Block disturbance observer-based sliding mode control for ship-mounted crane

Abstract

The ship-mounted cranes are widely used to transfer heavy cargos between ships and other transportation devices. This paper proposes a block disturbance observer-based sliding mode control method for suppressing the payload sway and achieving accurate positioning. In particular, the proposed block disturbance observer availably estimates external disturbances due to ship rolling, pitching and heaving for ship-mounted crane systems, and which are fed back to the input as compensation signals. By designing nonlinear sliding surface, the payload can arrive at their desired positions quickly while suppressing the payload swing simultaneously. Simulation and experimental results show the effectiveness of the proposed control scheme. Using this method, oscillations are reduced to less than 10 cm during transfer.

Introduction

Ship-mounted cranes are specialized lifting devices installed on ships to handle cargo. As ocean engineering becomes more and more popular and important, the use of ship-mounted cranes has been growing over the past few decades. Consequently, the control problem for ship-mounted cranes has attracted considerable attention.^1–5 Ship-mounted cranes belong to the class of underactuated systems, whose control input dimension is less than the degree of freedom dimension. Thus, the mechatronics and control community of underactuated systems have become research hotspots.^6–10

The main control objective for ship-mounted cranes is to transport the cargos to the desired location quickly, with negligible payload sway during the transportation process and no residual swing at the end. Due to the inherently underactuated nature of cranes, this problem is very challenging. Over the past few decades, the control issues of land-fixed crane systems are extensively studied by scholars all over the world. Specifically, various closed-loop control methods have been developed, such as delayed position-feedback control), ¹¹ quasi-PID control, ¹² partial feedback linearization control, ¹³ sliding mode control (SMC), ¹⁴ and adaptive nonlinear control. ¹⁵ Besides, the open-loop control methods including the intelligent optimal control ¹⁶ and input shaping control^17,18 were often used to satisfy various demands from the industry (e.g., low-cost and user convenience).

Moreover, unlike land-fixed cranes, the ship-mounted cranes are adversely affected by various external disturbances (such as sea waves, sea winds, etc.) due to harsh (sea) working conditions,^4,5,19 which makes the control problem extremely challenging. For the past few decades, the study of ship-mounted cranes has achieved great progress.^20–25 For ship-mounted tower cranes, a nonlinear feedback controller was investigated in Qian and Fang. ²⁶ For container cranes, observer-based nonlinear robust control and continuous integral sliding mode control approaches were proposed.^27–30 For offshore boom cranes, several methods were proposed: time-delayed feedback control algorithm, ³¹ various energy-based control law,^26,32,33 and adaptive control method. ³⁴ What is more, for the unmodeled dynamics and other disturbances, several researchers have presented the active disturbances suppression methods that introduce bounded constants to compensate them.^4,19

Due to the analysis above, the control of ship-mounted cranes remains a challenging and unresolved issue. The ship’s motion that caused by sea waves inevitably degrades control performance. Currently, some researchers use a separation methodology for offshore boom cranes,^19,22 which involves dividing the crane model into two parts: the crane component and the disturbance component. Unfortunately, this approach fails to address disturbances in the unactuated parts of the crane dynamics, resulting in suboptimal control performance. In addition, the existing literature studies mainly focus on the influence of ship roll and heave motion on ship-mounted cranes. To get closer to the engineering practice, this paper meanwhile considers ship roll, pitch and heave, and does not treat the ship motion as disturbances, and as the new state variables and performing intricate mathematical manipulations, the disturbance component is successfully integrated into the crane dynamics. This eliminates the need for direct analysis of ship motion-related disturbances, greatly simplifying the controller design process. Using this new model, a block disturbance observer-based sliding mode controller is developed for ship-mounted cranes, and the stability of the closed-loop system is analyzed by Lyapunov techniques. Finally, the simulation and experimental results validate the effectiveness of the proposed control method.

The main contributions of this paper can be summarized as follows.

• (1) This paper regards the ship roll, pitch and heave motions as a part of the crane dynamics. This process is closer to the actual working conditions of ship-mounted cranes, based on this idea, a novel 5-DOF mathematical model of ship-mounted crane has been established.

• (2) For ship-mounted cranes, a new compensated sliding mode control (CSMC) algorithm, which combines block disturbance observer(BDOB) with sliding mode controller, is developed to restrain the payload sway based on decoupling the original mathematical model of ship-mounted crane, and the BDOB is used to address the persistent disturbances caused by ship roll, pitch and heave motions.

The rest of this paper is organized as follows. In Section “Modeling of a ship-mounted crane,” the dynamic model of the ship-mounted crane system is established. In Section “Controller design,” the system dynamics are further reformulated to analyze conveniently, and the sliding mode controller design and corresponding stability are carried out. Section “Design of block disturbance observer” presents the block disturbance observer to compensate the external disturbances. Some comparative experiments are presented to validate the effectiveness of the proposed method in Section “Simulation and experiment.” Section “Conclusions” draws the paper to a conclusion.

Notations

Throughout the paper, $S_{\alpha }$ and $C_{\alpha }$ are utilized to denote functions $\sin \alpha$ and $\cos \alpha$ , respectively.

Modeling of a ship-mounted crane

Equation of motion

In the mathematical modeling of the ship-mounted crane, the following assumptions are followed:

• The crane is regarded as a rigid body and a particle, and the cable is an inextensible massless cable.

• Ignoring the friction torque between various parts of the crane and ignore the air resistance.

• Considering the external interference, it is assumed that the ship is a uniform rigid body.

The schematic diagram of ship-mounted crane system is shown in Figure 1, where there are three frames [i.e. the inertial coordinate XOY, the ship coordinate x_so_sy_s and crane coordinate xoy], and the corresponding system parameters are illustrated in Table 1. For ship coordinate, x_s is parallel to ship deck and points toward the bow, and z_s is vertical to ship deck down. The origin of the inertial coordinate O coincides with the origin of the ship coordinate o_s.

Figure 1.

Coordinate frames of a ship-mounted crane.

Table 1.

Parameters of ship-mounted cranes.

Symbols	Variables /Parameters	Units
α	Jib slewing angle	rad
β	Jib luffing angle	rad
Lb	Length of jib	m
L c	Rope length	m
Hc	Crane base height	m
θx	Payload swing angle projection on the xz plane	rad
θy	Payload swing angle projection on the yz plane	rad
g	Gravitational acceleration	m/s2
m	Payload mass	kg
Mb	Jib mass	kg

As illustrated in Figure 1, the position of boom tip P $(x_{p1},y_{p1},z_{p1})$ in the crane coordinate system is:

$$\begin{array}{c}{x}_{p1}=L_b{C}_{\beta }{C}_{\alpha }\hfill \\ y_{p1}=L_b{C}_{\beta }{S}_{\alpha }\hfill \\ z_{p1}=-L_b{S}_{\beta }\hfill \end{array}$$

(1)

where $\alpha$ and $\beta$ denote the slew angle and luff angle of crane boom.

The ship motions mainly include the rolling angle $\varphi$ , the pitching angle $\eta$ and the heaving $h$ , through coordinate transformation, which are introduced to the crane dynamics. Thereby, the position of the boom tip in the inertial coordinate is as follows:

$$\begin{array}{c}\multicolumn{1}{c}{\left[\begin{array}{c}{x}_p\hfill \\ \multicolumn{1}{c}{y_p}\\ \multicolumn{1}{c}{z_p}\end{array}\right]=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{-h}\end{array}\right]+\left[\begin{array}{ccc}1\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& C_{\varphi }\hfill & -S_{\varphi }\hfill \\ \multicolumn{1}{c}{0}& S_{\varphi }\hfill & C_{\varphi }\hfill \end{array}\right]}\\ \left[\begin{array}{ccc}{C}_{\eta }\hfill & 0\hfill & S_{\eta }\hfill \\ \multicolumn{1}{c}{0}& 1\hfill & 0\hfill \\ \multicolumn{1}{c}{-S_{\eta }}& 0\hfill & C_{\eta }\hfill \end{array}\right]\left\{\left[\begin{array}{c}{x}_{p1}\hfill \\ \multicolumn{1}{c}{y_{p1}}\\ \multicolumn{1}{c}{z_{p1}}\end{array}\right]\right\}\hfill \end{array}$$

(2)

The position of the payload Q $(x_q,y_q,z_q)$ in the inertial coordinate system is given by

$$\begin{array}{c}{x}_q=x_P+L_c{S}_{{\theta }_x}\hfill \\ y_q=y_p+L_c{C}_{{\theta }_x}{S}_{{\theta }_y}\hfill \\ z_q=z_p+L_c{C}_{{\theta }_x}{C}_{{\theta }_y}\hfill \end{array}$$

(3)

Therefore, the kinetic energy of payload swing caused by system motion and the potential energy of payload are given as follow:

$$T=\frac{1}{2}m({\stackrel{·}{x}}_q^2+{\stackrel{·}{y}}_q^2+{\stackrel{·}{z}}_q^2)$$

(4)

$$V=\mathrm{mg}{z}_q$$

(5)

According to the kinetic energy and potential energy of the payload, the Lagrangian L is defined as

$$L=T-V$$

(6)

For the anti-swing control of the ship-mounted crane, the swing angle and of the crane ${\theta }_x$ and ${\theta }_y$ are selected as the generalized coordinates, according to Lagrange’s equation, there are:

$$\{\begin{array}{c}\frac{d}{\mathrm{dt}}\left(\frac{\partial L}{\partial {\stackrel{·}{\theta }}_x}\right)-\frac{\partial L}{\partial {\theta }_x}=0\hfill \\ \frac{d}{\mathrm{dt}}\left(\frac{\partial L}{\partial {\stackrel{·}{\theta }}_y}\right)-\frac{\partial L}{\partial {\theta }_y}=0\hfill \end{array}$$

(7)

After calculation and sorting, the swing equation of the payload Q is:

$$\begin{array}{c}\multicolumn{1}{c}{m_{11}(·){\stackrel{··}{\theta }}_x+m_{13}(·)\stackrel{··}{\alpha }+m_{14}(·)\stackrel{··}{\beta }+c_{11}(·){\stackrel{·}{\theta }}_x+c_{13}(·)\stackrel{·}{\alpha }}\\ +c_{14}(·)\stackrel{·}{\beta }+c_{15}(·){\stackrel{·}{L}}_c+g_1(·)=0\hfill \end{array}$$

(8)

$$\begin{array}{c}\multicolumn{1}{c}{m_{22}(·){\stackrel{··}{\theta }}_y+m_{23}(·)\stackrel{··}{\alpha }+m_{24}(·)\stackrel{··}{\beta }+c_{21}(·){\stackrel{·}{\theta }}_x+c_{22}(·){\stackrel{·}{\theta }}_y}\\ +c_{23}(·)\stackrel{·}{\alpha }+c_{24}(·)\stackrel{·}{\beta }+c_{25}(·){\stackrel{·}{L}}_c+g_2(·)=0\hfill \end{array}$$

(9)

where

$$\begin{array}{l}{m}_{11}(·)=L_c\\ m_{13}(·)=-L_b{C}_{\beta }{C}_{\alpha }{S}_{{\theta }_x}{S}_{{\theta }_y}-L_b{C}_{\beta }{C}_{\eta }{S}_{\alpha }{C}_{{\theta }_x}\\ m_{14}(·)=-L_b{C}_{\alpha }{S}_{\beta }{C}_{{\theta }_x}+L_b{S}_{\beta }{S}_{\alpha }{S}_{{\theta }_x}{S}_{{\theta }_y}+L_b{C}_{\beta }{S}_{{\theta }_x}{C}_{{\theta }_y}\\ m_{22}(·)=L_c{C}_{{\theta }_x}\\ m_{23}(·)=L_b{C}_{{\theta }_y}{C}_{\beta }{C}_{\alpha }\\ m_{24}(·)=L_b{S}_{{\theta }_y}{C}_{\beta }-L_b{C}_{{\theta }_y}{S}_{\beta }{S}_{\alpha }\\ c_{11}(·)={\stackrel{.}{L}}_c-0.5L_b{C}_{\beta }{C}_{\alpha }\stackrel{.}{\alpha }{C}_{{\theta }_x}{S}_{{\theta }_y}\\ +0.5L_b\stackrel{.}{\beta }{S}_{\beta }{S}_{\alpha }{C}_{{\theta }_x}+0.5L_b{C}_{\beta }\stackrel{.}{\beta }{C}_{{\theta }_x}{C}_{{\theta }_y}\\ +0.5L_b{C}_{\beta }{S}_{\alpha }\stackrel{.}{\alpha }{S}_{{\theta }_x}+0.5L_b{C}_{\alpha }{S}_{\beta }\stackrel{.}{\beta }{S}_{{\theta }_x}\\ c_{13}(·)=0.5L_b{C}_{\beta }{C}_{\alpha }{C}_{{\theta }_x}{S}_{{\theta }_y}{\stackrel{.}{\theta }}_x+L_b\stackrel{.}{\beta }{C}_{\alpha }{S}_{\beta }{S}_{{\theta }_x}{S}_{{\theta }_y}\\ -0.5L_b{C}_{\beta }{S}_{\alpha }{S}_{{\theta }_x}{\stackrel{.}{\theta }}_x+L_b{S}_{\alpha }{S}_{\beta }\stackrel{.}{\beta }{C}_{{\theta }_x}\\ c_{14}(·)=L_b\stackrel{.}{\alpha }{C}_{\alpha }{S}_{\beta }{S}_{{\theta }_x}{S}_{{\theta }_y}+0.5L_b{S}_{\beta }{S}_{\alpha }{C}_{{\theta }_x}{\stackrel{.}{\theta }}_x\\ +0.5L_b{C}_{\beta }{S}_{\alpha }{S}_{{\theta }_x}{\stackrel{.}{\theta }}_x+L_b{S}_{\alpha }{S}_{\beta }\stackrel{.}{\beta }{C}_{{\theta }_x}\\ c_{15}(·)={\stackrel{.}{\theta }}_x\\ c_{21}(·)=-L_c{S}_{{\theta }_x}{\stackrel{.}{\theta }}_y\\ c_{22}(·)=-L_c{S}_{{\theta }_x}{\stackrel{.}{\theta }}_x-0.5L_b{C}_{\beta }{C}_{\alpha }{S}_{{\theta }_y}\stackrel{.}{\alpha }+0.5L_b\stackrel{.}{\beta }{S}_{\beta }{S}_{\alpha }{S}_{{\theta }_y}\\ +C_{{\theta }_x}{\stackrel{.}{L}}_c+0.5L_b{C}_{\beta }\stackrel{.}{\beta }{C}_{{\theta }_y}\\ c_{23}(·)=-L_b\stackrel{.}{\beta }{C}_{\alpha }{S}_{\beta }{C}_{{\theta }_y}-0.5L_b{S}_{{\theta }_y}{\stackrel{.}{\theta }}_y{C}_{\beta }{C}_{\alpha }\\ c_{24}(·)=-L_b{C}_{\alpha }{S}_{\beta }\stackrel{.}{\alpha }{C}_{{\theta }_y}+0.5L_b{\stackrel{.}{\theta }}_y{S}_{\beta }{S}_{\alpha }{S}_{{\theta }_y}+0.5L_b{C}_{\beta }{\stackrel{.}{\theta }}_y{C}_{{\theta }_y}\\ c_{25}(·)={\stackrel{.}{\theta }}_y{C}_{{\theta }_x}\\ g_1(·)=g{S}_{{\theta }_x}{C}_{{\theta }_y}-L_b{C}_{\beta }{\stackrel{.}{\beta }}^2{S}_{\alpha }{S}_{{\theta }_x}{S}_{{\theta }_y}-L_b{C}_{\beta }{S}_{\alpha }{\stackrel{.}{\alpha }}^2{S}_{{\theta }_x}{S}_{{\theta }_y}\stackrel{.}{\alpha }\\ +L_b{C}_{\beta }{\stackrel{.}{\beta }}^2{C}_{\alpha }{C}_{{\theta }_x}{S}_{{\theta }_y}+L_b{C}_{\beta }{C}_{\alpha }{\stackrel{.}{\alpha }}^2{C}_{{\theta }_x}+L_b{\stackrel{.}{\beta }}^2{S}_{\beta }{S}_{{\theta }_x}{C}_{{\theta }_y}\\ -L_c{S}_{{\theta }_x}{C}_{{\theta }_x}{\stackrel{.}{\theta }}_y^2+d_1(·)\\ g_2(·)=g{S}_{{\theta }_y}+L_b{C}_{\beta }\stackrel{.}{\beta }{S}_{\alpha }{C}_{{\theta }_y}+L_b{C}_{\beta }{S}_{\alpha }{\stackrel{.}{\alpha }}^2{C}_{{\theta }_y}+L_b{\stackrel{.}{\beta }}^2{S}_{\beta }{S}_{{\theta }_y}+d_2(·)\\ d_1(·)=S_{{\theta }_x}{C}_{{\theta }_y}\stackrel{··}{h}-L_b{S}_{{\theta }_x}\stackrel{··}{\varphi }{S}_{{\theta }_x}{S}_{\beta }-L_b\stackrel{··}{\eta }{C}_{{\theta }_x}{S}_{\beta }-L_b{S}_{\alpha }{S}_{{\theta }_x}\stackrel{··}{\varphi }{C}_{{\theta }_y}{C}_{\beta }\\ +L_b\stackrel{··}{\eta }{S}_{{\theta }_x}{C}_{{\theta }_y}{C}_{\beta }{C}_{\alpha }+C_{{\theta }_x}{\stackrel{.}{\theta }}_x{C}_{{\theta }_y}\stackrel{.}{h}-2L_b{S}_{{\theta }_x}\stackrel{.}{\varphi }{S}_{{\theta }_y}{C}_{\beta }\stackrel{.}{\beta }\\ -L_b{C}_{{\theta }_x}{\stackrel{.}{\theta }}_x\stackrel{.}{\varphi }{S}_{{\theta }_y}{S}_{\beta }+L_b\stackrel{.}{\eta }{S}_{{\theta }_x}{\stackrel{.}{\theta }}_x{S}_{\beta }-2L_b\stackrel{.}{\eta }{C}_{{\theta }_x}{C}_{\beta }\stackrel{.}{\beta }\\ -2L_b{S}_{\alpha }\stackrel{.}{\alpha }\stackrel{.}{\eta }{S}_{{\theta }_x}{C}_{{\theta }_y}{C}_{\beta }+2L_b{S}_{\alpha }{S}_{{\theta }_x}\stackrel{.}{\varphi }{C}_{{\theta }_y}\stackrel{.}{\beta }{S}_{\beta }\\ -2L_b\stackrel{.}{\alpha }{S}_{{\theta }_x}\stackrel{.}{\varphi }{C}_{{\theta }_y}{C}_{\beta }{C}_{\alpha }-2L_b\stackrel{.}{\eta }{S}_{{\theta }_x}{C}_{{\theta }_y}\stackrel{.}{\beta }{C}_{\alpha }{S}_{\beta }\\ -L_b{S}_{\alpha }{C}_{{\theta }_x}{\stackrel{.}{\theta }}_x\stackrel{.}{\varphi }{C}_{{\theta }_y}{C}_{\beta }+L_b\stackrel{.}{\eta }{C}_{{\theta }_x}{\stackrel{.}{\theta }}_x{C}_{{\theta }_y}{C}_{\beta }{C}_{\alpha }\\ d_2(·)=-L_b{C}_{\beta }{S}_{\alpha }\stackrel{··}{\varphi }{S}_{{\theta }_y}+L_b{C}_{\beta }{C}_{\alpha }{S}_{{\theta }_y}\stackrel{··}{\eta }+S_{{\theta }_y}\stackrel{··}{h}+L_b{S}_{\beta }\stackrel{··}{\varphi }{C}_{{\theta }_y}\\ -2L_b{C}_{\beta }{S}_{\alpha }\stackrel{.}{\alpha }{S}_{{\theta }_y}\stackrel{.}{\eta }-2L_b{C}_{\beta }{C}_{\alpha }\stackrel{.}{\alpha }\stackrel{.}{\varphi }{S}_{{\theta }_y}+2L_b\stackrel{.}{\beta }{S}_{\alpha }{S}_{\beta }\stackrel{.}{\varphi }{S}_{{\theta }_y}\\ -2L_b\stackrel{.}{\beta }{S}_{\beta }{C}_{\alpha }{S}_{{\theta }_y}\stackrel{.}{\eta }-L_b{C}_{\beta }{S}_{\alpha }\stackrel{.}{\varphi }{C}_{{\theta }_y}{\stackrel{.}{\theta }}_y+L_b{C}_{\beta }{C}_{\alpha }{C}_{{\theta }_y}{\stackrel{.}{\theta }}_y\stackrel{.}{\eta }\\ +2L_b{C}_{\beta }\stackrel{.}{\beta }\stackrel{.}{\varphi }{C}_{{\theta }_y}-L_b{S}_{\beta }\stackrel{.}{\varphi }{S}_{{\theta }_y}{\stackrel{.}{\theta }}_y+C_{{\theta }_y}{\stackrel{.}{\theta }}_y\stackrel{.}{h}\end{array}$$

In order to obtain the tension in the lift wire, the balance equation of the payload is established in the vertical direction as:

$$m{\stackrel{··}{z}}_q=\mathrm{mg}-T_q{C}_{{\theta }_x}$$

(10)

In (10), $T_q$ is the tension of the lift rope.

Substituting $z_q$ of (3) into (10) yields:

$$\begin{array}{c}\multicolumn{1}{c}{T_q=m{L}_b{C}_{\beta }\sec {\theta }_x\stackrel{··}{\beta }-m{C}_{{\theta }_y}{\stackrel{··}{L}}_c+m{L}_c{C}_{{\theta }_y}\tan {\theta }_x{\stackrel{··}{\theta }}_x}\\ +m{L}_c{S}_{{\theta }_y}{\stackrel{··}{\theta }}_y-2m{L}_c\tan {\theta }_x{\stackrel{·}{\theta }}_x{S}_{{\theta }_y}{\stackrel{·}{\theta }}_y+2m{\stackrel{·}{L}}_c{S}_{{\theta }_y}{\stackrel{·}{\theta }}_y\hfill \\ +2m{\stackrel{·}{L}}_c{C}_{{\theta }_y}\tan {\theta }_x{\stackrel{·}{\theta }}_x+v(·)+d(·)\hfill \end{array}$$

(11)

where

$$v(·)=m[\mathrm{gsec}{\theta }_x-L_b{\stackrel{·}{\beta }}^2\sec {\theta }_x{S}_{\beta }+L_c{\stackrel{·}{\theta }}_y^2{C}_{{\theta }_y}+2L_c{\stackrel{·}{\theta }}_x^2{C}_{{\theta }_y}]$$

$$\begin{array}{c}d(·)=msec{\theta }_x[L_b{C}_{\beta }{C}_{\alpha }\stackrel{··}{\eta }-L_b{C}_{\beta }{S}_{\alpha }\stackrel{··}{\varphi }+\stackrel{··}{h}+2L_b{S}_{\beta }\stackrel{.}{\beta }{S}_{\alpha }\stackrel{.}{\varphi }\hfill \\ -2L_b{C}_{\beta }{C}_{\alpha }\stackrel{·}{\alpha }\stackrel{·}{\varphi }-2L_b{S}_{\beta }\stackrel{·}{\beta }{C}_{\alpha }\stackrel{·}{\eta }-2L_b{C}_{\beta }{S}_{\alpha }\stackrel{·}{\alpha }\stackrel{·}{\eta }]\hfill \end{array}$$

Motor drive equation

The jib slewing, luffing and the payload lifting movements of the ship-mounted crane are driven by the slewing motor, luffing motor and lifting motor respectively. According to the torque balance equation on the motor output shaft, the differential equation of the motor with torque as input and motor output shaft angle as output is expressed as follows:

$$J_i\frac{d^2{\chi }_i}{d{t}^2}+f_i\frac{d{\chi }_i}{\mathrm{dt}}={\tau }_{\mathrm{mi}}-M_{\mathrm{ci}}(i=1,2,3)$$

(12)

In (12), $i=1,2,3$ refer to slewing motor, luffing motor and lifting motor respectively; ${\chi }_i$ is the rotation angle of motor output shaft; $J_i$ is the moment of inertia of the motor; $f_i$ is the damping coefficient; ${\tau }_{\mathrm{mi}}$ is input torque for motor; $M_{\mathrm{ci}}$ is the load torque.

For slewing motor, the differential equation is:

$$J_1\frac{d^2{\chi }_1}{d{t}^2}+f_1\frac{d{\chi }_1}{\mathrm{dt}}={\tau }_{m1}-M_{c1}$$

(13)

In the crane system, the slewing motor is connected to the boom support through a reducer, and the reduction ratio of the reducer is 160, that is ${\chi }_1=160\alpha$ , substituted into (13) to obtain:

$$\frac{J_1}{160}\frac{d^2\alpha }{d{t}^2}+\frac{f_1}{160}\frac{d\alpha }{\mathrm{dt}}={\tau }_{m1}-M_{c1}$$

(14)

For luffing motor, the differential equation is:

$$J_2\frac{d^2{\chi }_2}{d{t}^2}+f_2\frac{d{\chi }_2}{\mathrm{dt}}={\tau }_{m2}-M_{c2}$$

(15)

It is seen obtained from the geometric relationship of l in Figure 2.

$$l=\sqrt{L_b^2+H_c^2-2L_b{H}_c{S}_{\beta }}$$

(16)

It can be seen from Figure 2 that the relationship between rope length change and motor output angle is

$$l=-\frac{R_1}{2}{\chi }_2$$

(17)

where $R_1$ is the rope sheave radius of the luffing motor.

Figure 2.

The structure of crane.

Take the derivative of (16) with regard to time and combine (17) to obtain the differential equation of luffing motor expressed in luffing angle $\beta$ as:

$$J_2{f}_1(\beta )\frac{d^2\beta }{d{t}^2}+f_2{f}_1(\beta )\frac{d\beta }{\mathrm{dt}}+J_2{f}_2(\beta ){\left(\frac{d\beta }{\mathrm{dt}}\right)}^2={\tau }_{m2}-M_{c2}$$

(18)

where

$$f_1(\beta )=\frac{2L_b{H}_c{C}_{\beta }}{R_1\sqrt{L_b^2+H_c^2-2L_b{H}_c{S}_{\beta }}},$$

$$\begin{array}{c}\multicolumn{1}{c}{f_2(\beta )=\frac{2L_b{H}_c{S}_{\beta }}{R_1\sqrt{L_b^2+H_c^2-2L_b{H}_c{S}_{\beta }}}}\\ -\frac{2L_b^2{H}_c^2{\cos }^2\beta }{R_1{[L_b^2+H_c^2-2L_b{H}_c{S}_{\beta }]}^{3/2}}\hfill \end{array}$$

For lifting motor

$$J_3\frac{d^2{\chi }_3}{d{t}^2}+f_3\frac{d{\chi }_3}{\mathrm{dt}}={\tau }_{m3}-M_{c3}$$

(19)

The relationship between the rope length and the output angle of the lifting motor is as follows·

$$\{\begin{array}{c}{\stackrel{·}{\chi }}_3=\frac{{\stackrel{·}{L}}_c}{R_2}\hfill \\ {\stackrel{··}{\chi }}_3=\frac{{\stackrel{··}{L}}_c}{R_2}\hfill \end{array}$$

(20)

where $R_2$ is the rope sheave radius of the lifting motor.

Substituting (20) into (19), the differential equation of the rotating part of the hoisting motor represented by $L_c$ is:

$$\frac{J_3}{R_2}\frac{d^2{L}_c}{d{t}^2}+\frac{f_3}{R_2}\frac{d{L}_c}{\mathrm{dt}}={\tau }_{m3}-M_{c3}$$

(21)

The load torque of crane slewing, luffing and lifting motors can be expressed as:

$$M_{c1}=[J_b\mathrm{co}{s}^2\beta +M_b{\left(\frac{L_b}{2}\cos \beta \right)}^2]\stackrel{··}{\alpha }$$

(22)

$$M_{c2}=\frac{2J_{b\beta }\left(\beta \right)\stackrel{··}{\beta }+L({\theta }_x,{\theta }_y,\beta )T_q}{f_1(\beta )}$$

(23)

$$M_{c3}=T_q{R}_2$$

(24)

where $J_b$ is the moment of inertia of the boom to the centroid axis, $M_b$ is the mass of the boom, $J_{b\beta }(\beta )=J_b+M_b{\left(\frac{L_b}{2}\right)}^2{\cos }^2\beta$ , $L({\theta }_x,{\theta }_y,\beta )=L_b{C}_{{\theta }_x}{C}_{{\theta }_y}{C}_{\beta }$

Substituting (22)-(24) into (14), (18) and (21) respectively, and sorting them out:

$$m_{33}(·)\stackrel{··}{\alpha }+g_3(·)={\tau }_{m1}$$

(25)

$$\begin{array}{c}\multicolumn{1}{c}{m_{41}(·){\stackrel{··}{\theta }}_x+m_{42}(·){\stackrel{··}{\theta }}_y+m_{44}(·)\stackrel{··}{\beta }+m_{45}(·){\stackrel{··}{L}}_c+c_{41}(·){\stackrel{·}{\theta }}_x}\\ +c_{42}(·){\stackrel{·}{\theta }}_y+c_{44}(·)\stackrel{·}{\beta }+c_{45}(·){\stackrel{·}{L}}_c+g_4(·)={\tau }_{m2}\hfill \end{array}$$

(26)

$$\begin{array}{c}\multicolumn{1}{c}{m_{51}(·){\stackrel{··}{\theta }}_x+m_{52}(·){\stackrel{··}{\theta }}_y+m_{54}(·)\stackrel{··}{\beta }+m_{55}(·){\stackrel{··}{L}}_c+c_{51}(·){\stackrel{·}{\theta }}_x}\\ +c_{52}(·){\stackrel{·}{\theta }}_y+c_{55}(·){\stackrel{·}{L}}_c+g_5(·)={\tau }_{m3}\hfill \end{array}$$

(27)

where

$$\begin{array}{c}{m}_{33}(·)=J_1/160+J_{\beta }\mathrm{co}{s}^2\beta +M_b{L}_b^2{C}_{\beta }/4\hfill \\ m_{41}(·)=m{L}_c{f}_1^{-1}\left(\beta \right)L({\theta }_x,{\theta }_y,\beta )C_{{\theta }_y}\tan {\theta }_x\hfill \\ m_{42}(·)=m{L}_c{f}_1^{-1}\left(\beta \right)L({\theta }_x,{\theta }_y,\beta )S_{{\theta }_y}\hfill \\ m_{44}(·)=J_2{f}_1(\beta )+2f_1^{-1}\left(\beta \right)J_{b\beta }\left(\beta \right)\hfill \\ +f_1^{-1}\left(\beta \right)L({\theta }_x,{\theta }_y,\beta )m{L}_b{C}_{\beta }\sec {\theta }_x\hfill \\ m_{45}(·)=-{\mathrm{mf}}_1^{-1}\left(\beta \right)L({\theta }_x,{\theta }_y,\beta )C_{{\theta }_y}\hfill \\ m_{51}(·)=m{L}_c{C}_{{\theta }_y}\tan {\theta }_x{R}_2\hfill \\ m_{52}(·)=m{L}_c{S}_{{\theta }_y}{R}_2\hfill \\ m_{54}(·)=m{L}_b{C}_{\beta }\sec {\theta }_x{R}_2\hfill \\ m_{55}(·)=J_3/R_2-m{C}_{{\theta }_y}{R}_2\hfill \\ c_{33}(·)=f_1\stackrel{·}{\alpha }/160\hfill \\ c_{41}(·)=f_1^{-1}\left(\beta \right)L({\theta }_x,{\theta }_y,\beta )m[-L_c{\stackrel{·}{\theta }}_y{S}_{{\theta }_y}\tan {\theta }_x+{\stackrel{·}{L}}_c{C}_{{\theta }_y}\tan {\theta }_x]\hfill \\ c_{42}(·)=f_1^{-1}\left(\beta \right)L({\theta }_x,{\theta }_y,\beta )m[-L_c{\stackrel{·}{\theta }}_x{S}_{{\theta }_y}\tan {\theta }_x+{\stackrel{·}{L}}_c{S}_{{\theta }_y}]\hfill \\ c_{44}(·)=f_2{f}_1(\beta )\hfill \\ c_{45}(·)=f_1^{-1}\left(\beta \right)L({\theta }_x,{\theta }_y,\beta )m[{\stackrel{·}{\theta }}_y{S}_{{\theta }_y}+{\stackrel{·}{\theta }}_x{C}_{{\theta }_y}\tan {\theta }_x]\hfill \\ c_{51}(·)=R_{2}m[-L_c{\stackrel{·}{\theta }}_y{S}_{{\theta }_y}\tan {\theta }_x+{\stackrel{·}{L}}_c{C}_{{\theta }_y}\tan {\theta }_x]\hfill \\ c_{52}(·)=R_{2}m[-L_c{\stackrel{·}{\theta }}_x{S}_{{\theta }_y}\tan {\theta }_x+{\stackrel{·}{L}}_c{S}_{{\theta }_y}]\hfill \\ c_{55}(·)=R_{2}m[{\stackrel{·}{L}}_c({\stackrel{·}{\theta }}_y{S}_{{\theta }_y}+{\stackrel{·}{\theta }}_x{C}_{{\theta }_y}\tan {\theta }_x)]+f_3/R_2\hfill \\ g_3(·)=0\hfill \\ g_4(·)=f_1^{-1}\left(\beta \right)L({\theta }_x,{\theta }_y,\beta )v(·)+J_2{f}_2\left(\beta \right){\stackrel{·}{\beta }}^2+d_4(·)\hfill \\ d_4(·)=f_1^{-1}\left(\beta \right)L({\theta }_x,{\theta }_y,\beta )d(·)\hfill \\ g_5(·)=v(·)R_2+d_5(·)\hfill \\ d_5(·)=d(·)R_2\hfill \end{array}$$

Rewriting (8), (9), (25), (26), and (27) in matrix form, the overall dynamic equation of ship-mounted crane system is obtained as follows:

$$M\left(\mathit{q}\right)\stackrel{··}{\mathit{q}}+C(·)\stackrel{·}{\mathit{q}}+\mathrm{G}(·)=\tau$$

(28)

where $\mathit{q}$ , $M\left(\mathit{q}\right)$ , $C(\mathit{q},\stackrel{·}{\mathit{q}})$ , $\mathrm{G}(·)$ and $\mathrm{\tau }$ are the state vectors, the symmetric mass matrix, the Coriolis and centrifugal matrix, the gravitational force vector and input vector, respectively, defined as follows:

$$\begin{array}{c}\multicolumn{1}{c}{\mathit{q}={\left[{\theta }_x{\theta }_y\alpha \beta L_c\right]}^T\in R^5,\mathrm{G}(·)}\\ ={\left[g_1(·)g_2(·)g_3(·)g_4(·)g_5(·)\right]}^T\in R^5\hfill \end{array}$$

$$\begin{array}{c}M(q)=\left[\begin{array}{ccccc}{m}_{11}\hfill & 0\hfill & m_{13}\hfill & m_{14}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& m_{22}\hfill & m_{23}\hfill & m_{24}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & m_{33}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{m_{41}}& m_{42}\hfill & 0\hfill & m_{44}\hfill & m_{45}\hfill \\ \multicolumn{1}{c}{m_{51}}& m_{52}\hfill & 0\hfill & m_{54}\hfill & m_{55}\hfill \end{array}\right]\in R^{5\times 5},\hfill \\ C(·)=\left[\begin{array}{ccccc}{c}_{11}\hfill & 0\hfill & c_{13}\hfill & c_{14}\hfill & c_{15}\hfill \\ \multicolumn{1}{c}{c_{21}}& c_{22}\hfill & c_{23}\hfill & c_{24}\hfill & c_{25}\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & c_{33}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{c_{41}}& c_{42}\hfill & 0\hfill & c_{44}\hfill & c_{45}\hfill \\ \multicolumn{1}{c}{c_{51}}& c_{52}\hfill & 0\hfill & 0\hfill & c_{55}\hfill \end{array}\right]\in R^{5\times 5},\hfill \end{array}$$

$$\begin{array}{cc}\multicolumn{1}{c}{\tau }& ={\left[\begin{array}{cc}{\tau }_1\hfill & {\tau }_2\hfill \end{array}\right]}^T\in R^5,{\tau }_1={\left[\begin{array}{cc}0\hfill & 0\hfill \end{array}\right]}^T\in R^2,\hfill \\ \multicolumn{1}{c}{{\tau }_2}& ={\left[\begin{array}{ccc}{\tau }_{m1}\hfill & {\tau }_{m2}\hfill & {\tau }_{m3}\hfill \end{array}\right]}^T\in R^3.\hfill \end{array}$$

Controller design

Considering the actual working conditions of the ship-mounted crane, the payload will not reach above the boom tip, so there is no special description. The following reasonable assumptions will be made:

• Assumption 3.1. During the operation of the ship-mounted crane, the payload is always below the boom tip of the boom, that is $\left|{\theta }_x\right|<\pi /2$ , and $\left|{\theta }_y\right|<\pi /2$ .

• Assumption 3.2. During the operation of ship crane, the length of lifting rope $L_c>0$ .

• Lemma 1. According to the above assumptions, it can be proved that, when $L_c>0$ and $\left|{\theta }_x\right|<\pi /2$ , the inertia matrix $M(q)$ is a positive definite matrix.

The vector $Q\in R^5$ is introduced to represent $q(t)$ all sets that can be valued. Here, the Q value is determined by the structural parameters and operation mode of the ship crane.

• Proof 1. $|M(q)|\ne 0\forall q\in Q$ .

• Proof 2. $\Vert M^{-1}(q)\Vert \le M_0\forall q\in Q$ , here $\Vert ·\Vert$ is the Euclidean norm and $M_0$ is a positive constant.

According to lemma 1, the inertia matrix $M(q)$ is a positive definite matrix, so proof 1 is true, and because $M(q)$ is bounded, proof 2 is also true.

Decoupling of ship-mounted crane dynamic equations

The dynamic equation of ship-mounted crane is shown in (28). It can be seen from the mathematical model that the ship-mounted crane is a typical underactuated system. Generally, the control of underactuated system is complex. Therefore, the mathematical model of ship-mounted crane needs to be decoupled to generate a form more suitable for the controller design of underactuated system. The ship-mounted crane has five control quantities, which are related to five output components respectively. As an underactuated system, five output variables are driven by three input control signals, in which the vector related to the underactuated state is ${\mathbf{q}}_{\mathrm{u}}=[{\theta }_x{\theta }_y]$ and the state vector directly related to the driving signal is ${\mathbf{q}}_{\mathrm{a}}=[\begin{array}{ccc}\alpha \hfill & \beta \hfill & L_c\hfill \end{array}]$ . According to the underactuated and driving states, (28) can be decomposed into the following two subsystems:

$$\begin{array}{c}\multicolumn{1}{c}{M_{11}\left(\mathit{q}\right){\stackrel{··}{q}}_u+M_{12}\left(\mathit{q}\right){\stackrel{··}{q}}_a+C_{11}(\mathit{q},\stackrel{·}{\mathit{q}}){\stackrel{·}{q}}_u+C_{12}(\mathit{q},\stackrel{·}{\mathit{q}}){\stackrel{·}{q}}_a}\\ +G_1\left(\mathit{q}\right)={\tau }_1\hfill \end{array}$$

(29)

$$\begin{array}{c}\multicolumn{1}{c}{M_{21}\left(\mathit{q}\right){\stackrel{··}{q}}_u+M_{22}\left(\mathit{q}\right){\stackrel{··}{q}}_a+C_{21}(\mathit{q},\stackrel{·}{\mathit{q}}){\stackrel{·}{q}}_u+C_{22}(\mathit{q},\stackrel{·}{\mathit{q}}){\stackrel{·}{q}}_a}\\ +G_2\left(\mathit{q}\right)={\tau }_2\hfill \end{array}$$

(30)

where,

$$\begin{array}{c}{M}_{11}(q)=\left[\begin{array}{cc}{m}_{11}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& m_{22}\hfill \end{array}\right],M_{12}(q)=\left[\begin{array}{ccc}{m}_{13}\hfill & m_{14}\hfill & 0\hfill \\ \multicolumn{1}{c}{m_{23}}& m_{24}\hfill & 0\hfill \end{array}\right],\hfill \\ M_{21}(q)=\left[\begin{array}{cc}0\hfill & 0\hfill \\ \multicolumn{1}{c}{m_{41}}& m_{42}\hfill \\ \multicolumn{1}{c}{m_{51}}& m_{52}\hfill \end{array}\right],M_{22}(q)=\left[\begin{array}{ccc}{m}_{33}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& m_{44}\hfill & m_{45}\hfill \\ \multicolumn{1}{c}{0}& m_{54}\hfill & m_{55}\hfill \end{array}\right],\hfill \\ C_{11}(q,\stackrel{·}{q})=\left[\begin{array}{cc}{c}_{11}\hfill & 0\hfill \\ \multicolumn{1}{c}{c_{21}}& c_{22}\hfill \end{array}\right],C_{12}(q,\stackrel{·}{q})=\left[\begin{array}{ccc}{c}_{13}\hfill & c_{14}\hfill & c_{15}\hfill \\ \multicolumn{1}{c}{c_{23}}& c_{24}\hfill & c_{25}\hfill \end{array}\right],\hfill \\ C_{21}(q,\stackrel{·}{q})=\left[\begin{array}{cc}0\hfill & 0\hfill \\ \multicolumn{1}{c}{c_{41}}& c_{42}\hfill \\ \multicolumn{1}{c}{c_{51}}& c_{52}\hfill \end{array}\right],C_{22}(q,\stackrel{·}{q})=\left[\begin{array}{ccc}{c}_{33}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& c_{44}\hfill & c_{45}\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & c_{55}\hfill \end{array}\right],\hfill \end{array}$$

$$G_1(·)=[g_1(·)g_2(·){]}^T,G_2(·)=[g_3(·)g_4(·)g_5(·){]}^T.$$

The feedback signal of the control system includes state variable $\mathit{q}={\left[q_1{q}_2{q}_3{q}_4{q}_5\right]}^T\in R^5$ and their differential $\stackrel{·}{\mathit{q}}={\left[{\stackrel{·}{q}}_1{\stackrel{·}{q}}_2{\stackrel{·}{q}}_3{\stackrel{·}{q}}_4{\stackrel{·}{q}}_5\right]}^T\in R^5$ . In order to reduce the cost, only five sensors are configured in the actual system to measure five state variables q. The other feedback state $\stackrel{·}{q}$ is obtained by numerical differentiation of the measured signal, and a low-pass filter is used in the numerical differentiation to reduce the noise.

The control scheme is mathematically designed to make the driving state $q_a$ asymptotically close to the reference position $q_{\mathrm{ad}}$ , and make the underactuated state $q_u$ approach the expected value $q_{\mathrm{ud}}$ . According to Lemma 1, $M_{11}(q)$ is positive definite for each q, the dynamic equation (29) of the underactuated part can be rewritten as

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{··}{q}}_u=M_{11}^{-1}\left(\mathit{q}\right)[{\tau }_1-M_{12}\left(\mathit{q}\right){\stackrel{··}{q}}_a-C_{11}(\mathit{q},\stackrel{·}{\mathit{q}}){\stackrel{·}{q}}_u}\\ -C_{12}(\mathit{q},\stackrel{·}{\mathit{q}}){\stackrel{·}{q}}_a-G_1\left(q\right)]\hfill \end{array}$$

(31)

The simplified dynamic model of the system can be obtained by substituting (31) into (30):

$$\overline{M}\left(\mathit{q}\right){\stackrel{··}{q}}_a+{\overline{C}}_1(\mathit{q},\stackrel{·}{\mathit{q}}){\stackrel{·}{q}}_u+{\overline{C}}_2(\mathit{q},\stackrel{·}{\mathit{q}}){\stackrel{·}{q}}_a+\overline{G}\left(\mathit{q}\right)=u$$

(32)

where,

$$\begin{array}{cc}\multicolumn{1}{c}{\overline{M}\left(\mathit{q}\right)}& =M_{22}\left(\mathit{q}\right)-M_{21}\left(\mathit{q}\right)M_{11}^{-1}\left(\mathit{q}\right)M_{12}\left(\mathit{q}\right)\hfill \\ \multicolumn{1}{c}{{\overline{C}}_1(\mathit{q},\stackrel{·}{\mathit{q}})}& =C_{21}(\mathit{q},\stackrel{·}{\mathit{q}})-M_{21}\left(\mathit{q}\right)M_{11}^{-1}\left(\mathit{q}\right)C_{11}(\mathit{q},\stackrel{·}{\mathit{q}})\hfill \\ \multicolumn{1}{c}{{\overline{C}}_2(\mathit{q},\stackrel{·}{\mathit{q}})}& =C_{22}(\mathit{q},\stackrel{·}{\mathit{q}})-M_{21}\left(q\right)M_{11}^{-1}\left(\mathit{q}\right)C_{12}(\mathit{q},\stackrel{·}{\mathit{q}})\hfill \\ \multicolumn{1}{c}{\overline{G}\left(\mathit{q}\right)}& =G_2\left(\mathit{q}\right)-M_{21}\left(\mathit{q}\right)M_{11}^{-1}\left(\mathit{q}\right)G_1\left(\mathit{q}\right)\hfill \\ \multicolumn{1}{c}{u}& ={\tau }_2\hfill \end{array}$$

Take $q_a$ as the system output and modify the driving dynamics (32) as follows

$${\stackrel{··}{q}}_a={\overline{M}}^{-1}\left(\mathit{q}\right)[u-{\overline{C}}_1(\mathit{q},\stackrel{·}{\mathit{q}}){\stackrel{·}{q}}_u-{\overline{C}}_2(\mathit{q},\stackrel{·}{\mathit{q}}){\stackrel{·}{q}}_a-\overline{G}\left(\mathit{q}\right)]$$

(33)

where $\overline{M}(q)$ is a positive definite matrix.

Sliding surface design

Let ${\alpha }_r$ , ${\beta }_r$ and $L_{\mathrm{cr}}$ be the desired values of slewing angle $\alpha$ , luffing angle $\beta$ and rope length $L_c$ of ship-mounted crane respectively. Obviously, during the working process of the ship crane system, it is expected that the payload will not swing, so the desired values of ${\theta }_x$ and ${\theta }_y$ are 0, that is $q_{\mathrm{ud}}=[00{]}^T\in R^2$ , and $q_{\mathrm{ad}}=[{\alpha }_r{\beta }_r{L}_{\mathrm{cr}}{]}^T\in R^3$ .

Define the tracking error vectors $e_a\in R^3$ and $e_u\in R^2$ as follows:

$$\{\begin{array}{c}{e}_a=q_a-q_{\mathrm{ad}}=[\alpha -{\alpha }_r\beta -{\beta }_r{L}_c-L_{\mathrm{cr}}{]}^T\hfill \\ \multicolumn{1}{c}{e_u=q_u-q_{\mathrm{ud}}=[{\theta }_x{\theta }_y{]}^T}\end{array}$$

(34)

In (34), $e_a$ is the error vector corresponding to the driving state, $e_u$ is the error vector associated with the underactuated state.

Design the sliding surface $s\in R^3$ in the following form:

$$s={\stackrel{·}{e}}_a+{\lambda }_1{e}_a+{\lambda }_3{\stackrel{·}{e}}_u+{\lambda }_2{e}_u$$

(35)

where,

$$\begin{array}{c}{\lambda }_1=\left[\begin{array}{ccc}{\lambda }_{11}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& {\lambda }_{12}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & {\lambda }_{13}\hfill \end{array}\right],{\mathrm{\lambda }}_3=\left[\begin{array}{cc}{\lambda }_{31}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& {\lambda }_{32}\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \end{array}\right],\hfill \\ {\lambda }_2=\left[\begin{array}{cc}{\lambda }_{21}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& {\lambda }_{22}\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \end{array}\right]\hfill \end{array}$$

The derivative of (35) shows that the change rate of sliding surface is:

$$\stackrel{·}{s}={\stackrel{··}{e}}_a+{\lambda }_1{\stackrel{·}{e}}_a+{\lambda }_3{\stackrel{··}{e}}_u+{\lambda }_2{\stackrel{·}{e}}_u$$

(36)

It can be seen from the sliding mode surface that when s and ⋅ both approach zero, the control system should make e_a, e_u, ė_a, and ė_u approach zero, that is, when the system is in sliding mode motion, the control goal is achieved.

Design of sliding mode control law

In order to ensure that the system can asymptotically converge to the sliding mode surface, the sliding mode control law can be designed according to Lyapunov theorem.

Let Lyapunov function be:

$$V(t)=\frac{1}{2}{s}^{T}s$$

(37)

The derivative of V(t) with regard to time is defined as:

$$\stackrel{·}{V}(t)=s^T\stackrel{·}{s}$$

(38)

According to Lyapunov theorem, the condition of system stability is $\stackrel{·}{V}(t)=s^T\stackrel{·}{s}\le 0$ , so choose an appropriate sliding mode control law to meet the above conditions, that is, complete the design of sliding mode control law.

Substituting (34) into (36) yields the expression of the change rate of the sliding mode function with regard to each order derivatives of the state vector:

$$\stackrel{·}{s}={\stackrel{··}{e}}_a+{\lambda }_1{\stackrel{·}{e}}_a+{\lambda }_3{\stackrel{··}{e}}_u+{\lambda }_2{\stackrel{·}{e}}_u={\stackrel{··}{q}}_a+{\lambda }_1{\stackrel{·}{q}}_a+{\lambda }_3{\stackrel{··}{q}}_u+{\lambda }_2{\stackrel{·}{q}}_u$$

(39)

Substituting (31) into (39) yields:

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{s}=[{\mathrm{I}}_3-{\lambda }_3{M}_{11}^{-1}\left(\mathit{q}\right)M_{12}\left(\mathit{q}\right)]{\stackrel{··}{q}}_a+({\lambda }_1-{\lambda }_3{M}_{11}^{-1}\left(\mathit{q}\right)C_{12}(\mathit{q},\stackrel{·}{\mathit{q}})){\stackrel{·}{q}}_a}\\ +({\lambda }_2-{\lambda }_3{M}_{11}^{-1}\left(\mathit{q}\right)C_{11}(\mathit{q},\stackrel{·}{\mathit{q}})){\stackrel{·}{q}}_u-{\lambda }_3{M}_{11}^{-1}\left(\mathit{q}\right)G_1\left(\mathit{q}\right)\hfill \end{array}$$

(40)

Substituting (33) into (40), it can be further simplified as:

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{s}=[{\mathrm{I}}_3-{\lambda }_3{M}_{11}^{-1}\left(\mathit{q}\right)M_{12}\left(\mathit{q}\right)]{\overline{M}}^{-1}\left(\mathit{q}\right)}\\ [u-{\overline{C}}_1(\mathit{q},\stackrel{·}{\mathit{q}}){\stackrel{·}{q}}_u-{\overline{C}}_2(\mathit{q},\stackrel{·}{\mathit{q}}){\stackrel{·}{q}}_a-\overline{G}\left(\mathit{q}\right)]\hfill \\ +({\lambda }_1-{\lambda }_3{M}_{11}^{-1}\left(\mathit{q}\right)C_{12}(\mathit{q},\stackrel{·}{\mathit{q}})){\stackrel{·}{q}}_a\hfill \\ +({\lambda }_2-{\lambda }_3{M}_{11}^{-1}\left(\mathit{q}\right)C_{11}(\mathit{q},\stackrel{·}{\mathit{q}})){\stackrel{·}{q}}_u\hfill \\ -{\lambda }_3{M}_{11}^{-1}\left(\mathit{q}\right)G_1\left(\mathit{q}\right)\hfill \end{array}$$

(41)

By selecting the control input signal u, the control target can be achieved.

$$\stackrel{·}{V}\left(t\right)=s^T\stackrel{·}{s}\le -K_i\left|s_i\right|\forall i\in \{1,2,3\}$$

(42)

where $K_i$ is a positive constant.

Note that by imposing the condition in (42) on s, we guarantee that s converges to zero asymptotically. The control u is chosen as follows:

$$\begin{array}{c}\multicolumn{1}{c}{\mathit{u}={\overline{C}}_1(\mathit{q},\stackrel{·}{\mathit{q}}){\stackrel{·}{q}}_u+{\overline{C}}_2(\mathit{q},\stackrel{·}{\mathit{q}}){\stackrel{·}{q}}_a+\overline{G}\left(\mathit{q}\right)-\overline{M}\left(\mathit{q}\right)}\\ {[{\mathrm{I}}_3-{\lambda }_3{M}_{11}^{-1}\left(\mathit{q}\right)M_{12}\left(\mathit{q}\right)]}^{-1}\times [({\lambda }_1-{\lambda }_3{M}_{11}^{-1}\left(\mathit{q}\right)C_{12}(\mathit{q},\stackrel{·}{\mathit{q}})){\stackrel{·}{q}}_a\hfill \\ +({\lambda }_2-{\lambda }_3{M}_{11}^{-1}\left(\mathit{q}\right)C_{11}(\mathit{q},\stackrel{·}{\mathit{q}})){\stackrel{·}{q}}_u-{\lambda }_3{M}_{11}^{-1}\left(\mathit{q}\right)G_1\left(\mathit{q}\right)]\hfill \\ -\mathrm{Ksgn}(s)\hfill \end{array}$$

(43)

In (43), the matrix $K=\mathrm{diag}\{K_1,K_2,K_3\}$ is a positive definite matrix, which is the parameter to be designed for the control input, and its value determines the speed of s convergence to 0; $\mathrm{sgn}(·)$ is a symbolic function of $\mathrm{sgn}(s_i)$ , and its functional relationship of $\mathrm{sgn}(·)$ is:

$$\mathrm{sgn}(s_i)=\{\begin{array}{c}+1\mathrm{if}{s}_i>0\hfill \\ \multicolumn{1}{c}{0\mathrm{if}{s}_i=0}\\ \multicolumn{1}{c}{-1\mathrm{if}{s}_i<0}\end{array}$$

(44)

Stability analysis

Since the controller of (43) ensures that all state variables can reach the sliding mode surface s = 0, the tracking trajectory has also reached the sliding mode surface s = 0. Then, when ${\stackrel{·}{q}}_{\mathrm{ad}}=0$ and s = 0, (35) is rewritten as

$${\stackrel{·}{q}}_a+{\lambda }_1[q_a-q_{\mathrm{ad}}]+{\lambda }_3{\stackrel{·}{q}}_u+{\lambda }_2{q}_u=0$$

(45)

which implies that

$${\stackrel{·}{q}}_a=-{\lambda }_1[q_a-q_{\mathrm{ad}}]-{\lambda }_3{\stackrel{·}{q}}_u-{\lambda }_2{q}_u$$

(46)

For the underactuated state, combined with (31), (32), and (43), when s = 0, the underactuated acceleration equation can be written in compact form as

$${\stackrel{··}{q}}_u=P_1(\mathit{q},\stackrel{·}{\mathit{q}}){\stackrel{·}{q}}_a+P_2(\mathit{q},\stackrel{·}{\mathit{q}}){\stackrel{·}{q}}_u+P_3\left(\mathit{q}\right)$$

(47)

where

$$\begin{array}{c}\multicolumn{1}{c}{P_1(\mathit{q},\stackrel{·}{\mathit{q}})=M_{11}^{-1}\left(\mathit{q}\right)[M_{12}\left(\mathit{q}\right){({\mathrm{I}}_3-{\lambda }_3{M}_{11}^{-1}\left(\mathit{q}\right)M_{12}\left(\mathit{q}\right))}^{-1}}\\ \times ({\lambda }_1-{\lambda }_3{M}_{11}^{-1}\left(\mathit{q}\right)C_{12}(\mathit{q},\stackrel{·}{\mathit{q}}))-C_{12}(\mathit{q},\stackrel{·}{\mathit{q}})\hfill \end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{P_2(\mathit{q},\stackrel{·}{\mathit{q}})=M_{11}^{-1}\left(\mathit{q}\right)[M_{12}\left(\mathit{q}\right){({\mathrm{I}}_3-{\lambda }_3{M}_{11}^{-1}\left(\mathit{q}\right)M_{12}\left(\mathit{q}\right))}^{-1}}\\ \times ({\lambda }_2-{\lambda }_3{M}_{11}^{-1}\left(\mathit{q}\right)C_{11}(\mathit{q},\stackrel{·}{\mathit{q}}))-C_{11}(\mathit{q},\stackrel{·}{\mathit{q}})\hfill \end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{P_3\left(\mathit{q}\right)=M_{11}^{-1}\left(\mathit{q}\right)[M_{12}\left(\mathit{q}\right){({\mathrm{I}}_3-{\lambda }_3{M}_{11}^{-1}\left(\mathit{q}\right)M_{12}\left(\mathit{q}\right))}^{-1}}\\ \times {\lambda }_3{M}_{11}^{-1}\left(\mathit{q}\right)+M_{11}\left(\mathit{q}\right)]G_1\left(\mathit{q}\right)\hfill \end{array}$$

Given the three state variables $x_1=q_u\in R^2$ , $x_2={\stackrel{·}{q}}_u\in R^2$ , $x_3=(q_a-q_{\mathrm{ad}})\in R^3$ , the system can be rewritten in first-order form as follows:

$$\{\begin{array}{c}{\stackrel{·}{x}}_1=x_2\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{x}}_2={\stackrel{··}{q}}_u}\\ \multicolumn{1}{c}{{\stackrel{·}{x}}_3={\stackrel{·}{q}}_a}\end{array}=g(x)$$

(48)

Substituting the state variable $x_3$ into (46) yields:

$${\stackrel{·}{x}}_3=-{\lambda }_1{x}_3-{\lambda }_3{x}_2-{\lambda }_2{x}_1$$

(49)

If the acceleration function ${\stackrel{··}{q}}_u=g(x)$ of the underactuated state is assumed, the expression (47) has the following form:

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{x}}_2=-P_1\left(x\right){\lambda }_2{x}_1+(P_2\left(x\right)-P_1\left(x\right){\lambda }_3)x_2}\\ -P_1\left(x\right){\lambda }_1{x}_3+P_3\left(x\right)=g(x)\hfill \end{array}$$

(50)

For the convenience of subsequent calculation, an expression $f(x)$ about $x$ is introduced. According to (48) - (50), the $f(x)$ form can be obtained as follows:

$$\stackrel{·}{x}=\left[\begin{array}{c}{x}_2\hfill \\ \multicolumn{1}{c}{g(x)}\\ \multicolumn{1}{c}{-{\lambda }_2{x}_1-{\lambda }_3{x}_2-{\lambda }_1{x}_3}\end{array}\right]=f(x)$$

(51)

The expression (51) is an independent system, and the system can converge to zero asymptotically by choosing appropriate ${\lambda }_2$ , ${\lambda }_3$ , and ${\lambda }_1$ .

Linearizing equation (51) near the equilibrium point of x = 0, the following linear system can be obtained

$$\stackrel{·}{x}=\mathrm{A}x$$

(52)

where

$$\begin{array}{c}\multicolumn{1}{c}{\mathbf{A}={\frac{\partial \mathbf{f}(\mathbf{x})}{\partial \mathbf{x}}|}_{\mathbf{x}=0}={\left[\begin{array}{ccc}{0}_{2\times 2}\hfill & {\mathbf{I}}_{2\times 2}\hfill & 0_{2\times 3}\hfill \\ \multicolumn{1}{c}{\frac{\partial \mathrm{g}(\mathbf{x})}{\partial {\mathbf{x}}_1}}& \frac{\partial \mathrm{g}(\mathbf{x})}{\partial {\mathbf{x}}_2}\hfill & \frac{\partial \mathrm{g}(\mathbf{x})}{\partial {\mathbf{x}}_3}\hfill \\ \multicolumn{1}{c}{-{\mathbf{\lambda }}_2}& -{\mathrm{\lambda }}_3\hfill & -{\mathbf{\lambda }}_1\hfill \end{array}\right]|}_{\mathbf{x}=0}}\\ =\left[\begin{array}{ccc}{0}_{2\times 2}\hfill & {\mathbf{I}}_{2\times 2}\hfill & 0_{2\times 3}\hfill \\ \multicolumn{1}{c}{{\mathbf{A}}_{21}}& {\mathbf{A}}_{22}\hfill & {\mathbf{A}}_{23}\hfill \\ \multicolumn{1}{c}{-{\mathbf{\lambda }}_2}& -{\mathrm{\lambda }}_3\hfill & -{\mathbf{\lambda }}_1\hfill \end{array}\right]\hfill \end{array}$$

By calculation, the coefficient matrices ${\mathbf{A}}_{21}$ , ${\mathbf{A}}_{22}$ , and ${\mathbf{A}}_{23}$ can be obtained as follows:

$$\begin{array}{c}{\mathbf{A}}_{21}=-\left[\begin{array}{cc}\frac{{\lambda }_{11}{\lambda }_{21}+g}{L_{\mathrm{cr}}-{\lambda }_{31}}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& \frac{{\lambda }_{12}{\lambda }_{22}+g}{L_{\mathrm{cr}}-{\lambda }_{32}}\hfill \end{array}\right],\hfill \\ {\mathbf{A}}_{22}=-\left[\begin{array}{cc}\frac{{\lambda }_{11}{\lambda }_{31}-{\lambda }_{21}}{L_{\mathrm{cr}}-{\lambda }_{31}}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& \frac{{\lambda }_{12}{\lambda }_{32}-{\lambda }_{22}}{L_{\mathrm{cr}}-{\lambda }_{32}}\hfill \end{array}\right],\hfill \\ {\mathbf{A}}_{23}=-\left[\begin{array}{ccc}\frac{{\lambda }_{11}^2}{L_{\mathrm{cr}}-{\lambda }_{31}}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& \frac{{\lambda }_{12}^2}{L_{\mathrm{cr}}-{\lambda }_{32}}\hfill & 0\hfill \end{array}\right].\hfill \end{array}$$

In order to ensure the stability of the linearized system given by (52), the linearized state matrix A should be a hurwitz term.

According to the optimal control theory, if the form of a matrix E is:

$$\mathbf{E}=\left[\begin{array}{cc}{\mathbf{E}}_{11}\hfill & {\mathbf{E}}_{12}\hfill \\ \multicolumn{1}{c}{{\mathbf{E}}_{21}}& {\mathbf{E}}_{22}\hfill \end{array}\right]$$

then the determinant of E, $\left|E\right|$ , can be computed such that $\left|\mathbf{E}\right|=\left|{\mathbf{E}}_{22}\right|\times |{\mathbf{E}}_{11}-{\mathbf{E}}_{12}{\mathbf{E}}_{22}^{-1}{\mathbf{E}}_{21}|$ . Therefore, the characteristic polynomial of the linearized system (52) can be expressed as:

$$\begin{array}{c}\mathrm{\Delta }(s)=det(s{\mathbf{I}}_7-\mathbf{A})=det\left[\frac{\begin{array}{cc}s{\mathbf{I}}_2\hfill & -{\mathbf{I}}_2\hfill \\ \multicolumn{1}{c}{-{\mathbf{A}}_{21}}& s{\mathbf{I}}_2-{\mathbf{A}}_{22}\hfill \end{array}}{\begin{array}{cc}{\mathbf{\lambda }}_2\hfill & {\mathbf{\lambda }}_3\hfill \end{array}}|\frac{\begin{array}{c}{0}_{2\times 3}\hfill \\ \multicolumn{1}{c}{-{\mathbf{A}}_{23}}\end{array}}{s{\mathbf{I}}_3+{\mathbf{\lambda }}_1}\right]\hfill \\ =|s{\mathbf{I}}_3+{\mathbf{\lambda }}_1||\left[\begin{array}{cc}s{\mathbf{I}}_2\hfill & -{\mathbf{I}}_2\hfill \\ \multicolumn{1}{c}{-{\mathbf{A}}_{21}}& s{\mathbf{I}}_2-{\mathbf{A}}_{22}\hfill \end{array}\right]-\left[\begin{array}{c}{0}_{2\times 3}\hfill \\ \multicolumn{1}{c}{-{\mathbf{A}}_{23}}\end{array}\right]{(s{\mathbf{I}}_3+{\mathbf{\lambda }}_1)}^{-1}\left[{\mathbf{\lambda }}_2{\mathbf{\lambda }}_3\right]|\hfill \\ =(s+{\lambda }_{13})P(s)Q(s)\hfill \end{array}$$

where

$$\begin{array}{c}P(s)=s^3+\frac{{\lambda }_{11}{\lambda }_{21}+g}{L_{\mathrm{cr}}-{\lambda }_{32}}{s}^2+\frac{{\lambda }_{12}{\lambda }_{22}+g}{L_{\mathrm{cr}}-{\lambda }_{32}}s+\frac{{\lambda }_{12}{\lambda }_{22}+g}{L_{\mathrm{cr}}-{\lambda }_{32}}\hfill \\ Q(s)=s^3+\frac{{\lambda }_{11}{L}_{\mathrm{cr}}-{\lambda }_{21}}{L_{\mathrm{cr}}-{\lambda }_{31}}{s}^2+\frac{g}{L_{\mathrm{cr}}-{\lambda }_{31}}s+\frac{g{\lambda }_{11}}{L_{\mathrm{cr}}-{\lambda }_{31}}\hfill \end{array}$$

According to the hurwitz criterion, the necessary and sufficient condition for the stability of the linear system (52) is that the characteristic polynomials $P(s)$ and $Q(s)$ are hurwitz polynomials.

The condition for the characteristic polynomial of the system (3.25) to be the hurwitz polynomials is:

$$\{\begin{array}{c}{\lambda }_{31}<L_{\mathrm{cr}},{\lambda }_{21}<(L_{\mathrm{cr}}-1){\lambda }_{11}\hfill \\ \multicolumn{1}{c}{{\lambda }_{32}<L_{\mathrm{cr}},{\lambda }_{22}<(L_{\mathrm{cr}}-1){\lambda }_{11}}\end{array}$$

(53)

It can be seen from (53) that the necessary and sufficient conditions for the stability of the sliding surface are determined by the parameters $\{{\lambda }_{31},{\lambda }_{32},{\lambda }_{21},{\lambda }_{22}\}$ .

For the sake of formula concise, the saturation function is defined as

$$\mathbf{sat}\left(\frac{s_i}{\epsilon }\right)=\{\begin{array}{c}\mathrm{sgn}\left(\frac{s_i}{\epsilon }\right)\mathrm{if}\left|s_i\right|>\epsilon \hfill \\ \frac{s_i}{\epsilon }\mathrm{if}\left|s_i\right|\le \epsilon \hfill \end{array}$$

(54)

where, sign(s) is the sign function in (44), $\epsilon$ is the width of the boundary layer.

Design of block disturbance observer

When the ship is subjected to waves rolling, pitching and heaving, it is equivalent to introducing an uncertain disturbance term d in the system parameters.

• Assumption 4.1. The disturbance term $\mathit{d}\in R^n$ , $\Vert \mathit{d}\Vert <\hat{D}$ , $\hat{D}\in R_{+}$ is unknown; the first derivative of d is bounded. The formula (28) can be rewritten as:

$$\mathit{M}\left(\mathit{q}\right)\stackrel{··}{\mathit{q}}+C(·)\stackrel{·}{\mathit{q}}+\mathrm{G}(·)=\tau +\mathit{d}$$

(55)

where, $\mathit{d}={\left[{\mathit{D}}_1{\mathit{D}}_2\right]}^T$ , ${\mathit{D}}_1={\left[d_1{d}_2\right]}^T$ , ${\mathit{D}}_2={\left[0d_4{d}_5\right]}^T$ , and $d_1,d_2,d_4,d_5$ , are the disturbances caused by sea waves.

The control system diagram of ship-mounted crane is shown in Figure 3. The proposed block disturbance observer availably compensates external disturbances due to ship rolling, pitching and heaving for ship-mounted crane systems, and the sliding mode controller drives the payload to the desired positions precisely and reduce vibrations.

Figure 3.

Control system diagram of ship-mounted crane.

According to (29), (30), when the external disturbances exists, after separating the driving and underactuated components, the expression is rewritten as:

$$\begin{array}{c}\multicolumn{1}{c}{M_{11}\left(\mathit{q}\right){\stackrel{·}{x}}_{21}+M_{12}\left(\mathit{q}\right){\stackrel{·}{x}}_{22}+C_{11}(\mathit{q},\stackrel{·}{\mathit{q}})x_{21}}\\ +C_{12}(\mathit{q},\stackrel{·}{\mathit{q}})x_{22}+G_1\left(x_1\right)={\tau }_1+D_1\hfill \end{array}$$

(56)

$$\begin{array}{c}\multicolumn{1}{c}{M_{21}\left(\mathit{q}\right){\stackrel{·}{x}}_{21}+M_{22}\left(\mathit{q}\right){\stackrel{·}{x}}_{22}+C_{21}(\mathit{q},\stackrel{·}{\mathit{q}})x_{21}}\\ +C_{22}(\mathit{q},\stackrel{·}{\mathit{q}})x_{22}+G_2\left(x_1\right)={\tau }_2+D_2\hfill \end{array}$$

(57)

where, $x_1=\mathit{q},x_2=\stackrel{·}{\mathit{q}},x=[x_1{x}_2{]}^T,x_{11}=q_u,x_{21}={\stackrel{·}{q}}_u,x_{12}=q_a,x_{22}={\stackrel{·}{q}}_a.$

Design of underactuated partial disturbance observer

The disturbance in the underactuated state represented by (56) is estimated, and the designed disturbance observer is:

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{\hat{d}}}_1=-L_1(x_1,x_2){\hat{d}}_1+L_1(x_1,x_2)[M_{11}\left(x_1\right){\stackrel{·}{x}}_{21}}\\ +M_{12}\left(x_1\right){\stackrel{·}{x}}_{22}+C_{11}(x_1,x_2)x_{21}\hfill \\ +C_{12}(x_1,x_2)x_{22}+G_1(·)-{\tau }_1]\hfill \end{array}$$

(58)

where, ${\hat{d}}_1$ is the estimated value of the underactuated partial disturbance observer, $L_1(x_1,x_2)$ is the gain matrix of underactuated partial disturbance observer.

It can be seen from equation (58) that the designed disturbance observer contains not only the acceleration information of underactuated part ${\stackrel{·}{x}}_{21}$ , but also the acceleration information of driven part ${\stackrel{·}{x}}_{22}$ . According to (45), there are:

$${\stackrel{·}{x}}_{22}={\overline{M}}^{-1}\left(x_1\right)[{\tau }_2-{\overline{C}}_1(x_1,x_2)x_{21}-{\overline{C}}_2(x_1,x_2)x_{22}-\overline{G}\left(x_1\right)]$$

(59)

Substituting (59) into (58) yields:

$$\begin{array}{l}{\widehat{d}}_1=-L_1(x_1,x_2){\widehat{d}}_1+L_1(x_1,x_2)\{M_{11}\left(x_1\right){\stackrel{.}{x}}_{21}\\ +M_{12}\left(x_1\right){\overline{M}}^{-1}\left(x_1\right)[{\tau }_2-{\overline{C}}_1\left(x_1,x_2\right)x_{21}\\ -{\overline{C}}_2\left(x_1,x_2\right)x_{22}-\overline{G}\left(x_1\right)]+C_{11}\left(x_1,x_2\right)x_{21}\\ +C_{12}\left(x_1,x_2\right)x_{22}+G_1(·)-{\tau }_1]\}\end{array}$$

(60)

The auxiliary function $f_1(x_1,x_2)$ is introduced to make

$${\hat{d}}_1=z_1+f_1(x_1,x_2)$$

(61)

where, $z_1$ is the internal state vector of the BDOB, $f_1(x_1,x_2)$ is the introduced nonlinear function to be designed.

Let

$${\stackrel{·}{f}}_1(x_1,x_2)=L_1(x_1,x_2)M_{11}\left(x_1\right){\stackrel{·}{x}}_{21}$$

(62)

Substituting (62) and (60) into (61) produces:

$$\begin{array}{c}\frac{d{z}_1}{\mathrm{dt}}=-L_1(x_1,x_2)z_1+L_1(x_1,x_2)\{M_{12}(x_1){\overline{M}}^{-1}(x_1)\hfill \\ [{\tau }_2-{\overline{C}}_1(x_1,x_2)x_{21}-{\overline{C}}_2(x_1,x_2)x_{22}-\overline{G}(x_1)]\hfill \\ +C_{11}(x_1,x_2)x_{21}+C_{12}(x_1,x_2)x_{22}+G_1(·)\hfill \\ -{\tau }_1-f_1(x_1,x_2)\}\hfill \end{array}$$

(63)

Therefore, the disturbance observer without acceleration information can be obtained as:

$$\{\begin{array}{c}\frac{d{z}_1}{\mathrm{dt}}=-L_1(x_1,x_2)z_1+L_1(x_1,x_2)\{M_{12}(x_1){\overline{M}}^{-1}(x_1)\hfill \\ [{\tau }_2-{\overline{C}}_1(x_1,x_2)x_{21}-{\overline{C}}_2(x_1,x_2)x_{22}-\overline{G}(x_1)]+C_{11}(x_1,x_2)x_{21}\hfill \\ +C_{12}(x_1,x_2)x_{22}+G_1(·)-{\tau }_1-f_1(x_1,x_2)\}\hfill \\ {\hat{d}}_1=z_1+f_1(x_1,x_2)\hfill \end{array}$$

(64)

The estimation error of the disturbance observer is defined as:

$$\mathrm{\Delta }{d}_1=D_1-{\hat{d}}_1$$

(65)

In order to accurately estimate the external disturbance $D_1$ of ship-mounted crane, the derivative of equation (65) can obtain the error dynamic equation of disturbance observer as:

$$\mathrm{\Delta }{\stackrel{·}{d}}_1={\stackrel{·}{D}}_1-\stackrel{=}{{\hat{d}}_1}{\stackrel{·}{D}}_1-L_1(x_1,x_2)[D_1-{\hat{d}}_1]$$

(66)

When the disturbance is a slow time-varying signal that is ${\stackrel{·}{D}}_1=0$ , therefore, the dynamic equation of disturbance observer error is:

$$\mathrm{\Delta }{\stackrel{·}{d}}_1=-L_1(x_1,x_2)[D_1-{\hat{d}}_1]=-L_1(x_1,x_2)\mathrm{\Delta }{d}_1$$

(67)

Let

$$L_1(x_1,x_2)=X^{-1}{M}_{11}^{-1}(x_1)$$

(68)

where X is the undetermined gain matrix.

Substituting (68) into (62) and integrating both sides of (62) obtain:

$$f_1(x_1,x_2)=-X^{-1}{x}_{21}$$

(69)

The designed Lyapunov function is:

$$V_{11}(t)=\mathrm{\Delta }{d}_1^T{X}^T{M}_{11}(x_1)X\mathrm{\Delta }{d}_1$$

(70)

Since the inertia matrix $M_{11}(x_1)$ is a positive definite matrix and X is a reversible matrix, it is also a positive definite matrix. Therefore, V₁₁(t) is positive definite.

The derivative of (70) can be obtained (please refer to the Appendix for details)

$${\stackrel{·}{V}}_{11}\left(t\right)=-\mathrm{\Delta }{d}_1^T(X+X^T-X^T{\stackrel{·}{M}}_{11}\left(x_1\right)X)\mathrm{\Delta }{d}_1$$

(71)

In order to guarantee ${\stackrel{·}{V}}_{11}\left(t\right)<0$ , there is a positive definite matrix $\phi >0$ , which satisfies the inequality $X+X^T-X^T{\stackrel{·}{M}}_{11}\left(x_1\right)X\ge \phi$ . The value of X can be obtained by solving linear matrix inequality.

Design of disturbance observer for driving part

The design process of the disturbance observer in the driving part is similar to that in the underactuated part, and the proof process is omitted. The nonlinear disturbance observer is designed as follows:

$$\{\begin{array}{c}\frac{d{z}_2}{\mathrm{dt}}=-L_2(x_1,x_2)z_2+L_2(x_1,x_2)\hfill \\ \{M_{21}(x_1)M_{11}^{-1}(x_1)[{\tau }_1-M_{12}(x_1){\overline{M}}^{-1}(x_1)\hfill \\ [{\tau }_2-{\overline{C}}_1(x_1,x_2)x_{21}-{\overline{C}}_2(x_1,x_2)x_{22}-\overline{G}(x_1)]\hfill \\ -C_{11}(x_1,x_2)x_{21}-C_{12}(x_1,x_2)x_{22}-G_1(x_1)]\hfill \\ +C_{21}(x_1,x_2)x_{21}+C_{22}(x_1,x_2)x_{22}+G_2(·)\hfill \\ -{\tau }_2-f_2(x_1,x_2)\}\hfill \\ {\hat{d}}_2=z_2+f_2(x_1,x_2)\hfill \end{array}$$

(72)

Simulation and experiment

This section will verify the performance of the proposed approach by a series of simulation and hardware experiments.

Simulation results

For all the simulations, the system parameters are set as M_b = 5 kg, m = 1 kg, L_b = 1.25 m, H_c = 0.47 m, R₁ =R₂ = 0.015 m, J₁ = J₃ = 0.054 kg.cm², J₂ = 0.025 kg.cm².

Without loss of generality, the initial and desired positions of ship-mounted crane system are shown in Table 2, and the initial velocities and accelerations are zero.

Table 2.

The initial and desired positions of ship-mounted crane.

Initial position	Values (unit)	Desired position	Values (unit)
${\theta }_x^0$	0 rad	${\theta }_x^d$	0 rad
${\theta }_y^0$	0 rad	${\theta }_y^d$	0 rad
${\alpha }^0$	2 rad	${\alpha }^d$	π rad
${\beta }^0$	0.2 rad	${\beta }^d$	π/6 rad
$L_c^0$	0.8 m	$L_c^d$	1 m

When the ship-mounted crane works, it needs to be anchored and positioned, which limits the three degrees of freedom of sway, surge and yaw. Therefore, the hull is mainly affected by roll, pitch and heave of waves. In this paper, the three-level wind speed and sea state in China’s coastal area are taken as the research object to simulate the motion law of ships under such waves. According to the semi-empirical wave spectrum formula (73), which is recommended by the State Oceanic Administration of China, the wave spectrum is shown in Figure 4.

$$S_{\zeta }(\omega )=\frac{0.74}{{\omega }^5}\exp (-\frac{96.2}{U^2{\omega }^2})$$

(73)

where ω is the wave frequency, and U is the wind speed.

Figure 4.

Wave spectrum.

From the Figure 4, the main energy distribution of waves is [0.5,2], so the wave frequencies are set to 0.2 π rad/s in the simulation, the amplitude of roll and pitch is 0.1 rad, and the amplitude of heave is 0.02 m, that is,

$$\{\begin{array}{c}\varphi =0.1\sin 0.2\pi t\hfill \\ \eta =0.1\sin 0.2\pi t\hfill \\ h=0.02\sin 0.2\pi t\hfill \end{array}$$

(74)

For different working environments, two sets of simulation experiments are carried out, and compared with the ZV approach. ³⁵ The first one, when the external disturbances do not exist, the traditional SMC method is used to control the ship-mounted crane. The second one, when the disturbances of (73) exist, the proposed CSMC method is adopted.

Taking the slew angle of the ship-mounted crane as an example, the ZV controller is shown in Figure 5, and the angular velocity and acceleration are 0.3 rad/s and 0.2 rad/s² respectively.

Figure 5.

ZV controller.

Selecting the initial parameters of the controller according to (52), and after careful tuning, the control gains of the proposed controller are shown as in Table 3.

Table 3.

Sliding mode controller parameters.

Control gains	Value	Control gains	Value
K 1	0.3	λ 21	−0.1
K 2	5	λ 22	−1
K 3	12	λ 31	0.1
λ 11	1	λ 32	0.1
λ 12	0.8	ε	0.002
λ 13	0.9

The first group simulation results without disturbances are shown in Figure 6. It is seen from Figure 6 that, the proposed SMC can effectively restrain the swing of the payload compared with ZV controller and uncontrolled, and drive the payload to the desired position within 10 s and no residual swing.

Figure 6.

Simulation results without disturbances: (a) Swing angles of the payload, (b) Positions of the actuators, (c) Positions of the payload in the inertial coordinate, and (d) Three-dimensional trajectory of the payload.

The second group simulation results with disturbances are shown in Figure 7. It can be seen that, due to the disturbances of the sea waves, the payload has slight residual swing, but the amplitude is smaller compared with ZV controller and uncontrolled, indicating that the BDOB can address the external disturbances exerted by the sea waves. It is seen from Figure 7(b)that, the length of the payload rope has been adjusted periodically to counteract the wave heaving disturbance.

Figure 7.

Simulation results with disturbances: (a) Swing angles of the payload, (b) Positions of the actuators, (c) Positions of the payload in the inertial coordinate, and (d) Three-dimensional trajectory of the payload.

Hardware experimental results

In this subsection, hardware experiments are implemented on a self-built ship-mounted crane platform, which is a boom crane, together with a wave generator by using the 3-RPS parallel mechanism as shown in Figure 8. The boom angles are detected by the encoders (2²⁰ PPR), and the sway angles of the payload are measured by a self-built angle sensor, which is composed of two incremental encoders (1000 PPR). A Googol GUS-600-TG02-HD six-axis Embedded Motion Controller is used to control the 3-RPS parallel mechanism to generate the waves and collects data from the sensors. The experiment model of ship-mounted crane is built in the MATLAB/Simulink environment.

Figure 8.

Self-built ship-mounted crane experiment platform.

To better verify the control performance of the proposed controller, two sets of experiment tests are carried out, and the experimental conditions are the same as the simulation.

Experiment 1

The first group experiment results without disturbances are shown in Figure 9. It is seen from Figure 9 that, the proposed SMC can effectively restrain the swing of the payload compared with ZV controller and uncontrolled, and drive the payload to the desired position quickly and almost no residual swing.

Figure 9.

Experiment results without disturbances: (a) Swing angles of the payload, (b) Positions of the actuators, (c) Positions of the payload in the inertial coordinate, and (d) Three-dimensional trajectory of the payload.

Experiment 2

The second group experiment results with disturbances that are introduced by wave simulator, are shown in Figure 10. It is seen from Figure 10 that, the control process is more complicated due to the existence of external disturbances and the uncertainty of actual system parameters. However, it can be seen from the comparison experiment results that the proposed CSMC method shows better performance, which can effectively restrain the swing of the payload and with slight residual swing.

Figure 10.

Experiment results with disturbances: (a) Swing angles of the payload, (b) Positions of the actuators, (c) Positions of the payload in the inertial coordinate, and (d) Three-dimensional trajectory of the payload.

To sum up, the proposed control method realizes the effective control of the ship-mounted crane and has strong robustness to different types of external disturbances.

Conclusions

In order to solve the payload swing of ship-mounted crane in the process of transportation, this paper proposes a block disturbance observer-based sliding mode control method. Firstly, the dynamic model of ship-mounted crane is established, and the sliding mode controller is designed by decoupling the original mathematical model. Secondly, a CSMC that combines block disturbance observer and sliding mode control is proposed, and the BDOB is used to address the external disturbances caused by the sea wave. Finally, a series of simulation and experimental are carried out, and the comparative experimental results show that the control performance of the proposed method is better than the traditional method, and produced a reduction of 93% in residual oscillations. In the future, in addition to studying the stability of proposed methods, the convergence of proposed methods should also be considered and compared with recently advanced methods.