## 5 Experimental Validations of the Proposed Controller
The comparison demonstrates that the proposed controller allowed the references to be tracked with better quality and with oscillations with smaller amplitudes compared to the traditional constant gain controller. The same characteristic was observed for all the trajectories proposed, which had the aim of forcing the controller to track more quickly-changing references. Figure 4: (**a**) Photograph of the electromechanical configuration for the first articulation \(\theta_{1}\); (**b**) comparison of the controlled motion for the first articulation (\(\theta_{1}\)) when the PID and adaptive control with state constraints are considered. Figure 5: (**a**) Photograph of the electromechanical configuration for the second articulation \(\theta_{2}\); (**b**) comparison of the controlled motion for the second articulation (\(\theta_{2}\)) when the PID and adaptive control with state constraints are considered. The comparison of tracking errors using the mean square evaluation confirms the benefits of using the proposed controller, exhibiting faster convergence to the invariant zone characterized in the theorem presented in this study. In addition, the averaged value obtained at the steady state period of time is smaller with the state constraint controller (Figure 7). Figure 6: (**a**) Photograph of the electromechanical configuration for the third articulation \(\theta_{3}\); (**b**) comparison of the controlled motion for the third articulation (\(\theta_{3}\)) when the PID and adaptive control with state constraints are considered. The control action produced with the implemented controller exhibits different behavior compared to that generated with the constant gains variant. The obtained control action with the restricted controller has a smaller amplitude. Nonetheless, when the trajectories grow closer to the boundaries, the control action increases its amplitude, as predicted by the gains laws presented in Expression (11). Figure 8 compares the evolution of the control actions using the proposed laws for the state constraint variant and its comparison with constant values. This comparison confirms that the obtained controllers produce smaller amplitudes for the control action, which consumes less power during trajectory tracking of the reference states. The proposed controller demonstrates significant advantages concerning similar state feedback options. The set of benefits can be observed for both the effectiveness of the tracking trajectory and for the motion produced for the  
Figure 8: Comparison of control action between the proposed state constraint and the traditional state feedback control form. Figure 7: Comparison of the mean square error between the proposed state constraint and the traditional state feedback control form. trolled action of all the joints simultaneously. This study calculated the averaged integral for the norm of the control actions both using the proposed barrier-based control and traditional state feedback (PID control). The integral of the control action is presented when the number of samples corresponding to the end of the experimental evaluation was 3000. After carrying out averaged integration, we estimated the state feedback formulation as \(1.3\times 10^{9}\), while that obtained with the barrier-based controllers was \(5.8\times 10^{8}\). Moreover, we calculated the ratio of the average mean square error obtained with both controllers (estimated with the information presented in Figure 8) concerning the corresponding averaged value of the integral for the norm of the control actions. These values evaluate a measure of efficiency in tracking performance for the control power invested in attaining such tracking. For the PID controller, the proposed ratio was \(16.92\times 10^{-9}\), while the value with the barrier controller was \(0.89\times 10^{-8}\). This comparison explains the benefit of introducing the proposed controller with respect to traditional formulations. Figure 9a demonstrates a detailed view of the gripper structure based on the selected piezoelectric actuators.