## 2 Spatial Similarity Analysis and Sampling Method of Robot Errors in Configuration Space
### Experimental Analysis of Robot Error Spatial Similarity in Configuration Space
As \(Dq\) increases, \(De\) also increases significantly, with a more dispersed distribution, indicating that, when the joint angles differ greatly in configuration space, the positioning errors also differ significantly. Meanwhile, with the robot unloaded and the pose fixed, the spatial correlation of robot errors in Cartesian space can be represented by the distribution of \(De\) on \(Dp\), as shown in Equation (7), and the scatter plot of \(De\) on \(Dp\) is illustrated in Figure 4b. As \(Dp\) increases gradually, \(De\) also increases from small values, with the distribution becoming more dispersed. In the comparison between Figure 4a,b, it can be observed that, with the robot unloaded and pose fixed, the spatial similarity of robot errors is significant in both configuration space and Cartesian space, consistent with the trends shown in Equations (6) and (9). For the scenario where the robot is unloaded and the pose varies, the scatter plot of \(De\) on \(Dq\) is shown in Figure 5a. When \(Dq\) is small, \(De\) is also small and the distribution is tightly clustered. As \(Dq\) increases, \(De\) also increases significantly, and the distribution becomes more dispersed. The scatter plot of \(De\) on \(Dp\) is shown in Figure 5b, showing that changes in \(Dp\) have a smaller impact on \(De\), and the distribution of \(De\) remains uniform but dispersed. Clearly, when the robot is unloaded and the pose varies, the spatial similarity of robot errors remains significant in configuration space but is not as apparent in Cartesian space. Figure 4: Scatter plots of the positioning error distance on the spatial distances from the experimental result with the robot unloaded and the pose fixed, 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. Figure 5: Scatter plots of the positioning error distance on the spatial distances from the experimental result with the robot unloaded 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. For the scenario where the robot is under a 120 kg load and the pose is fixed, the scatter plot of _De_ on _Dq_ is shown in Figure 6a, and the scatter plot of _De_ on _Dp_ is shown in Figure 6b. As _Dq_ or _Dp_ gradually increases, _De_ shows a gradual increase, with a distribution that is more dispersed. Clearly, under these conditions, the spatial similarity of robot positioning errors is significant in both configuration space and Cartesian space. However, it is also observed that, compared to the scenario where the robot unloaded and the pose fixed, when the robot is under a heavy load, _De_ is higher at the same distances, indicating a noticeable decrease in the robot accuracy. For the scenario where the robot is under a 120 kg load and the pose varies, the scatter plot of _De_ on _Dq_ is shown in Figure 7a. As _Dq_ gradually increases, _De_ also increases significantly, and the distribution becomes more dispersed. The scatter plot of _De_ on _Dp_ is shown in Figure 7b, showing that changes in _Dp_ make little impact on _De_, and the distribution is uniform and dispersed, indicating that changes in _Dp_ have a minimal effect on _De_. Clearly, when the robot has a heavy load and the pose varies, significant error spatial similarity can still be observed in configuration space, while this characteristic almost disappears in Cartesian space. Figure 6: Scatter plots of the positioning error distance on the spatial distances from the experimental result with the robot of under a 120 kg load and the pose fixed, 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.