Abstract:
A small-scale positioning device employing a platform, a levering mechanism and a floating actuator device. The platform is movably attached to a fixed frame by a lever, a pair of levers, or more. The floating actuator device is coupled between at least one lever and the platform. When the actuator device is activated, it generates a force on the platform and an equal but opposite force on the levering mechanism, causing one or more levers to rotate around their respective fulcrums, thereby controlling the position of the movable platform relative to the fixed frame. The amount of displacement of the platform is dependent upon the effective expansion or contraction of the actuator device and the lever ratio. If the pair of levers are symmetrical, then motion is created in only a single degree of motion. Flexures may be included to prevent motion in unwanted directions.

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
   This application is entitled to the benefit of, and claims priority to, provisional U.S. Patent Application Ser. No. 60/441,219 filed Jan. 21, 2003 and entitled “A POSITIONING DEVICE EMPLOYING AN ACTUATOR PLATFORM AND LEVERING MECHANISM,” the entirety of which is incorporated herein by reference. 

   BACKGROUND OF THE PRESENT INVENTION 
   1. Field of the Present Invention 
   The present invention relates generally to devices for precision motion control, and in particular, to actuated devices for measuring, manufacturing or positioning objects in mesoscale, microscale or nanoscale technologies. 
   2. Background 
   Mechanisms or devices for use in positioning, measuring and manufacturing small objects are well known. Such devices conventionally include some sort of actuator device, such as a piezoelectric actuator made from such materials as PZT or, more recently, relaxor materials such as PZN-PT or PMN-PT. Some of the key benefits of piezoelectric actuator devices relative to other actuator device types are their ability to provide a relatively high amount of work while occupying a small volume. Unfortunately, because of how such devices are manufactured, and because of problems associated with the integration of such actuator devices into macroscopically dimensioned mechanisms, such devices produce only a limited displacement range for a given actuator volume, and thus the practical applicability of such mechanisms is limited. 
   In order to overcome the limited displacement ranges afforded by such mechanisms, some mechanisms amplify the displacement created by the actuator through the use of one or more lever devices to position or move a portion of the mechanism in one or more degrees of freedom. As used herein, motion in a single “degree-of-freedom” shall be understood to mean a single motion along a generally straight line or a rotation about an axis. In general, the term “single-degree-of-freedom” is used to represent a motion that can be defined in terms of a single value relative to a defined coordinate system. This might be a linear displacement along a linear coordinate or an angular rotation about an rotational coordinate (usually referred to as an axis). In practice this restriction is not necessary. For example, a point along a wire of arbitrary shape can be uniquely defined by its distance from one end of the wire and hence only a single value is required to define its location given that the wire represents the coordinate. 
   By supporting the object of interest on the movable portion of the mechanism (sometimes referred to herein as a “platform”), or integrating it with the platform, the object may likewise be moved or positioned in one or more degrees of freedom as desired. Such designs are desirable in that they provide the ability to position, measure, or manufacture objects with high range-to-resolution ratios. 
   Generally, mechanisms that utilize only a single lever may be designed for small (i.e., mesoscale, microscale or nanoscale) platforms, but are particularly inherent to mechanical losses, which are typically referred to as fractional loss of motion. Assuming a platform of finite stiffness, the fractional loss is significantly increased as the lever ratios are increased. Therefore, lever mechanisms are generally limited to small lever ratios in order to avoid high fractional losses, and this, in turn, limits the range offered at a particular resolution. A second disadvantage to levered mechanisms is the occurrence of unwanted degrees of freedom deviating from the principal axis of motion of the platform. For example, a single-degree-of-freedom platform translating along a principal axis will commonly generate a yaw and pitch motion as well. In addition, the motion produced at the end of a simple single lever is arcuate, rather than linear, in nature, thus resulting in addition errors. These errors, coupled with the yaw and pitch described above, are all undesirable for most precision motion applications. 
   One prior art approach to minimizing the effects of platform motion error is the use of symmetrical lever arrangements. For example, U.S. Pat. No. 6,467,761 to Amatucci et. al. discloses a positioning mechanism of a design employing symmetrical pairs of levers to reduce the effects of yaw motion. The design includes a fixed frame and a moving platform connected to the fixed frame by two pairs of lever arms, wherein each lever arm is free to rotate slightly around a flexure, attached to the fixed frame, that serves as a fulcrum for the lever. One end of each lever arm is attached to the platform, while the opposite end of two of the lever arms makes contact with an actuator that is mounted on the fixed frame. When the actuator is activated, the levers rotate around their respective fulcrums, thus moving the platform by an amplified amount (where the amount of amplification is dependent upon the lever ratio). As a result, the platform&#39;s width is dependent upon the length of the arms, and the platform size is effectively large for a high lever ratio. Moreover, the platform footprint cannot be any less than the combined length of the symmetrical lever arms. Consequently, a large platform with large lever ratios will result in a low natural frequency of the system. 
   SUMMARY OF THE PRESENT INVENTION 
   The present invention comprises a positioning device for precision controlled systems. One or more actuator devices are mounted to a platform and generate work upon one or more levers. Each lever is connected to the platform by a fulcrum and to a fixed frame by way of a rotating hinge. Each actuator device pushes against at least one lever and the lever rotates about the fulcrum and additionally rotates about a pivot point located on the frame. The rotation of the lever, in turn, displaces the platform, with the magnitude of the displacement being amplified relative to the displacement between the each end of the actuator. A pair of flexures symmetrically disposed about the platform&#39;s axis of motion provides a constraint that will guide the linear motion of the platform. The relative displacement of the platform with respect to the fixed frame is dependent upon the lever ratio. 
   For some applications it may be necessary or useful to incorporate more than one pair of levers to increase the displacement range without introducing significant loss of motion. For example, a multiple lever mechanism may aid in exceeding lever ratios of 5:1 and while maintaining low fractional loss. This type of design may be used in compact devices which may move over mesoscale ranges. Additionally, at least one actuator device mounted to the platform may be used to. generate work against at least one lever mechanism. Also, this platform design uses symmetrical levers but may be substantially decreased in size compared to prior art. A significant benefit is a substantial increase in mechanical bandwidth for the positioning device. 
   The same principle of operation may also be utilized for multiple-degree-of-freedom positioning systems. For example, a two-degree-of-freedom positioning device may utilize two nested platforms. Each platform attaches to its base through two levers attached to the platform by way of a fulcrum. Each platform has a dedicated actuator device affixed thereto where the actuator device pushes against the levers and thus the platform. 
   In another arrangement, a positioning device may utilize cascading levers which are cascaded along the principal axis of motion, wherein each actuator device generates work upon two separate levers. The levers may be cascaded in a way that each successive lever added in the cascade provides further amplification of motion or rotations at the output of the previous lever. 
   The present invention also includes different arrangements of the lever fulcrum relative to the point at which the force generated by the actuator device is applied. For example, the fulcrum may be disposed between the force application point and the lever&#39;s connection to the fixed frame, or the relative positions of the fulcrum and the application point may be reversed, such that the force application point is disposed between the fulcrum and the lever&#39;s connection to the fixed frame. This may lead, for example, to the fulcrum being placed at the very end of the lever. 
   This design provides the ability to extend the displacement range while maintaining high bandwidth motion control. Such features may find a wide range of applicability. For example, in the life sciences, longer scanning stages may be created for atomic force microscopy. In the ophthalmic lens industry, the devices of the present invention could be used in high-bandwidth fast tool servos. Any number of additional applications will be apparent to those of ordinary skill in the specific industry or technology area. 
   Another significant benefit is the reduction in overall size relative to many positioning devices. Although the positioning devices of the present invention use pairs of symmetrical levers, which in conventional devices results in a considerably wider footprint, the floating actuator design permits the same displacement to be provided by smaller devices than were previously possible. 
   Broadly defined, the present invention according to one aspect includes a small-scale positioning device including: a fixed frame; a platform, movably attached to the fixed frame via at least one lever; and a floating actuator device, coupled between the at least one lever and the platform, that when activated generates a force on the platform and an equal but opposite force on the at least one lever, thereby controlling the position of the movable platform relative to the fixed frame. 
   In features of this aspect, each lever is coupled to the platform at a respective fulcrum; each lever is further pivotably connected to the platform via a respective flexure, the flexure being separate from the fulcrum; the floating actuator device is of an automated type, such as a piezoelectric type (which may include an actuator formed from a relaxor material), an electrostrictive type, an electromagnetic type, a hydraulic type, a pneumatic type or a magnetostrictive type; is of a manual type, such as a fine adjustment screw or a micrometer; the at least one lever includes a pair of levers; the pair of levers are arranged symmetrically to one another; the pair of levers are arranged slightly asymmetrically to one another to achieve a yaw or pitch motion; the positioning device further includes at least one flexure, coupled between the platform and the fixed frame, that guides the motion of the platform; the at least one flexure includes a pair of symmetrical flexures; and the platform may be repositioned in only a single degree of freedom of motion. 
   In other features of this aspect, the platform may be repositioned in at least two degrees of freedom of motion; the floating actuator device is a first floating actuator device, the positioning device further includes at least a second floating actuator device, and the first floating actuator device repositions the platform in a first degree of motion and the second actuator device repositions the platform in a second degree of motion; the platform includes a first portion, coupled to the fixed frame by at least a first lever, and a second portion, coupled to the first portion by at least a second lever; the second floating actuator device moves the second portion of the platform relative to the first portion, and the first floating actuator device moves the platform relative to the fixed frame; the positioning device further includes at least one control system that controls the operation of the floating actuator device; the control system includes an open loop feedback controller; the control system includes a closed loop feedback controller; the floating actuator device is coupled to at least a first lever at a lever interface, and the fulcrum of the first lever lies generally between the lever interface and the flexure, or the floating actuator device is coupled to at least a first lever at a lever interface, and the lever interface lies generally between the fulcrum of the first lever and the flexure of the first lever; the at least one lever includes a cascaded lever arrangement; and the cascaded lever arrangement includes a first lever, coupled between the platform and the fixed frame, and a second lever, coupled between the first lever and the platform, wherein the floating actuator device is coupled between the platform and the second lever, wherein activation of the actuator device causes the second lever to rotate about a first fulcrum, thereby applying a force to the first lever, and wherein the application of the force to the first lever causes the first lever to rotate about a second fulcrum, thereby controlling the position of the movable platform relative to the fixed frame. 
   The present invention, according to another aspect of the present invention, is a method of positioning a platform relative to a fixed frame in a small-scale positioning device, including: providing a small-scale positioning device having a fixed frame, a platform that is movably attached to the fixed frame via at least one lever, and a floating actuator device, coupled between the at least one lever and the platform; activating the floating actuator device; and upon activating the floating actuator device, applying a force on the platform and an equal but opposite force on the at least one lever, thereby controlling the position of the movable platform relative to the fixed frame. 
   In features of this aspect, controlling the position of the movable platform relative to the fixed frame includes controlling motion of the movable platform relative to the fixed frame in one degree of freedom; the floating actuator device is a first floating actuator device and the method further includes providing a second floating actuator device, activating the second floating actuator device, and upon activating the second floating actuator device, controlling motion of at least a portion of the movable platform relative to the fixed frame in a second degree of freedom; and the step of providing a small-scale positioning device includes providing a small-scale positioning device having at least one flexure coupled between the platform and the fixed frame, and the method further includes guiding the motion of the platform in one degree of freedom via the at least one flexure. 
   Further areas of applicability of the present invention will become apparent from the detailed description provided hereinafter. It should be understood that the detailed description and specific examples, while indicating the preferred embodiment of the invention, are intended for purposes of illustration only and are not intended to limit the scope of the invention. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     Further features, embodiments, and advantages of the present invention will become apparent from the following detailed description with reference to the drawings, wherein: 
       FIG. 1  is a prototype of a single-degree-of-freedom positioning device in accordance with a first preferred embodiment of the present invention; 
       FIG. 2A  is a top cross-sectional view of the prototype device of  FIG. 1 , showing the platform in a first position; 
       FIG. 2B  is a top cross-sectional view of the prototype device of  FIG. 1 , showing the platform in a second position; 
       FIG. 3  is a top cross-sectional view of a two-degree-of-freedom positioning device in accordance with a second preferred embodiment of the present invention; 
       FIG. 4  is a top cross-sectional view of a positioning device utilizing two levers in accordance with a third preferred embodiment of the present invention; 
       FIG. 5  is a top cross-sectional view of a positioning device utilizing levers having an alternative arrangement of the force application point relative to the fulcrum, in accordance with a fourth preferred embodiment of the present invention; 
       FIG. 6  is a top cross-sectional view of a positioning device having a single actuator device and two pairs of levers, in accordance with a fifth preferred embodiment of the present invention; 
       FIG. 7  is a graphical illustration of the displacement of the platform of the prototype positioning device of  FIG. 1  versus the voltage applied to the actuator device; and 
       FIG. 8  is a top cross-sectional view of a positioning device utilizing cascaded levers in accordance with a sixth preferred embodiment of the present invention. 
   

   DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
   Referring now to the drawings, in which like numerals represent like components throughout the several views, the preferred embodiments of the present invention are next described. The following description of the preferred embodiment(s) is merely exemplary in nature and is in no way intended to limit the invention, its application, or uses. 
     FIG. 1  is a prototype of a single-degree-of-freedom positioning device  10  in accordance with a first preferred embodiment of the present invention. As shown in  FIG. 1 , the positioning device  10  includes a movable platform  12  attached to a fixed frame  11  by two symmetrical levers  13 ,  14 . The positioning device  10  may be monolithically fashioned or engineered from separate components. Force may be imparted to the levers  13 ,  14  by a floating actuator device  25 , thus causing the platform  12  to move. The actuator device  25  may be said to be “floating” in that it is not constrained to the fixed frame  11 , but only to the platform  12  and the levers  13 ,  14 . The actuator device  25  is preferably of an automated type but may also be of a manual type, such as fine adjustment screws and micrometers. Automated actuator device types suitable for use in the preferred embodiments of the present invention include, but are not limited to, electrostrictive, electromagnetic, piezoelectric, hydraulic, pneumatic and magnetostrictive, and combinations thereof. 
     FIG. 2A  is a top cross-sectional view of the prototype device  10  of  FIG. 1 , showing the platform  12  in a first position. The actuator device  25  is mounted to the platform  12  (optionally under preload, as will be apparent to those of ordinary skill in the art) and is coupled to the levers  13 ,  14  by a lever interface segment  19  and a first pair of symmetrical flexures  17 ,  18 . Each lever  13 ,  14  is rotatable about a respective fulcrum  15 ,  16 , and the opposite end of each lever  13 ,  14  is constrained to a respective pivot structure  20 ,  21 . The platform  12  is further constrained to a second pair of symmetrical flexures  22 ,  23 , which are symmetrically disposed about a principal axis of motion  40 . These flexures  22 ,  23  generate an additional stiffness to the platform  12  and provide a constraint that will guide the linear motion of the platform  12 . 
   In use, the actuator device  25  generates a force on the platform  12  and an equal but opposite force on the levers  13 ,  14 , which causes the levers  13 ,  14  to be rotated about respective fulcrums  15 ,  16 . Additionally, the levers  13 ,  14  are also rotated about respective pivot structures  20 ,  21  located on the fixed frame  11 . At the same time, the platform  12  is translated along the axis of motion  40  by an amplified amount which is dependent upon the lever ratio (the ratio of the distance between the respective pivot structure  20 ,  21  and the respective fulcrum  15 ,  16  to the distance between the fulcrum  15 ,  16  and the respective flexure  17 ,  18 ) of the two levers  13 ,  14 .  FIG. 2B  is a top cross-sectional view of the prototype device  10  of  FIG. 1 , showing the platform  12  in a second position. The amount of movement of the platform  12  relative to the fixed frame  11  is dependent upon the common (in this case) lever ratio. 
   The invention may likewise be implemented to obtain more than one degree of freedom of motion.  FIG. 3  is a top cross-sectional view of a two-degree-of-freedom positioning device  110  in accordance with a second preferred embodiment of the present invention. The two-degree-of-freedom positioning device  110  incorporates a fixed frame  111  and two nested platforms  112 ,  113  that are each translated in a linear path. The first platform  112  is wholly supported by the second platform  113 , which in turn is wholly supported by the fixed frame  111 . Put another way, the fixed frame  111  serves as a base for the second platform  113 , and the second platform  113  serves as a base for the first platform  112 . Each platform  112 ,  113  attaches to its respective base through two levers, and each lever is attached to the respective platform by way of a fulcrum. 
   More specifically, the first platform  112  is attached to the second platform  113  by a first pair of levers  114 ,  115  and a pair of flexures  122 ,  123  arranged symmetrically to a first axis of motion  140 . A first floating actuator device  119  is constrained to the first platform  112  and coupled to the first pair of levers  114 ,  115  by a first lever interface segment  118 , and the levers  114 ,  115  are also constrained to respective pivot structures  120 ,  121  that are supported by the second platform  113 . When activated, the first actuator device  119  generates a force on the first platform  112  and an equal but opposite force on the first pair of levers  114 ,  115 , which causes one lever  114  to rotate about a first fulcrum  116  and the other lever  115  to rotate about a second fulcrum  117 . 
   In like manner, the second platform  113  is constrained to the fixed frame  111  by a second pair of levers  124 ,  125  and a pair of flexures  132 ,  133  arranged symmetrically to a second axis of motion  141 . A second floating actuator device  129  is constrained to the second platform  113  and is coupled to the second pair of levers  124 ,  125  by a second lever interface segment  128 , and the levers  124 ,  125  are also constrained to respective pivot structures  130 ,  131  that are supported by the fixed frame  111 . When activated, the second actuator device  129  generates a force on the second platform  113  and an equal but opposite force on the second pair of levers  124 ,  125 , which causes the levers  124 ,  125  to be rotated about respective fulcrums  126 ,  127 . 
   In this configuration, the first platform  112  may thus translate orthogonally with respect to the motion of the second platform  113 . When the second levers  124 ,  125  are rotated, the second platform  113  is translated along the second axis of motion  141 . Similarly, when the first levers  114 ,  115  are rotated, the first platform  112  is translated along the first axis of motion  140 . However, because translation of the second platform  113  causes the first platform  112  to be translated as well, it should be apparent that the lateral position of the first axis of motion is subject to the translation of the second platform  113 . 
   Although  FIG. 3  illustrates a positioning device  110  for imparting motion in only two degrees of freedom, it will be apparent to anyone familiar with the design of such mechanisms that positioning devices offering still additional degrees of freedom may likewise be created without departing from the scope of the present invention. 
     FIG. 4  is a top cross-sectional view of a positioning device  210  utilizing two levers  213 ,  214  in accordance with a third preferred embodiment of the present invention. This positioning device  210  includes a platform  212  attached to a fixed frame  211  by two levers  213 ,  214 . The first lever  213  is directly fixed to the platform  212  by way of a first fulcrum  215 . Additionally, the second lever  214  is attached to the platform  212  by way of a second fulcrum  216 . A first floating actuator device  221  is mounted between the platform  212  and the first lever  213  at a first pivot location  217 , and a second floating actuator device  222  is similarly mounted between the platform  212  and the second lever  214  at a second pivot location  218 . 
   When activated, each actuator device  221 ,  222  generates a force on the platform  212  and an equal but opposite force on a respective lever  213 ,  214 . This causes the levers  213 ,  214  to rotate about a respective fulcrum  215 ,  216 . At the same time, the platform  212  is translated by an amplified amount which is dependent upon the lever ratio of the two levers  213 ,  214 . Essentially, this approach may provide reduction in fractional lost motions or rotations due to expansion of the actuator devices  221 ,  222 . Also, although each actuator device  221 ,  222  is only shown coupled to a single respective lever  213 ,  214 , it should be apparent to those of ordinary skill in the art that each actuator device may instead be coupled to a pair of symmetrical levers, such as those shown in  FIGS. 1 ,  2 A and  2 B, or on even more levers, such as those shown in  FIG. 6  (discussed below). 
   Alternative arrangements of a lever&#39;s fulcrum, relative to the point on the lever at which the force generated by the actuator device is applied, are also possible.  FIG. 5  is a top cross-sectional view of a positioning device  310  utilizing levers having an alternative arrangement of the force application point relative to the fulcrum, in accordance with a fourth preferred embodiment of the present invention. In this embodiment, like the first embodiment, a platform  312  is constrained to a fixed frame  311  by two symmetrical levers  313 ,  314  and two symmetrical flexures  323 ,  324 . The first lever  313  is constrained to the platform  312  by a first fulcrum  317  which is located close to the principal axis of motion  340 . Likewise, the second lever  314  is constrained to the platform  312  using a second fulcrum  318  which is located close to the principal axis of motion  340 , symmetrically to the first fulcrum  317 . 
   When activated, a first floating actuator device  321  generates a force on the platform  312  and an equal but opposite force on the first lever  313  at location  315 . Similarly, a second floating actuator device  322  may generate a force on the platform  312  and an equal but opposite force on the second lever  314  at location  316 . This causes the levers  313 ,  314  to rotate about their respective fulcrum  317 ,  318 . At the same time, the platform  312  is translated by an amplified amount which is dependent upon the lever ratio of the two levers  313 ,  314 . In this configuration, the two actuator devices  321 ,  322  may be displaced equally to generate a linear displacement of the platform  312  along the principal axis of motion  340 , or unequally to generate a yaw rotation in the platform  312 . 
   A positioning device may also utilize a single floating actuator device to operate more than one pair of levers.  FIG. 6  is a top cross-sectional view of a positioning device  410  having a single floating actuator device  425  and two pairs of levers  415 ,  416  and  417 ,  418 , in accordance with a fifth preferred embodiment of the present invention. The positioning device  410  further includes a platform  412  having a plurality of platform sections  432 ,  433 ,  434  that are rigidly attached to one another to form a rigid body. As shown, the actuator device  425  is connected to two lever interface segments  423 ,  424 . The first lever interface segment  423  is connected to a first pair of levers  415 ,  416 , and the second lever interface segment  424  is connected to a second pair of levers  417 ,  418 . Each lever  415 ,  416 ,  417  and  418  is constrained to the platform  412  by a respective fulcrum  419 ,  420 ,  421  and  422 . 
   A force generated by the actuator device  425  is applied to the first pair of levers  415 ,  416  via the first lever interface segment  423 . Thus, effectively, the fulcrum  419 ,  420  for each of the first pair of levers  415 ,  416  is located between the actuator device  425  and a respective fixed pivot  426 ,  427 . On the other hand, an equal but opposite force, also generated by the actuator device  425 , is applied to the second pair of levers  417 ,  418  via the second lever interface segment  424 , and thus the actuator device.  425  effectively contacts each of the second pair of levers  417 ,  418  between a respective fulcrum  421 ,  422  and a respective fixed pivot  428 ,  429 . The net result is that the platform  412  may be translated along a principal axis of motion  440  by activating only the single actuator device  425 . 
   The prototype positioning device  10  shown in  FIG. 1  was devised to assess the validity of the invention. The levers  13 ,  14  of the prototype positioning device  10  of  FIG. 1  have a 6:1 ratio, and the various flexures include leaf flexures for the fulcrum hinges and actuator device couplings and additional leaf flexures to constrain the carriage platform  12  to a linear motion along the principal axis of motion  40 . A study was done in which the actuator device  25  was a relaxor-type piezoelectric actuator comprising a single crystal PZN-PT stacked actuator device with an overall size of 5.0×5.0×5.0 mm 3 . The actuator device  25  may be controlled using either open or closed loop feedback control. Suitable actuator devices and closed loop control systems therefor are more fully described in commonly-assigned U.S. patent application Ser. No. 10/157,095, the entirety of which is incorporated herein by reference. The total stiffness of the mechanism in the prototype was determined to be approximately 1.6×10 7  N/m, and the effective stiffness was approximately 3.12×10 5  N/m. 
     FIG. 7  is a graphical illustration  500  of the displacement of the platform  12  of the prototype positioning device  10  of  FIG. 1  versus the voltage applied to the actuator device  25 . A laser interferometer was employed to measure the displacement of the platform  12  versus the applied field to the actuator device  25 . As shown, the total range exhibits an overall displacement of 115 μm with an amplification ratio of 4:1. (The actual amplification ratio of 4:1 is less than the 6:1 lever ratio due to fractional lost motion, this approach still represents a substantial increase in mechanical bandwidth over known positioning device designs.) Advantageously, although the use of symmetrical levers would appear to increase the overall size as compared to single-lever designs, the floating actuator approach allows smaller, more efficient levers to be used, thus effectively decreasing by a substantial amount (for the same -performance) the overall size of positioning devices of this design. 
   A positioning device with a floating actuator device may also utilize cascaded levers. In other words, the invention may employ levers which are cascaded in a way that each successive lever added in the cascade provides further amplification of motion or rotations at the output of the previous lever.  FIG. 8  is a top cross-sectional view of a positioning device  810  utilizing cascaded levers in accordance with a sixth preferred embodiment of the present invention. This positioning device  810  includes an actuator device  825  mounted to a platform  812  and coupled to a first pair of levers  813 ,  814  by a lever interface segment  829  and a first pair of flexures or pivot structures  817 ,  818 . Each lever  813 ,  814  is rotatable about a respective fulcrum  815 ,  816 , and the opposite end of each lever  813 ,  814  connects through a hinge to an intermediate solid rod  843 ,  844 . Each of the intermediate rods  843 ,  844  connects to one of a second pair of levers  823 ,  824  via another hinge. The second levers  823 ,  824  are rotatable about respective fulcrums  825 ,  826  with each fulcrum  825 ,  826 , in turn, attached to the platform  812 . The other end of this second set of levers  823 ,  824 , is attached to a fixed frame  811  by hinges  819 ,  820 . The platform  812  is further constrained to the fixed frame  811  by a second pair of symmetrical flexures  821 ,  822 , which are symmetrically disposed about a principal axis of motion  840 . These flexures  821 ,  822  generate an additional stiffness to the platform  812  and provide a constraint that will guide the linear motion of the platform  812 . 
   In use, the actuator device  825  generates a force on the platform  812  and an equal but opposite force on the first levers  813 ,  814 , which causes the levers  813 ,  814  to be rotated about their respective fulcrums  815 ,  816 . Additionally, the first levers  813 ,  814  are also rotated about the respective hinges on the intermediate solid rods  843 ,  844 . At the same time, the respective ends of the levers  813 ,  814  are translated along the axis of motion  840  by an amplified amount, relative to the platform  812 , which is dependent upon the lever ratio (the ratio of the distance between the respective pivot structure  817 ,  818  and the respective fulcrum  815 ,  816  to the distance between the fulcrum  815 ,  816  and the respective intermediate rod  843 ,  844 ) of the two levers  813 ,  814 . The amount of movement of the intermediate rods  843 ,  844  relative to the platform  812  is dependent upon the common (in this case) lever ratio corresponding to the first pair of levers  813 ,  814 . The motion of the intermediate rods  843 ,  844  is transmitted to the second levers  823 ,  824  that pivot about respective fulcrums  825 ,  826 , with each fulcrum  825 ,  826  being attached to the platform  812 . This amplified motion of the intermediate rods  843 ,  844  results in a force on the second levers  823 ,  824 , which causes the second levers  823 ,  824  to rotate about their fulcrums  825 ,  826  so that their ends are translated along the axis of motion  840  by an amplified amount relative to the platform  812 . The amount of amplification is dependent upon the lever ratio (the ratio of the distance between the respective intermediate rod  843 ,  844  and the respective fulcrum  825 ,  826  to the distance between the fulcrum  825 ,  826  and the respective hinges  819 ,  820  at the fixed frame  811 ) of the two levers  823 ,  824 . The amount of movement of the intermediate rods  843 ,  844  relative to the platform  812  is dependent upon the common (in this case) lever ratio corresponding to the second pair of levers  823 ,  824 . Because the input displacement to the second set of levers  823 ,  824  has been amplified by the first set of levers  813 ,  814 , the motion of the platform  812  relative to the fixed frame  811  is given by the product of both lever ratios. 
   Based on the foregoing information, it is readily understood by those persons skilled in the art that the present invention is susceptible of broad utility and application. Many embodiments and adaptations of the present invention other than those specifically described herein, as well as many variations, modifications, and equivalent arrangements, will be apparent from or reasonably suggested by the present invention and the foregoing descriptions thereof, without departing from the substance or scope of the present invention. Accordingly, while the present invention has been described herein in detail in relation to its preferred embodiment, it is to be understood that this disclosure is only illustrative and exemplary of the present invention and is made merely for the purpose of providing a full and enabling disclosure of the invention. The foregoing disclosure is not intended to be construed to limit the present invention or otherwise exclude any such other embodiments, adaptations, variations, modifications or equivalent arrangements; the present invention being limited only by the claims appended hereto and the equivalents thereof. Although specific terms are employed herein, they are used in a generic and descriptive sense only and not for the purpose of limitation.