# Motion Response of the Submersible Underwater Towed System as Cable Length Changes
## 1 Introduction  
A submersible underwater towed system must execute a range of maneuvers in accordance with the operational conditions and mission requirements during its mission. A towed system must maintain a consistent depth, heading, speed, and motion attitude to ensure navigation safety and gather seabed geological and marine hydrographic data. Additionally, it should be capable of quickly adjusting its depth, heading, speed, and motion attitude to comply with composite commands, such as obstacle avoidance and swift information retrieval. Special towed systems with lengthy cables and those requiring little impact from environmental forces will have more stringent control needs [1, 2]. This requires not only a submersible underwater towed system that can flexibly and reliably transition between steady-state and maneuvering motions but also a towed system that can dynamically adjust the length of the towed cable. Researchers have conducted studies on various maneuvering motions of underwater towed systems, including accelerated motion [3; 4], circular rotational motion [5; 6], and horizontal sinusoidal motion [7]. Researchers have conducted these studies using sea trials and numerical simulations to collect experimental data and create fundamental models. [PERSON] [8] developed a mathematical model that represents the underwater towed cable in three dimensions. This model is used to analyze the linear motion of the towed system when the speed varies. [PERSON] [9] examined the tension variation curves and shape of a towed cable in three distinct working conditions: no current, current in the same direction, and current in the opposite direction. These conditions were observed while the towed mother ship was experiencing accelerated motion. [PERSON] [10] conducted multiple sets of computational simulations to analyze the accelerated motion and U-turn motions. The numerical results were then compared with sea trial data to validate the accuracy and dependability of the simulations. [PERSON] [11] classified circular rotational motion into three states based on the ratio of the radius of gyration to the length of the towed cable in circular rotational towed motion: high-speed towed, low-speed towed, and a transient state between high and low speed. [PERSON] [12] suggested additional dimensionless coefficients to better describe the kinematic response phenomena that occur during circular rotational towed. These coefficients include the ratio of the mass of the towed cable to the mass of the towed body and the ratio of the unit mass length of the towed cable to the hydrodynamic force, based on [PERSON] [11]. [PERSON] [13] conducted a study on an underwater towed system that involved circular rotational motion at various radii, towed lengths, and speeds. The study revealed that the speed of rotation decreased from 9.85 m/s to 3.5 m/s, resulting in a significant decrease in the tension of the towed cable by 95 kN. [PERSON] [14] developed a hydrodynamic mathematical model for an underwater towed system that takes into account the changing length of the cable. The study confirmed that adjusting the length of the towed cable can effectively reduce fluctuations in the depth of the towed body and the tension in the cable when the mother ship is moving up and down. [PERSON] [15] utilized the lumped mass method to model the cable and successfully simulated the process of cable retraction and deployment by adjusting the number of mass points in the cable model. [PERSON] [16] developed a mathematical model for an underwater towed system of a submersible that takes into account the changing length of the cable. The model was used to analyze four different working conditions, each with varying release speeds and towing speeds. The results showed that reducing the towing speed or using an appropriate release length can enhance the stability of the towed body on the horizontal plane.