## 2 Materials and Methods
### Model-Free High-Order SMC with Finite-Time Convergence  
The dynamics of the UUV (Equation (4)) can be expressed in the Earth-fixed frame applying the following kinematic transformations [30]  
\[\dot{\eta}=J(\eta_{2})\;\
u\;\leftrightarrow\;\
u=\;J^{-1}(\eta_{2})\;\dot{\eta}, \tag{8}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Symbol** & **Value** & **Unit** \\ \hline Mass & m & 10 & kg \\ \hline Buoyancy & B & 100.06 & N \\ \hline Weight & W & 98.1 & N \\ \hline Center of gravity & \(r_{G}=(x_{g},y_{g},z_{g})\) & \(r_{G}=(0,0,0)\) & m \\ \hline Center of buoyancy & \(r_{B}=(x_{b},y_{b},z_{b})\) & \(r_{B}=(0,0,0.02)\) & m \\ \hline Inertia moment & I = diagonal \((I_{x},I_{y},I_{z})\) & I = diagonal \((0.16,0.16,0.16)\) & kg\(\cdot\)m\({}^{2}\) \\ \hline \multirow{6}{*}{Added mass parameters} & \(X_{\dot{u}}\) & \(-5.5\) & kg \\ \cline{2-4}  & \(Y_{\dot{v}}\) & \(-12.7\) & kg \\ \cline{2-4}  & \(Z_{\dot{w}}\) & \(-14.57\) & kg \\ \cline{2-4}  & \(K_{\dot{p}}\) & \(-0.12\) & kg\(\cdot\frac{m^{2}}{\text{rad}}\) \\ \cline{2-4}  & \(M_{\dot{q}}\) & \(-0.12\) & kg\(\cdot\frac{m^{2}}{\text{rad}}\) \\ \cline{2-4}  & \(N_{\dot{r}}\) & \(-0.12\) & kg\(\cdot\frac{m^{2}}{\text{rad}}\) \\ \hline \multirow{6}{*}{Linear Damping parameters} & \(X_{u}\) & \(-4.03\) & N \(\cdot\frac{s}{\text{m}}\) \\ \cline{2-4}  & \(Y_{\dot{v}}\) & \(-6.22\) & N \(\cdot\frac{s}{\text{m}}\) \\ \cline{2-4}  & \(Z_{w}\) & \(-5.18\) & N \(\cdot\frac{s}{\text{m}}\) \\ \cline{2-4}  & \(K_{p}\) & \(-0.07\) & N \(\cdot\frac{s}{\text{rad}}\) \\ \cline{2-4}  & \(M_{q}\) & \(-0.07\) & N \(\cdot\frac{s}{\text{rad}}\) \\ \cline{2-4}  & \(N_{r}\) & \(-0.07\) & N \(\cdot\frac{s}{\text{rad}}\) \\ \hline \multirow{6}{*}{Quadratic Damping parameters} & \(X_{u|u|}\) & \(-18.18\) & N \(\cdot\frac{s^{2}}{\text{m}^{2}}\) \\ \cline{2-4}  & \(Y_{v|v|}\) & \(-21.66\) & N \(\cdot\frac{s^{2}}{\text{m}^{2}}\) \\ \cline{2-4}  & \(Z_{w|w|}\) & \(-36.99\) & N \(\cdot\frac{s^{2}}{\text{m}^{2}}\) \\ \cline{2-4}  & \(K_{p|r|}\) & \(-1.55\) & N \(\cdot\frac{s^{2}}{\text{rad}^{2}}\) \\ \cline{2-4}  & \(M_{q|q|}\) & \(-1.55\) & N \(\cdot\frac{s^{2}}{\text{rad}^{2}}\) \\ \cline{2-4}  & \(N_{r|r|}\) & \(-1.55\) & N \(\cdot\frac{s^{2}}{\text{rad}^{2}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: BlueROV2 parameters. \[\ddot{\eta}=J(\eta_{2})\;\dot{v}+\;\dot{J}(\eta_{2})\;v\;\;\leftrightarrow\;\dot{v} =\;J^{-1}(\eta_{2})\;\big{[}\ddot{\eta}-\;J(\eta_{2})\;v\big{]}, \tag{9}\]  
where the transformation matrix \(J(\eta_{2})\) is defined as  
\[J(\eta_{2})=\;\left[ \begin{array}{cc}J_{1}(\eta_{2})&0_{3\times 3}\\ 0_{3\times 3}&J_{2}(\eta_{2})\end{array} \right], \tag{10}\]  
with  
\[J_{1}(\eta_{2})=\left[ \begin{array}{ccc}c_{\psi}c_{\phi}&-s_{\psi}c_{\phi}+s _{\phi}s_{\phi}c_{\psi}&s_{\psi}s_{\phi}+s_{\phi}c_{\psi}c_{\phi}\\ s_{\psi}c_{\phi}&c_{\psi}c_{\psi}+s_{\phi}s_{\psi}s_{\psi}&-c_{\psi}s_{\phi}+s_{ \psi}s_{\psi}c_{\phi}\\ -s_{\theta}&s_{\phi}c_{\theta}&c_{\phi}c_{\theta}\end{array} \right],\]  
\[J_{s}(\eta_{2})=\;\left[ \begin{array}{ccc}1&s_{\phi}t_{\phi}&c_{\phi}t_{\theta }\\ 0&c_{\phi}&-s_{\phi}\\ 0&s_{\phi}/c_{\theta}&c_{\phi}/c_{\phi}\end{array} \right],\;\text{and}\]  
\[\eta_{2}=[\phi,\theta,\psi]^{T},\]  
where \(c_{angle}=cos(angle),s_{angle}=sin(angle),\) and \(t_{angle}=tan(angle).\)  
After some mathematical manipulation, the resulting expression is  
\[M_{\eta}(\eta)\ddot{\eta}+C_{\eta}(\
u,\eta)\dot{\eta}+D_{\eta}(\
u,\eta)\dot{ \eta}+g_{\eta}(\eta)=\;\tau_{\eta}, \tag{11}\]  
where  
\[M_{\eta}(\eta)=\;J^{-T}(\eta)MJ^{-1}(\eta),\]  
\[C_{\eta}(\
u,\eta)=\;J^{-T}(\eta)[C(\
u)-MJ^{-1}(\eta)\dot{J}(\eta)]J^{-1}( \eta),\]  
\[D_{\eta}(\
u,\eta)=\;J^{-T}(\eta)D(\
u)J^{-1}(\eta),\]  
\[g_{\eta}(\eta)=\;J^{-T}(\eta)g(\eta)\]  
\[\tau_{\eta}(\eta)=\;J^{-T}(\eta)\tau \tag{12}\]  
[PERSON] et al.