# Design Approaches of an Exoskeleton for Human Neuromotor Rehabilitation
## 4 Dynamic Analysis of the Proposed Exoskeleton
### Exoskeleton Mathematical Modeling for Dynamic Analysis
From Equation (7), it can be written:  
\[M\overset{\bullet\bullet}{q}+J_{q}^{T}\cdot\lambda=Q^{a} \tag{8}\]  
The Lagrange multipliers will be determined as:  
\[\lambda=J_{q}^{-1}\Big{[}Q^{a}-M\overset{\bullet\bullet}{q}\Big{]} \tag{9}\]  
The connection forces from the \(k\) kinematic joint, depending on the axis global coordinates system can be determined with:  
\[F_{k}=\lambda_{k} \tag{10}\]  
Thus, the knee actuating mechanism is made from 1 to 6 kinematic links according to Figure 8. For these links, the proper generalized coordinates are:  
\[q_{1}=\{X_{E},Y_{E},\phi_{06},X_{F},Y_{F},\phi_{65},X_{B},Y_{B},\phi_{63},X_{ A},Y_{A},\phi_{01},\phi_{32}\}^{T} \tag{11}\]  
For an inverse dynamic analysis, the kinematic constraint equations in the case of knee actuating mechanism are:  
Thus, the kinematic joints connection forces are:  
\[F_{\rm I1}^{k}=\lambda_{I}^{1k} \tag{18}\]  
In the case of the ankle actuation mechanism, by having insight to Figure 8, this consists of 7 to 11 links. Thus, the proper generalized coordinate vector is:  
\[q_{2}=\left\{X_{M},Y_{M},\phi_{65},X_{N},Y_{N},\phi_{511},X_{L},Y_{L},\phi_{1110 },X_{J},Y_{J},\phi_{59},X_{G},Y_{G},\phi_{57},\phi_{98}\right\}^{T} \tag{19}\]  
For this, the kinematic constraint equations in the case of ankle actuation mechanism are:  
\[\left[ \begin{array}{c}X_{M}-\cos\phi_{65}X_{M}^{5}+\sin\phi_{65}Y_{M}^{5}-X_{ E}\\ \\ Y_{M}-\sin\phi_{65}X_{M}^{5}-\cos\phi_{65}Y_{M}^{5}-Y_{E}\\ \\ X_{N}-\cos\phi_{5\ 11}X_{N}^{11}+\sin\phi_{5\ 11}Y_{N}^{11}-X_{M}\\ \\ Y_{N}-\sin\phi_{5\ 11}X_{N}^{11}-\cos\phi_{5\ 11}Y_{N}^{11}-Y_{M}\\ \\ X_{L}-\cos\phi_{11\ 10}X_{L}^{10}+\sin\phi_{11\ 10}Y_{L}^{10}-X_{N}\\ \\ Y_{L}-\sin\phi_{11\ 10}X_{L}^{10}-\cos\phi_{11\ 10}Y_{L}^{10}-Y_{N}\\ \\ X_{J}-\cos\phi_{59}X_{J}^{9}+\sin\phi_{59}Y_{J}^{9}-X_{K}\\ \\ Y_{J}-\sin\phi_{59}X_{J}^{9}-\cos\phi_{59}Y_{J}^{9}-Y_{K}\\ \\ X_{G}-\cos\phi_{57}X_{G}^{7}+\sin\phi_{57}Y_{G}^{7}-X_{E}\\ \\ Y_{G}-\cos\phi_{57}X_{G}^{7}+\sin\phi_{57}Y_{g}^{7}-X_{E}\\ \\ \cos\phi_{11\ 10}X_{L}^{10}-\sin\phi_{11\ 10}Y_{L}^{10}+X_{N}-\cos\phi_{59}X_{L}^{ 9}+\sin\phi_{59}Y_{L}^{9}-X_{K}\\ \\ \sin\phi_{11\ 10}X_{L}^{10}+\cos\phi_{11\ 10}Y_{L}^{10}+Y_{N}-\sin\phi_{59}X_{L}^{ 9}-\cos\phi_{59}Y_{L}^{9}-Y_{K}\\ \\ \cos\phi_{98}X_{G}^{8}-\sin\phi_{98}Y_{G}^{8}+X_{J}-\cos\phi_{57}X_{G}^{7}+ \sin\phi_{57}Y_{G}^{7}-\cos\phi_{06}X_{E}^{6}-X_{0}\\ \\ \cos\phi_{98}X_{G}^{8}-\sin\phi_{98}Y_{G}^{8}+X_{J}-\cos\phi_{57}X_{G}^{7}+ \sin\phi_{57}Y_{G}^{7}-\cos\phi_{06}X_{E}^{6}-X_{0}\\ \\ \phi_{65}-0.05572857950+3.917643736\cdot t-46.09421696\cdot t^{2}+142.2026539 \cdot t^{3}-\\ 183.2410652\cdot t^{4}+106.6799384\cdot t^{5}-23.24793765\cdot t^{6}\\ \\ \phi_{5\ 11}-0.07468922928-4.125887507\cdot t+46.18350162\cdot t^{2}-139.7502390 \cdot t^{3}+\\ 185.4096885\cdot t^{4}-113.2636093\cdot t^{5}+25.99845833\cdot t^{6}\\ \end{array} \right]\]  
The corresponding Jacoby for generalized coordinates that satisfy Equation (20) is:  
\[J_{q2}=\frac{\partial q_{2i}}{\partial q_{2j}};\ \ i,j=\overline{1,16} \tag{21}\]  
The mass matrix, which corresponds to the ankle actuation mechanism, is:  
\[M_{2}=diag\left[ \begin{array}{cccccccccccccccccccc}m_{5}&m_{5}&J_{5}&m_{11} &m_{11}&J_{11}&m_{10}&m_{10}&J_{10}&m_{9}&J_{9}&m_{7}&m_{7}&J_{7}&J_{8}\\ \end{array} \right] \tag{22}\]For this, the linear and angular accelerations matrix corresponding to generalized coordinates given by Equation (19):  
\[a_{q2}=q_{2}^{\bullet\bullet}=\left[ \begin{array}{cccccccccccccccc}aX_{M}&aY_{M} &\varepsilon_{5}&aX_{N}&aY_{N}&\varepsilon_{11}&aX_{L}&aY_{L}&\varepsilon_{10}&aX_ {I}&aY_{I}&\varepsilon_{9}&aX_{G}&aY_{G}&\varepsilon_{7}&\varepsilon_{8}\end{array}  \right]^{T} \tag{23}\]  
\[Q_{2}^{d}=\left[ \begin{array}{cccccccccccc}0&-m_{5}g&0&0&-m_{11}g&0&0&-m_{10}g &0&0&-m_{9}g&0&0&-m_{7}g&0&-\frac{1}{2}m_{8}g\cdot I_{AB}\end{array}