3D PRINTED HYBRID ROBOT
 This application claims the benefit of priority to co-pending United States Application Ser. No. 62/191, 172, filed July 10, 2015, the contents of which is incorporated by reference.
STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR
 This invention was made with United States government support under Grant No. DMR-1402570 awarded by the National Science Foundation and Grant No. W91 lNF-09-1- 0476 award by the United States Army. The United States government has certain rights in this invention.
 This technology relates generally to soft robots. In particular, this invention relates to a soft robots integrated with hard or rigid components.
 Robots are typically composed of rigid components to promote high precision and controllability. Frequently constructed from hard metals such as aluminum and steel, these robots require large machining equipment and an intricate assembly process.
 Recent work has explored the possibility of creating soft-bodied robots inspired by invertebrates such as cephalopods and insect larvae, as well as vertebrates, including snakes and fish. The use of compliant materials facilitates the development of biologically inspired robotic systems that are more adaptable, safer, and more resilient than their fully rigid counterparts. The design and fabrication of soft robotic systems, however, present significant engineering challenges. The bodies of soft robots are typically fabricated
in custom-designed molds and require multiple assembly steps or lost wax techniques to embed actuation. The molds used to create these soft robots are complex and time-consuming to make, especially for prototype designs that are fabricated in small numbers and are constantly evolving. Additionally, some applications (such as ones requiring untethered robots) require rigid components to power and control the soft body.
 The functionality of soft robotics is dependent on effective bonding of stiff components to soft flexible substrates. This is because, under thermal or mechanical loading, large stresses are localized at such interfaces between dissimilar materials, which can cause permanent deformation, cracking or complete delamination of the interface. The magnitude of the localized interfacial stress increases with the disparity in material properties, but can be reduced by introducing a functionally graded interface. A functionally graded interface enhances the bonding strength between mismatched materials and reduces fracture driving forces, thus it can enable comparatively less failure-prone structures that are desirable for applications that simultaneously require the physical properties of both flexible and rigid materials.
 Many such functional gradient interface design solutions to minimize interfacial stresses between adjoining stiff and soft tissues are available in nature that can be mimicked to engineer wear-resistant high-strength multifunctional structures. For example, the toothed rings that line the suckers of squid (sucker ring teeth) are a perfect example of unmineralized functionally graded structure that is designed by nature for high-strength and wear-resistance. The tip of the tooth is rigid to effectively grip the prey. The stiffness decreases along the length of the tooth, i.e., from the rigid tip to the soft basal ring where it is attached to the soft sucker muscle mass. The modulus gradient within the tooth 1) enhances bonding between the tooth and soft sucker muscle mass, 2) provides optimum bending stiffness to the tooth to be able to bend under the contraction of the circular muscles to stab
the prey, and 3) reduces the local stress concentration by diffusing stress within the tooth structure and hence provides high strength.
 It is a major challenge, however, to design and optimize a functionally graded interface using a given set of mismatched constituent materials.
 In one aspect, a hybrid robot is fabricated with soft and rigid components that can apply high-localized stress using rigid parts for load-bearing, grasping, or cutting objects in addition to having a soft body structure capable of actuation for locomotion or other motions. These and other aspects and embodiments of the disclosure are illustrated and described below.
 A hybrid robot containing integrated soft and rigid components includes at least one soft component, the soft component made up of a first flexible material; at least one rigid component, the hard component made up of a second rigid material; and an expandable chamber capable of actuation upon pressurization, wherein the hard component is integrally connected to the soft component by a functionally graded connector, and wherein the functionally graded connector includes a compositional gradient comprised of the first soft material and the second rigid material.
 In one embodiment or more embodiments, the compositional gradient is a step gradient, or the compositional gradient is a continuous gradient.
 In any preceding embodiment, the functionally graded connector has a modulus of elasticity that ranges over one to four orders of magnitude and can be, for example, that the functionally graded connector has a modulus of elasticity that ranges over three orders of magnitude.
 In any preceding embodiment, the soft component, hard component and functionally graded connector are 3D printed components.
 In any preceding embodiment the rigid component includes a structural member, and for example the structural component is configured to bear a load or the structure component is configured to define an inextensible region of the expandable chamber.
 In any preceding embodiment the rigid component includes a tool, and for example the tool is selected from the group consisting of a gripper or cutter.
 In any preceding embodiment the hybrid robot further includes a control module secured to the rigid component of the soft robot, and for example, the control module comprises one or more of a power source, a high voltage source, an air compressor, a solenoid valve, a pressure regulator, combustible gases, oxygen gas, and fluid channels.
 In any preceding embodiment, the expandable chamber makes up at least a portion of the soft component.
 In any preceding embodiment, the expandable chamber is pneumatically or hydraulically pressurizable.
 In one aspect, the hybrid robot includes radially aligned graded teeth printed directly on a soft circular base, and for example, the hybrid robot further a restraining wall extending from the soft base and disposed around the radially aligned graded teeth, the teeth and the restraining wall defining a cavity for housing the expandable chamber.
 In another aspect, a hybrid robot containing integrated soft and rigid components, includes at least one soft component, the soft component made of a first flexible material; at least one rigid component, the hard component made of a second rigid material, wherein the rigid component includes a structural element of the robot; and an expandable chamber capable of actuation upon pressurization, wherein the hard component is integrally connected to the soft component by a functionally graded connector, wherein the functionally graded connector comprises a compositional gradient comprised of the first soft material and the second rigid material.
 In one or more embodiments, the structural element includes one or more of an arch, beam, column, plate, platform, cavity or brace.
 In one or more embodiments, the structural element is a load bearing element.
 In one or more embodiments, the load bearing element includes a plate or base configured to support a cargo, load or equipment needed for the locomotion, actuation or operation of the soft robot.
 In one or more embodiments, the soft component of the hybrid robot includes at least a portion of the expandable chamber.
 In one or more embodiments, the rigid component is configured to define an inextensible region of the expandable chamber.
 In one aspect, the hybrid robot includes two nested hemispheroids, including a lower flexible hemispheroid and an upper hemispheroid having a rigid component and a functionally graded connector having a modulus of elasticity that ranges over three orders of magnitude through a stepwise gradient creating a structure that transitions from highly flexible to fully rigid structure, and for example the nested hemispheroids define the expandable chamber.
 In any preceding embodiment, the hybrid robot is secured to at least one additional expandable chamber capable of actuation, and for example, the additional expandable chamber is secured mechanically or by glue.
 These and other aspects and embodiments of the disclosure are illustrated and described below.
BRIEF DESCRIPTION OF THE DRAWINGS
 The invention is described with reference to the following figures, which are presented for the purpose of illustration only and are not intended to be limiting.
 In the Drawings:
 FIG. l shows the schematic illustration of tooth models on soft substrates and having a tooth made from soft material, rigid material, and a compositionally graded material composed of n materials with modulus ranging from εΐ (softest), ε2, ε3... εη (rigid). .
 FIG. 2A shows a 3D printed ring of radially arranged functionally graded teeth used in the gripper.
 FIG. 2B shows a close up of an individual tooth, highlighting the gradient from rigid to flexible phases according to one or more embodiments.
 FIG. 2C shows a fully 3D printed biologically inspired hybrid gripper with 6 radially aligned graded teeth printed directly on fully soft base according to one or more embodiments.
 FIG. 2D is another view of the 3D printed biologically inspired hybrid gripper of FIG. 2C, demonstrating an internal tube-based actuator that when inflated, bends the teeth inward.
 FIG. 2E shows a cross-section through the gripper, revealing the internal cavity that houses the pneumatic actuator (not shown).
 FIG. 2F shows a 3D printed ring of radially arranged functionally graded teeth used in the gripper according to one or more embodiments.
 FIG. 3 A shows the schematic illustration of a shear test set up, in which the sample is held inside the 3D printed conical profiled hole in the substrate and the sample is displaced by uy in y-axis relative to the substrate.
 FIG. 3B shows the experimental (solid lines) and simulated (dashed lines) of fully rigid (upper curve), graded (middle curve) and soft (lower curve) tooth.
 FIG. 3C shows photographs of the samples having soft, rigid and graded teeth at displacement uy = 13 mm (failure point for rigid and soft tooth models) showing the position of the tip with respect to the substrate (top images) and the effect of shear force on the
interface adjoining tooth and soft base (bottom images). The tip of the soft tooth comes out of the hole and shows the worst gripping ability, however the rigid sample, while having good gripping ability, shows the maximum damage at the interface among three tooth models.
 FIG. 3D displays the simulated maximum principal strain distribution within the samples at shear displacement of uy = 2 mm, 4 mm and 6 mm.
 FIG. 4A shows the schematic illustration of a compression test set-up, in which the conical sample truncated at 5mm below the tip is compressed between two parallel plated by applying displacement uy in negative y-axis.
 FIG. 4B shows the experimental (solid lines) and simulated (dashed lines) compression response of fully rigid, graded and soft tooth.
 FIG. 4C displays simulated maximum principal strain distribution within the samples at displacement -uy = 0.5 mm, 1 mm and 1.5 mm are shown in (c).
 FIG. 5A is a schematic illustration of the combustion actuation of a soft robot having a functionally graded transition between a rigid component and a soft component, according to one or more embodiments.
 FIG. 5B shows the ignition sequence of fuel delivery, mixing, and sparking, in which butane and oxygen are alternately delivered to the combustion chamber (to promote mixing); after a short delay to promote additional mixing of the fuels, the gaseous mixture is ignited, resulting in combustion, and leg inflation occurs concurrently with fuel delivery, and leg deflation begins shortly after landing.
 FIG. 5C is a computer-aided design model of the entire robot, consisting of the main explosive actuator surrounded by three pneumatic legs, in which a rigid core module that contains power and control components sits atop the main body, protected by a semisoft shield, according to one or more embodiments.
 FIG. 6A is a schematic of the driving components and core module showing functional dependencies of the control hardware; and FIG. 6B is a CAD model of the core module with components from FIG. 6A labelled.
 FIG. 7 illustrates qualitative twisting analysis comparing 3D-printed beams that are fully flexible (top), half rigid (middle) and half flexible, or transition gradually from rigid to flexible (bottom), demonstrating material response to twisting force, as well as to validate the numerical values of the material properties used in simulation, which shows (Left) material distribution of the beams; (Middle) beams under torsion.; and (Right) simulation of beams under torsion.
 FIG. 8 is a jumping simulation of a soft robot model according to one or more embodiments, which shows (Left) ground reaction force as internal gases expand; (Middle) Pressure evolution inside the robot body as internal gases expand; and (Right) deformation state of rigid top, gradient top, and flexible top robot bodies at the initial state and the point of maximum simulated gas expansion, in which line thicknesses indicate material stiffness.
 FIGS. 9A-9B are impact simulations of a soft robot striking the ground at 45° according to one or more embodiments (this angle was chosen as a particularly extreme loading condition and because it correlated with observations from jumping experiments), in which FIG. 9A shows reaction forces experienced by the three robots upon striking a solid plane under simulated conditions representative of actual testing conditions; and FIG. 9B shows FEA results of rigid top, gradient top, and flexible top robots, compared at 50 N.
 A hybrid robot containing rigid components integrated into a soft bodied robot is provided. The hybrid robot includes at least one soft component made of a first flexible material and at least one rigid component made from a second rigid material. To provide a connection between the two components that has a strong bond and that is capable of
withstanding stresses, the rigid component is integrally connected to the soft component by a functionally graded connector. The functionally graded interface enhances the bonding strength between mismatched materials and reduces fracture driving forces, thus it can enable comparatively less failure-prone structures that are desirable for applications that
simultaneously require the physical properties of both flexible and rigid materials. The functionally graded connector is made up of a compositional gradient of the first soft material and the second rigid material.
 The rigid component is more rigid, e.g., stiffer, than the soft component. As used herein, a rigid component is considered "rigid" if it is capable of resisting deformation in response to an applied force typically encountered during the intended use or operation of the soft robot.
 Rigidity can be measured using a number of different parameters, such as
Young's modulus (Pa) or durometer. Young's modulus is an appropriate measure for at least some hybrid robot systems. Young's modulus of the rigid component is greater than the Young's modulus of the soft component, e.g., ει<εη. In one exemplary embodiment, a fully rigid material has a Young's modulus of ~1 GPa, whereas materials suitable for the soft, flexible component have a Young's Modulus of below -100 MPa. However, what is "rigid" is more appropriately defined in terms of the application rather than a strict value of Young's modulus. In one or more embodiments, there is a 1000-fold difference in Young's modulus between the soft component and the rigid compound. In other embodiments, the durometer of the hard component is greater than the durometer of the soft component. The two most common scales for durometer, using slightly different measurement systems, are the ASTM D2240 type A and type D scales. The A scale is for softer plastics, while the D scale is for harder ones. However, the ASTM D2240-00 testing standard calls for a total of 12 scales, depending on the intended use; types A, B, C, D, DO, E, M, O, OO, OOO, OOO-S, and R.
Each scale results in a value between 0 and 100, with higher values indicating a harder material. In one or more embodiments, there is a 1000-fold difference durometer between the soft component and the rigid compound.
 In one or more embodiments, the functionally graded connector has a modulus of elasticity that ranges over one order of magnitude and can range up to over four orders of magnitude. In one or more embodiments, the functionally graded connector has a modulus of elasticity that ranges over three orders of magnitude (from approximately lMPa to 1 GPa) through a functional gradient creating a structure that transitions from highly flexible (rubberlike) to fully rigid (thermoplastic-like). In one or more embodiments, the functionally graded connector ranges from a composition that is substantially 100% of the first soft material in a plurality of step gradients to a composition that is substantially 100% of the second rigid material. In one or more embodiments, the gradient is a continuous gradient. In one or more embodiments, the gradient is a stepwise gradient and the stepwise gradient can have between 5 and 100 layers of different combinations of materials. In one or more embodiments, the stepwise gradient can include n layers ranging from more than 2 to up to 100, e.g., n is 5, 6, 7, 8, 9, 10, 15, 20, 25, 30, 40 , 50, 60 , 70, 80, 90 or 100 different layers, or the number or layers can be in a range bounded by any of the values provided hereinabove. While the number of stepwise gradients can 100 or higher, practical results shown herein demonstrate that lower n values of about 10 provide a significant advantage in stress distribution.
 In one or more embodiments, the rigid component can be integrated into the soft robot as a structural element, that is, to provide structure or support to the soft components of the hybrid robot. For example, the rigid component can serve as a support in the robot and take the form of arches, beams, columns, plates, platforms, cavities and/or braces. In one or more embodiments, the rigid component can be configured to serve as a load bearing element. The load bearing element can be a plate or base that can carry the cargo, load or
equipment needed for the locomotion, actuation or operation of the soft robot. By way of example, the rigid component can be a platform that houses an on-board air compressor for generation of actuation pressure of the device. In other embodiments, the rigid component can be a platform that houses diagnostic equipment and/or other functional equipment used by a soft robot.
 In other embodiments, the rigid component can be a tool that is integral with the soft body of the hybrid robot. Thus, for example, the soft robot can integrate tools such as hard grippers, cutting implements, and the like. Because the rigid tools are bonded to the soft body using a functionally graded interface, the bond is strong and the stresses that arise during its use can be distributed through the soft body.
 The hybrid robot includes at least one pneumatic or hydraulic chamber, e.g., an expandable chamber, that can be pressurized for actuation of a desired motion by the robot. In one or more embodiments, the pneumatic or hydraulic chamber includes a flexible or expandable region that enlarges or expands on pressurization and a stiffer, less expandable region that resists expansion or enlargement on pressurization. The location and arrangement of the expandable and inextensible regions help to define the actuation motion of the hybrid robot. The relative flexibility of the expandable region and stiffness of the inextensible region can be a function of the relative wall thickness in the pneumatic or hydraulic chamber (a thicker wall will be relatively less extensible than a thinner wall of the same composition). The relative flexibility of the expandable region and stiffness of the inextensible region can be derived by positioning of the flexible chamber walls against a rigid element of the hybrid robot (the flexible component will expand, while the rigid component does not deform). In one or more embodiments, the rigid component can make up the inextensible portion of the pneumatic or hydraulic chamber. The expandable chamber can be 3D printed and integral with the soft and rigid components of the hybrid robot. See, e.g,. the jumping robot discussed
herein below. In other embodiments, the expandable chamber can be an additional element that is combined with the hybrid robot to provide motion. Elements of the hybrid robot can be 3D printed to create a cavity or housing for securing the expandable chamber. See, e.g., the radially gripping hybrid robot disclosed herein below.
 In one or more embodiments, the hybrid robot is manufactured using a multimaterial 3D printer (e. g. Connex500, Stratasys) to directly print the rigid component of a robot and the soft material components for actuation, obviating the need for complex molding techniques or assembly. The fabrication of soft robots using multimaterial 3D printing has numerous advantages over traditional molding techniques. This strategy promotes high-throughput prototyping by enabling rapid design iteration with no additional cost for increased morphological complexity. By allowing designers greater freedom, 3D printing also facilitates the implementation of modularity and the separation of power and control actuators. Beyond soft robotics specifically, the ability to print a single structure composed of multiple materials enables investigation into mechanically complex designs, without the drawbacks of complicated assembly or inconsistent manufacturing repeatability. One such design is a modulus gradient that eases the transition from soft to rigid components through stress reduction at the interface of materials mismatched in compliance. Although the materials available to this fabrication strategy are currently limited, and perhaps best suited to the fabrication of prototype devices, future development of materials compatible with 3D printing will only enhance the relevance of this approach. Variable elasticity graded fabrication of polymer materials is discussed in "Functionally Graded Rapid Prototyping" available at http.;//T £^
prototyping (downloaded July 6, 2016), which is incorporated by reference.
 It should be apparent to one of skill in the art that both the hybrid robot shapes as well as the nature of the chemical compositions and the gradient steepness can be varied to allow a stable hybrid robots over a range of performance conditions.
Hybrid Robot using Rigid Component as a Tool
 An embodiment, in which the rigid component is a tool is exemplified by the fabrication of a soft robot incorporating a 'tooth'. A tooth serves as a gripping or punching tool that can, for example, assist a soft robot in locomotion or object handling. The seamless functionally graded synthetic structure was inspired by the squid sucker ring tooth that has high strength and is compliant and resilient. The fabrication uses a multi -material 3D printer (e.g. Connex500, Statasys Ltd.) to directly print simplified functionally graded cone-shaped tooth geometry, using 10 different materials with an elastic modulus ranging from 0.36 MPa to 1 GPa (Shore S 27 (softest) to Shore D 83 (hardest).
 FIG. 1 shows the schematic illustration of tooth models on soft substrates corresponding 3D printed samples used in the study. All 30 mm tooth models are built on 10 mm thick substrate fabricated out of the softest material (0.36 MPa) to imitate the soft muscular mass of squid suckers.
 The printed samples of a functionally graded tooth are then tested under shear and compressive force, and are compared against fully rigid tooth and fully flexible tooth models for contact deformation, gripping ability and resilience. In order to validate the design a commercial finite element package (Abaqus/Standard) can be used to simulate and compare the mechanical response of the conical samples under shear and compressive loading with experimental results.
 To mimic the shear forces applied on the tooth, shear tests on synthetic tooth models were performed by holding the tip of the tooth in a 3D printed substrate pre- punctured with a conical depth profiled hole and shearing the samples by applying
displacement (uy) perpendicular to the longitudinal axis of the tooth using a custom designed holder. The cross-sectional view of the substrate, displaying depth profile of the hole and position of the synthetic tooth in the substrate is shown in FIG. 3 A. The objective of this test is to simultaneously measure the flexibility and gripping ability of the tooth under shear stress and the durability of the interface adjoining the tooth and base. The graph in FIG. 3B shows the shear response of the synthetic rigid (upper), functionally graded (middle), and soft (lower) conical tooth. The solid line plots and the surrounding shaded region represents the experimental data and standard deviation obtained from four different samples, respectively, for each tooth model. For physical comparison, the position of the tip and the magnified image of the tooth-base interface at the lateral displacement of x = 13 mm for three tooth models is also shown in FIG. 3C. Among three synthetic tooth models, the soft tooth shows the lowest stiffness, requiring the least amount of force per unit displacement. However at a displacement of only uy = 12.84 mm, the tip comes out of the punctured hole, and hence the soft tooth has the least gripping ability compared to the rigid and gradient tooth models, also shown in FIG. 3C (top images). In contrast, the rigid tooth shows the maximum visible deformation at the contact adjoining the tooth and base, and fails at the tooth-base interface at a similar displacement (uy = 12.91 mm), which is also evident in FIG. 3C (bottom images). Ultimately, the functionally graded tooth is the optimal model for lateral gripping and shearing. It utilizes the rigid tip for good grip and soft-soft tooth-base connection for reducing the interfacial stress. The modulus gradient from rigid tip to soft base optimizes the bending stiffness and diffuses the stress within the structure that reduces the local stress concentration at the tooth-base interface. The simulated mechanical response for
displacements ranging from uy = 0 mm to uy = 6 mm represented by dashed lines matches closely with the experimental results, thus verifying the experimental observations and results.
 Images of the simulated maximum principal strain distribution within the three tooth models at displacements ranging from uy = 2mm, 4mm and 6 mm are shown in FIG. 3D. The strain maps are generated by clamping the tooth at 11 mm below the tip and applying the displacement perpendicular to the length of the tooth, similar to the experimental set up shown in FIG. 3 A. These simulations further support the experimental observations. In all tooth models, the strain increases with increasing displacement. Clearly, the maximum principal strain at any displacement is highest in the rigid tooth structure (~ 2.5 fold at x = 6 mm in comparison with soft tooth), and is concentrated at the interface that leads to deformation at tooth-base interface. In contrast the strain is uniformly distributed in the soft tooth from tip to base, providing flexibility but at the cost of gripping ability. The gradient tooth design helps to diffuse the strain at the interface towards the tip yet maintains low strain at the tip, resulting in a stronger interface and good grip with some compromise in flexibility.
 In order to understand the mechanics of tooth penetration into prey, compression tests were run on the designed tooth models. To apply uniform compressive force on the structure and avoid a stress singularity on the conical tips in the simulations, a truncated conical tooth geometry was used for all three models. The truncated geometry is achieved by cutting the cone down by only 5 mm from the top. See, e.g., FIG. 3D. The sample is sandwiched between two compression plates and stress is applied by displacing the top plate by uy = 0 mm to uy = - 2.5 mm. The schematic of experimental set-up for compressive loading and the direction of the plate displacement are shown in FIG. 4A. FIG. 4B shows the experimental (solid line) and simulated (dashed line) response of the rigid conical tooth (upper curve), functionally graded tooth (middle curve), and soft tooth (lower curve) conical tooth under compressive loading. In each example, the experimental and simulated results were similar enough that the dashed and solid lines appear superimposed on one another. The mechanical response recorded from four different samples for each tooth model shows a
small deviation, which is characterized by the colored shaded areas. The three curves show a linear relationship between force and displacement, with the slope defining the stiffness of the samples. As expected, the rigid tooth has the largest stiffness, while the stiffness of soft tooth is 98% lower compared to rigid cones. Indeed, the soft tooth design is not desirable, since the teeth are too flexible to penetrate into the prey. Compared with other tooth designs, the gradient design has a moderate stiffness, one third of the value of rigid design. To further explore the best design, the contour map of the principal strain was plotted for the three tooth models at compressive displacement of uy = 0.5 mm, 1 mm, and 1.5 mm, shown in FIG. 4C. Although the applied strain is not transferred to the tooth-base interface in the soft design, the high concentrated strain of 40% at uy = 1.5 mm around the tip leads to bending of the tip. In contrast, the strain is localized around the interface for the rigid tooth design with a maximum value of 40% at the same displacement. This suggests the probability of permanent deformation at the cone- substrate interface during compression (i.e. during prey penetration) and hence indicates low resilience. Finally, in the gradient design the strain is more evenly distributed, although not optimized in this case. The maximum local principal strain at displacement, uy = 1.5 mm, is 11%, which is significantly lower than the maximum strain in the other two cases. Hence, the gradient design is most attractive in terms of failure resistance, while still possessing a moderate stiffness.
 While testing described above was conducted using a cone angle of 15° (see, FIG. 1 for description f cone angle), other angles were evaluated. The cone angle provides a structural gradient, in addition to the compositional gradient, and was found to also affect stress distribution. Tooth samples having a cone angle of 8° (and same functional gradient as described above) provided too steep a structural gradient and the hard material tip could not withstand the applied strain and bent out of the experimental set up of FIG. 3 A. On the other
hand, tooth samples having cone angles of 20° (and same functional gradient as described above) were not able to bend and failed more readily at the tooth/base interface.
 In other embodiments, the steepness of the functional gradient was found to affect the hybrid robot performance. Similar mechanical testing on tooth samples having 8 layers in the same 30 mm height responded differently under shear force. The fewer number of steps required a larger step difference, which causes the top two more rigid layers to bend independently of the remaining 6 layers.
 It should be apparent to one of skill in the art therefore that both the hybrid robot shapes as well as the nature of the chemical compositions and the gradient steepness can be varied to allow a stable hybrid robots over a range of performance conditions.
 The mechanical modeling of an individual tooth provided guidance to the design of a radially gripping hybrid robot. The hybrid robot includes an array of functionally graded teeth connected to a pneumatic ring like actuator. The soft robot mimicked the circular muscles in a squid sucker. The flexibility of the tooth structure and its capability to withstand high-localized stresses is demonstrated. FIG. 2A shows a 3D printed ring of radially arranged functionally graded teeth used in the gripper integral with a base made of the soft component material. In application, the soft base can be part of a larger robot so that the tool (radial gripper) is integrated into a soft robot. The soft body of the robot can be advantageous in many applications, such in medical environments where soft tissue is exposed. FIG. 2B shows a close up of an individual tooth, highlighting the gradient from rigid to flexible phases. FIG. 2C shows the fully 3D printed biologically inspired hybrid gripper with 6 radially aligned graded teeth printed directly on fully soft base according to one or more embodiments. There is an internal tube-based actuator that when inflated, bends the teeth inward as seen in FIG. 2D. FIG. 2E shows a cross-section through the gripper, revealing the internal cavity that houses the pneumatic actuator (not shown). The internal cavity is defined
by an outer restraining wall that serves as the stiffer, less expandable region that resists expansion or enlargement on pressurization. The restraining wall is also 3D printed with the formation of the hybrid robot and is made of a material having a durometer intermediate to the soft and rigid component materials.
 In one or more embodiments, the individual teeth can be secured to the soft component base using a hinge made up of the soft material. A soft material hinge may be desired when low actuation pressures are used or a greater range of actuation displacement is desired. An exemplary device is shown in FIG. 2F.
Hybrid Robot using Rigid Component as a Structural Element
 In one embodiment, the rigid component serves as a structural element. An exemplary robot having a rigid component structural element integral with a soft component is shown in FIGs. 5A-5C, which illustrates a 3D printed fabrication of a soft robot 500 that incorporates a robot body 510 having rigid cavity 520 that can serve as a load bearing element. The load bearing element can carry cargo, load or equipment needed for the locomotion, actuation or operation of the soft robot. In one embodiment, element 520 serves as a load bearing platform to carry an air compressor, battery and supporting
 In one or more embodiments, the robot is capable of jumping by combustion expansion of gases in the robot interior. The robot body 510 , shown in FIGS. 5A-5C, is composed primarily of two nested hemispheroids 520, 530, that together form an internal volume of the robot. The flexible bottom hemispheroid 530 features a small depression 540 that provides an initial volume into which oxygen and butane are injected. Ignition 550 of the gases causes a volumetric expansion 560, launching the robot into the air (FIGS. 5A). FIG. 5B shows the ignition sequence of fuel delivery, mixing, and sparking, in which butane and oxygen are alternately delivered to the combustion chamber (to promote mixing); after a
short delay to promote additional mixing of the fuels, the gaseous mixture is ignited, resulting in combustion, and leg inflation occurs concurrently with fuel delivery, and leg deflation begins shortly after landing. The top hemispheroid 520 has a modulus of elasticity that ranges over three orders of magnitude (from approximately lMPa to 1 GPa) through a stepwise gradient of nine different layers, creating a structure that transitions from highly flexible (rubber-like) to fully rigid (thermoplastic-like). The gradient spans the entire top hemisphere. Looking at FIG. 5A, in the last image (in which the robot is in the air), everything above the center of the explosion is graded (i.e. everything that is part of the top hemisphere) whereas everything below (i.e. the bottom hemisphere) is not graded (it is totally flexible). This applies to the large central explosive actuator as well as the pneumatic legs. The materials used were digital combinations of commercial 3D printing materials offered by Stratasys, specifically VeroWhitePlus RGD835 (rigid) and TangoPlus FLX930 (flexible). The Young's moduli of the materials used in the gradient are presented in Table 1.
Table 1. Young's Modulus for Material Layers
 In addition to providing a mechanical interface for the rigid control components, the rigid portion of the top hemispheroid also prevents undesired expansion locally and focuses the energy of combustion into the ground, enhancing the jumping efficiency.
Pneumatic legs 570, which use a nested hemi-ellipsoid components having and upper rigid hemi-ellipsoid 580 and a lower soft hemi-ellipsoid 585 connected through functionally graded connector similar to that of the main body, surround the central explosive actuator and are used to tilt the body before a jump, controlling the direction of locomotion. The legs are also 3D printed and they can have the same gradient as the body. The pneumatic legs can be secured to the body by glue (adhesive) or mechanical fasteners. This separation of power and control actuators simplifies actuation and gives greater control over direction.
 In order to simplify prototyping, a modular design was used with a rigid core module 590 containing the control components (which are expensive and change infrequently during design iteration of the body), connected through a predefined interface to the body of the robot (FIG. 5C). This modularity enables efficient iteration of the robot body design, as well as rapid replacement in the case of destructive testing. The core module contains a custom circuit board, high-voltage power source, battery, miniature air compressor, butane fuel cell, bank of six solenoid valves, oxygen cartridge, pressure regulator, and an internal network of channels to facilitate interfacing between the components as necessary. FIGS. 6A and 6B provide a schematic diagram and illustration of the components of the core module and their connections to the soft robot body. The core module is mechanically attached to the rigid portion of the body with a layer of high-strength mushroom-head fasteners. Otherwise, it interfaces with the body only through four tubes (three pneumatic tubes for the legs and one tube for fuel delivery to the combustion chamber) and two wires (which produce the spark in
the combustion chamber). The core module can be protected from impact by a protective shield 595.
 Characterization of nine 3D-printed materials with a set of mechanical tests informed the design of the 3D-printed rigid/soft robot. Qualitative twisting experiments were performed to gain an intuitive understanding of the response of the various materials (FIG. 7). Mechanical testing on a universal testing machine (Instron 5544, Instron) yielded quantitative values of material properties (supplementary text). This information was used to simulate the operation of the robot using finite element analysis (FEA) software, which allowed comparison of the relative efficiency of jumping robots with different material distributions. Further simulations allowed examination of the differences in stress concentrations as a function of material distribution. Simulations were done with (undeformed) beam dimensions of 25.4 mm x 152.4 mm x 1.0 mm. The maximum stresses in each of these beams are 0.35 MPa, 0.54 MPa, and 0.37 MPa, respectively. Compared to the half rigid and half flexible beam, the fully flexible and gradient beams experience maximum stresses of 64.8% and 68.5%, respectively. The results from these studies revealed that, when compared to an abrupt material transition, the incorporation of a graded interface could achieve a 30% reduction in maximum stress upon tensile loading, reaching a value comparable to the maximum stress observed in a soft, single material model. Although a perfectly smooth gradient from rigid to flexible would have been ideal, the capability of the fabrication technique was limited to a stepwise gradient of at most nine materials. Additional motivation for the use of a gradient was derived from considerations of the effect of stress concentrations on interfacial failure in multi -material systems.
 The actuation strategy necessitated a flexible bottom hemispheroid, whereas the off-the-shelf control components required a rigid housing; however, the stiffness distribution of the top hemispheroid was unconstrained. Thus, to determine how the material properties
of the top hemispheroid would affect jumping, three cases were simulated: (i) a flexible top with a small rigid portion to mount control hardware, (ii) a top fully rigid, and (iii) a fully rigid top (FIG. 8). Simulations showed that the flexible top was inefficient at directing the energy of combustion into the ground and propelling the robot, suggesting weak jump performance. As expected, the simulated rigid top robot produced the highest ground reaction force, whereas the gradient top robot exhibited a performance between the two extremes.
 Additional simulations were carried out to investigate the behavior of the three designs during the impact of landing (FIG. 9A, showing force vs. displacement curves for a rigid top robot (left curve), gradient top robot (middle curve) and soft top robot (right curve). The results (also shown in the simulation in FIG. 9B) indicate that the rigid top robot experiences a given reaction force (50 N) at a much smaller deformation than either the gradient or flexible top robots. Immediately upon impact, the rigid top robot experiences an abrupt increase in force, whereas the gradient top robot experiences a more moderate increase. The flexible top robot sees almost no increase, until the small rigid portion strikes the ground, initiating a rapid increase akin to that of the rigid top robot. Integrating the force displacement curves (up to 50 N), the rigid and flexible top robots only absorb 13 and 73% (respectively) of the impact energy that the gradient top robot absorbs. The increased energy absorbed by the gradient top robot during impact suggests that it will be most successful at distributing the impulse over a longer duration, therefore reducing peak stresses and providing the least violent landing.
 These simulation results were experimentally verified by 3D printing the different test cases. A jumping robot with a completely rigid top was able to jump 1.12 m untethered using 40 ml of butane and 120 ml of oxygen. Identical testing conditions on a gradient top robot produced a jump of 0.25 m. A flexible top robot was deemed impractical to print
because of the predictions from FEA. As predicted by the simulations, the gradient top robot was less efficient at jumping. However, the gradient top robot was better able to withstand the impact of landing. In one test, the body of the rigid top robot shattered upon landing, surviving a total of just five jumps; the gradient top robot survived more than twice that number of jumps and remained operational. Other nearly identical gradient top robots survived over 100 jumps (in 81% of these tests, we removed the core module from the body and delivered the combustion products and ignition sparks through a tether to simplify testing, reducing the system mass to about 50% that of the untethered system). To provide a direct comparison in landing behavior, the gradient top robot was additionally dropped from the maximum height achieved by the rigid top robot and successfully survived 35 falls (supplementary text). The stiffness gradient provides the necessary rigidity to transfer the impulse of combustion to generate effective jumping, and the compliance of the base absorbs and dissipates the energy of the landing impact. By trading the jumping efficiency of the rigid robot for an improved ability to survive landings, the gradient top robot demonstrated a greater overall robustness. Further testing on the gradient top robot showed high resilience and good performance. This robot autonomously jumped up to 0.76 m (six of its body heights) high and demonstrated directional jumping of up to 0.15m (0.5 body lengths, 20% of jump height) laterally per jump. Unlike previous combustion-powered soft jumpers that were either tethered or achieved only a few untethered jumps due to inconsistent connection of electrical and mechanical components at the interface of the rigid and soft components, this design allowed for many successful jumps with a single soft robot (21 untethered jumps and 89 tethered jumps). Another jumper design has also shown the ability to perform multiple jumps, can operate on uneven terrain, and can even recover from landing in any orientation, although at the sacrifice of directional control. The high energy density of the fuels theoretically allows onboard storage of sufficient fuel for 32 consecutive jumps
(supplementary text). The bodies were extremely robust, surviving dozens of jumps before they became unusable. The monolithic design has no sliding parts or traditional joints that can be fouled or obstructed by debris or rough terrain, and the nested design requires minimal deformation for actuation. There was no significant damage to the soft (or rigid) body materials due to the brief exposure to elevated combustion temperatures and flames.
Materials and Methods
 The Stratasys website provided general information regarding the 3D printed material used in this project (specifically, the "Poly Jet Materials Data Sheet" and the "Digital Materials Data Sheet"). Published data sheets indicate that the materials used in the robot ranged in hardness from Shore A 27 to Shore D 83. We performed additional analysis of the 3D printed material through different tests on a universal testing machine (Instron 5544, Instron). Cyclic testing indicated that at high rates of extension, significant hysteresis was present due to the viscoelastic properties of the 3D printed material. However, at rates below 0.03125 mm/s, all viscous effects were negligible and the material behaved elastically. Each of the nine different materials used in the stiffness gradient was tested in a standard tensile test (ASTM D 638, Type IV), performed at 0.03125 mm/s to eliminate any rate dependent behavior. From this test, we obtained values for the shear and Young's moduli of each material, which were subsequently used in the simulations.
 We conducted extension tests on samples that featured either an abrupt transition from the softest to the most rigid material or a more gradual step-wise transition by incorporating materials of intermediate moduli. In fatigue tests in which we repeatedly stretched the samples to an extension of 5 mm (20% of the test section) at 0.03125 mm/s, the discrete samples failed after an average of 436 cycles, whereas the gradient samples lasted an order of magnitude longer (most samples were discontinued after 24 hours of testing, over 8640 cycles).
 We designed the robots using SolidWorks, a 3D computer aided design (CAD) software, and printed them with a multimaterial 3D printer (Connex500, Stratasys Ltd.). The body was printed as a single piece. We cleared residual support material from the 3D printing process through a small excavation hole using a high pressure washer (Powerblast High Pressure Water Cleaner, Balco UK). After clearing all of the support material, we sealed the excavation hole by attaching a custom 3D printed cap to the body with
cyanoacrylate adhesive (Loctite 416, Henkel AG & Company, KGaA).
 In addition to the custom fabricated body, the robot consisted of a number of off- the-shelf components. These include a lithium polymer battery (E-flite 180 mAh 2S 7.4V 20C, Horizon Hobby Inc.), a mini-diaphragm pump (KPV-14A, Clark Solutions), six miniature pneumatic solenoid valves (X- Valve, Parker Hannifin Corp.), a butane fuel cell (RC-31, Master Appliance Corp.), and a pressure regulator (PRD-2N1-0-V, Beswick
Engineering Co.). Oxygen was stored in a repurposed 16g C02 cartridge, outfitted with a piercing fitting (GCP-1038-3V, Beswick Engineering Co.) and ball valve (MBV-1010-303- V, Beswick Engineering Co.). The high voltage source was obtained from components of a continuous ignition gas lighter (57549 Olympian GM-3X Gas Match, Cameo Manufacturing Inc.). The circuit board was custom designed with an Atmel ATmegal68 microcontroller, and was programmed using the Arduino IDE.
 The timing sequence on the solenoid valves determined how much butane and oxygen was delivered to the combustion chamber. After fixing the settings on both the pressure regulator (which was in-line with the oxygen cartridge) and the valve of the butane fuel cell, we determined flow rates of oxygen and butane by opening the respective valves for a predetermined amount of time and measuring the amount of gas delivered by bubbling into an inverted graduated cylinder. This procedure was repeated throughout the testing period to ensure consistency.
 The first step in the testing procedure was refilling the butane fuel cell (if necessary) and refilling the oxygen cartridge. The oxygen cartridge was filled from a supply tank of oxygen regulated to 90 psi, and then sealed using the ball valve. It was then threaded into the regulator on the robot, keeping the ball valve closed until the initiation of a new test. Due to the rapid use of oxygen, five oxygen cartridges were filled and used during each testing cycle.
 We explored the space of butane to oxygen ratios extensively during testing, and found a baseline mixture of 50 mL of oxygen and 24 mL of butane per jump to be the most consistent. The volume of the oxygen cartridges and the filling pressure limited the number of jumps on a single cartridge to two (or three if the amount of fuel delivered was reduced appropriately).
 The circuit board was designed to run the same program each time the robot was turned on. Adjustments to the program required plugging the circuit board directly into a computer and opening the Arduino IDE.
 Experiments were recorded using both a DSLR camera (D600, Nikon Inc.) and a high-speed camera (Phantom v710, Vision Research Inc.). The latter was run at 1000 or 2000 frames per second and operated using Phantom Camera Control (PCC) software.
 Non-linear finite element analysis was performed using the commercial package Abaqus/Explicit (v6.12) (Abaqus Unified FEA, Dassault Systemes). All materials were modeled using a Neo-Hookean material model (27), each with a specific initial shear modulus. The shear moduli were determined experimentally by performing uniaxial tension tests and fitting the stress-strain curves using a least squares approximation.
 To qualitatively show the effect of using materials with different moduli within the same structure, we deformed three beams with a different material distribution by twisting them 180 degrees. These beams were modeled using approximately 10,000 tetrahedral
elements (Abaqus element code C3D4), and quasi-static conditions were assured by using a relatively long simulation time, as well as a small damping factor.
 We simulated the behavior upon pressurizing the internal cavity of the robot, neglecting the dynamic effects that occur in experiments when actuating the robot. Instead, we ensured quasi-static conditions to generate smoother results that enable a better comparison between the different designs of the top hemispheroid. We modeled the hemispheroid using the same shear moduli as used for the beams, but to reduce computation time we used approximately 50,000 triangular shell elements (Abaqus element code S3R), instead of using solid tetrahedral elements. In the simulations, we fully account for contact between all faces of the model. We inflated the internal cavity by using the surface-based fluid cavity capability in Abaqus, and monitored the pressure during inflation. To determine the force that was generated during inflation, we fixed the top center of the top hemispheroid (1 cm diameter) and measured the vertical reaction force during inflation.
 To determine the forces generated during impact, we used the same conditions as those used in inflation. All dynamic effects were neglected; instead the robot was slowly forced into the ground by displacing the top center of the top hemispheroid (1 cm diameter) down towards the ground, while monitoring the reaction force in the upward direction.
 Unless otherwise defined, used or characterized herein, terms that are used herein (including technical and scientific terms) are to be interpreted as having a meaning that is consistent with their accepted meaning in the context of the relevant art and are not to be interpreted in an idealized or overly formal sense unless expressly so defined herein. For example, if a particular composition is referenced, the composition may be substantially, though not perfectly pure, as practical and imperfect realities may apply; e.g., the potential presence of at least trace impurities (e.g., at less than 1 or 2%) can be understood as being within the scope of the description; likewise, if a particular shape is referenced, the shape is
intended to include imperfect variations from ideal shapes, e.g., due to manufacturing tolerances. Percentages or concentrations expressed herein can represent either by weight or by volume.
 Although the terms, first, second, third, etc., may be used herein to describe various elements, these elements are not to be limited by these terms. These terms are simply used to distinguish one element from another. Thus, a first element, discussed below, could be termed a second element without departing from the teachings of the exemplary embodiments. Spatially relative terms, such as "above," "below," "left," "right," "in front," "behind," and the like, may be used herein for ease of description to describe the relationship of one element to another element, as illustrated in the figures. It will be understood that the spatially relative terms, as well as the illustrated configurations, are intended to encompass different orientations of the apparatus in use or operation in addition to the orientations described herein and depicted in the figures. For example, if the apparatus in the figures is turned over, elements described as "below" or "beneath" other elements or features would then be oriented "above" the other elements or features. Thus, the exemplary term, "above," may encompass both an orientation of above and below. The apparatus may be otherwise oriented (e.g., rotated 90 degrees or at other orientations) and the spatially relative descriptors used herein interpreted accordingly. Further still, in this disclosure, when an element is referred to as being "on," "connected to," "coupled to," "in contact with," etc., another element, it may be directly on, connected to, coupled to, or in contact with the other element or intervening elements may be present unless otherwise specified.
 The terminology used herein is for the purpose of describing particular embodiments and is not intended to be limiting of exemplary embodiments. As used herein, singular forms, such as "a" and "an," are intended to include the plural forms as well, unless the context indicates otherwise.
 It will be appreciated that while a particular sequence of steps has been shown and described for purposes of explanation, the sequence may be varied in certain respects, or the steps may be combined, while still obtaining the desired configuration. Additionally, modifications to the disclosed embodiment and the invention as claimed are possible and within the scope of this disclosed invention.