## 4 Experiments
### Coordinate Transformation Error Analysis  
To assess the accuracy of coordinate transformation from image space to the robotic workspace, a structured translation error test was conducted. Specifically, the end-effector of the manipulator was commanded to move along the \(x\) and \(y\) directions by fixed distances (50 mm and 60 mm, respectively). The positions of a selected rebar intersection point were recorded before and after each movement using the vision system. The measured displacement was then compared against the expected ground-truth motion to evaluate the spatial consistency of the transformation pipeline.  
This method simulates a known geometric displacement on a fixed local feature, effectively constructing an artificial ground truth, which is a widely adopted strategy in 3D vision system accuracy assessment. To further validate the transformation precision, a high-accuracy probe was mounted on the robotic arm and used to physically touch the center of each rebar intersection point. As shown in Figure 14, the actual end-effector coordinates recorded during probing are considered as the true reference positions in the robot coordinate frame.  
Figure 13: Calibration result analysis: (**a**) Reprojection error distribution of all calibration images; (**b**) Bar chart of the mean reprojection error for each calibration image. The error is measured in pixels and reflects the accuracy of camera calibration. Most images exhibit mean errors below 0.1 pixel, indicating high calibration precision and consistency; (**c**) Rotation estimation error (in radians) for each calibration image. The error is derived from the standard deviation of the optimized rotation vector, indicating the uncertainty in camera orientation estimation. As shown in the figure, most errors remain below 0.0025 radians, demonstrating high precision and stability in the pose calibration process; (**d**) Translation estimation error (in millimeters) for each calibration image. The error is computed from the standard deviation of the optimized translation vectors, indicating the uncertainty in position estimation. As shown in the figure, most errors lie between 0.6 mm and 0.8 mm with moderate variation, demonstrating reliable and consistent spatial localization accuracy in the calibration process.  
Following coordinate transformation, sensor-based alignment verification was performed to evaluate the consistency between the vision-derived coordinates and the robotic control system. By comparing the positions of the same intersection point across both coordinate systems before and after the predefined motion, the transformation error could be quantitatively assessed. This step is especially critical for precision tasks such as rebar binding, where millimeter-level accuracy is required.  
The experimental results are summarized in Table 3, which presents representative sample points along with their corresponding displacement errors. Statistically, the average absolute error along the \(x\)-axis is 0.757 mm, with a relative error of 1.514%, while, along  
Figure 14: Probe-based ground-truth acquisition system for coordinate transformation validation.  
the \(y\)-axis, the average absolute error is 0.914 mm, with a relative error of 1.523%. Most samples exhibit displacement errors of around 1 mm with a uniform distribution, indicating that the transformation from image coordinates to the robotic coordinate system is both accurate and robust, meeting the requirements for high-precision spatial localization. The \(z\)-coordinate in the camera frame is directly obtained from the depth value measured by the RGB-D sensor. By using the native depth measurement instead of computing the \(z\)-coordinate through coordinate transformation, this approach effectively minimizes projection errors and enhances the overall accuracy of 3D localization, particularly in the depth direction.