# Equivalent Dynamic Analysis of a Cable-Driven Snake Arm Maintainer
## 3 Methods
### Equivalent Joint Dynamics  
The equivalent joint dynamics formed by the decoupling of the cable traction can control the SAM's movement. The cable traction forces of each joint of the SAM are dynamically coupled. Moreover, each joint involves some acceleration, meaning the external force and external torque received are also unbalanced. Solving the cable traction force using traditional force analysis and dynamic equations is highly complicated. In this paper, we propose an equivalent dynamics modelling method for the strongly coupled dynamic characteristics of each joint cable. The effects of all joint cable traction forces are forcibly equated to joint torques to create a dynamic model. Then, the joint torque can be converted into the traction force of the cable through the forcibly equivalent inverse transformation, and the real-time decoupling calculation of the cable traction force can be realized. First, the equivalent joint dynamics of the end joint of the SAM are decoupled in any given pose. As shown in Figure 6, using force analysis, the equivalent transformation of the cable traction force (\(\mathbf{F_{1}^{10}},\mathbf{F_{2}^{10}},\mathbf{F_{3}^{10}}\)) and the joint driving torque (\(T_{1}^{10},T_{2}^{10}\)) can be realized. Then, the forcibly equivalent SAM dynamics equations can be obtained by traditional dynamics modelling methods (Lagrangian, Newtonian-Euler, etc.). As shown in Figure 6, assume that the equivalent joint torques (\(T_{1}^{10},T_{2}^{10}\)) are known, and solve for the cable traction force (\(\mathbf{F_{1}^{10}},\mathbf{F_{2}^{10}},\mathbf{F_{3}^{10}}\)). First, solve the direction vectors of \(\mathbf{F_{1}^{10}},\mathbf{F_{2}^{10}},\mathbf{F_{3}^{10}}\) and decompose them to the coordinate systems \((x_{2},y_{2},z_{2})\) and \((x_{3},y_{3},z_{3})\) of point \(\mathbf{A_{2}}\). Then, we solve the value of \(\mathbf{F_{1}^{10}},\mathbf{F_{2}^{10}},\mathbf{F_{3}^{10}}\) via the forcibly equivalent method. Because a single joint is controlled by three cables and has the characteristic of redundant driving, countless solutions exist. To limit the number of solutions and avoid reverse tension, set the minimum cable preload force to \(10\) N. By analyzing the cable layout of the end joints, the following is obtained: \(\mathbf{F_{3}^{10}}=10\) N.  
Figure 5: Traction force distribution of each cable in the initial pose. The following could be used to solve the direction vectors of \(\mathbf{F_{1}^{10}},\mathbf{F_{2}^{10}}\), \(\mathbf{F_{3}^{10}}\) in the coordinate system: \((x_{2},y_{2},z_{2})\) and \((x_{3},y_{3},z_{3})\) of point \(\mathbf{A_{2}}\). The coordinates of point \(B\) and \(\mathbf{E}\) in the coordinate system \((x_{2},y_{2},z_{2})\) are:  
\[\mathbf{B}(x_{B2},y_{B2},z_{B2})=\mathbf{Tr}(0,-h,0) \begin{bmatrix}-rsin\alpha_{1}^{10 }\\ 0\\ rcos\alpha_{1}^{10}\\ 1\end{bmatrix}  \tag{9}\]  
\[\mathbf{E}(x_{E2},y_{E2},z_{E2})=\mathbf{R}(x_{2},\gamma_{d})\mathbf{R}(z_{3},\delta_{d}) \mathbf{Tr}(0,h,0) \begin{bmatrix}-rsin\alpha_{1}^{10}\\ 0\\ rcos\alpha_{1}^{10}\\ 1\end{bmatrix}  \tag{10}\]  
where \(2h\) is the length of the universal joint, \(\gamma_{d}\) is the rotation angle of the universal joint around the \(x_{2}\) axis, \(\delta_{d}\) is the rotation angle of the universal joint around the \(z_{3}\) axis, and \(\alpha_{1}^{10}\) is the rotation angle of the cable hole around the axis of the joint.