## 2 Methods
### Error Similarity Analysis with TCP Calibration Uncertainties
\begin{bmatrix}\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\  
where \(\theta_{\ell,k}\) and \(\Delta\theta_{k}\) are the elements of \(\theta_{\ell}\) and \(\Delta\theta\) respectively. Similarly, by substituting Equation (9) and Equation (10) into Equation (8), Equation (8) can be rewritten as  
\[\begin{bmatrix}\Delta\mathbf{P}^{\prime}(\theta_{\ell})\\ 1\end{bmatrix}=\left(\prod\limits_{k=1}^{6}\text{Rot}(\theta_{\ell,k},0,0)- \left(\prod\limits_{k=1}^{6}\text{Rot}(\theta_{\ell,k}+\Delta\theta_{k},0,0) \right)\Delta^{W}\,\mathbf{T}_{\text{T}}\right)\cdot\begin{bmatrix}\mathbf{O}_ {3\times 1}\\ 1\end{bmatrix}\ell=i,\,j \tag{12}\]  
By combining Equations (5), (11) and (12), one can obtain  
\[\begin{split}&\gamma^{*}(h)-\hat{\gamma}^{*}(h)=\frac{1}{2 \sum\limits_{k=1}^{n}N(h_{k})}\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{N(h_{i}) }\left\|\left(2\Xi_{i}-\left(\Delta^{W}\mathbf{T}_{\text{T}}+\mathbf{I}_{4\times 4} \right)\Delta\[PERSON]_{i}\right)\left(\left(\Delta^{W}T_{\text{T}}-\mathbf{I}_{4 \times 4}\right)\Delta\[PERSON]_{i}\right)\\ &+\left(2\[PERSON]_{j}-\left(\Delta^{W}\mathbf{T}_{\text{T}}+\mathbf{I}_{4 \times 4}\right)\Delta\[PERSON]_{j}\right)\left(\left(\Delta^{W}\mathbf{T}_{\text{T}}- \mathbf{I}_{4\times 4}\right)\Delta\[PERSON]_{j}\right)\\ &+2\left(\Delta\[PERSON]_{i}\Big{(}\mathbf{I}_{4\times 4}-\Delta^{W}\mathbf{T}_{ \text{T}}\Big{)}\[PERSON]_{j}+\[PERSON]_{i}\[PERSON]_{j}\Big{(}\mathbf{I}_{4\times 4}-\Delta^{W} \mathbf{T}_{\text{T}}\Big{)}\\ &+\left(\Delta\[PERSON]_{i}\Delta^{W}\mathbf{T}_{\text{T}}\Delta\[PERSON]_{j}\Big{)} \left(\Delta^{W}\mathbf{T}_{\text{T}}-\mathbf{I}_{4\times 4}\right)+\Delta\[PERSON]_{i} \Big{(}\Delta^{W}\mathbf{T}_{\text{T}}-\mathbf{I}_{4\times 4}\Big{)}\Delta\Xi_{j}\right) \end{bmatrix}\begin{bmatrix}\mathbf{O}_{3\times 1}\\ 1\end{bmatrix} \lVert_{2}^{2}\end{split} \tag{13}\]  
where  
\[ \begin{split}\Xi_{\ell}=\prod\limits_{k=1}^{6}\text{Rot}(\theta_{ \ell,k},0,0)\\ \Delta\Xi_{\ell}=\prod\limits_{k=1}^{6}\text{Rot}(\theta_{\ell,k}+\Delta \theta_{k},0,0)\end{split} \qquad\qquad\qquad\qquad\qquad\qquad\ell=i, \,j \tag{14}\]  
In Equation (13), \(\gamma^{*}(h)\) and \(\hat{\gamma}^{*}(h)\) represent the situation that the semivariances with and without TCP frame calibration error, respectively. When TCP frame calibration error is ignored,  
\[\Delta^{W}\,\mathbf{T}_{\text{T}}-\mathbf{I}_{4\times 4}=\mathbf{O}_{4\times 4} \tag{15}\]  
However, due to the calibration error, Equation (15) cannot hold strictly, which leads to Equation (13) not being equal to 0. Therefore, the similarity of TCP positioning error is affected by TCP calibration error. According to Equation (8), 500 sets of robot positioning errors are randomly generated, and the random errors of each joint angle of the robot are set to a maximum of \(\pm 0.1^{\circ}\). The calibration error of the TCP frame in each direction is set to a maximum of 0 mm, \(\pm 1\) mm and \(\pm 5\) mm, respectively. The similarity between the positioning errors and the Euclidean distances of the robot joints is shown in Figures 3, 4 and 10. In the figures, the horizontal axis represents the Euclidean distance between corresponding joints in 2 different robot configurations, while the vertical axis represents the deviation of the positioning error between the two corresponding robot configurations. From Figure 3, it can be seen that without considering the TCP frame establishment error, there is a correlation between the positioning errors of any two points in the Cartesian space and the distance in the joint space. That is, the smaller the distance in joint space (the horizontal axis in the figure), the smaller the difference in positioning errors in the Cartesian space (the vertical axis). Therefore, the unbiased optimal estimation can be used to estimate positioning errors in the Cartesian space. However, when TCP frame establishment errors are introduced, joint angle errors are coupled with the establishment error.