## 3 Kinematic Analysis
### Inverse Kinematics Analysis of the 3-DOF Upper-limb Exoskeleton  
Given an arbitrary end-effector position of the 5-DOF upper-limb exoskeleton in the workspace, the corresponding joint angles of the 3-DOF upper-limb exoskeleton can be calculated using inverse kinematics. In case the obtained angles are in the motion range of the corresponding joint, the 3-DOF upper-limb exoskeleton can reach the position of the 5-DOF upper-limb exoskeleton in the workspace, i.e., the given position is in the workspace of the 3-DOF upper-limb exoskeleton.  
According to the previous results, the kinematics of the 3-DOF upper-limb exoskeleton are analyzed with the Denavit-Hartenberg parameters (DH), where 0 is the initial coordinate, as shown in Figure 3. The DH parameters are listed in Table 6.  
The extrema of \(\theta_{1}^{3}\), \(\theta_{2}^{3}\) and \(\theta_{3}^{3}\) obtained by inverse kinematics are listed in Table 7[26]. The range of listed extremum \(\theta_{3}^{3}\) shown in Table 7 is \(5^{\circ}\) greater than the range of elbow external-flexion/internal-extension \(\left(\theta_{3}^{3}\right)\) shown in Table 4. The motion range of the elbow joint actuator in the 5-DOF upper-limb exoskeleton reaches \(130^{\circ}\). It is feasible to reference this parameter as the motion range of the elbow external-flexion/internal-extension \(\left(\theta_{3}^{3}\right)\), thus \(\theta_{3}^{3}\) derived from inverse kinematics meets the motion range of the 3-DOF upper-limb exoskeleton.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Links** & \(\mathbf{\alpha}_{i}\) & \(\mathbf{a}_{i}\) & \(\mathbf{\theta}_{i}^{3}\) & \(\mathbf{d}_{i}\) \\ \hline
1 & 0 & 0 & \(\theta_{1}^{3}\) & 0 \\
2 & \(-90^{\circ}\) & \(a_{2}\) & \(\theta_{2}^{3}\) & 0 \\
3 & 0 & \(a_{3}\) & \(\theta_{3}^{3}\) & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 6: DH parameters of the 3-DOF upper-limb exoskeleton.  
Figure 7: The workspace of the right arm of the 5-DOF upper-limb exoskeleton (mm): (**a**) Isometric view; and (**b**) right view.  
Based on the motion range of the 3-DOF upper-limb exoskeleton in Table 4, the workspace of the 5-DOF upper-limb is divided into two parts: Part of the workspace that the 3-DOF upper-limb exoskeleton can reach called valid workspace; and part of the workspace that the 3-DOF upper-limb exoskeleton cannot reach called invalid workspace. The inverse kinematics workspace consists of valid workspace and invalid workspace (Figure 8). The area consisting of the blue points is the valid workspace, and the area consisting of the red points is the invalid workspace. Here, the x-axis is in the same direction as the exoskeleton, and point (0,0,0) is the center of shoulder flexion/extension.  
A total of 5001 points are placed in the inverse kinematics workspace, with 4621 points placed in the valid workspace, accounting for approximately 92.4%, and 380 points placed in the invalid workspace, accounting for approximately 7.6%. The invalid workspace is located behind the coronal plane of the shoulder joint. The end-effector of the 5-DOF upper-limb exoskeleton will not reach this area, while the valid workspace is the actual workspace of the 5-DOF upper-limb exoskeleton when working. The existence of invalid workspace does not affect the accessibility of the 3-DOF upper-limb exoskeleton, i.e., the 3-DOF upper-limb exoskeleton can reach all points in the actual workspace of the 5-DOF upper-limb exoskeleton.