Dielectric elastomer actuators for octopus inspired suction cups

In the present work, we focused on mimicking the acetabular radial muscles that are responsible for the production of suction in the octopus sucker (section 1.2). The development of an artificial bioinspired device that mimics a biological system requires the use of a compact actuation method. An actuation unit must be embedded in each suction cup and must be activated independently from all other suction units. Dielectric elastomer (DE) actuators were identified as proper actuators to replicate the function of the acetabular radial muscles. Compared to traditional actuation technologies (e.g., electrical, piezoelectric motors, etc), DE actuators do not require a transmission, are lighter and are more compact. Among the soft actuators, DE actuators have the best trade-off between the actuation strain rate and force generated [20]. DE actuators meet the specifications of being soft and generating high pressures that can be exploited for attachment of the artificial prototypes combined with a low electrical power consumption [14]. The DE actuator consists of two compliant electrodes separated by a dielectric medium. DE actuators exploit the electric field applied between the electrodes to generate mechanical work. The opposite charges on the two electrodes are attracted to each other, thus producing a compressive stress on the compliant dielectric layer that is consequently deformed. The principle at the base of the DE actuators, in a planar configuration, is described by the following equation [21]:

$$\begin{eqnarray}&&P=\varepsilon {{\varepsilon }_{0}}\frac{{{V}^{2}}}{{{t}^{2}}},\end{eqnarray} \tag{ 1 }$$

which relates the voltage V applied to the two electrodes and the electrostatic pressure P generated on the dielectric in the direction of the electric field. The parameter t is the thickness of the dielectric membrane interposed between the two electrodes, ε is the relative dielectric constant of the material, and ε₀ is the dielectric constant of vacuum. The actuation unit was designed as shown in figure 2. The DE actuator (active membrane, am, in figure 2) is hydrostatically coupled with a passive membrane (pm in figure 2) via a chamber filled with fluid ((f) in figure 2). Water was chosen as the first solution, as it was effective as a proof of concept. The two membranes (active and passive membranes) are sealed by two rigid circular frames (rf in figure 2). The application of a voltage to the DE actuator results in a thickness variation and an area expansion of the active membrane. This configuration induces a pressure change in the fluid inside the chamber defined by the two membranes, as it has already been demonstrated for an actuator with a single layer [22]. In the present work, this concept is adopted and designed to maximize the pressure difference inside the chamber during actuation.

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The membrane can be modelled as a thin diaphragm, inflated with an internal pressure given by the injected fluid. The internal pressure is equilibrated by the stress given by the traction of the membrane during expansion [23]. A relation between pressure and internal volume can be found, equilibrating the force given by the internal pressure times the area of the membrane and the force generated by the membrane at the rim of connection with the rigid frame (figure 3). The force F $_{p,z}$ acting on the membrane is calculated as the pressure difference times the area of the diaphragm. The reaction force to the stretch of the membrane F_F at the frame is calculated as the total membrane stress ${{\sigma }_{{\rm m}}}$ times the cross-sectional area of the membrane at the perimeter of the frame. The equilibrium is calculated in the vertical axis (figure 3):

$$\begin{eqnarray}&&{{F}_{p,z}}=\vartriangle p\pi {{R}^{2}}={{F}_{F,z}}={{\sigma }_{M}}t2\pi R\;{\rm sin} \alpha ,\end{eqnarray} \tag{ 2 }$$

where F $_{p,z}$ is the force at the frame of the pressure, R is the radius of the membrane in rest state, F $_{F,z}$ is the force of the membrane in vertical direction and t is the thickness of the membrane. The stress of the membrane ${{\sigma }_{{\rm m}}}$ has two components: an initial stress σ₀ due to prestretch of the membrane and a component ${{\sigma }_{D}}$ generated by the stretch of the membrane during inflation. The latter depends, according to Hookeʼs law, on the strains in radial and tangential direction and on the Young modulus of the material (Y_m) [23]. The radial strain is considered constant over the entire membrane, because the bending moments are neglected, an assumption that is valid for the thin membrane that we are considering. The radial strain is calculated from the length of the parabola that describes the deflection curve of the inflated membrane respect to the base length with no pressure applied:

$$\begin{eqnarray}&&{{\varepsilon }_{0}}\approx \frac{2w_{0}^{2}}{3{{R}^{2}}},\end{eqnarray} \tag{ 3 }$$

where w₀ is the height of the centre of the diaphragm. Assuming that the tangential strain is equal to radial strain through the entire membrane [23], the total stress resulting from the deflection of the membrane is:

$$\begin{eqnarray}&&{{\sigma }_{{\rm m}}}={{\sigma }_{0}}+{{\sigma }_{D}}={{\sigma }_{0}}+\frac{2w_{0}^{2}{{Y}_{{\rm m}}}}{3{{R}^{2}}(1-{{\nu }_{{\rm m}}})},\end{eqnarray} \tag{ 4 }$$

where ν_m is the Poissonʼs ratio. The angle α in figure 3 is derived from the slope of the diaphragm at the rim, which is the derivative of the parabolic curve that describes the deflection of the membrane, calculated at the rim:

$$\begin{eqnarray}&&\alpha ={{{\rm tan} }^{-1}}\left( \frac{{\rm d}w}{{\rm d}r}\left| _{r=R} \right. \right)={{{\rm tan} }^{-1}}\left( \frac{2{{w}_{0}}}{R} \right).\end{eqnarray} \tag{ 5 }$$

Combining (5) with (2) it is possible to obtain a relation between the pressure and the central height of the diaphragm:

$$\begin{eqnarray}&&\vartriangle p={{\sigma }_{M}}\frac{2t}{R}{\rm sin} \left( {{{\rm tan} }^{-1}}\left( \frac{2{{w}_{0}}}{R} \right) \right).\end{eqnarray} \tag{ 6 }$$

The thickness t of the diaphragm decreases as the surface of the membrane increases for the effect of the inflation. Considering that the material is incompressible, the following relation for the thickness as function of the height of the parabola was found, equating the volume of the membrane in rest state and its volume during expansion, calculated as the surface of the paraboloid times the thickness of the diaphragm:

$$\begin{eqnarray}&&t=\frac{6R{{t}_{0}}w_{0}^{2}}{\left[ {{({{R}^{2}}+4w_{0}^{2})}^{3/2}}-{{R}^{3}} \right]},\end{eqnarray} \tag{ 7 }$$

where t₀ is the initial thickness of the membrane. Equation (7) can be then integrated in (6). In order to find a direct relation between the volume of the fluid contained by the membrane and the pressure generated, the volume of the paraboloid is inserted in (6) and (7), replacing the height of the paraboloid w₀ with:

$$\begin{eqnarray}&&{{w}_{0}}=\frac{2{{V}_{{\rm ol}}}}{\pi {{R}^{2}}},\end{eqnarray} \tag{ 8 }$$

where V_ol is the volume enclosed by the paraboloid with base radius R and height w₀. The effect of the actuation was then considered and included in the equilibrium equation to have an exhaustive description of the electro-mechanical interaction. The application of high voltage between the electrodes produces an electrostatic stress that acts in opposite direction respect to the stress ${{\sigma }_{{\rm m}}}$. The electrostatic stress is defined by Maxwellʼs stress tensor [24]:

$$\begin{eqnarray}&&\sigma _{ij}^{E}=\epsilon {{\epsilon }_{0}}{{E}_{i}}{{E}_{j}}-\frac{1}{2}\epsilon {{\epsilon }_{0}}{{E}^{2}}{{\delta }_{ij}},\end{eqnarray} \tag{ 9 }$$

where E is the electric field and δ_ij the Kronecker delta. The electric field was considered uniform over the entire membrane, as in the case of a capacitor with parallel plates positioned at a distance t. With this simplification, the electrostatic stress in radial direction, perpendicular to the electric field can be expressed by (1) and inserted in (4) as a third term of the total stress ${{\sigma }_{{\rm m}}}$, that takes the final form:

$$\begin{eqnarray}&&{{\sigma }_{{\rm m}}}={{\sigma }_{0}}+\frac{2w_{0}^{2}{{Y}_{{\rm m}}}}{3{{R}^{2}}(1-{{\nu }_{{\rm m}}})}-\epsilon {{\epsilon }_{0}}\frac{{{V}^{2}}}{{{t}^{2}}}.\end{eqnarray} \tag{ 10 }$$

This linearized model considers a linear elastic material, that is an acceptable approximation only for a limited range of the stress–strain curve of the acrylic tape that was used in the experiments [25], but is sufficient to qualitatively evaluate the effect of the geometric parameters on the actuation performance.

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The DE actuator was designed with a widely available acrylic material (VHB 4905, 3 M), with electrodes made of silver conductive grease (Circuit Works CW7100, Chemtronics). The acrylic tape was uniformly prestrained to 300%, resulting in a final thickness of approximately 50 μm. A simple device was built to uniformly stretch the membrane in a radial direction, following the design of a folding mechanism [26]. Actuators with several layers were obtained to construct a sandwich structure of single-layer actuators, exploiting the adhesiveness of the tape to bond one layer to another. A specific procedure was used to avoid the formation of air bubbles in between the layers. The two membranes, which must be bonded, were first attached to two circular frames with different diameters. At this stage, the two frames aligned concentrically were placed inside a vacuum bag and the bag was sealed. Air was removed, and the collapsing bag pushed the membranes together, so that the air was not trapped between the two adjacent layers. This method dramatically reduced the formation of air bubbles, aiding in the reproducibility of the design (as described in section 3).

Six actuation units with a varying number of layers, from one to six, were developed and fabricated. The passive membrane was made with the same procedure as the active membrane but without the electrodes. Therefore, the active and passive membranes had comparable mechanical characteristics. When fluid was injected inside the actuation unit, the two membranes expanded outward symmetrically. The diameter of the actuator was set at 12 mm based on the specification of the biological sucker, as defined in a previous work [27].

The aim of the experiments was to measure the pressure difference that the actuation unit was able to generate. To test this capability, the actuation unit ((B) in figure 4) was bonded on a rigid cylindrical chamber ((A) in figure 4) and hydrostatically coupled with the passive membrane of (B). During the experiments, the pressure in the actuation unit (B) and in the rigid chamber (A) was recorded. The initial pressure inside the actuation unit was dependent on the volume of the fluid enclosed and the number of layers that composed the actuator. The experimental setup consisted of two main parts (figure 4) : (1) the actuation unit (B) sealed on a chamber filled with a fluid (A) and (2) a hydraulic piston (C) that was used to inject the fluid in the actuation unit coupled with a microslide stage (D). The microslide stage was equipped with a laser displacement sensor (E) that was used to calculate the fluid volume injected in (B). There were two differential pressure sensors (26pc series, Honeywell) (F): one connected to the actuation unit (B) and the other connected to the lower chamber (A). A board equipped with a high voltage dc/dc converter (EMCO corporation) that was able to generate up to 5 kV was used to supply the high voltage for the actuation unit. The data acquisition board, the high voltage dc/dc board, and the laser were connected to a PC to record the data synchronously. The procedure for the experiments included four steps:

• (i)  
  Chamber (A) was filled with water and the actuation unit was sealed on this chamber. Syringe (G) was used to remove the air remaining inside chamber (A).
• (ii)  
  The actuation unit (B) was filled with liquid injected by hydraulic piston (C) connected to the microslide stage (D). The volume was calculated from the displacement measured by laser (E). The piston was used to set the initial volume inside the actuation unit and was maintained fixed during actuation.
• (iii)  
  The pressure inside chamber (A) was adjusted, removing or injecting fluid with syringe (G). The pressure before activating the actuator was regulated to be equal to the atmospheric pressure, simulating the condition in which the actuation unit was surrounded by fluid at the same pressure.
• (iv)  
  A high voltage was applied to the actuator, beginning at 1 kV and increasing in steps of 250 V, up to 2.5 kV.

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Steps (ii)–(iv) were repeated, injecting five different amounts of liquid, and the pressure was measured by pressure sensors and was recorded. The pressure difference was calculated as the maximum difference between the average value of the pressure, measured in a time window of one second, before actuation and the minimum value recorded during actuation. The pressure difference has a positive sign, as the pressure during actuation is always lower than the pressure in non-actuated condition.

In figure 5, an overview of the complete artificial device is provided. The artificial suction cup consists of an actuation unit ((a) in figure 5) bonded to a passive component. The actuation unit (described in section 2.1.1) mimics the acetabular radial muscles of the octopus sucker, generating the pressure difference respect to the external environment; the passive component mimics the infundibulum of the octopus sucker, allowing the device to be sealed on the substrate during attachment (this passive component is called the artificial infundibulum throughout the rest of the present work).

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The passive component is made using a soft material and is bonded directly to the rigid frames of the actuation unit. The liquid enclosed in the passive component, when it is sealed on the substrate, defines a second chamber that is hydrostatically coupled with the active unit. The actuation unit can be used to generate a change in the pressure in both chambers ((a) and (i) in figure 5) due to the hydrostatic coupling.

In the octopus sucker, when acetabular radial muscles are contracting, the internal volume of the sucker tends to increase. Because of the incompressibility of water, the internal volume does not change (the muscular contraction is in fact isometric), but a pressure drop is generated instead. In order to have attachment, the sucker must be sealed on the substrate before contraction of the radial muscles, and the contraction is associated with a small variation of volume due only to the amount of water enclosed in the sealed infundibulum, flowing into the acetabulum through the grooves. As the acetabular muscular hydrostat, the actuation unit is incompressible. When the active membrane is actuated, its thickness decreases and the volume enclosed by the membranes tends to increase. In case the sucker is not attached to a substrate, the effect is an increase of the volume of the infundibular chamber ((i) in figure 5). In case the suction cup is attached to the substrate, the water inside the infundibulum resists the volume expansion, and thus the pressure of the water inside the suction cup decreases, generating attachment force. The attachment process proposed using the DE actuators can be summarized as illustrated in figure 6. First, the suction cup is pushed on the substrate (a) and a seal is formed (b). The pressure inside the infundibulum in this phases equals the external pressure P_ext. Then, the activation of the actuation unit induces a reduction of the interior pressure, both in the active unit ((a) in figure 5) from P₀ to P_a and in the chamber below it ((i) in figure 5) from P_ext to P_att. (c). The reduction of pressure is the same in both chambers. The attachment force is determined by the difference between the pressure of the water inside the infundibular part and the environmental pressure multiplied by the area of the artificial infundibulum that is sealed by the rim on the substrate.

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A suction cup prototype was developed based on the design shown in figure 5 to provide a proof of concept of the design described in section 2.2.1. In particular, the prototype was exploited to verify the simplification made in tests on the actuation unit (where a rigid chamber was used). The artificial infundibulum must, in fact, match the hypothesis to perfectly seal on the substrate and to be rigid enough to sustain the difference of pressure without collapsing. Based on the gathered results on the actuation unit performance presented in section 3.2, a qualitative test was carried out to prove the feasibility of the actuator design proposed in this work. The artificial infundibulum was made of Dragon-Skin silicone (Smooth-On, shore hardness A10) (as shown in figure 7, left) using a mould that replicates the morphology of a real octopus sucker [27]. The artificial infundibulum was glued to the rigid frame that holds the membranes of the actuation unit (figure 7, right). The actuator was fabricated with the procedure described in section 2.1.3 and was composed of six layers. The size of the artificial infundibulum was defined in accordance with the geometrical proportions that have been observed in the octopus sucker. In fact, the natural ratio (10 : 1) between the action area of the radial muscles (responsible of the suction generation) and the orifice area (which is proportional to the diameter of the infundibulum) in an octopus sucker was maintained in the case of the artificial prototype.

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The prototype of the active suction cup, consisting of the actuation unit and the artificial infundibulum, was tested in water. A setup equipped with a pressure sensor was used to measure the pressure generated by the actuation unit inside the artificial infundibulum. A small plexiglass box with a hole in the bottom face connected to the pressure sensor (26pc series, Honeywell) was filled with water, and the suction cup was immersed in the water. Plexiglass was chosen as substrate for the preliminary test for its low surface roughness.

The silicone infundibulum was pressed on the bottom surface with the pressure sensor beneath it, and a constant force of 1 N was applied to maintain close contact with the substrate. The force is necessary to deform the silicone infundibulum and have a close contact between the infundibulum and the substrate.

The aim of the test was to prove the concept of the suction cup and to verify that a pressure difference could be generated in a non-ideal condition of a deformable infundibulum.