## 2 Configuration Design of the Spatial 8R Mechanism Configuration
\begin{array}{c}\underset{6}{\overset{0}{\overset{0}{\right.}}}T_{1}= \begin{bmatrix}n_{x}&r_{x}&t_{x}&p_{x}\\ n_{y}&r_{y}&t_{y}&p_{y}\\ n_{z}&r_{z}&t_{z}&p_{z}\\ 0&0&0&1\end{bmatrix}\end{array}  \tag{6}\]  
where  
\[ \begin{array}{c}n_{x}=c_{1}c_{2}(c_{34}c_{5}c_{6}-s_{34}c_{6}s_{5})-s_{1}(c_ {34}c_{6}s_{5}+s_{34}c_{5}c_{6})+c_{1}s_{2}s_{6}\\ n_{y}=c_{2}s_{6}-(c_{34}c_{5}c_{6}-s_{34}c_{6}s_{5})\\ n_{z}=-c_{1}(c_{34}c_{6}s_{5}+s_{34}c_{5}c_{6})-c_{2}s_{1}(c_{34}c_{5}c_{6}-s_{ 34}c_{6}s_{5})-s_{1}s_{2}s_{6}\\ r_{x}=s_{1}(c_{34}s_{5}s_{6}+s_{34}c_{5}s_{6})-c_{1}c_{2}(c_{34}c_{5}s_{6}-s_{ 34}c_{5}s_{6})+c_{1}c_{6}s_{2}\\ r_{y}=s_{2}(c_{34}c_{5}s_{6}-s_{34}s_{5}s_{6})+c_{2}c_{6}\\ r_{z}=c_{1}(c_{34}s_{5}s_{6}+s_{34}c_{5}s_{6})+c_{2}s_{1}(c_{34}c_{5}s_{6}-s_{ 34}s_{5}s_{6})-c_{6}s_{1}s_{2}\\ t_{x}=s_{1}(c_{34}c_{5}-s_{34}s_{5})+c_{1}c_{2}(c_{34}s_{5}+s_{34}c_{5})\\ t_{y}=-s_{2}(c_{34}s_{5}+s_{34}c_{5})\\ t_{z}=c_{1}(c_{34}c_{5}-s_{34}s_{5})-c_{2}s_{1}(c_{34}s_{5}+s_{34}c_{5})\\ p_{x}=g-s_{1}[s_{34}(m+l_{2}s_{5})+hs_{3}-l_{2}c_{34}c_{5}]-l_{1}s_{1}+c_{1}c _{2}[c_{34}(m+l_{2}s_{5})+hc_{3}+l_{2}s_{34}c_{5}]\\ p_{y}=-s_{2}[c_{34}(m+l_{2}s_{5})+hc_{3}+l_{2}s_{34}c_{5}]\\ p_{z}=-c_{1}[s_{34}(m+l_{2}s_{5})+hs_{3}-l_{2}c_{34}c_{5}]-l_{1}c_{1}-c_{2}s_ {1}[c_{34}(m+l_{2}s_{5})+hc_{3}+l_{2}s_{34}c_{5}]\\ \end{array} \]  
Equation (6) is the pose of the terminal link coordinate system R\({}_{26}\)-\(x_{26}\)\(z_{26}\) relative to the static coordinate system \(O_{b}\)-\(xyz\), which is the basic equation for kinematic analysis of 6R branch chains. To verify the obtained results, the value of the 6R branch chain transformation matrix was calculated when \(\theta_{1}=\theta_{2}=\theta_{3}=0^{\circ}\), \(\theta_{4}=90^{\circ}\), \(\theta_{5}=90^{\circ}\) and \(\theta_{6}=0^{\circ}\), and the calculation result is  
\[ \begin{array}{c}\underset{6}{\overset{0}{\overset{0}{\right.}}}T_{1}= \begin{bmatrix}1&0&0&g-h\\ 0&1&0&h\\ 0&0&1&l_{2}-m-l_{1}\\ 0&0&0&1\end{bmatrix}\end{array}  \tag{7}\]  
The calculation results show that the obtained \(\underset{6}{\overset{0}{\overset{0}{\right. }}}T_{1}\) is correct. According to the D-H parameters of the 2R branch chain in Table 2 and Equation (3), we can obtain the transformation matrix of each revolute pair of the 2R branch chain. For the 2R branch chain, the transformation matrix of each link is multiplied, and the transformation matrix of the 2R branch chain is:  
\[ \begin{array}{c}\underset{2}{\overset{0}{\overset{0}{\right.}}}T_{2}= \underset{1}{\overset{0}{\overset{0}{\right.}}}T_{2}^{1}T=\begin{bmatrix}c_{7 }c_{8}&-c_{7}s_{8}&s_{7}&k-l_{3}s_{7}\\ s_{8}&c_{8}&0&0\\ -c_{8}s_{7}&s_{7}s_{8}&c_{7}&-c_{7}l_{3}\\ 0&0&0&1\end{bmatrix}\end{array}  \tag{8}\]  
By substituting \(\theta_{7}=\theta_{8}=0^{\circ}\) into \(\underset{2}{\overset{0}{\right. }}T_{2}\), we can verify that it is correct. In conclusion, the basic equations for the kinematic analysis of the 8R mechanism in this space are obtained. The pose matrix of the moving platform is expressed by the \(Z\)-\(Y\)-\(X\) Euler angle (\(\alpha\), \(\beta\), \(\gamma\)):  
\[ \begin{array}{c}\mathbf{T}=\begin{bmatrix}c\alpha c\beta&\alpha s\beta s \gamma-s\alpha c\gamma&\alpha s\beta c\gamma+s\alpha c\gamma&x\\ s\alpha c\beta&s\alpha s\beta s\gamma+c\alpha c\gamma&s\alpha s\beta c\gamma- \alpha s\gamma&y\\ -s\beta&c\beta s\gamma&c\beta c\gamma&z\\ 0&0&0&1\end{bmatrix}\end{array}  \tag{9}\]  
where \(s_{i}\) stands for \(\sin(\theta_{i})\), \(c_{i}\) stands for \(\cos(\theta_{i})\), \(\mathbf{T}\) is the pose matrix, \((x,y,z)\) is the position coordinates of the origin \(O_{m}\) of the moving coordinate system in the static coordinate  
According to the Jacobian condition number of the obtained velocity, when \(\lambda_{1}=1\), the mechanism has a single abduction and adduction movement, indicating that the mechanism has good dexterity in this movement and has completely isotropic kinematics.