## Appendix A Simulation Data of the AUV  
For the torpedo-type AUV, its center of buoyancy (CB) is \((0,0,0.02)\) m, its center of gravity (CG) is \((0,0,0)\) m and its moment of inertia is \((0.69,16.82,16.82)\) kg \(\cdot\) m\({}^{2}\). The AUV is equipped with an underwater manipulator, and the manipulator's base position is \(\mathbf{p}_{0}^{\text{B}}=(0.6,0,0.2)^{T}\) m. Some model parameters of the AUV given in (3) are composed of rigid-body terms and hydrodynamic terms [42], as shown in the following equations. \begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{**Control Schemes**} & \multicolumn{2}{c}{**Linear Trajectory**} & \multicolumn{4}{c}{**Circular Trajectory**} \\ \cline{2-7}  & **ISE\({}_{\mathbf{p}}\)** & **ISE\({}_{\mathbf{o}}\)** & \(\int||\mathbf{F}_{\mathbf{td}}||\mathbf{dt}\) & **ISE\({}_{\mathbf{p}}\)** & **ISE\({}_{\mathbf{0}}\)** & \(\int||\mathbf{F}_{\mathbf{td}}||\mathbf{dt}\) \\ \hline Proposed control\({}_{\text{c2}}\) & 0.3083 & 22.5597 & 1161.6 & 0.3318 & 32.2834 & 1532.6 \\ \(\text{Ho}-\text{EKF}_{\text{c1}}\) & 0.6869 & 34.5853 & 607.2053 & 1.0775 & 63.1945 & 1420.4 \\ Proposed control\({}_{\text{c1}}\) & 0.2905 & 26.1319 & 603.8734 & 0.3708 & 40.1401 & 1428.3 \\ \hline \end{tabular} Note : \(\mathbf{\epsilon}_{k}=[(\tilde{\mathbf{\eta}}_{\text{E1}})^{T},(\tilde{\mathbf{\eta}}_{ \text{E2}})^{T}]^{T}\), \(\text{ISE}_{p}=\int||\tilde{\mathbf{\eta}}_{\text{E1}}||dt\), \(\text{ISE}_{\text{o}}=\int||\tilde{\mathbf{\eta}}_{\text{E2}}||dt\).  
\end{table} 
Table 6: Performance analysis of the UVMS for the linear and circular trajectories tracking. \[\mathbf{M}_{\rm{v}}=\mathbf{M}_{\rm{RB}}+\mathbf{M}_{\rm{A}},\ \ \ \mathbf{C}_{\rm{v}}(\mathbf{v}_{r})=\mathbf{C}_{\rm{RB} }(\mathbf{v}_{r})+\mathbf{C}_{\rm{A}}(\mathbf{v}_{r}),\ \ \mathbf{D}_{\rm{v}}(\mathbf{v}_{r})=\mathbf{D}_{\rm{NL}}{\rm diag }\left(|\mathbf{v}_{\rm{r}}|\right)+\mathbf{D}_{\rm{L}}{\rm diag}(|\mathbf{v}_{r}|)\]  
where \(\mathbf{M}_{\rm{RB}}\) and \(\mathbf{C}_{\rm{RB}}\) are the rigid-body terms which represent the inertia matrix and the Coriolis and centripetal matrix. \(\mathbf{M}_{\rm{A}}\), \(\mathbf{C}_{\rm{A}}\), \(\mathbf{D}_{\rm{NL}}\) and \(\mathbf{D}_{\rm{L}}\) are the matrices related to the hydrodynamic forces. \(\mathbf{M}_{\rm{A}}\) and \(\mathbf{C}_{\rm{A}}\) are the added mass matrix and the added Coriolis and centripetal matrix. \(\mathbf{D}_{\rm{NL}}\) and \(\mathbf{D}_{\rm{L}}\) are the quadratic damping matrix and the lift matrix. \(\mathbf{M}_{\rm{A}}\), \(\mathbf{C}_{\rm{A}}\), \(\mathbf{D}_{\rm{NL}}\) and \(\mathbf{D}_{\rm{L}}\) are given in the following equations [15]. \[\mathbf{M}_{\rm{A}}=\!\!\!-\!\! \begin{bmatrix}X_{\rm{u}|\rm{u}|}&0&0&0&0&0\\ 0&Y_{\rm{v}|\rm{v}}&0&0&0&Y_{\rm{r}|\rm{r}}\\ 0&0&Z_{\rm{w}}&0&Z_{\rm{q}|\rm{q}}&0\\ 0&0&0&K_{\rm{p}|\rm{p}}&0&0\\ 0&0&M_{\rm{w}}&0&M_{\rm{q}}&0\\ 0&N_{\rm{v}}&0&0&0&N_{\rm{r}}\end{bmatrix} \!\!=\!\! \begin{bmatrix}\mathbf{A}_{11}& \mathbf{A}_{12}\\ \mathbf{A}_{21}&\mathbf{A}_{22}\end{bmatrix} ,\mathbf{C}_{\rm{A}}=\!\! \begin{bmatrix}\mathbf{0}_ {3\times 3}&-\!\!\!-\!\!S(\mathbf{A}_{11}\mathbf{v}_{1}\!+\!\mathbf{A}_{12}\mathbf{v}_{2})\\ -\!\!\!S(\mathbf{A}_{11}\mathbf{v}_{1}\!+\!\mathbf{A}_{12}\mathbf{v}_{2})&-\!\!\!S(\mathbf{A}_{21} \mathbf{v}_{1}\!+\!\mathbf{A}_{22}\mathbf{v}_{2})\end{bmatrix} \]  
\[\mathbf{D}_{NL}=\!\!\!-\!\! \begin{bmatrix}X_{\rm{u}|\rm{u}|}&0&0&0&0&0\\ 0&Y_{\rm{v}|\rm{v}}&0&0&0&Y_{\rm{r}|\rm{r}}\\ 0&0&Z_{\rm{w}|\rm{w}}&0&Z_{\rm{q}|\rm{q}}&0\\ 0&0&0&K_{\rm{p}|\rm{p}|}&0&0\\ 0&0&M_{\rm{w}|\rm{w}}&0&M_{\rm{q}|\rm{q}}&0\\ 0&N_{\rm{v}|\rm{v}}&0&0&0&N_{\rm{r}|\rm{r}}\end{bmatrix} ,\mathbf{D}_{\rm{L}}=- \begin{bmatrix}0&0&0&0&0&0\\ 0&Y_{\rm{uv}}&0&0&0&Y_{\rm{ur}}\\ 0&0&Z_{\rm{uw}}&0&Z_{\rm{uq}}&0\\ 0&0&0&0&0&0\\ 0&0&M_{\rm{uw}}&0&M_{\rm{uq}}&0\\ 0&N_{\rm{uv}}&0&0&0&N_{\rm{ur}}\end{bmatrix}