## Appendix A  
Let there be any two vectors \(\mathbf{a}\),\(\mathbf{b}\) in the space, where the vector \(\mathbf{a}\) translates the distance \(I\) along the p axis, and then rotates \(\theta\) to become the vector \(\mathbf{b}\). Define the dual angle notation, \(\hat{\mathbf{\theta}}=\theta+\varepsilon\mathbf{I}\), which relates an arbitrary 3D spatial line \(\mathbf{a}\) to a given 3D spatial line \(\mathbf{b}\) by a rotation \(\theta\) about a unique axis and with a translation \(\mathbf{I}\) along the same axis. Here, \(\mathbf{r}_{\mathbf{a}}\mathbf{r}_{\mathbf{b}}\) represent the vectors from any arbitrary point in space, denoted as \(\mathbf{O}\), to points \(\mathbf{a}\) and \(\mathbf{b}\), respectively. The symbols \(\mathbf{a}^{\prime}\), \(\mathbf{b}^{\prime}\) signify the cross products. The cross products \(\mathbf{a}^{\prime}\) and \(\mathbf{b}^{\prime}\) can be represented by \(\mathbf{a}\) and \(\mathbf{b}\) and their corresponding \(\mathbf{r}_{\mathbf{a}}\) and \(\mathbf{r}_{\mathbf{b}}\), as shown in Equation (A1). \[\mathbf{a}^{\prime}=\mathbf{r}_{\mathbf{a}}\times\mathbf{a},\mathbf{b}^{\prime}=\mathbf{r}_{\mathbf{b}} \times\mathbf{b}\] (A1)  
We can write the vectors \(\mathrm{a}\) and \(\mathrm{b}\) representing line segments \(0A\) and \(0B\) in Plucker form as \(\hat{\mathbf{a}}\) and \(\hat{\mathbf{b}}\), respectively. \[\hat{\mathbf{a}}=\mathbf{a}+\varepsilon(\mathbf{r}_{\mathbf{a}}\times\mathbf{a}),\hat{\mathbf{b}}=\bm {b}+\varepsilon(\mathbf{r}_{\mathbf{b}}\times\mathbf{b})\] (A2) \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Model** & **Optimization Algorithm** & **Actual Position** & **Iterations** & **Time(s)** \\ \hline \multirow{4}{*}{DH} & Genetic algorithm & [369.8147, 345.8322, 702.9025] & 25 & 3.43 \\  & Simulated annealing algorithm & [369.8150, 345.8317, 702.9021] & 5500 & 62.15 \\  & Numerical solution Algorithm & [369.8143, 345.8315, 702.9017] & 32 & 1.15 \\  & Particle Swarm Optimization & [369.8125, 345.8306, 702.9003] & 225 & 3.36 \\ \hline \multirow{4}{*}{DH Taylor expansion} & Genetic algorithm & [372.2827, 347.7579, 706.3873] & 400 & 4.50 \\  & Simulated annealing algorithm & [369.8165, 345.8329, 702.9044] & 7700 & 52.30 \\  & Numerical solution Algorithm & [369.8146, 345.8319, 702.9018] & 31 & 0.41 \\  & Particle Swarm Optimization & [369.8101, 345.8311, 702.8985] & 380 & 4.60 \\ \hline \multirow{4}{*}{dual quaternions} & Genetic algorithm & [369.8161, 345.8352, 702.9070] & 350 & 2.39 \\  & Simulated annealing algorithm & [369.8118, 345.8293, 702.8986] & 3900 & 28.50 \\ \cline{1-1}  & Numerical solution Algorithm & [369.8146, 345.8319, 702.9018] & 32 & 0.45 \\ \cline{1-1}  & Particle Swarm Optimization & [366.5362, 345.7464, 701.7661] & 85 & 1.18 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of calculation time and quantity under different models. Decomposing the vector \(\mathbf{b}\) along the orthogonal directions of \(\mathbf{a}\) and \(\mathbf{p}\times\mathbf{a}\) gives Equation (A3). \[\mathbf{b}=\cos\theta\mathbf{a}+\sin\theta(\mathbf{p}\times\mathbf{a})\] (A3)  
Considering the relationship depicted in the figure, where \(\mathbf{r_{b}}=\mathbf{r_{a}}+\mathbf{l}\), and combining it with the above equation, we can simplify it to Equation (A4). \[\hat{\mathbf{b}}=(\cos\theta-\varepsilon\text{lsin }\theta)(\mathbf{a}+\varepsilon\mathbf{r_{a} }\times\mathbf{a})+(\sin\theta+\varepsilon\text{lcos }\theta)\{[\mathbf{p}+\varepsilon(\mathbf{r_{a}}\times\mathbf{p})]\times\mathbf{a}\}\] (A4)  
The inverse Taylor series transformation is utilized for the change process, as shown in Equation (A5), which ultimately simplifies Equation (A6).