## 5 Calibration Experiment and Algorithm Comparison
### Measurement System Uncertainty Analysis  
1. Vision Positioning Measurement Uncertainty  
The uncertainty in visual positioning is composed of both the camera calibration uncertainty and the hand-eye calibration uncertainty.  
Camera Calibration Uncertainty:  
The camera calibration uncertainty can be calculated as follows: the pixel error of the camera is \(\mu_{B_{1}}\) = \(\pm\)0.02 mm; average measurement error of circular points on the calibration board is \(\mu_{B_{2}}\) = \(\pm\)0.03 mm; the re-projection error obtained from calibrating the camera using 20 sets of checkerboard calibration patterns is 0.12 pixels, which translates to \(\mu_{A_{1}}\) = 0.1198 mm based on focal length and pixel size. The combined uncertainty from the camera calibration process can be expressed as  
\[\mu_{1}=\sqrt{\mu_{A_{1}}^{2}+\mu_{B_{1}}^{2}+\mu_{B_{2}}^{2}}=0.1251\text{ mm} \tag{11}\]  
Hand-eye calibration uncertainty:  
Figure 12: Calibration accuracy verification experiment using a ruby probe.  
In the hand-eye calibration model, the error of the hand-eye calibration algorithm for determining \(X\) can be represented by the deviation between each calculated Xi and the true X. This deviation is expressed as the distance between the origins of the i camera coordinate systems under the same robot end-effector coordinate frame. The maximum deviation observed is 0.5235 mm, while the minimum deviation is 0.0062 mm. Therefore, the uncertainty of the hand-eye calibration algorithm is taken as \(\mu_{A_{2}}\) = 0.5235 mm. Additionally, the robot end-effector repeatability error is \(\mu_{B_{3}}\) = 0.025 mm. Combining these factors, the calibration uncertainty of the X can be determined.  
\[\mu_{2}=\sqrt{\mu_{1}^{2}+\mu_{A_{2}}^{2}+\mu_{B_{3}}^{2}}=0.5388\text{mm} \tag{12}\]  
(2) Contact Calibration Uncertainty  
In the contact probe calibration model, the error in determining the TCP (Tool Center Point) is represented by the average residual value after the convergence of the LM-D algorithm. This is calculated as the average of the accumulated values obtained by substituting the convergence results into the objective function. The maximum residual value from the experimental data is used as the reference, resulting in a calibration algorithm uncertainty of \(\mu_{A_{3}}\) = 0.0838 mm.  
The contact probe calibration relies on the robot's absolute accuracy. After absolute accuracy calibration, the absolute positioning error of the robot end effector is \(\mu_{B_{4}}\) = 0.1 mm, and the intrinsic error of the contact probe itself is \(\mu_{B_{5}}\) = 0.001 mm. Combining these factors, the calibration uncertainty of the contact probe can be determined.  
\[\mu_{3}=\sqrt{\mu_{A_{3}}^{2}+\mu_{B_{4}}^{2}+\mu_{B_{5}}^{2}}=0.1305\text{mm} \tag{13}\]  
In summary, the calibration uncertainty of the visual measurement system is 0.5388 mm, while the calibration uncertainty of the contact measurement system is 0.1305 mm. The uncertainty of the visual measurement system is greater than that of the contact probe. Therefore, the combined calibration method proposed in this paper plays an important role in improving the positioning accuracy of parts.