﻿ 3_UPU/PU_RRP海上稳定廊桥运动学建模与仿真
 舰船科学技术  2023, Vol. 45 Issue (1): 78-82    DOI: 10.3404/j.issn.1672-7649.2023.01.014 PDF
3_UPU/PU_RRP海上稳定廊桥运动学建模与仿真

Kinematics modeling and simulation of 3_UPU/PU_RRP offshore gangway
QIU Jian-chao, NIU Anqi, QIU Wei-han, CHEN Hai-quan, WANG Sheng-hai
College of Marine Engineering, Dalian Maritime University , Dalian 116026, China
Abstract: The ship will produce multidimensional swaying montion due to the disturbance of wind,wave swell and other loads during the voyage. which poses a serious threat to the safe transfer of people and cargo on board. For this reason, this paper designs a offshore gangway with motion compensation function. The system is analogous to the series-parallel hybrid mechanism, and it is maximized and discretized into 3_UPU/PU parallel mechanism and RRP series mechanism. The DH parameter method and Euler angle coordinate conversion method are used to analyze the inverse kinematics of the two respectively. And use the spatial geometry method to calculate the kinematics equation of theoffshore gangway. Calculate the inverse kinematics equations of the model with the help of Matlab, use Adams to verify the correctness of the kinematics modeling, obtain the displacement and angle change curves of each actuator of the model in its motion space, and design the control system for the offshore gangway, dynamic analysis and trajectory planning provide theoretical basis and basis.
Key words: offshore gangway     kinematics modeling     simulation verification
0 引　言

1 海上廊桥系统原理及结构

 图 1 3_UPU/PU_RRP海上稳定廊桥 Fig. 1 3_UPU/PU_RRP offshore gangwaay

2 海上廊桥系统运动学建模

2.1 空间几何模型（坐标系）建立

 图 2 海上稳定廊桥空间几何模型 Fig. 2 Spatial geometry model of offshore gangway

2.2 3_UPU/PU并联稳定平台坐标转换矩阵

 $_n^{n + 1}R = {R_x}({\theta _{n + 1x}}){R_y}({\theta _{n + 1y}})。$ (1)

 $\begin{gathered} {}_{{O_o}}^{O'}R = {}_{{O_o}}^{O'}{R_x}{}_{{O_o}}^{O'}{R_y} = \left[ {\begin{array}{*{20}{c}} {{\rm{c}}{\beta _o}}&0&{{\rm{s}}{\beta _o}} \\ {{\rm{s}}{\beta _o}s{\gamma _o}}&{c{\gamma _o}}&{ - {\rm{s}}{\gamma _o}c{\beta _o}} \\ { - {\rm{s}}{\beta _o}c{\gamma _o}}&{{\rm{s}}{\gamma _o}}&{{\rm{c}}{\gamma _o}c{\beta _o}} \end{array}} \right] \\ \end{gathered}。$ (2)

 $\begin{gathered} {}_{{O_s}}^{{O_e}}R = {}_{{O_s}}^{{O_e}}{R_x}{}_{{O_s}}^{{O_e}}{R_y} = \left[ {\begin{array}{*{20}{c}} {{\rm{c}}{\beta _s}}&0&{{\rm{s}}{\beta _s}} \\ {{\rm{s}}{\beta _s}s{\gamma _s}}&{{\rm{c}}{\gamma _s}}&{ - {\rm{s}}{\gamma _s}c{\beta _s}} \\ { - {\rm{s}}{\beta _s}c{\gamma _s}}&{{\rm{s}}{\gamma _s}}&{{\rm{c}}{\gamma _s}c{\beta _s}} \end{array}} \right]，\\ \end{gathered}$ (3)

 ${}_{{O_o}}^{{O_e}}R = {}_{{O_s}}^{{O_e}}R{}_{{O_x}}^{{O_s}}R{}_{O'}^{{O_x}}R{}_{{O_o}}^{O'}R ，$ (4)

 ${}_{{O_x}}^{{O_e}}R = {}_{{O_s}}^{{O_e}}R{}_{{O_x}}^{{O_s}}R。$ (5)
2.3 RRP串联舷梯坐标转换矩阵

 ${}_i^{i + 1}T = \left[ {\begin{array}{*{20}{c}} {{\rm{c}}{\theta _i}}&{ - {\rm{s}}{\theta _i}}&0&{{\alpha _{i - 1}}} \\ {{\rm{s}}{\theta _i}c{\alpha _{i - 1}}}&{{\rm{c}}{\theta _i}c{\alpha _{i - 1}}}&{ - {\rm{s}}{\alpha _{i - 1}}}&{ - {\rm{s}}{\alpha _{i - 1}}{d_i}} \\ {{\rm{s}}{\theta _i}s{\alpha _{i - 1}}}&{{\rm{c}}{\theta _i}{\rm{s}}{\alpha _{i - 1}}}&{{\rm{c}}{\alpha _{i - 1}}}&{{\rm{c}}{\alpha _{i - 1}}{d_i}} \\ 0&0&0&1 \end{array}} \right] ，$ (6)

 ${}_1^0T = \left[ {\begin{array}{*{20}{c}} {c{\theta _1}}&{ - s{\theta _1}}&0&0 \\ {s{\theta _1}}&{c{\theta _1}}&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \end{array}} \right]，$ (7)

 ${}_2^1T = \left[ {\begin{array}{*{20}{c}} {c\left( {{\theta _2}{\text{ + }}90^\circ } \right)}&{ - s\left( {{\theta _2}{\text{ + }}90^\circ } \right)}&0&0 \\ 0&0&{ - 1}&0 \\ {s\left( {{\theta _2}{\text{ + }}90^\circ } \right)}&{c\left( {{\theta _2}{\text{ + }}90^\circ } \right)}&0&0 \\ 0&0&0&1 \end{array}} \right]，$ (8)

 ${}_3^2T = \left[ {\begin{array}{*{20}{c}} 1&0&0&0 \\ 0&0&{ - 1}&{- \left( {{l_1}{\text{ + }}{d_3}} \right)} \\ 0&1&0&0 \\ 0&0&0&1 \end{array}} \right]，$ (9)

 ${}_3^0T = {}_1^0T{}_2^1T{}_3^2T 。$ (10)
2.4 海上稳定廊桥系统位姿分析

 $\left\{ {\begin{array}{*{20}{l}} {{}^{{O_o}}{P_{{A_1}}} = {{\left[ {\begin{array}{*{20}{c}} { - {K_1}\cos 30^\circ }&{ - {K_1}\sin 30^\circ }&0 \end{array}} \right]}\;^{\rm{T}}}}，\\ {{}^{{O_o}}{P_{{A_2}}} = {{\left[ {\begin{array}{*{20}{c}} {{K_1}\cos 30^\circ }&{ - {K_1}\sin 30^\circ }&0 \end{array}} \right]}\;^{\rm{T}}}} ，\\ {{}^{{O_o}}{P_{{A_3}}} = {{\left[ {\begin{array}{*{20}{c}} 0&{{K_1}}&0 \end{array}} \right]}\;^{\rm{T}}}} 。\end{array}} \right.$ (11)

 $\left\{ {\begin{array}{*{20}{l}} {{}^{{O_x}}{P_{{B_1}}} = {{\left[ {\begin{array}{*{20}{c}} { - {K_2}\cos 30^\circ }&{ - {K_2}\sin 30^\circ }&0 \end{array}} \right]}\;^{\rm{T}}}}，\\ {{}^{{O_x}}{P_{{B_2}}} = {{\left[ {\begin{array}{*{20}{c}} {{K_2}\cos 30^\circ }&{ - {K_2}\sin 30^\circ }&0 \end{array}} \right]}\;^{\rm{T}}}}，\\ {{}^{{O_x}}{P_{{B_3}}} = {{\left[ {\begin{array}{*{20}{c}} 0&{{K_2}}&0 \end{array}} \right]}\;^{\rm{T}}}} 。\end{array}} \right.$ (12)

${A_i}$ 在惯性坐标系中的表示为：

 ${}^{{O_e}}{P_{{A_i}}} = {}_{{O_o}}^{{O_e}}R{}^{{O_o}}{P_{{A_i}}} ，$ (13)

${B_i}$ 在惯性坐标系中的表示为：

 ${}^{{O_e}}{P_{{B_i}}} = {}_{{O_x}}^{{O_e}}R{}^{{O_x}}{P_{{A_i}}}，$ (14)

UPU支链长度的求解公式为：

 ${l_i} = \sqrt {{{\left( {{}^{{O_e}}{P_{{A_i}}} - {}^{{O_e}}{P_B}} \right)}^2}}。$ (15)

 $P = {\left[ {\begin{array}{*{20}{c}} x&y&z \end{array}} \right]\;^T} ，$ (16)

$P$ 点在其参考坐标系中的表示为：

 ${}^0P = {}_1^0T{}_2^1T{}_3^2TP 。$ (17)

 $L = \sqrt {{{\left( {{l_1} + {l_2}} \right)}^2} - {{\left( {z - {h_2}} \right)}^2}}，$ (18)

 $\left\{ {\begin{array}{*{20}{l}} {{\theta _1}=\arctan \left( {\dfrac{y}{x}} \right)} ，\\ {{\theta _2} = {\text{π}} - \arccos \left( {\dfrac{{{l_1} + {l_2}}}{{z - {h_2}}}} \right)} ，\\ {{l_2} = \sqrt {{x^2} + {y^2} + {{\left( {z - {h_2}} \right)}^2}} - {l_1}}。\end{array}} \right.$ (19)
3 海上廊桥系统运动学仿真及分析 3.1 船体运动学方程

 $\left\{ {\begin{array}{*{20}{l}} {\psi \left( t \right) = 10^\circ \sin \left( {\dfrac{{2\text{π} t}}{8}} \right)} ，\\ {\theta \left( t \right) = 6^\circ \sin \left( {\dfrac{{2\text{π} t}}{4}} \right)}，\\ {h\left( t \right) = - 500\sin \left( {\dfrac{{2\text{π} t}}{8}} \right){\rm{mm}}}。\end{array}} \right.$ (20)
3.2 海上稳定廊桥运动学逆解

 ${}^{{O_e}}P = {\left[ {\begin{array}{*{20}{c}} 0&{1640}&{750} \end{array}} \right]^{\rm{T}}}{\rm{mm}}。$ (21)

 图 3 $\psi \left( t \right)$ ， $\theta \left( t \right)$ ， $h\left( t \right)$ 变化曲线 Fig. 3 Change curve of $\psi \left( t \right)$ , $\theta \left( t \right)$ , $h\left( t \right)$

 图 4 UPU支链长度变化曲线 Fig. 4 UPU branch chain length change curve

 图 5 ${\theta _2}$ 变化曲线 Fig. 5 Change curve of ${\theta _2}$

 图 6 ${l_2}$ 变化曲线 Fig. 6 Change curve of ${l_2}$
3.3 海上稳定廊桥运动学方程验证

 图 7 海上稳定廊桥Adams仿真模型 Fig. 7 Virtual prototype of offshore gangway in Adams

 图 8 UPU支链长度变化曲线 Fig. 8 UPU branch chain length change curve

 图 9 ${\theta _2}$ 变化曲线 Fig. 9 Change curve of ${\theta _2}$

 图 10 ${l_2}$ 变化曲线 Fig. 10 Change curve of ${l_2}$

4 结　语

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