## 2 Materials and Methods
### Model-Free High-Order SMC with Finite-Time Convergence
[26; 27] established that Equation (10) is linearly parametrizable by the product of a regressor \(Y(\eta,\dot{\eta},\ddot{\eta})\in\mathbb{R}^{n\times p}\) composed of known nonlinear functions and a vector \(\theta\in\mathbb{R}^{p}\) containing constant parameters. Then, this parametrization can be rewritten in terms of a nominal reference \(\dot{\eta}_{r}\) and its time derivative \(\ddot{\eta}_{r}\) as follows  
\[M_{\eta}(\eta)\ddot{\eta}_{r}+C_{\eta}(\
u,\eta)\dot{\eta}_{r}+D_{\eta}(\
u, \eta)\dot{\eta}_{r}+g_{\eta}(\eta)=\;Y\big{(}\eta,\dot{\eta},\dot{\eta}_{r}, \ddot{\eta}_{r}\big{)}\theta, \tag{13}\]  
Subtracting Equation (12) from both sides of Equation (10) leads to the open-loop error dynamics expression  
\[M_{\eta}(\eta)\dot{S}_{r}+C_{\eta}(\
u,\eta)S_{r}+D_{\eta}(\
u,\eta)S_{r}=\tau _{\eta}-\;Y\big{(}\eta,\dot{\eta},\dot{\eta}_{r},\ddot{\eta}_{r}\big{)}\theta, \tag{14}\]  
where \(S_{r}=\dot{\eta}-\ddot{\eta}_{r}\) is known as the extended error. Now, consider the following nominal reference  
\[\dot{\eta}_{r}=\;\dot{\eta}_{d}-\alpha\widetilde{\dot{\eta}}+S_{d}-K_{i}\int_ {0}^{t}sign\big{(}S_{\eta}\big{)}d\sigma, \tag{15}\]  
where \(\widetilde{\dot{\eta}}=\eta-\eta_{d}\) represents the tracking error of the position, \(\eta_{d}\) is the desired trajectory, \(\alpha\) and \(K_{i}\) are diagonal positive definite \(n\;\times\;n\) gain matrices, \(sign(x)\) stands for the \(signum\)\(function\) of vector \(x\) and  
\[M_{\eta}(\eta)\ddot{\eta}+C_{\eta}(\
u,\eta)\dot{\eta}+D_{\eta}(\
u,\eta)\dot{ \eta}+g_{\eta}(\eta)=\;\tau_{\eta}, \tag{11}\]0  
\[M_{\eta}(\eta)\ddot{\eta}+C_{\eta}(\
u,\eta)\dot{\eta}+D_{\eta}(\
u,\eta)\dot{ \eta}+g_{\eta}(\eta)=\;\tau_{\eta}, \tag{11}\]1  
\[M_{\eta}(\eta)\ddot{\eta}+C_{\eta}(\
u,\eta)\dot{\eta}+D_{\eta}(\
u,\eta)\dot{ \eta}+g_{\eta}(\eta)=\;\tau_{\eta}, \tag{11}\]2with \(\kappa>0\). So that the extended error \(S_{r}\) can be rewritten as follows  
\[M_{\eta}(\eta)\ddot{\eta}+C_{\eta}(\
u,\eta)\dot{\eta}+D_{\eta}(\
u,\eta)\dot{ \eta}+g_{\eta}(\eta)=\;\tau_{\eta}, \tag{11}\]3  
The result is the model-free high-order SMC with the following control law  
\[M_{\eta}(\eta)\ddot{\eta}+C_{\eta}(\
u,\eta)\dot{\eta}+D_{\eta}(\
u,\eta)\dot{ \eta}+g_{\eta}(\eta)=\;\tau_{\eta}, \tag{11}\]4  
Notice that the so-called model-free control law does not depend on any parameter, matrix, or vector describing the hydrodynamics of the vehicle, it is defined by a diagonal definite positive \(n\ \times\ n\) gain matrix \(K_{d}\), and the extended error \(S_{r}\). Furthermore, since the _signum function_ is not used directly but through its integral, the chattering effect common in conventional SMC is not present. A stability analysis of the controller just described can be found in [34] where the existence of the sliding mode was proven. A TBG will be used to achieve finite-time convergence. A TBG is a scalar time function that provides a smooth transition from 0 to 1, with a duration controllable by a time-base \(t_{b}\). Its derivative provides a bell-shaped speed profile. The TBG implemented in this work was introduced by [PERSON] [26] and [PERSON] [27] for finite-time convergence of a robotic arm. Consider the fifth-order polynomial  
\[M_{\eta}(\eta)\ddot{\eta}+C_{\eta}(\
u,\eta)\dot{\eta}+D_{\eta}(\
u,\eta)\dot{ \eta}+g_{\eta}(\eta)=\;\tau_{\eta}, \tag{11}\]5  
and its time derivative  
\[M_{\eta}(\eta)\ddot{\eta}+C_{\eta}(\
u,\eta)\dot{\eta}+D_{\eta}(\
u,\eta)\dot{ \eta}+g_{\eta}(\eta)=\;\tau_{\eta}, \tag{11}\]6  
with \(t_{0}\) representing the initial time and subjected to the following conditions: \(\xi(t_{0})=\dot{\xi}(t_{0})=\dot{\xi}(t_{b})=\dot{\xi}(t_{b})=0\). The time-base \(t_{b}\) can be chosen arbitrarily and does not depend on the initial condition. Now, \(\xi(t)\) will be used to parameterize a time-varying gain \(\alpha(t)\) as follows  
\[\alpha(t)=\left\{ \begin{array}{cc}\alpha_{0}\frac{\dot{\xi}}{(1-\xi)+\xi},& 0\leq t\leq t_{b}\\ \alpha_{c},&t>t_{b}\end{array}