# Orbital-Based Automatic Multi-Layer Multi-Pass Welding Equipment for Small Assembly Plates
## 3 Equipment Debugging and Simulation Analysis  
### Debugging of the Robotic Arm  
Before commissioning, it must first be specified that the movement direction of the external axis of the rail is the Z direction in the world coordinate system. The robotic arm, wire feeder, tool flange, quick-plug mechanism, welding torch, and laser sensor are installed onto the bearing mechanism. After the mechanical installation is completed, TCP calibration and laser calibration of the robotic arm are performed. This study adopts the hand-eye calibration method and recommends maintaining a unified calibration posture reference before calibration to prevent precision issues and mechanical arm failures such as lock-up. The starting point of calibration is the flange, which is the end part of the robotic arm. The coordinate system relationships comply with the Cartesian right-hand coordinate system rule shown in Figure 17. During calibration, the coordinate system coincides with the earth coordinate system at the base of the robotic arm, placed at the central position of the robotic arm base, and has the same orientation as the robotic arm's world coordinate system. The upper computer processes the hand-eye coordinate system relationship through calibration algorithms and the internal and external parameter matrices of the laser sensor. It converts the relative relationship between the origin of the robotic arm's Cartesian coordinate system relative to the robotic arm flange into the relative spatial relationship between the origin of the robotic arm base's Cartesian coordinate system and the sensor's laser scanning point, as shown in Figure 17. The calibration process requires completing the transformations among the four coordinate systems B, T, C, and CL to obtain the coordinate conversion relationships between two-dimensional pixel coordinates and the spatial coordinate system. The visual recognition results are transferred to the robotic arm's coordinate system, thereby achieving precise robotic arm motion control and visual guidance. In Figure 17, {B} represents the robotic arm base coordinate system, {T} represents the laser sensor coordinate system, {C} represents the laser sensor coordinate system, and [Cl] represents the calibration object coordinate system. Figure 17: Relative diagram of the coordinate system for the calibration process. For the coordinates of the same point p in any two Cartesian coordinate systems {M\({}_{1}\)} and {M\({}_{2}\)} in three-dimensional space, the transformation matrix (1) is a 4 \(\times\) 4 matrix, which can be represented as the combination \(H\) of rotation \(R\) and translation \(T\). \[H= \begin{bmatrix}R&T\\ 0&1\end{bmatrix} = \begin{bmatrix}r_{11}&r_{12}&r_{13}&t_{x}\\ r_{21}&r_{22}&r_{23}&t_{y}\\ r_{31}&r_{32}&r_{33}&t_{z}\\ 0&0&0&1\end{bmatrix}  \tag{1}\]  
In the Equation (1), the submatrix composed of R represents the rotation matrix, and \(T(t_{x},t_{y},t_{z})\) represents the translation vector. The coordinates of point p in {\(M_{1}\)}, \((x_{1}\,,y_{1}\,,z_{1})\) are transformed into homogeneous coordinates \((x_{2}\,,y_{2}\,,z_{2},1)\), thereby obtaining the coordinates of point p in {\(M_{2}\)}, \((x_{2}\,,y_{2}\,,z_{2})\). \[ \begin{bmatrix}x_{2}\\ y_{2}\\ z_{2}\\ 1\end{bmatrix} = \begin{bmatrix}r_{11}&r_{12}&r_{13}&t_{x}\\ r_{21}&r_{22}&r_{23}&t_{y}\\ r_{31}&r_{32}&r_{33}&t_{z}\\ 0&0&0&1\end{bmatrix}\begin{bmatrix}x_{1}\\ y_{1}\\ z_{1}\\ 1\end{bmatrix}  \tag{2}\]  
The following coordinate system transformation relationships are expressed as Euclidean transformation matrices: BHCl represents the transformation from the base coordinate system {B} to the calibration object coordinate system {Cl}. CHT represents the transformation from the laser sensor coordinate system {C} to the TCP coordinate system {T}.