## 3 Experimental Results
### Multi-Axis Robotic Tracking Experiment under Rehabilitation Trajectory  
Based on PID controller theory, a better output response can be obtained by adjusting the Kp, Ki, and Kd parameters, but the adjustment of the parameters often requires experience and a lot of time. Additionally, the robotic arm driven by PAMs in this study has a total of 12 controller parameters in four degrees of freedom, and considering different load effects and different trajectory commands, it is necessary to make adjustments one by one to achieve better control performance, which is not efficient. Therefore, in the multi-axis robotic arm tracking experiment, the LMS-PID controller was first used to track each rotary axis for a certain period of time, and then a set of available PID parameters was obtained. Figure 20 shows the controller parameter curve obtained at the 20 th cycle of the four rotary axes tracking, based on its parameters, and then the iterative learning control was imported to improve its output response. Using an LMS-PID controller in each axis of rotation, tracking after a certain period can obtain a set of PID parameters from this experiment. Figure 20 illustrates the controller parameter curve obtained with four axes of rotation traced in the 20 th period. Based on this parameter, the iterative learning control was inputted to improve its output response. In terms of the RMSE values of the initial and optimal tracking periods of each axis before and after the subsequent use of ILC, for example, the angular RMSE of the first axis decreases from the initial \(1.1112^{\circ}\) to \(0.2444^{\circ}\) after the 12 th learning.  
Figure 19: Motion trajectory 2 of each axis of the robot arm is planned by kinematics.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Rehabilitation** & **x-Axis** & **y-Axis** & **z-Axis** \\
**Trajectory 2** & & & \\ \hline Trajectory equation & \(0.06\cdot\sin\left(\frac{2\pi k}{n}\right)\) & \(0.45\) & \(-0.14-0.06\cdot\cos\left(\frac{2\pi k}{n}\right)\) \\ \hline Initial state & Joint1: \(0^{\circ}\), Joint2: \(39^{\circ}\), Joint3: \(0^{\circ}\), Joint4: \(57^{\circ}\) \\ \hline Circle radius & & \\ (one way) & & \(0.06\) (m) \\ \hline Number of splits & & \\ \((n)\) & & \(1500\) \\ \hline Number of & & \\ samples \((k)\) & & \(1\)-\(1500\) \\ \hline \hline \end{tabular}
\end{table}
Table 8: Rehabilitation trajectory 2—curvilinear equation in space.  
The curves of the RMSE of each axis during the whole learning process are shown in Figure 20. In order to observe the trajectory of the robotic arm end point in space before and after learning, the trajectory of each axis before learning and after learning for a certain number of times is substituted into the forward kinematics to determine the trajectory in space, and the RMSE of the trajectory after the 15 th learning is 0.007753 (m), which is better than the RMSE of the trajectory before learning, at 0.01421 (m). Rehabilitation trajectory 2 was also compared to the control method of trajectory 1, and the result is shown in Figure 21. The LMS-PID controller has different trajectory commands, and even if the same convergence factor and the same trajectory period are used, the error caused by the trajectory experiment is not the same for different trajectories, so the obtained internal controller parameters are not exactly the same. The initial and optimal RMSE values of each axis before and after the use of ILC for track 2 were compiled. Taking the first axis as an example, its angle RMSE decreased from the initial 1.4162\({}^{\circ}\) to 0.2853\({}^{\circ}\) after the 14 th study, which is an improvement of 1.1309\({}^{\circ}\).