}{cc}\hfill W\left(t\right)=& {\displaystyle \frac{1}{2}}{\dot{\zeta }}^T{M}_{\zeta }\left(\zeta \right)\dot{\zeta }+M_{2}g{\zeta }_2(1-cos{\zeta }_3)\hfill \\ \hfill & +m_{2}g{L}_2(1-cos{\zeta }_4)+{\displaystyle \frac{1}{2}}{k}_{1p}{e}_1^2+{\displaystyle \frac{1}{2}}{k}_{2p}{e}_2^2\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}{\tilde{\omega }}_{\mathbf{j}}^{\mathbf{T}}{\Pi }_{\mathbf{j}}^{-\mathbf{1}}{\tilde{\omega }}_{\mathbf{j}}+{\displaystyle \frac{1}{2}}{\tilde{\omega }}_{\mathbf{l}}^{\mathbf{T}}{\Pi }_{\mathbf{l}}^{-\mathbf{1}}{\tilde{\omega }}_{\mathbf{l}}\hfill \end{array}$$

(15)

Where, $k_{1p},k_{2p}\in R^{+}$ represent the positive control gain, ${\Pi }_{\mathbf{j}}=diag\{{\pi }_{j1},{\pi }_{j2}\}\in R^{2\times 2}$ represents a positive definite diagonal matrix, ${\Pi }_{\mathbf{l}}\in R^{1\times 1}$ represents positive control gain, ${\pi }_{j1},{\pi }_{j2}\in R^{+}$ are positive adjustable gains, ${\tilde{\omega }}_{\mathbf{j}}\left(\mathbf{t}\right)\in R^{2\times 1},{\tilde{\omega }}_{\mathbf{l}}\left(\mathbf{t}\right)\in R^{1\times 1}$ represent error estimation vectors.

$$\begin{array}{c}\hfill {\tilde{\omega }}_{\mathbf{j}}={\widehat{\omega }}_{\mathbf{j}}-{\omega }_{\mathbf{j}},{\tilde{\omega }}_{\mathbf{l}}={\widehat{\omega }}_{\mathbf{l}}-{\omega }_{\mathbf{l}}\Rightarrow {\dot{\tilde{\omega }}}_{\mathbf{j}}={\dot{\widehat{\omega }}}_{\mathbf{j}},{\dot{\tilde{\omega }}}_{\mathbf{l}}={\dot{\widehat{\omega }}}_{\mathbf{l}}\end{array}$$

(16)

where, ${\widehat{\omega }}_{\mathbf{j}}\left(\mathbf{t}\right),{\widehat{\omega }}_l\left(t\right)\in R^{2\times 1}$ are the online estimation signals of ${\omega }_{\mathbf{j}},{\omega }_{\mathbf{l}}$, respectively.

By taking the derivative of Formula (15) and combining with Formulas (1)–(4), (12) and (16), we can get:

$$\begin{array}{cc}\hfill \dot{W}\left(t\right)=& \left(F_j+k_{1p}{e}_1-{\gamma }_{\mathbf{j}}^{\mathbf{T}}{\omega }_{\mathbf{j}}-M_{1}g{L}_{j}cos{\zeta }_1\right){\dot{\zeta }}_1\hfill \\ \hfill & +\left(F_l+k_{2p}{e}_2-{\gamma }_{\mathbf{l}}{\omega }_{\mathbf{l}}+M_{2}g\right){\dot{\zeta }}_2\hfill \\ \hfill & +{\tilde{\omega }}_{\mathbf{j}}^{\mathbf{T}}{\Pi }_{\mathbf{j}}^{-\mathbf{1}}{\dot{\widehat{\omega }}}_{\mathbf{j}}+{\tilde{\omega }}_{\mathbf{l}}^{\mathbf{T}}{\Pi }_{\mathbf{l}}^{-\mathbf{1}}{\dot{\widehat{\omega }}}_{\mathbf{l}}\hfill \end{array}$$

(17)

Therefore, the following preliminary control law is given as:

$$\begin{array}{cc}\hfill & F_{j0}=-k_{1p}{e}_1-k_{1d}{\dot{\zeta }}_1+M_{1}g{L}_{j}cos{\zeta }_1+{\gamma }_{\mathbf{j}}^{\mathbf{T}}{\widehat{\omega }}_{\mathbf{j}}\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill & F_{l0}=-k_{2p}{e}_2-k_{2d}{\dot{\zeta }}_2-M_{2}g+{\gamma }_{\mathbf{l}}{\widehat{\omega }}_{\mathbf{l}}\hfill \end{array}$$

(19)

where, $k_{1d},k_{2d}\in R^{+}$, represent positive control gains, ${\widehat{\omega }}_j\left(t\right),{\widehat{\omega }}_l\left(t\right)$ represent the updating laws, are designed as follows:

$$\begin{array}{c}\hfill {\dot{\widehat{\omega }}}_{\mathbf{j}}=-{\Pi }_{\mathbf{j}}{\gamma }_{\mathbf{j}}^{\mathbf{T}}{\dot{\zeta }}_1,{\dot{\widehat{\omega }}}_{\mathbf{l}}=-{\Pi }_{\mathbf{l}}{\gamma }_{\mathbf{l}}{\dot{\zeta }}_2\end{array}$$

(20)

Then, the controller (18)-(19) do not incorporate relevant information about the double pendulum angles. Furthermore, like most feedback control methods, the control law may be overshot. Therefore, the adaptive controller is ultimately proposed as follows:

$$\begin{array}{cc}\hfill F_j=& -k_{1p}{e}_1-k_{1d}{\dot{\zeta }}_1+M_{1}g{L}_{j}cos{\zeta }_1+{\gamma }_{\mathbf{j}}^{\mathbf{T}}{\widehat{\omega }}_{\mathbf{j}}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & -k_{1j}({\dot{\zeta }}_3^2+{\dot{\zeta }}_4^2){\dot{\zeta }}_1-{\lambda }_j{\displaystyle \frac{{({\zeta }_{1d}+{\epsilon }_{\zeta 1})}^2-{\zeta }_1^2+{\zeta }_1{e}_1}{{\left[{({\zeta }_{1d}+{\epsilon }_{\zeta 1})}^2-{\zeta }_1^2\right]}^2}}{e}_1\hfill \\ \hfill F_l=& -k_{2p}{e}_2-k_{2d}{\dot{\zeta }}_2-M_{2}g+{\gamma }_l{\widehat{\omega }}_l\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill & -k_{2j}({\dot{\zeta }}_3^2+{\dot{\zeta }}_4^2){\dot{\zeta }}_2-{\lambda }_l{\displaystyle \frac{{({\zeta }_{2d}+{\epsilon }_{\zeta 2})}^2-{\zeta }_2^2+{\zeta }_2{e}_2}{{\left[{({\zeta }_{2d}+{\epsilon }_{\zeta 2})}^2-{\zeta }_2^2\right]}^2}}{e}_2\hfill \end{array}$$

(22)

where, $k_{1j},k_{2j},{\lambda }_j,{\lambda }_l\in R^{+}$ represent the positive control gains, ${\epsilon }_{\zeta 1},{\epsilon }_{\zeta 2}\in R^{+}$, ${\epsilon }_{\zeta 1},{\epsilon }_{\zeta 2}$ are incorporated to limit the maximum overshoot of ${\zeta }_1\left(t\right)$ and ${\zeta }_2\left(t\right)$, respectively. The flowchart of the overall control system is shown in Figure 2.

4. Closed-Loop System Stability Analysis

In this section, we conduct a stability analysis of the proposed controller. By using Lyapunov function and LaSalle’s invariance theorem, the asymptotic stability of the system is proved.

$\mathbf{Theorem}\mathbf{1}:$ According to the designed adaptive controllers in the Formula (21) and the Formula (22) and the updating laws in the Formula (20), it can ensure that the jib and rope attain the desired position while preventing the hook and load from swinging. These results are described mathematically as follows:

$$\begin{array}{cc}& \underset{t\to \infty }{lim}{\left[\begin{array}{cccc}{\zeta }_1\left(t\right)& {\zeta }_2\left(t\right)& {\dot{\zeta }}_1\left(t\right)& {\dot{\zeta }}_2\left(t\right)\end{array}\right]}^T\hfill \\ & ={\left[\begin{array}{cccc}{\zeta }_{1d}& {\zeta }_{2d}& 0& 0\end{array}\right]}^T\hfill \end{array}$$

(23)

$$\begin{array}{cc}& \underset{t\to \infty }{lim}{\left[\begin{array}{cccc}{\zeta }_3\left(t\right)& {\zeta }_4\left(t\right)& {\dot{\zeta }}_3\left(t\right)& {\dot{\zeta }}_4\left(t\right)\end{array}\right]}^T\hfill \\ & ={\left[\begin{array}{cccc}0& 0& 0& 0\end{array}\right]}^T\hfill \end{array}$$

(24)

Synchronously, the angle of the jib and the length of the lifting rope should not exceed ${\zeta }_{1d}+{\epsilon }_{\zeta 1}$ and ${\zeta }_{2d}+{\epsilon }_{\zeta 2}$, and the overshoot of ${\zeta }_1,{\zeta }_2$ is less than ${\epsilon }_{\zeta 1}$ and ${\epsilon }_{\zeta 2}$, respectively.

$$\begin{array}{c}\hfill {\zeta }_1\left(t\right)<{\zeta }_{1d}+{\epsilon }_{\zeta 1}\end{array}$$

(25)

$\mathbf{Proof}:$ Initially, create the Lyapunov function as follows:

$$\begin{array}{cc}\hfill \mathrm{V}=& {\displaystyle \frac{1}{2}}{\dot{\zeta }}^T{M}_{\zeta }\left(\zeta \right)\dot{\zeta }+M_{2}g{\zeta }_2\left(1-cos{\zeta }_3\right)\hfill \\ \hfill & +m_{2}g{L}_2\left(1-cos{\zeta }_4\right)+{\displaystyle \frac{1}{2}}{k}_{1p}{e}_1^2+{\displaystyle \frac{1}{2}}{k}_{2p}{e}_2^2\hfill \\ \hfill & +{\displaystyle \frac{{\lambda }_j{e}_1^2}{2\left[{\left({\zeta }_{1d}+{\epsilon }_{\zeta 1}\right)}^2-{\zeta }_1^2\right]}}+{\displaystyle \frac{{\lambda }_l{e}_2^2}{2\left[{\left({\zeta }_{2d}+{\epsilon }_{\zeta 2}\right)}^2-{\zeta }_2^2\right]}}\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}{\tilde{\omega }}_{\mathbf{j}}^{\mathbf{T}}{\Pi }_{\mathbf{j}}^{-\mathbf{1}}{\tilde{\omega }}_{\mathbf{j}}+{\displaystyle \frac{1}{2}}{\tilde{\omega }}_{\mathbf{l}}{\Pi }_{\mathbf{l}}^{-\mathbf{1}}{\tilde{\omega }}_{\mathbf{l}}\hfill \end{array}$$

(26)

Then, taking the derivative of V in Equation (26) and substituting it into Equations (13), (16) and (20), we can get the following result as:

$$\begin{array}{cc}\hfill \dot{\mathrm{V}}=& (F_j+k_{1p}{e}_1-{\gamma }_j^T{\omega }_j-M_{1}g{L}_{j}cos{\zeta }_1\hfill \\ \hfill & +{\lambda }_j{\displaystyle \frac{{({\zeta }_{1d}+{\epsilon }_{\zeta 1})}^2+{\zeta }_1^2-{\zeta }_1{e}_1}{{\left[{({\zeta }_{1d}+{\epsilon }_{\zeta 1})}^2-{\zeta }_1^2\right]}^2}}{e}_1){\dot{\zeta }}_1+{\tilde{\omega }}_j^T{\gamma }_j{\dot{\zeta }}_1\hfill \\ \hfill & +(F_l+k_{2p}{e}_2-{\gamma }_l{\omega }_l+M_{2}g\hfill \\ \hfill & +{\lambda }_l{\displaystyle \frac{{({\zeta }_{2d}+{\epsilon }_{\zeta 2})}^2+{\zeta }_2^2-{\zeta }_2{e}_2}{{\left[{({\zeta }_{2d}+{\epsilon }_{\zeta 2})}^2-{\zeta }_2^2\right]}^2}}{e}_2){\dot{\zeta }}_2+{\tilde{\omega }}_l{\gamma }_l{\dot{\zeta }}_1\hfill \end{array}$$

(27)

By substituting Equations (20)–(22) into Equations (27), we get:

$$\begin{array}{cc}\hfill \dot{\mathrm{V}}=& -k_{1d}{\dot{\zeta }}_1^2-k_d{\dot{\zeta }}_2^2-k_{1j}\left({\dot{\zeta }}_3^2+{\dot{\zeta }}_4^2\right){\dot{\zeta }}_1^2\hfill \\ & -k_{2j}\left({\dot{\zeta }}_3^2+{\dot{\zeta }}_4^2\right){\dot{\zeta }}_2^2\le 0\hfill \end{array}$$

(28)

This shows that:

$$\begin{array}{c}\hfill \mathrm{V}\left(t\right)\le \mathrm{V}\left(0\right)\le +\infty \end{array}$$

(29)

Since $\left|{\zeta }_1\left(0\right)\right|\ne {\zeta }_{1d}+{\epsilon }_{\zeta 1},\left|{\zeta }_2\left(0\right)\right|\ne {\zeta }_{2d}+{\epsilon }_{\zeta 2}$, suppose that ${\zeta }_1\left(t\right)$ and ${\zeta }_2\left(t\right)$ tend to go beyond the boundaries of ${\zeta }_{1d}+{\epsilon }_{\zeta 1}$ and ${\zeta }_{2d}+{\epsilon }_{\zeta 2}$, respectively, then it is clear from (26) that $\mathrm{V}\to \infty$, which is in contradiction with $\mathrm{V}\le \infty$, therefore,

$$\begin{array}{c}\hfill \left|{\zeta }_1\left(t\right)\right|<{\zeta }_{1d}+{\epsilon }_{\zeta 1},\left|{\zeta }_2\left(t\right)\right|<{\zeta }_{2d}+{\epsilon }_{\zeta 2}\end{array}$$

(30)

From (28), it is evident that the closed-loop system is Lyapunov stable. We can further conclude that:

$$\begin{array}{c}\hfill \mathrm{V}\in L_{\infty }\Rightarrow {\dot{\zeta }}_1,{\dot{\zeta }}_2,{\dot{\zeta }}_3,{\dot{\zeta }}_4,e_1,e_2,e_2,{\tilde{\omega }}_j,{\widehat{\omega }}_j,{\tilde{\omega }}_l,{\widehat{\omega }}_l\Rightarrow L_{\infty }\end{array}$$

(31)

To prove the Formulas (23) and (24), let $\Phi$ be the largest invariant set contained in $\Theta$, and define $\Theta$ as:

$$\begin{array}{c}\hfill \Theta =\left\{\left(\zeta ,\dot{\zeta }\right):\dot{\mathrm{V}}\left(t\right)=0\right\}\end{array}$$

(32)

As can be seen from (14),(28) and (30), we can obtain that in $\Phi$,

$$\begin{array}{cc}\hfill {\dot{\zeta }}_1=0,& {\dot{\zeta }}_2=0\Rightarrow {\ddot{\zeta }}_1=0,{\ddot{\zeta }}_2=0,f_j=0,f_l=0\hfill \\ \hfill \Rightarrow & e_1={\eta }_1,e_2={\eta }_2\Rightarrow {\zeta }_1={\zeta }_{1d}+{\eta }_1,{\zeta }_2={\zeta }_{2d}+{\eta }_2\hfill \end{array}$$

(33)

where, ${\eta }_1$, ${\eta }_2$ are constants that need to be established. Combined with (21)-(22) and (33), we can get that

$$\begin{array}{cc}& F_j={\eta }_3,F_l={\eta }_4\hfill \end{array}$$

(34)

$$\begin{array}{cc}& {\eta }_3=-k_{1p}{\eta }_1+M_{1}g{L}_{j}cos{\zeta }_1\hfill \\ & -{\lambda }_j{\displaystyle \frac{{({\zeta }_{1d}+{\epsilon }_{\zeta 1})}^2-{\zeta }_{1d}({\zeta }_{1d}+{\eta }_1)}{{\left[{({\zeta }_{1d}+{\epsilon }_{\zeta 1})}^2-{({\zeta }_{1d}+{\eta }_1)}^2\right]}^2}}{\eta }_1\hfill \end{array}$$

(35)

$$\begin{array}{cc}& {\eta }_4=-k_{2p}{\eta }_2+M_{2}g\hfill \\ & -{\lambda }_l{\displaystyle \frac{{({\zeta }_{2d}+{\epsilon }_{\zeta 2})}^2-{\zeta }_{2d}({\zeta }_{2d}+{\eta }_2)}{{\left[{({\zeta }_{2d}+{\epsilon }_{\zeta 2})}^2-{({\zeta }_{2d}+{\eta }_2)}^2\right]}^2}}{\eta }_2\hfill \end{array}$$

(36)

That is, $F_j$ and $F_l$ remain the same in $\Phi$. Substituting Equation (33) into Equation (1) yields:

$$\begin{array}{cc}& M_2{L}_j{\zeta }_2\left({\ddot{\zeta }}_{3}sin\left({\zeta }_3-{\zeta }_1\right)+{\dot{\zeta }}_3^{2}cos\left({\zeta }_3-{\zeta }_1\right)\right)\hfill \\ & +m_2{L}_2{L}_j({\ddot{\zeta }}_{4}sin\left({\zeta }_4-{\zeta }_3\right)\hfill \\ & -{\dot{\zeta }}_3{\dot{\zeta }}_{4}cos\left({\zeta }_4-{\zeta }_3\right)+{\dot{\zeta }}_4^{2}cos\left({\zeta }_3-{\zeta }_4\right))\hfill \\ & ={\eta }_3-M_{1}g{L}_{j}cos{\zeta }_1\hfill \end{array}$$

(37)

That is

$$\begin{array}{cc}& {\displaystyle \frac{d}{dt}}\left[M_2{L}_j{\zeta }_2{\dot{\zeta }}_{3}sin\left({\zeta }_3-{\zeta }_1\right)+m_2{L}_2{L}_j{\dot{\zeta }}_{4}sin\left({\zeta }_4-{\zeta }_3\right)\right]\hfill \\ & ={\eta }_3-M_{1}g{L}_{j}cos{\zeta }_1\hfill \end{array}$$

(38)

The integral of (38) is calculated as follows:

$$\begin{array}{c}\hfill \begin{array}{c}{M}_2{L}_j{\zeta }_2{\dot{\zeta }}_{3}sin\left({\zeta }_3-{\zeta }_1\right)+m_2{L}_2{L}_j{\dot{\zeta }}_{4}sin\left({\zeta }_4-{\zeta }_3\right)\hfill \\ =\left({\eta }_3-M_{1}g{L}_{j}cos{\zeta }_1\right)t+{\eta }_5\hfill \end{array}\end{array}$$

where, ${\eta }_5$ is a constant to be determined. If ${\eta }_3\ne M_{1}g{L}_{j}cos{\zeta }_1$ , then when $t\to \infty$,

$$\begin{array}{c}\hfill \begin{array}{c}|M_2{L}_j{\zeta }_2{\dot{\zeta }}_{3}sin\left({\zeta }_3-{\zeta }_1\right)+m_2{L}_2{L}_j{\dot{\zeta }}_{4}sin\left({\zeta }_4-{\zeta }_3\right)|\to \infty \hfill \end{array}\end{array}$$

(39)

which contradicts with $sin\left({\zeta }_3-{\zeta }_1\right),sin\left({\zeta }_4-{\zeta }_3\right)\in L_{\infty }$ and ${\dot{\zeta }}_3,{\dot{\zeta }}_4\in L_{\infty }$. According to Equations (33)–(36) and (39), in $\Phi$,

$$\begin{array}{cc}& {\eta }_3=M_{1}g{L}_{j}cos{\zeta }_1\hfill \end{array}$$

(40)

$$\begin{array}{cc}& {\eta }_5=m_2{L}_2{L}_j{\dot{\zeta }}_{4}sin\left({\zeta }_4-{\zeta }_3\right)\hfill \\ & +M_2{L}_j{\zeta }_2{\dot{\zeta }}_{3}sin\left({\zeta }_3-{\zeta }_1\right)\hfill \end{array}$$

(41)

$$\begin{array}{cc}& F_j=M_{1}g{L}_{j}cos{\zeta }_1\hfill \end{array}$$

(42)

Then, by combining ${\zeta }_1={\zeta }_{1d}+{\eta }_1$ and ${\eta }_3=0$, we can rewrite (35) as follows:

$$\begin{array}{c}\hfill \left(k_{1p}+{\lambda }_j{\displaystyle \frac{{\left[\right({\zeta }_{1d}+{\epsilon }_{\zeta 1})}^2-{\zeta }_1{\zeta }_{1d}]}{{\left[{({\zeta }_{1d}+{\epsilon }_{\zeta 1})}^2-{({\zeta }_{1d}+{\eta }_1)}^2\right]}^2}}\right){\eta }_1=0\end{array}$$

(43)

Since ${\zeta }_1<{\zeta }_{1d}+{\epsilon }_{\zeta 1}$, we have ${({\zeta }_{1d}+{\epsilon }_{\zeta 1})}^2>\left|{\zeta }_1{\zeta }_{1d}\right|>{\zeta }_1{\zeta }_{1d}$, indicating that the parentheses in Equation (43) are always positive. Thus, from (33) and (43) :

$$\begin{array}{c}\hfill {\eta }_1=0\Rightarrow e_1=0,{\zeta }_1={\zeta }_{1d}\end{array}$$

(44)

By substituting (33) into (2)-(4), and simplifying the process, we can get:

$$\begin{array}{cc}& m_2{L}_2{\ddot{\zeta }}_{4}sin\left({\zeta }_3-{\zeta }_4\right)-M_2{\zeta }_2{\dot{\zeta }}_3^2-m_2{L}_2{\dot{\zeta }}_4^{2}cos\left({\zeta }_3-{\zeta }_4\right)\hfill \\ & =M_{2}g\left(1-cos{\zeta }_3\right)-k_{2p}{e}_2-\Delta \hfill \end{array}$$

(45)

$$\begin{array}{cc}& M_2{\zeta }_2^2{\ddot{\zeta }}_3+m_2{L}_2{\zeta }_2{\ddot{\zeta }}_{4}cos\left({\zeta }_3-{\zeta }_4\right)\hfill \\ & +m_2{L}_2{\zeta }_2{\dot{\zeta }}_4^{2}sin\left({\zeta }_3-{\zeta }_4\right)=-M_{2}g{\zeta }_{2}sin{\zeta }_3\hfill \end{array}$$

(46)

$$\begin{array}{cc}& m_2{L}_2{\zeta }_2{\ddot{\zeta }}_{3}cos\left({\zeta }_3-{\zeta }_4\right)+m_2{L}_2^2{\ddot{\zeta }}_4\hfill \\ & -m_2{L}_2{\zeta }_2{\dot{\zeta }}_3^{2}sin\left({\zeta }_3-{\zeta }_4\right)=-m_{2}g{L}_{2}sin{\zeta }_4\hfill \end{array}$$

(47)

where, $\Delta ={\lambda }_l{\displaystyle \frac{{({\zeta }_{2d}+{\epsilon }_{\zeta 2})}^2-{\zeta }_2^2+{\zeta }_2{e}_2}{{\left[{({\zeta }_{2d}+{\epsilon }_{\zeta 2})}^2-{\zeta }_2^2\right]}^2}}{e}_2$.

Then, by multiplying both sides of (45) with ${\zeta }_{2}sin\left({\zeta }_3-{\zeta }_4\right)$, multiplying both sides of (46) with $cos\left({\zeta }_3-{\zeta }_4\right)$, and calculating the sum of the two new equations, it can be concluded that

$$\begin{array}{cc}& m_2{L}_2{\zeta }_2{\ddot{\zeta }}_4+M_2{\zeta }_2^2{\ddot{\zeta }}_{3}cos\left({\zeta }_3-{\zeta }_4\right)-M_2{\zeta }_2^2{\dot{\zeta }}_3^{2}sin\left({\zeta }_3-{\zeta }_4\right)\hfill \\ & =-\left(M_{2}g+\Delta +k_{2p}{e}_2\right){\zeta }_{2}sin\left({\zeta }_3-{\zeta }_4\right)-M_{2}g{\zeta }_{2}sin{\zeta }_4\hfill \end{array}$$

(48)

Further, by multiplying both sides of (48) with $m_2{L}_2$, multiplying both sides of (47) with $M_2{\zeta }_2$. Then, after mathematical calculation, we can get that

$$\begin{array}{c}\hfill sin\left({\zeta }_3-{\zeta }_4\right)={\displaystyle \frac{m_1{L}_2}{M_{2}g+\Delta +k_{2p}{e}_2}}{\ddot{\zeta }}_4\end{array}$$

(49)

By plugging (49) into (41) and integrating both sides of (41), we get the result:

$$\begin{array}{cc}& -M_2{L}_j{\zeta }_{2}cos\left({\zeta }_3-{\zeta }_1\right)+m_2{L}_2{L}_j\hfill \\ & \times {\displaystyle \frac{m_1{L}_2}{2\left(M_{2}g+\Delta +k_{2p}{e}_2\right)}}{\dot{\zeta }}_4^2={\eta }_{5}t+{\eta }_6\hfill \end{array}$$

(50)

Where, ${\eta }_6$ is a constant. Similarly, if supposing ${\eta }_5\ne 0$, then the left-hand side of (50) will tend to infinity, which contradicts with the result of (31). Consequently, the following conclusions can be drawn by combining (41):

$$\begin{array}{c}\hfill {\eta }_5=0\Rightarrow {\dot{\zeta }}_3=0,{\dot{\zeta }}_4=0,{\ddot{\zeta }}_3=0,{\ddot{\zeta }}_4=0\end{array}$$

(51)

By substituting (51) into (46), (47) we get:

$$\begin{array}{c}\hfill {\zeta }_3=0,{\zeta }_4=0\end{array}$$

(52)

Finally, by plugging (51) and (52) into (45), we get:

$$\begin{array}{c}\hfill e_2=0,{\zeta }_2={\zeta }_{2d}\end{array}$$

(53)

Combining the conclusions of Equations (33), (44) and (51)–(53), we can conclude that the largest invariant set $\Phi$ contains only the equilibrium point, that is

$$\begin{array}{cc}& \underset{t\to \infty }{lim}{\left[\begin{array}{cccccccc}{\zeta }_1& {\zeta }_2& {\zeta }_3& {\zeta }_4& {\dot{\zeta }}_1& {\dot{\zeta }}_2& {\dot{\zeta }}_3& {\dot{\zeta }}_4\end{array}\right]}^T\hfill \\ & ={\left[\begin{array}{cccccccc}{\zeta }_{1d}& {\zeta }_{2d}& 0& 0& 0& 0& 0& 0\end{array}\right]}^T\hfill \end{array}$$

(54)

Therefore, using LaSalle’s invariance theorem, theorem 1 is proved.

5. Simulation and Experiment Results

To validate the control effectiveness of the proposed controller, we have constructed the dynamics model as well as the proposed adaptive controller conducted in the MATLAB/Simulink environment. And the experiment is carried out on the hardware experiment platform of self-made double-swing crane

5.1. Simulation Results and Analysis

In the simulation test, the parameters of the crane system are configured as follows:

$$\begin{array}{c}\mathit{M}=0.5\mathrm{kg},m_1=0.4\mathrm{kg},m_2=0.2\mathrm{kg},L_j=0.6\mathrm{m}\hfill \\ L_2=0.2\mathrm{m},g=9.8\mathrm{m}/{\mathrm{s}}^2,\rho =0.4sin(4.2t)\hfill \end{array}$$

Without losing generality, in all simulation tests, the initial value and desired equilibrium point of the states are set as follows:

$$\begin{array}{c}{\zeta }_1\left(0\right)=0\mathrm{rad},{\zeta }_2\left(0\right)=0.5\mathrm{m},{\zeta }_3\left(0\right)=0\mathrm{rad},{\zeta }_4\left(0\right)=0\mathrm{rad}\hfill \\ {\zeta }_{1d}=\pi /9\mathrm{rad},{\zeta }_{2d}=0.3\mathrm{m},{\zeta }_{3d}=0\mathrm{rad},{\zeta }_{4d}=0\mathrm{rad}\hfill \end{array}$$

To validate the proposed method, we performed several sets of simulations:

$\mathit{SimulationGroup}\mathit{1}:$ We chose the PD control method and the adaptive nonlinear antiswing controller in the article [34] as the comparison methods for this group of experiments. After careful tuning, the control gains for the PD controller are selected as $k_{1d}=80,k_{2d}=10,k_{1p}=70,k_{2p}=15.$ And the nonlinear controller is as follows:

$$\begin{array}{c}{f}_1=-k_{p1}{e}_1-k_{d1}{\dot{e}}_1-{\Phi }_1^T{\widehat{\omega }}_1-{\Phi }_3^T{\widehat{\omega }}_3\hfill \\ f_2=-k_{p2}{e}_2-k_{d2}{\dot{e}}_2-{\Phi }_2^T{\widehat{\omega }}_2\hfill \end{array}$$

And its control gains are:

$$k_{d1}=10,k_{d2}=10,k_{p1}=60,k_{p2}=60$$

The results of the comparative experiment are shown in Figure 3a. To make things more intuitive, the units of angle variable change from radians to degrees. As can be seen from Figure 3a, the PD controller and the nonlinear controller cannot eliminate positioning errors and do not suppress the wave interference. In addition, for PD controller, the peak angles of the hook and the load are 3.09 deg and 8.91 deg respectively. Furthermore, there is static error in the control of jib and rope length, and there are fluctuations exceeding 2 deg even after reaching the target position. For nonlinear controller, the peak angles of the hook and the load are 1.55 deg and 2.94 deg, respectively. Additionally, there is static error and residual oscillation, and an overshoot of 9.47% in the control of jib. In contrast, the control method proposed in this paper can accurately reach the target position under continuous wave interference, and the peak angles of the hook and the load are 0.13 deg and 0.12 deg respectively, and the double pendulum angles can basically converge to 0 at last, which shows that the control method has good control ability.

To visually illustrate the results, we conduct a quantitative analysis of the simulation results for three methods in Table 1, where $t_1$ and $t_2$ represent the arrival times of the jib and rope, respectively.

$\mathit{SimulationGroup}\mathit{2}:$ To assess the robustness of the proposed controller, we added an external interference at $T=10s$ on the basis of the second group of experiments, and the experimental results are shown in Figure 3b. When external interference is added at $T=10s$, the PD control method and the nonlinear controller cannot make the swing angle of the hook and the load return to a small range. In contrast, the proposed controller can show good robustness.

$\mathit{SimulationGroup}\mathit{3}:$ In this group of simulations, we increased the load mass to 1 kg and compared it with PD controller and the mentioned nonlinear controller. The simulation results are shown in the Figure 4a. It can be seen that neither the PD controller nor the nonlinear controller is able to eliminate the positioning errors of the actuators. For the pendulum angles of the hook and the load, the peak angle of the load with the PD controller is 4.12 deg, with residual swinging of over 1 deg. The peak load angle with the nonlinear controller also is 2.11 deg, and cannot eliminate positioning errors and suppress the wave interference. By contrast, the proposed controller delivers precise positioning performance without any error. Additionally, it confines the pendulum angles of the hook and load within 0.1 deg. Simulation results demonstrate that the proposed controller maintains good control performance even after increasing the load mass.

$\mathit{SimulationGroup}\mathit{4}:$ In this group of simulations, we set the initial and target length of the rope to 0.1 m and 0.4 m, respectively, and compared them with the nonlinear controller and PD controller. The simulation results are shown in Figure 4b. It can be seen that there is an overshoot of 24.25% in the nonlinear controller when the rope length becomes longer. Due to the adaptive term and overshoot limit term, the proposed controller achieves accurate positioning performance without any overshoots. Additionally, the fluctuations in the swing angles of the hook and load under the control of PD controller and nonlinear controller become more severe, with peak values of 4.58 deg, 4.05 deg, 4.59 deg and 4.33 deg, respectively. In contrast, the proposed controller can suppress the load swing angle to within 0.06 deg. Simulation results demonstrate that the proposed controller maintains good control performance even after changing the target position.

6. Experiment Results and Analysis

In order to verify the actual control performance of the proposed active disturbance rejection controller, experiments are carried out on the hardware experiment platform of self-made double-swing crane, and the simulation results are compared. As shown in Figure 5, the device consists of personal computer, motion control board, encoders, motors, jib, rope, hook and payload. Two motors are equipped to actuate jib moving and suspension rope varying, respectively. Four encoders are used to measure the jib displacements, suspension rope lengths and swing angles of hook and payload, respectively. The physical parameters of the hardware platform are shown as $\mathit{M}=1.92\mathrm{kg},m_1=0.3\mathrm{kg},m_2=0.5\mathrm{kg},L_j=0.7\mathrm{m},L_2=0.2\mathrm{m},g=9.8\mathrm{m}/{\mathrm{s}}^2$. The initial and target positions are set as ${\zeta }_1\left(0\right)=16\mathrm{deg},{\zeta }_2\left(0\right)=0.5\mathrm{m},{\zeta }_3\left(0\right)=0\mathrm{rad},{\zeta }_4\left(0\right)=0\mathrm{rad},{\zeta }_{1d}=35\mathrm{deg},{\zeta }_{2d}=0.3\mathrm{m},{\zeta }_{3d}=0\mathrm{rad},{\zeta }_{4d}=0\mathrm{rad}$.

The experiment result is shown in Figure 6. It can be seen that the proposed controller can drive the jib and the rope to reach the target position quickly and accurately, and at the same time, the swing angle of the hook and the load can be suppressed within a small range, in which the peak swing angle of the hook is 1.35 deg, and the residual swing angle is within 0.2 deg. The swing angle of the load is within 0.05 deg. This shows that the controller has a good control effect.

7. Conclusion

In this paper, an active disturbance rejection controller based on energy coupling method is proposed. The proposed method can guarantee the accurate control of the jib and rope while suppressing the oscillation of the hook and load in the case of continuous wave disturbance and external disturbance. On the theoretical side, the asymptotic stability of the system is demonstrated through the Lyapunov method and LaSalle’s invariance theorem through strict mathematical analysis, on the application aspect, through simulation and experiment, it is proved that the control method has good control performance. In future work, we consider designing robust control methods for unknown deep-sea dynamics to cope with the underwater working environment.