﻿ 两转动一平动波浪补偿并联机构的运动学及奇异性分析
 舰船科学技术  2023, Vol. 45 Issue (1): 57-63    DOI: 10.3404/j.issn.1672-7649.2023.01.011 PDF

Kinematics and singularity analysis of two rotation and one translation parallel mechanism for wave compensation
LIU Zhi-lin, HU Yi-fei, WU Jin-bo
School of Ship and Ocean Engineering, Huazhong University of Science and Technology, Wuhan 430070, China
Abstract: A new type of parallel mechanism with two rotation and one translation was proposed, and its kinematics analysis, mechanism optimization and singularity analysis were carried out. Firstly, the forward and inverse kinematics of the parallel mechanism configuration were solved by the homogeneous transformation matrix of the upper and lower platforms. Then, taking the minimum force and stroke of the position and attitude adjusting cylinder as the objective function, the optimal design and analysis of the mechanism were carried out, and the layout form of the gravity balance cylinder of the mechanism was determined. finally, according to the static equilibrium condition of the moving platform, five singular configurations of the mechanism were found by using the Grassmann line geometry method, and the whole workspace was traversed through numerical simulation to eliminate the singularity of the mechanism.
Key words: parallel manipulator     wave compensation     the forward and inverse kinematics     optimal design     singular configuration
0 引　言

1 运动学正反解

1.1 机构结构分析

 图 1 机构原理图 Fig. 1 3 DOF Parallel platform manipulator
1.2 反　解

 图 2 机构简图及坐标系的建立 Fig. 2 The establishment of mechanism diagram and coordinate system
 $T = Trans( - k{D_1}/2,0,Z) \cdot Rot({Y'},\beta ) \cdot Rot({X'},\alpha )，$ (1)
 $Trans( - k{D_1}/2,0,Z) = \left[ {\begin{array}{*{20}{c}} 1&0&0&{ - k{D_1}/2} \\ 0&1&0&0 \\ 0&0&1&Z \\ 0&0&0&1 \end{array}} \right]，$ (2)
 $Rot(X',\alpha ) = \left[ {\begin{array}{*{20}{c}} 1&0&0&0 \\ 0&{\cos \alpha }&{ - \sin \alpha }&0 \\ 0&{\sin \alpha }&{\cos \alpha }&0 \\ 0&0&0&1 \end{array}} \right] ，$ (3)
 $Rot(Y',\beta ) = \left[ {\begin{array}{*{20}{c}} {\cos \beta }&0&{\sin \beta }&0 \\ 0&1&0&0 \\ { - \sin \beta }&0&{\cos \beta }&0 \\ 0&0&0&1 \end{array}} \right] 。$ (4)

${\rm{Trans}}( - k{D_1}/2,0,Z)$ ${\rm{Rot}}(X',\alpha )$ ${\rm{Rot}}(Y',\beta )$ 计算式代入式（1），可得：

 ${\boldsymbol{T}} = \left[ {\begin{array}{*{20}{c}} {\cos \beta }&{\sin \alpha \cdot \sin \beta }&{\cos \alpha \cdot \sin \beta }&{ - {{k}}{D_1}/2} \\ 0&{\cos \alpha }&{ - \sin \alpha }&0 \\ { - \sin \beta }&{\sin \alpha \cdot \cos \beta }&{\cos \alpha \cdot \cos \beta }&Z \\ 0&0&0&1 \end{array}} \right]。$ (5)

 $({x_{{B_1}}},{y_{{B_1}}},{z_{{B_1}}}) = ( - {D_1}/2, - {H_1}/2,0)，$ (6)
 $({x_{{B_2}}},{y_{{B_2}}},{z_{{B_2}}}) = ({D_1}/2, - {H_1}/2,0)，$ (7)
 $({x_{{B_3}}},{y_{{B_3}}},{z_{{B_3}}}) = (0,{H_1}/2,0)。$ (8)

 ${({x_{{A_1}}},{y_{{A_1}}},{z_{{A_1}}},1)^{\rm{T}}} = T \cdot {(0, - k{H_1}/2,0,1)^{\rm{T}}}，$ (9)
 ${({x_{{A_2}}},{y_{{A_2}}},{z_{{A_2}}},1)^{\rm{T}}} = T \cdot {(k{D_1}, - k{H_1}/2,0,1)^{\rm{T}}}，$ (10)
 ${({x_{{A_3}}},{y_{{A_3}}},{z_{{A_3}}},1)^{\rm{T}}} = T \cdot {(k{D_1}/2,k{H_1}/2,0,1)^{\rm{T}}}。$ (11)

 $\begin{split} ({x_{{A_1}}},{y_{{A_1}}},{z_{{A_1}}}) =\;& ( - k{H_1}/2\sin \alpha \cdot \sin \beta - k{D_1}/2,\\ &- k{H_1}/2\cos \alpha , - k{H_1}/2\sin \alpha \cdot \cos \beta +Z) ，\end{split}$ (12)
 $\begin{split} ({x_{{A_2}}},{y_{{A_2}}},{z_{{A_2}}}) =\;& (k{D_1}\cos \beta - k{H_1}/2\sin \alpha \cdot \sin \beta - kD/2, \\&- k{H_1}/2\cos \alpha , - k{D_1}\sin \beta -k{H_1}/ 2 \times\\& \sin \alpha \cdot \cos \beta + Z)，\\[-12pt] \end{split}$ (13)
 $\begin{split} ({x_{{A_3}}},{y_{{A_3}}},{z_{{A_3}}}) =\;& (k{D_1}/2\cos \beta + k{H_1}/2\sin \alpha \cdot \sin \beta - k{D_1}/2,\\&k{H_1}/2\cos \alpha , - k{D_1}/ 2\mathrm{sin}\beta +\\&k{H}_{1}/2sin\alpha cos\beta +Z)，\\[-12pt] \end{split}$ (14)

3个电动缸的长度为：

 ${L_1}^2 = {({x_{{A_1}}} - {x_{{B_1}}})^2} + {({y_{{A_1}}} - {y_{{B_1}}})^2} + {({z_{{A_1}}} - {z_{{B_1}}})^2} ，$ (15)
 ${L_2}^2 = {({x_{{A_2}}} - {x_{{B_2}}})^2} + {({y_{{A_2}}} - {y_{{B_2}}})^2} + {({z_{{A_2}}} - {z_{{B_2}}})^2} ，$ (16)
 ${L_3}^2 = {({x_{{A_3}}} - {x_{{B_3}}})^2} + {({y_{{A_3}}} - {y_{{B_3}}})^2} + {({z_{{A_3}}} - {z_{{B_3}}})^2}，$ (17)

 $\begin{split} {L}_{1}{}^{2}=\;&{\left(k{H}_{1}/2\mathrm{sin}\alpha \cdot \mathrm{sin}\beta \text+\frac{\left(1-k\right)}{2}{D}_{1}\right)}^{2}+( - k{H_1}/2\cos \alpha +\\& {H_1}/2)^2 + \left( - k{H_1}/2\sin \alpha \cdot \cos \beta + Z\right)^2 ，\\[-12pt] \end{split}$ (18)
 $\begin{split} {L}_{2}{}^{2}=\;&{\left(k{D}_{1}\mathrm{cos}\beta -k{H}_{1}/2\mathrm{sin}\alpha \cdot \mathrm{sin}\beta -\frac{\left(1+k\right)}{2}{D}_{1}\right)}^{2}+\\ &{\left( - k{H_1}/2\cos \alpha + {H_1}/2\right)^2} + \left( - k{D_1}\sin \beta - \right.\\&\left. k{H}_{1}/2sin\alpha cos\beta +Z\right)^{2}，\end{split}$ (19)
 $\begin{split} {L_3}^2 =\;& {(k{D_1}/2\cos \beta + k{H_1}/2\sin \alpha \cdot \sin \beta - k{D_1}/2)^2} +\\& {(k{H_1}/2\cos \alpha - {H_1}/2)^2} + ( - k{D_1}/2\sin \beta +\\& k{H}_{1}/2sin\alpha cos\beta +Z)^{2}。\end{split}$ (20)

1.3 正　解

 ${L_1}^2 = {\left({x_{{A_1}}} + \frac{{{D_1}}}{2}\right)^2} + {\left({y_{{A_1}}} + \frac{{{H_1}}}{2}\right)^2} + {z_{{A_1}}}^2 ，$ (21)
 ${L_2}^2 = {\left({x_{{A_2}}} - \frac{{{D_1}}}{2}\right)^2} + {\left({y_{{A_2}}} + \frac{{{H_1}}}{2}\right)^2} + {z_{{A_2}}}^2，$ (22)
 ${L_3}^2 = {x_{{A_3}}}^2 + {\left({y_{{A_3}}} - \frac{{{H_1}}}{2}\right)^2} + {z_{{A_3}}}^2 ，$ (23)

 ${(k{D_1})^2} = {({x_{{A_1}}} - {x_{{A_2}}})^2} + {({y_{{A_1}}} - {y_{{A_2}}})^2} + {({z_{{A_1}}} - {z_{{A_2}}})^2} ，$ (24)
 $\begin{split} {\left(k{H_1}\right)^2} =\;& {\left({x_{{A_3}}} - \frac{{{x_{{A_1}}} + {x_{{A_2}}}}}{2}\right)^2} + {\left({y_{{A_3}}} - \frac{{{y_{{A_1}}} + {y_{{A_2}}}}}{2}\right)^2} + \\& {\left({z_{{A_3}}} - \frac{{{z_{{A_1}}} + {z_{{A_2}}}}}{2}\right)^2} ，\end{split}$ (25)
 $\begin{split} &\left({x}_{{A}_{3}}-\frac{{x}_{{A}_{1}}+{x}_{{A}_{2}}}{2}\right)\cdot ({x}_{{A}_{1}}-{x}_{{A}_{2}}）+\left({y}_{{A}_{3}}-\frac{{y}_{{A}_{1}}+{y}_{{A}_{2}}}{2}\right)\times\\ &({y}_{{A}_{1}}-{y}_{{A}_{2}})+\left({z}_{{A}_{3}}-\frac{{z}_{{A}_{1}}+{z}_{{A}_{2}}}{2}\right)\cdot ({z}_{{A}_{1}}-{z}_{{A}_{2}})=0 ，\end{split}$ (26)

 ${y_{{A_1}}} = {y_{{A_2}}} = - {y_{{A_3}}} ，$ (27)
 $3{x_{{A_1}}} - {x_{{A_2}}} + 2{x_{{A_3}}} = - 2k{D_1}，$ (28)
 \left\{ \begin{aligned} k{D_1}\cos \beta =\;& {x_{{A_2}}} - {x_{{A_1}}} ，\\ k{D_1}\sin \beta =\;& {z_{{A_1}}} - {z_{{A_2}}} ，\end{aligned} \right. (29)
 \left\{ \begin{aligned} & - k{H_1}/2\cos \alpha = {y_{{A_1}}}，\\ & k{H_1}\sin \alpha \cos \beta - k{D_1}/2 \cdot \sin \beta = {z_{{A_3}}} - {z_{{A_1}}}，\end{aligned} \right. (30)
 $- kH/2\sin \alpha \cdot \cos \beta + Z = {z_{{A_1}}}。$ (31)

${D_1} = 1\;{\rm{m}},\;{H_1} = 1\;{\rm{m}},\;k = 0.5$ ，运算结果如表1所示。

2 机构优化设计分析

2.1 目标函数

 图 3 上平台俯视图 Fig. 3 Top view of upper platform

 $f（X）=\frac{1\;000}{{\tau }_{\mathrm{max}}}+\frac{1}{{L}_{\mathrm{max}}} 。$ (32)

 ${F^{\rm{T}}}\delta x = {\tau ^{\rm{T}}}\delta l 。$ (33)

 $\delta x = {\boldsymbol{J}}\delta l，$ (34)

 ${F^{\rm{T}}}J\delta l = {\tau ^{\rm{T}}}\delta l，$ (35)

 ${F^{\rm{T}}}J = {\tau ^{\rm{T}}}，$ (36)

 $\tau = {J^{\rm{T}}}F。$ (37)
2.2 优化结果

${D_1} = 1\;{\rm{m}},\;{H_1} = 1\;{\rm{m}},\;k = 0.5$ ，变量约束条件为 $0 \leqslant a \leqslant 0.5,0 \leqslant b \leqslant 0.5,0 \leqslant a + c \leqslant 0.5$ ，旋转运动约束为 $- 15^\circ \leqslant \alpha \leqslant 15^\circ ,\; - 6^\circ \leqslant \beta \leqslant 6^\circ$ ，升沉运动约束为 $- 0.69\;{\rm{m}} \leqslant Z \leqslant 0.91\;{\rm{m}}$ ，上平台初始高度为0.8 m，上平台质量为100 kg，单个重力平衡缸的承载力为100 kg，有效载荷质量为100 kg，依次布置在上平台四周，偏距为1 m。利用遗传算法[8]以优化目标函数为目标来求解优秀个体，算法控制参数如下：

3 奇异性分析 3.1 Grassmann线几何法[11]

Grassmann为了揭示线簇的几何特性，对线几何进行深入的研究。Grassmann线几何法详尽说明了空间线矢之间的几何关系，并利用线矢之间的关系，得出了不同线簇的秩。根据线簇的秩从1到5分别介绍，如图4所示。

 图 4 Grassmann线几何 Fig. 4 Grassmann line geometry

3.2 静力学分析

 图 5 机构受力分析图 Fig. 5 Force analysis diagram of manipulator

 ${F_{O'}} = [{u_1},{u_2},{u_3},{f_1},{f_2},{f_3}]F，$ (38)
 ${M_{O'}} = [{r_1} \times {u_1},{r_2} \times {u_2},{r_3} \times {u_3},{d_1} \times {f_1},{d_2} \times {f_2},{d_3} \times {f_3}]F。$ (39)

 $Q={{\boldsymbol{J}}_p} \cdot F 。$ (40)

 ${{\boldsymbol{J}}_p} = {\left[ {\begin{array}{*{20}{c}} {{u_1}^{\rm T}}&{{{({r_1} \times {u_1})}^{\rm T}}} \\ {{u_2}^{\rm T}}&{{{({r_2} \times {u_2})}^{\rm T}}} \\ {{u_3}^{\rm T}}&{{{({r_3} \times {u_3})}^{\rm T}}} \\ {{f_1}^{\rm T}}&{{{({d_1} \times {f_1})}^{\rm T}}} \\ {{f_2}^{\rm T}}&{{{({d_2} \times {f_2})}^{\rm T}}} \\ {{f_3}^{\rm T}}&{{{({d_3} \times {f_3})}^{\rm T}}} \end{array}} \right]^{\rm T}} 。$ (41)

3.3 奇异形位分析

3.3.1 线簇秩为1的奇异位形

3.3.2 线簇秩为2的奇异位形

 图 6 线簇秩为2的奇异位形 Fig. 6 Singular configurations with rank 2 of line clusters
3.3.3 线簇秩为3的奇异位形

 图 7 线簇秩为3的奇异位形 Fig. 7 Singular configurations with rank 3 of line clusters

 图 8 静力转换矩阵的行列式 Fig. 8 Determinant of static transformation matrix
4 结　语

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