Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 PROGRAMMABLE SOFT ACTUATORS FOR DIGITAL AND ANALOG CONTROL RELATED APPLICATIONS [0001] This application claims the benefit of priority to co-pending United States Provisional Application Serial No.63/375,960, filed September 16, 2022, the contents of which is incorporated by reference. COPYRIGHT NOTICE [0002] This patent disclosure may contain material that is subject to copyright protection. The copyright owner has no objection to the facsimile reproduction by anyone of the patent document or the patent disclosure as it appears in the U.S. Patent and Trademark Office patent file or records, but otherwise reserves any and all copyright rights. FIELD OF THE INVENTION [0003] This application relates to non-electronic control of soft robotic actuators. In particular, this application relates to pneumatically-actuated digital and analog control. STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT [0004] This invention was made with government support under 2011754, 2025158, and 2144809 awarded by National Science Foundation (NSF) and under DE-SC0000989 awarded by U.S. Department of Energy (DOE). The government has certain rights in this invention. BACKGROUND OF THE TECHNOLOGY [0005] Soft robotic actuators can be used in applications including navigating uncertain environments, object manipulation, rehabilitation, prosthetics, and non-invasive surgery. However, many soft robotic actuator systems use hard valves and electrical components. Therefore, it is challenging to develop soft pneumatic control systems capable of both digital and analog operation and to develop soft pneumatic control systems that are not limited by speed of actuation, range of pressure, time required for fabrication, or loss of power through pull-down resistors. Integration of control systems into soft devices may enable complex - 1 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 devices and difficult actuation sequences while maintaining the advantages that soft materials provide. SUMMARY [0006] In one aspect, an actuator includes a first inextensible shell; a second inextensible shell; a first flexible membrane, wherein the second inextensible shell is attached to the first inextensible shell via the first flexible membrane and configured to be displaced relative to the first inextensible shell between a first position and a second position; wherein the second inextensible shell is configured to be in a first position when a pressure below a first threshold is applied to the first inextensible shell, and the second inextensible shell is configured to be displaced from the first position to the second position when a pressure above the first threshold is applied to the first inextensible shell and acts on the first flexible membrane; and a flexible fluid channel system operably linked to the second inextensible shell and including a first fluid input channel, a second fluid input channel, and a fluid output channel, wherein the flexible fluid channel system is configured to restrict flow of fluid through the first fluid input channel and allow flow of fluid through the second fluid input channel when the second inextensible shell is in the first position, and to allow flow of fluid through the first fluid input channel and restrict flow of fluid through the second fluid input channel when the second inextensible shell is in the second position. [0007] In some embodiments, the second inextensible shell is configured to return to the first position from the second position when the pressure applied to the first inextensible shell is below a second threshold. [0008] In some embodiments, the actuator includes a piston actuator. [0009] In some embodiments, the actuator includes a mechanically actuated switch. [0010] In some embodiments, the second inextensible shell is disposed within the first inextensible shell. [0011] In some embodiments, the flexible fluid channel system is configured to restrict flow of fluid through the first fluid input channel by kinking the first fluid input channel when the second inextensible shell is in the first position, and to restrict flow of fluid through the second fluid input channel by kinking the second fluid input channel when the second inextensible shell is in the second position. [0012] In some embodiments, displacement of the second inextensible shell from the first position to the second position causes opening of the first fluid input channel and kinking of the second fluid input channel. - 2 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 [0013] In some embodiments, displacement of the second inextensible shell from the second position to the first position causes kinking of the first fluid input channel and opening of the second fluid input channel. [0014] In some embodiments, the flexible fluid channel system is configured in bistable configuration. [0015] In some embodiments, the actuator further includes a restraint configured to apply a force opposing the pressure applied to the first inextensible shell, wherein the restraint increases the first threshold. [0016] In some embodiments, the second inextensible shell is configured to return to the first position from the second position when the pressure applied to the first inextensible shell is below a second threshold, and wherein the restraint increases the second threshold. [0017] In some embodiments, the restraint includes an elastic material. [0018] In some embodiments, the actuator further includes a pressure input fluidically connected to the first inextensible shell. [0019] In some embodiments, the actuator further includes a first pressure source fluidically connected to the first fluid input channel. [0020] In some embodiments, the actuator further includes a second pressure source fluidically connected to the second fluid input channel. [0021] In some embodiments, the actuator further includes a second pressure source fluidically connected to the second fluid input channel, wherein the second pressure source is held at a constant pressure and wherein the first fluid input channel is fluidically connected to atmospheric pressure. [0022] In some embodiments, the actuator further includes a first pressure source fluidically connected to the first fluid input channel, wherein the second fluid input channel is fluidically connected to atmospheric pressure. [0023] In some embodiments, the actuator further includes a first pressure source fluidically connected to the first fluid input channel, wherein the first pressure source is held at a constant value, and a second pressure source fluidically connected to the second fluid input channel. [0024] In some embodiments, the actuator further includes a second pressure source fluidically connected to the second fluid input channel, wherein the first fluid input channel is fluidically connected to atmospheric pressure. - 3 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 [0025] In some embodiments, the actuator further includes a first pressure source fluidically connected to the first fluid input channel, and a second pressure source fluidically connected to the second fluid input channel, wherein the second pressure source is held at a constant pressure. [0026] In some embodiments, the actuator further includes a first pressure source fluidically connected to the first fluid input channel, and a second pressure source fluidically connected to the second fluid input channel. [0027] In some embodiments, the actuator further includes a third inextensible shell, wherein the second inextensible shell is attached to the first inextensible shell via the first flexible membrane at a first end and attached to the third inextensible shell via a second flexible membrane at a second end, wherein the second inextensible shell is configured to be displaced relative to the third inextensible shell when a pressure above a third threshold is applied to the third inextensible shell and acts on the second flexible membrane, wherein the fluid output channel is fluidically connected to the third inextensible shell and a second fluid source is fluidically connected to the third inextensible shell. [0028] In some embodiments, the second inextensible shell is configured to return to the first position from the second position when the pressure applied to the third inextensible shell is below a fourth threshold. [0029] In one aspect, a method of actuating the actuator includes applying a pressure to the first inextensible shell; displacing the second inextensible shell relative to the first inextensible shell between the first position and the second position. [0030] In some embodiments, the method further includes applying the pressure above the first threshold to the first inextensible shell and displacing the second inextensible shell from the first position to the second position. [0031] In some embodiments, displacing the second inextensible shell from the first position to the second position causes opening of the first fluid input channel and kinking of the second fluid input channel. [0032] In some embodiments, the method includes displacing the second inextensible shell from the second position to the first position. [0033] In some embodiments, the method includes applying the pressure below a second threshold to the first inextensible shell and displacing the second inextensible shell from the second position to the first position. - 4 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 [0034] In some embodiments, displacing the second inextensible shell from the second position to the first position causes kinking of the first fluid input channel and opening of the second fluid input channel. [0035] In some embodiments, a restraint configured to apply a force opposing the pressure applied to the first inextensible shell increases the first threshold. [0036] In some embodiments, the method further includes applying the pressure below a second threshold to the first inextensible shell and displacing the second inextensible shell from the second position to the first position, wherein the restraint increases the second threshold. [0037] In some embodiments, the restraint includes an elastic material. [0038] In one aspect, actuator includes a first inextensible shell; a second inextensible shell; a flexible membrane, wherein the second inextensible shell is attached to the first inextensible shell via the flexible membrane and configured to be displaced relative to the first inextensible shell between a first position and a second position; wherein the second inextensible shell is configured to be in a first position when a pressure below a first threshold is applied to the first inextensible shell, the second inextensible shell is configured to be displaced from the first position to the second position when a pressure above the first threshold is applied to the first inextensible shell and acts on the flexible membrane; and a flexible fluid channel system operably linked to the second inextensible shell and including a fluid input channel and a fluid output channel; wherein the flexible fluid channel system is configured to allow flow of fluid through the fluid input channel when the second inextensible shell is in the first position, and to restrict flow of fluid through the fluid input channel when the second inextensible shell is in the second position, and wherein the fluid output channel is fluidically connected to the first inextensible shell. [0039] In some embodiments, the second inextensible shell is configured to return to the first position from the second position when the pressure applied to the first inextensible shell is below a second threshold. [0040] In some embodiments, the second inextensible shell is disposed within the first inextensible shell. [0041] In some embodiments, the actuator further includes a fluid source fluidically connected to the fluid input channel. [0042] In some embodiments, the fluid output channel is configured to apply pressure to the first inextensible shell and act on the flexible membrane. - 5 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 [0043] In some embodiments, the actuator further includes a restraint configured to apply a force opposing the pressure applied to the first inextensible shell, wherein the restraint increases the first threshold. [0044] In some embodiments, the second inextensible shell is configured to return to the first position from the second position when the pressure applied to the first inextensible shell is below a second threshold; and wherein the restraint increases the second threshold. [0045] In some embodiments, the restraint is inextensible. [0046] In some embodiments, the restraint is elastic. [0047] In some embodiments, a force applied by the restraint is configured to be varied to vary the pressure applied to the first inextensible shell. [0048] In some embodiments, wherein a force applied by the restraint is configured to be varied to vary an output pressure at the fluid output channel. [0049] In one aspect, method of actuating the actuator includes applying a pressure to the fluid input channel; displacing the second inextensible shell relative to the first inextensible shell between the first position and the second position. [0050] In some embodiments, displacing the second inextensible shell from the first position to the second position causes kinking of the fluid input channel. [0051] In some embodiments, the method includes displacing the second inextensible shell from the second position to the first position. [0052] In some embodiments, displacing the second inextensible shell from the second position to the first position causes opening of the fluid input channel. [0053] In some embodiments, the method further includes applying a force to a restraint attached to the first inextensible shell and the second inextensible shell. [0054] In some embodiments, applying the force to the restraint modifies an output pressure through the fluid output channel. [0055] In one aspect, a system includes at least one of a valve and a regulator, wherein each valve includes a first inextensible shell; a second inextensible shell; a first flexible membrane; wherein the second inextensible shell is attached to the first inextensible shell via the first flexible membrane and configured to be displaced relative to the first inextensible shell the between a first position and a second position; wherein the second inextensible shell is configured to be in a first position when a pressure below a first threshold is applied to the first inextensible shell, the second inextensible shell is configured to be displaced from the first position to the second position when a pressure above the first threshold is applied to the - 6 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 first inextensible shell and acts on the first flexible membrane; and a first flexible fluid channel system operably linked to the second inextensible shell and including a first fluid input channel, a second fluid input channel, and a first fluid output channel, wherein the first flexible fluid channel system is configured to restrict flow of fluid through the first fluid input channel and allow flow of fluid through the second fluid input channel when the second inextensible shell is in the first position, and to allow flow of fluid through the first fluid input channel and restrict flow of fluid through the second fluid input channel when the second inextensible shell is in the second position; and wherein each regulator includes a first inextensible shell; a second inextensible shell; a flexible membrane, wherein the second inextensible shell is attached to the first inextensible shell via the flexible membrane and configured to be displaced relative to the first inextensible shell between a first position and a second position; wherein the second inextensible shell is configured to be in a first position when a pressure below a first threshold is applied to the first inextensible shell, the second inextensible shell is configured to be displaced from the first position to the second position when a pressure above the first threshold is applied to the first inextensible shell and acts on the flexible membrane; and a flexible fluid channel system operably linked to the second inextensible shell and including a fluid input channel and a fluid output channel, wherein the flexible fluid channel system is configured to allow flow of fluid through the fluid input channel when the second inextensible shell is in the first position, and to restrict flow of fluid through the fluid input channel when the second inextensible shell is in the second position, wherein the fluid output channel is fluidically connected to the first inextensible shell. [0056] In some embodiments, the second inextensible shell of the valve is configured to return to the first position from the second position when the pressure applied to the first inextensible shell of the valve is below a second threshold. [0057] In some embodiments, the second inextensible shell of the regulator is configured to return to the first position from the second position when the pressure applied to the first inextensible shell of the regulator is below the second threshold. [0058] In some embodiments, the system further includes one or more soft actuators, each soft actuator including at least one fluidic chamber, wherein supply of pressure to the one or more soft actuators is controlled by the valve, the regulator, or a combination thereof. [0059] In some embodiments, the system includes two or more valves. [0060] In some embodiments, the system includes two or more regulators. - 7 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 [0061] In some embodiments, the system includes at least one valve and at least one regulator. [0062] In some embodiments, the system includes at least one valve configured in a logic circuit including at least valve configured as at least one of a NOT gate, an AND gate, an OR gate, an SR latch, an INHIB gate, an IMPLY gate, and an XOR* gate. [0063] In some embodiments, each regulator includes a restraint configured to apply a force opposing the pressure applied to first inextensible shell of the regulator, wherein the restraint increases the first threshold. [0064] In some embodiments, the restraint is inextensible. [0065] In some embodiments, the restraint is extensible. [0066] In some embodiments, a force applied by the restraint is configured to be varied to vary the pressure applied to the first inextensible shell of the regulator. [0067] In some embodiments, the fluid output channel of each regulator is fluidically connected to a soft actuator including at least one fluidic chamber and wherein each regulator controls supply of pressure to the soft actuator. [0068] In some embodiments, the system includes a control glove, the control glove including one or more inextensible restraints, each inextensible restraint attached to the first inextensible shell of one of the regulators and configured to be attached to a finger of a user, wherein a force applied by each restraint is configured to be varied to vary the pressure applied to the first inextensible shell of the corresponding regulator. [0069] In some embodiments, the system includes at least three valves arranged in a ring and forming a ring oscillator system. [0070] In some embodiments, the first fluid input of each valve is fluidically connected to atmospheric pressure; the second fluid input channel of each valve is fluidically connected to a common fluid source; and the fluid output channel of each valve is fluidically connected to the first inextensible shell of the subsequent valve in the ring oscillator system. [0071] In some embodiments, the system further includes at least one regulator upstream of the ring oscillator system. [0072] In some embodiments, the ring oscillator system is fluidically connected to a soft actuator including at least one fluidic chamber and the ring oscillator system controls supply of pressure to the soft actuator. [0073] In some embodiments, the system further includes a counter circuit including a buffer, a pneumatic capacitor, and an SR latch, wherein the SR latch is configured to switch - 8 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 when the ring oscillator system has completed a threshold number of cycles, wherein the regulator includes a restraint attached to the first inextensible shell of the regulator and to the second inextensible shell of the regulator. [0074] In some embodiments, the threshold number of cycles is set by the regulator by selecting a force applied by the restraint. [0075] In some embodiments, system further includes an untethered soft robot. [0076] In some embodiments, the untethered soft robot is capable of locomotion. [0077] In some embodiments, the system includes a counter circuit and a ring oscillator system. [0078] In some embodiments, a distance traveled by the untethered soft robot is set by the counter circuit. [0079] In some embodiments, wherein the untethered soft robot includes a soft robotic actuator configured to grip objects. [0080] In some embodiments, the untethered soft robot receives a single pressure input. [0081] In some embodiments, the fluid output channel of each valve is fluidically connected to a soft actuator including at least on fluidic chamber and wherein each valve controls supply of pressure controls the soft actuator. [0082] In some embodiments, the system further includes a cushion matrix including a plurality of soft actuators. [0083] In some embodiments, the system is connected to a single pressure source. [0084] In some embodiments, the system includes one or more switches. [0085] In some embodiments, one or more of the valves are configured as OR gates to control which of the plurality of soft actuators are supplied with a pressure. [0086] In some embodiments, the system includes a ring oscillator system. [0087] In some embodiments, the system is configured to supply pressure to a portion of the soft actuators arranged in a column. [0088] In some embodiments, the system is configured to supply pressure to a portion of the soft actuators arranged in a row. [0089] In some embodiments, the system is configured to sequentially supply pressure to each soft actuator. [0090] In some embodiments, the system includes a control glove, the control glove including one or more inextensible restraints, each inextensible restraint attached to the first inextensible shell of one of the regulators and configured to be attached to a finger of a user, - 9 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 wherein a force applied by each restraint is configured to be varied to vary the pressure applied to the first inextensible shell of the corresponding regulator and to control supply of pressure to each soft actuator. [0091] Any one of the embodiments disclosed herein may be properly combined with any other embodiment disclosed herein. The combination of any one of the embodiments disclosed herein is expressly contemplated. BRIEF DESCRIPTION OF THE DRAWINGS [0092] The objects and advantages will be apparent upon consideration of the following detailed description, taken in conjunction with the accompanying drawings, in which like reference characters refer to like parts throughout, and in which: [0093] FIG.1A shows a schematic of a soft piston actuator in an actuated state, according to certain embodiments. [0094] FIG.1B shows a schematic of a soft piston actuator in a partially actuated state, according to certain embodiments. [0095] FIG.1C shows a schematic of a soft piston actuator in an unactuated state, according to certain embodiments. [0096] FIG.1D shows a photograph of a soft piston actuator in an actuated state, according to certain embodiments. [0097] FIG.1E shows a photograph of a soft piston actuator in an unactuated state, according to certain embodiments. [0098] FIG.2A shows a schematic of a mechanically actuated switch in an unactuated state, according to certain embodiments. [0099] FIG.2B shows a schematic of a mechanically actuated switch in an actuated state, according to certain embodiments. [0100] FIG.2C shows a photograph of a mechanically actuated switch in an unactuated state, according to certain embodiments. [0101] FIG.2D shows a photograph of a mechanically actuated switch in an actuated state, according to certain embodiments. [0102] FIG.3A shows a schematic of a valve in an unactuated state, according to certain embodiments. [0103] FIG.3B shows a schematic of a valve in an actuated state, according to certain embodiments. - 10 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 [0104] Fig.3C shows a photograph of a valve, according to certain embodiments. [0105] FIG.3D shows a free body diagram of forces acting on a piston during snap- through, according to certain embodiments. [0106] FIG.3E shows a free body diagram of forces acting on a piston during snap-back, according to certain embodiments. [0107] FIG.4A shows a diagram of a NOT gate, according to certain embodiments. [0108] FIG.4B shows a truth table for a NOT gate, according to certain embodiments. [0109] FIG.4C shows a schematic of a valve configured as a NOT gate, according to certain embodiments. [0110] FIG.4D shows the measured input and output pressures of a valve configured as a NOT gate, according to certain embodiments. [0111] FIG.5A shows a diagram of an AND gate, according to certain embodiments. [0112] FIG.5B shows a truth table for an AND gate, according to certain embodiments. [0113] FIG.5C shows a schematic of a valve configured as an AND gate, according to certain embodiments. [0114] FIG.5D shows the measured input and output pressures of a valve configured as an AND gate, according to certain embodiments. [0115] FIG.6A shows a diagram of an OR gate, according to certain embodiments. [0116] FIG.6B shows a truth table for an OR gate, according to certain embodiments. [0117] FIG.6C shows a schematic of a valve configured as an OR gate, according to certain embodiments. [0118] FIG.6D shows the measured input and output pressures of a valve configured as an OR gate, according to certain embodiments. [0119] FIG.7A shows a truth table for an SR latch, according to certain embodiments. [0120] FIG.7B shows a schematic of a valve configured as an SR latch, according to certain embodiments. [0121] FIG.7C shows the measured input and output pressures of a valve configured as SR latch, according to certain embodiments. [0122] FIG.8A shows a diagram of an INHIB gate, according to certain embodiments. [0123] FIG.8B shows a truth table for an INHIB gate, according to certain embodiments. [0124] FIG.8C shows a schematic of a valve configured as an INHIB gate, according to certain embodiments. - 11 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 [0125] FIG.8D shows the measured input and output pressures of a valve configured as an INHIB gate, according to certain embodiments. [0126] FIG.9A shows a diagram of an IMPLY gate, according to certain embodiments. [0127] FIG.9B shows a truth table for an IMPLY gate, according to certain embodiments. [0128] FIG.9C shows a schematic of a valve configured as an IMPLY gate, according to certain embodiments. [0129] FIG.9D shows the measured input and output pressures of a valve configured as an IMPLY gate, according to certain embodiments. [0130] FIG.10A shows a diagram of an XOR* gate, according to certain embodiments. [0131] FIG.10B shows a truth table for an XOR* gate, according to certain embodiments. [0132] FIG.10C shows a schematic of a valve configured as an XOR* gate, according to certain embodiments. [0133] FIG.10D shows a diagram of a circuit logically equivalent to an XOR* gate, according to certain embodiments. [0134] FIG.11A shows a diagram of an XOR gate formed by a combination of AND, OR, and NOT gates, with possible gate reductions outlined, according to certain embodiments. [0135] FIG.11B shows a diagram of an XOR gate formed by a combination of AND, OR, and NOT gates, with possible gate reductions outlined, according to certain embodiments. [0136] FIG.11C shows a diagram of an XOR gate with a reduced number of gates by use of INHIB and IMPLY gates, according to certain embodiments. [0137] FIG.11D shows a diagram of XOR gate with a reduced number of gates by use of a three-input gate XOR*, according to certain embodiments. [0138] FIG.12A shows a schematic of a regulator in a first, low output pressure state, with flow allowed, according to certain embodiments. [0139] FIG.12B shows a schematic of a regulator at second, high output pressure state, with flow blocked, according to certain embodiments. [0140] FIG.12C shows a graph of regulator output pressure as a function of force applied by a restraint, according to certain embodiments. [0141] FIG.13A shows a diagram of a ring oscillator, according to certain embodiments. - 12 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 [0142] FIG.13B shows a schematic of a ring oscillator formed by three valves and a regulator, according to certain embodiments. [0143] FIG.13C shows a graph of output pressure in a ring oscillator over time, according to certain embodiments. [0144] FIG.13D shows a graph of oscillation period as a function of supply pressure in a ring oscillator, according to certain embodiments. [0145] FIG.14A shows a schematic of a side view of alignment of TPU and parchment paper during fabrication of a soft piston, according to certain embodiments. [0146] FIG.14B shows a schematic of a top view of alignment of TPU and parchment paper during fabrication of a soft piston, according to certain embodiments. [0147] FIG.14C shows a schematic of a flexible membrane or diaphragm after heat pressing during fabrication of a soft piston, according to certain embodiments. [0148] FIG.14D shows a schematic of alignment of a first inextensible shell, an input tube, and heat shrink tubing during fabrication of a soft piston, according to certain embodiments. [0149] FIG.14E shows a schematic of a soft piston chamber after applying heat to attach a first inextensible shell and an input tube during fabrication of a soft piston, according to certain embodiments. [0150] FIG.14F shows a schematic of a completed soft piston, according to certain embodiments. [0151] FIG.15A shows a schematic of an inner inextensible shell of a mechanical switch during fabrication of a mechanical switch, according to certain embodiments. [0152] FIG.15B shows a schematic of an inner inextensible shell of a mechanical switch with a flexible fluid channel system attached during fabrication of a mechanical switch, according to certain embodiments. [0153] FIG.15C shows a schematic of an outer inextensible shell of a mechanical switch during fabrication of a mechanical switch, according to certain embodiments. [0154] FIG.15D shows a schematic of a completed mechanical switch, according to certain embodiments. [0155] FIG.16A shows a schematic of soft piston and a mechanical switch before attachment during fabrication of a valve, according to certain embodiments. [0156] FIG.16B shows a schematic of a completed valve formed by a soft piston and a mechanical switch, according to certain embodiments. - 13 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 [0157] FIG.17A shows a micro-computed tomography image of a bistable valve configured as a pneumatic switch, according to certain embodiments. [0158] FIG.17B shows force-distance curves of different components of a bistable, pneumatic valve, according to certain embodiments. [0159] FIG.17C shows snap-through and snap-back pressures of a bistable, pneumatic valve with different elastic stiffness, according to certain embodiments. [0160] FIG.17D shows experimentally measured valve snap-through and snap-back pressures for a bistable, pneumatic valve (solid black diamonds) compared with predictions (lines), according to certain embodiments. [0161] FIG.18 shows characterization of snap-through of pneumatic valves, according to certain embodiments. [0162] FIG.19A shows a schematic of a valve with an elastic restraint, according to certain embodiments. [0163] FIG.19B shows force-displacement curves for 1, 2, 3, or 4 elastic bands, according to certain embodiments. [0164] FIG.19C shows modeled force-displacement curves for valves with 1, 2, 3, or 4 elastic bands, according to certain embodiments. [0165] FIG.20A shows a graph of measured force-displacement curves for components of a valve with one elastic and 0 kPa input pressure, according to certain embodiments. [0166] FIG.20B shows a graph of measured force-displacement curves for components of a valve with four elastics and 80 kPa input pressure, according to certain embodiments. [0167] FIG.20C shows a graph of measured force-displacement curves for the isolated piston of valves at different input pressures, according to certain embodiments. [0168] FIG.20D shows a graph of measured force-displacement curves for the valves at different input pressures, according to certain embodiments. [0169] FIG.21A shows a schematic of a soft piston and a mechanical switch before attachment during fabrication of a regulator, according to certain embodiments. [0170] FIG.21B shows a schematic of a completed regulator formed by a soft piston and a mechanical switch, according to certain embodiments. [0171] FIG.22A shows a schematic of two soft pistons and a mechanical switch before attachment during fabrication of an SR latch, according to certain embodiments. [0172] FIG.22B shows a schematic of a completed SR-latch formed by two soft pistons and a mechanical switch, according to certain embodiments. - 14 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 [0173] FIG.23A shows a top view of a five-finger control glove and soft robotic hand with one soft actuator for each finger, according to certain embodiments. [0174] FIG.23B shows a pressure regulator used as an analog human input with an analog control glove to control movement of fingers of a soft robotic hand, according to certain embodiments. [0175] FIG.23C shows a robotic hand being held at various degrees of actuations (between fully actuated and unactuated states) with functionality, according to certain embodiments. [0176] FIG.23D shows the output pressure as a function of time during operation of an actuator of a soft robotic hand, according to certain embodiments. [0177] FIG.23E shows angles of a finger actuator of a soft robotic hand corresponding to a user’s finger angle, according to certain embodiments. [0178] FIG.24A shows a photograph of the hook-and-loop fasteners used to secure a string restraint of a control glove to a finger, according to certain embodiments. [0179] FIG.24B shows a top view of the control glove on a human hand, according to certain embodiments. [0180] FIG.25 shows a photograph of a soft robotic hand, according to certain embodiments. [0181] FIG.26A shows photographs comparing the angle of a robotic finger to the angle of a human finger using a control glove, according to certain embodiments. [0182] FIG.26B shows a graph of robotic finger angle and human finger angle, according to certain embodiments. [0183] FIG.27A shows a diagram of a counter circuit to count the number of oscillations in a ring oscillator, according to certain embodiments. [0184] FIG.27B shows a graph of the number of cycles before a counter switches state at different volumes and input pressures, according to certain embodiments. [0185] FIG.28A shows a diagram of a cascading counter comprising multiple JK-latches, according to certain embodiments. [0186] FIG.28B shows a diagram of a JK-latch comprising gates constructed with valves, according to certain embodiments. [0187] FIG.29A shows an untethered robot, according to certain embodiments. [0188] FIG.29B shows an untethered robot retrieving an object at a distance based on a regulator setting of 38 kPa, according to certain embodiments. - 15 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 [0189] FIG.29C shows an untethered robot retrieving an object at a distance based on a regulator setting of 24 kPa, according to certain embodiments. [0190] FIG.30A shows a diagram of a control circuit of an untethered robot, according to certain embodiments. [0191] FIG.30B shows key modes for forward displacement during operation of an untethered robot, according to certain embodiments. [0192] FIG.30C shows key modes for reverse displacement during operation of an untethered robot, according to certain embodiments. [0193] FIG.31A shows a logical circuit for a cushion matrix, according to certain embodiments. [0194] FIGs.31B-31C show schematics of activation of columns of cushions in a cushion matrix for a rolling operation, according to certain embodiments. [0195] FIG.31D shows a photograph of activation of a column of cushions in a pneumatic cushion matrix for a rolling operation, according to certain embodiments. [0196] FIG.31E shows experimental pressure measurements of activation of a column of cushions in a cushion matrix for a rolling operation, according to certain embodiments. [0197] FIGs.31F-31G show schematics of activation of rows of cushions in a cushion matrix for a lifting operation, according to certain embodiments. [0198] FIG.31H shows a photograph of activation of a row of cushions in a cushion matrix for a lifting operation, according to certain embodiments. [0199] FIG.31I shows experimental pressure measurements of activation of a row of cushions in a cushion matrix for a lifting operation, according to certain embodiments. [0200] FIG.31J shows experimental pressure measurements of activation of cushions during an oscillation function, according to certain embodiments. [0201] FIGs.31K-31N show photographs of analog control of a pneumatic cushion using a control glove, according to certain embodiments. DETAILED DESCRIPTION [0202] In some embodiments, this application describes actuators and systems that form pneumatically-actuated logic circuits for analog and digital control. In some embodiments, the actuators and systems disclosed herein are components of soft robotic control systems. In some embodiments, the actuators and systems disclosed herein can be further assembled into a higher-complexity control system. In some embodiments, the systems disclosed herein use - 16 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 nonlinear mechanics of structures comprising soft materials—for example, by leveraging membrane inversion and fluid channel (e.g., tube) kinking—to form modular soft components, including a piston actuator and a bistable pneumatic switch. These components may be combined to create valves and regulators capable of analog pressure regulation, simplified digital logic, controlled oscillation, nonvolatile memory storage, linear actuation, and interfacing with human users in both digital and analog formats. In some embodiments, these components can be combined to form systems, including (i) a wearable glove capable of analog control of a soft artificial robotic hand based on input from a human users’ fingers; (ii) a human-controlled cushion matrix designed for use in medical care; and (iii) an untethered robot which travels a distance dynamically programmed at the time of operation to retrieve an object. In some embodiments, the actuators and systems disclosed herein can provide complementary digital and analog control of soft robots using a unified valve architecture. [0203] In some embodiments, the soft pneumatic actuators disclosed herein allow complex sequences of actuation and precise control of forces in both digital and analog formats. Non-limiting examples of these actuators include valves, regulators, pistons, and switches. In some embodiments, these actuators leverage the nonlinear mechanics of soft materials. In some embodiments, based on this design, systems can be created that are capable of analog pressure regulation, linear actuation, digital logic, pressure amplification, controlled oscillation, nonvolatile memory storage, and interfacing with human users. In some embodiments, the programmability of these actuators (e.g., with tunable pressures of actuation and output) simplifies the operation of multi-pressure systems, enabling untethering of robotics and creating intuitive, human-friendly devices. [0204] In some embodiments, the actuators disclosed herein can be assembled into logic circuits to control actuation of pneumatic soft robotic actuators. In some embodiments, soft robotic actuators can be actuated by applying pressure to the soft robotic actuators, e.g., by supplying a fluid to the soft robotic actuators to inflate or by removing a fluid from the soft robotic actuators to deflate. In some embodiments, the pressure is a positive pressure (e.g., pressure higher than external pressure), while in other embodiments, the pressure is a negative pressure (e.g., pressure lower than external pressure). In some embodiments, pressure is applied by supplying a fluid. In some embodiments, pressure is applied by removing a fluid or applying a vacuum. The fluid can be a liquid or a gas. In some embodiments, the actuators disclosed herein can control the supply of pressure or fluid to a - 17 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 soft robotic actuator. In some embodiments, the actuators disclosed herein can control the removal of pressure or fluid from a soft robotic actuator. [0205] The operation pressure of the actuators disclosed herein can be tuned by altering the materials or geometry of components of the actuators. In some embodiments, the operating pressure can be tuned by altering the stiffness of the flexible channels (e.g., to alter the kinking force). In some embodiments the operating pressure can be tuned by altering the stiffness of a restraint (e.g., to alter the force exerted by the restraint). In some embodiments, the operating pressure can be tuned by altering friction (e.g., to alter the force of friction). In some embodiments, the operating pressure can be tuned by altering the size or geometry of the piston head (e.g., to alter the piston pressure because pressure is equal to force divided by area). In some embodiments, actuators disclosed herein can operate at pressures between about 20 and 170 kPa. In some embodiments, the valves disclosed herein can operate at about 20 kPa, 30 kPa, 40 kPa, 50 kPa, 60 kPa, 70 kPa, 80 kPa, 90 kPa, 100 kPa, 110 kPa, 120 kPa, 130 kPa, 150 kPa, 160 kPa, 170 kPa, 180 kPa, 190 kPa, 200 kPa, 250 kPa, 300 kPa, 350 kPa, 400 kPa, 450 kPa, 500 kPa, 550 kPa, 600 kPa, or at any pressure in any range bounded by any two values disclosed herein. Throughout the specification, pressure is expressed as the net value relative to atmospheric pressure. For example, 0 kPa corresponds to 1atm of pressure, and 80 kPa corresponds to 80 kPa above 1 atm. [0206] Many soft device control systems still rely on hard valves and electrical control components to enable complex functions and difficult actuation sequences and therefore lose certain advantages that soft materials provide. Fluid logic implemented through microfluidics are limited to small force scales by slow flow rates (0.1 to 10 mL/min), and these microfluidic systems lose energy at steady state due to pull-down resistors. Many soft valves are limited by pressure loss introduced by pull-down resistors, low operational pressure (which in turn limits oscillation frequency), long fabrication times, high costs, and the lack of ability to function in cascaded logic circuits. [0207] Without wishing to be limited to any theory, it was surprisingly discovered that developing the general pressure regulator for soft devices disclosed herein enables at least four significant advances in soft control capabilities. First is the development of combined analog and digital control circuits. Digital circuits, which can provide a binary output based on a binary input, can achieve high levels of computational complexity at the cost of space and time but provide a binary response: on (1) or off (0). Analog circuits, which can provide analog output based on analog input, realize simple functions like actuation delays with few - 18 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 components but are not generalizable to a range of desired behaviors. The addition of an analog regulator to a suite of digital valves enables creation of highly functional, complex and generalizable combined digital-analog circuits, minimizing the time, space and pressure requirements of a control system. Second is the simplification of control systems. Multiple input lines are often necessary when logical elements and actuators of a soft robotic system require different operating pressures. A pressure regulator can simplify design by enabling multiple, separate output pressures from the same input line. Third, a pressure regulator can be used for untethered soft robots. Fully untethered soft pneumatic robots must carry their pressure supply. Reaching acceptable energy densities can be achieved through chemical reactions or pressurized fluids, but for pressurized fluids, pressure regulators composed of rigid materials are generally required to lower the supply pressure. In contrast, a compliant pressure regulator can replace this hard component. Fourth, soft analog pressure regulators can be used to obtain variable or analog human input (e.g., by manually adjusting a control knob to change the behavior or state of a soft robot). In this way, the development of a lightweight regulator composed of compliant materials would allow safer and easier interaction with users, especially in wearables. [0208] In some embodiments, the actuators and systems disclosed herein include modular components that take advantage of nonlinear mechanics of soft materials, including membrane inversions and fluid channel (e.g., tube kinking). These components may include a piston actuator and a mechanically actuated switch, which can be assembled into actuators, for example, valves, regulators, and SR latches. In some embodiments, these actuators can be assembled into systems with digital logic control, analog pressure control, nonvolatile memory storage, and interpretation of human and environmental inputs in both digital and analog formats. In some embodiments, two examples of useful combined digital-analog circuits described herein include a counter circuit and a ring oscillator with the built-in capability for an adjustable oscillation frequency. [0209] Without wishing to be limited to any theory, it was surprisingly discovered that the actuators and systems disclosed in certain embodiments herein provide the advantage of the following unique combination of achievements: (i) introducing analog control capabilities; (ii) implementing CMOS-type logic to avoid energy loss at a steady state, e.g., by simultaneously opening one channel and closing another channel to allow switching between two different pressure sources without use of pull-down resistors; (iii) high pressure tolerances, e.g., by increasing maximum operating pressure to more than 165 kPa for digital - 19 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 CMOS-type valves; (iv) high oscillation frequencies, e.g., by increasing operating pressure to consequently decrease the response time of the valves, corresponding to a 10x increase in ring oscillation frequency (from 0.3 Hz to 3 Hz when considering CMOS-type valves); (v) fast, inexpensive fabrication, e.g., by reducing the fabrication time to less than 12 minutes and reducing the material cost to less than $0.40 USD per valve; (vi) easy repair and reconfiguration of valves, e.g., due to fast, inexpensive fabrication and modular design; (vii) non-volatile memory, e.g., via set-reset (SR) latches; (viii) programmable snap-through and switching of valves; and (iv) reducing the number of physical valves needed to build digital logic circuits by introducing both previously unreported pneumatic logic gates (i.e., INHIB, IMPLY, and a 3-input gate) and combined digital and analog control circuits. [0210] In some embodiments, in addition to improved logic gate functionality (greater than 165 kPa maximum pressure and ten times faster oscillation frequencies compared to previous CMOS-type valves), the actuators disclosed herein bring several higher-level advantages relative to other approaches. In some embodiments, the modularity of these actuators valves means that a wide range of functions (digital and analog control, nonvolatile memory, linear actuation, and human operated pneumatic switches) can all be achieved using largely the same soft device design and fabrication process. In some embodiments, this simplicity leads to faster prototyping times—existing components can easily be recombined or reconfigured into new devices—but it is also an important step towards enabling large- scale production and adoption of complex soft devices. [0211] Beyond the aforementioned advantages, analog pressure regulation is a unique benefit of actuators and systems disclosed in certain embodiments herein. For example, analog pressure regulation expands the capabilities of soft devices by reducing the number of necessary input lines to multi-pressure systems, helping untether soft robots, creating intuitive human-robot interactions, and enabling combined digital-analog circuits. The programmability of these actuators and systems represents another important advantage. In some embodiments, at a component or actuator level, the snap-through pressures of a valve and output pressure of an analog regulator can easily be set. In some embodiments, at a system level, the kinking and membrane inversion mechanisms disclosed herein are general concepts that can be transferred to other materials (e.g., plastics and fabrics) and scales depending on application needs. [0212] The actuators and systems disclosed herein address critical limitations in actuation of soft robotic systems. These actuators and systems introduce a valve architecture capable - 20 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 of simultaneous digital and analog control. Implementation of CMOS-type logic avoids energy loss at a steady state, and the fabrication time and cost have been reduced to less than 12 minutes and less than $0.40 USD, respectively. In some embodiments, compared to other valves capable of CMOS-type logic, the maximum operating pressure is increased by more than 300%; these changes consequently improve the response time of the valves, which corresponds to a 10x increase in oscillation frequency when configured as a ring oscillator. In some embodiments, the approach disclosed herein also allows valves to be repaired and reconfigured easily and reduces the number of valves needed to build digital logic circuits by introducing previously unreported pneumatic logic gates. In some embodiments, these features lead to new capabilities of soft robots and devices by expanding the suite of onboard control functionality without the need for tethering or incorporation of hard, electronic controllers. I. Modular Components [0213] In some embodiments, disclosed herein are soft actuators that can be assembled using modular components made of soft materials (e.g., polymeric materials). Exemplary components include piston actuators and mechanical actuators. In some embodiments, these components can be combined to form actuators. In some embodiments, piston actuators and mechanical actuators are combined to form valves. In some embodiments, piston actuators and mechanical actuators are combined to form regulators. A. Piston Actuator [0214] In some embodiments, the systems disclosed herein include a piston-like actuator. In some embodiments, a piston actuator includes a rolling diaphragm or flexible membrane that takes advantage of the nonlinear mechanics of membrane inversion. [0215] In some embodiments, an actuator includes a piston actuator that includes a first inextensible shell, a second inextensible shell, and a flexible membrane, wherein the second inextensible shell is attached to the first inextensible shell via the flexible membrane and configured to be displaced relative to the first inextensible shell between a first position and a second position; wherein the second inextensible shell is configured to be in the second position when a pressure above a first threshold is applied to the first inextensible shell and acts on the first flexible membrane and the second inextensible shell is configured to be in the first position when the pressure applied to the first inextensible shell is below the first threshold. In some embodiments, the first inextensible shell receives an input pressure and - 21 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 acts as a piston chamber. In some embodiments, the second inextensible shell acts as a piston. In some embodiments, the flexible membrane forms a surface at one end of the second inextensible shell and thereby acts as a piston head, such that the second inextensible shell is displaced relative to the first inextensible shell when a pressure is applied to the first inextensible shell, e.g., to the interior of first inextensible shell, and acts on the flexible membrane. I [0216] FIGs.1A-1E show an exemplary piston actuator 101. As shown in FIG.1A, a piston actuator includes a first inextensible shell 111, a second inextensible shell 112, and a flexible membrane 113. In the example shown in FIGs.1A-1E, the first inextensible shell and the second inextensible shell are tubes. However, the first and second inextensible shells can include any shape or configuration where the flexible membrane forms a surface between the first inextensible shell and the second inextensible shell, and so that first inextensible shell forms a piston chamber 131, the membrane forms a piston head 132, and the second inextensible shell forms a piston. For example, the first and second inextensible shells can have any hollow geometry, including cylindrical or non-cylindrical geometry. In some embodiments, the piston chamber of the first inextensible shell together with the piston head formed by the flexible membrane are airtight, sealed by the membrane. In the example shown in FIGs.1A-1E, the second inextensible tube is attached to the first inextensible tube via the flexible membrane so that the membrane forms a surface between the first inextensible tube and the second inextensible tube, and so that the first inextensible tube forms a piston chamber 131, the membrane forms a piston head 132, and the second inextensible tube forms a piston. [0217] In the example shown in FIGs.1A-1C, the piston actuator 101 is in a first position in an unactuated state (Fig.1C) and in a second position in an actuated state (Fig.1A). In some embodiments, the piston actuator is actuated by applying a force, e.g., by applying a pressure to the first inextensible shell or by supplying a fluid to the first inextensible shell (e.g., supplying a pressurized fluid into chamber 131), which act as a piston chamber. FIG. 1A shows a schematic of the piston actuator with a high pressure applied to the first inextensible shell, e.g., to the interior of the first inextensible shell, which actuates the piston actuator and displaces the second inextensible shell to the right. At sufficiently high pressures, e.g., above a threshold pressure, as shown in FIG.1A, the flexible membrane becomes inverted. In some embodiments, the flexible membrane can invert from one configuration to another when a force is applied, e.g., a pressure difference between the two - 22 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 sides of the flexible membrane is applied. In some embodiments, the shape, thickness, stiffness, or surface roughness of the flexible membrane can be used to tune the threshold pressure at which the flexible membrane becomes inverted. FIG.1D shows a photograph of an exemplary actuated piston actuator made using straws for the inextensible shells and thermoplastic polyurethane for the flexible membrane. FIG.1B shows the piston with an intermediate pressure applied to the first inextensible shell (e.g., by applying a pressure to the first inextensible shell or by supplying a fluid to the first inextensible shell (e.g., supplying a pressurized fluid into chamber 131)), which partially actuates the piston and displaces the second inextensible shell a lesser distance to the right. FIG.1C shows the piston with no pressure applied to the first inextensible shell, demonstrating an unactuated state, where the second inextensible shell is not displaced. FIG.1E shows a photograph of an exemplary unactuated piston actuator made using straws for the inextensible shells and thermoplastic polyurethane for the flexible membrane. [0218] Although the first and second inextensible shells are shown in FIGs.1A-1C as an inner and outer tube respectively, other configurations can be possible and are contemplated. In some embodiments, a piston actuator can include a plurality of inner shells within an outer shell such that a single input pressure can displace each of the plurality of inner shells, resulting in switching of each inner shell. In some embodiments, the inner shells switch simultaneously. In some embodiments, a portion of inner shells can be switched when a pressure is applied, e.g., if individual pistons formed by the inner shells and their corresponding flexible membranes have different threshold pressures. In some embodiments, the first and second inextensible shells are arranged linearly. In some embodiments, the first and second inextensible shells are arranged non-linearly. In some embodiments, the shells are arranged in a branched configuration, e.g., like a tree diagram. In some embodiments, the flexible membrane can be attached to the first and second inextensible shells at different positions, such that the actuations can occur in different directions or at different positions along the length of the first inextensible shell, in addition to the linear actuation described in FIGs.1A-1C. For example, the second inextensible shell can be configured so that it is at an angle to the first inextensible shell such that actuation occurs non-linearly, in a direction other than parallel to the first inextensible shell. In another example, the piston has a branched configuration, with an inner inextensible shell connected to the outer inextensible shell at each branch via a flexible membrane. In another example, the piston includes an outer inextensible shell with a plurality of inner inextensible shells arranged in a series along the - 23 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 length of the outer inextensible shell, each inner inextensible shell connected to the outer inextensible shell via a flexible membrane. In this example, application of pressure to the piston can result in simultaneous actuation of the series of inner inextensible shells. Alternatively, application of pressure to the piston can result in actuation of a portion of the series of inextensible shells, e.g., if individual pistons formed by the inner shells have different threshold pressures. [0219] The inextensible shells can have a cross-section of any shape, and the first and second inextensible shells can have the same or different cross-section shapes. Non-limiting examples of cross-sections include circular, oval, square, rectangular, triangular, or polygonal cross-sections, or irregular cross-sections. B. Mechanically Actuated Switch [0220] In some embodiments, the systems disclosed herein include a mechanical switch. In some embodiments, a mechanical switch takes advantage of non-linear mechanics of tube kinking of soft materials. [0221] In some embodiments, an actuator includes a mechanically activated switch that includes a first inextensible shell; a second inextensible shell; wherein the second inextensible shell is configured to be displaced relative to the first inextensible shell between a first position and a second position; wherein the second inextensible shell is configured to be in a first position when a force below a first threshold is applied to the second inextensible shell, and the second inextensible shell is configured to be displaced from the first position to the second position when a force above the first threshold is applied to the second inextensible shell and acts on the first flexible membrane; a flexible fluid channel system operably linked to the second inextensible shell and configured to restrict flow of fluid through a portion of the flexible fluid channel system when the second inextensible shell is in the first position, and to allow flow of fluid through the portion of the flexible fluid channel system when the second inextensible shell is in the second position. A fluid channel system is a system of fluidically connected channels capable of delivering a fluid or pressure. In some embodiments, a fluid channel system includes one or more fluid channels or tubes. The cross section of the fluid channels can be in any shape, and non-limiting examples of the cross-section shapes include circular, oval, square, rectangular, triangular, or polygonal cross- sections, or irregular cross-sections. In some embodiments, a flexible fluid channel system comprises a flexible material capable of being kinked or tubes capable of being kinked. - 24 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 [0222] FIGs.2A-2D show an exemplary mechanical switch 202. As shown in FIG.2A, a mechanical switch includes a first inextensible shell 211, a second inextensible shell 212, and a flexible fluid channel system 214. The flexible fluid channel system can include a first fluid input channel 215, a second fluid input channel 216, and an output channel 217. In the example shown in FIGs.2A-2D, the first inextensible shell and the second inextensible shell are tubes. However, the first and second inextensible shells can include any shape or configuration where the flow is restricted through the first fluid input channel and allowed through the second fluid input channel in a first, unactuated state, and the flow is allowed through the first fluid input channel and restricted through the second fluid input channel in a second, actuated state. For example, the first and second inextensible shells can have any hollow geometry, including cylindrical or non-cylindrical geometry. As shown, in the FIG. 2A, when the switch is in a first, unactuated state, the second inextensible shell 212 is in a first position, and flow of fluid through the first fluid input channel 215 is restricted, while flow of fluid through the second fluid input channel 216 is allowed. In contrast, as shown in FIG.2B, when the switch is in a second, actuated state, e.g., by application of force, the second inextensible shell is displaced right to a second position, and flow of fluid through the first fluid input channel 215 is allowed, and flow of fluid through the second fluid input channel 216 is restricted. [0223] In the example shown in FIGs.2A-2B, restriction of fluid flow is accomplished by kinking the respective fluid channel (e.g., tube) and the fluid channel system is associated with, linked to, or attached to the second inextensible shell so that displacement of the second inextensible shell introduces or removes kinking of the fluid channel system. In some embodiments, the fluid channel system is associated with, linked to, or attached to the first inextensible shell. In the first position (or “first state”), shown in FIG.2A, the first fluid input channel 215 is kinked, and the second fluid input channel 216 is open. When a force above a threshold force is applied to the second inextensible shell 212, the second inextensible shell is displaced relative to the first inextensible shell 211, causing a snap- through phenomenon where the kinking of the fluid channel system alternates. Application of the threshold or snap-through force causes the switch to switch to a second position (or “second state”). The transition from the first position to the second position is referred to as “snap-through.” The threshold force applied to cause the switch to transition from the first position to the second position (or “snap-through”) is referred to as the “snap-through” force. In the second state, shown in FIG.2B, the first fluid input channel 215 is open, and the - 25 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 second fluid input channel 216 is kinked. In some embodiments, kinking of the flexible channel is the result of a non-linear mechanical instability of the flexible channel system. [0224] In this embodiment, the switch is bistable such that, after snap-through, the flexible channel will remain in a kinked position after the force is released. A bistable switch is a switch that has two stable states, in which an applied force may be used to cause switching between those states. For example, a bistable switch remains in a first position until a first threshold force or above is applied to switch to a second position, then remains in the second position after the force is released and does not return to the first position until a second threshold force is applied to return the switch to the first position. In this example, the switch will not remain in an intermediate position between the first position and the second position without an applied force to hold the switch in the intermediate position. For example, if the second threshold force is in the direction opposite the first threshold force, the switch will remain in the second position when the force is released. For example, if the second threshold force is in the same direction as the first threshold force, the switch will not remain in the second position when the force is released. [0225] The transition from the second position to the first position is referred to as “snap- back.” The threshold force applied to cause the switch to transition from the second position to the first position (or “snap-back”) is referred to as the “snap-back” force. In some embodiments, the snap-through and snap-back forces have the same magnitude. In some embodiments, to return a bistable switch to the first position from the second position, the first threshold force is applied to the second inextensible shell in the opposite direction. In some embodiments, the snap-through and snap-back forces are different. In some embodiments, to return a bistable valve to the first position from the second position, a second threshold force (snap-back force) is applied to the second inextensible shell. In some embodiments, the snap-through force is high enough to maintain switch bistability but low enough to allow easy transitions between states. In some embodiments, the force can be tuned by altering flexible channel material and geometry. In some embodiments, the snap- through and snap-back forces depend on a kinking force of the flexible channel. In some embodiments, the kinking force depends on one or more of (1) the stiffness of the flexible channel, (2) the dimensions of the flexible channel (e.g., wall thickness or cross-sectional geometry), and (3) the position or arrangement of the flexible channel relative to the inner and outer shell (e.g., bending angles of the flexible channels or length of shells). - 26 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 [0226] In some embodiments, the second threshold force is applied in the direction opposite of the first threshold force. Where the second threshold force is applied in the opposite direction of the first threshold force, the first threshold force is expressed with a positive sign, and the second threshold force is expressed with a negative sign. In these embodiments, the switch remains in a first position until a force above the first threshold is applied to switch to a second position. In this embodiment, the switch will remain in the second position when no force is applied, when a positive force greater than or less than the first threshold is applied, and when a negative force with a magnitude less than the second threshold is applied. In this embodiment, the switch will remain in the second position until a force below the second threshold force is applied to return the switch to the first position, e.g., until a force in a direction opposite the first threshold force and with a magnitude greater than the second threshold force is applied. [0227] In some embodiments, the second threshold force is applied in the same direction as the first threshold force. Where the second threshold force is applied in the same direction as the first threshold force, both the first threshold force and the second threshold force are expressed with a positive sign. In this embodiment, the switch will not remain in the second position when no force is applied. In this embodiment, the switch will remain in the second position when a positive force above the first and second thresholds or between the first threshold and the second threshold is applied. In this embodiment, the switch will remain in the second position until a force below the second threshold force is applied to return the switch to the first position, e.g., until a force in the same direction as the first threshold force and with a magnitude less than the second threshold force is applied. [0228] In some embodiments, the first threshold (e.g., snap-through) force is between 0.1N and 5N. In some embodiments, the first threshold force is about 0.1N, 0.2N, 0.3N, 0.5N, 1N, 1.5N, 2N, 2.5N, 3N, 3.5N, 4N, 4.5N, 5N or any force in any range bounded any two values disclosed herein. In some embodiments, the second threshold (e.g., snap-back) force is between about 0.1N and 5N. In some embodiments, the second threshold force is about 0.1N, 0.2N, 0.3N, 0.5N, 1N, 1.5N, 2N, 2.5N, 3N, 3.5N, 4N, 4.5N, 5N or any force in any range bounded any two values disclosed herein. [0229] FIG.2C shows a photograph of an exemplary switch made using straws for the inextensible shells and elastic tubing in an unactuated state. FIG.2D shows a photograph of an exemplary switch made using straws for the inextensible shells and elastic tubing in an actuated state. - 27 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 [0230] Although the first and second inextensible shells are shown in FIGs.2A-2B as an inner and outer tube respectively, other configurations can be possible. In some embodiments, a mechanically activated switch can include a plurality of inner shells, each inner shell associated with a flexible fluid channel system, within an outer shell such that a single input force can displace each of the plurality of inner shells, resulting in switching of each inner shell. In some embodiments, the inner shells switch simultaneously. In some embodiments, a portion of inner shells can be switched when a force is applied, e.g., if individual switches formed by the inner shells have different threshold forces. In some embodiments, the first and second inextensible shells are arranged linearly. In some embodiments, the first and second inextensible shells are arranged non-linearly. In some embodiments the shells are arranged in a branched configuration, e.g., like a tree diagram. In some embodiments, the flexible channels can be attached to the first and second inextensible shells at different positions, such that the actuations can occur in different directions or at different positions along the length of the first inextensible shell, in addition to the linear actuation describe in FIGs.2A-2B. For example, the second inextensible shell can be configured so that it is at an angle to the first inextensible shell such that actuation occurs non-linearly, in a direction other than parallel to the first inextensible shell. In another example, the switch has a branched configuration, with an inner inextensible shell connected to the outer inextensible shell at each branch via an associated flexible fluid channel system. In another example, the switch includes an outer inextensible shell with a plurality of inner inextensible shells arranged in a series along the length of the outer inextensible shell, each inner inextensible shell connected to the outer inextensible shell and having one or more associated fluid channel systems. In this example, application of force to the switch can result in simultaneous actuation of the series of inner inextensible shells. Alternatively, application of force to the switch can result in actuation of a portion of the series of inextensible shells, e.g., if individual switches formed by the inner shells have different threshold forces. [0231] The inextensible shells can have a cross-section of any shape, and the first and second inextensible shells can have the same or different cross-section shapes. Non-limiting examples of cross-sections include circular, oval, square, or rectangular cross-sections. [0232] In some embodiments, the flexible fluid channel system includes a first fluid input channel, a second fluid input channel, and an output fluid channel that are integral with each other. In some embodiments, the flexible fluid channel system includes one or more separate - 28 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 fluid channels that are assembled to form a fluid channel system. The channel of the flexible fluid channel system can have a cross-section of any shape, and the first fluid input channel, second fluid input channel, and output fluid channel can have the same or different cross- section shapes. Non-limiting examples of cross-sections include circular, oval, square, rectangular, triangular, or polygonal cross-sections, or irregular cross-sections. C. Valves [0233] In some embodiments, a piston and a switch can be combined to form a valve. In some embodiments, a piston translates pneumatic input to a mechanical force that acts on the switch. [0234] In one aspect, a valve includes a first inextensible shell; a second inextensible shell; a first flexible membrane; wherein the second inextensible shell is attached to the first inextensible shell via the first flexible membrane and configured to be displaced relative to the first inextensible shell between a first position and a second position; wherein the second inextensible shell is configured to be in a first position when a pressure below a first threshold is applied to the first inextensible shell, and the second inextensible shell is configured to be displaced from the first position to the second position when a pressure above the first threshold is applied to the first inextensible shell and acts on the first flexible membrane; a flexible fluid channel system operably linked to the second inextensible shell and including a first fluid input channel, a second fluid input channel, and a fluid output channel; and wherein the flexible fluid channel system is configured to restrict flow of fluid through the first fluid input channel and allow flow of fluid through the second fluid input channel when the second inextensible shell is in the first position, and to allow flow of fluid through the first fluid input channel and restrict flow of fluid through the second fluid input channel when the second inextensible shell is in the second position. In some embodiments, the first inextensible shell receives an input pressure and acts as a piston chamber. In some embodiments, the flexible membrane forms a surface at one end of the second inextensible shell and thereby acts as a piston head, such that the second inextensible shell is displaced relative to the first inextensible shell when a pressure is applied to the first inextensible shell, e.g., to the interior of the first inextensible shell, and acts on the flexible membrane. In some embodiments, the second inextensible shell acts as a piston. In some embodiments, the valve allows switching between the first fluid input channel and the second fluid input channel. [0235] FIGs.3A-3B show a schematic of an exemplary valve 303. FIG.3C shows a photograph of an exemplary valve made using straws for the inextensible shells and - 29 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 thermoplastic polyurethane for the flexible membrane. As shown in FIG.3A, the valve includes a first inextensible shell 311, a second inextensible shell 312, a flexible membrane 313, and a flexible fluid channel system 314. The flexible fluid channel system can include a first fluid input channel 315, a second fluid input channel 316, and an output channel 317. In the example shown in FIGs.3A-3B, the first inextensible shell and the second inextensible shell are tubes. However, the first and second inextensible shells can include any shape or configuration where the flexible membrane forms a surface between the first inextensible shell and the second inextensible shell, and so that first inextensible shell forms a piston chamber, the membrane forms a piston head, and the second inextensible shell forms a piston and where the flow is restricted through the first fluid input channel and allowed through the second fluid input channel in a first, unactuated state, and the flow is allowed through the first fluid input channel and restricted through the second fluid input channel in a second, actuated state. For example, the first and second inextensible shells can have any hollow geometry, including cylindrical or non-cylindrical geometry. In some embodiments, the piston chamber of the first inextensible shell together with the piston head formed by the flexible membrane are air-tight, sealed by the membrane. In the example shown in FIGs.3A-3B, the second inextensible tube is attached to the first inextensible tube via the flexible membrane so that the membrane forms a surface between the first inextensible tube and the second inextensible tube, and so that first inextensible tube forms a piston chamber, the membrane forms a piston head, and the second inextensible tube forms a piston. [0236] In the example shown in FIGs.3A-3B, the first fluid input channel 315 is fluidically connected to a constant pressure source, and the second fluid input channel 316 is fluidically connected to atmospheric pressure. As shown, in the FIG.3A, when the valve is in a first, unactuated state, the second inextensible shell 312 is in a first position, and flow of fluid from the pressure source through the first fluid input channel 315 is restricted, while flow of fluid through the second fluid input channel 316, which is connected to atmospheric pressure, is allowed. In this example, the first, unactuated state is a closed valve (output pressure is zero). In contrast, as shown in FIG.3B, when a pressure is applied to the first interior shell, e.g., to the interior of the first inextensible shell 311, and acts on the flexible membrane 313, the valve is in a second, actuated state. In this second state, the second inextensible shell 312 is displaced right to a second position, and flow of fluid from the pressure source through the first fluid input channel 315 is allowed, and flow of fluid through the second fluid input channel 316 is restricted. In this example, the second, actuated state is - 30 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 an open valve (output pressure is provided by the first fluid input channel 315). Although in this example, fluid flow is restricted through the first fluid input channel in the unactuated state and fluid flow is restricted through the second fluid input channel in the actuated state, selection of which fluid input channel is restricted in the unactuated state and which fluid input channel is restricted in the actuated state is arbitrary and interchangeable. [0237] In the example shown in FIGs.3A-3B, restriction of fluid flow is accomplished by kinking the respective fluid channel and the fluid channel system is linked to the second inextensible shell so that displacement of the second inextensible shell introduces or removes kinking of the fluid channel system. In some embodiments, the fluid channel system is linked to the first inextensible shell. In the first position (or “first state”), shown in FIG.3A, the first fluid input channel 315 is kinked, and the second fluid input channel 316 is open. When a pressure above a threshold pressure is applied to the first inextensible shell 311, that pressure acts on the flexible membrane 313 of the piston, causing the second inextensible shell to be displaced relative to the first inextensible shell 311. As with the mechanical switch, applying a pressure or force above the threshold force causes a snap-through phenomenon where the kinking of the fluid channel system alternates. Application of a first threshold or snap-through force causes the valve to switch to a second position (or “second state”). The transition from the first position to the second position is referred to as “snap- through.” The threshold force applied to cause the switch to transition from the first position to the second position (or “snap-through”) is referred to as the “snap-through” force. In the second state, shown in FIG.3B, the first fluid input channel 315 is open, and the second fluid input channel 316 is kinked. Kinking of the flexible channel is the result of a non-linear mechanical instability. [0238] In this embodiment, the switch is bistable such that, after snap-through, the flexible channel will remain in a kinked position after the force is released. A bistable valve is a valve that has two stable states, in which an applied force may be used to cause switching between those states. For example, a bistable valve remains in a first position until a first threshold force is applied to switch to a second position, then remains in the second position after the force is released and does not return to the first position until a second threshold force is applied to return the valve to the first position. In this example, the valve will not remain in an intermediate position between the first position and the second position without an applied force to hold the valve in the intermediate position. For example, if the second threshold force is in the direction opposite the first direction, the switch will remain in the - 31 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 second position when the force is released. For example, if the second threshold force is in the same direction as the first threshold force, the switch will not remain in the second position when the force is released. [0239] The transition from the second position to the first position is referred to as “snap- back.” The threshold force applied to cause the switch to transition from the second position to the first position (or “snap-back”) is referred to as the “snap-back” force. In some embodiments, the snap-through and snap-back forces have the same magnitude. In some embodiments, to return a bistable valve to the first position from the second position, the first threshold force can be applied in the opposite direction. In some embodiments, the snap- through and snap-back forces are different. In some embodiments, to return a bistable valve to the first position from the second position, a second threshold force (snap-back force) is applied to the second inextensible shell. In some embodiments, the snap-through force is high enough to maintain switch bistability but low enough to allow easy transitions between states. In some embodiments, the force can be tuned by altering flexible channel material and geometry. In some embodiments, the snap-through and snap-back forces depend on a kinking force of the flexible channel. In some embodiments, the kinking force depends on one or more of (1) the stiffness of the flexible channel, (2) the dimensions of the flexible channel (e.g., wall thickness or cross-sectional geometry), and (3) the position or arrangement of the flexible channel relative to the inner and outer shell (e.g., bending angles of the flexible channels or length of shells). [0240] In some embodiments, the second threshold force is applied in the direction opposite of the first threshold force. Where the second threshold force is applied in the opposite direction of the first threshold force, the first threshold force is expressed with a positive sign, and the second threshold force is expressed with a negative sign. In these embodiments, the valve remains in a first position until a force above the first threshold is applied to switch to a second position. In this embodiment, the valve will remain in the second position when no force is applied, when a positive force greater than or less than the first threshold is applied, and when a negative force with a magnitude less than the second threshold is applied. In this embodiment, the valve will remain in the second position until a force below the second threshold force is applied to return the valve to the first position, e.g., until a force in a direction opposite the first threshold force and with a magnitude greater than the second threshold force is applied. - 32 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 [0241] In some embodiments, the second threshold force is applied in the same direction as the first threshold force. Where the second threshold force is applied in the same direction as the first threshold force, both the first threshold force and the second threshold force are expressed with a positive sign. In this embodiment, the valve will not remain in the second position when no force is applied. In this embodiment, the valve will remain in the second position when a positive force above the first and second thresholds or between the first threshold and the second threshold is applied. In this embodiment, the valve will remain in the second position until a force below the second threshold force is applied to return the valve to the first position, e.g., until a force in the same direction as the first threshold force and with a magnitude less than the second threshold force is applied. [0242] In some embodiments, the valve also includes a restraint 318 that opposes the forces applied to the flexible membrane 313, which acts as a piston head, when pressure is applied to the first inextensible shell 311, which acts as a piston chamber. In some embodiments, the restraint is attached to the first inextensible shell and the second inextensible shell. In some embodiments, the restraint is attached to the first inextensible shell and the flexible membrane. In some embodiments, the restraint is attached to the second inextensible shell and a component external to the valve (e.g., a component of a system that incorporates the valve). By applying a force opposite the force applied to the flexible membrane by the applied pressure, a restraint can be used to modify the threshold pressure of a valve for snap-through or snap-back. In some embodiments, the restraint increases the threshold pressures for snap-through or snap-back. In some embodiments, addition of a restraint converts the valve from a bistable valve to a monostable valve when no pressure is applied. A monostable valve is a valve that is in a first state when no pressure is applied or when the pressure is below a threshold pressure and a second state when a pressure above the threshold pressure is applied, but then returns to the first state when pressure is no longer applied. In some embodiments, the valve includes a second restraint applying a force opposite the first restraint such the net force of the two restraints opposes the applied force. In this embodiment, the second restraint can be used to further modify the threshold pressure of the valve. In some embodiments, the restraint is an elastic band. In some embodiments, the restraint is a shape-memory alloy or shape-memory polymer. [0243] In some embodiments, the threshold or snap-through pressure of a valve with a restraint can be tuned based on the stiffness of the restraint. When pressure is applied to the first inextensible shell, e.g., to the interior of the first inextensible shell (which acts as a - 33 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 piston chamber), the pressure acts on the flexible membrane (which acts as a piston head). The resulting force on the flexible membrane can be referred to as the “piston force.” The valve will snap forward (snap-through) to a second state when the piston force exceeds a snap-through force. If the valve is bistable, the valve will remain in the second state until a snap-back force is applied to cause the valve to snap-back to the first position. When a valve snaps forward, from the actuated state to the unactuated state, this piston force is opposed by the following forces: restraint tension, the force of the kinked tube, and frictional forces. ^^ _^^^௧^^ > ^^ _{^^^௧^^^^௧} + ^^ _{^^^^^ௗ ௧௨^^} + ^^ _^^^^௧^^^ For example, as shown in the free-body diagram in FIG.3D, in the valve shown in FIGs.3A- 3B, during snap forward (displacement of the second inextensible shell to the right) from the first state (FIG.3A) to the second state (FIG.3B), the force applied to the piston head is to the right, while the forces applied by the restraint, kinked tube, and friction are all to the left. Snap-through occurs when the force exerted on the piston head exceeds the sum force applied by the restraint, kinked tube, and friction. In contrast, during snap-back (displacement of the second inextensible tube to the left), the restraint tension opposes the piston, frictional, and tube kinking forces: For example, as shown in the free-body diagram in FIG 3E, in the valve shown in FIGs.3A- 3B, during snap back from the second state (FIG.3B) to the first state (FIG.3A), the force applied by the restraint is to the left, and the forces applied to the piston, by the kinked tube, and by friction are to the right. Snap-back occurs when the force exerted by the piston is less than the difference between the restraint tension and the sum of the forces of the kinked tube and friction. In some embodiments, by changing the stiffness of the restraint, and thereby changing restraint tension, the snap-through and snap-back pressures can be selected. [0244] In some embodiments, a valve can be configured as a logic gate. In some embodiments, a valve can be configured as a logic gate by using at least one of the first inextensible shell 311, the first fluid input channel 315, or the second fluid input channel 316 to define a binary pressure or fluid input. In some embodiments, one or more valves can form a logic circuit. Because the valves disclosed herein are capable of switching between two pressure sources, each connected to a flexible fluid input channel, these valves can compute Boolean computer logic and CMOS-type logic is possible. In some embodiments a constant “high pressure” (e.g., greater than the threshold or snap-through pressure) can - 34 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 correspond to a Boolean 1 and atmospheric pressure can correspond to Boolean 0. In this way, the valves disclosed herein can provide digital control. [0245] FIGs.4A-4D show an exemplary NOT gate using the valves disclosed herein. FIG.4A shows a diagram of a NOT gate with one input and one output. FIG.4B shows a truth table for a NOT gate, with each column indicating the output for a given input. For the output to be 1, the input must be 0. FIG.4C shows a valve 403 configured as a NOT gate. A NOT gate can be constructed using a single valve by supplying atmospheric pressure to the initially kinked channel 415, supplying a constant pressure to the initially unkinked channel 416, and treating the piston chamber formed by the first inextensible shell 411 as an input. In this example, the first fluid input channel 415 is connected to atmospheric pressure, the second fluid input channel 416 is connected to a constant pressure. The first inextensible shell 411 is connected to a pressure source that acts as an input to the NOT gate, supplying an input of 0 (atmospheric pressure) or 1 (pressure above the snap-through pressure). When the pressure input to the first inextensible shell is 0 (top), the second fluid input channel 416 is open, allowing flow from the pressure source to the outlet 417, resulting in an output of 1. When the pressure input to the first inextensible shell is 1 (bottom), the second inextensible shell 412 is displaced, and the second fluid input channel 416 is kinked, restricting flow from the constant pressure source to the outlet 417, resulting in an output of 0. FIG.4D shows a graph of the input and output pressures over time. As shown in FIG.4D, the output is 1 when the input is 0. [0246] FIGs.5A-5D show an exemplary AND gate using the valves disclosed herein. FIG.5A shows a diagram of an AND gate with two inputs (Input 1, Input 2). FIG.5B shows a truth table for an AND gate, with each column indicating the output for a given combination of inputs. For the output to be 1, both Input 1 and Input 2 must be 1. FIG.5C shows a valve 503 configured as an AND gate in the state corresponding to the fourth column of the truth table (both Input 1 and Input 2 are 1). An AND gate can be constructed using a single valve by supplying atmospheric pressure to the initially unkinked channel 516 and treating the initially kinked channel 515 and the piston chamber formed by the first inextensible shell 511 as inputs. In this example, the second fluid channel 516 is connected to atmospheric pressure. The first inextensible shell 511 is connected to a pressure source that acts as Input 2 to the AND gate, supplying an input of 0 (atmospheric pressure) or 1 (pressure above the snap-through pressure). The first fluid input channel 515 is connected to a pressure source that acts as Input 1 to the AND gate, supplying an input of 0 (atmospheric - 35 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 pressure) or 1 (high pressure). When the pressure input to the first inextensible shell is 0, the first fluid input channel 515 is kinked, restricting flow from the high-pressure source to the outlet 517, resulting in an output of 0. As shown in FIG.5C, when the pressure input to the first inextensible shell is 1 (Input 2 =1), the second inextensible shell 512 is displaced, and the first fluid input channel 515 is opened, allowing flow from the first fluid input channel to the outlet 517. If, as shown in FIG.5C, the first fluid input channel also provides an input of high pressure (Input 1 =1), the resulting output is 1. FIG.5D shows a graph of the input and output pressures over time. As shown in FIG.5D, the output remains 0 when only Input 1= 1 or only Input 2 = 1, but the output is 1 when Input 1 = 1 and Input 2 = 1. The bracket on the graph corresponds to the fourth column of the truth table and the valve configuration shown in FIG.5C. [0247] FIGs.6A-6D show an exemplary OR gate using the valves disclosed herein. FIG. 6A shows a diagram of an OR gate with two inputs (Input 1, Input 2). FIG.6B shows a truth table for an OR gate, with each column indicating the output for a given combination of inputs. For the output to be 1, at least one of Input 1 and Input 2 must be 1. FIG.6C shows a valve 603 configured as an OR gate in the state corresponding to the second column of the truth table (Input 1 =1 and Input 2 = 0). An OR gate can be constructed using a single valve by supplying constant pressure to the initially kinked channel 615 and treating the initially unkinked channel 616 and piston chamber formed by the first inextensible shell 611 as inputs. In this example, the first fluid input channel 615 is connected to a constant pressure. The second fluid input channel 616 is connected to a pressure source that acts as Input 1 to the OR gate, supplying an input of 0 (atmospheric pressure) or 1 (high pressure). The first inextensible shell 611 is connected to a pressure source that acts as Input 2 to the OR gate, supplying an input of 0 (atmospheric pressure) or 1 (pressure above the snap-through pressure). As shown in FIG.6C, when the pressure input to the first inextensible shell is 0, the first fluid input channel 615 is kinked, restricting flow from the constant pressure source to the outlet 617. However, in this configuration the second fluid input channel 616 is open, allowing flow from the pressure source connected to the second fluid input channel 616. If, as shown in FIG.6C, the second fluid input channel 616 also provides an input of pressure (Input 1 =1), the resulting output is 1. When the pressure input to the first inextensible shell is 1 (Input 2 =1), the second inextensible shell 612 is displaced, and the first fluid input channel 615 is opened, allowing flow from the constant pressure source to the outlet 617, resulting in an output of 1. FIG.6D shows a graph of the input and output pressures over - 36 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 time. As shown in FIG.6D, the output is 1 if Input 1= 1, if Input 2 = 1, and if Input 1 = 1 and Input 2 = 1. The brackets on the graph correspond to the second column of the truth table and the valve configuration shown in FIG.6C. [0248] FIGs.7A-7C show an exemplary set-reset (SR) latch using the valves disclosed herein. FIG.7A shows a truth table for an SR-latch. Input 1 is “SET” and Input 2 is “RESET.” An SR-latch can switch between a first state and a second state, retaining its state after pressure is removed. The output is undefined when both inputs are 1. When both inputs are 0, the SR latch outputs its previous state, creating memory. FIG.7B shows a valve 703 configured as an SR-switch in the state corresponding to the third column of the truth table (Input 1 (SET) = 0 and Input 2 (RESET) = 1). In this example, a third inextensible shell 711b is added to the side of the second inextensible shell 712 opposite the first inextensible shell 711a, and the second inextensible shell 712 is attached to the third inextensible shell 711b via a second flexible membrane 713b. The first inextensible shell 711a is connected to a pressure source that acts as Input 1 (SET) to the SR-latch gate, supplying an input of 0 (atmospheric pressure) or 1 (pressure above the snap-through pressure of the first flexible membrane 713a). The third inextensible shell 711b is connected to a pressure source that acts as Input 2 (RESET) to the SR-latch gate, supplying an input of 0 (atmospheric pressure) or 1 (pressure above the snap-through pressure of the second flexible membrane 713b). The first fluid input channel 715 is connected to a constant pressure source, and the second fluid input channel 716 is connected to atmospheric pressure. As shown in FIG.7B, when no pressure is applied to the first inextensible shell 711a (Input 1 =0), the SR-latch is in a first state. In this state, the first fluid input channel 715 is kinked, restricting flow from the constant pressure source to the outlet 717, and the second fluid output channel 716 connected to atmospheric pressure is open, resulting in an output of 0. When pressure is applied to the first inextensible shell 711a (Input 1 = 1), the second inextensible shell 712 is displaced, and the SR-latch is in a second state. In this state, the first fluid input channel 715 is opened, allowing flow from the constant pressure source to the outlet 717, resulting in an output of 1. The SR-latch will remain in the second state if the pressure in the first inextensible shell 711a is removed. The SR-switch can then be reset to the first state by applying a pressure to the third inextensible shell 711b (Input 2 = 1) and displacing the second inextensible shell 712 to its original position. In this state, the first fluid input channel 715 is kinked again, restricting flow from the constant pressure source to the outlet 717, resulting in an output of 0. FIG.7C shows a graph of the input and output pressures over time. - 37 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 [0249] FIGs.8A-8D show an exemplary INHIB gate using the valves disclosed herein. FIG.8A shows a diagram of an INHIB gate with two inputs (Input 1, Input 2). An INHIB gate is equivalent to chaining a NOT gate to one input of an AND gate. FIG.8B shows a truth table for an INHIB gate, with each column indicating the output for a given combination of inputs. FIG.8C shows a valve 803 configured as an INHIB gate in the state corresponding to the third column of the truth table (Input 1 = 0 and Input 2 = 1). An INHIB gate can be constructed using a single valve by supplying atmospheric pressure to the initially kinked channel 815 and treating the initially unkinked channel 816 and the piston chamber formed by the first inextensible shell 811 as inputs. In this example, the first fluid input channel 815 is connected to atmospheric pressure. The second fluid input channel 816 is connected to a pressure source that acts as Input 1 to the INHIB gate, supplying an input of 0 (atmospheric pressure) or 1 (high pressure). The first inextensible shell 811 is connected to a pressure source that acts as Input 2 to the INHIB gate, supplying an input of 0 (atmospheric pressure) or 1 (pressure above the snap-through pressure). When the pressure input to the first inextensible shell is 0, the first fluid input channel 815 is kinked. However, in this configuration the second fluid input channel 816 is open, allowing flow from the pressure source connected to the second fluid input channel. If the second fluid input channel 816 also provides an input of pressure (Input 1 =1), the resulting output is 1. As shown in FIG.8C, when the pressure input to the first inextensible shell 811 is 1 (Input 2 =1), the second inextensible shell 812 is displaced, and the first fluid input channel 815 is opened, and the second fluid input channel 816 is kinked, restricting flow from the pressure source to the outlet 817, resulting in an output of 0. FIG.8D shows a graph of the input and output pressures over time. The brackets on the graph correspond to the third column of the truth table and the valve configuration shown in FIG.8C. [0250] FIGs.9A-9D show an IMPLY gate using the valves disclosed herein. FIG.9A shows a diagram of an IMPLY gate with two inputs (Input 1, Input 2). An IMPLY gate is equivalent to chaining a NOT gate to one input of an OR gate. FIG.9B shows a truth table for an IMPLY gate, with each column indicating the output for a given combination of outputs. FIG.9C shows a valve 903 configured as an IMPLY gate in the state corresponding to the first column of the truth table (Input 1 = 0 and Input 2 = 0). An IMPLY gate can be constructed using a single valve by supplying a constant pressure to the initially unkinked channel 916 and treating the initially kinked channel 915 and piston chamber formed by the first inextensible shell 911 as inputs. In this example, the second fluid input channel 916 is - 38 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 connected to a constant pressure. The first fluid input channel 915 is connected to a pressure source that acts as Input 1 to the IMPLY gate, supplying an input of 0 (atmospheric pressure) or 1 (high pressure). The first inextensible shell 911 is connected to a pressure source that acts as Input 2 to the IMPLY gate, supplying an input of 0 (atmospheric pressure) or 1 (pressure above the snap-through pressure). As shown in FIG.9C, when the pressure input to the first inextensible shell is 0 (Input 2 = 0), the first fluid input channel 915 is kinked. However, in this configuration, the second fluid input channel 916 is open, allowing flow from the constant pressure source connected to the second fluid input channel, resulting in an output of 1. When the pressure input to the first inextensible shell 911 is 1 (Input 2 =1), the second inextensible shell 912 is displaced, and the first fluid input channel 915 is opened, and the second fluid input channel 916 is kinked, restricting flow from the pressure source connected to the second fluid input channel 916 to the outlet 917. However, if the first fluid input channel 915 also provides an input of constant pressure (Input 1 =1), the resulting output is 1. In contrast, if the pressure to the first inextensible shell is 1 (Input 2 =1) and the first fluid input channel instead provides an input of atmospheric pressure (Input 1= 0), the resulting output is 0. FIG.9D shows a graph of the input and output pressures over time. [0251] FIGs.10A-10D show an XOR* gate using the valves disclosed herein. FIG.10A shows a diagram of an XOR* gate with three inputs (Input 1, Input 2, Input 3). FIG.10B shows a truth table for an XOR* gate, with each column indicating the output for a given combination of inputs. FIG.10C shows a valve 1003 configured as an XOR* gate in the state corresponding to the third column of the truth table (Input 1 = 0, Input 2 = 1, and Input 3 = 0). An XOR* gate can be constructed using a single valve by treating the initially kinked channel 1015, the initially unkinked channel 1016, and the piston chamber formed by the first inextensible shell 1011 as inputs. In this example, the first fluid input channel 1015 is connected to a pressure source that acts as Input 1 to the XOR* gate, supplying an input of 0 (atmospheric pressure) or 1 (high pressure). The first inextensible shell 1011 is connected to a pressure source that acts as Input 2 to the XOR* gate, supplying an input of 0 (atmospheric pressure) or 1 (pressure above the snap-through pressure). The second fluid input channel 1016 is connected to a pressure source that acts as Input 3 to the XOR* gate, supplying an input of 0 (atmospheric pressure) or 1 (high pressure). When the pressure input to the first inextensible shell is 0 (Input 2 = 0), the first fluid input channel 1015 is kinked. However, in this configuration, the second fluid input channel 1016 is open, allowing flow from the constant pressure source connected to the second fluid input channel, resulting in an output of - 39 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 1 if pressure is applied to the second fluid input channel (Input 3 =1) and an output of 0 if no pressure is applied to the second fluid input channel (Input 3 = 0). When the pressure input to the first inextensible shell 1011 is 1 (Input 2 =1), the second inextensible shell 1012 is displaced, and the first fluid input channel 1015 is opened, allowing flow from the pressure source connected to the first fluid input channel, and the second fluid input channel 1016 is kinked, restricting flow from the pressure source connected to the second fluid input channel 1016 to the outlet 1017. However, if the first fluid input channel 1015 also provides an input of constant pressure (Input 1 =1), the resulting output is 1. In contrast, if the pressure to the first inextensible shell is 1 (Input 2 =1) and the first fluid input channel instead provides no pressure (Input 1= 0), the resulting output is 0. FIG.10D shows four-valve circuit that is equivalent to this one-valve XOR* gate, made of one OR gate, two AND gates, and a NOT gate. In this way, a XOR* valve can be used to reduce the number of valves in a logic circuit. [0252] In some embodiments, the valves disclosed herein can be used to create an XOR gate (exclusive OR) that outputs a Boolean 1 if exactly one input is a 1. FIGs.11A-11D show the minimum number of valves needed to construct an XOR circuit. As shown in FIGs. 11A-11B, to form an XOR gate using a combination NOT, AND, and OR gates, requires a minimum of five valves (one OR gate, two AND gates, and two NOT gates). Potential gates for reduction are outlined in FIGs.11A-11B. The two sets of NOT and AND gates outlined in FIG.11A can be replaced with INHIB gates, reducing the minimum number of valves needed to form an XOR gate from five to three. This reduced circuit is shown in FIG.11C. Alternatively, the OR, AND, and NOT gates outlined in FIG.11B can be replaced with an XOR* gate, reducing the minimum number of valves needed to form an XOR gate from five to two. This reduced circuit is shown in FIG.11D. Reducing the number of gates in a circuit using configurations of the valves disclosed herein can provide benefits including faster actuation, inexpensive fabrication, and reduced size. For example, faster actuation provides the benefit of faster response time and faster calculations. In another example, reducing the size of a circuit reduces the volume of air required for actuation, which reduces energy losses and reduces the weight of a system, enabling untethered soft robotic devices that support their own control circuit and pressure source while moving. These benefits enable development of soft, smart devices with complex circuits in small devices with useful functionality at a low cost. - 40 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 D. Regulator [0253] In some embodiments, a piston and a switch can be combined to form a regulator. In some embodiments, a regulator can provide analog control. In some embodiments, a piston translates pneumatic input to a mechanical force that acts on the switch. [0254] In yet another aspect, a regulator is described, including a first inextensible shell; a second inextensible shell; a flexible membrane; wherein the second inextensible shell is attached to the first inextensible shell via a flexible membrane and configured to be displaced relative to the first inextensible shell between a first position and a second position; wherein the second inextensible shell is configured to be in a first position when a pressure below a first threshold pressure is applied to the first inextensible shell, and the second inextensible shell is configured to be displaced from the first position to the second position when a pressure above a first threshold is applied to the first inextensible shell and acts on the first flexible membrane; and a flexible fluid channel system operably linked to the second inextensible shell and comprising a fluid input channel and a fluid output channel; wherein the flexible fluid channel system is configured to allow flow of fluid through the fluid input channel when the second inextensible shell is in the first position, and to restrict flow of fluid through the fluid input channel when the second inextensible shell is in the second position; and wherein the fluid output channel is fluidically connected to the first inextensible shell. In some embodiments, the first inextensible shell receives an input pressure and acts as a piston chamber. In some embodiments, the flexible membrane forms a surface at one end of the second inextensible shell and thereby acts as a piston head, such that the second inextensible shell is displaced relative to the first inextensible shell when a pressure is applied to the first inextensible shell, e.g., to the interior of the first inextensible shell, and acts on the flexible membrane. In some embodiments, the second inextensible shell acts as a piston. [0255] FIGs.12A-12B show an exemplary regulator 1204. As shown in FIG.12A, the regulator includes a first inextensible shell 1211, a second inextensible shell 1212, a flexible membrane 1213, and a flexible fluid channel system 1214. The flexible fluid channel system can include a fluid input channel 1216, an output channel 1217, and a channel 1215 connecting the fluid output channel 1217 to the first inextensible shell 1211. In the example shown in FIGs.12A-12B, the first inextensible shell and the second inextensible shell are tubes. However, the first and second inextensible shells can include any shape or configuration where the flexible membrane forms a surface between the first inextensible shell and the second inextensible shell, and so that first inextensible shell forms a piston - 41 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 chamber, the membrane forms a piston head, and the second inextensible shell forms a piston and where the flow is allowed through the fluid input channel in a first, unactuated state, and the flow is restricted through the input fluid channel in a second, actuated state. For example, the first and second inextensible shells can have any hollow geometry, including cylindrical or non-cylindrical geometry. In some embodiments, the piston chamber of the first inextensible shell together with the piston head formed by the flexible membrane are air- tight, sealed by the membrane. In the example shown in FIGs.12A-12B, the second inextensible tube is attached to the first inextensible tube via the flexible membrane so that the membrane forms a surface between the first inextensible tube and the second inextensible tube, and so that first inextensible tube forms a piston chamber, the membrane forms a piston head, and the second inextensible tube forms a piston. In this example, the output of the regulator provides an input to the piston chamber. [0256] In the example shown in FIGs.12A-12B, the fluid input channel 1216 is fluidically connected to a pressure source. As shown in the FIG.12A, when a low input pressure is applied to the first inextensible shell 1211 via the fluid input channel 1216, the regulator is in a first, unactuated state. In this first, unactuated state, the second inextensible shell 1212 is in a first position, and flow of fluid through the fluid input channel 1216 is allowed. In contrast, as shown in FIG.12B, when the output pressure is high, a higher pressure is applied to the first inextensible shell 1211 via the fluid input channel 1216 and acts on the flexible membrane 1213, and the regulator is in a second, actuated state. In this second state, the second inextensible shell 1212 is displaced right to a second position, and flow of fluid from the pressure source through the fluid input channel 1216 is restricted. [0257] In the example shown in FIGs.12A-12B, restriction of fluid flow is accomplished by kinking the fluid input channel and the fluid channel system is linked to the second inextensible shell so that displacement of the second inextensible shell introduces or removes kinking of the fluid input channel. In the first state, shown in FIG.12A, the fluid input channel 1216 is open. When a threshold pressure is applied to the first inextensible shell 1211 via the fluid input channel 1216, that pressure acts on the flexible membrane 1213 of the piston, causing the second inextensible shell 1212 to be displaced relative to the first inextensible shell 1211. As with the mechanical switch, applying a pressure or force above the threshold force causes a snap-through phenomenon where the kinking of the fluid channel system alternates. Application of the threshold or snap-through force causes the regulator to - 42 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 switch to a second state. In the second state, shown in FIG.12B, the fluid input channel 1216 is kinked. Kinking of the flexible channel is the result of a non-linear mechanical instability. [0258] In some embodiments, regulator includes a restraint 1218 that opposes the forces applied to the flexible membrane 1213 of the piston when pressure is applied to the first inextensible shell 1211. In some embodiments, the restraint is attached to the first inextensible shell and the second inextensible shell. In some embodiments, the restraint is attached to the first inextensible shell and the flexible membrane. In some embodiments, the restraint is attached to the second inextensible shell and a component external to the valve (e.g., a component of a system that incorporates the valve). In some embodiments, the restraint is attached to the second inextensible shell and external control or source of force input (e.g., to a hand that provides a pulling or releasing motion). By applying a force opposite the force applied to the flexible membrane by the applied pressure, a restraint can be used to modify the threshold pressure of a regulator (which causes kinking of the fluid input channel) and thereby modify the output pressure. In some embodiments, the restraint increases the threshold pressure. In some embodiments, the regulator includes a second restraint applying a force opposite the first restraint such the net force of the two restraints opposes the applied force. In this embodiment, the second restraint can be used to further modify the threshold pressure of the regulator. In some embodiments, a restraint can be used to apply force to the piston, for example, to transfer force from a user to the piston. In some embodiments, the restraint is inextensible. FIG.12C shows an example where increasing a pulling force applied to an inextensible restraint linearly increases the output pressure of the regulator. As shown in FIG.12C, as the force applied by the restraint increases, the output pressure increases, and as the force applied by the restraint decreases, the output pressure decreases. In this way, a regulator restraint can be used to provide analog control of the output pressure by altering the threshold pressure at which the regulator switches between a kinked and unkinked state. By tuning the threshold pressure for switching, an average desired analog pressure can be achieved. [0259] In some embodiments, the restraint is an extensible material. For example, the restraint can include one or more elastic bands. In this embodiment, unlike in FIG.12C, the relationship between applied force and displacement is non-linear. In some embodiments, an extensible restraint provides a similar pressure-force curve compared to a non-extensible restraint, but it will give different (tunable) force-displacement and pressure-displacement curves. For example, a similar magnitude of force can be applied by pulling an extensible - 43 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 restraint further away than would be necessary to pull an inextensible restraint to regulate to a given pressure. In this example, displacement would depend on the spring constant or the stiffness of the extensible restraint. In some embodiments, the restraint includes extensible segments and inextensible segments. [0260] In some embodiments, the restraint is a shape-memory alloy or shape-memory polymer. [0261] In some embodiments, the force applied via the restraint can be increased to increase the output pressure. [0262] In some embodiments, a regulator includes a pull-down resistor to reduce output pressure. In some embodiments, the pull-down resistor is a long, thin tube connected to atmospheric pressure. In some embodiments, a pulldown resistor connects the fluid output channel to atmospheric pressure. II. Materials [0263] In some embodiments, the systems and actuators disclosed herein include one or more inextensible shells. In some embodiments, the inextensible shells include an inextensible material that resists deformation. In some embodiments, the inextensible shells avoid permanent deformation. In some embodiments, avoiding permanent deformation allows the inextensible shells to function as components of a piston when pressure is applied. In some embodiments, the inextensible shells are stiffer than the flexible membrane. In some embodiments, the inextensible shells are stiffer than the flexible fluid channel system. In some embodiments, the inextensible shells include polymer, foam, textile, paper, carbon fiber materials, composite materials, or any combination thereof. Non-limiting examples of polymers include polypropylene, polyethylene, polystyrene, polymethyl methacrylate, polyethylene terephthalate, polyvinyl chloride, polyamides (e.g., Nylon), polyimides (e.g., Kapton), polyesters, polytetrafluoroethylene (e.g., Teflon), polydimethylsiloxane (PDMS), or combinations thereof. [0264] In some embodiments, the systems and actuators disclosed herein include one or more flexible membranes. In some embodiments, the flexible membrane material is selected to have non-linear mechanical behavior, including membrane inversion. In some embodiments, the flexible membrane is elastomeric. In some embodiments, the flexible membrane is a flexible film. In some embodiments, the flexible membrane is a thin, inextensible film. In some embodiments, the flexible membrane includes a polymer, foam, or textile, or any combination thereof. Non-limiting examples of polymers include rubber (e.g., - 44 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 natural or synthetic), silicone elastomer, latex, polyurethanes, polyethylene (PE), plasticized polyvinyl chloride (PVC), other plasticized polymers or combinations thereof. In some embodiments, the flexible membrane is a thermoplastic polyurethane (TPU). In some embodiments, the flexible membrane includes a flexible biopolymer. Non-limiting examples of biopolymers include collagen, gelatin, chitosan, alginate, cellulose, starch, lignin, carbohydrates, and combinations thereof. [0265] In some embodiments, the systems and actuators disclosed herein include one or more flexible channels. In some embodiments, the flexible channel is capable of being kinked. In some embodiments, the flexible channel is non-extensible. In some embodiments, the flexible channel is extensible. In some embodiments, the flexible channel is air-tight. In some embodiments, the flexible channel is elastomeric. In some embodiments, the flexible channel recovers deformation caused by movement of the inextensible shells and returns to its initial configuration after actuation. In some embodiments, the flexible channel recovers deformation by a restoring force. In some embodiments, the flexible channel recovers elastically. In some embodiments, the flexible channel includes a polymer, foam, or textile, or any combination thereof. Non-limiting examples of polymers include rubber (e.g., natural or synthetic), silicone elastomer, latex, polyurethanes, polyvinyl chloride, polyester, or combinations thereof. In some embodiments, the flexible channel is silicone. [0266] In some embodiments, the systems and actuators disclosed herein include on or more restraints. In some embodiments, a restraint opposes a force applied to the flexible membrane. In some embodiments, the restraint applies a tensile force. In some embodiments, the restraint includes an elastic material, e.g., an elastic band. Non-limiting examples of elastic materials include rubber (e.g., natural or synthetic), silicone elastomer, latex, polyurethanes, elastomers, polyurethane, polybutadiene, neoprene, stretchable fabrics, polyether-polyurea copolymers, or combinations thereof. In some embodiments, the restraint includes an inextensible material, e.g., an inextensible string made of natural or synthetic fibers. Non-limiting examples of inextensible materials include natural or synthetic fibers and non-extensible polymers. Non-limiting examples of natural or synthetic fibers include cotton, wool, silk, linen, rayon, polyester, nylon, and combinations thereof. Non-limiting examples of inextensible polymers include polypropylene, polyethylene, polystyrene, polymethyl methacrylate, polyethylene terephthalate, polyvinyl chloride, polyamides (e.g., Nylon), polyimides (e.g., Kapton), polyesters, polytetrafluoroethylene (e.g., Teflon), polydimethylsiloxane (PDMS), or combinations thereof. In some embodiments, the restraint - 45 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 includes a shape-memory alloy (SMA) or a shape-memory polymer (SMP). In these embodiments, an external stimulus can be applied to cause the restraint to return from a deformed state to an original state. Non-limiting examples of SMAs and SMPs include nickel-titanium (NiTi, also known as nitinol), copper-aluminum nickel, polyurethanes, block co-polymers of polyethylene terephthalate (PET) and polyethyleneoxide (PEO), and combinations thereof. Non-limiting examples of external stimuli include light, temperature, electric field, magnetic field, moisture, chemical stimuli, and combinations thereof. Non- limiting examples of chemical stimuli include pH, ion concentration, and combinations thereof. [0267] In some embodiments, the systems disclosed herein include one or more soft robotic actuators. In some embodiments, soft robotic actuator includes an inextensible material. In some embodiments, the soft robotic actuator includes an elastomeric material. In some embodiments, the soft robotic actuator includes a polymer, foam, or textile, or any combination thereof. Non-limiting examples of polymers include rubber (e.g., natural or synthetic), silicone elastomer, latex, polyurethanes, or combinations thereof. In some embodiments, the soft robotic actuator includes silicone, e.g., polydimethylsiloxane (PDMS). In some embodiments, the soft robotic actuator includes a strain-limiting, inextensible layer along one side of the actuator to enable a bending motion of the actuator. In some embodiments, the strain-limiting layer has a higher stiffness than other materials of the soft robotic actuator. Non-limiting examples of a strain-limiting layer include silicone, paper, polymers, woven fiber mesh, and combinations thereof. Non-limiting examples of polymers that can be used in a strain-limiting layer include paper, cellulose, polyamides, polyesters, polyimides, carbon fiber, nylon, non-stretchable fibers (e.g., cotton or silk), polyethylene, polystyrene, polymethyl methacrylate, polyethylene terephthalate, and other polymers used below their glass transition temperature. [0268] The operation pressure of the actuators disclosed herein can be tuned by altering the materials or geometry of components of the actuators. In some embodiments, the operating pressure can be tuned by altering the stiffness of the flexible channels (e.g., to alter the kinking force). In some embodiments the operating pressure can be tuned by altering the stiffness of a restraint (e.g., to alter the force exerted by the restraint). In some embodiments, the operating pressure can be tuned by reducing friction (e.g., to alter the force of friction). In some embodiments, the operating pressure can be tuned by altering the size or geometry of the piston head (e.g., to alter the piston pressure because pressure is equal to force divided by - 46 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 area). For example, higher operating pressure can be achieved by better controlling fabrication. In some embodiments, controlling fabrication improves sealing between components, enabling higher operating pressures. In another example, lower operating pressure can be achieved by using softer flexible channels or reducing friction. In some embodiments, actuators disclosed herein can operate at pressures between about 20 and 170 kPa. In some embodiments, the valves disclosed herein can operate at about 20 kPa, 30 kPa, 40 kPa, 50 kPa, 60 kPa, 70 kPa, 80 kPa, 90 kPa, 100 kPa, 110 kPa, 120 kPa, 130, 150 kPa, 160 kPa, 170 kPa, 180 kPa, 190 kPa, 200 kPa, 250 kPa, 300 kPa, 350 kPa, 400 kPa, 450 kPa, 500 kPa, 550 kPa, 600 kPa, or at any pressure in any range bounded by any two values disclosed herein. III. Systems [0269] In some embodiments, the systems disclosed herein include at least one valve or regulator as disclosed herein. In some embodiments, systems including valves, regulators, or combinations of valves and regulators can be used to control soft pneumatic systems. For example, systems of valves, regulators, or combinations of valves and regulators can control delivery of fluid or pressure to a soft pneumatic system or soft robotic system. In some embodiments, valves can provide digital control. In some embodiments, the valves are configured as logic gates. In some embodiments, valves can function as logic gates in a control circuit. Exemplary logic gates include, NOT gates, AND gates, OR gates, SR- latches, INHIB gates, IMPLY gates, and XOR* gates. In some embodiments, regulators can provide analog control. In some embodiments, a combination of valves and regulators provides integrated digital and analog control. Although this application includes examples of control circuits, valves and regulators can be arranged in any configuration to form a control circuit. A. Ring Oscillator [0270] In some embodiments, the systems disclosed herein include a ring oscillator (e.g., a ring oscillator system) formed using the valves disclosed herein. In some embodiments, a ring oscillator converts a constant input into a time-varying output. In some embodiments, a ring oscillator includes a plurality of NOT gates in series such that the output fluid channel of a first NOT gate serves an input to the first inextensible shell of a second NOT gate. In some embodiments, the output of the n ^th NOT gate serves as the input of the first inextensible shell of the n ^th +1 NOT gate. In an embodiment with N NOT gates, the output of the last (N ^th ) - 47 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 NOT gate serves as the input of the first inextensible shell of the first NOT gate. In some embodiments, this configuration results in alternating output pressure associated each NOT gate. In some embodiments, a ring oscillator converts a constant input to a periodic, oscillating output. In some embodiments, a ring oscillator operates with an output at the following frequency f: where T is the time delay for a single inverter and n is the number of NOT gates in series. In some embodiments, an oscillator includes three or more NOT gates. In some embodiments, a ring oscillator has any odd number of NOT gates. In some embodiments, an odd number of NOT gates leads to instability and therefore oscillation. [0271] In some embodiments, the system includes a regulator as disclosed herein upstream of the ring oscillator. In some embodiments, the output of the regulator provides the single input of the ring oscillator. In some embodiments, the regulator provides analog control of the ring oscillator by providing analog control of the input pressure to the ring oscillator. In this way, input pressure of the oscillator can be controlled using soft components. [0272] FIGs.13A-13D show an exemplary system including a ring oscillator with three valves and one regulator upstream of the ring oscillator. As shown in FIG.13A, the ring oscillator includes 3 NOT gates and one output associated with each NOT gate. FIG.13B shows a schematic of the three-ring oscillator. A regulator 1304 provides an input to the ring oscillator, which includes three valves 1303a, 1303b, 1303c, each configured as a NOT gate. The first fluid input channel 1315a, 1315b, 1315c of each valve 1303a, 1303b, 1303c in the ring oscillator is connected to atmospheric pressure. A constant pressure is applied to the fluid input channel 1316d of the regulator 1304, and the fluid output channel 1317d of the regulator 1304 acts as an input the second fluid input channel 1316a, 1316b, 1316c of each valve 1303a, 1303b, 1303c. The fluid output channel 1317a, 1317b, 1317c of each valve 1303a, 1303b, 1303c is fluidically connected to and provides an input to the piston formed by the first inextensible shell 1311a, 1311b, 1311c of the subsequent valve. In this example, the fluid output channel 1317a of the first valve 1303a, is fluidically connected to the first inextensible shell 1311b second valve 1303b; the fluid output channel 1317b of the second valve 1303b, is fluidically connected to the first inextensible shell 1311c third valve 1303c; and the fluid output channel 1317c of the third valve 1303c, is fluidically connected to the first inextensible shell 1311a first valve 1303a. - 48 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 [0273] FIG.13C shows a graph of output pressure from each valve in the exemplary ring oscillator as a function of time. As shown in FIG.13C, this arrangement of NOT gates can create instability in the system that results in an oscillatory output 1317a, 1317b, 1317c from each valve, even though a constant pressure is supplied to the valves of the ring oscillator. In this figure, the input pressure is shown by a dotted line. [0274] FIG.13D shows a graph of the period of oscillation of the exemplary oscillator as a function of input or supply pressure. As shown in FIG 13D, the oscillation period decreases (i.e., the oscillation frequency increases) as the pressure increases. If, as shown in FIG.13B, the ring oscillator includes a regulator 1304 with a restraint 1318, the regulator can provide analog control of the ring oscillator’s frequency by tuning the force exerted by the restraint. In some embodiments, the regulator 1304 includes an inextensible restraint 1318 that can be used to set the output pressure from the regulator, thereby tuning the oscillator frequency. In some embodiments, using the regulators disclosed herein, the input pressure and therefore the oscillation frequency of the ring oscillator can be controlled using all soft components. B. Counter [0275] In some embodiments, the systems disclosed herein include a counter to count the number of oscillations of a ring oscillator. In some embodiments, the system includes a digital-analog circuit that counts the number of oscillations. FIGs.27A-27B show an exemplary counter. FIG.27A shows a schematic of a logic circuit for the exemplary counter. In this example, the output of a ring oscillator is connected to a buffer which provides positive pressure when actuated without allowing any backwards flow to the oscillator. In this example, the buffer is a pneumatic equivalent of an electronic relay. When positive pressure is applied to this buffer from the oscillator, a separate pressure line is connected to the output of the buffer such that no back flow is possible. In some embodiments, the buffer is a soft component that uses the valve structures disclosed above. For example, the buffer can be a “YES” valve where an input of 0 leads to an output of 0 and an input of 1 leads to an output of 1. In these embodiments, the oscillator is connected to the first inextensible shell of the valve, and the ends of the initially kinked channel are connected to the separate pressure line and buffer output so that when the buffer valve receives an input from the oscillator, the buffer valve snaps-through, unkinking the channel and allowing flow to the separate buffer output. This buffer incrementally pressurizes a pneumatic capacitor (e.g., a sealed, fixed volume that stores pressure) through a thin pneumatic resistor during each oscillation period. A pneumatic resistor is a component that requires a pressure drop for a flow of air to pass - 49 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 through. In some embodiments, a pneumatic resistor includes a portion of a thin, long tube (e.g., an elastic or non-elastic tube). In these embodiments, the resistance of the pneumatic resistor can be tuned by the inner diameter of the tube or the length of the tube. In some embodiments, the pneumatic resistor pressurizes the capacitor by connecting the fluidic outlet of the ring oscillator and the fluidic inlet of the capacitor. The capacitor in turn connects to the “set” input of an SR latch, which functions as a pressure comparator in this circuit. When the pressure on the side of the SR latch connected to the capacitor exceeds the pressure applied to the other side (which is provided by a regulator), the latch will snap through, indicating that a preset number of cycles has been reached. By changing the setting of the analog regulator, e.g., by selecting the force applied by a restraint, a human user can change the pressure at which the latch will snap through and thereby set the number of cycles (or length of time) counted. C. Soft Robotic Systems [0276] In some embodiments, the systems disclosed herein include one or more soft robotic actuator. In some embodiments, the actuators disclosed herein can be assembled into logic circuits to control actuation of pneumatic soft robotic actuators. In some embodiments, the valves disclosed herein can provide digital control of one or more soft robotic actuators. In some embodiments, the regulators disclosed herein can provide analog control of one or more soft robotic actuators. In some embodiments, the systems disclosed herein include both valves and regulators to provide integrated digital and analog control of one or more soft robot actuators. EXAMPLES [0277] Certain embodiments will now be described in the following non-limiting examples. I. Soft Piston Actuator [0278] FIGs.1A-1E show an exemplary soft piston actuator 101. FIG.1A shows the soft piston actuator in an actuated state, FIG.1B shows the soft piston actuator in a partially actuated state, and FIG.1C shows the soft piston actuator in an unactuated state. The piston- like linear actuator employs a rolling diaphragm fabricated from a flexible film. Rolling diaphragms can be used in both linear and rotational hard pistons to reduce energy loss from friction while maintaining a fluid-tight seal, but methods to create long rolling diaphragms are intensive and none are commercially available. By heat sealing two layers of - 50 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 thermoplastic polyurethane separated by parchment paper, rolling diaphragms can be fabricated with arbitrary geometry. Using lightweight polypropylene straws 111, 112 (e.g., used as tubes for drinking consumer beverages) as inner 112 and outer 111 piston chambers secured to the diaphragm 113, a simple pneumatic piston 101 can be created. This piston functions independently as a linear actuator, or as a modular component to be combined with other soft devices explored in this work. FIGs.1D-1E show a photo of the exemplary piston actuator in an actuated state (FIG.1D) and an unactuated state (FIG.1E). A. Piston Actuator Fabrication [0279] FIGs.14A-14F show fabrication of an exemplary piston actuator. This piston actuator is assembled from three components: a flexible thermoplastic polyurethane (TPU) rolling diaphragm 1413 and two compliant polypropylene piston chambers of different diameters: an outer inextensible shell 1411 and an inner inextensible shell 1412 (e.g., sections of a drinking straw). To fabricate the diaphragm, as shown in FIGs.14A-14C, two layers of TPU 1413 (Stretchlon 200, Airtech International Inc.) were heat pressed on either side of a non-bonding layer of parchment paper (Heavy Duty Parchment, Katbite). FIGs.14A-14B show alignment of the TPU and parchment paper before heat pressing. This non-bonding layer is cut at 30% power and 70% speed using a laser cutter (14000 Laser System, Epilog). The heat press (DK20SP, Geo Knight and Co Inc.) bonds the TPU layers to each other at 145 ℃ and 60 psi for 30 seconds. After pressing, the diaphragms were cut out from the surrounding TPU. The non-bonding layer was manually removed to form a diaphragm 1413, as shown in FIG.14C. [0280] In this example, the piston chamber includes a 5 cm length of wide polypropylene straw 1411 (outer inextensible shell, ST12100-Clear, Alink) connected to a silicone input tube 1421 (5054K321, McMaster-Carr) using two diameters of heat shrink tubing (6363K213, McMaster-Carr) and (6363K214, McMaster-Carr). The input tube can provide an input of pressure or fluid. After aligning the components, as shown in FIG.14D, the heat shrink is activated with a heat gun (SDL-8622, Yome) on the low setting, as shown in FIG 14E. As shown in FIG.14F, the piston is then assembled by taping the diaphragm 1413 to the outside of the piston chamber 1411 with tape (Matte Finish Magic Tape, Scotch). If the piston will function independently, a plain, unmodified drinking straw 1412 (inner inextensible shell, HS-1711272052, Hulless) is inserted as the piston rod. Alternatively, if the piston is a component in another device (e.g., a valve, SR latch, or regulator), the piston rod can function as an interface and be fabricated and added later. - 51 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 II. Switch [0281] FIGs.2A-2D show an exemplary mechanical switch 202. To develop the mechanically actuated pneumatic switch shown in FIGs.2A-2B, an elastic tube 214 with two polypropylene drinking straws forming an outer inextensible shell 211 and inner inextensible shell 212 can be used. The elastic tubing 214 is threaded between the two straws 211, 212 and secured with fast-drying glue. In the initial state, shown in the schematic of FIG.2A and the photograph of FIG.2C, one soft tube 215 is kinked while the other 216 is extended to allow flow. As shown in the schematic of FIG.2B and the photograph of FIG.2D, a net force above a critical value of 0.5 N causes the inner straw 212 to translate with respect to the outer straw 211 and alternate which of the two soft tubes 215, 216 is kinked. The kinking of the tube is the result of a non-linear mechanical instability: this ‘snap-through’ phenomenon is bistable, meaning that the tube will remain in the kinked position when the force is released. Likewise, a net force of 0.5 N applied in the opposite direction will return the switch to its original state. Preferably, the snap-through force is large enough to maintain switch bistability but also low enough that the switch easily transitions between states. This force is theoretically tunable by altering tube material and geometry, but a force of 0.5 N satisfies both conditions in this example. The switch can be used alone as a button for humans to reprogram and control pneumatic circuits, but it could also be used to detect obstacles or obtain other environmental information. A. Switch Fabrication [0282] FIGs.15A-15D show fabrication of an exemplary switch. The pneumatic switch 1502 includes two laser-patterned straws 1511, 1512 and silicone tubing (51135K11, McMaster-Carr). As shown in FIG.15A, the inner drinking straw 1512 (HS-1711272052, Hulless) is laser cut (14000 Laser System, Epilog) at 10% speed and 30% power with four holes 1522a, 1522b, 1522c, 1522d for the silicone tubing 1515, 1516. As shown in FIG.15C, the outer straw 1511 (ST12100-Clear, Alink) is laser cut at 30% speed and 70% power with two offset slits 1523a, 1523b for tubing 1515, 1516 to pass through. As shown in FIG.15B, to assemble the switch, silicone tubing 1515, 1516 (51135K11, McMaster-Carr) is fed through each pair of holes 1522a, 1522b, 1522c, 1522d in the inner straw 1512 and secured with a fast-drying glue, e.g., superglue, (45198, Loctite). As shown in FIG.15D, the inner straw 1512 and tubing 1515, 1516 are inserted into the outer straw 1511 and diagonally opposite tubes are secured with tape 1524a, 1524b (Matte Finish Magic Tape, Scotch) and - 52 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 glue, e.g., superglue (45198, Loctite). 14 mm of tubing was left between the inner straw 1512 and the edge of the offset slit 1523a, 1523b where the tubing is secured. [0283] Minor modifications to this fabrication process can be made if the switch is used as a component in another device (e.g., valve, regulator, or SR latch). These modifications are described below. The inner straw can include holes 1519 to attach restraints. III. Valve [0284] Instead of using two separate interior polypropylene straws—one as the piston head and one inside of the switch—the two components can be combined by using a single straw as an interface. The resulting device is a valve, as shown in FIGs.3A-3C, comprising lightweight, compliant materials. The piston translates a pneumatic input into a mechanical force acting on the switch. Once pressure reaches the critical snap-through force per unit area, the switch rapidly changes states from the unactuated state, shown in FIG.3A to the actuated state in FIG.3B. The switch shown in FIGs.3A-3B is restrained with elastic bands 318, which converts the bistable system to a monostable one when no pressure is applied. Therefore, the resulting device is a valve 303 that switches between two states depending on applied pressure in the first straw 311. The snap-through pressure is determined by the stiffness of elastic bands used, introducing programmability to the valves. [0285] Three valves were tested to 10,000 cycles each before an elastic band failed on one valve. This externally attached elastic band can be easily swapped without refabrication of the entire device. The method of fabrication is rapid (requiring less than 12 minutes) and inexpensive (less than $0.40 USD per valve) compared to molded silicone and 3D printed valves, which require hours to cure or print. A. Valve Fabrication [0286] FIGs.16A-16B shows fabrication of a valve. A valve can be fabricated using a piston 1601 (including the outer inextensible shell 1611a and flexible diaphragm 1613, with the inner inextensible shell 1612 omitted) and a switch 1602 (including the inner inextensible shell 1612, outer inextensible shell 1611b, and flexible tubing 1614). FIG.16A shows the piston and switch before attachment. To make a switch for valve, a second set of smaller holes 1619 are laser cut into inner inextensible shell 1612 at a 90-degree angle with respect to the holes intended for flexible silicone tubing 1614. This second set of holes 1619 is connected with a slit to which up to four rubber bands (12205T75, McMaster-Carr) can be connected. Two additional slits are also laser cut on the outer inextensible shell chamber - 53 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 1611b, through which the rubber bands may pass. Finally, two notches 1625a, 1625b are laser cut on the end of the outer inextensible shell 1611b, allowing easy connection to a piston. [0287] As shown in FIG.16B, the outer inextensible shell 1611a of the piston 1601 is fastened to outer inextensible tube 1611b of the switch 1602 using tape (Matte Finish Magic Tape, Scotch). Rubber bands (12205T75, McMaster-Carr) are attached through the laser cut holes 1619 and are restrained via attachment to the outer inextensible shell 1611a such that they are extended by 5 mm when the piston is unactuated. As shown in Table 1, the material cost of one valve is approximately $0.40. Table 1. Estimated cost of materials (in USD) used for the fabrication of valves proposed in this work for digital and analog control. Description Supplier Amount Unit Cost Amount/valve Cost/valve Airtech International TPU Inc. 1.35 sqm $3.50 0.002 $0.005 parchment paper Katbite 247 sqm $19.00 0.002 $0.000 PP straws, thick Amazon 500 pcs $10.00 0.30 $0.006 PP straws, thin Amazon 500 pcs $10.00 0.30 $0.006 silicone tube, thick McMaster-Carr 0.30 m $1.20 0.02 $0.080 silicone tube, thin McMaster-Carr 0.30 m $1.00 0.05 $0.167 heat shrink tube, thick McMaster-Carr 3.0 m $15.00 0.015 $0.075 heat shrink tube, thin McMaster-Carr 3.0 m $7.50 0.015 $0.038 rubber band McMaster-Carr 1000 pcs $10.00 1.0 $0.010 Krazy glue Amazon 5 tubes $8.00 0.005 $0.008 Scotch tape Amazon 99 m $14.00 0.04 $0.006 Total cost / valve $0.40 B. Valve tunability [0288] As shown in FIGs.17A-17D, when configured as a valve, multiple forces act on the inner “switching” straw 1712. FIG.17A shows a micro-computed tomography image of a bistable valve configured as a pneumatic switch. FIG.17B shows force-distance curves of different components of a bistable, pneumatic valve, including tube kinking, elastic restraint, piston, and total device. The summation of the force-distance curves for the individual components is similar to the force-distance curve for the total device. FIG.17B shows the force-distance curves for the elastic band, the piston, and tube kinking, as well as the sum of the force-distance curve of these components and force-distance curve of the total device. As shown in FIG.17B, pressure applied to the piston head balances the opposing elastic tension - 54 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 and friction from the system in the base state. For the valve shown in FIG.17A, when these forces sum to the critical value of 0.5 N, the switch snaps through (i.e., the inner straw 1712 moves to the right). Returning to the base state requires a net force of 0.5 N in the opposite direction, creating a region of hysteresis. FIG.17C shows snap-through and snap-back pressures of a bistable, pneumatic valve with different elastic stiffness. As shown in FIG. 17C, changing elastic stiffness of the elastic band 1718 affects the pressure at which this net force is reached and shifts the snap-through and snap-back pressures. Alternatively, the stiffness of the flexible tube can be changed to alter snap-through and snap-back pressures. Consequently, the operating pressure of the valve can be tuned based on the intended application. For example, the valve can control pressures significantly higher than the input pressures it receives. This behavior enables logic cascading and pressure amplification. [0289] By individually characterizing the force on each component, a model can be created to predict valve snap-through pressures as they depend on elastic stiffness. Elastic, piston, and frictional forces are modeled analytically as functions of switch displacement, elastic stiffness, and input pressure. As discussed below, tube-kinking is modeled empirically by measuring the switch in isolation. When the valve snaps forward, frictional, elastic, and tube-kinking forces all oppose the applied piston force, which yields a force balance equation and allows us to determine snap-through pressures. When the valve snaps back, the elastic force opposes piston, frictional, and tube kinking forces, which gives a separate force-balance equation, such that snap-through and snap-back can be different. FIG. 17D shows experimentally measured valve snap-through and snap-back pressures for a bistable, pneumatic valve (solid black diamonds) compared with predictions (lines), according to certain embodiments. As shown in FIG.17D, the semi-empirical physical model applied in this work agrees with experimental results. Such quantitative correlation enables programmability with precision and control. In implementations of digital logic, the model can guide design of snap-through pressure between two threshold values corresponding to Boolean zero and one; for analog systems, on the other hand, this tunability allows real-time regulation of setpoint pressures. [0290] FIG.18 shows snap-through characterization. For consistency with experimental measurements, a positive force value corresponds to an applied force in the negative x- direction (e.g., a positive tip force acts to push the switch towards the left). In practice, snap- through and stability characteristics can be tuned by choosing the elastic stiffness of restraints used in the valve. As shown in FIG.18, an isolated switch that is not attached to a piston or - 55 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 elastic restraint is bistable and has three equilibria, each corresponding to a location where the piston tip or end of the inner shell exerts no force. These points shift position depending on the direction of travel of the switch due to system friction. The first and third equilibria are stable and correspond to the on and off states of the switch. Conversely, the second equilibrium is unstable, occurring in the middle of snap-through when forces from tube kinking act towards the right under a slight perturbation to the right (causing runaway in the positive x-direction) and act towards the left under a slight perturbation to the left (causing runaway in the negative x-direction). Perturbations in this state therefore cause the switch to snap to either the on or off positions, both of which are stable. Due to the two points of stable equilibrium, the isolated switch is bistable. Note that the necessary snap-through force required to transition between the two stable states is equal to the maximum (or minimum) force along force-distance curve between stable equilibria for forward (or backward) switching. In contrast, a monostable switch has an equilibrium only at a force of zero. In some embodiments, a monostable valve with snap-through can be well-suited for digital control. In some embodiments, a monostable valve without snap-through could be well- suited for analog control. For example, without snap-through, it is possible to ensure that the pressure being used as a threshold is high enough to fully cause the switch to transition states. C. Modeling Snap-Through Pressure [0291] FIGs.19A-19C show an elastic friction model of a valve. FIG.19A shows a schematic of a valve 1903 with inputs to the model labeled. Three component forces contribute to determining the pressure exerted on the internal membrane 1913 at which a valve snaps through: the switch snap-through force caused by tube kinking, the restoring force exerted by the elastic restraints (e.g., elastic bands) 1918, and friction inherent to the system. Switch snap-through occurs when the force exerted on the piston head 1932 exceeds the sum of all of these components. To predict the force contribution of each component, a semi-empirical model was developed based on a combination of experimental and analytical techniques. [0292] Tube-Kinking Force: Because tube kinking is difficult to accurately describe with a closed-form analytical model, the force exerted by the tip of the switch as it transitions states using a universal testing machine (68SC-2, Instron) was measured. The maximum force recorded is considered the required snap-through force. The mean snap-forward and snap-backward forces across three valves were taken to determine the snap-through forces and positions to be used in the model. In this example, snap-forward requires 0.5 N of force - 56 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 and occurs when the inner switch 1912 is displaced 1.3 mm with respect to the outer casing 1911, while snap-backward also requires 0.5 N of force (applied in the opposite direction), but occurs at a displacement of 10.9 mm. [0293] Elastic Stiffness: Each set of elastic bands 1918 was modeled as an ideal spring, with stiffness coefficients determined experimentally. FIG.19B shows the force- displacement curves of different numbers of elastic bands (1 elastic band, 2 elastic bands, 3 elastic bands, 4 elastic bands), with zero extension set to be 7.62 cm of elastic band length. Sharp deviations in force are artifacts from measurement. These measurements were made using a universal testing machine (68SC-2, Instron). The slope of each curve is the calculated stiffness coefficient for that number of bands. It is desirable to calculate a length- independent property of the elastic material; and this is achieved by multiplying each stiffness coefficient by the length of elastic material tested (7.62 cm). [0294] Elastic Force: Due to the complex geometry of the valve, the elastic force applied to the piston tip cannot be described directly with Hooke’s law. Instead, it is necessary to account for several system parameters. Critically, it is assumed that there is a constant frictional “sticking” force ( ^^ _^ ) between the rubber band 1918 and edge of the outer switch chamber 1911. This sticking force separates the elastic into two sections with lengths ^^ _^ and ^^ _ଶ ( ^^) respectively (note that the length is constant, while ^^ _ଶ ( ^^) varies with tip extension, ^^). These lengths are shown in FIG.19A. If the difference in elastic tension between the two sides of this point, | ^^ _^ ( ^^) – ^^ _ଶ ( ^^)|, is less than ^^ _^ , then the amount of material in each section of elastic will remain constant. Otherwise, the elastic will slide until the difference in tension is ^^ _^ . Other parameters shown in FIG.19A are the distance between the inner piston 1912 and outer chamber 1911 (ℎ), the initial horizontal distance between the sticking point and the rubber band ( ^^ _^ ), and the length-independent stiffness constant calculated previously ( ^^ _^ ) from the measurements shown in FIG.19B. Additionally, ^^ _^ ( ^^) and ^^ _ଶ ( ^^) are defined as the unstretched lengths of and ^^ _ଶ ⁽ ^^ ⁾ , and ^^ is defined as extension of the elastic when the switch is retracted. Using this setup, it is possible to determine a set of analytically solvable equations for each state of the switch (sliding while extending, sliding while retracting, and the region where elastic sliding does not occur). The equations are solved iteratively while incrementally changing x, and using the results of the previous step to determine which state the switch is in. In this example, 4000 steps are used. [0295] If the switch is sliding while extending ( ^^ is increasing, and ^^ _ଶ ( ^^) − ^^ _^ ( ^^) ≥ ^^ _^ ), the difference in tensions is set equal to ^^ _^ . - 57 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 ^^ _ଶ ( ^^) − ^^ _^ ( ^^) = ^^ _^ (1) The tension in each section of elastic is a function of the stretched and unstretched lengths of that section, as well as the length independent stiffness constant. The second stretched length of elastic is determined by extension. A final equation follows from the conservation of elastic material (assuming the portion of the elastic band inside the inner straw does not stretch). ^{Equations 1-5 can be solved analytically to find the five unknown values: ^^^( ^^), ^^ଶ( ^^), ^^^( ^^),} _{Finally, ^^ଶ௫} ⁽ _^^ ⁾ _{, the component of ^^ଶ} ⁽ _^^ ⁾ _{which acts in parallel to the piston, can be determined.} [0296] If the switch is sliding while retracting ( ^^ is decreasing and ^^ _^ ( ^^) − ^^ _ଶ ( ^^) ≥ ^^ _^ ), the same process is followed, except Equation 1 reflects the change in direction: ^^ _ଶ ( ^^) − ^^ _^ ( ^^) = − ^^ _^ (8) Therefore, Equation 6 holds with the substitution ^^ _^ → − ^^ _^ . Additionally, since the elastic band cannot support a compressive load, ^^ _^ ( ^^) and ^^ _ଶ ( ^^) are set as 0 if they are ever calculated as negative. - 58 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 [0297] If the band is not sliding (| ^^ _ଶ ( ^^) − ^^ _^ ( ^^)| ≤ ^^ _^ ), the length ^^ _^ ( ^^) retains its value from the previous calculation step. Then ^^ _ଶ ⁽ ^^ ⁾ can be solved directly using Equation 5. The _{tension values ^^^} ⁽ _^^ ⁾ _{, ^^ଶ} ⁽ _^^ ⁾ _{, and ^^ଶ௫} ⁽ _^^ ⁾ _{are then found with Equations 2, 3, 4 and 7.} [0298] To determine the force from the elastic band and friction felt by the piston, the algorithm is run as described above, incrementing through two cycles of extension and retraction (each 18 mm in length) with the initial conditions ^^ _ଶ ( ^^) − ^^ _^ ( ^^) = 0 and ^^ _^ ( ^^) = ^^ _^ and using measured values from the valve, ^^ _^ = 60 mm, ℎ = 2.5 mm, ^^ _^ = 8 mm. Additionally, ^^ _^ is 2.10, 4.24, 6.52, or 8.44 N for 1-4 elastic bands, respectively. Finally, ^^ _^ is a fitting parameter that varies with the number of elastics. For one elastic, ^^ _^ = 0.2 N. For more elastics, this value is multiplied by the number of elastics used due to the increased force applied to the sticking point (e.g., for three elastics, ^^ _^ = 0.6 N). This value is on the same order of magnitude as the rest of the forces present in the system. ^^ _ଶ௫ ( ^^) is recorded at each step in the second cycle as the force experienced by the inner piston. FIG.19C shows the output of this model for a full cycle of extension and retraction for 1, 2, 3, and 4 elastic bands. The final value for the elastic force ( ^^ _^^^^௧^^ ) is taken from the second cycle, at the step with displacement equal to the snap-through or snap-back position (1.3 mm during extension for snap-through or 10.9 mm during retraction for snap-back, shown by vertical lines). [0299] Piston Force: The predicted piston force required to snap the switch through is determined by the component force balance described previously. FIGs.20A-20D show force-displacement curves of gate components, as well as the experimentally measured force- displacement curve for the complete device, and a force-displacement curve based on a component-based model (summing the forces of the components in the model). FIG.20A shows a graph of measured force-displacement curves for a valve with one elastic and 0 kPa input pressure to the piston chamber formed by the outer inextensible shell. FIG.20B shows a graph of measured force-displacement curves for a valve with four elastics and 80 kPa input pressure. FIGs.20A-20B demonstrate that the model based on the sum of the measured force-displacement curve for the components (piston, elastic with friction, flexible tube) is in agreement with the measured force-displacement curve for the total device. If the switch is snapping forward, the elastic force ( ^^ _^^^^௧^^ ) and tube-kinking force ( ^^ _௧௨^^ ) oppose the piston force [0300] If the switch is snapping back, ^^ _௧௨^^ and ^^ _^^^௧^^ oppose ^^ _^^^^௧^^ . - 59 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 [0301] Piston Pressure: A linear relationship was determined to convert piston force ( ^^ _^^^௧^^ ) to an equivalent input pressure ( ^^). To determine this relationship, piston tip force of an isolated piston without a switch component was recorded at varying pressures using a universal testing machine (68SC-2, Instron), and two linear trendlines were fit (one during extension, and one during retraction). These measurements of piston force at different pressures are shown in FIG.20C. Piston force is set to be negative as it opposes the force from elastic bands when configured as a gate. The gate force of a valve at different pressures (each with four elastic bands) is shown in FIG.20D. In contrast to the piston force, which measures the force on the piston of an isolated piston, the gate force is the force on the piston tip when the piston is assembled with a switch to form a valve. [0302] During extension (with ^^ given in kPa and ^^ _^^^௧^^ in N): ^ ^{^^^^௧^^ + 0.157 (11)} 0 _.049 [0303] During retraction: ^ _{^ =} ^{^^^^^௧^^ − 0.140 (12)} 0 _.054 [0304] The difference between extension and retraction is due to hysteresis caused by friction. The snap-through pressures predicted by this model are shown and compared to experimental results in FIG.17D above. D. Logic Gates [0305] The bistable nature of the valves disclosed herein allows them to manipulate digital values without ambiguity. After assigning logical values to high- and low-pressure states, this valve can compute Boolean computer logic. Since pressure outputs can switch between two entirely different sources, CMOS-type logic is possible (eliminating energy loss through pulldown resistors). In the example in FIGs.3A-3B, the valve is configured with a high-pressure line connected to the initially kinked channel while the initially unkinked channel is open to atmospheric pressure. If “high pressure” is considered to represent a Boolean 1, and atmospheric pressure is considered to represent a Boolean 0, then this valve mimics a non-inverting Schmitt trigger. For example, the value of the “high pressure” representing a Boolean 1 is greater than the snap-through pressure of the valve. Since this snap-through pressure is tunable, the pressure value representing a Boolean 1 might change depending on the intended application. In this example, a Boolean 1 will refer to pressures - 60 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 between 30-75 kPa (used in different applications and demonstrations), while a Boolean 0 will refer to atmospheric pressure. [0306] Under this framework, varying inputs to the valve creates a variety of useful logic gates. For example, as shown in FIG.4A-4D, connecting constant atmospheric pressure (P ₀ ) to the initially kinked channel with a constant source of high pressure (P ₁ ) connected to the initially unkinked channel is analogous to a NOT gate. In this example, the piston chamber is the sole variable input to the gate; however, multiple tubes can be set as inputs to create two or three input logic gates. The set of AND and OR gates, shown in FIGs.5A-5D and 6A-6D, respectively, along with the NOT gate is a functionally complete set of logic gates, meaning that any Boolean operation can be computed with enough gates and time. [0307] In practice, lowering the number of physical valves in a logic system ensures rapid actuation and low system volume. This goal is achieved with additional logic gates previously unreported in soft systems. A single INHIB gate, shown in FIGs.8A-8C is equivalent to chaining a NOT gate to one input of an AND gate. Similarly, an IMPLY gate, shown in FIGs.9A-9D, is equivalent to chaining a NOT gate into one input of an OR gate. Adding these gates to the set lowers the number of valves needed to create an XOR gate from five to three. Additionally, considering all tubes on the gates as variable inputs creates a three-input logic gate, as shown in FIGs.10A-10D. This gate is labeled XOR*, as adding it to the previous set allows XOR and XNOR gates to be constructed with just two valves. All gates were tested with a pneumatic “1” between 60-75 kPa. E. Valve Failure Testing [0308] The soft valve can fail in at least two critical ways: the soft diaphragm bursting, and pressure leaking through the kinked tubing. These two failure modes were tested individually across three valves. To test the breakthrough pressure of the kinked channel, input pressure was slowly increased until an increase in output pressure was measured with a pressure sensor (MGA-300-A-9V-R, SSI Technologies). The mean breakthrough pressure was 168 kPa. However, at the kink breakthrough pressure, the leaking flow rate is small. In cases where the output of the valve allows some amount of flow– such as when the valve is configured as a pressure regulator– this leakage may not impact performance. As input pressure increases past the breakthrough point, valve functionality will degrade. [0309] To test the bursting pressure of the diaphragm of a soft valve, piston input pressure was slowly increased until bursting was observed. The mean bursting pressure across three trials was 215 kPa. During these trials, the tube input was secured with an - 61 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 adhesive glue (45198, Loctite) in addition to heat shrink tubing (6363K213, McMaster-Carr) and (6363K214, McMaster-Carr). This modification was made to reach higher piston input pressures but does not otherwise change valve functionality. [0310] The upper limits for breakthrough and bursting are related to how well components of the valves are sealed together. For example, in the valves tested in this example, glue and tape were used to seal components, and these materials are not yet optimized for sealing plastic components). With better sealing (and controlled fabrication), the upper pressure limit is expected to double or even triple. IV. Regulator [0311] In addition to digital logic, the systems disclosed herein are capable of analog pressure control. As shown in FIGs.12A-12B, removing (or blocking) the initially kinked channel 1215 present in the valve described above and feeding the output 1217 to the piston chamber formed by the outer inextensible shell 1211 creates a regulator 1204. If the output pressure is high, the piston formed by the inner inextensible shell 1212 kinks the initially open internal tube 1216, limiting flow. FIG.12A shows the regulator 1204 in a low output pressure state, when the tube 1216 is open. FIG.12B shows the regulator 1204 in a high output pressure state, when the tube 1216 is kinked. The pressure at which a kink forms is determined by the balancing of force between the piston and opposing force components (from the valve restraint 1218, tube-kinking, and friction). In the example shown in FIGs. 12A-12B, an inextensible string as the valve restraint instead of an elastic band to directly transfer a human user’s force input to the piston head. Accordingly, as shown in FIG.12C, by increasing the external pulling force applied to a string, one can raise the output pressure of the regulator. A. Regulator Fabrication [0312] FIGs.21A-21B show fabrication of a regulator. A regulator can be fabricated using a piston 2101 (including the outer inextensible shell 2111a and flexible diaphragm 2113, with the inner inextensible shell 2112 omitted) and a switch 2102 (including the inner inextensible shell 2112, outer inextensible shell 2111b and flexible tubing 2114). FIG.21A shows the piston and switch before attachment. FIG.21B shows the complete regulator after attachment. To make a switch for regulator, the same modifications to a switch intended for a valve can also be applied to a switch intended for a regulator. However, strings can be used instead of rubber bands in regulators to directly transfer force to the piston head. - 62 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 Additionally, one of the silicone tubes (and associated holes in the inner and outer inextensible shells) can be omitted to further simplify fabrication. [0313] As shown in FIG.21A, the first inextensible shell 2111a piston 2101 is fastened to the first inextensible 2111b shell of the switch 2102 with tape (Matte Finish Magic Tape, Scotch). A restraint (elastic band or string) is attached to the outer inextensible shell 2111a, e.g., through the attachment holes 2119, (thereby putting horizontal tension on the inner piston), and the pneumatic output 2117 is connected to the piston input 2121, creating the self-regulating feedback loop. [0314] Apart from improving human-device interactions, a regulator is an important step towards fully untethered soft robots. A regulator can separate control systems and different actuators, allowing them to function at different pressures while all using a common source. Additionally, onboard high-pressure sources can be regulated down to more appropriate levels for actuation and control. A pressure of up to 168 kPa was achieved in this exemplary regulator; however higher pressures can be achieved with optimized fabrication (e.g., improved sealing of components). V. SR latch [0315] FIGs.7A-7C show a set-reset latch (SR-latch). As shown in FIG.7B, a latch includes components of a valve, including an outer inextensible shell 711a, inner inextensible shell 712, flexible diaphragm 713a, and flexible tubing 714. Adding a second pressure chamber 711b to the opposite side of a valve 703 via a second flexible diaphragm 713b and removing the elastic band causes the snap-back pressure to be negative, recovering the bistability observed in the elementary switch shown in FIGs.2A-2B. The result is a single- valve SR latch. This SR latch retains its mechanical state even after the pressure source is removed, enabling nonvolatile memory using the same physical framework as other logic gates disclosed herein. A. SR-Latch Fabrication [0316] FIGs.22A-22B show fabrication of an SR latch. An SR-latch can be fabricated using two pistons 2201a, 2201b (each including the outer inextensible shell 2211a, 2211b and flexible diaphragm 2213a, 2213b, with the inner inextensible shell 2212 omitted) and a switch 2202 (including the inner inextensible shell 2212, outer inextensible shell 2211c, and flexible tubing 2214). FIG.22A shows the pistons and switch before attachment. To make a switch for an SR Latch, four notches 2225a, 2225b, 2225c, 2225d were laser cut, two on each - 63 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 end of the outer switch chamber 2211c, to enable easy connection with pistons 2201a, 2201b. As shown in FIG.22B, the pistons 2201a, 2201b are fastened on either end of the switch 2202 using tape (Matte Finish Magic Tape, Scotch). VI. Systems A. Control Glove for Soft Robotic Hand [0317] Because the regulators disclosed herein are lightweight, durable, and compliant and because they output variable pressures, these regulators are good candidates for a wearable method of pneumatic control. To demonstrate suitability in a wearable device, a glove with five embedded regulators was fabricated, each regulator connected to a different finger of a soft robot. This glove, shown in FIG.23A-23E controls a soft robotic hand. FIG. 23A shows top view of a five-finger control glove on a human hand (left) and soft robotic hand with one soft actuator for each finger (right). Each regulator includes an inextensible string as a restraint that is connected to the user’s finger to allow the user to vary the force exerted by the string on the piston of the regulator. FIG.23B shows photographs of the pressure regulator used as an analog human input in an analog control glove that controls the soft robotic hand. As shown in FIG.23B, when the wearer of the control glove bends a finger, force is exerted on a string attached to a regulator, and the corresponding finger of the robotic hand is actuated. For example, in the first photograph, when the wearer of the control glove bends the pinky finger, the pinky finger of the soft robotic hand bends. The second photograph similarly shows bending of the middle finger of the wearer and the soft robotic hand, the third photograph shows bending of the pointer finger of the wearer and the soft robotic hand, and the fourth photo shows bending of the thumb of the wearer and the soft robotic hand. [0318] As shown in FIG.23C, the glove can be used to hold any finger of the soft robot hand at any position between its maximum and minimum displacement, a continuous form of control that digital pneumatic systems alone cannot achieve, nor even emulate without an infeasible number of logic elements. FIG.23C shows a robotic hand being held at four different positions (1, 2, 3, 4) between fully actuated and unactuated states during pulling and releasing of a finger by a user. FIG.23D shows the output pressure as a function of time during operation of an actuator of a soft robotic hand, with the four positions (1, 2, 3, 4) from FIG.23C labeled. As the user pulls the finger (increasing the force on the sting) the output pressure increases, and as the user releases the finger (decreasing the force on the string) the - 64 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 output pressure decreases. By tailoring the length of the inextensible string, the deflected angle of each robotic finger actuator could be matched to a human user’s finger within 2° of deviation over a range of 0° to 130°. As shown in FIG.23E, when the finger of the control glove is at 38°, the corresponding finger of the soft robotic hand is also at 38° (top), and when the finger of the control glove is at 129°, the corresponding finger of the soft robotic hand is also at 129° (bottom). Additional results are shown below in FIGs.26A-26B. [0319] FIGs.24A-24B show a control glove. FIG.24A shows a photograph of the hook- and-loop fasteners used to secure a string restraint of a control glove to a finger, and FIG. 24B shows a top view of the control glove on a human hand, according to certain embodiments. The control glove includes five pressure regulators, fabricated as discussed above and shown in FIGs.21A-21B. As shown in FIG.24B, the five regulators are secured to a user’s forearm using two textile strips with hook-and-loop fasteners adhered to the ends. Each regulator input is connected to a common high-pressure line. As shown in FIG.24A, the string restraint attached to each regulator interfaces with individual fingers of the user by way of additional hook-and-loop fasteners. One loop secures the string at the fingertip, while another loop—which holds a guiding tube—secures the string at the proximal phalanges (lower finger). Each string is connected to the regulator with an adjustable bend knot, which allows the string lengths to be adjusted without device refabrication. [0320] FIG.25 shows a soft robotic hand. The robotic hand includes 5 individual bending PneuNet (pneumatic network) actuators which act as fingers. In this example, each PneuNet actuator includes a pneumatic network of interconnected chambers embedded in an elastomeric material and a strain-limiting, inextensible layer along one side of the actuator to enable a bending motion of the actuator. Silicone elastomer (Dragon Skin 30, Smooth-On) was used as the actuator material. Each finger actuator was formed using a silicone mold. Each finger actuator is then attached to a housing. The housing for the finger actuators was 3D printed using a desktop printer (CR-10S Pro V2) with PLA (PLA+, eSUN). Two housing components (body and lid) were glued to each other (45198, Loctite). To interface the soft robotic hand with a control glove, the input of each finger actuator is connected to the output of a regulator on the control glove. The output of each finger actuator is also connected to atmospheric pressure through a pneumatic pulldown resistor, i.e., 60cm of thin tubing. [0321] FIGs.26A-26B show tuning of a control glove so that the finger angle of a user matches the finger actuator angle of a soft robotic hand. This demonstrates the utility of the control glove (and by extension, the analog regulator) as an intuitive human input for soft - 65 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 devices. Using an embedded adjustable bend knot to modify the length of string connecting the regulator to a users’ finger, displacement response of the actuator is matched to the finger. The results of this experiment are shown in FIGs.26A-26B. FIG.26A shows frames from a video that were analyzed to determine the deviation between the actuator and user finger angles at finger angles of 0°, 22°, 38°, 60°, 82°, 90°, 106°, and 129°. A summary of these results is shown in Table 2 below, and a graph of measured actuator angle as a function of measured human finger angle is shown in FIG.26B. As shown in Table 2 and FIG.26B, the measured maximum deviation of two degrees (i.e., 2°) was observed. Table 2. Comparison of human and actuator finger angle [0322] Alternatively, in some embodiments, analog control can be provided by a component other than a glove. For example, in addition, to control gloves, a control stick, a control mat, or a control panel can be used. In any of these examples, a force externally applied to each restraint can be translated to vary the pressure applied to the inextensible tube and flexible membrane. B. Combined Digital-Analog Logic Circuits [0323] The component-level modularity disclosed herein enables development of devices for both digital logic and analog control to be fabricated using the same architecture. In addition, using device-level modularity, combined analog-digital circuits with complementary functions can be constructed. Two examples of combined digital-analog circuits are a ring oscillator with a continuously varying output frequency, shown in FIGs. 13A-13D, and a reprogrammable counter circuit, shown in FIGs.27A-27B and 28A-28B. Both circuits cannot be integrated into soft devices using only digital components (without any analog components) due to device size restrictions. For example, while a digital counter - 66 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 circuit would involve 30 or more physical gates, the analog regulator disclosed herein reduces the number of gates for fabricating a counter to just six. [0324] FIGs.13A-13D show an exemplary ring oscillator. This ring oscillator is fabricated by arranging three NOT gates in a ring. FIG.13A shows a diagram of this ring oscillator, while FIG.13B shows a schematic of this ring oscillator formed by three valves 1303a, 1303b, 1303c and a regulator 1304. The output 1317a, 1317b, 1317c of the flexible tube of each valve is connected to the input of the outer inextensible shell 1311a, 1311b, 1311c of the next valve in the ring. FIG.13C shows the output pressure as a function of time for the three valves in the ring with a constant input pressure of 78 kPa. Maximum pressure output is 76 kPa, indicating only a 3% reduction in pressure. As shown in FIG.13C, this ring oscillator configuration creates an instability in the system that results in an oscillatory output 1317a, 1317b, 1317c from each valve with a constant pressure input. FIG.13D shows the period of the oscillation as a function of input pressure. Because the frequency of oscillation is highly dependent on input pressure, an analog pressure regulator 1304 placed upstream from the oscillator can control oscillation period and frequency. In practice, one could use a regulator to change the speed of an untethered robot or adjust the oscillation frequency and force output of a mechano-therapeutic device to maximize comfort and function. [0325] Soft ring oscillators using CMOS-type architectures can reach frequencies of about 0.3 Hz. However, there is a demonstrated need for higher frequency oscillation in applications including haptics, communication, grasping, and reaching. Ring oscillators assembled from inverters that require pulldown resistors to function can reach higher frequencies (10 to 15 Hz). However, the loss of pressure inherent to architectures utilizing resistors limits the ratio of output pressure to input pressure ( ^^ _^ / ^^ _^ ); oscillators with pulldown resistors have demonstrated ^^ _^ / ^^ _^ ratios of 0.76 and 0.24. Locomotive soft robots can use either (i) an oscillatory pressure input or (ii) a constant pressure input that generates oscillation, which then translates to movement. Improvements in oscillation speed can directly increase locomotive speed. Due to the high operational pressures, quick snap- through times, and CMOS-type architecture, ring oscillators built with the valves and regulators disclosed herein can reach a ^^ _^ / ^^ _^ ratio of 0.97 as well as frequencies of up to 3.6 Hz, over ten times faster than other CMOS-type designs. [0326] Building on the ring oscillator, an integrated digital-analog circuit was designed to count the number of oscillations in order to program time-dependent robot operation. FIG. 27A shows a diagram of an exemplary counter circuit. The output of the ring oscillator - 67 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 connects to a buffer, which provides positive pressure when actuated without allowing any backwards flow to the oscillator. This buffer incrementally pressurizes a pneumatic capacitor (a sealed, fixed volume) through a thin pneumatic resistor during each oscillation period. The capacitor in turn connects to the “set” input of an SR latch, which functions as a pressure comparator in this circuit. When the pressure on the side of the SR latch connected to the capacitor exceeds the pressure applied to the other side (which is provided by a regulator), the latch will snap through, indicating that a preset number of cycles has been reached. FIG.27B shows a graph of the number of cycles before a counter switches state at different volumes and regulator input pressures. The number of cycles increases as the pressure increases and as the volume increases. By changing the setting of the analog regulator, a human user can change the number of cycles (or length of time) counted. This combined digital-analog counter is “infinitely” adjustable, in the sense that the only limitation to the number of distinct settings is the quality of fabrication. [0327] FIGs.28A-28B shows an example circuit of a stable cascading counter using the gates available with the valves disclosed herein. In comparison to the combined digital- analog count described above (which requires only six valves), this digital-only circuit would require additional valves. An exemplary digital counter, shown in the diagram of FIG.28A, employs chained JK flip-flops, which are an extension of SR latches. JK flip-flops are designed to avoid the possibility for an invalid input (where both the set and reset signals are high), accept a clock input, and toggle the output when both the J and K inputs are high and the clock input changes from low to high. FIG.28B shows a diagram of a JK-latch comprising gates that are constructed with the valves disclosed herein. As shown in FIG. 28B, even with the reductions provided by INHIB gates, 30+ valves are required to physically construct this circuit. Functionally, this device would count the number of times the input switches from a pneumatic 0 to a pneumatic 1, up to a maximum of eight (i.e., binary “111”). [0328] The components and actuators disclosed herein can be used to enable a human user to program the circuit while reducing the number of gates required to do so. Using a large number of gates introduces many practical fabrication challenges, including compounded pneumatic resistance, actuation delays, and significant space requirements, making the circuit impractical for use, especially on an untethered device. In contrast, the combined digital-analog circuit can count an arbitrary number of cycles using just six physical valves. - 68 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 C. Untethered robot [0329] The ring oscillator and counter circuit were combined to create a dynamically programmable untethered soft robot, shown in FIG.29A. Control schemes for soft devices enable robots which can walk, grab, and navigate environments. Such robots are generally inexpensive to produce, safe for interaction with humans and animals, and durable in harsh environments. The combination of these traits makes soft robotic technology appealing for use in search and rescue or exploration scenarios. However, many of these robots either require a tethered pressure source to function or are limited in capabilities due to weight and size constraints. Because the valves developed in this work are light (less than 5g), they are a good candidate for use in an untethered robot. [0330] The untethered robot in FIG.29A includes a pneumatic network gripper, an extendable frame, silicone feet, and piston actuators. FIGs.29B-29C show frames of a video of a robot retrieving an object at two different distances based on counter settings programmed dynamically at the time of deployment. This untethered locomotive robot can move forward, grasp an object, and return to its initial position. The forward distance traveled can be dynamically programmed by a human user at the time of deployment by adjusting the net tension in the rubber bands (elastic restraints) attached to the regulator portion of the analog counter circuit described above, which keeps track of the number of steps taken by the robot. In practice, the user would choose how many rubber bands attached to the regulator to leave in tension; different numbers of rubber bands correspond to different pre-calibrated distances. [0331] FIG.29B shows an untethered robot with a regulator setting of 38 kPa (at t = 0s, 4 s, 8s, 10s, 12s), while FIG.29C shows an untethered robot with a regulator setting of 24 kPa (at t = 0s, 5s, 10s, 15s, 17s). As shown in FIG.29B, at t = 0s, the untethered robot is 9 in away from the object. The untethered robot in FIG.29B is moving forward at t = 4s. At t = 8s, the untethered robot in FIG.28B has reached the object and used the gripper to grip the object. The untethered robot in FIG.29B is moving backward at t = 10s and has returned to its original position at t = 12s. As shown in FIG.29C, at t = 0s, the untethered robot is 5.5 in away from the object. The untethered robot in FIG.29C is moving forward at t = 5s. At t = 10s, the untethered robot in FIG.29C has reached the object and used the gripper to grip the object. The untethered robot in FIG.29C is moving backward at t = 15s and has returned to its original position at t = 17s. The internal capacitance of this untethered robot is 80 cm ³ , - 69 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 yielding 11 and 7 stepping cycles for the 39 kPa and 24 kPa deployments, respectively, based on the cycle programming data shown in FIG.27B. [0332] This demonstration marks an untethered soft robot which can be programmed dynamically at the time of deployment. Furthermore, robot locomotion is achieved by using the piston actuator as a standalone device, demonstrating its versatility and practicality outside of use as a soft valve. [0333] The locomotive robot shown in FIG.29A includes three main components: a flexible frame, actuators (locomotive pistons and gripper), and the combined digital and analog logic control circuit (shown in FIG.30A). The frame includes laser-cut straws assembled in a rectangular prism. Holes were laser cut in the straws for ease of construction. Two different diameters of straws are used for construction (ST12100-Clear, Alink). Along the length of the frame, these two diameters slide freely within each other, allowing the frame to vary in length based on actuation by piston actuators. Rubber bands restrain the frame to a minimum length when no opposing force is applied. On each of the four “legs” of the robot, low-friction fluorinated ethylene propylene sheets (3782, Fibre Glast) are affixed to enable sliding. [0334] Piston actuators interface with the frame to symmetrically extend it when pressure is applied. A modified pneumatic network (PneuNet) gripper is attached to the front of the robot. The gripper allows the robot to retrieve objects and is molded with curable silicone (Dragon Skin 30, Smooth-On) with 3D printed molds. Piston actuators are also used to cyclically push a high-friction silicone (Dragon Skin 30, Smooth-On) foot into the ground. The silicone foot is also molded using 3D printed molds. Actuating one of the two feet during frame extension introduces an asymmetrical force that biases the robot to move in one direction. Robot motion can be controlled by switching which foot is actuated. This process is shown in FIGs.30B-30C. As shown in FIG.30B, during forward displacement, the actuation of the back foot biases froward extension, and the actuation of the front foot biases forward retraction. As shown in FIG.30C, when the counter reaches the limit, the gripper closes and reverse displacement begins. During reverse displacement, the actuation of the front foot biases backward extension and actuation of the back foot biases backward retraction. [0335] The robot is pneumatically powered with a single onboard CO ₂ pressure vessel (17557, Fluval) regulated to a pressure of 138 kPa. This single, constant pressure is regulated to two separate lower levels using the soft analog regulators. The full control circuit is - 70 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 illustrated in FIG.30A. One pressure line runs to the PneuNet gripper, which uses a lower pressure compared to the piston actuators. A second pressure line runs to a regulator- controlled counter circuit that is designed to be set at the time of robot operation to control distance travelled. A human user chooses how many rubber bands to tension on the regulator, thereby setting the robot counter. The main component of the logical control circuit is a central ring oscillator, which runs off of a third pressure line (at 138 kPa). The oscillator converts a constant pressure into an oscillating output, which enables robot locomotion. One output of the oscillator connects to the analog counter circuit through a pneumatic resistor and 80 cubic centimeter capacitor. Once the robot has walked for the pre- set number of steps, the pressure in the pneumatic capacitor will exceed the output of the human-set regulator, changing the output of the SR-latch from low-pressure to high-pressure. Simultaneously, another output of the ring oscillator connects to the two pistons on the robot frame, as well as one of the silicone-foot pistons. The final oscillator output connects to the other silicone foot. During each oscillation cycle, the robot frame extends, and the silicone feet bias it to move forward, as shown in FIG.30B. Once the SR-latch in the timer circuit snaps through, the high-pressure output connects to three components in two steps. First, a pneumatic buffer switches on, allowing the down-regulated pressure line to actuate the PneuNet gripper. Second, two soft valves function as two-way relays and alternate which foot is connected to each section of the ring oscillator (thereby reversing the robot’s direction, as shown in FIG.30C). The robot will continue moving in the reversed direction until it is retrieved by a human user, or until the onboard pressure source is exhausted. The untethered robot was able to travel a total distance of at least 60 feet (over 50 body lengths) before depleting a single CO2 canister. D. Multifunctional Cushion Matrix [0336] To demonstrate another capability of the pneumatic control system developed in this work, a prototype of an exemplary cushion matrix was fabricated. The cushion matrix is fabricated in a similar manner to the piston actuator described above. Parchment paper (Heavy Duty Parchment, Katbite) is heat-pressed between two layers of heat-sealable textile. The parchment paper is patterned with a laser cutter (14000 Laser System, Epilog) at 30% power and 70% speed. The heat-press (DK20SP, Geo Knight and Co Inc.) is activated at 171 ℃ and 60 psi for 30 s. Excess textile and parchment paper are manually cut away, leaving six separate pouches. Each pouch has an input connection to which tubing can be glued. - 71 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 [0337] This cushion matrix is shown in FIGs.31A-31N. FIG.31A shows the logical circuit of the cushion matrix. The circuit includes five switches (1, 2, A, B, X). The cushion is connected to a single pressure source, yet depending on which pneumatic switch is turned on, lifting, rolling, or oscillatory mechano-therapeutic functions can be activated. [0338] FIGs.31B-31E show the rolling function. As shown in FIGs.31B-C, the rolling function activates a column of cushions, which could lift one side of a human and aid in, for example, patient repositioning. FIG.31B shows the first left column of cushions activated when switch 1 is turned on. FIG.31C shows the right column of cushions activated when switch 2 is turned on. FIG.31D shows a photograph of a cushion matrix with the left column activated. FIG.31E shows experimental pressure measurements of a cushion matrix with the left column activated. [0339] FIGs.31F-31I show the lifting function. As shown in FIGs.31F-31G, the lifting function inflates a row of cushions, e.g., near the head or feet. FIG.31F shows the third row of cushions activated when switch A is turned on. FIG.31G shows the second row of cushions activated when switch B is turned on. FIG.31H shows a photograph of a cushion matrix with the first row activated. FIG.31I shows experimental pressure measurements of a cushion matrix with the first row activated. [0340] FIG.31J shows the oscillation function. The oscillation function sequentially inflates and deflates pairs of pouches using the ring oscillator when switch X is turned on. FIG.31J shows experimental pressure measurements of the cushion while the oscillation function is activated at different time points in the oscillation cycle. [0341] Control of each function is realized by connecting pouches with a series of OR gates, which prevents flow between pouches that should remain uninflated. A regulator controls the actuation pressure of each function as well as the frequency of oscillation. The prototype was operated with an input pressure of 70 kPa. The cushion is restrained with a transparent acrylic sheet to simulate the weight of a user. [0342] As an alternative to this integrated digital-analog control unit, a human user can directly control different sets of pouches using an analog control glove, shown in FIGs.31K- 31N. In this example, each regulator of the control glove is connected to a pair of cushions. FIGs.31K-31N show a use case in which the thumb, index finger, and middle finger are used to control the top, middle, and bottom pairs of actuators, respectively, on the cushion matrix. FIG.31K shows the unactuated cushion matrix, with no fingers bent. FIG.31L shows the first row of cushions activated when the thumb is bent. FIG.31M shows the second row of - 72 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 cushions bent when the index finger is bent. FIG.31N shows the third row of cushions bent when the middle finger is bent. The degree of deflection of the thumb and fingers serves as a method to achieve continuous control of the pressure to which each pair of pouches is inflated. [0343] In addition to applications in patient lifting and mechanotherapy, the incorporation of direct analog control of these three regions on the cushion matrix offers promise for emotional interaction with patients in quarantine or under other restrictions regarding interaction with medical professionals. Haptic approaches have been shown to be preferred by humans compared to rigid alternatives. The glove and cushion matrix can be used in systems which also apply heat in human interaction and emotional connection. [0344] A scaled-up version of this cushion matrix can aid medical professionals who require physical assistance in repositioning patients in medical settings, including elevating body parts or moving. Such a device can reduce the number of work-related injuries in hospitals. Because hospital rooms are typically equipped with an in-house pressure source, they are a logical candidate for use of a scaled-up version of this device. However, the proposed cushion matrix can be used on any bed with a suitable supply of pressurized air (e.g., in nursing homes or for at-home comfort), and is not limited to hospitals. VII. Experimental Measurements A. Pressure Measurement [0345] Experimental pressure measurements were taken when recording valve responses, snap-through pressures, regulator function, control glove pressures, piston force response, etc. In each experiment, a digital pressure sensor (ADP5151, Panasonic) was connected to every pressure location of interest (a valve input or output). To measure vacuum pressures, a separate sensor (ADP5101, Panasonic) was used. The electronic output of each sensor was measured using a central data acquisition device (USB-6002, National Instruments) before being converted to pressure values. When measuring the snap-through pressures of valves, the pressure input was gradually increased using a flow regulator (ITV0050-2BL, SMC). In the case of measuring negative snap-through pressures, a manually-operated syringe was used to gradually create a vacuum to apply negative pressure. When measuring a decreasing pressure with no flow output (e.g., the decreasing output of a pressure regulator), a pneumatic pulldown resistor (60 cm of thin tubing) was connected to allow gradual pressure release. - 73 - ActiveUS 200968734 Attorney Docket Number: 0042697.00591WO1 Date of Electronic Deposit: September 15, 2023 B. Force-Displacement Characterization [0346] The force-displacement relationships of internal valve components (shown in FIG. 17B) and the regulator input force (FIG.12C) were acquired with a universal testing machine (68SC-2, Instron). Each valve component was secured in the testing machine before starting two cycles of extension and retraction, each 18 mm in length (or until 6 N was reached in the case of the regulator). The force exerted by the component on the testing machine was recorded at each point in the second cycle. C. Pressure Mat Measurement [0347] The external pressure exerted by the cushion matrix (shown in FIGs.31A-31N) was measured using a pressure-mapping sensor (5250, TekScan). The cushion matrix was placed on top of the pressure-mapping sensor and constrained with flat surfaces on both sides before the spatial-pressure distribution was recorded. [0348] It will be appreciated that while one or more particular materials or steps have been shown and described for purposes of explanation, the materials or steps may be varied in certain respects, or materials or steps may be combined, while still obtaining the desired outcome. Additionally, modifications to the disclosed embodiment and the invention as claimed are possible and within the scope of this disclosed invention. - 74 - ActiveUS 200968734