Review of soft fluidic actuators: classification and materials modeling analysis

Pneumatic artificial muscles (PAMs) [112], also known as McKibben actuators, are one of the first generations of SPFAs. This soft actuator is composed of hollow elastomer tubes reinforced by fiber stiffness layers. Depending on their design, they will either expand or contract when pressure is applied. The invention of this artificial muscle is generally attributed to Richard H. Gaylord (1958), but it was popularized at the beginning of the 1960s by Joseph L McKibben [113]. The first SFA gripper, with four fingers, was demonstrated by Suzumori et al [104] in 1989 (figure 3(a)). These fingers include three chambers that give them three DOFs and can bend in any direction. This gripper can grasp a wide range of objects. There is a lot of research on soft robots that can be found using this actuation mechanism. For instance, Polygerinos et al [100] suggested a flexible glove for robot-assisted rehabilitation. The device utilized the McKibben mechanism not only to support precise functional grasping but also to remain light and low profile (figure 4(a)). Some approaches tried to combine multiple McKibben actuators to increase SFA functionality with more complex motions. As an example, Al Abeach et al [114] developed McKibben muscles for a three-fingered gripper. Both extensor and contractor McKibben designs were deployed to provide the form and efficient force for grasping ability, respectively.

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PAM elastomer actuators exhibit complex nonlinear snap-through instabilities. This behavior allows the actuator to gradually store elastic energy, before releasing it suddenly to exert rapid motion or high force [115]. As shown in figure 4(b), Overvelde et al [116] developed this kind of nonlinear mechanism to exert high force and trigger large geometrical changes by sequential steps. Rothemund et al [128] designed a bistable soft valve. They calculated the required switching pressure as a function of the geometry and valve's material McKibben's muscles are also employed in the actuation of continuum robots. Tsukagoshi et al [117], presented an elephant trunk-type manipulator named Active Hose, consisting of a spiral tube turned around the manipulator backbone like a coil, to generate bending moment. This can be useful in rescue operations.

The other type of manipulator which benefited from PAM actuators is OctArm. It was first presented by Grissom et al [106] (figure 3(b)) and consists of three serial sections that are actuated separately. By applying pressure inside the chamber of each section, the arm starts simultaneously to bend and extend longitudinally for the whole-arm grasping of objects [118]. A large manipulator continuum robot with McKibben actuators consisting of six sections is reported by [119] (figure 4(c)). Applying air pressure of around four bars causes a 66% extension in section and 380° rotation in less than 0.5 s. Walker et al [118] in 2005 developed cephalopod robots incorporating 12 McKibben actuators. The considerable length of the robots (120 cm), acting like a manipulator, achieves more kinematic DOFs than in previous pneumatic arms and is more similar to the real biological inspiration. SPFAs can be made using highly extensible elastomer materials such as silicones. With these materials, highly deformable and adaptable soft actuators appeared. In these kinds of actuators, one or more embedded chambers are actuated and deformed by applying pressurized fluid, which can be operated pneumatically [94, 120,121,129], or hydraulically [122–124]. On account of their light weight and cleanness, pneumatic systems in most cases are preferred over hydraulic designs especially in gripper design (figure 4(d)) [125].

Pneumatic networks (PneuNets) are a famous pneumatic version of these actuators working as a gripper. This was first presented by Needleman [130] in 1977. He demonstrated that PneuNets, comprising a series of channels in an elastomer, can inflate like balloons for actuation. This mechanism was later developed and used as a soft gripper by Ilievski in 2011 [107] (figure 3(c)). This gripper consists of six legs for grasping soft fragile objects like an egg or even a live small animal like a mouse. In an interesting work, Mosadegh et al [126] developed the PneuNets architecture, achieving rapid response and more durable actuation cycles by proposing a gap layer between the walls of each chamber. Inextensible fibers are added to the FEAs to boost local stiffness and consequently the weight-object ratio in the grasping application (figure 4(e)). In [131], they employed polyaramid fibers to prevent the local weakening of the elastomer during repeated actuations. Deimel and Brock [132] developed a SPFA with a three-fingered hand and flexible palm. The fingers are made of fiber-reinforced silicone, and the palm has substantial passive compliance. The RBO Hand shows the capacity to grasp a wide variety of objects, including water bottles, eyeglasses, and sheets of fabric. Later, they presented the RBO Hand 2, composed of a five-finger and palm configuration with similar fiber-reinforced actuation technology to develop an SPFA hand [133]. It demonstrated dexterity similar to a human hand with the ability to perform most human grasping tasks.

Various types of PAMs have been developed in recent years. A famous one is a Peano-fluidic muscle presented by Veale et al [134]. It consists of flat layers of thermoplastic, textile reinforced plastic, or textile/silicone composite. The intervals of these layers are bonded perpendicularly in the direction of contraction. When air pressure is applied, the shapes of tubes become round with a contract ratio between 15% and 30%. The geometries of the tube affect the static and dynamic behavior of Peano-muscles [127] (figure 4(f)). The optimum channel should not exceed 20% for maximizing performance. The narrower channels increase flow restriction, subsequently, a damping force model was applied to Peano's muscle for high-accuracy controllability and further suitability in uncontrolled environments [135].

As discussed in the previous section, a similar mechanism to the Peano-muscle is the HASEL actuator. It was introduced in 2018 [64] and designed to produce linear contraction with stack (figure 3(g)). Peano-HASEL is one type of HASEL actuator, exhibits fast and precise linear motion that closely resembles muscle-mimetic activation without stack, prestretch, or rigid frames. It was developed by Kellaris et al [69] and made of a rectangular shell formed by flexible polymer films filled with a liquid dielectric, and planting a pair of electrodes on either side of the shell . When a charge opposes the electrodes zip together due to the electrostatic force, hence the fluid squeezes into the volume of the shell which is not surrounded by the electrodes and creates linear contraction of the actuator. This linear actuator can lift more than 200 times its weight with a strain rate of 900% per second at 10 kV. The fast response speed, self-sensing and self-healing advantages of HASEL actuators make them a promising candidate for applications in different soft robotic mechanisms such as untethered soft robots for manipulation and continuum applications [111], tubular pump [136], and prosthetic finger driven by Peano-HASEL [137]. Rothemund et al [67] reviewed the latest advances and future opportunities of HASEL in the soft actuators field.

Vacuum mechanisms have also been widely employed in soft robots as actuators. Negative-pressure operations are safer, more compact, and more robust compared to pressurized actuators. They cannot burst when the actuator collapse. Moreover, decreasing their volume enables them to go through congested or narrow areas compared to their nominal sizes. One of the representative examples of SVFAs is the universal soft gripper developed by Brown et al [108] as shown in figure 3(d). Because of its simple structure, it is one of the earliest and most famous soft vacuum grippers. Unlike other soft robot actuation mechanisms, it is simply composed of a membrane filled with granular materials; the stiffness of the bag is changed by evacuating air and provides sufficient force for lifting and holding objects. It shows promising performance, especially when the shape or material properties of the object are unknown or when precise grasping is not required. This gripper was able to pick up a wide variety of objects of different sizes and shapes, such as a wooden hemisphere, spring, small LED, tube, cups, raw egg, shock absorber, etc. The device can rapidly grasp and release a wide range of objects; however, it is not appropriate for grasping flat or soft objects. The universal gripper was commercialized in [138] and has inspired several research applications, such as a prosthetic jamming terminal device [139], human collaborative robot [140], universal hand for position adjusting and assembly tasks [141], deep-sea sample-collecting device [142], flexible endoscope [143] (figure 5(a)), and soft multi-modulus manipulator for minimally invasive surgery [144] (figure 5(b)). Amend and Lipson [145] presented two simple two-fingered configurations with pockets of granular material used as end-effectors at the fingertips. This design enables each of the fingertips to work separately as independent universal grippers, or to work together like a finger and a thumb. The variable stiffness, lightweight, and energy efficiency of granular jamming make it popular for use in the soft robotics field [146]. The granule particles can be coffee, glass, plastic, or beans. The application determines the grain size; for example, powder-like granular size is generally utilized in soft robotic grippers [143, 147]. Soft manipulators, which require greater stiffness, normally employ larger grains [148]. Sayyadan et al [149] studied the impact of various mechanical parameters (stiffness, curvature radius, applied moment, internal stresses, and defection) on the behavior of cantilever membrane beam samples by presenting a simplified formulation under different vacuum pressure conditions. They designed various experimental tests with latex membranes filled with granular materials such as hemp, sun-dried barberries, black peppers and datura seeds.

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Another important class of SVFAs was created by Yang et al [110] as a vacuum-actuated muscle-inspired pneumatic structure (VAMPs) (figure 3(f)). It uses the buckling of elastomeric beams to generate muscle-like motions when negative pressure is applied. Its mechanism differs from those of previous elastomeric pneumatic actuators such as PneuNets or McKibben. They can generate a linear motion similar to biological muscles. This mechanism is very similar to the performance of human muscles. Unlike other pneumatic actuation such as McKibben and PneuNets, the deformation is not obtained from area expansion and occurs inside the structure. The VAMP actuator made by Yang was able to lift 400 g. Figure 5(d) shows the performance of the VAMP. They also built a muscle-like actuator to simulate a skeleton arm moving similarly within the human body. It can contract up to 40% of its length, with loading stresses up to 65 KPa. The final displacement of the muscle is nearly five times the primary length of the VAMP. With this design, the gripper can pick up a volleyball weighing 274 g. VAMPs actuators are fast, with low cost, are easy to fabricate, lightweight, and operate safely with human interactions [150] (figure 5(c)). Verma extended Yang's works by combining a pressurized and vacuum actuator for a soft robot climbing in a tube application [154]. This climbing robot is composed of a VAMP actuator for linear motion and two ring-shaped pneumatic actuators at its extremities to hold the robot in position inside the tube. These linear actuators integrated one DOF and provided one single motion. While Jiao et al [151] proposed a multi-task actuator to offer many different types of motion at the same time, such as twisting, radial and linear movement (figure 5(d)). Their design included seven SVFAs to provide five crawl deformations.

Origamis are new innovative structures that have large potential use in soft robotics because of their lightweight, low-cost, easily available materials, and simple design for complex motions. They do not need hinges or joints and are actuated by applying positive or negative pressure. Therefore, according to their design and application, they can be SPFA or SVFA. Origami is the art of generating 3D structures by folding 2D sheets [155]. In [109], Martinez et al proposed a wide range of origami soft actuators by combining a stretchable elastomer with a non-stretchable but easily bendable sheet (figures 3(e) and 6(a)). These actuators can perform a range of complex motions that would be difficult to achieve with hard robots. Figure 6(b) shows the origami-based robotic grippers proposed by Chen et al [152]. It is inspired by a paper fortune teller origami design. Its reconfigurable mechanism makes the gripper be able to pick up flat-surface or non-planar objects. Li et al [156] suggested fluid-driven origami-inspired artificial muscles with multiaxial complex motions . Their origami actuator is fast and powerful, with a very low manufacturing cost. A soft active origami robot with a self-actuation design without the assistance of any external actuators is reported in [157]. In [153] Lee et al suggested a mathematical model for predicting the energy efficiency of a soft origami actuator and connected pump together. This model can be applied for the automation of low-cost off-grid operations and human-robot collaboration (figure 6(c)). Paez et al [158] presented a lightweight origami shell-reinforced bending module within the desired range of displacement and force requirements. Rus et al reviewed the design, fabrication, actuation, sensing, and control of origami robots with their applications in the different robotic areas [159].

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SFAs have been combined with other techniques to improve their performance, including constructability, variable stiffness, and operational range criteria. Table 2 summarizes various novel hybrid actuation approaches to address potential advances in the performance of SFAs. SMPs [160] and low melting point alloys (LMPAs) [161, 178] are deployed with SFAs to enhance shape configurability by changing and controlling the position and bending angle (figure 7(a)). Particle jamming and layer jamming can be integrated by the SFA to increase the stiffness of soft robots [162, 163, 185] (figure 7(b)). Adhesion technology such as electro-adhesive material [164] (figure 7(c)) and Gecko adhesion technique [188] (figure 7(d)) are added to SFA grippers to enhance grasping performance by increasing the lifting weight ratio and object shape diversity. Combining soft and rigid robot characteristics can build new capabilities for soft robots. For instance, Stokes et al proposed a hybrid soft robot consisting of a wheeled robot (hard robot part) and PneuNet SPFA (soft robot part) to manipulate and grasp an object at the same time. Pagoli et al [182] (figure 7(e)) introduced the innovative variable stiffness soft finger. Its soft pneumatic sliding joint can move and rotate along with the finger by using two electric motors. The changing position of the bending point increases the capability of the finger in terms of shape control and variable configuration. The stiffness, and consequently the applied force, at the tipping point of the finger is controlled by the pneumatic pressure inside the soft silicone link. As explained in the previous section, the climbing robot designed by Verma et al [154] includes two kinds of pressurized (SPFA) and vacuumed actuators (SVFA) and thus can also be classified in the hybrid design domain. Hybrid design can also be found in origami soft robots by using simultaneously positive (SPFA) and negative pressure (SVFA) to increase the actuating capability. A hybrid crawling soft robot is illustrated in [187] utilizing these characteristics for mobility. Li et al [181] suggested a pre-charged hybrid gripper with a combination of SPFA and tendon-driven mechanisms. The pushing/pulling cable controls the bending angle of the SPFA, the advantage of the proposed mechanism being that controlling cable movement is much easier and more accurate than pneumatic pressure. Kim et al [165] integrated SPFA with an origami pump which is controlled by a tendon-driven mechanism (figure 7(f)). The main advantage of the proposed gripper is that it can work without needing an external pneumatic source such as a compressor. This design helps to miniaturize soft robot actuators.

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Advances in the field of soft robotics largely depend on the knowledge of material behavior in the design of soft robotic structures and the control of these robots. Silicone rubbers are the most common material used in soft robotic systems, because of their hyper-elastic properties, lightweight, low cost, and fast and simple fabrication. Several SFA design architectures that can be found in the literature use silicone rubbers. They produce high power-to-weight ratios, requiring small input air pressures yet generating large deformations. Furthermore, they can easily be shaped into different configurations which makes them suitable for building soft actuators with a complex design. The actuation performance, such as response time, stiffness and the amount of generated force, is dependent on the type of silicone. The mechanical properties of widespread types of silicones used in soft robotic systems are listed in table 3. Several companies producing elastomer silicone can be found on the market; the most well-known brands are Smooth-On [166], Gelest [189], Dow Corning [167], and Wacker [168]. Most applications of these materials in soft robotic systems, especially in SFAs, are reviewed in this section.

EcoFlex is one of the popular silicones that are frequently used. It is commercialized by Smooth-On with several Shore hardness ratings, ranging from 00–10–00–50. The mechanical properties of three widely-used EcoFlex Shores in soft robotic applications are listed in table 3. They include hyperelasticity capability, which enables them to be streched several times their original size without rupturing. This characteristic makes them convenient in soft applications. For instance, EcoFlex 00–50 and 00–30 are very useful for developing different types of soft sensors, including prosthetic strain sensors [169], hyperelastic pressure sensors [170], flexible and wearable pressure sensors [171], healthcare biomedical wearable sensors [172], and piezoresistive sensors for human motion detection applications [173]. Furthermore, their large elongation properties make them appropriate for actuation mechanisms. Elsayed et al [174] studied the material properties of silicones and their effects on the bending angle of a soft pneumatic actuator. They designed and built the same geometry module with two different silicone materials, EcoFlex 00–30 and 00–50. Their experimental tests showed that the softer EcoFlex 00–30 module required a lower pressure of 0.1 bar, while the other material needed 0.32 bar to reach 90 degrees. Their approach shows that the behavior of the soft actuator is dependent on the type of silicone used. Studying and comparing different types of silicon in this review paper can thus help to select the proper material for SFA actuators. Several works on the use of EcoFlex materials in SPFAs can be mentioned. For instance, Calisti et al [175] proposed an octopus with six flexible limbs made from EcoFlex 00–30 with the dual capability of locomotion and grasping objects. Flexible limbs are responsible for the stability and correct balancing of the octopus in water. They also provide an effective pushing force to move the robot forward and to grasp the object by wrapping themselves around it. Tian et al [176] developed an SPFA human hand made of EcoFlex 00–30. It consisted of five fingers and a palm, with two joints in the thumb and three joints in the other four fingers. This soft hand can reach any point in a 3D workspace, using a variety of shapes and configurations. It also produces low resistance and carries fragile objects without damage [176].

Dragon Skin is another range of silicone commercialized by Smooth-on. Unlike the EcoFlex series, Dragon Skins have a higher Young's modulus and require more fluid pressure to actuate as SFAs. On the other hand, their greater hardness enables them to apply a larger force during actuation. Yap et al [190] studied and characterized the curvature radius and the force in SPFAs with different material stiffnesses. They fabricated four types of silicone rubber (EcoFlex 00–30, EcoFlex 00–50, Dragon Skin 10, and Dragon Skin 20). They defined a ratio coefficient to compare the behavior of these materials in terms of stiffness and output force by dividing the curvature radius by the original length. Their experimental results showed that for SPFAs with a 10 mm thickness, EcoFlex 00–30 achieved a minimum ratio of 0.088 at 42 kPa, while EcoFlex 00–50 reached this ratio at 52 kPa. The required pressure for Dragon Skin 10 to attain the minimum ratio of 0.092 was 180 kPa, and for Dragon Skin 20, the minimum ratio because of higher hardness was not lower than 0.199, when applying 380 kPa. On the other hand, the maximum force of the SPFA increased when the stiffness of the material increased. For example, the maximum force output for EcoFlex 00–10 and 00–50 was 2.33 at 42 kPa and 3.98 at 52 kPa respectively. For Dragon Skin 10 and Dragon Skin 20, the output force ratio was higher, reaching 8.82 at 180 kPa and 9.96 at 380 kPa, respectively.

Another popular silicone rubber in soft robot applications is Sylgard 184, due to its characteristics, including optical transparency, low viscosity, average tear resistance, and the ability to be sealed by plasma-activated surface bonding [196]. It is commercialized by Dow Corning [197]. The viscosity and Young's modulus of this silicone are 3500 cp and 3.9 MPa respectively. The high modulus of Sylgard 184 makes it a stiff and inappropriate choice for SFA applications, since it requires a higher pressure than EcoFlex to function as an actuator. However, it can be useful, especially in soft grippers if integrated with other actuation methods such as DEA [198] or Gecko adhesion [199]. It is also found in a wide range of sensor products, e.g. capacitive strain sensor [200], pressure sensor [201], and tactile sensor [202]. White et al [203] fabricated Sylgard 184 silicone layers with gallium–indium alloy as a resistant sensor to measure the geometry changes due to deformations. They were deployed to build a sensor for soft robots. This sensor was able to measure uniaxial strain and curvature, and could be applicable in soft skin sensors. In a similar approach, Markvicka et al [204] studied the mechanical behavior of an elastomer composite with four different blends of Sylgard 184 and Sylgard 527 containing liquid metal droplets. It ruptured when mechanical damage occurred and could be a suitable sensor for damage detection in soft robots with self-healing properties.

In recent years, self-healing materials have been developed to recover their structure entirely from mechanical damage, without using external stimuli [205]. This autonomous capability increases the commerciality of SFAs, especially in unstructured environments. On the other hand, self-healing polymers are usually more expensive and require more synthetic steps and chemical modification processes [206]. Diels–Alder networks are popular thermo-reversible polymers deployed by Terryn et al [207] to heal SFAs ripped, perforated, or scratched by sharp objects. Later they have shown the safe healing ability of SFA's applications in safe human-robot interactions such as social robots, household robots, and hand rehabilitation devices. Shepherd et al [131] developed a soft fluid actuator integrated with polyaramid fibers (Kevlar) reinforcement. After actuating with positive pressure, this SFA could seal itself after being punctured with a 14-gauge needle. Even after removing the needle, the pressure was retained inside the chamber [208]. Bilodeau et al [209] reviewed recent advanced and future self-healing applications and damage-resilient materials in soft robotic systems.

Elastosil M4601 is commercialized by the Wacker Chemical company and can be found in some of the soft robotics literature. As shown in table 3, its Shore hardness and Young's modulus are very similar to Dragon Skin 30, but unlike EcoFlex and the Dragon Skin series, it has low optical transparency, which limits its applications as a soft optoelectronic sensor. Nevertheless, its higher stiffness makes it a good option for soft actuators, especially in soft gripper applications. Galloway et al [210] developed an underwater two-opposing-pairs soft robotic gripper made using M4601 silicone to manipulate fragile and delicate samples on deep reefs. By applying a 310 kPa pressure, the gripper can produce a 52.9 N lift force. Mosadegh et al [126] replaced the soft EcoFlex with a stiffer Elastosil M4601 and the actuation pressure increased eight times for the same bending angle. Robertson et al [211] suggested four parallel SPFAs made of M4601 to produce a higher force, around 112 N. This is 23% more than the volumetrically equivalent single SPFA. These experiments demonstrated the interest of utilizing a multiple SPFA for high-performance soft robotic applications rather than existing uniform and non-optimal SPFA designs. At room temperature, EcoFlex 00–30 (with a Shore hardness 00–30) has the shortest pot life of 45 min, while this value for Elastosil M4601 with a Shore hardness of 28 A and Sylgard 184 with Shore hardness of 43 A are 90 min. Wienzek et al [212] studied the increase in long-term storage of mixed silicone liquid at low temperature for the strain-limiting top layer of a soft gripper. They tested three types of silicone samples (Elastosil M4601, EcoFlex 00–30, and Sylgard 184). They mixed and maintained the samples at −25 °C for 12 weeks. Viscosity was measured weekly to determine the curing characteristics. The results show that EcoFlex 00–30 solidified after 14 d, while the mixed sample solutions of Elastosil M4601 and Sylgard 184 were still liquid and usable for casting processes after period of 8.7 and 12 weeks, respectively. This study helps to separate the mixing and molding process and increase the fabrication options for silicones.

As listed in table 3 for the production of SFAs, some approaches utilize other types of silicone, such as translucent RTV615 [213] with Shore hardness 44 A and commercialized by Momentive, translucent KE-1603 with Shore hardness 28 A [214, 215], blue color Mold Star 30 A [216], and ExSil 100 with Shore hardness 15 A. ExSil 100 was first introduced by Goff et al [217] and later commercialized by Gelest. Although it has high elongation up to 5000%, its Young's modulus is 0.02 MPa, which makes it too soft to use as an actuator or gripper. It is normally used in diaphragms, microfluidics, vibration damping, high-performance seals, optics and electrical interconnectors [189].

As explained before, the stiffness and generated force in SFAs are dependent on the type of silicon. Considering this, some approaches combine different types of silicone materials to attain the desired stiffness. Shepherd et al [94] developed a multigate walking robot with different silicone layers. Due to its high extensibility under low stresses, EcoFlex 00–30 was used as the actuating layer, and Sylgard 184 was selected as a strain-limiting layer. This combination not only enables the soft robot to operate at low pressures (7 psi), but also provides the desired stiffness. In their next approach [129], these authors replaced the actuation layer of EcoFlex 00–30 by M4601 to increase to larger loads such as the weight of the robot body and components for untethered operation; inevitably, material of this hardness requires higher pressure actuation (22 psi). Hassan et al [218] proposed a tendon-actuated soft three-finger gripper made by using three different types of soft materials: Dragon Skin 30, Smooth-Sil 950, and a third type manufactured by combining Smooth-Sil 950 with EcoFlex 00–30. The intrinsic properties of Dragon Skin 30 make it sticky compared to Sil 950. Thus, the first soft gripper made using Dragon Skin 30 shows a better performance with respect to slipping than the second one. To overcome this limitation in the second gripper, they suggested attaching silicone strips made of Smooth-Sil 950 with EcoFlex 00–30 on the surface of the third gripper to guarantee stable grasping for lateral bending. Subramaniam et al [216] developed a multi-material SVFA gripper with an active palm for grasping applications. They used different types of silicones such as Mold Star30, Smooth Sil 940, Smooth Sil 960, and EcoFlex 00–30 to achieve the desired stiffness, Mold Star30 was selected for the skin layer because of its high deformation at low pressures.

All the silicone materials presented in the previous paragraphs use molding techniques, while in recent years additive manufacturing (AM) techniques such as 3D printing have also been employed to directly fabricate SFAs. The most successful fused deposition modeling (FDM) material for soft robotics is NinjaFlex (Shore hardness of 85 A) made of thermoplastic polyurethanes (TPU), which can withstand strains above 700% with a Young's modulus of around 10 MPa. The SPFAs that are printed using this method can produce a blocking force of up to 75 N [102]. FilaFlex [219] and Agilus30 [220, 221] are the other two types of TPUs employed to print SFAs. The mechanical properties of these materials and their suppliers are listed in table 3. The manufacturing methods of SFAs, and especially 3D printing technology, will be discussed in a dedicated section.

The classical molding method can be used to fabricate different designs of the SFA actuator [222]. Thanks to the latest developments in 3D printing technology, the design of mold parts has improved significantly, which enables the designer to make complex soft components with more accuracy. Normally, catalyzed silicone rubber consists of two parts that should be mixed homogeneously with the specified ratio according to the manufacturer's instructions. In most cases vacuum degassing for 4–5 min is suggested to avoid air entrapment. An alternative and more effective way is putting the mixed silicone into a centrifuge machine. Cure time is variable and differs from 30 min to 1 d at room temperature, depending on the silicone viscosity. This time can be reduced to less than an hour by putting the mixed liquid in an oven at a temperature of around 70 °C [223]. Molding complex structures, especially with undercuts and internal architectures, is very difficult [224]. To overcome this problem, AM methods have also been proposed [225].

SFAs can be printed directly using 3D printers. The FDM method is one of the most widely-used techniques for material fabrication using 3D printers, at low cost and eliminating any supporting molding material, easing changes in the material, and also reducing the fabrication time. The working principle is based on a heating filament and horizontally depositing molten materials via extrusion nozzle onto a surface, layer by layer. NinjaFlex is the most common material used in the 3D printing of SFAs, due to its high strain and force-producing ability when used as an actuator [102, 226]. Peele et al [227] used the stereolithography (SL) technique to produce a SPFA layer by layer from an elastomeric precursor material. The proposed DMP-SL printing process is a promising way to fabricate a monolithic actuator in one single process. In the SL approach, the solidification of liquid resin is controlled by photo-polymerization by a laser beam or a digital light projector. In SL, unlike FDM, one resin can be printed at one time, and this is the major potential challenge of using this technique. For more information about 3D printing methods, the reader is referred to the review articles [228–230].

The earliest analytical model for SPFAs is Euler-Bernoulli's beam theory. In this theory, SPFAs are assumed to be cantilever beams with a fixed support on one side and a moment on the other side. The model is useful when SFAs have simple (particularly symmetric) structures. Several works can be found in the literature using this theory, such as bi-bellow actuators with three chambers developed by Shapiro [231], pneumatic bending joints with anisotropic rigidity [232], and soft biomimetic robotic fish [233]. This theory is not applicable for hyperelastic material with large bending deformations such as silicone, where cross-sectional planes do not remain perpendicular to the bending moment axis. Some approaches have been tried to improve the result of this method. In most of the previous works, Young's modulus is assumed to be constant, while experimental results show that the stress–strain behavior of these materials is more complex, and the relation between cross-section and curvature radius cannot be found easily [234]. The analytical method approach is more successful in continuum robot modeling, especially when the material is not hyperelastic. The backbone curve approach [235] was the first kinematic model for continuum robots. Later, the constant curvature model (CCM) [54] was suggested for the kinematics of multi-section soft robots. Trivedi et al [236] deployed the work-energy principle to develop a geometric model for SPFA manipulators, and showed that their model is more accurate than the CCM. Polygerinos et al [237] used analytical methods to model SPFA with fiber-reinforced bending pneumatic actuators. Wang et al [238] presented a simplified model of a soft pneumatic gripper with simple line links connected by a set of viscoelastic joints. In conclusion, SPA analytical models come with a lot of approximations and simplification in terms of shape and material properties, which make them inaccurate and require a robust controller to compensate for this lack of accuracy.

5.2.1. Off-line FEM simulation.

Due to the highly nonlinear responses of silicone rubbers, modeling and analyzing SPFAs is quite challenging. The finite element method (FEM) has widely been considered to predict the behavior of SPFAs. Material properties, configuration cross-sections, compressibility effects of the pneumatic cavity, and actuation boundary conditions can be defined in the FEM software, and contribute to increasing simulation accuracy. Because of the powerful FEM tools available to model hyperelastic materials, various FEM solutions have been introduced in the literature. Optimal design is the other advantage of using FEM simulation to meet specific performance criteria such as reducing the geometry dimensions [239], improving actuating speed [126], or enhancing the performance of soft actuators by maximizing the bending angle [221, 240]. Particularly in a commercial application, it is necessary to use FEM optimization once and then produce the SPFAs and prerequisites such as molding devices to reduce the production costs and time. In table 4, we summarize the different FEM solvers and the material properties which are used to predict the hyperelastic characteristics of silicone. Silicone rubber is modeled as an isotropic, incompressible and hyperplastic material. The mechanical behavior of hyperelastic materials is characterized by the strain energy function U, which is then given by [241]:

$$\begin{equation}U = {\sum\limits_{i + j = 1}^N {{C_{ij}}{{({{\bar I}_1} - 3)}^i}({{\bar I}_2} - 3)} ^j} + \sum\limits_{i = 1}^N {\frac{1}{{{k_i}}}{{({J_{el}} - 1)}^{2i}}}\\[-10pt] \end{equation} \tag{ 1 }$$

Table 4. Material modeling for various types of silicone for soft fluidic.

Material	Treatment	Software	Model	Coefficient (MPa)	References
EcoFlex 00–10	—	ABAQUS	Ogden (N = 1)	${\mu _1} = 12.605 \times {10^{ - 3}},{\alpha _1} = 4.32$	Sparks et al [249]
EcoFlex 00–30	Curing at 120 °C for 60 min after vacuum degassing (EcoFlex) Strain rate: 300 mm min−1	ABAQUS	Yeoh	${C_{10}} = 5.072 \times {10^{ - 3}},{C_{20}} = - 3.31 \times {10^{ - 4}},$ ${C_{30}} = - 1.5 \times {10^{ - 5}}$	Elsayed et al [174]
	ASTM D638, strain rate: 500 mm min−1	ABAQUS	Yeoh	${C_{10}} = 0.012662$	Polygerinos et al [237]
	ASTM D412, curing at room temperature	ABAQUS	Ogden (N = 3)	${\mu _1} = 0.024361,{\alpha _1} = 1.7138,$ ${\mu _2} = 6.6703 \times {10^{ - 5}},{\alpha _2} = 7.0679,$ ${\mu _3} = 4.5381 \times {10^{ - 4}},{\alpha _3} = - \,3.3659$	Moseley et al [240]
	—	ABAQUS	Ogden (N = 3)	${\mu _1} = 1.887 \times {10^{ - 3}},{\alpha _1} = - 3.848,$ ${\mu _2} = 2.225 \times {10^{ - 2}},{\alpha _2} = 0.6632,$ ${\mu _3} = 3.574 \times {10^{ - 3}}\,,{\alpha _3} = \,4.225,\,{D_1} = 2.9259$	Agarwal et al [250]
	Curing at 60 °C for 15 min after vacuum degassing	ABAQUS	Arruda-Boyce	$\mu = 0.03,\lambda = 3.9$	Martinez et al [251]
	Curing at 55 °C for 20 min after vacuum degassing	ABAQUS	Yeoh	${C_{10}} = 7.61 \times {10^{ - 3}},{C_{20}} = 2.42 \times {10^{ - 4}},\,$ ${C_{30}} = - 6.2 \times {10^{ - 7}}$	Sareh et al [246]
	ISO 527–3	ABAQUS	Ogden (N = 3)	${\mu _1} = 2.2 \times {10^{ - 2}},{\alpha _1} = 1.3,\,{\mu _2} = 4 \times {10^{ - 4}},$ ${\alpha _2} = 5,{\mu _3} = - 2 \times {10^{ - 3}},{\alpha _3} = - 2$	Steck et al [252]
		ABAQUS	Yeoh	${C_{10}} = 1.7 \times {10^{ - 2}},{C_{20}} = - 2 \times {10^{ - 4}},$ ${C_{30}} = 2.3 \times {10^{ - 5}}$
	—	ABAQUS	Neo-Hookean	${C_{10}} = 0.01$	Subramaniam et al [216]
	ASTM D412, the strain rate of 450 mm min−1, curing at room temperature with vacuum degassing	Developed Python code	Ogden (N = 3)	${\mu _1} = - 0.322,{\alpha _1} = 3.31,\,{\mu _2} = 0.19,$ ${\alpha _2} = 3.115,{\mu _3} = 0.145,{\alpha _3} = 3.468$	Marechal et al [245]
EcoFlex 00–50	Curing at 120 °C for 60 min after vacuum degassing (EcoFlex)	ABAQUS	Ogden (N = 3)	${\mu _1} = 107.9 \times {10^{ - 3}},{\alpha _1} = 1.55,\,$ ${\mu _2} = 21.47 \times {10^{ - 6}},{\alpha _2} = 7.86,$ ${\mu _3} = - 87.1 \times {10^{ - 3}},{\alpha _3} = - \,1.91$	Elsayed et al [174]
	—	ABAQUS	Yeoh	${C_{10}} = 1.9 \times {10^{ - 2}},{C_{20}} = 9 \times {10^{ - 4}},$ ${C_{30}} = - 4.75 \times {10^{ - 6}}$	Runge et al [253]
	—	ANSYS	Hookean	—	Nasab et al [254]
	Strain rate: 0.2 mm s−1	—	Mooney-Rivlin	${C_1} = 10.401 \times {10^{ - 3}},{C_2} = 21.362 \times {10^{ - 3}}$	Pineda et al [255], Lee et al [256]
	ASTM D412, the strain rate of 450 mm min−1, curing at room temperature with vacuum degassing	Developed Python code	Ogden (N = 3)	${\mu _1} = 1.97,{\alpha _1} = 2.911,\,{\mu _2} = - 3.671,$ ${\alpha _2} = 3.008,{\mu _3} = 1.740,{\alpha _3} = 3.096$	Marechal et al [245]
Dragonskin 10	—	ABAQUS	Yeoh	${C_{10}} = 36 \times {10^{ - 3}},{C_{20}} = 2.58 \times {10^{ - 5}},$ ${C_{30}} = - 5.6 \times {10^{ - 7}}$	Sareh et al [246]
	ISO 37, strain rate: 450 mm min−1, curing at room temperature with vacuum degassing	ABAQUS	Ogden (N = 3)	${\mu _1} = - 1.8261,{\alpha _1} = 1.613,\,{\mu _2} = 1.12,$ ${\alpha _2} = 2.0184,{\mu _3} = 0.7951,{\alpha _3} = 0.9386$	Byrne et al [257]
	—	ANSYS	Mooney–Rivlin	${C_{10}} = 0.04,{C_{01}} = - 0.033,\,{C_{11}} = 1.2 \times \,{10^{ - 3}}$	Basturen et al [258]
	ASTM D412, Strain rate of 500 mm/min	—	Yeoh	${C_{10}} = 7.61 \times {10^{ - 3}},{C_{20}} = 2.42 \times {10^{ - 4}},$ ${C_{30}} = - 6.2 \times {10^{ - 7}}$	Low et al [247]
	ASTM D412, the strain rate of 450 mm min−1, curing at room temperature with vacuum degassing	Developed Python code	Ogden (N = 3)	${\mu _1} = 1.971 \times {10^{ - 19}},{\alpha _1} = 18.341,$ ${\mu _2} = 1.03,{\alpha _2} = 2.729,{\mu _3} = - 1.059,{\alpha _3} = 2.649$	Marechal et al [245]
Dragon Skin 30	Strain rate: 300 mm/min	ABAQUS	Yeoh	${C_{10}} = 1.19 \times {10^{ - 3}},{C_{20}} = 2.3 \times {10^{ - 2}}$	Elsayed et al [174]
	—	ANSYS	Ogden (N = 1)	${\mu _1} = 75.449 \times {10^{ - 3}},{\alpha _1} = 5.836$	Heung et al [259]
	—	ANSYS	Ogden (N = 1)	${\mu _1} = 0.1581,{\alpha _1} = 2.7172$	Al-Rubaiai et al [248]
	ASTM D412, the strain rate of 450 mm min−1, curing at room temperature with vacuum degassing	Developed Python code	Mooney–Rivlin	${C_{10}} = 0.247,{C_{01}} = - 0.33,\,{C_{20}} = 2.09 \times {10^{ - 4}}$	Marechal et al [245]
Elastosil M4601	ASTM D638, strain rate: 500 mm/min	ABAQUS	Yeoh	${C_{10}} = 0.11\,,{C_{20}} = 0.02$	Polygerinos et al [237], Zhang et al [260]
	—	ABAQUS	Hookean	E = 0.54 MPa	Yang et al [261]
	ASTM D638	ABAQUS	Yeoh	${C_{10}} = 0.125,{C_{20}} = 0.0075$	Wang et al [262]
	Curing at 70 °C for 20 min	ANSYS	Hookean	$E = 0.387$	Hu et al [263]
	ASTM D638	—	Mooney-Rivlin	${C_1} = 10.401 \times {10^{ - 9}},{C_2} = 21.362 \times {10^{ - 9}}$	Ogura et al [215]
Silica gel	—	ABAQUS	Yeoh	${C_{10}} = 0.036\,,{C_{20}} = 0.007$	Zhang et al [264]
Agilus30		ABAQUS	Mooney-Rivlin	${C_{10}} = - 0.4889,{C_{01}} = 0.7147,$ ${C_{11}} = \, - 0.2704,\,{C_{20}} = 0.07929,$ ${C_{02}} = 0.4709,\,{D_1} = 0.4574,\,{D_2} = 0$	Pasquier et al [220], Chen et al [221]
SmoothSil 960	Curing at room temperature for 24 h	ABAQUS	Neo-Hookean	${C_{10}} = 0.17$	Subramaniam et al [216]
SmoothSil 940	—	ABAQUS	Neo-Hookean	${C_{10}} = 0.12$	Subramaniam et al [216]
MoldStar 30	—	ABAQUS	Neo-Hookean	${C_{10}} = 0.055$	Subramaniam et al [216]
Elastomeric Precursor	—	ANSYS	Hookean	—	Peele et al [227]
RTV-KE1603	Curing at room temperature after vacuum degassing	MSC Marc	Mooney–Rivlin	${C_{10}} = 8.635 \times {10^{ - 2}},{C_{01}} = 6.213 \times {10^{ - 2}},$ ${C_{11}} = - 1.2896 \times {10^{ - 2}},\,{C_{20}} = 3.425 \times {10^{ - 3}},$ ${C_{02}} = - 6.577 \times {10^{ - 1}}$	Wakimoto et al [214], Ogura et al [215]
NinjaFlex	Printing temperature: 245 °C	ABAQUS	Ogden (N = 3)	${\mu _1} = - 30.921,{\alpha _1} = 0.508,\,{\mu _2} = 10.342,$ ${\alpha _2} = 1.375,{\mu _3} = 26.791\,,{\alpha _3} = \, - 0.482$	Yap et al [102]
	ISO 37, strain rate: 100 mm s−1, printing Temperature 240 °C	ANSYS	Mooney–Rivlin	${C_{10}} = - 2.33 \times {10^{ - 1}},{C_{01}} = 2.562,$ ${C_{11}} = - 0.561,\,{C_{20}} = 0.9$	Tawk et al [265, 266]
FilaFlex	Printing temperature: 235 °C	ANSYS	Mooney–Rivlin	${C_{10}} = 1.594,{C_{01}} = 0.44,\,{C_{11}} = - 4.4 \times {10^{ - 3}}$	Hu et al [219]

where $U$ is the strain energy potential per unit volume, $N$is the polynomial order, ${\bar I_1}$and ${\bar I_2}$ are the deviatoric strain invariants, ${C_{ij}}$is a material-specific parameter, ${J_{el}}$ is the elastic volume ratio and ${k_i}$ expresses compressibility. Considering the silicone as an incompressible material, the term ${k_i}$ is omitted, which simplifies the general polynomial form of the strain energy potential. (1) can be fitted by different hyperelastic models, i.e. Mooney–Rivlin, Yeoh, Ogden, or Neo-Hookean models.

• Mooney–Rivlin material model

This model was one of the first hyperelastic models used to predict the nonlinear behavior of isotropic hyperelastic materials [242]. The strain-energy function for this material model is:

$$\begin{equation}U = \sum\limits_{i = 1}^{N = 2} {C_i}{({\bar I_i} - 3)^i}.\end{equation} \tag{ 2 }$$

• Ogden material model

Based on the theory of elasticity, the Ogden model was developed first time by Ogden [243] and has the general form:

$$\begin{equation}U = \sum\limits_{i = 1}^N \frac{{2{\mu _i}}}{{\alpha _i^2}}(\lambda _1^{{\alpha _i}} + \lambda _2^{{\alpha _i}} + \lambda _3^{{\alpha _i}} - 3).\end{equation} \tag{ 3 }$$

where ${\mu _i}$ and ${\alpha _i}$ are material constants and λ, are principal stretches.

• Yeoh material model

This model was first presented in 1990 for incompressible materials [244]:

$$\begin{equation}U = \sum\limits_{i = 1}^3 {C_i}{({\bar I_1} - 3)^i}.\end{equation} \tag{ 4 }$$

As shown in this equation, the strain-energy function in this model relies only on the first strain invariant (${\bar I_1}$).

• Neo–Hookean material model

It was presented by Holzapfel [267]. For the Neo–Hookean material model, the function of the strain energy is related to a linear equation for the principal strains

$$\begin{equation}U = {C_1}({\bar I_1} - 3).\end{equation} \tag{ 5 }$$

Table 4 summarizes the coefficients of these equations based on previous approaches in the literature. These constant parameters are calculated by stress–strain experiments. The uniaxial test is more widespread and typical than biaxial and planning tests. Selecting and designing the most appropriate test for the specimen of silicone increases the accuracy of the model parameters. Several approaches were studied to predict the nonlinear elastic behavior of silicone rubber under different loading conditions in order to understand the mechanics by finding the best least square curve fitting the potential strain energy function. Marechal et al [245] provided a database of the best constitutive models and the values of the coefficients according to uniaxial tensile tests recommended in the ASTM D412 for elastomers. Each silicone specimen was cured at room temperature with the nominal mixing ratio recommended by the manufacturers. We deployed this database to compare different suggested constitutive models in the previous approaches listed in table 4. The results for different silicone materials are shown in figure 8. Abaqus is used as the framework to reproduce the curve fitting of the suggested constitutive model in each reference. Although the treatment conditions, such as degassing, natural aging or the addition of pigment may affect the mechanical properties of the hyperelastic materials in this simulation, we assumed that these models were extracted in general conditions, such as the mixing ratio recommended by the manufacturer and curing at room temperature, without considering differences in the testing process and measurement equipment. Furthermore, most of the reviewed articles do not mention which type of test data, true or engineering strain-stress, were used to predict the material models. Note that engineering stress, also known as nominal stress, is calculated by dividing the applied force by the primary cross-section area of the material, while in true stress this area is changed and calculated with respect to time. We extracted the true and engineering strain-stress data from the proposed constitutive models in these articles using ABAQUS software. These figures help to compare the models by assuming that the experimental protocol is the same and based on ASTM D412. We take Marechal's test data as the reference and compare it with the other models for each type of silicone, by true and engineering stress versus strain results, as presented in figure 8. This figure shows the experimental data compared with the best-fitting FE models results for the various silicone rubbers. As shown in (figures 8(a) and (b)), for EcoFlex 00–30 in a small stress–strain range, most of the models are fitted with acceptable divergence. The Yeoh model suggested by Sareh et al [246] fits the experimental data with few differences. In the EcoFlex 00–50 graphs (figures 8(c) and (d)) the variation between the proposed models and raw experimental data is obvious even for small stress–strain values. The Yeoh model by Low et al [247] (figures 8(e) and (f)) and the first-order Ogden model by Al-Rubaiai et al [248] (figures 8(g) and (h)) predict the behavior of Dragon skin 10 and 00–30 respectively with minimum divergence, even in large stress values.

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5.2.2. Real-time FEM simulation.

The obstacles we have discussed to developing analytical and numerical models have led to research attempts being made to control soft robots using nonparametric methods based on learning or vision. These aim to be a more efficient alternative. Lee et al [279] proposed a nonparametric local learning technique to learn the inverse kinematics and control of SFAs. The model is able to predict the end-effector position of the robot accurately in the presence of an external dynamic disturbance. They utilized FEM to generate a sample of kinematic data to pre-train the initial control. A neural network was applied in [280] to control a 1-DOF SPFA, with a vision-based motion capture system acquiring unknown soft actuator parameters. A feedforward neural network to learn the 3D nonlinear inverse kinematic model of a soft octopus-arm was implemented and tested in [281]. The potential challenges of this method are the accuracy of the training-based kinematic computation, dependent on properly selected datasets. Several works can be found in the literature concerning the visual servoing of soft actuators. Li et al [282] proposed an adaptive Kalman filter for continuum robot path tracking. They used pressures and tip position as input data. Then they estimated the robot's Jacobian of deformation by gathering the required data from the vision system. Zhang et al [283] used real-time FEM simulation using SOFA to predict the Jacobian matrix of the robot. The correct position of the tipping point was modified in the feedback control law using a visual servoing system. Although the vision-based methods are efficient to reduce the number of sensors required to provide the state-space variables of the robots, the hardware requirements and the complex calibration process are the main remaining challenges to using this method in soft robot control scenarios [284, 285].

To summarize this section, figure 9 shows the steps of the SFA fabrication procedure considering all the design parameters, including geometry, materials, constitutive law, and pressure which affect each other during the analysis and manufacturing of soft actuators. It should be noted that selecting the proper material for soft actuators depends on the different factors calculated during analysis.

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