Sway and disturbance rejection control for varying rope tower cranes suffering from friction and unknown payload mass

Abstract

Tower cranes are well-known underactuated systems, where the design of controllers for them with time-varying rope length was weak in the past because of their complex dynamic characteristic. The payload oscillation will become worse when the jib slew angle, the trolley position and the rope length are changed simultaneously. The proposed method is designed based on robust adaptive sliding mode control via tracking nonzero initial reference trajectories, in which frictions and lumped disturbances in the crane system are eliminated, and unknown payload mass is effectively estimated online. Lyapunov technique is combined with LaSalle’s invariance theorem to design controller and analyze stability. Various and strict simulations are applied, which validate the effectiveness and extreme robustness of the proposed method.

Appendices

Appendix A

The detailed expressions of (8) are as follows:

$$\begin{aligned}&\varvec{M}( {\varvec{q}})\ddot{ {\varvec{q}}} + {\varvec{C}}( {\varvec{q}}, \dot{ {\varvec{q}}})\dot{ {\varvec{q}}} + {\varvec{G}}( {\varvec{q}}) = {\varvec{U}} - {{\varvec{F}}}_s(\dot{ {\varvec{q}}}) + {\varvec{D}}, \nonumber \\&\varvec{M}( {\varvec{q}}) = \left[ \begin{array}{l} {m_{11}}\;{m_{12}}\;{m_{13}}\;{m_{14}}\;{m_{15}}\\ {m_{21}}\;{m_{22}}\;{m_{23}}\;{m_{24}}\;{m_{25}}\\ {m_{31}}\;{m_{32}}\;{m_{33}}\;{m_{34}}\;{m_{35}}\\ {m_{41}}\;{m_{42}}\;{m_{43}}\;{m_{44}}\;{m_{45}}\\ {m_{51}}\;{m_{52}}\;{m_{53}}\;{m_{54}}\;{m_{55}} \end{array} \right] \; ,\nonumber \\&{m_{11}} = {J_0} + m{x^2} + {M_t}{x^2} + m{l^2} - m{l^2}{C_1}^2{C_2}^2\nonumber \\&+ 2mxl{C_2}{S_1},{m_{12}} = - ml{S_2},{m_{13}} = mx{S_2},\nonumber \\&{m_{14}} = - m{l^2}{C_1}{C_2}{S_2},{m_{15}} = m{l^2}{S_1} + mxl{C_2},\nonumber \\&{m_{21}} = - ml{S_2},{m_{22}} = {M_t} + m,{m_{23}} = m{C_2}{S_1},\nonumber \\&{m_{24}} = ml{C_1}{C_2},{m_{25}} = - ml{S_1}{S_2},{m_{31}} = mx{S_2},\nonumber \\&{m_{32}} = m{C_2}{S_1},{m_{33}} = m,{m_{34}} = {m_{35}} = 0,\nonumber \\&{m_{41}} = - m{l^2}{C_1}{C_2}{S_2},{m_{42}} = ml{C_1}{C_2},{m_{43}} = 0,\nonumber \\&{m_{44}} = m{l^2}{C_2}^2,{m_{45}} = 0,{m_{51}} = m{l^2}{S_1} + mxl{C_2},\nonumber \\&{m_{52}} = - ml{S_1}{S_2},{m_{53}} = {m_{54}} = 0,{m_{55}} = m{l^2},\nonumber \\&{\varvec{C}}( {\varvec{q}},\dot{ {\varvec{q}}}) = \left[ \begin{array}{l} {c_{11}}\;{c_{12}}\;{c_{13}}\;{c_{14}}\;{c_{15}}\\ {c_{21}}\;{c_{22}}\;{c_{23}}\;{c_{24}}\;{c_{25}}\\ {c_{31}}\;{c_{32}}\;{c_{33}}\;{c_{34}}\;{c_{35}}\\ {c_{41}}\;{c_{42}}\;{c_{43}}\;{c_{44}}\;{c_{45}}\\ {c_{51}}\;{c_{52}}\;{c_{53}}\;{c_{54}}\;{c_{55}} \end{array} \right] ,\nonumber \\&{c_{11}} = mx\dot{x} + {M_t}x\dot{x} + ml{C_2}{S_1}\dot{x} + ml\dot{l}\nonumber \\&- ml{C_1}^2{C_2}^2\dot{l} + mx{C_2}{S_1}\dot{l} + m{l^2}{C_1}{C_2}^2{S_1}{{{\dot{\theta }} }_1}\nonumber \\&+ mxl{C_1}{C_2}{{{\dot{\theta }} }_1} + m{l^2}{C_1}^2{C_2}{S_2}{{{\dot{\theta }} }_2} - mxl{S_1}{S_2}{{{\dot{\theta }} }_2},\nonumber \\&{c_{12}} = mx{\dot{\alpha }} + {M_t}x{\dot{\alpha }} + ml{C_2}{S_1}{\dot{\alpha }} ,{c_{13}} = ml{\dot{\alpha }} \nonumber \\&- ml{C_1}^2{C_2}^2{\dot{\alpha }} + mx{C_2}{S_1}{\dot{\alpha }} - ml{C_1}{C_2}{S_2}{{{\dot{\theta }} }_1}\nonumber \\&+ mx{C_2}{{{\dot{\theta }} }_2} + ml{S_1}{{{\dot{\theta }} }_2},{c_{14}} = m{l^2}{C_1}{C_2}^2{S_1}{\dot{\alpha }} \nonumber \\&+ mxl{C_1}{C_2}{\dot{\alpha }} - ml{C_1}{C_2}{S_2}\dot{l} + m{l^2}{C_2}{S_1}{S_2}{{{\dot{\theta }} }_1}\nonumber \\&+ m{l^2}{C_1}{{{\dot{\theta }} }_2} - m{l^2}{C_1}{C_2}^2{{{\dot{\theta }} }_2},\nonumber \\&{c_{15}} = m{l^2}{C_1}^2{C_2}{S_2}{\dot{\alpha }} - mxl{S_1}{S_2}{\dot{\alpha }} + mx{C_2}\dot{l}\nonumber \\&+ ml{S_1}\dot{l} + m{l^2}{C_1}{{{\dot{\theta }} }_1} - m{l^2}{C_1}{C^2}_2{{{\dot{\theta }} }_1}\nonumber \\&- mxl{S_2}{{{\dot{\theta }} }_2},{c_{21}} = - mx{\dot{\alpha }} - Mx{\dot{\alpha }} \nonumber \\&- ml{C_2}{S_1}{\dot{\alpha }} - m{S_2}\dot{l} - ml{C_2}{{{\dot{\theta }} }_2},{c_{22}} = 0,\nonumber \\&{c_{23}} = - m{S_2}{\dot{\alpha }} + m{C_1}{C_2}{{{\dot{\theta }} }_1} - m{S_1}{S_2}{{{\dot{\theta }} }_2},\nonumber \\&{c_{24}} = m{C_1}{C_2}\dot{l} - ml{C_2}{S_1}{{{\dot{\theta }} }_1} - ml{C_1}{S_2}{{{\dot{\theta }} }_2},\nonumber \\&{c_{25}} = - ml{C_2}{\dot{\alpha }} - m{S_1}{S_2}\dot{l} - ml{C_1}{S_2}{{{\dot{\theta }} }_1}\nonumber \\&- ml{C_2}{S_1}{{{\dot{\theta }} }_2},{c_{31}} = - ml{\dot{\alpha }} + ml{C_1}^2{C_2}^2{\dot{\alpha }} \nonumber \\&- mx{C_2}{S_1}{\dot{\alpha }} + m{S_2}\dot{x} + ml{C_1}{C_2}{S_2}{{{\dot{\theta }} }_1} - ml{S_1}{{{\dot{\theta }} }_2},\nonumber \\&{c_{32}} = m{S_2}{\dot{\alpha }} ,{c_{33}} = 0,{c_{34}} = ml{C_1}{C_2}{S_2}{\dot{\alpha }} \nonumber \\&- ml{C_2}^2{{{\dot{\theta }} }_1},{c_{35}} = - ml{S_1}{\dot{\alpha }} - ml{{{\dot{\theta }} }_2},\nonumber \\&{c_{41}} = - m{l^2}{C_1}{C_2}^2{S_1}{\dot{\alpha }} - mxl{C_1}{C_2}{\dot{\alpha }} \nonumber \\&- ml{C_1}{C_2}{S_2}\dot{l} - m{l^2}{C_1}{C_2}^2{{{\dot{\theta }} }_2},{c_{42}} = 0,\nonumber \\&{c_{43}} = - ml{C_1}{C_2}{S_2}{\dot{\alpha }} + ml{C_2}^2{{{\dot{\theta }} }_1},\nonumber \\&{c_{44}} = ml{C_2}^2\dot{l} - m{l^2}{C_2}{S_2}{{{\dot{\theta }} }_2},\nonumber \\&{c_{45}} = - m{l^2}{C_1}{C_2}^2{\dot{\alpha }} - m{l^2}{C_2}{S_2}{{{\dot{\theta }} }_1},\nonumber \\&{c_{51}} = - m{l^2}{C_1}^2{C_2}{S_2}{\dot{\alpha }} + mlx{S_1}{S_2}{\dot{\alpha }} + ml{C_2}\dot{x}\nonumber \\&+ ml{S_1}\dot{l} + m{l^2}{C_1}{C_2}^2{{{\dot{\theta }} }_1},{c_{52}} = ml{C_2}{\dot{\alpha }} ,\nonumber \\&{c_{53}} = ml{{{\dot{\theta }} }_2} + ml{S_1}{\dot{\alpha }} ,{c_{54}} = m{l^2}{C_1}{C_2}^2{\dot{\alpha }} \nonumber \\&+ m{l^2}{C_2}{S_2}{{{\dot{\theta }} }_1},{c_{55}} = ml\dot{l} ,\nonumber \\&{\varvec{G}}( {\varvec{q}}) = {[g_1\;g_2\; g_3\;g_4\;g_5]^T},\nonumber \\&g_1=0,g_2=0,g_3= - mg{C _1}{C _2},g_4=mgl{C _2}{S _1},\nonumber \\&g_5=mgl{C _1}{S _2}. \end{aligned}$$

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Appendix B

The detailed terms in (13) and (14) are shown as follows:

$$\begin{aligned}&{\varvec{\varGamma }} ( {\varvec{q}},\dot{ {\varvec{q}}}) = {[{a_1}\;{a_2}\;{a_3}\;{a_4}\;{a_5}]^T},\nonumber \\&{a_1} = \frac{{ - 2{M_t}{\dot{\alpha }} \dot{x}x}}{{{M_t}{x^2} + {J_0}}},{a_2} = x{{{\dot{\alpha }} }^2},\nonumber \\&{a_3} =\frac{1}{{{{M_t}{x^2} + {J_0}}}} ({M_t}{{{\dot{\theta }} }_2}^2l{x^2} + {M_t}{{{\dot{\alpha }} }^2}l{x^2} + {J_0}{{{\dot{\theta }} }_2}^2lm\nonumber \\&+ {J_0}{{{\dot{\alpha }} }^2}l + {M_t}{{{\dot{\theta }} }_1}^2l{x^2}{C_2}^2 + {J_0}{{{\dot{\theta }} }_1}^2l{C_2}^2\nonumber \\&- 2{J_0}{\dot{\alpha }} \dot{x}{S_2} - 2{M_t}{{{\dot{\theta }} }_1}{\dot{\alpha }} l{x^2}{C_1}{C_2}{S_2}\nonumber \\&+ 2{M_t}{{{\dot{\theta }} }_2}{\dot{\alpha }} l{x^2}{S_1} + 2{J_0}{{{\dot{\theta }} }_2}{\dot{\alpha }} l{S_1}\nonumber \\&- {M_t}{{{\dot{\alpha }} }^2}l{x^2}{C_1}^2{C_2}^2 - {J_0}{{{\dot{\alpha }} }^2}l{C_1}^2{C_2}^2\nonumber \\&- 2{J_0}{{{\dot{\theta }} }_1}{\dot{\alpha }} l{C_1}{C_2}{S_2}),\nonumber \nonumber \\&{a_4} =\frac{1}{{{l{C_2}({M_t}{x^2} + {J_0})}}}( - {M_t}g{x^2}{S_1} - {J_0}g{S_1} \nonumber \\&- 2{M_t}{{{\dot{\theta }} }_1}\dot{l}{x^2}{C_2}- 2{J_0}{{{\dot{\theta }} }_1}\dot{l}{C_2} + 2{M_t}{\dot{\alpha }} \dot{l}{x^2}{C_1}{S_2}\nonumber \\&+ 2{J_0}{\dot{\alpha }} \dot{l}{C_1}{S_2} + 2{M_t}{{{\dot{\theta }} }_1}{{{\dot{\theta }} }_2}l{x^2}{S_2}\nonumber \\&+ 2{J_0}{{{\dot{\theta }} }_1}{{{\dot{\theta }} }_2}l{S_2} - 2{M_t}{\dot{\alpha }} \dot{x}lx{C_1}{S_2}\nonumber \\&+ {M_t}{{{\dot{\alpha }} }^2}l{x^2}{C_1}{C_2}{S_1} + {J_0}{{{\dot{\alpha }} }^2}l{C_1}{C_2}{S_1}\nonumber \\&+ 2{M_t}{{{\dot{\theta }} }_2}{\dot{\alpha }} l{x^2}{C_1}{C_2} + 2{J_0}{{{\dot{\theta }} }_2}{\dot{\alpha }} l{C_1}{C_2}),\nonumber \\&{a_5} =\frac{1}{ {{l({M_t}{x^2} + {J_0})}}} ( - 2{M_t}{{{\dot{\theta }} }_2}\dot{l}{x^2}- 2{J_0}{\dot{\alpha }} \dot{x}{C_2}\nonumber \\&- 2{M_t}{{{\dot{\theta }} }_1}{\dot{\alpha }} l{x^2}{C_1}{C_2}^2 - {M_t}{{{\dot{\theta }} }_1}^2l{x^2}{S_2}{C_2}\nonumber \\&- {J_0}{{{\dot{\theta }} }_1}^2l{S_2}{C_2} - 2{M_t}{\dot{\alpha }} \dot{l}{x^2}{S_1}\nonumber \\&- 2{J_0}{{{\dot{\theta }} }_1}{\dot{\alpha }} l{C_1}{C_2}^2\nonumber \\&- 2{J_0}{{{\dot{\theta }} }_2}\dot{l} + {M_t}{{{\dot{\alpha }} }^2}l{x^2}{C_1}^2{C_2}{S_2}\nonumber \\&- {M_t}g{x^2}{C_1}{S_2} + 2{M_t}{\dot{\alpha }} \dot{x}lx{S_1}\nonumber \\&- 2{J_0}{\dot{\alpha }} \dot{l}{S_1} + {J_0}{{{\dot{\alpha }} }^2}l{C_1}^2{C_2}{S_2}) , \end{aligned}$$

(55)

$$\begin{aligned}&{\varvec{\varPhi }} ( {\varvec{q}}) = {\left[ \begin{array}{l} {b_{11}}\;{b_{21}}\;{b_{31}}\;{b_{41}}\;{b_{51}}\\ {b_{12}}\;{b_{22}}\;{b_{32}}\;{b_{42}}\;{b_{52}}\\ {b_{13}}\;{b_{23}}\;{b_{33}}\;{b_{43}}\;{b_{53}} \end{array} \right] ^T},\nonumber \\&{b_{11}} = \frac{1}{{{M_t}{x^2} + {J_0}}},{b_{12}} = 0,{b_{13}} = \frac{{ - x{S_2}}}{{{M_t}{x^2} + {J_0}}},\nonumber \\&{b_{21}} = 0,{b_{22}} = \frac{1}{{{M_t}}},{b_{23}} = \frac{{ - {C_2}{S_1}}}{{{M_t}}},\nonumber \\&{b_{31}} = \frac{{ - x{S_2}}}{{{M_t}{x^2} + {J_0}}},{b_{32}} = \frac{{ - {S_1}{C_2}}}{{{M_t}}},\nonumber \\&{b_{33}} = \frac{1}{{{{M_t}m({M_t}{x^2} + {J_0})}}}( {M_t}^2{x^2} + {J_0}m{C_2}^2\nonumber \\&- {J_0}m{C_1}^2{C_2}^2 - {M_t}m{x^2}{C_1}^2{C_2}^2\nonumber \\&+ {J_0}{M_t} + {M_t}m{x^2}),\nonumber \\&{b_{41}} = \frac{{{C_1}{S_2}}}{{{C_2}({M_t}{x^2} + {J_0})}},{b_{42}} = \frac{{ - {C_1}}}{{{M_t}l{C_2}}},\nonumber \\&{b_{43}} = \frac{C_1}{{{{M_t}l{C_2}({M_t}{x^2} + {J_0})}}}( - {M_t}lx + {J_0}{C_2}{S_1}\nonumber \\&+ {M_t}lx{C_2}^2 + {M_t}{x^2}{C_2}{S_1})\nonumber \\&{b_{51}} = \frac{{ - x{C_2} - l{S_1}}}{{l({M_t}{x^2} + {J_0})}},{b_{52}} = \frac{{{S_1}{S_2}}}{{{M_t}l}},\nonumber \\&{b_{53}} = \frac{{S_2}}{{{{M_t}l({M_t}{x^2} + {J_0})}}}(- {J_0}{C_2} + {J_0}{C_1}^2{C_2}\nonumber \\&+ {M_t}{x^2}{C_1}^2{C_2} + {M_t}lx{S_1}). \end{aligned}$$

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