## 2 System Modeling of MATC  
To achieve steady straight-line sailing in an MATC system, the first step is to establish a dynamic model of the system. Force analysis, as the foundation of system modeling, plays a critical role in understanding and designing the system. By conducting a thorough force analysis, the interactions and effects of various forces acting on each AUV unit during its movement in water can be identified, providing a theoretical basis for subsequent dynamic modeling and control strategy development. Steady straight-line sailing refers to the entire MATC system moving in a straight line at a constant speed while maintaining stable relative distances between the AUV units. Achieving this requires applying appropriate thrust to each AUV unit, ensuring the system moves uniformly and that the inter-unit distances remain within the desired range. This balance is essential for the stability and efficiency of the MATC system during its operations. For the purposes of analysis, the following assumptions are made. The shape, center of mass, center of buoyancy, and thrust position of each AUV unit in the MATC system are identical. The initial state assumes that the gravitational force is equal to the buoyant force. The vertical motion is neglected, and the MATC system maintains horizontal straight-line motion during operation. The gravitational and buoyant forces of the inter-unit connection structure are negligible. The relationship between spring deformation and the force between units is accurately described. The thrust applied to each side of the units is synchronized and equal in magnitude. The distances between units are equal when the springs are undeformed. The thrust direction of each unit is forward, and all units share identical performance characteristics. These assumptions simplify the analysis while providing a foundation for the dynamic modeling and control strategy development. The MATC system is subject to several interconnected forces during straight-line sailing. These forces include thrust, drag, and reactive forces from adjacent units. The thrust is generated by the propellers on both sides of each AUV unit. The drag is caused by fluid resistance. To simplify the coupling effects between units, it is assumed that the reaction forces are caused by spring deformation. Figure 1 shows an MATC system composed of three AUV units. Figure 2 shows a schematic of the forces acting on the MATC system. In this figure: \(n\) represents the number of AUV units in the MATC system; \(L_{a}\) represents the length of each AUV unit; \(u_{i}\), \(l_{i}\), and \(r_{i}\) represent the total thrust, the left thrust, and the right thrust of the \(i\)-th unit, respectively; \(w_{i}\) represents the fluid drag acting on the \(i\)-th unit; \(f_{i(i+1)}\) represents the reactive force that unit \(i\) exerts on unit \(j\), and \(f_{(i+1)i}\) represents the reactive force that unit \(j\) exerts on unit \(i\). During straight-line sailing, the following relations are satisfied: \(l_{i}=r_{i}\); \(u_{i}=l_{i}+r_{i}\). Based on [PERSON]'s second law, the dynamic equations for each AUV unit are as follows:  
\[\left\{ \begin{aligned} m_{1}\dot{v}_{1}&=u_{1}-w_{1}- f_{21}\\ &\vdots\\ m_{i}\dot{v}_{i}&=u_{i}-w_{i}+f_{(i-1)i}-f_{(i+1)i}i=2, \cdots,n-1\\ &\vdots\\ m_{n}\dot{v}_{n}&=u_{n}-w_{n}+f_{(n-1)n}\end{aligned} \right. \tag{1}\]  
where \(m_{i}\), \(v_{i}\), and \(\dot{v}_{i}\) represent the mass, velocity, and acceleration of the \(i\)-th unit, respectively. Figure 1: An MATC system composed of 3 AUV units. Figure 2: Schematic of the forces acting on the MATC system. If the spring coefficient between units is \(h_{i(i+1)}\), the above equation can be rewritten as:  
\[\left\{ \begin{aligned} m_{1}\dot{v}_{1}=u_{1}-w_{1}-h_{12}(x_{12}-L_{ b})\\ m_{i}\dot{v}_{i}=u_{i}-w_{i}+h_{(i-1)i}\Big{(}x_{(i-1)i}-L_{b}\Big{)}\\ -h_{i(i+1)}\Big{(}x_{i(i+1)}-L_{b}\Big{)},i=2,\cdots,n-1\\ m_{n}\dot{v}_{n}=u_{n}-w_{n}+h_{(n-1)n}\Big{(}x_{(n-1)n}-L_{b}\Big{)} \end{aligned} \right.