## 2 Materials and Methods
### Controller Design  
The control objective in this work is to ensure that the generalized states in the dynamics in (1) track a sufficiently smooth desired trajectory denoted by \(x_{d}:\mathbb{R}_{\geq 0}\to\mathbb{R}^{n}\). The desired trajectory \(x_{d}\) and its time derivative \(\dot{x}_{d}\) are known and bounded such that \(x_{d},\dot{x}_{d}\in\mathcal{L}_{\infty}\). A known tracking error, denoted as \(e:\mathbb{R}_{\geq 0}\to\mathbb{R}^{n}\), is defined as  
\[e(t)\triangleq x(t)-x_{d}(t). \tag{4}\]  
Another objective of this work is to modify the error system to inject an SM term into the open-loop error system (i.e., the SM term is not in the controller). This task is accomplished through the design of an auxiliary tracking error, denoted by \(r:\mathbb{R}_{\geq 0}\to\mathbb{R}^{n}\), that is defined as  
\[r(t)\triangleq e(t)+\alpha e_{f}(t), \tag{5}\]  
where \(e_{f}:\mathbb{R}_{\geq 0}\to\mathbb{R}^{n}\) represents an auxiliary filtered error signal and \(\alpha\in\mathbb{R}_{>0}\) represents a user-defined control gain. The time derivative of the auxiliary filtered error is defined as  
\[\dot{e}_{f}(t)\triangleq-k_{1}\text{sgn}(r(t))-k_{2}r(t)-\beta e_{f}(t), \tag{6}\]  
where \(k_{1},k_{2},\beta\in\mathbb{R}_{>0}\) are user-defined control gains and \(\text{sgn}(\cdot)\) represents the signum function. Notice that the filter dynamics in (6) can be solved to determine \(e_{f}(t)\). Taking the time derivative of (5) and using (1), (4) and (6) yields  
\[ \begin{array}{lll}\dot{t}(t)&=&f_{\sigma}(x(t))+d_{\sigma}(t)+g_{\phi}(x(t),t)u(t)-\dot{x}_{d}(t)\\ &&-\alpha k_{1}\text{sgn}(r(t))-\alpha k_{2}r(t)-\alpha\beta e_{f}(t).\end{array}  \tag{7}\]  
For notational brevity, an auxiliary term \(\chi_{\sigma}:\mathbb{R}_{\geq 0}\to\mathbb{R}^{n}\) is defined as  
\[\chi_{\sigma}(t)\triangleq f_{\sigma}(x(t))+d_{\sigma}(t). \tag{8}\]  
The auxiliary term can be upper-bounded using Assumptions 1 and 2 and Equations (4) and (5) such that  
\[\|\chi_{\sigma}(t)\|\leq\Phi+\rho(\|z(t)\|)\|z(t)\|, \tag{9}\]where \(\rho(\cdot):\mathbb{R}\rightarrow\mathbb{R}_{>0}\) is a known, radially unbounded, and strictly increasing positive function, \(\Phi\in\mathbb{R}_{>0}\) is a known constant, and \(z:\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}^{2n}\) is a composite error vector defined as  
\[z(t)\triangleq\Big{[} \begin{array}{cc}r^{T}(t)&e_{f}^{T}(t)\end{array} \Big{]} ^{T}. \tag{10}\]  
Substituting (8) into (7) yields an open-loop error function as  
\[ \begin{array}{lll}\dot{r}(t)&=&\chi_{\sigma}(t)+g_{\phi}(x(t),t)u(t)-\dot{x} _{d}(t)\\ &&-\alpha k_{1}\mathrm{sgn}(r(t))-\alpha k_{2}r(t)-\alpha\beta e_{f}(t).\end{array}  \tag{11}\]  
Since the control effectiveness matrix \(g_{\phi}(x(t),t)\) is unknown, the matrix must be estimated using an adaptive approach. In this work, the estimate of the generalized torque/force input is expressed as  
\[\dot{g}_{\phi}(x(t),t)u(t)=Y_{\phi}(x(t),t,u(t))\dot{\theta}(t), \tag{12}\]  
where \(\dot{g}_{\phi}:\mathbb{R}^{n}\times\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}^{n \times m}\) is an estimate of \(g_{\phi}(x(t),t)\) and \(\dot{\theta}:\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}^{q}\) is an estimate of the unknown vector of constants \(\theta\). The difference between the uncertain parameters and their estimates results in a parameter estimation error, denoted as \(\bar{\theta}:\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}^{q}\), that is defined as  
\[\bar{\theta}(t)\triangleq\theta-\hat{\theta}(t). \tag{13}\]  
**Assumption 5**. : _The estimate of the control effectiveness matrix \(\dot{g}_{\phi}(x(t),t)\), \(\forall t\geq t_{0}\), is a full-row rank matrix. The right pseudo inverse of \(\dot{g}_{\phi}(x(t),t)\) is denoted as \(\hat{\xi}_{\phi}^{+}:\mathbb{R}^{n}\times\mathbb{R}_{\geq 0}\rightarrow \mathbb{R}^{m\times n}\), where \(\hat{\xi}_{\phi}^{+}(\cdot)\triangleq\hat{\xi}_{\phi}^{T}(\cdot)\Big{(}\hat{g }_{\phi}(\cdot)\hat{\xi}_{\phi}^{T}(\cdot)\Big{)}^{-1}\).