Flapping Soft Fin Deformation Modeling using Planar Laser-Induced Fluorescence Imaging

Summary

The present protocol involves the measurement and characterization of 3D shape deformation in underwater flapping fins built with polydimethylsiloxane (PDMS) materials. Accurate reconstruction of these deformations is essential for understanding the propulsive performance of compliant flapping fins.

Abstract

Propulsive mechanisms inspired by the fins of various fish species have been increasingly researched, given their potential for improved maneuvering and stealth capabilities in unmanned vehicle systems. Soft materials used in the membranes of these fin mechanisms have proven effective at increasing thrust and efficiency compared with more rigid structures, but it is essential to measure and model the deformations in these soft membranes accurately. This study presents a workflow for characterizing the time-dependent shape deformation of flexible underwater flapping fins using planar laser-induced fluorescence (PLIF). Pigmented polydimethylsiloxane fin membranes with varying stiffnesses (0.38 MPa and 0.82 MPa) are fabricated and mounted to an assembly for actuation in two degrees of freedom: pitch and roll. PLIF images are acquired across a range of spanwise planes, processed to obtain fin deformation profiles, and combined to reconstruct time-varying 3D deformed fin shapes. The data are then used to provide high-fidelity validation for fluid-structure interaction simulations and improve the understanding of the performance of these complex propulsion systems.

Introduction

In nature, many fish species have evolved to use a variety of body and fin motions to achieve locomotion. Research to identify the principles of fish locomotion has helped drive the design of bioinspired propulsion systems, as biologists and engineers have worked together to develop capable next-generation propulsion and control mechanisms for underwater vehicles. Various research groups have studied fin configurations, shapes, materials, stroke parameters, and surface curvature control techniques^{1,2,3,4,5,6,7,8,9,10,11,12}. The importance of characterizing tip vortex generation and wake inclination to understand thrust generation in single- and multi-fin systems has been documented in numerous studies, both computational and experimental^{13,14,15,16,17,18}. For fin mechanisms made of compliant materials, shown in various studies to reduce wake inclination and increase thrust¹⁷, it is also essential to capture and accurately model their deformation time-history to pair with the flow structure analysis. These results can then be used to validate computational models, inform fin design and control, and facilitate active research areas in unsteady hydrodynamic loading on flexible materials, which need validation¹⁹. Studies have used direct high-speed image-based shape tracking in shark fins and other complex objects^20,21,22, but the complex 3D fin shape often blocks optical access, making it difficult to measure. Thus, there is a pressing need for a simple and effective method to visualize flexible fin motion.

A material widely used in compliant fin mechanisms is polydimethylsiloxane (PDMS) due to its low cost, ease of use, ability to vary stiffness, and compatibility with underwater applications²³, as described extensively in a review by Majidi et al.²⁴. In addition to these benefits, PDMS is also optically transparent, which is conducive to measurements using an optical diagnostic technique such as planar laser-induced fluorescence (PLIF). Traditionally within experimental fluid mechanics²⁵, PLIF has been used to visualize fluid flows by seeding the fluid with dye or suspended particles or taking advantage of quantum transitions from species already in the flow that fluoresce when exposed to a laser sheet^26,27,28,29. This well-established technique has been used to study fundamental fluid dynamics, combustion, and ocean dynamics^{26,30,31,32,33}.

In the present study, PLIF is used to obtain spatiotemporally resolved measurements of shape deformation in flexible fish-inspired robotic fins. Instead of seeding the fluid with dye, the underwater kinematics of a PDMS fin are visualized at various chordwise cross-sections. Although planar laser imaging can be performed on regular cast PDMS without additional fluorescence, modifying PDMS to enhance fluorescence can improve the signal-to-noise ratio (SNR) of the images by reducing the effects of background elements, such as the fin mounting hardware. PDMS can be made fluorescent by employing two methods, either by fluorescent particle seeding or pigmentation. It has been reported that, for a given part ratio, the former alters the stiffness of the resultant cast PDMS³⁴. Therefore, a nontoxic, commercially available pigment was mixed with transparent PDMS to cast fluorescent fins for the PLIF experiments.

To provide an example of using these fin kinematics measurements for computational model validation, the experimental kinematics are then compared with values from the coupled fluid-structure interaction (FSI) models of the fin. The FSI models used in the computations are based on the first seven eigenmodes computed using the measured material properties for the fins. Successful comparisons validate fin models and provide confidence in using the computational results for fin design and control. Further, the PLIF results demonstrate that this method can be used to validate other numerical models in future studies. Additional information about these FSI models can be found in prior work^35,36 and in fundamental texts of computational fluid dynamics methods^37,38. Future studies can also allow for simultaneous measurements of solid deformations and fluid flows for improved experimental studies of FSI in robotic fins, bioinspired soft robots, and other applications. Furthermore, because PDMS and other compatible elastomers are widely used in various fields, including sensors and medical devices, visualizing deformations in flexible solids using this technique can benefit a larger community of researchers in engineering, physics, biology, and medicine.

Protocol

1. Fin fabrication

1. Build a fin mold based on the desired shape design.
   1. Design and build a custom 3D-printed gloss-finished mold of fin shape (Figure 1). See STL files for fabricating the mold in Supplementary Coding Files 1-4.
   2. Insert structural elements into the mold, such as a 3D-printed rigid plastic leading-edge spar. See the STL file of the spar in Supplementary Coding File 2.
2. Mix PDMS (see Table of Materials) in the desired part ratio.
   1. Select the part ratio of base elastomer to curing agent (i.e., 10:1 or 20:1) to achieve higher or lower elastic modulus, respectively. Weigh the corresponding amounts of base and hardener.
      NOTE: Both 10:1 and 20:1 (elastomer to curing agent) were used in the present study.
   2. Measure the fluorescent pigment (see Table of Materials) such that the total mixture contains 0.1%-1% pigment by weight, depending on the desired brightness of pigmentation. Add the pigment to the PDMS blend.
   3. Pour the measured amounts of elastomer, hardener, and pigment into a planetary centrifugal mixer (mixing at 423 x g for 30 s and de-aerating at 465 x g for 30 s) and mix accordingly.
3. Cast the fin in the mold.
   1. Degas and pour the PDMS mixture into the mold for the fin. Place the mold into the oven at 70 °C for 45 min, and let it cure overnight at 37 °C.
   2. Once the curing is complete, remove the cast fin from the mold (Figure 2).
4. Perform tensile testing following the ASTM standard³⁹.
   1. For each fin cast in Step 1.3., cast one Type IV specimen using the same PDMS and pigment blend in a Type IV shaped mold using the previously described Steps 1.1.-1.3.
      NOTE: See the STL files to cast the Type IV specimen in Supplementary Coding File 5 (mold shown in Figure 1C), and see Figure 3 for examples of the Type IV samples tested.
   2. Clamp the test specimen into the tensile testing machine (see Table of Materials). Measure the initial length, width, and thickness (mm) of the narrow sample section.
   3. Subject the test specimen to tension in 5 mm increments, ensuring that the sample remains stretched in the elastic region only, not overstretched. Decrease the tension in 5 mm increments until the total sample displacement is 0 mm (original position). Record the lengths (mm) and forces (N) of the narrow section at each increment.
   4. To calculate the elastic modulus of the sample, plot the stress-strain curve and determine the best linear fit and R² value.

2. Experimental setup and trials

1. Mount the PLIF hardware (see Table of Materials) to a rectangular glass water tank (2.41 m x 0.76 m x 0.76 m).
   1. Mount and use a pulsed laser system (see Table of Materials) to generate a planar light sheet intersecting the tank at its mid-plane at a specified frequency (30 Hz), as shown in Figure 4.
   2. Mount and use a 4 MP charge-coupled device (CCD) camera equipped with a lens (35 mm) and a longpass fluorescence filter (560 nm) (see Table of Materials).
   3. Calibrate the micrometer-to-pixel conversion by taking a single image from the CCD camera with a ruler placed in the laser sheet plane (Figure 5). Select two positions on the camera and divide the distance in micrometers by separating pixels. Ensure this micrometer-to-pixel ratio is small enough (sub-millimeter) for the application.
2. Synchronize the laser pulses and camera images with the flapping fin using trigger outputs from the fin software and signals from a delay generator and associated software (see Table of Materials) to coordinate the camera, laser heads, and fin motion. See Supplementary Figure 1 for an example of the delay generator software interface settings.
   1. Set the laser system.
      NOTE: Ensure all laser safety measures are in accordance with institutional guidelines.
      1. Turn the laser system on by rotating the power key to the right to run the chiller that cools the laser heads. The fault light blinks until the system is ready to power the lasers. Do not press the power button that turns the lasers on until all laser modes are set correctly.
      2. Set the Trigger Source to EXT LAMP/EXT Q-SW (external lamp/external Q-switch).
      3. For both laser heads, set the laser energy to the desired level (i.e., approximately 60%-80% of the full power) and ensure that the Q-switch is turned on by pressing each Q-switch button.
      4. Turn the lasers on by pressing the Power button.
         NOTE: As the Trigger Source is set to EXT LAMP/EXT Q-SW, the laser heads are ready to fire, but only fire after the system receives an external trigger from the software.
   2. Set the camera.
      1. Plug in the power cables to the camera and ensure proper connections to the computer and software.
      2. Open the camera settings software and select the proper port.
         1. Under Trigger > Settings, set "Trigger in:" to External and "Mode:" to Fast.
         2. Under Exposure, set "Exposure Control" to Off.
      3. Open the camera capture software and select the proper camera card.
         1. Click on the Grab Sequence button.
         2. Click on the Capture Settings button, select TIFF images, select Series of frames ..., and select the desired file path, 6 Digit Number, Continuous, and Accept.
         3. Click on Start Capture.
            NOTE: As the camera settings are set to an external trigger, the camera is ready to collect images but only captures these images after the system receives an external trigger from the software.
   3. Set the delay generator.
      1. Turn on the delay generator, and connect the External Gate Channel to the fin trigger, Channels A-D to the laser (A: laser head 1, B: Q-switch to laser 1, C: laser head 2, and D: Q-switch to laser 2), and Channel E to the camera.
      2. Open the delay generator software.
      3. Select the "Pulse Mode" to Burst and "System Resolution" to 4 ns.
      4. Set the "Period (s)" to 0.033333352.
      5. Set the "External Trigger/Gate Mode" to Triggered, "Threshold (V)" to 0.20, and "Trigger Edge" as Rising.
      6. On Channels > Ch A, click on the Enabled checkbox. Set the "Delay (s)" to 0.000000004, "Width (s)" to 0.005000000, "Amplitude (V)" to 5.00, "Channel Mode" to Duty Cycle, "Wait Count" to 0, "Sync Source" to T0, "Polarity" to Normal, "Multiplexer" to A, "Duty Cycle On" to 1, "Duty Cycle Off" to 1, and "Gate Mode" to Disabled.
      7. On Channels > Ch B, click on the Enabled checkbox. Set the "Delay (s)" to 0.000138000, "Width (s)" to 0.005000000, "Amplitude (V)" to 5.00, "Channel Mode" to Duty Cycle, "Wait Count" to 0, "Sync Source" to Ch A, "Polarity" to Normal, "Multiplexer" to B, "Duty Cycle On" to 1, "Duty Cycle Off" to 1, and "Gate Mode" to Disabled.
      8. On Channels > Ch C, click on the Enabled checkbox. Set the "Delay (s)" to 0.033333304, "Width (s)" to 0.005000000, "Amplitude (V)" to 5.00, "Channel Mode" to Duty Cycle, "Wait Count" to 0, "Sync Source" to Ch A, "Polarity" to Normal, "Multiplexer" to C, "Duty Cycle On" to 1, "Duty Cycle Off" to 1, and "Gate Mode" to Disabled.
      9. On Channels > Ch D, click on the Enabled checkbox. Set the "Delay (s)" to 0.000138000, "Width (s)" to 0.005000000, "Amplitude (V)" to 5.00, "Channel Mode" to Duty Cycle, "Wait Count" to 0, "Sync Source" to Ch C, "Polarity" to Normal, "Multiplexer" to D, "Duty Cycle On" to 1, "Duty Cycle Off" to 1, and "Gate Mode" to Disabled.
      10. On Channels > Ch E, click on the Enabled checkbox. Set the "Delay (s)" to 0.000000004, "Width (s)" to 0.005000000, "Amplitude (V)" to 5.00, "Channel Mode" to Normal, "Wait Count" to 0, "Sync Source" to T0, "Polarity" to Normal, "Multiplexer" to E, and "Gate Mode" to Disabled.
3. Align the fin so the laser sheet passes through one chordwise section of the fin at a selected spanwise position and secure the fin platform with the mounting hardware.
4. Connect the power to the fin control hardware and fin motors (see Table of Materials) to begin fin flapping with the selected kinematics, and turn off all ambient lights.
5. Press Run in the delay generator software to begin the synchronized experiments and acquire images of the intersection of the laser sheet with the fin throughout the stroke cycle. This needs to be conducted over 200+ stroke cycles.
6. Press Stop in the delay generator software and disconnect the fin from the power source.
7. Move the fin platform so the laser sheet crosses at a new spanwise position and perform experiments to acquire the images again. Repeat Steps 2.3.-2.6. for the number of desired measurements (eight different spanwise positions, as shown by the black dashed lines in Figure 2A).
8. Replace the fin with additional desired fin membranes (two fin stiffnesses, PDMS 10:1 and PDMS 20:1) and repeat the experiments.

3. Image analysis

1. For each experimental trial conducted in Step 2.4., locate the file where the images are stored and create a subfolder for each fin position or phase throughout the stroke cycle. Sort the image files into their corresponding subfolders.
2. For each fin phase subfolder, read the 200+ images as pixel-value arrays (imread.m). Sum the pixel-value arrays for all the images and divide by the number of images to generate a mean image. Write the image to a new file (imwrite.m). Repeat this step for each fin position throughout the stroke cycle (30 positions).
3. Perform a histogram enhancement on each mean image (imadjust.m) to extend the dynamic intensity range of the images to the full available range to improve the contrast between the fin and background.
4. Set the intensity thresholds and binarize each image to obtain a black-and-white image (imbinarize.m). The resulting white shapes should correspond to pieces of the fin cross-section.
5. Extract all white objects (fin pieces) from the binary image (bwareafilt.m), and display the image (imshow.m). Create a trace of the binary image boundary for each image to obtain a 2D shape by selecting all the fin (white) pixels that touch the background (black) pixels (bwboundaries.m).
   NOTE: Due to imposed fin kinematics, the view of the PLIF measured cross-section in some frames may be occluded by another part of the fin. In such cases, either there is no coherent fin shape apparent from the images, or only the leading edge (LE) remains visible (Figure 6).
6. Perform Steps 3.1.-3.5. for each fin cross-section.

4. Reconstruction of 3D deflection

1. Assuming that the LE position (at least closer to the stroke axis) in the flexible cases is the same as that of the LE in a rigid fin of the same shape, line the plane cuts along their LE for the same time-step, and compare with the results from the corresponding rigid fin shape.
2. Use a least-squares fit to approximate the resulting centerline shape of the fin cross-section for all plane cuts and reconstruct the 3D fin shape using a simplified convex-hull from these fitted profiles.
3. Compare the resulting fin shapes with 3D FSI models (generated from their centerlines) to showcase how this process can be used as high-fidelity validation.
   1. Generate a surface triangulation of the partially rigid nylon and partially flexible PDMS fin.
   2. Use a commercial structural dynamics software (see Table of Materials) to obtain the eigenmodes of the hybrid material.
      1. Perform scaling studies to match the steady-state displacement obtained using uniform pressure differential on the fin surfaces.
      2. Scale the modes to match the displacement obtained from the software.
   3. With the proper scale factor, use the first few dominant modes (usually 7 or 8) employed in the coupled FSI solver to simulate the unsteady flow over the flexible fin.
      1. Treat the body as an embedded entity in a background mesh.
         NOTE: The coupled solver was validated for the Turek-Hron problem of flow over a circular cylinder with a flexible sting at the back³⁵ and extended for flapping fin simulations³⁶.
      2. Prescribe the kinematics of the fin motion from the experiments.
      3. Monitor the time history of the force production and the shape of the fin along several plane cuts throughout the flapping cycle, and compare with experiments.

Results

A trapezoidal fish-inspired artificial pectoral fin was cast in two different materials (PDMS 10:1 and 20:1, both mixed with fluorescent dye) out of a mold, each with a rigid leading-edge spar inserted into the leading quarter chord (Figure 2 and Figure 3). Tensile testing of the two fin materials (Figure 3) yielded elastic moduli of 0.38 MPa and 0.82 MPa for the PDMS 20:1 and PDMS 10:1 fins, respectively, with an R² of 0...

Discussion

Planar laser-induced fluorescence is typically used to visualize aqueous flows by seeding the fluid with dye, which fluoresces when exposed to a laser sheet^25,26. However, using PLIF to visualize deformations in compliant materials has not been previously reported, and this study describes an approach for obtaining time history measurements of high-resolution shape deformation in flexible solid fins using PLIF. Comparing these fin measurements with FSI simulation...

Materials

Name	Company	Catalog Number	Comments
ADMET controller	ADMET	MTESTQuattro
Axon II	Society of Robots		Microcontroller for the fin hardware
Berkeley Nucleonics Delay Generator	Berkeley Nucleonics Corp	Model 525	BNC delay generator and software
BobCat Cam Config	Imperx		Camera settings software
CCD camera	Imperx	B2340	4 MegaPixel
COMSOL	COMSOL Inc		Commercial structural dynamics software for fluid-structure interaction modeling
D646WP Servo	Hitec	36646S	32-Bit, Digital, High Torque, Waterproof Servo for the fin pitch rotation
D840WP Servo	Hitec	36840S	32-Bit, Multi Purpose, Waterproof, Steel Gear Servo for the fin stroke rotation
Electric Pink fluorescent pigment	Silc Pig	PMS812C
EverGreen (532 nm dual pulsed Nd:YAG laser system)	Quantel	EVG00070	Laser head and power supply, 70 mJ
Force transducer	ADMET	SM-10-961	10 lbf load cell
FrameLink Express	Imperx		Camera capture software
Longpass fluorescence filter	Edmund Optics		560 nm
MATLAB	MathWorks		Software for image analysis
Planetary centrifugal mixer	THINKY MIXER	AR-100
Silicone rubber compounds	Momentive	RTV615	Clear PDMS
Stratasys J750	Stratasys		3D printer, polyjet
Universal testing machine	ADMET	eXpert 2611	Table top model
VeroBlack	Stratasys		3D printer material to build the molds
VeroGray	Stratasys		3D printer material to build the molds