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Adaptive Hierarchical Sliding Mode Control Design for 3D Ship-Mounted Container Crane with Saturating Actuators

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Abstract

The study proposes a dynamic model and an adaptive controller for a 3D offshore container crane based on a hierarchical sliding mode control structure and artificial neural network. An overhead-type container crane mounted on a ship with actuator saturation is considered, and its dynamic model is derived from the Lagrange formulation. Hierarchical sliding mode control (HSMC) is utilized to construct the controller structure, while radial basis function (RBF) neural networks are integrated to estimate unknown dynamics and compensate for saturation nonlinearity. The control law is designed to ensure the stability of sliding surfaces, and an updated law for neural network’s weight matrices is constituted from a candidate of Lyapunov function. Simulations and experiments are provided to verify the effectiveness and robustness of the proposed control scheme in offshore conditions containing sea wave excitation and wind force.

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Acknowledgements

This research is funded by International School, Vietnam National University, Hanoi, Vietnam. We would like to thank Viet-Anh Le (Department of Mechanical Engineering, University of Delaware, USA) and Dr. Trieu Van Pham (Vietnam Maritime University) for assisting this study.

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  1. International School, Vietnam National University, Hanoi, 100000, Vietnam

    Thai Dinh Kim, Linh Ngoc Nguyen & Hai Xuan Le

  2. Faculty of Electrical Engineering, Hanoi University of Industry, Hanoi, 100000, Vietnam

    Xuan Minh Dinh

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  1. Thai Dinh Kim
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6 Appendices

6 Appendices

1.1 6.1 Appendix A

The components of the \({\mathbf{M}}({\mathbf{q}})\) are given as follows:

$$\begin{aligned}&{{m}_{11}}={{m}_{1}}+{{m}_{2}}+{{m}_{c}};\\&{{m}_{12}}={{m}_{21}}=0;\\&{{m}_{13}}={{m}_{31}}={{m}_{c}}\left( c\beta s\alpha c\theta +s\beta s\varphi s\theta +c\alpha c\beta c\varphi s\theta \right) ;\\&{{m}_{14}}={{m}_{41}}={{m}_{c}}lc\beta \left( c\alpha c\theta -c\varphi s\alpha s\theta \right) ;\\&{{m}_{15}}={{m}_{51}}=-{{m}_{c}}l\left( s\alpha s\beta c\theta -c\beta s\varphi s\theta +c\alpha c\varphi s\beta s\theta \right) ;\\&{{m}_{22}}={{m}_{2}}+{{m}_{c}};\\&{{m}_{23}}={{m}_{32}}={{m}_{c}}\left( c\varphi s\beta -c\alpha c\beta s\varphi \right) ; \\&{{m}_{24}}={{m}_{42}}={{m}_{c}}lc\beta s\alpha s\varphi ;\\&{{m}_{25}}={{m}_{52}}={{m}_{c}}l\left( c\beta c\varphi +c\alpha s\beta s\varphi \right) ; \\&{{m}_{33}}={{m}_{c}};{{m}_{34}}={{m}_{43}}=0; {{m}_{35}}={{m}_{53}}=0; \\&{{m}_{44}}={{m}_{c}}{{l}^{2}}co{{s}^{2}}\beta ;\\&{{m}_{45}}={{m}_{54}}=0; \\&{{m}_{55}}={{m}_{c}}{{l}^{2}} \end{aligned}$$

The elements on the \(\mathbf {C}(\mathbf {q},\dot{\mathbf {q}})\) matrix are expressed as:

$$\begin{aligned}&{{c}_{11}}=0;\quad {{c}_{12}}=-\dot{\varphi }({{m}_{2}}+{{m}_{c}})s\theta ; \\&{c_{13}} = - {m_c}\left( \begin{array}{l} \dot{\beta }\left( {s\alpha s\beta c\theta - c\beta s\varphi s\theta + c\alpha c\varphi s\beta s\theta } \right) + \\ \dot{\alpha }\left( {c\beta c\varphi s\alpha s\theta - c\alpha c\beta c\theta } \right) \end{array} \right) ;\\&{c_{14}} = - {m_c}\left( \begin{array}{l} \dot{l}c\beta \left( {c\varphi s\alpha s\theta - c\alpha c\theta } \right) + \\ \dot{\alpha }lc\beta \left( {s\alpha c\theta + c\alpha c\varphi s\theta } \right) + \\ \dot{\beta }ls\beta \left( {c\alpha c\theta - c\varphi s\alpha s\theta } \right) \end{array} \right) ;\\&{c_{15}} = - {m_c}\left( \begin{array}{l} \dot{l}\left( {s\alpha s\beta c\theta - c\beta s\varphi s\theta + c\alpha c\varphi s\beta s\theta } \right) + \\ \dot{\alpha }ls\beta \left( {c\alpha c\theta - c\varphi s\alpha s\theta } \right) + \\ \dot{\beta }l\left( {c\beta s\alpha c\theta + s\beta s\varphi s\theta + c\alpha c\beta c\varphi s\theta } \right) \end{array} \right) ;\\&{{c}_{21}}=\dot{\varphi }({{m}_{2}}+{{m}_{c}})s\theta ; {{c}_{22}}=0;\\&{{c}_{23}}={{m}_{c}}(\dot{\beta }\left( c\beta c\varphi +c\alpha s\beta s\beta \right) +\dot{\alpha }c\beta s\alpha s\varphi ); \\&{{c}_{24}}={{m}_{c}}s\varphi \left( \dot{l}c\beta s\alpha +\dot{\alpha }lc\alpha c\beta -\dot{\beta }ls\alpha s\beta \right) ;\\&{c_{25}} = {m_c}\left( \begin{array}{l} \dot{l}\left( {c\beta c\varphi + c\alpha s\beta s\varphi } \right) + \dot{\beta }l\left( {c\alpha c\beta s\varphi - c\varphi s\beta } \right) + \\ \dot{\alpha }ls\alpha s\beta s\varphi \end{array} \right) ;\\&{c_{31}} = {m_c}\left( \begin{array}{l} \dot{\varphi }\left( {c\varphi s\beta s\theta - c\alpha c\beta s\varphi s\theta } \right) + \\ \dot{\theta }\left( {s\beta c\theta s\varphi + c\alpha c\beta c\varphi c\theta - c\beta s\alpha s\theta } \right) \end{array} \right) ;\\&{{c}_{32}}=-\dot{\varphi }{{m}_{c}}(s\beta s\varphi +c\alpha c\beta c\varphi ); {{c}_{33}}=0;\\&{{c}_{34}}=-\dot{\alpha }{{m}_{c}}l{{c}^{2}}\beta ; {{c}_{35}}=-\dot{\beta }{{m}_{c}}l;\\&{{c}_{41}}=-{{m}_{c}}lc\beta \left( \dot{\theta }\left( c\alpha s\theta +c\varphi s\alpha c\theta \right) -\dot{\varphi }s\alpha s\varphi s\theta \right) ;\\&{{c}_{42}}=\dot{\varphi }{{m}_{c}}lc\beta c\varphi s\alpha ; {{c}_{43}}=\dot{\alpha }{{m}_{c}}l{{c}^{2}}\beta ;\\&{{c}_{44}}={{m}_{c}}lc\beta \left( \dot{l}c\beta -\dot{\beta }ls\beta \right) ;{{c}_{45}}=-\dot{\alpha }{{m}_{c}}{{l}^{2}}s\beta c\beta ; \\&{c_{51}} = {m_c}l\left( \begin{array}{l} \dot{\varphi }\left( {c\beta c\varphi s\theta + c\alpha s\beta s\varphi s\theta } \right) + \\ \dot{\theta }\left( {c\beta c\theta s\varphi + s\alpha s\beta s\theta - c\alpha c\varphi s\beta c\theta } \right) \end{array} \right) ;\\&{{c}_{52}}=-\dot{\varphi }{{m}_{c}}l(c\beta s\varphi -c\alpha c\varphi s\beta ); \quad {{c}_{53}}=\dot{\beta }{{m}_{c}}l; \\&{{m}_{54}}=\dot{\alpha }{{m}_{c}}{{l}^{2}}s\beta c\beta ; \quad {{c}_{55}}=\dot{l}{{m}_{c}}l. \end{aligned}$$

The components of the gravitational force vector \(\mathbf {G}(\mathbf {q})\) are presented as:

$$\begin{aligned}&{{g}_{1}}=-gc\varphi s\theta ({{m}_{1}}+{{m}_{2}}+{{m}_{c}});\\&{{g}_{2}}=gs\varphi ({{m}_{2}}+{{m}_{c}}); \\&{{g}_{3}}=-g{{m}_{c}}c\alpha c\beta ; \\&{{g}_{4}}=g{{m}_{c}}lc\beta s\alpha ; {{g}_{5}}=g{{m}_{c}}lc\alpha s\beta . \end{aligned}$$

The elements of the vector of wave-wind disturbance \(\mathbf {W}\) is determined as follows:

$$\begin{aligned}&{{w}_{1}}=-{{m}_{16}}\ddot{\varphi }-{{m}_{17}}\ddot{\theta }-{{m}_{18}}\ddot{z}-{{c}_{16}}\dot{\varphi }-{{c}_{17}}\dot{\theta }-{{c}_{18}}\dot{z};\\&{{w}_{2}}=-{{m}_{26}}\ddot{\varphi }-{{m}_{27}}\ddot{\theta }-{{m}_{28}}\ddot{z}-{{c}_{26}}\dot{\varphi }-{{c}_{27}}\dot{\theta }-{{c}_{28}}\dot{z};\\&{{w}_{3}}=-{{m}_{36}}\ddot{\varphi }-{{m}_{37}}\ddot{\theta }-{{m}_{38}}\ddot{z}-{{c}_{36}}\dot{\varphi }-{{c}_{37}}\dot{\theta }-{{c}_{38}}\dot{z};\\&{{w}_{4}}={{f}_{wx}}-{{m}_{46}}\ddot{\varphi }-{{m}_{47}}\ddot{\theta }-{{m}_{48}}\ddot{z}-{{c}_{46}}\dot{\varphi }-{{c}_{47}}\dot{\theta }-{{c}_{48}}\dot{z};\\&{{w}_{5}}={{f}_{wy}}-{{m}_{56}}\ddot{\varphi }-{{m}_{57}}\ddot{\theta }-{{m}_{58}}\ddot{z}-{{c}_{56}}\dot{\varphi }-{{c}_{57}}\dot{\theta }-{{c}_{58}}\dot{z}. \end{aligned}$$

where

$$\begin{aligned}&{{m}_{16}}=-{{s}_{y}}s\theta ({{m}_{2}}+{{m}_{c}});\\&{{m}_{17}}=h({{m}_{1}}+{{m}_{2}}+{{m}_{c}});\\&{{m}_{18}}=-c\varphi s\theta ({{m}_{1}}+{{m}_{2}}+{{m}_{c}});\\&{{m}_{26}}=-\left( {{m}_{2}}+{{m}_{c}} \right) \left( hc\theta -{{s}_{x}}s\theta \right) ; \\&{{m}_{27}}=0; {{m}_{28}}=s\varphi ({{m}_{2}}+{{m}_{c}}); \\&{m_{36}} = - {m_c}\left( \begin{array}{l} {s_y}\left( {s\beta s\varphi + c\alpha c\beta c\varphi } \right) + \\ h\left( {c\varphi s\beta c\theta - c\alpha c\beta c\theta s\varphi } \right) + \\ {s_x}s\theta \left( {c\alpha c\beta s\varphi - c\varphi s\beta } \right) \end{array} \right) ;\\&{m_{37}} = {m_c}\left( \begin{array}{l} h\left( {c\beta s\alpha c\theta + s\beta s\varphi s\theta + c\alpha c\beta c\varphi s\theta } \right) + \\ {s_x}\left( {s\beta c\theta s\varphi + c\alpha c\beta c\varphi c\theta - c\beta s\alpha s\theta } \right) \end{array} \right) ;\\&{{m}_{38}}=-{{m}_{c}}\cos \alpha \cos \beta ;\\&{{m}_{46}}={{m}_{c}}lc\beta s\alpha ({{s}_{y}}c\varphi -hc\theta s\varphi +{{s}_{x}}s\varphi s\theta ); \\&{{m}_{47}}\!=\!-{{m}_{c}}lc\beta \left( {{s}_{x}}\left( c\alpha s\theta \!+\! c\varphi s\alpha c\theta \right) \!+\!h\left( c\varphi s\alpha s\theta \!-\! c\alpha c\theta \right) \right) ;\\&{{m}_{48}}={{m}_{c}}l\cos \beta \sin \alpha ; \\&{m_{56}} = - {m_c}l\left( \begin{array}{l} {s_y}\left( {c\beta s\varphi - c\alpha c\varphi s\beta } \right) +\\ h\left( {c\beta c\varphi c\theta + c\alpha s\beta c\theta s\varphi } \right) - \\ {s_x}\left( {c\beta c\varphi s\theta + c\alpha s\beta s\varphi s\theta } \right) \end{array} \right) ; \\&{m_{57}} = - {m_c}l\left( \begin{array}{l} h\left( {s\alpha s\beta c\theta - c\beta s\varphi s\theta + c\alpha c\varphi s\beta s\theta } \right) + \\ {s_x}\left( {c\alpha c\varphi s\beta c\theta - c\beta c\theta s\varphi - s\alpha s\beta s\theta } \right) \end{array} \right) ; \\&{{m}_{58}}={{m}_{c}}lc\alpha s\beta ;\\&{{c}_{16}}\!=\!\dot{\varphi }s\theta ({{m}_{1}}\!+\!{{m}_{2}}\!+\!{{m}_{c}})\left( hc\theta \!+\!{{s}_{x}}s\theta \right) \!-\!{{\dot{s}}_{y}}({{m}_{2}}\!+\!{{m}_{c}})s\theta ;\\&{{c}_{17}}=-\dot{\theta }{{s}_{x}}({{m}_{1}}+{{m}_{2}}+{{m}_{c}});\\&{{c}_{26}}=({{m}_{2}}+{{m}_{c}})\left( {{{\dot{s}}}_{x}}s\theta -\dot{\varphi }{{s}_{y}}+\dot{\theta }\left( hs\theta +{{s}_{x}}c\theta \right) \right) ;\\&{{c}_{27}}=\dot{\varphi }\left( {{m}_{2}}+{{m}_{c}} \right) \left( hs\theta +{{s}_{x}}c\theta \right) ; \\&{c_{36}} = - {m_c}\left( \begin{array}{l} {{\dot{s}}_y}\left( {s\beta s\varphi + c\alpha c\beta c\varphi } \right) + \\ {{\dot{s}}_x}s\theta \left( {c\alpha c\beta s\varphi - c\varphi s\beta } \right) + \\ \dot{\varphi }\left( \begin{array}{l} {s_y}\left( {c\varphi s\beta - c\alpha c\beta s\varphi } \right) + \\ {s_x}s\theta \left( {s\beta s\varphi + c\alpha c\beta c\varphi } \right) - \\ hc\theta \left( {c\alpha c\beta c\varphi + s\beta s\varphi } \right) \end{array} \right) + \\ \dot{\theta }\left( \begin{array}{l} hs\theta \left( {c\alpha c\beta s\varphi - c\varphi s\beta } \right) + \\ {s_x}c\theta \left( {c\alpha c\beta s\varphi - c\varphi s\beta } \right) \end{array} \right) \end{array} \right) ;\\&{c_{37}} = - {m_c}\left( \begin{array}{l} {{\dot{s}}_x}\left( {c\beta s\alpha s\theta - s\beta c\theta s\varphi - c\alpha c\beta c\varphi c\theta } \right) + \\ \dot{\varphi }\left( \begin{array}{l} hs\theta \left( {c\alpha c\beta s\varphi - c\varphi s\beta } \right) + \\ {s_x}c\theta \left( {c\alpha c\beta s\varphi - c\varphi s\beta } \right) \end{array} \right) \\ + \dot{\theta }\left( \begin{array}{l} h\left( \begin{array}{l} c\beta s\alpha s\theta - s\beta c\theta s\varphi - \\ c\alpha c\beta c\varphi c\theta \end{array} \right) + \\ {s_x}\left( \begin{array}{l} c\beta s\alpha c\theta + s\beta s\varphi s\theta + \\ c\alpha c\beta c\varphi s\theta \end{array} \right) \end{array} \right) \end{array} \right) ; \\&{c_{46}} = {m_c}lc\beta s\alpha \left( \begin{array}{l} {{\dot{s}}_y}c\varphi + {{\dot{s}}_x}s\varphi s\theta + \\ \dot{\varphi }\left( {{s_x}c\varphi s\theta - {s_y}s\varphi - hc\varphi c\theta } \right) + \\ \dot{\theta }s\varphi \left( {hs\theta + {s_x}c\theta } \right) \end{array} \right) ;\\&{c_{47}} = - {m_c}lc\beta \left( \begin{array}{l} {{\dot{s}}_x}\left( {c\alpha s\theta + c\varphi s\alpha c\theta } \right) - \\ \dot{\varphi }s\alpha s\varphi \left( {hs\theta - {s_x}c\theta } \right) \\ + \dot{\theta }\left( \begin{array}{l} h\left( {c\alpha s\theta + c\varphi s\alpha c\theta } \right) + \\ {s_x}\left( {c\alpha c\theta - c\varphi s\alpha s\theta } \right) \end{array} \right) \end{array} \right) ; \\&{c_{56}} = {m_c}l\left( \begin{array}{l} {{\dot{s}}_y}\left( {c\alpha c\varphi s\beta - c\beta s\varphi } \right) + \\ {{\dot{s}}_x}s\theta \left( {c\beta c\varphi + c\alpha s\beta s\varphi } \right) + \\ \dot{\varphi }\left( \begin{array}{l} - {s_y}\left( {c\beta c\varphi + c\alpha s\beta s\varphi } \right) + \\ hc\theta \left( {c\beta s\varphi - c\alpha c\varphi s\beta } \right) + \\ {s_x}s\theta \left( {c\alpha c\varphi s\beta - c\beta s\varphi } \right) \end{array} \right) \\ \dot{\theta }\left( \begin{array}{l} hs\theta \left( {c\beta c\varphi + c\alpha s\beta s\varphi } \right) + \\ {s_x}c\theta \left( {c\beta c\varphi + c\alpha s\beta s\varphi } \right) \end{array} \right) \end{array} \right) ;\\&{c_{57}} = {m_c}l\left( \begin{array}{l} {{\dot{s}}_x}\left( {c\beta c\theta s\varphi + s\alpha s\beta s\theta - c\alpha c\varphi s\beta c\theta } \right) \\ + \dot{\varphi }\left( \begin{array}{l} hs\theta \left( {c\beta c\varphi + c\alpha s\beta s\varphi } \right) + \\ {s_x}c\theta \left( {c\beta c\varphi + c\alpha s\beta s\varphi } \right) \end{array} \right) + \\ \dot{\theta }\left( \begin{array}{l} h\left( \begin{array}{l} c\beta c\theta s\varphi + s\alpha s\beta s\theta - \\ c\alpha c\varphi s\beta c\theta \end{array} \right) + \\ {s_x}\left( \begin{array}{l} s\alpha s\beta c\theta - c\beta s\varphi s\theta + \\ c\alpha c\varphi s\beta s\theta \end{array} \right) \end{array} \right) \end{array} \right) ;\\&{c_{18}} = {c_{28}} = {c_{38}} = {c_{48}} = {c_{58}} = 0; \end{aligned}$$

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Kim, T.D., Nguyen, L. ., Dinh, X. . et al. Adaptive Hierarchical Sliding Mode Control Design for 3D Ship-Mounted Container Crane with Saturating Actuators. J Control Autom Electr Syst 33, 1643–1658 (2022). https://doi.org/10.1007/s40313-022-00939-6

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