Real-time trajectory planning for ship-mounted rotary cranes considering continuous sea wave disturbances

Abstract

The increasing use of ship-mounted rotary cranes in marine trade has complicated operations, as the varying rope length increases the system’s underactuation. Additionally, these cranes are often subject to wave disturbances during load transportation. This paper proposes a trajectory planning method based on disturbance observer to address these challenges. To begin, coordinate transformations are used to couple continuous yaw and roll disturbances with the original state variables, creating new state variables. A disturbance observer is then used to observe heave disturbance while combining the designed sway suppression trajectory with the reference trajectory to achieve precise positioning of the load and suppress the swaying angle. The stability of the proposed method is demonstrated through the use of theoretical techniques such as Lyapunov, LaSalle’s invariance principle, and Barbalat’s lemma. Furthermore, the effectiveness of the proposed approach is confirmed through experiments conducted on a constructed experimental platform.

Appendix

The detailed elements in (1) are as follows:

$$\begin{aligned} M_s= & {} \left[ \begin{array}{ccccc} m_{11} &{} m_{12} &{} m_{13} &{} m_{14} &{}m_{15}\\ m_{21} &{} m_{22} &{} m_{23} &{} m_{24} &{}m_{25}\\ m_{31} &{} m_{32} &{} m_{33} &{} m_{34} &{}m_{35}\\ m_{41} &{} m_{42} &{} m_{43} &{} m_{44} &{}m_{45}\\ m_{51} &{} m_{52} &{} m_{53} &{} m_{54} &{}m_{55}\\ \end{array} \right] ,\\ C= & {} \left[ \begin{array}{ccccc} c_{11} &{} c_{12} &{} c_{13} &{} c_{14} &{} c_{15}\\ c_{21} &{} c_{22} &{} c_{23} &{} c_{24} &{} c_{25}\\ c_{31} &{} c_{32} &{} c_{33} &{} c_{34} &{} c_{35}\\ c_{41} &{} c_{42} &{} c_{43} &{} c_{44} &{} c_{45}\\ c_{51} &{} c_{52} &{} c_{53} &{} c_{54} &{} c_{55}\\ \end{array} \right] , G(q){=}\left[ \begin{array}{ccccc} g_{1}&g_{2}&g_{3}&g_{4}&g_{5} \end{array} \right] ^{T},\\ \tau= & {} \left[ \begin{array}{ccccc} \tau _{1}&\tau _{2}&\tau _{3}&0&0 \end{array} \right] ^{T}, \tau _f=\left[ \begin{array}{ccccc} \tau _{1f}&\tau _{2f}&\tau _{3f}&0&0 \end{array} \right] ^{T},\\ q= & {} \left[ \begin{array}{ccccc} q_{1} &{} q_{2} &{} q_{3} &{} q_{4} &{} q_{5} \\ \end{array} \right] ^{T}, d^*=\left[ \begin{array}{ccccc} d_1^*&d_2^*&d_3^*&d_4^*&d_5^*\end{array} \right] ^{T}, \\ \end{aligned}$$

$$\begin{aligned} {m_{11}}= & {} m{L^2} + \frac{1}{3}{M}{L^2}, {m_{12}} = mL{q_3}{S_1}{S_5},\\ {m_{13}}= & {} - mL{C_{1 - 4}}{C_5},{m_{14}} = mL{q_3}{S_{1 - 4}}{C_5}, \\ {m_{15}}= & {} mL{q_3}{C_{1 - 4}}{S_5},{m_{21}} = {m_{12}},\\ {m_{22}}= & {} \frac{1}{3}{M}{L^2}{C_1}^2\\{} & {} \quad + m{L^2}{C_1}^2 + m{q_3}^2{S_4}^2{C_5}^2 \\{} & {} \quad + m{q_3}^2{S_5}^2+ 2mL{q_3}{C_1}{S_4}{C_5},\\ {m_{23}}= & {} mL{C_1}{S_5}, {m_{24}} = - m{q_3}^2{C_4}{S_5}{C_5},\\ {m_{25}}= & {} m{q_3}^2{S_4}+ mL{q_3}{C_1}{C_5},\\ {m_{31}}= & {} {m_{13}}, {m_{32}} = {m_{23}}, {m_{33}} = m, \\ {m_{34}}= & {} 0, {m_{35}} = 0,{m_{41}} = {m_{14}}, \\ {m_{42}}= & {} {m_{24}}, {m_{43}} = {m_{34}},\\ {m_{44}}= & {} m{q_3}^2{C_5}^2, {m_{45}} = 0, \\ {m_{51}}= & {} {m_{15}}, {m_{52}} = {m_{25}},\\ {m_{53}}= & {} {m_{35}}, {m_{54}} = {m_{45}}, \\ {m_{55}}= & {} m{q_3}^2, {c_{11}} = 0, \\ {c_{12}}= & {} m{L^2}{S_1}{C_1}{{\dot{q}}_2}+ mL{q_3}{S_1}{S_4}{C_5}{{\dot{q}}_2}\\{} & {} \quad + \frac{1}{3}{M}{L^2}{S_1}{C_1}{{\dot{q}}_2} + mL{{\dot{q}}_3}{S_1}{S_5} \\{} & {} \quad + mL{q_3}{S_1}{C_5}{{\dot{q}}_5},\\ {c_{13}}= & {} mL{S_1}{S_5}{{\dot{q}}_2} - mL{S_{1 - 4}}{C_5}{{\dot{q}}_4} + mL{C_{1 - 4}}{S_5}{{\dot{q}}_5},\\ {c_{14}}= & {} mL{q_3}{C_{1 - 4}}{C_5}{{\dot{q}}_4}- mL{{\dot{q}}_3}{S_{1 - 4}}{C_5} \\{} & {} + mL{q_3}{S_{1 - 4}}{S_5}{{\dot{q}}_5},\\ {c_{15}}= & {} mL{q_3}{C_{1 - 4}}{C_5}{{\dot{q}}_5} + mL{q_3}{S_1}{C_5}{{\dot{q}}_2} \\{} & {} \quad + mL{{\dot{q}}_3}{C_{1 - 4}}{S_5} + mL{q_3}{S_{1 - 4}}{S_5}{{\dot{q}}_4}, \\ {c_{21}}= & {} -m{L^2}{S_1}{C_1}{{\dot{q}}_2}- mL{q_3}{S_1}{S_4}{C_5}{{\dot{q}}_2} \\{} & {} \quad +mL{q_3}{C_1}{C_5}{{\dot{q}}_1} - \frac{1}{3}{M}{L^2}{S_1}{C_1}{{\dot{q}}_2},\\ {c_{22}}= & {} - m{L^2}{S_1}{C_1}{{\dot{q}}_1} - \frac{1}{3}{M}{L^2}{S_1}{C_1}{{\dot{q}}_1}\\{} & {} \quad - mL{q_3}{S_1}{S_4}{C_5}{{\dot{q}}_1} + m{q_3}^2{S_4}{C_4}{C_5}^2{{\dot{q}}_4}\\{} & {} + mL{q_3}{C_1}{C_4}{C_5}{{\dot{q}}_4} +m{q_3}^2{C_4}^2{S_5}{C_5}{{\dot{q}}_5} \\{} & {} \quad - mL{q_3}{C_1}{S_4}{S_5}{{\dot{q}}_5}+ m{q_3}{{\dot{q}}_3}{S_4}^2{C_5}^2 \\{} & {} \quad + m{q_3}{{\dot{q}}_3}{S_5}^2 + mL{{\dot{q}}_3}{C_1}{S_4}{C_5},\\ {c_{23}}= & {} m{q_3}{{\dot{q}}_2}{S_4}^2{C_5}^2+ m{q_3}{{\dot{q}}_2}{S_5}^2 \\{} & {} \quad + mL{C_1}{S_4}{C_5}{{\dot{q}}_2} - m{q_3}{C_4}{S_5}{C_5}{{\dot{q}}_4} \\{} & {} \quad + mL{C_1}{C_5}{{\dot{q}}_5} + m{q_3}{S_4}{{\dot{q}}_5},\\ {c_{24}}= & {} m{q_3}^2{S_4}{S_5}{C_5}{{\dot{q}}_4}+ m{q_3}^2{S_4}{C_4}{C_5}^2{{\dot{q}}_2} \\{} & {} + mL{q_3}{C_1}{C_4}{C_5}{{\dot{q}}_2}+ m{q_3}^2{C_4}{S_5}^2{{\dot{q}}_5}\\{} & {} \quad - m{q_3}{{\dot{q}}_3}{C_4}{S_5}{C_5},\\ {c_{25}}= & {} - mL{q_3}{C_1}{S_5}{{\dot{q}}_5}+ m{q_3}^2{C_4}^2{S_5}{C_5}{{\dot{q}}_2}\\{} & {} \quad + m{q_3}{{\dot{q}}_3}{S_4}- mL{q_3}{C_1}{S_4}{S_5}{{\dot{q}}_2}\\{} & {} \quad + m{q_3}^2{C_4}{S_5}^2{{\dot{q}}_4} + mL{{\dot{q}}_3}{C_1}{C_5},\\ {c_{31}}= & {} mL{S_{1 - 4}}{C_5}{{\dot{q}}_1}- mL{S_1}{S_5}{{\dot{q}}_2},\\ {c_{32}}= & {} - m{q_3}{S_4}^2{C_5}^2{{\dot{q}}_2}- m{q_3}{S_5}^2{{\dot{q}}_2}\\{} & {} - mL{C_1}{S_4}{C_5}{{\dot{q}}_2}- mL{S_1}{S_5}{{\dot{q}}_1}\\{} & {} \quad + m{q_3}{C_4}{S_5}{C_5}{{\dot{q}}_4}- m{q_3}{S_4}{{\dot{q}}_5},\\ {c_{33}}= & {} 0, \\ {c_{34}}= & {} - m{q_3}{C_5}^2{{\dot{q}}_4}+ m{q_3}{C_4}{S_5}{C_5}{{\dot{q}}_2},\\{} & {} - m{q_3}{{\dot{q}}_5} -m{q_3}{S_4}{{\dot{q}}_2},\\ {c_{41}}= & {} -mL{C_{1 - 4}}{{\dot{q}}_1}, \\ {c_{42}}= & {} - m{q_3}^2{S_4}{C_4}{C_5}^2{{\dot{q}}_2} - mL{q_3}{C_1}{C_4}{C_5}{{\dot{q}}_2} \\{} & {} \quad - m{q_3}^2{C_4}{C_5}^2{{\dot{q}}_5}\\{} & {} - m{q_3}{{\dot{q}}_3}{C_4}{S_5}{C_5},\\ {c_{43}}= & {} m{q_3}{C_5}^2{{\dot{q}}_4} - m{q_3}{C_4}{S_5}{C_5}{{\dot{q}}_2},\\ {c_{44}}= & {} m{q_{3}}{{\dot{q}}_3}{C_5}^2 - m{q_3}^2{S_5}{C_5}{{\dot{q}}_5},\\ {c_{45}}= & {} - m{q_3}^2{C_4}{C_5}^2{{\dot{q}}_2}- m{q_3}^2{S_5}{C_5}{{\dot{q}}_4}, \\ {c_{51}}= & {} - mL{q_3}{S_{1 - 4}}{S_5}{{\dot{q}}_1}- mL{q_3}{S_1}{C_5}{{\dot{q}}_2},\\ {c_{52}}= & {} - m{q_3}^2{C_4}^2{S_5}{C_5}{{\dot{q}}_2}+ mL{q_3}{C_1}{S_4}{S_5}{{\dot{q}}_2} \\{} & {} - mL{q_3}{S_1}{C_5}{{\dot{q}}_1}+ m{q_3}{{\dot{q}}_3}{S_4} + m{q_3}^2{C_4}{C_5}^2{{\dot{q}}_4},\\ {c_{53}}= & {} m{q_3}{S_4}{{\dot{q}}_2} + m{q_3}{{\dot{q}}_5}, \\ {c_{54}}= & {} m{q_3}^2{S_5}{C_5}{{\dot{q}}_4} + m{q_3}^2{C_4}{C_5}^2{{\dot{q}}_2},\\ {c_{55}}= & {} m{q_3}{{\dot{q}}_3},\\ {g_1}= & {} (m + \frac{1}{2}{M})gL{C_1}, {g_2} = 0, {g_3} = - mg{C_4}{C_5},\\ {g_4}= & {} mg{q_3}{S_4}{C_5}, {g_5} = mg{q_3}{C_4}{S_5},\\ d_1^{*}= & {} - (m + \frac{1}{2}{M}){\ddot{hL}}{C_1}, \\ d_2^{*}= & {} 0, d_3^{*} = m{\ddot{h}}{C_4}{C_5}, \\ d_4^{*}= & {} - m{\ddot{h}}{q_3}{S_4}{C_5}, \\ d_5^{*}= & {} - m{\ddot{h}}{q_3}{C_4}{S_5},\\ q_1= & {} \theta _1-\alpha , q_{2}=\theta _2-\beta , q_{3}=l(t), \\ q_{4}= & {} \theta _3-\alpha , q_{5}=\theta _4-\beta , \end{aligned}$$

where $\tau _{1}$ and $\tau _{2}$ are the torque of the boom, g is the acceleration of gravity, and $d^{*}$ is the force of sea disturbances on the crane system, respectively. We abbreviate $\sin {q}_1$ and $\cos {q}_1$ as S1 and C1 for readability, respectively.