## 3 Kinematic Analysis
### Forward Kinematics  
The wearable upper limb rehabilitation robot is a typical human-machine cooperation system. The robot is consistent with the movement of the affected limb of the human body. Therefore, to accurately obtain the motion curve of the affected limb, a forward kinematic analysis is required for the wearable upper limb rehabilitation robot. To ensure that the designed wearable upper limb rehabilitation robot has good applicability and practical applications, the D-H parameter model of the wearable upper limb rehabilitation robot based on the D-H coordinate system method needs to \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Parts**} & \multirow{2}{*}{**Degrees of Freedom**} & \multirow{2}{*}{**Movement Range of Human**} & **Movement Range of Robot** \\  & & & & [34; 35] \\ \hline Shoulder & Flexion/extension & 0\({}^{\circ}\)-90\({}^{\circ}\)/0\({}^{\circ}\)–45\({}^{\circ}\) & 0\({}^{\circ}\)-90\({}^{\circ}\)/0\({}^{\circ}\)–10\({}^{\circ}\) \\ Shoulder & Abduction/adduction & 0\({}^{\circ}\)-90\({}^{\circ}\)/0\({}^{\circ}\)–45\({}^{\circ}\) & 0\({}^{\circ}\)–75\({}^{\circ}\)/0\({}^{\circ}\)–45\({}^{\circ}\) \\ Shoulder & Internal/external rotation & 0\({}^{\circ}\)-80\({}^{\circ}\)/0\({}^{\circ}\)–30\({}^{\circ}\) & 0\({}^{\circ}\)–80\({}^{\circ}\)/0\({}^{\circ}\)–30\({}^{\circ}\) \\ Elbow & Flexion/extension & 0\({}^{\circ}\)–135\({}^{\circ}\) & 0\({}^{\circ}\)–105\({}^{\circ}\) \\ Elbow & Internal/external rotation & 0\({}^{\circ}\)–45\({}^{\circ}\)/0\({}^{\circ}\)–45\({}^{\circ}\) & 0\({}^{\circ}\)–45\({}^{\circ}\)/0\({}^{\circ}\)–45\({}^{\circ}\) \\ Wrist & Flexion/extension & 0\({}^{\circ}\)–75\({}^{\circ}\)/0\({}^{\circ}\)–75\({}^{\circ}\) & 0\({}^{\circ}\)–45\({}^{\circ}\)/0\({}^{\circ}\)–45\({}^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Free distribution of human upper limbs and robot degrees of freedom. Figure 5: A 3D model of the wearable upper limb rehabilitation robot. set related parameters, which including describing the connecting rod, which used to describe the geometric characteristic parameters of connecting rods, the connection parameter relationship between two connecting rods and the parameters that define the relationship between connecting rods. Lastly, the parameters can be brought into the correlation transformation matrix to get the corresponding results by setting the parameters. In order to obtain the relationship between the rotation and translation among adjacent members of the wearable upper limb rehabilitation robot and its terminal pose, a reference coordinate system should be assigned to each joint [36]. In this paper, according to kinematics theory, Denavit-Hartenberg (D-H) coordinate system method is adopted to establish the D-H parameter model of the upper limb rehabilitation robot ontology (see Figure 6), as shown in Figure 7. The geometric size of each connecting rod of the rehabilitation robot can be described by four parameters. \(a_{i}\) and \(\alpha_{i}\) are used to describe the geometric features of the connecting rod itself. The numerical values are determined by the distance and included angle between the axes \(z_{i-1}\) and \(z_{i}\) The other parameters offset \(d_{i}\) and joint angle \(\theta_{i}\) represent the connection relationship between the two connecting rods, and the values are determined by the distance and included angle between the axes \(x_{i-1}\) and \(x_{i}\).The parameters of the rehabilitation robot D-H model are listed in Table 2.  
According to the spatial coordinate system established for each joint of the wearable upper limb rehabilitation robot, forward kinematics analysis is carried out, i.e., the coordinate transformation from