# Dynamic Modeling and Simulation of a Cyclic Towing System for Underwater Vehicles
## 2 Model Development
### System Dynamics Modeling
\begin{array}{l}U=\frac{1}{2}k_{0}x_{1}^{2}+\frac{1}{2}k_{a1}(x_{a1}-x_{2}- r_{2}\theta_{b})^{2}+\frac{1}{2}k_{a2}(x_{3}-x_{a1})^{2}\\ +\frac{1}{2}k_{a3}(x_{a2}-x_{3})^{2}+\frac{1}{2}k_{a4}(x_{1}+x_{a2}-r_{1}\theta _{a})^{2}+\frac{1}{2}k_{a5}(x_{1}+x_{a3}+r_{1}\theta_{a})^{2}\\ +\frac{1}{2}k_{a6}(x_{2}-x_{a3}-r_{2}\theta_{b})^{2}+\frac{1}{2}k_{7}x_{2}^{2 }\end{array}  \tag{3}\]  
The system's dissipation term \(D\) is given by:  
\[ \begin{array}{l}D=\frac{1}{2}\epsilon_{0}\dot{x}_{1}^{2}+\frac{1}{2}c_{a1} (\dot{x}_{a1}-\dot{x}_{2}-r_{2}\dot{\theta}_{b})^{2}+\frac{1}{2}c_{a2}(\dot{ x}_{3}-\dot{x}_{a1})^{2}\\ +\frac{1}{2}c_{a3}(\dot{x}_{a2}-\dot{x}_{3})^{2}+\frac{1}{2}c_{a4}(\dot{x}_{1} +\dot{x}_{a2}-r_{1}\dot{\theta}_{a})^{2}+\frac{1}{2}c_{a5}(\dot{x}_{1}+\dot{ x}_{a3}+r_{1}\dot{\theta}_{a})^{2}\\ +\frac{1}{2}c_{a6}(\dot{x}_{2}-\dot{x}_{a4}-r_{2}\dot{\theta}_{b})^{2}+\frac{ 1}{2}c\gamma\dot{x}_{2}^{2}\end{array}  \tag{4}\]  
In the process of analyzing the system, only the lateral vector of the model is considered, without taking into account the longitudinal vector of the model, so the influence of gravity and buoyancy in the system are not considered. Based on Equations (1)-(4), the equations of motion are derived from the Lagrange equation as a system of differential equations:  
\[\mathbf{M}\ddot{\mathbf{x}}+\mathbf{C}\dot{\mathbf{x}}+\mathbf{K}\mathbf{x}= \mathbf{F} \tag{5}\]  
In the equation:  
\[\mathbf{M}=diag(m_{1}\quad m_{2}\quad m_{3}\quad m_{a1}\quad m_{a2}\quad m_{ a3}\quad J_{1}\quad J_{2})\]  
Figure 3: Structural force analysis of the cyclic towing system. Figure 2: Dynamic model of the cyclic towing system. \[\mathbf{K}=\left[ \begin{array}{cccccccc}k_{0}+k_{a4}+k_{a5}&0&0&0&k_{a4}&k_{a5}&(- k_{a4}+k_{a5})r_{1}&0\\ 0&k_{a1}+k_{a6}+k_{7}&0&-k_{a1}&0&-k_{a6}&0&(k_{a1}-k_{a6})r_{2}\\ 0&0&k_{a2}+k_{a3}&-k_{a2}&-k_{a3}&0&0&0\\ 0&-k_{a1}&-k_{a2}&k_{a1}+k_{a2}&0&0&0&-k_{a1}r_{2}\\ k_{a4}&0&-k_{a3}&0&k_{a3}+k_{a4}&0&-k_{a4}r_{1}&0\\ k_{a5}&-k_{a6}&0&0&0&k_{a5}+k_{a6}&k_{a5}r_{1}&k_{a6}r_{2}\\ (-k_{a4}+k_{a5})r_{1}&0&0&0&-k_{a4}r_{1}&k_{a5}r_{1}&(k_{a4}+k_{a5})r_{1}^{2}&0 \\ 0&(k_{a1}-k_{a6})r_{2}&0&-k_{a1}r_{2}&0&k_{a6}r_{2}&0&(k_{a1}+k_{a6})r_{2}^{2} \end{array} \right]\]  
\[\mathbf{C}=\left[ \begin{array}{cccccccc}c_{0}+c_{a4}+c_{a5}&0&0&0&c_{a4}&c_{ a5}&(-c_{a4}+c_{a5})r_{1}&0\\ 0&c_{a1}+c_{a6}+c_{7}&0&-c_{a1}&0&-c_{a6}&0&(c_{a1}-c_{a6})r_{2}\\ 0&0&c_{a2}+c_{a3}&-c_{a2}&-c_{a3}&0&0&0\\ 0&-c_{a1}&-c_{a2}&c_{a1}+c_{a2}&0&0&0&-c_{a1}r_{2}\\ c_{a4}&0&-c_{a3}&0&c_{a3}+c_{a4}&0&-c_{a4}r_{1}&0\\ c_{a5}&-c_{a6}&0&0&0&c_{a5}+c_{a6}&c_{a5}r_{1}&c_{a6}r_{2}\\ (-c_{a4}+c_{a5})r_{1}&0&0&0&-c_{a4}r_{1}&c_{a5}r_{1}&(c_{a4}+c_{a5})r_{1}^{2}& 0\\ 0&(c_{a1}-c_{a6})r_{2}&0&-c_{a1}r_{2}&0&c_{a6}r_{2}&0&(c_{a1}+c_{a6})r_{2}^{2} \end{array} \right]\]  
\[\mathbf{x}=\left[ \begin{array}{cccccccc}x_{1}&x_{2}&x_{3}&x_{a1}&x_{a2}&x_{a 3}&\theta_{a}&\theta_{b}\end{array} \right]^{\mathrm{T}}\]  
\[\dot{\mathbf{x}}=\left[ \begin{array}{cccccccc}\dot{x}_{1}&\dot{x}_{2}&\dot{x }_{3}&\dot{x}_{a1}&\dot{x}_{a2}&\dot{x}_{a3}&\dot{\theta}_{a}&\dot{\theta}_{b} \end{array} \right]^{\mathrm{T}}\]  
\[\tilde{\mathbf{x}}=\left[ \begin{array}{cccccccc}\ddot{x}_{1}&\ddot{x}_{2}& \ddot{x}_{3}&\ddot{x}_{a1}&\ddot{x}_{a2}&\ddot{x}_{a3}&\ddot{\theta}_{a}&\ddot {\theta}_{b}\end{array} \right]^{\mathrm{T}}\]  
\[\mathbf{F}=\left[ \begin{array}{cccccccc}F_{1}&F_{2}&F_{3}&F_{4}&F_{5}&F_{6}& T_{1}&T_{2}\end{array}