## 3 Mathematical Model of Lower-Limb Robotic Orthosis  
The general configuration of a two degree of freedom (2 DOF) robot is shown in Figure 3a where the subscript \(i=1\) (hip), 2 (knee). \(\theta_{i}\), \(l_{i}\) and \(l_{ci}\) are joint angles, length of links and the distance from joints to the corresponding center of mass, respectively. These parameters of the AIRGAIT robotic orthosis are provided in Table 2 in detail.  
The dynamics of 2 DOF robotic is described by the following Euler-Lagrange equation  
\[D(\theta)\ddot{\theta}+C(\theta,\dot{\theta})\dot{\theta}+G(\theta)=T \tag{1}\]  
where \(T=\begin{bmatrix}T_{1}&T_{2}\end{bmatrix}^{T}\) and \(\theta=\begin{bmatrix}\theta_{1}&\theta_{2}\end{bmatrix}^{T}\) are the vector of applied torques and joint angles, respectively. The system mass \(D(\theta)\), the coriolis matrix \(C(\theta,\dot{\theta})\) and the vector of gravity \(G(\theta)\) are  
\[D(\theta) =\begin{bmatrix}D_{11}&D_{12}\\ D_{21}&D_{22}\end{bmatrix} \tag{2}\] \[C(\theta,\dot{\theta}) =\begin{bmatrix}C_{11}&C_{12}\\ C_{21}&C_{22}\end{bmatrix}\] (3) \[G(\theta) =\begin{bmatrix}G_{1}&G_{2}\end{bmatrix}^{T} \tag{4}\]  
with  
\[D_{11} =m_{1}l_{c1}^{2}+m_{2}(l_{1}^{2}+l_{c2}^{2}+2l_{1}l_{c2}cos\theta _{2})+I_{1}+I_{2} \tag{5a}\] \[D_{12} =D_{21}=m_{2}(l_{c2}^{2}+l_{1}l_{c2}cos\theta_{2})\] (5b) \[D_{22} =m_{2}l_{c2}^{2}+l_{2} \tag{5c}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline
**DOFs** & \(\mathbf{m_{i}}\) **[kg]** & \(\mathbf{I_{i}}\) **[kg m\({}^{2}\)]** & \(\mathbf{L_{i}}\) **[m]** & \(\mathbf{L_{ci}}\) **[m]** \\ \hline Hip & 1.34 & 0.052 & 0.4 & 0.2 \\ Knee & 0.97 & 0.032 & 0.35 & 0.15 \\ \hline \end{tabular}
\end{table}
Table 2: Parameters of the AIRGAIT robotic orthosis.  
Figure 3: **(a)** Typical 2 DOF robotic **(b)** Robotic Orthosis with two mono-articular and one bi-articular muscles. The 1, 2, 3 subscripts denote hip, knee and bi-articular muscle. A and P subscripts denote the anterior and posterior PAMs.  
\[C_{11} = -m_{2}l_{1}l_{2}sin\theta_{2}\theta_{2} \tag{6a}\] \[C_{12} = -m_{2}l_{1}l_{c2}sin\theta_{2}(\theta_{1}+\dot{\theta_{2}})\] (6b) \[C_{21} = m_{2}l_{1}l_{c2}sin\theta_{2}\theta_{1}\] (6c) \[C_{22} = 0 \tag{6d}\]  
\[G_{1} = (m_{1}l_{c1}+m_{2}l_{1})gcos\theta_{1}+m_{2}l_{c2}gcos(\theta_{1 }+\theta_{2}) \tag{7a}\] \[G_{2} = m_{2}l_{c2}gcos(\theta_{1}+\theta_{2}) \tag{7b}\]  
As seen in Figure 3b, in the AIRGAIT orthosis configuration, the hip and knee joints are actuated by two mono-articular muscles, whereas the bi-articular which connects between the hip and knee joints has influence on both joints simultaneously. Hence, the relation between the torques \(\tau_{1}\), \(\tau_{2}\) and \(\tau_{3}\) generated by the corresponding pair of PAMs and the joints torques can be described by  
\[T=W\tau \tag{8}\]  
where  
\[\tau=\begin{bmatrix}\tau_{1}&\tau_{2}&\tau_{3}\end{bmatrix}^{T} \tag{9}\]  
and \(W\in\mathbb{R}^{2\times 3}\) is the transformation from the muscle to joint space:  
\[W=\begin{bmatrix}1&0&1\\ 0&1&1\end{bmatrix} \tag{10}\]  
Since \(W\) is not a square matrix, \(\tau\) can be derived from Equations (8) and (10) by using the Moore-Penrose pseudo-inverse matrix \(W^{+}\)[34] of \(W\) as  
\[\tau=W^{+}T \tag{11}\]  
From Equations (1) and (11), the dynamic model of the lower-limb robotic orthosis considering the additional bi-articular muscle can be expressed by the following equation  
\[\tau=D(\theta)\ddot{\theta}+\bar{C}(\theta,\dot{\theta})\dot{\theta}+\bar{G} (\theta) \tag{12}\]  
where \(\bar{D}(\theta)=W^{+}D(\theta)\), \(\bar{C}(\theta,\dot{\theta})=W^{+}C(\theta,\dot{\theta})\), and \(\bar{G}(\theta)=W^{+}G(\theta)\), respectively.