# Design and Isokinetic Training Control Method of Leg Press Training Device
## 3 Human Dynamics Model of Resistance Training
### Dynamic Model of Human Leg Connecting Rod in Training Process  
The human legs (thighs, legs, and feet) are regarded as rigid link. The foot on the pedal is always fixed in the same position (fixed pedal angle). Thus, the integral mechanism of the \"human leg-slider\" can be regarded as a crank-connecting rod mechanism. As shown in Figure 5, the human leg is simplified to a three-link rigid mechanism. In Figure 5, \(m_{1}\), \(m_{2}\), \(m_{3}\), and \(m_{p}\) mean the mass of thigh, leg, foot, and slider, respectively, \(F_{I}\) is the impedance force, and \(F_{h}\) represents the pedal thrust along the slide. Based on the principle of virtual work, the dynamic mathematical model of the mechanism is established. The mechanism is a system of mass points with complete constraints, which consists of three components' mass centers and the end point of the human leg under external force. Before listing the virtual work equation, the centroid parameters of the system of mass point of a man-machine system are first described, as shown in Tables 2-4. In Tables 2-4, \(S=AC\), and the values of \(f_{1}\), \(f_{2}\), \(cg_{1}\), \(g_{2}\) are as follows:  
\[f_{1}(\theta_{1},\theta_{2})=\frac{\sin(\theta_{2}-\phi)}{l_{1}\sin(\theta_{2 }-\theta_{1})} \tag{4}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Link Number** & \(\mathbf{X}\)**-Directional Velocity \(\frac{\frac{\partial\mathbf{R}^{i}}{\partial\mathbf{t}}}{\partial\mathbf{t}}\) & \(\mathbf{Y}\)**-Directional Velocity \(\frac{\frac{\partial\mathbf{R}^{i}}{\partial\mathbf{t}}}{\partial\mathbf{t}}\) \\ \hline \(l_{1}\) & \(-\frac{1}{2}f_{1}l_{1}\hat{\Im}\sin\theta_{1}\) & \(\frac{1}{2}f_{1}l_{1}\hat{\Im}\cos\theta_{1}\) \\ \(l_{2}\) & \(\left(\cos\phi+\frac{1}{2}f_{2}l_{2}\sin\theta_{2}\right)\hat{\Im}\) & \(-\left(\sin\phi+\frac{1}{2}f_{2}l_{2}\cos\theta_{2}\right)\hat{\Im}\) \\ \(l_{3}\) & \(\hat{S}\cos\phi\) & \(-\hat{S}\sin\phi\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Centroid location description of components. Figure 5: Human leg model during impedance training. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Link Number** & \(\mathbf{X}\)**-Coordinate \(\mathbf{R}^{i}_{x}\)** & \(\mathbf{Y}\)**-Coordinate \(\mathbf{R}^{i}_{y}\)** \\ \hline \(l_{1}\) & \(\frac{1}{2}l_{1}\cos\theta_{1}\) & \(\frac{1}{2}l_{1}\sin\theta_{1}\) \\ \(l_{2}\) & \(S\cos\phi-e\sin\phi-\frac{1}{2}l_{2}\cos\theta_{2}\) & \(-S\sin\phi-e\cos\phi-\frac{1}{2}l_{2}\sin\theta_{2}\) \\ \(l_{3}\) & \(S\cos\phi-e\sin\phi+\frac{1}{2}l_{3}\cos\left(\theta_{3}+\phi\right)\) & \(-S\sin\phi-e\cos\phi+\frac{1}{2}l_{3}\sin\left(\theta_{3}+\phi\right)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Description of centroid velocity of components. \[g_{1}(\theta_{1},\theta_{2})=\frac{l_{2}\cos(\theta_{1}+\theta_{2})\cos^{2}(\theta_ {2}+\phi)+l_{1}\cos(2\theta_{2})\cos^{2}(\theta_{1}+\phi)}{l_{1}^{2}l_{2}\cos \theta_{2}\sin^{3}(\theta_{2}-\theta_{1})} \tag{5}\]  
\[f_{2}(\theta_{1},\theta_{2})=\frac{\sin(\theta_{1}-\phi)}{l_{2}\sin(\theta_{1} -\theta_{2})} \tag{6}\]  
\[g_{2}(\theta_{1},\theta_{2})=\frac{l_{1}\cos(\theta_{1}+\theta_{2})\cos^{2}( \theta_{1}+\phi)+l_{1}\cos(2\theta_{1})\cos^{2}(\theta_{2}+\phi)}{l_{1}l_{2}^{ 2}\cos\theta_{1}\sin^{3}(\theta_{1}-\theta_{2})} \tag{7}\]  
According to the principle of virtual work, it can be obtained as follows:  
\[\delta W_{e}=-m_{1}g\delta R_{y}^{1}-m_{2}g\delta R_{y}^{2}-\left(m_{3}+m_{p} \right)g\delta R_{y}^{3}+M_{2}\delta\theta_{2}-F_{l}\delta S \tag{8}\]  
where \(m_{1}\), \(m_{2}\), \(m_{3}\), and \(m_{p}\) mean the weight of thighs, legs, feet, and slider, respectively, \(F_{l}\) means resistance force, \(M_{2}\) is a function showing the relationship between \(\theta_{2}\), \(\frac{d\theta_{2}}{dt}\), and quadriceps stimulation rate \(Z\), and its equation is expressed as:  
\[M_{2}=-KZ\theta_{2}-BZ\frac{d\theta_{2}}{dt}, \tag{9}\]  
where \(KZ\) means the extension of knee-joint muscle elasticity, and \(BZ\) means the knee-joint extension damping. \[