# Configuration Design of an Upper Limb Rehabilitation Robot with a Generalized Shoulder Joint
## 6 Comprehensive Kinematic Performance Analysis of the Shoulder Joint Configuration  
### Inverse Kinematic Solution and Flexibility Analysis of Two Configurations  
Referring to the size of the Chinese adult human body (GB/T10000-1988) [36] and combining the physical drive module size of two different configurations, the dimensions to be used are shown in Table 2.  
At the same time, the patient's upper arm lift angle range is set during the rehabilitation robot-assisted movement process as 0\({}^{\circ}\)-150\({}^{\circ}\). The internal rotation/external rotation angle of the upper arm is set to \(\gamma=30^{\circ}\), and the upper arm's natural sag position is set as the lift angle \(\beta=0^{\circ}\). In other lifting planes, this configuration is also selected as the initial lifting angle to determine the initial movement posture in each lifting plane. Combining the size parameters listed in Table 2 and the inverse solution method of human-machine closed-chain kinematics given above, when the upper arm of the human body is lifted in the 0\({}^{\circ}\), 30\({}^{\circ}\), 60\({}^{\circ}\), 90\({}^{\circ}\) and 120\({}^{\circ}\) lifting planes and the relationship between the turning angle \(\theta_{i}^{1}(i=4,5,6)\) of the \(2P_{a}1P3R_{a}\) turning pair \(R_{i}^{1}(i=4,5,6)\) and the lifting angle \(\beta\) are shown in Figure 10b-f. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value/(mm)** & **Parameter** & **Value/(mm)** \\ \hline Scapula belt length \(l_{sg}\) & 150 & Distance \(l_{1}\) from human sternum to rotation pair \(R_{a1}^{2}\) & 300 \\ Distance \(l_{1}\) from \(P_{1}\) to \(P_{2}\) & 200 & Distance from \(R_{a1}^{2}\) to \(R_{a2}^{2}\) & 100 \\ Distance \(l_{2}\) from \(P_{2}\) to \(P_{3}\) & 200 & Length of parallel four-bar mechanism & 150 \\ Distance \(l_{3}\) from \(P_{3}\) to \(R_{4}\) & 200 & Distance \(l_{2}\) from \(R_{a5}^{2}\) to \(P_{6}^{2}\) & 150 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Dimension parameter table of the \(2P_{a}1P3R_{a}\) and \(5R_{a}1P\) man machine closed chain. It can be seen from Figure 10 that when the upper arm performs a lifting motion in different lifting planes, then the rotational angle \(\theta_{i}^{1}(i=4,5,6)\) of each joint of the mechanical arm moves smoothly, and the corresponding rotational angle changes similarly in different lifting planes. Among them, \(\theta_{4}\) and \(\theta_{6}\) increase with the elevation of the lift angle \(\beta\); \(\theta_{5}\) first increases and then decreases, and in different lift planes, the change trend and movement range of \(\theta_{6}\) and \(\theta_{5}\) remain unchanged. \(\theta_{4}\) continuously increases the initial change angle as the lift angle increases. The change in the movement pair follows the piecewise function of the shoulder blade belt and the lift angle, showing the same changing law, and the movement distance changes smoothly. The relationship between the moving pair \(R_{i}^{2}(i=1,2)\) of the \(5R_{4}1P\) configuration and the lift angle \(\beta\) is shown in Figure 11a. The relationship between the turning angle \(\theta_{i}^{2}(i=3,4,5)\) of the turning pair \(R_{i}^{2}(i=3,4,5)\) and the lifting angle \(\beta\) is shown in Figure 11b-f. It can be seen from Figure 11 that when the upper arm performs a lifting motion in different lifting planes, the rotation angles of each joint of the manipulator arm move smoothly, and the corresponding rotation angles move similarly in different lifting surfaces. The changes in \(\theta_{1}\) and \(\theta_{2}\) follow the piecewise function of the scapula belt and the lift angle, respectively, showing the same change law, and the rotation angle changes smoothly. Among them, \(\theta_{3}\) and \(\theta_{4}\) decrease with increasing \(\beta\), while \(\theta_{5}\) first increases and then decreases. In different uplift planes, the changing trend and range of motion of \(\theta_{3}\) remain unchanged, which is closely related to the range of \(\beta\).