2.1. Structural design

Pneumatic actuator function by networks of elastomeric channels that inflate upon pressurization. Though multiple soft actuators exist, they have not been utilized to their full extent in a soft robot due to inherent flexibility. A solution was presented herein that relies on a rigidity change in a granular material jamming that allows a material to transition from a liquid-like state to a solid-like state. As shown in Fig. 1(a), a two-layer pneumatic actuator comprises a driving layer and a jamming layer. The driving layer can bend by air pressure, and the jamming layer is to achieve stiffness variation. As shown in Fig. 1(a), the driving layer structure is an extendable layer consisting of several equally spaced inflatable airbags. The inflated airbag is an arched air chamber with semicircular air channels. The bending principle of the actuator is illustrated in Fig. 1(b). When the driving layer is inflated and pressurized, the airbag expands and deforms under the action of air pressure. Furthermore, the jamming layer does not produce telescopic deformation. The actuator can achieve a bending deformation. At rest state, there is no pressure difference between the inside and the outside of the airbag. When actuated by air pressure, the bending angle of the driving layer is dependent on the pressure difference. Although the actuator can achieve good bending properties, the stiffness of the soft actuators cannot be varied a lot given a low pneumatic. So, a jamming layer sealed with a pack of particles is designed to achieve it.

Fig. 1. Structure and working principle of the soft actuator. (a) Design of soft actuator. (b) Bending principle. (c) Stiffness variation principle.

The jamming layer is comprised of the flexible membrane that encloses many granules. The flexible membrane contains particles that can be jammed by applying a vacuum or unjammed by releasing it. The jamming granules loosely distribute in the cavity under vacuum off. The jamming granules are freely movable in the niche due to external force, exhibiting a flowing state. The rigidity of the jamming layer is low. In the vacuum state, the jamming granules densely distribute in the cavity. The jamming granules move to a small extent as subjecting to external force. In this case, the actuator exhibits high stiffness. As shown in Fig. 1(c), there are two working modes. The first mode is the bending way. When the jamming layer is vacuum off, and the driving layer is inflated and pressurized, the actuator could be bending. In this way, the actuator exhibits low stiffness and flexibility to achieve significant bending. The second way is called rigidity mode. In this way, the actuator exhibits high load capacity. Via switching between jamming states, the actuator can behave either like a flexible member with low bending stiffness or like a stiff one with high stiffness.

2.2. Fabrication of actuator

The actuator was fabricated from highly elastic silicone rubber using a casting molding process. Since the driving layer has a multi-cavity structure, a driving and jamming layers were fabricated separately. The driving layer and the jamming layer are bonded to form an actuator. The actuator was made of 30-degree hardness and ordinary AB-type two-component silicone rubber. The specific manufacturing process of the actuator is shown in Fig. 2.

Fig. 2. A fabrication process of the soft actuator. (a) The manufacturing process of the driving layer. (b) Manufacturing of jamming layer. (c) Combination process of the actuator.

3.1. Theoretical model

To illustrate the large-deformation capability of the actuator, we established the model describing the relationship between the air pressure and the bending angle. The soft actuator was fabricated using silicone rubber. We employed an incompressible Neo-Hookean model to describe its mechanical behavior. The strain energy is given by:

(1) \begin{align}W = {C_{10}}({I_1} - 3)\end{align}

where ${I_1}$ is the first invariant of the stress tensor, ${I_1}{\rm{ = }}\lambda _1^2 + \lambda _2^2 + \lambda _3^2$ , ${\lambda _1}$ , ${\lambda _2}$ , ${\lambda _3}$ is axial, circumferential, and radial principal stretch ratio, respectively. ${C_{10}} = \frac{G}{2}$ , where $G$ is shear modulus of the material. The principal stress could be expressed as a function of $W$ , ${\lambda _i}$ , and the Lagrange multiplier $p$ :

(2) \begin{align}{\sigma _i} = \frac{{\partial W}}{{\partial {\lambda _i}}} - p\lambda _i^{ - 1}\end{align}

Due to constraint of the jamming layer, the circumferential principal stretch is negligible, so that ${\lambda _2} = 1$ . Furthermore, the material is incompressible, thus ${\lambda _1}{\lambda _2}{\lambda _3} = 1$ . The principal stretch along each direction can be obtained that

(3) \begin{align} {\lambda _1}{\rm{ = }}\lambda ,\,\,{\lambda _2}{\rm{ = 1}},\,\,\,{\lambda _3}{\rm{ = }}{1 \over \lambda }\end{align}

The principal stress can be given by combining Eqs. (1)–(3).

(4) \begin{align} \left\{ \begin{array}{l}{\sigma _1} = \dfrac{{\partial W}}{{\partial {\lambda _1}}} - p\lambda _1^{ - 1} = \dfrac{G}{2}\left(\lambda - \dfrac{1}{{{\lambda ^3}}}\right)\\{\sigma _2} = \dfrac{{\partial W}}{{\partial {\lambda _2}}} - p\lambda _2^{ - 1} = \dfrac{G}{2}\left(\lambda - \dfrac{1}{{{\lambda ^2}}}\right)\\{\sigma _3} = \dfrac{{\partial W}}{{\partial {\lambda _3}}} - p\lambda _3^{ - 1} = 0\\p = \dfrac{{{C_{10}}}}{2}\lambda _3^2 = \dfrac{{{C_{10}}}}{{2{\lambda ^2}}}\end{array} \right.\end{align}

The stretch $\lambda $ is within the range of $1 \le \lambda \le 1.5$ considered in bending deformation of the actuator. The principal stress ${\sigma _2}$ is significantly smaller than ${\sigma _1}$ . Therefore, the stress ${\sigma _1}$ is the only principal stress to be considered.

Here, we only consider the static mechanical behavior of the soft actuator and set the air pressure inside the air chamber to be the same. Therefore, one of the air cavities was selected for force analysis as shown in Fig. 3. The torque at the inner wall of the air cavity generated by the air pressure denotes ${M_p}$ . ${M_{{\sigma _1}}}$ is the torque from internal stress at the top of the driving layer and the bottom restraining layer. ${M_{{\sigma _2}}}$ represents the torque induced by the internal stress of the driving layer at the top of the airbag. According to the torque balance, the following equation can be obtained as:

(5) \begin{align} {M_p} = {M_{{\sigma _1}}} + {M_{\sigma 2}}\end{align}

Fig. 3. Force analysis of one of air cavity.

The torque ${M_p}$ can be expressed as:

(6) \begin{align} {M_p} = 2P\int_0^{\frac{{\rm{\pi }}}{2}} {({R_c}\sin \varphi + b)R_c^2{{\cos }^2}} \varphi d\varphi \end{align}

where ${R_c}$ is the radius of the air chamber, $\varphi $ is the rotation angle, $b$ is the thickness of jamming layer, and $P$ is the air pressure.

Similarly, the torque ${M_{{\sigma _1}}}$ and the torque ${M_{{\sigma _2}}}$ can be expressed as:

(7) \begin{align} {M_{\sigma 1}}{\rm{ = }}\int_0^b {2{\sigma _1}} ({R_c} + t)L\beta d\beta \end{align}

(8) \begin{align} {M_{\sigma 2}}{\rm{ = }}2\int_0^t {\left\{ {\int_0^{\dfrac{{\rm{\pi }}}{2}} {{\sigma _\tau }[{{({R_c} + \tau )}^2}\sin \varphi + b({R_c} + \tau )]Ld\varphi } } \right\}} d\tau \end{align}

where $t$ is the thickness of the driving layer and ${\sigma _\tau }$ is the tensile stress.

The longitudinal stretch and strain in the jamming layer can be obtained as (see Fig. 3):

(9) \begin{align} {\lambda _\beta }{\rm{ = }}\frac{{\beta + R}}{R} = \frac{{\beta + L/\theta }}{{L/\theta }} = \frac{{\beta \theta }}{L} + 1\quad {\sigma _1}{\rm{ = }}\frac{{{C_{10}}}}{2}\left({\lambda _\beta } - \frac{1}{{\lambda _\beta ^3}}\right)\end{align}

Similarly, the stretch and strain for the top layer is

(10) \begin{align} {\lambda _\tau }{\rm{ = }}\frac{{R + b + ({R_c} + \tau )\sin \varphi }}{R}\quad {\sigma _2}{\rm{ = }}\frac{{{C_{10}}}}{2}\left({\lambda _\tau } - \frac{1}{{\lambda _\tau ^3}}\right)\end{align}

By substituting Eqs. (6)–(8) into Eq. (5), a relationship between the input air pressure and the bending angle can be obtained:

(11) \begin{align} P{\rm{ = }}\frac{{6\left[ {{M_{\sigma 1}}(\theta ) + {M_{\sigma 2}}(\theta )} \right]}}{{4{R_c}^3 + 3{\rm{\pi }}{R_c}^2 + b}}\end{align}

3.2. Motion performance analysis

To demonstrate the motion performance of the actuator, we applied a pressure range of [0–45] kPa to test its bending angle. The theoretical, simulated, and experimental results of the bending angle under various air pressures are shown in Fig. 4(b). Our experiments and theoretical results show agreement and indicate the potential role of predictive motion characteristics for soft actuator design. The experimental measurement results are relatively accurate compared to theoretical and simulation results, because the experiment reflects the bending angle of the actuator under real conditions. It can be seen from the Fig. 4(b) that the theoretical and simulation results are different from the experimental values. The main reasons for the difference between simulation, theoretical models, and experimental results are as follows: (1) the silicone material parameters used in the simulation and theoretical model are obtained by fitting the experimental data measured through the tensile test, and there are errors with actual material; (2) the incompressible Neo-Hookean model adopted in this paper cannot accurately describe the super-elastic properties of silica gel; and (3) the gravity do not take into account in the simulation and theoretical model. The bending angle varies linearly with the air pressures. Furthermore, the bending angle is 224° at the air pressure of only 45 kPa. It was shown that relatively low pressure could gain a significant bending deformation, exhibiting the excellent motion feature of the presented soft actuator.

Fig. 4. Relationship between bending angle and air pressure, structural parameters, and volume fraction.

Fig. 5. Bending angle and end force of the soft actuator with different air pressure levels.

To capture the effects of the structural parameters on the motion performance of the actuator, we analyzed the bending angle variation from analytical and FEM modeling results (using software ANSYS). Poisson’s ratio is set as 0.48, Young’s modulus 2.25 MPa, and the density 1200 kg/m [Reference Laschi, Mazzolai and Cianchetti3]. The Mooney–Rivlin second-order model in simulation software ANSYS is used to describe the mechanical properties of the silicone hyper-elastic material. The material constants associated with silica gel C ₁₀ is 0.18 MPa and C ₀₁ is 0.09 MPa. The tetrahedral element is applied to divide the mesh. The air pressures are exerted perpendicularly to the air channel. According to Eq. (8), the bending angle is associated with these parameters, such as radius of the air chamber, thickness of jamming layer, and thickness of driving layer. We measured the bending angle for parameters ${R_c}$ , ${t_{\rm{r}}}$ , and ${t_{\rm{a}}}$ ranging from 10 to 24 mm, 1.2 to 2.6 mm, 1.2 to 2.6 mm, and maintaining the air pressure 20 kPa. The results showed that as the air chamber radius increases, the bending angle increases. The reason is that the larger the air cavity is, the greater the tensile deformation is under the same air pressure. Similarly, thicker driving and jamming layers limit the tensile deformation, resulting in a smaller bending angle. Therefore, the desired bending angle and motion range can be obtained by reasonably designing its parameters.

Fig. 6. Comparison of four actuators for bending performance. (a) Structural. (b) Bending angles of various chamber section shapes. (c) The relationship between bending and air pressure. (d) The relationship between end forces and air pressure.

Fig. 7. Influence of air pipeline radius. (a) Influence of the pipeline radius on bending angle. (b) Bending angle over time with various radius. (c) Demonstration of bending morphology variation.

Due to the jamming layer bonded with the driving layer, we investigated how the compactness of particles inside the jamming structure affects the actuator’s bending deformation. The compactness can be described with the volume fraction of granules. We experimentally tested the bending angle with volume fraction ranging from 0 to 100%, and the results are illustrated in Fig. 4(f). To demonstrate only the influence of the volume fraction, we set it to the same certain air pressure 25 kPa that does not affect the analytical results. The results showed that the maximum margin of the bending angle is up to ${58^0}$ in no particles and wholly filled with granules. It reveals that the filling degree of particles significantly affects the motion characteristics of the soft actuator. Because the compactness of particles represents the higher stiffness of the jamming layer, it is difficult for the driving layer to bend. Thus, the stiffness and bending performance of the actuator are contradictory and need to be a trade-off.

Figure 5(a) and (b) show the bending angle and end force with different air pressure levels under different thickness of driving layer. From these figures, we can find that as the thickness increases, the bending angle decreases, but the end force increases. This shows that these mechanical properties are contradictory. To obtain a specific mechanical performance for the soft actuator, we need to tailor the structural parameters. When the thickness is 1.8 mm, the end force of the actuator has no significant change, but the force still has a significant increase when it is 2.0 mm. Meanwhile, we consider that the bending angle can be up to above 70⁰ under the air pressure 100 kPa; thus, the thickness of the driving layer is determined as 2 mm. Similarly, the Fig. 5(c) and (d) illustrate the performance difference with the radius of the air chamber. The radius has less influence on end force but has significant effect on the bending angle. Thus, the radius is set as 18 mm. The number of air chamber can also result in the difference between bending angle and end force as shown in Fig. 5(e) and (f). The number of the air chamber has little effect on the bending angle, but great effect on the end force. With the number increases, the end force decreases. The number of air chamber is determined as 13. Thus, the bending angle and output force of the actuator are contradictory and need to be a trade-off. A comparative analysis method was employed to optimize the structural parameters of the soft actuator for functional specifications.

We demonstrated good motion performance by comparing three actuators with various chamber section shapes, such as rectangular, triangular, and square. We assumed the same structural parameters (the lengths, air channel radius, thickness, and volume) of the four actuators. Because the air pressure in each air chamber is greater than that in the outside, the difference of air pressure causes the expansion of the air chamber. This enables adjacent air chambers to press against each other, resulting in tensile deformation of the driving layer. But the jamming layer does not stretch, causing the actuator bend. Therefore, the bending angle of the soft actuator depends on expansion deformation of each air chamber. Figure 6 illustrates the expansion deformation with different section shapes. The air pressure is 20 kPa for measuring the corresponding bending angle (see Fig. 6(b)). The experimental results demonstrated that the designed actuator with a semicircular air chamber has a significantly larger bending angle than the other actuators. Figure 6(c) and (d) show the bending angle and end force with different air pressure levels. The actuator with semicircular section shape has better bending and force output compared with other three section shapes. This further demonstrates that the proposed soft actuator has better mechanical performance.

Furthermore, we investigated the dependence of gas pipeline radius $r$ (see Fig. 3) on bending angle and bending morph over time. Figure 7(a) shows the bending angle with the small pipeline radius of 0.5, 0.25, and 0.1 mm. As can be seen from the figure, the bending angle could decrease as the radius decreases. The smaller pipeline radius increases the flow resistance inside air chamber, resulting in bending morph variation. Figures 7(b) and 6(c) illustrated the bending morph changes over time with various pipeline radii. The actuator has a different motion pattern at a certain time under the same pressure. For example, the morphology varies at time 1 s as shown in Fig. 7(c). Therefore, the actuator has various morphologies during its motion, especially in the case of relatively small air pipe diameters.

4.1. Influence factors of actuator stiffness

As shown in Fig. 8, the jamming layer is equivalent to a cantilever structure composed of multilayer granules as it is subjected to vacuum pressure. When an external force ${F_{\rm{i}}}$ is applied at the end of the jamming layer, the displacement of the end ${d_{\rm{o}}}$ is generated, and the stiffness of the jamming layer can be expressed as:

(12) \begin{align} k = \frac{{{F_{\rm{i}}}}}{{{d_{\rm{o}}}}}\end{align}

Fig. 8. Equivalent model of jamming layer.

It can be seen from formula (12) that when the external force ${F_{\rm{i}}}$ is constant, the end displacement ${d_{\rm{o}}}$ is smaller, the higher the jamming layer stiffness is. From the energy point of view, the work done by the external force is transformed into the elastic strain energy of the jamming layer and the energy loss caused by the friction between the granules of each layer, which can be expressed as:

(13) \begin{align} {W_{\rm{F}}} = {F_{\rm{i}}}{d_{\rm{o}}} = U + {W_{\rm{s}}}\end{align}

where ${W_{\rm{F}}}$ represents the work done by the external force, ${W_{\rm{s}}}$ is the energy loss due to the friction between the granules, and $U$ is the elastic strain energy.

Equation (13) shows that the larger the energy loss is, the smaller the elastic strain energy is. And the smaller the displacement of the end of the jamming layer is, the higher the rigidity of the jamming layer is. The stiffness of the jamming layer is mainly affected by the energy loss caused by the friction between granules, and the friction energy loss between the granules of each layer can be expressed as:

(14) \begin{align} {W_{\rm{s}}} = nf{d_s} = n\mu {F_n}{d_s}\end{align}

where $f$ is the friction between the layers of granules, ${d_s}$ is the sliding displacement of each layer of granules, and $n$ is the number of layers of granules. $\mu $ and ${F_n}$ are the friction factor and the normal force between the granule layers. When the jamming layer is in a vacuum state, the normal force between the granules is

(15) \begin{align} {F_n} = \Delta Ps\end{align}

where $\Delta P$ is the vacuum inside the flexible strip, and $s$ is the surface area between the jamming granules and the flexible membrane.

Therelationship between the number of granule layers $n$ and the volume fraction of jamminggranules is

(16) \begin{align} n = {c_1}{\varphi _z}\end{align}

where ${c_1}$ is the proportionality factor.

The jamming granule volume fraction ${\varphi _z}$ represents the ratio of the jamming granule volume ${V_z}$ to the flexible strip volume ${V_r}$ in the normal state:

(17) \begin{align} {\varphi _z} = \frac{{{V_z}}}{{{V_r}}}\end{align}

Substituting Eqs. (12)–(15) into Eq. (11), the friction energy loss is

(18) \begin{align} {W_{\rm{s}}} = \frac{{{c_1}\mu \Delta Ps{d_s}{V_z}}}{{{V_r}}}\end{align}

The friction factor $\mu $ is related to the size, shape, and surface roughness of the jamming granules[Reference Mosadegh, Polygerinos and Keplinger28, Reference Abeach, Nefti-Meziani and Theodoridis29]. According to Eq. (16), the stiffness of the actuator is related to the size of the jamming granules, the volume fraction, the shape, the roughness of the surface, and the internal vacuum of the jamming layer.

4.2. Stiffness performance analysis

We aim to investigate the actuator’s stiffness variation with granule magnitude, shape, surface roughness, and volume fraction. The experimental setup is shown in Fig. 9(a) to measure the stiffness of the soft pneumatic actuator. We used a weight of 300 g acting on the end of the actuator and measured the corresponding displacement using a laser interferometer. The stiffness at the position can denote the ratio between the load and the displacement. The smooth spheroidal glass sand (see Fig. 9(b)) with diameters of 1, 3, and 5 mm are used to analyze the effect of granule magnitude on the stiffness (Fig. 9(c)). The double star indicates that different particle diameters have a significant difference in stiffness. The experimental results showed the smaller the granule diameter is, the greater the stiffness of the actuator is. The smaller granules increase their contact area, resulting in greater friction and stiffness. To illustrate the influence of granule shapes, we compared the stiffness of two conditions of particles with spherical and cube. Because of the large contact area between cubic granules, the actuator filled with cubic granules exhibits higher stiffness than spherical granules. Granule surface roughness has impact on the contact friction, and two types of jamming layers filled with particles of different relative roughness were used to show their varied stiffness. Obviously, the actuator with rough granules has a higher stiffness, because the coarse granules have a higher friction factor.

Fig. 9. Stiffness test of the actuator. (a) Experimental setup. (b) Granules filled in the actuator. (c) Grab testing of different objects. (d) Relationship between stiffness and volume fraction. (e) Relationship between stiffness and negative pressure. (f) Stiffness under various bending angle.

To illustrate the effect of granule volume fraction on the stiffness, we filled the jamming layer with volume fractions of 20%, 40%, 60%, 80%, and 100%, respectively. Figure 9(d) shows that the increased stiffness is up to 100 N/m, compared with the volume fraction of 20% and 100%. This indicates that the compactness of the jamming layer has a significant influence on the stiffness because of the increase of its friction. As mentioned earlier, jamming of granular media needs to subject to a specific level of vacuum pressure. We explored how various levels of stress impact the stiffness and set the negative pressure range of [0 ∼ -25] kPa. Figure 9(e) shows the stiffness variation under different vacuum pressure. With the increase in vacuum pressure, the stiffness increases rapidly, but the stiffness varies slowly once the pressure is above 15 kPa. Therefore, the stiffness cannot be entirely regulated by vacuum pressure.

Furthermore, we want to understand whether the stiffness varies under a specific negative pressure when the actuator bends to different angles. The air pressure range of [10–50] kPa is applied to the driving layer to bend different angles. As can be seen from Fig. 9(f), the stiffness changes slightly at different bending angles, indicating that the actuator’s stiffness has a good consistency during its motionprocess.

To demonstrate the dynamic feature of the soft actuator, we constructed an experimental setup (Fig. 10(a)) to test its dynamic response subjected to a square-wave air pressure, which is obtained by using relays and solenoid valve sets. It is inflated at period ${t_1}$ , and it is deflated at time slot ${t_2}$ . The square-wave air pressure (duty ratio 50%) with different frequencies indicates various inflating and deflating time. Figure 10(b) described the vibrational process of the soft actuator as the square-wave air pressure with period of 1, 2, 4, and 8 s are applied. By comparing these figures in Fig. 10(b), we can find the following differences as the period T increases. (1) The number of oscillations of the actuator reduces with the increase of the period. (2) Because the square-wave air pressure is applied, the steady-state response is bending back and forth. A red dotted box indicates the steady-state response in Fig. 10(b). The transient process time decreases with the increase of the period. (3) The amplitude of steady-state vibration increases with the increase of period. The reason for this difference is that the rheological properties of the soft actuator vary from different dynamic air pressure. Therefore, the soft actuator exhibits mechanical properties at dynamic pressure that are different from those at static pressure. The soft actuator exhibits different transient and steady vibration behavior when the jamming layer is vacuum off or vacuum on. From the transient response, the actuator with greater stiffness needs more time to reach steady state, which means it is more responsive. But when it stabilizes, a more rigid actuator has a smaller range of motion (see also from Fig. 10(c)), that is, the amplitude of its swing is smaller. Furthermore, we analyzed the vibrational magnitude with various frequency and stiffness variation. The larger the vibration frequency (the smaller the period), the smaller the amplitude of the actuator’s vibration. The difference of vibration magnitude in vacuum and non-vacuum state is determined by the vibration frequency. This dynamic performance provides support for us to design soft robot using this soft actuator.

Fig. 10. Vibration performance of the soft actuator under square-wave air pressure. (a) Experimental setup. (b) Vibration process of the soft actuator with various periods. (c) Comparison of effects of stiffness on vibrational magnitude.

4.3. Variable stiffness test

To show the potential advantage of stiffness variety, we carried out two sets of experiments to demonstrate the carrying capacity and adapt autonomously to different shapes. We compared the maximum mass that can be grabbed using the soft actuator with vacuum on and off (see Fig. 11(a)). Figure 11(b) shows the load-bearing capability of the actuator with different negative pressure. As the negative pressure increases, the weight that can be sustained increases. The actuator can bear up to 700 g when the negative pressure is −60 kPa. The actuator can sustain the weight up to 1.39 kg, and it showed that the effect of stiffness change is evident. It shows that the particle jamming method has better load-bearing capacity compared to SMP [Reference Yang, Chen, Li, Michael and Wei30]. Meanwhile, our proposed variable stiffness method can achieve the change of stiffness by applying different negative pressure. We demonstrate that the actuator grasps a variety of objects (Fig. 11(c)) by this way of pinching grasp and conforming to objects. It shows that the actuator has strong adaptability and can adjust the stiffness in terms of different shapes, sizes, and masses of objects. The soft actuator exhibits the ability to manipulate delicate objects and to undergo large deformation. And the compliance of the actuator allows it conforms to different shapes.

Fig. 11. Demonstration of stiffness variation of the actuator. (a) Maximum weight that can be grabbed. (b) Load-bearing capacity test. (c) Grabbing different objects.