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Magnetic crack-based piezoinductive mechanical sensors: way to extreme robustness and ultra-sensitivity
基于磁性裂纹的压电感应机械传感器：实现极致稳健性和超高灵敏度的方法

Received: 20 January 2025
收到日期：2025年1月20日
Accepted: 1 July 2025
接受日期：2025年7月1日
Published online: 10 July 2025
在线出版日期：2025年7月10日

Abstract  抽象的

Yulian Peng 1 , 3 1 , 3 ^(1,3){ }^{1,3}, Zhengyan Wang 1 , 3 1 , 3 ^(1,3){ }^{1,3}, Houping Wu 1 , 3 1 , 3 ^(1,3){ }^{1,3}, Junchen Luo 2 2 ^(2){ }^{2}, Xinxin Chang 1 1 ^(1){ }^{1}, Yufeng Wang 1 1 ^(1){ }^{1}, Shiwu Zhang ( 1 ) ( 1 ) ^((1)){ }^{(1)}, Zhihua Feng 1 1 ^(1){ }^{1}, Unyong Jeong ( 1 ) 2 ⊠ & ( 1 ) 2 ⊠ & ^((1))^(2)⊠&{ }^{(1)}{ }^{2} \boxtimes \& Hongbo Wang ( 1 ) 1 ⊠ ( 1 ) 1 ⊠ ^((1))^(1)⊠{ }^{(1)}{ }^{1} \boxtimes
彭玉莲 1 , 3 1 , 3 ^(1,3){ }^{1,3} 、王正艳 1 , 3 1 , 3 ^(1,3){ }^{1,3} 、吴厚平 1 , 3 1 , 3 ^(1,3){ }^{1,3} 、罗俊辰 2 2 ^(2){ }^{2} 、常欣欣 1 1 ^(1){ }^{1} 、王玉峰 1 1 ^(1){ }^{1} 、张世武 ( 1 ) ( 1 ) ^((1)){ }^{(1)} 、冯志华 1 1 ^(1){ }^{1} 、郑云勇 ( 1 ) 2 ⊠ & ( 1 ) 2 ⊠ & ^((1))^(2)⊠&{ }^{(1)}{ }^{2} \boxtimes \& 王洪波 ( 1 ) 1 ⊠ ( 1 ) 1 ⊠ ^((1))^(1)⊠{ }^{(1)}{ }^{1} \boxtimes

Soft mechanical sensors with high performance, mechanical robustness, and manufacturing reproducibility are crucial for robotics perception, but simultaneously satisfying these criteria is rarely achieved. Here, we suggest a magnetic crack-based piezoinductive sensor (MC-PIS) which exploits the strain modulation of magnetic flux in cracked ferrite films. The MC-PIS is insensitive to fatigue-induced crack propagation and environmental changes, showing same performance even when scratched in half or run over by a car. It can detect bidirectional bending with a precision of 0.01 ∘ 0.01 ∘ 0.01^(@)0.01^{\circ} from − 200 ∘ − 200 ∘ -200^(@)-200^{\circ} to 327 ∘ 327 ∘ 327^(@)327^{\circ}, allowing for real-time reconstruction of dynamic shape changes of a flexible ribbon. We demonstrate an artificial finger recognizing surface topology and musical notes via vibrations, a crawling robot responding appropriately to external stimuli, a tree-planting gripper performing consecutive tasks from digging soil, removing stones, to placing trees. The MC-PIS opens a new paradigm to develop ultrasensitive yet highly robust sensors in real-world robotics applications.
高性能、高机械强度和高制造可重复性的软机械传感器对于机器人感知至关重要，但同时满足这些标准的情况却很少见。本文，我们提出了一种基于磁裂纹的压电电感传感器 (MC-PIS)，它利用了裂纹铁氧体薄膜中磁通量的应变调制。MC-PIS 对疲劳引起的裂纹扩展和环境变化不敏感，即使被划伤一半或被汽车碾压也能保持同样的性能。它可以检测从 − 200 ∘ − 200 ∘ -200^(@)-200^{\circ} 到 327 ∘ 327 ∘ 327^(@)327^{\circ} 的 0.01 ∘ 0.01 ∘ 0.01^(@)0.01^{\circ} 精度的双向弯曲，从而可以实时重建柔性带的动态形状变化。我们展示了一个通过振动识别表面拓扑和音符的人造手指、一个对外部刺激做出适当响应的爬行机器人，以及一个可以执行从挖土、移除石头到种植树木的连续任务的植树夹持器。MC-PIS 为在现实世界的机器人应用中开发超灵敏且高强度的传感器开辟了新的范式。

Soft mechanical sensors that can detect strain, pressure, and/or bending curvature have wide applications, including artificial tactile skin for humanoids 1 , 2 1 , 2 ^(1,2){ }^{1,2}, wearable activity monitoring 3 − 5 3 − 5 ^(3-5){ }^{3-5}, perception of soft robotic motions 6 6 ^(6){ }^{6}, medical instruments 7 7 ^(7){ }^{7}, on-skin and implantable healthcare systems 8 , 9 8 , 9 ^(8,9){ }^{8,9}, and human-machine interfaces 10 − 12 10 − 12 ^(10-12){ }^{10-12}. Driven by increasing demands of soft mechanical sensors, remarkable progress has been made in materials, device structures, and fabrication processes 13 , 14 13 , 14 ^(13,14){ }^{13,14}. For practical uses, however, it remains a great challenge to simultaneously meet the following requirements; signal reliability under repeated stimuli, mechanical robustness, long-term stability without fatigue, high sensitivity in a wide range of stimulation, simple structure, multimodality, fast data collection, and fabrication reproducibility. Piezoresistive and piezocapacitive soft mechanical sensors have won the majority due to their simple structure, high sensitivity, fast data acquisition as well as the strain-pressure multimodality when
能够检测应变、压力和/或弯曲曲率的软机械传感器具有广泛的应用，包括人形机器人的人造触觉皮肤 1 , 2 1 , 2 ^(1,2){ }^{1,2} 、可穿戴活动监测 3 − 5 3 − 5 ^(3-5){ }^{3-5} 、软机器人运动感知 6 6 ^(6){ }^{6} 、医疗器械 7 7 ^(7){ }^{7} 、皮上和植入式医疗保健系统 8 , 9 8 , 9 ^(8,9){ }^{8,9} 以及人机界面 10 − 12 10 − 12 ^(10-12){ }^{10-12} 。在对软机械传感器日益增长的需求的推动下，材料、器件结构和制造工艺 13 , 14 13 , 14 ^(13,14){ }^{13,14} 取得了显著进展。然而，对于实际应用来说，同时满足以下要求仍然是一个巨大的挑战：重复刺激下的信号可靠性、机械鲁棒性、长期稳定性而不疲劳、在各种刺激下的高灵敏度、结构简单、多模态、快速数据采集和制造可重复性。压阻和压电容软机械传感器因其结构简单、灵敏度高、数据采集速度快以及在施加压力时的应变-压力多模态性而赢得了大多数应用。
properly deconvoluted 15 − 17 15 − 17 ^(15-17){ }^{15-17}. Unfortunately, soft piezoresistive sensors have limitations in terms of fabrication reproducibility and consistent performance, which become deteriorated as the sensitivity increases 18 , 19 18 , 19 ^(18,19){ }^{18,19}. Soft piezocapacitive sensors are usually specialized for pressure sensing, but strain sensitivity is limited 20 , 21 20 , 21 ^(20,21){ }^{20,21}. Although intrinsic deformability of ion gels has been explored for fabrication of ionic mechanical sensors on the basis of impedance change or capacitance change 22 , 23 22 , 23 ^(22,23){ }^{22,23}, the sensors exhibit a sluggish response to dynamic stimulation 24 , 25 24 , 25 ^(24,25){ }^{24,25}. Very recently, soft piezoinductive sensors have been suggested for noncontact movement recognition and excellent bending angle resolution 26 , 27 26 , 27 ^(26,27){ }^{26,27}, but their resolutions for pressure and strain are relatively low. New approaches are required to simultaneously meet the above-mentioned requirements.
正确解卷积 15 − 17 15 − 17 ^(15-17){ }^{15-17} 。遗憾的是，软压阻传感器在制造可重复性和一致性能方面存在局限性，并且随着灵敏度的增加，这些局限性会逐渐恶化 18 , 19 18 , 19 ^(18,19){ }^{18,19} 。软压电容传感器通常专用于压力传感，但应变灵敏度有限 20 , 21 20 , 21 ^(20,21){ }^{20,21} 。尽管人们已经探索了离子凝胶的固有变形能力，以基于阻抗变化或电容变化制造离子机械传感器 22 , 23 22 , 23 ^(22,23){ }^{22,23} ，但这些传感器对动态刺激的响应较慢 24 , 25 24 , 25 ^(24,25){ }^{24,25} 。最近，软压电电感传感器已被提出用于非接触式运动识别和出色的弯曲角度分辨率 26 , 27 26 , 27 ^(26,27){ }^{26,27} ，但它们对压力和应变的分辨率相对较低。需要新的方法来同时满足上述要求。
Electrically resistive cracks have been used to produce highly sensitive piezoresistive sensors, either in the ranges of very small
电阻裂纹已被用来生产高灵敏度的压阻传感器，无论是在非常小的范围内
tensile strain ( ε ε epsi\varepsilon ) (ultrasensitive when ε ≤ 2 % ε ≤ 2 % epsi <= 2%\varepsilon \leq 2 \% ) by using metallic cracks 28 , 29 28 , 29 ^(28,29){ }^{28,29} or in the large strain ranges ( ε > 10 % ε > 10 % epsi > 10%\varepsilon>10 \% ) by using deformable composites 30 , 31 30 , 31 ^(30,31){ }^{30,31}. Even though hierarchical crack structures have been introduced to bridge the two strain ranges 32 32 ^(32){ }^{32} and some progress has been made 33 − 35 33 − 35 ^(33-35){ }^{33-35}, the resistive crack-based sensors often have the technological issues such as performance degradation, signal hysteresis, and large signal drift by fatigue during repeated operation. Those issues are attributed to the intrinsic characteristics of the ON-or-OFF electrical current by crack formation and the inevitable propagation of the cracks by fatigue (Supplementary Fig. 1). Unlike the electric current, the magnetic flux can pass through the air gap (with a low magnetic permeability, μ r = 1 μ r = 1 mu_(r)=1\mu_{\mathrm{r}}=1 ) between magnetic domains (with a high magnetic permeability, μ r > 100 ) 36 − 38 μ r > 100 36 − 38 mu_(r) > 100)^(36-38)\left.\mu_{\mathrm{r}}>100\right)^{36-38}. This gradual change of the magnetic flux in through-thickness cracks may result in a gradient piezoinductive response of an inductive coil, hence making the sensor insensitive to crack configuration, crack-propagation related issues, and allowing high-precision detection in a small-to-large strain range. This air gap-modulated magnetic flux change has been used for sensing displacement in bulk industry systems 39 , 40 39 , 40 ^(39,40){ }^{39,40}, however the concept has not been explored for soft electronics.
通过使用金属裂纹 28 , 29 28 , 29 ^(28,29){ }^{28,29} ，可以在拉伸应变（ ε ε epsi\varepsilon ）（ ε ≤ 2 % ε ≤ 2 % epsi <= 2%\varepsilon \leq 2 \% 时超灵敏）下实现电阻式传感器，或通过使用可变形复合材料 30 , 31 30 , 31 ^(30,31){ }^{30,31} 在大应变范围内（ ε > 10 % ε > 10 % epsi > 10%\varepsilon>10 \% ）实现电阻式传感器。尽管已经引入了分层裂纹结构来弥合这两个应变范围 32 32 ^(32){ }^{32} ，并且取得了一些进展 33 − 35 33 − 35 ^(33-35){ }^{33-35} ，但基于裂纹的电阻式传感器在重复操作过程中通常存在性能下降、信号滞后和疲劳引起的大信号漂移等技术问题。这些问题归因于裂纹形成导致的电流开或关的固有特性，以及裂纹因疲劳而不可避免的扩展（补充图 1）。与电流不同，磁通量可以穿过磁畴（具有高磁导率， μ r > 100 ) 36 − 38 μ r > 100 36 − 38 mu_(r) > 100)^(36-38)\left.\mu_{\mathrm{r}}>100\right)^{36-38} ）之间的气隙（具有低磁导率， μ r = 1 μ r = 1 mu_(r)=1\mu_{\mathrm{r}}=1 ）。贯穿厚度的裂纹中磁通量的逐渐变化可能导致感应线圈的梯度压电感应响应，从而使传感器对裂纹结构、裂纹扩展相关问题不敏感，并允许在小到大应变范围内进行高精度检测。这种气隙调制的磁通量变化已用于感测大宗工业系统中的位移 39 , 40 39 , 40 ^(39,40){ }^{39,40} ，但该概念尚未在软电子学中进行探索。
In this study, we present a deformable magnetic crack-based piezoinductive sensor (MC-PIS), which exploits the magnetic flux variation of a cracked ferrite film (CFF) and its resultant effect on the inductance of an electric coil placed below the ferrite film. This sensor shows ultrahigh mechanical robustness without fatigue behavior and achieves ultra-sensitivity in a small-to-large strain range for both
本研究提出了一种基于可变形磁裂纹的压电电感传感器 (MC-PIS)，该传感器利用了裂纹铁氧体薄膜 (CFF) 的磁通量变化及其对置于铁氧体薄膜下方的电线圈电感的影响。该传感器表现出超高的机械强度，且无疲劳行为，并且在从小到大的应变范围内均实现了超高灵敏度。
compressive and tensile stimuli. The simple structure allows excellent fabrication reproducibility and fast data acquisition. By utilizing the intrinsic deconvolution for compression and extension, we demonstrate a soft robot which monitors its dynamic shape changes, obtains a height profile of microstructure patterns, detects an obstacle and moves over it, digs soil and plants a tree, and pretends dead or runs away accordingly to external stimuli.
压缩和拉伸刺激。简单的结构使其具有出色的制造可重复性和快速的数据采集能力。通过利用压缩和拉伸的固有反卷积，我们展示了一个软体机器人，它可以监测其动态形状变化，获取微结构图案的高度轮廓，检测障碍物并越过障碍物，挖土和植树，并根据外部刺激装死或逃跑。

Results  结果

Strain modulated magnetic reluctance of the CFF
叠层磁场滤波器的应变调制磁阻

We used a commercial flexible ferrite film which consists of a 100 μ m 100 μ m 100 mum100 \mu \mathrm{~m}-thick ferrite film with a 50 μ m 50 μ m 50 mum50 \mu \mathrm{~m}-thick adhesive layer, a 30 μ m 30 μ m 30 mum30 \mu \mathrm{~m} thick poly (ethylene terephthalate) (PET) film on top, and a 100 μ m 100 μ m 100 mum100 \mu \mathrm{~m} thick backing paper. The ferrite film was cut into a strip (ex. 5 mm × 25 mm 5 mm × 25 mm 5mmxx25mm5 \mathrm{~mm} \times 25 \mathrm{~mm} ), and rolled on a cylindrical rod (diameter = 4 mm = 4 mm =4mm=4 \mathrm{~mm} ) to generate cracked flakes (Supplementary Fig. 2). After delaminating the backing paper, the CFF can be directly stuck on other substrate through the adhesive layer. Figure 1a shows scanning electron microscope (SEM) images of the non-cracked ferrite film, the CFF without strain ( ε = 0 % ε = 0 % epsi=0%\varepsilon=0 \% ), and the uniaxially-stretched CFF at ε = 8 % ε = 8 % epsi=8%\varepsilon=8 \%. The air-gap was variable depending on magnitude of the applied strain, and the relative magnetic flux gradually decreased with increasing air-gap (Supplementary Fig. 3 and Movie S1). Figure 1b exhibits the variation of the magnetic flux in the ferrite film as a function of the air-gap. The magnetic flux decreased considerably under tensile strain; B / B 0 = 0.61 B / B 0 = 0.61 B//B_(0)=0.61B / B_{0}=0.61 at ε = 0 % ε = 0 % epsi=0%\varepsilon=0 \%, and B / B 0 = 0.40 B / B 0 = 0.40 B//B_(0)=0.40B / B_{0}=0.40 at ε = 3 % ε = 3 % epsi=3%\varepsilon=3 \%.
我们使用了商用柔性铁氧体薄膜，它由 100 μ m 100 μ m 100 mum100 \mu \mathrm{~m} 厚的铁氧体薄膜、 50 μ m 50 μ m 50 mum50 \mu \mathrm{~m} 厚的粘合剂层、 30 μ m 30 μ m 30 mum30 \mu \mathrm{~m} 厚的聚对苯二甲酸乙二醇酯 (PET) 薄膜和 100 μ m 100 μ m 100 mum100 \mu \mathrm{~m} 厚的背纸组成。将铁氧体薄膜切成条状（例如 5 mm × 25 mm 5 mm × 25 mm 5mmxx25mm5 \mathrm{~mm} \times 25 \mathrm{~mm} ），然后卷在圆柱形棒（直径 = 4 mm = 4 mm =4mm=4 \mathrm{~mm} ）上以产生破裂的薄片（补充图 2）。剥离背纸后，CFF 可以通过粘合剂层直接粘贴在其他基板上。图 1a 显示了无裂纹铁氧体薄膜、无应变的 CFF（ ε = 0 % ε = 0 % epsi=0%\varepsilon=0 \% ）和单轴拉伸 CFF 在 ε = 8 % ε = 8 % epsi=8%\varepsilon=8 \% 的扫描电子显微镜 (SEM) 图像。气隙根据施加的应变的大小而变化，并且相对磁通量随着气隙的增加而逐渐减小（补充图 3 和电影 S1）。图 1b 展示了铁氧体薄膜中磁通量随气隙的变化。在拉伸应变下，磁通量显著下降； B / B 0 = 0.61 B / B 0 = 0.61 B//B_(0)=0.61B / B_{0}=0.61 处为 ε = 0 % ε = 0 % epsi=0%\varepsilon=0 \% ， B / B 0 = 0.40 B / B 0 = 0.40 B//B_(0)=0.40B / B_{0}=0.40 处为 ε = 3 % ε = 3 % epsi=3%\varepsilon=3 \% 。
Fig. 1 | Concept of the magnetic crack-based piezoinductive sensor (MC-PIS) and its uniqueness. a SEM images and magnetic flux density ( B B BB ) of a ferrite strip with no cracks, and with cracks at 0 % 0 % 0%0 \% and 8 % 8 % 8%8 \% tensile strains ( ε ) ( ε ) (epsi)(\varepsilon). b 2D simulation of B B BB in the cracked ferrite film (CFF) strip with different air-gap. B B BB was compared to those in a non-cracked ferrite film ( B 0 B 0 B_(0)B_{0} ) and in the air ( B air B air  B_("air ")B_{\text {air }} ). c Schematic illustration
图 1 | 基于磁性裂纹的压电电感传感器 (MC-PIS) 的概念及其独特性。a 无裂纹的铁氧体条带的 SEM 图像和磁通密度 ( B B BB )，以及在 0 % 0 % 0%0 \% 和 8 % 8 % 8%8 \% 拉伸应变 ( ε ) ( ε ) (epsi)(\varepsilon) 下出现裂纹的铁氧体条带。b 具有不同气隙的裂纹铁氧体薄膜 (CFF) 条带中 B B BB 的 2D 模拟。 B B BB 与无裂纹铁氧体薄膜 ( B 0 B 0 B_(0)B_{0} ) 和空气 ( B air B air  B_("air ")B_{\text {air }} ) 中的模拟进行了比较。c 示意图
of the CFF with the rigid flakes on the elastic substrate, and the resulting magnetic reluctance ( R m R m R_(m)R_{\mathrm{m}} ) modulation under tensile strain. d d d\mathbf{d} Schematic performance of the MC-PIS and the resistive crack-based sensor (RCS) under compressive and tensile strains. The MC-PIS is responsive to a wide range of compression and extension.
弹性基底上带有刚性薄片的 CFF，以及由此产生的拉伸应变下的磁阻（ R m R m R_(m)R_{\mathrm{m}} ）调制。 d d d\mathbf{d} MC-PIS 和基于裂纹的电阻式传感器（RCS）在压缩和拉伸应变下的示意图。MC-PIS 对各种压缩和拉伸应变均有响应。
Fig. 2 | Working principle and characteristics of the MC-PIS for bi-directional bending sensing. a Illustrations showing the working principle for bi-directional bending and camera images of the stretchable liquid metal coil and the flexible Cu planar coil. b b b\mathbf{b} Inductance variation of the MC-PIS at a bending angle from − 327 ∘ − 327 ∘ -327^(@)-327^{\circ} to 327 ∘ 327 ∘ 327^(@)327^{\circ}. The inset emphasizes no hysteresis of inductance changes during a bendingreleasing cycle. c Inductance variation of the MC-PIS during an incremental bending and releasing tests with a step of 1 ∘ 1 ∘ 1^(@)1^{\circ}. The inset shows inductance variation with a small bending angle change ( 0.01 ∘ 0.01 ∘ 0.01^(@)0.01^{\circ} ), indicating the high precision detection limit of
图 2 | 用于双向弯曲传感的 MC-PIS 的工作原理和特性。a 图示为双向弯曲的工作原理以及可拉伸液态金属线圈和柔性 Cu 平面线圈的相机图像。 b b b\mathbf{b} MC-PIS 在弯曲角度从 − 327 ∘ − 327 ∘ -327^(@)-327^{\circ} 到 327 ∘ 327 ∘ 327^(@)327^{\circ} 时的电感变化。插图强调了在弯曲-释放循环过程中电感变化没有滞后现象。c MC-PIS 在增量弯曲和释放测试过程中的电感变化，步骤为 1 ∘ 1 ∘ 1^(@)1^{\circ} 。插图显示了电感在弯曲角度变化较小 ( 0.01 ∘ 0.01 ∘ 0.01^(@)0.01^{\circ} ) 时的变化，表明其具有较高的精度检测限。
the MC-PIS. d d d\mathbf{d} Long-term stability of the sensor undergoes 50,000 bendingreleasing cycles between 110 ∘ 110 ∘ 110^(@)110^{\circ} and 120 ∘ 120 ∘ 120^(@)120^{\circ}. e Negligible performance degradation of the MC-PIS while the CFF layer was half-scratched (1) and full-scratched (split the CFF in half) with a tweezer. f f f\mathbf{f} Inductance changes of the MC-PIS when it was heated from 27 ∘ C 27 ∘ C 27^(@)C27^{\circ} \mathrm{C} to 70 ∘ C 70 ∘ C 70^(@)C70^{\circ} \mathrm{C}, and submerged in water of different temperatures ( 20 ∘ C 20 ∘ C (20^(@)C:}\left(20^{\circ} \mathrm{C}\right., 82 ∘ C , 2.8 ∘ C ) 82 ∘ C , 2.8 ∘ C {:82^(@)C,2.8^(@)C)\left.82^{\circ} \mathrm{C}, 2.8^{\circ} \mathrm{C}\right). g Comparison of the MC-PIS with respect to the state-of-the-art bending sensors in aspects of detection resolution (data points/degree), bending angle range, and mechanical robustness.
MC-PIS。 d d d\mathbf{d} 传感器在 110 ∘ 110 ∘ 110^(@)110^{\circ} 和 120 ∘ 120 ∘ 120^(@)120^{\circ} 之间经历了 50,000 次弯曲释放循环后仍能保持长期稳定性。e 用镊子半划伤 (1) 和全划伤（将 CFF 分成两半）CFF 层时，MC-PIS 的性能下降可忽略不计。 f f f\mathbf{f} 当 MC-PIS 从 27 ∘ C 27 ∘ C 27^(@)C27^{\circ} \mathrm{C} 加热到 70 ∘ C 70 ∘ C 70^(@)C70^{\circ} \mathrm{C} 并浸入不同温度的水中时，其电感变化为 ( 20 ∘ C 20 ∘ C (20^(@)C:}\left(20^{\circ} \mathrm{C}\right. , 82 ∘ C , 2.8 ∘ C ) 82 ∘ C , 2.8 ∘ C {:82^(@)C,2.8^(@)C)\left.82^{\circ} \mathrm{C}, 2.8^{\circ} \mathrm{C}\right) 。g 在检测分辨率（数据点/度）、弯曲角度范围和机械强度方面，MC-PIS 与最先进的弯曲传感器的比较。
It is noteworthy that once the ferrite film was cracked that nearly null air-gap condition can be achieved under compression.
值得注意的是，一旦铁氧体薄膜破裂，在压缩下就可以实现几乎零气隙状态。
As illustrated in Fig. 1c, the CFF has a structure of rigid-flakes-on-elastic-substrate, in which these ferrite flakes are completely separated by randomly distributed through-thickness cracks. The ferrite flakes were random in size and shape (Supplementary Fig. 4). The average dimension of the flakes was 500 μ m × 820 μ m 500 μ m × 820 μ m 500 mumxx820 mum500 \mu \mathrm{~m} \times 820 \mu \mathrm{~m}, and the average air-gap between the flakes was 6.7 μ m 6.7 μ m 6.7 mum6.7 \mu \mathrm{~m} at ε = 0 % ε = 0 % epsi=0%\varepsilon=0 \% and 49.4 μ m 49.4 μ m 49.4 mum49.4 \mu \mathrm{~m} at ε = 8 % ε = 8 % epsi=8%\varepsilon=8 \%. Figure 1 c also presents two-dimensional simulation on magnetic flux ( B ) ( B ) (B)(B) distributions in the CFF with a configuration extracted from a microscopic image, showing dynamic variation accordingly to the air-gaps under mechanical deformation. The principle of the magnetic circuit theory 39 39 ^(39){ }^{39} and details of the simulation results are provided in Supplementary Note 1. Since the ferrite flakes are rigid with very limited stretchability, when the CFF is stretched (or compressed), only air gaps between flakes vary with the applied strain, whereas the ferrite flakes only move apart from (/close to) each other (Supplementary Fig. 5). In addition, through-thickness cracking air-gaps and submillimeter rigid flakes make the CFF mechanically compliant to be compressed and stretched, and resilient to crack-propagation and fatigue. Benefiting
如图 1c 所示，CFF 具有弹性基底上刚性薄片的结构，其中这些铁氧体薄片完全被随机分布的贯穿厚度的裂纹隔开。铁氧体薄片的尺寸和形状随机（补充图 4）。薄片的平均尺寸为 500 μ m × 820 μ m 500 μ m × 820 μ m 500 mumxx820 mum500 \mu \mathrm{~m} \times 820 \mu \mathrm{~m} ，薄片之间的平均气隙在 ε = 0 % ε = 0 % epsi=0%\varepsilon=0 \% 处为 6.7 μ m 6.7 μ m 6.7 mum6.7 \mu \mathrm{~m} ，在 ε = 8 % ε = 8 % epsi=8%\varepsilon=8 \% 处为 49.4 μ m 49.4 μ m 49.4 mum49.4 \mu \mathrm{~m} 。图 1c 还展示了 CFF 中磁通量 ( B ) ( B ) (B)(B) 分布的二维模拟，其中提取了从显微图像中提取的结构，显示了机械变形下气隙的动态变化。磁路理论 39 39 ^(39){ }^{39} 的原理和模拟结果的细节在补充说明 1 中提供。由于铁氧体薄片具有刚性，拉伸性非常有限，因此当 CFF 被拉伸（或压缩）时，只有薄片之间的气隙会随施加的应变而变化，而铁氧体薄片之间只会彼此远离（/靠近）（补充图 5）。此外，贯穿厚度的开裂气隙和亚毫米级的刚性薄片使 CFF 具有机械柔顺性，能够承受压缩和拉伸，并且能够抵御裂纹扩展和疲劳。受益
from this structure and the magnetic induction mechanism, the MCPIS can operate at a much wider strain range from compressive strain to tensile strain ( − 5 % − 5 % -5%-5 \% to 10 % 10 % 10%10 \% ) and would only “saturate” at extreme conditions. Whereas cracked resistive sensors are insensitive to compressive strain and would completely disconnect and stop functioning above a small tensile strain ( ε =∼ 2 % ε =∼ 2 % epsi=∼2%\varepsilon=\sim 2 \% ). Figure 1d illustrates the distinct characteristics between the MC-PIS and a representative cracked metal film-based resistive sensor 28 28 ^(28){ }^{28}. Comparison of strain sensing range, resolution, materials, fabrication technologies, and stability between the MC-PIS and some representative resistive crack sensors are summarized in Supplementary Table 1.
从这种结构和磁感应机制来看，MCPIS 可以在从压缩应变到拉伸应变（ − 5 % − 5 % -5%-5 \% 到 10 % 10 % 10%10 \% ）的更宽应变范围内工作，并且只会在极端条件下“饱和”。而破裂的电阻传感器对压缩应变不敏感，并且会在较小的拉伸应变（ ε =∼ 2 % ε =∼ 2 % epsi=∼2%\varepsilon=\sim 2 \% ）以上完全断开并停止工作。图 1d 说明了 MC-PIS 和代表性的破裂金属薄膜电阻传感器 28 28 ^(28){ }^{28} 之间的不同特性。补充表 1 总结了 MC-PIS 和一些代表性电阻裂缝传感器之间的应变传感范围、分辨率、材料、制造技术和稳定性的比较。

Robust yet highly precise bidirectional bending sensing
坚固且高度精确的双向弯曲传感

Flexible sensors that can detect bidirectional bending curvatures/ angles are in high demand for proprioception and tactile sensing of robotics. A bidirectional bending sensor was fabricated by combining the CFF on a planar inductive coil (Fig. 2a and Movie S2). The soft planar coil was produced either by filling a microchannel in an elastomer substrate (Ecoflex 00-30) with stretchable liquid metal (eutectic In-Ga alloy) or by depositing a Cu trace pattern on a highly-
能够检测双向弯曲曲率/角度的柔性传感器在机器人本体感觉和触觉感知方面需求巨大。通过将 CFF 与平面电感线圈组合，制作了一个双向弯曲传感器（图 2a 和视频 S2）。柔性平面线圈的制作方法有两种：在弹性体基底（Ecoflex 00-30）中用可拉伸液态金属（共晶 In-Ga 合金）填充微通道；或者在高密度导电材料（例如，In-Ga 合金）上沉积 Cu 走线图案。
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  2. 1 1 ^(1){ }^{1} Department of Precision Machinery and Precision Instrumentation, University of Science and Technology of China, Hefei, China. 2 2 ^(2){ }^{2} Department of Materials Science and Engineering, Pohang University of Science and Technology, Pohang, Republic of Korea. 3 3 ^(3){ }^{3} These authors contributed equally: Yulian Peng,
    1 1 ^(1){ }^{1} 中国科学技术大学精密机械与精密仪器系，中国合肥。 2 2 ^(2){ }^{2} 韩国浦项科技大学材料科学与工程系，韩国浦项。 3 3 ^(3){ }^{3} 以下作者贡献相同：Yulian Peng，
    Zhengyan Wang, Houping Wu. e-mail: ujeong@postech.ac.kr; wangh@ustc.edu.cn
    王正彦，吴厚平.电子邮件： ujeong@postech.ac.kr ； wangh@ustc.edu.cn