## Abstract

Soft pneumatic actuators (SPAs) play an important role in leading the development of soft robotics. However, due to the inherent characteristics of soft materials, the low driving force limits the application of SPAs. This study presents a high-force soft pneumatic bending actuator (SPBA) that consists of a spring, an eccentric silicone cylinder, and a limiting fiber. Based on the Neo-Hookean model, a theoretical model is established to predict the relationship between the bending angle and the pressure of SPBA. Furthermore, we characterize the performance of SPBA in terms of the bending capability, tip force, as well as response time. The results demonstrate the effectiveness of the theoretical model, as well as the high tip force (10.2 N) and fast response capability of SPBA. Finally, SPBAs are used to construct a three-finger soft gripper. The load capacity of the gripper is proofed, which indicates that the gripping force of the gripper increases with the pressure of the fingers and the diameter of the object. The gripping test of the gripper is performed. The result shows that the gripper with the pinching mode can grip objects of various sizes and shapes in the air and underwater, and the gripper with enveloping mode can grip objects with weight up to 1.25 kg.

## 1 Introduction

Soft robots have received much attention owing to their new range of capabilities involving degrees-of-freedom and movement not achievable by their rigid counterparts [1]. The adaptability [2], agility [3], as well as sensitivity [4] of soft robots using highly deformable and flexible materials have been investigated deeply, where a soft actuator is one of the research hotspots. Soft actuators have served in diverse applications such as manipulation [5], locomotion [6], and medical applications [7] due to their compliance, low impedance, and ability to achieve complex motions with single control inputs.

Soft actuators are commonly constructed as monolithic structures from compliant materials such as elastomers [8], shape memory alloys [9], electroactive polymers [10], hydrogels [11], or composites that undergo a solid-state phase transition [12]. Their actuation can be achieved by a variety of stimuli, including chemical reactions [13], electrical charges [14], and pressurized fluids [15]. In particular, pneumatic powered soft actuators fabricated by elastomers are promising candidates for robotics applications because of their lightweight, high power-to-weight ratio, low material cost, and ease of fabrication with emerging digital fabrication techniques [16,17]. The soft pneumatic actuators (SPAs) can be integrated into the soft robotic systems as both actuators and structural elements [18,19].

Although SPAs have been extensively studied, they are mainly divided into two categories in the structure, namely, pneumatic network (penu-net) actuators [20] and fiber-reinforced actuators [21]. Pneu-net actuators are characterized by their multiple connected chambers, functioned as networks [22]. As internally pressurized, the chambers of a pneu-net actuator are inflated, and the actuator achieves extension motion. Then, a stiffer layer is added to limit the extension of the bottom side, and the pneu-net actuator realizes bending motion with large amplitude. By improving the structure of the slow pneu-net actuator, a novel pneu-net actuator with fast response performance has been proposed by Mosadegh et al. [23]. Based on the fast pneu-net actuator, a variety of improved pneu-net actuators have been designed. Wang et al. [24] and Hu and Alici [25] changed the chamber arrangement from the vertical direction to the oblique direction to generate coupled bending and twisting motions. A novel pneu-net actuator with adsorption function was developed to design a soft gripper for food packaging [26]. However, due to the limitation of the inherent characteristics of soft materials, most pneu-net actuators tended to exhibit a lower driving force and poor stability, which seriously hindered the improvement of the load capacity of soft robots as well as the expansion of their applications.

Furthermore, the fiber-reinforced actuator is mainly composed of helically arranged fibers wound on the outer wall of an elastomer body with a monolithic chamber [21]. By adding an alternative inextensible layer at the bottom side of the elastomer body, upon pressure, the asymmetric elongation of the top and bottom sides of the elastomer body causes bending deformation of the actuator. The fiber-reinforced actuator has been used to construct various soft robots, such as an eel-inspired underwater soft robot [27], a modular soft manipulator [28], and a variable stiffness soft gripper [29]. Although the large stiffness of fibers could increase the output force of the fiber-reinforced actuator [30], the output force was still limited by soft materials. Besides the low output force, the complex manual winding process of the fiber has extremely increased the fabrication difficulty and time cost, which is not conducive to the industrial application and promotion of fiber-reinforced actuators.

Different methods have been proposed to improve the output force of SPAs. Li et al. [31] wrapped fabric on the outer surface of SPA to increase the input pressure. Although the driving force was increased, the compliance of SPA was reduced. In addition, variable stiffness is another method to increase the stiffness of SPAs. Guo et al. [32] and Sun et al. [33] designed a ratchet mechanism and a pangolin scale-inspired mechanism to improve the load capability of SPAs, respectively. However, these mechanisms increase the structural complexity and manufacturing difficulty of SPAs. The layer jamming and particle jamming methods were always used to develop SPAs with variable stiffness [34,35], which could reduce the compliance and increase the structural complexity of SPAs. Besides, improving the manufacturing process and using high elastic modulus materials to fabricate SPAs to increase the output force have also been studied [36,37]. Similarly, the high flexibility of SPAs cannot be achieved. Therefore, there is still much room for the improvement in increasing the output force of SPAs.

In this article, we propose a novel high-force SPBA. The simple structure enables us to establish the theoretical model based on the Neo-Hookean model to accurately predict the deformation of SPBA. In addition, the performance of SPBA is studied, and the high tip force characteristic is demonstrated. Finally, a three-finger soft gripper is designed based on SPBA. The gripping force and gripping performance of the gripper are investigated, which demonstrates the potential applications of SPBA.

The remainder of this article is organized as follows. In Sec. 2, the concept design, fabrication details, and theoretical model of SPBA are presented. In Sec. 3, the performance of SPBA is characterized, including the bending angle, tip force, and response time. A three-finger soft gripper is designed in Sec. 4, and conclusion and future work are presented in Sec. 5.

## 2 Design, Fabrication, and Modeling of SPBA

### 2.1 Design and Fabrication.

Figure 1(a) shows the schematic of the proposed SPBA, which is composed of a spring, an eccentric silicone cylinder, and a limiting fiber. The spring is wound on the outer wall of the silicone cylinder, and the fiber is fixed at the bottom side of the silicone cylinder. The length, out diameter, inner diameter, and eccentric distance of the silicone cylinder are L = 100 mm, Do = 20 mm, Di = 10 mm, and e = 2.5 mm, respectively. The specification of the spring is presented in Table 1. On pressure, the spring restrains the radial expansion of the silicone cylinder, and the fiber limits the extension of the bottom side of the silicone cylinder. Therefore, the extension of the top side will force the silicone cylinder to bend to the bottom side, as shown in Fig. 1(b).

Fig. 1
Fig. 1
Close modal
Table 1

Specification of the spring

ParameterValue
MaterialSUS304WPB
Wire diameter0.6 mm
Outer diameter20.0 mm
Pitch1.2 mm
ParameterValue
MaterialSUS304WPB
Wire diameter0.6 mm
Outer diameter20.0 mm
Pitch1.2 mm

Figure 2 represents the fabrication process of SPBA. First, the spring is inserted into the assembled mold fabricated by 3D printing, and the degassed liquid silicone (Tianying950, Dongguan Tianying Craft Material Co., Ltd.) is poured. The liquid silicone is cured for 24 h at room temperature (20 °C). After curing, the silicone cylinder is released from the mold, and the lower end of the silicone cylinder is inserted into a rigid cup filled with degassed liquid silicone to seal the silicone cylinder. Then, an inelastic nylon fiber is inserted through the small hole at the bottom side of the silicone cylinder, and the liquid silicone is injected into the small hole to fix the fiber. Finally, a thin air tube is employed to connect the cavity of the silicone cylinder with an air compressor for pneumatic actuation. In addition, it is worth noting that the convenient casting process of SPBA eliminates the complicated manual winding process of the fiber of the fiber-reinforced SPA, which extremely facilitates the fabrication process.

Fig. 2
Fig. 2
Close modal

### 2.2 Theoretical Model.

To analyze the static deformation characteristic, a theoretical model is established to predict the relationship between the air pressure and the bending angle of SPBA. Figures 3(a) and 3(b) show the schematic diagram of SPBA under the initial and bending states. Because the spring limits the radial expansion of the silicone cylinder, we assume that the axial cross section of the silicone cylinder cannot be changed after deformation. Then, we select a sector-shaped micro-element with a central angle Θ, a thickness T, and an outer radius R on the axial cross section of the silicone cylinder in Fig. 3(a). After bending in Fig. 3(b), the central angle, thickness, and outer radius of the micro-element are θ, t, and r, respectively. According to the volume incompressibility of the silicone rubber, the volume of the silicone strip with the cross section of sector-shaped micro-element is constant. Therefore, the following relation can be obtained:
$(R+Ri)2ΘTL=(r+ri)2θtl$
(1)
where L and l are the length of the silicone cylinder before and after deformation, respectively. Ri and ri are the inner radius of the silicone cylinder before and after deformation.
Fig. 3
Fig. 3
Close modal
Simplifying Eq. (1) to yield:
$(r2−ri2)l=(R2−Ri2)L$
(2)
Then, because radial expansion of the soft cylinder is limited, we assume that the outer radius of the sector-shaped micro-element is constant before and after deformation, which can be expressed as r/R = 1. l/L = λ is defined. The deformation gradient F of the silicone cylinder can be expressed as follows:
$F=(∂r∂RrR∂r∂Θ∂r∂Zr∂θ∂RrR∂θ∂Θr∂θ∂Z∂z∂RrR∂z∂Θ∂z∂Z)=(1λ0001000λ)$
(3)
Therefore, the radial elongation ratio λr, circumferential elongation ratio $λθ$, and axial elongation ratio λz of the silicone cylinder can be obtained from Eq. (3):
$(λr=1λλθ=1λz=λ$
(4)
Because the deformation process of SPBA can be regarded as a quasi-static process, we conduct a moment balance analysis on the point O that is located at the end of SPBA shown in Fig. 4. Then, the following relation can be obtained:
$Ma=Mb+Mc$
(5)
where Ma, Mb, and Mc are the moments about the point O generated by the air pressure applied to the inner end face of SPBA, the axial restoring force of the silicone cylinder, and the axial restoring force of the spring, respectively. The elastic coefficient of the spring is just 10 N/m measured by the static tension method. Compared with the silicone cylinder, the spring is so soft that the axial restoring force is very small, which results in that the moment Mc in Eq. (5) can be ignored. Therefore, Eq. (5) can be rewritten as follows:
$Ma=Mb$
(6)
Fig. 4
Fig. 4
Close modal
To calculate Mb, we select a sector-shaped micro-element with a central angle and a height dr on the axial cross section of the silicone cylinder shown in Fig. 4. The distance between the micro-element and the center of the axial cross section is r. The angle between the micro-element and the vertical centerline of the axial cross section is φ. The axial stress of the silicone cylinder is represented by sz. Therefore, Mb can be calculated as follows:
$Mb=2(∫0π∫0Roszhrdrdφ−∫0π∫0Risz(h+e)rdrdφ)$
(7)
where Ro and Ri are the outer radius and inner radius of the axial cross section of the silicone cylinder.
Similar to the selection of the sector-shaped micro-element on the axial cross section of the silicone cylinder, if we select a same micro-element on the inner end surface of the silicone cylinder, Ma can be calculated as follows:
$Ma=2∫0π∫0RiΔP(h+e)rdrdφ$
(8)
where ΔP is the pressure in SPBA.
Next, to calculate the axial stress sz of the silicone cylinder in Eq. (7), we use Neo-Hookean model to describe the nonlinear large deformation characteristic of the silicone rubber, which can be expressed as follows:
$W=μ2(I1−3)$
(9)
where W is the strain energy density of the silicone rubber and μ is the shear modulus of the silicone rubber with a value of 0.16 MPa, which was measured by a tensile test. I1 is the first principal invariant of the right Cauchy-Green strain tensor, which can be expressed as follows:
$I1=λθ2+λr2+λz2=λ2+1λ2+1$
(10)
The nominal principal stress si of the silicone rubber can be obtained from Eq. (9):
$si=∂W∂λi−pλi$
(11)
where p is the Lagrange multiplier.
Substituting Eq. (4) into Eq. (11), the axial nominal principal stress sz can be obtained:
$sz=∂W∂λz−pλz=μ(λz−1λz3)$
(12)
Because the bottom side of the silicone cylinder cannot be extended under the limitation of the limiting fiber, the axial elongation ratio λz can also be expressed as follows:
$λz=λ=1+hαL$
(13)
where α is the bending angle of SPBA.
Substituting Eqs. (7), (8), (12), and (13) into Eq. (6), the relation between pressure ΔP and bending angle α can be obtained:
$ΔP=∫0π∫0Roμhr((L+hα)4−L4)drdφL(L+hα)3−∫0π∫0Riμr((L+hα)4−L4)(h+e)drdφL(L+hα)3∫0π∫0Ri(h+e)rdrdφ$
(14)

## 3 Characteristics of SPBA

### 3.1 Bending Angle.

To verify the accuracy of the theoretical model, an experiment has been conducted to measure the bending angle of SPBA under different pressures. The experimental setup is shown in Fig. 5(a). SPBA was mounted at the end of the manipulator (ZU3, JAKA) by a rigid flange fabricated by 3D printing (ABS, F170, Stratasys). A digital pressure reducing valve (IR2000, Atuosi) was used to adjust the output pressure of the air compressor. A graph paper was pasted behind the actuator as a reference. To avoid the burst of SPBA, a pretest has been carried out to determine that the maximum pressure supplied to SPBA is 0.2 MPa. The pressure was supplied to SPBA from 0 to 0.2 MPa with a step size of 0.01 MPa. The experimental result was recorded by a camera, and the bending angle was obtained using image analysis software (Image J, National Institute of Health, MD). Five trials were conducted under the same conditions to confirm the repeatability.

Fig. 5
Fig. 5
Close modal

Figure 5(c) shows the comparison between the theoretical (line) and experimental (circles) results for the bending angle of SPBA under different pressures. The error bar represents the standard deviation of five measurements of the bending angle. The small standard deviation of the five measurements at each pressure indicates the high repeatability of the experiment. The bending angle increases nonlinearly with the increase of pressure due to the nonlinear deformation characteristic of silicone rubber. The maximum bending angle of SPBA is 229.8 deg under the pressure of 0.2 MPa. Most importantly, with the increase of the pressure, the high consistency between the theoretical and experimental results indicates the high accuracy of the theoretical model.

### 3.2 Tip Force.

To measure the tip force exerted by SPBA, a blocked force experiment was conducted. As shown in Fig. 5(b), the proximal end of SPBA was mounted on the platform by a fixture and connected to the digital pressure reducing valve. The distal end was in contact with a compression load cell (SBT-20, SBT). A constraining platform was positioned on the top of SPBA. Upon pressure, the deformation of SPBA was constrained by the constraining platform, and the tip force was captured by the load cell. The experiment was repeated five times, and the results were averaged and shown in Fig. 5(d). The tip force increases with the increase of the pressure. The maximum tip force of SPBA is 10.2 N at the pressure of 0.2 MPa.

Table 2 shows a comparison of the tip force of our SPBA with existing SPAs. Table 2 shows that only the two SPAs proposed by Li et al. [31] and Yap et al. [36] have a larger tip force than SPBA. Apart from the large actuated pressures, the use of low-elastic materials can also increase the contact force of the two SPAs. Especially for the SPA proposed by Yap et al. [36], although the SPA fabricated by 3D printing of thermoplastic elastomer has a great tip force, the flexibility, tightness, and service life are extremely reduced. In addition, compared with the other SPAs, SPBA has a larger tip force, which indicates the excellent force characteristic.

Table 2

Comparison of the tip force of SPBA with existing SPAs

SPATypeFabrication methodPressure/MPaTip force/N
SPBASpring reinforcedCasting0.210.2
Wang et al. [38]Fiber reinforcedCasting and winding fiber0.1781.5
Kandasamy et al. [39]Fiber reinforcedCasting and winding fiber0.081.5
Li et al. [31]Penu-net and fabric reinforcedCasting and wrapping fabric0.415
Yap et al. [36]Penu-net3D printing0.377.36
Huang et al. [40]Penu-netCasting0.093.3
Alici et al. [41]Penu-netCasting0.123.6
Park et al. [42]Penu-netCasting0.082.4
Abondance et al. [43]Penu-netCasting0.252.3
Wang et al. [17]Penu-netCasting0.080.8
SPATypeFabrication methodPressure/MPaTip force/N
SPBASpring reinforcedCasting0.210.2
Wang et al. [38]Fiber reinforcedCasting and winding fiber0.1781.5
Kandasamy et al. [39]Fiber reinforcedCasting and winding fiber0.081.5
Li et al. [31]Penu-net and fabric reinforcedCasting and wrapping fabric0.415
Yap et al. [36]Penu-net3D printing0.377.36
Huang et al. [40]Penu-netCasting0.093.3
Alici et al. [41]Penu-netCasting0.123.6
Park et al. [42]Penu-netCasting0.082.4
Abondance et al. [43]Penu-netCasting0.252.3
Wang et al. [17]Penu-netCasting0.080.8

### 3.3 Response Time.

Step signal response is an important dynamic performance of SPBA, which reflects the speed from sleep to work. An experiment has been conducted to measure the response time of SPBA. The measurement device of response time is the same as that of the bending angle described in Sec. 3.1. In this experiment, a step signal was used to instantly open the digital pressure reducing valve. After SPBA reached steady state, the digital pressure reducing valve was closed instantly. Then, SPBA returned to the initial state. The measurement result is shown in Fig. 5(e). Under the action of the step signal, the bending time and recovery time of SPBA are 0.5 s and 0.8 s, respectively. The fast response speed indicates the superior dynamic performance of SPBA, which can meet the requirements of fast gripping.

## 4 Soft Gripper

### 4.1 Prototype.

SPAs are often used to construct finger-type soft grippers. To demonstrate the potential application of the proposed SPBA, a three-finger soft gripper is designed in Fig. 6. The soft gripper is composed of three SPBAs as the soft fingers, three rigid fixtures, and a rigid flange as the palm. The fixtures and palm are fabricated by 3D printing. The three soft fingers are evenly installed under the palm through the three fixtures. The installation angle between the three fingers and the palm is 120 deg. Therefore, the gripping size of the soft gripper can be increased from the diameter of the palm (120 mm) to the diameter of the fingertip inscribed circle (220 mm). Upon pressure, the three soft fingers can bend inward simultaneously to grip objects.

Fig. 6
Fig. 6
Close modal

### 4.2 Gripping Force.

Figure 7(a) shows the experimental setup for the gripping force measurement of a soft gripper. The soft gripper was mounted at the end of the manipulator. The 3D-printed rigid ball with four diameters (200, 150, 100, and 50 mm) in Fig. 7(b) was connected to the end of the tension meter (SH-100, Handpi). The tension meter was fixed on the optical table through a fixture. After the ball was gripped by the soft gripper, the manipulator moved vertically upward at a constant speed of 18 mm/s until the gripper was completely detached from the ball. In this process, the peak pulling force defined as the gripping force was recorded by the tension meter. When gripping each ball, the pressure of the fingers was increased from 0 to 0.2 MPa at a step size of 0.01 MPa. The experiment was repeated five times, and the results were averaged and shown in Fig. 7(c).

Fig. 7
Fig. 7
Close modal

### 4.3 Gripping Test.

The soft gripper proposed in this article can grip objects by pinching or enveloping. To investigate the gripping performance of the gripper under the two gripping mode, a gripping test has been conducted in Fig. 8 to grip objects with different shapes, weights, and sizes. Generally, gripping the object for 10 s without dropping is considered an effective grip [27]. As shown in Figs. 8(a)8(d), under the pinching mode, the gripper could easily grip objects with different diameters, such as a flowerpot, nut box, clip box, and liquid glue bottle. In Figs. 8(e)8(g), under the enveloping mode, a dumbbell, solder, and silicone bottle were gripped. In addition to working in the air, the waterproof of the SPBA enables the soft gripper to grip objects underwater. As shown in Figs. 8(h) and 8(i), a conch and crab were gripped underwater, which indicates the broad potential applications of the soft gripper in underwater sampling. The gripping tests were recorded by a camera (see Supplemental Movie2). The properties of the aforementioned gripped objects and input pressures required to grip them are listed in Table 3.

Fig. 8
Fig. 8
Close modal
Table 3

Results of the gripping test of the soft gripper

ObjectDiameter/mmWeight/gInput pressure/MPaGripping mode
Flowerpot2002030.11Pinching
Nut box901780.12Pinching
Clip box55900.13Pinching
Glue bottle20100.15Pinching
Dumbbell11012500.2Enveloping
Solder655400.16Enveloping
Silicone bottle755600.16Enveloping
Conch40520.13Pinching
Crab801820.14Pinching
ObjectDiameter/mmWeight/gInput pressure/MPaGripping mode
Flowerpot2002030.11Pinching
Nut box901780.12Pinching
Clip box55900.13Pinching
Glue bottle20100.15Pinching
Dumbbell11012500.2Enveloping
Solder655400.16Enveloping
Silicone bottle755600.16Enveloping
Conch40520.13Pinching
Crab801820.14Pinching

From Table 3, we can see that the soft gripper can successfully grip objects with a diameter from 20 mm to 200 mm. Furthermore, the soft gripper can hold a larger weight of the object under the enveloping-gripping mode than that under the pinching mode. The main reason is that under the pinching-gripping mode, the friction between the fingers and the object is the hold force for the soft gripper. However, under the enveloping mode, the hold force mainly includes the supporting force of the soft fingers, which is much larger than the friction. Therefore, the gripper can hold the dumbbell with a mass of 1.25 kg under the enveloping mode.

## 5 Conclusion and Future Work

The work focuses on the development of a novel high-force SPBA. The simple casting method was used for the fabrication of SPBA, which avoided the complicated manual winding process of fibers of the fiber-reinforced SPA. A theoretical model based on the Neo-Hookean model was developed to predict the bending angle of SPBA, and the controllable deformation of SPBA was achieved. In addition, the tip force and response time of SPBA were experimentally characterized. The results showed the large tip force and fast response capability of SPBA. Finally, SPBAs were used to mimic soft fingers to construct a three-finger soft gripper to demonstrate their potential application. A series of experimental tests were conducted to explore the gripping force and gripping performance of the soft gripper. The results indicated that the soft gripper could successfully grip objects of different sizes and shapes, as well as grip objects with weight up to 1.25 kg, which demonstrates the potential application of SPBA.

This work can serve as a useful reference for researchers working on SPAs, particularly those interested in developing high-force SPAs. In the future, we will perform finite element modeling-based simulations to analyze the deformation of SPBA. In addition, we plan to develop an inchworm-inspired soft crawling robot that uses SPBA as its body. The structure of SPBA will be optimized to achieve a larger output force with a smaller pressure. Finally, we also want to increase the stiffness of SPBA without changing the soft contact with the environment to increase the load of the soft gripper.

## Acknowledgment

This work was supported by the National Natural Science Foundation of China under Grant Nos. 52175125 and 51975537.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request. The authors attest that all data for this study are included in the paper.

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