## Appendix C  
The coordinate system [1] relative to [F] is obtained by rotating the coordinate system along the axis \(l_{z1}\) by an angle \(\delta\).  
Then, \(q_{F1,r}=\left[\cos\ \frac{\delta}{2},I_{z1}\text{sin }\frac{\delta}{2}\right]\) and \(q_{F1,d}=\frac{1}{2}\big{(}p_{i}-q_{i}\otimes p_{i}\otimes q_{i}^{*}\big{)} \otimes q_{i}\), as shown in Equation (A14), the dual quaternion representation.  
\[\hat{q}_{F1}=\left[\cos\ \frac{\delta}{2}\ 0\ 0\sin\ \frac{\delta}{2}\ 0\ 0\ 0\ 0\right]\] (A14)  
The coordinate system [2] is translated relative to coordinate system [1] along the axis \(I_{x1}\), given by \([\rho\cos\ \delta\ \rho\sin\ \delta\ 0]\).  
We have \(\mathbf{q}_{12,r}=[1,0,0,0]^{\rm T}\) and \(\mathbf{q}_{12,d}=[0,\rho\text{cos }\delta,\rho\text{sin }\delta,0]\). Thus, the dual quaternion representation \(\hat{\mathbf{q}}_{12}\) is given by Equation (A15).  
\[\hat{\mathbf{q}}_{F1}=\left[\text{cos }\frac{\delta}{2}\;0\;0\;\text{sin }\frac{\delta}{2}\;0\;0\;0\;0\right]\] (A15)  
The coordinate system [3] relative to [2] is obtained by rotating the coordinate system along the axis \(\mathbf{l}_{\mathbf{y}1}\) by an angle \(\varphi\).  
Then, \(\mathbf{q}_{23,r}=\left[\text{cos }\frac{\varphi}{2},\mathbf{l}_{\mathbf{y}1}\;\text{sin }\frac{\varphi}{2}\right]\) and \(\mathbf{q}_{23,d}=\left[0,0,0,\rho\text{sin }\frac{\varphi}{2}\right]\), as shown in Equation (A16), the dual quaternion representation.  
\[\hat{\mathbf{q}}_{23}=\left[\text{cos }\frac{\varphi}{2},-\text{sin }\delta\text{sin }\frac{\varphi}{2},\text{cos }\delta\text{sin }\frac{\varphi}{2},0,0,0,\frac{\rho}{2}\text{sin }\frac{\delta}{2}\text{sin }\frac{\varphi}{2}\right]\] (A16)  
The coordinate system [4] is translated relative to coordinate system [3] along the axis \(\mathbf{l}_{\mathbf{3}3}\), given by \(\left[\rho\;\text{cos }\delta,\rho\;\text{sin }\delta,0\right]\).  
We have \(\mathbf{q}_{34,r}=\left[1,0,0,0\right]^{\rm T}\) and \(\mathbf{q}_{34,d}=\left[0,-\rho\;\text{cos }\delta\text{cos }\varphi,-\rho\text{sin }\delta\text{sin }\varphi,\rho\text{sin }\varphi\right]\). Thus, the dual quaternion representation \(\hat{\mathbf{q}}_{34}\) is given by Equation (A17).  
\[\hat{\mathbf{q}}_{34}=\left[1,0,0,0,0,-\frac{\rho}{2}\text{cos }\delta\text{cos }\varphi,-\frac{\rho}{2}\text{sin }\delta\text{cos }\varphi,\frac{\rho}{2}\text{sin }\varphi\right]\] (A17)  
The coordinate system [E] relative to [4] is obtained by rotating the coordinate system along the axis \(\mathbf{l}_{\mathbf{z}4}\) by an angle \(-\delta\).  
Then, \(\mathbf{q}_{4E,r}=\left[\text{cos }\left(\frac{-\delta}{2}\right),\mathbf{l}_{\mathbf{z}4} \text{sin }\left(\frac{-\delta}{2}\right)\right]\) and \(\mathbf{q}_{4E,d}=\left[0,-\rho\text{sin }\delta(\text{cos }\varphi-1)\;\text{sin }\frac{\delta}{2},\right.\)  
\(\rho\;\text{cos }\delta(\text{cos }\varphi-1)\text{sin }\frac{\delta}{2},0\right]\), as shown in Equation (A18), the dual quaternion representation.  
\[\hat{\mathbf{q}}_{4E}=\left[\text{cos }\frac{\delta}{2},-\text{cos }\delta\text{sin }\varphi\;\text{sin }\frac{\delta}{2},-\text{sin }\delta\text{sin }\varphi\;\text{sin }\frac{\delta}{2},-\text{cos }\varphi\;\text{sin }\frac{\delta}{2},0,-\rho\;\text{sin }\delta\text{sin }\frac{\delta}{2}(\text{cos }\varphi-1),\right.\] (A18) \[\left.\rho\;\text{cos }\delta\text{sin }\frac{\delta}{2}(\text{cos }\varphi-1),0\right]\]  
By concatenating the coordinate transformations described in the above equations, we can obtain the transformation dual quaternion between the two coordinate systems, represented as Equation (A19).  
\[\hat{\mathbf{q}}_{EF}=\left[\text{cos }\frac{\varphi}{2},-\text{sin }\delta\text{sin }\frac{\varphi}{2},\text{cos }\delta\text{sin }\frac{\varphi}{2},0,0,0,\rho\cdot\text{sin }\frac{\varphi}{2}\right]\] (A19)