## 5 Conclusions  
In this research, an inertial linear ultrasonic motor with a double-stator structure was proposed, and higher performance and resolution were realized by utilizing the bend  
Figure 13: Displacement resolution measurement: (**a**) schematic of the applied voltage signal in resolution measurement: pulse sawtooth wave voltage signal; (**b**) step response of the motors under pulse sawtooth wave. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameters** & **Ultransonic Motor** & **Ultransonic Motor** & **TULA70**[26] & **Piezoelectric Motor** & **Proposed ILUM-D** \\  & [17] & [2] & & [27] & \\ \hline Size (mm\({}^{3}\)) & 2.8 \(\times\) 2.8 \(\times\) 10 & 4 \(\times\) 11 \(\times\) 22 & 7 \(\times\) 7 \(\times\) 22 & 3 \(\times\) 6 \(\times\) 18 & 14.6 \(\times\) 14.6 \(\times\) 29 \\ Frequency (kHz) & 152 & 50 & 30–70 & 0–30 & 36.5–39 \\ Voltage (V\({}_{\text{P-P}}\)) & 40 & 10 & 20–35 & 50 & 500 \\ PZT Type & Multilayer & Multilayer & Multilayer & Multilayer & Single-layer \\ Maximum thrust force (N) & 0.08 & 0.12 & 0.392 & 3.3 & 1.645 \\ Maximum velocity (mm/s) & 20 & 14.8 & 10 & 16 & 8.746 \\ Resolution (nm) & Not mentioned & Not mentioned & 100 & 800 & 27 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison between the proposed ILUM-D and the previous works. ing mode and the sawtooth wave signal. The working mechanism of the ILUM-D was illustrated and a finite element model was put forward. Modal analyses and the transient analyses were conducted to determine the feasibility and investigate the vibration characteristics of the driving shaft and the output performance of the motor. A prototype was fabricated and the experimental setups were built up to obtain the mechanical characteristics of the motor. The experimental results indicated that the higher resonance frequency and symmetrical structure made it possible to produce significant asymmetrical vibrations under the sawtooth wave signal excitation condition. The maximum output velocity and thrust force of the ILUM-D were measured as 8.746 mm/s and 1.645 N (frequency: 37.5 kHz, voltage: 500 V\({}_{\text{p-p}}\)), respectively, while those of the ILUM-S were 4.798 mm/s and 0.644 N (frequency: 35.0 kHz, voltage: 420 V\({}_{\text{p-p}}\)). A displacement resolution of 27 nm (frequency: 37.5 kHz, voltage: 300 V\({}_{\text{p-p}}\)) for the ILUM-D and that of 26 nm (frequency: 35.0 kHz, voltage: 300 V\({}_{\text{p-p}}\)) for the ILUM-S were achieved. Furthermore, the spatial attitude of the driving shaft was not affected by the axial position of the moving element during the motor operation, which will led to uniform contact at the driving interface. In future research, we will focus on improving the output performance of the ILUM-D and its application of parallel mechanisms and positioning methods. Conceptualization, L.Y. and Y.X. ; methodology, L.Y. and Y.X. ; software, Y.X. and J.W. ; validation, Y.X., X.H., J.W., J.Z. and R.Z. ; formal analysis, Y.X. ; investigation, Y.X. ; resources, L.Y. ; data curation, Y.X. and R.Z. ; writing--original draft preparation, L.Y. ; writing--review and editing, Y.X. ; visualization, Y.X. ; supervision, L.Y. ; project administration, L.Y. ; funding acquisition, L.Y. All authors have read and agreed to the published version of the manuscript. This paper was supported by the National Natural Science Foundation of China (Grant No. 52375058). Not applicable. The raw data supporting the conclusions of this article will be made available by the authors on request. Author [PERSON] was employed by the NUAA Super Control Technology Co., Ltd. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. Not applicable. The raw data supporting the conclusions of this article will be made available by the authors on request. Author [PERSON] was employed by the NUAA Super Control Technology Co., Ltd.