# Dynamic Modeling and Simulation of a Cyclic Towing System for Underwater Vehicles
## 3 Model Validation  
To ensure the validity of the dynamic model and the simulation of the cyclic towing system, it is necessary to validate the dynamic model discussed earlier. This section primarily focuses on conducting simulation experiments on the system model using MATLAB simulation software and comparing the results with those obtained from the commercial software ADAMS. The ADAMS/cable module offers two methods for modeling pulley cable systems: the simplified method and the discrete method [17]. The simplified method provides a quick approach to modeling, where the effects of gravity, inertia, bending stiffness, and torsional stiffness on the cable during transmission are not considered; the discrete method, on the other hand, discretizes the cable into small spheres, which are interconnected by generalized forces. Compared to the simplified method, the discrete method incorporates the mass, inertia, bending stiffness, and torsional stiffness of the cable into the model, allowing for a more accurate representation of the transmission process \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Velocity** & **Drag** & **Drag Coefficient** & **Average Drag Coefficient** & **Water Density** & **Projected Area** \\
**(m/s)** & **(N)** & **Cd** & **Cofficient** & \(\rho(\text{kg/m}^{3})\) & **A (m\({}^{2}\))** \\ \hline
1 & 16.572 & 0.266 & & & \\ \hline
2 & 61.902 & 0.249 & & & \\ \hline
3 & 134.789 & 0.241 & & & \\ \hline
4 & 234.509 & 0.235 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Calculated results of the underwater vehicle’s drag and drag coefficient. between the cable and pulley. To improve the accuracy of the calculations, the discrete method was used for modeling in this study. For validation, the method presented in this paper and the ADAMS software applied the same torque at the drive wheel, and the distance between the towing wheel and the guiding wheel was set to 40 m, while a tension of 3500 N was applied to the cable. The displacement, velocity, and acceleration were compared to analyze the errors between the two methods. The parameters used in the validation of the Lagrangian equation are shown in Table 2. These parameters are derived from an actual physical project and represent data obtained through on-site measurements during project implementation. In this study, the ODE45 function in MATLAB was used to solve differential equations. ODE45 employs a fourth- to fifth-order Runge-Kutta algorithm, where the fourth-order method provides a candidate solution and the fifth-order method controls the error. This adaptive (variable-step) numerical solution method is used to integrate the differential equations of the variable coefficient cyclic towing system. The calculation flowchart is shown in Figure 4. In the commercial software ADAMS, the discrete model of a cyclic towing system is established using the parameterized integration module cable. The parameters for the cable in the model are set as follows: diameter of 17 mm, length of 40 m, Young's modulus of 6 \(\times\) 10\({}^{9}\) N/m\({}^{2}\), density of 970 kg/m\({}^{3}\), and an element count of 200. The contact parameters between the cable and pulley are defined as: contact stiffness of 10\({}^{4}\) N/mm, force exponent of 1, damping coefficient of 0.1, and friction coefficient of 0.6. The dimensions of the pulleys are set as follows: the driving pulley has a diameter of 0.8 m, and the guiding pulley has a diameter of 0.7 m with a width of 50 mm and depth of 20 mm. When applying external load fluid resistance, although ADAMS software lacks a dedicated module for fluid resistance calculations, this can be achieved by treating the fluid resistance as a unidirectional force applied to the center of mass of the underwater vehicle [18]. Since fluid resistance is related to the vehicle's speed, the software can monitor the speed of the underwater vehicle, substitute it into the fluid resistance formula, and then apply the calculated force to the model's center of mass.