## 2 Structure and Dynamics Model of Lower Limb Rehabilitation Exoskeleton Robot
### Robot Dynamic Model  
According to the dynamic model of the lower limb rehabilitation exoskeleton robot, the mathematical relationship between the motion of the hip, knee joints, and the control torque can be obtained. Ignoring the influence of environmental interference on the lower limb rehabilitation exoskeleton robot, a simplified structure diagram of the unilateral lower limb was established in the sagittal plane, as shown in Figure 2. The physical parameters of the simplified structure are shown in Table 2.  
The Lagrange method is used to establish the dynamic model of the lower limb rehabilitation exoskeleton robot, which can be expressed by a second-order nonlinear differential Equation [27]. \[\mathbf{M}(\mathbf{\theta})\mathbf{\ddot{\theta}}+\mathbf{C}(\mathbf{\theta},\mathbf{\dot{\theta}})\mathbf{ \dot{\theta}}+\mathbf{G}(\mathbf{\theta})=\mathbf{\tau} \tag{1}\]  
where \(\mathbf{\theta}=\left[ \begin{array}{cc}\theta_{1}&\theta_{2}\end{array} \right]^{T}\) is the angular displacement of the hip and knee joints; \(\mathbf{\dot{\theta}}\) is the angular velocity; \(\mathbf{\dot{\theta}}\) is the angular acceleration; \(\mathbf{M}(\mathbf{\theta})=[M_{ij}]\) is an inertia matrix; and \(\mathbf{C}(\mathbf{\theta},\mathbf{\dot{\theta}})=[C_{ij}]\) \begin{table}
\begin{tabular}{c c} \hline \hline
**Nonenclature** & **Description** \\ \hline \(\theta_{i}\) & The rotation angles of the hip and knee joints in the sagittal plane \\ \(l_{i}\) & The lengths of the thigh and shank segments \\ \(l_{i}\) & The distance from the driving joint of the segment to the centroid of the segment \\ \(h_{i}\) & The position offset of the centroid of thigh and shank segments \\ \(m_{i}\) & The masses of the thigh segment and shank segments \\ \(l_{i}\) & The inertia of the thigh segment and shank segments \\ \(\tau_{i}\) & The driving torque of the hip and knee joints \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The physical parameters of the simplified structure. Figure 2: Simplified structural diagram of a unilateral lower limb. is a centrifugal force and Coriolis force matrix, and it has a strong coupling term, \(i=1\cdots 2\), \(j=1\cdots 2\). \(G(\theta)=\left[ \begin{array}{cc}G_{11}&G_{21}\end{array} \right]^{\rm T}\) is a gravity term [27,31], and \(\mathbf{\tau}=\left[ \begin{array}{cc}\tau_{11}&\tau_{21}\end{array} \right]^{\rm T}\) is the control torque of the hip and knee joints. Among them,  
\[ \begin{array}{l}M_{11}=I_{1}+I_{2}+m_{1}(h_{1}^{2}+lc_{1}^{2})+m_{2}(h_{2}^ {2}+l_{1}^{2}+lc_{2}^{2}+2l_{1}lc_{2}\cos\theta_{2}-2l_{1}h_{2}sin\theta_{2}) \\ M_{12}=M_{21}=I_{2}+m_{2}(h_{2}^{2}+lc_{2}^{2}+l_{1}lc_{2}\cos\theta_{2}-l_{ 1}h_{2}sin\theta_{2})\\ M_{22}=I_{2}+m_{2}(h_{2}^{2}+lc_{2}^{2})\\ C_{11}=-2m_{2}l_{1}(h_{2}\cos\theta_{2}+lc_{2}sin\theta_{2})-m_{2}l_{1}(h_{2} \cos\theta_{2}+lc_{2}sin\theta_{2})\dot{\theta}_{1}\dot{\theta}_{2}\\ C_{12}=C_{21}=0\\ C_{22}=m_{2}l_{1}(h_{2}\cos\theta_{2}+lc_{2}sin\theta_{2})\\ G_{11}=m_{2}g(lc_{2}\cos(\theta_{1}+\theta_{2})-h_{2}sin(\theta_{1}+\theta_{2} )+l_{1}\cos\theta_{1})+m_{1}g(h_{1}sin\theta_{1}+lc_{1}\cos\theta_{1})\\ G_{21}=m_{2}g(lc_{2}\cos(\theta_{1}+\theta_{2})-h_{2}sin(\theta_{1}+\theta_{2} ))\\ \end{array}  \tag{2}\]  
where \(l_{i}(i=1,2)\) is the length of the thigh and shank links; \(lc_{i}(i=1,2)\) and \(h_{i}(i=1,2)\) is the centroid position of the links; \(m_{i}(i=1,2)\) is the mass of the links; \(l_{i}(i=1,2)\) is the inertia of the links. The parameters in Equation (2) were obtained from the 1:1 three-dimensional model constructed by the commercial software Solidworks. For Equation (1), there are three properties of the dynamic model of the lower limb rehabilitation exoskeleton robot as follows [13,32]. **Property 1**. : \(M(\theta)\)_is a symmetric positive definite matrix and is bounded._  
**Property 2**.