Abstract:
The needle-sized surgical tools used in arthroscopy, otolaryngology, and other surgical fields could become even more valuable to surgeons if endowed with the ability to navigate around sharp corners to manipulate or visualize tissue. A needle-sized bendable joint design that grants this ability. It can be easily interfaced with manual tools or concentric tube robots and is straightforward and inexpensive to manufacture. The bendable joint includes of a nitinol tube with several asymmetric cutouts, actuated by a tendon.

Description:
RELATED APPLICATIONS 
       [0001]    This application claims the benefit of U.S. Provisional Application Ser. No. 62/166,310, filed May 26, 2015. This application also claims the benefit of U.S. Provisional Application Ser. No. 62/296,620, filed Feb. 18, 2016. The subject matter of these provisional applications is hereby incorporated by reference in its entirety. 
     
    
     GOVERNMENT RIGHTS 
       [0002]    This work was funded in part by the National Science Foundation (NSF) under CAREER award U.S. Pat. No. 1,054,331 and three Graduate Research Fellowships. It was also funded in part by the National Institutes of Health (NIH) under award numbers R01 EB017467 and R21 EB017952. The U.S. Government may have certain rights to the invention. 
     
    
     FIELD OF THE INVENTION 
       [0003]    The present invention relates to surgical tools for performing surgical operations. More specifically, the present invention relates to small diameter surgical tools for navigating the patient&#39;s anatomy in order to deliver therapy to a target location in the patient&#39;s body. In particular, the present invention relates to a surgical device with a bendable joint, such as a bendable tip that has an arc length varying curvature for implementation in small-diameter microsurgical tool devices, such as hand-operated catheter-like manipulators or concentric tube robots. 
       BACKGROUND 
       [0004]    There is a pressing need in robotic or remotely controlled hand-operated surgery for small-diameter surgical tools with bendable tips, which are sometimes referred to as bendable tip  12  joints or bendable tips  12  because they act as the bending joint proximate to the tool at the distal end or tip much in the manner that the human bendable tip  12  serves in relation to the hand. Most existing small-diameter surgical tools devices do not include bendable tips  12  and thus cannot navigate the sharp corners encountered in surgery, such as those at the skull base, in the middle ear, and in the ankle. Moreover, dexterity driven tasks, such as tissue resection and suturing, can be difficult to perform without a bendable tip  12 , especially through the small openings characteristic of natural orifice or percutaneous procedures. 
         [0005]    In one particular surgical field, small-diameter bendable tips  12  are needed to augment the capabilities of microsurgical devices, such as small-diameter catheter-like or concentric tube surgical robots, which can have diameters on the needle-sized order (e.g., having diameters as small as 1.0 mm or less). The performance of these small-diameter robotic systems, which are devised for delicate and intricate surgical procedures such as pituitary tumor resection, neurosurgery, and intracardiac surgery, among others, can be significantly enhanced with the addition of small bendable tips  12  for aiding in manipulating their end-effectors. 
         [0006]    Many small bendable tips  12  based on traditional mechanical linkages have been devised in the past. For example, previous small bendable tips  12  have been designed to incorporate the use of ball joints, universal joints, cable/pulley mechanisms, lead screws, parallel serial chains, and flexures. These designs range from 2.4 to 15.0 mm. Although it could be possible to downscale each of these designs to some degree, designs with a continuum structure, i.e., those in which are machined or otherwise engineered directly into the shaft structure of the catheter, needle, concentric tube, etc. would be easier to miniaturize than those containing multiple components. Among these continuum structures, those with the fewest components are most desirable for downscalability, making designs that involve machining the shaft of the device itself particularly appealing. Examples of continuum structures involve cutting nitinol tubes to create rectangular or triangular cutouts that form a compliant region for bending. In these instances, however, the diameters remain comparatively large, e.g., on the order of 6-10 mm. 
         [0007]    Additionally, known bendable structures suffer from limitations that relate to the manner in which their bending takes place. An example of this is shown in  FIG. 13 , which illustrates schematically a commercially available bendable tip catheter  300 . The catheter  300  is illustrated in various stages of actuation, beginning at a zero actuated condition  302 , fully actuated condition  308  and intermediate actuated conditions  304  and  306 . As shown in  FIG. 13 , as the actuation proceeds, the most proximal sections of the catheter  300 , indicated generally at  310 , bend first, followed by intermediate sections  312  and distal sections  314 . As a result, in the fully actuated condition  308 , the proximal sections  310  achieve full bend, and the degree of bending is reduced through the intermediate sections  312  and distal sections  314 . This behavior can be highly unfavorable in small spaces where these devices can be operated, because a high degree of bending at the distal-most portions is not only desired, but can be critical to performing a successful procedure. 
       SUMMARY 
       [0008]    A small-diameter bendable tip allows a user to design a tip with arc length varying curvature. The user can select properties, such as which portion of the tip bends first, in which order subsequent portions bend, how far each section is able to bend, and the general motion of the bending). This design improves the performance of commercial bendable catheter tips and microsurgical devices by enabling the user to specify the motion of the bendable tip and enhancing the dexterity of the tip in small spaces during surgery, an important characteristic of small steerable surgical devices. 
         [0009]    According to one aspect, the invention relates to a bendable joint including a tubular structure including a tubular side wall that extends along an axis and defines an inner lumen. At least one cutout is positioned along the length of the sidewall. Each cutout includes an axial portion of the sidewall that is removed and provides communication with the inner lumen. Each cutout helps to define a bend joint and at least one bend section. The bend joint includes the remaining portion of the side wall left along the length of the cutout. The at least one bend section includes complete tubular portions of the sidewall on opposite sides of the cutout. Each bend joint can deflect so that adjacent bend sections move relative to each other and assume a curved configuration. 
         [0010]    According to another aspect, the bend sections associated with each bend joint an move toward each other in response to deflection of the bend joint. 
         [0011]    According to another aspect, the bendable joint can include a tendon cable that extends within the inner lumen and has a connection with a distal one of the bend sections, Tension on the tendon cable can be applied to the distal one of the bend sections, which causes the bend joints proximal of the connection to deflect and causes the associated bend sections to move towards each other and assume a curved configuration. 
         [0012]    According to another aspect, the cutouts can have geometries selected such that the physical properties of the bend joints differ from each other, which causes the curvature of the bend joint to vary along its length. The cutouts can have geometries selected such that the physical properties of the bend joints differ from each other, which causes the bend joints to deflect in a predetermined order in response to tension applied to the tendon cable. 
         [0013]    According to another aspect, the tubular structure can be an inner tube of a concentric tube robot. 
         [0014]    According to another aspect, the cutouts can have rectangular geometries. The cutouts can be aligned with each other along the axis of the tubular structure. The cutouts can be rotated relative to each other along the axis of the tubular structure. 
         [0015]    According to another aspect, the geometries of the bend sections defined by the cutouts can be configured to define the amount of deflection that each bend joint can undergo. The geometries of the bend sections defined by the cutouts can be configured to collectively define the range of bending motion that can be achieved by the bendable joint. 
         [0016]    According to another aspect, the cutouts can define the joint along a tip portion of the tubular structure. 
         [0017]    According to another aspect, the tubular structure can be a nitinol tube. 
         [0018]    According to another aspect, the tubular structure can include a needle structure. The terminal end portion of the tubular structure can include a needle tip comprising a sharpened point. A cutout can be positioned adjacent to the needle tip. The bend joint defined by the cutout can allow the tip to deflect relative to the remainder of the tubular structure. The needle tip can include a beveled lead surface that is angled relative to a longitudinal axis of the tubular structure. The lead surface can be configured such that when the needle tip is advanced longitudinally through tissue, the tissue acting on the lead surface urges the needle tip to deflect relative to the remainder of the tubular structure through bending of the bend joint. 
         [0019]    According to another aspect, the bendable joint can include a tendon cable that extends within the inner lumen and has a connection with the needle tip. Tension on the tendon cable can be applied to the needle tip, which causes the bend joint adjacent the needle tip to deflect. Deflection of the needle tip relative to the remainder of the tubular structure can cause the tubular structure to follow a curved path when advanced through tissue. 
         [0020]    According to another aspect, the bendable joint can include an end effector for performing a surgical function positioned at the distal end of the tubular structure distal of the bend joint. A tendon cable can extends through the tubular structure and be connected to the end effector. The tendon cable can be actuatable to cause actuation of the end effector. 
         [0021]    According to another aspect, the cutouts can have non-rectangular geometries. The cutouts can have geometries that are generally key-shaped when viewed in profile. The key-shaped cutout geometries result in the bend sections having a generally tapered configuration, and the bend joints having semicircular edge portions. Adjusting the geometry of a circular portion of the key-shaped cutouts can affect the force required to deflect the bend joints. Adjusting the spacing and angle of tapered edges of the cutouts can affect the range of motion permitted between adjacent bend sections. 
     
    
     
       DRAWINGS 
         [0022]      FIGS. 1A and 1B  illustrate an apparatus including a small diameter bendable tip, according to an example embodiment. 
           [0023]      FIGS. 2A-2D  illustrate the bending of the small diameter bendable tip portion of the apparatus of  FIG. 1 . 
           [0024]      FIGS. 3A-D  illustrates example configurations of the small diameter bendable tip. 
           [0025]      FIGS. 4A and 4B  are schematic diagrams illustrating certain forces and moments acting on an example configuration of the small diameter bendable tip. 
           [0026]      FIG. 5  illustrates certain parameters and relevant kinematic values for a portion of the small diameter bendable tip. 
           [0027]      FIGS. 6A-6C  illustrate geometric layout and parameters for portions of the small diameter bendable tip. 
           [0028]      FIG. 7  is a schematic illustration of a portion of the small diameter bendable tip depicting certain kinematic properties. 
           [0029]      FIG. 8  illustrates one example configuration in which an example small diameter bendable tip with a distally mounted surgical instrument. 
           [0030]      FIG. 9  illustrates the example configuration of  FIG. 8  in different operational positions. 
           [0031]      FIG. 10  is a magnified view of a portion of the bendable tip  12  apparatus illustrated in  FIGS. 8 and 9 . 
           [0032]      FIG. 11  is a graph that illustrates a modeled versus experimental spatial trajectories of the example bendable tip  12  apparatus configuration of  FIGS. 8-10 . 
           [0033]      FIG. 12  is a graph that illustrates tendon force versus bendable tip  12  rotation for the example bendable tip  12  apparatus configuration of  FIGS. 8-10 . 
           [0034]      FIG. 13  illustrates a known bendable tube device. 
           [0035]      FIGS. 14A and 14B  illustrate alternative configurations of a small diameter bendable tip apparatus. 
           [0036]      FIGS. 15A and 15B  illustrate the operation of an alternative configuration of a small diameter bendable tip apparatus. 
           [0037]      FIGS. 16A and 16B  illustrate an alternative configuration of a small diameter bendable tip apparatus. 
           [0038]      FIGS. 17A-17D  illustrate the operation of the small diameter bendable tip apparatus of  FIGS. 16A and 16B . 
           [0039]      FIGS. 18A and 18B  illustrate the operation of the small diameter bendable tip apparatus of  FIGS. 16A and 16B  featuring an alternative actuator. 
       
    
    
     DESCRIPTION 
     Device Design 
       [0040]    The present invention relates to a surgical device with a bendable tip that has an arc length varying curvature for implementation in small-diameter surgical tool devices. Referring to  FIGS. 1A and 1B , according to one example embodiment, a surgical system  100  includes an apparatus  10  in the form of a small-diameter surgical tool device includes a small-diameter bendable joint  12 . In the example embodiment of  FIGS. 1A and 1B , the bendable joint  12  forms a bendable tip of the apparatus  10  and is therefore referred to herein as a bendable tip  12 . The position of the bendable joint, however, is not limited and could be located at any location along the length of the apparatus  10 . Additionally, in the example embodiment of  FIGS. 1A and 1B , the small-diameter surgical tool  10  comprises a concentric tube robot  20  comprising at least two concentric tubes. The surgical tool  10  could have other configurations, such as a tubular catheter configuration. 
         [0041]    As shown in  FIGS. 1A and 1B , the surgical system  100  can also include a drive system  102  for actuating the various components of the apparatus  10 , and a control system  104  for controlling the actuation system. The drive system  102  includes various actuation components, such as motors, solenoids, actuators, linkages, drive mechanisms, transmissions, etc. that supply the motive forces for operating the apparatus  10 . The control system  104  includes the input, processing, and signal generating components that generate the drive signals for controlling operation of the actuation components of the drive system  102 . 
         [0042]    In the example embodiment, there are two concentric tubes: a straight, typically stainless steel, outer tube  22  and a curved, typically nitinol, inner tube  24 . The outer tube  22  and inner tube  24  are individually and independently movable both axially along and rotationally about a longitudinal axis  28 . In the retracted position illustrated in  FIGS. 1 and 2 , the inner tube  24  is retracted within the outer tube  22  such that the portion protruding from the outer tube is substantially straight. As known in the art, the inner tube  24  assumes its curved configuration as it is extended or telescoped out of the outer tube  22 . This, in combination with the axial and rotational manipulation of the outer and inner tubes  22 ,  24 , defines a work space or volume within which the concentric tube robot  20  can deliver its tip to any location. 
         [0043]    Referring to  FIGS. 8 and 9 , the surgical tool  10  also includes an end effector or instrument  30  that is located at the distal end of the tool. In the example embodiment, the instrument  30  is a surgical instrument in the form of a curette that is typically used to cut, scrape or otherwise remove tissue. The surgical instrument  30  could, however, be any surgical tool for which delivery via a small-diameter surgical tool  10 . For example, the surgical instrument could comprise grippers, surgical lasers, graspers, retractors, scissors, imaging tips, cauterizing tips, ablation tips, morcelators, knives/scalpels, cameras, irrigation ports, suction ports, needles, or any other suitable surgical instrument. 
         [0044]    The surgical tool  10  can deliver the surgical instrument  30  to any location within the work space of the concentric tube robot  20 . The surgical instrument  30  itself can be further manipulated, for example, via a flexible rod or cable  32  that extends through inner lumens of the concentric tubes  22 ,  24 . The cable  32  can be manipulable, for example, to cause rotation (arrow B in  FIG. 8 ) or linear translation (arrow A in  FIG. 8 ) of the surgical instrument  30  about an instrument axis  34 , as indicated generally by arrows in  FIGS. 1A and 1B . In  FIG. 1A , the instrument axis  34  is coaxial with the tool axis  28 . In  FIG. 1B , the instrument axis  34  is transverse to the tool axis  28 . For the alternative surgical instruments  30 , the cable  32  may serve other purposes, such as actuating a mechanical linkage of the instrument or delivering energy to the instrument. 
         [0045]    The small-diameter bendable tip  12  is positioned at the distal end of the inner tube  24  just proximal of the instrument  30  and thereby connects the surgical instrument  30  to the concentric tube robot  20 . Conveniently, in the example embodiment of  FIGS. 1A and 1B , the bendable tip  12  is formed integrally as the distal end portion of the nitinol inner tube  24 . The bendable tip  12  has a non-actuated condition, shown in  FIG. 1A , in which the bendable tip is not bent and extends essentially or substantially along the tool axis  28 . The bendable tip  12  has an actuated condition, shown in dashed lines at  12 ′ in  FIG. 1B , in which the tip is bent along an arc, thus positioning the instrument axis  34  transverse to the tool axis  28 . The bendable tip  12  is selectively actuatable to place the tip at any intermediate position along the arc of travel between the actuated and non-actuated positions. 
         [0046]    Actuation of the bendable tip  12  is effectuated through the actuation of a tendon cable  40  which extends through the inner lumen of the concentric tubes  22 ,  24  and is connected to the bendable tip. A motor or other suitable drive mechanism (not shown) applies and varies the tension on the tendon cable  40  in order to effectuate the desired degree of bend in the tip. The drive mechanism for the tendon cable  40  can be integrated into the drive unit that operates the concentric tube robot  20  and the surgical instrument  30 . Actuation of the concentric tube robot  20 , surgical instrument  30 , and bendable tip  12  can thus be controlled via a single controller or control system that integrates and coordinates the control of all of these devices. 
         [0047]      FIGS. 2A-2D  illustrate the distal end of the inner tube  24 , including the bendable tip  12 , in greater detail. In  FIGS. 2A-3D , the illustrated bending of the tip  12  is effectuated through the application of tension to the tendon cable, which is omitted from these figures in order to illustrated the structure of the tip in greater detail. 
         [0048]    Referring to  FIGS. 2A-2D , the bendable tip  12  is formed by a series of cutouts  50  in which tube material (e.g., nitinol) is removed from the inner tube  24 . The cutouts  50  define bend joints  52 , which are the portions of the inner tube  24  that remain after the removal of the cutout material. The cutouts  50  also define tubular bend sections  54  that extend between the bend joints  52 . In the example embodiment illustrated in  FIGS. 2A-2D , the bendable tip  12  includes three cutouts  50  that define three bend joints  52  and three bend sections  54 . The bendable tip  12  could include a greater number of cutouts  50  or fewer cutouts, depending on factors, such as the desired performance characteristics of the tip and the particular application in which the tip is to be implemented. 
         [0049]    As shown in  FIGS. 2A-2D , the bendable tip  12  proceeds from a non-actuated or zero bend/deflection condition ( FIG. 2A ) to a fully actuated, full bend/deflection condition ( FIG. 2D ). Between these extremes, the bendable tip  12  proceeds through intermediate degrees of bending ( FIGS. 2B and 2C ). As the tip  12  bends, the bend joints  52  deflect and the bend sections  54  move and rotate/pivot. This movement is blocked when adjacent bend sections  54  engage each other or, in the case of the most proximally located bend section (the rightmost in  FIGS. 2A-2D ), engage the remainder of the inner tube  24 . The degree of movement that each bend section  54  is permitted to undergo is thus defined and limited by the geometry of the cutout(s)  50  that define it. 
         [0050]    In the example of  FIGS. 2A-2D , the bendable tip  12  includes cutouts  50  that are generally rectangular when viewed in profile, resulting in generally rectangular bend sections  54  and bend joints  52  with flat edge portions. Adjusting the geometry of the cutouts  50  can affect the bending action of the bend joints  52  and bend sections  54 . For example, adjusting the depth of the cutouts  50  can alter the cross-section of the bend joints  52  and can thereby affect the force required to deflect the bend joints. Adjusting the width of the cutouts  50  affects the spacing of the bend sections  54 , which can affect the range of motion permitted between adjacent bend sections. 
         [0051]    Examples of different geometries for the cutouts  50  are illustrated in  FIGS. 3A-3D . Note that in  FIGS. 3A-3D , the illustrated bend joint  12  includes five cutouts  50 , as compared to the three cutouts in the embodiment of  FIGS. 2A-2D . Referring to  FIGS. 3A and 3B , the bendable tip  12  includes cutouts  50  that are generally key-shaped when viewed in profile, resulting in generally tapered or spade shaped bend sections  54  and bend joints  52  with semicircular edge portions. Adjusting the geometry of the circular portion of the cutouts  50  can affect the force required to deflect the bend joints  52 . Adjusting the spacing and angle of the tapered edges of the cutouts  50  can affect the range of motion permitted between adjacent bend sections. 
         [0052]    The pattern of the cutouts  50  could be arranged in patterns that differ from the straight line pattern in the example embodiment of  FIGS. 1-2 . For example, referring to  FIGS. 3C and 3D , the bendable tip  12  includes cutouts  50  that are generally rectangular in shape when viewed in profile, and therefore the bend joints  52  and bend sections  54  act in a similar or identical manner to those illustrated and described with respect to the bendable tip of  FIGS. 2A-2D . In  FIGS. 3C and 3D , however, the cutouts  50  are progressively rotated relative to each other so that the bend joints  52  are arranged in a helical pattern along the length of the bend joint  12  and the bend sections are rotated with respect to each other. This configuration causes the tip  12  to assume a helical shape when deflected, which can provide the tip with an added degree of dexterity. From this, it can be seen that, according to the invention, the pattern and spacing of the cutouts  50  and, thus, the bend joints  52  and bend sections  54  can be selected so that the tip  12  assumes a desired bent configuration when actuated. These patterns are not limited to the linear or helical patterns illustrated in the figures, as those figures are illustrative of possible configurations and are not meant to limit or otherwise restrict other possible configurations. The configuration of the bendable tip  12  can be selected to any desired configuration through the selection of appropriately configured and spaced cutouts  50 . 
         [0053]    According to the invention, the construction of the bendable tip  12  allows the tip to be designed with an arc length varying curvature that is customizable to meet the demands of the user. By “arc length varying curvature,” it is meant that the properties or characteristics of under which each bend joint  52  and bend section  54  act during bending of the tip  12  can be configured individually. For each bend joint  52  and bend section  54 , the geometry of the cutout  50  can be configured to allow a user to select bend characteristics, such as the amount of force required to deflect each bend joint  52 , the order in which each bend joint/section of the tip bends, the range of deflection for each bend joint/section, and the general motion, i.e., straight vs. curved/helical, of the bending. The design of the bendable tip  12  offers improved performance by enabling the user to specify the motion of the bendable tip and enhancing the dexterity of the tip in small spaces during surgery. 
         [0054]      FIGS. 2A-2D  illustrate this construction. For convenience in describing the construction and operation of the bend joint  12 , the bend joints  52  and bend sections  54  in  FIGS. 2A-2D  are referred to as first, second and third as viewed from distal to proximal, i.e., left to right in the figures. 
         [0055]    Viewing  FIGS. 2A-2D , the bending motion of the of the bendable tip  12  is configured so that the first bend joint  52  deflects first and the first bend section  54  moves/pivots/rotates first. This is shown in  FIG. 2B , where the first bend section  54  moves/pivots/rotates into engagement with the second bend section before the second and third bend sections undergo significant movement. This is not to say that there is not some deflection/movement in the second or third bend joints  52  and bend sections  54 . Indeed, some deflection or movement in these joints and sections can be expected and is therefore illustrated in  FIG. 2B . This deflection, however, is minimal compared to the full deflection of the first bend joint  52 . 
         [0056]    Referring to  FIG. 2C , after the first bend joint  52  undergoes full deflection such that the first bend section  54  engages the second bend section, the second bend joint  52  deflects and the second bend section  54  moves/pivots/rotates into engagement with the third bend section. This occurs while there is minimal deflection in the third bend joint  52 . Again, this is not to say that there is not some deflection/movement in the third bend joint  52  and bend section  54 . Indeed, some deflection or movement in these joints and sections can be expected and is therefore illustrated in  FIG. 2B . This deflection is minimal compared to the full deflection of the second bend joint  52 . 
         [0057]    Referring to  FIG. 2D , after the second bend joint  52  undergoes full deflection such that the second bend section  54  engages the third bend section, the third bend joint  52  deflects and the third bend section  54  moves/pivots/rotates into engagement with the inner tube  24 . From this, it can be seen that the bendable tip  12  is configured to function so that the bend sections  54  move in succession, from tip to base, i.e., from distal to proximal. This particular motion can be advantageous, for example, in permitting the apparatus  10  to navigate sharp turns within the patient&#39;s anatomy. For instance, in  FIGS. 2A-2D , it can be seen that the overall length of the apparatus  10  is shortened as the tip  12  undergoes bending. If, however, the inner tube  24  is controlled to advance axially at the same rate that the length is shortened due to the bending, the net result is that the tip, i.e., the surgical instrument, will navigate a sharp turn that otherwise could not be navigated if all of the bend joints  52  deflect at the same time. 
         [0058]    The arc length varying curvature of the bendable tip  12  is customizable through selection of the geometry of the cutouts  50 , which define the geometries of the bend joints  52  and bend sections  54 . Each cutout  50  can have a uniquely configured geometry that defines the amount of force required to deflect the bend joint  52 , the direction in which the joint deflects, and the geometry of the bend sections  54 , which define the limit of angular deflection. In this manner, the behavior of each segment of the tip  12 , i.e., the bend joint  52  and adjacent bend sections  54  defined by a cutout  50 , can be tailored so that the motion profile of the tip, and the attached surgical instrument  30 , is suited to perform the desired tasks. The tip  12 , so designed, can access the target anatomical structures while avoiding others. 
       Alternative Configurations 
       [0059]    The bendable tip could have additional configurations that lend to its ability to provide a desired degree of reach and dexterity. For example, referring to  FIG. 14 , the apparatus  10  could include multiple tendon cables  40  that are actuatable independently to effectuate bending of the tip. In the examples of  FIG. 14 , the apparatus  10  includes two tendon cables  40   a ,  40   b . The apparatus  10  could, however, include a greater number of tendon cables  40 . 
         [0060]    In Configuration A in  FIG. 14 , tendon cable  40   a  is connected to the bend section  54  at the terminal end of the bendable tip  12 . The tendon cable  40   b  is connected to a bend section  54  at about the midpoint of the bendable tip  12 . The tendon cable  40   b  can be actuated to bend a proximal section  12   b  of the bendable tip  12 . The tendon cable  40   a  can be actuated to bend a distal section  12   a  of the bendable tip  12 . In operation, the tendon cable  40   b  can be manipulated to bend the proximal section  12   b  in order to adjust the position and attitude of the distal section  12   a , which can then be actuated to complete the task. 
         [0061]    Configuration B in  FIG. 14  is similar to Configuration A, except that the radial positions of the bend sections  12   a  and  12   b  are rotated 180 degrees from each other about the axis  28 . In this example configuration, the bend sections  12   a  and  12   b  bend in opposite directions, and the tendon cables  40   a  and  40   b  are configured to effectuate this bending. Of course, the apparatus  10  could be configured to include multiple bend sections arranged in varying radial positions with corresponding tendon cables providing actuating capabilities for those sections individually. 
         [0062]    Referring to  FIG. 15 , the apparatus  10  could include multiple nested concentric tubes  24   a ,  24   b , each of which includes its own corresponding bendable tip  12   a ,  12   b . In this configuration, each bendable tip  12   a ,  12   b  can operate in accordance with any of the example embodiments described herein. For example, each bendable tip  12   a ,  12   b  can include one or more bend sections and corresponding tendon cables. As another example, the cutouts of either bendable tip  12   a ,  12   b  can be arranged in any radial configuration along the length of their corresponding tubes  24   a ,  24   b.    
         [0063]    Referring to  FIG. 15 , in operation, the nested concentric tubes  24   a ,  24   b  can be manipulated for (1) translation along the axis  28  and (2) rotation about the axis. Through this axial and translational manipulation, the tubes  24   a ,  24   b  can be advanced, rotated, and bent in order to follow a desired path and also to achieve a desired shape. 
       Device Modeling 
       [0064]    To design the bendable tip  12  that exhibits an arc length varying curvature tailored to specific anatomical target structures and workspaces, kinematics and statics models are required. The kinematic model predicts the operation or motion of the bendable tip  12 . The statics model predicts how forces acting on the bendable tip  12 , i.e., the forces applied by the tendon cable  40 , affect the bending of the bend joints  52  and sections  54 . 
         [0065]    Referring to  FIGS. 4A and 4B , the cutouts  50  can be either symmetric ( FIG. 4A ) or asymmetric ( FIG. 4B ). Of course, the asymmetric cutout  50  has a much longer moment arm for the tendon force. The designs illustrated in  FIGS. 1-3  are asymmetric. One advantage of using asymmetric cutouts  50  is the longer moment arm between the tendon anchor point and the neutral bending plane, which enables significantly lower tendon cable  40  actuation forces for devices of comparable diameter. Another advantage is the ability to achieve a tighter radius of curvature, since the radius of curvature is measured about the center of the bendable tip  12 , whereas the tip bends about an offset neutral bending plane. Other advantages of the asymmetric geometry include single wire actuation and simplified tendon routing, since the tendon will naturally conform to the inside wall of the tube when pulled, and one need not design mechanisms to hold it in place (e.g., the use of two nitinol tubes with the tendon sandwiched between). 
         [0066]    One potential limitation of an asymmetric design is that it can bend in only one direction in the plane, rather than two. However, provided axial rotation of the entire device is possible (which it typically is for such devices), the impact of any potential drawback is minimized. Another potential limitation of an asymmetric bendable tip  12  is that while it can readily apply pulling forces, it can only apply pushing forces if the tissue being pushed is more compliant than the bendable tip  12  itself. It can, however, be possible to stiffen the bendable tip  12  to assist with pushing by inserting a wire through the central lumen. 
         [0067]    In addition to being able to be manufactured and assembled at small diameters, the continuum cutout design also offers a large design space. In the kinematics and statics modeling, the cutouts  50  are restricted to rectangular cutouts because they are straightforward to machine. With this restriction, the design parameters available are the height, depth, and spacing between cutouts  50 , as well as the number of cutouts. The models and design principles set forth below allow the designer to use these parameters to select the device&#39;s overall radius of curvature, total maximum bend angle, and required tendon force for actuation. 
       Kinematic Modeling 
       [0068]    We begin by modeling the kinematics of a single cutout of the asymmetric continuum bendable tip  12 . We assume that the portion of the tube that undergoes bending deforms in a constant curvature arc. This is a good assumption for small cut heights h, because the tendon follows an approximately circular path in this case. Following the direction of R. J. Webster III and B. A. Jones, “Design and kinematic modeling of constant curvature continuum robots: a review,”  The International Journal of Robotics Research , vol. 29, no. 13, pp. 1661-1683, 2010, we map tendon displacement (actuator space) to arc parameters (configuration space) then map arc parameters to task space. 
         [0069]    Arc parameters and relevant kinematic values for single cutout are shown in  FIG. 5 . The arc parameters we seek are curvature (K) and arc length (s). The actuator space to configuration space mapping is largely dependent on the location  y  of the neutral bending plane. The neutral bending plane experiences no strain in bending and intersects the centroids of the axial cross sections of the cut portions of the tube. 
         [0070]      FIG. 6  illustrates the geometric parameters (a, b, c, g, h, r i , r o ) that the designer is free to choose. The tendon  40  is looped through the top cutout  50 . The regions of the uncut portion of the tube used for the calculation of the neutral bending plane location are shown at A i  and A o . 
         [0071]    The location of the neutral bending plane is dependent on the depth of cut g and the inner and outer radii of the tube (r i  and r o  shown in  FIG. 6 ) and is given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     y 
                     _ 
                   
                   = 
                   
                     
                       
                         
                           
                             y 
                             _ 
                           
                           o 
                         
                          
                         
                           A 
                           o 
                         
                       
                       - 
                       
                         
                           
                             y 
                             _ 
                           
                           i 
                         
                          
                         
                           A 
                           i 
                         
                       
                     
                     
                       
                         A 
                         o 
                       
                       - 
                       
                         A 
                         i 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     1 
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    where Ao and Ai are the areas defined in  FIG. 6  and  y   o  and  y   i  are their respective centroids. They are given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         y 
                         _ 
                       
                       o 
                     
                     = 
                     
                       
                         4 
                          
                         
                           r 
                           o 
                         
                          
                         
                           
                             sin 
                             3 
                           
                            
                           
                             ( 
                             
                               
                                 1 
                                 2 
                               
                                
                               
                                 φ 
                                 o 
                               
                             
                             ) 
                           
                         
                       
                       
                         3 
                          
                         
                           ( 
                           
                             
                               φ 
                               o 
                             
                             - 
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 φ 
                                 o 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         y 
                         _ 
                       
                       i 
                     
                     = 
                     
                       
                         4 
                          
                         
                           r 
                           i 
                         
                          
                         
                           
                             sin 
                             3 
                           
                            
                           
                             ( 
                             
                               
                                 1 
                                 2 
                               
                                
                               
                                 φ 
                                 i 
                               
                             
                             ) 
                           
                         
                       
                       
                         3 
                          
                         
                           ( 
                           
                             
                               φ 
                               i 
                             
                             - 
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 φ 
                                 i 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       A 
                       o 
                     
                     = 
                     
                       
                         
                           r 
                           o 
                           2 
                         
                          
                         
                           ( 
                           
                             
                               φ 
                               o 
                             
                             - 
                             
                               sin 
                                
                               
                                 ( 
                                 
                                   φ 
                                   o 
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                       
                       2 
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       A 
                       i 
                     
                     = 
                     
                       
                         
                           r 
                           i 
                           2 
                         
                          
                         
                           ( 
                           
                             
                               φ 
                               i 
                             
                             - 
                             
                               sin 
                                
                               
                                 ( 
                                 
                                   φ 
                                   i 
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                       
                       2 
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       φ 
                       o 
                     
                     = 
                     
                       2 
                        
                       
                         arccos 
                          
                         
                           ( 
                           
                             
                               ( 
                               
                                 g 
                                 - 
                                 
                                   r 
                                   o 
                                 
                               
                               ) 
                             
                             / 
                             
                               r 
                               o 
                             
                           
                           ) 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       φ 
                       i 
                     
                     = 
                     
                       2 
                        
                       
                         arccos 
                          
                         
                           ( 
                           
                             
                               ( 
                               
                                 g 
                                 - 
                                 
                                   r 
                                   o 
                                 
                               
                               ) 
                             
                             / 
                             
                               r 
                               i 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     2 
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    which are valid for cuts that are at least as deep as the outer radius of the tube. 
         [0072]    Now we can use  y  to find the mapping from curvature to tendon displacement (Δl), noting  FIG. 5  and using the chord function and arc geometry: 
         [0000]    
       
         
           
             
               
                 
                   
                     Δ 
                      
                     
                         
                     
                      
                     l 
                   
                   = 
                   
                     h 
                     - 
                     
                       2 
                        
                       
                         ( 
                         
                           
                             1 
                             k 
                           
                           - 
                           
                             r 
                             i 
                           
                         
                         ) 
                       
                        
                       
                         sin 
                          
                         
                           ( 
                           
                             
                               κ 
                                
                               
                                   
                               
                                
                               h 
                             
                             
                               2 
                                
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   
                                     
                                       y 
                                       _ 
                                     
                                      
                                     κ 
                                   
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     3 
                   
                   ) 
                 
               
             
           
         
       
     
         [0073]    Since we want the mapping of tendon displacement to curvature, we need to invert (3). Since it has no analytic inverse, numerical techniques can be used, or, for small angles, we can use a first-order approximation to yield: 
         [0000]    
       
         
           
             
               
                 
                   κ 
                   ≈ 
                   
                     
                       Δ 
                        
                       
                           
                       
                        
                       l 
                     
                     
                       
                         h 
                          
                         
                           ( 
                           
                             
                               r 
                               i 
                             
                             + 
                             
                               y 
                               _ 
                             
                           
                           ) 
                         
                       
                       - 
                       
                         Δ 
                          
                         
                             
                         
                          
                         l 
                          
                         
                           y 
                           _ 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     4 
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    Once κ is known, s can be found using: 
         [0000]    
       
         
           
             
               
                 
                   s 
                   = 
                   
                     h 
                     
                       1 
                       + 
                       
                         
                           y 
                           _ 
                         
                          
                         κ 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     5 
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    Once the arc parameters κ and s are known, the homogeneous transformation between frames j and j+1 (as defined in  FIG. 5 ) can be found using: 
         [0000]    
       
         
           
             
               
                 
                   
                     T 
                     j 
                     
                       j 
                       + 
                       1 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           
                             cos 
                              
                             
                               ( 
                               
                                 κ 
                                  
                                 
                                     
                                 
                                  
                                 s 
                               
                               ) 
                             
                           
                         
                         
                           
                             - 
                             
                               sin 
                                
                               
                                 ( 
                                 
                                   κ 
                                    
                                   
                                       
                                   
                                    
                                   s 
                                 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               ( 
                               
                                 
                                   cos 
                                    
                                   
                                     ( 
                                     
                                       κ 
                                        
                                       
                                           
                                       
                                        
                                       s 
                                     
                                     ) 
                                   
                                 
                                 - 
                                 1 
                               
                               ) 
                             
                             / 
                             κ 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                               ( 
                               
                                 κ 
                                  
                                 
                                     
                                 
                                  
                                 s 
                               
                               ) 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                               ( 
                               
                                 κ 
                                  
                                 
                                     
                                 
                                  
                                 s 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                 ( 
                                 
                                   κ 
                                    
                                   
                                       
                                   
                                    
                                   s 
                                 
                                 ) 
                               
                             
                             / 
                             κ 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     6 
                   
                   ) 
                 
               
             
           
         
       
     
         [0074]    Due to the rectangular cutout geometry of the bendable tip  12 , the kinematic transformation from the base of the bendable tip  12  to the tip can be obtained. The kinematics of the entire bendable tip  12  are given by repeatedly applying the transformation (6) in conjunction with translations to account for the portions of the bendable tip  12  that do not bend: 
         [0000]    
       
         
           
             
               
                 
                   
                     T 
                     o 
                     t 
                   
                   = 
                   
                     
                       
                         T 
                         
                           z 
                           , 
                           a 
                         
                       
                        
                       
                         ( 
                         
                           
                             ∏ 
                             
                               j 
                               = 
                               1 
                             
                             n 
                           
                            
                           
                               
                           
                            
                           
                             
                               T 
                               j 
                               
                                 j 
                                 + 
                                 1 
                               
                             
                              
                             
                               T 
                               
                                 z 
                                 , 
                                 c 
                               
                             
                           
                         
                         ) 
                       
                     
                      
                     
                       T 
                       
                         z 
                         , 
                         
                           b 
                           - 
                           c 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     7 
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    where n is the number of cutouts and T z-a , T z,b-c , and T z,c  are translations along the z-axis by a, b-c, and c, respectively, as defined in  FIG. 6 . In addition, the angle of rotation of each section can be found explicitly as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       θ 
                       j 
                     
                      
                     
                       ( 
                       κ 
                       ) 
                     
                   
                   = 
                   
                     
                       s 
                        
                       
                           
                       
                        
                       κ 
                     
                     = 
                     
                       
                         ( 
                         
                           h 
                           
                             1 
                             + 
                             
                               
                                 y 
                                 _ 
                               
                                
                               κ 
                             
                           
                         
                         ) 
                       
                        
                       κ 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     8 
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    And thus the maximum angle of rotation for a single cutout is given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     θ 
                     
                       j 
                       , 
                       max 
                     
                   
                   = 
                   
                     
                       
                         θ 
                         j 
                       
                        
                       
                         ( 
                         
                           1 
                           / 
                           
                             r 
                             o 
                           
                         
                         ) 
                       
                     
                     = 
                     
                       h 
                       
                         
                           r 
                           o 
                         
                         + 
                         
                           y 
                           _ 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     9 
                   
                   ) 
                 
               
             
           
         
       
     
         [0075]    Two important bendable tip  12  characteristics, maximum bending angle and minimum radius of curvature, as shown in  FIG. 7 , can be calculated from geometry as: 
         [0000]    
       
         
           
             
               
                 
                   
                     θ 
                     max 
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           0 
                         
                         n 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         
                           j 
                           , 
                           max 
                         
                       
                     
                     = 
                     
                       n 
                        
                       
                         h 
                         
                           
                             r 
                             o 
                           
                           + 
                           
                             y 
                             _ 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     10 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     ρ 
                     min 
                   
                   ≈ 
                   
                     
                       r 
                       o 
                     
                     + 
                     
                       
                         
                           ( 
                           
                             n 
                             - 
                             1 
                           
                           ) 
                         
                          
                         c 
                       
                       
                         θ 
                         max 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     11 
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    where the approximately circular arc that defines ρ min  has length: 
         [0000]    
       
         
           
             
               
                 
                   S 
                   = 
                   
                     
                       n 
                        
                       
                         ( 
                         
                           
                             
                               
                                 r 
                                 o 
                               
                                
                               h 
                             
                             
                               
                                 r 
                                 o 
                               
                               + 
                               
                                 y 
                                 _ 
                               
                             
                           
                           + 
                           c 
                         
                         ) 
                       
                     
                     - 
                     c 
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     12 
                   
                   ) 
                 
               
             
           
         
       
     
       Statics Modeling 
       [0076]    Modeling the static behavior of the bendable tip  12  is more challenging than modeling the kinematic behavior, yet with the assumption of constant curvature bending, it is tractable. Based on the constant curvature assumption, strain along the length of the bendable tip  12  varies in a cross section of the portion of the tube in bending according to: 
         [0000]    
       
         
           
             
               
                 
                   
                     ε 
                      
                     
                       ( 
                       
                         y 
                         , 
                         κ 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       κ 
                        
                       
                         ( 
                         
                           y 
                           - 
                           
                             y 
                             _ 
                           
                         
                         ) 
                       
                     
                     
                       1 
                       + 
                       
                         
                           y 
                           _ 
                         
                          
                         κ 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     13 
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    and thus is linearly distributed about the neutral bending plane. This assumed relationship between the geometry and the material deformation allows for a simple computation of the strain energy, after which we use Castigliano&#39;s first theorem to determine the reaction force at the tendon. In general, the behavior of nitinol under applied stresses is complex and highly nonlinear, and depends on thermomechanical history. In this work we assume a simplified material model that represents the stress-strain behavior of nitinol as a piecewise linear stress-strain curve, so that the stress may be written as a function of strain as: 
         [0000]    
       
         
           
             
               
                 
                   
                     σ 
                      
                     
                       ( 
                       ε 
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             σ 
                             lp 
                           
                         
                         
                           
                             ε 
                             &lt; 
                             
                               
                                 σ 
                                 lp 
                               
                               / 
                               E 
                             
                           
                         
                       
                       
                         
                           
                             E 
                              
                             
                                 
                             
                              
                             ε 
                           
                         
                         
                           
                             
                               
                                 σ 
                                 lp 
                               
                               / 
                               E 
                             
                             ≤ 
                             ε 
                             ≤ 
                             
                               
                                 σ 
                                 up 
                               
                               / 
                               E 
                             
                           
                         
                       
                       
                         
                           
                             σ 
                             up 
                           
                         
                         
                           
                             ε 
                             &gt; 
                             
                               
                                 σ 
                                 up 
                               
                               / 
                               E 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     14 
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    where σ lp  is the lower plateau stress (corresponding to compression), σ up  is the upper plateau stress (corresponding to tension), and E is Young&#39;s modulus. Since we are modeling the material deformation as a one-dimensional stretching and compression of axial fibers, the strain energy density is the area under the stress-strain curve, given by the integral: 
         [0000]        W (ε)=∫ 0   ε σ( e ) de   (Eq. 15)
 
         [0077]    The total strain energy stored in the bendable tip  12  as a function of the curvature κ of a single cutout is given by: 
         [0000]        U (κ)= n∫   V     c     W ( y ,κ)) dV   (Eq. 16)
 
         [0000]    where V c  is the volume defined by the “Top View Cut” cross section of  FIG. 6  and cutout height h. We use Castigliano&#39;s first theorem to find the relationship between rotation θ of the bendable tip  12  and force F applied by the tendon to the bendable tip  12  tip: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       
                         U 
                          
                         
                           ( 
                           κ 
                           ) 
                         
                       
                     
                     
                       ∂ 
                       θ 
                     
                   
                   = 
                   
                     M 
                     = 
                     FL 
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     17 
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    where L is the moment arm length and θ=nsK. When the tendon is looped around the top flexure as shown in  FIG. 6 , the moment arm has length 
         [0000]    
       
         
           
             L 
             = 
             
               
                 ( 
                 
                   
                     
                       r 
                       o 
                     
                     + 
                     
                       r 
                       i 
                     
                   
                   2 
                 
                 ) 
               
               + 
               
                 
                   y 
                   _ 
                 
                 . 
               
             
           
         
       
     
         [0078]    Due to friction, the force the tendon applies to the tip of the bendable tip  12  will be a fraction of the actuator force applied to the tendon. Friction between the tendon and the tube wall becomes increasingly significant as cut height and angle of bending increase. To model this effect, we first find the angle γ (shown in  FIG. 5 ) that the tendon is required to navigate at a single corner of a cutout section at a given angle of deflection. We assume that the friction that occurs at these corners dominates friction elsewhere along the tendon path. Writing the static balance equations for a single corner, with _s as the static friction coefficient, we find that: 
         [0000]    
       
         
           
             
               
                 
                   F 
                   = 
                   
                     
                       η 
                        
                       
                           
                       
                        
                       
                         F 
                         tendon 
                       
                     
                     = 
                     
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               γ 
                               / 
                               2 
                             
                           
                           - 
                           
                             
                               μ 
                               s 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               γ 
                               / 
                               2 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               γ 
                               / 
                               2 
                             
                           
                           + 
                           
                             
                               μ 
                               s 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               γ 
                               / 
                               2 
                             
                           
                         
                       
                        
                       
                         F 
                         tendon 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     18 
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    where η&lt;1 accounts for the force lost due to friction at a corner. We can substitute (18) into (17) to yield: 
         [0000]    
       
         
           
             
               
                 
                   
                     F 
                     tendon 
                   
                   = 
                   
                     
                       1 
                       
                         
                           η 
                           
                             2 
                              
                             n 
                           
                         
                          
                         L 
                       
                     
                      
                     
                       
                         ∂ 
                         
                           U 
                            
                           
                             ( 
                             κ 
                             ) 
                           
                         
                       
                       
                         ∂ 
                         θ 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     19 
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    where 2n is included to account for the two corners of each cutout. This expression can be evaluated numerically using a finite difference method to relate F tendon  and θ. This statics model is experimentally validated in the following paragraphs. 
       Prototype and Experimental Validation 
       [0079]    A prototype of the bendable tip  12  is shown in  FIGS. 8, 9, and 10 . The prototype bendable tip  12  carries a curette as the surgical instrument. The curette is connected to a nitinol wire that runs through the tube. The wire can be rotated to effectuate rotation of the curette. 
         [0080]    The prototype bendable tip  12  was built using a MicroProto Systems MicroMill 2000 CNC mill (a small tabletop CNC mill) with aluminum titanium nitride coated, two flute, carbide, long flute, 0.02″ diameter square end mills. The tube was fixtured by gluing it in a channel drilled in an aluminum block. The nitinol tube had an outer diameter 1.16 mm and inner diameter of 0.86 mm. A cut depth of g=0.97 mm was chosen, which corresponds to a required tendon force for full bending of F tendon =5N and a maximum outer-fiber strain of 10.4% (Note that this is slightly higher than the 8-10% recoverable strain typically quoted for nitinol, but that it has been found to work well in practice, since only a small amount of the material at the very outside edge of the bendable tip  12  undergoes this strain, and then only at maximum articulation). The cut height was h=0.51 mm. The spacing between cuts was c=0:51 mm. The number of cuts was h=5 cuts in order to achieve greater than 90 degrees of bending. A summary of the design parameters and resulting design characteristics is shown in Table I: 
         [0000]    
       
         
               
               
               
               
               
             
           
               
                   
                   
               
               
                   
                 Parameter 
                 Value 
                 Characteristic 
                 Value 
               
               
                   
                   
               
             
             
               
                   
                 D o   
                 1.16 mm 
                 θ max   
                 138.6° 
               
               
                   
                 D i   
                 0.86 mm 
                 ρ min   
                 1.42 mm 
               
               
                   
                 g 
                 0.97 mm 
                 ε max   
                  10.4% 
               
               
                   
                 h 
                 0.51 mm 
                 F tendon   
                 5N 
               
               
                   
                 c 
                 0.51 mm 
               
               
                   
                 n 
                 5 
               
               
                   
                   
               
             
          
         
       
     
         [0081]      FIG. 7  illustrates the motion of the bendable tip  12  motion from 0 to 90 degrees bending angle. Note that the ring curette is being rotated during the bending motion of the bendable tip  12 . The prototype was experimentally validated without the curette instrument attached (see  FIG. 10 ). The actuation tendon held open the last cutout of the bendable tip  12  during the experimental trials, and the kinematics and statics models were calculated with n=4 cutouts. 
         [0082]    An experiment was conducted to validate the kinematic relationship of Equation 7 and the static relationship of Equation 19 concurrently. The experimental setup included a linear slide (Velmex A2512Q2-S2.5) with 0.01 mm resolution to displace the tendon and a force sensor (ATI Nano 17) with 3.125 mN resolution to record tendon force. The tendon was rigidly fixed to an acrylic plate that was then mounted onto the force sensor. The tendon and sensor assembly were then rigidly fixed to the linear slide carrier. 
         [0083]    The nitinol tube with cutout bendable tip  12  was mounted into a test fixture that was rigidly mounted to an optical table, such that the tube remained stationary while the bendable tip  12  was deflected with the linear slide. A 1 mm resolution grid was placed below the bendable tip  12 , and a camera mounted directly above the bendable tip  12  was used to capture images of the bendable tip  12  as it deflected. The bendable tip  12  was deflected in tendon displacements of 0.2 mm, and a picture of the bendable tip  12  deflection and the tendon force were recorded at each increment. 
         [0084]    Using image processing, the tip position was determined for each incremental deflection of the tendon. At full articulation, it was observed that the distal cutout was held open by the tendon that was routed through it (see  FIG. 10 ). For this reason, the plots in  FIGS. 11 and 12  were made based on n=4 cutouts. Alternative tendon attachment methods can address this issue. Results of the experiment are shown in  FIGS. 11 and 12 . 
         [0085]    Referring to  FIG. 11 , the bendable tip  12  starts at top of the figure and rotates counterclockwise from 0 to 110 degrees. These results show that the constant curvature assumption is a reasonable approximation for this geometry, since the bendable tip  12  tip follows the path predicted by the model. 
         [0086]    An experimental validation of the statics model is shown in  FIG. 12 . The model captures the superelastic behavior of the material, with the change in the slope of the graph indicating the transition of some of the volume of material into the stress plateau region. For the material properties, note that nitinol has an asymmetric stress strain relationship in tension and compression. We assume plateau stresses of σ lp , =−750 MPa and cσ up =500 MPa and a Young&#39;s modulus of E=60 GPa, which fall within ranges reported by the manufacturer and in the literature. The model is shown with a coefficient of friction of 0.36, which was chosen through nonlinear least squares optimization. Note that the superelastic, nonlinear behavior of the material is clearly captured by the model. 
       Discussion 
       [0087]    The prototype represents one set of viable design choices. With the rectangular cut profile described in previous sections, the designer must choose the depth of cut g (see  FIG. 6 ), height of cut h, number of cuts n, and axial spacing between cuts c. Moreover, the designer also has some freedom to select the tube radii, though this is likely to be from among a finite set of options due to material availability. The tube radii and the depth of cut are the most important parameters in determining bendable tip  12  behavior, because they determine the location of the neutral bending plane, which strongly affects the kinematics, strain in the bending material, and the required actuation force. A cut depth g&gt;r o  is desirable to achieve substantial bending compliance. The allowable depth of cut is bounded by the maximum allowable strain, where the maximum strain at full bending is given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     ε 
                     max 
                   
                   = 
                   
                     
                       ε 
                        
                       
                         ( 
                         
                           
                             r 
                             o 
                           
                           , 
                           
                             1 
                             / 
                             
                               r 
                               o 
                             
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           r 
                           o 
                         
                         - 
                         
                           y 
                           _ 
                         
                       
                       
                         
                           r 
                           o 
                         
                         + 
                         
                           y 
                           _ 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     20 
                   
                   ) 
                 
               
             
           
         
       
     
         [0088]    Cut height is not as significant as cut depth in determining bendable tip  12  behavior, but it is a factor in the bending radius (Eq. 10 and 11). Moreover, if cut height becomes too large, the constant curvature assumption will no longer hold, risk of buckling-like failure will increase, and frictional losses will increase (Eq. 18 and 19). 
         [0089]    The portions of uncut tube between the cutouts serve as hard stops which limit strain, allow large forces to be applied in the bendable tip&#39;s fully deflected state, and route the tendon in a curve that approximates a circular arc. The height of the uncut portions, parameter c in  FIG. 6 , should be as small as possible to minimize radius of curvature. However, as it decreases, risk of damaging the bendable tip  12  during actuation and environmental interaction increases. 
         [0090]    Additionally, if uniform curvature in multiple cutout sections is desired, it is essential to use a highly repeatable cutting process, as slightly deeper cutouts deflect much further for a given force than shallower cutouts do. That being said, it may be advantageous in future work to take advantage of non-uniform cut depths (and/or heights) to compensate for factors like non-constant tendon tension (due to frictional losses) along the bendable tip  12 , or application-specific design objectives. 
         [0091]    The experimental results show that the constant curvature assumption is a reasonable, though not perfect, approximation for our bendable tip  12 . We believe that tendon elongation was the primary source of error in the kinematics, which resulted in the model and experimental tip points not aligning perfectly in  FIG. 11 . The coefficient of friction is likely the least well known of all the parameters, since the amount of friction depends on factors such as surface roughness and geometry. Another potential source of error is the implicit assumption that cross sections do not deform during bending, which is a common assumption in beam bending analysis. 
         [0092]    In the future, we plan to study the significance of hysteresis in our statics model and develop a three-dimensional stiffness model in order to characterize the forces that the bendable tip  12  can exert. We also plan to conduct finite element modeling to characterize torsional properties and fatigue life and to explore strain profiles of non-rectangular cutouts. Another area of future work is to explore non-square cutout geometries to optimize bendable tip  12  performance for specific tasks. 
       Steerable Needle Bendable Tip 
       [0093]    Another embodiment employing the same principles described above is illustrated in  FIGS. 16-18 . Referring to  FIGS. 16A and 16B , in this embodiment, the bendable tip surgical device  12  is a bendable tip steerable needle  110 . The steerable needle  110  is a needle constructed of a flexible tube  112  with a one or more cutouts  114  that create a compliant bending region  116  of the tube. The number of cutouts  114  can vary depending on the desired bending performance characteristics for the needle  110 . In the embodiment illustrated in  FIGS. 16-18 , the needle  110  includes a single cutout  114 . The needle  110  could, however, include multiple cutouts  114  and perform in accordance with the descriptions of embodiments set forth above in which the bendable tip includes multiple cutouts. As an example, the flexible tube  112  can be constructed of a nickel-titanium, i.e., “nitinol,” alloy. 
         [0094]    The cutout  114  defines the boundary between an elongated body portion  120  and a tip  122  of the steerable needle  110 . The body portion  120  can have any desired length, which can, for example, vary depending on the procedure in which the steerable needle  110  is implemented. The tip  122  is formed by a beveled cut of the tube  112  that is filled or closed off, for example, by welding, soldering, or brazing, to form an angled or beveled lead surface  124 . Alternative fillers, such as a polymer, can be used to fill the tip  122  and form the lead surface  124 . 
         [0095]    The cutout  114  extends into the tube  112  in a direction normal to the tube axis  130 , entering the tube from opposite the lead surface  124 . In the embodiment illustrated in  FIGS. 16-18 , the cutout  114  has a generally rectangular cross-section. It is the dimensions of the cutout  114  and the diameter of the tube  112  that determine the range, indicated generally by angle θ, that the tip  122  can bend relative to the body portion  120 . This range of bending can be controlled or configured through the selection of the dimensions of the cutout  114 , e.g., the width of the cutout as measured along the axis  130 . The range of bending can also be controlled or configured through the selection of the shape of the cutout  114 , e.g., a V-shaped cutout as opposed to a rectangular cutout. 
         [0096]    The elongated tubular configuration of the bendable tip needle  110  advantageously includes a long inner lumen that defines a channel  126  within the body portion  120  of the tube  112  that extends to the opening, i.e., the cutout  114 , adjacent the needle tip  122 . This channel  126  can serve as a large working channel from the base of the needle to the tip, for example, to perform biopsy or drug delivery therapies. Further facilitating this is the fact that the bend is facilitated by the cutout  114  in the tube  112 , which eliminates the need for any mechanical joint components that would consume space in the channel  126 . 
         [0097]    Referring to  FIGS. 17A-17D , in operation, the steerable needle  110  is advanced toward a body of tissue  132 , such as human body tissue, in a direction indicated generally by arrow A. As the needle  110  enters the tissue  132 , the tissue offers resistance to needle advancement, as indicated generally by the arrow B in  FIG. 17B . A component of these resistance forces act normal to the lead surface  124  of the needle tip  122 , indicated generally by arrow C. These component forces cause the tip  122  to bend relative to the body portion  120  in a manner described above in regard to  FIGS. 16A and 16B . 
         [0098]    Referring to  FIG. 17C , the bent tip  122  causes the needle  110  to follow a curved path, as indicated generally by dashed line D in  FIG. 17C . As a result, a portion  134  of the body portion immediately trailing the tip  122  follows this curved path and assumes a curved configuration. 
         [0099]    Referring to  FIG. 17D , when the tip  122 , following the curved path D, reaches a desired trajectory, the body portion  120  can be rotated, as indicated generally by the arrow F in  FIG. 17D , which causes the tip to resume its non-bent configuration, extending along a path E that is coaxial with the body portion. Maintaining this rotation while the needle  110  is advanced (arrow A) can cause the needle to follow a straight path. When rotation is stopped, the tip  122  will again bend and follow a curved path as the needle is advanced further. Thus, by selecting the rotational orientation of the lead surface  124 , the curved path, i.e., the curved direction of needle advancement, can be selected. 
         [0100]    Referring to  FIGS. 18A and 18B , the large working channel  126  of the steerable needle  110  can be used to house an actuating member  140 , such as a cable or wire (e.g., nitinol wire), that acts as a tendon for actuating the bendable tip  122 . The tendon cable  140  is connected to the interior of the tip  122  at a connection point  142  formed, for example, via weld, solder, brazing, or adhesive bond. Tension on the tendon cable  140 , as indicated generally by the arrow G in  FIG. 18B , causes the tip  122  to bend at the bending portion  116 . 
         [0101]    In the configuration of  FIGS. 18A and 18B , the tube material can be selected and the bendable tip  122  can be configured such that the tip will not bend in response to tissue forces acting on the lead surface  124  as described above. Instead, bending of the tip  122  can be controlled primarily or exclusively through tension applied via the tendon cable  140 . In this manner, the degree of tip deflection, and thus the amount of curvature with which the needle  110  responds, can be selected through the displacement of the tendon cable  140 . Thus, not only does the tendon cable  140  allow for precise control of when the tip  122  bends, but also the degree to which it bends. As a result, the configuration of the bendable tip steerable needle  110  in  FIGS. 18A and 18B  can allow for a higher degree of precision in steering the needle. 
         [0102]    The bendable tip steerable needle  110  is suited for any needle-based procedure that requires accurate targeting and also provides the ability to reposition/retarget without full removal of the needle. This feature can be particularly useful, for example, for correcting needle misalignment or unforeseen deflection of the needle during insertion. 
         [0103]    The design of the bendable tip steerable needle  110  is straightforward and simple to build from a manufacturing perspective, while advantageously leaving the center working channel open all the way to the tip of the needle. Tip deflection can be achieved in a simple, accurate, and repeatable manner through tension on the tendon cable  140 . Though simple in design, the steerable needle  110  can exhibit a high degree of steerability with minimized tissue damage and a high degree of curvature.