# Configuration Design of an Upper Limb Rehabilitation Robot with a Generalized Shoulder Joint
## 2 Human Upper Limb Kinematics
### Human-Machine Compatible Configuration Design of the Shoulder Joint  
In an ideal state, the human body's affected limb and mechanical arm form a closed kinematic chain [33]. Based on the Hunt formula [34], the DOF of the human-machine closed chain is calculated by Equation (4). \[F=\sum_{i=1}^{n}f_{i}-d\times l \tag{4}\]  
where \(F\) is the number of DOFs, \(n\) is the number of joints in the closed-chain system, \(f_{i}\) is the number of DOFs of the joints, \(d\) is the dimension of the motion space of the member (generally taken as 6), and \(l\) is the number of closed loops. In the human generalized shoulder joint, the AC, SC, ST and GH are simplified into one joint; the motion of the joint is three rotations; the drift of the GH movement axis is in space. According to the above, the corresponding mechanical arm has 5 DOFs at the shoulder joint. Then, a closed chain of the man-machine can be formed, including the closed chain of the shoulder joint \(\Theta_{s}\). By analogy, the elbow joint closed-chain \(\Theta_{e}\) and the wrist joint closed-chain \(\Theta_{w}\) are shown in Figure 7.  
For mechanisms with unknown DOFs, Formula (4) is expressed as Formula (5). \[F=f_{k}+f_{uk}-d\times l=\sum_{i=1}^{j}f_{i}+\sum_{m=j+1}^{n}f_{m}-d\times l \tag{5}\]  
where \(f_{k}\) and \(f_{uk}\) are the DOFs of the known joints and the unknown joints of the human-machine closed chain, respectively, and \(j\) is the number of known joints in the closed chain. According to Figure 5, a simplified, equivalent mechanism model can be divided into three human-machine closed chains, namely, \(\Theta_{s}\), \(\Theta_{se}\) and \(\Theta_{seu}\). It is known that there are five DOFs of the human shoulder joint and five DOFs of the shoulder joint mechanism in the closed shoulder joint chain, so \(f_{ks}=10\). The same is true for \(f_{ks,se}=13\) and \(f_{ks,seu}=18\). The institutional design requirements are \(F_{s}=5\), \(F_{se}=6\) and \(F_{seu}=8\). Therefore, the unknown joint DOFs \(f_{uk,s},f_{uk,se}\) and \(f_{uk,seu}\) can be solved by Equation (5). \[f_{uk,s}=F_{s}-f_{ks,s}+d\times l=5-10+6=1 \tag{6}\]  
\[f_{uk,se}=F_{se}-f_{ks,se}+d\times l=6-13+12=5 \tag{7}\]  
Figure 7: Diagram of the upper limb man machine closed kinematic chains. \[f_{uk,eww}=F_{eww}-f_{k,eww}+d\times l=8-18+18=8 \tag{8}\]  
From Equations (6) and (7), it can be seen that the shoulder joint closed loop chain introduces at least one passive DOF and the shoulder-elbow closed loop chain must introduce five passive DOFs, and Equation (8) shows that the overall human-machine closed chain must introduce eight passive DOFs. Due to the introduction of five or more passive degrees of freedom in the independent closed-loop chain, the passive drive chain is longer. The efficiency of energy and force transmission will be lower [35], the overall robot control will become particularly complicated, the passive DOF introduced in the closed chain of the manipulator wrist joint is first eliminated, and the spatial flexibility of the handle is reduced to achieve better manipulator control accuracy. Second, the human-machine closed chain of the shoulder joint \(\Theta_{s}\) is required to be the just-constrained subsystem, and the passive DOF introduced by the human-machine closed chain should be:  
\[f_{s}=1;f_{e}=4;\ f_{w}=0 \tag{9}\]  
According to the generalized shoulder joint motion mechanism model, the main motion of the GH will be realized by the RRR mechanism with three rotation axes converging at one point, and there are many combinations to meet the configuration of the auxiliary motion branch chain.