# Validation of Cable-Driven Experimental Setup to Assess Movements Made with Elbow Joint Assistance
## 2 Methodology
### Static Model
The moment produced by the cable, \(\tau_{s}\), with the known distance between the actuation point and the joint, can be expressed as shown in Equation (9)  
\[\tau_{s}=T_{s}\|OD\|\sin(\alpha+\gamma). \tag{9}\]  
where \(\gamma=\tan^{-1}\!\left(\frac{\|AD\|}{d_{1}}\right)\) and \(\|OD\|=\frac{d_{1}}{\cos(\gamma)}\). To maintain a static equilibrium, \(\tau\) and \(\tau_{s}\) should be equal. Now, upon equating Equations (8) and (9), we obtain  
\[T_{s}=\frac{(mgl+m_{f}g\frac{l}{2}+m_{lc}gd_{1})\sin\theta}{\|OD\|\sin(\alpha+ \gamma)}. \tag{10}\]  
With \(\theta-\frac{\pi}{2}=\delta+\gamma\), the coordinates of points D and E can be written as in Equation (11),  
\[\left. \begin{array}{c}D_{x}=\|OD\|\sin(\gamma-\theta)\\ D_{y}=-\|OD\|\cos(\gamma-\theta)\\ E_{x}=\|CE\|\\ E_{y}=d_{2}\end{array} \right\} \tag{11}\]  
Let \(I\) be the intersection of lines \(\overline{EI}\) and \(\overline{DI}\)--refer to Figure 3--the coordinates \(I_{x}\) and \(I_{y}\) can be defined as in Equation (12), where \(a=-\cot(\theta)\) represents the slope and \(c=D_{y}-aD_{x}\). \[\left. \begin{array}{c}I_{x}=E_{x}\\ I_{y}=aE_{x}+c\end{array} \right\} \tag{12}\]  
The value of \(\alpha\) can be obtained as follows, from Figure 3(I), using the sine equality  
\[\frac{l_{s}}{\sin\theta}=\frac{d_{2}^{{}^{\prime}}}{\sin\alpha}. \tag{13}\]  
From distances \(d_{1}^{{}^{\prime}}\) and \(d_{2}^{{}^{\prime}}\), the length of the cable \(l_{s}\) can be calculated  
\[l_{s}=\sqrt{{d_{1}^{{}^{\prime}}}^{2}+{d_{2}^{{}^{\prime}}}^{2}-2d_{1}^{{}^{ \prime}}d_{2}^{{}^{\prime}}\cos\theta}. \tag{14}\]Using Equations (13) and (14), the relation between \(\theta\) and \(\alpha\) is represented as in Equation (15), which is true where the inverse sine function is monotonous. \[\alpha=\sin^{-1}\left(\frac{d_{s}^{\prime}\sin\theta}{\sqrt{{d_{1}^{\prime}}^{2 }+{d_{2}^{\prime}}^{2}-2d_{1}^{\prime}d_{2}^{\prime}\cos\theta}}\right) \tag{15}\]  
Using the cable tension \(T_{s}\), the moments in the anchor points A and C can be evaluated using Equation (16), where \(\overrightarrow{EMot}\) represents the direction of point \(E\) to motor location \(Mot\). Segments \(AD\) and \(CE\) are the height of the anchor points on the lower bar and upper bar, respectively. \[\left. \begin{array}{c}M_{A}=\overrightarrow{AD}\times\left(\frac{ \overrightarrow{DE}}{\|DE\|}\right)T_{s}\\ \\ M_{C}=\overrightarrow{CE}\times\left(\frac{\overrightarrow{EMot}}{\|EMot \|}+\frac{\overrightarrow{ED}}{\|ED\|}\right)T_{s}\end{array} \right\} \tag{16}\]  
Although Equation (15) can take the values of the joint angle from for \(\theta\in(0,\pi)\), the actual range of motion of a human arm is different. From the anthropometric data, the range of motion of elbow joint (in terms of \(\theta\)) is between \(52^{\circ}\) and \(180^{\circ}\)[38]. From Equation (10) the tension in the cable, \(T_{s}\), for different weights attached to the lower bar in the experimental setup can be calculated. Additionally, from Equation (14) the length of the cable to be actuated, \(l_{s}\), by the motor for any given joint angle '\(\theta\)' can be calculated. The tension in the cable calculated by the static model for a load at a certain joint position can be experimentally validated.