Mechanical Valves for On‐Board Flow Control of Inflatable Robots

Abstract Inflatable robots are becoming increasingly popular, especially in applications where safe interactions are a priority. However, designing multifunctional robots that can operate with a single pressure input is challenging. A potential solution is to couple inflatables with passive valves that can harness the flow characteristics to create functionality. In this study, simple, easy to fabricate, lightweight, and inexpensive mechanical valves are presented that harness viscous flow and snapping arch principles. The mechanical valves can be fully integrated on‐board, enabling the control of the incoming airflow to realize multifunctional robots that operate with a single pressure input, with no need for electronic components, cables, or wires. By means of three robotic demos and guided by a numerical model, the capabilities of the valves are demonstrated and optimal input profiles are identified to achieve prescribed functionalities. The study enriches the array of available mechanical valves for inflatable robots and enables new strategies to realize multifunctional robots with on‐board flow control.


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-Cut a segment with length of ∼ 40 mm out of a 10 ml plastic syringe with inner diameter 55 of 22 mm. This will form the chamber for the valve.

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-Cut a dome-shaped notch (with depth around 1 mm and diameter around 5mm) on the 57 internal surface of the cylindrical segment using a rotary tool workstation (220-01 Dremel). 58 When the piston moves across the notch, the on/off state of the valve is changed.

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• Bistable valve: The bistable valve is fabricated following the same steps used for the hysteretic 63 valve. The only differences between the two designs are the thickness of the plate (for the 64 bistable valve we use a metallic plate with thickness t plate = 0.05 mm) and the mounting angle θ plate (for the bistable valve we use θ plate = 0 • ).

Kirigami actuators.
To realize the robotic arm and climbing robot shown in Fig. 2 and 3 of the 67 main text, we use cylindrical kirigami actuators fabricated following the procedure described in our 68 previous work (1) -the only difference is that here we add another thin layer (thickness ∼ 1 mm) 69 of Ecoflex 00-50 inside the actuator to enhance its robustness and durability. Specifically, we use  Table S1. Note that Actuators II and III comprise identical units 73 for the entire structure, whereas to enable bending in Actuator I we introduce one column of unit 74 cells with increased ligament size δ 2 (shown as purple units in Fig. S2).   Table S1. Geometric parameters defining our kirigami actuators. All parameters are defined in Fig. S2. 1.3. Robotic arm. The robotic arm is constructed by connecting two bending kirigami actuators 76 (Actuator I in Table S1 and Fig. S2) through a viscous valve or a one-way viscous valve. Specifically, 77 we fabricate the robotic arm using the following steps: 78 • Fabricate two identical kirigami actuators (Actuator I in Table S1 and Fig. S2) following the 79 procedure described in our previous work (1). The only difference is that here we add another 80 thin layer (thickness t ∼1 mm) of Ecoflex 00-50 inside the actuator to enhance its robustness 81 and durability. More specifically, we pour additional Ecoflex inside the kirigami actuator and 82 rotate it slowly to achieve a uniform coating.

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• Glue perforated acrylic plates at the ends of the actuator to facilitate connection with other 84 actuators (see Fig. S3). Further, we 3D print a mold to cast elastomeric layers with thickness 85 ∼ 1 mm and the same shape as the acrylic plates. Such elastomeric layers are placed between 86 the acrylic plates and the valves to avoid leakages (see Fig. S3).

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• Arrange the two identical bending actuators so that the columns of unit cells with increased 88 ligament size δ 2 are diametrically opposed (Fig. S3).

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• Introduce a viscous valve (or one-way viscous valve) between the two actuators to regulate the 90 flow between them (as shown in Fig. S3). Secure the connection between the actuators and 91 the valve by threading a screw through the acrylic plates and the hole places in the center of 92 the valve.

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• Connect a tube to the top actuator to provide the pressure input.  I in Table S1 and Fig. S2) separated by a viscous valve or one-way viscous valve.
1.4. Tube climbing robot. As shown in Fig. S4A our climbing robot comprises two expanding kirigami actuators (Actuator III in Table S1) and an extending one (Actuator II in Table S1) 96 connected via a one-way viscous valve and a viscous valve. Further, to enable grasping, a hysteretic 97 valve and a gripper consisting of two PneuNets (2) is added at the top (Fig. S4B). To fabricate the 98 robot, we follow the steps below:

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• Fabricate two identical expanding kirigami actuators (Actuator III in Table S1 and Fig. S2) 100 and one extending actuator (Actuator II in Table S1 and Fig. S2) following the procedure 101 described in our previous work (1). The only difference is that here we add another thin 102 layer (thickness t ∼1 mm) of Ecoflex 00-50 inside the actuator to enhance its robustness and 103 durability. More specifically, we pour additional Ecoflex inside the kirigami actuator and rotate 104 it slowly to achieve a uniform coating.

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• Glue perforated acrylic plates at the ends of the actuators to facilitate connection with other 106 actuators (see Fig. S4A). Further, we 3D print a mold to cast elastomeric layers with thickness 107 ∼ 1 mm and the same shape as the acrylic plates. Such elastomeric layers are placed between 108 the acrylic plates and the valves to avoid leakages (see Fig. S4A).

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• Introduce a viscous valve between the top extending actuator and the elongating one and 110 a one-way viscous valve between the bottom extending actuator and the elongating one (as 111 shown in Fig. S4). Secure the connection between the actuators and the valve by threading a 112 screw through the acrylic plates and a hole places in the center of the valve.

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• Connect a tube to the bottom actuator to provide the pressure input.

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Additionally, to realize a robot capable of grasping an object, the following steps are also needed:

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• 3D print the two-part mold shown in Fig. S5A. This mold enables casting of PneuNets with 116 length l gripper = 30 mm, width w gripper = 10 mm and thickness t gripper = 5 mm.

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• Cure the Ecoflex for about an hour at room temperature.  The uncured Ecoflex 00-50 enables bonding between the two pieces.

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• Remove the soft actuator from the mold.

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• Insert a short tube in one end of the PneuNets.
for the connection with the top actuator via a screw; and (b) the other two holes, located at 131 the margin of the plate, are used to pass through the short tubes that connect to the Pneunets.

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• Apply glue (Sil-Poxy Silicone Adhesive, Smooth-On) to secure the tube and Pneunets to the 133 acrylic plate.

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• Attach a hysteretic valve to the top expanding actuator and connect the gripper to its chamber.

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Apply glue (Sil-Poxy Silicone Adhesive, Smooth-On) to fix and seal the valve and the gripper.

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• The robot is ready to be tested.  Table S1) and an extending one (Actuator II in Table S1) connected via a one-way viscous valve and a viscous valve. (B) A hysteretic valve and a gripper consisting of two PneuNets (2) is added to the top actuator to enable grasping.  Double 32 onto the half cured skeleton (to reach the full thickness) and then place the structure 158 from Step d on top of it to seal the frame and the chambers.

Experiments
In order to characterize the effect of the valves on the response of the robotic systems, we use 178 pressurized air of the order of a few kPa. To achieve such levels of pressure, we decrease the pressure 179 from the wall air-outlet (∼200 psi) using two pressure regulators (B74G-4AK-AD3-RMN by IMI connect each of them to an extension actuator (Actuator II in Table S1) and monitor the pressure

Characterization of the kirigami actuators.
As part of this study we also experimentally of this study we use the kirigami actuators in an unconstrained environment for the robotic arm and constrained within a tube for the climbing robot, we test them both in unconstrained and 202 constrained conditions. More specifically, we test Actuator I and II under unconstrained conditions 203 and find a liner dependence of the pressure from the supplied volume (Fig. S9B). Further, we test 204 Actuator II and III when placed in a tube with diameter of 35 mm (identical to the tube used to test 205 the climbing robot). As shown in Fig. S9C, in this case we find a highly non-linear pressure-volume 206 curve, as the pressure abruptly increases when the actuators get in contact with the tube.

Robotic Arm.
To characterize the deformation of our robotic arm, we monitor with a high-208 resolution camera (SONY RX100V) recording at a frame rate of 30 fps (i) 17 black circular markers 209 uniformly placed along the length of the kirigami actuators and (ii) a white circular marker placed 210 at the end of an acrylic rob connected to the bottom cap. We extract the coordinates of all markers 211 using an open-source digital image correlation and tracking package (3). We then use the coordinates 212 of the white marker to determine the trajectory followed by the robotic arm. Further, we determine 213 the radius R of the circle that best fits the black markers via a direct least-square algorithm (4, 5) 214 and calculate the average curvature of the two actuators as 215 κ = 1/R. [S1] 216 Finally, we note that during the tests the evoultion of the pressure inside the actuators is monitored 217 by pressure sensors (MPXV7025DP, Freescale Semiconductor Inc).

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In Figs. S16 and S17 we consider the robotic arm with a viscous valve and one-way viscous 219 valve, respectively, and report the recorded tip trajectory (left), pressure evolution in both actuators 220 (middle) and curvature evolution of both actuators (right) for different pressure inputs.

Tube climbing robot.
To test the climbing robot, we place the system inside a vertically oriented 222 acrylic tube with inner diameter of 35 mm and supply many pressure pulses, while recording its 223 motion using a high-resolution camera (SONY RX100V).

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Further, we conduct additional tests to characterize both the frictional force between the expanding 225 actuator and the tube and the axial force exerted by the extending actuator interaction between the 226 robot and the tube, as well as the extension force generate by the extending actuator.

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As shown in Fig. S10A, to characterize the frictional force, we place an expanding actuator inside

Modeling
To get a better understanding of the behavior of the proposed valves, we use numerical analyses.

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For the viscous valves, we simplify the Navier-Stokes equations to calculate the pressure drop. For Differently, when, as in this study, we use a compressible fluid (air) to inflate the robotic system, Note that, by introducing the ideal gas law n i andñ i can be written as where V i denotes the volume for the [i]-th actuator, R g is the ideal gas constant and Θ is the 301 temperature. Moreover, [S10] 316 For a system comprising N fluidic actuators interconnected via viscous valves Eq. (S10) results in 317 a system of N coupled differential equations, which given a pressure-volume relationship for the 318 actuators that can be numerically solved to determine the normalized change in volume for the [i]-th 319 actuator as a function of time. Finally, we note that for the first and last tube in the array Eq.

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(S10) needs to be modified as to account for the pressure input and the end of the array.  Note that these values for ξ are also summarized in Table S2.   Table S2. Equivalent conductance of the viscous valves.  Table S2, whereas for the connection between the input source and the 349 first actuator we use ξ 1 = 472.7. Finally, to connect P and V , we use the experimentally measured   Table S2, whereas for the connection between the input source and the first actuator we use ξ 1 = 472.7. Finally, to connect P and V , we use the experimentally measured pressure-volume relationship shown in Fig. S9C. As shown in 363 Fig. S20. we find that the model (dashed lines) agree well with the experimental measurements 364 (solid lines) for all considered pressure inputs. where r is the radius of the chamber.

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In Fig. S14C, we report the numerically predicted evolution of the equivalent pressure p as a [S18] In order to design a valve with a prescribed on/off pressure, in Figs. S14D and E we report the 405 phase diagrams of p on and p of f for a valve with t plate = 0.075 mm as a function of the mounting angle θ plate and length l plate . Instructed by the model, we choose θ plate = 45°and l plate = 17.5 mm as 407 design parameters, which enables the on/off switch of the valve at 20.8 kPa and 2.1 kPa, respectively.

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For the bistable valve, we reduce the thickness of the plate to t plate = 0.05 mm in order to operate 409 the valve at around 20 kPa. Similarly, we report the response of the bistable valve (with θ plate = 0°410 and l plate = 17.5 mm ) in Fig. S14F S21. Phase diagram of the minimum pressure inside the climbing robot's top actuator when a hysteretic valve is embedded into it. The embedded hysteretic valve affects the cavity of the top actuator and, in turn, the pressure-volume evolution. The area of the diagram that is bounded by the two dashed lines identifies the input parameters for which the robot will achieve climbing. Guided by our model, we choose pinput = 48 s and pinput = 15 kPa as the input for the grasping tests. Movie S1. Characterization of the mechanical valves.

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To demonstrate the capabilities of our mechanical valves, we connect each valve to an extending 423 actuator and compare the pressure evolution at the inlet and outlet of the valve.

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Movie S2. Robotic arm with different trajectories. 426 The robotic arm consists of two bending actuators and a viscous valve (or one-way viscous valve).

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It is capable of achieving multiple trajectories with a single pressure input.