# Orbital-Based Automatic Multi-Layer Multi-Pass Welding Equipment for Small Assembly Plates
## 3 Equipment Debugging and Simulation Analysis
### Debugging of the Robotic Arm
The calibration transformation relationship is shown in Figure 20.  
Using the laser five-point calibration method, a calibration starting center point \(u_{c},v_{c}\) and four calibration points at different positions with coordinates \(u_{i},v_{i}\) on the target plane are selected, along with the pose parameters of the base coordinate system, with coordinates \((x_{i}\,y_{i}\,z_{i})\) and angles \((a_{i}\,b_{i}\,c_{i})\), where \(i=1,2,3,4,5\). Then the offset matrix \(M\) is prepared. \[M= \begin{bmatrix}u_{1}&v_{1}&1\\ u_{2}&v_{2}&1\\ u_{3}&v_{3}&1\\ u_{4}&v_{4}&1\\ u_{5}&v_{5}&1\end{bmatrix}  \tag{16}\]  
Calculation of \(\textbf{BHT}-\textbf{T}\) is performed five times according to Equation (13) to obtain \(\textbf{T}_{i}(T_{i,x}\,T_{i,y}\,T_{i,z})\). Based on \(M\) and \(\textbf{T}\), the rotation and translation information is extracted from their product. Then, according to the required orthogonality, the three orthogonal vectors are computed (17). \[\left\{\left(R_{1}=M^{+} \begin{bmatrix}T_{1,x}\\ T_{2,x}\\ T_{3,x}\\ T_{4,x}\\ T_{5,x}\end{bmatrix} = \begin{bmatrix}r_{11}\\ r_{13}\\ r_{13}\\ r_{4,x}\\ T_{5,y}\end{bmatrix} \right)\left(R_{2}=M^{+} \begin{bmatrix}T_{1,y}\\ T_{2,y}\\ T_{3,y}\\ T_{4,y}\\ T_{5,y}\end{bmatrix} = \begin{bmatrix}r_{21}\\ r_{23}\\ r_{24}\\ r_{4}\\ \end{bmatrix} \right)\left(R_{3}=M^{+} \begin{bmatrix}T_{1,z}\\ T_{2,z}\\ T_{3,z}\\ T_{3,z}\\ T_{4,z}\\ T_{5,z}\end{bmatrix} = \begin{bmatrix}r_{31}\\ r_{33}\\ r_{33}\\ z_{r}\end{bmatrix} \right) \tag{17}\] \[M^{+}=\left(M^{T}M^{-1}\right)^{-1}\cdot M^{T}\]  
\[\textbf{V}_{1}= \begin{bmatrix}r_{11}\\ r_{21}\\ r_{31}\end{bmatrix} \textbf{V}_{3}= \begin{bmatrix}r_{13}\\ r_{23}\\ r_{33}\end{bmatrix} \textbf{V}_{2}= \begin{bmatrix}r_{11}\\ r_{21}\\ r_{31}\end{bmatrix} \times \begin{bmatrix}r_{13}\\ r_{23}\\ r_{33}\end{bmatrix} = \begin{bmatrix}r_{12}\\ r_{22}\\ r_{23}\end{bmatrix}  \tag{18}\]  
In (17), \(M^{+}\) represents the pseudoinverse (Moore-Penrose inverse) of \(M\). Based on the properties of rotation matrices, we determine that the cross products of the row vectors conform to the right-hand rule, thereby obtaining the relationships among V1, V2, and V3 in (18). Through precise calculations using five calibration points, we derive the optimal  
Figure 20: Calibration conversion relationship diagram. approximate solution. In (19), \(r_{ij}(i,j=1,2,3)\) and \(x_{r}y_{r}z_{r}\) are variables used to record the results. Subsequently, the variables are normalized. \[\hat{r}_{ij}=\frac{r_{ij}}{\sqrt{\sum\limits_{i=1}^{3}r_{ij}^{2}}}(j=1,2,3) \tag{19}\]  
The calibration result matrix \(X\) is constructed, which is the required \(CHT^{-1}\). \[X= \begin{bmatrix}r_{11}&r_{12}&r_{13}&t_{x}\\ r_{21}^{2}&r_{22}&r_{23}&t_{y}\\ r_{31}^{3}&r_{32}&r_{33}^{3}&t_{z}\\ 0&0&0&1\end{bmatrix}  \tag{20}\]  
For any measurement point \(\mathbf{P}\) with coordinates \(\mathbf{p}(\mathbf{u_{0}},\mathbf{v_{0}})\), it is homogenized to obtain Equation (21). \[p_{\{B\}}=BHT\quad\cdot\quad CHT^{-1}\cdot p=R\cdot X\cdot \begin{bmatrix}u_{0 }\\ 0\\ v_{0}\\ 1\end{bmatrix} = \begin{bmatrix}x_{0}\\ y_{0}\\ z_{0}\\ 1\end{bmatrix}  \tag{21}\]  
The sensor performs repeated emission scanning of laser beams, calculates the time difference and angles of the round trip, and integrates the data to form a more comprehensive two-dimensional data image. Using Equation (21), any two-dimensional coordinate \(p(u_{0}\,,v_{0})\) can be converted into a three-dimensional coordinate \(p(x_{0}\,,y_{0}\,,z_{0})\), facilitating further point cloud construction. In practical operation, the calibration object is placed under the robotic arm's world coordinate system within the working conditions. A central point and marking points in the four quadrants at different positions within the sensor's visual range are specified and extracted. As shown in Figure 21, when adjusting the central point, the laser locator is activated to initiate origin position calibration. The laser beam is superimposed with the positioning beam along the Z and Y axes to obtain the central positioning point.