Patent Publication Number: US-10760597-B2

Title: Soft robots, soft actuators, and methods for making the same

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This Utility Patent Application claims priority under 35 U.S.C. § 371 to International Application Serial No. PCT/US2016/029584, filed Apr. 27, 2016, which claims the benefit of Provisional Patent Application No. 62/153,165, filed Apr. 27, 2015; which are both incorporated herein by reference in their entirety. 
    
    
     BACKGROUND 
     Conventional robots are sometimes stiff and might not be suitable for moving through passages, such as passages that might have bends. Moreover, such robots might not be suitable for use in arteries in a living organism, such as a human body, in that such robots could possibly puncture artery walls. 
     More recently, fiber-reinforced elastomeric enclosure (FREE) actuators have been contemplated for possible use with or as a “soft” robot. FREEs are described in U.S. Application Publication No. 2015/0040753 (Bishop-Moser et al.), and entail one or more continuous fibers applied to a length of a hollow elastomeric cylinder in a helical pattern having a constant or uniform pitch. In the presence of a fluid medium, the FREE actuator experiences uniform change in shape. While the described FREE actuators appear promising for potential soft robot applications, the physical constraints presented by many soft robot usage environments may not be fully addressed. 
     SUMMARY 
     The inventors of the present disclosure recognized that a need exists for actuators, soft robots, and methods of designing and making actuators and soft robots that overcome one or more of the above-mentioned problems. 
     Some aspects of the present disclosure are directed toward a material-mapped actuator useful as or as part of a soft robot. The actuator exhibits mechanical properties that spatially vary along a coordinate system of the actuator. The actuator includes an actuator body that has an initial shape with a corresponding initial map of mechanical attributes consisting of locally-varying stiffness at each point in a volume of the actuator body. Further, the actuator body is configured to change to a new, different shape or different distribution of mechanical properties upon being activated by an actuation medium. The initial spatially-varying map of mechanical attributes influences and determines the new shape or distribution. In some embodiments, the actuator includes a material applied to a tubular body, such as locally-oriented fibers, meshes, threads, etc. that induce desired material anisotropies and strain limiting behaviors where the fiber orientation of pitch is free to vary in any direction along the actuator body. In other embodiments, the material-mapped actuator incorporates a spatially-varying distribution of mechanical properties that dictates multiple desired shapes as the actuation medium is applied, including, for example, an actuation sequence in which the actuator transitions from a first shape to one or more desired intermediate shapes, and from the desired intermediate shape(s) to a desired final shape. 
     Other aspects of the present disclosure are directed toward a method for making a soft robot for performing a specified procedure or task. The method include receiving procedure-related information indicative of at least a desired initial shape and a desired final shape, and optionally one or more desired intermediate shapes, of the soft robot in preforming the procedure. Design parameters for one or more material-mapped actuators are determined based upon the received procedure-related information, and one or more material-mapped actuators are formed as a function of the determined design parameters. The inverse design techniques of the present disclosure can optionally further include generating a manufacturing blueprint based upon material mapping, for example by operating a dithering algorithm. 
     Yet other aspects of the present disclosure are directed toward systems for manufacturing a soft robot consisting of one or more actuators. The system includes a mapping module and a manufacturing module. The mapping module includes a computing device operating on computer instructions to generate material mapping and manufacturing blueprint information. The manufacturing module is operated in accordance with the manufacturing blueprint information to generate one or more material-mapped actuators, such as by an additive or subtractive process. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIGS. 1A-1F  illustrate examples of different configurations of a robot during motion of the robot. 
         FIGS. 2A and 2B  illustrate an example of twisting a section of a robot. 
         FIGS. 3A and 3B  illustrate an example of bending a section of a robot. 
         FIG. 4  is an example illustrating a robot moving through an opening. 
         FIG. 5  is another example illustrating a robot moving through an opening. 
         FIG. 6  illustrates an example of a tube that includes a structure that has an angle that varies with distance along the length of the tube. 
         FIG. 7  illustrates an example of a stiffness distribution that allows a tube to be transformed into a spiral in response to an internal pressure. 
         FIG. 8  illustrates an example a stiffness distribution that allows a tube to extend in response to an internal pressure. 
         FIG. 9  illustrates an example a stiffness distribution that allows a tube to twist in response to an internal pressure. 
         FIG. 10  illustrates an example a stiffness distribution that allows a tube to bend in response to an internal pressure. 
         FIG. 11  illustrates an overview of a mathematical approach to designing a material-mapped actuator in accordance with principles of the present disclosure. 
         FIG. 12  represents conversion of material mapping information to manufacturing blueprint information using a dithering algorithm in accordance with principles of the present disclosure. 
         FIG. 13  is a schematic illustration of a system for manufacturing a soft robot in accordance with principles of the present disclosure. 
         FIG. 14  illustrates an example of a robot having a pilot line in a main flow passage. 
         FIG. 15  illustrates an example of a robot having a pilot line in a wall. 
         FIG. 16  illustrates an example of a robot having pilot lines at different locations in a wall. 
         FIGS. 17A-17C  illustrate an example of a valve at different states. 
         FIG. 17D  is a perspective view of a bushing useful with the valve of  FIGS. 17A-17C . 
         FIG. 18A  illustrates an example of a rotary valve. 
         FIG. 18B  illustrates an example of an inner portion of a rotary valve. 
         FIG. 18C  illustrates an example of an outer portion of a rotary valve. 
         FIG. 18D  illustrates a particular state of a rotary valve. 
         FIG. 19A  illustrates an example of an axially sliding valve. 
         FIG. 19B  illustrates an example of an inner portion of an axially sliding valve. 
         FIG. 19C  illustrates an example of an outer portion of an axially sliding valve. 
         FIG. 19D  illustrates a particular state of an axially sliding valve. 
         FIG. 20A  is a cross-sectional diagram of an asymmetric passive valve useful with soft robots of the present disclosure. 
         FIG. 20B  is a table of forward and reverse direction flow pressure of the valve of  FIG. 20A  at different cone angles and thicknesses. 
         FIG. 20C  is a contour plot of pressure ratios provided by the valve of  FIG. 20A  at different cone angles and thicknesses. 
         FIG. 21  is a simplified view of another soft robot in accordance with principles of the present disclosure traversing through various types of soil. 
     
    
    
     DETAILED DESCRIPTION 
       FIGS. 1A-1F  illustrate different configurations of a robot  100  in accordance with principles of the present disclosure and during motion of robot  100 . In  FIG. 1A , robot  100  is in an initial, unactuated state and is at an initial location, where a distal end  102  of robot  100  is at a point A- 1  and a proximal end  104  of robot  100  is at a point A- 2 . In general,  FIGS. 1A-1F  show the different configuration states of robot  100  as robot  100  moves from its initial location in  FIG. 1A  to a location in  FIG. 1F , where distal end  102  is at a point D and proximal end  104  is at a point C- 2 . 
     In some examples, robot  100  might be referred to as a “soft” (e.g., a soft catheter) robot that can conform to the shape of a flow passage, e.g., the flow passage in an artery in a living organism, such as a human body, e.g., without damaging the wall of the artery. For example, robot  100  might be formed from elastomers (e.g., silicone, thermoplastic elastomers, isoprenes, rubber, latex, etc.) that might be reinforced with fibers (nylon, woven carbon fibers, etc.) or that might include elastomers having different stiffnesses and/or different thicknesses that allow robot  100  to conform to the different shapes of different flow passages. 
     Robot  100  includes a section or actuator  110  that has a hollow core, e.g., a flow passage. Section  110  may be a circular tube, for example. Section  110  is configured to become a spiral (e.g., a section  110  spiral), as shown in  FIG. 1B , in response to a pressure of a fluid (e.g., that might be called a working fluid), such as water, saline, etc., within the hollow core of section  110 . For example, section  110  becomes a spiral in response to the hollow core of section  110  selectively receiving the working fluid from a fluid pressure source (e.g., an upstream pressure source), such as a pump (e.g. a piston-cylinder pump, a syringe, etc.) through a valve  111 - 1 , e.g., when valve  111 - 1  is actuated to an open (e.g., a fully open) state. 
     Valve  111 - 1  is between proximal end  104  and section  110 . Valve  111 - 1  selectively fluidly couples the hollow core of section  110  to the fluid pressure source that may be upstream of valve  111 - 1 . Note that section  110  in  FIG. 1A  is transformed into the spiral in  FIG. 1B  in response to the hollow core of section  110  selectively receiving the fluid from the pressure source. Selectively receiving a fluid as used herein, for example, means receiving the fluid in response to an action, such as the opening of valve  111 - 1 . 
     In some examples, valve  111 - 1  might be actuated by the working fluid, whereas in other examples valve  111 - 1  might be actuated by a pilot fluid (e.g., having the same composition as the working fluid) being directed to valve  111 - 1  by a pilot line. Note that the term “pilot fluid” is used to denote fluid that flows through pilot lines. 
     Valve  111 - 1  might be a multi-stage pressure-relief valve, for example, that is actuated into an open state in response to the pressure of the working fluid upstream of valve  111 - 1  reaching a certain pressure level, at which point the working fluid enters section  110 . In some examples, where valve  111 - 1  is opened by the pressure of the working fluid upstream of valve  111 - 1  reaching a certain pressure level, section  110  might become a spiral in response to that certain pressure level. 
     Valve  111 - 1  might be actuated into a closed state in response to a further increase in the pressure of the working fluid upstream of valve  111 - 1 , or, alternatively, in response to a decrease in the pressure of the working fluid upstream of valve  111 - 1 . In some examples, valve  111 - 1  might be opened in response to a pressure of the pilot fluid in a pilot line reaching a certain pressure level and might be closed in response to a pressure of the pilot fluid in the pilot line reaching a certain other pressure level, e.g., that might be greater than or less than the pressure of the pilot fluid that opens valve  111 - 1 . 
     In some examples, valve  111 - 1  might open (e.g., fully open) in response to the pressure of the working fluid upstream of valve  111 - 1  fluctuating (e.g., oscillating) at a certain frequency (e.g., at a resonant frequency of valve  111 - 1 ) and might close in response to the pressure of the working fluid upstream of valve  111 - 1  fluctuating at a certain other frequency (e.g., at a non-resonant frequency of valve  111 - 1 ). In other examples, valve  111 - 1  might open in response to the pressure of the pilot fluid in the pilot line fluctuating at a certain frequency (e.g., at a resonant frequency of valve  111 - 1 ) and might close in response to the pilot fluid in the pilot line fluctuating at a certain other frequency (e.g., at a non-resonant frequency of valve  111 - 1 ). 
     A section or actuator  112  has a hollow core, e.g., a flow passage, that is selectively fluidly coupled in series with the hollow core of section  110 . Section  112  may be a circular tube, for example. A valve  111 - 2  that is between section  110  and section  112  selectively fluidly couples the hollow core of section  110  in series with the hollow core of section  112 . 
     As used herein “fluidly coupled” means to allow the flow of fluid. For example, fluid is allowed to flow between the fluidly coupled hollow cores of sections  110  and  112 , e.g., from the hollow core of section  110  to the hollow core of section  112 . For selectively fluidly coupled hollow cores, fluid flows from the hollow core of section  110  to the hollow core of section  112  in response to an action, such as the opening of the valve  111 - 2  between the hollow core of section  110  and the hollow core of section t  112 . That is, for example, when a valve is between two hollow cores, the two hollow cores are selectively fluidly coupled to each other. 
     Section  112  is configured to extend, as shown in  FIG. 1C , in response to a pressure of a fluid within the hollow core of section  112 . For example, section  112  extends in response to the hollow core of section  112  selectively receiving the working fluid from section  110  through a valve  111 - 2 , e.g., when valve  111 - 2  is actuated to an open (e.g., a fully open) state. Section  112  in extends from the length in  FIG. 1B  to the length in  FIG. 1C  in response to the hollow core of section  112  selectively receiving the working fluid from the hollow core of section  110 . 
     In some examples, valve  111 - 2  might be actuated by the working fluid, whereas in other examples valve  111 - 2  might be actuated by the pilot fluid being directed to valve  111 - 2  by a pilot line. Valve  111 - 2  might be a multi-stage pressure-relief valve, for example, that is actuated into an open state in response to the pressure of the working fluid upstream of valve  111 - 2  in section  110  reaching a certain pressure level, e.g., that might be greater than the pressure level that opened valve  111 - 1 , at which point the working fluid enters section  112 . 
     In some examples, where valve  111 - 2  is opened by the pressure of the working fluid upstream of valve  111 - 2  reaching a certain pressure level, section  112  might extend in response to that certain pressure level. Valve  111 - 2  might be actuated into a closed state in response to a further increase in the pressure of the working fluid upstream of valve  111 - 2 , or, alternatively, in response to a decrease in the pressure of the working fluid upstream of valve  111 - 2 . In some examples, valve  111 - 2  might be opened in response to a pressure of the pilot fluid in a pilot line reaching a certain pressure level and might be closed in response to a pressure of the pilot fluid in the pilot line reaching a certain other pressure level, e.g., that might be greater than or less than the pressure of the pilot fluid that opens valve  111 - 2 . 
     In some examples, valve  111 - 2  might open (e.g., fully open) in response to the pressure of the working fluid upstream of valve  111 - 2  fluctuating (e.g., oscillating) at a certain frequency (e.g., at a resonant frequency of valve  111 - 2 ) and might close in response to the pressure of the working fluid upstream of valve  111 - 2  fluctuating at a certain other frequency (e.g., at a non-resonant frequency of valve  111 - 2 ). In other examples, valve  111 - 2  might open in response to the pressure of the pilot fluid in the pilot line fluctuating at a certain frequency (e.g., at a resonant frequency of valve  111 - 2 ) and might close in response to the pilot fluid in the pilot line fluctuating at a certain other frequency (e.g., at a non-resonant frequency of valve  111 - 2 ). 
     A section or actuator  114  has a hollow core, e.g., a flow passage, that is selectively fluidly coupled in series with the hollow core of section  112 . Section  114  may be a circular tube, for example. A valve  111 - 3  that is between section  112  and section  114  selectively fluidly couples the hollow core of section  112  in series with the hollow core of section  114 . 
     Section  114  is configured to become a spiral (e.g., a section  114  spiral), as shown in  FIG. 1D , in response to a pressure of a fluid within the hollow core of section  114 . For example, the hollow core of section  114  selectively receives the working fluid from the hollow core of section  112  through a valve  111 - 3 , e.g., when valve  111 - 3  is actuated to an open (e.g., a fully open) state. Note that section  114  in  FIG. 1C  is transformed into the spiral in  FIG. 1D  in response to the hollow core of section  114  selectively receiving the working fluid from section  112 . 
     In some examples, valve  111 - 3  might be actuated by the working fluid, whereas in other examples valve  111 - 3  might be actuated by the pilot fluid being directed to valve  111 - 3  by a pilot line. Valve  111 - 3  might be a multi-stage pressure-relief valve, for example, that is actuated into an open state in response to the pressure of the working fluid upstream of valve  111 - 3  in section  112  reaching a certain pressure level, e.g., that might be greater than the pressure level that opened valve  111 - 2 , at which point the working fluid enters section  114 . 
     In some examples, where valve  111 - 3  is opened by the pressure of the working fluid upstream of valve  111 - 3  reaching a certain pressure level, section  114  might become a spiral in response to that certain pressure level. Valve  111 - 3  might be actuated into a closed state in response to a further increase in the pressure of the working fluid upstream of valve  111 - 3 , or, alternatively, in response to a decrease in the pressure of the working fluid upstream of valve  111 - 3 . In some examples, valve  111 - 3  might be opened in response to a pressure of the pilot fluid in the pilot line reaching a certain pressure level and might be closed in response to a pressure of the pilot fluid in the pilot line reaching a certain other pressure level, e.g., that might be greater than or less than the pressure of the pilot fluid that opens valve  111 - 3 . 
     In some examples, valve  111 - 3  might open (e.g., fully open) in response to the pressure of the working fluid upstream of valve  111 - 3  fluctuating (e.g., oscillating) at a certain frequency (e.g., at a resonant frequency of valve  111 - 3 ) and might close in response to the pressure of the working fluid upstream of valve  111 - 3  fluctuating at a certain other frequency (e.g., at a non-resonant frequency of valve  111 - 3 ). In other examples, valve  111 - 3  might open in response to the pressure of the pilot fluid in the pilot line fluctuating at a certain frequency (e.g., at a resonant frequency of valve  111 - 3 ) and might close in response to the pilot fluid in the pilot line fluctuating at a certain other frequency (e.g., at a non-resonant frequency of valve  111 - 3 ). 
     A section or actuator  116  has a hollow core, e.g., a flow passage, that is selectively fluidly coupled in series with the hollow core of section  114 . A valve  111 - 4  that is between section  114  and section  116  selectively fluidly couples the hollow core of section  114  in series with the hollow core of section  116 . 
     Section  116  is configured to twist in response to a pressure of a fluid within the hollow core of section  116 . For example, section  116  twists in response to the hollow core of section  116  selectively receiving the fluid from section  114  through valve  111 - 4 , e.g., when valve  111 - 4  is actuated to an open state. 
       FIGS. 2A and 2B  illustrate the twisting of section  116 .  FIG. 2A  shows section  116  in an untwisted state, and  FIG. 2B  illustrates section  116  after twisting.  FIGS. 2A and 2B  show that twisting in the direction of arrow  202  about the central axis  204  of section  116  displaces open circles  210  and  220  from their respective locations in  FIG. 2A  to their respective locations in  FIG. 2B . Note that the twisting does not affect the orientation of the central axis  204 . 
     In some examples, valve  111 - 4  might be actuated by the working fluid, whereas in other examples valve  111 - 4  might be actuated by the pilot fluid being directed to valve  111 - 4  by a pilot line. Valve  111 - 4  might be a multi-stage pressure-relief valve, for example, that is actuated into an open (e.g., a fully open) state in response to the pressure of the working fluid upstream of valve  111 - 4  in section  114  reaching a certain pressure level, e.g., that might be greater than the pressure level that opened valve  111 - 3 , at which point the working fluid enters section  116 . 
     In some examples, where valve  111 - 4  is opened by the pressure of the working fluid upstream of valve  111 - 4  reaching a certain pressure level, section  116  might twist in response to that certain pressure level. Valve  111 - 4  might be actuated into a closed state in response to a further increase in the pressure of the working fluid upstream of valve  111 - 4 , or, alternatively, in response to a decrease in the pressure of the working fluid upstream of valve  111 - 4 . In some examples, valve  111 - 4  might be configured to be partially opened by different amounts, e.g., between the closed and fully open states, in response to respectively varying the pressure of the working fluid upstream of valve  111 - 4 . 
     In some examples, valve  111 - 4  might be opened (e.g., fully opened) in response to a pressure of the pilot fluid in the pilot line reaching a certain pressure level and might be closed in response to a pressure of the pilot fluid in the pilot line reaching a certain other pressure level, e.g., that might be greater than or less than the pressure of the pilot fluid that opens valve  111 - 4 . The valve  111 - 4  might be configured to be partially opened by different amounts, e.g., between the closed and fully open states, in response to respectively varying the pressure of the pilot fluid. 
     In some examples, valve  111 - 4  might fully open in response to the pressure of the working fluid upstream of valve  111 - 4  fluctuating (e.g., oscillating) at a certain frequency (e.g., at a resonant frequency of valve  111 - 4 ) and might close in response to the pressure of the working fluid upstream of valve  111 - 4  fluctuating at a certain other frequency (e.g., at a non-resonant frequency of valve  111 - 4 ). In other examples, valve  111 - 4  might fully open in response to the pressure of the pilot fluid in the pilot line fluctuating at a certain frequency (e.g., at a resonant frequency of valve  111 - 4 ) and might close in response to the pilot fluid in the pilot line fluctuating at a certain other frequency (e.g., at a non-resonant frequency of valve  111 - 4 ). In some examples, the valve  111 - 4  might be configured to be partially opened by different amounts between the fully open and closed states in response to the working fluid or the pilot fluid respectively fluctuating by different frequencies that might be between the frequency that closes valve  111 - 4  and the frequency that fully opens valve  111 - 4 . 
     A section or actuator  118  has a hollow core, e.g., a flow passage, that is selectively fluidly coupled in series with the hollow core of section  116 . A valve  111 - 5  that is between section  116  and section  118  selectively fluidly couples the hollow core of section  116  in series with the hollow core of section  118 . 
     Section  118  is configured to bend in response to a pressure of a fluid within the hollow core of section  118 . For example, section  118  bends in response to the hollow core of section  118  selectively receiving the fluid from section  116  through valve  111 - 5 , e.g., when valve  111 - 5  is actuated to an open (e.g., a fully open) state. 
       FIGS. 3A and 3B  illustrate the bending of section  118 .  FIG. 3A  shows section  118  in a straight, unbent state, and  FIG. 3B  illustrates section  118  after bending. It is seen that the central axis  310  of section  118  follows the bend in  FIG. 3B . 
     In some examples, valve  111 - 5  might be actuated by the working fluid, whereas in other examples valve  111 - 5  might be actuated by the pilot fluid being directed to valve  111 - 5  by a pilot line. Valve  111 - 5  might be a multi-stage pressure-relief valve, for example, that is actuated into a fully open state in response to the pressure of the working fluid upstream of valve  111 - 5  in section  116  reaching a certain pressure level, e.g., that might be greater than the pressure level that opened valve  111 - 4 , at which point the working fluid enters section  118 . 
     In some examples, where valve  111 - 5  is fully opened by the pressure of the working fluid upstream of valve  111 - 5  reaching a certain pressure level, section  118  might bend in response to that certain pressure level. Valve  111 - 5  might be actuated into a closed state in response to a further increase in the pressure of the working fluid upstream of valve  111 - 5 , or, alternatively, in response to a decrease in the pressure of the working fluid upstream of valve  111 - 5 . In some examples, valve  111 - 5  might be configured to be partially opened by different amounts, e.g., between the closed and fully open states, in response to respectively varying the pressure of the working fluid upstream of valve  111 - 5 . 
     In some examples, valve  111 - 5  might be fully opened in response to a pressure of the pilot fluid in the pilot line reaching a certain pressure level and might be closed in response to a pressure of the pilot fluid in the pilot line reaching a certain other pressure level, e.g., that might be greater than or less than the pressure that opens valve  111 - 5 . The valve  111 - 5  might be configured to be partially opened by different amounts, e.g., between the closed and fully open states, in response to respectively varying the pressure of the pilot fluid. 
     In some examples, valve  111 - 5  might fully open in response to the pressure of the working fluid upstream of valve  111 - 5  fluctuating (e.g., oscillating) at a certain frequency (e.g., at a resonant frequency of valve  111 - 5 ) and might close in response to the pressure of the working fluid upstream of valve  111 - 5  fluctuating at a certain other frequency (e.g., at a non-resonant frequency of valve  111 - 5 ). In other examples, valve  111 - 5  might fully open in response to the pressure of the pilot fluid in the pilot line fluctuating at a certain frequency (e.g., at a resonant frequency of valve  111 - 5 ) and might close in response to the pilot fluid in the pilot line fluctuating at a certain other frequency (e.g., at a non-resonant frequency of valve  111 - 5 ). In some examples, the valve  111 - 5  might be configured to be partially opened by different amounts between the fully open and closed states in response to the working fluid or the pilot fluid respectively fluctuating by different frequencies that might be between the frequency that closes valve  111 - 5  and the frequency that fully opens valve  111 - 5 . 
     A section or actuator  120  is between section  118  and distal end  102 . For example section  120  might be referred to as a tip  120  of robot  100 , as shown in  FIGS. 1A-1F . Section  120  might be configured as a sensor and/or an actuator in some examples. The sensor, for example, might be exposed at distal end  102 . For example, section  120  might be configured to sense the presence of a target material that may or may not be distinct from a wall, such as a wall of an artery (e.g., in the living organism, such as the human body), and/or might be configured to remove the target material. For example, the target material might be plaque on an artery wall. For example, section  120  might include sensing fibers, e.g., that may be exposed at distal end  102 , that can sense the presence of the target material, such as plaque, and/or fiber optics, e.g., that may be exposed at distal end  102 , that can deliver laser (excimer laser) pulses to the sensed target material to remove the sensed target material. Distal end  102  may be brought into contact, with the target material on a wall of an artery for sensing and/or removing the target material. 
     In some examples, section  120  might include a vibrator that while vibrating facilitates movement (e.g., burrowing) of robot  110  through a medium, such as dirt, sand, debris, etc. Distal end  102  may be closed to (e.g., may be configured to block) the flow of the working fluid. 
     The examples of  FIGS. 1A to 1F  show the movement of robot  100  through an opening  130 , such as the opening in an artery. The opening  130  is bounded by a wall  135 , as shown in  FIG. 1A , having internal bounding surface  138 , as shown in  FIGS. 1A-1F . Alternatively, for examples where robot  100  is configured to burrow through a medium, opening  130  and its bounding wall  135  would be replaced by the medium, so the medium surrounds robot  100 . 
     In  FIG. 1B , section  110  is transformed from its shape in  FIG. 1A  to a spiral (e.g., a section  110  spiral) in response to the hollow core of section  110  selectively receiving the working fluid so that the spiral exerts a force on bounding surface  138  sufficient to prevent the spiral from moving. For examples involving burrowing through the medium, the section  110  spiral might extend into the surrounding medium to prevent the section  110  spiral from moving. In some examples, a fluid, such as blood, might be flowing through opening  130 . The section  110  spiral allows the fluid to flow past the exterior of the spiral, e.g., though the center of the coils of the section  110  spiral, without stopping the flow of the fluid. 
     Section  110  is transformed from its initial shape in  FIG. 1A  to the section  110  spiral in  FIG. 1B  in response to valve  111 - 1  being actuated to an open state while valves  111 - 2  and  111 - 3  are closed, for example. For example, section  110  is transformed into the section  110  spiral in response to the pressure of the received fluid within the hollow core of section  110 . Note, for example, that the transformation of section  110  into the section  110  spiral can cause section  110  to extend to a length that is greater than the length of section  110  in  FIG. 1A , thereby causing distal end  102  to move from point A- 1  in  FIG. 1A  to point B- 1  in  FIG. 1B . For example, section  110  might extend while section  110  is being transformed into the section  110  spiral. 
     In  FIG. 1C , while the section  110  spiral is prevented from moving, section  112  is extended from its length in  FIG. 1B  to its length in  FIG. 1C  in response to the hollow core of section  112  selectively receiving the working fluid from the hollow core of section  110 . Note that extending section  112  causes distal end  102  to move from point B- 1  in  FIG. 1B  to point C- 1  in  FIG. 1C . 
     Section  112  extends in response to valve  111 - 2  being actuated to an open state, while valve  111 - 1  remains open and valve  111 - 3  is closed, for example. For example, section  112  extends in response to the pressure of the working fluid received in the hollow core of section  112  from the hollow core of section  110 . In some examples, the pressure of the working fluid received in the hollow core of section  112  might be greater than the pressure of the working fluid received in the hollow core of section  110 . For example, the pressure of the working fluid in the hollow core of the section  110  spiral in  FIG. 1B  might be increased until valve  111 - 2  opens at which point the working fluid flows into the hollow core of section  112 , causing section  112  to extend, as shown in  FIG. 1C . 
     In  FIG. 1D , while the section  110  spiral is prevented from moving and segment  112  is extended, section  114  is transformed from its shape in  FIG. 1C  to the section  114  spiral in response to the hollow core of section  114  selectively receiving the fluid from the hollow core of section  112  so that the section  114  spiral exerts a force on bounding surface  138  sufficient to prevent the section  114  spiral from moving. For examples involving burrowing through the medium, the section  114  spiral might extend into the surrounding medium to prevent the section  114  spiral from moving. In the examples where a fluid, such as blood, might be flowing through opening  130 , the section  114  spiral allows the fluid to flow past the exterior of the section  114  spiral, e.g., though the center of the coils of the section  114  spiral, without stopping the flow of the fluid. 
     Section  114  is transformed from its shape in  FIG. 1C  to the section  114  spiral in response to valve  111 - 3  being actuated to an open state while valves  111 - 1  and  111 - 2  are open, for example. Note, for example, that the transformation of section  114  into the section  114  spiral can cause section  114  to extend to a length that is greater than the length of section  114  in  FIG. 1C , thereby causing distal end  102  to move from point C- 1  in  FIG. 1C  to point D in  FIG. 1D . For example, section  114  might extend while section  114  is being transformed into the section  114  spiral. 
     In some examples, the pressure of the fluid received in the hollow core of section  114  might be greater than the pressure of the fluid received in the hollow core of section  112 . For example, the pressure of the working fluid in the hollow core of section  112  in  FIG. 1C  might be increased until valve  111 - 3  opens at which point the working fluid flows into the hollow core of section  114 , causing section  114  to transform into the section  114  spiral. 
     In  FIG. 1E , section  110  is returned to its initial, non-spiral shape of  FIG. 1A  while the section  114  spiral is prevented from moving and section  112  remains extended. For example, section  110  contracts when it is returned to its initial shape, causing proximal end  104  to move from point A- 2  in  FIG. 1D  to point B- 2  in  FIG. 1E  while distal end  102  remains at point D. 
     For example, section  110  might return to its initial shape in response closing valve  111 - 2  and reducing the pressure of the fluid in the hollow core of section  110  until section  110  returns to its initial shape while valve  111 - 1  is kept open. 
     In  FIG. 1F , section  112  is contracted to its initial, non-extended length, in  FIGS. 1A and 1B  while the section  114  spiral is prevented from moving and section  110  remains as in  FIG. 1E . Contracting section  112  causes proximal end  104  to move from point B- 2  in  FIG. 1E  to point C- 2  in  FIG. 1F  while distal end remains at point D. 
     For example, section  112  might contract in response to closing valve  111 - 3  and reducing the pressure of the fluid in the hollow cores of sections  110  and  112  until section  112  contracts to its initial length while valves  111 - 1  and  111 - 2  are kept open. 
     In some examples, section  120  might include a camera that allows robot  100  to identify branches  410  and  420  in an opening  430 , e.g., an opening in an artery, as shown in  FIG. 4 . This allows an operator to select which of the branches, branch  410  or branch  420 , such as branch  420 , to enter. For example, the configuration in  FIG. 4  is the configuration of  FIG. 1D  in the opening  430  that has a bend  435  and the branches  410  and  420 . Note that  FIG. 4  shows that section  112  can be extended through the bend  435 . 
     In some examples, section  118  might be bent from its straight configuration in the configuration of  FIG. 1C  in response to the hollow core of section  118  selectively receiving the fluid from the hollow core of section  112  through the hollow cores of sections  114  and  116 . For example, section  118  might be bent in response to valves  111 - 3  to  111 - 5  being actuated to an open state while valves  111 - 1  and  111 - 2  are open. For example, the pressure of the fluid received in the hollow core of section  118  might be greater than, but could be less than, the pressure received in the hollow core of section  112 , e.g., that causes section  112  to extend. In some examples, section  118  might be configured to bend as robot  100  goes from the configuration in  FIG. 1C , while residing in opening  430 , to the configuration shown in  FIG. 4 . Distal end  102 , and thus section  120 , may be brought into contact with a target material  450 , e.g., that may or may not be distinct from the bounding wall of branch  420  of opening  430 , for example, using the bending feature of section  118  or using the bending feature of section  118  in combination with the twisting feature of section  116 , for example. That is, for example, section  120  may sense target material  450 , such as plaque, and/or might remove target material  450 . 
       FIG. 5  shows robot  100  in an opening  530 , e.g., of an artery. Opening  530  has a tapered bend  540 , and  FIG. 5  shows the section  110  spiral conforming to the tapered bend  540 . For example, section  110  might be transformed into the section  110  spiral, as described in conjunction with  FIG. 1B , while in tapered bend  540 . Note that  FIG. 5  shows distal end  102 , and thus section  120 , being brought into contact with target material  450 , e.g., using the bending feature of section  118  or using the bending feature of section  118  in combination with the twisting feature of section  116 . 
       FIG. 6  illustrates a section or actuator  1000 . Section  1000  may be a circular tube having a hollow core, for example. For example, section  1000  may include an elastomeric (e.g., a silicone, rubber, latex, etc.) tube  1020  that includes a structure  1030 . In some examples, structure  1030  might be a fiber (a nylon, a woven carbon fiber, etc.), a thickness (e.g., a rib) of an elastomer (e.g., a silicone, rubber, latex, etc.), etc. 
     For example, an elastomeric rib causes the wall of tube  1020 , and thus the wall of section  1000 , to be thicker in locations where the elastomeric rib is located. As such, the thickness of the wall of section  1000  will vary. The stiffness of section  1000  will vary accordingly with the thickness of the wall of section  1000  to produce a local anisotropic stiffness distribution in section  1000 . 
     Alternatively, the structure  1030  might be a groove formed by removing a portion of the wall of tube  1020 . As such, the wall of tube  1020 , and thus the wall of section  1000 , will be thinner in locations where the groove is located, meaning that the stiffness of section  1000  will again vary, due to the variation in the thickness of the wall of section  1000 , to produce a local anisotropic stiffness distribution in section  1000 . 
     The structure  1030  has a pitch P, as shown in  FIG. 6 . The structure  1030  makes an angle α, measured clockwise from the central axis  1040  of section  1000  that varies with distance along the length of section  1000 , and thus along the length of the central axis  1040 . For example, the angle α might be an angle α 1  at a distance d 1  along the length of the central axis  1040  and a different angle α 2  at a distance d 2  along the length of the central axis  1040 . For example, the angle α varies over the pitch P. 
     The local anisotropic stiffness distribution in section  1000  can be changed by causing the angle α to vary differently along the length of section  1000 . For example, the variation of the angle α along the length of section  1000  can be respectively varied to respectively give respective local anisotropic stiffness distributions that respectively allow section  1000  to respectively become the sections  110 ,  112 ,  114 ,  116 , and  118  of robot  100 . For example, the variation of the angle α along the length of section  1000  might be selected to produce a local anisotropic stiffness distribution that allows section  1000  to become a spiral in response to a fluid pressure in the hollow core of section  1000 , to produce a local anisotropic stiffness distribution that allows section  1000  to extend in response to a fluid pressure in the hollow core of section  1000 , to produce a local anisotropic stiffness distribution that allows section  1000  to twist in response to a fluid pressure in the hollow core of section  1000 , or to produce a local anisotropic stiffness distribution that allows section  1000  to bend in response to a fluid pressure in the hollow core of section  1000 . 
       FIG. 7  illustrates a section or actuator  1100  that can be used for sections  110  and  114  of robot  100 . Section  1100  may be a circular tube having a hollow core, for example. Section  1100 , for example, may include an elastomeric (e.g., a silicone, rubber, latex, etc.) tube  1120  that includes a structure represented by the distribution of dots  1130 . The dots  1130  represent discrete portions that have a different stiffness than tube  1120  where there are no dots  1130  so that section  1100  has a local anisotropic stiffness distribution. 
     For example, dots  1130  might correspond to discrete locations where the wall of tube  1120  might be thicker or thinner than the thickness of the wall of tube  1120  where there are no dots. Alternatively, discrete portions represented by dots  1130  might be of a material (e.g., that might be formed in the wall of tube  1120 , such as by molding, three-dimensional printing, etc.) having a different stiffness than the wall of tube  1120  where there are no dots  1130 . In the example of  FIG. 11 , the local anisotropic stiffness distribution may allow section  1100  to become a spiral in response to a fluid pressure in the hollow core of section  1100 , meaning that section  1100  can be used for sections  110  and  114  of robot  100 . 
       FIG. 8  illustrates a section or actuator  1200  that can be used for section  112  of robot  100 . Section  1200  may be a circular tube having a hollow core, for example. For example, section  1200  may include an elastomeric (e.g., a silicone, rubber, latex, etc.) tube  1220  that includes a structure represented by the distribution of dots  1230 . The dots  1230  represent discrete portions that have a different stiffness than tube  1220  where there are no dots  1230  so that section  1200  has a local anisotropic stiffness distribution. 
     For example, dots  1230  might correspond to discrete locations where the wall of tube  1220  might be thicker or thinner than the thickness of the wall of tube  1220  where there are no dots. Alternatively, discrete portions represented by dots  1230  might be of a material (e.g., that might be formed in the wall of tube  1220 , such as by molding, three-dimensional printing, etc.) having a different stiffness than the wall of tube  1220  where there are no dots  1230 . In the example of  FIG. 12 , the local anisotropic stiffness distribution may allow section  1200  to extend in response to a fluid pressure in the hollow core of section  1200 , meaning that section  1200  can be used for section  112  of robot  100 . 
       FIG. 9  illustrates a section or actuator  1300  that can be used for section  116  of robot  100 . Section  1300  may be a circular tube having a hollow core, for example. For example, section  1300  may include an elastomeric (e.g., a silicone, rubber, latex, etc.) tube  1320  that includes a structure represented by the distribution of dots  1330 . The dots  1330  represent discrete portions that have a different stiffness than tube  1320  where there are no dots  1330  so that section  1300  has a local anisotropic stiffness distribution. 
     For example, dots  1330  might correspond to discrete locations where the wall of tube  1320  might be thicker or thinner than the thickness of the wall of tube  1320  where there are no dots  1330 . Alternatively, discrete portions represented by dots  1330  might be of a material (e.g., that might be formed in the wall of tube  1320 , such as by molding, three-dimensional printing, etc.) having a different stiffness than the wall of tube  1320  where there are no dots  1330 . In the example of  FIG. 13 , the local anisotropic stiffness distribution may allow section  1300  to twist in response to a fluid pressure in the hollow core of section  1300 , meaning that section  1300  can be used for section  116  of robot  100 . 
       FIG. 10  illustrates a section or actuator  1400  that can be used for section  118  of robot  100 . Section  1400  may be a circular tube having a hollow core, for example. For example, section  1400  may include an elastomeric (e.g., a silicone, rubber, latex, etc.) tube  1420  that includes a structure represented by the distribution of dots  1430 . The dots  1430  represent discrete portions that have a different stiffness than tube  1420  where there are no dots  1430  so that section  1400  has a local anisotropic stiffness distribution. 
     For example, dots  1430  might correspond to discrete locations where the wall of tube  1420  might be thicker or thinner than the thickness of the wall of tube  1420  where there are no dots. Alternatively, discrete portions represented by dots  1430  might be of a material (e.g., that might be formed in the wall of tube  1420 , such as by molding, three-dimensional printing, etc.) having a different stiffness than the wall of tube  1420  where there are no dots  1430 . In the example of  FIG. 10 , the local anisotropic stiffness distribution may allow section  1400  to bend in response to a fluid pressure in the hollow core of section  1400 , meaning that section  1400  can be used for section  118  of robot  100 . 
     Actuators  1000  ( FIG. 6 ),  1100  ( FIG. 7 ),  1200  ( FIG. 8 ),  1300  ( FIG. 9 ),  1400  ( FIG. 10 ) described above are non-limiting examples of actuators with mechanical properties that spatially vary along the actuator body coordinate system in accordance with some embodiments of the present disclosure. With this in mind, some aspects of the present disclosure relate to methods of designing and manufacturing one or more actuators having mechanical properties that spatially vary along the actuator body useful as or with a soft robot and mapped to performance parameters entailed or required by a particular end-use application. These constructions can be referred to as a “material-mapped actuator”. As a point of reference, many envisioned soft robot end-use applications will require the soft robot to transition or articulate through multiple different shapes, elongations, point stiffnesses, etc. throughout the procedure. Material-mapped actuators can be uniquely designed and manufactured in accordance with methods of the present disclosure to satisfy essentially any end-use design constraint, including achieving desired, differing shapes (or stiffness or other property such as applying a force on to an external body or object) throughout the end-use procedure. 
     In general terms, the material-mapped actuators of the present disclosure are premised upon the actuator designs described above in which a material (e.g., any of the “dots” described above) is applied to or in an elastomeric tubular body or hollow core. A discernible or macro shape associated with the material-mapped actuator is primarily generated by the tubular body. In the presence of a fluid medium (or “actuation medium”) passing through the tubular body, a property (e.g., shape, elongation, stiffness, etc.) of the tubular body (and thus of the material-mapped actuator as a whole) is caused to change as a function of the applied material. With this in mind, the material-mapped actuator is formed to provide an initial shape having a corresponding initial map of mechanical attributes consisting of locally-varying stiffness at each point in a volume of the tubular body. The material-mapped actuator is further configured such that in the presence of a fluid medium distributed within the tubular body, the material-mapped actuator changes to a new, different shape or different distribution of mechanical properties with the initial spatially-varying map of mechanical attributes influencing or dictating the new shape or distribution. 
     In some embodiments, the material-mapped actuator consists of an elastomer or combination of elastomers (either by locally continuous mixtures or locally discrete combinations) such that local mechanical properties (e.g., stiffness) are specified and possibly distinct at each point in the actuator body. In other embodiments, the material-mapped actuators of the present disclosure includes or consists of an elastomeric tube or other shape, or is constructed on top of an elastomeric tube (or other shape). In yet other embodiment material-mapped actuators of the present disclosure, the material applied to the elastomeric tubular body includes local, spatially-varying additives such as locally-oriented fibers, meshes, threads, fiberglass, carbon black or carbon fiber, knits, woven materials, or similar materials that induce desired material anisotropies and strain limiting behaviors where the fiber orientation or pitch is free to vary in any direction along the actuator body. That is to say, the applied fibers (or other shaped material) need not extend an entire length of the tubular body and need not exhibit a constant pitch (e.g., can be akin to chopped fibers or small meshes applied and oriented locally). In related embodiments, the fiber- or mesh-like materials can be pre-stretched, pre-compressed, or pre-bunched prior to application to the tubular body so as to influence different regions of the local material stress-strain curve (e.g., have little effect on mechanical properties during small stretches (or elongations) and begin to have a dominant effect at higher stretches). 
     Commensurate with the descriptions above, the material-mapped actuators of the present disclosure provide or define one or more cavities (e.g. flow passage(s) described above) that accepts an actuation media (e.g., fluid medium). The actuation media can be a changing volume fluid (i.e., liquid or gas), such as water, oil, air, etc., which fills the cavity (or cavities) with this change in volume effecting a change in the shape of the material-mapped actuator and/or change in spatial distribution of mechanical properties. In some embodiments, the material-mapped actuator is configured such that the work done by the actuation media induces a stretching force along a surface of the material-mapped actuator and the spatially varying stiffness induces a desired (or pre-determined) change in the overall shape of, and/or in external forces generated by, the material-mapped actuator. 
     In other embodiments, a property mapped into the material-mapped actuator is different than or includes attributes other than material stiffness. For example, local volumes of active materials that grow or shrink in size in spatially-dependent magnitudes due to an applied stimulus such as electric, magnetic, electromagnetic, thermal, mechano-vibrational, acoustic or optic stimuli (e.g., made of electro-active polymers or thermally- or electrically-responsive hydrogels) can be specified. With these optional mapping techniques, the combination of local growth from multiple sites can result in a desired final body shape or external force application. 
     In other embodiments, the local, spatially-varying mechanical properties of the material-mapped actuators of the present disclosure depend on time and/or stress. With this in mind, the material-mapped actuator can be configured such that the time and/or stress dependent properties influence the overall sequence or evolution of the shape of the actuator body over time and/or stress. For example, with embodiments in which the tubular body of the material-mapped actuator is hyperelastic, the material-mapped actuator can be configured to exploit non-linear hyperelastic characteristics such as nonlinearity, hysteresis, memory, and non-monotonic stress-strain curves to better ensure that actuation begins at lower pressures in less than all of the actuator (e.g., only one part or segment of the actuator is actuated or experiences a change at a lower pressure). 
     In yet other embodiments, the material-mapped actuator is configured such that the local spatially-varying mechanical properties (e.g., stiffness) change in response to an external stimulus (e.g., glass transition due to applied temperature, change in viscoelastic properties due to vibration in the medium, non-Newtonian characteristics of material such as a shear-thickening elastomer). With these optional constructions, a single material-mapped actuator can be configured to assume a plurality of different orientations for a given activation of the actuation medium based on the stimulus. 
     In yet other embodiments, the material-mapped actuator is configured such that the local material stiffness in each location of the actuator body is, at least in part, determined or dictated by the local thickness of the material. 
     By incorporating one or more or all of the above local spatially-varying mechanical properties, the material-mapped actuators of the present disclosure can be configured to assume multiple desired shapes in the presence of an applied actuation medium. With these constructions, the material-mapped actuator can be uniquely configured to exhibit a pre-determined actuation sequence from an initial state (e.g., shape) to a first final state having at least one property (e.g., shape) differing from that of the initial state and corresponding with a desired attribute of the end-use procedure; the first final state then because a “new” initial state for subsequent actuation to a new, second final state having at least one property (e.g., shape) differing from that of the first final state, etc. Thus, a single material-mapped actuator of the present disclosure can be configured to achieve a plurality (e.g., three or more) of different, desired shapes as part of the actuation sequence. Notably, while example actuators have been described as including a tubular actuator body, the material-mapped actuators and corresponding methods of mapping, designing and manufacturing of the present disclosure are in no way limited to tubular shape. The techniques of the present disclosure can be utilized with virtually any initial shape, including spherical, planar, amorphous, etc. 
     The performance or operational characteristics incorporated into the material-mapped actuators can be determined in various fashions. In some embodiments, methods of the present disclosure apply continuum mechanics (e.g., deformation mapping, deformation gradients, stress-strain tensors, etc.) as part of an inverse design technique. Inverse design is in reference to generalizable computational methods that map directly from arbitrary task requirements to the optimal design of more universal soft robots and actuators and mechanisms, independent of the actual manufacturing techniques employed. For a particular soft robot end-use application, task or procedure, at least a desired initial state or shape and final state or shape (or other performance parameter, such as exertion of force on an external body or object) of the soft robot will be known or can be determined. In many instances, one or more desired, intermediate, procedure-specific states or shapes will also be known or can be determined. This information, in turn, is utilized to provide an optimal mapping of elasticity and materials to construct a soft robot (including one or more material-mapped actuators) well-suited to perform the identified end-use application or procedure, conforming to the procedure-required states or shapes when fluid power (or other actuation medium) is applied. 
     The inverse design methods can be embodied by a computer “toolbox” (continuum deformation mappings) employed as part of a system for manufacturing a soft robot (and/or one or more material-mapped actuators to be used with a soft robot). The systems of the present disclosure can include a computing device including one or more processors and a memory as known in the art. The computing device incorporates the inverse design toolbox as processing software, optionally stored in the memory, configured to receive information indicative of one or more end-use application performance requirements and/or kinematic constraints, such as desired initial shape, final shape, and optionally one or more intermediate shapes, and generate optimal mapping of elasticity and materials for constructing the soft robot (or individual material-mapped actuator(s) thereof). The computing device can further be electronically connected to (or provided as part of) a machine adapted to form a material-mapped actuator based upon, or dictated by, information generated by the inverse design toolbox as described in greater detail below, using a generated digital “blueprint” that informs of the best choice of manufacturing techniques. 
     The inverse design toolbox can include or operate on a mathematical formulism that precisely describes the continuum displacements, and in some embodiments is sufficiently universal to handle virtually any soft shape. By way of example, a material-mapped actuator formed to generate a spiral shape can be described by continuum mechanics equations as generally reflected by  FIG. 11  (that otherwise shows a tubular actuator of the present disclosure in an initial or reference shape and in a deformed shape (spiral)). The equations below represent one non-limiting example of a description of the mechanics of the motion in accordance with principles of the present disclosure. First, a parametric curve, r(t), is used to define the “spine” of each segment of the deformed shape whose dimensions are directly dictated by the task or procedure to be performed. An envelope function, f env (·), can be applied to ensure the ends of the spiral line up with its axis and connect smoothing with adjacent segments to result in the desired shape. This approach can generalize traditional robotic concepts such as generalized joint angles q for rigid body transforms (e.g., in DH parameters), for example, L for actuator length (traditional prismatic joint extension) and R for spiral expansion (not available in traditional robotics) and q={L, R}. Both can be dictated by procedure specific geometry. Analytical forms for the tangent T(t), normal N(t), and binormal B(t) vectors dictated by classic Frenet-Serret geometry can be derived from r spine  as:
 
 T ( t )= r ′( t )/∥ r ′( t )∥;
 
 N ( t )= T ′( t )/∥ T ′( t )∥;
 
B(t)=T(t)×N(t) provided that it is twice differentiable.
 
     This demonstrates that a universal actuator design can be generated by a curve fitting method like cubic splines instead of synthesized a priori from analytical functions. Thus, the inverse design toolbox and corresponding methods of the present disclosure can operate on empirical data constraints. Optionally, additional variables can be implemented that parameterize a desired surface along r spine ( ) for the soft robot body. For example, a cylindrical surface, S spiral ( ), extended axially by t about r spine ( ) and wrapped radially by Θ can have a radius R i  dictated by the end-use application task or procedure (e.g., not more than 1 cm diameter). That is, S spiral (Θ, t, R, R i , f env )=r spine (t)+R i (N(t)cos(Θ)+B(t)sin(Θ)). This allows S spine ( ) to fully describe the desired kinematics of this link, starting from a straight cylinder and ending in a spiral with no twist or change in length (attributes not possible with existing soft robot design methods). 
     The inventors of the present disclosure have observed that the S spiral ( ) derived for soft actuators as above is equivalent to a deformation mapping φ(·) in continuum mechanics, commonly denoted as x=φ(X) where X=[X Y Z] and x=[x y z] correspond to undeformed and deformed shapes, respectively. If φ(·)=S spiral (·), the tools of continuum mechanics can be invoked to describe soft robot kinematics. For example, the deformation gradient: 
             F   =       ∂     φ   ⁡     (   X   )           ∂   X             
leads to the right Cauchy Deformation Gradient:
 
 C=F   T   F  
 
which can yield the stress tensors calculated as:
 
λ i =√{square root over ( C   ii )},  i= 1, 2, 3
 
The diagnolization of λ is:
 
               λ   _     =     [           λ   1         0       0           0         λ   2         0           0       0         λ   3           ]           
The angle of deformation can be expressed via the Polara Decomposition of C ij  or alternatively as:
 
     
       
         
           
             
               
                 θ 
                 ij 
               
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     This produces 3×3 symmetric matrices, where the diagonals are all 0 and can be reduced down to three rotations. The above observations allow the methods of the present disclosure to take any point X 0  on the initial (reference) shape and analyze the local anisotropic deformation. For example, a point X 0  can be selected in the reference shape and a unit cube can be plotted to show the reference and deformed cube configuration as:
 
 P   cube deformed   = λ R   ZXY   P   cube reference  
 
R ZYX  can be represented by the Euler Angle Rotation Θ 12 , Θ 13 , Θ 23  as:
 
 R   ZYX   =R   Z (Θ 12 ) R   Y (Θ 13 ) R   X (Θ 23 )
 
The above methodology or process is shown graphically in  FIG. 12 . It will be recognized that significant anisotropy and elongation of nearly 300% is considered. This map provides the required material stiffness at each point on the actuator&#39;s ski required to make the actuator (and thus the resultant soft robot) move from an initial shape to a desired shape.
 
     This approach can also account for kinematic forces. An approximation for the peak stress at each point on the “skin” of the actuator is: 
                 E   .     =       σ   ɛ     =         P   max     ⁢     r   max         ɛ   ⁢           ⁢     t   h             ,         
using circumferential hoop stress of a thin walled cylinder. The maximum internal cavity pressure, P max , and maximum desired tube diameter, r max , at this pressure can both be selected in accordance with desired performance parameters of the end-use application or procedure. The wall thickness, t h , can be left as a free design variable provided t h &lt;R max /10 to meet hoop equation assumptions. E maps the magnitude of local change in strain required at each point on the surface (Θ, t) to achieve the desired motion. For the non-limiting example spiral actuator, this can range from a 0% to 300% change from strain at rest in some embodiments. Thus, for an approximately linear material in this range (such as Kraton D1161), Hooke&#39;s law applies and can yield the magnitude of change in material stiffness required at each point on the surface of the initial shape to achieve the desired shape and meet static pressure and force requirements (e.g., the “blueprint” for manufacturing). Nonlinear materials in this elongation region such as latex may entail a model that is accurate in the regime of large elongation such as Gent models to determine a specific material&#39;s stiffness and hence a more accurate “blueprint”.
 
     Optionally, the stiffness mapping systems and methods of the present disclosure can apply the Cauchy-Stress tensor so as view stress over multiple planes. It can be possible to solve for the Cauchy-Stress tensor as a function of a material model, but may leave too many unknowns. Instead, internal pressure can be applied as boundary condition that is related to the Cauchy-Stress tensor; this would eliminate a set of unknowns and more easily identify or solve for the material property. The Cauchy-Stress tensor can be broken into hydrostatic stress (pressure) and devatoric stress (shear) as:
 
σ= s−pl  
 
The hydrostatic pressure can then be solved for material constant c 1  as:
 
             p   =       -     1   3       ⁢     tr   ⁡     (   σ   )                       c   1     =       -       3   ⁢   P       2   ⁢           ⁢     I   1     ⁢     tr   ⁡     (   B   )             =     -       3   ⁢   P       2   ⁢       tr   ⁡     (   B   )       2                   
This can be performed for each point and a mapping created of the material parameter as a function of position. A dithering algorithm can then be run to reduce the value of c 1  to two. The resultant Cauchy-Stress equation can be expressed as:
 
σ=2 W   J     1     F+ 2 W   J     2   ( I   1   F−B   2 )− c   0   I  
 
     By utilizing the stiffness map, E(Θ,t) as described above, systems and methods of the present disclosure can provide an optimized blueprint for constructing a corresponding material-mapped actuator. This can allow a free choice of suitable manufacturing methods or, if no ideal methods are available, the blueprint dictates what an optimal manufacturing method should strive to achieve. In some embodiments, a digital manufacturing blueprint is automatically generated based upon the desired stiffness map that otherwise corresponds to one or more shapes implicated by the end-use application or procedure. For example, a discretizing algorithm can be applied to the continuum desired material mapping such that the global effect of the discretized mapping is equivalent to the original continuum desired material mapping. In some embodiments, the discretized mapping entails an algorithm akin to the Floyd-Steinberg dithering algorithm, modified to have error diffusion minimize the global stiffness in error (possibly mapped through nonlinear stress-strain for certain materials such as latex) and implemented to minimize error for multiple anisotropic directions. In one exemplary format, the dithering algorithm accounts for two materials having different Young&#39;s modulus. The first material is selected in accordance with the lowest stiffness bound established by the stiffness mapping. The peak stiffness from the stiffness mapping will dictate selection of the second material. For example, the two different materials could be two different durometers of silicone, two different thicknesses of latex, two different durometers of a thermoplastic elastomer, etc. Regardless, the dithering algorithm maps an arrangement of the second material on to the first material in accordance with stiffness mapping.  FIG. 16  provides an example of a Young&#39;s Modulus mapping provided by the dithering algorithms of the present disclosure required near point X 0  to realize desired overall deformation. 
     Optionally, the discretized algorithm can be a volumetric dithering algorithm capable of discretizing a continuous material property mapping to a desired spatial resolution. In some embodiments, a set of stress maps is created with a single parameter being varied. For example, in the spiral parameterization the helix radius R can be varied and a specific pressure assigned to each radius R. A stress map is created for each iteration of the helix radius and pressure. The set of stress maps is then run through filtering and dithering algorithms to yield a single stress map that is able to actuate through several precision points that are created by each iteration. 
     Other algorithms can be utilized for formatting or generating the manufacturing blueprint from the stiffness mapping, for example as a function of the machines or equipment available for generating the material-mapped actuator. As a point of reference,  FIG. 13  schematically illustrates one embodiment of a system  1450  in accordance with principles of the present disclosure for manufacturing a material-mapped actuator useful as, or as part of, a soft robot configured to perform a desired or pre-determined end-use application, procedure or task. The system  1450  includes a mapping module  1452  and at least one manufacturing module  1454 . The mapping module  1452  includes a computing device programmed (or operating on software) to generate a stiffness mapping for a selected end-use application, procedure or task as described above, and to generate a manufacturing blueprint in accordance with the descriptions above. The mapping module  1452  can include one or more user input devices (e.g., touch screen, keyboard, etc.) by which a user can enter or select (e.g., in response to a prompt generated by the mapping module) one or more end-use application, procedure or task parameters. The user-entered or selected parameter(s) can include one or more of desired initial and final (and optionally intermediate) shapes, size, external forces, force applied to an external body, peak pressures, motion, etc. The mapping module  1452  can further be programmed to consider capabilities or techniques embodied by the manufacturing module  1454  in formulating the manufacturing blueprint. 
     The manufacturing module  1454  can assume a wide variety of forms capable of producing or generating a material-mapped actuator of the present disclosure, operating upon the manufacturing blueprint generated by the mapping module  1452 . In other embodiments, the mapping module  1452  can be programmed or configured to generate a stiffness mapping as described above, and the manufacturing module  1454  can include appropriate computing device hardware and/or software for generating an appropriate manufacturing blueprint based upon the stiffness mapping. Regardless, the manufacturing module  1454  includes or comprises one or more machines or devices for forming material-mapped actuator useful as or with a soft robot. For example, the manufacturing module  1454  is configured to be capable of imparting a material-mapping pattern onto or throughout a core body that is either provided to, or formed by, the manufacturing module. The material-mapping pattern can be generated with either an additive or subtractive manufacturing technique. 
     For example, the manufacturing module  1454  can be or include an additive-type device utilizing one or more of inkjet deposition, aerosol jet deposition, or extrusion deposition (e.g., 3D printer as known in the art). In some embodiments, the deposition process effectuated by the manufacturing module  1454  can be controlled in terms of one or both of the type of material being deposited and the material thickness at specified locations. In some related embodiments, the manufacturing module  1454  can be configured to perform digitally dithered material placement to achieve discrete elastomeric properties and/or durometers across the material-mapped actuator (and/or the resultant soft robot). In other related embodiments, the manufacturing module  1454  can be configured to perform continual mixing of elastomers where the mixing ratio of the base elastomers is varied throughout the deposition process to achieve continuous variation of elastomeric properties or durometers. By way of one non-limiting example, the manufacturing module  1454  can be akin to a CNC lathe with additional mechanical extrusion and piezo-jet nozzles for depositing droplets of high-viscosity elastomer resins with a surface deposition resolution of 0.5 mm or less. This allows printing at higher precision using established materials (e.g., aqueous latex, various silicones) and curing techniques (air dry, thermal vulcanization, UV initiated, mixed catalyst platinum cure, etc.). Moreover, new materials can be printed, such as thermoplastic elastomers that exhibit very high elongation rates (greater than 1000%) but have a more linear stress-strain response than latex. 
     Another additive manufacturing technique or step can include dip molding, with the manufacturing module  1454  optionally configured to control a material wall thickness of the actuator in response the time each region spends in the dip solution. 
     Alternatively or in addition, the manufacturing module  1454  can be or include a modified braiding machine, manual, or real-time robotic weaving, sewing or knitting of fibers to integrated into the structure of the material-mapped actuator. With these optional embodiments, pre-tensioning or pre-compression of the fibers during or upon deposition can be provided as desired. 
     Alternatively, the manufacturing module  1454  can be configured to perform a subtractive manufacturing process. Non-limiting examples include laser ablation, waterj et ablation (aquablation), precision machining, abrasion, etc. In optional, related embodiments, the manufacturing module  1454  can include or operate upon a concentric arrangement of tubular bodies. The tubular bodies can be subtractively manufactured using laser ablation, cutting or dissolution to remove a specified number of tube layers over a programmable region to change the elastomeric properties and/or thickness of the concentric arrangement to achieve desired deformations. 
     In yet other embodiments, the manufacturing module  1454  can be or include a pick and place-type robot that selects one, or a combination of, pre-assembled stiffness element(s) onto a hollow core. The elements can be combined or overlaid to achieve strain responses not possible with a single fiber. Exemplary stiffness elements include pre-tuned fabric/fiber meshes, mesh patches, chopped or milled fibers, stretchable fibers, pre-tensioned or pre-compressed fiber elements, etc. In related embodiments, long fibers or meshes wrapped helically around a tubular body can be would such that the period or pitch of the helix changes along the axis of the helix such that regions with large helical spacing actuate at lower pressures as compared to regions of denser helical spacing. In other embodiments, the stiffness element(s) can include active elements such as magnetically-active, electro-active, thermally-active, or frequency-dependent polymers. 
     In yet other embodiments, the manufacturing module  1454  can include a device configured to actuate (e.g., inflate or deflate) the tubular body of the actuator during manufacturing to pre-tension or pre-compress mapped regions in the actuator as material is deposited or subtracted. Alternatively or in addition, machine(s) of the manufacturing module  1454  actuate or deflate the tubular body to a desired shape and a layer of stiff strain-limiting material is added to the surface to limit actuation in the local material region to that desired configuration. 
     In other optional embodiments, the manufacturing module  1454  is capable of assembling two (or more) material-mapped actuators in a desired fashion for completing a soft robot. For example, the manufacturing module  1454  can be configured to assemble two (or more) material-mapped actuators in series, optionally assembling or forming a valve (or valves) between the serially-connected material-mapped actuators. The valve(s) can take any of the forms described elsewhere in the present disclosure. In other embodiments, the manufacturing module  1454  is configured to integrally form a series of differently-configured material-mapped actuators each fluidly separated from one another by one or more valves using any of the manufacturing techniques described above. The so-formed valve can be a passive or active valve. In some non-limiting embodiments, the manufacturing module  1454  is configured to integrally or homogenously form, based upon the blueprint mapping described above two or more serially connected material-mapped actuators fluidly separated from one another by an asymmetric valve formed to exhibit a desired cracking pressure. 
     Using the foregoing specification, aspects of the present disclosure can be implemented as a machine, process or article of manufacture by using standard programming and/or engineering techniques to produce programming software, firmware, hardware or any combination thereof. 
     Any resulting program(s), having computer-readable instructions, may be stored within one or more computer-readable media such as memory devices or transmitting devices, thereby making a computer program product or article of manufacture according to the present disclosure. As such, the term “software” as used herein is intended to encompass a computer program existent as instructions on any non-transitory computer-readable medium such as on any memory device that are to be executed by a processor. Examples of memory devices include hard disk drives, optical disks, magnetic tape, semiconductor memories such as FLASH, RAM, ROM, PROMS, and the like. 
     A machine embodying aspects of the present disclosure may involve one or more processing systems including, for example, CPU, memory/storage devices, communication links, communication/transmitting devices, servers, I/O devices, or any sub-components or individual parts of one or more processing systems, including software, firmware, hardware, and any combination or subcombination thereof. Using the descriptions provided herein, those skilled in the art will be readily able to combine software created as described with appropriate general purpose or special purpose computer hardware to create a computer system and/or computer subcomponents embodying aspects of the present disclosure, and to create a computer system and/or computer subcomponents for carrying out methods of the present disclosure. 
     As mentioned above, in some embodiments, soft robots of the present disclosure can include or incorporate two or more actuators connected in series and fluidly separated by one or more valves. The valves of the present disclosure can assume various forms, several non-limiting example of which are described below. For example,  FIG. 14  is a cross-sectional view (e.g., with cross-hatching omitted for clarity) of an example of robot  100 , showing a main flow passage  1500  that includes hollow cores  1510 ,  1512 ,  1514 ,  1516 , and  1518  respectively of sections  110 ,  112 ,  114 ,  116 , and  118 . In the example of  FIG. 14 , valve  111 - 1  selectively fluidly couples hollow core  1510  to the upstream pressure source, valve  111 - 2  selectively fluidly couples hollow cores  1510  and  1512 , valve  111 - 3  selectively fluidly couples hollow cores  1512  and  1514 , valve  111 - 4  selectively fluidly couples hollow cores  1514  and  1516 , and valve  111 - 5  selectively fluidly couples hollow cores  1516  and  1518 . Note that the working fluid flows through main flow passage  1500 . 
     Main flow passage  1500  is bounded by a tube wall  1525  that includes the walls of sections  110 ,  112 ,  114 ,  116 , and  118 . In the example of  FIG. 14 , a pilot line  1530  interconnects valves  111 - 1  to  111 - 5  and supplies the pilot fluid to valves  111 - 1  to  111 - 5  for actuating valves  111 - 1  to  111 - 5 . In some examples, pilot line  1530  might include (e.g., might be configured as) a rotary valve, such as a rotary valve  1911  discussed below in conjunction with  FIGS. 18A-18D , or an axially sliding valve, such as axially sliding valve  2011  discussed below in conjunction with  FIGS. 19A-19D . Note that for examples where the working fluid actuates valves  111 - 1  to  111 - 5 , pilot line  1530  might be omitted from  FIG. 14 . 
       FIG. 15  is a cross-sectional view (e.g., with cross-hatching omitted for clarity) of another example of robot  100 . Common numbering is used in  FIGS. 14 and 15  to denote elements common to  FIGS. 14 and 15 . In the example of  FIG. 15 , a pilot line  1630  is located in tube wall  1525 . For example, pilot line  1630  might be configured as a manifold, e.g., having an opening (e.g., an inlet)  1638  and openings (e.g., outlets)  1640 - 1  to  1640 - 5  respectively coupled to valves  111 - 1  to  111 - 5  for directing the pilot fluid to valves  111 - 1  to  111 - 5 . Alternatively, in some examples, pilot line  1630  might be configured as a rotary valve, such as the rotary valve  1911  discussed below in conjunction with  FIGS. 18A-18D , or an axially sliding valve, such as the axially sliding valve  2011  discussed below in conjunction with  FIGS. 19A-19D . 
       FIG. 16  is a cross-sectional view (e.g., with cross-hatching omitted for clarity) of another example of robot  100 . Common numbering is used in  FIGS. 14 and 16  to denote elements common to  FIGS. 14 and 16 . In the example of  FIG. 16 , pilot lines  1730 - 1  and  1730 - 2  are located in different portions of (e.g., at different circumferential locations in) tube wall  1525 . For example, pilot line  1730 - 1  might be configured as a manifold, e.g., having an opening (e.g., an inlet)  1738 - 1  and openings (e.g., outlets)  1740 - 1  to  1740 - 3  respectively coupled to valves  111 - 1  to  111 - 3  for directing the pilot fluid to valves  111 - 1  to  111 - 3 . Pilot line  1730 - 2 , for example, might be configured as a manifold, e.g., having an opening (e.g., an inlet)  1738 - 2  and openings (e.g., outlets)  1740 - 4  and  1740 - 5  respectively coupled to valves  111 - 4  and  111 - 5  for directing the pilot fluid to valves  111 - 4  and  111 - 5 . Alternatively, in some examples, each of pilot lines  1730 - 1  and  1730 - 2  might be configured as a rotary valve, such as the rotary valve  1911  discussed below in conjunction with  FIGS. 18A-18D , or an axially sliding valve, such as the axially sliding valve  2011  discussed below in conjunction with  FIGS. 19A-19D . 
       FIGS. 17A-17C  illustrate an example of a valve  1811  at different states. Valve  1800  might be used for any of the valves  111 - 1  to  111 - 5 . Valve  1811  is in a closed state in  FIG. 17A . Valve  1811  includes a spool  1814  mechanically coupled to a spring  1820  that is mechanically coupled in series with a damper  1822 . Valve  1811  includes an inlet port  1825  configured to receive the working fluid from upstream of valve  1811 . For example, inlet port  1825  might receive the working fluid from the upstream pressure source or the hollow core of one of sections  110 ,  112 ,  114 , or  116  of robot  100 . Valve  1811  includes an outlet port  1830 . In the closed state, spool  1814  completely covers (e.g., blocks) outlet port  1830 . In some examples, valve  1811  includes a pilot port  1835  that receives the pilot fluid from a pilot line. However, pilot port  1835  might be omitted for examples where valve  1811  is actuated by the working fluid. 
     Outlet port  1830  might output the working fluid to the hollow core of section  110  when inlet port  1825  receives the working fluid from the upstream pressure source, to the hollow core of section  112  when inlet port  1825  receives the working fluid from the hollow core of section  110 , to the hollow core of section  114  when inlet port  1825  receives the working fluid from the hollow core of section  112 , to the hollow core of section  116  when inlet port  1825  receives the working fluid from the hollow core of section  114 , or to the hollow core of section  118  when inlet port  1825  receives the working fluid from the hollow core of section  116 . 
     In the example of  FIG. 17A , valve  1811  receives either the working fluid with a fluctuating (oscillating pressure) or the pilot fluid with a fluctuating (oscillating pressure) at a non-resonant frequency of the spring  1820 , spool (e.g., mass)  1814 , damper  1822  system. For example, spool  1814  may remain stationary in the closed position shown in  FIG. 17A  while covering outlet port  1830  in response to the non-resonant frequency. That is, for example, valve  1811  remains closed in response to the non-resonant frequency. 
     In the example of  FIG. 17C , valve  1811  receives either the working fluid with a fluctuating (oscillating pressure) at a resonant frequency of the spring  1820 , spool (e.g., mass)  1814 , damper  1822  system or the pilot fluid with a fluctuating (oscillating pressure) at a resonant frequency of the spring  1820 , spool (e.g., mass)  1814 , damper  1822  system. For example, valve  1811  may fully open in response to the resonant frequency. 
     For example, the resonant frequency causes spool  1814  to oscillate between the fully open position, where outlet port  1830  is completely uncovered by spool  1814 , and, for example, the closed position of  FIG. 17A , as indicated by the dashed lines  1850  in  FIG. 17C . That is, for example, valve  1811  may oscillate between the fully closed and fully open states in response to the resonant frequency, where spool  1814  oscillates between the position where spool  1814  fully covers outlet port  1830  and the position where outlet port  1830  is fully uncovered by spool  1814  in response to the resonant frequency. 
     In the example of  FIG. 17B , valve  1811  receives either the working fluid with a fluctuating (oscillating pressure) or the pilot fluid with a fluctuating (oscillating pressure) at a certain frequency of the spring  1820 , spool (e.g., mass)  1814 , damper  1822  system sufficient to cause spool  1814  to oscillate with a lower displacement than in response to the resonant frequency. For example, valve  1811  may partially open in response to the certain frequency. 
     For example, the certain frequency causes spool  1814  to oscillate between the partially open position, where outlet port  1830  is partially uncovered by spool  1814 , and, for example, the closed position of  FIG. 17A , as indicated by the dashed lines  1852  in  FIG. 17B . That is, for example, valve  1811  may oscillate between the fully closed and partially open states in response to the certain frequency, where spool  1814  oscillates between the position where spool  1814  fully covers outlet port  1830  and the position where outlet port  1830  is partially uncovered by spool  1814  in response to the certain frequency. The certain frequency might be between the non-resonant frequency, e.g., in response to which spool  1814  remains covering outlet port  1830 , and the resonant frequency, e.g., in response to which spool  1814  oscillates between where outlet port  1830  is completely uncovered by spool  1814  and where spool  1814  completely covers outlet port  1830 . 
     One non-limiting embodiment of a bushing  1850  useful with the optional spool valves of the present disclosure is shown in  FIG. 17D . Bushing  1580  forms or defines one or more ports  1852 - 1856  that can be selectively covered and uncovered by an internally-carried spool (not shown). 
       FIG. 18A  illustrates an example of a rotary valve  1911  that includes an inner tube  1920  ( FIG. 18B ) and an outer tube  1925  ( FIG. 18C ). Inner tube  1920  is within outer tube  1925  so that inner tube  1920  and outer tube  1925  are coaxial about central axis  1930 . Inner tube  1920  is configured to rotate within outer tube  1925 , as indicated by arrows  1927  in  FIGS. 18A and 18B , e.g., in response to rotational motion imparted to inner tube  1920 , such as by a motor, a user (e.g., manually), etc. 
     Openings  1935 - 1  to  1935 - 5  (e.g., that might be circular openings) are distributed over the length of outer tube  1925 , in the direction of central axis  1930 , as shown in  FIG. 18C , and pass through a wall  1937  of outer tube  1925 . Openings  1940 - 1  to  1940 - 5  are distributed over the length of inner tube  1920 , in the direction of central axis  1930 , as shown in  FIG. 18B , and pass through a wall  1942  of inner tube  1920 . Openings  1940 - 1  to  1940 - 5  are elongated in a direction around the circumference of inner tube  1920 . Openings  1940 - 1  to  1940 - 5  respectively correspond to openings  1935 - 1  to  1935 - 5  and respectively align with openings  1935 - 1  to  1935 - 5  when inner tube  1920  is respectively rotated to different angular locations. 
     For example,  FIG. 18D  (a cross section of portions of tube wall  1937  of outer tube  1925  and of tube wall  1942  of inner tube  1920  with cross-hatching omitted for clarity) shows a particular state of valve  1911 , where openings  1935 - 1  and  1940 - 1  and openings  1935 - 2  and  1940 - 2  are in alignment at a particular angular position of inner tube  1920 . The concurrent alignment of openings  1935 - 1  and  1940 - 1  and openings  1935 - 2  and  1940 - 2  is facilitated by the elongation of openings  1940 - 1  and  1940 - 2 , for example. 
     In some examples, openings  1935 - 1 ,  1935 - 2 ,  1935 - 3 ,  1935 - 4 , and  1935 - 5  may be respectively fluidly coupled to the hollow cores of sections  110 ,  112 ,  114 ,  116 , and  118 . For example, when openings  1935 - 1  and  1940 - 1  are in alignment and openings  1935 - 2  and  1940 - 2  are in alignment at the particular angular location of inner tube  1920 , working fluid flows through inner tube  1920 , through aligned openings  1935 - 1  and  1940 - 1 , and into the hollow core of section  110 , and through aligned openings  1935 - 2  and  1940 - 2  and into the hollow core of section  112 , as indicated by arrow  1950  in  FIG. 18D . 
     In other examples, valve  1911  might be used to selectively direct pilot fluid to valves, such as valves  111 - 1 ,  111 - 2 ,  111 - 3 ,  111 - 4 , and  111 - 5 , so that the pilot fluid can actuate the valves. For example, openings  1935 - 1 ,  1935 - 2 ,  1935 - 3 ,  1935 - 4 , and  1935 - 5  may be respectively fluidly coupled to valves  111 - 1 ,  111 - 2 ,  111 - 3 ,  111 - 4 , and  111 - 5 . When openings  1935 - 1  and  1940 - 1  are in alignment and openings  1935 - 2  and  1940 - 2  are in alignment at the particular angular location of inner tube  1920 , for example, pilot fluid flows through inner tube  1920 , through aligned openings  1935 - 1  and  1940 - 1 , and to valve  111 - 1 , and through aligned openings  1935 - 2  and  1940 - 2  and to valve  111 - 2 , as indicated by arrow  1950  in  FIG. 18D . 
     Valve  1911 , and thus inner tube  1920  and outer tube  1925 , may be within the main flow passage of robot  100  and may extend the entire length of robot  100 . For example, the portions of valve  1911  respectively corresponding to openings  1935 - 1 ,  1935 - 2 ,  1935 - 3 ,  1935 - 4 , and  1935 - 5  and openings  1940 - 1 ,  1940 - 2 ,  1940 - 3 ,  1940 - 4 , and  1940 - 5  might be respectively in the hollow cores of sections  110 ,  112 ,  114 ,  116 , and  118  and might respectively conform to the actuated shapes of sections  110 ,  112 ,  114 ,  116 , and  118 . That is, for example, the portion of valve  1911  corresponding to openings  1935 - 1  and  1940 - 1  might conform to the spiral shape of section  110 ; the portion of valve  1911  corresponding to openings  1935 - 2  and  1940 - 2  might extend with portion  112 ; the portion of valve  1911  corresponding to openings  1935 - 3  and  1940 - 3  might conform to the spiral shape of section  114 ; the portion of valve  1911  corresponding to openings  1935 - 4  and  1940 - 4  might twist with section  116 ; and the portion of valve  1911  corresponding to openings  1935 - 5  and  1940 - 5  might bend with section  118 . 
       FIG. 19A  illustrates an axially sliding valve  2011  that includes an inner tube  2020  ( FIG. 19B ) and an outer tube  2025  ( FIG. 19C ). Inner tube  2020  is within outer tube  2025  so that inner tube  2020  and outer tube  2025  are coaxial about central axis  2030 . Inner tube  2020  is configured to slide in an axial direction (e.g., along central axis  2030 ) within outer tube  2025 , as indicated by arrow  2027  in  FIG. 19B , e.g., in response to axial motion imparted to inner tube  2020 , such as by a motor, a user (e.g., manually), etc. 
     Openings  2035 - 1  to  2035 - 5  (e.g., that might be circular openings) are distributed over the length of outer tube  2025 , in the direction of central axis  2030 , as shown in  FIG. 19C , and pass through a wall  2037  of outer tube  2025 . Openings  2040 - 1  to  2040 - 5  are distributed over the length of inner tube  2020 , in the direction of central axis  2030 , as shown in  FIG. 19B , and pass through a wall  2042  of inner tube  2020 . Openings  2040 - 1  to  2040 - 5  are elongated in the axial direction. Openings  2040 - 1  to  2040 - 5  respectively correspond to openings  2035 - 1  to  2035 - 5  and respectively align with openings  2035 - 1  to  2035 - 5  when inner tube  2020  is respectively slid to different axial locations. 
     For example,  FIG. 19D  (a cross section of portions of tube wall  2037  of outer tube  2025  and of tube wall  2042  of inner tube  2020  with cross-hatching omitted for clarity) shows a particular state of axially sliding valve  2011 , where openings  2035 - 1  and  2040 - 1  and openings  2035 - 2  and  2040 - 2  are in alignment at a particular axial position of inner tube  2020 . The concurrent alignment of openings  2035 - 1  and  2040 - 1  and openings  2035 - 2  and  2040 - 2  is facilitated by the elongation of openings  2040 - 1  and  2040 - 2 , for example. 
     In some examples, openings  2035 - 1 ,  2035 - 2 ,  2035 - 3 ,  2035 - 4 , and  2035 - 5  may be respectively fluidly coupled to the hollow cores of sections  110 ,  112 ,  114 ,  116 , and  118 . For example, when openings  2035 - 1  and  2040 - 1  are in alignment and openings  2035 - 2  and  2040 - 2  are in alignment at the particular axial location of inner tube  2020 , working fluid flows through inner tube  2020 , through aligned openings  2035 - 1  and  2040 - 1 , and into the hollow core of section  110 , and through aligned openings  2035 - 2  and  2040 - 2  and into the hollow core of section  112 , as indicated by arrow  2050  in  FIG. 19D . 
     In other examples, valve  2011  might be used to selectively direct pilot fluid to valves, such as valves  111 - 1 ,  111 - 2 ,  111 - 3 ,  111 - 4 , and  111 - 5 , so that the pilot fluid can actuate the valves. For example, openings  2035 - 1 ,  2035 - 2 ,  2035 - 3 ,  2035 - 4 , and  2035 - 5  may be respectively fluidly coupled to valves  111 - 1 ,  111 - 2 ,  111 - 3 ,  111 - 4 , and  111 - 5 . When openings  2035 - 1  and  2040 - 1  are in alignment and openings  2035 - 2  and  2040 - 2  are in alignment at the particular axial location of inner tube  2020 , for example, pilot fluid flows through inner tube  2020 , through aligned openings  2035 - 1  and  2040 - 1 , and to valve  111 - 1 , and through aligned openings  2035 - 2  and  2040 - 2  and to valve  111 - 2 , as indicated by arrow  2050  in  FIG. 19D . 
     Valve  2011 , and thus inner tube  2020  and outer tube  2025 , may be within the main flow passage of robot  100  and may extend the entire length of robot  100 . For example, the portions of valve  2011  respectively corresponding to openings  2035 - 1 ,  2035 - 2 ,  2035 - 3 ,  2035 - 4 , and  2035 - 5  and openings  2040 - 1 ,  2040 - 2 ,  2040 - 3 ,  2040 - 4 , and  2040 - 5  might be respectively in the hollow cores of sections  110 ,  112 ,  114 ,  116 , and  118  and might respectively conform to the actuated shapes of sections  110 ,  112 ,  114 ,  116 , and  118 . That is, for example, the portion of valve  2011  corresponding to openings  2035 - 1  and  2040 - 1  might conform to the spiral shape of section  110 ; the portion of valve  2011  corresponding to openings  2035 - 2  and  2040 - 2  might extend with portion  112 ; the portion of valve  2011  corresponding to openings  2035 - 3  and  2040 - 3  might conform to the spiral shape of section  114 ; the portion of valve  2011  corresponding to openings  2035 - 4  and  2040 - 4  might twist with section  116 ; and the portion of valve  2011  corresponding to openings  2035 - 5  and  2040 - 5  might bend with section  118 . 
     Other valve constructions are also envisioned by the present disclosure. For example, an asymmetrical passive valve can be employed. One non-limiting example of an asymmetrical passive valve  2200  is shown in simplified form in  FIG. 20A . The valve  2200  includes a cone  2202  arranged at a cone angle α relative to an outer wall  2204 . The cone  2202  has a thickness T. The valve  2200  is configured to allow viscous flow restriction in addition to a cracking pressure in one direction that differs from cracking pressure in the opposite direction (i.e., the directions associated with the arrows “Forward Flow” and “Reverse Flow” in  FIG. 20A ). By varying one or both of the cone angle α and the cone thickness T, selection asymmetry is available. For example,  FIG. 20B  is a table of representative pressures for the valve  2200  ( FIG. 20A ) in the Forward Flow direction (P fwd ) and the Reverse Flow direction (P rev ) at various cone angles α and cone thicknesses T.  FIG. 20C  is a contour plot of the ratio (P rev /P fwd −1) for the valve  2200  ( FIG. 20A ). With cross-references between  FIGS. 20A-20C , the valve  2200  can be constructed from a single elastomeric material where the design parameters such as thickness and cone angle can be used to tune the amount or level of asymmetry. 
     Further, in some embodiments, the valve structures and actuation thereof can incorporate pressure level indexing features (e.g., each valve open/closes at different ranges of pressure, like a multi-level relief valve). Valves that open at different differential pressures (e.g., sequencing valves, relief valves, etc.) can be used to connect the different sections or actuators; alternatively or in addition, two way differential pressure valves can be used to connect the different section or actuators. Alternatively, valves controlling each segment can have a linear or rotary sliding design, formed with an inner/outer sleeve arrangement in a pilot or main channel. The fit between the inner and outer sleeve can be a slight interference to a slight clearance, minimizing any fluid flow between the sleeves. The outer sleeve will have a series of holes in the wall that connect to the various actuation segments. The inner sleeve will also have a series of holes through the wall of the sleeve. The holes in the inner sleeve are arranged such that by translating and/or rotating the inner sleeve, the holes in the outer sleeve and inner sleeve will be aligned, opening the valve to one or more actuation segments. Both the inner and outer sleeves can flexible to the motion of the soft robot, but maintain axial alignment. This linear/rotary valve arrangement could be applied from a single pilot line (pressure supply) or two pilot lines (pressure supply and return). Regardless, each valve will be triggered at at-least-one different pressure differential, allowing each section to be actuated or unactuated sequentially and for a variety of pre-selected ranges. 
     For example, the first valve may open with a 10 psi pressure differential and the second valve at a 20 psi pressure differential. When the differential pressure across the first valve is greater than 10 psi fluid will fill the first actuator section and realize the spiral (or other) shape. Then, once a 20 psi threshold is reached across the second valve the second actuator section will fill with fluid and create the extension (or other) action. Each sequential valve will experience the same processes, but with the differential pressure value being increased for each valve. Since each valve can have multiple trigger thresholds, as the pressure increases some sections may experience a decrease in fluid level relative to others (e.g. the first spiral can collapse after the second has expanded and anchored in the tissue). 
     The valves may be opened/closed through temporal sequencing in some embodiments. This can be accomplished through dynamic or frequency variations of the input fluid. For example, time-dynamic input pressures conforming to a step or square-wave like input with passive flow restrictor valves can ensure that segments nearer the base actuate before segments further from the base. Temporal sequencing allows a valve to be toggled via pressure waves allowing for independent actuation of each section. Independent actuation allows for precise control of the soft robot (or other assembly incorporating three or more actuators of the present disclosure in series), each section can actuate or un-actuate to close in on the desired target. For example, the first valve can be latched partially open by experiencing a 50 Hz pressure wave, latched fully open by experiencing a 100 Hz pressure wave and latched closed by experiencing a 120 Hz pressure wave. The second valve will experience similar, selective actuation/latching via a different set of fluid pressure waves (e.g.  140 ,  160 , and  180  or, if two sections need simultaneous valve actuation,  100 ,  160  and  180 ). It should be noted that many pressure waves can applied in superposition, thus enabling the simultaneous, independent control of all valves and actuation states. Additionally, valves can be designed not to latch but to continuously change state as a function of the frequency or amplitude of the waves. For example, a valve will fully close at 50 Hz and fully open at 100 Hz, but can partially open to a desired level at a frequency in between 50 and 100 Hz. For example, a valve will be 50% open at 75 Hz. Similarly, the 50 Hz drives a valve to close and 100 Hz drives it to open (e.g., a binary open/close) then the presence of both waves at equal amplitude may effectively cause the valve to behave as if it were partially closed/open. 
     With the optional pressure level indexing embodiments described above, a pilot line can be used to control each valve state. Pressure signals can be sent through the pilot line to control the state of each valve. Each valve will respond to a unique pressure or frequency, which will result in a change of flow in the main channel. For example, the pressure signal(s) can be constant amplitude pressure commands to act as a pilot signal to valves controlling various actuators. Alternatively, the frequency of the pressure in the pilot line can be used to open and close the valves controlling the various actuators. In yet other embodiments, the pilot line valves can be combined with an inflow and outflow channel, with the pilot line controlling four states of the inflow/outflow channel as: State 1) inflow and outflow both closed; State 2) inflow open and outflow closed; State 3) inflow and outflow both open; and State 4) inflow closed and outflow open. 
     The soft robots and actuators (including material-mapped actuators) of the present disclosure can be highly beneficial in a plethora of different end-use applications, procedures or tasks. For examples, the soft robots and actuators (including material-mapped actuators) can be designed and employed to perform procedures within the human body, for example through or within blood vessels (e.g., ranging in diameter from 10 mm (femoral artery) to 1 mm (stenosed coronary artery)). The soft robots and actuators of the present disclosure can safely traverse blood vessels via the locomotion-type shape change features described above without hindering blood flow and exerting minimal force on vessel walls. By incorporating the optional worm-like design or locomotion effect, the soft robots and actuators of the present disclosure can “pull” themselves through arteries at a lower force than stiffer, “pushed” catheters and can conform to the tightest, most tortuous regions of a patient&#39;s vasculature. 
     The soft robots and actuators (including material-mapped actuators) can alternatively be configured or designed for other end-use applications, procedures or tasks, such as burrowing though soils and other materials. Soft robots of the present disclosure can be capable of generating multiple burrowing motions as describe above, and thus can entail adaptive gait change and burrow penetration mechanisms to enable efficient and precision burrowing through a wide variety of soil types. As a point of reference,  FIG. 21  illustrates, in simplified form, the locomotion gait and penetration mechanism utilized by nature to burrow through differing soil types (dry sand, wet sand, mud, and gravel). A soft robot  2500  in accordance with principles of the present disclosure can be constructed to replicate or mimic each of the discrete locomotion gaits implicated by a particular end-use application. For example, the soft robot  2500  can include one or more material-mapped actuators (referenced generally at  2502 ) described above designed to sequentially or responsively replicate a desired locomotive gait, along with one or more heads  2504  configured to provide or effectuate a desired penetration mechanism. The material-mapped actuators  2502  can be independently controlled by internal valves  2506  fed from a common pressure rail  2508 . The soft robot  2500  can be constructed to effectuate one or more of blunt penetration (to burrow through dry sand), fluidization (to burrow through wet sand), crack propagation (to burrow through mud), and/or excavation (to burrow through gravel, compact dirt, etc.). The locomotion gait(s) incorporated into the soft robot  2500  can be premised upon the two dominant burrowing gaits observed in nature, undulation and alternate anchoring, into the fundamental motion primitives. 
     In some embodiments, the soft robot  2500  (configured for burrowing through multiple types of soil) can include one or more sensors that provide information indicative of position, soil type, impediments to movement, etc. For example, carbon-nanotube elastomer sensors can be included that provide accurate measurements for links of the soft robot  2500  (position errors under 5% of the surface length using 2D multiplexed ratiometric; force of contact error &lt;20% using contact resistance and &lt;5% using piezo resistive effects after viscoelastic calibration). Sensing can be provided of the internal robot state, location of contact along the outer surface or skin of the soft robot  2500  and/or position of this contact. An accurate dynamic material analyzer (e.g., 300 Hz Bandwidth analyzer available under the trade designation Bose ElectroForce 3200 Series III) can be employed to characterize the uniaxial piezo-resistive pseudoelastic dynamic response of the carbon-nanotube sensor to construct a tractable nonlinear dynamic model for calibrating the sensor and quantify its accuracy and repeatability. Mathematical models can be implemented as code on an embedded processor in the sensor electronics to allow for repeatable, accurate real-time measurement of forces or pressures applied to the sensor (that is otherwise located at a link of the soft robot  2500 ). 
     Although specific examples have been illustrated and described herein, this application is intended to cover any adaptations or variations of these examples. It is manifestly intended that the scope of the claimed subject matter be limited only by the following claims and equivalents thereof.