Mechanical multiaxis force sensor for directly bridging sensing and fluidic actuation

Abstract

Robots are the intelligent systems that connect sensors and actuators. Many sensorimotor architectures use close-loop control codes that arbitrate sensing signals, entailing processing modules. Reactive architectures minimize computational demands by establishing direct sensor-actuator connections and have proven to be effective and robust. Here, we introduce a mechanical analogy of such reactive systems: a fluidic-based multiaxis mechanical soft force sensor (ME-SOFS) that directly couples sensory signals with fluidic actuation, eliminating the need for external computation or energy input. The ME-SOFS can be easily reconfigured and integrated to endow robots with somatosensory multiaxis force sensing capabilities. Based on fluid transduction, ME-SOFS converts applied force into mechanical output for fluidic actuators. We demonstrate this sensing-actuation loop in three scenarios: directional droplet manipulation, unified bending of cilia-like array guided by detected force vectors, and haptic feedback system that accelerates robotic grasping learning. ME-SOFSs demonstrate how fluidic approach can realize multiaxis force sensing for soft robots, and enable simplified, closed sensing-actuation loops and haptic human-machine interfaces.

Mechanical soft sensor enables direct sensing-actuation connection and multiaxis force detection for soft robotics.

INTRODUCTION

Actuators and sensors are the backbone of robotic systems (1). During interactive tasks, robots rely on sensory information to perceive and generate corresponding active responses, such as related actuations (2) and human haptic feedback (3). Turing machines and computers were proposed in 1936, providing a computing paradigm and platform. In robotics, such computing is used in higher-level controllers for sensing-actuation loops. However, the computational burden tends to increase in high-density sensing scenarios. Appropriately implementing reactive architectures that directly connect sensors and actuators can enable remarkably simple yet effective behaviors, with minimal computation (4).

In soft robotics, similarly to biological structures such as human fingertips (5), cat pads (6), and octopus suckers (7), strategies that equip soft robots with somatosensory multiaxis force sensing and direct responding abilities are valuable for reducing complexity in environmental interactions, enhancing sensing-actuation loops and sharing perception and action in human-machine systems (8, 9). Sensors and actuators often function in distinct physical domains. For instance, despite advancements in multiaxis soft force sensors based on piezoresistive (10–12), magnetic (13–15), capacitive (16–20), or triboelectric (21) mechanism, these sensors typically transduce external stimuli into electrical signals that are not inherently compatible with the fluidic actuation widely used in soft robotics. While such separate designs between sensors and actuators offer flexibility in their combination, the domain mismatch necessitates additional signal processing and conversion to establish a sensing-processing-commanding-actuation connection between the two. As a result, bridging sensing and actuation still relies heavily on external circuitry and computation. Taking pneumatic robot system as an example, a usual sensing-actuation connection requires data processing and controlling circuitry, pneumatic pump and valve, and their power supplies for computation and actuation. Matching the sensing and actuation in the same physical domain introduces a fundamentally different and simplified sensing-actuation connection in robotics, shedding light for robotic integration and bringing intelligence into the minimalist robotic system such as the electronics-free liquid circuit (22–25).

In this work, we introduce a multiaxis mechanical soft force sensor (ME-SOFS) that directly couples sensory signals with fluidic actuation, without the need for external computation or energy input (Fig. 1A). ME-SOFS uses a purely mechanical structure to encode external force stimuli into separate inner fluidic flows, effectively decoupling the force components. With this fluidic approach, ME-SOFS can create an effective mechanical sensing-actuation loop guided by the detected force vectors and dynamics, directly translating applied force into actuation and a haptic system to provide feedback onto a human fingertip (Fig. 1, Ac and D). A signal-generating circuit could be included into the fluidic loop for readable electrical signals. In such case, the ME-SOFS–induced fluidic flows could displace magnets in the fluid to produce flow-based magnetic fields that pass through three-dimensionally (3D) printed alloy arcs, evoking sequential voltage pulses, following Faraday’s law of electromagnetic induction (Fig. 1Ab). The ME-SOFS is highly scalable with tunable force-sensing sensitivities, enabling easy design, integration, and multiaxis force sensing capabilities in a 3D-printed sensorized soft glove and a soft arm.

Fig. 1. Design and working principle of the ME-SOFS for multiaxis force detection coordinated actuation.

(A) Illustrations showing the detailed structures and principles of the sensing-actuation loop that directly translate applied force into a haptic system to provide feedback onto a human fingertip. Images of the 3D-printed ME-SOFS (B), signal-generating circuit (C), and haptic device (D) are given. (E) The ME-SOFS is settled on a rotatable base; a signal-generating circuit is put in a water tank and connected to the ME-SOFS with soft tubes for water to flow through. (F) The responses of the ME-SOFS when different forces were applied: (a) 1-axis force was applied, and the signal-generating circuit only generates voltage in the X− channel; (b) 2-axis force was applied, and voltages were generated in X− and Y+ channels; (c) multiaxis force was applied, and voltages were generated in X−, Y+, and Z channels (see movie S1).

The results presented in this study demonstrate a unique hydraulic approach for developing ME-SOFS that simplify the integrated robotic sensing-actuation loop for soft robots. ME-SOFS paves the way for enabling embodied intelligence with rapid sensing-actuation response in robotic systems and facilitates the development of closed-loop haptic systems.

RESULTS

ME-SOFS hydraulic approach

Designed as a purely mechanical structure for direct force contact and decoupling (Fig. 1, Aa and B), the ME-SOFS features a porous architecture comprising a central soft pillar connected to five adjacent enclosed fluid-filled chambers (Fig. 1Aa). Four horizontal chambers (blue) detect forces in the x and y directions, while the vertical chamber (red) senses z-axis loading (Fig. 1B). The porous structure stabilizes the pillar and mechanically decouples directional forces. Upon force application, the pillar tilts in the force direction, translating the applied force into chamber deformations and displacing water through soft tubing toward soft actuators at the ends.

Integrating a signal-generating circuit into the sensor-actuator network encodes external force stimuli into readable sequential voltage pulses (Fig. 1, Ab and C). Each fluidic channel, composed of a polytetrafluoroethylene (PTFE) soft tube embedded in polydimethylsiloxane (PDMS) substrate, with an internal cylindrical magnet, independently measures transduction from its corresponding deformed chambers. Low–melting point alloy 3D-printed metallic arcs were positioned above each tube (fig. S1A) (26). One end of the channel was connected to the ME-SOFS via a fluid tube, while the other either linked directly to actuators or opened to the external water environment, enabling free water flow (Fig. 1E). Detailed fabrication methods are provided in Materials and Methods and fig. S1.

External force on the ME-SOFS translates into magnet movements within the fluidic channels. As the magnetic flux passes through the metallic arcs, Faraday’s law of electromagnetic induction generates an electromotive pulse, producing a sinusoidal electromotive force signal (Fig. 1Ab). The number of signal peaks (n) corresponds to the arcs traversed (n−1) (fig. S2A), directly linking water displacement volume to force input on the ME-SOFS. Figure 1E provides an overview of using ME-SOFS to detect and decouple multiaxis forces (Fig. 1F and movie S1).

ME-SOFS sensing characterization

The sensing performance of the ME-SOFS is primarily governed by the volume of water displaced in response to applied force. To explore the influence of design parameters, we conducted finite element analysis (FEA; fig. S3 and Materials and Methods). Among various structural features, three parameters emerged as the most critical: hole diameter (Φ), which determines the overall flexibility of the structure (fig. S3B), and the thickness (t) and inclination angle (α) of the rotationally generated slope, which primarily affect vertical force sensitivity (fig. S3, C and D). By systematically tuning these parameters, the ME-SOFS can be customized for adjustable force sensitivities in both horizontal and vertical directions (Fig. 2A).

Fig. 2. Functional verification of the ME-SOFS.

(A) Color map of loaded ME-SOFS with different design parameters for adjustable force sensitivities in both horizontal and vertical directions. (B) Sensing multiaxis force with the ME-SOFS: (a) profiles of peak numbers of the circuit evoked signals with respect to the applied force induced by a 45° oblique press. The applied forces are components along the horizontal and vertical directions versus the horizontal (blue) and vertical (red) channels’ signals, respectively. Solid lines represent Finte Element Method results. An instance of 17.5-N press force is shown: (b) In the color map of the ME-SOFS’s deformation, red shadows show that both the horizontal chamber and the vertical chamber are squeezed under an oblique press. (c) Movements of the magnets in horizontal and vertical circuit channels, showing the number of alloy coils the magnets passed n_H = 24, n_V = 21, respectively. (d) The evoked signals of horizontal and vertical directions during the press with respect to time; the dashed line indicates the applied resultant force. Signals in both the loading stage and the releasing stage are given. (C) The multiaxis soft sensor is scalable: (a) images of the ME-SOFSs of different sizes, foam structures, and materials; (b) the measuring sensitivities of different sensors in (a). (D) Obtained different measuring resolutions by using signal-generating circuits with different channel sizes. An ME-SOFS of 0.5× size was used for the test. (E) Performance of the sensor in challenging environments: high temperature, and underwater pressure environment. A soft pneumatic actuator was used to press an ME-SOFS of 0.75× size for the tests. Each data point in (B), (D), and (E) is an average of three different measurements (N = 3) in the same situation; error bars indicate ±SD.

An ME-SOFS with parameters Φ = 5 mm, t = 1.5 mm, and α = 30° was fabricated, and its force sensing performance was evaluated under both single-axis and multiaxis loading conditions (figs. S5, S6A, and S7 and Materials and Methods). The number of electromagnetically induced signal peaks showed strong linear correlations with decoupled force components in both directions. This linearity remained robust across varying force angles and loading rates, closely aligning with simulation predictions. As an example, Fig. 2B presents ME-SOFS sensing response under a 45° applied force. The corresponding horizontal and vertical force components were recorded, and the associated number of signal peaks was plotted against each force component (Fig. 2Ba). Under an applied force of 17.5 N (horizontal: 10.7 N; vertical: 13.8 N), both horizontal and vertical chambers deformed (Fig. 2Bb), translating into magnet displacement through respective fluidic channels. The magnets in the channels corresponding to the horizontal and vertical directions passed through 24 (n_H) and 21 (n_V) arcs, respectively (Fig. 2Bc), generating the electromotive force signals shown in Fig. 2Bd.

Since the channel is open to the external water environment, the magnet returns back upon force release. This reverse movement generates symmetric sinusoidal electromotive force signals, enabling differentiation between loading and unloading phases (Fig. 2Bd, fig. S8, movie S2, and Materials and Methods). Due to the inherent viscoelasticity of soft materials, ME-SOFS exhibits hysteresis (fig. S9), limiting its effective bandwidth to a frequency below 2.5 Hz (fig. S10 and Materials and Methods)—sufficient for most soft robotic manipulation tasks and can be further improved by using a stiffer material (fig. S9C). Notably, because of the viscoelasticity, when pressing ME-SOFS, the maximum transient force could be higher than the steady-state force (fig. S9B) and increase with the pressing speed. While the signal peak number for a given press indentation, that is, same steady-state force but with various loading speed, are not much changed (fig. S11B), the signals’ amplitude and time interval vary with the loading speed (fig. S11A) and thus could be used to identify the maximum transient force (fig. S11C).

ME-SOFS exhibits excellent repeatability in cyclic tests (fig. S12). The number of signal peaks, inherently integer values, displayed strong linear relationship with applied forces in both axes. By linearly fitting the peak number-force relationship (R² > 0.9), we defined the slope as ME-SOFS’ sensitivities (peak/N). Over 92-fold variability in sensitivity was achieved across different designs (Fig. 2Cb). As shown in Fig. 2Ca, ME-SOFS variants with different sizes (0.5× to 1.25× the original design; fig. S13), porous architecture (spherical, cubic, and tetrahedral cell types; fig. S6, D to I) (27), and materials (fig. S14) were systematically evaluated. Sensitivity can be further enhanced by modifying the signal-generating circuit. For instance, the sensitivity of a 0.5× ME-SOFS increased by 5.7 times (to 5.72 peak/N) when using a thinner channel (0.6-mm inner diameter) and smaller magnets (0.5-mm diameter and length) (Fig. 2D and fig. S15).

The limit of detection (LOD) of ME-SOFS is remarkably low. Based on Faraday’s law of electromagnetic induction, even minor displacement of the magnet—without fully traversing an arc—can result in a changing magnetic flux and instantaneously evoke a signal in the circuit (fig. S2B). As a result, the ME-SOFS could detect the grasping of a 10-g LEGO brick in force estimation tasks such as lightweight object manipulation (fig. S16 and Materials and Methods).

The open-ended channel also renders ME-SOFS insensitive to ambient water pressure, enabling stable force detection in underwater environments. As shown in Fig. 2E, even with underwater pressure of 110 kPa in a pressure tank (fig. S17C), the number of signal peaks remained highly linear with pressure loaded by a pneumatic soft actuator (fig. S17A). The observed shift in the peak number-actuator air pressure scatter plot (~110 kPa) reflected the additional environmental pressure required to achieve equivalent deformation on ME-SOFS. Furthermore, ME-SOFS sensitivity remained stable under high-temperature conditions, with sensitivity invariant in 90°C water (Fig. 2E, fig. S17B, and Materials and Methods). These results highlight the robustness of ME-SOFS for use under challenging conditions involving elevated pressure and temperature.

Robotic integration

To broaden the design and fabrication flexibility of soft robotic system with somatosensory capabilities, we developed an approach of scaling down, rapidly reconfiguring ME-SOFS at arbitrary positions on the surface of a soft robot, as demonstrated in two application scenarios.

Mechanically sensorized 3D-printed soft glove

A soft glove with integrated ME-SOFS units (scaling down to 1 cm³) was designed and fabricated with sensors at all five fingertips. Soft and flexible force sensors at the fingertips enable direct measurement of contact force between the hand and external objects, offering great potential in prosthetic hands and human-machine interactions (28–30). Unlike prior achievements that mostly require manual assembly, long fabrication time, and inconsistent sensor integration, our sensorized glove was 3D printed in a single continuous process using one soft material (Fig. 3Aa). Each ME-SOFS was monolithically integrated at the fingertip with a customized design for multiaxis force detection (Fig. 3Ab). The glove features a completely soft skeleton, free of rigid electronics, providing great conformity for safe interaction.

Fig. 3. Mechanically sensorized glove.

(A) An electronic-free printed soft glove with ME-SOFS (1 cm³) integrated on all five fingers’ tips: (a) overview of wearing the mechanically sensorized glove to grasp an orange; (b) side view of the glove’s finger part showing the integration of the soft sensor. (B) Grasping different objects with the glove (see movie S3): (a) image of grasping a cup, (b) the normal and tangential signals of the sensors on the thumb and middle finger during one grasping trial and multiple grasping trials, (c) calibration of the force-peak number relationships of the two fingers in the normal and tangential directions, and (d) predicting the weight of the grasped objects with the average peak number of the tangential signals from the two fingers.

We evaluated the glove’s grasping and sensing performance by grasping objects of varying weights (Fig. 3Ba, fig. S18, movie S3, and Materials and Methods). Voltage signals on the thumb and middle fingers were recorded in both normal and tangential directions over multiple grasping trials. The signals exhibited high repeatability, with grasping and lifting phases clearly distinct within individual trials (Fig. 3Bb). Following the ME-SOFS characterization on both fingers (Fig. 3Bc), object weights were predicted based on the average number of peaks in the tangential signals (Fig. 3Bd). In addition, by sensing force applied to the fingertip during sliding, the glove could differentiate between surfaces with varying shapes (fig. S19 and movie S4).

ME-SOFS integrated soft arm

A soft robotic arm integrated with an ME-SOFS at its tip was developed to demonstrate force-guided interaction (fig. S20A). Enabled by multiaxis force sensing capability of the ME-SOFS, the arm could directly detect contact with its surroundings and adapt its movement accordingly. For example, it successfully explored and navigated through an unknown maze (fig. S20, B to D; movie S5; and Materials and Methods). As shown in fig. S20B, the soft arm was programmed to step into the maze and contact the walls. When the ME-SOFS fluid chamber opposite the soft arm’s moving direction generated a sensing signal with the least peak (fig. S20D), contact with the wall was detected. In response, the soft arm adjusted its navigation path (fig. S20C) and eventually exited the maze. Beyond navigation, the ME-SOFS equipped soft arm also demonstrated the ability to distinguish targets with varying material moduli (fig. S20E), showcasing its potential for adaptive interaction in complex and unstructured environments.

Mechanical sensing-actuation loop

Taking advantage of fluidic transduction and self-decoupling characteristics of ME-SOFS, we created an integrated sensing-actuation system capable of sensing multiaxis applied forces and directly translating them into decoupled actuation (Fig. 4, figs. S21 and S22, and Materials and Methods), without the need for additional computation or energy input. As illustrated in Fig. 4Aa, the ME-SOFS was connected in series with a signal-generating circuit and an actuation component at the end, forming a hermetically sealed system. Channels and connecting tubes are prefilled with fluid, while the actuation module was to be actuated by both water and air. Upon mechanical loading on ME-SOFS, the displaced fluid was routed through the signal-generating circuit and into the actuation component, inducing a pressure change within the channel that triggered actuation. Concurrently, as fluid passed through the signal-generating circuit, it displaced the magnets, generating a series of voltage signals that correspond to the magnitude and direction of the applied force (Fig. 4Ab and fig. S22).

Fig. 4. Creating sensing-actuation loop with the ME-SOFS.

(A) Sensing-actuation system that can sense multidirectional applied force and straightforwardly traverse the force into decoupled actuation: (a) overview of the sensing actuating system, the fluid-driven (water or air) actuating part is connected to an ME-SOFS via the signal-generating circuit, forming a hermetic system. (b) The system generating signals with different peak number under different applied horizontal force. Two actuating parts: (c) Soft microfluidic controller and (d) soft mini-actuator array were shown to be controlled by the sensing-actuating system. (B) The system sensing the recurrent applied force to actuate the soft microfluidic controller for manipulating droplets (see movie S6): (a) External force was applied on the ME-SOFS and (b) squeezed inside water through the signal-generating circuit to inflate corresponding chamber of the microfluidic controller. (c) Top row: Illustrations showing how the microchamber’s deformation pushes the droplet forward. Bottom row: Images of microchamber under relaxed and inflated states. (d) The microfluidic controller responding to sequential force applied on the ME-SOFS. (e) Images of three different droplets being manipulated forward step by step. (C) The system sensing the external force and responding in the bending motion of the soft mini-actuator array (see movie S7): (a) Images of applied force on the ME-SOFS and the corresponding bending responses of the mini-actuator array. (b) The sensing signals along the x and y axes generated by the ME-SOFS; yellow shadows show the time periods when force was applied on the corresponding directions, and multiaxes sensing and actuation occurred when the shadows overlapped. Inset imaging and arrows indicate the directions of external force applied on the ME-SOFS.

Here, to demonstrate the versatility of this sensing-actuation loop, we integrated two different actuators: a soft microfluidic controller (Fig. 4Ac) and a soft mini-cilia array (Fig. 4Ad), to showcase how ME-SOFS can serve as both the sensing and control hub in soft robotics systems, enabling responsive actuation without reliance on centralized control units. Last, we demonstrated the practical utility of the sensing-actuation loop by accelerating robotic grasping task learning through a fluidic haptic interface.

Microfluidic controller

Active manipulation of droplets with physical or logical information in microfluidics is of importance for analytical chemistry and biosciences, or building up logic circuits (31). Attempts have been made to manipulate droplets by magnetics (32), electrowetting (33), and light (34). Here, by connecting a soft microfluidic controller to the ME-SOFS controlled sensing-actuation loop, we enabled fully mechanical droplet manipulation guided by external applied force (Fig. 4B). As illustrated in Fig. 4Ba, when an external force was applied, the ME-SOFS detected the force (Fig. 4Bb) and displaced the fluid to inflate the corresponding chamber within the controller. The inflation-induced deformation pushed the underlying droplet forward, driving it toward a direction that minimized its surface energy (Fig. 4Bc). When force was applied in sequential manner (fig. S22C), the four paired channels inside the microfluidic controller responded to the sensing signals, allowing the droplet to be transported step by step (Fig. 4Bd). Owing to the ME-SOFS’ decoupling multiaxis sensing capability, three different droplets could be manipulated independently using a single ME-SOFS unit (Fig. 4Be and movie S6). This integrated droplet manipulation system is compact, versatile, and software-free, making it especially suitable for applications in biochemistry, health care study, and environmental monitoring where portability, simplicity, and safety are essential.

Mini-cilia array

Inspired by aquatic animals that use cilia-like structures to control surrounding flows for locomotion or predation (35), researchers have been developing artificial cilia arrays across micrometer to millimeter scales toward microfluidic applications in flow manipulation (36, 37). Here, we presented a 1 × 3 soft mini-cilia array, with each cilium integrated with four orthometric channels corresponding to the decoupled sensing directions of the ME-SOFS. The array responds by bending in both the direction and magnitude of the force applied to the ME-SOFS, enabling actuation in response to single axis, multiaxes, and dynamically changing forces (Fig. 4C and movie S7). This ME-SOFS–enabled sensing-actuation strategy allows for qualitatively matched mechanical responses to external stimuli, realizing a fully mechanical and intuitive control of the robotic system without the need for electronic computation.

Haptic human-machine interface

To demonstrate how the sensing-actuation loop could facilitate robotic tasks, we connected the ME-SOFS to actuate a soft finger haptic pad composed of five orthogonally oriented chambers. This configuration provided directional haptic feedback on the human finger, corresponding to the force vectors detected by the sensor. Using the haptic system alone, the operator can help accelerate the robot learning for grasping tasks (Fig. 5, fig. S23, and movie S8). A robotic arm equipped with ME-SOFS–integrated gripper was connected in series to the signal-generating circuit and haptic device via fluidic flows, creating a close-loop haptic human-machine interface. As the eyes-covered operator closed their finger, the robotic arm and gripper tracked and mirrored the movement via a motion capture system, bringing the ME-SOFS into contact with target objects (Fig. 5Aa). The sensory feedback of ME-SOFS was directly converted to inflation at the haptic end, providing the operator with intuitive awareness of the grasping state (Fig. 5Ab). Simultaneously, the generated sensing voltages during a successful grasping trial were recorded and used to guide the robot in autonomously replicating the grasping motion (Fig. 5B).

Fig. 5. Haptic based teaching of robotic grasping.

(A) The ME-SOFS was integrated into a soft gripper and mounted on a robotic arm for object grasping, an operator with eyes covered could use fingers to control the gripper’s grasping process: (a) A motion capture system tracked the movement of the operator’s fingers, enabling control of the robotic arm and the soft gripper’s grasping motion. (b) The ME-SOFS was connected to a signal-generating circuit and then to a soft finger haptic pad and created haptic feedback on the operator’s fingers. (B) Related signals during the teaching and repeating process of grasping an egg (a) and a bottle (b) (see movie S8).

DISCUSSION

As demonstrated by the results above, ME-SOFS introduces an original fluidic approach that directly connects multiaxis force sensing with actuation, enabling self-computation and reactive actions through an embodied sensing-actuation loop. In a sense, we introduce a mechanical analogy, for purely mechanical, electronics-free computation. We envision this integration to substantially reduce control complexity in robotic systems when responding to the physical interaction of the body with the environment, shedding light for embodied intelligence development in soft sensing and actuation system design (38–40). By implementing the ME-SOFS system with structure such as soft valve (41), the actuation force could be further increased for enhanced capabilities. Unlike state-of-the art multiaxis soft force sensors that rely on embedded electrical components (table S1) (13–21), ME-SOFS uses a fully mechanical, parameterized 3D-printed structure. This characteristic unlocks an exceptionally broad design space, enables on-demand customization of sensor sizes, measuring ranges and sensitivities, making ME-SOFS adaptable to diverse applications. Its split design and unique fluidic mechanism create an electronic-free sensing element and remote signal-generating circuit, thus offering distinct advantages for operation in challenging environments, such as high-temperature, underwater pressure conditions, and possible electromagnetic interference. Our findings demonstrate that ME-SOFS can be easily reconfigured and integrated into robotic systems, supporting rapid and scalable manufacturing. These capabilities can be further enhanced by ongoing advances in 3D printing technologies (42, 43), which will enable the creation of more sophisticated and miniaturized sensors. Notably, the visibly trackable movement of magnets in the signal-generating circuit allows the sensing method to be implemented using camera-based systems, further compensating for possible sensing noises and broadening application scopes in the field of soft robotics. Last, beyond the peak number, the generated signals of ME-SOFS contain far more information about the dynamic applied force, which is worth investigating for applications in intelligent sensing during dynamic soft interactions.

MATERIALS AND METHODS

Parametric design of the ME-SOFS

The ME-SOFS structure is primarily composed of two parts. The first part is a central soft pillar with diameter Φ_p (fig. S3Ai) and a hemispherical bottom. The second part is a soft, porous architecture, with five enclosed chambers, distributed on the four sides and the bottom (fig. S3Ab). The pillar was placed at the center of the porous architecture (fig. S3Ac). As shown in the cross-sections C₁-C₁ (fig. S3Ac) and C₂-C₂ (fig. S3Af) of the ME-SOFS, each chamber was separated from the inner porous architecture by a membrane of thickness t_m. Inward hemispherical knobs were integrated on the membranes (height h₁ from the top and projected red area $S_{\mathrm{k}}$ on the membrane; fig. S3Ae) to contact the pillar. The horizontal knob has an elliptical projection with a fixed x axis and variable y axis to induce change of $S_{\mathrm{k}}$, while the vertical knob featured a circular projection with different radius. When a force is applied to the pillar, it tilts and presses against the knobs and the membranes, displacing water out of the corresponding deformed chambers. The chamber volume is designed to be larger than the maximum displacement caused by the pillar, while the inner walls were designed to avoid contact with each other during force application. The pillar also contacts the bottom knob, positioned h_p above the bottom membrane. The porous architecture synergistically affects the sensing performance of ME-SOFS in both horizontal and vertical directions.

To expand the design space and adjust vertical measuring range and sensitivity, a rotational generated slope is designed to support the pillar (fig. S3Ac). The slope has a thickness of t and an inclination angle of α. The porous architecture itself has a length l, height h_c, and hole diameter Φ (fig. S3Ai), with a pitch d_h between adjacent holes. Areas around the knobs are free from porous architecture (marked by red dotted lines in fig. S3Af), ensuring that the pillar’s movement will only induce deformation of the chambers in the corresponding directions (Fig. 1B). While several design parameters affect sensing performance, we set most parameters to fixed values (Φ_p = 12 mm, h_c = 18 mm, l = 20 mm, d_h = 4 mm, h_p = 2 mm, h₁ = 5 mm, horizontal membrane: 16 mm–by–12 mm ellipse, vertical membrane: 16-mm diameter circle) and focused on five variable parameters (Φ, t, α, t_m, and S_k) to evaluate their impact (fig. S3).

As shown in fig. S3B, the hole diameter Φ influences both the horizontal and vertical sensitivity. A smaller Φ value (4.5 to 5.5 mm, with increments of 0.1 mm, t = 2.1 mm, α = 30°, t_m = 0.7 mm, $S_{\mathrm{k}}/S_{\mathrm{m}}$ = 4/12) increases the force required to displace the same water volume in either horizontal or vertical direction. The absence of porous architecture reduces force resistance of the ME-SOFS. Parameters t and α primarily tune the vertical sensitivity. With Φ = 5 mm, α = 30°, t_m = 0.7 mm, and $S_{\mathrm{k}}/S_{\mathrm{m}}$ = 4/12, increasing t (1.1 to 2.1 mm, with increments of 0.2 mm) raised the vertical force required without affecting horizontal sensitivity (fig. S3C). Similarly, when Φ = 5 mm, t = 2.1 mm, t_m = 0.7 mm, and $S_{\mathrm{k}}/S_{\mathrm{m}}$ = 4/12, increasing α (15° to 40°, with increments of 5°) made vertical compression more difficult but had minor horizontal effects (fig. S3D).

Thickness of the membrane t_m (0.5 to 1 mm, with increments of 0.1 mm, Φ = 5 mm, t = 2.1 mm, α = 30°, $S_{\mathrm{k}}/S_{\mathrm{m}}$ = 4/12) had negligible effect on sensing properties in both horizontal and vertical directions (fig. S3E). Given that membrane area $S_{\mathrm{m}}$ differs between orientations, we also investigated how the $S_{\mathrm{k}}/S_{\mathrm{m}}$ ratio influences response. Results shown in fig. S3F indicate that $S_{\mathrm{k}}/S_{\mathrm{m}}$ (2/12 to 7/12, with increments of 1/12, Φ = 5 mm, t = 2.1 mm, α = 30°, t_m = 0.7 mm) had minimal effect in both directions.

Therefore, by primarily adjusting the three parameters Φ, t, and α, ME-SOFSs with different mechanical properties can be designed, as shown in the FEA results in Fig. 2A. For example, ME-SOFSs with different horizontal force sensing performances were shown in Fig. 2 (Aa and Ab). Under Φ = 4.8 mm, t = 2.1 mm, and α = 30°, a 10.2 N horizontal force displaced 57 μl of water (Fig. 2Aa). Increasing Φ to 5.3 mm under the same conditions reduced the force to 7.0 N and increased displacement to 106 μl (Fig. 2Ab). Similar trends were observed in vertical force sensing performance. With Φ = 5 mm, t = 2.1 mm, and α = 30°, 19.8 N vertical force produced 61 μl of water displacement (Fig. 2Ac), while reducing t to1.3 mm decreased the required force to 19.2 N for 121 μl of water displaced (Fig. 2Ad).

In addition, we evaluated three different porous architectures with identical foam unit cells (4 mm by 4 mm by 4 mm) (fig. S6). We chose spherical foam cell in this work because it simplifies sensor parameterization process and improves 3D printability, requiring minimal support structures compared to other metamaterials (few suspended porous architecture, so minimal printed supports).

Finite element simulation

To explore the design space of the ME-SOFS, static nonlinear finite element simulations were performed using Abaqus/Standard (SIMULIA, Providence, RI). The models used Neo-Hookean hyperelastic material behavior with 10-node quadratic tetrahedral elements (Abaqus element type C3D10H). To characterize the mechanical response of the materials used in ME-SOFS fabrication, we conducted uniaxial tensile tests on dog bone–shaped samples (American Society for Testing and Materials standard) made from four different materials using a single-axis Instron testing machine (model 5566, Instron, Inc.). The stress-strain behaviors are reported in fig. S14A. Considering that the ME-SOFS typically experiences strain below 30% under applied force, only the material behaviors within this range were considered. The four materials were set with Young’s moduli of 6.3 MPa (Flexible 80A), 1.3 MPa (Silicone 40A), 3 MPa (Elastic 50A), and 19 MPa (IBT) respectively, and all were modeled with Poisson’s ratio of 0.5.

To measure the volume of water displaced from the chamber under applied force, we filled the chambers with an extremely soft and compressible solid material, with a Young’s modulus of 10⁻⁶ Pa and a Poisson’s ratio of 0. A square plate, acting as a rigid shell, was used to contact and apply force to the pillar. Two sets of boundary conditions were applied: All four sides and bottom of the ME-SOFS were fixed, while the rigid shell was displaced in horizontal, vertical, or oblique directions (fig. S6Ba for horizontal, fig. S6Ca for vertical, and fig. S7 for oblique pressing). The applied forces on the rigid shell and the volume change of the compressible solid were analyzed to determine the relationship between the displaced water volume by the compressed ME-SOFSs with various design parameters.

Fabrication of the ME-SOFS prototypes

To fabricate the ME-SOFS, we used stereolithography (SLA) 3D printing using Form 3B (Formlabs, Inc.). Flexible 80A resin was used to fabricate the sensor with different foam structures and dimensions, as well as the mechanically sensorized 3D-printed glove. ME-SOFS was also fabricated with alternative materials, Silicone 40A (Material 2, 1.3 MPa), Elastic 50A (Material 3, 3 MPa), and IBT (Material 4, 19 MPa), and compared with Flexible 80A (Material 1, 6.3 MPa).

The signal-generating circuit was fabricated via 3D printing using a low–melting point alloy, Field’s metal, following the process described in (26). The printed alloy circuit has a width of 1.6 mm and a total length of 80 mm, with 32 semicircular arcs spaced 2.5 mm apart (fig. S1A). After printing, the alloys were integrated into the circuit through molding and casting steps (fig. S1B). First, a soft PDMS substrate (10:1 Sylgard 184) with molded channel grooves was prepared (fig. S1, Ba and Bb). PTFE soft tubes (1.6-mm outer diameter and 1.2-mm inner diameter) were embedded into these grooves, and an N35 neodymium magnet (1-mm outer diameter and 1.5-mm length) was inserted into each tube (fig. S1Bc). The low–melting point alloy arc, prewelded with thin copper wires (80 μm in diameter) at two ends for signal output, were placed above the tubes (fig. S1Bd). The assembly was then encapsulated with another layer of PDMS (fig. S1Be). Following curing and demolding, excess tubes were trimmed, and short copper rods (0.3 mm outer diameter) were vertically inserted at both ends of each channel to prevent magnet displacement outside the channel by the fluid flow (fig. S1Bf). The fabricated signal-generating circuit was then submerged in an open water environment and connected to the ME-SOFS using soft tubes (Fig. 1E).

ME-SOFS evaluation

Sensing signal-force characterization

As shown in fig. S5A, we characterized the response of ME-SOFSs to applied forces along different directions by using a vertical linear stage (Newmark) to indent the sensor. A force gauge (Tekscan) was mounted on the indenter of linear stage, which contacted the ME-SOFS via a 3D-printed clamp. Instead of altering the direction of the applied force, we maintained the pressing direction vertically and adjusted the orientation of the ME-SOFS. To precisely control the orientation, we mounted the ME-SOFS on a 3D-printed rotatable base capable of rotating in two directions. During testing, we applied a series of displacements (from 0 mm until the maximum range of ME-SOFS, with increments of 0.1 mm) to the central pillar of the ME-SOFS and measured the applied force using the force gauge. For single-axis evaluation, the applied force recorded by the force gauge was directly used for characterization. For multiaxis evaluation, the recorded force was decomposed into the ME-SOFS’s horizontal and vertical components according to the rotated angle of the base and then characterized with the ME-SOFS sensing signal along the horizontal and vertical directions, respectively. According to the schematic diagram shown in fig. S5B, the signal conditioning circuit comprises a power module, two ultralow-noise amplifiers, and a low-pass filter. A source meter (Keithley 2450) provides a ±6 V power supply, which powers two operational amplifiers (Analog Devices) to amplify the voltage signals at the two ends of the signal-generating circuit. A low-pass filter was implemented to attenuate high-frequency noise in the amplified voltage. The conditioned signals were then recorded using an oscilloscope (PicoScope 3406D, Pico Technology). The peaks of recorded sensing signals was recognized in Matlab with “Findpeaks” function featured with a threshold of 10% maximum amplitude (fig. S5C). Each test was repeated three times (N = 3), and the applied force and corresponding signals were recorded. The average applied force and peak number were calculated, and the standard deviation (SD) of the peak number was used as the error bar. Because the peak numbers are downward rounding integral as eq. S9 indicates (Supplementary Text), we consider the sensing accuracy as 1/n_max (n_max represents the maximum arc number of the signal-generating circuit); for a circuit with smaller channel, the accuracy should be 2.5%. After characterization, the peak number-force relationships were linearly fitted (with coefficient of determination R² > 0.9). Linear fitted lines of both horizontal and vertical sensing across all evaluated sensors are shown in figs. S7, S13, and S14. The slopes of the fitted lines are regarded as sensor sensitivities, in units of peak/N.

Differentiating the loading and unloading processes

As shown in Fig. 2Bd, the last peak of the forward signal was centrally symmetric to the first peak of the backward signal, as shown by both the horizontal (blue arrow) and vertical (red arrow) signals. This characteristic can be leveraged to detect transition between loading and unloading processes. Specifically, in fig. S8A, we demonstrated the detection of a “loading-loading” sequence (movie S2). The sensor was first vertically indented by 1.6 mm, followed by a brief pause of a few seconds, and then further indented to 2.2 mm (fig. S8Ai). This motion caused the magnet to pass through 17 alloy arcs initially, followed by an additional nine arcs (fig. S8Ab). The corresponding sensing signals of the two loading events are shown in fig. S8Ac, where the last peak of the first indentation was axially symmetric to the first peak of the second indentation. In contrast, a “loading-unloading” process produced different signal patterns (fig. S8B and movie S2). As demonstrated in fig. S8Ba, the sensor was first indented vertically by 1.6 mm and then released to 1 mm. The magnet first traversed 17 alloy arcs and then reversed backward through seven arcs (fig. S8Bb). The first peak of the unloading signal was centrally symmetric to the last peak of the loading signal (fig. S8Bc). Furthermore, we observed that at the moments at which the magnet changed direction, its velocity reached zero, indicated by the signal peak attenuation until reaching minimum voltage (see green arrows in fig. S8, Ac and Bc). This feature enables clear distinction of the loading and unloading transitions in ME-SOFS operation.

Hysteresis and creep performance of the ME-SOFS

The hysteresis curves of the ME-SOFS for both horizontal direction and vertical direction are shown in fig. S9A. For each direction, three cycles of loading and unloading were performed: 0 ∼ 10 N for horizontal direction and 0 ∼ 18 N for vertical direction. Hysteresis was observed in both cases under relatively large external force subjected to ME-SOFS. Notably, vertical direction exhibited less hysteresis than the horizontal direction. This difference may be attributed to the support provided by the rotationally generated slope, suggesting a potential strategy for reducing hysteresis through structural optimization in future designs. In addition to hysteresis, creep was also observed during loading process, as shown in fig. S9B. To ensure accurate sensor characterization, the system was held at target displacement for at least 1 s after loading to allow for force and corresponding sensor signal to stabilize. Similar hysteresis evaluation was conducted on an ME-SOFS made of stiffer material (IBT, Young’s Module of 19 MPa); results in fig. S9C show that when using stiffer material, the hysteresis of the sensor is largely reduced. While using stiffer material effectively relieved the problem of hysteresis, the sensitivity is largely reduced (Fig. 2Cb) and, consequently, larger measurement range and LOD (fig. S14B). Such trade-off should be considered for different tasks.

Bandwidth test of the ME-SOFS

A 1× size ME-SOFS was tested under vertical loading and unloading cycles with frequencies ranging from 0.5 to 2.5 Hz, with a fixed displacement of 1 mm (corresponding to a static force of ~5.5 N). The results are shown in fig. S10. As the loading frequency increased, dynamic force also increased. However, the peak number in the sensor signals remained unchanged because it mainly depends on the force deformation on the sensor. It can be seen that for loading frequencies from 0.5 to 1.8 Hz, the sensor signals mainly feature eight peaks; when the loading frequency increase to 2.5 Hz, the sensor still produced distinguishable responses during both loading and unloading phases, but the peak number decreased to seven. This reduction is attributed to the limited relaxation time at higher frequencies; the flexible ME-SOFS material requires time to return to its original shape during unloading, which becomes insufficient at 2.5 Hz. The bandwidth of ME-SOFS can be enhanced by using stiffer materials that allow for faster elastic recovery.

Repeatability test of the ME-SOFS

ME-SOFS demonstrated excellent repeatability over extended cycles, as shown in fig. S12A. After 10,000 vertical loading cycles with a displacement of 1.3 mm (corresponding to static force of ~6.7 N), the sensor response consistently exhibited 10 signal peaks. In addition, repeatability tests across three different ME-SOFS were conducted (fig. S12B); the slopes of the linear calibration curves of the three ME-SOFS are 3.21, 3.83, and 3.79 peak/N (coefficient of variation: 9.6%) in horizontal direction and 2.07, 2.28, and 2.24 peak/N (coefficient of variation: 5.1%) in vertical direction. These results confirm the robustness and reproducibility of ME-SOFS across multiple devices. For all this, the performance of ME-SOFS depends largely on the material properties, although the ME-SOFS tested in fig. S12A was fabricated 10 months ago and did not show significant change in performance. Recalibration is needed if material fatigue has resulted from much longer time or environmental damage.

Challenging environment testing

A 0.75× size ME-SOFS was tested in high temperature and underwater pressure environments. We used a pneumatic soft actuator that can function reliably in both tested environments to indent on ME-SOFS. The pneumatic soft actuator is made of Dragon Skin 20 and reinforced by Kevlar fibers (44) and can elongate to indent on ME-SOFS vertically when actuated with different air pressure (fig. S17A). For high-temperature testing, the assembled soft actuator and ME-SOFS were immersed in hot water (around 90°C) (fig. S17B), the soft actuator was inflated with air pressure ranging from 0 to 200 kPa, ME-SOFS vertical sensing signals were recorded, and signal peak numbers were counted and plotted to assess sensor performance. For underwater pressure testing, the integrated ME-SOFS, soft actuator, and signal-generating circuit were submerged in a water-filled pressure tank. After sealing, the tank was pressurized to simulate a depth of 11 m underwater (equivalent to 110 kPa gauge pressure) (fig. S17C). The soft actuator was inflated with air pressure ranging from 0 to 300 kPa to vertically indent on ME-SOFS. The resulting sensor signals were recorded and compared to baseline measurements taken in ambient air.

Estimating weight of objects with ME-SOFS

To show how the signals of ME-SOFS could be calibrated for force estimation, we used the ME-SOFS to recognize the weights of grasped objects. Two 0.75× size ME-SOFSs were mounted on a rigid gripper (Robotiq 2F-140 Adaptive Gripper) that is attached to a robotic arm (UR10e), as shown in fig. S16A. ME-SOFS were used to grasp and lift a range of standard weights, and peak numbers in z (grasping) and x (lifting) directions were recorded, with x-direction peak numbers correlating to force applied during lifting. By lifting different standard weights, the applied force in the x direction can be estimated by the peak number. The actual arc traveled was captured via camera, providing a reference to ME-SOFS peak numbers.

The gripper, initially set to an open position (140 mm), was operated at 50% speed and 0% force for all subsequent commands. The robot arm was moved to a starting position above the object holder, where the gripper closed to a 130-mm position to grasp the object, holding it for 2 s. The arm then lifted the object at a constant speed before halting and releasing the weight by returning the gripper to the 140-mm open position. Signal peaks during both grasping and lifting motions were recorded in the Z and X channels of the ME-SOFS sensors.

Thirteen different weights ranging from 10 to 530 g were lifted, with the corresponding sensor peak numbers and traveled arcs recorded. The peak numbers can be characterized and linearly correlated to standard weights with R² = 0.972, thus allowing for the estimation of the applied force in the x direction. The dotted line shows the linear regression model fitted to predict peak number in relation to lifted weight (fig. S16B).

The model was further validated using seven different objects (fig. S16C) to test the predicted peak number-weight correlation, where the predicted weights, derived from the peak number-force correlation, were compared against the actual weights. Results show that for the seven objects, predicted weights did not show large deviations from the actual weights (fig. S16D).

Distinguishing between different surfaces

By detecting the force applied to the fingertip, the mechanically sensorized glove can distinguish between different surface profiles during sliding. As shown in fig. S19A, a control test was conducted by pressing and then sliding the index finger (wearing the glove) across a flat surface. During this process, the normal and tangential signals were recorded. The normal signal primarily increased during the pressing stage, resulting in a rapidly evoked signal (gray area), while the tangential signal was mainly generated during the sliding phase (pink area). For the surface with raised structures (height difference h = 2 mm) (fig. S19B), a similar pattern was observed during the initial contact with the flat portion; normal signal mainly increased in the gray area and tangential signal in the pink area. When the sensor reached the relief portion of the surface profile (green area), both the normal and tangential signals increased. Subsequently, the sensor fell from the peak of the raised structure (blue arrow) and immediately contacted the next raised structure (red arrow), forming a rapid, distinct “unloading-loading” process. This transition produced a signal featuring central symmetry between the unloading (downhill, blue area) and loading (uphill, red area) stages, with the lowest signal amplitude occurring at the switching point (red dashed line). While increasing the surface height difference to h = 4 mm and sliding on the undulating surface (fig. S19C), the sensor’s response during the downhill stage was observed to be stronger than during the uphill stage. This asymmetry, attributed to the increased surface height, contrasts with the more symmetric signals observed at h = 2 mm depicted in fig. S19B. The detailed sliding interactions on the three surfaces and the corresponding magnet movements are shown in movie S4.

Distinguish targets with varying material moduli

Sensitivity along the z direction could be applied to distinguish targets with varying material moduli. We controlled the soft arm to press the targets vertically by fixed displacement and recorded the resulting ME-SOFS signals (fig. S20Ea). While the applied forces varied significantly for different material’s modulus (fig. S20Eb), the peak number-force relationships for the sensor signals maintained similar trends but different respective ranges (fig. S20Ec). Signals from indenting materials with Young’s moduli of 55 kPa (fig. S20Ed) and 662 kPa (fig. S20Ee) over the same displacement (d = 4 mm) exhibited markedly different signal features including peak number, amplitudes, and peak-to-peak time duration. These results suggest that ME-SOFS may offer a promising approach for detecting material modulus.

Fabrication of the actuation parts

Soft microfluidic controller

The soft microfluidic controller was fabricated according to steps illustrated in fig. S21A. First, individual layers corresponding to different functional components including inflating, membrane, and fluidic channel were molded and casted separately (fig. S21Aa); after molding, all the functional components were treated with Oxygen Plasma (Tergeo-Plus, PIE Scientific LLC, USA) and bonded together. Soft tubes were then inserted (fig. S21Ab) to appropriate ports. After that, oil (Novec 7500) and three water droplets (deionized water dyed with red, green, and blue food coloring) were injected into the fluidic channel (fig. S21Ac). Upon completion of fluid injection, the soft microfluidic controller was fully assembled and ready for actuation (fig. S21Ad).

Soft mini-cilia array

As fig. S21Ba shows, the soft mini-cilia array consists of two main parts: three cilia-like soft actuators and a 3D-printed base. The cilia-like soft actuators were fabricated via molding and casting steps (fig. S21Bb). To fabricate the actuator of millimeter scale, we used high precision 3D printing with Micro 2 system (MicroSLA, Inc., USA) to create detailed molds. Needles of 23-gauge (diameter of 0.573 mm) were used as cores to form the internal inflating chambers. After assembling the molds, a mixture of Ecoflex 00-10 elastomer was injected into the cavities. Once cured, the cilia-like soft actuator was demolded, and one end was sealed with silicone adhesive (Sil-Poxy, Smooth-on Inc.). Each actuator featured four orthometric channels, as shown in the cross-section view. The three fabricated cilia-like soft actuators were then mounted onto a 3D-printed base. Figure S21Bc shows images of the 3D-printed base from multiple perspectives. The 3D-printed base served as a connector to integrate individual channels of the actuators into a unified pneumatic system for simultaneous actuation.

Evaluation setup of the mechanical sensing-actuation loop

As fig. S22A shows, the mechanical sensing-actuation loop integrates the ME-SOFS with the actuation component. Unlike previous ME-SOFS setups that were open to atmosphere, this configuration is fully enclosed by the actuation module, altering force-signal relationship. In this setup, the ME-SOFS was 3D printed using Elastic 50A material. To evaluate the applied force-signal response of the system, we mounted a Nano 17 Force/Torque (F/T) sensor (ATI Inc., USA) above the ME-SOFS. The F/T sensor was constrained to move along a predefined orthometric path. We moved the F/T sensor to indent the ME-SOFS along Y+ direction and recorded the signal of both the F/T sensor and ME-SOFS. The results show that ME-SOFS still generated different sensing signals corresponding to varying applied force (fig. S22B).

In addition, we calibrated the haptic force that was created in Fig. 5Ab). As fig. S24A illustrates, we applied increasing vertical force on the ME-SOFS used in Fig. 5 and measured the haptic force generated by the soft finger haptic pad. Results show that a minimum haptic force of 0.005 N [higher than the threshold value of haptic generating on human fingertips; (45)] was generated under an applied force of 0.4 N, and a haptic force of 0.3 N was generated under an applied force of 12.7 N. The latencies between the applied force and the haptic force when they are connected with a 3-m (fig. S24B) and a 15-m fluidic tube (fig. S24C) are 30 and 160 ms, respectively, representing an acceptable level for haptic-based teleoperation (46–48).

Supplementary Materials

The PDF file includes:

Supplementary Text

Figs. S1 to S24

Table S1

Legends for movies S1 to S8

References

Other Supplementary Material for this manuscript includes the following:

Movies S1 to S8