Synthetic materials trained tactile sensing enables quantitative perception of Young’s modulus and Poisson’s ratio

Abstract

Accurately perceiving object softness remains challenging for tactile sensors, as current approaches estimate Young’s modulus from object deformation in response to an applied pressure while overlooking the pivotal role of Poisson’s ratio in shaping that deformation. Here we present a tactile sensor that quantitatively and accurately detects both Young’s modulus and Poisson’s ratio, achieved by constructing a broad materials library spanning diverse Young’s modulus–Poisson’s ratio and capturing materials’ surface bulging height–pressure trajectories. Using these trajectories as machine-learning features, our sensor achieved 84.7% anti-error rate assessment (ARA) in predicting Young’s modulus, markedly higher than 73.3% using pressure-bulging height pairs, and, for the first time, enabled quantitative inference of Poisson’s ratio with 87.7% ARA. Kendall’s Tau analysis verified a Young’s modulus ranking ARA of 91.3% for our sensor, outperforming conventional approaches (66.8%). The mechanistic influence of Poisson’s ratio on bulging morphology was revealed via a materials-library-enabled univariate experimental strategy combined with finite element simulations. When deployed on a robotic manipulator, the sensor generalizes to unseen samples, enabling accurate perception of Young’s modulus across diverse materials and heralding a new paradigm for next-generation tactile sensing.

Accurately perceiving object softness remains challenging for tactile sensors. Here, the authors present a tactile sensor that quantitatively and accurately detects both Young’s modulus and Poisson’s ratio.

Introduction

Touch is indispensable for intelligent interaction with the physical world. Human skin, through its dense network of mechanoreceptors^1–4, not only detects contact force but also deciphers objects’ material properties critical for manipulation and adaptation^5–7. Inspired by this capability, decades of research have pursued artificial tactile systems for robotics, leading to substantial progress beyond force detection, including qualitative assessments of objects’ shape^5,8–13, softness^8,12,14–25, roughness^22,26,27, texture^{21–23,28–30}, and material types^{13,22,27,31–34}. Despite these advances, current tactile sensors remain largely qualitative. Quantitative mechanical perception^14–17,20, essential for enabling robots to perform dexterous manipulation tasks comparable to those of humans, remains in its infancy.

Among mechanical parameters, Young’s modulus (E) has received particular attention in tactile sensing because it directly reflects a material’s stiffness, a key determinant of how objects deform under contact and how robotic systems should adapt their grip^35–37. Most tactile sensors designed for evaluating E operate on the principle of inducing and quantifying surface deformation of objects under a contact pressure. Building on this principle, two major architectural strategies have been developed: one employs stiff protrusions to indent the objects’ surface and infer E from the correlation between indentation depth and applied force^{14,16,17,21,23}, while the other employs concave geometries that drive the objects’ surface to bulge under compression, with bulging height under a given load serving as a proxy for stiffness^18–20,25. In addition to these sensor designs, emerging vibration-based sensors that actively excite and detect surface oscillations offer a promising route for quantitative assessment of E¹⁵.

Despite their success in the detection of E for specific objects, once facing broader material types, these approaches will share a critical limitation: they neglect the role of Poisson’s ratio (ν) which, together with E, fundamentally governs how a material deforms under stress. This omission is non-trivial for accurate E detection for broader types of materials. Variations in ν reshape surface deformation^38,39, thereby modulating the tactile sensor’s electrical response and introducing systematic biases into stiffness estimation. The root challenge lies in the coupled dependence of surface deformation on both E and ν which cannot be disentangled by conventional strategies relying on a single bulging or indentation metric at a fixed load^40,41. Consequently, existing tactile systems yield an incomplete and potentially inaccurate representation of material mechanics for materials with diverse ν. Addressing this limitation demands a paradigm shift in tactile sensing—through structural innovations and advanced signal processing—to not only improve the ARA of E detection across diverse materials, but ideally also enable simultaneous and quantitative perception of ν.

Here, we present a concave-surfaced tactile sensor that achieves this paradigm shift. By constructing a broad materials library spanning diverse E–ν combinations via a sacrificial templating method, and leveraging the sensor’s high stability to record complete bulging height–stress trajectories, we transform the sensing paradigm from static single-point metrics to trajectory-informed analysis. Coupled with machine learning, this strategy enables quantitative detection of both E and ν, with ARAs of 84.7% and 87.7%, respectively, substantially surpassing 73.3% (E) achieved by single-point metrics (bulging height-pressure pairs). Kendall’s Tau analysis further corroborates our tactile sensor’s high ARA in ranking objects’ E (91.3%), well above conventional methods (66.8%). The materials library also allowed univariate experiments which, together with finite element simulations, elucidated the mechanistic influence of ν on bulging morphology. Implemented on a robotic manipulator, the sensor generalized to unseen samples through simple touch. By simultaneously resolving E and ν, this work overcomes a long-standing limitation in tactile perception of material softness, achieving human-skin-like capability to discern stiffness across diverse materials, and makes a key advance toward dexterous robotic manipulation.

Results

Poisson’s ratio-dependent precision in Young’s modulus detection

Human skin perceives the softness of objects through force-induced deformation (Fig. 1a, upper panel). When the fingertip contacts an object, the applied pressure causes surface indentation and increases the contact area—both of which vary with the object’s E and serve as key tactile cues for softness perception⁴². Effective assessment of softness, therefore, relies on applying pressure to induce deformation and measuring both the applied force and the resulting shape change. These mechanical cues are detected by cutaneous mechanoreceptors, such as Merkel discs (SA-I) and Ruffini endings (SA-II), which transduce them into neural signals for processing in the central nervous system.

Fig. 1. Poisson’s ratio–dependent precision in Young’s modulus detection.

a Schematic of a representative sensing architecture inspired by human skin’s perception of softness. b Deformation comparison of materials with different ν under pressure: PDMS cube (positive ν) expands laterally, while PU sponge (negative ν) contracts. c Surface deformation of PDMS cubes (base-to-curing-agent ratios from 10:1 to 40:1; ν = 0.473, 0.474, 0.481, and 0.488; E = 1499.5, 571.1, 100.1, and 16.9 kPa) under compression over a circular aperture. d Surface deformation of pink PU sponge (PUS-P, ν = −0.128, E = 107.7 kPa), yellow PU sponge (PUS-Y, ν = −0.046, E = 68.9 kPa), and white PU sponge (PUS-W, ν = −0.036, E = 48.9 kPa).

Inspired by this mechanism, tactile sensors with bilayer sensing architectures have been developed to mimic human perception^{18–20,25,43}. In these designs, the bottom layer measures the applied pressure while the top layer captures the surface deformation—together providing the necessary force-deformation data necessary to infer material softness (Fig. 1a, bottom panel). Under an identical pressure, greater surface deformation typically indicates lower stiffness. This strategy has been widely adopted in various sensor platforms^{17–20,25,43}, allowing for accurate estimation of E across diverse soft materials. Although inspired by human tactile perception, the sensing configuration differs from the deformation mode of a finger contacting an object. In the present design, the object protrudes into the interior sensing layer, whereas in human touch, the object indents the compliant fingertip. Despite this geometric difference, both mechanisms rely on detecting deformation characteristics, such as indentation depth and contact area⁴², together with the associated contact force to infer the mechanical properties of the object.

Despite their success, current tactile sensing strategies suffer from a critical limitation: the implicit assumption that the ν of the target material is constant, or that all tested materials have similar ν. For instance, studies using the Hertzian Contact Model^14,16 typically fixed ν and inferred E of objects solely from surface deformation. More generally, silicone rubbers (0.45 <ν < 0.5) are widely adopted as model materials in softness detection^{14,16,17,19,20}, yielding high accuracy when softness is inferred solely from surface deformation. These simplifications overlook the wide diversity of ν found in real-world materials. Our observations show that materials with negative ν, such as polyurethane (PU) sponges, exhibit deformation behaviors that deviate markedly from those of high-ν materials like Polydimethylsiloxane (PDMS) (Fig. 1b and Supplementary Fig. 1). Relying solely on deformation amplitude to evaluate the E can thus significantly compromise accuracy.

To verify the impact of ν on deformation-based softness sensing, we measured the surface deformation of PDMS samples and PU sponges under pressure using the above-mentioned sensor architecture (Fig. 1a, bottom panel, and Supplementary Fig. 2a). PDMS samples with higher E exhibit smaller deformations (Fig. 1c) under identical pressure—a trend consistent with previous reports^19,20,43. In contrast, for PU sponges with negative ν, the trend reverses: deformation increases with increasing E (Fig. 1d). This inversion fundamentally challenges the reliability of current tactile sensors that assume monotonic relationships between E and deformation. The flaw becomes even more apparent when comparing a PDMS sample and a PU sponge with similar E but different ν (Supplementary Fig. 2b). PDMS samples show a near-linear increase in surface elongation with pressure, while the PU sponge (ν < 0) displays a nonlinear trend, underscoring the failure of deformation-only metrics in such cases. These findings demonstrate that ν strongly modulates deformation behavior and must be considered in sensor design or signal processing. Addressing this limitation requires a mechanistic understanding of ν-dependent deformation and new sensing strategies that maintain accuracy across materials with diverse mechanical responses.

Influence of Poisson’s ratio on surface bulging morphology

A key challenge in elucidating the impact of ν on E detection accuracy is the lack of materials with independently tunable ν and E, as the two are often intrinsically coupled. To overcome this, we adopted a “shotgun” strategy: fabricating a large number of porous PDMS samples with varied processing parameters, then selecting subsets to construct a model PDMS samples library spanning a wide range of ν and E. This approach enabled the identification of sample sets with similar E but different ν, and vice versa. In brief, the porous PDMS samples were fabricated using citric acid (CA) particles as a sacrificial template to control their pore structure. ν was mainly modulated by adjusting the size and mass fraction of CA particles, while E was mainly tuned by varying the base-to-curing-agent ratio, curing temperature, and mass fraction of CA (Fig. 2a and Supplementary Figs. 3–7). This strategy yielded 99 distinct samples, including 94 porous PDMS samples and 5 non-porous PDMS samples (Fig. 2b, and Supplementary Fig. 8), covering ν values ranging from 0.090 to 0.488 and E values from 16.9 kPa to 601.5 kPa (Fig. 2c, and Supplementary Figs. 9 and 10, and Supplementary Table 1). As negative ν PDMS samples could not be achieved, three types of commercial PU sponges with negative ν were incorporated into the materials library to further expand the mechanical diversity (Supplementary Figs. 10a and 11).

Fig. 2. Influence of Poisson’s ratio on surface bulging morphology.

a Fabrication process of porous PDMS sample, and main factors influencing ν and E. b Optical images of the materials library, including 94 porous PDMS, 5 non-porous PDMS, and 3 PU sponge cubes. c Distribution of ν and E across all samples (mean ± SD from 5 trials). d Diagram of the setup used to capture surface profile; inset shows typical surface profiles under varying pressures. e A typical simulated surface protrusion and corresponding meridian profile. f Surface profiles (height versus position) of porous PDMS samples with similar E but different ν under 25 kPa compression, showing both experimental and simulated results. g Surface profiles of PDMS samples with similar ν but different E under compressive pressures of 27, 24, 21, and 22 kPa, experimental and simulated results shown. h Surface profiles of PU sponges with different E and ν under 33 kPa pressure, experimental and simulated results shown. i, j Summary of the deformation trends.

To quantitatively characterize surface deformation, we developed a custom setup capable of applying controlled vertical loads while optically resolving the resulting bulging profile (Fig. 2d). A thin square steel plate with a central aperture was used to support the samples during compression. The four edges of the steel plate were fixed to a polymethyl methacrylate (PMMA) frame that contained a viewing channel for camera access. This configuration minimized undesired out-of-plane displacement of the platform while ensuring an unobstructed optical view of the deforming surface. To enhance imaging contrast without interfering with deformation, a thin and soft carbon black/Polydimethylsiloxane (CB/ PDMS) composite film (Supplementary Figs. 12–14) was laminated onto the steel plate, conformally adapting to the sample’s surface. The aperture diameter was set to 5 mm, optimized to balance deformation amplitude, edge-induced stress effect, and mechanical stability under cyclic loading (Supplementary Figs. 15–17). To eliminate interfacial slippage between the steel surface and CB/PDMS film during loading, a thin PDMS adhesion layer was spin-coated and thermally cured onto the steel plate surface. This coating provided robust interfacial adhesion between the CB/PDMS film and the steel plate, preventing lateral displacement during compression (Supplementary Fig. 18). Upon incremental loading, the sample surface exhibited progressively increased protrusion (inset of Fig. 2d and Supplementary Figs. 19–22). Surface profiles were extracted using the image-processing software, and due to their mirror symmetry, only half-profiles were analyzed (Supplementary Fig. 23).

To eliminate potential artifacts from minor variations in sample properties, we conducted finite element method (FEM) simulations using an axisymmetric model (Fig. 2e and Supplementary Figs. 24–27). In the model, a thin film was placed over a circular aperture (radius = 2.5 mm) and loaded against a bulk material with a E of 100 kPa and a ν of 0.15. A representative contour map of the film under 55 kPa pressure is shown in Fig. 2e.

We first examined porous PDMS samples with similar E but varying ν (Fig. 2f and Supplementary Fig. 28a). Surface protrusion decreased markedly as ν increased, indicating that materials with lower ν deform more prominently despite having comparable softness. This result again underscores the key limitation of conventional approaches that infer E solely from bulging amplitude. Such approaches become unreliable across materials with different ν. Next, we analyzed PDMS samples with similar positive ν values (0.477, 0.297, 0.224, and 0.144, respectively) but varying E. Softer materials exhibited greater surface protrusions under identical pressure (Fig. 2g and Supplementary Fig. 28b–e), consistent with previously reported mechanical behavior^19,20. However, this trend was inverted in porous PU sponges with negative ν (Fig. 2h and Supplementary Fig. 28f), where stiffer materials showed greater surface deformation. This counterintuitive response highlights the complex mechanics of auxetic materials and exposes the limitations of current tactile sensing systems, which typically assume 0 <ν < 0.5 and treat ν as a fixed input parameter^19,20. We further observed that although some samples exhibited comparable peak bulging heights under different pressures, their overall surface profiles differed (Supplementary Figs. 29–31), indicating that the central displacement alone is insufficient to represent the full deformation behavior. These results suggest that a complete deformation profile, rather than a single point metric, is critical for accurate E estimation. Collectively, these findings demonstrate that E and ν jointly determine deformation behavior, and both parameters must be considered in tactile sensing aimed at generalized material property estimation.

The simulations were systematically performed across a matrix of ν and E covering the full range of experimental conditions. The simulation results (Fig. 2f–h, curves) quantitatively reproduce the three deformation trends observed experimentally. First, for samples with constant E but varying ν, the protrusion amplitude increases as ν decreases (Fig. 2f and Supplementary Fig. 32a), consistent with experimental results. Second, for samples with fixed ν (0.477, 0.297, 0.224, and 0.144, respectively) and variable E, the protrusion height decreases with increasing E (Fig. 2g and Supplementary Fig. 32b–e), in agreement with experimental data. Third, the simulations of PU sponge-like materials reveals that stiffer auxetic samples exhibit greater bulging deformations (Fig. 2h, curves), supporting experimental observations.

To validate the experimental results, we quantitatively compared them with simulations across 15 PDMS samples. As the experimental and simulated position-height curves for each sample exhibit monotonic decay (Fig. 2f and g, Supplementary Fig. 33), data points at all positions along each curve (Supplementary Fig. 34a) were selected for comparison. The corresponding experimental and simulated height values were substituted into Eq. (2) (Supplementary Materials and Methods) to calculate the average relative errors. The overall mean of the per-sample average relative errors across 15 samples is 12.5%, indicating good agreement between the experiment and simulation (Supplementary Fig. 34b).

In contrast, larger discrepancies are observed for the polyurethane sponges (Fig. 2h and Supplementary Fig. 35a–c), with average relative errors of 46.1% (PUS-P), 53.5% (PUS-Y), and 48.0% (PUS-W), respectively. This deviation arises from limitations of the hyperfoam constitutive model, which cannot fully reproduce the material’s nonlinear deformation of porous structures, including cell wall buckling, collapse, and densification. In the simulation of PU sponges, the strain range is approximately 0–30%. A comparison of the input uniaxial test data with the simulated uniaxial stress-strain curves (Supplementary Fig. 35d–f) highlights this inherent fitting error, particularly in the small-strain region where the hyperfoam model is softer and deviates from the experimental values.

In summary, surface bulging increases with decreasing ν at constant E (Fig. 2i); for materials with positive ν, bulging increases as E decreases (Fig. 2j); for negative ν materials, bulging increases with increasing E, meaning stiffer materials exhibit greater deformation. These findings demonstrate that ν exerts a significant influence on deformation behavior. Consequently, conventional bulging-based tactile sensing, which typically neglects the influence of ν, is inherently limited. Overcoming this constraint calls for a new sensing strategy that accurately quantifies E and ideally enables simultaneous estimation of ν.

Dependence of surface bulging height on Poisson’s ratio

Although the full bulging profile clearly reflects the influence of E and ν, in principle, it allows for the derivation of E and ν (e.g., via analytical models such as Hertzian contact theory), capturing this information remains technically impractical for a compact tactile system. In contrast, the central bulging height is easier to detect using a displacement sensor, but as mentioned above, it alone does not fully represent the deformation characteristics. Given this trade-off, we reasoned that while E and ν influence the overall bulging profile, they should also impact distinct trends in how the central bulging height evolves with pressure. Thus, we hypothesized that by tracking this pressure-dependent central height evolution, it may be possible to infer both E and ν, offering a more compact, scalable sensing strategy without full-profile acquisition.

To test this hypothesis, we designed a custom setup to measure bulging height under varying pressures. The setup is adapted from a full-profile imaging platform, with the camera replaced by a laser displacement sensor for direct height readout (Fig. 3a and Supplementary Fig. 36). Using this setup, we first examined porous PDMS samples with similar E but varying ν. Results show that bulging height increased with decreasing ν (Fig. 3b), consistent with trends observed in full-profile measurements (Fig. 2f). For porous PDMS samples with similar positive ν but different E, bulging heights decreased as E increased (Fig. 3c), again aligning with the full-profile measurement results (Fig. 2g). As the contact angle increases, the shear force acting on the sample increases accordingly. Nevertheless, the qualitative bulging height trend remains consistent across different contact angles (0°, 30° and 60°; Supplementary Figs. 36–40). Specifically, for samples with similar Young’s modulus, the bulging height decreases as the Poisson’s ratio increases; while for samples with similar Poisson’s ratio, the height decreases with increasing Young’s modulus.

Fig. 3. Dependence of surface bulging height on Poisson’s ratio.

a Schematic of the laser displacement sensor-based setup for measuring central bulging height. Bulging height versus pressure for porous PDMS samples (b) with similar E but different ν, and (c) with similar ν but various E. d Bulging height versus pressure for PU sponges with varying ν and E; insets show the corresponding slope evolution. e Simulated bulging height-pressure curves and corresponding slope evolution for PU sponges mentioned in (d) under compression. f Slapshots of simulated structural evolution in PU sponges with different stiffness. g Comparison of bulging curves for a non-porous PDMS elastomer and a PU sponge with similar E but different ν. Data represent the mean of four independent measurements for (b), (c), and PDMS 30:1 in (g), and five for (d) and PUS-P in (g), with shaded error bands indicating the standard deviation.

Interestingly, PU sponges exhibited a non-monotonic behavior. At low pressures, softer sponges showed greater bulging, resembling PDMS samples with positive ν (Fig. 3d and Supplementary Fig. 41). However, at higher pressure, this trend reversed—stiffer sponges displayed larger bulging. The rate of bulging height increase, plotted in the inset of Fig. 3d, peaked at different pressures depending on softness: softer sponges transitioned earlier from rapid to slower height increase. This inversion is attributed to pressure-dependent changes in ν. As compressive strain increased, the sponges’ ν shifted from positive to negative (Supplementary Fig. 42), driven by structural transitions such as elastic bending, buckling, and eventual collapse of the three-dimensional porous networks (Supplementary Figs. 43–47). This previously underexplored behavior is difficult to capture via camera-based profile imaging but becomes clearly observable through central height evolution. These results underscore that the pressure-dependent height also encodes rich mechanical signatures and offers a practical and informative metric for inferring both E and ν.

We developed a numerical model to simulate the sponge mechanics under compression. The simulated bulging-height versus pressure curves (Fig. 3e) closely matched these experimental results in Fig. 3d, validating the distinctive deformation behavior compared with materials with positive ν. To uncover the mechanism underlying this distinctive deformation behavior from a microstructural perspective, we further constructed two honeycomb structural models with E and ν comparable to those of the white (PUS-W) and yellow (PUS-Y) sponges, and simulated their microscale structural deformation evolution. The slapshots of structural deformation in two sponges (Fig. 3f and Supplementary Figs. 48–50 and Movie 1) shows that under low pressure, both sponges expand laterally—characteristic of materials with positive ν—exhibiting bulging-height trends similar to those in Fig. 3c. As pressure increases, the pores begin to collapse, lateral contraction dominates, resembling negative ν behavior and reversing the height-pressure trend. The softer white sponge exhibits a faster initial bulging response due to its lower E and positive ν (inset plots of Fig. 3e). However, its weak skeleton collapses earlier under pressure, triggering a transition to the negative ν regime, where bulging growth slows. In contrast, the stiffer yellow sponge maintains its porous structure longer, delaying this transition. In short, stiffer sponges better resist collapse at the bulging part, sustaining greater protrusion. Conversely, in the softer sponge, pore collapse suppresses further expansion, resulting in a slower increase in bulging height despite lower E.

Pressure–height evolution curves display distinct patterns across materials with varying E and ν (Fig. 3b–d), particularly for non-porous PDMS cubes and PU sponges with similar E but different ν (Fig. 3g). Despite similar stiffness, their pressure-height trajectories diverge significantly, suggesting that such curves encode rich mechanical information relevant to both E and ν. While constructing an analytical mechanical model to extract E and ν from pressure-height data would be ideal, it is both time-consuming and beyond the current scope of our work. Instead, machine learning offers a powerful alternative for capturing the hidden correlations between pressure-height characteristics and underlying material properties. Although the physical interpretability is limited, such models can still provide accurate predictions.

As a proof of concept, we collected force-height data (Supplementary Fig. 51) from 61 PDMS samples using the setup described in Fig. 3a and Supplementary Fig. 36, and trained an FCNN model. The model achieved ARAs of 90.2% (and 87.7%), 82.4% (and 76.2%), and 83.5% (and 84.0%) for training, validation, and testing datasets (Supplementary Figs. 52–57), respectively, in predicting E (and ν). While the dataset remains relatively small and risks overfitting, these results demonstrate the potential feasibility of data-driven E and ν estimation. Importantly, our PDMS sample library with broad coverage of E and ν provides a strong foundation for model generalization. Scaling up the dataset and refining the model architecture could eventually enable accurate, robust joint prediction of both material parameters.

Tactile sensor enabling simultaneous detection of pressure and surface bulging height

We adopted the sensing architecture illustrated in Fig. 1a, which emulates human skin in capturing surface deformation characteristics of objects. Accurate inference of E and ν thus hinges on the precise and repeatable measurement of surface bulging dynamics under pressure. Conventional approaches, such as piezoresistive strain sensors⁴⁴, often suffer from baseline drift, compromising the reliability of deformation tracking and limiting their effectiveness in deriving materials properties, even when the bottom pressure sensor itself remains functional well. To address this, we implemented a magnetic displacement-sensing system that enables high-fidelity, drift-free measurement of bulging height with excellent repeatability. The pressure was measured using a hierarchical-structured piezoresistive interface integrated into a Wheatstone bridge circuit, which enhances sensitivity and signal stability.

The sensor structure is illustrated in Fig. 4a (Supplementary Fig. 58). The upper part is the magnetic displacement sensing module, consisting of a soft, elastic, thin, and cross-shaped CB/PDMS film (Supplementary Figs. 12–14 and Supplementary Fig. 59), back-adhered with a NdFeB/PDMS composite magnetic film (Supplementary Figs. 60 and 61), and a Hall sensor positioned below the CB/PDMS film. The CB/PDMS film is highly compliant, minimizing mechanical interference with surface bulging (Supplementary Fig. 22). Upon compressing, the sample surface protrudes, reducing the distance between the soft magnet and the Hall sensor, thereby producing an increase in output voltage (Fig. 4b). This configuration effectively transduces the bulging heights into voltage changes (Supplementary Fig. 62). The rightmost panel of Fig. 4b shows the voltage change (ΔV_D) during compression of nonporous PDMS samples with different stiffness. As expected, softer PDMS samples induce steeper increases in voltage changes, consistent with the bulging behavior observed in Fig. 3c.

Fig. 4. Tactile sensor enabling simultaneous detection of pressure and surface bulging height.

a Exploded 3D schematic and a photograph of the tactile sensor, alongside optical, microscopic, and SEM images of key components showing material composition and structural geometry. b Sensing principle of the magnetic displacement sensor, and typical voltage change—pressure curves for non-porous PDMS samples; V_D,0 denotes the baseline voltage of the Hall sensor; ΔV_D = V_D - V_D,0, where V_D is the real-time voltage of the Hall sensor. c Schematic of the pressure sensing module and working mechanism; the right panel shows resistance and voltage response upon compression. V_P,0 denotes the baseline voltage of the pressure sensing module; ΔV_P = V_P - V_P,0, where V_P is the real-time voltage of the pressure sensing module. d Voltage changes of the upper displacement sensor during 1000 cycles of loading/unloading at a peak pressure of 167 kPa. e Voltage changes of the bottom pressure sensor during 2000 cycles of loading/unloading at a peak pressure of 62 kPa. Voltage changes of displacement sensor versus that of pressure sensor during compression by f porous PDMS with similar E but differing ν; g the porous PDMS with similar ν but different E; h PU sponges with different ν and E; and (i) non-porous PDMS and pink PU sponge with similar E; All plots represent average results from five independent measurement. f–i The applied pressure is equal to 33 kPa.

The bottom part serves as the pressure sensing module, consisting of a porous CB/PDMS piezoresistive layer with a molded pyramidal surface (Supplementary Fig. 63), and circular interdigital electrodes (lower right in Fig. 4a). Upon compression, pore collapse and surface protrusion flattening in the piezoresistive layer reduce the resistance between electrode fingers (Fig. 4c and Supplementary Fig. 64a and b). When integrated into a Wheatstone bridge with three reference resistors, the pressure-dependent resistance change (ΔR) is converted into a voltage change (ΔV_P) with amplified signal magnitude (the rightmost plot in Fig. 4c). Replace the pressure in Fig. 4b with ΔV_P, and a ΔV_P-ΔV_D plot can be obtained (Supplementary Fig. 64c and d). In this way, all data originate from the tactile sensor and can be further processed to infer E and ν.

The magnetic displacement sensing architecture displays excellent stability and repeatability over cyclic loading-unloading tests (Fig. 4d and Supplementary Fig. 65), and significantly outperforms conventional resistance strain sensing structures, which suffer from baseline drift and signal degradation (Supplementary Fig. 66). The elastic piezoresistive layer of pressure-sensing module is inherently stable (Supplementary Fig. 67). When it was integrated into a Wheatstone bridge, the resulting voltage signal remains stable under cyclic loading test (Fig. 4e). The high stability and reliability of both displacement and pressure sensing modules of our tactile sensor are critical for acquiring high-fidelity surface deformation characteristics, which is an essential prerequisite for accurate derivation of E and ν. Leveraging these two integrated sensing functions, our tactile sensor simultaneously captures bulging height and pressure as voltage signals during contact with an object under varying loads.

We systematically tested a range of PDMS cubes and PU sponges using our tactile sensor under varying pressures, simultaneously recording voltage outputs from the upper displacement and bottom pressure sensing modules. The results (Fig. 4f–h, Supplementary Figs. 68 and 69a) show that ΔV_D-ΔV_P relationships closely recapitulate the bulging height-pressure curves (Fig. 3b–d) obtained with the setup in Fig. 3a. Notably, despite having similar E, the pink PU sponge and its PDMS counterpart followed distinct voltage trajectories (Fig. 4i and Supplementary Fig. 69b, corresponding Fig. 3g). We further examined two sets of samples with similar E but different ν, and our tactile sensor unambiguously resolved their differences (Supplementary Fig. 70). Together with the sensor’s high sensing repeatability (Supplementary Figs. 71 and 72), these ΔV_D-ΔV_P signatures offer a rich and discriminative feature space for machine-learning inference of both E and ν, especially when coupled with our materials library that spans a broad range of mechanical spectra.

Simultaneous prediction of Young’s modulus and Poisson’s ratio via machine learning

To enable large-scale data acquisition, we developed a customized automated system (Supplementary Fig. 73). Preliminary validation on five representative samples showed highly repeatable ΔV_P-ΔV_D trajectories over five independent trials (Supplementary Fig. 74), with minimal variation across tests. Extended cyclic testing (105 cycles) further confirmed the excellent repeatability of sensor outputs (Fig. 5a), and similar results were observed across 101 additional samples (Supplementary Fig. 75). These results again established a reliable foundation for machine learning-based material property prediction. To reduce computational load while preserving representative signal characteristics, we uniformly sampled 52 points along each ΔV_P-ΔV_D curve (ΔV_{D, n} input; corresponding ΔV_{P, n} output) for model training (Fig. 5b). The FCNN model was selected for its superior predictive performance. The FCNN was trained on 10,163 labeled datasets collected from the 102 materials in our material library, with a typical 8:1:1 split for training, validation, and testing sets, respectively. The loss curves of average relative error (ARE) converged stably for both E and ν (Fig. 5c and d), confirming robust model optimization. Error distribution analysis revealed that most predictions in the training and validation sets achieved anti-error rate assessments (ARAs) exceeding 85% (Fig. 5e and Supplementary Figs. 76–79). On the test set, the model reached ARAs of 87.7% for ν and 84.7% for E, exhibiting strong correlations between predictions and ground truth (Fig. 5f and g).

Fig. 5. Simultaneous prediction of Young’s modulus and Poisson’s ratio via machine learning.

a Representative demonstration of the tactile sensor’ high reproducibility over 105 loading/unloading cycles for a single sample. b Feature extraction strategy and architecture of the FCNN model. Training convergence curves for predicting (c) ν and (d) E; ARE: average relative error. e Relative error distribution of predicted ν and E on the training set. Correlation plots between predicted and true values of (f) ν and (g) E on the testing set; ARA: anti-error rate assessment. h Schematic of parameters used in the deflection model. i Correlation plots between predicted E derived from the deflection model and true E. j Fitting curve derived from five non-porous PDMS samples, used to predict E across 67 PDMS samples, yielding an anti-error rate assessment of 69.8%. Kendall’s Tau correlation analysis (Methods) of E predictions: k traditional fitting method (τ = 66.8%); l FCNN model (τ = 91.3%); τ: Kendall’s Tau coefficient. m Benchmark comparison with previous studies; x-axis: dividing accuracy by the number of sensor units; y-axis: multiplying the sample quantity by the accuracy rate divided by the number of sensing units.

To evaluate the influence of contact angle, additional sensing tests were performed at 10° and 30° (Supplementary Fig. 80). Across 21 samples, the ARAs for predicting ν and E are 87.1% and 87.3% at 0°, 85.1% and 85.2% at 10°, but decrease to 44.5% and 52.1% at 30°, respectively (Supplementary Figs. 81–84). The progressive decline in accuracy with increasing contact angle indicates the growing influence of shear forces on the deformation response. Nevertheless, within a limited angular deviation ( ≤ 10°), the quantitative sensing performance remains largely preserved, with ARAs of ~85% for both ν and E. In practical robotic applications, the sensor posture can be approximately controlled using machine-vision-assisted alignment, enabling near-vertical contact with angle variation below 10°.

We also evaluated other machine-learning models, including support vector regression, linear regression, and gradient boosting regression, using the same feature set (52 ΔV_P values) and datasets. All performed markedly worse than FCNN (Supplementary Fig. 85). To benchmark against conventional approaches, we further trained models with only a pressure-deformation pair (ΔV_P and ΔV_D) as features (Supplementary Fig. 86a). The ARE loss curves for both training and validation started low and plateaued within ten epochs with minimal reduction, while the validation loss fluctuated more strongly than the training loss, indicating weak or noisy correlations between (ΔV_P, ΔV_D) and (E, ν) that hinder reliable prediction (Supplementary Figs. 86b and c). The instability of the validation loss arises from a fundamental “many-to-one” mapping conflict: distinct deformation-pressure states can correspond to the same target value (E, ν), forcing the model to oscillate between contradictory gradient signals and preventing stable convergence. Consequently, the test set yielded maximum prediction ARAs of 66.8% for ν and 73.3% for E (Supplementary Fig. 87), far below the accuracy obtained when the surface protrusion trajectory (52 ΔV_P values) was used. These findings again highlight that relying solely on surface deformation under a given pressure (ΔV_P, ΔV_D) substantially compromises the ARA of E estimation.

We further applied a theoretical model⁴¹ to evaluate the physical validity of our tactile sensor output. The measured ν values of the samples were treated as known inputs (Fig. 5h and Methods). Using compression force F, bulging height h, edge length of the cubic sample a, and constant coefficient γ, the model yielded E with an anti-error rate assessment of 81.4% (Fig. 5i and Supplementary Fig. 88). This result confirms that our tactile sensor’s output signals are physically meaningful and conform to mechanical theory, supporting the generalizability of our FCNN model. Crucially, it also highlights the challenge of estimating E in the absence of ν information. For example, using a fitted curve derived from non-porous PDMS samples (Fig. 5j), assuming a fixed ν (traditional methods), the prediction anti-error rate assessment drops to only 69.8% (Supplementary Fig. 89), substantially lower than that achieved by our sensor-enabled FCNN model.

We compared the predictive ARA of the traditional method and our FCNN model using Kendall’s Tau Coefficient (τ), which evaluates rank correlation between predicted and actual E. The traditional fitting approach yielded τ of 66.8% (Fig. 5k), while our FCNN model achieved significantly higher τ of 91.3% (Fig. 5l), underscoring its superior performance, especially when ν varies across samples. Notably, our sensor enables quantitative, simultaneous estimation of both E and ν—demonstrated here for the first time in a tactile sensing platform—offering a decisive advantage over previously reported methods (Fig. 5m and Supplementary Table 2).

Robotic integration of a tactile sensor for predicting Young’s modulus and Poisson’s ratio via compressive touch

we integrated the sensor onto a robotic arm to demonstrate the practical application of our tactile sensor for direct estimation of E and ν through the touching process (Fig. 6a). Voltage signals from the displacement and pressure sensing modules were acquired via an NI data acquisition (DAQ) device. The data was processed using LabVIEW software to extract key features, and then transmitted to the Python-based FCNN model for the prediction of E and ν. The predicted values were sent back to LabVIEW for visualization. The performance of the tactile system was first validated using representative samples from the materials library, including yellow PU sponge, porous PDMS cubes, and non-porous PDMS cubes. The robotic arm was programmed to perform three consecutive pressing-releasing cycles per sample at a constant compression rate. Figure 6b shows the temporal evolution of voltage signals corresponding to bulging height and pressure. The extracted voltage data of each cycle were independently fed into the trained FCNN model, and the final predictions of E and ν were averaged across the three tests to enhance prediction robustness. Across eight tested samples, the prediction ARAs reached 86.7% for E and 88.9% for ν (Fig. 6c, Supplementary Table 3, and Supplementary Movies 2 and 3).

Fig. 6. Robotic integration of a tactile sensor for predicting Young’s modulus and Poisson’s ratio via compressive touch.

a Schematic of robotic tactile sensing system for the quantitative measurement of materials’ ν and E. b Time-resolved voltage outputs from the tactile sensors during testing of eight standard samples. Scale bar: 5 mm. c Predicted ARA of the samples in (b), averaged over three independent tests. d Time-resolved voltage outputs from the tactile sensors during compression of melamine sponge and boiled beef with different durations, along with corresponding true values, predictions, and ARA. Scale bar: 2 cm.

The practical utility of our tactile sensor was further demonstrated by evaluating its ability to identify the E and ν of objects with arbitrary shapes and sizes, which fall outside our materials library. This capability is essential for real-world deployment, where prior knowledge of object properties is often unavailable. Melamine sponge and boiled beef samples with varying dimensions were selected as representative “unknown” objects for validation (Fig. 6d and Supplementary Movie 4). Each object was compressed three times by the robotic arm equipped with our tactile sensor, following the same data acquisition and processing protocol as previous tests. Sample dimensions were randomly selected: melamine sponge (8.1 × 5.1 × 2.9 cm³), beef boiled for 70 minutes (6.2 × 2.0 × 1.5 cm³), and beef boiled for 40 minutes (7.3 × 2.0 × 1.5 cm³). Voltage signals reflecting both bulging height and pressure were recorded in real-time and processed by our trained FCNN model, yielding highly accurate predictions of E and ν (Supplementary Fig. 90). In particular, the temporal evolution of E and ν captured by our tactile sensor enables clear differentiation of food doneness. This mechanical sensing approach provides a robust complement to vision-based systems^17,45, which often face challenges in reliably assessing doneness levels. This suggests potential applications of our sensor in domestic or food-service robotics, such as automated assessment of bedding softness or food doneness.

Discussion

We demonstrate a tactile sensor that, for the first time, quantitatively resolves both Young’s modulus and Poisson’s ratio across a broad spectrum of materials. This capability relies on a comprehensive materials library spanning a wide range of Young’s modulus–Poisson’s ratio combinations, which underpins the sensor’s high dual-parameter sensing accuracy and provides a systematic framework for elucidating how Young’s modulus and Poisson’s ratio jointly shape surface bulging. By enabling controlled comparisons—such as materials with similar Young’s modulus but different Poisson’s ratio, or vice versa—this library offers mechanistic insights critical for guiding the design of next-generation tactile sensors. Beyond immediate performance gains, this dual-parameter perception establishes a new paradigm for tactile sensing, overcoming the constraints of conventional sensors restricted to narrow material classes. The approach is broadly generalizable, suggesting that training other sensors on such diverse datasets could potentially improve their Young’s modulus detection accuracy across varied materials. Despite the demonstrated capabilities, several limitations remain. The quantitative measurement of Young’s modulus and Poisson’s ratio can be influenced by factors such as sample thickness, complex surface geometries, and heterogeneity. Additionally, the current sensor imposes certain constraints on the minimum size and thickness of the objects that can be reliably characterized. Furthermore, the material composition and fabrication process limit the conformal adhesion of the sensor to highly curved surfaces. Looking forward, miniaturized, arrayed, and fully flexible implementations promise conformal, high-resolution tactile skins, unlocking transformative capabilities in dexterous robotic manipulation.

Methods

Calculation method for Poisson’s ratio of samples

To determine the Poisson’s ratio of the 102 samples, each sample was mounted on an electronic tensile testing machine and subjected to a uniaxial load corresponding to ~20% strain along the y-axis (Supplementary Fig. 9). Images of the undeformed and deformed states were recorded using an optical imaging system consisting of a microscope camera mounted on a height-adjustable stage. The widths (x₀, x₁) and heights (y₀, y₁) of the samples before and after deformation were extracted using ImageJ. Poisson’s ratio was then calculated from measured transverse and axial strains according to its definition.

Mechanical Characterization (Young’s modulus) of test samples and CB/PDMS film

Young’s modulus was obtained from standard compression tests using an electronic universal testing machine (EM2.502, 10 N/500 N, Shenzhen Tesmate Instruments). Each sample (102 samples in total) was compressed from 0 to 25% strain at a loading rate of 9 mm min⁻¹. The Young’s modulus was calculated from the stress–strain curves at ~20% strain, which lies within the elastic deformation regime of the PDMS samples (both porous and non-porous) and allows consistent comparison across all samples. The Young’s modulus of the CB/PDMS film was obtained from uniaxial tensile tests. CB/PDMS films were cut into standard dumbbell-shaped specimens and stretched quasi-statically, with the Young’s modulus extracted from the linear regime of the stress-strain curve (0-30% strain).

Finite element analysis

The deformation of the CB/PDMS films and samples under compression (Fig. 2e–h) was simulated using a two-dimensional (2D) axisymmetric model in the commercial software ABAQUS. A CB/PDMS film with a thickness of 41.9 μm was placed at the bottom of the samples and contacted a rigid support with a 5 mm circular hole, filleted (radius, 0.2 mm) to reduce stress concentration. The friction coefficient was set to 0.5 for the film-support interfaces. The sample and film were meshed with CAX8RH elements, while the support was modeled as a discrete rigid body. A displacement load was imposed at the top boundary of the samples. Both the porous PDMS samples and the CB/PDMS film were described using a Mooney-Rivlin hyperplastic model with material parameters: (C10 = 0.0230, C01 = –0.00419, D1 = 28.5664) and (C10 = 0.0182, C01 = –0.00187, D1 = 3.0588), respectively. A hyperfoam model (N = 2) was used to simulate the deformation of PU sponges (Figs. 2h, 3e), calibrated with uniaxial data from the three kinds of PU sponges (pink, white, yellow).

Honeycomb models were used to investigate the nonlinear deformation of the PU sponge by explicitly visualizing microstructure evaluation (Fig. 3f, Supplementary Figs. 48–50, and Supplementary Movie 1). Two hexagonal frameworks (9.6 mm in length,10 mm in width, Supplementary Fig. 48) were considered: a Y-honeycomb (cell 0.31 mm, wall 0.03 mm, from white foam) and a Y-honeycomb (cell 0.31 mm, wall 0.06 mm, from yellow foam). The honeycombs were meshed with CPE4R elements and compressed against a discrete rigid support (friction coefficient = 0.5) using the explicit dynamic module. The framework material was modeled as ideal plastic (E = 3 MPa, yield stress = 0.13 MPa). A displacement-controlled load (8 mm in 10 s, smooth-step function) was applied to the top boundary, following the configuration in Fig. 2e.

Theoretical model for Young’s modulus calculation

Sakamoto et al.⁴¹ developed a theoretical model to describe axisymmetric indentation mechanics of an elastic layer supported by a rigid foundation with a circular opening. This configuration closely resembles our sensor structure. Two distinctions exist between their model and our experimental system: (1) the model assumes an infinite elastic layer, whereas our system uses an elastic cubic sample; and (2) a thin film covers the elastic cubic sample in our system, which was excluded in the reference. Given that the edge length of our elastic cubic sample (15 mm) is three times the diameter of the circular opening (5 mm) and the film thickness is only 41.9 μm, these differences exert a negligible influence, allowing us to reasonably apply the theoretical model of Sakamoto et al.

To adapt the mode to our measurement system, we introduce a constant coefficient γ and simplify Equation 29 in the reference as follows:

$$E=\gamma \left(1-{\nu }^2\right)\frac{F}{h{a}^2}$$ 1

where, γ is a geometry-dependent constant related to the experimental configuration; F is the applied force to the elastic cubic sample; h is the bulging height, and a is the edge length of the elastic cubic sample. In our experiments, h was fixed at 310 μm for all samples to ensure high measurement accuracy while maintaining the compressive strain of the samples below 30%, where E remains close to the intrinsic E measured using standard compression tests (within 20% compressive strain). Since a is constant, h, a, and γ can be combined into a single coefficient k, leading to the simplified form:

$$E=k\left(1-{\nu }^2\right)F$$ 2

The coefficient k was determined by fitting the experimental data (Supplementary Fig. 91). For the quantitative tactile sensing, the displacement voltage is constrained to a fixed range (0–0.1 V, Supplementary Figs. 73–75), which correspondingly limits the CB/PDMS film’s strain to between 0 and 1.06% (Supplementary Fig. 92). The strain range is sufficiently small and the CB/PDMS film only brings a small additional pressure (71.4 Pa) to the PDMS samples, less than 1% of their Young’s modulus (see Table S1). So that the effect of the CB/PDMS film can be ignored.

Kendall’s Tau coefficient

Kendall’s Tau Coefficient (τ) was used to quantify the ordinal association between predicted and true Young’s moduli by comparing all possible sample pairs. For any two samples, concordance is assigned when the predicted ranking matches the ground truth, and discordance when the order is reversed. τ thus captures the degree of rank consistency, ranging from −1 (complete inversion) to 1 (perfect agreement). It is calculated as:

$$\tau =\frac{C-D}{n\left(n-1\right)/2}$$ 3

Where: C represents the number of concordant pairs; D denotes the number of discordant pairs; n is the sample size.

Author contributions

J.G. conceived the project. J.X. and J.G. designed the experiments. J.X. constructed the experimental platform, conducted most of the experiments, and visualized the results. G.M. built the theoretical model for Young’s modulus calculation. X.L. and G.M. performed mechanical simulations and contributed to the visualization of the results. Y.W. prepared the porous PDMS samples, performed the experiments, analyzed the results, and visualized the data. J.G., J.X., and G.M. wrote the manuscript with input from all the authors. J.G. developed the hardware-software system for data acquisition and a real-time visualization interface for demonstrations. Y.Y. conducted part of the mechanical simulation and discussed the results. M.G. contributed significantly to the electrical experiment test. X.M. contributed to the development of machine learning models and data visualization. L.Z. collected 3D images. J.G., Y.L., and J.C. discussed the results and initial draft. J.G. and G.M. supervised the project. All the authors approved the final version of the manuscript.

Peer review

Peer review information

Nature Communications thanks Zhi Liu, and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Funding

This work was supported by the National Natural Science Foundation of China (22475242, 92461302, 12372167, and 12321002), the Guangdong Basic and Applied Basic Research Foundation (2025A1515010271), and NKRD Program of China (2021YFA1500401).

Data availability

All data needed to support the conclusions of this manuscript are included in the main text and Supplementary Materials. The source data is available. Source data are provided with this paper.

Code availability

The Python code for the simultaneous quantitative measurement of Young’s modulus and Poisson’s ratio via tactile sensing is available at: 10.5281/zenodo.19973897.

Competing interests

All the authors conducted this research responsibly and ethically under an inclusive research environment. The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-026-73216-8.