Patent Publication Number: US-2002005679-A1

Title: Method and apparatus for improving a flexure stage

Description:
BACKGROUND OF THE INVENTION  
       [0001] 1. Field of the Invention  
       [0002] The invention relates to flexure stages and, more particularly, relates to a method of adjusting at least one operational parameter of a flexure stage to at least substantially prevent motion of the flexure stage&#39;s free end out of the desired path of free end travel upon operation of its actuator and hence to reduce out-of-plane or off-axis movement of a workpiece mounted on the free end to within acceptable parameters. The invention additionally relates to a flexure stage incorporating measures to at least substantially prevent at least one component of motion of its free end out of the above-described desired path.  
       [0003] 2. Discussion of the Related Art  
       [0004] Flexure stages are widely used for effecting precisely-controlled motion of a workpiece. Flexure stages are also used in a variety of applications to magnify the effect of a piezo stack or other expandable actuator so that the ratio of movement of the workpiece to actuator expansion is on the order of 10:1 or 20:1. Applications are myriad. Flexure stages can be used to support sensors for hard drive testers, drive elements in hard drives, or workpieces or tooling elements in machining applications such as diamond turning machines, grinding machines, milling machines, etc. A flexure stage can also be used to transport a sensor, or sample for use in a measurement application such as a scanning probe microscope or the like.  
       [0005] A flexure stage comprises one or more flexure elements, each of which is designed to translate a workpiece along a specific axis. Each flexure element is typically constructed from rigid components connected by flexible joints called flexures. The flexures are constructed to only allow bending in a single plane. The typical flexure element consists of a flexure frame on which is mounted an expandable actuator. The flexure frame includes a fixed end mounted on a support, and a free end spaced from the fixed end and supporting a sensor, a tool holder, and a tool or some other workpiece to be translated. The actuator acts on the free end to cause the free end to translate in a desired direction (hereafter the “X” direction).  
       [0006] The simplest type of flexure stage, generally known as a “single axis,” employs a single flexure element. Alternatively, two flexure elements can be coupled together (known as an “X-Y flexure stage”) capable of effecting movement of the desired workpiece in both the X direction and the Y direction. A third flexure element could be added to also effect movement of the workpiece in the vertical “Z” direction, thereby producing an “X-Y-Z flexure stage.” Flexure elements can be fabricated in a variety of ways. Two common geometries include a single ended flexure element and a double-ended flexure element. In the simplest case, a single ended flexure element consists of one fixed end and one free/moving end connected via a single set of flexures. For the double-ended flexure element there exist two fixed ends on either side of the free/moving end connected via two sets of flexures, usually in a symmetric fashion. Flexure stages of these and other types are commercially available, e.g., from Piezosystem Jena, Polytec, and Physik Instrumente, and are sold on instruments such as the metrology scanner on Digital Instruments&#39; atomic force microscope.  
       [0007] The typical flexure element is designed to promote motion of the free end in the XY plane and to inhibit motion in the Z plane. Towards this end, the flexure frame of the flexure element is much thicker (more resistant to bending) in the Z direction than in the X direction. Flex points or flex notches are often formed at the corners of the flexure frame and possibly at other locations to promote parallelogram-type movement of the free end of the flexure stage relative to the fixed end within the XY plane. However, it has been discovered that these measures are imperfect. Piezo stacks and other actuators seldom move perfectly axially but instead bow or twist during operation. By applying net torque out of the XY plane, this bowing or twisting usually results in movement of the flexure stage free end out of the XY plane in a simple or complex motion (either linearly or nonlinearly), in a pitching motion (i.e., along the X axis in a fore-to-aft manner), in a rolling motion (i.e., along the Y axis in a side-to-side manner), or in a combination of two or more of these motions.  
       [0008] The magnitude of motion out of the XY plane compared to the magnitude of motion within the XY plane is typically relatively small and is considered acceptable error in many applications. However, it has been discovered that the magnitude of outof-plane motion of many commercially available flexure stage free ends is not acceptable in all applications. For instance, in scanning probe microscopes and other high-accuracy instruments, it is often desirable to limit out-of-plane (Z) motion of the sensor serving as the workpiece to within a few nanometers upon a free end displacement on the order of 100 micrometers to 140 micrometers in the XY plane. That is comparable to seeking less than a centimeter of out-of-plane motion in over one mile of in-plane motion! Currently-available flexure stages are incapable of achieving this degree of precision.  
       [0009] While the necessary precision is being discussed largely in terms of motion out of the XY plane, it is important to note that the techniques discussed herein may be applied to restricting motion to a single axis or path rather than an entire plane. The invention is intended to encompass these applications as well.  
       OBJECTS AND SUMMARY OF THE INVENTION  
       [0010] It is therefore a principal object of the invention to provide a method of tuning or trimming a flexure stage so as to at least substantially prevent movement of a workpiece carried by the flexure stage free end out of its desired path of operation, be it an axis, a plane or a path in one two or all three dimensions, throughout the operational range of the flexure stage. The method includes imposing a force on a flexure stage to attempt to drive a free end of the flexure stage to move in a desired path relative to a fixed end of the flexure stage, detecting motion of a portion of the flexure stage out of the desired path, and then adjusting an operational parameter of the flexure stage to at least substantially prevent motion of the free end out of the desired path upon subsequent imposition of the force on the flexure stage.  
       [0011] Preferably, adjustment includes first moving the piezo stack or other actuator relative to the flexure frame in order to minimize as much as possible simple out-of-plane motion, followed if necessary with altering a physical characteristic of the flexure stage to eliminate any remaining simple out-of-plane motion as well as roll and/or pitch. These physical characteristics can be adjusted in a number of ways including: (1) adding material to or removing material from the flexure frame, (2) removing material from the piezo end flexure, and (3) adding a second actuator that imparts net out-of-plane force components on the flexure stage that at least partially offset net out-of-plane force components imposed on the flexure stage by the first actuator.  
       [0012] A particularly useful solution to the problem of free end twisting and of the resulting roll and pitch resides in attaching a structure stiff in the Z direction, such as a guide wire attached to the flexure stage. The preferred guide structure comprises at least one (and even more preferably two) wires each of which has a generally central portion attached to the free end and has a pair of fixed ends. The wires are tensioned to impart a counterbalancing torque on the free end upon free end twisting.  
       [0013] In the guide, if the flexure stage is a so-called X-Y flexure stage formed from two interconnected flexure elements extending at an angle from a common vertex that is located between the fixed end and the free end, then the guide preferably includes at least first and second wires extending (1) in parallel with one another, and (2) orthogonally with respect to the designated plane. The first wire is operatively coupled to the flexure stage proximate the free end, and the second guide wire is operatively coupled to the flexure stage proximate the vertex. Preferably, the guide further comprises a third wire extending in parallel with the first and second wires and operatively coupled to the flexure stage proximate the free end.  
       [0014] Wire guides or other similar guides have been found to increase the resonant frequency of the overall flexure stage and hence to increase the available speed at which they can be operated accurately.  
       [0015] Still another possible technique for tuning a flexure stage is to position a second actuator on the flexure stage in order to impose an out-of-plane force component on the flexure frame that at least substantially offsets a net out-of-plane force component imposed on the flexure frame by the first actuator so that the free end moves substantially solely in the designated plane. The second actuator could be used to solely correct for errors created by the first, either closed loop or open loop, or may be used to both correct error and to apply force to assist motion in the intended direction.  
       [0016] Another object of the invention is to provide an improved flexure stage that incorporates measures to at least significantly reduce motion of a workpiece carried by the free end of the flexure stage out of a desired path of motion upon operation of the actuator. For instance, the method used for coupling the actuator to the flexure system could be reduced in size or replaced by a ball bearing or some other structure approximating a point contact, or materials could be added to or removed from selected portions of the flex notches in the flexure frame.  
       [0017] These and other objects, features, and advantages of the invention will become apparent to those skilled in the art from the following detailed description and the accompanying drawings. It should be understood, however, that the detailed description and specific examples, while indicating preferred embodiments of the present invention, are given by way of illustration and not of limitation. Many changes and modifications may be made within the scope of the present invention without departing from the spirit thereof, and the invention includes all such modifications. 
     
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
     [0018] Preferred exemplary embodiments of the invention are illustrated in the accompanying drawings in which like reference numerals represent like parts throughout and in which:  
     [0019]FIG. 1 is a side elevation view of an X-Y flexure stage to which the present invention is applicable;  
     [0020]FIG. 2 is a sectional elevation view taken generally along the lines  2 - 2  in FIG. 1;  
     [0021]FIG. 3 is a perspective view of a piezo stack serving as an actuator for the flexure stage of FIG. 1 and of the associated piezo end flexure for connecting the piezo stack to the flexure frame;  
     [0022]FIG. 4 schematically represents an X flexure stage to which the present invention is applicable;  
     [0023]FIG. 5 is a side elevation view schematically representing the motion of the flexure stage of FIG. 4;  
     [0024]FIG. 6 is a flow chart representing a preferred technique for tuning or trimming a flexure stage in accordance with the present invention;  
     [0025]FIGS. 7 and 8 are a right side elevation view and a sectional front side elevation view, respectively, illustrating the tuning or trimming of a flexure stage by movement of a flexure-stage&#39;s actuator within the associated flexure frame;  
     [0026]FIG. 9 is a perspective view illustrating twisting of a flexure stage as well as the removal of material from a portion of the flexure frame so as to counteract twisting tendencies and the resulting roll and pitch;  
     [0027]FIGS. 10 and 11 are a right side elevation view and a sectional front side elevation view, respectively, of an X flexure stage incorporating a guide structure to reduce out-of-plane motion and to increase the flexure stage&#39;s resonant frequency;  
     [0028]FIG. 11A is a sectional plan view taken along the lines  11 A- 11 A in FIG. 11;  
     [0029]FIG. 12 is a perspective view of an X-Y flexure stage incorporating a guide structure similar to that employed by the X flexure stage of FIGS. 10, 11, and  11 A;  
     [0030]FIGS. 13 and 14 are a right side elevation view and a sectional front side elevation view, respectively, of a flexure stage in which a piezo end flexure of the flexure stage is thinned to reduce out-of-plane motion;  
     [0031]FIGS. 15 and 16 are a right side elevation view and a sectional front side elevation view, respectively, of a flexure stage in which a piezo end flexure is replaced by a ball-and-socket mechanism to cause the contact area between the piezo stack and the flexure frame to approximate a point to reduce out-of-plane motion;  
     [0032]FIGS. 17 and 18 are a right side elevation view and a front side elevation view, respectively, of a flexure stage in which material is removed or thinned from selected portions of the flexure frame to reduce out-of-plane motion;  
     [0033]FIGS. 19 and 20 are a right side elevation view and a partially cut-away front side elevation view, respectively, of a flexure stage in which material is added to selected portions of the flexure frame to reduce out-of-plane motion;  
     [0034]FIGS. 21 and 22 are a right side elevation view and a sectional front side elevation view, respectively, of a flexure stage which incorporates a second actuator that imposes net out-of-plane forces on the flexure stage free end that offsets net out-of-plane forces imposed on the flexure stage free end by the first actuator;  
     [0035]FIGS. 23 and 24 are a right side elevation view and a sectional front side elevation view, respectively, of a flexure stage and a sensor arrangement usable to detect out-of-plane motion of the flexure stage; and  
     [0036] FIGS.  25 - 27  are right side elevation views of alternative flexure stages with which the invention is applicable. 
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS  
     [0037] 1. Resume  
     [0038] Pursuant to the invention, a method is provided of tuning or trimming a flexure stage to constrain the movement of a workpiece carried by the flexure stage&#39;s free end to a desired path of free end travel . For example, in the case of an XY flexure stage, measures are incorporated into the XY flexure stage to prevent simple out-of-XY plane motion (either linear or nonlinear), rolling, pitching, or combinations of all three. A preferred tuning technique begins with initially aligning the flexure stage&#39;s actuator in reference to the flexure frame so that its highest off-axis force component extends as much as possible within the designated plane of motion, followed by positioning the actuator on the flexure stage so as to eliminate as much as possible simple out-of-plane motion and accompanied if necessary by incorporating additional measures to eliminate as much as possible roll or pitch motion and residual simple out-of-plane motion. These additional measures may include attaching a guide structure to the flexure stage, removing materials from one or more piezo end flexures, adding materials to or removing materials from one or more flexure points or other locations of the flexure frame, and/or adding a second actuator that imposes net out-of-plane forces on the flexure frame free end that at least partially offset those imposed by the first actuator. The resulting tuned flexure stage exhibits substantially less out-of-plane motion throughout the operational range of its actuator than the flexure stage prior to tuning.  
     [0039] 2. Flexure Stage Construction and Characteristics  
     [0040] The need for and manner of fine tuning a flexure stage can be better appreciated from an understanding of the structure of a typical flexure stage and its operational tendencies. Referring initially to FIGS.  1 - 3 , an X-Y flexure stage  30  is illustrated to which the present invention is applicable. The flexure stage  30  may be used to translate any workpiece for which precisely controlled motion within a plane is desirable. It is particularly-well suited for translating a sensor  32  of the type used, for example, in scanning probe microscopy. The flexure stage  30  can be used to translate a sensor  32  in either the X direction or the Y direction without moving the sensor  32  in the Z direction, i.e., out of the page as seen in FIG. 1.  
     [0041] The flexure stage  30  includes an X flexure element  34  and a Y flexure element  36  which are connected integrally to one another and which are designed to effect movement of the sensor  32  in the Y direction and X direction, respectively (the “X direction” and “Y direction” are defined below). Each flexure element  34  or  36  includes a parallelogram-type flexure frame  38  or  40  one end of which is designed to move relative to the other. Not all flexure frames are of parallelogram type, and the invention is intended to apply to other types, but the parallelogram type is used throughout for demonstrative purposes. For the sake of convenience, the end of the flexure frame that is designed to move will hereafter be referred to as the “free end,” and the end that serves as a frame of reference for this movement will hereafter be referred to as the “fixed end.” A fixed end  50  of the Y flexure element  36  is rigidly attached to an underlying support  51 , such as a Z stage of a scanning probe microscope (not shown), or a course positioner for machine tools. In the example, the X flexure element  34  extends perpendicularly from the Y flexure element  36  such that its fixed end merges with and is constituted by the free end of the Y flexure element at a vertex  52 . The free end  54  of the X flexure element supports, or is mechanically linked to, the sensor  32  in a conventional manner.  
     [0042] A first actuator  42  is mounted on the flexure frame  38  for effecting movement the free end  54  of the X flexure element  34  relative to the fixed end  52 . A second actuator  44  is mounted on the flexure frame  40  for effecting movement of the free end  52  of the Y flexure element  36  relative to the fixed end  50 . Both actuators  42  and  44  are supplied with electrical power via a cable  46  composed of several wires  48 .  
     [0043] Apart from the connections of their fixed and free ends to other components, the X and Y flexure elements  34  and  36  are of essentially identical construction, and the problems presented by their operation are also essentially identical. Hence, the details of only the X flexure element  34  will described but will pertain to both for the sake of conciseness.  
     [0044] In the illustrated embodiment, the X flexure element  34  comprises the flexure frame  38  which is formed from a relatively rigid material such as metal, plastic or ceramic. The flexure frame  38  presents the fixed end  52 , the free end  54 , and a cavity  56  formed therein between the free end  54  and the fixed end  52 . The cavity  56  is bordered at its inner end by the fixed end  52  and at its outer end and the free end  54 . The cavity  56  is also bordered at its sides by a pair of parallel legs  58  and  60  extending generally longitudinally from the inner edge surface of the cavity  56  to the outer edge surface. The legs  58  and  60 , as well as the remainder of the flexure frame  38 , are much thicker in the Z direction (i.e., in the direction perpendicular to the XY plane) than in their other dimensions to inhibit movement out of the XY plane. Flex notches  62 ,  64 ,  66 , and  68 , sometimes called flexures, are formed at the opposite ends of each of the legs  58  and  60  to promote parallel movement of the legs  58  and  60  relative to one another and to promote parallel movement of the free end  54  relative to the fixed end  52 . These flex notches  62 ,  64 ,  66 , and  68  preferably extend the entire thickness of the legs  58  and  60  in the Z direction.  
     [0045] The actuator  42  could comprise any suitable expandable device but preferably comprises a conventional piezo stack. Although the illustrated piezo stack  42  is rectangular in cross-section, it could also be round or another shape. The piezo stack  42  is mounted in the cavity  56  at an acute angle with respect to the longitudinal axis of the flexure frame  38  so as to impose a force on the free end  54  that has components in both the X direction and the Y direction. Mounting the piezo stack  42  at an angle in this manner amplifies the output so that the ratio of flexure stage free end motion in the X direction to piezo stack expansion is on the order of 10:1 or 20:1.  
     [0046] As best seen in FIGS. 2 and 3, the ends  41  and  43  of the piezo stack  42  are mounted in the cavity  56  by gluing them to metal inserts  70  and  72  that in turn are bonded in the ends of the cavity  56 . The inserts  70  and  72  are machined to have a shape that is generally triangular when viewed in side elevation as seen in FIG. 1. The resulting line of contact between the piezo stack  42  and the flexure frame  38  extends in the Z direction across at least a substantial width of the piezo stack  42 . This area of contact, being thinner in the direction orthogonal to the Z direction than the corresponding width of the piezo stack  42 , represents another flex point  74  or  76  hereafter referenced a “piezo end flexure.” As shown in FIGS. 2 and 3, the flexures  70  and  72  have been machined further to reduce the contact to a near point as discussed above.  
     [0047] The problems arising during operation of the piezo stack  42  that result in out-of-plane motion of the free end  54  and sensor  32  can occur whether the flexure stage is an X-Y flexure stage of the type illustrated in FIGS.  1 - 3 , a flexure stage in which two flexure elements are mounted end-to-end in a mirror image such that their free ends abut one another, or a simple X flexure stage. Because the problems encountered by the invention are most easily understood with respect to a single or X flexure stage, much of the discussion that follows will center on a single or X flexure stage, it being understood that the invention is equally applicable to an X-Y flexure stage or other flexure stages (examples of some of which are discussed in Section  5  below) that are designed to permit motion along a single axis or along some other path. Thus, while the invention will be discussed primarily with respect to constraining the flexure stage free end from movement out of a designated plane, the invention is also applicable to any application in which it is desired to constrain the free end of a flexure stage to movement along at least one axis or path. Hence, references herein to constraining movement to a desired path should be construed broadly to include constraint of motion according to the number of axes of motion the flexure stage is designed to allow. For example, where a flexure frame is designed to permit motion in two axes, X and Y for example), thereby describing a plane, motion out of the desired path is most simply described as “out-of-plane” motion. Where a flexure frame is designed to permit motion in a single axis, motion out of the desired path is most simply described as “out-of-axis” motion. In all cases, the potential range of motion is limited by the ordinary mechanical constraints of typical flexure frames. The function of the invention is to constrain the free end from at least one component of motion out of a desired path whether in one, two, or three dimensions. Hence, references herein to detecting or reducing motion out of a desired path should be construed to mean detecting, reducing, or otherwise addressing at least one component of motion out of the desired path. Other components of motion out of the desired path could also exist and could also but not necessarily be addressed.  
     [0048] Referring now to FIGS. 4 and 5, a single or X flexure stage  134  is illustrated that is identical to the X flexure element  34  of FIGS.  1 - 3  with the exception that the fixed end  152  is adapted to be connected to the underlying support  151  directly rather than through an intermediate flexure element. Elements of the flexure stage  134  corresponding to the flexure element  34  therefore are designated by the same reference characters, incremented by 100. These elements include (1) a frame  138  having a fixed end  152 , a free end  154 , and legs  158  and  160  in which are formed flex notches  162 ,  164 ,  166 , and  168 , and (2) an actuator  142  having ends  141  and  143  and associated piezo end flexures  174  and  176 . The primary range of motion of the flexure stage  134  is in the X direction, with motion in the Y direction arising because of parabolic shortening of the flexure stage  134  during movement. This type of parabolic shortening is characteristic to many single ended flexures, but does not usually occur in double ended flexures. The free end  154  and sensor  132  therefore move in the XY plane and are constrained from motion in the Z direction (i.e., into and out of the page in FIGS. 4 and 5).  
     [0049] It is important to note at this point that the “X” axis is the axis of intended motion rather than the axis of the free end  154  itself. It should also be noted at this point that, as discussed above and as can be seen in FIGS. 4 and 5, the sensor  132  is necessarily offset from the centroid of the free end  154  due to physical limitations of the system. Were it not for this offset, roll and pitch (as defined below) might not present as great a problem, as they do in the current example. In the general case, roll and pitch and yaw may lead to out-of-plane or out of axis motion of the sensor  132  and usually must be addressed if elimination of out-of-plane and/or out-of-axis motion is desired. Further, such out-of-plane and out of axis motion may in many cases effect over-all performance of the application, for reasons other than relative placement of the workpiece.  
     [0050] It has been discovered that, in practice, expandable actuators such as the piezo stack  142  do not move perfectly axially upon expansion, leading to various types of out-of-plane motion of the free end  154  of the flexure stage  134  with respect to the XY plane. Each of these out-of-plane motions will now be described.  
     [0051] The first type of out-of-plane motion encountered by flexure stages is simple out-of-plane motion resulting from bowing of the piezo stack  142  in a single plane that is not in the XY plane. This bowing is best understood if the piezo stack  142  is imagined outside the flexure such that when a voltage is applied, the stack bends or twists as it elongates. This bowing gives rise to so-called “first order” error in which the free end  154  moves either linearly or nonlinearly out of the XY plane. Thus, the free end  154  can be forced out of the intended plane of motion due to the out-of-plane forces created by the bowing. For example, bowing in the form of movement of the ends of the piezo stack  142  out of the page and towards the viewer in FIG. 5 would generally tend to cause the free end  154  to move into the page or away from the viewer.  
     [0052] Twisting may also occur either alone or combine with bowing to give rise to second order errors. Twisting occurs when the far end of the piezo stack  142  turns relative to the near end. This twisting generates moments that lead to pitching and rolling of the flexure stage free end  154 . “Roll” may be defined as movement in the direction of the Y axis or about the X axis from side-to-side. Roll is represented by the arrow  180  in FIGS. 4 and 5. “Pitch” may be defined as movement in the direction of the X axis or about the Y axis in a fore-to-aft manner. Pitch is represented by the arrow  182  in FIGS. 4 and 5. Roll and/or pitch may also occur due to bowing of the piezo stack in more than one plane. Of course, these types of motion are not mutually exclusive, and the free end  154  may encounter a combination of two or all three types of motion during operation of the piezo stack  142 . In the general case, the geometry of the contact between an actuator and a flexure frame will control what sort of undesirable motion is created the characteristics of the expanding actuator.  
     [0053] Referring again to FIGS. 4 and 5, roll and pitch can be better understood by recognizing that, due to the parallelogram structure of the flexure frame  138 , and due to the canted or inclined orientation of the piezo stack  142  within the cavity  156 , roll and pitch actually (though they are defined relative to the axis of motion) occur about axes A 1  and A 2  that extend diagonally between the flexure notches  164 ,  166 , and  162 ,  168  as illustrated. In the illustrated embodiment, roll and pitch due to piezo stack bowing will be most severe along the axis A 1  because the piezo stack  142  is pushing more perpendicularly to that axis A 1  than to the axis A 2 . Conversely, roll and pitch due to piezo stack twisting will be most severe along the axis A 2  which is more nearly parallel with the axis of the piezo stack  142 . However both roll and pitch can and do occur simultaneously about both axes A 1  and A 2 .  
     [0054] The complexity of the coupling between X and Y motion (and Z motion if included) and the flex of the actuator defies use of a single model to discuss all of the potential combinations and effects, thus the discussion here uses a single model to demonstrate that complexity and methods of attacking it rather than attempting to detail all possible solutions.  
     [0055] 3. Description of Basic Adjustment Technique  
     [0056] Having defined and explained the concepts of “out-of-plane,” and “out-of-axis,” motion, and deviation from a desired path as the more general case, as well as describing the factors contributing to out-of-plane motion in the example, the manner in which operational characteristics of the flexure stage  134  can be adjusted to eliminate out-of-plane motion or at least to reduce it to within acceptable parameters may now be described. A preferred general technique for this adjustment is described first, followed by a discussion of several non-mutually exclusive alternative mechanisms for its implementation.  
     [0057] Stated in its most basic concept, the invention relates to the identification of flexure stage out-of-plane motion and to the adjustment of one or more of the flexure stage&#39;s operational parameters to reduce out-of-plane motion to within acceptable parameters. The term “operational parameter” as used herein is broadly construed to include the physical structure of one or more components of the flexure stage as well as reaction of the flexure stage to internal or external stimuli. The adjustment can take the form of rearranging the flexure stage components relative to one another, physically altering the structure of the flexure stage  134 , adding external guides, and/or adding an additional actuator that offsets out-of-plane forces imposed by the actuator  142 .  
     [0058] Referring now to FIG. 6, a preferred closed-loop process for effecting this adjustment is illustrated. The process may be performed entirely manually or partially manually with some measurements and even some adjustments being performed automatically. Thus, it should be understood that the invention is intended to include both manual and automatic completion of the iterative processes. The process begins with Step  200  in which the piezo stack  142  (or other actuator if another comparable actuator is used instead of a piezo stack) is tested prior to its mounting in the flexure stage  134  to determine in which plane its bowing motion is the greatest. The piezo stack  142  is then positioned in the cavity  156  such that the greatest component of its bowing motion is located within the XY plane, thus reducing the need for subsequent adjustments.  
     [0059] Next, the out-of-plane motion of the flexure stage free end  154  is observed in step  202  over part or all of the range of motion of the piezo stack  142 , preferably using a sensor arrangement such as that illustrated in FIGS. 23 and 24. This sensor arrangement includes (1) a sensor jig  300  partially surrounding the free end  154  of the flexure stage  134 , and (2) a plurality of capacitance sensors extending between the flexure stage free end  154  and the sensor jig  300 . Four sensors  302 ,  304 ,  306 , and  308  are provided in the illustrated embodiment. The sensors  302 ,  306  and the sensors  304 ,  308  are located on opposite sides of the longitudinal centerline of the free end  154 . The sensors  302 ,  304  and the sensors  306 ,  308  are located above and below the lateral centerline of the free end  154 , respectively. The signal from any one sensor provides an indication of the direction and magnitude of simple out-of-plane motion. The direction and magnitude of pitch can be ascertained by comparing the signal from either of the sensors  302  or  306  to the signal from either of the sensors  304  or  308 . The direction and magnitude of roll can similarly be ascertained by comparing the signal from either of the sensors  302  or  304  to the signal from either of the sensors  306  or  308 . These signals can then be combined in a readily ascertainable manner to identify the type of motion in Step  204  in FIG. 6 as either simple out-of-plane motion or roll and/or pitch motion. The sensing could alternatively be accomplished with an interference microscope or other sensors to view the Z motion of the free end of the flexure over a large enough area to detect pitch, roll, and/or out-of-plane motion.  
     [0060] In the preferred and illustrated embodiment, different adjustment techniques are used to reduce simple out-of-plane motion or other types of out-of-plane motion. When simple out-of-plane motion is identified as indicated in Step  206 , the non-linear Z motion is measured over either the full range or part of the range of piezo stack motion in Step  208 . The process then determines in Step  210  whether or not the measured simple out-of-plane motion is within an acceptable parameter such as + 2 . 5  nanometers at the free end. If so, the process proceeds to Step  212 . If not, an operational parameter of the flexure stage  154  is adjusted in Step  214 . This adjustment preferably comprises changing the geometry or the position of the piezo stack  142  as detailed in Section  4 (a) below. The process then returns to Step  208 , where the motion is again measured, and proceeds through a closed loop consisting of the Steps  208 ,  210 , and  214  until the measured outof-plane motion is within accepted parameters.  
     [0061] If it is determined in Step  204  that the flexure stage free end  154  undergoes motion other than simple out-of-plane motion, the process proceeds to Step  216  and then to Step  218  in which the roll or pitch is measured over a part or full range of motion of the piezo stack  142 . The process then determines in Step  220  whether or not the measured roll or pitch is within accepted parameters, such as one to two arc seconds. If so, the process proceeds to Step  212 . If not, the process proceeds to Step  222  where the shape or geometry of the flexure frame  138  is changed or adjusted, preferably using one or more of the techniques described in Section 4(b) below, in an attempt to reduce the roll and pitch. Steps  218  through  222  are repeated in a reiterative, closed loop manner as many times as are necessary to reduce the roll and pitch to within accepted operational parameters.  
     [0062] It should be noted at this time that if both simple out-of-plane motion and roll or pitch motion are detected as represented by Step  224 , the Steps  206 - 214  and  216 - 222  are performed independently of one another to eliminate both types of motion.  
     [0063] Step  212  recognizes the fact that independently addressing simple out-of-plane motion and roll or pitch motion may adversely affect adjustments made in response to the other type of motion. For instance, depending upon the adjustment technique chosen, reducing simple out-of-plane motion may actually tend to exasperate rolling or pitching motion. Hence, after the independent adjustments identified above are completed, the process determines in Step  212  whether or not the flexure stage free end  154  moves entirely within acceptable parameters. If not, an adjustment deemed most likely to correct the detected defect is performed in Step  226 , and the process returns to Step  202  via Step  228  for further measurement and possible further adjustment. If so, no further adjustment is required, and the process ends in Step  230 .  
     [0064] 4. Preferred Adjustment Techniques  
     [0065] Several preferred techniques for eliminating or at least reducing out-of-plane motion will now be described, it being understood that these techniques are neither mutually exclusive nor all-inclusive.  
     [0066] a. Mechanisms for adjusting the geometry or position of driving elements  
     [0067] i. Actuator Position  
     [0068] As discussed in Section 3 above, the effects of bowing of a piezo stack  142  or another actuator on out-of-plane motion will vary depending upon the orientation of the piezo stack with respect to the cavity  156 . Simple out-of-plane motion and even some components of roll and pitch can be reduced by repositioning the piezo stack  142  within the cavity  156 . Repositioning the piezo stack  142  may encompass movement of one or both of its ends in the Z direction, the X direction, or a combination of both. This repositioning is illustrated schematically by FIGS. 7 and 8, which illustrate the preadjusted position of the piezo stack  156  in solid lines and the post-adjusted position in phantom lines.  
     [0069] In the preferred embodiment, after the direction and magnitude of out-of-plane motion are detected as discussed in Section 3 above, the piezo stack position is adjusted in the Y direction by an amount designed or estimated to eliminate this motion. For instance, if a detected bowing motion of the piezo stack results in simple out-of-plane motion of the free end  154  out of the page as seen in FIG. 7 or to the right as seen in FIG. 8, the piezo stack  142  is repositioned so that at least its upper end  141  is repositioned away from this direction of bowing to counteract the bowing effects. Repositioning is preferably performed before the final gluing or soldering of the piezo stack  142  in place, with the motion of the free end  154  being measured after each successive repositioning operation in a closed loop fashion as described in Section  3  above. Only after it is determined that the piezo stack  142  is optimally positioned will it be soldered or glued in place. This adjustment may also be made automatically, prior to or during use.  
     [0070] The positions of both ends  141  and  143  are adjusted in the illustrated embodiment. However, in some instances, it may be desirable to adjust the position of only one end  141  or  143 .  
     [0071] ii. Piezo end Flexure Modification  
     [0072] Another technique for compensating for out-of-plane motion involves modifying the contact area between the piezo stack  142  and the flexure frame  138 , thereby modifying the piezo end flexure  174  or  176 . As discussed above, the typical piezo stack  142  contacts the surface of the flexure frame cavity  156  along the entire width of the piezo stack  142 . This relatively thin but long area of contact may translate undesired out-of-plane forces to the flexure stage free end  154  that occur due to bow and/or twist of the piezo stack  142 . Conversely, if this contact were just a point contact, the unwanted forces imposed on the free end  154  due to piezo stack bowing and twisting would be substantially less on the free end  154 . However, a true point contact is impractical because piezo stack  142  typically imposes from five to fifteen pounds of force on the flexure frame  138  upon expansion. If this force were imposed on a very small point, the resulting pressures could crush the point. The line contact also may act to equalize net torques placed on the flexure frame  138 . Hence, alternative solutions that equalize unwanted torques or at least partially obtain the same effect are desirable.  
     [0073] One solution is illustrated schematically in FIGS. 13 and 14. This solution involves the removal of some of the contact surface between the piezo stack  142  and the flexure frame  138  to decrease the contact length and/or change the net torque applied by the piezo end flexure  174  and/or  176 . For instance, one or both sides of the piezo end flexure  174  could be partially drilled out to shorten the effective contact length of the upper end  141  of the piezo stack  142  with the flexure frame  138 . The decision as to the amount of material to remove, whether or not to remove material from one or both piezo end flexures  174  and  176 , or whether to remove material from side of the selected piezo end flexure(s), the other side, or both sides will depend upon the magnitude and direction of out-of-plane forces detected in Step  208  as discussed in Section  3  above. Typically, a relatively small amount of material should be removed from one or both sides of the piezo end flexure  174  and/or  176  at one or both ends of the piezo stack  142  first, the effects of this removal is measured, and then more material is drilled out incrementally until the desired effects are achieved as represented by the loop of Steps  208 - 214  in FIG. 6. In the illustrated embodiment, material is removed from one side of the piezo end flexure  174  located at the upper end  141  of the flexure stack to create a bore  175 . A similar bore is shown in FIG. 12 at the near end of piezo stack  42 .  
     [0074] An alternative technique with similar results, illustrated in FIGS. 15 and 16, approximates a point relationship by mounting one or both ends of the piezo stack  142  in the cavity  146  using a ball and socket mechanism  175 ′. This technique is at least theoretically more effective than reducing the size of the piezo end flexure by drilling but is more difficult and expensive to implement. Further, the torque applied by a line contact may be useful to accomplish the objects of this invention by allowing tuning of those torques by the above methods.  
     [0075] b. Mechanisms for Changing Flexure Geometry or Shape  
     [0076] Adjusting the position of the piezo stack  142  within the flexure stage  134  and/or adjusting the geometry of piezo stack contact with the flexure stage  134  significantly reduces out-of-plane motion due to piezo stack bowing and partially reduce pitching or rolling. However, they are not completely effective. Several techniques for further tuning the flexure stage to further reduce pitching, rolling, and, if necessary, residual simple out-of-plane motion, will now be detailed.  
     [0077] i. Mechanisms for Altering Flexure Frame Geometry  
     [0078] Twist resistance as well as resistance to piezo stack bowing can be enhanced either by removing material from the flexure frame  138  or by adding material to the flexure frame  138 . Removal of material lowers the spring constant associated with the flex notch where the material is removed, it also decreases resistance to compression and expansion under tension. There are many ways to remove material to accomplish reduction of spring constant and/or resistance to compression and tension. Two methods of removal of material are illustrated in FIGS. 17 and 18. The first is by cutting material to increase the depth by one or more of the flex notches  162 ,  164 ,  166 , or  168 . Flex notch depth can be increased by removing material from the inside of a flex notch (as seen at  164 ′). A similar effect may be accomplished by removing material from the leg at a location adjacent a flex notch (as seen at  168 ′). The amount of thinning, the length of the area to be thinned, and the location of this thinning will depend upon the desired results. If the goal is to reduce residual simple out-of-plane motion due to piezo stack bowing, one end of two or all four of the flex notches  162 ,  164 ,  166 , or  168  should be thinned. The end of each notch to be thinned is determined by the direction of bowing. Generally speaking, the end towards which the ends of piezo stack  142  bows is the end that is thinned. Hence, if the piezo stack  142  is bowing away from the page and towards the viewer in FIG. 17, material is removed from that end of one or more of the flex notches  162 ,  164 ,  166 , or  168 . One effect of this thinning is to effectively move the flex notch in the Z direction.  
     [0079] If, on the other hand, one wishes to reduce flexure frame twisting caused either by piezo stack twisting or non-simple piezo stack bowing, then material is removed from some of the flex notches but not others to resist twisting forces and the resultant rolling and pitching motions. FIG. 9 provides an illustrative example. In this example, piezo stack twisting, bowing, or a combination of both compress one side of the flexure frame  138  and expand the other side as represented by the arrows  190  and  192 , thereby imparting a counterclockwise twisting motion to the flexure stage  134  as represented by the arrow  194 . Cutting the notch  162  deeper (including all the way through as shown in FIG. 9) at its end  162 ′ in the illustrated manner gives rise to a countervailing torque as represented by arrow  196  that encourages flexure frame movement towards that notch portion  162 ′ or in a direction opposite to that in which flexure frame twisting tends to occur.  
     [0080] Of course, as with the previously-described techniques, parameter adjustment through material removal preferably is performed in an incremental, closed-loop fashion as represented by the Steps  216 - 222  in FIG. 6 with the effects of each incremental amount of material on pitch and roll being monitored before additional material is removed.  
     [0081] The effects of material removal or thinning can also be done by adding material to the flexure frame  138  and thereby increasing the flexure&#39;s resistance to stretching and compression and increasing its spring constant instead of or in addition to removing material from it. Hence, referring to FIGS. 19 and 20, material could be added to the flexure frame  138  either by filling in part or all of a flex notch as illustrated at  164 ″ or by welding or otherwise attaching material to the flexure frame  138  adjacent a desired flex notch as illustrated at  168 ″.  
     [0082] ii. Guide Arrangement  
     [0083] Considerable time and effort are required to remove material from the flexure frame  138  or to add material to the flexure frame  138 . Moreover, removing material necessarily weakens the flexure frame  138  to the point that there may be concern about fatigue limit on the material. Hence, it may be desirable in many applications to employ a guide structure instead of or in addition to thinning or thickening flex notches.  
     [0084] A preferred guide structure  400  for use in a single element or flexure stage  134  is illustrated in FIGS. 10,11, and  11 A. Guide arrangement  400  preferably includes a set of guide wires  402  and  404  attached to the free end  154  of the flexure stage  134 . Two parallel guide wires  402  and  404  are provided in the illustrated embodiment—one adjacent each lateral end of the upper horizontal edge of the free end  154 . Both guide wires  402  and  404  extend orthogonally to the XY plane or in the Z direction. The wires  402  and  404  are attached to the free end  154  by way of a T-shaped support bar  406  that extends in parallel with the X axis. Specifically, the wires  402  and  404  extend through respective slots  408  and  410  in the bar  406  and are fixed in the slots by depositing a bead  424  or  426  of epoxy or some other adhesive into the slot  408  or  410 . The free ends of each wire  402  or  404  are attached to a rigid support structure  422 , preferably the same structure to which the fixed end  152  of the flexure stage  134  is attached, by way of tubular wire holders  414 ,  416  and set screws  418 ,  420 . The set screws  418  and  420  preferably comprise allen screws threaded radially through the wire holders  414  and  416  and into locking engagement with the ends of the wires  402  and  404 . The wires  402  and  404  are placed under tension so that they impart considerable resistance to both pitching and rolling of the flexure stage free end  154 . However, they do not impart significant resistance to movement in the XY plane because they extend in the Z direction and are flexible. The wire holder could be piezoelectric or controllable in some other manner and could be driven by a voltage from the scan controller, or a feedback system of another kind, to keep the out-of-plane and/or off-axis motion to a minimum by changing the tension on individual wires. The appropriate voltage waveform to perform this correction could be learned by putting sensors on the free end, as mentioned above, and making the voltage such that over a scan the free end&#39;s out-of-plane and/or off-axis motion is minimized. The wire itself may also be electrostrictive or thermostrictive, rather than or in addition to the wire holder.  
     [0085] The flexure stage  134  could be attached to the nominal center of each wire  402  or  404  without giving further consideration to additional flexure stage tuning or trimming. However, if desired, additional tuning or trimming could be achieved by moving the clamping point longitudinally with respect to the center of the wire  402  or  404  so that, upon flexure stage twisting, unequalized forces are imposed on the flexure stage  134  due to the unequal distance between the flexure stage  134  and the ends of the wires  402  and  404 . This fine tuning could be achieved by changing the location at which the wire  402  or  404  is glued to the bar  406  and/or by changing the position of the pinning points at which the ends of the wires  402  and  404  are pinned or screwed to the wire holders  414  and  416 .  
     [0086] In addition to the other advantages described above, employing a guide structure to reduce flexure stage pitch and roll also has the advantage of increasing the stiffness of the moving portion of the flexure stage. This added stiffness increases the resonant frequency of the flexure stage  134  and hence increases the speed at which the instrument may be operated.  
     [0087] Different guide configurations could be used in different applications. For instance, in a “mirror-image” arrangement of the type described in Section 2 above, in which the free ends of two flexure elements are mounted end to end, wires could be attached to the opposite lateral sides of the common free end of the mirrored flexure elements. Any guide structure could be used, as long as it is stiff in the Z direction and flexible in the X direction.  
     [0088] Alternatively, and referring to FIG. 12, a wire guide structure  500  that incorporates an additional wire  503  could be attached to the vertex  52  of an X-Y flexure stage  30  in parallel with the wires  502  and  504  attached to the free end  54  of the X stage  34 . This additional wire  503  arrests the vertex  52  of the flexure stage  30  from Z displacement and pitch and roll. FIG. 12 also illustrates an alternative technique for attaching the wires  502  and  504  to the bar  506 . In this embodiment, the wires  502  and  504  extend through bores  508  and  510  extending through the bar  506  rather than being received in slots. It is not shown, but set screws or other adjustable means could be used to fix wires  502  and  504  to bar  506 . Of course, the bar  506  could be replaced with the bar  406  of the previous embodiment or any other suitable structure for attaching the wires  502  and  504  to the flexure stage  30 . A similar bar could be used to attach wire  503  to the vertex  52 .  
     [0089] The wire guide structure  400  has proven very effective at reducing flexure stage pitch and roll, but it still is capable of removing only a percentage of total out-of-plane motion. Even in those applications where it is preferred over varying the geometry of the flexure stage  134 , it is still desirable to first optimize the flexure stage  134  by changing the orientation of the piezo stack  142  within the flexure frame  138  and by then further tuning the flexure stage  134  using the wire guide structure  400 .  
     [0090] iii. Multiple Piezo Stacks  
     [0091] Referring now to FIGS. 21 and 22, still another way of eliminating at least some out-of-plane motion components is to install a second piezo stack  142 ′ or other actuator on the flexure frame  138  to act in concert with the piezo stack  142  to reduce net out-of-plane force components imparted by the first piezo stack  142 . (The term “net” out-of-plane force components is employed to reflect the fact that the piezo stack  142  could impose force components solely within the XY plane but that misalignments between the piezo stack  142  and the flexure frame  138  and other considerations could combine to cause the net forces as experienced by the free end  154  of the flexure to include out-of-plane components.)  
     [0092] According to this technique, the first piezo stack  142  is first mounted on the flexure frame  138  in a manner so as to minimize the out-of-plane force components imposed by it, i.e., by orientating it such that its worst bowing component is within the XY plane and possibly by altering its orientation and/or geometry to reduce other bowing components. A second piezo stack  142 ′ or other actuator then is mounted on the flexure frame  138  and positioned so that out-of-plane force components imposed by it tend to offset the net out-of-plane force components imposed by the first piezo stack  142  so that the net out-of-plane force components imposed by both piezo stacks  142  and  142 ′ is approximately zero. In the illustrated embodiment in which the piezo stack  142  is mounted in a cavity  156  of the flexure stage  134 , the second piezo stack  142 ′ is typically mounted in the same cavity  156  at a different orientation than the first piezo stack  142  with respect to the Z axis and possibly in a different orientation with respect to the X axis. As needed, and as space allows, more than two actuators could be used in this manner.  
     [0093] Like the other techniques described above, determining the optimum location of the second piezo stack  142 ′ is typically performed by trial and error in a closed-loop fashion.  
     [0094] This technique offers the advantage of being somewhat retrofittable because it may not require any alteration to the orientation or structure of either the piezo stack  142  or the flexure frame  138  after initial assembly. It exhibits the disadvantage, however, of being somewhat expensive to implement because it requires a second piezo stack  142 ′. This technique also usually proves less than fully effective at removing all net out-of-plane force components and hence in practice still would likely have to be combined with one or more of the other techniques described above including the use of guide wires, flexure stage thinning or thickening, etc.  
     [0095] Although the invention has been disclosed and described with respect to several preferred embodiments, many changes and modifications could be made to the invention without departing from the spirit thereof.  
     [0096] For instance, as discussed above, the various techniques for trimming or tuning a flexure stage are not mutually exclusive, but in practice would be combined with each other and possibly with other techniques to reduce out-of-plane motion to within acceptable parameters.  
     [0097] 5. Alternative Applications  
     [0098] As discussed above, the invention is not limited to use with flexure stages of the disclosed type. It instead is applicable to virtually any positioning apparatus in which an actuator is used to translate a free end of the apparatus with respect to a fixed end within a desired path.  
     [0099] For instance, an example of a double-ended flexure  230  with which the invention is applicable is illustrated in FIG. 25. Double-ended flexure  230  includes first and second flexure elements  234  and  236  mounted end to end such that they share a common free end  254  disposed intermediate their fixed ends  250  and  252 . An external piezo stack  242  acts on the free end  254  to drive it to effect linear movement along the X-axis as represented by the arrow in FIG. 25.  
     [0100]FIG. 26 illustrates a single ended flexure stage  334  configured for a single X-axis of motion. The flexure stage  334  includes a flexure frame  338  having a fixed end  352  and a free end  354 . An external piezo stack  342  acts on the free end  354  to effect linear movement along the X-axis as represented by the arrow in FIG. 26.  
     [0101]FIG. 27 illustrates a double ended flexure stage  430  configured to effect motion along both the X-axis and the Y-axis. The flexure stage  430  includes a first flexure element  434  and a second flexure element  436  disposed within the first flexure element  434 . The first flexure element  434  includes a center free end  452  (formed from a generally rectangular frame in the illustrated embodiment) and a pair of opposed fixed ends  450  and  450 ′. The second flexure element  436  is of similar construction but extends orthogonally with respect to the first flexure element  434  so that its fixed ends  451  and  451 ′ are fixed to sidewalls of the free end  452  of the first flexure element  434  and such that its free end  454  is disposed between the fixed ends  451  and  451 ′. A sensor or other workpiece is mounted on the free end  454  of the second flexure element  436 . Movement of this workpiece along the Y-axis is effected by way of a first, external piezo stack  442  engaging the sidewall of the first flexure element free end  452 . Movement of the workpiece along the X-axis is effected by way of a second piezo stack  442 ′ disposed within the first flexure element free end  452  and acting on the bottom surface of the second flexure element free end  454 .  
     [0102] Virtually all of the flexure tuning techniques described in Section 4 above are usable either alone or in combination with one another on each of the flexure stages of FIG. 25, FIG. 26, and FIG. 27 or on virtually any other flexure stage.  
     [0103] The scope of these and other changes will become apparent from the appended claims.