# Performance Analysis of Hybrid Kinematic Mechanism for Fusion Reactor Maintenance
## 2 HKM Kinematic Analysis
### HKM Forward Kinematics
\begin{bmatrix}\frac{\partial f_{i}}{\partial\alpha}&\frac{\partial f_{i}}{ \partial\beta}&\frac{\partial f_{i}}{\partial z}\\ \frac{\partial f_{i}}{\partial\alpha}&\frac{\partial f_{i}}{\partial\beta}& \frac{\partial f_{i}}{\partial z}\\ \frac{\partial f_{i}}{\partial\alpha}&\frac{\partial f_{i}}{\partial\beta}& \frac{\partial f_{i}}{\partial z}\\ \end{bmatrix}\begin{bmatrix}\gamma_{1}\\ \gamma_{2}\\ \gamma_{3}\end{bmatrix} =- \begin{bmatrix}f_{1}\\ f_{2}\\ f_{3}\end{bmatrix} . \tag{6}\]  
The forward kinematics of this parallel mechanism can be solved by using the Newton-Raphson iterative method [36]. The constraint equations for the kinetic platform \([\alpha,\beta,z]\) with respect to the static platform and the rod length \(l_{i}\) can be obtained as  
\[ \begin{cases}f_{1}(\alpha,\beta,z,l_{1})=0\\ f_{2}(\alpha,\beta,z,l_{2})=0.\\ f_{3}(\alpha,\beta,z,l_{3})=0\end{cases}  \tag{7}\]  
Solving the forward kinematics of the parallel mechanism is the process of finding the kinetic platform position \([\alpha,\beta,z]\) by giving the rod length \(l_{i}\). Thus, when \(l_{i}\) is taken as a known quantity, the system of equations can be abbreviated as  
\[\mathbf{f}(\mathbf{x})=0, \tag{8}\]  
where \(\mathbf{x}=[\alpha,\beta,z]^{T}\). \[\mathbf{f}(\mathbf{x})=[f_{1}(x),f_{2}(x),f_{3}(x)]^{T}. \tag{9}\]  
The iterative format for solving the nonlinear system of equations by [PERSON]'s method is as follows:  
\[\mathbf{x}_{k+1}=\mathbf{x}_{k}+[J(\mathbf{x}_{k})]^{-1}\mathbf{f}(\mathbf{x}_{k}),\ k=0,1,2\cdots, \tag{10}\]  
\[\mathbf{J}(\mathbf{x})= \begin{bmatrix}\frac{\partial f_{1}(x)}{\partial\alpha}& \frac{\partial f_{1}(x)}{\partial\beta}&\frac{\partial f_{1}(x)}{\partial z}\\ \frac{\partial f_{2}(x)}{\partial\alpha}&\frac{\partial f_{2}(x)}{\partial \beta}&\frac{\partial f_{2}(x)}{\partial z}\\ \frac{\partial f_{3}(x)}{\partial\alpha}&\frac{\partial f_{3}(x)}{\partial \beta}&\frac{\partial f_{3}(x)}{\partial z}\\ \end{bmatrix} . \tag{11}\]When the static platform of the parallel mechanism is locked in the position of the vertical ports of the vacuum vessel, the whole HKM can be equated to a 6-DOF series robot based on the forward kinematic solution \([\alpha,\beta,z]\). In order to represent the relationship between the motion of each joint, linkage and the end-effector, we use the Denavit-Hartenberg (DH) parameter method to establish the HKM forward kinematic model. The equivalent HKM kinematics equation established by DH parameters is as follows:  
\[\mathbf{T}_{6}^{0}= \begin{bmatrix}n_{x}&\alpha_{x}&a_{x}&p_{x}\\ n_{y}&\alpha_{y}&a_{y}&p_{y}\\ n_{z}&\alpha_{z}&a_{z}&p_{z}\\ 0&0&0&1\end{bmatrix} =\mathbf{T}_{1}^{0}(\beta)\mathbf{T}_{2}^{1}(\alpha)\mathbf{T}_{3}^{2}(z )\mathbf{T}_{4}^{3}(\delta_{4})\cdots\mathbf{T}_{6}^{5}(\delta_{6}), \tag{12}\]  
where \([\mathbf{n},\mathbf{o},\mathbf{a}]\) represents the end orientation of the HKM, \(\big{[}p_{x},p_{y},p_{z}\big{]}\) represents the end position, and \(\mathbf{T}_{i}^{t-1}\) represents the coordinate transformation of adjacent joints. Based on the coordinate system in Figure 2, the DH parameters for the HKM equivalent to a series robot are shown in Table 2.