# Performance Analysis of Hybrid Kinematic Mechanism for Fusion Reactor Maintenance
## 2 HKM Kinematic Analysis
### HKM Inverse Kinematics
\begin{aligned} \Phi_{1}&=p_{x}+a_{x}l_{MN}+l_{ LM}(a_{x}c\delta_{6}+n_{x}s\delta_{6})-(l_{OL}+\mathbf{z})(o_{x}s\delta_{5}-c \delta_{5}(a_{x}c\delta_{6}+n_{x}s\delta_{6}))\\ \[PERSON]_{2}&=p_{y}+a_{y}l_{MN}+l_{LM}(a_{y}c\delta_{6}+n_{y}s\delta_{6})-(l_{ OL}+\mathbf{z})(o_{y}s\delta_{5}-c\delta_{5}(a_{y}c\delta_{6}+n_{y}s\delta_{6})) \\ \[PERSON]_{3}&=p_{z}+a_{y}l_{MN}+l_{LM}(a_{z}c\delta_{6}+n_{z}s\delta_{6})-(l_{ OL}+\mathbf{z})(o_{z}s\delta_{5}-c\delta_{5}(a_{z}c\delta_{6}+n_{z}s\delta_{6})) \end{aligned} \right. \tag{18}\]  
The numerical and analytical methods can be used to solve the nonlinear equation system with three unknowns. Since we do not need the full analysis solution of the whole mechanism, we used the numerical method to solve the inverse kinematic solution of a simplified HKM. The numerical solution of Equation (18) was solved using the  
Figure 3: HKM simplified as a series mechanism. Newton-Raphson iterative method. The iterative format for solving the nonlinear system of equations by [PERSON]'s method is shown as follows:  
\[x_{k+1}=x_{k}+[J(x_{k})]^{-1}f(x_{k}),\ k=0,1,2\cdots, \tag{19}\]  
\[[\mathbf{J}(\mathbf{x})]= \begin{bmatrix}\frac{\partial f_{1}(x)}{\partial\delta_{5}}& \frac{\partial f_{1}(x)}{\partial\delta_{6}}&\frac{\partial f_{1}(x)}{\partial \delta_{2}}\\ \frac{\partial f_{2}(x)}{\partial\delta_{5}}&\frac{\partial f_{2}(x)}{\partial \delta_{6}}&\frac{\partial f_{2}(x)}{\partial\delta_{2}}\\ \frac{\partial f_{3}(x)}{\partial\delta_{5}}&\frac{\partial f_{3}(x)}{\partial \delta_{6}}&\frac{\partial f_{3}(x)}{\partial\delta_{2}}\end{bmatrix} . \tag{20}\]  
By iterative calculation of Equation (19), we can obtain the partial solution of the HKM inverse kinematics \([\delta_{5},\delta_{6},z]\). The rest of the solutions \([\alpha,\beta,\delta_{4}]\) we calculated by the analytical method. Any point \(\mathbf{U}=[x_{U},0,0,1]^{T}\) on the \(x_{4}\) axis can be chosen within the coordinate system of point \(\mathbf{O}\) in Figure 3. The kinematic equations of the point U are established from the points \(\mathbf{P}\) and \(\mathbf{N}\), respectively, based on the DH method. From the equality of the coordinates of the point \(\mathbf{U}\) in the forward (from point \(\mathbf{P}\)) and reverse (from point \(\mathbf{N}\)) kinematic equations, we can obtain:  
\[\mathbf{U_{p}}=\mathbf{D_{12}}(\beta)\mathbf{D_{23}}(\alpha)\mathbf{D_{34}}(z)\bm {D_{54}}(\delta_{4})\ \mathbf{U}\] \[= \begin{bmatrix}x_{U}(\wp\wp\delta_{4}+s\alpha s\beta s\delta_{4 })+z\alpha s\beta\\ x_{U}\alpha s\delta_{4}-(z-l_{OL})s\alpha\\ -x_{U}(\wp\wp\delta_{4}-s\alpha c\beta s\delta_{4})+z\alpha c\wp\\ 1\end{bmatrix} , \tag{21}\]  
\[\mathbf{U_{n}}=\mathbf{D_{N}}\mathbf{D_{76}}(\delta_{6})\mathbf{D_{65}}(\delta_{5})\mathbf{U}. \tag{22}\]  
The parameters \([\delta_{5},\delta_{6},z]\) in the coordinate equation of the point \(\mathbf{U}\) are known. By letting \(x_{U}=0\), the coordinates of the HKM forward kinematics \(\mathbf{U_{p}}\) and inverse kinematics \(\mathbf{U_{n}}\) at the point \(\mathbf{U}\) are equal, and we can obtain  
\[\alpha=-arcs\bigg{(}\frac{\mathbf{U_{n}}(2,1)}{z-l_{OL}}\bigg{)}, \tag{23}\]  
\[\beta=arcs\bigg{(}\frac{\mathbf{U_{n}}(1,1)}{z\alpha}\bigg{)}. \tag{24}\]  
Letting \(x_{U}\
eq 0\), we can obtain  
\[\delta_{4}=arcs\bigg{(}\frac{\mathbf{U_{n}}(2,1)+(z-l_{OL})s\alpha}{x_{U}c\alpha }\bigg{)}. \tag{25}\]  
Third, we calculated the HKM parallel mechanism inverse kinematic solution. Through the above calculations, we obtain the inverse kinematic solution \([\alpha,\beta,z,\delta_{4},\delta_{5},\delta_{6}]\) of the HKM equivalent to the series mechanism. The inverse kinematic solution of the HKM parallel mechanism is the process of solving for the stroke \((l_{1},l_{2},l_{3})\) of each drive piston, by giving the position vector \((\alpha,\beta,z)\) of the end-effector.