# Configuration Design of an Upper Limb Rehabilitation Robot with a Generalized Shoulder Joint
## 4 2P\({}_{a}\)1P3R\({}_{a}\) Shoulder Joint Mechanism Configuration  
The configuration of \(2P_{a}1P3R_{a}\) is shown in Figure 8. In the figure, \(R_{a4}^{1}\), \(R_{a5}^{1}\) and \(R_{a6}^{1}\) are the three active rotation pairs for the main motion. The three rotational axes are orthogonal to each other. \(P_{a1}^{1}\) and \(P_{a2}^{1}\) are the two active moving pairs, and the moving axes are orthogonal to each other, which can realize the upward/sinking movement of the GH motion axis in the coronal plane of the human body. \(P_{3}^{1}\) is the passive moving pair, which is orthogonal to \(P_{a1}^{1}\), \(P_{a2}^{1}\) and each other, and the passive sliding axis can realize forward/backward movement of the GH motion axis in the sagittal plane of the human body and can combine with \(P_{a1}^{1}\), \(P_{a2}^{1}\) to realize the tracking of the GH motion axis at any position in space, thereby improving the human-machine compatibility. \begin{table}
\begin{tabular}{c c c c} \hline
**Types of Sports Pair** & **Joint Number** & **Number of Combinations** & **Branch Configuration** \\ \hline
1-DOF & 3 & 8 & RRR, RRP, PRR, PRP, PPR, PRP, PRP, RPP \\ With 2-DOF & 2 & 8 & RU, PU, RC, PC, UR, UP, CR, CP \\ With 3-DOF & 1 & 1 & S \\ \hline \end{tabular}
\end{table} 
Table 1: Configuration combination of the auxiliary movement branch chain. (P-movement pair, R-rotation pair, U-Hooke’s hinge, C-Cylindrical pair, S-ball pair). A fixed coordinate system \(O_{b}^{1}-x_{y}^{1}y_{b}^{1}z_{b}^{1}\) is established in the SC \(U_{SC}\), a human shoulder joint equivalent mechanism connection system \(O_{s}^{1}-x_{y}^{1}y_{s}^{1}z_{b}^{1}\) is established in the GH \(S_{GH}\), and a connection system \(O_{g}^{1}-x_{g}^{1}y_{s}^{1}z_{g}^{1}\) is established at the same time here. The coordinate axes of the two coordinate systems coincide in the initial pose. A local coordinate system \(O_{w}^{1}-x_{w}^{1}y_{w}^{1}z_{w}^{1}\) is established on the frame of the robot, and a joint coordinate system \(O_{i}^{1}-x_{i}^{1}y_{i}^{1}z_{i}^{1}\) (\(i=1,2,\dots,6\)) is established for each movement pair of the generalized shoulder joint mechanism, where the \(R_{a4}^{1}\),\(R_{a5}^{1}\) and \(R_{a6}^{1}\) motion axes intersect at the axis of the shoulder joint. In the initial configuration of the mechanical arm, the coordinate origin \(O_{i}^{1}\) (\(i=4,5,6\)) coincides with the origin \(O_{s}^{1}\) of the GH joint coordinate system. The origin \(O_{w}^{1}\) of the gantry fixed coordinate system coincides with the movement centre \(O_{1}^{1}\) of the mobile pair \(P_{a1}^{1}\), as shown in Figure 8. The origin \(O_{w}^{1}\) of the frame local fixed coordinate system in the human body fixed coordinate system \(O_{b}^{1}-x_{y}^{1}y_{b}^{1}z_{b}^{1}\) is \(\left(X_{w}^{1},Y_{w}^{1},Z_{w}^{1}\right)\). According to the pose transformation matrix, the pose transformation matrix \({}^{b}T_{6}^{1}\) of the end coordinate system \(O_{6}^{1}-x_{6}^{1}y_{6}^{1}z_{6}^{1}\) of the mechanical arm branch chain relative to the human body fixed coordinate system \(O_{b}^{1}-x_{b}^{1}y_{b}^{1}z_{b}^{1}\) is:  
\[{}^{b}T_{6}^{1} ={}^{b}T_{w}^{1}T_{1}^{1}T_{2}^{1}T_{3}^{1}T_{4}^{1}T_{5}^{1}T_{6}^ {1} \tag{10}\] \[=\left( \begin{array}{cccc}n_{6x}^{1}&s_{6x}^{1}&a_{6x}^{1}&X_{6 }^{1}\\ n_{6y}^{1}&s_{6y}^{1}&a_{6y}^{1}&Y_{6}^{1}\\ n_{6x}^{1}&s_{6z}^{1}&a_{6z}^{1}&Z_{6}^{1}\\ 0&0&0&1\end{array} \right)\]  
where \({}^{b}T_{w}^{1}\), \({}^{w}T_{1}^{1}\) and \({}^{i}T_{i+1}^{1}\) (\(i=1,2,3,4,5\)) are the pose transformation matrices between adjacent coordinate systems; \(n_{6}^{1}=\left[n_{6x}^{1},n_{6y}^{1},n_{6z}^{1}\right]^{T}\), \(s_{6}^{1}=\left[s_{6x}^{1},s_{6y}^{1},s_{6z}^{1}\right]^{T}\) and \(a_{6}^{1}=\left[a_{6x}^{1},a_{6y}^{1},a_{6z}^{1}\right]^{T}\) are the direction vectors of each axis end coordinate system \(O_{6}^{1}-x_{6}^{1}y_{6}^{1}z_{6}^{1}\) in \(O_{b}^{1}-x_{b}^{1}y_{b}^{1}z_{b}^{1}\); \(P_{6}^{1}=\left[X_{6}^{1},Y_{6}^{1},Z_{6}^{1}\right]^{T}\) is the position vector of the origin \(O_{6}^{1}\).