## 2 Preliminaries and Problem Statement
### ROVs Kinematic and Dynamic Models  
According to International Towing Tank Conference (ITTC) recommendations and the Society of Naval Architecture and Marine Engineers (SNAME) Terminology Bulletin System, as illustrated in Figure 1, the motions of ROVs are described by the following two co-ordinate frames: the center of mass of the ROVs is set as the origin of the inertial frame, and the positive direction of the inertial frame follows the positive direction of motion of the ROVs. The following equation describes the kinematic model of an ROV:  
\[\dot{\eta}=J(\eta)v \tag{4}\]  
Figure 1: The inertial frame and the body-fixed frame of the ROV. where \(\eta=[x,y,z,\phi,\theta,\psi]^{T}\) are the position and attitude vectors in the inertia frame, \([x,y,z]\in\mathbb{R}\) are the position coordinates, and \([\phi,\theta,\psi]\in\mathbb{R}\) are the attitude coordinates. The linear and angular velocity vectors in the body-fixed frame are represented by \(v=[u,v,w,p,q,r]^{T}\), where \([u,v,w]\in\mathbb{R}\) are the surge, sway, and heave velocities, and \([p,q,r]\in\mathbb{R}\) are the roll, pitch, and yaw velocities. \(J(\eta)\in\mathbb{R}^{6\times 6}\) is the transition matrix between the inertial and the body-fixed frame, which can be written as follows:  
\[J(\eta)= \begin{bmatrix}I_{1}(\eta)&0_{3\times 3}\\ 0_{3\times 3}&J_{2}(\eta)\end{bmatrix}  \tag{5}\]  
where \(J_{1}(\eta)\) and \(J_{2}(\eta)\) are denoted, respectively, as follows:  
\[J_{1}(\eta)= \begin{bmatrix}cos\psi cos\theta&-sin\psi cos\phi+cos\psi sin\phi& sin\psi sin\phi+cos\psi sin\theta cos\phi\\ sin\psi cos\theta&cos\psi cos\phi+sin\psi sin\phi sin\phi&-cos\psi sin\phi+ sin\psi sin\theta cos\phi\\ -sin\theta&cos\theta sin\phi&cos\theta cos\phi\end{bmatrix}  \tag{6}\]  
\[J_{2}(\eta)= \begin{bmatrix}1&sin\phi tan\theta&cos\phi tan\theta\\ 0&cos\phi&-sin\phi\\ 0&sin\phi/cos\theta&cos\phi/cos\theta\end{bmatrix}  \tag{7}\]  
The ROVs' dynamics model [15] can be expressed as follows:  
\[M\psi+C(v)\psi+D(v)\psi+G(\eta)=\tau+w \tag{8}\]  
where \(M\in\mathbb{R}^{6\times 6}\) is the inertia matrix including the added fluid inertia mass and rigid body inertia mass, \(C(v)\in\mathbb{R}^{6\times 6}\) is the Coriolis force and centripetal force matrix, \(D(v)\in\mathbb{R}^{6\times 6}\) represents the hydrodynamic damping matrix due to the surround fluid, also including the effects of added mass, \(G(\eta)\in\mathbb{R}^{6\times 1}\) is a restoring force and moment vector consisting of gravity and buoyancy, \(\tau(\eta)\in\mathbb{R}^{6\times 1}\) is the vector of the control forces and moments generated by the thruster acting on the ROVs and \(w\in\mathbb{R}^{6\times 1}\) denotes the external disturbance vector. The inertia matrix is \(M=M_{RB}+M_{A}\), where \(M_{RB}\) is the rigid body mass matrix and \(M_{A}\) is the added mass matrix. The inertia matrix \(M\) can be expressed simply as follows:  
\[M= \begin{bmatrix}m-X_{\dot{u}}&0&0&0&0\\ 0&m-Y_{0}&0&0&0&0\\ 0&0&m-Z_{\dot{u}}&0&0&0\\ 0&0&0&I_{x}-K_{\dot{p}}&0&0\\ 0&0&0&0&I_{y}-M_{\dot{q}}&0\\ 0&0&0&0&0&I_{z}-N_{r}\end{bmatrix}  \tag{9}\]  
Considering the low cruising speed of the ROVs and that it can be approximated as a rigid-body symmetric structure, the Coriolis force and the centripetal force matrix \(C(v)\) can be simplified to the following:  
\[C(v)=\begin{bmatrix}0&0&0&0&(m-Z_{\dot{u}})w&-(m+Y_{\dot{v}})v\\ 0&0&0&-(m-Z_{\dot{u}})w&0&(m-X_{\dot{u}})u\\ 0&0&0&(m-Y_{\dot{v}})v&-(m-X_{\dot{u}})u&0\\ 0&(m-Z_{\dot{u}})w&-(m-Y_{\dot{v}})v&0&I_{z}r&I_{y}q\\ -(m-Z_{\dot{u}}  
\[D(v)=- \begin{bmatrix}X_{u}+X_{u|u|}|u|&0&0&0&0&0\\ 0&Y_{v}+Y_{v|v|}|v|&0&0&0&0\\ 0&0&Z_{w}+Z_{w|w|}|w|&0&0&0\\ 0&0&0&K_{p}+K_{p|p|}|p|&0&0\\ 0&0&0&0&M_{q}+M_{q|q|}|q|&0\\ 0&0&0&0&0&N_{r}+N_{r|r|r|}|r|\end{bmatrix}  \tag{11}\]  
Set the gravitational force W of the ROVs as equal to the buoyant force \(B\), namely, \(W=B\). Define the center of gravity coordinates as \([X_{W},Y_{W},Z_{W}]\) and the coordinates of the center of buoyancy as \([X_{B},Y_{B},Z_{B}]\), where \(X_{W}=X_{B}=0\), \(Y_{W}=Y_{B}=0\), \(Z_{B}>Z_{W}=0\).