# Equivalent Dynamic Analysis of a Cable-Driven Snake Arm Maintainer
## 3 Methods
### Static Cable Force  
To summarize the traction laws of SAMs, the force balance and torque balance methods can be used to analyze the cable traction characteristics of each joint of the snake arm along the horizontal direction of the \(Y\)-axis. As shown in Figure 4, we took the \(i\)-th joint as an example for static force analysis and decoupling calculations. This joint involves a total of seven external forces, where \(\mathbf{F_{1}^{i}},\mathbf{F_{2}^{i}},\mathbf{F_{3}^{i}}\) are the cable traction force; \(\mathbf{F_{4}^{i}}\) is the supporting force of the joint; \(\mathbf{F_{4}^{i+1}}\) is the reverse supporting force of the next joint; and \(\mathbf{G_{1}^{i}},\mathbf{G_{2}^{i}}\) are the gravity of the joint and the universal joint, respectively. Joint \(i\) is also in a state of force balance and torque balance.  
Torque balance analysis of \(\mathbf{F_{1}^{i}},\mathbf{F_{2}^{i}},\mathbf{F_{3}^{i}}\) indicates that \(\mathbf{F_{3}^{i}}\) is in a state of reaction torque. To avoid \(\mathbf{F_{3}^{i}}\) being negative, set \(\mathbf{F_{3}^{i}}=10\) N. From the torque balance around the \(Z\)-axis and \(X\)-axis, the following can be obtained:  
\[\mathbf{F_{1}^{i}}r_{i}s\alpha_{1}^{i}+\mathbf{F_{3}^{i}}r_{i}s\alpha_{3}^{i}+\mathbf{F_{2 }^{i}}r_{i}s\alpha_{2}^{i}=0 \tag{3}\]  
\[\mathbf{F_{1}^{i}}r_{i}c\alpha_{1}^{i}+\mathbf{F_{2}^{i}}r_{i}c\alpha_{2}^{i}+\mathbf{F_{3 }^{i}}r_{i}c\alpha_{3}^{i}+l\left(\mathbf{G_{2}^{i}}+\mathbf{F_{4}^{i+1}s}\beta_{i+1}+ \frac{\mathbf{G_{1}^{i}}}{2}\right)=0 \tag{4}\]  
where \(r_{i}\) is the distance from the cable-fixing hole to the axis of the joint, \(\alpha_{1}^{i}\) is the rotation angle of the first cable-fixing hole around the axis of the joint, and \(\alpha_{2}^{i}=\alpha_{1}^{i}+\frac{2\pi}{3}\), \(\alpha_{3}^{i}=\alpha_{1}^{i}-\frac{2\pi}{3}\). \(\beta_{i}\) is the angle between the joint reaction force and the axis of the joint. Therefore:  
\[\mathbf{F_{1}^{i}}=\frac{F_{3}^{i}r_{i}\big{(}s\alpha_{2}^{i}c\alpha_{3}^{i}- \alpha_{2}^{i}s\alpha_{3}^{i}\big{)}+\left(G_{2}^{i}+F_{4}^{i+1}s\beta_{i+1}+ \frac{G_{2}^{i}}{2}\right)ls\alpha_{2}^{i}}{r_{i}\big{(}s\alpha_{2}^{i}\alpha_ {1}^{i}-c\alpha_{2}^{i}s\alpha_{1}^{i}\big{)}} \tag{5}\]  
\[\mathbf{F_{2}^{i}}=\frac{F_{3}^{i}r_{i}\big{(}s\alpha_{1}^{i}c\alpha_{3}^{i}- \alpha_{1}^{i}s\alpha_{3}^{i}\big{)}+\left(G_{2}^{i}+F_{4}^{i+1}s\beta_{i+1}+ \frac{G_{2}^{i}}{2}\right)ls\alpha_{1}^{i}}{r_{i}\big{(}s\alpha_{2}^{i}\alpha_ {1}^{i}-c\alpha_{2}^{i}s\alpha_{1}^{i}\big{)}} \tag{6}\]  
Figure 4: Force analysis of any joint \(i\) balance state.  
From the force balance relationship, the following is obtained:  
\[\mathbf{F}_{4}^{i}=\sqrt{\left(F_{1}^{i}+F_{2}^{i}+F_{3}^{i}+F_{4}^{i+1}c\beta_{i+1} \right)^{2}+\left(G_{1}^{i}+G_{2}^{i}+F_{4}^{i+1}s\beta_{i+1}\right)^{2}} \tag{7}\]  
\[\beta_{i}=atan\frac{G_{1}^{i}+G_{2}^{i}+F_{4}^{i+1}s\beta_{i+1}}{F_{1}+F_{2}+F_ {3}+F_{4}^{i+1}c\beta_{i+1}} \tag{8}\]  
Set the initial pose of the SAM as horizontally forward along the \(Y\)-axis, and set the end load \(\mathbf{G}\) to \(0\) N. Using the above algorithm to solve the cable traction force of each joint in the initial state, the cable traction force distribution map is obtained, as shown in Figure 5.