Modeling and anti-sway control of ship-mounted crane

Abstract

This article analyzes the dynamics of a ship-mounted crane and establishes the mathematical model of the load. Based on the mathematical model, the time-delayed feedback control algorithm is proposed to anti-load sway. The Lambert W function and Newton–Raphson method are used to prove the stability of control system. The contour plot of the system damping is obtained; by selecting appropriate time delay and system gain, the system damping increases and load sway rapidly decays, thereby the system is stable. The co-simulation and experimental results show that the delayed feedback controller has better performance in anti-sway and robustness.

Keywords

Ship-mounted crane anti-sway time-delayed feedback stability MATLAB–Adams

Introduction

Ship-mounted cranes, which transfer loads using a rotary crane attached to the ship’s deck, are widely used to transfer cargos from ships to the harbor, or between two ships. The rotary cranes transfer loads by three motions: rotation, luffing, and load hoisting. In addition, there are still many issues in the performance of rotary cranes, such as load sway, positioning, and obstacle avoidance. The ship-mounted crane is more difficult to operate under the influence of the ship sway. Ship sway is generated by disturbances such as waves and winds, as well as movements of the load. During the transfer operation, the operator has to deal with the sway of the ship.

The main objectives of crane control research are to reduce or eliminate residual vibrations at the end of maneuvers, reduce maneuvering time, and enhance safe operating conditions. Researchers used many techniques to achieve these goals.¹ Input-shaping is one of the common control techniques. Parker et al.² present command-shaping approaches to reduce the operator-induced payload swing for a ship crane. Alhazza and colleagues,^3,4 focusing on the overhead crane, design a novel wave form command shaper of adjustable maneuvering time, which can effectively reduce load vibrations in intermittent maneuvers. Shen et al.⁵ provide a straight transfer transformation (STT) model to control the rotary crane. The proposed STT model can effectively eliminate the influence of centrifugal force. However, the input shaping and optimum control algorithm mentioned are open-loop control. This kind of method is sensitive to the change of system parameters. Besides, it also orders an end previously, but in actual engineer, the end is often adjusted by the operators.

With the rapid development of automatic control and artificial intelligence, various intelligent control techniques are applied to the robotic system.^6–11 For the open-loop control strategy is sensitive to initial disturbances, the closed-loop control and advanced intelligent techniques are also designed to control the vibrations of crane.^12,13

Sun et al.¹⁴ propose a nonlinear stabilizing control law for a ship-mounted crane; it is the first closed-loop control method without simplifying the original complex nonlinear dynamics, which can effectively control a ship-mounted crane with ship rolling and heaving movement. Tuan et al.¹⁵ design a nonlinear feedback controller for a shipboard container crane, in which the influence of wave and the flexibility of the cable are considered.

Park et al.¹⁶ design an adaptive sliding-mode controller and a fuzzy observer for an overhead crane with high-speed hoisting motion. The results show that the proposed control law is valid. Tuan et al.¹⁵ propose a back-stepping sliding-mode controller for a ship-mounted container crane. The dynamic model of the crane system considers the seawater disturbance and wire flexibility. The simulation results show that the proposed control strategy can significantly reduce the cargo vibration. Aksjonov et al.¹⁷ use Euler–Lagrange state-space approach to derive the mathematical model of a three-dimensional crane and design an anti-swing fuzzy logic controller to the crane. The results show that the positioning in fuzzy logic controller has an error, and it is still more stable and accurate compared to traditional proportional–integral (PI) controller.

This article, first, abstracts the actual crane system and then derivates the mathematical model of the rotary crane. It gets a mathematical model of the ship-mounted crane by coordination transformation. To simplify the nonlinear system into a linear system and optimize the process of stability analysis, this article proposes a time-delay feedback anti-swing control algorithm of the ship-mounted crane. Applying Lambert’s W function and Newton–Raphson algorithm to the stability analysis of the crane, it can get the right eigenvalue of the system’s differential equation and draw the damping contour of the system. From this chart, reasonable delay time and gain make swing of load decline quickly and stabilize the system as soon as possible.

Modeling of ship-mounted crane

The model of the ship-mounted crane comprised two parts: the load sway model and the mathematical model.

Load sway model

For simplicity, the following assumptions were made:

The body of the boom crane can be regarded as a rigid body, and the backlash of the boom and crane is neglected.

The load can be seen as a particle.

Frictional torque that may exist in torque-transfer mechanisms can be neglected.

Weight and elongation of ropes, due to tensile force, are neglected.

In order to derive the load sway model, the swing angle of the load is observed. A schematic diagram of the boom crane is shown in Figure 1. ( $x_{s}, y_{s}, z_{s}$ ) is the ship coordinate system, and the crane coordinate system ( $x, y, z$ ) is fixed to the rotary crane; the origin O is based on the intersection of the crane and the boom. Points Q and P, respectively, represent the load and the boom tip. The length of load cable is $L_{c}$ , the length of boom is $L_{b}$ , and the height of boom base is $H_{c}$ . $α (rad)$ is the slew angle, $β (rad)$ is the boom hoisting angle, and P $(x_{p} (t), y_{p} (t), z_{p} (t)) (m)$ is the boom tip inertial position.

Figure 1.

Schematic diagram of a boom crane.

The in-plane and out-of-plane angles of the payload are, respectively, represented by $θ_{x}$ and $θ_{y}$ in Figure 1. $(x_{p} (t), y_{p} (t), z_{p} (t))$ is the position of P in the inertial coordinate system, and the position of point Q is given by

\begin{array}{l} x_{Q} (t) = x_{P} (t) + \sin θ_{x} (t) \cos θ_{y} (t) L_{c} \\ y_{Q} (t) = y_{p} (t) + \sin θ_{y} (t) L_{c} \\ z_{Q} (t) = z_{p} (t) + \cos θ_{x} (t) \cos θ_{y} (t) L_{c} \end{array}

(1)

The inertial position $r_{Q}$ of Q is

r_{Q} (t) = x_{Q} (t) \vec{i} + y_{Q} (t) \vec{j} + z_{Q} (t) \vec{k}

(2)

The position of P in the crane coordinate system $({\tilde{x}}_{p} (t), {\tilde{y}}_{p} (t), {\tilde{z}}_{p} (t))$ is given by

\begin{matrix} {\tilde{x}}_{P} (t) = L_{b} \cos β (t) \cos α (t) \\ {\tilde{y}}_{P} (t) = L_{b} \cos β (t) \sin α (t) \\ {\tilde{z}}_{P} (t) = - L_{b} \sin β (t) \end{matrix}

(3)

We use the Lagrange approach to derive the equations of motion. It follows from Figure 1 that the Lagrange’s equations are given by

\frac{d}{dt} (\frac{\partial E_{k}}{\partial {\overset{\cdot}{q}}_{j}}) - \frac{\partial E_{k}}{\partial q_{j}} = - \frac{\partial E_{p}}{\partial q_{j}}, j = 1, 2, \dots, N

(4)

where $E_{k}$ is the potential energy of the payload, $E_{p}$ is the kinetic energy, $q_{j}$ is the particle generalized coordinate, and ${\overset{\cdot}{q}}_{j}$ is the rate of the particle change, that is, generalized velocity. For ship-mounted crane, $θ_{x}$ and $θ_{y}$ are the particle generalized coordinates.

In this article, combined with the dynamic model of the boom crane, set lifting point mass is M, and the load mass is m, we have

E_{k} (t) = E_{kx} (t) + E_{ky} (t) + E_{kz} (t)

(5)

Substituting equation (5) into equation (4), the swing equation of the load Q is expressed as

{\begin{matrix} {\overset{\cdot\cdot}{θ}}_{x} \cos θ_{y} - 2 {\overset{\cdot}{θ}}_{x} {\overset{\cdot}{θ}}_{y} \sin θ_{y} + [g \sin θ_{x} + {\overset{\cdot\cdot}{x}}_{P} \cos θ_{x}] \\ / l - \sin θ_{x} {\overset{\cdot\cdot}{z}}_{P} / l = 0 \\ {\overset{\cdot\cdot}{θ}}_{y} + \sin θ_{y} \cos θ_{y} {\overset{\cdot}{θ}}^{2}_{x} + [g \cos θ_{x} \sin θ_{y} - \sin θ_{x} \cdot \sin θ_{y} {\overset{\cdot\cdot}{x}}_{P} - \cos θ_{y} {\overset{\cdot\cdot}{y}}_{P}] \\ / l - \cos θ_{x} \sin θ_{y} {\overset{\cdot\cdot}{z}}_{P} / l = 0 \end{matrix}

(6)

where ${\overset{\cdot\cdot}{x}}_{p}$ , ${\overset{\cdot\cdot}{y}}_{p}$ , and ${\overset{\cdot\cdot}{z}}_{p}$ are the accelerations of the suspension point P in x-, y-, and z-directions.

Ship-mounted crane model

To accurately describe the dynamic model of the ship-mounted crane, the inertial coordinate system $I^{E}$ and the ship frame $I^{S}$ are chosen in Figure 2. On the sea, the ship-mounted crane has three linear and three rotational freedoms: the surge, sway, heave, roll, pitch, and yaw.

Figure 2.

Ship-mounted crane coordinate system.

Taking into account the ship’s dynamic positioning, we can ignore the surge, sway, and yaw, while only consider the roll, pitch, and heave.^12,18 So, the coordinates of suspension point P with respect to the inertial moment are given by equation (7)

\begin{matrix} [\begin{matrix} x_{p} \\ y_{p} \\ z_{p} \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ - h \end{matrix}] + [\begin{matrix} 1 & 0 & 0 \\ 0 & \cos ϕ & - \sin ϕ \\ 0 & \sin ϕ & \cos ϕ \end{matrix}] \\ [\begin{matrix} \cos θ & 0 & \sin θ \\ 0 & 1 & 0 \\ - \sin θ & 0 & \cos θ \end{matrix}] {[\begin{matrix} R_{x} \\ R_{y} \\ R_{z} \end{matrix}] + [\begin{matrix} {\tilde{x}}_{P} \\ {\tilde{y}}_{P} \\ {\tilde{z}}_{P} \end{matrix}]} \end{matrix}

(7)

where $ϕ$ is the rolling angle; $θ$ is the pitching angle; $R_{x}$ , $R_{y}$ , and $R_{z}$ are the positions of the boom base to the point $O'$ of ship centroid; and $h$ is the heave displacement.

Figure 3 shows the ship-mounted crane model diagram, where crane model is equation (3), $P^{S}$ is the position of boom tip P in the crane coordinate $({\tilde{x}}_{p} (t), {\tilde{y}}_{p} (t), {\tilde{z}}_{p} (t))$ , the coordinate transformation matrix is equation (7), $P^{E}$ is the position of boom tip P in the inertial coordinate $(x_{p} (t), y_{p} (t), z_{p} (t))$ , and the load sway equation is equation (6).

Figure 3.

Ship-mounted crane model diagram.

Time-delayed feedback control system

Time-delayed feedback

Time-delayed feedback anti-sway control is a nonlinear feedback control method by introducing a time delay to the output of the system,¹⁹ and it will produce damping in the system response; consequently, that can significantly suppress the oscillation amplitude of payload for a ship-mounted crane, and the time-delay control law takes the following form

\begin{array}{l} x_{p r e f} (t) = x_{i} (t) + k_{x} L_{c} \sin (θ_{x} (t - τ_{x})) \cos (θ_{y} (t - τ_{x})) \\ y_{p r e f} (t) = y_{i} (t) - k_{y} L_{c} \sin (θ_{y} (t - τ_{y})) \end{array}

(8)

where $τ_{x}$ and $τ_{y}$ are the time delays of $θ_{x}$ and $θ_{y}$ , respectively; and $k_{x}$ and $k_{y}$ are the gains of controller.

Stability analysis

While the roots of the characteristic equation of the linear system are in the left half s-plane, the system is asymptotic stable. For the relationship between the stability of the nonlinear system and the linearization system, Lyapunov proved very important conclusions:

If the linear approximation of nonlinear system is strictly stable, namely, all the roots are in the left half s-plane, then the nonlinear system will stabilize at a certain neighborhood of the equilibrium point of the linear approximation.

If the linear approximation of nonlinear systems has one root in the right half s-plane, then the nonlinear system cannot be stable at any neighborhood of the equilibrium point of the linear approximation.

According to this corollary, the article linearized and decoupled the control system nearby the equilibrium point, and analyzed the stability of the ship-mounted crane using the characteristic root method. From equation (6), the load swing has the characteristic of strongly couple and nonlinear; to linear it, we scale the system variables into fast-varying and slow-varying terms, where $ε$ is a small dimensionless quantity, which is the amplitude of the load movement. The fast-varying terms are

\begin{matrix} θ_{x} (t) = ε θ_{x} (t) \\ θ_{y} (t) = ε θ_{y} (t) \\ z_{p} (t) = ε z_{p} (t) \end{matrix}

(9)

and the slow-varying terms are

\begin{matrix} x_{i} (t) = ε^{2} x_{i} (t) \\ y_{i} (t) = ε^{2} y_{i} (t) \end{matrix}

(10)

Substituting equations (8)–(10) into equation (6), and letting the coefficient of $ε$ equal to 0, we obtain

\begin{matrix} {\overset{\cdot\cdot}{θ}}_{x} (t) + 2 μ {\overset{\cdot}{θ}}_{x} (t) + Ω^{2} θ_{x} (t) \\ + k_{x} {\overset{\cdot\cdot}{θ}}_{x} (t - τ_{x}) + 2 μ k_{x} {\overset{\cdot}{θ}}_{x} (t - τ_{x}) = 0 \end{matrix}

(11)

\begin{matrix} {\overset{\cdot\cdot}{θ}}_{y} (t) + 2 μ {\overset{\cdot}{θ}}_{y} (t) + Ω^{2} θ_{y} (t) \\ + k_{y} {\overset{\cdot\cdot}{θ}}_{y} (t - τ_{y}) + 2 μ k_{y} {\overset{\cdot}{θ}}_{y} (t - τ_{y}) = 0 \end{matrix}

(12)

where $Ω = \sqrt{g / L_{c}}$ is the natural frequency of the payload. Since both equations (11) and (12) are mathematically identical, we only solve equation (11).

From equation (11), the highest-order derivative of the differential equation also includes time-delay; this equation is called the neutral delay differential equation, which is of the form

\overset{\cdot}{x} (t) + C \overset{\cdot}{x} (t - τ) = Ax (t) + bx (t - τ), x \in R^{n}

(13)

To analyze the characteristic roots of delay differential equations, while DDE-Biftool mathematical tools were used, we can get its characteristic roots. But neutral differential equations are different, and its rightmost characteristic roots cannot be directly solved using the most of the existing mathematical tools. Usually, we set the solution of equation (11) as follows

θ_{x} (t) = A e^{σ t} \cos (ω t + θ_{0})

(14)

Substituting equation (14) into equation (11) and setting each coefficient of $\sin (ω t + θ_{0})$ and $\cos (ω t + θ_{0})$ equal to zero independently, we obtain an equation set, and then its characteristic roots can be solved using the numerical solution. However, this method cannot determine whether the root is the extreme right one in the complex plane.

This article presents a novel algorithm, which applied the combination of the Lambert W function¹⁹ and Newton–Raphson algorithm to analyze the stability of ship-mounted crane. This method not only simplifies the calculation process but also directly solves differential equations’ rightmost characteristic roots, and further judging the system stability. The characteristic equation of (13) is

\begin{matrix} Δ (λ) = (1 - p e^{- λ τ}) λ^{n} \\ + a_{1} (e^{- λ τ}) λ^{n - 1} + \dots + a_{n} (e^{- λ τ}) \end{matrix}

(15)

where $| p | < 1$ , and let $λ$ be equal to the rightmost characteristic root, we obtain

a = max {Re λ : Δ (λ) = 0, λ \in C}

(16)

If $a < 0$ , equation (11) is asymptotic stability.

Numerical solution

Based on the Lambert W function, it can give stability boundary of equation (11). The Lambert W function cannot be represented by elementary functions, but represented by the following transcendental equation

w e^{w} = z, z \in C

(17)

where $w = W_{k} (z)$ , $k = 0, \pm 1, \pm 2, \dots$ , $W_{k} (z)$ is called the Lambert W function. The different branches are denoted by $W_{k} (z)$ for any k, particularly when $k = 0$ , $W_{0} (z)$ is the principal branch of the Lambert W function, which includes the rightmost root for any $z \in C$ , and we obtain

max_{k = 0, \pm 1, \pm 2, \dots} Re W_{k} (z) = Re W_{0} (z)

(18)

By means of the above nature, the biggest real characteristic roots can be solved. Three mathematical software (MATLAB, Maple, and Mathematica) can directly calculate the Lambert W function. The calculation of solving the rightmost root’s real part of equation (11) is as follows.

We let $k = k_{x}$ and $τ = τ_{x}$ , then the characteristic equation of (11) is given by

Δ (λ) = λ^{2} + 2 μ λ + Ω + k λ^{2} e^{- λ τ} + 2 μ k λ e^{- λ τ}

(19)

Using Lambert W function, the implicit solution of $λ$ is given by

{\begin{matrix} W_{k} (y) = 2 u λ + Ω \\ y = - (λ^{2} + k λ^{2} e^{- λ τ} + 2 μ k λ e^{- λ τ}) e^{2 u λ + Ω} \end{matrix}

(20)

Substituting the gain $k$ and delay time $τ$ into equation (20), and only the characteristic root $λ$ is unknown number. From the above reasoning, if $k = 0$ , $W_{k} (y)$ takes a real part of the maximum

λ_{i + 1} = λ_{i} - \frac{F (λ_{i})}{F' (λ_{i})}

(21)

The iterative formula is the second-order convergence. The Halley method can also be used, and the core iterative formula of the method is given by

λ_{i + 1} = λ_{i} - (1 - \frac{1}{2} \frac{F (λ_{i}) F ″ (λ_{i})}{{(F' (λ_{i}))}^{2}})^{- 1} \frac{F (λ_{i})}{F' (λ_{i})}

(22)

The iterative formula is the third-order convergence, and the higher the order of convergence, the faster the convergence. Setting tolerance $ε$ , when the condition satisfies

| λ_{N} - λ_{N - 1} | < ε

(23)

The iteration is terminated, and $λ_{N}$ is the rightmost characteristic root, for any B satisfies the following equation

Re λ \leq Re λ_{N}

(24)

Substituting different $k$ into Lambert W function, and other characteristic roots can be obtained. In this case, setting initial value $λ_{0} = 0 + 0 i$ , iteration accuracy 10⁻⁶, $τ = 2$ , and $k = 0.1$ .

Figure 4 is the iterative process of real part and imaginary part of characteristic root $λ$ . After five iterations, the maximum real and imaginary parts of characteristic roots are obtained, and the iteration result is −0.0523 + 0.9024i.

Figure 4.

The real part and imaginary part of the characteristic root $λ$ .

Let $τ = 3$ , and discussing the relationship of the gain $k$ and the largest real part of the characteristic root. Figure 5 is the change process of the largest real part of characteristic root. When $k < 0.8$ , the largest real parts of all the roots are negative, that is, the system is stable. Instead, the system is unstable. When $k = 0.4$ , the largest real part of characteristic root gets the minimum, t is −0.1717.

Figure 5.

The largest real part of characteristic root for $τ = 3$ .

Let $k = 0.1$ , and discussing the relationship of the delay time and the largest real part of the characteristic root. Figure 6 is the change process of the largest real part of characteristic root with $τ$ . When $τ < 5$ , the largest real parts of all the roots are negative, that is, the system is stable. When $τ = 3$ , the largest real part of characteristic root $λ$ gets the minimum, t is −0.0345.

Figure 6.

The largest real part of characteristic root for $k = 0.1$ .

The damping of the time-delay control system of ship-mounted crane cannot be directly obtained from equation (11), but its value can be judged by the real part of the largest characteristic root $Re (λ_{N})$ . When $Re (λ_{N}) < 0$ , the smaller the $Re (λ_{N})$ , the larger the system damping, that is, the faster the swing angle response attenuates. When the characteristic root takes the minimum, the system damping is largest. Figure 7 shows the damping three-dimensional map as a function of the controller gain k and the time delay $τ$ .

Figure 7.

The three-dimensional map of damping.

Figure 8 shows the damping contours about the controller gain k and the time delay $τ$ . The figure is the projection of Figure 7 to $k - τ$ plane, and the darker the shade, the smaller the $Re (λ_{N})$ .

Figure 8.

The damping contour about k and τ.

Simulation results

To verify the performance of the proposed control strategy, we carry out some simulations. The dynamic model of ship-mounted crane is developed in section “Modeling of ship-mounted crane” and simulated under complex conditions included the seawater effect and the flexibility of the wire. The dynamic model is established in the MATLAB–Adams environment.

Simulation results without crane active movements

The boom length is $L_{b} = 50 m$ , the cable length is $L_{c} = 20 m$ in simulation, the initial slew angle is $- π / 2 rad$ , and the initial luffing angle is $π / 4 rad$ . Setting the rotary and luffing velocity equal to 0 rad/s. The load swing angular frequency is $ω_{n} = \sqrt{g / L_{c}} = 0.6283 rad / s$ , that is, 0.1 Hz. Considering practical sea conditions, we select the pitch motion of ship as $θ = \sin 0.2 π t$ . According to Figure 8, choose $τ = 3.5$ and $k = 0.1$ ; the virtual prototype of crane ship built in Adams 2013 is shown in Figure 9.

Figure 9.

Virtual prototype of crane ship.

Figure 10 is the reference rotary and luffing angles of crane without active movement. When the initial rotary and luffing angles are $- π / 2$ and $π / 4 rad$ , respectively, the solved reference rotary angle error is less than 0.006 rad and the luffing less than 0.02 rad.

Figure 10.

Reference luff and slew angles of chip-mounted crane.

Figure 11 is the response of the ship-mounted crane in the presence of ship pitching movement, wherein the solid line is the enabled controller, while the dotted line is not. It is seen from Figure 11 that even in the case of no crane active movements, the position error of payload is larger than that of the proposed method.

Figure 11.

Simulation result of the ship-mounted crane with ship pitching.

Simulation results with crane luffing or slewing movements

Ship-mounted crane slewing and luffing movement can be divided into three processes: acceleration, constant speed, and deceleration. In the slewing movement case, the acceleration is 0.05 rad/s², and the maximum speed reaches 0.1 rad/s after 2 s of acceleration. The parameters of the crane movement are shown in Figure 12.

Figure 12.

Motion parameters of ship-mounted crane slewing movement.

Figure 13 shows the solved reference rotary and luffing angles of the crane actuators by the time-delay controller, where all contain slight concussions for artificially introducing time-delay feedback of $θ_{x}$ and $θ_{y}$ .

Figure 13.

Reference angles of chip-mounted crane.

Figure 14 is the response of the ship-mounted crane with slewing and ship pitching, wherein the solid line is the enabled controller, while the dotted line is not. The simulation result shows that the cargo vibration of the controlled crane is reduced significantly while the uncontrolled result shows a large vibration.

Figure 14.

Simulation result of the ship-mounted crane with ship pitching.

Experimental result

To further verify the performance of the designed controller. The parameters of scale ship-mounted crane system are as follows: the boom length $L_{b} = 1.55 m$ , the initial cable length $L_{c} = 1 m$ , and the height of the boom base $H_{c} = 0.4 m$ . The experiment states are the same as simulation: the initial slew angle is $- π / 2 rad$ and the initial luffing angle is $π / 4 rad$ . Setting the rotary and luffing velocity equal to zero rad/s. The load swing frequency is 0.5 Hz. We select the ship pitching motion as $θ = \sin π t$ .

Figure 15 is the experiment result of the ship-mounted crane in the presence of ship pitching movement, wherein the solid line is the enabled controller, while the dotted line is not. It is seen from Figure 15 that as in the two simulation cases, the cargo vibration of the controlled crane is reduced significantly to unnoticeable amplitude within 5 s while the uncontrolled experiment result shows a large vibration.

Figure 15.

Experimental result of the ship-mounted crane with ship pitching.

Conclusion

This article proposed a time-delayed feedback control strategy for anti-sway control of a ship-mounted crane, proved the stability of the designed system by the combination of Lambert W function and Newton–Raphson methods, drawled the contours of the damping, and selected the appropriate controller parameters. Several test cases were simulated by MATLAB–Adams and validated experimentally on a scaled model of a ship-mounted crane. The co-simulation and experimental results show that the time-delay feedback control law is an effective technique of eliminating cargo vibration for ship-mounted cranes.

Footnotes

Academic Editor: Chenguang Yang

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (no. 51375140).

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