## 2 Modeling  
The UVMS investigated in this paper is composed of an underwater vehicle with a 3 DOFs manipulator. The coordinate system of the UVMS is shown in Figure 1. In the body-fixed frame \(\{B\}\), we define that the vectors of vehicle's linear and angular velocities are \(\
u_{1}\) and \(\
u_{2}\), where \(\
u_{1}=[u,v,w]^{T}\), \(\
u_{2}=[p,q,r]^{T}\) and \(\
u=[\
u_{1}^{T},\
u_{2}^{T}]^{T}\). The vector of joint positions is assumed to be \(\mathbf{q}=\left[q_{1},q_{2}\cdots q_{n}\right]^{T}\), where \(n\) is the number of manipulator's joints. The position and orientation vector of the UVMS relative to the body-fixed frame is assumed to be \(\mathbf{\zeta}=[\
u_{1}^{T},\
u_{2}^{T},\mathbf{q}^{T}]^{T}\). In the inertial frame \(\{I\}\), the vectors of end-effector's position and orientation are defined as \(\mathbf{\eta}_{E1}\) and \(\mathbf{\eta}_{E2}\), and assume \(\mathbf{x}_{E}=[\mathbf{\eta}_{E1}^{T},\mathbf{\eta}_{E2}^{T}]^{T}\).  
The kinematic model of the UVMS [21] can be obtained as shown in (1), where the velocities of the UVMS in the body-fixed frame (\(\mathbf{\zeta}\)) are mapped into end-effector velocities (\(\dot{\mathbf{x}}_{E}\)) via \(\mathbf{J}(R_{B}^{I},\mathbf{q})\).  
\[\dot{\mathbf{x}}_{E}{=}\begin{bmatrix}R_{B}^{I}&S(R_{B}^{I}p_{0}^{B}{+}R_{0}^{I}p_ {E}^{0})\mathbf{R}_{B}^{I}&J_{\text{pos}}^{I}\\ \mathbf{0}_{3\times 3}&R_{B}^{I}&J_{\text{orig}}^{I}\end{bmatrix}\left[\
u_{1}^{T} \
u_{2}^{T}\mathbf{q}^{T}\right]^{T}{=}J(R_{B}^{I},\mathbf{q})\mathbf{\zeta} \tag{1}\]  
Figure 1: Coordinate systems of the underwater vehicle manipulator system (UVMS).  
where \(\mathbf{J}(R_{B}^{I},\mathbf{q})\in\mathbb{R}^{6\times(6+n)}\), \(R_{0}^{I}=R_{B}^{I}R_{0}^{B}\), \(J_{\text{pos,q}}^{I}=R_{0}^{I}J_{\text{pos,q}}^{0}\) and \(J_{\text{ori,q}}^{I}=R_{0}^{I}J_{\text{ori,q}}^{0}\). \(R_{0}^{B}\) is the rotational matrix describing the transformation from the manipulator's base frame to the body-fixed frame, and \(\mathbf{p}_{E}^{0}\) is the position vector from manipulator's base to the center of the body-fixed frame. \(\mathbf{p}_{E}^{0}\) is the position vector from end-effector to the manipulator's base. \(J_{\text{pos,q}}^{0}\) is the manipulator's linear Jacobian matrix, and \(J_{\text{ori,q}}^{0}\) is the manipulator's angular Jacobian matrix. \(S(\cdot)\) is the cross-product operator.  
The vectors of vehicle's position and attitude relative to the inertial frame are defined as \(\eta_{1}\) and \(\eta_{2}\), where \(\eta_{1}=[x,y,z]^{T},\eta_{2}=[\phi,\theta,\psi]^{T}\) and \(\mathbf{\eta}=[\eta_{1}^{T},\eta_{2}^{T}]^{T}\). The velocity vector of the UVMS defined in the body-fixed frame (\(\xi\)) can be obtained by (2).  
\[\xi=\begin{bmatrix}\mathbf{R}_{I}^{B}&\mathbf{0}_{3\times 3}&\mathbf{0}_{3\times 3}\\ \mathbf{0}_{3\times 3}&\mathbf{J}_{v}&\mathbf{0}_{3\times 3}\\ \mathbf{0}_{3\times 3}&\mathbf{0}_{3\times 3}&\mathbf{I}_{3\times 3}\\ \end{bmatrix}\begin{bmatrix}\dot{\eta}_{1}\\ \dot{\mathbf{\eta}}_{2}\\ \dot{\mathbf{q}}\\ \end{bmatrix}=J_{\xi}(\eta_{2})\dot{\xi} \tag{2}\]  
where \(\mathbf{v}_{1}=\mathbf{R}_{I}^{B}\dot{\eta}_{1}\), \(\mathbf{v}_{2}=\mathbf{J}_{v}\dot{\eta}_{2}\) and \(\mathbf{\xi}=[\mathbf{\eta}_{1}^{T},\mathbf{\eta}_{2}^{T},\mathbf{q}^{T}]^{T}\). \(\mathbf{R}_{I}^{B}\) is the linear rotational matrix describing the transformation from the inertial frame to the body-fixed frame, and \(J_{v}\) is the angular rotational matrix. The values of \(\mathbf{R}_{I}^{B}\) and \(J_{v}\) can be referred to the literature [41]. \(\mathbf{J}_{\xi}(\eta_{2})\) is the Jacobian matrix which relates the vehicle velocities with respect to the inertial frame and the body-fixed frame.