Bioinspired Kirigami Structure for Efficient Anchoring of Soft Robots via Optimization Analysis

Abstract

Kirigami-inspired geometries offer a lightweight, bioinspired strategy for friction enhancement and anchoring in soft robotics. This study presents a bioinspired kirigami structure designed to enhance the anchoring performance of soft robotic systems through systematic geometric and actuation parameter optimization. Drawing inspiration from the anisotropic friction mechanisms observed in reptilian scales, we integrated linear, triangular, trapezoidal, and hybrid kirigami cuts onto flexible plastic sheets. A compact 12 V linear actuator enabled cyclic actuation via a custom firmware loop, generating controlled buckling and directional friction for effective locomotion. Through experimental trials, we quantified anchoring efficiency using crawling distance and stride metrics across multiple cut densities and actuation conditions. Among the tested configurations, the triangular kirigami with a 4 × 20 unit density on 100 µm PET exhibited the most effective performance, achieving a stride efficiency of approximately 63% and an average crawling speed of ~47 cm/min under optimized autonomous operation. A theoretical framework combining buckling mechanics and directional friction validated the observed trends. This study establishes a compact, tunable anchoring mechanism for soft robotics, offering strong potential for autonomous exploration in constrained environments.

1. Introduction

The field of soft robotics has emerged as a transformative domain in automation, offering compliance, adaptability, and resilience to navigate unstructured and constrained environments—capabilities that rigid robotic systems often lack [1,2,3]. Among nature’s most efficient movers, snakes exhibit remarkable terrain adaptability, navigating tight crevices or uneven surfaces using a repertoire of locomotion strategies such as rectilinear progression and lateral undulation [4,5,6]. Central to their locomotion efficiency is the anisotropic friction generated by ventral scales, which provide directional anchoring and propulsion with minimal energy expenditure [4,7]. These scales interlock with ground asperities and generate higher resistance to backward or lateral motion than forward, as experimentally shown by Hu et al. [4], who quantitatively modeled how belly scale anisotropy enables effective slithering on flat terrain.

Drawing inspiration from biomechanical principles, researchers have sought to emulate this anchoring capability in soft robotic platforms, targeting applications such as confined space navigation, biomedical systems, and terrain-adaptive exploration where traditional robots are not an appropriate option [8,9,10,11]. One promising approach involves the use of kirigami, the ancient art of patterned cuts in thin sheets—reimagined as a tool to engineer deformable surfaces with tunable mechanical and frictional properties to engage the surface in a scale-like fashion [12,13,14].

The morphological diversity found in reptilian scales offers a bioinspired template for the design of kirigami skins capable of controlled deformation and surface interaction. In this study, five distinct kirigami patterns are explored to optimize anchoring behavior: linear cuts that provide minimal frictional interaction but maintain structural stability; triangular cuts inspired by the sharp-edged ventral scales of Puff Adders, enhancing anisotropic friction and grip; trapezoidal patterns derived from the whiptail lizard, offering a balance between flexibility and mechanical support; and two hybrid variants, linear–triangular and linear–trapezoidal, that combine stability with directional anchoring features [6,15,16,17,18]. These structures enable tunable anisotropic friction under compression, echoing the engagement–release anchoring mechanics observed in snakes during slithering and concertina locomotion [19]. Prior implementations of kirigami-based soft robotic systems have commonly relied on pneumatic actuation to induce the necessary buckling deformation for locomotion. However, these systems often suffer from bulk, noise, and the need for sealed chambers and external pumps, limiting their compactness and autonomy [20,21]. This study utilizes a permanent magnet (PM) linear actuator as the actuation source. This actuator provides a silent, energy-efficient, and compact solution, enabling controlled axial compression and repeatable deformation cycles of the kirigami skin [22,23].

Overall, this work is to optimize the anchoring performance of a kirigami-based soft robotic crawler through systematic variation of both structural and actuation parameters. Anchoring efficiency is assessed via crawling distance, stride efficiency, and net propulsive force under varying conditions of cut geometry, PET sheet thickness, cut density, actuator stroke, and actuation speed. The combination of experimental testing with analytical modeling allows for a comprehensive investigation of how geometric and mechanical factors contribute to effective directional anchoring. To support the design process, a theoretical model is developed that integrates classical buckling mechanics with a friction-based analytical framework, enabling prediction of pop-up height and anisotropic friction forces. This dual perspective offers not only validation of observed results but also a foundation for further optimization of kirigami-based anchoring systems.

Further, this paper is structured as follows: Section 2 presents the design principles, fabrication methodology, and experimental setup for the kirigami-skinned robot. Section 3 provides detailed results from parametric studies and theoretical modeling. Section 4 discusses the implementation of an autonomous crawling system and its performance metrics. Finally, the conclusion summarizes key findings and outlines future research directions in sensor integration and terrain-adaptive anchoring strategies.

2. Design, Composer, and Experimental Setup

This section outlines the design principles, fabrication methods, and control strategies behind the kirigami-based soft robotic crawler, integrating bio-inspired kirigami structures with a PM linear actuator to optimize locomotion efficiency. We detail the kirigami skin’s geometry, material selection, actuator integration, experimental protocols, and a theoretical framework predicting performance, all validated by experimental outcomes.

2.1. Kirigami Structure Design

The kirigami structures were fabricated with arrays of patterns into polyester plastic sheets (PET) ($E=4.33\hspace{0.17em}\mathrm{GPa}$, $\nu =0.4$), chosen for their balance of flexibility and durability. The intricate kirigami patterns enabled the creation of designs tailored for specific frictional and locomotive roles: linear cuts as a baseline with high stability but minimal friction; triangular cuts, inspired by the Puff Adder’s sharp-edged scales (~5–8 mm [7,18,24]), for enhanced anisotropic friction and anchoring; and trapezoidal cuts, inspired by whiptail lizard ventral scales (~4–7 mm [25,26]), for balanced flexibility and support (Figure 1a). Prior biological studies have shown that snakes exploit directional friction coefficients (forward vs. backward vs. lateral) to optimize motion, often producing passive anchoring at specific body segments through contact angle modulation and scale placement [4]. This behavior has been quantitatively linked to scale geometry and surface interaction. Our asymmetrical kirigami cuts are explicitly designed to emulate these biomechanical strategies by deforming asymmetrically under compression, thereby maximizing anchoring in the desired direction of motion. Proposed design parameters for kirigami pattern analysis for optimization included a lattice unit defined by width $L=2l\mathrm{c}\mathrm{o}\mathrm{s}\left(\pi /6\right)$ and height $H=2l\mathrm{s}\mathrm{i}\mathrm{n}\left(\pi /6\right)$, where $l$ is the cut length and $\delta$ is the hinge width separating cuts at an angle $\gamma =\pi /3$ (Figure 1b). Fabrication involved cutting with the Silhouette Cameo 5 (Silhouette America, Inc., Lehi, UT, USA), producing expandable kirigami structures (Figure 1e,f).

2.2. Composer of the Kirigami-Skinned Crawlers for Parameters Analysis

To systematically evaluate the functional impact of kirigami structures on locomotion, a modular soft robotic crawler was composed, integrating mechanical, structural, and electronic subsystems. The crawler comprises a permanent magnet (PM) linear actuator, a kirigami-skinned silicone body, and a microcontroller-based control unit, designed to enable precise and repeatable testing under varied experimental conditions. The PM linear actuator (Model: LA-T8, GoMotorWorld, Shenzhen, China; 12 V, 60 N thrust, 15 mm/s rated speed, 30 mm strokewas selected for its compactness, silent operation, and higher precision compared to traditional pneumatic actuators [23,27]. The actuator provides a maximum stroke length of 30 mm, sufficient for evaluating different deformation regimes and crawling behaviors (Figure 2a). It is mechanically coupled to a compliant soft body formed from Ecoflex 00-30, silicone elastomer ((Smooth-On, Inc., Macungie, PA, USA; shear modulus μ_a ≈ 30 kPa), cast into a 1-millimeter-thick sheet, and degassed to eliminate voids. This flexible substrate offers low stiffness and high stretchability, allowing seamless integration with kirigami layers. The kirigami skin is fabricated from PET sheets (Changzhou Huisu Qinye Plastic Group, China) with kirigami cut patterns and is firmly bonded to the silicone layer from both sides, which ensures consistent attachment and mechanical transfer (Figure 2b). Upon actuation, the sheet undergoes buckling-induced pop-up deformation, translating the actuator’s linear motion into anisotropic surface contact and friction-driven propulsion. The closed configuration of the actuator maintains tension on the skin, which rebounds during retraction, facilitating the cyclic crawling motion.

A microcontroller-based control circuit governs the actuation cycle of the composed crawler. The control system features an Arduino and an LN298 motor driver, powered by dual 12 V lithium batteries (Kuongshun Electronic Limited, Guangzhou, China), delivering both driving power and regulated logic voltage. The actuator’s bidirectional movement is controlled via a polarity-switching algorithm, enabling fine-tuned extension and retraction sequences. A standard test consisted of five actuation cycles at 15 mm/s extension speed, unless otherwise varied, with displacement tracked across a polyurethane (PU) foam substrate that simulates compliant terrain (Figure 2c,d). To further understand frictional dynamics during locomotion, forward and backward friction forces were independently measured using a J14014 mechanical sensor (Model: Phyphox, Xiangsheng Science and Education Equipment Co., Ltd., Guang’an, China). In a separate experimental setup, the crawler was placed on PU foam while the sensor recorded resistance during controlled to-and-from motion (Figure 2e,f). This allowed for the quantification of anisotropic friction generated by various kirigami configurations, critical for optimizing propulsion mechanics. To systematically optimize the locomotion efficiency of the kirigami-based soft robotic crawler, a series of controlled experiments were conducted. Each test was designed to evaluate the influence of specific structural or actuation parameters on crawling performance. Each experiment was repeated five times to ensure repeatability, and the mean crawling distance along with standard deviation was recorded to quantify the effect of pattern geometry on traction, deformation behavior, and net displacement. The overarching objective was to understand the complex interplay between kirigami mechanics, surface interaction, and actuation dynamics. The following subsections describe each experiment and its purpose:

• Kirigami Structure Type vs. Crawling Distance: This experiment investigates how different kirigami geometries affect anisotropic friction and locomotion. Inspired by the morphological diversity of snake scales, five distinct patterns were tested: linear, trapezoidal, triangular, hybrid linear–triangular, and hybrid linear–trapezoidal. The primary aim was to analyze how each pattern’s deformation mechanics and surface contact profile contribute to directional friction and, thus, crawling performance. By comparing the average crawling distance over five actuation cycles, this study aims to identify the most efficient pattern for generating propulsive force (Figure 1c).
• PET Sheet Thickness vs. Crawling Distance: In this test, the effect of substrate flexibility and stiffness on locomotion was examined by using PET sheets of varying thicknesses: 50 μm (red), 100 μm, 150 μm, 200 μm (transparent), and 250 μm (green). The experiment evaluates the average crawling distance to determine the optimal balance between mechanical compliance and anchoring effectiveness, both of which are critical for reliable propulsion (Figure 1d).
• Kirigami Cuts Density vs. Crawling Distance: To assess the impact of surface coverage and unit repetition on locomotion performance, we analyzed five different kirigami cut densities, i.e., 2 × 14, 3 × 15, 4 × 20, 5 × 28, and 6 × 28, implemented over a standardized contact area of 51 mm × 140 mm. In this notation, each density value M×N represents a grid of M cut units along the longitudinal (lengthwise) axis and N units along the transverse (widthwise) axis, covering the entire kirigami sheet. This structured arrangement modulates the number, spacing, and distribution of pop-up units, thereby influencing inter-unit mechanical interference, surface anchoring behavior, and crawling performance.
• By systematically varying the cut densities, we evaluated how increasing the number of units affects deformation dynamics and effective engagement with the ground surface. This approach enabled identification of an optimal density configuration that maximizes propulsion while minimizing performance losses due to structural interference.
• Stroke Length vs. Crawling Distance: This test explores how actuator stroke amplitude influences the mechanical engagement of the kirigami skin with the substrate. The following four actuation protocols were examined: 10 mm (St10), 20 mm (St20), 30 mm (St30), and 30 mm with a 1-second hold before retraction (St30h1s). By measuring the crawling distance over five actuation cycles, the experiment evaluates how increased extension—combined with dwell time—affects unit deformation, surface interaction, and net displacement.
• Actuation Speed vs. Crawling Distance: This experiment evaluates how different extension/retraction speeds—10/10, 15/15, 20/20, and 30/30 mm/s—influence crawling performance. Actuation speed directly affects the contact time and engagement quality between kirigami structures and the surface. By comparing average crawling distances, the test aims to determine a speed regime that optimally balances actuation dynamics with surface interaction mechanics for efficient locomotion.

2.3. Theoretical and Analytical Design

To guide the design and optimization of the kirigami-based soft robotic crawler, we developed a theoretical framework to predict the pop-up height ($h$) and anisotropic friction properties of the kirigami skin, integrating buckling mechanics with tribological analysis. This model was applied to a polyethylene terephthalate (PET) sheet ($E=4.33\hspace{0.17em}\mathrm{GPa}$, $\nu =0.4$) and validated using experimental data to quantify how cut geometry and actuation translate into propulsive forces driving locomotion. The framework provides a predictive tool for design refinement, with predictions for kirigami design configurations to optimize traction features.

2.3.1. Pop-Up Height Analysis

The pop-up height of the kirigami skin emerges from localized buckling triggered by axial compression applied via a PM linear actuator with a 30 mm stroke. Drawing upon classical buckling theory for thin elastic sheets [13,21,28], the pop-up height is evaluated by the relation:

$$h=kt\sqrt{{\displaystyle \frac{ϵ}{{ϵ}_c}}}$$

(1)

where $t$ is the sheet thickness (100 µm); $ϵ=\Delta L/L_0=0.214$ is the applied compressive strain; ${ϵ}_c$ is the critical buckling strain; and $k$ is a geometric amplification factor determined via experimental calibration. Assuming $l=10\hspace{0.17em}\mathrm{mm}$ with $L_{\mathrm{cut}}=9\hspace{0.17em}\mathrm{mm}$, $t=0.1\hspace{0.17em}\mathrm{mm}$, and $\nu =0.4$ yields. The critical strain for uniaxial buckling is given by:

$${ϵ}_c={\displaystyle \frac{{\pi }^2}{12\left(1-{\nu }^2\right)}}{\left({\displaystyle \frac{t}{L_{\mathrm{cut}}}}\right)}^2$$

(2)

Substituting values with an initial geometric factor $k=1.0$ yields:${ϵ}_c\approx 1.207\times {10}^{-4}$ and $h\approx 4.21\hspace{0.17em}\mathrm{mm}$. Experimental validation across six measured locations (Figure 3b) evaluated via ImageJ (version 1.54 g) for samples with $l=10\hspace{0.17em}\mathrm{mm}$, $L_{\mathrm{cut}}=9\hspace{0.17em}\mathrm{mm}$, yielded average pop-up heights of 8.99 mm for triangular cuts and 7.85 mm for trapezoidal cuts.

To adjust the theoretical predictions with experimental data, the factor $k$ was calibrated to 2.14 for triangular patterns and 1.87 for trapezoidal ones. This yielded refined predictions of $h\approx 9.00\hspace{0.17em}\mathrm{mm}$ for triangular and $h\approx 7.87\hspace{0.17em}\mathrm{mm}$ for trapezoidal geometries. These elevated $k$ values reflect the additional geometric amplification introduced by kirigami patterns, which lead to more complex tent-like deformations beyond simple Euler buckling, with sharper triangular cuts exhibiting greater vertical displacement. For further comparison, when $L_{\mathrm{cut}}=6\hspace{0.17em}\mathrm{mm}$, the critical buckling strain increases to ${ϵ}_c\approx 2.72\times {10}^{-4}$, and using the previously calibrated $k=2.14$, the predicted pop-up height becomes approximately 5.61 mm for triangular and 4.58 mm for trapezoidal cuts. This prediction corresponds closely with experimental results for triangular and trapezoidal patterns as observed in Figure 3a, which further confirms the robustness of the model in capturing the influence of cut length and geometry on vertical deformation.

2.3.2. Anisotropic Friction Analysis

Directional locomotion in the crawler is enabled by anisotropic friction forces generated by the interaction of kirigami pop-ups with the substrate. As the actuated kirigami structures buckle upward, they form inclined flaps that generate differential friction in the forward and backward directions. This behavior was analytically modeled by expressing the effective forward and backward friction coefficients such as:

$${\mu }_{\mathit{forward}}={\mu }_0+k_{f}h\rho , {\mu }_{\mathit{backward}}={\mu }_0-k_{f}h\rho$$

(3)

where ${\mu }_0$ is the intrinsic base friction coefficient of the PET material against the substrate; $k_f$ is a friction amplification constant that encapsulates both the inclination angle and surface engagement; $h$ is the pop-up height as described earlier; and $\rho$ is the number of kirigami units of density. Using experimental results for triangular patterns at $l=7\hspace{0.17em}\mathrm{mm}$, configuration—where $h=5.61\hspace{0.17em}\mathrm{mm}$, $\rho =0.0112$ for 4 × 20 kirigami units over 51 mm × 140 mm, $N=0.58\hspace{0.17em}\mathrm{N}$, ${\mu }_{\mathrm{forward}}=2.241$ and ${\mu }_{\mathrm{backward}}=0.776$. Based on the experimental result, the base coefficient and enhancement factors were computed as follows:

$${\mu }_0=1.5085, k_f\approx 11.66\hspace{0.17em}{\mathrm{mm}}^{-1}$$

(4)

The corresponding net propulsive force can be calculated as follows:

$$F_{\mathrm{prop}}=N\left({\mu }_{\mathrm{forward}}-{\mu }_{\mathrm{backward}}\right)\approx 0.850\hspace{0.17em}\mathrm{N}$$

(5)

This propulsive force aligns closely with experimental locomotion performance and clearly demonstrates the functional contribution of directional friction enabled by different kirigami structures (Section 3). Overall, this integrated theoretical framework provides a valuable predictive tool for rational design and iterative optimization of kirigami-based soft robotic systems.

3. Results and Analysis

3.1. Effect of Kirigami Structure on Crawling Efficiency

3.1.1. Experimental Conditions

To investigate the influence of kirigami pattern geometry on locomotion performance, five distinct cut configurations—linear, triangular, trapezoidal, hybrid linear–triangular, and hybrid linear–trapezoidal—were fabricated on 100 μm transparent PET sheets (Young’s modulus E = 4.33 GPa, Poisson’s ratio ν = 0.4). Each pattern was designed with a cut density of 3 × 14 units, uniformly distributed across a 51 mm × 140 mm area and aligned longitudinally with the direction of actuation. The crawling performance of each kirigami configuration was evaluated on a compliant polyurethane (PU) foam substrate, chosen to replicate soft, friction-sensitive terrain. A precision PM linear actuator with a 30 mm stroke and a constant extension–retraction speed of 15 mm/s was employed to drive locomotion. For each test, the crawler underwent five full actuation cycles, and the average crawling distance was recorded to quantify the effect of pattern geometry on traction, deformation behavior, and net displacement.

3.1.2. Performance Analysis

Crawling distances varied significantly across patterns (Figure 4). The linear pattern, designed as a baseline with high stability but minimal friction, achieved only 5 mm due to its isotropic friction. In contrast, the triangular pattern, inspired by the Puff Adder’s sharp-edged scales for enhanced anisotropic friction, covered 65 mm. The trapezoidal pattern, inspired by whiptail lizard ventral scales for balanced flexibility and support, reached 51 mm. The hybrid linear–triangular pattern, combining stability with improved directional grip, recorded 59 mm, while the hybrid linear–trapezoidal pattern, merging friction and stability, achieved 60 mm. These results highlight the superior anisotropic friction of triangular and trapezoidal geometries, which enhance surface engagement through pop-up structures, as predicted by our buckling model. Better results of the triangular pattern stem from its ability to maximize forward friction while minimizing backward slip. The evaluated friction coefficients were ${\mu }_{\mathrm{forward}}=2.241$ and ${\mu }_{\mathrm{backward}}=0.776$, generating a propulsive force of $F_{\mathrm{prop}}\approx 0.850\hspace{0.17em}\mathrm{N}$. This aligns with the observed crawling distance of 65 mm (13 mm per cycle), yielding a crawling efficiency of ~43% (13 mm per 30 mm stroke). The pop-up height for this configuration was calculated as $h=5.61\hspace{0.17em}\mathrm{mm}$, which facilitates strong surface anchoring. The trapezoidal pattern, with measured friction coefficients (${\mu }_{\mathrm{forward}}=2.414$, ${\mu }_{\mathrm{backward}}=0.862$), produced a slightly higher propulsive force ($F_{\mathrm{prop}}\approx 0.900\hspace{0.17em}\mathrm{N}$), but its crawling distance of 51 mm (10.2 mm per cycle, efficiency ~34%) suggests that the broader pop-up geometry increases backward friction as well, reducing overall efficiency. The hybrid linear–triangular pattern exhibited the highest anisotropy (${\mu }_{\mathrm{forward}}=5.000$, ${\mu }_{\mathrm{backward}}=1.897$, $F_{\mathrm{prop}}\approx 1.800\hspace{0.17em}\mathrm{N}$), yet covered only 59 mm (11.8 mm per cycle, efficiency ~ 39%). This indicates that while the hybrid design maximizes directional grip, factors such as pop-up stability or dynamic surface interactions may limit its performance. The hybrid linear–trapezoidal pattern achieved 60 mm (12 mm per cycle, efficiency ~40%), with ${\mu }_{\mathrm{forward}}=2.586$, ${\mu }_{\mathrm{backward}}=1.552$ and $F_{\mathrm{prop}}\approx 0.600\hspace{0.17em}\mathrm{N}$, reflecting a balance between friction and stability.

The linear pattern’s negligible crawling results from its lack of directional pop-up structures, leading to isotropic friction (${\mu }_{\mathrm{forward}}\approx {\mu }_{\mathrm{backward}}$, $F_{\mathrm{prop}}\approx 0\hspace{0.17em}\mathrm{N}$). The small distance is due to minor surface interactions and actuator momentum. Compared to the linear pattern, the bioinspired designs amplify locomotion efficiency, aligning with snake scale functionality. Hybrid kirigami cuts ingeniously integrate linear cuts to introduce additional directional buckling points, enhancing the structural support of trapezoidal and triangular patterns. This synergy in hybrids explains their strong performance, with the hybrid trapezoidal pattern nearly matching the triangular pattern due to enhanced directional control. Hybrid designs illustrate the benefit of integrating linear elements with triangular or trapezoidal shapes, offering a synergistic effect that enhances both frictional engagement and structural integrity, proving particularly effective for varied terrain navigation. These results highlight the potential of kirigami cut design to enhance the efficiency of kirigami-based soft robotic locomotion. The complex geometries of the triangular and hybrid trapezoidal patterns are instrumental in achieving high-performance crawling by maximizing frictional forces and enhancing surface adaptability.

3.2. Impact of Sheet Thickness

3.2.1. Experimental Conditions

In this experiment, the influence of PET sheet thickness on the crawling performance of kirigami-patterned soft robotic crawlers is evaluated. Five different thicknesses—50 µm (red), 100 µm, 150 µm, 200 µm (transparent), and 250 µm (green)—were tested using both triangular and trapezoidal kirigami patterns. Each sample incorporated a 3 × 14 cut configuration over a fixed contact area of 51 mm × 140 mm, ensuring consistent cut density across all variants. To assess locomotion, each crawler underwent five complete extension–retraction cycles driven by a PM linear actuator with a 30 mm stroke at a constant speed of 15 mm/s on a compliant polyurethane (PU) foam substrate. The goal was to determine the optimal sheet thickness that provides an effective trade-off between material flexibility for out-of-plane deformation and mechanical stability for surface anchoring across both triangular and trapezoidal kirigami configurations.

3.2.2. Performance Analysis

Sheet thickness significantly influenced crawling distance for both triangular and trapezoidal patterns, as observed in Figure 5. The 100 µm sheet achieved the best results: 62 mm for the triangular pattern and 53 mm for the trapezoidal pattern. In comparison, the triangular pattern recorded distances of 54 mm (50 µm), 48 mm (150 µm), 44 mm (200 µm), and 41 mm (250 µm), while the trapezoidal pattern achieved 58 mm (50 µm), 42 mm (150 µm), 38 mm (200 µm), and 37 mm (250 µm). These results suggest that 100 µm offers an ideal compromise: sufficient flexibility for kirigami pop-ups and structural support for effective friction. As discussed in Section 3.1, the triangular pattern at 100 µm showed better crawling performance than trapezoidal cuts due to increased backward friction reducing overall efficiency. The 50 µm sheets, while more deformable, exhibited reduced distances, likely due to insufficient rigidity to sustain optimal contact and pop-up stability. Conversely, thicker sheets (150–250 µm) showed a consistent decline in performance for both patterns, with the 250 µm sheet achieving the lowest distances (41 mm for triangular and 37 mm for trapezoidal). This decline is attributed to increased stiffness, which limits pop-up formation and reversion post-contraction, retaining buckled shapes and diminishing anisotropic friction, as observed in Figure 5. Although the theoretical pop-up height model $h=kt\sqrt{ϵ/{ϵ}_c}$ suggests that thickness $t$ appears in both the numerator and denominator (through ${ϵ}_c$), and thus cancels out, experimental results reveal that sheet thickness still significantly affects real-world deformation and recovery behavior. While thinner sheets offer greater compliance and flexibility, excessively thin materials (e.g., 50 µm) may lack the structural integrity needed for consistent buckling and traction. Conversely, thicker sheets (e.g., 250 µm) exhibit higher stiffness and risk of plastic deformation, reducing recovery efficiency after repeated cycles. The 100 µm PET sheet achieves a balance between flexibility and structural resilience, highlighting its suitability for kirigami-based directional frictional properties. These findings underscore material thickness as a critical design parameter, complementing kirigami geometry for locomotion efficiency, with the triangular pattern slightly showing better results than the trapezoidal pattern at optimal thickness due to its higher friction anisotropy.

3.3. Kirigami Cuts Density vs. Crawling Distance

3.3.1. Experimental Setup

To investigate the impact of kirigami structure on locomotion, a systematic experimental study was conducted to evaluate how varying the kirigami cut density influences the crawling efficiency of the soft robot. The density of the kirigami structure was modified by altering the unit lattice length ($l$) and adjusting the number of unit cells within the fixed area. These two parameters are inherently interrelated; a higher density is achieved by reducing the unit length while increasing the number of cuts within a fixed surface area. For this study, four distinct density configurations were fabricated using triangular and trapezoidal kirigami patterns, all constructed from 100 μm thick PET sheets. The tested configurations included 2 × 14 units (l = 10 mm), 3 × 15 units (l = 9 mm), 4 × 20 units (l = 7 mm), 5 × 28 units (l = 5 mm), and 6 × 28 units (l = 4.8 mm). Each sample was subjected to five actuation cycles on a compliant polyurethane foam substrate to simulate crawling on soft terrain.

3.3.2. Performance Analysis

Figure 6a illustrates the relationship between kirigami cut density and crawling distance for both triangular and trapezoidal patterns. The experimental results reveal a nonlinear trend, indicating that increasing cut density initially enhances crawling performance up to an optimal point, after which further increases lead to degradation. Among the tested samples, the 4 × 20 triangular kirigami configuration achieved the highest crawling distance of 74 mm, followed by 65 mm for the trapezoidal counterpart. This performance peak is attributed to an optimal balance between unit spacing and surface interaction. At this density, individual kirigami units have sufficient room to buckle vertically, achieving a favorable pop-up height while maintaining effective contact with the surface. This combination enhances anisotropic friction and enables forward propulsion. In contrast, both lower and higher densities resulted in reduced crawling performance.

For instance, the 2 × 14 configuration exhibited limited surface interaction due to fewer anchoring units and limited deformation engagement with the surface, achieving only 55 mm (triangular) and 48 mm (trapezoidal) of crawling distance. Although the 3 × 15 configuration showed modest improvement (61 mm and 54 mm), the performance dropped notably in the 5 × 28 and 6 × 28 configurations, with crawling distances decreasing to 42–44 mm and further to 34–36 mm, respectively.

The primary cause of reduced crawling performance at higher cut densities can be attributed to two interrelated factors. First, although individual kirigami units do deform across all tested configurations, the buckling height notably decreases in samples with smaller and more densely packed cuts. This lower vertical deformation limits the capacity of each unit to generate the anisotropic friction necessary for effective surface anchoring. As a result, despite undergoing buckling, these compact units fail to produce enough vertical engagement with the substrate, weakening their contribution to forward propulsion. Secondly, overcrowding from excessive unit density reduces the effective interaction area between the kirigami skin and the surface. With less spacing between adjacent units, the deformation of kirigami units may interfere with its nearby units, diminishing the structure’s ability to conform and grip the surface effectively. This leads to less consistent contact and reduced anchoring force per unit during the actuation cycle. Collectively, these two effects—lower pop-up height and compromised surface engagement—cause diminished frictional force and, consequently, reduced locomotion efficiency at high cut densities.

To further validate these findings, we conducted frictional force analysis using a friction measuring setup (Section 2.2). As shown in Figure 7, the 4 × 20 configuration exhibited the highest stick–slip friction peaks (~3 N). In contrast, 5 × 28 and 6 × 28 configurations demonstrated lower peak values, confirming that excessive density reduces frictional effectiveness. Figure 6b includes side-by-side deformation images of kirigami samples at each density, clearly demonstrating the decreasing buckling amplitude and effective contact area. These results collectively confirm that an optimal configuration represents a balance between deformation, friction, and effective anchoring of kirigami units. These findings provide valuable insight into the bioinspired design of functional kirigami skins, particularly for use in confined environments where surface adaptability, frictional modulation, and efficient motion are essential.

3.4. Stroke Length Variation in Microcontroller-Controlled Experiments

3.4.1. Experimental Conditions

To further evaluate the locomotion performance of the soft robotic crawler, a series of experiments was conducted to analyze the stroke-to-crawl efficiency across varying actuation strategies. In this context, efficiency was defined as the ratio between the total crawling distance achieved by the robot and the total extension performed by the actuator over the duration of the test. The actuator extension was calculated as the product of the actuator’s stroke length and the number of actuation cycles completed within the 30-second experiment window. The goal of this study was to understand how different combinations of stroke length and cycle timing influence the conversion of actuator motion into forward locomotion, thus complementing the density-based structural study detailed in Section 3.3. Four configurations were tested, all using triangular kirigami skin patterns fabricated from 100 μm thick PET sheets (with a Young’s modulus of 4.33 GPa and Poisson’s ratio of 0.4) and operated on a compliant polyurethane (PU) foam surface at a constant actuation speed of 15 mm/s. The tested stroke configurations were as follows: St10, which employed a 10 mm stroke and completed 15 cycles; St20, with a 20 mm stroke and 12 cycles; St30, which used a 30 mm stroke across 7 cycles; and St30h1s, which also used a 30 mm stroke but with an added 1-second hold time at the end of each stroke, completing 5 total cycles. These configurations resulted in total actuator extensions of 150 mm (St10), 240 mm (St20), 210 mm (St30), and 150 mm (St30h1s), respectively.

3.4.2. Performance Analysis

The results revealed significant variation in stroke-to-crawl efficiency. The St30h1s configuration achieved the highest crawling distance, recording 79 mm of forward displacement from just 150 mm of actuator motion—resulting in an efficiency of approximately 52.7%. This markedly outperformed the St30 configuration, which reached 67 mm of displacement from 210 mm of actuator extension (an efficiency of 31.0%). The lower stroke configurations, St20 and St10, exhibited reduced efficiencies of 19.2% and 17.3%, achieving crawling distances of 46 mm and 28 mm, respectively. These findings indicate that shorter stroke lengths, while allowing for more actuation cycles within a given time frame, result in lower overall displacement due to limited deformation amplitude and weaker surface engagement (Figure 8).

The superior performance of St30h1s is attributed to the strategic 1-second hold time incorporated at the end of each stroke. This pause enhances the robot’s interaction with the substrate by allowing the kirigami skin to fully engage and conform to the surface, thus maximizing anisotropic friction during the transition between extension and retraction. The added hold time in St30h1s appears to exploit this frictional asymmetry more effectively, yielding a stride length of 15.8 mm per cycle, compared to 9.3 mm per cycle in St30. The data demonstrate that stroke length alone does not guarantee efficient locomotion. While St30 outperformed the shorter-stroke configurations, it still required greater actuator extension than St30h1s to approach similar displacement levels. This implies that timing and mechanical behavior during each cycle—particularly features like dwell time—can have a substantial impact on propulsion efficiency. By extending contact duration during the extension phase, the crawler benefits from improved anchoring and energy transfer to the ground, leading to more effective locomotion per unit of actuator work.

3.5. Optimization of Actuation Speed Parameters

3.5.1. Experimental Conditions

As discussed in the previous sections, not only does the applied strain on the kirigami skin play a critical role in deformation behavior, but dynamic actuation parameters also significantly influence the effectiveness of directional anchoring and overall locomotion performance. To investigate this, the present study evaluates the impact of actuation speed—specifically extension and retraction rates—on the crawling efficiency of a soft robotic system powered by a PM linear actuator. Experiments were conducted using a triangular kirigami pattern with a 4 × 20 cut density fabricated on a 100 µm PET sheet over a 51 mm × 140 mm contact area. The actuator was programmed to perform five extension–retraction cycles at a constant stroke length of 30 mm across four speed conditions defined by matched extension and retraction rates: 10/10, 15/15, 20/20, and 30/30 mm/s. These conditions were selected to evaluate how increasing the frequency of cyclic deformation affects the ability of the kirigami pop-ups to effectively anchor and re-engage with the substrate during locomotion. The goal was to determine whether higher actuation speeds reduce surface interaction time and compromise traction or whether they can maintain or even enhance performance through faster cycling.

3.5.2. Performance Analysis

Results shown in Figure 9 show that the crawling distances increased with actuation speed, peaking at the highest speed ratio. The 10/10 condition yielded 55 mm (5 cycles), averaging 11 mm per cycle. The 15/15 condition achieved 71 mm, averaging 14.2 mm per cycle. The 20/20 condition reached 91 mm, averaging 18.2 mm per cycle, while the 30/30 condition recorded 95 mm, averaging 19 mm per cycle. These results show that speed also impacts dynamic engagement rather than static friction anchoring force $F_{\mathrm{prop}}$. This corresponds to a predicted per-cycle distance of ~19 mm (based on the model’s scaling with $F_{\mathrm{prop}}$), aligning with the observed 19 mm per cycle for 30/30.

These trends suggest that higher actuation speeds promote more complete deployment of the kirigami structures during each stroke, improving vertical pop-up height and surface engagement. In contrast, lower speeds result in slower, incomplete deployment, limiting anisotropic friction and reducing overall locomotion performance. To model this effect, a deployment factor was introduced—defined as the fraction of the actuator’s stroke effectively converted into forward crawling. At 10 mm/s, the deployment factor is ~37%, consistent with the observed 11 mm stride. As speed increases, the factor improves to ~47%, 61%, and 63% at 15, 20, and 30 mm/s, respectively, aligning closely with experimental results. At higher speeds (20–30 mm/s), the actuator imparts greater momentum, allowing the kirigami units to fully buckle and anchor. This enhances anisotropic friction, resulting in more effective energy transfer and improved mechanical efficiency. These findings confirm that faster actuation cycles enable more complete kirigami engagement, thereby maximizing traction and stride length.

4. Autonomous Crawling of the Robot

Following a comprehensive experimental investigation of kirigami geometries, actuation speeds, and locomotion metrics, the triangular kirigami pattern with a 4 × 20 configuration ($l=7 \mathrm{mm}$, 100 µm PET sheet) was identified as the optimal design parameters for the autonomous soft robotic crawler. This configuration demonstrated superior crawling performance in experiments, leveraging enhanced pop-up deformation and pronounced anisotropic friction properties to generate effective surface anchoring and directional propulsion. The autonomous system builds upon microcontroller-driven experimentation and integrates a fully embedded control strategy. The prototype consists of an Arduino Nano microcontroller, a TB6612FNG motor driver (PCBSINO Technologies Limited, Shenzhen, China), and 12 V lithium battery (YG 12V, Zhejiang Yuyao Guotai Electronic Co., Ltd., Yuyao, China), with a 5 V regulated supply delivered via an LM7805 converter. The complete system is housed in a lightweight 150 g platform (Figure 10). To support untethered and repeatable operation, we developed a custom firmware loop on the Arduino Nano that manages actuator timing, direction control, and polarity switching. This autonomous loop enables cyclic crawling strokes with precise dwell and transition intervals, eliminating the need for external computation or command input. Actuation was programmed for a 30 mm stroke, using an extension speed of 30 mm/s, a 0.5-second dwell at full extension, and a 30 mm/s retraction speed. A polarity-switching control algorithm ensures smooth and repeatable actuation cycles, enabling the kirigami structure to engage the substrate during extension and retract cleanly without compromising the pop-up geometry. This compact and energy-efficient configuration leverages the low current draw of the PM actuator and the voltage regulation for stable performance under load. Moreover, the modularity of the electronics makes the system easily adaptable for future integration of sensors or control extensions [29]. Under autonomous operation, the crawler achieved a crawling speed of ~47 cm/min, with an average forward displacement of ~18 mm per 30 mm actuator cycle. This corresponds to a stride efficiency of approximately ~60%, confirming that nearly two-thirds of each actuator stroke contributed to productive locomotion (see Supplementary Video S1). These results validate the system’s mechanical and structural optimization and reinforce the potential of kirigami-enhanced soft robotics for untethered and adaptable movement in constrained environments.

5. Conclusions

This study presents a bioinspired kirigami structure engineered to enhance the anchoring performance of a soft robotic system through systematic design and parameter optimization. Drawing inspiration from the anisotropic friction mechanisms observed in snake and lizard scales, we integrated patterned kirigami skins with a soft silicone body actuated by a compact linear actuator. The kirigami cuts comprising linear, triangular, trapezoidal, and hybrid geometries were evaluated for their ability to produce controlled buckling-induced deformation that translates into effective directional surface anchoring.

Through a comprehensive set of experiments, we analyzed how variations in kirigami geometry, sheet thickness, cut density, stroke length, and actuation speed impact anchoring efficiency, as quantified by crawling distance and stride performance. Our results show that triangular and hybrid designs offer significant improvements in propulsive force and surface engagement over baseline linear patterns. From a series of experiments, the triangular kirigami cuts on a 100 µm PET sheet with a 4 × 20 density were identified as the optimal configuration. Using these optimized design features and refined actuation parameters derived from experimentation, the kirigami-based crawler achieved a stride efficiency of approximately 63% and a maximum autonomous crawling speed of ~47 cm/min, confirming the configuration’s effectiveness as a soft robotic anchoring mechanism.

Complementing the experimental results, we developed a theoretical model that combines buckling mechanics with frictional analysis to predict pop-up heights and directional friction coefficients. The strong agreement between predicted and observed performance metrics validates the model’s utility for guiding future kirigami-based anchoring designs. Overall, this work demonstrates that kirigami-engineered skins, when coupled with efficient linear actuation and parameter tuning, provide a scalable and tunable approach to improving anchoring performance in soft robots. The proposed system offers a compact alternative to traditional pneumatic actuators and holds promise for applications requiring autonomous locomotion and high friction control in constrained or unstructured environments. Future work will focus on extending this platform through sensor integration and terrain-adaptive control to further enhance anchoring robustness and functional autonomy.