## 2 Materials and Methods
### Stability Analysis  
The \(i^{th}\) time when \(\phi\) switches to \(w\in\mathcal{S}\) can be represented by \(\left\{t_{i}^{w,\text{on}}\right\}\), \(\forall i\in\mathbb{N}\), and the \(i^{th}\) time when \(\phi\) switches from \(w\in\mathcal{S}\) can be represented by \(\left\{t_{i}^{w,\text{off}}\right\}\), \(\forall i\in\mathbb{N}\). Let \(V_{L}:\ \mathcal{D}\times\mathbb{R}_{\geq 0}\ \to\mathbb{R}_{\geq 0}\) be a positive definite common Lyapunov-like candidate function that is continuously differentiable in \(y\), where \(y\) is a concatenated state vector that is defined as  
\[y\triangleq\left[\ z^{T}\quad\bar{\theta}^{T}\ \right]^{T}. \tag{18}\]  
The Lyapunov candidate function \(V_{L}\) is defined within the domain \(\mathcal{D}\subseteq\mathbb{R}^{2n+q}\) as  
\[V_{L}(y,t)\triangleq\frac{1}{2}r^{T}r+\frac{1}{2}e_{f}^{T}e_{f}+\frac{1}{2} \bar{\theta}^{T}\Gamma^{-1}\tilde{\theta}, \tag{19}\]  
where \(\mathcal{D}\triangleq\left\{y\in\mathbb{R}^{2n+q}\|\|y\|\leq\gamma\right\}\), \(\gamma\in\mathbb{R}_{>0}\) is a known constant that satisfies the inequality \(\gamma\leq\inf\{\rho^{-1}\big{(}\big{(}\sqrt{2 ak_{2}\delta},\infty\big{)} \big{)}\}\), \(\delta\in\mathbb{R}\) is a known constant defined as \(\delta\triangleq\min\left(k_{3},\ \beta-\frac{ck_{1}}{2}\right)\), and \(\varepsilon\in\mathbb{R}_{>0}\) is a selectable constant. Using the concatenated state vector \(y\) defined in (18), the Lyapunov-like function \(V_{L}\) can be upper- and lower-bounded as  
\[\lambda_{1}\|y\|^{2}\leq V_{L}\leq\lambda_{2}\|y\|^{2}, \tag{20}\]  
where the constants \(\lambda_{1},\lambda_{2}\in\mathbb{R}_{\geq 0}\) are defined as \(\lambda_{1}\triangleq\min\left(\frac{1}{2},\frac{1}{2}\lambda_{\min}\big{\{} \Gamma^{-1}\big{\}}\right)\) and \(\lambda_{2}\triangleq\max\left(\frac{1}{2},\frac{1}{2}\lambda_{\max}\big{\{} \Gamma^{-1}\big{\}}\right)\), \(\lambda_{\min}\) is defined as the minimum eigenvalue \(\{\cdot\}\), and \(\lambda_{\max}\) is defined as the maximum eigenvalue \(\{\cdot\}\). Additionally, based on Assumption 4 and (18), \(V_{L}\) can be further bounded as  
\[\frac{1}{2}\|z\|^{2}\leq V_{L}\leq\frac{1}{2}\|z\|^{2}+v_{1}. \tag{21}\]  
In (21), the function \(v_{1}:\mathbb{R}^{q}\to\mathbb{R}_{\geq 0}\) is defined as  
\[v_{1}\triangleq\frac{1}{2}\lambda_{\max}\Big{\{}\Gamma^{-1}\Big{\}}\|\bar{ \theta}\|_{\max}^{2}, \tag{22}\]  
where \(\big{\|}\bar{\theta}\big{\|}_{\max}\in\mathbb{R}_{>0}\) is the maximum value of \(\big{\|}\bar{\theta}(t)\big{\|}\) such that \(\big{\|}\bar{\theta}(t)\big{\|}\leq\big{\|}\bar{\theta}\big{\|}_{max}\forall t \geq t_{0}\), resulting from the projection algorithm in (16) and the bounds defined in Assumption 4. Moreover, let \(\mathcal{S}_{\mathcal{D}}\subseteq\mathbb{R}^{2n+q}\) be a set of allowable initial conditions that maintain system stability, defined as  
\[\mathcal{S}_{\mathcal{D}}\triangleq\Bigg{\{}y\in\mathbb{R}^{2n+q}\bigg{|}\|y \|\leq\sqrt{\frac{\lambda_{1}}{\lambda_{2}}}\gamma\Bigg{\}}. \tag{23}\]  
**Theorem 1**. : _Consider the uncertain, nonsmooth, and nonlinear dynamics in (1) that satisfy Assumptions 1-5. The control input in (15) and the parameter update law in (16) ensure semi-global exponential trajectory tracking to an ultimate bound in the sense that_  
\[\|y(t)\|^{2} \leq \frac{\lambda_{2}}{\lambda_{1}}\|y(t_{0})\|^{2}\exp(-\delta(t-t_{ 0}))+\frac{v}{\lambda_{1}}(1-\exp(-\delta(t-t_{0}))), \tag{24}\]  
_for all \(t\in[t_{0},\infty)\), provided that \(y(t_{0})\in\mathcal{S}_{\mathcal{D}}\), where \(v\triangleq\delta v_{1}+\frac{nk_{1}}{2\varepsilon}\), and provided that the following gain conditions are satisfied:_  
\[\beta>\frac{ek_{1}}{2},\quad\alpha k_{1}\geq\Phi,\quad\sqrt{\frac{v}{\lambda_{ 1}\delta}}\leq\gamma. \tag{25}\]  
Proof. : Since the closed-loop error system in (17) contains discontinuous terms, the time derivative of (19) exists almost everywhere (a.e.) for all \(t\in[t_{0},\infty)\).