# Design Approaches of an Exoskeleton for Human Neuromotor Rehabilitation
## 4 Dynamic Analysis of the Proposed Exoskeleton  
### Exoskeleton Mathematical Modeling for Dynamic Analysis  
To accomplish a dynamic analysis of the proposed exoskeleton concept, a kinematic scheme for both mechanisms that were to actuate knee and ankle joints was elaborated, as can be seen in Figure 8. This was performed in accordance with [28, 29, 30]. From this kinematic scheme, the following components can be identified: **O**--hip revolute joint, a direct motor joint; **E**--knee revolute joint which also assures support for the knee and foot actuated mechanism; **M**--knee revolute joint; **A**, **B**, **C**, **D**, **F**--revolute joints for knee mechanism actuation; **G**, **J**, **L**, **N**, **K**--revolute joints for ankle mechanism actuation. Link no. 1 represents the drive link for knee mechanism actuation and this mechanism is made from the connecting rod no. 2, support link no. 3 articulated in **CBD** revolute joints, and connection link no. 4 which actuates the tibia equivalent link no. 5 through revolute joint F. The equivalent link of the femur will assure the entire support for the knee actuation mechanism and this can be identified as link no. 6. By having insight into the experimental analysis database, we considered a case subject with the proper motion laws of hip, knee, and joints which will serve as input data through \(\phi_{06}\), \(\phi_{65}\), \(\phi_{57}\) variation angles. This will be used also as input data for an inverse kinematic analysis presented in [31] and the aim of this is to identify the drive joints' proper angular speeds. For this dynamic analysis, the Newton-Euler method completed with Lagrange multipliers will be used. Thus, the kinematic constraints equations for a mobile mechanical system are:  
\[\varphi(q,t)=0 \tag{1}\]  
Figure 8: Kinematic scheme of the exoskeleton lower limb. where: \(t\) is time and \(q\) represents the generalized coordinates. By differentiating the Equation (1) depending on time, we will obtain:  
\[\overset{\bullet}{\varphi}(q,t)=J_{q}\overset{\bullet}{q}+\frac{\partial\varphi _{i}}{\partial t} \tag{2}\]  
\[\overset{\bullet\bullet}{\varphi}(q,t)=J_{q}\overset{\bullet\bullet}{q}+\Big{(} J_{q}\overset{\bullet}{q}\Big{)}_{q}\overset{\bullet}{q}+2\frac{\partial^{2} \varphi_{i}}{\partial t\partial q_{j}}\overset{\bullet}{q}+\frac{\partial^{2} \varphi_{i}}{\partial t^{2}} \tag{3}\]  
\[\Big{(}J_{q}\overset{\bullet}{q}\Big{)}_{q}=\sum_{k=1}^{n_{c}}\frac{\partial ^{2}\varphi_{i}}{\partial q_{j}\partial q_{k}}\overset{\bullet}{q}_{k} \tag{4}\]  
where: \(i=\overline{1,n_{h}}\) represents the kinematic constraints equations number and \(j=\overline{1,n_{c}}\) represents the generalized coordinates number. The following notation will be performed:  
\[J_{q}\overset{\bullet}{q}=\
u=-\frac{\partial\varphi}{\partial t} \tag{5}\]  
\[J_{q}\overset{\bullet\bullet}{q}=-\Big{(}J_{q}\overset{\bullet}{q}\Big{)}_{q }\overset{\bullet}{q}-2\frac{\partial^{2}\varphi_{i}}{\partial t\,\partial q _{j}}\overset{\bullet}{q}-\frac{\partial^{2}\varphi_{i}}{\partial t^{2}}=a \tag{6}\]  
The Newton-Euler formalism motion equations, completed with Lagrange multipliers method, are:  
\[\left[ \begin{array}{cc}M&J_{q}^{T}\\ J_{q}&O\end{array} \right]\left[ \begin{array}{c}\overset{\bullet}{q}\\ a\end{array} \right]=\left[ \begin{array}{c}Q^{a}\\ a\end{array} \right] \tag{7}\]  
where: \(M\) is the mass matrix, \(\lambda\)--Lagrange multipliers, \(\tilde{q}\)--accelerations, \(Q^{a}\)--active generalized forces vector, \(a\)--is a term from Equation (6).