## 3 Theoretical Model
Decomposing the three-dimensional six degrees of freedom motion into horizontal plane motion and vertical plane motion, and neglecting forces and pitching moments on the vertical plane of the submarine during the study of horizontal plane motion, we can obtain the dynamic equations of motion for the submarine in the horizontal plane (as shown in Equation (4)). \[\left\{ \begin{array}{l}X=m\left(\overset{\bullet}{u}+qw-rv\right)\\ Y=m\left(\overset{\bullet}{v}+ru-pw\right)\\ Z=m\left(\overset{\bullet}{w}+pv-qu\right)\\ K=I_{x}\overset{\bullet}{p}+\left(I_{z}-I_{y}\right)q\tau\\ M=I_{y}\overset{\bullet}{q}+\left(I_{x}-I_{z}\right)rp\\ N=I_{z}\overset{\bullet}{r}+\left(I_{y}-I_{x}\right)pq\\ \left\{\begin{array}{l}X=m\left(\overset{\bullet}{u}+qw-rv\right)\\ Y=m\left(\overset{\bullet}{v}+ru-pw\right)\\ N=I_{z}\overset{\bullet}{r}+\left(I_{y}-I_{x}\right)pq\end{array}\right.\end{array} \right. \tag{3}\]  
During motion, the submarine experiences forces and moments that are related to its own motion parameters and navigation state. Specifically, these include the velocities and corresponding accelerations in three main directions (\(u\), \(v\), \(\omega\), \(\partial u/\partial t\), \(\partial v/\partial t\), \(\partial\omega/\partial t\)), angular velocities and corresponding angular accelerations in three directions (\(p\), \(q\), \(r\), \(\partial p/\partial t\), \(\partial q/\partial t\), \(\partial r/\partial t\)), the deflection angles and rotational speeds of the vector nozzles (\(\delta_{r}\), \(\delta_{b}\), \(\delta_{s}\), \(\partial\delta_{r}/\partial t\), \(\partial\delta_{b}/\partial t\), and the rotation speed \(n\) of the spiral mixed-flow pump. The definitions of relevant parameters can be referenced in Table 2, where the deflection angle of the vector nozzle is defined as the angular movement of the thrust vector along the rotational axis, and the rotational speed is defined as the angular velocity. When considering the submarine hull and the pump-jet vector propulsor as a system and neglecting the effect of the deflection velocity of the vector nozzles, the hydrodynamic forces and moments for the submarine's horizontal yaw motion can be simplified, as shown in Equation (5) [45]. Neglecting the bending velocity of the vector nozzle in the analysis of the submarine hull and pump-jet system can be justified due to its minimal impact compared to other dominant factors affecting hydrodynamic behavior, established negligible effects through sensitivity analysis, a focus on steady-state conditions, and the need for computational efficiency without compromising research objectives. The hydrodynamic forces acting on the pump-jet vector thrust submarine can be considered as functions of small motions. By applying the principle of expanding a multivariable function at a point, the multivariable function of the horizontal motion of the pump-jet vector thrust submarine expands using a Taylor series at a point to obtain a polynomial. Choosing the equilibrium motion of the submarine performing constant straight-linemotion for expansion results in the simplest polynomial, as shown in Equation (6) [46]. In Equation (6), the order of expansion is second-order, with an expected error of around 7%. \[\left\{ \begin{array}{l}X=f_{x}\left(u,v,r,\overset{\bullet}{u},\overset{ \bullet}{v},\overset{\bullet}{r},\overset{\bullet}{\delta_{r}}\right)\\ Y=f_{y}\left(u,v,r,\overset{\bullet}{u},\overset{\bullet}{v},\overset{ \bullet}{r},\overset{\bullet}{\delta_{r}}\right)\\ N=f_{N}\left(u,v,r,\overset{\bullet}{u},\overset{\bullet}{v},\overset{ \bullet}{r},\overset{\bullet}{\delta_{r}}\right)\end{array} \right. \tag{5}\]  
\[\left\{