## 3 Kinematic Analysis
### Forward/Inverse Kinematics  
The human-robot hybrid linkage coordinate system is built as shown in Figure 5; the global coordinate system \(\{O-x_{0}y_{0}z_{0}\}\) is located on the far end of the main ball screw while the plane \(x_{0}Oz_{0}\) is parallel to the sagittal plane. The linkage chain is divided into the linkage \(ABC\) and the linkage \(FED\) (right leg), and the forward/inverse kinematics of two parts are solved through the Denavit-Hartenberg (D-H) method in the global coordinate system \(\{O-x_{0}y_{0}z_{0}\}\). As the shape change of the parallelogram mechanism cannot change the relative direction of the opposite link, the coordinate transformation of \(\theta_{2}\) is divided into \(\frac{1}{2}T\) and \(\frac{2}{3}T\). The D-H parameters of the linkage \(ABC\) are listed in Table 2. The directions of \(\theta_{1}\), \(\theta_{3}\) and \(\theta_{5}\) are defined against the arrow directions shown in Figure 5.  
Figure 4: Human-robot hybrid kinematic model. The transformation matrix \(i^{-1}T\), which represents the coordinate transformation from frame \(i-1\) to \(i\), is obtained from Equation (1):  
\[ \begin{array}{c}{i-1}T=Rot(x,\alpha_{i-1})Trans(x,\alpha_{i-1})Rot(z,\theta_{i} )Trans(z,d_{i})\\ =\left[\begin{array}{cccc}c\phi_{i}&-s\theta_{i}&0&a_{i-1}\\ s\theta_{i}ca_{i-1}&c\theta_{i}ca_{i-1}&-s\alpha_{i-1}&-d_{i}s\alpha_{i-1}\\ s\theta_{i}\alpha_{i-1}&c\theta_{i}s\alpha_{i-1}&c\alpha_{i-1}&d_{i}c\alpha_{i- 1}\\ 0&0&0&1\end{array}\right]\end{array}  \tag{1}\]  
where \(c*=cos(*)\) and \(s*=sin(*)\). The coordinate that transforms from \(\{x_{0}y_{0}z_{0}\}\) to \(\{x_{4}y_{4}z_{4}\}\) is calculated as follows:  
\[ \begin{array}{c}{{}^{0}_{4}T}={{}^{0}_{1}T^{1}_{2}T^{2}_{3}T^{3}_{4}T}= \left[\begin{array}{cccc}n_{1x}&\ o_{1x}&a_{1x}&p_{1x}\\ n_{1y}&\ o_{1y}&a_{1y}&p_{1y}\\ n_{1z}&\ o_{1z}&a_{1z}&p_{1z}\\ 0&0&0&1\end{array}\right]\\ =\left[\begin{array}{cccc}cos(\theta_{1}+\theta_{3})&sin(\theta_{1}+\theta_{3 })&0&l_{2}cos\theta_{1}cos\theta_{2}+l_{1}cos\theta_{1}+a_{0}\\ 0&0&1&l_{2}sin\theta_{2}\\ sin(\theta_{1}+\theta_{3})&-cos(\theta_{1}+\theta_{3})&0&l_{2}sin\theta_{1} cos\theta_{2}+l_{1}sin\theta_{1}\\ 0&0&0&1\end{array}\right]\end{array}  \tag{2}\]  
where \(\theta_{i}\) represents the rotation angle; \(a_{0}\) is the linear sliding motion of the prismatic joint. The forward/inverse kinematics of the linkage \(ABC\) could be expressed as \({{}^{0}_{4}T}\) and Equation (3):  
\[\left\{ \begin{array}{l}\theta_{1}=arcsin\frac{p_{1z}}{l_{1}+2 cos\theta_{2} }\\ \theta_{2}=arcsin\frac{p_{1z}}{l_{1z}}\\ \theta_{3}=arctan(\frac{p_{1z}}{l_{1z}})-\theta_{1}\\ a_{0}=p_{1x}-l_{1}cos\theta_{1}-l_{2}cos\theta_{1}cos\theta_{2}\end{array} \right.. \tag{3}\]  
Figure 5: Human-robot hybrid linkage coordinate system. \begin{table}
\begin{tabular}{c c c c} \hline \(a_{i-1}\) & \(a_{i-1}\) & \(d_{i}\) & \(\theta_{i}\) \\ \hline \(a_{0}\) & \(-90^{\circ}\) & 0 & \(-\theta_{1}\) \\ \(l_{1}\) & \(90^{\circ}\) & 0 & \(\theta_{2}\) \\ \(l_{2}\) & \(0^{\circ}\) & 0 & \(-\theta_{2}\) \\ \(0\) & \(-90^{\circ}\) & 0 & \(-\theta_{3}\) \\ \hline \end{tabular}
\end{table} 
Table 2: Denavit-Hartenberg (D-H) parameters of the linkage \(ABC\). Similarly, the transformation of the linkage \(FED\) could be calculated using the D-H parameters in Table 3. To analyze the linkage \(ABC\) and the linkage \(FED\) in one coordinate system, a coordinate transformation matrix \({}^{0}_{F}T\) between \(\{x_{0}y_{0}z_{0}\}\) with \(\{x_{F}y_{F}z_{F}\}\) is built. The forward/inverse kinematics of the linkage \(FED\) is expressed as follows:  
\[