Geometric Confined Pneumatic Soft–Rigid Hybrid Actuators

Abstract

In this work, we propose a new kind of soft–rigid hybrid actuator composed mainly of soft chambers and rigid frames. Compared with the well-known fiber-reinforced soft actuators, the hybrid actuators are able to ensure the design of noncircular cross-sectional shapes. It is demonstrated that rigid frames are capable of providing geometric constraints, reducing the ineffective deformation, and improving the energy utilization for the hybrid actuators with noncircular cross-sections. The essential characteristics of rigid constraints and flexible constraints are obtained by simulation and experiments on specimens with three different cross-sectional shapes. Furthermore, a spring-fluid film model is introduced to characterize the behavior of a representative hybrid linear actuator and a bending actuator with a rectangular cross-section, and it is also proved by the corresponding experiments. The change of the cross-sectional shape of fiber-reinforced soft actuators under pressurization is also explained theoretically as a contrast. Then, two application examples, namely, a robotic gripper and a caudal fin formed from linear actuators, are designed and demonstrated, showing the advantages and potential applications of the proposed geometric confined hybrid actuators. The proposed soft–rigid hybrid actuators combine the properties of soft and rigid materials, expand the design scope of the compliant actuators, and provide new solutions for robotics, especially for soft robots with specific requirements for their shapes or profiles.

Introduction

Actuators have always been an important enabling factor for the development of robotic systems. In recent decades, extensive research on soft actuators has greatly promoted the rapid development of soft robotic technology that provides new robotic solutions that can conquer complex and unstructured environments, design bionic robot systems and achieve safe human–objects–machine interactions.^1–3 Compared with rigid actuators, soft actuators take advantage of the mechanical properties of soft materials and passive dynamics to achieve reduced design complexity, compliance, safety, and passive adaptability.

In particular, a large group of soft pneumatic actuators (SPAs) consisting of soft inflatable chambers and various reinforcements has been widely studied and used in a variety of applications.⁴ The soft inflatable chambers with different structures serve as key components converting the work done by internal pressure into expansion deformation. To obtain a certain kind of motion profile of a SPA, the material deformations in the directions and positions that contribute to the target deformation of the chamber should be enhanced, while those that do not should be constrained. So far, several kinds of reinforcements have been reported in the literature to constrain the ineffective deformation of a chamber, which results in more efficient desired deformations of the SPAs.

Fiber reinforcements have been widely introduced into the SPAs to achieve a wide range of deformation modes, and have been intensely studied in the robotic community.^5–7 The basic function of the fiber reinforcements is to restrict the strain along the fiber direction, and the most representative cases are the fibers that are used as inextensible layers or to constrain the circumferential expansion of the SPAs.^8–10 Using fabric reinforcements to tune the deformation of inflatable chambers for the SPAs is another method.^11–14 The well-known and widely used McKibben actuator belongs to this category. The fabric wrapping the soft chambers prevents bloating behaviors and is easily cut into various patterns for a variety of requirements. Additionally, paper-based structures have also been introduced into the SPAs to enhance their performance.^15,16

In addition, a fishbone-like structure (FLS) which is three-dimensionally (3D) printed with ductile material has been introduced to restrict the radial expansion as well as the extension at the bottom layer of a SPA.¹⁷ The ductile material is flexible so that the backbone of the FLS can bend along the robotic finger, and in addition, the single bone deforms during the actuation of the SPA. The FLS cannot restrict the cross-section of the SPA and it belongs to the flexible constraint. Taking advantage of the current reinforcements, SPAs are able to generate various deformations, including bending, twisting, linear extension, and contraction.

However, ineffective deformation still appears inevitable when undesirable deformation occurs during the actuation because the flexible reinforcements themselves generally have very small bending stiffness, and therefore the cross-sectional shapes of the SPAs will be difficult to constrain under pressurization. Moreover, in situations where there are specific requirements for the cross-sectional shapes of soft actuators, those constrained by flexible reinforcements will be difficult to use. Recently, a soft robotic pad constrained by fiber matrix was developed, showing the difficulty to use flexible constraint to achieve pad structure.¹⁸ By contrast, rigid reinforcements are able to constrain the deformation more effectively and result in soft–rigid hybrid actuators. For tasks that require delicacy and strength at the same time, such as human–machine interaction, agriculture, or robotic surgery, soft–rigid hybrid actuators possess great advantages and potential for applications.

Through various forms of combinations of soft and rigid components, many soft–rigid hybrid robotic systems have been reported in the literature. For example, a pneumatic soft–rigid hybrid actuator consisting of half-bellow-shaped soft sections between block shape rigid sections is developed.¹⁹ The rigid sections serve as connections between soft actuator sections and enhance force transfer. A lobster-inspired hybrid actuator consisting of a soft inflatable chamber and a rigid shell with joints and links is presented,²⁰ showing a practical approach to design robots capable of achieving good accuracy and compliance. Several hybrid soft–rigid robotic manipulators are presented to show some design principles for building hybrid robotic manipulators that incorporate soft and rigid materials.^21,22 A hybrid system comprising a hard-wheeled robot and a four-legged soft robot is developed showing the advantages of combining these two kinds of robots.²³

In this study, instead of the specific or case-by-case designs reported in the current literature, we propose a new kind of geometric confined compliant soft–rigid hybrid actuators consisting of soft inflatable chambers and rigid frames. Similar to the fiber-reinforced soft actuators, this kind of actuator is general and applicable to many kinds of robotic systems. The effect of rigid reinforcements on soft inflatable chambers with various cross-sectional shapes is studied by comparing with that of the flexible fibers. The performance of a representative hybrid linear actuator and a bending actuator with rectangular cross-section is investigated through modeling and experiments to show the basic response characteristics of the proposed hybrid actuators. Then, two application examples, including a flat two-finger robotic gripper and a flat dual-chamber caudal fin are designed and demonstrated, showing the advantages of the proposed hybrid actuators for application scenarios involving specific requirements for the shapes or profiles of the actuators or the soft robots.

Performance of the Proposed Hybrid Actuators

Essential characteristics of the rigid constraints and flexible constraints

The soft–rigid hybrid actuators proposed in this study mainly consist of two parts: one part is the soft inflatable chambers working as deformable bodies and the other part is the rigid frames working as reinforcements. One significant feature of the rigid reinforcements is that they are able to provide a geometric constraint, that is, the cross-sectional shapes of the soft chambers can be kept under pressurization. From the view point of cross-sectional shapes of the soft chambers, as shown in Figure 1, three cross-sectional shapes, including circular cross-section, cuboid cross-section, and trefoil cross-section are used to investigate the effect of rigid constraints (provided by rigid frames) on the deformation of cross-sections during the inflation of pneumatic chambers. These three cross-sections represent a large range of cross-sectional shapes from convex polygons to concave polygons of the hybrid actuators.

FIG. 1.

Several designs of the proposed hybrid actuators (bottom) compared with the fiber-reinforced soft actuators (middle). They are with three different cross-sectional shapes (top) covering a range of cross-sectional shapes from convex polygon to concave polygon of the hybrid actuators. Simulation and experimental results of their deformation characteristics are presented, showing the differences between these two types of compliant actuators qualitatively and as well as the advantages of the proposed hybrid actuators.

To illustrate the characteristics and advantages of the hybrid design, the widely used fiber-reinforced (represent flexible constraints) soft actuators are designed for comparison, which use the same inflatable chambers as the hybrid ones. Based on these specimens, the deformation of the proposed frame-reinforced soft–rigid hybrid actuators compared with the fiber-reinforced soft actuators are presented through simulation and experimental results, as shown in Figure 1.

It can be seen from Figure 1 that, for the actuator with a circular cross-section, the flexible fibers and rigid frames both have the same effect on constraining the deformation of the soft inflatable chamber. The radial deformation is constrained and the cross-sectional shape remains unchanged, resulting in the same deformation along the axial direction. However, for the actuator with a noncircular cross-section, the flexible fibers are not able to effectively constrain the cross-sectional shape, which changes toward a circular cross-section owing to the radial deformation, whereas the cross-sectional shapes of the actuators constrained by rigid frames are always retained during the deformation. These facts indicate that the design of pneumatic actuators with noncircular cross-sections cannot be ensured by the flexible reinforcements that are able to constrain the circumferential deformation but not the cross-sectional shape.

Therefore, compared with the fiber constraint, it can be obtained that the essential characteristic of the rigid constraint is that it can ensure the design of noncircular cross-sections of pneumatic actuators, and in situations where there are certain requirements for the cross-sectional shapes of the pneumatic actuators, the rigid reinforcements will play an important role. For pneumatic actuators with noncircular cross-sections, in addition to the change of cross-sectional shape, the ineffective deformation is inevitable to a certain extent when constrained by flexible fibers. The work done by the input pressure changes into two parts: one part is the deformation energy of the inflatable chamber and the other is the output work. For linear actuators, in this study, the energy consumed by the change of the cross-sectional shape reduces the proportion of useful work and should be avoided. Thus, the actuators constrained by rigid frames minimize the ineffective deformation and become more efficient.

Theoretical modeling of a representative hybrid linear actuator with rectangular cross-section

To further explore the characteristics of the proposed soft–rigid hybrid actuators quantitatively, an analytical model is required to capture the relationship between the input pressure and the deformation, as well as the relationship between the input pressure and the output force, which are important aspects of an actuator. In this study, we develop a quasistatic model of a representative hybrid linear actuator with a rectangular cross-section that helps reflect the differences between the effects of flexible and rigid reinforcements. The schematic diagram of the hybrid linear actuator in common working condition is shown in Figure 2. One end of the actuator is fixed (e.g., to a mobile platform), and the other end interacts with the environment by an external force. The actuator mainly consists of a rectangular soft chamber and rigid frames used to constrain the outward expansion of the chamber under pressurization. Thus, the actuator is only able to deform along the length freely or produce force when it encounters a hindrance.

FIG. 2.

Schematic diagram of the representative hybrid linear actuator in working condition and the modeling process. The sidewall of the soft chamber can be unfolded to a sheet of rubber, which can be equivalent to a system of springs filled with incompressible fluid. The springs are along the axes of a rectangular coordinate system forming a number of cubic infinitesimal elements, one of which is enlarged. The dimension in the initial (unloaded) and current states (loaded) of the rubber sheet are shown as well as the forces. This is inspired by the work done by Patrick et al.²⁴

For the hybrid linear actuator, all the sidewalls deform consistently along the direction of elongation. Therefore, the deformation of the sidewalls (i.e., the deformation of the actuator) is equivalent to the deformation of the rubber sheet obtained by unfolding the sidewalls, as shown in Figure 2 (bottom). Furthermore, the force analysis of the actuator is equivalent to the analysis of the rubber sheet. The pressure and external force produce normal stresses on the rectangular cross-section of the actuator, which corresponds to the normal stress $σ_{l}$ acting on the sheet in the lengthwise direction. Similarly, the internal pressure and external rigid frames produce normal stress $σ_{t}$ in the thickness direction of the unfolded rubber sheet. The deformation of the film in the width direction is constrained, which reflects the condition that the circumferential deformation of the actuator is constrained by rigid frames. The quasistatic model can be obtained through force equilibria across the sidewalls of the unfolded rubber sheet. Based on the above analysis, the external normal stresses acting on the sheet are expressed as follows: $σ_{l} = \frac{P (A - 2 t) (B - 2 t) - F}{2 (A + B) t - 4 t^{2}}, σ_{t} = - P$ (1)

where P is the internal pressure, F is the external force acting in the direction of elongation, A and B are the width and height of the soft chamber, respectively, and t is the wall thickness of the soft chamber or the unfolded rubber sheet.

To model the soft cuboid rubber sheet, there are two main mechanical properties that need to be considered. One is the hyperelastic material characteristics in all three directions, and the other is the mechanical coupling of the spatial deformations based on the incompressibility of the material. According to the work done by Patrick et al.²⁴ the rubber sheet can be assumed to be a 3D framework composed of a system of springs, which are embedded in a cuboid filled with incompressible fluid. The spring-fluid equivalent model of a basic material element is shown in Figure 2 (partial enlarged detail). The springs are assumed to have an equal free length in each direction and their stiffnesses are taken to be constant. The spring behavior in each space direction is coupled mechanically by the hydrostatic pressure p_h of the fluid and the cuboid shape of the rubber sheet or the framework is maintained during its deformation. The equilibrium equations are developed for the description of the mechanical stress versus stretch ratio of the sheet in the following three directions: the length direction (i.e., the elongation direction of the actuator), the width direction (i.e., the circumferential direction of the soft chamber), and the thickness direction. When under the external normal stress in each space direction $σ_{i} (i = l, c, t)$ , the force equilibria across the sidewalls of the unfolded rubber sheet are²³: $σ_{i} = K_{i} λ_{i} (λ_{i} - 1) - p_{h},$ (2)

where K_i represents the stiffness of the material model, which can be obtained from calibrations. Since the material in this study is homogeneous and isotropic, the stiffness is the same for all three directions, that is, $K_{i} = K$ . p_h represents the hydrostatic pressure of the fluid. Then, the deformation of the film is given by the stretch ratios $λ_{i}$ , which are expressed as: $λ_{l} = \frac{l}{l_{0}}, λ_{t} = \frac{t}{t_{0}}, λ_{c} = 1$ (3)

where l and t represents the current dimensions of the rubber sheet, and l₀ and t₀ represents the corresponding original dimensions of the rubber sheet. Because of the constraint in circumferential direction of the soft chamber, the stretch ratio $λ_{c}$ equals one. In addition, considering the volume conservation condition, another equation can be added: $λ_{l} λ_{c} λ_{t} = 1 .$ (4)

Substituting Equations (1), (3), and (4) into Equation (2), we can obtain the relationship of elongation, pressure, and external force as follows: $\begin{matrix} (\frac{(A - 2 t) (B - 2 t)}{2 (A + B) t - 4 t^{2}} + 1) P - \frac{1}{2 (A + B) t - 4 t^{2}} \\ F = K \frac{l}{l_{0}} (\frac{l}{l_{0}} - 1) - K \frac{l_{0}}{l} (\frac{l_{0}}{l} - 1), (t = \frac{l_{0} t_{0}}{l}) . \end{matrix}$ (5)

The terms on the left side of the above equation show the coupling relationship between the deformation and internal pressure and between the deformation and external force, respectively, and the terms on the right-hand side show the resulting deformation.

Based on the above spring-fluid modeling approach, we also theoretically analyze the deformation behavior of a cuboid soft linear actuator constrained by flexible reinforcements under the internal pressure. Through this analysis, the difference between the effects of flexible and rigid reinforcements on soft inflatable chambers has been further illustrated with regard to the chambers with noncircular cross-sections. Figure 3 shows one segment of the cuboid soft linear actuator. The entire actuator can be seen as a series connection of many of the segments, so the deformation behavior of the segment is able to reflect that of the entire actuator. As the flexible reinforcements are bendable and unable to provide geometric constraints, the soft chamber of the actuator will expand outward when inflated with pressure. In this study, it is assumed that the cross-sectional shape of the inflated chamber maintains an ellipse during expansion, as shown in the Figure 3 (loaded). The perimeter of the ellipse is constant, which is equal to the circumferential length of the flexible reinforcements (the flexible reinforcements are inextensible).

FIG. 3.

Schematic diagram of the deformation of a flexible-constrained (e.g., fiber-reinforced soft actuators) pneumatic linear actuator during pressurization. The actuator has the same dimensions as that of the aforementioned hybrid linear actuator used for modeling, except for the reinforcements, which is to provide a better comparison between these two kinds of actuators. Because of the uniform deformation in the direction of elongation, a section of the actuator is taken for modeling.

2 π b + 4 (a - b) = 2 (A - 2 t_{0}) + 2 (B - 2 t_{0})

(6)

where a and b are the semimajor axis and the semiminor axis of the assumed ellipse, respectively. Based on the above assumption, we are able to determine the deformation behavior of the cuboid soft linear actuator under constant pressure by establishing the potential energy expression of the segment. According to the above spring-fluid modeling approach, the soft segment can also be modeled as a 3D framework of a system of springs. Thus, the potential energy of the segment can be obtained by subtracting the work done by pneumatic pressure from the elastic energy of the spring system, which is expressed by: $\begin{matrix} Π & = V_{m} \frac{1}{2} K {(λ_{l} - 1)}^{2} + V_{m} \frac{1}{2} K {(λ_{t} - 1)}^{2} \\ - P (π a b l - (A - 2 t_{0}) (B - 2 t_{0}) l_{0}) \end{matrix}$ (7)

where V_m is the material volume of the segment and it is constant. Considering the incompressibility and the circumferential constraint of the flexible reinforcements, we have: $λ_{l} λ_{t} = 1 (λ_{c} = 1)$ (8)

Substituting the Equations (6) and (8) into the Equation (7), the following is obtained: $\begin{matrix} Π & = V_{m} \frac{1}{2} K {(λ_{l} - 1)}^{2} + V_{m} \frac{1}{2} K {(\frac{1}{λ_{l}} - 1)}^{2} \\ - P (\frac{π a (2 A + 2 B - 8 t_{0}) - 4 π a^{2}}{(2 π - 4)} λ_{l} l_{0} - (A - 2 t_{0}) (B - 2 t_{0}) l_{0}) \end{matrix}$ (9)

It can be seen that the potential energy of the segment under constant pressure is related to the axial deformation represented by $λ_{i}$ and the deformation of the cross-sectional shape represented by a. To illustrate the change of the cross-sectional shape, the partial derivative of potential energy function $Π$ with respect to a is calculated as: $\frac{\partial Π}{\partial a} = - P (\frac{π (2 A + 2 B - 8 t_{0}) - 8 π a}{(2 π - 4)} λ_{l} l_{0})$ (10)

According to the principle of minimum potential energy, the cross-sectional shape under static state can be obtained by setting the above equation to zero, and the result is: $a = \frac{(A + B - 4 t_{0})}{4} .$ (11)

Note that the cross-sectional shape of the flexible constraint actuator will change under pneumatic pressure. In addition, the calculated semimajor axis a is not the radius of the cross-section when it deforms to a circle $(A + B - 4 t_{0}) π$ , and this is because the formula of ellipse perimeter is chosen as $2 π b + 4 (a - b)$ , which is a simple approximation of the ellipse perimeter.

From the above analysis, it can be seen that for the hybrid linear actuator, the work done by pneumatic pressure $W_{p r e s s u r e}$ is converted to elastic energy of elongation $E_{e l o n g a t i o n}$ and output work $W_{o u t p u t}$ , which is expressed by: $W_{p r e s s u r e} = E_{e l o n g a t i o n} + W_{o u t p u t} .$ (12)

But for the flexible constraint linear actuator, the work done by pneumatic pressure $W_{p r e s s u r e}$ is converted to three parts: elastic energy of cross-section change $E_{c r o s s}$ , elastic energy of elongation $E_{e l o n g a t i o n}$ , and output work $W_{o u t p u t}$ , which is expressed by: $W_{p r e s s u r e} = E_{c r o s s} + E_{e l o n g a t i o n} + W_{o u t p u t} .$ (13)

In this study, for the two kinds of linear actuators, the elongation is effective deformation for actuation but the cross-sectional change is ineffective deformation for the actuation. Therefore, the hybrid linear actuator has a higher energy utilization in that it has no energy consumption due to ineffective deformation.

Theoretical modeling of a representative hybrid bending actuator with rectangular cross-section

In this section, we develop a quasistatic model of a representative hybrid bending actuator with a rectangular cross-section that helps illustrate the relationship between the input pressure and bending angle. The schematic diagram of the hybrid bending actuator is shown in Figure 4. The bending actuator is obtained by adding a strain-limiting layer to the bottom wall of the linear actuator. The other side walls will deform until their elastic torque balance the torque produced by the input pressure. Because the chamber wall is a thin layer, we ignore the influence of the thickness change. The pressure torque $T_{p r e}$ can be obtained as follows:

FIG. 4.

Schematic diagram of the hybrid bending actuator. The bending actuator is obtained by adding a strain-limiting layer to the bottom wall of the linear actuator. The elastic torque produced by the other side walls balances the torque produced by the input pressure.

T_{p r e} = P (A - 2 t_{0}) (B - 2 t_{0}) \frac{B}{2}

(14)

According to Equation (2), the elastic torque produced by the top wall $T_{t o p w}$ can be obtained as follows: $T_{t o p w} = B t_{0} A (K (\frac{B θ}{l_{0}} + {(\frac{B θ}{l_{0}})}^{2}) + K \frac{B θ l_{0}}{{(l_{0} + B θ)}^{2}} + P)$ (15)

The elastic torque produced by the side wall $T_{s i d w}$ can be obtained as follows: $T_{s i d w} = \int_{0}^{B - t_{0}} h t_{0} (K (\frac{h θ}{l_{0}} + {(\frac{h θ}{l_{0}})}^{2}) + K \frac{h θ l_{0}}{{(l_{0} + h θ)}^{2}} + P) d h$ (16)

Then, according to the torque balance, it can be obtained as follows: $T_{p r e} = T_{t o p w} + 2 T_{s i d w}$ (17)

where l₀ is the initial length of the chamber and it is also the length of the strain-limiting layer. t₀ is the thickness of the chamber wall. $θ$ is the bending angle of the actuator. P is the input pressure. According to the above equations, we can obtain the relationship between the bending angle and the input pressure.

Experimental results and analysis

To verify the validity of the spring-fluid theoretical model for characterizing the performance of the proposed hybrid actuators, the corresponding experiments were conducted on the hybrid actuator with rectangular cross-section used for modeling and on the fiber-reinforced soft actuator for comparison. Both types of actuators used the same soft inflatable chamber. The rigid frames were made of photosensitive resin by ultraviolet-curing 3D printing technology, and the soft chambers are made of silicon rubber (Elastosil M4601; Wacher Chemie AG) and fabricated through casting, one of the most common ways to fabricate SPAs. As the above fabrication procedures are well known, they will not be described in detail in this study. The rigid frames are mounted equidistantly on the soft chamber just like wearing a ring. The internal surface of the frames directly contacts with the outside surface of the soft chamber.

When pressurized, the soft chamber cannot expand in the radial direction due to the constraint of the rigid frames, and there will be force between the chamber and frames. Then, friction arises between the chamber and frames, which makes the frames move synchronously as the chamber elongates. The width of the frame is chosen to ensure better constraints based on the material (photosensitive resin), and frames with smaller width cannot fully constrain the radial deformation and frames with larger width will occupy more space. The aspect ratio of the frame is the same as the soft chamber.

The model parameter K is determined by fitting the theoretical curve of the uniaxial tensile deformation of a specimen to the corresponding experimental data, as shown in Figure 5a. The uniaxial tensile test was carried out under a quasistatic state in the experiment because, on the one hand, it conformed to the characteristics of the spring-fluid modeling approach, and on the other hand, viscoelasticity is not notable for the silicon rubber used in this study. Theoretically, for a uniaxial tensile test, that the following can be obtained:

FIG. 5.

Theoretical predictions and experimental results. (a) Fitting of the theoretical model to the mechanical behavior of M4601 in a uniaxial tensile test. (b) The relationships between the output force and actuation pressure for the hybrid linear actuator with three different initial stretch ratios. (c) The relationship between the longitudinal stretch ratio and actuation pressure for the hybrid actuator. (d) The radial deformation of the fiber-reinforced soft actuator with the same inflatable chamber as the hybrid one. The test prototypes are shown in the figure and as well the meaning of the quantities.

σ_{l} = K [λ_{l} (λ_{l} - 1) - \frac{1}{\sqrt{λ_{l}}} (\frac{1}{\sqrt{λ_{l}}} - 1)], (σ_{t} = σ_{c} = 0)

(18)

where $σ_{l}$ is the tensile stress and $λ_{l}$ is the corresponding stretch ratio. A good match between the experimental tensile test of M4601 and the theoretical model was achieved for a stiffness of $K \approx 3.037 \times 1 0^{- 1} M P a$ (Fig. 5a). Note that the stretch ratio in the tensile test reached 2.3, which covers the working range of most pneumatic actuators.

Figure 5c shows the relationship between the longitudinal stretch ratio and the actuation pressure for the hybrid actuator under the free condition. It can be seen that with an increase in pressure, the stretch ratio increases. In general, the experimental results denoted by points agree well with the theoretical prediction denoted by a solid line. However, note that the experimental data are slightly lower than that in the theoretical prediction because there are intervals without constraint between the rigid frames. This discrepancy is easy to compensate for, as it is consistent in the entire region. Furthermore, within the scope of sixty percent of the elongation $(λ_{l} \approx 1.6)$ of the actuator, the relationship is nearly linear, which means that the control of the proposed hybrid actuator will be easy.

The relationships between the output force and actuation pressure for the hybrid actuator under three stretch ratios were shown in Figure 5b. The output force is proportional to the actuation pressure when the elongation is hindered by the environment, because in this case all the pressure increments act onto the environmental objects through the contact surface. The experimental data agree well with the theoretical prediction. Figure 5d shows the radial deformation of the fiber-reinforced actuator when pressurized. The rectangular cross-section changes into a circular one under very small pressures, and then the cross-sectional shape does not change owing to the constraint of the fibers. Theoretically, the diameter of the circular cross-section becomes $(2 A + 2 B - 8 t_{0}) π + 2 t_{0} = 21.735 m m$ , which is close to the experimental data. The difference is because that the bending property of the chamber wall is not uniform, that is, the bending stiffness at the corner of the rectangular chamber is larger than that of the chamber wall.

Figure 6 shows the relationship between the input pressure and bending angle of the hybrid bending actuator under the free condition. It can be seen that with an increase in pressure, the bending angle increases. In general, the experimental results denoted by points agree well with the theoretical prediction denoted by a solid line, and within the scope of 0.14 MPa, the relationship is nearly linear. In addition, the rigid frames may limit the range of bending angle of the actuator under large deformation, as shown in Figure 7. When one end of the frames touches the adjacent ones, it can be obtained that:

FIG. 6.

Theoretical predictions and experimental results of the bending actuator.

FIG. 7.

Effect of the rigid frames on the range of bending angle of the hybrid bending actuator.

l_{0} - H β = N W

(19)

where H is the thickness of the frames. W is the width of the frames. N is the number of the frames. $β$ is the current bending angle, and it can be expressed as follows: $β = \frac{l_{0} - N W}{H}$ (20)

This means that the largest bending angle for hybrid bending actuators is related to the initial length of the soft chamber, the number of the frames and the size of the frames.

Application Demonstrations of the Proposed Hybrid Actuators

Usually in soft robotics, linear actuators are seldom used directly in practical applications, and they are often converted to different forms of bending actuators through certain mechanisms, such as antagonism, joints, strain limiting, and so on. Therefore, the linear actuator is used to show the basic characteristics of the proposed hybrid actuators through modeling and experiments. Then two kinds of bending actuators (the robotic gripper with two flat fingers and a flat caudal fin) are used to demonstrate the advantages in some specific applications, where actuators need to be able to maintain certain shapes designed previously during the actuation, and it is difficult to achieve by current flexible constraint actuators.¹⁷

For example, soft manipulators should adapt their shapes to the working environment and ensure a suitable way of contact with the objects, and bioinspired soft structures achieve functions of creatures by mimicking their shapes as the first step. In this study, a flat robotic gripper with two fingers (Fig. 8a) and a flat caudal fin (Fig. 8b) were designed and built based on the proposed soft–rigid hybrid actuators. The robotic fingers as well as the caudal fin have a flat structure, that is, the thickness is much lower than the in-plane dimensions. For the robotic gripper, this configuration provides line contact between the fingers and objects, which enables it to grip objects with various shapes using only two fingers. For the robotic caudal fin, the flat structure during actuation is a basic requirement for mimicking the shape of a real caudal fin (Supplementary Video S1).

FIG. 8.

Demonstrations of two kinds of applications: a flat two-finger gripper (a) and a flat robotic caudal fin (b). The design of the actuators is shown in the upper left corner of each graph. Both the actuators are able to maintain a flat structure during actuation. The flat gripper is able to provide line contact with objects and it can grip them easily from the sides (a3–5). For the robotic caudal fin, it can propel a plastic fish body forward at a speed of 16 mm/s with the actuation frequency of 0.5 Hz.

In these two application cases, two kinds of soft–rigid hybrid bending actuators were designed. One is composed of a linear actuator and a strain-limiting layer on one side, achieving a unidirectional bending actuator, which is used for the robotic gripper. The other is composed of two linear actuators connected in parallel, obtaining a bidirectional bending actuator, which is used for the robotic caudal fin. Thus, the linear actuator is the basic component of the two bending actuators. The robotic gripper is able to grip a cuboid and a cylinder from different angles easily because of the line contact, as shown in Figure 8a. The robotic caudal fin is mounted on a plastic fish and propels it forward with input pressure of half-sine wave, as shown in Figure 8b.

In these two demonstrations, two kinds of rectangular bending actuators based on the proposed hybrid actuator technique are designed, and the cross-sectional shapes are maintained during the actuation, which is difficult to be achieved when using fiber-reinforced soft actuators. The proposed soft–rigid hybrid actuators have the advantages of maintaining the original design shapes and profiles of the actuators and the soft robots as well. This feature makes the hybrid actuators more appropriate for some bioinspired soft robots, especially for those having specific requirements for the shapes and sizes. In addition, some working environment imposes restrictions on the shapes or profiles of the soft robots, which is also an aspect of application of the proposed soft–rigid hybrid actuators.

Discussion and Conclusion

Compared with the rigid actuators and soft actuators based on rigid materials and soft materials respectively, the soft–rigid hybrid actuators proposed in this study provide a new way to combine soft and rigid materials together; the use of rigid materials to constrain the ineffective deformation of the soft materials results in higher energy utilization. As seen from this study, fiber-reinforced soft actuators are difficult for the designs of noncircular cross-sections, whereas the proposed hybrid actuators are easy in such scenarios and can expand the design scope of cross-sections for compliant pneumatic actuators. Furthermore, the rigid reinforcements are able to protect the inflatable chambers from explosion or punctures to a certain extent. The output force and the response speed of a soft–rigid hybrid actuator can be high and efficient because of the restriction of ineffective deformation of soft materials. Then, more diversified frames or frames with hinges can be developed or embedded in the soft material in future studies for more diverse applications.

For modeling the hybrid linear actuators, the deformation of the actuator is fundamentally the deformation of the sidewalls along the elongation direction, and the 3D deformation of the sidewalls of the actuator is the same as the two-dimensional deformation of the unfolded sidewalls. But in a more general case, some conditions should be considered such as whether the wall thickness is consistent, whether the deformation is uniform or regular, and so on. In this study, the spring-fluid film model is suitable and agrees well with the experimental results because the core deformation part of a hybrid linear actuator is the soft inflatable chamber, which is a thin-walled structure in general and meets the requirements of the theoretical model.

Besides, the spring-fluid film modeling method may be adjusted to adapt directly to the walls of pneumatic chambers in 3D deformation of hybrid actuators. Moreover, for a specific application of hybrid actuators, the acting force from environment can be introduced into the model, then the quasistatic interaction process can be predicted. In addition, for specific applications, where there are requirements for the cross-sectional shape of actuators, the model can be used to determine the parameters of a hybrid actuator or to verify whether flexible constraint actuators are still competent. How to use the model in specific applications and to combine the model with environmental interactions remains a great challenge for soft robotic communities, which needs more study in the future.

Two application cases have been demonstrated experimentally, showing the ability of the proposed hybrid actuators to achieve the design geometry of the actuators or robots. Although the design of hybrid actuators contains rigid materials, they still have the properties of large deformation and compliance, because the rigid frames are able to move with the deformation of the soft chamber while constraining the ineffective deformations. Therefore, the proposed hybrid actuators can be applied to many fields of soft robotics and provide new solutions for bioinspired soft robots, especially for those with specific requirements for the shapes and sizes. In addition, the proposed hybrid actuators provide some insights for developing hybrid robots, given that most creatures in nature have various skeleton structures and soft tissues.

Footnotes

Author Disclosure Statement

No competing financial interests exit.

Funding Information

This work was supported by the National Natural Science Foundation of China (No. 51675413).

Supplementary Material

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Supplementary Material

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