# Bioinspired Kirigami Structure for Efficient Anchoring of Soft Robots via Optimization Analysis
## 3 Results and Analysis
### Effect of Kirigami Structure on Crawling Efficiency
#### 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 sur face 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 \(\mu_{\text{forward}}=2.241\) and \(\mu_{\text{backward}}=0.776\), generating a propulsive force of \(F_{\text{prop}}\approx 0.850\) N. This aligns with the observed crawling distance of 65 mm (13 mm per cycle), yielding a crawling efficiency of \(\sim\)43% (13 mm per 30 mm stroke). The pop-up height for this configuration was calculated as \(h=5.61\) mm, which facilitates strong surface anchoring. The trapezoidal pattern, with measured friction coefficients (\(\mu_{\text{forward}}=2.414\), \(\mu_{\text{backward}}=0.862\)), produced a slightly higher propulsive force (\(F_{\text{prop}}\approx 0.900\) N), but its crawling distance of 51 mm (10.2 mm per cycle, efficiency \(\sim\)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 (\(\mu_{\text{forward}}=5.000\), \(\mu_{\text{backward}}=1.897\), \(F_{\text{prop}}\approx 1.800\) N), yet covered only 59 mm (11.8 mm per cycle, efficiency \(\sim 39\)%). 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 \(\sim\)40%), with \(\mu_{\text{forward}}=2.586\), \(\mu_{\text{backward}}=1.552\) and \(F_{\text{prop}}\approx 0.600\) 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 (\(\mu_{\text{forward}}\approx\mu_{\text{backward}}\), \(F_{\text{prop}}\approx 0\) 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.  
Figure 4: Crawling distance comparison of different kirigami patterns demonstrating the impact of geometry on locomotion.