## 2 Spatial Similarity Analysis and Sampling Method of Robot Errors in Configuration Space
### Experimental Analysis of Robot Error Spatial Similarity in Configuration Space
Figure 7: Scatter plots of the positioning error distance on the spatial distances from the experimental result with the robot under 120 kg load and the pose varied, including dotted lines to indicate the error distribution range. (**a**) shows the distribution of _De_ on _Dq_ in configuration space, and (**b**) shows the _De_ on _Dp_ in Cartesian space. To quantitatively analyze the distribution characteristics of robot error similarity in Cartesian space and configuration space, the correlation between error similarity and spatial distance for different measurement samples is calculated utilizing the Pearson Correlation Coefficient (PCC), which is a statistical method to analyze the correlation between continuous variables [33]. The PCC between the error similarity and spatial distance for a set of points can be calculated as follows [34]:  
\[\text{PCC}=\frac{cov(\{De\},\{D\})}{std(\{De\})\cdot std(\{D\})},\text{PCC}\in[ -1,1] \tag{10}\]  
where \(\{De\}\) is the set of error distances between any two points in the measurement set, and \(\{D\}\) represents the set of distances between any two points. When analyzing the error spatial similarity in configuration space, each element in \(\{D\}\) is the \(Dq\) between every two sampling points, while, in Cartesian space, each element in \(\{D\}\) is the \(Dp\) of sampling points. When the PCC between two sets of variables is positive, a higher PCC value indicates a stronger positive correlation; when the PCC value approaches 0, the correlation between the two variables sets is almost non-existent. The PCC values between the positioning errors \(De\) and the joint angle distance \(Dq\) for different robot load and pose conditions in configuration space are shown in Table 1. When the robot is unloaded, the PCC between the \(De\) and \(Dq\) remains above 0.4, indicating a significant spatial similarity of positioning errors, regardless of whether the pose varies or not. When the robot is under a load of 120 kg, the PCC between \(De\) and \(Dq\) decreases but still remains above 0.4. Evidently, in configuration space, the robot error spatial similarity is pronounced and stable, is unaffected by pose variations, and only slightly decreases with the increase in load. The PCC values between the positioning errors \(De\) and the joint angle distance \(Dp\) for different robot load and pose conditions in Cartesian space are shown in Table 2. When the end-effector pose is fixed, the PCC between the \(De\) and \(Dp\) is 0.44 while the robot is unloaded, and it decreases to 0.381 when the robot is loaded with 120 kg. However, when the robot pose varies, the PCC between the \(De\) and \(Dp\) rapidly decreases to below 0.226, regardless of whether the end-effector is loaded or not. Evidently, in Cartesian space, when the end-effector pose is fixed, the spatial similarity of robot errors remains relatively significant, although it decreases with the load; however, when the pose varies, this characteristic significantly drops to an insignificant level, regardless of the load condition. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Load Status** & **Pose Variation Status** & **PCC** & \(p\)**-Value** \\ \hline
0 kg & Pose fixed & 0.469 & 1.236 \(\times\)\(10^{-22}\) \\ \hline
120 kg & Pose fixed & 0.424 & 3.498 \(\times\)\(10^{-12}\) \\ \hline
0 kg & Pose varied & 0.472 & 3.671 \(\times\)\(10^{-26}\) \\ \hline
120 kg & Pose varied & 0.403 & 1.261 \(\times\)\(10^{-26}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: PCC between the error similarity and distance among sampling points in configuration space. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Load Status** & **Pose Variation Status** & **PCC** & \(p\)**-Value** \\ \hline
0 kg & Pose fixed & 0.440 & 1.435 \(\times\)\(10^{-14}\) \\ \hline
120 kg & Pose fixed & 0.381 & 1.854 \(\times\)\(10^{-12}\) \\ \hline
0 kg & Pose varied & 0.214 & 1.021 \(\times\)\(10^{-12}\) \\ \hline
120 kg & Pose varied & 0.226 & 1.414 \(\times\)\(10^{-14}\) \\ \hline \hline \end{tabular}
\end{table}