Force Characterization of Insect-Scale Aquatic Propulsion Based on Fluid-Structure Interaction

TL;DR

This work tackles thrust generation at insect scale by leveraging fluid-structure interaction of soft tails driven by shape-memory alloy actuators. It combines a reactive-force model based on added-mass concepts with a newly developed µN-resolution force sensor to quantify instantaneous and cycle-averaged thrust for single-tail and dual-tail propulsors. The single-tail device achieves a peak thrust around $0.45$ mN and a cycle-averaged thrust up to $2.97\ \mu$N, while the dual-tail device reaches a peak of $0.61$ mN and a cycle-averaged thrust up to $22.6\ \mu$N, highlighting wake-structure interactions as a key driver of enhanced propulsion in multi-tail configurations. The reactive model aligns well with the single-tail measurements but underpredicts the dual-tail thrust, pointing to wake dynamics as a critical factor to incorporate in future models for accurate, control-oriented predictions.

Abstract

We present force characterizations of two newly developed insect-scale propulsors--one single-tailed and one double-tailed--for microrobotic swimmers that leverage fluid-structure interaction (FSI) to generate thrust. The designs of these two devices were inspired by anguilliform swimming and are driven by soft tails excited by high-work-density (HWD) actuators powered by shape-memory alloy (SMA) wires. While these propulsors have been demonstrated to be suitable for microrobotic aquatic locomotion and controllable with simple architectures for trajectory tracking in the two-dimensional (2D) space, the characteristics and magnitudes of the associated forces have not been studied systematically. In the research presented here, we adopted a theoretical framework based on the notion of reactive forces and obtained experimental data for characterization using a custom-built micro-N-resolution force sensor. We measured maximum and cycle-averaged force values with multi-test means of respectively 0.45 mN and 2.97 micro-N, for the tested single-tail propulsor. For the dual-tail propulsor, we measured maximum and cycle-averaged force values with multi-test means of 0.61 mN and 22.6 micro-N, respectively. These results represent the first measurements of the instantaneous thrust generated by insect-scale propulsors of this type and provide insights into FSI for efficient microrobotic propulsion.

Figures (7)

• Figure 1: Image composites of frames, taken at $\boldsymbol{5}$-s intervals, showing three swimming tests performed using three different insect-scale microrobots driven by FSI-based aquatic propulsors.(a) The single-tail $45$-mg VLEIBot swimmer presented inVLEIBot_2024, while operating excited by a $1$-Hz PWM signal with a DC of $8$ %.~(b) The dual-tail $90$-mg VLEIBot+ swimmer presented inVLEIBot_2024, while operating excited by a $1$-Hz PWM signal with a DC of $8$ %. (c) The dual-tail $900$-mg autonomous VLEIBot++ swimmer presented inLongwellCR2024, while operating excited by a $1$-Hz PWM signal with a DC of $5$ %. These experiments can be viewed in the attached supplementary movie.
• Figure 2: Kinematic definitions and geometry of the tail planform used for analysis and model formulation.(a) Idealized kinematics of the tail undulation during an actuation cycle. The inertial and body-fixed frames of reference are defined as $\boldsymbol{\mathcal{N}} = \{\boldsymbol{n}_1,\boldsymbol{n}_2, \boldsymbol{n}_3\}$ and $\boldsymbol{\mathcal{B}} = \{\boldsymbol{b}_1,\boldsymbol{b}_2,\boldsymbol{b}_3\}$, respectively. To describe the motion of the tail as a traveling wave in the $\boldsymbol{b}_1$--$\boldsymbol{b}_2$ plane, we define the Lagrangian coordinate, $s$, and the centerline relative to $\boldsymbol{\mathcal{N}}$ as $\boldsymbol{r}(s,t) = \left[x(s,t)\,\,y(s,t)\,\,0 \right]^T$, where $t \geq 0$ denotes time. According to this model, the traveling wave travels from left to right and the undulating tail travels at a constant speed $U$ from right to left; therefore, the centerline relative to $\boldsymbol{\mathcal{B}}$ can be described as $\boldsymbol{r}_{\boldsymbol{\mathcal{B}}}(s,t) = \left[x(s,t)-Ut\,\,y(s,t)\,\,0 \right]^T$. To track the local forces acting on the tail, we define the Lagrangian frame $\left\{ \boldsymbol{u}_{\text{t}},\boldsymbol{u}_{\text{n}} \right\}$, where $\boldsymbol{u}_{\text{t}}(s,t)$ and $\boldsymbol{u}_{\text{n}}(s,t)$ are the unit vectors tangent and normal to the tail's centerline. (b) The planform of the undulating tails used to drive the propulsors considered in this paper have a parabolic shape with a local height given by $h(s) = 0.694\sqrt{l-s}$.
• Figure 3: Idealization and dynamic modeling of the DCS constituting the µN-resolution force sensor used to measure thrust.(a) Simplified dynamic model of the DCS made of Invar-$36$. (b) Magnitude of the normalized sensing structure's response, $\left| \delta(\omega_0)\right|/\delta(0)$ in decibels (dB), to a periodic excitation of the form $F(t) = F_0 e^{j\omega_0t}$, for $j = \sqrt{-1}$.
• Figure 5: Experimental setup used for characterization of the force generated by a propulsor during operation.(a) A Mathworks Simulink Real-Time host--target system, equipped with a National Instruments PCI-$6229$ AD/DA board, is used to generate, process, and record signals at a rate of $5$ kHz. The PWM signal required for actuation is generated using the AD/DA board of the host--target system; then, this signal is power-amplified with a MOSFET-based circuit that acts as a switch that opens and closes the electrical paths to the actuator from a power supply. After the tested propulsor is attached to the sensing structure, as depicted in the inset, the target plate (see Fig. \ref{['FIG03']}(a)) is aligned beneath the probe of the capacitive sensor (Physik Instrumente D-$519.021$) that measures the deflection of the DCS. Then, the voltage outputted by the capacitive sensor is filtered through a signal conditioner (Physik Instrumente E-$852$ PISeca) and recorded using the AD/DA board of the host--target system. During testing, the propulsor's tail remains submerged from its leading edge downward in an acrylic pool filled with water, as depicted in the inset. (b) Photograph of the tested single-tail propulsor attached to the DCS aligned beneath the probe of the capacitive displacement sensor. (c) Photograph of the tested dual-tail propulsor attached to the DCS aligned beneath the probe of the capacitive displacement sensor.
• Figure 6: Data and photographic sequences corresponding to two force characterization experiments.(a) Measured instantaneous force generated by the tested single-tail propulsor over $5.5$ s of steady-state operation, using a $1$-Hz PWM actuation signal with a DC of $5$ %. In this case, the means of the peak and cycle-averaged forces, $\bar{F}_{\text{p}}$ and $\bar{F}_{\text{a}}$, are $0.48$ mN and $3.6$ µN, respectively.~(b) Measured instantaneous force generated by the tested dual-tail propulsor over $5.5$ s of steady-state operation, using a $1$-Hz PWM actuation signal with a DC of $5$ %. In this case, the means of the peak and cycle-averaged forces, $\bar{F}_{\text{p}}$ and $\bar{F}_{\text{a}}$, are $0.58$ mN and $12.6$ µN, respectively.~(c) Measured instantaneous force generated by the tested single-tail propulsor over $5.5$ s of steady-state operation, using a $2$-Hz PWM actuation signal with a DC of $10$ %. In this case, the means of the peak and cycle-averaged forces, $\bar{F}_{\text{p}}$ and $\bar{F}_{\text{a}}$, are $0.084$ mN and $-0.72$ µN, respectively.~(d) Measured instantaneous force generated by the tested dual-tail propulsor over $5.5$ s of steady-state operation, using a $2$-Hz PWM actuation signal with a DC of $10$ %. In this case, the means of the peak and cycle-averaged forces, $\bar{F}_{\text{p}}$ and $\bar{F}_{\text{a}}$, are $0.39$ mN and $11.4$ µN, respectively.~(e) Sequence of video frames---taken at intervals of $0.1$ s---of the full actuation cycle corresponding to the first second of the experimental data presented in~(a).~(f) Sequence of video frames---taken at intervals of $0.1$ s---of the full actuation cycle corresponding to the first second of the experimental data presented in~(b). Video footage of these experiments can be viewed in the accompanying supplementary movie.