Bi‐Shell Valve for Fast Actuation of Soft Pneumatic Actuators via Shell Snapping Interaction

Abstract Rapid motion in soft pneumatic robots is typically achieved through actuators that either use a fast volume input generated from pressure control, employ an integrated power source, such as chemical explosions, or are designed to embed elastic instabilities in the body of the robot. This paper presents a bi‐shell valve that can fast actuate soft actuators neither relying on the fast volume input provided by pressure control strategies nor requiring modifications to the architecture of the actuator. The bi‐shell valve consists of a spherical cap and an imperfect shell with a geometrically tuned defect that enables shell snapping interaction to convert a slowly dispensed volume input into a fast volume output. This function is beyond those of current valves capable to perform fluidic flow regulation. Validated through experiments, the analysis unveils that the spherical cap sets the threshold of the snapping pressure along with the upper bounds of volume and energy output, while the imperfect shell interacts with the cap to store and deliver the desired output for rapid actuation. Geometry variations of the bi‐shell valve are provided to show that the concept is versatile. A final demonstration shows that the soft valve can quickly actuate a striker.


Bi-shell valve for fast actuation of soft pneumatic actuators via shell snapping interaction
Chuan Qiao, Lu Liu, Damiano Pasini* S1. Fabrication S1.1. Fabrication of the input chamber Figure S1 illustrates the manufacturing process for the input chamber encompassing six layers laser cut (CM1290 laser cutter, SignCut Inc., Canada) from a 6-mm-thick acrylic plate (McMaster-Carr, USA). The first layer at the bottom is fully solid with geometry parallel to the external profile of the two shells. From the second to the fifth layer the acrylic plates consists of a 15.5-mm-wide ring with an external profile identical to the first layer, forming the wall of the input chamber. Moreover, the third and fourth layers have an opening at both ends along their long axis, a feature that allows connection to the PVC plastic tubing for volume input and pressure sensor. The sixth layer at the top covers the input chamber enclosed by the first five layers, with two large circular holes that connect the imperfect shell and the spherical cap. In addition, six smaller holes on the top layer are used to fasten the input chamber with other parts of the bi-shell valve. Adhaero SuperGlue (Dollarama, Canada) is used to join all pieces of the input chamber assembly, and silicone rubber Elite double 32 (Zhermack, Italy) is applied to seal off its internal walls.

S1.3. Fabrication of the elastic shells
Shell fabrication for both the imperfect shell and the spherical cap follows a procedure previously used. [1,2] Elite Double 32 (Zhermack, Italy) is casted on the surface of 3D-printed molds ( Figure S3). For each shell, we fabricated a 1 mm-thick mold with Onyx filament using fused deposition modeling (FDM). The surface of the molds has geometry identical to that of   Figure S4 shows the components and assembly of our bi-shell valve system. The imperfect shell and the spherical cap are mounted on the input chamber, with their inner volume connected through the two large holes on the top surface of the input chamber. Any leakage that may exist between the elastic shells and the input chamber are sealed with Elite Double 32. Above the elastic shells is a fixture plate, which can be fastened to the input chamber with screws through six small holes along its edges. This fixture plate has two large circular holes that allow the elastic shells to freely deform without entering in contact with the fixture plate.

S1.4. Assembly of the bi-shell valve
Since the base of the spherical cap is thicker than that of the imperfect shell, the fixture plate can tightly clamp the base of the spherical cap on the input chamber to ensure airtight connection to the output chamber. The output chamber is mounted on the fixture plate with a square gasket made of Elite Double 32 that can tolerate the deformation of the fixture plate due to clamping. The leaks that may occur between the fixture plate and the output chamber are sealed with Elite Double 32 and Adhaero 5 minute epoxy (Dollarama, Canada).

Figure S4.
Components and assembly of the bi-shell valve system.

S1.5. Fabrication of the pneumatic striker
Our pneumatic striker consists of two main components: a miniature airbag to convert volume input into motion, and a paper stick to hit the table tennis ball. The airbag is cut from a 0.02mm-thick compostable kitchen bag (no name, Loblaws Inc, Canada) as a folded bi-layer film with an area of 28x28 mm. A segment of tubing is glued on the bottom surface of the cut film with Adhaero SuperGlue. A hole is then cut through the film to enable airflow between the airbag and the tubing. Finally, the airbag is sealed with Elite Double 32. The paper stick is cut from a 0.29-mm-thick paper and then glued on the top surface of the airbag with Elite Double 32. The length of the stick is 56 mm, and the width is 7 mm.

S2.1. Characterization of a single shell
To characterize the pressure-volume response of a single shell, we assemble the experiment apparatus shown in Figure S5, which includes an acrylic fixture to hold the sample, a  Figure 2E and 2F.

S2.2. Characterization of the bi-shell valve
Our experiment with the bi-shell valve is performed with the set-up described above for

S3. Effect of air compressibility
Since air is a compressible fluid, here we study the influence of air compressibility. We assume that the air in the bi-shell-valve, the tubing, and the syringe follows the ideal gas law: where p is the pressure of the gas, V is the volume of the gas, n is the amount of substance of gas, g R is the ideal gas constant, and T is the absolute temperature of the gas. In both the initial state and loaded states. the gas in the pneumatic system (the bi-shell valve, syringe and connecting tubing) must satisfy where the subscript stands for initial (0) and loaded states (1).
We assume the gas undergoes an isothermal process ( 0 The volume change due to air compressibility can be expressed as This value is proportional to the initial volume of the system 0 sys V , and increases with the pressure change from the initial state. In our bi-shell valve, the maximum change in pressure from the initial state (atmosphere pressure 1.3 10 mm  for the bi-shell valve and 3 3 7.5 10 mm  for a single shell), and corroborate the assumption made in this work that air compressibility can be neglected.
Hence, the volume change of the syringe can be assumed as the volume change of the bi-shell valve and the separate shells.

S4. Finite element analysis
To further investigate the mechanical performance of the bi-shell valve, we conduct a set of finite element method (FEM) simulations with the commercial software package ABAQUS/STANDARD. The shell material is modelled as an incompressible neo-Hookean solid. The Young's modulus and Poisson's ratio (1.23 MPa and 0.5) are determined by fitting the simulation results with the experimental data within a range previously used in the literature. [1,3] This leads to the adoption of the following coefficients for our neo-Hookean model: C10 0.205 MPa  and 1 D1 0 MPa   . We employ the modified Riks method to simultaneously solve for pressure and shell deformation. Since in our experiments we observe that both the imperfect shell and the spherical cap exhibit only an axisymmetric mode of deformation, we build our numerical model with axisymmetric elements (the two-node linear shell element SAX1 or the four-node bilinear quadrilateral element CAX4RH) to avoid the expensive computational cost of three-dimensional simulations. Although in some cases the imperfect shell may exhibit non-axisymmetric deformations, our previous study shows that an axisymmetric analysis can still be sufficient to retain a high level of accuracy. [2] We impose a fixed boundary condition at the bottom of the shells and a uniform pressure at their surfaces.
The volume change V  is calculated with the pressure p and the total external work done by the pressure p U , which is given by As described below, our computational analysis for each separate shell as well as for the bishell valve is conducted into two steps. First, we consider an as-designed (ideal) model that is free from any manufacturing imperfections and does not account for any initial deformation caused by the clamping of the bottom ring. In this scenario, we systematically explore the geometric space of the bi-shell valve to unveil its sensitivity to a varying shell geometry.
Second, to validate our numerical model with experimental results, we develop a set of realistic models, one for the spherical cap and the other for the imperfect shell. These models enable to capture the effect of the initial deformation due to clamping in spherical cap, and to incorporate as-manufactured imperfections, in particular thickness variations, in fabricated imperfect shells.

S4.1. Modelling of the spherical cap
As-designed model. The as-designed spherical cap is modelled with axisymmetric line element SAX1. The geometry of the spherical cap is 1 0.05 . A mesh convergence study shows that 51 elements are sufficient to model the spherical cap ( Figure S7A). In this work, around 51 elements are used for the spherical cap.
To systematically study the response of the spherical cap with varying geometry, we explore the geometry space defined by the normalized thickness 1 t R ranging from 0.01 to 0.1 and the normalized height h R spanning from 0.1 to 0.5; the radius at the base is fixed as Realistic model. To capture cross-section variation in a representative sample of the spherical cap, we use a digital camera EOS 800D (Canon, Japan). Our observations show that the spherical cap has a uniform thickness profile ( Figure S8A), hence our analysis of the spherical cap studies only the role of the initial deformation due to clamping.
In Figure S9A, the spherical cap is modelled with CAX4RH elements, whereas the acrylic plate that clamps the base of the cap is modelled with rigid body line elements RAX1. For the spherical cap, our mesh convergence study shows that four elements through the thickness are sufficient ( Figure S7B). Hence, we employ here at least four elements through the thickness.
The interaction between the cap and the plate is set as "hard" contact with a friction coefficient of 0.5. To investigate the effects of the initial deformation due to clamping, we first apply a downward displacement on the plate, and then apply a pressure on the shell to deflate the shell. The displacement is systematically varied from 0 (no clamping) to 0.3 mm (tight clamping). In Figure S9B, when the cap is clamped through the plate for 0.3 mm, an upward displacement occurs at the top of the cap. In Figure S10A, we find that the buckling pressure increases monotonically with the displacement of the clamping plate over a wide range of values from 403 Pa (no clamping) to 587 Pa (0.3 mm of clamping). From this set of results for the spherical cap, we decide to include the initial deformation due to clamping in our realistic numerical model. To minimize the difference of results between experiments and simulations, the displacement due to clamping is set as 0.1026 mm. This enables to yield a buckling pressure close to that of the representative sample of the spherical cap ( Figure 2E), and to bring below 0.6% the relative error in the buckling pressure between simulation and experiment. In addition, we find that the buckling pressure of the unclamped case ( Figure   S10A) is slightly lower than the results in Figure S7. The reason for this is that in the simulations that study the effect of clamping, the thick band at the base of the shell is included; this provides an elastic support to the shell which is dissimilar to the fixed boundary condition employed in other simulations.

As-designed model
We model the as-designed imperfect shell using the axisymmetric shell element SAX1. elements are sufficient to model the imperfect shell ( Figure S11A). In this work, an average of 81 elements are used for the imperfect shell.

Realistic model
As described above, our fabrication process, in particular the mould we used to produce our samples, had the following outcome on the shell geometries. The thickness profile of the spherical cap is uniform since the mould has no change in curvature, as opposed to that of the imperfect shell, which has variations due to abrupt changes in the curvature of the mould. To develop a realistic model of the imperfect shell with response that parallels that of the asmanufactured geometry, we investigate separately the role of non-uniform thickness as well as that of the initial deformation due to clamping, as described below.
Non-uniform thickness profile. Figure S8B shows that the as-manufactured shell features a thickness build up at locations above and below the elliptical arc; it is at those points that sudden changes of curvature appear in the mould. To obtain precise measurement of the thickness profile of the imperfect shell, we determine the distance between the inner and outer surfaces of the shell first from digital images (e.g. Figure S8B), and then by rectifying the data with measurements taken through a digital caliper ( Figure S8C). This set of results is used to generate a numerical model that captures thickness variations along the shell profile; in particular the thickness profile of the shell is partitioned into five sections ( Figure S8D), and to each of these portions the pertinent thickness is assigned. The SAX1 element is used to generate the model.
Initial deformation due to clamping. Here we solely study the role of the initial deformation due to clamping on an imperfect shell with uniform thickness. Figure S9C shows an imperfect shell clamped to the fixture plate. The plate is modelled as a rigid body with the axisymmetric rigid two-node line element RAX1. Since the clamped base of the imperfect shell is too thick to be considered as a shell, we use the axisymmetric quadrilateral element CAX4RH instead of the SAX1 element. Our mesh convergence study shows that four elements through the thickness are sufficient for the imperfect shell ( Figure S11). Hence, at least four elements through the thickness are adopted. The interaction between the shell and the clamp is set as "hard" contact with a friction coefficient of 0.5. In our simulations, we first simulate the initial deformation by imposing a vertical displacement on the plate, and then apply a pressure on the shell to deflate the shell. The displacement of the plate is varied from 0 to 0.3 mm. Figure S9D shows that the deformation due to clamping is localized at the base, while the body of the shell is not affected. In Figure S10B, the response of the imperfect shell is also not sensitive to the initial deformation caused by clamping. From these results, we conclude that it is reasonable to neglect the initial deformation of the imperfect shell due to clamping.

S4.3. Modelling of the bi-shell valve
We now study the collective response of the shells forming our valve system by combining the models of each individual shell into one model. SAX1 elements are used for both constituent shells of the as-designed model, which does not feature any variation in thickness or initial deformation due to clamping. On the other hand, for the realistic model, the thickness variation in the imperfect shell and the initial deformation of the spherical cap are modelled as described above for the realistic model of each shell.
We impose a uniform pressure on both shells, and trace the equilibrium path of the bi-shell system with the modified Riks method. The released energy is calculated as the negative of the area under the pressure-volume curve where S is the equilibrium path from the pre-snapping state (i) to the post-snapping state (ii) ( Figure 1).

S5. Buckling modes of imperfect shell
While the buckling of the spherical cap subject to uniform pressure has been extensively studied in literature, [1,4,5] only recently we unveil that of an imperfect shell with a large axisymmetric defect away from the pole. [2] Therein, we investigated an individual imperfect shell with a circular defect, and demonstrated the existence of additional three buckling modes, besides to the classical bifurcation, that can be programmed on demand through geometry tuning.
In the current work, we amend the defect geometry to an elliptical arc for the convenience of manufacturing, and specify two defining parameters ( Figure 4B): the meridional angles at the upper and lower boundary of the defect U  and L  . By varying U  and L  , we can show the emergence of four possible buckling modes ( Figure S12). For a small defect ( Figure S12A), the shell defect undergoes the classical bifurcation buckling, which is characterized by a downward dimple at the pole of the hemisphere. The pressure increases rapidly to a high buckling pressure (bifurcation point) before dropping immediately to a low plateau. We name this mode as mode 1. [2] When the defect size increases, Figure S12B shows that the buckling mode changes from the classical bifurcation mode to a snap-through buckling mode, which is characterized by a localized deformation that evolves mainly within the defect (mode 2).
Similar to mode 1, the pressure attains the maximum at a small volume change (limit point 1).
For further increase of defect size ( Figure S12C), the maximum pressure is reached when the main deformation localizes below the defect (mode 3). Dissimilar from mode 1 and mode 2, the pressure in mode 3 gradually increases to the maximum at a much larger volume change (limit point 2). Depending on the shell geometry, the pressure may also show a plateau before the attainment of the maximum pressure ( Figure 1E). In a special case ( Figure S12D), the shell buckles with a mixed mode that combines mode 2 and mode 3. The pressure shows a lower peak at a small volume change (limit point 1) before finally attaining the maximum value at a large change in volume (limit point 2).
The four buckling modes identified above can be overlaid onto the map of the attainable valve output illustrated in Figure 4. The result is shown in Figure S13.

S6.1. Interaction between shells
In our bi-shell valve, the total volume change of the imperfect shell and the spherical cap is determined by the input volume change dispense by the syringe. Since the air compressibility can be neglected (see S3), the sum of the volume change of each shell ( 1 V  and 2 V  ) should balance that of the syringe ( in V  ) such that When the imperfect shell and the spherical cap are slowly deflated by the syringe, their pressure at the quasi-equilibrium state must be equal are the pressure of each shell as a function of its own volume change. Substituting Equation S10 into Equation S11 yields At the pre-snapping state (i), Equation S11 and S12 are rewritten as where the subscript (i) denotes the pre-snapping state. While the pre-snapping state of the spherical cap is determined by its own buckling point, the pre-snapping state of the imperfect shell can be determined by finding the value of At the post-snapping state (ii), the balance of pressure is rewritten as When the pre and post-snapping states have been determined, the released energy can be calculated from the separate response of each shell as are the volume changes at the postsnapping states.
Since Equation S15 does not require handling the simulation of the whole valve, rather it allows to determine the global performance of the valve from those of the constituents. For this reason, the approach that we propose in this work enables a sizeable reduction of the computational cost, and can be readily used to compute the attainable valve output that is plotted in Figure 4.

S6.2. Relation between shell interaction and valve output
Equation S10 to S15 provide a mathematical description of the general interaction between the shells. However, they have been applied neither to assess the pressure volume curves of the individual shells nor to calculate the valve output. In this section, we apply these equations to analyze a group of pressure-volume curves with changing thickness so as to further explain the relation between shell interaction and valve output. Our specific focus is on understanding the reason for the small yellow regions in Figure 4, where the volume change and released energy reach the maximum values. Figure S13 shows a set of the pressure-volume responses of the shells illustrated in Figure 4B.
The curves pertain to a spherical cap (red line) and a an array of imperfect shells with varying thickness 2 t R (green and black lines). A range of responses can be observed. have a maximum pressure that is lower than the buckling pressure of the spherical cap. In this case, the bi-shell system cannot snap because the pressure is unable to reach the buckling pressure of the cap.
A comparison of Figure S12 with Figure S14 highlights that only mode 3 and 4 are suitable for the bi-shell valve. In mode 1 and 2, the pressure of the imperfect shell quickly reaches the maximum with a small volume change. On one hand, if the maximum pressure is larger than the buckling pressure of the spherical cap, the shell interaction will be similar to the case in Figure S14 for thick imperfect shells with 2 / 0.055 t R  (black lines on the left). The bi-shell valve can only snap for a small volume change and released energy. On the other hand, if the maximum pressure is less than the buckling pressure of the spherical cap, there will be no snapping at all, as in the case with the thinnest imperfect shell ( 2 / 0.02 t R  ). In mode 3 and 4, the imperfect shell can undergo a large deformation before attaining the maximum pressure, a response that can potentially form a plateau of pressure before collapse. The green lines in Figure S14 therefore identify the set of bi-shell valves that can attain a large volume change when the plateau pressure is between the pre and post-snapping states; in addition, their maximum values of released energy can be obtained when the plateau pressure is just below the pre-snapping pressure.

S7. Upper bounds of valve output
Here we study the upper bounds of our valve output for both volume and released energy. Figure S15 shows two representative curves of pressure-volume, each representing one individual shell, the spherical cap (A) and the imperfect shell (B). Upon snapping, the pressure of both shells decreases from i p to ii p . In ideal conditions, we could assume the post-snapping pressure of the imperfect shell retains the pre-snapping pressure i p (dashed line in Figure S15B). Thus, the pressure of the spherical cap could also retain i p , thus leading to a post-snapping volume change of Figure S15A). Since it is unrealistic for the imperfect shell to have a post-snapping pressure higher than i p , the post-snapping volume change of the spherical cap will never get larger than the upper bound for the volume change output. As per the released energy, we can use Equation S15 to rewrite the energy released from state p are function of the volume change of each shells (blue lines in Figure S15).
Since the total volume change is constant during snapping, according to Equation S10 we The first two terms in the bracket of the last row of Equation S16 are Similarly since  and i 2 p p  , the last two terms in the bracket of the last row of Equation S16 satisfies Finally, Substituting Equation S19 and S20 into S16, we have The above demonstrate that

S9. Alternative designs of the bi-shell valve
The bi-shell valve introduced in the main text operates through deflation. Here we introduce two design variations to achieve alternative functions: a pneumatic volume fuse and a rapid inflation valve.
 Figure S17A shows a pneumatic volume fuse. This concept modifies the original bi-shell valve operating in a deflation mode in the position of the output chamber, which is here moved on the top of the imperfect shell. In this configuration, before snapping, the imperfect shell can be deflated to generate a continuous volume output 2 V  ( Figure   S17B). When the imperfect shell is in the pre-snapping state (i), further deflation will trigger the snapping of the volume fuse, which reduces the volume change of the imperfect shell from state (i) to state (ii). The outcome is a pneumatic fuse: the volume change of the imperfect shell at state (i) sets the threshold of volume output that the fuse cannot exceed.
 Figure S17C shows a rapid inflation valve. The original bi-shell valve concept is here altered by flipping the two elastic shells upside down. This valve works in inflation mode in a way similar to that of the original valve that operation in a deflation mode. When slowly inflated at the inlet ( Figure S17D), the imperfect shell first inflates to store energy and volume change. Upon snapping, the imperfect shell deflates from state (i) to state (ii) so as to release energy and volume change, while the spherical cap snaps upward. The advantage of this design is the provision of a fast volume output for the rapid inflation of any actuator that may be connected to the outlet.

S10. Effect of elastic modulus on the performance of the bi-shell valve
To study the effect of material elasticity on the performance of the bi-shell valve, we perform a set of numerical simulations, where the shell material is assumed linear elastic with a Young's modulus ranging between 1 and 10 MPa. In Figure S18A, the pressure increases linearly with the Young's modulus, while the volume change is not affected by a change in the Young's modulus values. In Figure S18B, the released energy of the bi-shell valve increases linearly with the Young's modulus from 1 to 10 MPa with discrete step of 1 MPa. On the other hand, the volume output of the valve is not affected by a change in the Young's modulus values ( Figure S18C).

S11. Integration of the bi-shell valve integrated into a soft robot or actuator.
To integrate our valve within a soft robot, the output chamber can be merged with the interior of the soft actuator, while the whole system of the bi-shell valve and the soft actuator can be controlled from the input chamber. A possible layout of the valve-actuator integration is given in Figure S19. Both the rigid input and output chambers can be replaced with thick soft walls by molding [6] . The integration would only require the merging of both molds, that of the bishell valve and that of the soft actuator. The exterior of the robot is purposely left as undefined because it can be shaped by design to deliver a certain function. For example to achieve motion, the exterior body can be designed as a partially corrugated cylinder with grips in contact with the ground. [6] Figure S19. Schematic of the bi-shell valve integrated into a soft robot or actuator.

S12. Scenario of a similar valve with only one shell
If there is only the spherical cap ( Figure S20), there can be no fast transfer of air volume between shells. This means that the total volume of the input and output chambers remains constant. As a result, the flow rate provided at the valve input equals that of the valve output.
In our experiment, the flow rate at the valve inlet is 3 mL/min, a value that cannot provide fast actuation. On the other hand, if only the imperfect shell is present, fast actuation cannot yet be achieved. The reason is as for the above. No fast transfer of air volume between shells can occur, hence we cannot convert a low flow rate (input) to a fast flow rate (output). Figure S20. Scenario with only the spherical cap.