## 3 Kinematic Analysis  
Forward kinematics indicate that the input \(\theta_{1}\) may determine the end-effector's final position and the orientation of the end effector. In this scenario, the novel hybrid manipulator with 9-DOF (\(\theta_{1}\)... \(\theta_{9}\)) is characterized as having a serial manipulator with six joint angles and a parallel manipulator with three joint angles. Where \(\theta_{1}\), \(\theta_{2}\), \(\theta_{3}\), \(\theta_{4}\), \(\theta_{5}\), \(\theta_{6}\) link \(O_{0}O_{1}\), \(O_{1}O_{2}\), \(O_{2}O_{3}\), \(O_{3}O_{4}\), \(O_{4}O_{5}\), \(O_{5}O_{6}\), respectively, \(\theta_{7}\), \(\theta_{8}\), and \(\theta_{9}\) link \(O_{6}O_{7}\), as illustrated in Figure 5.  
The new forward kinematics of the hybrid manipulator can be solved by (1). This describes the transformation matrix between the frame \(O_{0}\) and frame \(O_{7}\). \[T_{7}^{0}=T_{1}^{0}\ T_{2}^{1}\ T_{3}^{2}\ T_{4}^{3}\ T_{5}^{4}\ T_{6}^{5}\ T_{7}^{6} \tag{1}\]  
Therefore, the position and orientation transformation matrix between \(O_{0}\) and \(O_{6}\) can be determined using the Denavit-Hartenberg (D-H) parameter and can be obtained as in Table 2. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Link \(i\)**} & **Link Twist** & **Link Length** & **Joint Offset** & **Joint Angle** \\  & \(\mathbf{\alpha_{i}}\) **(Degree)** & \(\mathbf{a_{i}}\) **(mm)** & \(\mathbf{d_{i}}\) **(mm)** & \(\mathbf{\theta_{i}}\) **(Degree)** \\ \hline
**0-1** & 90 & -N & M & \(\theta_{1}\) \\
**1-2** & 0 & -H & 0 & \(\theta_{2}+90\) \\
**2-3** & \(-90\) & -I & 0 & \(\theta_{3}\) \\
**3-4** & 90 & 0 & -G & \(\theta_{4}\) \\
**4-5** & \(-90\) & 0 & 0 & \(\theta_{5}\) \\
**5-6** & 180 & 0 & -L & \(\theta_{6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: KUKA KR6 R900 (Serial Manipulator) D-H Parameters. Figure 5: Kinematic Diagram for the New Serial-Parallel Manipulator. Therefore, \(T_{1}^{0}\), \(T_{1}^{1}\), \(T_{2}^{2}\), \(T_{3}^{2}\), \(T_{4}^{3}\), \(T_{5}^{4}\), and \(T_{6}^{5}\) can be obtained by (2)-(8), where \(C\) and \(S\) represent \(Cos\) and \(Sine\). \[T_{1}^{0}= \begin{bmatrix}C\theta_{1}&0&S\theta_{1}&-N.C\theta_{1}\\ S\theta_{1}&0&-C\theta_{1}&-N.S\theta_{1}\\ 0&1&0&M\\ 0&0&0&1\end{bmatrix}  \tag{2}\]  
\[T_{2}^{1}= \begin{bmatrix}-S\theta_{2}&-S\theta_{2}&0&H.S\theta_{2}\\ C\theta_{2}&C\theta_{2}&0&-H.C\theta_{2}\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix}  \tag{3}\]  
\[T_{3}^{2}= \begin{bmatrix}C\theta_{3}&0&-S\theta_{3}&-I.C\theta_{3}\\ S\theta_{3}&0&C\theta_{3}&-I.5\theta_{3}\\ 0&-1&0&0\\ 0&0&0&1\end{bmatrix}  \tag{4}\]  
\[T_{4}^{3}= \begin{bmatrix}C\theta_{4}&0&S\theta_{4}&0\\ S\theta_{4}&0&-C\theta_{4}&0\\ 0&1&0&-G\\ 0&0&0&1\end{bmatrix}  \tag{5}\]  
\[T_{5}^{4}= \begin{bmatrix}C\theta_{5}&0&-S\theta_{5}&0\\ S\theta_{5}&0&C\theta_{5}&0\\ 0&-1&0&0\\ 0&0&0&1\end{bmatrix}  \tag{6}\]  
\[T_{6}^{5}= \begin{bmatrix}C\theta_{6}&S\theta_{6}&0&0\\ S\theta_{6}&-C\theta_{6}&0&0\\ 0&0&-1&-L\\ 0&0&0&1\end{bmatrix}  \tag{7}\]  
While (8) shows the transformation matrix between \(O_{6}\) and \(O_{7}\), where \(u_{x},~{}u_{y},~{}u_{z},~{}v_{x},~{}v_{y},~{}v_{z},~{}w_{x},~{}w_{y},~{}\) and \(w_{z}\) are for orientation and \(O_{7,x},~{}O_{7,y},~{}\) and \(O_{7,z}\) are for the position:  
\[T_{7}^{6}= \begin{bmatrix}u_{x}&v_{x}&w_{x}&O_{7,x}\\ u_{y}&v_{y}&w_{y}&O_{7,y}\\ u_{z}&v_{z}&w_{z}&O_{7,z}\\ 0&0&0&1\end{bmatrix}  \tag{8}\]  
The transformation between the moving platform and the base can be described by a position vector: \(P=\overline{O_{6}O_{7}}\) and a \(3\times 3\) rotation matrix \(R_{B}^{A}\), which describes the orientation of the moving platform with coordinate system B, with respect to the base with coordinate system A, as in Equation (9). \[R_{B}^{A}= \begin{bmatrix}u_{x}&v_{x}&w_{x}\\ u_{y}&v_{y}&w_{y}\\ u_{z}&v_{z}&w_{z}\end{bmatrix}