# Validation of Cable-Driven Experimental Setup to Assess Movements Made with Elbow Joint Assistance
## 2 Methodology
### Static Model  
A static model is developed for the above experimental setup--refer to Figure 1--wherein the position of the upper bar is vertical, and the lower bar flexes and extends with respect to it. The schematic of a representative position is shown in Figure 3(I). The joint flexion is described by considering the internal angle of the joint and is denoted as '\(\theta\)'. The angle between the cable and the lower bar is '\(\alpha\)'. The mass of the load at the end of the lower bar in the experimental setup is denoted as '\(m\)', while the mass of the lower bar itself is  
Figure 2: (**a**) Experimental setup to test cable under cyclic loading [30]. (**b**) Bar graph highlighting the number of cycles to failure for different cables under different weights. \({}^{\prime}m_{f}\)', the mass of the load cell attachment is '\(m_{lc}\)', and the length of the forearm bar is '\(l\)'. The corresponding notations are highlighted in Figure 3.  
Let the geometric parameter be as follows:  
\[d_{1}=\|OA\|,\ d_{2}=\|OC\|,\ l_{s}=\|DE\|,\ d_{1}^{\prime}=\|DI\|,\ d_{2}^{ \prime}=\|EI\|,\ l=\|OB\| \tag{2}\]  
From Figure 3(II), \(\beta+\pi-\theta=\dfrac{\pi}{2}\). The static equilibrium of forces and moments can be defined as in Equations (3) and (4)  
\[\overrightarrow{F}+\overrightarrow{T_{s}}+\overrightarrow{(m+m_{f}+m_{lc}) }\widehat{g}=\overrightarrow{0} \tag{3}\]  
\[\left(\overrightarrow{OD}\times\overrightarrow{T_{s}}\right)+\left( \overrightarrow{OB}\times\overrightarrow{mg}\right)+\left(\overrightarrow {OG}\times\overrightarrow{m_{f}}\widehat{g}\right)+\left(\overrightarrow {OA}\times\overrightarrow{m_{lc}}\widehat{g}\right)=\overrightarrow{0} \tag{4}\]  
where \(\overrightarrow{F}\) represents the reaction force generated at the joint under the action of cable tension, \(\overrightarrow{T_{s}}\), and the gravity forces are \(\overrightarrow{m_{f}g}\), \(\overrightarrow{m_{lc}g}\), and \(\overrightarrow{mg}\). Here, \(g\) represents the acceleration due to gravity and \(G\) represents the center of mass of the lower bar relative to the joint \(O\). Let \(T_{s}\) be the magnitude of \(\overrightarrow{T_{s}}\). The components of the reaction force \(\overrightarrow{F}\) in the \(x\) and \(y\) directions are represented in Equation (5). \[\left. \begin{array}{c}F_{x}-T_{s}\text{cos}\left(\alpha+\beta\right)=0,\\ F_{y}+T_{s}\text{sin}\left(\alpha+\beta\right)=\left(m+m_{f}+m_{lc}\right)g, \end{array} \right\} \tag{5}\]  
With \(\beta=\theta-\dfrac{\pi}{2}\) we obtain  
\[\left. \begin{array}{c}F_{x}-T_{s}\text{cos}\left(\alpha+\theta-\dfrac{\pi}{ 2}\right)=0,\\ F_{y}+T_{s}\text{sin}\left(\alpha+\theta-\dfrac{\pi}{2}\right)=\left(m+m_{f}+ m_{lc}\right)g.\end{array} \right\} \tag{6}\]  
Figure 3: Schematic representation of the experimental setup. **(I)** highlights the geometrical parameters and **(II)** highlights the angles. Angles are positive in an anti-clockwise direction, with **(III)** highlighting the various forces on the experimental setup and moment at the elbow joint. The reaction forces at the joint can also be written as  
\[F_{x}-T_{s}\sin(\alpha+\theta)=0,\] \[F_{y}-T_{s}\cos(\alpha+\theta)=\left(m+m_{f}+m_{lc}\right)g. \tag{7}\]  
The anchor points on both bars and the cable between them form a triangle, as shown in Figure 3(I). Subsequently, the triangle properties can be used to calculate the unknown variables \(\alpha\), \(\gamma\), \(\delta\), \(d_{1}^{\prime}\), \(d_{2}^{\prime}\), and \(l_{s}\). With the weight loaded on the experimental setup, the turning moment \(\tau\) required to hold the joint at any particular joint position \(\theta\in(0,\pi)\) can be formulated as shown below. \[\tau=(mgl+m_{f}g\frac{l}{2}+m_{lc}gd_{1})\sin\theta, \tag{8}\]  
Equation (8) determines the minimum moment to be produced by the pulling cable for the actuation of the forearm.