^{2} Tianjin Key Laboratory of Intelligent Robotics, Nankai University, Tianjin 300350, China
Recently, a great surge in efforts have been seen to tackle the control problem of underactuated mechatronic systems, such as translational oscillator with rotational actuator (TORA) systems^{[1]}, underactuated vehicles^{[2, 3]}, underactuated surface vessels^{[4, 5]}, rotationaltranslational actuator (RTAC) systems^{[6]}, underactuated robotic systems^{[7]}, underactuated cranes^{[8–10]}, doublependulum crane systems^{[11]}, etc.^{[12–19]}, which have less applicable control inputs than the tobecontrolled system degrees of freedom (DOFs). Being a typical underactuated nonlinear system, in marine industries, offshore cranes have been playing increasingly important roles as strong transportation tools primarily due to their merits of high transport capacity, super operation flexibility, less energy consumption, etc. Suffering from external disturbances such as sea waves, sea winds or currents, the ultimate regulation control objective of offshore shipmounted crane systems is the accurate and efficient positioning of payloads with small swing against ship′s motion disturbances. However, during the transferring processes under harsh sea conditions, there is always unexpected payload swing induced by inertia and numerous external disturbances, which are dangerous and may cause severe impacts to the people or freights around. Hence, investigation of an effective control approach for an offshore shipmounted crane is of significant importance in terms of both theoretical value and practical applications.
To effectively control an underactuated offshore shipmounted crane, which contains a trolley moving along the boom and a payload connected by an inelastic wire, a preferred technique is to appropriately control the trolley motion by fully analyzing the inherent coupling mechanism between trolley movement, payload swing and ship′s motion. During past decades, many researches regarding offshore crane control approaches have been developed and reported in the literature, which can be roughly classified as linear methodology as well as nonlinear and intelligent strategies. For the former category, to simplify control design, the complex nonlinear crane model has been linearized around its equilibrium points, then a number of linear control approaches, such as linear quadratic regulator (LQR) control^{[20]}, optimal control^{[21]}, input shaping method^{[22]}, etc., can be utilized. However, as a means to achieve better control performance, the system nonlinearities cannot be neglected and should be strictly tackled, for which, nonlinear and intelligent strategies, including antiswing nonlinear controller^{[23]}, slidingmode control^{[24]}, adaptive boundary controller^{[25]}, etc.^{[26]}, are proposed.
Additionally, although the mechanical structure of an offshoremounted crane resembles that of a landfixed overhead crane, which has the same characteristic of underactuation, the control problems for those systems are fairly different mainly due to the unmatched interference acting on the shipmounted crane system induced by persistent sea wave disturbances, which, largely upgrade the difficulty in controller design for offshore shipmounted cranes. Therefore, many ambitious control schemes proposed for landfixed cranes covering doublependulum cranes^{[11, 27–29]}, overhead cranes^{[8, 10, 30–32]}, tower cranes^{[9, 33]}, gantry cranes^{[34]}, etc., cannot be applied to offshore shipmounted cranes directly. Moreover, to tackle the unmatched interference for a great variety of control systems, a number of methods like adaptive control algorithm^{[35]}, output tracking control^{[36]}, neural networkbased control^{[37]}, are also investigated.
So far, it is still a quite open problem for offshore shipmounted crane control due to some issues that remain unsolved, such as accurate positioning of payloads with lower swing in the marine environment. To this end, many attempts have been devoted to the improvement of highperformance control schemes for such systems. For example, besides the existing methods mentioned above^{[23–26]}, recently, two antiswing nonlinear controllers considering ship roll disturbances for an offshore boom crane are proposed by Lu et al.^{[38]}, which contain a full state feedback controller as well as an output feedback controller. Moreover, in dealing with some unknown periodic sea wave disturbances, Qian et al.^{[39]} design an efficient nonlinear adaptive learning control, which presents good robustness against everlasting disturbances and unknown parameters. For flexible marine installations, through analyzing the vessel dynamics, He et al.^{[40]} develop a robust adaptive boundary control to achieve underwater positioning operations. By constructing an elaborate storage function, Sun et al.^{[41]} provide a complete Lyapunovbased nonlinear antiswing control method for an offshore crane, while for shipmounted boom cranes with ship roll and heave movements, a nonlinear controller is proposed also by the same authors^{[42]}. Moreover, for some systems with submerged payload hanging from offshore crane vessel, the nonlinear dynamic response has been investigated by Hannana and Bai^{[43]}, while a fuzzy sliding mode control approach has been proposed^{[44]}. However, either linearized dynamic model or accurate knowledge is needed for the aforementioned control strategies, which are restrictive for the control issues of most offshore shipmounted crane systems.
As a brief review, by examining the existing control approaches for offshore shipmounted cranes, the following essential issues remain open and unresolved.
1) Most existing methods for complex offshore shipmounted crane systems need to linearize the nonlinear crane dynamics or neglect some specific nonlinear terms around the equilibrium point, which may not obtain high control performance when the system states exceed the small ranges due to external disturbances, e.g., under harsh sea conditions.
2) For presently available closedloop control methods for offshore cranes, they cannot theoretically guarantee the asymptotic stability of the equilibrium point, which is also a very important control problem for such underactuated systems. Moreover, since an offshore shipmounted crane always suffers from some unmatched external disturbances, which are mainly caused by sea waves or currents, the accurate positioning of the payload with lower swing angle during transportation process is an imperative requirement for the control system in the industrial field.
3) Almost all existing control strategies are sensitive to external disturbances, which lack robustness for such offshore crane systems. Therefore, proposing of a high performance control law against undesired extraneous perturbations is of great importance.
Generally speaking, for control of mechanical systems, an energybased controller designing approach is feasible and effective owing to its efficacy of energy elimination. In this regard, for instance, an energybased control method for the regulation of overhead cranes is proposed by Sun and Fang^{[45]}, while an energybased control of double pendulum cranes has also been developed^{[46]}.
Considering the previously mentioned facts, in this paper, we present a novel nonlinear energybased coupling control for an offshore shipmounted crane, which guarantees the asymptotic stability of the equilibrium point and achieves satisfactory control performance for various transportation tasks. Specifically, via introducing an elegant composite error signal, which enhances the inner coupling between system states and disturbed motion, an offshore shipmounted crane system is then transformed into an interconnected system, which brings much convenience in controller design and stability analysis for the reconstructed crane model. Based on which, as is generally known that energy can reflect and describe the motion and status of a dynamical system, through choosing a proper Lyapunov candidate function, an energybased controller is then designed. The convergence of the closedloop system′s equilibrium point is proven by Lyapunov techniques and LaSalle′s invariance theorem. Finally, to illustrate the promising application prospect of the proposed control method, numerical simulation and experiments are both implemented, which clearly show its effectiveness as well as the robustness against external disturbances.
In summary, as the main contribution of this paper, the proposed method successfully achieves an improved control performance, which is presented as follows:
1) The newly defined coupling system states involve the inherent mechanism of the nonlinear dynamics, which, along with the straightforward stability analysis, guarantees the superior control performance of the offshore shipmounted crane system.
2) Unlike traditional energybased control methods, the proposed controller takes full account of the state coupling of the nonlinear dynamics, which takes a much simpler structure independent of the system parameters.
3) The proposed controller is robust against numerous external disturbances, which is demonstrated by numerical/experimental results.
The remaining sections of the presentation are structured in the following manner. In Section 2, the original offshore crane dynamics is briefly introduced, and for the kernel part, a novel composite signal is developed and then the model transformation is implemented. Section 3 refers to the processing of controller design and stability analysis for the coupled crane system. Simulation results are given in Section 4, while experimental results are implemented in Section 5 to further demonstrate the superior performance of the proposed control scheme. Section 6 draws the main conclusion regarding this work.
2 Crane dynamics and model trans formation 2.1 Dynamics and control objectiveConsidering the horizontal transportation stage of an offshore shipmounted crane, see Fig. 1, the control issue can be depicted as precise positioning and swing elimination of payloads under external disturbances caused by sea waves or sea winds. Specifically, two frames are involved, which include the landfixed frame
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Fig. 1. Schematic illustration of an offshore shipmounted crane 
In our previous work, the dynamics of the 2dimensional (2D) offshore shipmounted crane, which contains the disturbing effect of ship′s motion, can be given as follows^{[47]}:
$\begin{split}& \left( {{m_1} + {m_2}} \right){{\ddot L}_x} + {m_2}{C_\theta }L\ddot \theta  {m_2}L{S_\theta }{{\dot \theta }^2}=\\&\;\;\;\;\;\;\; {F_x}  {f_{rx}} + {m_1}g{S_\alpha }  {m_2}g{S_\alpha }  {f_{{L_x}}}\end{split}$  (1) 
${m_2}L{C_\theta }{\ddot L_x} + {m_2}{L^2}\ddot \theta + {m_2}gL{S_{\theta  \alpha }} =  {f_\theta }$  (2) 
where
Besides,
$\begin{split}{f_{{L_x}}} = & \,\left( {{m_1}h + {m_2}h  {m_2}L{C_\theta }} \right)\ddot \alpha +\\& \left( {{m_1}{L_x} + {m_2}{L_x} + {m_2}L{S_\theta }} \right){{\dot \alpha }^2}+\\& 2{m_2}L\dot \theta {S_\theta }\dot \alpha\end{split}$  (3) 
$\begin{split}{f_\theta } = & \,{m_2}L\left( {h{C_\theta }  L  {L_x}{S_\theta }} \right) \ddot \alpha + \\& {m_2}L\left( {{L_x}{C_\theta } + h{S_\theta }} \right){{\dot \alpha }^2}\quad\quad\quad\\& 2{m_2}L{{\dot L}_x}{S_\theta }\dot \alpha\end{split}$  (4) 
where h is a constant denoting the height of the boom.
In this paper, we mainly consider the transportation tasks from deck to deck, whose control objective is to position the payloads accurately with lower payload swing. Above all, before the controller design, the target position
${{y}_{d}}={{L}_{xd}}+L{{S}_{{{\theta }_{d}}}}$  (5) 
where
${{L}_{xd}}={{y}_{d}}L{{S}_{\alpha }}$  (6) 
${{\theta }_{d}}=\alpha.\quad\quad\quad\quad$  (7) 
Then, the error signals
${{e}_{1}}\left( t \right)={{L}_{x}}{{L}_{xd}}={{L}_{x}}{{y}_{d}}+L{{S}_{\alpha }}$  (8) 
${{e}_{2}}\left( t \right)=\theta {{\theta }_{d}}=\theta \alpha.\quad\quad\quad\quad\quad\;\;$  (9) 
Hereto, the basic control tasks of an offshore shipmounted crane can be summarized as follows:
1) To regulate the suspended payload from its initial position to the desired location, denoted as
2) To suppress the residual payload swing (i.e., the swing after the cargo arrives at
3) To eliminate the effect induced by external sea wave disturbances within finite time.
More specifically, to stabilize the offshore shipmounted crane system at the desired equilibrium point, the main control objective is given by
${\left[{{e_1}(t)} \;\; {{{\dot e}_1}(t)} \;\; {{e_2}(t)} \;\;{{{\dot e}_2}(t)}\right]^{\rm T}} ={\left[0 \;\; 0 \;\; 0 \;\; 0\right]^{\rm T}}.$  (10) 
From an arbitrary initial status in the presence of the unexpected additive disturbances
In order to enhance the coupling behavior of system states and facilitate the subsequent controller development and analysis, in this subsection, the model transformation is performed. Substituting formulas (3) and (4) into the original dynamics (1) and (2), respectively, and after some mathematical manipulations, one can obtain:
$\begin{split}&\left( {{m_1} + {m_2}} \right)\left( {{{\ddot L}_x} + h\ddot \alpha } \right) + {m_2}{C_\theta }L\left( {\ddot \theta  \ddot \alpha } \right)\\ & \quad\quad {m_2}L{S_\theta }{\left( {\dot \theta  \dot \alpha } \right)^2} = \\ & \quad\quad {F_x}  {f_{rx}} + {m_1}g{S_\alpha }{m_2}g{S_\alpha }  \left( {{m_1} + {m_2}} \right){L_x}{{\dot \alpha }^2}\end{split}$  (11) 
$\begin{split}& {m_2}L{C_\theta }\left( {{{\ddot L}_x} + h\ddot \alpha } \right) + {m_2}{L^2}\left( {\ddot \theta  \ddot \alpha } \right) + {m_2}gL{S_{\theta  \alpha }}=\\& \quad\quad {m_2}L{S_\theta }{L_x}\ddot \alpha  {m_2}L\left( {{L_x}{C_\theta } + h{S_\theta }} \right){{\dot \alpha }^2}+ \\ & \quad\quad 2{m_2}L{S_\theta }{{\dot L}_x}\dot \alpha .\end{split}$  (12) 
Herein, based on the form of (11) and (12), we define the novel composite error signals
${{\xi }_{1}}={{{\dot{e}}}_{1}}+{{\lambda }_{\alpha }}\varphi \left( \alpha \right)+{{\lambda }_{\xi }}\phi \left( {{e}_{2}} \right)$  (13) 
${{\xi }_{2}}=\dot{\theta }\dot{\alpha }={{{\dot{e}}}_{2}} \quad\quad\quad\quad\quad\quad$  (14) 
where
Taking the time derivative of (13) and (14), respectively, we have
${{{\dot{\xi }}}_{1}}={{{\ddot{e}}}_{1}}+{{\lambda }_{\alpha }}\dot{\varphi }\left( \alpha \right)+{{\lambda }_{\xi }}\frac{\partial \phi \left( {{e}_{2}} \right)}{\partial {{e}_{2}}}{{\xi }_{2}}$  (15) 
${{{\dot{\xi }}}_{2}}=\ddot{\theta }\ddot{\alpha }={{{\ddot{e}}}_{2}} \quad\quad\quad\quad\quad\quad\quad\;\;$  (16) 
while integrating those functions of (13) and (14), one obtains:
$\begin{split}\int_0^t {{\xi _1}} \left( \tau \right) {\rm d}\tau = & \, {e_1} + {\lambda _\alpha }\displaystyle\int_0^t {\varphi \left( {\alpha \left( \tau \right)} \right){\rm d}\tau }+\\& {\lambda _\xi }\displaystyle\int_0^t {\phi \left( {{e_2}\left( \tau \right)} \right){\rm d}\tau } \end{split}$  (17) 
$\int_{0}^{t}{{{\xi }_{2}}}\left( \tau \right) {\rm d}\tau={{e}_{2}}.\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$  (18) 
Then, substituting (13)–(18) into (11) and (12), the coupled dynamics model can be rewritten as
$\begin{split}& \left( {{m_1} + {m_2}} \right){{\dot \xi }_1} + {m_2}{C_\theta }L{{\dot \xi }_2}  {m_2}L{S_\theta }\xi _2^2=\\& \quad {F_x} \!\! {f_{rx}} \!+\! {m_1}g{S_\alpha }  {m_2}g{S_\alpha } \! \!\left( {{m_1} \!+\! {m_2}} \right){L_x}{{\dot \alpha }^2}+\\& \quad \left( {{m_1} + {m_2}} \right)(  {{\ddot L}_{xd}}  h\ddot \alpha )+\\& \quad \left( {{m_1} + {m_2}} \right)({\lambda _\alpha }\dot \varphi \left( \alpha \right) + {\lambda _\xi }\frac{{\partial \phi \left( {{e_2}} \right)}}{{\partial {e_2}}}{\xi _2})\end{split}$  (19) 
$\begin{split}& {m_2}L{C_\theta }{{\dot \xi }_1} + {m_2}{L^2}{{\dot \xi }_2} + {m_2}gL{S_{\theta  \alpha }}=\\ & \quad {m_2}L({S_\theta }{L_x}\ddot \alpha  ({L_x}{C_\theta } + h{S_\theta }){{\dot \alpha }^2} + 2{S_\theta }{{\dot L}_x}\dot \alpha )+\\ &\quad {m_2}L{C_\theta }(  {{\ddot L}_{xd}}  h\ddot \alpha ) + {\lambda _\alpha }{m_2}L{C_\theta }\dot \varphi (\alpha )+\\ &\quad {\lambda _\xi }{m_2}L{C_\theta }\frac{{\partial \phi \left( {{e_2}} \right)}}{{\partial {e_2}}}{\xi _2}.\end{split}$  (20) 
To facilitate the following analysis, based on (19) and (20), the matrix form of an offshore shipmounted crane system is compacted as
${{{M}}_\xi }\dot\xi + {{{V}}_\xi }\xi = {{{F}}_c} + {{{f}}_r} + {{{F}}^*} + {{{F}}_a} $  (21) 
where the coupled system error vector is defined as
$\xi \left( t \right) = {\left[{\begin{array}{*{20}{c}}{{\xi _1}\left( t \right)} & {{\xi _2}\left( t \right)}\end{array}} \right]^{\rm T}} \in {{\bf{R}}^2}$  (22) 
the inertia matrix
${{{M}}_\xi } = \left[{\begin{array}{*{20}{c}}{{m_1} + {m_2}} & {{m_2}{C_\theta }L}\\{{m_2}L{C_\theta }} & {{m_2}{L^2}}\end{array}} \right]$  (23) 
and
${{{V}}_\xi } = \left[{\begin{array}{*{20}{c}}0 & {{m_2}L{S_\theta }\dot \theta }\\0 & 0\end{array}} \right].$  (24) 
In (21),
${{{F}}_c} = {\left[{\begin{array}{*{20}{c}}{{F_x}} & 0\end{array}} \right]}^{\rm T}$  (25) 
with
${{{f}}_r} = {\left[{\begin{array}{*{20}{c}}{{f_{rx}}} & 0\end{array}} \right]^{\rm T}}.$  (26) 
In addition, for the coupling disturbance vectors,
${{{F}}^*} = {\left[{\begin{array}{*{20}{c}}{f_1^{\rm{*}}} & {{m_2}Lf_2^{\rm{*}}}\end{array}} \right]}^{\rm T}$  (27) 
and
${{{F}}_a} = {\left[{\begin{array}{*{20}{c}}{{f_{a1}}} & {{f_{a2}}}\end{array}} \right]}^{\rm T}$  (28) 
respectively, the details are as follows:
$f_1^{\rm{*}} = {m_1}g{S_\alpha }  {m_2}g{S_\alpha }  \left( {{m_1} + {m_2}} \right){L_x}{\dot \alpha ^2} \quad$  (29) 
$\begin{split}f_2^{\rm{*}} = & \, {S_\theta }{L_x}\ddot \alpha  \left( {{L_x}{C_\theta } + h{S_\theta }} \right){{\dot \alpha }^2} + 2{S_\theta }{{\dot L}_x}\dot \alpha +\\& {m_2}L{C_\theta }(  {{\ddot L}_{xd}}  h\ddot \alpha )\end{split}$  (30) 
and
$\begin{split}{f_{a1}} = & \left( {{m_1} + {m_2}} \right)(  {{\ddot L}_{xd}}  h\ddot \alpha )+\\& \left( {{m_1} + {m_2}} \right)\left({\lambda _\alpha }\dot \varphi \left( \alpha \right) + {\lambda _\xi }\frac{{\partial \phi \left( {{e_2}} \right)}}{{\partial {e_2}}}{\xi _2}\right)\end{split}$  (31) 
${{f}_{a2}}={{\lambda }_{\alpha }}{{m}_{2}}L{{C}_{\theta }}\dot{\varphi }\left( \alpha \right)+{{\lambda }_{\xi }}{{m}_{2}}L{{C}_{\theta }}\frac{\partial \phi \left( {{e}_{2}} \right)}{\partial {{e}_{2}}}{{\xi }_{2}}.$  (32) 
Analogous to many other EulerLagrange systems^{[1–14]}, the following properties and assumption are also valid for the transformed model (21) as:
Property 1. The inertia matrix
Property 2. The matrix of
${{{x}}^{\rm T}}\left( {\frac{1}{2}{{{\dot{ M}}}_\xi }  {{{V}}_\xi }} \right){{x}} = 0, \;\;\;\forall {{x}} \in {{\bf{R}}^2}.$  (33) 
Assumption 1. Without loss of generality, considering the movement range of payload swing subject to the physical constraints, we assume that
$\frac{\pi }{2}\le \theta \alpha \le \frac{\pi }{2}.$  (34) 
Remark 1. It is noteworthy that this assumption is widely used in the cranerelated literature^{[6, 8, 11]}. From an industrial perspective, considering that the payloads are heavy enough in practice and it is almost impossible to go above the trolley, this assumption is reasonable.
3 Controller design and stability analysisIn this section, a novel energybased nonlinear coupling controller is derived on the basis of the coupled offshore shipmounted crane model, which achieves simultaneous precise positioning and effective load swing suppression.
3.1 Controller designFirst of all, instead of using feedforward or predictive control frameworks, we determine to derive an energy analysisbased controller. Based on this, for the control objective of (6), (7) and on the basis of the coupled error signal (22), the system mechanical energy is represented as follows:
$E = \frac{1}{2}{\xi ^{\rm T}}{{M}}\xi + mgL\left( {1  {C_{\theta  \alpha}}} \right)$  (35) 
whose time derivative can be calculated, by utilizing (21)–(32) and the property of (33), as
$\dot E = {\xi ^{\rm T}}({{M}}\dot \xi + \frac{1}{2}{\dot{ M}}\xi ) + {m_2}gL{S_{\theta  \alpha }}(\dot \theta  \dot \alpha )$  (36) 
which derives:
$\begin{split}\dot E = & \,{\xi _1}\left( {{F_x}  {f_{rx}} + f_1^* + ({m_1} + {m_2})(  {{\ddot L}_{xd}}  h\ddot \alpha )} \right.+\\& \left. { \left( {{m_1} + {m_2}} \right){\lambda _\alpha }\dot \varphi \left( \alpha \right) + \left( {{m_1} + {m_2}} \right){\lambda _\xi }\frac{{\partial \phi \left( {{e_2}} \right)}}{{\partial {e_2}}}{\xi _2}} \right)+\\& {m_2}L{\xi _2}({\lambda _\alpha }{C_\theta }\dot \varphi \left( \alpha \right) + f_2^* + {\lambda _\xi }{C_\theta }\frac{{\partial \phi \left( {{e_2}} \right)}}{{\partial {e_2}}}{\xi _2}).\end{split}$  (37) 
Then, define the energybased controller as
$\begin{array}{l}{F_x} =  {\lambda _\alpha }\left( {{m_1} + {m_2}} \right)\dot \varphi \left( \alpha \right)  {\lambda _\xi }\left( {{m_1} + {m_2}} \right)\displaystyle\frac{{\partial \phi \left( {{e_2}} \right)}}{{\partial {e_2}}}{\xi _2}\\\;\;\;\;\;\;\; {k_\xi }{\xi _1}  {k_p}\displaystyle\int_{0}^{t} {{\xi _1}\left( \tau \right){\rm d}\tau } + {f_{rx}}\\\;\;\;\;\;\;\; f_1^* + \left( {{m_1} + {m_2}} \right)\left( {{{\ddot L}_{xd}} + h\ddot \alpha } \right)\end{array}$  (38) 
where
Moreover, in order to decrease the closedloop system energy, based on the form of (36), we choose
${{\lambda }_{\alpha }}{{C}_{\theta }}\dot{\varphi }\left( \alpha \right)=f_{2}^{*}$  (39) 
and
$\frac{\partial \phi \left( {{e}_{2}} \right)}{\partial {{e}_{2}}}=1\le 0$  (40) 
where the yettoconstruct functions defined in (13) can be consequently selected as
$\varphi \left( \alpha \right)=\frac{1}{{{\lambda }_{\alpha }}}\int_{0}^{t}{\frac{f_{2}^{*}\left( \tau \right)}{{{C}_{\theta }}}}{\rm d}\tau$  (41) 
and
$\phi \left( {{e}_{2}} \right)=\left( \theta \alpha \right).$  (42) 
Substituting (41) and (42) into (38), the ultimate form of controller can be given as
$\begin{split}{F_x} = &  {k_\xi }{\xi _1}  {k_p}\int_0^t {{\xi _1}\left( \tau \right){\rm d}\tau } + {f_{rx}}\\& f_1^* + \left( {{m_1} + {m_2}} \right)\left( {{{\ddot L}_{xd}} + h\ddot \alpha } \right)+\\& \left( {{m_1} + {m_2}} \right)\frac{{f_2^*}}{{{C_\theta }}} + {\lambda _\xi }\left( {{m_1} + {m_2}} \right){\xi _2}\end{split}$  (43) 
where
The stability of the overall closedloop system is analyzed subsequently by using Lyapunov techniques and LaSalle′s invariance theorem.
Theorem 1. Under the proposed nonlinear controller (43), the payload is driven to the desired position while the swing can be damped out globally in the sense that
$\mathop {\lim }\limits_{t \to \infty } {\mkern 1mu} {\left[{\begin{array}{*{20}{c}}{{e_1}} & {{{\dot e}_1}} & {{e_2}} & {{{\dot e}_2}}\end{array}} \right]^{\rm T}} = {\left[{\begin{array}{*{20}{c}}0 & 0 & 0 & 0\end{array}} \right]^{\rm T}}.$  (44) 
Proof. Let a positive definite Lyapunov candidate function
$\begin{split}V = & \frac{1}{2}{\xi ^{\rm T}}{{M}}\xi + mgL\left( {1  {C_{\theta  \alpha }}} \right)+\\& \frac{1}{2}{k_p}{\left( {\int_0^t {{\xi _1}\left( \tau \right){\rm d}\tau } } \right)^2} \ge 0.\end{split}$  (45) 
Taking the time derivative of (45), and substituting the controller of (43) as well as the results of (36), (41) and (42), one leads to
$\begin{split}\dot V = & {\xi _1}\left( {  {k_\xi }{\xi _1}  {k_p}\int_0^t {{\xi _1}\left( \tau \right){\rm d}\tau + {k_p}\int_0^t {{\xi _1}\left( \tau \right){\rm d}\tau } } } \right)\\& {\lambda _\xi }{m_2}L{C_\theta }\xi _2^2 \end{split}$  (46) 
implying that
$\dot{V}={{k}_{\xi }}\xi _{1}^{2}{{\lambda }_{\xi }}{{m}_{2}}L{{C}_{\theta }}\xi _{2}^{2}\le 0.$  (47) 
According to (47), it is easily shown that the closedloop system states around the equilibrium points are stable in the Lyapunov sense, namely that
${{\xi }_{1}}\left( t \right), {{\xi }_{2}}\left( t \right)\in {{L}_{\infty }}$  (48) 
then, from (13) and (14), we have
${{{\dot{L}}}_{x}}\left( t \right), \dot{\theta }\left( t \right)\in {{L}_{\infty }}$  (49) 
which, together with (43) and (45), indicates
${L_x}\left( t \right), \int_0^t {{\xi _1}\left( t \right){\rm d}\tau }, \int_0^t {{\xi _2}\left( t \right){\rm d}\tau } \in {L_\infty }$  (50) 
as well as
${{e}_{1}}\left( t \right), {{e}_{2}}\left( t \right)\in {{L}_{\infty }}\Rightarrow$  (51) 
$ {{F}_{x}}\left( t \right)\in {{L}_{\infty }}.\quad\quad\quad\quad$  (52) 
Furthermore, to facilitate further analysis of the closedloop system, let
${{S}}=\left\{ {{e}_{1}}, {{{\dot{e}}}_{1}}, {{e}_{2}}, {{{\dot{e}}}_{2}}:\dot{V}=0 \right\}$  (53) 
in which, it follows from (47) and (53) that
$\begin{split}& {\xi _1} = 0,\; {\xi _2} = 0 \Rightarrow \\& {{\dot \xi }_1} = 0, \;{{\dot \xi }_2} = 0.\end{split}$  (54) 
Then, putting (54) into (21), by utilizing (8), (9) and (13)–(16), in the set
$\begin{split}& {{\dot e}_2} = {\xi _2} = 0 \\& {F_x}  {f_{rx}} + {f_{a1}} + f_1^* = 0 \\& {f_{a2}} + {m_2}Lf_2^* = 0 \\& g{S_{\theta  \alpha }} = 0\end{split}$  (55) 
implying that
$\begin{split}& {e_2} = \theta  \alpha = 0 \\& \int_0^t {{\xi _1}} \left( \tau \right){\rm d}\tau = 0\Rightarrow\end{split}$  (56) 
$ {{e}_{1}}=0\Rightarrow\quad\quad\quad\quad\quad\;$  (57) 
$ {{{\dot{e}}}_{1}}={{{\dot{L}}}_{x}}{{{\dot{L}}}_{xd}}=0\Rightarrow$  (58) 
$ {{L}_{x}}={{L}_{xd}}\quad\quad\quad\quad\quad\;$  (59) 
where it is not difficult to show that the largest invariant set
${\left[\!\!\!{\begin{array}{*{20}{c}}{{e_1}\left( t \right)} & {{{\dot e}_1}\left( t \right)} & {{e_2}\left( t \right)} & {{{\dot e}_2}\left( t \right)}\end{array}}\!\!\! \right]^{\rm T}} = {\left[\!\!\!{\begin{array}{*{20}{c}}0 & 0 & 0 & 0\end{array}} \!\!\right]^{\rm T}}.$  (60) 
Finally, using LaSalle′s invariant theorem, the conclusion of Theorem 1 can be obtained as
$\mathop {\lim }\limits_{t \to \infty } {\mkern 1mu} {\left[{\begin{array}{*{20}{c}}{{e_1}} & {{{\dot e}_1}} & {{e_2}} & {{{\dot e}_2}}\end{array}} \right]^{\rm T}} = {\left[{\begin{array}{*{20}{c}}0 & 0 & 0 & 0\end{array}} \right]^{\rm T}}.$  (61) 
□
Therefore, a direct application of LaSalle invariance theorem approves (44), which achieves the main control objective of accurate positioning of payload with lower swing angle under persistent external disturbances.
4 Numerical simulationIn this section, we implement some groups of numerical simulation tests in the environment of Matlab/Simulink and provide the simulation results to verify the performance of the designed offshore shipmounted crane control system.
As stated previously, the control objective is to position the payload of the offshore shipmounted crane to the expected accurate point against some external disturbances. Before the simulation/experiment, the dynamic model depicted in (21) is implemented in the Matlab/Simulink environment, and the parameters of the offshore shipmounted crane system are chosen as
$\begin{split}& {m_1} = 4.3\;{\rm{kg}}, \;{m_2} = 0.5\;{\rm{kg}}\\& L = 0.3\;{\rm{m}}, \;h = 0.58\;{\rm{m}}\\& g = 9.8\;{\rm{m}}/{{\rm{s}}^2}.\end{split}$ 
In this group of simulation, high control performance of the proposed controller (43) subjected to sea wave disturbances has been fully testified.
Firstly, the initial states of the system are selected as
${L_x}(0) = 0\;{\rm{m}}, \;\theta (0) = 0\;{\rm{deg}}$ 
and the desired position of payload and the ship motion disturbance are set as
$\begin{array}{l}{y_d} = 0.15\;{\rm{m}}\\\alpha = 0.5\cos \left( {0.1t} \right)  0.5\;{\rm{deg}}.\end{array}$ 
To obtain proper performance, through carrying out abundant numerical simulations, the control gains for the proposed energybased controller in this group are selected as follows:
$\begin{array}{l}{k_p} = 8, \;{k_\xi } = 5 \\{\lambda _\xi } = 1.5, \;{\lambda _\alpha } = 5.5.\end{array}$ 
Under unmatched sea wave disturbances, the following results of the designed shipmounted crane control system are presented in Fig. 2, together with the results of the newly designed coupling states
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Fig. 2. Simulation 1: Trolley displacement error

It can be seen from these results that, the proposed controller achieves satisfactory performance under external disturbances, in the sense that the trolley position error
Moreover, the new composite signals are also depicted in Fig. 2 to show more transient response of the offshore shipmounted crane system, which greatly improves the efficiency of the control strategy with rapid convergence of system states.
4.2 Simulation 2To further show the superiority of the proposed energybased control method, in this group of simulation, some comparative numerical tests with linear quadratic regulator (LQR) controller are then implemented.
Generally speaking, a traditional LQR controller has the following form as^{[48]}
${F_x} =  {K_1}{e_{Lx}}  {K_2}{\dot e_{{L_x}}}  {K_3}{e_\theta }  {K_4}{\dot e_\theta }$ 
where to implement fair experiment, the proper control gains of LQR controller are solved by Matlab as follows:
$\begin{array}{l}{K_1} = 8.1, \;{K_2} = 37.1\\{K_3} =  24.0, \;{K_4} = 1.8.\end{array}$ 
In this group of simulation, the desired position of payload and the ship motion disturbance are set as
$\begin{array}{l}{y_d} = 0.5\;{\rm{m}}\\\alpha = 0.18\sin \left( {0.1t} \right)\;{\rm{deg}}\end{array}$ 
with the initial states both selected as
${L_x}(0) = 0\;{\rm{m}}, \;\theta (0) = 0\;{\rm{deg}}.$ 
After careful tuning, to achieve fair simulations, the control gains for the proposed energybased controller are selected as
$\begin{array}{l}{k_p} = 4.5, \;{k_\xi } = 2.5, \\{\lambda _\xi } = 1, \;{\lambda _\alpha } = 5.\end{array}$ 
Corresponding simulation results are depicted in Fig. 3, which give the trolley position error
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Fig. 3. Simulation 2: Trolley displacement error

It can be obviously seen from the comparative simulation results that, driven by the proposed controller, the error of trolley displacement and the payload swing angle both converge to zero within 10 s under persistent sea wave disturbances, which performs better than that of the LQR controller. Additionally, asymptotic convergence of system states can be guaranteed under the proposed controller, which shows a better performance than a simple linear controller such as LQR control.
5 Hardware experimentFrom a practical perspective, this section exhibits some hardware experimental results to further test the performance of the proposed control method.
Based on a selfbuilt offshore shipmounted crane test bed^{[9, 39]}, some actual experiments are conducted to further verify the actual performance of the proposed approach. As shown in Fig. 4, this system consists of four parts, including a kernel control component, an actuating device, a mechanical framework and a chassis.
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Fig. 4. Selfbuilt offshore shipmounted crane hardware experiment platform 
In the system, the trolley moving along the boom and the steel rope connected with the suspending payload are driven by two SYNTRON alternating current (AC) servo motors, respectively. The swing angle of the payload is measured by angular sensors installed beneath the boom. Besides, the function of the chassis in this platform is to imitate the motion of sea waves and other unknown disturbances.
For realtime control algorithm implementation, Matlab/Simulink realtime Windows target running under Windows XP operating system is established for the control system. For the actuating device, seven actuators are communicated with seven AC servo motors so as to control the mechanical system. Moreover, one Google technology GT2800ACC2V2.0V motion control board with I/O interfaces is employed as the data acquisition unit to collect data from the encoders and convey control signals to the driver so that the control voltage can be applied to the AC servo motors successfully.
Beforehand, regarding the utilized offshore shipmounted crane hardware platform (see Fig. 4), the physical parameters of the testbed are configured as
$\begin{array}{l}{m_1} = 3.5\;{\rm{kg}}, \;{m_2} = 0.5\;{\rm{kg}}\\L = 0.3\;{\rm{m}}, \;h = 0.58\;{\rm{m}}\\g = 9.8\;{\rm{m}}/{{\rm{s}}^2}.\end{array}$ 
For sufficient verification, based on these conditions, we will carry out two groups of hardware experiments. Specifically, we will first demonstrate the satisfactory performance of the proposed control method under harsh sea wave disturbances. Then, in experiment Group 2, additional experiments will be implemented to verify the robustness against unexpected disturbances such as payload swing perturbations.
5.1 Experiment 1In this subsection, the first group verifies the control performance of our approach under external sea wave disturbances, with the target value of payload set as
${y_d} = 0.15\;{\rm{m}}$ 
and the external disturbance chosen as
$\alpha (t) = 0.3\sin (\frac {\pi}{4})\;{\rm{deg}}.$ 
To obtain better performance, the control gains for the experiment are tuned as
$\begin{array}{l}{k_p} = 15, \;{k_\xi } = 25 \\{\lambda _\xi } = 10, \;{\lambda _\alpha } = 10.\end{array}$ 
As the control objective of this group of experiment is to drive the trolley to arrive at its target destination denoted as
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Fig. 5. Experiment 1: trolley displacement error

Besides, the ship roll disturbance
Therefore, the accurate positioning and antiswing performance of the proposed nonlinear controller can be fully demonstrated in this group of experiments.
5.2 Experiment 2Furthermore, another test in this group is implemented to verify the robustness of the closedloop control system against transient payload swing perturbation during transportation process, whose experimental results are shown in Fig. 6.
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Fig. 6. Experiment 2: trolley displacement error

Specifically, for practical purposes, we set the target value of payload as
${y_d} = 0.5\;{\rm{m}}$ 
and choose the external disturbance as
$\alpha (t) = 0.3\sin (\frac{\pi} {4})\;{\rm{deg}}.$ 
Here, the control gains for the experiment are selected the same as in Experiment 1 as follows:
$\begin{array}{l}{k_p} = 15, \;{k_\xi } = 25 \\{\lambda _\xi } = 10, \;{\lambda _\alpha } = 10.\end{array}$ 
As clearly shown in Fig. 6, some unexpected external disturbances are added to the payload swing angles on purpose at about the 8th second (see the red marked places), while the control performance is still satisfactory in terms of asymptotic convergence, which demonstrates superior performance of the proposed controller.
Overall, with these simulation/experimental results, it can be concluded that the proposed energybased control method allows for robustness, which is important for practical use.
6 ConclusionsThis paper provides energybased coupling control approach for an offshore shipmounted crane. Successfully settling the challenges associated with underactuated property and unmatched external disturbances of such kind of systems, the proposed novel controller enhances the coupling behavior between trolley motion, payload swing and ship roll motion and then leads to an improved control performance. Based on the coupled model, a novel energybased nonlinear controller is developed on the basis of system′s mechanical energy, then Lyapunov techniques as well as LaSalle′s invariant theorem are coherently utilized to further analyze the asymptotic stability of the closedloop crane system. Simulation and experimental results are then included to demonstrate the superior control performance even in terms of transient payload swing perturbations. In forthcoming efforts, the control of 3D offshore shipmounted crane which includes rotation motion of the boom should be taken into consideration, while more complex control problems such as tracking control are also involved in our future work.
AcknowledgementsThis work was supported by National Natural Science Foundation of China (No. 11372144), National Science Fund for Distinguished Young Scholars of China (No. 61325017), and National Science Foundation of Tianjin.
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