# Dynamic Modeling and Simulation of a Cyclic Towing System for Underwater Vehicles
## 1 Introduction
proposed a control strategy based on F+PID fuzzy switching controls for underwater motion control of a double flexible cable towing-type catapult test carriage [7]. [PERSON] et al. proposed a single flexible cable towing-type catapult test carriage system based on a self-submersible tension leg platform, developed the mathematical model for the single flexible cable towing carriage system, and conducted research on high-precision speed control technology for the track carriage [8]. [PERSON] et al. developed a joint space-based dynamic model for an underwater tension leg platform driven by hydraulic winches, proposed a dual-loop control strategy, and conducted research on the dynamic modeling and control issues of the test platform [9]. [PERSON] et al. designed a double winch structure for use in a cyclic towing system, calculated the structural parameters of the winch, and conducted a finite element analysis of the structure using ANSYS, demonstrating the reliability of the optimized structural design compared to traditional winches [10]. [PERSON] et al. calculated the operating parameters of a carriage truss in a self-propelled towing system and conducted a finite element analysis of the main beam of the carriage and the central testing platform's lifting rod, providing a foundation for the system's operational stability [11,12]. [PERSON] et al. used ANSYS finite element software to perform vibration and mechanical analysis on the key parts of the towing system's carriage chassis, verifying its structural characteristics [13]. [PERSON] et al. proposed  
Figure 1: Schematic diagram of the towing system. a two-part underwater towing system for near-surface towing experiments, obtained hydrodynamic coefficients through experimental methods, and developed a mathematical model of the two-part towing system, demonstrating that the lift compensation device can improve system stability by reducing depth fluctuations of the ballast [14]. This study establishes a dynamic model for an underwater cyclical test platform system using the Lagrangian equation and implements a program in the MATLAB (2023b) environment, employing the fourth- to fifth-order Runge-Kutta algorithm for dynamic simulation analysis. To verify the accuracy of the proposed method, a comparison is made between the results for displacement, velocity, acceleration, and tension obtained using the multibody dynamics software ADAMS (v.24.1) and the Lagrangian equation for the same example. Moreover, compared to ADAMS, the Lagrangian method offers unique advantages. For instance, the cable model, which consists of a spring and damper, can address cable elongation--a problem that ADAMS cannot solve--and is analyzed separately in Section 4 of this paper. The focus of this study is to investigate the impact of key parameters, such as different input torques and tension forces, on the speed fluctuations, cable deformation, and tension of underwater vehicles. Cyclical systems are widely used in modern society across various devices and platforms, such as cyclical test stands and elevators. This research aims to develop an accurate dynamic model for such cyclical systems, providing scientific guidance for their design and optimization to meet the needs of practical engineering applications.