## 3 Results and Analysis
### Error Compensation Experiments  
According to the results of the numerical simulation, the machining process of the HLTST is optimized and the error experimental platform is built based on the optimized table, as shown in Figure 10, in which the hydraulic cylinder is a custom-made product with a built-in displacement sensor, whose range is 50 mm, and the measurement accuracy is 0.01 mm; the table is installed with the Rion RNA826T inclination, with a measurement range of \(\pm 15^{\circ}\), a resolution of 0.001\({}^{\circ}\), and a measurement accuracy of 0.02\({}^{\circ}\). In order to verify the coupling error compensation model, the error compensation model of the table was first solved through parameter identification experiments. Subsequently, this model was employed to control the table's movement to various orientations, allowing for the calculation of pitch and roll angle deviations. In the absence of a work-bench load, and with \(g_{5}=0\) and \(\hat{x}_{4}=0\) set, the pitch layer and roll layer were separately controlled to perform a reciprocating motion at specific angles, and the strokes of the hydraulic cylinders \(s_{4}\) and \(s_{5}\) were recorded. The experimental results are presented in Tables 3 and 4. \begin{table}
\begin{tabular}{c c c c} \hline
**Rotation Layer** & \(\delta\)**(mm)** & \(\mu_{E}\) & \(\sigma_{E}\) \\ \hline \multirow{6}{*}{Pitch layer} & \(-0.03\) & \(0.0067\%\) & \(0.0577\) \\  & \(-0.02\) & \(0.0045\%\) & \(0.0384\) \\  & \(-0.01\) & \(0.0022\%\) & \(0.0192\) \\  & \(0.01\) & \(-0.0022\%\) & \(0.0192\) \\  & \(0.02\) & \(-0.0045\%\) & \(0.0385\) \\  & \(0.03\) & \(-0.0067\%\) & \(0.0678\) \\ \hline \multirow{6}{*}{Roll layer} & \(-0.03\) & \(0.0121\%\) & \(0.0518\) \\  & \(-0.02\) & \(0.0081\%\) & \(0.0345\) \\ \cline{1-1}  & \(-0.01\) & \(0.004\%\) & \(0.0172\) \\ \cline{1-1}  & \(0.01\) & \(-0.004\%\) & \(0.0172\) \\ \cline{1-1}  & \(0.02\) & \(-0.0081\%\) & \(0.0344\) \\ \cline{1-1}  & \(0.03\) & \(0.0121\%\) & \(0.0516\) \\ \hline \end{tabular}
\end{table} 
Table 2: Volatility analysis of independent errors. Figure 9: (**a**) Heat map of pitch angle coupling errors; (**b**) heat map of roll angle coupling errors. In the MATLAB R2020a fitting toolbox, Equations (12) and (13) were used as the fitting functions to fit the nonlinear least squares curves [32] to the data points in Tables 3 and 4 to solve for the undetermined coefficients \(\Delta r_{4}\) and \(\Delta r_{5}\). The nonlinear model was first developed:  
\[y=f(\mathbf{x},\mathbf{\beta})+\varepsilon \tag{24}\]  
where \(\mathbf{x}=(x_{1},x_{2},\cdots,x_{k})\) is the multivariate input vector, \(\mathbf{\beta}=\left(\beta_{1},\beta_{2},\cdots,\beta_{p}\right)\) is the parameter to be estimated, \(y\) is the observation, \(\varepsilon\) is the fitting error, and the objective of the optimization is to minimize the sum of squared residuals (SSE):  
\[SSE(\mathbf{\beta})=\sum_{i=1}^{n}[y_{i}-f(\mathbf{x}_{i},\mathbf{\beta})]^{2} \tag{25}\]  
Figure 10: Error experimental platform. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Experiment** & **Targeted** & **Targeted** & **Actual** & **Actual** & **Error in \(\hat{\boldsymbol{x}}_{4}\)** & \(\boldsymbol{s}_{4}\)**/(mm)** \\
**No.** & \(\hat{\boldsymbol{x}}_{4}\)**/(rad)** & \(\hat{y}_{g}\)**/(rad)** & \(\hat{\boldsymbol{x}}_{4}\)**/(rad)** & \(\hat{y}_{g}\)**/(rad)** & **Error in \(\hat{\boldsymbol{x}}_{4}\)** & \(\boldsymbol{s}_{4}\)**/(mm)** \\ \hline
1 & 0 & 0 & 0 & 0 & \(-0.0003\pi\) & 0 & 180.021 \\  
& 0 & 0 & \(-0.0003\pi\) & 0 & 180.017 \\ \hline \end{tabular}
\end{table} 
Table 3: Data of pitch layer error parameter identification experiments. The optimal parameter estimates \(\hat{\beta}\) were obtained based on the Levenberg-Marquardt algorithm, and Jacobi matrix \(\mathbf{J}\) was calculated according to Equation (26).