## 3 Results and Analysis
### Error Compensation Experiments
\[\mathbf{J}_{ij}=\left.\frac{\partial f(\mathbf{x}_{i},\mathbf{\beta})}{\partial \hat{\beta}_{j}}\right|_{\mathbf{\beta}=\hat{\beta}} \tag{26}\]  
The sensitivity matrix \(\mathbf{S}\) is defined as  
\[\mathbf{S}=\left(\mathbf{J}^{\mathrm{T}}\mathbf{J}\right)^{-1}\mathbf{J}^{ \mathrm{T}} \tag{27}\]  
The effect of the perturbation of the observed data on the parameters to be estimated can be expressed as follows:  
\[\Delta\beta\approx\mathbf{S}\cdot\Delta\mathbf{y} \tag{28}\]  
Taking \(R^{2}\geq 0.99\) as the accuracy index, the fitting leads to \(\overline{\Delta r_{4}}=-0.25165\) and \(\overline{\Delta r_{5}}=-0.65232\), with corresponding SSE values of 0.0104 and 0.0216. The matrix \(\mathbf{J}\) was derived using Equations (12) and (13), and the value of \(\mathbf{J}^{\mathrm{T}}\mathbf{J}\) was subsequently calculated as follows:  
\[\mathbf{J}^{\mathrm{T}}\mathbf{J}= \begin{bmatrix}0.0475&0.0002\\ 0.0002&0.0453\end{bmatrix}  \tag{29}\]  
Equation (28) shows that matrix \(\mathbf{J}^{\mathrm{T}}\mathbf{J}\) is non-singular, indicating that the model parameters are less sensitive to data perturbations, and thus the error compensation model after parameter identification can be expressed as  
\[s_{4}=\sqrt{180729.5-179119.5\cos(\hat{x}_{4}+\arctan(0.0026\cos\hat{y}_{5})+ 0.1894\pi)} \tag{30}\]  
\[s_{5}=\sqrt{187117.4-184584.7\cos(\hat{y}_{5}+\arctan(0.0019\cos\hat{x}_{4})+ 0.1550\pi)} \tag{31}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Experiment** & **Targeted** & **Targeted** & **Actual** & **Actual** & **Error in \(\hat{y}_{5}\)** & **\(s_{5}\)/(mm)** \\
**No. ** & \(\hat{x}_{d}\)**/(rad)** & \(\hat{y}_{g}\)**/(rad)** & \(\hat{x}_{d}\)**/(rad)** & \(\hat{y}_{g}\)**/(rad)** & **Error in \(\hat{y}_{5}\)** & **\(s_{5}\)/(mm)** \\ \hline
1 & 0 & 0 & \(-0.0001\pi\) & 0 & 0 & 155.387 \\  
The error compensation model was deployed to regulate the movement of the unloaded table, aligning it with the attitude set via the uniform test design method. Subsequent to each movement, the angular error was recorded and the pitch and roll layers returned to their original positions before each movement. The variations in pitch and roll angles before and after compensation under no-load conditions are shown in Figure 11. The reductions in pitch and roll angle errors after the error compensation of the HLTST at no load and a rated load (2.5 t) are shown in Figure 12.  
As demonstrated in Figure 11, the error compensation method achieved significant improvements in angular measurement accuracy. Specifically, the mean pitch angle error was reduced from 3.27% to \(-\)0.10%, while the mean roll angle error showed a similar improvement, decreasing from 0.64% to \(-\)0.22%. Notably, the maximum pitch angle deviation exhibited a substantial reduction from 10.71% to \(-\)1.78%, and the roll angle's maximum error dropped from 3.60% to \(-\)3.57%. In addition, the root mean square error (RMSE) of pitch and roll angles were reduced by 84.52% and 23.56%, respectively, after the implementation of error compensation. Figure 11: (**a**) Error compensation for pitch angle under no load; (**b**) error compensation for roll angle under no load. Figure 12: (**a**) Error experiment under no load; (**b**) error experiment under rated load (2.5 t). Figure 12 demonstrates that while the error compensation model exhibits slight performance degradation under rated load conditions compared with no-load operation, it remains effective in enhancing the HLTST's positioning accuracy. Quantitative analysis revealed that the mean pitch angle error reduction decreased from 2.77% to 2.27% under the rated load, while the roll angle improvement maintained stability with mean error reduction changing from 0.36% to 0.35%. In addition, the maximum positioning errors of the HLTST in the pitch and roll angles under heavy-load conditions are 4.91% and \(-\)3.99%, respectively, which satisfy the positioning accuracy requirements.