## 3 Overall System
### Coordinate Transformation System
 \tag{11}\]  
The camera coordinates are extended to homogeneous coordinates by adding a fourth component of \(1\), resulting in \((X_{C},Y_{C},Z_{C},1)\). As shown in Equation (12), this can be transformed into the three-dimensional point \((X_{e},Y_{e},Z_{e},1)\) in the robotic arm's end-effector coordinate system using the matrix \(T_{CE}\). \[ \begin{bmatrix}X_{e}\\ Y_{e}\\ Z_{e}\\ 1\end{bmatrix} =T_{CE} \begin{bmatrix}X_{e}\\ Y_{e}\\ Z_{e}\\ 1\end{bmatrix}  \tag{12}\]  
To convert the camera coordinates to the robotic arm's base coordinate system, we also need to consider the transformation between the end-effector and the base. Let the transformation matrix from the end-effector coordinate system to the base coordinate system be \(T_{EB}\), which is typically obtained from the kinematics of the robotic arm. Given the point \((X_{e},Y_{e},Z_{e},1)\) in the end-effector coordinate system, as shown in Equation (13), it can be transformed into the three-dimensional point \((X_{b},Y_{b},Z_{b},1)\) in the base coordinate system using \(T_{EB}\). \[ \begin{bmatrix}X_{b}\\ Y_{b}\\ Z_{b}\\ 1\end{bmatrix} =T_{EB} \begin{bmatrix}X_{e}\\ Y_{e}\\ Z_{e}\\ 1\end{bmatrix}  \tag{13}\]  
Therefore, the transformation from the camera coordinate system to the base coordinate system can be expressed as Equation (14). \[ \begin{bmatrix}X_{b}\\ Y_{b}\\ Z_{b}\\ 1\end{bmatrix} =T_{EB}\cdot T_{CE}\cdot \begin{bmatrix}X_{c}\\ Y_{c}\\ Z_{c}\\ 1\end{bmatrix}  \tag{14}\]  
The transformation of pixel coordinates of intersection points to the robotic arm's base coordinate system may be affected by errors resulting from inaccuracies in hand-eye calibration and kinematic calibration of the robot. These errors are mainly caused by the limitations of calibration algorithms, measurement inaccuracies in the robot's joints and links, flexible deformations, measurement noise, and environmental fluctuations. Inaccurate hand-eye calibration can lead to misalignment between the camera and the robotic arm coordinate systems, while kinematic calibration errors can further magnify these deviations, ultimately compromising the end-effector's positioning accuracy [17]. Therefore, it is essential to provide the robotic arm with precise three-dimensional coordinates of the intersection points. In this study, a chessboard calibration plate and a probe are used to implement a chessboard-based coordinate transformation method that converts pixel coordinates into the robotic arm's coordinate system. This method simplifies the coordinate transformation by aligning the origin and axes of the robotic arm's coordinate system with those of the chessboard calibration plate, effectively making the two coordinate systems coincide. This alignment offers several advantages: it reduces computational complexity and avoids unnecessary coordinate conversions, thereby minimizing error accumulation; it enhances operational accuracy, improving system performance in high-precision tasks; and it streamlines the calibration and verification process, making the system more intuitive and efficient. This approach is particularly well-suited for industrial scenarios that demand high precision and repeatability. First, the origin of the robotic arm coordinate system is aligned with that of the calibration plate using a probe. The robotic arm is positioned so that its coordinate origin coincides with the predefined location of the calibration plate placed at the end-effectorprobe, ensuring that both origins match as closely as possible. Next, the coordinate axes (x, y, z) of both systems are aligned to ensure directional consistency. Once the origins and axes are properly aligned, external parameter calibration is performed using a chessboard pattern to obtain the extrinsic matrix, as shown in Equation (15). The resulting intrinsic and extrinsic matrices are presented in Figure 11. The extrinsic matrix is then used to transform intersection point coordinates from the camera coordinate system to the chessboard coordinate system, as described in Equation (16).