## 3 Overall System
### Coordinate Transformation System  
Since we obtain the intersection points in pixel coordinates from the image, to enable the robotic arm to perform the tying task at these intersections, the three-dimensional coordinates of the intersection points in the robot's coordinate system are required. This involves a key problem in robotics known as hand-eye calibration [34]. The purpose of  
Figure 10: When the length spacing is 3 and the width spacing is 1, the total path length obtained by optimizing the greedy algorithm is shorter than the total path length planned by the unoptimized algorithm (with the starting point at the lower-left corner). Figure 9: When the length spacing is 1 and the width spacing is 3, the path planning methods of the two algorithms are the same, and the total path length is identical (with the starting point at the lower-left corner). hand-eye calibration is to determine the spatial relationship between the camera and the robot's end-effector (such as the robotic arm), thus aligning the camera's pixel coordinate system with the robot's working coordinate system. This process is commonly used in robotic vision applications, such as object grasping and visual guidance. In recent years, several optimized hand-eye calibration methods have been proposed based on classical theories. [PERSON] et al. [35] proposed an optimization method based on minimizing re-projection errors. By converting the calibration problem into the minimization of re-projection errors, they achieved a more accurate calibration of the spatial relationship between the camera and the robotic arm. This method is not limited to specific scenarios and is applicable to general hand-eye calibration needs. [PERSON] et al. [36] proposed a globally optimal symbolic hand-eye calibration method. By introducing symbolic computation methods, they developed an algorithm capable of globally optimizing the hand-eye calibration process. This method ensures a global optimal solution during the calibration process, significantly improving calibration accuracy while reducing computational complexity, making it suitable for complex robotic vision systems. The process of converting pixel coordinates to the robotic arm's coordinates typically involves multiple steps of coordinate transformation. Starting from the image plane coordinates, the transformation proceeds through camera calibration, including both intrinsic and extrinsic parameters, and ultimately results in the three-dimensional position in the robotic arm's coordinate system. First, the pixel coordinates \((u,v)\) in the image are converted to the three-dimensional coordinates \((X_{c},Y_{c},Z_{c})\) in the camera coordinate system. This transformation requires the camera's intrinsic matrix \(K\), as given in Equation (9), along with the depth information \(Z_{c}\). \[K= \begin{bmatrix}f_{x}&0&c_{x}\\ 0&f_{y}&c_{y}\\ 0&0&1\end{bmatrix}  \tag{9}\]  
where \(f_{x}\) and \(f_{y}\) are the focal lengths of the camera, and \((c_{x},c_{y})\) are the coordinates of the principal point (the center of the optical axis). Based on the perspective projection model, given the pixel coordinates \((u,v)\) and the depth \(Z_{c}\), the three-dimensional coordinates \((X_{c},Y_{c},Z_{c})\) in the camera coordinate system can be obtained, as shown in Equation (10). \[ \begin{bmatrix}X_{c}\\ Y_{c}\\ Z_{c}\end{bmatrix} =K^{-1} \begin{bmatrix}u\cdot Z_{c}\\ v\cdot Z_{c}\\ Z_{c}\end{bmatrix}  \tag{10}\]  
where \(K^{-1}\) is the inverse of the camera's intrinsic matrix. Through hand-eye calibration, the transformation matrix \(T_{CE}\) between the camera coordinate system and the robotic arm's end-effector coordinate system can be obtained. This transformation matrix is typically acquired through calibration experiments and consists of the rotation matrix \(R_{CE}\) (\(3\times 3\) matrix) and the translation vector \(t_{CE}\) (\(3\times 1\) vector), as shown in Equation (11). \[ \begin{bmatrix}X_{c}\\ Y_{c}\\ Z_{c}\end{bmatrix} =K^{-1} \begin{bmatrix}u\cdot Z_{c}\\ v\cdot Z_{c}\\ Z_{c}\end{bmatrix}