A Rolling Soft Robot Driven by Local Snap-Through Buckling

Abstract

Previous rolling soft robots have difficulty in balancing the locomotion speed with energy efficiency and have limited terrain adaptability. This work proposes a rolling soft robot driven by local snap-through buckling, which employs the fast response and configuration maintenance of the bistable structure to enhance the locomotion performance of the soft robot. A theory based on bifurcation and the energy principle is established to analyze the rolling mechanism. The influences of loading position and geometric parameters on the rolling performance are investigated and verified experimentally. The soft robot shows good locomotion speed (0.95 body length per second, BL/s) and small energy loss due to the almost unchanged configuration during the rolling process. The soft robot adapts to complex terrains, including a step with the height of 15 mm, a slope with the angle of 18.36°, and a broken bridge with the gap length of 90 mm (0.443 BL). The proposed rolling soft robot not only has good application prospects in land exploration missions and medical applications but also provides inspiration for the development of rolling soft robots.

Introduction

Soft robots made of flexible materials can change their shapes by bending, tension, compression, and torsion to adapt to various confined spaces.¹ Compared with rigid robots, soft robots have infinite degrees of freedom, safe interaction with humanity, and good flexibility,² thus expanding the application fields of robots, such as pipeline inspection,³ disaster relief and rescue,⁴ underwater surveys,⁵ medical services,⁶ and social services.⁷ Terrestrial locomotion of the soft robot includes crawling, rolling, walking, jumping, and so on.^8,9 Rolling locomotion has attracted much attention due to its good potential in high-speed locomotion and good adaptability to unknown terrain.^10,11

The rolling mechanisms of soft robots include gravity-based rolling,^12–15 ballistic rolling,^10,16 passive rolling,^17–19 and so on. Gravity-based rolling changes the robot’s shape and shifts the center of gravity to achieve rolling, but this rolling mechanism usually has a slow rolling velocity and poor adaptability to complex terrains. Ballistic rolling utilizes the fast deformation of the robot’s body to cause fast rolling locomotion, although the drastic deformation leads to poor locomotion stability, large energy consumption, and limited terrain adaptability. Passive rolling is achieved by deformation induced by external stimuli such as humidity,¹⁷ magnetic fields,¹⁸ and light,¹⁹ but it is sensitive to environmental disturbance. Conventional rolling mechanisms usually have a large deformation level²⁰ (the ratio of the maximum size to the minimum size of the soft robot in one loading cycle), low energy efficiency²¹ (the ratio of effective locomotion energy to the total energy input), and limited terrain adaptability. Moreover, previous rolling soft robots failed to balance the rolling velocity with the rolling stability. Therefore, it is necessary to study new rolling mechanisms to improve the rolling performance of soft robots.

Recently, bistable structures have offered a promising actuation method for increasing the locomotion speed of soft robots. A bistable structure has two stable equilibrium states and does not require continuous energy input to maintain the stable state, which can achieve fast locomotion, improve the driving efficiency, and amplify the force output through the bistable transition.²² The bistable structure improves the locomotion performance of the soft robot by using its fast response and antidisturbance characteristics. Examples include a TMP (Tachi-Miura polyhedron) origami-shell-reinforced bistable soft robot,²³ a soft swimming robot powered by shape memory polymer muscles,²⁴ and a high-speed cheetah-like running soft robot.²⁵ However, these soft robots driven by bistable structures usually suffer from excessive deformation during locomotion. A large amount of input energy is transformed into strain energy instead of the desired locomotion energy, which leads to low energy efficiency. Besides, using bistable structures to drive the rolling soft robot may have the potential to improve the rolling performance of soft robots, but it is still a challenge for the robot design since previous bistable structures are hard to balance these rolling performances (rolling velocity, energy efficiency, rolling stability, and terrain adaptability).

This article proposes a novel rolling soft robot and a rolling mechanism with balanced rolling performance, which adopts two nested elastic rings as the robot structure and uses local snap-through buckling to drive the rolling of the soft robot. The pneumatic actuation is used to drive the local bistable transition due to its large driving force, fast response, and precise control.¹ This localized bistable design not only ensures fast response and high rolling speed but also avoids excessive overall deformation, thus improving energy efficiency. In addition, the two-nested ring structure enhances the stability of the robot’s locomotion and improves the adaptability to complex terrain, which helps to solve the challenges faced by traditional rolling soft robots. This article is organized as follows. Firstly, the design and fabrication of soft robots and soft actuators are introduced. Then, to explore the rolling mechanism driven by local snap-through buckling, a buckling model of static configuration of the two nested elastic rings considering gravity is established. A critical curvature theory based on the bifurcation and energy principle is proposed to theoretically analyze the effects of loading positions and geometrical parameters on the critical loading curvature and the critical rolling velocity, which are in agreement with the experimental results. Finally, the rolling experiments of the soft robot on different terrains, such as the flat surface, steps, the slope, and the broken bridge, are demonstrated, which proves that the soft robot has good environmental adaptability as well as locomotion performance.

Materials and Methods

Structure design

The proposed soft robot consists of an elastic inner ring, an elastic outer ring, and eight soft actuators, as shown in Figure 1a. The elastic inner ring is made of a thin polyvinyl chloride sheet with a length of 509 mm, width of 50 mm, and thickness of 0.381 mm, while the elastic outer ring is made of printed paper with a length of 487 mm, width of 50 mm, and thickness of 0.104 mm. The bending stiffness ratio of the inner ring to the outer ring is 919, and the length ratio of the inner ring to the outer ring is 1.0452. The eight soft actuators are glued equidistantly on the inner side of the inner ring. How to determine the number of soft actuators is analyzed in Supplementary Data S1. The two ends of the inner and outer rings are connected, respectively, and then the inner ring is put inside the outer ring. Due to the mismatched length of the inner and outer rings, the inner and outer rings are subjected to compression force and tensile force, respectively, resulting in an inward “blister” buckling structure in the inner ring. The overall configuration of two nested rings resembles the tank track due to the loading of gravity. The fabricated soft robot is shown in Supplementary Figure S1, with a length of 202.5 mm, a height of 73.5 mm, a width of 50 mm, and a mass of 83.85 g.

FIG. 1.

Design of the soft robot and performance test of the soft actuator. (a) Schematic of fabrication process for the soft robot. (b) The relationship between loading pressure and output force for soft actuators. (c) The relationship between loading pressure and bending angle for a soft actuator. For the actuator characterization, each experiment’s data are the average value of five measurements.

To drive the rolling of a soft robot, a compact bending soft actuator is designed based on the “Pneu-nets” structure.²⁶ The soft actuator is made of silicone rubber (Dragon Skin™ 20) by using a 3D printed mold. The length, the width, and the thickness of the soft actuator are 26 mm, 50 mm, and 7 mm, respectively (Supplementary Fig. S2). Compared with traditional “Pneu-nets” structure,²⁶ the soft actuator replaces the integrated nonstretchable layer with the elastic inner ring, simplifying the robot’s structure and manufacturing process and improving the overall integration of the soft robot.

Figures 1b and 1c show the effects of loading pressure on the bending performance of the soft actuator. The results show that with the increase of loading pressure, the output force and bending angle of the soft actuator increase. When the loading pressure is 70 kPa, the bending angle of the soft actuator reaches 90° and the output force reaches 3 N. Considering the service life of the soft actuator, the loading pressure under 70 kPa will be used in the subsequent experiments. The output force in Figure 1b is related to the driving force of the robot rolling on complex terrains, which will be discussed in the “Locomotion performance tests” section. The angle-pressure curve in Figure 1c will be used to calculate the experimental critical loading curvature k_c in Figure 6.

Locomotion analysis

The locomotion of the soft robot is realized by loading the soft actuator in the specific sequence. To control the soft actuator, we built an air pressure control system. The details of the air pressure control system are shown in Supplementary Data and Supplementary Figure S3. The rolling locomotion of the proposed soft robot is shown in Figure 2. When a pressure of 40 kPa is applied to the soft actuator A1 (Fig. 2a), the actuator produces bending deformation and thus applies a local loading curvature to the inner ring. Note that the applied curvature of the soft actuator A1 has an opposite sign with the intrinsic curvature of the inner ring at the initial loading position in Figure 2a.

FIG. 2.

Schematic of the locomotion when loading soft actuator A1. (a) Initial state, (b) snap-through intermediate state, and (c) end state of soft actuator A1. The yellow wireframe in the right figure denotes the soft actuator being loaded. The soft robot rolled 45 mm under a loading pressure of 40 kPa.

When the loading curvature exceeds a critical value, the soft actuator A1 rapidly moves rightward along the inner ring (Fig. 2b) and drives the overall locomotion of the soft robot until it arrives at a stable state with the opposite curvature (Fig. 2c). Meanwhile, the soft actuator A2 reaches the initial position of the soft actuator A1. The soft actuator A1 can maintain the stable state and the position after it is unloaded. Then, if an identical pressure is applied to the soft actuator A2, the soft actuator A2 will produce the same locomotion as the soft actuator A1. Therefore, if soft actuators are loaded with the pressure in the sequence (A1–A2–…–A8), the soft robot will realize rightward continuous rolling. Conversely, if soft actuators are loaded in the order of (A8–A7–…–A1), the soft robot will realize leftward continuous rolling. For the two-nested ring structure of the robot, there is sufficient compression force and friction between the two rings, which prevents the slippage between the two rings during the locomotion, and no slippage was observed in the experiments.

Static configuration analysis

Because the two nested ring configuration of the proposed soft robot is different from previous rolling soft robots, it is necessary to establish a corresponding mechanics theory to provide a theoretical basis for the structure design and the rolling mechanism analysis. Here, based on the principle of minimum potential energy, a buckling model is established to analyze the static configuration of two nested rings by considering gravity loading. Figure 3 shows the force diagram of the soft robot with the Cartesian coordinate oxy. Because the soft robot configuration is symmetric with respect to the y-axis, the right half of the configuration is taken as an example. The natural coordinate is used to represent the arc length of the configuration curve. Here the superscripts in, out, and com represent the inner ring, the outer ring, and the contact region of the two rings, respectively. The vertex of the blister of the elastic inner ring is set as the origin of the inner ring with sⁱⁿ = 0. The top middle point of the outer ring is set as the origin of the elastic outer ring with s^out = 0. The separation point between the contact region of two rings and the ground is set as the origin of the contact region of two rings with s^com = 0. The arc lengths of the inner ring, the outer ring, and the contact region of the two rings at the separation point of the two rings are ${\bar{s}}^{in}$ , ${\bar{s}}^{out}$ , and ${\bar{s}}^{com}$ , respectively.

FIG. 3.

Force diagram of soft robot configuration. The orange triangle represents the separation point of inner and outer rings (blister separation point). The red dots represent the origins of the inner ring, the outer ring, and the contact region of the two rings.

Take the inner ring as the example to derive the static equilibrium equation. The Cartesian coordinates of any point P on the inner ring can be expressed as r (sⁱⁿ) = x(sⁱⁿ) e _x + y(sⁱⁿ) e _y , where e _x and e _y are the unit vectors on the x-axis and y-axis, respectively. The tangent vector of point P on the inner ring can be expressed as τ (sⁱⁿ) = cosθⁱⁿ(sⁱⁿ) e _x + sinθⁱⁿ(sⁱⁿ) e _y , where θⁱⁿ(sⁱⁿ) is the deflection angle of point P relative to the x-axis. The blister satisfies the nonstretchable condition r $'$ (sⁱⁿ) = τ (sⁱⁿ), and the relationship between the Cartesian coordinate and the deflection angle can be obtained^27,28 $x^{'} (s^{in}) = \cos θ^{in} (s^{in}), y^{'} (s^{in}) = \sin θ^{in} (s^{in})$ (1)

Thus, the energy functional of the inner ring containing the bending energy, gravitational potential energy, and the constraint terms can be expressed as $\begin{array}{l} W^{in} = \int_{0}^{{\bar{s}}^{in}} w^{in} ({x^{'}}^{in}, {y^{'}}^{in}, y^{in}, {θ^{'}}^{in}, θ^{in}) d s^{in} \\ = \int_{0}^{{\bar{s}}^{in}} (\underset{Bending energy}{\underset{︸}{\frac{K^{in}}{2} {({θ^{'}}^{in} (s^{in}))}^{2}}} + \underset{\begin{array}{l} Gravitational \\ energy \end{array}}{\underset{︸}{ρ g y^{in}}} - \underset{x constraint}{\underset{︸}{F_{x}^{in} ({x^{'}}^{in} - \cos θ^{in})}} + \underset{y constraint}{\underset{︸}{F_{y}^{in} ({y^{'}}^{in} - \sin θ^{in})}}) d s^{in} \end{array}$ (2)where Kⁱⁿ = EⁱⁿIⁱⁿ denotes the bending stiffness, and Eⁱⁿ and Iⁱⁿ denote the Young’s modulus and the moment of inertia, respectively. θ $^{'}$ ⁱⁿ(sⁱⁿ) = dθⁱⁿ/dsⁱⁿ denotes the curvature. $F_{x}^{in}$ denotes the internal force in the x direction. $F_{y}^{in} =$ ρⁱⁿgsⁱⁿ denotes the internal force in the y direction. $F_{x}^{in}$ and $F_{y}^{in}$ have the meaning of Lagrange multipliers. ρⁱⁿ denotes the density. Two constraint terms of the energy functional come from the nonstretchable condition.

Assuming an infinitesimal virtual deformation of the inner ring curve θ_ξ(sⁱⁿ) = θ (sⁱⁿ) + ξη(sⁱⁿ), x_ξ(sⁱⁿ) = x(sⁱⁿ) + ξu(sⁱⁿ), y_ξ(sⁱⁿ) = y(sⁱⁿ) + ξv(sⁱⁿ), where ξ is a small positive parameter, the first order variation of Wⁱⁿ can be written as $δ W^{in} = \int_{0}^{{\bar{s}}^{in}} [\frac{\partial w^{in}}{\partial θ^{in}} - (\frac{\partial w^{in}}{\partial {θ^{'}}^{in}})^{'}] \cdot η + [\frac{\partial w^{in}}{\partial x^{in}} - (\frac{\partial w^{in}}{\partial {x^{'}}^{in}})^{'}] \cdot u + [\frac{\partial w^{in}}{\partial y^{in}} - (\frac{\partial w^{in}}{\partial {y^{'}}^{in}})^{'}] \cdot v d s$ (3)

Considering the arbitrariness of η(sⁱⁿ), u(sⁱⁿ), v(sⁱⁿ), the inner ring needs to satisfy the following equilibrium equations ${\begin{matrix} K^{in} {θ^{″}}^{in} (s^{in}) - F_{x}^{in} \sin θ^{in} (s^{in}) + F_{y}^{in} \cos θ^{in} (s^{in}) = 0 \\ F_{x} {^{'}}^{in} = 0 \\ F_{y} {^{'}}^{in} = ρ^{in} g \end{matrix}$ (4)where $F_{x} {^{'}}^{in} =0$ indicates that the internal force $F_{x}^{in}$ in the x direction is a constant. From this $F_{y} {^{'}}^{in} = ρ^{in} g$ , we can get $F_{y}^{in} = ρ^{in} g s^{in}$ , which indicates that the internal force $F_{y}^{in}$ in the y direction varies with arc length sⁱⁿ.

Similarly, the static equilibrium equations for the outer ring and the contact region of two rings can be obtained as follows: ${\begin{matrix} K^{out} {θ^{″}}^{out} (s^{out}) - F_{x}^{out} \sin θ^{out} (s^{out}) + F_{y}^{out} \cos θ^{out} (s^{out}) = 0 \\ K^{com} {θ^{″}}^{com} (s^{com}) - F_{x}^{com} \sin θ^{com} (s^{com}) + F_{y}^{com} \cos θ^{com} (s^{com}) = 0 \end{matrix}$ (5)where K^out = E^outI^out and K^com = E^comI^com denote the bending stiffness. E^out and E^com denote the Young’s modulus. I^out and I^com denote the moment of inertia. $F_{x}^{out}$ and $F_{x}^{com}$ denote the constant internal forces in the x direction. $F_{y}^{out} = ρ^{out} g s^{out}$ and $F_{y}^{com} = [ρ^{in} {\bar{s}}^{in} + ρ^{out} {\bar{s}}^{out} + ρ^{com} ({\bar{s}}^{com} - s^{com})]$ denote the varying internal forces in the y direction. ρ^out and ρ^com denote the density, where ρ^com = ρⁱⁿ + ρ^out. In addition, the internal forces of the inner ring, the outer ring, and the contact region of the two rings satisfy the static force equilibrium at the separation point, that is, $F_{x}^{in} + F_{x}^{out} + F_{x}^{com} =0$ , $F_{y}^{i n} + F_{y}^{out} + F_{y}^{com} =0$ .

The structure of the soft robot is symmetric and the boundary conditions θⁱⁿ(0) = 0, θ^out(0) = 0 are satisfied at the symmetry axis. The boundary conditions θ $^{'}$ ⁱⁿ(0) = 0, θ^com (0) = 0 are satisfied at the separation point between the contact region of the two rings and the ground. For two nested elastic rings, the deflection angle and curvature are continuous along the curve, so the continuity condition is satisfied at the separation point of the inner and outer rings. ${\begin{cases} θ^{com} ({\bar{s}}^{com}) = θ^{in} ({\bar{s}}^{in}) = θ^{out} ({\bar{s}}^{out}), \\ {θ^{'}}^{com} ({\bar{s}}^{com}) = {θ^{'}}^{in} ({\bar{s}}^{in}) = {θ^{'}}^{out} ({\bar{s}}^{out}), \end{cases}$ (6)

By combining the continuity conditions and boundary conditions, the numerical solution of the soft robot configuration can be obtained by using the built-in solver “bvp4c” in MATLAB. One example of the numerical solution of the theory is shown as the black dotted line in Supplementary Figure S1, which is in good agreement with the experiment and verifies the correctness and feasibility of the above buckling model.

According to Equations (4) and (5), the configuration of the soft robot is determined by the stiffness ratio Kⁱⁿ/K^out, the length ratio Lⁱⁿ/L^out, and the gravity of the soft robot, as shown by the configuration map in Supplementary Figure S4. The results show that the stiffness ratio dominates the overall configuration of the soft robot. The large Kⁱⁿ/K^out corresponds to the low gravity center, the large contact area with the ground, good terrain adaptability, and good rolling stability. It should be noted that when Kⁱⁿ/K^out > 10, the effect of Kⁱⁿ/K^out becomes slight. The length ratio dominates the shape of the blister, which may affect the rolling performance of the soft robot and will be shown in Figure 7e. Therefore, the appropriate stiffness ratio and length ratio of the soft robot can be selected to meet the diverse task requirements. In this study, we pursue the balanced performance of the soft robot in terms of rolling velocity, rolling stability, and adaptability to complex terrains, and thus we chose a high stiffness ratio Kⁱⁿ/K^out of 919 and a suitable length ratio Lⁱⁿ/L^out of 1.0452 for the terrain adaptability test.

Rolling mechanism analysis

As shown in the “Locomotion analysis” section, the locomotion of the proposed soft robot is essentially a passive motion driven by the bistable transition motion of the local loading region. Due to the bistable characteristics of the proposed soft robot, which is significantly different from the traditional rolling mechanism, it is necessary to establish a corresponding theoretical model to analyze the critical loading curvature of the soft robot that triggers rolling.

Due to the complexity of directly analyzing the locomotion of the overall configuration, the analysis of the soft robot is simplified to the analysis of the local loading region, that is, to study the bistable mechanism and the critical loading curvature of the local loading region. Taking the motion of the soft actuator A1 in Figure 2 as an example, the initial state and the end state of the soft actuator A1 are represented in Figure 4a. Note that the blister has varied curvature at different positions. By using the zero curvature point (blue dot in Fig. 4a) as the boundary, set the curvature of the green blister region as negative and the curvature of the yellow blister region as positive. Ignore the global displacement and rotation of the loading region, and put the two ends of the loading region of two stable states together (Fig. 4b). It is observed that the local loading region transits between two stable states with opposite curvature signs, and there is an asymmetrical unstable state and a critical state. This feature is similar to the bistable beam driven by symmetrical boundary actuation.²⁹ The difference is that the bistable structure in this work has a “movable boundary,” and the distance d_l between two ends of the loading region is not fixed during the bistable transition.

FIG. 4.

(a) Schematic of the locomotion principle for one loading cycle. The red line denotes the loading region. The yellow dots denote the ends of the loading region. The blue dot denotes the zero curvature point. (b) Schematic of snap-through transition of the loading region. The green solid line and yellow solid line denote stable states I and II, respectively. The black dashed line denotes the unstable state. The red dashed line denotes the critical state. d_l presents the distance between two ends of the loading region. s_l presents the length of the local loading region. β denotes the symmetric boundary loading angle. β_c denotes the critical boundary loading angle. (c) Schematic of the curvature distribution of the blister and the loading positions P₁∼P₅. Blue dot presents zero curvature point. (d) Schematic of the static configuration for the same inner ring length Lⁱⁿ (509 mm) with different outer ring lengths L^out (480 mm, 487 mm, 495 mm, and 500 mm).

Since the length of the local loading region s_l is much smaller than that of the blister, it can be assumed that the distance d_l between two ends of the loading region is fixed, which will be verified by the experiments in the “Analysis of critical loading curvature” section. Based on this assumption, the snap-through buckling phenomenon in the local loading region can be regarded as the saddle-node bifurcation.²⁹ The method of applying boundary loading angles at two ends can be used to analyze the snap-through transition of the loading region under curvature loading. The change of midpoint deflection ω in the loading region can be used to represent the locomotion of the loading region under boundary loading angle β (Fig. 4b). In this way, our problem becomes to explore the minimum boundary loading angle of the snap-through buckling behavior in the loading region under the fixed distance d_l constraint, that is, to study the critical boundary loading angle β_c.

According to the static equilibrium equations in the “Static configuration analysis” section, the distance d_l between two ends of any loading region on the blister can be solved numerically, where the length of the loading region is fixed at s_l = 0.026 m. Then, the equilibrium configurations of the stable and unstable states of the loading region subjected to any boundary loading angle β can be numerically solved by combining Equations (1) and (4) with the distance d_l and the boundary condition. $θ^{in} (s_{1}^{in}) = β, θ^{in} (s_{2}^{in}) = - β$ (7)where $s_{1}^{in}$ denotes the arc coordinate of the left end of the loading region and $s_{2}^{in}$ denotes the arc coordinate of the right end of the loading region.

Based on the equilibrium configuration of the loading region, the midpoint deflection ω and normalized strain energy can be obtained for any boundary loading angle β. As the boundary loading angle β increases, the midpoint deflection ω of the loading region changes. When the midpoint deflection of the stable state I is equal to that of the unstable state, the system instantly transits to another stable state (stable state II) with lower energy, and the critical boundary loading angle β_c is obtained.

The position of the local loading region influences the critical boundary loading angle, and five loading positions are shown in Figure 4c. Here these loading positions denote the midpoint of the loading region or the soft actuator. The point of zero curvature is denoted as P₀ and the point of maximum curvature in the green blister region is denoted as P₅. The normalized loading position can be defined as $s_{p l} = \frac{s_{pi} - s_{p0}}{s_{p5} - s_{p0}}$ (8)where s_pi is the arc coordinate of the midpoint of the loading region, s_p5 is the arc coordinate of point P₅, and s_p0 is the arc coordinate of point P₀. Four points P₁∼P₄ with different normalized loading positions (0.2845, 0.4282, 0.5718, and 0.7155), are taken between P₀ and P₅.

The theoretical critical loading curvature can be expressed as $k_{c} = λ \frac{β_{c}}{s_{l}} {(\frac{L^{in}}{L^{out}})}^{6} + σ {(1 - s_{p_{l}})}^{2}$ (9)where λ and σ are scaling factors. λ = 0.5275 and σ = 4.1615 for the loading region with loading length s_l = 0.026 m. In contrast, the experimental critical loading curvature of the soft actuator at different loading positions for different length ratios can be calculated as follows: $k_{c e} = \frac{π α}{180 s_{_{l}}}$ (10)where α is the bending angle of the soft actuator at different loading positions for different length ratios. In the experiment, the critical loading pressure of the soft actuator at different loading positions for different length ratios can be obtained, and then the corresponding bending angle α of the soft actuator could be obtained by the angle-pressure curve in Figure 1c.

The theoretical critical rolling velocity can be expressed as: $v_{c} = τ s_{p l} {(\frac{L^{in}}{L^{out}})}^{10} - γ \frac{β_{c}}{T}$ (11)where τ and γ are scaling factors, and T is the duration time of actuation for each actuator. τ = 0.47, γ = 0.235, and T = 1 s for the robot with Kⁱⁿ/K^out = 919.

According to Equations (9) and (11), the theoretical critical loading curvature k_c and the theoretical critical rolling velocity v_c of the soft robot are affected by the length ratios Lⁱⁿ/L^out, the loading position s_pl, and the critical boundary loading angle β_c. The length ratios and the loading position determine the initial mean curvature of the loading region, which in turn affects the critical boundary loading angle and the theoretical critical loading curvature. The magnitude of the theoretical critical loading curvature determines the loading pressure, which in turn determines the driving force and rolling velocity of the soft robot. Therefore, different application scenarios, rolling velocity, and driving forces can be considered when selecting appropriate length ratios and loading positions.

Results and Discussion

Analysis of critical loading curvature

Take the soft robot configuration with Kⁱⁿ/K^out = 919 and Lⁱⁿ/L^out = 1.0452 as an example, and the loading position is at P₃. The midpoint deflection ω and normalized strain energy of the loading region can be obtained, as shown in Figures 5a and 5b. With the increase of the boundary loading angle β, the midpoint deflection ω of the stable state I of the loading region decreases (Fig. 5a), and the normalized strain energy accumulates (Fig. 5b). When the midpoint deflection and the strain energy of the stable state I are equal to those of the unstable state, the loading region transitions from the stable state I to the stable state II instantaneously, and the critical boundary loading angle β_c = 0.514 rad can be determined. After the snap-through transition, the strain energy of the loading region plummets (Fig. 5b).

FIG. 5.

(a) Schematic of the midpoint deflection ω at the loading position P₃ versus the boundary loading angle β where Kⁱⁿ/K^out = 919, Lⁱⁿ/L^out = 1.0452, and s_l = 0.026 m. (b) Schematic of the normalized strain energy at the loading position P₃ versus the boundary loading angle β, where Kⁱⁿ/K^out = 919, Lⁱⁿ/L^out = 1.0452, and s_l = 0.026 m. (c) Schematic of the relationship between the midpoint deflection ω and the boundary loading angle β for different loading positions of the same configuration with Kⁱⁿ/K^out = 919, Lⁱⁿ/L^out = 1.0452, and s_l = 0.026 m. (d) Schematic of the relationship between the midpoint deflection ω and the boundary loading angle β at loading position P₅ for different length ratios Lⁱⁿ/L^out (1.0604, 1.0452, 1.0283, 1.0180) where Kⁱⁿ/K^out = 919 and s_l = 0.026 m.

In order to investigate the effect of the loading position on the critical boundary loading angle β_c, the relationship between the midpoint deflection ω and the boundary loading angle β for loading positions P₁∼P₅ is calculated, as shown in Figure 5c. The loading position closer to the zero curvature point has the smaller critical boundary loading angle β_c and the smaller critical loading curvature k_c.

The length ratio Lⁱⁿ/L^out affects the geometry of the blister and therefore influences the critical loading curvature k_c. By fixing other parameters, the critical boundary loading angle β_c of the soft robot with different length ratios Lⁱⁿ/L^out is shown in Figure 5d. Note that the length ratio is tuned by changing L^out and fixing Lⁱⁿ as shown by configurations in Figure 4d. As the length ratio Lⁱⁿ/L^out decreases, the blister structure becomes elongated, the initial mean curvature of the loading region decreases, and the critical boundary loading angle β_c decreases. Therefore, the critical loading curvature k_c is smaller for slender blister structures with smaller length ratios Lⁱⁿ/L^out.

In order to provide useful guidance on the soft robot design and loading position selection, the relationship between the critical loading curvature k_c of the blister and the loading position s_pl is calculated for three different length ratios Lⁱⁿ/L^out, as shown in Figure 6. For the same loading position s_pl, the configuration with a larger length ratio Lⁱⁿ/L^out requires a larger critical loading curvature k_c. In order to verify the proposed theory, corresponding experiments were conducted by fabricating soft robots with fixed Lⁱⁿ (509 mm), fixed Kⁱⁿ/K^out (919), and varied L^out (480 mm, 487 mm, and 495 mm), as shown by the experiment results in Figure 6. The experimental critical loading curvature of the soft actuator at different loading positions for different length ratios is calculated by Equation (10). In Figure 6, the experimental results are consistent with the proposed theory, which also proves the correctness of the assumption of fixed d_l.

FIG. 6.

The critical loading curvature k_c with different loading position s_pl for three length ratios Lⁱⁿ/L^out (1.0604, 1.0452, 1.0283).

In the above analysis, it can be seen that the loading position closer to the zero curvature point shows smaller critical loading curvature of the soft robot and smaller loading pressure in the experiment, which corresponds to a smaller driving force to the soft robot. For the loading position selection of the soft robot, appropriate driving force needs to be considered. For example, for complex terrains such as steps and the slope, a larger driving force is required, so the loading position away from the zero curvature point can be selected. In addition, a larger driving force can be achieved by increasing length ratios Lⁱⁿ/L^out with the loading position fixed. However, large length ratios Lⁱⁿ/L^out increase the height of the blister, which is unfavorable to the rolling of the soft robot. Therefore, the application scenarios and the required driving force need to be comprehensively considered in the design and loading position selection of the soft robot to ensure optimum performance in practical applications.

Locomotion performance tests

The adaptability to complex terrain and the rolling performance are important indicators to evaluate a soft robot’s potential for application. A series of experiments were carried out to investigate the rolling performance on flat ground and the locomotion performance in various terrains (steps, the slope, and the broken bridge).

The rolling velocity of the soft robot is related to the loading pressure and loading timing sequence. By using the loading pressure of 55 kPa and the loading timing diagram in Figure 7a, the soft robot moved 500 mm in 2.6 s, with an average speed of 192.3 mm/s or a body length velocity of 0.95 BL/s, as shown in Figure 7b and Supplementary Movie S1. The loading control parameters of the loading timing include the loading pressure P for each actuator, the duration time T of actuation for each actuator, and the overlapping time T_o between neighboring actuators. Note that if not specified, the soft robot in this section is configured with Kⁱⁿ/K^out = 919 and Lⁱⁿ/L^out = 1.0452, with the loading position at the point of maximum curvature.

FIG. 7.

(a) Loading timing diagram for soft actuator control of flat ground rolling with P = 55 kPa, T = 0.3 s, and T_o = 0 s. (b) The soft robot’s rolling on flat ground. (c) The rolling velocity (one step of rolling) with the loading pressure P. In the experiment, only one actuator is loaded to drive one step of rolling, and the duration time T is fixed at 1 s. (d) Effects of control parameters (duration time T and overlapping time T_o) on the average rolling velocity of the soft robot. In the experiment, eight actuators are loaded with different loading timing. For the test of duration time (yellow line), P = 55 kPa, T_o = 0 s. For the test of the overlapping time (blue line), P = 55 kPa, T = 0.5 s. The negative value for T_o denotes the loading interval time between neighboring actuators. (e) The critical rolling velocity v_c with different loading position s_pl for three length ratios Lⁱⁿ/L^out (1.0604, 1.0452, 1.0283).

In Figure 7c, the effect of the loading pressure P on the rolling velocity (one step of rolling) is shown where only one actuator is loaded to drive one step of rolling and the duration time T is fixed at 1 s. For the tested robot and tested actuator, there is a critical rolling velocity (0.45 BL/S) and a maximum rolling velocity (0.91 BL/S) for one step of rolling. The robot cannot roll when the loading pressure is smaller than the critical value of 40 kPa. When P ≥ 40 kPa, the rolling velocity increases with the increase of the loading pressure until the robot reaches the maximum rolling velocity with P ≥ 55 kPa. Therefore, for each actuator, a loading pressure P between the critical value and the maximum value should be selected for actuation.

In Figure 7d, the effect of the duration time T (yellow line) on the average rolling velocity (continuous rolling) is shown where eight actuators are loaded with P = 55 kPa and T_o = 0 s. The average rolling velocity decreases with the increase of T. There is a maximum average rolling velocity (0.95 BL/S), which corresponds to a minimum T (0.3 s), as shown in Figure 7b and Supplementary Movie S1. The robot cannot roll when the duration time T < 0.3 s because the loading time for each actuator is too short to drive one step of rolling.

The effect of the overlapping time T_o (blue line) on the average rolling velocity (continuous rolling) is also shown in Figure 7d, where eight actuators are loaded with P = 55 kPa and T = 0.5 s. The average rolling velocity increases with the increase of T_o. There is a maximum average rolling velocity (0.85 BL/S), which corresponds to a maximum T_o (0.2 s). The robot cannot roll when the overlapping time T_o is over the maximum value as the loading time for each actuator is too short to drive one step of rolling. Therefore, for the fast rolling velocity, small T and large T_o should be selected, but the control parameters should ensure that the loading duration time for each actuator has a minimum of 0.3 s.

The proposed robot has the critical rolling velocity under the critical loading pressure, which is affected by the length ratio and the loading position. According to Equation (11), the relationship between the theoretical critical rolling velocity v_c and the loading position s_pl is calculated for three length ratios Lⁱⁿ/L^out (1.0604, 1.0452, and 1.0283), as shown in Figure 7e. In order to verify the proposed theory, corresponding experiments were conducted, and the experimental results are shown as the pentagons in Figure 7e. The results show that the critical rolling velocity increases with the increase of the length ratio and the loading position.

In order to further validate the locomotion performance of the soft robot, a series of experimental scenarios were designed to simulate the complex terrains that might be encountered in applications. In the terrain experiments, a large driving force is required for the robot to navigate complex terrains. Therefore, the loading positions of the soft robot are close to the maximum curvature point to obtain a large driving force. The loading timing diagram used in three terrain experiments is shown in Supplementary Figure S5.

The three-stage steps were built with each stage measuring 1.5 cm in height and 20 cm in length. Under the loading pressure of 40 kPa, the soft robot successfully climbed the steps (Fig. 8a), and the average climbing speed was 61.22 mm/s (Supplementary Movie S2). During the climbing process, the configuration of the soft robot hardly changed, and the whole robot body moderately tilted when rolled over the steps tip, which is similar to that of a tank. Compared with the deformation-driven soft robot,³⁰ our soft robot has a small deformation, low deformation energy consumption, and a smooth locomotion trajectory, which demonstrates its good obstacle-crossing ability in complex terrain.

FIG. 8.

Schematics of the soft robot navigating different terrains. (a) Climbing steps. (b) Crossing a broken bridge. (c) Climbing a slope. (d) Traversing cluttered terrain.

The maximum gap length of the broken bridge that the soft robot can cross is 90 mm (Fig. 8b), which is nearly half of the robot’s body length (0.443 BL). Under the loading pressure of 40 kPa, the soft robot crossed the broken bridge with an average speed of 56.25 mm/s (Supplementary Movie S3). Most of the previous rolling soft robot configurations are wheel-shaped, and they easily slip into the broken gap and get stuck. In contrast, the proposed soft robot has a large contact area with the ground due to its two-nested ring structure, and the locomotion stability is improved. Due to the novel rolling mechanism driven by local snap-through buckling, it can keep its configuration stable without significant deformation and does not get stuck when crossing the broken bridge.

The slope with the slope angle of 18.36° was built. Under the loading pressure of 57 kPa, the soft robot climbed the slope with an average speed of 52.63 mm/s (Fig. 8c and Supplementary Movie S4). The slope angles of 10° and 15° were also tested to show the effect of the slope angle on the loading pressure, where the loading timing diagram has the same T and T_o. The result shows that the larger slope angle requires a larger loading pressure to provide a larger driving force. In literatures, the maximum slope angle that can be traveled by the circular rolling soft robot developed was 5°,³¹ and that of the triangular rolling soft robot was 8°.¹² In comparison, our soft robot shows a significant advance in terms of slope climbing ability due to the tank-track-like configuration and the novel rolling mechanism driven by local snap-through buckling.

In addition, a more realistic outdoor scenario was established. The soft robot coherently passed through the cluttered environment, including tree branches, gravel paths, rugged slopes, and the broken bridge (Fig. 8d and Supplementary Movie S5).

The load-bearing capacity of the robot was tested as shown in Supplementary Movie S7, where the robot carried the payload and rolled one step. The experiment result shows that the maximum load capacity of the present soft robot is 42.5 g, and more load capacity can be achieved by using elastic rings with larger stiffness.

Comparison of locomotion performance

The locomotion performance of the proposed rolling soft robot is compared with that of previous rolling soft robots, as shown in Figure 9a. The proposed rolling soft robot has a maximum rolling speed of 192.3 mm/s or 0.95 BL/s, a weight of 83.85 g, and a deformation level of 1.039. In addition, the energy efficiency of the proposed rolling robot is 0.098% compared with 0.016% for the GoQbot¹⁶ and 0.0041% for the DECSO prototype A.³⁰ The detailed calculation process of energy efficiency is shown in Supplementary Data S1. It is shown that the soft robot in this article has an advantage in terms of the rolling speed, energy efficiency, and mass, which is mainly attributed to the novel rolling mechanism driven by local snap-through buckling that reduces the deformation level of the soft robot during the rolling and thus decreases the energy loss and improves the energy efficiency so that the proposed soft robot can balance the energy efficiency with the rolling speed. In addition, the unique two-nested ring configuration and the bistable characteristics enable smooth two-way rolling on flat ground (Supplementary Movie S6) and successful navigation through complex terrains (the slope, steps, and the broken bridge), which can maintain the current configuration and position when the pressure is unloaded. Therefore, compared with previous rolling soft robots, the proposed soft robot has good locomotion stability and good terrain adaptability.

FIG. 9.

(a) Comparison of rolling velocity, mass, and deformation level between this work and previous rolling soft robots.^{12,13,15,17,30,32–37} (b) Comparison of adapted terrains.^{10–12,14,30,31,35,38}

In order to evaluate the adaptability of rolling soft robots to complex terrains, the different terrains that rolling soft robots can pass are compared, as shown in Figure 9b. Most of the rolling soft robots in the literature can only adapt to one or two terrains due to their locomotion mechanism. Few previous rolling soft robots can cross the broken bridge. In contrast, the rolling soft robot in this study can adapt to complex terrains, including the broken bridge, and can maintain good locomotion speed and energy efficiency during the navigation process. This performance not only shows the adaptability of the proposed rolling soft robot to complex environments but also reflects the advantages of the proposed rolling mechanism.

Conclusion

This article proposes a novel rolling soft robot driven by local snap-through buckling. By applying local curvature to the local loading region, the loading region generates a local snap-through buckling transition, which drives the deformation and rolling of the entire soft robot. As verified by theoretical analysis and experiment, the shape of the soft robot remains almost unchanged during the rolling locomotion, and the energy loss is low. The maximum speed of continuous rolling on flat ground is 192.3 mm/s or 0.95 BL/s, which is faster than most of the previous rolling soft robots. The proposed soft robot can navigate through complex terrains, including a slope with a maximum angle of 18.36°, steps with each stage height of 15 mm, and a broken bridge with a maximum gap length of 90 mm (0.443 BL). The maximum load capacity of the proposed soft robot is 42.5 g, and more load capacity can be achieved by using elastic rings with larger stiffness.

The good rolling performance and excellent terrain adaptability of the soft robot are attributed to the novel rolling mechanism and unique two-nested ring configuration. The proposed rolling mechanism balances high rolling speed with energy efficiency by using the snap-through transition of the local loading region, which enhances the stability and controllability of rolling. The soft robot only drives one actuator to achieve one step of rolling, which simplifies the driving method compared with some previous rolling soft robots^10,17,30 loading multiple actuators at each step of rolling. The two-nested-ring configuration of the soft robot resembles the tank track and ensures the high efficiency and stability of the soft robot in the locomotion process. Besides, the designed soft actuator simplifies the nonstretchable layer and realizes the lightweight design of the soft robot. The proposed robot cannot change its height while rolling, but an alternate simple method to change the robot’s height is replacing the outer ring with different lengths and stiffness while the inner ring remains the same.

The proposed rolling soft robot with tethers has good application prospects in medical services³⁰ by carrying the inspection device and rolling on the body of the patient on the bed, which is similar to the medical application of the reported soft robot DECSO.³⁰ Other potential applications, such as land exploration, may require an untethered version of the robot. As preliminary research, this article mainly studies the rolling mechanism of the proposed robot with the tethered pneumatic control. However, the pneumatic actuation method and the big number of actuators make it difficult to integrate the untethered soft robot with all components on board. Using other suitable actuation methods with small size and small mass, for example, cable actuation³⁹ and electric actuation⁴⁰ can facilitate the untethered integration of the proposed rolling robot in future works. The proposed new rolling mechanism with the bistable structure is anticipated to bring new inspiration to the development of soft robots.

Footnotes

Author Disclosure Statement

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this article.

Funding Information

This work was supported by the National Natural Science Foundation of China (12302132) and the Natural Science Foundation of Fujian Province (2021J05114 and 2023J05249).

Supplementary Material

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Supplementary Material

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