## 6 Discussion  
Robotic-assisted rehabilitation is a boon to those with upper and lower limb disabilities, as well as the medical community. Better control methods used in these devices to carry out rehabilitative activities may physically improve the patient's condition. Trajectory tracking is one such activity that assists patients in their early recovery by allowing them to train their targeted joints continually. Trajectory monitoring improves muscular mobility through training. The researchers are driven to advance an exoskeleton in medicine because it can treat patients more conveniently and efficiently than standard rehabilitation approaches. Control engineering plays a crucial role in increasing the capacity of these devices using various efficient control algorithms and allowing the system to track or behave as per the required performance, which is a significant challenge. This article attempts to enhance one such performance, i.e., the ULRRE trajectory tracking. This research presents a simulation study that addresses the trajectory tracking control problem for a two-degrees of freedom ULRRE that models the human upper limb, including the shoulder and elbow joints. The simulation tests the effectiveness of ESO-based methods used for rehabilitation tracking applications [40,51,52]. The investigated disturbance and parameter variation case studies on ULRRE determine the controllers' robustness as mentioned above while performing trajectory tracking. When a disturbance is given to the system, the error plots for the ADRC, NLADRC, IADRC, and proposed method reveal that ESO-based controllers exhibit disturbance rejection capabilities. The results show precise trajectory tracking in no disturbance and with disturbance case, where all ESO-based methods requires nearly identical control signal levels. The proposed control method utilizes the combination of LESO and FTSTD, whereas IADRC [52] uses ESO nonlinear state error feedback (NLSEF) and tracking differentiator (TD), which is complex in design as the number of parameters for tuning increases. NLADRC [51] uses nonlinear ESO and NLSEF, which has fewer parameters to tune than IADRC and the proposed method. ADRC [40] uses the ESO and has the minor tuning parameters among all controllers. From the results, the performance indices indicate that the proposed methods enhance the system's performance by tracking accurate trajectory and reducing the trajectory tracking error than IADRC, NLADRC, and ADRC. The proposed controller is less complex than IADRC, as it only uses ESO and FTSTD to achieve better \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{6}{c}{**Elbow Joint**} \\ \hline  & ITSE (Deg.) & & ISE (Deg.) & & ITAE (Deg.) & & IAE (Deg.) \\ \hline Control & \(-20\%\) & \(+20\%\) & \(-20\%\) & \(+20\%\) & \(-20\%\) & \(+20\%\) & \(-20\%\) & \(+20\%\) \\ \hline Proposed & 7.501 & 7.522 & 1.51 & 1.516 & 16.88 & 16.93 & 3.383 & 3.395 \\ \hline IADRC [52] & 13.59 & 13.60 & 2.721 & 2.726 & 23.50 & 23.51 & 4.703 & 4.707 \\ \hline NLADRC [51] & 15.71 & 15.72 & 3.144 & 3.149 & 25.23 & 25.24 & 5.047 & 5.052 \\ \hline ADRC [40] & 17.97 & 17.99 & 3.596 & 3.602 & 26.98 & 27 & 5.398 & 5.403 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Performance indices of the elbow joint \(\pm 20\%\) parameter variation and constant disturbance. accuracy than IADRC. In ESO-based methods, choosing observer bandwidth is critical. It affects the tracking error, which tends to decrease with the increase in observer bandwidth, but it inversely affects the control effort, increasing bandwidth. The observer bandwidth is chosen as a trade-off between the tracking performance and control signal requirement and is one of the limitations of this model and ESO-based methods. The proposed method will compare the presented ESO-based approaches to the proposed method in the future. The test technique will be designed and put to the test in a controlled laboratory environment.