CROSS REFERENCE TO RELATED APPLICATIONS 
     This application is a continuation of U.S. application Ser. No. 12/965,362 filed Dec. 10, 2010, now assigned U.S. Pat. No. 8,470,051, which claims priority from and benefit of the filing date of U.S. provisional application Ser. No. 61/286,345 filed Dec. 14, 2009, and the entire disclosure of each of said prior applications is hereby expressly incorporated by reference into the present specification. 
    
    
     GOVERNMENT INTEREST 
     This invention was made with government support under Contract No. N66001-06-C8005, awarded by Defense Advanced Research Projects Agency. The government has certain rights in the invention. 
    
    
     BACKGROUND 
     A “mechanical finger” refers to an elongated, articulating, mechanical appendage. Like a human finger, a mechanical finger has one end joined to a structure that acts as a base and an opposite end that is not anchored or connected. A mechanical finger used for grasping typically has two or more rigid sections, and preferably at least three, connected end to end by articulating joints. Terminology used to describe the anatomy of a human finger is used to describe a mechanical finger. As in the human finger, each section of the finger is referred to as a “phalanx.” A finger extends from a base and is comprised of at least two, and preferably three, phalanges joined end to end by pivoting or articulating joints. A first articulating joint joins a proximal phalanx to a base, such as a palm of a hand. A second articulating joint joins the proximal phalanx to an intermediate or middle phalanx, and a third articulating joint joins the intermediate phalanx to a distal phalanx. The first joint is referred to as the metacarpophalangeal (MCP) joint, the second as the proximal interphalangeal (PIP) joint, and the third as the distal interphalengeal (DIP) joint. 
     In a mechanical finger, the phalanges are coupled to one or more motors to cause flexion and extension of the finger. When using a kinematic mechanism for coupling a single motor to the phalanges, the position of the actuator fully determines the position of the joints, but the torque at each joint is unknown. With a differential mechanism, the torque at the actuator determines the torque at each of the driven joints, but neither the velocity nor the position of the individual joints are specified by the actuator velocity or position alone. A kinematic mechanism produces consistent, predictable motion of the finger joints, but it does not allow the finger to curl around an object. Differential mechanisms allow curling and grasping, but often deviate from the desired motion due to forces at the fingertip, causing buckling, or due to friction in the joints, causing undesirable curling behavior when not conforming. 
     BRIEF DESCRIPTION 
     According to one aspect of an exemplary embodiment of a mechanical finger comprising at least two phalanges driven by a single actuator, and a differential transmits torque in parallel from the actuator to the MCP joint and the PIP joint. 
     According to another aspect, the mechanical finger further includes a variable stop that limits rotation of the PIP joint based on the angle of rotation of the MCP joint. Such a mechanical finger is capable of exhibiting consistent predictable motion when moving in free space or when contacting an object at the fingertip, and curling in order to conform to an object when the contact is at other locations on the finger. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1A  is a schematic illustration of a mechanical finger driven by a single actuator. 
         FIG. 1B  is a schematic illustration of an alternate embodiment of a mechanical finger driven by a single actuator. 
         FIG. 1C  is a schematic illustration of an alternate embodiment of a mechanical finger driven by a single actuator. 
         FIG. 1D  is a schematic illustration of an alternate embodiment of a mechanical finger driven by a single actuator. 
         FIG. 1E  is a schematic illustration of an alternate embodiment of a mechanical finger driven by a single actuator. 
         FIG. 2  is a perspective view of an example of a prosthetic finger, partially constructed and without a covering, embodying a coupling mechanism according to the principles of the mechanical finger of  FIG. 1 . 
         FIG. 3  is an exploded view of the prosthetic finger of  FIG. 2 . 
         FIG. 4A  is a side view, rendered with perspective, of proximal and medial phalanges of the prosthetic finger of  FIG. 2 , which is only partially constructed to reveal a differential linkage. 
         FIG. 4B  is a side view, not rendered with perspective, of the partially constructed proximal and medial phalanges of the prosthetic finger of  FIG. 2 , in an extended position. 
         FIG. 4C  is a side view, not rendered with perspective, of the partially constructed proximal and medial phalanges of the prosthetic finger of  FIG. 4B , in a fully flexed position. 
         FIG. 5A  is a side view, rendered with perspective, of proximal and medial phalanges of an alternate embodiment of a prosthetic finger that is partially constructed to reveal a differential linkage. 
         FIG. 5B  is a side view, not rendered with perspective, of the partially constructed proximal and medial phalanges of an alternate embodiment of the prosthetic finger of  FIG. 2 , in an extended position. 
         FIG. 5C  is a side view, not rendered with perspective, of the partially constructed proximal and medial phalanges of an alternate embodiment of the prosthetic finger of  FIG. 5B , in a fully flexed position. 
         FIG. 6  is a side view, rendered in perspective of proximal and medial phalanges of alternate embodiment of a mechanical partially constructed to reveal a differential linkage. 
         FIG. 7  is a side, perspective view of the partially constructed prosthetic finger of  FIG. 2 , with certain elements removed to reveal a linkage. 
         FIG. 8A  is a side, non-perspective view of the prosthetic finger of  FIG. 2 , with several parts removed to illustrate a stop linkage. 
         FIG. 8B  is a perspective view of  FIG. 7B . 
     
    
    
     DETAILED DESCRIPTION 
     In the following description of a mechanical finger, like numbers refer to like parts. 
       FIGS. 1A-1E  schematically illustrate several alternative embodiments of mechanisms for driving a mechanical finger  100  using a single motor. The mechanism combines a differential, a kinematic linkage and a PIP linkage for coupling the torque and position of a drive output to a mechanical finger  100  having at least two sections in order to control its flexion and extension in a manner that permits it to be used in connection with grasping or other applications in which a curling action is desirable. Such applications include, but are not limited to, robotic hands and prosthetic hands. 
     The illustrated examples of mechanical finger  100  comprise at least a proximal phalanx  102 , a medial or middle phalanx  104 , and, in the embodiments of  FIGS. 1A to 1E , a distal phalanx  106 . “Phalanx” refers to an elongated, rigid section of the finger, and “phalanges” to multiple sections of the finger. The phalanges are sometimes also referred to herein as first, second and third sections, respectively, of the mechanical finger. Articulating joints, which are not expressly indicated in the figure, permit joined phalanges to pivot with respect to each other around an axis of the joint. The X-axis  108  of the figure represents the angle of extension and flexion of the phalanges relative to each other and to a reference ground  110 . A greater angle indicates flexion of the finger and a smaller angle indicates extension of the finger. The length of arrow  112  represents the angle, designated by the variable Θ PF  between the proximal phalanx  102  and a ground  110 . Similarly, the lengths of arrows  114  and  116  represent the relative angles between the proximal phalanx and the middle phalanx, and between the middle phalanx and the distal phalanx, respectively. These angles are designated in the figure by the variables Θ MF  and Θ DF , respectively. 
     The angular position and torque transmitted by an output of a single actuator or drive, which output is represented by line  118 , controls the flexion and extension of the finger. Any type of suitable motor can power the actuator or drive. The type of the motor will depend on the application. The angular position of the output is represented by line  120  and is designated by the variable Θ m . Torque applied to an object by a joint is represented as a linear force in the figure. The torque delivered by the output of the drive is represented by line  122 . Variable T m  represents the magnitude of the torque from a motor connected to the drive. Note that the motor is not expressly illustrated in the figures. Torque on the metacarpophalangeal (MCP) joint (not shown), designated T mcp , which is generated by force applied to the proximal phalanx, is represented by line  103 . Similarly, torque on the proximal interphalangeal (PIP) joint (not shown) is designated T pip  and is represented by line  105 . Torque on the distal interphalangeal (DIP) joint (not shown) is designated T pip  and is represented by line  105 . 
     A hybrid mechanism comprising a kinematic linkage and differential enables conformal grasping by the finger due to the differential, but at the same time curling behavior can be precisely defined during application of forces to the distal phalanx only. In the examples illustrated by the schematics of  FIGS. 1A-1E , a differential  124  coupled to ground  110  applies the torque T m  from the motor to the proximal phalanx  102 . The differential also applies the torque to linkage  130  in the embodiments of  FIGS. 1A ,  1 B,  1 D, or to medial phalanx  104  in the two-phalanx embodiment of  FIG. 1C , or to a second differential  125  in the embodiment of  FIG. 1E . The differential  124  couples the drive output with the MCP joint and the PIP joint. Thus, the drive applies torque to both the PIP and MCP joints in the embodiments of  FIGS. 1A-1C  and  1 E. In the embodiment of  FIG. 1D , the combination of differential  124  and differential  125  applies torque applied to the MCP, PIP and DIP joints. 
     Linkage  130  in  FIGS. 1A ,  1 B, and  1 E functions as a kinematic linkage, coupling the motion of PIP and DIP joints through an algebraic relationship. Linkage  130  couples the PIP and DIP joints (not shown), so that both joints rotate together, in a fixed relationship, resulting in the medial and distal phalanges curling together in a natural curling motion. Movement of link  130  relative to the proximal phalanx  102  causes the middle phalanx to rotate about the PIP joint (not shown), and the distal phalanx to rotate with respect to the middle phalanx around the DIP joint (not shown). This coupled curling relative to the proximal phalanx  102  occurs even while motion of proximal phalanx  102  is blocked, such as when conformal grasping is occurring. 
     As shown in the embodiment illustrated only in  FIG. 1B , the linkage  124  may, optionally, include a compliant element  128 , in series with ground, represented in the figure by spring  128 . The compliant element is, for example, comprised of an elastic element that generates a spring force. The spring provides compliance for series elasticity and shock mitigation by allowing linkage  124  to stretch a little when forces are applied to it. Elasticity and shock mitigation or dampening can be desirable in certain applications, such a prosthetics. Movement of the linkage  130  relative to the proximal phalanx  102 , such as during curling when the proximal phalanx  102  is blocked, also results in compression of a compliant member represented in the figure by a spring  132  coupled between the proximal phalanx and the link  130 . The spring acts to extend the PIP joint. 
     Referring only to  FIGS. 1A-1D , in each of the illustrated examples a linkage  126  adjusts the position of stop  134  based on rotation of the MCP joint. Stop  134  limits the range of motion of the PIP joint. The linkage sets the position of the hard stop based on the degree of rotation of the MCP joint from ground. Stopping rotation of the PIP joint limits extension of the medial phalanx, as well as the distal phalanx, beyond a predetermined angle relative to the proximal phalanx. The angle of rotation of the MCP joint is represented in the figure as the distance between ground  110  and the proximal phalanx  102 . The angle of the PIP joint relative to the phalanx is indicated by the length of line  114  in the figure. The stop rotates with respect to the PIP joint as the MCP joint rotates, and thus it depends on the angle of the MCP joint. When the proximal phalanges motion is not blocked, the stop linkage  126  enforces natural, simultaneous curling of all three joints, the MCP, PIP and DIP joints. Linkage  126  also enables the finger to resist forces on the distal phalanx without the differential allowing the PIP and DIP joints to straighten and the MCP joint to flex. Despite the system having a differential, the posture of all three joints can thus remain fixed (not against stops) irrespective of the magnitude of a single external force applied to the distal phalanx. 
     Because of the use of a differential linkage to couple torque from the drive to the MCP and PIP joints, the positions of the MCP and PIP joints are not fully determined by the position of the drive. For any given position of the drive output, the finger mechanism has one free motion available, which is an extension of the proximal phalanx and a flexing of the PIP and DIP joints. Preferably, linkage dimensions and moment arms are chosen so that external forces applied to the finger distal to a point near the fingertip act to straighten the finger, and forces applied proximal to this point act to curl the finger. The point at which the behavior changes from straightening to curling is referenced as the “focal point” of the differential. For external forces that act proximal to the focal point, the MCP joint will extend and the PIP joint will flex. 
     Referring now to  FIGS. 1A-1E , the linkage  126  is also used to move the endpoint for return spring  132 . The return spring  132  acts to straighten the finger and to keep the mechanism pushed over to one side of this free range of motion. In the absence of any external forces pushing on the finger, the return spring makes the finger act as though the differential  124  is not present. The return spring can also provide some resistance to curling of the fingers when forces are applied to the dorsal side of the finger. Any compliance in the differential  124  will result in some motion, but this will occur in all three joints and is not due to the differential coupling. 
     As illustrated by the embodiment of  FIG. 1E , adjustable stop  134  for the PIP joint may be omitted for an application not requiring it, or in which it is desirable not to have it. In this example, the linkage  126  controls only the position of the end point of the PIP joint return spring  132 . The linkage  126  thus becomes a spring centering linkage. 
     Referring now only to  FIG. 1D , this embodiment of a mechanical finger includes a differential  125  comprising differential linkage  136  in place of a kinematic linkage. The differential couples the medial and distal phalanges using a differential relationship. This embodiment also optionally includes an adjustable stop  138  for the DIP joint and return spring  140  for placing a torque on the DIP joint that tends to extend the distal phalanx relative to the medial phalanx. Linkage  142  is connected to proximal phalanx  102  and adjusts the position of DIP stop  138  based on the angle of rotation of the PIP joint. It also sets the endpoint of return spring  140 . 
       FIGS. 2 ,  3 ,  4 A- 4 C,  5 A- 5 C,  6 ,  7  and  8 A-B illustrate various aspects of an exemplary embodiments of mechanical finger  100  for use in a prosthetic application. The prosthesis comprises at least one prosthetic finger  200 . The prosthesis may also include, depending on the needs of the patient, a prosthetic hand, comprising a prosthetic palm to which the mechanical finger is attached, and a prosthetic arm, to which the prosthetic hand is attached. Only the internal structure of the prosthetic finger is illustrated in the figures. 
     Prosthetic finger  200  is comprised of proximal phalanx  202 , medial phalanx  204 , and distal phalanx  206 . Distal phalanx  206  has been omitted from  FIGS. 4A-4F  for purposes of illustration. Metacarpophalangeal (MCP) joint  208  connects the finger to a base element, for example, an artificial palm or hand, which is not shown. Proximal interphalangeal (PIP) joint  210  joins the proximal and medial phalanges. Distal interphalangeal (DIP) joint  212  joins the medial and distal phalanges. 
     In the embodiment shown in FIGS.  3  and  4 A- 4 C, the proximal phalanx  202  houses a differential linkage comprised of a connecting rod  214 , a pivot link  216 , and another connecting rod  218 . Connecting rod  214  is joined by pin  220  to an arm extending from drive output  222 , and thus connects the output drive to one end of the pivot link  216 . Although not shown, a motor—a stepper motor, for example—located in the base element rotates a drive input, which in this example is pin  223 , which in turn rotates the drive output. Drive output  222  is fixed to the pin  223 . Pin  221  joins the connecting rod to the spring. Connecting rod  218  connects the other end of the pivot link to plate  228  of the medial phalanx  204 . Pin  224  joins the pivot link to the connecting rod  218 , and pin  226  joins the connecting rod to the plates  228   a  and  228   b , which comprise the primary structural elements for medial phalanx  204 . The midpoint of the pivot link is fixed by pin  230  to plates  232   a  and  232   b . The pivot link will rotate within the proximal phalanx, about the axis of pin  230 , as indicated by comparing  FIGS. 4B and 4C , when the drive output rotates. During flexion, rotation of the drive output  222  pulls the connecting rod  214 , which pulls on the pivot link  216 , which pulls on a second connecting rod  218 , which pulls on plates  228   a  and  228   b  of the medial phalanx. 
     In an alternate embodiment shown in  FIGS. 5A-5C , the pivot link  216  ( FIGS. 4A-4C ) is replaced by an in series compliant element for giving the finger compliance for series elasticity and shock mitigation. In this example, the compliant element comprises spring  217 . Except for the added compliance and elasticity provided by the spring, the differential with spring performs in a substantially similar manner as the pivot link  216 . In another alternate embodiment shown in  FIG. 6 , the pivot link  216  and the connecting rods  214  and  218  are replaced with a linkage comprising a single connecting rod  219  that is connected by pins  220  and  226  to the drive housing  220  and plate  228   b  of the medial phalanx  204 . As can be seen in the figure, the connecting rod must extend beyond the envelope of the proximal phalanx  204 . 
     In each of the embodiments shown in  FIGS. 2 to 8B , plates  228   a  and  228   b  are the primary structural elements of medial phalanx  204 . Plates  232   a  and  232   b  are the primary structural elements comprising the proximal phalanx  202 . The differential linkage of  FIGS. 4A-4C  and  5 A- 5 C described above is housed between the plates. To these plates can be attached shells to give the proximal phalanx its desired exterior shape in the particular prosthetic or other application. 
     The pins used to join components in the differential linkage, as well as in other linkages described below, permit relative rotation of the joints that are joined. The location of pin  226  is eccentric to the axis of the PIP joint to form a moment arm. The axis of the PIP joint is defined by pin  236 , which pivotally connects the clevis formed by plates  232   a  and  232   b  of the proximal phalanx with plates  228   a  and  228   b  of the medial phalanx. For a given rotation of the drive output, either the MCP joint or the PIP joint can rotate. Rotation of the drive output not only applies torque to the MCP joint by causing the pivot link to push against pin  230 , but it also rotates the link, causing the other part of the link to transmit a force that is applied to pin  226 . Even if the proximal phalanx is blocked, the link will nevertheless pivot and apply torque to the PIP joint. Thus, torque from the drive is applied to both the MCP joint and the PIP joint. 
     Referring now to  FIGS. 2 ,  3 , and  7 , the medial phalange  204  houses a kinematic linkage for coupling rotation of the PIP joint to the DIP joint so that both curl simultaneously. The kinematic linkage comprises a connecting rod  238  that spans between the proximal phalanx  202  and the distal phalanx  206 . Pin  240  at a proximal end of the connecting rod engages hole  242  on plate  232   b  of the proximal phalanx. Pin  244  on the distal end of the connecting rod engages hole  246  in the distal phalanx. The distal phalanx is linked to the medial phalanx by a hinge formed by pins  248   a  and  248   b . These pins cooperate respectively, with a hole  250   a  on plate  228   a  and hole  250   b  on plate  228   b  of the medial phalanx, and with holes  250   a  and  250   b  on opposite forks of a clevis extending from a shell forming distal phalanx  252 . Although in this embodiment the linkage is comprised of a single connecting rod, it could comprise multiple links. Furthermore, a differential could be substituted for the kinematic linkage, as described in connection with  FIG. 1D . 
     Referring now only to  FIGS. 2 ,  3 ,  8 A and  8 B, the mechanical finger  200  includes, in this embodiment, fixed stop  253  that stops rotation of the PIP joint to prevent hyperextension of the medial phalanx. In this embodiment, a movable PIP stop part  254  rotates on the same axis as the PIP joint to reduce the permitted range of motion of the medial phalanx by limiting further rotation of the PIP joint based on the degree of flexion of the MCP joint. The centerline of pin  236  defines the axis of rotation. The PIP stop part stop includes a stop portion  255  that interferes with  257  of plate  228   a  of the medial phalanx to prevent the medial phalanx from extending. The position of the PIP stop part  254  is based on the degree of rotation of the MCP joint, and is accomplished in this embodiment by a linkage comprising connecting rod  260  between a housing  256  for a drive (not shown) and PIP stop part  254 . The linkage may also be implemented using multiple links. A pin connects the distal end of connecting rod  260  to arm portion  264  of the PIP stop part  254 . The proximal end of connecting rod  258  is connected by another pin to the drive housing. As the MCP joint rotates due to flexion of the proximal phalanx  202 , the connecting rod pulls on the arm  264 , causing the PIP stop part to rotate in the same direction. 
     With the PIP-stop linkage, the medial phalanx  204  is stopped either by the fixed stop  253  on the proximal phalanx when the proximal phalanx is fully extended, or by the movable stop of PIP-stop part  254  when the MCP joint is rotated during flexion of the proximal phalanx. If the MCP joint rotates, then the PIP joint is forced to rotate as well by the PIP-stop part. During free motion, or when forces are applied to the fingertip, movement of the PIP-stop part helps to produce predictable curling like a fully kinematic mechanism. 
     In this embodiment, the rotational position of the PIP-stop part  254  also controls the endpoint  270  of the return spring  266 . This spring, which is normally compressed, has the effect of extending the medial phalanx, thus pushing the PIP joint against the PIP-stop. If no external forces act on the finger, the force generated by the spring causes the motion of the finger joints to be controlled by the PIP-stop. If, however, an object blocks the motion of the proximal phalanx, then the differential linkage continues applying torque to the PIP joint, causing PIP and DIP joints to curl and further compressing the return spring. 
     The kinematic linkage for controlling the position of the PIP stop based on the motion of the MCP joint could also be used to limit or affect the motion of the PIP and DIP joints in other ways. For example, the PIP stop can be removed, permitting the linkage to be for controlling the end point of the return spring without limiting the motion of the PIP joint. 
     Although not necessary for operation of the finger as described above, joint positions can be measured using potentiometers coupled with the joints and feedback to a controller for the drive motor in order to drive the finger to desired position, subject to the limitations of being able to do so caused by the differential. Similarly, strain gauges can be placed on, for example, the drive housing  256  to measure torque on the finger and feed the measured torque back to a controller to change the impedance of the finger. 
     Although the particular components forming the linkages and the phalanges illustrated in  FIGS. 2-7  have advantages when used in a prosthetic application, the structures are intended to be illustrative only of the linkage mechanisms illustrated by  FIG. 1 . These components can be adapted or substituted for when implementing a differential mechanism in parallel with a kinematic mechanism in accordance with  FIG. 1 . For example, linkages may be replaced with belts or cables or other passive mechanical mechanisms to achieve the same general purpose. Although it is common to use linkages for kinematic mechanisms and cables for differential mechanisms, but either type can be used for either purpose. In addition to being implemented as a linkage, as exemplified by  FIGS. 2-8B , the differentials described above may also be implemented using a belt or cable, for example one linking the drive output to a drum or pulley at the PIP joint, a gear train, or a toggle. 
     Furthermore, applications in which a mechanical finger in accordance with  FIGS. 1A-1E  can be used include any type of application involving grasping, and include many different types of robotic applications that are not limited to those attempting to mimic a human hand or prosthetic applications. For instance, an anthropomorphic grip may have benefits in many diverse or unstructured or unforeseen contexts just as human hands are so successfully versatile, including industrial grippers, rovers or mobile robots, entertainment, home robots, surgery or minimally invasive surgery, massage, patient transfer or stabilization, and many others. 
     The foregoing description is of exemplary and preferred embodiments. The invention, as defined by the appended claims, is not limited to the described embodiments. Alterations and modifications to the disclosed embodiments may be made without departing from the invention. The meaning of the terms used in the claims are, unless expressly stated otherwise, intended to have ordinary and customary meaning and are not intended to be limited to the details of the illustrated structures or the disclosed embodiments.
A mechanical finger comprises a plurality of phalanges coupled to a single actuator using a kinematic linkage and a differential linkage arranged in parallel. The mechanical finger is capable of exhibiting consistent predictable motion when moving in free space or when contacting an object at the fingertip, and of curling in order to conform to an object when the contact is at other locations on the finger.