## 4 Discussion  
By referencing Figure 2, it is observed that the tracking errors decreased with increasing time for both ARISE and SM, which is likely due to the improving estimate of \(\theta\) due to the adaptive update laws. However, the convergence in the proposed ARISE controller was better than in the SM controller, which is likely due to the ARISE controller implementing an integral SM term rather than a discontinuous SM term. Referencing Figure 3, it is clear that the RMS errors across different disturbance values for both controllers were consistent. However, the ARISE controller exhibited considerably smaller RMS errors than the SM controller, which indicates that the ARISE controller has a better tracking performance than SM control. When the process to generate Figure 3 was repeated using the constant disturbances in Table 1 ('e'-'h'), similar results were obtained for both controllers (i.e., the results for 'a' and 'e', 'b' and 'f', and so on were nearly identical), and thus the figure is omitted. Similarly, Figure 4 depicts that the RMS errors of the ARISE controller were smaller than those of the SM controller even across varying control effectiveness parameters, thereby indicating that the ARISE controller performs better on a switched system. Furthermore, there was a notable decrease in the RMS errors of the second state for both controllers once the system reached SS, as illustrated in Figure 4. This result may be due to the second state having fewer nonlinearities associated with its dynamic model. Likewise, referencing Figure 5, the RMS errors of the ARISE controller increased negligibly with increasing values of \(\mu\). Comparatively, the RMS errors of the SM controller significantly increased with increasing values of \(\mu\). This consistent performance of the ARISE controller despite changing the dynamic parameter suggests that ARISE is more robust than SM control when implemented on a switched system. It is also noticeable from Figure 5 that the magnitude of the RMS errors associated with ARISE was always smaller than that of the errors from the SM controller, further demonstrating the ARISE controller's effectiveness over SM control.  
Referencing Figure 7, the SM controller exhibits a rapid switching within the control input, which induces the chattering effect. Conversely, the ARISE controller displays a  
Figure 8: The tracking errors of the ARISE controller for the soft robot dynamic model.  
smoother control input, effectively mitigating the chattering effect. A key contribution of the proposed controller is that it is able to control a nonsmooth system, unlike traditional RISE controllers. Even though RISE controllers also use an integral SM term similar to ARISE, it can only be implemented on differentiable or smooth dynamic systems. The ARISE controller, on the other hand, obtains the benefits of RISE while being implementable on a wider class of dynamic systems. Referencing Figure 8, it can be seen that the proposed ARISE controller maintained a bounded tracking performance, reinforcing the applicability of the controller in a real-world scenario.  
It should furthermore be noted that SM control is capable of achieving exponential tracking of the considered dynamic system, whereas ARISE control is only able to achieve uniformly ultimately bounded tracking of the same system. In particular, while using the ARISE controller, the system converges to a local neighborhood around the target trajectory, whereas an SM controller can theoretically converge to the exact trajectory. Despite the theoretical limitations of the ARISE approach, its tracking performance was superior to the SM control in simulation. It is anticipated that the improved performance of the ARISE controller compared to SM control will be more substantial in experimentation since real actuators are unable to perfectly implement SM control signals due to their requirement for infinite control frequency.