Thermomigration and interfacial reaction in Cu/Sn-58Bi/Cu micro solder joint under temperature gradient
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摘要:
探究了焊点平均温度为110 ℃(时效)及180 ℃(回流)时，Cu/Sn-58Bi/Cu微焊点在温度梯度作用下的原子热迁移行为及界面反应行为. 结果表明，在时效过程中，因Bi相的网状结构，Cu/Sn-58Bi/Cu微焊点冷、热两端界面金属间化合物(intermetallic compound, IMC)呈现对称性生长. 在温度梯度为1 000 ℃/cm时，未发生明显的Bi原子迁移现象，但当温度梯度达到或超过1 300 ℃/cm时，Bi原子会由热端向冷端界面迁移，并在冷端界面处偏聚. 在回流过程中，温度梯度驱动Cu原子由微焊点热端向冷端界面迁移，导致两端界面IMC呈非对称性生长，而并未发现Bi原子的热迁移行为. 因此，当Cu/Sn-58Bi/Cu微焊点中钎料呈液态时，温度梯度仅驱动了Cu原子的热迁移，并未致使Bi原子发生热迁移；当钎料呈固态时，在较低温度梯度下Cu和Bi原子均不发生明显的热迁移，但较高的温度梯度会引发Bi原子的热迁移.
Abstract:In this paper, the atomic thermomigration behavior and interfacial reaction behavior of Cu/Sn-58Bi/Cu micro solder joints under temperature gradient (TG) were explored at average temperatures of 110 ℃ (aging) and 180 ℃ (soldering). During the aging process, the symmetrical growth of the interfacial intermetallic compound (IMC) was clearly observed in Cu/Sn-58Bi/Cu micro solder joints due to the net structure of Bi phase. There were no obvious Bi atoms thermomigration under the TG of 1 000 ℃/cm. While when the TG reached or exceeded 1 300 ℃/cm, Bi atoms migrated from the hot end to the cold end and then significantly segregated near the cold end interface. During the soldering process, with the driven of TG, the migration of Cu atoms from the hot end to the cold end resulting in the asymmetrical growth of IMC between the hot and cold ends. And there were no obvious thermomigration of Bi atoms. In conclusion, the TG had caused the mass Cu atoms and no Bi atoms to migrate from the hot end toward the cold end, resulting in asymmetrical growth of IMC with the solder in liquid-state. However, with the solder in solid-state, there was no obvious thermomigration of Cu and Bi atoms under lower TG and only thermomigration of Bi atoms under higher TG.
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Keywords:
- micro solder joints /
- Sn-58Bi /
- thermomigration /
- interface reaction /
- Bi phase segregation
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0. 序言
管道插接形成的相贯线焊缝在航天航空、石油化工、建筑、船舶等工程领域广泛存在，是一种典型的焊缝形式[1]. 但管管相贯结构所形成的焊缝为空间曲线焊缝，坡口形状不规则且有不均匀的变化[2]. 为实现其焊接自动化，国内外学者进行了大量的研究工作. 早期的管管焊接设备的研究主要是针对大型的管道对接焊接，通过使用周向运动的焊接小车，环绕焊缝运动实现大型管道的相贯线焊接[3]. 或者采用分段的方式将圆周焊道细分为小段，再平滑过渡分段函数拟合为连续函数[4]，以此来研究管道对接全位置焊接. 康少杰等人[5]开发了一种可变式圆弧轨道的焊接机器人装置，能进行相贯线焊缝全方位不同宽度焊缝的焊接. 随着管管焊接的需求趋于复杂化、差异化，管管焊接设备的研究也朝着精细化智能化发展. 高利军等人[6]设计了一种新型钢制暖气片相贯线焊缝专用焊机，并建立了相贯线焊缝数学模型，推导出了基于此模型的焊机自步角算法. 刘永滨等人[7]则采用MSC.Marc软件对天然气管道在役修补焊接过程进行数值模拟，其管道残余变形峰值出现在焊缝的收弧区域，最大收缩变形量为1.79 mm. 上述关于管道插接的相贯线焊接的研究，多是针对于单一管道或者同一类型的多个不同管道插接焊缝，对于同时需要焊接两个及以上不同直径乃至不同插接形式的焊接任务，相关研究并不多见.
针对筒体内壁的管道插接焊接任务，设计了一种能提取支管空间位置和尺寸信息的传感器，并在此基础上建立管道插接焊缝位置模型和焊缝特征与焊枪姿态矩阵，最后利用MATLAB结合初值对该数学模型进行了数值模拟验证.
1. 管道插接焊缝位置识别传感器
管道插接有管-板插接、管-管插接、管-锥插接和管-球插接等种类，其中每个种类有各种插接形式，例如管-管插接又包含有管管正交、管管斜交、管管正交偏置、管管斜交偏置4种形式，所以管道插接焊缝不仅曲线复杂，而且种类繁多. 然而，在大部分管道插接焊缝的焊接任务中，其主件(即主板、主管、锥体或球体)位置均为已知，只需确定支管位置及尺寸，便能得到焊缝位置.
如图1所示，焊缝位置识别传感器通过提取支管开口截面内圆的空间位置和尺寸(即图1a加粗的空间圆)，来确定支管空间位置，提取方法为图1b所示通过传感器获取过支管开口截面内圆的同轴的两锥面A和 B，联立两锥面得到的空间圆即为支管开口截面内圆. 实际操作时，通过获取锥面A和锥面B上各4条相同方位的母线，取交点达到相同效果. 此传感器对于不同支管直径，不同管道插接种类和管道插接形式具有通用性. 焊缝位置识别传感器如图1c所示，由支架、曲柄、连杆、滑块、角度传感器和铰链底座组成，共含有4组曲柄滑块机构.
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图 1 传感器识别目标、方法和三维模型Figure 1. Sensor identification target, identification and 3D model. (a) open section inner circle; (b) identification method; (c) sensor 3D model如图2所示，当进行采样识别工作时，传感器整体沿支架轴线下降伸入支管内部，由于受支管对曲柄的反作力，4组曲柄滑块机构将分别运动到某一位置，此时进行第一次采样，由4个角度传感器获取此时曲柄对支架的夹角，由编码器反馈此时传感器高度，每组曲柄滑块的位置和角度信息可以得到一条空间直线；采集完毕后，传感器继续下降一定位置，进行第二次采样. 将第二次采样得到的空间直线与第一次采样得到的空间直线联立，可以得到4个交点，所得的交点均在支管开口截面内圆上. 空间上三点可确定空间圆的尺寸与位置，第4点用来防止干扰点与减小误差.
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图 2 传感器信号采集Figure 2. Sensor signal acquisition. (a) first sampling; (b) second sampling2. 管道插接焊缝的数学模型
以圆筒内壁的管道插接为研究对象，包含管管正交、管管斜交、管管正交偏置、管管斜交偏置等插接形式，建立基于焊缝位置识别传感器的焊缝数学模型，及焊缝焊枪特征矩阵. 如图3所示，传感器增加小车支架、小车移动副、转盘转动副、电动缸移动副，并移动到图3所示的初始位置，以转盘转动副轴线与传感器支架轴线交点为原点O1，重力方向为z1轴，圆筒轴线方向为x1轴，建立O1x1y1z1初始坐标系，其中小车移动副的移动方向与x1轴方向重合，转盘轴线方向与x1轴重合，在初始位置传感器轴线与z1轴重合.
2.1 管道插接焊缝位置模型
如图4所示，小车从初始位置沿x1轴方向平移
$ l $ ，转盘以x1轴为轴线顺时针旋转$\alpha $ 角,得到O2x2y2z2工作坐标系，则坐标系O1x1y1z1到坐标系O2x2y2z2的变换矩阵.$$ {}_2^1T = \left[ {\begin{array}{*{20}{c}} 1&0&0&l \\ 0&{\cos \alpha }&{ - \sin \alpha }&0 \\ 0&{\sin \alpha }&{\cos \alpha }&0 \\ 0&0&0&1 \end{array}} \right] $$ (1) 圆筒内壁半径为R，轴线在x1O1z1面内，与x1轴平行且距离为
$ {l_0} $ . 圆筒内壁在坐标系O2x2y2z2下的方程为$$ {(y + {l_0}\sin \alpha )^2} + {({\textit{z}} - {l_0}\cos \alpha )^2} = {R^2} $$ (2) 在O2x2y2z2坐标系下，已知传感器铰链底座的4个铰链轴线所在平面到x2O2y2面距离为Z0，4组曲柄的方位角为
$$ {\phi _i} = \left\{ \begin{gathered} 0,\;i = 1 \\ \frac{1}{2}{\text{π}} ,\;i = 2 \\ {\text{π}} ,\;i = 3 \\ \frac{3}{2}{\text{π}} ,\;i = 4 \\ \end{gathered} \right. $$ (3) 第一次采样得到的直线参数方程为
$$ \left\{ \begin{gathered}p_i =(x_i,y_i,z_i,) \\ x = (t + {r_{\rm{0}}}) \cos {\phi _i} \\ y = (t + {r_{\rm{0}}}) \sin {\phi _i} \\ {\textit{z}} = - t \cot {\beta _{1i}}+ {Z_0} + {Z_1} \\ \end{gathered} \right. ,\;i=1,2,3,4 $$ (4) 第二次采样得到的直线参数方程为
$$ \left\{ \begin{gathered} x = (t + {r_{\rm{0}}}) \cos {\phi _i} \\ y = (t + {r_{\rm{0}}}) \sin {\phi _i} \\ {\textit{z}} = - t \cot {\beta _{2i}} + {Z_0} + {Z_1} + {Z_2} \\ \end{gathered} \right. $$ (5) 式中：
$ {r_0} $ 为铰链轴线到支架轴线距离；$ {\beta _{1i}} $ ，$ {\beta _{2i}} $ 分别为第一次采样和第二次采样得到的曲柄与传感器轴线夹角的角度数据；$ {Z_1} $ 为第一次采样时电动缸移动距离；$ {Z_2} $ 为第一次采样移动到第二次采样时电动缸移动距离，联立(4)和(5)，可得$$ {t_i} = \frac{{{Z_2}}}{{\cot {\beta _{2i}} - \cot {\beta _{1i}}}},\;i = 1,2,3,4 $$ (6) 代入式(4)得到两组直线的4个交点，此4点即为支管开口截面内圆上4点，设
$ {P_i} = ({x_i},{y_i},{z_i}), i = 1,2,3,4 $ ，则$$ \left\{ \begin{gathered} {x_i} = ({t_i} + {r_{\rm{0}}}) \cos {\phi _i} \\ {y_i} = ({t_i} + {r_{\rm{0}}}) \sin {\phi _i} \\ {{\textit{z}}_i} = - {t_i} \cot {\beta _{1i}} + {Z_0} + {Z_1} \\ \end{gathered} \right. ,\;i = 1,2,3,4 $$ (7) 取支管开口截面内圆所在平面内的两个向量为
$$ \left\{\begin{gathered} {{ {{\boldsymbol{n_1}}}}} = {P_1} - {P_2} = ({x_1} - {x_2},{y_1} - {y_2},{{\textit{z}}_1} - {{\textit{z}}_2}) \\ {{ {{\boldsymbol{n_2}}}}} = {P_1} - {P_3} = ({x_1} - {x_3},{y_1} - {y_3},{{\textit{z}}_1} - {{\textit{z}}_3}) \\ \end{gathered} \right.$$ (8) 则支管开口截面内圆所在平面的法向量为
$$ \left\{ \begin{gathered} {\boldsymbol{n}} = { {\boldsymbol{n_1}}} \times { {\boldsymbol{n_2}}} = ({{{x}}_{ {{n}}}},{{{y}}_{ {{n}}}},{{{{{{{\textit{z}}}}}}}_{ {{n}}}}) \\ {{{x}}_{ {{n}}}} = \left| {\begin{array}{*{20}{c}} {{y_1} - {y_2}}&{{{\textit{z}}_1} - {{\textit{z}}_2}} \\ {{y_1} - {y_3}}&{{{\textit{z}}_1} - {{\textit{z}}_3}} \end{array}} \right| \\{{{y}}_{ {{n}}}} = \left| {\begin{array}{*{20}{c}} {{{\textit{z}}_1} - {{\textit{z}}_2}}&{{x_1} - {x_2}} \\ {{{\textit{z}}_1} - {{\textit{z}}_3}}&{{x_1} - {x_3}} \end{array}} \right| \\{{{{\textit{z}}}}_{ {{n}}}} = \left| {\begin{array}{*{20}{c}} {{x_1} - {x_2}}&{{y_1} - {y_2}} \\ {{x_1} - {x_3}}&{{y_1} - {y_3}} \end{array}} \right| \end{gathered}\right.$$ (9) 直线
$ {l_{{P_1}{P_2}}} $ 、直线$ {l_{{P_1}{P_3}}} $ 的中点坐标为$$ \left\{\begin{gathered} {P_{12}} = \frac{{{P_1} + {P_2}}}{2} = ({x_{{P_{12}}}} , {y_{{P_{12}}}} ,{{\textit{z}}_{{P_{12}}}}) = \left(\frac{{{x_1} + {x_2}}}{2} , \frac{{{y_1} + {y_2}}}{2} , \frac{{{{\textit{z}}_1} + {{\textit{z}}_2}}}{2}\right) \\ {P_{13}} = \frac{{{P_1} + {P_3}}}{2} = ({x_{{P_{13}}}} , {y_{P_{13}}} ,{{\textit{z}}_{{P_{13}}}}) = \left(\frac{{{x_1} + {x_3}}}{2} , \frac{{{y_1} + {y_3}}}{2} , \frac{{{{\textit{z}}_1} + {{\textit{z}}_3}}}{2}\right) \\ \end{gathered} \right.$$ (10) 直线
$ {l_{{P_1}{P_2}}} $ 、直线$ {l_{{P_1}{P_3}}} $ 的中垂线方向向量为$$ \left\{\begin{gathered} {{ {{\boldsymbol{n_{{\boldsymbol{12}}}}}}}} = {\boldsymbol{ n}} \times {{ {\boldsymbol{n_1}}}} =\left(x_{12}, y_{12}, {\textit{z}}_{12}\right) \\ {{ {{\boldsymbol{n_{{\boldsymbol{13}}}}}}}} = {\boldsymbol{n}} \times {{ {\boldsymbol{n_2}}}} =\left(x_{13}, y_{13}, {\textit{z}}_{13}\right) \\ \end{gathered}\right. $$ (11) 由于直线
$ {l_{12}} $ 与直线$ {l_{13}} $ 的中垂线都经过空间圆的圆心c，有$P_c=\left(x_c, y_c, {\textit{z}}_c\right)$ ，即$$ {P_{12}} + {\lambda _1}{ {{\boldsymbol{n_{{\boldsymbol{12}}}}}}} = {P_{13}} + {\lambda _2}{ {{\boldsymbol{n_{{\boldsymbol{13}}}}}}} $$ (12) $$ \left\{\begin{array}{l} x_{P_{12}} + \lambda_1 x_{ 12}=x_{P_{13}} + \lambda_2 x_{ 13} \\ y_{P_{12}} + \lambda_1 y_{12}=y_{P_{13}} + \lambda_2 y_{13} \end{array}\right. $$ (13) $$ \lambda_2=\frac{\left(y_{P_{12}}-y_{P_{13}}\right) x_{ {12}}-\left(x_{P_{12}}-x_{P_{13}}\right) y_{ 12}}{y_{P_{13}} x_{P_{12}}-y_{P_{12}} x_{P_{13}}} $$ (14) $$ ({x_c},{y_c},{{\textit{z}}_c}) = {P_{13}} + {\lambda _2}{ {{\boldsymbol{n_{{\boldsymbol{13}}}}}}} $$ (15) 由距离公式得半径r为
$$ r = \left| {{P_1} - {P_c}} \right| = \sqrt {{{({x_1} - {x_c})}^2} + {{({y_1} - {y_c})}^2} + {{({{\textit{z}}_1} - {\textit{z}_c})}^2}} $$ (16) 设x2轴单位矢量为
$ {\boldsymbol{i}} = (1,0,0) $ ，支管开口截面与y2O2z2面的交线的单位矢量为$ {\boldsymbol{a}} $ ，则$$ \left\{\begin{array}{l} {\boldsymbol{a}} = \dfrac{{ {\boldsymbol{n}} \times ( - {\boldsymbol{i}})}}{{\left| { {\boldsymbol{n}} \times ( - {\boldsymbol{i}})} \right|}} =\left(x_a, y_a, {\textit{z}}_a\right) \\ {\boldsymbol{b }}= \dfrac{{ {\boldsymbol{n}} \times {\boldsymbol{a}}}}{{\left| { {\boldsymbol{n}} \times {\boldsymbol{a}}} \right|}}=\left(x_b, y_b, {\textit{z}}_b\right) \end{array}\right. $$ (17) 由圆心
$ c $ 、圆上正交单位矢量$ {\boldsymbol{a}}$ 和$ {\boldsymbol{b}}$ 、半径r可得空间圆参数方程.$$ \left\{\begin{array}{l} x=x_c + r x_{ {{{a}}}} \cos \theta\, + r x_{ {{{b}}}} \sin \theta \\ y\,=y_c + r y_{ {{{a}}}} \cos \theta \,+ r y_{ {{{b}}}} \sin \theta \\ {\textit{z}}\,=\,{\textit{z}}_c + r {\textit{z}}_{ {{{a}}}} \cos \theta\, + r {\textit{z}}_{ {{{b}}}} \sin \theta \end{array}\right.$$ (18) 式(18)为支管开口截面内圆所在空间位置方程，参数
$ \theta $ 取值范围为$ [0,\;2{\text{π}} ] $ ，此时参数$ \theta $ 的物理意义为圆上一点到圆心c连成的直线与$ {\boldsymbol{a}}$ 的夹角.支管内壁曲面参数方程为
$$ \left\{\begin{array}{l} x^{\prime}=x-\lambda x_{ {{n}}} \\ y^{\prime}=y-\lambda y_{ {{n}}} \\ {\textit{z}}^{\prime}={\textit{z}}-\lambda {\textit{z}}_{ {{n}}} \end{array}\right. $$ (19) $$ \left\{\begin{array}{l} x^{\prime} = x_c + r x_{ {{{a}}}} \cos \theta + r x_{ {{{b}}}} \sin \theta - \lambda x_{ {{{n}}}} \\ y^{\prime}= y_c + r y_{ {{{a}}}} \cos \theta \,+ r y_{ {{{b}}}} \sin \theta - \lambda y_{ {{{n}}}} \\ \,{\textit{z}}^{\prime} ={\textit{z}}_c + r {\textit{z}}_{ {{{a}}}} \cos \theta \,+ r {\textit{z}}_{ {{{b}}}} \sin \theta- \lambda {\textit{z}}_{ {{{n}}}} \end{array}\right. $$ (20) 式中：x，y，z为空间圆参数方程联立支管内壁曲面参数方程与主管方程.
$$ y^{\prime 2} + {\textit{z}}^{\prime 2}=R^2 $$ (21) $$ \lambda_\theta=\frac{\sqrt{R^2 y_{ {{n}}}^2 + R^2 {\textit{z}}_{ {{n}}}^2-{\textit{z}}^2 y_{ {{n}}}^2-y^2 {\textit{z}}_{ {{n}}}^2 + 2 y {\textit{z}} y_{ {{n}}} {\textit{z}}_{ {{n}}}}-y y_{ {{n}}} + {\textit{z}} {\textit{z}}_{ {{n}}}}{y_{ {{n}}}^2 + {\textit{z}}_{ {{n}}}^2} $$ (22) 带入(20)得到焊缝方程，即
$$ \left\{\begin{array}{l} x=x_c + r x_{ {{{a}}}} \cos \theta + r x_{ {{{b}}}} \sin \theta-\lambda_\theta x_{ {{{n}}}} \\ y\,=y_c + r y_{ {{{a}}}} \cos \theta \,+ r y_{ {{{b}}}} \sin \theta-\lambda_\theta y_{ {{{n}}}} \\ \,{\textit{z}}=\,{\textit{z}}_c + r {\textit{z}}_{ {{{a}}}} \cos \theta \,+ r {\textit{z}}_{ {{{b}}}} \sin \theta\,-\lambda_\theta {\textit{z}}_{ {{{n}}}} \end{array}\right. $$ (23) 2.2 管道插接焊缝特征模型
焊缝坐标系及焊缝辅助坐标系的定义参照文献[4]，如图5所示，焊缝方程为关于
$ \theta $ 的参数方程，取$ \theta = {\theta _w} $ ，有焊缝上一点$ {P_w}({x_0},{y_0},{z_0}) $ ，空间圆上一点$ P_w^{'|}(x{'_w},y{'_w},z{'_w}) $ ，设$ {P_w} $ 点为当前焊点，则圆筒内壁在该焊点处的切平面A的法向量，即$$ { {{\boldsymbol{S}}_{ \boldsymbol{1}}}} = ({X_1},{Y_1},{Z_1}) = (0,{y_w} + {l_0}\sin \alpha ,{{\textit{z}}_w} - {l_0}\cos \alpha ) $$ (24) 支管在焊点处的切平面B的法向量为
$$ {{ {\boldsymbol{S}}_{ \boldsymbol{2}}}} = ({X_2},{Y_2},{Z_2}) = (x{'_w} - {x_c},y{'_w} - {x_c},{\textit{z}}{'_w} - {{\textit{z}}_c}) $$ (25) 平面A和平面B的夹角为二面角
$ \varphi $ ，即$$ \cos ( {{ {\boldsymbol{S}}_{ \boldsymbol{1}}}},{ {{\boldsymbol{S}}_{ \boldsymbol{2}}}}) = \frac{{{{{{\boldsymbol{S}}_{ \boldsymbol{1}}}}} \cdot {{ {{\boldsymbol{S}}_{ \boldsymbol{2}}}}}}}{{\left| {{{ {{\boldsymbol{S}}_{ \boldsymbol{1}}}}}} \right| \cdot \left| {{{ {{\boldsymbol{S}}_{ \boldsymbol{2}}}}}} \right|}} $$ (26) $$ \varphi = {\text{π}} - \arccos (\frac{{{{{{\boldsymbol{S}}_{ \boldsymbol{1}}}}} \cdot {{ {{\boldsymbol{S}}_{ \boldsymbol{2}}}}}}}{{\left| {{{{{\boldsymbol{S}}_{ \boldsymbol{1}}}}}} \right| \cdot \left| {{{ {{\boldsymbol{S}}_{ \boldsymbol{2}}}}}} \right|}}) $$ (27) 由于相贯线上任意点的切线,始终与过该点的两个圆柱切平面的交线重合，因此XW(XW1)坐标轴向量为
$$ { {{\boldsymbol{S}}_{ \boldsymbol{3}}}} = { {{\boldsymbol{S}}_{ \boldsymbol{2}}}} \times { {{\boldsymbol{S}}_{ \boldsymbol{1}}}} = ({X_3},{Y_3},{Z_3}) $$ (28) ZW1 坐标轴向量为
$$ { {{\boldsymbol{S}}_{ \boldsymbol{4}}}} = {{{\boldsymbol{S}}_{ \boldsymbol{2}}}} \times {{{\boldsymbol{S}}_{ \boldsymbol{3}}}} = ({X_4},{Y_4},{Z_4}) $$ (29) YW1坐标轴的向量为
$$ {{{\boldsymbol{S}}_{ \boldsymbol{5}}}} = { {{\boldsymbol{S}}_{ \boldsymbol{4}}}} \times { {{\boldsymbol{S}}_{ \boldsymbol{3}}}} = ({X_5},{Y_5},{Z_5}) $$ (30) 坐标系O2x2y2z2到坐标系Ow1xW1yW1zW1变换矩阵为
$$ {\boldsymbol{{}_{{\boldsymbol{w1}}}^2T}} = \left[ {\begin{array}{*{20}{c}} {{X_3}}&{{X_4}}&{{X_5}}&{{x_0}} \\ {{Y_3}}&{{Y_4}}&{{Y_5}}&{{y_0}} \\ {{Z_3}}&{{Z_4}}&{{Z_5}}&{{{\textit{z}}_0}} \\ 0&0&0&1 \end{array}} \right] $$ (31) 坐标系Ow1xW1yW1zW1绕xW1轴顺时针旋转
$ \varphi /2 $ ，得焊缝坐标系OwxWyWzW，即$$ {\boldsymbol{{}_w^{{\boldsymbol{w1}}}T}} = \left[ {\begin{array}{*{20}{c}} 1&0&0&0 \\ 0&{\cos \dfrac{\varphi }{2}}&{ - \sin \dfrac{\varphi }{2}}&0 \\ 0&{\sin \dfrac{\varphi }{2}}&{\cos \dfrac{\varphi }{2}}&0 \\ 0&0&0&1 \end{array}} \right] $$ (32) 焊缝坐标系相对于初始坐标系的特征矩阵为
$$ {}_{\boldsymbol{w}}^{\boldsymbol{1}}{\boldsymbol{T}} ={}_{\boldsymbol{2}}^{\boldsymbol{1}}{{\boldsymbol{T}}}\;\;{\boldsymbol{{}_{{\boldsymbol{w1}}}^2T}}\;\; {\boldsymbol{{}_w^{{\boldsymbol{w1}}}T}} = \left[ {\begin{array}{*{20}{c}} 1&0&0&0 \\ 0&{\cos \dfrac{\varphi }{2}}&{ - \sin \dfrac{\varphi }{2}}&0 \\ 0&{\sin \dfrac{\varphi }{2}}&{\cos \dfrac{\varphi }{2}}&0 \\ 0&0&0&1 \end{array}} \right] $$ (33) 2.3 焊枪姿态模型
焊缝坐标系到焊枪坐标系为
$$ {}_{\boldsymbol{r}}^{\boldsymbol{w}} {\boldsymbol{T}}= \left[ {\begin{array}{*{20}{c}} {\cos \gamma \cos \beta + \sin \gamma \sin \beta \sin \alpha }&{ - \sin \gamma \cos \beta + \cos \gamma \sin \beta \sin \alpha }&{\sin \beta \cos \alpha }&{{p_{twx}}} \\ {\sin \gamma \cos \alpha }&{\cos \gamma \cos \alpha }&{ - \sin \alpha }&{{p_{twy}}} \\ { - \cos \gamma \sin \beta + \cos \beta \sin \alpha \sin \gamma }&{\sin \gamma \sin \beta + \cos \beta \sin \alpha \cos \gamma }&{\cos \beta \cos \alpha }&{{p_{tw{\textit{z}}}}} \\ 0&0&0&1 \end{array}} \right] $$ (34) 通过焊枪坐标系相对焊缝坐标系的位姿矩阵，可以用数学算法计算出焊枪的3个角.
$$ 设 \;\; {}_{\boldsymbol{r}}^{\boldsymbol{w}}{\boldsymbol{T}} = \left[ {\begin{array}{*{20}{c}} {{C_{00}}}&{{C_{01}}}&{{C_{02}}}&{{p_{twx}}} \\ {{C_{10}}}&{{C_{11}}}&{{C_{12}}}&{{p_{twy}}} \\ {{C_{20}}}&{{C_{21}}}&{{C_{22}}}&{{p_{tw{\textit{z}}}}} \\ 0&0&0&1 \end{array}} \right] $$ (35) 由式(34)和式(35)可得
$$ \left\{\begin{gathered} \alpha = - \arcsin {C_{12}} \\ \beta = a\tan 2({C_{02}},{C_{22}}) \\ \gamma = a\tan 2({C_{10}},{C_{11}}) \\ \end{gathered}\right. $$ (36) 3. 仿真试验
在Creo平台上搭建筒体内壁管管插接焊缝自动焊接机器人的三维模型，并模拟工作环境，采集传感器角度信息，将数据带入上述模型与算法当中，获得管管插接焊缝空间位置数学模型.利用MATLAB软件对数学模型进行仿真验证，并与设定的工件焊缝曲线进行对比分析.设定同一空间坐标点为起始点，以相同的比例将工件焊缝曲线和传感器得到的数据曲线划分为64等份，得到各点的坐标值，部分数据如表1所示.
表 1 实际坐标点与采样坐标点(mm)Table 1. Actual coordinate point and the sampling coordinate point实际坐标点 采样坐标点 X Y Z X Y Z 10.000
9.950
9.800
9.553
9.210
8.775
8.253
7.648
6.967
6.2160
0.998
1.986
2.955
3.894
4.794
5.646
6.442
7.173
7.8336.633
6.707
6.924
7.261
7.691
8.184
8.711
9.246
9.770
10.26410.000
10.033
9.800
9.536
9.227
8.789
8.217
7.631
6.994
6.2370
1.065
2.035
2.914
3.911
4.912
5.760
6.420
7.295
7.9566.633
6.666
7.097
7.440
7.884
8.267
8.750
9.289
9.671
10.237对全部64个坐标数据进行分析比较，可得标定坐标点与采样坐标点在x轴和y轴方向的误差较小，其最大值为0.12 mm，而z轴方向的偏差较大，所以主要分析z轴方向的误差. 图6为z轴方向的焊缝偏差图，最大的焊缝偏差为0.25 mm，其精度满足实际焊接精度要求.
图7为传感器识别曲线与工件焊缝曲线整体对比图. 从图7可以看到，虽然二者存在一定偏差，但是整体偏差不大，传感器数据曲线基本接近工件焊缝曲线，证明了传感器模型和焊缝轨迹算法的准确性.
基于ABB公司开发的机器人仿真软件RobotStudio为平台，建立ABB机器人仿真系统. 通过通讯的方式将传感器所采集的焊缝位置，作为机器人运动轨迹传输给仿真平台，结合焊缝姿态模型与焊枪姿态模型，由运动学逆解求出机械臂关节量，再把求得的变量带入运动学求正解，进行轨迹与姿态仿真.仿真结果如图8所示，焊枪的运动轨迹与焊缝的特征完全重合，焊枪运动姿态也始终位于焊点两切面的夹角平分线上，验证了传感器所建立的数学模型正确性.
4. 结论
(1) 设计了适用于筒体内壁的管-管、管-板、管-球和管-锥等管道插接焊缝任务的焊缝位置识别传感器，该传感器能识别管道插接焊缝的空间位置，具有很强的实用性，为管道插接焊接任务的智能化提供支持.
(2) 提出了基于此传感器的针对筒体内壁管-管插接焊接任务的焊缝位置计算方法，并给出了焊缝特征矩阵与焊枪姿态矩阵，利用MATLAB软件对该方法获得的管管插接焊缝空间位置数学模型进行仿真验证，标定坐标点与采样坐标点在x轴和y轴方向的误差最大值仅为0.12 mm，在z轴方向上最大的焊缝偏差为0.25 mm，其精度满足实际焊接精度要求，表明了该方法的有效性和准确性.
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图 1 试验装置示意图
Figure 1. Schematics of experimental device. (a) the structure for immersion soldering; (b) the initial solder joint; (c) experimental device
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图 2 温度梯度下钎料层中的温度分布模拟
Figure 2. Simulation results of temperature distribution in the solde under temperature gradient experiment. (a) 1 000 ℃/cm(L + S); (b) 1000 ℃/cm(S + S); (c) 1 300 ℃/cm(S + S); (d) 1 500 ℃/cm(S + S)
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图 3 Cu/Sn-58Bi/Cu初始微焊点局部微观组织
Figure 3. Microstructure of Cu/Sn-58Bi/Cu initial solder image. (a) SEM; (b) EBSD; (c) phase distribution; (d) Bi phase EBSD orientation
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图 4 Cu/Sn-58Bi/Cu微焊点在1 000 ℃/cm温度梯度下回流不同时间后微观组织
Figure 4. Morphology of Cu/Sn-58Bi/Cu solder reflow for different time under 1 000 ℃/cm temperature gradient. (a) 5 min; (b) 15 min; (c) 30 min; (d) 60 min
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图 5 Cu/Sn-58Bi/Cu微焊点在1 000 ℃/cm温度梯度下时效不同时间后微观组织
Figure 5. Morphology of Cu/Sn-58Bi/Cu solder aging for different time under 1 000 ℃/cm temperature gradient. (a) 100 h; (b) 200 h; (c) 400 h
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图 6 Cu/Sn-58Bi/Cu微焊点在不同温度梯度下时效100 h后微观组织
Figure 6. Morphology of Cu/Sn-58Bi/Cu solder after aging for 100 h under different temperature gradients. (a) 0 ℃/cm; (b) 1 000 ℃/cm; (c) 1 300 ℃/cm; (d) 1 500 ℃/cm
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图 7 不同条件下微焊点中Bi元素分布
Figure 7. Distribution of Bi element in solder under different conditions
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图 8 不同条件下微焊点中Sn元素分布
Figure 8. Distribution of Sn element in solder under different conditions
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图 9 Cu/Sn-58Bi/Cu微焊点在不同温度梯度下时效200 h后微观组织
Figure 9. Morphology of Cu/Sn-58Bi/Cu solder after aging for 200 h under different temperature gradients. (a) 0 ℃/cm；(b) 1 000 ℃/cm；(c) 1 300 ℃/cm； (d) 1 500 ℃/cm
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图 10 Cu/Sn-58Bi/Cu微焊点在不同温度梯度下时效400 h后微观组织
Figure 10. Morphology of Cu/Sn-58Bi/Cu solder after aging for 400 h under different temperature gradients. (a) 0 ℃/cm; (b) 1 000 ℃/cm;(c) 1 300 ℃/cm; (d) 1 500 ℃/cm
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图 11 不同温度梯度下Bi原子迁移示意图
Figure 11. Schematic image of Bi atom migration at different temperature gradients. (a) 0 ℃/cm; (b) 1 000 ℃/cm; (c) 1 300 ℃/cm; (d) 1 500 ℃/cm
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图 12 不同温度梯度下Cu原子迁移示意图
Figure 12. Schematic image of Cu atom migration at different temperature gradients
表 1 Cu/Sn-58Bi/Cu微焊点温度梯度下试验条件
Table 1 Experimental conditions for temperature gradient of Cu/Sn-58Bi/Cu solder
试验
条件焊点高度
H/μm平均温度
T/℃反应时间
t/h温度梯度
△T/(℃·cm−1)时效 100 110 100，200，400 1 000，1 300，1 500 回流 100 180 0.083，0.25，0.5，1 1 000 
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