# Performance Analysis of Hybrid Kinematic Mechanism for Fusion Reactor Maintenance
## 2 HKM Kinematic Analysis
### HKM Inverse Kinematics  
The inverse kinematics of the whole HKM are derived using following steps: First, the HKM geometry model is simplified. The HKM is a hybrid structure combining a parallel mechanism and a series mechanism. If both parallel and series mechanisms are considered in the inverse kinematic derivation, they will both determine the position and orientation of the end-effector. The position solution contains 3 independent equations, the orientation solution contains 9 non-independent equations, and 12 equations are solved for six unknowns, leading to a very complex solution process. To solve this problem, we simplify the parallel mechanism of HKM into a series mechanism, and equate the driving functions of the three pistons as the rotation around the \(x\) and \(y\) axes of the point \(\mathbf{P}(x_{1},y_{1},z_{1})\) and the translational motion around the \(z\) axis of the point \(\mathbf{O}(x_{2},y_{2},z_{2})\). The equivalent model is shown in Figure 3. The parallel mechanism is equated to a series mechanism, and its degrees of freedom are decomposed to a rotational motion around point \(\mathbf{P}\)\(y_{1}\) axis and \(x_{2}\) axis, respectively, which constitute the universal joint. The prismatic motion of the kinetic platform is equated to the translational motion around the \(z_{3}\) axis of point \(\mathbf{B}\). After simplification, the whole HKM can be regarded as a 6-DOF series robot. Second, analytical and numerical methods were combined to calculate the HKM inverse kinematics. By analyzing the simplified HKM in Figure 3, we find that joints 1-4 share the same axis, so we can simplify the solution by reducing the number of unknowns in the system of equations by solving in the reverse direction from point \(\mathbf{N}\) to point \(\mathbf{P}\). We assume that the position and orientation of the HKM endpoint \(\mathbf{N}\) is:  
\[\mathbf{D}_{N}= \begin{bmatrix}n_{x}&\alpha_{x}&a_{x}&p_{x}\\ n_{y}&\alpha_{y}&a_{y}&p_{y}\\ n_{z}&\alpha_{z}&a_{z}&p_{z}\\ 0&0&0&1\end{bmatrix} , \tag{13}\] \begin{table}
\begin{tabular}{c c c c} \hline
**Rod.\(i\)** & **Variable** & **Rotation Angle (\({}^{\circ}\))** & **Distance (mm)** \\ \hline
1 & \(\alpha\) & (14.83, 0, 0) & (0, 0, 0) \\
2 & \(\beta\) & (0,10.04, 0) & (0, 0, 0) \\
3 & \(z\) & (0,0) & (0, 0, 2740) \\
4 & \(\delta_{4}\) & (0,0, 5.5) & (0, 0, 0) \\
5 & \(\delta_{5}\) & (0, 8.56, 0) & (0, 0, \(-1800\)) \\
6 & \(\delta_{6}\) & (15.72, 0, 0) & (0, 0, \(-1250\)) \\ \hline \end{tabular}
\end{table} 
Table 2: HKM modified DH parameters. where \((n_{i_{i}},\ o_{i},a_{i})\) denotes the end orientation and \(\left(p_{x},\ p_{y},p_{z}\right)\) denotes the end position. The reverse direction kinematic equation of the HKM is established by the DH method:  
\[\mathbf{P_{P}}=\mathbf{D_{N}}*\mathbf{D_{76}}*\mathbf{D_{65}}*\mathbf{P_{51}}, \tag{14}\]  
\[\mathbf{D_{76}}= \begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&l_{MN}\\ 0&0&0&1\end{bmatrix} * \begin{bmatrix}c\delta_{6}&0&s\delta_{6}&0\\ 0&1&0&0\\ -s\delta_{6}&0&c\delta_{6}&0\\ 0&0&0&1\end{bmatrix} , \tag{15}\]  
\[\mathbf{D_{65}}= \begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&l_{LM}\\ 0&0&0&1\end{bmatrix} * \begin{bmatrix}1&0&0&0\\ 0&c\delta_{5}&-s\delta_{5}&0\\ 0&s\delta_{5}&c\delta_{5}&0\\ 0&0&0&1\end{bmatrix} , \tag{16}\]  
\[\mathbf{P_{51}}= \begin{bmatrix}0\\ 0\\ l_{OL}+\mathbf{z}\\ 1\end{bmatrix} , \tag{17}\]  
where \(c\) denotes \(cos\) and \(s\) denotes \(sin\). This convention will be adopted in future studies as well. Therefore, we simplify the HKM non-independent inverse kinematic equation system to an independent equation system containing only three unknowns \(\mathbf{\Phi}(\delta_{5},\delta_{6},\mathbf{z})\), as follows:  
\[\left\{