## 4 Experimental Investigation  
The prototype of the framed piezoelectric actuator with a weight of 83.8 g was manufactured and fabricated based on the finite element simulation results, as shown in Figure 7a. And the assembled railed carrying system prototype is illustrated in Figure 7b. At first, we conducted the vibration measurement tests on the prototype of the framed piezoelectric actuator by using a 3D Doppler laser Vibrometer (PSV-500-3D, Polytec, Waldbrunn, Germany). The vibration shapes and amplitude-frequency curves of the actuator prototype were measured, as shown in Figure 8. The two operating vibration modes were obtained, and their corresponding resonant frequencies are 20.539 kHz and 20.345 kHz, respectively. In the two operating vibration modes, the frequency differences between the simulated and measured resonant frequencies are 41 Hz and 163 Hz, respectively, thereby verifying the validation of the simulated results. The reason for the resonant frequency difference between the FEM and the experiment is mainly caused by the machining and assembly errors of the actuator prototype. The glue layers between the piezoelectric plates and the frame straight beam, which were ignored in simulation analysis, also contribute on the resonant frequency discrepancy. Figure 8: _Cont._  
Figure 7: The prototypes of (**a**) the framed piezoelectric actuator and (**b**) the assembled piezoelectric carrier robot. To evaluate the mechanical performances of the assembled framed piezoelectric actuator prototype, an experimental setup was built, as shown in Figure 9, and then a series of tests were carried out. Although the resonant frequencies of the two excited vibration modes were determined for the actuator prototype, its optimal driving frequency has to be measured due to the fact that the installation of the preload mechanism changed its mechanical boundary. Under the excitation voltage of 200 V\({}_{\text{PP}}\), the frequency sensitivity characteristic of the actuator prototype was tested as the driving frequency was varied from 20.2 kHz to 21.2 kHz, as shown in Figure 10. The measured mean velocity of the actuator prototype increased first and then decreased with the driving frequency increased. The optimal driving frequency of the actuator prototype was 20.75 kHz, corresponding to its maximum mean velocity of 99.8 mm/s. Figure 8: Vibration measurement results of the actuator prototype using the 3D Doppler laser vibrometer. (**a**) Mode A. (**b**) Mode B: the measured third order bending vibration shape presented in the long piezoelectric composite beams. (**c**) Mode B: the tested first order bending vibration exhibited in the short piezoelectric composite beams. (**d**) Amplitude-frequency curves. The motion characteristic of the actuator prototype was then measured at the optimal driving frequency when the excitation voltage was varied from 50 V\({}_{\text{PP}}\) to 400 V\({}_{\text{PP}}\) with a step size of 50 V\({}_{\text{PP}}\), as shown in Figure 11. The mean velocity of the actuator prototype increased with the excitation voltage increased. At the driving voltage of 400 V\({}_{\text{PP}}\), the maximum mean velocity of the actuator prototype reached up to 384.9 mm/s. These results present the expected maneuverability of the actuator prototype. Figure 11: Measured motion characteristic of the framed piezoelectric actuator prototype. Figure 9: Experimental setup for measure mechanical performances of the actuator prototype. To verify the load capacity, we conducted measurements on the actuator prototype at different excitation voltages. Experimental results are shown in Figure 12. The driving frequency was still 20.75 kHz. As the load weight increased, the mean velocity of the actuator prototype decreased at each excitation voltage. Moreover, the maximum load weights were close to 600 g and 900 g when the excitation voltages were 200 V\({}_{\text{pp}}\) and 300 V\({}_{\text{pp}}\), respectively.