## 5 Conclusions and Future Work  
This paper presents an initial design for impact-based, dynamic amplification of small piezoelectric transducer displacement, with some capabilities for high-to-low resonant frequency conversion and output frequency tuning. In Table 3, major performance attributes of the proposed mechanism are compared to other amplification strategies with similar size reported in the literature. Table 3 highlights the unique large amplitude, low-frequency output of an impact-based approach to piezoelectric transducer amplification. Such functionality may provide opportunities for generating periodic appendage motions in small-scale robots where bio-inspired dynamics call for motion in much lower frequency ranges than are typical for piezoelectric transducers.  
However, there remains substantial future work to complete a comprehensive study of this amplification strategy, both to fully understand dynamic behaviors and evaluate practical applications to micro-robotics. While the simple dynamic model introduced here captures several of the qualitative features of the proposed amplification mechanism, the impact event model, in particular, is too simplistic to fully predict experimental behaviors. Meanwhile, substantial prototype robot development, focusing on fluid coupling with oscillatory fin motion and packaging to avoid impeding mechanism motion, would be required to evaluate whether propulsion can be generated. Similarly, feasibility for generating legged locomotion would depend on gaining a better understanding of mechanism  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Ref.** & **Type** & **Amplitude (\(\upmu\)m)** & **Operating Frequency (Hz)** & **Dimension (mm\({}^{3}\))** & **Voltage (V)** \\ \hline
[24] & Leverage & 10 & \(\approx\)50,000 & \(5\times 5\times 2\) & 500 \\
[25] & Leaf springs & 600–700 & NaN & \(\approx\)10 \(\times\) 10 \(\times\) 30 & 1000 \\
[25] & Planar compliant & 1200 & NaN & \(\approx\)10 \(\times\) 10 \(\times\) 30 & 1000 \\  & mechanism & Compact & & & \\
[8] & out-of-plane & 6.3 & 0.5 & 2 \(\times\) 2\(\times\) NaN & 170 \\  & transmission & & & & \\
[26] & Extensional & 8.7 & 265 kHz & \(\approx\)4.66 \(\times\) 2.54 \(\times\) 0.005 & 100 \\  & microactuator & & & & \\
[12] & Piezo wiggle: & 6000 & NaN & \(\approx\)31.55 \(\times\) 1.55 \(\times\) 6 & 20–40 \\  & Linear Motor & & & & \\  & Amplifying & & & & \\
[10] & transmission and & 243 & 507 & 15 \(\times\) 3 \(\times\) 0.6 & 120 \\  & compact & & & & \\  & suspension & & & & \\  & Impact to & & & & \\ This & compliant & 1194 & 12–17 & 55 \(\times\) 22 \(\times\) 1 & 40 \\ work & structure & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 3: Features of previously reported piezoelectric amplification mechanisms at millimeter–centimeter scales.  
modeling and design for robustness over changing gravitational loads. Nonetheless, the approach may allow robot designers to generate near-periodic motion in unique operating regions for frequency, amplitude, and power in small-scale robotic systems.  
Amplification conceptualization: C.S. and K.R.O.; Modeling: K.R.O., C.S., and S.Y.; Experimental validation: S.Y. and C.S.; Integration Studies: J.L. and K.R.O. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
Data Availability Statement: Data collected during this study may be obtained by contacting the corresponding author at [EMAIL_ADDRESS].  
The authors thank members of the Microdynamics and Waves Laboratory: Microsystems at the University of Michigan for their assistance with this work. Partial financial support was provided by the University of Michigan.  
The authors declare no conflict of interest.