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Tactile Perception through Fluid-Solid Interaction

Arman Goshtasbi, Minke Berghuis, Aida Parvaresh, Saravana Prashanth Murali Babu, Robert W. Style, Ahmad Rafsanjani

arXiv:2610.03773v1cond-mat.softcs.RO

TL;DR

Integrating tactile sensing into soft materials is challenging, and embedded electronics can be vulnerable to strain-related problems. This paper uses viscous-fluid channels and two external pressure sensors to estimate touch position and force, reporting localization error figures and force estimation across 1D and 2D configurations.

  • Problem

    Integrating tactile sensors into soft or deformable materials is challenging because of fabrication complexity, integration constraints, and environmental sensitivity.

    Source: INTRODUCTION · p. 2

  • Method

    The approach infers touch position and force from pressure signatures in viscous-fluid channels using machine learning and two external pressure sensors.

    Source: INTRODUCTION · p. 3

  • Results

    Localization is reported at Mean Average Error (MAE): 4 mm, <7% error, with robust force estimation across both 1D and 2D configurations.

    Source: DISCUSSION · p. 12

  • Takeaways & Limitations

    The study establishes a baseline for viscous fluids enhancing tactile sensing with minimal hardware and computation, while noting material compatibility, non-uniform pressure, and slow response challenges.

    Source: DISCUSSION · p. 14

    PDMS permeability restricted usable fluids to a specific silicone oil formulation and prevented systematic exploration of intermediate viscosities.

    Source: DISCUSSION · p. 13

Abstract

from arXiv · show

Soft tactile sensors elevate robotic touch through enhanced flexibility and adaptability, yet most existing designs depend on embedded electronics that are susceptible to interference and environmental limitations. In this work, we leverage fluid-solid interactions to develop a class of soft tactile sensors that operate entirely without electronics at the sensing site. The sensor comprises a fluid-filled elastomeric channel connected to only two external pressure sensors. Touching different regions of the elastomeric surface displaces the viscous fluid, producing distinct pressure patterns that encode both touch position and force. These signals are decoded through a machine learning framework that integrates feature extraction, soft clustering, and adaptive neuro-fuzzy inference to achieve accurate localization and force estimation. We validate this concept through single-point touch localization and force estimation in a linear (1D) sensor and extend the same sensing principle to 2D tactile mapping by routing the channel across the surface using space-filling curves, while maintaining the same minimal hardware setup. This simple approach remains effective in environments where conventional electronic sensors often fail, such as underwater or in the presence of magnetic interference.

INTRODUCTION

Tactile perception matters when vision is unreliable during physical interaction, but integrating high-resolution tactile sensing into soft materials is challenging. This work proposes a fluid–solid sensing approach that uses two external pressure sensors for touch localization and force estimation.

INTRODUCTION

  • Tactile perception is important for direct interaction when vision is unreliable under occlusion, poor illumination, or reflective and cluttered scenes.Robots entering homes, healthcare settings, and collaborative workspaces need to perceive and physically interact with their surroundings.Source: INTRODUCTION · p. 2
  • Embedded electronics in soft or deformable substrates face fabrication and integration challenges, and repeated strain can cause debonding, conductor fatigue, or signal drift.These limitations constrain established capacitive, resistive, optical, and hybrid approaches.Source: INTRODUCTION · p. 2
  • Soft tactile sensing explores compliant materials and multiple transduction mechanisms, while some designs relocate or minimize electronics at the contact interface.The passage lists magnetic, barometric, resistive, capacitive, optical, and vision-based approaches.Source: INTRODUCTION · p. 2
  • Prior electronics-free approaches include detecting deformation through acoustic waves and inferring load from fluid displacement observed by an external camera.Acoustic sensing can use a wave through a channel or a Lamb wave through a thin elastomeric layer.Source: INTRODUCTION · p. 2
  • Pneumatic tactile designs have used six sensors for nine contact areas, 32 sensors for a 16×16 array, or nine pressure readings for qualitative touch localization.These examples use sensor arrays or multiple channel readings to expand spatial coverage.Source: INTRODUCTION · p. 2
  • The proposed viscous-fluid channel uses two external pressure sensors and machine learning to infer touch position and force in 1D, then extends to 2D mapping along space-filling curves.The study focuses on sensing position and strength; damage tolerance and long-term durability are not quantitatively evaluated.Source: INTRODUCTION · p. 3

RESULTS

The results characterize the two-pressure-sensor fluidic tactile design across 1D and Hilbert-curve 2D configurations and evaluate learning-based touch estimation under varied conditions, including environmental challenges. They report position and force estimation across scaled sensors, curved configurations, and human–robot interaction tasks, and discuss factors associated with sensing performance and its limitations.

Sensor design and working mechanism

  • The sensor uses a fluid-filled elastomeric channel and two end pressure sensors; a Hilbert-curve path extends the same principle to 2D, with machine learning estimating touch position and force.The pipeline extracts pressure maxima, and for 2D also rise time, then combines soft-cluster memberships with an adaptive neuro-fuzzy inference system to map sensor outputs to touch parameters.Source: Sensor design and working mechanism · pp. 3–4

Tactile response of the 1D sensor

  • In 1D tests, pressure difference increased with indentation in the working range, while larger channels and nearly incompressible fluids produced larger signals.At shallow indentations, backflow made the pressure difference nearly zero; after channel collapse, pressure saturated. Off-channel sensitivity diminished around y ≈ ±5 mm. Oil produced informative, viscosity-dependent rise times but responded more slowly; the study selected 100 cSt oil. Increasing channel length reduced pressure, with a roughly 52% drop in the 50 mm channel and about 70% in the 150 mm channel between x/l = 0.1 and 0.5.Source: Tactile response of the 1D sensor · pp. 4–5

Understanding the sensing mechanism using theory and simulation

  • The theoretical model fit oil-channel measurements, while simulations showed that backflow weakens pressure differences, especially for lighter touches and less viscous fluids.Stronger indentation restricts backflow, and when backflow is allowed, higher oil viscosity produces a larger pressure difference. The model’s nearly linear pressure–indentation behavior holds under channel closure or for high-viscosity fluids.Source: Understanding the sensing mechanism… · pp. 6–7

Tactile response of the 2D Sensor

  • In the 2D sensor, opposing end-pressure trends encode touch position, while oil-filled channels improve slight-touch sensitivity and spatial resolution by limiting flowback.Pressure saturates after channel collapse; the delay between pressure peaks was about 0.85 s, making rise time an additional estimation feature. Air and water were less sensitive to slight and moderate touches because of rapid flowback.Source: Tactile response of the 2D Sensor · p. 8

Learning touch position and force

  • For the 2D sensor, FCM estimated on-channel position at R& = 99.18% and RMSE = 0.026, and force at R& = 84.74% and RMSE = 0.66 N.With on- and off-channel touches included, force estimation reached R& = 82.04% and RMSE = 0.71 N; off-channel touches can weaken pressure signals and lead to force underestimation. Position predictions were slightly less accurate at shallow indentations, and force estimates varied more at higher forces as pressure saturated.Source: Learning touch position and force · pp. 9–10

Adaptability to Environmental Constraints

  • Underwater, touch responses remained effectively unchanged after accounting for hydrostatic baseline shifts, with dry-to-submerged differences below 5 Pa.The sensor also produced nearly identical pressure patterns with plastic and magnetic indenters, including in the 2D configuration.Source: Adaptability to Environmental… · p. 10
  • Halved sensors retained comparable force estimation and about 5% and 7% mean spatial error in x and y, while a quarter-sized sensor was slightly less accurate; flat-trained sensors also generalized to curved surfaces.The curved-substrate tests detected touches on convex and concave regions.Source: Adaptability to Environmental… · pp. 10–11
  • With training limited to an 8 mm indenter, mean positional error stayed below 5 mm across three tip shapes and below 2.5 mm for HOS diameters from 4 to 10 mm, including real-time operation.Smaller tips weakened pressure signals through more localized deformation and flowback, especially at shallow depths; flat tips sealed channels less effectively than curved tips.Source: Adaptability to Environmental… · p. 11

Human-Robot Interaction

  • In human–robot interaction tests, 1D position estimates controlled the robot and force-position mapping stayed accurate in nearly all tested cells, while 2D quadrant touches moved the arm to corresponding positions.The highest-force 1D touch was slightly underestimated; for a robot-mounted 2D sensor, baseline correction compensated for motion-induced pressure drift and enabled reliable estimates despite flat-only training.Source: Human-Robot Interaction · pp. 11–12

DISCUSSION

  • The 1D sensor achieved 1.8 mm position accuracy and force estimation over 1–10 N; the 2D Hilbert sensor achieved 3.3–4.2 mm positional errors and 0.7 N force accuracy using two sensors.The 2D sensor mapped a 60×60 mm² area, using substantially fewer sensing elements than the cited technologies; the work also reports operation underwater and under magnetic interference.Source: DISCUSSION · p. 12
  • Higher fluid viscosity and incompressibility strengthened pressure response, and fluid–solid interaction simulations showed that early pressure equilibrium arises from flowback suppressed at higher viscosities.Baseline adjustment compensated for diffusion-, motion-, and ambient-pressure-related drift, supporting consistent position estimation across experiments months apart without model retraining.Source: DISCUSSION · pp. 12–13
  • The discussion identifies fluid–elastomer diffusion, pressure loss along longer channels, non-uniform 2D pressure–force behavior, and response times on the order of seconds as limitations.It points to water-based fluids, viscosity tuning, denser or improved channel fabrication, richer temporal features, and recalibration methods as possible improvements; the current minimal feature approach enables real-time processing.Source: DISCUSSION · pp. 13–14

METHODS

  • The evaluation used a fuzzy C-means and ANFIS framework to predict position and force from 1D pressure inputs and, for 2D, pressure rise times, with shuffled data split 80/20 for training and validation.The experimental setup used two differential pressure sensors and a robotic arm with an indenter and load cell.Source: METHODS · pp. 15–16

DATA AVAILABILITY / CODE AVAILABILITY

FUNDING

The research was supported by an Independent Research Fund Denmark grant awarded to A.R.

FUNDING

  • The Independent Research Fund Denmark supported the research through a Sapere Aude: DFF-Research Leader Grant awarded to A.R.Grant number: 1051-00075B.Source: FUNDING · p. 21

TABLE

Table 1 compares the proposed method with state-of-the-art soft tactile sensors that are electronic-free at the sensing site.

TABLE

  • Table 1 compares the proposed method with state-of-the-art soft tactile sensors that are electronic-free at the sensing site.Source: Table 1 · p. 22

FIGURES

The figures trace the fluidic sensor from pressure-response characterization through 1D and 2D touch estimation, environmental tests, and human-robot interaction. The 2D sensor extends the channel with a Hilbert curve, while fabrication figures document how the 1D and 2D structures are formed and connected.

FIGURES

  • Figure 1 shows a fluid-filled elastomer channel and a Hilbert-curve 2D extension; pressure features are processed through soft clustering and ANFIS to estimate touch position and force.The figure contrasts signals across positions, media, and touch depths, and shows maximum pressure features for both sensors plus rise time for the 2D sensor.Source: Figure 1 · p. 23
  • Figure 2 compares 1D pressure and force responses across fluid media, channel diameters, touch locations, and channel lengths.Its panels include on-channel and off-channel touches and pressure time response.Source: Figure 2 · p. 24
  • Figure 4 samples 16 positions on the 2D channel across indentation depths from 1 to 4 mm and shows saturation after 3.6 mm.The figure also compares quadrant-specific pressure responses and left-right rise times.Source: Figure 4 · p. 26
  • Figures 5 and 6 present feature spaces and position and force estimates for the 1D and 2D sensors, with the 2D figure also mapping positional error and full-dataset force estimates.The 1D results use oil at 100 cSt and a 1.5 mm channel diameter.Source: Figure 5 · p. 27, Figure 6 · p. 28
  • Figures 7 and 8 show environmental adaptability, sensor-size comparisons, varied-indent​er and curved-surface tests, and robot-control demonstrations using 1D and 2D sensors.The 1D demonstrations control the robot’s X-axis; the 2D demonstrations map touches to robot XZ positions.Source: Figure 7 · p. 29, Figure 8 · p. 30

S1. Sensor fabrication

  • The 1D sensor is molded, cured, fluid-filled, and connected to tubing; the 2D sensor uses a removable 3D-printed Hilbert curve to form its channel before sealing and filling.A pre-cured silicone layer is placed over the tacky brushed layer to bond the 2D structure without bubbles.Source: S1. Sensor fabrication · p. 32, Figure S1 · p. 32

S2. Experimental setup

The supplementary experiments characterize indentation with synchronized force and pressure measurements, then examine how fluid, channel geometry, and viscosity affect the sensor’s usable response. They also assess silicone-oil diffusion and the response trade-offs of PVP solutions.

S2. Experimental setup

  • The UR5 applies position-controlled indentation while a load cell measures force and two differential pressure sensors record pressure at 100 Hz through a synchronized DAQ.The load cell supplies force feedback because the UR5 does not provide it.Source: S2. Experimental setup · p. 33, Figure S2 · p. 33

S3. Data acquisition for the 1D sensor

  • At low indentation depths, pressure differences can be too small for localization; high-viscosity oil improves pressure distinction but can settle slowly, while smaller channels respond earlier and saturate sooner.Incompressible water and oil produce larger overall pressure responses than compressible air.Source: S3. Data acquisition for the 1D sensor · pp. 34–35, Figure S4 · p. 35

S4. Diffusion of silicone oil into Ecoflex 00-30

S5. Effect of working fluid viscosity on sensor response

  • PVP solutions at approximately 20 and 50 cSt provide an intermediate-viscosity alternative used to reduce diffusion effects while tuning the fluid response.Silicone oil diffusion into Ecoflex 00-30 limited experiments with silicone oil to relatively high viscosities.Source: S5. Effect of working fluid viscosity… · p. 37
  • 0.4–0.7 s (PVP) versus 0.6–0.9 s (silicone oil) rise times accompany intermediate PVP pressure contrast and faster recovery; water-based solution viscosity can vary with temperature.Higher viscosity increases pressure contrast and extends the effective working range, while silicone oil’s larger steady-state response relaxes slowly.Source: S5. Effect of working fluid viscosity… · p. 37

S6. Effect of touch frequency on the pressure response of 1D sensors

The 1D sensor’s pressure response varies with touch timing: higher repetition frequencies slightly increase peak pressure, while short contacts may end before the signal reaches its steady-state maximum.

Varying touch intervals along the length

Varying touch intervals at fixed positions

Fixed touch intervals with varying holding times

  • Shorter contacts reduce peak and accumulated pressure because fluid redistribution takes time, even when applied force reaches comparable maxima.Baseline correction does not remove this transient effect; contacts shorter than the characteristic rise time may be underestimated by features relying mainly on maximum pressure and rise time.Source: S6. Effect of touch frequency on the… · pp. 41–42, Figure S9 · p. 41

S7. Theoretical model

The theoretical model treats a touch as a fluid-volume displacement that pressurizes the channel on both sides of the indenter. It relates the resulting pressures to touch position and normalized bulk modulus.

S7. Theoretical model

  • A touch at distance x divides the chamber into left and right domains, with half of the displaced volume assigned to each side and no flowback assumed.The displacement deforms both the tube and the embedded elastomeric channel.Source: S7. Theoretical model · p. 43
  • The model derives left- and right-side pressures from added volumes, effective bulk moduli, and the channel volumes on each side of the touch.It defines normalized bulk modulus κ as the ratio of the elastomer and tube-system moduli.Source: S7. Theoretical model · p. 43
  • At the channel midpoint, the two end pressures are equal; the resulting pressure relations are expressed using normalized bulk modulus κ and midpoint pressure P*.The model uses these relations to connect displaced volume and measured pressure.Source: S7. Theoretical model · pp. 43–44

S8. Numerical simulations

The numerical model represents the sensor as coupled fluid and solid domains under indentation. Its boundary condition captures elastomeric restoring effects, while simulations show that fluid viscosity changes the transient response.

S8. Numerical simulations

  • The axisymmetric boundary condition couples in-plane displacement to hoop stress, with a virtual radius controlling the coupling strength.As the virtual radius increases, the stabilizing effect diminishes; an infinite radius recovers plane strain.Source: S8. Numerical simulations · p. 45, Figure S11 · p. 45
  • Higher fluid viscosity slows the simulated response and leaves a larger residual channel gap under the same load, although indenter displacement is nearly unchanged.The gap is the shortest distance between the channel walls, and closure is defined using a numerical offset.Source: S8. Numerical simulations · p. 45, Figure S12 · p. 46
  • Gravity is neglected because hydrostatic and self-weight stresses are small relative to the system’s pressure-driven and elastic stresses.The cited hydrostatic pressure difference for a 2 mm water-filled channel is approximately 20 Pa, compared with typical driving pressures in the kilopascal range.Source: S8. Numerical simulations · pp. 45–46
  • Transient simulations use an implicit second-order BDF scheme and a segregated solver with Anderson acceleration.A damping factor of 0.7 is applied to the fluid-solid interaction update.Source: FSI Solver details · p. 46

S9. Data acquisition for the 2D sensor

The 2D sensor was tested at 16 channel locations and 16 indentation depths using air, water, and oil. Oil produced the largest working range and a distinct pressure rise-time feature.

S9. Data acquisition for the 2D sensor

  • Oil-filled channels show a larger working range than air- and water-filled channels in the 2D tests.The comparison used 16 locations and 16 indentation depths.Source: S9. Data acquisition for the 2D sensor · p. 47
  • Faster flowback in air and water makes left and right pressures more similar, while intricate Hilbert-curve channels may damp flow more in water than in air.The passage presents the water-versus-air explanation as likely.Source: S9. Data acquisition for the 2D sensor · p. 47
  • Oil shows a clear difference in left-right pressure rise time, whereas air and water do not show a significant difference.The distinct oil rise-time behavior could provide an additional feature for learning and estimation.Source: S9. Data acquisition for the 2D sensor · p. 47

S10. Repeatability test for 2D sensor

Baseline correction reveals a stable, reproducible 2D sensor response over two weeks despite raw baseline drift. The section also describes the FCM–ANFIS model used to estimate touch position and force.

S10. Repeatability test for 2D sensor

S11. FCM–ANFIS modeling approach

  • FCM assigns samples membership across overlapping clusters, with the fuzzification coefficient controlling cluster overlap.Cluster count was selected by comparing training and testing RMSE and mean error.Source: S11. FCM–ANFIS modeling approach · p. 49
  • The five-layer ANFIS maps pressure inputs to predicted position or force, combining adaptive fuzzification with fixed rule, normalization, and output layers.Its adaptive defuzzification layer uses first-order Sugeno functions whose parameters are estimated by least squares.Source: S11. FCM–ANFIS modeling approach · p. 50

S12. Learning results

FCM generally outperforms conventional regression methods for position and force estimation, although MLP can match or slightly exceed it for position in some cases. Its position estimates remain accurate across tested media and channel diameters, while force estimates are better for air- and water-filled channels than oil-filled ones.

S12. Learning results

  • FCM consistently outperforms MLP, SVR, LR, and PR for position and force estimation overall, though MLP sometimes matches or slightly exceeds it for position.MLP performs significantly worse for force estimation; SVR is sensitive to viscosity and channel diameter, and LR is weaker at capturing nonlinear behavior.Source: Figure S16 · p. 51, S12. Learning results · p. 51, Figure S17 · p. 52
  • R² > 99% and RMSE generally below 2 mm for FCM position estimation across all tested media and channel diameters.The oil-filled 1 mm channel shows somewhat higher error, particularly at low indentation depths.Source: S12. Learning results · p. 52, Figure S18 · p. 53
  • Air- and water-filled channels yield better FCM force estimates than oil-filled channels, whose working range starts at lower indentation depths and forces.The text attributes this difference to more linear pressure–force relationships in air and water and a more nonlinear response in oil.Source: S12. Learning results · pp. 52–53, Figure S19 · p. 54
  • Increasing cluster count initially improves estimation accuracy, but gains saturate as computation time and potential overfitting rise.The analysis covers both 1D and 2D models.Source: Figure S20 · p. 55, S12. Learning results · p. 55

S13. Additional results for the pressure response of the 2D sensor

The 2D sensor’s pressure magnitude and rise-time features vary along the Hilbert curve, while prediction errors depend on indentation depth and location. The reported spatial patterns include weaker signals in two quadrants and higher errors where pressure histories are difficult to distinguish.

S13. Additional results for the pressure response of the 2D sensor

  • Pressure maps are nearly symmetric across the two transducers, with a gradient along the Hilbert curve and weaker signals in the second and third quarters.The pressure decay is more pronounced at lower pressures.Source: Figure S21 · p. 56, S13. Additional results for the… · p. 56
  • Rise time has a clear spatial gradient along the Hilbert curve, but varies less across indentation depths than pressure magnitude.The authors suggest combining pressure peaks with rise-time information could improve position and indentation-intensity prediction.Source: S13. Additional results for the… · p. 56, Figure S22 · p. 57
  • Position and force prediction errors decrease at higher indentation depths; X-direction errors are consistently smaller than Y-direction errors.The authors associate the depth trend with stronger pressure signals and more constrained flowback, and suggest outlet locations may explain the X–Y difference.Source: S13. Additional results for the… · pp. 56–57, Figure S23 · p. 58, Figure S24 · p. 58
  • Some regions with longer Hilbert-curve distances between adjacent points have higher prediction errors and pressure histories that are indistinguishable at certain points.The authors note that the complex, nonlinear system makes a clear relationship between pressure-history characteristics and prediction error difficult to establish.Source: S13. Additional results for the… · p. 57, Figure S25 · p. 59

S14. Demonstration

Baseline correction compensates for water-depth offsets, producing nearly identical sensor responses across dry and submerged conditions. The 2D sensor also produces identical pressure signals with plastic and magnetic indenters, and the experiments use HOS indenters except when testing indenter shapes.

S14. Demonstration

  • Baseline correction removes the water-depth-dependent offset from the 1D sensor and yields nearly identical response profiles underwater.Increasing water-column height raises the raw baseline through hydrostatic pressure.Source: Figure S26 · p. 60, S14. Demonstration · p. 60
  • For the 2D sensor, corrected dry and submerged pressure responses remain nearly identical through 100 mm water height, with differences below 5 Pa.Without baseline removal, the offset increases with water level.Source: S14. Demonstration · p. 60, Figure S27 · p. 61
  • The 2D sensor produces identical pressure signals with plastic and magnetic indenters at 16 locations.The tests used an indentation depth of 2 mm.Source: Figure S28 · p. 61, S14. Demonstration · p. 61

Indenter Shape

  • The indenter-shape experiments use different shapes; all other experiments use HOS indenters.Figure S29 depicts the indenter shapes used for the experiments reported in Fig. 6D.Source: Indenter Shape · p. 62, Figure S29 · p. 62
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