﻿ 脱插机构的误差灵敏度分析与优化设计
 舰船科学技术  2023, Vol. 45 Issue (11): 171-175    DOI: 10.3404/j.issn.1672-7619.2023.11.036 PDF

1. 海军装备部武汉局驻郑州地区军事代表室，河南 郑州 450015;
2. 中国船舶集团有限公司第七一三研究所，河南 郑州 450015

Insert-draw sensitivity analysis based on the geometric error transformation model
SUN Ji-hong1, WU Ling-wei2, CHEN Bing2
1. Military Representative Office of Haizhuang Wuhan Bureau in Zhengzhou, Zhengzhou 450015, China;
2. The 713 Research Institute of CSSC, Zhengzhou 450015, China
Abstract: In this paper, the geometric error transmission model of the launch canister's insert-draw is established, and the sensitivity coefficient of the processing and assembly errors of various components to the spatial error of the insert-draw is systematically analyzed. It is determined that the rotation errors between the lower connecting rod base and the insert-draw base, between the lower connecting rod base and the translational rod, and between the translational rod and the upper connecting rod seat along the y axis are the key error sources, and the improvement measures are put forward and implemented. Finally, the effectiveness of the error sensitivity analysis and the correctness of the improvement measures are verified by experiments. This method provides a reliable theoretical support for the error optimization analysis of the whole system, and greatly improves the docking success rate of the insert-draw.
Key words: error     insert-draw     sensitivity analysis     error transformation model
0 引　言

1 多体系统几何误差传递模型 1.1 误差描述

 图 1 配合面误差专递模型 Fig. 1 Matching surface error transformation model

 $\begin{split} T =& \left[ {\begin{array}{*{20}{c}} 1 & 0 & 0 & 0 \\ 0 & {\cos \Delta \alpha } & { - \sin \Delta \alpha } & 0 \\ 0 & {\sin \Delta \alpha } & {\cos \Delta \alpha } & 0 \\ 0 & 0 & 0 & 1 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\cos \Delta \beta }&0&{\sin \Delta \beta }&0 \\ 0&1&0&0 \\ { - \sin \Delta \beta }&0&{\cos \Delta \beta }&0 \\ 0&0&0&1 \end{array}} \right]\times\\ &\left[ {\begin{array}{*{20}{c}} {\cos \Delta \gamma }&{ - \sin \Delta \gamma }&0&0 \\ {\sin \Delta \gamma }&{\cos \Delta \gamma }&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \end{array}} \right] \left[ {\begin{array}{*{20}{c}} 1&0&0&{\Delta x} \\ 0&1&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \end{array}} \right]\times\\ &\left[ {\begin{array}{*{20}{c}} 1&0&0&0 \\ 0&1&0&{\Delta y} \\ 0&0&1&0 \\ 0&0&0&1 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 1&0&0&0 \\ 0&1&0&0 \\ 0&0&1&{\Delta z} \\ 0&0&0&1 \end{array}} \right]= \\ &\left[ \begin{array}{*{20}{c}} {\cos \Delta \beta \cos \Delta \gamma } \\ {\cos \Delta \alpha \sin \Delta \gamma + \sin \Delta \alpha \sin \Delta \beta \cos \Delta \gamma }\\ {\sin \Delta \alpha \sin \Delta \gamma - \cos \Delta \alpha \sin \Delta \beta \cos \Delta \gamma }\\ 0 \end{array}\right.\\ &\left. \begin{array}{*{20}{c}} { - \cos \Delta \beta \sin \Delta \gamma } \\ {\cos \Delta \alpha \cos \Delta \gamma - \sin \Delta \alpha \sin \Delta \beta \sin \Delta }\\ {\sin \Delta \alpha \cos \Delta \gamma + \cos \Delta \alpha \sin \Delta \beta \sin \Delta \gamma }\\ 0\end{array} \right.\\ &\left. \begin{array}{*{20}{c}} {\sin \Delta \beta }&{\Delta x} \\ { - \sin \Delta \alpha \cos \Delta \beta }&{\Delta y} \\ {\cos \Delta \alpha \cos \Delta \beta }&{\Delta z} \\ 0&1 \end{array} \right]。\\[-30pt] \end{split}$ (1)

 $T \approx \left[ {\begin{array}{*{20}{c}} 1&{ - \Delta \gamma }&{\Delta \beta }&{\Delta x} \\ {\Delta \gamma }&1&{ - \Delta \alpha }&{\Delta y} \\ { - \Delta \beta }&{\Delta \alpha }&1&{\Delta z} \\ 0&0&0&1 \end{array}} \right] 。$ (2)

 $p{a'} = T \cdot pa 。$

 ${T_A} = \left[ {\begin{array}{*{20}{c}} 1&{ - \Delta {\gamma _A}}&{\Delta {\beta _A}}&{\Delta {x_A}} \\ {\Delta {\gamma _A}}&1&{ - \Delta {\alpha _A}}&{\Delta {y_A}} \\ { - \Delta {\beta _A}}&{\Delta {\alpha _A}}&1&{\Delta {z_A}} \\ 0&0&0&1 \end{array}} \right]。$ (3)

 $T_A^B = {T_A} \cdot T_{{A'}}^{{B'}} \cdot {T_B} = \left[ {\begin{array}{*{20}{c}} 1&{ - \Delta \gamma _A^B}&{\Delta \beta _A^B}&{\Delta x_A^B} \\ {\Delta \gamma _A^B}&1&{ - \Delta \alpha _A^B}&{\Delta y_A^B} \\ { - \Delta \beta _A^B}&{\Delta \alpha _A^B}&1&{\Delta z_A^B} \\ 0&0&0&1 \end{array}} \right]。$ (4)

 $pa = T_A^B \cdot pb = {T_A} \cdot T_{{A'}}^{{B'}} \cdot {T_B} \cdot pb 。$ (5)
1.2 脱插机构多体系统坐标系定义和变换

 图 2 发射箱脱插机构结构简化图 Fig. 2 Simplified structure diagram of launch canister insert-draw

 图 3 发射箱脱插机构拓扑图 Fig. 3 Topological diagram of launch canister insert-draw

${T_{{U_1}}}$ ${T_{{U_{12}}}}$ ${T_{{U_{23}}}}$ ${T_{{U_{34}}}}$ 分别看作为脱插机构连杆座（刚体1）相对于脱插机构基座及刚体1与刚体2、刚体2与刚体3、刚体3与刚体4对应的理想配合面，坐标齐次变换矩阵分别为 ${T_{{U_1}}}$ ${T_{{U_{12}}}}$ ${T_{{U_{23}}}}$ ${T_{{U_{34}}}}$ ；由脱插机构各个零件的几何尺寸确定各个零件自身2个理想配合面的相对位置关系，用 ${T_1}$ ${T_2}$ ${T_3}$ ${T_4}$ 分别表示刚体1、刚体2、刚体3和刚体4自身2个理想配合面的坐标齐次变换矩阵。

 $m = {T_{{U_0}}} \cdot {T_1} \cdot {T_{{U_{12}}}} \cdot {T_2} \cdot {T_{{U_{23}}}} \cdot {T_3} \cdot {T_{{U_{34}}}} \cdot {T_4} \cdot {m_4}。$ (6)

 $m = {T_1} \cdot {T_2} \cdot {T_3} \cdot {T_4} \cdot {m_4} 。$ (7)

 $\begin{split} \Delta m =& {T_{{U_0}}} \cdot {T_1} \cdot {T_{{U_{12}}}} \cdot {T_2} \cdot {T_{{U_{23}}}} \cdot {T_3} \cdot {T_{{U_{34}}}} \cdot {T_4} \cdot {m_4}- \\ & {T_1} \cdot {T_2} \cdot {T_3} \cdot {T_4} \cdot m =\\ &({T_{{U_0}}} \cdot {T_1} \cdot {T_{{U_{12}}}} \cdot {T_2} \cdot {T_{{U_{23}}}} \cdot {T_3} \cdot {T_{{U_{34}}}} \cdot {T_4}-\\ &{T_1} \cdot {T_2} \cdot {T_3} \cdot {T_4}) \cdot m = {T_{{U_{04}}}} \cdot m 。\end{split}$ (8)

${T_{{U_{04}}}}$ 为脱插机构组装后的总体误差坐标齐次变换矩阵，可表示为：

 ${T_{{U_{04}}}} = \left[ {\begin{array}{*{20}{c}} 1&{ - \Delta \gamma _0^4}&{\Delta \beta _0^4}&{\Delta x_0^4} \\ {\Delta \gamma 4}&1&{ - \Delta \alpha _0^4}&{\Delta y_0^4} \\ { - \Delta \beta _0^4}&{\Delta \alpha _0^4}&1&{\Delta z_0^4} \\ 0&0&0&1 \end{array}} \right] 。$ (9)
1.3 脱插机构灵敏度分析模型

 $X = f({U_1},{U_2} \cdots \cdots {U_k})，$ (10)
 ${U_k} = [\begin{array}{*{20}{c}} {\Delta {x_k}}&{\Delta {y_k}}&{\Delta {z_k}}&{\Delta {\alpha _k}} &{\Delta {\beta _k}}&{\Delta {\gamma _k}} \end{array}] 。$ (11)

 $E = \frac{{\partial F\left( x \right)}}{{\partial x}} 。$ (12)

 $\Delta X = \frac{{\partial f}}{{\partial {U_k}}}\Delta {U_k} = {E_k}\Delta {U_k}。$ (13)

 $E = \left[ {\begin{array}{*{20}{c}} {\dfrac{{\partial {f_1}}}{{\partial {U_1}}}}&{\dfrac{{\partial {f_1}}}{{\partial {U_2}}}}& \cdots & \cdots & \cdots &{\dfrac{{\partial {f_1}}}{{\partial {U_k}}}} \\ {\dfrac{{\partial {f_2}}}{{\partial {U_1}}}}&{\dfrac{{\partial {f_2}}}{{\partial {U_2}}}}& \cdots & \cdots & \cdots &{\dfrac{{\partial {f_2}}}{{\partial {U_k}}}} \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ {\dfrac{{\partial {f_6}}}{{\partial {U_1}}}}&{\dfrac{{\partial {f_6}}}{{\partial {U_2}}}}& \cdots & \cdots & \cdots &{\dfrac{{\partial {f_6}}}{{\partial {U_k}}}} \end{array}} \right]，$ (14)
 ${E_k} = \dfrac{{\partial {f_l}}}{{\partial {U_k}}} = \left[ {\dfrac{{\partial {f_l}}}{{\partial \Delta {x_k}}}}\;\;{\dfrac{{\partial {f_l}}}{{\partial \Delta {y_k}}}}\;\; {\dfrac{{\partial {f_l}}}{{\partial \Delta {z_k}}}}\;\; {\dfrac{{\partial {f_l}}}{{\partial \Delta {\alpha _k}}}}\;\; {\dfrac{{\partial {f_l}}}{{\partial \Delta {\beta _k}}}}\;\;{\dfrac{{\partial {f_l}}}{{\partial \Delta {\gamma _k}}}} \right] 。$ (15)

 ${C_k} = \left| {{E_k}} \right|{{} \mathord{\left/ {\vphantom {{} {\sum {\left| {{E_k}} \right|} }}} \right. } {\sum {\left| {{E_k}} \right|} }} 。$ (16)

2 脱插机构总体误差分析 2.1 误差矢量计算

2.2 关键因素确定

3 措施制定及试验验证 3.1 措施制定

3.2 试验验证

 图 4 发射箱上电插头和雷弹插座相对位置示意图 Fig. 4 Schematic diagram of relative position of electric plug and torpedo-missile on launch canister

4 结　语

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