## 2 Materials and Methods
### Simulation
To ensure a fair comparison between the ARISE controller and the SM controller, the dynamic parameters and adaptive update law were kept the same during each sim \begin{table}
\begin{tabular}{c c c} \hline \hline
**Disturbances** & \(\mathbf{d_{1}}\) & \(\mathbf{d_{2}}\) \\ \hline a & 0.5 cos(t) & 0.5 sin(t) \\ b & 0.1 cos(t) & 0.1 sin(t) \\ c & 0.05 cos(t) & 0.05 sin(t) \\ d & 0.01 cos(t) & 0.01 sin(t) \\ e & 0.5 & 0.5 \\ f & 0.1 & 0.1 \\ g & 0.05 & 0.05 \\ h & 0.01 & 0.01 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Continuous and constant disturbances used in simulation. Figure 1: Control framework for ARISE controller. ulation. The control law itself is the only difference (note that the proposed ARISE controller is derived in Section 2.2, while the SM controller is introduced below). The SM controller that is used for comparative purposes in the simulations is defined as \(u=\hat{g}_{\phi}^{+}(x(t),t)(-k_{\phi}(t)-k_{5}\text{sgn}(e)+\hat{x}_{d}(t))\), where \(k_{4}\) is a robust control gain for the SM controller and \(k_{5}\) is an SM control gain. The SM control gains were selected as \(k_{4}=90\) and \(k_{5}=3.5\). The listed gains for both controllers were adjusted using an empirical-based method for a single set of dynamic parameters and disturbances (i.e., \(\mu=1.5\), control effectiveness parameters = set 'GA', and disturbances = set 'a'). First, the control gains for the ARISE controller were selected and then adjusted to achieve an improved performance (i.e., minimum tracking errors). After that, the control gains for the SM controller were selected and then adjusted similar to the ARISE controller gains. While selecting the control gains for the SM controller, similar gain values were considered to ensure a fair comparison between the two controllers. The selected control gain values were then used across all other dynamic sets during the simulations. Note: The control gains for the ARISE controller could be adjusted using a variety of optimization techniques, including Particle Swarm Optimization, Genetic Algorithms, and the Nelder-Mead Simplex Method. However, a manual empirical gain tuning (i.e., trial-and-error) method was employed in this work. The use of optimization methods to tune the gains can be investigated in future research to further enhance the control performance. Similar to the development of the ARISE adaptive update law, an adaptive update law was developed for the SM controller as \(\hat{\theta}(t)\triangleq\text{Proj}\Big{(}\Gamma\gamma_{\phi}^{T}(x(t),t,u(t) )\epsilon(t)\Big{)}\). The initial estimates of \(\theta\) and the gain \(\Gamma\) were selected to be the same for the ARISE and SM controllers. The RMS of the tracking errors were evaluated for both controllers to compare their effectiveness, thereby providing valuable insights into their robustness when implemented on a switched system. To further demonstrate the practical use of the proposed ARISE controller, another simulation was carried out on a soft robotic arm. A general nonlinear dynamic model of a 2D steering robotic arm without switching is presented. The dynamic parameters for the simulation were selected as \(f(x)=[-0.5x_{1};-0.5x_{2}]\), \(d(t)=[0.05\cos(t);0.05\sin(t)]\), \(g=[1,0.5,-0.5;0.2,-1,0.7]\). Similarly, the control parameters were selected as \(k_{1}=1.5\), \(k_{2}=100\), \(k_{3}=20\), \(\alpha=5\), \(\beta=10\), and \(\Gamma=\text{diag}\{0.1,0.2,0.05,0.15,0.1,0.2\}^{-1}\). To start the simulation, we needed to define the initial conditions for the robot's states and the unknown parameters of the control effectiveness matrix \(g\). These initial conditions were selected as \(x[0]=[0.1;0.1]\) and \(\hat{\theta}(0)=[0.5;0.5;0.5;0.5;0.5;0.5]\). Similarly, the desired trajectories were selected as \(x_{d}=[0.3\sin(t),0.3\cos(t)]^{T}\). Note that the dynamic model presented above is a generic model for soft robotic end-effectors that use 3 actuators to control motion in 2 dimensions.