## 3 Theoretical Model  
The yaw and heading movements of the submarine will cause a shift in the coordinate system defined with the submarine as a reference. The spatial absolute coordinate system in the Fluent platform based on finite volume methods is fixed, and relative and rotating coordinate systems can only be defined through specific motion laws, unable to accommodate unpredictable real-time changes in coordinates. For the six degrees of freedom motion of the pump-jet vector thrust submarine, the International Towing Tank Conference (ITTC) recommended that a coordinate system can be used, as shown in Figure 3. The ITTC coordinate system includes an absolute coordinate system and a body-fixed coordinate system. The absolute coordinate system is denoted as \(E\)-\(\xi\eta\zeta\), while the body-fixed coordinate system fixed on the submarine is represented by \(O\)-\(xyz\). The origin of the body-fixed coordinate system coincides with the centroid of the submarine, with the longitudinal axis \(O_{x}\) parallel to the submarine's baseline, the transverse axis \(O_{y}\) parallel to the submarine's waterline, pointing to the starboard side, and the vertical axis \(O_{z}\) perpendicular to the \(O_{xy}\) plane, with a positive direction towards the submarine's bottom. The body-fixed coordinate system _O-xyz_ forms a right-handed Cartesian coordinate system [44]. The motion parameters for both the absolute and body-fixed coordinate systems are detailed in Table 2.  
At the initial moment when the fixed coordinate system and body-fixed coordinate system overlap, they form an angle as the body-fixed coordinate system moves with the submarine's motion, resulting in linear velocity and angular velocity relative to the fixed coordinate system. Therefore, the six degrees of freedom equations of motion for the pump-jet vector thrust submarine are first established in the fixed coordinate system and are then projected onto the body-fixed coordinate system to obtain the kinematic equations of motion in the body-fixed coordinate system. The transformation equations between the fixed coordinate system and the body-fixed coordinate system are shown in Equations (1) and (2) [44], where \(\psi\) represents the roll angle, \(\theta\) represents the pitch angle, and \(\varphi\) represents the yaw angle. \[\left[ \begin{array}{c}x\\ y\\ z\end{array} \right]=A\left[ \begin{array}{c}\xi\\ \eta\\ \zeta\end{array} \right] \tag{1}\]  
\[A=\left( \begin{array}{ccc}\cos\psi\cos\theta&\cos\psi\sin\theta\sin\varphi- \sin\psi\cos\varphi&\cos\psi\sin\theta\cos\varphi+\sin\psi\sin\varphi\\ \sin\psi\cos\theta&\sin\psi\sin\theta\sin\varphi+\cos\psi\cos\varphi&\sin\psi \sin\theta\cos\varphi-\cos\psi\sin\varphi\\ -\sin\theta&\cos\theta\sin\varphi&\cos\theta\cos\varphi\end{array} \right) \tag{2}\]  
The pump-jet vector thrust submarine exhibits three-dimensional six degrees of freedom (6 DOF) motion characteristics, making theoretical research challenging due to the complexity of multi-dimensional spatial movements. By combining hydrodynamic characteristics of the submarine with laws of physics that are independent of the motion process and only related to the motion state, the general equations of motion for the submarine can  
Figure 3: Pump-jet vector propulsor submersible coordinate system. \begin{table}
\begin{tabular}{c c c c} \hline
**Vector Quantity** & _x-Axis_ & _y-Axis_ & _z-Axis_ \\ \hline Speed \(U\) & \(u\) & \(v\) & \(w\) \\ Angular velocity \(\Omega\) & \(p\) & \(q\) & \(r\) \\ Force \(F\) & \(X\) & \(Y\) & \(Z\) \\ Torque _Mall_ & \(K\) & \(M\) & \(N\) \\ \hline \end{tabular}
\end{table} 
Table 2: Projection of motion parameters and random coordinates of force. be derived based on [PERSON]'s equations, the theorem of momentum for the center of mass, and the theorem of momentum about an axis (as shown in Equation (3) [45]), where \(I_{x}\), \(I_{y}\), and \(I_{z}\) represent the moments of inertia of the submarine about the \(O_{x}\), \(O_{y}\), and \(O_{z}\) axes of the body-fixed coordinate system and \(m\) is the mass of the submarine.