## 2 Description of Dynamic Modelling of AUV  
This section presents a system model of a well-known underactuated underwater robot with six degrees of freedom (6-DOF), as shown in Figure 1.  
Underwater vehicles operate in fluids and are subject to a series of forces called hydrodynamics. The ocean currents are more likely to affect underwater vehicles' dynamics and control performance due to their unpredictable speed and direction. For the control problems of underactuated underwater vehicles, it is generally necessary to study their velocity, position, attitude, heading, and degrees of freedom. It is essential to establish a suitable reference frame. A 6-DOF underwater robot model is introduced below, and the corresponding positions of its coordinate system, \((O_{0}X_{0}Y_{0}Z_{0})\) and the fixed coordinate system \((OXYZ)\), are given. Since the influence of the earth's rotation on low-speed moving objects is negligible, it is considered that the acceleration motion of objects on the earth's surface can be ignored. This is regarded as an inertial coordinate system \((O_{0}X_{0}Y_{0}Z_{0})\), and [PERSON]'s laws of motion can be applied. Figure 1: Body-fixed frame and earth-fixed reference frame for an underwater robot. Due to its unique nature, it will be complicated to implement if the controller is designed using the 6-DOF model. Therefore, when creating the controller, the propeller design layout causes the vehicle not to have any lateral and vertical propellers to control its roll and heave motions, nor does it allow active control of the pitch and roll motions. Trajectory tracking management on the horizontal plane can generally be decomposed into two motion models, horizontal and vertical, thereby simplifying the AUV model. This paper only considers the motion of the AUV on the horizontal plane, that is, the lateral dynamics, including surge, sway, and yaw. [PERSON]'s laws establish the kinematic equations in a fixed coordinate system [11]:  
\[M\dot{v}+C(v)v+D(v)v+g(\eta)=F \tag{1}\]  
where \(F=[F_{u},0,F_{r}]\) are the surge force \(F_{u}\) and yaw moment \(F_{r}\) generated by the drive. Taking into account that balance is considered in the structural design of the robot, we can assume that the vehicle is symmetrical about the three planes \(O_{0}X_{0}Y_{0}\), \(O_{0}X_{0}Z_{0}\), and \(O_{0}Y_{0}Z_{0}\), as is the inertia matrix, including the additional mass of the vehicle. \(C(v)\) denotes the Coriolis and centripetal matrix having the following form: \(F=[F_{u},0,F_{r}]\) where the surge force \(F_{u}\) and yaw moment \(F_{r}\) are generated by the drive. \(C(v)\) represents Coriolis and the centripetal matrices of the form:  
\[C(v)= \begin{bmatrix}0&0&-M_{\dot{v}}v\\ 0&0&M_{\dot{u}}u\\ M_{\dot{v}}v&-M_{\dot{u}}u&0\end{bmatrix}  \tag{2}\]  
\(D(v)\) is the damping matrix, which has the form:  
\[D(v)= \begin{bmatrix}X_{u}+D_{u}|u|&0&0\\ 0&Y_{v}+D_{v}|v|&0\\ 0&0&N_{r}+D_{r}|r|\end{bmatrix}  \tag{3}\]  
\(g(\eta)\) represents the restoring force composed of gravity and buoyancy. \(\dot{v}= \begin{bmatrix}\dot{u}&\dot{v}&\dot{r}\end{bmatrix} ^{T}\) is the linear acceleration and angular acceleration of the object (motion) frame. \(\dot{u}\) and \(\dot{v}\) are linear accelerations in surge and sway direction, respectively. \(\dot{r}\) is the angular acceleration in the yaw direction. \(v= \begin{bmatrix}u&v&r\end{bmatrix} ^{T}\) are the linear and angular velocities relative to the object (motion) coordinate system, \(u\)--surge velocity, \(v\)--sway velocity, and \(r\)--yaw rate. \(\eta=[x&y&\psi]^{T}\) are positions and orientations concerning the inertial (fixed) frame, \(x\)--surge position, \(y\)--sway position, and \(\psi\)--yaw angle. The mass parameters of underwater vehicles are given at the beginning of the design: mass \(m\) and inertia \(I_{z}\). For the additional mass of the underwater vehicle, it is necessary to use software to draw the AUV 3D model, and then use the fluid dynamics software to simulate it \(X_{\dot{u}}\), \(Y_{\dot{v}}\), \(N_{r}\).