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Submitted to DOI: 10.1002/adma.201600660R2 Rotary Actuators Based on Pneumatically-Driven Elastomeric Structures Xiangyu Gong, Ke Yang, Jingjin Xie, Yanjun Wang, Parth Kulkarni, Alexander S. Hobbs, and Aaron D. Mazzeo* Department of Mechanical and Aerospace Engineering Rutgers, the State University of New Jersey 98 Brett Road Piscataway, NJ 08854 E-mail: [email protected] Keywords: soft robots; elastomers; locomotion; pneumatics; wheels

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  • Submitted to DOI: 10.1002/adma.201600660R2

    Rotary Actuators Based on Pneumatically-Driven Elastomeric Structures

    Xiangyu Gong, Ke Yang, Jingjin Xie, Yanjun Wang, Parth Kulkarni, Alexander S. Hobbs,

    and Aaron D. Mazzeo*

    Department of Mechanical and Aerospace Engineering

    Rutgers, the State University of New Jersey

    98 Brett Road

    Piscataway, NJ 08854

    E-mail: [email protected]

    Keywords: soft robots; elastomers; locomotion; pneumatics; wheels

  • Submitted to

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    This paper describes a unique mechanism a soft rotary actuator based on peristaltic motion

    of elastomeric materials, which consists of an inflatable stator paired with a rotor. Timed inflation

    and deflation of the air-filled bladders in the stator enable controllable rotational speed of the rotor.

    With two configurations, these rotary actuators are capable of having either an internal rotor for

    winch/joint-like applications or an external rotor that can serve as a wheel. Fabrication of these

    actuators employs the use of 3D-printed molds and millimeter-scale soft lithography.

    Characterization of the relationships between speed, torque, and power for these actuators provides

    a baseline for potential uses in locomotion and transportation of payloads. A squishy, four-wheeled

    vehicle enabled by these actuators traveled at a speed of 3.7 cm/s, negotiated irregular terrain, and

    endured mechanical impact from a drop eight times its height. This class of actuators extends the

    potential functionality of soft robotic systems by providing rotational torque without requiring

    bending or twisting.

    Relying on the simple motion of bending, soft pneumatic actuators[14] have manipulated

    fragile objects with pressurized fingers[4,5] and have crawled/undulated by changing their gaits[6].

    Demonstrations of bending inflatable actuators have also included manta rays[3], camouflage[7],

    resistance to puncture[8] and impact[9], elastomeric origami-like devices[10], power through

    controlled explosions[1113], fast flexing with low strain[14], endurance in adverse environments[15],

    octopus arms[1618], and robotic tentacles[19]. However, rotation plays a significant role in the design

    of simple machines, as conventional robots often do not depend on bending actuators but employ

    joints and torque-providing motors.

    Recent advances in pneumatic, elastomeric actuators are not capable of delivering torque

    through pure rotation. Soft belt-like robots[20,21] and a spherical robot[22] rolled, but the employed

    class of structures circular belts and spheres required the locomotors to rotate completely upon

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    themselves. In addition, belt and sphere-based soft robots have yet to be optimized for transporting

    payloads and handling pneumatic plumbing that can twist on itself.

    While many soft robots have their basis in bio-inspired design or evolution[18,23,24], wheels on

    axles distinct from rolling mechanisms employed by animals (e.g., three-banded armadillos,

    pangolins, wheel spiders, and pebble toads) are a notable exception, as fully rotating components

    do not exist in nature at the scale of millimeters[25,26]. In nature, the competitively advantageous

    modes for soft creatures appear to resemble walking, crawling, and jumping, as verified through

    genetic and evolutionary algorithms for both soft and hard virtual creatures[2729]. Nonetheless,

    wheel-and-axle assemblies are absent from nature, even though wheels have proven more

    advantageous in certain scenarios. ,

    Future wheeled robots with elastomeric components might find uses in search and rescue

    missions in extreme environments or over varied terrains, such as an irregular tunnel that requires

    a compliant body with large deformation, or a rugged lakebed under water. With the low sensitivity

    of elastomers to magnetic fields, wheeled robots or rotary joints made of elastomers might

    facilitate travel in space with limited exposure to radiation and cold. Another application might

    include the manipulation of objects during magnetic resonance imaging (MRI). Lacking sharp or

    metal components, soft rotary actuators may also serve as safe motors and be compatible with

    future human-friendly robots or vehicles.

    Peristalsis is common in nature. Peristaltic motion generated by the circular muscles of the

    human esophagus helps push a bolus of food toward the stomach[30]. A variety of limbless crawlers,

    such as earthworms, snails and snakes, use peristaltic mechanics and have inspired multiple worm-

    like robots[3133]. The design in this work combines peristaltic motion with the layout of a spinning

    motor, which consists of a rotor and a stator. The stator is a circular, elastomeric structure with

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    embedded, hollow bladders. Each bladder has an inlet that provides pressurized air. We divide or

    distribute the flow of pressurized air into multiple subgroups of bladders with each subgroup being

    inflated and deflated sequentially. The sequential inflations and deflations generate a peristaltic

    wave offset a radial distance from the rotational center and exert a set of circumferential forces to

    provide torque on the rotor.

    This work demonstrates two configurations of rotary actuators that enable the conversion of

    peristaltic motion to torque. In the first configuration (Type 1), the inflatable stator surrounds a

    hard plastic rotor. In the second configuration (Type 2), the stator sits inside a rotor made of

    another elastomer. Figure 1a shows one complete cycle of sequential inflation of four subgroups

    of bladders on stators without rotors for both types. These two configurations lend themselves to

    unique sets of applications, such as winches (Type 1) and squishy wheels (Type 2).

    As shown in Figure 1b-c, each time one subgroup of the air-filled bladders receives pneumatic

    pressure, the four growing bladders in the stator exert force/torque on the rotor to make it turn. The

    rotation stops when the bladders make contact with the internal corners of the rotor. This stop or

    locked position is a stable state capable of maintaining its position. To move from this stable

    state, we exhaust the inflated subgroup and inflate an adjacent subgroup of bladders to transition

    the rotary actuator through another metastable step. The repeated sequence of inflation and

    deflation generates the staggered turning of the rotor. With increases in the frequency of sequential

    actuations, the continuous nature of the rotary motion increases.

    Using computer-based, geometric models created in SolidWorks (Dassault), a three-

    dimensional (3D) printer (FlashForge Creator) produced Type 1 rotors and molds for elastomeric

    components. The material used for the 3D printing is a thermoplastic acrylonitrile butadiene

    styrene (ABS). The Supporting Information (Figure S1) discusses more details concerning

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    Figure 1. Basic configurations and sequential actuation of subgroups of embedded bladders for

    two types of rotary actuators. (a) Subgroups of bladders inflating around fixed stators in peristaltic

    fashion (12341). Four different colors mark the four subgroups, respectively. (b) An

    actuator with an internal rotor (Type 1) showing a step angle of 22.5, along with rotation between

    stable states (enclosed by the dashed lines). (c) An actuator with an external rotor (Type 2) showing

    a step angle of 22.5 and rotation between stable states (enclosed by the dashed lines).

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    fabrication. Briefly, we made stators of a silicone-based rubber Ecoflex 00-50 (Smooth-On Inc.)

    by bonding the cured top layer to the partially cured bottom layer without applying an intermediate

    layer of adhesive (Figure S1 a). With these two layers, a stator contains closed rectangular

    bladders (length: 5 mm, width: 1.5 mm, depth: 5 mm, and membrane thickness: 2.5 mm) inside of

    it. In order to reduce the friction and smooth displacement between the stator and the rotor, we

    applied lubricant (Super Lube, Synco Chemical Corp.) between the sliding interfaces of the two

    components. To understand how the pressurized air acted on the bladders, we calculated the strains

    on an inflated stator using finite-element analysis (FEA) in ANSYS with data from tensile testing

    of Ecoflex 00-50. The maximum principal strain of the inflated bladders (~ 200%) did not exceed

    the maximum strain at break of the Ecoflex 00-50 (980%) provided by the manufacturer.

    Experiments have shown the sufficient bonding between the two layers of the stator. However,

    failure occurred occasionally at a pressure higher than 62.1 kPa due to delamination after low-

    cycle fatiguing (see Supporting Information).

    A programmable pneumatic system with solenoid valves (Figure S4) allowed us to select

    groups for inflation and to change the flow rate of the pressurized air indirectly with a pressure

    regulator. Figure 2a depicts the staggered inflation and deflation of four subgroups of bladders

    embedded in a stator. The blue solid lines represent the command signals for the opening and

    closing of each valve that was associated with a group of bladders. The red dashed lines indicate

    the change in the internal pressure within the bladders. A higher pressure set on the regulator,

    which permitted greater flow, resulted in a shorter inflation time tinflation the time needed to inflate

    a group of bladders to a volume that was just large enough to enable a rotation of the rotor. Before

    the actual internal pressure within the bladders reached the pressure that might otherwise cause a

    larger deformation or delaminate the bonded layers, we halted the inflation by switching the

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    position of the valve and exhausted the air in the bladders to the surrounding environment. Unlike

    inflation times, which were dependent on the pressure set on the regulator, the deflation times

    tdeflation remained consistent for a given volume of the inflated bladders. To measure the time of

    deflation, we used a camera with a moderate speed (120 frames/second) to record the deflation of

    each group (Figure S5, Movie S1). In this way, we found the average deflation times of Type 1

    and Type 2 actuators were approximately 130 msec and 190 msec, respectively. Meanwhile, we

    determined the static internal pressure in these inflated bladders (~ 48.3 kPa, 7 psi, for both Types)

    with the minimum inflation that could enable the continual rotation of a rotor. Knowing this static

    pressure, we estimated the corresponding minimum volumes of the inflated bladder for both

    designs (Type 1: 8.0 cm3, Type 2: 3.3 cm3) with a gridded background (Figure S3 a, c).

    The rotational speed of the actuators relied on both times of inflation and deflation for each

    group of bladders. With an approach similar to that of controlling an electromagnetic motor by

    changing the applied voltage, we manipulated the speed of the rotary actuator by varying the

    pressure set on the regulator. With a specific inflation time corresponding to a varied set pressures,

    we calculated rotational speeds with the following formula:

    Rotational Speed (RPM) = (60 sec/min)(1000 msec/sec)

    (16 groups/rotation)(tinflation

    + tdeflation

    ) (1)

    The units of tinflation and tdeflation are milliseconds/group.

    To understand the capabilities (i.e., rotational speeds and load capacity) of the actuators, we

    characterized them in similar fashion to that of an electromagnetic motor. Figure 2b-c represent

    the Type 1 and Type 2 actuators connected to the pneumatic system, respectively. By varying the

    set pressure and timing as shown in Figure 2a, we experimentally mapped the set pressure

    inflation time (Figure 3a) and rotational speedset pressure relationships (Figure 3b) by

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    Figure 2. (a) The sequential inflation and deflation of groups of bladders. (b) Experimental setup

    for characterizing set pressure rotational speed for Type 1 actuators. (c) A Type 2 actuator

    attached with a string to an external load. (d) A winch-like setup used to profile the relationships

    between rotational speeds of Type 1 actuators and applied loads. The scale bars are 2 cm.

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    Figure 3. (a) Measured relationships between the set pressure on the regulator and the inflation

    time for both types of actuators without any external load. (b) Measured rotational speed as a

    function of set pressures for both types of actuators based on the measurements in (a). (c) Torque-

    speed characteristics of a Type 1 actuator in a winch (paired value divided by two for the

    symmetric setup) and the estimated power based on its linearized fit and the product of the torque

    and speed. (d) Torque-speed characteristics of a single Type 2 actuator.

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    converting the inflation times to rotational speeds with Equation 1 and previously measured

    deflation times.

    For the Type 2 actuators, we directly attach an external load to the string (Figure 2c). For the

    Type 1 actuators, however, we built a symmetric experimental setup (Figure 2d) to permit the

    application of external loads (see Figure S6). We then inserted the shaft with rotor-shaped ends

    into the stators. This winch-like setup permitted varying external loads and kept the rotors

    balanced inside the stators when profiling the torquerotational speed curve (Figure 3c). With

    this setup, we divided the weight in half to get the external torque acting on each of the Type 1

    actuators. For both actuators, we initially set a pressure of 68.9 kPa (10 psi) on the regulator to

    make the actuators turn without any external loading. With increasing external loads on the

    actuators, we observed larger inflation times to overcome increasing pressure-based forces and

    friction between the rotor and stator. Thus, in similar fashion to conventional electromagnetic

    rotors, higher torques resulted at lower speeds of rotation.

    Figure 3a profiles the set pressures on the regulator corresponding to the desired inflation

    times for each subgroup of bladders with no external load on the rotors. Figure 3b shows the

    relationships between the rotational speed and the set pressures from the data shown in Figure 3a,

    the previously measured deflation times, and Equation 1. In general, increasing the set pressure

    on the regulator lead to a higher rotational speed, because the inflation time decreased due to an

    increased flow rate. According to the measured results, Type 1 actuators achieved higher rotational

    speed than Type 2 actuators with the same set pressures, which reflects the facts that the Type 1

    actuators were less massive and had less friction with their stators than Type 2 actuators. The

    bladders in both types inflated by approximately the same amount at the same rate, and the higher

    speeds of Type 1 actuators than those of Type 2 actuators were also because the bladders acted at

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    a distance closer to the axis of rotation than in the Type 2 actuators. There was also a remarkable

    change in the speed and release of kinetic energy for the Type 1 actuators at approximately 55.2

    kPa (8 psi) caused by the snap-through behavior of the inflatable elastomers[3437]. To verify the

    snap-through behavior of the bladders embedded in the elastomeric stators, we monitored the real-

    time internal pressure of a bladder that was under inflation (Figure S7 and Movie S9).

    Figure 3c-d characterized the rotational speeds of the two types of actuators, respectively, as

    a function of the torque at a specified pressure set on the regulator. We calculated the torque based

    on the applied, hanging loads and their distance from the axis of rotation (diameter of the shaft of

    the winch: 4.1 cm; outer diameter of the Type 2 actuator: 9.2 cm). At a fixed pressure of 69 kPa

    (10 psi), starting with no load, the winch for Type 1 actuators rotated at a speed of 17 RPM, and

    the Type 2 actuators had a rotational speed of 10.2 RPM. Generally, the rotational speeds of the

    actuators decreased when the external loads increased, until the torque reached 6.5 mNm for Type

    1 actuators and 38 mNm for Type 2 actuators. At that point, the actuators were no longer able to

    bear the external torque or turn continually, with the rotors sliding backwards inside the stators.

    Using linear regression, we fit the relationships between rotational speed and external torque to a

    linear model (solid lines). In this way, we also estimated the power of the actuators as a function

    of the speed of rotation. According to the calculated power rotational speed curves (dashed lines)

    in Figure 3c-d, the maximum values the actuators achieved were 8.4 mW (Type 1) and 26 mW

    (Type 2), when the rotational speeds were at a minimum. The overall power of the Type 1 actuator

    was lower than that of Type 2, but the Type 1 actuator achieved a higher rotational speed. In

    addition to the capability of lifting payloads, we also demonstrated how the soft rotary actuators

    were compatible with previously published pneumatic soft actuators (i.e., a soft gripper embedded

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    with PneuNets)[4]. A winch equipped with a soft gripper going up and down was able to grasp and

    lift an object (Figure S8, Movie S3).

    To demonstrate the application of the rotary actuators powered by the pneumatic system, we

    built prototypes of squishy vehicles equipped with the actuators working as wheels with a diameter

    of 9.2 cm. Combining multiple controllable motions, a two-wheeled vehicle demonstrated the

    capability of navigating around an obstacle and had a top speed of 4.9 cm/s (Figure S9, Movie

    S4). We also extended the two-wheeled device to a rover-like vehicle with four wheels (Figure

    4a-b). A structure made of Ecoflex 00-50 served as a chassis to connect two two-wheeled modules.

    At the same set pressure of 86.2 kPa (12.5 psi) used to drive the two-wheeled vehicle, we needed

    longer inflation times (300 msec) to compensate for the extra mass of the chassis total mass of

    1.23 kg and friction between stators and rotors. The rotational speed of each wheel was 7.7 RPM,

    and therefore, the speed of the vehicle was 3.7 cm/sec (Figure 4c, Movie S5).

    This four-wheeled vehicle puts forth several potential benefits to endurance and navigation

    over varied terrains. First, there were not any rigid components in the entire vehicle, and the

    elastomeric wheels/components helped absorb mechanical impact. A drop test from eight times its

    height (0.72 m) did not cause any damage to the body or the actuators (Figure 4d, Movie S6) of

    the vehicle. After bouncing and landing on the ground safely, the vehicle moved forward.

    Second, navigation over difficult terrain with a naturally compliant set of wheels and

    suspension suggests the potential of eliminating the complexity associated with rigid members and

    multi-component suspensions for allowing wheels to maneuver over obstacles. In Figure 4e (and

    Movie S7), the squishy vehicle negotiated a rocky terrain with a puddle. As shown, the chassis

    made of Ecoflex 00-50 bent to comply with the rugged landscape. As an aside, the slight climb

    from the puddle back to the higher ground was at a reduced linear speed, associated with a lower

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    Figure 4. A squishy, four-wheeled vehicle. (a-b) Assembled components of a four-wheeled

    vehicle. (c) The vehicle traveled on a flat surface without any obstacles at a speed of 3.7 cm/s

    (Movie S5). (d) A drop test on a four-wheeled vehicle showed its capability of withstanding

    mechanical impact. The vehicle fell from a height of 0.72 m eight times the height of the vehicle.

    After bouncing, the vehicle remained intact and moved forward (Movie S6). (e) A four-wheeled

    vehicle negotiated a rocky terrain with a puddle in its path. The vehicle started navigating over a

    rocky terrain. It went down into the puddle with its body bending and moved up to the other side

    of the rocks (Movie S7).

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    angular velocity of the rotary actuators (i.e., a longer inflation time) with a higher pressure set on

    the regulator.

    Finally, the vehicle equipped with the actuators functioned not only in a dry environment but

    also under water (Figure S10, Movie S8). In this example, the vehicle pushed a buoy and carried

    a few rocks to compensate for the buoyancy of the underwater, air-filled cavities. With no metal

    components and moisture-sensitive electronics on the vehicle itself, the tethered locomotor was

    waterproof. Follow-up work might explore the resistance of the adhesives and silicones to long-

    term exposure to water, while inspiring future low-cost amphibious vehicles.

    To conclude, the soft rotary actuators in this work generated motion through biologically-

    inspired peristalsis that used alternating inflation and deflation of pneumatic bladders. The two

    types of rotary actuators employed complementary configurations of stators and rotors. For the

    chosen geometries, the speed and torque of the soft actuators depended on three factors: pressure

    and flow of the compressed air delivered to the bladders, timing of the solenoid valves, and the

    load on the actuators. These rotary actuators have torque-speed characteristics similar to those

    of conventional electromagnetic motors. The quantitative characterization might also serve as an

    initial baseline for future soft, rotary devices. Unlike conventional electromagnetic motors, they

    are also capable of locking at a fixed angle without the application of additional power (i.e., a

    sealed inflatable bladder is able to maintain pressure and apply reactive forces to a rotor to maintain

    a position).

    The Type 1 actuators with a rigid, 3-D printed internal rotor demonstrated their use in winch-

    like applications. Consisting of entirely soft components, the Type 2 actuators enabled the

    fabrication of a squishy, wheeled vehicle with the ability to withstand mechanical impact, negotiate

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    rocky and wet terrains, and work under water as an amphibious vehicle. These actuators provided

    rotational torque without bending and twisting.

    Future uses of this work might include the design and fabrication of soft robotic joints and soft

    wheels for enhanced locomotion. An on-board control system, such as a micropneumatic

    manifold[38], and a compact air compressor used by resilient, untethered robots[15] might also make

    it possible to fabricate future autonomous and untethered soft wheeled devices. For these potential

    applications, outstanding technical challenges to be addressed include the development of

    materials and mechanical sciences to understand the large deformation and cyclic fatigue of the

    elastomeric bladders, delamination at the bonded interfaces, the role of rapid releases of kinetic

    energy as the bladders pass through metastable snap-through instabilities, and modest outputs of

    rotary, mechanical power. This work marks initial steps toward designing soft material-based

    systems capable of millimeter-scale, continuous rotary actuation.

    Acknowledgements

    The authors acknowledge funding from Rutgers University through the School of Engineering,

    the Department of Mechanical and Aerospace Engineering, the University Research Council, and

    an A. Walter Tyson Assistant Professorship Award.

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  • Submitted to DOI: 10.1002/adma.201600660R2

    Rotary Actuators Based on Pneumatically-Driven Elastomeric Structures

    Supporting Information

    Xiangyu Gong, Ke Yang, Jingjin Xie, Yanjun Wang, Parth Kulkarni, Alexander S. Hobbs,

    and Aaron D. Mazzeo*

    Department of Mechanical and Aerospace Engineering

    Rutgers, the State University of New Jersey

    98 Brett Road

    Piscataway, NJ 08854

    E-mail: [email protected]

    Keywords: soft robots; elastomers; locomotion; pneumatics; wheels

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    Rotary Actuators Based on Pneumatically-Driven Elastomeric Structures

    Supporting Information

    Fabrication of Soft Rotary Actuators

    Fabrication of the stators (Figure S1 a) required the molding and assembly of two separate

    components: one for the structure with groups of inflatable bladders (top layer) and one for a layer

    to seal these chambers (bottom layer). We fabricated all the molds from a thermoplastic

    acrylonitrile butadiene styrene (ABS) with a 3D printer (FlashForge Creator) using computer-

    aid design. As shown in Figure S1a, we removed the top layer of the stator from the mold, cleaned

    the surface of the top layer with isopropyl alcohol, and then put it on the partially-cured, adhesive-

    bottom layer that still stayed in the mold. It required approximately 10 minutes for the bottom

    layers to be partially cured at room temperature. After the bottom layer was fully cured, the

    complete stator was ready for removal.

    The designated holes printed on the bottom layer indicate the positions of the inlets for

    compressed air. After punching 16 holes on the bottom layer with needles, we inserted silicone

    rubber tubing (OD:1/16, ID:1/32) into each bladder through the holes. As the final step, we

    sealed the gap between the tubing and the holes with silicone rubber adhesive Sil-Poxy (Smooth-

    On Inc.). In general, the fabrication for the two types of stators was the same, and both of them

    consisted of silicone-based elastomer. However, the fabrication and the materials of the two types

    of rotors (Type 1: 3.65-cm diameter, Type 2: 9.2-cm diameter) are different. We fabricated the

    hard internal rotor (Type 1) from ABS directly through the 3D printer (FlashForge Creator), using

    computer-aid design (Figure S1 b). Meanwhile, Type 2 rotors were molded from a thermosetting

    elastomer Mold Star 30 (Smooth-On Inc.).

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    For Type 2 actuators, the 3D printer (FlashForge Creator) produced the molds for the different

    layers of the external rotor, and then we molded these layers (Figure S1 c). We bonded the

    structured layers with Sil-Poxy. There are three layers: one layer with the main structure designed

    to interact with the stator inside of it, one PDMS layer to constrain the slide on the axial direction

    between the rotor and the stator, and one layer that constrained and fixed the stator on the other

    side. The transparency of the PDMS layer also showed the internal structure and helped us monitor

    how the rotor and the stator interacted when controlling an actuator. The material for the other two

    layers (blue) is Mold Star 30 (Smooth-On Inc.). Mold Star 30 is a type of platinum silicone rubber,

    which is also an elastomer, but it is harder than the Ecoflex series.

    Step Angle

    Similar to a conventional stepper motor driven by a series of pulsed signals, the inflation of

    each group of bladders drives the rotor of the soft rotary actuator to turn by a fixed angle, which

    we call a step angle. According to this principle of design, the rotor turns in discrete increments

    from subgroup of embedded bladders to subgroup of embedded bladders. Thus, the number of

    equally spaced bladders in the stator determines the step angle of the rotary actuator. In this paper,

    there are 16 bladders in the stators. Thus, in each step (from one subgroup to the next subgroup),

    the rotor turns by 22.5 (360/16).

    Friction between Sliding Components

    There is friction between the stator and the rotor. In order to reduce the friction and smooth the

    rotation, we applied lubricant between the sliding interfaces of the rotor and stator. After

    qualitative experiments with different lubricants, including dish soap, water, silicone oil, and Super

    Lube grease (Synco Chemical Corp.), we concluded that Super Lube grease provided the

    smoothest rotation among these options. Super Lube grease is a nontoxic synthetic lubricant with

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    Figure S1. Millimeter-scale soft lithography of the rotary actuators. (a) A 3D-printed mold

    patterns cast Ecoflex into the thick portion of a stator. After curing the thick portion of the stator,

    it bonds to a thin cover of Ecoflex that is partially cured and adhesive. The tubing then goes into

    the molded bladders of the stator with Sil-Poxy providing an adhesive sealing. (b) A rotor of Type

    1, which was fabricated with ABS by a 3D printer. (c) A rotor of Type II consists of three layers,

    which surrounded an internal stator (not pictured).

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    PTFE, and it does not drip or dry out easily. Moreover, it does not react with or significantly alter

    the texture of the elastomers.

    Uniaxial tensile tests of Ecoflex 00-50 and the Yeoh model

    To relate the static internal pressure to the deformation of the bladders on a stator, and whether

    the corresponding principal strain would exceed the maximum allowable strain of the elastomeric

    materials (Ecoflex 00-50), we performed finite-element simulations of the inflations in ANSYS

    by importing the designed geometry of the stators. Figure S2 shows the engineering stress-strain

    curves of Ecoflex 00-50 from tensile data of five degassed and cured samples. The green curve is

    plotted with the mean values of the five samples, and the blue curve is the stress-strain curve of

    one of these five samples. We considered the material as a hyper-elastic solid and fitted the test

    data to the Yeoh hyper-elastic material model with the module for curve fitting in ANSYS

    Workbench, and then plotted the curves in MATLAB (shown as red curves with marks). The strain

    energy density of nearly-incompressible Yeoh model (also shown by Mosadegh, et al.[14]) has the

    following form:

    (2)

    where , , and is the deformation gradient and and are the

    materials parameters. When we set N = 3 (3rd-ordered Yeoh model), the curve fitting module in

    ANSYS calculated the parameters. For the mean data of the five samples, we have ,

    , , . For the data of the sample used in this study,

    the parameters were , , , .

    U = Ci0I

    1-3( )

    i

    i=1

    N

    +1

    Di

    J -1( )2i

    i=1

    N

    I1= tr dev FF( )

    T

    J = det F( ) F Ci0 Di

    C10

    =1.9010-2

    C20

    = 9.010-4 C30

    = -4.7510-6 D1= D

    2= D

    3= 0

    C10

    =1.7510-2 C20

    = 6.7010-4 C30

    = -2.6510-6 D1= D

    2= D

    3= 0

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    Figure S2. Stress-strain curve and Yeoh model of Ecoflex 00-50. (Top) Uniaxial tensile test (green

    and blue lines without marks) of the elastomer (Ecoflex 00-50) used for simulations of the stators

    inflated by varied static pressures. We fitted the stress-stain curve with the Yeoh model of

    hyperelasticity (red lines with marks). (Bottom) The region of most interest (strain: 0 to 250%)

    since the principal strains were less than 2, according to the simulations.

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    By varying the applied pressure inside of embedded bladders, we witnessed the deformation of

    the bladders and the estimated principal strains shown in Figure S3 b, d. According to the results

    of the simulations, the maximum principal strain of the inflated bladders was 2, which means the

    region of most interest on the curve is the strain of 0-200% indicated in Figure S2 (Bottom).

    Simulated and Experimental Behavior of Inflation with Static Internal Pressures

    We performed 3D finite element analyses for both types of actuator with Mechanical APDL

    R15 (ANSYS Inc., PA). There were 50600 and 58292 elements in the mesh for Type 1 and Type

    2 actuators, respectively. By using a hex-dominant method, we were able to create combined types

    of element, including Hex20, Tet10, Wed15, and Pyr13. In both models, 96% of the total elements

    were comprised of Hex20 elements. There was a fixed-support boundary condition on the inner

    circumferential surface for Type 1 actuator and the outer circumferential surface for Type 2

    actuator, respectively. As for the material model, we adopted a three-parameter Yeoh model for

    the hyperelastic material (Ecoflex 50) in a similar fashion to the work completed by Mosadegh, et

    al.[14] (Figure S2). The load applied on the bladders gradually increased from 0 psi to 9 psi in both

    models.

    Given a specified static internal pressure, the simulations showed the volumes of a subgroup

    of bladders and their corresponding maximum principal strains when pressurized. The maximum

    principal strain (~200%) did not exceed the maximum strain at break of the Ecoflex 00-50 (980%)

    provided by the manufacturer, which means the bladders should not have yielded at an applied

    pressure up to 62.1 kPa. We then validated the FEA qualitatively by comparing the deformation

    of the stators caused by the applied pressures with experimentally acquired images (Figure S3 b,

    d). For the stators consisting of two separate bonded layers, these experiments also demonstrated

    that the adhesion between bonded parts was generally sufficient for pressures associated with

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    Figure S3. (a, c) Experimental tests on the behavior of two types of stators with static internal

    pressures up to 55 kPa (8 psi). (b, d) Simulated behavior of the stators showing the deformation

    and maximum principal strain as a function of internal pressure up to 53 kPa (7.7 psi). The tubing

    that provided pressurized air was hidden behind the gridded background.

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    rotation. That said, failures did occur occasionally at higher pressures than 62.1 kPa due to

    delamination after low-cycle fatiguing.

    In addition, the experimentally acquired images of the stators with one subgroup of bladders

    inflated (Figure S3 a, c) suggested that the simulated results were similar to the experiments at

    corresponding pressures. According to Figure S3 a, c, increasing the static internal pressure

    enlarged the deformation of the bladders. With 41.4 kPa (6 psi) of applied pressure, the

    deformation was not conspicuous. A large expansion of the bladders occurred when the static

    internal pressure was between 48.3 kPa (7 psi) and 55.2 kPa (8 psi). We hypothesize that the large

    deformation of the embedded bladders appeared to force the rotor to turn. The rapid expansions

    with their sudden releases of kinetic energy also facilitated the forcing of the actuators through

    their metastable states. Furthermore, the experiments revealed that the increasingly enlarged

    deformation of the inflated bladders (after passing through the snap-through instability) did not

    show any delamination between the two layers of the stator.

    Programmable Pneumatic System and Speed Control

    We built a programmable pneumatic system (Figure S4) to control and direct pressurized air

    to the inflatable bladders in the rotary actuators. A Type 700 high-flow pressure regulator

    (ControlAir Inc.) regulated the pressure for the inflation, with a pressure gauge (Ashcroft, Inc.)

    monitoring the pressure. To control the timing of the solenoids accurately, we used an FPGA-

    based NI cRIO-9076 with two 8-channel, 1 s high-speed digital output modules NI 9474 from

    National Instruments. The two NI 9474 modules provided 16 digital outputs that could drive up

    to 16 directional control (3-port, 2-position) solenoid valves (VKF Series, SMC Inc.). The software

    is similar and the LabVIEW Virtual Instruments are nearly identical to the pneumatic system

    employed for multi-gait, reconfigurable, and camouflaging robots[6,7]. This system allows us not

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    Figure S4. Pneumatic System. (a) Schematic of the programmable pneumatic system used for soft

    rotary actuators. (b) A photo of the setup carried on a cart. (The pressure regulator and the camera

    are not shown in the photo.)

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    only to control manually each valve through the keyboard but also to import a spreadsheet with

    the information of the control pattern (timing and a specific sequence) of the actuation of the valves

    and achieve the automatic control.

    Once assembled, inflation of the bladders in the stator caused the rotor to turn. To reduce the

    fatigue and large deformation caused by high, alternating pressure-based loads on the bladders,

    and to avoid yielding of the material, we inflated the bladders to a constant volume the minimum

    inflation that made the rotors turn consistently. With the pneumatic system equipped with the

    specified solenoid valves, pressure regulator, and tubing, we changed the flow rate of the

    pressurized air filling in the bladders by regulating the pressure with the regulator. Higher flow

    rates of air induced higher inflation rate of the bladders, thus, a shorter time was needed to

    inflate each subgroup of bladders to the minimum volume required. While this minimum volume

    had an associated static internal pressure, the use of a standard regulator meant that we would

    typically set a pressure on the regulator that might otherwise cause a larger static deformation. In

    this way, the bladders inflated quickly to the minimum volume when a solenoid then cut off the

    airflow before the actual internal pressure in the bladders reached the pressure set on the regulator.

    Estimating Deflation Times with a High-Speed Camera

    As discussed in the main text, increasing pressure lead to a higher rotational speed due to an

    increased inflation time (Equation 1). However, the deflation times remained relatively

    consistent for a given volume of the inflated bladders. To estimate the rotational speed of a rotary

    actuator, we used a modest high-speed camera (Figure S5, Movie S1).

    Instead of sending command signals (Figure 2a) to the valves, we manually open a valve for

    one subgroup of bladders with a keyboard. Starting from a very low pressure, we gradually

    tinflation

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    Figure S5. Estimation of deflation times using high-speed videos. (a) Four groups of deflation of

    a Type 1 stator suggested an average deflation time msec. (b) Four groups of deflation

    of a Type 2 stator gave an average deflation time msec. The red boxes highlight the

    LED light that was about to turn off (Movie S1).

    tdeflation

    =125

    tdeflation

    =190

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    increased the set pressure on the regulator, until the inflation of these bladders just reached the

    minimum that could enable a rotation from this subgroup to an adjacent subgroup. Thus, the

    pressure on the regulator (~ 48.3 kPa (7 psi)) reflected the static internal pressure of these inflated

    bladders. And then, we closed the valve to exhaust the bladders. By observing and counting

    the frames (acquired from high-speed videos) between closing a valve (indicated by the LED lights

    on the valves) and a complete exhaustion of a group of bladders, we were able to calculate a

    deflation time for this group. We did nine groups of experiments for Type 1 actuators and four

    groups of experiments for Type 2 actuators.

    Figure S5 summarizes four groups of experiments for each type of actuator. For each group,

    we picked the frame showing the LED light on a valve as it was about to turn off and the frame

    showing the shape of the stator when completely restored. The speed of the camera we used was

    120 frames/sec. By counting the number of total frames between the two selected frames, we were

    able to calculate the deflation time . We indicated the calculated for each experiment

    in the figure. The average for Type 1 was 125 msec (130 msec with nine experiments), and

    the average for Type 2 was 190 msec.

    Prototyping the winch with Type 1 Actuators

    In order to produce a torque-speed curve for Type 1 actuators, we designed and prototyped

    a winch equipped with Type 1 actuators (Figure S6). A 3D-printed cylindrical shaft coupled two

    Type 1 rotors through a cross-shaped hole. A string attached to the shaft carried external loads for

    characterization (Figure S6 a). Two 3D-printed brackets held the internal portions of the two

    stators, and the tubing on the stators easily passed though the 16 small inlets on each bracket

    tdeflation

    tdeflation

    tdeflation

    tdeflation

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    Figure S6. A prototype of the winch designed for Type 1 actuators. (a) A 3D-printed cylindrical

    shaft coupled with two Type 1 rotors. (b) Two 3D-printed brackets fixed on a wooden stand.

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    (Figure S6 b). A stand made of wooden boards attached with the brackets formed the whole

    frame of the winch. After equipping the stators in the brackets and inserting the shaft into the

    stators, we profiled the torque-speed relationship for a Type 1 actuator, by varying the load

    added onto the winch (see Figure 3c).

    Snap-Through Instability of a Pressurized Bladder

    To provide the direct evidence for the snap-through instability of a bladder with a release of

    kinetic energy, we monitored the real-time internal pressure in the bladder corresponding to the

    change in supplied volume. We used a syringe pump as the air supply, and monitored the internal

    pressure with a pressure sensor (ASDX, Honeywell) and an Arduino-based (Arduino, Mega) data

    acquisition (DAQ) system (Figure S7 a). We observed the inflation of a Type 2 bladder. Pumping

    the bladder with a syringe at four plunger velocities (v = 1, 2, 3, 4 mm/s), we plotted four pressure

    time curves (Figure S7 b). The supplied volume was simply the product of the syringes

    displacement and the internal cross-sectional area (113 mm2 from an inner diameter of 6 mm) of

    the syringe. In Figure S7 c, we noticed there was a significant non-linear change in the pressure-

    volume relationship at a supplied volume of ~1500 mm3. This change in the slope of the curve is

    characteristic of an inflatable elastomeric bladder with a snap-through instability[3437].

    A Winch with a Soft Gripper

    We attached an elastomeric gripper (16.4 grams) with embedded PneuNets[4] on the winch,

    showing that it was possible to combine the rotary actuators with a soft gripper to build a multi-

    functional soft system (Movie S3). With a 3D printer (FlashForge Creator) producing the molds,

    we fabricated a soft robotic gripper from Ecoflex 00-30 (Smooth-On Inc.) for the strain-limiting

    layer, and Mold Star 15 (Smooth-On Inc.) for the inextensible layer. Silicone tubing (independent

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    Figure S7. Verifying the snap-through behavior of an inflated bladder. (a) Experimental setup for

    monitoring the real-time internal pressure of a bladder. (b) Pressure-time curves at four different

    plunger velocities. (d) Pressure- supply volume curves for Type 2 actuators.

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    of the pneumatic system) supplied the compressed air to the gripper through a manually controlled

    syringe.

    The external torque on the winch was Nm, which was estimated with the mass of the

    gripper. Thus, the torque acting on each stator was Nm. Starting at the top (Height 22

    cm), the winch lowered the gripper at a rate of 1.7 cm/sec (Figure S8 a-b). After the shaft turned

    through 11 steps, the gripper reached the position (Height 13 cm) shown in Figure S8 b. When

    the gripper contacted the object, we actuated the gripper to grasp the object by compressing the air

    into the embedded PneuNets through a syringe. Then, the winch started to lift the gripped object.

    The external torque on the winch increased to Nm due to the mass of the object. In order

    for the winch to carry heavier load at the same speed, we increased the pressure slightly on the

    regulator. In Figure S8 c-d (Height: from 0 to 14 cm), the winch raised the gripper and the object

    at a rate of 1.6 cm/sec, turning in around 17 steps.

    A Squishy, Two-Wheeled Vehicle

    The basic design consisted of an elastomeric body, two axles made of Mold Star 30 (Smooth-On

    Inc.), and two Type 2 actuators (Figure S9 a). In case the parallel-wheeled device would lose its

    balance similar to a unicycle and not be able to function as a real vehicle, we attached a

    paperclip to each end of the body as supports (Figure S9 b). Directed by the designed command

    pattern for each wheel, the two-wheeled vehicle demonstrated its ability to navigate around an

    obstacle (Figure S9 c-h, Movie S4). This vehicle presented the capability of going forward, going

    backward, and turning. The set pressure on the regulator was 86.2 kPa (12.5 psi) and the

    corresponding inflation time was approximately 180 msec. Based on Equation 1, the rotational

    3.210-3

    1.610-3

    9.210-3

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    Figure S8. A winch equipped with two Type 1 stators acting in parallel and a soft gripper (16.4

    grams) that could grasp and lift an object weighted 30.7 grams. (a) The gripper started at the top.

    (b) The winch lowered the gripper at a speed of 1.6 cm/sec. (11 steps of the rotor). (c) The gripper

    inflated and grasped the object. (d) The winch raised the gripper and grasped object at a rate of 1.6

    cm/sec (17 steps of the rotor) (Movie S3).

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    Figure S9. (a) Design of the squishy two-wheeled vehicle, (b) The highlighted pieces are two

    paperclips used to balance the vehicle. (c-h) To navigate around an obstacle in its way, the two-

    wheeled soft vehicle moved forward, stopped, moved backward, made a left turn, went forward,

    turned back to straight and then went forward. The arrows indicate the moving directions of the

    vehicle (Movie S4).

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    speed was 10.3 RPM. Considering the diameter of the wheels was 9.2 cm, the speed of the two

    wheeled vehicle was 4.9 cm/s.

    A Squishy, Four-Wheeled Vehicle Working Under Water

    Lacking any electronic components or metal members, the prototype of a four-wheeled vehicle

    was able to work under water (Figure S10, Movie S8). To compensate for the buoyancy of the

    air-filled cavities under water and enhance the traction between the elastomeric wheels (Type 2

    rotors) and the water tank, we loaded rocks on the body of the vehicle. At the pressure of 86.2 kPa

    (12.5 psi) set on the regulator, the vehicle traveled around 37 cm under water at a speed of 3.6

    cm/sec.

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    Figure S10. A squishy, four-wheeled vehicle carried rocks and moved forward under water at a

    speed of 3.6 cm/sec, driven by the pneumatic system. A floating plastic ball/buoy indicated the

    level of the water in tank (Movie S8).

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    Movie S1. Experiments to estimate the deflation time for Type 1 and Type 2 actuators (see Figure

    S5).

    Movie S2. A Type 2 actuator rotating at ~13 RPM.

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    Movie S3. A winch equipped with two Type 1 stators acting in parallel to lower and lift a soft

    gripper (see Figure S8).

    Movie S4. A two-wheeled soft vehicle navigating around an obstacle (see Figure S9).

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    Movie S5. A squishy, four-wheeled vehicle traveling at a speed of 3.7 cm/s (see Figure 4c).

    Movie S6. A drop test on a four-wheeled vehicle (see Figure 4d).

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    Movie S7. A squishy, four-wheeled vehicle navigating a rocky terrain with a puddle in its path

    (see Figure 4e).

    Movie S8. A four-wheeled soft vehicle working under water (see Figure S10).

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    Movie S9. Inflation of a bladder in a Type 2 stator with a syringe pump. The velocity of the plunger

    is 3mm/s. (see Figure S7, a)