## 5 Experimental Validations of the Proposed Controller  
An experimental prototype was designed, constructed, instrumented, and controlled to confirm the proposed controller's effectiveness. The proposed technique considers the inclusion of the experimental robotic manipulator carrying the gripper. The proposed methodology for testing the developed controller includes the practical realization of the distributed parameter representation for the manipulator, considering the application of the selected constructive strategy, the instrumentation using DC motors as actuators, and optical encoders for detecting an angular position at the joints. The engines selected for performing the movement used a 12 volt DC motor (Servocity), including an encoder (32 RPM). The encoder information was used to close the loop in position control. Figure 3 shows the entire structure of the developed micromanipulator with all the components considered in the design, including the DC motors, the piezoelectric actuators, and the mechanical sections as well as the image processing element to detect the current microgripper configuration. This discretized form of the controller was implemented on an ATMega microcontroller 2560, allowing the development of the proposed controller for the dynamics of the piezoelectric actuator. This board has 54 digital I/O pins (of which 15 provide PWM output that can be used for controlling DC motors), 16 analog input pins, and 40 mA current per I/O pin. The presence of perturbations, modeling uncertainties, and measurement noise appear as part of device operation under the selected conditions. The perturbations and uncertainties result from the mechanical configuration and the unconsidered dynamics of the actuators. This study used a PID controller for comparison, considering that this is the controller which is mainly used for robotic manipulators. The comparative study evaluated the application of PID control in general, and did not consider the independent effect of each gain in its structure. These parameters were adjusted using the self-tunning process inserted in the PID formulation proposed in the Matlab algorithm. This algorithm performs a frequency-response estimation experiment that injects signals into the plant and measures the plant output with a closed or open feedback loop. The method uses the resulting estimated frequency response to tune PID gains for the plant. The application of such an algorithm allowed us to create a fair comparison with the controller proposed as the main contribution of this study. Even when other parameters produce lesser mean square errors, they may increase the power related to the controller's application. The selected gains for the PID controller and the initial conditions for the adaptive control considering the state restrictions are presented in Table 1, with \(K_{c}=[K_{P},K_{D},K_{I}]\). Figures 4a, 5a, and 6a show the structural and motion components at all joints of the manipulator carrying the piezoelectric manipulator. These photographs demonstrate the composition of the mechanical structure and the DC motors as well as the relative distribution of the encoders. In addition, the mechanical engines located at each joint are depicted in order to demonstrate how the relative motion at each joint is obtained. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Joint / Gain** & \(K_{P}\) & \(K_{D}\) & \(K_{I}\) \\ \hline First & 90.92 & 5.21 & 3.4 \\ Second & 80.92 & 15.21 & 4.6 \\ Third & 72.34 & 24.12 & 7.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Selected values of the PID controller and initial conditions for the adaptive controller. Figure 3: Entire structure of the micromanipulator using the piezoelectric actuators. Figures 4b, 5b, and 6b depict the reference and the experimental trajectories for all the articulations in the robotic arm. All figures show the angular motion in degrees.