Tuning Stiffness with Granular Chain Structures for Versatile Soft Robots

Abstract

Stiffness variation can greatly enhance soft robots' load capacity and compliance. Jamming methods are widely used where stiffness variation is realized by jamming of particles, layers, or fibers. It is still challenging to make the variable stiffness components lightweight and adaptive. Besides, the existing jamming mechanisms generally encounter deformation-induced softening, restricting their applications in cases where large deformation and high stiffness are both needed. Herein, a multifunctional granular chain assemblage is proposed, where particles are formed into chains with threads. The chain jamming can be classified into two types. Granular chain jamming (GCJ) utilizes typical particles such as spherical particles, which can achieve both high stiffness and great adaptability while keeping jamming components relatively lightweight, while by using cubic particles, a peculiar deformation-induced stiffening mechanism is found, which is termed as stretch-enhanced particle jamming (SPJ). The versatility of GCJ and SPJ mechanisms in soft robots is demonstrated through soft grippers, soft crawlers, or soft bending actuators, where great passive adaptability, high load capacity, joint-like bending, friction enhancement, or postponing buckling can be realized, respectively. This work thus offers a facile and low-cost strategy to fabricate versatile soft robots.

Introduction

Soft robotics generally composed of mainly soft materials is an emerging field that aims to develop compliant robots capable of safely interacting with humans or surroundings, as well as effectively performing tasks in unstructured environments.^1–3 One major challenge for versatile soft robots involves developing variable stiffness components, which can generate high stiffness when manipulating heavy objects or low stiffness during human-machine interaction.^4,5

Various methods for stiffness variation have been presented, including jamming mechanisms, shape memory polymers (SMPs), low melting point alloys (LMPAs), electrorheological (ER) materials, and magnetorheological (MR) materials.^6–9 Each method has its own specific features. For example, SMPs^10–12 and LMPAs^13,14 have shown a wide range of stiffness variation, but with slow responses, and ER/MR materials^15,16 exhibit a fast response speed, but with poor ability to resist bending moments. Among these methods, jamming strategies are still the most widely used methods in numerous applications owing to their simple, fast, low-cost, and robust properties.

The jamming mechanisms for stiffness variation are generally acquired by assembling a number of discrete media in a confined chamber and applying a pressure gradient by vacuum,^17–20 tendon-driven forces,^21,22 electrostatic forces,^23,24 or pressurized air.^25–27 According to different discrete media, three main types of jamming technologies have been proposed, that is, particle jamming, layer jamming, and fiber jamming. Particle jamming structures exhibit great adaptability in the natural state and once jamming is induced, it can achieve drastic stiffness improvement, which have been used in soft robotic gripper design,^28,29 minimally invasive surgery,^30,31 and wearable applications.^32,33 Both layer jamming and fiber jamming structures are lightweight and compact; however, the former can achieve a larger range of bending stiffness variation, and the latter possess more degrees of freedom and are great at tensile stiffening.^34,35 These two jamming technologies have been mainly applied to robotic manipulators^36,37 and wearable devices.^38–40

However, it is still challenging to realize wide-range bending stiffness variation for jamming methods when lightweight requirement or great adaptability is important. The particle jamming structure generally requires a substantial volume of particles to realize significant stiffness, resulting in a bulkier and weightier system. And the layers are difficult to form complex shapes, thus the layer jamming structures cannot achieve great adaptability. Besides, the layers are unable to deal with large lateral compression because permanent creasing of the layers can be caused, which affects their performance severely.

Moreover, the layer jamming structures cannot realize lateral bending, restricting their applications.³⁸ Fiber jamming structures can deal with the drawbacks of layer jamming, whereas they lack in the modulation range of bending stiffness.⁴¹ Recently, hybrid jamming principle combining the particle jamming and layer jamming has been presented to alleviate the stated drawbacks; however, it requires multiple vacuum inputs to realize stiffness variation, complicating the soft robotic systems.⁴²

Besides, the stiffness induced by jamming typically decreases when the deformation becomes large, that is, the phenomenon of the deformation-induced softening. The mechanisms of stiffness improvement for the jamming-based methods rely on the dramatically increased interlocking forces (or maximum static friction forces) among the discrete media. In the presence of loads for the jamming structures, the corresponding active forces among the discrete media are also generated.

Once the active forces surpass the interlocking forces, generally at the large deformation of the jamming structures, the contact interfaces among a part of discrete media experience a transition from cohesive to slip, which leads to a remarkable decrease in the stiffness, namely deformation-induced softening.^41,43,44 As a result, in the presence of relatively large force or large deformation, the jamming structures may not provide sufficient stiffness, resulting in failure to execute tasks effectively.³⁶ Thus, the deformation-induced stiffening behavior, that is, the stiffness enlargement with the elevated deformation, is more applicable.

In this article, we develop a novel hybrid granular chain assemblage fabricated by fixing numerous granular chains on a block, with each chain being formed by a number of perforated particles with a flexible string (Fig. 1A). The flexible strings can be stretched during the bending of granular chain assemblages, and the magnitude of the stretch can be affected by particle shapes. According to whether the deformation-induced stiffening phenomenon exists, two different ways of stiffness improvement are classified. Granular chain jamming (GCJ) uses typical spherical particles where the strings' stretches are negligible. On the other hand, stretch-enhanced particle jamming (SPJ) employs typical cubic particles, where strings' stretches are significant due to the particles' shapes.

FIG. 1.

Fundamental stiffness-tunable principles and mechanical behaviors of GCJ and SPJ structures. (A) Illustration of a GCJ structure and an SPJ structure. (B) Demonstration of the compliant-stiff duality of a GCJ structure induced by transforming ΔP = 0 kPa to ΔP = 80 kPa. (C) Demonstration of the deformation-induced stiffening behavior of an SPJ structure caused by increasing weight from 50 to 200 g. (D) Schematic of a bent GCJ structure under a low load and a high load. The granular chains of the GCJ structure keep cohesive with low loads, while for higher loads, longitudinal shear stress along the interfaces among the granular chains is large enough to induce relative slip among those granular chains. (E) Schematic of an SPJ structure without and with bending. For the unbent state, no gap exists between the adjacent particles of a granular chain, but once the SPJ structure bends, the gaps appear obviously. (F) Schematic showing the bending distinction of the four-particle granular chains formed by spherical particles and cubic particles. (G) Summary of mechanical behavior. As for GCJ, when vacuum is off, the structure has low stiffness; when vacuum is on, the stiffness of the structure is enhanced severely, whereas, as the load increases, the structure experiences stiffness decrease (i.e., deformation-induced softening) due to the slip of granular chains. In contrast, through SPJ, the stiffness of the structure increases as the load increases (i.e., deformation-induced softening), owing to the elevated tensions in flexible strings. All scale bars are 2 cm. GCJ, granular chain jamming; SPJ, stretch-enhanced particle jamming.

The advantages of the proposed two types of variable stiffness approaches are as follows: the GCJ structures need less volumes and lighter weights to obtain higher stiffness compared to particle jamming, possess greater ability to resist bending moments, inducing a larger stiffness modulation compared to particle and fiber jamming, and exhibit greater adaptability compared to layer jamming. The adaptability for GCJ structures is due to the great mobility of granular chains, while the SPJ structures can accomplish deformation-induced stiffening, which provides a preferable capacity of loads in the case of large deformation. We present the versatility of GCJ and SPJ mechanisms through soft grippers, soft crawlers, or soft bending actuators, where great passive adaptability, high load capacity, joint-like bending, friction enhancement, or postponing buckling can be realized, respectively. These demonstrations substantiate the feasibility of utilizing the GCJ or SPJ mechanisms to devise versatile soft robots.

Results

Distinct stiffness-tunable and mechanical behaviors of GCJ and SPJ

The GCJ and SPJ structures are fabricated by a five-step process (Section S1 and Supplementary Fig. S1, Supporting Information). A granular chain assemblage is composed of multilayered granular chains involving the alternate arrangement of two kinds of layers, one of which contains six granular chains and the other has five granular chains (Section S2 and Supplementary Fig. S2, Supporting Information). We test the GCJ and SPJ structures in a three-point bending configuration for various conditions (Section S3 and Supplementary Fig. S3, Supporting Information), where the results are also verified by the finite element (FE) simulations using ABAQUS.

The stiffness of a GCJ structure can be well tuned by applying vacuum. When vacuum is off, the granular chains bend separately, and the GCJ structure shows a compliant property. When vacuum is on, the independent granular chains become a cohesive part and the bending stiffness of the GCJ structure increases sharply. In Figure 1B, we demonstrate the compliant-stiff duality of a GCJ structure. When vacuum is off, the structure is compliant and a 50 g weight results in large bending; while vacuum is applied, the structure becomes stiff enough to be capable of supporting a 1000 g weight. The GCJ structures also exhibit the mechanical behavior of deformation-induced softening, which is similar to the conventional particle jamming, layer jamming, and fiber jamming. This phenomenon is mainly caused by the slip of granular chains (Fig. 1D, G and Section S5, Supporting Information).

The SPJ structures, however, can show deformation-induced stiffening behavior. The tunable stiffness by SPJ relies on the bending of jamming structures. In Figure 1C, the deformation-induced stiffening of an SPJ structure is demonstrated. When a 50 g weight is placed on the SPJ structure, it can support the weight with negligible bending. When the weight is increased to 200 g, the structure can support such a weight with slightly increased bending. Specifically, as shown in Figure 1E and F, the bending of an SPJ structure leads to the emergence of obvious gaps between the adjacent cubic particles on a granular chain.

The appearance of those gaps implies that the strings are stretched. In turn, these stretched strings make the cubic particles more jammed, resulting in the mechanical behavior of deformation-induced stiffening (Fig. 1G). The granular chain assemblages with cubic particles show a remarkable deformation-induced stiffening behavior, while the granular chain assemblages with spherical particles are flexible. The physics underlying this difference is explained by a simple mechanical model (Fig. 1F, Section S4 and Supplementary Fig. S4, Supporting Information).

Characterization on GCJ structures

The experimental and numerical studies on GCJ and SPJ structures are conducted to further understand the stiffness-tunable mechanisms of GCJ and SPJ and investigate how the structural parameters of granular chain assemblages or vacuum pressure affect the stiffness variation. In this study, we first conduct the characterization on GCJ structures.

The effect of vacuum pressure on the mechanical behavior of GCJ structures is shown in Figure 2A. As expected, the unjammed structure exhibits a roughly linear force-deflection curve with a small slope throughout the bending test, because the whole granular chains slip immediately with bending. While jammed, the slopes of the force-deflection curves are enlarged sharply at low deflection where the granular chains are cohesive, and the maximal stiffness range (defined by the ratio of the force-deflection curves' slopes between the vacuum on and vacuum off states) is measured to be up to 162.

FIG. 2.

Mechanical behavior of GCJ structures. (A) Effect of vacuum pressure on the force-deflection relationship of GCJ structures. (B) Effect of the number of granular chain layers on the force-deflection relationship of GCJ structures. (C) Effect of the tensile rigidity of flexible strings on the force-deflection relationship of GCJ structures. For each experimental curve, the shaded region denotes 1 standard deviation, and the solid line represents mean value.

However, as the deflection further increases, the granular chains begin to slip and the slopes of the force-deflection curves decline; thus, the deformation-induced softening mechanical behavior is exhibited. In addition, we observe that by increasing the vacuum pressure from 20 to 80 kPa, the force at slip increases, representing the ability of GCJ structures to resist bending moments is further improved.

Then, the effects of the number of granular chain layers and the rigidity of flexible strings on the mechanical behavior of GCJ structures are presented in Figure 2B and C, respectively. The flexible strings made of nylon with three groups of tensile rigidity EA are adopted, where E and A are the Young's modulus and the cross-sectional area of a nylon string, respectively. It is apparent that the stiffness can be enhanced by increasing the number of granular chain layers or the rigidity of flexible strings. In this study, during the bending of the jammed GCJ structures, the tensions in the strings can be induced (Fig. 1F), which also contribute to the overall stiffness of jammed structures. Increasing the rigidity of flexible strings can enlarge the tensions in the strings, which in turn increases the stiffness of the jammed GCJ structures.

To investigate the relationship between stiffness modulation range of GCJ and the rigidity of flexible strings, we test the force-deflection relationship of GCJ structures with different flexible strings under ΔP = 0 kPa and ΔP = 80 kPa (Supplementary Fig. S6, Supporting Information). The measured maximal stiffness ranges are 68, 76, and 84 for $E A = 65 P a \cdot m^{2}$ , $E A = 294 P a \cdot m^{2}$ , and $E A = 768 P a \cdot m^{2}$ , respectively. Thus, increasing the tensile rigidity of strings can improve the variable stiffness range. Besides, the comparison of GCJ and particle jamming structures without strings indicates that the flexible strings play an important role (Section S3 and Supplementary Fig. S6, Supporting Information).

The experiments regarding the effects of vacuum pressure and the structural parameters of granular chain assemblages on the mechanical behavior of GCJ structures can be well captured by the FE simulations. And according to the FE models, we discover that the slip of adjacent granular chains occurs at high loads (Supplementary Fig. S5, Supporting Information).

Characterization on SPJ structures

The SPJ structures with typical cubic particles are featured as the deformation-induced stiffening mechanical behavior, which is quite different from that of the GCJ structures. Figure 3A–C show the effects of the number of granular chain layers, the rigidity of flexible strings, and the number of flexible strings on the mechanical behavior of SPJ structures. The slopes of the force-deflection curves become larger with the increased deflection (or force), indicating that the deformation-induced stiffening mechanical behavior is induced. Besides, the increase in the number of granular chain layers, the rigidity of flexible strings, or the number of flexible strings can further increase the stiffness of SPJ structures, which are caused by the ascending tensions in the strings. In this study, the cubic particles with four perforations are fabricated (Supplementary Fig. S2, Supporting Information).

FIG. 3.

Mechanical behavior of SPJ structures. (A) Effect of the number of granular chain layers on the force-deflection relationship of SPJ structures. (B) Effect of tensile rigidity of flexible strings on the force-deflection relationship of SPJ structures. (C) Effect of the number of particle's perforations on the forced-deflection relationship of SPJ structures. (D) Comparison of the force-deflection relationships of anisotropic SPJ structures loaded in opposite directions. Taking the bent SPJ structure in Figure 1E for reference, the insets in (C) and (D) represent the side views of a cubic particle and the flexible strings inside the particle. For each experimental curve, the shaded region denotes 1 standard deviation, and the solid line represents mean value.

As shown in Figure 3D, we can obtain a novel anisotropic SPJ structure, that is, when loaded in opposite directions, the SPJ structure exhibits different stiffening behaviors (Section S5, Supporting Information). This is due to the use of cubic particles with four perforations and two kinds of strings with different stiffness.

The simulated results regarding the dependence of stiffness of SPJ structures on the structural parameters agree well with the experimental results. For the large bending, we find that for the GCJ structure, there are no obvious gaps between the adjacent particles of a granular chain and the tensions in the flexible strings are smaller, while for SPJ, the obtained gaps and tensions are larger (Supplementary Fig. S5, Supporting Information). According to Figures 2 and 3, the experimental results are smaller than the simulated results, and such deviations are mainly due to the assumption of uniform structures for simulation. In the real situation, the granular chains near the distal ends of the jamming structures collapse due to gravity; thus, the stiffness of the jamming structures is decreased. Besides, the simulated force-deflection curves are fluctuant, which are mainly caused by the relative slip between the adjacent particles.

Demonstrations for stiffness tuning with GCJ structures

The typical soft fiber-reinforced bending actuators (Section S6 and Supplementary Fig. S7, Supporting Information) and the soft grippers are used to show the applications of GCJ structures in tuning stiffness. The GCJ structure is adhered to the bottom of the soft actuator by using Sil-Poxy (Smooth-On, Inc.) as the silicone adhesive. Four different functions are realized as follows:

First, passive adaption function is demonstrated (Fig. 4A, B). A two-fingered soft robotic gripper is developed by assembling two soft actuators to a 3D printed base. Since the granular chains are capable of moving freely, the unjammed granular chains can function as a soft interface for the soft actuator, and such a high-compliance property confers the two-fingered robotic gripper with great passive adaptability. When grasping an object, sink generally appears on the contact surfaces of the robotic gripper through passive adaptation, which facilitates the robotic gripper creating better contact with the objects (Fig. 4A). And if the objects are required to be manipulated, vacuum is applied and the soft gripper is stiffened by GCJ to promote the rigidity of grasping. Specially, we further show the advantage of passive adaptation in grasping the objects with nonround contours (Fig. 4B, Section S7 and Supplementary Fig. S8, Supporting Information).

FIG. 4.

Demonstrations of passive adaptation and enhanced load capacity based on GCJ. (A, B) A two-fingered soft gripper integrated by GCJ structures has great shape adaptability. The two-fingered soft gripper is capable of well conforming to objects with both round contour (such as a disk and a scissor) and square contour (such as a card). While for the robotic gripper without GCJ structures, the round contours of pressurized soft actuators are incompatible with the nonround contour of the square card, which leads to the separation between the robotic gripper and the card and is easy to make the grasping state failed. (C) GCJ improves the load capacity of a three-fingered soft gripper. The maximum weight that the three-fingered soft gripper can carry is increased from 48 g (without GCJ structures) to 1000 g (with GCJ structures and ΔP = 80 kPa). All scale bars are 2 cm.

Figure 4 shows that passive adaption is a great property, which can not only broaden the scope of grasping objects but also greatly improve the stability of grasping. In previous jamming mechanisms, only particle jamming can be exploited to endow soft grippers with passive adaptability. In this study, GCJ can also achieve great passive adaptability, indicating that the proposed granular chain structures are useful in designing versatile soft robots. Note that the shape adaptability for GCJ structures is due to the great mobility of granular chains, which is barely influenced by the mechanical properties of flexible strings. Thus, the GCJ structures formed by different flexible strings can still exhibit great shape adaptability.

Second, enhanced load capacity function is demonstrated (Fig. 4C). A three-fingered soft robotic gripper fabricated by assembling three soft actuators to a 3D printed base is used. In Figure 2A, the measured maximal stiffness range based on GCJ is up to 162. Such wide-range stiffness modulation allows the soft robotic gripper to grasp high-load objects. More importantly, increasing the level of vacuum pressure can enlarge the capacity of GCJ structures to resist external loads, which is also vital to achieve high bearing capacity for the soft robotic gripper.

As shown in Figure 4C, for the soft gripper without granular chain assemblages, the maximal load it can carry is only 48 g; when the GCJ structures are introduced, the maximal load that the soft gripper can carry is improved from 140 g (ΔP = 0 kPa) to 1000 g (ΔP = 80 kPa), indicating the load capacity of the soft robotic gripper can be enhanced severely.

Third, joint-like bending function is demonstrated by altering the form of granular chains. For the joint-like bending function, a three-part granular chain assemblage structure consisting of short granular chains and short flexible strings is required (Fig. 5A). When a GCJ structure formed by the three-part granular chain assemblage is jammed, the regions with particles become stiff, behaving like rigid links, and the regions with only flexible strings can keep flexible acting like joints. Thus the bending of such a jammed structure is discrete, which is entirely distinct from the continuous bending of soft fiber-reinforced bending actuators. We adhere the jamming structure to a soft actuator, and the soft actuator can accomplish joint-like bending with applied vacuum (Fig. 5B).

FIG. 5.

Demonstrations of joint-like bending and friction enhancement based on diversiform GCJ structures. (A) Schematic of a three-part granular chain assemblage. (B) Photograph of achieving joint-like bending for a soft fiber-reinforced bending actuator. (C) A human finger-inspired soft gripper with two kinds of grasping modes. Based on the contours of the targeted objects, human fingers can achieve embracement-like grasp or nip-like grasp. When no vacuum is applied, the soft gripper bends continuously and can perform embracement-like grasp on the large round object. When vacuum is applied, the soft gripper bends in a joint-like form and performs nip-like grasp on the small nonround object. (D) Schematic of a jagged granular chain assemblage. (E) Photograph of realizing rough surface for a soft fiber-reinforced bending actuator. (F) The soft gripper integrated by the jagged granular chain assemblages is endowed with rough surfaces; thus, it can perform stable and reliable grasping for objects with various shapes such as a mobile power, a coil, a tape, a remote control, a piece of glass, and a cup of yogurt. All scale bars are 2 cm.

Human fingers can be regarded as multi-joint structures, which possess excellent grasping performance such as the active transformation of grasping modes according to the contours of targeted objects (Supplementary Fig. S9, Supporting Information). In this study, using the three-part GCJ structures, a human finger-inspired soft gripper with tunable grasping modes is developed. As shown in Figure 5C, when grasping a tape, depending on the placement states of the tape, the proposed gripper or the human fingers can perform different grasping modes (Section S8, Supporting Information).

Finally, the friction enhancement function is demonstrated as follows. We make a jagged granular chain assemblage structure composed of the alternating arrangement of small particles and large particles (Fig. 5D). By adhering this jagged GCJ structure, the soft fiber-reinforced bending actuator is endowed with the rough surface (Fig. 5E). Friction enhancement is necessary for soft robotics.^45,46 Note that the proposed jagged jamming structure can not only provide indispensable friction for the soft fiber-reinforced bending actuators but also implement stiffness adjustment, which further promotes the versatility of soft actuators (Fig. 5F).

Demonstrations for stiffness tuning with SPJ structures

Two practical function cases for SPJ mechanism in soft robotics are demonstrated. First, enhanced load capacity function is demonstrated. To this end, we design a novel inchworm-inspired crawler with stiffness enhancement, which consists of two parallel soft fiber-reinforced bending actuators, an SPJ structure, a rubber layer, rear feet, and front feet (Fig. 6A and Supplementary Fig. S10, Supporting Information). The movement of the crawler is described in Section S9 (Supporting Information).

FIG. 6.

Demonstrations of enhanced load capacity and postponing buckling occurrence based on SPJ. (A) The movement process of an inchworm-inspired soft crawler integrated by an SPJ structure. (B) Comparison of soft crawler's load-bearing performance with and without SPJ structure. The contact between the feet of the soft crawler and the horizontal plane can be well held under a 500 g weight with SPJ structure (Left), while the contact can be severely broken without SPJ structure (Right) under the same weight. (C) Experimental setup for displaying the performance of soft actuator in pressing the switch that controls a bulb. (D) Comparison of performance in pressing a switch for soft actuators with and without SPJ structure. The soft actuator with SPJ structure can press the switch successfully (Top), while another without SPJ structure cannot press the switch (Bottom). (E) Schematic of experimental setup for buckling characterization of soft actuator. (F) Comparison of the relationship between tip output force and tip displacement for soft actuators with and without SPJ structure. The shaded region denotes 1 standard deviation, and the solid line represents mean value. All scale bars are 2 cm.

It can be observed that the close contact between the feet and the horizontal plane in the bending state is important for the movement of the crawler. To embody the effect of stiffness improvement through SPJ for soft crawlers, as shown in Figure 6B, in the bending state, we place a 500 g weight on the proposed crawler and its feet still keep well contact with the horizontal plane due to the enhanced load capacity (Supplementary Video S1, Supplementary Materials). In contrast, for the soft crawler without the SPJ structure, the same weight leads to severe separation of feet from the horizontal plane (Supplementary Video S2, Supplementary materials). Therefore, by virtue of the SPJ mechanism, we can increase the load capacity of the soft crawlers and broaden their practical applications.

We then demonstrate buckling postponing function (Supplementary Videos S3 and S4, Supplementary materials). For this objective, we install a soft fiber-reinforced bending actuator on a vertical guide (Chengdu Fuyu Technology Co., Ltd) and drive its distal end to press a switch that controls a bulb (Fig. 6C). As shown in Figure 6D, the soft actuator with the SPJ structure turns on the switch successfully. In this study, a buckling characterization setup is established to investigate the buckling behavior (Fig. 6E).

As shown in Figure 6F, for the soft actuator without the SPJ structure, it exhibits an undesirable discontinuous buckling behavior, that is, the slope of the force-displacement curve become negative in the post-buckling regime.⁴⁷ When the SPJ structure is introduced, the discontinuous buckling is eliminated. Besides, due to the existence of the SPJ structure, the output force of the distal end of the soft actuator is also improved sharply. It should be noted that, compared with the GCJ, the enhanced load capacity and the buckling postponing based on the SPJ do not require any vacuum equipment, which can simplify the design of multifunctional soft robots and are useful in untethered robots, portable assistant devices, and so on.

Discussions

Granular chain assemblages, a new class of variable stiffness structures for soft actuators, are simple and inexpensive to fabricate. Two types of excellent variable stiffness principles are presented in this study, that is, GCJ and SPJ, which are quite different from the previous jamming mechanisms using chain-like structures, including chain-like granular jamming²¹ and hybrid jamming combining granules and a chain structure.⁴⁸ The characteristics of these two jamming methods and the proposed jamming mechanisms are summarized in Table 1 and in Section S13 in Supporting Information.

Table 1.

Comparison of Conventional Jamming Mechanisms Using Chain-Like Structures and the Proposed Jamming Mechanisms

	Chain-like granular jamming²¹	Hybrid jamming⁴⁸	Granular chains assemblages
	Chain-like granular jamming²¹	Hybrid jamming⁴⁸	Granular chain jamming	Stretch-enhanced particle jamming
Actuation methods	Active cable-driven actuation	Active vacuum-based actuation	Active vacuum-based actuation	Passive bending-based actuation
Constituents	cubic or cylindrical particles, flexible strings	spherical particles, rigid chain structures	spherical particles, flexible strings	cubic particles, flexible strings
Resistance to bending moments	Large	Large	Large	Medium
Stiffness change	Large	Large	Large	Medium
Passive adaptability	Void	Void	Great	Void
Deformation-induced stiffening	Void	Void	Void	Great

As for GCJ, the discrete media are granular chains, which are different from traditional particle jamming, layer jamming, and fiber jamming. Compared with particle jamming and fiber jamming, the existence of strings in the granular chain assemblages enables the GCJ structures to resist larger bending moments and achieve larger stiffness modulation ranges. The physical explanations for this behavior are provided in Section S10 in Supporting Information. In comparison to layer jamming structures, the GCJ structures exhibit a preferable deformability, enabling them to adapt to the contact objects well and overcome the large lateral compression (Supplementary Fig. S11, Supporting Information). And the granular chains are fiber-like structures, which possess more degrees of freedom than the laminar structures.

As for SPJ, its main advantage lies in the deformation-induced stiffening behavior induced by the flexible strings, which is suitable to tune stiffness for the soft actuators with large deformations. The actuation of SPJ structures is executed by the bending; thus, additional extra actuation equipment is not required. Recently, researchers proposed passive particle jamming methods.^25,49,50 The difference between the passive particle jamming and SPJ is provided in Section S11 in Supporting Information.

The tangling behavior among granular chains is inevitable (Supplementary Fig. S1, Supporting Information), especially for a granular chain assemblage containing larger numbers of granular chains and longer granular chains. Yet, the tangled behavior for the granular chains is an interesting phenomenon, which can generate different jamming behaviors (Section S12, Supporting Information). How to actively use the tangling behavior to further enhance stiffness modulation needs more investigation in the future.

As shown in Figures 2 and 3, the force-deflection curves are nonlinear, indicating that the stiffness of granular chain structures is strongly coupled with deflection, which makes it difficult to obtain analytical stiffness models. In fact, the theoretical models of granular chain structures require the analysis for relative movements between particles, particles and strings, and particles and skin, which involves the analytical relationships for the particle-particle contact, particle-skin contact, and particle-string contact. These models are complex and deserve an in-depth investigation. Note that in this work, we mainly show the variable stiffness capabilities of GCJ and SPJ, and demonstrate their applications in soft robots. Thus, in our future investigation, we will carry out a systematic theoretical framework for the stiffness models of GCJ and SPJ structures.

Conclusions

This article presents a novel rigid-flexible hybrid granular chain assemblage structure that can be exploited to achieve two kinds of variable stiffness principles: GCJ and SPJ. Both experiments and simulations reveal that the GCJ structures can achieve an excellent stiffness variation range (162-fold in experiments) and the SPJ structures exhibit stiffening phenomenon where stiffness increases with bending deformation.

Specifically, the GCJ structures present the great passive adaptability in the compliant state and the strong resistance to moments in the rigid state, which are demonstrated by the soft grippers integrated with GCJ structures that can grasp objects with irregular shapes and large weights. Furthermore, we demonstrate the soft actuator is able to achieve joint-like bending and friction enhancement using the special forms of GCJ structures. And based on these two properties, the human finger-inspired soft grippers with tunable grasping modes and the soft grippers with rough contact surfaces are proposed, respectively.

SPJ benefiting from the stretch of flexible strings shows deformation-induced stiffening behavior. Through this peculiar variable stiffness mechanism coupled with deformation, an inchworm-inspired soft crawler with enhanced load capacity is devised and the buckling postponing for soft actuators is also demonstrated. Our results show that the mechanisms of GCJ and SPJ have great prospects to fabricate versatile soft robots.

Footnotes

Author Disclosure Statement

No competing financial interest exists.

Funding Information

This research was supported by the National Natural Science Foundation of China (Grant Nos. 12072266, 11972290) and the Natural Science Foundation of Shaanxi Province of China (2020JM-105).

Supplementary Material

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