## 2 Materials and Methods  
### Dynamics  
Consider a nonsmooth, uncertain, and control-affine nonlinear dynamic system  
\[\dot{x}(t)=f_{\sigma}(x(t))+d_{\sigma}(t)+\underbrace{g_{\phi}(x(t),t)u(t)}_{ \tau_{\phi}(x(t),t)}, \tag{1}\]  
where \(x:\mathbb{R}_{\geq 0}\to\mathbb{R}^{n}\) represents the generalized states, \(f_{\sigma}:\mathbb{R}^{n}\to\mathbb{R}^{n}\) represents the uncertain, nonlinear, and nonsmooth drift dynamics, and \(d_{\sigma}:\mathbb{R}_{\geq 0}\to\mathbb{R}^{n}\) represents the nonsmooth and unknown disturbances. The signal \(\sigma(x(t),t)\), denoted by \(\sigma:\mathbb{R}^{n}\times\mathbb{R}_{\geq 0}\to\mathcal{P}\triangleq\{1,2, \ldots,p\}\), represents an uncertain switching signal that indicates the index of \(f_{\sigma}(x(t))\) and \(d_{\sigma}(t)\), where the number of unique forms for \(f_{\sigma}(x(t))\) and \(d_{\sigma}(t)\) is represented by \(p\in\mathbb{N}\). The generalized input torque or force in (1) is represented as \(\tau_{\phi}:\mathbb{R}^{n}\times\mathbb{R}_{\geq 0}\to\mathbb{R}^{n}\) and is defined as the product of the control input \(u:\mathbb{R}_{\geq 0}\to\mathbb{R}^{m}\) and the unknown and nonsmooth control effectiveness matrix \(g_{\phi}:\mathbb{R}^{n}\times\mathbb{R}_{\geq 0}\to\mathbb{R}^{n\times m}\). The signal \(\phi(x(t),t)\), denoted by \(\phi:\mathbb{R}^{n}\times\mathbb{R}_{\geq 0}\to\mathcal{S}\triangleq\{1,2, \ldots,s\}\), represents a known switching signal that indicates the index of \(\tau_{\phi}(x(t),t)\) and \(g_{\phi}(x(t),t)\), where the number of unique forms for \(\tau_{\phi}(x(t),t)\) and \(g_{\phi}(x(t))\) is represented by \(s\in\mathbb{N}\). **Remark 1**. : _Every unique form of \(f_{\sigma}(x(t))\) and \(d_{\sigma}(t)\) is represented by the signal \(\sigma\in\mathcal{P}\), which switches values whenever \(f_{\sigma}(x(t))\) and \(d_{\sigma}(t)\) have a discrete jump. Likewise, every unique form of \(\tau_{\phi}(x(t),t)\) and \(g_{\phi}(x(t),t)\) is represented by the signal \(\phi\in\mathcal{S}\), which switches values whenever \(\tau_{\phi}(x(t),t)\) and \(g_{\phi}(x(t),t)\) have a discrete jump. Whenever \(\sigma\in\mathcal{P}\) and \(\phi\in\mathcal{S}\) are constant, the dynamics in (1) are continuous (i.e., \(f_{\sigma}(x(t))\), \(d_{\sigma}(t)\), \(g_{\phi}(x(t),t)\), and \(\tau_{\phi}(x(t),t)\) are continuous for constant \(\sigma\) and \(\phi\))._  
The following assumptions must be satisfied to facilitate the control development:  
**Assumption 1**. : _The drift function \(f_{\sigma}(x(t))\) is upper-bounded by a known constant \(c_{F}\in\mathbb{R}_{>0}\) such that \(\|f_{\sigma}(x(t))\|\leq c_{F}\), \(\forall t\geq t_{0}\), and \(\forall\sigma\in\mathcal{P}\) whenever \(\|x(t)\|\in\mathcal{L}_{\infty}\), where the term \(t_{0}\in\mathbb{R}_{\geq 0}\) represents the initial time and \(\|\cdot\|\) represents the standard Euclidean norm._  
**Assumption 2**. : _The disturbance \(d_{\sigma}(t)\) is upper-bounded by a known constant \(c_{D}\in\mathbb{R}_{>0}\) such that \(\|d_{\sigma}(t)\|\leq c_{D}\), \(\forall t\geq t_{0}\), and \(\forall\sigma\in\mathcal{P}\)._  
**Assumption 3**. : _The control effectiveness matrix \(g_{\phi}(x(t),t)\) is a full-row rank matrix for all \(t\geq t_{0}\)._  
**Assumption 4**. : _The generalized torque/force input \(\tau_{\phi}(x(t),t)=g_{\phi}(x(t),t)u(t)\) can be linearly parameterized for all \(t\geq t_{0}\) such that_  
\[g_{\phi}(x(t),t)u(t)=Y_{\phi}(x(t),t,u(t))\theta, \tag{2}\]  
_where \(\theta\in\mathbb{R}^{q}\) represents a vector of the unknown constants in \(g_{\phi}(x(t),t)\) and \(q\in\mathbb{N}\) represents the number of uncertain constants in \(g_{\phi}(x(t),t)\). The matrix \(Y_{\phi}:\mathbb{R}^{n}\times\mathbb{R}_{\geq 0}\times\mathbb{R}^{m}\to \mathbb{R}^{n\times q}\) represents a measurable regression matrix. Also, the portions of \(g_{\phi}(x(t),t)\) that vary explicitly with time are assumed to be bounded such that \(g_{\phi}(x(t),t)\in\mathcal{L}_{\infty}\) whenever \(x(t)\in\mathcal{L}_{\infty},\forall t\geq t_{0}\).