## 2 Spatial Similarity Analysis and Sampling Method of Robot Errors in Configuration Space
### Spatial Distribution Patterns of Robot Errors in Configuration Space  
In robot configuration space, the theoretical position \(\mathbf{P}\) of the Tool Center Point (TCP) for any joint angles \(\mathbf{Q}\) can be solved using the robot kinematic model as shown in Equation (3), denoted as \(\mathbf{P}=f\) (\(\mathbf{Q}\)). The robot positioning error \(\mathbf{e}\) is defined as the inconsistency  
Figure 1: Definition of the base coordinate frame and tool coordinate frame of the robot. Figure 2: Parameterization method for a single robot joint based on screw motion principles. between the theoretical position and the actual position of the TCP, and the distribution function of robot errors in configuration space is defined as  
\[\mathbf{e}=E(\mathbf{Q}) \tag{4}\]  
According to the theoretical analysis and experimental validation by [PERSON] et al. [18; 31], for two reachable points of the robot, whether in Cartesian space or configuration space, the smaller the distance between the two coordinates, the more similar the robot errors. This phenomenon is referred to as the spatial similarity of robot errors. Let \(\mathbf{Q}_{1}\) be the joint angles where the TCP's theoretical position is \(\mathbf{P}_{1}\) and the corresponding positioning error is \(\mathbf{e}_{1}\) and \(\mathbf{Q}_{2}\) be the joint angles where the TCP's theoretical position is \(\mathbf{P}_{2}\) with an error of \(\mathbf{e}_{2}\). The distance between the joint angles in the configuration space is defined as \(Dq=\|\mathbf{Q}_{1}-\mathbf{Q}_{2}\|\), the Euclidean distance between the TCP positions is \(Dp=\|\mathbf{P}_{1}-\mathbf{P}_{2}\|\), and the Euclidean distance between the positioning errors is \(De=\|\mathbf{e}_{1}-\mathbf{e}_{2}\|\). In configuration space, the spatial correlation function of robot errors is defined as  
\[De=g(Dq) \tag{5}\]  
To reveal the spatial similarity of robot errors, a definition of the spatial correlation function in Equation (5) is given as  
\[\frac{d(De)}{d(Dq)}=\frac{\
abla\|E(\mathbf{Q}_{1})-E(\mathbf{Q}_{2})\|}{\
abla\|\mathbf{ Q}_{1}-\mathbf{Q}_{2}\|}=\frac{De}{Dq}\;\frac{(E(\mathbf{Q}_{1})-E(\mathbf{Q}_{2}))(dE(\mathbf{Q}_ {1})-dE(\mathbf{Q}_{2}))}{(\mathbf{Q}_{1}-\mathbf{Q}_{2})(d\mathbf{Q}_{1}-d\mathbf{Q}_{2})} \tag{6}\]  
Due to the continuous differential manifold of robot positioning errors in the configuration space [30], the error distribution function will be sufficiently smooth and exhibit local monotonic properties similar to those in Euclidean space when the distance \(Dp\) between \(\mathbf{Q}_{1}\) and \(\mathbf{Q}_{2}\) is sufficiently small. At this point, the differential function of the spatial correlation shown in Equation (6) is always positive, which means that the robot errors exhibit significant spatial similarity within any local region in the configuration space. Meanwhile, in Cartesian space, the spatial correlation function of robot positioning errors is defined as  
\[De=h(Dp) \tag{7}\]  
And a definition of the spatial correlation function in Cartesian space as shown in Equation (7) is given as  
\[\frac{d(De)}{d(Dp)}=\frac{\
abla\|\mathbf{e}_{1}-\mathbf{e}_{2}\|}{\
abla\|\mathbf{P}_{1 }-\mathbf{P}_{2}\|} \tag{8}\]  
Although any reachable joint angles of a six-DOF robot correspond to a unique TCP position according to the kinematic model shown in Equation (3), solving for the joint angles corresponding to any given TCP position is a multi-solution problem due to the redundant degrees of freedom. Only when the robot end-effector orientation is fixed does the TCP position correspond to a unique set of joint angles, and only under this condition can Equation (8) be expressed as  
\[\frac{d(De)}{d(Dp)}=\frac{\
abla\|\mathbf{e}_{1}-\mathbf{e}_{2}\|}{\
abla\|\mathbf{P}_{1 }-\mathbf{P}_{2}\|}=\frac{\
abla\|E(\mathbf{Q}_{1})-E(\mathbf{Q}_{2})\|}{\
abla\|f(\mathbf{Q} _{1})-f(\mathbf{Q}_{2})\|} \tag{9}\]  
At this point, in Cartesian space, the distribution pattern of robot errors shows significant spatial similarity.