# Development of a Wearable Finger Exoskeleton for Rehabilitation
## 3 Exoskeleton Hand Analysis
### Dynamics Analysis  
The approach followed in this paper applies the Newton-Euler dynamic formulation to solve the dynamics of the exoskeleton hand. The main advantages of this technique are the facility of implementation and obtaining models with a reduced number of operations. The Newton-Euler formulation is derived by the direct interpretation of [PERSON]'s second law of motion, which describes system dynamics in terms of force and momentum. The equations incorporate all the forces and momentum acting on individual exoskeleton links, including the coupling forces and momentum between the links. The equations obtained from the Newton-Euler method include the constraint forces acting between adjacent links. Thus, additional arithmetic operations are required to eliminate these terms and obtain explicit relations between the joint torques and the resultant motion in terms of joint displacements. For the development of dynamic analysis through the Newton-Euler formulation, the finger exoskeleton mechanism is simplified and shown in Figure 5. The diagram represents only the links of the proximal phalanx and middle phalanx of the fingers; the main reason why only these two phalanxes are contemplated is because of the type of movement that the distal phalanx performs concerning the middle phalanx, which is an underactuated type, which, when exerting a force on the middle phalanx in the same way will generate movement in the distal phalanx. \(d_{1}\) and \(d_{2}\) are the lengths of the section of the mechanism that moves the proximal phalanx and the middle phalanx; \(d_{c1}\) and \(d_{c2}\) are the lengths of each joint of the phalanges (MCF, IFP) with its center of mass; \(m_{1}\) and \(m_{2}\) are the masses of each section of the mechanism; \(q_{1}\) and \(q_{2}\) are the joint coordinates of the proximal phalanx and the middle phalanx, respectively, and \(I_{1}\) and \(I_{2}\) represent the inertia tensor for each link. Using Figure 5, the Newton-Euler equations for the dynamic model for link \(d_{1}\) are:  
\[\mathbf{f}_{0,1}-\mathbf{f}_{1,2}+m_{1}\mathbf{g}-m_{1}\dot{\mathbf{V}}_{c1}=0 \tag{31}\]  
\[\mathbf{N}_{0,1}-\mathbf{N}_{1,2}+\mathbf{r}_{1,c1}\times\mathbf{f}_{1,2}- \mathbf{r}_{0,c1}\times\mathbf{f}_{0,1}-\mathbf{I}_{1}\dot{\omega}_{1}=0\] (32  
\[\mathbf{N}_{1,2}-\mathbf{r}_{1,\angle 2}\times\mathbf{f}_{1,2}-\mathbf{I}_{2} \dot{\omega}_{2}=0 \tag{34}\]  
where \(\boldsymbol{V}_{i}\) represents the velocity of the centroid for each link, \(\omega_{i}\) represents the angular velocity for link \(i\), \(\dot{\omega}_{i}\) represents the angular acceleration for link \(i\). To obtain a closed-form for dynamic equations, first, eliminate the constraint forces and separate them from the joint torques, to explicitly involve the joint torques in the dynamic equations. For Figure 5, joint torques \(\tau_{1}\) and \(\tau_{2}\) are equal to the coupling moments:  
\[N_{i-1,i}=\tau_{i},i=1,2 \tag{35}\]  
Substituting Equations (34) and (35) into Equation (32) gets  
\[\tau_{1}-\tau_{2}-\mathbf{r}_{0,c1}\times m_{1}\dot{\mathbf{V}}_{c1}-\mathbf{r }_{0,1}\times m_{2}\dot{\mathbf{V}}_{c2}+\mathbf{r}_{0,c1}\times m_{1}\mathbf{ g}+\mathbf{r}_{0,1}\times m_{2}\mathbf{g}-I_{1}\dot{\omega}_{1}=0 \tag{36}\]  
\[\tau_{2}-\mathbf{r}_{1,c2}\times m_{2}\dot{\mathbf{V}}_{c2}+m_{2}\mathbf{g}- I_{2}\dot{\omega}_{2}=0 \tag{37}\]  
Rewriting Equations (36) and (37) in terms of \(q_{i}\) and linear velocity \(\boldsymbol{V}_{ci}\) is  
\[V_{c1}=\left( \begin{array}{c}-d_{c1}\dot{q}_{1}\sin q_{1}\\ -d_{c1}\dot{q}_{1}\cos q_{1}\end{array} \right) \tag{38}\]  
\[V_{c2}=\left( \begin{array}{c}-\{d_{1}\sin q_{1}+d_{c2}\sin(q_{1}+q_{2})\} \dot{q}_{1}-d_{c2}\{\sin(q_{1}+q_{2})\}\dot{q}_{2}\\ \{\dot{d}_{1}\cos q_{1}+d_{c2}\cos(q_{1}+q_{2})\}\dot{q}_{1}+d_{c2}\{\cos(q_{1 }+q_{2})\}\dot{q}_{2}\end{array}