## 3 Dynamics Model of the APK Joint with VSA
### Dynamic Model of Human-Machine Hybrid System  
The APK is installed below the residual thigh of the transfemoral amputees and is connected with the thigh receiving cavity of the patient. Then, a strongly coupled human-machine hybrid system  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Maximum knee torque & 87 Nm \\ Range of motion & \(-\)5\({}^{\circ}\)\(\sim\)115\({}^{\circ}\) \\ Range of length & 48\(\sim\)52 cm \\ Total mass & \(\leq\)2.3 kg \\ \hline \hline \end{tabular}
\end{table}
Table 1: The main parameters of APK.  
Figure 1: Mechanical structure of APK: (**a**) the mechanical structure of the VSA joint; (**b**) virtual prototype; and (**c**) physical prototype.  
is formed between the human body and the APK during walking. The diagram of a human-machine hybrid system in the sagittal plane is exhibited in Figure 2.  
By using Euler-Lagrange method, the dynamic equation of unilateral human-machine hybrid model can be obtained as follows:  
\[\text{M}(\theta)\ddot{\theta}+\text{C}(\theta,\dot{\theta})\dot{\theta}+\text{G }(\theta)+\text{\boldmath$\omega$}_{e}=\tau+\text{\boldmath$\omega$}_{G} \tag{1}\]  
where \(\theta=[\theta_{h}\,\theta_{k}]^{T}\in\mathbb{R}^{2}\) denotes joint angle, input torque is represented by \(\tau=[\tau_{h}\,\tau_{k}]^{T}\) with \(\tau_{h}\) the torque of biological hip joint, and \(\tau_{k}\) denotes the torque of prosthetic knee. \(\tau_{e}=[\tau_{eh}\,\tau_{ek}]^{T}\) represent the interactive torque and unknown disturbance caused by movement of other parts of the upper body to lower limb. \(\tau_{G}=[\tau_{Gh}\,\tau_{Gk}]^{T}\) denotes the torque generated by ground reaction force (GRF) on each joint during support phase. \(\text{M}(\theta)=[M_{hh}\,M_{hk};M_{kh}\,M_{kk}]\) denotes the inertia matrix. \(\text{C}(\theta,\dot{\theta})=[\text{C}_{hh}\,\text{C}_{hk};\text{C}_{kh}\, \text{C}_{kk}]\) denote the Coriolis and Centripetal forces matrix. \(\text{G}(\theta)=[\text{G}_{h}\,\text{G}_{k}]\) represents the gravitational force vector. The dynamic equations and model parameters are detailed as follows:  
\[\text{M}(\theta) =\left[\begin{array}{cc}M_{hh}&M_{hk}\\ M_{kh}&M_{kk}\end{array}\right] \tag{2}\] \[M_{hh} =m_{t}p_{1}^{2}+m_{s}l_{s}^{2}+m_{s}p_{s}^{2}+2m_{s}l_{t}p_{s}cos( 2\theta_{h})cos\theta_{k}\] \[M_{hk} =-m_{s}p_{s}^{2}-m_{s}l_{t}p_{s}cos(2\theta_{h})cos\theta_{s}\] \[M_{kh} =-m_{s}p_{s}^{2}-m_{s}l_{t}p_{s}cos(2\theta_{h})cos\theta_{k}\] \[M_{kk} =m_{s}p_{s}^{2}\] \[\text{C}(\theta,\dot{\theta}) =\left[\begin{array}{cc}\text{C}_{hh}&\text{C}_{hk}\\ \text{C}_{kh}&\text{C}_{kk}\end{array}\right]\] \[\text{C}_{hh} =-2m_{s}l_{t}p_{s}sin(2\theta_{h})cos\theta_{k}\dot{\theta}_{h}-m _{s}l_{t}p_{s}cos(2\theta_{h})sin\theta_{k}\dot{\theta}_{k}\] \[\text{C}_{hk} =m_{s}l_{t}p_{s}cos(2\theta_{h})(\dot{\theta}_{k}-\dot{\theta}_{h})\] \[\text{C}_{kh} =m_{s}l_{t}p_{s}(2 sin(2\theta_{h})cos\theta_{k}+cos(2\theta_{h}) sin\theta_{k})\dot{\theta}_{h}\] \[\text{C}_{kk} =0\] \[\text{G}(\theta) =\left[\begin{array}{c}G_{h}\\ G_{k}\end{array}\right]\] \[G_{h} =(m_{t}gp_{t}+m_{s}g\dot{l}_{t})sin\theta_{h}-m_{s}gp_{s}sin(\theta _{k}-\theta_{h})\] \[G_{k} =m_{s}gp_{s}sin(\theta_{k}-\theta_{h})\]  
Figure 2: The diagram of a human–machine hybrid system in sagittal plane.  
where \(m_{s},l_{s},p_{s}\) are the mass of shank, length of shank, and distance between the knee and the centroid of the shank, respectively, and \(m_{t},l_{t},p_{t}\) are the mass of thigh, length of thigh, and distance between the hip and the centroid of the thigh, respectively.