Computational design of ultra-robust strain sensors for soft robot perception and autonomy

Soft robots undergo large, high-DOF deformations and dynamic actuation, demanding compliant strain sensors that can provide reliable, real-time body state estimation and environmental feedback for autonomous navigation. Existing strain sensors are difficult to design predictively due to diverse sensing requirements (e.g., tunable sensitivity and linear range) and reliance on iterative, empirical prototyping. Further, real-world operation involves noisy, intermittent, and frequency-varying motions that often degrade sensor signals and stability. This work addresses these challenges with a computationally guided design of piezoresistive strain sensors that incorporate user-programmed interdigital crack arrays within micro-crumpled SWNT films. The approach enables predictive tuning of sensitivity and working window, accurate physics-based modelling of sensing behavior, and exceptional robustness to noise, intermittent cycling, and dynamic frequencies. Integrated with soft robots from macro to micro scales, these sensors support machine-learning-enabled trajectory prediction and terrain altitude awareness for autonomous navigation.

Prior efforts to customize strain sensor characteristics (e.g., sensitivity, linear window) have generally relied on distinct design rules and multiple trial-and-error iterations, hampering rapid, predictive development. Alternative paths using mathematical/statistical models or physical simulations struggle because conventional soft strain sensors exhibit unpredictable material/structural evolution under deformation, limiting model fidelity. Stability has also been underexplored: most studies evaluate sensors under monotonic, repetitive, controlled conditions that do not reflect noisy, intermittent, and variable-frequency environments typical of soft robot operation. Under such conditions, common strain sensors experience material/structure failures and signal distortions, undermining closed-loop control and autonomy. Review literature highlights the need for robust sensors tolerant to multiaxial disturbances, intermittent loading, and evolving working speeds (0.1–20 Hz).

Sensor design and fabrication: A two-stage, computationally guided design produces PCAM (programmed cracks array within micro-crumples) strain sensors. Stage 1 uses computer-programmed laser processing (0.08 mW) to etch user-defined interdigital crack arrays into a ~600 nm SWNT layer deposited on PS shrink film, leaving the PS intact; crack width ~20 µm. The crack density ρ is defined as the cumulative crack length divided by SWNT area. Stage 2 thermally shrinks the PS above Tg (~100 °C); at 140 °C the SWNT layer undergoes biaxial compression forming isotropic micro-crumples that overlay the programmed cracks. After shrinkage, a 2 mm Ecoflex (00-35) overcoat is applied and the PS substrate is dissolved, yielding the Ecoflex-coated SWNT PCAM sensor. Fabrication parameters ρ (via patterning) and φ (shrinkage ratio, φ=(D0−Dafter)/D0, 0–55% tuned by heating duration) program sensor sensitivity and linear working window. Planar and crumpled control sensors were also fabricated for comparison. Characterization: SEM and in situ SEM observed structural evolution under strain; tensile testing measured stress–strain and extracted Young’s modulus of SWNT layers for different φ. General sensing tests included uniaxial strains, cyclic bending and twisting, and hysteresis quantification (difference between stretching and releasing signals at given strain). Mechanical robustness was assessed under: (i) noisy, interrupted deformations (sequences of stretching, twisting, bending), (ii) intermittent cyclic loading (20 rounds × 5000 cycles at 85% strain, 1 h rest between rounds; total 100,000 cycles), and (iii) varying operation frequencies (static 0 Hz for >30 min; dynamic 1–10 Hz and up to 23 Hz). Signal deviation was defined as ((δi−δ1)/A), where δi is the relative resistance change of cycle i, δ1 that of the first cycle, and A the first-cycle amplitude. FEA modelling: 3D COMSOL multiphysics models of the bilayer (SWNT/Ecoflex) structure were constructed with input parameters ρ and φ (affecting crack number and SWNT Young’s modulus). Dual-field simulations (mechanical + electrical) applied uniaxial stretching by moving one boundary while fixing the other, and a constant current to extract surface potential and compute resistance changes versus strain. Contact used penalty formulation; cohesive zone used displacement-based damage; mesh minimal element size 150×150×100 μm. Electrical equations included ∇·J=0, J=σE, ∇×E=0, and −∇V=E. Models reproduced sensing curves for various ρ and φ; unloading was not modelled due to nanoscale, non-steady elastic relaxation in the polymer substrate. Robot integration and ML: PCAM sensors were integrated on (i) a 15 cm magnetically actuated origami crawler (head/tail Nd-Fe-B magnets; automated actuation platform) for multimodal locomotion; (ii) a soft pneumatic crawler (dual elastomeric tubes with friction feet and automated pumping); and (iii) a tetrapod microrobot (~700 μm) built from shrunk interdigital SWNT patterns on Nd-Fe-B–Ecoflex. For trajectory prediction, an ANN with 4 fully connected layers (ReLU, BatchNorm, MSE loss, 10-fold CV) was trained on multi-channeled sensor signals plus actuation instructions (crawling and turning directions) as inputs and time-series (x,y) locations (from camera and Tracker software) as outputs. Training used 38 datasets; 5 unseen datasets tested performance. For terrain altitude prediction, a similar ANN used on-body sensor data (with robot trajectory context) to predict hill altitudes along the path; 30 training datasets and 2 unseen test datasets were collected. Data smoothing used adjacent-averaging with window m=100.

- Programmable sensing characteristics: - Increasing crack density ρ from 300 to 600 to 1200 μm mm−2 increased gauge factor (GF) from 2.3 to 8.4 to 35 and slightly expanded linear working window from 17–70% to 15–70% to 10–70%. - By tuning shrinkage ratio φ (25%, 40%, 55%), sensors exhibited shifting linear windows from 5–50% to 10–70% to 20–120%, with representative GFs up to 42 (φ=25%) and 20 (φ=55%). - Further GF improvement (>200) achievable by controlling laser etching depth of the SWNT layer (supplementary data). - Accurate physical modelling: - FEA dual-field models (mechanical + electrical) reproduced experimental sensing curves across ρ and φ variations; higher ρ yielded larger surface potential drops at fixed strain, explaining higher GF. - Measured SWNT-layer Young’s modulus decreased from 33.8 MPa (φ=25%) to 5.5 MPa (φ=55%), consistent with larger micro-crumples storing elastic energy and facilitating deformation; models explained working-window shifts with φ via lower stress concentration and slower crack propagation. - Robust mechanical performance: - Tolerated noisy, interrupted multiaxial deformations (stretch–twist–bend sequences) with stable signals; planar controls exhibited large fluctuations. Noise tolerance ~50% strain. - Withstood 100,000 intermittent cycles at 85% strain with stable baselines/peaks; signal deviation ~6% at 25,000 cycles versus ~40% (planar) and ~120% (crumpled without programmed cracks). - Stable operation across frequencies: steady at 0 Hz >30 min; consistent 1–10 Hz (minor 10 Hz shift from substrate hysteresis); functional up to 23 Hz. - Hysteresis (stretch vs release) increased with strain: 0.51, 1.75, 1.91, 2.90, 3.54, 3.63 for 20–70% strain (Ecoflex contribution). Sandwiched structures can reduce hysteresis (supplementary). - Cross-scale robot integration: - Origami robot provided reliable on-body sensing for multimodal motion, obstacle detection, and surface roughness discrimination; pneumatic robot maintained sensing over long trajectories with ~6% signal deviation across repeated runs; microrobot (~700 μm) showed reversible transformations with on-board sensing under magnetic actuation. - Machine intelligence: - ANN trajectory prediction achieved <4% relative error and <3 cm absolute error across 5 unseen test datasets, vastly outperforming an actuation-only benchmark (≈10× higher errors). - ANN terrain altitude prediction achieved mean relative errors of 9% and 10% and MAE of 0.2 mm and 0.3 mm on two unseen test trajectories; predicted altitude profiles closely matched ground truth.

The study addresses the dual challenge of predictive design and robust operation of soft strain sensors. By prescribing sensor microstructures—interdigital crack density and micro-crumple morphology—the authors create a tunable platform whose sensitivity and linear working window can be preselected. Importantly, these structural parameters enable accurate forward modelling of sensing curves using coupled mechanical–electrical FEA, bridging a long-standing gap between empirical sensor development and physics-based prediction. Mechanistically, higher crack density increases potential drop and relative resistance change under strain, boosting GF, while increased shrinkage ratio introduces larger micro-crumples, lowering effective modulus, reducing stress concentrations, and expanding the working window. The deterministic crack propagation within micro-crumpled films underpins the sensor’s exceptional robustness to noisy, intermittent, and dynamic deformations typical of soft robots, preserving consistent signals necessary for closed-loop control and learning. Integrated with soft robots across scales, the sensors provide high-quality proprioceptive data that, when fused with actuation inputs, enable accurate, data-efficient ML models for trajectory tracking and terrain awareness. Collectively, the results demonstrate that computationally designed, physically modelled, and mechanically resilient strain sensors can substantially advance soft robot perception and autonomy.

This work introduces PCAM strain sensors whose crack density and shrinkage-controlled micro-crumples enable programmable sensitivity and linear range, accurate FEA-based performance prediction, and ultra-robust operation under noisy, intermittent, and high-frequency conditions. The sensors integrate seamlessly with soft robots from macro to micro scales and support machine-learning pipelines for precise trajectory prediction and real-time terrain altitude awareness. These contributions establish a unified, customizable sensor platform that reduces empirical iteration and enhances reliability for autonomous soft systems. Future research should: (i) extend modelling to capture unloading and viscoelastic relaxation, (ii) further reduce hysteresis via structural and material optimization, (iii) generalize the design to broader material systems and multimodal sensing, and (iv) scale to multi-robot networks and more advanced ML for cooperative, swarm-level autonomy in complex environments.

- The FEA modelling framework accurately predicts sensing during loading but does not capture unloading due to nanoscale, non-steady elastic relaxation in the polymer substrate. - Sensor hysteresis increases with strain (attributed to Ecoflex), potentially shifting signals at higher frequencies; a sandwiched architecture can mitigate but not fully eliminate hysteresis. - Robustness to mechanical noise demonstrated up to ~50% strain; behavior beyond this threshold was not validated. - Modelling accuracy depends on appropriate parameterization (e.g., crack contact stickiness, SWNT–Ecoflex interface, SWNT modulus), which may require calibration per device/material batch.