## 3 Results and Analysis
### Numerical Simulation and Process Optimization  
In order to analyze the distribution characteristics, sensitivity, and volatility of the geometric errors of the pitch and roll angles, numerical simulations of the independent and coupling errors were carried out based on Equations (9), (10), (20), and (21) using MATLAB R2020a on a PC with a 3.2 GHz CPU. The results of the effect of contour error \(\delta\) on the independent errors of pitch and roll angles at different strokes of the hydraulic cylinder are shown in Figure 8.  
The error heat map shows that the absolute value of the angular independence error \(\left|\,E\,\right|\) is symmetrically distributed up and down with \(\delta\) = 0 as the symmetry axis. Within the normal range of motion of the hydraulic cylinder, the absolute value of the pitch angle independence error \(\left|\,E_{x}\,\right|\) decreases with an increasing depression angle and increases with an increasing elevation angle. The absolute value of the roll angle independence error \(\left|\,E_{y}\,\right|\) increases with an increasing right roll angle and decreases with an increasing left roll angle. In Figure 8a, the independent error \(E_{x}\) exhibits two distinct boundaries when the hydraulic cylinder stroke \(s_{4}\) at the pitch level approaches 180 mm. These boundaries display an antisymmetric distribution on their left and right sides. The horizontal coordinate corresponding to this boundary represents the initial travel after error correction. When \(\delta_{4}>0\), the initial travel is shifted 1 mm to the right; conversely, it is shifted 1 mm to the left. Similarly, the error distribution of the roll layer also exhibits such characteristics. When \(\delta\) lies between \(-0.5\) mm and \(0.5\) mm, the extreme values of \(E_{x}\) and \(E_{y}\) are 4.28% and 4.04% and the extreme deviations are 8.68% and 7.90%, respectively. The error trends in the  
Figure 7: Linear fitting of pitch layer and roll layer forward kinematics model. heat map illustrate that the roll angle is slightly more sensitive to contour errors than the pitch angle. To ensure the precise manufacturing of equipment, it is imperative to establish strict tolerance limits for parts during the machining process design. Assuming that the absolute value of the error \(\left|\,E\,\right|\leq 0.26\%\) for both pitch and roll angles across any stroke, the contour errors must adhere to the following relationships:  
\[ \begin{cases}-0.03\;\text{mm}\leq\delta_{4}\leq 0.03\;\text{mm}\\ -0.03\;\text{mm}\leq\delta_{5}\leq 0.03\;\text{mm}\end{cases}  \tag{22}\]  
In order to investigate the fluctuation of the independent error \(E\) in the stroke range of the hydraulic cylinder [31], the standard deviation of the independent error \(E\) corresponding to the different face profile errors \(\delta\) is calculated according to Equation (23). \[\sigma_{E}=\sqrt{\frac{1}{N}\sum_{i=1}^{N}(E_{i}-\mu_{E})^{2}} \tag{23}\]  
where \(\sigma_{\text{E}}\) is the standard deviation, \(E_{i}\) is the sample value, \(\mu_{\text{E}}\) is the mean value, \(N\) is the total number of samples, and the calculation results are shown in Table 2. The magnitude of \(\sigma_{\text{E}}\) shows that when \(\delta\in[-0.01,\,0.01]\) mm, the independent errors \(E\) in the pitch and roll angles are less volatile. Therefore, in order to reduce the size and volatility of the errors in the pitch and roll angles, the dimensional deviation of the radius of the circular arc slot should be controlled within 0.01 mm as far as possible while balancing the processing difficulty and cost. The results of the influence of the difference \(\Delta r\) of the contour error of the pitch and roll layers on the coupling error \(\varepsilon\) at different strokes of the hydraulic cylinder are shown in Figure 9.  
The heat map shows that the coupling error of the pitch and roll angles increases in parallel with the increase in the contour error difference \(\Delta r\).