# Performance Analysis of Hybrid Kinematic Mechanism for Fusion Reactor Maintenance
## 2 HKM Kinematic Analysis
### HKM Forward Kinematics  
The HKM parallel mechanism forward kinematics is solved by using both numerical and analytical methods. The analytical method mainly uses the elimination method to remove the unknown factors from the constraint equations of the mechanism and obtain a higher-order equation containing only the input and output parameters. This method can obtain all solutions of the mechanism, but the disadvantage is that the elimination process is very complicated, requires high computational accuracy, and is not universally applicable. Therefore, this paper uses the numerical method to solve the forward kinematics of the parallel mechanism. As shown in Figure 2, the nonlinear constraint equations for the parallel mechanism link length \(l_{i}\) are as follows:  
\[ \begin{cases}f_{1}(\alpha,\beta,z)=l_{1}^{2}-(\overrightarrow{\mathbf{P}\mathbf{O}}+ \overrightarrow{\mathbf{O}\mathbf{B}_{1}}-\overrightarrow{\mathbf{P}\mathbf{A}_{1}})^{T}( \overrightarrow{\mathbf{P}\mathbf{O}}+\overrightarrow{\mathbf{O}\mathbf{B}_{1}}-\overrightarrow {\mathbf{P}\mathbf{A}_{1}})\\ f_{2}(\alpha,\beta,z)=l_{2}^{2}-(\overrightarrow{\mathbf{P}\mathbf{O}}+ \overrightarrow{\mathbf{O}\mathbf{B}_{2}}-\overrightarrow{\mathbf{P}\mathbf{A}_{2}})^{T}( \overrightarrow{\mathbf{P}\mathbf{O}}+\overrightarrow{\mathbf{O}\mathbf{B}_{2}}- \overrightarrow{\mathbf{P}\mathbf{A}_{2}}),\\ f_{3}(\alpha,\beta,z)=l_{3}^{2}-(\overrightarrow{\mathbf{P}\mathbf{O}}+ \overrightarrow{\mathbf{O}\mathbf{B}_{3}}-\overrightarrow{\mathbf{P}\mathbf{A}_{3}})^{T}( \overrightarrow{\mathbf{P}\mathbf{O}}+\overrightarrow{\mathbf{O}\mathbf{B}_{3}}- \overrightarrow{\mathbf{P}\mathbf{A}_{3}})\end{cases}  \tag{1}\]  
where \(\alpha\), \(\beta\) are the rotation angles of the kinetic platform around the \(x_{1}\) axis and \(y_{1}\) axis of point \(\mathbf{P}\), respectively, and \(z\) is the displacement of the follower prismatic chain. \begin{table}
\begin{tabular}{c c c} \hline \hline
**No.** & **Variable** & **Value** \\ \hline
1 & \(l_{PAi}\) & 1000 mm \\
2 & \(l_{A1B1}\) & 2510.68 mm \\
3 & \(l_{A2B2}\) & 3032.78 mm \\
4 & \(l_{A3B3}\) & 2814.66 mm \\
5 & \(l_{OBi}\) & 475 mm \\
6 & \(l_{OL}\) & 1800.00 mm \\
7 & \(l_{LM}\) & 1250.00 mm \\
8 & \(l_{MN}\) & 1025.00 mm \\
9 & \(\delta_{4}\) & 5.5\({}^{\circ}\) \\
10 & \(\delta_{5}\) & 8.56\({}^{\circ}\) \\
11 & \(\delta_{6}\) & 15.72\({}^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The structure design parameters of the HKM. Figure 2: HKM structural features (**a**); HKM parallel mechanism coordinate system (**b**); HKM series mechanism coordinate system (**c**). The coordinates of the point \(O\) are the following:  
\[O=Rot(y,\beta)Rot(x,\alpha)Trans(0,\ 0,\ z), \tag{2}\]  
where \(Rot\) is the rotation matrix and \(Trans\) is the translation matrix. Taking \(X= \begin{bmatrix}\alpha&\beta&z\end{bmatrix} ^{T}\), the following result is obtained after simplification. \[f_{i}(X)=l_{i}^{2}-(PO+OB_{i}-PA_{i})^{T}(PO+OB_{i}-PA_{i}). \tag{3}\]  
Let the theoretical solution of this system of equations be \(X^{*}= \begin{bmatrix}\alpha^{*}&\beta^{*}&z^{*}\end{bmatrix} ^{T}\), then \(X^{*}=X_{k}+\gamma\), \(X_{k}= \begin{bmatrix}\alpha_{k}&\beta_{k}&z_{k}\end{bmatrix} ^{T}\) is the initial value of this nonlinear system of equations, and \(\gamma= \begin{bmatrix}\gamma_{1}&\gamma_{2}&\gamma_{3}\end{bmatrix} ^{T}\) is the error vector. Expanding the Equation (3) at \(X_{k}\) according to the Taylor series expansion, neglecting the higher-order partial derivatives and letting \(f_{i}(X_{k}+\gamma)\) converge to zero, we obtain  
\[f_{i}(X_{k}+\gamma)=f(X_{k})+\frac{\partial f_{i}}{\partial\alpha}\gamma_{1}+ \frac{\partial f_{i}}{\partial\beta}\gamma_{2}+\frac{\partial f_{i}}{\partial z }\gamma_{3}. \tag{4}\]  
When \(f_{i}(X_{k}+\gamma)=f(X^{*})=0\), a linear system of equations is obtained. \[ \begin{bmatrix}\frac{\partial f_{i}}{\partial X_{i}}\end{bmatrix} [\gamma]=-[f( X_{i})]. \tag{5}\]  
Its expanded form is  
\[