## 4 Conclusions  
This study developed a multi-functional redundant robotic arm with four degrees of freedom using PAM as the main actuator to allow patients with upper limb mobility impairment to perform rehabilitation trajectory tasks in the future while wearing the robotic arm. The exoskeleton robot arm is designed based on the structure of the joints of the human arm due to the need to wear it on the human body. Since the mechanism may collide with the user when it is in motion, an actuator and control commands are used to limit the space available for each rotary axis during the movement of the robot arm so as \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**No-Load—Rehabilitation Trajectory 1** & **1 st Axis** & **2 nd Axis** & **3 rd Axis** & **4 th Axis** \\ \hline Optimal learning period & 12 & 17 & 19 & 15 \\ \hline Initial RMSE (degrees) & 1.1112 & 1.2123 & 1.4021 & 0.9831 \\ \hline Optimal RMSE (degrees) & 0.2444 & 0.8017 & 0.2395 & 0.6953 \\ \hline
**Load-On—Rehabilitation Trajectory 1** & **1 st Axis** & **2 nd Axis** & **3 rd Axis** & **4 th Axis** \\ \hline Optimal learning period & 12 & 20 & 20 & 15 \\ \hline Initial RMSE (degrees) & 1.2315 & 1.1253 & 1.3931 & 1.1458 \\ \hline Optimal RMSE (degrees) & 0.2384 & 0.9381 & 0.2337 & 0.6061 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Trajectory 1—before and after applying ILC—table comparing the tracking results of each axis with and without loading effect. Figure 26: Trajectory 1—4 th-axis (load-on) ILC trial 1 and trial 15 tracking and error diagrams. not to exceed the limit of human arm joints. The rotary axis control part is controlled by PAM. Therefore, the mathematical model of the single-axis robot arm driven by two PAMs was firstly developed, followed by the design of the PID, feedback (LMS-PID), and feed-forward (ILC) controllers. Because of the complexity of the nonlinear characteristics of the PAM actuator, the model and control parameters are uncertain when performing the tracking experiment of the PID controller by simulating the parameters, which leads to the ineffective control of the parameters. Therefore, this study used a self-adaptive (LMS-PID) controller and an iterative learning controller (ILC) to compensate; the ILC performs the feed-forward compensation by iteratively modifying the control commands, while the LMS-PID controller is an adaptive feedback controller using the LMS algorithm. The parameters obtained from the LMS-PID combined with the iterative learning control can produce a better control effect, especially the phase lag and transient parts, which cannot be solved by the PID controller. Iterative learning control can reduce the error of the system to a certain range within a limited number of learning times, which can greatly improve the control effect of the robot arm. Overall, in the single-axis robotic arm controller experiment, the results showed that LMS-PID was superior to the conventional PID control method. In the control experiment of the robotic arm under multi-axis recovery trajectory, the experimental results showed that the study can plan the training of drawing a circle on the wall, which can effectively train the shoulder joint area. In the multi-axis robotic tracking experiment under the rehabilitation trajectory, the results showed that the LMS-PID controller reduced the RMSE of the tracking trajectory to 0.2444 and 0.2853. In the robotic arm joining/loading experiment, the results showed that the tracking accuracy was still accurate after the loading. Finally, the robotic arm in the rehabilitation mode was only used by people without impairments, but the purpose of this development is for patients with arm injuries. It is hoped that we can cooperate with hospitals in the future so that patients with arm disabilities can use the robotic arm developed by this study to perform rehabilitation exercises. Conceptualization, C.-J.L. ; methodology, C.-J.L. ; software, Y.-Y.C. ; validation, C.-J.L. and Y.-Y.C. ; formal analysis, C.-J.L. ; data curation, C.-J.L.