## 4 Experimental Validation
### Dynamic Model Experimental Results  
In this experiment, the VSL robot was given a pose, followed by the recording of its movement from its pose to the given pose at a frame rate of 30 fps. First, the second joint (\(\theta_{2}\)) was moved from \(90^{\circ}\) to \(16^{\circ}\), then, the third joint (\(\theta_{3}\)) was moved from \(0^{\circ}\) to \(-45^{\circ}\), and, finally, the first joint was moved from \(0^{\circ}\) to \(-43^{\circ}\). Then, using the SOLIDWORKS program, the photos of each frame were inserted into a 2D drawing, the angles of the robot's movement were measured manually, and the final position of the tip was tracked, frame by frame, from the beginning until it came to a complete stop. Examples of these images are shown in Figures 13-15. The same pose was then applied to the model in SIMULINK and its motion was also recorded. Finally, the collected experimental and simulated data were recorded, plotted, and compared. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & \(\theta_{1}\) & \(\theta_{2}\) & \(\theta_{3}\) & \multicolumn{3}{c}{**Experimental (mm)**} & \multicolumn{3}{c}{**Theoretical (mm)**} & \multicolumn{3}{c}{**Error (mm)**} \\ \cline{4-11}  & & & \(x\) & \(y\) & \(z\) & \(x\) & \(y\) & \(z\) & \(x\) & \(y\) & \(z\) \\ \hline
1 & 88.69 & 71.61 & \(-\)86.82 & - & 326.45 & 262  
Figure 14: Sample of the recorded video frames in which the angles and the tip position are measured using SOLIDWORKS, moving \(\theta_{3}\) in XZ-axis plane. Figure 13: Sample of the recorded video frames in which the angles and the tip position are measured using SOLIDWORKS, moving \(\theta_{2}\) in XZ-axis plane. Figure 15: Sample of the recorded video frames in which the angles and the tip position are measured using SOLIDWORKS 2020, moving \(\theta_{1}\) in XY-axis plane. The results of the experiment show that, although the model's responses were adequate, they did not have the degree of accuracy we would have liked. The average error of the angular displacement for the first joint was \(0.63^{\circ}\), for the second joint \(1.06^{\circ}\), and for the third joint \(1.27^{\circ}\). In order to achieve greater precision, it is essential that we carry out further investigations into the damping and stiffness coefficients. The plots for the three angles of the joints versus time for the experimental and simulated responses of the manipulator are illustrated in Figure 16, Figure 17, and Figure 18, respectively. In general, the plots show that the simulation model captured the whole trends of the 3 DOF flexible link manipulator experimental joint angles data. However, there were some discrepancies between the experimental and simulated responses that could be attributed to various factors such as modeling assumptions (e.g., damping coefficient), measurement  
Figure 16: Plotting for the angles of the joints versus time for the manipulator: experimental and simulation responses for \(\theta_{1}\). Figure 17: Plotting for the angles of the joints versus time for the manipulator: experimental and simulation responses for \(\theta_{2}\). noise, or unmodeled dynamics (e.g., motors). The three recorded data plots for the three joint angles showed some degree of oscillation, which was due to the flexibility of the links. For \(\theta_{3}\) in Figure 18, there were more oscillations just before the steady state than for \(\theta_{2}\) in Figure 19, which was due to the second link flexibility effect on the third link, while for \(\theta_{2}\) in Figure 19, there was only the second link flexibility effect. The oscillations of the links decrease or disappear with higher rotational speed when the centrifugal force exceeds the flexibility torque but increases the overshoot when it stops. The stronger the acceleration or deceleration, the more oscillations occur at the beginning and end of the movement.